The Pearson Series in Economics

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2 The Pearson Series in Economics Abel/Bernanke/Croushore Macroeconomics Bade/Parkin Foundations of Economics Berck/Helfand The Economics of the Environment Bierman/Fernandez Game Theory with Economic Applications Blanchard Macroeconomics Blau/Ferber/Winkler The Economics of Women, Men and Work Boardman/Greenberg/Vining/ Weimer Cost-Benefit Analysis Boyer Principles of Transportation Economics Branson Macroeconomic Theory and Policy Brock/Adams The Structure of American Industry Bruce Public Finance and the American Economy Carlton/Perloff Modern Industrial Organization Case/Fair/Oster Principles of Economics Caves/Frankel/Jones World Trade and Payments: An Introduction Chapman Environmental Economics: Theory, Application, and Policy Cooter/Ulen Law & Economics Downs An Economic Theory of Democracy Ehrenberg/Smith Modern Labor Economics Ekelund/Ressler/Tollison Economics Farnham Economics for Managers Folland/Goodman/Stano The Economics of Health and Health Care Fort Sports Economics Froyen Macroeconomics Fusfeld The Age of the Economist Gerber International Economics Gordon Macroeconomics Greene Econometric Analysis Gregory Essentials of Economics Gregory/Stuart Russian and Soviet Economic Performance and Structure Hartwick/Olewiler The Economics of Natural Resource Use Heilbroner/Milberg The Making of the Economic Society Heyne/Boettke/Prychitko The Economic Way of Thinking Hoffman/Averett Women and the Economy: Family, Work, and Pay Holt Markets, Games and Strategic Behavior Hubbard/O Brien Economics Money, Banking, and the Financial System Hughes/Cain American Economic History Husted/Melvin International Economics Jehle/Reny Advanced Microeconomic Theory Johnson-Lans A Health Economics Primer Keat/Young Managerial Economics Klein Mathematical Methods for Economics Krugman/Obstfeld/Melitz International Economics: Theory & Policy Laidler The Demand for Money Leeds/von Allmen The Economics of Sports Leeds/von Allmen/Schiming Economics Lipsey/Ragan/Storer Economics Lynn Economic Development: Theory and Practice for a Divided World Miller Economics Today Understanding Modern Economics Miller/Benjamin The Economics of Macro Issues Miller/Benjamin/North The Economics of Public Issues Mills/Hamilton Urban Economics Mishkin The Economics of Money, Banking, and Financial Markets The Economics of Money, Banking, and Financial Markets, Business School Edition Macroeconomics: Policy and Practice Murray Econometrics: A Modern Introduction Nafziger The Economics of Developing Countries O Sullivan/Sheffrin/Perez Economics: Principles, Applications and Tools Parkin Economics Perloff Microeconomics Microeconomics: Theory and Applications with Calculus Perman/Common/ McGilvray/Ma Natural Resources and Environmental Economics Phelps Health Economics Pindyck/Rubinfeld Microeconomics Riddell/Shackelford/Stamos/ Schneider Economics: A Tool for Critically Understanding Society

3 Population Growth in the Solow Model CHAPTER 6 THE SOURCES OF GROWTH AND THE SOLOW MODEL Assume that low standards of living (i.e., low capital-labor ratios, in the Solow model's terms) are the reason that some countries exhibit high fertility rates. Comment on the effect of population control policies to reduce fertility rates. Explain why you think such policies will be effective or not. 7. One of the most well-known population control policies is the one-child policy implemented by China since the late 1970s. Productivity Growth in the Solow Model 9. Start by graphing the U.S. steady state capitallabor ratio and label it k 1900 (draw only the investment and the depreciation and capital dilution curves). a) On the same graph show the effects of the following: The massive immigration wave of the early 1900s Summing up the Solow Model 10. Start by drawing a given country s steady state, using only the investment and the depreciation and capital dilution curves. On the same graph do the following: a) Consider the effects of an immigration wave of individuals that exhibit both Comment on the side effects of such a policy. This policy has been dubbed a success, since fertility rates dropped by a considerable amount. Do you agree with this type of population control? 8. Based on the Solow model s conclusions about population growth, comment on the effects of immigration on a country s a) aggregate output level. b) capital-labor ratio. The increase in productivity derived from new technologies that occurred at the same time b) Find the new steady state capital-labor ratio considering that output per worker in the United States grew at a positive rate ss between 1900 and 1910 and label it. k 1910 higher saving and fertility rates than the current population. Draw the new curves. b) Identify the new steady state capital-labor ratio level. Is it necessarily higher or lower than the previous steady state? MYECONLAB CAN HELP YOU GET A BETTER GRADE. If your exam were tomorrow, would you be ready? For each chapter, MyEconLab Practice Test and Study Plans pinpoint which sections you have mastered and which ones you need to study. That way, you are more efficient with your study time, and you are better prepared for your exams. To see how it works, turn to page 17 and then go to

4 Chapter 6 Appendix A The Golden Rule Level of the Capital-Labor Ratio We have seen that a higher saving rate s leads to a higher level of output per worker. Does this conclusion mean that more saving is always better? Clearly not, since s of 100% would leave no income left over for buying food or shelter. To identify the saving rate that produces the highest level of economic well-being, we use the Solow model to compare steady states at different saving rates. Steady States at Different Capital-Labor Ratios Consumption per worker is a natural choice for maximizing the economic well-being of workers in an economy in the steady state. 1 After all, people care more about the amount of goods and services they consume than the amount of capital or output in the economy. Policy makers may try to influence the national saving rate to maximize consumption per worker in the steady state. But what saving rate achieves this? To answer this question, the policy maker needs to identify the steady-state level of the capital-labor ratio that maximizes consumption per worker, which is known as the Golden Rule capital-labor ratio. The name is a reference to the Bible, which states that you should do unto others, as you would have them do unto you. In economic terms, policy makers should weigh the well-being of all future generations equally. In this case, you would want all generations to have the same maximum level of consumption, which is achieved when the steady state capitallabor ratio leads to the highest level of consumption per worker. 1 There is clearly more to life than just consumption. People also care about the amount of leisure they have and we are ignoring such considerations here. 168

5 Golden Rule Capital-Labor Ratio CHAPTER 6 THE SOURCES OF GROWTH AND THE SOLOW MODEL 169 To determine the Golden rule capital-labor ratio, which we will denote by, we need to compare steady-state levels of consumption per worker, c, for different capital-labor ratios, k. We start by recognizing that since y = c + i, we can write consumption per worker in the steady state as follows: k G c = y - i (1) where the indicate values at the steady state. In other words, consumption per worker at the steady state is just income minus investment per worker at the steady state. The value of income per worker at the steady state is straightforward: it comes directly out of substituting k into the production function: y = Ak (2) At the steady state, the capital stock is not changing. As we saw in Equation 12 in the chapter, investment must be equal to depreciation plus capital dilution, i.e., i = (d + n)k (3) We plot the steady-state levels of y and i against the steady-state level of k in panel (a) of Figure 6A1.1. Substituting Equations 2 and 3 into Equation 1, c = Ak - (d + n)k (4) The difference between y and i, which equals the steady state level of consumption per worker, c, is shown by the colored area between the y and i curves in panel (a) of Figure 6A1.1, and is also shown in panel (b). Notice in panel (a) that c is at its maximum value when the slope of the production function curve, which is the marginal product of capital, is equal to the slope of the depreciation plus capital dilution line, which is d + n, the depreciation rate plus the rate of population growth. The Golden Rule level of the capital-ratio, k G, occurs when 2 MPK = d + n (5) Marginal Product of Capital = Depreciation Rate + Population growth rate 2 We can also derive this condition using calculus. From Equation 1, c = y i. To maximize c, differentiate this equation with respect to k and set it to zero. That is, dc >dk = dy >dk - (d + n) = 0. This equation thus produces the condition dy >dk = d + n. Since dy /dk is the marginal product of capital, this condition is the same as in the text. Note that we need to modify this condition when adding productivity growth to the Solow model.

6 170 PART THREE LONG-RUN ECONOMIC GROWTH FIGURE 6A1.1 The Golden Rule Level of the Capital-Labor Ratio In panel (a), steadystate consumption per worker, c (shown in panel (b)), is at its maximum value at k G when the slope of the production function curve, the marginal product of capital, equals the slope of the depreciation plus capital dilution line, d + n. Output per Worker, y Steady-state level of consumption per worker. slope = ( + n) (a) Solow Diagram ( + n)k Per Worker Production Function, y t = Ak t Golden Rule level. k G Capital-Labor Ratio, k (b) Steady-state Consumption Consumption per Worker, c Steady-state level of consumption per worker. k G Golden Rule level. Capital-Labor Ratio, k To see why the condition in Equation 5 makes sense, let s consider what happens when k 6 k G. In this case, the marginal product of capital is higher than the depreciation rate plus the population growth rate. Adding another unit of k adds more output, MPK, than the increase in investment, d + n, so that consumption, c, must rise. On the other hand, if k 7 k, MPK 6 d + n, adding another unit of k G increases output by

7 CHAPTER 6 THE SOURCES OF GROWTH AND THE SOLOW MODEL 171 MPK, which is less than the increase in investment, d + n, so that consumption, c, falls. Consumption per worker in the steady state, c, therefore reaches a maximum at k = k G. Implications of the Golden Rule Capital-Labor Ratio What saving rate would a policy maker choose if he or she wants the economy to get to the Golden Rule capital-labor ratio? Figure 6A1.2 provides the answer. The Golden Rule level of the saving rate, s G, is the one that causes the investment function, s G Ak t, to intersect with the depreciation plus capital dilution line, (d + n)k t, at the Golden Rule level of the capital-labor ratio, k G. If the saving rate were higher than s G, then the intersection of the investment curve and the depreciation line would be at a steady-state level of k that would be higher than k G, and so the steady-state level of consumption per worker would fall. Similarly, if the saving rate was lower than s G, then the intersection of the investment curve and the depreciation line would be at k 6 k G, and the steady-state level of consumption per worker would also fall. When the saving rate is at s G, consumption per worker is at its highest value. FIGURE 6A1.2 The Golden Rule Level of the Saving Rate The Golden Rule level of the saving rate, s G, leads to an investment function, s G Ak t, that intersects with the depreciation plus capital dilution line, (d + n)k t, at the Golden Rule level of the capitallabor ratio, k G. Investment and Depreciation plus Capital Dilution i = k G The Golden Rule level occurs where the investment curve intersects the depreciation line plus capital dilution. y t = Ak 1 ( + n)k t s G Ak t i G k G Capital-Labor Ratio, k t