$4.$ Consider the consumption$-$savings problem for a two period asset choice problem. The preferences of the consumer are $\backslash ($ u$\backslash$left$($c$\_\{1\}\backslash$right$)+\backslash$beta u$\backslash$left$($c$\_\{2\}\backslash$right$)\backslash )$ for $\backslash (\backslash$beta $\backslash$in$(0,1)\backslash )$ where $\backslash ($ c$\_\{1\}\backslash )$ is consumption and $\backslash ($ c$\_\{2\}\backslash )$ is consumption tomorrow. Assume utility is $\backslash ($ u$($c$)\backslash )$ is strictly increasing, strictly concave, and continuously differentiable. There is a single asset $\backslash ($ b $\backslash )$ that pays rate of return $\backslash ($ r $\backslash )$ on principle $($so if you invest $\backslash ($ b $\backslash )$ today, you get $\backslash ($ b$(1+$r$)\backslash )$ tomorrow. Also, the consumer gets wage income of $\backslash ($ w$\_\{1\}>0\backslash )$ today and $\backslash ($ w$\_\{2\}\backslash$geq $0\backslash )$ tomorrow.

a$.$ Write down the Marshallian demand problem for a consumer using "sequential budget constraits".

b$.$ Now, rewrite the Marshallian demand problem as a "single budget constraint" following ideas in class.

c$.$ What are the prices in this "single budget constraint" case.

d$.$ Draw a picture of the Marshallian choice problem for consumption today and tomorrow. Explain if given endowments, and given preferences, your Marshallian Consumer is a saver or a borrower.

e$.$ Now, can you develop a "Hicksian" demand problem for this consumer when the budget constraint can be represented by a single budget constraint.

f$.$ Discuss the duality between the two representations of the consumer's choice problem.