Managerial Analysis FINA 6387
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Chapter 10 Market Power Monopoly and Monopsony T III FIRM STRUCTURE CHAPTER 10 MARKET POWER MONOPOLY AND MONOPSONY I TEACHING NOTES In most textbooks the title of this chapter would be Monopoly and monopsony would be found in a section of the chapter on factor markets This text however gives monopoly and monopsony parallel treatment There is an initial discussion of monopoly Sections 14 a briefer discussion of monopsony Section 5 and a joint consideration of the two Section 6 Exercises 1 through 5 focus on the monopolist s determination of a profitmaximizing output Exercises 6 and 7 explore the multiplant firm Exercise 8 examines the decision in the US antitrust case against Alcoa Exercises 10 and 12 examine monopsony power Exercises 9 13 14 and 15 focus on price regulation Although previous chapters have presented the rule for profit maximization you should brie y review marginal revenue and price elasticity of demand through a careful derivation of Equation 101 A discussion of the derivation of Equation 101 will elucidate the geometry of Figure 103 illustrate that because the monopolist chooses a quantity such that marginal revenue is positive demand at that quantity is elastic Equation 101 also leads directly to the Lerner Index in Section 102 This provides fruitful ground for a discussion of a monopolist s market power For example if Ed is large eg because of close substitutes then 1 the demand curve is at 2 the marginal revenue curve is at although steeper than the demand curve and 3 the monopolist has little power to raise price above marginal cost To reinforce these points introduce a nonlinear demand curve by for example showing the location of the marginal revenue curve for a unitelastic demand curve Once this concept has been clearly presented the discussion of the effect of an excise tax on a monopolist with nonlinear demand Figure 105 will not seem out of place The social response to market power provides a good topic for class discussion and this topic can be introduced by comparing the deadweight loss with the analysis of market intervention given in Chapter 9 For example compare Figure 109 with Figure 96 Because Exercises 9 13 and 15 involve kinked marginal revenue curves you should present Figure 1010 if you plan to assign those problems Although Figure 1010 is complicated exposure to it here will help when it reappears in Chapter 12 I REVIEW QUESTIONS 1 A monopolist is producing at a point where its marginal cost exceeds its marginal revenue How should it adjust its output level to increase its pro t When marginal cost is greater than marginal revenue the incremental cost of the last unit produced is greater than incremental revenue The firm would increase its profit by not producing the last unit It should continue to reduce production thereby decreasing marginal cost and increasing marginal revenue until marginal cost is equal to marginal revenue 2 We write the percentage markup of prices over marginal cost as P MCP For a pro t maximizing monopolist how does this markup depend on the elasticity of demand Why can this markup be viewed as a measure of monopoly power We can show that this measure of market power is equal to the negative inverse of the price elasticity of demand P 7 MC 7 1 P ED 112 Chapter 10 Market Power Monopoly and Monopsony The equation implies that as the elasticity increases demand becomes more elastic the inverse of elasticity decreases and the measure of market power decreases Therefore as elasticity increases decreases the firm has less more power to increase price above marginal cost 3 Why is there no market supply curve under monopoly The monopolist s output decision depends not only on marginal cost but also on the demand curve Shifts in demand do not trace out a series of prices and quantities that we can identify as the supply curve for the firm lnstead shifts in demand lead to changes in price output or both Thus there is no onetoone correspondence between the price and the seller s quantity therefore a monopolized market lacks a supply curve 4 Why might a rm have monopoly power even if it is not the only producer in the market The degree of monopoly power or market power enjoyed by a firm depends on the elasticity of the demand curve that it faces As the elasticity of demand increases ie as the demand curve becomes atter the inverse of the elasticity approaches zero and the monopoly power of the firm decreases Thus if the firm s demand curve has any elasticity less than infinity the firm has some monopoly power 5 What are some of the sources of monopoly power Give an example of each The firm s exploitation of its monopoly power depends on how easy it is for other firms to enter the industry There are several barriers to entry including exclusive rights eg patents copyrights and licenses and economies of scale These two barriers to entry are the most common Exclusive rights are legally granted property rights to produce or distribute a good or service Positive economies of scale lead to natural monopolies because the largest producer can charge a lower price driving competition from the market For example in the production of aluminum there is evidence to suggest that there are scale economies in the conversion of bauxite to alumina See US 1 Aluminum Company ofAmerica 148 F2d 416 1945 discussed in Exercise 7 below 6 What factors determine how much monopoly power an individual firm is likely to have Explain each one brie y Three factors determine the firm s elasticity of demand 1 the elasticity of market demand 2 the number of firms in the market and 3 interaction among the firms in the market The elasticity of market demand depends on the uniqueness of the product ie how easy it is for consumers to substitute away from the product As the number of firms in the market increases the demand elasticity facing each firm increases because customers may shift to the firm s competitors The number of firms in the market is determined by how easy it is to enter the industry the height of barriers to entry Finally the ability to raise the price above marginal cost depends on how other firms react to the firm s price changes If other firms match price changes customers will have little incentive to switch to another supplier 7 Why is there a social cost to monopoly power If the gains to producers from monopoly power could be redistributed to consumers would the social cost of monopoly power be eliminated Explain brie y When the firm exploits its monopoly power to raise the price above marginal cost consumers buy less at the higher price Consumers enjoy less surplus the difference between the price they are willing to pay and the market price on each unit consumed Some of the lost consumer surplus is not captured by the seller and is a deadweight loss to society Therefore if the gains to producers were redistributed to consumers society would still suffer the deadweight loss 113 Chapter 10 Market Power Monopoly and Monopsony 8 Why will a monopolist s output increase if the govermnent forces it to lower its price If the government wants to set a price ceiling that maximizes the monopolist s output what price should it set By restricting price below the monopolist s profitmaximizing price the government can change the shape of the firm s marginal revenue ZMR curve When a price ceiling is imposed MB is equal to the price ceiling for all quantities lower than the quantity demanded at the price ceiling If the government wants to maximize output it should set a price equal to marginal cost Prices below this level induce the firm to decrease production assuming the marginal cost curve is upward sloping The regulator s problem is to determine the shape of the monopolist s marginal cost curve This task is difficult given the monopolist s incentive to hide or distort this information 9 How should a monopsonist decide how much of a product to buy Will it buy more or less than a competitive buyer Explain brie y The marginal expenditure is the change in the total expenditure as the purchased quantity changes For a firm competing with many firms for inputs the marginal expenditure is equal to the average expenditure price For a monopsonist the marginal expenditure curve lies above the average expenditure curve because the decision to buy an extra unit raises the price that must be paid for all units including the last unit All firms should buy inputs so that the marginal value of the last unit is equal to the marginal expenditure on that unit This is true for both the competitive buyer and the monopsonist However because the monopsonist s marginal expenditure curve lies above the average expenditure curve and because the marginal value curve is downward sloping the monopsonist buys less than a firm would buy in a competitive market 10 What is meant by the term monopsony power Why might a firm have monopsony power even if it is not the only buyer in the market Monopsony power is the power in the factor market held by the buyer A buyer facing an upwardsloping factor supply curve has some monopsony power In a competitive market the seller faces a perfectlyelastic market curve and the buyer faces a perfectly elastic supply curve Thus any characteristic of the market eg when there is a small number of buyers or if buyers engage in collusive behavior that leads to a lessthan per39fectlyelastic supply curve gives the buyer some monopsony power 11 What are some sources of monopsony power What determines how much monopsony power an individual rm is likely to have The individual firm s monopsony power depends on the characteristics of the buying side of the market There are three characteristics that enhance monopsony power 1 the elasticity of market supply 2 the number of buyers and 3 how the buyers interact The elasticity of market supply depends on how responsive producers are to changes in price If in the short run supply is relatively fixed then supply is relatively inelastic For example since tobacco farmers can sell their crop to only a handful of tobacco product producers the power to buy at a price below marginal value is increased 12 Why is there a social cost to monopsony power If the gains to buyers from monopsony power could be redistributed to sellers would the social cost of monopsony power be eliminated Explain brie y With monopsony power the price is lower and the quantity is less than under competitive buying conditions Because of the lower price and reduced sales sellers lose revenue Only part of this lost revenue is transferred to the buyer as consumer surplus and the net loss in total surplus is deadweight loss Even if the consumer surplus could be redistributed to sellers the deadweight loss persists This inefficiency will remain because quantity is reduced below a level where price is equal to margin cost 114 Chapter 10 Market Power Monopoly and Monopsony 13 How do the antitrust laws limit market power in the United States Give examples of the major provisions of the laws Antitrust laws which are subject to interpretation by the courts limit market power by proscribing a firm s behavior in attempting to maximize profit Section 1 of the Sherman Act prohibits every restraint of trade including any attempt to fix prices by buyers or sellers Section 2 of the Sherman Act prohibits behavior that leads to monopolization The Clayton Act with the RobinsonPatman Act prohibits price discrimination and exclusive dealing sellers prohibiting buyers from buying goods from other sellers The Clayton Act also limits mergers when they could substantially lessen competition The Federal Trade Commission Act makes it illegal to use unfair or deceptive practices 14 Explain brie y how the US antitrust laws are actually enforced Antitrust laws are enforced in three ways 1 through the Antitrust Division of the Justice Department whenever firms violate federal statutes 2 through the Federal Trade Commission whenever firms violate the Federal Trade Commission Act and 3 through civil suits The Justice Department can seek to impose fines or jail terms on managers or owners involved or seek to reorganize the firm as it did in its case against ATamp T The FTC can seek a voluntary understanding to comply with the law or a formal C 39 39 order T quot quot or J 39 can sue in federal court for awards equal to three times the damage arising from the anticompetitive behavior EXERCISES 1 Will an increase in the demand for a monopolist s product always result in a higher price Explain Will an increase in the supply facing a monopsonist buyer always result in a lower price Explain As illustrated in Figure 104b in the textbook an increase in demand need not always result in a higher price Under the conditions portrayed in Figure 104b the monopolist supplies different quantities at the same price Similarly an increase in supply facing the monopsonist need not always result in a higher price Suppose the average expenditure curve shifts from AE1 to AEZ as illustrated in Figure 101 With the shift in the average expenditure curve the marginal expenditure curve shifts from ME1 to MEZ The ME1 curve intersects the marginal value curve demand curve at Q1 resulting in a price of P When the AE curve shifts the ME2 curve intersects the marginal value curve at Q2 resulting in the same price at P 115 Chapter 10 Market Power Monopoly and Monopsony Price ME 1 AE1 ME2 I I P I I I 39 39 MV I I Q1 Q2 Quantity Figure 101 2 Caterpillar Tractor one of the largest producers of farm tractors in the world has hired you to advise them on their pricing policy One of the things the company would like to know is how much a 5 percent increase in price is likely to reduce sales What would you need to know to help the company with their problem Explain why these facts are important As a large producer of farm equipment Caterpillar Tractor has market power and should consider the entire demand curve when choosing prices for its products As their advisor you should focus on the determination of the elasticity of demand for each product There are three important factors to be considered First how similar are the products offered by Caterpillar s competitors If they are close substitutes a small increase in price could induce customers to switch to the competition Secondly what is the age of the existing stock of tractors With an older population of tractors a 5 percent price increase induces a smaller drop in demand Finally because farm tractors are a capital input in agricultural production what is the expected profitability of the agricultural sector lf farm incomes are expected to fall an increase in tractor prices induces a greater decline in demand than one would estimate with information on only past sales and prices 3 A monopolist rm faces a demand with constant elasticity of 20 The firm has a marginal cost of 20 per unit and sets a price to maximize profit If marginal cost should increase by 25 percent would the price charged also rise by 25 percent Yes The monopolist s pricing rule as a function of the elasticity of demand for its product is P MC 7 1 P E d or alternatively 116 Chapter 10 Market Power Monopoly and Monopsony In this example Ed 2 20 so 1Ed 12 price should then be set so that MC P MC 2 Therefore if MC rises by 25 percent price then price will also rise by 25 percent When MC 20 P 40 When MC rises to 20125 25 the price rises to 50 a 25 increase 4 A rm faces the following average revenue demand curve P 100 001Q where Q is weekly production and P is price measured in cents per unit The firm s cost function is given by C 50Q 30000 Assulning the firm maximizes profits a What is the level of production price and total profit per week The profitmaximizing output is found by setting marginal revenue equal to marginal cost Given a linear demand curve in inverse form P 100 001Q we know that the marginal revenue curve will have twice the slope of the demand curve Thus the marginal revenue curve for the firm is MR 100 002Q Marginal cost is simply the slope of the total cost curve The slope of TC 30000 50Q is 50 So MC equals 50 Setting MR MC to determine the pro tmaximizing quantity 100 002Q 50 or Q 2500 Substituting the profitmaximizing quantity into the inverse demand function to determine the price P 100 0012500 75 cents Profit equals total revenue minus total cost 1 752 500 30000 502500 or 1 325 per week b The government decides to levy a tax of 10 cents per unit on this product What will the new level of production price and profit be as a result Suppose initially that the consumers must pay the tax to the government Since the total price including the tax consumers would be willing to pay remains unchanged we know that the demand function is Pquot T 100 001Q or Pquot100001Q T where 13 is the price received by the suppliers Because the tax increases the price of each unit total revenue for the monopolist decreases by TQ and marginal revenue the revenue on each additional unit decreases by T MR 100 002Q T where T 10 cents To determine the profitmaximizing level of output with the tax equate marginal revenue with marginal cost 100 002Q 10 50 or Q 2000 units Substituting Q into the demand function to determine price Pquot 100 0012000 10 70 cents 117 Chapter 10 Market Power Monopoly and Monopsony Profit is total revenue minus total cost 7 agooog 00009 30000i 100000 cents or 100 per week Note The price facing the consumer after the imposition of the tax is 80 cents The monopolist receives 70 cents Therefore the consumer and the monopolist each pay 5 cents of the tax If the monopolist had to pay the tax instead of the consumer we would arrive at the same result The monopolist s cost function would then be TC 5OQ 30000 TQ 50 TQ 30000 The slope of the cost function is 50 T so MC 50 T We set this MC to the marginal revenue function from part a 100 00262 50 10 or Q 2000 Thus it does not matter who sends the tax payment to the government The burden of the tax is re ected in the price of the good 5 The table below shows the demand curve facing a monopolist who produces at a constant marginal cost of 10 Price Quantity 27 0 24 2 21 4 18 6 15 8 1 2 10 9 1 2 6 14 3 16 0 18 a Calculate the rm s marginal revenue curve To find the marginal revenue curve we first derive the inverse demand curve The intercept of the inverse demand curve on the price axis is 27 The slope of the inverse demand curve is the change in price divided by the change in quantity For example a decrease in price from 27 to 24 yields an increase in quantity from 0 to 2 Therefore the slope is a and the demand curve is P 27 7 15Q The marginal revenue curve corresponding to a linear demand curve is a line with the same intercept as the inverse demand curve and a slope that is twice as steep Therefore the marginal revenue curve is MR27 3Q 118 Chapter 10 Market Power Monopoly and Monopsony What are the fir39m s pro t maximizing output and price What is the fir39m s profit The monopolist s maximizing output occurs where marginal revenue equals marginal cost Marginal cost is a constant 10 Setting MR equal to MC to determine the profit maXimiZing quantity 27 3Q 10 or Q567 To find the profitmaximizing price substitute this quantity into the demand equation P 27 7 B5 67g 185 Total revenue is price times quantity TR Da567g 10483 The profit of the firm is total revenue minus total cost and total cost is equal to average cost times the level of output produced Since marginal cost is constant average variable cost is equal to marginal cost Ignoring any fixed costs total cost is 10Q or 5667 and profit is 10483 7 5667 4817 What would the equilibrium price and quantity be in a competitive industry For a competitive industry price would equal marginal cost at equilibrium Setting the expression for price equal to a marginal cost of 10 27 7 15Q 10 or Q 1133 Note the increase in the equilibrium quantity compared to the monopoly solution What should the social gain be if this monopolist were forced to produce and price at the competitive equilibrium Who would gain and lose as a result The social gain arises from the elimination of deadweight loss Since deadweight loss under monopoly is equal to the difference between the price under monopoly minus the price under competition 185 10 85 times the difference between the quantity under competition minus the quantity under monopoly 113 567 567 times one half the deadweight loss is a triangle under the demand curve 0585567 2410 Furthermore consumers gain this deadweight loss plus the monopolist s profit of 48 The monopolist s profits are reduced to zero and the consumer surplus increases by 72 6 A rm has two factories for which costs are given by Factory1 ClalT 10Q Factor39y2 CZaZT 20Q The firm faces the following demand curve P700 5Q where Q is total output ie Q Q1 Q2 a On a diagram draw the marginal cost curves for the two factories the average and marginal revenue curves and the total marginal cost curve ie the marginal cost of producing Q Q1 Q2 Indicate the profit maximizing output for each factory total output and price The average revenue curve is the demand curve P 700 5Q 119 Chapter 10 Market Power Monopoly and Monopsony For a linear demand curve the marginal revenue curve has the same intercept as the demand curve and a slope that is twice as steep MR 700 10Q Next determine the marginal cost of producing Q To find the marginal cost of production in Factory 1 take the first derivative of the cost function With respect to Q dC1 la 9 W 7 20Q Similarly the marginal cost in Factory 2 is M 40 dQ 239 Rearranging the marginal cost equations in inverse form and horizontally summing them we obtain total marginal cost MCT MC MCZ SMCT 7 7 20 40 40 MCT g QQ1Q2 or Profit maximization occurs Where MCT MR See the Figure 106a for the profit maximizing output for each factory total output and price Price 800 700 MC2 MC1 MCT 600 PM 500 400 300 200 MR D 100 T 7390 110 Quantity 9 0 Figure 106a Calculate the values of Q1 Q2 Q and P that maximize pro t Calculate the total output that maximizes pro t ie Q such that MCT MR 700710Q or QZBO 120 Chapter 10 Market Power Monopoly and Monopsony Next observe the relationship between MC and MB for multiplant monopolies MR MCT MC1MC2 We know that at Q 30 MR 700 1030 400 Therefore MC1 400 20Q1 or Q1 20 and MC2 400 40622 or Q2 10 To find the monopoly price PM substitute for Q in the demand equation PM 700 530 or PM 550 c Suppose labor costs increase in Factory 1 but not in Factory 2 How should the rm adjust ie raise lower or leave unchanged Output in Factory 1 Output in Factory 2 Total output Price An increase in labor costs will lead to a horizontal shift to the left in MCl causing MCT to shift to the left as well since it is the horizontal sum of MC1 and M02 The new MCT curve intersects the MR curve at a lower quantity and higher marginal revenue At a higher level of marginal revenue Q2 is greater than at the original level for MR Since QT falls and Q2 rises Q1 must fall Since QT falls price must rise 7 A drug company has a monopoly on a new patented medicine The product can be made in either of two plants The costs of production for the two plants are MC1 20 2Q1 and MC2 10 5Q2 The rm s estimate of the demand for the product is P 20 3Q1 Q2 How much should the rm plan to produce in each plant and at what price should it plan to sell the product First notice that only MC2 is relevant because the marginal cost curve of the first plant lies above the demand curve Price MC2 10 5Q2 MC1 20 2Q1 Figure 10 7 This means that the demand curve becomes P 20 SQZ With an inverse linear demand curve we know that the marginal revenue curve has the same vertical 121 8 Chapter 10 Market Power Monopoly and Monopsony intercept but twice the slope or MR 20 6Q2 To determine the profitmaximizing level of output equate MR and MCZ 20 6Q2 10 5Q2 or Q Q2 091 Price is determined by substituting the profitmaximizing quantity into the demand equation P 2073691g173 One of the more important antitrust cases of this century involved the Aluminum Company of America Alcoa in 1945 At that time Alcoa controlled about 90 percent of primary aluminum production in the United States and the company had been accused of monopolizing the aluminum market In its defense Alcoa argued that although it indeed controlled a large fraction of the primary market secondary aluminum ie aluminum produced from the recycling of scrap accounted for roughly 30 percent of the total supply of aluminum and many competitive rms were engaged in recycling Therefore Alcoa argued it did not have much monopoly power a Provide a clear argument in favor of Alcoa s position Although Alcoa controlled about 90 percent of primary aluminum production in the United States secondary aluminum production by recyclers accounted for 30 percent of the total aluminum supply Therefore with a higher price a much larger proportion of aluminum supply could come from secondary sources This assertion is true because there is a large stock of potential supply in the economy Therefore the price elasticity of demand for Alcoa s primary aluminum is much higher than we would expect given Alcoa s dominant position in prim aluminum production In many applications other metals such as copper and steel are feasible substitutes for aluminum Again the demand elasticity Alcoa faces might be lower than we would otherwise expect Provide a clear argument against Alcoa s position While Alcoa could not raise its price by very much at any one time the stock of potential aluminum supply is limited Therefore by keeping a stable high price Alcoa could reap monopoly profits Also since Alcoa had originally produced the metal reappearing as recycled scrap it would have considered the effect of scrap reclamation on future prices Therefore it exerted effective monopolistic control over the secondary metal supply The 1945 decision by Judge Learned Hand has been called one of the most celebrated judicial opinions of our time Do you know what Judge Hand s ruling was Judge Hand ruled against Alcoa but did not order it to divest itself of any of its United States production facilities The two remedies imposed by the court were 1 that Alcoa was barred from bidding for two primary aluminum plants constructed by the government during World War II they were sold to Reynolds and Kaiser and 2 that it divest itself of its Canadian subsidiary which became Alcan 9 A monopolist faces the demand curve P 11 Q where P is measured in dollars per unit and Q in thousands of units The monopolist has a constant average cost of 6 per unit a Draw the average and marginal revenue curves and the average and marginal cost curves What are the monopolist s pro t maximizing price and quantity and what is the resulting pro t Calculate the rm s degree of monopoly power using the Lerner index Because demand average revenue may be described as P 11 Q we know that the marginal revenue function is MR 11 2Q We also know that if average cost is constant then marginal cost is constant and equal to average cost MC 6 122 Chapter 10 Market Power Monopoly and Monopsony To find the profitmaximizing level of output set marginal revenue equal to marginal cost 11 2Q6 orQ25 That is the profitmaximizing quantity equals 2500 units Substitute the profit maXimiZing quantity into the demand equation to determine the price P 11 25 850 Profits are equal to total revenue minus total cost 7t TR TC ARQ ACXQ 01 1 8525 625 625 or 6250 The degree of monopoly power is given by the Lerner Index P MC M 0294 P ACMC DAR I I 8 10 12 Q Figure 109a A govermnent regulatory agency sets a price ceiling of 7 per unit What quantity will be produced and what will the rm s pro t be What happens to the degree of monopoly power To determine the effect of the price ceiling on the quantity produced substitute the ceiling price into the demand equation 7 11 Q or Q4000 This quantity is the profitmaximizing level of output for the monopolist because at that level MR MC Profits are equal to total revenue minus total cost 1 74 000 64 000 4000 The degree of monopoly power is w 0143 123 Chapter 10 Market Power Monopoly and Monopsony What price ceiling yields the largest level of output What is that level of output What is the firm s degree of monopoly power at this price If the regulatory authority sets a price below 6 the monopolist would prefer to go out of business instead of produce because it cannot cover its average costs At any price above 6 the monopolist would produce less than the 5000 units that would be produced in a competitive industry Therefore the regulatory agency should set a price ceiling of 6 thus making the monopolist face a horizontal effective demand curve up to Q 5000 To ensure a positive output so that the monopolist is not indifferent between producing 5000 units and shutting down the price ceiling should be set at 6 5 where 5 is small Thus 5000 is the maximum output that the regulatory agency can extract from the monopolist by using a price ceiling The degree of monopoly power is ip MCi65 6 0 as5gt0 P 6 6 10 Michelle s Monopoly Mutant Turtles MMMT has the exclusive right to sell Mutant Turtle t shirts in the United States The demand for these t shirts is Q 10000P2 The rm s short run cost is SRTC 2000 5Q and its long run cost is LRTC 6Q a What price should MMMT charge to maximize pro t in the short run What quantity does it sell and how much profit does it make Would it be better off shutting down in the short run MMMT should offer enough tshirts such that MR MC In the short run marginal cost is the change in SRTC as the result of the production of another tshirt ie SRMC 5 the slope of the SRTC curve Demand is 10000 Q 727 P or in inverse form P 1000 Total revenue PQ is 100Q12 Taking the derivative of TR with respect to Q MR 50Q 12 Equating MR and MC to determine the profitmaximizing quantity 5 5002 or Q 100 Substituting Q 100 into the demand function to determine price 12 P 10010039 10 The profit at this price and quantity is equal to total revenue minus total cost 1 10100 2000 5100 1500 Although profit is negative price is above the average variable cost of 5 and therefore the firm should not shut down in the short run Since most of the firm s costs are fixed the firm loses 2000 if nothing is produced If the profitmaximizing quantity is produced the firm loses only 1500 What price should MMMT charge in the long run What quantity does it sell and how much pro t does it make Would it be better off shutting down in the long run In the long run marginal cost is equal to the slope of the LRTC curve which is 6 Equating marginal revenue and long run marginal cost to determine the profit maximizing quantity 12 50Q 6 or Q6944 124 Chapter 10 Market Power Monopoly and Monopsony Substituting Q 6944 into the demand equation to determine price P 10050623912 100650 12 Therefore total revenue is 83333 and total cost is 41667 Profit is 41667 The firm should remain in business c Can we expect MMMT to have lower marginal cost in the short run than in the long run Explain why In the long run MMMT must replace all fixed factors Therefore we can expect LRMC to be higher than SRMC 11 You produce widgets which you sell in a perfectly competitive market at a market price of 10 per widget The widgets are manufactured in two plants one in Massachusetts and the other in Connecticut Because of labor problems in Connecticut you are forced to raise wages in that plant so marginal costs in that plant increase In response to this should you shift production and produce more in the Massachusetts plant No production should not shift to the Massachusetts plant although production in the Connecticut plant should be reduced In order to maximize profits a mulitplant firm will schedule production at all plants so that the following two conditions are met Marginal costs of production at each plant are equal Marginal revenue of the total amount produced is equal to the marginal cost at each plant These two rules can be summarized as MR MC1 MC2 MCS where the subscript indicates the plant The firm in this example has two plants and is in a perfectly competitive market In a perfectly competitive market P MR To maximize profits production among the plants should be allocated such that P MCJQC MCmQm where the subscripts denote plant locations c for Connecticut etc The marginal costs of production have increased in Connecticut but have not changed in Massachusetts Since costs have not changed in Massachusetts the level of Qm that sets MCmQm P has not changed Figure 1011 12 The employment of teaching assistants TAs by major universities can be characterized as a monopsony Suppose the demand for TAs is W 30000 125n where W is the wage as an annual salary and n is the number of TAs hired The supply of TAs is given by W 1000 75n 125 Chapter 10 Market Power Monopoly and Monopsony If the university takes advantage of its monopsonist position how many TAs will it hire What wage will it pay The supply curve is equivalent to the average expenditure curve With a supply curve of W 1000 75n the total expenditure is Wn 1000n 75n2 Taking the derivative of the total expenditure function with respect to the number of TAs the marginal expenditure curve is 1000 150n As a monopsonist the university would equate marginal value demand with marginal expenditure to determine the number of TAs to hire 30000 125n 1000 150n or n 1055 Substituting n 1055 into the supply curve to determine the wage 1000 751055 8909 annually If instead the university faced an infinite supply of TAs at the annual wage level of 10000 how many TAs would it hire With an infinite number of TAs at 10000 the supply curve is horizontal at 10000 Total expenditure is 10000n and marginal expenditure is 10000 Equating marginal value and marginal expenditure 30000 125n 10000 or n 160 13 Dayna s Doorstops Inc DD is a monopolist in the doorstop industry Its cost is C 100 5Q Q2 and demand is P 55 2Q a What price should DD set to maximize profit and what output does the rm produce How much pro t and consumer surplus does DD generate To maximize profits DD should equate marginal revenue and marginal cost Given a demand of P 55 2Q we know that total revenue PQ is 55Q 2Q2 Marginal revenue is found by taking the first derivative of total revenue with respect to Q or MR d 55 7 4Q dQ Similarly marginal cost is determined by taking the first derivative of the total cost function with respect to Q or 7 dTC 7 dQ Equating MC and MB to determine the profitmaximizing quantity 554Q2Q5 or Q 10 MC 2Q75 Substituting Q 10 into the demand equation to determine the profitmaximizing price P 55 210 35 Profits are equal to total revenue minus total cost 1 3510 100 510 102 200 126 Chapter 10 Market Power Monopoly and Monopsony Consumer surplus is equal to onehalf times the profitmaximizing quantity 10 times the difference between the demand intercept the maximum price anyone is Willing to pay and the monopoly price CS 2 051055 35 100 What would output be if DD acted like a perfect competitor and set MC P What profit and consumer surplus would then be generated ln competition profits are maximized at the point Where price equals marginal cost Where price is given by the demand curve 55 2Q52Q or Q 15 Substituting Q 15 into the demand equation to determine the price P 55 215 25 Profits are total revenue minus total cost or 1 2515 100 515 152 125 Consumer surplus is CS 0555 2515 225 What is the deadweight loss from monopoly power in part a Price MC2Q5 I 5 10 15 20 25 30 Quanmy Figure 1013c The deadweight loss is equal to the area of the triangle ABC DWL 0535 1515 10 50 Suppose the government concerned about the high price of doorstops sets a maximum price for doorstops at 27 How does this affect price quantity consmner surplus and DD s pro t What is the resulting deadweight loss 127 Chapter 10 Market Power Monopoly and Monopsony With the imposition of a price ceiling the maximum price that DD may charge is 2700 Note that When a ceiling price is set above the competitive price the ceiling price is equal to marginal revenue for all levels of output sold up to the competitive level of output Substitute the ceiling price of 2700 into the demand equation to determine the effect on the equilibrium quantity sold 27 55 2Q or Q 14 Consumer surplus is CS 0555 2714 196 Profits are 1 2714 100 514 142 152 The deadweight loss is 200 This is equivalent to a triangle of 0515 1427 23 2 Now suppose the government sets the maximum price at 23 How does this affect price quantity consumer surplus DD s profit and deadweight loss With a ceiling price set below the competitive price DD Will decrease its output Equate marginal revenue and marginal cost to determine the profitmaximizing level of output 23 52Q orQ14 With the governmentimposed maximum price of 23 profits are 1 2314 100 514 142 96 Consumer surplus is realized on only 14 doorsteps Therefore it is equal to the consumer surplus in part d ie 196 plus the savings on each doorstep ie CS 27 2314 56 Therefore consumer surplus is 252 Deadweight loss is the same as before 200 Finally consider a maximum price of 12 What will this do to quantity consumer surplus profit and deadweight loss With a maximum price of only 12 output decreases even further 12 5 2Q or Q 85 Profits are 1 1285 100 585 852 2775 Consumer surplus is realized on only 85 units Which is equivalent to the consumer surplus associated With a price of 38 38 55 285 ie 0555 3885 7225 plus the savings on each doorstep ie 38 1285 221 Therefore consumer surplus is 29325 Total surplus is 26550 and deadweight loss is 8450 14 There are 10 households in Lake Wobegon Minnesota each with a demand for electricity of Q 50 P Lake Wobegon Electric s LWE cost of producing electricity is TC 500 Q a If the regulators of LWE want to make sure that there is no deadweight loss in this market what price will they force LWE to charge What will output be in that case Calculate consumer surplus and LWE s profit with that price The rst step in solving the regulators problem is to determine the market demand for electricity in Lake Wobegon We horizontally sum each household s demand for 128 Chapter 10 Market Power Monopoly and Monopsony electricity to arrive at market demand For example if the regulator announced a price of 50 no one would purchase electricity If the regulator announced a price of 25 each household would purchase 25 units with a market demand of 250 Therefore the market demand curve has an intercept at 50 the same intercept as the individual demand curves and a slope of 7 or P 50 01Q To avoid deadweight loss the regulators will set price at marginal cost Given TC 500 Q we know that MC 1 the slope of the total cost curve Setting price equal to marginal cost and solving for quantity 50 01Q 1 or Q 490 Profits are equal to total revenue minus total costs 1 1490 500 490 500 Total consumer surplus is CS 0550 1490 12005 or 120050 per household If the regulators want to make sure that LWE doesn t lose money what is the lowest price they can impose Calculate output consumer surplus and pro t in that case Is there any deadweight loss To guarantee that LWE does not lose money regulators will allow LWE to charge the average cost of production where TC 500 i 7 Q Q To determine the equilibrium price and quantity under average cost pricing set price equal to average cost AC 50701Q1 Q Solving for Q yields the following quadratic equation 01Q2 49Q 500 0 Note if Q2 bQ 0 0 then Using the quadratic formula Q 7 49 i 492 7 401500 201 there are two solutions 104 and 4796 Only the latter satis es the secondorder condition that marginal cost is rising when it intersects marginal revenue Therefore Q 4796 and P 204 At this quantity and price profit is zero Consumer surplus is 7 CS 2 0550 2044796 11500 129 Chapter 10 Market Power Monopoly and Monopsony Deadweight loss is DWL 204 1490 479605 540 c Kristina knows that deadweight loss is something that this small town can do without She suggests that each household be required to pay a xed amount just to receive any electricity at all and then a per unit charge for electricity Then LWE can break even while charging the price you calculated in part a What fixed amount would each household have to pay for Kristina s plan to work Why are you sure that no household will choose instead to refuse the payment and go without electricity Fixed costs are 500 If each household pays 50 the fixed costs are covered and the utility can charge marginal cost for electricity Because consumer surplus per household under marginal cost pricing is 120050 each would be willing to pay the 50 15 A monopolist faces the following demand curve Q 144P2 where Q is the quantity demanded and P is price Its average variable cost is AVG Q12 y and its fixed cost is 5 a What are its profit maximizing price and quantity What is the resulting profit The monopolist wants to choose the level of output to maximize its profits We must remember that the level of output affects price For simplicity we invert the demand function to find the price P as a function of output Q This allows us to express total revenue as a function of Q 144 Q 7 P or inverse form 144 144 12 P27orP 77 Q Q do To find the profitmaximizing level of output we set marginal revenue equal to marginal cost 6 3 Q 4 7 7 or x 2E From the demand function P 12412 6 Profit total revenue minus total cost is equal to n 64 5 4 2413 11 b Suppose the government regulates the price to be no greater than 4 per unit How much will the monopolist produce and what will its profit be The price ceiling truncates the demand curve that the monopolist faces at P 4 or 144 Q 7 16 4 Because of the regulation the demand curve now has two parts 4 if Q g 9 P 12Q39 ij gt 9 130 9 Therefore if the monopolist produces 9 units or less the price must be Chapter 10 Market Power Monopoly and Monopsony Thus total revenue and marginal revenue also should be considered in two parts 4Q if Q g 9 TR m and 126 if Q gt 9 41ngg 712 MR 6Q To find the profitmaximizing level of output set marginal revenue equal to marginal cost Thus for P 4 ifQgt9 43rorr om2vu If the monopolist produces an integer number of units the profitmaximizing production level is 7 units price is 4 and revenues are 28 7 4 Costs are 2352 5 7 32 Profits are thus 448 There is a shortage of two units since the quantity demanded at the price of 4 is 9 units Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output What price will do this To maximize output the regulated price should be set so that demand equals marginal cost The regulated price becomes the monopolist s marginal revenue curve Which is a horizontal line With an intercept at the regulated price To maximize profits the firm produces Where marginal cost is equal to demand at the regulated price Here P MC implies 12Q 2 32Q12 or Q 8 Substituting the profitmaximizing quantity into the demand equation we find P 1283912 P 424 or 131 Chapter 13 Game Theory and Competitive Equilibrium CHAPTER 13 GAME THEORY AND COMPETITIVE STRATEGY I TEACHING NOTES Chapter 13 continues the discussion of competitive firms in the context of twoplayer games with the first three sections covering all topics introduced in Chapter 12 If you did not present Section 125 you should do so after discussing Sections 131 and 132 Sections 134 through 138 introduce advanced topics The presentation throughout the chapter focuses on the intuition behind each model or strategy The exercises focus on relating Chapter 13 to Chapter 12 and on behavior in repeated games Two concepts pervade this chapter rationality and equilibrium Assuming the players are rational means that each player maximizes his or her own payoff whether it hurts or helps other players Rationality underlies many of the equilibria in the chapter Underlying all these models is the definition of a Nash equilibrium which the students will find esoteric When presenting each model ask whether a unique Nash equilibrium exists If there is more than one discuss the conditions that will favor each equilibrium The analysis in the last five sections of the chapter is more demanding but the examples are more detailed Section 134 examines repeated games and it will be important to discuss the role of rationality in the achievement of an equilibrium in both finite and infinitehorizon games Example 132 points out conditions that lead to stability in repeated games while Example 133 presents an unstable case Sections 135 136 and 137 introduce strategy in the context of sequential games To capture the students attention discuss the phenomenal success of WalMart in its attempt to preempt the entry of other discount stores in rural areas see Example 134 First define a strategic move second discuss the advantage of moving first third present Example 134 and fourth continue with other forms of strategic behavior including the use of new capacity and RampD to deter entry see Examples 135 and 136 You may wish to reintroduce the case of bilateral monopoly during the discussion of strategic behavior in cooperative games which concludes this chapter I REVIEW QUESTIONS 1 What is the difference between a cooperative and a noncooperative game Give an example of each In a noncooperative game the players do not formally communicate in an effort to coordinate their actions They are aware of one another s existence but act independently The primary difference between a cooperative and a noncooperative game is that a binding contract ie an agreement between the parties to which both parties must adhere is possible in the former but not in the latter An example of a cooperative game woul be a formal cartel agreement such as OPEC or a joint venture An example of a noncooperative game would be a race in research and development to obtain a patent 2 What is a dominant strategy Why is an equilibrium stable in dominant strategies A dominant strategy is one that is best no matter what action is taken by the other party to the game When both players have dominant strategies the outcome is stable because neither party has an incentive to change 3 Explain the meaning of a Nash equilibrium How does it differ from an equilibrium in dominant strategies A Nash equilibrium is an outcome where both players correctly believe that they are doing the best they can given the action of the other player A game is in equilibrium if neither player has an incentive to change his or her choice unless there is a change by the other player The key feature that distinguishes a Nash equilibrium from an equilibrium in dominant strategies is the dependence on the opponent s behavior An equilibrium in dominant strategies results if each player has a best choice regardless 186 Chapter 13 Game Theory and Competitive Equilibrium of the other player s choice Every dominant strategy equilibrium is a Nash equilibrium but the reverse does not hold 4 How does a Nash equilibriuln differ from a game s maximin solution In what situations is a maximin solution a more likely outcome than a Nash equilibrium A maximin strategy is one in which each player determines the worst outcome for each of the opponent s actions and chooses the option that maximizes the minimum gain that can be earned Unlike the Nash equilibrium the maximin solution does not require players to react to an opponent s choice If no dominant strategy exists in which case outcomes depend on the opponent s behavior players can reduce the uncertainty inherent in relying on the opponent s rationality by conservatively following a maximin strategy The maximin solution is more likely than the Nash solution in cases where there is a higher probability of irrational nonoptimizing behavior 5 What is a tit for tat strategy Why is it a rational strategy for the infinitely repeated Prisoners Dilemma A player following a titfortat strategy will cooperate as long as his or her opponent is cooperating and will switch to a noncooperative strategy if their opponent switches strategies When the competitors assume that they will be repeating their interaction in every future period the longterm gains from cooperating will outweigh any short term gains from not cooperating Because the titfortat strategy encourages cooperation in infinitely repeated games it is rational 6 Consider a game in which the Prisoners Dilemlna is repeated 10 times and both players are rational and fully informed Is a tit for tat strategy optimal in this case Under what conditions would such a strategy be optimal Since cooperation will unravel from the last period back to the first period the titfor tat strategy is not optimal when there is a finite number of periods and both players anticipate the competitor s response in every period Given that there is no response possible in the eleventh period for action in the tenth and last period cooperation breaks down in the last period Then knowing that there is no cooperation in the last period players should maximize their selfinterest by not cooperating in the secondto last period This unraveling occurs because both players assume that the other player has considered all consequences in all periods However if there is some doubt about whether the opponent has fully anticipated the consequences of the titfortat strategy in the final period the game will not unravel and the titfortat strategy can be optimal 7 Suppose you and your competitor are playing the pricing game shown in Table 138 Both of you must announce your prices at the same time Might you improve your outcome by promising your competitor that you will announce a high price If the game is to be played only a few times there is little to gain If you are Firm 1 and promise to announce a high price Firm 2 will undercut you and you will end up with a payoff of 50 However next period you will undercut too and both firms will earn 10 If the game is played many times there is a better chance that Firm 2 will realize that if it matches your high price the longterm payoff of 50 each period is better than 100 at first and 10 thereafter 8 What is meant by first mover advantage Give an example ofa gaming situation with a rst mover advantage A firstmover advantage can occur in a game where the first player to act receives the highest payoff The firstmover signals his or her choice to the opponent and the opponent must choose a response given this signal The firstmover goes on the offensive and the secondmover responds defensively In many recreational games from chess to football the firstmover has an advantage In many markets the first firm to introduce a product can set the standard for competitors to follow In some 187 Chapter 13 Game Theory and Competitive Equilibrium cases the standardsetting power of the first mover becomes so pervasive in the market that the brand name of the product becomes synonymous with the product eg Kleenex the name of Kleenexbrand facial tissue is used by many consumers to refer to facial tissue of any brand 9 What is a strategic move How can the development ofa certain kind of reputation be a strategic move A strategic move involves a commitment to reduce one s options The strategic move might not seem rational outside the context of the game in which it is played but it is rational given the anticipated response of the other player Random responses to an opponent s action may not appear to be rational but developing a reputation of being unpredictable could lead to higher payoffs in the long run Another example would be making a promise to give a discount to all previous consumers if you give a discount to one Such a move makes the firm vulnerable but the goal of such a strategic move is to signal to rivals that you won t be discounting price and hope that your rivals follow suit 10 Can the threat ofa price war deter entry by potential competitors What actions might a firm take to make this threat credib e Both the incumbent and the potential entrant know that a price war will leave their firms worse off Normally such a threat is not credible Thus the incumbent must make his or her threat of a price war believable by signaling to the potential entrant that a price war will result if entry occurs One strategic move is to increase capacity signaling a lower future price and another is to engage in apparently irrational behavior Both types of strategic behavior might deter entry but for different reasons While an increase in capacity reduces expected profits by reducing prices irrational behavior reduces expected profits by increasing uncertainty hence increasing the rate at which future profits must be discounted into the present 11 A strategic move limits one s exibility and yet gives one an advantage Why How might a strategic move give one an advantage in bargaining A strategic move in uences conditional behavior by the opponent If the game is well understood and the opponent s reaction can be predicted a strategic move leaves the player better off Economic transactions involve a bargain whether implicit or explicit In every bargain we assume that both parties attempt to maximize their selfinterest Strategic moves by one player provide signals to which another player reacts If a bargaining game is played only once so no reputations are involved the players might act strategically to maximize their payoffs lf bargaining is repeated players might act strategically to establish reputations for expected negotiations EXERCISES 1 In many oligopolistic industries the same firms compete over a long period of time setting prices and observing each other s behavior repeatedly Given that the number of repetitions is large why don t collusive outcomes typically result If games are repeated indefinitely and all players know all payoffs rational behavior will lead to apparently collusive outcomes ie the same outcomes that would result if firms were actively colluding All payoffs however might not be known by all players Sometimes the payoffs of other firms can only be known by engaging in extensive and costly information exchanges or by making a move and observing rivals responses Also successful collusion encourages entry Perhaps the greatest problem in maintaining a collusive outcome is that changes in market conditions change the collusive price and quantity The firms then have to repeatedly change their agreement on price and quantity which is costly and this increases the ability of one firm to cheat without being discovered 188 Chapter 13 Game Theory and Competitive Equilibrium 2 Many industries are often plagued by overcapacity firms simultaneously make major investments in capacity expansion so total capacity far exceeds demand This happens in industries in which demand is highly volatile and unpredictable but also in industries in which demand is fairly stable What factors lead to overcapacity Explain each brie y In Chapter 12 we found that excess capacity may arise in industries with easy entry and differentiated products In the quot 39 J 39 39 mo e I p39 g demand curves for each firm lead to output with average cost above minimum average cost The difference between the resulting output and the output at minimum longrun average cost is defined as excess capacity In this chapter we saw that overcapacity could be used to deter new entry that is investments in capacity expansion could convince potential competitors that entry would be unprofitable Note that although threats of capacity expansion may deter entry these threats must be credible 3 Two computer firms A and B are planning to market network systems for of ce information management Each firm can develop either a fast high quality system H or a slower low quality system L Market research indicates that the resulting pro ts to each rm for the alternative strategies are given by the following payoff matrix Firm B H L H 30 30 50 35 WA L 4060 20 20 a If both rms make their decisions at the same time and follow maximin low risk strategies what will the outcome be With a maximin strategy a firm determines the worst outcome for each option then chooses the option that maximizes the payoff among the worst outcomes If Firm A chooses H the worst payoff would occur if Firm B chooses H A s payoff would be 30 If Firm A chooses L the worst payoff would occur if Firm B chooses L A s payoff would be 20 With a maximin strategy A therefore chooses H If Firm B chooses L the worst payoff would occur if Firm A chooses L the payoff would be 20 If Firm B chooses H the worst payoff 30 would occur if Firm A chooses L With a maximin strategy B therefore chooses H So under maximin both A and B produce a highquality system b Suppose both firms try to maximize pro ts but Firm A has a head start in planning and can commit rst Now what will the outcome be What will the outcome be if Firm B has a head start in planning and can commit first If Firm A can commit first it will choose H because it knows that Firm B will rationally choose L since L gives a higher payoff to B 35 vs 30 This gives Firm A a payoff of 50 If Firm B can commit first it will choose H because it knows that Firm A will rationally choose L since L gives a higher payoff to A 40 vs 30 This gives Firm B a payoff of 60 c Getting a head start costs money you have to gear up a large engineering team Now consider the twostage game in which first each rm decides how much money to spend to speed up its planning and second it announces which product H or L it will produce Which rm will spend more to speed up its planning How much will it spend Should the other rm spend anything to speed up its planning Explain In this game there is an advantage to being the first mover If A moves first its profit is 50 If it moves second its profit is 40 a difference of 10 Thus it would be willing to spend up to 10 for the option of announcing first On the other hand if B moves first its profit is 60 If it moves second its profit is 35 a difference of 25 and thus would be 189 Chapter 13 Game Theory and Competitive Equilibrium willing to spend up to 25 for the option of announcing first Once Firm A realizes that Firm B is willing to spend more on the option of announcing first then the value of the option decreases for Firm A because if both firms were to invest both firms would choose to produce the highquality system Therefore Firm A should not spend money to speed up the introduction of its product if it believes that Firm B is spending the money However if Firm B realizes that Firm A will wait Firm B should only spend enough money to discourage Firm A from engaging in research and development which would be an amount slightly more than 10 the maximum amount Ais willing to spend 4 Two rms are in the chocolate market Each can choose to go for the high end of the market high quality or the low end low quality Resulting pro ts are given by the following payoff matrix Firm 2 Low High Low 20 30 900 600 Fmquot 1 High 100 800 5050 a What outcomes if any are Nash equilibria lf Firm 2 chooses Low and Firm 1 chooses High neither will have an incentive to change 100 gt 20 for Firm 1 and 800 gt 50 for Firm 2 If Firm 2 chooses High and Firm 1 chooses Low neither will have an incentive to change 900 gt 50 for Firm 1 and 600 gt 30 for Firm 2 Both outcomes are Nash equilibria b If the manager of each firm is conservative and each follows a maximin strategy what will be the outcome lf Firm 1 chooses Low its worst payoff 20 would occur if Firm 2 chooses Low lf Firm 1 chooses High its worst payoff 50 would occur if Firm 2 chooses High Therefore with a conservative maximin strategy Firm 1 chooses High Similarly if Firm 2 chooses Low its worst payoff 30 would occur if Firm 1 chooses Low lf Firm 2 chooses High its worst payoff 50 would occur if Firm 1 chooses High Therefore with a maximin strategy Firm 2 chooses High Thus both firms choose High yielding a payoff of 50 for both c What is the cooperative outcome The cooperative outcome would maximize joint payoffs This would occur if Firm 1 goes for the low end of the market and Firm 2 goes for the high end of the market The joint payoff is 1500 Firm 1 gets 900 and Firm 2 gets 600 d Which firm bene ts most from the cooperative outcome How much would that rm need to offer the other to persuade it to collude Firm 1 benefits most from cooperation The difference between its best payoff under cooperation and the next best payoff is 900 100 800 To persuade Firm 2 to choose Firm 1 s best option Firm 1 must offer at least the difference between Firm 2 s payoff under cooperation 600 and its best payoff 800 ie 200 However Firm 2 realizes that Firm 1 benefits much more from cooperation and should try to extract as much as it can from Firm 1 up to 800 190 Chapter 13 Game Theory and Competitive Equilibrium 5 Two major networks are competing for viewer ratings in the 800 900 RM and 900 1000 PM slots on a given weeknight Each has two shows to ll this time period and is juggling its lineup Each can choose to put its bigger show rst or to place it second in the 900 1000 PM slot The combination of decisions leads to the following ratings points results Network 2 First Second First 18 18 23 20 Network 1 Second 4 23 1616 a Find the Nash equilibria for this game assuming that both networks make their decisions at the same time A Nash equilibrium exists when neither party has an incentive to alter its strategy taking the other s strategy as given By inspecting each of the four combinations we find that First Second is the only Nash equilibrium yielding a payoff of 23 20 There is no incentive for either party to change from this outcome b If each network is risk averse and uses a maximin strategy what will be the resulting equilibrium This conservative strategy of minimizing the maximum loss focuses on limiting the extent of the worst possible outcome to the exclusion of possible good outcomes lf Network 1 plays First the worst payoff is 18 If Network 1 plays Second the worst payoff is 4 Under maximin Network 1 plays First Here playing First is a dominant strategy If Network 2 plays First the worst payoff is 18 If Network 2 plays Second the worst payoff is 16 Under maximin Network 2 plays First The maximin equilibrium is First First with a payoff of 1818 c What will be the equilibrium if Network 1 can makes its selection rst If Network 2 goes first If Network 1 plays First Network 2 will play Second yielding 23 for Network 1 If Network 1 plays Second Network 2 will play First yielding 4 for Network 1 Therefore if it has the first move Network 1 will play First and the resulting equilibrium will be First Second lf Network 2 plays First Network 1 will play First yielding 18 for Network 2 If Network 2 plays Second Network 1 will play First yielding 20 for Network 2 If it has the first move Network 2 will play Second and the equilibrium will again be First Second d Suppose the network managers meet to coordinate schedules and Network 1 promises to schedule its big show rst Is this promise credible and what would be the likely outcome A move is credible if once declared there is no incentive to change Network 1 has a dominant strategy play the bigger show First In this case the promise to schedule the bigger show first is credible Knowing this Network 2 will schedule its bigger show Second The coordinated outcome is likely to be First Second 191 Chapter 13 Game Theory and Competitive Equilibrium 6 Two competing firms are each planning to introduce a new product Each rm will decide whether to produce Product A Product B or Product C They will make their choices at the same time The resulting payoffs are shown below We are given the following payoff matrix which describes a product introduction game Firm 2 A B C A 1010 010 1020 Firm 1 B 100 2020 515 2010 155 3030 a Are there any Nash equilibria in pure strategies If so what are they There are two Nash equilibria in pure strategies Each one involves one firm introducing Product A and the other firm introducing Product C We can write these two strategy pairs as A C and C A where the first strategy is for player 1 The payoff for these two strategies is respectively 1020 and 2010 b If both firms use maximin strategies what outcome will result Recall that maximin strategies maximize the minimum payoff for both players For each of the players the strategy that maximizes their minimum payoff is A Thus AA will result and payoffs will be 1010 Each player is much worse off than at either of the pure strategy Nash equilibrium c If Firm 1 uses a maximin strategy and Firm 2 knows what will Firm 2 do If Firm 1 plays its maximin strategy of A and Firm 2 knows this then Firm 2 would get the highest payoff by playing C Notice that when Firm 1 plays conservatively the Nash equilibrium that results gives Firm 2 the highest payoff of the two Nash equilibria 7 We can think of the US and Japanese trade policies as a Prisoners Dilemma The two countries are considering policies to open or close their import markets Suppose the payoffmatrix is Japan Open Close Open 10 10 5 5 US Close 100 5 1 1 a Assume that each country knows the payoff matrix and believes that the other country will act in its own interest Does either country have a dominant strategy What will be the equilibrium policies if each country acts rationally to maximize its welfare Choosing Open is a dominant strategy for both countries If Japan chooses Open the US does best by choosing Open lf Japan chooses Close the US does best by choosing Open Therefore the US should choose Open no matter what Japan does If the US chooses Open Japan does best by choosing Open If the US chooses Close Japan does best by choosing Open Therefore both countries will choose to have Open policies in equilibrium 192 Chapter 13 Game Theory and Competitive Equilibrium b Now assuIne that Japan is not certain that the US will behave rationally In particular Japan is concerned that US politicians Inay want to penalize Japan even if that does not maximize US welfare How might this affect Japan s choice of strategy How might this change the equilibrium The irrationality of US politicians could change the equilibrium from Close Open If the US wants to penalize Japan they Will choose Close but Japan s strategy Will not be affected since choosing Open is still J apan s dominant strategy 8 You are a duopolist producer of a homogeneous good Both you and your competitor have zero marginal costs The market demand curve is P 30 Q where Q Q1 Q2 Q1 is your output and Q2 is your competitor s output Your competitor has also read this book a Suppose you are to play this game only once If you and your competitor must announce your output at the same time how much will you choose to produce What do you expect your profit to be Explain These are some of the cells in the payoff matrix Firm 2 s Output Firm 1 s 0 5 10 15 20 25 30 Output 0 00 0125 0200 0225 0200 0125 00 5 1250 100100 75150 50150 25100 00 00 10 2000 15075 100100 5075 00 00 00 15 2250 10050 7550 00 00 00 00 20 2000 10025 00 00 00 00 00 25 1250 00 00 00 00 00 00 30 00 00 00 00 00 00 00 If both firms must announce output at the same time both firms believe that the other firm is behaving rationally and each firm treats the output of the other rm as a fixed number a Cournot equilibrium Will result For Firm 1 total revenue Will be 7R 2 30 Q Q2Q or TR1 30Q17Qf inQZ Marginal revenue for Firm 1 Will be the derivative of total revenue With respect to Q1 am 73072 7 aQ Q1 Q2 Because the rms share identical demand curves the solution for Firm 2 Will be symmetric to that of Firm 1 TR 7 30 7 2Q 7 Q QZ 2 1 To find the profitmaximizing level of output for both firms set marginal revenue equal to marginal cost Which is zero Q1 157 722 and Q1 15 Q2 2 193 Chapter 13 Game Theory and Competitive Equilibrium With two equations and two unknowns we may solve for Q1 and Q2 Q1157 b57p or Q1 10 Substitute Q1 and Q2 into the demand equation to determine price By symmetry Q2 10 P 30 10 10 orP 10 Since no costs are given profits for each firm will be equal to total revenue 11 7R11010 100 and 12 7R2 1010 100 Thus the equilibrium occurs when both firms produce 10 units of output and both firms earn 100 Looking back at the payoff matrix note that the outcome 100 100 is indeed a Nash equilibrium neither firm will have an incentive to deviate given the other firms choice Suppose you are told that you must announce your output before your competitor does How much will you produce in this case and how much do you think your competitor will produce What do you expect your profit to be Is announcing rst an advantage or disadvantage Explain brie y How much would youpay to be given the option of announcing either rst or second If you must announce first you would announce an output of 15 knowing that your competitor would announce an output of 75 Note This is the Stackelberg equilibrium 2 TRIwib1QZgQI30Q17Q57Q1 7k15Q1739 Therefore setting MR MC 0 implies 15 Q1 0 orQ1 15 and Q2 75 At that output your competitor is maximizing profits given that you are producing 15 At these outputs price is equal to 30 15 75 75 Your profit would be 1575 1125 Your competitor s profit would be 7575 5625 Announcing first is an advantage in this game The difference in profits between announcing first and announcing second is 5625 You would be willing to pay up to this difference for the option of announcing first Suppose instead that you are to play the rst round of a series of 10 rounds with the same competitor In each round you and your competitor announce your outputs at the same time You want to maximize the sum of your pro ts over the 10 rounds How much will you produce in the first round How much would you expect to produce in the tenth round The ninth round Explain brie y Given that your competitor has also read this book you can assume that he or she will be acting rationally You should begin with the Cournot output and continue with the Cournot output in each round including the ninth and tenth rounds Any deviation from this output will reduce the sum of your profits over the ten rounds 194 Chapter 13 Game Theory and Competitive Equilibrium d Once again you will play a series of 10 rounds This time however in each round your competitor will announce its output before you announce yours How will your answers to c change in this case If your competitor always announces first it might be more pro table to behave by reacting irrationally in a single period For example in the first round your competitor will announce an output of 15 as in Exercise 7b Rationally you would respond with an output of 75 If you behave this way in every round your total profits for all ten rounds will be 56250 Your competitor s pro ts will be 1125 However ifyou respond with an output of 15 every time your competitor announces an output of 15 profits will be reduced to zero for both of you in that period If your competitor fears or learns that you will respond in this way he or she will be better off by choosing the Cournot output of 10 and your profits after that point will be 75 per period Whether this strategy is profitable depends on your opponent s expectations about your behavior as well as how you value future profits relative to current profits Note A problem could develop in the last period however because your competitor will know that you realize that there are no more longterm gains to be had from behaving strategically Thus your competitor will announce an output of 15 knowing that you will respond with an output of 75 Furthermore knowing that you will not respond strategically in the last period there are also no longterm gains to be made in the ninth period from behaving strategically Therefore in the ninth period your competitor will announce an output of 15 and you should respond rationally with an output of 75 and so on 9 You play the following bargaining game Player A moves first and makes Player B an offer for the division of 100 For example Player A could suggest that she take 60 and Player B take 40 Player B can accept or reject the offer If he rejects the amount of money available drops to 90 and he then makes an offer for the division of this amount If Player A rejects this offer the amount of money drops to 80 and Player A makes an offer for its division If Player B rejects this offer the amount of money drops to 0 Both players are rational fully informed and want to maximize their payoffs Which player will do best in this game Solve the game by starting at the end and working backwards lfB rejects A s offer at the 3rd round B gets 0 When A makes an offer at the 3rd round B will accept even a minimal amount such as 1 So A should offer 1 at this stage and take 79 for herself In the 2nd stage B knows that A will turn down any offer giving her less than 79 so B must offer 80 to A leaving 10 for B At the first stage Aknows B will turn down any offer giving him less than 10 So A can offer 11 to B and keep 89 for herself B will take that offer since B can never do any better by rejecting and waiting The following table summarizes this Round Money Offering Party Amount to A Amount to B Available 1 100 A 89 1 1 2 90 B 80 10 8 80 A 79 1 End 0 0 0 10 Defendo has decided to introduce a revolutionary video game and as the first rm in the market it will have a monopoly position for at least some time In deciding what type of manufacturing plant to build it has the choice of two technologies Technology A is publicly available and will result in annual costs of CAq 10 Sq 195 Chapter 13 Game Theory and Competitive Equilibrium Technology B is a proprietary technology developed in Defendo s research labs It involves higher fixed cost of production but lower marginal costs CBq 60 2q Defendo s CEO must decide which technology to adopt Market demand for the new product is P 20 Q where Q is total industry output a Suppose Defendo were certain that it would maintain its monopoly position in the market for the entire product lifespan about five years without threat of entry Which technology would you advise the CEO to adopt What would be Defendo s profit given this choice Defendo has two choices Technology A With a marginal cost of 8 and Technology B With a marginal cost of 2 Given the inverse demand curve as P 20 Q total revenue PQ is equal to 20Q Q2 for both technologies Marginal revenue is 20 2Q To determine the profits for each technology equate marginal revenue and marginal cost 20 ZQA 8 or QA 6 and 202QB 2 orQB 9 Substituting the profitmaximizing quantities into the demand equation to determine the profitmaximizing prices we find PA20614 and PB 20 911 To determine the profits for each technology subtract total cost from total revenue M 146 10 86 26 and 13 119 60 29 21 To maximize profits Defendo should choose technology A b Suppose Defendo expects its archrival Offendo to consider entering the market shortly after Defendo introduces its new product Offendo will have access only to Technology A If Offendo does enter the market the two rms will play a Cournot game in quantities and arrive at the Cournot Nash equilibrium i If Defendo adopts Technology A and Offendo enters the market what will be the profits of both rms Would Offendo choose to enter the market given these profits If both firms play Cournot each Will choose its best output taking the other s strategy as given Letting D Defendo and O Offendo the demand function Will be P 20 QB Q0 Profit for Defendo Will be nD iQD 7620913 7 Eb8QD or nD 12QD Q iQDQO 10 To determine the profitmaximizing quantity set the first derivative of profits With respect to QB equal to zero and solve for QB 122QDQ00orQD605QO D This is Defendo s reaction function Because both firms have access to the same technology hence the same cost structure Offendo s reaction function is analogous QO605QD 196 Chapter 13 Game Theory and Competitive Equilibrium Substituting Offendo s reaction function into Defendo s reaction function and solving for D QB 6 056 05QD 4 Substituting into Defendo s reaction function and solving for Q0 Q0 6 054 4 Total industry output is therefore equal to 8 To determine price substitute QB and Q0 into the demand function P204412 The profits for each firm are equal to total revenue minus total costs nD 412 10 84 6 and quoto 412 10 84 6 Therefore Offendo would enter the market ii If Defendo adopts Technology B and Offendo enters the market what will be the profit of each rm Would Offendo choose to enter the market given these pro ts Profit for Defendo Will be nD bagQD foogD 7 Iao2QDg or nD 18QD 7Q 7QDQ0 760 The change in profit With respect to QB is ag 265420 D To determine the profitmaximizing quantity set this derivative to zero and solve for QB 18 2QD Q0 0 or QB 9 05620 This is Defendo s reaction function Substituting Offendo s reaction function into Defendo s reaction function and solving for QB QB 9 056 05QD or QB 8 Substituting QB into Offendo s reaction function yields Q0 6 058 or Q0 2 To determine the industry price substitute the profitmaximizing quantities for Defendo and Offendo into the demand function P208210 The profit for each firm is equal to total revenue minus total cost or HD 108 60 28 4 and quoto 102 10 82 6 With negative profit Offendo should not enter the industry iii Which technology would you advise the CEO of Defendo to adopt given the threat of possible entry What will be Defendo s pro t given this choice What will be consumer surplus given this choice 197 Chapter 13 Game Theory and Competitive Equilibrium With Technology A and Offendo s entry Defendo s profit would be 6 With Technology B and no entry by Defendo Defendo s pro t would be 4 I would advise Defendo to stick with Technology A Under this advice total output is 8 and price is 12 Consumer surplus is 0520 128 32 c What happens to social welfare the smn of consumer surplus and producer pro t as a result of the threat of entry in this Inarket What happens to equilibrimn price What might this imply about the role of potential competition in limiting market power From 10a we know that under monopoly Q 6 and profit is 26 Consumer surplus is 0520 146 18 Social welfare is the sum of consumer surplus plus pro ts or 18 26 44 With entry social welfare is 32 consumer surplus plus 12 industry profit or 44 Social welfare changes little with entry but entry shifts surplus from producers to consumers The equilibrium price falls with entry and therefore potential competition can limit market power Note that Defendo has one other option to increase quantity from the monopoly level of 6 to discourage entry by Offendo lf Defendo increases output from 6 to 8 under Technology A Offendo is unable to earn a positive profit With an output of 8 Defendo s profit decreases from 26 to 802 10 88 22 As before with an output of 8 consumer surplus is 32 social welfare is 54 In this case social welfare rises when output is increased to discourage entry 11 Three contestants A B and C each have a balloon and a pistol From xed positions they re at each other s balloon When a balloon is hit its owner is out When only one balloon remains its owner is the winner and receives a 1000 prize At the outset the players decide by lot the order in which they will re and each player can choose any remaining balloon as his target Everyone knows that A is the best shot and always hits the target that B hits the target with probability 9 and that C hits the target with probability 08 Which contestant has the highest probability of winning the 1000 Explain why lntuitively C has the highest probability of winning though A has the highest probability of shooting the balloon Each contestant wants to remove the contestant with the highest probability of success By following this strategy each improves his chance of winning the game A targets B because by removing B from the game As chance of winning becomes much greater B s probability of success is greater than C s probability of success C will target A because if C targets B and hits B then A will target C and win the game B will follow a similar strategy because if B targets C and hits C then A will target B and will win the game Therefore both B and C increase their chance of winning by eliminating A first Similarly A increases his chance of winning by eliminating B first A complete probability tree can be constructed to show that As chance of winning is 8 percent B s chance of winning is 32 percent and Cs chance of winning is 60 percent 198 Chapter 7 The Cost of Production CHAPTER 7 THE COST OF PRODUCTION TEACHING NOTES In this chapter it is easy for the students to concentrate too much on definitions and geometry and lose focus on the economics Therefore keep in mind the key concepts opportunity cost shortrun average and marginal cost cost minimization and longrun average cost These concepts can be illuminated with the supplementary material provided at the end of the chapter which includes sections on economies of scope learning curves and estimating and predicting costs The Appendix presents the calculus of constrained optimization as applied to cost minimization All exercises involve some algebra or geometry Exercises 12 and 13 are time consuming but rewarding Opportunity cost is the conceptual base of this chapter While most students think of costs in accounting terms they must develop an understanding of the distinction between accounting economic and opportunity costs One source of confusion is the opportunity cost of capital ie why the rental rate on capital must be considered explicitly by economists It is important for example to distinguish between the purchase price of capital equipment and the opportunity cost of using the equipment The opportunity cost of a person s time also leads to some confusion for students Following the discussion of opportunity cost the chapter diverges in two directions one path introduces types of cost and cost curves and the other focuses on cost minimization Both directions converge with the discussion of longrun average cost The geometry of total fixed variable average and marginal costs can prove to be tedious An emphasis on the following issues helps students master this topic 1 the relationship between the production function diminishing returns in the short run input prices and the shapes of the various cost curves 2 the distinction between total average and marginal and 3 the reasonableness of the assumption of constant input prices note that this assumption will be relaxed in Chapter 10 s discussion of monopsony The determination of the costminimizing quantity is crucial to understanding Chapters 8 and 10 The concept of duality minimizing cost subject to a given level of 1 1s o 39 output subject to a given level of total cost clarifies this concept for students A clear understanding of shortrun cost and cost minimization is necessary for the derivation of longrun average cost With longrun costs stress that firms are operating on shortrun cost curves at each level of the fixed factor and that longrun costs do not exist separately from shortrun costs Exercise 6 illustrates the relationship between longrun cost and cost minimization with an emphasis on the importance of the expansion path Stress the connection between the shape of a longrun cost curve and returns to scale While Section 77 is starred it does not require calculus Example 75 Cost Functions for Electric Power gives students another view of longrun average cost and allows for discussion of minimum efficient scale an important determinant of industry structure REVIEW QUESTIONS 1 A rms pays its accountant an annual retainer of 10000 Is this an explicit or implicit cost Explicit costs are actual outlays They include all costs that involve a monetary transaction An implicit cost is an economic cost that does not necessarily involve a monetary transaction but still involves the use of resources When a firm pays an annual retainer of 10000 there is a monetary transaction The accountant trades his or her time in return for money Therefore an annual retainer is an explicit cost 2 The owner of a small retail store does her own accounting work How would you measure the opportunity cost of her work Opportunity costs are measured by comparing the use of a resource with its alternative uses The opportunity cost of doing accounting work is the time not spent in other 66 Chapter 7 The Cost of Production ways ie time such as running a small business or participating in leisure activity The economic cost of doing accounting work is measured by computing the monetary amount that the time would be worth in its next best use 3 Suppose a chair manufacturer finds that the marginal rate of technical substitution of capital for labor in his production process is substantially greater than the ratio of the rental rate on machinery to the wage rate for assembly line labor How should he alter his use of capital and labor to minimize the cost of production To minimize cost the manufacturer should use a combination of capital and labor so the rate at which he can trade capital for labor in his production process is the same as the rate at which he can trade capital for labor in external markets The manufacturer would be better off if he increased his use of capital and decreased his use of labor decreasing the marginal rate of technical substitution MRTS He should continue this substitution until his MRTS equals the ratio of the rental rate to the wage rate 4 Why are isocost lines straight lines The isocost line represents all possible combinations of labor and capital that may be purchased for a given total cost The slope of the isocost line is the ratio of the input prices of labor and capital lf input prices are fixed then the ratio of these prices is clearly fixed and the isocost line is straight Only when the ratio or factor prices change as the quantities of inputs change is the isocost line not straight 5 If the marginal cost of production is increasing does this tell you whether the average variable cost is increasing or decreasing Explain Marginal cost can be increasing while average variable cost is either increasing or decreasing lf marginal cost is less greater than average variable cost then each additional unit is adding less more to total cost than previous units added to the total cost which implies that the AVG declines increases Therefore we need to know whether marginal cost is greater than average cost to determine whether the AVG is increasing or decreasing 6 If the marginal cost of production is greater than the average variable cost does this tell you whether the average variable cost is increasing or decreasing Exp in If the average variable cost is increasing decreasing then the last unit produced is adding more less to total variable cost than the previous units did on average Therefore marginal cost is above below average variable cost If marginal cost is above average variable cost average variable cost is also increasing 7 If the firm s average cost curves are U shaped why does its average variable cost curve achieve its minimmn at a lower level of output than the average total cost curve Total cost is equal to fixed plus variable cost Average total cost is equal to average fixed plus average variable cost When graphed the difference between the Ushaped total cost and average variable cost curves is the average fixed cost curve lf fixed cost is greater than zero the minimum of average variable cost must be less than the minimum average total cost 8 If a firm enjoys increasing returns to scale up to a certain output level and then constant returns to scale what can you say about the shape of the firm s long run average cost curve When the firm experiences increasing returns to scale its longrun average cost curve is downward sloping When the firm experiences constant returns to scale its longrun average cost curve is horizontal If the firm experiences increasing returns to scale then constant returns to scale its longrun average cost curve falls then becomes horizontal Chapter 7 The Cost of Production 9 How does a change in the price of one input change the rm s long run expansion path The expansion path describes the combination of inputs for which the firm chooses to minimize cost for every output level This combination depends on the ratio of input prices if the price of one input changes the price ratio also changes For example if the price of an input increases less of the input may be purchased for the same total cost The intercept of the isocost line on that input s axis moves closer to the origin Also the slope of the isocost line the price ratio changes As the price ratio changes the firm substitutes away from the now more expensive input toward the cheaper input Thus the expansion path bends toward the axis of the now cheaper input See Exercise 76 10 Distinguish between economies of scale and economies of scope Why can one be present without the other Economies of scale refer to the production of one good and occur when proportionate increases in all inputs lead to a morethanproportionate increase in output Economies of scope refer to the production of more than one good and occur when joint output is less costly than the sum of the costs of producing each good or service separately There is no direct relationship between increasing returns to scale and economies of scope so production can exhibit one without the other See Exercise 13 for a case with constant productspecific returns to scale and multiproduct economies of scope EXERCISES 1 Assume a computer firm s marginal costs of production are constant at 1000 per computer However the xed costs of production are equal to 10000 a Calculate the rm s average variable cost and average total cost curves The variable cost of producing an additional unit marginal cost is constant at 1000 1000 so the average variable cost is constant at 1000 T 7 1000 Average fixed cost is w Average total cost is the sum of average variable cost and average fixed cost ATC 1000 b If the rm wanted to minimize the average total cost of production would it choose to be very large or very small Explain The firm should choose a very large output because average total cost decreases with increase in Q As Qbecomes infinitely large ATC will equal 1000 2 If a firm hires a currently unemployed worker the opportunity cost of utilizing the worker s service is zero Is this true Discuss From the worker s perspective the opportunity cost of his or her time is the time not spent in other ways including time spent in personal or leisure activities Certainly the opportunity cost of hiring an unemployed mother of preschool children is not zero While it might be difficult to assign a monetary value to the time of an unemployed worker we can not conclude that it is zero From the perspective of the firm the opportunity cost of hiring the worker is not zero and the firm could purchase a piece of machinery rather than hiring the worker Chapter 7 The Cost of Production 3a Suppose that a rm must pay an annual franchise fee which is a xed sum independent of whether it produces any output How does this tax affect the firm s xed marginal and average costs Total cost TC is equal to fixed cost FC plus variable cost VC Fixed costs do not vary with the quantity of output Because the franchise fee FF is a fixed sum the firm s fixed costs increase by this fee Thus average cost equal to w and average fixed cost equal to increase by the average franchise fee Note that the franchise fee does not affect average variable cost Also because marginal cost is the change in total cost with the production of an additional unit and because the fee is constant marginal cost is unchanged 3b Now suppose the rm is charged a tax that is proportional to the number of items it produces Again how does this tax affect the firm s xed marginal and average costs Let 1 equal the per unit tax When a tax is imposed on each unit produced variable costs increase by tQ Average variable costs increase by t and because fixed costs are constant average total costs also increase by It Further because total cost increases by t with each additional unit marginal costs increase by t 4 A recent issue ofBusiness Week reported the following During the recent auto sales slump GM Ford and Chrysler decided it was cheaper to sell cars to rental companies at a loss than to lay off workers That s because closing and reopening plants is expensive partly because the auto makers current union contracts obligate them to pay many workers even if they re not working When the article discusses selling cars at a loss is it referring to accounting profit or economic pro t How will the two differ in this case Explain brie y When the article refers to the car companies selling at a loss it is referring to accounting profit The article is stating that the price obtained for the sale of the cars to the rental companies was less than their accounting cost Economic profit would be measured by the difference of the price with the opportunity cost of the cars This opportunity cost represents the market value of all the inputs used by the companies to produce the cars The article mentions that the car companies must pay workers even if they are not working and thus producing cars This implies that the wages paid to these workers are sunk and are thus not part of the opportunity cost of production On the other hand the wages would still be included in the accounting costs These accounting costs would then be higher than the opportunity costs and would make the accounting profit lower than the economic profit 5 A chair manufacturer hires its assembly line labor for 22 an hour and calculates that the rental cost of its machinery is 110 per hour Suppose that a chair can be produced using 4 hours of labor or machinery in any combination If the firm is currently using 3 hours of labor for each hour of machine time is it minimizing its costs of production If so why If not how can it improve the situation If the firm can produce one chair with either four hours of labor or four hours of capital machinery or any combination then the isoquant is a straight line with a slope of 1 and intercept at K 4 and L 4 as depicted in Figure 75 Chapter 7 The Cost of Production The isocost line TC 22L 110Khas a slope o 7 702 when plotted with capital on the vertical axis and has intercepts at K and L g The cost minimizing point is a corner solution where L 4 and K 0 At that point total cost is 88 Capital lsocost slope 020 Costminimizing 1 corner solution I I I39 I L b 1 2 3 4 5 a or Figure 75 6 Suppose the economy takes a downturn and that labor costs fall by 50 percent and are expected to stay at that level for a long time Show graphically how this change in the relative price of labor and capital affects the rm s expansion path Figure 76 shows a family of isoquants and two isocost curves Units of capital are on the vertical axis and units of labor are on the horizontal axis Note In drawing this figure we have assumed that the production function underlying the isoquants exhibits constant returns to scale resulting in linear expansion paths However the results do not depend on this assumption If the price of labor decreases while the price of capital is constant the isocost curve pivots outward around its intersection with the capital axis Because the expansion path is the set of points where the MRTS is equal to the ratio of prices as the isocost curves pivot outward the expansion path pivots toward the labor axis As the price of labor falls relative to capital the firm uses more labor as output increases Chapter 7 The Cost of Production Capital Expansion path before wage fall 4 Expansion path after wage fall 3 2 1 I I I 1 2 3 4 5 Labor Figure 76 7 You are in charge of cost control in a large metropolitan transit district A consultant you have hired comes to you with the following report Our research has shown that the cost of running a bus for each trip down its line is 30 regardless of the number of passenger s it carries Each bus can carry 50 people At rush hour when the buses are full the average cost per passenger is 60 cents However during off peak hours average ridership falls to 18 people and average cost soars to 167 per passenger As a result we should encourage more rush hour business when costs are cheaper and discourage off peak business when costs are higher Do you follow the consultant s advice Discuss The consultant does not understand the definition of average cost Encouraging ridership always decreases average costs peak or offpeak If ridership falls to 10 costs climb to 300 per rider Further during rush hour the buses are full How could more people get on Instead encourage passengers to switch from peak to offpeak times for example by charging higher prices during peak periods 8 An oil refinery consists of different pieces of processing equipment each of which differs in its ability to break down heavy sulfurized crude oil into nal products The refinery process is such that the marginal cost of producing gasoline is constant up to a point as crude oil is put through a basic distilling unit However as the unit lls up the firm nds that in the short run the amount of crude oil that can be processed is limited The marginal cost of producing gasoline is also constant up to a capacity limit when crude oil is put through a more sophisticated hydrocracking unit Graph the marginal cost of gasoline production when a basic distilling unit and a hydrocracker are used The production of gasoline involves two steps 1 distilling crude oil and 2 refining the distillate into gasoline Because the marginal cost of production is constant up to the capacity constraint for both processes the marginal cost curves are mirror L shapes Chapter 7 The Cost of Production Marginal Cost MC2 I I I Q1 Q2 Figure 7 8 Quantity Total marginal cost MCT is the sum of the marginal costs of the two processes ie MCT MC1 MCZ where MC1 is the marginal cost of distilling crude oil up to the capacity constraint Q1 and MC2 is the marginal cost of refining distillate up to the capacity constraint Q2 The shape of the total marginal cost curve is horizontal up to the lower capacity constraint If the capacity constraint of the distilling unit is lower than that of the hydrocracking unit MCT is vertical at Q1 If the capacity constraint of the hydrocracking unit is lower than that of the distilling unit MCT is vertical at Q2 9 You manage a plant that mass produces engines by teams of workers using assembly machines The technology is summarized by the production function Q4KL where Q is the number of engines per week K is the number of assembly machines and L is the number of labor teams Each assembly machine rents for r 12000 per week and each team costs w 3000 per week Engine costs are given by the cost of labor teams and machines plus 2000 per engine for raw materials l u p oflo Your plant has a fixed a as part of its design What is the cost function for your plant namely how much would it cost to produce Q engines What are average and marginal costs for producing Q engines How do average costs vary with output K is fixed at 10 The shortrun production function then becomes Q 40 L This implies that for any level of output Q the number of labor teams hired will be L Q 40 The total cost function is thus given by the sum of the costs of capital labor and raw materials TCQ rK wL 2000Q 1200010 3000Q40 2000 Q 120000 2075Q The average cost function is then given by ACQ TCQQ 120000Q 2075 and the marginal cost function is given by 6 TCQ la Q 2075 Marginal costs are constant and average costs will decrease as quantity increases due to the fixed cost of capital Chapter 7 The Cost of Production b How many teams are required to producing 80 engines What is the average cost per engine To produce Q 80 engines we need L Q40 labor teams or L 2 Average costs are given by ACQ 120000Q 2075 or AC 2 3575 c You are asked to make recommendations for the design of a new production facility What would you suggest In particular what capitallabor KL ratio should the new plant accommodate If lower average cost were your only criterion should you suggest that the new plant have more production capacity or less production capacity that the plant you currently manage We no longer assume that K is fixed at 10 We need to find the combination of K and L which minimizes costs at any level of output Q The costminimization rule is given by MPK MPL I W I To find the marginal product of capital observe that increasing K by 1 unit increases Q by 4L so MPK 4L Similarly observe that increasing L by 1 unit increases Q by 4K so MPL 4K Mathematically MPK AQAK 4L and MPL AQAL 4K Using these formulas in the costminimization rule we obtain 4Lr 4Kw or KL W r 3000 12000 14 The new plant should accommodate a capital to labor ratio of 1 to 4 The firm s capitallabor ratio is currently 102 or 5 To reduce average cost the firm should either use more labor and less capital to produce the same output or it should hire more labor and increase output 10 A computer company s cost function which relates its average cost of production AC to its cumulative output in thousands of computers CQ and its plant size in terms of thousands of computers produced per year Q within the production range of 10000 to 50000 computers is given by AC 10 0ICQ 03Q a Is there a learning curve effect The learning curve describes the relationship between the cumulative output and the inputs required to produce a unit of output Average cost measures the input requirements per unit of output Learning curve effects exist if average cost falls with increases in cumulative output Here average cost decreases as cumulative output CQ increases Therefore there are learning curve effects b Are there increasing or decreasing returns to scale To measure scale economies calculate the elasticity of total cost TC with respect to output Q ATC ATC E TC AQ MC C Q E Ac39 Q Q If this elasticity is greater less than one then there are decreasing increasing returns to scale because total costs are rising faster slower than output From average cost we can calculate total and marginal cost TC QAC 10Q 01CQQ 03Q2 therefore 73 Chapter 7 The Cost of Production dTC MC Q 10 7 01CQ 0662 Because marginal cost is greater than average cost because 06Q gt 03Q the elasticity EC is greater than one there are decreasing returns to scale The production process exhibits a learning effect and decreasing returns to scale During its existence the firm has produced a total of 40000 computers and is producing 10000 computers this year Next year it plans to increase its production to 12000 computers Will its average cost of production increase or decrease Explain First calculate average cost this year AC1 10 01CQ03Q 10 0140 0310 9 Second calculate the average cost next year AC2 10 0150 0312 86 Note Cumulative output has increased from 40000 to 50000 The average cost Will decrease because of the learning effect 11 The short run cost function of a company is given by the equation C 190 53Q where C is the total cost and Q is the total quantity of output both measured in tens of thousands a What is the company s fixed cost When Q 0 C 190 0r 1900000 Therefore xed cost is equal to 190 0r 1900000 If the company produced 100000 units of goods what is its average variable cost With 100000 units Q 10 Variable cost is 5362 5310 530 or 5300000 TVC 530 53 10 What is its marginal cost per unit produced Average variable cost is With constant average variable cost marginal cost is equal to average variable cost 53 What is its average fixed cost TFC 7 190 At Q 10 average fixed cost is 10 19 Suppose the company borrows money and expands its factory Its fixed cost rises by 50000 but its variable cost falls to 45000 per 10000 units The cost of interest I also enters into the equation Each one point increase in the interest rate raises costs by 30000 Write the new cost equation Fixed cost changes from 190 to 195 Variable cost decreases from 53 to 45 Fixed cost also includes interest charges 31 The cost equation is C19545QSI 12 Suppose the long run total cost function for an industry is given by the cubic equation TC a bQ cQ2 dQ3 Show using calculus that this total cost function is consistent with a U shaped average cost curve for at least some values ofa b c d To show that the cubic cost equation implies a Ushaped average cost curve we use algebra calculus and economic reasoning to place sign restrictions on the parameters of the equation These techniques are illustrated by the example below Chapter 7 The Cost of Production First if output is equal to zero then TC a where 1 represents fixed costs In the short run fixed costs are positive a gt 0 but in the long run where all inputs are variable a 0 Therefore we restrict a to be zero Next we know that average cost must be positive Dividing TC by Q39 AC b 0Q dQ2 This equation is simply a quadratic function When graphed it has two basic shapes a U shape and a hill shape We want the U shape ie a curve with a minimum minimum average cost rather than a hill shape with a maximum To the left of the minimum the slope should be negative downward sloping At the minimum the slope should be zero and to the right of the minimum the slope should be positive upward sloping The first derivative of the average cost curve with respect to Q must be equal to zero at the minimum For a Ushaped AC curve the second derivative of the average cost curve must be positive The first derivative is c 2dQ the second derivative is 2d If the second derivative is to be positive then d gt 0 If the first derivative is equal to zero then solving for c as a function of Q and d yields 0 2dQ If d and Q are both positive then 0 must be negative 0 lt 0 To restrict b we know that at its minimum average cost must be positive The minimum occurs when 0 2dQ 0 We solve for Q as a function of c and d Q 7 gt 0 Next substituting this value for Q into our expression for average cost and simplifying the equation ACbcQdQ2 bcE kdE llt or 2 2 o2 02 3o2 202 02 AC b b b gt 0 2d 3d 6d 6d 6d 2 implying b gt Sid Because 02 and d gt 0 19 must be positive In summary for Ushaped longrun average cost curves 1 must be zero In and d must be positive 0 must be negative and 4db gt 02 However the conditions do not insure that marginal cost is positive To insure that marginal cost has a U shape and that its minimum is positive using the same procedure ie solving for Q at minimum marginal cost 76 3d and substituting into the expression for marginal cost b 20Q 3dQ2 we find that 02 must be less than 319d Notice that parameter values that satisfy this condition also satisfy 4db gt 02 but not the reverse Chapter 7 The Cost of Production Costs I 017 033 050 067 083 100 Quantlty 1nDozens Figure 712 For example let a 0 b 1 c 1 d 1 Total cost is Q Q2 Q3 average cost is 1 Q Q2 and marginal cost is 1 2Q SQZ Minimum average cost is Q 12 and minimum marginal cost is 13 think of Q as dozens of units so no fractional units are produced See Figure 712 13 A computer company produces hardware and software using the same plant and labor The total cost of producing computer processing units H and software programs S is given by TC aH bS cHS where a b and c are positive Is this total cost function consistent with the presence of economies or diseconomies of scale With economies or diseconomies of scope There are two types of scale economies to consider multiproduct economies of scale and productspecific returns to scale From Section 75 we know that multiproduct economies of scale for the twoproduct case S H S are 7C ac S H39s CH CS where MCH is the marginal cost of producing hardware and MC S is the marginal cost of producing software The productspecific returns to scale are CH TC a S TC am SS abch where TC0S implies no hardware production and TCH0 implies no software production We know that the marginal cost of an input is the slope of the total cost with respect to that input Since TC Di csgbs aH I97 Hg 76 Chapter 7 The Cost of Production we haveMCH a CS andMCS b CH Substituting these expressions into our formulas for S H S SH and S S S 7 1H bS 7 cHS or His Hbe csgs CHE SHS aHerSi OHS gt 1 because cHS gt 0 Also HaSb72cHS S bHbS70HS9bS or H H ich S 7 DHicHSQ bichl d 1 a S Y S am bS cHS aH1 S S cH 39 There are multiproduct economies of scale SHS gt 1 but constant productspecific returns to scale SH SC 1 Economies of scope exist if SC gt 0 Where from equation 78 in the text S TC awn TC aST TC ac S1 7 or TC 3 SI aH bS him I cHSI SE or TC 32 SI OHS S gt 0 TC 3 SI Because cHS and TC are both positive there are economies of scope Chapter I 3Pricing and Employment of Inputs Topics to be Discussed n Competitive Factor Markets n Factor Markets with Monopsony Power n Factor Markets with Monopoly Power Competitive Factor Markets n 5 5 5 Characteristics 1 Large number of sellers and buyers of the factor of production 2 The buyers and sellers of the factor of production are price takers We will illustrate the demand for a factor input and assume only one input is variable Demand for factor inputs is a derived demand 7 Depends on output demand The Decision to Hire or Fire Assume 7 Two inputs are needed to produce output Capital K and Labor L 7 Cost ofK is r and the cost oflabor is w 7 K is fixed andL is variable Problem 7 How much labor to hire Measuring the Value of a Worker s Output The owner of donut shop wants to know whether it is profitable to hire an additional worker It will be pro table if the additional revenue from an additional worker is greater than its cost ie MRPL gt wage Marginal Revenue Produce MRPL MRPL MR MPL Assume perfect competition in the product market Then MR P o MRPL P MPL Competitive Factor Markets n Question When more workers are hired what will happen to the value of MRPL Marginal Revenue Product Wages per Hour Competitive Output NIarket lVIonopolistic Output Blarket Hours of Work MRPL is downward sloping because MPL decreases due to diminishing marginal returns n Question Why is the MRPL for the monopoly output market below the MRPL for the competitive output market In a competitive output market MRP In a monopolistic output market MR lt P n Choosing the pro tmaximizing amount of labor If MRPL gt w the marginal cost of hiring a worker hire the worker If MRPL lt w hire less labor If MRPL w pro t maximizing amount of labor Hiring by a Firm in the LaborMarket with Capital Fixed The pro t maximizing firm will hire L units of labor at the point where the marginal revenue product of labor is equal to the wage rate Hiring by a Firm in the Labor Market with Capital Fixed Price of I Labor MRPLDL L Quantity of Labor Competitive Factor Markets n If the market supply of labor increased relative to demand baby boomers or female entry a surplus of labor would exist and the wage rate would fall n Question How would this impact the quantity demanded for labor Competitive Factor Markets n Comparing Input and Output Markets MRPL MPL MR and at profit maximizing number of workers MRPL w MPL MR w MR 2 wMPL wMPL 2 MC of production Competitive Factor Markets n Comparing Input and Output Markets In both input and output choices occur where MR MC 7 MR from the sale of the output 7 MC from the purchase of the input Competitive Factor Markets n The Supply of Inputs to a Firm Determining how much of an input to purchase 7 Assume a perfectly competitive factor market A Firm 5 Input Supply in a Competitive Factor Market wage Market Supply S of Labor Supply of Labor Facing Firm Market Demand for Labor 10 MRP Demand for Labor D 1 of Workers of Workers n Observations The rm is a price taker at 10 SAEME10 ME MRP 50 units n Question Why is 50 units the pro t maximizing quantity Competitive Factor ilarket n A competitive factor market is in equilibrium when the price of the input equates the quantity demanded to the quantity supplied Labor AIarketEquilibrium Labor Market Equilibrium wage Competitive Output NIarket wage Monopolistic Output Market sL AE DL MRPL DL MRPL i L C Number of Workers L1 Number of Workers n Equilibrium in a Competitive Output Market D LMRPL SL WC MRPL MRPL PXMPL Markets are ef cient n Equilibrium in a Monopolistic Output Market MR lt P MRP MRMPL Hire LM at wage wM vM marginal bene t to society w M marginal cost to the rm Pro ts maximized Using less than the ef cient level of input Equilibrium in a Competitive FactorMarket n Economic Rent For a factor market economic rent is the ali erence between the payments made to a factor of production and the minimum amount that must be spent to obtain the use of that factor Economic Rent Economic Rent I l I L Number of Workers Economic Rent n Question What would be the economic rent if S L is perfectly inelastic n Land A Perfectly Inelastic Supply With land inelastically supplied its price is determined entirely by demand at least in the short run
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