CHAPTER 12 MONOPOLISTIC COMPETITION AND OLIGOPOLY 一、QUESTIONS FOR REVIEW 1.What are the characteristics of a monopolistically competitive market?What happens to the equilibrium price and quantity in such a market if one firm introduces a new,improved product? The two primary characteristics ofa monopolistically competitive marketare(1) that firm compete differntiated products which are highly,but not perfectly,substitutable and (2)that there is free entry and exit from the market. When a new firm enters a monopolistically competitive market (seeking positive profits),the demand curve for each of the incumbent firms shifts inward,thus reducing the price and quantity received by the incumbents.Thus the introduction of a new product by a firm will reduce the price received and quantity od ofexistin products. 2.Why is the firm's demand curve flatter than the total market demand curve in monopolistic competition? Suppose a monopolistically competitive firm is making a profit in the short run.What will happen to its demand curve in the long run? be firm's pro oduct The of the fi curve is greater than the elasticity of market demand because it is easier fo consumers to switch to another firm's highly substitutable product than to switch consumption to an entirely different product.Profit in the short run induces other Grms to enter as firms enter the incumbent firm's demand and marginal revenue the profit-maximizing quantity .leaving no incentive for morefm Eventually,pr Some experts have argued that too many brands of breakfast cereal are on the market.Give an argument to support this view.Give an argument against it. Pro:Too many brands of any single product signals excess capacity,implying an output level smaller than one that would minimize average cost. Con:Consumers value the freedom to choose among a wide variety of competing products. (Note:In 1972 the Federal Trade Commission filed suit against Kellogg,General Mills,and General Foods.It that these firms attempted tos ppress entry into the ce real market by introducing 150 heavily advertised brands between 1950 and 1970.crowding competitors off grooers'shelves. This case was eventually dismissed in 1982.)
CHAPTER 12 MONOPOLISTIC COMPETITION AND OLIGOPOLY 一、QUESTIONS FOR REVIEW 1. What are the characteristics of a monopolistically competitive market? What happens to the equilibrium price and quantity in such a market if one firm introduces a new, improved product? The two primary characteristics of a monopolistically competitive market are (1) that firms compete by selling differentiated products which are highly, but not perfectly, substitutable and (2) that there is free entry and exit from the market. When a new firm enters a monopolistically competitive market (seeking positive profits), the demand curve for each of the incumbent firms shifts inward, thus reducing the price and quantity received by the incumbents. Thus, the introduction of a new product by a firm will reduce the price received and quantity sold of existing products. 2. Why is the firm’s demand curve flatter than the total market demand curve in monopolistic competition? Suppose a monopolistically competitive firm is making a profit in the short run. What will happen to its demand curve in the long run? The flatness or steepness of the firm’s demand curve is a function of the elasticity of demand for the firm’s product. The elasticity of the firm’s demand curve is greater than the elasticity of market demand because it is easier for consumers to switch to another firm’s highly substitutable product than to switch consumption to an entirely different product. Profit in the short run induces other firms to enter; as firms enter the incumbent firm’s demand and marginal revenue curves shift inward, reducing the profit-maximizing quantity. Eventually, profits fall to zero, leaving no incentive for more firms to enter. 3. Some experts have argued that too many brands of breakfast cereal are on the market. Give an argument to support this view. Give an argument against it. Pro: Too many brands of any single product signals excess capacity, implying an output level smaller than one that would minimize average cost. Con: Consumers value the freedom to choose among a wide variety of competing products. (Note: In 1972 the Federal Trade Commission filed suit against Kellogg, General Mills, and General Foods. It charged that these firms attempted to suppress entry into the cereal market by introducing 150 heavily advertised brands between 1950 and 1970, crowding competitors off grocers’ shelves. This case was eventually dismissed in 1982.)
4.Why is the Cournot equilibrium stable (ie.,why don't firms have any incentive to change their output levels once in equilibrium)?Even if they can't collude why don't firms set their outputs at the joint profit-maximizing levels (i.e.,the levels they would have chosen had they colluded)? A Cournot equilibrium is stable because each firm is producing the amount that maximizes its profits,given what its competitors are producing.If all firms behave this way,no firm has an incentive to change its output Without collusion. firms find it difficult to agree tacitly to reduce ouput.Once one firm reduces its output,other firms hay in entive t expense of the firm that is limiting its sales 5.In the Stackelberg model,the firm that sets output first has an advantage Explain why. The Stackelberg leader gains the advantage because the second firm must accept the leader's large output as given and produce a smaller output for itself.If larger quantity,this would reduce price and profit.The first firm knowe that the snd frm will have no produce a smaller output in order to maximize profit,and thus,the first firm is able to capture a larger share ofindustry profits. 6.What do the Cournot and Bertrand models have in common?What is different about the two models? Both are oligopoly models in which firms produce a homogeneous good.In the Cournot model,each firm assumes its rivals will not change the quantit yproduced and model, each not change the price the charge In both models.each firm takes some aspect of its rivals behavior(either quantity or price)as fixed when making its own decision.The difference between the two is that in the Bertrand model firms end up producing where price equals marginal cost,whereas in the Cournot model the firms will produce more than the monopoly output but less than the competitive outpu 7.Explain the meaning of a Nash equilibrium when firm competing with respect to price. Why is the equilibrium stable? Why don't the firms rais prices to the level that maximizes joint profits? A Nash equilibrium in price competition occurs when each firm chooses its price assuming its competitor's price as fixed.In equilibrium.each firm does the best it can.conditional on its competitors'prices.The equilibrium is stable because firms are maximizing profit and no firm has an incentive to raise or lower its price each firm has an incentive to cheat.By bwering price,the cheating firm can increase its market share and profits.A second reason that firms do not collude is that such collusion violates antitrust laws.In particular,price fixing violates Section 1 of the Sherman Act.Of course.there are attempts to circumvent antitrust laws through tacit colusion
4. Why is the Cournot equilibrium stable (i.e., why don’t firms have any incentive to change their output levels once in equilibrium)? Even if they can’t collude, why don’t firms set their outputs at the joint profit -maximizing levels (i.e., the levels they would have chosen had they colluded)? A Cournot equilibrium is stable because each firm is producing the amount that maximizes its profits, given what its competitors are producing. If all firms behave this way, no firm has an incentive to change its output. Without collusion, firms find it difficult to agree tacitly to reduce output. Once one firm reduces its output, other firms have an incentive to increase output and increase profits at the expense of the firm that is limiting its sales. 5. In the Stackelberg model, the firm that sets output first has an advantage. Explain why. The Stackelberg leader gains the advantage because the second firm must accept the leader’s large output as given and produce a smaller output for itself. If the second firm decided to produce a larger quantity, this would reduce price and profit. The first firm knows that the second firm will have no choice but to produce a smaller output in order to maximize profit, and thus, the first firm is able to capture a larger share of industry profits. 6. What do the Cournot and Bertrand models have in common? What is different about the two models? Both are oligopoly models in which firms produce a homogeneous good. In the Cournot model, each firm assumes its rivals will not change the quantity produced. In the Bertrand model, each firm assumes its rivals will not change the price they charge. In both models, each firm takes some aspect of its rivals behavior (either quantity or price) as fixed when making its own decision. The difference between the two is that in the Bertrand model firms end up producing where price equals marginal cost, whereas in the Cournot model the firms will produce more than the monopoly output but less than the competitive output. 7. Explain the meaning of a Nash equilibrium when firms are competing with respect to price. Why is the equilibrium stable? Why don’t the firms raise prices to the level that maximizes joint profits? A Nash equilibrium in price competition occurs when each firm chooses its price, assuming its competitor’s price as fixed. In equilibrium, each firm does the best it can, conditional on its competitors’ prices. The equilibrium is stable because firms are maximizing profit and no firm has an incentive to raise or lower its price. Firms do not always collude: a cartel agreement is difficult to enforce because each firm has an incentive to cheat. By lowering price, the cheating firm can increase its market share and profits. A second reason that firms do not collude is that such collusion violates antitrust laws. In particular, price fixing violates Section 1 of the Sherman Act. Of course, there are attempts to circumvent antitrust laws through tacit collusion
8.The kinked demand curve describes price rigidity.Explain how the model works.What are its limitations?Why does price rigidity arise in oligopolistic markets? According to the kinked-demand curve model each firm faces a demand curve that is kinked at the currently prevailing price.If a firm raises its price,mostof its customers would shift their purchases to its competitors.This reasoning implies a highly elastic demand for price increases.If the firm lowers its price er its etitor ould also wer thei prices.This This kink in the demand curve implies a discontinuity in the marginal revenue curve,so only large changes in marginal cost lead to changes in price.However accurate it is in pointing to price rigidity.this model does not explain how the rigid price is determined.The ofthe by other firmsdesire to avoid mutually de ructive price competition Why does price leadership sometimes evolve in oligopolistic markets? Explain how the price leader determines a profit-maximizing price. Since firms cannot explicitly coordinate on setting price,they use implicit means.One form of implicit collusion is to follw a price leader.The price leader often the dominant firm in the industry,determines its profit-maximizing price by calculating the demand curve itfaces:it subtracts the quantity supped at each price bya other firms from the market demand,and the residu al is its deman curve.The leader chooses the quantity that equates its marginal revenue with marginal cost The market price is the price at which the leader's profit-maximizing quantity sells in the market.At that price,the followers supply the remainder ofthe market. 10.Why has the OPEC oil cartel succeeded in raising prices substantially,while the CIPEC e pp cartel has not?What conditions are ne essary fo cartelization?What organizational problems must a cartel overcome? Successful cartelization requires two characteristics: demand should be inelastic.and the cartel must be able to control most of the supply.OPEC succeeded in the short run because the short-run demand and supply of oil were both inelastic.CIPEC has not been successful because both demand and non-CIPEC upp were highly price faces organizational proble ms agreement on a price and a division of the market among cartel members;and monitoring and enforcing the agreement. 二、EXERCISES 1.Suppose all firms in a monopolistically competitive industry were merged into one large firm.Would that new firm produce as many different brands?Would it produce only a single brand?Explain
8. The kinked demand curve describes price rigidity. Explain how the model works. What are its limitations? Why does price rigidity arise in oligopolistic markets? According to the kinked-demand curve model, each firm faces a demand curve that is kinked at the currently prevailing price. If a firm raises its price, most of its customers would shift their purchases to its competitors. This reasoning implies a highly elastic demand for price increases. If the firm lowers its price, however, its competitors would also lower their prices. This implies a demand curve that is more inelastic for price decreases than for price increases. This kink in the demand curve implies a discontinuity in the marginal revenue curve, so only large changes in marginal cost lead to changes in price. However accurate it is in pointing to price rigidity, this model does not explain how the rigid price is determined. The origin of the rigid price is explained by other models, such as the firms’ desire to avoid mutually destructive price competition. 9. Why does price leadership sometimes evolve in oligopolistic markets? Explain how the price leader determines a profit -maximizing price. Since firms cannot explicitly coordinate on setting price, they use implicit means. One form of implicit collusion is to follow a price leader. The price leader, often the dominant firm in the industry, determines its profit-maximizing price by calculating the demand curve it faces: it subtracts the quantity supplied at each price by all other firms from the market demand, and the residual is its demand curve. The leader chooses the quantity that equates its marginal revenue with marginal cost. The market price is the price at which the leader’s profit-maximizing quantity sells in the market. At that price, the followers supply the remainder of the market. 10. Why has the OPEC oil cartel succeeded in raising prices substantially, while the CIPEC copper cartel has not? What conditions are necessary for successful cartelization? What organizational problems must a cartel overcome? Successful cartelization requires two characteristics: demand should be inelastic, and the cartel must be able to control most of the supply. OPEC succeeded in the short run because the short-run demand and supply of oil were both inelastic. CIPEC has not been successful because both demand and non-CIPEC supply were highly responsive to price. A cartel faces two organizational problems: agreement on a price and a division of the market among cartel members; and monitoring and enforcing the agreement. 二、EXERCISES 1. Suppose all firms in a monopolistically competitive industry were merged into one large firm. Would that new firm produce as many different brands? Would it produce only a single brand? Explain
Monopolistic competition is defined by product differentiation.Each firm in advertising. If these competitors merge into a single firm,the resulting monopolist would not produce as many brands.since too much brand competition is internecine (mutually destructive).However,it is unlikely that only one brand would be produced afer the merger.Producing several brands withdifferen prices and c ha ristics is one method of spli tting the market into setso customers with different price elasticities,which may also stimulate overal demand. 2.Consider two firms facing the demand curve P=50-5Q where Q=Q +Q2.The firms'cost functions are C1(Q)=20+10Q1 and C2(Q2)=10+ 12Q4 a.Suppose both firms have entered the industr What is the ioint profit-maximizing level ofo output?How m uch will each firm produce? How would your answer change if the firms have not yet entered the industry？ If both firms enter the market,and they collude,they will face a marginal revenue curve with twice the slope of the demand curve: MR=50.10Q. Setting marginal revenue equal to marginal oost(the marginal cost of Firm 1 since it is lower than that of Firm 2)to determine the profit-maximizing quantity. 50.10Q=10,orQ=4 Substituting=4into the demand function to determine price P=50-5*4=$30 The question now is bow the firms will divide the total output of 4 among themselves Since the two firms have different oost functions,it will not be optimal for them to split the output evenly between them.The profit maximizing solution is for firm 1 to produce all of the output so that the profit for Firm 1 will be: x1=(30)(④·.(20+(104)=$60. The profit for Firm 2 will be: 2=(300)-(10+(12)(0》=-$10. Total industry profit will be: r=元1+元2=60.10=S50. If they split the them then total profit would be6(20 for firm 1 and $26 for firm 2).If firm 2 prefe rred to earn a profit of $26 as oppo to $25 then firm 1 could give $1 to firm 2 and it would still have profit of $24,which is higher than the $20 it woul earn if they split output.Note that if firm 2
Monopolistic competition is defined by product differentiation. Each firm earns economic profit by distinguishing its brand from all other brands. This distinction can arise from underlying differences in the product or from differences in advertising. If these competitors merge into a single firm, the resulting monopolist would not produce as many brands, since too much brand competition is internecine (mutually destructive). However, it is unlikely that only one brand would be produced after the merger. Producing several brands with different prices and characteristics is one method of splitting the market into sets of customers with different price elasticities, which may also stimulate overall demand. 2. Consider two firms facing the demand curve P = 50 - 5Q, where Q = Q1 + Q2 . The firms’ cost functions are C1 (Q1 ) = 20 + 10Q1 and C2 (Q2 ) = 10 + 12Q2 . a. Suppose both firms have entered the industry. What is the joint profit-maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? If both firms enter the market, and they collude, they will face a marginal revenue curve with twice the slope of the demand curve: MR = 50 - 10Q. Setting marginal revenue equal to marginal cost (the marginal cost of Firm 1, since it is lower than that of Firm 2) to determine the profit-maximizing quantity, Q: 50 - 10Q = 10, or Q = 4. Substituting Q = 4 into the demand function to determine price: P = 50 – 5*4 = $30. The question now is how the firms will divide the total output of 4 among themselves. Since the two firms have different cost functions, it will not be optimal for them to split the output evenly between them. The profit maximizing solution is for firm 1 to produce all of the output so that the profit for Firm 1 will be: 1 = (30)(4) - (20 + (10)(4)) = $60. The profit for Firm 2 will be: 2 = (30)(0) - (10 + (12)(0)) = -$10. Total industry profit will be: T = 1 + 2 = 60 - 10 = $50. If they split the output evenly between them then total profit would be $46 ($20 for firm 1 and $26 for firm 2). If firm 2 preferred to earn a profit of $26 as opposed to $25 then firm 1 could give $1 to firm 2 and it would still have profit of $24, which is higher than the $20 it would earn if they split output. Note that if firm 2
supplied all the output then it would set marginal revenue equal to its marginal cost or 12 and earn a profit of 62.2.In this case,firm 1 would earn a profit of-20,so that total industry profit would be 42.2. If Firm 1 were the only entrant.its profits would be60 and Firm2s would be 0. If Firm 2 were the only entrant,then it woul equate marginal revenue with its marginal cost to determine its profit-maximizing quantity: 50-10Q2=12.0rQ2=3.8 Substituting into the demand equation to determine price: P=50-5*3.8=$31. The profits for Firm 2 will be: 2=3103.8)·(10+(123.8》=s62.20. b.What is each firm's equilibrium output and profit if they behave noncooperatively?Use the Cournot model Draw the firms'reaction curves and show the equilibrium In the Cournot model Firm 1 takes Firm 2's output as given and maximizes profits.The profit function derived in 2.a becomes 元1=(60.5Q-5Q2)1(20+101o π=40g-5g-5Q22-20. Setting the derivative of the profit function with respect toto zero.we find Firm I's reaction function: ag =40-100-50,=0m2=4-(9》 Similarly,Firm 2s reaction functionis g=38-(》 To find the Cournot equilibrium.we substitute Firm 2's reaction function into Firm I's reaction function: g=4-⑤3.8-)rg=28 1 Substituting this value for into the reaction function for Firm 2,we fin Q2=2.4 Substituting the values for and into the demand function to determine the equilibrium price: P=50-52.8+2.4)=$24
supplied all the output then it would set marginal revenue equal to its marginal cost or 12 and earn a profit of 62.2. In this case, firm 1 would earn a profit of –20, so that total industry profit would be 42.2. If Firm 1 were the only entrant, its profits would be $60 and Firm 2’s would be 0. If Firm 2 were the only entrant, then it would equate marginal revenue with its marginal cost to determine its profit-maximizing quantity: 50 - 10Q2 = 12, or Q2 = 3.8. Substituting Q2 into the demand equation to determine price: P = 50 – 5*3.8 = $31. The profits for Firm 2 will be: 2 = (31)(3.8) - (10 + (12)(3.8)) = $62.20. b. What is each firm’s equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms’ reaction curves and show the equilibrium. In the Cournot model, Firm 1 takes Firm 2’s output as given and maximizes profits. The profit function derived in 2.a becomes 1 = (50 - 5Q1 - 5Q2 )Q1 - (20 + 10Q1 ), or = 40Q1 − 5Q1 2 − 5Q1Q2 − 20. Setting the derivative of the profit function with respect to Q1 to zero, we find Firm 1’s reaction function: Q1 = 40 −10Q1 - 5Q2 =0, or Q1 = 4 - Q2 2 . Similarly, Firm 2’s reaction function is Q2 = 3.8 − Q1 2 . To find the Cournot equilibrium, we substitute Firm 2’s reaction function into Firm 1’s reaction function: Q1 = 4 − 1 2 3.8 − Q1 2 , or Q1 = 2.8. Substituting this value for Q1 into the reaction function for Firm 2, we find Q2 = 2.4. Substituting the values for Q1 and Q2 into the demand function to determine the equilibrium price: P = 50 – 5(2.8+2.4) = $24
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