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Unformatted text preview: Notes on Graduate Macroeconomics: Part 2. Basic Neoclassical Model Yongsung Chang January 27, 2011 1 Basic NeoClassical Model: Prescott, 1986; King, Plosser, Rebelo 1988 1.1 Economic Environment 1.1.1 Preferences: Household Economy is populated by many identical infinitelylived 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 constantreturnsto 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 firstorder 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.
 Fall '07
 Chang
 Macroeconomics

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