mphil macroeconomics: economic growth
TRANSCRIPT
MPhil Macroeconomics: Economic Growth
Roland Meeks
Nuffield [email protected]
Michaelmas 2006
Contents
1 Introduction 3
2 Solow’s ‘shocker’ 5
2.1 Details of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Steady state growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Technological change . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Growth accounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.5 Decentralised economy . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 Normative aspects of growth . . . . . . . . . . . . . . . . . . . . . . . 20
3 Convex growth model with optimisation 22
3.1 Cass-Koopmans-Ramsey . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.1 Continuous time . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.1.2 Discrete time . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2 Growth rates and convergence . . . . . . . . . . . . . . . . . . . . . . 30
3.2.1 Steady state growth . . . . . . . . . . . . . . . . . . . . . . . 30
3.2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4 First generation endogenous growth 37
4.1 The AK model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Knowledge externalities . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2.1 Pareto optimality . . . . . . . . . . . . . . . . . . . . . . . . . 45
1
2 Economic Growth
4.3 Human capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.1 Lucas (1988) . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 Generalisations of Lucas . . . . . . . . . . . . . . . . . . . . . 50
4.3.3 A tractable two-sector model . . . . . . . . . . . . . . . . . . 50
5 Cross sections and convergence 53
5.1 Mankiw, Romer, and Weil . . . . . . . . . . . . . . . . . . . . . . . . 55
5.2 Some critics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.3 Institutions, constitutions and geography . . . . . . . . . . . . . . . . 63
5.4 Time series empirics . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6 R&D based endogenous growth 68
6.1 Better goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.1.1 Endogenous growth . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Monopolistic competition . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.1 Demand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.2.2 Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.3 New goods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3.1 R&D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.3.2 General equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3.3 Endogenous growth . . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Semi-endogenous growth . . . . . . . . . . . . . . . . . . . . . . . . . 83
I thank Florin Bilbiie, Chris Bowdler, Matthias Ederer, Simon Quinn and Bruno
Strulovici for helpful comments and discussion on material in these notes. Please
direct feedback on content, errors, and omissions to the email address above.
Roland Meeks 3
1 Introduction
Economic growth occurs whenever people take resources and rearrange
them in ways that are more valuable.
– Paul Romer (‘Economic Growth’, econlib.org)
In these lectures we will study how and why standards of living change over time.
Much of this work is aggregative, and therefore properly the remit of macroeco-
nomics. The broad facts of economic growth are well known, and arguably defy
any single explanation. Economists have had to look to other fields for ideas in
building models of the growth process, particularly political economy, development
and economic history. We will be mainly concerned with the tools needed to build
analytical models, and the statistical methods needed to test them. A big part
of the divide between growth economics and the other, complementary, studies of
economic growth, is in fact methodological.
One issue to mention at the outset is whether measured output is capturing the
notion of economic wellbeing that we want to explore1. Helpman (2004) comments
that although people do care about income, they also care about freedom, liberty
and health. His view is that ‘a good measure of living standards has to account
for many factors. But most of them are hard to measure. And it is even harder
to decide how much weight to give each one. As a result, real income per capita
is often used as a rough measure of a country’s standard of living’ (p. 1). The
UN Human Development Index is one such attempt to create a composite living
standards index2. The view that ‘other’ measures of wellbeing can be safely ignored
has come in for some recent criticism, notably from economists such as Layard (2005)
working on ‘happiness’3. But economists mostly agree that when we want to explain
the big differences in living standards that have occurred over time, and the yawning
gaps that still exist between countries, GNP is the right place to start.
Figures 1 and 2 show data for US GNP growth during the first half of the 20th
century. This is the kind of data that growth economists had at their disposal when
the neoclassical theory of growth that we will study was developed, during the 1950s.
1I will use ‘output’ and ‘income’ interchangeably, unless there is risk of confusion. The two areequivalent in a closed economy.
2See http://hdr.undp.org/reports/global/2005/3See http://cep.lse.ac.uk/research/ happiness/
4 Economic Growth
These are the observations we would hope to explain, in the first instance, with the
model we will develop in the first few lectures. A celebrated list of ‘stylised facts’
Figure 1: Trends in US output and capital, 1909-1949
1910 1915 1920 1925 1930 1935 1940 1945 1950
0.0
0.5
1.0
Log GNP/man hour Log capital/man hour
1910 1915 1920 1925 1930 1935 1940 1945 1950
−0.05
0.00
0.05
Growth of GNP/man hour
Source: Solow (1957, Table I)
was constructed by Kaldor [see Snowdon and Vane (2005, p. 595)], summarising the
features of the data a model should be consistent with:
K1: Output per worker grows continuously, with no secular tendency for the rate of
growth of productivity to decline.
K2: The capital-labour ratio shows continuous growth.
K3: The rate of return on capital is stable.
K4: The capital-ouput ratio is stable.
K5: The shares of labour and capital in GDP remain stable.
K6: We observe significant variation in the rate of growth of productivity across
countries.
Not all of these are uncontroversial today, and the items in the list aren’t inde-
pendent. Nevertheless it is a useful benchmark.
Roland Meeks 5
Figure 2: Capital Share and Employment Rate
1910 1915 1920 1925 1930 1935 1940 1945 1950
0.325
0.350
0.375
0.400Capital share
1910 1915 1920 1925 1930 1935 1940 1945 1950
70
80
90
100Employment rate (%)
Source: Solow (1957, Table I)
2 Solow’s ‘shocker’
The neoclassical growth model contains two surprises: first, steady state
growth is independent of the savings rate; second, the main source of growth
is technological change, rather than capital accumulation.
Please keep in mind we are dealing with a drastically simplified story, a
‘parable’ ... You ask of a parable not if it is literally true, but if it is well
told.
– Solow (1970)
The quotation I’ve given here are a good way to summarise the style of research that
we’re mainly going to be dealing with in at least the first few lectures of this course.
In spite of the vintage nature of many of the papers I reference, they are important
to know as they have been enormously influential in macroeconomics, development
and political economy over the last half-century. So you will better understand why,
6 Economic Growth
say, a real business cycle theorist writes down an aggregate production function
and talks about total factor productivity disturbances driving business cycles, if
you have understood the problem that Solow solved, and how he solved it. In this
lecture, we’ll look at a particular model of long run growth, but just as important,
we’ll encounter some of the key concepts used for talking about growth models in
general.
To begin our formal study of economic growth, we are going to look in detail at
the neoclassical growth model, as formulated by Solow (1956). Keynes [quoted in
Snowdon and Vane (2005, p. 593)] wrote in 1930 that the lack improvement in living
standards throughout most of human history was due to ‘the remarkable absence
of important technical improvements and the failure of capital to accumulate’. The
neoclassical growth model gives pride of place to the process of capital accumulation.
As we will see, it does not explain why technical improvements are produced; that
topic is the subject of a later lecture; but it does make important testable predictions.
2.1 Details of the model
The aggregate production function is only a little less legitimate concept
than, say, the aggregate consumption function, and for some kinds of
long-run macro-models it is almost as indispensable... As long as we
insist on practicing macro-economics we shall need aggregate relation-
ships.
– Solow (1957)
In this economy, there is a single commodity, whose rate of production is Y (t). Part
of this output is consumed, and the rest is saved. As the economy is closed, all
saving must take the form of additions to domestic stocks. These homogeneous
stocks are labelled capital K(t), and you can see that saving and gross investment
are the same thing. It is assumed that the fraction of output saved is a constant s,
so there is a basic identity at every instant of time
K(t) = sY (t) (1)
where the dot denotes a time derivative. You can see that the assumption of a con-
stant savings rate pins down investment, and has as its mirror image a particularly
Roland Meeks 7
simple consumption function (a standard Keynesian one in fact). Output is pro-
duced under constant returns to scale using two factors, capital and labour4. The
technical possibilities for production are summarised in the function
Y (t) = F [K(t), L(t)] (2)
which you can think of as combining the service flows from employed capital and
labour to produce an output flow5.
Combining (1) and (2), we find that capital accumulates according to
K(t) = sF [K(t), L(t)] (3)
which is a single differential equation with two unknowns. As the object at this stage
is to track the growth of the capital stock, we will make labour into an exogenous
or forcing variable. Specifically, let there be a constant participation rate (call it 1
for simplicity), and allow growth in the population to be constant at rate n. Given
an initial population L0, population at time t is therefore L(t) = L0ent. That’s
the supply side, and in our model, we will take the full employment case that the
labour input employed in (2) is equal to this L(t) (either by asserting that there is
full employment - of both factors - or by assuming that factor markets clear, with
supply in both cases being inelastic). Finally, we get that
K(t) = sF [K(t), L0ent] (4)
which has the makings of an object that we know how to solve, if given a specific
form for F .
Actually we can get some qualitative results without assuming anything about
the production function. Denote by lower case letters variables that have been
divided through by the workforce L(t); then k is the capital-labour ratio, and y
is the output-labour ratio, or output per worker. The capital-labour ratio is also
4Recall that constant returns corresponds to linear homogeneity (i.e. of degree 1) of F . Thatmeans that a proportionate increase in all factors raises output by the same factor. Individually(holding the other input fixed), factors have diminishing returns. Stepping outside the model,decreasing returns might not literally be true, but merely reflect the practice of bringing on streamthe most efficient machines/plants first.
5Solow makes this net output; you can also think of there being zero capital depreciation.Textbooks invariably write gross output, and make explicit the depreciation allowance, and thatis what we will do in these lectures too.
8 Economic Growth
known as the capital intensity, and we’ll use the terms interchangeably. We can
write the production function in a form that gives output per worker as a function
of capital per worker
y(t) = F [k(t), 1] := f [k(t)],
or to put it another way, the output that results from employing a single worker
with varying quantities of capital. The key point about this form of the production
function, which is referred to as the intensive form, is that higher capital intensity
means more output per worker. Because of diminishing returns, this function will
be concave.
Differentiate k = K/L with respect to time and you get
k
k= [quotient rule] =
K
K−L
L(5)
The objects on the RHS are growth rates. By assumption L/L = n, and K is given
by (3), hence
k = sF (K,L)
Kk − nk (6)
Multiply and divide the first RHS term by L to find the differential equation de-
scribing the capital-labour ratio
k = sf(k) − nk (7)
You can interpret sf(k) as savings per worker and nk as the replacement capital
needed to maintain a constant capital intensity (which is being reduced by popu-
lation growth and capital consumption). The most important properties of (7) can
be determined immediately; the production function and a ray from the origin must
intersect (twice, obviously one time at the origin!) under the ‘Inada conditions’ [see
Romer (1996, p. 9)]. When savings per worker exceed break-even investment, capital
intensity grows, and conversely. This renders the stationary point k = 0 stable. The
bottom line, to quote Solow (1956, p. 70) is: ‘[w]hatever the initial capital-labour
ratio, over time the system will develop toward a state of balanced growth at the
natural rate.’6 That means that a country with an initially low capital intensity
will experience an increased level of living standards but a declining rate of growth
in output as it approaches the steady state.
6The other equilibrium I mentioned, with no capital, is unstable.
Roland Meeks 9
Exact solution: the Cobb-Douglas case
If we’re willing to restrict (2) to be Cobb-Douglas then we can find the promised
solution to (4). That differential equation becomes
K = sKαL1−α0 e(1−α)nt (8)
You will notice that this is non-linear, and usually we only try to solve linear dif-
ferential equations. Fortunately, this one has a special form called ‘separability’
[Chiang (1984, pp. 489–90) but watch the typos]. To see this, write the equation in
differential form, and multiply through by the function of K appearing on the RHS
to get:
K−αdK = sL1−α0 e(1−α)ntdt (9)
The object appearing on the LHS is in K alone, and the object on the RHS is in t
alone. That means we can integrate directly, taking K(0) = K0:
∫ K(z)
K0
K−αdK = sL1−α0
∫ z
0
e(1−α)ntdt (10)
which after some work is:
1
1 − α[K(z)1−α −K1−α
0 ] = sL1−α0
[
1
(1 − α)ne(1−α)nz −
1
(1 − α)n
]
(11)
This simplifies to:
K(z)1−α =s
nL1−α
0 e(1−α)nz −s
nL1−α
0 +K1−α0 (12)
which is an equation describing the path of K as a function of initial endowments
and time alone (there are no endogenous variables in this equation, which is the
meaning of ‘solution’ in this context). If we know K - and L is given - we know Y ,
game over. I have plotted convergence paths for this model with different values for
the savings rate, in Figure 3 (note scale carefully).
Question: As z → ∞, what is the rate of growth of K? What is the equilibrium
capital intensity?
10 Economic Growth
Figure 3: Convergence Paths for Per Capita Output and Capital in the Solow Model
0 50 1000.5
1
1.5
2
2.5s = 0.1
Time0 50 100
0
2
4
6
8s = 0.3
Time
0 50 1000
2
4
6
8
10
12s = 0.5
Time0 50 100
0
5
10
15s = 0.7
Time
CapitalOutput
CapitalOutput
CapitalOutput
CapitalOutput
Source: Author’s calculations. n = .02, g = .025, δ = .025, α = 1/3
Roland Meeks 11
2.2 Steady state growth
It’s useful to distinguish between national income growth due to increased capital
intensity, known as capital deepening, which ceases at the point where k = 0, and
the growth in national income due to population increase known as capital widening
(more workers, same capital intensity) [Snowdon and Vane (2005, p. 607)]. Similarly,
growth towards the steady state is referred to as intensive growth, whereas growth
in the steady state is referred to as extensive, being due to the rise in the workforce.
What does Solow mean by ‘balanced growth at the natural rate’? The term
balanced growth means all variables growing at a constant, but possibly different
rate. Of course that includes the situation where quantities are increasing at the
same constant rates. I emphasise that in models with optimisation, the balanced
growth path is part of the optimal path. The balanced growth path is also
where you are once transition dynamics – if any – are played out. If transition
is reasonably rapid, that will be most of the time, which is why they are worth
studying (see turnpike theorems). For obvious reasons, the alternative terminology
steady state growth also gets used. The term natural rate is explained in §2.6 below.
In the Solow model, the three main quantities are output, capital and labour.
We know that the workforce is growing at rate n, so we need to check that Y and
K are also growing at rate n. That’s simple since we know the steady state exists
and that the meaning of steady state is k = 0, so by (5) K/K = n. Then:
Y = [total derivative] = FKK + FLL (13)
= FKKK
K+ FLL
L
L(14)
divide through by Y to convert to growth rates
Y
Y=
FKK
Y
K
K+FLL
Y
L
L(15)
gY =FKK
YgK +
FLL
YgL (16)
Using knowledge of gK and gL,
gY =
[
FKK + FLL
Y
]
n (17)
= [Euler’s theorem] = n (18)
12 Economic Growth
which confirms the natural supposition in a constant-returns world that propor-
tionate increases in inputs leads to an equivalent increase in output. The central
implication is that output per worker does not grow on the balanced growth path.
What’s missing?
2.3 Technological change
Perfectly arbitrary changes over time in the production function can be
contemplated in principle, but are hardly likely to lead to systematic
conclusions.
– Solow (1956)
What is missing from the story of growth told thus far is an explanation for why
our historical experience has been continuous growth in both output and capital
per worker for the best part of two centuries, particularly in the USA and Western
Europe. Solow (1970, p. 33) gives two possibile explanations: technological progress
and increasing returns. We come back to increasing returns in the context of en-
dogenous technological improvements in another lecture, so for now let’s concentrate
as Solow does on exogenous technical change. The quote about ‘perfectly arbitrary’
changes in the production function is nice because it highlights the fact that we need
to be very specific about how we permit technological change to occur in order to
get strong results. That is the subject of this section.
You need to be aware of some terms used in the literature for this part. Techno-
logical change in an aggregate CRS production function F (K,L) is Harrod neutral
if it is labour augmenting. Technological change is Hicks neutral if it does not affect
the optimal choices of capital and labour7. Uzawa (1961) proves that technological
invention represented by F (K,L, t) is Harrod neutral if and only if the production
function is of the form G(K,A(t)L). Similarly it is Hicks neutral if and only if the
production function is of the formA(t)F (K,L). Graphically, that can be represented
by an upward shift in the production function. A corollary of these observations is
that the two types of technical progress are equivalent only for the Cobb-Douglas
form.
7More precicely, the ratio of the marginal products of capital and labour are unchanged at aconstant capital-labour ratio. Harrod neutral change is the situation where the marginal productof capital alone is unchanged at a constant capital-output ratio.
Roland Meeks 13
It’s a good idea to get straight what these terms mean. Labour augmenting
technological change means that raw labour units – the number of aggregate hours
worked – are being augmented by a coefficient of labour productivity over time,
to produce an increase in effective labour input. This happens independently from
what is happening to capital, so with the same equipment a worker today produces
more than a worker yesterday. In general, by productivity of any input we will
always mean the size of the coefficient that converts from natural units, be it man-
hours or acres or whatever, into effective units of the input. That should be after
we have accounted for changes in the quality of these inputs (see the next section).
On the other hand, Hicks neutral technological change means that we are able to
produce more today than we did yesterday without any change in the combined
effective factor inputs.
The terms get used in different ways, even amongst economists. For example,
labour productivity often means ‘output per man-hour’. Another example is that
technologists might say that a shift to a more IT intensive mode of production is a
change in technology, whereas economists see this as a substitution of IT capital for
labour (or other capital) and therefore a shift along the same production function.
‘Substitution takes place if the introduction of computer-intensive equipment pro-
duces benefits that are fully captured or internalised by the users of IT and their
suppliers. Technical change occurs only if more output is produced from the same
inputs (e.g. if some of the benefits spill over to third parties not involved in the
transaction)’ (Jorgenson and Stiroh, 1999, p. 109). Finally, total factor productivity
(TFP) measures the combined effects of all factor-augmenting and Hicks neutral
technological change on output.
It turns out that technical progress has to take the labour augmenting form in
order to be consistent with the existence of steady state growth [a proof of this is
given in Barro and Sala-i-Martin (2004, ch. 1.5.3)]. Some intuition for this is given
by Solow (1970); reason by contradiction: the steady state requires output to grow
at the same rate as capital in natural units. Labour and capital have to grow at the
same rate in efficiency units. If so, then output also grows at that rate, contradicting
the requirement that it grows at the same rate as capital in natural units.
14 Economic Growth
Figure 4: Solow’s Estimates of TFP
1910 1915 1920 1925 1930 1935 1940 1945 1950
1.00
1.25
1.50
1.75A
1910 1915 1920 1925 1930 1935 1940 1945 1950
−0.05
0.00
0.05
g_A
Source: Solow (1957, Table I). Top panel: level. Bottom panel: growth.
2.4 Growth accounting
Growth accounting decomposes income growth into growth in inputs plus a
residual. The residual is often attributed to total factor productivity (TFP).
In an accounting sense, growth in TFP probably causes between a quarter
and a half of economic growth in rich countries.
Measuring the contribution of inputs to output growth is the goal of growth account-
ing. It breaks down observed changes in GDP into components, hence the label.
The issue is precisely how to separate moves along a production function from shifts
to a new production function (which may happen simultaneously), a problem orig-
inally solved by Solow (1957). Suppose that production is constant-returns with
Hicks neutral technical change, given by:
Y = AF (K,L)
Roland Meeks 15
Figure 5: Cross Country TFP Growth 1961-1988
1960 1965 1970 1975 1980 1985 1990
0.000
0.025
0.050
France
1960 1965 1970 1975 1980 1985 1990
0.00
0.02
0.04
Germany
1960 1965 1970 1975 1980 1985 1990
−0.025
0.000
0.025
0.050
0.075 Japan
1960 1965 1970 1975 1980 1985 1990
−0.02
0.00
0.02
U.S.
Source: Jones (1995b).
and there is perfect competition. Then as we already saw:
Y
Y=A
A+FKK
Y
K
K+FLL
Y
L
L
Under competition, factors receive their marginal products, so we have the co-
efficients on factor growth rates being identified with factors’ share of income,
FLL/Y = wL/Y for example. Denoting these shares by si, we can then write
TFP growth as a function of the gap between output growth and observed growth
in factor inputs:
g :=A
A=Y
Y− sK
K
K− sL
L
L
If all income goes to either labour or capital then sL + sK = 1, so calling st the
labour share at time t, we can write a discrete-time approximation to g:
g ≈ ∆ lnYt − (1 − st)∆ lnKt − st∆ lnLt (19)
16 Economic Growth
where “∆” means ‘the change in’. Given observersations on shares and effective
inputs, we can calculate g, which is often referred to as the ‘Solow residual’.
We can recover the (normalised) level of TFP from the growth rate by ‘integrat-
ing’ the differenced series, as in Figure 4, which gives 40 years of data from the first
half of the C20th. There was steady growth in productivity over the period, with
output growth uncorrelated to capital. Given that y = Af(k), it is evident that a
plot of y/A against k gives us f , i.e. it removes the shift factor leaving just move-
ments along the production function. This is shown in Figure 6. There is evidence
Figure 6: Solow’s Aggregate Production Function
2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3
0.64
0.66
0.68
0.70
0.72
0.74
Source: Solow (1957, Table I). Abscissa: capital stock per man hour. Ordinate: Private non-farmGNP per man hour divided by TFP. The points lying above the main cluster correspond to theyears 1943–49.
there for diminishing returns, first. Second, a Cobb-Douglas production function is
not a bad fit, with the slope of an OLS line through these points being .35, close
to capital’s average share (.34). Third, over the 1909-1949 period, technical change
was approximately Hicks-neutral.
Solow was aware that technical change could take other forms. In general, if we
Roland Meeks 17
have:
Y (t) = A(t)F [K(t), B(t)L(t)]
then the ‘Solow residual’ will be:
R(t) := gA +FBB
YgB = gY −
FKK
YgK −
FLL
YgL
The validity of all this depends on whether inputs be accurately measured. First,
different capital and labour inputs have to be adjusted for quality, as discussed in
Jorgenson and Griliches (1967). The idea is to categorise inputs and then weight
each category by its payments (so wages in the case of labour, and rental rates
for capital, taken as a proxy for their marginal products). Labour quality might
improve as a result of better education or health, and that results in more output
for a given man-hour of work with the same capital stock. That increase would be
attributed to TFP if a quality adjustment is not made. Second, firms may vary
the rate at which they utilize inputs. This matters because we are interested in the
service flows coming from factor inputs, and by assuming that the flow is propor-
tional to the stock, we could be making an error in our estimation of their input to
production8. For example, utilization of equipment is likely to vary cyclically (this
has been approximated by energy use). Similarly, firms may hoard labour if there
are costs to varying the size of the workforce, or non-convexities in the work-leisure
choice (as seems clear). During busy times, man-hours at work are spent more in-
tensively, and in slow times, workers are kept on 9-5 even if there isn’t much to do.
These considerations lead to the probable conclusion that actual inputs are a good
deal more procyclical than their stocks (Burnside and Eichenbaum, 1996). A final
addition is that if firms have market power, they are able to compensate factors at
less than their marginal products (we will talk about this again much later) [see
Hall (1988)]. The conclusion of this work is that the contribution of TFP growth
to output growth, which Solow put at around 7/8th, is far lower once these adjust-
ments are made (see Table 1). An immediate point is that TFP growth has been
slower in the post-1960 period than in the immediate pre-war epoch (I don’t know
why non-overlapping dates weren’t chosen)9.
8Solow (1957, fn. 3) recognises this.9The latest edition of the Barro and Sala-i-Martin book (2004) has a whole chapter on growth
accounting; the previous edition (1995) has a part chapter.
18 Economic Growth
Table 1: Growth accounting
Country gY gK gL gA
1947–73France .0542 .0225 .0021 .0296
(42%) (4%) (54%)Germany .0661 .0269 .0018 .0374
(41%) (3%) (56%)U.K. .0373 .0176 .0003 .0193
(47%) (1%) (52%)U.S. .0402 .0171 .0095 .0135
(43%) (24%) (34%)
1960–95France .0358 .0180 .0033 .0130
(53%) (10%) (38%)Germany .0312 .0177 .0014 .0132
(56%) (4%) (42%)U.K. .0221 .0124 .0017 .0080
(56%) (8%) (36%)U.S. .0318 .0117 .0127 .0076
(37%) (40%) (24%)
Source: Barro and Sala-i-Martin (2004, p. 439)
A final and quite subtle point, made by Helpman (2004, pp. 25–8), Barro and
Sala-i-Martin (2004, ch. 10.5) and Romer (2005, p. 31), is that finding a high contri-
bution of TFP in overall growth says nothing about the causes of economic growth.
For example, high investment rates are at least in part a response to TFP growth,
since TFP growth makes investment more profitable. It therefore induces capital
accumulation, and ‘as a result, fast accumulation of capital is often a reflection of
high TFP, or an expected high rate of productivity growth.’ Of course, to establish
causal links you need a model. What you should take away from this is that TFP
increases are a proximate cause of growth, likely to be endogenous, and so growth
accounting is a first step in identifying the causes of productivity change.
Roland Meeks 19
2.5 Decentralised economy
You might be wondering about what exactly is going on inside the Solow economy.
Implicit in the story we told is the idea of a decentralised competitive economy,
in which there are competitive factor markets. Competition means capital and
labour are paid their marginal products. Let’s assume that technical progress is
of the Harrod form, so that aggregate output is given by F (K,AL). Rescaling all
variables by efficiency units of labour we have that the intensive production function
is f(k), from which factor payments are:
r =∂F
∂K:=
∂
∂K[ALf(k)] = ALf ′(k)
1
AL= f ′(k) (20)
w =∂F
∂L:=
∂
∂L[ALf(k)] = Af(k) − ALf ′(k)
K
AL2= A[f(k) − f ′(k)k] (21)
(repeated use of linear homogeneity). Under our assumptions, factor payments
exhaust national income:
rK + wL = f ′(k)K + A[f(k) − f ′(k)k]L = ALf(k) := F (K,AL) (22)
Stylised fact K3 said that the return on capital is stable; we also know that real
wage growth (the return to labour) shows secular increase. Supposing the economy
is growing in the steady state, we can take total differentials of (20) and (21) to find:
dr = f ′′(k) dk (23)
dw = [f(k) − f ′(k)k] dA+ Af ′(k) dk − A[f ′′(k) + f ′(k)] dk
= [simplify] = wdA
A− Af ′′(k) dk
= [defn of dr] = wdA
A−Adr (24)
or in terms of growth rates:
gr :=r
r=
f ′′(k)k
f ′(k)
k
k:=
f ′′(k)k
f ′(k)gk (25)
gw :=w
w=
A
A− A
r
w
r
r:= gA − A
r
wgr (26)
This means that the return on capital grows in proportion to the growth in capital
intensity. On the balanced growth path, there is no change in capital intensity, so
20 Economic Growth
the return on capital is stable. Wage growth is then simply equal to productivity
growth, which is what we set out to show.
Question: As k approaches the steady state from below, what happens to the
return on capital? What happens to wages?
2.6 Normative aspects of growth
Solow’s approach was to ask how fast the economy would grow, given the primatives
of technology growth, technical possibilities and the rate of population increase.
Phelps (1961) asked instead, what is the most desirable long run savings rate?10
Phelps places certain constraints on the problem. First, he is interested only
in choosing amongst constant savings rates. The analytical framework is that of
Solow, with the economy taken to be on a balanced growth path; capital and output
therefore grow at the same exponential rate, their ratio is constant, and there are
no transition dynamics. Secondly, the economy grows at its natural rate, the rate
which depends upon values taken by technical parameters, but not upon the choice
of savings rate.
The first result is that each generation would want to choose the same savings
rate. Because the savings rate is fixed, consumption paths never cross. The result
embodies the notion of reciprocity inherent in the Golden Rule of ethics, in that all
generations are treated equally: hence the term Golden Rule of capital accumulation.
The second result, which states the condition for an economy to be following the
Golden Rule, is that the savings rate should be set equal to the share of capital,
also known as the profit rate. An equivalent way of stating this, which I ask you
to derive below, is that the return on capital or interest rate be set equal to the
economy’s rate of growth. One way to understand this is the following. Suppose
growth in output comes from population growth at rate n. The marginal unit of
consumption sacrificed today for investment is repaid tomorrow according to the
interest rate r; on the other hand, the amount of capital per head is being eroded
by population growth. This is a cost to holding capital. The total return to holding
capital r−n, for people with no time preference should be zero. If it’s negative, you
could increase future consumption by saving less. If it’s positive, you could increase
10Edmund S. Phelps is the 2006 Nobel Prize winner. For links to his citation, and that of Kuznetsand Solow, see my teaching page http://www.nuff.ox.ac.uk/users/meeks/teaching.htm.
Roland Meeks 21
future consumption by saving more. So you should choose the savings rate so that
total returns are zero: r = n. (When people discount the future, meaning that
their preferences weight consumption in future time periods lower than consumption
today, we will see that the Golden Rule needs to be modified accordingly).
When the Golden Rule is followed, said Phelps, ‘each generation invests on behalf
of future generations that share of income which, subject to [the law of motion for
capital], it would have had past generations invest on behalf of it.’ Later, the condi-
tions for what came to be known as dynamic efficiency were elucidated more fully.
An economy is dynamically efficient if it is impossible to increase the consumption
of future generations except by reducing the consumption of the current generation.
This is the situation of undersaving. Dynamic inefficiency is the opposite situa-
tion of oversaving, whereby all generations would benefit from a reduction in the
savings rate and corresponding increase in consumption. What to do in the first
case is a purely normative question, and was addressed in the context of Rawlsian
(antiutilitarian) justice by Phelps and Riley (1978).
Question: Derive the alternative version of the Golden rule, that the interest rate
(net of depreciation if you include it in your model) should equal the growth rate of
output.
22 Economic Growth
3 Convex growth model with optimisation
Mathematicians, leading the quest for a growth strategy, grappled with
extremals, functionals and Hamiltonians.
Yet nothing practicable emerged.
– Phelps’ Fable for growthmen (1961)
In this lecture we will examine the growth pattern experienced by a population that
optimises its allocation of resources optimally over time. In Solow (1956), the pro-
portion of saving out of current income s is a constant, which at the same time means
that the proportion consumed is also a constant 1−s. But research on consumption,
which is covered in detail by John Muellbauer later in the course, emphasises the
intertemporal nature of the consumption/savings decision. An integrated treatment
of consumption, saving and growth is the goal for this section. What we get over and
above Solow’s treatment is a better look at how economic incentives affect growth,
and a way to think about welfare. Since Phelps’s tongue-in-cheek remark at the
expense of intertemporal optimisation, many practical problems have in fact been
solved, and the tools have gained widespread use.
Usually, the formal analysis of this topic is given in continuous time. There
are many good textbook treatments of this, of which you could pick one or two to
look at: Aghion and Howitt (1998, §1.2), Barro and Sala-i-Martin (1995/2004, ch. 2),
Blanchard and Fischer (1989, ch. 2), Lucas (2002, ch. 1.2) and Romer (1996, ch. 2A).
We can also do the analysis in discrete time. This might seem like unneccesary
repetition, but it will be helpful to practice the analysis in the first instance; second,
under some special assumptions analytic results are available; third, there will be
a bridge to Florin Bilbiie’s lectures on the RBC model. There are fewer textbook
treatments in this case, but Sargent (1987, pp. 24–27) and Ljungqvist and Sargent
(2004, ch. 14) should help, as should the survey article by McCallum (1996, §§1–3).
Throughout, we will assume there is no finite horizon to economic decisions made
by agents; Christopher Bliss will lecture on overlapping generations models later in
the term.
Roland Meeks 23
3.1 Cass-Koopmans-Ramsey
Agents choose their savings and consumption in response to the returns to
capital. Low levels of capital mean high returns, so agents postpone consump-
tion and accumulate capital. Under standard assumptions, the economy is
saddle path stable.
3.1.1 Continuous time
Consider the case of a competitive economy where measured time t ∈ R+. Take the
economy to be populated by many identical families of constant size, whose welfare
depends on household consumption c according to
U0 =
∫
∞
0
e−ρtu[c(t)]dt (27)
where u is known as the instantaneous utility function11. The coefficient ρ > 0 cap-
tures the rate of time preference, again assumed common across households12. The
individual household owns some amount of capital k, and supplies labour services
inelatically (leisure does not appear in the utility function, notice). Output is given
by the CRS production function
y(t) = f [k(t)]
and on the expenditure side, it has consumption and gross investment (additions to
the stock of capital):
c(t) + k(t) + δk(t) = y(t) (28)
(everything in per capita terms). We are going to solve this maximisation problem
using optimal control theory.
To solve the maximisation problem, the most straightforward method is to ap-
ply the results you have learned about the maximum principle (MP). Recall the
Hamiltonian H(c, k, µ, t) as the sum of the integrand function u and the product of
a costate variable µ and the equation of motion for k:
H = e−ρtu(c) + µ[f(k) − δk − c]
11This formulation, which is equivalent to ‘additive separability’ in discrete time, already imposessubstantial structure on preferences; see Deaton (1992, ch. 1).
12Exponential discounting is also a special assumption, used because it is one way of ensuringthe time consistency of decisions; see Blanchard and Fischer (1989, ch. 2, fn. 4 and pp. 70–72)
24 Economic Growth
or in current value terms
H = u(c) + λ[f(k) − δk − c] (29)
where λ := eρtµ. The MP says choose controls to maxc H for all t. It’s worth
articulating the assumptions we are going to make before trying to find the maximum
of this function. (a) The control variable in this case is consumption. The control
set defines the feasible choices of c at each point in time. If at any time we hit
the boundaries of this control set (assuming it’s a compact set), then the usual first
order conditions won’t hold. In our case, consumption is bounded below by zero,
and above by the entire stock of undepreciated capital plus output. So we’ll need
to think about whether, along the optimal path, we will hit the boundaries or not
(it turns out not). (b) Sufficiency. You will recall that for the first order conditions
for a maximum of a function in one variable to be sufficient, we require concavity.
Similarly here, we require the functions u(c) and f(k)−c−δk to be jointly concave in
c, k [i.e. the Hessian of each function needs to be negative semi-definite; Simon and
Blume (1994, ch. 17.3)]. The objective function is concave in c by assumption, and
doesn’t depend on k; the constraint is linear, and therefore automatically concave,
in c, and concave also in k by the assumed concavity of the production function.
More details are in Chiang (1992, ch. 8).
The first order necessary conditions for the problem maxc H are:
Hc = u′(c) − λ = 0 (30)
Hk = λ[f ′(k) − δ] = −λ+ ρλ (31)
Hλ = f(k) − c− δk = k (32)
limt→∞
e−ρtλk = 0 and k(0) = k0 (33)
The first point to note is that (30) says that the costate is equal to the marginal
utility of consumption. The natural interpretation of the costate is that it is the
shadow price of a marginal unit of capital at time t, or generally the increase in
the objective function that results from a marginal relaxation of the resource con-
straint. This condition therefore says a unit of resource should be equally valuable
as consumption or investment. I stress that this is an equilibrium condition, not
an identity. The transversality condition in (33) is the infinite horizon analogue to
the straightforward proposition that, at the end of a planning period, either stocks
should be exhausted or they should have zero value (again, in terms of the objective).
Roland Meeks 25
Let’s now try to characterise the solution in terms of {c, k}. We can eliminate
the costate using (30) by differentiating with respect to time:
λ = [chain rule] = u′′(c)c
whence:
λ
λ=u′′(c)c
u′(c)
The RHS gives the elasticity of marginal utility (Blanchard and Fischer, 1989, p. 40);
if preferences are of the CRRA form defined by u′(c) = c−σ, then:
u′′(c)c
u′(c)= −σ
c
c
(log preferences correspond to σ = 1 in this equation). From (31) using this special-
isation:
c =1
σ[f ′(k) − δ − ρ]c (34)
You should think of this as a condition that has to hold on an optimal path for
c; it is the continuous time version of the consumption Euler equation that you
will see time and again. A verbal explanation is given by Blanchard and Fischer
(1989, pp. 41–3); an alternative in Aghion and Howitt (1998, p. 19). For now,
note that if you invest, rather than consume a unit of resource, the return on that
investment is f ′(k) − δ. Whenever the returns to saving are high, and exceed the
consumer’s rate of time preference ρ, the consumption stream tilts upwards - you
prefer to save now and consume later, and conversely. Consumption growth is less
sensitive to returns when σ - which indexes the desire for smoothness over time -
is high (unsurprisingly)13. People talk about the marginal product of capital and
the interest rate (on any asset) interchangeably, since by arbitrage in a frictionless
setting they must be the same.
[phase diagram]
Note that the CRRA or isoelastic preferences are the only form which permit
balanced growth. The reason is that constant growth is consistent with a constant
elasticity of substitution between time periods, which in turn is consistent with an
invariant rate of return on capital.
13We could also read this equation the other way, so that consumption determines returns; that’sgeneral equilibrium!
26 Economic Growth
An alternative in terms of consumption/investment shares
An occasionally useful rephrasing of the optimisation problem given above involves
us choosing the share of income that goes to consumption or investment. Noting
that gross investment is k + δk, and calling its expenditure share out of current
income s, we have that:
c+ i = f(k)
c+ k + δk = (1 − s)f(k) + sf(k)
You can verify that the first order conditions from the Hamiltonian:
maxs
H = u[(1 − s)f(k)] + λ[sf(k) − δk]
are identical to those above.
3.1.2 Discrete time
The setup is the same as before, except now measured time t ∈ N. Households
maximise the discrete time analogue of (27), namely
U0 =∞∑
t=0
βtu(ct) (35)
where β := 1/(1 + ρ) is known as the discount factor. The budget constraint is now
ct + kt+1 − kt + δkt = f(kt) (36)
and has an obvious parallel to (28). The initial capital given to this economy will
be k0. The discrete time transversality condition is
limt→∞
βtu′(ct)kt = 0
The problem is to maximise (35) by choice of the control variable c. Actually, we
have some flexibility in how the control is chosen. Looking at the budget constraint,
it is clear that choosing consumption is equivalent to choosing next period’s capital
stock. (I will use the notation that x denotes next period’s value of x). The state
variable is the current stock of capital. Given the structure of the problem, and the
definition of the state and control variables, we can write the Bellman equation as:
V (k) = maxk∈[0,k]
{u(c) + βV (k)} (37)
Roland Meeks 27
where maximisation is subject to (36). The object V is the value function for this
problem, that is, the maximum attainable value of the objective given the state
k. The value function for a log-utility Cobb-Douglas model is plotted in Figure 7.
Notice that it is increasing and concave in k. In (37) the value function appears as
an unknown. It appears on both sides of the Bellman equation, but on the right it is
inside the max function. To obtain Figure 7 I had to make a numerical approxima-
tion to V and have the computer solve the maximisation problem. A constructive
method of finding V is provided by theorems due to Bellman and Blackwell, which
say that we can iterate from a reasonable guess for V , and these iterations will
converge geometrically to the optima.
Figure 7: Optimal Value Function
0 1 2 3 4 5 6 7 8 9 10 11 12
10.0
12.5
15.0
17.5
20.0
22.5
25.0Value function
Source: Author’s calculations. Parameter values: α = .33, δ = .05, β = .98, σ = 1.
I can write the value function in the form (37) – without time subscripts on the
value function – because the problem is time invariant, in the sense that shifting the
calendar date forwards or backwards makes no difference to how the problem looks.
If, on the other hand, the problem had a finite horizon, that would not be true.
Notice also that the control set is bounded (meaning bounded above and bounded
28 Economic Growth
below); in the maximisation, the upper limit of the capital stock, k = f(k)+(1−δ)k.
One cannot have a negative capital stock, neither can the capital stock exceed the
amount that would result from zero consumption today. Substitute in for c on the
RHS of (37), and differentiate with respect to k to find the first order condition:
−u′[f(k) + (1 − δ)k − k] + βV ′(k) = 0 or u′(c) = βV ′(k) (38)
The solution to this equation is going to be a policy function that maps from the
state into a setting for the control, which in this case is a function h : k → k. Assume
that this h exists, is unique, and is differentiable. Substituting the maximiser into
the RHS of (37) gives:
V (k) = u[f(k) + (1 − δ)k − h(k)] + βV [h(k)] (39)
Away from corners, the derivative of this function is:
V ′(k) = u′[f(k) + (1 − δ)k − h(k)]{f ′(k) + 1 − δ − h′(k)} + βV ′[h(k)]h′(k)
= [using (38)] = u′(c){f ′(k) + 1 − δ − h′(k)} + u′(c)h′(k)
V ′(k) = u′(c){f ′(k) + 1 − δ} (40)
This result can be found directly, using the Benveniste-Scheinkman formula (see
Sargent, 1987, §1.3). What we have now is the derivative of the value function,
which appears in the first order necessary condition for an optima (38). It holds
every period, so shifting time forward by one period in (40) (I reintroduce time
subscripts) and using this to eliminate V ′ from the RHS of (38):
u′(ct) = βu′(ct+1){f′(kt+1) + 1 − δ} (41)
which is the familiar Euler equation.
The unknowns are the pair of functions V and h. We saw what the value function
looked like for a particular example before, and I have calculated the optimal policy
function h for the same example. It is plotted in Figure 8. Notice that it is very
linear in k in the region of the steady state, which suggests that the usual linear or
log-linear approximations are probably good. In order to get a closed-form solution
to the problem, we’re going to use the guess-and-verify method, and we’re going to
have to make some strong assumptions on functional forms. There are essentially
only two forms for u and f that yield solutions. The one we’ll use is log utility,
Roland Meeks 29
Figure 8: Optimal Policy Function
0 1 2 3 4 5 6 7 8 9 10 11 12
2.5
5.0
7.5
10.0
12.5
kt
kt+1
Optimal kt+1 policy 45o line
0 1 2 3 4 5 6 7 8 9 10 11 12
2.5
5.0
7.5
10.0
12.5
kt
initial guess
Source: Author’s calculations. Panel (a) shows optimal choice of kt+1 as a function of kt. Panel(b) shows iterates of a modified version of Howard’s policy improvement algorithm. See Ljungqvistand Sargent (2004). Parameter values: α = .33, δ = .05, β = .98, σ = 1. Steady state k∗ = 10.03.
Cobb-Douglas with full depreciation (LCD hereafter). The reasonableness of the
last restriction depends on the time period considered. If it is decades, rather than
months, much of the capital stock probably is economically obsolete. So u′(c) = c−1,
δ = 1 and f(k) = Akα. Now suppose:
V (k) = E + F ln k (42)
If this were the value function, then the policy function would be:
h(k) =βF
1 + βFAkα
30 Economic Growth
On eliminating unknown constants, the optimal policy rule turns out to be14:
kt+1 = βαAkαt
which has a unique stable equilibrium when α < 1 (take logs and iterate from any
k0 > 0).
3.2 Growth rates and convergence
There is monotonic convergence to the steady state, but at a rate faster
than predicted by the Solow model. Steady state growth again depends on
exogenous technical progress.
3.2.1 Steady state growth
Let’s take the model of §3.1.1. The standard proceedure for finding the rate of
steady state growth is to just assume that there is a balanced growth path (BGP),
and that one of your variables is growing at some constant rate. You then try to
solve out for what that rate is, and how fast the other variables are growing. In that
spirit, say consumption growth on the BGP of the Ramsey model is g, then (34)
tells us that:
σg + ρ = f ′(k) − δ (43)
It’s apparent that if we are looking for g ≥ 0, then there will come a point when
the net marginal product of capital will fall below the rate of time preference (recall
the consumption tilting argument above). That suggests that there is a stationary
point, in other words, a zero growth equilibrium. In the Cobb-Douglas case it is
easy to invert (43), or by similar logic (41), to find the steady-state capital stock:
k∗ =
(
α
ρ+ δ
)1/(1−α)
This is the capital stock that agents willingly maintain forever, in the steady state.
Contrast this with the capital stock corresponding to the maximum sustainable level
14Substitute the maximiser into the Bellman equation – which now holds with equality, i.e.without the max operator on the RHS – and use the form of the value function given in (42). Onequating coefficients, you can obtain the answer given in the main text.
Roland Meeks 31
of steady state consumption. The latter is found by observing that:
c = f(k) − δk
which achieves a maximum at the point:
k+ =(α
δ
)1/(1−α)
This value of k+ is the ‘golden rule’ capital stock encountered earlier, and is larger
than k∗ due to the positive discount factor (k∗ is known as the ‘modified golden
rule’). By the logic of intertemporal optimality, an economy with k+ could increase
welfare by consuming part of the capital stock immediately. See McCallum (1996,
pp. 48–9) and §2.6 of these notes.
This way of finding steady state growth rates extends to a world in which there is
exogenous labour-augmenting technical change. In that case, one finds that growth
in variables rescaled by effective labour is zero in the steady state. So for example,
if c := C/AL then with zero population growth c/c = C/C − A/A = 0, so the
growth rate of the level of consumption is equal to the rate of technical change. In
endogenous growth models, this exercise produces more interesting results.
3.2.2 Convergence
So far, we have said quite a bit about growth in the steady state, i.e. after all
the convergence dynamics associated with capital accumulation have played out.
We can justify this focus by arguing that, with very long horizons, the economy
spends most of its time on the BGP. On the other hand, the model is probably
not the literal truth over every epoch. In this section, we will see how to calculate
the rate of convergence of a single economy to its steady state. Later, we will see
how empirical work motivated by these findings has found evidence for convergence
between different economies.
We will begin with the Solow (1956) model studied last time. Let’s suppose there
is Harrod neutral technical change so that the fundamental differential equation15
is:
K = sF (K,AL) − δK (44)
15Now in terms of gross output, as in Barro and Sala-i-Martin (2004, ch. 1).
32 Economic Growth
What happens when we divide through by effective labour? Well k := K/AL so we
can figure out K:
k/k = K/K − A/A− L/L = K/K − x− n
Substitute into (44) to find:
k = sf(k) − (x+ n+ δ)k
So now we have the ODE in terms of capital per unit of effective labour when there
is exogenous labour augmenting technological change and population growth (this
will be useful to us in later lectures too). The key technique is nothing more that a
Figure 9: Convergence Speed for the Solow Model
0 0.05 0.1 0.15 0.20.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
Population Growth Rate
Con
verg
ence
Rat
e
Source: Author’s calculations.
first order Taylor approximation; say we expand the RHS around a point k∗, then:
k = [sf ′(k∗) − (x+ n + δ)](k − k∗) + higher order terms (45)
Roland Meeks 33
As this is linear in k, it is straightforward to solve. For example, if f is Cobb-
Douglas, and we approximate at the steady state:
k = const + (α− 1)(x+ n+ δ)k + higher order terms (46)
giving a convergence rate of (1−α)(x+ n+ δ). The empirical work on convergence
we look at in a later lecture makes heavy use of this formula, so make sure you
understand it. What quantitative prediction does the model provide? Mankiw et al.
(1992) pick x+δ = .05, and on the basis of the capital share series we saw in lecture 1,
α ≈ 1/3. Then the relationship between convergence speed and population growth
is shown in Figure 9. What does this mean? Well if population growth is 5%, then
convergence to steady state is about 6% per annum. Say an economy is initially at
1/5 of its steady state income level (roughly the differential between Japan and the
US in 1950), then it will take 27 years to reach the steady state. If the population
is stable, it would take 46 years. In the Ramsey model, we have two interdependent
differential equations, but we can use the same technique of linear approximation.
This is doubly helpful, as we can explicitly solve the system for its local behaviour
[we won’t do that here; see Blanchard and Fischer (1989, p. 47)]. The other sense of
convergence is the one that happens between different economies. This is referred
to as β convergence. We saw that differences in the savings rate affect growth given
the capital stock because of changes in the steady state; also, that differences in the
initial capital stock lead to different growth rates for identical savings rates. This
leads to the conclusion that levels and growth rates can be uncorrelated in cross
sections (this notion is - a lack of - ‘unconditional’ convergence), unless of course
countries have identical steady states. We’ll return to this topic in a later lecture.
N.B. Barro and Sala-i-Martin use a log-linear approximation, which gets you the
same result. This technique is covered by Florin Bilbiie.
Question: Can you derive the convergence rate for output in the Solow (1956)
economy?
Application: King and Rebelo
Part of the methodological contribution made by Sargent, Lucas, Kydland and
Prescott and other scholars to macroeconomics has been the quantification of the-
oretical models. The idea is to construct a kind of laboratory economy, which can
be simulated under different assumptions on how the world might be. Using this
34 Economic Growth
laboratory, a researcher can distinguish between quantitatively important and unim-
portant predictions. We have already done this ourselves (in Figure 9 for example),
but as you can guess there is disagreement between econometricians who want to
see some evidence of fit to data, and theorists who are interested in the robustness
of underlying mechanisms.
In this quantitative theory tradition, King and Rebelo (1993) ask how well the
transition dynamics of the neoclassical model match up to experience. If ‘technology’
is primarily a public good, which is one end of a continuum of possibilities, then
cross country differences in growth rates must be due to their different positions
relative to steady state. The experiment carried out by King and Rebelo is to start
the model economy from a below steady state level of capital, such that half of the
growth over a 100 year period is due to accumulation and half to technology (in a
growth accounting sense). To do this, a model similar to that of §3.1.2 is calibrated
to match the US economy.
With standard values for the capital share, some odd things happen. First,
convergence is very rapid early on, with 3/4 done within 10 years; second, investment
follows a hump-shaped pattern, so that it first overshoots its long-run value then
converges to it from above; third, the real interest rate (return on capital) initially
exceeds 100% per annum. Of course, these observations are related. It can easily
be seen that the high rates of return in the low capital state come from the strong
curvature of the production function close to zero, and fast convergence follows from
this incentive. King and Rebelo show that this message is preserved even when
agents are very unwilling to intertemporally substitute, and when vintage capital
effects or adjustment costs are taken into account. The possibility of international
capital flows only exacerbate the problem, as return differentials between rich and
poor countries in the simplest neoclassical story are so high that investors would be
overwhelming them with investment. To get a feel for the likely range of variation
in returns, Table 2 gives some summary statistics for the U.S.; the model predicts
numbers that are orders of magnitude too large. In Figure 10, the convergence
paths for various values of the capital share are plotted; higher capital shares mean
a more nearly linear production function, and therefore lower returns and slower
convergence.
How could high capital shares, which imply more reasonable convergence dy-
namics, be justified? One answer, which becomes a theme in much of the later work
Roland Meeks 35
Table 2: Annual Real Rates of Return, U.S. 1926–1987
Series Average Std. Err.Common stocks 6.65 0.40Corporate bonds 1.83 0.17Treasuries 0.42 0.04Long-term govt 1.18 0.23
Source: King and Rebelo (1993, Table 3)
we look at, is given by Barro et al. (1995). They suggest a ‘broad’ capital defini-
tion that includes human capital. This has the added advantage of answering the
near-fatal problem of convergence in the small open economy as well. By allowing
physical capital to be collateralisable, but human capital not, they prevent broad
capital being instantly financed by borrowing abroad, and a ‘jump’ to its steady
state level. They find reasonable convergence rates for both types of capital in the
extended model.
36 Economic Growth
Figure 10: Convergence in the Cass-Koopmans-Ramsey Model
Source: King and Rebelo (1993, p. 919).
Roland Meeks 37
4 First generation endogenous growth
4.1 The AK model
A model with convex production technology can produce steady state growth
without exogenous technical progress. A necessary condition is for the mar-
ginal product of capital to remain positive and greater than the inverse of
the discount rate as the quantity of capital increases, since there is always an
incentive to defer some consumption in favour of continued investment.
So far we’ve studied models for which the accumulation of capital was the main en-
gine of growth in per capita income, in the absence of exogenous technical change.
Eventually, diminshing returns mean that growth peters out. The reason is that
capital deepening eventually causes the interest rate to fall below the (inverse of)
the discount rate. As we saw in §3.1.1 of these notes, the growth rate of consump-
tion (I refered to the ‘tilt’ or ‘slope’ of the consumption path) depends upon the
economy’s net rate of return. Returns eventually fall to the point where people are
indifferent between consumption today or tomorrow. That was the consequence of
the Inada conditions, which in the Solow model had the effect of guaranteeing that
an equilibrium capital intensity existed. If you turn this on its head, evidently by
preventing marginal productivity falling too far, you can dispense with the steady
state too, and this turns out to be precisely what we need to explain never-ending
balanced growth.
This idea was explored by Jones and Manuelli (1990). Consider a representative
consumer with preferences U0 =∑
∞
t=0 βtu(ct). In this economy, resources are either
consumed or invested in physical capital, so the period budget constraint she faces
is:
ct + kt+1 − (1 − δ)kt = f(kt)
where u and f are concave and twice differentiable. So far so standard. The as-
sumption (Inada condition) that limk→∞ f ′(kt) = 0 is normally made to ensure the
boundedness (above) of kt, and therefore the absence of long-run increases in per-
capita income. Jones and Manuelli note that this condition is not necessary for
the concavity of f , and that it is sufficient to assume that marginal productivity is
bounded below by a positive number. The simplest example of such a production
38 Economic Growth
function is f(k) = bk (their notation, but you can see why such a model gets called
‘AK’; but be careful, there is no productivity growth of any kind in this model).
Jones and Manuelli have to prove first that a solution exists, and second that
there is endogenous growth. I explain in a footnote the conditions for existence,
which you may skip. If:
β(b+ 1 − δ) > 1 (Condition G)
then the economy displays long run growth. The term in parentheses is the return
on capital, so this condition says loosely that the values taken by the technical
parameters of the economy are such that investment in capital is always attractive.
To find the necessary conditions for an optimum of the problem posed above,
form the Lagrangean16:
L =
∞∑
t=0
βt{u(ct) + λt(bkt + [1 − δ]kt − ct − kt+1)}
16To guarantee the maximisation problem is well-defined, lifetime utility needs to be finite. Thisproperty holds if the growth of period utility can’t be too fast. With CRRA preferences, theexistence of the infinite sum evaluated along some consumption path {c∗
t}∞
t=0 can be establishedas follows. Write lifetime utility as:
U =∞∑
t=0
βtc∗(1−σ)t
− 1
1 − σ
Suppose the growth rate of consumption θ − 1 is constant on the path, with c∗0 < ∞. Write thepart of the sum which depends upon t as follows:
ST =
T∑
t=0
βt[c∗0θt]1−σ = c
∗(1−σ)0
1 − (βθ1−σ)T
1 − βθ1−σ
If |βθ1−σ| < 1 then:
limT→∞
βT θ(1−σ)T = 0 and hence
limT→∞
ST = c∗(1−σ)0
1
1 − βθ1−σ< ∞
What kind of magnitudes are we talking about? If we pick β = 0.96 and σ = 1/2, then themaximum growth rate of consumption is 0.082. A more usual value in the literature is σ = 2,which gives a minimum growth rate of −0.039. But the key point is that the growth of period
utility does not exceed 1/β, as this places a bound on welfare effects of growth. It’s possible,although not easy, to choose among paths which yield nonfinite objectives (Leonard and Long,1992, ch. 9).
Roland Meeks 39
Differentiating the sum, we find first order conditions that hold at every t:
∂L
∂ct= u′(ct) − λt = 0
∂L
∂kt+1
= −λt + βλt+1(b+ 1 − δ) = 0
making the consumption Euler equation (CRRA case):
θt :=ct+1
ct= {β[b+ 1 − δ]}1/σ (47)
You can see that the solution displays the property of positive long-run growth
when Condition G holds, without the need for exogenous technical change. The
significance of the existence condition which limits period utility growth is that
in spite of sustained growth, there are not necessarily huge welfare consequences to
policies that affect the growth rate. Principal amongst these are taxation rates which
impact the rate of return on capital. International differences in capital taxation
lead in this model to permanently different growth rates. Jones and Manuelli point
out that this means that it’s not necessarily the case that ‘growth’ effects are more
important than ‘levels’ effects in their model.
The other thing to notice about (47) is that on the optimal path, as defined by
the Euler equation, θt is a constant; so we must always be on the balanced growth
path. Crucially, growth does not depend upon the stock of capital. Steady state
growth now depends upon the productivity of capital (+), depreciation (–), and
the elasticity of intertemporal substitution (+), whereas before it was independent
of the economy’s ‘deep’ parameters. A second important lesson from this paper is
that long-run growth is possible in the presence of fixed factors; what we need is
that some accumulable factor is produced with constant returns (Ljungqvist and
Sargent, 2004, ch. 14.7). For discussion of a continuous time version, see Barro and
Sala-i-Martin (2004, ch. 4).
A problem with this story is that with fixed labour supply, all income goes to
capital. Jones and Manuelli suggest adding human capital as a second capital good,
and calibrating the model in such a way that its share does not fall to zero. What
exactly human capital might mean will be taken up below, but first we will look at a
class of endogenous growth models that break decreasing returns through a second
important mechanism: externalities.
Question: What’s the growth rate in the Jones and Manuelli setup if u(c) = −e−νc?
40 Economic Growth
4.2 Knowledge externalities
Knowledge is fundamentally different from physical capital as it has some
public good characteristics. Production of new knowledge has spillover effects
that can drive long run growth. Because the market does not internalise these
spillovers, growth in a decentralised economy is not Pareto optimal.
If the marginal product of knowledge were truly diminishing, this would
imply that Newton, Darwin, and their contemporaries mined the rich-
est vein of ideas and that scientists now must sift through the tailings
and extract ideas from low-grade ore. That knowledge has an important
public-good characteristic is generally recognised. That the production
of new knowledge exhibits some form of diminishing marginal produc-
tivity should not be controversial.
– Romer (1986, p. 1020)
You may have noticed that governments subsidise the acquisition of new knowledge,
exactly through places like Oxford University! In reality, there is a distinction
between basic scientific research, which adds to the public stock of knowledge, and
research undertaken within companies that adds to applied knowledge specific to
particular competancies. Outside the world of chemical engineering, microprocessors
and the like, there are the more mundane innovations that add to productivity in
retail (self service shops for example). There are also what Romer refers to as meta-
ideas, knowledge about the best way to structure the production of new knowledge.
Leading examples are patent and copyright laws, and peer-reviewed government
research grants. Thinking more broadly, changes to the organisational form of the
corporation itself is a type of meta-knowledge, studied in detail by Chandler (1990).
We’ve had a lot to say about physical capital and its role in economic growth
in the past lectures. In this section, we’re going to go to the opposite extreme and
have output depend only upon a stock of ‘knowledge’, and not on physical capital
at all. In this way, we zero in on the role that technology plays in production. The
starting point is that knowledge has some special properties that physical capital
does not possess. It will be helpful at this point to review some definitions:
Excludable good: an owner’s ability to prevent others from using her good without
payment. The degree of excludeability of an idea depends, inter alia, on the strength
of patent protection.
Roland Meeks 41
Rival good: the consumption of a rival good incurs the marginal cost of producing
it. Standard economic goods are excludable and rival. By contrast, once the cost of
producing new knowledge has been incurred, it can be used over without additional
cost.
Public good: A good which is both non-exclusive and non-rival17.
Knowledge is non-rival, and depending upon the institutional structures surround-
ing it may be partly non-excludeable. For example, pure mathematics is a non-rival
good that is also non-excluded. On the other hand, the knowledge in your textbook
is non-rival but partially excludable, thanks to the law of copyright. The Solow ap-
proach to technology was to treat it as a public good, as already commented. The
new dimension added by Romer (1986) was the purposeful production of new knowl-
edge by self-interested agents, which requires a degree of private control. In order
to capture this, he used a Marshallian production externality, so the acquisition of
new knowledge has unintended and unremunerated benefits for others. The most
obvious examples of this are the ability to dismantle or reverse engineer an innova-
tion, and the mobility of researchers between firms. The externality device allowed
Romer to study increasing returns within a competitive (price-taking) setting. This
‘approximation’ was necessary at the time, he says, because ‘methodological and
formal issues had been holding everything up’ (Snowdon and Vane, 2005, p. 682).
We will look into a simplified version of the Romer model here18. The economy
is populated by a large number of firms and households. For simplicity, assume
that the number of each is equal to S, with each firm being run by a household.
The production technology available to an individual firm i combines knowledge ki
with a general economy-wide stock of aggregate knowledge19 K =∫
kidi. Assume
that the production function has the general form F (k,K), and that given the
state of aggregate knowledge, F is concave in the knowledge of a particular firm;
also, F is increasing in K. Given that K is just the sum over the knowledge of
individual firms, it is an externality, or unremunerated external effect. We can
distinguish between private and social returns in this context. The private return
17Intermediate goods that may be partly excludable and/or partly nonrival are referred to asclub goods, e.g. the right to swim lengths in a pool.
18A textbook treatment is given by Aghion and Howitt (1998, ch. 1.4.2) and Barro and Sala-i-Martin (2004, ch. 4)
19Romer also includes a vector of other inputs; perhaps representing capital, land, etc.; whichare in fixed supply. An alternative assumption is that the average rather than aggregate stock ofknowledge matters for production; this eliminates the scale effect, discussed below.
42 Economic Growth
is that which the firm sees as the consequence of its investment decision; the social
returns are those which are the unintended consequence of investment spilling over
to all firms. If the increasing returns were internalised, factors would not be paid
their marginal products, otherwise the sum of the shares of income going to each
factor would exceed one! When firms have market power, they ‘mark up’ their
output, or conversely ‘mark down’ factors, paying them less than their marginal
products (see §6.2 of these notes). By making increasing returns an external effect,
Romer (1986) skirts the problem. He returned to it in his later ‘varieties’ paper.
Romer focuses on the case where there are increasing returns to knowledge from
the social standpoint. We will take the sum of private and social returns to be
unity, or exactly constant. Once again, that means F is linearly homogeneous. This
choice is to stress that non-convex technologies are neither necessary nor sufficient
for endogenous growth, as explained by Jones and Manuelli (1990, p. 1033).
Once production has taken place, the resulting goods can be used for consump-
tion, or transformed into new knowledge. However, knowledge cannot be turned
back into goods, so individually and in aggregate K ≥ 0, and knowledge does not
depreciate over time. Household-firms have the usual preferences, so the Hamil-
tonian and first order conditions for the optimisation problem are:
H = u(ci) + λ[F (ki, K) − ci] (48)
Hc = u′(ci) − λ = 0 (49)
Hk = λF1(ki, K) = −λ + ρλ (50)
Hλ = F (ki, K) − ci = k (51)
plus transversality:
limt→∞
e−ρtλk = 0
Given symmetry (ki = k ∀i) and the identity between firm and population size, the
aggregate quantity of knowledge K = Sk. There will be one initial condition, that
k(0) = k0. Differentiating (49) with respect to time gives us λ/λ = u′′(c)c/u′(c),
and rearranging (50) gives the rate of growth in marginal utility as a function of the
difference between the rate of time preference and returns to knowledge:
λ
λ= ρ− F1(k, Sk)
Roland Meeks 43
It’s now useful to rewrite things a little, so using the property of linear homogeneity:
F (k,K) := kF (1, K/k) := kf(K/k)
By this token:
d
dkF (k,K) :=
d
dkkf(K/k) = f(K/k) + kf ′(K/k)(−K/k2)
= f(S) − Sf ′(S) (52)
which is very interesting from our point of view, as it says that the marginal product
of knowledge does not depend on the level of knowledge, and therefore that the Euler
equation is:
−λ
λ= f(S) − Sf ′(S) − ρ
If we specialise to CRRA utility, then:
c
c=
1
σ[f(S) − Sf ′(S) − ρ] (53)
and the law of motion for knowledge (51) becomes:
k
k= f(S) −
c
k(54)
This is a useful pair of equations since the growth rate of c doesn’t depend on k,
and that will make it easier to find a solution.
So what can we get out of this? The difficulty of finding a complete solution (i.e.
{c∗(t)}∞t=0) to general versions of these problems means that the usual procedure
would be to assume that there is a balanced growth path on which c/c = g, then to
find what g is in terms of the parameters of the model (Aghion and Howitt, 1998,
pp.27–8). This works if we specialise to a CRS Cobb-Douglas with F (k,K) = kνKγ,
which implies:
g =1
σ(νSγ − ρ) (55)
From (54) that must be the growth rate of knowledge too (or else c/k would change
on the BGP). First, it’s good that the long term growth rate depends on the fun-
damental features of the economy, such as the rate of time preference. That didn’t
44 Economic Growth
happen in the neoclassical model. Second, many researchers have commented on
the fact that growth is increasing in the scale of the economy S; in other words, the
more firms/households there are, the more the economy benefits from individuals’
spillover effects. That has the immediate implication that integrating markets, for
example through increased trade, has the potential to raise growth rates. That is
potentially highly significant, as there is a belief stemming from empirical research
and from case studies that economic integration is beneficial. For an extensive the-
oretical treatment see Grossman and Helpman (1991a).
What about the transition to the BGP? It turns out that there is no transition;
the economy jumps right onto the BGP and stays there. You can show this in three
steps (Barro and Sala-i-Martin, 2004, ch. 4.1.4): (i) use (53) to find the time path
of consumption; (ii) substitute into (54) and solve for k; (iii) use the transversality
condition find the unique optimal path given initial knowledge. In fact, we already
have enough information to draw a phase diagram.
To partition up the phase space, we set each of (54) and (53) to zero. In the first
case,
k = 0 =⇒ c = Sγk (56)
which is a positive linear function of k. For a given k, any c above this line results
in knowledge falling (something we ruled out in fact, although we didn’t deal with it
formally), whereas c below the line results in knowledge rising. In the second case:
c = 0 =⇒1
σ[νSγ − ρ]c = 0 (57)
so the term inside brackets could only ever be zero by a chance combination of
parameter values, so it must be that c = 0. Everywhere above the horizontal axis,
c is rising, so long as the (constant) marginal product of knowledge exceeds the
discount rate (assume it does). The stable manifold/saddle path must therefore
be described by a line sandwiched between the horizonal axis and the nullcline for
knowledge.
[figure]
This is endogenous growth, and it happens because growth in the public stock
of knowledge can outweigh diminishing private returns. In our case, the spillovers
exactly offset these diminishing returns, a feature which has drawn some criticism
because it restricts the region of the parameter space in which balanced endogenous
growth can take place to a measure zero subset (McCallum, 1996).
Roland Meeks 45
4.2.1 Pareto optimality
An issue to be aware of here is that the competitive equilibria of economies with
unremunerated external effects are, perhaps unsurprisingly, not Pareto optimal. To
show this, we can figure out what the hypothetical planner’s solution looks like. Her
problem is to maximise the same objective functional as the agent, but now instead
of taking K as fixed, we impose K = Sk from the outset. The consequence of this
is that output once again has the ‘AK’ form. Thus:
H = u(ci) + λ[f(S)k − ci]
Hc = u′(ci) − λ = 0
Hk = λ[f(S)] = −λ + ρλ
Using our assumptions, consumption growth is:
gp =1
σ(Sγ − ρ)
Compared with (55), growth is higher, since ν < 1; the average product of knowledge
and the marginal social product are equal (constant returns), and exceed the private
marginal product (52). So by internalising the spillover, the planner correctly sets
growth according to the former rather than the latter.
4.3 Human capital
When all factors of production are reproduceable, endogenous growth can
result. Human capital may be accumulated by people making purposeful
investment decisions. Spillovers in the acquision of human capital increase
the growth rate of the economy, but lead market outcomes to be inefficient.
I will begin by considering an alternative, or at least a complementary,
engine of growth to the ‘technological change’ that serves this purpose in
the Solow model ... what Schultz and Becker call ‘human capital’. The
theory of human capital focuses on the fact that the way an individual
allocates his time over various activities in the current period affects his
productivity ... in future periods.
– Lucas (1988, p. 17)
46 Economic Growth
The idea of this section is to look at a model in which there are two accumulable
factors – physical and human capital – one of which is produced with a constant
returns technology, and to show that this results in endogenous growth. Whereas
disembodied knowledge as conceived by Romer does not depreciate, human capital
does not last forever, most obviously because lives are finite. That suggests that
human capital cannot be an engine for permanent growth. As we see in §5.1, years
of schooling are often used as an empirical proxy for human capital. On the other
hand, the model we study in this section conceives of human capital as a stock of
worker skills that can increase without limit as the product of societal activity. (It
is then unclear as to how this concept differs from knowledge, as both are to some
extent disembodied.) It turns out that endogenous growth is possible because all
factors are reproduceable. This property is sufficient, but not necessary, for
endogenous growth [see also Jones and Manuelli (1990)].
The main focus is Lucas’ highly influential paper, which is reproduced in the
book Lucas (2002). A textbook treatment is given in continuous time by Barro and
Sala-i-Martin (2004, ch. 5.2.2) and discrete time by Ljungqvist and Sargent (2004,
ch. 14.5.1–2). An account of the human capital topic in more detail can be found
in Aghion and Howitt (1998, ch. 10).
4.3.1 Lucas (1988)
Lucas identifies human capital with skill or worker ‘effectiveness’. This means that
a worker with human capital h(t) is half as productive as one with human capital
2h(t), so h is an index on [0,∞). He wants us to think of h as loosely the human
capital of an infinitely-lived family, so that h can be accumulated without bound.
He argues ‘that human capital is a social activity, involving groups of people in a
way that has no counterpart in the accumulation of physical capital’ (p. 19).
The acquisition of human capital depends upon the purposeful investment of
scarce resources, in this case time, plus existing human capital20. Each worker
budgets for a fraction u of non-leisure time to be spent in the production of final
output, and a fraction 1−u to be spent in the acquisition of new skills. If there are N
workers in total, then the number at each skill level is N(h), and the skill-weighted
20Notice that the fixed time endowment introduces a nonreproduceable factor. However, themultiplicative structure assumed means the effect is trivial.
Roland Meeks 47
man hours in production is:
N e =
∫
∞
0
u(h)N(h)h dh = [symmetry] = uhN (58)
The technology by which human capital is produced is:
h(t) = h(t)δ[1 − u(t)] (59)
where δ is a parameter of controlling how the rate of growth of the LHS depends on
time spent in skill acquisition; if all time is devoted to skill acquisition, the fastest
rate at which skills will grow is δ. There are no diminishing returns to h, notice.
There is a second component to human capital accumulation that Lucas wants
to capture, which along the same lines as Romer is the idea that there is an external
or spillover effect. In contrast to Romer’s assumption that the sum of all knowledge
is the source of the external effect, Lucas assumes that the average level of skill
matters, but you can see that it’s in the same spirit:
ha =
∫
∞
0hN(h)dh
∫
∞
0N(h)dh
= [symmetry] = h (60)
Again, the individual is assumed to be sufficiently small that she does not take the
external effect of her decisions into account when making her plan. It turns out that
the spillover isn’t needed to generate long run growth, but we will be interested in
welfare consequences, so I’ll leave it in.
Final (net) aggregate output is produced by the technology:
Y (t) = AK(t)β[u(t)h(t)N(t)]1−βha(t)γ (61)
and the aggregate resource constraint is:
N(t)c(t) + K(t) = Y (t) (62)
where c is per capita consumption. Unlike Lucas, I’ll take population to be constant
(it’s a trivial extension to make it variable).
We will now find the growth rate of this economy along the balanced growth path
[for transition dynamics see Barro and Sala-i-Martin (2004, ch. 5.2.2)]. When an
externality is present (γ > 0) the competitive equilibrium and the planner’s solution
will differ, as in the Romer model. There are two controls, c and u, and two states,
48 Economic Growth
K and h; notice that choice of u is intratemporal (better ‘instantaneous’), so I would
think of this as being chosen by reference to the current states and shadow prices
at each point in time. In forming the Hamiltonian, we will need to introduce two
costates, one for the state variable K denoted θ1, and one for the state variable h,
denoted θ2. Formally, the planner’s Hamiltonian and FOCs are given by:
Hp =N
1 − σ(c1−σ − 1) + θ1[AK
β(uNh)1−βhγ −Nc] + θ2δh(1 − u) (63)
Hpc = Nc−σ −Nθ1 = 0 (64)
Hpu = θ1(1 − β)u−1Y − θ2δh = 0 (65)
Hpk = θ1βK
−1Y = −θ1 + ρθ1 (66)
Hph = θ1(1 − β + γ)h−1Y + θ2δ(1 − u) = −θ2 + ρθ2 (67)
In a competitive equilibrium, replace (67) with:
Hceh = θ1(1 − β)h−1Y + θ2δ(1 − u) = −θ2 + ρθ2 (68)
To find the rate of growth of this economy on the balanced growth path, let κ = c/c;
then using (64) in (66) gives Lucas’s (19):
βAKβ−1(uhN)1−βhγ = ρ+ σκ (69)
We know what the growth rate of h is given by (59), which we will label ν. Differ-
entiating the last equation with respect to time gives:
(β − 1)K
K+ (1 − β)
u
u+ (1 − β + γ)
h
h= 0 (70)
Since u is bounded between zero and unity, it must not grow in steady state. So the
growth rate of consumption per head is:
κ =
(
1 − β + γ
1 − β
)
ν (71)
which is also the growth rate of the capital stock since c/K is constant. Therefore
the equilibrium value of u ultimately determines the growth rate of this economy,
via (59), its effect on the growth rate of human capital.
To find this growth rate, differentiate (64) and (65) with respect to time to
obtain the usual expression for θ1 in terms of the elasticity of marginal utility, and
Roland Meeks 49
the expression for θ2:
θ1θ1
+ βK
K− β
u
u+ (1 − β + γ)
h
h=θ2θ2
+h
h
θ2θ2
= −σκ+ βκ− (β − γ)ν (72)
On the optimal path, the central planner sets from (67):
(
θ2θ2
)p
= ρ−θ1θ2
(1 − β + γ)h−1Y − δ(1 − u) (73)
and agents in the competitive equilibrium set from (68):
(
θ2θ2
)ce
= ρ−θ1θ2
(1 − β)h−1Y − δ(1 − u) (74)
Since the ratio of the shadow price of physical to human capital is in both instances:
θ1θ2
=δh
(1 − β)u−1Y(75)
the Y ’s cancel, and we can write u in terms of ν to get:(
θ2θ2
)p
= ρ− (δ − ν)γ
1 − β− δ (76)
(
θ2θ2
)ce
= ρ− δ (77)
Equate each of these to (72) and solve for ν:
νp = σ−1
[
δ −1 − β
1 − β + γρ
]
(78)
νce =(1 − β)(δ − ρ)
σ(1 − β + γ) − γ(79)
Lucas calls the former the efficient rate of human capital growth [his (24) and (26)].
The two are equal in the absence of an external effect of human capital acquisition
(with σ = 1), as expected, and the growth rate of consumption is equal to ν; with
the externality, consumption growth is faster. In both cases, there is no tendency
for growth to peter out.
50 Economic Growth
4.3.2 Generalisations of Lucas
The Lucas model generalised the ideas of Romer (1986) by having the production
of broadly defined ‘knowledge’ or human capital take place in a different sector to
the production of physical goods for consumption or additions to capital. But it’s
special because the production of human capital requires no physical capital. In the
absence of spillovers, we saw that both types of capital grow at the same rate, and
that makes sense in a constant returns world because diminishing returns to one or
the other factor cannot set in when their relative intensities remain the same. A
generalisation followed in the literature was to allow for different returns to scale,
and different factor intensities in the production of each type of capital. It turns
out that the only specification for which both endogenous growth and stable factor
intensities are possible is constant returns in both sectors. It is possible for human
capital not to be useful in the production of physical capital, which since its demand
was instrumental (in the sense that it is useful as an intermediate step to getting
consumption goods) means that no resources will be put into acquiring skills.
4.3.3 A tractable two-sector model
At the cost of removing some of the subtleties of variable production technologies
for physical and human capital, one can obtain some simple results from a model
similar to Lucas’. Again, we will treat all factors as reproduceable, but now there is
a single production technology. Suppose that physical and human capital obey the
following laws of motion:
k = sky − δk
h = shy − δh
where the shares of income going to gross investment in each type of capital are
denoted sk and sh repectively. Output is a Cobb-Douglas combination of capital
types y = Akαh1−α, where we will take A to be a constant scale factor. There is a
representative agent who wishes to maximise the standard functional subject to the
adding up restriction c = (1 − sk − sh)y. The Hamiltonian for this problem is:
H = u[(1 − sk − sh)Akαh1−α] + λ(skAkαh1−α − δk) + µ(shAkαh1−α − δh)
Roland Meeks 51
from which you can deduce the first order conditions for the shares imply:
u′(c) = λ
u′(c) = µ
the shadow prices of each type of capital are identical, and equal to the marginal
utility of consumption. The first order conditions with respect to physical capital
gives:
λ[
(1 − sk − sh)αy
k+ skα
y
k− δ]
+ µshαy
k= −λ + ρλ
into which you can substitute the restriction on the shadow prices to obtain:
λ(
αy
k− δ)
= −λ + ρλ
with a similar Euler equation for human capital:
λ[
(1 − α)y
h− δ]
= −λ+ ρλ
Returns on the two types of asset must be equal. By equating the LHS of each of
these conditions, one obtains the result that the ratio of human to physical capital
is a constant [compare to (75)]:
h
k=
1 − α
α:= ψ (80)
One implication of this is that the expenditure shares must be related by ψsk = sh
by (80) and (80). The Euler equation (specialise to log utility for example) tells us
the growth rate of consumption is a constant:
c/c = αAψ1−α − δ + ρ (81)
Notice that this holds at all points on the optimal path (we didn’t say anything about
the BGP). Given that the production technology is now expressible as y = Aψ1−αk,
the growth rate of output is:
y/y = skAψ1−α − δ (82)
An implication here is that changes in the share of income going to investment affect
the growth rate [this forms the basis of the econometric testing of Jones (1995b)].
52 Economic Growth
But since we have reduced the two state variable problem to effectively a single state
variable problem, we can solve out to find that the equilibrium growth rates are all
constant, and equal to the growth rate of consumption.
One way to get interesting dynamics out of this model is to prevent instantaneous
adjustment of the capital ratio to its steady state value ψ. Barro and Sala-i-Martin
(2004, ch. 5.1.2) use a non-negativity constraint on investment, so that a capital
imbalance cannot be rectified by simply transferring some of the stock of human
capital over to physical capital. That seems pretty reasonable, and yields a transition
path similar to the neoclassical model, as the imbalance is eliminated. Another
approach could be to impose a general adjustment cost function, which says the
costs to increasing the flow of investment into a capital stock are increasing in the
rate of flow.
Roland Meeks 53
5 Cross sections and convergence
It is surely an irony that one of the lasting contributions of endogenous
growth theory is that it stimulated empirical work that demonstrated
the explanatory power of the neoclassical growth model.
– Barro (1997, p. x)
The subject of this section is some aspects of the ‘new growth evidence’ that started
to emerge in the early 1990s. The best-known work in this literature is probably due
to Barro and co-authors, but one of the most widely cited papers is by Mankiw et al.
(1992), so we will examine it in detail. Temple (1999) summarises the distinguishing
features of the literature as follows:
First, researchers have often tried to integrate developing and developed
countries in a single empirical framework ... Second, the research makes
intensive use of the cross-section variation in growth rates and other vari-
ables. Finally, the research questions are often inspired at least in part
by recent growth theory. There is a renewed emphasis on human capital,
and to a lesser extent, research and development (R&D) as important
variables in explaining differing growth experiences.
One of the questions that Temple identifies is whether countries converge toward
their steady states, and if so, how fast (our focus here). The neoclassical model
predicts that an economy that is far from its steady state will grow faster than an
economy that is close to its steady state. The reason is that diminishing returns
to reproduceable factors causes returns to capital to fall over time, weakening the
intertemporal substitution motive. But at first gasp there is a problem. In Figure 11,
I have plotted the level of GDP in 1960 for 104 countries, against their cumulative
growth (the log difference) up to 1985. Evidently, there is not much raw correlation
there. But this may not be a good ‘test’ of the model, since if economies differ
along dimensions other than their initial capital intensities, then convergence is to
be expected only in a conditional sense, i.e. after controlling for the steady state.
What this points to is that the neoclassical growth model is now supposed to predict
income levels, on the basis of parameters such as the savings rate, the population
growth rate, et cetera. An interesting comment on the uses of the neoclassical model
in the empirical literature is made by McCallum (1996, p. 63):
54 Economic Growth
Figure 11: Unconditional Convergence
5.5 6 6.5 7 7.5 8 8.5 9 9.5−100
−50
0
50
100
150
200
Log 1960 GDP
Cum
ulat
ive
GD
P G
row
th 1
960−
1985
Source: Mankiw et al. (1992) data.
...the [neoclassical] model was designed to provide understanding about
growth, not about international differences in income levels. In support
of this last assertion, it may be noted that there is no mention of using
the model for the latter purpose in Solow (1956, 1957, 1994), or Meade
(1962), or Hahn (1987). That use seems to have been discovered by
Mankiw, Romer, and Weil (1992).
Finally, part of the stimulus for this work was precisely because certain simple
versions of the endogenous growth story predict that economies jump straight to
the steady state, so have no convergence dynamics at all. An empirical result that
says economies do converge is then seen as supportive of the neoclassical model.
In that sense, it was development of endogenous growth theory that affirmed the
neoclassical model, if you believe that this is what the research actually shows. Let’s
turn to that now.
Roland Meeks 55
5.1 Mankiw, Romer, and Weil
The Solow growth model is reinterpreted as a model for levels. Regressions
on a large cross section of countries show a significant role for human capital.
They also indicate conditional convergence at a rate of about 2% per annum.
Robustness checks and alternative controls spawned an extensive literature.
The paper asks two related questions. First, is the Solow growth model consistent
with international variation in the standard of living? Second, is the model consis-
tent with estimated rates of convergence to the steady state? To address this, the
authors propose an ‘augmented’ version of the model, that includes human capi-
tal as an additional explanatory factor, as an appropriate levels specification. The
economy is then predicted to converge to a steady state ratio of human to physical
capital, a ratio which can nevertheless vary in the initial period. If so, ‘a country
that starts with a high ratio of human to physical capital (perhaps because of a
war that destroyed mainly physical capital) tends to grow rapidly because physical
capital is more amenable than human capital to rapid expansion’ (Barro, 1997, p. 3).
Suppose output Y = KαHβ(AL)1−α−β with L/L = n, A/A = g, then letting
lower case letters indicate rescaling by efficiency units of labour AL, we can write
two differential equations describing the evolution of k and h:
k = skkαhβ − (n + g + δ)k (83)
h = shkαhβ − (n + g + δ)h (84)
where sk and sh are the (fixed) savings rates of physical and human capital. Whereas
in Lucas (1988) human capital is produced in a separate sector, here all reproduce-
able factors share the same production function.
The implied steady state (lecture 2) when there are diminishing returns to ac-
cumulable factors are:
k∗ =
(
s1−βk sβ
h
n + g + δ
)1/(1−α−β)
h∗ =
(
sαks
1−αh
n+ g + δ
)1/(1−α−β)
An implication of the augmented model is therefore that there is more saving overall
in this economy when β > 0, and higher saving means higher income; but because
56 Economic Growth
reproduceable factors overall have a larger share in production, income per head
is reduced more by population growth than in the standard model. Substitute the
steady states into the production function, and take logs, and you get:
log
[
Y (t)
L(t)
]
= logA(0) + gt+α
1 − α− βlog sk +
β
1 − α− βlog sh
−α + β
1 − α− βlog(n + g + δ)
To operationalise this relation, Mankiw et al. assume that the unobserved variable
logA(0) = a+ε. They interpret A(0) broadly as differences in ‘resource endowments,
climate, institutions, and so on’ (p. 411). Back in levels, the effect of ε is a
multiplicative scaling of productivity. In a given sample of countries, they take this
disturbance to be independently and identically distributed (most important is that
it’s independent of the explanatory variables, i.e. rule out situations where savings
rates respond to higher productivity).
The estimating equation is:
log
[
Yi(t)
Li(t)
]
= a+ gt+α
1 − α− βlog sk,i +
β
1 − α− βlog sh,i
−α+ β
1 − α− βlog(ni + g + δ) + εi (85)
The specification in (85) embodies some parameter restrictions, as the sum of the
first two coefficients (not including the constant) is seen to equal (minus) the third,
but it might not be a good idea to impose this from the outset. Rather, they first
estimate without restrictions and test whether their sum is statistically different
from zero (they find not). The estimated coefficients can be used to find the factor
shares; a ‘test’ of the model – albeit an indirect one – is whether these estimated
factor shares are ‘reasonable’ on the basis of what is known from direct estimates of
factor shares.
What are reasonable values for the shares? We saw previously that capital’s
share on US data was reasonably stable at around 1/3 during the 40 years from
1909, and it has remained close to this figure subsequently. Human capital is more
problematic. Mankiw et al. argue that workers receiving the minimum wage (in the
US) have zero human capital, so the difference between the minimum and average
(manufacturing) wages is the return on human capital. Therefore 1/2 to 1/3 is
Roland Meeks 57
Table 3: Mankiw et al. Augmented Solow Model
SampleIntermediate OECD
LHS: log GDP per working age person in 1985CONSTANT 7.97 8.71
(0.15) (0.47)log(I/GDP) - log(n+ g + δ) 0.71 0.29
(0.14) (0.33)log(SCHOOL) - log(n+ g + δ) 0.74 0.76
(0.09) (0.28)R2 0.77 0.28s.e. of regression 0.45 0.32Observations 75 22Implied α 0.29 0.14
(0.05) (0.15)Implied β 0.30 0.37
(0.04) (0.12)
Source: Mankiw et al. (1992, Table II). Standard errors in parentheses.
taken as a guess for β (as always, assuming factors are paid their marginal value
products).
One immediately striking aspect of equation (85) is that the αs and βs are
the same across the sample, so all countries are treated as possessing the same
factor shares/production function. A possible justification for this has been given
by Mankiw [quoted in Snowdon and Vane, p. 604]:
The production function should not be viewed literally as a description
of a specific production process, but as a mapping from quantities of
inputs into a quantity of output. To say that different countries have
the same production function is merely to say that if they had the same
inputs, they would produce the same output. Different countries with
different levels of inputs need not rely on exactly the same processes for
producing goods and services. When a country doubles its capital stock,
it does not give each worker twice as many shovels. Instead, it replaces
shovels with bulldozers. For the purposes of modelling economic growth,
this change should be viewed as a movement along the same production
58 Economic Growth
function, rather than a shift to a completely new production function.
The data includes that displayed in Figure 11, and is from the Summers-Heston
database (also known as the Penn World Tables, or PWT). It covers the period
1960 to 1985; the dependent variable is the log of 1985 GDP divided by working age
population. Population growth ni is the average over the sample period. The savings
rate sk is proxied by total investment as a proportion of GDP, again averaged over the
sample. The human capital savings rate is taken to be proportional to the percentage
of working age population in secondary school. Two main groups of countries are
considered: OECD, and a sample labelled ‘intermediate’ which excludes very small
states, and those with poor quality data. The restricted model is not rejected;
Table 4: Mankiw et al. Tests for Conditional Convergence in an Augmented Solow-Model
SampleIntermediate OECD
LHS: log diff, GDP per working age person 1960–1985CONSTANT 3.09 3.55
(0.53) (0.63)log(Y60) -0.372 -0.402
(0.067) (0.069)log(I/GDP) - log(n+ g + δ) 0.506 0.396
(0.095) (0.152)log(SCHOOL) - log(n+ g + δ) 0.266 0.236
(0.080) (0.141)R2 0.44 0.66s.e. of regression 0.30 0.15Observations 75 22Implied λ 0.0186 0.0206
(0.002) (0.002)Implied α 0.44 0.38
(0.07) (0.13)Implied β 0.23 0.23
(0.06) (0.11)
Source: Mankiw et al. (1992, Table VI). Standard errors in parentheses.
coefficient esimates are given in Table 3. For the intermediate sample, the implied
Roland Meeks 59
Figure 12: Conditional Convergence - Intermediate
−1.25 −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50
−0.75
−0.50
−0.25
0.00
0.25
0.50
0.75
Source: Author’s calculations on Mankiw et al. (1992) data. The ordinate is the residual from aregression of log Y 85− logY 60 on log SCHOOL, log (I/GDP ) and log (n+ g + δ). The abscissa isthe residual from a regression of log Y 60 on the same explanatory variables. The slope of the lineis the regression coefficient reported in Table 4.
capital shares are in the rough area delineated from US data. For the OECD, the
prior values just about fall into a 1 s.e. region from the point estimates. It appears
that human capital is an extremely important explanatory factor. The implied
shares are large, and the coefficients statistically significant.
Now consider the second part of the hypothesis, that each country converges to
its own steady state, as determined by savings rates on broad capital, and population
growth. By calculating the convergence rate in the ‘augmented’ Solow economy, and
substituting for the steady state, we arrive at the estimating equation:
log
[
Yi(t)
Li(t)
]
− log
[
Yi(0)
Li(0)
]
= θ log a + gt+ θα
1 − α− βlog sk,i
+ θβ
1 − α− βlog sh,i − θ
α + β
1 − α− βlog(ni + g + δ) − θ log
[
Yi(0)
Li(0)
]
+ εi (86)
where θ = 1− exp{λt}, and λ = (1−α− β)(n+ g+ δ) is the parameter of interest.
60 Economic Growth
Figure 13: Conditional Convergence - OECD
−0.8 −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 0.8 1.0
−0.4
−0.3
−0.2
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Source: Author’s calculations on Mankiw et al. (1992) data. The ordinate is the residual from aregression of log Y 85− logY 60 on log SCHOOL, log (I/GDP ) and log (n+ g + δ). The abscissa isthe residual from a regression of log Y 60 on the same explanatory variables. The slope of the lineis the regression coefficient reported in Table 4.
We now need to assume that εi is independent of initial income per capita in order
to obtain consistent parameter estimates. I can’t see how that can be true.
The number to focus on in Table 4 is the estimated coefficient on Y60. It is
statistically significant in both subsamples, and of the expected sign. From (86),
λ = − log(1+θ)/t for t = 25 years yields the implied rates of convergence, which are
about 2% per annum. The standard error on these estimates is tiny, and therefore
this convergence finding, replicated across regions as well as countries, is taken by
some as a ‘stylised fact’ (Barro and Sala-i-Martin, 2004, chs. 11, 12). The small
standard error also means that the OECD converges faster, on average, than the
sample as a whole. I have plotted this finding in Figures 12 and 13, so that you can
see how the sample is distributed around the regression line (N.B. Barro tends to
plot my ordinate value against the level of log start GDP; that means the slope of
the line shown is not the same as the regression coefficient. In my figures, the line
Roland Meeks 61
is the proper partial correlation between the dependent and independent variable).
The finding of faster OECD convergence, under the assumption that technology is
a public good, means that they must start out further from their steady state than
the sample as a whole.
Second, notice that the measure of human capital is strongly significant in the in-
termediate sample, but weakly significant in the OECD, although the implied shares
come out the same. The technologies in the two samples look more similar than in
the levels results, with more weight on physical capital. The obvious conclusion is
that the omitted variable in the levels equation is biasing the coefficients on the
savings rate terms, because initial income is in fact correlated with average savings
rates over the sample period. That is perfectly permissible, but it does mean that
the support to the theoretical model given by the first regression is contradicted by
the second regression21, something that does not come through in the paper.
5.2 Some critics
... I keep asking myself, do I really believe that there is a surface out
there in space whose axes are labelled with all the things Robert Barro
and company put on them? Do I believe that there is such a surface, and
countries or points on that surface could in principle move from one place
to another on it and then move back to where they began by changing
their form of government or by having more or fewer assassinations? ...
I would not bet anything on the existence of that surface.
– Solow, interview in Snowdon and Vane (2005, p. 667)
The findings of Mankiw, Romer, and Weil have been widely scrutinised. One cri-
tique focuses on the reliability of the econometrics. Temple (1998) looks both at the
influence of unrepresentative observations, and at sensitivity to measurement error.
As he discusses in more length in his survey [Temple (1999, §2.3-2.4)], a number of
difficulties need to be taken into account. For example, the use of GDP per head
rather than GDP per worker-hour (the quantity that theory uses) is an obvious
shortcut in the absence of reliable hours data. But for Japan this changes the in-
come differential with the US from 20% on the former measure to 40% on the latter,
21Just to be clear, if you add inital GDP per capita to both sides of (86) you get (85) with anextra variable, the coefficient on which is 1 − θ, so they are directly comparable.
62 Economic Growth
and with developing economies the differential may simply be unknown. Another
example is the measure of growth itself. The PWT data shows growth rates that
differ from those reported in countries’ own national accounts, because of variations
in domestic relative prices. That has lead some researchers to recommend using
national accounts data for growth and the PWT for levels. Next, the human capital
variable has been found to be problematic, particularly because of a disconnect be-
tween macro and micro studies (Temple, 1999, §7.2). The quality of human capital
may be captured by life expectancy, which other studies find to be strongly positive
(Barro, 1997, pp. 19–22). However, the appropriate response to measurement error
is ‘to try and identify which variables are particularly badly measured, and hence
isolate the findings that are least reliable’ (Temple, 1999, §4.5). Temple (1998) con-
structs bounds on the point estimate of convergence, under different assumptions on
the measurement error in the data, shown in Table 5 (note this is not the same as
sampling error). The bounds are wide enough to indicate that either the sample as
Table 5: Convergence Rates Allowing for Measurement Error
Intermediate OECDLower Bound 0.005 0.015Upper Bound 0.063 0.036
Source: Temple (1998, Table IV).
a whole does not converge, or that it does so faster than the OECD. The mid-point
of these bounds is above 2% in both cases, but that figure is perfectly consistent
with the evidence. Temple concludes that ‘the measurement error problem is almost
certainly too great to draw firm conclusions about convergence from cross-section
regressions ... even if fixed effects and parameter heterogeneity are negligible diffi-
culties, and even when the assumptions made about measurement error are rather
optimistic’ (pp. 372–3). In this context see also Romer (2005, pp. 31–37).
Another important point to mention is the endogeneity of the regressors, par-
ticularly when long time averages are used. Another way of saying this is that ‘the
more right-hand-side variables go into those regressions, the more they seem to me
to be just as likely the consequences of success or failure of long-term growth as the
cause’ [Solow quoted in Snowdon and Vane (2005, p. 667)]. We would expect that
investment rates would tend to be higher when growth is high, say due to rising
Roland Meeks 63
TFP. Equally, population growth may be related to incomes (Barro, 1991). The
standard response to such issues is to use an instrumental variables estimator, but
there is a dearth of suitable instruments out there; in a panel setting, one might be
able to get away with lagged values of the regressors.
5.3 Institutions, constitutions and geography
Under the heading of ‘convergence’ studies, Barro and Sala-i-Martin (2004, ch. 12)
shows results for models similar to those above, but with vastly more ‘control’ vari-
ables. For example, democratic freedoms, indexed according to electoral rights, enter
with a possible non-linear effect. Their finding is that ‘democratization appears to
enhance growth for countries that are not very democratic but to retard growth for
countries that have already achieved a substantial amount of democracy’ (p. 529).
Similar exercises are undertaken for civil liberties, the rule of law, fraction Muslim
or Buddhist, whether you were a British colony (bad news), openness to trade et
cetera. Many of these variables are very sensible – a failure of the rule of law cannot
be beneficial to growth. Formally, the intercept in (85) is modified to be a function
of these controls. The rationale is that a better control for level of initial income
will reduce bias in coefficient estimates22.
As you might expect, this literature has been greeted with skepticism from a
number of angles. Endogeneity bias is likely to contaminate results when ‘explana-
tory’ variables such as education themselves depend on growth, for example because
the returns to education are greater in a growing economy (or an economy which is
expected to grow). A difficult issue, which you have to make your own mind up on,
is whether research is really ‘just science’, or whether the questions it asks presup-
pose a certain answer. You may think that a paper that finds countries are poor
because they have a high rate of population growth and insufficient thrift is making
an ill disguised point. If so, you would need to find an answer for the findings. On
the other hand, you might think that such a paper has neatly summarised some
22It’s important to distinguish between factors that are broadly growth effects, and those thatprimarily influence income levels. Lucas warns ‘[t]he removal of an inefficiency that reduced outputby five percent (an enormous effect) spread out over 10 years is simply a one-half of one percentannual growth rate stimulus. Inefficiencies are important and their removal certainly desirable, butthe familiar ones are level effects, not growth effects. (This is exactly why it is not paradoxical thatcentrally planned economies, with allocative inefficiencies of legendary proportions, grow about asfast as market economies)’ (1988, p. 12).
64 Economic Growth
proximate causes of economic wellbeing. In which case you would need to convince
yourself that plausible alternative explanations were fairly ruled out.
The reverse causation issue has been seriously taken on by Acemoglu et al. (2001),
who try to explain cross country income levels by focusing on institutions. They use
colonial history as a source of exogenous variation in institutional arrangements. In
a nutshell, what type of institutions a colony ended up with depended on whether
European colonialists were ‘settlers’ or ‘exploiters’. The former type wanted to set
up new states, typically with institutions protective of rights and weary of excessive
government power. The latter type wanted to extract resources from their colonies,
and ruled in accordance. The choice of strategy was in part influenced by the ease
of settlement of these lands, which in turn was a function of the settlers’ mortal-
ity rate. Given enough persistence in institutional structure, current institutional
arrangements can be instrumented by this variable. They findings are for a ‘sub-
stantial, but not implausible large’ effect of institutions on growth, but little role
for geography variables (latitude, distance from coast, mean temperature). Jeffrey
Sachs has been a strong advocate of the importance of geography variables, partic-
ularly disease burden. In his 2004 OXONIA lecture23, he talked at length about the
role malaria plays in retarding development. The trap here is a kind of geographical
determinism. The argument is over whether geography affects growth directly, or
through institutions.
5.4 Time series empirics
Time series tests of endogenous growth models do not find the predicted
relationship between investment rates and growth rates.
Earlier, I mentioned that a simple two-sector endogenous growth model, with phys-
ical and human produced using identical technologies, had the implication that
output growth and investment rates s should be positively correlated:
gy = sAψ1−α − δ (87)
This suggests that a test of the ‘AK’ version of the endogenous growth model could
be based upon time series data for output growth and investment rates. Jones
(1995b) takes data on two types of investment good from the PWT, for a panel of
23See http://www.oxonia.org/events 2004 sachs.html and Sachs (2005).
Roland Meeks 65
15 OECD countries24 (separating total investment from producer durables invest-
ment). He finds that OECD investment rates have trended upwards in the post
war period, especially in producer durables, whereas growth rates have remained
constant, or fallen slightly (see Figure 14). It’s possible that a persistent increase in
Figure 14: Growth and Investment Rates – UK and W.Germany
1950 1955 1960 1965 1970 1975 1980 1985 1990
0.00
0.05
0.10 GR_DEU GR_GBR
1950 1955 1960 1965 1970 1975 1980 1985 1990
10
20
30
GBR_CD DEU_CD
GBR_CI DEU_CI
Source: Author’s calculations on Jones (1995b) data. UK (GBR) and W. Germany (DEU) areshown. The top panel gives log difference of GDP per worker, the bottom panel investment ratesfor total gross investment (CI) and producer durables investment (CD).
investment rates was accompanied by a persistent decrease in some other determi-
nant of long term growth. Jones is skeptical that such a variable exists, noting that
‘[h]uman capital investment and openness, two leading possibilities, both certainly
trend upward in the postwar period’ (p. 508).
The figures are prima facie evidence against the endogenous growth model be-
ing literally true. It could be that the investment rate does have a dynamic effect
on growth, but one which is stretched out in time (you could think of lots of rea-
sons why investment spending might boost growth only with a lag). To investigate
24Australia, Austria, Belgium, Canada, Denmark, Finland, France, W.Germany, Italy, Japan,Netherlands, Norway, Sweden, UK, USA.
66 Economic Growth
Figure 15: Dynamic and Cumulative Growth/Output Response to a PermanentUnit Increase in Investment
0 5 10 15 20−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
Total InvestmentProducer Durables Investment
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Total Investment
Producer Durables Investment
Source: Jones (1995b, Table V). The left panel shows the effect on growth, the right panel showsthe cumulated effect.
this possibility, Jones formulates a univariate ‘autoregressive distributed lag’ (ADL)
model, of the kind described in Hendry (1995). I will write Jones, eq. 4 & 5 without
the lag operator notation, for the case of 2 lags (not the case used in estimation):
gt = a1gt−1 + a2gt−2 + b0st + b1st−1 + b2st−2 + εt
You can manipulate this equation by adding and subtracting terms, without chang-
ing its stochastic properties to obtain the equation:
gt = a1gt−1 + a2gt−2 + (b0 + b1 + b2)st − (b1 + b2)(st − st−1)− b2(st−1 − st−2) + εt
With the parameter estimates in hand, dynamic multipliers can be calculated for any
horizon. In particular, the long-run multiplier is seen to be (b0+b1+b2)/(1−a1−a2).
Roland Meeks 67
There are two potential things to think about when estimating this. First, some
countries’ st appear to have a stochastic trend, whereas the g are stationary. I say
appear, because a share is obviously bounded between 0 and 1, so cannot literally
trend anywhere. Nevertheless, if in the sample one series trends up stochastically,
while the other is stationary, we wouldn’t expect much correlation between them.
Second, the current investment share has to be correlated with the regression error,
since it has yt in the denominator, and yt is linked to investment through the na-
tional accounts identity. There are a couple of responses to this, but Jones rejects
the common instrumental variables approach because instruments are ‘notoriously
hard to come by in the investment literature’ (p. 522). Instead, he constructs an
upper and lower bound for the coefficient on the contemporaneous variable ∆st.
He imposes that the sum of coefficients on st is zero (so the long run multiplier is
zero), then estimates an OLS equation on lagged growth rates and lagged changes
in the investment share, for a range of values for the ∆st coefficient. Also included
are a constant and time trend for each country. The latter is intended ‘to capture
any exogenous movements in growth rates that are omitted from the specification’.
That strikes me as quite optimistic – most of the correlation has to come from the
fluctuations around the level of trend growth that Jones wants to explain, and so
the omission of recognised business cycle variables is more likely to bias the results
than the omission of a trend. You would have to investigate this yourselves to find
if the effect was quantitatively significant.
In Figure 15 I have plotted the response of output growth to a unit increase in in-
vestment, calculated by Jones ignoring endogeneity bias on the coefficient estimates.
The total investment response is weaker than the producer durables response, which
may be explicable due to changes in the composition of investment, that caused the
depreciation component to rise over time. But an increase in durable investment
does raise growth, but only over a short horizon; Jones says ‘five to eight years’
(p. 513), it looks to me like there is not much action to speak of beyond year one.
Therefore, ‘not only does it appear that a permanent increase in the investment rate
has only a transitory effect on the growth rate, but also it appears that the horozon
over which that effect occurs is sufficiently short to make the predictions of the AK
model misleading’ (Jones, 1995b, p. 514).
68 Economic Growth
6 R&D based endogenous growth
To incentivise firms to undertake research, new designs must yield private
profits. That necessitates a departure from competitive market structures.
The National Science Foundation estimates that the United States spent $276 billion
on research and development (R&D) in 2002, of which around 70% was performed
by industry25. This expenditure accounts for 44% of the total for the entire OECD,
which itself performs almost all of global R&D. Trends in R&D intensity are shown
in Figure 16. These numbers are small compared with investment in physical cap-
ital, but of course increases in applied knowledge drive investment in new capital.
Estimates of the contribution of R&D to TFP growth are unfortunately very diffi-
cult to construct (see Grossman and Helpman 1991a, ch. 1), but a strong prior is
that it is substantial.
In this section we turn to explicit treatments of R&D as the driving force behind
technological change. The key mechanism we will study is the invention of new and
better product lines as a result of purposeful R&D expenditures by firms. The up-
front cost of R&D will be financed by the monopoly profits that are later earned on
sales of new products, i.e. we will exclude entry by competitor firms to the market
for existing varieties or qualities of good26. In our previous discussion on knowledge
spillovers in §4.2, we allowed knowledge to be perfectly nonexcludable. Crucially,
we retain this assumption for production of new inventions: there will be nothing to
stop others looking at new patents, and using the technology as a basis for their own
research. Meanwhile, nonrivalry implies increasing returns to all factors considered
jointly, and so that markets must be noncompetitive (Romer, 1990a).
We won’t distinguish between the activities of research and development. There
is an important difference between fundamental discoveries, and the transformation
of ideas into workable production processes for new or improved products. In par-
ticular, the fundamental discoveries, for example those of basic science, can be made
widely available whereas innovations in production processes are possible to exclude,
and may well come about inside the firm through learning-by-doing [see Aghion and
25http://www.nsf.gov/statistics/seind04/c4/c4h.htm. Figures are based on reported R&D in-vestment, converted into US dollars at PPP exchange rate.
26The classic tension therefore exists between static and dynamic optimality: for static optimalitywe would demand marginal cost pricing; but without profits, innovation would cease, and so wouldgrowth.
Roland Meeks 69
Howitt (1998, ch. 6)].
The key papers in this literature are Romer (1990b), Grossman and Helpman
(1991b), Aghion and Howitt (1992) and Jones (1995a). We will follow the textbook
treatment of Grossman and Helpman (1991a) and Aghion and Howitt (1998).
Figure 16: Non-Defence R&D as a Proportion of GDP
1980 1985 1990 1995 2000
2.0
2.5
3.0US Japan
1980 1985 1990 1995 2000
1.0
1.5
2.0
2.5
Germany France
1980 1985 1990 1995 2000
1.00
1.25
1.50
1.75UK Italy
1980 1985 1990 1995 2000
0.75
1.00
1.25
1.50
1.75 Canada Russia
Source: National Science Foundation. See http://www.nsf.gov/statistics/.
6.1 Better goods
In this section, we will look at a simple model in which R&D leads to technological
innovations that improve the quality of an existing good. The name given to this
process is a quality ladder, which may have rungs that are equally spaced, or may
have rungs that are endogenously spaced depending on R&D intensity. If the rungs
of a quality ladder are far apart, innovations are referred to as ‘drastic’. These models
capture features of the real world in a very pleasing way, for example, the idea that
at each point in time there is a product leader who can earn monopoly profits from
producing a particular good, until some other firm develops a better product, taking
70 Economic Growth
over the technological frontier. Such a process is labelled ‘Schumpeterian’ by Aghion
and Howitt, as it captures the notion that growth entails ‘creative destruction’, i.e.
the obsolescence of old methods through innovation. Compare to the view expressed
by Mankiw in §5.1 earlier, in the context of the neoclassical production function,
that ‘replac[ing] shovels with bulldozers’ should be thought of as a ‘movement along
the same production function’.
Suppose the economy is populated by agents possessing a linear utility function
over final goods at each date y(τ), who discount the future at rate r:
U =
∫
∞
0
y(τ)e−r τdτ
where τ indexes time. Final goods will be a Cobb-Douglas function of an interme-
diate good x, with a productivity factor A representing the highest quality level yet
attained:
y = Axα
At any point in time, only the best quality intermediate good is used in production.
The production of intermediate goods takes a simple form: one unit of labour
produces one unit of intermediate goods, according to a blueprint owned by the
most recent innovator. Labour is also used in the research lab. To capture inherent
uncertainty in research, model the arrival of new innovations as a Poisson process
with intensity proportional to the quantity of research labour.
The probability of a single researcher making one innovation in a tiny interval of
time dt is λ dt (I figure this by taking an approximation to the distribution function
at x = 1 around v = 0, whereupon I get this term plus a term which is o(dt), i.e.
dominated by a linear function. See insert). The probability of n researchers making
a single innovation in the interval dt is modelled simply as proportionally higher,
nλ dt. When an innovation occurs, we move up to the next rung on the quality
ladder. Rungs are equidistant in log terms at γ, so Ai = γi describes the technology
ladder, with i indexing innovations27.
27For some reason, Aghion and Howitt (1998) index innovations by t. Set γ > 1 so that weascend, not descend the ladder!
Roland Meeks 71
Definition: Poisson process Let the probability of x changes in each in-
terval of length w be:
f(x, v) =(λv)xe−λv
x!x ∈ {1, 2, ...}
Then the random variable X has a Poisson distribution with parameter m =
λv. This distribution is derived from the Poisson postulates, which say:
i. The probability of one change in a short interval is approximately pro-
portional to the length of the interval.
ii. The number of changes in nonoverlapping intervals are independent.
iii. The probability of two or more changes in a short interval is essentially
zero.
The market for labour clears when the fixed aggregate supply of labour L is
equal to the demand for labour from the research and manufacturing sectors:
L = x+ n
Households allocate labour between research and production. They choose to work
on the activity that has the higher return. This gives an arbitrage relation between
wages earned in manufacturing innovation i and the expected discounted payoff to
discovering innovation i+ 1:
wi = λVi+1 (88)
Here V is the discounted profit obtained from owning the blueprint for, and produc-
ing, the intermediate good of quality i + 1. The expected profit is the probability
that a particular worker will find this innovation, times the payoff if she does.
To find the value of a blueprint, think of it as an asset that provides a divi-
dend stream πi but which disappears with probability ni+1λ (the rate at which the
next innovation is found; remember the index tracks the current product, so this
is research labour involved in finding the next product). Instead of discounting the
profit flow at rate r, you will make an adjustment for the obsolescence rate to get:
Vi+1 = πi+1/(r + λni+1)
72 Economic Growth
For maximum welfare, it would be better if this overdiscounting did not take place.
The benefits of innovation are permanent in the sense that we do not slip down the
quality ladder. The monopolist does not internalise this benefit, and so tends to do
too little research. In the current model, the quality leader does no research, since
knowledge is public (although excludable). Other entrepreneurs will find it more
profitable to develop the next innovation than the incumbent. The fact that they
do innovate, and steal the incumbent’s business causing him a welfare loss, is also
not internalised, and turns out to lead to too much research [see Aghion and Howitt
(1998, §2.3)].
Final goods producers are competitive. This means they take the price of out-
put and inputs as given, and maximise their profits. Demand for intermediates is
therefore:
xd = arg maxx
{y − px} = [sub production fn] = arg maxx
{Axα − px}
= (αAp)1/(α−1)
Intermediate goods are produced by a monopoly. Taking the demand curve of final
goods producers as given, the monopolist maximises his profit:
xs = arg maxx
{p(x) − wx} = [use inverse dd fn] = arg maxx
{αAxα − wx}
=
(
α2
w/A
)1/(1−α)
All of these calculations implicitly refer to innovation i, and naturally the interme-
diate goods market clears.
Bring together these results in the following definitions:
ωi := wi/Ai
This is the real, productivity-adjusted, production wage for innovation i.
x(ωi) = (α2)1/(1−α)ω−α/(1−α)i , x′ < 0
This is the supply curve, which is decreasing in the real wage.
π(ωi) =
(
1
α− 1
)
x(ωi)ωi, π′ < 0
This is the productivity-adjusted profit level. The unadjusted level of profit on
innovation i is just πi = Ai π(ωi).
Roland Meeks 73
Using these definitions, the behaviour of the model can be summarised in two
equations in the real wage and the number of workers in the research sector:
ωi =γλ π(ωi+1)
r + λni+1
L = ni + x(ωi)
To get the real wage equation, use the arbitrage relation (88) and the expressions
for V and π.
6.1.1 Endogenous growth
Now we can find the balanced growth path. On a BGP, productivity-adjusted
wages and research employment must be constant. Drop the i subscripts in the two
equation system above and denote with a hat the steady state values of research
labour and wages:
ω =γλ π(ω)
r + λnL = n + x(ω)
As Aghion and Howitt explain, the labour market equation slopes up in ω−n space,
and the real wage equation slopes down. The steady state exists for some reasonable
parameter values. You can verify that:
n =γλ(1/α− 1)L− r
λ(1 − γ[1/α− 1])
(The reader should check this). The comparative static implications of these equa-
tions are intuitive: the more drastic are innovations, the more research is done since
research is potentially more profitable; likewise the lower is discounting, the higher
are the discounted future profits; the greater the likelihood of innovation, the more
cost effective is research, but the shorter the interval over which firms expect to be
able to enjoy monopoly profits (so generally this effect is ambiguous).
Let’s find the growth rate of the economy. Final output yi = Aix(ω)α is a
function of changing productivity and an unchanging quantity of intermediate goods.
Growth in final output therefore comes from growth in technology. How do we find
the growth rate of technology?
We can use the unchanging quantity of research labour n in conjunction with
information on the stochastic process for innovations to find the expected growth
74 Economic Growth
rate of A over a short interval of time. Over a short interval of time v, there will
be one innovation with probability nλv by Poisson postulate (i). The change in
productivity is either dA = γi+1 − γi with probability nλv, or it doesn’t change. So
the expected change in A is:
E[dA] = γi(γ − 1)nλv =⇒E[dA]
A= (γ − 1)nλv (89)
Take v to be the small time increment dt and you have that the expected growth
rate of productivity is gA = nλ(γ − 1). The bigger the increment to quality γ, the
faster the expected growth of productivity. The more workers in research n, and
the more effective research is λ, the faster is expected growth. This isn’t exactly the
expression written in Aghion and Howitt (1998, p. 60); but note that if γ is close to
1, then log(γ) ≈ γ − 1. My way avoids introducing an approximation. Later, you
will want to compare this equation to (103) and to (104). The key implication to
take away is that never-ending endogenous growth is possible, and is increasing in
the number of research workers.
6.2 Monopolistic competition
We could get away with a simple one-good model in the quality ladders model,
but for the varieties approach it is essential that we be able to handle multiple
goods. So before looking at the varieties model, we will review the main features of
monopolistic competition in general equilibrium, within the standard Dixit-Stiglitz
framework. Conceptually, the problem that this framework solves is that on one
hand, we want markets to be thin enough that firms retain some market power; on
the other, we don’t want to have so few firms that we are forced to deal with the
complication of strategic interaction between them. The solution is to treat firms as
local monopolists in the production of particular goods, but to have the number of
goods (and the number of agents) be large so each is small relative to the economy
as a whole [see Romer (1990a)].
6.2.1 Demand
We will need consumers to have CES preferences over a ‘large number’ n of goods.
Each commodity will be labelled according to an index x(j) with j ∈ [0, n]. Prefer-
Roland Meeks 75
ences are of the form:
D =
[∫ n
0
x(j)αdj
]1/α
(90)
Think what would happen in the usual static consumer choice problem, where the
budget constraint is:
M ≥
∫ n
0
x(j)p(j)dj (91)
This budget constraint says nothing different from usual, i.e. that income M should
exceed expenditure on all commodities. Naturally, the agent will spend her entire
budget on commodities at a maximum. Form the Lagrangean:
L =
[∫ n
0
x(j)αdj
]1/α
+ λ
(
M −
∫ n
0
x(j)p(j)dj
)
(92)
There is a problem here which I will just gloss over. The problem is that for a
continuum, changing a particular value of x has no effect on the integral. What is
the area of a vertical line? Zero. So, differentiating the Lagrangean with respect to
x will produce zero. But this is how the problem is usually cast, so we should think
about how to get around it. One way would be to consider the sum with intervals
h between elements, with small but not infinitessimal h. Another way would be to
mentally substitute ‘sum’ for ‘integral’ throughout, in which case the maths works
fine. In any case, we will take:
∂L
∂x(j)=
[∫ n
0
x(i)αdi
](1−α)/α
x(j)α−1 − λp(j) = 0 (93)
The marginal rate of substitution (the ratio of marginal utilities) between any two
commodities j and k is then, as usual, equated to the relative prices at an optimum:
x(j)α−1
x(k)α−1=p(j)
p(k)=⇒ x(j) =
(
p(j)
p(k)
)
−1/(1−α)
x(k) (94)
where the exponent is the elasticity of substitution ǫ. This is nice because there is a
symmetry we can exploit, as we know the demand for any j relative to this particular
k is always the same. Using this information, the budget constraint becomes:∫ n
0
p(j)x(j)dj =
∫ n
0
p(j)
(
p(j)
p(k)
)
−ǫ
x(k)dj
= [x(k), p(k) independent of index j]
M = x(k)p(k)ǫ
∫ n
0
p(j)1−ǫdj
76 Economic Growth
We can use this to substitute x(k) out of the demand curve (94) to obtain, for any
j:
x(j) = Mp(j)−ǫ
∫ n
0p(i)1−ǫdi
(95)
which says that demand for each commodity is a function of relative price.
6.2.2 Production
Let output of commodity j be produced with one factor, labour, according to the
concave production function f(L). Consider the problem of the monopolistic firm
facing a competitive labour market in which the wage is w. The firm wants to
maximise profits, which are revenue p(j)x(j) minus costs wL(j). Costs themselves
obviously depend on output, since more output requires more labour, and more
labour means a higher wage bill. Depending on the form of f (so far I just said
f ′ ≥ 0, f ′′ ≤ 0), adding a marginal unit of labour might not produce a constant
increment of output, in which case marginal cost might be rising with x.
There are two ways to see what the firm will do. The first way is to write
profit, recognising the dependence of L on x, and recognising that the firm faces the
demand curve (95), and so can independently choose either output or price:
π(j) = p(j)x(j) − wL(j) =p(j)1−ǫ
∫ n
0p(i)1−ǫdi
− wf−1[x(j)] (96)
Say f were linear, then x(j) = γL(j) and L(j) = x(j)/γ; if x(j) = L(j)ξ, with
0 < ξ < 1, then L(j) = x(j)1/ξ. We will take the former case as the example from
now on (this section and the those following), and ‘by choice of units’ have one
unit of labour produce one unit of output; you can confirm that the FOC for profit
maximisation requires:
p(j) =ǫ
ǫ− 1w (97)
To get this result, I had to assume that the integral in the denominator of the profit
function is a constant with respect to p(j). That’s odd mathematically, even if we
mentally substitute a sum rather than an integral over a continuum, as the sum
does depend on p(j); but the argument goes that each firm is a small enough player
that its choice of price doesn’t impact the aggregate, and is completely standard. I
suppose I think of it as a good approximation.
Roland Meeks 77
Since ǫ > 1 in (97), this means that prices are set as a markup on marginal cost,
with the size of the markup decreasing in the elasticity of substitution. Intuitively,
when products are not close substitues, a firm can charge more without loosing too
much demand.
To see that it is marginal cost on the RHS in general, we can take the second
route, which I’ll call the ‘two-step’ route. Consider the dual problem of cost minimi-
sation, subject to producing some quantity x(j). Introduce a Lagrange multiplier µ
that has the interpretation of marginal cost (why?):
L = wL+ µ[x− f(L)]
Then by choice of L, the necessary condition for a minimum implies:
µ =w
f ′(L)(98)
Step two, recall from elementary micro that the monopolist sets price so that MC
= MR, and solve (write revenue as a function of x).
Profits, and aggregate profits, can be found by substituting the maximiser back
into (96). Noting that (p− w)/p = 1/ǫ, under symmetry:
π =M
ǫn= [definition of ǫ] = M
1 − α
n(99)
Profits increase with aggregate nominal income (just a rescaling), and decrease with
the degree of substitutability of commodities, and with the number of commodities
themselves; intuitively, more goods means more competition. Aggregate profits are
n times this quantity.
In an equilibrium, the goods and labour markets will clear, aggregate income
– the sum of labour income and profits – will equal aggregate expenditure, and
relative prices are determined. By symmetry, the relative price of all commodities
is unity, a result which means that in spite of monopolistic competition there is no
allocative inefficiency in this economy. The relative price of labour is determined by
the equilibrium markup.
Question: Can you solve for the equilibrium allocations, given a fixed labour supply
of L?
78 Economic Growth
6.3 New goods
When R&D leads to the discovery of new varieties of products, endogenous
growth can result from spillovers in the research process. However, the spec-
ification of increasing returns has been criticised as arbitrary. An alternative
‘semi-endogenous’ growth model is proposed.
One aspect of economic growth is the increasing variety of available products.
Consumers with‘a taste for diversity’, as in the previous section, can be made better
off through such expansion [consider differenitating (90) with respect to n]28. We will
assume that in addition to the manufacturing of commodities described above, firms
in the economy also devote resources to R&D. This activity leads to the discovery
of ‘blueprints’ for new varieties, which we will assume can be patented for ever (an
alternative assumption is that imitation is sufficiently costly that no firm would
do it). That means n can grow over time, which with a suitable reinterpretation
of the CES aggregator means that TFP is also growing. This section is based on
Grossman and Helpman (1991a, ch. 3); see also Barro and Sala-i-Martin (2004, ch. 6)
and Romer (2005, ch. 3), and the appendix to Jones (1995a).
To see this, first note that the ‘lifetime’ utility of the representative agent in
this economy will be taken to be the discounted sum of a log-transformation (unit
elasticity of intertemporal substitution) of the utility index (90). The bundle of
goods created by the CES aggregator might not be individual goods that the agent
consumes, but rather a bunch of intermediate goods that someone else has com-
bined into one composite commodity (as in the previous section; the approaches are
equivalent). The production function of this composite, or ‘final’ goods firm is just
the CES aggregator, and notice that it has CRS (multiply each commodity by a
constant to check for linear homogeneity). Under symmetry, the aggregator is:
D =
[∫ n
0
xα
]1/α
= xn1/α
The resources embodied in final goods is X = nx, so the ratio of D to X is final
output per unit of input, or, TFP= n(1−α)/α (Grossman and Helpman, 1991a, pp. 46–
47).
Because of monopolistic competition, and having sole rights to produce a par-
ticular variety, the firm is able to capture the profits from its blueprint, and it is
28For a discussion on defining preferences that display taste for variety, see Benassy (1996).
Roland Meeks 79
this possibility that renders the R&D process worthwhile. Will this process be self-
sustaining? We know from (99) that more varieties means more competition means
lower profit. As profitability falls, we will see that the reduction in the stockmarket
value of firms that this entails, renders the up-front costs of R&D unfinanceable. At
that point, R&D ceases and so does TFP growth. We now turn to look at this in
more detail.
6.3.1 R&D
An entrepreneur engages in R&D to produce a blueprint for a new good and to
start manufacturing it. The market for manufactures is as described in the previous
section. The production of blueprints requires the input of labour, which has pro-
ductivity 1/a. We will think of this process taking place in continuous time. Then
to invent an incremental variety dn, the entrepreneur must input l/a units of labour
over the interval dt. You can label this the ‘production function’ for new designs:
n =l
a=⇒
n
n=
l
an
clearly this is CRS in l. An implication of the specification is that a given growth
rate of new designs is going to require an increasing resource input.
There are no credit market frictions, and no aggregate uncertainty, so we may as
well have entrepreneurs finance the up-front costs of development by issuing equity.
In each period, all profits will be paid out in the form of a dividend. The value
of a blueprint is v, which we will take to be the present discounted value of future
profits:
v(t) =
∫
∞
t
e−ρ(τ−t)π(τ)dτ
For investors to want to hold shares in the company formed by an entrepreneur,
the payoff has to exceed the opportunity cost of funds, a condition known as ‘no
arbitrage’:
π(t) + v(t) = ρv(t)
where profit will be given by (99).
We will take entry into R&D to be free, and that creates the restriction that
excess returns are zero. Even though the invention of a new variety will yield
80 Economic Growth
a positive stream of profits for the entrepreneur, and therefore dividends for the
investor, the up-front costs are large enough to swamp these gains. Specifically, the
value of an incremental variety is vdn = vl/a dt and the cost is wldt. Research
goes ahead if vl/a ≥ wl or v ≥ wa. However, the case of a strict inequality cannot
be an equilibrium, as it implies unbounded labour demand by entrepreneurs. An
equilibrium requires:
v ≤ wa
{
equality n > 0
inequality n = 0
meaning equilibrium with positive research requires v = wa with equality, whereas
a no-research equilibrium (zero employment in R&D) occurs at any moment when
v < wa.
6.3.2 General equilibrium
Our goal now is to use what we know about the momentary equilibrium to pin down
the dynamics of the system in terms of firm value and the number of varieties. The
aggregate supply of labour is L units, of which l ≥ 0 units are employed by the
R&D sector. Using the production relation, we can write l = an. Employment
in manufacturing depends on aggregate production; from (95), the production of
the jth commodity in a symmetric equilibrium is x = M/pn. Normalise nominal
expenditure to unity, then the aggregate∫
xdj = 1/p, which by one-person one-
unit-output is also employment. The labour market clearing condition is thus:
an+1
p= L (100)
Combined with the no-arbitrage condition described previously:
v(t) = ρv(t) −1
ǫn(101)
We now have a pair of differential equations in v and n, but still need to eliminate
p.
If there is positive R&D so that n > 0, it must be profitable for firms to devote
resources to it. By the free entry condition described above, the value of a blueprint
will be just equal to the cost of acquiring it v = wa. Under monopolistic competition,
we already saw that the price charged per brand is a constant markup over the
Roland Meeks 81
wage, so we can subsititute for w in terms of p in any momentary equilibrium, to
reason that v = αap. What then is the boundary value of profits, below which no
R&D takes place? At the point where there is no research, all employment is in
manufacturing and p = 1/L. Substituting for p we find:
v =aα
Lwith n > 0 when v > v. Also, by a rearrangement of (100):
n =1
aL−
1
ap
from which ap can be substituted out for v/α to yield an equation in n and v alone:
n =
{
La− α
vif v > v
0 otherwise(102)
This gives us enough information, along with (101), to draw a phase diagram in
{n, v} space.
[phase diagram]
There are two main phases of economic development associated with this model.
First, there is a phase with positive research and an increasing number of varieties.
This results in increased TFP, and therefore growth in output. As more firms
manufacturing more commodities compete for a fixed supply of labour, wages have
to go up; since markups are constant, that means sales per brand have to fall.
The depressing effect of lower sales per brand eventually causes research into new
brands to become unprofitable. At this point, R&D ceases, no new varieties are
introduced, and the economy as a whole stops growing. There is an analogy here
with the cessation of growth in the neoclassical model: in both cases, private returns
to some reproduceable factor (capital, the stock of varieties) eventually get small
enough that people prefer to consume now rather than invest for a low return. As
suggested previously in the context of the neoclassical model, the way to prevent
this happening is to keep returns high. As with the Romer model of §4.2, a way to
achieve this is to recognise the spillovers that occur when new knowledge is produced.
6.3.3 Endogenous growth
We now modify the production technology for new blueprints to reflect spillovers
in the research process. Specifically, an incremental design is produced using re-
search labour and a general stock of pre-existing knowledge that can be tapped
82 Economic Growth
at no expense. Letting K denote aggregate knowledge, and Ln aggregate research
employment, the rate of increase in new designs is:
n =LnK
a(103)
Following Grossman and Helpman, we specify the link between individual knowledge
production and the aggregate stock of knowledge as simply proportional to past
R&D. Later, we will see the consequences of that decision, but for now simply choose
units so that K = n (unit coefficient of proportionality). How does this modify the
dynamic equilibrium described above? First, we recall that for R&D to go ahead,
we need the profit from an incremental design vdn to be at least as great as its cost
of production wLndt. With our new research technology, and recalling that in any
momentary equilibrium free entry limits excess returns, we have a modified version
of (100):
v ≤wa
n
{
equality n > 0
inequality n = 0
We also have a change to the boundary value of a new design at which it becomes
unprofitable to do research:
v′ =aα
nL
so (102) becomes:
g :=n
n=
{
La− α
vnif v > v′
0 otherwise
It turns out that by defining a new variable with the interpretation of the inverse
of aggregate profits V := 1/nv, we can write the whole system in terms of one
differential equation. From (101), and the definition of V :
V
V= −
n
n−v
v=⇒
V
V=
1
εV − g − ρ
which we can draw in {g, V } space. Setting V = 0, we see that the nullcline for
V is an upward sloping straight line of slope ε. To the right of this line, growth
of varieties is faster; this means R&D employment is higher. Higher employment
in R&D means lower employment in manufacturing, and therefore a lower supply
Roland Meeks 83
of goods, higher prices, and higher firm values (lower V ). To the left of the stable
manifold, the rate of growth in varieties is falling, as is the value of firms; however,
the value of firms is falling faster, so even when g = 0, profits per brand continue to
fall.
[phase diagram]
Ultimately, expectations are only fulfulled at one point. If profits are stable, the
steady state growth rate of innovation is:
g =L
aε−ε− 1
ερ
In this economy, there are exactly offsetting forces of profit erosion due to brand
proliferation on one hand, and falling costs to innovation because of research exter-
nalities on the other.
6.4 Semi-endogenous growth
To use an analogy from the computer software industry, are scale effects
a bug or a feature? ... [o]verall they are a feature [but] in the first
generation of idea-based growth models, these scale effects appeared in
an especially potent way ... This strong form of scale effects – in which
the long-run growth rate of the economy depends on its scale – is a bug.
– Jones (2004)
An objection to the specification above was made by Jones (1995a). The essence
of it is that (103) implies a scale effect, i.e. the prediction that an increase in
the resources devoted to R&D should raise the growth rate of the economy. That
prediction comes through in the expression we derived for the growth rate of the
varieties economy via dependence on L. As employment in R&D has been trending
upward since the 1960s, this implies faster growth rates; as Jones (1995b) shows
however, this is inconsistent with the evidence.
To map between what we just did and what Jones does, remember that expand-
ing varieties cause TFP growth when there is a CES aggregator, so increasing the
number of varieties is like expanding the quantity of technical knowledge available
for producing final goods. Then (using the standard notation for TFP of A) we wish
to contrast:
A = δAφLλA (104)
84 Economic Growth
(LA is employment in research) with φ = 1 and λ = 1 – the specification we just
looked at – with some alternatives. The parameter φ controls the strength of research
spillovers, while the parameter λ ≤ 1 controls the degree of duplication externalities
that can occur in research.
Start with the case of λ = 1 (no duplication). If it were the case that φ < 0,
the rate of innovation decreases with the level of knowledge (so Newton and Darwin
really did have the best ideas). When φ > 0, there are positive spillovers to research,
but note that φ = 1 is a ‘completely arbitrary degree of increasing returns’ (p. 766).
In the first pass at the varieties model, we had φ = 0, the intermediate case of
constant returns to scale. Then there was no long lasting growth. In the second
case, we had φ = 1 and endogenous growth.
To get some intuition, consider what would happen in a toy model with output
depending on labour input (with constant returns) and knowledge (with elasticity
σ ≥ 0):
Yt = Aσt LY,t
Then on a BGP, the growth rate of knowledge is a constant, so (104) implies that:
0 = δAφ−1LnLn
Ln
+ (φ− 1)δAφ−1LnA
Aor gL = (1 − φ)gA
the growth rate of ideas is proportional to the growth rate of the labour force. That
means that output growth is also proportional to the growth rate of the labour force
on the BGP:
gy =σ
1 − φgL
and the scale effect has been eliminated. Why has it been eliminated? Because
before the growth rate of ideas depended only on the number of people employed
in research, and now it depends on the growth rate of the research workforce. New
ideas are proportional to the number of people looking for them, but the percentage
increase in new ideas a given number of people can find is inversely related to the
current stock of ideas. On a BGP, there must be a constant percentage increase in
the stock of ideas, and so knowledge advances in proportion to the rate of increase
in the number of people looking. Jones (2004, §6.2) suggests the following thought
experiment: holding choice variables constant, is there any reason why doubling the
Roland Meeks 85
value of a state variable should double its rate of change? In the case of R&D, there
seems little compelling reason why knowing twice as much should double the rate
at which we learn.
Jones’ conclusion is that with his semi-endogenous growth model, although long-
run income per capita growth is infeasible, in practice the transition to steady state
is long enough to be significant. Although it has the same predictions as Solow, the
result now derives from a model in which there are private incentives to knowledge
acquisition.
86 Economic Growth
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