Matching and Price Competition: Comment - Kojima, Fuhito (American Economic Review, 2007)
The theory of two-sided matching markets has interested researchers for its theoretical appeal and relevance to real-life applications. The matching of medical residents and hospitals in the United States has been studied extensively by Alvin E. Roth (1984) and others. The National Resident Matching Program (NRMP), the matching authority in the US hospital-resident matching market, runs a centralized matching mechanism that is a variant of the deferred acceptance algorithm of David Gale and Lloyd S. Shapley (1962), redesigned by Roth and Elliott Peranson (1999). A recent antitrust case against the NRMP charged that the centralized matching mechanism suppressed wages of residents. Although the lawsuit itself was dismissed, it sparked discussion about the effect of centralized matching on wages and efficiency. Jeremy Bulow and Jonathan Levin (2006, BL henceforth) investigate a matching market with price competition where each firm can hire only one worker and show that (a) the average wage is lower, (b) profit of each firm is higher, (c) wages are more compressed, and (d) the resulting matching is slightly less efficient in the presence of the matching mechanism than in any competitive equilibrium. Although BL declare “we have chosen our assumptions for analytical simplicity and transparency, not as the most realistic possible model of the residency match” (654), these results were often interpreted as an argument against the NRMP and led to discussion about potential changes of the matching mechanism. Vincent P. Crawford (forthcoming), for instance, proposes the “Flexible-Salary Match,” in which hospitals are allowed to indicate several levels of possible wages, and medical students are allowed to express preferences over pairs of hospitals and wages. We show that conclusions (a) and (b) above do not necessarily hold when firms may hire more than one worker and the number of workers in different firms are different. More specifically, we present an example with multiple positions in which the average worker wage is higher in the equilibrium with the matching mechanism than in a competitive equilibrium; and profit of each firm is lower in the equilibrium with the matching mechanism than in a competitive equilibrium. Our findings may explain why some of the results of BL disagree with empirical findings of Muriel Niederle and Roth (2003, 2004), who find little or no effect of the centralized matching on wages in some medical matching markets in the United States. Note that different firms hire different numbers of workers in many labor markets like the NRMP. There are other reasons why the conclusions and policy implications of BL may not be applicable to markets like the NRMP. For example, a competitive equilibrium may not be a