# Setup and solve utility maximization

A consumer wants to maximize his utility function $U(X_1,X_2)=Min(X_1,X_2)$. The price of $X_1$ is 2 and the price of $X_2$ is 4 and his income is 40. Setup the utility maximization problem and solve for $(X_1,X_2)$.

## My attempt

$U(X_1,X_2)=MIN(X_1,X_2)$

s.t $2X_1+4X_2=40$

$MRS= \frac{MU_{x2}}{MU_{x1}}=\frac{P2}{P1}$

I know that $P_2=4$ and $P_1=2$ but I am not sure where to get the partial derivatives for $MU_{x2}$ and $MU_{x1}$.

• Plot some indifference curves, and the budget line. Then try and see how to reach the highest possible indifference curve in the budget. This video might be helpful: youtube.com/watch?v=S4v03C39jAI – Amit Feb 10 '17 at 7:51

Instead of trying to find the MRS, consider the optimality condition to be $X_1 = X_2$, along with the budget constraint. Think of the equality I just gave as replacing the condition on the MRS.
• Okso $x_1 = x_2 = 20/3 ?$ and that's it? – combo student Feb 10 '17 at 2:41