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I am exploring the Muth-Mills monocentric city model covered in Brueckner's article.

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 but it has not been answered, which is why this post exists. Any help would be appreciated!

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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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