Multidimensional Sorting: Comparative Statics

TL;DR

This paper develops a comprehensive theory of comparative statics for multidimensional assignment models, decomposing technological change into components that explain changes in earnings and labor reallocation, with empirical application to U.S. data.

Contribution

It introduces a novel decomposition of technological change into gradient and divergence-free components in multidimensional sorting models.

Findings

Quantifies the effects of cognitive skill-biased technological change on sorting and earnings in U.S. data.

Provides a complete theoretical framework for comparative statics in multidimensional assignment models.

Characterizes labor reallocation and marginal earnings changes through a Poisson equation.

Abstract

In sorting literature, comparative statics for multidimensional assignment models with general output functions and input distributions is an important open question. We provide a complete theory of comparative statics for technological change in general multidimensional assignment models. Our main result is that any technological change is uniquely decomposed into two distinct components. The first component (gradient) gives a characterization of changes in marginal earnings through a Poisson equation. The second component (divergence-free) gives a characterization of labor reallocation. For U.S. data, we quantify equilibrium responses in sorting and earnings with respect to cognitive skill-biased technological change.