a. To find the maximum fee the commission could charge, we need to determine the number of taxis that would serve the market at an average price of $15. Given the demand function Q = 10 - 0.5P, we can substitute P = $15 into the demand function:
Q = 10 - 0.5 * 15
Q = 10 - 7.5
Q = 2.5
So, at an average price of $15, the quantity demanded is 2.5 trips per week. To determine the number of taxis that would serve the market, we need to find the quantity supplied at this price. The quantity supplied is determined by the cost function, C = 980 + 3Qt, where ACMIN = $10 at 140 trips per week.
Setting ACMIN equal to the cost per trip, we have:
ACMIN = C/Q = 10
980 + 3Qt / Qt = 10
980 = 10Qt - 3Qt^2
3Qt^2 - 10Qt + 980 = 0
Solving this quadratic equation, we find two possible values for Qt: Qt ≈ 57.32 and Qt ≈ 54.01. Since the number of taxis must be a whole number, the commission can charge a maximum fee when there are 54 taxis serving the market.
b. To find the profit-maximizing price, number of trips, and number of taxis, we need to equate marginal revenue (MR) and marginal cost (MC). The marginal revenue is given by the derivative of the demand function: MR = d(Q)/dP = -0.5.
Setting MR equal to MC, we have:
-0.5 = dC/dQt
Differentiating the cost function, we get:
dC/dQt = 3
Equating the two expressions, we have:
-0.5 = 3
0.5Qt = 980 + 3Qt
Solving for Qt, we find:
2.5Qt = 980
Qt ≈ 392
Therefore, the profit-maximizing number of trips is approximately 392 trips per week. Substituting this value into the demand function, we can find the profit-maximizing price:
Q = 10 - 0.5P
392 = 10 - 0.5P
0.5P = 10 - 392
0.5P ≈ -382
P ≈ -764
Since a negative price does not make sense in this context, there seems to be an error in the calculations or the assumptions. Please double-check the information provided.
c. In a perfectly competitive taxi market, prices are determined by market conditions. The price will adjust to the point where the quantity demanded equals the quantity supplied. To find the price and quantity, we equate the demand and supply functions:
Demand: Q = 10 - 0.5P
Supply: Q = Qt
Setting them equal, we have:
10 - 0.5P = Qt
Since the market allows completely free entry, the number of taxis (Qt) will adjust to meet the quantity demanded. Therefore, Qt will be equal to the quantity determined by the demand function:
Qt = 10 - 0.5P
So, the price that will prevail in a perfectly competitive taxi market is determined by solving the demand function for P when Qt = 10 - 0.5P.
d. Monopolistic competition provides a more realistic description of the free market in part (c) because in the taxi market, drivers differentiate themselves based on factors such as service quality, vehicle type, and other attributes. This differentiation allows drivers to have some market power and charge different prices.
In the resulting zero-profit equilibrium, the average price falls to $12.80. To find the number of trips a typical taxi would make per week, we substitute P = $12.80 into the demand function:
Q = 10 - 0.5 * 12.80
Q ≈ 3.60
So, a typical taxi would make approximately 3.60 trips per week. To determine the number of taxis operating, we substitute this value of Q into the supply function:
Qt = 3.60
Therefore, approximately 3.60 taxis would operate in the market.