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Chapter 16 Game Theory and Oligopoly

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Title: Chapter 16 Game Theory and Oligopoly


1
Chapter 16 Game Theory and Oligopoly
2
Figure 16.1 The monopoly equilibrium
3
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.

4
Duopoly as a Prisoners Dilemma
  • A Duopoly is an oligopoly in which there are only
    two firms in the industry.

5
Table 16.1 Duopoly profit matrix
6
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.

7
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.

8
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.

9
Figure 16.2 Finding a Cournot best-response
function
10
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.

11
Figure 16.3 The Cournot equilibrium
12
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.

13
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.

14
Figure 16.4 Isoprofit or indifference curves
15
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.

16
Figure 16.5 Joint profit not maximized in Nash
equilibrium
17
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.

18
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.

19
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.

20
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.

21
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.

22
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.

23
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).

24
Figure 16.6 Finding a Bertrand best-response
function
25
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.

26
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.

27
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.

28
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.

29
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.

30
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.

31
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.

32
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

33
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.

34
Figure16.7 The inducement to entry
35
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

36
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.

37
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)

38
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

39
Figure 16.8 Identifying the limit price and the
limit output
40
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.

41
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.

42
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).

43
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.

44
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.
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