Econ 302
Assignment 3 — Solution
1.
(a) The monopolist solves:
max Π(Q) = Q(a − bQ) − cQ.
Q
The first order condition is
a − 2bQ − c = 0,
or equivalently,
a−c
,
2b
which is the monopolist’s optimal quantity; the associated price is
Q=
P (Q) = a − b
a−c
2b
=
a+c
.
2
(b) Fix a symmetric strategy profile in which every firm produces q units.
The profit of any firm (say firm 1) when he produces q1 and the others each produce q
is:
Π1 (q1 ) = q1 P (q1 + (n − 1)q) − cq1 = q1 (a − b(q1 + (n − 1)q)) − cq1 .
Given q, firm 1 wants to maximize Π1 (q1 ). The first order condition of firm 1 with
respect to q1 is
a − b(n − 1)q − 2bq1 − c = 0,
which gives firm 1’s best-response to q of others:
q1 =
a − b(n − 1)q − c
.
2b
At a symmetric Nash equilibrium, we must have q1 = q, i.e.,
q=
a − b(n − 1)q − c
,
2b
1
or equivalently,
q=
a−c
.
(n + 1)b
The market price given these quantities is
P (nq) = a −
n
1
n
(a − c) =
a+
c.
n+1
n+1
n+1
a−c
is the symmetric pure-strategy Nash equilibIn summary, each firm produces q = (n+1)b
1
n
rium. The equilibrium price is n+1 a + n+1 c.
Figure 1: Graph for Problem 1, part (c).
(c) In the graph, P M and QM are the monopolist’s price and quantity, P C and QC are the
price and total quantity from the Nash equilibrium in the quantity competition game
(Cournot competition), and QP C = a−c
is the quantity from perfect competition:
b
P (QP C ) = a − bQP C = c.
2
The deadweight loss for monopoly is
1 M
1
(P − c)(QP C − QM ) =
2
2
a+c
−c
2
a−c a−c
−
b
2b
=
(a − c)2
.
8b
The deadweight loss for Cournot competition is
1 C
1
(P − c)(QP C − QC ) =
2
2
1
n
a+
c−c
n+1
n+1
a − c n(a − c)
−
b
(n + 1)b
=
(a − c)2
.
2(n + 1)2 b
Notice that the deadweight loss for Cournot competition (n ≥ 2) is decreasing with n,
and is always smaller than the deadweight loss for monopoly (and becomes the same
when n = 1). In other words, competition reduces the deadweight loss.
(d) Assume Cournot/quantity competition. As the number n of firms tends to infinity, in
the Nash equilibrium the price tends to c and the deadweight loss tends to 0, i.e., we get
convergence to perfect competition.
2. The profit of firm 1 when he produces q1 while firm 2 produces q2 is:
Π1 (q1 ) = q1 (10 − (q1 + q2 )) − 2q1 .
Given q2 , firm 1 wants to maximize Π1 (q1 ). The first order condition of firm 1 with
respect to q1 is
10 − q2 − 2q1 − 2 = 0,
which gives firm 1’s best-response to q2 :
8 − q2
.
2
Likewise, the profit of firm 2 when he produces q2 while firm 1 produces q1 is:
q1 =
(1)
Π2 (q2 ) = q2 (10 − (q1 + q2 )) − (q2 )2 .
Given q1 , firm 1 wants to maximize Π2 (q2 ). The first order condition of firm 2 with
respect to q2 is
3
10 − q1 − 2q2 − 2q2 = 0,
which gives firm 2’s best-response to q1 :
10 − q1
.
(2)
4
At a Nash equilibrium, we must have mutually best responses, i.e., a solution to Equations
(1) and (2). It is easy to show that the solution, and hence the Nash equilibrium, is
q2 =
22
12
, q2 = .
7
7
The market price given these (Nash equilibrium) quantities is
q1 =
P (q1 + q2 ) = 10 −
34
36
= .
7
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3. There are many pure-strategy Nash equilibria in this game. Below I describe all
possible equilibria.
(a) For any p1 ∈ [2.01, 3], it is a Nash equilibrium for firm 1 to set p1 as his price, and firm
2 to set p2 = p1 + 0.01; in this equilibrium, firm 1 captures the whole market and sells
Q(p1 ) = 10 − p1 units.
In the Nash equilibrium (p1 , p2 = p1 + 0.01) where p1 + 0.01 < 3, firm 2 sets his price
below his marginal cost of 3, which seems irrational. However, this p2 = p1 + 0.01 is a
best response to p1 because firm 2 never gets a chance to actually sell at this price, so
he gets a profit of 0 in any case.
You should check that for firm i, any pi < ci is weakly dominated by pi = ci ; this is a good
exercise in reviewing the concept of dominance. This means that a weakly dominated
strategy can still be a part of a Nash equilibrium, in contrast to strict dominance.
However, a Nash equilibrium with a weakly dominated strategy is not very plausible in
practice.
(b) For any p1 ∈ [2.01, 3], it is a Nash equilibrium for firm 1 to set p1 as his price, firm j to
set pj = p1 + 0.01 for some j between 2 and 5, and the rest of the firms to set pi > p1 ,
where i 6= j and i 6= 1; in this equilibrium, firm 1 captures the whole market and sells
Q(p1 ) = 10 − p1 units.
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(c) For this case it makes sense to distinguish between two classes of Nash equilibria:
Class 1. It is a Nash equilibrium for firm 1 to set p1 = 6, and firm 2 to set any p2 > 6;
in this equilibrium, firm 1 captures the whole market and sells Q(6) = 4 units.
Class 2. For any p1 ∈ [2.01, 5.99], it is a Nash equilibrium for firm 1 to set p1 as his
price, and firm 2 to set p2 = p1 + 0.01; in this equilibrium, firm 1 captures the whole
market and sells Q(p1 ) = 10 − p1 units.
Note that p1 = 6 is firm 1’s optimal price when he is the monopolist:
max q(10 − q) − 2q
⇐⇒
q
q = 4, P (q) = 6.
In part (a) and (b), firm 1 also wants to have a price of 6, but he is forced to sell at a
price of at most 3 because given p1 = 6 > 3 = c2 firm 2 would undercut this price, so
p1 = 6 is not a sustainable price.
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