Oligopoly Theory
STRATEGIC SITUATION
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 S_{i} be the set of all possible
strategies for player i. Let s_{i} denote an
element of S_{i}, and let
s_{i} denote the strategies of all players other
than i. The payoff to individual i is represented by
y_{i}(s_{1}, ...,
s_{N}). The group of strategies
(s_{1}^{*}, ...,
s_{N}^{*}) constitute a Nash
equilibrium if, for each player i,
y_{i}(s_{i}^{*},
s_{i}^{*}) >=
y_{i}(s_{i},s_{i}^{*})
for all s_{i} in S_{i}.
A Nash equilibrium
 need not be Pareto efficient
 is "selfenforcing"
 need not be unique
 need not exist in pure strategies
COURNOT MODEL
Assumptions:
 quantity is the strategic variable (Nash equilibrium in quantities)
 price clears the market
 homogeneous goods (not essential)
Let P = P(q_{1},q_{2}) be the
inverse demand function. Let
C_{1}(q_{1}) and
C_{2}(q_{2}) be the cost
functions of firms 1 and 2. If
(q_{1}^{*},
q_{2}^{*}) is a Nash
equilibrium, then
q_{1}^{*} maximizes
P(q_{1},q_{2}^{*})q_{1}
 C_{1}(q_{1})
q_{2}^{*} maximizes
P(q_{1}^{*},q_{2})q_{2}
 C_{2}(q_{2})
These two conditions can be written in terms of two first order conditions:
P + q_{1}^{*}P' 
C_{1}' = 0
P + q_{2}^{*}P' 
C_{2}' = 0
 price exceeds marginal costs
 marginal costs need not be equalized across firms, i.e., production is
inefficient
 compare the monopoly pricing equation:
P + Q^{M}P' 
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 rearrange the first order conditions to get
(P  MC_{i})/P = a_{i}/e
where a_{i} is the market share of firm i and e is
the industry's elasticity of demand.
DIAGRAMATIC REPRESENTATION
The best response function (also known as reaction function)
shows the optimal (i.e.,
profitmaximizing) output q_{1} as a function of
q_{2}. It is implicitly defined by the first order
condition for firm 1:
P'(q_{1} +
q_{2})q_{1} +
P(q_{1} +
q_{2}) 
C_{1}'(q_{1}) = 0.
We can write q_{1} =
BR_{1}(q_{2}). Apply a
similar reasoning to firm 2, we can write q_{2} =
BR_{2}(q_{1}).
The Nash equilibrium can alternatively be defined as the solution to the two
equations
q_{1} =
BR_{1}(q_{2}).
q_{2} =
BR_{2}(q_{1}).
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/dq_{j}(dy_{i}/dq_{i})
tends to be negative.)
When best response functions are downward sloping, we call the strategic
variables strategic substitutes.
We
can also draw isoprofit curves for firm 1 that shows combinations of
q_{1} and q_{2} 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 BR_{1} curve
Putting the best response functions and the isoprofit curves together, we
can see
 there is potential for profitable collusion
 there are enforcement problems associated with collusion
EXAMPLES
1. Let
 D = 1  P,
 C_{1} = c_{1}q_{1},
 C_{2} = c_{2}q_{2},
Firm 1 maximizes
(1  q_{1}  q_{2})q_{1} 
c_{1}q_{1}.
The first order conditions is
(1  q_{1}  q_{2})  q_{1} 
c_{1} = 0.
Rearrange this to
q_{1} =
(1  q_{2}  c_{1})/2
=BR_{1}(q_{2}).
Note
 the BR function is downward sloping
 the BR function will shift out when marginal cost falls
Write a similar first order condition for firm 2:
(1  q_{1}  q_{2}) 
q_{2} 
c_{2} = 0.
Solve the two first order equations simultaneously to get the Nash
equilibrium output:
q_{1}^{*}
= (1  2c_{1} + c_{2})/3.
q_{2}^{*}
= (1 + c_{1}  2c_{2})/3.
Substitute these equilibrium output back into the original profit function
to get the equilibrium profits
y_{1}^{*}
= [(1  2c_{1} +
c_{2})/3]^{2}.
y_{2}^{*}
= [(1 + c_{1} 
2c_{2})/3]^{2}.
Notice that firm 1's output and profits depends (positively) on firm 2's
cost.
2. Let
 D = 1  P
 n firms, each with the same marginal cost = c
Consider a typical firm i. Firm i maximizes
(1  q_{1}  ... 
q_{n})q_{i} 
cq_{i}.
The first order condition is
(1  q_{1}  ...  q_{n}) 
q_{i} 
c = 0.
In a symmetric equilibrium, q_{1} = ... =
q_{n} = 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)
Note
 dP/dn < 0
 P approaches c when n approaches infinity
STACKELBERG MODEL
Consider a twostage game where firm 1 (the leader) chooses its output level
q_{1} in the first stage. In stage 2, firm 2 (the
follower) chooses its output level q_{2}, knowing
what q_{1} is.
The Stackelberg leader is implicitly assumed to have some ability to commit
to its chosen output level q_{1}. It will not change
q_{1} after knowing what q_{2}
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 q_{1} as given. The
optimal q_{2} is given by
q_{2} =
BR_{2}(q_{1}).
This BR function is just like the one in the Cournot model.
In stage 1, firm 1 chooses q_{1} to maximize its
profits, taking into account firm 2's optimal response:
maximize P(q_{1} +
BR_{2}(q_{1}))q_{1}
 C_{1}(q_{1}).
The first order condition is
(1 + BR'_{2})P'q_{1} + P =
MC_{1}.
Compare with the Cournot outcome, where P'q_{1} + P =
MC_{1}, we can see that
q_{1}^{S} >
q_{1}^{C}.
Moreover, since BR'_{2} < 0,
q_{2}^{S} =
BR_{2}(q_{1}^{S})
<
BR_{2}(q_{1}^{C})
= q_{2}^{C}.
EXAMPLE
 D = 1  P
 C_{1} =
c_{1}q_{1}
 C_{2} =
c_{2}q_{2}
As in Example 1 in the section on Cournot model, Firm 2's best response
function is
BR_{2}(q_{1} = (1 
q_{1} 
c_{2})/2.
Firm 1 maximizes
(1  q_{1}  (1 
q_{1} 
c_{2})/2)q_{1} 
c_{1}q_{1}.
The solution gives
q_{1}^{S} =
(1 + c_{2}  2c_{1})/2.
Substitute this back into the best response function for firm 2, we get
q_{2}^{S} =
(1  3c_{2} + 2c_{1})/4.
BERTRAND MODEL
The Bertrand model can be interpreted as a game in which prices are
the strategic variables.
Example 1.
Assume
 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_{1}(p_{1},p_{2})=
D(p_{1}) if p_{1} <
p_{2};
D(p_{1})/2 if p_{1} =
p_{2};
0 if p_{1} >
p_{2}.
Notice that this demand function is discontinuous.
 price = c_{1} (minus a very small number) if
c_{1} >
c_{2}
 with more than 2 firms, price is equal to the second lowest marginal
cost
 if firms have the same marginal cost, price = marginal cost.
 pure strategies equilibrium may not exist if marginal costs are not
horizontal
Example 2.
Assume
 differentiated products; e.g.,
D_{1}(p_{1},p_{2})
is the demand curve for firm 1. D_{1} is decreasing
in the first argument and increasing in the second
 cost functions are
C_{1}(q_{1})
and C_{2}(q_{2})
Suppose the demand function takes the form D_{1} = a
 bp_{1} + dp_{2}. Then
profits for firm 1 is
y_{1} =
(a  bp_{1} +
dp_{2})p_{1} 
C_{1}(a  bp_{1} +
dp_{2})
We have

dy_{1}/dp_{1} =
(a  2bp_{1} +
dp_{2}) +
bC'_{1}(a  bp_{1} +
dp_{2})

d/dp_{2}(dy_{1}/dp_{1})
=
d +
bdC''_{1}(a  bp_{1} +
dp_{2})
The incentive the raise p_{1} is higher when
p_{2} 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
 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
BR_{1} function
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(q_{1} + q_{2}). Define a
parameter L_{ij} to be i's conjecture of j's output
response when q_{i} increases. I.e.,
L_{ij} =
dq_{j}/dq_{i}.
Firm i maximizes P(q_{i} +
q_{j})q_{i} 
C_{i}(q_{i}). The first order
condition is:
P'q_{i}(1 + L_{ij}) + P 
C'_{i} = 0.
 if L = 1, then P = C' (perfect competition)
 if L = 0, then P'q_{i} + P = C' (Cournot model)
 if L = 1/s  1 (s is market share), then P'Q + P = C' (monopoly)
Another possibility is to impose the condition that
L_{ij} has to be equal to the equilibrium
dq_{j}/dq_{i}. This is known
as the consistent conjectural variations model.

Lecture
Notes

Klemperer
 Fat cats