1

CHAPTER 4 Gauging an Economy’s Productive Capacity In this chapter you will learn to:

LO 4.1 Explain the production function and the assumptions that determine how factors of production are combined to produce an output.

LO 4.2 Show how the Cobb-Douglas Production Function can reflect some important features of an

economy. LO 4.3 Show the relationship over time between a country’s capital stock and its saving rate, using the

dynamic law of motion for the capital stock. LO 4.4 Show how growth of a country’s capital stock, its labor force, and its total factor productivity

each contribute to output growth, according to the Solow Decomposition. LO 4.5 Explain why output, investment, and consumption will converge to a steady state in the long run.

Chapter Overview

We noted at the beginning of this text that growth and productivity was a major issue in

macroeconomics. What makes an economy grow? Why do some economies grow more rapidly than

others? What policies will help our country grow more?

You may be skeptical about placing so much importance on these questions. After all, just to produce

and consume more things at an ever faster rate cannot be life’s ultimate goal. A broader goal, and one

that would be agreeable to most of us, would be to raise everyone's standard of living. Of course, the

standard of living is a multi-faceted concept that is difficult to measure. Economists have long

recognized that a country’s output is a highly imperfect indicator of their well-being. We might look at

how much people consume – what they enjoy consuming -- rather than what they produce.

2

Moreover, the term consumption still does not adequately convey the broader notion of that the term

standard of living implies, much less other more deeply fulfilling aspects of life. Even so, we also cannot

deny that more economic growth can help to raise the standard of living dramatically for substantial

segments of the world’s population. As we will see later in this chapter, most of this progress has

occurred during the past two centuries.

Why did economic growth – and with it the standard of living -- stagnate for 1700 years and

then, in very little time, head for the stratosphere? A critical factor has been technological progress.

(key term: technological progress: inventions and innovations that permit more or improved production

of goods and services). The period from 1820 to 2000 included both the second industrial revolution

during the mid-19th and early 20th centuries, and the revolution in computers and information

technology that took place during the latter part of the 20th century. Prior to that, technology did

advance, but only very slowly. 1 One development that took place early in the Industrial Revolution, the

use of steam engines, opened the door for factory productions and motorized transportation. Later,

harnessing of water and fossil fuels to produce electricity, the internal combustion engine that

permitted us to build machines that not only transported us more rapidly (on land, sea and air), were all

developments that formed foundations of modern industrial economies.

However, it isn’t just technological progress that helps us grow. Economists typically focus on two main

factors of production. First, in order to produce anything, human beings themselves must contribute

some effort. They summarize this factor as labor – measured either in terms of the number of people

who work, or the number of total hours that people dedicate to work in a given period of time (a month,

a year, etc.). Second, we typically require some kind of plant and or equipment—something to help

produce goods and services -- is typically necessary. Economists summarize this factor as the capital

stock -- the accumulation of capital investment that took place in previous periods. It is an economy’s

savings (the difference between its disposable income and its consumption) that will be available to

firms so they can invest. In this way, the economy accumulates even more capital that will be used to

produce other goods. If we consume more today, we save less, we invest less, and we have less

productive capital in the future.

1 One major development during this time was the printing press, invented by Johannes Guttenberg around the year 1440; this brought about the publication of more books and, as a complement, higher literacy rates.

3

This chapter develops several tools that will help us understand more clearly how output, investment,

and consumption are related to one another. We will use an important analytical tool in economics, the

production function, which will reveal how factors like labor and capital are combined to yield output.

We will also be able to pin down more precisely how investment, capital goods, and output are linked

together over time. Our framework will be based on the Solow/Swan growth model (named after

economists Robert Solow (1924– ) of the Massachusetts Institute of Technology and Trevor Swan

(1918–1989) of the Australian National University). We will use the model to show how capital, labor,

and technological progress have all contributed to economic growth. We will also see that two other

factors have played a key role: abundant energy, including fossil fuels, and the level of education and

training of the work force.

4.1 The Production Function

LO 4.1 Explain the production function and the main assumptions that determine how factors of production are combined to produce an output. The production of even the simplest good – be it a box of fruit or a shirt – can be a complex process and

may require many inputs. We typically need some land on which to place our workplace; the workplace

may need some furniture, an electrical hook-up, some machinery (a hammer and nails to construct to

put nails in for the fruit box, a sewing machine and some scissors to transform the fabric into a

garment), and of course we need the input of human beings who produce something – labor. As goods

become more complex, the production process itself becomes more complex and the number of inputs

typically rises.

Even so, simple models of the real world can help us understand otherwise complex processes. To think

about an economy’s capacity to produce goods and services, economists often begin with a simple

production function that summarizes how certain input factors are transformed into outputs. [key term:

production function; an expression that summarizes the production process of goods or services] The

production function is governed by some simple but sensible assumptions about the production

process.

LO 2.1 LO 2.1

LO 4.1

4

The General Form and Assumptions of a Production Function

To produce any good or service, we assume that two factors of production are required. Capital (K)

summarizes the infrastructure required for production: buildings, machines, computers, and so on. 2

Labor (L) summarizes the human effort brought to bear on the production process, measured as the

number of workers.3 Typically, the owners of capital hire workers in a labor market. However, the

owners of capital themselves – individuals called entrepreneurs – may also contribute labor to the

production process. 4

We will analyze an economy that has no government and is closed – no imports or exports. Implicitly,

we will add up, or aggregate, all firms and sectors into a single unit. The aggregate production function

for that economy tells us the volume of total output for the economy. In its most general form, an

aggregate production function is:

Here, Y is total real GDP. There are two factors of production: the stock of capital K and the number of

workers L (labor). The component total factor productivity A captures other aspects of production. One

example might be the level of technology. If the citizens of an economy implement a new and more

productive technology, A will rise. For now, we will assume A to be constant at unity. Later in the

chapter, we will analyze the effect of changes in A.

We must now make some assumptions about how K and L are combined to produce output Y. We will

first assume that when we use more of either, we produce more. That is, the change in output that

results from a one-unit increase in either K or L is positive -- that is, the marginal products of both capital

2 As we discussed in Chapter 3, two kinds of capital are often reported: nonresidential and residential. The latter RES represents the houses and apartments where people reside. Thus, in real data, the total capital stock KTOT

is the sum of nonresidential capital K and residential capital KRES. The latter provides another service: shelter. 3 It would be more accurate to say that L measures person-hours: the number of workers times the number of hours worked. In our first model, we will assume that each worker works for same number of hours. 4 We can also introduce land (“T” for terra) into the production function. However, for the analysis here we are assuming that the amount of land in use is fixed.

Assumed initially to be co

Total Total Capital - LaborOutput Factor value of number of

Producitivity mach

n

inery worker

sta

s

nt

Y = A * f( K , L ) (4.1)

5

and labor, mpk and mpl respectively, are positive. The more we use of either factor, the more output

rises. We can formalize this idea with the marginal products of capital and labor, mpk and mpl, defined

as the change in output that results from employing one additional unit of each input holding the other

input constant:

Here, the term A gives us important information: if A rises, the marginal products of both capital and

labor must also rise. As an example, consider the evolution of mobile phones – from the first cell phones

to the latest smart phone. Mobile phones have increased productivity for many professions. For

example, imagine a salesperson that visits potential customers. Before cell phones, she might have

needed to find a public pay telephone to update meeting plans. With a cell phone, she can make and

revise plans on the spot. As these devices became more sophisticated, more complex information –

photographs of products, detailed proposals, even audiovisual presentations – she is able to send this to

clients anytime, anywhere. Hence, in this example the marginal product of both capital (the mobile

phone) and labor (the salesperson) have risen.

Second, we assume that marginal products are diminishing: as we use more and more of each input,

their marginal products fall. (This assumption should not be new to you: remember that the assumption

of diminishing marginal products is nearly universal in economic analysis.)

Third, we assume that the production function exhibits constant returns to scale: if we multiply all

factors by x our output will increase also by x:

To see why this assumption makes sense, consider an example. Suppose that a firm’s capital -- plant and

equipment – is valued at 4500 (K=4500), that the firm employs 1000 workers (L=1000) and that it

MarginalProduct All else equal, an

of Capital extra unit of capital will producemore oPositi utve put.

ΔYmpk = > 0

ΔK

MarginalProduct All else equal, an of Labor extra unit of labor will produce

more output.Positive

ΔYmpl = > 0

ΔL

Total Total Capital LaborOutput

Constant returns to s

FactorProducitiv

cale

ity

Y = A * f( K , L )x x x

(4.2)

(4.3)

(4.4)

6

produces 1500 units (Y=1500). Under the assumption of constant returns to scale, if we doubled the

scale of the factory to 9000 and the number of workers to 2000, total output would also double to 3000.

Suppose now that doubling inputs of K and L more than doubles output. This would mean that the

average cost per unit of output would also be falling. As you may remember from your first course in

economics, when there is room for average costs to fall, firms will expand their scale or output; any

individual firm will be forced to do so by the other competing firms in that market. But, by making the

assumption of constant returns to scale, we’ve effectively assumed that those firms have already

entered the market and they have driven average costs down to their minimum – the point at which all

economies of scale have been exhausted.

An Example of a Production Function

Figure 4.1 illustrates a production function. Here, we assume that both total factor productivity (A) and

the labor supply (L) are constant. In this way, we see how output increases when we increase capital. If

there is no capital, there is no output (K=Y=0). As capital increases by increments of 100 -- that is,

K=4500, 4600, 4700, and so on -- output increases. Thus we have confirmed that the marginal product of

capital is positive.

We see also that the marginal product of capital is diminishing. Note that the slope of the line, while

positive, becomes progressively less and less steep as the usage of capital K increases. As a matter of

geometry (you may remember this from high school), the slope of a line is the “rise” of the line divided

by the “run” of the line. In this case, the rise divided by the run is Y divided by K – the marginal

product of capital. Hence we know that the marginal product of capital is diminishing.

The table in Figure 4.1 will also confirm that the marginal product of capital is diminishing. We have

calculated the interval estimate of mpk=Y/K. For example, when K increases by 100 from 4500 to

4600, output Y increases from 1570.2 to 1580.6; here Y=10.39 and the marginal product of capital is

mpk=Y/K= 0.1039. Then, as capital usage increases, the interval estimate of mpk gets progressively

smaller and smaller.

7

Notice also that the slope continually changes – even between the increments of 100 units of K. Since

the slope is changing within the interval, the interval estimate that we got is only an approximation – it

is not very precise. A more precise estimate of the mpk is a point estimate of the slope for a very small

increase in capital. In the next section, we will learn to compute a more precise estimate.

In our example we have also assumed labor L to be constant. However, the analysis would then be

symmetric. Sometimes, what we want to know is not the total output Y but the output per worker Y/L.

Our assumption of constant returns to scale assures us that we can express Y/L as a function of the

capital stock per worker K/L – this is sometimes known as “capital-to-labor ratio.”

How do we know that we can do this? In Equation (4.4) we just set x=1/L. Thus we would have:

*Total ConstantOutput per Capital per

FactorWorker WorkerProducitivity

Y / L = A f( K / L , 1 ) (4.5)

Figure 4.2 The Production Function

1610.9

1600.9

1590.9

1580.6

1570.2

4500 4600 4700 4800 4900

K Y K Y mpk=Y/K

4500 1,570.2

4600 1,580.6 100 10.39 0.1039

4700 1,590.9 100 10.23 0.1023

4800 1,600.9 100 10.08 0.1008

4900 1,610.9 100 9.93 0.0993

NOTE: Total Factor Productivy (A) and Labor (L) are both constant.

output Y = Af(K,L);o

utp

ut Y

= A

f(K

,L)

capital (K)

point estimate of mpk=slope of line

Interval estimates of the marginal product of capital (mpk): positive and diminshing.

Figure 4.1 The Production Function

8

Note that when we divide the number of workers by itself, we get L/L = 1. This is a constant term that

simply ‘drops out’ of the analysis. This becomes a convenient trick that will help us in our calculations.

4.2 Getting More Specific: The Cobb-Douglas Production Function

LO 4.2 Show how the Cobb-Douglas Production Function can reflect some important features of an economy.

Above, we considered a general idea of a production function and we discussed some widely used

assumptions. A general production function permits us to discuss issues on a broad level. But,

sometimes we need to be more specific. Often, we must phrase our ideas in a way that we can plug in

some numbers and make some calculations ourselves.

Economists frequently use a functional form to help them implement specific assumptions. [key term:

functional form: a specific form of an economic model that permits the analyst to implement their

specific assumptions] This is a convenient way for us to plug in numbers and make our own calculations,

even if we give up a bit of generality. But, when we choose a specific functional form, we want to make

sure that we have retained some assumptions of our more general formulation.

When economists discuss output and productivity, and they need to have a specific functional form to

help them make illustrative calculations, they frequently turn to one that is named after two pioneering

researchers -- Charles Wiggans Cobb (1875–1949), who was both an economist and an mathematician,

and Paul Howard Douglas (1892–1976), who also served as a U.S. Senator from Illinois. Jointly, these two

economists conducted pioneering research about the relationship between factors of production and

output that we see in actual macroeconomic data. They developed a way of looking at production that

has since become known as the Cobb-Douglas production function, written as:

The links between the general production function, Equation (4.1), and the Cobb-Douglas production

function, Equation (4.6), are readily apparent. Both tell us that total output is a function of capital and

labor. Both also include a measurement of “other factors” – total factor productivity. But, unlike the

*Total Total Capital Labor

Output FactorProduciti

α

vity

α 1-Y = A K *L (4.6)

LO 4.2

9

general production function, the Cobb-Douglas function allows us to plug actual numbers into it. For

example, one important feature of the Cobb-Douglas is its exponential form: we raise capital K to the

power (alpha) and labor L to the power 1- (one minus alpha). In our first example, shown in Table

4.1, we will assume that A=1 and that alpha () is 0.3.

We will treat our assumption about A as arbitrary for now. Later, when we apply a Cobb Douglas

function to real data, we will also learn how to calculate A. But, our assumption that =0.3 is an

important one. The research by Cobb and Douglas suggests that should be approximately 0.3. (This

means that 1- would have to be approximately 0.7.) Below, we will see why this assumption is

reasonable and remains widely used, even today.

More importantly, as the simple example in Table 4.1 illustrates, the Cobb-Douglas production function

will give us the three properties of the general production function that we discussed above: (i) marginal

products of capital and labor will always be positive, (ii) marginal products are diminishing, and (iii) there

are constant returns to scale.5

Part A of Table 4.1 shows what happens if we keep the number of workers constant at 1000 (L=1000),

but we vary the amount of capital used: K=4500, 4600, 4700, and so on. We see many of the same

numbers that appeared in Figure 4.1. As we increase the amount of capital that we use (see second

column from the left), we see that output Y also rises (leftmost column). In the fourth and fifth columns

from the left, we calculate the change in output Y and the change in capital K – which is always 100.

For our initial level of capital, 4500, our output is 1570.2. When we boost our capital usage to 4600,

output rises to 1580.6. As before, the marginal product of capital, mpk, equals 0.1039. As K increases,

mpk goes down. Hence, for capital we have diminishing marginal productivity.

Part B of Table 4.1 shows what happens if we keep the amount of capital K constant at the initial value

(4500), but we vary the amount of labor used by increments of 100: L=1000, 1100, 1200, and so on.

5 In reality, this is a bit restrictive: as long as 0<<1, marginal products will be positive and diminishing and there will be constant returns to scale.

Table 4.1 Cobb-Douglas Production Function:An Example

10

A= 1

= 0.3

A. Labor constant, capital variable

Y K L Y K mpk=Y/K payment to

capital=mpk*K

Capital's

share=payment to

capital/GDP

1570.2 4500 1000

1580.6 4600 1000 10.39 100.0 0.104 477.84 0.30

1590.9 4700 1000 10.23 100.0 0.102 480.85 0.30

1600.9 4800 1000 10.08 100.0 0.101 483.82 0.30

1610.9 4900 1000 9.93 100.0 0.099 486.75 0.30

1600.9 4800 1000

1601.0 4801 1000 0.10 1.00 0.100 480.34 0.30

B. Capital constant, labor variable

Y K L Y L mpl=Y/L payment to

laborl=mpl*L

Labor's

Share=payment to

capital/GDP

1570.2 4500 1000.000

1678.6 4500 1100.000 108.34 100.0 1.083 1191.69 0.71

1784.0 4500 1200.000 105.42 100.0 1.054 1264.99 0.71

1987.3 4500 1400.000 203.27 200.0 1.016 1422.90 0.72

2085.6 4500 1500.000 98.33 100.0 0.983 1474.95 0.71

1570.2 4500 1000

1571.3 4500 1001 1.10 1.00 1.099 1100.10 0.70

C. Both capital and labor variable

Y K L

1570.2 4500 1000

3140.5 9000 2000

4710.7 13500 3000

4710.7 13500 3000

7851.2 22500 5000

9421.4 27000 6000

Interval estimtes of mpk: positive and diminishing.

(ii) constant returns to scale.

Shares for capital and labor -using point estimates of mpk and mpl.

Point estimate of mpk.

Interval estimates of mpl: positive and diminishing.

Point estimate of mpl.

*Total Total Capital Labor

Output FactorProduciti

α

vity

α 1-Y = A K *L

11

As was the case with capital, as we increase the number of workers we use (second column from the

left), output Y also rises (leftmost column). In the fourth and fifth columns from the left, we calculate the

change in output Y and the change in labor at L at 100. When we boost our labor usage from 1000 to

1100, output rises from 1570.2 to 1678.6; this is an increase in output of Y 108.34. The marginal

product of labor, mpl, is simply the amount of extra output that we obtain when we use an additional

worker. As was the case with capital, as we increase the number of workers we see that the mpl, while

still positive, falls. Hence, for labor, as for capital, the marginal product is diminishing.

Part C of Table 4.1 shows what happens when both capital K and labor L are allowed to vary. If K=4500

and L=1100, Y=1570.2. If we double both factors (to K=2000 and L=9000), output also doubles to 3140.5.

More generally, if we multiply initial usage of capital and labor by some number x, the output will also

increase x times. Thus, we confirm: the Cobb Douglas production function exhibits constant returns to

scale.

Keep in mind that the calculations of mpk=Y/K for increments of K=100 and mpl=Y/L for

increments L=100 are interval estimates -- not point estimates. As we discussed above, such estimates

are imprecise. To obtain a more precise point estimate, we would use very small changes in capital and

labor. We thus plug very small changes for K and L into the formula. We do this in Parts A and B of Table

4.1; we assume that both K and L increase by 1 unit. In Part A, when we increase capital from K = 4800

to 4801 units, the mpk will be 0.10; this result lies between 0.101 and 0.099. In Part B, as L is increased

from 1000 to 1001, the mpl will be 1.099; this is somewhat above the nearest interval estimate, 1.083.

To see why point estimates might be useful, we have to recall a key principle from our microeconomic

toolbox: factors are paid their marginal product.6 Each unit of capital will receive as compensation its

mpk; each additional laborer (or person-hour of labor) is paid mpl.

Our units need not be discrete integers. For example, in the case of labor, we might think of increasing

the amount of persons that we use, and that increase need not be discrete: we might hire a half-time

worker (i.e., 0.5 worker) or even a quarter-time worker (i.e. 0.25 worker). For this reason, we need to

have a continuous or point estimate of the marginal product of labor.

6 We assume that firms are free to enter and exit into all markets. Such an assumption is consistent with a perfectly competitive economy.

12

Using point estimates will help us better understand our assumption that =0.3. First, let’s calculate the

total payments that each of the two factors will receive:

Now, we can calculate the shares that capital and labor will each receive:

Here, if we use the more accurate measure of mpk and mpl, we find that capital’s share and labor’s

share are 30% and 70%, respectively. In Table 4.1, these calculations appear in the rightmost column, in

the last lines of Part A (capital) and Part B (labor), respectively. The calculations reflect the assumption

that =0.3 and (1-)=0.7.

This last result is not an accident. For a Cobb-Douglas production function, when we use point estimates

of the mpk and mpl, the shares of capital and labor (mpk*K/Y, mpl*L/Y) must be and 1- respectively.

This is a very attractive feature of the Cobb-Douglas production function – and one of the reasons why

this functional form is so widely used in economic analysis.

Further to the Point: Evidence Favoring the Cobb-Douglas Parameters.

A glance at the data makes clear why the assumption that =0.3 is so widely

used: as Figure 4.2 shows, for the United States, between 1932 and 2011 capital’s share averaged 31%

and the remainder for labor was about 69%. This also shows why it is so important that the exponents in

the Cobb-Douglas function add up to unity. If they did not, the shares of capital and labor would not sum

Amount ofMarginal product capital used of capitalin production

Total Payment To Capital = mpk * K

Number Marginal product of workers used of labor

in production

Total Payment To Labor = mpl * L

*Amount ofMarginal product

capital used of capitalin production

Capital's share = Total Payment To Capital / Y = [ mpk K ] / Y

Amount ofMarginal product labor used of labor

in production

Labor's share = Total Payment To Labor / Y = [ mpl * L ] / Y

(4.7)

(4.8)

(4.9)

(4.10)

Further to the Point….

13

to 100%. This is not possible: the compensation to factors of production must correspond exactly to

total output. End Further To the Point

4.3 Savings and the Capital Stock over Time: The Law of Motion

LO 4.3 Show the relationship over time between a country’s capital stock and its saving rate, using the dynamic law of motion for the capital stock.

As we have previously discussed, when households save, they accumulate assets. To a large degree,

such assets are financial: money, bank deposits, bonds, or equity claims. But, we also know how such

savings flows were intermediated through the financial system to firms, to help them make investments

in physical capital. In turn, capital goods are used to produce other goods in the future – goods that are

both consumed and invested.

Now, let's see more precisely how saving and investment help boost economic activity over time. To do

so, we will use the Solow/Swan growth model introduced above. 7

7 Solow obtained more recognition for his work than Swan, including a Nobel Prize, but Solow himself publicly acknowledged the key role of Swan. See Solow, Robert, "A Contribution to the Theory of Economic Growth." Quarterly Journal of Economics 70 (1): 65–94 (1956); and Swan, Trevor, “Economic Growth and Capital Accumulation,” Economic Record (November 1956) 32:2, 334–361.

0

10

20

30

40

50

60

70

80

19

32

19

35

19

38

19

41

19

44

19

47

19

50

19

53

19

56

19

59

19

62

19

65

19

68

19

71

19

74

19

77

19

80

19

83

19

86

19

89

19

92

19

95

19

98

20

01

20

04

20

07

20

10

In p

erc

en

t

Shares of Capital and Labor

Labor Share (mean=69%)

Capital Share (mean=31%)

Source: US Bureau of Econ Analysis

Figure 4.2: United States Shares of Income for Capital and Labor

LO 4.3

14

We’ve assumed a very simple economy: it is closed (no imports or exports) and there is no government.

Thus, there are only two alternative uses for our output: consumption or investment. We summarize

this as:

That is, some of our output will be goods that can be consumed today, while the remainder is capital

goods that are used to produce other goods in the future. In addition, saving (the difference between

output and consumption) must equal investment. We summarize this as:

Using the Solow/Swan model, we have to make assumptions about how much people (or households)

save. In later chapters, we will discuss some of the determinants of consumption and savings. We will

learn that families may increase or decrease their savings rates in response to a number of different

factors: their wealth and their income (today and in the future), interest rates, age, how many

dependents are in a family, and uncertainty.

However, in this model we abstract from these considerations to make a simplifying assumption:

households in this economy save a constant fraction of total output. Hence, we write the savings

function as:

To illustrate, consider the data in Table 4.2. These data apply a savings rate of 15% (=0.15) to the

previous Cobb-Douglas example. The data show that if = 0.15 (15%) and Y=1570.2, as calculated from

the Cobb Douglas production function, total savings would be $235.5.

Enjoy today Build capital for future

Total Consumption - Investment -Output-

toda

Two uses for output

t t

productiony

tY = C + I

Enjoy today Build capital for future production

The familiar savings - investment identity (cl

Total Consumption - Saving Investment -O

osed economy, no government)

t

utpu

t t

t

tY - C = S = I

, )ProductionSaving Fixed fraction functionby households of output

t tS = σY σAf(K L=

(4.11)

(4.12)

(4.13)

15

The Stock-Flow Distinction

When someone makes an investment expenditure, they build up their productive capital. The

relationship between investment expenditures and capital is the first of many relationships that involve

the relationship between stocks and flows:

A stock is a value that is measured at a single point in time and that exists independently of any unit of

time. [key term: stock; a value that is measured at a single point in time and that exists independently of

any unit of time. ] Consider a building that is used for business. It makes sense to say that “The building

is worth $1 Million dollars.” This is the stock value. It does not make sense to say “The building is worth

$1 Million per year.” A building is not something that ‘happens’ during a year. In common usage, a

“stock” can sometimes refer to the financial assets that are more correctly known as equity shares of a

company. Hence, in a “stock market” – for example the New York Stock Exchange – it is more correct to

say that people trade equity claims on firms. 8

A flow is the value of some transaction or other economic event that takes place during a specific period

of time. [key word: flow; the value of some transaction or other economic event that takes place during

a specific period of time.] For example, it is correct to talk about how much a firm sells each year. It is

8 However, we might say that a building generates $1 Million of income each year. In this case, in calculating the value of the building, we would have to recognize the income in both now and in the future. We would have to calculate the present value of the building. This concept is discussed in Chapter 4.

Table 4.2

The Savings Function, An Example

Total Factor Productivity A 1

Capital K 4500.0

Labor L 1000

Capital Share 0.3

Savings Rate 0.15

Output Y 1570

Savings S=Y 235.5

Note: Production function is Cobb-Douglas

Assumptions

Results

16

sensible to say that a firm“…sells $500,000 per year.” Since flows take place at a specific time, we must

specify that what that period of time is – week, month, year, or decade. A person that receives $100,000

in income each year for a decade $1,000,000 in total.

The value of a stock may change between one point in time and another. If so, there must be a

corresponding flow that explains the change in the stock. This is how we explain the evolution of the

capital stock over time. The capital stock at the end of the current period tK reflects the fact that we

began the period with a capital stock of t-1K (end of previous period, t-1); during the period there were

expenditures on capital goods whose amount was tI and the estimated depreciation on the capital stock

was tDEP . Hence, we write the equation that describes the evolution of the capital stock over time as:

Here is an example of investment and capital. Suppose that we want to know the value of plant and

equipment for a firm at the end of the current year (t). At the end of the previous year (t-1), that stock

was $1 Million. During year t, the company added a new facility that meant more productive capacity.

It paid $120,000 to add on these additions. Thus, the company’s gross non-residential investment was

$120K. At the same time, over the year, through normal wear-and-tear, the value of the existing capital

fell – that is, it depreciated by some amount. This was estimated to be $15,000. Thus, the company’s

capital stock at the end of the year is $1,105,000. This is its initial capital stock at the end of the previous

period ($1 Million) plus the gross (non-residential) investment – a flow of $120 K – minus the

depreciation – a flow of $15K. Another frequently used term is “net investment” which is simply gross

investment minus depreciation. In our example, net investment would be $120K - $15K equals $105K.

When a firm (perhaps a construction company) makes expenditures to build places where people live

(for example, houses or apartments), it makes residential investment expenditures. In so doing, they

increase the stock of residential capital RESK . And, like non-residential capital, there is a corresponding

stock-flow equation for the stock of residential capital.

Capital Capital Investment Depreciation Current Period Previous Period Expenditures

Current Period

Stock

t t-1

Current Perio

StockFlow

Flow

t

d

tK = K + I - DEP (4.14)

17

Stocks and Flows in the Solow/Swan Model

In the Solow/Swan model, we assume that each period, a fixed percentage of the capital stock δ

depreciates away – the wear and tear mentioned earlier. Thus, the law of motion for the capital stock

may be written:

In words, this equation tells us that today’s capital stock equals the capital stock that we inherited from

the previous period, minus depreciation and plus new capital investment. Note that we have put a

subscript of t-1 on the investment term. This reflects the assumption that it takes one period for newly

capital expenditures to become productive – to come "online." We can rewrite this equation in another

way:

The term t-1δK is the volume of depreciation. This is simply the total value of the wear and tear on the

existing stock of capital. To visualize this, consider trucks that are used to haul goods from their factory

to a point of sale. The performance of certain parts on the truck – tires, fan belts, engine, radiator, water

pump, and so on – will deteriorate or cease altogether as a result of normal use of the truck.

Periodically, we have to replace these parts in order to keep the truck useful. Importantly, the volume of

depreciation is a linear function: stays constant. To see what this means, suppose we have one truck

with four wheels. Periodically, we’d have to replace all 4 tires. If we had two such trucks, we’d have to

replace all 8 tires, and so on.

Net of depreciationToday's capital Yesterday's capital stock - Gros

Law of motion of capit

s stoc

al sto

k investmen

c

t t-1 t-1

k

t

K = K (1 -δ) + I

Total saving (identically equal to

t t-1 t-1 tVolToday's capital Yesterday's

stock capitume of depreciaton --

the "wear and tea inv

r" oal stock n theexiestm stine g capitant l s

-

. k

1

t) oc

K = K + σY δK

(4.16)

(4.17)

RES RES RES RESt t-1 t tK = K +I -DEP (4.15)

18

Table 4.3 shows how to calculate the capital stock, using the example from the previous section. At the

end of the previous period, the capital stock was $4500. During the period, total savings was $235.5,

which we add to the capital stock. The depreciation rate δ is assumed to be 0.4 (4%). Hence the volume

of depreciation will be $180 – a number that we subtract from the initial capital stock. Hence, the capital

stock at the end of the period will be $4555.5 (=4500+235-180).

4.4 Accounting for Economic Growth: A Production Function Approach

LO 4.4 Show how growth of a country’s capital stock, its labor force, and its total factor productivity each contribute to output growth, according to the Solow Decomposition.

As we discussed above, an advantage of using a specific functional form is that we are able to plug in

some numbers and make some calculations. Our goal here is to learn something real about the

economy. In this chapter, we have been using the Cobb-Douglas functional form. Until now, we have

focused our efforts on some preliminary exercises that used artificial data.

Total Factor Productivity A 1

Labor L 1000

Capital, end of previous period Kt-1 4500.0

Output Y 1570

Savings Rate 0.15

Savings =Investment S=Y=I 235.5

Depreciation Rate 0.04

Volume of depreciation Kt-1 180.0

Capital, end of current period Kt-1 4555.5

Note: Production function is Cobb-Douglas

Capital Share 0.3

Initial value

Add

Subtract

Final value

Table 4.3: The Evolution of the Capital Stock.

LO 4.4

19

Now, we are ready "get our hands dirty" with data from a real economy, the United States. We will

apply U.S. data to a Cobb-Douglas function. Our aim in doing so is to learn more about why we observed

a certain growth rate in the economy during a given period of time. We want to know the extent to

which two main factors -- namely, capital per worker and total factor productivity -- help explain the

growth of output on a per-worker basis. Our analysis will take place on a "medium-term" basis: we will

look at the rate of economic growth that took place on a decade-by-decade (rather than year-by-year)

basis.

We begin with data for the United States that are indicated by the Cobb Douglas production function

(equation 4.6 above). In Table 4.4, the two left columns of data output, Y and the capital stock K, both

measured in billions of chained U.S. dollars (base year = 2005). Directly to the right, the number of

workers, L, is measured in thousands. Directly to the right, we show our first calculations: output per

worker and capital stock per worker, Y/L and K/L, respectively. We must be careful about the units here:

output and capital are measured in billions while workers are measured in thousands. If we multiply the

former terms by 1000, we will be measuring Y/L and K/L in thousands of dollars per worker.

Thus, this table confirms that the average level of output per worker in the United States has almost

tripled over a 60-year period, rising from $34 thousand dollars per worker in 1950 (just after the end of

World War II) to almost $95 thousand dollars in 2010.

This last number may seem to be out of line with the average earnings of the people that you may know.

Such a number appear to be too "high" when compared with some other, more familiar, concepts. First,

this number is on a per worker basis – not in terms of the total population (per capita). The total

population in an economy is greater than the number of people working. Hence, output per worker will

always exceed output per capita.

Second, we are measuring output, not household disposable income. Indeed, median household

income in recent years in the United States has been considerably less – around $50 Thousand dollars.

(According to the definition of ‘median’, half of the population earns more than $50 Thousand and the

other half earns less.)

20

Output (Y) Capital (K) Labor (L) Y/L K/L A=(Y/L)/[(K/L)] (K/L)

1950 2004.3 5990.1 58,921 34.0 101.7 8.50 4.00

1960 2828.5 8573.4 65,776 43.0 130.3 9.98 4.31

1970 4266.3 12162.5 78,678 54.2 154.6 11.95 4.54

1980 5834.0 16666.1 99,303 58.7 167.8 12.63 4.65

1990 8027.0 22520.4 118,793 67.6 189.6 14.01 4.82

2000 11216.4 29084.2 136,891 81.9 212.5 16.42 4.99

2010 13088.0 33471.4 139,064 94.1 240.7 18.16 5.18

Ratio 2010/1950 6.53 5.59 2.36 2.77 2.37 2.14 1.30

= 0.3

(Y/L)t/(Y/L)t-10 (K/L)t/(K/L)t-10 At/At-10 [(K/L)t/(K/L)t-10]

1950-1960 1.264 1.282 1.173 1.08 0.000

1960-1970 1.261 1.186 1.198 1.05 0.000

1970-1980 1.083 1.086 1.057 1.02 0.000

1980-1990 1.150 1.130 1.109 1.04 0.000

1990-2000 1.213 1.121 1.172 1.03 0.000

2000-2010 1.149 1.133 1.106 1.04 0.000

Average all decades 1.187 1.156 1.136 1.04 0.001

Cobb-Douglas Parameter (capital's share)

Step 1: Calculate outputper worker (Y/L), and capital per worker (K/L).

Step 4: Check

Begin with these data:

Thousands of workers

Billions of Dollars

Billions of Dollars

Thousands of dollars/worker

Step 2: Calculate elementsof production function: total factor productivity and the per

worker contribution of capital.

Step 3: Calculate growth factorsby decade

for Y/L, K/L, A, (K/L)

INFERREDvalue of TFPYear

Decade

α

t t t

t-i t-1 t-i

(Y / L) A (K / L)= *

(Y / L) A (K / L)

Growth factor, output per

worker

Contribution of total factor productivity

Contribution of capital /worker

Sources: Bureau of Economic Analysis, author's calculations.

Table 4.4: Growth Accounting Exercise United States

21

Finally, such a simple calculation does not tell us how income is distributed amongst the population. A

relatively small portion of the population appears to be earning an ever greater share of the economies

income in the US. Hence, while $95 thousand may be the output per worker in the United States, it

must be more than disposable income per worker – a concept that may be less abstract to you.

We can also see that, over the same 60-year period, market participants have accumulated a substantial

amount of productive capital. On a per-worker basis, the economy’s capital in 2010 was about 2.4 times

what it was in 1950. The Cobb-Douglas production function tells us by how much this additional capital

contributed to the growth in additional output per worker between 1950 and 2010.

To express the Cobb-Douglas result in terms of output per worker, we divide both sides of equation 4.6

by L. In so doing, we learn that output per worker depends on both total factor productivity and output

per worker.

Note that subscripts “t” appear with the terms Y/L, A, and K/L. These tell us that the variable in question

may change over time. By contrast, the Cobb-Douglas parameter is assumed to remain constant.

One of the main stories that the Cobb-Douglas production function tells us is how different factors

therein contribute to additional output. To see what this means, let’s compare two economies that have

different values for K/L but are otherwise identical. Suppose that K/L in economy A is twice that of

economy B. We know that output per worker Y/L in economy A will exceed that of economy B, but by

less than a factor of 2. This must be so since the marginal product of capital is diminishing: that is, < 1.

In a growth accounting, total factor productivity is not directly observed. Instead, we may infer its value

by simply inverting equation (4.18) to obtain:

*Total Output Total Capital stockper worker Factor per worker

Produc

αt t t

itivity

(Y / L) = A (K / L)

=Total Total Output Capital stock

Factor per worker per workerProducitivit DIRECTLY OBSERVED DIRECTLY OBSERVED

INF

αt

y

t t

ERRED

A (Y / L) (K / L)

(4.18)

(4.19)

22

Step 2 of Table 4.4 shows the inferred value of total factor productivity for the United States. For

example, in 1950, output per worker was $34 thousand and capital per worker was $101.7 thousand.

Hence, applying the exponent, we find that for 1950, (K/L) was 4.0. Therefore, we infer that the value

of A for 1950 is 8.50. We also see that, by 2010, that factor had more than doubled – to 18.16.

Digression: What IS Total Factor Productivity Anyway?

It is important to recognize that an increase in total factor productivity is associated with increases in

the marginal products of capital and labor (mpk, mpl) of equal proportions. Table 4.5 provides an

example. If we return to the hypothetical example from the beginning of the chapter, Table 4.1 shows a

comparison of the marginal product estimated in Table 4.5 with the assumption that A=1, and an

alternative assumption in which total factor productivity has doubled to A=2.

Table 4.5 shows that marginal products for both capital and labor have also exactly doubled: mpk rises

from 0.1 to 0.2, while mpl rises from 1.099 to 2.098. Hence, applying this logic to the United States, we

conclude that the marginal products of both labor and capital had more than doubled over the 60 year

period between 1950 and 2010. But, what might have brought about such increases in these marginal

products? It would be tempting to say that technological innovations, like the ones that we discussed at

the beginning of the chapter, were a key factor.

In this sense, it would be tempting to conclude that the increase in A in the United States, by a factor of

2.4 from 8.5 in 1950 to 18.16 in 2010, was due to technological innovations. However, we cannot say

that our measure of total factor productivity exclusively reflects technological innovation. As we will

soon learn, other factors can also be important in determining total factor productivity.

23

One of the limitations of our method for calculating total factor productivity is that we do not observe it

directly. Instead, we infer – we "back it out" from observable variables, as occurs in equation (4.19).

Such a technique, frequently used by economists, is known as calculating a residual – what’s left over.

Such residual calculations can be a useful part of the scientific process since they tell us how much we

don’t know. Assessing our ignorance is not a bad thing. Instead, residual calculations can help point

economists in the right direction in future research – things that we need to know. In this case, a future

step in assessing productivity is to try to directly measure the effect of technology.9 In the next chapter,

we will consider the notion that public policy can also affect an economy’s productivity. Productivity

may differ dramatically amongst countries because their policies differ dramatically.

Further to the Point: Economic growth in the United States: Decade-by-Decade

Over the period 1950-1960, the performance of the U.S. economy was uneven in

the sense that growth was higher in some decades than others. We can better

9 For example, we can directly measure the extent to which a given technology – indoor plumbing, electricity, automobiles, high speed internet, or mobile telephones -- has been adopted in an economy.

= 0.3

A. Labor constant, capital variable

Y K L Y K mpk=Y/K

1600.9 4800.0 1000 0.00

1601.0 4801.0 1000 0.10 1.0 0.100

3201.9 4800 1000

3202.1 4801 1000 0.20 1.00 0.200

B. Capital constant, labor variable

Y K L Y L mpl=Y/L

1570.2 4500 1000 0.00

1571.3 4500 1001 1.10 1.0 1.099

3140.5 4500 1000

3142.7 4500 1001 2.20 1.00 2.198

A=1(From previous table)

A=2

A=1(From previous table)

A=2

Increase in marginal products of capital, labor proportional to increase in total factor producivity.

Here, doubling TFP means that mpk and mpl also double.

*Total Total Capital Labor

Output FactorProduciti

α

vity

α 1-Y = A K *L

Table 4.5: Total factor productivity and marginal products of capital and labor

Further to the Point….

24

understand this performance by examining the contributions of total factor productivity and the

capital/worker ratio to growth.

To make this calculation, we divide equation (4.6) evaluated in some year t by that same equation

evaluated i=10 years ago. That is, our calculation is:

On the left-hand side is the growth factor: the value of output-per-worker today compared to what it

was 10 years ago. Note that this number is simply one plus the percentage change. (Hence, for some

variable X, the growth factor is Xt/Xt-i – a number that is identically equal to 1 + %X). On the right-hand

side, the first term the contribution of total factor productivity; it is identical to the growth factor of A.

The second term is the contribution of capital/worker; this is the growth factor of K/L elevated to the

power .

For the years 1950 - 1960 the growth factor was 1.264. This is simply another way of saying that output

per worker grew by 26.4% over the period. If we divide this term by 10, we see that the average yearly

growth was approximately 2.6%. For this same period, the growth factor of A was 1.173. That is, out of

the 26.4% total growth of Y/L, most -- about 17.3% - was explained by total factor productivity growth.

The remainder, about 8%, was explained by the fact that the U,S, economy had accumulated more

capital. The capital-worker ratio increased 28.2 percent (growth factor was 1.282). However, since we

have assumed that the marginal product of capital is diminishing, as reflected in our assumption that

=0.3 (less than one), we must conclude that the 28.2 percent increase in the capital/labor ratio alone

could not have explained the 26.4% increase in output/worker over the decade. Rather, total factor

productivity also played a role.

Another way to think about the growth decomposition is to consider a counterfactual: what might have

happened if circumstances were different. Hence, we can see that if there had been no total factor

productivity growth, output per worker would have grown over the decade by only 8% -- not by the

26.4% we observed. Likewise, if the United States had accumulated just enough capital to keep pace

α

t t t

t-i t-1 t-i

(Y / L) A (K / L)= *

(Y / L) A (K / L)

Growth factor, output per

worker

Contribution of total factor productivity

Contribution of capital /worker

(4.20)

25

with the growth of the labor force (about 12%), the growth of output/worker would have been 8% less

than what we actually observed.

If we look at the bottom line of this portion of Table 4.4, we see the average growth factors and

contributions by decade. This line shows that, for the six decades, the average growth factor for output

per worker was 1.187; that is, the average growth rate of Y/L was about 18.7% per decade, or about

1.8% per year. We can see that both capital accumulation and total factor productivity are important for

growth. It might be tempting to say that TFP is "more important" than K/L: if the K/L ratio had stayed

constant, growth per decade would have dropped to about 13.6%; if there had been no total factor

productivity growth, average growth per decade would have fallen much further than to about 4%.

However, this calculation does not imply that economies should stop investing. If they were to stop

investing altogether, the capital/labor ratio would fall – and with it the output / labor ratio. For example,

suppose that capital stock were to fall by 5%, but TFP stayed the same. In this case, the counterfactual

growth factor for output/worker would have been: 1.136*[(.95)0.3]=1.119. Thus, average growth in any

decade would have been under 12%, rather than the 18.7% that we observed.

We are now in a position to compare economic performance during the decades. Calculations from the

bottom part of Table 4.4 are shown graphically in Figure 4.3: growth of Y/L is shown as the blue solid

bars, growth of total factor productivity is shown by the red striped bars. The highest growth rates of

output occurred during the first two decades after World War II -- the 1950s and the 1960s -- each at

about 26% (about 2.6% per year).

The period of poorest economic growth was the 1970s (about 8.3%, or less than 1% per year). In the

1980s, economic growth improved; in the 1990s growth rates approached levels achieved during the

1950s and 1960s, whereas during first decade of the 21st century, economic growth declined. (Growth of

Y/L during this time, while poor, still exceeded that of the 1970s. However, the economic downturn that

occurred during the 2008-2010 was more severe in many ways than anything that had occurred during

the 1970s. For example, the unemployment rates observed during the early 21st century were far more

severe than those during the 1970s.)

Not surprisingly, a casual look at the Figure 4.3 confirms that output growth and total factor productivity

growth tend to move up and down with one another. During the 1950s and the 1960s, when output

growth was high, so was TFP growth. During the 1970s, when Y/L growth was meager, so was TFP. As

26

economic growth improved during the 1980s and 1990s, so did TFP growth. And, TFP growth did fall in

the first decade of the 21st century.

This does not mean that capital accumulation was irrelevant. To the contrary, the capital/labor ratio

grew most rapidly during the 1950s (28%) and the 1960s (18%). During the more sluggish 1970s, this

growth rate dropped substantially to about 8.6%. However, during the decades that followed, while

income growth rates recuperated, the growth rate of the K/L remained lower than it was during the

1960, remaining in the range of 12-13.5 percent per decade. In this sense, the robust growth of the

earlier period (the 1950’s and the 1960s) relied more on capital accumulation, the growth rates of the

1990s – which approached those of the earlier decades, relied more on TFP growth than capital

accumulation. END Further to the Point.

What’s Your View? Should we be optimistic or pessimistic about the future?

Now, let’s take a longer view of the data. Figure 4.4 shows per capita growth for

the United States (including estimates for the pre-colonial and colonial periods) in

solid blue, and an average of selected European countries in red slanted lines. The top pair of bars

shows that, in both regions, average economic growth between the years 1 and 1700 was very low,

when compared to more recent years.

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

#NUM!

0.95

1.00

1.05

1.10

1.15

1.20

1.25

1.30

1950-1960 1960-1970 1970-1980 1980-1990 1990-2000 2000-2010

US: Growth DecompositionSource: Bureau of Economic Analysis, Author's Calculations

Growth, output per worker Growth, total factor productivity

'90s: Robust output growth ;High TFP growth

'80s: Improved output growth

higher TFP growth

'00s: Weaker output growth ;

lower TFP growth

Gro

wth

fact

or:

on

e p

lus

pe

rce

nta

ge c

han

geC

alcu

alte

d b

y d

eca

de

.

'50s and '60s: Superior output growth;HIgh TFP growth

'70s: Poorest output growth;

Lowest TFP growth

Figure 4.3: Growth Decomposition United States

Note: these figures correspond to those from Table 4.4

What’s Your View?

27

The innovations introduced during the past century or so also relieved humans of the need to do many

forms of brute and dangerous labor. There are clear links here between economic growth and the

standard of living. The indoor toilets that we take for granted were not widely installed in homes before

the 19th century. As per capita income rose, so did sanitary conditions and thus life expectancy.

Thus we see: most of the technological innovations that we now enjoy have been introduced during the

past two centuries. For this reason, most people in the industrialized world – and especially the United

States – have long held a secure belief that standards of living would be continually improving, with each

generation better off than the previous one.

In recent years, this belief has been shaken. The bottommost bars of Figure 4.1 represent the first years

of the 21st century – before the severe downturn of 2007-2009, but include the recession of 2000-2003.

These bars indicate that growth during this short period has been lower in both regions than in the

previous years.

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%

1-1700

1700-1820

1820-1850

1850-1900

1900-1950

1950-1975

1975-2000

2000-2006

US and Europe: Output per capitaAverage yearly growth, In percent; Source: Maddison (2009)

United States

Selected European Countries (average)

First 1700 years of common era: pre-industrial times, scant economic growth

Mid-18th through Mid-20th century: later stages of industrial revoultion, growth in US consistently above European levels.

18th and early 19th Century: Early industrial revolution , US independence.

Higher growth in Europe,partly reflecting post-World War II recuperation.

Information revolution: lower rates of growth in industrial world, US grows more than Europe.

Growth, percent per year

Year

, Co

mm

on

Era

Figure 4.4

28

In 2012, noted economist Robert Gordon of Northwestern University bluntly voiced a more pessimistic

view:10 “There was virtually no growth before 1750, and thus there is no guarantee that growth will

continue indefinitely...the past 250 years could well turn out to be a unique episode in human history.”

Gordon also argues that the innovations of the late 19th and early 20th century (electricity, the internal

combustion engine, running water, indoor toilets, communications, petroleum) were far more

important in improving our standard of living than the innovations of the recent information age

(computers, the web, mobile phones). He emphasizes that the United States and other industrialized

countries will continue to grow, but they should not expect that growth to come from explosive

technological progress. What’s your view? Can you think of events and discoveries that you read about

in the newspapers – the wide spread use of hydraulic fracturing, the use of robots, ever-increasing

computer speed, innovations in education (including how colleges and universities operate) that can

boost our economic prospects? Is it possible that we will enjoy another "boom" of TFP-driven growth?

Or, as Professor Gordon suggests, should we scale down our expectations?

END What’s Your View.

Further To the Point: Oil Prices and Information Technology: How Do They

Affect Economic Growth? As we discussed above, a growth accounting exercise

is often just a first step in learning about what makes an economy grow. The

purpose of our residual calculation is to help us proceed in a more structured way. For example, we

might ask about the role of oil prices – a variable that we previously found may be an element of total

factor productivity.

In fact, the data in Table 4.6 suggest that oil prices may have played a part. Two decades of strong

economic growth, the 1960s and the 1990s, are also associated with oil price declines. During the 1970s,

when economic growth was weakest, oil prices posted their largest increases. When the economy

improved during the 1980s, oil prices rose less than in the 1970s on an average basis – and began to

decline around the middle of the decade. Much in the same way, growth during the 2000s slowed when

compared to the 1990s, and oil prices increased.

10 Gordon, Robert J., “Is U.S. Economic Growth Over? Faltering Innovation Confronts the Six Headwinds,” NBER Working Paper No. 18315 (August 2012).

Further to the Point….

29

What about the role of information technology? As Harvard economist Dale W. Jorgenson and his co-

authors point out, even though computers for large-scale data processing had been available on the

private market since the 1950s, when the Remington-Rand Corporation introduced the UNIVAC,

computers initially did not appear to make the economy more productive – according to the data. 11 In

1987, Robert Solow concluded that “you can see the computer age everywhere but in the productivity

statistics.” However, using methods that extend our (simple) TFP calculations, Jorgenson (2008) and

other authors do conclude that between (roughly) 1995 and 2000, computers and information

technology did become an important source of productivity growth. Then, during the 2000s, the

importance of computers for productivity growth waned. End Further to the Point

Further to the Point/Online Feature. An Extended Analysis of Human Capital, Education, and

Growth In an extended online analysis, we examine how a work force that is more skilled and

educated will bring about higher economic growth. End Further to the Point Online Callout.

11 These authors are referring here not only to total factor productivity but also to independent changes in the marginal product of labor – a concept that we will discuss in a later chapter.

(Y/L)t/(Y/L)t-10 Poilt/Poil

t-10

1960s 1.261 0.771 Strong economic growth, oil price declines.

1970s 1.083 3.733 Weakest economic growth, largest oil price increases

1980s 1.150 1.074 Improving economic growth, smaller oil price increases

1990s 1.213 0.538 Strong economic growth, oil price decline

2000s 1.149 2.340 Lower economic growth, substantial oil price increase

Growth factors: Output/Worker and Oil Prices

DecadeOutput / Worker

Growth factorOil Price

Growth FactorRemarks

Table 4.6: Economic Growth in the United States Do oil prices play a role?

Further to the Point*….

*Online Feature.

30

4.5 Capital Accumulation in the Long-Run: The Concept of a Steady State

LO 4.5 Explain why output, investment, and consumption will converge to a steady state in the long run.

Previously, we encountered the Cobb-Douglas Production Function, which told us that the amount we

produce was determined by two factors of production and their productivity level:

Note the addition of the time subscript “t.” We know that the capital that we are able to use in the

current period (t) is only that which we had accumulated in the previous period (t-1). It takes a period

for capital to "come online." We also introduced this equation:

We must now ask: Is our total savings Y sufficiently large to cover the volume of depreciation K?

Figure 4.5 helps us to think about this question. As in the previous figures, the blue line is output; as

before, labor employed is fixed at 1000 and only the capital stock is allowed to vary. And, as before, the

blue line has a positive but diminishing slope, confirming that the marginal product of capital is

diminishing.

(4.17)

*α

t-1 ttTotal Capital LaborTotal

Output FactorProducit

1-

ivity

αtY = A K * L (4.6)

Total saving (identically equal to

t t-1 t-1 tVolToday's capital Yesterday's

stock capitume of depreciaton --

the "wear and tea inv

r" oal stock n theexiestm stine g capitant l s

-

. k

1

t) oc

K = K + σY δK

LO 4.5

31

The green line shows that total savings is 0.15 (15%). Note that the slope of this line is simply the fixed

ratio times output. The slope of the green line, like that of the blue line, is positive but diminishing. As

capital stock increases, the rise in our output is progressively lower and lower.

For this reason, the increase rise in savings (Y) for a given increase in the capital stock (K) must also

progressively decline. The red line shows the volume of depreciation K. Unlike either the blue or green

lines, this one has a constant slope: this reflects our assumption above that is a constant, and hence K

is a linear function.

For low levels of K, Y exceeds K. This means that there is sufficient savings to purchase new capital

goods, in addition to replacing the portion of the capital stock that has worn away. To extend the

example from above, so long as Y> K, firms are able not only to maintain their current trucks (new

tires, water pumps, and so on) but also to purchase new trucks.

1611

1601

241.6

240.1

4800 4900

output Y = Af(K,L);

capital (K)

Af(KSS,L)=0.15*YSS

SS,=0.15depreciation= K;

=0.05

Af(K,L)=0.15*Y

Af(K,L)=0.152*Y Af(KSS,L)=0.152*YSS

SS,=0.152

KSS,

=0.15

KSS,

=0.152

output Y = Af(K,L)

Figure 4.5: The Notion of a Steady State

32

But, as the capital stock increases and the volume of wear and tear increases, the difference between

Y and K progressively becomes smaller and smaller. This means K, while also growing, is increasingly

used just to replace the capital stock that is wearing away; ever fewer resources are available to grow

the capital stock.

At some point Y = K; this is where the green line and the red line cross. At this point, since we’ve

assumed that is fixed, it is impossible to accumulate more capital. Instead, the capital stock will

converge to what is known as the steady state level of capital, KSS.

This notion of a steady state, widely used in economics, is a long-run equilibrium condition.[key term:

steady state; a long-run equilibrium in which certain key variables remain constant unless they are

shocked by some external factor.] Certain variables in an economic model may converge toward their

steady state values, from which they will not move – unless they are shocked by some factor that is

outside the model.

In Figure 4.5, the steady state stock of capital (Kss) is approximately 4800 units, steady state output (YSS)

is approximately 1601, and steady state savings (the minimum amount of savings that is required to

replace the capital that has worn away) is YSS=KSS=240.1.

We can approach the steady state from the other direction. Suppose that the capital stock K were to

exceed its steady state value of 4800. How might this come about? We would have to assume that the

extra capital appeared like “manna from heaven” or came from outer space. If that were the case, the

economy would have “too much capital” – more than it would be able to maintain with the given

savings rate. Since total savings would not be sufficient to maintain the capital stock (Y < K), the

capital stock would instead decline. As it declines, the marginal product of capital increases; the capital

stock would continue to fall, but at a slower rate – until it once again converges to KSS (in the chart, KSS=

4800 for =0.15). At this point, total savings Y would once again be driven into equality with volume of

depreciation K.

If households increase their savings rate, the steady state capital stock will also increase, as confirmed in

Figure 4.5. When the savings rateincreases by a small amount, from 15.0% to 15.2%, the savings line

shifts – the new relevant line is the now the purple line (instead of the green line). Steady state capital

stock increases to 4900, steady state output increases to 1611, and steady state savings – the amount

required to just offset the depreciation on the steady state stock of capital – now equals 241.6.

33

Extension: An Increase in Total Factor Productivity (A)

Figure 4.6 illustrates how an increase in A will affect output and savings. Initially there is an impact

effect: when A increases, the blue line for output Y=A*f(K,L) shifts up. (As before, we hold the number of

workers constant). Such an increase in A means more output for the country with the same level of

input. For example, if A increases from 1 to 1.03, output will initially increase by 3% -- from 1601 to

1601*1.03=1649. Thus, there are more resources for households to save (and hence invest). In this

example, the additional income is just about 48. Initially, consumers will save a fraction (1-) of that

additional income. In this case, since =15%, consumption C will rise by 0.85* 48=41 units. The

remaining 7 units (7=0.15*48) are saved and invested.

We can see that the capital stock will also rise by 7 units in the next period. This increase must be true

since our initial position was one of a steady state. If A had not changed, there would have been just

enough saving and investment to cover the volume of depreciation K. After the increase in A,

households save more than before: in period 1, Y> K.

Figure 4.6

An Increase in Total Factor Productivity (A)

1670

1601

250

240

4800 5526

output Y = Af(K,L);

capital (K)

YSS,A=1

depreciation= K;=0.05

Af(K,L)=0.15*Y

Af(K,L)=0.152*Y

KSS,

A=1KSS,

A=1.03

output Y = Af(K,L)Increase in A

from A=1 to A= 1.03

saving Y = Af(K,L)Increase in A

from A=1 to A= 1.03

YSS,A=1.03

YSS,A=1.03

sYSS,A=1

34

Table 4.7 shows that in successive periods, as the capital stock grows the gap betweenY and K

becomes ever smaller. Period 0 is the period just before the improvement in productivity (A = 1). The

steady state capital stock equals 4803. (Note that in Figure 4.6, the figure for KSS, 4800, is approximate).

When A rises to 1.03, output, consumption, and savings all rise initially, in Period 1, by 3%.

For Period 2 and afterward, the capital stock K begins to grow by a very small amount every period. As

the capital stock grows, so too must output Y, along with consumption and savings. In addition, the gap

between Y and K jumps from 0 in period 0 to 7.2 in Period 1, but falls thereafter.

How long does the capital stock keep growing? As long as Y > K. By contrast, at the point where

Y=K, the economy has arrived at the new steady state. Precisely when does that happen? In our

example, the correct answer is "never." Instead, as more capital is accumulated, and the marginal

product of capital continues to fall, the difference between Y and K gets ever smaller—it approaches

zero. However, we can exactly compute the steady state using our Cobb-Douglas example, as we will do

in the next section.

Table 4.7 Impacts of an Increase in Total Factor Productivity (A)

35

Calculating Steady States: The Cobb-Douglas Case

Until now, we have relied on graphing and intuition to generate several key insights behind the

Solow/Swan growth model. However, the model becomes even more useful if we can plug in some

numbers. To do so we will extend the Cobb-Douglas example from above. For most of our calculations, it

will be easier to express key economic variables like output, capital, consumption, savings, and the

volume of depreciation on a per-worker basis.

As a first step, we will want to know what determines capital per worker in the steady state, KSS/L. For

the Cobb-Douglas case, this turns out to be quite intuitive. Remember first that we can write output per

worker as a simple function of capital per worker:

Hence, if the steady state capital to labor ratio is 5, A=1 and =0.3, steady state output per worker must

be about 1.62. We can also very easily compute the steady state savings per worker:

In the steady state, when capital is constant at KSS, total savings are just enough to offset depreciation:

A bit of algebra will now get us our first firm results: the steady state values of capital and output per

worker.

in the steady state

α

Output per worker Capital per workein the steady state

ss ssr

(Y / L) = A(K / L)

in the steady stateSaving per worker Fixed fraction of

output per wo

αSS SS

rker

S / L = σA(K /L)

Saving: Fixed fraction Volume of depreciation of output Fixed frac

Steady state relathionshp between tion of capital

saving and volu

α

me of deprec

SS

iati

S

on

SσA(K /L) = δ(K /L)

Steady state

capital pe

1/(α-1

r wor

)

ker

SS

δK / L =

Aσ

Steady state

outp

α/(α-1)

SS SS

ut

δY / L = A(K / L) = A

Aσα

Steady state capital and output will be higher if

savings increases or depreciation decreases.

(4.21)

(4.24)

(4.25)

(4.22)

(4.23)

36

These two equations should be very intuitive. The first thing to note is that since <1, 1/(-1). Keeping

this in mind, we can confirm our intuition; both the capital/worker and output/worker ratio in the

steady state will be higher if either total factor productivity A rises or the savings rate rises, but they

will be lower if depreciation rises.

Consider the following example. Assume that =0.3, =.15, and =0.05. Prior to our innovation, we will

assume that A = 1. As a result of the innovation, A will rise 3% to 1.03. The results are shown in Table

4.8. The top part of the table expresses all of the results on a per worker basis. The bottom part of the

table computes values based on the assumption that L=1000. We confirm the numbers in Figure 4.6. A

3% increase in productivity will bring about increases in capital, output, and consumption by about 4.3%.

The capital/worker ratio rises from (about) 4.8 to 5.0. The output/per worker ratio increases from 1.60

to 1.67 and the consumption/worker ratio increases from 1.36 to 1.42. Finally, in the steady state, we

also save 4.3% more. This must be so, since the steady state volume of deprecation (“maintenance

costs”) has also risen by 4.3%.

The diagrams in Figure 4.7 illustrate how key variables in the economy evolve over time – before and

after the shock. Figure 4.7.a (upper left diagram) shows that the capital/labor (or capital/worker) ratio

K/L begins at some steady state. It jumps up precisely when total factor productivity increases, since

that productivity itself provides the economy with extra resources to accumulate new capital.

Thereafter, a kind of positive feedback takes place: as more capital is accumulated, more output is

produced and even more resources become available to continue adding to the capital stock. However,

in each successive period, the capital stock increases at an ever slower rate – as reflected in the K/L

line’s declining slope, as the K/L ratio approaches its new, higher, steady state.

We have already discussed why this slowdown takes place: while the rate of depreciation δ is assumed

to be constant, the marginal product of capital (mpk) declines as the stock of capital increase (holding all

else equal including A). This is shown in Figure 4.7.e (bottom left diagram). Thus, we can see that the

productivity increase raises total saving per workerσY /L , as shown in the Figure 4.7.c (middle left

diagram).

37

Table 4.8 Long Run Effects of an Increase in A

Productivity

A KSS/L YSS/L KSS/L YSS/L CSS/L

Before Innovation 1.000 4.800 1.601 0.2400 0.2400 1.361

After Innovation 1.030 5.007 1.670 0.2504 0.2504 1.420

Percent Diff: Alt minus Base 3.0 4.3 4.3 4.3 4.3 4.3

KSS YSS KSS YSS CSS

Before Innovation 4800.0 1600.9 240.0 240.0 1360.9

After Innovation 5007.0 1670.0 250.4 250.4 1419.6

4.3 4.3 4.3 4.3 4.3

Steady State Values -- L = 1000

Steady State Values -Per Worker

Measured as percent change:100*(Alt-Base)/Base

Figure 4.7: Gradual Convergence to a new steady state after a total factor productivity shock.

*

Time Time

Time Time

Starting from steady state, capital stock jumps when total factor productivity increases, then continues to rise (but at an ever slower pace), ultimately approaching its new, higher steady state.

Starting from steady state, output per worker starts to rise when total factor productivity increases; as like the capital stock, output then continues to rise, but at an ever slower pace as it approaches its new, higher steady state.

Rise in total factor productivity provides more resources for saving and capital accumulation.... ... but the proportion of savings devoted to simply maintaining the capital stock (volume of depreciation

K/L) continues to rise as the capital stock approaches its new steady state.

After the shift in total factor productivity, the marginal product of capital jumps up, but then gradually falls toward its previous steady state value -- reflecting the gradual increase in the capital stock.

After the shift in total factor productivity, the marginal productof jumps upward and continues to increase, gradually approaching a new, higher, steady state value -- boosted by the gradual increase in the capital stock

1.59

1.60

1.61

1.62

1.63

1.64

1.65

1.66

1.67

1.68

0 10 20 30 40 50 60 70 80 90 100

Ra

tio

K/L

Periods of Time

a. Capital / Labor Ratio (K/L)

4.75

4.80

4.85

4.90

4.95

5.00

0 10 20 30 40 50 60 70 80 90 100

Ra

tio

Y/L

Periods of Time

b. Output / Labor Ratio (Y/L)

0.24

0.24

0.24

0.24

0.25

0.25

0.25

0.25

0 10 20 30 40 50 60 70 80 90 100

Ra

tio

Sa

vin

gs/L

ab

or

Periods of Time

c. Savings / Labor Ratio (Y/L)

0.00

0.00

0.00

0.01

0.01

0.01

0 10 20 30 40 50 60 70 80 90 100

Ra

tio

Ca

pit

al A

ccu

mu

lati

on

/ L

ab

or

Periods of Time

d. Capital Accumulation / Labor Ratio ((Y-K)/L)

9.9%

10.0%

10.0%

10.1%

10.1%

10.2%

10.2%

10.3%

10.3%

10.4%

10.4%

0 10 20 30 40 50 60 70 80 90 100

Ma

rgin

al P

rod

uct

of C

ap

ita

l

Periods of Time

e. Marginal Product of Capital (mpk)

1.10

1.11

1.12

1.13

1.14

1.15

1.16

1.17

1.18

1.19

1.20

0 10 20 30 40 50 60 70 80 90 100

Ma

rgin

al P

rod

uct

of L

ab

or

Periods of Time

f. Marginal Product of Labor (mpl)

38

However, the amount of this saving that is used simply to maintain the existing capital stock – the

volume of depreciation δK /L also rises. Hence, as shown in the Figure 4.7.d (middle right diagram), the

amount of saving that is used for new capital accumulation, (σY -δK) /L initially jumps when the

productivity increase occurs, but then falls rapidly, approaching again its steady state value of zero.

The figures also show how workers benefit from the productivity shock – on a permanent basis. In

Figure 4.7.b (the top right diagram), output per worker (Y/L) reflects the pattern of K/L: it also rises at a

rate that is initially rapid but declines as it approaches its new, higher steady state. Hence the slope of

the Y/L curve also falls over time. We can also see, in Figure 4.7.f (the bottom right diagram), that the

marginal product of labor (mpl) also rises (albeit at a declining rate) as it approaches a new, higher,

steady state value.

Extension: A More Durable Capital Stock

Figure 4.8 and Table 4.19 illustrate that a decrease in the depreciation rate will boost steady state

output and capital. That is, if we have longer and more durable capital, we can use more of our savings

to accumulate new capital. We might think of a decrease in as a kind of technological progress.

We may have invented longer lasting machines or parts – for example, putting steel belts in tires

to reduce wear and tear. In our example, the depreciation rate falls just slightly from 5% to 4.9%. A drop

of this magnitude is comparable to the 3% increase in A, insofar as both result in an increase of steady

state capital KS from 4800 to 5007. The increase in steady state output is less than the productivity

increase (from 4800 to 5526).

Dep. Rat

KSS/L YSS/L KSS/L YSS/L CSS/L

Before Innovation 0.050 4.800 1.601 0.2400 0.2400 1.361

After Innovation 0.049 5.007 1.621 0.2431 0.2431 1.378

Percent Diff: Alt minus Base -2.9 4.3 1.3 1.3 1.3 1.3

KSS YSS KSS YSS CSS

Before Innovation 4800.0 1600.9 240.0 240.0 1360.9

After Innovation 5007.0 1621.3 243.1 243.1 1378.3

4.3 1.3 1.3 1.3 1.3

Steady State Values -Per Worker

Steady State Values -- L = 1000

Measured as percent change:100*(Alt-Base)/Base

Table 4.9

A Decrease in the Rate of Depreciation ()

39

Chapter Summary

This chapter dealt with the long-run determinants of economic growth. The ultimate goal of

humans is to raise their standard of living and their overall well-being, and not simply to

produce and consume as much as possible. While two main determinants of economic welfare

are consumption and leisure, other factors (including clean water, clean air, and access to

medical services) also matter. Even so, better economic performance – higher levels of output –

can help us attain these goals. In this sense, a country’s economic growth is closely linked to the

well-being of its citizens.

The chapter introduced some important tools to analyze economic growth. The production

function summarizes the relationship between the inputs that are used to produce goods and

services – capital and labor – and output. The efficiency with which these inputs are

transformed into output is known as total factor productivity. We also reviewed some basic

assumptions: the marginal products of both capital and labor are positive, but diminishing; we

assume constant returns to scale. (LO 4.1)

As a further aid to our analysis, we learned how to use a specific production function: the Cobb

Douglas production function, although very simple, has some implications that correspond to

1621

1601

243

240

4800 5007

output Y = Af(K,L); A=1

capital (K)KSS,

d=0.05KSS,

d=0.048

saving Y = Af(K,L)YSS,=0.05

depreciation= K;decreaase in from

=0.05 to =0.049

YSS,=0.049

YSS,=0.05

YSS,=0.049

Figure 4.8

A Decrease in the Rate of Depreciation ()

40

what we observe in the real world. Armed with the Cobb Douglas, we could calculate the level

of output, given levels of total factor productivity, capital, and labor. (LO 4.2)

Our next step was to consider how the capital stock grew over time. We began with some

assumptions taken from the Solow/Swan model of economic growth. During each period, a

country’s residents save a fixed proportion of their output rather than consuming it. At the same

time, the capital stock depreciates – suffers wear and tear – by a rate that is proportional to the

size of the capital stock itself. Therefore, we found that today’s capital stock equals yesterday’s

capital stock, plus today’s investment, minus the volume of depreciation. (LO 4.3)

With these tools, we were able to find out the sources of an economy’s growth by performing

what is known as a Solow Decomposition. As part of our analysis, we inferred total factor

productivity on the basis of observed levels of output, capital, and labor. That is, we learned

how to "back out" total factor productivity as a residual calculation. On a per-worker basis, we

calculated the contribution of capital accumulation-- how much of our economic growth was

due to the fact that we save and invest. We then attributed the remainder of our growth to

changes in total factor productivity. Over the past five decades in the United States, we found

that, while both factors were important, the periods of highest growth were those when the

growth of total factor productivity was the highest. (LO 4.4)

Finally, we introduced the concept of a steady state, a condition in which certain key economic

variables remain constant. In the case of the Solow growth model, we found that as we

accumulated ever more capital, it also became ever costlier to maintain it. Since the

depreciation rate is constant, as the capital stock grows, so does the volume of depreciation.

Thus, as an ever increasing portion of savings provided by households must be used to merely

maintain the existing capital stock, rather than expand it, the capital stock is growing ever closer

to its upper limit – the steady state. In turn, so long as the savings rate, total factor productivity,

and the rate of depreciation remain unchanged, output and consumption, both on a per-worker

basis, will approach upper limits at the steady state. This finding is an alternative way of

illustrating a previous point: without continual increases in total factor productivity, we will not

continue to enjoy continual economic growth. (LO 4.5)

41

Key Terms

production function, 3

entrepreneurs, 4

factors of production, 4

diminishing, 5

functional form, 8

volume of depreciation, 15

steady state, 16

Questions

1. True or false (Explain): “Economist’s focus on economic growth is misplaced because the purpose of

life should not be just to make more things.”

2. Consider some of the main assumption behind the aggregate production function. Do marginal

products increase or decrease as we use more of a factor of production? If capital and labor are the only

factors of production and we increase the use of both by 10%, by how much should output increase by?

3. Suppose that we know (a) total output, (b) the amount of labor that is used in production, and (c ) the

marginal product of labor, but we don’t know anything else about the production function. How can we

find out capital’s share of total output?

4. Suppose that a new and better fertilizer is discovered. How might the productivity of farmers be

affected? What are the elements of the production function that might change?

5. True/False (explain) “An increase in total factor productivity of 1% will mean that steady state

consumption/worker will correspondingly rise by 1%.”

6. True/False (explain) “If capital goods become more durable, society becomes poorer because fewer

people will be employed in the jobs required to build capital goods.”

TO BE REDONE BY MGH

42

Problems and Applications

1 .

K Y mpk

1040 20000

2000 25000

Consider the following data for output and capital. Assume

that labor is fixed. Compute the interval estimate for the marginal product of capital (mpk).

2 .

L Y mpl

100 5000

101 5005

Consider the following data for output and labor.

Assume that capital is fixed. Compute the interval estimate for the marginal product of labor (mpl).

3 .

A 1.2

0.35

K L Y Y K mpk K*mpk Share, K

5000 100

5001 100

Share, L

mpl

Consider the data the table below. The production function is Cobb-Douglas.

Compute output for K=5000 and 5001. Compute the marginal product of capital, including intermediate calculations for the change in output and the change in

capital . For K=5001, compute the payment to capital (K*mpk). Compute capital's share of total output. Compute also labor's share. Use the fact that labor's share is L*mpl/Y to compute the marginal product of labor when K=5001.

4 .

A 1.5

0.37

Y K L Y L mpl Payment L Share, L

986.09 2500

2500

mpk=

Consider the data the table below. The production function is Cobb-Douglas. Compute the

amount of labor that is initially used when output Y=986.09. Then, suppose ten workers are added. Compute output for this new level of workers. Compute the marginal product of labor,

including intermediate calculations for the change in output and the change in labor. Compute the payment to labor (L*mpl). Compute labor's share of total output. Compute also capital's share. Use the fact that capital's share is K*mpk/Y to compute the marginal product of capital,

assuming the final number of workers (original plus 10).

LO 2.2

LO 4.1

LO 4.2

LO 4.1

LO 4.2

43

5 .

A 1.5 0.37

Y K L Y L mpl mpk

2500 190

2500 191

A 1.7 0.37

Y K L Y L mpl mpk

2500 190

2500 191

A mpl mpk

Consider the data the table below. The production function is Cobb-Douglas. Part A of

the table shows data before an increase in total factor productivity (A) takes place. Part B of the table shows data after an increase in total factor productivity (A) takes place.

For both parts A and B, compute output, marginal product of labor (including intermediate calculations for interval estimate). Also, use the fact that capital's share is K*mpk/Y to compute the marginal product of capital, assuming the final number of

workers (191). Then in part C, compute the percentage changes of A, mpl, and mpk,

A. Before change in total factor productivity

B. After change in total factor productivity

C. Growth, in percent: Part B versus part A.

6 .

Total Factor Productivity A 1

Capital K 4000.0

Labor L 300

Capital Share 0.3

Savings Rate 0.15

Output Y

Savings S=Y

Assumptions

Results

Use the data from the table below to compute output and savings.

Assume that the production function is Cobb-Douglas.

LO 4.2

LO 4.3

44

7 .

Total Factor Productivity A 1.3

Labor L 12000

Capital, end of previous period Kt-1 20000

Output Y

Savings Rate 0.17

Savings =Investment S=Y=I

Depreciation Rate 0.05

Volume of depreciation Kt-1

Capital, end of current period Kt-1

Use the data from the table below to compute the following: output,

savings, volume of depreciation, and end period capital stock. Assume that the production function is Cobb-Douglas.

8 .

Output (Billions of dollars) Y 7723

Capital stock (Billions of dollars) K 17724

Labor (Millions of workers) L 103.437

Output per worker (Thousand dollars/worker) Y/L

Capital per worker (Thousand dollars/worker) K/L

Total Factor Productivity A

Per-worker contribution of capital (K/L)

Capital's share 0.3

Using the data provided, calculate total factor productivity (A), output per

worker, capital per worker, total factor productivity, and the per-worker contribution of capital. Assume that the production function is Cobb-Douglas.

LO 4.3

LO 4.4

45

9 .

Year 0 Year 10 Growth Factor

Year 10/Year 0

Output (Billions of dollars) Y 87211 90274

Capital stock (Billions of dollars) K 18764 19817

Labor (Millions of workers) L 101.7 102.1

Output per worker (Thousand dollars/worker) Y/L

Capital per worker (Thousand dollars/worker) K/L

Total Factor Productivity A

Per-worker contribution of capital (K/L)

Capital's share 0.3

Growth of output per worker Year 0 -Year 10 percent

Growth of TFP Year 0 -Year 10 percent

Per-worker contribution of capital Year 0 -Year 10 percent

Using the data provided, calculate total factor productivity (A), output per worker, capital per

worker, total factor productivity, and the per-worker contribution of capital. Assume that the production function is Cobb-Douglas.

10 .

Total Factor Productivity A 1.20

Capital's Share 0.33

Savings Rate 0.14

Depreciation Rate 0.05

Intermediate Term

Exponent

Steady State Capital/Labor Ratio KSS/L

Steady State Output /Labor Ratio YSS/L

Steady State Savings/Labor Ratio SSS/L=YSS/L

Steady State Consumption/Labor Ratio CSS/L=YSS/L-SSS/L

Consider the following data for a Cobb-Douglas Production function and the savings rate in

the upper part of the table. First, calculate the intermediate term and the exponent. Then, compute steady state capital, output, savings, and consumption on a per-worker basis: KSS/L,

YSS/L, SSS/L and CSS/L..

δ

Aσ1/(α-1)

LO 4.4

LO 4.4

46

LO 4.5

11 .

Previous SS Impact New SS

Total Factor Productivity A 2.00 2.06 2.06

Capital's Share 0.33 0.33 0.33

Savings Rate 0.12 0.12 0.12

Depreciation Rate 0.05 0.05 0.05

Intermediate Term …

Exponent …

Capital/Labor Ratio KSS/L …

Output /Labor Ratio Y/L

Savings/Labor Ratio S/L=Y/L

Consumption/Labor Ratio C/L=Y/L-S/L

Consider the following data for a Cobb-Douglas Production function, the savings rate, and the

depreciation rate in the upper part of the table. The leftmost column provides data prior to a change in total factor productivity, while the second and third columns provide data that

apply after a change in total factor productivity. Calculate the Previous and new steady states in the first and third columns. To do so, first, calculate the intermediate term and the exponent. Then, compute steady state capital, output, savings, and consumption on a per-

worker basis: KSS/L, YSS/L, SSS/L and CSS/L. In the middle column, calculate output, savings, and consumption ratios that we see just after the change in total factor productivity (but before

any initial investment takes place).

δ

Aσ1/(α-1)

LO 4.5

LO 4.4