# Finding Utility Function for Optimal Allocation in Consumer Choice Model

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.

• Hi! Is $\alpha$ positive or negative? Seems like unless $-1<\alpha<0$, you are going to run into trouble with extreme values of $p$. Commented Mar 23 at 18:46
• Also, do you have reason to believe that such a utility function exists, or do you just assume it? Commented Mar 23 at 18:49
• Hi, thank you for your comment. I have no proof that such a utility function exists. I have also added that $-1 < \alpha < 0$. Commented Mar 24 at 20:23

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 $$\alpha = -1$$.
• 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. Commented Mar 27 at 13:34