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 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.
We show that a greedy algorithm finds the optimal set. The optimal choices are “less aggressive” than the sequentially optimal ones, but “more aggressive” than the best singletons. The optimal set in general contains gaps. We provide a comparative static on the chosen set.
Keywords: College application, Submodular optimization, Greedy algorithm, Directed search
We study infinitely repeated games with observable actions, where players have present-biased (so-called beta-delta) preferences. We give a two-step procedure to characterize Strotz–Pollak equilibrium payoffs: compute the continuation payoff set using recursive techniques, and then use this set to characterize the equilibrium payoff set U(beta,delta). While Strotz–Pollak equilibrium and subgame perfection differ here, the generated paths and payoffs nonetheless coincide.
We then explore the cost of the present-time bias. Fixing the total present value of 1 util flow, lower beta or higher delta shrinks the payoff set. Surprisingly, unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set U(beta,delta) is not separately monotonic in beta or delta. While U(beta,delta) is contained in payoff set of a standard repeated game with smaller discount factor, the present-time bias precludes any lower bound on U(beta,delta) that would easily generalize the beta = 1 folk-theorem.