Having trouble with how to put this together. I have an answer key, but the individual steps I am struggling with.
Two period economy with a representative consumer that maximizes the utility function $$ U(C_1,C_2)=\ln(C_1)+\beta\ln(C_2) $$
Subject to the lifetime budget constraint $$ \frac{C_1+C_2}{(1+r)}=W $$ Where $0<\beta<1$ and $C_1$ and $C_2$ are consumption levels in period 1 and 2 respectively. $r$ is the real interest rate and total wealth $W$ is the sum of housing wealth H and the present discounted value of after tax life time income.
Derive the levels of optimal consumption in the two periods as a function of $W$. Provide economic intuition for the optimality condition.
In the solution, we have: Let $R=(1+r)$. Maximize $U(C_1,C_2)$ subject to budget constrain. Substitute $C_2$ as a function of $C_1$ into the utility function. Take the FOC, derive the optimality condition $$ U'(C_1)=\beta RU'(C_2) $$
Can anyone step this out for me? Giving me some pointers on how to start the problem would be ideal, so I can work through the rest on my own, and perhaps ask for further assistance if I need it.
You have the wrong life time budget constraint, it should be $$ \mathbf{C_1+\dfrac{C_2}{R}=W}\Rightarrow C_2=RW-RC_1$$
So the utility is $u(C_1)+\beta u(RW-RC_1)$. Take derivatives with respect $C_1$ to get the FOC.
Please do not use $U$ for the life-time utility and for the period utility, it will only confuse you more: $$U(C_1,C_2)=u(C_1)+\beta u(C_2)\text{ where } u(x)=\ln(x).$$