Question

A monopolist faces the demand curve \(P=11-Q\) where \(P\) is measured in dollars per unit and \(Q\) in thousands of units. The monopolist has a constant average \(\operatorname{cost}\) of \(\$ 6\) per unit. a. Draw the average and marginal revenue curves and the average and marginal cost curves. What are the monopolist's profit-maximizing price and quantity? What is the resulting profit? Calculate the firm's degree of monopoly power using the Lerner index b. A government regulatory agency sets a price ceiling of \(\$ 7\) per unit. What quantity will be produced, and what will the firm's profit be? What happens to the degree of monopoly power? c. What price ceiling yields the largest level of output? What is that level of output? What is the firm's degree of monopoly power at this price?

Short answer

For part a, the monopolist's profit-maximizing price and quantity are $2.5 and 8.5k units respectively, resulting in a profit of $-29.75k and a Lerner index of -1.4. For part b, with the price ceiling of $7 per unit, the quantity produced is 4k units, the profit is $4k, and the degree of monopoly power becomes 0.14. For part c, the price ceiling that yields the largest level of output is $6 per unit, which produces 5k units, and reduces the firm's degree of monopoly power to zero.

Step-by-step solution

Calculate the profit maximising quantity and price

The marginal revenue (MR) curve is obtained by taking the derivative of the total revenue (TR = P*Q), which comes from the demand curve. For P = 11 - Q, we have Q = 11 - P, thus TR = P*(11-P). The marginal revenue (MR) is then calculated by taking the derivative of TR with respect to Q, providing MR = 11 - 2P. Setting MR equal to MC gives the profit-maximizing quantity: 11 - 2P = 6, then P = (11 - 6) / 2 = 2.5. Substituting P into the demand curve gives the quantity: Q = 11 - P = 11 - 2.5 = 8.5

Calculate the firm's profit

The profit is given by \(\pi\) = TR - TC = P*Q - AC*Q = (P-AC)*Q = (2.5 - 6) * 8.5 = -29.75 (since \(AC = 6\))

Calculate the firm's degree of monopoly power using the Lerner index

The Lerner index, L, is calculated as L = (P - MC) / P = (2.5 - 6) / 2.5 = -1.4

Determine the quantity produced and the firm's profit under a price ceiling of $7

At the price ceiling of P = 7, we substitute this into the demand function to obtain Q = 11 - P = 11 - 7 = 4. The profit in this case is \(\pi\) = TR - TC = P*Q - AC*Q = (P-AC)*Q = (7 - 6) * 4 = 4. The Lerner index is L = (P - MC) / P = (7 - 6) / 7 = 0.14

Determine the price ceiling for the largest level of output

To maximize output, the price should be set equal to marginal cost, PC = MC = 6. The quantity in this case will be Q = 11 - P = 11 - 6 = 5. The firm's degree of monopoly power at this price (using the Lerner index) is L = (P - MC) / P = 0 since P = MC in this scenario

Key concepts

Marginal Revenue Curve

The marginal revenue (MR) curve is a visual representation showing how the revenue from selling one additional unit changes as the quantity of units sold increases. For a monopolist, this curve is crucial for determining the profit-maximizing level of output. The MR can be calculated by taking the derivative of total revenue (TR), which is itself a function of the demand curve. In our exercise, for a demand curve given by P = 11 - Q, the total revenue function becomes TR = P * Q = (11 - Q) * Q. The marginal revenue is then derived as MR = 11 - 2Q.

Understanding that the MR curve slopes downward is important because it reflects the fact that a monopolist faces a downward sloping demand curve. This implies that the firm must reduce its price to sell more units, which in turn decreases the additional revenue generated from selling each subsequent unit. In our case, the MR curve will intersect the marginal cost (MC) curve to determine the optimum quantity for profit maximization. This intersection is critical because it signals that beyond that point, the cost of producing an additional unit (MC) will exceed the revenue gained (MR) from selling it.

Marginal Cost Curve

The marginal cost (MC) curve represents the change in total costs associated with producing one more unit of a good or service. This is where the concept of diminishing returns often comes into play—which is not the case for our monopolist, who has a constant average cost of \(6 per unit. Under such circumstances, the marginal cost is equal to the average cost and remains constant. Thus, our monopolist's MC curve is horizontal at the price of \)6.

Profit maximization occurs at the level of output where the marginal revenue (MR) equals the marginal cost (MC). Hence, in our exercise, we've used the condition MR = MC to find the profit-maximizing price and quantity. The intersection of MR and MC curves provides two valuable pieces of information: the quantity of goods a monopolist should produce and the cost of producing that last unit.

Lerner Index

The Lerner index is a measure of a firm's monopoly power, defined as the extent to which a firm can mark up the price above marginal cost. The formula is L = (P - MC) / P, where P is the price charged by the monopolist, and MC is the marginal cost of production. The value of the Lerner index ranges between 0 and 1, where 0 represents a perfectly competitive market (no pricing power) and 1 indicates a pure monopoly (maximum pricing power).

In our exercise, the initial Lerner index is negative due to the firm’s average cost being higher than the price—implying that the firm is not generating profit at the calculated price of \(2.5. The Lerner index becomes positive under a price ceiling of \)7, as the firm is now able to make a profit. It's a helpful tool for understanding the degree of control a monopolist has over pricing in relation to its costs.

Price Ceiling

A price ceiling is a legal maximum price set by the government on a good or service, with the intent to prevent prices from reaching levels that are too high for consumers to afford. When a price ceiling is imposed on a monopolist, it can impact the firm's production decisions and profitability. In our exercise, with the price ceiling set at \(7, we observe that the monopoly is constrained to produce a quantity where the price is equal to \)7. This reduces the quantity produced to 4 units from the initial unconstrained profit-maximizing quantity of 8.5 units.

The imposition of a price ceiling also affects the monopolist's degree of monopoly power. As the price approaches the marginal cost, the firm's pricing power diminishes, which is reflected in a lower Lerner index. However, if the price ceiling is set below the average cost, the firm could sustain losses. Consequently, price ceilings can be a double-edged sword, aiming to protect consumers but potentially leading to underproduction or losses for the producer.

Monopoly Power

Monopoly power refers to the ability of a monopolist to set and maintain prices higher than in competitive markets, where firms have little to no influence over the market price. This power comes from the lack of close substitutes for the monopolist's product and barriers to entry that prevent other firms from entering the market. Strong monopoly power can result in higher prices for consumers, less choice, and can be indicative of inefficiency within an industry.

In our example, the monopolist initially has the power to set the price at $2.5, which is below the monopolist’s average cost. The exercise shows that monopoly power is not just about setting high prices; it's also about the firm's capacity to cover its costs and earn a profit. The Lerner index and the effect of price ceilings on profitability are closely related to the extent of monopoly power. When analyzing monopolies, it's crucial to consider both the demand side (consumers) and the supply side (producers) to understand the implications of monopoly power on different market outcomes.