Question

Suppose the government wished to combat the undesirable allocation 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 wishes 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 it 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 lump-sum subsidy is not effective because it provides no incentive for a monopolist to change its production level, as it's just a fixed amount of money given regardless of the quantity produced. On the other hand, a per-unit-of-output subsidy encourages the monopolist to produce more units, as it provides them with a certain amount of money for each unit produced. This moves production closer to the socially optimal level, reducing deadweight loss and combating undesirable allocation effects. The government can further maximize the difference between the total value of the good to consumers and the good's total cost by using the formula \(t=-P \cdot \frac{1}{e_{Q, P}}\) to set the per-unit subsidy, where \(t\) is the subsidy and \(P\) is the competitive price.

Step-by-step solution

Part a: Why lump-sum subsidy is not effective

A lump-sum subsidy is a fixed amount of money given to the monopolist regardless of the quantity produced. Given that this amount does not depend on the production level, it does not provide any incentive for the monopolist to change its production level. In other words, a lump-sum subsidy merely transfers wealth from the government to the monopolist but does not affect the undesirable allocation effects of a monopoly, which include underproduction and deadweight loss.

Part b: Graphical proof for a per-unit-of-output subsidy

A per-unit-of-output subsidy is a certain amount of money given to the monopolist for each unit produced. This creates an incentive to produce more units. Let's illustrate this with a graphical proof. 1. Draw the demand curve, which shows the relationship between the quantity demanded and the price. 2. Draw the marginal cost (MC) curve, which represents the additional cost for producing each additional unit. 3. Draw the marginal revenue (MR) curve, which shows the additional revenue from selling each additional unit. 4. Identify the monopolist's original equilibrium point (Qm, Pm): This is where MR = MC. 5. Now, add the per-unit subsidy to the MC curve, which will create a new MC curve (MC + Subsidy). 6. Identify the new equilibrium point with the subsidy (Qs, Ps): This is where the new MC curve (MC + Subsidy) intersects the MR curve. 7. Observe that the new equilibrium point is closer to the socially optimal level of output (where MC = Demand). The graphical proof shows that a per-unit-of-output subsidy encourages the monopolist to produce at a level closer to the socially optimal level, which reduces deadweight loss and combats the undesirable allocation effects of a monopoly.

Part c: Setting the per-unit subsidy to maximize consumer surplus

To maximize the difference between the total value of the good to consumers and the good's total cost, the government must set the subsidy in a way that ensures maximum consumer surplus: \(\frac{t}{P}=-\frac{1}{e_{Q, P}}\) Where \(t\) is the per-unit subsidy, and \(P\) is the competitive price. 1. We must calculate the price elasticity of demand (PED), denoted by \(e_{Q, P}\): PED is the percentage change in quantity demanded as a result of a percentage change in price, which can be represented as \(e_{Q, P} = \frac{ΔQ}{ΔP} \times \frac{P}{Q}\). 2. PED is negative, indicating that as price increases, quantity demanded decreases (and vice versa). As a result, the negative sign in the given formula ensures that the subsidy (\(t\)) remains positive. 3. Now, we can rearrange the formula and solve for the subsidy (\(t\)): \(t=-P \cdot \frac{1}{e_{Q, P}}\). This formula shows that the subsidy is inversely proportional to the price elasticity of demand. Intuitively, when demand is more price-elastic (i.e., price changes result in larger changes in quantity demanded), a smaller subsidy can accomplish the government's goal. By using this formula, the government can effectively set the per-unit subsidy to maximize the consumer surplus and combat the undesirable allocation effects of a monopoly.

Key concepts

Lump-sum subsidy

A lump-sum subsidy is essentially a fixed amount of money given to a monopolist, regardless of how much they produce. The idea behind this is to provide financial assistance, but it unfortunately doesn't address the key issue of monopolistic inefficiency. This type of subsidy doesn't incentivize increased production, as it's not linked to the quantity of goods produced.

In the context of a monopoly, production levels are usually lower than optimal, leading to a deadweight loss. This means that the total welfare or benefit that could be achieved if more goods were produced and sold at a lower price is not realized. A lump-sum subsidy fails to motivate the monopolist to alter this output because the subsidy is not tied to changes in production levels.

As a result, the monopolist's motivation to remain at a lower production level persists, continuing the cycle of inefficiency and preventing any real change in resource allocation.

Per-unit-of-output subsidy

A per-unit-of-output subsidy directly ties financial assistance to the amount of goods produced. This means that for every additional unit the monopolist produces, they receive an extra payment from the government.

The benefit of this type of subsidy is that it aligns the monopolist's interests with social welfare improvements. As the monopolist receives more earnings per unit, they are encouraged to increase production levels.

• This shifts the equilibrium closer to where marginal cost equals demand, a point known as the socially optimal output level.
• It reduces the inefficiencies typically associated with monopoly power.
• Consumers benefit from increased availability of goods and potentially lower prices.

Graphically, when we introduce a per-unit subsidy, the marginal cost curve effectively shifts downwards, encouraging the monopolist to increase production until the marginal cost matches the decreased effective cost.

This brings the monopolist’s production and pricing closer to competitive market levels, decreasing deadweight loss and promoting better allocation of resources.

Price elasticity of demand

The concept of price elasticity of demand (PED) is crucial when optimizing the effects of a subsidy in a monopolistic market. PED measures how much the quantity demanded of a good responds to changes in price. Specifically, it is calculated as:

\[ e_{Q, P} = \frac{\Delta Q}{\Delta P} \times \frac{P}{Q} \]This value is typically negative since an increase in price leads to a decrease in quantity demanded.

When setting a per-unit subsidy, understanding PED helps determine the subsidy level needed to achieve the desired social benefit. A lower elasticity suggests that consumers are less responsive to price changes, requiring a different subsidy approach than a market with high elasticity.

• A high elasticity means a small price change causes a large quantity change.
• Low elasticity implies a smaller reaction in quantity to price changes.

Based on elasticity, the government's subsidy goal of maximizing consumer welfare can be achieved by setting:

\[ \frac{t}{P} = -\frac{1}{e_{Q, P}} \]This formula indicates that the necessary subsidy is inversely proportional to the price elasticity. In essence, if demand is more elastic, a smaller subsidy is needed. This careful consideration ensures that the government effectively uses subsidies to balance monopoly power and improve market efficiency.