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Chapter 16 Game Theory and Oligopoly

Figure 16.1 The monopoly equilibrium

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.

Duopoly as a Prisoners Dilemma

- A Duopoly is an oligopoly in which there are only

two firms in the industry.

Table 16.1 Duopoly profit matrix

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.

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.

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.

Figure 16.2 Finding a Cournot best-response

function

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.

Figure 16.3 The Cournot equilibrium

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.

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.

Figure 16.4 Isoprofit or indifference curves

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.

Figure 16.5 Joint profit not maximized in Nash

equilibrium

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.

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.

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.

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.

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.

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.

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).

Figure 16.6 Finding a Bertrand best-response

function

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.

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.

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.

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.

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.

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.

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.

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

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.

Figure16.7 The inducement to entry

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

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.

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)

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

Figure 16.8 Identifying the limit price and the

limit output

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.

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.

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).

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.

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.