Lecture 3 - Solow Model

Transcription

1 Lecture 3 - Solow Model EC308 Advanced Macroeconomics 16/02/2016 (EC308) Lecture 3 - Solow Model 16/02/ / 26

2 Introduction Solow Model Sometimes known as Solow-Swan Model: Solow (1956): General Production Function Swan (1956): Cobb-Douglas Antecedent of modern neoclassical growth models In contrast to Harrod-Domar, Solows model is centered on a neoclassical aggregate production function with substitutability between inputs and diminishing returns to each input. A dynamic general equilibrium model even though preferences and dynamic HH optimization are missing Full employment of capital and labor is assumed (EC308) Lecture 3 - Solow Model 16/02/ / 26

3 Technology Technology Neoclassical Production Function Y = F (K, L, A) A: Technology/knowledge (Assumed to be constant for the basic model) (EC308) Lecture 3 - Solow Model 16/02/ / 26

4 Technology Assumptions(that makes F neoclassical) 1 Positive and diminishing marginal products F K, F L > 0 and F KK, F LL < 0 2 F is linearly homogeneous in K and L CRTS in K and L: F (zk, zl, A) = zf (K, L, A) (1) (EC308) Lecture 3 - Solow Model 16/02/ / 26

5 Technology Digression on homogeneity Definition: f(x, y) is homogeneous of degree r ( h.o.d. r ) if f(zx, zy) = z r f(x, y) for all r > 0. Proposition 1: Let f(x, y) be h.o.d. r. Then each of its partial derivatives, i.e. f x, f y, are h.o.d. r 1. Proof: DERIVE! Proposition 2 - Euler s Theorem: f(x, y) is homogeneous of degree r if and only if rf(x, y) = f x x + f y y Proof ( ): DERIVE! (EC308) Lecture 3 - Solow Model 16/02/ / 26

6 Technology Digression on homogeneity Q: What does Euler s Theorem imply for the neoclassical production function? A: Y = F K K + F L L Neoclassical Theory of Distribution Proof of Euler s via Y = F (K, L): DERIVE! (EC308) Lecture 3 - Solow Model 16/02/ / 26

7 Technology Assumptions(that makes F neoclassical) Differentiate (1) w.r.t. K: F K is h.o.d 0 in K and L Differentiate (2) w.r.t. z: zf K (zk, zl, A) = zf K (K, L, A) (2) Do the same for L: F K (zk, zl, A) z = F KK K + F KL L = 0 (3a) F KL K + F KL L = 0 (3b) (EC308) Lecture 3 - Solow Model 16/02/ / 26

8 Technology Assumptions(that makes F neoclassical) (3a) and (3b) imply: i. F KL > 0 ii. F KK F LL = F 2 KL (Derive!) (EC308) Lecture 3 - Solow Model 16/02/ / 26

9 Technology Assumptions(that makes F neoclassical) 3 Inada Conditions: } lim K 0 F K =, lim L 0 F L = shape of the production function! lim K F K = 0, lim L F L = 0 These 3 conditions are necessary for a production function to be called Neoclassical. 4 Essentiality: Each input is essential (implied by conditions (1)-(3), hence not a necessary condition to make it neoclassical) F (0, L) = F (K, 0) = 0 (EC308) Lecture 3 - Solow Model 16/02/ / 26

10 Technology Per Capita Variables Using CRTS property of F: Let z = 1 L and expand F by z: Y L = F (K, 1, A) L Then the production function in per capita form: y = f(k) where k K L and y Y L Note that A is constant for now, just like a parameter of a function (EC308) Lecture 3 - Solow Model 16/02/ / 26

11 Technology Properties of f(k) (Use F (K, L) = Lf(K/L) and derive!) Marginal Products f (k) = F K, F L = f(k) kf (k) Inada conditions lim k 0 f (k) = lim k f (k) = 0 Essentiality f(0)=0 Graph! (EC308) Lecture 3 - Solow Model 16/02/ / 26

12 Rest of the Model Rest of the Model Just like Harrod-Domar: S(t) = sy (t) (4) Y (t) = C(t) + I(t) (5) I(t) = δk(t) + K(t) (6) L(t) L(t) = n L (7) (EC308) Lecture 3 - Solow Model 16/02/ / 26

13 Dynamic Equilibrium Equilibrium S(t) = I(t) (8) Note the similarity with H-D model: K(t) = sy (t) δk(t) (9) K K = s Y K δ (EC308) Lecture 3 - Solow Model 16/02/ / 26

14 Dynamic Equilibrium Fundamental equation of Solow Model Converted to per capita terms, (9) gives us the fundamental equation of the Solow model: (DERIVE!) k = sf(k) (n L + δ)k (10) sf(k): per capita saving per gross investment level: GRAPH 2 nl + δ: Effective depreciation rate of K L (EC308) Lecture 3 - Solow Model 16/02/ / 26

15 Dynamic Equilibrium Fundamental equation of Solow Model First note how Inada conditions imply that there is only one intersection point between s f(k) and (n L + δ) k: Slope of f(k) starts from and continuously decreases as k increases. Since the slope of (n L + δ) k is constant, there can be only one point where two curves intersect. (EC308) Lecture 3 - Solow Model 16/02/ / 26

16 Dynamic Equilibrium Steady State (Long Run Equilibrium) SS: LR Equilibrium where per capita variables are constant. k = 0 gives k = k (SS value of k) Solve for k sf(k ) = (n L + δ)k (EC308) Lecture 3 - Solow Model 16/02/ / 26

17 Dynamic Equilibrium Steady State (Long Run Equilibrium) Note that if k is constant y = f(k ) and c = (1 s)y are also constant. However levels of aggregate variables are growing: (Why? Derive?) k k = 0 K K = L L = n L ẏ y = 0 Ẏ Y = L L = n L ċ c = 0 Ċ C = L L = n L (EC308) Lecture 3 - Solow Model 16/02/ / 26

18 Transition to SS and Stability Transition to Steady State The model has interesting applications about how the economy reaches SS GRAPH! Divide (10) k = sf(k) (n L + δ)k by k: γ k = k k = sf(k) (n L + δ) k (γ usually used to denote growth rate of a variable) (EC308) Lecture 3 - Solow Model 16/02/ / 26

19 Transition to SS and Stability Steady State (Long Run Equilibrium) Note that at SS γ k = 0 by definition, so LR growth is zero. But growth rate during the transition is non-zero. Here we see that γ k > 0 if s f(k) k γ k = 0 if s f(k) k γ k < 0 if s f(k) k > n L + δ = n L + δ < n L + δ (EC308) Lecture 3 - Solow Model 16/02/ / 26

20 Transition to SS and Stability Main Results First main result Therefore, all aggregate variables grow at the same rate: population growth rate, n L Explains why Solow model is considered to be one of the exogenous growth models (EC308) Lecture 3 - Solow Model 16/02/ / 26

21 Transition to SS and Stability Transitional Dynamics Note how we get to SS: GRAPH (EC308) Lecture 3 - Solow Model 16/02/ / 26

22 Transition to SS and Stability Transitional Dynamics Suppose at k 0 < k :sf(k 0 ) > (n L + δ)k 0 k > 0 Economy is saving(investing) more capital than required to compensate for effective depreciation, therefore the amount of per capita capital is increasing. Continues till we reach k At k 1 > k ; the opposite reasoning (EC308) Lecture 3 - Solow Model 16/02/ / 26

23 Transition to SS and Stability Transitional Dynamics So graphically; GRAPH (EC308) Lecture 3 - Solow Model 16/02/ / 26

24 Transition to SS and Stability Transitional Dynamics First lets see mathematically why s f(k) k ] Intuition? f(k) k [ s f(k) k k = s f (k) k is decreasing s k 2 = (f (k) f(k) k ) s k < 0 (f (k) f(k) k ) = F L/k = MP L/k < 0 is the average productivity of k and due to diminishing MP assumption, it decreases as k increases: For low levels of capital,there would be a high AP k, but due to diminishing MP, higher k levels will lead to low AP k (EC308) Lecture 3 - Solow Model 16/02/ / 26

25 Transition to SS and Stability Transitional Dynamics So: Suppose, again, we start from k 0 < k : Graph! What if economy stars from k > k? (EC308) Lecture 3 - Solow Model 16/02/ / 26

26 Transition to SS and Stability Global Stability So, it doesn t matter where the economy starts, it always converges to the k SS, steady state. We call such systems Globally Stable (EC308) Lecture 3 - Solow Model 16/02/ / 26