Nought Venture, Nought Have: Risk Taking and Efficient Investment in Matching Markets

We study a two-sided assortative matching market in which agents attract partners through costly investment and strategic risk taking. Each agent chooses an investment level and an arbitrary distribution of nonnegative quality that is a mean-preserving spread of investment. We allow for unequal populations and different investment costs across populations. Finite markets have a unique nontrivial semi-symmetric equilibrium, with investment below the Pareto frontier. We show that this inefficiency disappears as the economy grows with the population ratio held fixed and characterise its rate of convergence. The limit provides a foundation for the analysis of large matching markets, by selecting a unique efficient equilibrium from the continuum economy's infinitely many equilibria, which feature efficient investments but different risk-taking profiles.