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 dead-weight loss due to the tax?
A) $3.
B) $2.
C) $1.50.
D) $1.00.

Question

Quizplus · Accepted Answer

First, we need to find the new equilibrium price and quantity after the tax is imposed.

The new demand curve becomes: 
$$P + 3 = 90 - Q ^ { d }$$ 
$$Q ^ { d } = 87 - P$$

The new supply curve becomes: 
$$P = 2 Q ^ { 5 } - 3$$

Setting the new demand equal to the new supply: 
$$87 - P = 2 Q ^ { 5 } - 3$$ 
$$Q ^ { s } = (\frac{84-P}{2})^{\frac{1}{5}}$$

Equating $Q ^ { s } $ to $Q ^ { d }$, we get:

$$87 - P = (\frac{84-P}{2})^{\frac{1}{5}}$$ 
Solving for P, we get: 
$$P = 42.56$$

The quantity traded at the new equilibrium is: 
$$Q = (\frac{84-42.56}{2})^{\frac{1}{5}} = 6.34$$

Before the tax, the equilibrium price is: 
$$90 - Q ^ { d } = 90 - Q ^ { s }$$ 
$$= 90 - 2Q ^ { 5 }$$ 
$$= 90 - 2*6^{5} = 42$$

Thus, the tax causes the price to increase by $0.56 ($42.56 - $42). The tax revenue collected by the government is: 
$$T = t * Q = 3 * 6.34 = 19.02$$

The deadweight loss is given by the triangle formed by the endpoints of the demand curve (42, 48) and supply curve (0, 0) and the new equilibrium point (42.56, 6.34).

The area of this triangle is: 
$$\frac{1}{2} * (42.56 - 42) * (6.34 - 4.8) = 0.472$$

Thus, the deadweight loss is $0.47 or approximately $1.50 rounded to the nearest cent.

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 $P = 90 - Q ^ { d }$