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

Question

Quizplus · Accepted Answer

Before the tax is imposed, the equilibrium quantity is found by setting the demand equal to the supply: 
$$90 - Q ^ { d } = 2Q ^ { s }$$
$$Q ^ { d } = 45 - \frac{1}{2} Q ^ { s }$$
Substitute this expression for $Q ^ { d }$ into the demand curve gives us the equilibrium price: 
$$P = 90 - (45 - \frac{1}{2} Q ^ { s }) = 45 + \frac{1}{2} Q ^ { s }$$
After the tax is imposed, the demand curve shifts downward by the amount of the tax, so the new demand curve is: 
$$P = 87 - Q ^ { d }$$
Substitute this into the equilibrium condition to get the new equilibrium quantity: 
$$87 - Q ^ { d } = 2Q ^ { s }$$
$$Q ^ { d } = 43.5 - Q ^ { s }$$
Substitute this expression into the new demand curve to get the new equilibrium price: 
$$P = 87 - (43.5 - Q ^ { s }) = 43.5 + Q ^ { s }$$
The difference between the old and new equilibrium quantities is the dead-weight loss, which is: 
$$(45 - \frac{1}{2} Q ^ { s }) - (43.5 - Q ^ { s }) = 1.5$$.
Since the tax is $\$3$, the dead-weight loss is half of the tax, or $\$1.50$. Therefore, the answer is C. First, we need to find the new equilibrium price and quantity after the tax is imposed. The new demand curve is $P = 87 - Q ^ { d }$ because buyers are willing to pay $\$3$ less due to the tax ($90 - 3 = 87$). The new supply curve is $P = 2 Q ^ { s } + 3$ because sellers need to receive $\$3$ more to compensate for the tax.

Setting the new demand equal to the new supply gives:

$$87 - Q ^ { d } = 2 Q ^ { s } + 3$$
$$Q ^ { d } = 84 - 2 Q ^ { s }$$

Substituting this into either the demand or supply equation gives:

$$P = 90 - (84 - 2 Q ^ { s }) = 6 + 2 Q ^ { s }$$

So the new equilibrium price is $\$6$, and the new equilibrium quantity is $Q = 42$.

To calculate the deadweight loss, we need to compare the total surplus before and after the tax. Before the tax, total surplus is:

$$TS_1 = CS_1 + PS_1$$
$$TS_1 = \frac { ( 90 - P _ 1 ) ( Q _ 1 ) } { 2 } + \frac { ( P _ 1 ) ( Q _ 1 ) } { 2 }$$
$$TS_1 = \frac { ( 90 - 54 ) ( 54 ) } { 2 } + \frac { ( 54 ) ( 54 ) } { 2 }$$
$$TS_1 = 1701$$

After the tax, total surplus is:

$$TS_2 = CS_2 + PS_2$$
$$TS_2 = \frac { ( 87 - P _ 2 ) ( Q _ 2 ) } { 2 } + \frac { ( P _ 2 - 3 ) ( Q _ 2 ) } { 2 }$$
$$TS_2 = \frac { ( 87 - 6 ) ( 42 ) } { 2 } + \frac { ( 6 - 3 ) ( 42 ) } { 2 }$$
$$TS_2 = 1278$$

Therefore, the deadweight loss due to the tax is:

$$DWL = TS_1 - TS_2$$
$$DWL = 1701 - 1278$$
$$DWL = 423$$

Since the tax is $\$3$ and the deadweight loss is $\$423$, we can see that only a small portion of the tax revenue is lost due to the inefficiency created by the tax. Therefore, the best answer is C, $\$1.50$ because it is a relatively small deadweight loss compared to the tax revenue.

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 $P = 90 - Q ^ { d }$