Chapter 6, Problem 33RE

To determine

To Find: The consumer surplus when the sales level is 100.

Answer to Problem 33RE

The value of the consumer surplus is $\$7166.67$ .

Explanation of Solution

Given:

The demand function for the commodity is $p=2000-0.1x-0.01x^2$ .

Calculation:

Consider the given function is,

  $p=2000-0.1x-0.01x^2$

Then, for the surplus is $x$ ,

  $\begin{array}{c}P=p\left(100\right)\\ =2000-0.01\left(100\right)-0.01{\left(100x\right)}^2\\ =2000-10-100\\ =\$1890\end{array}$

Then, the consumer surplus is,

  $\begin{array}{c}{\displaystyle {\int }_0^{100}\left[p\left(x\right)-P\right]}dx={\displaystyle {\int }_0^{100}\left(2000-0.1x-0.01x^2-1890\right)}\\ ={\left[110x-\frac{0.1}{2}{x}^3-\frac{0.01}{3}{x}^3\right]}_0^{100}\\ =\left[11000-500-\frac{10000}{3}\right]\\ =\frac{21500}{3}\\ =\$7166.67\end{array}$