Chapter 6.7, Problem 10E

To determine

To Calculate: The consumer surplus at a certain selling price for a certain demand function.

Answer to Problem 10E

The demand function is $p\left(x\right)=13.21-\frac{0.5}{35}x$ and the consumer surplus at a selling price of $\$6$ is $\$4850$ .

Explanation of Solution

Given Information:

• When the price is $\$7.50$ , the sales quantity is $400$ .
• The sales amount increases by $35$ for every price decrease of $\$0.5$

It means $\frac{d\left[p\left(x\right)\right]}{dx}=\frac{0.5}{35}$

Method Used: 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 $P$

Calculation: Let the demand function be $p\left(x\right)=a-bx$ . Differentiate the demand function to get

  $\frac{dp\left(x\right)}{dx}=-b=-\frac{0.5}{35}$

  $\Rightarrow b=\frac{0.5}{35}$

Now put the value of $b$ and the given values of price and sales quantity to find $a$ as

  $7.50=a-\frac{0.5}{35}400$

  $\Rightarrow a=13.21$

Therefore, the demand function is $p\left(x\right)=13.21-\frac{0.5}{35}x$

Now, put the value of $p=6$ in the demand function to write

  $6=p(X)=13.21-\frac{0.5}{35}X$

  $\Rightarrow X=(13.21-6)\times \frac{35}{0.5}$

  $\Rightarrow X\approx 505$

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

  $ConsumerSurplus=\underset{0}{\overset{505}{{\displaystyle \int }}}\left(\left[13.21-\frac{0.5}{35}x\right]-6\right)dx$

  $={13.21x-\frac{0.5}{35}\times \frac{x^2}{2}]}_0^{505}=6671-1821.6$

  $\approx 4850$

Therefore, the consumer surplus is approximately $\$4850$ .