Chapter 6.7, Problem 5E

To determine

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

Answer to Problem 5E

The consumer surplus is $\$407.25$

Explanation of Solution

Given Information: The demand function is $p(x)=\frac{450}{x+8}$ and the selling price is $\$10$ .

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: Put the value of $p=10$ in the demand function to write

  $p(X)=\frac{450}{X+8}=10$

  $\Rightarrow X+8=\frac{450}{10}=45$

  $\Rightarrow X=37$

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

  $ConsumerSurplus=\underset{0}{\overset{37}{{\displaystyle \int }}}\left(\frac{450}{x+8}-10\right)dx$

  $={450\ln (x+8)-10x]}_0^{37}=450\ln \left(\frac{45}{8}\right)-10\times 37$

  $=407.25$

Therefore, the consumer surplus is $\$407.25$ .