1. Prepared by: Fernando Quijano and Yvonn Quijano And Modified by Gabriel Martinez 11 C H A P T E R Saving, Capital Accumulation, and Output

2. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Saving and Growth Country Germany1,2052,3205,00518,01423,2472.33.1 Japan9631,8252,21616,89924,7722.54.8 United States2,8436,74511,92122,48032,6291.92.0 Annual % change 1950-2000 Annual % change 1870-2000 18701913195019792000 Since 1950 the US saving rate (S/Y) has averaged 18%. The German saving rate averaged 24% and the Japanese, 34%. Can this explain the growth differences? Probably not.

3. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier BlanchardPreview  Suppose output rises.  A proportion of that, sY, is saved.  Assume S=I. Then I = sY.  Investment increases the capital stock.  With a greater capital stock, we can produce more.  Output rises.

4. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier BlanchardPreview  Greater output leads to more saving, capital accumulation, and therefore more output.  Is this a perpetual cycle?  No. There are diminishing returns to capital. Output Rises S and I Rise Capit al Stock rises

5. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier BlanchardPreview Diminishing Returns A Fixed Proportion I =  K Capital Accumulation

6. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier BlanchardPreview  There are diminishing returns to capital.  More capital will increase output, but at a decreasing rate.  Suppose population growth is positive.  Eventually, the increase in output won’t be enough to increase output-per-capita.  So there’s a “dynamic equilibrium”, a steady state of capital accumulation in the long run.

7. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Interactions Between Output and Capital  Two important relations in the long run are: –The amount of capital determines the amount of output produced. –The amount of output determines the amount of saving and investment, and so the amount of capital accumulated. 11-1

8. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard  The aggregate production function is a specification of the relation between aggregate output and the inputs in production. Y = aggregate output. K = capital — the sum of all the machines, plants, and office buildings in the economy. buildings in the economy. N = labor — the number of workers in the economy. The function F, tells us how much output is produced for given quantities of capital and labor. The Effects of Capital on Output

9. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Capital on Output  Constant returns to scale implies that we can rewrite the aggregate production function as:  The amount of output per worker, Y/N depends on the amount of capital per worker, K/N.  As capital per worker increases, so does output per worker.

10. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Capital on Output  Under constant returns to scale, we can write the relation between output and capital per worker as follows: If we define Simplifying: More capital per worker produces more output per worker.

11. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Output per Worker and Capital per Worker Output and Capital per Worker Increases in capital per worker lead to smaller and smaller increases in output per worker. An increase in capital per worker, K/N, causes a move along the production function.

12. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Capital on Output  Focus on the role of capital accumulation: 1.The size of the population, the participation rate, and the unemployment rate are all constant.  Because this is the long run, it’s natural to assume that u t =u t-1.  This fit the concerns of the Classical economists: What determines the Wealth of Nations in the long run, when all prices are flexible?

13. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Capital on Output  Focus on the role of capital accumulation: 2.There is no technological progress.  Or, at least, we don’t know where it comes from. 3.Saving equals investment  Financial markets work perfectly.

14. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Capital on Output  Under these assumptions, the first important relation we want to express is between output and capital per worker: In words, higher capital per worker leads to higher output per worker.

15. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Output and Investment:  The equations below describe the relation between private saving and investment:

16. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation  The saving rate (s) is the proportion of income that is saved (sY).

17. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Output and Investment:  In the long run, private saving is equal to investment, and proportional to income.  Therefore, investment is proportional to output:  Higher output → higher saving → higher investment.

18. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Investment and Capital Accumulation:  Suppose that capital is eternal (once installed, it’s there forever)  Then the evolution of the capital stock is given by:  Investment adds to capital.

19. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Investment and Capital Accumulation:  More realistically, capital depreciates.  The evolution of the capital stock is given by:  Investment adds to capital.   denotes the rate of depreciation.  A proportion (1-  ) of capital remains from the previous period.

20. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Investment and Capital Accumulation:  Combine the relation from output to investment,, and the relation from investment to capital accumulation, we get

21. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation Investment and Capital Accumulation:  If we divide this equation by N, we get

22. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation  Can we use this equation to know what will happen to capital per worker over time? Output and Capital per Worker:

23. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation  We can articulate the change in capital per worker over time by rearranging terms in the equation above. In words, the change in the capital stock per worker (left side) is equal to saving per worker minus depreciation (right side). Output and Capital per Worker:

24. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of Output on Capital Accumulation The stock of capital per worker will increase if The total amount of saving in the economy exceeds the part of the capital stock that is worn out. Output and Capital per Worker:

25. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Implications of Alternative Saving Rates  Our two main relations are:  Combining the two relations, we can study the behavior of output and capital over time. First relation: Capital determines output. Second relation: Output determines capital accumulation 11-2

26. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output  From our main relations above, we express output per worker (Y/N) in terms of capital per worker to derive the equation below: change in capital from year t to year t+1 investment during year t depreciation during year t

27. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output  If investment per worker (sY) exceeds depreciation per worker, the change in capital per worker is positive: Capital per worker increases.  If investment per worker is less than depreciation per worker, the change in capital per worker is negative: Capital per worker decreases. change in capital from year t to year t+1 investment during year t depreciation during year t

28. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output Capital and Output Dynamics  Output per worker increases with capital per worker, but by less and less as capital per worker increases.  Investment per worker increases by a proportion of output per worker.  More capital per worker leads to more investment, but at a diminishing rate.  Suppose a country made a great effort to produce K/N and to save a bunch of output to invest it. The “extra bang for the extra buck” diminishes.

29. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output Capital and Output Dynamics When capital and output are low, investment exceeds depreciation, and capital increases. When capital and output are high, investment is less than depreciation and capital decreases.  At low levels of K/N, the “extra bang for the extra buck” invested is large, larger than what is being taken away by depreciation.  If the country already has a very large K/N, the “extra bang for the extra buck” invested is small and is overwhelmed by depreciation.

30. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output   At K 0/N, capital per worker is low, investment exceeds depreciation, thus, capital per worker and output per worker tend to increase over time.

31. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output   At K*/N, output per worker and capital per worker remain constant at their long-run equilibrium levels.

32. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output   At K 1/N, capital per worker is too high. Here investment is overwhelmed by depreciation. Therefore K/N and Y/N will fall over time. A D C B

33. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output  This model suggests convergence: –Take a poor country (one with low K/N) and a rich country (that has a high K/N). –The poor country will probably be farther away from K*/N than the rich country. –Then the poor country should grow faster than the rich country and catch up.

34. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output  This model suggests convergence: –Given the same level of technology and human capital, same institutions, etc., … –This model says that all countries should converge to the same level.

35. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Dynamics of Capital and Output  This model suggests convergence: –High capital accumulation is not enough to sustain growth forever: if K/N is too high, diminishing returns will make investment to be lower than depreciation.  Compare Singapore with Hong Kong (case study).

36. Steady-State Capital and Output

37. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Steady-State Capital and Output  The state in which output per worker and capital per worker are no longer changing is called the steady state of the economy. In steady state, the left side of the equation above equals zero, then:  The important point is to notice that there is one and only one value of K*/N that satisfies this equation.

38. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Steady-State Capital and Output  There is only one value of K*/N that satisfies this equation:  Given the steady state of capital per worker (K*/N), the steady-state value of output per worker (Y*/N), is given by the production function:  There is only one steady-state value of Y*/N.

39. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard  The steady-state values of K*/N and of Y*/N are uniquely determined by the saving rate.  A higher s will lead to a higher K*/N and a higher Y*/N.  But a higher s will leave the growth rate of Y*/N unaffected. Steady-State Capital and Output  There is only one value of K*/N that satisfies this equation:

40. The Saving Rate and Output

41. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output  The effects of the saving rate on the growth rate of output per worker: 1.The saving rate has no effect on the long run growth rate of output per worker, which is equal to zero. 1.Output per worker and capital per worker are constant in the steady state. 2.If an economy wanted to increase the steady state K*/N every year it would have to increase savings/output every year.

42. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output 2.Nonetheless, the saving rate determines the level of output per worker in the long run. Other things equal, countries with a higher saving rate will achieve higher output per worker in the long run.

43. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output The Effects of Different Saving Rates A country that raises its saving rate achieves a higher level of output per worker in steady state. But, in the steady state, output/worker does not grow.

44. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output 3.Therefore, an increase in the saving rate will lead to higher growth of output per worker for some time, but not forever.  The saving rate does not affect the long-run growth rate of output per worker.  After an increase in the saving rate, growth will end once the economy reaches its new steady state.

45. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output The Effects of an Increase in the Saving Rate on Output per Worker An increase in the saving rate leads to a period of higher growth until output reaches its new higher steady-state level. The economy takes some time to reach the new steady state as it accumulates capital.

46. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Output The Effects of an Increase in the Saving Rate on Output per Worker in an Economy with Technological Progress If there’s technological progress, the growth rate of Y/N is positive in the steady state. An increase in the saving rate leads to a period of higher growth until output reaches a new, higher path. (This is a logarithmic scale, so the slope of the steady-state path is equal to the growth rate of Y/N.)

47. The Optimal Saving Rate and Optimal Consumption

48. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption  The previous section suggests that a country could choose its steady state level of capital per worker and output per worker.  What steady state should a country choose?  I’d think that the steady state that maximizes consumption. –(Output is nice, but I’d focus on eating).

49. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption Steady state capital / worker, K*/N Steady state output / worker, Y*/N Steady state depreciation / worker,  K*/N Steady state output / worker, f(K*/N) Steady state investment / worker, sf(K*/N) ?

50. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption  Because people care about consumption (and not about investment or output), society should maximize steady-state consumption. Max consumption Golden-rule income Golden-rule level of capital/ worker Golden- rule saving rate

51. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption  The level of capital that causes the value of the saving rate that yields the highest level of consumption for all generations in steady state is known as the golden-rule level of capital. Max consumption Golden-rule saving Golden-rule level of capital/ worker Golden- rule saving rate

52. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption  In the steady state, investment has to be equal to depreciation. –(If investment and depreciation were not equal, capital would accumulate, which violates the definition of the steady state.)  So adding capital means more depreciation (more capital has to be replaced).

53. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption  But adding capital also means more output  Consumption, in a closed economy with no government, is C = Y – I  From the above we know that

54. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption Steady state capital / worker, K*/N Steady state output / worker, Y*/N Steady state depreciation / worker,  K*/N C*/N Steady state output / worker, f(K*/N) Consumption is given by the vertical distance between the production function and the depreciation line. Since steady state K*/N is given by the intersection of the depreciation line and the investment curve, steady state consumption is the difference between Y*/N and K*/N.

55. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption The Effects of the Saving Rate on Consumption per Worker in Steady State  At very low levels of steady-state saving, steady- state capital and output per worker are very low and consumption is very low.  At very high levels of steady-state saving, steady- state capital per worker is very high. But because of diminishing returns, a very large proportion of output has to be devoted to replacing depreciated capital, and consumption is low.

56. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption Steady state capital / worker, K*/N Steady state output / worker, Y*/N Steady state depreciation / worker,  K*/N Steady state output / worker, f(K*/N) Steady state investment / worker, sf(K*/N)

57. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption   Evidently, if s=0, there’s no investment. Capital depreciates to zero, and output disappears.   For s smaller than s G, increases in the saving rate (which increase I*/N), lead to higher capital and output per worker. Y*/N increases so much that C*/N = Y*/N – I*/N increases.  How do we find the Golden-Rule level of saving?  Golden Rule: set saving so that this generation’s consumption equals that of all future generations.

58. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Saving Rate and Consumption   For s larger than s G, increases in the saving rate still lead to higher capital and output per worker, but lower consumption per worker because Y*/N increases by less than I*/N.   For s=1, capital and output per worker are high, but all of the output is used to replace depreciation, so C*=0.  Golden Rule: set saving so that this generation’s consumption equals that of all future generations.  How do we find the Golden-Rule level of saving?

59. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Steady state capital / worker, K*/N Steady state output / worker, Y*/N Steady state depreciation / worker,  K*/N C*/N Steady state output / worker, f(K*/N) The Saving Rate and Consumption  How do we find the Golden-Rule level of capital/worker?  Set K*/N such that the slope of f(K*/N) is equal to .  Set marginal product of capital/worker = depreciation

60. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Social Security, Social Security Reform, and Capital Accumulation in the United States   One way to run a social security system is the pay-as-you-go system, where the taxes that workers pay are the benefits that current retirees receive. – –There is little or no saving or investment.   Another is the fully-funded system, where workers are taxed, their contributions invested in financial assets, and when workers retire, they receive the principal plus the interest payments on their investments. – –All contributions are saved and invested.

61. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Social Security, Social Security Reform, and Capital Accumulation in the United States   The US has a pay-as-you-go system, so that retirees’ benefits depend on current workers’ contributions.   But the ratio of workers to retirees has been falling.   In anticipation of demographic changes, the Social Security tax rate has been increased, and contributions are now larger than benefits, leading to the accumulation of a Social Security trust fund.   But the trust fund is expected to be depleted in mid- century, so there’s need for more reform. – –Raise the tax rate further, increase the retirement age, reduce benefits. – –Shift to a fully-funded system (more pain, but more saving).

62. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Getting a Sense of the Magnitudes  To get an idea of how to connect the model with reality, we need to do a numerical simulation of the model.  That is, we need to define the functions precisely, and we need to give numbers for all the parameters. 11-3

63. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Getting a Sense of the Magnitudes  Assume the production function is:  Output per worker is:  That is, the first relation of the model (capital/worker determines output) is  And the second relation of the model (output determines capital accumulation) is  Then,

64. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of the Saving Rate on Steady-State Output  In steady state, the left side equals zero:  Squaring both sides,  Dividing by (K/N) and rearranging,  Steady-State Output per worker is given by: The steady state capital per worker is equal to the square of the ratio of the saving rate to the depreciation rate.

65. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Effects of the Saving Rate on Steady-State Output  Steady-state output per worker is equal to the ratio of the saving rate to the depreciation rate.  A higher saving rate and a lower depreciation rate both lead to higher steady-state capital per worker and higher steady-state output per worker.

66. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The Dynamic Effects of an Increase in the Saving Rate Dynamic Effects of an Increase in the Saving Rate from 10 to 20% on the Level and the Growth Rate of Output per Worker Suppose  =0.10 and s=0.10. Now suppose s rises to 0.20. K/N quadruples and Y/N doubles. It takes a long time for output to adjust to its new higher level after an increase in the saving rate. Put another way, an increase in the saving rate leads to a long period of higher growth.

67. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The U.S. Saving Rate and the Golden Rule  In steady state, consumption per worker is equal to output per worker minus depreciation per worker.  Knowing that: then: and

68. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The U.S. Saving Rate and the Golden Rule  We saw above that the golden rule level of capital is found by setting K*/N such that the slope of f(K*/N) is equal to .  =0.10

69. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard The U.S. Saving Rate and the Golden Rule Table 11-2 The Saving Rate and the Steady-state Levels of Capital, Output, and Consumption per Worker-  =10% Saving Rate, s Capital per worker, K/N Output per worker, Y/N Consumption per worker, C/N 0.00.00.00.0 0.11.01.00.9 0.24.02.01.6 0.39.03.02.1 0.416.04.02.4 0.525.05.02.5 0.636.06.02.4 :::: 1.0100.010.00.0

70. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Physical Versus Human Capital  The set of skills of the workers in the economy is called human capital.  An economy with many highly skilled workers is likely to be much more productive than an economy in which most workers cannot read or write.  The conclusions drawn about physical capital accumulation remain valid after the introduction of human capital in the analysis. 11-4

71. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Extending the Production Function  When the level of output per workers depends on both the level of physical capital per worker, K/N, and the level of human capital per worker, H/N, the production function may be written as:  An increase in capital per worker or the average skill of workers leads to an increase in output per worker.

72. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Human Capital, Physical Capital, and Output  An increase in how much society “saves” in the form of human capital—through education and on- the-job-training—increases steady-state human capital per worker, which leads to an increase in output per worker.  In the long run, output per worker depends not only on how much society saves but also how much it spends on education. –How do we generate the institutions to improve the quality of education? –How do we reduce spending on current consumption to increase spending on education?

73. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Human Capital, Physical Capital, and Output  In the United States, spending on education comprises about 6% of GDP, compared to 16% investment in physical capital. This comparison: –Accounts for the fact that education is partly consumption. –Does not account for the opportunity cost of education. –Does not account for the opportunity cost of on-the-job- training. –Considers gross, not net investment. Depreciation of human capital is slower than that of physical capital.

74. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Endogenous Growth  A recent study has concluded that output per worker depends roughly equally on the amount of physical capital and the amount of human capital in the economy.  Still, models with human and physical capital conclude that higher levels of accumulation of H/N or K/N (that is, more education or saving) lead to higher levels of output/worker but not higher growth rates. –Saving/Income can’t be higher than 100%. –The same holds for expenditure on education / income.

75. © 2003 Prentice Hall Business PublishingMacroeconomics, 3/e Olivier Blanchard Endogenous Growth  Models that generate steady growth even without technological progress are called models of endogenous growth, where growth depends on variables such as the saving rate and the rate of spending on education (Lucas and Romer). –These models note that higher levels of education lead to better technology. Continuous technological improvement can lead to continuously positive growth rates.