Chapter 6.7, Problem 8E

To determine

To Calculate: The consumer surplus and producer surplus when demand and supply curves are given.

Answer to Problem 8E

The consumer surplus is $\$1000$ and the producer surplus is $\$2000$ .

Explanation of Solution

Given Information: The demand curve is $p_d\left(x\right)=50-\frac{1}{20}x$ and the supply curve is $p_s\left(x\right)=20+\frac{1}{10}x$ .

Concept Used:

• In a purely competitive market, the selling price and quantity pair are at the intersection point of the supply and demand curves.
• The consumer surplus at a particular selling price $P$ and a demand function $p\left(x\right)$ is

$ConsumerSurplus=\underset{0}{\overset{X}{{\displaystyle \int }}}\left[p\left(x\right)-P\right]dx$

where, $X$ is the amount of goods sold when selling price is $P$ .

• The producer surplus at a particular selling price $P$ and a supply function $p_s\left(x\right)$ is

  $ProducerSurplus=\underset{0}{\overset{X}{{\displaystyle \int }}}\left[P-p_s(x)\right]dx$

where, $X$ is the amount of goods sold when selling price is $P$

Calculation: To find the selling price $P$ and the sale quantity $X$ , equate the given $p_d\left(x\right)$ and $p_s\left(x\right)$ to write

  $50-\frac{1}{20}X=20+\frac{1}{10}X$

  $\Rightarrow \frac{3}{20}X=30$

  $\Rightarrow X=200$

Put this value of $X$ in either $p_d\left(x\right)$ or $p_s\left(x\right)$ to get the value of $P$ as

  $P=50-\frac{1}{20}200=40$

Now put $P,p_d\left(x\right)$ and $X$ in the integral for the consumer surplus to get

  $ConsumerSurplus=\underset{0}{\overset{200}{{\displaystyle \int }}}\left[\left(50-\frac{1}{20}x\right)-40\right]dx=\underset{0}{\overset{200}{{\displaystyle \int }}}\left(10-\frac{1}{20}x\right)dx$

  $={10x-\frac{x^2}{40}]}_0^{200}=2000-1000$

  $=1000$

Therefore, the consumer surplus is $\$1000$ .

Next, to calculate the producer surplus, put $P,p_s\left(x\right)$ and $X$ in the integral for producer surplus and evaluate

  $ProducerSurplus=\underset{0}{\overset{200}{{\displaystyle \int }}}\left(40-[20+\frac{1}{10}x]\right)dx=\underset{0}{\overset{200}{{\displaystyle \int }}}(20-\frac{1}{10}x)dx$

  $={20x-\frac{x^2}{20}]}_0^{200}=4000-2000$

  $=2000$

Therefore, the producer surplus is $\$2,000$

Graph: In the plot shown below, consumer surplus is the area enclosed by the upper red triangle and producer surplus is the area enclosed by the lower red triangle.