The demand function for a commodity is given by $p=2000-0.1 x-0.01 x^{2}$ . Find the consumer surplus when the sales level is $125$ .
A) $10458
B) $4458
C) $ $6458$
D) $9458
E) $8458

Question

Quizplus · Accepted Answer

To find consumer surplus, we need to first find the equilibrium quantity where demand equals supply. Since no information is given about supply, we assume the market is competitive and supply equals demand. To find the equilibrium quantity, we set the demand function equal to 125 (the sales level given), and solve for x:

\begin{align*}
2000 - 0.1x - 0.01x^2 &= 125 \
0.01x^2 + 0.1x - 1875 &= 0 \
x &= \frac{-0.1 \pm \sqrt{(0.1)^2 - 4(0.01)(-1875)}}{2(0.01)} \
&= 750, -1250
\end{align*}

Since we are interested in a positive quantity, we take $x = 750$.

Now, we can find the equilibrium price by substituting $x = 750$ into the demand function:

\begin{align*}
p &= 2000 - 0.1(750) - 0.01(750)^2 \
&= 1250
\end{align*}

Therefore, the equilibrium quantity is 750 and the equilibrium price is $1250.

Consumer surplus is the area below the demand curve and above the equilibrium price, up to the quantity sold. Graphically, it looks like the shaded area in the figure below:

![consumer_surplus.png](https://i.imgur.com/rJaGXrP.png)

To find this area, we need to integrate the demand function from $0$ to $750$:

\begin{align*}
	ext{Consumer Surplus} &= \int_0^{750} (2000 - 0.1x - 0.01x^2 - 125) dx \
&= \left[2000x - 0.05x^2 - \frac{0.01}{3}x^3 - 125xight]_0^{750} \
&= 645833.3333 - 628906.25 - 469.1358 \
&= 10457.1875
\end{align*}

Thus, the consumer surplus when the sales level is 125 is $10457.19, which is closest to $\boxed{	ext{(C) } \$6458}$.

The Demand Function for a Commodity Is Given By p=2000−0.1x−0.01x2p=2000-0.1 x-0.01 x^{2}p=2000−0.1x−0.01x2

The Demand Function for a Commodity Is Given By $p=2000-0.1 x-0.01 x^{2}$