Question

1. A water supply company faces the following inverse demand function: P = 100 – 2Q. Its cost function is C = 100+2Q. Note P is the water rate per thousand liters of water and Q is the quantity measured in thousand liters of water.
   1. Calculate the firm’s profit-maximizing output and price. What profit it will make? Will there be a deadweight loss to society? How much?
   2. Suppose a government regulator regulates the price following the marginal cost pricing rule. What price and water consumption will result? How much profit the monopoly will now make? What cost will the regulator incur? What deadweight loss will result due to this regulation?
   3. Now suppose the regulator wants the monopoly to follow the average cost pricing rule. What price and quantity will result? What will be the deadweight loss?

Answer 1

a) Under perfect copetition, the profit maximizing condition requires the equality of price and marginal cost.

The cost function of the firm

C= 100+2Q

The marginal cost function becomes

MC= 2

Thus, for profit maximization,

100- 2Q =2

Q= 49

The correspoding price= 100- 2*49= 2

The profit is given by the difference between total revenue and total cost of the monopoly.

Total revenu

PQ= 100Q- 2Q²= 100*49- 2(49)²= 4900- 4802= 98

There is no deadweight loss under perfect competition.

b) Under a monopoly, the consition of profit maximization of the monoply requires the equality of marginal cost and marginal revenue.

The inverse demand function faced by the monopoly is

P= 100-2Q

The total revenue function becomes

PQ= 100Q- 2Q²

The MR function becomes

MR= 100- 4Q

The cost function is given as

C= 100+ 2Q

The marginal cost function is

MC= 2

When the monopoly would be maximizing its profit

MR= MC

100- 4Q=2

4Q= 98

Q= 24.5

The price charged by the monopoly = 100- 2*24.5= 51

The profit of the monopoly is the difference by the total revenue and total cost

Total revenue= 100*24.5 - 2(24.5)² =2450- 1200.5 = 1249.5

The total cost of the monoply= 100+ 2*24.5= 149

The total profit of the monopoly becomes= 1249.5- 149= 1100.5

Under a monopoly there is always a deadweight loss.