Tractable Staggered Bargaining

Abstract

Many supply markets feature staggered bargaining, yet prevailing models assume simultaneous contract formation. Assuming simultaneity risks misattributing price dynamics to bargaining weights, but explicitly modeling staggering introduces explosive state space growth. I show that if bilateral shares are fixed, which corresponds to a property called step-by-step, then one can eliminate future states via a finite dependence-type cancellation. I propose a model of Nash equilibrium in Kalai proportional bargains, the only bargaining family satisfying the step-by-step property in general. I establish that the resulting model, which I call Nash-in-Kalai, yields a tractable GMM estimation framework that nests transferable-utility Nash-in-Nash. I test the importance of timing using results from a companion paper studying hospital–insurer contracting in West Virginia. I find that contracts are pervasively staggered, negotiators are substantially forward-looking, and a static model produces bargaining weights that are biased and sometimes incoherent.