Road Map to this Lecture

Transcription

1 Economic Growth 1

2 Road Map to this Lecture 1. Steady State dynamics: 1. Output per capita 2. Capital accumulation 3. Depreciation 4. Steady State 2. The Golden Rule: maximizing welfare 3. Total Factor Productivity slide 1 2

3 Economic Growth: The Questions What determines output per capita? How can we improve a country s rate of growth? Will inequality between countries disappear? Or will it increase? Welfare, poverty, health what does economic growth teach us about these issues? slide 2 3

4 Millennium Development Goals Progress Report slide 3 4

5 Millennium Development Goals Progress Report slide 4 5

6 Millennium Development Goals Progress Report slide 5 6

7 Economic Growth: Some Observations Economic Growth is a very recent phenomenon: less than 300 years old! Growth in OECD countries seems to be converging to each other Growth in Sub-Saharan countries has fallen in the last 25 years slide 6 7

8 Growth is a Recent Phenomenon: American Real GDP per Capita, slide 7 8

9 Convergence: The Irish Tiger slide 8 9

10 Convergence among the G-7 G 7 Economies: Output per Capita as a Share of U.S. Level slide 9 10

11 The Argentinian Conudrum: Divergence? slide 10 11

12 Estimated effects of economic growth A 10% increase in income is associated with a 6% decrease in infant mortality Income growth also reduces poverty. Example: Growth and Poverty in Indonesia change in income per capita +76% -12% change in # of persons living below poverty line -25% +65% slide 11 12

13 Income and poverty in the world % of population living on $2 per day or less selected countries, 2000 Madagascar Kenya India Nepal Bangladesh China Botswana 40 Peru Mexico 30 Thailand Brazil Russian Chile 0 Federation S. Korea $0 $5,000 $10,000 $15,000 $20,000 Income per capita in dollars slide 12 13

14 Huge effects from tiny differences annual growth rate of income per capita 25 years percentage increase in standard of living after 50 years 100 years 2.0% 64.0% 169.2% 624.5% 2.5% 85.4% 243.7% 1,081.4% slide 13 14

15 The Solow Model due to Robert Solow, won Nobel Prize for contributions to the study of economic growth a major paradigm: widely used in policy making benchmark against which most recent growth theories are compared looks at the determinants of economic growth and the standard of living in the long run slide 14 15

16 Features of the Solow Growth Model 1. K is no longer fixed: investment causes it to grow, depreciation causes it to shrink. 2. L is no longer fixed: population growth causes it to grow. 3. The consumption function is simpler. 4. No G or T only to simplify presentation; we can still do fiscal policy experiments) slide 15 16

17 The production function In aggregate terms: Y = F (K, L ) Define: y = Y/L = output per worker k = K/L = capital per worker Assume constant returns to scale: zy = F (zk, zl ) for any z > 0 Pick z = 1/L. Then Y/L = F (K/L, 1) y = F (k, 1) y = f(k) where f(k) = F (k, 1) slide 16 17

18 The production function Output per worker, y f(k) 1 MPK =f(k +1) f(k) Note: this production function exhibits diminishing MPK. Capital per worker, k slide 17 18

19 The Cobb-Douglas Example Y = AK α L 1 - α Y/L = A(K/L) α (L/L) 1-α = Ak α slide 18 19

20 What Determines the Level of Capital Stock? The capital stock increases when we invest. The capital stock decreases due to depreciation In per capita terms: k = i - δ k δ is the depreciation rate slide 19 20

21 Depreciation Rate: An Example Question: If a machine lasts for 25 years, what is its depreciation rate? Answer: δ = 4% per year. In year 1 we have 100% of the machine s services, in year 25, we have 0. Hence, 100/25 = 4 slide 20 21

22 The national income identity Y = C + I (remember, no G ) In per worker terms: y = c + i where c = C/L and i = I/L slide 21 22

23 The consumption function s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) Note: s is the only lowercase variable that is not equal to its uppercase version divided by L Consumption function: c = (1 s)y (per worker) slide 22 23

24 Saving and investment saving (per worker) = y c = y (1 s)y = sy National income identity is y = c + i Rearrange to get: i = y c = sy (investment = saving) Using the results above, i = sy = sf(k) slide 23 24

25 Output, consumption, and investment Output per worker, y f(k) y 1 c 1 sf(k) i 1 k 1 Capital per worker, k slide 24 25

26 Depreciation Depreciation per worker, δk δ = the rate of of depreciation = the fraction of of the capital stock that wears out each period δk 1 δ Capital per worker, k slide 25 26

27 Capital accumulation Change in capital stock = investment depreciation k = i δk Since i = sf(k), this becomes: k = sf(k) δk slide 26 27

28 The equation of motion for k k = sf(k) δk the Solow model s central equation Determines behavior of capital over time which, in turn, determines behavior of all of the other endogenous variables because they all depend on k. E.g., income per person: y = f(k) consump. per person: c = (1 s) f(k) slide 27 28

29 The steady state k = sf(k) δk If investment is just enough to cover depreciation [sf(k) = δk ], then capital per worker will remain constant: k = 0. This constant value, denoted k *, is called the steady state capital stock. slide 28 29

30 Capital Accumulation over Time K t+ 1 = K t + investment depreciation K t+ 1 = K t + sy t δk t sy = δ t K t K Y t t = s δ slide 29 30

31 The steady state Investment and depreciation δk sf(k) k * Capital per worker, k slide 30 31

32 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) investment k depreciation k 1 k * Capital per worker, k slide 31 32

33 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) k k 1 k * Capital per worker, k slide 32 33

34 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) k k 1 k 2 k * Capital per worker, k slide 33 34

35 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) investment k depreciation k 2 k * Capital per worker, k slide 34 35

36 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) k k 2 k * Capital per worker, k slide 35 36

37 Moving toward the steady state Investment and depreciation k = sf(k) δk δk sf(k) k k k * 2 k 3 Capital per worker, k slide 36 37

38 Moving toward the steady state Investment and depreciation Summary: As As long as as k < k *, *, investment will exceed depreciation, and k will continue to to grow toward k *. *. k = sf(k) δk k 3 k * δk sf(k) Capital per worker, k slide 37 38

39 Two forces at work With low levels of k, there is little depreciation, hence Investment > depreciation Capital accumulates k moves toward k * At high levels of k, depreciation overwhelms investment, capital decumulates and k moves toward k * slide 38 39

40 An Explanation At low levels of k, the MPK is high, so an extra unit of k produces a lot of output. Hence, per unit of k, there is more marginal consumption and saving. If k > k * MPK is low, and the reverse is true slide 39 40

41 Now you try: Draw the Solow model diagram, labeling the steady state k *. On the horizontal axis, pick a value greater than k * for the economy s initial capital stock. Label it k 1. Show what happens to k over time. Does k move toward the steady state or away from it? slide 40 41

42 A numerical example Production function (aggregate): Y = F ( K, L) = K L = K L 1/2 1/2 To derive the per-worker production function, divide through by L: 1/2 1/2 1/2 Y K L K = = L L L Then substitute y = Y/L and k = K/L to get y = f ( k) = k 1/2 slide 41 42

43 A numerical example, cont. Assume: s = 0.3 δ = 0.1 initial value of k 0 = 4.0 slide 42 43

44 A numerical example, cont. C = (1 s)y = (1 s)f(k) = I = sf(k) = δ k = 0.1 x 4 = 0.4 k = i - δ k = = 0.2 k 1 = k 0 + k = = 4.2 slide 43 44

45 Approaching the Steady State: A Numerical Example Assumptions: y = k; s = 0.3; δ = 0.1; initial k = 4.0 Year Year kk yy cc i i δk δk Dk Dk slide 44 45

46 Exercise: solve for the steady state Continue to assume s = 0.3, δ = 0.1, and y = k 1/2 Use the equation of motion k = sf(k) δk to solve for the steady-state values of k, y, and c. slide 45 46

47 k = Solution to exercise: 0 def. of steady state sf( k*) = δ k * eq'n of motion with k = k * = 0.1 k * using assumed values k * 3 = = k * k * Solve to get: k * = 9 and y* = k * = 3 Finally, c* = (1 s) y* = = 2.1 slide 46 47

48 An increase in the saving rate An increase in the saving rate raises investment causing the capital stock to grow toward a new steady state: Investment and depreciation dk s 2 f(k) s 1 f(k) * k 1 * k 2 k slide 47 48

49 Prediction: Higher s higher k *. And since y = f(k), higher k * higher y *. Thus, the Solow model predicts that countries with higher rates of saving and investment will have higher levels of capital and income per worker in the long run. slide 48 49

50 Income per person in 1992 (logarithmic scale) 100,000 International Evidence on Investment Rates and Income per Person Canada U.S. Denmark Germany Japan 10,000 Egypt Pakistan Ivory Coast Mexico Brazil Peru U.K. Israel France Italy Finland Singapore 1,000 Chad Uganda India Cameroon Indonesia Zimbabwe Kenya Investment as percentage of output (average ) slide 49 50

51 The Golden Rule: introduction Different values of s lead to different steady states. How do we know which is the best steady state? Economic well-being depends on consumption, so the best steady state has the highest possible value of consumption per person: c * = (1 s) f(k * ) An increase in s leads to higher k * and y *, which may raise c * reduces consumption s share of income (1 s), which may lower c * So, how do we find the s and k * that maximize c *? slide 50 51

52 The Golden Rule Capital Stock * k gold = the Golden Rule level of capital, the steady state value of k that maximizes consumption. To find it, first express c * in terms of k * : c * = y * i * = f(k * ) i * = f(k * ) δk * In general: i = k + δk In the steady state: i * = δk * because k = 0. slide 51 52

53 The Golden Rule Capital Stock Then, graph f(k * ) and δk *, and look for the point where the gap between them is biggest. y = f ( k ) * * gold gold steady state output and depreciation * k gold * c gold i = δk * * gold gold δk * f(k * ) steady-state capital per worker, k * slide 52 53

54 The Golden Rule Capital Stock c * = f(k * ) δk * is biggest where the slope of the production func. equals the slope of the depreciation line: MPK = δ * k gold * c gold δk * f(k * ) steady-state capital per worker, k * slide 53 54

55 Why is MPK = δ the right equilibrium? MPK = the benefit of an extra unit of capital (per worker) δ = the cost of an extra unit of k If MPK > δ then I should be trying to accumulate k If MPK < δ then I should be trying to get rid of some k slide 54 55

56 The transition to the Golden Rule Steady State The economy does NOT have an automatic tendency to move toward the Golden Rule steady state. Achieving the Golden Rule requires that policymakers adjust s. This adjustment leads to a new steady state with higher consumption. But what happens to consumption during the transition to the Golden Rule? slide 55 56

57 Starting with too much capital If k > k * * gold then increasing c * requires a fall in s. In the transition to the Golden Rule, consumption is higher at all points in time. y c i t 0 time slide 56 57

58 What policies could bring down k? Objective: reduce the savings rate (classic example: Japan in the 1990s) Taxation: Increase capital gains tax Not fix double taxation of dividends Govt. Expenditure: crowding out investment with public expenditure slide 57 58

59 Starting with too little capital If k < k * * gold then increasing c * requires an increase in s. Future generations enjoy higher consumption, but the current one experiences an initial drop in consumption. y c i t 0 time slide 58 59

60 What policies could increase k? Here we want to give incentives to savers and investors Responsible fiscal policy: Rein-in Govt. Expenditure to avoid crowding out investment Reduced taxation on savings (increase taxation on consumption) Easier said than done in poor countries slide 59 60

61 Total Factor Productivity Empirical studies a Cobb-Douglas production function with α = 1/3 approximates the data very well Using y = k 1/3, how well does this model explain output per capita in different countries? slide 60 61

62 Disparities in Income per capita Predicted versus Actual GDP per capita (as a fraction of U.S. values) GDPpc Predicted Burundi China Brazil U.K. Italy Japan Country per capita capital stock U.S. slide 61 62

63 Improving the fit with TFP The differences between countries could depend on differences in overall productivity (TFP) E.g.: A Ferrari can go 200 MPH in the Autobahn but only 25MPH in a dirt road Recall: y = Ak 1/3. A is called total factor productivity or the Solow residual Let s allow for differences in A across countries slide 62 63

64 Revisiting Income per capita differences with TFP (w.r.t( w.r.t.. US) Country GDPpc k 1/3 TFP Japan Italy U.K Brazil China Burundi slide 63 64