Games with continuous payoffs.
The Cournot game
In all the games discussed so far, firms had a discrete set
of choices (high – medium – low, enter – not enter). In
real markets, firms choose from a range of prices and
quantities.
One of the game-theoretical models dealing with
continuous set of choices is the Cournot model of a
duopoly.
“Duopoly” = a market with two firms.
The game:
Two identical firms producing identical goods enter the
same market simultaneously and choose the quantities
they are going to produce. Each firm knows what the
market demand is but does not know the rival’s choice of
quantity.
Market demand is
Q = 130 – P
There are two firms, therefore
Each firm has TCi = 10 qi
Q = q1 + q2
(i = 1, 2)
Firm 1’s profit = P q1 – 10 q1 = (P – 10) q1 =
= (130 – q1 – q2 – 10) q1 =
Firm 1’s profit = P q1 – 10 q1 = (P – 10) q1 =
= (130 – q1 – q2 – 10) q1 = (120 – q1 – q2) q1 =
= 120 q1 – q12 – q2q1
Firm 1’s profit depends on its own choices AND on Firm 2’s
choices; therefore, it is a game.
To solve this game, we are going to once again use the
best response technique: one firm assumes certain action
on the other firm’s part, and finds the best response to that
hypothetical action.
Let’s assume q2 = 30. What is firm 1’s best response to that?
130
100
DMARKET
D1
MC1
10
MR1
q1* , our best response
30
The same in mathematical terms:
The residual demand for firm 1 is
The inverse residual demand is
q1 = 100 – P
P = 100 – q1
Hence MR = 100 – 2 q1
If MC = 10, then firm 1’s best response to firm 2’s
decision to produce 30 units is
MR = MC
100 – 2 q1 = 10
90 = 2 q1
q1* = 45
The general case:
If firm 2 chooses to produce quantity q2, then firm 1’s
inverse residual demand is
P = (130 – q2) – q1
Hence MR1 = (130 – q2) – 2 q1
If MC = 10, we set MR = MC to get
(130 – q2) – 2 q1 = 10, or
2 q1 = 120 – q2,
or
q1* = 60 – 0.5 q2
Alternatively, using differentiation:
The formula for firm 1’s profit is
Π1 = (P – 10) q1 = (130 – q1 – q2 – 10) q1
If firm 2 produces 30 units, then
Π1 = (90 – q1) q1 = 90 q1 – q12
We can find our best response by using the profitmaximization approach:
•Our control variable is q1.
•Take the derivative of our profit with respect to q1,
•Set it equal to zero and solve for q1.
In the more general case,
Π1 = (120 – q2 – q1) q1 = (120 – q2) q1 – q12
Even though notation-wise variables q2 and q1 look
very similar, we have to keep in mind that Firm 1
has control only over q1 and therefore treats q2 as
given, or a constant. All the rules are applied
accordingly.
d 1
 (120 – q2) – 2 q1
dq1
(120 – q2) – 2 q1 = 0
q1 = 60 – 0.5 q2
Either way, we can see that for every q2 chosen by Firm
2, Firm 1 has a unique best response,
q1* = 60 – 0.5 q2
The smaller the q2, the greater the q1* chosen in
response. (Makes sense)
It is easy to guess that, due to the symmetry of the
game,
Firm 2’s best response to Firm 1’s choice is
q2* = 60 – 0.5 q1
Let us plot the best response functions in the (q1, q2)
space.
q1*(q2), or BRF of Firm 1
q2
The point where strategies
(quantity choices) are best
responses to each other
70
60
50
q2*(q1),
or BRF of Firm 2
40
30
20
10
0
10
20
30
40
50
60
70
80
q1
The equilibrium can also be found algebraically, by solving
the two best response equations together:
q1* = 60 – 0.5 q2
(Eq.1)
q2* = 60 – 0.5 q1
(Eq.2)
(Since we are looking for best responses of both firms, we
can ignore the asterisks or assume all q’s in the equations
appear with asterisks.)
From (Eq.1), q2 = 120 – 2 q1. Plugging it into (Eq.2) yields
120 – 2 q1 = 60 – 0.5 q1
Rearranging gives
1.5 q1 = 60, or
q1 = 40.
By plugging q1= 40 into either of the two original equations,
we get q2 = 40 (which is not surprising, given the symmetry
of the game).
We can also calculate resulting equilibrium price(s) and
quantities.
P = 130 – Q =
By plugging q1= 40 into either of the two original equations,
we get q2 = 40 (which is not surprising, given the symmetry
of the game).
We can also calculate resulting equilibrium price(s) and
quantities.
P = 130 – Q = 130 – (q1 + q2) = $50
Profit of each firm is
Π = (P – ATC) ∙ q = (50 – 10) ∙ 40 = $1600
Questions for discussion:
1. How will the outcome of the Cournot game change
if the firms are no longer identical (the marginal costs
of the two firms are different)?
2. How does the outcome of the Cournot game
compares to the case when the firms are able to
collude?
If a monopoly faces demand given by P = 130 – Q,
and has MC = $10, then…
If a monopoly faces demand given by P = 130 – Q,
and has MC = $10, then…
TR = P∙Q = (130 – Q) Q = 130 Q – Q2
Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q,
Profit-maximization condition, MR = MC, or 130 – 2 Q = 10
results in 2 Q = 120, or Q = 60
From the demand equation, P = $70, and the resulting
profit is
(P – ATC) Q
If a monopoly faces demand given by P = 130 – Q,
and has MC = $10, then…
TR = P∙Q = (130 – Q) Q = 130 Q – Q2
Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q,
Profit-maximization condition, MR = MC, or 130 – 2 Q = 10
results in 2 Q = 120, or Q = 60
From the demand equation, P = $70, and the resulting
profit is
(P – ATC) Q = ($70 - $10) 60 =
If a monopoly faces demand given by P = 130 – Q,
and has MC = $10, then…
TR = P∙Q = (130 – Q) Q = 130 Q – Q2
Marginal Revenue, MR = d(TR) / dQ = 130 – 2 Q,
Profit-maximization condition, MR = MC, or 130 – 2 Q = 10
results in 2 Q = 120, or Q = 60
From the demand equation, P = $70, and the resulting
profit is
(P – ATC) Q = ($70 - $10) 60 = $3600
We can summarize our findings in a table.
Market price
Market quantity
Profits
Noncooperative
(Cournot) duopoly
Collusion
(monopoly)
$50
$70
40 + 40 = 80
60
$1600 + $1600 =
= $3200
$3600
The higher profit earned in the monopoly case
explains why firms prefer to collude, or at least to
coordinate their prices and outputs in some way.