<p>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).</p>

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2017-02-10T02:41:09.893Z
Okso $x_1 = x_2 = 20/3 ?$ and that's it?
combo student
CC BY-SA 3.0
2017-02-10T02:31:03.760Z
2017-02-10T02:31:03.760Z
Theoretical Economist
<p>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 $(X_1,X_2)$.</p>

<h2>My attempt</h2>

<p>$U(X_1,X_2)=MIN(X_1,X_2)$</p>

<p>s.t $2X_1+4X_2=40$</p>

<p>$MRS= \frac{MU_{x2}}{MU_{x1}}=\frac{P2}{P1}$</p>

<p>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 $MU_{x1}$.</p>

CC BY-SA 3.0
2017-02-10T07:51:51.887Z
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:   https://www.youtube.com/watch?v=S4v03C39jAI
Amit
<macroeconomics><utility>
Setup and solve utility maximization
combo student
# Setup and solve utility maximization

(Asked by: username_0 on 2017-02-10)

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 $MU\_{x1}$.

**Tags:** macroeconomics, utility

> **Comments:**
> **username_1**: 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:   https://www.youtube.com/watch?v=S4v03C39jAI  

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## 1 answers

### Answer by username_2 on 2017-02-10. Score: 2

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.

> **Comments:**
> **username_0**: Okso $x_1 = x_2 = 20/3 ?$ and that's it?  

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