A city must decide whether to build a downtown parking garage and what rate to charge. It is

considering two rates: a flat $1.50-per-hour rate or an all-day rate averaging $1 per hour (based on a $10 daily rate and an average 10-hour stay). Parking demand is given by Q =
900-300P, where Q is the number of cars in the garage each hour and P is the hourly rate. The
capital cost of the garage is estimated to be $20 million and its annual operating cost to be $0.62 million (regardless of the number of cars utilizing it) over its estimated 40-year life. The city's discount rate is 8%.
a) Sketch the demand curve (per hour) and calculate total benefits—the sum of
consumer surplus and revenue—from the garage under either rate. (Multiply by 10
hours per day and 260 working days per year to find annual values). Should the
facility be built? If so, which of the two rates should be charged?
b) Could a private developer profitably build and operate the garage? Which of the two
rates would it set? (Assume it faces the same demand, costs, and discount rate as the city).

a) The demand curve is given by Q = 900 - 300P, where Q is the number of cars in the garage each hour and P is the hourly rate.

Under the flat $1.50-per-hour rate, the total benefits would be equal to the consumer surplus plus revenue, which is equal to (900 - 300*1.50)*1.50 = $1125. Under the all-day rate averaging $1 per hour, the total benefits would be equal to (900 - 300*1)*1 = $600.

Given that the capital cost of the garage is estimated to be $20 million and its annual operating cost to be $0.62 million, the facility should be built. The all-day rate averaging $1 per hour should be charged, as it yields higher total benefits than the flat $1.50-per-hour rate.

b) A private developer could profitably build and operate the garage. It would set the all-day rate averaging $1 per hour, as it yields higher total benefits than the flat $1.50-per-hour rate.

a) Since Q = 900-300P, the demand curve is downward sloping. Let's calculate the total benefits under each rate.

For the $1.50-per-hour rate:
Consumer Surplus = 0.5*(900-300*1.50)*10*260 = $1,170,000
Revenue = $1.50*10*260 = $3,900

Total Benefits = Consumer Surplus + Revenue = $1,170,000 + $3,900 = $1,173,900

For the all-day rate:
Consumer Surplus = 0.5*(900-300*(10/10))*10*260 = $975,000
Revenue = $10*260 = $2,600

Total Benefits = Consumer Surplus + Revenue = $975,000 + $2,600 = $977,600

Comparing the two rates, the total benefits are higher for the $1.50-per-hour rate. Therefore, the facility should be built and the rate of $1.50 per hour should be charged.

b) To determine if a private developer can profitably build and operate the garage, we need to consider the costs and revenue.

For the $1.50-per-hour rate:
Revenue = $1.50*10*260 = $3,900
Operating Cost = $0.62 million

Profit = Revenue - Operating Cost = $3,900 - $0.62 million = -$616,100 (Loss)

For the all-day rate:
Revenue = $10*260 = $2,600
Operating Cost = $0.62 million

Profit = Revenue - Operating Cost = $2,600 - $0.62 million = -$617,400 (Loss)

Both rates result in losses for a private developer. Therefore, a private developer would not find it profitable to build and operate the garage, regardless of the rate chosen.

a) To sketch the demand curve, we can start by substituting the given formula Q = 900-300P into a table to calculate the number of cars in the garage at different hourly rates. Let's assume 5 different hourly rates ranging from $0 to $2.

At P = $0: Q = 900 - 300(0) = 900 cars per hour
At P = $0.50: Q = 900 - 300(0.50) = 750 cars per hour
At P = $1.00: Q = 900 - 300(1.00) = 600 cars per hour
At P = $1.50: Q = 900 - 300(1.50) = 300 cars per hour
At P = $2.00: Q = 900 - 300(2.00) = 0 cars per hour

Now we can plot these points on a graph with P (hourly rate) on the x-axis and Q (number of cars) on the y-axis.

The plot will show a downward-sloping line starting at (0,900) and ending at (2,0). Let's label this line as the "demand curve".

To calculate the total benefits, we need to find the consumer surplus and revenue for each rate.
Consumer Surplus: Consumer surplus is the difference between what consumers are willing to pay (based on their demand) and what they actually pay for a good or service.
Revenue: Revenue is the total amount of money collected from consumers.

For the flat $1.50-per-hour rate:
Consumer Surplus = (1/2) x (300)(1.50) = $225 per hour
Revenue = (300)(1.50) = $450 per hour

For the all-day rate averaging $1 per hour:
Consumer Surplus = (1/2) x (600 - 10 - 1)(1) + (1/2) x (300 - 10 - 1)(2) = $285 per hour
Revenue = (600 - 10 - 1)(1) + (300 - 10 - 1)(2) = $885 per hour

To find annual values, we multiply the hourly values by 10 hours per day and 260 working days per year.

For the flat $1.50-per-hour rate:
Consumer Surplus = $225 x 10 hours x 260 days = $585,000 per year
Revenue = $450 x 10 hours x 260 days = $1,170,000 per year
Total Benefits = Consumer Surplus + Revenue = $585,000 + $1,170,000 = $1,755,000 per year

For the all-day rate averaging $1 per hour:
Consumer Surplus = $285 x 10 hours x 260 days = $741,000 per year
Revenue = $885 x 10 hours x 260 days = $2,301,000 per year
Total Benefits = Consumer Surplus + Revenue = $741,000 + $2,301,000 = $3,042,000 per year

To determine whether the facility should be built and which rate to charge, we compare the total benefits to the costs of building and operating the garage.

Costs:
Capital Cost = $20 million
Annual Operating Cost = $0.62 million

Discount Rate = 8%

Present Value of Costs = Capital Cost + (Annual Operating Cost / Discount Rate)
Present Value of Costs = $20 million + ($0.62 million / 0.08) = $27.75 million

Comparing the total benefits of each rate:
For the flat $1.50-per-hour rate, Total Benefits = $1,755,000 per year
For the all-day rate averaging $1 per hour, Total Benefits = $3,042,000 per year

Since both rates' total benefits exceed the present value of costs ($27.75 million), the facility should be built. The higher total benefits of $3,042,000 per year suggest that the all-day rate averaging $1 per hour should be charged.

b) To determine if a private developer could profitably build and operate the garage, we need to calculate the present value of benefits.

Present Value of Benefits = Total Benefits / Discount Rate
For the flat $1.50-per-hour rate: Present Value of Benefits = $1,755,000 / 0.08 = $21,937,500
For the all-day rate averaging $1 per hour: Present Value of Benefits = $3,042,000 / 0.08 = $38,025,000

Comparing the present value of benefits to the present value of costs ($27.75 million), both rates generate higher present value of benefits. Therefore, a private developer could profitably build and operate the garage.

If a private developer were to operate the garage, the rate that it would set should be the one that maximizes the present value of benefits. In this case, the all-day rate averaging $1 per hour generates a higher present value of benefits ($38,025,000), making it the more profitable option for a private developer.

To answer these questions, we need to follow a step-by-step approach:

a) Sketching the demand curve:
The demand curve is given by Q = 900 - 300P, where Q is the number of cars in the garage each hour and P is the hourly rate. To sketch this curve, we can assign some values to P and calculate the corresponding values for Q. Let's assume P values of $0, $1, $2, $3, and $4 per hour:

When P = $0, Q = 900 - 300(0) = 900
When P = $1, Q = 900 - 300(1) = 600
When P = $2, Q = 900 - 300(2) = 300
When P = $3, Q = 900 - 300(3) = 0
When P = $4, Q = 900 - 300(4) = -300

Since negative values for Q are not meaningful in this context, we can ignore the $4 per hour rate. Plotting the remaining P and Q pairs on a graph will give us the demand curve.

To calculate the total benefits, we need to find the consumer surplus and revenue for each rate and multiply them by the number of hours in a day (10) and the number of working days in a year (260).

For the flat $1.50-per-hour rate:
Consumer Surplus = (1/2) * Q * (Pflat - 0) = (1/2) * (900 - 300P) * (1.5 - 0)
Revenue = Pflat * Q = 1.5 * (900 - 300P)

For the all-day rate averaging $1 per hour:
Consumer Surplus = (1/2) * Q * (Pall-day - 0) = (1/2) * (900 - 300P) * (1 - 0)
Revenue = Pall-day * Q = 10 * (900 - 300P)

To calculate the total benefits, we sum the consumer surplus and revenue:

Total Benefits for the flat $1.50-per-hour rate = Consumer Surplus + Revenue = (1/2) * (900 - 300P) * (1.5 - 0) + 1.5 * (900 - 300P)
Total Benefits for the all-day rate averaging $1 per hour = (1/2) * (900 - 300P) * (1 - 0) + 10 * (900 - 300P)

Next, we compare the total benefits for each rate and see which one is higher. If the total benefits are positive, it indicates that the facility should be built.

b) To determine whether a private developer can profitably build and operate the garage, we compare the costs with the total benefits. If the total benefits exceed the costs, it would be profitable. We use the same total benefits calculated in part a) and compare them with the costs, which include the capital cost and annual operating costs over the estimated 40-year life. Additionally, we need to apply the discount rate of 8% to find the present value of costs and benefits.

If the private developer decides to charge a different rate, they need to set it in a way that maximizes total benefits.

By following these steps, we can find the answers to both parts of the question.