Question

Suppose that the (inverse) demand curve for Cranberries is given by P = 40 − 6Q and TC = $4Q + $3Q²

1. What is equilibrium Price and Quantity and Profit if the market is competitive?
2. What is equilibrium Price and Quantity and Profit if there are two firms in the market (note Q = q₁ + q₂)?  
3. What is equilibrium Price and Quantity and Profit if there are monopoly in the market (note Q = Q)?  
4. If there were 3 firms, where do you estimate the output and the price would be—this does not require a mathematical calculation—it is based on the expectations created by the prior three answers (1-b, 1-c, and 1-d).  

Answer 1

Marginal cost (MC) = dTC/dQ = 4 + 6Q

(a) In competitive market, P = MC.

40 - 6Q = 4 + 6Q

12Q = 36

Q = 3

P = 40 - (6 x 3) = 40 - 18 = 22

(b) When there are two firms, assuming Cournot duopoly, P = 40 - 6q1 - 6q2

MC1 = 4 + 6(q1 + q2) = 4 + 6q1 + 6q2

MC2 = 4 + 6(q1 + q2) = 4 + 6q1 + 6q2

For firm 1,

Total revenue (TR1) = P x q1 = 40q1 - 6q1² - 6q1q2

Marginal revenue (MR1) = TR1/q1 = 40 - 12q1 - 6q2

Equating MR1 and MC1,

40 - 12q1 - 6q2 = 4 + 6q1 + 6q2

18q1 + 12q2 = 36

3q1 + 2q2 = 6..........(1) (Best response, firm 1)

For firm 2,

Total revenue (TR2) = P x q2 = 40q2 - 6q1q2 - 6q2²

Marginal revenue (MR2) = TR2/q2 = 40 - 6q1 - 12q2

Equating MR2 and MC2,

40 - 6q1 - 12q2 = 4 + 6q1 + 6q2

12q1 + 18q2 = 36

2q1 + 3q2 = 6..........(2) (Best response, firm 2)

Cournot equilibrium is obtained by solving (1) and (2).

(1) x 2 yields: 6q1 + 4q2 = 12...........(3)

(2) x 3 yields: 6q1 + 9q2 = 18...........(4)

(4) - (3) yields: 5q2 = 6

q2 = 1.2

q1 = (6 - 3q2)/2 [From (2)] = [6 - (3 x 1.2)]/2 = (6 - 3.6)/2 = 2.4/2 = 1.2

Q = 1.2 + 1.2 = 2.4

P = 40 - (6 x 2.4) = 40 - 14.4 = 25.6

(c) A monopolist will equate MR with MC.

P = 40 - 6Q

Total revenue (TR) = P x Q = 40Q - 6Q²

MR = dTR/dQ = 40 - 12Q

40 - 12Q = 4 + 6Q

18Q = 36

Q = 2

P = 40 - (6 x 2) = 40 - 12 = 28

(d) The higher the number of firms, the lower the market price and the higher the market quantity. Therefore, if there are 3 firms, price (P*) will lie between the price in duopoly and perfect competition, and quantity (Q*) will lie between the price in duopoly and perfect competition. So,

25.6 > P* > 22, and

3 > Q* > 2.4.