Chapter 9

Suppose an economy described by the Solow model has the following production function: Y = K^1/2(LE)^1/2. a. For this economy, what is f(k)? b. Use your answer to part (a) to solve for the steady-state value of y as a function of s, n, g, and δ. c. Two neighboring economies have the above production function, but they have different parameter values. Atlantis has a saving rate of 28 percent and a population growth rate of 1 percent per year. Xanadu has a saving rate of 10 percent and a population growth rate of 4 percent per year. In both countries, g = 0.02 and δ = 0.04. Find the steady-state value of y for each country.

a. In the Solow model with technological progress, y is defined as output per effective worker, and k is defined as capital per effective worker. The number of effective workers is defined as L E (or LE), where L is the number of workers, and E measures the efficiency of each worker. To find output per effective worker y, divide total output by the number of effectiv e workers: b. To solve for the steady-state value of y as a function of s, n, g, and δ, we begin with the equation for the change in the capital stock in the steady state: Δk = sf(k) - (δ + n + g)k = 0. The production function Y=sqroot k can also be rewritten as y^2 = k. Plugging this production function into the equation for the change in the capital stock, we find that in the steady state: sy - (δ + n + g)y^2 = 0. Solving this, we find the steady-state value of y: y* = s/(δ + n + g). C. The question provides us with the following information about each country: Atlantis: s = 0.28 n = 0.01 g = 0.02 δ = 0.04 Xanadu S=.1 n=.04 g=.02 δ = .04 Using the equation for y* that we derived in part (a), we can calculate the steady-state values of y for each country. Developed country: y* = 0.28/(0.04 + 0.01 + 0.02) = 4 Less-developed country: y* = 0.10/(0.04 + 0.04 + 0.02) = 1

This question asks you to analyze in more detail the two-sector endogenous growth model presented in the text. a. Rewrite the production function for manufactured goods in terms of output per effective worker and capital per effective worker. b. In this economy, what is break-even investment (the amount of investment needed to keep capital per effective worker constant)? c. Write down the equation of motion for k, which shows Δk as saving minus break-even investment. Use this equation to draw a graph showing the determination of steady-state k. (Hint: This graph will look much like those we used to analyze the Solow model.) d. In this economy, what is the steady-state growth rate of output per worker Y/L? How do the saving rate s and the fraction of the labor force in universities u affect this steady-state growth rate? e. Using your graph, show the impact of an increase in u. (Hint: This change affects both curves.) Describe both the immediate and the steady-state effects. f. Based on your analysis, is an increase in u an unambiguously good thing for the economy? Explain.

a. In the two-sector endogenous growth model in the text, the production function for manufactured goods is Y = F [K,(1 - u) EL]. We assumed in this model that this function has constant returns to scale. As in Section 3-1, constant returns means that for any positive number z, zY = F(zK, z(1 - u) EL). Setting z = 1/EL, we obtain y/EL = F(K/EL, (1-U)) Using our standard definitions of y as output per effective worker and k as capital per effective worker, we can write this as y = F[k,(1 - u)] b. To begin, note that from the production function in research universities, the growth rate of labor efficiency, ΔE/E, equals g(u). We can now follow the logic of Section 9-1, substituting the function g(u) for the constant growth rate g. In order to keep capital per effective worker (K/EL) constant, break-even investment includes three terms: δk is needed to replace depreciating capital, nk is needed to provide capital for new workers, and g(u) is needed to provide capital for the greater stock of knowledge E created by research universities. That is, break-even investment is [δ + n + g(u)]k. c. Again following the logic of Section 9-1, the growth of capital per effective worker is the difference between saving per effective worker and break-even investment per effective worker. We now substitute the per-effective-worker production function from part (a) and the function g(u) for the constant growth rate g, to obtain Δk = sF [k,(1 - u)] - [δ + n + g(u)]k In the steady state, Δk = 0, so we can rewrite the equation above as sF [k,(1 - u)] = [δ + n + g(u)]k. As in our analysis of the Solow model, for a given value of u, we can plot the left and right sides of this equation The steady state is given by the intersection of the two curves. d. The steady state has constant capital per effective worker k as given by Figure 9-2 above. We also assume that in the steady state, there is a constant share of time spent in research universities, so u is constant. (After all, if u were not constant, it wouldn't be a "steady" state!). Hence, output per effective worker y is also constant. Output per worker equals yE, and E grows at rate g(u). Therefore, output per worker grows at rate g(u). The saving rate does not affect this growth rate. However, the amount of time spent in research universities does affect this rate: as more time is spent in research universities, the steady-state growth rate rises. e. An increase in u shifts both lines in our figure. Output per effective worker falls for any given level of capital per effective worker, since less of each worker's time is spent producing manufactured goods. This is the immediate effect of the change, since at the time u rises, the capital stock K and the efficiency of each worker E are constant. Since output per effective worker falls, the curve showing saving per effective worker shifts down. At the same time, the increase in time spent in research universities increases the growth rate of labor efficiency g(u). Hence, break-even investment [which we found above in part (b)] rises at any given level of k, so the line showing breakeven investment also shifts up. Figure 9-3 shows these shifts. In the new steady state, capital per effective worker falls from k1 to k2. Output per effective worker also falls. f. In the short run, the increase in u unambiguously decreases consumption. After all, we argued in part (e) that the immediate effect is to decrease output, since workers spend less time producing manufacturing goods and more time in research universities expanding the stock of knowledge. For a given saving rate, the decrease in output implies a decrease in consumption. The long-run steady-state effect is more subtle. We found in part (e) that output per effective worker falls in the steady state. But welfare depends on output (and consumption) per worker, not per effective worker. The increase in time spent in research universities implies that E grows faster. That is, output per worker equals yE. Although steady-state y falls, in the long run the faster growth rate of E necessarily dominates. That is, in the long run, consumption unambiguously rises. Nevertheless, because of the initial decline in consumption, the increase in u is not unambiguously a good thing. That is, a policymaker who cares more about current generations than about future generations may decide not to pursue a policy of increasing u. (This is analogous to the question considered in Chapter 8 of whether a policymaker should try to reach the Golden Rule level of capital per effective worker if k is currently below the Golden Rule level.) .

How can policymakers influence a nation's saving rate?

Economic policy can influence the saving rate by either increasing public saving or providing incentives to stimulate private saving. Public saving is the difference between government revenue and government spending. If spending exceeds revenue, the government runs a budget deficit, which is negative saving. Policies that decrease the deficit (such as reductions in government purchases or increases in taxes) increase public saving, whereas policies that increase the deficit decrease saving. A variety of government policies affect private saving. The decision by a household to save may depend on the rate of return; the greater the return to saving, the more attractive saving becomes. Tax incentives such as tax-exempt retirement accounts for individuals and investment tax credits for corporations increase the rate of return and encourage private saving.

How does endogenous growth theory explain persistent growth without the assumption of exogenous technological progress? How does this differ from the Solow model?

Endogenous growth theories attempt to explain the rate of technological progress by explaining the decisions that determine the creation of knowledge through research and development. By contrast, the Solow model simply took this rate as exogenous. In the Solow model, the saving rate affects growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. By contrast, many endogenous growth models in essence assume that there are constant (rather than diminishing) returns to capital, interpreted to include knowledge. Hence, changes in the saving rate can lead to persistent growth.

In the Solow model, what determines the steady-state rate of growth of income per worker?

In the Solow model, we find that only technological progress can affect the steady-state rate of growth in income per worker. Growth in the capital stock (through high saving) has no effect on the steady-state growth rate of income per worker; neither does population growth. But technological progress can lead to sustained growth.

Give an example of an institutional difference between countries that might explain the differences in income per person.

The legal system is an example of an institutional difference between countries that might explain differences in income per person. Countries that have adopted the English style common law system tend to have better developed capital markets, and this leads to more rapid growth because it is easier for businesses to obtain financing. The quality of government is also important. Countries with more government corruption tend to have lower levels of income per person.

Choose two countries that interest you—one rich and one poor. What is the income per person in each country? Find some data on country characteristics that might help explain the difference in income: investment rates, population growth rates, educational attainment, and so on. (Hint: The Web site of the World Bank, http://www.worldbank.org, is one place to find such data.) How might you figure out which of these factors is most responsible for the observed income difference? In your judgment, how useful is the Solow model as an analytic tool for understanding the difference between the two countries you chose?

On the World Bank Web site (www.worldbank.org), click on the data tab and then the indicators tab. This brings up a large list of data indicators that allows you to compare the level of growth and development across countries. To explain differences in income per person across countries, you might look at gross saving as a percentage of GDP, gross capital formation as a percentage of GDP, literacy rate, life expectancy, and population growth rate. From the Solow model, we learned that (all else the same) a higher rate of saving will lead to higher income per person, a lower population growth rate will lead to higher income per person, a higher level of capital per worker will lead to a higher level of income per person, and more efficient or productive labor will lead to higher income per person. The selected data indicators offer explanations as to why one country might have a higher level of income per person. However, although we might speculate about which factor is most responsible for the difference in income per person across countries, it is not possible to say for certain given the large number of other variables that also affect income per person. For example, some countries may have more developed capital markets, less government corruption, and better access to foreign direct investment. The Solow model allows us to understand some of the reasons why income per person differs across countries, but given it is a simplified model, it cannot explain all of the reasons why income per person may differ.

The amount of education the typical person receives varies substantially among countries. Suppose you were to compare a country with a highly educated labor force and a country with a less educated labor force. Assume that education affects only the level of the efficiency of labor. Also assume that the countries are otherwise the same: they have the same saving rate, the same depreciation rate, the same population growth rate, and the same rate of technological progress. Both countries are described by the Solow model and are in their steady states. What would you predict for the following variables? a. The rate of growth of total income b. The level of income per worker c. The real rental price of capital d. The real wage

How do differences in education across countries affect the Solow model? Education is one factor affecting the efficiency of labor, which we denoted by E. (Other factors affecting the efficiency of labor include levels of health, skill, and knowledge.) Since country 1 has a more highly educated labor force than country 2, each worker in country 1 is more efficient. That is, E1 > E2. We will assume that both countries are in steady state. a. In the Solow growth model, the rate of growth of total income is equal to n + g, which is independent of the work force's level of education. The two countries will, thus, have the same rate of growth of total income because they have the same rate of population growth and the same rate of technological progress. b. Because both countries have the same saving rate, the same population growth rate, and the same rate of technological progress, we know that the two countries will converge to the same steady-state level of capital per effective worker k*. This is shown in Figure 9-1. Hence, output per effective worker in the steady state, which is y* = f(k*), is the same in both countries. But y* = Y/(L E) or Y/L = y* E. We know that y* will be the same in both countries, but that E1 > E2. Therefore, y*E1 > y*E2. This implies that (Y/L)1 > (Y/L)2. Thus, the level of income per worker will be higher in the country with the more educated labor force. c. We know that the real rental price of capital R equals the marginal product of capital (MPK). But the MPK depends on the capital stock per efficiency unit of labor. In the steady state, both countries have k*1 = k*2 = k* because both countries have the same saving rate, the same population growth rate, and the same rate of technological progress. Therefore, it must be true that R1 = R2 = MPK. Thus, the real rental price of capital is identical in both countries. d. Output is divided between capital income and labor income. Therefore, the wage per effective worker can be expressed as w = f(k) - MPK • k. As discussed in parts (b) and (c), both countries have the same steady-state capital stock k and the same MPK. Therefore, the wage per effective worker in the two countries is equal. Workers, however, care about the wage per unit of labor, not the wage per effective worker. Also, we can observe the wage per unit of labor but not the wage per effective worker. The wage per unit of labor is related to the wage per effective worker by the equation Wage per Unit of L = wE. Thus, the wage per unit of labor is higher in the country with the more educated labor force.

In the steady state of the Solow model, at what rate does output per person grow? At what rate does capital per person grow? How does this compare with the U.S. experience?

In the steady state, output per person in the Solow model grows at the rate of technological progress g. Capital per person also grows at rate g. Note that this implies that output and capital per effective worker are constant in steady state. In the U.S. data, output and capital per worker have both grown at about 2 percent per year for the past half-century.

Labor productivity is defined as Y/L, the amount of output divided by the amount of labor input. Start with the growth-accounting equation and show that the growth in labor productivity depends on growth in total factor productivity and growth in the capital-labor ratio. In particular, show that LOOK ON CHEG for ? Hint: You may find the following mathematical trick helpful. If z = wx, then the growth rate of z is approximately the growth rate of w plus the growth rate of x. That is,

LOOK at Answer key

Suppose an economy described by the Solow model is in a steady state with population growth n of 1.8 percent per year and technological progress g of 1.8 percent per year. Total output and total capital grow at 3.6 percent per year. Suppose further that the capital share of output is 1/3. If you used the growth-accounting equation to divide output growth into three sources—capital, labor, and total factor productivity—how much would you attribute to each source? Compare your results to the figures we found for the United States in Table 9-2.

Look at answer key

What data would you need to determine whether an economy has more or less capital than in the Golden Rule steady state?

To decide whether an economy has more or less capital than the Golden Rule, we need to compare the marginal product of capital net of depreciation (MPK - δ) with the growth rate of total output (n + g). The growth rate of GDP is readily available. Estimating the net marginal product of capital requires a little more work but, as shown in the text, can be backed out of available data on the capital stock relative to GDP, the total amount of depreciation relative to GDP, and capital's share in GDP.

In the United States, the capital share of GDP is about 30 percent, the average growth in output is about 3 percent per year, the depreciation rate is about 4 percent per year, and the capital-output ratio is about 2.5. Suppose that the production function is Cobb-Douglas and that the United States has been in a steady state. (For a discussion of the Cobb-Douglas production function, see Chapter 3.) a. What must the saving rate be in the initial steady state? [Hint: Use the steady-state relationship, sy = (δ + n + g)k.] b. What is the marginal product of capital in the initial steady state? c. Suppose that public policy alters the saving rate so that the economy reaches the Golden Rule level of capital. What will the marginal product of capital be at the Golden Rule steady state? Compare the marginal product at the Golden Rule steady state to the marginal product in the initial steady state. Explain. d. What will the capital-output ratio be at the Golden Rule steady state? (Hint: For the Cobb-Douglas production function, the capital-output ratio is related to the marginal product of capital.) e. What must the saving rate be to reach the Golden Rule steady state?

To solve this problem, it is useful to establish what we know about the U.S. economy: • A Cobb-Douglas production function has the form y = k^α, where α is capital's share of income. The question tells us that α = 0.3, so we know that the production function is y = k^0.3. • In the steady state, we know that the growth rate of output equals 3 percent, so we know that (n + g) = 0.03. • The depreciation rate δ = 0.04. The capital-output ratio K/Y = 2.5. Because k/y = [K/(LE)]/[Y/(LE)] = K/Y, we also know that k/y = 2.5. (That is, the capital-output ratio is the same in terms of effective workers as it is in levels.) a. Begin with the steady-state condition, sy = (δ + n + g)k. Rewriting this equation leads to a formula for saving in the steady state: s = (δ + n + g)(k/y). Plugging in the values established above: s = (0.04 + 0.03)(2.5) = 0.175. The initial saving rate is 17.5 percent. b.We know from Chapter 3 that with a Cobb-Douglas production function, capital's share of income α = MPK(K/Y). Rewriting, we have MPK = α/(K/Y). Plugging in the values established above, we find MPK = 0.3/2.5 = 0.12. c. We know that at the Golden Rule steady state: MPK = (n + g + δ). Plugging in the values established above: MPK = (0.03 + 0.04) = 0.07. At the Golden Rule steady state, the marginal product of capital is 7 percent, whereas it is 12 percent in the initial steady state. Hence, from the initial steady state we need to increase k to achieve the Golden Rule steady state. d. We know from Chapter 3 that for a Cobb-Douglas production function, MPK = α (Y/K). Solving this for the capital-output ratio, we find K/Y = α/MPK. We can solve for the Golden Rule capital-output ratio using this equation. If we plug in the value 0.07 for the Golden Rule steady-state marginal product of capital, and the value 0.3 for α, we find K/Y = 0.3/0.07 = 4.29. In the Golden Rule steady state, the capital-output ratio equals 4.29, compared to the current capital-output ratio of 2.5. e. We know from part (a) that in the steady state s = (δ + n + g)(k/y), where k/y is the steady-state capital-output ratio. In the introduction to this answer, we showed that k/y = K/Y, and in part (d) we found that the Golden Rule K/Y = 4.29. Plugging in this value and those established above: s = (0.04 + 0.03)(4.29) = 0.30. To reach the Golden Rule steady state, the saving rate must rise from 17.5 to 30 percent. This result implies that if we set the saving rate equal to the share going to capital (30 percent), we will achieve the Golden Rule steady state.

An economy has a Cobb-Douglas production function: Y = Kα(LE)^1−α. (For a review of the Cobb-Douglas production function, see Chapter 3.) The economy has a capital share of a third, a saving rate of 24 percent, a depreciation rate of 3 percent, a rate of population growth of 2 percent, and a rate of labor-augmenting technological change of 1 percent. It is in steady state. a. At what rates do total output, output per worker, and output per effective worker grow? b. Solve for capital per effective worker, output per effective worker, and the marginal product of capital. c. Does the economy have more or less capital than at the Golden Rule steady state? How do you know? To achieve the Golden Rule steady state, does the saving rate need to increase or decrease? d. Suppose the change in the saving rate you described in part (c) occurs. During the transition to the Golden Rule steady state, will the growth rate of output per worker be higher or lower than the rate you derived in part (a)? After the economy reaches its new steady state, will the growth rate of output per worker be higher or lower than the rate you derived in part (a)? Explain your answers.

a. In the steady state, capital per effective worker is constant, and this leads to a constant level of output per effective worker. Given that the growth rate of output per effective worker is zero, this means the growth rate of output is equal to the growth rate of effective workers (LE). We know labor grows at the rate of population growth n and the efficiency of labor (E) grows at rate g. Therefore, output grows at rate n+g. Given output grows at rate n+g and labor grows at rate n, output per worker must grow at rate g. This follows from the rule that the growth rate of Y/L is equal to the growth rate of Y minus the growth rate of L. b. First find the output per effective worker production function by dividing both sides of the production function by the number of effective workers LE: LOOK ON CHEGGG To solve for capital per effective worker, we start with the steady state condition: Δk = sf(k) - (δ + n + g)k =0. Now substitute in the given parameter values and solve for capital per effective worker (k): 0.24k^(1/3)=(0.03+0.02+0.01)k k^(2/3)=4 k=8. Substitute the value for k back into the per effective worker production function to find output per effective worker is equal to 2. The marginal product of capital is given by MPK=1/(3k^(2/3) ) . Substitute the value for capital per effective worker to find the marginal product of capital is equal to 1/12. c. According to the Golden Rule, the marginal product of capital is equal to (δ + n + g) or 0.06. In the current steady state, the marginal product of capital is equal to 1/12 or 0.083. Therefore, we have less capital per effective worker in comparison to the Golden Rule. As the level of capital per effective worker rises, the marginal product of capital will fall until it is equal to 0.06. To increase capital per effective worker, there must be an increase in the saving rate. d. During the transition to the Golden Rule steady state, the growth rate of output per worker will increase. In the steady state, output per worker grows at rate g. The increase in the saving rate will increase output per effective worker, and this will increase output per effective worker. In the new steady state, output per effective worker is constant at a new higher level, and output per worker is growing at rate g. During the transition, the growth rate of output per worker jumps up, and then transitions back down to rate g.

Prove each of the following statements about the steady state of the Solow model with population growth and technological progress. a. The capital-output ratio is constant. b. Capital and labor each earn a constant share of an economy's income. [Hint: Recall the definition MPK = f(k + 1) − f(k).] c. Total capital income and total labor income both grow at the rate of population growth plus the rate of technological progress, n + g. d. The real rental price of capital is constant, and the real wage grows at the rate of technological progress g. (Hint: The real rental price of capital equals total capital income divided by the capital stock, and the real wage equals total labor income divided by the labor force.)

a. In the steady state, we know that sy = (δ + n + g)k. This implies that k/y = s/(δ + n + g). Since s, δ, n, and g are constant, this means that the ratio k/y is also constant. Since k/y = [K/(LE)]/[Y/(LE)] = K/Y, we can conclude that in the steady state, the capital-output ratio is constant. b. We know that capital's share of income = MPK (K/Y). In the steady state, we know from part (a) that the capital-output ratio K/Y is constant. We also know from the hint that the MPK is a function of k, which is constant in the steady state; therefore the MPK itself must be constant. Thus, capital's share of income is constant. Labor's share of income is 1 - [Capital's Share]. Hence, if capital's share is constant, we see that labor's share of income is also constant. c. We know that in the steady state, total income grows at n + g, defined as the rate of population growth plus the rate of technological change. In part (b) we showed that labor's and capital's share of income is constant. If the shares are constant, and total income grows at the rate n + g, then labor income and capital income must also grow at the rate n + g. d. Define the real rental price of capital R as R = Total Capital Income/Capital Stock = (MPK K)/K = MPK. We know that in the steady state, the MPK is constant because capital per effective worker k is constant. Therefore, we can conclude that the real rental price of capital is constant in the steady state. To show that the real wage w grows at the rate of technological progress g, define TLI = Total Labor Income L = Labor Force Using the hint that the real wage equals total labor income divided by the labor force: w = TLI/L. Equivalently, wL = TLI. In terms of percentage changes, we can write this as Δw/w + ΔL/L = ΔTLI/TLI. This equation says that the growth rate of the real wage plus the growth rate of the labor force equals the growth rate of total labor income. We know that the labor force grows at rate n, and, from part (c), we know that total labor income grows at rate n + g. We, therefore, conclude that the real wage grows at rate g.

In the economy of Solovia, the owners of capital get two-thirds of national income, and the workers receive one-third. a. The men of Solovia stay at home performing household chores, while the women work in factories. If some of the men started working outside the home so that the labor force increased by 5 percent, what would happen to the measured output of the economy? Does labor productivity—defined as output per worker—increase, decrease, or stay the same? Does total factor productivity increase, decrease, or stay the same? b. In year 1, the capital stock was 6, the labor input was 3, and output was 12. In year 2, the capital stock was 7, the labor input was 4, and output was 14. What happened to total factor productivity between the two years?

a. The growth in total output (Y) depends on the growth rates of labor (L), capital (K), and total factor productivity (A), as summarized by the equation ΔY/Y = αΔK/K + (1 - α)ΔL/L + ΔA/A, where α is capital's share of output. We can look at the effect on output of a 5-percent increase in labor by setting ΔK/K = ΔA/A = 0. Since α = 2/3, this gives us ΔY/Y = (1/3)(5%) = 1.67%. A 5-percent increase in labor input increases output by 1.67 percent. Labor productivity is Y/L. We can write the growth rate in labor productivity as ΔY/Y =((Δ(Y/L)/Y/l))-(ΔL/L) Substituting for the growth in output and the growth in labor, we find Δ(Y/L)/(Y/L) = 1.67% - 5.0% = -3.34%. Labor productivity falls by 3.34 percent. To find the change in total factor productivity, we use the equation ΔA/A = ΔY/Y - αΔK/K - (1 - α)ΔL/L. For this problem, we find ΔA/A = 1.67% - 0 - (1/3)(5%) = 0. Total factor productivity is the amount of output growth that remains after we have accounted for the determinants of growth that we can measure. In this case, there is no change in technology, so all of the output growth is attributable to measured input growth. That is, total factor productivity growth is zero, as expected. b. Between years 1 and 2, the capital stock grows by 1/6, labor input grows by 1/3, and output grows by 1/6. We know that the growth in total factor productivity is given by ΔA/A = ΔY/Y - αΔK/K - (1 - α)ΔL/L. Substituting the numbers above, and setting α = 2/3, we find ΔA/A= (1/6) - (2/3)(1/6) - (1/3)(1/3) = 3/18 - 2/18 - 2/18 = - 1/18 = -0.056. Total factor productivity falls by 1/18, or approximately 5.6 percent

Two countries, Richland and Poorland, are described by the Solow growth model. They have the same Cobb-Douglas production function, F(K, L) = A KαL1−α, but with different quantities of capital and labor. Richland saves 32 percent of its income, while Poorland saves 10 percent. Richland has population growth of 1 percent per year, while Poorland has population growth of 3 percent. (The numbers in this problem are chosen to be approximately realistic descriptions of rich and poor nations.) Both nations have technological progress at a rate of 2 percent per year and depreciation at a rate of 5 percent per year. a. What is the per-worker production function f(k)? b. Solve for the ratio of Richland's steady-state income per worker to Poorland's. (Hint: The parameter α will play a role in your answer.) c. If the Cobb-Douglas parameter α takes the conventional value of about 1/3, how much higher should income per worker be in Richland compared to Poorland? d. Income per worker in Richland is actually 16 times income per worker in Poorland. Can you explain this fact by changing the value of the parameter α? What must it be? Can you think of any way of justifying such a value for this parameter? How else might you explain the large difference in income between Richland and Poorland?

a. The per worker production function is F(K, L)/L = AK^α L^1-α/L = A(K/L)^α = Ak^α b. LOOK ON ANSWER SHEET c. If α equals 1/3, then Richland should be 41/2, or two times, richer than Poorland. d. IF 4^(a/1-a)= 16, then it must be the case that , which in turn requires that α equals 2/3. Hence, if the Cobb-Douglas production function puts 2/3 of the weight on capital and only 1/3 on labor, then we can explain a 16-fold difference in levels of income per worker. One way to justify this might be to think about capital more broadly to include human capital—which must also be accumulated through investment, much in the way one accumulates physical capital.