“Search with Multilateral Bargaining,” M. Elliott (2010)

Matthew Elliott is another job market candidate making the rounds this year, and he presented this nice paper on matching today here at Northwestern. In the standard bilateral search model (due to Hosios), firms and workers choose whether or not to enter the job market (paying a cost), then meet sequentially with some probability and bargain over the wage. In these models, there can either be too much entry or too little; an additional unemployed worker entering makes it easier for firms to find an acceptable worker but harder for other unemployed workers to find a job. This famous model (itself an extension of Diamond-Mortensen-Pissarides, they of the 2010 Nobel) has been extended to allow for search costs, on-the-job search and trilateral bargaining, where two firms fight over one worker. Extending it to the most general case, where n firms and m workers, perhaps varying by worker and firm, constructed stochastically, is a problem which required much more advanced tools in network theory.

Elliott provides those results, remaining in the framework of negotiated rather than posted wages; as he notes, this theory is perhaps more applicable to high-skill labor markets where wages are not publicly posted, and workers are heterogeneous. Workers and firms simultaneously decide whether to enter the job market (paying a cost) and how hard to search (in an undirected manner). Workers match stochastically with each firm (who desires to hire one worker) depending on the level of search. Firms then negotiate how to split the surplus the match will generate, and some agent is hired.

If this sounds like Shapley-Shubik assignment to you, you’re on the right track. Because we’re in a Shapley-Shubik world, pairwise stability of the final assignment places us in the core; there are no deviations, even coalitional deviations, available. In a companion paper, Elliott shows that the assignment can be decomposed into the actual assigned links and the “best outside option” link for each agent. The minimum pairwise stable payoff can be found by adding and subtracting the values of each agent’s chain of outside option links.

The results for the labor market are these: there is never too much entry, search can sometimes be too heavy (though never too light), and that the labor market is “fragile”; it can unravel quickly. Entry is efficient because a new entrant will only change payoffs if he forces an old link to sever. By the definition of pairwise stability, the firm and the worker from that link must collectively be getting a higher payoff if they sever, since otherwise they would just reform their old link. That is, new entrants only thicken the market. Unlike in Hosios, since entering firms in Elliott have to bid up the wage of a worker they want in order to “steal” him from his current match, their effect on other firms when they enter does not cause a negative externality: they pay for causing the externality. The same argument in reverse applies to workers. Search is too heavy because having more outside options allows you to, in some sense, negotiate away more of the surplus from your current match. Labor market fragility occurs because, mathematically, everyone is getting payoffs that result from a weighted, connected graph. If one agent decides not to enter (his entry costs rise by epsilon), then the outside option of other agents is lowered. Their own current links are therefore willing to give them less of the surplus. Because of this, they may choose not to enter, and so on down the line. That is, minor shocks to the search process can create the necessary amplifications seen in the business cycle.

It would be nice to extend this type of model of the labor market to its dynamic setting – it’s not clear to me how sensible it is to talk about labor markets unraveling when all choices are being made simultaneously. Nonetheless, this paper does provide continuing metaproof about the usefulness of network theory and matching to a wide range of economic problems. The operations research types still no doubt have a lot to teach economists.

(EDIT: Forgot the link to the working paper initially. http://www.stanford.edu/~mle1/search_with_multilateral_bargaining.pdf)

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