Kaldor Stylized facts

Transcription

1 Kaldor Stylized facts 1. Y/L grows overt ime 2. K/L grows over time 3. MPK roughly constant (debated, decreasing) over time 4. K/Y roughly constant over time 5. rk/y and wl/y roughly constant over time 6. g of y/l differ substantially across countries Solow Growth Model (Ch. 1 of B/S book) This model will provide a framework to think about determinants of economic growth, convergence behavior, long-run growth, etc.

2 Neoclassical production function: Y = F(K, L, A). A: non-rival, non-excludible technology. One-sector technology: Y goes for C or ΔK (gross investment). K, capital, is cumulated net investment. Can interpret K to include human capital. K depreciates at rate δ > 0 (exogenous). L = labor supply (market clears with full employment). Labor supply = population 2

3 (or proportional to population). No laborleisure choice and labor-force-participation rate is constant. Can extend model to make these endogenous. Population grows exogenously at rate n 0: (1/L) (dl/dt) = n L(t) = L(0) e nt. Can extend model to make fertility, mortality, immigration endogenous then n is endogenous. 3

4 Properties of neoclassical production function (see Ch. 1 of Economic Growth). 1. Constant returns to scale (CRS) in K, L: F(λK, λl, A) = λ F(K, L, A) for all λ > Positive and diminishing marginal products: F K, F L > 0; F KK, F LL < Limiting (Inada) conditions: Lim (F K ) =, Lim (F L ) = ; K 0 L 0 Lim (F K ) = 0, Lim (F L ) = 0; K L These conditions imply each input is essential: 4

5 F(0, L) = F(K, 0) = 0. CRS, with λ = 1/L, implies F(K/L, 1, A) = F(K, L, A) (1/L) = Y/L. Y/L depends on K/L, not scale of inputs. An example of a neoclassical production function is the Cobb-Douglas one: Y = AK α L 1-α, A > 0; 0 < α < 1. If factors paid marginal products, factor shares of income are constant at α for K and 1-α for L. 5

6 Basic model assumes exogenous technological progress A depends only on time, t: Y = F(K, L, t). Common assumption to get nice steady-state results, when n = (1/L) (dl/dt) is constant, is that technical progress takes laboraugmenting form. Y depends on K and effective labor, Lˆ = L φ(t): Y = F(K, Lˆ ). If technical change takes place at constant rate x 0, 6

7 Lˆ = Le xt, Lˆ grows at rate n + x. Technology might instead augment K (Solow) or F(K, L) overall (Hicks). If production function is Cobb-Douglas, the three forms are indistinguishable. Standard model has labor-augmenting technical change at constant rate, x. CRS property (with λ = 1/ Lˆ ) implies Y/ Lˆ = F[K/ Lˆ, 1]. Define 7

8 ŷ Y/ Lˆ, kˆ K/ Lˆ. Then ŷ = f( kˆ ). Output per unit of effective labor depends only on capital per unit of effective labor. Can show: Y/ K = f ( kˆ ), Y/ L = [f( kˆ ) kˆ f ( kˆ )] e xt. Solow equation 8

9 Change in capital stock is dk/dt = I δk, where I is gross investment. Assume closed economy, so that net investment, dk/dt, = S, net saving. Let s be net saving rate; ratio of S to net income or net domestic product, Y - δk. Then dk/dt = s (Y δk). Convenient to focus on kˆ : (1/ kˆ ) (d kˆ /dt) = (1/K) (dk/dt) (n+x). Substituting for dk/dt, (1/ kˆ ) (d kˆ /dt) = s (Y/K δ) (n+x). 9

10 Y/K is average product of capital, which equals f( kˆ )/ kˆ. Therefore, (1/ kˆ ) (d kˆ /dt) = s f( kˆ )/ kˆ - (n+x+sδ). In level form, after multiplying by kˆ : d kˆ /dt = s f( kˆ ) (n+x+sδ) kˆ. Solow equation holds whether s is constant or variable. Basic model has s exogenous and constant (0 < s < 1). Ramsey model (Ch. 2 of Economic Growth) makes s endogenous household 10

11 choice. Cannot usually write solution as closed form, s = s( kˆ ). Solow diagram. Steady state is at kˆ *, where d kˆ */dt = 0. Note stability. kˆ * satisfies s f( kˆ *) = (x+n+sδ) kˆ *. An increase in s raises kˆ *. An increase in f( ) raises kˆ *. An increase in x, n, or δ lowers kˆ *. 11

12 (x+n+sδ) kˆ f( kˆ ) s f( kˆ ) kˆ (0) kˆ * kˆ Solow diagram in level form 12

13 In steady state, kˆ is constant and ŷ is constant. Since Lˆ grows at rate x+n, K grows in steady state at rate x+n. Since K and Lˆ grow at same rate, CRS implies Y grows at rate x+n. Per capita quantities k K/L and y Y/L grow in steady state at rate x. Long-run per capita growth rates are independent of s, n, δ, and f( ). Growth rate in transition Growth rate of kˆ is (1/ kˆ ) (d kˆ /dt) = s f( kˆ )/ kˆ - (x+n+sδ). 13

14 See Solow diagram in growth-rate form. Key is diminishing average product of capital, f( kˆ )/ kˆ, which comes from neoclassical properties. Important transition property is that a rise in kˆ reduces (1/ kˆ ) (d kˆ /dt). Transition can change if s, n, f( ) not constant. For example, rising s can offset tendency for growth rate to fall. Same with falling n and rising f( ) (perhaps due to improving institutions). 14

15 (1/ kˆ) (d kˆ /dt) x+n+sδ kˆ (0) kˆ* s f( kˆ )/ kˆ kˆ Solow diagram in growth-rate form 15

16 For given kˆ, an increase in s raises (1/ kˆ ) (d kˆ /dt) on impact. Same with a rise in f( ). A rise in n or δ lowers (1/ kˆ ) (d kˆ /dt). A rise in x lowers (1/ kˆ ) (d kˆ /dt) but leaves unchanged the growth rate of k. The growth rate of ŷ is (1/ ŷ ) (d ŷ /dt) = [ kˆ f ( kˆ )/f( kˆ )] [(1/ kˆ ) (d kˆ /dt)] = α [(1/ kˆ ) (d kˆ /dt)]. 16

17 If α constant (Cobb-Douglas case), effects of s, f( ), n, δ work by affecting (1/ kˆ ) (d kˆ /dt). An increase in x lowers (1/ ŷ ) (d ŷ /dt) by α therefore, raises growth rate of y on impact. Summary implications. NGM implies that growth rate takes the form: (1/ ŷ ) (d ŷ /dt) = Φ( ŷ / ŷ *). (-) Growth rate of y adds x. This form for growth rate of y is the one used in crosscountry empirical work. ŷ * in Solow 17

18 model depends on s, n, and f( ). Institutions (tax systems, corruption, religious values, etc.) might affect productivity, f( ). Other variables affect convergence speed of ŷ toward ŷ *. For example, if we allow for human capital, h. ĥ (0)/ kˆ (0) high generates fast growth of ŷ toward ŷ * in some models. Recovery from wartime destruction? Long-term growth rate in Solow model depends only on x, which is exogenous. 18

19 Simplest approach to long-term endogenous growth is to eliminate diminishing returns to K, at least asymptotically. AK model is Y = AK (not a neoclassical production function). Can then dispense with exogenous technical progress, x = 0. Figure shows Solow growth diagram with AK production function. Growth rate of k is constant. Positive if A > n+sδ. 19

20 Other models have AK property asymptotically but convergence behavior in transition. (1/k) (dk/dt) sa n+sδ k AK Growth Model 20

21 Main thrust of endogenous growth models is to have theories that explain rate of technological progress x in the Solow model. Idea of conditional convergence. Lower real gdp per person (y) implies faster per capita growth rate, conditional on factors (individual characteristics, policies, institutions) that determine long-run position, y*: Dy/y = F(y, y*) (-) (+) 21

22 y* includes things like saving rate, population growth rate, quality of governance (e.g. rule of law), regulatory policies, international openness, macroeconomic policy, nature of educational and health institutions, etc. Are these exogenous? See Figures on convergence relationships across countries and U.S. states. 22

23 .08 growth rate, real GDP per person in 1960 Growth Rate versus Level of Real GDP per Person for a Broad Group of Countries The horizontal axis shows real GDP per person in 1960 in 2000 U.S. dollars on a proportionate scale for 112 countries. The vertical axis shows the growth rate of real GDP per person for each country from 1960 to The red line is the straight line that provides a best fit to the relation between the growth rate of real GDP per person (the variable on the vertical axis) and the level of real GDP per person (on the horizontal axis). Although this line slopes upward, the slope is in a statistical sense negligibly different from zero. Hence, the growth rate is virtually unrelated to the level of real GDP per person. Thus, this broad group of countries does not display convergence. 23

24 .045 growth rate, POR GRE SPA IRE ITA AUS BEL FRA NLD NOR ICE UK CAN LUX US DEN SWE SWZ real GDP per person in 1960 Growth Rate versus Level of Real GDP per Person for OECD Countries The horizontal axis shows real GDP per person in 1960 in 2000 U.S. dollars on a proportionate scale for 18 of the 20 founding members of the OECD (excluding Germany and Turkey). 24

25 Growth Rate of Personal Income Per Person, NC SC FL VA GA TN TX AL WV AR MS KY NM MO KS UT ME IN LA WI NE IA MI MD MN NH VT OH SD DE IL NJCT PA WA MA ND OR NY RI ID WY CO CA AZ MT NV Personal Income Per Person in 1880 Growth Rate versus Level of Income per Person for U.S. States, The horizontal axis shows real personal income per person in U.S. dollars on a proportionate scale for 47 U.S. states. The two-letter abbreviation identifies the state. (Alaska, the District of Columbia, Hawaii, and Oklahoma are excluded.) The vertical axis shows the growth rate of real personal income per person for each state from 1880 to The solid line is the straight line that provides a best fit to the relation between the growth rate of income per person (the variable on the vertical axis) and the level of income per person (on the horizontal axis). The line has a clear negative slope therefore, a lower level of income per person in 1880 matches up with a higher growth rate of income per person from 1880 to Thus, the U.S. states exhibit convergence. 25

26 growth rate real GDP per person Growth Rate versus Level of Real GDP per Person: Conditional Convergence for a Broad Group of Countries The horizontal axis shows real GDP per person in 2000 U.S. dollars on a proportionate scale. The data are for 71 countries in 1965, 85 countries in 1975, and 82 countries in (The sample was based on the availability of data 86 countries appear at least once.) The vertical axis shows the corresponding growth rates of real GDP per person for , , and Each of the growth rates filters out (and, therefore, holds constant) the estimated effects of the variables discussed in the text. The red line is the straight line that provides a best fit to the relation between the growth rate of real GDP per person (the variable on the vertical axis) and the level of real GDP per person (on the horizontal axis). The line has a clear negative slope. Therefore, once we hold constant the other variables, a lower level of real GDP per person matches up with a higher growth rate of real GDP per person. This relation is called conditional convergence. 26