We develop a model where risk-averse workers can costly invest in their skills before matching with heterogenous firms. At the investment stage, workers face multiple sources of risk. They are uncertain about how skilled they will turn out and also about their income shock realizations at the time of employment. We analyse the equilibria of two versions of the model that depend on when uncertainty resolves, which determines the available risk-sharing possibilities between workers and firms. We provide a thorough analysis of equilibrium comparative statics regarding changes in risk, worker and firm heterogeneity, and technology. We derive conditions on the match output function and risk attitudes under which these shifts lead to more investment and show how this affects matching and wages. To illustrate the applied relevance of our theory, we provide a stylized quantitative assessment of the model and analyse the sources (risk, heterogeneity, or technology) of rising U.S. wage inequality. We find that changes in risk were the most important driver behind the surge in inequality, followed by technological change. We show that these conclusions are significantly altered if one neglects the key feature of our model, which is that educational investment is endogenous.

In many economic applications of matching, the teams that form compete later in market structures with strategic interactions or with knowledge spillovers. Such post-match competition introduces externalities at the matching stage: a team’s payoff depends not only on their members’ attributes but also on those of other matched teams. This article develops a large market model of matching with externalities, in which first teams form, and then they compete. We analyse the sorting patterns that ensue under competitive equilibrium as well as their efficiency properties. Our main results show that insights substantially differ from those of the standard model without externalities: there can be multiple competitive equilibria with different sorting patterns; both optimal and competitive equilibrium matching can involve randomization; and competitive equilibrium can be inefficient with a matching that can drastically deviate from the optimal one. We also shed light on the economic relevance of our matching model with externalities. We analyse two economic applications that illustrate how our model can rationalize the trend in within- and between-firm inequality, and also the evolution of markups of sectors where firms have market power.

While many real-world principal-agent problems have both moral hazard and adverse selection, existing tools largely analyze only one at a time. Do the insights from the separate analyses survive when the frictions are combined? We develop a simple method—decoupling—to study both problems at once. When decoupling works, everything we know from the separate analyses carries over, but interesting interactions also arise. We provide simple tests for whether decoupling is valid. We develop and numerically implement an algorithm to calculate the decoupled solution and check its validity. We also provide primitives for decoupling to work and analyze several extensions. (JEL D82, D86)