Oligopoly Theory


Individuals face a strategic situation if the payoff to each person depends directly on the actions chosen by other persons, and each person is aware of this fact when he chooses his action.

Game theory is a formal way of modeling strategic situations. The most widely used solution concept in game theory is the Nash equilibrium.

A set of strategies constitute a Nash equilibrium if, holding the strategies of all other players unchanged, no single player can attain a higher payoff by choosing a different strategy.

Let Si be the set of all possible strategies for player i. Let si denote an element of Si, and let s-i denote the strategies of all players other than i. The payoff to individual i is represented by yi(s1, ..., sN). The group of strategies (s1*, ..., sN*) constitute a Nash equilibrium if, for each player i,

yi(si*, s-i*) >= yi(si,s-i*) for all si in Si.

A Nash equilibrium



Let P = P(q1,q2) be the inverse demand function. Let C1(q1) and C2(q2) be the cost functions of firms 1 and 2. If (q1*, q2*) is a Nash equilibrium, then

q1* maximizes P(q1,q2*)q1 - C1(q1)
q2* maximizes P(q1*,q2)q2 - C2(q2)

These two conditions can be written in terms of two first order conditions:

P + q1*P' - C1' = 0
P + q2*P' - C2' = 0


The best response function (also known as reaction function) shows the optimal (i.e., profit-maximizing) output q1 as a function of q2. It is implicitly defined by the first order condition for firm 1:

P'(q1 + q2)q1 + P(q1 + q2) - C1'(q1) = 0.
We can write q1 = BR1(q2). Apply a similar reasoning to firm 2, we can write q2 = BR2(q1).

The Nash equilibrium can alternatively be defined as the solution to the two equations

q1 = BR1(q2).
q2 = BR2(q1).
Diagramatically, the Nash equilibrium can be represented by the intersection of the best response curves.

In the Cournot model, the best response function is often assumed to be downward sloping because an increase in other firms' output will tend to reduce the incentive for each firm to increase output. (That is, d/dqj(dyi/dqi) tends to be negative.)

When best response functions are downward sloping, we call the strategic variables strategic substitutes.

We can also draw iso-profit curves for firm 1 that shows combinations of q1 and q2 that give firm 1 equal profits. These curves have the following characteristics:

Putting the best response functions and the iso-profit curves together, we can see


1. Let

Firm 1 maximizes
(1 - q1 - q2)q1 - c1q1.
The first order conditions is
(1 - q1 - q2) - q1 - c1 = 0.
Rearrange this to
q1 = (1 - q2 - c1)/2 =BR1(q2).
Note Write a similar first order condition for firm 2:
(1 - q1 - q2) - q2 - c2 = 0.
Solve the two first order equations simultaneously to get the Nash equilibrium output:
q1* = (1 - 2c1 + c2)/3.
q2* = (1 + c1 - 2c2)/3.
Substitute these equilibrium output back into the original profit function to get the equilibrium profits
y1* = [(1 - 2c1 + c2)/3]2.
y2* = [(1 + c1 - 2c2)/3]2.
Notice that firm 1's output and profits depends (positively) on firm 2's cost.

2. Let

Consider a typical firm i. Firm i maximizes

(1 - q1 - ... - qn)qi - cqi.
The first order condition is
(1 - q1 - ... - qn) - qi - c = 0.
In a symmetric equilibrium, q1 = ... = qn = q. We can then rewrite the first order condition in the form:
1 - nq - q - c = 0.
(Note it is important to impose symmetry only after deriving the first order condition.) Solve the above to get
q = (1 - c)/(n + 1).
Q = nq = n(1 - c)/(n + 1)
P = 1 - Q = (1 + nc)/(n + 1)


Consider a two-stage game where firm 1 (the leader) chooses its output level q1 in the first stage. In stage 2, firm 2 (the follower) chooses its output level q2, knowing what q1 is.

The Stackelberg leader is implicitly assumed to have some ability to commit to its chosen output level q1. It will not change q1 after knowing what q2 is, even though such a change might be profitable ex post.

To solve the Stackelberg game, we proceed backwards. In stage 2, firm 2 maximizes profits taking q1 as given. The optimal q2 is given by

q2 = BR2(q1).
This BR function is just like the one in the Cournot model.

In stage 1, firm 1 chooses q1 to maximize its profits, taking into account firm 2's optimal response:

maximize P(q1 + BR2(q1))q1 - C1(q1).
The first order condition is
(1 + BR'2)P'q1 + P = MC1.

Compare with the Cournot outcome, where P'q1 + P = MC1, we can see that

q1S > q1C.
Moreover, since BR'2 < 0,
q2S = BR2(q1S) < BR2(q1C) = q2C.


As in Example 1 in the section on Cournot model, Firm 2's best response function is

BR2(q1 = (1 - q1 - c2)/2.

Firm 1 maximizes

(1 - q1 - (1 - q1 - c2)/2)q1 - c1q1.
The solution gives
q1S = (1 + c2 - 2c1)/2.
Substitute this back into the best response function for firm 2, we get
q2S = (1 - 3c2 + 2c1)/4.


The Bertrand model can be interpreted as a game in which prices are the strategic variables.

Example 1.


Let D(p) be the market demand function. Then the demand function for firm 1 is D1(p1,p2)=

D(p1) if p1 < p2;
D(p1)/2 if p1 = p2;
0 if p1 > p2.
Notice that this demand function is discontinuous.

Example 2.


Suppose the demand function takes the form D1 = a - bp1 + dp2. Then profits for firm 1 is

y1 = (a - bp1 + dp2)p1 - C1(a - bp1 + dp2)
We have The incentive the raise p1 is higher when p2 is higher. This suggests that the best response function is upward sloping.

When the best response function is upward sloping, we call the strategic variables strategic complements.

In a Bertrand model with differentiated goods


The conjectural variations model cannot be interpreted as the Nash equilibrium of a game. Yet the model is useful for describing the degree of monopoly power in oligopolistic markets.

Consider a duopoly with homogeneous goods. Let P = P(q1 + q2). Define a parameter Lij to be i's conjecture of j's output response when qi increases. I.e., Lij = dqj/dqi.

Firm i maximizes P(qi + qj)qi - Ci(qi). The first order condition is:

P'qi(1 + Lij) + P - C'i = 0.

Another possibility is to impose the condition that Lij has to be equal to the equilibrium dqj/dqi. This is known as the consistent conjectural variations model.

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