Question

3) For a certain good we have LaTeX: q=f\left(p\right)=200e^{-0.4p}q = f ( p ) = 200 e − 0.4 p.

a) Find the elasticity of demand at price p = $50.

b) At p = $50, is the demand elastic, inelastic, or does it have unit elasticity? Explain what this means for this product.

c) Find the elasticity of demand at price p = $20.

d) At p = $20, is the demand elastic, inelastic, or does it have unit elasticity? Explain what this means for this product.

Answer 1

(a)20

(b)Elastic

(c)8

(d) Elastic

Explanation:

Elasticity of demand(E) indicates the impact of a price change on a product's sales.

The general formula for an exponential demand curve is given as:

`y=ae^(-bp)`

Given the demand curve formula

`q=f\left(p\right)=200e^(-0.4p)`

The formula for Elasticity of demand, E

`E = -(p)/(q)\frac{\text{d}q}{\text{d}p}`

(a)When Price, p = $50

p=50

`q=200e^(-0.4*50)=200e^(-20)`

`\frac{\text{d}q}{\text{d}p}=-0.4*200e^(-0.4p)=-80e^(-0.4p)`

Therefore:

`E = -(50)/(200e^(-20))*-80e^(-0.4*50)\\=(1)/(4e^(-20))*80e^(-20)\\\\E=20`

(b)At p = $50, Since elasticity is greater than 1, the demand is elastic.

An elasticity value of 20 means that a 1% increase in price causes a 20% decrease in demand.

(c)At p=$20

p=20

`q=200e^(-0.4*20)=200e^(-8)`

`\frac{\text{d}q}{\text{d}p}=-0.4*200e^(-0.4p)=-80e^(-0.4p)`

Therefore:

`E = -(20)/(200e^(-8))*-80e^(-0.4*20)\\=(1)/(10e^(-20))*80e^(-20)\\\\E=8`

(d)At p = $20, the demand is elastic.

An elasticity value of 8 means that a 1% increase in price causes a 8% decrease in demand.