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Advanced Macro (Subject) / VL8 (Lesson)

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• cake-eating problem formulation X: size of the cake Ct: Consumption in period t - consumption constraint is sum(Ct)=X Ut(Ct):period utility, when UC>0 and UCC<0 (1-ρ): discount rate
• Utility maximizing funciton max U(C1) + (1-δ)U(C2)+(1-δ)2U(C3)  s.t. C1+C2+C3=X
• Lagrangian of the cake problem L = U(C1) + (1-ρ)U(C2)+(1-ρ)2U(C3) + λ ( X-C1-C2-C3) FOC: UC1= λ (1-ρ)UC2=λ (1-ρ)2UC3=λ X=C1+C2+C3
• What follows for consumption follows for more than 3 C Ct=( X(1-ρ)t-1 )/( SUMt=1 bis T (1-ρ)t-1) when T →∞ Ct=ρX(1-ρ)t-1
• What does the Euler-Equation ensure Euler-Equation comes from FOC and is: U'(Ct)=βV't+1(Xt+1) The Euler-Equation ensures that the allocation of consumption over time is optimal
• Stochastic formulation The size of the cake in period 2 is uncertain With probability p the cake size is reduced by γ, with 1-p it is increased by γ We assume that γ is small relative to the cake size and that the consumption smoothing motive ensures that the next-period cake size is positive Stochastic Belmann function: Vt(Xt)=max U(Ct)+β[E{Vt+1(Xt+1)}] where E{.} is an expectations operator E{V2(U2)}=p*ln(X2L)+(1-p)ln(X2H) X2L=X1-C1-γ X2H=X1-C1+γ
• 2 period capital model kt+1=f(kt)-ct+(1-δ)kt kt+1≥0 yt=f(kt) k0=k_

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