Chapter 6.7, Problem 4E

To determine

To find: The consumer surplus when the sales when the sales level is 300. Illustrate this by drawing the demand curve and identifying the consumer surplus as an area.

Answer to Problem 4E

The consumer surplus is $\$2250$ and the required diagram is shown in Figure 1

Explanation of Solution

Given:

The given function is $p=20-0.05x$ .

Calculation:

Consumer the numbers of product sold is $X=300$ , the corresponding price is,

  $\begin{array}{c}P=20-0.05\left(300\right)\\ =5\end{array}$

Then, the consumer surplus is,

  $\begin{array}{c}{\displaystyle {\int }_0^{300}\left[p\left(x\right)-P\right]}dx={\displaystyle {\int }_0^{300}\left[20-0.05x-5\right]}dx\\ ={\left[15x-0.02x^2\right]}_0^{300}\\ =\left(15\left(300\right)-0.025{\left(300\right)}^2\right)\\ =\$2250\end{array}$

The interpretation of the consumer surplus and area under the demand curve above the line $P=56$ as shown in Figure 1.

The required sketch is shown in Figure 1.

  

Figure 1

Thus, the consumer surplus is $\$2250$ and the required diagram is shown in Figure 1