## Figure 133

Monopoly Price Regulation: Optimal Price/Output Decision Making

Monopoly regulation imposes a price ceiling at P2 just sufficient to provide a fair return (area PnAECn) on investment. Under regulation, price falls from P1 to P2 and output expands from Q1 to Q2.

Price and cost

Price and cost

Quantity per time period

TC = \$3,750,000 + \$70Q + 0.00002Q2 MC = ATC/AQ = \$70 + \$0.00004Q

where cost is expressed in dollars.

To find the profit-maximizing level of output, demand and marginal revenue curves for annual service must be derived. This will give all revenue and cost relations a common annual basis. The demand curve for annual service is 12 times monthly demand:

Total and marginal revenue curves for this annual demand curve are

TR = \$270Q - \$0.00048Q2 MR = ATR/AQ = \$270 - \$0.00096Q

The profit-maximizing level of output is found by setting MC = MR (where Mn = 0) and solving for Q:

MC = MR \$70 + \$0.00004Q = \$270 - \$0.00096Q \$0.001 Q = \$200 Q = 200,000

The monthly service price is

This price/output combination generates annual total profits of n = \$270Q - \$0.00048Q2 - \$3,750,000 - \$70Q - \$0.00002Q2 = -\$0.0005Q2 + \$200Q - \$3,750,000 = -\$0.0005(200,0002) + \$200(200,000) - \$3,750,000 = \$16,250,000

If the company has \$125 million invested in plant and equipment, the annual rate of return on investment is v u t ^ 4. \$16,250,000 ni_ 1Q0, Return on Investment = = 0.13, or 13%

\$125,000,000

Now assume that the State Public Utility Commission decides that a 12 percent rate of return is fair given the level of risk taken and conditions in the financial markets. With a 12 percent rate of return on total assets, Malibu Beach would earn business profits of n = Allowed Return X Total Assets = 0.12 X \$125,000,000 = \$15,000,000

To determine the level of output that would generate this level of total profits, total profit must be set equal to \$15 million:

n = TR - TC \$15,000,000 = -\$0.0005Q2 + \$200Q - \$3,750,000

This implies that

which is a function of the form aQ2 + bQ - c = 0. Solving for the roots of this equation provides the target output level. We use the quadratic equation as follows:

_ -200 ± V 2002 - 4(-0.0005)(18,750,000) 2(-0.0005)

= 150,000 or 250,000