Chapter 24: Problem 6

Suppose a monopoly produces a harmful externality. Use the concept of consumer surplus to analyze whether an optimal tax on the polluter would necessarily be a welfare improvement.

Short Answer

Expert verified

Answer: Yes, an optimal tax imposed on a monopoly that produces a harmful externality is considered a welfare improvement if the reduction in externality outweighs the loss in consumer surplus.

Step by step solution

Determine the monopoly's demand, supply, and external cost curves

To assess the impact of a tax on consumer surplus, we first need to establish the demand, supply, and external cost curves associated with the monopolist's production. Let's denote the demand curve as D(p), the monopolist's supply curve (also known as marginal cost curve) as MC(p), and the external cost curve as EC(p), where p is the price of the product.

Calculate the consumer surplus without the tax

Before imposing an optimal tax, we should determine the initial consumer surplus without the tax. Consumer surplus is the difference between the maximum amount consumers are willing to pay for a good and the actual amount they pay. Mathematically, it can be expressed as: Consumer Surplus (CS) = ∫ [D(p) - p] dp where the integral is calculated between the original equilibrium price and the highest price that the consumers are willing to pay (i.e., the price at which the demand curve intersects the price-axis).

Determine the optimal tax

Next, we need to find the optimal tax that would internalize the harmful externality. The optimal tax can be found by setting the marginal external cost equal to the marginal benefit from the tax, which is the reduction in the quantity produced. Mathematically, the optimal tax (t) can be expressed as: t = EC'(q) where q is the quantity produced, and EC' is the derivative of the external cost curve, representing the marginal external cost.

Calculate the consumer surplus with the tax

Once the optimal tax is determined, we should calculate the new consumer surplus after the tax is imposed. The new demand and supply curves will now be D(p+t) and MC(p+t), respectively. We can calculate the new consumer surplus as follows: New Consumer Surplus (CS') = ∫ [D(p+t) - (p+t)] dp where the integral is calculated between the new equilibrium price due to the tax and the highest price that the consumers are willing to pay.

Compare the change in consumer surplus to the reduction in externality

Lastly, we need to compare the change in consumer surplus due to the tax (∆CS = CS' - CS) with the reduction in externality. If the reduction in externality outweighs the loss in consumer surplus, it can be concluded that the optimal tax is a welfare improvement. If ∆CS is less than the reduction in externality, the tax is considered a welfare improvement, as the benefits from reducing the externality are greater than the loss in consumer surplus. If ∆CS is greater than the reduction in externality, the tax is not considered a welfare improvement, as the loss in consumer surplus is larger than the benefits from reducing the externality.