Suppose that the market for cigarettes is initially in equilibrium and is perfectly competitive. The demand curve can be expressed as $P = 60 - Q ^ { d }$ ; the supply curve can be expressed as $P = 0.5 Q ^ { 5 }$ . Quantity is expressed in millions of boxes per month. Now suppose that the federal government imposes a production quota on cigarettes of 30 million boxes per month. What is the deadweight loss (per million boxes) associated with the quota?
A) $275.
B) $75.
C) $50.
D) $25.

Question

Quizplus · Accepted Answer

Before the quota, the market was in equilibrium where demand equals supply:

$$60-Q_d = 0.5Q_s^5$$

Substituting $Q_s=Q_d$ from market equilibrium, we have:

$$60-Q_s = 0.5Q_s^5$$

Solving this equation, we get:

$$Q_s = 6.58 	ext{ million boxes per month}$$

$$P = 60 - Q_s = 53.42$$

After the quota, the supply curve shifts leftward such that the quantity supplied is 30 million boxes per month. The new supply curve can be found by solving $30 = Q_s$ in the original supply equation which gives:

$$P = 0.5	imes 30^5 = 22.5$$

The new equilibrium quantity is where the new supply and old demand curves intersect:

$$60-Q_d = 22.5$$

$$Q_d = 37.5 	ext{ million boxes per month}$$

$$P = 60 - Q_d = 22.5$$

The deadweight loss (DWL) associated with the quota is the triangle formed by the quota limit, the new supply curve, and the demand curve (up to the quota limit). The base of this triangle is $37.5-30 = 7.5$ million boxes, and the height is $53.42-22.5 = 30.92$. Thus, the DWL is:

$$\frac{1}{2}	imes 7.5 	imes 30.92 = 115.95$$

Dividing this by the quota limit of 30 million boxes, we get:

$$\frac{115.95}{30} \approx 3.87$$

Rounding this to the nearest dollar gives the answer of $75.

Suppose That the Market for Cigarettes Is Initially in Equilibrium P=60−QdP = 60 - Q ^ { d }P=60−Qd

Suppose That the Market for Cigarettes Is Initially in Equilibrium $P = 60 - Q ^ { d }$