• Assume two firms—Firm 1 and Firm 2
• The price and output decision of Firm 1 affects the behaviour of Firm 2
• If Firm 1 decides to increase output, prices will become lower for the two firms and their individual and aggregate profits will fall.
• Thus, there is interdependence between the two firms
• The profit of Firm 1 would be, say, \(\Pi_{1}=\Pi_{1}(q_{1}, q_{2})\) and that of Firm 2 \(\Pi_{2}=\Pi_{2}(q_{1}, q_{2})\).

• Augustin Cournot (1838)
• A duopoly model
• Two spring water firms
• How much would each produce and sell?

• Assumptions:
  • Homogeneous products
  • Firms choose output or quantities
  • Firms compete with each other just once
  • Firms make their production decisions simultaneously
  • No entry by other firms
  • Each firm treats the other’s quantity as a fixed number, one that will not respond to its own production decisions.

• Each firm chooses its quantity of output to maximize its profits, taking the other firm’s output as given
• Given the rival’s quantity, the firm maximizes its profits by picking the quantity where \(MR = MC\).
• Each firm’s quantity depends on the quantity of its rival’s—a relationship known as a reaction curve.
• The equilibrium quantity for the two firms occur at the intersection of the two reaction curves.

• The residual demand curve is the portion of the market demand curve that remains for the first firm after the second firm has sold its output.
• For the linear demand curve \(P=a-bQ\), the residual demand curve facing the first firm would be
• \[P=a-bQ\] \[ =a-b(Q_{1}+ Q_{2})\] \[=a-bQ_{2}-bQ_{1}\]
• where \(a-bQ_{2}\) is the price intercept.

• The marginal revenue curve for the first firm is found as if the first firm were a monopolist facing the residual demand curve \(P=a-bQ_{2}-bQ{1}\).
• Marginal revenue has the same intercept but twice the slope. \(MR=a-bQ_{2}-2bQ_{1}\).
• To maximize profits, set \(MC = MR\) and solve for \(Q_{1}\):
• \[=a-bQ_{2}-2bQ_{1}\]
• \[2bQ_{1}=a-bQ_{2}\]
• \[Q_{1}=\frac{a}{2b}-\frac{1}{2}Q_{2}\].

• The reaction function is a curve that shows the profit-maximizing level of output for one firm for each amount supplied by another.
• for Cournot duopoly with linear demand and zero marginal costs, the reaction function for Firm 1 is:
• \[R_{1}(Q_{2})=Q_{1}=\frac{a}{2b}-\frac{1}{2}Q_2\].
• The corresponding reaction function of Firm 2 is
• \[R_{2}(Q_{1})=Q_{2}=\frac{a}{2b}-\frac{1}{2}Q_{1}\].

• On a graph of output of Firm 1 versus Firm 2, the reaction curve for Firm 1
• \[R_{1}(Q_{2})=Q_{1}=\frac{a}{2b}-\frac{1}{2}Q_{2}\]
• has vertical intercept \(Q_{1}=\frac{a}{2b}\) when \(Q_{2}=0\)
• horizontal intercept \(Q_{2}=\frac{a}{b}\) when \(Q_{1}=0\)

• and similarly for the reaction function of Firm 2.
• To find the equilibrium output of Firm 1 and Firm 2
• where the two reaction functions intersect, solve
• \[Q_{2}=\frac{a}{2b}-\frac{1}{2}Q_{1}\] for \[Q_{1}=\frac{a}{b}-2Q_{2}\]

• then set equal to \[Q_{1}=\frac{a}{2b}-\frac{1}{2}Q_{2}\]
• \[\frac{a}{b}-2Q_{2}=\frac{a}{2b}-\frac{1}{2}Q_{2}\]
• \[\frac{3}{2}Q_{2}=\frac{a}{2b}\].
• \[Q_{1}=Q_{2}=\frac{a}{3b}\].

• Or alternatively insert \(Q_{2}=\frac{a}{2b}-\frac{1}{2}Q_{1}\) into
• \[Q_{2}=\frac{a}{2b}-\frac{1}{2}Q_{2}\]
• \[Q_{1}=\frac{a}{2b}-\frac{1}{2}(\frac{a}{2b}-\frac{1}{2}Q_{1})\]
• \[\frac{3}{4}Q_{1}=\frac{a}{4b}\]
• \[Q_{1}=Q_{2}-\frac{a}{3b}\].

• Thus, with regard to the quantity of each firm, \(Q_{1}=Q_{2}=\frac{a}{3b}\)
• Total quantity, \[Q=Q_{1}+Q_{2}=\frac{2a}{3b}\]
• Price, \[P=a-bQ=a-b(\frac{2a}{3b})=\frac{a}{3}\]

• Profit, \[TR-TC=PQ-0\]
• \[PQ=\frac{a}{3b}(\frac{a}{3})=\frac{a^{2}}{9b}\]
• Industry profit, \(\Pi =\frac{2a^{2}}{9b}\).

Cournot assumes firms are price takers, however it is more realistic to assume that firms set prices and allow the consumers decide how much to buy at each price. This is the Bertrand idea.

Assumptions

• two firms
• prices are set by firms simultaneously
• homogeneous products
• marginal cost equals \(c\) for both firms
• firms satisfy all the demand (i.e. there is no capacity constraint)

• demand for firm \(i\) depends on the price set by its rival:
• \[D(p_{i}, p_{j})={D(p_{i})}\] if \(p_{i}<p_{j}\) a firm captures all demand
• \[D(p_{i}, p_{j})={\frac{1}{2}D(p_{i})}\] if \(p_{i}=p_{j}\) firms share demand equally
• \[D(p_{i}, p_{j})={0}\] if \(p_{i}>p_{j}\) a firm losses all demand.

• The objective is to find the reaction functions and the Nash equilibrium.
• The Nash equilibrium \((p_{i}^{*}, p_{j}^{*})\) maximizes profits given the action of the other firm.
• \[\Pi^{i}(p_{i}^{*}, p_{j}^{*})\ge\Pi^{i}(p_{i}, p_{j}^{*})\] for all \(p_{i}\)
• \[\Pi^{i}(p_{i}^{*}, p_{j}^{*})\ge\Pi^{i}(p_{i}^{*}, p_{j})\] for all \(p_{j}\)
• The Bertrand paradox is that the unique equilibrium is \(p_{i}^{*}=p_{j}^{*}=c\) and therefore \(\Pi^{i*}=\Pi^{j*}\)

• To prove that this is a unique equilibrium, proceed as follows.
• Assume \(p_{i}^{*}>p_{j}^{*}>c\) is a Nash equilibrium.
• It can be shown that this would not be possible.
• Firm 1 will not have demand \(D_1=0\) which implies \(\Pi^{1}=0\)
• Firm 2 will have all the demand of the market
• \(D_2=D(p_2^{*})\) and \(\Pi^{2}=(p_2^{*}-c)D(p_2^{*})>0\)
• This is not an equilibrium because Firm 1’s best response to \(p_2^{*}\) is not \(p_1^{*}\) but \(p_1^{1}=p_2^{*}-\epsilon\)
• where \(\epsilon\) is very small and this causes \(\Pi^{1}>0\)

• thus the situation where \(p_1^{*}>p_2^{*}>c\) is not a Nash equilibrium
• Assume that \(p_1^{*}=p_2^{*}>c\) is an equilibrium
• In this case firms share the market equally. Thus:
• \[\Pi^{1}=(p_1^{*}-c)(\frac{1}{2}D(p_1^{*}))>0\]
• \[\Pi^{2}=(p_2^{*}-c)(\frac{1}{2}D(p_2^{*}))=\Pi^{1}>0\]
• This is however not an equilibrium because the best response of, say, Firm 1 to \(p_2^{*}\) is not \(p_1^{*}\) but \(p_1^{*}=p_2^{*}-\epsilon\).

• In this case, Firm 1 will win all the demand
• \[\Pi^{1'}=(p_1^{'}-c)D(p_1^{'})=(p_1^{*}-c)D(p_1^{*})\]>\[\Pi^{1}=(p_1^{*}-c)(\frac{1}{2}D(p_1^{*}))>0\].
• Therefore, \(p_1^{*}=p_2^{*}>c\) is not an equilibrium in the Bertrand model.

• Assume that \(p_1^{*}>p_2^{*}=c\) is an equilibrium
• In this case Firm 1 has no demand to begin with. That is \(\Pi^{1}=0\) and \(\Pi^{2}=(p_2^{*}-c)D(p_2^{*})=0\) meaning all the demand goes to Firm 2.
• This is not an equilibrium because the best response of, say, Firm 2 to \(p_1^{*}\) is not \(p_2^{*}\) but \(p_2^{'}=p_1^{*}-\epsilon\), which allows Firm 2 to keep all the demand in the market. That is \(\Pi^{2'}=(p_2^{'})D(p_2^{'})>0\).
• Again, \(p_1^{*}>p_2^{*}=c\) is not an equilibrium.

• The only possible equilibrium is \(p_1^{*}=p_2^{*}=c\)
• It has to be shown that this is indeed an equilibrium
• To show that something is an equilibrium is to show that there is no incentive to deviate from it.
• In this case, firms share the market but have zero profits. That is
• \(\Pi^{1}=0\) and \(\Pi^{2}=0\)

• If Firm 1 reduces it price \(p_1\), this implies \(\Pi^{1}=(p_1^{*}-\epsilon-c)D(p_1^{*}-\epsilon)=-D(p_1^{*}-\epsilon)<0\)
• Therefore, Firm 1 has not incentive to lower its price \(p_1\)
• If Firm 1 increases its price \(p_1\), this implies \(\Pi^{1}=(p_1^{*}+\epsilon-c).0=0\)
• Firm 1, therefore, has no incentive to increase its price \(p_1\) either.
• Therefore, Firm1 has no incentive to deviate. Thus, \(p_1^{*}\) is its best response to \(p_2^{*}\)
• Same goes to Firm 2.