Multidimensional Sorting: Comparative Statics

2025-12-11

AI Summary

Comparative statics of technological change in multidimensional assignment models—under general production functions and input distributions—has remained an open problem. Method: This paper introduces a novel orthogonal decomposition framework that uniquely decomposes any technology shift into a gradient component (capturing changes in marginal returns, governed by a Poisson equation) and a divergence-free component (representing labor reallocation). Integrating vector field decomposition, multidimensional optimal transport theory, and quantitative equilibrium response modeling, we develop the first comprehensive comparative statics theory for general settings. Contribution/Results: Applying this framework, we precisely identify and quantify the equilibrium effects of U.S. cognitive-skill-biased technological change on occupational sorting and income distribution. Our approach provides a unified analytical paradigm for research on multidimensional matching and technical progress.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Search and Optimization: Distributed Search
• Multiagent Systems: Mechanism Design

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Develops comparative statics for multidimensional assignment models
• Decomposes technological change into gradient and divergence-free components
• Quantifies sorting and earnings responses to skill-biased technological change

Innovation

• Decomposes technological change into gradient and divergence-free components
• Characterizes marginal earnings changes via Poisson equation
• Quantifies equilibrium responses to skill-biased technological change