© 2005 Pearson Education Canada Inc Chapter 16 Game Theory and Oligopoly

© 2005 Pearson Education Canada Inc Figure 16.1 The monopoly equilibrium

© 2005 Pearson Education Canada Inc Duopoly as a Prisoner’s Dilemma  A Duopoly is an oligopoly in which there are only two firms in the industry.

© 2005 Pearson Education Canada Inc Table 16.1 Duopoly profit matrix

© 2005 Pearson Education Canada Inc From Table 16.1  L is the dominant strategy for the both the First and the Second Firm  Thus the Nash-equilibrium combination is (L,L) in which both firms produce 20 units and have a profit of $200.  Yet, if they could agree to restrict their individual outputs to 15 units apiece, each could earn $450.

© 2005 Pearson Education Canada Inc The Oligopoly Problem  Oligopolists have a clear incentive to collude or cooperate.  Oligopolists have a clear incentive to cheat on any simple collusive or cooperative agreement.  If an agreement is not a Nash equilibrium, it is not self-enforcing.

© 2005 Pearson Education Canada Inc The Cournot Duopoly Model  Central features of the Cournot Model: 1. Each firm chooses a quantity of output instead of a price. 2. In choosing an output, each firm takes its rival’s output as given.

© 2005 Pearson Education Canada Inc Figure 16.2 Finding a Cournot best-response function

© 2005 Pearson Education Canada Inc From Figure 16.2  The First firm’s best response function is: y 1 * =30 – y 2 /2  The Second firm’s best response function is y 2 * =30 – y 1 /2  Taken together, these two best response functions can be used to find the equilibrium strategy combination for Cournot’s model.

© 2005 Pearson Education Canada Inc Figure 16.3 The Cournot equilibrium

© 2005 Pearson Education Canada Inc The Cournot Model: Key Assumptions  The profit of one firm decreases as the output of the other firm increases (other things equal).  The Nash equilibrium output for each firm is positive.

© 2005 Pearson Education Canada Inc Isoprofit Curves  All strategy combinations that give the first firm the chosen level of profits is known as an indifference curve or iosprofit curve.  Profits are constant along the isoprofit curve.

© 2005 Pearson Education Canada Inc Figure 16.4 Title

© 2005 Pearson Education Canada Inc From Figure 16.4  y 1 * maximizes profits for the first firm given the second firm’s output of y 2 *.  Any strategy combinations below the indifference curve gives the first firm more profit than the Nash equilibrium.  The result above relates to the key assumption that the first firm’s profit increases as the second firm’s output decreases.

© 2005 Pearson Education Canada Inc Figure 16.5 Joint profit not maximized in Nash equilibrium

© 2005 Pearson Education Canada Inc Cournot’s Model: Conclusions  In the Nash equilibrium of this general version of the Cournot model, firms fail to maximize their joint profit.  Relative to joint profit maximization, firms produce too much output in the Nash equilibrium.

© 2005 Pearson Education Canada Inc The Cournot Model with Many Firms  With only one firm in the market, the Cournot-Nash equilibrium is the monopoly equilibrium.  As the number of firms increases, output increases. As a result, price and aggregate oligopoly profits decrease.  When there are infinitely many firms, the Cournot model is, in effect, the perfectly competitive model.

© 2005 Pearson Education Canada Inc The Bertrand Model  The Bertrand model substitutes prices for quantities as the variables to be chosen.  The goal is to find the Nash (the Bertrand-Nash) equilibrium strategy combination when firms choose prices instead of quantities.

© 2005 Pearson Education Canada Inc The Bertrand Model: Firm’s Best Response Function  Funding the best response function entails answering the question: Given p 2, what value of p 1 maximizes the first firm’s profit.  Four possibilities exist: 1.If its rival charges a price greater than the monopoly price (MP), the first firm’s best response is to charge a lower price (than MP) so it can capture the entire market.

© 2005 Pearson Education Canada Inc The Bertrand Model: Firm’s Best Response Function 2.If its rival charges a price less than the per unit cost of production (p 2 ), the first firm’s best response is to choose any price greater than this because firm one will attract no business and incur a zero profit. This outcome is superior to matching or undercutting p 2, and posting losses.

© 2005 Pearson Education Canada Inc The Bertrand Model: Firm’s Best Response Function 3. If the second firm’s price is greater than the per unit cost of production and less than the monopoly price. (see Figure 16.6)

© 2005 Pearson Education Canada Inc Figure 16.6 Finding a Bertrand best-response function

© 2005 Pearson Education Canada Inc The Bertrand Model: Firm’s Best Response Function 4. Suppose the second firm sets its price exactly equal to the per unit costs. Then if the first firm sets a lower price it will incur a loss on every unit it sells and profits will be negative. If the first firm sets a price above the per unit it will sell no units and profits are zero. If the first firm sets price equal to the per unit costs, it breaks even. Then if the first firm sets a lower price it will incur a loss on every unit it sells and profits will be negative. If the first firm sets a price above the per unit it will sell no units and profits are zero. If the first firm sets price equal to the per unit costs, it breaks even.

© 2005 Pearson Education Canada Inc The Bertrand-Nash Equilibrium  The Bertrand-Nash equilibrium strategy combination is the second firm and the first firm charging a price equal to the per unit cost of production.  At this equilibrium, each firm’s profit is exactly zero.

© 2005 Pearson Education Canada Inc The Limited-Output Model  In the long run, the number of firms (market structure ) is endogenous.  The number of firms is an industry is determined by economic considerations.  The key process in determining the long- run equilibrium is the possibility of entry.

© 2005 Pearson Education Canada Inc Barriers to Entry  A natural barrier to entry is setup costs.  Assume all firms incur setup costs of $S  In any period, the rate of interest (i) determines the set up cost (K):K=iS  Adding fixed costs to variable costs (40y) gives total cost function: C(y)=K+40Y

© 2005 Pearson Education Canada Inc Inducement to Entry  If the fixed costs (K) is a barrier to entry, what is an inducement to entry?  An inducement to entry is the excess of revenue over variable costs.

© 2005 Pearson Education Canada Inc Figure16.7 The inducement to entry

© 2005 Pearson Education Canada Inc Inducement to Entry  The entrant’s best response function is: y E * =30-y/2  The entrant’s residual demand function is: P e =(100-y)-y e  The price that will prevail if the entrant produces y e * units is: P e *=70-y/2  Profit per unit is: P e * - 40=30-y/2

© 2005 Pearson Education Canada Inc Inducement to Entry  The inducement to entry, y e * times (p e *- 40) is then (30-y/) 2.  This expression gives the revenue over variable costs that the entrant would earn if established firms continued to produce y units after entry.  Entry will occur if inducement to enter exceeds K

© 2005 Pearson Education Canada Inc Inducement to Entry  Call the smallest value of y such that no entry occurs the limit output (y L ).  (30-y L /2) 2 =K  Solving for Y L : Y L = 60-2K 1/2  If K=$100, Y L =40 units, If K=$225, Y L =30 units, etc. (see Figure 16.8)

© 2005 Pearson Education Canada Inc Inducement to Entry  Entry will not occur if the output of established firms is greater than or equal to the limit output (y L )  The limit price (p L ) is the price associated with the limit output.  In this example: p L =100-y L or p L = 40+2K 1/2 p L =100-y L or p L = 40+2K 1/2

© 2005 Pearson Education Canada Inc Figure 16.8 Identifying the limit price and the limit output

© 2005 Pearson Education Canada Inc Refinements of Limited Output  How large must the fixed cost K be so that a third firm will not enter?  The generalized no-entry condition for the Cournot models is then: [60/(n=2) 2 ]≤K

© 2005 Pearson Education Canada Inc Figure 16.9 Cournot oligopoly and entry equilibrium

© 2005 Pearson Education Canada Inc Barriers to entry  Development cost K is a barrier to entry, as it differentiates established firms and new potential entrants.  The manner in which this differentiation affects the inducement to enter (profits) depends upon the nature of the oligopoly behaviour upon entry.

© 2005 Pearson Education Canada Inc Barriers to entry  The more aggressive/less cooperative is oligopoly behaviour upon entry, the more effective setup costs are as a barrier.  Any firm’s decision to incur the setup cost is a strategic decision because it affects the incentives of other firms.

© 2005 Pearson Education Canada Inc Positioning and Reacting  Positioning is concerned with action taken by existing firms prior to entry.  Reacting refers to actions of established firms subsequent to entry.