Rec3_Macro.pdf

Rec3_Macro.pdf - Macroeconomic Theory and Analysis(Spring...

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Macroeconomic Theory and Analysis, (Spring 2018) Recitation 3 James Graham Peter Paz New York University Economics (GSAS) Two period competitive equilibrium with variable labor supply Here we briefly describe a two-period competitive equilibrium model with variable labor supply. We follow the model in the lecture notes, with the addition of labor supply choices in each period. Otherwise, we keep the same household and firm structure as before. Household decisions The household chooses consumption and labor in each period, and chooses savings in period 1. The household earns wages on its labor supply, and since it owns the firms also receives firm profits. Household savings earn a gross return R . The household problem is: max C 1 ,C 2 ,N 1 ,N 2 ,S h C 1 - σ 1 1 - σ - b N 1 - ' 1 1 - ' + β C 1 - σ 2 1 - σ - b N 1 - ' 2 1 - ' ! s.t. C 1 = 1 + N 1 W 1 - S h C 2 = 2 + N 2 W 2 + RS h We can substitute the two budget constraints into the utility function, leaving us with an uncon- strained problem in 3 variables: labor choices and savings: max N 1 ,N 2 ,S h ( 1 + N 1 W 1 - S h ) 1 - σ 1 - σ - b N 1 - ' 1 1 - ' + β ( 2 + N 2 W 2 + RS h ) 1 - σ 1 - σ - b N 1 - ' 2 1 - ' ! The three first order conditions are: S h : C - σ 1 = β RC - σ 2 N 1 : bN ' 1 = W 1 C - σ 1 N 2 : bN ' 2 = W 2 C - σ 2 1
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Rewriting, we have: R = C - σ 1 β C - σ 2 (0.1) W 1 = bN ' 1 C - σ 1 (0.2) W 2 = bN ' 2 C - σ 2 (0.3) where ( 0.1 ) sets the interest rate equal to the Inter-temporal marginal rate of substitution (i.e. between consumption in period 1 and 2); and ( 0.2 ) and ( 0.3 ) set wages equal to the Intra-temporal marginal rate of substitution (i.e. between consumption and labor within the same period).
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