Chapter 13

Imperfect compeTiTION: A game-theoretic approach
Boiling Down Chapter 13
The world seemed rather artificial in the market models of perfect competition and monopoly. It was a world of little variety and no complicating problems of interdependence, transportation costs, advertising, or product differentiation. We now must consider this more complex and realistic world.
The theory of imperfect competition explores strategies that help explain behavior where interdependence is present and firms must make assumptions about their opponent’s actions. Game theory is the tool used to explore many of these relationships. As in the prisoner's dilemma of Chapter 7 in your text, firms would be more profitable if they priced together above marginal cost than if they competed price down to marginal cost. Alternatively, the firm that cheated on the price agreement while the other firm cooperated would have the most profit possible. When the payoffs are such that a firm will be better off with one strategy no matter what the other firm does, a dominant strategy exists and will likely be followed. Table 13-2 in the text shows a duopoly situation where defecting is the dominant strategy. However, if additional price cuts would be damaging to both firms, the final outcome could be zero profits for both firms and the defecting strategy would not be dominant in the long run. In cases like cigarette advertising, the dominant strategy is to advertise even though it dissipates profit and leaves firms with less than they would have with a no-advertising strategy.
If both firms have a dominant strategy, a clear equilibrium exists. If only one firm has a dominant strategy and the other does not, an equilibrium still exists because the firm without the dominant strategy can predict what the other firm will do and choose its best action accordingly. This kind of equilibrium is called a Nash equilibrium.
When interaction among firms is ongoing, as it usually is, firms will be better off if they cooperate than if they defect, because if one defects the other feels compelled to defect and the payoff will be minimal for both. Guaranteeing cooperation is easiest when a defector can be punished in future transactions. One way to punish defectors is to promise to respond in kind each time defection occurs. If the promise is believed, a firm will lose over time if it defects, and so cooperation is likely. This strategy is called tit-for-tat and has been shown to effectively lead to cooperation in wartime circumstances as well as business situations. However, when more than two parties are involved in a game situation, tit-for-tat becomes difficult because it is hard to punish one defector without hurting cooperators
When action by players in a game is sequential, the first player will act based on her prediction of what the other player will do. If the most desirable action of the initial actor is undesirable to the opponent, but better than the result of retaliatory behavior, then the rational choice is not to retaliate. However, if retaliation would be a worst-case situation for the initiator, then certain retaliation would be the victim's best strategy. If doomsday actions can be guaranteed, the threat of doomsday is diminished. The example of the former USSR and U.S. foreign policy is an example of sequential strategy at work. If a doomsday device is installed that guarantees retaliation when a missile attack is launched, then the likelihood of an attack is diminished. In corporate strategy the same kind of automatic deterrence can convince competitors that a confrontation would be counterproductive. The Sears Tower case in the text is an example of this strategy in action. Competitors are not likely to enter a battle they know they will lose.
Oligopoly considers the market structure where only a few players are in the market game. They are highly interdependent since each firm is strongly influenced by what competitors do and must therefore make some assumption about how opponents will act and react in all market situations. Because of this interdependence, corporate strategy develops like a game based on anticipated behavior.
A Cournot duopolist assumes that the competitor will keep output constant even when faced with changes from its counterpart. This model develops a demand curve for a given firm by taking the difference between the total market demand and the output level of the competitor who maintains a constant output. This residual demand curve is then used to find the MC = MR profit-­maximizing solution in the same manner that a monopolist would profit maximize. Cournot assumed MC to be zero for simplicity, so if the marginal revenue function is set equal to zero and solved for Q, an equation results that can be used to find the amount a firm should produce for all possible levels of an opponent's output. This equation is called a reaction curve.
In like manner, the reaction curve of the competitor can be derived. When the two reaction curves are plotted on a graph showing the ideal output for each firm after the opponent's quantity is given, an equilibrium can be seen where the two reaction curves intersect. The final output will be identical for each firm, and the total output will be one-third greater than the output that would have resulted from a single-price profit-maximizing monopolist.
Another attempt to explain duopoly behavior assumes that firms can alter price without changing a competitor's price decision. Given this as­sumption, it becomes desirable to price just below the competitor's price and sell to the entire market. Since both firms proceed in this fashion, they both end up pricing at marginal cost, if MC is constant, and the final outcome is the same as a perfectly competitive model. This Bertrand model seems similar to the Cournot model, in that it assumes a key decision of the competitor will remain constant no matter what a firm does, but its predictions are considerably different. However, both models have assumptions that appear unrealistic and the predictions of the models do not consistently match oligopoly outcomes.
The next step in exploring interdependence is to allow one of the duopolists to behave strategically. Stackelberg assumes that one of the Cournot firms, the Stackelberg leader, will anticipate that the other firm, the follower, will move along its reaction curve, thus changing the optimal output level for the initial firm. Instead of moving down its own reaction curve, the firm develops its own demand curve by substituting the opponent's reaction curve into the market demand curve in place of the opponent's quantity of output. Once the firm's own demand curve is specified, it is a simple matter to select the profit-maximizing price and quantity. This results in an equilibrium that has higher profits for the firm that behaved strategi­cally. The model breaks down if both firms choose to behave strategically, in which case the second-guessing of a competitor's behavior is never-ending and therefore indeterminate. When compared with each other for efficiency and consumer benefit, the Bertrand model fares best since it results in the perfect competitive equilibrium point. The Stackelberg model rates second and Cournot third with the collusion shared monopoly model being the most harmful to consumers.
Economies of scale in oligopoly industries mean that the market may be too small to accommodate two firms at their lowest cost output. In this case the two firms sharing the market would have incen­tive to try to drive each other out and gain monopoly control. The fact that any aggressor may end up the loser makes aggressive price cutting less likely than a live-and-let-live strategy.
The first model of monopolistic competition developed is the Chamberlin model which focuses on the representative firm in a symmetrical market to show that product differentiation leads to higher price, excess capacity, and no economic profit. However, it does provide more variety in the marketplace because of the constant attempt for each of the many firms in the industry to differentiate their products from those of their competitors. These attempts to create special differentiated market niches can include product specification differences, spatial location differences or time variations. 
The spatial model developed in this chapter weighs transportation costs against the increased costs of producing small batches in order to have more locations served. Having many smaller stores with higher per unit costs will be desirable only if the cost of getting to one large vendor more than dissipates the economies of scale that the larger vendor has. The optimal number of vendors occurs when the transportation savings from adding one more vendor exactly equals the extra cost per unit of production that results from having each vendor operate at a slightly smaller size.
It is a short jump from spatial considerations to time considerations. A plane ride involves the cost of flying and the cost of waiting for departure. The more often planes leave, the less waiting time is involved but the harder it is to fill the planes to capacity. As in the spatial cases, two costs are being weighed against each other. The ideal number of flights exists when the sum of these costs is minimized.
The value of this spatial model is apparent when one realizes that it can be applied to almost any case of product differentiation. The desire for variety must be constantly compared with the costs of differentiation. Wherever various product characteristics are possible, a tradeoff exists that cannot be ignored. Generic white shirts would be cheaper than shirts with 100 different colors and labels, but then people could not express their individuality. One grade of gasoline would be easier to produce, thus cheaper than four types, but then performance and economy lovers could not both be satisfied. If variety is the spice of life, then monopolistic models like the spatial model will be useful tools in analyzing how far product differentiation will go.
The cost of variety is borne more heavily by the rich than the poor because the rich have a higher demand for variety than do the poor. Vendors price the Cadillacs and exotic restaurant food higher relative to cost because they know that the desire for such differentiated goods is expressed mostly by the rich.
So far, it has been assumed that vendors respond to consumer demand for variation. This has been called the traditional sequence of markets. However, John Galbraith's "revised sequence" suggests that advertising is the vendors' way of programming consumers to feel demand for differen­tiated products. This programming is intended to develop product loyalty and enhance demand for a given firm's product. The flaw in the revised sequence is that consumers will buy only a product that pleases them. They will not likely be fooled twice if a product does not live up to its hype. Consequently, it serves little purpose to advertise a product that does not fit a perceived need in the consumer's market basket, especially in the area of experience-type goods. Thus, this revised sequence is weak, and, where it is effective, will work only briefly. It is likely that few vendors will spend great sums for a minimal, short-run impact. Only when the vendor is sure of success will a large advertising budgets be mobilized.
Chapter Outline

• Game theory helps to explain how firms will relate in an interdependent situation.
• The prisoner's dilemma example illustrates how dominant and non-dominant strategies affect choices.
• In joint profit-maximizing situations the game is between the cooperators and the defectors to the agreement.
• Advertising strategy provides an example of how interdependence leads to decisions predicted by game theory.
• A Nash equilibrium can occur when one firm has a dominant strategy, even though the other does not.
• Risk averse firms in an industry that does not have a Nash equilibrium may select a maximin strategy in which they look at the worst that can happen with each action and then choose the best of those options.
• The tit-for-tat strategy helps to promote stability in decisions where interdependence exists.
• Sequential games, as in the MAD defense or the tallest building strategy, require that credibility be established or doomsday devices be implemented.
• The Cournot model assumes that competitors keep quantity fixed.
• A demand and reaction curve can be derived for each firm.
• Total output will be greater than a monopolist would produce, and price will be lower than a monopoly price.
• The Bertrand model assumes that competitors keep price fixed, but price competition will inevitably lead to a competitive solution.
• The Stackelberg model assumes that a firm expects its competitor to act like a Cournot duopolist, and then the firm responds strategically.
• In terms of price and quantity the consumer is best served by the Bertrand model followed by the Stackelberg and Cournot models with the shared monopoly market form being the most restrictive in output and the highest in price.
• Increasing returns to scale may or may not lead to a monopoly supplier in the industry.
• The Chamberlin model of product differentiation assumes perfect symmetry of firms in the market and concludes that entry eliminates economic profit. In this model producers are too small for lowest cost production. 
• The spatial model of monopolistic competition focuses on distance as a characteristic of a product that influences demand.
• The example of providing the optimal number of restaurants in a cir­cular community shows how distance and meal costs interact to form a package purchase.
• The optimal number of restaurants occurs where the increased cost per meal, because of smaller output per establishment, is just equal to the cost reduction in travel to the consumer because the food is closer.
• The product characteristic approach applies to characteristics other than distance such as time as in the airline scheduling case.
• The cost of variety is likely to be borne by those most able to bear the higher cost of differentiated production as in the case of Cadillacs and lavish restaurant food.
• The role of advertising as a determinate of consumer preference has been overplayed by some who accept Galbraith's revised sequence.

Important Terms

game theory oligopoly-duopoly dominant strategy collusion	antitrust guidelines monopolistic competition product differentiation free entry and exit
payoff matrix Nash equilibrium	contestable markets preemptive price cut
prisoners dilemma tit-for-tat	spatial modeling optimal number of locations
sequential games	product characteristics
strategic entry deterrence	revised sequence
Cournot model	traditional sequence
residual demand curve	firms that are perfectly symmetrical
reaction function	Chamberlin model
Bertrand model	DD and dd demand curves
Stackelberg model

A Case to Consider
1. Megan and Matt have recently recognized that the action of one of them sig­nificantly affects the business of the other. Therefore each has developed a strategy that takes the other person's output as given. Then the profit­ maximizing price and output are determined from the market that is left. This case is designed to help you find out where the computer market equi­librium will be, assuming that the market demand is   P = 1,800 - Q, where Q is equal to the sum of the quantity produced by both Megan and Matt. If Megan believes that Matt will sell 500 computers per day, what is Megan's demand curve? (Answer in both numerical terms and with nota­tion assuming that the demand curve notation is P = a - bQ and where Q = Qm + Qg represents the quantities of both Matt and Megan, respective­ly.)

 

2. What will Megan's marginal revenue curve be? (Again, use both notation and numerical forms.)

 

3. If Megan has constant marginal costs of 600 dollars per computer and acts as a Cournot duopolist, what will her output be if Matt keeps producing at 500 units?

 

4. Since Megan would have a different profit-maximizing price and quantity for each possible quantity of output produced by Matt, the equation showing her profit-maximizing quantity is called her reaction curve. What would that reaction curve be? (Hint: Derive it in the same way that Equation 13.3 is derived in the text, except that marginal cost is positive rather than zero. This equation should be in notational form.)

 

5. Sketch a graph below, with the quantity for Matt on the horizontal axis and the quantity for Megan on the vertical axis. Draw the reaction curve for Megan using the numbers of this case.

 

6. In a manner similar to question 4 above, derive Matt's reaction curve to Megan and plot it on the graph above.

7. If Matt develops a strategy like Megan's and they both act consistently on that strategy, what will the equilibrium price and quantity be for each ven­dor? (Hint: The quantity will occur where the two reaction curves inter­sect, and the price can then be read from the individual demand curves.)

 

8. What will happen in this market if Matt and Megan change their strategy and assume that the other one will not change price? Describe the process in words, and give a numerical price and quantity solution. As­sume for this question that there are no fixed costs in computer vending. Use the same 600 dollar marginal computer cost that existed above.

9. If Matt and Megan secretly colluded to fix prices as if they were a profit-max­imizing cartel, what would their price and joint output be? How much profit would each make?

Multiple-Choice Questions

• In a prisoner's dilemma with prisoners A and B, if they both confess, A gets 5 years and B gets 8 years. If both remain silent, A gets 2 years and B goes free. If one confesses and the other does not, the one who confesses gets 1 year and the other gets 15 years. Which statement is true of this case?
• There is a dominant strategy for both A and B.
• There is no dominant strategy for either A or B.
• There is a dominant strategy for A but not for B.
• There is a dominant strategy for B but not for A.

• With reference to question 1 above, which of the following is a true statement?
• A Nash equilibrium exists as the question stands.
• A Nash equilibrium does not exist as the question stands, but if the penalties for both remaining silent were doubled, a Nash equilibrium would then exist.
• If the penalties were changed so that if both parties confessed, they would go free, then it would be the dominant strategy for both to confess.
• Both b and c are correct statements, but a is incorrect.
• The tit-for-tat strategy for repeated play in oligopoly theory means that firm 1 will
• defect from an agreement to collude the first time only.
• defect if the opponent defected last time and cooperate if the op­ponent cooperated last time.
• cheat every time to reap as big and quick a profit as possible.
• cooperate every time no matter what the opponent does.
• Research on the tit-for-tat strategy shows that it will be most successful when
• the penalty for cheating is low.
• players are willing to discard the strategy temporarily in an unpre­dictable manner.
• it is followed consistently over a long period of time.
• the players approach a clearly defined end to their transactions.
• Author Frank critiques the MAD defense philosophy as an inadequate ex­planation of peace because
• it is not rational to strike back if a nation is worse off by doing so.
• there was no doomsday retaliating device in place during the cold war time.
• interdependence requires that the commitment problem be solved even in international affairs.
• of all the above.
• of a reason other than any listed above.
• One who follows the Cournot duopoly strategy assumes that competing firms
• treat each other's price as fixed when making an output decision.
• treat each other's quantity as fixed when making an output decision.
• will pick the strategy most damaging to each other.
• will collude informally rather than cut price.
• In a Cournot duopoly model, one characteristic of the equilibrium posi­tion is that
• both firms produce the same amount of output.
• one firm drives the other out of the market.
• the output is less than the single-price monopoly output would be if only one firm were in the market.
• output is the same as would exist in a competitive market.

8.   When a deep recession hits an economy and an oligopoly industry like the auto industry, what happens to the reaction curves of these oligopoly firms?
a.   They shift to the right.
b.   They shift to the left.
c.   They are unaffected by recession pressures.
d.   They are effected but it is impossible to tell how they are effected because any outcome would be possible because of the high level of interdependence.

 

9.   Which oligopoly model would be most desirable from the point of view of the consumer?

• Cournot
• Bertrand
• Stackelberg
• Monopoly collusion
• A weakness of the Stackelberg model is that
• both firms behave naively rather than strategically.
• both firms behave strategically.
• one firm behaves strategically while the other behaves naively.
• there are no assumptions about how the firms will react to each other.
• Which of the following is most likely to be both a fixed and a sunk cost of your auto repair business?
• The air compressor for your garage
• The prescription safety glasses you wear while welding
• The computer that does your inventory and billing
• The tow truck that brings in much of your business
• In the Chamberlin model
• the firms maximize sales rather than profits since market share is important for survival.
• new firms do not enter the market in the long run because they fear price wars.
• existing firms perceive their demand curve to be less elastic than the market demand.
• products in an industry under consideration are considered to be close substitutes by consumers.
• non of the above are true.
• Which is not a criticism of the Chamberlin model?
• The concept of an industry group is hard to define.
• It attempts to be more realistic than perfect competition, but it does not predict behavior in the marketplace with any more accuracy than perfect competition.
• The model presumes that any firm can attract buyers from other firms so that the products of each firm in the industry are perfect substitutes.
• It assumes that entry is easy but exit is difficult for the firms.
• Which of the following does not act as a doomsday device?
• The practice of programming into a computer the price above which an alternative supplier will be used.
• A bank robber demanding the money in five minutes and then lighting a five minute fuse on a bomb.
• Promising to marry someone and giving a $5000 engagement ring.
• A student promising not to cheat on a take-home test.
• In a spatial model of monopolistic competition, the transportation cost share of the product cost increases
• when more evenly spaced vendor locations are offered.
• when fewer evenly spaced vendor locations are offered.
• as the distance between vendors is decreased.
• as the per mile cost of transportation decreases.
• Beginning from a base of one vendor, the optimal number of vendor locations is reached when the _____ in the transportation cost per unit of product exactly equals the _____ in the production cost per unit of product.
• rise, rise
• rise, fall
• fall, fall
• fall, rise
• As transportation costs fall, we can expect
• the optimal number of vendors to be increased.
• the optimal number of vendors to be reduced.
• the optimal number of vendors to stay the same.
• product price to be driven upward.
• If economies of scale are more substantial than originally thought, the optimal number  of vendors will
• be greater than originally thought.
• be smaller than originally thought.
• not be affected by the change in thinking.
• be indeterminate because economies of scale have nothing to do with the optimal number of vendors.
• Which statement is true? The optimal number of vendors will be greater, other things being equal, the
• higher the start-up cost of an establishment.
• less dense the population of the area involved.
• lower the start-up cost of an establishment.
• steeper the slope of the total-cost-of-meals curve.

20. An island has a 100 mile road around its perimeter and there are 10 restaurants located equidistant from each other on the road. What is the average round trip mileage that each customer will have to travel for a restaurant meal? Assume the people are located equidistant from each other on the island road.
a.   10 miles
b.   5 miles
c.   2.5 miles
d.   20 miles

• If each product had only one quality or characteristic that mattered to consumers, the monopolistic competition models would
• be more like the monopoly models of an earlier chapter.
• be more like the competitive models of an earlier chapter.
• predict that average production costs would be higher than they are with multi-characteristic products.
• not be modified in any of the above ways.
• The "revised sequence" concept implies that
• producers are more sensitive to consumer demand than they once were.
• advertising is more informative than persuasive.
• products that might fail under the traditional sequence might now be successful.
• all the above are true.
•  The cost of variety tends to be borne
• more heavily by those who desire the variety.
• more heavily by those who dislike the variety.
• in an unpredictable random fashion.
• by the producer.
• Which of the following is not a true statement about oligopoly and monopolistic competition?
• Entry and exit is easy in monopolistically competitive firms but it is not easy in oligopoly firms.
• In the long run, monopolistically competitive firms do not earn economic profits, but oligopoly firms might in many situations.
• Both types of firms have a marginal revenue function that lies below the demand function.
• The tit for tat strategy applies to both market forms.
• Which of the following products is most likely to fit the oligopoly models?
• Rolled steel
• Soap
• Medical services
• College education
• An overarching theme of this chapter is that
• imperfect competition does not go by normal economic ways of thinking.
• product differentiation is inefficient and undesirable.
• Interdependence leads to less certainty in modeling than earlier market forms.
• People are less self-centered in big organizations.

Problems

• If OPEC consisted of only Saudi Arabia and Nigeria and if they both stuck to an agreement to fix oil prices, then Saudi Arabia would make $100 profit and Nigeria would make $60 profit. If both cheated on the agree­ment, Saudi Arabia would make $70 and Nigeria would make only $30. If Saudi Arabia cheated and Nigeria did not, then Nigeria would make $25 and Saudi Arabia would make $80. Likewise, if Nigeria cheated and Saudi Arabia did not, then Nigeria would make $75 and Saudi Arabia would make $50.
• Does either country have a dominant strategy?

 

• Does a  Nash equilibrium exist in this case? Explain.

 

• If Nigeria knew that Saudi Arabia had a tit-for-tat strategy that it would definitely follow, what would be the outcome of the market interaction?

 

• In class I sometimes allow students the option of forming a cartel on an exam. If all students hand in a blank take-home exam, they will all get an A- for the exam. If one person answers the questions as best he can, he will get an A and everyone who handed in a blank paper gets a D. Assume there are only two persons in the class. Also assume for this exercise that both students would get Bs if they took the test normally. Finally, assume that I grade on a curve that makes class standing important in the final grade when all other grades are averaged.
• Set up the payoff matrix for this problem with Neil as one student and Laura as the other student.

 

• Is there a dominant strategy for both Neil and Laura? If so, what is it? If not, is there a Nash equilibrium?

• The graph below depicts a market demand curve that is shared by two in­terdependent firms. Their MCs are constant at 10.

 

• What is the market demand equation?

 

• What is the equation for the demand curve of firm 1 if it assumes that firm 2 will produce an output of 10?

 

• What is the equation for firm 1’s marginal revenue?

• Derive the reaction curves for both firms.

• Graph the reaction curves of both firms.
• What is the equilibrium output and price for each firm?

• If one firm controlled the entire market, what output would it sell and what price would it charge?

• If firm 1 operated like a Stackelberg leader, what output would it produce in this case?

• If firm 2 is the Stackelberg follower, how much output will it produce?

4. Figure 13.16 in your text shows the two cost functions that must be considered when the optimal number of restaurants is sought. Using a similar graph, sketch in the two cost functions that would need to be considered by an airline company that was trying to determine how many flights a day to make to a given city. One function will be the costs of flying the plane and its passengers. The other function will be the cost to passengers in waiting time. After sketching these costs, vertically sum them and show where the optimal number of flights will be. Make up your own numbers, but the general slopes of the functions will be the important part of this problem. Finally, if there were equations given for both of the functions you draw, how would you go about solving for the optimal number of flights? List the steps required and the equilibrium condition necessary for the optimal number of flights to be attained.

                                                                                   Number of flights

 

• The graph below shows a demand structure for a monopolistically competitive firm with assumptions like the Chamberlin model. Sketch in a set of additional curves needed to show a firm in SR equilibrium making economic profits. Which curves will change in the long run as adjustments to equilibrium are made? Explain why the curves change.

                                                                                                        

• In the movie “A Beautiful Mind”, John Nash, for whom the Nash equilibrium is named, is seated in the university snack shop with friends observing several girls as potential dates. Recognizing that one girl would be the first choice of all the men Nash says that Adam Smith was wrong when he claimed that the group welfare will be maximized when everyone pursues his own self interest. Assume that the payoff matrix for Thomas and David is as shown below. The choices involve asking the top choice girl for a date or choosing one of the other girls as a second choice. Assume also, as Nash did, that if everyone asks the top choice and one is successful, the other girls will refuse to accept dates since they will know they are second choices. Anyone asking the second choice first will not be rejected since the girl will not know she was second choice.

  

David	Top choice for a date	Second choice for a date
Top choice for a date	A = 20     B = 0	A = 8     B = 18
Second choice for a date	A = 18     B = 10	A = 15    B = 15

     

• Do Thomas and David have dominant strategies? Why or why not?

 

• Is there a Nash equilibrium in this case? Why or why not?
• If each person adopted a maximin strategy what would the outcome be?
• Using the payoff matrix, explain why John Nash felt Adam Smith was wrong in his assertion about social optimal outcomes from the pursuit of self interest.

 

• If both Thomas and David were close friends and each seeks the best interest of the other, what will happen and how does this outcome compare with the socially optimal outcome?

 

7.   True or false:  A Nash equilibrium will also be the most desirable point for the firms to be in.  Explain.

 

answers to Questions for Chapter 13
Case Questions

• Megan's demand curve is P = (a _ bQm) _ bQc in notation terms and P = 1300 _ Q. (Qm and Qg are quantities of output for Matt and Megan respectively.)The vertical intercept term shows that 500 output is being conceded to Matt.
• Megan's MR is the same as 1 above except it has twice slope of the demand curve.
• MC = MR at a quantity of 350.
• Take the notational MR curve in question 2 above, set it equal to MC because Megan is a profit maximizer, and then solve for Qg. The result is the formula Qg = (a - bQm - MC)/2b.  It can be formulated into Qg = (a - MC)/2b - 1/2Qm. The first term becomes the intercept of the reaction curve and the 1/2 is the slope of the reaction curve.
• See the sketch labeled Case 13-5 below for the answers to question 5.
• See the sketch labeled Case 13-5 below for the answers to question 6.

Matt’s Reaction Curve

Quantity for Megan
600                       1200

• Equate the two reaction curves as they are sketched in the previous question.  The equation 1200 – 2Qm = 600 – .5Qg becomes Qm = 400. Qg also will be 400 so an industry output of 800 will be sold at a price of 1000.
• The model reverts to the Bertrand model where price competition drives price down to MC. The price will equal 600 and each will sell 600 units of output.
• As joint monopolists, both Matt and Megan take the MR from the market demand and equate it with MC. Thus 1800 –2Q = 600. Total output will be 600 with each producing 300. Price will then be 1200.  Each have total revenue of 360,000 and Total costs of 180,000 and a profit of 180,000 each.

Multiple-Choice Questions

• c, A should confess no matter what B does. B should confess only if A does.
• a, With both confessing, neither one has an incentive to change strategy.
• b, A commitment device of some sort is necessary here or it is little more than MAD.
• c, Trust builds and repeat business gives everyone incentive to stick to the strategy.
• d, Commitment devices are not easy to put in place internationally.
• b, Assuming this in the face of repeated counter-evidence weakens this model a bit.
• a, The reaction curves are symmetrical, leading to identical output for each firm.
• B, they shift left because the demand curve they face has shifted left lowering the intercept of the reaction curve.
• b, The Bertrand model has the highest output and the lowest prices.
• c, This is more realistic than the Cournot model, but still one firm out-thinks the other.
• b, No one else would be willing to buy or lease your prescription glasses but they would be able to use the other items in the problem.
• d, The competitive part of the model assumes that the products are close substitutes.
• d, both entry and exit are easy in order for the model to bid away long run economic profits.
• d, Nothing will happen if the promise is broken in secret. In each of the other options there are automatic consequences programmed into the circumstances. A doomsday device is a way of solving the commitment problem we learned about earlier in the text.
• b, A longer trip adds to both time and travel costs. 
• d, The marginal benefit in transportation will equal the marginal cost of new facilities .
• b, The cost of building restaurants has not changed so it now becomes a least cost approach to drive further and build fewer restaurants.
• b, Lower costs from larger output can finance larger serve area transportation costs.
• b, the longest round trip will be 10 miles and the shortest 0 for an average of 5 miles.
• c, Lower start-up costs can mean lower prices which make increased mileage expense feasible.
• b, It is the differentiation of products that makes this market form meaningful. Products with only one characteristic would be homogenous like perfectly competitive products.
• c, The producer rather than the consumer is "king" under the revised sequence and that means that advertising and marketing tactics alter the consumer’s real preferences.
• a, Those who don't mind generic homogeneity can avoid the high cost of brand names so those seeking variety are the ones that must pay the higher price for it.
• d, Tit for tat strategies only apply to a few firms that are interdependent.
• a, Rolled steel is a fairly generic product produced by relatively few large interdependent firms. The other products can be differentiated considerably.
• c, When one firm’s behavior effects another firm’s operations it is hard to proceed with certainty.

Problems

• a) Nigeria has a dominant strategy to cheat. Saudi Arabia has no dominant strategy.

1.   b) Yes,  Nigeria cheats so the Saudis will cheat and neither have incentive to fix once they are in the cheating mode.
1.   c) Nigeria would fix if they could be sure that the Saudis would follow.
2.   a) Put Neil on the left and Laura on the top of a four quadrant matrix. Put an A‑ for each in the northwest cell, a B for each in the southeast cell, an A for Neil and a D for Laura in the southwest cell, and a D for Neil and an A for Laura in the northeast cell. The upper row shows Neil handing in a blank paper as does the first column for Laura.
2.   b) Both Laura and Neil have a dominant strategy to cheat, and this is why the students always fail in their effort to get easy A’s in my class.
3.   a)  P = 50 _ QT 

• b)  P = 40 – Q1  
• c)  MR1 = 40 – 2Q
• d)  Q1 = 20 – Q2/2 and Q2 = 20 – Q1/2

• f)  The equilibrium output of each firm will be 13.33  and the price will be $23.33.
• g) A monopolist in this market would sell 20 units and charge a price of 30.
• h) Firm 1, as a Stackelberg leader, would produce 20 units. Behaving strategically, this firm will not follow its reaction curve down to the Cournot equilibrium, because it knows it will be better off staying at an output of 20. This answer is arrived at by taking the market demand equation in the beginning of this question and adding the reaction curve of the Stackelberg follower to the independent variable. The demand becomes P = 50 - (Q1 + 20 - 1/2 Q1) or P = 30 - .5Q1.  By doubling the slope to get marginal revenue and equating MR with MC of 10, the result is 20 = Q1.
• i)  The Stackelberg follower will take the 20 as given from the Stackelberg leader. When 20 is put into its reaction curve of Q2 = 20 - .5Q1, the result is an output of 10.

 

4.   See the sketch labeled Problem 13-4 for a sketch of the optimal number of flights. The first step is to find the first derivatives of each of the two cost functions. That will tell you how fast the one cost is falling and how rapidly the other cost is rising. Then set the two derivatives equal to each other and solve for the number of flights. As long as one cost falls faster than the other cost is rising the total costs are falling. When the rising cost rises faster than the falling cost is falling the total costs are rising. Only when the two rates of change in cost are the same will total costs be minimized. Since the derivatives measure the rates of change in costs, setting them equal will be the equilibrium condition.           

 

•  a) Thomas has a dominant strategy to go for his first choice. David doesn’t not have a dominant strategy since he should go for his second choice if Thomas goes for his first and he should take the top choice if Thomas takes his second choice.
•  b) Since David knows Thomas will follow his dominant strategy, David will go for his second choice so a Nash equilibrium does exist.
•  c) The worst that can happen to Thomas if he takes his first choice is 18 and the worst possible if he takes his second choice is 8 so he takes his first choice. David, using the same logic, chooses 10 over 0 so he goes for his second choice.
•  d) The optimal social outcome is the sum of 15 + 15 or 30 which came from each choosing their second choice. If both pursue their first choice without considering their interdependence, the total social welfare is only 20.
•  e) Both will let the first choice for the other and each will have 15 which is the socially optimal outcome. This of course assumes that each knows the other’s preferences.
• A Nash equilibrium will not always be the most desirable point. If one views the payoff chart for the firms in tables 13.2 to 13.4 the dominant strategies have lower rewards than would be possible if both firms employed the opposite strategy.

 

• The demand curve for education in a community with only two colleges is P = 100 - Q. The cost of educating an additional student is 8. Complete the table below for the four scenarios described in a-d. There are no fixed costs.
• Both colleges get together and collude to price fix at a profit maximizing level.
• Both colleges assume that the other will hold the present tuition constant despite what the competition does.
• Both colleges assume that the other will hold the present enrollment constant despite what the competition does.
• College 1 believes college 2 will adjust to its new profit maximizing solution each time college 1 adjusts its enrollment strategy. (Show calculations below.)

 

Assumption	Quantity of A	Quantity of B	Price	Profit of A	Profit of B
a
b
c
d

• Goodyear and Bridgestone face the following profit and price relationships.

                                                                 Bridgestone           

Goodyear	$100	$60
$100	F = 20:  G = 36	F = 33:  G = 10
$60	F = 12:  G = 35	F = 11:  G = 30

• If there is a dominant strategy, is it a Nash equilibrium?

• 3.  True or False   If neither one of two interdependent firms in the same industry has a dominant strategy, then a Nash equilibrium cannot exist.  Explain.

 

3.  True or False   If neither one of two interdependent firms in the same industry has a dominant strategy, then a Nash equilibrium cannot exist.  Explain.

• Each person’s boredom with drab shirts goes down as the number of colors in their shirt increases. The value of the boredom per person occurs according to the formula 1/N where N is the number of colors in the shirt. One thousand people buy shirts and it costs 250 to add a new color to shirt production.
• Sketch a simple graph showing how many colors shirt manufacturers should include in their shirts. Plot points for N = 1,2,3,and 4.

• Calculate the number of colors that should be included in shirts, given the information above. Show your work so the process you are using is clear.

5.  The revised sequence of J.K. Galbriath has considerable intuitive appeal, but may be lacking in real world applicability. Describe the concept briefly and explain why it may not be accurate.