# Is it possible to get back the consumer’s utility function from their demand functions?

I am curious about if it’s possible to reverse the utility maximization process, i.e. given the consumer’s Marshallian demand functions, find their utility function.

I was thinking of trying to find the utility function for a consumer whose Marshallian demands for two goods are the classic Micro 101 linear demand curves:

$$x^\star = \alpha_x - \beta_x p_x$$

$$y^\star = \alpha_y - \beta_y p_y$$

Where $$\beta_x,\beta_y > 0$$ are constants and $$\alpha_x, \alpha_y > 0$$ could be functions of the consumer’s budget (and maybe the other good’s price indicating if the goods are complements/substitutes).

Or let $$\alpha_x, \alpha_y$$ be functions depending solely on the consumer’s income and adding an interaction term (like we do in Econometrics) that indicates if the goods are complements/substitutes like this:

$$x^\star = \alpha_x - \beta_x p_x + \gamma_x p_y$$

$$y^\star = \alpha_y - \beta_y p_y + \gamma_y p_x$$

where $$\gamma_x, \gamma_y$$ are constants (or maybe even functions of the agent’s income).

In my idea, the $$\gamma_i$$ could be either positive, negative or zero, meaning the goods are substitutes, complements or unrelated, respectively.

Then I’d like to try to generalize this idea for other neat functional forms such as power, exponential and logarithm demand as in this question: Examples of bounded, positive, inverse demand curves

I’d also like to try it for Cobb-Douglas demands (They’re one of the most common functional forms in Economics, yet I’ve never seen them as demand functions):

$$x^\star = C I^\epsilon {p_x}^\eta {p_y}^\zeta$$

where

• $$\epsilon$$ would tell me if the good is luxury/normal/neutral/inferior.
• $$\eta$$ would tell me if the good’s demand is elastic/unit elasticity/inelastic.
• $$\zeta$$ would tell me if the goods are substitutes/unrelated goods/complements.
• $$C > 0$$ is a constant.

Is this idea possible or even widely known? If so, I would appreciate any orientation.

• This is known as the integrability problem. See this note for some examples on recovering utility from some of the demand functions you mentioned. Commented May 11, 2023 at 14:09
• Thank you @HerrK.! This is just what I wanted. Do you know which is the support theorem they mention at the bottom of page 7? It looks like Shephard’s lemma to me, however, Shephard’s lemma is for Hicksian demands, not Marshallian ones. Commented May 12, 2023 at 16:16
• Note that it's support function theorem, and there are various formulations. In this context, it's essentially Prop 3.F.1 in MWG, since $e(p,v)$ is a support function of the set $\{x\in\mathbb R_{++}^n:u(x)\ge v\}$. Commented May 12, 2023 at 18:47