Quantitative Comparative Statics for a Multimarket Paradox
Comparative statics is a well established research field where one analyzes how marginal changes in parameters of a strategic game affect the resulting equilibria. While classic comparative statics is mainly concerned with qualitative approaches (e.g., deciding whether a parameter change improves or hurts equilibrium profits or welfare), we provide a framework to expose the extend (not monotonicity) of a discrete (not marginal) parameter change, with the additional benefit that our results can even be used when there is uncertainty about the exact model instance. We apply our quantitative approach to the multimarket oligopoly model introduced by Bulow, Geanakoplos and Klemperer (1985). They describe the counterintuitive example of a positive price shock in the firm's monopoly market resulting in a reduction of the firm's equilibrium profit. We quantify for the first time the worst case profit reduction for multimarket oligopolies with an arbitrary number of markets exhibiting arbitrary positive price shocks. For markets with affine price functions and firms with convex cost technologies, we show that the relative 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 consumer surplus. We find tight bounds also for these measures showing that total profit and consumer surplus decreases by at most 25% and 16.6%, respectively.
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