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Econ117_Top-3

Econ117_Top-3 - Ec 117 GROWTH THEORY Foster UCSD TOPIC 3...

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Ec 117 – GROWTH THEORY LECTURE NOTES Foster, UCSD 1-Jul-09 TOPIC 3 – HARROD-DOMAR MODEL A. Production Functions and Returns to Scale 1. Introduction: a) The Harrod-Domar and Solow-Swan growth models incorporate production and well as consumption, so we will need a macroeconomic production function. b) Aggregate production function: Q = F(K, L). 1) K and L are capital and labor employed. 2) Q = total output (GDP). 3) Technology reflected in algebraic form of F(●). c) Average and marginal factor products. 2. Homogeneity and Returns to Scale: a) For some λ ≥ 0, let Q 0 = F(K 0 , L 0 ) and Q 1 = F(K 1 , L 1 ) = F(λK 0 , λL 0 ). b) A production function is said to be “homogeneous of degree h ” if: Q 1 ≡ λ h Q 0 That is, F(λK 0 , λL 0 ) ≡ λ h F(K 0 , L 0 ) Note the identity (≡). c) If h = 1, the production function is “homogeneous of degree 1 ” and is said to exhibit constant returns to scale (CRS). 1) If it is CRS, then F(λK 0 , λL 0 ) ≡ λ F(K 0 , L 0 ). 2) For λ = 2, this means that doubling all inputs (K and L) will result in exact doubling of output Q. 3. Properties of CRS Production Functions: a) If Q = F(K, L) is CRS, then average and marginal products are “scale invariant” ( i.e. , “homogeneous of degree zero”). 1) Average product proof (divide thru identity by λK and by λL): Notation (Production Functions) Q Output, GDP L Labor force employed K Capital stock employed ) , ( ; ) , ( ; L K F K Q MPK K Q APL L K F L Q MPL L Q APL K L = = = = = = 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 ) , ( ) , ( ) , ( APL L Q L L K F L L K F L L K F L Q APL = = = = = λ λ λ λ λ

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Ec 117 HARROD-DOMAR p. 2 2) Marginal product proof (differentiate thru identity wrt K and wrt L): b) Euler’s Theorem. 1) If F(K, L) is CRS, then: F(K, L) = F K (K, L) K + F L (K, L) L = MPK K + MPL L. 2) Proof (differentiate thru identity wrt λ): c) Input elasticities add up to 1. 1) If F(K, L) is CRS, then: η(Q, K) + η(Q, L) ≡ 1 2) Proof (divide thru Euler Theorem identity by Q): 0 1 0 0 0 0 1 0 0 0 0 0 ) , ( ) ( ) ( ) , ( MPL MPL MPL L L K F MPL dL L d L L K F = = = λ λ λ λ λ λ λ [ ] . 1 ) , ( ) , ( ) , ( : ) , ( ) , ( ) ( ) ( ) , ( ) ( ) ( ) , ( ) , ( 0 0 proof the complete to at Evaluate L K F L L K F K L K F is That L K F L K F d L d L L K F d K d K L K F L K F L K = = + = + = λ λ λ λ λ λ π λ λ λ λ λ λ λ λ λ λ λ λ λ 1 ) , ( ) , ( ) , ( = 2245 + = + = + Q L K F L Q K Q Q L L Q Q K K Q Q L F Q K F L K η η
Ec 117 HARROD-DOMAR p. 3 B. Some Common Production Functions 1. Fixed Proportions Production Function: a) Q p = F(K, L) = min {σK, βL}, σ > 0 and β > 0 are constants. b) Isoquant map. Isoquants are right-angled lines. At vertex, σK = βL, and along line of vertices, K = βL/σ and L/L = β/σ. Above this line, Q= βL. Below it, Q = σK. c) Marginal products.

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