Sufficient Decisions in Multi‐Sided and Multiproduct Markets

Abstract

We show that in many models where firms make multiple decisions, analysis can be made more tractable by re‐formulating the model into one in which each firm makes a single choice, which we call a sufficient decision. The transformation allows application of standard techniques in these settings, including pass‐through for tax incidence and upward pricing pressure for merger analysis. The transformation works because the assumption of profit maximization links the firms’ decisions together. Examples include models of monopoly and oligopoly in two‐sided markets, where a natural sufficient decision may be the number of transactions that the firm facilitates, and multiproduct markets.

Pass-through of subsidies to prices under limited competition: Evidence from Canada’s Nutrition North program

“…The Weyl and Fabinger (2012) formula applies to a single-product setting but may also apply to multi-product firms. Alexandrov and Spulber (2017) show that under relatively weak conditions (continuity of profit and consumer surplus functions and strict quasi-concavity of profit functions) one can reduce high dimensional multiproduct firm decisions to a single "sufficient decision" for profits -"transactions" can serve this purpose, and equilibrium in oligopoly and monopoly markets can be analyzed using the formula above by replacing price and cost with average revenue and marginal cost per transaction. Armstrong and Vickers (2017) show that for multi-product Cournot oligopoly, the analysis of own and cross-product cost passthrough can be greatly simplified when consumer surplus is homothetic.…”

“…Under conditions satisfied by many demand systems used in applied work (e.g. CES utility and quadratic utility/linear demand) there is no "cross-cost" pass-through in prices (despite potentially large cross-price effects in the demand system) and price for each product depends only on its own marginal cost; in this case the pass-through analysis above applies product by product and not just for the "black-box" of transactions analyzed by Alexandrov and Spulber (2017).…”

Platform economics and antitrust enforcement: A little knowledge is a dangerous thing

“…What, if anything, can be said in general about when it is a sufficient statistic? Alexandrov and Spulber () have made a start in answering this question by examining the more general question of when multiple dimensions of firm conduct can be collapsed to a single sufficient statistic. However, as the authors note, their approach does not apply to discrete changes in conduct (e.g., imposition of an exclusive dealing requirement) of the sort that typically are at issue in antitrust litigation.…”

“…However, as the authors note, their approach does not apply to discrete changes in conduct (e.g., imposition of an exclusive dealing requirement) of the sort that typically are at issue in antitrust litigation. Although Alexandrov and Spulber () identify conditions under which their approach can be applied to mergers (e.g., the Spence, distortion is not strong enough to induce consumer surplus to fall as output rises), they implicitly make the very strong assumption that the relationship between a firm's price level and price structure does not change when the firm merges. I would expect this assumption to be frequently violated in practice.…”