<p>Define the marginal rate of subsitution <span class="math-container">$MRS = \frac{v_2}{v_1}$</span>. This gives the slope of the indifference curve at the optimal choice. For fixed utility level <span class="math-container">$u$</span>, this is a function of <span class="math-container">$q$</span> alone. The value of <span class="math-container">$q$</span> itself is a function of all exogenous parameters (say <span class="math-container">$\gamma$</span>).</p>
<p>Given this, the first order condition can be written as:
<span class="math-container">$$
\left.MRS(q(\gamma))\right|_u = p(\gamma).
$$</span>
Taking the derivative of both sides with respect to <span class="math-container">$\gamma$</span> gives:
<span class="math-container">$$
\left.\frac{\partial MRS(q)}{\partial q}\right|_{u} \frac{\partial q}{\partial \gamma} = \frac{\partial p}{\partial \gamma}.
$$</span>
From this, it follows that:
<span class="math-container">$$
\frac{\partial q}{\partial \gamma} = \eta \frac{\partial p}{\partial \gamma},
$$</span>
where <span class="math-container">$\eta = \left(\left.\frac{\partial MRS(q)}{\partial q}\right|_{u}\right)^{-1}
$</span> as given in footnote 3 of the linked article of Brueckner.</p>
<p>Notice that generically, <span class="math-container">$\eta$</span> will be function of <span class="math-container">$q$</span> and therefore of all exogenous variables (including <span class="math-container">$\gamma$</span>). As such, it is not necessarily a fixed number.</p>

CC BY-SA 4.0
2021-04-26T06:54:29.170Z
2021-04-26T06:54:29.170Z
tdm
<p>I am exploring the Muth-Mills monocentric city model covered in Brueckner's <a href="http://www.socsci.uci.edu/%7Ejkbrueck/course%20readings/handbook%20chapter.pdf" rel="nofollow noreferrer">article</a>.</p>
<p>It is given that consumers earn the same income <span class="math-container">$y$</span> and buy <span class="math-container">$q$</span> housing for a price <span class="math-container">$p$</span> at distance <span class="math-container">$x$</span> from the centre, while incurring transport costs of <span class="math-container">$tx$</span>.</p>
<p>Consumers also have a utility function</p>
<p><span class="math-container">$$v(c,q)=v(y−tx−p(ϕ)q(ϕ),q(ϕ))=u$$</span>
where <span class="math-container">$$ϕ=x,y,t,u$$</span>
which they maximise with respect to <span class="math-container">$q$</span> subject to the budget constraint</p>
<p><span class="math-container">$$c=y−tx−pq$$</span></p>
<p>The first-order condition implies <span class="math-container">$\frac{v_2(y−tx−pq,q)}{v_1(y−tx−pq,q)}=p$</span>.</p>
<p>I am able to work out the derivations for <span class="math-container">$\frac{∂p}{∂γ},γ=x,y,t$</span> but I was wondering how Brueckner was able to derive that <span class="math-container">$\frac{∂q}{∂γ}=η\frac{∂p}{∂γ}$</span>.</p>
<p>I believe this question is identical to the second part of the question <a href="https://economics.stackexchange.com/questions/26182/brueckners-basic-urban-model-comparative-statics">here</a> but it has not been answered, which is why this post exists. Any help would be appreciated!</p>

<urban-economics><comparative-statics>
Muth-Mills Model
kékszajkók
# Muth-Mills Model

(Asked by: username_0 on 2021-04-25)

I am exploring the Muth-Mills monocentric city model covered in Brueckner's [article](http://www.socsci.uci.edu/%7Ejkbrueck/course%20readings/handbook%20chapter.pdf).

It is given that consumers earn the same income $y$ and buy $q$ housing for a price $p$ at distance $x$ from the centre, while incurring transport costs of $tx$.

Consumers also have a utility function

$$v(c,q)=v(y−tx−p(ϕ)q(ϕ),q(ϕ))=u$$
where $$ϕ=x,y,t,u$$
which they maximise with respect to $q$ subject to the budget constraint

$$c=y−tx−pq$$

The first-order condition implies $\frac{v\_2(y−tx−pq,q)}{v\_1(y−tx−pq,q)}=p$.

I am able to work out the derivations for $\frac{∂p}{∂γ},γ=x,y,t$ but I was wondering how Brueckner was able to derive that $\frac{∂q}{∂γ}=η\frac{∂p}{∂γ}$.

I believe this question is identical to the second part of the question [here](https://economics.stackexchange.com/questions/26182/brueckners-basic-urban-model-comparative-statics) but it has not been answered, which is why this post exists. Any help would be appreciated!

**Tags:** urban-economics, comparative-statics

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## 1 answers

### Answer by username_1 on 2021-04-26. Score: 2 | Accepted answer

Define the marginal rate of subsitution $MRS = \frac{v\_2}{v\_1}$. This gives the slope of the indifference curve at the optimal choice. For fixed utility level $u$, this is a function of $q$ alone. The value of $q$ itself is a function of all exogenous parameters (say $\gamma$).

Given this, the first order condition can be written as:
$$
\left.MRS(q(\gamma))\right|\_u = p(\gamma).
$$
Taking the derivative of both sides with respect to $\gamma$ gives:
$$
\left.\frac{\partial MRS(q)}{\partial q}\right|\_{u} \frac{\partial q}{\partial \gamma} = \frac{\partial p}{\partial \gamma}.
$$
From this, it follows that:
$$
\frac{\partial q}{\partial \gamma} = \eta \frac{\partial p}{\partial \gamma},
$$
where $\eta = \left(\left.\frac{\partial MRS(q)}{\partial q}\right|\_{u}\right)^{-1}
$ as given in footnote 3 of the linked article of Brueckner.

Notice that generically, $\eta$ will be function of $q$ and therefore of all exogenous variables (including $\gamma$). As such, it is not necessarily a fixed number.


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