<p>This is a rather indirect way.</p>
<p>For <span class="math-container">$\omega, z \in \mathbb{R}_{++}$</span>, define the (demand) correspondence:</p>
<p><span class="math-container">$$
D(\omega, z) = \left\{(x,y) \in \mathbb{R}^2_+| MRS(x,y) = \omega \text{ and } \omega x + y = z.\right\}
$$</span>
The idea is that <span class="math-container">$D(\omega, z)$</span> gives the demand correspondence for (relative prices <span class="math-container">$\omega = MRS(x,y)$</span> and (normalized) income <span class="math-container">$z = \omega x + y$</span>.</p>
<p>If <span class="math-container">$D(\omega, z)$</span> would turn out to be a (smooth) function, one could then check if it satisfies the usual integrability conditions: Slutsky symmetry and negative definiteness. (Alternatively, one could check if <span class="math-container">$D(\omega, z)$</span> satisfies the SARP (strong axiom of revealed preference).)</p>

CC BY-SA 4.0
2024-04-08T15:57:29.577Z
Thanks so much. That was the trick I needed for my application.
John Sturm
CC BY-SA 4.0
2024-04-08T06:12:53.063Z
2024-04-08T06:12:53.063Z
tdm
<p>Is it known when a marginal-rate-of-substitution function can be rationalized by some utility function?</p>
<p>More precisely, and focusing on the case of two goods, what conditions are required on <span class="math-container">$M: (\mathbb R_{\geq 0})^2 \to \mathbb R$</span> in order for there to exist <span class="math-container">$u: (\mathbb R_{\geq 0})^2 \to \mathbb R$</span> such that for all <span class="math-container">$x,y \geq 0$</span>,
<span class="math-container">$$
M(x,y)=\frac{u_x(x,y)}{u_y(x,y)}
$$</span></p>

CC BY-SA 4.0
2024-04-05T16:57:21.373Z
Here is one thought:

One way to look at this condition is as a linear, homogeneous, partial differential equation
$$ u_x(x,y) = M(x,y) u_y(x,y) $$
The question then boils down to what conditions on $M(x,y)$ guarantee that such a PDE has a solution. I am unfortunately not knowledgeable on this topic.
John Sturm
<microeconomics><consumer-theory><real-analysis>
When are marginal rates of substitution consistent with a utility function?
John Sturm
# When are marginal rates of substitution consistent with a utility function?

(Asked by: username_0 on 2024-04-03)

Is it known when a marginal-rate-of-substitution function can be rationalized by some utility function?

More precisely, and focusing on the case of two goods, what conditions are required on $M: (\mathbb R\_{\geq 0})^2 \to \mathbb R$ in order for there to exist $u: (\mathbb R\_{\geq 0})^2 \to \mathbb R$ such that for all $x,y \geq 0$,
$$
M(x,y)=\frac{u\_x(x,y)}{u\_y(x,y)}
$$

**Tags:** microeconomics, consumer-theory, real-analysis

> **Comments:**
> **username_0**: Here is one thought:  One way to look at this condition is as a linear, homogeneous, partial differential equation $$ u_x(x,y) = M(x,y) u_y(x,y) $$ The question then boils down to what conditions on $M(x,y)$ guarantee that such a PDE has a solution. I am unfortunately not knowledgeable on this topic.  

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## 1 answers

### Answer by username_1 on 2024-04-08. Score: 2

This is a rather indirect way.

For $\omega, z \in \mathbb{R}\_{++}$, define the (demand) correspondence:

$$
D(\omega, z) = \left\{(x,y) \in \mathbb{R}^2\_+| MRS(x,y) = \omega \text{ and } \omega x + y = z.\right\}
$$
The idea is that $D(\omega, z)$ gives the demand correspondence for (relative prices $\omega = MRS(x,y)$ and (normalized) income $z = \omega x + y$.

If $D(\omega, z)$ would turn out to be a (smooth) function, one could then check if it satisfies the usual integrability conditions: Slutsky symmetry and negative definiteness. (Alternatively, one could check if $D(\omega, z)$ satisfies the SARP (strong axiom of revealed preference).)

> **Comments:**
> **username_0**: Thanks so much. That was the trick I needed for my application.  

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