<p>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.</p>
<p>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:</p>
<p><span class="math-container">$x^\star = \alpha_x - \beta_x p_x$</span></p>
<p><span class="math-container">$y^\star = \alpha_y - \beta_y p_y$</span></p>
<p>Where <span class="math-container">$\beta_x,\beta_y &gt; 0$</span> are constants and <span class="math-container">$\alpha_x, \alpha_y &gt; 0$</span> could be functions of the consumer’s budget (and maybe the other good’s price indicating if the goods are complements/substitutes).</p>
<p>Or let <span class="math-container">$\alpha_x, \alpha_y$</span> 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:</p>
<p><span class="math-container">$x^\star = \alpha_x - \beta_x p_x + \gamma_x p_y$</span></p>
<p><span class="math-container">$y^\star = \alpha_y - \beta_y p_y + \gamma_y p_x$</span></p>
<p>where <span class="math-container">$\gamma_x, \gamma_y$</span> are constants (or maybe even functions of the agent’s income).</p>
<p>In my idea, the <span class="math-container">$\gamma_i$</span> could be either positive, negative or zero, meaning the goods are substitutes, complements or unrelated, respectively.</p>
<p>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: <a href="https://economics.stackexchange.com/questions/40855/examples-of-bounded-positive-inverse-demand-curves">Examples of bounded, positive, inverse demand curves</a></p>
<p>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):</p>
<p><span class="math-container">$x^\star = C I^\epsilon {p_x}^\eta {p_y}^\zeta$</span></p>
<p>where</p>
<ul>
<li><span class="math-container">$\epsilon$</span> would tell me if the good is luxury/normal/neutral/inferior.</li>
<li><span class="math-container">$\eta$</span> would tell me if the good’s demand is elastic/unit elasticity/inelastic.</li>
<li><span class="math-container">$\zeta$</span> would tell me if the goods are substitutes/unrelated goods/complements.</li>
<li><span class="math-container">$C &gt; 0$</span> is a constant.</li>
</ul>
<p>Is this idea possible or even widely known? If so, I would appreciate any orientation.</p>

CC BY-SA 4.0
2023-05-11T14:09:11.403Z
This is known as the [integrability problem](https://en.wikipedia.org/wiki/Integrability_of_demand). See [this note](https://healy.econ.ohio-state.edu/kcb/Ec121a/Lecture14.pdf#page=7) for some examples on recovering utility from some of the demand functions you mentioned.
Herr K.
CC BY-SA 4.0
2023-05-12T16:16:14.010Z
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.
Nicolas Torres
CC BY-SA 4.0
2023-05-12T18:47:44.357Z
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 $\{x\in\mathbb R_{++}^n:u(x)\ge v\}$.
Herr K.
<microeconomics><utility><preferences><optimization><demand>
Is it possible to get back the consumer’s utility function from their demand functions?
Nicolas Torres
# Is it possible to get back the consumer’s utility function from their demand functions?

(Asked by: username_0 on 2023-05-11)

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](https://economics.stackexchange.com/questions/40855/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.

**Tags:** microeconomics, utility, preferences, optimization, demand
> **Comments:**
> **username_1**: This is known as the [integrability problem](https://en.wikipedia.org/wiki/Integrability_of_demand). See [this note](https://healy.econ.ohio-state.edu/kcb/Ec121a/Lecture14.pdf#page=7) for some examples on recovering utility from some of the demand functions you mentioned.  
> **username_0**: 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.  
> **username_1**: 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 $\{x\in\mathbb R_{++}^n:u(x)\ge v\}$.  

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## 0 answers

