Calculus and Indifference Curves in an Urban Economics Example I am reading the paper 'The Structure of Urban Equilibria' by Jan Brueckner.

It uses a monocentric city model, where all consumers earn income $y$ at the centre of the city. They buy $q$ housing for a price $p$ at distance $x$ from the centre, incurring transport costs $tx$.

Consumers have a utility function:

$v(c,q)=v(y - tx - p(\phi)q(\phi),q(\phi))=u$

where $\phi=x,y,t,u$

The budget constraint is:

$c = y - tx - pq$



The tangency condition implies:

$\frac{v_1(y - tx - pq, q)}{v_2(y - tx - pq, q)} = p$   

where the subscript 1 denotes partial differentiation w.r.t. the first argument etc.

The paper then discusses how $p$ and $q$ vary with $x, y, t$ and $u$.

If $\phi=x,y,t$, we stay on the same indifference curve. I find it relatively straightforward to find $\frac{\partial{p}}{\partial{x}},\frac{\partial{p}}{\partial{y}}$ and $\frac{\partial{p}}{\partial{t}}$.

If $\eta$ is the slope of the income-compensated demand curve, then $\frac{\partial{q}}{\partial{\phi}} = \eta\frac{\partial{p}}{\partial{\phi}}$.

Now to allow $u$ to vary. The budget constraint swings out to meet a new indifference curve, determining the new $p$ and $q$.

I can find $\frac{\partial{p}}{\partial{u}}$. Totally differentiate the utility function w.r.t u:

$\frac{d}{du}[v(y - tx - p(\phi)q(\phi),q(\phi))= u] = v_1(-\frac{\partial{p}}{\partial{u}}q-p\frac{\partial{q}}{\partial{u}})+v_2(\frac{\partial{q}}{\partial{u}})=1$ 

Since, by the tangency condition $v_2=pv_1$:

$v_1(-\frac{\partial{p}}{\partial{u}}q-p\frac{\partial{q}}{\partial{u}}+p\frac{\partial{q}}{\partial{u}})=v_1(-\frac{\partial{p}}{\partial{u}}q)=1$

So $\frac{\partial{p}}{\partial{u}} = \frac{-1}{qv_1}$.

The paper then quotes:

$\frac{\partial{q}}{\partial{u}} = [\frac{\partial{p}}{\partial{u}}-\frac{\partial{MRS}}{\partial{c}}\frac{1}{v_1}]\eta$

I don't know how to derive this. I'm guessing the first term in the square brackets is a substitution effect and the second term is an income effect.

Please help me understand this last expression $\frac{\partial{q}}{\partial{u}} = [\frac{\partial{p}}{\partial{u}}-\frac{\partial{MRS}}{\partial{c}}\frac{1}{v_1}]\eta$ and how to derive it.
The utility function under consideration is $v(c,q)$ and then
$$MRS(c,q) = \frac{\partial v/\partial q}{\partial v/\partial c} = v_2/v_1$$
make the functional denpendency of on $u$ explicit then you have
$$\frac{\partial}{\partial u}MRS(c(u),q(u)) = \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
where the last identity follows because you know that $p = MRS$. Now simply rearrange the identity
$$\frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u} + \frac{\partial MRS(c(u),q(u))}{\partial q} \frac{\partial q(u)}{\partial u} = \frac{\partial p(u)}{\partial u},$$
to get
$$ \frac{\partial q(u)}{\partial u} = \frac{\left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right]}{\frac{\partial MRS(c(u),q(u))}{\partial q} },$$
then use that Brueckner has defined $\eta := \left[\frac{\partial MRS(c(u),q(u))}{\partial q}\right]^{-1}$ in footnote(3) to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{\partial c(u)}{\partial u}\right] \eta ,$$
and finally apply the rule that $\frac{\partial c(u)}{\partial u} = \frac{1}{\partial v/\partial c} = 1/v_1$ to get
$$ \frac{\partial q(u)}{\partial u} = \left[\frac{\partial p(u)}{\partial u} - \frac{\partial MRS(c(u),q(u))}{\partial c} \frac{1}{v_1}\right] \eta.$$