We study competition in matching markets with random heterogeneous preferences and an unequal number of agents on either side. First, we show that even the slightest imbalance yields an essentially unique stable matching. Second, we give a tight description of stable outcomes, showing that matching markets are extremely competitive. Each agent on the short side of the market is matched with one of his top choices, and each agent on the long side either is unmatched or does almost no better than being matched with a random partner. Our results suggest that any matching market is likely to have a small core, explaining why small cores are empirically ubiquitous.

We propose a new model of two-sided matching markets, which allows for complex heterogeneous preferences, but is more tractable than the standard model, yielding rich comparative statics and new results on large matching markets. We simplify the standard Gale and Shapley (1962) model in two ways. First, following Aumann (1964) we consider a setting where a finite number of agents on one side (colleges or firms) are matched to a continuum mass of agents on the other side (students or workers). Second, we show that, in both the discrete and continuum model, stable matchings have a very simple structure, with colleges accepting students ranked above a threshold, and students demanding their favorite college that will accept them. Moreover, stable matchings may be found by solving for thresholds that balance supply and demand for colleges. We give general conditions under which the continuum model admits a unique stable matching, in contrast to the standard discrete model. This stable matching varies continuously with the parameters of the model, and comparative statics may be derived as in competitive equilibrium theory, through the market clearing equations. Moreover, given a sequence of large discrete economies converging to a limit economy, the set of stable matchings of the discrete economies converges to the stable matching of the limit economy. We bound the rate of convergence of the set of stable matchings of large discrete economies to the continuum approximation, and show that comparative statics regarding the unique stable matching of the continuum model extend to strong set ordering of the sets of stable matchings of approximating discrete economies. We model the transferrable utility case, as in Becker (1973). We characterize the limit of school choice mechanisms used in practice, generalizing previous results of Che and Kojima (2010). Finally, we illustrate the model's applicability by quantifying how competition induced by school choice gives schools incentives to invest in quality. Specifically, we show that schools have muted, and possibly even negative incentives to invest in quality dimensions that benefit lower ranked students.

Owned by nobody and controlled by an almost immutable protocol the Bitcoin payment system is a platform with two main constituencies: users and profit seeking miners who maintain the system's infrastructure. The paper seeks to understand the economics of the system: How does the system raise revenue to pay for its infrastructure? How are usage fees determined? How much infrastructure is deployed?What are the implications of changing parameters in the protocol?A simplified economic model that captures the system's properties answers these questions. Transaction fees and infrastructure level are determined in an equilibrium of a congestion queueing game derived from the system's limited throughput.The system eliminates dead-weight loss from monopoly, but introduces other inefficiencies and requires congestion to raise revenue and fund infrastructure. We explore the future potential of such systems and provide design suggestions. * We are grateful to Campbell Harvey, Refael Hassin, Seth Stephens-Davidowitz and Aviv Zohar for helpful conversations and to seminar participants at the Central Bank of Finland, Columbia, EIEF, MSR-NYC, NYCE and Stanford for helpful comments.

We propose a new model of two-sided matching markets, which allows for complex heterogeneous preferences, but is more tractable than the standard model, yielding rich comparative statics and new results on large matching markets. We simplify the standard Gale and Shapley (1962) model in two ways. First, following Aumann (1964) we consider a setting where a finite number of agents on one side (colleges or firms) are matched to a continuum mass of agents on the other side (students or workers). Second, we show that, in both the discrete and continuum model, stable matchings have a very simple structure, with colleges accepting students ranked above a threshold, and students demanding their favorite college that will accept them. Moreover, stable matchings may be found by solving for thresholds that balance supply and demand for colleges. We give general conditions under which the continuum model admits a unique stable matching, in contrast to the standard discrete model. This stable matching varies continuously with the parameters of the model, and comparative statics may be derived as in competitive equilibrium theory, through the market clearing equations. Moreover, given a sequence of large discrete economies converging to a limit economy, the set of stable matchings of the discrete economies converges to the stable matching of the limit economy. We bound the rate of convergence of the set of stable matchings of large discrete economies to the continuum approximation, and show that comparative statics regarding the unique stable matching of the continuum model extend to strong set ordering of the sets of stable matchings of approximating discrete economies. We model the transferrable utility case, as in Becker (1973). We characterize the limit of school choice mechanisms used in practice, generalizing previous results of Che and Kojima (2010). Finally, we illustrate the model's applicability by quantifying how competition induced by school choice gives schools incentives to invest in quality. Specifically, we show that schools have muted, and possibly even negative incentives to invest in quality dimensions that benefit lower ranked students.

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