A certain commodity has demand curve p = $\frac { 20 } { x + 5 }$

Question

A certain commodity has demand curve p = $\frac { 20 } { x + 5 }$ - 1 at sales level x. What is the consumers' surplus if 5 units are currently being sold?
A) 20(ln 10) - 10
B) 20(ln 15 - ln 5) - 10
C) 20(ln 15 + ln 5) - 10
D) $\frac { 1 } { 20 }$ (ln 3) - 20
E) none of these

Quizplus · Accepted Answer

The consumer surplus is calculated as the area under the demand curve above the price level, from 0 to the quantity sold. For the given demand curve $p = \frac{20}{x + 5} - 1$, and with 5 units sold, the price $p$ at $x = 5$ is $\frac{20}{5 + 5} - 1 = 2 - 1 = 1$. The consumer surplus is the integral of the demand curve from 0 to 5 minus the area of the rectangle formed by the quantity (5) and the price (1), which is $5 \times 1 = 5$. The integral of the demand curve is $\int_0^5 \left(\frac{20}{x + 5} - 1\right) dx$, which simplifies to $20 \ln(x + 5) - x $ evaluated from 0 to 5. This gives $20 \ln(10) - 5 - (20 \ln(5) - 0)$, which simplifies to $20 \ln(2) - 5$. Since this specific result is not listed among the options, the correct answer is E, none of these.

A Certain Commodity Has Demand Curve P = 20x+5\frac { 20 } { x + 5 }x+520​