Quantitative spatial models are just Rosen-Roback with market access

UPDATE (31 Jul 21): This version kills the mixed land use nonsense (explored previously) to get straight to the heart of the matter. An old version of the post is available here.

UPDATE (24 Dec 20): Nick Tsivanidis’ JMP generates reduced-form equations for a QSM that omits the mixed-use land market. The contribution of this post, then, is to say these are useful equations, and provide a dumb method for dealing with mixed use.

The computable general equilibrium quantitative spatial model of Ahlfeldt et al. (2015) (ARSW) basically just gives Rosen-Roback equations (in the style of Glaser and Gottlieb (2008)) that have market access terms. That’s to say, the model’s urban spatial equilibrium gives some log-linear equations of the endogenous prices as functions of the fundamentals and market access.

We start with the standard setup: a discrete geography where consumers choose a location to live $i$ and to work $j$. They each inelastically supply a unit of labor for which they receive income $w_j$. Commuting from $i$ to $j$ incurs iceberg utility costs $d_{ij}$. Cobb Douglas preferences over a freely traded numeraire good with share $1-\beta$ and housing $h_i$ with share $\beta$ and price $r_i$, with shifters from residential amenities $B_i$ and an idiosyncratic Frechet term $z_{ijk}$ for consumer $k$ give rise to indirect utility functions of the form,

\[V_{ijk} = z_i B_i (d_{ij})^{-1} \times {w_j \over r_i^\beta}\]

Integrating over the Frechet draws, which we assume to have location parameter $1$ and scale parameter $\varepsilon$, the commuting probabilities are,

\[\pi_{ij} = { ({ d_{ij} r_i^\beta})^{-\varepsilon} (B_i w_j)^\varepsilon \over \sum_{i',j'} ({ d_{i'j'} r_{i'}^\beta})^{-\varepsilon} (B_{i'} w_{j'})^\varepsilon } = { ({ d_{ij} r_i^\beta})^{-\varepsilon} (B_i w_j)^\varepsilon \over \Phi}\]

For consumer $k$, demand for housing is,

\[h_{ijk} = \beta {w_j \over r_i}\]

Which aggregates to,

\[h_i = \sum_j \pi_{ij} \beta {w_j \over r_i}\]

First, defining “resident market access,” i.e., access to well-paying jobs,

\[RMA_i = \sum_j (d_{ij})^{-\varepsilon} w_j^{1+\varepsilon}\]

we can rewrite the aggregate demand equation. We assume each location is endowed with a fixed supply of land, $\bar H_i$, so equating supply and demand,

\[\beta \frac1\Phi{RMA_j}r_i^{-(\beta\varepsilon+1)}B_i^{\varepsilon} = \bar H_i\]

Taking logs and rearranging,

\[\log r_i = \frac1{1+\varepsilon\beta}\bigg( \log \beta + \log RMA_i + \varepsilon \log B_i - \log \Phi - \log \bar H_i\bigg)\]

Now suppose the first order condition on profit maximization for a firm at location $j$ is,

\[A_j (L_j)^{-\alpha} = w_j\]

If we define “firm market access” as access to cheap housing,

\[FMA_j = \sum_i (d_{ij})^{-\varepsilon} (B_i / r_i^\beta)^{\varepsilon}\]

Then we can write labor supply as,

\[L_j = \sum_i \pi_{ij} = (w_j)^\varepsilon \times {FMA_j \over \Phi}\]


\[A_j w_j^{-(1+\varepsilon \alpha)} \times (FMA_j/\Phi)^{-\alpha} = 1\]


\[\log w_j = \frac1{(1+\varepsilon \alpha)}\bigg( \log A_j -\alpha \log FMA_j + \alpha \log \Phi\bigg)\]

The point

The results are pretty intuitive, despite the algrebra it took to get to them:

The point is that market access in the model is a demand shifter for residences and a supply shifter for productive factors. If the good is no longer freely traded, then output market market access is a labor demand shifter for firms, and so the impact of geography of wages is ambiguous and depends on the correlation between input and output market access.