When contracts are formed at different times, the prevailing simultaneous-bargaining approach risks introducing bias, but explicitly modeling staggering introduces explosive state space growth. I show the step-by-step property implies a finite dependence-type cancellation of future states. I propose the Nash-in-Kalai model, which satisfies step-by-step in general, and which I show yields a tractable GMM framework. Applying the model to healthcare data, I find a static model produces biased, and sometimes incoherent, bargaining weights.
Nash weight bargaining weights are not identified in general; among bargaining solutions that satisfy IIA, only Kalai proportional bargaining weights are identified without knowledge of information timing. When contracts are multiperiod, bargaining weights may once again become unidentified, but identification can be restored under plausible conditions.
I quantify the impact of predictable increases in benchmark-linked prices between negotiations. There can be real effects in the presence of staggered contracting and time discounting. Using panel data on hospital–insurer contracts from West Virginia, I find both occur.
I show that even when the density of propensity scores may be unbounded near zero, t-statistics based on a thresholded Augmented IPW estimator can remain well-calibrated. I characterize the necessary conditions in terms of black-box rates and minimal smoothness orders (including new results for global regression rates) and use the conditions to propose rules of thumb for clipping or trimming rates.
I investigate vertical market contract dynamics by documenting and analyzing a novel panel dataset of hospital–insurer contracts in West Virginia. The largest insurer typically formed three- and five-year contracts. In contrast, smaller insurers generally formed long-lived contracts with faster price growth. By documenting a unique dataset and stark dynamic implications, this research contributes to a larger understanding of vertical market dynamics and helps set the stage for future work.
We extend Dorn and Guo (2022)'s characterization of bounds for Tan's marginal sensitivity model to considerably more general assumptions and estimands.