Question

Suppose the government wishes to combat the undesirable allocational effects of a monopoly through the use of a subsidy. a. Why would a lump-sum subsidy not achieve the government's goal? b. Use a graphical proof to show how a per-unit-of-output subsidy might achieve the government's goal. c. Suppose the government wants its subsidy to maximize the difference between the total value of the good to consumers and the good's total cost. Show that, to achieve this goal, the government should set \\[\frac{t}{P}=-\frac{1}{e_{Q, P}},\\] where \(t\) is the per-unit subsidy and \(P\) is the competitive price. Explain your result intuitively.

Short answer

Answer: A per-unit subsidy reduces the marginal cost of production for every additional unit produced, which encourages the monopolist to produce more and drives the price down. This moves the market closer to the competitive equilibrium price and quantity. The relationship between the per-unit subsidy (t), competitive price (P), and price elasticity of demand (e_{D, P}) is given as: \(\frac{t}{P} = -\frac{1}{e_{D, P}}\) This means that the government should set the per-unit subsidy inversely proportional to the price elasticity of demand. When demand is inelastic (price elasticity is low), a smaller subsidy is required, while a larger subsidy is needed when demand is elastic (price elasticity is high). This policy helps maximize welfare by reducing allocational inefficiencies of monopolies while considering the price responsiveness of consumers.

Step-by-step solution

a. Understanding Lump-sum Subsidy

A lump-sum subsidy is a fixed amount given to the monopolist regardless of the quantity produced. This kind of subsidy will have no effect on the monopolist's marginal cost, meaning that it will not alter the monopolist's production decision. Consequently, the monopolist will continue to produce the same quantity and charge the same price as before, and the allocational inefficiency of the monopoly will remain unaffected.

b. Graphical Proof for Per-unit Subsidy

A per-unit subsidy, on the other hand, reduces the marginal cost of production for every additional unit produced. This pushes the monopolist to produce more, driving the price down, which moves the market closer to the competitive equilibrium price and quantity. To analyze this, follow these steps in the graph: 1. Draw the demand curve (D) and monopoly's marginal revenue (MR) curve. 2. Draw the monopoly's initial marginal cost curve (MC1) and identify the initial monopoly equilibrium point (E1, where MR = MC1). 3. Now, add the per-unit subsidy to the initial marginal cost curve, resulting in a new marginal cost curve (MC2 = MC1 - t). 4. Identify the new equilibrium point (E2, where MR = MC2) and observe that the produced quantity has increased and the price has decreased compared to the initial equilibrium point E1. As shown, the per-unit subsidy shifts the marginal cost curve downwards and hence moves the market closer to the competitive equilibrium, which is the government's goal.

c. Relationship between Per-unit Subsidy, Competitive Price, and Price Elasticity of Demand

We are asked to find the relationship between the per-unit subsidy (t), competitive price (P), and price elasticity of demand (e_{D, P}) to maximize the difference between the total value of the good to consumers and the total cost of producing the good. To maximize this difference, we need to equate the marginal benefit (MB) to the marginal cost (MC) of producing the good, which is the competitive equilibrium condition. Since the demand curve represents the marginal benefit, we have: \(MB = MC\) At competitive equilibrium, the price which the consumers are willing to pay is equal to the price which the monopolist is incurring, thus incorporating the per-unit subsidy. In other words: \(P = MC + t\) Rearranging the equation gives: \(\frac{t}{P} = 1 - \frac{MC}{P}\) Now let's relate the price elasticity of demand with the ratio \(\frac{MC}{P}\). Price elasticity of demand is defined as the percentage change in quantity demanded in response to a one percent change in price, or: \(e_{D, P} = \frac{\% \Delta Q_{D}}{\% \Delta P}\) Applying the definition for elasticity in terms of the demand curve's slope: \(e_{D, P} = \frac{P}{Q_{D}} \cdot \frac{\partial Q_{D}}{\partial P}\) We can rewrite the elasticity of demand as: \( - \frac{\partial Q_{D}}{\partial P} = e_{D, P} \cdot \frac{Q_{D}}{P}\) Differentiating the demand curve, which equates marginal benefit with price: \(\frac{\partial MB}{\partial P} = - \frac{\partial Q_{D}}{\partial P}\) Use this identity to relate price elasticity to the ratio \(\frac{MC}{P}\): \(\frac{MC}{P} = 1 - \frac{1}{-e_{D, P}}\) Finally, we get the desired relationship: \(\frac{t}{P} = -\frac{1}{e_{D, P}}\) Intuitively, this result suggests that the government should set the per-unit subsidy inversely proportional to the price elasticity of demand. When the demand is inelastic (price elasticity is low), the government should give a smaller subsidy, whereas when the demand is elastic (price elasticity is high), the government should give a larger subsidy. This policy maximizes the welfare gain by reducing the allocational inefficiencies of a monopoly while accounting for the price responsiveness of consumers.

Key concepts

Allocational Effects

In economics, allocational effects refer to how resources are distributed in a market. When a monopoly exists, allocational inefficiencies are common. This happens because a monopolist maximizes profit by setting the quantity produced where marginal cost equals marginal revenue, rather than at the point where marginal cost equals price, which is the condition for allocational efficiency. As a result, fewer goods are produced at a higher price than in a competitive market, leading to a loss in consumer surplus and overall welfare.

By implementing a per-unit subsidy, the government aims to correct these allocational effects. The subsidy lowers the monopolist's effective marginal cost, encouraging more production. This leads to an increase in output closer to what would occur in a perfectly competitive market, thereby improving the allocation of resources. The government’s goal with such a subsidy is to bring the quantity produced and the price of the good closer to what they would be in a competitive equilibrium.

Marginal Cost

Marginal cost is the additional cost incurred to produce one more unit of a good. For monopolists, the marginal cost is crucial in determining the optimal level of output. A profit-maximizing monopolist produces where marginal cost equals marginal revenue, not at the point where price equals marginal cost, unlike in competitive markets. Consequently, monopolists produce less output at a higher market price.

A per-unit subsidy affects a monopolist’s marginal cost by effectively lowering it. When a subsidy is applied per unit, the new marginal cost curve shifts downward by the amount of the subsidy. This incentivizes the monopolist to produce more because the additional cost for producing one more unit is now reduced. Thus, it helps move market conditions closer to what they would be under perfect competition, increasing total welfare and addressing allocative inefficiency.

Price Elasticity of Demand

Price elasticity of demand measures how sensitive the quantity demanded is to a change in price. Specifically, it is the percentage change in the quantity demanded that results from a one-percent change in price. Understanding this concept is important for determining the appropriate level of subsidy the government should set.

With price elasticity of demand, the more elastic the demand, the more responsive consumers are to price changes. If demand is elastic, significant changes in quantity demanded occur with minimal price adjustments. Conversely, inelastic demand signifies that quantity demanded is less responsive to price changes. The formula \(\frac{t}{P} = -\frac{1}{e_{Q, P}}\) suggests that a higher elasticity (more responsive demand) would require a larger subsidy to maximize welfare benefits, as it means the monopolist is more likely to increase output substantially with reduced costs.

This relationship provides an intuitive guideline for subsidy policies, suggesting that governments can optimize resource allocation and enhance consumer surplus by adjusting subsidies in accordance with the price elasticity of demand.

Competitive Equilibrium

Competitive equilibrium is a foundational concept in economics where supply meets demand in a perfectly competitive market such that quantity supplied equals quantity demanded at the equilibrium price. At this point, resources are distributed efficiently, and there is no tendency for change in output or price.

In such a market, price equals marginal cost, which is the condition for allocational efficiency. However, monopolies disrupt this balance, resulting in less output and higher prices. By applying a per-unit subsidy to a monopolistic market, the objective is to push production closer to competitive equilibrium levels.

The subsidy adjusts the monopolist’s production decisions by lowering the effective marginal cost. This makes them operate similarly to firms in a competitive market, producing more quantities that consumers demand at lower prices, enhancing welfare. The closer production gets to competitive equilibrium, the more resources are optimally allocated, minimizing wastage and maximizing total economic welfare.