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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 $MU_{x1}$.

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  • $\begingroup$ 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 $\endgroup$ – Amit Feb 10 '17 at 7:51
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The MRS = price ratio condition works only if the utility function is differentiable, and the solution is interior (the consumer consumes strictly positive amounts of all goods).

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.

To see where the above mentioned equality comes from, consider the case where it is not satisfied, and convince yourself that there must be a way to reallocate consumption in a way that strictly increases utility.

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  • $\begingroup$ Okso $x_1 = x_2 = 20/3 ?$ and that's it? $\endgroup$ – combo student Feb 10 '17 at 2:41

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