Question

The private marginal benefit for commodity X is given by 10-X, where Z is the number of units consumed. The private marginal cost of producing X is constant at $5. For each unit of X produced, an external cost of $2 is imposed on member of society.

a. In the absence of any government intervention, how much X is produced?

b. What is the efficient level of production of X?

c. What is the gain to society involved in moving from the inefficient to the efficient level of production?

d. Suggest a Pigouvian tax that would lead to the efficient level. How much revenue would the tax raise?

Answer 1

a) Private Marginal Benefit (PMB) is a demand curve.

$PMB = 10-X$

Private Marginal Cost (PMC) = 5

PMB = PMC

$10 - X = 5 \\ X = 10 - 5\\ X = 5$

= 5 units

b)

At socially efficient equilibrium, PMB is equal to Social Marginal Cost (SMC).

$= 5 + 2\\ = 7$

Efficient level of production of X will be $10 - 7 = 3$

= 3 units

c)

Gain to society is equal to deadweight loss

$=\frac{1}{2} \times(5-3) \times (7-5)\\ = \frac{1}{2} \times 2 \times 2 \\ = 2$

d)

At socially efficient equilibrium marginal external cost is equal to Pigouvian tax

Revenue is equal to socially equilibrium product multiply by tax.

$= 2\times3\\ = \$6$