Question

Consider a market with market demand P(Q) = 70 -8Q and each firm in the market faces a total cost TC(Q) = 22Q.

Suppose there is only one firm in the market.

(a) What is the profit-maximizing price and quantity in the market?

(b) What are the profits and consumer surplus?

Now suppose we have a Cournot duopoly where firms choose quantities.

(c) What is the equilibrium price and market quantity?

(d) What is the consumer surplus and profits for each firm?

Answer 1

a)

A firm maximizes by equation Marginal Revenue and Marginal Cost

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