Multidimensional Sorting: Comparative Statics

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.

💡 Research Summary

This paper addresses a fundamental gap in the existing literature on multidimensional assignment models: the lack of a comprehensive theory for comparative statics. In complex labor markets where agents possess multiple dimensions of skills, understanding how equilibrium states—specifically earnings and assignment patterns—respond to exogenous technological shocks is a significant mathematical and economic challenge. The authors successfully develop a complete theory that provides a rigorous framework for analyzing these dynamics.

The core contribution of the research is a novel decomposition theorem. The authors demonstrate that any technological change in a multidimensional assignment model can be uniquely decomposed into two distinct, orthogonal components. The first component, termed the “gradient” component, characterizes the changes in marginal earnings. By employing a Poisson equation, the authors provide a mathematical characterization of how technological shifts alter the potential function of earnings, effectively capturing how the value of specific skill dimensions increases or decreases. This component focuses on the change in the “magnitude” of rewards for given inputs.

The second component, the “divergence-free” component, characterizes the reallocation of labor. This part of the decomposition captures the structural shifts in the assignment of agents to tasks. It describes the “flow” of workers across the multidimensional skill space, representing how technological progress drives labor from one set of tasks to another, even in the absence of direct changes in marginal productivity. This distinction is crucial, as it separates changes in the “value” of skills from changes in the “structure” of the labor market.

To demonstrate the empirical relevance of this theoretical framework, the authors apply it to U.S. longitudinal data, specifically focusing on Cognitive Skill-Biased Technological Change (SBTC). They quantify how the rise of technologies that favor cognitive abilities has reshaped the equilibrium in the U.S. labor market. The analysis reveals that technological change does not merely shift the wage distribution upward for certain groups but fundamentally reconfigures the allocation of labor across different skill-intensive roles.

In conclusion, this paper provides a powerful analytical toolset for economists and policymakers. By decomposing technological change into gradient and divergence-free components, the research offers a precise way to distinguish between technology that increases the returns to specific skills and technology that structurally transforms the labor market. This framework is particularly vital for understanding the long-term implications of modern advancements such as Artificial Intelligence and automation on income inequality and structural unemployment.