A reaction curve RC, also called reaction function or best-reply function, is the locus of optimal, i.e. profit-maximizing, actions that a firm may undertake for any given action chosen by a rival firm. The RC diagram is the standard tool for the graphical analysis of duopoly. In the diagram the market equilibrium is at the intersection of the RCs, one for each firm. The commonest case of RC diagram is that of the Cournot duopoly model.reaction curves; duopoly; Cournot

Giocoli, Nicola

English

Research Papers in Economics

REACTION CURVES 
 
Basic concepts 
A reaction curve RC, also called reaction function or best-reply function, is the locus of optimal, 
i.e. profit-maximizing, actions  ( ) i j R a  that a firm i may undertake for any given action  j a  chosen 
by a rival firm j. The RC diagram is the standard tool for the graphical analysis of duopoly. In the 
diagram the market equilibrium is at the intersection of the RCs, one for each firm. The commonest 
case of RC diagram is that of the Cournot duopoly model. 
Consider two firms 1 and 2 producing a homogeneous product with output levels  1 q  and  2 q  and 
aggregate output  1 2 Q q q = + . Provided invertibility conditions are met, the inverse demand function 
gives the market price associated with aggregate output  ( ) ( ) 1 2 p Q p q q = + . Assume each firm has a 
cost function  ( ) i i c q ,  1,2 i = , and take the strategic variable for both firms to be the output level, so 
that  firm  1’s  maximization  problem  is:  ( ) ( ) ( )
1 1 1 2 1 1 1 max , q q q p Q q c q π = − .  Given  that  firm  1’s 
profit also depends on firm’s 2 output, firm 1’s optimal choice must also take into account firm 2’s 
choice. A similar problem can be formulated for firm 2. 
Following the so-called Cournot assumption, we model each firm as taking as given the rival’s 
quantity. The first order condition (FOC) for each firm is therefore: 
  ( ) ( ) ( ) ( )
1 2 ,
0
i
i i i
i
q q
p Q p Q q c q
q
π ∂
′ ′ = + − =
∂
,  1,2 i =  
Firm 1’s FOC determines the optimal, i.e., profit-maximizing, output choice by firm 1 as a function 
of either its belief about firm 2’s expected output choice or its observation of firm 2’s actual choice 
(see below for an explanation of these two possible interpretations). As in Cournot (1971 [1838], 
Fig.2), we depict the pair of FOCs with the RC diagram, where the RC of firm 1 (RC1) is implicitly 
defined by the FOC  ( ) ( ) 1 1 2 2
1
,
0
R q q
q
π ∂
=
∂
 , and that of firm 2 (RC2) by  ( ) ( ) 2 2 1 1
2
,
0
R q q
q
π ∂
=
∂
. The 
name reaction curve captures the idea that a firm will optimally modify its choice following the 
change in its belief about (or its observation of) the rival’s choice. The slope of each RC indicates 
the size of a firm’s optimal reaction to such a change. For example, the slope of RC1 is: 
( )
2
1
1 2
1 2 2
1
2
1
q q
R q
q
π
π
∂
∂ ∂ ′ = −
∂
∂
 , and similarly for  ( ) 2 1 R q ′ .  The diagram below is based on the simplifying assumption of linear demand and constant and 
identical marginal cost (see Fulton 1997, Martin 2002 for a step-by-step derivation of the diagram).  
In such a case the RCs are straight lines with negative slope. The equilibrium pair ( )
* *
1 2 , q q  lies at 
the intersection of the RCs.  
 
Figure 1 
Two interpretations 
In the belief-based version of the model, two conditions need be satisfied for the firms’ choices to 
constitute an equilibrium (Kreps 1990). First, no firm, on the basis of its own beliefs, must wish to 
modify its own choice. Second, the firms’ equilibrium actions must be consistent with the beliefs 
upon which they act. Hence, a Cournot equilibrium in a basic duopoly model is given by the output 
pair ( )
* *
1 2 , q q  such that: i) each firm is choosing its profit maximizing output given the beliefs about 
the other firm’s choice, and, ii) each firm’s beliefs are correct at equilibrium. The similarity with the 
game-theoretic notion of Nash equilibrium, and with the static fixed point concept underlying it, is 
well-known. Yet, the association with non-cooperative game theory came just with Shubik 1959.  
Before that, Cournot model was mostly considered as a dynamic analysis involving a sequential 
adjustment process undertaken by two firms who ignored that their own choices could influence the 
rival’s behavior. The outcome of the process – the equilibrium position – was viewed as the end 
state of a trial-and-error sequence of actual output choices along the RCs. An instance of such a 
sequence is showed in the diagram above. Assume firm 2 observes output  1A q  chosen by firm 1. Its 
profit-maximizing reaction to the rival’s choice may be read on RC2 and amounts to producing  2A q . 
Then firm 1 in its turn optimally reacts to 2’s choice by modifying its production to  1B q , and so on. The process actually converges to equilibrium (more on this convergence below), i.e., to the pair 
( )
* *
1 2 , q q  such that neither firm has any further incentive to modify its own choice.  
The dynamic reading of Cournot model is highly questionable because it requires the two firms to 
adopt a myopic, almost irrational behavior. As William Fellner put it, at equilibrium the duopolists 
turn out to be <<… right for the wrong reasons.>> (1949, p.58). Each firm is in fact assumed to go 
on  making  its  output  choice  by  always  taking  as  given  the  rival’s  quantity  –  this  despite  the 
evidence clearly showing that the rival is actually reacting to one’s own choice. Indeed, the only 
consequence of the rival’s reaction is taken to be the modification of the quantity that each firm still 
takes as given in its own RC. Though not 100% faithful to Cournot’s own words (he actually 
assumed that duopolists only looked at the direct influence of each other’s output choices, while 
disregarding the indirect ones – viz., the reactions – because that would demand too much from 
their reasoning power: see Giocoli 2003), the interpretation of RCs as the illustration of a myopic 
sequence of actual actions and reactions has been dominant in the first half of the 20
th century, 
starting at least from Fisher 1898.     
However, also the belief-based version of Cournot model has a long history. As early as 1924 
Arthur Bowley argued that in order to solve the FOCs of a standard duopoly problem <<...we 
should need to know [ 2 q ] as a function of [ 1 q ], and this depends on what each producer thinks the 
other is likely to do. >> (Bowley 1924, 38). A new notion, the conjectural variation  i j i j v q q = ∂ ∂ , 
1,2 i = ,  was  introduced,  to  represent  firm  i’s  arbitrary  conjecture  about  how  its  rival  j  would 
respond to a (small) variation of i’s output. The conjectural variation then entered both FOCs to 
highlight the fact that the solution of the duopoly model should depend upon the exact value of each 
firm’s conjecture over the rival’s reaction. The Cournot model became just a special case – that 
with  12 21 0 v v = = , to indicate that each firm’s believed the rival would not react to its own choice – 
but an infinite array of  i j v  values was equally possible. A stream of literature elaborated Bowley’s 
intuition, to emphasize the intrinsic indeterminacy of duopolistic interaction. 
Despite their early formulation, a clear distinction between the two approaches has only been 
achieved in the 1970s  (see e.g. Friedman 1977). In the conjectural approach the RCs are viewed as 
purely mental tools to be used in the reasoning process that underlies the solution of any static game 
where both firms simultaneously make their choices. A belief-based RC is thus a decision rule 
giving the a firm’s optimal choice as a function of the conjectured action of the rival. This is the 
only interpretation of RCs which makes economic sense in a simultaneous game, where no room 
exists for interpreting RCs in terms of the firms’ actual choices since, by definition, each firm 
selects its action before observing the rival’s one and thus has no chance at all to react. In such a game each RC indicates what a firm would do in case it happened to know of a change in the rival’s 
action, which of course it does not because no RC point outside the Nash equilibrium can ever be 
observed. Therefore, if we stick to the actual-choice interpretation of Cournot competition even in a 
simultaneous setup, we must admit that the construction of the RC diagram is a mere technical 
device,  only  used  for  expositional  purposes.  On  the  contrary,  actual-choice  RCs  have  a  real 
economic content in the dynamic setup typical of sequential games. The RC may now be read as a 
function determining a firm’s (myopic) reaction in a given period in terms of the other firm’s action 
during the preceding period; moreover, each firm may well exploit the rival’s RC to calculate how a 
change in its behavior may affect the rival’s (myopic) choice. 
 
Extensions 
The simple, textbook version of the RC diagram enjoys some useful properties. Assuming quantity 
competition, a linear inverse demand and constant marginal costs, the RCs are straight lines with 
constant  negative  slopes  and  intercepts  with  the  axis  equal  to,  respectively,  the  perfectly 
competitive and the monopoly quantities (see figure 2).  
 
Figure 2 
Consider RC1 and adopt for definiteness the actual-choice interpretation. RC1’s intercept with the 
X-axis identifies the optimal choice when  2 0 q = , i.e., when firm 1’s residual demand (the demand 
left after firm 2 has chosen its output level) coincides with market demand. But when firm 1 faces 
the whole demand curve, its optimal choice is to produce the monopoly quantity  m q . Conversely, 
RC1’s intercept with the Y-axis captures the situation when firm 2’s output choice has been such 
that firm 1’s optimal reaction is to produce nothing. Thus, firm 2 must have chosen an output level 
capable of inducing the rival to abstain from production (so-called limit quantity). This may only happen when  2 q  is so large to push market price down to equal the common average and marginal 
cost, i.e., when firm 2 offers the perfectly competitive quantity  pc q  (because in such a case any 
positive output by firm 1 would push price further down, below cost).  A similar argument explains 
the intercepts of RC2.  
Given that  pc q  is larger than  m q , it follows that whenever  1 q  is measured on the X-axis RC1 
always crosses RC2 from above. This, in turn, warrants the stability of Cournot equilibrium as 
implemented through the above-mentioned trial-and-error process. Remarkably, Cournot himself 
acknowledged that, though stability might not be warranted in general, it was actually implied by 
the relative size of RCs’ intercepts (1971, 81-2). In a more general setting, the equilibrium may 
either be nonexistent or multiple (see Martin 2002). A non-existence result may emerge when RCs 
are discontinuous (as in Amoroso 1921’s counterexample or because firms’ profit functions are not 
concave due to the “excessive” convexity of demand). Multiplicity may ensue when the slope of the 
RCs (say, of  ( ) 1 2 R q ′ ) is larger than 1 in absolute value. Both cases are obviously excluded in the 
standard, linear version of the model.  
Figure 2 also shows a simple graphical technique for comparing the market performance of a 
Cournot equilibrium with that of the two extreme cases, monopoly and perfect competition. Given 
the RCs’ intercepts with the axes, we may draw a dashed line representing the  ( ) 1 2 , q q  pairs that 
jointly give the perfectly competitive output  pc q  and a dotted line for the pairs giving the joint 
monopoly output  m q . The two lines allow an immediate understanding of the  intermediate nature 
of the Cournot equilibrium pair ( )
* *
1 2 , q q : total output 
* *
1 2 d q q q = +  is lower than  pc q  and larger than 
m q . It follows that the price level and the firms’ profit are also intermediate between those in the 
two extreme cases.  
The standard RC diagram may also encompass the case of a difference in the firms’ efficiency. 
The thick line in Figure 2 is RC2bis , that is, firm 2’s RC when it enjoys a cost advantage over firm 
1. RC2bis is higher because firm 2 optimally produces more than before for any quantity offered by 
firm 1. Hence, also 
*
2 q  will be larger (and 
*
1 q  smaller) than in the traditional Cournot equilibrium. 
Finally, take the most general form of the slope of the RC, namely,  ( )
2
1
1 2
1 2 2
1
2
1
a a
R a
a
π
π
∂
∂ ∂ ′ = −
∂
∂
, where 
i a   is  a  generic  strategic  variable,  or  action.  Given  that  the  second  order  condition  for  the maximum profit requires 
2
1
2
1
0
a
π ∂
<
∂
, it turns out that the sign of  ( ) 1 2 R a ′  depends on that of the 
cross partial derivative of firm 1’s profit function, i.e., on how  1 π  varies as an effect of a change in 
the  rival’s  action.  RC1  is  negatively  sloped  when,  as  in  the  Cournot  model,  the  cross  partial 
2
1
1 2 a a
π ∂
∂ ∂
 is negative and positively sloped when it is positive. In the former case, firms’ actions are 
said to be strategic substitutes (e.g. quantities), in the latter, strategic complements (e.g. prices; see 
Bulow et al. 1985).  
 
Figure 3 
The  RC  diagram  for  the  case  of  strategic  complements  is  therefore  as  in  figure  3,  which 
represents two firms competing in prices and producing a homogeneous product (so-called “pure 
Bertand” case). The equilibrium price pair is  ( ) , pc pc p p , where  pc p  is the perfectly competitive 
price, namely a price equal to marginal cost c. In case of partial product differentiation, both RCs 
would  shift  upward  and  the  equilibrium  pair  ( )
* *
1 2 , p p   would  lie  somewhere  between  the 
competitive and the monopoly price pairs.  If the differentiation is complete, the RCs become 
orthogonal to the axes, with intercepts at the monopoly price  m p , to suggest the independence of 
each firm’s price decision from the rival’s choice. 
Nicola Giocoli 
University of Pisa 
 
 
 References 
Amoroso L. 1921, Lezioni di Economia Matematica, Bologna: Zanichelli 
Bowley A.L. 1924, The Mathematical Groundwork of Economics, Oxford: OUP. 
Bulow J. Geanakoplos J. and Klemperer P.D. 1985, “Multimarket oligopoly: strategic substitutes 
and complements”, Journal of Political Economy 93 (3), 488-511 
Cournot A.A. 1971 [1838], Researches into the Mathematical Principles of the Theory of Wealth, 
New York: Augustus M. Kelley Publishers. 
Fellner W. 1949, Competition among the few, New York: A.A. Knopf. 
Fisher I. 1898, “Cournot and mathematical economics”, Quarterly Journal of Economics, 12, 119-
138. 
Friedman J. 1977, “Cournot, Bowley, Stackelberg and Fellner, and the evolution of the reaction 
function”, in: Balassa B. & Nelson R. (eds.), Economic Progress, Private Values and Public Policy: 
Essays in Honor of William Fellner, Amsterdam: NorthHolland, 139-160. 
Fulton M. 1997, “A graphical analysis of the Cournot-Nash and Stackelberg models”, Journal of 
Economic Education, 28 (1), 48-57. 
Giocoli N. 2003, “Conjecturizing Cournot: the conjectural variations approach to duopoly theory”, 
History of Political Economy, 35 (2), 175-204 
Kreps D. 1990, A Course in Microeconomic Theory, Harvester-Wheatsheaf. 
Martin S. 2002, Advanced Industrial Economics. Second Edition, Oxford: Blackwell 
Shubik M. 1959, Strategy and Market Structure. Competition, Oligopoly and the Theory of Games, 
New York: John Wiley & Sons. 
 
 

1898, “Cournot and mathematical economics”,

A graphical analysis of the Cournot-Nash and Stackelberg models”,

Advanced Industrial Economics. Second Edition,

Competition among the few,

Conjecturizing Cournot: the conjectural variations approach to duopoly theory”,

Cournot, Bowley, Stackelberg and Fellner, and the evolution of the reaction function”,

Lezioni di Economia Matematica,

Multimarket oligopoly: strategic substitutes and complements”,

Reaction curves

Reaction curves

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