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Games with continuous payoffs. The Cournot game In all the games discussed so far, firms had a discrete set of choices (high – medium – low, enter – not enter). In real markets, firms choose from a range of prices and quantities. One of the game-theoretical models dealing with continuous set of choices is the Cournot model of a duopoly.

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“Duopoly” = a market with two firms. The game: Two identical firms producing identical goods enter the same market simultaneously and choose the quantities they are going to produce. Each firm knows what the market demand is but does not know the rival’s choice of quantity. Market demand is Q = 130 – P There are two firms, therefore Q = q 1 + q 2 Each firm has TC i = 10 q i (i = 1, 2)

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Firm 1’s profit = P q 1 – 10 q 1 = (P – 10) q 1 = = (130 – q 1 – q 2 – 10) q 1 =

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Firm 1’s profit = P q 1 – 10 q 1 = (P – 10) q 1 = = (130 – q 1 – q 2 – 10) q 1 = (120 – q 1 – q 2 ) q 1 = = 120 q 1 – q 1 2 – q 2 q 1 Firm 1’s profit depends on its own choices AND on Firm 2’s choices; therefore, it is a game. To solve this game, we are going to once again use the best response technique: one firm assumes certain action on the other firm’s part, and finds the best response to that hypothetical action.

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Let’s assume q 2 = 30. What is firm 1’s best response to that? 130 30 10 100 q 1 *, our best response D MARKET D1D1 MR 1 MC 1

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The same in mathematical terms: The residual demand for firm 1 is q 1 = 100 – P The inverse residual demand is P = 100 – q 1 Hence MR = 100 – 2 q 1 If MC = 10, then firm 1’s best response to firm 2’s decision to produce 30 units is MR = MC 100 – 2 q 1 = 10 90 = 2 q 1 q 1 * = 45

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The general case: If firm 2 chooses to produce quantity q 2, then firm 1’s inverse residual demand is P = (130 – q 2 ) – q 1 Hence MR 1 = (130 – q 2 ) – 2 q 1 If MC = 10, we set MR = MC to get (130 – q 2 ) – 2 q 1 = 10, or 2 q 1 = 120 – q 2, or q 1 * = 60 – 0.5 q 2

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Alternatively, using differentiation: The formula for firm 1’s profit is Π 1 = (P – 10) q 1 = (130 – q 1 – q 2 – 10) q 1 If firm 2 produces 30 units, then Π 1 = (90 – q 1 ) q 1 = 90 q 1 – q 1 2 We can find our best response by using the profit- maximization approach: Our control variable is q 1. Take the derivative of our profit with respect to q 1, Set it equal to zero and solve for q 1.

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In the more general case, Π 1 = (120 – q 2 – q 1 ) q 1 = (120 – q 2 ) q 1 – q 1 2 Even though notation-wise variables q 2 and q 1 look very similar, we have to keep in mind that Firm 1 has control only over q 1 and therefore treats q 2 as given, or a constant. All the rules are applied accordingly. (120 – q 2 ) – 2 q 1 (120 – q 2 ) – 2 q 1 = 0 q 1 = 60 – 0.5 q 2

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Either way, we can see that for every q 2 chosen by Firm 2, Firm 1 has a unique best response, q 1 * = 60 – 0.5 q 2 The smaller the q 2, the greater the q 1 * chosen in response. (Makes sense) It is easy to guess that, due to the symmetry of the game, Firm 2’s best response to Firm 1’s choice is q 2 * = 60 – 0.5 q 1 Let us plot the best response functions in the (q 1, q 2 ) space.

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0 10 20 30 40 50 60 70 80 q 1 70 60 50 40 30 20 10 q2q2 q 1 * (q 2 ), or BRF of Firm 1 q 2 * (q 1 ), or BRF of Firm 2 The point where strategies (quantity choices) are best responses to each other

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The equilibrium can also be found algebraically, by solving the two best response equations together: q 1 * = 60 – 0.5 q 2 (Eq.1) q 2 * = 60 – 0.5 q 1 (Eq.2) (Since we are looking for best responses of both firms, we can ignore the asterisks or assume all q’s in the equations appear with asterisks.) From (Eq.1), q 2 = 120 – 2 q 1. Plugging it into (Eq.2) yields 120 – 2 q 1 = 60 – 0.5 q 1 Rearranging gives 1.5 q 1 = 60, or q 1 = 40.

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By plugging q 1 = 40 into either of the two original equations, we get q 2 = 40 (which is not surprising, given the symmetry of the game). We can also calculate resulting equilibrium price(s) and quantities. P = 130 – Q =

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By plugging q 1 = 40 into either of the two original equations, we get q 2 = 40 (which is not surprising, given the symmetry of the game). We can also calculate resulting equilibrium price(s) and quantities. P = 130 – Q = 130 – (q 1 + q 2 ) = $50 Profit of each firm is Π = (P – ATC) ∙ q = (50 – 10) ∙ 40 = $1600

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Questions for discussion: 1. How will the outcome of the Cournot game change if the firms are no longer identical (the marginal costs of the two firms are different)? 2. How does the outcome of the Cournot game compares to the case when the firms are able to collude?

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If a monopoly faces demand given by P = 130 – Q, and has MC = $10, then…

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If a monopoly faces demand given by P = 130 – Q, and has MC = $10, then… TR = P∙Q = (130 – Q) Q = 130 Q – Q 2 Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q, Profit-maximization condition, MR = MC, or 130 – 2 Q = 10 results in 2 Q = 120,or Q = 60 From the demand equation, P = $70, and the resulting profit is (P – ATC) Q

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If a monopoly faces demand given by P = 130 – Q, and has MC = $10, then… TR = P∙Q = (130 – Q) Q = 130 Q – Q 2 Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q, Profit-maximization condition, MR = MC, or 130 – 2 Q = 10 results in 2 Q = 120,or Q = 60 From the demand equation, P = $70, and the resulting profit is (P – ATC) Q = ($70 - $10) 60 =

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If a monopoly faces demand given by P = 130 – Q, and has MC = $10, then… TR = P∙Q = (130 – Q) Q = 130 Q – Q 2 Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q, Profit-maximization condition, MR = MC, or 130 – 2 Q = 10 results in 2 Q = 120,or Q = 60 From the demand equation, P = $70, and the resulting profit is (P – ATC) Q = ($70 - $10) 60 = $3600

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We can summarize our findings in a table. Noncooperative (Cournot) duopoly Collusion (monopoly) Market price$50$70 Market quantity40 + 40 = 8060 Profits $1600 + $1600 = = $3200 $3600 The higher profit earned in the monopoly case explains why firms prefer to collude, or at least to coordinate their prices and outputs in some way.

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