<p>Rather than solving the tangency condition for <span class="math-container">$x_1$</span> and substituting into the budget constraint, it is much easier to solve the tangency condition for <span class="math-container">$x_2$</span> and substitute into the budget constraint.</p>
<p>As you have found, the tangency condition requires that</p>
<p><span class="math-container">$$x_2=\frac{p_1x_1(x_1+1)}{p_2(2x_1+1)}$$</span></p>
<p>Substituting into the budget constraint gives</p>
<p><span class="math-container">$$p_1x_1+p_2\frac{p_1x_1(x_1+1)}{p_2(2x_1+1)}=W.$$</span></p>
<p>Multiplying through by <span class="math-container">$(2x_1+1)$</span> gives</p>
<p><span class="math-container">$$p_1x_1(2x_1+1)+p_1x_1(x_1+1)=(2x_1+1)W$$</span></p>
<p>or</p>
<p><span class="math-container">$$p_1x_1(3x_1+2)=(2x_1+1)W$$</span></p>
<p>Now just rearrange to get a quadratic equation in <span class="math-container">$x_1$</span> and solve using the quadratic formula.</p>
<hr />
<p>Note that you should also argue why it is not  optimal for <span class="math-container">$x_1$</span> or <span class="math-container">$x_2$</span> to be zero (i.e. why a corner solution is not optimal).</p>

CC BY-SA 4.0
2023-09-08T00:50:05.063Z
But how to find the expenditure function? $u=(x_1^2+x_1)x_2=(x_1^2+x_1)\frac{x_1p_1(x_1+1)}{p_2(2x_1+1)}\implies p_1 x_1^4+2p_1x_1^3+p_1x_1^2-2up_2x_1=p_2u$. How do I solve for $x_1$?
Ludwig Gershwin
CC BY-SA 4.0
2023-09-08T04:35:52.950Z
Once you have found the solution $x_1^*(p,W)$, $x_2^*(p,W)$ to the utility maximization problem, you find the indirect utility function $v$, which is given by $v(p,W)=u(x_1^*(p,W),x_2^*(p,W))$, by substituting the solution $(x_1^*,x_2^*)$  into $u$. Then, to find the expenditure function $e$, use the equation $v(p,e(p,\bar{u}))=\bar{u}$ (which amounts to solving $v(p,e)=\bar{u}$ for $e$).
smcc
CC BY-SA 4.0
2023-09-07T10:14:00.003Z
2023-09-07T10:14:00.003Z
smcc
<p>The utility function seems innocent: <span class="math-container">$u(x_1,x_2)=(x_1^2+x_1)x_2$</span>. I want to find the indirect utility function and the expenditure function, but I've encountered some problems. Here's what I've got:
<span class="math-container">$$MU_1=2x_1x_2+x_2,MU_2=x_1^2+x_1$$</span>
<span class="math-container">$$\frac{MU_1}{MU_2}=\frac{p_1}{p_2}\implies \frac{2x_1x_2+x_2}{x_1^2+x_1}=\frac{p_1}{p_2}$$</span>
<span class="math-container">$$\implies x_1^2p_1+x_1(p_1-2x_2p_2)-x_2p_2=0$$</span>
<span class="math-container">$$\implies x_1=\frac{2x_2p_2-p_1+\sqrt{p_1^2+4x_2^2p_2^2}}{2p_1}$$</span>
<span class="math-container">$$x_2=\frac{x_1^2p_1+x_1p_1}{2x_1p_2+p_2}$$</span>
Substitute back into the budget constraint, I got
<span class="math-container">$$W=\frac{2x_2p_2-p_1+\sqrt{p_1^2+4x_2^2p_2^2}}{2}+p_2x_2$$</span>
but I have no idea how to solve for <span class="math-container">$x_2$</span>. Maybe I'm going in the wrong direction. Any ideas are greatly appreciated.</p>

<utility>
How to find the indirect utility function and the expenditure function through this interesting utility function?
Ludwig Gershwin
# How to find the indirect utility function and the expenditure function through this interesting utility function?

(Asked by: username_0 on 2023-09-07)

The utility function seems innocent: $u(x\_1,x\_2)=(x\_1^2+x\_1)x\_2$. I want to find the indirect utility function and the expenditure function, but I've encountered some problems. Here's what I've got:
$$MU\_1=2x\_1x\_2+x\_2,MU\_2=x\_1^2+x\_1$$
$$\frac{MU\_1}{MU\_2}=\frac{p\_1}{p\_2}\implies \frac{2x\_1x\_2+x\_2}{x\_1^2+x\_1}=\frac{p\_1}{p\_2}$$
$$\implies x\_1^2p\_1+x\_1(p\_1-2x\_2p\_2)-x\_2p\_2=0$$
$$\implies x\_1=\frac{2x\_2p\_2-p\_1+\sqrt{p\_1^2+4x\_2^2p\_2^2}}{2p\_1}$$
$$x\_2=\frac{x\_1^2p\_1+x\_1p\_1}{2x\_1p\_2+p\_2}$$
Substitute back into the budget constraint, I got
$$W=\frac{2x\_2p\_2-p\_1+\sqrt{p\_1^2+4x\_2^2p\_2^2}}{2}+p\_2x\_2$$
but I have no idea how to solve for $x\_2$. Maybe I'm going in the wrong direction. Any ideas are greatly appreciated.

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2023-09-07. Score: 1

Rather than solving the tangency condition for $x\_1$ and substituting into the budget constraint, it is much easier to solve the tangency condition for $x\_2$ and substitute into the budget constraint.

As you have found, the tangency condition requires that

$$x\_2=\frac{p\_1x\_1(x\_1+1)}{p\_2(2x\_1+1)}$$

Substituting into the budget constraint gives

$$p\_1x\_1+p\_2\frac{p\_1x\_1(x\_1+1)}{p\_2(2x\_1+1)}=W.$$

Multiplying through by $(2x\_1+1)$ gives

$$p\_1x\_1(2x\_1+1)+p\_1x\_1(x\_1+1)=(2x\_1+1)W$$

or

$$p\_1x\_1(3x\_1+2)=(2x\_1+1)W$$

Now just rearrange to get a quadratic equation in $x\_1$ and solve using the quadratic formula.

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Note that you should also argue why it is not optimal for $x\_1$ or $x\_2$ to be zero (i.e. why a corner solution is not optimal).

> **Comments:**
> **username_0**: But how to find the expenditure function? $u=(x_1^2+x_1)x_2=(x_1^2+x_1)\frac{x_1p_1(x_1+1)}{p_2(2x_1+1)}\implies p_1 x_1^4+2p_1x_1^3+p_1x_1^2-2up_2x_1=p_2u$. How do I solve for $x_1$?  
> **username_1**: Once you have found the solution $x_1^*(p,W)$, $x_2^*(p,W)$ to the utility maximization problem, you find the indirect utility function $v$, which is given by $v(p,W)=u(x_1^*(p,W),x_2^*(p,W))$, by substituting the solution $(x_1^*,x_2^*)$  into $u$. Then, to find the expenditure function $e$, use the equation $v(p,e(p,\bar{u}))=\bar{u}$ (which amounts to solving $v(p,e)=\bar{u}$ for $e$).  

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