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About the article
Published Online: 2013-10-09
“Supporting Memorandum of Points and Authorities,” In Re Apple & AT&TM Anti-trust Litigation, U.S. District Court, Northern District of California, San Jose Division, Case no. C 07–5152 JW, 12 September 2008. In 2011, Apple signed contracts allowing AT&T’s competitors Verizon and Sprint to start selling iPhones.
We do not claim that the equilibrium is unique, merely that the beliefs described in the text are consistent with equilibrium, and that the equilibrium action given those beliefs (and our tie-breaking assumption) is unique. A discussion of uniqueness of equilibrium beliefs would require a description of the consequences of neither firm making a bid and would take us far afield.
The quadratic utility function produces linear demand functions. However, symmetry of the cross partials of the Hicksian demand functions requires (Singh and Vives 1984).
Gabszewicz and Thisse (1979), Shaked and Sutton (1982), and Bonanno (1986) used a model of consumer behavior that generates a system of linear demand functions similar to this model, though with a slightly different interpretation of the demand system parameters. Products have a physical characteristic (location) that measures quality. Consumers have identical preferences but different incomes and buy at most one unit of a product.
Moner-Colonques, Sempere-Monerris, and Urbano (2004) examine a market in which two upstream firms decide whether to sell their products through one or both of the downstream vendors. Their model allows the upstream firms to charge only a per-unit price, whereas our upstream firm also uses a transfer. In addition, we allow the cross-price coefficients c and C to differ. The difference in these coefficients is key to our results.
Given our earlier assumptions, we have three free parameters, and F, but only c and C have a direct effect on the equilibrium quantities for a given number of downstream vendors. We set and in all our simulations. Given inequalities , these parameter choices imply that c and C must each be less than . For specificity, we set and and examine how the results vary with C.
This claim can be verified immediately using the equation for the difference in generic prices, eq.  in the Appendix. Similarly, the claim regarding the difference in new-product price can be verified by using eq. .
We use the same parameters as above to produce this figure. This figure uses the change in consumer surplus only to illustrate the change in consumer welfare. Neither our heuristic argument in the text nor the formal statement and proof in “Consumer welfare” in Appendix use consumer surplus to determine the change in consumer welfare.
The working paper on which this article is based shows that most of our qualitative results, with the exception of Proposition 8, continue to hold if we alter the original game by constraining m so that equilibrium duopoly profits do not fall below .