EC 205 Macroeconomics I

Transcription

1 EC 205 Macroeconomics I

2 Macroeconomics I Chapter 8 & 9: Economic Growth

3 Why growth matters In 2000, real GDP per capita in the United States was more than fifty times that in Ethiopia. Over the period , real GDP per capita in China grew at a rate of 7.6% annually, while, in Argentina, real GDP per capita grew at a rate of only 0.1%. 76 times slower! From 1960 to 2000, the fastest growing country in the world was Taiwan, which grew at 6%. The slowest growing country was Zambia which grew at - 1.8%. That is, people in Zambia were markedly worse off in 2000 than they were in 1960.

4 Why growth matters The theory of economic growth seeks to address these issues and provide explanations. Is there some action a government of India could take that would lead the Indian economy to grow like Indonesia's or Egypt's? If so, what exactly? If not, what is it about the nature of India that makes it so? The consequences for human welfare involved in questions like these are simply staggering: Once one starts to think about them, it is hard to think about anything else. - Robert Lucas Jr., Nobel Laureate (1995)

5 links to prepared Gapminder.org notes: circle size is proportional to population size, color of circle indicates continent, press play on bottom to see the cross section graph evolve over time click here for one-page instruction guide Income per capita and Life expectancy Infant mortality Malaria deaths per 100,000 Adult literacy Cell phone users per 100 people

6 Why growth matters Anything that effects the long-run rate of economic growth even by a tiny amount will have huge effects on living standards in the long run. annual growth rate of income per capita 25 years 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%

7 Why growth matters If the annual growth rate of U.S. real GDP per capita had been just one-tenth of one percent higher from , the average person would have earned $2,782 more during the decade. If the annual growth rate of Turkish real GDP had been 1% higher between , Turkey would have generated an additional ~1000TL of income per person measured in 1998 prices during the period.

8 The lessons of growth theory can make a positive difference in the lives of hundreds of millions of people. These lessons help us understand why poor countries are poor design policies that can help them grow learn how our own growth rate is affected by shocks and our government s policies

9 Kaldor s Stylized Facts In 1957, Nicholas Kaldor documented some key facts on economic growth by empirical investigation. Kaldor did not claim that any of the variables he examines would be constant at all times; they tend to be constant when averaging the data over long periods of time.

10 Kaldor s Stylized Facts Six key facts: Output per worker grows at a roughly constant rate that does not diminish over time. Capital per worker grows over time. The rate of return to capital is constant. The capital/output ratio is roughly constant. The share of capital and labor in net income are nearly constant. Growth rate of output per worker differs substantially across countries.

11 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

12 The Solow Model

13 How Solow model is different from Chapter 3 s model 1. K is no longer fixed (exogenous), it is endogenized: investment causes it to grow, depreciation causes it to shrink & whole discussion is over the behavior of the evolution of the capital stock 2. L is no longer fixed: population growth causes it to grow 3. the consumption function is simpler

14 How Solow model is different from Chapter 3 s model 4. no G or T (only to simplify presentation; we can still do fiscal policy experiments) 5. cosmetic differences

15 Representative Firm Notation: Y denotes output, L denotes labor, K denotes capital, and A denotes a constant measure of productivity (also known as the Solow-residual) Solow model assumes a country s output is produced by a representative neoclassical production function. A function is called a neoclassical production function iff it satisfies the following four properties: 1. Constant returns to scale (CRS or CRTS) 2. Positive and diminishing marginal products of capital and labor 3. Essentiality of inputs 4. Inada (limiting conditions)

16 Constant Returns to Scale Let the production function be of the form Y=F(K,L) A function satisfies CRTS in inputs if for all λ>0 F(λK,λL)=λF(K,L) If both inputs double, so does output/production Does the Cobb-Douglas production function satisfy CRTS? Y=F(K,L)=AK α L 1-α

17 Positive & Diminishing Marginal Products A production function F(K,L) displays positive and diminishing marginal products in capital and labor iff F K (K,L) > 0 & F L (K,L) > 0 F KK (K,L) < 0 & F LL (K,L) < 0 Additional labor and capital increase production, only at a decreasing rate Both the first and the second units of labor increase production, but the first additional unit of labor contributes more than the second for the same constant level of capital Does the Cobb-Douglas production function satisfy positive and diminishing marginal products?

18 Essentiality of Inputs A production function F(K,L) satisfies essentiality of inputs property iff F (K,0) = 0 F (0,L) = 0 Both inputs are essential for production Does the Cobb-Douglas production function satisfy positive and diminishing marginal products?

19 Inada Conditions A production function F(K,L) satisfies Inada conditions iff & & limf K (K,L) = K 0 limf K (K,L) = 0 K lim L 0 F L (K,L) = limf L L (K,L) = 0 In the absence of capital or labor, the incremental increase of the absent input boosts output substantially (from zero to some positive amount) As we increase one of the inputs so much while keeping the other constant, after a point (infinite units to be exact) the additional unit of the vastly abundant input does not increase production at all. Does the Cobb-Douglas production function satisfy positive and diminishing marginal products?

20 Solow Model Assumptions Output is produced with a neoclassical production function We are interested in not aggregate but per-capita measures, since the latter is a better conventional measure of standard of living, hence our focus. We assume that everybody works, hence the number of workers is equal to the number of citizens of a country. Therefore L is both the number of workers, AND the number of citizens, Thus y=y/l is the income/output per capita. L t = L t+1 =... = L Population is constant so that

21 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)

22 Properties of the Per-Capita Production Function Y df (k) K = F(K,L) = K dk = f '(k) MPK of F( ) is equal to marginal product of per-capita capital f '(k) > 0& f ''(k) < 0 As per-capita capital increases, so does per-capita output, yet only at a diminishing rate f (0) = 0 lim k 0 Capital is essential in production f '(k) = &limf '(k) = 0 k Initial per-capita capital increase boosts production from zero units to a positive amount, and when there is too much percapita capital, additional units of capital do not contribute

23 The production function Output per worker, y f(k) (with A 2 >A 1 ) f(k) (with A 1 ) 1 MPK = f(k +1) f(k) f(k) (with A 3 <A 1 ) Capital per worker, k

24 The Basics of the Solow Model Households own the factor of production K t & L t, hence the national output Y t =F(K t, L t ) Households consume a fraction of income and save the rest s = the saving rate, the fraction of income that is saved (s is an exogenous parameter) (1-s) as marginal propensity to consume Y t =F(K t, L t )=C t + S t

25 The Rule of Motion for Capital Capital depreciates at a constant rate δ so that in period t, δk t units of capital becomes useless (1-δ) K t carried over to the next period, t+1 Savings are used to finance investment and investment contributes to the stock of capital Next period s capital increases by I t =S t =sy t =sf(k t, L t ) The connection between K t & K t+1 then K t+1 =(1-δ) K t +I t =(1-δ) K t + sy t =(1-δ) K t + sf(k t, L t ) K t+1 =(1-δ) K t + sf(k t, L t ) is the rule of motion for capital

26 The Rule of Motion for Per-Capita Capital K t+1 =(1-δ) K t + sf(k t, L t ) Divide both sides by L t K t+1 /L t =(1-δ) K t /L t + sf(k t,l t ) /L t Rearranging yields K t+1 /L t =(1-δ) k t + sf(k t ) Since L t = L t+1 =... = L k t+1 =(1-δ) k t + sf(k t ) we can rewrite this expression as k t+1 =(1-δ) k t + sf(k t ) is the rule of motion for per-capita capital

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

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

29 The Steady State Alternative way of writing the rule of motion Δk t+1 = s f(k t ) δk t where Δk t+1 =k t+1 k t If investment is just enough to cover depreciation [sf(k) = δk ], then capital per worker will remain constant, i.e. Δk = 0 or k t =k t+1 = = k * This one value of k, denoted k * is called the steady state capital stock.

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

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

32 Moving toward the steady state Investment and depreciation Δk = sf(k) - δ k δk sf(k) Δk k 1 k 2 k * Capital per worker, k

33 Moving toward the steady state Investment and depreciation Δk = sf(k) - δ k δk sf(k) investment Δk depreciation k 2 k * Capital per worker, k

34 Moving toward the steady state Investment and depreciation Δk = sf(k) - δ k δk sf(k) Δk k 2 k * Capital per worker, k

35 Moving toward the steady state Investment and depreciation Δk = sf(k) - δ k δk sf(k) Δk k k * 2 k 3 Capital per worker, k

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

37 A numerical example Production function (aggregate): Y t = F (K t,l t ) = K t L t = K t 1/2 L t 1/2 Converting into-per capita terms y t = f (k t ) = k t 1/2

38 A numerical example, cont. Assume: s = 0.3 δ = 0.1 initial value of k = 4.0

39 Approaching the steady state: A numerical example Year k y c i δ k Δ k

40 Example: Solve for the steady state Continue to assume s = 0.3, = 0.1, and y = k 1/2 Imposing the steady state condition on the rule of motion for per-capita-capital to solve for the steadystate values of k, y, and c. 40

41 Example: Solve for the steady state s f ( k *) 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 *

42 An increase in the saving rate An increase in the saving rate raises investment causing k to grow toward a new steady state: Investment and depreciation δk s 2 f(k) s 1 f(k) * k 1 * k 2 k

43 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.

44 International evidence on investment rates and income per person Income per person in 2009 (log scale) 100,000 10,000 1, Investment as percentage of output (average )

45 The Golden Rule: Introduction Different values of s lead to different steady states. How do we know which is the best steady state? The best steady state has the highest possible consumption per person: c* = (1 s) f(k*). An increase in s leads to higher k* and y*, which raises c* reduces consumption s share of income (1 s), which lowers c*. So, how do we find the s and k* that maximize c*?

46 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 * ) - δ k * In the steady state: i * = δ k * because Δk = 0.

47 The Golden Rule capital stock Then, graph f(k * ) and δk *, 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 k * * gold gold δ k * f(k * ) steady-state capital per worker, k *

48 The Golden Rule capital stock c * = f(k * ) - δ k * is biggest where the slope of the production function equals the slope of the depreciation line: MPK = δ * k gold * c gold δ k * f(k * ) steady-state capital per worker, k *

49 The transition to the Golden Rule steady state The economy does NOT have a 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?

50 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

51 Starting with too little capital * * If k kgold 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

52 Population growth Assume the population and labor force grow at rate n (exogenous): L L n L t+1 =(1+n) L t, L t+2 =(1+n) L t+1, EX: Suppose L 1 = 1,000 in year 1 and the population is growing at 2% per year (n = 0.02). Then ΔL 2 = n L 1 = ,000 = 20, so L 2 = 1,020 in year 2.

53 Solow Model with Population Growth K t+1 =(1-δ) K t + sf(k t, L t ) Divide both sides by L t K t+1 /L t =(1-δ) K t /L t + sf(k t,l t ) /L t Rearranging yields K t+1 /L t =(1-δ) k t + sf(k t ) Since expression as L t = L t+1 (1+ n) for any t we can rewrite this k t+1 (1+n) =(1-δ) k t + sf(k t )

54 Solow Model with Population Growth We cannot use the former diagram to look at the effects of population growth on the steady state variables Instead we ll define a draw a diagram with k t & k t+1 on the axis Let k t+1 = g(k t ) = sf (k t ) + (1- d )k t 1+ n

55 Solow Model with Population Growth k t+1 k * 4 k 3 k 2 Steady-State g(k t ) = sf (k t ) + (1- d )k t 1+ n 45 k 1 k 2 k 3 k * k t

56 Solow Model with Population Growth k t+1 n 2 > n 1 k * 1 k * 2 g(k t ) = sf (k t ) + (1- d )k t 1+ n 1 g(k t ) = sf (k t ) + (1- d )k t 1+ n 2 45 k * 2 k * 1 k t

57 Solow Model with Population Growth k t+1 k * k * s 2,d 1 s 1,d 1 k * s 1,d 2 s 2 > s 1 &d 2 > d 1 g(k t ) = s 2 f (k t ) + (1- d 1 )k t 1+ n g(k t ) = s 1 f (k t ) + (1- d 1 )k t 1+ n g '(k t ) = s 1 f (k t ) + (1- d 2 )k t 1+ n 45 k * s 1,d 2 k * s 1,d 1 k * s 2,d 1 k t

58 Prediction: Higher n lower k*. And since y = f(k), lower k* lower y*. Thus, the Solow model predicts that countries with higher population growth rates will have lower levels of capital and income per worker in the long run.

59 International evidence on population growth and income per person Income per person in 2009 (log scale) 100,000 10,000 1, Population growth (percent per year, average )

60 The Golden Rule with population growth To find the Golden Rule capital stock, express c * in terms of k *, first rewrite impose the steady state restrictions on the rule of motion: k* = sf (k*) + (1- d )k * 1+ n which yields (n + d )k* = sf (k*) Since c * = f(k * ) - sf(k * )= f(k * )-(n+δ)k* c * is maximized when f (k * )-(n+δ)=0 or MPK = δ + n

61 Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. If true, then the income gap between rich & poor countries would shrink over time, causing living standards to converge. (Also known as Absolute Convergence ) In real world, many poor countries do NOT grow faster than rich ones. Does this mean the Solow model fails?

62 Growth empirics: Convergence Solow model predicts that, other things equal, poor countries (with lower Y/L and K/L) should grow faster than rich ones. No, because other things aren t equal. In samples of countries with similar savings & pop. growth rates, income gaps shrink about 2% per year. In larger samples, after controlling for differences in saving, pop. growth, and human capital, incomes converge by about 2% per year.

63 Growth empirics: Convergence What the Solow model really predicts is conditional convergence - countries converge to their own steady states, which are determined by saving, population growth, and education. This prediction seems to be true in the real world.

64 Growth empirics: Convergence Not controlling for other things: Controlling for other things:

65 Growth empirics: Production efficiency and free trade Since Adam Smith, economists have argued that free trade can increase production efficiency and living standards. Research by Sachs & Warner: Average annual growth rates, developed nations developing nations open 2.3% 4.5% closed 0.7% 0.7%

66 Growth empirics: Production efficiency and free trade To determine causation, Frankel and Romer exploit geographic differences among countries: Some nations trade less because they are farther from other nations, or landlocked. Such geographical differences are correlated with trade but not with other determinants of income. Hence, they can be used to isolate the impact of trade on income. Findings: increasing trade/gdp by 2% causes GDP per capita to rise 1%, other things equal.

67 Some Strengths and Weaknesses of the Solow Model The Solow-model predicts that Countries converge to their steady-state levels of percapita capital and output in the long-run, and if they have different saving rates, depreciation rates, or population growth rates, or productivity levels, the steady-states may differ. Poor countries (countries with lower levels of per-capita capital and output) grow faster. Countries with high saving rates, low depreciation rates, high productivity, and low population growth reach higher levels of steady states. Solow model somewhat successfully addresses the some of the stylized facts a-la Kaldor.

68 Solow Model & Data

69 Some Strengths and Weaknesses of the Solow Model The model has clear weaknesses, as well Naïve modeling of consumption limits the reliability of predictions It does not explain why different countries have different saving and productivity rates It focuses on investment and capital, while the much more important factor of TFP is still unexplained The model does not provide a theory of sustained long-run economic growth The role of institutions, cultures, resources, are not ignored The dissatisfaction gave rise to endogenous growth models

70 Alternative perspectives on population growth The Malthusian Model (1798) Predicts population growth will outstrip the Earth s ability to produce food, leading to the impoverishment of humanity. Since Malthus, world population has increased sixfold, yet living standards are higher than ever. Malthus neglected the effects of technological progress.

71 Chapter Summary 1. The Solow growth model shows that, in the long run, a country s standard of living depends: positively on its saving rate negatively on its population growth rate 2. An increase in the saving rate leads to: higher output in the long run faster growth temporarily but not faster steady-state growth

72 3. Chapter Summary If the economy has more capital than the Golden Rule level, then reducing saving will increase consumption at all points in time, making all generations better off. If the economy has less capital than the Golden Rule level, then increasing saving will increase consumption for future generations, but reduce consumption for the present generation.