Question

Consider a market for a good characterized by an inverse market demand P(Q) = 200−Q. There are two firms, firm 1 and firm 2, which produce a homogeneous output with a cost function C(q) =q2+ 2q+ 10.

1. What are the profits that each firm makes in this market?

2. Suppose an advertising consultant approaches firm 1 and offers to increase consumers’ value for the good by $10. He offers this in exchange for payment of $200. Should the firm 1 undertake advertising?

3. In this market, is there any potential for the free rider problem of advertising? Explain.

Answer 1

Price is 200-Q. So revenue is

P*Q=200Q-Q2

Marginal Revenue=200-2Q

Cost function is Q2+2Q+10

Marginal Cost=2Q+2

Profit maximization will happen at

MR=MC

200-2Q=2Q+2

198=4Q

Q=49.5.

P at Q=49.5 would be 200-49.5=150.5.

Revenue=150.5*49.5=7449.75

Cost=49.52+2*49.5+10

=2559.25

Profit=7449.5-2559.25

=4890.25

Each firm will make=4890.25/2=2445.125

2. If the value of the product goes up by 10, it means the new price that the consumer percieves is

P=210-Q. So,

MR=210-2Q

Marginal Cost=2Q+2

Profit maximization will happen at

MR=MC

210-2Q=2Q+2

Q=208/4=52

At Q=52,

P=210-52=158

Revenue=52*158=8216

Cost=522+2*52+10

=2818

Total profit=8216-2818

=5398

Each firm's profit=2699.

Increase for firm 1 from previous scenario=2699-2445.125

=253.975.

Since this is higher than the cost of 200 the consultant is asking for, firm 1 should take the offer.

3. Yes there is potential for free rider problem because the product of both firms is homogenous. Which means advertising for one will benefit the other too. And it will not cost the other firm anything, hence the other firm will free ride.