Suppose that a market is initially in equilibrium. The initial demand curve is $P = 90 - Q ^ { d }$ . The initial supply curve is $P = 2 Q ^ { 5 }$ . Suppose that the government imposes a $3 tax on this market. What is the change in consumer surplus due to the tax?
A) $450.
B) $420.50.
C) $29.50.
D) $0.50.

Question

Quizplus · Accepted Answer

To find the change in consumer surplus, we need to calculate the consumer surplus before and after the tax and find the difference.

Before the tax, the market is in equilibrium, so we can set demand equal to supply:

$$90 - Q^{d} = 2Q^{5}$$

Solving for Q, we get:

$$Q^{d} = (2Q^{5} - 90)^{1/5}$$

Plugging this back into the demand curve, we get:

$$P = 90 - (2Q^{5} - 90)^{1/5}$$

At the original equilibrium quantity, Q = 4.

$$P = 90 - (2(4^{5}) - 90)^{1/5} = 58.84$$

Consumer surplus before the tax is:

$$CS = \frac{1}{2}(90 - 58.84)(4) = 60.64$$

After the tax, the supply curve shifts up by $3:

$$P = 2Q^{5} + 3$$

Setting demand equal to this new supply curve, we get:

$$90 - Q^{d} = 2Q^{5} + 3$$

Solving for Q, we get:

$$Q^{d} = (2Q^{5} + 87)^{1/5}$$

Plugging this back into the demand curve, we get:

$$P = 90 - (2Q^{5} + 87)^{1/5}$$

At the new equilibrium quantity, Q = 3.67.

$$P = 90 - (2(3.67^{5}) + 87)^{1/5} = 61.27$$

Consumer surplus after the tax is:

$$CS = \frac{1}{2}(90 - 61.27)(3.67) = 31.14$$

The change in consumer surplus is:

$$\Delta CS = CS_{after} - CS_{before} = 31.14 - 60.64 = -29.50$$

Therefore, the answer is (C) $29.50.

Suppose That a Market Is Initially in Equilibrium P=90−QdP = 90 - Q ^ { d }P=90−Qd

Suppose That a Market Is Initially in Equilibrium $P = 90 - Q ^ { d }$