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Recitation 13

Externalities: Taxation & Subsidies

Problem 1: Externalities — Taxation

A company emits pollutants into the air, causing a negative externality. The marginal cost of production for the company is given by \( MC = 50 + 0.5q \). The marginal social cost, accounting for the pollution's externalities, is \( MSC = 50 + 1.5q \). The market price is constant at $75.

  1. Determine the socially efficient quantity of production.
  2. Given the company's marginal cost function, what quantity would the company produce in the absence of regulation?
  3. Calculate the Pigovian tax rate necessary to achieve the socially efficient quantity.

(a) Socially efficient quantity

To determine the socially efficient quantity, set \( MSC = 75 \):

\( 50 + 1.5 \times q = 75 \)

\( 1.5 \times q = 25 \)

\( q \approx 16.67 \)

(b) Quantity without regulation

To find the company's production in the absence of regulation, set \( MC = 75 \):

\( 50 + 0.5 \times q = 75 \)

\( 0.5 \times q = 25 \)

\( q = 50 \)

(c) Pigovian tax rate

The Pigovian tax rate is calculated by finding the marginal external cost at the socially efficient quantity:

\( MSC(16.67) - MC(16.67) = (50 + 1.5 \times 16.67) - (50 + 0.5 \times 16.67) \approx 16.67 \)

Therefore, the Pigovian tax rate required to achieve the socially efficient quantity is approximately \( \$16.67 \).

Problem 2: Positive Externalities — Subsidies

A local government is considering subsidizing a public transportation system to encourage use and reduce traffic congestion, which has positive externalities. The marginal social benefit (MSB) of the transportation system is given by \( MSB = 200 - 2q \), where \( q \) represents the number of users. The marginal private benefit (MPB), representing user demand, is \( MPB = 150 - 2q \). The marginal cost of operating the system is constant at 40.

  1. Determine the socially optimal number of users (socially efficient quantity).
  2. What quantity will occur without any subsidy?
  3. Calculate the subsidy per user required to achieve the socially optimal quantity.

(a) Socially optimal number of users

To find the socially optimal number of users, set \( MSB = 40 \):

\( 200 - 2 \times q = 40 \)

\( 2 \times q = 160 \)

\( q \approx 80 \)

(b) Quantity without any subsidy

The quantity that occurs without any subsidy is determined by setting \( MPB = 40 \):

\( 150 - 2 \times q = 40 \)

\( 2 \times q = 110 \)

\( q \approx 55 \)

(c) Subsidy per user

The subsidy per user required to achieve the socially optimal quantity:

\( MSB(80) - MPB(80) = (200 - 2 \times 80) - (150 - 2 \times 80) = 40 - (-10) = 50 \)

Thus, the subsidy per user required to achieve the socially optimal quantity is \( \$50 \).

Problem 3: Positive Externalities — College Education

Consider the market for college education in the United States. College degrees have become quite expensive in the U.S., but around 1 million people in the U.S. earn a degree each year. Suppose the yearly supply and demand functions for degrees in the U.S. are:

\( S(p) = 20p \)
\( D(p) = 1,200,000 - 10p \)

  1. Using these supply and demand functions, find the equilibrium price and quantity in the private market for 4-year degrees.
  2. Suppose each person with a college degree creates an external benefit for society that can be valued at \(\$10,000\). Graph the private supply and demand curves, and label the equilibrium that you found in part a. Then, add to your graph the social demand curve for college degrees.
  3. Label the section of your graph showing the deadweight loss from the externality of a college degree. Is this DWL coming from over- or underproduction?
  4. If the government were to try to correct this market failure, would it be better to use a tax to do so, or a subsidy? What is the correct value for the tax or subsidy they should use?
  5. When taking the external benefits of the externality into account, the social demand function for college degrees is:

    \( D(p) = 1,300,000 - 10p \)

    Using this new demand curve, calculate the socially optimal price and quantity for a 4-year college degree.
  6. If the government does not intervene in the market for college degrees, what is the dollar value of the deadweight loss in the private equilibrium?

(a) Equilibrium price and quantity in the private market

Set supply equal to demand:

\( 1,200,000 - 10p = 20p \)

\( 1,200,000 = 30p \)

\( p^* = \$40,000 \)

Substitute into the supply function:

\( Q^* = 20 \times 40,000 = 800,000 \)

(b) Graph of private and social demand curves

Each person with a college degree creates an external benefit valued at \(\$10,000\). The social demand curve is shifted rightward by this external benefit.

Supply and demand graph for college education
(c) Deadweight loss and underproduction

DWL is present in the triangle bounded on the left by the quantity of 800,000, on top by the social demand line, and on bottom by the private supply line. Here, the market is underproducing the good, as buyers of college degrees do not take into account the external benefit their education produces.

(d) Tax versus subsidy

With positive externalities, the government should use a subsidy to encourage consumption. The correct value for the subsidy is always exactly equal to the marginal external benefit: here, \(\$10,000\).

(e) Socially optimal price and quantity

Using the social demand function \( D(p) = 1,300,000 - 10p \), set it equal to supply:

\( 1,300,000 - 10p = 20p \)

\( 1,300,000 = 30p \)

\( p^* = \$43,333 \)

Substitute into the supply function:

\( Q^* = 20 \times 43,333 = 866,667 \)

(f) Dollar value of deadweight loss

The level of underproduction is \( 866,667 - 800,000 = 66,667 \) units. The "height" of the DWL triangle is equal to the marginal external benefit (\(\$10,000\)). Using the area of a triangle formula:

\( \text{DWL} = \frac{1}{2} \times 66,667 \times 10,000 = \$333,335,000 \)