Question

A general exponential demand function has the form q = Ae−bp (A and b nonzero constants). (a) Obtain a formula for the price elasticity E of demand at a unit price of p.

Answer 1

E = (p/q)(dq/dp)

dq/dp = -bAe^(−bp)
(p/q)(dq/dp) = [p/Ae^(−bp)] (-bAe^(−bp))
= -pb

Answer 2

`E = -pb`

Explanation:

The Elasticity(E) of demand at a unit price of p is given by:

`E= ((p)/(q)) \cdot ((dq)/(dp))`

As per the statement:

A general exponential demand function has the form :

`q = Ae^(-bp)`

where, A and b is non zero constants.

Using derivative formula:

`(d)/(dx)(e^(-x))= -e^(-x)`

First find the derivative of q with respect to p.

`(dq)/(dp) = -Ab \cdot e^(-bp)`

⇒
`(dq)/(dp) = -b \cdot Ae^(-bp)`

Using
`q = Ae^(-bp)`

⇒
`(dq)/(dp) = -bq`

then;

`E = (p)/(q) \cdot (-bq) = -pb`

⇒
`E = -pb`

Therefore, a formula for the price elasticity E of demand at a unit price of p is,
`E = -pb`