The consumer demand for a certain commodity is $D ( p ) = 5,000 e ^ { - 0.03 p }$ units per month when the market price is p dollars per unit. Determine the market price that will result in the greatest consumer expenditure.
A) $30.31
B) $31.31
C) $33.33
D) $34.33

Question

Quizplus · Accepted Answer

To find the market price that results in the greatest consumer expenditure, we need to maximize the expenditure function, which is the product of the demand function and the price, $E(p) = pD(p) = p \cdot 5,000e^{-0.03p}$. To maximize this, we differentiate it with respect to $p$ and set the derivative equal to zero. The derivative of $E(p)$ with respect to $p$ is $E'(p) = 5,000e^{-0.03p} - 150pe^{-0.03p}$. Setting $E'(p) = 0$ and solving for $p$ gives us the value that maximizes consumer expenditure. Solving this equation leads to $p = 33.33$, which corresponds to choice C.

The Consumer Demand for a Certain Commodity Is D(p)=5,000e−0.03pD ( p ) = 5,000 e ^ { - 0.03 p }D(p)=5,000e−0.03p

The Consumer Demand for a Certain Commodity Is $D ( p ) = 5,000 e ^ { - 0.03 p }$