Question $1$: Consumption and the Permanent Income Hypothesis Model $(80$ points$).$

Consider the problem $of$ the representative consumer who lives for two periods. $In$ periods $t$ and

$t+1,$ the individual receives income $(an$ endowment$)Y_t$ and $Y_{t+1},$ respectively. The individual

can borrow $or$ save $at$ the market interest rate, $r.$ The individual also pays lump$-$sum taxes $to$ the

government: $T_t$ and $T_{t+1}.$ The individual only derives utility from consuming the consumption

basket, $C,$ and the utility function takes $on$ the following form:

subject $to$ the constraints:

$U(C_t,C_{t+1})=ln(C_t)+ln(C_{t+1})$

$C_t+S=Y_t-T_t$

$C_{t+1}=Y_{t+1}-T_{t+1}+(1+r)S$

where $Ct$ denotes consumption $in$ period $t,S$ denotes savings, and $is$ the discount factor.

Combining the two budget constraints, $we$ derived the single inter$-$temporal budget constraint:

$C_t+\frac{C_{t+1}}{1+r}=Y_t-T_t+\frac{Y_{t+1}}{1+r}-\frac{T_{t+1}}{1+r}(1+r)=MRSct,ct+1$