Matching in closed-form: equilibrium, identification, and comparative statics

Abstract

This paper provides closed-form formulas for a multidimensional two-sided matching problem with transferable utility and heterogeneity in tastes. When the matching surplus is quadratic, the marginal distributions of the characteristics are normal, and when the heterogeneity in tastes is of the continuous logit type, as in Choo and Siow (J Polit Econ 114:172–201, 2006), we show that the optimal matching distribution is also jointly normal and can be computed in closed form from the model primitives. Conversely, the quadratic surplus function can be identified from the optimal matching distribution, also in closed-form. The closed-form formulas make it computationally easy to solve problems with even a very large number of matches and allow for quantitative predictions about the evolution of the solution as the technology and the characteristics of the matching populations change.

Appendix: Matrix differentiation

The Kronecker product and the vectorization operation are extremely useful when it comes to studying asymptotic properties involving matrices. The idea is that matrices $m\times n$, where m stands for the number of rows and n for the number of columns, can be seen as $mn\times 1$ vectors in $R^{mn}$, and linear operation on such matrices can be seen as higher order matrices. To do this, the fundamental tool is the vectorization operation, which vectorizes a matrix by stacking its columns. Introduce $\tau _{mn}$ a collection of invertible maps from $\left\{ 1,\ldots ,m\right\} \times \left\{ 1,\ldots ,n\right\}$ onto $\left\{ 1,\ldots ,mn\right\}$, such that $\tau _{mn}\left( i,j\right) =m\left( j-1\right) +i$.

Definition 1

For $\left( M\right)$ a $m\times n$ matrix, $\mathrm{vec}\left( M\right)$ is the vector $v\in \mathbb {R}^{mn}$ such that $v_{\tau _{mn}\left( i,j\right) }=M_{ij}$.

Next, we introduce the transposition tensor $\mathbb {T}_{m,n}$ as the $mn\times mn$ matrix such that

$$\begin{aligned} \mathbb {T}_{m,n}\mathrm{vec}\left( M\right) =\mathrm{vec}\left( M^{*}\right) . \end{aligned}$$

The matrix operator $T_{m,n}$ is a permutation matrix with zeros and a single 1 on each row and column. Note that $\mathbb {T}_{m,n}=\mathbb {T}_{n,m}^{-1}$, so $\mathbb {T}_{m,n}\mathbb {T}_{n,m}\mathrm{vec}\left( M\right) =\mathrm{vec}\left( M\right) .$ Furthermore, $\mathbb {T}_{m,n}=\mathbb {T}_{n,m}^{*}$. The next definition deals with Kronecker product, which is closely related to vectorization.

Definition 2

Let A be a $m\times p$ matrix and B an $n\times q$ matrix. One defines the Kronecker product $A\otimes B$ as the $mn\times pq$ matrix such that

$$\begin{aligned} \left( A\otimes B\right) _{n\left( i-1\right) +k,q\left( j-1\right) +l}=A_{ij}B_{kl}. \end{aligned}$$

The following fundamental property characterizes the Kronecker product.

Fact 1

For all $q\times p$ matrix X,

$$\begin{aligned} \mathrm{vec}\left( BXA^{*}\right) =\left( A\otimes B\right) \mathrm{vec}\left( X\right) \end{aligned}$$

The following important basic properties follow.

Fact 2

Let A be a $m\times p$ matrix and B an $n\times q$ matrix. Then:

• 1. (Associativity) $\left( A\otimes B\right) \otimes C=A\otimes \left( B\otimes C\right) .$

• 2. (Distributivity) $A\otimes \left( B+C\right) =A\otimes B+A\otimes C.$

• 3. (Multilinearity) For $\lambda$ and $\mu$ scalars, $\lambda A\otimes \mu B=\lambda \mu \left( A\otimes B\right)$

• 4. For matrices of appropriate size, $\left( A\otimes B\right) \left( C\otimes D\right) =\left( AC\right) \otimes \left( BD\right)$.

• 5. $\left( A\otimes B\right) ^{*}=A^{*}\otimes B^{*}$.

• 6. If A and B are invertible, $\left( A\otimes B\right) ^{-1}=A^{-1}\otimes B^{-1}$.

• 7. For vectors a and $b, a^{\prime }\otimes b=ba^{\prime }$ (in particular, $aa^{\prime }=a^{\prime }\otimes b$).

• 8. If A and B are square matrices of respective size m and n,

  $$\begin{aligned} \det \left( A\otimes B\right) =\left( \det A\right) ^{m}\left( \det B\right) ^{n}. \end{aligned}$$

• 9. $\mathrm{Tr}\left( A\otimes B\right) =\mathrm{Tr}\left( A\right) \mathrm{Tr}\left( B\right)$.

• 10. $\mathrm{rank}\left( A\otimes B\right) =\mathrm{rank}\left( A\right) \mathrm{rank}\left( B\right)$.

• 11. The singular values of $A\otimes B$ are the product of the singular values of A and those of B.

Let f be a smooth map from the space of $m\times p$ matrices to the space of $n\times q$ matrix. Define $\frac{df\left( A\right) }{dA}$ as the $\left( nq\right) \times \left( mp\right)$ matrix such that for an $m\times p$ matrix X,

$$\begin{aligned} \mathrm{vec}\left( \lim _{e\rightarrow 0}\frac{f\left( A+eX\right) -f\left( A\right) }{ e}\right) =\frac{df\left( A\right) }{dA}.\mathrm{vec}\left( X\right) . \end{aligned}$$

We use the notation $A^{-*}$ for $\left( A^{*}\right) ^{-1}$.

Fact 3

Let A be a $m\times p$ matrix and B an $n\times q$ matrix. Then:

• 1. $\frac{\mathrm{d}\left( AXB\right) }{\mathrm{d}X}=B^{*}\otimes A$.

• 2. $\frac{\mathrm{d}A^{*}}{dA}=\mathbb {T}_{m,p}$.

• 3. $\frac{\mathrm{d}A^{-1}}{\mathrm{d}A}=-\left( A^{-*}\otimes A^{-1}\right)$.

• 4. $\frac{\mathrm{d}A^{2}}{\mathrm{d}A}=I\otimes A+A^{*}\otimes I$

• 5. For A symmetric, $\frac{\mathrm{d}A^{1/2}}{\mathrm{d}A}=\left( I\otimes A^{1/2}+A^{1/2}\otimes I\right) ^{-1}$.