M06_ABEL4987_7E_IM_C06

Chapter 6Long-Run Economic Growth

Learning Objectives

I. Goals of Chapter 6

A. Identify forces that determine the growth rate of an economy

1. Changes in productivity are key2. Saving and investment decisions are also important

B. Examine policies governments may use to influence the rate of growth

II. Notes to Sixth Edition Users

A. This chapter remains largely unchanged, except for updated data in the applications

Teaching Notes

I. The Sources of Economic Growth (Sec. 6.1)

A. Production function

Y AF(K, N) (6.1)

1. Decompose into growth rate form: the growth accounting equation

Y/Y A/A aK K/K aN N/N (6.2)

2. The a terms are the elasticities of output with respect to the inputs (capital and labor)3. Interpretation

a. A rise of 10% in A raises output by 10%b. A rise of 10% in K raises output by aK times 10%c. A rise of 10% in N raises output by aN times 10%

4. Both aK and aN are less than 1 due to diminishing marginal productivity

B. Growth accounting

1. Four steps in breaking output growth into its causes (productivity growth, capital input growth, labor input growth)a. Get data on Y/Y, K/K, and N/N, adjusting for quality changesb. Estimate aK and aN from historical datac. Calculate the contributions of K and N as aK K/K and aN N/N, respectivelyd. Calculate productivity growth as the residual: A/A Y/Y – aK K/K – aN N/N

Numerical Problems 1 and 2 are growth accounting exercises.

©2011 Pearson Education, Inc. Publishing as Addison Wesley

106 Abel/Bernanke/Croushore • Macroeconomics, Seventh Edition

2. Growth accounting and the productivity slowdowna. Denison’s results for 1929–1982 (text Table 6.3)

(1) Entire period output growth 2.92%; due to labor 1.34%; due to capital 0.56%; due to productivity 1.02%

(2) Pre-1948 capital growth was much slower than post-1948(3) Post-1973 labor growth slightly slower than pre-1973(4) Productivity growth is major difference

(a) Entire period: 1.02%(b) 1929–1948: 1.01%(c) 1948–1973: 1.53%(d) 1973–1982: –0.27%

b. Productivity growth slowdown occurred in all major developed countries

Theoretical Application

Growth accounting provides the basis for the real business cycle (RBC) model of the economy, which we will discuss in greater detail in Chapter 10. The RBC model takes movements in total factor productivity to be the primary source of business cycle fluctuations.

3. Application: the post-1973 slowdown in productivity growthWhat caused the decline in productivity?a. Measurement—inadequate accounting for quality improvementsb. The legal and human environment—regulations for pollution control and worker safety,

crime, and declines in educational quality

Data Application

Mark Wynne’s article, “The Comparative Growth Performance of the U.S. Economy in the Postwar Period,” Federal Reserve Bank of Dallas Economic Review, First Quarter 1992, pp. 1–16, argues that the postwar period up to the mid-1970s showed extraordinary productivity growth. After the mid-1970s, productivity returned to more normal levels.

c. Oil prices—huge increase in oil prices reduced productivity of capital and labor, especially in basic industries

d. New industrial revolution—learning process for information technology from 1973 to 1990 meant slower growth

4. Application: the recent surge in U.S. productivity growtha. Labor productivity growth increased sharply in the second half of the 1990sb. Labor productivity and TFP have grown steadily over the past 26 years (text Fig. 6.1)c. Labor productivity growth has generally exceeded TFP growth since 1995 (Fig. 6.2)d. The gap between labor productivity growth and TFP growth can be seen in the equation

(6.3)

(1) Equation (6.3) suggests that labor productivity growth (the left-side term) exceeds TFP growth (the first right-side term) when capital growth exceeds labor growth


Chapter 6 Long-Run Economic Growth 107

e. The increase in labor productivity can be traced to the ICT (information and communications technologies) revolution(1) But other countries also had an ICT revolution, and their labor productivity did not

rise as much as in the United States(2) European labor productivity did not rise as much as in the U.S. because of

government regulationsf. Why is there such a lag between ICT investment and increases in productivity?

(1) Because productivity improvements require not just technological advances, but also investment in intangible capital—research and development, reorganization of firms, and worker training

g. Is the recent episode unique in U.S. history?(1) Not really: 1873–1890—steam power, trains, telegraph; 1917–1927—electrification

in factories; 1948–1973—transistor

II. Growth Dynamics: The Solow Model (Sec. 6.2)

A. Two basic questions about growth

1. What’s the relationship between the long-run standard of living and the saving rate, population growth rate, and rate of technical progress?

2. How does economic growth change over time? Will it speed up, slow down, or stabilize?

B. Setup of the Solow model

1. Basic assumptions and variablesa. Population and work force grow at same rate nb. Economy is closed and G 0c. Ct Yt It (6.4)d. Rewrite everything in per-worker terms: yt Yt/Nt; ct Ct/Nt; kt Kt/Nt

e. kt is also called the capital-labor ratio2. The per-worker production function

a. yt f(kt) (6.5)b. Assume no productivity growth for now (add it later)c. Plot of per-worker production function—text Figure 6.3d. Same shape as aggregate production function

Numerical Problem 3 and Analytical Problem 6 work with the per-worker production function.

3. Steady statesa. Steady state: yt, ct, and kt are constant over timeb. Gross investment must

(1) Replace worn out capital, dKt

(2) Expand so the capital stock grows as the economy grows, nKt

c. It (n d)Kt (6.6)d. From Eq. (6.4),

Ct Yt It Yt (n d)Kt (6.7)

e. In per-worker terms, in steady state

c f(k) (n d)k (6.8)



f. Plot of c, f(k), and (n d)k (Figure 6.1; identical to text Figure 6.4)

Figure 6.1

g. Increasing k will increase c up to a point(1) This is kG in the figure, the Golden Rule capital-labor ratio(2) For k beyond this point, c will decline(3) But we assume henceforth that k is less than kG, so c always rises as k rises

4. Reaching the steady statea. Suppose saving is proportional to current income:

St sYt, (6.9)

where s is the saving rate, which is between 0 and 1



b. Equating saving to investment gives

sYt (n d)Kt (6.10)

c. Putting this in per-worker terms gives

sf(k) (n d)k (6.11)

d. Plot of sf(k) and (n d)k (Figure 6.2; identical to text Figure 6.5)

Figure 6.2

e. The only possible steady-state capital-labor ratio is k*

f. Output at that point is y* f(k*); consumption is c* f(k*) (n d)k*

g. If k begins at some level other than k*, it will move toward k*

(1) For k below k*, saving > the amount of investment needed to keep k constant,so k rises

(2) For k above k*, saving < the amount of investment needed to keep k constant,so k falls

Numerical Problems 5, 6, and 7 look at equilibrium in the Solow model.

h. To summarize, with no productivity growth, the economy reaches a steady state, with constant capital-labor ratio, output per worker, and consumption per worker

C. The fundamental determinants of long-run living standards

1. The saving ratea. Higher saving rate means higher capital-labor ratio, higher output per worker, and higher

consumption per worker (shown in text Figure 6.6)b. Should a policy goal be to raise the saving rate?

(1) Not necessarily, since the cost is lower consumption in the short run(2) There is a trade-off between present and future consumption



2. Population growtha. Higher population growth means a lower capital-labor ratio, lower output per worker,

and lower consumption per worker (shown in text Figure 6.7)b. Should a policy goal be to reduce population growth?

(1) Doing so will raise consumption per worker(2) But it will reduce total output and consumption, affecting a nation’s ability to defend

itself or influence world eventsc. The Solow model also assumes that the proportion of the population of working age is

fixed(1) But when population growth changes dramatically this may not be true(2) Changes in cohort sizes may cause problems for social security systems and areas

like health care3. Productivity growth

a. The key factor in economic growth is productivity improvementb. Productivity improvement raises output per worker for a given level of the capital-labor

ratio (text Fig. 6.8)c. In equilibrium, productivity improvement increases the capital-labor ratio, output per

worker, and consumption per worker(1) Productivity improvement directly improves the amount that can be produced at

any capital-labor ratio(2) The increase in output per worker increases the supply of saving, causing the

long-run capital-labor ratio to rised. Can consumption per worker grow indefinitely?

(1) The saving rate can’t rise forever (it peaks at 100%) and the population growth rate can’t fall forever

(2) But productivity and innovation can always occur, so living standards can rise continuously

e. Summary: The rate of productivity improvement is the dominant factor determining how quickly living standards rise

Analytical Problems 1, 2, 3, and 4 look at how changes in the fundamentals affect an economy’s economic growth.

4. Application: The growth of Chinaa. China is an economic juggernaut

(1) Population 1.3 billion people(2) Starting with low level of GDP, but growing rapidly (text Fig. 6.10)

b. Fast output growth attributable to(1) Huge increase in capital investment(2) Fast productivity growth (in part from changing to a market economy)(3) Increased trade

c. Investment is huge in China; at the cost of current consumption, so saving is highd. Labor

(1) China has a huge labor force; comparative advantage in labor-intensive industries (and wages are low)

(2) As China grows, wages and standard of living will risee. Problems China faces

(1) Weak banking system(2) Increasing income inequality



(3) Much unemployment in rural areas

D. Endogenous growth theory—explaining the sources of productivity growth1. Aggregate production function

Y AK (6.12)

a. Constant MPK (1) Human capital

(a) Knowledge, skills, and training of individuals(b) Human capital tends to increase in same proportion as physical capital

Data Application

For more information and a look at the data on the returns to human capital, see Ellis W. Tallman and Ping Wang, “Human Capital Investment and Economic Growth: New Routes in Theory Address Old Questions,” Federal Reserve Bank of Atlanta Economic Review, September/October 1992, pp. 1–12. For an excellent and more detailed overview of new growth theory, see the symposium in the Journal of Economic Perspectives, Winter 1994.

(2) Research and development programs(3) Increases in capital and output generate increased technical knowledge, which offsets

decline in MPK from having more capital2. Implications of endogenous growth

a. Suppose saving is a constant fraction of output: S sAKb. Since investment net investment depreciation, I K dKc. Setting investment equal to saving implies:

K dK sAK (6.13)

d. Rearrange (6.13):

K/K sA d (6.14)

e. Since output is proportional to capital, Y/Y K/K, so

Y/Y sA d (6.15)

f. Thus the saving rate affects the long-run growth rate (not true in Solow model)

Theoretical Application

The Wall Street Journal discussed the theory of endogenous growth and the contributions of Stanford economist Paul Romer, in the article “Wealth of Notions,” January 21, 1997.

3. Summarya. Endogenous growth theory attempts to explain, rather than assume, the economy’s

growth rateb. The growth rate depends on many things, such as the saving rate, that can be affected by

government policies

Policy Application

For a good recent review of how government policy can contribute to economic growth, see Satyajit Chatterjee, “Making More Out of Less: The Recipe for Long-Term Economic Growth,” Federal Reserve Bank of Philadelphia Business Review, May/June 1994.



Data Application

Is the Solow model or the model of endogenous growth a better representation of how economic growth is determined? To find out, Ben S. Bernanke and Refet S. Gürkaynak of Princeton University examined data from many different countries from 1960 to 1998 (“Is Growth Exogenous? Taking Mankiw, Romer, and Weil Seriously,” NBER Macroeconomics Annual 2001, in Ben S. Bernanke and Kenneth Rogoff, eds., Cambridge, MA: MIT Press, 2002). They tested whether the Solow model or several alternative models of endogenous growth were more consistent with the data.

Bernanke and Gürkaynak tested a key implication of the Solow model: that the steady-state growth rate of a country does not depend on variables such as the rate of human capital accumulation and the saving rate. They found that, in fact, countries’ growth rates are closely correlated with both the saving rate and the rate of human capital accumulation, which suggests either that the Solow model does not work well or that the economies are not in steady state.

However, the Solow model implies that even if an economy is not in a steady state, the growth rate of total factor productivity (TFP) is exogenous: it does not depend on the saving rate or on any other behavioral variable, such as the level of education in a country or the growth rate of the labor force. After constructing measures of long-run TFP growth for about 50 countries, Bernanke and Gürkaynak examined the relationship between it and other variables. They found that there is, in fact, a strong relationship between TFP growth and the saving rate, some relationship between TFP growth and the growth rate of the labor force, and a weaker relationship between TFP growth and the level of education. Thus, the data do not support the Solow model.

Models of endogenous growth imply that human capital formation and physical capital accumulation should be related to the long-run growth rate of output. Bernanke and Gürkaynak found that there is indeed such a relationship in the data across many countries. However, the models they test are not perfect, as they cannot explain why savings rates and rates of human capital accumulation differ so much across countries.

Overall, the research of Bernanke and Gürkaynak suggests that models of endogenous growth hold more promise than the Solow model in explaining economic growth.

III. Government Policies to Raise Long-Run Living Standards (Sec. 6.3)

A. Policies to affect the saving rate

1. If the private market is efficient, the government shouldn’t try to change the saving ratea. The private market’s saving rate represents its trade-off of present for future consumptionb. But if tax laws or myopia cause an inefficiently low level of saving, government policy to

raise the saving rate may be justified2. How can saving be increased?

a. One way is to raise the real interest rate to encourage saving; but the response of saving to changes in the real interest rate seems to be small

b. Another way is to increase government saving(1) The government could reduce the deficit or run a surplus(2) But under Ricardian equivalence, tax increases to reduce the deficit won’t affect

national savingB. Policies to raise the rate of productivity growth

1. Improving infrastructurea. Infrastructure: highways, bridges, utilities, dams, and airportsb. Empirical studies suggest a link between infrastructure and productivity



c. U.S. infrastructure spending has declined in the last two decadesd. Would increased infrastructure spending increase productivity?

(1) There might be reverse causation: Richer countries with higher productivity spend more on infrastructure, rather than vice versa

(2) Infrastructure investments by government may be inefficient, since politics, not economic efficiency, is often the main determinant

2. Building human capitala. There’s a strong connection between productivity and human capitalb. Government can encourage human capital formation through educational policies,

worker training and relocation programs, and health programsc. Another form of human capital is entrepreneurial skill

Government could help by removing barriers like red tape3. Encouraging research and development

Government can encourage R and D through direct aid to research

Policy Application

Many issues relating to government policy and its effects on growth are discussed in a special issue of the Journal of Monetary Economics, December 1993. The articles were presented at a World Bank Conference on the research project, “How Do National Policies Affect Long-Run Growth?”

Additional Issues for Classroom Discussion

1. More on Measurement and Productivity

The textbook discusses some of the issues involved in measurement and productivity. If the quality of outputs isn’t accounted for accurately, then increases in quality go unmeasured, and real GDP is higher than is measured. But the implications go even further. To see this, discuss the issue of computers usedas capital with your class.

When computers are part of a firm’s capital, they contribute to production. Suppose, however, that the government counts more computers than there really are. This could happen, for example, if it was using a computer price that was several years old in constructing its price index. In this case, how does measured productivity differ from actual productivity? This is relevant, since, despite the increased use of computers, total factor productivity has barely budged over the past 20 years.

2. Financial Institutions and Growth

We often take for granted our well-developed financial system. But many countries have financial institutions that are much more primitive than ours. You may wish to discuss with your class the relationship between financial institutions and growth.

You could begin by asking how small, medium, and large companies in the United States obtain financing for investment. Your students will note that smaller companies more likely rely on banks or internally generated funds, while larger companies can borrow directly in the stock and bond markets. Then you can discuss how individuals borrow to buy houses and other durable goods. Again, there are a variety of financial sources. Next, ask them to think about whether such financial institutions are likely to be available in either war torn countries or newly industrializing countries. Finally, you might point out that even advanced countries like Japan have suffered in recent years from failing to have strong financial institutions in place.



3. Is Growth Good?

Students like to discuss the benefits versus the costs of growth. While it’s easy for the government to calculate output (GDP), it’s much harder to account for the quality of life. And everyone is aware of the tradeoffs between growth and quality of life, as seen in such side effects of growth as pollution, traffic, and suburban sprawl.

Ask your students whether they think our economy’s current growth rate is about right, or should it be slower to prevent some of the problems growth causes? Should we allow strip mines to rip giant holes in the earth, just to provide coal and minerals at lower prices? Should we allow the production of radioactive waste, just to have cheaper electricity? Should we allow species to become extinct, just to allow new housing developments? Should we destroy the rainforests, just to reduce the cost of doing business in South America?

Answers to Textbook Problems

Review Questions

1. The three sources of economic growth are capital growth, labor growth, and productivity growth. The growth accounting approach is derived from the production function.

2. A decline in productivity growth is the primary reason for the slowdown in output growth in the United States since 1973. Productivity growth may have declined because of deterioration in the legal and human environment, reduced rates of technological innovation, and the effects of high oil prices. To some extent the apparent decline in productivity may be due to measurement difficulties.

3. The rise in productivity growth in the 1990s occurred because of the revolution in information and communications technologies (ICT). Not only were there improvements in ICT, but also government regulations did not rein in the growth of productivity in the United States, as they did in other countries, such as those in Europe. In addition, intangible investment (research and development, reorganization of firms, and worker training) allowed the ICT improvements to boost productivity.

4. A steady state is a situation in which the economy’s output per worker, consumption per worker,and capital stock per worker are constant.

5. If there is no productivity growth, then output per worker, consumption per worker, and capital per worker will all be constant in the long run. This represents a steady state for the economy.

6. The statement is false. Increases in the capital-labor ratio increase consumption per worker in the steady state only up to a point. If the capital-labor ratio is too high, then consumption per worker may decline due to diminishing marginal returns to capital, and the need to divert much of output to maintaining the capital-labor ratio.

7. (a) An increase in the saving rate increases long-run living standards, as higher saving allows for more investment and a larger capital stock.

(b) An increase in the population growth rate reduces long-run living standards, as more output must be used to equip the larger number of new workers with capital, leaving less output available to increase consumption or capital per worker.

(c) A one-time increase in productivity increases living standards directly, by increasing output, and indirectly, since by raising incomes it also raises saving and the capital stock.



8. Endogenous growth theory suggests that the main sources of productivity growth are accumulationof human capital (the knowledge, skills, and training of individuals) and technological innovation (research and development, as well as learning by doing). The production function in an endogenous growth model does not exhibit diminishing marginal productivity of capital. This differs from the production function in the Solow model, which has diminishing marginal productivity of capital.

9. Government policies to promote economic growth include policies to raise the saving rate and policies to increase productivity. One way to increase the saving rate is to increase the real returnto saving by providing a tax break, as Individual Retirement Accounts did in the United States. Unfortunately, the response of saving to increases in the real rate of return is small. Another way to increase the saving rate is to reduce the government budget deficit. However, the theory of Ricardian equivalence suggests that this will not do much to increase national saving. Note that an increase in the saving rate will increase the steady-state capital-labor ratio, but will not increase the long-run rate of economic growth.

One way that government policy can increase productivity is by spending more on the economy’s infrastructure, which has been neglected over the past two decades in the United States. Another possibility is to support the creation of human capital by spending more on education and training programs, and reducing barriers to entrepreneurial activity. The issue is whether the government should intervene in the market to do these things, or whether the free market by itself provides an efficient outcome.

A one-time increase in productivity will increase the steady-state capital-labor ratio but will not increase the long-run rate of economic growth. To increase the long-run rate of economic growth,the growth rate of productivity must be permanently increased.

Endogenous growth theory modifies these conclusions to some extent. A rise in the saving rate leads to a higher long-run rate of economic growth in endogenous growth models.

Numerical Problems

1. Hare: $5000 (1.03)70 $39,589

Tortoise: $5000 (1.01)70 $10,034

2.

20 Years Ago Today Percent Change

Y 1000 1300 30%K 2500 3250 30%N 500 575 15%

(a) A/A Y/Y aK K/K aN N/N 30% (0.3 30%) 0.7 15% 30% 9% 10.5% 10.5%

Capital growth contributed 9% (aK K/K), labor growth contributed 10.5% (aN N/N), productivity growth was 10.5%.



(b) A/A 30% (0.5 30%) (0.5 15%) 30% 15% 7.5% 7.5%

Capital growth contributed 15% (aK K/K), labor growth contributed 7.5% (aN N/N), productivity growth was 7.5%.

3. (a)

Year K N Y K/N Y/N

1 200 1000 617 0.20 0.617

2 250 1000 660 0.25 0.660

3 250 1250 771 0.20 0.617

4 300 1200 792 0.25 0.660

This production function can be written in per-worker form since Y/N K.3N.7/N K.3/N.3 (K/N).3. Note that K/N is the same in years 1 and 3, and so is Y/N. Also, K/N is the same in years 2 and 4, and so is Y/N.

(b)

Year K N Y K/N Y/N

1 200 1000 1231 0.20 1.231

2 250 1000 1316 0.25 1.316

3 250 1250 1574 0.20 1.259

4 300 1200 1609 0.25 1.341

This production function can’t be written in per-worker form since Y/N K.3N.8/N K.3/N.2. Note that K/N is the same in years 1 and 3, but Y/N is not the same in these years. The same is true for years 2 and 4.

4. To answer this problem, an approximate solution can be found by finding the ratio GDP (2006)/GDP (1950), taking the natural logarithm of that ratio and dividing by 56. This is the answer given in the table below.

Real GDP Per Capital Growth1950 2006 Ratio Rate

Australia 7,412 24,343 3.28 2.1%

Canada 7,291 24,951 3.42 2.2%

France 5,186 21,809 4.21 2.6%

Germany 3,881 19,993 5.15 2.9%

Japan 1,921 22,462 11.69 4.4%

Sweden 6,739 24,204 3.59 2.2%

United Kingdom 6,939 23,013 3.32 2.1%

United States 9,561 31,049 3.25 2.1%

Germany and Japan had the highest growth rates because damage from World War II caused capital per worker to be lower than its steady-state level, and thus output per worker was temporarily low.



5. (a) sf(k) (n d)k

0.3 3k.5 (0.05 0.1)k0.9k.5 0.15k0.9/0.15 k/k.5

6 k.5

k 62 36y 3k.5 3 6 18c y (n d)k 18 (0.15 36) 12.6

(b) sf(k) (n d)k0.4 3k.5 (0.05 0.1)k1.2k.5 0.15k1.2/0.15 k/k.5

8 k.5

k 82 64y 3k.5 3 8 24c y (n d)k 24 (0.15 64) 14.4

(c) sf(k) (n d)k0.3 3k.5 (0.08 0.1)k0.9k.5 0.18k0.9/0.18 k/k.5 5 k.5 k 52 25y 3k.5 3 5 15c y – (n d)k 15 (0.18 25) 10.5

(d) sf(k) (n d)k0.3 4k.5 (0.05 0.1)k1.2k.5 0.15k1.2/0.15 k/k.5

8 k.5 k 82 64y 4k.5 4 8 32c y – (n d)k 32 (0.15 64) 22.4

6. (a) In steady state, sf(k) (n d)k

0.1 6k.5 (0.01 0.14)k0.6k.5 0.15k0.6/0.15 k/k.5

4 k.5

k 42 16 capital per workery 6k.5 6 4 24 output per workerc .9 y .9 24 21.6 consumption per worker(n d)k 0.15 16 2.4 investment per worker



(b) To get y 2 24 48, since y 6k.5, then 48 6k.5, so k.5 8, so k 64. The capital-labor ratio would need to increase from 16 to 64. To get k 64, since sf(k) (n d)k, s 48 0.15 64, sos 0.2. Saving per worker would need to double.

7. First, derive saving per worker as sy y c g [1 0.5(1 t) t] 8k.5 0.5(1 t)8k.5 4 (1 t)k.5

(a) When t 0, sy 4 (1 0)k.5 4k.5 national saving per workerInvestment per worker (n d)k 0.1kIn steady state, sy (n d)k, so 4k.5 0.1k, or 40k.5 k, so 1600k k2, so k 1600. Since k 1600, y 8 1600.5 320, c 0.5(1 0)320 160, and (n d)k 0.1 1600 160 investment per worker.

(b) When t 0.5, sy 4 (1 0.5)k.5 2k.5 national saving per workerInvestment per worker (n d)k .1kIn steady state, sy (n d)k, so 2k.5 0.1k, or 20k.5 k, so 400k k2, so k 400. Since k 400, y 8 400.5 160, c 0.5(1 0.5)160 40, and (n d)k 0.1 400 40 investment per worker, g ty 0.5 160 80.

Analytical Problems

1. (a) The destruction of some of a country’s capital stock in a war would have no effect on the steady state, because there has been no change in s, f, n, or d. Instead, k is reduced temporarily, but equilibrium forces eventually drive k to the same steady-state value as before.

(b) Immigration raises n from n1 to n2 in Figure 6.3. The rise in n lowers steady-state k, leading to a lower steady-state consumption per worker.

Figure 6.3



(c) The rise in energy prices reduces the productivity of capital per worker. This causes sf(k) to shift down from sf 1(k) to sf 2(k) in Figure 6.4. The result is a decline in steady-state k. Steady-state consumption per worker falls for two reasons: (1) Each unit of capital has a lower productivity, and (2) steady-state k is reduced.

Figure 6.4

(d) A temporary rise in s has no effect on the steady-state equilibrium.(e) The increase in the size of the labor force does not affect the growth rate of the labor force, so

there is no impact on the steady-state capital-labor ratio or on consumption per worker. However, because a larger fraction of the population is working, consumption per person increases.

2. (a) Solow model

The rise in capital depreciation shifts up the (n d)k line from (n d1)k to (n d2)k, as shown in Figure 6.5. The equilibrium steady-state capital-labor ratio declines. With a lower capital-labor ratio, output per worker is lower, so consumption per worker is lower (using the assumption that the capital-labor ratio is not so high that an increase in k will reduce consumption per worker). There is no effect on the long-run growth rate of the total capital stock, because in the long run the capital stock must grow at the same rate (n) as the labor force grows, so that the capital-labor ratio is constant.

Figure 6.5



(b) Endogenous growth modelIn an endogenous growth model, the growth rate of output is Y/Y sA d, so the rise in the deprecia-tion rate reduces the economy’s growth rate. Similarly, the growth rate of capital equals K/K sA d, which also declines when the depreciation rate rises. Since consumption is a constant fraction of output, its growth rate declines as well. So the increase in the depreciation rate reduces the long-run growth rate of the capital stock, as well as long-run capital, output, and consumption per worker.

3. (a) With a balanced budget T/N g. National saving is S s(Y – T) sN[(Y/N)(T/N)] sN( y – g). Setting saving equal to investment gives

S I,

sN(y – g) (n d)K,

s(y – g) (n d)k,

s[f(k) – g] (n d)k.

This equilibrium point k* is shown in Figure 6.6.

Figure 6.6

(b) If the government permanently increases purchases per worker, the s[ f(k) – g] curve shifts down from s[ f(k) – g1] to s[ f(k) – g2] in Figure 6.7. In steady-state equilibrium, the capital-labor ratio is lower. Output per worker, capital per worker, and consumption per worker are lower in the steady state. The optimal level of government purchases is not zero—it depends on the benefits of the government purchases as well as on the costs of these purchases.

Figure 6.7



4. St sYt – hKt Nt(syt – hkt). Setting St It yields Nt(syt – hkt) (n d)Kt. Dividing through by Nt and eliminating time subscripts for steady-state variables gives sy – hk (n d)k. Rearranging and using the expression y f(k) gives sf(k) (n d h)k.

The steady-state value of capital per worker, k*, is given by the intersection of the (n d h)k line with the sf(k) curve, as shown in Figure 6.8. Output per worker is f(k*). Since Ct Yt – St, c y –(sy – hk) (1 – s)f(k*) hk*. This expression gives consumption per worker.

Figure 6.8

A change in the steady-state value of h increases the slope of the (n d h)k line, as shown in Figure 6.9. This reduces the steady-state value of per-worker capital (k*), per-worker output [since y* f(k*)], and per-worker consumption [since c* (1 s)y* hk* and both y* and k* decline].

Figure 6.9

5. The initial level of the capital-labor ratio is irrelevant for the steady state. Two economies that are identical except for their initial capital-labor ratios will have exactly the same steady state.

Since the two economies must have the same growth rate at the steady state, and since the economy with the higher current capital-labor ratio has higher current output per worker, then the country with the lower current capital-labor ratio must grow faster.



The answer holds true regardless of which country is in a steady state. If the country with a higher initial capital-labor ratio is in a steady state at capital-labor ratio k*, then the other country’s capital-labor ratio will rise until it too equals k*. So the country with the lower capital-labor ratio grows faster than the one with the higher capital-labor ratio.

If the country with the lower initial capital-labor ratio is in a steady state at capital-labor ratio k*, then the other country’s capital-labor ratio is too high and it will decline until it equals k*. So the country with the higher capital-labor ratio must grow more slowly than the country with the lower capital-labor ratio. If the two countries are allowed to trade with each other, then their convergence to the same capital-labor ratio and output per worker will occur even faster.

6. The growth accounting equation is

Y/Y A/A (aK K/K) (aN N/N).

We are just increasing the amount of capital and labor, and there is no change in productivity, soA/A 0. If the production function can be written in per-worker terms, then total output must increase in the same proportion as the percentage increase in capital and labor, so K/K N/N Y/Y X. Plugging this into the growth accounting equation,

Y/Y A/A (aK K/K) (aN N/N),

X 0 aKX aNX,

X (aK aN)X,

aK aN 1.

7. Assume there are a constant number of workers, N, so that Ny Y and Nk K. Since y Akah1–a andh Bk, then y Ak a(Bk)1–a (AB1–a)k. Then Y Ny (AB1–a)K XK, where X equals AB1–a. This puts the production function in notation used in the chapter.

Investment is K dK sY national saving. Dividing through both sides of that expression by K and using the production function gives K/K d sXK/K sX, so K/K sX d, which is the long-run growth rate of physical capital. Since output and human capital are proportional to physical capital, they will all grow at that same rate.


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