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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!

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

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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).

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  • $\begingroup$ 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? $\endgroup$ Commented Mar 27 at 11:03
  • $\begingroup$ 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. $\endgroup$
    – Giskard
    Commented Mar 27 at 13:34

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