Consider a market with market
Consider a market with market demand P(Q) = 70 -8Q and each firmin the market faces a total cost TC(Q) = 22Q.
Suppose there is only one firm in themarket.
(a) What is the profit-maximizing price and quantity in themarket?
(b) What are the profits and consumer surplus?
Now suppose we have a Cournot duopoly where firms choosequantities.
(c) What is the equilibrium price and market quantity?
(d) What is the consumer surplus and profits for each firm?
Answer:
a)
A firm maximizes by equation Marginal Revenue and MarginalCost
P = 70 – 8Q
Total Revenue = P*Q = (70 – 8Q)*Q
Marginal Revenue = dTR/dQ = 70 – 16Q
Marginal Cost = dTC/DQ = 22
MR = MC
70 – 16Q = 22
16Q = 48
Q = 3
P = 70-8Q = 46
b)
Profit = TR – TC = 46*3 – 22*3 = 72
Consumer Surplus = 1/2*(70-46)*3 = 36
c)
Now there the two firms q1 and q2
P = 70 – 8(q1+q2)
Firm 1:
Total Revenue = P*q1 = (70 – 8(q1+q2))*q1
Marginal Revenue = dTR/dQ = 70 -16q1 – 8q2
Marginal Cost = dTC/Dq1 = 22
MR = MC
70 -16q1 – 8q2 = 22
16q1 = 48 – 8q2
q1 = 3 – 0.5q2
Similarly for firm 2 we get
q2 = 3 -0.5 q1
Put the value of q1 from above we get,
q2 = 3 -0.5(3 -0.5 q2)
q2 = 3 – 1.5 + 0.25q2
0.75q2 = 1.5
q2 = 2
q1 = 3 -0.5 q2 = 2
Q = q1+q2 = 2+2 = 4
Price = 70-8(2+2) = 38
d)
Profit of Firm 1 = TR – TC = P*q1 – TC = 38*2 – 22*2 = 32
Profit of Firm 2 = TR – TC = P*q2 – TC = 38*2 – 22*2 = 32
Total Profit = 32+32 = 64
Consumer Surplus = 1/2*(70-38)*4 = 64