Identifying reaction functions in differential oligopoly games

Luca Lambertini,Andrea Mantovani

doi:10.1016/j.mathsocsci.2006.06.004

Abstract

We investigate the issue of strategic substitutability/complementarity and define the concepts of best reply function and dominant strategy in deterministic differential games. We prove that if a player's Hamiltonian is additively separable w.r.t. controls, then players follow dominant strategies at every instant. Otherwise, if the Hamiltonian is not additively separable w.r.t. controls, instantaneous best replies can be properly characterised. However, under additive separability, we show that strategic interaction via best replies can still be characterised at the steady state. Illustrative examples are Ramsey and Solow's growth models, reformulated as oligopoly games, and a Cournot differential game with sticky price.

Highlights

The issue of super-/submodularity has been investigated mostly in static games, and refers to the slope of reaction functions in the game, as initially pointed out by Bulow, Geanakoplos and Klemperer (1985)
The analysis has been focused on games with strategic complementarities and their use in industrial economics (Vives, 1990; Milgrom and Roberts, 1990 and Amir, 1996) and in comparative statics analysis (Milgrom and Shannon, 1994)
The difference between the two cases lies in the fact that while in the first case we observe an instantaneous reaction function characterisng the optimal behaviour of player i at any time during the game, in the second case we only observe player i’s best reply at the steady state equilibrium, while i’s optimal behaviour during the transition to the steady state can be characterised in terms of states and costates only, regardless of any player j’s control

Summary

Introduction

The issue of super-/submodularity has been investigated mostly in static games, and refers to the slope of reaction functions in the (stage) game, as initially pointed out by Bulow, Geanakoplos and Klemperer (1985). To the best of our knowledge, only Jun and Vives (2004) carry out theoretical research on intertemporal strategic complementarity/substitutability They compare steady states of open loop and stable closed-loop equilibria in symmetric differentiated duopoly model with adjustment costs. More interestingly, we consider the SolowSwan model, where firms invest in order to accumulate capacity, and operate at full capacity at any time In this model, first order conditions do not produce instantaneous best reply functions, since optimal controls depend only on state variables at any time during the game, which is solved in dominant strategies. Jarmin (1994) raises the same question by looking at the early rayon industry and found empirical evidence of dynamic strategic behavior

Preliminaries

The Ramsey model

Cournot competition

Bertrand competition

The Solow model

Conclusion