曼昆 宏观经济经济学第九版 英文原版答案9

Chapter 9 Economic Growth II Technology Empirics and Policy 71 Answers to Textbook Questions and Problems CHAPTER 9 Economic Growth II Technology Empirics and Policy Questions for Review 1 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 2 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 3 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 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 4 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 5 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 6 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 Problems and Applications 1 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 effective workers Chapter 9 Economic Growth II Technology Empirics and Policy 72 YLE K12 LE 12YLE K12LE12YLE 1212E12YLE KLE 12y k 12 b To solve for the steady state value of y as a function of s n g and we begin with the equation for the change in the capital stock in the steady state k sf k n g k 0 The production function can also be rewritten as y2 k Plugging this production function y k into the equation for the change in the capital stock we find that in the steady state sy n g y2 0 Solving this we find the steady state value of y y s n g c The question provides us with the following information about each country Atlantis s 0 28 Xanadu s 0 10 n 0 01 n 0 04 g 0 02 g 0 02 0 04 0 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 2 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 Chapter 9 Economic Growth II Technology Empirics and Policy 73 YLE K13 LE 23YLE 13L2E23YLE K 131313YLE KLE 13y k 13 To solve for capital per effective worker we start with the steady state condition k sf k n g k 0 Now substitute in the given parameter values and solve for capital per effective worker k 0 24 13 0 03 0 02 0 01 23 4 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 1 3 23 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 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 3 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 where is capital s share of income The question tells us that 0 3 so we know that the production function is y k0 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 0 04 Chapter 9 Economic Growth II Technology Empirics and Policy 74 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 n g k Rewriting this equation leads to a formula for saving in the steady state s 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 MPK K Y Rewriting we have MPK 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 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 Y K Solving this for the capital output ratio we find K Y 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 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 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 Chapter 9 Economic Growth II Technology Empirics and Policy 75 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 4 a In the steady state we know that sy n g k This implies that k y s n g Since s 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 w w L L TLI 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 Chapter 9 Economic Growth II Technology Empirics and Policy 76 5 a The per worker production function is F K L L AK L1 L A K L Ak b In the steady state k sf k n g k 0 Hence sAk n g k or after rearranging k sA ng 1 Plugging into the per worker production function from part a gives y A 1 s ng 1 Thus the ratio of steady state income per worker in Richland to Poorland is y Richland y Porland sRichland ichland g sPorland orland g 1 0 320 5 1 0 2 0 10 5 3 0 2 1 c If equals 1 3 then Richland should be 41 2 or two times richer than Poorland d If 16 then it must be the case that which in turn requires that equals 2 3 4 1 1 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 6 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 Chapter 9 Economic Growth II Technology Empirics and Policy 77 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 7 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 YEL FKEL 1 u Chapter 9 Economic Growth II Technology Empirics and Policy 78 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 E 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 k 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 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 k sF k 1 u n g u k In the steady state k 0 so we can rewrite the equation above as sF k 1 u 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 Chapter 9 Economic Growth II Technology Empirics and Policy 79 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 8 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 learne