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 producer surplus due to the tax?
A) $900.
B) $841.
C) $59.
D) $29.50.

Question

Quizplus · Accepted Answer

To find the change in producer surplus due to the tax, we need to first find the new equilibrium price and quantity after the tax is imposed.

The new supply curve will be:

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

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

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

Solving for Q, we get:

$$Q = 4.79$$

Plugging this back into either the demand or supply equation, we can find the new equilibrium price:

$$P = 2 (4.79)^5 + 3 = 307.85$$

The producer surplus before the tax was:

$$\int_{0}^{90-2Q^{5}} (90-Q^{d}-2Q^{5})dQ = 6750$$

The producer surplus after the tax is:

$$\int_{0}^{4.79} (307.85 - 2Q^{5} - 3) dQ = 6690.62$$

The change in producer surplus due to the tax is:

$$6690.62 - 6750 = -59.38$$

Rounded to the nearest cent, this is a decrease of $59 in producer surplus. Thus, the best choice is C ($59).

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 $P = 90 - Q ^ { d }$