graduate_macro_notes2

graduate_macro_notes2 - Notes on Graduate Macroeconomics...

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Unformatted text preview: Notes on Graduate Macroeconomics: Part 2. Basic Neoclassical Model Yongsung Chang January 27, 2011 1 Basic Neo-Classical Model: Prescott, 1986; King, Plosser, Rebelo 1988 1.1 Economic Environment 1.1.1 Preferences: Household Economy is populated by many identical infinitely-lived individuals (worker or household) with preferences over goods and leisure: A worker maximizes his utility over consumption C t and leisure L t : max C t ,I t ,N t U = E ∞ X t =0 β t u ( C t , 1- N t ) , β < 1 (1) subject to C t + I t = W t N t + R t K t . (2) K t +1 = (1- δ ) K t + I t . (3) Households rent capital which yields the rate of return R t and supply labor N t at wage rate W t . Capital depreciates at the rate δ . 1.1.2 Technology: Firm The representative firm produces output according to a constant-returns-to- scale technology in capital and labor Y t = A t F ( K t ,N t X t ) 1 where A t is the level of total factor productivity and X t is labor augmenting (permanent) technological progress. Firm maximizes the profit by renting capital and hiring labor from competitive factor markets. π t = max K d t ,N d t Y t- R t K d t- W t N d t 1.1.3 Market Clearing • Goods market clears Y t = C t + I t • Labor market clears N t = N d t • Capital market clears K t = K d t 1.2 Competitive Equilibrium and Social Planner’s Problem Based on the First and Second Welfare Theorems, the competitive equi- librium is equivalent to the solution of the social planner’s problem. The Lagrangian associated with this planning problem is ˆL = E ‰ ∞ X t =0 β t u ( C t , 1- N t )+ ∞ X t =0 β t Λ t [ A t F ( K t ,N t X t )+(1- δ ) K t- K t +1- C t ] . (4) where K is given and Λ t is the Lagrange multiplier attached to the t period resource constraint. The first-order conditions are: u C ( C t , 1- N t ) = Λ t (5) u L ( C t , 1- N t ) = Λ t A t F N ( K t ,N t X t ) X t (6) E t [ β Λ t +1 [ A t +1 F K ( K t +1 ,N t +1 X t +1 ) + (1- δ )]] = Λ t (7) A t F ( K t ,N t X t ) = C t + K t +1- (1- δ ) K t . (8) for t = 0 , 1 , 2 ,...., ∞ and the transversality condition lim t →∞ β t Λ t K t +1 = 0 1.3 Steady State Most industrialized economies exhibit a steady growth in per capita output. Following Kaldor’s observation, we restrict our economy to exhibit a “bal- anced growth path” along the steady state. This imposes some restriction on the functional forms of technology and preferences. 2 1.3.1 Restriction on Technology Swan (1963) and Phelps (1966) show that permanent technical change must be expressible in a labor augmenting form. Suppose we write production function as Y t = A t F ( X Kt K t ,X Nt N t ), where X Kt represents capital aug- menting technical progress and X Nt represents labor augmenting technical progress: X Kt +1 /X Kt = γ XK , X Nt +1 /X Nt = γ XN . One of the Kaldor’s stylized fact requires the rate of return to capital constant. This implies that r t + δ = MPK t = AF K ( X Kt K t ,X Nt N t ) X Kt = AF K (1 ,Z t ) X Kt is con-...
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This note was uploaded on 09/06/2011 for the course ECO 476 taught by Professor Chang during the Fall '07 term at Rochester.

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graduate_macro_notes2 - Notes on Graduate Macroeconomics...

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