Notes on Econ 102 Sect. 2, Fall 2005

Transcription

1 Notes on Econ 102 Sect. 2, Fall 2005 José-Víctor Ríos-Rull,, University of Pennsylvania Chapter 4 Growth Accounting Evolved from notes written by Jesús Fernández-Villaverde October 3,

2 4 Growth Accounting Output is produced by inputs capital K and labor L, in combination with the available technology A Want: decompose the growth rate of output into the growth rate of capital input, the growth rate of labor input and technological progress. This exercise is called growth accounting. Why? 54

3 Aggregate production function Maps inputs into output: Y = F (A, K, L) A is called total factor productivity (TFP). Cobb-Douglas example: Y = A K α L 1 α Interpretation. 55

4 Discrete vs. Continuous Time In discrete time a variable is indexed by time: x t. In continuous time a variable is a function of time: x(t). We observe the world only in discrete time... but it is often much easier to work with continuous time! 56

5 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t 57

6 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t and that 1 + g x (t 1, t) = x t 57

7 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t and that 1 + g x (t 1, t) = x t Take logs log [1 + g x (t 1, t)] = log ( xt ) 57

8 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t and that 1 + g x (t 1, t) = x t Take logs log [1 + g x (t 1, t)] = log ( xt ) Taylor series expansion of log (1 + y) around y = 0: log (1 + y) y=0 = ln y + higher order terms y 1! 57

9 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t and that 1 + g x (t 1, t) = x t Take logs log [1 + g x (t 1, t)] = log ( xt ) Taylor series expansion of log (1 + y) around y = 0: log (1 + y) y=0 = ln y + higher order terms y 1! Then: log [1 + g x (t 1, t)] g x (t 1, t) log ( ) xt g x (t 1, t) log x t log = log x t 57

10 From Growth Rates and Logarithms Remember that g x (t 1, t) = x t and that 1 + g x (t 1, t) = x t ( ) Take logs log [1 + g x (t 1, t)] = log xt Taylor series expansion of log (1 + y) around y = 0: log (1 + y) y=0 = ln y + higher order terms y 1! Then: log [1 + g x (t 1, t)] g x (t 1, t) log ( xt ) g x (t 1, t) log x t log = log x t Remember from calculus that validity of Taylor series expansion is local: g small! 57

11 Moving between Continuous and Discrete Time Let x(t) be a variable that depends of t. We write x(t) dx(t). 58

12 Moving between Continuous and Discrete Time Let x(t) be a variable that depends of t. We write x(t) dx(t). Time derivative: x(t) = lim t 0 x(t + t) x(t) t 58

13 Moving between Continuous and Discrete Time Let x(t) be a variable that depends of t. We write x(t) dx(t). Time derivative: x(t) = lim t 0 x(t + t) x(t) t Take logs d log[x(t)] = x(t) x(t) = g x(t) 58

14 Moving between Continuous and Discrete Time Let x(t) be a variable that depends of t. We write x(t) dx(t). Time derivative: x(t) = lim t 0 x(t + t) x(t) t Take logs d log[x(t)] = x(t) x(t) = g x(t) Then: x(t) x(t) = lim t 0 x(t+ t) x(t) t x(t) 58

15 Moving between Continuous and Discrete Time Let x(t) be a variable that depends of t. We write x(t) dx(t). Time derivative: x(t) = lim t 0 x(t + t) x(t) t Take logs d log[x(t)] = x(t) x(t) = g x(t) Then: x(t) x(t) = lim t 0 x(t+ t) x(t) t x(t) When t is small (let s say a year): g x (t) = x(t) x(t) x(t + 1) x(t) x(t) = g x (t 1, t) log x t 58

16 Growth Rates of Ratios: Suppose k(t) = K(t) L(t). What is g k(t)? 59

17 Growth Rates of Ratios: Suppose k(t) = K(t) L(t). What is g k(t)? 1 Take logs log(k(t)) = log(k(t)) log(l(t)) 59

18 Growth Rates of Ratios: Suppose k(t) = K(t) L(t). What is g k(t)? 1 Take logs log(k(t)) = log(k(t)) log(l(t)) 2 Differentiate with respect to time d log((k(t)) k(t) k(t) = d log(k(t)) = K(t) K(t) L(t) L(t) d log(l(t)) g k (t) = g K (t) g L (t) 59

19 Growth Rates of Ratios: Suppose k(t) = K(t) L(t). What is g k(t)? 1 Take logs log(k(t)) = log(k(t)) log(l(t)) 2 Differentiate with respect to time d log((k(t)) k(t) k(t) = d log(k(t)) = K(t) K(t) L(t) L(t) d log(l(t)) g k (t) = g K (t) g L (t) 3 Growth rate of a ratio = the difference of the growth rates: g k (t) = g K (t) g L (t) 59

20 Growth Rates of Ratios: Suppose k(t) = K(t) L(t). What is g k(t)? 1 Take logs log(k(t)) = log(k(t)) log(l(t)) 2 Differentiate with respect to time d log((k(t)) k(t) k(t) = d log(k(t)) = K(t) K(t) L(t) L(t) g k (t) = g K (t) g L (t) d log(l(t)) 3 Growth rate of a ratio = the difference of the growth rates: g k (t) = g K (t) g L (t) 4 A constant ratio that both variables grow at same rate: g k (t) = 0 g K (t) = g L (t) 59

21 Growth Rates of Weighted Products. Suppose Y (t) = K(t) α L(t) 1 α What is g Y (t)? [1] Take logs log(y (t)) = α log(k(t)) + (1 α) log(l(t)) [2] Differentiate d log(y (t)) Ẏ (t) Y (t) = α d log(k(t)) = α K(t) K(t) + (1 α) d log(l(t)) + (1 α) L(t) L(t) g Y (t) = α g K (t) + (1 α) g L (t) Growth rate = weighted sum, (weights equal to share parameters) 60

22 Growth Accounting Observations in discrete time. Production Function: Y (t) = F (A(t), K(t), L(t)) Differentiating with respect to time and dividing by Y (t) Ẏ (t) Y (t) = F AA(t) Ȧ(t) Y (t) A(t) + F kk(t) Y (t) K(t) K(t) + F LL(t) Y (t) L(t) L(t) Useful benchmark: Cobb-Douglas Y (t) = A(t)K(t) α L(t) 1 α. Why? (factor shares) Taking logs and differentiating with respect to time gives g Y (t) = g A (t) + αg K (t) + (1 α)g L (t) g A is called TFP growth or multifactor productivity growth. 61

23 The Cobb-Douglas Production Function and its properties Under competition factors of production are paid their marginal productivities. w = F (K, L) L = (1 α)ak α L α r = F (K, L) K = α A K α 1 L 1 α Labor and capital shares of output in any year t are W t L t = (1 α)a tkt α L α t Y t Y t L t = 1 α r t K t Y t = α A t Kt α 1 Y t L 1 α t K t = α Cobb-Douglas production functions are consistent with the Kaldor facts. 62

24 Doing the Accounting Pick α (capital share). Measure g Y, g K and g L from the data. g A is the residual g A (t) = g Y (t) α g K (t) (1 α) g L (t) Therefore g A is also called the Solow residual. Severe problems if missmeasurement (g K is hard to measure). 63

25 Data for the US We pick α = 1 3 Per. g Y αg K (1 α)g L TFP (g A ) (32%) 0.2 (8%) 1.4 (56%) (30%) 0.2 (6%) 2.1 (64%) (47%) 0.3 (20%) 0.6 (33%) (32%) 0.3 (12%) 1.4 (56%) Key observation: Productivity Slowdown in the 70 s Note: the late 90 s look much better 64

26 Reasons for the Productivity Slowdown 1. Sharp increases in the price of oil in 70 s 2. Structural changes: more services and less and less manufacturing goods produced 3. Slowdown in resources spent on R&D in the late 60 s. 4. TFP was abnormally high in the 50 s and 60 s 5. Information technology (IT) revolution in the 70 s 65

27 Growth Accounting for Other Countries One key question: was fast growth in East Asian growth miracles mostly due to technological progress or mostly due to capital accumulation? Why is this an important question? Country Per. g Y α αg K (1 α)g L g A Germany % 8% 49% Italy % 3% 48% UK % 4% 52% Argentina % 26% 31% Brazil % 20% 29% Chile % 26% 40% Mexico % 23% 36% Japan % 14% 29% Hong Kong % 28% 30% Singapore % 31% 4% South Korea % 42% 12% Taiwan % 40% 20% 66