Selling to Cournot oligopolists: pricing under uncertainty & generalized mean residual life
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We study a classic Cournot market, which we extend to a two-stage game with endogenous cost formation: the retailers' marginal cost represents purchases from a price-setting, revenue-maximizing supplier. Any demand uncertainty falls to the supplier, who acts first and sets the wholesale price under incomplete information concerning the retailers' willingness to pay. We introduce the generalized mean residual life (GMRL) function of the supplier's belief distribution F and show that his revenue function is unimodal, if the GMRL function is decreasing - (DGMRL) property - and F has finite second moment. In this case, we characterize the supplier's optimal price as a fixed point of his MRL function and provide a bound on the market-inefficiency due to demand uncertainty, which is tight over the class of DMRL distributions. We then turn to the class of DGMRL random variables and study their moments, limiting behavior, and closure properties. Under the additional assumption that F has a density, we establish its relationship to the widely used class of increasing generalized failure rate (IGFR) random variables. If a random variable is IGFR, then it is DGMRL. We provide a sufficient condition for the converse to be true and show that the limiting behavior of the GMRL and GFR functions is closely linked under a reciprocal relationship.
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