A baseline toy AMM model

Recently, I wanted a baseline model of a city that is tractable enough to work with and doesn’t require me to lean too heavily on the computer so I can think about how deviations from standard assumptions affect urban geography. I picked the right functional forms and ended up with something tractable that delivers analytic solutions, and thought I’d share as a public good.

The (workhorse) quantitative spatial model of Ahlfeldt et al. (2015) is great for quantitative work but makes opaque the underlying urban economics, unless you make enough assumptions to make it (effectively) log-linear.

To make some progress on this, I’m going to work with a variation of the Alonso-Muth-Mills model. This model is indeed so workhorse that its exposition is all over youtube (see: Ed Glaeser or Kevin Murphy).

The basic insight of the model is right off the budget constraint. Suppose everyone gets wage $w$ and consumes a unit of housing at price $p(x)$, and pays commputing costs $\tau(x)$. Provided location $x$ (distance to the central business district) doesn’t directly enter preferences, the budget constraint,

\[\text{consumption} + p(x) = w - \tau(x)\]

implies that,

\[\frac{dp}{dx} = - \frac{d \tau}{dx}.\]

Which is usually heralded as the key insight: it’s all hedonics. The housing price gradient exactly offsets commuting costs; house prices in cities reflect compensating differentials.

The remainder of the treatment of this model in typical texts (see, e.g., Brueckner, 1987) relies on non-specific utility functions or graphical analysis to get the remainder of the comparative statics. I want some nice functional forms to make clear the role of preferences, supply, and spatial equilibrium in determining the economic geography of the city, and also make it clear how messing with the standard assumptions will affect the equilibrium allocation.

The toy model

Environment We’ve got a linear city where everyone works $x$ units from the central business district. There is a unit mass of workers distributed along the line with mass $n(x)$ at each point. Housing per unit floorspace $h(x)$ costs $p(x)$. Each location on the line has a unit of land with price $r(x)$. Workers commute to the city center to work at jobs that pay $w$. Everyone lives along $[0,\bar x]$, where $\bar x$ is determined by the reservation value of land, $\bar r$, (e.g., agriculture), so that $r(\bar x) = \bar r$.

Equilibrium is a set of prices $r(x),p(x)$ and quantities $h(x)$ and $n(x)$ such that households optimize, developers optimize, landlords optimize, and the market for floorspace clears.

I’ll solve the model out and make the necessary assumptions below:

Preferences Households have Cobb-Douglas preferences over consumption, $c$ (the numeraire good), and housing, $h$,

\[U(c,h) = \kappa c^{1-\beta} h^\beta\]

where $\kappa$ is an constant that will kill nuisance terms. Households have budget constraints,

\[c + p(x) h(x) = w-\tau x\]

Optimization means,

\[{d U / d H \over d U / d c} = p(x)\]

while spatial equilibrium means,

\[U(c,h) = \bar u.\]

The utility level, $\bar u$, is endogenous.

Housing Floorspace $f(x)$ is produced with an upward-sloping convex cost function and sold at price $p(x)$. Developers bid for land owned by landowners through a strategyproof mechanism such that the winning bid generates zero profits for the developer, and moreover the price of land $r(x)$ is equal to its profitability.

Developers solve,

\[p(x)f(x) - C( f(x) ) - r(x)\]

where the cost function is $C(f) = \frac\gamma{(1+\gamma)}f^{1+1/\gamma}$.

Maximization generates,

\[f(x) = p(x)^\gamma\]

while zero profits the difference of total marginal and average costs are zero,

\[r(x) = p(x)^{1+\gamma}(1+\gamma)^{-(1+\gamma)}\]

Let $n(x)$​ denote the number of people in a given location, so that when the floorspace market clears,

\[\beta y(x) n(x) = p(x)^{1+\gamma}\]

where $y(x) = w - \tau x$. Using the zero profit condition, we can now write,

\[r(x) = {\beta \over 1 + \gamma} y(x) n(x)\]

Landlord optimization means that,

\[r(\bar x) = \bar r.\]

Since spatial equilibrium holds at the border, we can use preferences to write,

\[\bar r \big( y(x)/ y(\bar x) \big)^{1+\gamma \over \beta} = \frac\beta{1+\gamma} y(x)n(x).\]

This allows us to solve for $n(x)$,

\[n(x) = \frac{1+\gamma}\beta \bar r \big( y(x)/ y(\bar x) \big)^{1+\gamma \over \beta} 1/y(x) .\]

Finally, the labor market must clear,

\[\int_0^{\bar x} n(x) dx = 1 .\]

We’ve now reduced the model to one equation in one unknown. Solving the labor market clearing condition allows me to back out,

\[\bar x = (w / \tau) \bigg( 1 - \bigg( 1 + \frac\tau{\bar r} \bigg)^{-\beta \over 1 + \gamma} \bigg) .\]

This gives some nice comparative statics on urban form. The city gets bigger as $w$ increases, and smaller as commuting costs change, though the effect of commuting costs is nonlinear. As the value of agricultural land grows, the city must get denser. When housing supply $\gamma$ is elastic, the city gets denser.