Robust Quantitative Comparative Statics for a Multimarket Paradox

TL;DR

This paper develops a quantitative method to bound the worst-case impact of exogenous parameter changes on equilibria in multimarket oligopoly, revealing limits on profit and welfare reductions under positive price shocks.

Contribution

It introduces a novel quantitative comparative statics approach that bounds the maximum effects of parameter changes in multimarket Cournot models, including worst-case profit and welfare impacts.

Findings

Profit loss of any firm is at most 25% under positive price shocks.

Total profit and social welfare decrease by at most 25% and 16.6%, respectively.

Bounds apply to various equilibrium concepts, including mixed and correlated equilibria.

Abstract

We introduce a quantitative approach to comparative statics that allows to bound the maximum effect of an exogenous parameter change on a system's equilibrium. The motivation for this approach is a well known paradox in multimarket Cournot competition, where a positive price shock on a monopoly market may actually reduce the monopolist's profit. We use our approach to quantify for the first time the worst case profit reduction for multimarket oligopolies exposed to arbitrary positive price shocks. For markets with affine price functions and firms with convex cost technologies, we show that the relative profit loss of any firm is at most 25% no matter how many firms compete in the oligopoly. We further investigate the impact of positive price shocks on total profit of all firms as well as on social welfare. We find tight bounds also for these measures showing that total profit and social welfare decreases by at most 25% and 16.6%, respectively. Finally, we show that in our model, mixed, correlated and coarse correlated equilibria are essentially unique, thus, all our bounds apply to these game solutions as well.