Question $5(15$ points$)$ Suppose Adam Smith's utility from consumption in periods $1$ and $2$ is given by $\backslash ($ u$\backslash$left$($c$\_\{1\},$ c$\_\{2\}\backslash$right$)=$c$\_\{1\}$ c$\_\{2\}\backslash ),$ and his endowment of income in the two periods is $\backslash ((\backslash$mathrm$\{$m$\}1,\backslash$mathrm$\{$~m$\}2)=(200,100)\backslash ).$ There is a positive interest rate $\backslash (\backslash$mathrm$\{$r$\}>0\backslash )$ at which consumers can borrow$/$save$,$ and there is no inflation. Assume Adam maximizes his utility subject to his budget.

a$)(3$ points$)$ Write Adam's intertemporal budget constraint.

b$)(3$ points$)$ Solve for Adam's optimal consumption bundle as a function of the interest rate r $.$

c$)(2$ points$)$ Suppose the interest rate is $\backslash (\backslash$mathrm$\{$r$\}=0\mathrm{.}25\backslash ).$ Find Adam's optimal bundle.

d$)(3$ points$)$ Suppose instead we are considering a lower interest rate $\backslash (\backslash$mathrm$\{$r Is it possible to set the interest rate such that Adam chooses to be a borrower in period $1?$ If so$,$ what is the rate? If not, explain why

e$)(4$ points$)$ Now, suppose there is a credit constraint for Adam and he can at most borrow $\backslash (\backslash$$ $30\backslash )$ in the first period. Draw his budget constraint in the graph and clearly label the necessary points on the axes.