- 1.6. Consider the Cournot duopoly model where inverse demand is P(Q) = a − Q but firms have asymmetric marginal costs: c₁ for firm 1 and c²₂ for firm 2. What is the Nash equilibrium if 0 < c; < a/2 for each firm? What if c₁ < c₂ a+q₁₂₁?
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- 1. Best responses in a Cournot Oligopoly Firm A and Firm B sell identical goods Total market demand for the good is: The inverse demand function is therefore 1 P(QM) = 780 -Q=780 -0.02222QM 45 QM is total market production (i.e., combined production of firm's A and B. That is: Q(P) = 35, 100- 45P 2M = A +QB As a result, the inverse demand curve for each firm is: P(QA, QB) = 780- -1/32₁-752 45 Unlike the example in class, the two firms have different costs. = 4000A TCA (QA) TCB (QB) = 260QB = 780 -0.022220A -0.02222QB a. Using the demand function and the cost functions above, what is firm A's profit function. b. Using the profit function above and assuming that firm B produces Qg, calculate what firm A's best response is to firm B’s decision to produce QB- Note: Firm A's best response should be a function of B2. An industry contains two firms that have identical cost functions C(q)=10+2q. The inverse demand function for the market is P=50-2Q where Q is the total industry output. Assuming the firms compete in quantities: Find the firms' best response functions. b. Solve for the Cournot Nash Equilibrium of the game. What is the total industry output in equilibrium? What is the equilibrium price? с. i. If both firms could collude, what would the industry output and price be? Suppose they decide that each firm produces half of the industry output found in part (i). Is this agreement self-enforcing? Explain. ii. a.Two firms with differentiated products are competing in price. Firm A and B face thefollowing demand curves: Q_A = 70 − 2P_A + P_B and Q_B = 120 − 2P_B + P_Arespectively. Assume production is costless.a. Give equations for and graph each firm’s reaction curve.b. If both firms set their prices at the same time, what is the Nash equilibriumprice, quantity, and profit for each firm?c. Suppose A sets its price first and then B responds. What price and quantitydoes each firm set now? Is it advantageous to move first?d. Compare the profits from part b and c. Which firm benefits more from thesequential price choosing? (Please do b-d, thanks :))
- 12. Two firms with differentiated products are competing in price. Firm A and B face the following demand curves: QA = 90 – 2PĄ + Pg and QB = 140 – 2Pg + PA respectively. Assume production is costless. a. Give equations for and graph each firm's reaction curve. b. If both firms set their prices at the same time, what is the Nash equilibrium price, quantity, and profit for each firm? c. Suppose A sets its price first and then B responds. What price and quantity does each firm set now? Is it advantageous to move first? d. Compare the profits from part b and c. Which firm benefits more from the sequential price choosing?1. In class, we showed that in a Cournot duopoly with demand p: ab(91 +92) and marginal costs of c₁ for firm 1 and c2 for firm two, the firms' best response functions were as follows: a C1 92 2b 91 a-c2 92 - 2b 91 822 and (a) Find equilibrium expressions for q₁ and q in terms of a, b, C1, and C2 only. (b) Show that the equilibrium price will be p": = a + c1 + €2 3Let ci be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by P=1-Q. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry prof- its, consumer surplus, and total welfare. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1’s cost would change the equilibrium. Draw a representative isoprofit for firm 1.
- Oligopoly Consider a market in which the market demand curve is given by P= 18 - Q, Firm 1 has a marginal cost of 3, while Firm 2 has amarginal cost of 6. Find the Cournot equilibrium output. What arethe Stackelberg equilibrium output if firm 1 act as the leader?Duopoly: 1. Consider a duopoly game with 2 firms. The market inverse demand curve is given by P(Q) = 120-Q, where Q = 9₁ +9₂ and q; is the quantity produced by firm i. The firm's long run total costs are given by C₁(9₁)=2q₁ and C₂(92)=92, respectively. a. Determine the Nash Equilibrium for Cournot competition, in which firms compete based on quantity. What is each firm's best response as a function of the other firm's output? Graph these best response functions in on the same graph. Compute the associated payoffs for each firm. b. Determine the Nash Equilibrium for Bertrand competition, in which firms compete based on price and stand ready to meet market demand at that price. What is each firm's best response as a function of the other firm's price? Graph these best response functions in on the same graph. Compute the associated payoffs for each firm. Game Th* Suppose there are two identical firms in an industry who compete by setting quantities. The output of firm 1 is denoted by q, and that of firm 2 is denoted by q Each firm faces a constant marginal cost of 3. Let Q denote total output. Le Q-q+42 The inverse demand curve in the market is given by (a) Find the Cournot-Nash equilibrium quantity produced by each firm and the market price. (b) If the firms could collude, what would be the total output in the mar ket? Assuming each firm produces half of the collusive output, what is the profit of each firm? (c) Suppose each firm produces half of the collusive output identified in part (b). Firm 1 considers a deviation from this arrangement. What would be the best deviating output of firm 1 and its deviation profit? (d) Suppose firms interact repeatedly over an infinite horizon, and firms have a common discount factor & € (0,1) Specify a trigger strategy for each firm to sustain the collusive arrangement as an equilibrium outcome Cal- culate…
- One of the predictions of the oligopoly model is that: non-price competition is uncommon and price-cutting competition among rivals is common. O prices tend to remain relatively stable despite short-run fluctuations in market demand. the firms' costs of production (raw material, labor, advertising) remain constant over time. only one buyer (monopsony) will result in the long run. MacBook Pro -> G Search or type URL %23 3 4. 7 8 W R YConsider the Cournot duopoly model. Suppose the inverse demand is given by P(Q) = 200 - Q where Q = q1 + q2, and production has no costs. %3! What are the payoffs for each firm? O 200 – q1 and 200 - q2 O (200 – q1 - q2)qı - 91 and (200- q1-92)92- 92 200 – q1 – q2 and 200 – q1 - 92 O (200 – q1 - q2)q1 and (200 - q1- 92)q2 O (200 – q1)qı and (200- q2)q2Consider a market in which there are two firms: A and B. Each firm produces a differentiated product and chooses its price Assume that each firm can set price equal to $60 or $70. The payoffs associated with each set of prices are shown If the firms choose price simultaneously, then the Nash equilibrium price for firm A If firm A chooses price first and can commit to that price, then firm A will Firm B's Price is set its price equal to $00 $70 OA. $60, $60 OB. $70, $70 $2700 $2475 $60 OC 570, $60 OD. $60, $70 $2700 $3375 Firm A's Price $3375 $3300 $70 $2475 $3300
