Question

Natural Monopoly: Suppose that PG&E is a natural monopoly. PG&E faces the following inverse demand curve for monthly demand for gas: P=260- 1/4Q. Suppose its marginal and average variable costs are a constant $10 per kilowatt hour.

1. Find the profit maximizing quantity and price if this natural monopolist was not regulated

2. Draw a graph for the monopolist, showing the demand curve, the marginal revenue curve, and the profit maximizing output and price

3. Now suppose that the natural monopolist is regulated as to price and quantity. Assume that average fixed costs at Q(reg) is $30. Find Q(reg) and P(reg). Show your answers in the graph above.

Answer 1

Given —

Inverse Demand Function : P = 260 - 0.25 Q

MC = 10

In monopoly, equilibrium condition is given by—

MR = MC ( Marginal Revenue = Marginal Cost)

We have , TR = price . Quantity

TR = (260-0.25Q) . Q = 260Q - 0.25Q^2

On differentiating TR with respect to Q we get —

dTR/dQ = 260- 0.5Q = MR

MR = 260 - 0.5 Q

Setting up the equilibrium condition-

MR = MC

260 - 0.5Q = 10

250 = 0.5Q

Q = 250/0.5 = 500

Q * = 500 .... Equilibrium quantity

Now substituting the equilibrium quantity into demand function to get equilibrium price —

P = 260 - 0.25(500) = 260-125=135

P* = 135 .....Equilibrium price

A natural monopolist will have the following -

EQUILIBRIUM PRICE (P*) = 135

EQULIBRIUM QUANTITY (Q*) = 500

For graph , kindly refer Image -1

Case -2 REGULATED MONOPOLY

Given , at the equilibrium quantity of regulated monopoly, the average fixed costs equal $30.

We know that , in a regulated monopoly, equilibrium condition is

P = AFC ( Price equals Average Fixed Costs)

That is , P = 30

When P = 30, quantity demanded is-

30 = 260 - 0.25Q

0.25 Q = 230

Q = 920

So a regulated monopoly has -

Equlibrium Quantity (Qr)= 920

Equilibrium price (Pr) = 30

With regulation, Quantity increases from Q* to Qr and price falls from P* to Pr such that monopolist now makes zero economic profits.

For graph, some important coordinates—

when p = 0 , Q = 1040

when q=0, P = 260 ,

When MR = 0 , q = 1040