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Chapter 4 The Theory of Economic Growth The Solow growth model allows us to analyze the determinants of the long-run trend value of the standard of living and its growth rate. When the economy is in its balanced-growth equilibrium, the growth rates of output per worker, the capital-labor ratio, and labor efficiency will all be the same. The balanced-growth equilibrium occurs when the capital-output ratio is constant. In balanced-growth equilibrium and assuming a Cobb-Douglas production function, the level of output per worker (Y/L) depends upon the saving rate (s), the rate of growth of the labor force (n), the rate of growth of labor efficiency (g), the rate of depreciation of capital (), the parameter of the production function (), and the current level of labor efficiency (E). In balanced-growth equilibrium, the growth rate of labor efficiency alone determines how fast output per worker grows. Learning Guide You may find this chapter challenging. There is only one model in this chapter the Solow growth model but it is abstract. Once you see the key point however, the Solow model becomes suddenly simple. Some professors believe the material covered in Chapter 4 is the most important material of the course. You will especially want to be able to answer the Section C and Section D questions in this chapter and in Chapter 5. You will need some math skills in this chapter. Math skills covered in Chapter B are indicated with Short on time? If you are short on time, you are in a bind. It is almost impossible to study this chapter quickly. At a minimum, learn the equations for really: do not try to save time on Chapter 4. K/Y and Y/L. But You must understand the Cobb-Douglas production function. The determinants of longrun economic growth and the determinants of balanced-growth equilibrium are very important. Distinguishing between events that change only the level of output per worker and those that also change the growth rate of output per worker is key. You need to acquire both a technical mastery of the topics and a conceptual one. A. BASIC DEFINITIONS Before you apply knowledge, you need a basic grasp of the fundamentals. In other words, there are some things you just have to know. Knowing the material in this section won't guarantee a good grade in the course, but not knowing it will guarantee a poor or failing grade. USE THE WORDS OR PHRASES FROM THE LIST BELOW TO COMPLETE THE SENTENCES. SOME ARE USED MORE THAN ONCE; SOME ARE NOT USED AT ALL. balanced-growth equilibrium capital stock capital-labor ratio capital-output ratio Cobb-Douglas Depreciation efficiency of labor growth rate investment Keynesian labor force output per worker saving rate Solow 1. ____________________ occurs when the various forces determining economic growth are balanced so that output per worker is increasing from period to period at the same rate as the capital-labor ratio. 2. The ____________________ production function is a particular algebraic form of the general abstract function . 3. The ____________________ has increased due to improvements in technology and in business organization. 4. The ____________________ production function states . 5. Total saving equals ____________________ spending. 6. ____________________ reduces the capital stock due to wear-and-tear and obsolescence. 7. Output per worker is a function of two inputs: the ____________________ and the ____________________. 8. The two major factors generating differences between economies' productive potential are the ____________________ and the ____________________. 9. When the economy is in ____________________, the growth rate of output per worker equals the growth rate of labor efficiency. 10. The ____________________ is the ratio of the sum of household saving, government saving, and foreign saving to total output. 11. The growth model is also called the ____________________ model, named after the economist who received the Nobel Prize for developing the model. 12. The best proxy we have for the material standard of living is ____________________. 13. The ____________________ is the total amount of business equipment, machinery and buildings available for producing goods and services. 14. When graphing the production function and the balanced-growth equilibrium line, the ____________________ goes on the horizontal axis and ____________________ goes on the vertical axis. 15. ____________________ refers to construction of buildings and purchases of business equipment and machinery. 16. The ____________________ is the average amount of physical capital available to workers for production. 17. The ____________________ is the share of real output (GDP) that is saved. 18. Real GDP per worker converges to its ____________________ path as the capital-output ratio converges to its equilibrium value. HINT: In Chapters 4 and 5 we are looking at the long-run economy. In the long run, the economy is always at full employment, always on the production possibilities frontier, always on the aggregate production function. Output and income (real GDP, Y) always equal their potential. CIRCLE THE CORRECT WORD OR PHRASE IN EACH OF THE FOLLOWING SENTENCES 19. In the very long run, a higher saving rate will / will not increase output per worker and will / will not permanently increase the growth rate of output per worker. 20. Increased obsolescence of capital will increase / decrease the depreciation rate. 21. The economy is / is not in balanced-growth equilibrium when %Δ(Y/L) = %Δ(K/L) = %(E). 22. The economy is / is not in balanced-growth equilibrium when K/Y is constant. 23. The balanced-growth equilibrium line is a straight line / nonlinear curve emanating from the origin. 24. Output per worker increases / decreases when capital per worker increases. 25. Output per worker increases / decreases when technology improves. 26. Output per worker increases / decreases when efficiency of labor increases. 27. The growth rate of the labor force is endogenous / exogenous to the growth model. 28. The growth rate of labor efficiency is endogenous / exogenous to the growth model. 29. The growth rate of output is endogenous / exogenous to the growth model. 30. The growth rate of the capital stock is endogenous / exogenous to the growth model. 31. An increase in government spending, all else constant, causes a(n) increase / decrease in government saving and a(n) increase / decrease in the saving rate. 32. An increase in consumption spending, all else constant, causes a(n) increase / decrease in household saving and a(n) increase / decrease in the saving rate. 33. An increase in imports, all else constant, causes a(n) increase / decrease in foreign saving and a(n) increase / decrease in the saving rate. _______________________________ SELECT THE ONE BEST ANSWER FOR EACH MULTIPLE-CHOICE QUESTION. 34. In balanced-growth equilibrium, A. the amount of capital per worker is constant. B. investment per worker is constant. C. the amount of output per worker is constant. D. the capital-to-output ratio is constant. 35. The aggregate production function tells us A. how a firm combines its inputs to produce its output. B. how the economy's total labor force, capital, and technology can be used to produce output. C. the balanced-growth equilibrium. D. the rate of diminishing returns. 36. To determine the balanced-growth equilibrium value of K/Y, the Solow growth model requires information about all of the following variables EXCEPT A. the rate of growth of the labor force. B. the size of the labor force. C. the saving rate. D. the rate of depreciation of capital. 37. Over the last 200 years, the U.S. standard of living has been A. smoothly increasing from year to year. B. increasing from decade to decade but not necessarily from year to year. C. constant. D. sometimes increasing and sometimes decreasing with no clear trend over time. TO THE CHALKBOARD Explaining Figure 4.8 Textbook Figure 4.8 shows how equilibrium output per worker (Y/L) and efficiency of labor (E) grow over time when the economy is in balanced-growth equilibrium. Remember that in that equilibrium, Y/L and K/L will grow at the same rate as labor efficiency: g. A constant rate of growth provides smooth growth but is not graphed as a straight line. A straight line would depict increases that were the same amount each period (such as, $5,000 per month) but would then be a declining rate of growth (percentage change) each period. In equilibrium, the rate of growth (percentage change) is constant from period to period, which means the amount of growth is increasing from period to period. The graphical result is above: a smooth curve whose slope continually increases. If we were to use a logarithmic scale to depict how equilibrium output per worker (Y/L) and efficiency of labor (E) grow over time when the economy is in balanced-growth equilibrium, then we would have a straight line graph. A constant percentage change over time produces a constant slope when log(Y/L) is shown. The graphical result is at the right: a curve with a constant slope. In balanced growth equilibrium, the slopes of the two curves will be equal. 38. The efficiency of labor can increase when A. workers acquire new and better skills. B. employers reorganize the work place to increase sales with fewer workers. C. scientific discoveries make machines more productive. D. all of the above. 39. Ultimately, the most important factor determining the growth of output per worker over time is the A. saving rate. B. level of output per worker. C. growth rate of the labor force. D. growth rate of labor efficiency. 40. A decrease in the growth rate of the labor force, all else constant, will permanently A. increase the level of Y/L and its growth rate. B. increase the level of Y/L but have no effect on its growth rate. C. change neither the level of Y/L nor its growth rate. D. decrease the level of Y/L but have no effect on its growth rate. _______________________________ TO THE CHALKBOARD The Key Equations There are a number of equations in this chapter. Here is a list of the key equations, with a brief description of each. Be sure you learn especially equations [4] and [5]. [1] [2] [3] [4] [5] The general production function. Output per worker depends upon the capitallabor ratio (also known as capital per worker) and the efficiency of labor. The Cobb-Douglas production function, which specifies the functional form of the relationship between output per worker, capital per worker, and labor efficiency. How capital stock changes from one period to the next. Capital stock at the beginning of period t + 1 equals capital stock at the beginning of the previous period plus the saving rate (s) times last period's output (Yt) minus the depreciation rate () times capital stock at the beginning of period t. The balanced-growth equilibrium value of the capital-output ratio equals the saving rate (s) divided by the sum of the labor force growth rate (n), the labor efficiency growth rate (g), and the depreciation rate (). The balanced-growth value of output per worker. When the economy is in equilibrium, output per worker is a constant proportion of labor efficiency E: Y/L equals the saving rate (s) divided by the sum of the labor force growth rate (n), the labor efficiency growth rate (g), and the depreciation rate (), all raised to the power ( divided by one minus ), and then multiplied by the value of labor efficiency, E. B. MANUPULATION OF CONCEPTS AND MODELS Most instructors expect you to be able to do basic manipulation of the concepts. Being able to do so often means you can earn a C in a course. But if you want a better grade, you'll need to be able to complete this section easily and move on to sections C and D. NOTE: The distinguishing feature of a Cobb-Douglas production function is that the exponents sum to one (1). For instance, Y = AKβL(1-β) is also Cobb-Douglas because + β (1 - β) = 1. The Cobb-Douglas production function is convenient to use because it has some very nice mathematical properties. A Cobb-Douglas production function exhibits constant returns to scale: if you double all of the inputs, output will also double. Mathematically, the function exhibits constant returns to scale because the exponents sum to 1. 1. The Cobb-Douglas production function is . A. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production function when = 0.2. (You might find it easier to do the graph using a spreadsheet package such as Quattro Pro or Excel, or using a graphing calculator.) B. Suppose labor efficiency, E, is 10,000. Graph the Cobb-Douglas production function when = 0.8. C. When does a change in the capital to labor ratio generate the larger change in output per worker, when = 0.2 or when = 0.8? Explain your answer, using the economic concept of diminishing returns. Many students are tempted to ask, "Is the Cobb-Douglas production function realistic?" That's the wrong question. The right question is, "Is the CobbDouglas production function a good enough approximation of reality to allow us to do reasonable analysis and come up with useful conclusions?" Yes. 2. A. Suppose E = 10,000 and = 0.4. Assume production can be described by the Cobb-Douglas production function. Compute the value of output per worker at each level of capital per worker shown at the right. B. When K/L doubles, does Y/L double? Why or why not? In Questions 3, 4, and 5, you will work with the definitions of labor force, labor efficiency, and capital stock, and see how their values change over time. 3. Compute the average value of n, the growth rate of the labor force, by decade, 19502000. Consult the Economic Report of the President to locate labor force data. Labor Force (Thousands age 16 & over) 1950 1960 1970 1980 1990 Average Annual Growth Rate 2000 The "efficiency of labor" is sometimes thought of as simply "technology" but includes business organization too. In general, E is telling us how much output each worker is able to produce with capital. If someone invents a faster or more reliable computer (a piece of capital), a worker will probably be able to produce more output with the same number of computers. Is the worker really more "efficient" in the usual sense of the word? Does he spend less time on the phone? Does she organize her work space so it is more productive? No. The "efficiency of labor" rises in this case not because of any "efficient" action by the worker, but because a technological advance that affected capital increased the worker's productivity. 4. Suppose that the labor force is initially 50,000,000 and labor efficiency is initially 10,000. Suppose g = 0.025 and n = 0.03. Complete the following table. Period 1 2 3 4 5 Et Lt (in thousands) 10,000 5. Use the relationship describing changes in the capital stock, Kt+1 = Kt + sYt Kt, to answer the following questions. A. Suppose Kt = 2,500, δ = 4 percent, the saving rate is 15 percent, and Yt = 1,000. What is the value of the capital stock at the end of year t + 1? B. Suppose Kt+1 = 8,000, Kt = 7,000, s = 10 percent, and δ = 0.03. What was the value of output in year t? C. Suppose Yt = 300, s = 0.20, Kt+1= 1,000, and the depreciation rate is 4 percent. What was the value of Kt? 6. Suppose s = 16% n = 2% g = 2% δ=4 A. What is the equilibrium level of the capital-to-output ratio, K/Y? B. For each of the following levels of K/L, calculate the balanced-growth equilibrium value of output per worker: K/L 1,000 2,000 3,000 4,000 equilibrium Y/L C. At the right, graph these four equilibrium combinations of K/L and Y/L. Connect the four points with a straight line. D. What is the intercept of the line you drew? What is its slope? TO THE CHALKBOARD Finding Equilibrium K/L and Y/L: The Graphical Approach Figures 4.5 and 4.9 show how to find the longrun equilibrium level of output per worker. Let's review. The production function is a behavioral relationship. We use the Cobb-Douglas production function: This function can be presented in a two-dimensional graph with , the dependent variable, on the vertical axis and an independent variable, on the horizontal axis. In general, the graph will look as shown at the right. The equilibrium condition be transformed to can , or. This condition can also be presented in a two-dimensional graph, again with on the vertical axis and on the horizontal axis. In general, the graph will look as shown at the right: a straight line with an intercept of 0 and a slope of . Combining these two curves onto one graph gives us the graph at the right. The point where the two curves cross is the one combination of output per worker and capital per worker that satisfies both the behavioral relationship and the equilibrium condition. It is the only combination of and that is both on the production function where the economy always is in the long run and on the balanced-growth equilibrium line. 7. Suppose s = 10% n = 2% A. What is the equation for the production function? Graph the production function at the right. B. What is the equilibrium capital-output ratio? At the right, graph the balanced growth equilibrium line, Y/K. C. From your graph, what is the balancedgrowth equilibrium level of output per worker? g = 1% δ=3 α=.5 E=500 What if the economy is not at balanced-growth equilibrium? Suppose the economy is initially at a combination of K/L and Y/L that is on the production function where the economy always is in the long run but above the balancedgrowth equilibrium line. Then the economy is producing more output than is needed for balanced growth, so there is more saving and thus more investment and thus faster growth of the capital stock than we would have at the balanced growth equilibrium. And when the capital stock grows, output per worker increases. So if the economy is initially at a point above the balanced-growth equilibrium line, K/L and Y/L increase until the economy converges to balanced-growth equilibrium. 8. Suppose s = 24% n = 2% g = 0% δ= 4% A. At right, graph the balanced-growth equilibrium line, Y/K. B. Suppose the capital-labor ratio is 2,000. What is the approximate value of output per worker? Label that point A. C. Answer this question simply by looking at your graph: Is point A a balancedgrowth equilibrium combination of K/L and Y/L? How do you know? D. When the capital-labor ratio is 2,000, what is the balanced-growth equilibrium level of output per worker? Is the actual level of output per worker greater than or less than the balanced-growth equilibrium level? E. From one period to the next, the level of output per worker determines the amount of saving per worker which determines investment per worker which determines the next periods capital-labor ratio. In the next period, will K/L still equal 2,000? Why? 9. Suppose s = 24% n = 2% A. At right, graph the balanced-growth equilibrium line, Y/K. B. Suppose the capital-labor ratio is 8,000. What is the approximate value of output per worker? Label that point B. g = 0% δ= 4% C. Answer this question simply by looking at your graph: Is point B a balanced-growth equilibrium combination of K/L and Y/L? How do you know? D. When the capital-labor ratio is 8,000, what is the balanced-growth equilibrium level of output per worker? Is the actual level of output per worker greater than or less than the balanced-growth equilibrium level? E. In the next period, will K/L still equal 8,000? Why? NOTE: In Questions 10, 11, and 12, you will see how capital, labor, and output change over time as the economy moves toward a balanced-growth equilibrium. In Question 10, we begin with the capital-output ratio below its equilibrium level. In Question 11, we begin with the capital-output ratio above its equilibrium level. In Question 12, we begin with the capital-output ratio at its equilibrium level. 10. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 15, and the production function is Cobb-Douglas, with = 2/3. Complete the following table. Use a separate sheet of paper for your calculations. Period 1 2 3 4 K 75,000 L 15 E 100 Y/L Y K/Y Is the economy at its balanced-growth equilibrium? Why or why not? Answer the question without computing the equilibrium value of the capital-output ratio. 11. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 5, and the production function is Cobb-Douglas, with = 2/3. Complete the following table. Use a separate sheet of paper for your calculations. Period 1 2 3 4 K 75,000 L 5 E 100 Y/L Y K/Y Is the economy at its balanced-growth equilibrium? Why or why not? Answer the question without computing the equilibrium value of the capital-output ratio. 12. Suppose n = 0.015, g = 0.008, = 0.03, s = 0.25, E1 = 100, K1 = 75,000, L1 = 7.14, and the production function is Cobb-Douglas, with = 2/3. Complete the following table. Use a separate sheet of paper for your calculations. Period 1 2 3 4 K 75,000 L 7.4 E 100 Y/L Y K/Y Is the economy at its balanced-growth equilibrium? Why or why not? Answer the question without computing the equilibrium value of the capital-output ratio. NOTE: So have you seen the powerful simplicity of the model yet? The standard of living output per worker depends upon a bunch of things: the saving rate, the rate at which the labor force is growing, the depreciation rate, the value of labor efficiency. But one thing and only one thing determines whether the standard of living increases from generation to generation: the rate of growth of labor efficiency. If efficiency isn't growing if the workforce isn't acquiring new and better skills, if kids aren't learning in school and becoming smarter than their parents, if businesses aren't finding ways to reorganize to improve worker efficiency, if scientists aren't making new discoveries then there will be no improvement in the standard of living over time. TO THE CHALKBOARD: Deriving Equilibrium Y/L=(s/(n + g + δ))(α/α-1)-Et K/Y is the capital-output ratio. Given values for s, n, g, and , its balanced-growth equilibrium value is a number. Y/L is output per worker. Given values for s, n, g, δ, and α, the balanced-growth equilibrium value of Y/L is a constant proportion of labor efficiency, E. Given a value for E as well, Y/L is a number. To derive the expression for Y/L requires that we remember two things: [1] manipulate K/Y until you have an expression with K/L in it and [2] use the Cobb-Douglas production function. Now we will substitute this last expression into the production function. 13. This question asks you to work with the equations for the balanced-growth equilibrium values of the capital-output ratio and output per worker. Use the formula for the balanced-growth equilibrium value of the capital-output ratio, . A. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent annually, labor efficiency is increasing by 2 percent each year, and capital depreciates 3.5 percent annually. What is the balanced-growth value of the capital-output ratio? Suppose = 0.6 and E = 100. What is the equilibrium value of output per worker? B. Suppose the saving rate is 25 percent, the labor force is increasing 3 percent annually, labor efficiency is increasing by 2 percent each year, and capital depreciates 3.5 percent annually. (That is, use the values from part A, but change the saving rate to 25 percent.) Now what is the balanced-growth value of the capital-output ratio? Suppose α= 0.6 and E = 100. What is the equilibrium value of output per worker? C. Suppose the saving rate is 20 percent, the labor force is increasing 4 percent annually, labor efficiency is increasing by 2 percent each year, and capital depreciates 3.5 percent annually. (That is, use the values from part A, but change the growth rate of the labor force to 4 percent.) Now what is the balanced-growth value of the capital-output ratio? Suppose α = 0.6 and E = 100. What is the equilibrium value of output per worker? D. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent annually, labor efficiency is increasing by 2.5 percent each year, and capital depreciates 3.5 percent annually. (That is, use the values from part A, but change the growth rate of labor efficiency to 2.5 percent.) Now what is the balancedgrowth value of the capital-output ratio? Suppose α = 0.6 and E = 100. What is the equilibrium value of output per worker? E. Suppose the saving rate is 20 percent, the labor force is increasing 3 percent annually, labor efficiency is increasing by 2 percent each year, and capital depreciates 5.5 percent annually. (That is, use the values from part A, but change the depreciation rate to 5.5 percent.) Now what is the balanced-growth value of the capital-output ratio? Suppose α = 0.6 and E = 100. What is the equilibrium value of output per worker? NOTE: Why does investment spending not income change when saving changes? We are looking at the economy in the long run when output is always equal to its potential; the economy is always on the production function. So a change in spending and saving by one sector of the economy say households doesn't . . . indeed can't . . . change total output. The economy remains on the production function. For reasons we'll explore in Chapter 7, a change in spending and saving by households, government agencies, or the rest of the world causes a change in investment spending by businesses. 14. Suppose C = $8,000 billion I = $1,000 billion G = $1,000 billion T = $1,000 billion GX = $2,000 billion IM = $2,000 billion Y = $10,000 billion A. What is the annual value of household saving SH? Of government saving SG? Of foreign saving SF? Of total saving? What is the saving rate s? B. Suppose consumption spending increases to $8,500 billion per year. What is the new annual value of household saving SH? Of total saving? What is the saving rate s? What is the new annual value of investment spending? 15. Suppose that each year C = $7,000 billion I = $1,500 billion G = $2,000 billion T = $1,500 billion GX = $3,000 billion IM = $3,500 billion Y = $10,000 billion A. What is the annual value of household saving SH? Of government saving SG? Of foreign saving SF? Of total saving? What is the saving rate s? b. Suppose government spending increases to $2,500 billion per year and taxes are cut to $1,000 billion per year. What is the new annual value of household saving SH? Of government saving SG? Of total saving? What is the saving rate s? What is the new annual value of investment spending? 16. Suppose s = 15% n = 2% g = 0% δ== 4% α== 2/3 E = 500 A. What is the equation for the production function? Graph the production function at the right. B. What is the equilibrium capital-output ratio? At the right, graph the balanced growth equilibrium line, Y/K. C. What is the balanced-growth equilibrium level of output per worker? Use the equation Confirm your answer by finding the balanced-growth equilibrium point on your graph. D. Suppose the saving rate increases to 19.5%. What is the new equilibrium capitaloutput ratio? Graph the new balanced-growth equilibrium line above. E. Answer this question simply by looking at your graph: Is the balanced-growth equilibrium combination of K/L and Y/L that you found in part C still the equilibrium? How do you know? F. What is the new balanced-growth equilibrium level of output per worker? What is the new equilibrium capital-labor ratio? Explain how the economy moves from the initial balanced-growth equilibrium level to this new level of output per worker. G. Once the economy has moved to the new balanced-growth equilibrium level of output per worker, will there be any further changes in K/L and Y/L? Why? 17. This question asks you to compute the change in the equilibrium level of output per worker as a result of changes in the parameters. Use the equations and A. Suppose the saving rate is 20 percent, the depreciation rate is 2.7 percent, the labor force is growing by 1.9 percent annually, and labor efficiency is growing by 2 percent a year. Suppose the production function is Cobb-Douglas with α= 0.7. Suppose E currently equals 4,000. What is the equilibrium value of output per worker? B. Suppose the saving rate is 21 percent, and the remainder of the parameters are the same as in part A. What is the new equilibrium value of output per worker? What is the percentage change in the equilibrium value of output per worker when the saving rate increases by one percentage point? C. Suppose the depreciation rate is 3.7 percent, and the remainder of the parameters are the same as in part A. What is the new equilibrium value of output per worker? What is the percentage change in the equilibrium value of output per worker when the depreciation rate increases by one percentage point? D. Suppose the labor force is growing by only 0.9 percent annually and the remainder of the parameters are the same as in part A. What is the new equilibrium value of output per worker? What is the percentage change in the equilibrium value of output per worker when the labor force growth rate decreases by one percentage point? TO THE CHALKBOARD: When Labor Efficiency Grows How do the graphs look when labor efficiency is increasing at a constant rate g over time? The balanced-growth equilibrium line is unaffected. The capital-output ratio in equilibrium equals s / (n + g + δ). Increases in E over time simply mean that g is positive, not that g is increasing. (Review Section B, Question 4.) But the production function is affected. When E increases, the production function shifts up. (Review Chapter 3, Section B, Question 9E.) So when E is increasing, K/L and Y/L increase along the economy's balanced-growth equilibrium line as shown here. 18. Suppose s = 15% n = 2% g = 5% δ=3% α=2/3 A. What is the balanced-growth equilibrium value of K/Y? B. When E = 10,000, what is the balanced-growth equilibrium value of Y/L? C. Fill in the table at right, showing how changes in E (remember: g = 5%) lead to changes in the balanced-growth equilibrium value of Y/L. D. Between periods 1 and 2, what is the growth rate of Y/L? Of E? C. APPLYING OF CONCEPTS AND MODELS Now we're getting to the good stuff. Being able to apply a specific concept or model to a real world situation -- where you are told which model to apply but you have to figure out how to apply it -- is often what you need to earn a B in a course. This is where macroeconomics starts to become interesting and the world starts to make more sense. 1. The personal saving rate in the United States averaged 8.3 percent in the 1960s but only 5.9 percent in the 1990s. According to the Solow growth model, what is the longrun effect on output per worker? On the long-run growth rate of output per worker? 2. Total saving includes saving by government agencies. When the government runs a budget surplus when government revenues exceed government outlays government is increasing the nation's total saving. When the government runs a budget deficit, government is decreasing the nation's total saving. During the Clinton Administration (1993 - 2000), the federal government budget changed from a deficit of $290 billion (about 4.7 percent of GDP) to a surplus of $230 billion (about 2.4 percent of GDP). If the surplus had not been subsequently eliminated, what would have been the long-run effect on standards of living? On the long-run growth rate of output per worker? 3. During the George W. Bush Administration (Bush 43, the younger George Bush, 2001 - 2008), the federal government's budget balance shifted from a surplus of $230 billion (about 2.4 percent of GDP) to a deficit of over $400 billion (about 3.5 percent of GDP). If there is no reversal of these fiscal policies, what is the long-run effect on the standard of living of the movement from budget surplus to deficit? On the longrun growth rate of output per worker? 4. If engineers were to develop a way to make machines last much longer, that would dramatically lower depreciation rates. According to the Solow growth model, what is the long-run effect on output per worker? On the long-run growth rate of output per worker? 5. Competition has led scores of U.S. corporations to reorganize, cutting costs without cutting output. What is the effect on labor efficiency? On output per worker? On the long-run growth rate of labor efficiency? On the long-run growth rate of output per worker? 6. Politicians concerned over the burgeoning trade deficit the excess of imports into the United States over exports from the United States have called upon Americans to stop buying foreign goods and "Buy American." If the people do as the politicians urge and permanently reduce imports, what is the long-run effect on foreign saving? On investment? On output per worker? On the long-run growth rate of output per worker? D. EXPLAINING THE REAL WORLD Most instructors are delighted when you are able to figure out which concept or model to apply to a real world situation. Being able to do so means you thoroughly understand the material and is often what you need to do to earn an A in a course. This is where you experience the power of macroeconomic theory. 1. What is the long-run effect of the increase in women's labor force participation on output per worker? On output per capita? On the long-run growth rate of output per 2. The United Nations sponsors several programs aimed at helping women in poor nations gain reproductive control. If fertility is lowered, what will be the long-run effect on standards of living in those nations? On the long-run growth rate of the standard of living? worker? 3. In the mid-1800s, steam power replaced water power in New England manufacturing. What should have been the long-run impact of this development on standards of living? On the long-run growth rate of the standard of living? Why? 4. In the late 1990s, everyone worried that the "Y2K Bug" would lead to collapse of computer-driven systems on January 1, 2000. As a result, most businesses replaced their computer equipment. What effect would the computer purchases have had on standards of living in the long run? On the long-run growth rate of the standard of living? 5. The government is choosing between spending $100 billion on funding scientific research and lowering personal taxes by $100 billion. Which action will increase longrun standards of living? Why? 6. A $1.3 trillion tax cut was approved by Congress in 2001. Consider three possible uses of the tax cut: [1] the recipients spend all of the tax cut, purchasing consumer goods and services; [2] the recipients save the entire tax cut, placing the funds into various financial assets; [3] the recipients save half of the tax cut and use the other half to pay off their credit card debt and other loans. In each case, what is the long-run effect of the tax cut? Are there long-run economic benefits of the tax cut? 7. Someone says to you, "There is no good reason to come up with policies that would raise the economy's saving rate. The growth rate of output per worker just winds up where it was initially." The statement is both right and wrong. Identify one thing about the statement that is right and explain why it is right. Identify one thing about the statement that is wrong and explain why it is wrong. E. POSSIBILITIES TO PONDER The more you learn, the more you realize you have more to learn. These questions go beyond the material in the text. They are the sort of questions that distinguish A+ or A work from A- work. Some of them may even serve as decent starting points for junior or senior year research papers. 1. If America and India share knowledge regarding production methods, technology, and organization of the workplace, will Indian standards of living equal American standards of living in the long run? 2. In the 1940s, many economic historians thought that a country had to have a railroad sector to experience economic growth. After all, the United States economy had boomed at the same time that the railroad was being constructed throughout and across America, replacing dirt roads and waterways. According to the Solow model, what aspects of railroad development might have been key to increasing economic growth? Would a railroad be necessary today for an industrializing nation to experience economic growth? 3. If you are advising the government on how it spends money and you want to increase long-run standards of living, what advice do you give regarding which projects the government should fund? What advice do you give regarding tax cuts? 4. Should a government have "increase the rate of economic growth" as one of its policy goals? Should this be a government's only policy goal? 5. The AIDS epidemic is killing thousands of people each day in sub-Saharan Africa. (http://www.avert.org/subaadults.htm) If the standard by which government decides whether to address a problem is "Does this government action contribute to economic growth?," will the government establish programs to end or at least reduce the number of deaths by AIDS? SOLITIONS SOLUTIONS SOLUTIONS A. Basic Definitions * indicates there are notes below related to this question. SOLUTIONS 1. Balanced-growth equilibrium 2. Cobb-Douglas 3. efficiency of labor 4. Cobb-Douglas 5. investment 6. Depreciation 7. Capital-labor ratio; efficiency of labor 8. Capital-output ratio; efficiency of labor 9. Balanced-growth equilibrium 10. Saving rate 11. Solow 12. Output per worker 13. Capital stock 14. Capital-labor ratio; output per worker 15. Investment* 16. Capital-labor ratio 17. Saving rate 18. Balanced-growth equilibrium *15. Remember: In economics, investment is not purchasing stocks and bonds. Investment always refers to purchases by business that add to the capital stock. 19. Will; will not 24. Increases 29. Endogenous 20. Increase 25. Increases* 30. Endogenous* 21. Is 26. Increases 31. Decrease; decrease 22. Is* 27. Exogenous* 32. Decrease; decrease 23. Straight line* 28. Exogenous 33. Increase; increase *22. The two definitions of balanced-growth equilibrium in questions 21 and 22 are equivalent. *23. The production function is the nonlinear curve in the graph that helps you determine the equilibrium values of K/L and Y/L. *25. Technology is synonymous with labor efficiency in the Solow model as developed in Textbook Chapter 4. *27. The growth rate of the labor force, n, is not determined by any of the factors that are part of the Solow growth model. Therefore, n is exogenous to the model. *30. The growth rate of the capital stock depends upon two exogenous factors s and and upon one endogenous factor output. Therefore the growth rate of K is itself endogenous. 34. D. The definition of "balanced-growth equilibrium" is that this is the point where the capital-output ratio is constant, which is equivalent to stating that capital per worker and output per worker are growing at the same rate. 35. B. The aggregate production function is usually expressed as . It is a statement about production of total output for the entire economy, not about production within one firm. 36. B. The balanced-growth equilibrium value is . We need information about the saving rate (s), the labor force growth rate (n), the growth rate of labor efficiency (g), and the depreciation rate () only. 37. B. Check out Textbook Figure 4.1. Real GDP per capita our proxy for the standard of living has increased a great deal since 1800; the value for 2000 is much higher than the value for 1800. But real GDP per capita does not increase smoothly. The Great Depression of the 1930s jumps out from that figure. But even ignoring the Great Depression, you can see that there are years when real GDP per capita falls. In Chapter 4, we are not concerned with the year-to-year movements in the standard of living, but in the determinants of that long-run decades-long trend. 38. D. The efficiency of labor refers only to how much output per worker is produced with a given amount of capital per worker. "Efficiency" can be thought of in its usual sense, as an action that someone takes to accomplish a goal with fewer resources. But in the context of the production function, anything that increases output given capital is said to increase "efficiency" of labor. So improving worker skills increases "efficiency of labor", as do reorganizing business to improve worker efficiency and technological improvement in physical capital. 39. D. When the economy is in equilibrium, it is the growth rate of labor efficiency and only the growth rate of labor efficiency that determines the growth rate of output per worker. 40. B. A decrease in the growth rate of the labor force will raise the level of output per worker (Y/L). While the economy is transitioning to its new balanced-growth path, the growth rate of Y/L will be temporarily higher than the growth rate of labor efficiency. ("Temporarily" may take many years this is a long-run analysis.) But the decrease in the growth rate of the labor force will have no permanent effect on the growth rate of Y/L. Only the growth rate of labor efficiency determines the permanent equilibrium growth rate of output per worker. B. Manipulation of Concepts and Models A. Y/L = (K/L)0.2E0.8 appears as shown at the right. Notice that at low levels of K/L, the returns to additional capital per worker are quite large; the curve is nearly vertical between K/L = 0 and K/L = 1000. But as capital per worker rises, the returns become smaller and smaller. B. Y/L = (K/L)0.8E0.2 appears as shown at the right. Notice that the returns to additional capital per worker do not change much as capital per worker changes. That is, the slope of the production function is nearly constant. C. At very low levels of K/L, a change in the capital to labor ratio generates a larger change in output per worker when α= 0.2; but at moderate and high levels of K/L, a change in the capital to labor ratio generates a larger change in output per worker when α = 0.8. Regardless of the value of α, increases in the amount of capital per worker generate additional output per worker (returns are positive), but the amount of additional output per worker gets smaller with each increase in the amount of capital per worker (returns are diminishing as K/L rises). When α = 0.2, diminishing returns to inputs set in quickly. When α = 0.8, diminishing returns are slow to appear. So over most of the range of K/L, increases in capital per worker generates a larger change in output per worker when α = 0.8. 2. A. Y/L = (5,000)0.4(10,000)0.6 = 7,578.58 Y/L = (10,000)0.4(10,000) 0.6 =10,000.00 Y/L = (20,000) 0.4(10,000) 0.6 = 13,195.08 B. When K/L doubles, Y/L does not double. With the Cobb-Douglas production function, additions to capital per worker do increase output per worker, but the size of increases in output per worker is not a constant proportion of additions to capital per worker. When K/L doubles, Y/L increases by 32 percent. If both E and K/L doubled, then Y/L would double. 3. In the Economic Report of the President 2005, the data for Labor Force are found in Table B35. The average annual growth rate is found by taking the 10th root of the ratio of, for instance, the 1960 to 1950 value, and subtracting 1 from that value. That is, (69,628/62,208)0.1 - 1 = 0.0113 = 1.13%. 1950 1960 1970 Labor Force Thousands age 16 & over 62,208 69,628 82,771 Average Annual Growth Rate 1.13% 1.74% 106,940 125,840 142,583 1980 1990 2000 2.60% 1.64% 1.26% 4. Period 1 2 3 4 5 Lt (in thousands) 50,000.00 51,500.00 53,045.00 54,636.35 56,275.44 Et 10,000.00 10,000.00 10,506.25 10,768.91 11,038.13 5. A. Kt+1 = 2,550 Kt+1 = Kt + sYt- δKt K t+1 = 2,500 + 0.15(1,000) 0.04(2,500) K t+1 = 2,500 + 150 100 = 2,550 B. Yt = 12,100 K t+1 = Kt + sYt -δKt =Yt = 12,100 C. Kt = 979.17 K t+1 = Kt + sYt –δKt K t+1 - sYt = Kt – δKt K t+1 - sYt = (1-δ)Kt = Kt = Kt = 979.17 6. A. B. In equilibrium, K/Y = 2. So multiplying both sides by Y/2 and then dividing by L we see that in equilibrium, C. The graph is at the right. D. The intercept is 0. The slope is the rise Δ(Y/L) over run Δ(K/L) which is , equal to 1 over the balanced-growth equilibrium capitaloutput ratio. This is always true. The slope of the line of equilibrium combinations of K/L and Y/L will always equal 1 over the balanced-growth equilibrium capital-output ratio. 7. A. The production function is Y/L = (K/L)0.5(900) 0.5= 30(K/L) 0.5. The graph is at the right. The balancedgrowth equilibrium line has a slope of 1/1.67 = 0.6. B. K/Y = s / (n + g + δ) = 0.10 / (0.02 + 0.01 + 0.03) = 0.10 / 0.06 = 1.67. The graph is at the right. C. In balanced-growth equilibrium, Y/L = 1,500. 8. A. In balanced growth equilibrium, K/Y = 4. Because we graph K/L on the horizontal (not vertical) axis and Y/L on the vertical (not horizontal) axis, the balanced-growth equilibrium line is a straight line from the origin with slope equal to 1 over the balanced-growth equilibrium capital-output ratio. So the balanced-growth equilibrium line has a slope of 1/4. As seen at the right, the balanced-growth equilibrium line begins at (0,0) and includes the point (4000, 1000). B. We read the value of output per worker off of the production function. When K/L is 2,000, Y/L is about 900. C. Point A is not a balanced-growth equilibrium combination of K/L and Y/L. We know this because point A is not on the balanced-growth equilibrium line. D. When K/L is 2,000, the balanced-growth equilibrium level of output per worker is 500 = 2000/4. So at point A the actual level of output per worker is greater than the balanced-growth equilibrium level of output per worker. E. Were Y/L equal to its balanced-growth equilibrium level of 500, the capital-labor ratio would not change from period to period. [Important: A constant value of K/L occurs in equilibrium in this question because we have assumed that labor efficiency is constant (g = 0).] Because Y/L is greater than the balanced-growth equilibrium level (900 > 500), then there will be more saving and thus more investment than is needed to maintain K/L at its existing level of 2000. Additional investment will add to the capital stock. So K/L will increase from one period to the next. 9. A. These values are the same as in Question 8. In balanced growth equilibrium, K/Y = 4. So the balanced-growth equilibrium line has a slope of 1/4. B. We read the value of output per worker off of the production function. When K/L is 8,000, Y/L is about 1,500. C. Point B is not a balanced-growth equilibrium combination of K/L and Y/L. We know this because point B is not on the balancedgrowth equilibrium line. D. When K/L is 8,000, the balanced-growth equilibrium level of output per worker is 2,000. So at point B the actual level of output per worker is less than the balancedgrowth equilibrium level of output per worker (1500 < 2000). E. Were Y/L equal to its balanced-growth equilibrium level of 2,000, the capital-labor ratio would not change from period to period. [Important: A constant value of K/L occurs in equilibrium in this question because we have assumed that labor efficiency is constant (g = 0).] Because Y/L is less than the balanced-growth equilibrium level (1,500 < 2,000), then there will be less saving and thus less investment than is needed to maintain K/L at its existing level of 2000. A shortage of investment will reduce the capital stock. So K/L will decrease from one period to the next. 10. Period 1 2 3 4 K 75,000.00 77,839.52 80,761.45 83,767.66 L 15.00 15.22 15.45 15.69 E 100.00 100.80 101.61 102.42 Y/L 1,357.21 1,381.18 1,405.27 1,429.50 Y 20,358.13 21,028.39 21,716.22 22,422.01 Sample calculations are for period 2. K2 = K1 + sY1 - δK1 = 75,000 + 0.25(20,358.13) - 0.03(75,000) = 77,839.52 L2 = L1(1 + n) = 15.00(1 + 0.015) = 15.22 E2 = E1 (1 + g) = 100.00(1 + 0.008) = 100.8 = = 1,381.18 K/Y 3.68 3.70 3.72 3.74 Y2 = (Y2 / L2) L2 = 1,381.18(15.22) = 21,028.39 (K/Y) 2 = K2 / Y2 = 77,839.53 / 21,028.39 = 3.70 The economy is not at its balanced-growth equilibrium; K/Y is increasing. If the economy was at its balanced growth equilibrium, K/Y would be constant. Because K/Y is increasing, K/L and Y/L must be increasing, so we are initially at a level of output per worker that is below equilibrium. 12. Period 1 2 3 4 K 75,000.00 76,723.87 78,487.54 80,291.93 L 7.14 7.25 7.36 7.47 E 100.00 100.80 101.61 102.42 Y/L 2,226.26 2,243.87 2,261.63 2,279.52 Y 15,895.48 16,261.55 16,636.08 17,019.26 K/Y 4.72 4.72 4.72 4.72 Sample calculations are for period 2. K2 = K1 + sY1 - δK1 = 75,000 + 0.25(15,895.48) - 0.03(75,000) = 76,723.87 L2 = L1 (1 + n) = 7.14(1 + 0.015) = 7.25 E2 = E1 (1 + g) = 100.00(1 + 0.008) = 100.8 = = 2,243.87 Y2 = (Y2 / L2) L2 = 2,243.87(7.25) = 16,261.55 (K/Y) 2 = K2 / Y2 = 76,723.87 / 16,261.55 = 4.72 The economy is at its balanced-growth equilibrium; K/Y is constant. Notice that at the balanced-growth equilibrium, capital, labor, labor efficiency, the capital per worker ratio, and the output per worker ratio are all increasing. Further calculations would show you that output per worker and capital per worker are both increasing at a constant rate of 0.8 percent per period. 13. A. K/Y = 0.20 / (0.03 + 0.02 + 0.035) = 2.35 Y/L = (2.35)(0.6/0.4)(100) = 360.92 B. K/Y = 2.94 and Y/L = 504.41 C. K/Y = 2.11 and Y/L = 305.46 D. K/Y = 2.22 and Y/L = 331.27 E. K/Y = 1.90 and Y/L = 262.88 14. A. SH = Y - T - C = 10,000 - 1,000 - 8,000 = $1,000 billion per year SG = T - G = 1,000 - 1,000 = $0 billion per year SF = IM - GX = 2,000 - 2,000 = $0 billion per year Total saving = SH + SG + SF = 1,000 + 0 + 0 = $1,000 billion per year s = Total saving / Y = 1,000 / 10,000 = 0.10 = 10% B. SH = Y - T - C = 10,000 - 1,000 - 8,500 = $500 billion per year Total saving = SH + SG + SF = 500 + 0 + 0 = $500 billion per year s = Total saving / Y = 500 / 10,000 = 0.05 = 5% I = sY = Total saving = $500 billion per year. Investment decreases when household saving falls. 15. A. SH = Y - T - C = 10,000 - 1,500 - 7,000 = $1,500 billion per year SG = T - G = 1,500 - 2,000 = $500 billion per year SF = IM - GX = 3,500 - 3,000 = $500 billion per year Total saving = SH + SG + SF = 1,500 + (500) + 500 = $1,500 billion per year s = Total saving / Y = 1,500 / 10,000 = 0.15 = 15% B. SH = Y - T - C = 10,000 - 1,000 - 7,000 = $2,000 billion per year SG = T- G = 1,000 2,500 = $1,500 billion per year Total saving = SH + SG + SF = 2,000 + (1,500) + 500 = $1,000 billion per year s = Total saving / Y = 1,000 / 10,000 = 0.10 = 10% I = sY = Total saving = $1,000 billion per year. Investment falls when government spending rises. 16. A. The production function is .The production function is graphed at the right. B. The balanced-growth equilibrium capital-output ratio is = 0.15 / 0.06 = 2.5. It is graphed as a straight line with a slope of 1/2.5 = 0.4. C. Y/L = 3,125. = (2.5)2(500) = 6.25(500) = 3,125. From the graph, it appears that the balanced-growth equilibrium level of Y/L the point where the balanced-growth equilibrium line and the production function cross is indeed 3,125. K/L equals about 8,000. D. The new balanced-growth equilibrium K/Y ratio is (0.195)/(0.02 + 0 + 0.04) = 3.25. The initial and new balanced-growth equilibrium lines are shown at the right. E. No, the old combination of K/L and Y/L (about 8,000 and 3,125) is no longer the balanced-growth equilibrium combination. When K/L equals about 8,000, output per worker which we read from the production function is greater than the balanced-growth equilibrium level of output per worker which we read from the second equilibrium line. F. The new balanced-growth equilibrium level of output per worker is Y/L = (3.25)2(500) = 5,281.25. To find the new equilibrium level of K/L we use the production function: . We know the values of everything but K/L. , so . The new higher saving level generates additional resources and funds for 3 investment, increasing the capital stock and thus the capital-labor ratio. As the capital-labor ratio increases, output per worker increases along the production function. Because of diminishing returns to investment ( is less than 1), the increases in Y/L are progressively smaller for each increase in K/L. Gradually the economy therefore converges to the new balanced-growth equilibrium combination of K/L = about 17,000 and Y/L = 5,281.25. G. Once the economy has moved to its new balanced-growth equilibrium level of output per worker, there will be no further changes in K/L or Y/L because we assumed g = 0. The production function is stationary from period to period when the efficiency of labor is constant (when g=0). And so once the economy grows to its new higher level of output per worker, there will be no further economic growth. Barring further changes in the saving rate, depreciation rate, labor force growth rate, or efficiency of labor, the economy will stagnate. 17. A. Equilibrium output per worker is 53,153.0. K/Y = 0.2 / (0.019 + 0.02 + 0.027) = 3.03 Y/L = (3.03)(0.7/0.3)(4,000) = 53,152.96 B. Equilibrium output per worker is now 56,562, an increase of 12.1 percent. K/Y = 0.21 / (0.019 + 0.02 + 0.027) = 3.18 Y/L = (3.18) (0.7/0.3) (4,000) = 56,561.98. %Δ(Y/L) = 56,561.98 / 53,152.96 - 1 = 0.121 = 12.1% C. Equilibrium output per worker is now 38,244, a decrease of 28.0 percent. K/Y = 0.20 / (0.019 + 0.02 + 0.037) = 2.63 Y/L = (2.63)(0.7/0.3)(4,000) = 38,244.14. %Δ(Y/L) = 38,244.14 / 53,152.96 - 1 = -0.280 = -28.0% D. Equilibrium output per worker is now 77,987, an increase of 46.7 percent. K/Y = 0.2 / (0.009 + 0.02 + 0.027) = 3.57 Y/L = (3.57) (0.7/0.3) (4,000) = 77,987.43. %Δ(Y/L) = 77,987.43 / 53,152.96 - 1 = 0.467 = 46.7% 18. A. The balanced-growth equilibrium capital-output ratio is B. Y/L = 22,500. C. See table. D. Between periods 1 and 2, Y/L increases by 5 percent (23,625/22,500 = 1.05). That is the same rate at which E is increasing: 5 percent. This is no surprise! When the economy moves from one balancedgrowth equilibrium to another, K/L, Y/L, and E all grow at exactly the same rate, here 5 percent. C. Applying Concepts and Models 1. A lower saving rate lowers K/Y in balanced-growth equilibrium, lowering Y/L for any given value of E. But in balanced-growth equilibrium, Y/L and K/L will grow at the rate g which is unchanged. So the value of Y/L for any value of E will be lower, but in equilibrium, standards of living will continue to grow at the rate g. 2. The increase in government saving will increase the saving rate and thus increase the balanced-growth value of the capital-output ratio. For a given value of labor efficiency, E, output per worker (the standard of living) will also increase. The equilibrium rate of growth of the standard of living will not change, however, unless there is a change in the growth rate of labor efficiency. 3. The decrease in government saving has the opposite effect of what we found in #2. The drop in government saving lowers the saving rate, decreasing in the long run the funds available for investment. The balanced-growth equilibrium capitaloutput ratio falls, lowering output per worker for a given value of labor efficiency. The equilibrium rate of growth of the standard of living will not change, however, unless there is a change in the growth rate of labor efficiency. 4. A fall in the depreciation rate increases the balanced-growth equilibrium capitaloutput ratio. Given the level of labor efficiency, E, equilibrium levels of output per worker will rise. The equilibrium rate of growth of output per worker will not change, however, unless there is a change in the growth rate of labor efficiency. 5. Changes in business organization can increase labor efficiency, which will increase output per worker. But unless there are ongoing year-after-year improvements in organization which generate annual increases in labor efficiency, there is no permanent long-run change in the growth rate of labor efficiency and thus no permanent change in the growth rate of output per worker. 6. If Americans reduce their purchases of imported goods and services, substituting domestically-produced goods and services in their place, foreign saving will fall. The resulting drop in total saving will reduce investment spending, and thus lower output per worker for a given level of labor efficiency as the balanced-growth equilibrium capital-output ratio declines. The equilibrium rate of growth of output per worker will not change, however, unless there is a change in the growth rate of labor efficiency. D. Explaining the Real World 1. The question is asking about the effect of a change in the labor force growth rate on equilibrium output per worker. The increase in women's labor force participation provides a period when the labor force is growing at a faster rate. For example, if women's labor force participation began at 20 percent and grew to 70 percent over a 50-year period, but thereafter remained steady at 70 percent, then during those 50 years the labor force will be growing at a higher rate than before women's participation began to grow. After the 50-year period, labor force growth would fall back to the rate of growth of the population. During the 50-year period, faster labor force growth lowers K/Y in balanced-growth equilibrium, lowering Y/L given E at equilibrium. Increased labor force participation increases the worker-population ratio the number of workers per capita. So even though output per worker will be lower (given E), output per capita will be higher. The equilibrium rate of growth of output per worker will not change, however, unless there is a change in the growth rate of labor efficiency. 2. The question is asking about the impact of changes in population growth and thus in labor force growth on balanced-growth equilibrium output per worker. A decrease in population growth rates should also lower labor force growth rates, with a lag of about 15 years. A decrease in labor force growth rates will increase the balanced-growth equilibrium capital-output ratio, increasing the equilibrium level of output per worker given efficiency. The equilibrium rate of growth of the standard of living will not change, however, unless there is a change in the growth rate of labor efficiency. 3. The question is asking about the effect of a change in labor efficiency on equilibrium output per worker. All else constant, the development of steam power should increase standards of living in the long run because it would increase labor efficiency. Whether or not the growth rate of output per worker increased permanently would depend upon the continuation of technological developments that subsequently increased labor efficiency further. 4. The question is asking about the effect of a change in depreciation on equilibrium output per worker. All else constant, replacing computer equipment because of the Y2K Bug did not increase the usable capital stock. The Y2K Bug essentially increased the rate of depreciation of computer equipment, rendering existing computer systems obsolete. That is, it increased the depreciation rate. An increase in the depreciation rate would lower the balanced-growth equilibrium capital-output ratio and, given a level of E, would lower output per worker. However if labor efficiency continued to grow at the same rate, g, then in the long run, equilibrium output per worker and capital per worker would also continue to grow at rate g. On the other hand, if the new computer equipment not only corrected the Y2K Bug but also enabled workers to produce more output with the same quantity of capital, it would have increased labor efficiency. The increase in labor efficiency would increase output per worker in equilibrium. 5. The question is asking about the different impacts of increasing labor efficiency and decreasing saving. Funding scientific research has the better chance of increasing standards of living in the long run if the research increases labor efficiency. Lowering taxes will lower government saving. All else constant, lower government saving decreases the equilibrium capital-output ratio and, given labor efficiency, lowers equilibrium output per worker. If consumers save the entire tax cut, then personal saving will rise to offset the drop in government saving, leaving total saving unchanged. In this case, however, there is still no long-run increase in the standard of living. 6. The question is asking about the long-run effect of changes in saving. In case [1], total saving declines, so the equilibrium capital-output ratio declines, as does the balanced-growth equilibrium level of output per worker. In case [2], total saving remains the same, so the balanced-growth equilibrium capital-output ratio and the equilibrium level of output per worker also remain the same. Case [3] is the same as case [2]! When consumers pay off debt, they are saving. Saving is simply "not spending on currently produced goods and services" so whether consumers put funds into their savings account or pay off their credit card bills, they are saving. In case [1], there are no long-run economic benefits. In cases [2] and [3], there are long-run economic benefits of the tax cut only if you assume that the government will lower its spending commensurate with the tax cut and that investment spending by the private sector is economically better than spending by the government. For instance, if investment spending by the private sector has a lower depreciation rate than government spending on infrastructure or other investment, then transferring funds from the government to the private sector can raise balanced-growth equilibrium levels of the capital-output ratio and output per worker. It all depends upon the types of spending cuts imposed by Congress and the type of new spending undertaken by the private sector. 7. The question is asking about the difference between changes in the level of output per worker and in the rate of growth of output per worker. It is true that an increase in the saving rate has no permanent effect on the growth rate of output per worker. In balanced-growth equilibrium, the growth rate of output per worker equals the growth rate of labor efficiency. A change in the saving rate has no effect on g, the growth rate of labor efficiency. But the statement is incorrect in saying "there is no good reason" to encourage saving. Granted, the growth rate of output per worker will return to its initial rate, but during the transition period from the initial balanced growth equilibrium to the new balanced-growth equilibrium, output per worker will be growing faster than labor efficiency. And it can take decades for the economy to adjust to a new balanced-growth equilibrium! Few people would truly scoff at this opportunity to enjoy 30, 40, or 50 years of faster growth in standards of living. E. Possibilities to Ponder No solutions are given to these questions. The questions are designed to be somewhat open ended. Each question draws on your understanding of the concepts covered in this chapter.