Working Paper: NBER ID: w20641
Authors: Allan Collard-Wexler; Gautam Gowrisankaran; Robin S. Lee
Abstract: A “Nash equilibrium in Nash bargains” has become a workhorse bargaining model in applied analyses of bilateral oligopoly. This paper proposes a non-cooperative foundation for “Nash-in-Nash” bargaining that extends the Rubinstein (1982) alternating offers model to multiple upstream and downstream firms. We provide conditions on firms’ marginal contributions under which there exists, for sufficiently short time between offers, an equilibrium with agreement among all firms at prices arbitrarily close to “Nash-in-Nash prices”—i.e., each pair's Nash bargaining solution given agreement by all other pairs. Conditioning on equilibria without delayed agreement, limiting prices are unique. Unconditionally, they are unique under stronger assumptions.
Keywords: Bargaining; Nash Equilibrium; Bilateral Oligopoly
JEL Codes: C78; D43; L13
Edges that are evidenced by causal inference methods are in orange, and the rest are in light blue.
| Cause | Effect |
|---|---|
| weak conditional decreasing marginal contributions (awcdmc) (D79) | Nash-in-Nash limit equilibrium (C72) |
| Nash-in-Nash limit equilibrium (C72) | agreements form at prices arbitrarily close to Nash-in-Nash solution (C78) |
| immediate formation of agreements at Rubinstein prices (D43) | Nash-in-Nash prices (E39) |
| marginal contributions of firms (D21) | bargaining outcomes (C78) |
| sum of marginal gains from individual agreements does not exceed the marginal gains from the entire set of agreements (C78) | uniqueness of equilibrium prices (D41) |
| interdependencies and externalities among firms (L14) | equilibrium existence (C62) |