An electric utility uses natural gas as fuel for a large mul

An electric utility uses natural gas as fuel for a large multi-unit power plant. With all units in-service, for a given hour the plant\'s fixed cost is $120,000 and its variable cost is $50 per megawatt demanded. Total revenue in dollars for a given hour is 150X-0.02X^2 where X is megawatts demanded for a given hour: What is the hourly demand in megawatts that will maximize total revenue? What is the hourly demand in megawatts that yields the maximum profit? What is the maximum profit? What is the range of hourly megawatts demanded that will result in the plant being profitable?

Solution

(a) Total Revenue will be maximized, whem MR = 0

TR = 150X - 0.02X²

MR = d(TR)/dX = 150 - 0.04X

150 - 0.04X = 0

0.04X = 150

   X = 150/0.04 = 3750

Therefore, the hourly demand 3750 megawatts.

(b) Total Cost = 120,000 + 50X

   MC = 50

    TR = 150X - 0.02X²

Profit will be maximized when MR = MC

150 - 0.04X = 50

0.04X = 100

X = 100/0.04 = 2500

Therefore, to maximize profit hourly demand is 2500 megawatts.

(c) Profit = TR - TC

             = 150X - 0.02X² - 120,000 - 50X

             = 100X - 0.02X² - 120,000

            = 100(2500) - 0.02(2500)² - 120,000

            = 250,000 - 125,000 - 120,000

            = $5000

Therefore, maximum profit is $5000.

(d) The plant will be profitable as long at TR > TC