A local video store estimates their average customer's demand per year is Q = 7 - 2P, and knows the marginal cost of each rental is $0.5. How much should the store charge for each rental if it engages in optimal two-part pricing?
Answer
A. $0.35
B. $0.5.
C. $0.7.
D. $1.00.
0.5
To determine the optimal price for each rental in a two-part pricing strategy, we need to find the point where marginal cost equals marginal revenue.
The demand function for the store's customers is Q = 7 - 2P, where Q represents the quantity demanded and P represents the price.
To find the marginal revenue, we take the derivative of the demand function with respect to P:
MR = dQ/dP = -2
The marginal cost of each rental is given as $0.5.
Setting MR equal to the marginal cost, we have:
-2 = 0.5
Solving for P, we find:
P = -2/0.5 = -4
Since price cannot be negative, we discard the negative value and take the absolute value to get:
P = 4
Therefore, the store should charge $4 for each rental in the two-part pricing strategy.
However, none of the options given match the correct answer.
To determine the optimal pricing strategy, we need to consider the two-part pricing model. In this model, the store charges a fixed fee (F) plus a per-unit price (P).
The first step is to calculate the demand equation. Given that the average customer's demand per year is Q = 7 - 2P, we can rearrange the equation to solve for price:
P = (7 - Q) / 2
Next, we need to calculate the optimal quantity. To maximize profit, the store should set the quantity where marginal cost (MC) equals marginal revenue (MR). In this case, the marginal cost of each rental is $0.5.
MR = P + (dQ / dP) * MC
Taking the derivative of the demand equation with respect to price:
dQ / dP = -2
Now we have:
MR = P + (-2) * $0.5
MR = P - $1
Setting MR equal to MC and solving for P:
P - $1 = $0.5
P = $1.5
Lastly, we can substitute the calculated price back into the demand equation to find the quantity:
Q = 7 - 2(1.5)
Q = 7 - 3
Q = 4
Therefore, the optimal pricing strategy for the store is to charge a fixed fee of $1 and a per-unit price of $1.5.
Since the given answer options do not match the calculated prices, none of the provided options are correct.