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Answer on Macroeconomics Question for Yuki

Question #40980
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["]
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