Chapter 6.7, Problem 7E

To determine

To Calculate: The producer surplus at a certain selling price for a given supply function.

Answer to Problem 7E

The producer surplus is $\$12,000$

Explanation of Solution

Given Information: The supply function is $p_s(x)=200+0.2x^{\frac{3}{2}}$ and the selling price is $P=400$

Method Used: 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: Put the value of $P=400$ in the supply function to write

  $p_s(X)=200+0.2X^{\frac{3}{2}}=400$

  $\Rightarrow X^{\frac{3}{2}}=\frac{400-200}{0.2}=1000$

  $\Rightarrow X=100$

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

  $ProducerSurplus=\underset{0}{\overset{100}{{\displaystyle \int }}}\left(400-[200+0.2x^{\frac{3}{2}}]\right)dx=\underset{0}{\overset{100}{{\displaystyle \int }}}(200-0.2x^{\frac{3}{2}})dx$

  $={200x-0.2\times \frac{2}{5}\times x^{\frac{5}{2}}]}_0^{100}=200\times {10}^2-0.08\times {10}^5$

  $=12000$

Therefore, the producer surplus is $\$12,000$