Question $2$:

Consider an economy which lasts for three periods, $t=1,2$ and $3.$ The nominal flow budget constraint of the representative household populating this economy is given by$,$

$\frac{A_t}{1+R_t}=A_{t-1}+Y_t-C_t$

where $A_t$ are the nominal bonds purchased in at time t$,R_t$ is the nominal rate of interest, $Y_t$ is the income of the household and $C_t$ what they spend on consumption. Since the economy ends in period $4,A_3=0,$ as the household has no incentive to save beyond their lifetime and no$-$one will lend to them expecting repayment in periods $t4.$ There is no uncertainty.

$($a$)$Derive the household's intertemporal budget constraint and re$-$write it in real terms using lower case letters to denote real variables obtained by dividing by the price level, e$.$g$.c_t=\frac{C_t}{P_t},$ and defining the real interest rate as$,1+r_t=\frac{1+R_t}{1+{}_{t+1}}$ where the rate of inflation is given by $1+{}_t=\frac{P_t}{P_{t-1}}.$ Explain what it means.

$($b$)$ The household's utility function is given by$,$

$U={\displaystyle \underset{t=1}{\overset{3}{}}}{}^{t-1}ln(c_t)$

where $0<1$ is the household's discount factor. Derive the household's consumption Euler equation. Explain what it means.

$($c$)$ Use the consumption Euler equation and the intertemporal budget constraint to derive the household's consumption function.

$($d$)$Assume $=1,r_t=0\%,{}_t=2\%,$ for $t=1,2$ and $3,y_1=y_2=0,y_3=3$ and $A_0=0.$ What is the household's optimal consumption$/$savings plan?

$($e$)$Now suppose the real interest rate is $5\%$ in every period, but all other parameters are as described in $($d$).$ What is the household $\text{'}$s optimal consumption$/$savings plan now? Give an intuitive explanation as to how your answer to part $($e$)$ differs from part $($d$).$