Cournot model in economics

2. ▪ Augustin Cournot, a French mathematician, published his model in 1836.
▪ Assume a single firm wishes to enter a market currently supplied by a monopoly.
▪ The entrant is able to offer a product that is identical in all respects to that of the
incumbent monopolist and to produce it at the same unit cost.
▪ Entry is attractive because under the assumption of constant and identical costs, we
know that the monopolist is producing where price is greater than marginal cost,
which means that the price also exceeds the marginal cost of the would-be entrant.
▪ Hence, the entrant firm will see that it can profitably sell some amount in this
market.
▪ However,as per Cournot, the new entrant will choose an output level that
maximizes its profit, after taking account of the output being sold by the
monopolist.

3. ▪ If entry occurred and the new firm produced its chosen output, the monopolist
would react.
▪ Before entry, the monopolist chose a profit-maximizing output assuming no other
rivals. Now, the former monopolist will have to re-optimize and choose a new level.
▪ In doing, the monopolist will (as did the new entrant previously) choose an output
level that maximizes profits given the output sold by the new rival firm.
▪ This process of each firm choosing an output conditional on the other’s output
choice is to be repeated.
▪ For every output choice by the incumbent, firm 1, the entrant, firm 2, is shown to
have a unique, profit-maximizing response and vise-versa.
▪ Cournot called the graph representations of these responses Reaction Curves.

4. ▪ The industry inverse demand curve is linear, and can be described by:
▪ P = A − BQ = A − B(q1 + q2) …….…(1)
▪ where Q is the sum of each firm’s production, i.e., the total amount sold on the market;
▪ q1 is the amount of output chosen by firm 1, the incumbent firm;
▪ q2 is the amount of output chosen by firm 2, the new competitor.
▪ Also assume that each firm faces the same, constant marginal cost of production, c.
▪ If we now consider firm 2 alone, and take firm 1’s output, q1, as given, the inverse
demand curve, facing firm 2 is:
▪ P = A − Bq1 − Bq2…………………… (2)
▪ Eq (2) is formally identical to Eq (1). However, from firm 2’s perspective, the first
two terms on the right-hand side are not part of its decision-making, and can be
taken as given.

5. ▪ In other words, those two terms
together form the intercept of firm
2’s perceived demand curve so that
firm 2 understands that the only
impact its output choice has on
price is given by the last term of the
equation, namely, −Bq2.
▪ Any change in the anticipated
output choice of the firm would be
communicated to firm 2 by means of
a shift in firm 2’s perceived demand
curve.The given Figure illustrates
this point.
▪ Firm 2’s demand curve in the
Cournot duopoly game depends on
firm 1’s output.
▪ An increase in q1 to q1’ shifts D2, the
demand curve facing Firm 2,
downwards.

6. ▪ Associated with each demand curve illustrated in Figure on previous slide, there is
a marginal revenue curve that is twice as steeply sloped. (slope of MR is always
twice that of AR)
▪ So, firm 2’s marginal revenue curve is also a function of q1 given by:
▪ MR2 = (A − Bq1) − 2Bq2 ………………………………………….. (3)
▪ Marginal cost for each firm is constant at c. Setting marginal revenue MR2 equal to
marginal cost c, as required for profit-maximization, and solving for q2* yields firm
2’s Reaction Curve.
▪ Thus we have MR2 = c, which implies that
▪ A − Bq1 − 2Bq2* = c or
▪ 2Bq2* = A − c − Bq1.

7. ▪ Further simplification then gives the Reaction Curve for firm 2:
▪ ………………………… (4)
▪ Equation (4) describes firm 2’s best output choice, q2* , for every choice of q1.
▪ The relationship is negative, i.e., every increase in firm 1’s output lowers firm 2’s demand
and marginal revenue curves and, with a constant marginal cost, also lowers firm 2’s profit-
maximizing output.
▪ We may symmetrically re-work the industry demand curve to show that firm 1’s individual
demand depends similarly on firm 2’s choice of output, so that as q2 changes, so does the
profit-maximizing choice of q1.
▪ Then, we may analogously derive firm 1’s Reaction Curve giving its best choice of q1 for
each alternative possible value of q2. By symmetry with firm 2, this is given by:
▪ …………………………….. (5)
▪ As was the case for firm 2, firm 1’s profit maximizing output level q1* falls as q2 increases.

8. ▪ The Reaction Curve for each firm is
shown in Figure in which the strategic
variables for each firm and firm
outputs are on the axes.
▪ Consider first, the Reaction Curve of
firm 1, the initial monopolist.
▪ This curve says that if firm 2
produces nothing, then firm 1 should
optimally produce quantity
(A − c)/2B
▪ This is the pure monopoly level
output, at which we assumed firm 1 to
be producing in the first place.

9. ▪ Now consider the Reaction Curve for firm 2.
▪ That curve shows that if firm 1 were
producing at the assumed level of
(A − c)/2B,
▪ then firm 2’s best bet is to produce at level
(A − c)/4B,
▪ that is, firm 2 should enter the market.
▪ However, if firm 2 does choose that level,
then firm 1 will no longer do best by
producing the monopoly level. Instead, firm
1 will maximize profits by selecting quantity
▪ q1 = 3(A − c)/8B

10. ▪ As Cournot understood, none of the output or strategy combinations described
above corresponds to an equilibrium outcome.
▪ In each case, the reaction of one firm is based upon a choice of output for the other
firm that is not, itself, that other firm’s best reaction to the initial chooser’s selection.
▪ For the outcome to be an equilibrium, it must be the case that each firm is
responding optimally to the (optimal) choice of its rival.
▪ For that to be the case though, requires that both firms be on their respective
Reaction Curves.This happens at only one point, namely, the intersection of the two
Reaction Curves.

11. ▪ Reaction Function for Firm 1 & 2 at point of intersection will be equal.
▪ For firm 2, it is given by and for firm 1 by
▪ Substituting this into firm 2’s Reaction Curve allows firm 2 (and also us) to solve for:
▪ In turn, this implies:
▪ Because the two firms are identical, it also implies that as well.
▪ Total output for this market is:
▪ Substituting this into the demand function gives the equilibrium price:
▪ Profit for each firm is total revenue less total cost, which can be solved as

12. ▪ Cournot duopoly model just presented has a unique Nash equilibrium.
▪ Unfortunately, Cournot’s original discussion somewhat obscures the underlying
Nash insight and suggests a kind of trial-by-error learning process by which the
two firms act and react and thereby move along their Reaction Curves until the
equilibrium is achieved.
▪ But the power of the Nash equilibrium is that it makes it unnecessary to play out
such an iterative procedure in real time.
▪ If the entrant, firm 2, is rational and strategic then in choosing its own production
level, it must anticipate that the incumbent, firm 1, will do whatever maximizes its
profits.
▪ Rationally, strategic firms can work through the Cournot model as a pure thought
experiment, and select the unique Nash equilibrium output

13. ▪ Go through Chapter 9 of Industrial Organization - Contemporary Theory and
Empirical Applications Book and solve the Practice Problem 9.2