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 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

The Lerner Index for the original situation is 0.2941, confirming some monopoly power. By introducing a price ceiling of $7, output increases to 4 thousand units, profit decreases to $4 thousand and the degree of monopoly power is reduced to 0.1429. A price ceiling of $6 results in maximized output (5 thousand units), no profit and no monopoly power (Lerner Index is zero).

Step-by-step solution

Identifying Average Revenue and Marginal Revenue

The average revenue (AR) and marginal revenue (MR) curves for a monopolist are derived from the demand curve. Given the demand curve as \(P=11-Q\), we can infer that the price acts as the average revenue (\(AR=P\)). So, \(AR=11-Q\). For the marginal revenue (MR), the equation is \(MR=11-2Q\) because marginal revenue slope is double that of average revenue.

Identifying Average Cost and Marginal Cost

The monopolist's average cost is given as a constant, which also acts as the marginal cost (since no specific cost function has been given separately). Therefore, \(AC=MC=$6\). The marginal cost (MC) is constant and equals the average cost.

Profit Maximizing Quantity and Price

To find the profit-maximizing quantity, equate the marginal cost (MC) to the marginal revenue (MR). Solving \(11-2Q=6\) gives \(Q=2.5\) (in thousands of units). Substituting \(Q=2.5\) in the demand curve gives \(P=8.5\) (profit-maximizing price in dollars per unit).

Calculating Profit

The profit can be calculated by subtracting total cost from total revenue. The profit per unit is price minus average cost. So, for \(Q=2.5\) and \(P=8.5\), the profit per unit is \(P-AC=8.5-6=$2.5\). The total profit is then the profit per unit times quantity: \(TP=2.5*2.5=$6.25\) thousand.

Calculating Lerner Index

The Lerner Index is calculated as \(L=(P-MC)/P\). Substituting the values we found, \(L=(8.5-6)/8.5=0.2941\). So the degree of monopoly power is 0.2941.

Investigating the Price Ceiling of $7

When the price ceiling is set to $7, the price itself is 7. From the demand function \(P=11-Q\), we find \(Q=4\). The profit per unit is \(P-AC=7-6=1\), and total profit is \(TP=1*4=$4\) thousand. The degree of monopoly power \(L=(7-6)/7=0.1429\), is reduced.

Determining the Price Ceiling for Largest Output

The largest output will arise when price equals average cost. Hence, the price that yields the highest output is $6 (equal to the cost of production). The resulting level of output, and thus the quantity, can be determined as \(Q=11-P=5\). As the firm would be generating zero economic profit, the Lerner Index would be zero.

Key concepts

Marginal Revenue

Marginal revenue is a vital concept in understanding monopolies. It refers to the extra revenue that a firm gains when it sells one additional unit of a product. For a monopolist, the marginal revenue is less than the price of the product due to the downward-sloping demand curve. This happens because the monopolist must lower the price for all units to sell more, not just the extra one. In terms of our problem, given the demand curve as \( P = 11 - Q \), the marginal revenue curve is derived as \( MR = 11 - 2Q \). This shows that marginal revenue decreases twice as fast as quantity compared to the decline in price. Understanding this relationship helps in determining the point where marginal revenue equals marginal cost, which is where profit is maximized.

Average Cost

Average cost refers to the total cost of production divided by the number of units produced, providing a per-unit cost of production. In this exercise, it is given as a constant \( \\(6 \) per unit. This is a unique situation because usually, costs change with the level of output. However, in this problem, both the average cost \( (AC) \) and marginal cost \( (MC) \) remain constant at \( \\)6 \). This simplifies the analysis since the cost per unit doesn't change with the number of units produced. Knowing the average cost is essential when calculating profit, as profit per unit is determined by subtracting this cost from the selling price.

Lerner Index

The Lerner Index is a measure of a firm's market power. It indicates how much a firm can mark up its prices over marginal cost. The formula for the Lerner Index is \( L = (P - MC) / P \), where \( P \) is the price of the product, and \( MC \) is the marginal cost. In our scenario, when the price is \( \\(8.5 \) and the marginal cost is \( \\)6 \), the Lerner Index is calculated as \( L = (8.5 - 6) / 8.5 = 0.2941 \). This value reflects the degree of monopoly power, showing the extent to which the monopolist can charge a price above the marginal cost. When a price ceiling is applied, for example \( \$7 \), the Lerner Index drops to \( 0.1429 \), indicating reduced market power.

Price Ceiling

A price ceiling is a regulatory measure that limits the maximum price that can be charged for a product, aimed at protecting consumers from high monopoly prices. In this scenario, a price ceiling of \( \$7 \) is introduced, impacting the monopolist's strategy. With a price ceiling, the market equilibrium adjusts to the new maximum price, producing more units than in the unregulated monopoly scenario. Specifically, the quantity produced rises to \( Q = 4 \) units, from the lower unregulated quantity. Under the price ceiling, the firm generates less profit per unit, and the overall monopoly power, as measured by the Lerner Index, is diminished to \( 0.1429 \). Price ceilings can reshape the competition dynamics and benefit consumers by increasing the quantity supplied.