The consumer demand for a certain commodity is $D ( p ) = 1,000.00 e ^ { - 0.33 p }$ units per month when the market price is p dollars per unit. Express consumers' total monthly expenditure for the commodity as a function of p and determine the market price that will result in the greatest consumer expenditure.
A) $45.45
B) $1,000.00
C) $3.03
D) $24.24

Question

Quizplus · Accepted Answer

The total monthly expenditure is given by the product of the price and the quantity demanded:$$E(p) = pD(p) = p(1,000.00 e ^ { - 0.33 p }) = 1,000.00 pe ^ { - 0.33 p }$$To find the market price that will result in the greatest consumer expenditure, we need to find the maximum of this function. We can do this by taking the derivative of the function and setting it equal to zero:$$\frac{dE}{dp} = 1,000.00 e ^ { - 0.33 p } - 333.33 pe ^ { - 0.33 p } = 0$$$$1,000.00 = 333.33 p$$$$p = 3.00$$Therefore, the market price that will result in the greatest consumer expenditure is $3.00 per unit.

The Consumer Demand for a Certain Commodity Is D(p)=1,000.00e−0.33pD ( p ) = 1,000.00 e ^ { - 0.33 p }D(p)=1,000.00e−0.33p

The Consumer Demand for a Certain Commodity Is $D ( p ) = 1,000.00 e ^ { - 0.33 p }$