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

First, we need to find the equilibrium price and quantity in the absence of the tax. Setting market supply equal to market demand:

$Q ^ { s } = Q ^ { d }$ 
$-2 + P = 30 - P$ 
$P^* = 16$
$Q^* = 14$

With a tax of $\$4$ per unit, the new market supply curve is $Q ^ { s } = -2 + P - 4$, or $Q ^ { s } = -6 + P$. The new equilibrium price and quantity are found by setting market supply equal to market demand:

$Q ^ { s } = Q ^ { d }$ 
$-6 + P = 30 - P - 4$ 
$P^* = 20$ 
$Q^* = 10$

To find consumer surplus, we need to calculate the area under the demand curve and above the equilibrium price, up to the quantity purchased. This is a triangle with base 6 (the difference between the original quantity of 14 and the new quantity of 10) and height 10 (the difference between the original equilibrium price of 16 and the new equilibrium price of 20), divided by 2, or $6 \times 10 / 2 = 30$.

To find producer surplus, we need to calculate the area above the supply curve and below the equilibrium price, up to the quantity sold. This is a triangle with base 6 and height 14 (the original quantity), divided by 2, or $6 \times 14 / 2 = 42$.

Total surplus is the sum of consumer surplus and producer surplus, or $30 + 42 = 72$. Before the tax, we need to find the equilibrium price and quantity by setting market demand equal to market supply:

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

After the tax of $\$ 4$ is imposed, the new supply curve will shift upward by $\$ 4$ (since sellers need to receive the same amount after the tax):

\begin{align*}
Q ^ { s } &= - 2 + P - 4 \
&= P - 6 \
\end{align*}

The new equilibrium price will be the price that makes the quantity demanded equal to the quantity supplied:

\begin{align*}
Q ^ { s } &= Q ^ { d } \
P - 6 &= 30 - P \
P &= \$18 \
\end{align*}

The quantity demanded and quantity supplied at this new equilibrium price are:

\begin{align*}
Q ^ { d } &= 30 - 18 = 12 \
Q ^ { s } &= 18 - 6 = 12 \
\end{align*}

To find the new consumer surplus, we need to find the area below the demand curve and above the new equilibrium price:

\begin{align*}
\text{New Consumer Surplus} &= \frac { ( \$ 30 - \$ 18 ) \times 12 } { 2 } \
&= \$72 \
\end{align*}

To find the new producer surplus, we need to find the area below the new equilibrium price and above the supply curve:

\begin{align*}
\text{New Producer Surplus} &= \frac { ( \$ 18 - ( \$ 16 + \$ 4 ) ) \times 12 } { 2 } \
&= \$72 \
\end{align*}

Therefore, the total surplus is:

\begin{align*}
\text{Total Surplus} &= \text{Consumer Surplus} + \text{Producer Surplus} \
&= \$72 + \$72 \
&= \$144 \
\end{align*}

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

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