Question

A monopolist firm faces a demand with constant elasticity of \(-2.0 .\) It has a constant marginal cost of \(\$ 20\) per unit and sets a price to maximize profit. If marginal cost should increase by 25 percent, would the price charged also rise by 25 percent?

Short answer

No, the price charged would not rise by 25%. Due to the inelastic demand, the monopolist can absorb part of the cost increase without losing all of its sales, so the price rise would be less than 25%.

Step-by-step solution

Understand the concept

A firm facing a demand elasticity of -2.0 signifies inelastic demand because the absolute value of elasticity is greater than 1. Marginal cost is $20. As the monopolist desires to maximize profit, it sets price where its marginal revenue equals marginal cost (MR=MC).

Analyze price change with increased marginal cost

Rearrange the formula MR=MC to derive the Lerner Index which is (P-MC)/P = -1/EL, where EL is the price elasticity of demand and P is price. When the marginal cost (MC) increases by 25%, the new MC is $25. But due to inelastic demand, the monopolist doesn't need to increase the price by the same percentage, so the price rise would be less than 25%.

Key concepts

Demand Elasticity

Demand elasticity measures how sensitive the quantity demanded of a good is to its price change. It's a key concept for monopolists in price setting because it determines how much they can change their price without significantly losing customers. An elasticity of -2.0, as mentioned in our example, indicates an inelastic demand, where a given percentage increase in price leads to a smaller percentage decrease in quantity demanded. This gives a monopolist room to increase prices without facing a proportional decline in sales volume.

Marginal Cost

Marginal cost is the cost of producing one additional unit of a product. For a monopolist, knowing the marginal cost is crucial since it's used to determine the price level that maximizes profit. In our exercise, the marginal cost is set at $20. When a monopolist sets a price, they need to consider this cost to ensure each unit sold generates a profit. Changes in the marginal cost, like the 25% increase discussed, impact the pricing decision—a higher marginal cost might lead to higher prices to maintain profit margins.

Profit Maximization

Profit maximization is a fundamental objective for any monopolist. To achieve this, the firm finds the balance where marginal cost (MC) equals marginal revenue (MR), the extra income from selling one more unit of a good. This point ensures the firm is not overproducing, which would drive costs above revenue, or underproducing, which would mean missed profit opportunities. A monopolist leverages demand elasticity to find that sweet spot in price that maximizes the difference between total revenue and total costs.

Lerner Index

The Lerner Index is a measure of a firm's market power—the ability to set prices above marginal costs. It is calculated as \( (P - MC) / P \), where \( P \) represents price, and \( MC \) stands for marginal cost. The larger the value, the greater the firm's price-setting power, suggesting less competition. In the context of our exercise, it implies that with an inelastic demand, the firm does not have to raise prices proportionally to the increased marginal cost. This index helps understand the impact of cost changes on pricing and how much of that change can be passed on to consumers.