Intermediate Macroeconomics, EC2201. L1: Economic growth I

Transcription

1 Intermediate Macroeconomics, EC2201 L1: Economic growth I Anna Seim Department of Economics, Stockholm University Spring / 44

2 Contents and literature Growth facts. Production. Literature: Jones (2014), Ch Klein (2016a), Durlauf et al. (2005). 2 / 44

3 It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts. -Sherlock Holmes (A Scandal in Bohemia, 1891). 3 / 44

4 Extracted from: Jones (2014). 4 / 44

5 Growth over the very long run Sustained growth a recent phenomenon. From 1700 onwards per-capita income has increased by a factor of almost 90 in the richest countries of the world. Countries have started to grow at different points in time. 5 / 44

6 Figure 3.1 reveals large cross-country differences: per-capita GDP in Ethiopia about 1/40 of that of the US. Striking given that standards of living differed by no more than a factor of 2 or 3 prior to The great divergence: the emergence of large cross-country differences in living standards from 1700 onwards. 6 / 44

7 Extracted from: Jones (2014). 7 / 44

8 Extracted from: Jones (2014). 8 / 44

9 Growth mid-1800 s onwards Substantial increases in GDP per capita in industrialised countries over the last 140 years. In terms of GDP per capita, the gap between the richest and the poorest countries has grown substantially. The richest countries of the world seem to approach parallel growth paths from the mid 1900s onwards while others lag behind (convergence clubs). 9 / 44

10 Extracted from: Acemoglu, D. (2007), Introduction to Modern Economic Growth, Princeton University Press. 10 / 44

11 Extracted from: Durlauf et al. (2005). 11 / 44

12 Growth Some major economies have maintained or improved their position relative to the US (e.g. France, Italy, Spain and Japan). In the poorer group, some have improved (Korea, India, China) while others have fallen further behind (Nigeria, Ethiopia). Great heterogeneity: growth miracles and growth disasters abundant. 12 / 44

13 Extracted from: Durlauf et al. (2005). The table lists the top 15 in a ranking of growth performance / 44

14 Extracted from: Durlauf et al. (2005). The table lists the bottom 15 in a ranking of growth performance / 44

15 Extracted from: Jones (2014). 15 / 44

16 Mathematical preliminaries: the natural logarithm Recall that lnx is the natural logarithm of x. By definition: x = e a a = lnx. Properties: ln(xy) = lnx + lny. ln(x/y) = lnx lny. lnx α = α lnx, where α is a constant. 16 / 44

17 Differentiation: notation Let y = f (x). The derivative of y with respect to x is written: dy dx = df dx = f (x). When y is a function of time, t, this derivative is often written ẏ(t) dy dt. (1) 17 / 44

18 Differentiation: notation Let y = f (x 1,x 2 ). The derivatives of y with respect to its arguments, x 1 and x 2, are written: y = f = f x x 1 x 1. 1 y = f = f x x 2 x / 44

19 Differentiating a composite function Let y = f (g) and g = g(x), so that y = f (g(x)). This implies: dy dx = df dg dg dx. (2) 19 / 44

20 Differentiating a logarithmic function The derivative of the ln-function is given by: d(lny) dy In particular, if y = y(t), (2) and (3) imply: d(lny(t)) dt = 1 y. (3) = ẏ(t) y(t). (4) 20 / 44

21 Differentiating a polynomial Consider a polynomial, y = x α. The derivative is given by: Examples: d dx d ( 1 dx x dy dx = αx (α 1). ( x + 0.5x 2 x 4) = 1 + x 4x 3. ) = d ( x 1) = x 2 = 1 dx x 2 21 / 44

22 Growth in discrete time Consider a variable y. In discrete time, the change in y over the interval [t,t + 1] is given by: y t = y t+1 y t. (5) The discrete-time growth rate of y measures the percentage change in y over the interval [t,t + 1] and is obtained as g y y/ t y t = y t+1 y t y t. (6) 22 / 44

23 Equation (6) suggests: y t+1 = (1 + g y )y t, t. Suppose that we start at t = 0, with y 0, and that there are 3 periods: y 1 = (1 + g y )y 0 y 2 = (1 + g y )y 1 y 3 = (1 + g y )y 2 23 / 44

24 Substituting recursively: The constant growth rule y 3 = (1 + g y )y 2 The constant growth rule: = (1 + g y )(1 + g y )y 1 }{{} y 2 = (1 + g y )(1 + g y )(1 + g y )y 0 }{{} y 1 = (1 + g y ) 3 y 0. y t = (1 + g y ) t y / 44

25 Growth in continuous time In continuous time, we study the change in y as t 0, i.e. compute the derivative of y with respect to t: dy dt = lim y(t + t) y(t) t 0 t (7) Using the notation (1), recall that (7) is often written ẏ(t). 25 / 44

26 The continuous-time growth rate of y is defined: Finally, note that (4) implies: γ y ẏ(t) y(t). (8) γ y = d lny(t). (9) dt 26 / 44

27 A model of production Notation: Y : output. K: capital input. L: labour input. A: Total Factor Productivity (TFP). α: the capital share. 27 / 44

28 A model of production Consider the following production function: Y = F (K,L). (10) The marginal product of capital, MPK: MPK = Y K = F(K,L) = F K. K The marginal product of labour, MPL: MPL = Y L = F(K,L) = F L. L 28 / 44

29 Extracted from: Jones (2014). 29 / 44

30 The Cobb-Douglas production function The Cobb-Douglas production function: where α (0,1). Y = AK α L 1 α, (11) Homogenous of degree 1 (constant returns to scale): F (κk,κl) = A(κK) α (κl) 1 α = Aκ (α+1 α) K α L 1 α = κf (K,L). 30 / 44

31 Profit maximisation under Cobb-Douglas A representative firm faces: max Π = AK α L 1 α K,L }{{} rk wl F (K,L) where Π denotes profits, w the wage and r the rental rate of capital. First-order conditions (FOCs) for profit maximisation: Π K = αak } α 1 {{ L 1 α } r = 0, (12) MPK Π L = (1 α)ak α L α w = 0. (13) }{{} MPL 31 / 44

32 The demand for capital and labour Re-arranging (12): Re-arranging (13): ( ) K (1 α) MPK = αa = α Y L K = r (14) ( ) K α MPL = (1 α)a = (1 α) Y L L = w. (15) 32 / 44

33 Note that (14) and (15) imply: and α = rk Y (1 α) = wl Y. Jones (2014) assumes that α = 1/3. Why? 33 / 44

34 Extracted from: Jones (2014). 34 / 44

35 Extracted from: Jones (2014). 35 / 44

36 Income per capita Can the model explain why some countries are so much richer than others? Cross-country comparisons require expressions in per-capita terms. Divide (11) by L: where k K/L. y Y L = AK α L 1 α L = AK α L α = Ak α, 36 / 44

37 Extracted from: Jones (2014). 37 / 44

38 Extracted from: Jones (2014). 38 / 44

39 Extracted from: Jones (2014). 39 / 44

40 Extracted from: Jones (2014). 40 / 44

41 Extracted from: Jones (2014). 41 / 44

42 Why does TFP differ across countries? The accounting exercise in Jones (2014) suggests that A is three times as important as k in explaining cross-country differences in y. Why does TFP differ across countries? Human capital. Technology. Institutions. Misallocation. 42 / 44

43 Extracted from: Jones (2014). 43 / 44

44 What we did Growth facts. Production. Literature: Jones (2014), Ch Klein (2016a), Durlauf et al. (2005). 44 / 44