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.

We introduce and solve a new class of “downward-recursive” static portfolio choice problems. An individual simultaneously chooses among ranked stochastic options, and each choice is costly. In the motivational application, just one may be exercised from those that succeed. This often emerges in practice, such as when a student applies to many colleges or when a firm simultaneously tries several technologies. We show that such portfolio choice problems quite generally entail maximizing a submodular function of finite sets—which is NP-hard in general. Still, we show that a greedy algorithm finds the optimal set, finding first the best singleton, then the best single addition to it, and so on. We show that the optimal choices are “less aggressive” than the sequentially optimal ones, but “more aggressive” than the best singletons. Also, the optimal set in general contains gaps. We provide some comparative statics results on the chosen set.

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.

A finite number of vertically differentiated firms simultaneously compete for and screen agents with private information about their payoffs. In equilibrium, higher firms serve higher types. Each firm distorts the allocation downward from the efficient level on types below a threshold, but upward above. While payoffs in this game are neither quasi‐concave nor continuous, if firms are sufficiently differentiated, then any strategy profile that satisfies a simple set of necessary conditions is a pure‐stategy equilibrium, and an equilibrium exists. A mixed‐strategy equilibrium exists even when firms are less differentiated. The welfare effects of private information are drastically different than under monopoly. The equilibrium approaches the competitive limit quickly as entry costs grow small. We solve the problem of a multi‐plant firm facing a type‐dependent outside option and use this to study the effect of mergers.

Toward understanding assortative matching, this is a self-contained introduction to research on search and matching. We first explore the nontransferable and perfectly transferable utility matching paradigms, and then a unifying imperfectly transferable utility matching model. Motivated by some unrealistic predictions of frictionless matching, we flesh out the foundational economics of search theory. We then revisit the original matching paradigms with search frictions. We finally allow informational frictions that often arise, such as in college-student sorting.

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.