The profits for Firms 1 and 2 are equal to 元1=(242.8)-20+(102.8)=19.20and 元2=2402.4)-(10+(122.4》=18.80. c.How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but the takeover is not? In order to determine how much Firm I will be willing to pay to purchase Firm 2.we must compare Firm I's profits in the monopoly situation versus those in an oligopoly.The difference between the two will be what Firm 1 is willing to pay for Firm 2 From part a,profit of firm 1 when it set marginal revenue equal to its marginl what the firm would arn if it was a From part b.profit was $19.20 for firm 1.Firm 1 would therefore be willing to pay up to $40.80 for firm 2. 3.A monopolist can produce at a constant average(and marginal)cost of AC=MC=5.It faces a market demand curve given by Q=53-P. a.Calculate the profit-maximizing price and quantity for this monopolist.Also calculate its profits. The monopolist wants to choose quantity to maximize its profits: max元=PQCQ =(53.Q(Q).5Q.or =480.Q. Todetermine the profit-maximizing quantity,set the change inwith respect to the change in equal toero and =-2Q+48=0,orQ=24 Substitute the profit-maximizing quantity,Q=24,into the demand function to find price: 24=53-P,0rP=$29. Profits are equal to 元=TR.TC=(29)24到·(6)24④=$576. b.Suppose a second firm enters the market.Let Q be the output of the first firm and Q,be the output of the second.Market demand is now given by Q1+Q2=53-P. Assur that this second firm has the osts as the first write the profits ofea When tho irmenters,price can be writt two firms:P=may write the profit functions for the two firms:
The profits for Firms 1 and 2 are equal to 1 = (24)(2.8) - (20 + (10)(2.8)) = 19.20 and 2 = (24)(2.4) - (10 + (12)(2.4)) = 18.80. c.How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but the takeover is not? In order to determine how much Firm 1 will be willing to pay to purchase Firm 2, we must compare Firm 1’s profits in the monopoly situation versus those in an oligopoly. The difference between the two will be what Firm 1 is willing to pay for Firm 2. From part a, profit of firm 1 when it set marginal revenue equal to its marginal cost was $60. This is what the firm would earn if it was a monopolist. From part b, profit was $19.20 for firm 1. Firm 1 would therefore be willing to pay up to $40.80 for firm 2. 3. A monopolist can produce at a constant average (and marginal) cost of AC = MC = 5. It faces a market demand curve given by Q = 53 - P. a. Calculate the profit-maximizing price and quantity for this monopolist. Also calculate its profits. The monopolist wants to choose quantity to maximize its profits: max = PQ - C(Q), = (53 - Q)(Q) - 5Q, or = 48Q - Q 2 . To determine the profit-maximizing quantity, set the change in with respect to the change in Q equal to zero and solve for Q: d dQ Q Q = −2 + 48 = 0, or = 24. Substitute the profit-maximizing quantity, Q = 24, into the demand function to find price: 24 = 53 - P, or P = $29. Profits are equal to = TR - TC = (29)(24) - (5)(24) = $576. b. Suppose a second firm enters the market. Let Q1 be the output of the first firm and Q2 be the output of the second. Market demand is now given by Q1 + Q2 = 53 - P. Assuming that this second firm has the same costs as the first, write the profits of each firm as functions of Q1 and Q2 . When the second firm enters, price can be written as a function of the output of two firms: P = 53 - Q1 - Q2 . We may write the profit functions for the two firms:
π=Pg-Cg)=53-0-Q3-5g,r 不1=5301-QQ2-5Q1 and 乃=Pg2-C(g)=(53-g-g23-5g2,orπ2=5322-Q号-822-502 Suppose (as in the Cournot model)that each firm chooses its profit-m cimizing level of output on the assumption that its co petitor's output is xed.Find curve"the rule thatgive its desired output in terms of its competitor's output). Under the Cournot assumption,Firm 1 treats the output of Firm 2 as a constant in its maximization of profits.Therefore.Firm 1 chooses Q to maximize in b with being treated as a constant.The change in with respect to a change in Q1 is 膏-9-20-0-3=0c8=24-号 ctionor Firm 1.which generates thep problem is symmetric,the reaction function for Firm 2 is =2-号 d.Calculate the Cournot equilibrium (ie.,the values of Q and Q2 for which both firms are doing as well as they can given their competitors'output). What are the resulting market price and profits ofeach firm? t for each firm m that resu functions by substituting the reaction function for Firm 2 into the one for Firm 1: g=24-（(24-号)rg=16 By symmetry,Q2=16. To determine the price,substitute and into the demand equation: P=53.16.16=$21 Profits are given by 元=PQ-C(Q)=元=(2116)·(616)=$256 Total profits in the industry are +x=$256+$256=$512
1 = PQ1 −C Q1 ( )= 53 −Q1 − Q2 ( )Q1 − 5Q1 , or 1 1 1 2 1 2 1 = 53Q −Q −Q Q − 5Q and 2 = PQ2 −C Q2 ( )= 53− Q1 − Q2 ( )Q2 − 5Q2 , or 2 2 2 2 1 2 2 = 53Q −Q −Q Q −5Q . c. Suppose (as in the Cournot model) that each firm chooses its profit-maximizing level of output on the assumption that its competitor’s output is fixed. Find each firm’s “reaction curve” (i.e., the rule that gives its desired output in terms of its competitor’s output). Under the Cournot assumption, Firm 1 treats the output of Firm 2 as a constant in its maximization of profits. Therefore, Firm 1 chooses Q1 to maximize 1 in b with Q2 being treated as a constant. The change in 1 with respect to a change in Q1 is 1 1 1 2 1 2 53 2 5 0 24 Q 2 Q Q Q Q = − − − = , or = − . This equation is the reaction function for Firm 1, which generates the profitmaximizing level of output, given the constant output of Firm 2. Because the problem is symmetric, the reaction function for Firm 2 is Q Q 2 1 24 2 = − . d. Calculate the Cournot equilibrium (i.e., the values of Q1 and Q2 for which both firms are doing as well as they can given their competitors’ output). What are the resulting market price and profits of each firm? To find the level of output for each firm that would result in a stationary equilibrium, we solve for the values of Q1 and Q2 that satisfy both reaction functions by substituting the reaction function for Firm 2 into the one for Firm 1: Q1 = 24 − 1 2 24 − Q1 2 , or Q1 = 16. By symmetry, Q2 = 16. To determine the price, substitute Q1 and Q2 into the demand equation: P = 53 - 16 - 16 = $21. Profits are given by i = PQi - C(Qi ) = i = (21)(16) - (5)(16) = $256. Total profits in the industry are 1 + 2 = $256 +$256 = $512
e.Suppose there are N firms in the industry,all with the same constant marginal cost,MC=5.Find the Cournot equilibrium.How much will each frm produce, what will be the market price,and ho ow much profi will each firm earn?Also,show that as N becomes large the market price approaches the price that would prevail under perfect competition. If there are Nidentical firms,then the price in the market willbe P=53-g,+g++Qx) Profits for the ith firm are given by =Pe-c(2.). 元=530-22-2,2-g2-0.2-50 Differentiating to obtain the necessary first-order condition for profit maximization, 0-朗-g-20-6s-6-0 Solving for Qi g=24-g,+.+g+g+.+0) If all firms face the same costs they will all produce the same kvel of output,1.e. Q=Q.Therefore, Q=24-0w-10*,or20*=48-w-1g,or W+12*=48org=N+ 8 We may substitute for Q=NO,totaloutput,in the demand function P=3-N) Total profits are T=PQ-C(Q)=P(NQ)-5(NQ*) (w(
e. Suppose there are N firms in the industry, all with the same constant marginal cost, MC = 5. Find the Cournot equilibrium. How much will each firm produce, what will be the market price, and how much profit will each firm earn? Also, show that as N becomes large the market price approaches the price that would prevail under perfect competition. If there are N identical firms, then the price in the market will be P = 53 − Q1 + Q2 + + QN ( ). Profits for the i’th firm are given by i = PQi − C Qi ( ), i = 53Qi − Q1Qi − Q2Qi − − Qi 2 − − QNQi − 5Qi . Differentiating to obtain the necessary first-order condition for profit maximization, d dQ Q Q Q i i N = 53 − − −2 − − − 5= 0 1 . Solving for Qi , Qi = 24 − 1 2 Q1 + + Qi −1 + Qi +1 + + QN ( ). If all firms face the same costs, they will all produce the same level of output, i.e., Qi = Q*. Therefore, Q* = 24 − 1 2 (N − 1)Q*, or 2Q* = 48 − (N −1)Q*, or (N +1)Q* = 48, or Q* = 48 (N + 1) . We may substitute for Q = NQ*, total output, in the demand function: P = 53 − N 48 N +1 . Total profits are T = PQ - C(Q) = P(NQ*) - 5(NQ*) or T = 5 3 − N 4 8 N + 1 ( N) 4 8 N + 1 − 5N 4 8 N +1 or
-[-(() or )G) N+1 Notice that with N firms e-() and that,as Nincreases (N) Q=48 Similarly.with 9-) asN→o, P=53-48=5. With P=5Q=53.5=48. Finally, =2300N+ 0asN→n m=$0. In perfect competition,we know that profits are zero and price equals marginal cost.Here,r=$0 and P=MC=5.Thus,when N approaches infinity,this market approaches a perfectly competitive one. 4.This exercise is a continuation of Exercise 3.We return to two firms with the same constant average and marginal cost,AC=MC=5,facing the market demand curve Q+Q=53-P.Now we will use the Stackelberg model to analyze what will happen ifone of the firms makes its output decision before the other. a.Suppose Firm 1 is the Stackelberg leader (i.e.,makes its output decisions before firm 2).Find the reaction curves that tell each firm how much to produce in terms of the output ofits competitor. Firm 1.the Stackelberg leader,will choose its output.to maximize its profits,subject to the reaction function of Firm2: max PQ-C(Q) subject to g=4-()
T = 48 − ( N) 48 N + 1 ( N) 48 N + 1 or T = (4 8) N + 1 − N N + 1 (4 8) N N +1 = (2, 304) N ( N + 1) 2 . Notice that with N firms Q = 48 N N + 1 and that, as N increases (N → ) Q = 48. Similarly, with P = 53 − 48 N N + 1 , as N → , P = 53 - 48 = 5. With P = 5, Q = 53 - 5 = 48. Finally, T = 2,304 N (N +1) 2 , so as N → , T = $0. In perfect competition, we know that profits are zero and price equals marginal cost. Here, T = $0 and P = MC = 5. Thus, when N approaches infinity, this market approaches a perfectly competitive one. 4. This exercise is a continuation of Exercise 3. We return to two firms with the same constant average and marginal cost, AC = MC = 5, facing the market demand curve Q1 + Q2 = 53 - P. Now we will use the Stackelberg model to analyze what will happen if one of the firms makes its output decision before the other. a. Suppose Firm 1 is the Stackelberg leader (i.e., makes its output decisions before Firm 2). Find the reaction curves that tell each firm how much to produce in terms of the output of its competitor. Firm 1, the Stackelberg leader, will choose its output, Q1 , to maximize its profits, subject to the reaction function of Firm 2: max 1 = PQ1 - C(Q1 ), subject to Q2 = 24 − Q1 2
Substitute for Q2 in the demand function and.after solving for P. substitute for Pin the profit function: mx元-(53-g-(24-g)e)-g To determine the profit-maximizing quantity,we find the change in the profit function with respect to a change in 胎-8-现-2+0-8 Set this expression equal to 0 to determine the profit-maximizing quantity 53-2Q1-24+Q1-5=0,0rQ1=24. Substituting Q1=24 into Firm 2's reaction function givesQ: 2=24-24=-12 Substituteand into the demand equation to find the price P=5324-12=$17 Profits for each firm are equal to total revenue minus total costsor 1=(1720-(6)20=$288and 2=(1712-(6(12)=$144 Total industry profit,=+=$288+$144 =$432. Compared to the Cournot equilibrium,total output has increased from 32 to 36.price has fallen from $21 to $17.and total profits have fallen from $512 to $432.Profits for Firm 1 have risen from S256 to $288.while the profits of Firm 2 have declined sharply from $256 to$144. b. How much will each firm produce,and what will its profit be? If eachfirm believes that it is the Stackelberg kader,while the other firm is the Cournot follower,they both will initially produce 24 units,so total output will be 48 units.The market price will be driven to $5.equal o marginal cost.It is impossible to specify exactly where the new equilibrium point will be,because no point is stable when both firms are trying to be the Stackelberg leader. ete in selling identical wi choose their output nd Q2 s ly and face the dem P=30-2, where Q=Qi+Qz Until recently,both firms had zero marginal costs.Recent environmental regulations have increased Firm 2's marginal cost to $15.Firm
Substitute for Q2 in the demand function and, after solving for P, substitute for P in the profit function: max 1 = 53 − Q1 − 24 − Q1 2 Q1 ( ) − 5Q1 . To determine the profit-maximizing quantity, we find the change in the profit function with respect to a change in Q1 : d dQ Q Q 1 1 1 1 = 53 − 2 −24 + −5. Set this expression equal to 0 to determine the profit-maximizing quantity: 53 - 2Q1 - 24 + Q1 - 5 = 0, or Q1 = 24. Substituting Q1 = 24 into Firm 2’s reaction function gives Q2 : Q2 24 24 2 = − = 12. Substitute Q1 and Q2 into the demand equation to find the price: P = 53 - 24 - 12 = $17. Profits for each firm are equal to total revenue minus total costs, or 1 = (17)(24) - (5)(24) = $288 and 2 = (17)(12) - (5)(12) = $144. Total industry profit, T = 1 + 2 = $288 + $144 = $432. Compared to the Cournot equilibrium, total output has increased from 32 to 36, price has fallen from $21 to $17, and total profits have fallen from $512 to $432. Profits for Firm 1 have risen from $256 to $288, while the profits of Firm 2 have declined sharply from $256 to $144. b. How much will each firm produce, and what will its profit be? If each firm believes that it is the Stackelberg leader, while the other firm is the Cournot follower, they both will initially produce 24 units, so total output will be 48 units. The market price will be driven to $5, equal to marginal cost. It is impossible to specify exactly where the new equilibrium point will be, because no point is stable when both firms are trying to be the Stackelberg leader. 5. Two firms compete in selling identical widgets. They choose their output levels Q1 and Q2 simultaneously and face the demand curve P = 30 - Q, where Q = Q1 + Q2 . Until recently, both firms had zero marginal costs. Recent environmental regulations have increased Firm 2’s marginal cost to $15. Firm