Suppose mountain spring water can be produced at no cost and the inverse demand
ID: 1204805 • Letter: S
Question
Suppose mountain spring water can be produced at no cost and the inverse demand for mountain spring water is P = 1200 – 0.2Q. Answer the following questions.
a. Suppose the market of mountain spring water is supplied by a monopoly firm that cannot price discriminate. Find the monopoly firm’s profit-maximizing price and quantity of production. (10 pts.) [Hint: MR = P + (P/Q)*Q]
b. Suppose the market of mountain spring water is supplied by two firms (Firm A and firm B) that behave like a Cournot duopoly. Find the Nash Equilibrium price and quantity of production for each firm. (10 pts.) [Hint: Q = QA + QB; MRA = P + ((P/QA)*QA and MRB = P + ((P/QB)*QB]
c. Suppose the market of mountain spring water is supplied by two firms (Firm A and firm B) that behave like a Stackelberg duopoly where firm A is the leader and firm B is the follower. Find the Nash Equilibrium price and quantity of production for each firm. (10 pts.) [Hint: Q = QA + QB; MRA = P + ((P/QA)*QA and MRB = P + ((P/QB)*QB]
Explanation / Answer
a. Profit maximizing price of a monopolist is when marginal cost equal marginal revenue.
P = 1200 - 0.2Q
TR = P*Q = 1200Q - 0.2Q2
MR = 1200 - 0.4Q
MC = 0
Q = 3000
P = 1200 - 0.2Q = $600
b. In a Cournot duopoly, it becomes an output setting situation. A duopoly means two firms, so that:
Q = Q1 + Q2
P = 1200 - 0.2Q
P = 1200 - 0.2Q1 - 0.2Q2
To find the marginal revenue of the first firm, you can rewrite it like this
P = 1200 - 0.2Q1 - 0.2Q2
TR = 1200Q1 - 0.2Q12 - 0.2Q1Q2
MR = 1200 - 0.4Q1 - 0.2Q2
MC = 0
=> 0.4Q1 = 1200 - 0.2Q2
Q1 = 3000 - 0.5Q2
That's the reaction function. Since marginal costs are the same for the two firms, the reaction functions are symmetrical.
Q2 = 3000 - 0.5Q1
Also, Q1 = Q2 when they have the same marginal costs.
Q1 = 3000 - 0.5Q1
1.5 Q1 = 3000
Q1 = 2000 = Q2
The total quantity: Q = Q1 + Q2 = 4000
Now find the price with that quantity:
P = 1200 - 0.2Q
P = $400
(c) Let, firm A : first mover
Firm B's reaction function is: Q2 = 3000 - 0.5Q1
Then, P = 1200 - 0.2Q1 - 0.2Q2
=1200 - 0.2Q1 - 0.2( 3000 - 0.5Q1)
= 1200 - 0.2Q1 - 600 - 0.1Q1
= 600 - 0.3Q1
TR1 = 600Q1 - 0.3Q12
MR1 = 600 - 0.6Q1
For equilibrium
MR1 = MC1
600 - 0.6Q1 = 0
=> 0.6Q1 = 600
Q1 = 600/0.6 = 1000 units
Q2 = Q2 = 3000 - 0.5Q1 = 3000 - 0.5(1000) = 2500 units
P = 1200 - 0.2Q1 - 0.2Q2 = 1200 - 0.2(1000) - 0.2(2500) = $500