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5 Cournot Suppose there are two departments selling economics degrees in one market competing fol- lowing the rules of the Cournot Oligopoly Model econ and man. econ. Suppose market demand for an economics degree is 7200-2p. Suppose both departments marginal cost is $3000 per degree. What is each department's residual demand curve? 5.1 5.2 What is each department's best response functions? 5.3 What is the Nash-Cournot equilibrium in this market? What is the price at this equilibrium? 5.4 What are profits for the two departments at this equilibrium? 5.5 Graph the marginal cost, demand, residual demand, and marginal revnue curve for the econ department. Show the equilibrium price and quantity for the econ department. 5.6

Answer 1

Two departments sell degree of economics in a market. The market demand function is, Q 7200 2p 7200 Q 2 -3600 2 Here, Q 1+q2 The marginal cost is $3,000 per degree. 5.1. The residual demand curve is the market demand curve minus quantity supplied by other firms Residual demand curve for department 1 (econ) ql 7200 2p-q2 Residual demand curve for department 2 (man. econ), q2 7200-2p-gl 5.2. To find the best responses for two firms, determine the profit function of each firm and derivate the function The profit function for department 1 is, - pgl-cql q2 ql-3000q1 3600 1 3600q1 2-3000ql Derivate the above profit function with respect to ql and find the best response for firm 1 as follows:

at 3600-ql--g2-3000 1 0 600-q ql 600-- Hence, the best response function for department 1 is ql 600-q2 2 The profit function for department 2 is, pql-cql = 1 q2 q2-3000q2 3600 1 = 3600q2-glq2 1 q22-3000q2 Derivate the above profit function with respect to ql and find the best response for firm 1 as follows: 3600 ql-q2-3000 0 600 gl-q2 q2 600 1 Hence, the best response function for department 2 is g2=600-gl 2 5.3 To determine Nash Equilibrium in this market, first, substitute q2 function in ql function

ql 600 600 ql 1 gl 600 300+-gl 1 gl 300 ql 300 x 3 - 400 Substitute the value of ql in function q2 and solve as foll s: q2 600(400) - -600-200 -400 gl=q2=400 Hence, the market equilibrium quantities are 5.4. The equilibrium price is, p 3600 (400+400) 2 -3600-400 =3200 Hence, the equilibrium price is 3200| 5.5 The profits for both departments Profits TR-TC - ( 3200 x 400)-(3000 x 400) 1,280,000-1,200,000 80,000 Hence, the profits for each department is $80,000

5.6 The plot of residual demand for department 1 (econ) with equilibrium is shown below 3200 MC 200 00 Residual demand MR. 3000 1000 2000 400 00 ql 400 In the above graph, the following equations plotted: are MC 3,000 Resi dual demand: ql 7200-2p-q2 _ 7200-2p-400 -6800-2p p 3400 2 MR 3400-l The best response function for both departments is plotted below P, MR, MC

q2 BR1 BR2 600 (400,400) 400 200 20 400 640 800 120 1400 1000 C ql 6. In Stackelberg equilibrium, the equilibrium takes place when the best response curve of the follower firm and profit curve of the leader intersects Here, department 2 (man. eco) moves first, thus department 2 is leader and department 1 (eco) is follower 6.1 The best response function of econ department (follower) is, The profit function for department 1 is, pql-cq 1 3600 q2 ql-3000q1 = 1 -3600q 1 -국91-9lg2-3000gl Derivate the above profit function with respect to ql and find the best response for firm 1 as follows

a 3600-l- 3000 = 0-7-10. 0 600-q q2 ql 600 1 Hence, the best response function for department econ is ql 600-q2 6.2 The residual demand curve man. eco department is, q2 7200-2p-q - 7200-2p - 600 1 p 3300 q2 _ 6.3 To determine equilibrium, first compute profit function of man. eco (department 2) department = pql-cql q2 q2-3000q2 3600 2-al22 3600q2-glq -3000q2 Substitute best response of econ department in above profit function, 1 q22-3000q2 q2 a2 3600q2 600 1 -3000q2 3600q2 300q 2+q2 2 Derivate the above equation with respect to q2

d 3600-300 2-q2-3000 dq2 0 300 2 q2-300x 2 600 Substitute value of q2 in the best response of econ department, 1 ql 600-(600) =600 300 -300 Hence, the market equilibrium quantities are ql 300,q2 600 6.4 The market equilibrium price is, p 3600-300+600) 2 -3600-450 -3150 Hence, the market equilibrium price is $3,150 6.5 The profits for both departments Profits (econ) TR -TC =(3200x300)-(3000x 300) 960,000-900,000 60,000 Profits (manecon) TR -TC (3200x 600)-(3000 x 600) - 1,920,000 1,800,000 = -120,000

6.6 The plot of residual demand for department 2 (man. econ) with equilibrium is shown below 4000 3,150 000 MC 2000- Demand 1000 MR 600 1000 2000 3000 4000 5000 Z000 8000 6000 92 In the above graph, the following equations are plotted: MC 3,000 Resi dual demand: q2 7200-2p-ql 1 -7200-2p-600 p = 3300 q2 MR 3300 q2 MR