Suppose that there are two firms competing in the market for taxi services. Big Ben Taxis has the marginal cost MCB = $9 per trip, and the fixed cost FCB = $3,000,000. While Whitehall Taxis has the marginal cost MCW = $15 per trip, and the fixed cost FCW = $1,000,000. Inverse demand for taxi trips in the market is given by the function, P = 75 â Q/10,000 . In this equation, P is the price of a taxi trip, and Q is the total quantity of taxi trips supplied by the two taxi companies.
Question 1: Find the equilibrium price and quantities for the case in which the two taxi companies engage in Cournot (quantity) competition. What profits will Big Ben Taxis and Whitehall Taxis earn. (11 Marks)
Question 2: Using your answers to question 1, determine which firm has the greater market power. (4 Marks)
Question 3: Now suppose that a firm can only supply taxi services if it purchases a licence from the government. What is the highest fee that the government can charge for a license, if the government wants both Big Ben Taxis and Whitehall Taxis to purchase a license? (Note: A licence does not place a limit on the number of taxi trips a company can supply. You should assume that both firms are charged the same fee.) (2 Marks)
Question 4: If, instead, the government wants to maximise the revenue it receives from taxi license fees, how many licenses should it sell, and what fee should it charge? (8 Marks)
Q1 and Q2 answered, can anyone help with Q3 and Q4?
Q1
)P = 75 - (Q / 10,000) = 75 - [(QB + QW) / 10,000] (
QB, QW: Quantity of Big Ben & Whitehall respectively)
Total revenue, Big Ben, TRB = P x QB = 75QB - [(QB2 + QW.QB) / 10,000]
Marginal revenue, Big Ben, MRB = dTRB / dQB = 75 - [(2QB + QW) / 10,000]
Equating MRB with MCB, 75 - [(2QB + QW) / 10,000] = 9 750,000 - 2QB - QW = 90,000 2QB + QW = 660,000 .......(1) [Reaction function of Big Ben]
Total revenue, Whitewall, TRW = P x QW = 75QW - [(QW.QB + QW2) / 10,000]
Marginal revenue, Whitehall, MRW = dTRW / dQW = 75 - [(QB + 2QW) / 10,000]
Equating with MCW, 75 - [(QB + 2QW) / 10,000] = 15 750,000 - QB - 2QW = 150,000 QB + 2QW = 600,000 .....(2) [Reaction function of Whitehall]
Cournot equilibrium is obtained by solving (1) and (2).
2QB + QW = 660,000 .......(1) (2) x 2 results: 2QB + 4QW = 1,200,000 ...(3)
(3) - (1) yields:
3QW = 540,000 QW = 180,000 QB = 600,000 - 2QW = 600,000 - (2 x 180,000) = 600,000 - 360,000 = 240,000
Q = QW + QB = 180,000 + 240,000 = 420,000 P = 75 - (420,000 / 10,000) = 75 - 42 = 33
Profit = Total revenue - Total cost = (P x Q) - [FC + (MC x Q)] = Q x (P - MC) - FC [FC: Fixed cost]
Profit, Big Ben = [240,000 x (33 - 9)] - 3,000,000 = (240,000 x 24) - 3,000,000 = 5,760,000 - 3,000,000 = 2,760,000
Profit, Whitehall = [180,000 x (33 - 15) - 1,000,000 = (180,000 x 18) - 1,000,000 = 3,240,000 - 1,000,000 = 2,240,000 (3)
The fixed fee charged by government is equal to consumer surplus (CS) for each company, where CS is the area between demand curve and market price.
From demand curve, When Q = 0, P = 75 [Reservation price] For Big Ben, CS = (1/2) x $(75 - 33) x 240,000 = (1/2) x $42 x 240,000 = $5,040,000
This is the maximum license fee for Big Ben. For Whitehall, CS= (1/2) x $(75 - 33) x 180,000 = (1/2) x $42 x 180,000 = $3,780,000
This is the maximum license fee for Whitehall. (4) Revenue is maximized when marginal revenue is zero.
Marginal revenue, Big Ben, MRB = dTRB / dQB = 75 - [(2QB + QW) / 10,000] So, 75 - [(2QB + QW) / 10,000] = 0 750,000 - 2QB - QW = 0 2QB + QW = 750,000 ......(4)
Marginal revenue, Whitehall, MRW = dTRW / dQW = 75 - [(QB + 2QW) / 10,000] So, 75 - [(QB + 2QW) / 10,000] = 0 750,000 - QB - 2QW = 0 QB + 2QW = 750,000 ....(5) Multiplying (5) by 2, 2QB + 4QW = 1,500,000 ......(6) 2QB + QW = 750,000 ...... (4) (6) - (4) gives: 3QW = 750,000 QW = 250,000 QB = 750,000 - (2 x 250,000) = 750,000 - 500,000 = 250,000 Q = QB + QW = 250,000 + 250,000 = 500,000 P = 75 - (500,000 / 10,000) = 75 - 50 = $25
