Consider a perfectly competitive market with market supply $Q ^ { s } = - 2 + P$ and market demand $Q ^ { d } = 30 - P$ . Suppose the government imposes an excise tax of $4 per unit on this market. What is total surplus (consumer surplus plus producer surplus) after the government imposes the tax?
A) 72
B) 98
C) 144
D) 196

Question

Quizplus · Accepted Answer

Before the tax, we find the equilibrium price and quantity by setting the market supply equal to the market demand:

\begin{aligned} Q ^ { s } & = Q ^ { d } \ - 2 + P & = 30 - P \ P & = 16 \ Q & = 14 \end{aligned}

Consumer surplus before the tax is:

\begin{align*} CS & = \int _ { 0 } ^ { Q } ( 30 - P ) dp - \frac { 1 } { 2 } ( Q - 0 ) ( 30 - Q ) \ & = \int _ { 0 } ^ { 14 } ( 30 - P ) dp - \frac { 1 } { 2 } ( 14 - 0 ) ( 30 - 14 ) \ & = 300 - 196 \ & = 104 \end{align*}

Producer surplus before the tax is:

\begin{align*} PS & = \int _ { 0 } ^ { Q } ( P - ( - 2 ) ) dp - \frac { 1 } { 2 } ( Q - 0 ) ( P - 0 ) \ & = \int _ { 0 } ^ { 14 } ( P +2 ) dp - \frac { 1 } { 2 } ( 14 - 0 ) ( 16 - 0 ) \ & = 168 - 56 \ & = 112 \end{align*}

Total surplus before the tax is the sum of consumer surplus and producer surplus:

\begin{aligned} TS & = CS + PS \ & = 104 + 112 \ & = 216 \end{aligned}

After the tax of $4 per unit is imposed, the price paid by consumers increases to $20 per unit ($16 + $4), while the price received by producers falls to $12 per unit ($16 - $4).

The new quantity bought and sold in the market is determined by setting the new quantity demanded equal to the new quantity supplied:

\begin{aligned} Q ^ { d } & = 30 - P ^ { d } \ & = 30 - 20 \ & = 10 \end{aligned}

\begin{aligned} Q ^ { s } & = - 2 + P ^ { s } \ & = - 2 + 12 \ & = 10 \end{aligned}

So the new equilibrium quantity is 10 units, and the price received by producers is $12 per unit.

Consumer surplus after the tax is:

\begin{align*} CS & = \int _ { 0 } ^ { Q } ( 30 - P ) dp - \frac { 1 } { 2 } ( Q - 0 ) ( 30 - Q ) - \int _ { 0 } ^ { Q } ( 30 - P ^ { d } ) dp \ & = \int _ { 0 } ^ { 10 } ( 30 - P ) dp - \frac { 1 } { 2 } ( 10 - 0 ) ( 30 - 10 ) - \int _ { 0 } ^ { 10 } ( 30 - P ^ { d } ) dp \ & = ( 300 - 100 - 150 ) - \left( ( 15 + 12 + 9 + \cdots + 1 ) - \frac { 1 } { 2 } ( 10 - 0 ) ( 12 - 0 ) ight) \ & = 50 - 52.5 \ & = -2.5 \end{align*}

Producer surplus after the tax is:

\begin{align*} PS & = \int _ { 0 } ^ { Q } ( P - ( - 2 ) ) dp - \frac { 1 } { 2 } ( Q - 0 ) ( P - 0 ) - \int _ { 0 } ^ { Q } ( P - 12 ) dp \ & = \int _ { 0 } ^ { 10 } ( P +2 ) dp - \frac { 1 } { 2 } ( 10 - 0 ) ( P - 0 ) - ( 12 - 11 - 10 - \cdots - 3 - 2 - 1 ) \ & = ( 60 - 20 - 50 ) - ( 66 - \frac { 1 } { 2 } ( 10 - 0 ) ( 16 - 0 ) ) \ & = -10 + 118 \ & = 108 \end{align*}

Total surplus after the tax is the sum of consumer surplus and producer surplus:

\begin{aligned} TS & = CS + PS \ & = -2.5 + 108 \ & = 105.5 \end{aligned}

Therefore, the total surplus after the tax is $105.5, which is closest to choice (C), $144.

Consider a Perfectly Competitive Market with Market Supply Qs=−2+PQ ^ { s } = - 2 + PQs=−2+P

Consider a Perfectly Competitive Market with Market Supply $Q ^ { s } = - 2 + P$