Multiproduct-Firm Oligopoly: An Aggregative Games Approach

Abstract

We develop an aggregative games approach to study oligopolistic price competition with multiproduct firms. We introduce a new class of IIA demand systems, derived from discrete/continuous choice, and nesting CES and logit demands. The associated pricing game with multiproduct firms is aggregative and a firm's optimal price vector can be summarized by a uni‐dimensional sufficient statistic, the ι‐markup. We prove existence of equilibrium using a nested fixed‐point argument, and provide conditions for equilibrium uniqueness. In equilibrium, firms may choose not to offer some products. We analyze the pricing distortions and provide monotone comparative statics. Under (nested) CES and logit demands, another aggregation property obtains: All relevant information for determining a firm's performance and competitive impact is contained in that firm's uni‐dimensional type. We extend the model to nonlinear pricing, quantity competition, general equilibrium, and demand systems with a nest structure. Finally, we discuss applications to merger analysis and international trade.

Media market concentration and pluralism

“…There is thus some similarity to the multiproduct firm literature (see, e.g., Anderson & De Palma, , ; Nocke & Schutz, ) where firms decide about their product scope. Typically, these models use some multinomial logit setups.…”

“…Both papers obtain similar result that the range of products per firm decreases as the number of firms in the market rises. Nocke and Schutz () develop an aggregative games approach, with constant elasticity of substitution and multinomial logit demands as special cases. They find an opposite effect and show that firms increase their scope of products as the industry becomes more competitive.…”

“…In the Anderson and De Palma (, ) models, firms hold back their product range strategically in order to relax competition. In the Nocke and Schutz () setup, the intuition is different. As the industry becomes more competitive, a firm more worries about losing sales to rivals but starts less worrying about cannibalizing its own sales.…”

“…The variants produced by any firm are symmetric substitutes, though, and hence, all firms set the same markup in equilibrium and benefit from relaxing competition. Nocke and Schutz () study multiproduct firm competition with arbitrary firm, portfolio, and product heterogeneity. In their framework, the share of demand, lost due to self‐cannibalization, is proportional to the initial market share, which is the lower, the higher is the intensity of competition.…”

Ordinal Potentials in Smooth Games

“…For a less obvious example, consider the interesting class of multiplicatively separable aggregative games for which an ordinal potential has been constructed explicitly (Kukushkin, 1994;Nocke and Schutz, 2016). Thus, assume that payo¤s admit the representation u i (x N ) = x i (x N ) for all i 2 N , where : X N !…”