TECHNICAL NOTES WITH DISCUSSION
These notes provide the formal underpinning for my paper "Tethering Vertical Merger Analsyis"
Section 4.1: N-Firm Bargaining [pdf]
This section of my 1989 doctoral dissertation examines a non-cooperative bargaining model between an upstream monopolist and N downstream Cournot competitors. The model provides one of the first non-cooperative foundations in the literature for simultaneous Nash bargaining that has come to be called "Nash-in-Nash bargaining" between an upstream monopolist and downstream oligopolists. [See also Horn and Wolinsky (RJE, 1988), Davidson (JLE, 1988); Jun (ReStud, 1989).] Unlike much of the recent literature on Nash-in-Nash bargaining, my model assumes that bargainers look ahead to effects of changes in negotiated wholesale prices on downstream prices and quantities, and disagreement profits are modeled by solving for equilibrium downstream prices conditional on the number of downstream firms that are active in the market while bargainers are in a state of disagreement. It also spells out the assumptions required to establish the relationship between Rubinstein-style alternating offer bargaining in this multi-player setting and the "Nash-in-Nash" solution. For more on that, see O'Brien (2014) on intermediate product third degree price discrimination under bargaining.
A finding that has important implications for vertical merger analysis is that as the number of downstream firms gets large, the equilibrium wholesale price remains strictly below the price a take-it-or-leave-it upstream monopolist would set. (This finding also appears in the O'Brien (RJE 2014), which has the same implications for vertical mergers and restraints.) This result does not depend on the shape of the demand or cost function apart from regularity that insures a well behaved solution. This means that if the number of downstream firms exceeds a critical value, a vertical merger raises price. This finding is related to (but predates) the findings of Hart & Tirole (Brookings, 1990), O'Brien and Shaffer (RJE, 1992), and McAfee and Schwartz (AER, 1994) that private bilateral contracting over nonlinear tariffs can lead to marginal transfer prices too low to maximize joint profits, and that vertical integration (possibly through merger) or restraints may restore or get closer to the fully integrated outcome by raising price. The general point is that bilateral bargaining over linear input prices involves bilateral contracting externalities similar to those that arise with bilateral contracting over nonlinear tariffs and leads to a similar result. In the linear price bargaining model with upstream monopoly, linear demand, and constant marginal cost, a vertical merger raises price when the number of downstream firms is 3 or greater.
It is important to recognize that this does not mean that vertical mergers are necessarily (or even often) anticompetitive in environments where the merging upstream firm has market power, sells to multiple downstream competitors, and negotiates supply terms with downstream firms bilaterally. There are a host of issues including: investment, market power, nature of upstream and downstream contracting, etc. Most critically, if the merging upstream firm's profit share is small, that would likely imply that downstream firms have outside options omitted from the model. Outside options might include the ability to seek (or threaten to seek) (i) alternative suppliers of the same product, (ii) a substitute product, or (iii) backward integration, among other possible options. When outside options are binding, the effects of a merger depend on whether and how the merger alters outside options. Both the economic and antitrust literature on mergers have paid too little attention to this point, which also plays a significant role in the effects of intermediate product price discrimination.
Section 4.4: Other Applications -- Vertical Integration [pdf]
This section of my 1989 doctoral dissertation explains the implications of vertical integration/mergers when an upstream monopolist negotiates linear input prices with downstream Cournot oligopolists.
Full Dissertation (copyright 1988) [pdf]
It's a bit embarrassing to post this old work in this form (pre-typesetting days; the electronic copy on a floppy I can't read), but there has been enough confusion and reinventing of wheels on bargaining in antitrust that I feel compelled. Please contact me with any questions about the implications of this analysis for vertical and horizontal issues in markets with negotiated prices. Some of the key implications are described above, but the devil is in the details.