If the inverse demand function for a monopoly’s product is p = 100 – 2Q,
then the firm’s marginal revenue function is
A -2
B 100-4Q.
C 200-4Q.
D 200-2Q.
The correct answer and explanation is :
The correct answer is B) 100 – 4Q.
Explanation:
In order to find the marginal revenue (MR) function for a monopoly, we need to understand the relationship between the price (p), quantity (Q), and total revenue (TR). The total revenue is calculated as:
[
TR = p \times Q
]
Step 1: Inverse Demand Function
Given the inverse demand function:
[
p = 100 – 2Q
]
This expresses the price at which the firm can sell a given quantity (Q). The monopoly sets its price based on the quantity sold, so to calculate the total revenue, we multiply the price by the quantity:
[
TR = (100 – 2Q) \times Q
]
Step 2: Simplifying Total Revenue (TR)
Now, expand the expression for total revenue:
[
TR = 100Q – 2Q^2
]
Step 3: Marginal Revenue (MR)
Marginal revenue is the derivative of total revenue with respect to quantity (Q). To find the marginal revenue, differentiate the total revenue function:
[
MR = \frac{d(TR)}{dQ} = \frac{d(100Q – 2Q^2)}{dQ}
]
Differentiate each term:
[
MR = 100 – 4Q
]
Thus, the marginal revenue function is MR = 100 – 4Q.
Conclusion:
The marginal revenue function for this monopoly is 100 – 4Q, which corresponds to option B.
The reason why the marginal revenue curve is steeper than the demand curve (with a coefficient of ( -4Q ) instead of ( -2Q )) is because, for monopolies, to sell an additional unit, they must lower the price not only on the new unit but also on all the previous units. Therefore, the marginal revenue decreases at a faster rate than the price.