<p>The condition is particularly true if utility is <em>quasi-linear</em>, i.e. <span class="math-container">$V(q,m)=u(q)+m$</span>, where <span class="math-container">$u(\cdot)$</span> captures the consumer's utility from the (only) good and <span class="math-container">$m$</span> is leftover money from consuming that good. Here <span class="math-container">$m$</span> can be treated as the numeraire whose price is <span class="math-container">$1$</span>.</p>
<p>Given income <span class="math-container">$I$</span> and price <span class="math-container">$p$</span> of the good, the consumer's utility maximization problem is
<span class="math-container">\begin{equation}
\max_q\; u(q)+(I-pq) 
\end{equation}</span>
The first order condition is then
<span class="math-container">\begin{equation}
u'(q)=p
\end{equation}</span></p>
<p>This is also a special case of the more general condition given in <a href="https://economics.stackexchange.com/a/53336">RegressForward's answer</a>. Just take <span class="math-container">$j$</span> as the numeraire, and then <span class="math-container">$\frac{MU_j}{p_j}=1$</span>, which makes <span class="math-container">$MU_i=p_i$</span>.</p>

CC BY-SA 4.0
2022-11-01T04:22:12.110Z
I didn't follow the last part. I think you are talking of the $j$ in RegressForward's answer. What exactly do we take as a numeraire? Thank you.
Rick_Morty
CC BY-SA 4.0
2022-11-01T11:09:52.160Z
@Citrus: $j$ being the numeraire implies $MU_j=p_j=1$ which in turn implies $MU_i=p_i$ (though the latter is not necessarily equal to $1$.
Herr K.
CC BY-SA 4.0
2022-11-01T02:34:06.803Z
2022-11-01T16:28:59.707Z
2022-11-01T16:28:59.707Z
Herr K.
<p>Good thinking. You're on track but not quite there.</p>
<p>You're thinking of the <em>optimal consumption rule</em> - you're correct it has to do with marginal utility.
Define: <span class="math-container">$\frac{\partial U(Q)}{\partial Q} = MarginalUtility = MU$</span></p>
<p>And in a two-good consumption problem, the optimal consumption is then found by:
<span class="math-container">$\frac{MU_i}{P_i}=\frac{MU_j}{P_j}$</span></p>
<p>and exhausting the agent's budget.</p>

CC BY-SA 4.0
2022-11-01T04:16:58.777Z
$\max[U(x_1,x_2)]$ subject to $p_1 x + p_2 x_2$ has the optimal condition $\frac{\text{MU}_1}{\text{MU}_2} = \frac{p_1}{p_2}$ (along with the budget condition) as you suggested. May I ask what you're referring to as $Q$?
Rick_Morty
CC BY-SA 4.0
2022-11-01T13:09:57.813Z
What does the $X$ represent in your equation there?
RegressForward
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2022-11-01T14:51:57.970Z
If $X_1, X_2$ are two different goods, then $x_1$ and $x_2$ (in my comment) are the quantities consumed by the agent. I thought, that's what you meant by $MU_i$ and $MU_j$ - the marginal utilities of two goods. And similarly, $p_i, p_j$ as the prices of the same two goods.
Rick_Morty
CC BY-SA 4.0
2022-11-01T15:06:58.480Z
Right, so $Q$ in these cases refers to the quantity of the good.
RegressForward
CC BY-SA 4.0
2022-11-01T16:39:48.190Z
I misunderstood one thing. Things make sense now; thank you very much!
Rick_Morty
CC BY-SA 4.0
2022-10-31T21:17:22.890Z
2022-10-31T21:17:22.890Z
RegressForward
<p>Is it true that <span class="math-container">$\frac{\partial U(Q)}{\partial Q} = P$</span> where <span class="math-container">$U$</span> is the utility function, <span class="math-container">$P$</span> is the output price  and <span class="math-container">$Q$</span> is the output quantity? Maybe not exactly this, but is something like this (possibly at the optimal value) true by any chance?</p>
<p>Is there any way to relate the output price and the utility function of the consumer?</p>

<microeconomics>
Is $\frac{\partial U(Q)}{\partial Q} = P$?
Rick_Morty
# Is $\frac{\partial U(Q)}{\partial Q} = P$?

(Asked by: username_0 on 2022-10-31)

Is it true that $\frac{\partial U(Q)}{\partial Q} = P$ where $U$ is the utility function, $P$ is the output price and $Q$ is the output quantity? Maybe not exactly this, but is something like this (possibly at the optimal value) true by any chance?

Is there any way to relate the output price and the utility function of the consumer?

**Tags:** microeconomics


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## 2 answers

### Answer by username_1 on 2022-11-01. Score: 2 | Accepted answer

The condition is particularly true if utility is *quasi-linear*, i.e. $V(q,m)=u(q)+m$, where $u(\cdot)$ captures the consumer's utility from the (only) good and $m$ is leftover money from consuming that good. Here $m$ can be treated as the numeraire whose price is $1$.

Given income $I$ and price $p$ of the good, the consumer's utility maximization problem is
\begin{equation}
\max\_q\; u(q)+(I-pq)
\end{equation}
The first order condition is then
\begin{equation}
u'(q)=p
\end{equation}

This is also a special case of the more general condition given in [username_2's answer](https://economics.stackexchange.com/a/53336). Just take $j$ as the numeraire, and then $\frac{MU\_j}{p\_j}=1$, which makes $MU\_i=p\_i$.

> **Comments:**
> **username_0**: I didn't follow the last part. I think you are talking of the $j$ in username_2's answer. What exactly do we take as a numeraire? Thank you.  
> **username_1**: @Citrus: $j$ being the numeraire implies $MU_j=p_j=1$ which in turn implies $MU_i=p_i$ (though the latter is not necessarily equal to $1$.  

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### Answer by username_2 on 2022-10-31. Score: 2

Good thinking. You're on track but not quite there.

You're thinking of the *optimal consumption rule* - you're correct it has to do with marginal utility.
Define: $\frac{\partial U(Q)}{\partial Q} = MarginalUtility = MU$

And in a two-good consumption problem, the optimal consumption is then found by:
$\frac{MU\_i}{P\_i}=\frac{MU\_j}{P\_j}$

and exhausting the agent's budget.

> **Comments:**
> **username_0**: $\max[U(x_1,x_2)]$ subject to $p_1 x + p_2 x_2$ has the optimal condition $\frac{\text{MU}_1}{\text{MU}_2} = \frac{p_1}{p_2}$ (along with the budget condition) as you suggested. May I ask what you're referring to as $Q$?  
> **username_2**: What does the $X$ represent in your equation there?  
> **username_0**: If $X_1, X_2$ are two different goods, then $x_1$ and $x_2$ (in my comment) are the quantities consumed by the agent. I thought, that's what you meant by $MU_i$ and $MU_j$ - the marginal utilities of two goods. And similarly, $p_i, p_j$ as the prices of the same two goods.  
> **username_2**: Right, so $Q$ in these cases refers to the quantity of the good.  
> **username_0**: I misunderstood one thing. Things make sense now; thank you very much!  

---

