Consider a two period intertemporal consumption $$ labor model. Suppose that the representative consumer $$ s lifetime utility function is given by vc l c l c l c l $(,,,)$ ln$()$ ln$()11221122=+++.$

Assume that the representative consumer begins period $1$ with zero assets. The period$1,$ period$-2,$ and lifetime budget constraints in this model, expressed in real terms, are

thus given, respectively, by

ca tw l

c a ra t w l

$11111$

$221222$

$11$

$111$

$+=$

$+=++$

$()(),$

$()()(),$

c

c

r tw l tw l

r $1$

$2$

$111$

$222$

$1$

$1111$

$1$

$+$

$+$

$=+$

$+$

$()()()().$

The tax rates in the two periods are t t $12==05.,$ and the real wages in the two periods

are w$1=20$ and w$2=22.($Note: You are not given a numerical value for the real interest rate; you will solve for this in part b below.$)$

a$.$ Solve for the representative consumer $$ s optimal choices of consumption in each

of the two periods $($i$.$e$.,$ solve numerically for the optimal c$1$ and optimal c$2).$ Be

clear about any important steps and arguments in your logic$/$computations$.($Note:

If you can solve without setting up and solving a Lagrangian, you may do so$.)$

b$.$ Using your solution in part a above, solve for the numerical value of the real interest rate r $.$ Show any important steps in your logic$/$computation$.($Note: If you were

unable to fully solve part a$,$ you can still work through this part by correctly and

fully describing how you would compute r as if you had fully solved part a$.)$

c$.$ Can you solve numerically for the optimal choices of leisure in this model? If

so$,$ do so$,$ showing any important steps in your logic$/$computation$.$ If not, briefly

describe the economic and$/$or mathematical issue$($s$)$ in this model that prevents

you from doing so$.$