Chapter 16 Game Theory and Oligopoly

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Chapter 16Game Theory and Oligopoly
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Figure 16.1 The monopoly equilibrium
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The monopoly equilibrium

• This chapter uses the following linear market
  demand curve p100-y
• Assume that each firm in the industry has a
  marginal/average/unit cost of 40.
• MR 100-2y
• Profits are maximized by charging the price
  associated with the optimal level of output -the
  level of output where MRMC.
• Total profits (TP) TR TC.

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Duopoly as a Prisoners Dilemma

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

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Table 16.1 Duopoly profit matrix
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From Table 16.1

• L is the dominant strategy for 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 a piece, each
  could earn 450.

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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.

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The Cournot Duopoly Model

• Central features of the Cournot Model
• Each firm chooses a quantity of output instead of
  a price.
• In choosing an output, each firm takes its
  rivals output as given.

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

• The First firms best response function is
  y130 y2/2
• The Second firms best response function is
  y230 y1/2
• Taken together, these two best response functions
  can be used to find the equilibrium strategy
  combination for Cournots model.

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Figure 16.3 The Cournot equilibrium
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The Cournot Model Key Assumptions

• The profit of one firm decreases as the output of
  the other firm increases (other things being
  equal).
• The Nash equilibrium output for each firm is
  positive.

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Isoprofit Curves

• All strategy combinations that give the first
  firm the chosen level of profits is known as an
  indifference curve or isoprofit curve.
• Profits are constant along the isoprofit curve.

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Figure 16.4 Isoprofit or indifference curves
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From Figure 16.4

• y1 maximizes profits for the first firm, given
  the second firms output of y2.
• Any strategy combinations below the isocost curve
  gives the first firm more profit than the Nash
  equilibrium.
• The result above relates to the key assumption
  that the first firms profit increases as the
  second firms output decreases.

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Figure 16.5 Joint profit not maximized in Nash
equilibrium
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Cournots 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.

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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 increase, 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.

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The Cournot Model with Compliments

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

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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.

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The Bertrand Model Firms Best Response Function

• Finding the best response function entails
  answering the question Given p2, what value of
  p1 maximizes the first firms profit.
• Four possibilities exist
• 1. If its rival charges a price greater than the
  monopoly price (MP), the first firms best
  response is to charge a lower price (than MP) so
  it can capture the entire market.

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The Bertrand Model Firms Best Response Function

• 2. If its rival charges a price less than the per
  unit cost of production (p2), the first firms
  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 p2, and posting
  losses.

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The Bertrand Model Firms Best Response Function

• 3. If the second firms price is greater than the
  per unit cost of production and less than the
  monopoly price.
• If p1lt p2, the first firm captures the entire
  market and its profits increase as its price
  increases.
• When p1 p2, the two firms split the profit.
• When p1gt p2, the first firms profit is zero
  because it sells nothing when its price exceeds
  the second firms price.
• (see Figure 16.6).

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Figure 16.6 Finding a Bertrand best-response
function
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The Bertrand Model Firms 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.

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The Bertrand-Nash Equilibrium

• The Bertrand-Nash equilibrium strategy
  combination has the second firm and the first
  firm charging a price equal to the per unit cost
  of production.
• At this equilibrium, each firms profit is
  exactly zero.

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The Collusive Model of Oligopoly

• The collusive model of oligopoly is when
  oligopolists decide to collude on a joint
  strategy.
• In the Cournot and Bertrand models, the
  equilibriums are individually rational but
  collectively irrational, as firms have a clear
  incentive to collude.
• However, if firms do manage to form a collusive
  agreement, there is a clear private incentive for
  each party to cheat.

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The Collusive Model of Oligopoly

• In the Cournot model, the individual incentive to
  cheat on the collusive agreement increases as the
  number of parties to the agreement increases.
• This means that the larger the number of firms in
  an industry, the less likely is a collusive
  equilibrium.
• If the number of firms is large enough, some firm
  or firms will succumb to the temptation to cheat,
  thereby destroying the collusive agreement.

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Experimental Evidence

• Taken together, experiments suggest that no
  single model is applicable to all oligopoly
  situations.
• Perhaps the most economists can hope for is a
  selection of oligopoly models, each applicable to
  a particular range of economic circumstances.

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The Limited-Output Model

• In the long run, the number of firms (market
  structure) is endogenous.
• The number of firms in an industry is determined
  by economic considerations.
• The key process in determining the long-run
  equilibrium is the possibility of entry.

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The Limited Output Model

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

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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)KiS
• Adding fixed costs to variable costs (40y) gives
  total cost function
• C(y)K40Y

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Inducement to Entry

• If the fixed costs (K) are a barrier to entry,
  what is an inducement to entry?
• An inducement to entry is the excess of revenue
  over variable costs.

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Figure16.7 The inducement to entry
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Inducement to Entry

• The entrants best response function is
  yE30-y/2
• The entrants residual demand function is
  Pe(100-y)-ye
• The price that will prevail if the entrant
  produces ye units is Pe70-y/2
• Profit per unit is Pe - 4030-y/2

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Inducement to Entry

• The inducement to entry, ye times (pe-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.

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Inducement to Entry

• Call the smallest value of y, such that no entry
  occurs, the limit output (yL).
• (30-yL/2)2K
• Solving for YL YL 60-2K1/2
• If K100, YL40 units, If K225, YL30 units,
  etc.
• (see Figure 16.8)

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Inducement to Entry

• The no entry condition says entry will not occur
  if the output of established firms is greater
  than or equal to the limit output (yL)
• The limit price (pL) is the price associated with
  the limit output.
• In this example
• pL100-yL or pL 402K1/2

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Figure 16.8 Identifying the limit price and the
limit output
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Strategic Choice of Industry Output

• The existing level of industry output (y) and
  development costs (K) are barriers to entry.
• If y is less than the limit output yL, the firm
  will enter the industry.
• If y is equal to or more than the limit output
  yL, the firm will not enter the industry.

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Strategic Choice of Industry Output

• We have calculated that if K225, then yL30
  (the monopoly output).
• Thus, if setup costs are 225 or higher, the
  monopoly output of 30 will successfully deter
  entry a natural monopoly scenario.

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Strategic Choice of Industry Output

• If Klt 225, the ordinary monopolist output will
  not deter entry (yLgt30).
• In this case the monopolist will produce exactly
  yL units of output.
• Since it has already incurred the setup cost, its
  objective is to maximize revenues over variable
  costs (gross profits).

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Critique of the Model

• The postulate that entrants take the current
  industry output as a given is the major weakness
  of the limited-output model.
• A potential entrants concern is not with present
  but the future output of the sitting (currently
  in the industry) monopolist.

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Critique of the Model

• When a sitting monopolist produces the limit
  output, its decision is intended as a credible
  warning to potential entrants that it will
  continue to produce the limit output in the
  future.
• If entrants take this warning seriously, they
  will stay out of the market.