© 2009 Pearson Education Canada 16/1 Chapter 16 Game Theory and Oligopoly.

© 2009 Pearson Education Canada16/1

Chapter 16Chapter 16

Game Theory and OligopolyGame Theory and Oligopoly


Figure 16.1 The monopoly equilibriumFigure 16.1 The monopoly equilibrium


The monopoly equilibriumThe monopoly equilibrium

This chapter uses the following linear market This chapter uses the following linear market demand curve: demand curve: p=100-yp=100-y

Assume that each firm in the industry has a Assume that each firm in the industry has a marginal/average/unit costmarginal/average/unit cost of $40. of $40.

MR= 100-2yMR= 100-2y Profits are maximized by charging the price Profits are maximized by charging the price

associated with the optimal level of output -associated with the optimal level of output -the level of output where MR=MC.the level of output where MR=MC.

Total profits (TP) = TR – TC.Total profits (TP) = TR – TC.


Duopoly as a Prisoner’s DilemmaDuopoly as a Prisoner’s Dilemma

A Duopoly is an oligopoly in which A Duopoly is an oligopoly in which there are only two firms in the there are only two firms in the industry.industry.


Table 16.1 Duopoly profit matrixTable 16.1 Duopoly profit matrix


From Table 16.1From Table 16.1

L L is the dominant strategy for both is the dominant strategy for both the First and the Second Firmthe First and the Second Firm

Thus, the Nash-equilibrium Thus, the Nash-equilibrium combination is (combination is (L, LL, L) in which both ) in which both firms produce 20 units and have a firms produce 20 units and have a profit of $200.profit of $200.

Yet, if they could agree to restrict Yet, if they could agree to restrict their individual outputs to 15 units a their individual outputs to 15 units a piece, each could earn $450.piece, each could earn $450.


The Oligopoly ProblemThe Oligopoly Problem

Oligopolists have a clear incentive to Oligopolists have a clear incentive to collude or cooperate.collude or cooperate.

Oligopolists have a clear incentive to Oligopolists have a clear incentive to cheat on any simple collusive or cheat on any simple collusive or cooperative agreement.cooperative agreement.

If an agreement is not a Nash If an agreement is not a Nash equilibrium, it is not self-enforcing.equilibrium, it is not self-enforcing.


The Cournot Duopoly ModelThe Cournot Duopoly Model

Central features of the Central features of the Cournot ModelCournot Model::

1.1. Each firm chooses a quantity of output Each firm chooses a quantity of output instead of a price.instead of a price.

2.2. In choosing an output, each firm takes In choosing an output, each firm takes its rival’s output as given.its rival’s output as given.


Figure 16.2 Finding a Cournot Figure 16.2 Finding a Cournot best-response functionbest-response function


From Figure 16.2 From Figure 16.2

The First firm’s best response The First firm’s best response function is: function is: yy11

**=30 – y=30 – y22/2/2 The Second firm’s best response The Second firm’s best response

function is function is yy22**=30 – y=30 – y11/2/2

Taken together, these two best Taken together, these two best response functions can be used to response functions can be used to find the find the equilibrium strategy equilibrium strategy combination combination for Cournot’s model. for Cournot’s model.


Figure 16.3 The Cournot equilibriumFigure 16.3 The Cournot equilibrium


The Cournot Model: Key AssumptionsThe Cournot Model: Key Assumptions

The profit of one firm decreases as the The profit of one firm decreases as the output of the other firm increases (other output of the other firm increases (other things being equal).things being equal).

The Nash equilibrium output for each firm The Nash equilibrium output for each firm is positive.is positive.


Isoprofit CurvesIsoprofit Curves

All strategy combinations that give All strategy combinations that give the first firm the chosen level of the first firm the chosen level of profits is known as an indifference profits is known as an indifference curve or curve or isoprofit curveisoprofit curve. .

Profits are constant along the Profits are constant along the isoprofit curve.isoprofit curve.


Figure 16.4 Isoprofit or indifference curvesFigure 16.4 Isoprofit or indifference curves


From Figure 16.4From Figure 16.4

yy11** maximizes profits for the first firm, maximizes profits for the first firm, given the second firm’s output of given the second firm’s output of yy22*.*.

Any strategy combinations below the Any strategy combinations below the isocost curve gives the first firm more isocost curve gives the first firm more profit than the Nash equilibrium.profit than the Nash equilibrium.

The result above relates to the key The result above relates to the key assumption that the first firm’s profit assumption that the first firm’s profit increases as the second firm’s output increases as the second firm’s output decreases.decreases.


Figure 16.5 Joint profit not Figure 16.5 Joint profit not maximized in Nash equilibriummaximized in Nash equilibrium


Cournot’s Model: ConclusionsCournot’s Model: Conclusions

In the Nash equilibrium of this In the Nash equilibrium of this general version of the Cournot general version of the Cournot model, firms fail to maximize their model, firms fail to maximize their joint profit.joint profit.

Relative to joint profit maximization, Relative to joint profit maximization, firms produce too much output in the firms produce too much output in the Nash equilibrium.Nash equilibrium.


The Cournot Model with Many FirmsThe Cournot Model with Many Firms

With only one firm in the market, the With only one firm in the market, the Cournot-Nash equilibrium is the monopoly Cournot-Nash equilibrium is the monopoly equilibrium. equilibrium.

As the number of firms increase, output As the number of firms increase, output increases. As a result, price and aggregate increases. As a result, price and aggregate oligopoly profits decrease.oligopoly profits decrease.

When there are infinitely many firms, the When there are infinitely many firms, the Cournot model is, in effect, the perfectly Cournot model is, in effect, the perfectly competitive model.competitive model.


The Cournot Model with ComplimentsThe Cournot Model with Compliments

The Cournot-Nash equilibrium in The Cournot-Nash equilibrium in which firms produce the same good which firms produce the same good is not Pareto-optimal, as the firms is not Pareto-optimal, as the firms produced too much.produced too much.

The Cournot-Nash equilibrium in The Cournot-Nash equilibrium in which firms produce complements is which firms produce complements is not Pareto-optimal, as the firms not Pareto-optimal, as the firms produced too little.produced too little.


The Bertrand ModelThe Bertrand Model

The The Bertrand model substitutes prices for substitutes prices for quantities as the variables to be chosen.quantities as the variables to be chosen.

The goal is to find the Nash (the Bertrand-The goal is to find the Nash (the Bertrand-Nash) equilibrium strategy combination Nash) equilibrium strategy combination when firms choose prices instead of when firms choose prices instead of quantities.quantities.


The Bertrand Model: Firm’s Best The Bertrand Model: Firm’s Best Response FunctionResponse Function

Finding the best response function entails Finding the best response function entails answering the question: Given answering the question: Given pp22, what value , what value of of pp11 maximizes the first firm’s profit. maximizes the first firm’s profit.

Four possibilities exist:Four possibilities exist:

1.1. If its rival charges a price greater than the If its rival charges a price greater than the monopoly price (monopoly price (MPMP), the first firm’s best ), the first firm’s best response is to charge a lower price (than response is to charge a lower price (than MPMP) ) so it can capture the entire market. so it can capture the entire market.



2.2. If its rival charges a price less than the per If its rival charges a price less than the per unit cost of production (unit cost of production (pp22), the first firm’s ), the first firm’s best response is to choose any price greater best response is to choose any price greater than this because firm one will attract no than this because firm one will attract no business and incur a zero profit. This outcome business and incur a zero profit. This outcome is superior to matching or undercutting is superior to matching or undercutting pp22, , and posting losses.and posting losses.



33. . If the second firm’s price is greater than the If the second firm’s price is greater than the per unit cost of production and less than the per unit cost of production and less than the monopoly price.monopoly price.

– If If pp11< < pp22,, the first firm captures the entire market the first firm captures the entire market and its profits increase as its price increases.and its profits increase as its price increases.

– When When pp11= = pp22, the two firms split the profit., the two firms split the profit.

– When pWhen p11> > pp22, the first firm’s profit is zero because , the first firm’s profit is zero because it sells nothing when its price exceeds the second it sells nothing when its price exceeds the second firm’s price.firm’s price.

(see Figure 16.6).(see Figure 16.6).


Figure 16.6 Finding a Bertrand Figure 16.6 Finding a Bertrand best-response functionbest-response function



44. . Suppose the second firm sets its price exactly Suppose the second firm sets its price exactly equal to the per unit costs.equal to the per unit costs.

Then if the first firm sets a lower price it will Then if the first firm sets a lower price it will incur a loss on every unit it sells and profits incur a loss on every unit it sells and profits will be negative. If the first firm sets a price will be negative. If the first firm sets a price above the per unit, it will sell no units and above the per unit, it will sell no units and profits are zero. If the first firm sets price profits are zero. If the first firm sets price equal to the per unit costs, it breaks evenequal to the per unit costs, it breaks even..


The Bertrand-Nash EquilibriumThe Bertrand-Nash Equilibrium

The Bertrand-Nash equilibrium strategy The Bertrand-Nash equilibrium strategy combination has the second firm and the first combination has the second firm and the first firm charging a price equal to the per unit cost firm charging a price equal to the per unit cost of production.of production.

At this equilibrium, each firm’s profit is exactly At this equilibrium, each firm’s profit is exactly zero.zero.


The Collusive Model of OligopolyThe Collusive Model of Oligopoly

The collusive model of oligopoly is when The collusive model of oligopoly is when oligopolists decide to collude on a joint oligopolists decide to collude on a joint strategy.strategy.

In the Cournot and Bertrand models, the In the Cournot and Bertrand models, the equilibriums are individually rational but equilibriums are individually rational but collectively irrational, as firms have a clear collectively irrational, as firms have a clear incentive to collude.incentive to collude.

However, if firms do manage to form a However, if firms do manage to form a collusive agreement, there is a clear private collusive agreement, there is a clear private incentive for each party to cheat.incentive for each party to cheat.


The Collusive Model of OligopolyThe Collusive Model of Oligopoly In the Cournot model, the individual incentive In the Cournot model, the individual incentive

to cheat on the collusive agreement increases to cheat on the collusive agreement increases as the number of parties to the agreement as the number of parties to the agreement increases.increases.

This means that the larger the number of firms This means that the larger the number of firms in an industry, the less likely is a collusive in an industry, the less likely is a collusive equilibrium.equilibrium.

If the number of firms is large enough, some If the number of firms is large enough, some firm or firms will succumb to the temptation to firm or firms will succumb to the temptation to cheat, thereby destroying the collusive cheat, thereby destroying the collusive agreement. agreement.


Experimental EvidenceExperimental Evidence

Taken together, experiments suggest Taken together, experiments suggest that no single model is applicable to that no single model is applicable to all oligopoly situations.all oligopoly situations.

Perhaps the most economists can Perhaps the most economists can hope for is a selection of oligopoly hope for is a selection of oligopoly models, each applicable to a models, each applicable to a particular range of economic particular range of economic circumstances.circumstances.


The Limited-Output ModelThe Limited-Output Model

In the long run, the number of firms In the long run, the number of firms (market structure) is (market structure) is endogenous.endogenous.

The number of firms in an industry is The number of firms in an industry is determined by economic considerations.determined by economic considerations.

The key process in determining the long-The key process in determining the long-run equilibrium is the run equilibrium is the possibility of entry.possibility of entry.


The Limited Output ModelThe Limited Output Model

Limited output modelsLimited output models or or limited limited price modelsprice models focus on the theory of focus on the theory of the oligopoly in the long run, where the the oligopoly in the long run, where the number of firms is determined number of firms is determined endogenously and there is the endogenously and there is the possibility of entry.possibility of entry.


Barriers to EntryBarriers to Entry

A natural barrier to entry isA natural barrier to entry is setup costs. setup costs. Assume all firms incur setup costs of Assume all firms incur setup costs of $S$S In any period, the rate of interest In any period, the rate of interest (i)(i)

determines the set up cost determines the set up cost (K):K=iS(K):K=iS Adding fixed costs to variable costs (40Adding fixed costs to variable costs (40yy) )

gives total cost function: gives total cost function:

C(y)=K+40YC(y)=K+40Y


Inducement to EntryInducement to Entry

If the fixed costs (K) are a barrier to If the fixed costs (K) are a barrier to entry, what is entry, what is an inducement to an inducement to entry?entry?

An inducement to entry is the excess An inducement to entry is the excess of revenue over variable costs. of revenue over variable costs.


Figure16.7 The inducement to entryFigure16.7 The inducement to entry


Inducement to Entry Inducement to Entry

The entrant’s best response function The entrant’s best response function is: is: yyEE

**=30-y/2=30-y/2 The entrant’s residual demand The entrant’s residual demand

function is: function is: PPee=(100-y)-y=(100-y)-yee

The price that will prevail if the The price that will prevail if the entrant produces entrant produces yyee** units is: units is: PPee*=70-*=70-y/2y/2

Profit per unit is: Profit per unit is: PPee* - 40=30-y/2* - 40=30-y/2



The inducement to entry, The inducement to entry, yyee* * timestimes (p (pee*-*-40) is then (30-y/)40) is then (30-y/)22..

This expression gives the revenue over This expression gives the revenue over variable costs that the entrant would earn variable costs that the entrant would earn if established firms continued to produce if established firms continued to produce yy units after entry. units after entry.

Entry will occur if inducement to enter Entry will occur if inducement to enter exceeds exceeds K.K.



Call the smallest value of Call the smallest value of y,y, such that no such that no entry occurs, the entry occurs, the limit outputlimit output (y (yLL).).

((30-y30-yLL/2)/2)22=K=K

Solving for YSolving for YLL: Y: YLL= 60-2K= 60-2K1/21/2

If If KK=$100, =$100, YYLL=40 units, If =40 units, If KK=$225, =$225, YYLL=30 =30 units, etc. units, etc.

(see Figure 16.8)(see Figure 16.8)



The The no entry conditionno entry condition says entry says entry will not occur if the output of will not occur if the output of established firms is greater than or established firms is greater than or equal to the limit output (equal to the limit output (yyLL))

The The limit price (plimit price (pLL)) is the price is the price associated with the associated with the limit outputlimit output..

In this example:In this example:

ppLL=100-y=100-yLL or or ppLL= 40+2K= 40+2K1/21/2


Figure 16.8 Identifying the limit Figure 16.8 Identifying the limit price and the limit outputprice and the limit output


Strategic Choice of Industry OutputStrategic Choice of Industry Output

The existing level of industry output The existing level of industry output (y)(y) and development costs and development costs (K)(K) are are barriers to entry. barriers to entry.

If If yy is less than the limit output is less than the limit output yyLL, , the firm will enter the industry.the firm will enter the industry.

If If yy is equal to or more than the limit is equal to or more than the limit output output yyLL, the firm will not enter the , the firm will not enter the industry.industry.



We have calculated that if K=$225, We have calculated that if K=$225, then then yyLL=30 (the monopoly output).=30 (the monopoly output).

Thus, if setup costs are $225 or Thus, if setup costs are $225 or higher, the monopoly output of 30 higher, the monopoly output of 30 will successfully deter entrywill successfully deter entry – a – a natural monopoly scenario.natural monopoly scenario.



If KIf K< $225< $225, , the ordinary monopolist the ordinary monopolist output will not deter entry (output will not deter entry (yyLL>30).>30).

In this case the monopolist will In this case the monopolist will produce exactly yproduce exactly yL L units of output.units of output.

Since it has already incurred the Since it has already incurred the setup cost, its objective is to setup cost, its objective is to maximize revenues over variable maximize revenues over variable costs (costs (gross profitsgross profits).).


Critique of the ModelCritique of the Model

The postulate that entrants take the The postulate that entrants take the current industry output as a given is current industry output as a given is the major weakness of the limited-the major weakness of the limited-output model.output model.

A potential entrant’s concern is not A potential entrant’s concern is not with present but the future output of with present but the future output of the the sitting (currently in the industry) sitting (currently in the industry) monopolistmonopolist..


Critique of the ModelCritique of the Model

When a sitting monopolist produces the When a sitting monopolist produces the limit output, its decision is intended as a limit output, its decision is intended as a credible warning to potential entrants credible warning to potential entrants that it will continue to produce the limit that it will continue to produce the limit output in the future.output in the future.

If entrants take this warning seriously, If entrants take this warning seriously, they will stay out of the market.they will stay out of the market.

© 2009 Pearson Education Canada 16/1 Chapter 16 Game Theory and Oligopoly.

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