Question

A monopolist faces the following demand curve: \\[ Q=144 / P^{2} \\] where \(Q\) is the quantity demanded and \(P\) is price. Its average variable cost is \\[ \mathrm{AVC}=Q^{1 / 2} \\] and its fixed cost is 5 a. What are its profit-maximizing price and quantity? What is the resulting profit? b. Suppose the government regulates the price to be no greater than \(\$ 4\) per unit. How much will the monopolist produce? What will its profit be? c. Suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. What price will accomplish this goal?

Short answer

The profit-maximizing price and quantity need to be calculated using calculus and profit maximization condition. When the government imposes a price ceiling, the quantity produced and profit will change accordingly. If the government wants to induce maximum output, the price ceiling should be set as the AVC.

Step-by-step solution

Establish the Monopolist's Profit Maximization Strategy

First, to solve for the profit-maximizing price and quantity, the cost function should be identified using the average variable cost (AVC) and fixed cost (FC). The total variable cost (TVC) can be derived by integrating the AVC (i.e., \( \int Q^{1 / 2} dQ \)), resulting in \( \frac{2}{3} Q^{3/2} \). Adding FC to TVC will give us the total cost (TC). TC= \( \frac{2}{3} Q^{3/2} \) + 5. The revenue function, which is the price times the quantity indicates \( R = P * Q = 144 / P \). Then, derive the profit function which is the difference between the revenue and cost function indicates \( \pi = R - TC \). The monopolist maximizes profit where Marginal Cost = Marginal revenue or where the profit function is at its highest point.

Calculate profit-maximizing quantity and price

Differentiate the profit function with respect to \( Q \) and set the result equal to zero. Solve for \( Q \) which represents the profit-maximizing quantity. From the demand equation, substitute \( Q \) in terms of \( P \) to find the profit-maximizing price.

Government Regulation: Maximum Price

Next, consider the case where the government imposes a maximum price of $4 per unit. Use the demand equation to substitute this price \( P = 4 \) and solve for \( Q \). Subsequently, substitute the quantity in the profit equation to find the new profit.

Government Regulation: Price Ceiling for Maximum Output

Finally, suppose the government wants to set a ceiling price that induces the monopolist to produce the largest possible output. One approach is to set the price equal to the AVC. When the price is equal to the AVC, the monopolist can cover its variable costs and will therefore not shutdown, leading to maximal output. In this case, use the AVC equation \( Q^{1 / 2} \) to solve for \( P \).