<p>Suppose utility function is <span class="math-container">$u(x, y) = kxy$</span> where <span class="math-container">$k&gt;0$</span>. Consider the following problem</p>
<p><span class="math-container">$\displaystyle \max_{x\geq 0, y\geq 0} \ \ kxy \\ \text{s.t.} \ p_Xx + p_Yy \leq I $</span></p>
<p>where <span class="math-container">$p_X &gt;0$</span>, <span class="math-container">$p_Y &gt; 0$</span>, and <span class="math-container">$I\geq 0$</span>.</p>
<p>Solving the above utility maximization problem we get the following solution (also known as demand):</p>
<p><span class="math-container">\begin{eqnarray*} (x^d,y^d)(p_X, p_Y, I) = \left(\frac{I}{2p_X},\frac{I}{2p_Y} \right) \end{eqnarray*}</span></p>
<p>and the optimal value of utility or the indirect utility function as:</p>
<p><span class="math-container">\begin{eqnarray*} V(p_X, p_Y, I) = u\left(\frac{I}{2p_X},\frac{I}{2p_Y} \right) = \frac{kI^2}{4p_Xp_Y} \end{eqnarray*}</span></p>
<p>Therefore, when <span class="math-container">$k = 4$</span> we get the desired indirect utility function. So, <span class="math-container">$u(x, y) = 4xy$</span>.</p>

CC BY-SA 4.0
2022-10-02T09:36:55.657Z
(-1) This is a guess and validate type of answer (ideal for cramming) that does not impart knowledge, while a general answer exists; see [Rumi's comment](https://economics.stackexchange.com/questions/52915/how-to-find-the-utility-function-using-indirect-utility-function#comment86346_52915).
Giskard
CC BY-SA 4.0
2022-10-02T09:53:32.507Z
That is funny. According to you memorising Shepherd's Lemma or Roy's identity is not cramming. I think a randomly selected Economics student is more likely to know demand for Cobb Douglas than these Lemmas :)
Amit
CC BY-SA 4.0
2022-10-02T14:28:23.853Z
You are right about the random student for sure! Is that a good thing though?
Giskard
CC BY-SA 4.0
2022-10-01T16:00:43.930Z
2022-10-01T16:00:43.930Z
Amit
<p>The indirect utility function looks like this:</p>
<p><span class="math-container">$v(p_x, p_y, I) = \frac{I^2}{p_x*p_y}$</span></p>
<p>where:
<span class="math-container">$I$</span> - income,
<span class="math-container">$p_i$</span> - price for good <span class="math-container">$i ∈ x,y$</span>.</p>
<p>Find the utility function <span class="math-container">$U(x,y)$</span></p>

CC BY-SA 4.0
2022-10-01T15:40:19.280Z
What have you tried?
Rumi
CC BY-SA 4.0
2022-10-01T15:48:32.463Z
I will help you get started. See if you can figure it out. Isolate Income in terms of V. Use partial derivatives with respect to each goods price to get Hicksian (Shephards Lemma). Get the price ratio in one of the Hicksians and plug it in the other Hicksian. Solve for V.
Rumi
CC BY-SA 4.0
2022-10-02T10:20:49.670Z
Also see this answer for an alternative approach: https://economics.stackexchange.com/a/16745/11824
Amit
<microeconomics><utility>
How to find the utility function using indirect utility function?
Laney
# How to find the utility function using indirect utility function?

(Asked by: username_0 on 2022-09-30)

The indirect utility function looks like this:

$v(p\_x, p\_y, I) = \frac{I^2}{p\_x\*p\_y}$

where:
$I$ - income,
$p\_i$ - price for good $i ∈ x,y$.

Find the utility function $U(x,y)$

**Tags:** microeconomics, utility

> **Comments:**
> **username_1**: What have you tried?  
> **username_1**: I will help you get started. See if you can figure it out. Isolate Income in terms of V. Use partial derivatives with respect to each goods price to get Hicksian (Shephards Lemma). Get the price ratio in one of the Hicksians and plug it in the other Hicksian. Solve for V.  
> **username_2**: Also see this answer for an alternative approach: https://economics.stackexchange.com/a/16745/11824  

---

## 1 answers

### Answer by username_2 on 2022-10-01. Score: 1 | Accepted answer

Suppose utility function is $u(x, y) = kxy$ where $k>0$. Consider the following problem

$\displaystyle \max\_{x\geq 0, y\geq 0} \ \ kxy \\ \text{s.t.} \ p\_Xx + p\_Yy \leq I $

where $p\_X >0$, $p\_Y > 0$, and $I\geq 0$.

Solving the above utility maximization problem we get the following solution (also known as demand):

\begin{eqnarray\*} (x^d,y^d)(p\_X, p\_Y, I) = \left(\frac{I}{2p\_X},\frac{I}{2p\_Y} \right) \end{eqnarray\*}

and the optimal value of utility or the indirect utility function as:

\begin{eqnarray\*} V(p\_X, p\_Y, I) = u\left(\frac{I}{2p\_X},\frac{I}{2p\_Y} \right) = \frac{kI^2}{4p\_Xp\_Y} \end{eqnarray\*}

Therefore, when $k = 4$ we get the desired indirect utility function. So, $u(x, y) = 4xy$.

> **Comments:**
> **username_3**: (-1) This is a guess and validate type of answer (ideal for cramming) that does not impart knowledge, while a general answer exists; see [username_1's comment](https://economics.stackexchange.com/questions/52915/how-to-find-the-utility-function-using-indirect-utility-function#comment86346_52915).  
> **username_2**: That is funny. According to you memorising Shepherd's Lemma or Roy's identity is not cramming. I think a randomly selected Economics student is more likely to know demand for Cobb Douglas than these Lemmas :)  
> **username_3**: You are right about the random student for sure! Is that a good thing though?  

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