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
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
sN). The group of strategies
sN*) constitute a Nash
equilibrium if, for each player i,
for all si in Si.
A Nash equilibrium
- need not be Pareto efficient
- is "self-enforcing"
- need not be unique
- need not exist in pure strategies
- quantity is the strategic variable (Nash equilibrium in quantities)
- price clears the market
- homogeneous goods (not essential)
Let P = P(q1,q2) be the
inverse demand function. Let
C2(q2) be the cost
functions of firms 1 and 2. If
q2*) is a Nash
These two conditions can be written in terms of two first order conditions:
P + q1*P' -
C1' = 0
P + q2*P' -
C2' = 0
- price exceeds marginal costs
- marginal costs need not be equalized across firms, i.e., production is
- compare the monopoly pricing equation:
P + QMP' -
C' = 0
Oligopolists only take into partial account the
effect of their output expansion on output price. Therefore equilibrium
price is lower and equilibrium quantity is higher than the monopoly solution.
- we can re-arrange the first order conditions to get
(P - MCi)/P = ai/e
where ai is the market share of firm i and e is
the industry's elasticity of demand.
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:
C1'(q1) = 0.
We can write q1 =
BR1(q2). Apply a
similar reasoning to firm 2, we can write q2 =
The Nash equilibrium can alternatively be defined as the solution to the two
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,
tends to be negative.)
When best response functions are downward sloping, we call the strategic
variables strategic substitutes.
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:
- the preference direction is downwards
- the isoprofit curve has an inverted U shape and it reaches a maximum on
the BR1 curve
Putting the best response functions and the iso-profit curves together, we
- there is potential for profitable collusion
- there are enforcement problems associated with collusion
Firm 1 maximizes
(1 - q1 - q2)q1 -
The first order conditions is
(1 - q1 - q2) - q1 -
c1 = 0.
Rearrange this to
(1 - q2 - c1)/2
- D = 1 - P,
- C1 = c1q1,
- C2 = c2q2,
Write a similar first order condition for firm 2:
(1 - q1 - q2) -
c2 = 0.
Solve the two first order equations simultaneously to get the Nash
= (1 - 2c1 + c2)/3.
= (1 + c1 - 2c2)/3.
Substitute these equilibrium output back into the original profit function
to get the equilibrium profits
= [(1 - 2c1 +
= [(1 + c1 -
Notice that firm 1's output and profits depends (positively) on firm 2's
- the BR function is downward sloping
- the BR function will shift out when marginal cost falls
- D = 1 - P
- n firms, each with the same marginal cost = c
Consider a typical firm i. Firm i maximizes
(1 - q1 - ... -
The first order condition is
(1 - q1 - ... - qn) -
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)
- dP/dn < 0
- P approaches c when n approaches infinity
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
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 +
The first order condition is
(1 + BR'2)P'q1 + P =
Compare with the Cournot outcome, where P'q1 + P =
MC1, we can see that
Moreover, since BR'2 < 0,
- D = 1 - P
- C1 =
- C2 =
As in Example 1 in the section on Cournot model, Firm 2's best response
BR2(q1 = (1 -
Firm 1 maximizes
(1 - q1 - (1 -
The solution gives
(1 + c2 - 2c1)/2.
Substitute this back into the best response function for firm 2, we get
(1 - 3c2 + 2c1)/4.
The Bertrand model can be interpreted as a game in which prices are
the strategic variables.
- homogeneous products
- constant marginal costs with no capacity constraint
Let D(p) be the market demand function. Then the demand function for firm 1 is
D(p1) if p1 <
D(p1)/2 if p1 =
0 if p1 >
Notice that this demand function is discontinuous.
- price = c1 (minus a very small number) if
- with more than 2 firms, price is equal to the second lowest marginal
- if firms have the same marginal cost, price = marginal cost.
- pure strategies equilibrium may not exist if marginal costs are not
- differentiated products; e.g.,
is the demand curve for firm 1. D1 is decreasing
in the first argument and increasing in the second
- cost functions are
Suppose the demand function takes the form D1 = a
- bp1 + dp2. Then
profits for firm 1 is
(a - bp1 +
C1(a - bp1 +
The incentive the raise p1 is higher when
p2 is higher. This suggests that the best
response function is upward sloping.
(a - 2bp1 +
bC'1(a - bp1 +
bdC''1(a - bp1 +
When the best response function is upward sloping, we call the strategic
variables strategic complements.
In a Bertrand model with differentiated goods
- we usually assume that the best response functions are upward sloping
- the isoprofit curves for firm 1 has the following characteristics
- the preference direction is upwards
- it has a U shape and it reaches its lowest point on the
CONJECTURAL VARIATIONS MODEL
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.,
Firm i maximizes P(qi +
Ci(qi). The first order
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.
- if L = -1, then P = C' (perfect competition)
- if L = 0, then P'qi + P = C' (Cournot model)
- if L = 1/s - 1 (s is market share), then P'Q + P = C' (monopoly)
- Fat cats