a private garage owner has identified two distinct market segments: short term parkers and all day parkers with respective demand curves of Ps = 3-(Qs/200) and Pc = 2-(Qc/200). Here Pis the average hourly rate and Q is the number of cars parked at this price. The garage owner is considering charging different prices (on a per hour basis) for short term parking and all day parking. the capacity of the garage is 600 cars and the cost of adding cars up to this limit is negligible.
What price should he charge for each type of parker? How many type of each parkerwill use the garage at this prices?
Take a shot, what do you think.
Hint: you have a price-discriminating monopolist. First determine the Marginal revenue equations for each market. Marginal Cost is zero in both. So, allocate the marginal car to the market where MR is highest. Since the two demand curves have the same slope, at some point the two MR in each market will be the same -- at that point allocate the cars 50/50.
2(qs/200)=1-2(qc/200)
To determine the optimal prices for each type of parker and the number of each type of parker who will use the garage at these prices, we need to find the equilibrium quantity and price for each market segment.
1. Equilibrium for Short Term Parkers:
The demand curve for short term parkers is Ps = 3 - (Qs/200).
- Set the demand equal to the capacity of the garage:
3 - (Qs/200) = 600
(Qs/200) = 3 - 600
(Qs/200) = -597
- Solve for Qs:
Qs = -597 * 200
Qs = -119,400
Since a negative quantity is not possible, this implies that the demand for short term parkers is 0 at a price of 3.
2. Equilibrium for All Day Parkers:
The demand curve for all day parkers is Pc = 2 - (Qc/200).
- Set the demand equal to the capacity of the garage:
2 - (Qc/200) = 600
(Qc/200) = 2 - 600
(Qc/200) = -598
- Solve for Qc:
Qc = -598 * 200
Qc = -119,600
Again, a negative quantity is not possible, so the demand for all day parkers is also 0 at a price of 2.
Based on these calculations, the optimal prices for each type of parker would be:
- Short Term Parkers: Price (Ps) = $3 per hour
- All Day Parkers: Price (Pc) = $2 per hour
However, since there is no demand for either segment at these prices, the garage owner may need to adjust the prices to attract customers.
To determine the prices the garage owner should charge for each type of parker and the number of each parker that will use the garage at these prices, we need to find the profit-maximizing equilibrium.
Step 1: Calculate the Total Revenue (TR) for each market segment:
- For short term parkers: TRs = Ps * Qs
- For all day parkers: TRc = Pc * Qc
Step 2: Calculate the Total Cost (TC) of running the garage:
Since the cost of adding cars up to the limit is negligible and the garage has a capacity of 600 cars, we can assume that the cost is constant. Let's denote the Total Cost as TC.
Step 3: Calculate the Total Profit (TP):
TP = TRs + TRc - TC
TP = (Ps * Qs) + (Pc * Qc) - TC
Step 4: Differentiate Total Profit with respect to Qs and Qc to find the number of cars that will use the garage for each market segment at the equilibrium:
dTP/dQs = d(Ps * Qs)/dQs + d(Pc * Qc)/dQs - 0
dTP/dQc = d(Ps * Qs)/dQc + d(Pc * Qc)/dQc - 0
Step 5: Set the derivatives equal to zero and solve for Qs and Qc to find the equilibrium quantities:
dTP/dQs = 0 => d(Ps * Qs)/dQs + d(Pc * Qc)/dQs = 0
dTP/dQc = 0 => d(Ps * Qs)/dQc + d(Pc * Qc)/dQc = 0
Step 6: Once the equilibrium quantities of cars for each market segment are found, substitute these values back into the demand curves to determine the equilibrium prices:
Ps = 3 - (Qs/200)
Pc = 2 - (Qc/200)
By following these steps and solving the equations, you will be able to find the optimal prices for each type of parker and the number of each parker that will use the garage at those prices.