Index

1. Classical Growth Theory: From Smith to Marx 4

1.1. Smith; Ricardo; Marx 41.1.1. Adam Smith 41.1.2. David Ricardo 51.1.3. Karl Marx 61.2. After Marx: Structural Change and Steady States 10

2. Keynesian Growth: The Cambridge Version 16

3. Neoclassical Growth 26

3.1. The Neoclassical Growth Theory 263.1.1. Introduction 263.1.2. The Solow-Swan Growth Model 273.1.3. Adding Depreciation 323.1.4. Solving the System 333.1.4. 1. The Cobb-Douglas Solution 333.1.4. 2. The General Solution 363.1.5. Adjustment Processes: Solow vs. Harrod 373.2. Empirical Implications 443.2.1. Introduction 443.2.2. The Solow Paradox 453.2.3. The Convergence Hypotheses 493.2.3.1. Absolute Convergence 493.2.3.2. Conditional Convergence 513.2.4. Poverty Traps 523.2.4.1. The Technological Trap 523.2.4.2. The Population Trap 583.3. Technical Progress 603.3.1.     Adding Technical Progress 613.3.2.     Empirical Implications 673.4. Selected References 69

4. Multisector Growth 73

4.1. The Uzawa Two-Sector Growth Model 734.1.1.      Basic Setup 744.1.2.      Diagrammatic Representation 794.1.3.      Analytical Solutions 904.1.3.1.   Analytical Solution I: Classical Hypothesis 904.1.3.2.   Analytical Solution II: Proportional Savings 974.1.4.      Indeterminacy, Instability and Cycles 1064.2. Optimal Two-Sector Growth 1114.2.1. The Uzawa-Srinivasan Model 1114.2.2. Case I: Consumer Goods are More Capital-Intensive 114 4.2.3. Case II: Investment Goods are More Capital-Intensive 122 4.2.4. Conclusion 1254.3. Selected References 126

5. Optimal Growth 1285.1. Optimal Growth: Introduction 1285.2. The Ramsey Exercise 1325.3. Golden Rule Growth 1425.4. Intertemporal Social Welfare 1465.4.1.     Intertemporal Social Welfare Functions 1475.4.2.     The Defense of Discounting 1525.4.2.1.   The Tastes Defense 1525.4.2.2.   The Dynastic Defense 1555.4.2.3.   The Decentralization Defense 1585.4.3.     The Koopmans Axiomatization 159 5.4.4.     Population Growth 1685.4.5.     Overlapping Generations 1705.4.6.     Varying Time Preference 1735.4.7.     Intertemporal Justice 1765.4.7.1.   Rawlsian Social Welfare 1765.4.7.2.   Rawlsian Altruism 1795.5. (The Cass-Koopmans Optimal Growth Model) -5.6. (Optimal Two-Sector Growth) 1115.7. Optimal Growth: Conclusion 1815.8. Selected References 184

2

1. Classical Growth Theory: From Smith to Marx 4

1.1. Smith; Ricardo; Marx 41.1.1. Adam Smith 41.1.2. David Ricardo 51.1.3. Karl Marx 61.2. After Marx: Structural Change and Steady States 10

1. Classical Growth Theory: From Smith to Marx

3

1.1 Smith; Ricardo; Marx 

1.1.1 Adam Smith

When Adam Smith wrote his famous 1776 treatise, he called it An Inquiry into Nature and Causes of the Wealth of Nations.  Some have taken this as indicating that he was concerned primarily with economic growth.   In this way, Smith moved away from the Cantillon-Physiocratic system which concentrated on "natural equilibrium" of circular flows, and brought back into economics what had been the Mercantilists' pet concern.

Smith posited a supply-side driven model of growth.  Succinctly we can lay out the story via the simplest of production functions:

Y = (L, K, T)

where Y is output,  L is labor, K is capital and T is land, so output is related to labor and capital and land inputs. Consequently output growth (gY) was driven by population growth (gL), investment (gK) and land growth (gT) and increases in overall productivity (g).   Succinctly:

gY = (g, gK, gL, gT)

Population growth, Smith proposed in the traditional manner of the time, was endogenous: it depends on the sustenance available to accommodate the increasing workforce. Investment was also endogenous: determined by the rate of savings (mostly by capitalists); land growth was dependent on conquest of new lands (e.g. colonization) or technological improvements of fertility of old lands. Technological progress could also increase growth overall: Smith's famous thesis that the division of labor (specialization) improves growth was a fundamental argument.  Smith also saw  improvements in machinery and international trade as engines of growth as they facilitated further specialization. Smith also believed that "division of labor is limited by the extent of the market" - thus positing an economies of scale argument. As division of labor increases output (increases "the extent of the market") it then induces the possibility of further division and labor and thus further growth. Thus, Smith argued, growth was self-reinforcing as it exhibited increasing returns to scale. Finally, because savings of capitalists is what creates investment and hence growth, he saw income distribution as being one of the most important determinants of how fast (or slow) a nation would grow. However, savings is in part determined by the profits of stock: as the capital stock of a country increases, Smith posited, profit declines - not because of decreasing marginal productivity, but rather because the competition of capitalists for workers will bid wages up. So lowering the living standards of workers was another way to

4

maintain or improve growth (although the counter-effect would be to reduce labor supply growth). Despite increasing returns, Smith did not see growth as eternally rising: he posited a ceiling (and floor) in the form of the "stationary state" where population growth and capital accumulation were zero.

1.1.2. David Ricardo Smith's model of growth remained the predominant model of Classical Growth. David Ricardo (1817) modified it by including diminishing returns to land. Output growth requires growth of factor inputs, but, unlike labor,  land is "variable in quality and fixed in supply".  This means that as growth proceeds, more land must be taken into cultivation, but land cannot be "created".  This has two effects for growth:  firstly, increasing landowner's rents over time (due to the limited supply of land) cut into the profits of capitalists from above; secondly, wage goods (from agriculture) will be rising in price over time and this then cuts into profits from below as workers require higher wages. This, then, introduces a quicker limit to growth than Smith allowed, but Ricardo also claimed (at first) that this decline can be happily checked by technological improvements in machinery (albeit, also with diminishing productivity) and the specialization brought by trade, although he also had stationary states.  However, in the third edition of his Principles, Ricardo modified his position on machinery.  He claimed that, in fact, machinery displaces labor and that the labor "set free" might not be reabsorbed elsewhere (because capital is not simultaneously "set free") and thus merely create downward pressure on wages and thus lower labor income.   In order to reabsorb this extra labor without this effect, then the rate of capital accumulation must be increased.  But there is no obvious mechanism for this to happen -- particularly given the tendency described above for profits and thus savings to decline over time.Ricardo's portrait is somewhat more pessimistic than Smith's. The ultimately dismal portrait, however, was painted by T.R. Malthus (1796) with his famous claim that population growth was not so easily checked and would quickly outstrip growth and cause increasing misery all around.  John Stuart Mill improved little upon Ricardo, perhaps only to emphasize the need for control of population growth to put a brake on declining growth and his view of stationary states as wonderful things to achieve.

1.1.3 Karl Marx Karl Marx (1867-1894) modified the Classical picture once again.  For "modern" growth theory, Marx's achievement was critical: he not only provided, through his famous "reproduction" schema, perhaps the most rigorous formulation to date of a growth model, but he did so in a multi-sectoral context and provided, in the process, such critical ingredients as the concept of a "steady-state" growth equilibrium.  We analyze Marx's theory in more detail elsewhere.  Here we are interested in merely sketching the "story" he sought to tell and how it differed from the earlier Classicals.  Firstly, unlike Smith or Ricardo, Marx did not believe that labor supply was endogenous to the wage.  As a result, Marx had wages determined not by necessity or "natural/cultural" factors but rather by bargaining between capitalists and workers and this process would be influenced by the amount of unemployed laborers in the economy (the "reserve

5

army of labor", as he put it).  Marx also saw profits and "raw instinct" as the determinants of savings and capital accumulation.  Thus, contrary to Smith, he saw a declining rate of profit doing nothing to stem capital accumulation and bring the stationary state about, but only as an inducement for capitalists to further reduce wages and thus increase the misery of labor. Like the Classicals, Marx believed there was a declining rate of profit over the long-term.  The long-run tendency for the rate of profit to decline is brought about not by competition increasing wages (as in Smith), nor by the diminishing marginal productivity of land (as in Ricardo), but rather by the "rising organic composition of capital".   Marx defined the "organic composition of capital" as the ratio of what he called constant capital to variable capital.  It is important to realize that  constant capital is not what we today call fixed capital, but rather circulating capital such as raw materials. Marx's "variable capital" is defines as  advances to labor, i.e. total wage payments, or heuristically, v = wL (where w is wages and L is labor employed).   The rate of profits, Marx claimed, are defined as:

    r = s/(v+c)

where r is the rate of profit, s is the surplus, and (v+c) are total advances (constant and variable).  Surplus, s, is the amount of total output produced above total advances, or s = y - (v+c), where y is total output.   It is important to note that for Marx only labor produces surplus value. This was to become a sore point of debate between the Neo-Ricardians and the Neo-Marxians in later years. Marx called the ratio of surplus to variable capital, s/v, the "exploitation rate" (surplus produced for every dollar spent on labor).   Marx referred to the ratio of constant to variable capital, c/v, as the organic composition of capital (which can be viewed as a sort of capital-labor ratio).  Notice that dividing numerator and denominator of r by v we obtain:         r = (s/v)(v/(v+c))) so the rate of profit can be expressed as a positive function of the exploitation rate (s/v) and a negative function of the organic composition of capital (c/v)). Marx then argued that the exploitation rate (s/v) tended to be fixed, while the organic composition of capital (c/v) tended to rise over time, thus the rate of profit has a tendency to decline.  Why?  The basic logic can be thought of as follows. For simplicity, assume a static economy (no labor supply growth).  As the surplus accrues to capitalists and, necessarily, capitalists invest that surplus into expanding production, then output will rise over time while the labor supply remains constant.  Thus, the labor market gets gradually "tighter" and so wages will rise. Thus, v (= wL) rises and r falls.   But this decline in r is temporary. There are forces at work which will restore profit rate   What are these forces?  In Cantillon, Smith, et al., a rise in wages would induce population growth which would then loosen the labor markets and bring wages back down again.  Marx does not accept this story.  For Marx, wages are set by "bargaining" in the labor market.  Thus, there is no "extra supply of labor" being encouraged by the higher wages.  However, Marx argued, capitalists can boost their profit rate back up by introducing labor-

6

saving machinery into production -- thereby releasing labor into unemployment.    There are two effects of this.  Firstly, notice that v declines because labor (L) is released. But, concurrently, the employment of machinery  implies that constant capital, c, rises.  Thus, the introduction in labor-saving machinery does not seem to change anything: the fall in v from less labor is counteracted by the rise in c, so it seems that c/v stays constant.  This is where the second effect kicks in: the concurrent expansion in the unemployed -- the "reserve army of labor" -- will, by itself, influence the labor bargaining process and reduce wages down to subsistence. Thus v declines further.  So, on the whole, the net effect of a labor-saving technology is to raise c/v, i.e. to reduce the rate of profit.   But notice that v declines further because labor is released. So, both the w and the L part of v = wL declines.  But, concurrently, the employment of machinery  implies that constant capital rises, thus c rises.  Thus, the fall in L is counteracted by the rise in c, so that, on the whole, v   So, in sum, the organic composition of capital, c/v, falls.  Profits, consequently, are increased.       Thus, the L part of v = wL declines and so r = s/(v+c) comes back up.  There is a double effect in that, of course, the release of labor is not automatically absorbed by higher investment so that a "reserve army of labor" is created.  In this manner, at the bargaining table, firms will be at an advantage relative to their employees, so that wages decline (or at least are prevented from rising further).    But this is merely a temporary respite.  Profits will be reinvested, output will grow again, labor markets will tighten once more and the whole process will repeat itself.  The problem is that the second time around, there is less labor to lay off.  Recall, L was already reduced in the first round.  Introducing more machinery reduces L further -- and, via several rounds,  further and further -- until there is hardly any more L that can be released.  When the system gets to the point that there are no more laborers to be fired, then there is nothing to bring s/v back up.   The profit rate declines and firms will begin going bankrupt. The bankruptcy of firms means a sudden release of even more labor and capital into the market, depressing prices tremendously.  Firms which remain active will thus be able to buy the bankrupt smaller firms and thus acquire more labor and capital at very cheap rates -- indeed, cheaper than their proper "value".  This increases the   The unemployed, thus, act as a "reserve army of labor" and bring wages back down to a manageable level. However, the introduction of labor-saving capital and laying off of workers means that c rises while v falls, i.e.  the organic composition of capital rises.  It is easy to notice that a constant s/v and a rising c/v  will necessarily reduce the profit rate (to see this, just notice that r can be rewritten as:  r = (s/v)(v/(v+c))).  Thus, there is a natural tendency for the rate of profit to fall.    One way to prevent this decline in r would be to increase the exploitation rate in proportion to which variable capital declines relative to constant capital. The manner of increasing the exploitation rate, Marx claimed, was up to the

7

devilish imagination of the capitalist. Technological progress in the form of machinery or division of labor were not wholly beneficial ways of improving growth either.  Marx took on Ricardo's idea that machinery is labor-saving and leads to a disproportional adjustment: the rate of release of labor does not accompany the rate of re-absorption of that labor, so that there tends to be permanent "technological" unemployment which can be used to bring down the wage. One does not even need to undertake it: technological improvement is also a way capitalists can increase their leverage over labor merely by threatening it with mechanization.  Whereas Marx contended that division of labor was a way of generating the "alienation" of the working classes and thus tie them more dependently to the production process - thereby, again, reducing the bargaining position of labor. The issue of trade, another possible check to the decline in profit rate, was seen by Marx as an inducement to produce on an even greater scale - thereby increasing the organic composition of capital further (and reducing profit quicker). The connection between trade with non-capitalist economies to prevent of the decline in profit rate was for later Marxians like Rosa Luxemburg (1913) to propose in their theories of imperialism.  However, despite all their efforts, Marx claimed that there were social limits to the extent to which capitalists could increase the exploitation rate, while no such thing limited the growing organic composition of capital. Consequently, Marx envisioned that greater and greater cut-throat competition among capitalists for that declining profit.  Then a crisis occurs: large firms buy up the small firms at cheaper rates (i.e. below , and thus the total number of firms declines.  This will boost the surplus value as firms can now purchase capital  As capital becomes more concentrated in fewer .  The increasing   increasing the tendency for capital to be concentrated in fewer and fewer hands, combined with the greater misery of labor would culminate in ever greater "crises" which would destroy capitalism as a whole. Marx had only temporary "stationary states", punctuating the secular tendency to breakdown.

1.2. After Marx: Structural Change and Steady States

Marx's frightening vision did not carry over into Neoclassical theory. But then, it is hard to say the early Neoclassicals had a substantial theory of growth at all. The possible exception was Marshall, but even he improved little upon the Classical system (of Smith and Ricardo, not Marx). That was only to be really developed in later years.

Concern with growth was then largely confined to the German and English Historical Schools, although these thinkers did little more than improve the recording and collection of facts on economic history. They did explore, for instance, institutional and cultural roots of productivity and factor changes (especially regarding population growth and the social-cultural habits that induced capital accumulation), but the essence of their system were only footnotes to the Classical theory. The American Institutionalists also did little beyond this - except that their massive empirical efforts on business cycles and national incomes accounts might have spurred new interest into the

8

phenomenon of growth. Simon Kuznets, in particular, was instrumental in this respect.

However, in the 1920s and 1930s, three new sets of stories emerged which improved upon the Classical theory substantially. They all drew, to a good extent, from Karl Marx's theoretical schema that had been channeled by a European tradition that ran through Tugan-Baranovsky, Spiethoff and Aftalion.   Specifically, two themes ran through the new stories: firstly, that the economy should be considered explicitly in its disaggregated, multi-sectoral structure; secondly, the concept of a steady-state growth path is introduced as a a reference point for such an economy. 

The first of these "structural" theories was that developed by Joseph A. Schumpeter in his 1911 classic, Theory of Economic Development and then further explored later on in his Business Cycles (1939) and his Capitalism, Socialism and Democracy (1942). His system was, again, supply driven: the main secular engine of growth was the increase in factor supplies. The difference, however, was Schumpeter's resurrection of Smith's concern with the entrepreneur as an innovator who improved growth by efficiently combining resources, adopting new technical improvements in machinery and conducting the division of labor.

Schumpeter's starting point is the steady state, or rather, a smoothly expanding economy. Unlike Smith, his population growth was exogenous and his savings rate rather constant or, at best, a residual and not a driver of growth - he was not very much concerned with distribution. In Schumpeter's view, the driver of "development" (as opposed to boring "growth") were discontinuous punctuated changes in the economic environment. These, he claimed, were brought about by a variety of things (e.g. sudden discoveries of new factor supplies), but entrepreneurial innovation was the central one.

The entrepreneur's innovations drive development but their motive, like Marx had argued, was "raw instinct" - profit-derived wealth being merely an "index" of that instinct. Innovation, again like Marx, was not wholly exogenous: quite the contrary, competition for small profits "induced" entrepreneurs to innovate, whereas uncompetitive periods with high profits were a brake on the rate of innovation.

It only takes a few leaders. From a steady economy, a technical innovation by a single entrepeneur opens up new profitable avenues - therefore, more entrepreneurs are induced to innovate, thereby increasing the profits in the economy as a whole, thereby driving growth. But as as the "supply of entrepreneurs" in any generation is numerically exhausted, capitalists turn upon each other and compete away the existing profits. Profits begin to decline and the economy slows down. However, the decline in profits will eventually, again, induce those with entrepreneurial inclinations to once again innovate.

One may think of this as more of a cycle theory than a growth theory, but Schumpeter claimed that there were ratchet effects in innovation so that entrepreneurial-driven spurts of economic activity led to progressively higher

9

levels of income. And there is no long run need to slow down: unlike Ricardo, Schumpeter claimed that there were no diminishing returns to innovation. The only reason one may be driven towards slower, steady-state is that all the entrepreurs in a generation might be already "used up".

There are also institutional preconditions for innovation: a capitalist system (private ownership of property) was one, existence and availability of plentiful credit is another. Like Wicksell, Schumpeter abandoned Say's Law and claimed that credit made present activity independent of past activity and thus enabled entrepreneurship. Hence, since entrepreneurial innovation could be arrested by lack of credit, then financial innovation was also an important factor for increasing growth.

Although he did not have diminishing returns to innovation, Schumpeter did have long-run elements in his theory which induced a breakdown in growth. These are rooted in social-cultural changes: enterprise may grow to the point entrepreneurial function may be replaced by bureaucratic managers who are less apt to innovate; growth uncovers economies of scale and may lead to permanently high industrial concentration and high profits (which again, are a brake on innovation); also, entrepreneurial activity will be progressively viewed as "bad" because capitalism leads to the breakdown of social and family relations and alienates the bourgeoisie and, in particular, is despicable to intellectuals who are highly influential upon public attitudes; this negative view of rapacious entrepreneurship will then conspire, culturally speaking, to diminish the supply of entrepreneurs. (Also, that same breakdown in the family may also take away from the "dynastic" aspirations which often lies behind the "raw instincts" of entrepreneurs).

The concept of "steady-state" was still primitive in Schumpeter.  It was given more precision in the second set of theories we consider, namely the "steady-state" multi-sectoral growth theories of Gustav Cassel (1918) and John von Neumann (1937).  Both Cassel and von Neumann presented growth models which are akin to Marx's reproduction scheme in many respects but differed essentially in the absence of "crisis".  One can argue that it was probably more inspired by the general equilibrium theory of  Léon Walras (1874), who also referred to the concept of steady state growth in his theory of capital.

John von Neumann, in particular, followed the Classical idea that surplus is the determinant of growth but, contrary to the Classicals, did not concern himself with any falling rates of profit. John von Neumann's concern was in the formalization of steady-state growth, but without reference to any Classical constraints that might bring the surplus down and bring the economy to a stationary state without growth. To some extent, this was due to the fact that, as a mathematician, von Neumann abstracted much from the "social considerations" that often went into the Classical theories, i.e. he did not concern himself with possible resource constraints presented by land, or changes in fertility, or "entrepreneurial" behavior or any other such concepts. His exercise was a thoroughly mathematical one -- foreshadowing the later formalization of the Classical theory by Sraffa and Leontief in many ways.  

10

As a result, the dynamic models of Gustav Cassel and John von Neumann have a perpetual multi-sectoral, steady-state growth rate which they saw as perpetual and constant.   They identified the rate of growth to be identical to the rate of profit - the "Golden Rule" already implicit in the Classicals, Schumpeter and Walras. For more on all this, see our reviews of the Walras-Cassel model and the von Neumann system.

The final set of growth theories that emerged in the 1920s and 1930s are the "structural" theories of growth developed by the Soviet economist Grigory Fel'dman (1928) and the Kiel School (e.g. Adolph Lowe (1926, 1954, 1976),  Fritz Burchardt (1928, 1931), Alfred Kähler (1933), Emil Lederer (1931), Hans Neisser (1933, 1942), Wassily Leontief (1941)).  They effectively take the story up where Cassel-von Neumann drop off.  They are more explicitly indebted to Marx's theory, particularly his schema of extened reproduction and his recognition of technological unemployment. 

The Kiel School was particularly interested in what happens off the steady-state path.  They focus on technological change as the big crucial variable that is constantly leading to increases in the rate of return on capital and thus higher investment.  The difference is that the resulting growth is not steady, but rather "disproportionate".  For instance, after technical progress, investment goods sector output increases while that of consumer goods industries lags behind, leading thereby to changes in relative prices during the process of adjustment.   These changes in relative prices can lead to technological unemployment in certain industries (e.g. consumer goods) while growth proceeds at bursting speed in others.  There is, as Neisser expressed it, "a race" betwen technical displacement of labor in some sectors and the rate of absorption of labor in other sectors from capital accumulation.  Notice that "traditional" recipes for curing unemployment, e.g. lowering wages or stimulating demand, will not affect the technological unemployment problem as these are aggregate measures, not designed for the specific sectoral problems.   As they Lederer noted:

"The primitive conception that, whenever unemployment exists one could always restore equilibrium by a reduction in wages belongs into the junk-room of theory" (E. Lederer, 1931: p.32)

Several aspects of the structural theories of growth of the Kiel School were absorbed by dynamic input-output models (e.g. Leontief , 1953;  Samuelson and Solow, 1953; Morishima, 1964, 1973).   They were more directly influential on the development of Friedrich von Hayek's (1928, 1931) theory of macrofluctuations and  John Hicks's (1973) theory of the disequilibrium growth "traverse".

11

2. Keynesian Growth: The Cambridge Version 16

12

2. Keynesian Growth: The Cambridge Version

The heroic entrepreneurs of Schumpeter are resurrected, only slightly less heroically, in The General Theory (1936) of J.M. Keynes. Investment, in the Keynesian system, is an independent affair contingent upon finance and the "animal spirits" of entrepreneurs.

The issue is that Keynes did not extend his theory of demand- determined equilibrium into a theory of growth. This was left for the Cambridge Keynesians to explore. The first to come up with an extension was Sir Roy F. Harrod who (concurrently with Evsey Domar) introduced the "Harrod-Domar" Model of growth (Harrod in 1939, Domar in 1946).

Recall, from Keynes, that investment is one of the determinants of aggregate demand and that aggregate demand is linked to output (or aggregate supply) via the multiplier. Abstracting from all other components, we can write that, in goods market equilibrium:

Y = (1/s)I

where Y is income, I investment, s the marginal propensity to save (and thus the multiplier is 1/s). But investment, note Harrod and Domar, increases the productive capacity of an economy and that itself should change goods market equilbrium.

For "steady state" growth, in the language of Harrod-Domar, aggregate demand must grow at the same rate as the economy's output capacity grows. Now, the investment-output ratio, I/Y, can be expressed as (I/K)(K/Y). Now, I/K is the rate of capital accumulation and K/Y is the capital-output ratio (call it "v"). Thus, the rate of capital accumulation, I/K, is the rate of capacity growth (call that "g"). Thus, for steady state it must be that I/K = (dY/dt)/Y = g (i.e. the rate of capital accumulation/capacity growth, I/K, and the real rate of output growth (dY/dt)/Y, must be at the same rate, g). Thus, plugging in our terms:

I/Y = (I/K)(K/Y) = gv

13

But recall our goods market equilibrium term from the multiplier, i.e. Y = (1/s)I which can be rewritten I/Y = s. Thus, the condition for full employment steady-state growth is gv = s, or simply:

g = s/v

Thus, s/v is the "warranted growth rate" of output. However, Harrod and Domar originally held s and v as constants - determined by institutional structures. This gives rise to the famous Harrodian "knife-edge": if actual growth is slower than the warranted rate, then effectively we are claiming that excess capacity is being generated, i.e. the growth of an economy's productive capacity it outstripping aggregate demand growth. This excess capacity will itself induce firms to invest less - but, then, that decline in investment will itself reduce demand growth further - and thus, in the next period, even greater excess capacity is generated.

Similarly, if actual growth is faster than the warranted growth rate, then demand growth is outstripping the economy's productive capacity. Insufficient capacity implies that entrepreneurs will try to increase capacity through investment - but that that itself is a demand increase, making the shortage even more acute. With demand always one step ahead of supply, the Harrod-Domar model guarantees that unless we have demand growth and output growth at exactly the same rate, i.e. demand is growing at the warranted rate, then the economy will either grow or collapse indefinitely.

The "knife-edge", thus, means that the steady-state growth path is unstable: the only stable growth path, the "knife-edge", is where the real growth rate is equal to s/v permanently. Any slight shock that will lead real growth to deviate from this path ensures that we will not gravitate back towards that path but will rather move further away from it.

It was up to Nicholas Kaldor (1955, 1957) to rescue this by proposing that savings are variable and would "jump" to the value necessary to bring the actual growth rate back into its warranted path. To justify this, Kaldor had to employ Classical considerations of income distribution with two classes: capitalists (who save a portion of their profits) and workers (who save from wages). Thus, letting s be the capitalists' propensity to save and s' be the workers', then total savings are:

S = sP + s'W

where P are profits and W are wages. Naturally, W + P = Y, total income is made up of profits and wages, so W = Y - P. As capitalists are assumed to save more than workers, s > s', then obviously savings are positively related to the share of profits in income, P/Y.

For goods market equilibrium, it must be that investment is equal to savings, I = S. Following the Keynesian axiom that investment is independent, then investment determines savings (or, to word it differently, aggregate demand

14

determines aggregate supply). However, as noted profits are positively related to savings. Hence, by substitution:

I = sP + s'(Y - P)

which rearranging yields:

P/Y = [1/(s-s')](I/Y) - s'/s-s'

In other words, given the marginal propensities to save of each class, the relative size of profits in income is dependent only on the investment decision, I/Y. Naturally, the more investment, the greater the necessary slice profit takes out of income.

If we assume workers save nothing, so that s' = 0, then we quickly reach the conclusion that:

P/Y = (1/s)I/Y

where P/Y depends on I/Y. Note that this is reminiscent of Keynes' famous "widow's cruse" remark:

"However much of profits entrepreneurs spend on consumption, the increment of wealth belonging to the entrepreneurs remains the same as before. Thus, profits, as a source of capital increment for entrepreneurs, are a widow's cruse which remains undepleted, however much be devoted to riotous living" (J.M.Keynes, Treatise on Money, 1930: p.139)

Or any attempt by capitalists to increase their consumption (and thus reduce savings), will merely result in increased profits - thereby generating the savings to make up for their initial decline. Or, as Kaldor (1955) reminds us, this is merely Kalecki's adage that "capitalists earn what they spend and workers spend what they earn".

What if we are not in goods market equilibrium? Suppose we have excess demand for goods so that I > S, then investment has generated a level of profits are too low for equilibrium, i.e. capitalists have not saved enough. Consequently, as pressure is placed on the goods market, prices will rise and, assuming wages are constant, real wages will fall, increasing the share of profits in income. Thus, P/Y rises, which in turn increases savings, and so on until equilibrium is re-established.

What about growth? Recall that I/Y = (I/K)(K/Y), where I/K is the rate of capital accumulation (equal to the rate of growth of productive capacity, g) and K/Y is the capital-output ratio (v). Thus, we can write I/Y = gv.

Now recall Kaldor's relationship, P/Y = (1/s)I/Y. Thus:

P/Y = gv/s

15

so that g = (s/v)P/Y. Recalling that v = K/Y, then this can be rewritten:

g = s(P/K)

But we should note that the ratio P/K is merely the rate of profit, r. Calling it thus, we can rewrite:

r = g/s

the rate of profit is equal to the growth rate divided by the savings rate of capitalists - which is also known as the "Cambridge rule" for growth. In a von Neumann model, recall, workers consumer everything (as here), but he also has it that capitalists save everything (so s = 1). But note that in this case, we have r = g, or "Golden Rule" growth. Thus, we immediately see the affinity between Cambridge growth models and von Neumann growth models. Morishima's (1960, 1964) extension of von Neumann models which allowed for capitalist consumption produces precisely this "Cambridge rule" for von Neumann.

Joan Robinson (1962) recommended a modification so as to understand the properties of this model better. We have not really discussed what determines investment: we simply posited a full employment relationship, i.e. I/Y = gv, so as to obtain Kaldor's steady-state. But surely, in a Keynesian world, an independent investment function should remain independent! Robinson (1962) posited a relationship I/Y = f(P/Y) or g = f(r), where investment decisions by firms were functions of (expected) profit. She argued that this was a concave function, based on Kalecki's (1937) principle of increasing risk: investment is positively related to expected profit, but at a decreasing rate - as every extra unit of investment means greater debt and thus greater risk to the firm.

However, we know from the Kaldor relationship, P/Y = (1/s)I/Y or r = g/s, that profits are themselves generated by investment. Thus, Robinson's question can be asked: when is it true that the profits generated by the investment in the Kaldor relationship will themselves generate investment decisions that, in turn, generate the original profits? Alternatively, what is there that guarantees that the profits generated by the Kaldor relationship will themselves generate the amount of investment needed to sustain them? This is a question of stability.

16

Robinson's (1962: p.48) diagram above of the concave Kalecki function and the linear increasing risk function is reproduced below. Assuming all is well, then we should have two equilibria where rs = g = f(r). Consider the rightmost equilibrium first. To the right of that equilibrium, Robinson posited that the economy was generating less profits than planned and thus investment plans will be shelved, inducing deaccumulation of capital and hence reducing growth. To the immediate left of it, the economy is generating more profits than planned, and thus firms will revise their expectations upwards and invest more, thereby increasing accumulation and growth. Hence, the right equilibrium is stable. A similar exercise will show that the left equilibrium is, for the same reasons, unstable.

Robinson (1962) went on to enrich her analysis by introducing labor growth and to consider the implications of including unemployment and inflation and the method of adjustment explicitly in the model. She discusses the various types of growth situations that could be encountered - Golden Rule and otherwise.

Another extension was provided by Luigi Pasinetti (1962). It is unlikely that workers do not save, as we have assumed. Originally, Kaldor (1955) proposed that workers did save out of wages, but less than capitalists - in which case, profits would be more sensitive to the investment decision than we have allowed. However, Pasinetti (1962) called this "a logical slip". If workers can save, we should conceive of two different "types" of capital falling under different ownership: "workers' capital" and "capitalists' capital". Let us call the former K' and the latter K. Thus total savings are S = sP + s'(P' + W), workers save out of both profits and wages.

It is necessary that workers be paid a rate of interest on their capital just in the same manner as capitalists receive a rate of profit on theirs. By competition and arbitrage, Pasinetti argued that the rate of profit/interest for both capitalist and workers on their capital is equalized. Or:

17

P/K = P'/K' = r

where P' is workers' profits. For savings, let S be capitalist savings and S' worker savings out of profits. Therefore, for steady state growth:

S/K = S'/K' = g

In the long-run, for steady-state, it must be that the rate of accumulation must be equal for both capitalists and workers, i.e.

P/S = P'/S'

otherwise, if the rate of wealth accumulation is faster for either of the classes, then there will be a change in distribution and, as a result, a change in the composition of aggregate demand. In long-run equilibrium, aggregate demand must be stable therefore this is a necessary assumption.

However, as a consequence of this assumption, we can note that:

P/sP = P'/s'(W + P')

where s and s' are the marginal propensity to save of capitalists and workers. Note again that workers also save out of wages, W, as well as profits, P', whereas capitalists only receive and save out of profits. Cross-multiplying:

s'(W + P') = sP'

Now, if investment (I) is equal to total savings which means that:

I = s'(W + P') + sP

then using our previous relationship:

I = sP' + sP = s(P + P')

Let us call total profits P* = P + P', then I = sP* or:

P* = (1/s)I

So it must be that:

r = (P*/K) = (1/s)I/K = g

i.e., for long run Golden Rule steady-state growth, only the capitalist's propensity to save needs to be considered - workers' saving propensities can be dropped by the wayside. Thus, even with worker savings, the "Cambridge rule" is iron-clad. Only capitalists' savings propensity matters. As Pasinetti notes:

18

"In the long run, workers' propensity to save, though influencing the distribution of income between capitalists and workers, does not influence the distribution of income between profits and wages. Nor does it have any influence on the rate of profit!" (L.Pasinetti, 1962)

But there were important assumptions in the model yet undiscussed. Pasinetti posits one of his conditions to guarantee existence to be:

s > I/Y > s'

so that profits cannot take "a null or negative share of wages" (Pasinetti, 1962). This, in essence, defines the mechanism for adjustment. If distribution can be somehow organized such that there will be a "correct" level of profits to give us the savings necessary to be in equilibrium: i.e. make I/K = s/v. The first question that must be asked here is not only whether you can calculate for a given investment level what the profit level will be but whether there will be pressures that might bring this into equilibrium. Within certain limits, Kaldor argues, variations can take place such that P/Y is a function of the change in the I/Y ratio. According to Kaldor, prices respond to relative money wage rates as a consequence of demand. Assume, for instance, that given an excess demand for goods, prices will increase but not wages. As a consequence there is a shift in distribution such that there will be an increase in the profit share. Since profits increase, this implies there will be a substantial growth in savings.

However, as J.E. Meade (1961) points out, if prices rise relative to wages, then the real wage decreases. By substitution between capital and labor, there will be a change in the capital-output ratio (v). Therefore, for Kaldorian adjustment to be applied, there is an implicit dependence on a constant capital-output ratio. However, a constant v necessarily means that we cannot be in long-run equilibrium since technique would otherwise be entirely flexible. One can perhaps regard at it as a vintage model, but here prices would have to change faster than wages. The greatest difficulty in this model, nevertheless, remains the adjustment towards the steady-state path. How do profits adjust so that one will achieve the steady-state savings rate? According to Kaldor, prices respond to relative money wage rates as a consequence of demand. Assume, for instance, that given an excess demand for goods, prices will increase but not wages. As a consequence there is a shift in distribution such that there will be an increase in the profit share. Since profits increase, this implies there will be a substantial growth in savings.

However, as J.E. Meade (1961, 1963, 1966) points out, if prices rise relative to wages, then the real wage decreases. By substitution between capital and labor, there will be a change in the capital-output ratio (v). Therefore, for Kaldorian adjustment to be applied, there is an implicit dependence on a constant capital-output ratio. However, a constant K/Y necessarily means that we cannot be in long-run equilibrium since technique would otherwise be entirely flexible.

But a more general criticism can be made. We can note that given a stock of capital, labor and output, if prices move faster than wages, then profits will

19

increase whereas if wages move faster than prices, then profits will fall - without changing techniques. The variety of consequences of this has led several economists, such as Meade (1961) and, later, Nell (1982), to argue that at least for a long-run model, Kaldor's theory has a rather poor price-adjustment mechanism. "Mr. Kaldor's theory of distribution is more appropriate for the explanation of short-run inflation than of long-run growth." (Meade, 1961: x).

3. Crecimiento Neoclásico 26

3.1. La Toría del Crecimiento Neoclásico 263.1.1. Introdución 263.1.2. El Modelo Solow-Swan 273.1.3. Adición de la Depreciación 323.1.4. Resolución del Sistema 33

20

3.1.4. 1. La Solución tipo Cobb-Douglas 333.1.4. 2. La Solución General 363.1.5. El Procerso de Ajuste: Solow vs. Harrod 373.2. Implicaciones Empíricas 443.2.1. Introdución 443.2.2. La Paradoja de Solow 453.2.3. La Hipótesis de la Convergencia 493.2.3.1. Convergencia Absoluta 493.2.3.2. Convergencia Condicional 513.2.4. Trampas de Pobreza 523.2.4.1. La Trampa Tecnológica 523.2.4.2. La Trampa de la Población 583.3. El Cambio Técnico 603.3.1.     Adición del Cambio Técnico 613.3.2.     Implicaciones Empíricas 673.4. Bibliografía 69

3. Neoclassical Growth

3.1 The Neoclassical Growth Theory

____________________________________________________________________

"I thought that the mere introduction of labor as another factor in a simple production function would be a rather minor improvement, and that any serious attempt to make these models more realistic should require a very complex production function with a great degree of disaggregation. On the whole this is true, and yet a recent article by Robert M. Solow, which appeared in print just as I was writing these lines, has shown how a growth model can be enriched by the use of a not very complex but less rigid production function."

(Evsey Domar, Essays in the Theory of Economic Growth, 1957: p.8)

21

____________________________________________________________________

3.1.1. IntroductionIn the Harrod-Domar growth model, steady-state growth was unstable. In the popular term of the day, it was a "knife-edge" in the sense that any deviation from that path would result in a further move away from that path. However, Robert M. Solow (1956), Trevor Swan (1956) and, a bit later, James E. Meade (1961) contested this conclusion. They claimed that the capital-output ratio of the Harrod-Domar model should not be regarded as exogenous. In fact, they proposed a growth model where the capital-output ratio, v, was precisely the adjusting variable that would lead a system back to its steady-state growth path, i.e. that v would move to bring s/v into equality with the natural rate of growth (n). The resulting model has become famously known as the "Solow-Swan" or simply the "Neoclassical" growth model. A brief word or two on historical precedence is warranted. James Tobin (1955) introduced a growth model similar to Solow-Swan which also included money (and thus a predecessor of the monetary growth theory). However, Tobin did not solve explicitly for the stability of the steady-state. Also, it is become increasingly common to credit Jan Tinbergen (1942) as presenting effectively the same model as Solow-Swan, including even empirical estimates of the relevant coefficients. Finally, Harold Pilvin (1953), before Solow, had argued that the Harrodian knife-edge problem could be resolved if a flexible capital-output ratio were introduced, but he did not formulate the concept of a steady-state. Harrod's (1953) response to Pilvin is quite instructive in stressing that flexible technology does not, in fact, resolve the Harrodian knife-edge as he originally conceived it. But more on this later.

3.1.2. The Solow-Swan Growth ModelFor the analysis, let us begin with the macroeconomic equilibrium condition that aggregate demand equal aggregate supply, Yd = Y. This translates, automatically, into claiming that investment equals savings, i.e. I = S. Now, according to the simplest of consumption functions, C = cY, where c is the marginal propensity to consume. Now, by definition, savings are S = Y - C = Y - cY or simply S = (1-c)Y. Letting s = (1-c), the marginal propensity to save, then we can express savings as some proportion of total output,

S = sY

so combining this with the macroeconomic equilibrium condition:

I = sY

Dividing through by L, the amount of labor in the economy, then:

I/L = s(Y/L)

so, letting i = I/L and y = Y/L, we see that the macroeconomic equilibrium condition becomes:

22

i = sy

Now, aggregate supply (output) is given by a production function of the general form:

Y = F(K, L)

which is assumed to vary continuously with K and L (i.e. smooth substitutability) and exhibits constant returns to scale. Dividing through by L:

Y/L = F(K/L, 1)

or, letting k = K/L, we can rewrite this as:

y = (k)

where (·) is the "intensive" or "per capita" form of the production function F(·). As a result, the macroeconomic equilibrium condition can be rewritten as:

i = s(k)

This can be thought as representing equilibrium investment per person. If we assume that macroeconomic equilibrium holds at all times (i.e. I = S always), then i = s (k) can also be referred to as the actual investment per person.Figure 1 depicts the intensive production function y = (k) and the actual (equilibrium) investment function, i = s (k). Notice that at any k, we can derive investment per person, i, output per person, y, and, residually, consumption per person (c = C/L = y - i). The slope of the intensive production function is k

= (k)/ k which happens to also be the marginal product of capital, i.e. k = FK (see our discussion of intensive production functions). Finally, notice that the capital-output ratio, v = K/Y = k/y, is captured as the slope of a ray from the origin to production function. Thus, changing k will change the ray and thus v. So, unlike the Harrod-Domar model, v is not exogenously fixed.

23

 

Fig. 1 - Intensive Production Function

So, let us turn to growth. By assumption, we assume population grows exogenously at the rate n, i.e.

gL = (dL/dt)/L = n

If there is no investment, then k = K/L will automatically fall as population grows. So, for k to be constant, there must be investment (i.e. capital must grow) at rate n:

gKr = (dK/dt)/K = n

where we have attached the superscript "r" to indicate that this is the required growth rate of capital to keep the capital-labor ratio, k, steady. As investment is defined as I = dK/dt, then we can rewrite this as:

Ir = nK

where Ir is required investment. Dividing through by labor, L, Ir/L = nK/L, or:

ir = nk

which is the required investment per person to maintain a steady k. Why are we obsessed with keeping k constant? Well, we are interested in "steady-state" growth which, as defined by Gustav Cassel (1918), means

24

"proportional" growth in a manner that there are no induced changes in relative prices over time. It is obvious (from Figure 1, for instance) that a change in k will change the marginal products of capital and labor. Assuming the marginal productivity theory of distribution holds, so that capital and labor are, in equilibrium, priced at their marginal products, then if we allow k to change over time in our solution, then we are also allowing relative factor prices to fluctuate over time. As Cassel's definition of a "steady-state" growth equilibrium does not allow this, then, consequently, we must focus on getting k to stay constant.We can depict the steady-state k in Figure 2, by superimposing the required investment function, ir = nk, on top of our old diagram. Notice that only at k* is actual investment equal to required investment, i = ir. At any other k, i ir.

Fig. 2 - Steady-State Growth

It is a simple matter to note that not only is k* our steady-state capital-labor ratio, it is also a stable capital-labor ratio. If our initial capital labor ratio is below k* (e.g. at k1), then actual investment is greater than required investment, i > ir, which means that capital is actually growing faster than labor, so k will increase. Conversely, if our initial k is above k* (e.g. at k2), then actual investment is below required, i < ir, so capital is growing slower than labor, so k will fall. Thus, the steady-state capital-labor ratio, k*, is stable in the sense that any other k will have the tendency to approach it over time. We can capture the entire stability story in terms of a simple differential equation as follows:

dk/dt = i - ir

or, plugging in our terms for i and ir:

25

dk/dt = s (k) - nk

which is the fundamental Solowian growth equation. At steady-state, k*, dk/dt = 0 and so (ignoring the trivial origin case), it must be that s (k*) = nk*, i.e. k* is the steady-state. In Figure 3, we depict the phase diagram of the Solowian differential equation. Notice that dk/dt = 0 at steady-state k*. If we are at k1 < k*, then dk/dt > 0, whereas at k2 > k*, then dk/dt < 0. [Note: if we start at the origin, k = 0, equilibrium holds trivially (i = i r = 0). However, we can ignore the origin solution safely, not only because it is not very interesting but also because it is unstable, i.e. if k > 0, then the underlying dynamics drive us away from the origin].

 

Fig. 3 - The Solow-Swan Phase Diagram

3.1.3. Adding DepreciationBefore proceeding, let us insert a slight modification in the model to account for capital depreciation. The macroeconomic equilibrium condition (i = s (k)) remains the same; it is the required investment rate that now becomes different. In order for k to remain constant, then capital must grow not only to accompany population growth but also to cover depreciation of old capital. Specifically, we now have:

ir = (n+ )k

26

as our required investment rate, where is the capital depreciation rate. The fundamental Solowian differential equation needs to be rewritten as:

dk/dt = s (k) - (n+ )k

In terms of Figure 2, all that will happen when we insert capital depreciation is that the required investment line ir will become steeper (with slope n+ ) and so the steady-state ratio k* will be lower (see Figure 4). However, notice that now the growth rates of the level variables -- capital stock, output and consumption -- all rise to (n+).

Fig. 4 - Steady-State Growth with Depreciation

3.1.4. Solving the System

- The Cobb-Douglas Solution- The General Solution

3.1.4. 1. The Cobb-Douglas Solution

The explicit solution to the fundamental Solowian differential equation, dk/dt = s (k) - nk, is particularly easy if we assume a specific form for the production function. Let us assume that F(K, L) is Cobb-Douglas, so that:

27

Y = K L(1- )

where is the elasticity parameter (0 1). In intensive form, the Cobb-Douglas production function can be written as:

y = k

This permits us to rewrite our fundamental Solowian differential equation (without depreciation) as:

dk/dt = sk - nk

which is a non-linear differential equation (see earlier Figure 3). To resolve this, we can linearize it by defining a new term z = k1- , which upon differentiation with respect to t yields:

dz/dt = (1-)(dk/dt)/k

Now, dividing the original Solowian equation by k, we obtain:

(dk/dt)/k = s - nk/k

so, plugging this into our dz/dt term:

dz/dt = (1- )[s - nk/k]

Now, as k/k = k1- = z by definition, then this equation can be expressed as:

dz/dt = (1-)s - (1- )nz

which is a simple linear first order differential equation in z. As we know from simple mathematics, the solution z(t) is:

z(t) = Ce-(1-)nt + z*

where z* is the equilibrium value of z and C is a constant, both of which have to be deciphered. The equilibrium z* is defined where dz/dt = 0, i.e.

(1-)s - (1- )nz* = 0

so:

z* = s/n.

Thus:

z(t) = Ce-(1- )nt + s/n

28

To translate this to obtain the solution k(t), we just re-transform it back, i.e. as z = k1- , then k = z1/(1- ). So:

k(t) = {Ce-(1- )nt + s/n}1/(1- )

To decipher C, we need to assume some initial value of k, call it k(0) = k 0. So, when t = 0:

k(0) = {C + s/n}1/(1- ) = k0

so, rearranging:

C = k01- - s/n.

thus, written out in full, the solution to the Solowian differential equation is:

k(t) = {[k01- - s/n]e-(1- )nt + s/n}1/(1- )

Now, as n > 0 and 0 < < 1 by assumption, then it is evident that this equation is stable, i.e. [k0

1- - s/n]e-(1-)nt 0 as as t , so that in the end, k(t) k*, where:

k* = (s/n)1/(1- )

is the steady-state capita-labor ratio.

3.1.4. 2. The General SolutionFor more general functional forms, the simpler mathematical method of checking the stability of the Solowian differential equation is via Lyapunov's method. For this, let us introduce a Lyapunov function:

V(t) = (k(t) - k*)2

where k* is capital-labor ratio. Notice that if k(t) = k*, when we are in steady-state, then V(t) = 0, but outside of steady-state, k(t) k*, so V(t) > 0. The system is considered "stable" if dV(t)/dt < 0, i.e. as time progresses, the difference between k(t) and k* is reduced. Now, let z = k(t) - k*, so that the function becomes V(t) = z2. So differentiating V(t) with respect to time:

dV(t)/dt = 2z·(dz/dt)

29

As dz/dt = dk(t)/dt, then by the fundamental Solow-Swan differential equation, dk/dt = s(k) - nk, then:

dV(t)/dt = 2z·[s (k(t)) - nk(t)]

or as k(t) = z + k*, then:

dV(t)/dt = 2z·[s (z + k*) - n(z +k*)]

Now, the concavity of the production function implies that [ (z+k*) - (k*)]/z (z), or (z+k*) (k*) + z(k*), so:

dV(t)/dt 2z·[s (k*) + sz (k*) - n(z +k*)]

Since s (k*) = nk* (by definition of steady-state equilibrium), then this reduces to:

dV(t)/dt 2z·[sz (k*) - nz]

Now, as s (k*) = nk*, then s = nk*/ (k*), so:

dV(t)/dt 2z·[znk* (k*)/ (k*) - nz]

factoring out nz/ (k*):

dV(t)/dt [2nz2/ (k*)]·[k* (k*) - (k*)]

or factoring out -1:

dV(t)/dt -[2nz2/ (k*)]·[ (k*) - k* (k*)]

Notice that as we have assumed constant returns to scale, then the term [ (k*) - k* (k*)] is the marginal product of labor, which is positive, while the term [2nz2/ (k*)] is unambiguously non-negative, thus:

dV(t)/dt < 0

and thus we know that the steady-state capital-labor ratio k* is globally stable. In other words, beginning from any capital-labor ratio (other than 0), we will converge to the steady-state ratio k*.

3.1.5. Adjustment Processes: Solow vs. Harrod

The argument of the Solow-Swan growth model, as we have presented it, seems straightforward enough. But it gives the impression that the reason k adjusts to steady-state k* comes out merely from the "technical" aspects of the model -- from the properties of a constant returns to scale production function and national income accounts identities. If there is an "economic theory" in there, it seems to be well-disguised.

30

But there is a lot of economics in it -- we just have to look at the details. From the outset, the first thing we need to fix in our minds is that the production function (k) is not merely a "technical" thing; it contains within it an intricate and completely Neoclassical economic theory.

To see the underlying argument, it is useful to compare the Solow-Swan model with the Harrod-Domar model. Let us recall that the Harrod-Domar problem exhibited two knife-edges: the balance between the actual and warranted rates of growth ("macroeconomic stability") and the balance between warranted and natural rates of growth ("employment stability"). The Solow-Swan model does not address macroeconomic stability but only employment stability.

The question of macroeconomic stability is ignored in Solow-Swan by the assumption that planned investment equals planned savings at all times. No consideration whatsoever is paid to the underlying "macroeconomic" adjustment process that makes this true. But the "knife-edge" Harrod and Domar focused on was precisely that one. As Frank Hahn notes:

"It will be noted straightaway that [Solow's] argument has no bearing on Harrod's knife-edge claim. Harrod had not proposed that warranted paths diverge from the steady state but that actual paths did. The latter are neither characterized by a continual equality of ex ante investment and savings nor by continual equilibrium in the market for labour. Thus although Solow thought he was controverting the knife-edge argument he had only succeeded in establishing the convergence of warranted paths to the steady-state." (F.H. Hahn, 1987)

The intricacies of macroeconomic adjustment, explicit theories of interest and expectations, which were the main concerns of Harrod and Domar, are completely missing in Solow-Swan. And the cost of ignoring them is high. As Frank Hahn (1960) has himself demonstrated, when even the slightest attention is paid to the underlying macroeconomic adjustment of a Solow-Swan model in a proper manner, the stability of its steady-state can be cast into serious doubt.Still, let us turn to the second Harrodian knife-edge, "employment instability". We can translate the Solow-Swan model into Harrod-Domar terms as follows. Recall that 1/v is the slope of a ray from the origin to the intensive production function. There is a different 1/v ray for every k. Let v* be the capital-output ratio associated with the steady-state k*. Now, at steady state, we know that s (k*) = nk*, or:

(k*)/k* = n/s

But notice that (k*)/k* = 1/v*, thus at k*:

s/v* = n

31

which is the Harrod-Domar equilibrium for the second knife-edge. Now, let v1

be the capital-output ratio associated with the disequilibrium capital-labor ratio k1 (in Figure 1). Obviously, (k1)/k1 = 1/v1. As n and s are constant and 1/v1 > 1/v* (slope of the ray associated with k1 is steeper than that associated with k*), then we know 1/v1 > n/s, or simply:

s/v1 > n.

which is a Harrod-Domar employment disequilibrium. However, we know k* is stable, so k1 approaches k*, implying that v1 falls to v*, so that the ratio s/v1

declines to meet n. Thus, it is through adjustments in the capital-output ratio, v, that the Solow-Swan model "solves" Harrod-Domar's second instability problem.

Fig. 1 - Adjustment to Steady-State

What is the exact economic mechanism that permits this? Nothing less than assuming that the factor markets clear at all times and that the marginal productivity theory of distribution holds. Underlying the Solowian production function is a detailed factor market adjustment process. To see this in action, examine Figure 2. For a constant returns to scale intensive production function, at any k, the slope of a tangent ray is k = Fk, the marginal product of

Fig. 2 - Factor Price Adjustment

capital while the intercept of the tangent ray on the vertical axis is merely y - kk = FL, the marginal product of labor.

32

Let r denote the real rate of return on capital and let w denote the real wage. By the Neoclassical theory of distribution, the marginal productivity of a factor will constitute the demand for that factor. In equilibrium, factor demand equals factor supply, and thus at the market-clearing factor prices, Fk = r and FL = w. Thus, in Figure 2, we denote the slope of the tangent ray as r and the intercept on the vertical axis as w. One more thing can be deciphered from the diagram: namely, the point where the tangent curve intersects the horizontal axis in the left quadrant yields the factor price ratio, = w/r.

So now let us turn to the Solow-Swan adjustment process. Suppose we begin at k1, as shown in Figure 2. The underlying market-clearing factor prices are r1

and w1, or in ratio form, 1. However, we know that k1 is not a steady state, so, eventually, k1 will rise to k2. In this movement, the supply of capital increases more than the supply of labor, implying that if factor markets are to continue to clear, then the rate of profit must fall (r1 declines to r2) and the wage must rise (w1 increases to w2). In the movement from k1 to k2, the factor-price ratio rises from 1 to 2. This is exactly what we see when comparing the intercepts of the tangent lines in Figure 2.To see the factor market adjustment process more clearly, it is useful to translate Figure 2 into isoquant form, which we do in Figure 3. Beginning with the given amounts of capital, K1 and L1, we have factor market clearing, as the profit-maximizing firm will produce Y1 and the wage-profit ratio, 1 is precisely what is needed for this. Thus, point 1 = (L1, K1) in Figure 3 corresponds to position k1 in Figure 2.Now, suppose capital supply increases to K2 and labor supply to L2. Obviously, as we see in Figure 3, capital increases a lot more than labor, so the new factor supply position will be at point 2 = (L2, K2), and thus the capital-labor ratio is higher (on ray k2). Now, suppose market prices do not adjust. In other words, suppose that the wage-profit ratio remains 1. In Figure 3, this is represented by the isocost line 1 , which has the same slope as the isocost curve 1, and thus represents the "old" factor prices, but passes through the new factor

33

supply point 2. Notice then, that at point 2, with the old factor prices ruling, the marginal rate of technical substitution exceeds the factor-price ratio, L/ K > w/r.

Fig. 3 - Factor Market Adjustment

At these old factor prices, the profit-maximizing firms in our economy will attempt to produce output Y2. As shown in Figure 3, their demand for capital will consequently be K2 and their demand for labor is L2 . So, with old factor prices 1, new factor supplies are at 2 while new factor demands are at 2 . Notice immediately that K2 < K2 and L2 > L2, so there is excess supply of capital and excess demand for labor. We have factor market disequilibrium. Consequently, assuming in a typical Neoclassical fashion that factor markets adjust automatically, then wages must rise and profits must fall, so the factor price ratio must increase from 1 to 2. Notice that at these new factor prices, 2, firms will produce Y2 and demand exactly what is supplied, K2 and L2. We have restored factor market equilibrium.We can also depict this adjustment process in the intensive production function diagram in Figure 2. Specifically, if k1 increases to k2 but factor prices remain constant at 1, then firms will be attempting to produce y2 in Figure 2, which lies above the intensive production function. At this (y2 , k2) position, we see that w/r = 1 and L/ K = 2, thus L/ K > w/r again. This is equivalent to the factor market disequilibrium we depicted in Figure 3. If we permit the wage

34

profit ratio to adjust freely so that w/r = 2, then L/ K = w/r is restored, and firms are now producing y2. This is equilibrium once again.In sum, we see that a Neoclassical factor price adjustment process is exactly what is captured by the "curvature" of the intensive production function. Thus, when we posited the straight-line production function in our depiction of the modern Harrod-Domar model, the critical feature is not so much that we were assuming a single technology, but rather that we are not assuming that there was an underlying Neoclassical factor market clearing process. This is the important point. Harrod-Domar did not believe that factor prices were driven by factor-market clearing, thus they did not incorporate a Solow-Swan type of CRS production function with flexible technology. Specifically, as Roy Harrod (1948, 1953, 1973) explains, following the Keynesian schema, the rate of interest, r, is governed by monetary phenomena; the real wage, well, by an assortment of other things, e.g. unions, etc. So, = w/r is not determined by factor market clearing as Neoclassical theory (and Solow-Swan) assume. As Harrod did not know how (or why) the monetary authorities, labor unions, firms, etc., would adjust r and w so that the economy could be guided to the steady-state capital-labor ratio, he therefore assumed that the capital-output ratio v was a constant. [Note: in a revision of his model, Harrod (1960) does attempt to account for the influence of changing interest, importing, incidentally, the supply of saving from Ramsey (1928).]Seen in this light, it is difficult to accept the common refrain that Solow-Swan "generalized" the Harrod-Domar model just because they allowed for flexible technology whereas Harrod-Domar did not. This is certainly what Solow insinuated, arguing that the "bulk of this paper is devoted to a model of long-run growth which accepts all the Harrod-Domar assumptions except that of fixed proportions." (Solow, 1956: p.66). But, as this discussion has hopefully made clear, it is not technology that is critically different. It is the adjustment process. From the perspective of Ockham's razor, it may very well be that the Harrod-Domar model is "more general" than Solow-Swan. Harrod-Domar make fewer restrictive assumptions. Firstly, they do not assume an instantaneously stable macroeconomic equilibrium (as Solow-Swan do). Secondly, they do not assume any particular factor price adjustment mechanism (as Solow-Swan do). The question of generality, then, can only be resolved by an empirical race between the two models. If Harrod-Domar can explain the same things as Solow-Swan, then, by Ockham's razor, Harrod-Domar is clearly "superior" because it does so with fewer assumptions. If Solow-Swan explains the data better than Harrod-Domar, then we would lean towards declaring Solow-Swan the "better" model. However, as we shall see, it turns out that Solow-Swan performs poorly when confronted with empirical evidence. Substantial modifications have to be added (particularly regarding technical progress) to make it comply with the data. Interestingly, the kind of modifications to the Solow-Swan growth model that "endogenous growth theory" has proposed in recent years turn out to generate a reduced-form dynamical system that is virtually identical to the Harrod-Domar model. Taking an analogy from astronomy, economists have had to add epicycles upon epicycles upon epicycles to the Solowian growth model in order to have it explain what could be more simply explained by the Harrod-Domar model. The conclusion imposes itself.

35

Finally, we should remind ourselves why Solow-Swan is a "Neoclassical" and not a "Keynesian" growth model. From the outset, we have a very Neoclassical factor market equilibrium adjustment process. But, perhaps more strikingly, the macroeconomics are very different. A complete Keynesian growth model would have investment as a function of financial conditions and savings derived from investment via the multiplier. In the Solow-Swan model, not only are all Keynesian "financial" factors omitted, but the direction of causality between savings and investment is reversed. This is equivalent to re-imposing Say's Law. Thus, the Solow-Swan growth model is "Neoclassical" in every respect, and not an extension of "Keynesian" macroeconomics, as has occasionally been advertised. For attempts at introducing more "Keynesian" features into a growth model, see our review of Cambridge growth theory and Keynes-Wicksell monetary growth models.

3.2 Neoclassical Growth: Empirical Implications 

________________________________________________________

"The so-called "underdeveloped" areas, as compared with the advanced, are underequipped with capital in relation to their population and natural resources. We shall do well to keep in mind, however, that this is by no means the whole story. Economic development has much to do with human endowments, social attitudes, political conditions -- and historical accidents. Capital is a necessary but not a sufficient condition of progress."

(Ragnar Nurkse, Problems of Capital Formation in Underdeveloped Countries, 1953: p.1)

________________________________________________________

3.2.1. IntroductionGrowth theory and development theory may seem like natural bedfellows. As it happens, however, they have had a rather tempestuous relationship over the past forty years. Growth theory focuses on how a nation's output-labor ratio grows. Development theory tries to explain why nations possess very different standards of living and what can be done about them. Of course, "standards of living" is not quite the same as national income per capita. Increasing the latter is not very meaningful if it is very unevenly distributed, does nothing to alleviate mass poverty, exacerbates structural problems or is unsustainable in the longer term.Be that as it may, during the 1950s and early 1960s, the consensus among economists was that development and growth were virtually one and the same thing. The essential difference between a developed nation and an underdeveloped nation, it was felt, was that one had a high income per capita

36

while the other had a lower ratio. In terms of growth models, the process of "development" was identified merely as the attempt by a nation to increase its capital-labor ratio, i.e. to accumulate capital at a faster rate than population, so that its income per capita would "catch up" with the industrialized world.In the 1960s, this view gradually disappeared. The idea that underdeveloped nations were merely pint-sized, antiquated versions of industrialized nations was seen as untenable. Underdeveloped nations in the modern world face many unique challenges and problems which industrialized nations never had to contend with when they were "growing up". For instance, a country trying to develop while, at the same time, integrating itself into an advanced international economic order is something quite unprecedented. As a result, development economists began focusing on the particular experiences of underdeveloped nations on their own terms, with all their peculiar features and structural problems.In recent years, however, the "development-as-growth" perspective has once again emerged to the fore. Today, economists and policy-makers repeatedly appeal to growth theory to explain the differences between the experiences of nations and to guide development policy. Apparently, the theory of choice in most studies is the Solow-Swan growth model (or one of its variants). Given the importance of the policy questions involved, it is worthwhile to spend some time on the implications of Neoclassical growth theory for economic development.

3.2.2. The Solow Paradox

What are the empirical implications of the Solow-Swan growth model? The first thing to notice is that the population growth rate "dictates" the steady-state growth rates of all the variables in the economy. In other words, at steady-state, all level variables -- output, Y, consumption, C, capital, K, and labor, L -- grow at the same natural rate n. If there is depreciation, they all grow at the same rate n+ . As a result, all the per capita ratio variables -- output per person, y, capital per person, k and consumption per person, c -- do not grow at all. They are constant over time.

The policy implications are intriguing. If, at steady-state, we wish to change the rate of growth of any of the level terms, then population growth must change. A change in n implies a change in the required investment line as we see in Figure 1 (n1 > n2) and thus an accompanying movement in the steady-state ratio. So, a change in n changes the growth rates of the level variables but the per capita ratios will, once the new steady state is achieved, remain constant.

37

Fig. 1 - A Rise in Population Growth

This leads us to our second point, the Solowian paradox of thrift. This claims that a permanent change in the rate of savings, s, will not permanently change the economy's growth rate. For instance, an increase in the savings rate (from s1 to s2 in Figure 2), will "swing" the investment curve up, so that we move from the steady-state ratio k1* to the new steady-state ratio k2*. Now, before this shift, all level variables were growing at the rate n. Immediately after the change in the savings rate, capital grows a little bit faster than n, so that k increases from k1* (and output and consumption grow a bit faster too). But as k approaches k2*, the growth rate of capital slows down. When we are at k2*, capital growth (and output and consumption growth) returns to n. So increasing the savings rate permanently will only increase growth rates temporarily. In the longer run, it will have no effect on growth rates.

38

Fig. 2 - Changing the Savings Rate

Associated with this is also the paradox of output, i.e. a country which has a "higher" production function, (k), will also fail to permanently increase growth rates. To see this, go through the same exercise in Figure 2, but leave s the same and change only (k) -- the results are almost geometrically identical.Intuitively, one can think of the Solow-Swan model as a person ("capital") attempting to run on a treadmill ("labor"). The growth of capital is the speed of the runner, while the growth of labor is the speed of the treadmill. We have a "steady-state" if the runner manages to stay in the same place. If he runs too slow, then he will fall behind (k declines); if he runs too fast, he will move forward (k rises). Thus, in order to stay in the same place, capital has to run exactly as fast as the treadmill. Anything that increases the speed of the treadmill (a rise in n or ), forces capital to run faster just to stay in the same place. Stretching the analogy, the savings rate merely determines where on the treadmill a person will be running-in-place (close to the front, in the middle, close to the back, etc.), but regardless of where he chooses to be on the treadmill, he still has to run at the same speed. This result is "paradoxical" because one of the old saws of development theory was that increasing the savings rate would accelerate growth. W. Arthur Lewis (1954, 1955) was one of the primary proponents of this idea. As he writes "the countries which are now relatively developed have at some time in the past gone through a rapid acceleration in the course of which their rate of net investment has moved from 5 per cent [of national income] or less to 12 per cent or more" (Lewis, 1955, p.208). Consequently, "The central problem in the

39

theory of economic growth is to understand the process by which a community is converted from being a 5 per cent to a 12 per cent saver." (1955, p.226),However, Lewis relied on Classical growth theory and the Harrod-Domar model to derive his conclusion. But the Solow-Swan model tells us is that the Lewis thesis is only temporarily correct, i.e. there is a short-run acceleration in growth, but in the long-run, growth settles down once again to its previous rate.A third thing that should be noticed from Figure 2 is that steady-state growth is, in general, consumption inefficient. In other words, the economy's steady-state does not necessarily yield the highest consumption per capita forever. In Figure 2, notice that c1* > c2*. So if we begin with savings propensity s2, steady state k2* rules and consumption per capita is c2*. Clearly, the steady-state k2* is not the maximum consumption possible. But recall that steady-states are stable, so there are no inherent economic reasons to move away from this inefficient position. If, and this is a big if, everyone in the economy could somehow collectively decide to decrease the savings rate from s2 to s1, then per capita consumption could increase permanently from c2* to c1*. Of course we obtain this because of the way we drew our diagram. It is conceivable that decreasing s might lead to lower c, for instance. But the principle should be clear: it is quite possible that the steady-state we end up at will be consumption inefficient in the sense that somehow changing the propensity to save will improve consumption per capita permanently. We shall return to this point when discussing "optimal growth". It shall also crop up again when discussing monetary growth models, as consumption-inefficiency makes growth theory amenable to government policy.

3.2.3. The Convergence Hypotheses

-. Absolute Convergence- Conditional Convergence

We should touch upon the convergence hypotheses of the Solow-Swan model, given that it has generated much empirical speculation in recent years. There are two versions of this. the absolute convergence and the conditional convergence hypotheses.

3.2.3.1. Absolute Convergence

The absolute convergence hypothesis, posits the following: consider a group of countries, all of which have have access to the same technology ( (·)), the same population growth rate (n) and the same savings propensity (s), and only differ in terms of their initial capital-labor ratio, k. Then, we should expect all countries to converge to the same steady-state capital-labor ratio, output per capita and consumption per capita ( k*, y*, c*) and, of course, the same growth rate (n).

40

Absolute convergence is depicted in Figure 1, where we can assume that k1

represents the capital-labor ratio of a poor country and k2 the capital-labor ratio of a rich country. As they are otherwise identical, the stability of the Solow-Swan model predicts that both the poor and rich countries will approach the same k*. Notice that this means that the poor country will grow relatively fast (capital and output grow faster than n), while the rich nation will grow quite slowly (capital and output grow slower than n). Stated differently in adjustment terms, as k1 < k2, then (k1) > (k2), so the marginal product of capital relative to labor is higher in the poor nations than in the rich ones, thus the poor will accumulate more capital and grow at a faster rate than the rich.

Fig. 1 - Absolute Convergence

This may seem a farfetched proposition -- but many Neoclassical argue that it is not entirely ludicrous. Consider, say, the end of World War II, when the capital stocks (but not the labor) of Japan and Germany were destroyed by Allied bombing and other war-related actions. Notice that the other features of the defeated nations, namely their technological possibilities, savings rates, population growth rates, etc. were pretty much still the same as before the war. Or, more pertinently, they were virtually the same as other countries in the industrialized world. So, relative to other industrialized countries with similar parameters, post-war Germany and Japan had exceptionally low capital-labor ratios, k (akin to k1 in Figure 1). In accordance with the absolute convergence hypothesis, the Solow-Swan model would predict that these two

41

nations would subsequently grow faster than other industrialized countries in the immediate post-war period -- as indeed they did.Of course, for the world as a whole, the absolute convergence hypothesis is bound not to hold as nations are not as similar to one another as in the aforementioned example. It is difficult to presume that, say, Mozambique and Denmark ought to "converge" to the same ratios and growth rates. Their savings propensities, technological possibilities and population growth rates are just so very different.

3.2.3.2. Conditional ConvergenceThe conditional convergence hypothesis states that if countries possess the same technological possibilities and population growth rates but differ in savings propensities and initial capital-labor ratio, then there should still be convergence to the same growth rate, but just not necessarily at the same capital-labor ratio. This is due to the paradox of savings outlined above. In short, the conditional convergence hypothesis asserts that countries can differ in the their steady-state ratios (e.g. k1* vs. k2* in Figure 2 below) and thus differ in consumption per capita, but as long as they have the same population growth rate, n, then all their level variables -- capital, output, consumption, etc. -- will eventually grow at that same rate.

Fig. 2 - Conditional Convergence

Of course, even the conditional convergence hypothesis should not necessarily hold when comparing the industrialized world with the underdeveloped world because the population growth rates between countries in these two groups are different (Denmark vs. Mozambique again). But the conditional convergence hypothesis ought to help explain why countries with similar population growth rates (e.g. India and Nigeria) can converge to the same

42

growth rates, albeit with different steady-state capital-labor ratios, and thus different income/consumption per capita.

3.2.4. Poverty Traps

- The Technological Trap       - The Population Trap

On the whole, the convergence hypotheses do not do very well empirically. As such, growth theorists have searched for ways of modifying the Solow-Swan growth model to explain why some countries do so well and others do so poorly. One of the favorite methods is by arguing that the difference can be explained by technological progress, exogenous or endogenous. However, technological change is not the only explanation. One set of stories that caught on quite early was the idea of poverty traps. In a sense, these are the exact opposites of the convergence hypotheses. They try to explain, within a Solow-Swan growth model, why some countries exhibit stagnant growth (i.e. growth with low-levels of income per capita) while others race on ahead.

There are two types of poverty traps: technologically-induced poverty traps and demographically-induced poverty traps. We shall consider each of them separately. Both cases involve the inclusion of a non-linearity into the system. Both, incidentally, were considered by Robert Solow (1956).

3.2.4.1. The Technological Trap

In the 1940s, it was quickly realized that poor nations are poor, that they are already saving what they can and still do not seem able to "accelerate" anything. A consensus emerged that underdeveloped countries might be caught in a "poverty trap", a vicious circle of low savings and few investment opportunities. How can this be explained?

Allyn Young (1928) recalled Adam Smith's old idea about how the "division of labor is limited by the extent of the market". Dynamized into a growth context, this highlights the importance of externalities and increasing returns to scale in generating and sustaining an accelerated rate of growth. Nations that did not manage to achieve increasing returns were left behind. Those that did would "take-off" into ever-increasing standards of living.

Paul Rosenstein-Rodan (1943, 1961), Hans Walter Singer (1949), Ragnar Nurkse (1953), Gunnar Myrdal (1957) and Walt Whitman Rostow (1960) appropriated this idea for development theory. They argued that increasing returns only set in after a nation has achieved a particular threshold level of output per capita. Poor countries, they argued, were caught in a poverty trap because they had been hitherto unable to push themselves above that threshold. In contrast, successful developing nations had benefited, at some earlier point, from a massive and wide-spread injection of capital, just enough to push them over the threshold and thereafter to "take-off". Their policy recommendations for underdeveloped nations therefore focused on recreating

43

this Big Push artificially, whether by inflows of foreign capital or debt-financed government investment.

We can capture this idea in a Solow-Swan model by allowing for non-linearities in the production function. Specifically, the argument is that the production function (·) has a middle portion where it exhibits increasing returns to scale, i.e.

   < 0

  for 0 < k < ka

(k) 

 > 0

  for ka< k < kb

   < 0

  for kb< k

Thus, the production function exhibits increasing returns to scale between critical values ka and kb Otherwise, there are constant returns. The result is depicted in Figure 1.

44

 

Fig. 1 - Technological Trap

The features of the technological trap can be read from the diagram directly. We have four steady-states -- 0, k1*, k2* and k3*. Of these, k1* and k3* are stable, while 0 and k2* are unstable. The implication is that if a country begins with a capital labor ratio that is below k2*, then it will inexorably approach the stagnant steady-state ratio k1*. If its initial capital-labor ratio is above k2*, then it will approach the much better steady-state k3*. The ratio k2*, then, is the "threshold" which a nation has to reach to "take-off" and achieve the higher steady-state.Here is where the "Big Push" story comes in. It is argued that developed nations had, at some point in their history, a "Big Push" in terms of massive capital investment (or a demographic collapse) which pushed them over the k2* edge, which then, by the regular forces of the Solow-Swan model, drove them further up to the high k3* steady-state. Underdeveloped nations failed to experience this "Big Push" and so remained stuck in the orbit of k1*. It is not that they did not try, of course. Efforts by developing nations to push the capital-labor ratio up with public and private investment schemes did not work

45

simply because these were not bold enough. They might have pushed themselves above k1*, but not enough to cross over the k2* threshold. For this to happen, a really big "Big Push" is needed.There are a few alternative policy options for stagnant nations in light of the technological trap. The first is that temporarily increasing the savings rate might actually serve as a policy option in this case. Specifically, consider Figure 2 and suppose that we have a country with savings rate s1 stuck at the stagnant steady-state ratio k1*. In order to manipulate itself into a Big Push, a rise in the savings rate from s1 to s2, will result in a situation where there is only one stable steady-state ratio -- the very high k4* in Figure 2. Maintaining the s2

savings rate for a while, the nation will enjoy a rapid rise in the capital-labor ratio from k1* towards k4*. However, it need not maintain this savings rate forever. Once the capital-labor ratio has gone past k2*, it can lower the savings rate back down to s1, and now the country is within the orbit of the high capital-labor ratio, k3*, and will move inexorably towards it by the standard properties of Solow-Swan adjustment. Thus, a temporary rise in the savings rate is one way for a nation to pull itself out of the technological trap.

Fig. 2 - Temporary Rise in the Savings Rate

Another way of escaping the technological trap is to temporarily lower the population growth rate. This is depicted in Figure 3. A nation stuck at k1* could swing the ir curve downwards by decreasing population growth from n1 to n2

temporarily, thereby leaving the very high k4* as the only steady-state ratio. The old population growth can be safely restored once the Solowian dynamics naturally pushed the economy over k2*.

46

Fig. 3 - Temporary Decline in Population Growth

One disappointing feature of the technological trap model is that, in the end, different countries may be at different steady-state ratios, but they still exhibit identical growth rates. In other words, in Figure 1, a poor economy at steady-state k1* and a rich economy at steady-state k3* would still experience the same growth rates of level variables and no growth in per capita variables. In a way, then, this result is similar to the conditional convergence case. This result can be easily circumvented if, following the arguments of the early development theorists, we decide to simply omit the upper diminishing returns portion of the production function, i.e. if we posit that: 

 < 0

  for 0 < k < ka

(k) 

   

   > 0

  for ka< k

as shown in Figure 4. In this case, increasing returns hold from ka onwards.

47

Fig. 4 - Increasing Returns with No Convergence

The consequence of this modification is that once a nation passes over the k2* threshold, it will grow forever. Thus, now, a poor nation is trapped in the sense that it will be stuck at its low steady-state and experience no growth in per capita ratios (the true meaning of "stagnation"), while a rich country's per capita income, constrained by no steady-state at all, will continue rising forever after. In this kind of situation, there is no "convergence" of growth rates between poor and rich nations.

3.2.4.2. The Population TrapAnother interesting kind of poverty trap is the one induced by population. This is a variation on the canonical model which has interesting lessons for development because it generates a poverty trap without having to assuming anything about technology.In the Solow-Swan model, the rate of population growth was given exogenously. However, recall that in Classical models of growth, population growth is endogenous. Following Robert Malthus (1798), it was posited that the rate of growth of population is dependent on income per capita. Specifically, as income per capita rises, then the population growth rate rises. This is known as the Malthusian theory of demographic transition.Solow (1956) introduced the Malthusian demographic transition into his model. He followed the Classicals in allowing that when income per capita was very low, then population declined i.e. n was negative. But as income per capita

48

increases, population growth would increase. Solow topped off the story by also accounting for fertility declines at very high income per capita. As population growth n is a function of y and y = (k), then population is indirectly function of the capital-labor ratio, i.e. n = n(k). We can summarize the demographic relationship by defining critical values ka and kb, where:

  <

0   for 0 < k < ka

n = n(k)  

> 0

  for ka < k < kb

  <

0  for kb < k

The implications of this demographic transition is that we obtain a non-linear required investment curve, ir = n(k)k as depicted in Figure 5. Beginning from the origin, we see that population declines until ka, after which it begins to increase, initially at an increasing rate, and then at a decreasing rate, until it hits kb, after which population begins to decline again.

Fig. 5 - Population Trap

Notice that in Figure 5, we have three steady-state equilibria: 0, k1* and k2*. However, of these, only k1* is stable; the origin and k2* are both unstable. Thus,

49

for any capital-labor ratio between 0 and k2*, the system will tend to bring it back to k1*. Once again, the interesting implication is if, by some "Big Push", the economy can be elevated to a capital-labor ratio above k2*, then there will be a constantly increasing income per capita thereafter. In this demographic transition model, we do not have convergence in levels or growth rates between poor and very rich countries. The Malthusian "population trap" story was emphasized in development theory by Harvey Leibenstein (1954, 1957) and R. Nelson (1956). However, we should note that, empirically, there is no relationship between population growth and income per capita. More precisely, it is argued that this relationship is no longer valid because of national and international health efforts of the past few decades have helped push down the death rates and improved birth rates in underdeveloped countries. So, even if this story could explain past experience, it is not really "policy-effective" anymore. If anything, population growth is today more correlated with income distribution rather than income levels.

3.3 Technical Progress

___________________________________________________________________

"Is the "residual factor" a measure of the contribution of knowledge or is it simply a measure of our ignorance of the causes of economic growth?"

J. Vaizey, The Residual Factor and Economic Growth, 1964: p.5

______________________________________________________________________

- Adding Technical Progress- Empirical Implications

A property of the Solow-Swan growth model which is a bit disturbing is the fact that, at steady-state, all ratios -- the capital-labor ratio, output per person and consumption per person -- remain constant. This is a bit of a disappointment for it implies that standards of living do not improve in steady-state growth. This is not only despiriting, it is also empirically dubious: it contradicts at least two of the "stylized facts" of industrialized economies laid out in Kaldor (1961) -- namely, that the capital-labor and output-labor ratios have been rising over time and that the real wage has been rising. O f course, just because industrialized countries, and others besides, have experienced ever-increasing per capita consumption and output over the past three centuries does not, by itself, "contradict" the Solow-Swan model. After all, out of steady-state, we can easily have changing ratios. So, one possible explanation for the "stylized facts" that is consistent with the Solow-Swan model is simply that industrialized nations are still in the process of adjusting and just have not reached their steady-state equilibrium yet. And why not? It is not unreasonable to assume that adjustment to steady state might take a very long time (cf. Sato, 1963; Atkinson, 1969). But economists are a rather impatient sort. They like to believe that economies tend to be at or around their steady-states most of the time (a noble exception

50

is Meade (1961)). As a consequence, in order to reconcile the Solow-Swan model with the stylized facts, it is tempting to argue that there has been some sort of "technical progress" in the interim that keeps pushing the steady-state ratios outwards.

3.3.1 Adding Technical ProgressRecall that when we write our production function as Y = F(K, L), we are expressing output as a function of capital, labor and the production function's form itself, F(·). If output is growing, then this can be due to labor growth (changes in L), capital growth (changes in K) and productivity growth/technical progress (changes in F(·)). We have thus far ignored this last component. It is now time to consider it.Technical progress swings the production function outwards. In a sense, all we need to do is simply add "time" into the production function so that:

Y = F(K, L, t).

or, in intensive form:

y = (k, t)

The impact of technical progress on steady-state growth is depicted in Figure 1, where the production function (·, t) swings outwards from (·, 1) to (·, 2) to (·, 3) and so on, taking the steady-state capital ratio with it from k1* to k2* and then k3* respectively.. So, at t =1, (·, 1) rules, so that beginning at k0, the capital-labor ratio will rise, approaching the steady-state ratio k1*. When technical progress happens at t = 2, then the production function swings to (·, 2), so the capital-labor ratio will continue increasing, this time towards k2*. At t =3, the third production function (·, 3) comes into force and thus k rises towards k3*, etc. So, if technical progress is happens repeatedly over time, the capital-labor ratio will never actually settle down. It will continue to rise, implying all the while that that the growth rates of level variables (i.e. capital, output, etc.) are higher than the growth of population for a rather long period of time. 

51

Fig. 1 - Technical Progress

Before proceeding, the first thing that must be decided is whether this is a "punctuated" or "smooth" movement. Is technical progress a "sudden" thing that happens only intermittently (i.e. we swing the production function out brusquely and drastically and then let it rest), or is it something that is happening all the time (and so we swing the production function outwards slowly and steadily, without pause). Joseph Schumpeter (1912) certainly favored the exciting "punctuated" form of technical progress, but modern growth theorists have adhered almost exclusively to its boring, "smooth" version. In other words, most economists believe that (·, t) varies continuously and smoothly with t.The simple method of modeling production by merely adding time to the production function may not be very informative as it reveals very little about the nature and character of technical progress. Now, as discussed elsewhere, there are various types of "technical progress" in a production function. The one we shall consider here is Harrod-neutral or labor-augmenting technical progress. In fact, as Hirofumi Uzawa (1961) demonstrated, Harrod-neutral

52

technical progress is the only type of technical progress consistent with a stable steady-state ratio k*. This is because, as we prove elsewhere, only Harrod-neutral technical progress keeps the capital-output ratio, v, constant over time. Formally, the easiest way to incorporate smooth Harrod-neutral technical progress is to add an "augmenting" factor to labor, explicitly:

Y = F(K, A(t)·L)

where A(t) is a shift factor which depends on time, where A > 0 and dA/dt > 0. To simplify our exposition, we can actually think of A(t)·L as the amount of effective labor (i.e. labor units L multiplied by the technical shift factor A(t)). So, output grows due not only to increases in capital and labor units (K and L), but also by increasing the "effectiveness" of each labor unit (A). This is the simplest way of adding Harrod-neutral technical progress into our production function. Notice also what the real rate of return on capital and labor become: as Y = F(K, A(t)·L) then the rate of return on capital remains r = F K, but the real wage is now w = A(t)·( F/ (A(t)·L)] = A·FAL. Modifying the Solow-Swan model to account for smooth Harrod-neutral technical progress is a simple matter of converting the system into "per effective labor unit" terms, i.e. whenever L was present in the previous model, replace it now with effective labor, A(t)·L (henceforth shortened to AL). So, for instance, the new production function, divided by AL becomes:

Y/AL = F(K/AL, 1)

so, in intensive form:

ye = (ke)

where ye and ke are the output-effective labor ratio and capital-effective labor ratio respectively. Notice that as F(K, AL) = AL· (ke), then by marginal productivity pricing, the rate of return on capital is:

r = FK = (AL· (ke))/ K

But as (ke) = (K/AL), then (AL· (ke))/ K = AL· (ke)·( ke/K), and since ke/ K = 1/AL, then (AL· (ke))/ K = (ke), i.e.

r = (ke)

the slope of the intensive production function in per effective units terms is still the marginal product of capital. What about the real wage? Well, continuing to let the marginal productivity theory rule, then notice that:

w = (F(K, AL)/dL

= (AL· (ke)) /dL

53

= A· (ke) + AL· (ke)·( ke/dL)

as dke/dL = -AK/(AL)2 = -K/AL2 = -ke/L then:

w = A· (ke) - AL· (ke)·ke/L

or simply:

w = A[ (ke) - (ke)·ke]

The macroeconomic equilibrium condition I = sY, becomes:

I/AL = s(Y/AL)

or:

ie = sye = s (ke)

where ie is the investment-effective labor ratio. Now, suppose the physical labor units, L, grow at the population growth rate n (i.e. gL = n) and labor-augmenting technical shift factor A grows at the rate (i.e. gA = ), then effective labor grows at rate + n, i.e.:

gAL = gA + gL = + n

Now, for steady-state growth, capital must grow at the same rate as effective labor grows, i.e. for ke to be constant, then in steady state gK = + n, or:

Ir = dK/dt = ( +n)K

is the required investment level. Dividing through by AL:

ire = ( +n)ke

where ire is the required rate of investment per unit of effective labor. The resulting fundamental differential equation is:

dke/dt = ie - ire

or:

dke/dt = s (ke) - (n+ )ke

which is virtually identical with the one we had before. The resulting diagram (Figure 2) will also be the same as the conventional one. The significant difference is that now the growth of the technical shift parameter, , is included into the required investment line and all ratios are expressed in terms of effective labor units.

54

Consequently, at steady state, dke/dt = 0, and we can define a steady-state capital-effective labor ratio ke* which is constant and stable. All level terms -- output, Y, consumption, C, and capital, K -- grow at the rate n+ .

 

Fig. 2 - Growth with Harrod-Neutral Technical Progress

If the end-result is virtually identical to before, what is the gain in adding Harrod-neutral technical progress? This should be obvious. While all the steady-state ratios -- output per effective capita, ye*, consumption per effective capita, ce*, and capital per effective capita, ke* -- are constant, this is not informative of the welfare of the economy. It is people -- and not effective people -- that receive the income and consume. In other words, to assess the welfare of the economy, we want to look at output and consumption per physical labor unit. Now, the physical population L is only growing at the rate n, but output and consumption are growing at rate n + . Consequently, output per person, y = Y/L, and consumption per person, c = C/L, are not constant; they are growing at the steady rate , the rate of technical progress. Thus, although steady-state growth has effective ratios constant, actual ratios are increasing: actual people are getting richer and richer and consuming more and more even when the economy is experiencing steady-state growth.

3.3.2.     Empirical Implications

55

How valid is this empirically? From the outset, notice that the Solow-Swan model with technical progress accounts for all of the Kaldorian stylized facts. Namely, at steady-state (we are dropping the asterisk):

(1) the investment-output ratio, I/Y = ie/ye = s (ke)/ (ke) = s is constant,

(2) the capital-output ratio K/Y = ke/ (ke) is constant,

(3) the capital-labor ratio k = K/L and the output-labor ratio y = Y/L are growing at rate ;

(4) the rate of return on capital, r = FK = (ke) is constant;

(5) the real wage is w = FL = A[ye - ke· (ke)] is growing at rate

(6) the relative share of capital rK/Y = (ke)·ke/ye is constant, and the relative share of labor wL/Y = w/Aye = A[ye - ke· (ke)]/Aye = 1 - ke· (ke)/ye is constant.

The main implication of all this is that the Solow-Swan growth model can only explain steadily-increasing standards of living (growing y and c) via technical progress. There is an entire body of empirical literature, known as "growth accounting", which attempts to address the empirical validity of this modified Solow-Swan model. Unlike the model just described, they usually assume that the technical progress factor A(t) is outside the production function, i.e.:

Y = A(t) (K, L).

where A > 0 and dA/dt > 0, is the technical progress parameter (in this context, A is referred to as the "Total Factor Productivity" or "TFP" parameter). Thus, unlike the model above, growth accounting literature assumes that technical progress is Hicks-neutral or TFP-augmenting rather than Harrod-neutral/labor-augmenting. [Note: Hello, does this not contradict Uzawa's (1961) proof? Not quite. It is possible for technical progress to be both Hicks-neutral and Harrod-neutral if the production function has constant unit elasticity of substitution, i.e. = 1. As we prove elsewhere, the Cobb-Douglas form of the production function is the only functional form that fulfills this. And you were wondering why it was so popular?]The growth accounting literature asks the simple question: given the history of output growth, how much of it was due to growth of capital inputs (gK), growth labor inputs (gL) and technical progress (gA)? Output growth, labor growth and capital growth are observable, but technical progress is not. How do we estimate it?Empirical growth accounting began with the famous studies of Moses Abramovitz (1956, 1962) and Robert Solow (1957). Their procedure in calculating gA was to deduct the growth rates of capital and labor (multiplied by

56

their respective factor prices) and ascribing the "residual" to technical progress. For example, if we assume Cobb-Douglas form, so that the production function is:

Y = AK L(1- )

where 0 1, the growth accounting literature is interested in the relationship:

gY = gA + gK + (1- )gL

where, as gY, gK, gL and are more or less observable, then gA can be imputed residually. In fact, total factor productivity growth, gA, is often referred to simply as the Solow residual.The striking feature of the early investigations of growth accounting was the size of the Solow residual. Solow (1957), for instance, calculates that only 12.5% of growth in output per capita in the 1909-1949 period in the United States was due to factor accumulation -- leaving 87.5% to be explained by technical progress! This is a bit dispiriting as it implies that the overwhelming majority of the growth that is empirically observed is "outside" the explanatory power of the Solow-Swan growth model!In a series of famous studies, Edward Denison (1962), Zvi Griliches (1963) and Dale W. Jorgensen and Zvi Griliches (1967) argued that the there were errors in measurement in the early growth accounting work. For instance, if we remind ourselves that technical progress usually arrives "embodied" in new capital goods, then a lot more of growth can be ascribed to the "qualitative growth" of capital inputs. Thus, the importance of the Solow residual -- the growth in "total factor productivity" -- was argued to be substantially less than that estimated by earlier researchers. We will turn to technical progress again when examining endogenous growth theory.

3.4. Selected References

M. Abramovitz (1956) "Resource and Output Trends in the United States since 1870", American Economic Review, Vol. 46

M. Abramovitz (1962) "Economic Growth in the United Sates", American Economic Review, Vol. 52, p.762-82.

A.B. Atkinson (1969) "The Time Scale of Economic Models: How long is the long run?", Review of Economic Studies, Vol. 36 (2), p.137-52.

E.F. Denison (1962) The Sources of Economic Growth in the United States and the Alternatives Before Us. New York: Committee on Economic Development.

Z. Griliches (1963) "The Sources of Measured Productivity Growth: United States Agriculture, 1940-60", Journal of Political Economy, Vol. 71, p.331-46.

57

F.H. Hahn (1960) "The Stability of Growth Equilibrium", Quarterly Journal of Economics, Vol. 74, p.206-26.

F.H. Hahn (1987) "`Hahn Problem'", in Eatwell, Milgate and Newman, editors, The New Palgrave: A dictionary of economics. London: Macmillan.

F.H. Hahn and R.C.O. Matthews (1964) "The Theory of Economic Growth: A survey", Economic Journal, Vol. 74, p.779-902. As reprinted in 1969 Surveys of Economic Theory: Vol. II - Growth and development. London: Macmillan.

R.F. Harrod (1948) Towards a Dynamic Economics: Some recent developments of economic theory and their application to policy. London: Macmillan.

R.F. Harrod (1953) "Full Capacity vs. Employment Growth: Comment", Quarterly Journal of Economics, Vol. 67 (4), p.553-9.

R.F. Harrod (1960) "A Second Essay in Dynamic Theory", Economic Journal, Vol. 70, p.277-93.

D.W. Jorgensen and Z. Griliches (1967) "The Explanation of Productivity Change", Review of Economic Studies, Vol. 34, p.249-83.

H.G. Johnson (1966) "The Neoclassical One-Sector Growth Model: A geometric exposition and extension to a monetary economy", Economica

H. Leibeinstein (1954) Theory of Economic-Demographic Development. Princeton, NJ: Princeton University Press.

H. Leibeinstein (1957) Economic Backwardness and Economic Growth. New York: Wiley.

W.A. Lewis (1955) The Theory of Economic Growth. Homewood, Ill: Irwin.

T.R. Malthus (1798) An Essay on the Principle of Population. 1960 reprint of 1798 and 1892 editions, New York: Modern Library.

J.E. Meade (1961) A Neo-Classical Theory of Economic Growth. 1983 reprint of 1962 edition, Westport, Conn: Greenwood.

G. Myrdal (1957) Economic Theory and Under-Developed Regions. London: Duckworth.

R.R. Nelson (1956) "A Theory of the Low Level Equilibrium Trap", American Economic Review, Vol. 46, p.894-908.

R. Nurkse (1953) Problems of Capital-Formation in Underdeveloped Countries. 1962 edition, New York: Oxford Univeristy Press.

58

H. Pilvin (1953) "Full Capacity versus Full Employment Growth", Quarterly Journal of Economics, Vol. 67 (4), p.545-52.

P. Rosenstein-Rodan (1943) "The Problem of Industrialization of Eastern and South-Eastern Europe", Economic Journal, Vol. 53, p.202-11.

P. Rosenstein-Rodan (1961) "Notes on the Theory of the Big Push", in H.S. Ellis and H.C. Wallich, editors, Economic Development in Latin America. New York: Macmillan.

W.W. Rostow (1960) The Stages of Economic Growth. Cambridge, UK: Cambridge University Press.

R. Sato (1963) "Fiscal Policy in a Neo-classical Growth Model: An analysis of time required for equilibrating adjustment", Review of Economic Studies, Vol. 30 (1), p.16-23.

J.A. Schumpeter (1911) The Theory of Economic Development: An inquiry into profits, capital, credit, interest and the business cycle. 1934 translation, Cambridge, Mass: Harvard University Press.

H.W. Singer (1949) "Economic Progress in Underdeveloped Countries", Social Research,

R.M. Solow (1956) "A Contribution to the Theory of Economic Growth" Quarterly Journal of Economics. Vol. 70 (1) pp. 65-94.

R.M. Solow (1957) "Technical Change and the Aggregate Production Function", Review of Economics and Statistics, Vol. 39, pp. 312-20.

R.M. Solow (1970) Growth Theory: An exposition. 1988 edition, Oxford: Oxford University Press.

T.W. Swan (1956) "Economic Growth and Capital Accumulation", Economic Record, Vol. 32 (2), p.334-61.

J. Tinbergen (1942) "Zur Theories der langsfristigen Wirschaftsentwicklung", Weltwirtschaftliches Archiv, Vol. 55, p.511-49. Translated 1959 in L.H. Klaassen, L.M. Koyck, and H.J. Witteveen, editors, Jan Tinbergen: Selected Papers. Amsterdam.

J. Tobin (1955) "A Dynamic Aggregative Model", Journal of Political Economy, Vol. 63 (2), p.103-15.

H. Uzawa (1961) "Neutral Inventions and the Stability of Growth Equilibrium", Review of Economic Studies, Vol. 28, p.117-24.

A.A. Young (1928) "Increasing Returns and Economic Progress", Economic Journal, Vol. 38, p.527-42.

59

4. Multisector Growth 73

4.1. The Uzawa Two-Sector Growth Model 734.1.1.      Basic Setup 744.1.2.      Diagrammatic Representation 794.1.3.      Analytical Solutions 904.1.3.1.   Analytical Solution I: Classical Hypothesis 904.1.3.2.   Analytical Solution II: Proportional Savings 974.1.4.      Indeterminacy, Instability and Cycles 1064.2. Optimal Two-Sector Growth 1114.2.1. The Uzawa-Srinivasan Model 1114.2.2. Case I: Consumer Goods are More Capital-Intensive 114 4.2.3. Case II: Investment Goods are More Capital-Intensive 122 4.2.4. Conclusion 1254.3. Selected References 126

IV - Multisector Growth

60

4.1. The Uzawa Two-Sector Growth Model

"[Uzawa] finds that his model economy is always stable...if the consumption-goods sector is more capital-intensive than the investment-goods sector. It seems paradoxical to me that such an important characteristic of the equilibrium path should depend on such a casual property of the technology. And since this stability property is the one respect in which Uzawa's results seem qualitatively different from those of my 1956 paper on a one-sector model, I am anxious to track down the source of this difference."

(Robert M. Solow, "Note on Uzawa", 1961, Review of Economic Studies)

"It is evident that in all these constructions the condition that the equilibrium at a moment in time be unique is crucial. The rest of the story is really concerned with ensuring that there is a steady state with positive factor prices. But the assumptions required to establish uniqueness of momentary equilibrium are all terrible assumptions."

(Frank H. Hahn, "On Two-Sector Growth Models", 1965, Review of Economic Studies)

________________________________________________________

 Two-sector extensions of the Solow-Swan growth model were introduced by Hirofumi Uzawa (1961, 1963), James E. Meade (1961) and Mordecai Kurz (1963). This led to an explosion of research in the 1960s, conducted primarily in the Review of Economic Studies, on the two-sector growth model. Then, as suddenly as it had appeared, this line of research evaporated in the 1970s.

4.1.1.      Basic Setup Hirofumi Uzawa's (1961, 1963) two-sector growth model considers a Solow-Swan type of growth model with two produced commodities, a consumer good and an investment good. Both these goods are produced with capital and labor. So we have two outputs and two inputs, of which the most interesting feature is that one of the outputs is also an input. To use the old Hicksian analogy, in the Uzawa two-sector model, we are using labor and tractors to make corn and tractors. For the following exposition, we have benefited particularly from Burmeister and Dobell (1970) and Siglitz and Uzawa (1970).Let us follow the basic setup of the Uzawa two-sector model. We begin with the following definitions:

Yc = output of consumer goodLc = labor used in consumer good sectorKc = capital used in consumer good sectorYi = output of investment good

61

p = price of investment good (in terms of consumer good)Li = labor used in investment good sectorKi = capital used in investment good sectorY = total output of economyL = total supply of laborK = total supply of capitalw = return to labor (wages)r = return to capital (profit/interest)

The principal equations of the two-sector model can thus be set out as follows:

Yc = Fc(Kc, Lc) - consumer sector production function (1)

Yi = Fi(Ki, Li) - investment sector production function

(2)

Y = Yc + pYi - aggregate output (3)

Lc + Li = L - labor market equilibrium (4)

Kc + Ki = K - capital market equilibrium (5)

w = dYc/dLc = p·(dYi/dLi) - labor market prices (6)

r = dYc/dKc = p·(dYi/dKi) - capital market prices (7)

gL = n - labor supply growth (8)

gK = Yi/K - capital supply growth (9)

These equations should be self-evident. The consumer goods sector and the investment goods sector each use both capital and labor to produce their output. We capture this with equations (1) and (2), where Fc(·) is the consumer goods industry production function and Fi(·) the investment goods industry production function. Both production functions Fc(·) and Fi(·) are nicely "Neoclassical", in the sense of exhibiting constant returns to scale, continuous technical substitution, diminishing marginal productivities to the factors, etc.Equation (3) is merely the definition of aggregate output, expressed in terms of the consumer good. Equations (4) and (5) are also self-evident: the market demand for labor is Lc + Li and the market demand for capital is Kc + Ki. As L

62

and K are the respective supplies, then (4) and (5) are merely the factor markets equilibrium conditions so that demand equals supply in each market.Now, we assume no barriers competition in the factor markets, so that there is free movement of labor and capital across sectors. This implies that the wage rate w and the profit rate r must be the same in both the consumer goods and investment goods industry. Neoclassical economic theory tells us that the marginal productivity schedules for each factor in each industry form those industries' demand functions for the factors. As such, in labor market equilibrium, the return to labor (w) must be equal to the marginal product of labor in the consumer goods sector (dYc/dLi) and the marginal product of labor in the investment goods sector p·(dYi/dLi). This is equation (6). Equation (7) asserts the analogous condition in capital market equilibrium, i.e. that the rate of return on capital (r) is equal to the marginal product of capital in both sectors.Finally, as the investment goods industry produces all the new capital goods in the economy, then, ignoring depreciation, we can define the change in the total stock of capital as that sector's output, i.e. dK/dt = Y i, so the growth rate of capital is gK = (dK/dt)/K = Yi/K, which is equation (8). Labor supply is assumed to grow exogenously at the exponential rate n, thus the growth rate of labor is gL = (dL/dt)/L = n, which is equation (9).We would now like to express everything in intensive form, i.e. in per capita or per labor unit terms. This gets a bit tricky. But defining:

yc = Yc/L c = Lc/Lkc = Kc/Lc

c(kc) = Fc(Kc, Lc)/Lc

yi = Yi/L i = Li/Lki = Ki/Li

i(ki) = Fi(Ki, Li)/Li

y = Y/Lk = K/L

Then equations (1)-(9) above can be converted to the following:

yc = cc(kc) - consumer sector intensive production function

(1 )

yi = ii(ki) - investment sector intensive production function

(2 )

y = yc + pyi - aggregate output per capita (3 )

c + i = 1 - labor market equilibrium (4 )

63

ckc + iki = k - capital market equilibrium (5 )

w = c - kcc = p·( i - kii )

- labor market prices (6 )

r = c  = p·i - capital market prices (7 )

gL = n - labor supply growth (8 )

gK = yi/k - capital supply growth (9 )

Equations (1 ) and (2 ) are the intensive production functions. These are derived as follows. Recall from (1) that Yc = Fc(Kc, Lc), then dividing through by Lc, we obtain Yc/Lc = Fc(Kc/Lc, 1) = c(kc). But Yc/Lc = (Yc/L)(L/Lc) = yc/ c. Thus yc

= cc(kc), which is (1 ). The transformation from (2) to (2 ) follows a similar procedure.Each of these intensive production functions have simple properties. For instance, their first derivatives are the marginal product of capital, i.e. Fc/dKi

= c (kc) and Fi/dKi = i (ki), so diminishing marginal productivity implies c (kc) < 0 and i (ki) < 0. The production functions also fulfill the famous "Inada conditions", formulated by Ken-Ichi Inada (1963). Specifically:

c(0) = 0, c( ) =

c (0) = , c ( ) = 0

for the intensive production function for the consumption good. The equivalent Inada conditions apply to the intensive production function for the investment good:

i(0) = 0, i( ) =

i (0) = , i ( ) = 0

(see also our discussion of production functions).Equations (3 ) and (4 ) are obtained merely by dividing (3) and (4) by L. Equation (5 ) is obtained by dividing (5) by L, which yields Kc/L + Ki/L = K/L = k, but as Kc/L = (Kc/Lc)(Lc/L) = kcc and Ki/L = (Ki/Li)(Li/L) = kii. So, ckc + iki = k, as we have in (5 ).Equations (6 ) and (7 ) use Euler's theorem. Now, it is a simple matter to show that dYc/dKc = c (kc) and dYi/dKi = i(ki). So, the competitive condition in (7) is converted to r = c  = p·i  . By constant returns to scale, we know from Euler's theorem that Yc = (dYc/dK)·K + (dYc/dLc)·Lc, thus dividing through by L and

64

rearranging: dYc/dLc = (Yc/L) - (K/L)·(dYc/dK) = yc - k·c . The corresponding transformation can be done for dY i/dLi. This is how we convert (6) to (6 ). Finally, equation (9 ) is obtained simply by multiplying (9) through by 1 = L/L, so gK = (Yi/L)/(K/L) = yi/k.Now, following Uzawa's notation, let us define (omega) as the wage-profit ratio, i.e. = w/r. Thus, combining equations (6 ) and (7 ):

= w/r = [ c(kc) - kc·c ]/c  = [ i(ki) - ki·i ]/i

or simply:

= ( c(kc)/c ) - kc = ( i(ki)/i ) - ki

Now, notice that:

d /kc = - c ·c(kc)}/(c (kc))2 > 0

d /ki = - i · i(ki)}/(i (ki))2 > 0

Thus, is positively related to kc and ki. It is not difficult to see that these are monotonic relationships. Consequently we can define the functions:

kc = kc( ) where kc = (c )2/(c · c) > 0

ki = ki( ) where ki = (i )2/(i · i) > 0

which will be used extensively as they will form the boundaries of our equilibrium path.The growth story can be quickly told. At steady-state, the capital-labor ratio k must be constant. As k = K/L, then:

gk = gK - gL

so, using our expression for gK and gL

(dk/dt)/k = yi/k - n

so:

dk/dt = yi - nk

Which is our fundamental differential equation. So, we have a steady-state where dk/dt = 0.Of course, this is not the end of the story, for we have yet to consider the question of macroeconomic equilibrium. Specifically, note that while we have laid out the supply of consumer and investment goods, we have said nothing so far about the demand for these outputs. As it turns out, this will depend crucially on the consumption-savings behavior of households. Specifically, the demand for consumer goods will depend on the amount of income households

65

consume, while the demand for investment goods will depend on the amount of savings. Now, we can follow the "Classical" economists and presume that all wages are consumed and all profits are saved (as Uzawa (1961) did); or we allow for some saving out of both wages and profits (as Uzawa (1963) allows) and we can even impose that the propensity to save out of these two categories of income is different (as Drandakis (1963) presumes).Whatever the case, the model will not be closed until we consider the demands for outputs explicitly. This is, after all, a Neoclassical model, which means that the imputation theory should hold: output demands will determine output supplies and consequently factor market equilibrium. Causality thus runs from preferences of households to factor market equilibrium

 4.1.2.      Diagrammatic RepresentationThe relationship between the factor intensities of the two sectors and the relative position of the kc() and ki() curves is simple to deduce diagrammatically. Consider Figure 1, where we represent the intensive production functions of the two sectors. Let us begin with a given capital-labor ratio, k*. Suppose we wish to specialize completely in the production of consumer goods, i.e. suppose yi = 0 so that y = yc = yc/c, where c = 1 (all labor allocated to consumer goods sector) and kc = k* (so all capital allocated to consumer goods sector). We see immediately that our entire economy is governed by the intensive production function c(kc).

So, with k*, we produce yc* as our aggregate output. As we know from our discussion of intensive production functions, the slope of the ray tangent to the production function is c (= rc), the marginal product of capital, the point where that ray intersects the vertical axis, c - kcc (= wc), the marginal product of labor. Most importantly, the point where the tangent ray intersects the horizontal axis is c = wc/rc, the ratio of marginal products. If we specialize in producing consumer goods, then the ratio c can be seen as the resulting equilibrium factor price ratio, i.e. the factor prices that clear the capital and labor markets, where the initial supply of capital and labor is captured by the given capital-labor ratio k*.

66

Fig. 1 - Two-Sector Model (for given k*)

Now, suppose we specialize completely in investment goods production completely. In this case, yc = 0, and so y = p·yi = p·yi/ i as i = 1 (all labor to investment goods production) and ki = k* (all capital to investment goods). Thus, aggregate per capita output is governed by the intensive production function i(ki) depicted in Figure 1. With the given k*, notice, we will produce y = p·yi* which (although this depends on p) seems to be more than yc*. This should alert us to the fact that the investment goods industry is less capital-intensive than the consumer goods industry.We can see this relative capital intensity diagrammatically in Figure 1 by remembering that the slope of a ray from the origin to the relevant point on an intensive production is 1/v, the reciprocal of the capital-output ratio, v. Thus, letting vc and vi denote the capital-output ratios in the consumer goods and investment goods industries respectively, we see immediately in Figure 1 by the rays connecting 0 to ec and ei, that 1/vi > 1/vc, so that vc > vi, in other words, you need more capital per unit of output in the consumer goods industry than in the investment goods industry. Relatively speaking, consumer goods are capital-intensive and investment goods are labor-intensive. Of course, we could have drawn this differently so that the factor-intensities were reversed.Now, continuing on our specialization into investment goods, we notice that a tangent line with slope i(ki) intersects the horizontal axis at i = wi/ri, the equilibrium factor price ratio. Notice immediately that i > c, which is another indication that the consumer goods industry is relatively capital-intensive. Specifically, note that relative factor shares can be denoted k/ = rK/wL. The higher this k/ ratio, the greater the capital-intensity. In terms of Figure 1, holding k = k* constant, we see that as i > c, then k*/ i < k*/ c, indicating, once again, that the investment goods sector is relatively capital intensive.Notice the implication of what we have just done: for a given k, the resulting equilibrium factor-price ratio will depend on our allocation between sectors. Specifically, if we allocation all factors to the consumer goods sector, the equilibrium will be c; if we allocated all factors into the investment goods

67

sector, the equilibrium will be i. The assumption of strict concavity of production functions guarantees this uniqueness. If we do not know the sectoral composition of output, we cannot determine what the actual equilibrium factor-price ratio is: it can range from a minimum of c (complete specialization in consumer goods) to a maximum of i (complete specialization in investment goods), i.e. for a given k,

min max

where min = c and max = i. The Inada conditions guarantee us that these exist.Notice also that varying the given k, these upper and lower boundaries for equilibrium factor prices will vary. Specifically, note that in Figure 1, if we increase k above k*, then both c and i will increase. We depict the resulting boundaries in Figure 2 as the upward-sloping curves c(k) and i(k). We draw them as straight lines, but this is not necessarily the case. The only things that are posited are (i) that the relationship between k and c and i is unique and monotonically increasing (by assumptions of strict concavity and constant returns to scale for the production function) and (ii) that the investment goods boundary i(k) will always lies above the consumer goods boundary c(k) (from the assumption that consumer goods industry is more capital-intensive than the investment goods industry). Naturally, if we change the assumption of

factor-intensity, so that investment goods are more capital-intensive than consumer goods, then c(k) would lie everywhere above i(k).

Fig. 2 - Factor Prices and Quantities

68

There is, of course, another way of depicting the curves. Specifically, as i(k) and c(k) are monotonically increasing and unique, then they are invertible, i.e. we can specify the identical curves ki() and kc(), where for a given factor price ratio (e.g. *), we have the resulting capital-labor ratios in both sectors (ki and kc respectively). This is identically depicted in Figure 2, but now we read as the independent variable and k as the dependent..To see this inversion in production function space, examine Figure 3. The given factor-price ratio * will set a point on the horizontal axis from which emanate two rays, one with slope rc and another with slope ri, corresponding to the marginal products of capital for the consumer goods and investment goods industries. These rays form tangencies with the intensive production functions of both the consumer goods and capital goods industries at points e c and ei

respectively, which translate into resulting capital-labor ratios kc and ki. Thus, this indicates the relationships kc() and ki() that we find depicted in Figure 2. (notice also that kc > ki, which is another indicator of the relative factor intensity of the sectors -- use the same formula, k/ , and notice that kc/ * > ki/ *).

Fig. 3 - Two Sector Model (for given *)

Notice in Figure 3 that although the factor-price ratio is the same for both sectors, so * = ri/wi = rc/wc, we have it that ri rc and wi wc, so it seems that the rates of return to capital and wages are not equal across sectors. But we must not forgot the price of investment goods, p. Specifically, p will be such that p·ri = rc and p·wi = wc.Finally, notice that the amount that will be produced when factor prices are * can be deciphered from yc/ c and yi/ i on the vertical axes. Notice that both industries have positive output per capita (yc, yi > 0), so we are not specializing exclusively in either of them. Both sectors are allowed to operate and the particular amounts they produce will be dictated by the equilibrium factor price ratio we begin with, *.

69

This, of course, does not end the story. If we allow both and k to vary, then the entire shaded region in Figure 2 becomes available as a solution. This is where the rest of the Uzawa model comes into play. Stepping ahead of ourselves a little bit, it might be worthwhile to sketch out what we are aiming for.We will proceed in reference to Figure 4. Suppose we are given an initial aggregate capital-labor ratio k0. As we say nothing about allocation between sectors, then there is a whole range of equilibrium factor prices which are consistent with that allocation (within some maximum/minimum range). Pick one of these factor price ratios. This will then determine a sectoral allocation of kc() and ki(). But this may not be necessarily market-clearing, i.e. it may be that ckc( ) + iki( ) k0, so that our demands for factors are not equal to our initial supplies of the factor, which implies that the factor price ratio we have chosen is not appropriate. So, for the initial k0, we must search for a market-clearing factor price ratio that makes demands equal to supplies.The line k() in Figure 4 maps out the locus of equilibrium factor prices for every aggregate capital-labor ratio. The fact that this is upward-sloping everywhere and lies between the boundaries is important. Suppose we begin at k0. The locus k( ) tells us that 0 is the market-clearing factor price ratio. Thus, the corresponding sectoral allocations kc( 0) and ki( 0) are equilibrium allocations, i.e. ckc(0) + iki(0) = k0. In contrast, for initial capital stock k0, the wage-profit ratio 1 is not market clearing, so ckc(1) + iki(1) k0. However, for initial capital stock k1, 1 is the market-clearing wage-profit ratio (i.e. ckc(1) + iki(1) = k1). So positions a and c represent factor market equilibrium, while positions such as b and d are factor market disequilibrium. The curve k() is merely the locus of equilibrium positions.

Fig. 4 - Factor Market Equilibrium LocusIndeed, all we have been concerned with so far is for an equilibrium at a moment in time -- what the literature calls momentary equilibrium. For obvious reasons, we call this simply a "factor market equilibrium". However, this says nothing about the long-run position of the system. Specifically, any capital-labor ratio k is in factor market equilibrium if we have the right factor prices for

70

it. But a factor market equilibrium does not presuppose or imply steady-state growth.Suppose (k0, 0) is a factor market equilibrium today but it is not a steady-state. Consequently, the production of goods will proceed, and factors will grow at their own rates (labor by the natural rate, n, capital by the sectoral allocation to investment goods production). Nothing we have said so far presupposes that capital and labor will grow at the same rate. Thus, it is likely that tomorrow the capital-labor ratio may be different, say, it may increase from k0 to k1. Consequently, tomorrow, new factor equilibrium prices will be obtained (1), which determines sectoral allocation, which in turn determines capital growth, etc.If labor is growing at an exogenous natural growth rate n and we posit some exogenous propensity to save, then (hopefully) there exists a capital-labor ratio k* consistent with steady-state. This is going to be one of the points along the horizontal axis in Figure 4. But, more importantly, as long as our factor equilibrium locus is monotonic, there will be associated with k* is a unique set of "steady-state" equilibrium factor prices, * and consequently, steady-state sectoral capital-labor ratios kc(*) and ki(*). If there is not, we have serious problems.The questions before us are several. Firstly, can we define a factor market equilibrium locus k( ) that possesses nice properties? By "nice" we mean that it sits "within" the shaded area of Figure 4, and is upward-sloping and monotonic, so that we can define a unique factor price equilibrium for every aggregate capital-labor ratio k. This is crucial. Suppose not. Suppose we have a situation like the one depicted in Figure 5, where we have a bizarre-looking k() locus. For capital-labor ratio k0, we have three factor market equilibrium prices, a, b and c, which means that from k0, we are not sure which factor market equilibrium will obtain. As each is associated with a different sectoral allocation (kc( ), ki( )), we do not know which direction we are heading in!

71

Fig. 5- Troubling Equilibria

Let us analyze this troubling case in a bit more detail. As we know, under competitive conditions, r = p·i = c. As a result, we can express p, the price of investment goods in terms of consumer goods, as the ratio of marginal products of capital:    p = c(kc)/i (ki)Now, as c < 0 and i < 0, then p is proportional to the relative capital-labor ratios ki/kc, i.e.

p ki/kc

so if ki is high relative to kc, then i(ki) is low relative to c(kc), which implies that p is relatively high. Thus, the higher ki/kc, the higher p will be.Now, consider Figure 5 again, where we have three factor market equilibrium. Each of these equilibria will be associated with a different price p. So, consider combination a = (k0, a). We can deduce diagrammatically that at this point, ki/kc is quite high and thus the corresponding price, call it pa, will be quite high. Conversely, consider combination c = (k0, c), which has a relatively low ki/kc, and thus the corresponding price pc will be relatively low. So, rather loosely, we can infer that pa > pb > pc.Plotting the production possibilities frontier associated with k0 (call it PPF0) in Figure 6, which plots the feasible combinations of yc and yi associated with the capital-labor ratio k0. We see that the three equilibrium prices pa, pb and pc

associated with the respective factor market equilibrium combinations a, b and c in Figure 5 are represented by three price lines with different slopes. In general, as total output is yc + pyi and k0 is fixed, then the slope of a price line

72

in PPF space is -p. Recalling that pa > pb > pc, then we see that pa is the steepest and pc is the flattest line. All three lines form equilibrium points at their tangency with the PPF, i.e. factor market equilibrium conditions a, b and c in Figure 5 have their corresponding equilibrium output allocations a, b and c in Figure 6.So, now let us note the following: as pa is the steepest, then equilibrium a corresponds to an output allocation where the output produced by investment goods sector is relatively high, while the output of the consumer goods sector is relatively low. Conversely, as pc is the flattest, then at equilibrium c, the output of investment goods relative to consumer goods is small. Thus, factor

market equilibrium c corresponds to the least amount of yi of all the equilibria, while factor market equilibrium a corresponds to the highest amount of yi of all equilibria.

Fig. 6 - Equilibrium Output Allocations

Now, suppose k0 is a "steady-state" ratio. What does it mean in this context? We have three possible factor-market equilibria associated with it -- and each of them is associated with a different level of investment goods production, y i. But, recall, gK = yi/k0, so each of these equilibria are associated with different capital growth rates. So if we grant that points a, b and c in Figures 5 and 6 are equilibrium allocations, they surely they cannot all be consistent with steady-state growth!Suppose that equilibrium b, with investment goods production y i

b, corresponds to steady-state growth, i.e. yi

b/k0 = gK = gL. Then, clearly, if we have equilibrium a, then as we see from Figure 6, yi

a > yib, it must be that gK > gL, and so the

aggregate capital-labor ratio increases. Thus the combination (k0, a) cannot be a steady-state. Similarly, if factor market equilibrium c holds, then y i

c < yib

73

and thus gK < gL, so the capital-labor ratio falls and we are not in steady-state either.Thus, in sum, multiplicity of factor market equilibria at a given aggregate capital-labor ratio k0 causes real problems. Even if k0 is a steady-state capital-labor ratio, it is only steady-state if factor equilibrium price b obtains. If, for some reason, the other factor market equilibrium prices a or c happen to hold at k0, then k0 is not a steady-state capital-labor ratio.The implications of this can be understood by examining the resulting differential equation in Figure 7. Notice that at k0, only equilibrium b corresponds to dk/dt = 0. Equilibrium a corresponds to dk/dt > 0 and equilibrium c corresponds to dk/dt < 0. So, at k0, we can have steady-state growth, increasing growth or decreasing growth. All three are possible.

Fig. 7 - Troubling Steady-States

A further issue made stark in Figure 7 is that we can also have multiple steady-state capital-labor ratios. There is no reason to assume that k0 and associated factor market equilibrium b is the only steady-state. A different capital-labor ratio, k1, with associated factor market equilibrium d can also be a steady-state. The implications for the stability of steady-state are knotty: it could be that some steady-states will be stable, some unstable, some may be only half-stable. In fact, as we shall see later, a differential equation as depicted in Figure 7 can actually yield us a limit cycle, so that we oscillate continuously, but never quite approach a steady-state.These sorts of trouble will implicate the ability of the economy to approach steady-state growth. We would like to place sufficient restrictions on our model such that most of these difficulties are ruled out. What we want to end up with a k( ) locus that looks more like the nice, monotonic one in Figure 4 rather than the squiggly one in Figure 5. However, as it turns out, these restrictions are actually quite severe.So, in sum, the two-sector model poses two essential questions: (1) can we guarantee a unique factor-market equilibrium for every capital-labor ratio k?

74

(2) is there a unique steady-state growth path k* and can we guarantee that, starting from any capital-labor ratio, the system will approach k* over time? Much of what follows is an attempt to establish sufficient conditions so that we can answer these questions in the affirmative.

4.1.3.  Analytical Solutions

- Analytical Solution I: Classical Hypothesis- Analytical Solution II: Proportional Savings

4.1.3.1.  Analytical Solution I: Classical Hypothesis

The solution to the two-sector model depends crucially on the kind of savings assumptions we make. In his first version, Uzawa (1961) considered the "Classical hypothesis" that workers consume all their income and capitalists save all their profits. In other words, wL is the demand for consumer goods and rK the demand for new capital goods. In equilibrium, demand equals supply for the consumer goods and investment goods markets, i.e.

wL = Yc

rK = pYi

or, in per capita form:

w = yc

rk = pyi

Recalling that yc = cc(kc) and yi = ii(ki) by definition and that w = c (kc) and r = pi (ki) by competition, these conditions become:

c (kc) = c c(kc)

pi (ki)·k = pi i(ki)

or, solving for c and i respectively:

c = c(kc)/c(kc)

i = k·i(ki)/i(ki)

But we also know from the wage-profit ratio equation that + kc = ( c(kc)/c(kc)) and + ki = (i(ki)/i(ki)), so:

c = 1/( + kc)

i = k/( + ki)

75

Now, recall that the capital-market clearing condition was k = ckc + iki. Consequently, rewriting this for c:

c = (k - iki)/kc

so, substituting in our expression for i:

c = [k - kik/( + ki)]/kc

or:

c = [k /( + ki)]/kc

Now, by the labor-market clearing condition, c + i = 1. Thus adding this c

to our previous i::

c + i = [k /( + ki)]/kc + k/( + ki) = 1

or multiplying through by kc and rearranging:

k( + kc)/( + ki) = kc

So, we can express k as:

k = kc·( + ki)/( + kc)

As kc = kc( ) and ki = ki( ), then this expresses k as a function of . This is the locus k() of factor market equilibria.We can decipher the slope of this function as:dk/d = {[(dkc/d )( + ki)( +kc) + kc(1+dki/d )·( +kc) - (1 + dkc/d ) kc·( + ki)}/( + kc)2

so, multiplying by 1/k = ( + kc)/kc·( + ki):(dk/d )·(1/k) = {[(dkc/d )( + ki)( +kc) + kc(1+dki/d )·( +kc) - (1 + dkc/d ) kc·( + ki)}/ kc·( + ki)·( + kc)so canceling terms:

(dk/d )·(1/k) = (dkc/d )/kc + (1+dki/d )/( +ki) - (1 + dkc/d )/( + kc)

and rearranging:

(dk/d )·(1/k) = (dkc/d )·[1/kc - 1/( +kc)] + (dki/d )/( +ki) + [1/( +ki) - 1/( +kc)]

Now, as dkc/d , dki/d > 0 and as [1/( +ki) - 1/( +kc)] = /( +kc) > 0, then the sign of this equation depends crucially on the sign of [1/( +ki) - 1/( +kc)] = (kc - ki)/[( +ki)·( +kc)]. So, if we assume that consumer goods are more capital-intensive than investment goods, so that kc > ki, then we are

76

guaranteed that dk/d > 0. If, in contrast, ki > kc, so that investment goods are more capital intensive, then dk/d is ambiguous.We shall refer to the assumption that kc > ki for all , i.e. that consumer goods are more capital-intensive that investment goods, as the "Uzawa capital-intensity condition". As we have seen, this is a sufficient condition for uniqueness of factor market equilibria, i.e. only by adopting the Uzawa capital-intensity condition can we conclude with certainty that dk/d > 0 for all [ min, max]. In other words, the Uzawa capital-intensity assumption makes the factor market equilibrium locus, k(), monotonically upward-sloping wherever it lies between the boundaries kc( ) and ki( ).[Note: Edwin Burmeister (1968) characterizes the determinacy of factor market equilibria in terms of the Jacobian determinant J( ) of the system, which is particularly useful. See Burmeister and Dobell, 1970: Ch. 4.]All this is for factor market equilibria -- or what the literature calls "momentary equilibria". We still have not touched upon the existence, uniqueness or stability of the a steady-state growth path. To prove stability of steady-state growth, Uzawa appeals to a theorem of Arrow, Block and Hurwicz (1959) regarding the limits of a differentiable equation. We shall skip the formal stability proof and refer to his article. Intuitively, the idea behind stability comes from recalling the basic underlying differential equation:

dk/dt = yi - nk

As we have made the "Classical hypothesis" that all profits are saved and all wages are spent, remember that this implies that rk/p = y i, and by the marginal productivity assumption, rk/p = i (ki), so this reduces to:

dk/dt = (ki) - nk

Is this stable? Stability of the steady-state capital-labor ratio, k*, requires that d(dk/dt)/dk < 0 around k*. But the capital-labor ratio of the investment goods sector, recall, is a function of equilibrium factor prices, , and these, in turn, are determined by the aggregate capital-labor ratio, k. So by intuitive chain rule logic:

d(dk/dt)/dk = (d(ki)/dki)·(dki/d )·(d /dk) - n

Determining the sign of this is the crucial step. Obviously, (ki) is negatively related to ki by simple diminishing marginal productivity, i.e. d(ki)/dki < 0. We know already that dki/d > 0. So the question boils down to d/dk. We have proved that if Uzawa's capital-intensity condition holds, then d /dk > 0, and thus we are home free because this implies that for values of k above n, d(dk/dt)/dk < 0 and thus our system is stable.This is interesting. The Uzawa capital-intensity condition was imposed to guarantee uniqueness of factor-market equilibrium for every k. But it also implies uniqueness and stability of the steady-state growth path. Why does relative capital-intensity matter for stability of growth equilibrium? Because of the infamous "Wicksell Effects". To see why, recall that by the Classical hypothesis, wL = Yc and rK = pYi, then rK/wL = p·(Yi/Yc), or:

77

K/L = (w/r)·p·(Yi/Yc)

or, in per capita terms:

k = ·p·(yi/yc)

So if rises, everything else constant, k rises. But everything else is not constant. Specifically, the price of investment goods, p, is a function of and relative outputs, yi/yc = c(ki)/c(kc), are also functions of sectoral allocation k i

and kc which are also functions of . So we should rewrite this as:

k( ) = ·p()·(yi()/yc())

Evidently, lots of things can happen now. So let us try it again: suppose rises, then k( ) must rise unless p() falls and/or yi()/yc() falls sufficiently, in which case k( ) falls. So, what rules this possibility out? Uzawa's capital-intensity assumption. To see this, let us begin by proving that p cannot fall in response to a rise in if the consumer goods sector is more capital-intensive. As we know the price of investment goods p can be expressed as:

p = c /i

But as the marginal products are themselves functions of kc and ki (which are, in turn, functions of ), then we can write p as a function of :

p() = c (kc()) /i(ki())

To decipher the relationship between p and , just differentiate:

dp/d = {c ·(dkc/d )·i - i ·(dki/d )·c )/[i ]2

which seems pretty ugly. But recall that, from before, dkc/d = -c 2/(c · c) and dki/d = -i 2/(i · i). So, plugging in:

dp/d = {-c ·(c 2/(c · c))·i + i ·(i 2/(i ·i))·c }/[i ]2

and canceling terms:

dp/d = {(i 2/i)·c - (c 2/ c)·i }/[i ]2

or:

dp/d = {(i 2/i)·c - (c 2/ c)·i }/[i ]2

dp/d = c /i - (c 2)/(c·i )

Recalling that p = c /i , then multiplying through by 1/p = (i /c ):

(dp/d )·(1/p) = (i /c )·c / i - (i /c )·(c 2)/(c·i )

78

or:

(dp/d )·(1/p) = i /i - c 2/c

so, finally, remembering from before that + ki = i/i and + kc = c/c , then:

(dp/d )·(1/p) = 1/( + ki) - 1/( + kc)

So, if kc > ki (consumer goods more capital-intensive), then dp/d > 0, while if ki > kc (investment sector more capital-intensive), then dp/d < 0. This should not be a surprising result. It is reminiscent of the famous Stolper-Samuelson theorem: specifically, that a rise in the price of a good is positively related with a rise in the return to the factor in which that good in intensive. So, if the investment-goods industry is capital-intensive, a relative rise in the return to capital (a fall in ) will be associated with a rise in the price of investment goods (p). Conversely, if the investment-goods industry is labor-intensive, then a relative rise in the wage (a rise in ) will be associated with a rise in p.So, Uzawa's capital-intensity assumption implies that dp/d > 0. That's half the problem solved. Now, we only need to make sure that yi/yc cannot fall sufficiently in response to a rise in to make k fall as well. We resort to basic intuition: a rise in is automatically related to an increase in the capital-intensity of both sectors, i.e. ki and kc rise and thus yi and yc rise. Furthermore, by Uzawa's capital-intensity assumption, then if yi/yc falls, we are releasing factors from a labor-intensive investment goods industry into a capital-intensive consumer goods industry, so we are increasing the average capital-intensity of the economy. Thus k simply cannot fall in response to a rise in because not only do both sectors become more capital-intensive, but we are transferring factors from labor-intensive to more capital-intensive industries. The aggregate capital-labor ratio k must rise.However, if we violate Uzawa's capital-intensity condition, and assume that investment goods are more capital intensive, then note that dp/d < 0 and if yi/yc falls, we are releasing factors from a capital-intensive sector into a labor-intensive one. Consequently, there is a strong countervailing tendency: it is quite possible that the fall in py i/yc more than outweighs the rise in so that k falls. In other words, an increase in the wage-profit rate can lead to a fall in the capital-labor ratio, i.e. we are employing more of the factor which has become relatively more expensive! In geometric terms, the market demand for capital is not everywhere negatively related to profit and/or the market demand for labor is not negatively related to wages. The market factor demand curves will have curious shapes with upward-sloping portions. This is the kind of thing that yields multiplicity of factor market equilibria and make our system indeterminate. In other words, at k, we do not know which factor prices will result, and thus we may end up at a higher or lower capital-labor ratio tomorrow, which means that we cannot tell whether we will be moving towards or away from the steady-state capital-labor ratio.Of course, the Uzawa capital-intensity condition is sufficient for stability, but not necessary. One can construct many examples where the investment goods sector is more capital-intensive than consumer goods and still have stability of the steady-state. But, and this is more important, without it, we can also

79

construct many reasonable examples with instability. And these are more reasonable.[Note: This intuition is derived largely from Solow (1961) and Hahn (1965). Uzawa (1963) traces the source of his capital-intensity condition to Knut Wicksell's discussion of "Åkerman's problem" (Wicksell, 1923). Morishima (1969: p.45) calls this the "Shinkai-Uzawa" condition, in recognition of Y. Shinkai's (1960) work on two-sector models with fixed coefficients of production, where he obtained the result that growth equilibrium is stable if and only if the consumer goods industry is more capital intensive than the investment goods sector. When we have flexible production functions, as in the Uzawa model, this is merely a sufficient, but not necessary, condition for stability. John Hicks (1965) finds this condition, but does not dwell much over it. He finds it again and gives it more attention in his Neo-Austrian model (Hicks, 1973).]

4.1.3.2.   Analytical Solution II: Proportional Savings Robert Solow (1961) criticized Uzawa's (1961) result on the basis that it seemed to depend to heavily on the "Classical hypothesis" that all wages are spent and all profits are saved. As a result, Hirofumi Uzawa (1963) hit the drawing board again and introduced a more flexible savings hypothesis.For flexibility, let us assume that sw is the average propensity to save out of wages and sr is the average propensity to save out of profits. [Note: this actually follows Drandakis (1963) -- in his original presentation, Uzawa (1963) assumes that sw = sr = s.] So total demand for investment goods is swwL + srrK, while total demand for consumer goods is (1-sw)wL + (1-sr)rK. In equilibrium, demand equals supply for the consumer goods and investment goods markets, i.e.

(1-sw)wL + (1-sr)rK = Yc

swwL + srrK = pYi

so, in per capita form:

(1-sw)w + (1-sr)rk = yc

sww + srrk = pyi

We shall concentrate on the second equation. Note that sww = swwr/r = swr as = w/r by definition. Thus substituting in:

(sw + srk)r = pyi

Dividing through by p and recalling that yi = ii(ki) by definition and r/p = i (ki) by competition then:

(sw + srk)i (ki) = ii(ki)

80

Now, recall that our capital-accumulation equation is:

dk/dt = yi - nk

which, in view of our expression above, we can write as:

dk/dt = (sw + srk)i (ki) - nk

which is our fundamental differential equation.Let us now concentrate on the factor market equilibrium locus. In factor market equilibrium:

i + c = 1

ckc + iki = k

which are the conditions for labor market clearing and capital market clearing respectively. Combining:

k = (1-i)kc + iki

So recalling our original macroeconomic equilibrium equation, we now have a system of simultaneous equations in i and k, specifically:

i(ki - kc) - k = -kc

ii(ki) - srk i (ki) = swi (ki)

or, in matrix form:

(ki - kc)

-1 i   -kc

      =  

i(ki)-sri (ki)

k   swi(ki

)

By Cramer's Rule, the solution k is:

k = |A2|/|A|

where A is the determinant of the matrix on the left, while A2 is the determinant of that matrix with the solution vector in the second column. Thus note that:

|A| = -sri (ki)(ki - kc) + i(ki)

and:

81

|A2| = swi (ki)(ki - kc) + i(ki)kc

so:

k = [swi (ki)(ki - kc) + i(ki)kc]/[i(ki) -sri (ki)(ki - kc)]

so, dividing both numerator and denominator by i :

k = [sw(i /i)(ki - kc) + (i/i )kc]/[(i/i ) - sr(i /i )(ki - kc)]

or:

k = [sw(ki - kc) + (i/i )kc]/[(i/i ) -sr(ki - kc)]

Finally, recalling from our competition condition that i/i = ki + , then:

k = [sw(ki - kc) + (ki + )kc]/[(ki + ) - sr(ki - kc)]

Now, adding to both sides:

k + = [sw(ki - kc) + (ki + )kc]/[(ki + ) - sr(ki - kc)] +

= [sw(ki - kc) + (ki + )kc + (ki + ) - sr(ki - kc)]/[(ki + ) - sr(ki

- kc)]

so:

k + = [(sw - sr)(ki - kc) + (kc + )(ki + )]/[(ki + ) - sr(ki - kc)]

Which is, effectively, the form of our factor market equilibrium locus k(). Notice that if we take Uzawa's (1963) special case that sw = sr = s (which we shall call the Uzawa savings assumption), then this reduces to:

k + = [(kc + )(ki + )]/[(ki + ) - s(ki - kc)]

or, noting that the denominator can be written as:

(ki+ ) - s(ki-kc) = s(ki+ ) + (1-s)(ki+ ) - ski + skc

= s(kc+ ) + (1-s)(ki+ )

then we can rewrite our whole equation as:

k + = [(kc + )(ki + )]/[s(kc+ ) + (1-s)(ki+ )]

which is exactly Uzawa's (1963) equation (21).Enough of algebra. The important question that emerges from all this is the following: is there a positive monotonic relationship between and k? In other

82

words, is it true that there is a unique equilibrium factor-price ratio for any given k? We shall not bother with existence (although that also ought to be proven) and concentrate on uniqueness of . Adopting Uzawa's savings assumption (sw = sr = s), then let us define zc = kc + and zi = ki + and g() = k, so:

g()= zczi/[szc + (1-s)zi] -

Then, differentiating with respect to :

dg()/d = [(zczi + zczi )·(szc + (1-s)zi) - (szc + (1-s)zi )·zczi] /[szc + (1-s)zi]2 - 1

or:

dg( )/d = [szc2zi + (1-s)zi

2zc ]/[szc + (1-s)zi]2 - 1

Now, recall that zc = kc + and zi = ki + , which implies that zc = kc + 1 and zi = ki + 1 where, as kc > 0 and ki > 0, implies then that zc > 1 and zi > 1. Thus:

dg()/d > [szc2 + (1-s)zi

2]/[szc + (1-s)zi]2 - 1

But examine the fraction in this expression. Now, we know that (zc - zi)2 0, so:

zc2 + zi

2 - 2zczi 0

thus, multiplying by s(1-s):

s(1-s)zc2 + s(1-s)zi

2 - 2s(1-s) zczi 0

or:

s(1-s)zc2 + s(1-s)zi

2 2s(1-s)zczi

Thus, adding s2zc2 + (1-s)2zi

2 to both sides:

s(1-s)zc2 + s2zc

2+ s(1-s)zi2 + (1-s)2zi

2 s2zc2 + (1-s)2zi

2 + 2s(1-s)zczi

Or as s(1-s) + s2 = s and s(1-s) + (1-s)2 = (1-s), and noticing that the term on the right is merely [szc + (1-s)zi]2, then:

szc2 + (1-s)zi

2 [szc + (1-s)zi]2

which implies that

dg()/d > [szc2 + (1-s)zi

2]/[szc + (1-s)zi]2 - 1 > 0

83

so dg()/d is positive and the factor equilibrium locus is everywhere upward-sloping, i.e. is a monotonically increasing function of k and there is a unique factor-market clearing associated with every k.This is intriguing. When we made the "Classical" savings hypothesis, we needed Uzawa's capital-intensity assumption to guarantee uniqueness of factor market equilibrium. But now no capital-intensity assumptions are needed. The Uzawa savings hypothesis, that sw = sr = s, is by itself sufficient to obtain uniqueness.What if we drop this special case? If sw sr, Drandakis (1963) has shown that either of the following are sufficient for uniqueness:

(i) sr > sw and kc > ki

(ii) sr < sw and kc < ki

We shall call these the Drandakis mixed conditions. Effectively, it states that either configuration (i) or configuration (ii) is sufficient to guarantee there is a unique for every k. This is striking. Note that in the Uzawa savings case, no capital-intensity hypothesis was needed to guarantee uniqueness. But by allowing sw sr, then not all configurations work: we must append a capital-intensity hypothesis to the savings propensity hypothesis to ensure uniqueness. Notice that the Drandakis mixed conditions (i) and (ii) are not necessary but rather sufficient conditions, so other configurations might work, but it is not guaranteed.So much for factor market equilibrium. Let us now turn to steady-state growth. Our fundamental differential equation is:

dk/dt = (sw + srk)i (ki) - nk

so, assuming steady-state equilibrium, dk/dt = 0 and so at the solution k (and associated factor-market clearing and sectoral allocation ki, kc):

(sw+ srk)i(ki)/k = n

Of course, there is nothing, in principle, that rules out multiple steady-states, i.e. there could be several solutions to this. How can we rule this out? Drandakis (1963) showed that if the first of the Drandakis mixed conditions are met (i.e. if sr sw and kc ki), then the steady-state k is unique and stable. So, for sufficiency, not only do we require the old Uzawa capital-intensity condition (consumer goods are more capital-intensive than investment goods), we also need to assume that proportionally more is saved out of profits than out of wages. To prove this, note that from before, condition (i) states that if s r > sw, and kc > kw, then dk()/d > 0. Now:

d(dk/dt)/dk = (sw(d /dk) + sr)i + (sw + srk)·i ·(dki/d )·(d /dk) - n

Now, recall that dki/d = -(i)2/ i·i , so i·(dk/d ) = -(i )2/i, so:

84

d(dk/dt)/dk = (sw(d /dk) + sr)i - (sw + srk)·(i)2·(d /dk)/i - n

or, rearranging a bit:

d(dk/dt)/dk = sw (d /dk)i + sri- (swi + srki )·i ·(d /dk)/i - nk

Now, from the macroeconomic equilibrium condition (sw + srk)i(ki) = ii(ki), so:

srki = ii - swi

and:

sri = (ii - swi)/k

so:

d(dk/dt)/dk = sw (d /dk)i + (ii - swi)/k - (swi + ii - swi)·i ·(d /dk)/ i - n

rearranging:

d(dk/dt)/dk = sw (d /dk)i + ii/k - swi( /k) - i·i·(d /dk) - n

or, factoring i:

d(dk/dt)/dk = [(sw - i)·(d /dk) - sw( /k)]i + i i/k - n

So the sign of d(dk/dt)/dk depends crucially on the sign of [(sw - i)·(d /dk) - sw( /k)]. If:

(sw - i)(d /dk) < sw( /k)

then d(dk/dt)/dk < 0 around equilibrium and we will have local stability.Let us now deduce the implications of this. We know that dk/dt = y i - nk, or simply:

dk/dt = ii(ki) - nk

So, if we have steady-state k*, then dk/dt = 0, so:

i = nk*/i(ki*)

Now, we know that d(dk/dt)/dk < 0 if (sw - i)(d /dk) < sw( /k). So, plugging in, we see that around steady-state

(sw - nk*/i(ki*))(d /dk) < sw( /k*)

85

So, if sw < nk*/i(ki*), then this inequality holds for certain, so we will be assured that dk/dt < 0 around the steady-state k*.Alternatively, consider the following. As (sw + srk)i(ki) = ii(ki) by macroeconomic equilibrium, then:

sw = ii(ki)/(i(ki)) - srk/

and as i/i= ki + by competition, then:

sw = i(ki + )/ - srk/

or:

sw - i = (i ki - srk)/

Thus, if iki < srk, then it must be that sw - i < 0. But we know that (sw - i)(d /dk) < sw( /k) is sufficient condition for (dk/dt)/k < 0. So, if iki < srk, then (dk/dt)/k < 0. Now, as we know, at steady-state, i = nk*/i(ki*), thus substituting in, if:

(nk*/i(ki*))ki* < srk*

then k* is stable. Rearranging, this condition becomes:

sr > nki*/i(ki*)

So, in sum if:

(i) sw nk*/i(ki*)

or:

(ii) sr nki*/i(ki*)

then we are guaranteed that (dk*/dt)/k* < 0, so the steady-state capital-labor ratio, k* is (at least locally) stable.Drandakis's (1963) sufficiency conditions may be examined more closely now. More precisely, if the Drandakis's mixed condition that sr > sw and kc < ki is met, then both (i) and (ii) will hold. To see this, note that if s r > sw, then srk* > swk*. Also, if kc > ki, then swkc* > swk* > swki*. So combining:

srk* > swki*

so, dividing through by * + ki*

srk*/( * + ki*) > swki*/( * + ki*)

But as ki*/( * + ki*) = 1 - */( *+ki*), then this can be written:

86

srk*/( * + ki*) > sw [1- */( * + ki*)]

Now, recall by competition that i(ki*)/i (ki*) = * + ki*. So:

srk*i (ki*)/i(ki*) > sw [1- *i(ki*)/i(ki*)]

rearranging:

[srk* + sw *]·i(ki*)/i(ki*) > sw

Now, in steady-state, we know that (sw * + srk*)i(ki*) = nk*. Thus, substituting in:

nk*/i(ki*) > sw

which is precisely the condition (i) for stability. Condition (ii) follows by extension. Thus, the first Drandakis mixed condition (sr > sw and kc > ki) is indeed sufficient for stability.An interesting observation is to realize that the sufficient condition for stability, (sw - i)(d /dk) < sw( /k), can be rewritten as:

sw [(d /dk) - ( /k)] - i(d /dk) < 0

But recall from our discussion of production theory the elasticity of substitution between factors is = (dk/d)·( /k). Notice, then that if 1, then this inequality is guaranteed. Thus, 1 is also a sufficient condition for stability.Finally, an even more famous sufficiency condition derived by Drandakis (1963) is that the elasticity of substitution of the consumer-goods sector be greater than 1, i.e. that

c = (dkc/d )·( /kc) 1.

which we shall not attempt to prove here.

4.1.4.  Indeterminacy, Instability and Cycles

 Let us summarize what we have found. Remember that the Uzawa two-sector growth model poses two essential questions: (1) can we guarantee a unique factor-market equilibrium for every capital-labor ratio k? (2) can we guarantee that k approximates a steady-state growth path k* over time? We have found sufficient conditions for (1) and (2).

In Uzawa's (1961) Classical hypothesis case (all wages spent, all profits saved), the sufficient condition for (1) was that the Uzawa capital-intensity condition hold true (i.e. that the consumer goods industry is more capital-intensive that the investment goods industry). This, as we saw intuitively, was also sufficient to guarantee uniqueness and stability of the steady state (i.e. sufficient for (2)).

87

In the more flexible savings case, we found that a sufficient condition for (1) was that sr = sw = s (Uzawa's (1963) savings condition). Another sufficient condition (stated, but not proved) was that sr > sw and kc > ki and still another was that sr < sw and kc < ki (the Drandakis's (1963) mixed conditions). To guarantee (2), we must add the Uzawa capital-intensity assumption to Uzawa's savings condition (not shown, but deducible). Alternatively, we have shown that the first of Drandakis' mixed conditions is also sufficient for (2).

Of course, all the conditions we have outlined thus far are all sufficient conditions for stability, but not necessary conditions. In other words, other configurations of savings and capital-intensity could yield us uniqueness and/or stability, but it these would by no means be guaranteed.

It might be useful, then, to consider cases where we have indeterminacy and instability with other configurations. Examples are given by, among others, Uzawa (1963) and Inada (1963). An interesting case is the possibility of cycles. Specifically, consider Figure 8, effectively a reproduction of the differential equation we had in Figure 7, but now with arrowheads included to indicate direction. Recall that this was obtained from a factor market equilibrium locus k() which was not monotonic (see Figure 6).

In Figure 8, there are two steady-states, b and d. Immediately, we have indeterminacy of steady-state equilibrium as either could obtain. It is immediately noticeable that both steady states b and d are locally unstable, in the sense that any slight nudge away from them and we will return to them. Thus, we have no stable steady-states. At k0, we also have indeterminacy of factor market equilibria: points a, b and c are all possible factor market equilibria at k0. There is no reason to accept one over the other.

Fig. 8 - Indeterminacy and Cycles

88

A cycle can be traced from the passage along the points (1, 2, 3, 4). Specifically, suppose we begin at point c, with capital stock k0 and factor price equilibrium represented by c. In this case, dk/dt < 0, and so the capital-labor ratio declines below k0. We continue declining until we reach kL (point 1 in Figure 8). At kL, we still have it that dk/dt < 0, so there will be a tendency to continue to reduce k. But notice that the slightest reduction in k below k L leads to an immediate jump from point 1 to point 2, i.e. there is a catastrophic jump from dk/dt < 0 to dk/dt > 0.

Thereafter the capital-labor ratio begins rising from kL towards k0. At k0, however, we will be at factor equilibrium point a: we have no reason to jump down towards b to get steady-state. But staying at a implies that dk/dt > 0, so k continues to rise. It will do so until we reach kU. At this capital-labor ratio, a slight increase in k will lead to an immediate catastrophic jump from point 3 to point 4, and thus a drastic reversal from dk/dt > 0 to dk/dt < 0 around kU. Now, the capital-labor begins falling, from kU towards k0. But at k0, we will have the factor market equilibrium implied by c, and not b. Thus, we continued declining towards kL.

Thus, as we see in Figure 8, we have a constant cycle fluctuating back and forth between kL and kU, passing over k0 in the process but never actually stopping there because the steady-state factor market equilibrium b never gets the chance to realize itself.

How anomalous is this case? Inada (1963) provides a precise example of the kind of cyclical behavior we see. Naturally, none of the uniqueness conditions we imposed above hold here -- e.g. Uzawa's capital-intensity assumption must have been violated in order to obtain this kind of differential equation. Uzawa (1963) considered the possibility of cycles. He noted that the kind of jumps required (from 1 to 2, from 3 to 4) implied discontinuous jumps in the factor price equilibria. What if these were not accommodated? In other words, what if we forced factor prices to adjust only slowly? This requires that we allow for unemployment of capital and/or labor for extended periods of time. This modification will not be pursued here.

An interesting consideration is the Uzawa savings hypothesis, sw = sr = s. This was, as we noted earlier, sufficient to rule out multiple factor-market equilibria. But, in and of itself, it does not rule out multiple steady-states. Specifically, we could still have the case depicted in Figure 9, where although we only have one factor-market equilibrium for every k, nonetheless we have three steady-state capital-labor ratios, k1, k2 and k3. We will have alternating stable/unstable capital-labor ratios (k1 and k3 are stable, and k2 unstable). But this might not really matter too much: from any initial k, we will approach some steady-state. For instance, in Figure 9, if we begin with a capital-labor ratio smaller than k2, we will approach k1 over time; if our initial k is greater than k2, then we will approach k3.

89

Fig. 9 - Multiple Steady-States

However, if we were to impose the Uzawa capital-intensity condition on top of the Uzawa savings assumption, then Figure 9 could no longer hold: we would have uniqueness and stability of steady-state equilibrium.

Figure 9 has an even more interesting implication as it is almost directly comparable to the Solow-Swan (one-sector) growth model. We obtain Figure 9 by assuming the same savings propensity out of wages and profits, so there is only one savings rate in the economy, s = sw = sr, just like in Solow-Swan! Now, we don't have the complications of multiple factor-market equilibria, but we do have the complications of multiple steady-state growth paths. In Solow-Swan this never happened: steady-state was always unique. However, even though all the main assumptions are the same, the two-sector model yields a qualitatively different result.

There are other modifications to the standard Uzawa two-sector model. For instance, Drandakis (1963) endogenized the labor supply by making it responsive to wages (thus getting actually closer to the Classical way of thinking). Morishima (1969) did the same, but found that the Uzawa capital-intensity condition was no longer sufficient for stability.

No less interestingly, Joseph Stiglitz (1967) has allowed capital to be owned by explicitly different capitalist-worker classes and analyzed the consequent implications. Adopting the savings assumptions from Pasinetti (1962), specifically that the savings propensity of capitalists exceeds that of workers, then he found that the Uzawa capital-intensity assumption is sufficient for uniqueness of factor market equilibrium and also sufficient for the uniqueness and stability of steady-state growth path. However, for existence of a steady-state path at all, it is necessary and sufficient that the savings rate of workers is less than the ratio of investment to output.

90

Other modifications include the following. Yasui and Uzawa (1964) make capital depreciation endogenous. Inada (1966) introduced fixed capital which cannot be transferred from one use to another. Two-sector models with fixed proportions technology, as noted, have also been considered by Y. Shinkai (1960) and John Hicks (1965). Inada (1963) has allowed for the savings rate itself to be a function of income, so s = s(y).

Technological change has been one of the bigger headaches because now we must ensure that technological change is such that all sectors grow at the same rate. Diamond (1965) and Takayama (1965) developed a classification scheme is analogous to the one-sector case where we have Harrod-neutral technical change where there is constant capital per effective labor unit in each sector as well as in the aggregate. Charles Kennedy (1962) proposed a different type of neutral technical progress where it is the value of capital per labor unit that remains constant, which has different implications. We consider this elsewhere.

Finally, we should try to decipher the main lesson from all this. Robert Solow claims that it "seems paradoxical to me that such an important characteristic of the equilibrium path [i.e. stability] should depend on such a casual property of the technology." (Solow, 1961). Perhaps it is not that paradoxical. As Frank Hahn notes, the Uzawa capital-intensity condition and attendant savings hypotheses "are all terrible assumptions" (Hahn, 1965) which probably never or only rarely hold in the real world. It is thus reasonable to conclude from this exercise that, if anything, these models have taught us that the real world is a bit more complicated than one-sector models let on. Rather than the smooth convergence to a single steady-state path we find in the Solow-Swan model, the Uzawa two-sector model indicates that we ought to expect a good amount of indeterminacy, instability and cyclical behavior in growth paths. Now that sounds a little bit more like the world we live in.

The famous "optimal growth" extension of the two-sector model by Uzawa (1964) and Srinivasan (1964), which tries to introduce some more determinacy into the result shall be considered in the next section.

4.2. Optimal Two-Sector Growth

4.2.1. The Uzawa-Srinivasan Model

The central result of the Uzawa (1961, 1963) two-sector growth model is that the uniqueness of factor market ("momentary") equilibria and the stability of the steady-state growth path depends crucially on the relative factor-intensities of the two sectors and attendant savings hypotheses. The most celebrated result is that we are only assured stability if the consumer goods sector is more capital-intensive than the investment goods sector. Alternative configurations are generally not sufficient to guarantee nice results. Consequently, Hirofumi Uzawa (1964) and T.N. Srinivasan (1964) sought to find a way of pinning things down without such a radical capital-intensity assumption. They did so looking for the "optimal" growth path, specifically the growth path that maximized the integral of the consumption path.

91

We shall not bother to repeat definitions and notations here. We shall merely set out the basic equations for the Uzawa two-sector model:

yc = cc(kc)- consumer sector intensive production function

(1)

yi = Ii(ki)- investment sector intensive production function

(2)

y = yc + pyi - aggregate output per capita (3)

c + I = 1 - labor market equilibrium (4)

ckc + iki = k - capital market equilibrium (5)

w = c - kcc         = p·(I - kii )

- labor market prices (6)

r = c = p·i - capital market prices (7)

gL = n - labor supply growth (8)

gK = yi/k - capital supply growth (9)

The derivation of these were given in an earlier section.Now, as announced, Uzawa (1964) proposed that we consider an economy run by a social planner who seeks to maximize the integral of consumption per capita subject to these constraints. Specifically, his program is:

max 0

yc e- t dt

s.t.

dk/dt = yi - nkk(0) = k0

yc = cc(kc)yi = ii(ki) c + i = 1 ckc + iki = kkc, ki, c, i 0

92

Notice that consumption per capita is merely the output of the consumer goods sector, yc. The term is the time preference of the social planner (and notice that there is no other "utility" element involved). The first constraint, dk/dt = y i

- nk is the fundamental differential equation and this is obtained by recognizing that gk = gK - gL, and then substituting in (8) and (9).So far so good. But things can be reduced further. Notice that combining (4) and (5), we obtain:

c = (k - ki)/(kc - ki)

i = (kc - k)/(kc - ki)

We can then plug these terms into (1) and (2) to obtain

yc = cc(kc) = [(k - ki)/(kc - ki)]·c(kc)

yi = ii(ki) = [(kc - k)/(kc - ki)]·i(ki)

Finally, we have not used (6) or (7) yet. These will yield (as we showed in the last section) the functions:

kc = kc(

where kc = (c )2/(c ·c) > 0

ki = ki()

where ki = (i )2/(i ·i) > 0

which will form boundaries of our factor market equilibrium. As we know, the boundaries imply that the wage-profit ratio [ min, max], where:min = c(k), max = i(k)     if     kc( ) ki( ) for all i.e. if the consumer goods sector is more capital-intensive. Alternatively:max = c(k), min = i(k)    if    kc( ) ki( ) for all i.e. if the investment goods sector is more capital-intensive. All this is explained in greater detail in the previous section.Plugging our new terms for yc and yi, omitting the equations already used and adding in our boundaries, our program becomes:

max 0

{[(k - ki)/(kc - ki)]· c(kc)}e- t dt

s.t.

dk/dt = [(kc - k)/(kc - ki)]· i(ki) - nkk(0) = k0

kc = kc()ki = ki()c(k) i(k)

The control variable is the wage-profit ratio, ; the state variable is k. Notice that the last line indicates that we have assumed that the consumer-goods

93

industry is more capital-intensive. If the investment-goods industry was more capital-intensive, we would replace that line with i(k) c(k). Therefore, let us divide our analysis into two parts, one for each case.

4.2.2. Case I: Consumer Goods are More Capital-Intensive Suppose the famous Uzawa capital-intensity condition holds, so that consumer goods are more capital intensive than investment goods, i.e. kc() > ki() for all admissable . Setting up the current-value Hamiltonian:

H = [(k - ki)/(kc - ki)]·c(kc) + {[(kc - k)/(kc - ki)]· i(ki) - nk}

where is the current-value costate variable. Notice that kc and ki are implicitly functions of . First order conditions are (after a lot of ugly algebra):dH/d = [ i - c ]·{(dki/d )·[(k - ki)( + ki)/(kc - ki)2] + (dkc/d )·[(kc - k)( + kc)/(kc - ki)2]} = 0recalling our definitions of c and i, this can be written as:

[i - c ]·[(dki/d )· c·( + ki)/(kc - ki) + (dkc/d )· i·( + kc)/(kc - ki)] = 0

Now, if we assume consumer goods are more capital-intensive, kc > ki, then (kc

- ki) > 0. Then as c, i 0 and dkc/d > 0 and dki/d > 0, then obviously the entire second term is positive. Alternatively, if we assume that investment goods are more capital intensive, then (kc - ki) < 0, and the entire second term is negative. In either case, it must be that:

i - c  = 0

(corner solutions would allow this to be different, but then i or c would be set to zero). Notice that this means:

= c /i

which should be familiar to us. Recall that p = c /i , thus the costate variable is nothing other than the (shadow) price of the investment goods.Continuing with our Hamiltonian, notice that:

-dH/dk = d /dt - = -[ c(kc)/(kc - ki) - i(ki)/(kc - ki) - n]

or, rearranging:

d /dt = (n + ) + [ i(ki) - c(kc)]/(kc - ki)

Now, recall that: i = (ki + )·i and c = (kc + )· c , so:

d /dt = (n + ) + [ (ki + )· i - (kc + )·c ]/(kc - ki)

Recall that from our first condition we obtained = c / i . So plugging in:

94

d /dt = (n + ) + [(ki + )·c - (kc + )·c ]/(kc - ki)

so rearranging:

d /dt = (n + ) + (ki - kc)·c /(kc - ki)

or simply:

d /dt = (n + ) - c

Finally, recognizing that c = i , this reduces to:

d /dt = (n + - i )

which is quite a neat expression. Now, as = p(), then differentiating with respect to time:

d /dt = (dp/d )(d /dt)

or:

d /dt = (d /dt)/(dp/d )

So, substiting in d /dt and remembering that = p:

d /dt = p·(n + - i )/(dp/d)

The question that emerges is what is p/(dp/d )? Well, we know from before that:

(dp/d )·(1/p) = 1/( + ki) - 1/( + kc)

Thus, we have it that:

 

d /dt = [n + - i ]/{1/( + ki) - 1/( + kc)}

          (10)

which seems ugly, but is actually quite innocuous. This is our first differential equation. Our second comes from the condition dH/d = dk/dt, and is merely the recovery of the constraint:

dk/dt = [(kc – k)/(kc - ki)]·i(ki) - nk (11)

Thus (10) and (11) are our two differential equations in (k, ) space.  This is plotted in Figure 1.

95

Let us begin with (10). The derivation of the isokine d /dt = 0 in Figure 1 quite simple. Recognize that if d /dt = 0, then:

n + = i [ki()]

Assuming i and ki are invertible, then:

* = ki-1[i -1(n + )]

As n+ and i (.) and ki(·) are given and do not vary with k, then there is a unique * for which this holds true. Thus, the d /dt = 0 is a horizontal line in ( , k) space. The implict dynamics can be found as follows. Defining = (dp/d )·(1/p), then notice that (10) can be rewritten as:

d /dt = (n + - i )/

so, differentiating with respect to :

d(d /dt)/d = [-i - (d /d )(n+ - i )}/ 2

But, evaluated near *, we know that n + = i [ki( *)], thus this reduces to:

d(d /dt)/d | * = -i / > 0

which is positive by assumption that i < 0 and by the Uzawa capital-intensity hypothesis, > 0. Thus, a small increase in above * will lead to a rise in , while a fall in below * will lead to a further fall. Thus, the vertical directional arrows moving away from the d /dt = 0 isokine in Figure 1.

Fig. 1 - The d /dt = 0 Isokine

96

What about (11)? The isokine dk/dt = 0 is established as follows. Note that dk/dt = 0 implies that:

[(kc - k)/(kc - ki)]·i(ki) = nk

We wish to solve this for k. Notice that:

i(ki)kc/(kc - ki) = {n + i(ki)/(kc - ki)}k

or:

i(ki)kc = {n(kc - ki) + i(ki)}k

so:

k() = i(ki)kc/{n(kc - ki) + i(ki)} (12)

This will form the shape of our dk/dt = 0 isokine. It is necessary to decipher what isokine looks like. Notice that we can rewrite (12) as:

(kc - k) = [nk/i(ki)]·(kc - ki)

Now, nk/i(ki) > 0 by assumption, thus the sign of (kc - k) depends critically on the sign of (kc - ki), i.e. on which sector is more capital-intensive. Now, by our assumption that consumer goods are more capital intensive than investment goods, then kc( ) > ki( ) for all admissable . Consequently, we necessarily have it that:

k() < kc( ) for all

so the isokine of dk/dt = 0 will lie everywhere to the left of the k c( ) cure (see Figure 2).

97

However, the dk/dt = 0 isokine does not necessarily lie everywhere to the right of ki(). Specifically, notice that the dk/dt = 0 isokine intersects the k i() line at n, so that for all > n, we have it that k() > ki().

Fig. 2 - The dk/dt = 0 Isokine

How do we know it intersects at n and not earlier or later? To see why, suppose that i(ki) > nk. If this is true, then (12) implies that:

(kc - k)/(kc - ki) = [nk/i(ki)] < 1

which implies (kc - k) < (kc - ki), or simply:

k() > ki()

So, k lies to the right of k i whenever it is the case that i(ki) > nk. Of course, it is not true that this holds for all . Nonetheless, we know that for low this will be true. To see why, let us proceed slowly. We first want to prove that if i(ki) > nki then i(ki) > nk. To see this, note that (12) can be rewritten as:

(i(ki) - nki)k = (i(ki) - nk)kc

98

so assuming neither k nor kc are zero, then necessarily, i(ki) > nki implies i(ki) > nk. The condition i(ki) > nki can be depicted in Figure 3 where we have the intensive production function for investment goods i(ki) depicted as well as nk, a ray from the origin with slope n. The point en depicts the intersection of the intensive production function and the nk ray. At this intersection, the capital-labor ratio in the investment goods industry is k i

n, thus i(ki) = nkin. So, for all ki

< kin, we have it that i(ki) > nki, but for all ki > ki

n, we have it that i(ki) < nki.

Fig. 3 - Maximum Factor Price Ratio n

Now, associated with this critical point is a factor price ratio n. We can obtain this by extending a curve tangent to the intersection point en to the horizontal axis. Where this tangent line intersects the axis is the maximum factor price ratio, n. If > n, then notice that this implies that the corresponding k i is greater than ki

n, or ki > kin, but then i(ki) < nki and thus the condition that

dk/dt = 0 isokine lies to the right of ki( ) no longer holds. Thus, for all factor price ratios up to n, we have it that k( ) > ki( ). For factor price ratios above n, we have it that k( ) < ki( ), thus the dk/dt = 0 locus has exceeded the left boundary. This is what we see in Figure 2. The point kn is the aggregate capital-labor ratio that corresponds to the maximum factor price ratio, n. Thus, we have established that dk/dt = 0 a locus k( ) where:

ki() k() kc()

for all n.Now, let us examine the dynamic properties. From equation (11), we can see immediately that:

d(dk/dt)/dk = -i/(kc-ki) - n < 0

unambiguously, as kc > ki by the Uzawa capital-intensity assumption. Thus, the horizontal directional arrows in Figure 2 are stable towards the dk/dt = 0 isokine.

99

We can now superimpose Figure 1 and Figure 2 to yield the dynamics in Figure 3. As we can see immediately, we have a general steady-state where the d /dt = 0 and the dk/dt = 0 isokines intersect, at e = ( *, k*) in Figure 3. As is obvious, the dynamics indicate that we have a saddlepoint stable system, with the d /dt = 0 isokine acting as the stable arm and dk/dt = 0 isokine as the unstable arm. So, all paths that begin off the stable arm will gradually move away from the equilibrium.

Fig. 4 -Dynamics of Optimum Growth -- kc( ) > ki( ) case

The stable arm is actually a bit more complex, due to the boundaries formed by the kc( ) and ki( ) loci. We can trace it as follows: for values of k between 0 to kL, the stable arm is the ki( ) locus; for k values between kL to kU, the stable arm is the d /dt = 0 isokine, and for k above kU, the stable arm is the kc( ) locus. Thus, the thick black line in Figure 5 denotes the full stable arm of the economy.The logic is the following. If k < kL, then the economy needs to grow quickly to catch up to k*. Consequently, it will specialize completely in the production of investment goods -- thus we "jump" to the ki( ) locus. In contrast, if k > kU, the economy needs to slow down on capital-accumulation so that k declines, thus it jumps to complete specialization in consumer goods and cuts production of investment goods to zero, thus for such high values of k we jump to the kc( ) locus.For capital-labor ratios in between kL and kU, we do not "jump" to complete specialization in either consumer goods or investment goods, but produce a little bit of both -- thus for k (kL, kU), we will choose points in the interior of the space in Figure 5.But why choose * in particular? Because from *, d /dt = 0, so there is no change in over time and the dynamics are such that we glide smoothly and asymptotically to the balanced growth point, e = ( *, k*). If we chose a higher than * but still in the interior of the area, notice that the underlying dynamics would push upwards over time, even if k approached k* over time (which it might not!). Eventually, when k finally hits k* (or if we hit a boundary), would be so far above * that there would have to be a sudden and drastic correction in , an enormous jump down to *. Similarly, if we initially choose a below *, would be pushed further downwards, so that there would have to be an eventual drastic correction in factor prices. Such late catastrophic jumps in factor prices are not necessarily "optimal" things. Far better to jump early onto * and just let the natural dynamics of the economy keep

100

constant at * while we gradually approach k*. That is why the stable arm will be chosen for k (kL, kU). This can be deduced from the transversality conditions of the solution.

4.2.3. Case II: Investment Goods are More Capital-Intensive Suppose we drop the Uzawa capital-intensity assumption. In other words, let us allow it that investment goods are more capital-intensive than consumer goods, so that ki( ) > kc( ) for all . As we know from before, allowing this in a two-sector model can mean that all hell breaks loose. But, with the optimality criterion keeping things in control, things can go quite smoother.The modifications on our previous case are as follows. Firstly, in diagrammatic terms, the kc( ) curve will lie everywhere above the k i( ) curve (see Figure 5), thus reversing the boundaries of our previous case. In the optimization problem, we reverse the range of our factor price ratio, so that now i(k) c(k). The rest of the program, the Hamiltonian and the conditions for a maximum are the same. Therefore, we end up with the same differential equations:

d /dt = [n + - i ]/{1/( + ki) - 1/( + kc)} (10 )

dk/dt = [(kc - k)/(kc - ki)]·i(ki) - nk (11 )

which are identical to (10) and (11) we had before.Let us proceed with the derivation of the isokines. For d /dt = 0, we still obtain the same result that:

n + = i [ki( )]

for which there is a unique solution *, thus the d /dt = 0 isokine is a horizontal line, just like before. However, notice now that evaluating dynamics at equilibrium, we have:

d(d /dt)/d |* = -i / < 0

because = (dp/d )·(1/p) < 0 when investment goods are more capital-intensive. Thus, unlike before, the d /dt = 0 isokine is stable in , so if > *, then declines, while if < *, then rises. The vertical directional arrows thus approach the d /dt = 0 isokine.How about the dk/dt = 0 isokine? Setting (11 ) to zero, we can resolve this for k to yield:

 

K() = i(ki)kc/{n(kc - ki) + i(ki)} (12 )

which identical to our (12) before. Recall that we could re-express this as:

(kc - k) = [nk/ i(ki)]·(kc - ki)

101

So, since nk/ i(ki) > 0 by assumption, and since investment goods are more capital-intensive than consumer goods, then (kc - ki) < 0 and thus (kc - k) < 0, i.e.

k( ) > kc( ) for all

so the isokine of dk/dt = 0 will lie everywhere to the right of the kc( ) curve (see Figure 5).However, like before, the isokine does not lie everywhere on one side of the ki( ) curve. There is an intersection point between the dk/dt = 0 isokine and the ki( ) locus at a critical wage-profit ratio n. This is in fact identical to before, i.e. n solves i(ki( n)) = nki( n), so if < n, then i(ki) > nki and so k( ) < ki( ), so that the dk/dt = 0 isokine lies to the left of the k i( ) locus and thus withing the bounds. In contrast, if > n, then i(ki) < nki and therefore k( ) > ki( ) so that the isokine lies to the right of the k i( ) locus and thus outside the bounds. We see this in Figure 5.The dynamics of the dk/dt = 0 isokine are easy to decipher. Specifically, note that:

d(dk/dt)/dk = - i/(kc-ki) - n > 0

unambiguously because ki > kc for all by the new capital-intensity assumption. Thus, a slight nudge in k above the isokine will lead to a further rise in k, and a slight movement below, will lead to a further reduction in k. Thus, the dk/dt = 0 isokine is unstable, as indicated by the unstable horizontal arrows in Figure 5.Combining the two isokines, d /dt = 0 and dk/dt = 0, we obtain the phase diagram in Figure 5. Once again, it is a saddlepoint, except now the stable arm is not any of the isokines, but off it. The stable arm is depicted by thick black line in Figure 5. The long-run equilibrium, the steady-state growth path, is the intersection point e = (k*, *) in Figure 5.

102

Fig. 5 -Dynamics of Optimum Growth -- ki() > kc() case

Effectively, the same analysis applies as before. Notice that the stable arm of the saddlepoint intersects the ki() locus at kL and the kc( ) locus at kU. Now, if k < kL, then k is so much below k* that it makes sense to specialize completely in the production of investment goods (thus jumps to the k i( ) locus) so that k climbs quickly. If, in contrast, k > kU, then k is so much higher than k*, that we want to stop accumulating capital and specialize completely in the production of consumer goods, so that k falls quickly. Finally, if we start at a k between kL

and kU, we will jump onto the saddlepoint stable arm, and glide slowly towards the steady-state equilibrium, e = (k*, *).

4.2.4. ConclusionAs we see from the Uzawa-Srinivasan exercise, adding optimality criterion removes many of the difficulties we found in the conventional Uzawa two-sector growth model. Specifically, we no longer have the Uzawa capital-intensity requirement for stability. Consumer goods can be more or less capital-intensive than the investment goods, but that will not affect the

103

"stability" of the system. The system, after all, is driven by the social planner, and his sole criterion is the optimality of the consumption path. Thus, the social planner will drive us straight to the balanced growth path, and by-pass all the "real-world" difficulties we had in our simpler two-sector growth model. As it happened, the Uzawa-Srinivasan attempt to find "optimal growth" in a two-sector model preceded and was in fact the impetus for the resurrection of the Ramsey one-sector optimal growth model by David Cass (1965) and Tjalling C. Koopmans (1965).However, before declaring victory, we should note some peculiarities about the social planner. Firstly, the social planner is maximizing consumption per capita and not utility. Thus, the traditional Benthamite justification of social utility is not really used (or, rather, we have replaced a diminishing marginal utility with constant marginal utility for the social welfare function). Secondly, we obtain "saddlepoint" stability, which is not quite "stability". In principal, beginning with any given k and , we will not go to balanced growth, but rather move away from it. Thus, there needs to be a guide to set initial wage-profit ratio on the stable arm to ensure that we go to steady-state.Before the lamentable rise of the "representative agent" reasoning we have today, it used to be argued that the government could perform many of the functions of the social planner for these intertemporal optimization problems. Specifically, by manipulating various fiscal, monetary and pricing policy instruments, the government could attempt to guide us to the steady-state growth path.In fact, the two-sector model lends itself rather nicely to treatment of government activity. As Hirofumi Uzawa (1969) and Kenneth J. Arrow and Mordecai Kurz (1970) demonstrate, we can think of a mixed economy as one where there is a private sector producing one kind of good and a public sector producing another (roads, bridge, dams, etc.) which can be used by the first sector and vice-versa. Add a government objective to the story, and this is effectively an optimal two-sector growth model.Models of monetary growth, stemming from the contributions of James Tobin (1965) onwards, for instance, can be considered to be a type of two-sector model with room for government activity -- but now "money creation" is our second "sector". However, before we proceed with these models, it is necessary to consider multi-sectoral models where we have more than one type of capital good. These "heterogeneous capital" growth models shall be taken up in our next section.

4.3. Selected ReferencesK.J. Arrow and M. Kurz (1970) Public Investment, the Rate of Return and Optimal Fiscal Policy. Baltimore: Johns Hopkins University Press.T.N. Srinivasan (1964) "Optimal Savings in a Two Sector Model of Growth", Econometrica, Vol. 32, p.358-73.H. Uzawa (1964) "Optimal Growth in a Two-Sector Model of Capital Accumulation", Review of Economic Studies, Vol. 31, p.1-24.H. Uzawa (1969) "Optimum Fiscal Policy in an Aggregative Model of Economic Growth", in I. Adelman and E. Thorbecke, editors, The Theory and Design of Economic Development. Balitmore: Johns Hopkins Press.

104

5. Optimal Growth 1285.1. Optimal Growth: Introduction 1285.2. The Ramsey Exercise 1325.3. Golden Rule Growth 1425.4. Intertemporal Social Welfare 1465.4.1.     Intertemporal Social Welfare Functions 1475.4.2.     The Defense of Discounting 1525.4.2.1.   The Tastes Defense 1525.4.2.2.   The Dynastic Defense 1555.4.2.3.   The Decentralization Defense 1585.4.3.     The Koopmans Axiomatization 159 5.4.4.     Population Growth 1685.4.5.     Overlapping Generations 1705.4.6.     Varying Time Preference 1735.4.7.     Intertemporal Justice 1765.4.7.1.   Rawlsian Social Welfare 1765.4.7.2.   Rawlsian Altruism 179

105

5.5. (The Cass-Koopmans Optimal Growth Model) -5.6. (Optimal Two-Sector Growth) 1115.7. Optimal Growth: Conclusion 1815.8. Selected References 184

5. Optimal Growth

5.1. Optimal Growth: Introduction

________________________________________________________

"For the end of economy is not the physical augmentation of goods but always the fullest possible satisfaction of human needs."

(Carl Menger, Principles of Economics, 1871: p.190)

___________________________________________________________At least as far back as Eugen von Böhm-Bawerk (1889), economists had entertained the idea that people are "myopic" in the sense that they tend to underestimate their future needs and desires and therefore "discount" their future utilities. This was seen by Böhm-Bawerk and many of his contemporaries as an irrationality, a result of a deficient cognitive process.From this proposition, the Cambridge economist Arthur C. Pigou (1920) posed an interesting conundrum: if, indeed, agents tend to underestimate their future utility, they will probably not make proper provision for their future wants and thus personally save less than they would have wished had they made the calculation correctly. In other words, Pigou proposed, the very fact that people possess defective "telescopic faculties" probably means that savings, as a whole, are less than what is "optimal". This, Pigou conjectured, implies that there is a "market failure" of sorts in the market for savings.

106

"Generally speaking, everybody prefers present pleasures or satisfactions of given magnitude to future pleasures or satisfactions of equal magnitude, even when the latter are perfectly certain to occur. But this preference for present pleasures does not -- the idea is self-contradictory -- imply that a present pleasure of given magnitude is any greater than a future pleasure of the same magnitude. It implies only that our telescopic faculty is defective, and that we, therefore, see future pleasures, as it were, on a diminished scale....This reveals a far-reaching economic disharmony. For it implies that people distribute their resources between the present, the near future, and the remote future on the basis of a wholly irrational preference."

(A.C. Pigou, Economics of Welfare, 1920: p.24-5)

Yet in order to confirm that the rate of savings thrown up by a market system with myopic agents was indeed suboptimal, one must first determine what the optimal savings rate might be. It is at this point that the Cambridge philosopher Frank P. Ramsey (1928) picked up on Pigou's pregnant suggestion. Ramsey applied standard Benthamite utilitarian calculus to derive the "optimal rate of savings" for a society. He proposed an intertemporal social welfare function and then tried to obtain the "optimal" rate of savings as the rate which maximized "social utility" subject to some underlying economic constraints. Of course, Ramsey deliberately excluded discounting of future utility form this social welfare function: just because people are individually short-sighted, does not mean that society should be similarly "short-sighted". This is a normative, not a positive exercise.Ramsey's conclusion was to confirm Pigou's suggestion: the optimal rate of savings is higher than the rate that myopic agents in a market economy would choose. Yet Ramsey's arguments fell largely on deaf ears for three reasons. Firstly, Ramsey's use of the calculus of variations in his argument was quite beyond the mathematical understanding of most contemporary economists. Secondly, even the economic parts of the argument were subtle and unfamiliar. Recall that Irving Fisher's Theory of Interest was only written in 1930, so most economists understanding of the concept of "intertemporality" was still rudimentary.  Thirdly, and perhaps more importantly, Ramsey's exercise was an unfashionable one. The 1930s were the years of the Paretian revival of ordinalism and the "unholy alliance" of economics and Benthamite utilitarianism was gradually unraveling. The "New Welfare Economics" stayed clear of anything which implied any sort of interpersonal comparisons of cardinal utility. Ramsey's social utility function was certainly regarded as a damnable construction. During the 1950s and 1960s, capital and growth theory was emerging into its own and questions about "efficient" programs of accumulation were being asked (e.g. Malinvaud, 1953). Some people seemed to recollect that Ramsey had a thing or two to say about this. In a rather anachronistic, but highly commendable effort, Paul Samuelson and Robert Solow (1956) brought forth an extension of Ramsey's original model to a multi-commodity scenario.

107

Overall, it was practical considerations that resurrected the question of the "optimal" savings rate. For development economists -- who were in those days obsessed with government planning and accumulation -- the question was both natural and urgent. Jan Tinbergen (1956) was perhaps the first to try his hand.  Yet, almost from the outset, the exercise was attacked. As Peter T. Bauer (1957) and Jan de Van Graaff (1957) argued, the determination of the "optimal" savings rate is irrelevant for policy on at least two grounds. Firstly, it is not implementable -- people save what they will save, period. Secondly, even if could be implemented (e.g. via Social Security schemes and what not), it is not up for a "social planner" (i.e. government) to dictate it according to some simple ethical criterion set up by economists. If anything, it is a political decision, and will be the outcome of the political culture of a nation. And the "social planner", envisaged in optimal growth theory, is frighteningly authoritarian.  However, these objections were largely overruled by the dirigiste spirit of the times.  The optimal savings question was first applied on Keynesian (Harrod-Domar) growth models by Jan Tinbergen (1956, 1960) and Richard Goodwin (1961). But the Neoclassical (Solow-Swan) growth model had recently become available too. This model was particularly interesting because its steady-state path was generally "consumption-inefficient". Since the rate of savings is one of the critical parameters in determining the Solow-Swan steady-state, the question of "what is the optimal savings rate?" emerged quite naturally. In the early 1960s, numerous researchers independently examined the question of optimal savings for the Neoclassical model. The answer seemed simple: the optimal rate of savings will be that which makes the rate of return on capital equal to the natural rate of population growth. This "Golden Rule" for efficient growth, as it has been called, was set forth simultaneously by Edmund S. Phelps (1961), Jacques Desrousseaux (1961), Maurice Allais (1962), Joan Robinson (1962), Christian von Weizsäcker (1962) and Trevor Swan (1963).The derivation of the Golden Rule did not employ Ramsey's old Benthamite trappings of "social utility" and all that. These were, however, brought back into prominence after the subtle but influential work of Tjalling Koopmans (1960). Taking Ramsey's construction seriously (and piling on generous coats of "ordinalist" polish), Koopmans made it hip to consider intertemporal social welfare functions once again. Across the hallway in the two-sector growth model world, Hirofumi Uzawa (1964) and T.N. Srinivasan (1964) demonstrated how intertemporal optimality and growth theory could be combined fruitfully. After some false starts and a flurry of activity, David Cass (1965), Tjalling Koopmans (1965), Edmond  Malinvaud (1965), James A. Mirrlees (1967), Karl Shell (1967) and others finally pieced together the canonical one-sector optimal growth model. Although this is sometimes (and erroneously) called the "Ramsey" model, we prefer to refer to it by its other name, the "Cass-Koopmans" optimal growth model.Optimal growth theory began to recede in the 1970s for a variety of reasons. Firstly, the inconsistencies in capital theory unearthed during the Cambridge Controversy were a source of despair for growth theorists across the board and the optimal growth theorists were not immune to it. Furthermore, economists realized that the "social planner" did not really exist and developments in microeconomic theory indicated that any appeal to "representative agents" should be greeted with suspicion. But, above everything, it was the

108

"saddlepoint dynamics" of optimal growth models made them seem inherently inapplicable. There was no good economic reason to suppose that an economy would "stumble" upon the optimal growth path. Consequently, as the 1970s progressed, optimal growth models were discarded as ultimately inapplicable constructions, however beautiful and utopian they may seem. The tune changed in the 1980s with the rise of the rational expectations revolution. Saddlepoint dynamics began being regarded as an asset rather than a liability of a model. Specifically, rational expectations were precisely the mechanism by which an economy would jump onto the stable arm of a saddlepoint. Indeed, saddlepoints were necessary if one were to obtain a precise solution to a model with rational expectations! It was also during the 1980s that the "decentralization" argument began being put forth more forcefully. As a result, optimal growth models stopped being conceived of as "normative" exercises about the way the economy should work, and started being regarded as a "positive" exercise about the way the economy does work. Real business cycle theory, one of the principal macroeconomic enterprises of the late 1980s and 1990s, built itself up precisely on that premise.Our brief survey of optimal growth theory concentrates almost exclusively on its connection with one-sector, Neoclassical growth theory. We begin with Ramsey's 1928 exercise and then jump a quarter-century to the 1960s  the "Golden Rule" of growth. We then  take a rather leisurely digression on intertemporal social welfare functions and the ethical implications of time preference. All this leads us to the Cass-Koopmans optimal growth model, the version of optimal growth theory that is closest to Solow-Swan. Finally, we turn to a brief discussion of turnpikes and the infamous "decentralization" argument.

5.2. The Ramsey Exercise

________________________________________________________

"[Ramsey's 1928 article] is, I think, one of the most remarkable contributions to mathematical economics ever made, both in respect of the intrinsic importance and difficulty of its subject, the power and elegance of the technical methods employed, and the clear purity of illumination with which the writer's mind is felt by the reader to play about it subject. The article is terribly difficult reading for an economist, but it is not difficult to appreciate how scientific and aesthetic qualities are combined in it together."

(John Maynard Keynes, "F.P. Ramsey", Economic Journal, 1930)

________________________________________________________

As outlined in our introduction, Arthur C. Pigou's (1920) assertion that myopic agents might "save too little" was taken up by the brilliant young Cambridge philosopher, Frank P. P. Ramsey (1928) -- long before growth theory came into

109

its own.  Ramsey's main concern was to determine the optimal rate of savings and then show how myopic agents would not achieve that optimum.But what is the optimal rate of savings?  Ramsey's exercise was explicitly grounded in Benthamite utilitarianism.  He sought to find the allocation across generations which maximized social welfare -- with "social welfare" defined as the sum of utilities of people in a society.  From the social welfare-maximizing allocation, we will be able to determine an "optimal" rate (or rather, path) of saving.However, this comes up against traditional Benthamite problem of defining exactly whom constitutes "society". This is particularly pertinent in a growth context as the issue of balancing the interests of current and the future members of society is a critical ingredient. It is obvious that we can maximize the social welfare of the current generation by having them simply consume all their income, but then there would be no savings and thus no capital to generate income for the next generation. Like Arthur Pigou (1920), Frank Ramsey argued that "society" is composed of everybody in every generation, current and future, and that they all should be given equal weight in the social welfare function. Now, as we outline later, a direct Benthamite sum of utilities yields the problem that this sum can be infinite -- and infinities cannot be compared, and thus an optimum might not be found. Adding a time discount factor would solve the problem, but Frank Ramsey considered time discounting as "a practice which is ethically indefensible and arises merely from the weakness of the imagination" (Ramsey, 1928). Time preference, as Pigou (1920: Pt I, Ch. 2) originally asserted, is a personal weakness which should not be imported into a normative exercise. (see our review of "intertemporal social welfare" for more details).However, Ramsey recognized that by omitting time preference, the problem of non-comparability of infinite sums emerged. In its stead, he introduced the ingenious device of "bliss points". Specifically, he defined a social unwelfare function of the following sort:

R = t=0 (B - U(Ct))

or, in continuous time:

R = 0 [B - U(Ct)] dt

where B is the "bliss" level of utility, assumed to be the maximum utility achievable for a generation. This is the same across generations. Note that [B - U(Ct)] is the distance away from bliss that generation t is if it consumes C t: the smaller [B - U(Ct)] is, the "better off" the generation is. We can see the utility function in Figure 1. When C = C*, then we are at "bliss" in the sense that marginal utility is zero, U (C*) = 0, so that increasing consumption beyond this reduces utility (or, more generally, does not increase it). At lower consumption levels, such as C0 in Figure 1, B > U(C0), and U (C0) > 0.

110

Fig. 1 - Utility Function with Bliss PointConsequently, for a social optimum, we wish to find the consumption allocation

across generations for which R is minimized. This, Ramsey asserted, side-steps the issue of "infinite sums", but, as we shall later, it is actually doubtful that the Ramsey device is foolproof.Let us proceed with the Ramsey exercise. Thus far, we have only some idea of what the Ramsey social welfare function is. Now, we need to add the rest of the economy. In macroeconomic equilibrium, aggregate demand is equal to aggregate supply, or Y = C + I. As I = dK/dt and Y = F(K, L), then:

dK/dt = F(K, L) - C

forms the "real economy" constraint which our society faces. Thus, consumption paths must be feasible in this manner. Now, Ramsey (1928) included the disutility of labor supply into his utility function, so letting U(C) be the utility of consumption and V(L) the disutility of labor supply, Ramsey combined them in an additively separable manner, i.e. u(C, L) = U(C) - V(L), where U > 0, U < 0 and V > 0 and V > 0. Consequently, defining B as "bliss", we mean that [U(Ct) - V(L)] B for all acceptable Ct and thus, note, U (C) = 0 when U(C) - V(L) = B (i.e. at bliss, marginal utility of further consumption is zero). The distance from bliss for any generation t is [B - U(C(t)) + V(L(t))]. Ramsey then set out his social welfare problem in minimization form as:

min 0 [B - U(C(t)) + V(L(t))] dt

s.t.

dK(t)/dt = F(K(t), L(t)) - C(t)

K(0) = K0

111

where K0 is the initial capital stock.The problem can be thought of as following. Social welfare is maximized if every generation is "at bliss". However the society's initial capital stock K0 (and thus the initial income) may be too low to achieve bliss levels of consumption immediately. Consequently, we might be interested in achieving bliss as quickly as possible by holding back consumption and producing a lot of capital today. The more we save now, the sooner society will reach bliss. But this is a bit unfair for the first few generations who must sacrifice their own consumption to ensure quicker convergence to bliss. Consequently, the "cost" of achieving bliss quicker is the consumption utility that the initial generations lose in their sacrifice. Balancing quick convergence to bliss with the utility cost of foregone consumption is the heart of the Ramsey problem.The solution follows standard calculus of variations techniques -- a mathematical tool which, incidentally, Ramsey (1928) was among the first to introduce into economics. For simplicity of notation, let K = dK/dt and L = dL/dt and let the intergrand be denoted:

I = B - U(C) + V(L)

So, substituting in our constraint:

I = B - U[F(K, L) - K ] + V(L)

Defining IL = dI/dL and IL = dI/dL and equivalently for IK and IK , then:

IL = -U ·FL + V

IL = 0

IK = -U ·FK

IK = U

Euler equations are the solution for calculus of variations problems. For our particular problem, we have the following pair of Euler equations:

IL + d(IL )/dt = -U ·FL + V = 0

IK + d(IK )/dt = -U ·FK + dU /dt = 0

which hold for all t 0. The first Euler equation reduces simply to:

V = U ·FL for all t

so that, at every t, the marginal disutility of labor must be equal to the marginal product of labor multiplied by the marginal utility of consumption. The second equation reduces to:

FK = -(dU /dt)/U for all t

112

so that at every t, the marginal product of capital is equal to the negative of the growth rate of marginal utility of consumption. Now, since I is time-autonomous, then we apply the well-known result that:

I - K IK = for all t

where is some constant. Thus:

[B - U(C) + V(L)] - K U =

But what is ? For this we need the transversality conditions. There are two of them. The first is merely:

limt (I - L IL ) = 0

which, substituting in, yields:

limt [B - U(C) + V(L)] = 0

so (net) utility approaches bliss in the limit, i.e. [U(C) - V(L)] B as t . The second transversality condition is that:

limt (I - K IK ) = 0

But from before, we know that I - K IK = , so by substitution:

limt = 0

But as is a constant, independent of time, then it must be that = 0 for all t. Thus, returning back to our equation we see:

[B - U(C) + V(L)] - K U = 0

so, solving for K :

K = [B - U(C) + V(L)]/U for all t

This is the famous Keynes-Ramsey Rule for dynamic efficiency. Specifically, this states that the optimal amount of investment at any time t (K ) is equal to the distance-from-bliss divided by the marginal utility of consumption. If we ignore labor L (or rather assume labor is supplied inelastically), then this reduces to:

K = [B - U(C)]/U

What does this mean? Intuitively, suppose that we decide to "speed up" society's convergence to bliss. In other words, let us force the current generation to save amount K so that one extra generation will be at bliss in the future. The net gain of that future generation is B (its new, blissful utility)

113

minus U(C) (the utility it would have otherwise), i.e. B - U(C). As this is positive, it might always seem worthwhile to do to so. But the cost of "speeding up", as we stipulated, was the consumption foregone by the initial generation. If they save amount K , then U ·K is the cost of speeding up convergence by one generation. Optimally, then, the marginal (net) benefit of speeding up, B - U(C), must be equal to the marginal cost of doing so, U ·K . That is the Keynes-Ramsey rule. This can be illustrated heuristically by appealing to Figure 2. We have depicted two utility paths I and II, both of which achieve bliss B after some time. Path I (solid line) converges to bliss at time T, but Path II (dashed line) converges to bliss one period earlier, at T-1. But notice that Path I also starts at a higher consumption and thus utility than Path II, U(C0

I) > U(C0II). Path II converges to

bliss faster than Path I, but it sacrifices the utility of the early generations. Thus, the "gain" of using Path II is the speeded-up convergence to utility and thus the utility gains of the generations around T-1 (captured by the lightly shaded area in Figure 2), while the "loss" of using Path II is the foregone higher utility of the earlier generations (captured by the darkly-shaded area in Figure 2). The Keynes-Ramsey rule asserts merely that the optimal path will be that where the marginal gain of speeding up is equal to the marginal cost of doing so.[Note: the prefix "Keynes" is added to this result because of Ramsey's gracious acknowledgment to John Maynard Keynes for coming up with the interpretation we just provided].

Fig. 2 - Converging to Bliss[Note: We could have solved the Ramsey problem via optimal control methods, setting up a Hamiltonian H = [B - U(C) + V(L)] + [F(K, L) - C] where is the costate variable, and so on. Karl Shell (1969) makes the curious assertion that this Ramsey problem does not fulfill the traditional transversality condition of infinite-horizon problems, limt (t) = 0. This is not quite right, however.]There are a few interesting features worth noting. Firstly, note that the Keynes-Ramsey rule is independent of the pricing of factors, specifically it is

114

independent of marginal product of capital (as Ramsey (1928) was quick to emphasize). Intuitively, all we need to know for the rule is the initial capital stock, K0, the shape of the utility function, U(C), and the bliss level, B, and we can derive the optimal consumption path. To see this, examine Figure 3, where we have redrawn our utility function. As K = Y - C, then the Keynes-Ramsey rule becomes

Y - C = [B - U(C)]/U

So, for a given Y, we can deduce the optimal consumption C* by finding the level of consumption for which this is true. This is shown in Figure 3.

Fig. 3 - Optimal ConsumptionNotice, from Figure 3, that at the bliss point, all output is consumed (YB = CB). This will be approached asymptotically. Specifically, suppose that in Figure 3, Y is the initial level of income (i.e. Y = F(K0), where K0 is the initial capital stock). As Y = F(K0) > C*, then K * > 0, so capital increases and thus pushing output further to the right along the axis (and consumption along with it). Thus, over time, Y YB and C CB asymptotically (as K 0, it declines in size during the process).Notice that the "bliss" point corresponds to what was later called Golden Rule growth. But note that now we obtain Golden Rule growth not by choosing among the Solowian steady-states, but starting from any arbitrary initial position. Thus, by endogenizing the propensity to save via the Benthamite social welfare function, the Golden Rule equilibrium becomes stable. We can represent the dynamics of the Ramsey model via the phase diagram in Figure 4. Omitting labor, we can rewrite our Euler equation for capital as:

dU /dt = -FK·U

115

which is a differential equation in U . Recovering our constraint, we obtain a second differential equation:

dK/dt = F(K) - C

For simplicity, let us assume that C = C(U ), so consumption is a function of marginal utility (so U must be smoothly invertible, as it is in Figure 3 above). Thus we have a system of two differential equations in U and K. Consider dU /dt = -FK·U first. In steady state, dU /dt = 0, so -FK·U = 0, which implies that either FK = 0 or U = 0. We assume that FK > 0 for all K (diminishing marginal productivity everywhere), then it must be that U = 0 (and, if U = 0, recall, we are at "bliss"). In (K, U )-space in Figure 4, then, the isokine for dU /dt = 0 will be the horizontal axis (where U = 0 and K is indeterminate). The disequilibrium dynamics are easy to decipher. Note that:

d(dU /dt)/dU = -FK < 0

so above the dU /dt = 0 isokine, U has a tendency to fall, whereas below the isokine, U will rise. Thus, the vertical directional arrows in Figure 4.

Fig. 4 - Ramsey DynamicsNow, consider the second differential equation. In steady-state, dK/dt = 0, which implies that F(K) = C. As C = C(U ) which is invertible by construction, then the isokine dK/dt = 0 will have a form U = C-1(F(K)) and slope dU /dC < 0 (diminishing marginal utility of consumption). This is shown by the downward-sloping dK/dt = 0 isokine in Figure 4. The disequilibrium dynamics for the dK/dt = 0 isokine can be deciphered by noting that:

d(dK/dt)/dK = FK > 0

116

so that to the right of the dK/dt = 0 isokine, K rises, while to the left of it, K falls. Thus the horizontal directional arrows in Figure 4. Together, the isokines dK/dt = 0 and dU /dt = 0 have underlying dynamics which resemble a "saddlepoint": there is a single stable arm going to the steady-state equilibrium (0, KB), but all other paths move away from it. Notice that the steady-state (U , K) = (0, KB) is "bliss" as U = 0, so KB is the level of capital corresponding to bliss. Saddlepoint dynamics imply that, beginning at some initial capital level K0, we can define a unique level of marginal utility, U 0

which puts us on the stable arm (and this chosen U 0 corresponds to choosing an optimal initial consumption, C0, as we had earlier in Figure 3). As time progresses, and we continue choosing consumption rates and thus utilities so that we stay on the stable arm, we move towards the steady-state (0, KB), so that K KB and U 0, i.e. we are approaching bliss asymptotically.The Ramsey (1928) exercise remained dormant for the next half-century.  It was resurrected in the 1950s and 1960s, when the emergence of growth theory resurrected some of the questions Ramsey had asked.   The first result, the simplest result, was the determination of the "Golden Rule" of growth.

5.3. Golden Rule Growth

________________________________________________________

"Then a policy-maker was heard to say, "Forget grand optimality. Solovians are a simple people. We need a simple policy...If we make investment a constant proportion of output, our search for the ideal investment policy reduces to finding the best value of s, the fixed investment ratio." "It's fair," Solovians all said. The King agreed. So he established a prize for discovery of the optimum investment ratio."

117

(Edmund S. Phelps, "Golden Rule of Accumulation", American Economic Review, 1961)

________________________________________________________

In the Solow-Swan growth model that the steady-state is consumption-inefficient, i.e. the steady-state consumption per person is not the highest that is attainable.   However,  the steady-state is determined by a handful of exogenous parameters - the savings rate, s, the population growth rate, n, the depreciation rate, , and the rate of technical change, . Changing any one of these will yield a different steady-state capital-labor ratio. But Solow-Swan assumed that s, n, and were exogenously fixed, and thus the steady-state position is also fixed.Richard Kahn (1959) suggested that if these parameters could be "chosen" somehow, then we could easily make the model consumption-efficient, i.e. obtain a steady-state solution where consumption per person (now and forever) is the highest possible.  But how is this to be done?  A few people recalled that, a half-century earlier, Frank P. Ramsey (1928) had determined the "optimal rate of savings" in a simple economy by using the fiction of a grand "social planner".   Could this be applied to growth models too?  The intricacies of Ramsey's utilitarian exercise seemed a little bit too complex,  but the notion was clear.  If  the parameters of a growth model could be "chosen" collectively by society -- or chosen by some grand planner who could impose it upon society -- then the "optimal" steady-state position could be determined.  Although, at least in principle, any of the parameters could be chosen, the focus was immediately placed on the savings rate, s.   The other parameters, n, and , were beyond the social planner's power. "The planner has no jurisdiction over these growth rates, which he has to take as given by God and the engineers." (Robinson, 1962: p.130).So what is the "optimal rate of savings" in a growing economy?  This issue was first broached in the context of a Harrod-Domar growth model by Jan Tinbergen (1956).  For the Solow-Swan growth model, the question was answered  independently by Edmund S. Phelps (1961, 1966), Jacques Desrousseaux (1961), Maurice Allais (1962), Joan Robinson (1962), Christian von Weizsäcker (1962) and Trevor Swan (1963).Recall that in the Solow model, we found that the resulting steady-state growth path was consumption inefficient. By this we meant that (generally) the steady-state capital-labor ratio did not give us the highest consumption per person. Richard Kahn (1959) suggested that if the savings rate could be chosen somehow, then we could easily make the model consumption-efficient, i.e. obtain a steady-state solution where consumption per person (now and forever) is the highest possible.Edmund Phelps (1961) posed the question in a rather amusing article about the kingdom of Solovia where a certain fellow, Oiko Nomos, won a prize by guessing rightly the best savings rate for the kingdom.  He termed the solution to this problem the "Golden Rule" of growth. This is in reference to the old Biblical adage to "do unto others as you would have them do unto you" -- where the "others", in this case, are the future generations of society. Obviously, if a society could choose a savings rate that maximized its own consumption, it would save nothing and consume everything. But that would

118

leave future generations in a lurch as no capital would have been built to enhance future output and consumption. If, conversely, the current generation saved so much that future generations would in fact be better off than the current, then we are also violating "Golden Rule" as we are not doing unto ourselves what we have done for posterity. Thus, the "Golden Rule" condition is that the collectively-chosen or policy-imposed savings propensity is such that future generations can enjoy the same level of consumption per capita as the initial one.Mathematically, finding the conditions for "Golden Rule" growth translates itself into finding the saving propensity that maximizes consumption per capita which is consistent with steady-state growth. The procedure is simple. Recall that consumption per capita is merely the difference between output per capita and investment/savings per capita, i.e. c = y - sy, or:

c = (k) - s (k)

where s is the propensity to save, c = C/L and k = K/L. In diagrammatic terms, as we saw earlier, this is merely the difference between the intensive production function and investment function in (y, k)-space. This difference is what we seek to maximize. The constraint is that we are in steady state, i.e. that dk/dt = 0, or s (k) = nk. Thus, society is confronting the following program:

max c = (k) - nk

The first order condition for a maximum is that:

dc/dk = k - n = 0

In other words, we are at the Golden Rule when the steady state capital-labor ratio, k*, is such that the marginal product of capital is equal to the natural growth rate, k = n.

119

Diagrammatically, we can see this immediately (Figure 1). Remember that our choice variable is the savings rate, s, thus the actual investment function i = s (k) is not imposed upon us but can be chosen. In Figure 1, we see two savings rates, s1 and s2, yielding two different steady-state capital-labor ratios, k1* and k2*. Which is better? Our criteria is to maximize consumption per capita at the steady-state, thus we seek to compare c1* and c2*. Diagrammatically, c1* > c2* so obviously choosing the savings rate s1 is superior to choosing s2.

<Fig. 1 - Golden Rule Growth>We know this is true because maximum consumption will be where the difference between the intensive production function y = (k) and required investment per capita line, ir = nk, is greatest. Thus, the Golden Rule exercise

is to choose s such that the steady-state k* will be such that these two curves are at their greatest difference. This can be found simply by placing a line parallel to the ir = nk line at a tangency with the y = (k) curve in Figure 1. In terms of Figure 1, this is at k1*, the steady-state capital-labor ratio associated with the savings rate s1. Any other savings rate, even those that yield higher output per capita (like s2), nonetheless yield a lower consumption per capita. Notice that as the slope of ir is equal to the slope of (k) at the Golden Rule capital-labor ratio, k1*, then k = n.If we interpret k as the rate of return on capital, then we see that the "Golden Rule" condition k = n is quite familiar. We encountered it, for instance, in the growth model of John von Neumann (1937). Joan Robinson (1962) referred to this as the "Neo-neoclassical Theorem".Although the Golden Rule of growth is simple to derive, it avoids some of the more intricate questions of the Ramsey exercise.  Firstly, note, we are determining the optimal rate by choosing between Solowian steady-states, not

120

the optimal rate from any initial position.  Secondly, it is not clear that the Golden Rule of growth is "socially optimal" in a wider sense.  We were concerned with maximizing steady-state consumption per person in every generation. But, in economics, a person's welfare is attached not to the quantity that he consumes but rather the utility that he attains.   How might the solution be different if we attempted an explicit utilitarian exercise?  This was what the Ramsey exercise was aiming at.  The marriage of the Solow-Swan growth model and Benthamite utilitarianism was accomplished by David Cass (1965) and Tjalling C. Koopmans (1965) -- what has become known as the Cass-Koopmans optimal growth model.  But a lot of philosophical groundwork on the meaning and construction of intertemporal social welfare had to be done beforehand. 

5.4. Intertemporal Social Welfare

________________________________________________________

"[Harrod] declared himself a "cardinalist" and pointed in particular to the fact that cardinal utility is necessary in dynamic analysis. With this I am in complete agreement. It is indeed the gospel I have tried to preach for nearly a generation. To me, the idea that cardinal utility should be avoided in economic theory is completely sterile. It is derived from a very special and indeed narrow part of the theory, viz., that of static equilibrium."

(Ragnar Frisch, 1964, "Dynamic Utility", Econometrica)

"In the case of the individual, pure time preference is irrational, it means that he is not viewing all moments as equally parts of one life. In the case of society, pure time preference is unjust: it means...that the living take advantage of their position in time to favor their own interests."

(John Rawls, A Theory of Justice, 1971:P.295)

________________________________________________________

Optimal growth theory was initiated by Frank Ramsey (1928) as an exercise in normative economics. As outlined in our introduction, the Ramsey exercise was geared to address Arthur Pigou's (1920: Part I, Ch. 2) concerns about the

121

implications for society's savings of the personal "irrationality" of discounting future utility. Effectively, Ramsey demonstrated that an economy in which people have positive time preference will save far below what is optimal -- exactly as Pigou predicted.The practical implications are self-evident: if time preference does lead people to "save too little", perhaps the government should step in and "force" them to save. As Pigou concluded, "there is wide agreement that the State should protect the interests of the future in some degree against the effects of our irrational discounting and of our preferences for ourselves over our descendants." (Pigou, 1920: p.29). Public pension systems such as the American "Social Security" program were designed precisely with this goal in mind. But Ramsey's (1928) "proof" that a society composed of people with positive time preference saved too little depended heavily on how he determined the "optimal" level of savings. Ramsey determined this with perfectly Benthamite instincts: he looked for the optimal allocation of consumption across generations which maximized intertemporal "social welfare", defined as the sum of individual utilities across generations. This has led to much debate and what follows is a rather lengthy, but by no means deep, digression on "intertemporal" social welfare functions, with particular attention paid to the concept of "time preference"

5.4.1.  Intertemporal Social Welfare Functions

Frank Ramsey (1928), following Pigou (1920), argued that "society" is composed of everybody in every generation, current and future, and that they all should be given equal weight in the social welfare function. So, suppose that every time period t = 0, 1, 2, 3, ..., a generation is born and another dies. Each generation has H people in it. Thus, Benthamite utilitarianism implies that the social utility of this society is:

S = t=0 ( h=1

H uth (ct

h))

where cth is the consumption of person h of generation t and u t

h(·) is his utility function. Thus we are defining social utility as the (unweighted) sum of utilities of all people, current and future. Let us make the traditional Benthamite "equal capacity for pleasure" assumption, so that all people, across generations and within generations, possess the same utility function, i.e. u(.) = ut

h(·) for all h = 1, 2, ... H and t = 1, 2, .... This then reduces the social welfare function to S = t=0 ( h=1

H u(cth)).

Furthermore, to get rid of the problem of allocation within a generation (or, given our assumption of "equality"), we can also assume that every household at time t gets the same consumption, i.e. ct

h = ct for all h = 1, 2, .., H, so that our social welfare function is further reduced to S = t=0 (H·u(ct)). Now, as ct is consumption per person, then we can define Ct = H·ct as the aggregate consumption of the generation at t and define U(·) as the "aggregate" utility of that generation, so U(Ct) = H·u(ct). After all these maneuvers, we end up with a new social welfare function:

S = t=0 U(Ct)

122

which can be conceived of as the sum of aggregate utilities of each generation. An allocation, then, can be defined as an infinite sequence of aggregate

consumption bundles, i.e. C ={C0, C1, C2, ..., }. An example is shown in Figure 1. The "social optimum" is the allocation or sequence of consumption bundles that maximizes the social welfare function S.

Fig. 1 - A Consumption AllocationHowever, the absence of time-preference implies incompleteness of the social preference orderings. Specifically, if we have two sequences, C and C , it may be impossible to "compare" them and say which one is "socially better". Why this is so should be immediately evident: with an infinite time horizon, it is quite possible that there are feasible consumption paths such that S = , even if u(·) is bounded above every step of the way and the economic constraints are doing their job. If two different consumption paths each yield an S which is infinite, then they become incomparable -- as two infinities cannot be arithmetically ordered. Alternative methods of comparing paths with infinite sums have been proposed. A famous one is the "overtaking criterion" of Christian von Weizsäcker (1965) and Hiroshi Atsumi (1965), and later refined by David Gale (1967). Specifically, this proposes to "convert" the infinite-horizon to a finite-horizon problem, and then check if one program dominates another for subsequent extensions of the horizon. Specifically, consider two paths C and C which are of infinite length. Now, impose a finite time, T, and compare the two paths up to that finite time. By imposing the final time period, the truncated social welfare measure becomes finite for any path, i.e. t=0

T U(Ct) < for all C if U(·) is bounded. Thus, all consumption paths become comparable up to this final time period. So, suppose that comparing two paths C and C , we find that for finite time T:

t=0T U(Ct) > t=0

T U(Ct )

In which case we will be tempted to argue that, at least up to T, the path C is socially better than the path C . Now, if we can prove that this continues to

123

hold true if we extend the end-period T to T+1, T+2, T+3,... etc., then we can actually come around to concluding that C is a socially better consumption allocation than C , even though both are infinitely long. Even if we cannot compare both infinite sums directly, we can compare their finite equivalents, and then approximate the infinite case by gradually increasing the horizon.Formally, by the "overtaking criterion", path C is said to be "better" than C if there is a time period T* such that for all T T*, t=0

T U(Ct) > t=0T U(Ct ). We

say a path C* is "socially optimal" if there is a T* such that, for all T T*, t=0T

U(Ct*) t=0T U(Ct) for all other feasible paths C.

However, the overtaking criterion does not solve all our problems. The issue of possible non-comparability of paths continues to lurk. For instance, it is quite possible that there is no T* such that the inequality holds for all T T*. For instance, suppose consumption path C yields the utility stream {1, 0, 2, 0, 3, 0, ..., } and path C yields utilities {0, 2, 0, 3, 0, 4, ...}. These are not comparable by the overtaking criterion: if we set the final horizon, T, at an odd time period, then C is better than C ; but if we set T at an even time period, then C is better than C. Thus, there is no T* for which one path will be consistently better than another for all T after T* (cf. Koopmans, 1965).The overtaking criterion only permits a partial ordering over consumption paths. However, a partial ordering might be enough for most purposes. Put more precisely, as our example has shown, the overtaking criterion may find us an "optimal" path C* in the sense that it cannot be bettered by another path, but that does not imply that C* is itself better than every other feasible path. [Note: A variation on this theme is the "agreeable criterion" proposed by Peter J. Hammond and James A. Mirrlees (1973). Loosely, given two infinite-horizon paths, C and C , if we can agree that whatever happens in these paths after a particular time period T is "inconsequential" to us, then we can order these paths according to their truncated values. A survey of criteria can be found in McKenzie (1986).]Partial ordering only does a partial job -- and when considering issues like "social optimality", that may not be good enough. We want a complete ordering. Tjalling Koopmans's (1965) suggestion was to introduce the notion of utility discounting -- what has become known as "time preference". By this he meant that the sum of social utility should be weighed so that earlier generations are "more socially valuable" than latter generations. Specifically, the social welfare function could be rewritten in the form first suggested by Paul Samuelson (1937):

S = t=0 t U(Ct)

where 0 < < 1 is the time discount factor. The further away a generation is, the less its utility matters for social utility. This maneuver yields the result that all consumption allocations over an infinite horizon will yield finite sums for S, i.e. S < for all possible paths C. With time discounting, then, all paths become arithmetically comparable and we can thus find a social optimum simply by comparing the sums.However, following Pigou, Frank Ramsey (1928) considered time discounting as "a practice which is ethically indefensible and arises merely from the weakness of the imagination" (Ramsey, 1928). But Ramsey recognized that by omitting time discounting, the problem of non-comparability of infinite sums emerged.

124

In its stead, as we have seen, he introduced the reverse social welfare function in the following manner:

R = t=0 (B - U(Ct))

where B is the "bliss" level of utility. An allocation C is "better" than an allocation C if:

t=0 (B - U(Ct)) < t=0

(B - U(Ct))

Consequently, for a social optimum, we wish to find the consumption allocation for which R is minimized.How does the Ramsey device solve our comparability problem? Well, suppose that path C achieves "bliss" after the tth generation and that bliss is maintained for every generation forever after. In contrast, suppose that a different path C achieves bliss at generation t+1, and then bliss is maintained forever after. As the utilities for our society are the same for both paths after t+1, so we can effectively ignore the utilities achieved after t+1. All that matters for comparison between C and C is the utilities that are achieved up to generation t+1. These are finite sums which can be arithmetically ordered.Of course, the question imposes itself: how do we know that "bliss" will be achieved for consumption paths at some point in time? We don't. This is the Achilles heel of the Ramsey device. On the one hand, it might not matter: as long as the paths converge asymptotically to bliss, that might be enough to compare two paths. On the other hand, they might not converge fast enough for comparisons to be made. Thus, the Ramsey criterion only permits a partial ordering of consumption allocations. As a result, it is common to impose "convergence to bliss" by either assuming "utility saturation" or, more indirectly, "capital saturation" in finite time. If these are imposed, then convergence to bliss within a finite time horizon will be possible and thus complete comparability. Let us now turn to continuous time. If we assume people are continuously being born and dying (at equal rates, so that the population stays constant), we can convert all our previous social welfare functions into continuous time form as shown in Table 1:

Type Discrete Time Continuous Time

Benthamite sum S = t=0 U(Ct) S = 0

U(C(t)) dt

Samuelson "discount"

S = t=0 t U(Ct) S = 0

U(C(t))e- t

dt

Ramsey "bliss" R = t=0 [B - R = 0

[B -

125

device U(Ct)] U(C(t))] dt

Table 1 - Intertemporal Social Welfare Functions

In continuous time, an "allocation" is no longer merely an infinite sequence, but can be characterized as a function over time, C is a function C: [0, ] R. However, for comparability, we are still interested in ensuring that our social welfare functions S or R achieve finite values for every consumption allocation C. As it turns out, the Samuelson social welfare function is always finite. Formally, if U(·) has an upper bound and > 0, then the integral S = 0

U(C(t))e- t dt will have a finite value.However, as shown by Sukhamoy Chakravarty (1962), we are not ensured that for a social welfare function without this time-discount factor, this will be true. In particular, we are curious about the Ramsey function R. Now, a consumption path acceptable to the Ramsey criteria cannot exceed bliss, so we know for certain that R 0 as t . However, this is not sufficient to ensure comparability, for R may not approach 0 fast enough. In other words, even if we assume that U(·) is bounded above, so that U(C(t)) B for all C(t), we can still have it that U(C(t)) approaches B asymptotically at too slow a rate so that in fact there is no consumption path that ensures that the integral converges. In other words, B may not be achieved for any finite C. Ramsey (1928), however, argued (without proof) that there would be at least one program that would ensure convergence. But neither Chakravarty (1962) nor Koopmans (1965) proves that time preference will do the trick either. They assumed that time preference in the social planner's function would be sufficient to yield an optimal solution, but offered up no proper "proof" of this. Intertemporal social welfare functions imply that we have to deal with infinite-dimensional commodity spaces. As we discuss elsewhere, the mathematics of infinite-dimensional spaces can be quite complicated. As a result, this question was left somewhat vague until the 1980s, when Donald J. Brown and Lucinda M. Lewis (1981), Michael Magill (1981) and A. Araujo (1985), addressed the issue again -- this time with a fully-developed mathematical arsenal. They confirmed the intuition: positive time preference is a sufficient (albeit not always necessary) condition for the existence of optimal consumption paths.

5.4.2.  The Defense of Discounting

- The Tastes Defense- The Dynastic Defense- The Decentralization Defense

Including the time discount rate into our social welfare function would certainly make things easier. But how are we to justify it? If we follow Pigou, Ramsey and company in their reasoning, there seems to be no "ethical" justification for putting a utility discount into the social welfare function. How might this be disputed?

126

5.4.2.1.  The Tastes Defense

The inventors of intertemporal preferences, Eugen von Böhm-Bawerk (1886) and Irving Fisher (1930), believed that discounting future utility was an irrationality. This was the line that Arthur C. Pigou (1920), Frank Ramsey (1928) and Roy Harrod (1948) took -- which is precisely why they felt that it should not be included in the intertemporal social welfare function. As Roy Harrod reiterates forcefully:

"After all, pure time preference is a weakness; a man may choose to sacrifice 2 units of utility -- of utility not money -- in 20 years from now for sake 1 unit now; but in 20 years' time he will presumably regret having done so. Unfortunately he will not then be able to reverse the process. On the assumption -- unwarranted, no doubt, some of you may think -- that a government is capable of planning what is best for its subjects, it will pay no attention to pure time preference, a polite expression for rapacity and the conquest of reason by passion." (R.F. Harrod, 1948: p.40).

But there are articulate defenses in the opposite direction as well (e.g. Ludwig von Mises, 1949: Ch. 18). One could say that positive time preference is just that: a preference and not a personal weakness or defect that ought to be "corrected". It need not be justified, it just "is" and de gustibus non est disputandum (we cannot quarrel over tastes). In this view, for someone to say it is an "irrational preference", as Pigou (1920: p.25) did, is oxymoronic.So the simplest, defense for including a discount factor in the social welfare function is that, well, people have positive time-preference -- and a preference is a preference is a preference. It should be respected by the social planner. Removing time preference from the social welfare function, far from being "ethical", can in fact be deemed unduly authoritarian as it disregards people's tastes. Arguments to this end were forwarded by Peter T. Bauer (1957) and Otto Eckstein (1957). Yet this is not a perfect argument for if we are going to stick to "preferences" argument, then should not the preferences of future generations be taken into account? If their opinion had any bearing on the present, then it would be precisely to discount the utility of the earlier generations. Clearly, we are at an impasse.Of course, all this is wordplay. As Maurice Allais (1947: Ch. 4) and Jan de Van Graaff (1957: p.103) note, the optimal level of savings is a political and ethical question, for which the market's solution is only one among many. Bauer (1957) would probably agree -- but would cast his vote in favor of non-interference nonetheless.But if we agree to this, then are we not conceding too much to posterity? If we are to explicitly consider the question as a political one, one must wonder whether future generations should have any claim at all! In effect, only living members are involved in the political process and those are, effectively, the only ones social welfare functions should respect. As Stephen Marglin explains:

"I, for one, do not accept the Pigovian formulation of social welfare. If I am going to play the neoclassical, or rather neo-

127

Benthamite, game, in which individuals are assumed to have well-defined preferences that are identical to their utilities, I want to play the rest of the bourgeois-democratic game of philosophical liberalism as well: in particular, I want the government's social welfare function to reflect only the preferences of present individuals. Whatever else a democratic society may or may not imply, I consider it axiomatic that a democratic government reflects only the preferences of the individuals who are presently members of the body politic." (Marglin, 1963).

And why not? After all, we cannot second-guess the desires of people who are not born yet and perhaps we should not even try. Who is to argue on their behalf and should we believe them? Why should we make room in our polity for current political representatives for future generations that do not exist? To do so might be as undemocratic as, say, allowing clergymen a dominant position in current political affairs because they are the "representatives" of supernatural beings and human afterlife -- concepts which are no more vague and speculative than "future generations". Of course, the clergy have had such power in the past, but it is clearly not part of the modern "bourgeois-democratic" conception of political life.If we were to agree that only "present members of the body politic" should count, this might seem to turn the balance towards the Eckstein-Bauer corner of the debate, restoring the ethical legitimacy of positive time preference. But perhaps future generations do have political representatives in the present -- namely, that the living individuals themselves are their advocates, however imperfect. In other words, current people do have "social tastes" which incorporate the interests of future generations. They actually want the social welfare function to reflect these. A strict behaviorist would contend that this is nonsense. If people's tastes incorporate this advocacy for future generations, their behavior should reflect this. People's high time preference rate demonstrates that they do not really care much about them -- and that is the only accurate measure of their concern.But what if, contends the opposite camp, living individuals have a discrepancy between their "personal tastes" and "social tastes". Might people really want zero (or at least low) discount rates for society, even while possessing a personal high time preference rate? This does not necessarily dismiss the behavioral argument, but rather sharpens it by dividing people's behavior into two: people's political choices (e.g. voting for recycling and environmental protection laws) reflect their "social tastes", but people's personal economic choices (e.g. how much recycling they themselves do) reflect their "personal tastes". Any behaviorist would be forced to admit that, indeed, people's political behavior usually does not match their personal behavior.But should not these two types of tastes be consistent with each other? Not necessarily. As outlined by Stephen  Marglin (1963), there are at least two ways to argue this. The first argument, credited to Gerhard Colm, is simply that the "frame of reference" is different in personal and social considerations. People wear two hats on the relevant time discount rates. Individuals may have defective telescopic faculties when making decisions about what they want to save individually, but when asked what "society" should save, the individual

128

might recommend a different (i.e. a much lower) discount rate. Again, think of the penchant for Westerners to individually generate enormous amounts of waste while condemning Western wastefulness at the same time.The second argument, articulated by William J. Baumol (1952) and Amartya Sen (1961), is only subtly different in that it emphasizes the "free-rider" aspects of the problem. Specifically, people will vote for policies with low discount rates (e.g. municipal recycling programs) in the expectation that others will comply with them, while personally they will perform actions which reflect their personal high discount rate (e.g. not bother to recycle their own garbage). They might not feel they are being "inconsistent" in their personal and social tastes because they expect others to comply with the laws that they have voted for.Both these cases reinforce Pigou's arguments for disregarding personal discounting. People's social tastes are for zero or very low discounting and this is what should be included in the intertemporal social welfare function. Their personal taste for high discounting, as revealed by their individual actions, should not be considered sufficient justification for its inclusion. Viewed from this prism, the Pigou-Ramsey social welfare function is not authoritarian at all, but complies with what people "really" want when they are in a "social" frame of mind or when they are voting.However, this puts us right back in our dilemma. The "tastes" defense does not seem, on its own, to be capable of justifying the inclusion of positive time preference in our social welfare function. Pigou, Ramsey, Harrod and company would be overjoyed.

5.4.2.2.  The Dynastic DefenseIs there a median position where we can include the "reality" of personal discounting into a social welfare function, without contaminating it with its "unethical" features? One maneuver consistent with utilitarian ethics is to transform our concern with "generations" into a concern for "dynasties". [although suggested much earlier, Robert Barro (1974) was perhaps the first prominent modern economist to perform this trick explicitly]. Specifically, let us eliminate all future generations from our social welfare function, so that the social planner is concerned with the utility of the current generation only, i.e. the social welfare function is simply:

S = h=1H u0

h(c0h)

where u0h(·) and c0

h are the utility function and consumption plan of the hth living household, H are the total number of households alive at the initial time period t = 0. Thus, backing away from Pigou and Ramsey and moving towards Marglin, we now have "society" defined merely as the living individuals and not future ones. Although Marglin's argument would imply future generations have no "legitimate claims" on the current generation, that does not mean that they cannot have "emotional claims". The trick is to take this to the extreme and argue that currently living individuals have "dynastic" utility functions. By this we mean that a living individual is altruistic towards his "dynasty", i.e. his utility takes into account the utility of his progeny. Thus, the future is

129

reintroduced into the story not because it is "ethical" to do so, but merely because that is what current living individuals do anyway.The implications of this become interesting. The utility of the current generation depends upon not only their own consumption but also on the utility of their children. But, by the same logic, the utility of their children depends, in turn, on the utility of their children and so on ad infinitum through the ensuing dynasty. To see this clearly, let h denote the "dynasty" stemming from the living agent h and suppose that there is only one child per adult (no population growth). Then we can stipulate that u0

h = u0h(c0

h, u1h), so the utility of the

household h living at t = 0 depends on the utility of their direct descendent, the household h that is living at t = 1. As u1

h itself is a function of consumption at t = 1 and the utility of their progeny (generation h at t = 2), then u1

h = u1h(c1

h, u2

h), which we can plug that back into the original generation's utility so u0h =

u0h (c0

h, u1h(c1

h, u2h)). Iterating further, the utility of generation t = 2 of dynasty

h is a function of their consumption (c2h) and the utility of their progeny, u3

h, i.e. u2

h = u2h(c2

h, u3h). We can proceed in this manner for all future generations of

dynasty h. Thus, recursing all the utilities of a dynasty into themselves, the utility of household h at the initial time period t = 0 is the stream of utilities achieved by the entire ensuing dynasty in the future, i.e.

u0h = u0

h(c0h, u1

h(c1h, u2

h(c2h, u3

h(c3h, ....))))

Since currently-living individuals are myopic (it is a "personal" weakness, as Pigou allowed), then a positive rate of time preference can be introduced without ethical implications. The discount factor indicates that a household is a little bit selfish (or more accurately, just plain short-sighted) in the sense that they do not consider the utility of their children to be quite as important as their own. Specifically, let us propose that the utility function u0

h is an additively separable function with positive time preference. Let us assume that every generation of dynasty h has the same utility function uh(·) which is dynasty-dependent but time-independent. Furthermore, from the perspective of the current living agent (living at t = 0), the utility of his descendent at time t = 1 is uh(·), where 0 < < 1 is the "discount factor" which we assume to be the same across dynasties and generations. Consequently, we can rewrite the utility of the current living member of household h as:

u0h = uh(c0

h) + uh(c1h) + 2 uh(c2

h) + ...

or simply:

u0h = t=0

t uh(cth)

The rest is simplicity itself. Taking our social welfare function, as Marglin (1963) suggests, over currently living people alone, we see that:

S = h=0H u0

h = h=0H t=0

t uh(cth)

so, switching summation signs:

130

S = t=0 h=0

H t uh(cth)

Finally, assuming that all households within a generation have the same capacity for pleasure, then uh(·) = u(·) for all h = 1, 2, .., H, and therefore (for Benthamite fairness) the same contemporaneous consumption allocation, c t

h = ct for all h = 1, 2, .., L, so our social welfare function becomes S = t=0

L·t

u(ct). Letting L·u(ct) = U(Ct), where U and Ct are the aggregate utility and aggregate consumption of generation t, then we obtain:

S = t=0 t U(Ct)

which is a simple infinite-horizon social welfare function with a time discount factor. In continuous time, this can be expressed as:

S = 0 U(Ct)e- t dt

where is the rate of time preference. These are exactly the Samuelson social welfare functions we were hoping for earlier.In sum, from the dynastic perspective, the burden of taking the utility of future generations into account is shouldered by currently-living individuals rather than the social planner. But since the social planner takes the utility of current-living individuals and because these take the utility of their descendants into account, then we can regard the resulting social welfare function with intertemporal discounting, S = 0

U(Ct)e- t dt, to be "ethically defensible". Including time preference into the social welfare function does not imply that our social planner is a moral desperado, but merely that our households have "defective telescopic faculties".

5.4.2.3.  The Decentralization DefenseA different argument in favor of discounting, which we shall consider later in more detail, is the "decentralization" defense, originally attributed to Robert Becker (1980). This is perhaps the best-known argument and, by far, the most popular today. But it requires an entire overhaul of our understanding of the intertemporal social welfare function.In a nutshell, the "decentralization" thesis argues that time preference should be included in the social welfare function because, well, it is really not a social welfare function at all! Instead of conceiving of the Ramsey problem as an exercise in normative economics, the decentralization argument considers it to be one of positive economics. They insist that a "properly" [sic] formulated model of the market, with infinitely-lived (!), intertemporally-optimizing, myopic (?) consumers and firms, with perfect foresight (!) and facing perfectly working capital markets (!), will yield exactly the same solution as the Ramsey social planner. In this view, the entire social welfare maximization exercise is merely a mathematical summary of the actual workings of the market. Consequently, adding time preference to the social welfare function is, in fact, necessary because consumers are myopic and the Ramsey social welfare function, over which we have been losing sleep, is really nothing other than the utility of the representative consumer. Although the decentralization argument stretches credulity to an enormous degree, it is the most widely accepted argument today for including time

131

preference. It has interesting implications for it has modified the nature and significance of optimal growth theory.

5.4.3.  The Koopmans Axiomatization

One of the main drawbacks of the "preference is a preference" defense of discounting was that it was not clear that this is so. If positive time-preference is to be introduced as an expression of people's natural tastes, it must be introduced as an axiom of preference itself or derived from more primitive axioms. Yet impatience is a general behavioral postulate. We are not talking about preferences over toothpastes which can vary immensely across people; we are referring to a time-discount factor that is positive and present, to a greater or lesser degree, in everybody. It has a generality and universality of applicability. To say that it "just a preference" does not explain its regularity.

The derivation of "impatience" from more basic axioms on preferences were attempted by Kelvin Lancaster (1963), Tjalling Koopmans (1960, 1972), Koopmans, Diamond and Williamson (1964), Peter Diamond (1965) and Peter Fishburn and Ariel Rubinstein (1982). Here we shall consider Koopmans's contributions, which were directly concerned with deriving a discount factor from the social planner's utility. This turns out to be a quite subtle and profound defense of discounting.

Koopmans's (1960) inspiration was John von Neumann and Oskar Morgenstern's (1944) trick with choice under uncertainty. Recall that they stipulated that people have preferences over lotteries and then found the necessary axioms on these preferences that would yield Bernoulli's "expected utility" form. Similarly, Koopmans argued that the social planner has preferences over infinite consumption paths and then searched for the axioms that would make such preferences representable by a Samuelson intertemporal utility function with discounting. Intuitively, letting S(·) represent the social planner's utility function over consumption paths, then, for any two path C and C, Koopmans (1960) sought to find axioms on S(·) such that:

if S(C) S(C ) then t=0 tU(Ct) t=0

tU(Ct )

where U(·) is an elementary, single-generation utility function and is a time-discount factor. So, rather than creating a social welfare function by adding up each generation's utility, Koopmans goes the other way and derives generational utility functions U(·) plus discount factor from the planner's social welfare function S(·). Notice there was a second purpose to Koopmans's construction: namely, the restoration of ordinality into the intertemporal utility construction.

132

Obviously, if we take the Samuelson function, t=0 t U(Ct), as our social

welfare function, then it is a cardinal function: the particular numerical values we give our realized generational utilities, U(Ct), U(Ct+1), etc. matter very much as they will be subsequently added up. But like von Neumann-Morgenstern, Koopmans was aiming at a "ordinal utility which is cardinal". The real and only utility function in the Koopmans world is S(·) and it is defined as a representation of social planner's preferences over infinite-horizon consumption paths. This utility function S(·) is ordinal, i.e. the numerical values we assign to S(C), S(C ), etc. do not matter, as long as the preference ordering over paths is maintained. So, in this way, Koopmans makes the whole intertemporal utility exercise "acceptable" to radical Paretians.Koopmans (1960) proceeded as follows. Let C denote an entire infinite-horizon consumption path and Ct denote the tth element of that path, so C = {C1, C2, C3, .., Ct, ...}. We will let the term 2C denote the path C excluding the first entry, i.e. 2C = {C2, C3, C4, ..., Ct, ..,}. Thus, the path C can be written as C = {C1, 2C}. Let denote the "commodity space", in this case the set of infinite-horizon consumption paths, so C . It is assumed that the social planner's preferences can be captured by a nice social utility function S: R, where if S(C) S(C ), then the social planner prefers path C to path C . Koopmans then imposes the following axioms (he calls them "postulates") on S(·):

(P.1) Continuity: for any given C, if M and M are values such that M < S(C) < M , then there is a such that S(C ) for any C within distance of C (i.e. d(C, C ) ), satisfies M < S(C ) M.

(P.2) Sensitivity: If for any two paths C and C , we have it that C = {C1, 2C} and C = {C1 , 2C}, i.e. the two paths are identical except for the first component, then there are values C1 and C1 such that S(C) S(C ).

(P.3) Non-Complementarity: If S(C1, 2C) S(C1 , 2C), then S(C1, 2C ) S(C1 , 2C ). Similarly, if S(C1, 2C) S(C1, 2C ), then S(C1 , 2C) S(C1 , 2C ).

(P.4) Stationarity: S(C1, 2C) S(C1, 2C ) if and only if S(2C) S(2C ).

(P.5) Boundedness: There exist paths Ca and Cb such that S(Ca) S(C) S(Cb).

Note that axioms (P.1)-(P.5) are assumed directly on the social planner's utility function S: R -- which is assumed to exist and somehow

133

represent the planner's "preferences" over intertemporal allocations. Of course, we should say something about whether the social utility function S represents these preferences, and how these utility postulates might be connected to other, more primitive axioms on preferences. Koopmans (1972) forges this connection, but we shall skip it here.The meaning of the axioms can be briefly explained. Postulate (P.1) is a simple continuity axiom: any slight variation in the path does not lead to drastic changes in the social planner's utility. It is expressed in this funny way for technical reasons. Axiom (P.2) means that every generation counts, i.e. if there are two paths that are identical in every respect except for the first period where it is drastically different, then the social welfare of those paths will be different. Postulate (P.3) is a kind of independence axiom. Specifically, it argues that if two paths are identical except in one period, then it does not matter what the identical part looks like when comparing them. There is non-complementarity between periods in the sense that the social planner's preference over what generation t achieves is independent of his preferences over what generation t+1 achieves. Axiom (4), stationarity, is the famous and most debatable one in the Koopmans array. What it says explicitly is that preferences between two events remain the same, even if we push these events forward. For instance, consider the baseline consumption path (1, 1, 1, 1..., ), so there is steady consumption of one unit of the consumption good every period. Now suppose that the following adjustments are possible. Namely, we can either add X to the first time period or add amount Y to the second time period, so that the two alternative paths are now: C = (1 + X, 1, 1, 1, ...) or C = (1, 1 + Y, 1, 1, ....). Suppose that the social planner's preferences are such that:

S(1 + X, 1, 1, 1, ...) > S(1, 1 + Y, 1, 1, ..)

he prefers the X adjustment today rather than the Y adjustment tomorrow. Now, suppose we push the event forward by a period, and construct path D = (1, 1+X, 1, 1, ...) and path D = (1, 1, 1+Y, 1, ...). Notice that D = (1, C) and D = (1, C ), so all we have done is shunted each of the "events" forward by one period. Stationarity says that if S(C) > S(C ) then it must be that S(D) > S(D ), i.e.

S(1, 1 + X, 1, 1, 1, ...) > S(1, 1, 1 + Y, 1, 1, ..)

and vice-versa. So, what the stationarity axiom effectively says is that intertemporal preferences between two periods remain the same even if we shunt those two periods further ahead in time. Intuitively, if the social planner prefers one apple today to two apples tomorrow then he will prefer one apple in 30 days to two apples in 31 days. Interestingly,

134

Diamond (1965) attempted to redo this analysis without the stationarity axiom.Koopmans (1960) shows that a social welfare function which possesses these five axioms will necessarily exhibit positive time preference. We shall not attempt to prove this here. But we can give an idea of why this is so with an example. Suppose there are four utility paths A, B, C and D, as depicted in Figure 1 (A and B are black, C and D are red). Let us suppose that the consumption path A is superior to consumption path B every step of the way, so, socially, S(A) > S(B). Now, suppose that paths C and D are constructed in the following manner. For the first period (t = 1), path C maintains consumption level c0, but subsequently, for t = 2, 3, 4, .. it follows precisely the same path A did. Thus, C is merely A with a one-period lag. Similarly, D keeps consumption c0 for the first period and follows path B with a one-period lag thereafter. Thus, C = {c0, A} and D = {c0, B}.

Figure 1 - Time Perspective

As far as the first period is concerned, paths C and D are identical and only thereafter does the difference begin. Obviously, then, S(C) > S(D) for the same reasons that S(A) > S(B). In fact, it is a straightforward application of the stationarity axiom. But what about the utility difference? Intuitively, the utility difference between C and D should be less than the utility difference between A and B because for a little while (i.e. during period t = 1), paths C and D yield the same consumption and thus utility. Because paths C and D are less different than paths A and B, their utility difference should be smaller. Heuristically, then, S(A - B) > S(C - D). This is what Koopmans,

135

Diamond and Williamson (1964) labelled "time perspective" in the sense that the difference between utility streams is smaller the further away in time it is..The necessity of time preference follows from this observation. Suppose, for example, that A = {3, 3, 3, ..., } and B = {2, 3, 3, 3....}, so (A - B) = {1, 0, 0, 0, ...}. At the same time, C = {c0, 3, 3, 3, ...} and D = {c0, 2, 3, 3, ...} so (C - D) = {0, 1, 0, 0, ...}. Then obviously the statement S(A-B) > S(C - D) implies that S(1, 0, 0, ...) > S(0, 1, 0, 0, ..). This is time preference. To see this explicitly, notice that from the third period onwards, (A-B) and (C-D) are identical. Let us just lop of all the remaining periods except the first two from consideration and so consider the plot in a simple indifference map (Figure 2). Now, by the non-complementarity axiom, S(A-B) > S(C - D) implies that {1, 0} is preferred to {0, 1}. In Figure 3, this translates into saying that the social indifference curve S(A-B) which passes through (1, 0) lies above the social indifference curve S(C - D) which passes through (0, 1).

Fig. 2 - Time Preference

The social indifference curves in Figure 2 exhibit time preference. We can deduce this by comparing indifference curve S(A-B) with an alternative indifference curve V. Notice that V (dashed curve in Figure 2) passes through both (1, 0) and (0, 1). We see then that V is an indifference curve which has no time preference -- it is indifferent between (1, 0) and (0, 1). Now, let us assume that S and V are additively separable. This means that we can decompose the utility of consuming bundle (C1, C2) into two utility components, i.e.

136

S(C1, C2) = S1(C1) + S2(C2)

V(C1, C2) = V1(C1) + V2(C2)

where S1, S2 are the utilities of consuming in periods 1 and 2 respectively for the social utility function S while V1, V2 are the corresponding utilities for social utility function V. Now, let s and v denote the points where S and V intersect the 45 line (cf. Figure 2). As these are on the 45 line, then, for simplicity and abusing notation a bit, let s = [s, s] and v = [v, v]. Notice also that s = v, where is a positive scalar. The slope of the indifference curves S and V at the 45 lines are captured by the marginal rates of substitution:

MRSS(s) = -dC2/dC1|S(s) = S1 (s)/S2 (s)

MRSV(v) = -dC2/dC1|V(v) = V1 (v)/V2 (v)

But notice, in Figure 2, that S at the 45 line is steeper than V at the 45 line. Thus:

MRSS(s) > MRSV(v)

But as MRSV(v) = 1, then it must be that MRSS(s) > 1, or S1 (s) > S2 (s). In English, if we are consuming the same amount in both periods, we would make a gain in utility by reallocating a unit of consumption from period 2 to period 1, i.e. the present is "preferred", in pure utility terms, to the future. Notice that the arguments in the indifference curve S are identical because we are evaluating it on the 45 line. So the difference between S1 (s) and S2 (s) must arise purely from the difference in the utility parts, S1(·) and S2(·), and not in the amounts consumed. There is pure time preference. By the Archimedean property of numbers, there is some factor [0, 1] such that S1 (·) = S2 (·). So, our social utility function can be rewritten as

S(·, ·) = S1(·) + S1(·)

or defining S1(·) = U(·):

S(·, ·) = U(·) + U(·)

Thus, we have decomposed our social welfare function into a single elementary utility function U(·) and a discount factor . We can proceed by iterative construction to show that:

S(·, ·, ·, ...) = t=1 t-1U(·)

137

i.e. our social planner's utility function defined over an infinite horizon is identical to the Samuelson (1937) discounted intertemporal social welfare function. The pure rate of time preference, call it , can be defined as = (1- )/ , implying that:

= 1/(1+ )

so, as long as > 0, then < 1. If = 0 (no time preference rate), then = 1 (no time discount factor).Let us step back and think about what we have just done. A positive time preference rate on the social planner's utility has been deduced (with the assistance of additive separability) from the simple observation of the property of "time perspective", i.e. that S(A - B) > S(C - D). This assertion was purely intuitive: C and D share the first time period, so C should be "less different" from D than A is from B. This, in a nutshell, is the crux of the argument. What exactly makes it work?Critical in this intuition is the sensitivity axiom (P.2). The implications of this axiom can be thought through as follows. Suppose that a consumption path can be permanently improved forever if we merely starve the first generation to death. In the grand scheme of things -- i.e. in an infinite horizon -- the utility loss of the generation in the first period seems to be negligible when compared to the increased utility of all the remaining generations. What Koopmans was seeking with his sensitivity axiom was to prevent the social planner from making such a calculation. He should not ignore the utility loss of one generation just because it is one period out of infinity. Sensitivity is thus not merely a mathematical condition, it is also an "ethical" condition.We begin the see how "sensitivity" is related to the time perspective property. In our simple example, if we could ignore the first period, then the difference between paths A and B could be considered identical to the difference between paths C and D. But, by sensitivity, we cannot ignore the first period. The fact that C and D share the same consumption in the first period must be accounted for in the social planner's preferences. It is from this assertion that we can conclude that S(A - B) > S(C - D) and, as we have seen, thereafter go all the way to positive time preference and representation of the planner's social utility by a Samuelson function. For details, consult Koopmans, Diamond and Williamson (1964).There is an observation worth making at this point. Koopmans's analysis suggests that if we attempt to derive a social welfare function without time preference, then we are implicitly violating sensitivity somehow. This makes sense. Without time preference, the utilities achieved by any finite number of generations can be "ignored" as the remaining infinity overwhelms them completely. Indeed, in continuous time, such negligibility is automatic.

138

But it is the ethical implications of this that are interesting. What Koopmans has highlighted with his exercise is the interlinkage between sensitivity with time preference. On the one hand, discounting is unethical because, say, it could be used to justify current environmentally-destructive activities whose effects would only be felt millions of years in the future. On the other hand, discounting is ethical because without it, the infinite future overwhelms the finite present. More acutely, without discounting, we could justify savagely destroying one generation now if it yielded a series of minuscule gains (no matter how small) that would accrue forever after. An ethical balance must be struck: either we count every generation equally (in which case, no single generation counts at all), or we allow discounting (in which case every generation counts, but some less than others). Which is more ethical? See Kenneth Arrow (1979) for some reflections on this.

5.4.4.  Population Growth

The original Ramsey model contained a stationary population: every generation was of the same size. Now, we shall add population growth, as this makes it more comparable to the Solow-Swan growth model, which is our ultimate objective. Let L(t) denote the number of people at time t. Let us suppose that population grows at the exponential rate n, so that population at time t can be expressed as:

L(t) = L0ent

where L0 is the initial population. The growth rate of labor can also be written as gL = (dL/dt)/L = n. As the population is now increasing, we must make appropriate adjustments to our social welfare function. Let us move away from Ramsey (1928) by supposing that labor is supplied inelastically, so that the labor supply at time t is L(t), which is also the number of living individuals in the economy. Thus, let us begin with the discounted social welfare function:

S = 0 [ h=1

L(t) uth (ct

h)]e- t dt.

Making the Benthamite "equal capacity for pleasure" assumption, so that u th(·)

= u(·) for all people h in every time period t, and that every person in the same generation receives the same consumption, i.e. ct

h = ct for each h = 1, 2, .. L(t). Then this social welfare function reduces to:

S = 0 [L(t)·u(ct)]e- t dt

But as L(t) = L0ent and normalizing so L0 = 1, then:

S = 0 [ent·u(ct)]e- t dt

139

or:

S = 0 u(ct)e-( -n)t dt

thus social welfare is discounted by the time preference rate adjusted by the rate of population growth. Thus, we can think of ( -n) as the "actual" or "net" discount rate. The logic for this is a bit subtle, but can be understood as follows: if we did not incorporate population growth into our discount factor then we would be punishing a single individual in the future twice -- once because he is in the future (and his forefathers were "myopic"), and twice because he belongs to a generation which is larger in number. In order to keep some sense of equal treatment across individuals in this social welfare function, we must adjust the discount rate for the population growth rate. We should note that this form is not generally adhered to. David Cass (1965), for instance, employed only in the discount, i.e. he used:

S = 0 u(ct)e- t dt

as his social welfare function in a model with growing population.. Although this treats "individuals" unjustly (by punishing people for being part of large generations), it treats "generations" justly. In contrast, the ( -n) discount rate treats individuals justly, but generations unjustly (larger generations have a relatively greater weight). Which to choose? This choice of discount factor is not entirely inconsequential as they yield different solutions for the optimal growth path. Still, we come down heavily in favor of the ( -n) discount as this is more consistent with Benthamite logic. In the construction of social welfare, we cannot really think of a good reason to accept the "generation" as the fundamental unit. The use of ( -n) as the discount rate is defended convincingly by Kenneth J. Arrow and Mordecai Kurz (1970: p.11-14). However, there is a downside to the Arrow-Kurz ( -n) formulation. Specifically, for S < , we need it that > n, i.e. the rate of time preference must exceed the rate of population growth for the integral to converge. This is a necessary assumption, but not necessarily a very reasonable or intuitive one. If, as it turns out in the solution, is equated with the rate of interest, then this convergence condition says that we need the rate of interest to exceed the natural rate of growth. Effectively, this implies that anyone who takes on debt at some point but whose real income grows at the natural rate will necessarily be in the quandary of never really being able to pay back his debt without loss of income.

5.4.5.  Overlapping Generations

One of the critical and most debatable assumptions we have maintained thus far in our arguments is the assumption of successive generations. In other words, we have assumed that, every period, a new generation arises and the old one dies off. Generations precede and follow each other, but they do not overlap at any point. This is a very restrictive and unrealistic assumption but one that, unfortunately, is difficult to dispose of.

140

Models which allow successive generations to overlap with each other were first proposed by Maurice Allais (1947) and, independently, Paul Samuelson (1958). They noticed immediately that such a structure has some intriguing implications for intertemporal social welfare.

There are many ways of modeling overlapping generations. The simplest is the "two-period-life" version. In this case, each generation lives for two periods -- call it "youth" and "old age". At any time period, one generation of youths coexists with one generation of the elderly. At the beginning of the next period, the elderly die off, the youths themselves become elderly and a new generation of youths is born. Thus, there are two "overlapping" generations of people living at any one time.

Although we cover this in more detail elsewhere, our interest is in the social welfare implications of overlapping generations. To see this, let us attempt to construct a social welfare function when generations overlap. We assume a generation born at time period t (call it "generation t") lives for two periods: t and t+1. Let ct

t and ct+1t denote the consumption in periods t and t+1

respectively by generation t. Let us denote by ut(ctt, ct+1

t) the intertemporal (two-period) utility function of generation t. Allowing for additive separability utility and personal myopia, we can write:

ut(ctt, ct+1

t) = ut(ctt) + ut(ct+1

t)

where is the personal discount factor. Now, this is for a single generation that is born at time t. As a new generation is born every time period t, then the intertemporal social welfare function is:

S = u0(0, c10) + t=1

ut(ctt, ct+1

t)

where u0(0, c10) is the utility of the first generation of elderly people (born at t

= 0), who have had no "youth". Notice that this is intertemporal, so every generation, present and future, is given equal weight in this social welfare function (there was a small controversy between Abba Lerner (1959) and Paul Samuelson (1959) over this). Thus, assuming the same personal discount rate across generations, we can plug in our explicit form:

S = u0(c10) + t=1

[ut(ctt) + ut(ct+1

t)]

or, rearranging:

S = t=1 ut(ct

t) + t=0 ut(ct+1

t)

By the Benthamite "equal capacity for pleasure" argument, let ut(·, ·) be the same across generations. This permits us to drop the t superscripts and rewrite the social welfare function simply as:

S = t=1 u(ct) + t=0

u(ct+1)

141

This is revealing. For any positive consumption path, this social welfare function S is not a finite sum, i.e. S = for any {ct} > 0. Thus, not only are paths "non-comparable", but we cannot find a "social optimum". The old problem re-emerges.The overlapping generations construction yields interesting implications. Firstly, even when we incorporate personal myopia, we do not end up with finite social welfare sums. We cannot appeal to the reality of individual discounting to solve the incomparability problem. To make the sums finite, to make consumption paths comparable, we require that the social planner start making evaluations of the relative social worth of different generations. Personal discounting will not do as a substitute. Thus, letting be the social planner's discount rate per generation, then we end up with:

S = t=1 t-1u(ct) + t=0

t-1u(ct+1)

where, assuming 0 < < 1, then S becomes finite and paths are now comparable. But is an explicitly unethical discount. There is nothing obvious we can pluck out of society that can justify it. We must simply accept that our social planner is "morally challenged".Secondly, the decentralization thesis does not hold in overlapping generations. Specifically, it can be easily shown that in an overlapping generations model, the competitive equilibrium is not Pareto-optimal. This means that a social planner (or a government) can achieve a superior allocation than that yielded by the market. The social planner's solution (if we can find one) will be different from the market solution. The decentralization thesis breaks down.However, there is a trick that is possible: namely, if we follow the "dynastic" logic employed earlier. Including intergenerational altruism and "bequests" in an overlapping generations model, as Robert Barro (1974) did, we can effectively replicate the traditional Ramsey-style infinite-horizon problem with successive generations and restore the decentralization thesis.

5.4.6.  Varying Time Preference

If we add time preference, why assume it is the same across dynasties and constant across time?

The problem of allowing heterogeneous discount rates for contemporaries was already anticipated by Frank Ramsey (1928). He argued that if two dynasties have different discount rates and a loan market is in operation, then "equilibrium would be attained by a division of society into two classes, the thrift enjoying bliss and the improvident at the subsistence level" (Ramsey, 1928). This is confirmed by R.A. Becker (1980). However, Robert Lucas and Nancy Stokey (1984) demonstrate that if the utility function is non-additively separable, then this result is not necessarily true.

Ramsey (1928) also argued that if a single person is discounting his future utility, he must discount all of it at the same constant rate. Otherwise, what he planned for the future will be changed when that future get closer. This has become known as dynamic inconsistency.

142

To see why, first we need to convince ourselves that a single discount rate and Koopmans's stationarity axiom are effectively the same thing. Recall that the stationarity axiom claims that preferences between two periods remain the same even if we shunt those periods forward. So, suppose that we have a baseline consumption path (1, 1, 1, .., 1) and one of the following two adjustments can happen: either we add X to consumption in time period t or we add amount Y to consumption to time period t + h. Suppose we have chosen adjustments X and Y in such a manner than the individual is indifferent between the two alternative paths, thus, assuming a constant discount rate, , then:

tU(1 + X) + t+hU(1) = tU(1) + t+hU(1+Y)

Notice that if we divide through by t and rearrange this, then::

U(1 + X) - U(1) = h[U(1 + Y) - U(1)]

Notice that t disappears from this expression; only the absolute time difference, h, remains. In other words, we remain indifferent between the two adjustments X and Y as long as these two adjustment happen with a difference of h periods of each other. It doesn't matter whether X happens in period 3 and Y in period 3 + h or whether X happens in period 45 and Y in period 45 + h.But now suppose that we have two different discount factors. Namely, let t be the discount for period t and t+h be the discount for period t+h. Then, once again, choose X and Y so that the agent is indifferent:

tU(1 + X) + t+hU(1) = tU(1) + t+hU(1+Y)

But now, when this is rearranged:

U(1 + X) - U(1) = ( t+h/ t)[U(1 + Y) - U(1)]

Now, notice that t remains in the expression. Thus, our preferences over the adjustments are dependent not only on the absolute time difference, h, but also on the actual reference time t when the events happen. Stationarity is broken. The absence of stationarity means that we can have dynamic inconsistency, i.e. plans that are made at one point in time, are contradicted by later behavior. The identification of this possibility is often credited to Robert Strotz (1956). Its implications are teased out in Bezalel Peleg and Menachem Yaari (1973).Intuitively, recall that if the stationarity axiom is violated, then we can have it that we prefer one apple today to two apples tomorrow, but, at the same time, prefer two apples in 31 days to one apple in thirty days. Why this leads to inconsistency is obvious. If I make a consumption plan according to these preferences, I will plan to receive two apples in 31 days, but then, as time passes and that day approaches, I'll change my mind and choose to get the one apple one day earlier. My initial plans are inconsistent with my subsequent actions.

143

Interestingly, the old economists who came up with time preference allowed for the discount rate to change over time. For instance, we find William Stanley Jevons arguing that:

"An event which is to happen a year hence affects us on the average about as much one day as another; but an event of importance, which is to take place three days hence will probably affect us on each of the intervening days more acutely than the last." (Jevons, 1871: p.34-5)

Or, even more explicitly in Eugen von Böhm-Bawerk:

"I should like to call special attention, further, to the fact, that the undervaluation which resutls from these causes is not at allgraduated harmoniously, in the subjective valuation of the individuals, according to the length of the time that intervenes. I mean, it is not graduated in this way, for example, that the man who discounts a utility due in one year by 5%, must discount a utility due in two years by 10%, or one due in three months by 1¼%. On the contrary, the original subjective undervaluations are, in the highest degree, unequal and irregular. In particular, so far as the undervaluation is caused by defects of the will, there may be a strong difference between an enjoyment hich offers itself at the very moment and one which does not; while, on the other hand, there may be a very small difference, or no difference at all, between an enjoyment which is pretty far away, and one which is further away." (Böhm-Bawerk, 1889: p.257-8).

However, the relevance of this question to our context depends on our interpretation of the social welfare function. If we maintain a "successive generations" interpretation, then it is not clear what this means because people do not live more than a period to begin with. If we argue on the basis of "dynastic" utility, changing time preference makes some sense (cf. Phelps and Pollack, 1968). Specifically, within a dynasty, different generations will have different time preference rates, e.g. the generation of time 0 discounts at rate , but the generation of time 1 discounts at rate , where . This means that the "plans" that generation 0 sets for generation 1 (and all subsequent generations) are not followed by generation 1 when the time arrives, who go on to develop their own distinct plans instead. The ethical implications of this "time inconsistency" in a dynasty are, however, a bit harder to distentangle.These discussions have kicked off a series of experimental studies on time discounting in recent years. Although the results are mixed, they suggest that people often use non-constant "hyperbolic" discounting rather than constant "exponential" discounting. See Shefrin (1998) and Rubinstein (2000) for surveys and critical evaluations of this literature.However, we should note that for normative purposes, changing time preferences are not necessarily a deep challenge. Ethically, there is no case for supporting hyperbolic discounting in the social planner's utility function and so, in that case, we can ignore whether people personally discount hyperbolically or not. However, if we derive the social welfare function via the "dynastic"

144

utility argument, then hyporbolic discounting matters very much indeed, even if the ethical implications are unclear. Changing time preference matters most if we were to adopt the positive "decentralization" thesis for our intertemporal social planner.Finally, there is no need to assume that time preference is completely exogenous. Tjalling Koopmans, Peter Diamond and Richard Williamson (1964) and Hirofumi Uzawa (1968a, 1968b), for instance, have argued quite persuasively that the time preference factor should be dependent on the levels of consumption. This is what we obtain with non-additively separable utility. But the outcome of allowing this is unclear. Harl Ryder and Geoffrey Heal (1973) have incorporated changing time preferences into an optimal growth model and shown that the resulting dynamics can be quite complicated.

5.4.7.     Intertemporal Justice

- Rawlsian Social Welfare- Rawlsian Altruism

5.4.7.1.   Rawlsian Social Welfare

So far, we have adhered to a (weighted) Benthamite structure for our social welfare function. But there are alternative structures. John Rawls's (1971) "maximin" social welfare function is one that comes to mind quite readily. We can adapt this to the intertemporal scenario quite simply. Instead of maximizing the sum of utilities of different generations, we maximize the minimum utility of any generation. Specifically, our social welfare function takes the form:

S = mint Ut(Ct}

So choosing consumption allocations that maximize social welfare function S translates into finding the allocation that maximizes the minimum utility, i.e.

max S = max{C} {mint Ut(Ct)}.

In other words, the social planner wants to improve the lot of the generation that is worst off.The Rawlsian social welfare function may be commendable for its highly egalitarian structure within a generation, but it is trickier in its intertemporal form. Recall that, in the Ramsey exercise, savings reduce the utility of the current generation, but, via capital accumulation and growth, that implies higher utility for future generations. Thus, a Benthamite social welfare function will lose on one end but gain in the other. The Rawlsian social welfare function, however, just loses. If the worst-off generation saves anything, then social welfare as a whole is lower because the consequent gains in utility by other better-off generations will not be counted. Thus, the main peculiarity of the intertemporal Rawlsian social welfare function is that it cannot balance the utilities of current and future generations. As a result, in the absence of

145

population growth or technical progress, it predicts that the optimal -- or "just" -- rate of savings will be zero.[Note: this result was already alluded to by Tinbergen (1960) and Solow (1974). Phelps and Riley (1978) create intergenerational externalities by considering a Rawlsian overlapping generations model. In this case, the savings of one generation when young can be partly compensated in the future (when they are old), so net savings will not be zero necessarily.]Now, Rawls (1971: p.284-93) was quite aware of this dilemma. Although he refrains from applying his "maximin" principle in the simple manner given above, he makes suggestive remarks to the effect that it might be applicable if "dynastic" utilities were incorporated. If "[t]he parties are regarded as representing family lines, say, with ties of sentiment across generations", then "the characterization of justice remains the same. The criteria for justice between generations are those that would be chosen in in the original position." (Rawls, 1971: p.292).So, incorporating dynastic considerations once again, we now have a social utility function:

S = mint Ut(Ct, Ut+1)

so the utility function of generation t includes utility function of generation t+1. Now, if we assume that generation t's utility function is additively separable and exhibits time preference -- and recalling that utility of the t+1 generation includes the utility of the t+2 generation -- then we can write this as:

Ut(Ct, Ut+1) = Ut(Ct) + Ut+1(Ct, Ut+1)

so generation t explicitly recognizes that generation t+1 is itself altruistic towards generation t+2, etc. The maximin social welfare function would thus be:

S = mint [Ut(Ct) + Ut+1(Ct+1, Ut+2)]

or, recursing further for Ut+2, etc., we obtain in the end:

S = mint =t U(C)

So, maximizing this social welfare function means maximizing the smallest discounted infinite stream of utility. Note that it is the starting point of the stream, t, that matters. Kenneth J. Arrow (1973) and Partha Dasgupta (1974), who started on this track, claimed that the dynastic Rawlsian form would yield dynamic inconsistency and "it is at least questionable that the sawtooth pattern [of dynastic inconsistency] corresponds to any intuitive idea of justice" (Arrow, 1973). We should note here that Arrow-Dasgupta result relies on the faulty manner in which they incorporate "dynastic" considerations. Their analysis was criticized by Guillermo Calvo (1978), who provided the form and analysis we use here.

146

To see dynamic inconsistency à la Calvo, examine Figure 1, where we have two consumption plans, C (in black) and C (in red). Obviously, the worst off generation in both cases is the first.

Figure 1 - Dynamic Inconsistency with Rawlsian SWF

Now, if we take the dynastic perspective, from the first generation's point of view, it may very well be that C is better than C. This is because the higher utility gained by generations 2 and 3 via C will, in generation 1's altruistic calculation, be weighted heavier than the relatively lower utility generations 4, 5, 6, etc. will consequently get (if you think this is not obvious in Figure 1, you know we can easily adjust the path so that it is so). So, by the maximin criteria, C has greater social welfare than C. But now examine the same paths from the perspective of generation 3 onwards. In generation 3's view, the utility of generations 4 and 5 are given much more weight than they had in generation 1's perspective. So, from generation 3's perspective, even though they themselves get higher utility with C than with C, the immediacy of the drop right after them means that C will be better than C . This is dynamic inconsistency. Generation 1 will plan for path C , but by generation 3, that plan will be dropped by generation 3, and path C will be adopted.Now, Calvo (1978) shows that while dynamic inconsistency is possible it is not necessarily the result. He proves that if we use the dynastic Rawlsian social welfare function with a standard Neoclassical optimal growth model, we can obtain a dynamically-consistent solution.

5.4.7.2.  Rawlsian AltruismWe should note that John Rawls (1971) rejected time preference immediately. In his famous "original position...there is no reason for the parties to give any weight to mere position in time." (Rawls, 1971: p.294). He recognizes the mathematical need for time-preference to provide complete comparability, but remains unimpressed. "Unhappily I can only express the opinion that these devices simply mitigate the consequences of mistaken principles" (1971: p.297).However, we find all this quite curious. Rawls abandoned the Benthamite structure, yet it is precisely the Benthamite structure that makes time preference "unjust". Time preference does not, in and of itself, imply

147

"selfishness". That arises in combination with other elements of the utility function -- specifically, the assumption of additive separability. To see this, suppose there is a cake that can be eaten by a father or a son, both of whom would receive 100 utils per slice of cake consumed. If we are maximizing the pure sum of utilities without discounting, then it would not be inconsistent for the father to split the cake equally with his son. However, with discounting, the father underestimates his child's utility, e.g. he believes that the son will only get 80 utils per slice consumed. Several consequent courses of action are conceivable. For instance, the father might divide the cake so that his son gets a greater proportion of the cake to compensate for his "lower" estimated utility per slice (e.g. father receives 30% and the son 70% of the cake). In this case, time preference did not cause "selfishness" at all, but quite the opposite! This is "Rawlsian altruism". In contrast, if the father is a pure Benthamite where only the "dynasty's aggregate utility" matters, then the father will consume a greater portion of the cake himself because his contribution to "dynasty utility" is greater than the son's. We can state Rawlsian altruism as follows. Let the father be generation t and the son be generation t+1, then instead of writing the father's utility as an additively separable utility function, we can write it as a Leontief discounted utility function:

Ut(Ct, Ut+1) = min {Ut(Ct), Ut+1(Ct+1, Ut+2)}

and then recurse future generations through this, yielding:

Ut(Ct, Ut+1) = min {Ut(Ct), Ut+1(Ct+1), 2 Ut+2(Ct+2), ...}

In this case, we actually get the result of infinite patience! Specifically, as 0 < < 1, then implies U(Ct+ ) 0, so the utility of the father at time t (i.e. the minimum of his dynasty) is the near-zero utility of the last descendent, far into the infinite future. He will thus consume nothing himself and allocate all of his income into the far future!We see, then, that the time discount factor, in and of itself, does not mean the father is "selfish". It is really only when we combine it with a Benthamite altruism, i.e. an additively separable dynastic utility function, that we achieve that "unjust" result. Time discounting can mean that the older generation is selfish (Benthamite altruism) or that it acts like a "mother hen" towards its progeny (Rawlsian altruism). Which case arises turns out to depend on the functional form of the dynastic utility function.One can counter, of course, that one should not incorporate time discounting into the Rawlsian maximin function as it doesn't yield impatience. In other words, when we place the discount factor in the Rawlsian utility function we are not really incorporating "time-preference" but something else. Perhaps. But the original definition of time-preference, as posited by Böhm-Bawerk (1889), was constant underestimation of future utility. That is enough justification to include , regardless of the functional form of the utility function. It yields impatience in the additively separable utility function, but altruistic "patience" in the Rawlsian maximin form.5.7. Optimal Growth: Conclusion

148

We have surveyed several of the important phases in the history of optimal growth theory, from its beginnings in the work of Ramsey, through the Golden Rule, through the painstaking ethical acrobatics of constructing intertemporal social welfare functions, through to the Cass-Koopmans construction, the revelation of the turnpike and, finally, the "decentralization" thesis. It was a necessarily superficial survey, and we hope you proceed to our references for a fuller account. We nonetheless feel that we have at least touched upon most of the major issues in the field.

We have been neglectful in one significant respect: namely, we have deliberately avoided touching on more recent developments in optimal growth theory, such as the rise of "Real Business Cycle" theory and "New Growth Theory" in the 1980s. We did so because we believe the interpretation and purpose of these latter constructions are so distinct from those of the original theory that they deserve an entirely separate treatment. However, we cannot end our discussion of optimal growth theory without at least some final reflections on this metamorphosis.

Optimal growth theory has changed more than we have indicated here. The "Real Business Cycle" research program, initiated in the early 1980s by Finn Kydland, Edward Prescott, Robert King and Sergio Rebelo, took Neoclassical growth theory -- and optimal growth theory in particular -- as its basic underlying model. It heavily and unabashedly relies upon the "decentralization thesis" for methodological justification. It prides itself in its tremendous efforts to "calibrate" the ghostly parameters of the Cass-Koopmans model -- time preference, utility functions, productivity, etc. -- so that the optimal solution path of this normative model is matched to the actual empirical data of economies around the world. Empirical accounts of growth and fluctuations of output, employment and growth are regarded as the optimal paths derived as solutions to appropriately-calibrated optimal growth models.

Today, optimal growth theory and other variations on the Ramsey exercise are marketed as actual representations of how the economy works -- despite the fact that reality tells us the social planner does not exist. And if we insist on interpreting him as a "representative agent", then we must keep in mind that microeconomic theory (specifically, the Sonnenschein-Mantel-Debreu theorem), tells us that he would misbehave. But, for some mystifying reason, modern economics persists with this fiction, despite the overwhelming theory and evidence against it. Appealing to "representative agents" is a deplorable, but sadly common and ancient habit in economics.

There are more paradoxes that could be drawn out of this -- which ought to be particularly delicious for Austrian School economists. For instance, the decentralization thesis basically argues that the centralized economy of (a benevolent) Stalin "represents" or "achieves the same solution as" the decentralized economy of wild capitalist markets. The credibility of this assertion is worth contemplating for a moment or two, particularly in light of the Socialist Calculation debates of the 1930s.

149

One may also wish to reflect upon the meaning of "prices" in such an economy. One of the remarkable justifications put forth for the use of an infinitely-lived, perfect foresight, representative agent in modern modeling is that agent heterogeneity, imperfect foresight, etc. would make the "optimal solutions" indeterminate. Put another way, modern theorists can only derive equilibrium prices when there are no incentives for exchange among agents; but when such an incentive exists, they cannot obtain equilibrium prices! A naughty wag could certainly get a lot of mileage out of that.

As this survey should make evident, this transformation in motivation for and application of optimal growth theory is one which the original constructors would find surprising, if not appalling. Frank Ramsey certainly conceived of his contribution more as an exercise in Benthamite utilitarian philosophy than in descriptive economics. At any rate, recall that his main point was to demonstrate that the market solution would be suboptimal. The early builders of the 1960s -- Tinbergen, Goodwin, Koopmans and others -- were eager to put it to good use in development planning, perhaps naively believing that the ought of optimal growth theory could be deliberately planned into becoming an is, after all.

As late as mid-1970s, Tjalling Koopmans was still arguing that the principal clientèle of optimal growth theory should be "policy economists who may find it useful to have the more abstract ideas of this field in the back of their mind when coping with the day-to-day pressures for outcomes rather than criteria." (Koopmans, 1977). One of the classics of optimal growth theory, the famous treatise of Kenneth J. Arrow and Mordecai Kurz (1970) was written almost as a handbook for policy economists. The interface between government policy and optimal growth was also explored by other pioneering spirits, such as Hirofumi Uzawa (1969) and Edmund S. Phelps (1974). The conclusion of Tjalling Koopmans's Nobel lecture captures the spirit behind the original construction of optimal growth theory:

"The economist as such does not advocate criteria of optimality He may invent them. He will discuss their pros and cons, sometimes before, but preferably after trying out their implications. He may also draw attention to situations where allover objectives, such as productive efficiency, can be served in a decentralized manner by particularized criteria such as profit maximization. But the ultimate choice is made, usually only implicitly and not always consistently, by the procedures of decision making inherent in the institutions, laws and customs of society. A wide range of professional competences enter into the preparation and deliberation of these decisions. To the extent that the economist takes part in this decisive phase, he does so in a double role, as economist, and as a citizen of his polity: local polity, national polity or world polity."

(T.C. Koopmans, 1977)

150

As such, for those interested in the sociology of doctrines, optimal growth theory turns out to be quite a curiosity. It was drastically transformed in the 1980s from a normative theory to a positive theory, from a theory which sought to find out what ought to be, to one which claims to discern what is. This has almost no parallel in the history of economic theory. It is striking in at least two respects. Firstly, this metamorphosis managed to do something that seems to be quite unprecedented: instead of falling into the classical fallacy of "deriving an ought from an is", as David Hume famously warned against, they actually managed to fall into the reverse fallacy of "deriving an is from an ought". Secondly, optimal growth theory has magically moved from one end of the ideological spectrum to almost the exact opposite -- the planning tool of the dirigiste economist has become the bible of the Panglossian.

"[Modern economic theory] proposes that the actual economy can be read as if is acting out or approximating the infinite-time discounted utility maximizing program of a single, immortal, "representative agent"....What Ramsey took to be a normative model, useful for working out what an idealized, omniscient planner should do, has been transformed into a model for interpreting last year's and next year's national accounts.

Of course that is the economics of Dr. Pangloss, and it bears little relation to the world...The consequence is this: no account has been given of how and why a decentralized economy could behave as if guided by a Ramsey maximizer. It is true that an Arrow-Debreu equilibrium is an allocation that maximizes a special social welfare function, but that is not the case, for instance, when some insurance markets are absent, or indeed when any even mildly realistic phenomena are added."

5.8. Optimal Growth: Selected References

A. Abel and O.J. Blanchard (1983) "An Intertemporal Equlibrium Model of Saving and Investment", Econometrica, Vol. 51 (3), p.675-92.

M. Allais (1947) Economie et Intérêt. Paris: Imprimerie Nationale.

M. Allais (1962) "The Influence of the Capital-Output Ratio on Real National Income", Econometrica, Vol. 30, p.700-28.

K.J. Arrow (1973) "Rawls's Principle of Just Saving", Swedish Journal of Economics, Vol. 75, p.323-35.

K.J. Arrow (1979) "The Trade-Off Between Growth and Equity", in H.I. Greenfield, A.M. Levenson, W. Hamovitch and E. Rotwein, editors, Theory for Economic Efficiency: Essays in honor of Abba P. Lerner. Cambridge, Mass: M.I.T. Press.

151

K.J. Arrow and M. Kurz (1970) Public Investment, the Rate of Return and Optimal Fiscal Policy. Baltimore: The Johns Hopkins University Press.

H. Atsumi (1965) "Neoclassical Growth and the Efficient Program of Capital Accumulation", Review of Economic Studies, Vol. 32, p.127-36.

P.T. Bauer (1957) Economic Analysis and Policy in Underdeveloped Countries. Durham, NC: Duke University Press.

R.J. Barro (1974) "Are Government Bonds Net Wealth?",  Journal of Political Economy, Vol. 82 (6), p.1095-1117.

W.J. Baumol (1952) Welfare Economics and the Theory of the State. Cambridge, Mass: Harvard University Press.

R.A. Becker (1980) "On the Long-Run Steady State in a Simple Dynamic Model of Equilibrium with Heterogeneous Households", Quarterly Journal of Economics, Vol. 95, p.;375-82.

R.A. Becker and M. Majumdar (1989) "Optimality and Decentralization in Infinite Horizon Economies", in G. Feiwel, editor, Joan Robinson and Modern Economics. New York: New York University Press.

O.J. Blanchard and S. Fischer (1989) Lectures in Macroeconomics. Cambridge, Mass: M.I.T. Press

E. v. Böhm-Bawerk (1889) The Positive Theory of Capital. 1923 reprint of 1891 Smart translation, New York: Stechert.

D.J. Brown and L.M. Lewis (1981) "Myopic Economic Agents", Econometria, Vol. 49, (2), p.359-68.

G. Calvo (1978) "Some Notes on Time Inconsistency and Rawls' Maximin Criterion", Review of Economic Studies, Vol. 45 (1), p.97-102.

D. Cass (1965) "Optimum Growth in an Aggregative Model of Capital Accumulation", Review of Economic Studeis, Vol. 32, p.233-40.

D. Cass (1966) "Optimum Growth in an Aggregative Model of Capital Accumulation: A turnpike theorem", Econometrica, Vol. 34, p.833-50.

S. Chakravarty (1962) "The Existence of an Optimum Savings Program", Econometrica, Vol. 30 (1), p.178-87.

P. Dasgupta (1974a) "On Some Problems Arising from Professor Rawls's Conception of Distributive Justice", Theory and Decision, Vol. 4, p.325-44.

P. Dasgupta (1974b) "On Some Alternative Criteria for Justice Between Generations", Journal of Public Economics, Vol. 3, p.

152

J. Desrousseaux (1961) "Expansion table et taux d'intérêt optimal", Annales de Mines, p.31-46.

P.A. Diamond (1965) "The Evaluation of Infinite Utility Streams", Econometrica, Vol. 33, p.170-7.

O. Eckstein (1957) "Investment Criteria for Economic Development and the Theory of Intertemporal Welfare Economics", Quarterly Journal of Economics,

P.C. Fishburn and A. Rubinstein (1982) "Time Preference", International Economic Review, Vol. 23 (3), p.677-94.

I. Fisher (1930) The Theory of Interest: As determined by impatience to spend income and opportunity to invest it. 1954 reprint, New York: Kelley and Millman.

R. Frisch (1964) "Dynamic Utility", Econometrica, Vol. 32, p.418-24.

D. Gale (1967) "On Optimal Development in a Multi-Sector Economy", Review of Economic Studies, Vol. 34, p.1-18.

R.M. Goodwin (1961) "The Optimal Growth Path for an Underdeveloped Economy", Economic Journal, Vol. 71, p.756-74.

J. de V. Graaff (1957) Theoretical Welfare Economics. 1971 edition, Cambridge, UK: Cambridge University Press.

F.H. Hahn (1985) Money, Growth and Stability. Cambridge, Mass: MIT Press.

F.H. Hahn and R.M. Solow (1995) A Critical Essay on Modern Macroeconomic Theory. Cambridge, Mass: M.I.T. Press.

P.J. Hammond and J.A. Mirrlees (1973) "Agreeable Plans", in J.A. Mirlees and N. Stern, editors, Models of Economic Growth. New York: Wiley.

R.F. Harrod (1948) Towards a Dynamic Economics: Some recent developments of economic theory and their application to policy. London: Macmillan.

J. Hirshleifer (1970) Investment, Interest and Capital. Englewood Cliffs, NJ: Prentice-Hall.

R.F. Kahn (1959) "Exercises in the Analysis of Growth", Oxford Economic Papers,

J.M. Keynes (1930) "F. P. Ramsey: An obituary", Economic Journal, Vol. 40, p.153-4. As reprinted in Keynes, 1933, Essays in Biography. London: Macmillan.

A.P. Kirman (1992) "Whom or What Does the Representative Agent Represent?", Journal of Economic Perspectives, Vol. 6 (2), p. 117-36.

153

T.C. Koopmans (1960) "Stationary Ordinal Utility and Impatience", Econometrica, Vol. 28, p.287-309.

T.C. Koopmans (1965) "On the Concept of Optimal Economic Growth", Pontificae Academiae Scientiarum Scripta Varia, Vol. 28, p.225-300.

T.C. Koopmans (1967) "Objectives, Constraints and Outcomes in Optimal Growth Models", Econometrica, Vol. 35, p.1-15.

T.C. Koopmans (1972) "Representation of Preference Orderings over Time", in C.B. McGuire and R. Radner, editors, Decision and Organization A volume in honor of Jacob Marschak. Amsterdam: North-Holland.

T.C. Koopmans (1977) "Concepts of Optimality and Their Uses", American Economic Review, Vol. 67 (3), p.261-74.

T.C. Koopmans, P.A. Diamond and R.E. Williamson (1964) "Stationary Utility and Time Perspective", Econometrica, Vol. 32, p.82-100.

K. Lancaster (1963) "An Axiomatic Theory of Consumer Time Preference", International Economic Review, Vol. 4, p.221-31.

A.P. Lerner (1959) "Consumption-Loan Interest and Money", Journal of Political Economy, Vol. 67 (5), p.512-8.

G. Loewenstain and J. Elster (1992), editors, Choice Over Time. New York: Russell Sage Foundation.

R.E. Lucas and N. Stokey (1984) "Optimal Growth with Many Consumers", Journal of Economic Theory, Vol. 32, p.139-71.

M.J.P. Magill (1981) "Infinite Horizon Programs", Econometrica, Vol. 49 (3), p.679-711.

E. Malinvaud (1965) "Croissances optimales dans un modèle macroéconomique", Pontificae Academiae Scientiarum Scripta Varia, Vol. 28, p.301-384.

S.A. Marglin (1963) "The Social Rate of Discount and the Optimal Rate of Investment", Quarterly Journal of Economics, Vol. 77 (1), p.95-111.

L.W. McKenzie (1986) "Optimal Economic Growth, Turnpike Theorems and Comparative Dynamics", in K.J. Arrow and M.D. Intriligator, Handbook of Mathematical Economics, Vol. III. Amsterdam: North-Holland.

J.A. Mirrlees (1967) "Opimum Growth when Technology is Changing", Review of Economic Studies, Vol. 34 (1), p.95-124.

154

L. v. Mises (1949) Human Action: A treatise on economics. 1963 edition, New Haven: Yale University Press.

B. Peleg and M. Yaari (1973) "On the Existence of a Consistent Course of Action when Tastes are Changing", Review of Economic Studies, Vol. 40 (3), p.391-401.

E.S. Phelps (1961) "The Golden Rule of Accumulation: A Fable for Growthmen", American Economic Review, Vol. 51, p.638-43.

E.S. Phelps (1966) Golden Rules of Economic Growth: Studies of efficient and optimal investment. New York: Norton.

E.S. Phelps (1974) Fiscal Neutrality Toward Economic Growth. New York: McGraw-Hill.

E.S. Phelps and R.A. Pollack (1968) "On Second-Best National Saving and Game-Equilibrium Growth", Review of Economic Studies, Vol. 35, p.

E.S. Phelps and J.G. Riley (1978) "Rawlsian Growth: Dynamic programming of capital and wealth for intergeneration "maximin" justice", Review of Economic Studies, Vol. 45 (1), p.103-20.

A.C. Pigou (1920) The Economics of Welfare. 1952 (4th) edition, London: Macmillan.

F.P. Ramsey (1928) "A Mathematical Theory of Saving", Economic Journal, Vol. 38, p.543-59.

J. Rawls (1971) A Theory of Justice. Cambridge, Mass: Harvard University Press.

J. Robinson (1962) "A Neo-classical Theorem", Review of Economic Studies, Vol. 29, p.219-26. Reprinted as "A Neo-neoclassical Theorem", in Robinson (1962) Essays in the Theory of Economic Growth, 1968 edition, London: Macmillan.

A. Rubinstein (2000)

H.E. Ryder and G.M. Heal (1973) "Optimum Growth with Intertemporally Dependent Preferences", Review of Economic Studies, Vol. 40 (1), p.1-32.

P.A. Samuelson (1937) "A Note on Measurement of Utility", Review of Economic Studies, Vol. 4, p.155-61.

P.A. Samuelson (1958) "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money", Journal of Political Economy, Vol. 66 (5), p.467-82.

P.A. Samuelson (1959) "Reply to Lerner", Journal of Political Economy, Vol. 67 (5), p.518-22.

155

P.A. Samuelson (1965) "A Catenary Turnpike Theorem Involving Consumption and the Golden Rule", American Economic Review, Vol. 55, p.486-96.

P.A. Samuelson and R.M. Solow (1956) "A Complete Capital Model Involving Heterogeneous Capital Goods", Quarterly Journal of Economics, Vol. 70, p.537-62.

J.A. Schumpeter (1912) The Theory of Economic Development. 1934 translation, Cambridge Mass: Harvard University Press.

A.K. Sen (1961) "On Optimising the Rate of Saving", Economic Journal, Vol. 71, p.479-96.

H. Shefrin (1998) "Changing Utility Functions" in S. Barberà, P.J. Hammond and C. Seidl, editors, Handbook of Utility Theory: Vol. I - Principles. Dordrecht: Kluwer.

K. Shell (1967) "Optimal Programs of Capital Accumulation for an Economy in which there is Exogenous Technical Change", in Shell, 1967, editor, Essays on the Theory of Optimal Economic Growth. Cambridge, Mass: M.I.T. Press.

K. Shell (1969) "Application of Pontryagin's Maximum Principle to Economics", in H.W. Kuhn and G.P. Szegö, editors, Mathematical Systems Theory and Economics, Vol. 1, p.241-92.

R.M. Solow (1974) "Intergenerational Equity and Exhaustible Resources", Review of Economic Studies, Vol. 41, p.29-45.

T.N. Srinivasan (1964) "Optimal Savings in a Two Sector Model of Growth", Econometrica, Vol. 32, p.358-73.

R.H. Strotz (1956) "Myopia and Inconsistency in Dynamic Utility Maximization", Review of Economic Studies, Vol. 23 (3), p.165-80.

T. W. Swan (1963) "Of Golden Ages and Production Functions", in K. Berrill, editor, Economic Development with Special Reference to East Asia: Proceedings of an International Economic Association conference. London: Macmillan.

J. Timbergen (1956) "The Optimum Rate of Saving", Economic Journal, Vol. 66, p.603-9.

J. Tinbergen (1960) "Optimum Savings and Utility Maximization Over Time", Econometrica, Vol. 28, p.481-9.

H. Uzawa (1964) "Optimal Growth in a Two-Sector Model of Capital Accumulation", Review of Economic Studies, Vol. 31, p.1-24.

156

H. Uzawa (1968) "Market Allocation and Optimum Growth", Australian Economic Papers, Vol. 7, p.17-27.

H. Uzawa (1968) "Time Preference, the Consumption Function and Optimum Asset Holdings", in J.N. Wolfe, editor, Value, Capital and Growth. Edinburgh: University of Edinburgh Press.

H. Uzawa (1969) "Optimum Fiscal Policy in an Aggregative Model of Economic Growth", in I. Adelman and E. Thorbecke, editors, The Theory and Design of Economic Development. Balitmore: Johns Hopkins Press.

C.C. von Weizsäcker (1962) Wachstum, Zins und Optimale Investifionsquote. Basel: Kyklos-Verlag.

C.C. von Weizsäcker (1965) "Existence of Optimal Programs of Accumulation for an Infinite Time Horizon", Review of Economic Studies, Vol. 32, p.85-104.

M.E. Yaari (1964) "On the Existence of an Optimal Plan in a Continuous-Time Allocation Process", Econometrica, Vol. 32 (4), p.576-90.

M.E. Yaari (1965) "Uncertain Lifetime, Life Insurance and the Theory of the Consumer", Review of Economic Studies, Vol. 32, p.137-50.

157