<p>I think your formula is still too general, so what you want will not be possible. Given <span class="math-container">$-1 &lt; \alpha &lt; 0$</span> and
<span class="math-container">$$
m^* = A \left(\frac{p}{\omega}\right)^\alpha,
$$</span>
we have
<span class="math-container">$$
\frac{m^*p}{w} = A \left(\frac{p}{\omega}\right)^{1+\alpha}.
$$</span>
The left hand side is the ratio of the consumer's money spent on this good <span class="math-container">$m$</span>. But
<span class="math-container">$$
\lim_{\omega \to 0} A \left(\frac{p}{\omega}\right)^{1+\alpha} = \infty,
$$</span>
so if you make the income of this consumer small enough, their optimal good allocation implies that they spend more than 100% of their income on this good <span class="math-container">$m$</span>, which should not be possible. If <span class="math-container">$1+\alpha$</span> were negative, you would run into a similar problem by increasing income.</p>
<p>In the special case of <span class="math-container">$1+\alpha = 0$</span>, a solution actually exists; Cobb-Douglas:
<span class="math-container">$$
U(m,x) = m^Ax^{1-A}
$$</span>
yields this exact demand function for <span class="math-container">$m$</span> with <span class="math-container">$\alpha = -1$</span>.</p>
<hr />
<p>A general lesson is that if demand is proportional to a power of income, problems arise if it is not specified that the formula only works in a certain range/interior cases, unless demand is linear in income (homothetic utility functions).</p>

CC BY-SA 4.0
2024-03-27T11:03:44.420Z
Thank you for your response. Your explanation is clear, especially since the empirical relationship has been identified within a specific income range. I plan to apply this utility function for numerical applications within the same income bracket. In this context, could you advise if it's possible to identify a utility function that results in a constant elasticity of the price-to-income ratio?
lucas_vivier
CC BY-SA 4.0
2024-03-27T13:34:40.617Z
What you describe is not entirely clear to me, but I never did any empirical work with utility functions. I recommend posting a new question with some additional details on parameters such as $\alpha$ and  the desired income range.
Giskard
CC BY-SA 4.0
2024-03-24T21:09:53.560Z
2024-03-27T13:36:45.387Z
2024-03-27T13:36:45.387Z
Giskard
<p>I'm working on a consumer choice model involving a consumer with one good and a numeraire. In this model, the price of the numeraire is assumed to be one. My objective is to identify the utility function that results in the following optimal allocation:</p>
<p><span class="math-container">$$m^* = A \left(\frac{p}{\omega}\right)^\alpha$$</span></p>
<p>Here, <span class="math-container">$\omega$</span> represents the consumer's income, and <span class="math-container">$A$</span> and <span class="math-container">$\alpha$</span> are constants and <span class="math-container">$-1 &lt; \alpha &lt; 0$</span>. The consumer's goal is to maximize their utility subject to the budget constraint:</p>
<p><span class="math-container">$$\max u(m,x) \quad \text{s.t.} \quad \omega = x + p \cdot m$$</span></p>
<p>The first-order conditions (FOCs) for this optimization problem are:</p>
<p><span class="math-container">$$u_x = \lambda$$</span></p>
<p><span class="math-container">$$u_m = \lambda \cdot p$$</span></p>
<p><span class="math-container">$$x^* = \omega - m^* \cdot p$$</span></p>
<p>Given this setup, I'm seeking assistance in determining the utility function <span class="math-container">$u(m,x)$</span> that leads to the optimal allocation <span class="math-container">$m^{*}$</span> as specified above.</p>
<p>Here's what I've tried so far:</p>
<ul>
<li>I've attempted to reverse-engineer the utility function starting from the FOCs but got stuck at integrating the conditions to find a cohesive function.</li>
<li>I also looked into some standard utility functions (like Cobb-Douglas), but I'm not sure how to adapt them to fit this particular problem.</li>
</ul>
<p>Thank you in advance for your help!</p>

CC BY-SA 4.0
2024-03-23T18:46:28.160Z
Hi! Is $\alpha$ positive or negative? Seems like unless $-1<\alpha<0$, you are going to run into trouble with extreme values of $p$.
Giskard
CC BY-SA 4.0
2024-03-23T18:49:06.683Z
Also, do you have reason to believe that such a utility function exists, or do you just assume it?
Giskard
CC BY-SA 4.0
2024-03-24T20:23:57.913Z
Hi, thank you for your comment. I have no proof that such a utility function exists. I have also added that $-1 < \alpha < 0$.
lucas_vivier
<microeconomics><utility><consumer-theory>
Finding Utility Function for Optimal Allocation in Consumer Choice Model
lucas_vivier
# Finding Utility Function for Optimal Allocation in Consumer Choice Model

(Asked by: username_0 on 2024-03-23)

I'm working on a consumer choice model involving a consumer with one good and a numeraire. In this model, the price of the numeraire is assumed to be one. My objective is to identify the utility function that results in the following optimal allocation:

$$m^\* = A \left(\frac{p}{\omega}\right)^\alpha$$

Here, $\omega$ represents the consumer's income, and $A$ and $\alpha$ are constants and $-1 < \alpha < 0$. The consumer's goal is to maximize their utility subject to the budget constraint:

$$\max u(m,x) \quad \text{s.t.} \quad \omega = x + p \cdot m$$

The first-order conditions (FOCs) for this optimization problem are:

$$u\_x = \lambda$$

$$u\_m = \lambda \cdot p$$

$$x^\* = \omega - m^\* \cdot p$$

Given this setup, I'm seeking assistance in determining the utility function $u(m,x)$ that leads to the optimal allocation $m^{\*}$ as specified above.

Here's what I've tried so far:

* I've attempted to reverse-engineer the utility function starting from the FOCs but got stuck at integrating the conditions to find a cohesive function.
* I also looked into some standard utility functions (like Cobb-Douglas), but I'm not sure how to adapt them to fit this particular problem.

Thank you in advance for your help!

**Tags:** microeconomics, utility, consumer-theory
> **Comments:**
> **username_1**: Hi! Is $\alpha$ positive or negative? Seems like unless $-1<\alpha<0$, you are going to run into trouble with extreme values of $p$.  
> **username_1**: Also, do you have reason to believe that such a utility function exists, or do you just assume it?  
> **username_0**: Hi, thank you for your comment. I have no proof that such a utility function exists. I have also added that $-1 < \alpha < 0$.  

---

## 1 answers

### Answer by username_1 on 2024-03-24. Score: 2

I think your formula is still too general, so what you want will not be possible. Given $-1 < \alpha < 0$ and
$$
m^\* = A \left(\frac{p}{\omega}\right)^\alpha,
$$
we have
$$
\frac{m^\*p}{w} = A \left(\frac{p}{\omega}\right)^{1+\alpha}.
$$
The left hand side is the ratio of the consumer's money spent on this good $m$. But
$$
\lim\_{\omega \to 0} A \left(\frac{p}{\omega}\right)^{1+\alpha} = \infty,
$$
so if you make the income of this consumer small enough, their optimal good allocation implies that they spend more than 100% of their income on this good $m$, which should not be possible. If $1+\alpha$ were negative, you would run into a similar problem by increasing income.

In the special case of $1+\alpha = 0$, a solution actually exists; Cobb-Douglas:
$$
U(m,x) = m^Ax^{1-A}
$$
yields this exact demand function for $m$ with $\alpha = -1$.

---

A general lesson is that if demand is proportional to a power of income, problems arise if it is not specified that the formula only works in a certain range/interior cases, unless demand is linear in income (homothetic utility functions).

> **Comments:**
> **username_0**: Thank you for your response. Your explanation is clear, especially since the empirical relationship has been identified within a specific income range. I plan to apply this utility function for numerical applications within the same income bracket. In this context, could you advise if it's possible to identify a utility function that results in a constant elasticity of the price-to-income ratio?  
> **username_1**: What you describe is not entirely clear to me, but I never did any empirical work with utility functions. I recommend posting a new question with some additional details on parameters such as $\alpha$ and  the desired income range.  

---

