Quantitative spatial models are just Rosen-Roback with market access
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.
This is almost visible in the text: recovering amenities stems from writing down equations that are log-linear in the endogenous variables. The text only presents two of these equations, even though the model has 5 endogenous variables (employment and residence headcounts $H_M$ and $H_R$, wages $w$, and prices of floorspace ($q$ for commercial, $Q$ for residential), plus the share of floorspace devoted to residential activities, $\theta$). The geography is discrete; sites are indexed by $i$ or $j$, and connected via the transportation network, $d_{ij}$, where $d_{ij}$ is an iceberg commuting cost. People pick residence $i$ and workplace $j$, pay commuting costs, $d_{ij}$ and have Frechet preference shocks with scale parameter $T_i E_j$ and shape parameter $\varepsilon$.
Ahlfeldt et al. are unable to write down five log-linear equations because the way the land market is modeled. They have a fixed supply of land $\bar K_i$, and density, $\varphi_i$ (which is for some reason exogenous too), and the share of floorspace ($L$) devoted to commercial ($M$) and residential ($R$) activity, is endogenous and split via $\theta_i$. The big problem with this is that they back out this object and equilibrium prices via,
\[\underbrace{\theta_i L_i}_{\text{commercial L demand}} + \underbrace{(1-\theta_i)L_i}_{\text{residential L demand}} = \underbrace{\varphi_i \bar K}_{\text{inelastic supply of L}}\]and the paper doesn’t really pay attention to wages since they don’t have them in the data.
If you change the model of the land market, so the market clearing condition is multiplicative in residential and commercial floorspace, you can recover an analytic expression for the price of floorspace. Then it’s relatively straightforward (though obscenely tiresome, algebraically) to write down log-linear equations for the endogenous variables as a function of the exogenous variables (amenities) and market access terms.
It makes sense (to me) to put these on the RHS since they’re external to the commuter and the firm; no agent ought to internalize their effect on equilibrium market access when optimizing their behavior.
Too much algebra
The land market
Starting off with the land market, we suppose there’s $\bar K_i$. Let “density” at a given site be endogenous and denoted $\varphi_i$.
Write “market clearing” in the following CRS way,
\[\varphi_i \bar K_i = L_{Mi}^\mu L_{Ri}^{1-\mu}\]So $\mu = \frac1{1- \epsilon_{L_M, L_R}}$, i.e., holding lot density and size fixed, if a 1% increase in $L_R$ means a 1% decrease in $L_M$, i.e., the elasticity of commercial floorspace with respect to residential floorspace is $\epsilon_{L_M,L_R}=-1$, then $\mu=1/2$. NB: this isn’t the same $\mu$ as in ARSW.
To see this, rearrange,
\[L_M = \varphi^{1/\mu} K^{1/\mu} L_R^{1-1/\mu}\]Assuming that commercial and residential floorspace isn’t substituted one-for-one is a bit weird, but it’s necessary to close the model analytically. Maybe this is due to zoning or firecode regulations. Key to stomaching this is noting that $\varphi$ doesn’t have normal units, like height.
Then, if you assume that density has convex costs, $C(\varphi) = \varphi_0 \varphi^\delta$, construction firms solve,
\[\max_{L_R, L_M, \varphi} qL_M + Q L_R - \varphi_0 \varphi^\delta \quad\text{s.t.}\quad \varphi \bar K = L_{M}^\mu L_{R}^{1-\mu}\]The Cobb-Douglasness of the problem means that the first order conditions on commercial and residential floorspace give an equilibrium relationship between commercial and residential floor prices:
\[q = Q {\mu \over 1-\mu} {L_R \over L_M}\]The first order condition on density allows you to solve for the lagrange multiplier, which gives you an equilibrium relationship between residential floor prices and the demands for floorspace.
\[Q = \varphi_0\delta(1-\mu) K^{-\delta}L_R^{(1-\mu)\delta-1}L_M^{\delta\mu}\]This is the object that’s missing in ARSW and is going to help solve out the whole model.
Technology
Start with production technology, which is also Cobb-Douglas,
\[y_j = A_j (H_{Mj})^\alpha (L_{Mj})^{1-\alpha}\]This is going to give us zero profits. The first order conditions for $H_M$ and $L_M$ give,
\[{\alpha \over 1-\alpha} {L_{Mj} \over H_{Mj}} = {w \over q}\]Zero equilibrium profits means,
\[y_j - w_j H_{Mj} - q_j L_{Mj} = 0 \\ A_j (L_{Mj}/H_{Mj})^{1-\alpha} - q_j L_{Mj} = w_j\]Plugging in the first order condition on labor,
\[(L_M / H_M)^{1-\alpha} = w_j / \alpha A_j\\ \implies w = q^{-{1-\alpha\over\alpha}} ((1-\alpha)A_j)^{1/\alpha}{\alpha\over1-\alpha}\]Then demand for floorspace is,
\[L_{Mj} = (1-\alpha)^{1/\alpha} q^{-{1\over\alpha}} (A_j)^{1/\alpha} H_{Mj}\]Preferences
First, defining,
\[\begin{align*} \psi_i &\equiv (T_i^{1/\varepsilon}B_i Q_i^{\beta-1})^\varepsilon \\ \omega_j &\equiv (E_j w_j)^\varepsilon \end{align*}\]the model’s gravity equation is,
\[\pi_{ij} \propto (d_{ij})^{-\varepsilon} \psi_i \omega_j\]If you sum across $i$s and $j$s workplace employment and residence population quantities, you get,
\[H_{Mj} = \omega_j WMA_j \\ H_{Ri} = \psi_i RMA_i\]Where $WMA$ is “workplace market access” (access to cheap homes via the transportation network) and $RMA$ is “resident market access” (access to high paying jobs).
The Cobb-Douglas structure here gives easy demand equations – the demand for residential floorspace is then,
\[L_{Ri} = (1-\beta)\mathbb E[w_s \mid i] H_{Ri} / Q_i\]Where $\mathbb E[w_s \mid i]$ is the expected income of a generic resident living at site $i$. A fun way to write this down is,
\[\begin{align*} E[w_s \mid i] H_{Ri} &= \sum_j \pi_{ij\mid j} w_j H_{Mj} \\ &= \psi_i \sum_j (d_{ij})^{-\varepsilon} {w_j H_{Mj} \over WMA_j} \\ &= \psi_i \widetilde{RMA}_i \end{align*}\]This is because the market access terms solve,
\[\begin{align*} RMA_i &= \sum_j (d_{ij})^{-\varepsilon} {H_{Mj} \over WMA_j} \\ WMA_j &= \sum_i (d_{ij})^{-\varepsilon} {H_{Rj} \over RMA_j} \end{align*}\]so the second line of the expected income calculation is a “wage-weighted” version of $RMA$.
Solving
First, solving for the demand of commercial floorspace (algrebraic “algorithm”):
- plug labor supply into commercial floorspace demand, so that it depends on wages and floor prices, and then use zero profits to replace wages so that commercial floorspace demand is a function of floor price only.
- Use construction firm profit maximization to replace the commercial price of floorspace with a function of the residential floor price, and the quantities demanded for commericial and residential floorspace.
- Replace the quantity demanded for residential floorspace with the consumer’s demand equation, which depends only on the price of residential floorspace and market access.
- Rearrange so that quantity of commercial floorspace is on one side of the equation, and the price, fundamentals, and market access on the other.
This should get you,
\[L_{Mj} = Q_i^{\varepsilon(\beta-1)\over 1-\alpha}\left(\tilde B_i \widetilde{RMA}_i\right)^{1/(1-\alpha)} E_j A_j^{(1+\varepsilon)\over\alpha-1} WMA_j \tilde \alpha\]where $\tilde B_i$ has a $T_i^{1/\varepsilon}$ multiplied into it, and $\tilde \alpha$ is a sinister constant with a bunch of $\alpha$ and $\mu$ terms.
Then, using, residential floorspace demand, $L_{Ri} = (1-\beta) \tilde B_i Q_i^{\varepsilon(\beta-1)-1}\widetilde{RMA}_i$, one can solve the construction firm equation that relates floorspace quantities and prices, which recovers $Q$ as a function of the fundamentals and market access terms alone,
\[\ln Q = \Psi_Q - \delta\Lambda \ln K + \Sigma\Lambda \ln B + \Sigma\Lambda \ln \widetilde{RMA} + \delta\mu\Lambda \ln \tilde A + \delta\mu\Lambda \ln WMA\]Where $\Psi_Q$ is some terrible constant, and all the greeks are positive, provided the term that makes density costs convex, $\delta$ is “big enough” ($\delta > \frac1{1-\mu}$). The greeks here are messy nonlinear functions of the elasticities, e.g.,
\[\Lambda^{-1} = 1 - ((1-\mu)\delta-1) (\varepsilon(\beta-1)-1) - {\delta\mu\varepsilon(\beta-1)\over 1-\alpha}\]So long as I can sign them, I would prefer not to think about them.
The zero equilibrium profits equation, with floorspace demand and $\ln Q$ plugged is informative about wages,
\[\ln w_j = \Psi_w + \Delta\ln K_j \Theta \ln \widetilde{RMA}_j - \Pi \ln WMA_j + \Upsilon \ln A_j + \Theta \ln \tilde B_j\]The sign on the greeks here is ambiguous, e.g., $\Pi >0$ but need, $\Lambda^{-1}>\varepsilon(1-\beta)\Sigma\implies \Theta >0$.
Then, with prices, we can go back to quantities,
\[\ln H_{Mj} = (1+\varepsilon\Pi)\ln WMA_j + \varepsilon \Theta \ln \widetilde{RMA}_j + \ln \hat A_j + \varepsilon \Theta \ln \tilde B_j\]Where $\hat A$ has some $E_j$s multiplied in, and is raised to some positive powers. Finally,
\[\small \ln H_{R} = \Psi_{H_R} + \Delta \ln K + \Theta \ln \tilde B + (\Theta-1)\ln \widetilde{RMA} + \ln RMA + (\Upsilon-1)\ln \tilde A + (\Upsilon-1) \ln WMA\]Where, e.g., $\Upsilon -1 = \delta\mu\varepsilon(\beta-1)< 0$ and $\Theta -1 = \varepsilon(\beta-1)\Sigma \Lambda < 0$.
The point
The results are pretty intuitive, despite the algrebra it took to get to them:
- Raising the resident market access of a location ought to increase its population, all else equal.
- Increasing firms’ productivity at a location will decrease residential population because it will increase demand for commercial floorspace, which, because of density costs, will bid up residential floorspace prices.
- Wages are increasing in productivity (intuitive) but decreasing in workplace market access, as wages will no longer have to compensate for costly commutes.
Some simplifications to this mess are probably key to nailing the signs and getting some clean empirical predictions.