Question

A monopolist can produce at constant average and marginal costs of \(A C=M C=5 .\) The firm faces a market demand curve given by \(Q=59-P\) a. Calculate the profit-maximizing price-quantity combination for the monopolist. Also calculate the monopolist's profits. b. What output level would be produced by this industry under perfect competition (where price \(=\text { marginal cost }) ?\) c. Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist's profits and the consumer surplus received in case (a). What is the value of the "deadweight loss" from monopolization?

Short answer

Step 1: Derive the MR curve P = 59 - Q TR = P x Q = (59 - Q) x Q MR = d(TR) / dQ = 59 - 2Q Step 2: Find the profit-maximizing quantity (Q*) MR = MC 59 - 2Q = 5 2Q = 54 Q* = 27 Step 3: Calculate the profit-maximizing price (P*) P = 59 - Q* P* = 59 - 27 = 32 Step 4: Calculate the monopolist's profit Profit = (P* - MC) x Q* Profit = (32 - 5) x 27 = 729 Output level under perfect competition: P = MC Q = 59 - 5 = 54 Consumer surplus under perfect competition: CS = 0.5 x base x height CS = 0.5 x (59 - 5) x 54 = 1,458 Consumer surplus under monopoly: CS = 0.5 x base x height CS = 0.5 x (59 - 32) x 27 = 364.5 Compare consumer surplus and profits between monopoly and perfect competition: 1,458 > 364.5 + 729 Deadweight loss: DWL = CS (perfect competition) - (CS (monopoly) + Monopolist's profit) DWL = 1,458 - (364.5 + 729) = 364.5 The profit-maximizing price-quantity combination for the monopolist is P* = 32 and Q* = 27, with profits of 729. The output level under perfect competition is 54. The consumer surplus under perfect competition is 1,458, which is greater than the sum of the monopolist's profits and consumer surplus (364.5 + 729) in the monopoly case. The deadweight loss from monopolization is 364.5.

Step-by-step solution

Derive the marginal revenue (MR) curve

To find the MR curve, start by expressing the demand curve in terms of price: P = 59 - Q. Then, find the total revenue (TR) by multiplying price and quantity: TR = P x Q = (59 - Q) x Q. Differentiate this expression with respect to Q to obtain the MR curve: MR = d(TR) / dQ.

Find the profit-maximizing quantity (Q*)

Profit maximization occurs when MR = MC. Set MR = MC and solve for Q*.

Calculate the profit-maximizing price (P*)

Find the profit-maximizing price by substituting Q* into the demand curve, P = 59 - Q*.

Calculate the monopolist's profit

Calculate the monopolist's profit by finding the difference between total revenue and total cost at the profit-maximizing price-quantity combination. #b. What output level would be produced by this industry under perfect competition (where price = marginal cost)?#

Output level under perfect competition

In perfect competition, price equals marginal cost (P = MC). Solve P = MC for quantity, Q. #c. Calculate the consumer surplus obtained by consumers in case (b). Show that this exceeds the sum of the monopolist's profits and the consumer surplus received in case (a). What is the value of the "deadweight loss" from monopolization?#

Consumer surplus under perfect competition

Find consumer surplus (CS) under perfect competition by calculating the area of the triangle formed by the demand curve, the perfectly competitive price, and the quantity produced under perfect competition. CS = 0.5 x base x height.

Consumer surplus under monopoly

Calculate the consumer surplus under the monopoly by finding the area of the triangle formed by the demand curve, the monopolistic price, and the quantity produced under the monopoly: CS = 0.5 x base x height.

Compare consumer surplus and profits between monopoly and perfect competition

Show that the consumer surplus in case (b) (perfect competition) is higher than the sum of the monopolist's profits and consumer surplus in case (a) (monopoly).

Calculate the deadweight loss

Find the deadweight loss (DWL) from the monopolization by calculating the difference between the consumer surpluses in case (b) and case (a). DWL = CS (perfect competition) - (CS (monopoly) + Monopolist's profit).

Key concepts

Marginal Revenue

Understanding marginal revenue (MR) is crucial for analyzing a firm's revenue patterns. By definition, MR is the additional revenue that a firm makes from selling one more unit of a product. To determine MR mathematically, we calculate the derivative of the total revenue (TR) with respect to quantity (Q). In the context of our monopolist example, TR is defined as the product of price (P) and quantity, which can be expressed as TR = (59 - Q) * Q. Through differentiation, we acquire the MR curve, indicating how much revenue is added with each additional unit sold. Knowing MR is instrumental for businesses to decide how much to produce to maximize their profits.

Profit Maximization

Profit maximization is a fundamental goal for most firms, achieved by finding the perfect balance between production costs and revenue. In our scenario, a monopolist determines this balance by equating marginal revenue (MR) to marginal cost (MC), which is given as a constant $5 in this case. Setting MR equal to MC and solving for the quantity, we get the profit-maximizing output level. Subsequently, the corresponding price can be found by plugging this quantity into the demand equation. This price-quantity pair ensures maximum possible profit for our monopolist under the given market conditions.

Consumer Surplus

Consumer surplus (CS) is the difference between what consumers are willing to pay for a good and what they actually pay. It represents the net benefit to consumers, effectively the area under the demand curve and above the price they pay. For our perfect competition scenario, the CS can be visualized as the triangular area below the demand curve and above the equilibrium price, which equals marginal cost. Mathematically, it's calculated as half of the base, which is the quantity sold, multiplied by the height, which is the difference between the highest price consumers are willing to pay and the market price. Consumer surplus is typically larger in perfect competition than in a monopoly, reflecting the higher prices and restricted output in monopolistic markets.

Deadweight Loss

Deadweight loss (DWL) is the loss of economic efficiency when the equilibrium for a good or a service is not achieved or is not achievable. In the case of monopolization, DWL occurs because the monopolist sets a higher price and lower quantity than what would prevail in a perfectly competitive market. This results in potential trades that do not occur—transactions that would have been mutually beneficial in a competitive market. The deadweight loss is quantified by calculating the difference in consumer surplus between the perfect competition and monopoly scenarios and then subtracting the monopolist's profit. This calculation indicates the cost of inefficiency imposed on the society due to the monopolist's price and output decisions.

Perfect Competition

Perfect competition is a market structure characterized by many firms selling identical products, where no single firm can influence the market price. Under perfect competition, firms are price takers, meaning they accept the market price determined by the forces of demand and supply. In our exercise, under perfect competition, the price equals marginal cost (P=MC), allowing us to find the output level which maximizes social welfare, as it aligns with consumers' demand perfectly. It's the ideal scenario for efficiency, leading to maximized consumer surplus and no deadweight loss—attributes starkly contrasting a monopolistic market structure.