A consumer lives for two periods (the present and the future). His income in period $1$ is $100$. His income in period $2$ is $200$. Prices of the single consumption good are $\$1$ per unit in both periods. The interest rate at which he can lend or borrower is $20\%$ per period. Consunption in the present and the future periods is $C_{1}$ and $C_{2}$. The utility function is $U(C_{1}, C_{2}) = C_{1}C_{2}$.
$1$. Solve the optimization problem of the consumer (non round number of units is okay, if that's the solution.
$2$. Is the consumer a borrower or a lender? Is the result intuitive to you?
I am confused on how to solve for $C_{1}$ and $C_{2}$ for $\#1$. I tried solving for $C_{1}$ and got $\$16,666.67$. I did $100(200 / 1.2)$. Is this correct? Also, the consumer is a borrower correct? Any and all help would be greatly appreciated. Thanks in advance.
In general the savings are income minus expenses: $s=y-e$. The expenses in period 1 and 2 are $1\cdot c_1$ and $1\cdot c_2$, where 1 is the price of the good. Over the two periods the savings has to be $0$. We only can compare/sum the two periods if we discount the second period by the factor $q=1+i=1.2$
The saving in period 1 is $s_1=100-c_1$. The discounted saving in period 2 is $s_2=\frac{200-c_2}{1.2}$.
Thus we maximize $U(c_1,c_2)=c_1\cdot c_2$ under the condition $100-c_1+\frac{200-c_2}{1.2}=0$
This can be done by applying the method of Lagrange multipliers.
$$L(c_1,c_2,\lambda)=c_1\cdot c_2+\lambda\left(100-c_1+\frac{200-c_2}{1.2}\right)$$
$$\frac{\partial L}{\partial c_1}=c_2-\lambda=0\Rightarrow c_2=\lambda \qquad (1)$$
$$\frac{\partial L}{\partial c_2}=c_1-\frac{\lambda}{1.2}=0\Rightarrow c_1=\frac{\lambda}{1.2} \qquad (2)$$
$$\frac{\partial L}{\partial \lambda}=100-c_1+\frac{200-c_2}{1.2}=0\qquad (3)$$
Dividing the first equation (1) by the second equation (2).
$\frac{c_2}{c_1}=1.2\Rightarrow c_2=1.2\cdot c_1$
Inserting the expression for $c_2$ in (3).
$100-c_1+\frac{200-1.2\cdot c_1}{1.2}=0$
This equation can be solved for $c_1$. You will see that $c_1=\$\frac{400}3=\$ 133\frac13$. And therefore $c_2=\$ 160$.