Allen and Arkolaks (2014) via Allen et al. (2024)
Allen and Arkolakis (2014) develops an economic geography model that’s been the backbone to much quantiative spatial work in the last ten years. The framework allows for externalities in the form of local agglomeration and congestion forces, and proves the existence and uniqueness of the equilibrium when agglomeration forces do not outweigh congestion forces.
This is a classic condition in basically every spatial paper. However, the proof in that paper relies on characterizing the system with a homogenous Hammerstein equation of the second kind and uses a bunch of old theorems from Jentzsch, Fredholm, and looking up stuff in Zabreyko et al. (1975)’s handbook of integral equations. This may be fun for some, but it’s not particularly helpful if you’re trying to work with some extension of the model that’s not isomorphic to the framework in the paper. Levi Crews has some guidance on this.
Along comes Allen et al. (2024) which provides a theorem that allows one to check for existence and uniqueness of equilibrium in a broad class of spatial models. The theorem says, suppose there are $N$ locations indexed by $i,j$, and $H$ “interactions” (i.e., types of endogenous variables) indexed by $h$. Then, if you can express the equilibrium as,
\[x_{ih} = \sum_{j=1}^N f_{ijh}\big( x_{j,\cdot} \big),\]you can write an $H\times H$ matrix of the supremum of the elasticities of $f$, $\bf A$ whose elements are,
\[({\bf A})_{h,h'} \equiv \sup_{ij} \bigg| {\partial \ln f_{ijh} \over \partial \ln x_{j,h'} } \bigg|\]and then check if its spectral radius, $\rho(\bf A) < 1$ to guarantee existence and uniqueness. However, in most of these economic models, one of the variables is idenfied only up-to-scale (say, prices, which are expressed in nominal terms), you’re going to find $\rho( \bf A)=1$. In this case, it’s sufficient to check if there’s some region $j$ for which all the elasticities are strictly bounded above by the corresponding entry in $\bf A$, or if all the model is constant-elasticity, in which case uniqueness returns.
Unfortunately few economic systems are exactly written in the form above. Generally, the equilibrium system of equations in spatial models is nonlinear in that nonlinear functions of $x$ appear on both the lefthandside and righthandside of the equilibrium system. Consequenrly, such systems cannot easily be reduced to the form above algebraically. The paper provides a change-of-variables formula to handle this. Essentially, if the matrix of righthandside elasticities is ${\bf B}$ , and the matrix of lefthandside elasticities is $\Gamma$, then, ${\bf A}$ is the absolute value of ${\bf B} \Gamma^{-1}$.
As an application and a bit of a public good cookbook, I’ll walk through this check for the original Allen and Arkolakis (2014) model.
The original Allen & Arkolakis (2014) setup
There are a discrete number of locations indexed $i,j$, each of which produces a unique traded variety over which consumers have constant-elasticity-of-substitution preferences with elasticity of substitution $\sigma$. Varieties are traded with iceberg trade costs $\tau_{ij}$. Labor is the only factor of production, and is subject to increasing returns, so that, output in location $i$ can be written as $y_i = Z_i L_i$ and $Z_i = \bar Z_i (L_i)^\alpha$, so $\alpha\geq0$ captures the extent of increasing returns. Workers earn their marginal product, so nominal wages reflect the marginal revenue product of labor, $w_i = p_i Z_i$.
Consumers are freely mobile, and so a spatial equilibrium holds, in that welfare $\mathcal W$ must be everywhere equalized across space. Location welfare depends on the amenity adjusted real wage, $A_i w_i / P_i$, where $P_i$ is the price index. Amenities are subject to congestion externalities, $A_i = \bar A_i (L_i)^{\beta}$ with $\beta < 0$ reflecting the demons of density.
The equilibrium is characterized by three equations:
Goods market clearing,
\[w_i L_i = \sum_j \tau_{ij}^{1-\sigma} \left( {w_i/Z_i \over P_j} \right)^{1-\sigma}w_j L_j\]Spatial equilibrium,
\[\mathcal W = A_i {w_i \over P_i}\]and the price index,
\[P_i = \left( \sum_j \tau_{ji}^{1-\sigma} (w_j/Z_j)^{1-\sigma} \right)\]To reduce this to a system of $NH$ nonlinear equations in the endogenous variables, ${ w_i, L_i}$, replace the price index in good market clearing with the spatial equilibrium equation,
\[\begin{align*} w_i L_i &= \sum_j \tau_{ij}^{1-\sigma} \left( {w_i/Z_i} \right)^{1-\sigma} P_j^{\sigma-1} w_j L_j \\ &= \sum_j \tau_{ij}^{1-\sigma} \left( {w_i/Z_i} \right)^{1-\sigma} \left( A_j w_j \mathcal W^{-1} \right)^{\sigma-1} w_j L_j \\ \end{align*}\]Then, substitute for $A_i$ and $Z_i$ with their definitions, and rearrange so endogenous variables with index $i$ are on the lefthandside, while the $j$-indexed variables are on the right,
\[\begin{align*} w_i L_i &= \mathcal W^{1-\sigma} \sum_j \tau_{ij}^{1-\sigma} \left( {w_i} \right)^{1-\sigma} \bar Z_i^{\sigma-1} L_i^{(\sigma-1)\alpha} \left( w_j \right)^{\sigma-1} \bar A_j^{\sigma-1} (L_j)^{(\sigma-1)\beta} w_j L_j \\ \implies w_i^\sigma L_i^{1+(1-\sigma)\alpha} &= \mathcal W^{1-\sigma} \sum_j \tau_{ij}^{1-\sigma} \bar Z_i^{\sigma-1} \bar A_j^{\sigma-1} w_j^\sigma L_j^{1-(1-\sigma)\beta} \\ \end{align*}\]Now, to get the second equation, substitute into the price index the spatial equilibrium condition, and then substitute for the externality terms and rearrange,
\[\begin{align*} A_i w_i \mathcal W^{-1} &= \left( \sum_j \tau_{ji}^{1-\sigma} (w_j/Z_j)^{1-\sigma} \right)^{1 \over 1 - \sigma} \\ \implies \bar A_i^{1-\sigma} w_i^{1-\sigma} L_i^{(1-\sigma)\beta} &= \mathcal W^{1-\sigma} \sum_j \tau_{ji}^{1-\sigma} (w_j)^{1-\sigma} \bar Z_j^{\sigma-1} L_j^{-(1-\sigma)\alpha} \end{align*}\]So, to recap, our two equations are,
\[\begin{align} w_i^\sigma L_i^{1+(1-\sigma)\alpha} &= \mathcal W^{1-\sigma} \sum_j \tau_{ij}^{1-\sigma} \bar Z_i^{\sigma-1} \bar A_j^{\sigma-1} w_j^\sigma L_j^{1-(1-\sigma)\beta} \\ w_i^{1-\sigma} L_i^{(1-\sigma)\beta} &= \mathcal W^{1-\sigma} \sum_j \tau_{ji}^{1-\sigma} (w_j)^{1-\sigma} \bar A_i^{\sigma-1} \bar Z_j^{\sigma-1} L_j^{-(1-\sigma)\alpha}. \end{align}\]We now need to construct $\bf A$ by constructing $\bf B$ and $\bf \Gamma$.
The matrix $\bf B$ contains the righthandside elasticities with respect to $w$ and $L$, and $\Gamma$ contains those on the lefthandside. So,
\[\bf B = \begin{pmatrix} \sigma & 1-(1-\sigma)\beta \\ 1-\sigma & -(1-\sigma) \alpha \end{pmatrix}, \quad \Gamma = \begin{pmatrix} \sigma & 1+(1-\sigma)\alpha \\ 1-\sigma & (1-\sigma)\beta \end{pmatrix}\]Inverting $\Gamma$ by hand is instructive, since we won’t need to multiply the determinant through when constructing eigenvalues, since the eigenvalues of $a \bf M$ are just $a \times$ the eigenvalues of $\bf M$.
Consequently,
\[1/\det \Gamma = \frac1{(\sigma-1)(1+(1-\sigma)\alpha - \sigma\beta)}\]We’ll assume for now that this obect is positive $1+\alpha>\sigma(\alpha+\beta)$, otherwise we’d need to flip the sign when taking absolute values. Dividing out $(\sigma-1)$ from $\Gamma^{-1}$,
\[{\bf A} = | {\bf B} {\bf \Gamma^{-1} } | = \bigg|\frac1{1+(1-\sigma)\alpha - \sigma\beta} \cdot \begin{pmatrix} \sigma & 1-(1-\sigma)\beta \\ 1-\sigma & -(1-\sigma) \alpha \end{pmatrix} \begin{pmatrix} -\beta & {-(1+(1-\sigma)\alpha) \over \sigma-1} \\ 1 & {\sigma \over \sigma-1} \end{pmatrix} \bigg|\]Multiplying matrices,
\[| {\bf B} {\bf \Gamma^{-1} } | = \bigg|\frac1{1+(1-\sigma)\alpha - \sigma\beta} \cdot \begin{pmatrix} 1-\beta & \sigma(\alpha+\beta) \\ (\sigma-1)(\alpha+\beta) & 1+\alpha \end{pmatrix} \bigg|\]We now have to find the spectral radius of this, which is the largest eigenvalue in absolute value. To compute the eigenvalues, I use the trace and determinant to compute the discriminant of the characteristic polynomial.
The trace and determinant of the matrix are,
\[\begin{align*} \operatorname{tr} &= 2 + \alpha - \beta \\ \det &= (1-\beta)(1+\alpha)-\sigma(\sigma-1)(\alpha+\beta)^2 \end{align*}\]The discriminant is,
\[\text{discriminant}= \text{tr}^2 - 4\det = (\alpha+\beta)^2(2\sigma-1)^2\]so the eigenvalues are,
\[\begin{align*} \lambda_1 &= \frac1{1+(1-\sigma)\alpha - \sigma\beta} \frac12 \left( 2 + \alpha - \beta + (\alpha+\beta)(2\sigma-1) \right) \\ \lambda_2 &= \frac1{1+(1-\sigma)\alpha - \sigma\beta} \frac12 \left( 2 + \alpha - \beta - (\alpha+\beta)(2\sigma-1) \right) \end{align*}\]Simplifying,
\[\lambda_1 = {1-\beta + \sigma(\alpha + \beta) \over 1 + (1-\sigma)\alpha - \sigma \beta}, \quad \lambda_2 = 1\]To guarantee existence and uniqueness, then, we must ensure that $\lambda_1 < 1$ so that the spectral radius remains 1 (as is the case in models with nominal endogenous variables). This holds if,
\[{1-\beta + \sigma(\alpha + \beta)} < 1 + (1-\sigma)\alpha - \sigma \beta \implies (2\sigma-1)(\alpha+\beta)< 0\]as $\sigma>1$, it must be that $\alpha+\beta < 0$ to guarantee existence and uniqueness. In short, congestion forces must dominate.
To recap, the cookbook is,
- Reduce the equilibrium to a system of $N H $ equations.
- Write the equations so that $i$ endogenous variables are on the lefthandside, and $j$ endogenous variables (which are summed over) are on the righthandside.
- Construct $B$ and $\Gamma$, and compute $B \Gamma^{-1}$.
- Apply the absolute value operator to $B \Gamma^{-1}$.
- Compute the eigenvalues, and see if there is a parameter restriction that ensures the largest in absolute value is less than (or equal to) 1.