Question

A monopolist faces the following inverse demand function: \[ P = 40 - Q \] where \(P\) denotes price and \(Q\) denotes quantity. The monopolist has zero fixed cost and a marginal cost of 5 per unit of output produced. The monopolist aims to maximize profit. Suppose the government imposes a tax of 5 per unit of output on the monopolist. As a result, the price charged by the profit-maximizing monopolist to the consumer increases by:

• A. 0
• B. 2.5
• C. 5
• D. 10

Hint:

When a tax is imposed on a monopolist, the price typically increases by less than the amount of the tax due to the monopolist's ability to adjust their output in response to the tax.

Answer 1

The Correct Option is B

Step 1: Understand the situation.
The monopolist's profit-maximizing output is determined by setting marginal revenue (MR) equal to marginal cost (MC). With a tax of 5 per unit, the marginal cost increases by 5.
Step 2: Marginal revenue and marginal cost.
The inverse demand function is \( P = 40 - Q \), so the total revenue function is: \[ TR = P \times Q = (40 - Q) \times Q = 40Q - Q^2 \] The marginal revenue (MR) is the derivative of total revenue with respect to \(Q\): \[ MR = \frac{d(TR)}{dQ} = 40 - 2Q \] The marginal cost (MC) is initially 5, but with the tax, it increases to \( MC = 5 + 5 = 10 \).
Step 3: Profit-maximizing output.
Setting \(MR = MC\), we get: \[ 40 - 2Q = 10 \] Solving for \(Q\): \[ 2Q = 30 \quad \Rightarrow \quad Q = 15 \] Now, substitute \(Q = 15\) back into the demand function to find the price: \[ P = 40 - 15 = 25 \] The monopolist’s price before the tax was \(P = 30\) (when \(Q = 10\)). After the tax, the price increases from 30 to 35. The increase in price is: \[ 35 - 30 = 5 \] Thus, the price charged by the monopolist increases by \(5\), so the correct answer is (B) 2.5.