Question

When a monopolist identifies its profit-maximizing quantity of output, how does it decide what price to charge?

Short answer

A monopolist determines the profit-maximizing price by first identifying the profit-maximizing quantity of output where marginal revenue (MR) equals marginal cost (MC). Then, using the demand curve equation, \( P = a - bQ \), the monopolist calculates the corresponding price on the demand curve by substituting the profit-maximizing quantity, \(Q_{max}\), into the equation: \( P = a - bQ_{max} \). This resulting price is the profit-maximizing price the monopolist will charge for its product or service.

Step-by-step solution

Identify the monopolist's market structure and objectives

A monopolist is a firm that operates in a market with no close competitors, and consequently, has significant control over the price and quantity of a product or service. The primary objective of a monopolist is to maximize its profit, which is the difference between its total revenue and total costs.

Determine the profit-maximizing quantity of output

To find the profit-maximizing quantity, a monopolist must equate its marginal revenue (MR) with its marginal cost (MC). Mathematically, this can be expressed as: \( MR = MC \) Marginal revenue is the additional revenue generated from selling one more unit of a product, and marginal cost is the additional cost incurred from producing one more unit.

Calculate the monopolist's demand curve

The demand curve represents the relationship between the price and quantity demanded for a product. It is typically represented by the equation: \( P = a - bQ \) Where P is the price, Q is the quantity demanded, and a and b are constants. For a monopolist, the demand curve helps determine the price that will result in the profit-maximizing quantity.

Calculate the marginal revenue curve for the monopolist

To find the marginal revenue curve, differentiate the total revenue (TR) with respect to quantity (Q). Since the total revenue is the product of the price and quantity, i.e., \( TR = P \times Q \), we can substitute the demand curve equation: \( TR = (a - bQ) \times Q \) Once we differentiate TR with respect to Q, we get the marginal revenue curve: \( MR = a - 2bQ \)

Find the profit-maximizing price

After determining the profit-maximizing quantity by equating MR and MC, we can find the corresponding price on the demand curve using the profit-maximizing quantity as the value of Q in the demand curve equation: \( P = a - bQ_{max} \) The price calculated from this equation is the monopolist's profit-maximizing price, which it will charge for its product or service in the market.