Answer to Question #277021 in Microeconomics for Claudia

Question

The private marginal benefit for commodity X is given by 10 - X, where X is the

number of units consumed. The private marginal cost of producing X is given by 4 +

x and producing this commodity creates a negative externality, the marginal external

cost is constant at $2.

(a) How much X is produced without government intervention and at what price?

What are producer and consumer surplus in this case?

(b) What is the efficient level of production of X?

(c) Suggest a Pigouvian tax that would lead to the efficient level of production.

(d) What is the producer surplus in this market with the optimal Pigouvian tax?

(e) If some of the government revenue is spent compensating taxed producers for

their lost surplus, how much remains?

Answer 1

Task #277021

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

$PMB = 10-X$

Private Marginal Cost (PMC) $4+X$

$PMB = PMC$

$10 - X = 4+X\\ 2X = 10 - 4\\ X = 3$

= 3 units .

Price

$10-3=\$7\\=\$7$

Consumer surplus.

$10\times 3-\frac{9}{2}- 7\times3=4.5\\=4.5$

Producer surplus

$7-2=5\\=5$

b)

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

$3+ 2= 5$

Efficient level of production of X will be $10 - 5 = 5$

= 5 units

c)

Pigouvian tax that will lead to effiecient level of production.

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

d)

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

Revenue is equal to socially equilibrium product multiply by tax.

$2\times5= \$10\\=\$10$

(e)

If some of the revenue is spent compensating taxed producers for their lost surplus. What remains will be the difference between the revenues and the producer surplus.

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