Answers to PS12

Econ252: Intermediate Microeconomic Theory
ANSWERS TO HOMEWORK ASSIGNMENT 12
Chp 10) 4) a) The marginal revenue of firm 1 is calculated from the demand equation using the
twice as steep rule.
P = 36 − 3q2 − 3q1 ⇒ MR1 = 36 − 3q2 − 6q1
To find the best response function of firm 1 we need to find the profit maximizing output level
*
q1 given the output level of firm 2 is fixed at q2 . Thus using the profit maximization rule we set
equal to
and solve for q1:
€
1
36 − 3q2 − 6q1 = 18 ⇒ q*1 = 3 − q2
€ 2
The marginal revenue of firm 2 can be calculated from the demand using the twice as steep rule.
€ ⇒ MR = 36 − 3q − 6q
P = 36 − 3q − 3q
€
1
2
2
1
2
To find the best response function of firm 2 we need to find the profit maximizing output level
€ q * given the output level of firm 1 is fixed at q . Thus using the profit maximization rule we set
1
2
equal to
and solve for q2 :
€
€
1
36 − 3q1 − 6q2 = 18 ⇒ q* 2 = 3 − q1
2
€
b) Consider ( q1 = 1, q2 = 2 ). To see if this pair of output levels is Nash equilibrium for this
€
quantity competition game we need to check if either firm has an incentive to change its output
level given the output level of its rival.
€ So if firm 1 knows that firm 2 is producing 2 units ( q2 = 2) its best response to this would be:
€
1€
1
q1 = 3 − q2 = 3 − 2 = 3 −1 = 2 . Thus firm 1 would not stick to producing 1 unit but in fact
2
2
would produce 2 units. Thus this pair can not be a Nash equilibrium.
€
€
c) In Nash equilibrium each firm is maximizing its profit given the output of its competitor so no
firm has an incentive to change its output. To find the Nash equilibrium output levels for firm 1
and firm 2, we need to solve both firms best response functions together (i.e. we need to find q1
and q2 which satisfy both equations simultaneously).
1
1
q1 = 3 − q2 and q2 = 3 − q1
2
2
The easiest way to proceed is to insert the first equation in to the second one and solve for q2.
€
€
1
1
1
3 1
q2 = 3 − q1 = 3 − (3 − q2 ) ⇒ q2 = 3 − + q2
2
2
2
2 4
To find€q1 we would insert q2 = 2 into the first equation:
1
q1 = 3 − 2 = 3 −1 = 2 ⇒ q1 = 2
2
⇒ q2 = 2
Thus firm 1 producing 1 unit of output, firm 2 producing 2 units of output ( q1 = 2 , q2 = 2 ) is the
Nash equilibrium.
€
€
€
5) a) If the two firms act like a cartel to maximize their joint profit, they would find the monopoly
output and share it equally among themselves.
P = 36 − 3Q ⇒ MR = 36 − 6Q
MR = MC
⇒ 36 − 6Q = 18 ⇒ Q = 3
Thus each firm would produce q1 = 1.5 , q2 = 1.5
€
b) The best response functions of the each firm is given by:
1
1
q1 = 3 − q2 and q2 = 3 − q1
€
€
2
2
Each firm has an incentive to cheat on their cartel agreement. For example given that firm 2 is
1
€ producing 1.5 units,
€ firm 1 would want to produce: q1 = 3 − 2 (1.5) = 2.25
€