Muth-Mills Model

Question

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!

Answer 1

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.