1. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-1 The Theory of Economic Growth: The Solow Growth Model Reading: DeLong/Olney: Macroeconomics; McGraw-Hill; 2006; Chapter 4

2. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-2 Questions What are the causes of long-run economic growth? What is the “efficiency of labor”? What is an economy’s “capital intensity”? What is an economy’s “balanced- growth path”? What can we say about convergence to the “balanced growth path”?

3. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-3 Questions How important is faster labor-growth as a drag on economic growth? How important is a high saving rate as a cause of economic growth? How important is technological and organizational progress for economic growth?

4. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-4 Long-Run Economic Growth Material standards of living and levels of economic productivity in the U.S. are more than 4 times what they are in Mexico due to differences in –the skills of the labor force –the value of the capital stock –the level of technology and organization currently used in production

5. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-5 Figure 4.1 – American Real GDP per Capita, 1800-2004 (in 2004 Dollars)

6. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-6 Long-Run Economic Growth We classify the factors that generate differences in productive potentials into two broad groups –differences in the efficiency of labor how technology is deployed and organization is used –differences in capital intensity how much current production has been set aside to produce useful machines, buildings, and infrastructure

7. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-7 The Efficiency of Labor The efficiency of labor has risen for two reasons –advances in technology –advances in organization Economists are good at analyzing the consequences of advances in technology but they have less to say about their sources

8. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-8 Capital Intensity There is a direct relationship between capital-intensity and productivity –a more capital-intensive economy will be a richer and more productive economy

9. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-9 Standard Growth Model Also called the Solow growth model Consists of –variables –behavioral relationships –equilibrium conditions The key variable is labor productivity –output per worker (Y/L)

10. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-10 Solow Growth Model Balanced-growth equilibrium –the capital intensity of the economy (K/Y) is stable, but (K/L) and (Y/L) grow –the economy’s capital stock and level of real GDP are growing at the same rate –the economy’s capital-output ratio is constant

11. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-11 Solow Growth Model The balanced-growth path: –if the economy is on its balanced-growth path, the present value and future values of output per worker will continue to follow the balanced-growth path –if the economy is not yet on its balanced- growth path, it will head towards it

12. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-12 The Production Function The production function tells us how the average worker’s productivity (Y/L) is related to the efficiency of labor (E) and the amount of capital at the average worker’s disposal (K/L) Cobb-Douglas production function

13. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-13 The Production Function  measures how fast diminishing marginal returns to investment set in –the smaller the value of , the faster diminishing returns are occurring

14. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-14 Figure 4.2 - The Cobb-Douglas Production Function for Different Values of 

15. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-15 The Production Function The value of the efficiency of labor (E) tells us about the placement of the production function –a higher level of E means that more output per worker is produced for each possible value of the capital stock per worker

16. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-16 Figure 4.3 - The Cobb-Douglas Production Function for Different Values of E

17. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-17 The Production Function: Example E = $10,000  = 0.3 K/L = $125,000

18. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-18 The Production Function: Example If K/L rises to $250,000 –the first $125,000 of K/L increased Y/L from $0 to $21,334 –the second $125,000 of K/L increased Y/L from $21,334 to $26,265

19. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-19 Saving, Investment, and Capital Accumulation The net flow of saving is equal to the amount of investment Remember from National Accounts that real GDP (Y) can be divided into four parts –consumption (C) –investment (I) –government purchases (G) –net exports (NX = GX - IM)

20. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-20 Saving, Investment, and Capital Accumulation

21. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-21 Saving, Investment, and Capital Accumulation The right-hand side shows the three pieces of total saving –household saving (S H ) –government saving (S G ) –foreign saving (S F )

22. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-22 Saving, Investment, and Capital Accumulation Let’s assume that total saving is a constant fraction (s) of real GDP Therefore, it must be true that

23. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-23 Saving, Investment, and Capital Accumulation We will refer to s as the economy’s saving rate –we will assume that it will remain at its current value as we look far into the future –s measures the flow of saving and the share of total production that is invested and used to increase the capital stock

24. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-24 Saving, Investment, and Capital Accumulation The capital stock is not constant We will let –K 0 will mean the capital stock at some initial year –K 2003 will mean the capital stock in 2003 –K t will mean the capital stock in the current year –K t+1 will mean the capital stock next year –K t-1 will mean the capital stock last year

25. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-25 Saving, Investment, and Capital Accumulation Investment will make the capital stock tend to grow Depreciation makes the capital stock tend to shrink –the depreciation rate is assumed to be constant and equal to 

26. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-26 Saving, Investment, and Capital Accumulation Next year’s capital will be The capital stock is constant when

27. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-27 Saving, Investment, and Capital Accumulation Suppose that the economy has no labor force growth and no growth in the efficiency of labor –if K/Y < s/, depreciation is less than investment so K and K/Y will grow until K/Y = s/ –if K/Y > s/, depreciation is greater than investment so K and K/Y will fall until K/Y = s/

28. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-28 Saving, Investment, and Capital Accumulation Thus, if the economy has no labor force growth and no growth in the efficiency of labor, the equilibrium condition of this growth model is

29. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-29 Figure 4.5 – Equilibrium with Just Investment and Depreciation Capital Stock per Worker Output per Worker K/L  /s (assumes that  and E are constant) (the reciprocal of the capital-output ratio) The capital stock is growing because K/L >  /s The capital stock is shrinking because K/L <  /s Equilibrium

30. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-30 Saving, Investment, and Capital Accumulation Remember that, in this particular case, we are assuming that –the economy’s labor force is constant –the economy’s capital stock is constant –there are no changes in the efficiency of labor Thus, equilibrium output per worker is constant Now we will complicate the model

31. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-31 Adding in Labor Force and Labor Efficiency Growth Growth in labor force (L) –assume that L is growing at a constant rate (n) –if this year’s labor force is equal to 10 million and the growth rate is 2% per year, next year’s labor force will be

32. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-32 Figure 4.6 - Constant Labor-Force Growth (at n = 2% per Year)

33. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-33 Adding in Labor Force and Labor Efficiency Growth Assume that the efficiency of labor (E) is growing at a constant proportional rate (g) –if this year’s efficiency of labor is $10,000 and the growth rate is 1.5% per year, next year’s efficiency of labor will be

34. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-34 Figure 4.7 - Efficiency-of-Labor Growth at g = 1.5% per Year

35. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-35 The Balanced-Growth Capital- Output Ratio When we assumed that the labor force and efficiency were both constant (so that n and g are equal to 0), the equilibrium condition was Since s and  are constant, K/Y is constant

36. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-36 The Balanced-Growth Capital- Output Ratio If we assume that the labor force and efficiency grow at n and g, the equilibrium condition still requires that K/Y is constant –the economy is in balanced growth –output per worker is growing at the same rate as the capital stock per worker both growing at the same rate as the efficiency of labor

37. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-37 The Balanced-Growth Capital- Output Ratio The economy will be in balanced growth equilibrium when This is the balanced-growth equilibrium capital-output ratio

38. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-38 Some Mathematical Rules The growth rate of a product is equal to the sum of the growth rates of its components The proportional growth rate of a quotient is equal to the difference between the proportional growth rates of its components

39. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-39 Some Mathematical Rules The proportional change of a quantity raised to a power is equal to the proportional change in the quantity times the power to which it is raised A quantity growing at k percent per year doubles in 72/k years

40. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-40 Balanced-Growth Output per Worker Suppose that the economy is on its balanced-growth path –K/Y is equal to its balanced-growth equilibrium value Let’s calculate the level of output per worker (Y/L)

41. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-41 Balanced-Growth Output per Worker Begin with the capital-output ratio version of the production function

42. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-42 Balanced-Growth Output per Worker Since the economy is on its balanced- growth path Since s, n, g, , and  are all constants, [s/(n+g+)] /1- is a constant

43. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-43 Balanced-Growth Output per Worker Along the balanced-growth path, output per worker is simply a constant multiple of the efficiency of labor Over time, the efficiency of labor grows at a constant rate g –Y/L must be growing at the same proportional rate

44. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-44 Figure 4.8 – Balanced Growth: Output per Worker and the Efficiency of Labor

45. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-45 Balanced-Growth Output per Worker Capital intensity (K/Y) determines what multiple Y/L is of the current labor efficiency –things that increase K/Y make the balanced-growth equilibrium Y/L a higher multiple of the efficiency of labor –things that reduce K/Y make the balanced- growth equilibrium Y/L a lower multiple of the efficiency of labor

46. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-46 Balanced-Growth Output per Worker Changes in the capital intensity shift the balanced-growth path up or down But the growth rate of Y/L along the balanced-growth path is simply the rate of growth of the efficiency of labor –the material standard of living grows at the same rate of labor efficiency –changes in K/L alone will not accomplish this

47. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-47 Figure 4.9 – Calculating Balanced-Growth Output per Worker Capital Stock per Worker Output per Worker Y/L Balanced-Growth Y/K (function of E t ) [K/Y = s/(n+g+  )] current output per worker along the balanced- growth path

48. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-48 Figure 4.9 – Calculating Balanced-Growth Output per Worker Capital Stock per Worker Output per Worker Y/L Balanced-Growth Y/K Anything that increases the balanced- growth capital-output ratio will rotate the capital-output line clockwise Y/K’ Y/L will rise

49. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-49 Figure 4.9 – Calculating Balanced-Growth Output per Worker Capital Stock per Worker Output per Worker Y/L Balanced-Growth Y/K Anything that decreases the balanced- growth capital-output ratio will rotate the capital-output line counter- clockwise Y/K’ Y/L will fall

50. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-50 Off the Balanced-Growth Path When K/Y > s/(n+g+) –K/Y is falling because investment is insufficient to keep K growing as fast as Y When K/Y < s/(n+g+) –K/Y is rising because the growth in K outruns the growth in Y K/Y and Y/L will converge to their balanced-growth paths

51. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-51 Figure 4.10 - Convergence to a Balanced- Growth Capital-Output Ratio of 4

52. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-52 How Fast the Economy Heads for Its Balanced-Growth Path A fraction (1-)(n+g+) of the gap between the economy’s current position and its balanced-growth path will be closed each year –assume that this fraction is 4% –according to the rule of 72,the economy will move halfway to equilibrium in 72/4 or 18 years –the convergence does not happen quickly

53. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-53 Figure 4.11 – The Return of the West German Economy to Its Balanced Growth Path

54. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-54 The Labor Force Growth Rate The faster the growth of the labor force, the lower will be the economy’s balanced-growth K/Y ratio –the larger the share of current investment that must go to equip new workers with the capital they need Thus, a sudden, permanent increase in labor force growth will also lower Y/L on the balanced-growth path

55. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-55 Figure 4.12 – The Labor Force Growth Rate Matters

56. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-56 The Saving Rate and the Price of Capital Goods The higher the share of real GDP devoted to saving and gross investment, the higher will be the economy’s balanced-growth K/L ratio –more investment increases the amount of new capital A higher saving rate also increases Y/L along the balanced-growth path

57. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-57 Figure 4.13 - Investment Shares of Output and Relative Prosperity

58. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-58 Growth Rate of the Efficiency of Labor An increase in g will reduce the economy’s balanced-growth K/L ratio –past investment will be small relative to current output Changes in g change the growth rate of Y/L along the balanced-growth path –these effects are overwhelmed by the direct effect of g on Y/L

59. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-59 Growth Rate of the Efficiency of Labor The growth rate of the standard of living can change if and only if g changes –other factors can shift Y/L up but do not permanently change the growth rate of Y/L

60. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-60 Table 4.2 – Effects of Increases in Parameters on the Solow Growth Model When there is an increase in the parameter… The Effect on… Equilibrium K/Y Level of Y Level of Y/L Permanent Growth Rate of Y Permanent Growth Rate of Y/L sIncreasesUp No change nDecreasesUpDownIncreasesNo change  DecreasesDown No change gDecreasesUp Increases

61. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-61 Summary One principal force driving long-run growth in output per worker is the set of improvements in the efficiency of labor springing from technological progress

62. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-62 Summary A second principal force driving long- run growth in output per worker is the increases in capital intensity – the ratio of the capital stock to output

63. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-63 Summary The balanced-growth equilibrium in the Solow growth model occurs when the capital output ratio K/Y is constant –when K/Y is constant, the capital stock and real output are growing at the same rate

64. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-64 Summary The Cobb-Douglas production function we use is Y/L = [K/L]  E (1-) –this is equivalent to Y/L = [K/Y] /1- (E) –an increase in  makes the production function steeper –an increase in E makes the production function shift up

65. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-65 Summary In equilibrium, investment equals saving: I = S = S H +S G +S F –we assume S/Y = s = saving rate is constant

66. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-66 Summary The balanced-growth equilibrium value of the capital output ratio K/Y is a constant equal to 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  –in balanced growth: K/Y = s/(n+g+)

67. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-67 Summary If the economy’s actual value of K/Y is initially greater than s/(n+g+), then K/Y will fall until it reaches its equilibrium value If the economy’s actual value of K/Y is initially less than s/(n+g+), then K/Y will rise until it reaches its equilibrium value

68. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-68 Summary It can take decades or generations for K/Y to reach its balanced-growth equilibrium value

69. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-69 Summary An increase in the saving rate s, a decrease in the labor force growth rate n, or a decrease in the depreciation rate  increases Y/L –the growth rate of Y/L will accelerate as the economy moves to its new higher balanced-growth path –once there, Y/L will grow at the same rate as it did initially

70. Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved. 4-70 Summary In balanced-growth equilibrium, the growth rate of output per worker equals the growth rate of labor efficiency g –only increases in g can produce a lasting increase in the growth rate of output per worker