Chapter 8

In the discussion of German and Japanese postwar growth, the text describes what happens when part of the capital stock is destroyed in a war. By contrast, suppose that a war does not directly affect the capital stock, but that casualties reduce the labor force. Assume the economy was in a steady state before the war, the saving rate is unchanged, and the rate of population growth after the war is the same as it was before. a. What is the immediate impact of the war on total output and on output per person? b. What happens subsequently to output per worker in the postwar economy? Is the growth rate of output per worker after the war smaller or greater than it was before the war?

a. The production function in the Solow growth model is Y = F(K, L), or expressed in terms of output per worker, y = f(k). If a war reduces the labor force through casualties, then L falls but k = K/L rises. The production function tells us that total output falls because there are fewer workers. Output per worker increases, however, since each worker has more capital. b. The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady state prior to the war, then after the war the economy has a capital stock that is higher than the steady-state level. This is shown in Figure 8-2 as an increase in capital per worker from k* to k1. As the economy returns to the steady state, the capital stock per worker falls from k1 back to k* , so output per worker also falls. Hence, in the transition to the new steady state, the growth of output per worker is slower than normal. In the steady state, we know that the growth rate of output per worker is equal to zero, given there is no technological change in this model. Therefore, in this case, the growth rate of output per worker must be less than zero until the new steady state is reached.

Many demographers predict that the United States will have zero population growth in the coming decades, in contrast to the historical average population growth of about 1 percent per year. Use the Solow model to forecast the effect of this slowdown in population growth on the growth of total output and the growth of output per person. Consider the effects both in the steady state and in the transition between steady states.

First, consider steady states. In Figure 8-7, the slower population growth rate shifts the line representing population growth and depreciation downward. The new steady state has a higher level of capital per worker, k* 2, and hence a higher level of output per worker. What about steady-state growth rates? In steady state, total output grows at rate n, whereas output perworker grows at rate 0. Hence, slower population growth will lower total output growth, but perworker output growth will be the same. Now consider the transition. We know that the steady-state level of output per worker is higher with low population growth. Hence, during the transition to the new steady state, output per worker must grow at a rate faster than 0 for a while. In the decades after the fall in population growth, growth in total output will transition to its new lower level while growth in output per worker will jump up but then transition back to zero.

In the Solow model, population growth leads to steady-state growth in total output, but not in output per worker. Do you think this would still be true if the production function exhibited increasing or decreasing returns to scale? Explain. (For the definitions of increasing and decreasing returns to scale, see Chapter 3, "Problems and Applications," Problem 3.)

If there are decreasing returns to labor and capital, then increasing both capital and labor by the same proportion increases output by less than this proportion. For example, if we double the amounts of capital and labor, then output increases by less than double. This may happen if there is a fixed factor such as land in the production function, and it becomes scarce as the economy grows larger. Then population growth will increase total output but decrease output per worker, since each worker has less of the fixed factor to work with. If there are increasing returns to scale, then doubling inputs of capital and labor more than doubles output. This may happen if specialization of labor becomes greater as population grows. Then population growth increases total output and also increases output per worker, since the economy is able to take advantage of the scale economy more quickly

Consider an economy described by the production function: Y = F(K, L)= K^0.4^L0.6 a. What is the per-worker production function? b. Assuming no population growth or technological progress, find the steady-state capital stock per worker, output per worker, and consumption per worker as a function of the saving rate and the depreciation rate. c. Assume that the depreciation rate is 15 percent per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for saving rates of 0 percent, 10 percent, 20 percent, 30 percent, and so on. (You might find it easiest to use a computer spreadsheet.) What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? d. Use information from Chapter 3 to find the marginal product of capital. Add to your table from part (c) the marginal product of capital net of depreciation for each of the saving rates. What does your table show about the relationship between the net marginal product of capital and steady-state consumption?

a. We follow Section 8-1, "Approaching the Steady State: A Numerical Example." The production function is Y = K^0.4 L^0.6. To derive the per-worker production function f(k), divide both sides of the production function by the labor force L: y/L =(K6.4L^.6)/L Rearrange to obtain: Y/l= K/L^.4 Because y = Y/L and k = K/L, this becomes: y = k^0.4 . b. Recall that Δk = sf(k) - δk. The steady-state value of capital per worker k* is defined as the value of k at which capital per worker is constant, so Δk = 0. It follows that in steady state 0 = sf(k) - δk, or, equivalently, K*/f(k*) = S/δ For the production function in this problem, it follows that: K*/f(k*)^.4 = S/δ Rearranging: (k*)^.6 = S/δ or (k*) = S/δ^1/.6 Substituting this equation for steady-state capital per worker into the per-worker production function from part (a) gives: y* =S/δ^.4/.6 . Consumption is the amount of output that is not invested. Since investment in the steady state equals δk* , it follows that c*=f(k*)-(k*)-δk* =(S/δ^.4/.6) - (S/δ^1/.6) c.The table below shows k*, y*, and c* for the saving rate in the left column, using the equations from part (b). We assume a depreciation rate of 15 percent (i.e., 0.1). (The last column shows the marginal product of capital, derived in part (d) below). Note that a saving rate of 100 percent (s = 1.0) maximizes output per worker. In that case, of course, nothing is ever consumed, so c* = 0. Consumption per worker is maximized at a rate of saving of somewhere between 21 and 22 percent—that is, where s equals capital's share in output. This is the Golden Rule level of s. D.The marginal product of capital (MPK) is the change in output per worker (y) for a given change in capital per worker (k). To find the marginal product of capital, differentiate the per-worker production function with respect to capital per worker (k): MPK = .4k^-.6 = .4/k^.6 To find the marginal product of capital net of depreciation, use the equation above to calculate the marginal product of capital and then subtract depreciation, which is 15 percent of the value of the steady-state level of capital per worker. These values appear in the table above. Note that when consumption per worker is maximized, the value of the marginal product of capital net of depreciation is zero.

Might a policymaker choose a steady state with more capital than in the Golden Rule steady state? With less capital than in the Golden Rule steady state? Explain your answers.

When the economy begins above the Golden Rule level of capital, reaching the Golden Rule level leads to higher consumption at all points in time. Therefore, the policymaker would always want to choose the Golden Rule level because consumption is increased for all periods of time. On the other hand, when the economy begins below the Golden Rule level of capital, reaching the Golden Rule level means reducing consumption today to increase consumption in the future. In this case, the policymaker's decision is not as clear. If the policymaker cares more about current generations than about future generations, he or she may decide not to pursue policies to reach the Golden Rule steady state. If the policymaker cares equally about all generations, then he or she chooses to reach the Golden Rule. Even though the current generation will have to consume less, an infinite number of future generations will benefit from increased consumption by moving to the Golden Rule.

Consider how unemployment would affect the Solow growth model. Suppose that output is produced according to the production function Y = K^α[(1 − u)L]^1−α, where K is capital, L is the labor force, and u is the natural rate of unemployment. The national saving rate is s, the labor force grows at rate n, and capital depreciates at rate δ. a. Express output per worker (y = Y/L) as a function of capital per worker (k = K/L) and the natural rate of unemployment (u). b. Write an equation that describes the steady state of this economy. Illustrate the steady state graphically, as we did in this chapter for the standard Solow model. c. Suppose that some change in government policy reduces the natural rate of unemployment. Using the graph you drew in part (b), describe how this change affects output both immediately and over time. Is the steady-state effect on output larger or smaller than the immediate effect? Explain.

a. To find output per worker y we divide total output by the number of workers: Y/L =K^a[(1-u)L]^-a Y=(k/l)^a (1-u)^1-a Y= k^a(1-u)^1-a where the final step uses the definition k =K/L . Notice that unemployment reduces the amount of output per worker for any given capital-labor ratio because some of the workers are not producing anything. The steady state is the level of capital per worker at which the increase in capital per worker from investment equals its decrease from depreciation and population growth: sy=(δ+n)k sk^a(1-u)^1-a =(δ +n)k K*=(1-u)(s/s+n)^1/1-a Finally, to get steady-state output per worker, plug the steady-state level of capital per worker into the production function: y*=((1-u*)(s/δ+n)^1/1-a)^a (1-u*)^1-a =(1-u*)(s/δ+n)^a/1-a Unemployment lowers steady-state output for two reasons: for a given k, unemployment lowers y, and unemployment also lowers the steady-state value k* b. The steady state can be graphically illustrated using the equations that describe the steady state from part (a) above. Unemployment lowers the marginal product of capital per worker and, hence, acts like a negative technological shock that reduces the amount of capital the economy can maintain in steady state. Figure 8-8 shows this graphically: an increase in unemployment lowers the sf(k) line and the steady-state level of capital per worker. c. Figure 8-9 shows the pattern of output over time. As soon as unemployment falls from u1 to u2, output jumps up from its initial steady-state value of y* (u1). The economy has the same amount of capital (since it takes time to adjust the capital stock), but this capital is combined with more workers. At that moment the economy is out of steady state: it has less capital than it wants to match the increased number of workers in the economy. The economy begins its transition by accumulating more capital, raising output even further than the original jump. Eventually the capital stock and output converge to their new, higher steady-state levels. .

In the Solow model, how does the saving rate affect the steady-state level of income? How does it affect the steady-state rate of growth?

In the Solow growth model, a high saving rate leads to a large steady-state capital stock and a high level of steady-state output. A low saving rate leads to a small steady-state capital stock and a low level of steady-state output. Higher saving leads to faster economic growth only in the short run. An increase in the saving rate raises growth until the economy reaches the new steady state. That is, if the economy maintains a high saving rate, it will also maintain a large capital stock and a high level of output, but it will not maintain a high rate of growth forever. In the steady state, the growth rate of output (or income) is independent of the saving rate.

Why might an economic policymaker choose the Golden Rule level of capital?

It is reasonable to assume that the objective of an economic policymaker is to maximize the economic well-being of the individual members of society. Since economic well-being depends on the amount of consumption, the policymaker should choose the steady state with the highest level of consumption. The Golden Rule level of capital represents the level that maximizes consumption in the steady state. Suppose, for example, that there is no population growth or technological change. If the steadystate capital stock increases by one unit, then output increases by the marginal product of capital MPK; depreciation, however, increases by an amount δ, so that the net amount of extra output available for consumption is MPK - δ. The Golden Rule capital stock is the level at which MPK = δ, so that the marginal product of capital equals the depreciation rate.

Country A and country B both have the production function Y = F(K, L) = K^1/3L^2/3. a. Does this production function have constant returns to scale? Explain. b. What is the per-worker production function, y = f(k)? c. Assume that neither country experiences population growth or technological progress and that 20 percent of capital depreciates each year. Assume further that country A saves 10 percent of output each year and country B saves 30 percent of output each year. Using your answer from part (b) and the steady-state condition that investment equals depreciation, find the steady-state level of capital per worker for each country. Then find the steady-state levels of income per worker and consumption per worker. d. Suppose that both countries start off with a capital stock per worker of 1. What are the levels of income per worker and consumption per worker? e. Remembering that the change in the capital stock is investment less depreciation, use a calculator (or, better yet, a computer spreadsheet) to show how the capital stock per worker will evolve over time in both countries. For each year, calculate income per worker and consumption per worker. How many years will it be before the consumption in country B is higher than the consumption in country A?

a. A production function has constant returns to scale if increasing all factors of production by an equal percentage causes output to increase by the same percentage. Mathematically, a production function has constant returns to scale if zY = F(zK, zL) for any positive number z. That is, if we multiply both the amount of capital and the amount of labor by some amount z, then the amount of output is multiplied by z. For example, if we double the amounts of capital and labor we use (setting z = 2), then output also doubles. To see if the production function Y = F(K, L) = K^1/3 L^2/3 has constant returns to scale, we write: F(zK, zL) = (zK)^1/3 (zL)^2/3 = zK^1/3 L^2/3 = zY. Therefore, the production function Y = K^1/3 L^2/3 has constant returns to scale. b. To find the per-worker production function, divide the production function Y = K1/3L2/3 by L: Y/L = (K^1/3 L^2/3)/ L If we define y = Y/L, we can rewrite the above expression as: y = K^1/3) / (L1/3 Defining k = K/L, we can rewrite the above expression as: y = k^1/3 c. We know the following facts about countries A and B: δ = depreciation rate = 0.20, sa = saving rate of country A = 0.1, sb = saving rate of country B = 0.3, and y = k1/3 is the per-worker production function derived in part (b) for countries A and B. The growth of the capital stock Δk equals the amount of investment sf(k), minus the amount of depreciation δk. That is, Δk = sf(k) - δk. In steady state, the capital stock does not grow, so we can write this as sf(k) = δk. To find the steady-state level of capital per worker, plug the per-worker production function into the steady-state investment condition, and solve for k*: sk^1/3 = δk. Rewriting this: k^2/3 = s/δ k = (s/δ)^3/2. To find the steady-state level of capital per worker k*, plug the saving rate for each country into the above formula: Country A: k = (sa/δ)^3/2 = (0.1/0.2)^3/2 =0.35. Country B: k = (sb/δ)3/2 = (0.3/0.2) 3/2 = 1.84. Now that we have found k* for each country, we can calculate the steady-state levels of income per worker for countries A and B because we know that y = k^1/3: y*a = (0.35)^1/3 = 0.71. y*b = (1.84)^1/3 = 1.22. We know that out of each dollar of income, workers save a fraction s and consume a fraction (1 -s). That is, the consumption function is c = (1 - s)y. Since we know the steady-state levels of income in the two countries, we find Country A: c = (1 - sa)y = (1 - 0.1)(0.71) = 0.64. Country B: c = (1 - sb)y = (1 - 0.3)(1.224) = 0.86. d. If capital per worker is equal to 1 in both countries, we find the following values for income per worker and consumption per worker in each country: Country A: y = 1 and c = 0.9 Country B: y = 1 and c = 0.7. e. Using the following facts and equations, we calculate income per worker y, consumption per worker c, and capital per worker k: sa = 0.1. sb = 0.3. δ = 0.2. ko = 1 for both countries. y = k^1/3 c = (1 - s)y. Note that it will take seven years before consumption in country B is higher than consumption in country A

Draw a well-labeled graph that illustrates the steady state of the Solow model with population growth. Use the graph to find what happens to steady-state capital per worker and income per worker in response to each of the following exogenous changes. a. A change in consumer preferences increases the saving rate. b. A change in weather patterns increases the depreciation rate. c. Better birth-control methods reduce the rate of population growth. d. A one-time, permanent improvement in technology increases the amount of output that can be produced from any given amount of capital and labor.

a. An increase in the saving rate will shift the saving curve upwards, as illustrated in Figure 8-3. Since actual investment is now greater than breakeven investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker. b. An increase in the depreciation rate will shift the break-even investment line upwards to (δ2 + n) as illustrated in Figure 8-4. Since actual investment is now less than break-even investment, the level of capital per worker will decrease and the steady-state level of capital per worker will be lower. The decrease in capital per worker will decrease output per worker. c. A reduction in the rate of population growth will shift the break-even investment line down and to the right to (δ + n2) as illustrated in Figure 8-5. Since actual investment is now greater than breakeven investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker. d. The technological improvement increases output f(k), and as a result the saving curve shifts upwards as illustrated in Figure 8-6. Since actual investment is now greater than break-even investment, the level of capital per worker will increase and the steady-state level of capital per worker will be higher. The increase in capital per worker will increase output per worker.

"Devoting a larger share of national output to investment would help restore rapid productivity growth and rising living standards." Do you agree with this claim? Explain, using the Solow model.

Suppose the economy begins with an initial steady-state capital stock below the Golden Rule level. The immediate effect of devoting a larger share of national output to investment is that the economy devotes a smaller share to consumption; that is, "living standards" as measured by consumption fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise. The productivity of workers is the average amount produced by each worker—that is, output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises. In the new steady state, output grows at rate n, while output per worker grows at rate zero. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden Rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher. Thus, an increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth, but not the sacrifice required to achieve it.

In the Solow model, how does the rate of population growth affect the steady-state level of income? How does it affect the steady-state rate of growth?

The higher the population growth rate is, the lower the steady-state level of capital per worker, and therefore there is a lower level of steady-state income per worker. For example, Figure 8-1 shows the steady state for two levels of population growth, a low level n1 and a higher level n2. The higher population growth n2 means that the line representing population growth and depreciation is higher, so the steady-state level of capital per worker is lower In a model with no technological change, the steady-state growth rate of total income is n: the higher the population growth rate n is, the higher the growth rate of total income. Income per worker, however, grows at rate zero in steady state and, thus, is not affected by population growth