Duopoly reaction curves

Duopoly reaction curves
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  Famous Figures andDiagrams inEconomics Edited by Mark Blaug Professor Emeritus, University of London and ProfessorEmeritus, University of Buckingham, UK  Peter Lloyd Professor Emeritus, Department of Economics, University of Melbourne, Australia Edward Elgar Cheltenham, UK • Northampton, MA, USA  © Mark Blaug and Peter Lloyd 2010All rights reserved. No part of this publication may be reproduced, stored in aretrieval system or transmitted in any form or by any means, electronic,mechanical or photocopying, recording, or otherwise without the priorpermission of the publisher.Published byEdward Elgar Publishing LimitedThe Lypiatts15 Lansdown RoadCheltenhamGlos GL50 2JAUKEdward Elgar Publishing, Inc.William Pratt House9 Dewey CourtNorthamptonMassachusetts 01060USAA catalogue record for this bookis available from the British LibraryLibrary of Congress Control Number: 2009941142ISBN 978 1 84844 160 6 (cased)Printed and bound by MPG Books Group, UK   137  17. Duopoly reaction curves Nicola Giocoli BASIC CONCEPTS A reaction curve (RC), also called reaction unction or best-reply unction,is the locus o optimal, that is, prot-maximizing, actions R i  ( a  j  )that a rm i  may undertake or any given action a  j  chosen by a rival rm  j  . The RCdiagram is the standard tool or the graphical analysis o duopoly. In thediagram, the market equilibrium is at the intersection o the RCs, one oreach rm. The commonest case o a RC diagram is that o the Cournotduopoly model.Consider two rms 1 and 2 producing a homogeneous product withoutput levels q 1 and q 2 and aggregate output Q 5 q 1 1 q 2 . Provided invert-ibility conditions are met, the inverse demand unction gives the marketprice associated with aggregate output  p ( Q ) 5  p ( q 1 1 q 2 ). Assume eachrm has a cost unction c i  ( q i  ), i  5 1, 2, and take the strategic variable orboth rms to be the output level, so that rm 1’s maximization problem is:max   q 1 p 1 ( q 1 , q 2 ) 5  p ( Q ) q 1 2 c 1 ( q 1 ). Given that rm 1’s prot also dependson rm’s 2 output, rm 1’s optimal choice must also take into account rm2’s choice. A similar problem can be ormulated or rm 2.Following the so-called Cournot assumption, we model each rm astaking as given the rival’s quantity. The rst order condition (FOC) oreach rm is thereore:  0 p i  ( q 1 , q 2 ) 0 q i  5  p ( Q ) 1  p r ( Q ) q i  2 c r i  ( q i  ) 5 0, i  5 1, 2Firm 1’s FOC determines the optimal, that is, prot-maximizing, outputchoice by rm 1 as a unction o either its belie about rm 2’s expectedoutput choice or its observation o rm 2’s actual choice (see below or anexplanation o these two possible interpretations). As in Cournot (1971[1838], Fig. 2), we depict the pair o FOCs with the RC diagram, where theRC o rm 1 (RC 1 ) is implicitly dened by the FOC 0 p 1 ( R 1 ( q 2 ), q 2 )  /  0 q 1 5 0,and that o rm 2 (RC 2 ) by 0 p 2 ( R 2 ( q 1 ), q 1 )  /  0 q 2 5 0. The name reac-tion curve captures the idea that a rm will optimally modiy its choice  138 Famous fgures and diagrams in economics ollowing the change in its belie about (or its observation o) the rival’schoice. The slope o each RC indicates the size o a rm’s optimal reactionto such a change. For example, the slope o RC 1 is:  R r 1 ( q 2 ) 52 0 2 p 1 0 q 1 0 q 2 0 2 p 1 0 q 21 ,and similarly or R r 2 ( q 1 ).The Figure 17.1 is based on the simpliying assumption o linear demandand constant and identical marginal cost (see Fulton 1997 and Martin2002 or a step-by-step derivation o the diagram). In such a case, the RCsare straight lines with negative slope. The equilibrium pair( q * 1 , q * 2 )lies atthe intersection o the RCs. TWO INTERPRETATIONS In the belie-based version o the model, two conditions need to be satis-ed or the rms’ choices to constitute an equilibrium (Kreps 1990). First, RC 1 RC 2 q 1 q 1A q 1B q 1* q 2A q 2B q 2* q 2 E Figure 17.1 Reaction curves – linear demand case with constant and identical costs    Duopoly reaction curves 139 no rm, on the basis o its own belies, must wish to modiy its own choice.Second, the rms’ equilibrium actions must be consistent with the beliesupon which they act. Hence, a Cournot equilibrium in a basic duopolymodel is given by the output pair( q * 1 , q * 2 )such that: i) each rm is choos-ing its prot maximizing output given the belies about the other rm’schoice, and, ii) each rm’s belies are correct at equilibrium. The similaritywith the game-theoretic notion o Nash equilibrium, and with the staticxed point concept underlying it, is well known. Yet the association withnon-cooperative game theory came only with Shubik (1959).Beore the belie-based version, the interpretation o the Cournot modelwas based on a dynamic analysis o actual rms’ choices. This involveda sequential adjustment process undertaken by two rms which ignoredthe act that their own choices could inuence the rival’s behavior. Theoutcome o the process – the equilibrium position – was viewed as the endstate o a trial-and-error sequence o actual output choices along the RCs.An instance o such a sequence is shown in Figure 17.1. Assume rm 2observes output q 1 A chosen by rm 1. Its prot-maximizing reaction to therival’s choice may be read on RC 2 and amounts to producing q 2 A . Thenrm 1 in its turn optimally reacts to 2’s choice by modiying its productionto q 1 B  , and so on. The process actually converges to equilibrium (more onthis convergence below), that is, to the pair( q * 1 , q * 2 ), such that neither rmhas any urther incentive to modiy its own choice.The dynamic reading o the Cournot model is highly questionablebecause it requires the two rms to adopt a myopic, almost irrationalbehavior. As William Fellner put it, at equilibrium the duopolists turnout to be ‘right or the wrong reasons’ (1949, p.58). Each rm is in actassumed to go on making its output choice by always taking as given therival’s quantity – this despite the evidence clearly showing that the rival isactually reacting to one’s own choice. Indeed, the only consequence o therival’s reaction is taken to be the modication o the quantity that eachrm still takes as given in its own RC. Though not 100 per cent aithulto Cournot’s own words, the interpretation o RCs as the illustration o amyopic sequence o actual actions and reactions was dominant in the rsthal o the 20th century, starting at least rom Fisher (1898). Cournot actu-ally assumed that duopolists only looked at the direct inuence o eachother’s output choices, while disregarding the indirect ones – the reactions – because that would demand too much rom their reasoning power, seeGiocoli 2003.However it would be a mistake dating the belie-based version o theCournot model to the game-theoretic turn o industrial economics becausethis version also has a long history. As early as 1924, Arthur Bowleyargued that in order to solve the FOCs o a standard duopoly problem ‘we
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