Advanced topics in managerial economics

In the final part of the module, we discuss several advanced topics in Managerial Economics:

  • Oligopoly theory: How do firms decide their prices and quantities in a situation of interdependence, taking into account the reactions of their competitors to their own decisions?
  • Cost-plus pricing: How do firms set their prices if they cannot identify their marginal revenue and marginal cost curve?
  • Price discrimination: How can firms can take advantage of differences between the demand curves of different consumers to increase their profit, by charging different prices to different consumers?

Oligopoly theory

An oligopoly is a market with a few large firms with medium-to-high market power, which recognise their interdependence.

A duopoly is a special case of oligopoly, where there are only two firms.

Successful managers in oligopoly firms must learn to anticipate the actions and reactions of the other firms in the market.

Strategic behaviour consists of the actions taken by firms, in order to plan for or react to the actions of competitors.

Each firm’s demand and marginal revenue curves depend on the pricing decisions and output decisions of every other firm in the same market.

The resulting interdependence and strategic behaviour make decisions more complicated and uncertain than they are in the markets analysed in part 3.

Game theory

Game theory is a useful tool for analysing how to behave in strategic situations involving interdependence.

A game is a situation in which two or more decision makers, or players, face choices between a number of possible actions at any stage of the game.

A game in which all players choose their actions simultaneously, before knowing the actions chosen by other players, is a simultaneous game.

A game in which the players choose their actions in turn, so that a player who moves later knows the actions that were chosen by players who moved earlier, is a sequential game.

We will consider simultaneous games only.

A simultaneous game that is played only once is a single-period game.

A game that is played more than once is a multiple-periodor repeated game. A multiple-period game can be repeated either infinitely, or a finite number of times.

A player’s strategyis a set of rules telling him which action to choose under each possible set of circumstances that might exist at any stage in the game.

Each player aims to select the strategy that will maximize his own payoff.

Interdependence is the key defining characteristic of a game. Each player knows that the decisions of other players affect his payoff, but when he decides what to do he does not know what decisions have been taken by the other players.

In most game theory examples that we will consider, the players are two or more oligopolistic firms.

Strategies are the decisions the firms have to take about price or output, or other commercial decisions (e.g. advertising, product characteristics, research and development, location).

Payoffs are the profits generated by implementing the chosen strategies.

For our first game theory example, Firms A and B both choose between two strategies: ‘set a high price’ and ‘set a low price’.

The entries in the table are the profits or payoffs to the two firms, dependent on the strategies chosen by both firms.

Within each cell, the number in bold is A’s profit, and the number in italics is B’s profit.

For example, if A selects High and B selects Low, A’s payoff is +3 and B’s payoff is +2.

B’s strategy

HighLow

High+4 +4+2 +3

A’s strategy

Low+3 +2+1 +1

Consider A’s choice between High and Low. First examine which of High and Low is best for A if B selects High; then examine which of High and Low is best for A if B selects Low:

  • If B selects High, High gives A a payoff of +4, while Low gives A a payoff of +3. It is best for A to select High.
  • If B selects Low, High gives A a payoff of +2, while Low gives A a payoff of +1. It is best for A to select High.

In this game, it is always best for A to choose High.

Now consider B’s choice between High and Low, using a similar approach:

  • If A selects High, Highgives B a payoff of +4, while Low gives B a payoff of +3. It is best for B to selectHigh.
  • If A selects Low, High gives B a payoff of +2, while Low gives B a payoff of +1. It is best for B to select High.

It is always best for B to choose High.

Dominant strategy equilibrium

A player’s dominant strategy is a strategy that is best for that player, no matter what the other player decides to do.

In the example, High is A’s dominant strategy, because it is the best strategy for A no matter what strategy B selects.

Similarly, Highis B’s dominant strategy, because it is the best strategy for B no matter what strategy A selects.

In this example, (High, High) is a dominant strategy equilibrium.

At the dominant strategy equilibrium, both firms earn a payoff of +4.

+4 is the best payoff achievable by either firm under any circumstances.

It seems natural that the players should choose the combination of strategies that produces this best-possible payoff.

However, we will see that not all games always produce such a good outcome for the players.

For our second game theory example, Firms A and B both choose between the same two strategies: ‘set a high price’ and ‘set a low price’.

B’s strategy

HighLow

High+3 +3+1 +4

A’s strategy

Low+4 +1+2 +2

As before, consider A’s choice between High and Low:

  • If B selects High, High gives A a payoff of +3, while Low gives A a payoff of +4. It is best for A to select Low.
  • If B selects Low, High gives A a payoff of +1, while Low gives A a payoff of +2. It is best for A to select Low.

In this game, it is always best for A to choose Low.

Now consider B’s choice between High and Low:

  • If A selects High, Highgives B a payoff of +3, while Low gives B a payoff of +4. It is best for B to selectLow.
  • If A selects Low, High gives B a payoff of +1, while Low gives B a payoff of +2. It is best for B to select Low.

It is always best for B to choose Low.

In this example, (Low, Low) is a dominant strategy equilibrium.

With (Low, Low), both firms earn a payoff of +2.

However, this time something appears to be wrong. If the firms had selected the other strategy (High, High), both firms would have earned a higher payoff of 3, rather than 2.

Prisoner’s dilemma

This second example (above) is a special type of game, known as a prisoner’s dilemma game.

In a prisoner’s dilemma game, there are dominant strategies for both firms that produces payoffs that are lower than the payoffs the firms could achieve by cooperating, and agreeing to choose a strategy different from the dominant strategy.

Why is this type of game known as a prisoner’s dilemma?

The police hold two prisoners, Jane and Bill, who are suspected of having committed a serious crime, which carries a sentence of 12 years in prison.

The police have insufficient evidence to secure a conviction unless one or both prisoners confesses.

However, the police have enough evidence to convict both prisoners of a minor crime, which carries a sentence of 2 years in prison.

The prisoners are separated physically and they cannot communicate. The police tell them the following:

  • If both confess, both receive a reduced sentence of 6 years in prison.
  • If both do not confess, both receive a sentence of 2 years for the minor crime.
  • If one confesses and the other does not confess, the one who confesses receives a reduced sentence of 1 year, and the one who does not confess receives the full sentence of 12 years.

Bill’s strategy

Don’t confessConfess

Don’t confess–2 –2–12 –1

Jane’s strategy

Confess–1 –12–6 –6

Consider Jane’s choice between Don’t confess and Confess.

  • If Bill selects Don’t confess, Don’t confessgives Jane a payoff of –2, while Confess gives Jane a payoff of –1. It is best for Jane to selectConfess.
  • If Bill selects Confess, Don’t confess gives Jane a payoff of –12, while Confess gives Jane a payoff of –6. It is best for Jane to select Confess.

It is always best for Jane to choose Confess.

Now consider Bill’s choice between Don’t confess and Confess:

  • If Jane selects Don’t confess, Don’t confessgives Bill a payoff of –2, while Confess gives Bill a payoff of –1. It is best for Bill to selectConfess.
  • If Jane selects Confess, Don’t confess gives Bill a payoff of –12, while Confess gives Bill a payoff of –6. It is best for Bill to select Confess.

It is always best for B to choose Confess.

Jane and Bill both confess, and both receive the 6-year sentence. But if they had been able to cooperate, they could have agreed not to confess, and both would have received the 2-year sentence.

Even acting independently, they might be able to reach this cooperative solution. Jane knows that if she does not confess, she gets the 2-year sentence as long as Bill does the same.

However, Jane is worried because she knows there is a big incentive for Bill to ‘cheat’ by confessing. By doing so Bill can receive a 1-year sentence, while Jane gets a 12-year sentence!

Bill is in a similar position. If he does not confess, he gets a 2-year sentence as long as Jane also does not confess. But Bill also knows there is a big incentive for Jane to ‘cheat’ in an obtain to get the 1-years sentence.

The cooperative solution might be achievable, especially if Jane and Bill can trust each other not to cheat, but it is also unstable and liable to break down.

Not all prisoner’s dilemma games generate bad outcomes for the players. Sometimes they might be able to achieve the cooperative solution, even if they act independently.

  • The cooperative solution might be achieved if there is good communication between the players. If oligopoly firms meet frequently, they can exchange information and monitor each other’s actions.
  • An important characteristic of any game is the time it takes for a player who has been deceived to retaliate. The longer the reaction lags, the greater the temptation for either player to ‘cheat’.
  • If the game is repeated, the players may eventually learn that cooperation is better ‘cheating’. Firms can change their prices continuously. They tend to learn over time that aggressive price-cutting leads to hostile reactions from competitors, that cancel out any short-term gains.

Nash equilibrium: Making mutually best decisions

At the dominant strategy equilibrium that is derived in the previous examples, neither firm or player can improve its payoff, given the current strategy of the other firm or player.

In the first example:

  • Given that B selects High, if A switches from High to Low, A’s payoff falls from +4 to +3.
  • Given that A selects High, if B switches from High to Low, B’s payoff also falls from +4 to +3.

In the second example:

  • Given that B selects Low, if A switches from Low to High, A’s payoff falls from +2 to +1.
  • Given that A selects Low, if B switches from Low to High, B’s payoff also falls from +2 to +1.

An equilibrium that has this property is known as a Nash equilibrium.In a Nash equilibrium, neither firm can improve its payoff given the strategy chosen by the other firm.

The manager chooses the strategy to give his firm the highest payoff, given what he believes or anticipates about the actions of the firm’s rivals.

The manager must believe that he is correctly anticipating the actions of his firm’s rivals.

Therefore his guess concerning their actions must be consistent with the assumption that they will also attempt to find the strategy that gives them the highest payoff, given what they believe or anticipate about the reactions of their rivals.

Strategic stability is a property of a Nash equilibrium. Strategic stability means no player can change his action individually and improve his payoff, while the other players stick to the same actions.

  • A dominant strategy equilibrium is always a Nash equilibrium.
  • But a Nash equilibrium is not always a dominant strategy equilibrium.

The next example refers to advertising decisions by Coke and Pepsi for the SuperBowl game.

The payoffs to both Coke and Pepsi of setting a ‘low’, ‘medium’ or ‘high’ advertising budget depend upon the actions of the other producer.

Pepsi’s budget

LowMediumHigh

Low60 4557.5 5045 35

Coke’s budgetMedium50 3565 3030 25

High45 1060 2050 40

Consider Coke’s choices:

  • If Pepsi chooses Low, Coke’s best choice is Low.
  • If Pepsi chooses Medium, Coke’s best choice is Medium.
  • If Pepsi chooses High, Coke’s best choice is High.

Therefore there is no dominant strategy for Coke.

Similarly, consider Pepsi’s choices:

  • If Coke chooses Low, Pepsi’s best choice is Medium.
  • If Coke chooses Medium, Pepsi’s best choice is Low.
  • If Coke chooses High, Pepsi’s best choice is High.

Therefore there is no dominant strategy for Pepsi.

However, (High, High) is a Nash equilibrium, with the property of strategic stability.

  • Coke expects Pepsi to choose High.If this is correct, High is Coke’s best choice.
  • Similarly, Pepsi expects Coke to choose High. If this is correct, High is Pepsi’s best choice.

Notice that Coke and Pepsi would both made be better off by cooperating or agreeing to choose (Low, Low).

However, this cooperative solution is strategically unstable. If Coke chooses Low, Pepsi has an incentive to ‘cheat’ and choose Medium instead of Low. Therefore the cooperative solution is unstable and likely to break down.

Best-response curves and continuous decision choices

All of the previous examples assume the firms face only two or three different choices, with payoffs that can be represented in a simple table.

However, in practice many decisions are continuous. The firm can choose any output levels and prices along a continuous scale.

A best-response curve indicates the firm’s best decision, taking account the decision the firm expects its rival to take.

In the next example, two airlines Arrow and Bravo have the following demand curves for services on a particular route:

QA = 4000 – 25PA + 12PB

QB = 3000 – 20PB + 10PA

where QA and QB are the numbers of tickets, and PA and PB are the prices charged by Arrow and Bravo, respectively.

For simplicity, we assume both airlines have constant returns to scale, and their LAC and LMC curves are horizontal:

LACA = LMCA = 160

LACB = LMCB = 180

Each firm wishes to calculate the best price for it to charge, for any price it might expect its rival to charge.

Consider the calculation from Arrow’s perspective.

First, suppose Bravo charges $100. Arrow’s demand is:

QA = 4000 – 25PA + (12100) = 5200 – 25PA

Rearranging, we can write25PA = 5200 – QAorPA = 208 – 0.04QA

PA = 208 – 0.04QA is the expression for Arrow’s demand curve.

Therefore MRA = 208 – 0.08QA is the expression for Arrow’s marginal revenue curve.

Setting MRA = LMC gives 208 – 0.08QA = 160

Solving for QA givesQA = 600and PA = 208 – 0.04600 = 184

Therefore when Bravo charges $100, Arrow should charge $184.

Figure 4.1A illustrates Arrow’s profit maximizing pricing decision when Bravo’s price is PB=100.

Second, suppose Bravo charges $200. Arrow’s demand is:

QA = 4000 – 25PA + (12200) = 6400 – 25PA

Rearranging, we can write25PA = 6400 – QAorPA = 256 – 0.04QA

PA = 256 – 0.04QA is the expression for Arrow’s demand curve.

Therefore MRA = 256 – 0.08QA is the expression for Arrow’s marginal revenue curve.

Setting MRA = LMC gives 256 – 0.08QA = 160

Solving for QA givesQA = 1200and PA = 256 – 0.041200 = 208

Therefore when Bravo charges $200, Arrow should charge $208.

Figure 4.1B illustrates the construction of Arrow’s best-response curve, obtained by running a straight line through the two points (PA=184, PB=100) and (PA=208, PB=200).

Figure 4.2 shows Arrow’s and Bravo’s best-response curves on the same diagram.

Bravo’s best-response curve is obtained in the same way, by substituting some values of PA into Bravo’s demand function and solving Bravo’s profit-maximizing output and pricing decision.

The mutually best prices are those at the point of intersection, point N in Figure 4.2 with (PA=212, PB=218).

At N, neither airline can increase its profit by individually changing its price. Therefore N is a Nash equilibrium, and N has the property of strategic stability.

It can be shown that there are other points on Figure 4.2 at which both airlines earn higher profits than at N.

For example, at point C with (PA=230, PB=235), the profits of both airlines are higher than at N.

However, C is strategically unstable because both airlines have an incentive to ‘cheat’ by reducing price in order to return to apoint on their own best-response curves.

Cost-plus pricing

Survey evidence suggests not all firms use the ‘marginal revenue equals marginal cost’ rule in order to set their prices.

  • Some managers believe it is difficult or impossible to obtain reliable estimates of demand and marginal cost.
  • Some firms rely on the experience of senior executives, whose knowledge allows them to use judgement or rules-of-thumb for setting prices.

Cost-plus pricing involves setting price equal to average total cost (ATC in the short run, or LAC in the long run) plus a percentage mark-up.

P = ATC + (m  ATC) = (1 + m)ATC(short-run case)

P = LAC + (m  LAC) = (1 + m)LAC(long-run case)

where m is the mark-up.

There are some potential practical difficulties associated with cost-plus pricing:

  • ATC or LAC usually varies with output. Therefore the manager has to make some assumptions about the value of Q and the value of ATC or LAC to use in the cost-plus pricing formula.
  • The theory does not indicate how the mark-up should be determined. The manager might target a ‘fair’ profit margin, or rely on an arbitrary rule or on past experience.
  • The theory does not incorporate any consideration of demand conditions. Although this simplifies the pricing decision, it makes it impossible to find the profit maximizing price, except by pure luck.

In Figure 4.3, the manager might assume that the firm can sell Q=5000, and can charge a mark-up of 50% on ATC=20, so m=0.5.

Therefore the manager expects to incur ATC=20, and charge P=30, and earn an economic profit of (30–20)5000 = 50000.

However, this plan does not take account of demand. At P=30, the firm can only sell Q=4000, but can charge a mark-up of 66.7% on ATC=18, so m=0.667. In this case, economic profit = (30–18)4000 = 48000.

An experienced or a lucky manager might decide to sell Q=3000, and charge a mark-up of 100% on ATC=20, so m=1. In this case, economic profit = (40–20)3000 = 60000.