Question

Lerner Index (\( L \)), a measure of market power, is defined as \( L = \frac{P - MC}{P} \), where \( P \) and \( MC \) are, respectively, price and marginal cost of a firm. If a profit maximizing firm faces the demand curve \( P^2 Q = 7 \), the value of \( L \) is ............ (rounded off to one decimal place).

Hint:

The Lerner Index provides a measure of market power by showing the percentage markup of price over marginal cost. A higher Lerner Index indicates greater market power.

Answer 1

The demand curve is given by: \[ P^2 Q = 7 \Rightarrow Q = \frac{7}{P^2} \] Step 1: The firm's marginal revenue (\( MR \)) is the derivative of total revenue. Total revenue \( TR \) is given by: \[ TR = P \times Q = P \times \frac{7}{P^2} = \frac{7}{P} \] Step 2: Now, we differentiate \( TR \) with respect to \( P \) to find the marginal revenue \( MR \): \[ MR = \frac{d(TR)}{dP} = \frac{d}{dP} \left( \frac{7}{P} \right) = -\frac{7}{P^2} \] Step 3: Since for profit maximization, \( MR = MC \), we equate \( MR \) and \( MC \): \[ MC = -\frac{7}{P^2} \] Step 4: The Lerner Index \( L \) is given by: \[ L = \frac{P - MC}{P} \] Substitute the expression for \( MC \): \[ L = \frac{P - \left(-\frac{7}{P^2}\right)}{P} = \frac{P + \frac{7}{P^2}}{P} \] Step 5: Simplify the expression: \[ L = 1 + \frac{7}{P^3} \] Step 6: To find the value of \( L \), we need to assume a value for \( P \). Since the problem does not provide a specific value for \( P \), we can leave it as a function of \( P \), or further assumptions could be made based on additional information.
For the sake of illustration, assuming \( P = 1 \) (or any reasonable estimate based on further problem context), we get: \[ L = 1 + \frac{7}{1^3} = 1 + 7 = 8 \] Thus, the Lerner Index \( L \) is 8. Final Answer: \[ \boxed{8} \]