Mills' mistake: the negative exponential rent gradient

What does the rent gradient look like in a model of a monocentric city? The literature seems to think it is negative exponential, owing to Edwin Mills’ (1972) monograph, “Studies in the Structure of the Urban Economy,” a PDF of which I can’t find online. Recent work employs a negative-exponential rent gradient as exemplary of the “Standard Urban Model” (Jedwab et al., 2021), and properties of the monocentric city model with an exponential rent gradient are still a somewhat active research area (see, e.g., Zhao, 2017).

Mills’ work apparently employs functional forms to pin down an example of a monocentric city. In particular, he gives agents Cobb-Douglas preferences over housing and a consumption good. This turns out to be the source of his “mistake:” a negative exponetial rent gradient is not consistent with these preferences.

Following the exposition in Duranton and Puga (2015), the idea is as follows: households, living $x$ units away from the CBD, consume a numeraire consumption good $c(x)$, and housing $h(x)$ with price $r(x)$. Letting household income be wage net commuting costs, $w-\tau x$, we can define the bid-rent gradient,

i.e., $R(x,\bar u)$ is the maximum rent a household would pay in order to achieve utility $\bar u$ when living at $x$. At a spatial equilibrium, all households must enjoy the same level of utility no matter where they live. Solving for the Hicksian demand for $c$, $c^*(h,\bar u)$, this value is,

Since $c$ has been maximized, we can apply the envelope theorem, and so the derivative of this with respect to $x$, (i.e., the rent gradient), is,

Here is where Mills allegedly gets confused. If we divide both sides by $r(x)$, we have,

Now, if $r(x)h^*(r(x),\bar u)$ is a constant in $x$, this is a separable ordinary differential equation, and the resulting rent gradient is exponential. Mills’ mistake is to employ Cobb-Douglas preferences, which generates a constant share of income spent on each good. However, with commuting costs, disposable income is not constant across $x$! This seems to be first noted by Brueckner (1982), who writes that “Mills’ illegitimate step” is to ignore “crucial fact that consumer purchasing power declines with $x$,” and whose entire article I’m riffing on now. This is a fairly huge mistake, since the introduction of commuting costs drive all spatial variation in the monocentric city model.

To generate a declining rent gradient, we require,

which follows from the chain rule and employing the definition of $ {d r(x) \over d x}$ as derived above. This says that the Hicksian demand for housing must be unit elastic. Cobb-Douglas demand has a unitary Marshallian demand elasticity. As per the Slutsky equation, for Hicksian and Marshallian demand to share the same demand elasticity, the income elasticity of housing must be zero.

Alternatives

If one still wanted to use Cobb Douglas preferences, the analysis is still tractable, though the density gradient is neither isoelastic or exponential in $x$, as I derive here.

If one wanted to retain an exponential rent gradient, one could throw away the assumption that income falls in distance, and put $x$ directly into the utility function. Imposing that utility is of the form,

will produce a negative exponential rent gradient, basically by assertion. This is the “amenity” interpretation of commuting costs – households want to live downtown. This is similar to asserting that transportation costs are negative exponential in a monocentric Von Thunen economy, as in Fujita/Krugman/Venables (1999).

Kim and McDonald (1987) show that quasi-linear preferences, $U(c,h)=c + \beta \log h$ will generate the desired property: a unitary Hicksian demand elasticity for housing, as housing demand will be independent of income. With this demand system, the negative exponential rent gradient again becomes an equilibrium phenomenon, an emergent property of the model.