The City of Berkeley is currently considering alternative ways of providing cable service to its citizens. Based on an econometric analysis of several recently awarded cable franchises in other cities, economists have determined that the total cost, TC, and inverse demand curves for a cable company in Berkeley would be:
TC = 2Q – 0.1Q2 + 0.005Q3 and
P = 20 – 0.5 Q
and where output, Q, is measured in thousands and P is the monthly basic tier price.
a. Given this information, what are the equations for the total and marginal revenue curves, and what do these curves look like on a graph?
b. City Councilor A believes the city should own and operate a cable system for the purpose of making as much profit as possible. The profit would be used to lower the city government’s deficit. If Councilor A gets her way, what will be the price and output of cable and by how much will the city-owned system be able to reduce the city’s deficit?
c. Councilor B believes the city should produce as much cable service as possible without losing money (i. e., the city should provide cable to its citizens on a nonprofit basis). If Councilor B gets his way, what output and price will result?
d. Councilor C believes that the private sector should provide cable to the city but that the single, private firm that gets the city’s franchise should pay 10 percent of its total revenue back to the city in the form of an annual franchise fee. If Councilor C gets her way and the franchise is awarded to the firm promising to pay the largest franchise fee, what price and output will result? What will be the size of the annual franchise fee?
SOLUTION
TR = PQ = (20 - 0.5Q)Q = 20Q - 0.5Q2 MR = 20 - Q The marginal revenue curve has a vertical intercept of $20 and slopes down with a slope of -1. The total revenue curve is shaped like an upside down U, equal to 0 at Q=0 and at Q=40, and reaching its peak at Q=20.
π = TR - TC = 20Q - .5Q2 - (2Q - .1Q2 +.005Q3). Solving for Q, we get Q = 17.05 and price of $11.5. Profit, which can be used to reduce the deficit, equals $165.84.
They would set price equal to average total cost, which yields an output of 32 and a price of $4.
The private firm’s total revenue would equal 20Q - .5Q2 - .1(20Q - .5Q2) = 18Q-.45Q2. Taking the derivative of the profit function using this new total revenue equation and solving for Q yields Q = 16.8 and P = $11.6. The franchise fee is $19.49 (10 percent of total revenue).