$(2)[24$ points$]$ Consider a discrete$-$time variation $of$ the Ramsey model with $no$ population growth. The

representative household maximizes the lifetime utility,

$U={\displaystyle \underset{i=0}{\overset{}{}}}(\frac{1}{1+}{)}^{i}u(c_{t+i},l_{t+i})$

where $is$ the subjective discount rate $(>0).$ The instantaneous utility function $is$ given $by$

$u(c_{t+i},l_{t+i})=ln{c}_{t+i}-\frac{l_{t+i}^{1+}}{1+},$AAi$=0,1,2,$cdots,$,$

where $>0.$ The instantaneous utility function has two arguments. The first term refers $to$

consumption per member $of$ household, $c,$ and the second $is$ the amount each worker works, $l.$

Finally, the household's lifetime budget constraint $is$

${\displaystyle \underset{i=0}{\overset{}{}}}(\frac{1}{1+r}{)}^i{c}_{t+i}={\displaystyle \underset{i=0}{\overset{}{}}}(\frac{1}{1+r}{)}^i{w}_{t_i}{l}_{t+i}$

where $ris$ the real interest rate $(constant)$ and $w_{t+i}is$ the real wage $in$ time $t+$i for $i=0,1,2,$cdots,$.$

Consider two$-$period problem $(i=0,1)$

$(a)[6$ points$]$ Set $up$ the Lagrangian $of$ the household for the two$-$period problem, and find the first$-$order conditions.

$(b)[8$ points$]$ Show how the relative labor supply $in$ the two periods depends $onr.In$ particular, explain why $affects$ the responsiveness $of$ labor supply $to$ the real interest rate.

$(c)[8$ points$]$ Now suppose that this household supplies one unit $of$ labor $in$ the first period and does not work $in$ the second period. That $is,`t=1$ and $`t+1=0.$ Find optimal consumption $in$ period $t,c*t,$ and determine whether the household increases $or$ decreases the optimal consumption $at$ time $t$ when $r$ rises. Justify why your answer economically makes sense.