Answer to Question #40980 in Macroeconomics for Yuki

Question

Let Ct denote the individual's consumption of nondurables on date t, and let Dt be the stock of durable goods the consumer owns at date t. A stock of durables yields its owner a proportional service flow each period it owned. Income process is deterministic. The representative consumer has a perfect foresight about his or her income process and maximizes.
U1= Σβt[θlogCt + (1 − θ)logDt] (∞　t=1). subject to Dt = (1−δ)Dt−1 +X and At = (1+r)(At−1 +Yt −Xt −Ct)
where A0 and D0 is initially given. Here At is a financial wealth at the end of date t, Yt is date t income and r is a market interest.
1) What are the choice variables in this maximization problem? Derive first order conditions.
2)Using FOC, show that
θ/Ct=(1-θ)/Dt+β(1−δ)(1-θ)/Dt+1+β^2(1−δ)^2(1-θ)/Dt+2+・・・ and interpret it.
3)Using FOC, show that
(1-θ)Ct/θDt=1-(1−δ)/(1+r)
4)Assume β(1+r)=1 and let ι ≡ 1-(1−δ)/(1+r), show that the optimal level of C1 and D1 are given by
C1=θr/(1+r)[(1+r)A0+(1−δ)D0+Σ(1/(1+r))^t-1Yt]
D1=(1-θ)r/(1+r)ι["]

Answer 1

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