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where the marginal utility of the composite good MC ⋆ t and marginal utility of the continuation utility next period MUt+1 are MC ⋆ t = (1 − β)U 1 ψ MUt+1 = βU 1 ψ t Et θ −1 t (C ⋆ t ) 1−γ � U 1−γ t+1 � 1 θ −1 U −γ t+1 In equilibrium, the pricing kernel Mt+1 is equal to the intertemporal marginal rate of substitution: Mt+1 = ∂Ut ∂Ct+1 Furthermore the marginal utility of consumption tomorrow and today can be expressed as: where ∂Ut ∂Ct+1 ∂Ut ∂Ct ∂Ut ∂Ct = MUt+1 · MC ⋆ t+1 · ∂C⋆ t+1 = MC ⋆ t · ∂C⋆ t ∂Ct ∂C ⋆ t ∂Ct Plugging (2),(3), and (7) into (5) and (6), we get: ∂Ut ∂Ct+1 ∂Ut ∂Ct Plugging (8) and (9) into (4), we get: = β(1 − β)U 1 � ψ t Et = C⋆ t Ct U 1−γ t+1 � 1 θ −1 = (1 − β)U 1 ψ t (C ⋆ t ) 1−γ θ −1 Ct � ⋆ Ct+1 Mt+1 = β 7.3 Derivation of Demand Schedule C ⋆ t � 1−γ θ Ct Ct+1 ∂Ct+1 U 1 ψ −γ � � 1−γ ⋆ θ −1 t+1 Ct+1 Ct+1 � 1−γ Ut+1 Et[U 1−γ t+1 ] � 1 1− θ The final goods firm solves the following profit maximization problem �� 1 max Pt {Xi,t} i∈[0,1] 0 X ν−1 ν i,t di � ν ν−1 − 31 � 1 0 Pi,tXi,t di (2) (3) (4) (5) (6) (7) (8) (9)
where Pt is the nominal price for the final good, Xi,t is intermediate good input i ∈ [0, 1], and Pi,t is the nominal price for the intermediate good. The first-order condition with respect to Xi,t is which can be rewritten as Pt �� 1 0 X ν−1 ν i,t di � ν ν−1 −1 Pi,t = Pt � Xi,t Yt − 1 ν Xi,t − Pi,t = 0 � − 1 ν Using the expression above, for any two intermediate goods i, j ∈ [0, 1], Xi,t = Xj,t � Pi,t Since markets are perfectly competitive in the final goods sector, the zero profit condition must hold: Substituting (11) into (12) gives PtYt = � 1 0 Xi,t = PtYt Substitute (10) into (13) to obtain the price index Pt = �� 1 0 Pj,t � −ν (10) (11) Pi,tXi,t di (12) P −ν i,t � 1 0 P1−ν j,t dj P 1−ν j,t dj � 1 1−ν Given the expression for the price index, (13) can be rewritten as Xi,t = Yt � Pi,t Pt Since each intermediate goods firm is atomistic, their actions will not effect Yt nor Pt. Thus, each of these � −ν firms will face an isoelastic demand curve with price elasticity ν. 32 (13)
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