Suppose that a market is initially in equilibrium. The initial demand curve is $P = 90 - Q ^ { d }$. The initial supply curve is $P = 2 Q ^ { s }$. 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 first find the post-tax equilibrium price and quantity.

After the tax is imposed, the demand curve shifts down by $3, becoming:

$$P = 87 - Q^d$$

The supply curve remains the same:

$$P = 2Q^s$$

At the new equilibrium, the quantity demanded and supplied will be equal:

$$87 - Q^d = 2Q^s$$

Solving for Q, we get:

$$Q = 21$$

Plugging this into either the demand or supply function gives us the post-tax equilibrium price:

$$P = 2 \times 21 = 42$$

The change in consumer surplus is then:

Initial CS - New CS

To find the initial consumer surplus, we can integrate the demand function:

$$\int_0^{90} (90-Q) dQ = 4050$$

To find the new consumer surplus, we need to find the area of the triangle formed by the new equilibrium price (42), the initial equilibrium price (45), and the quantity demanded at both equilibria (Q=15 at the initial equilibrium, Q=21 at the post-tax equilibrium):

$$New CS = \frac{1}{2} (45-42) (15+21) = 29.5$$

Therefore, the change in consumer surplus is:

$$4050 - 29.5 = 4020.5$$

or approximately $29.50. The correct answer is (C).

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 $P = 90 - Q ^ { d }$