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

Answer:

(a)20

(b)Elastic

(c)8

(d) Elastic

Step-by-step 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

$E = -\dfrac{p}{q}\dfrac{\text{d}q}{\text{d}p}$

(a)When Price,  p = $50

p=50

$q=200e^{-0.4*50}=200e^{-20}$

$\dfrac{\text{d}q}{\text{d}p}=-0.4*200e^{-0.4p}=-80e^{-0.4p}$

Therefore:

$E = -\dfrac{50}{200e^{-20}}*-80e^{-0.4*50}\\=\dfrac{1}{4e^{-20}}*80e^{-20}\\\\E=20$

(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

$q=200e^{-0.4*20}=200e^{-8}$

$\dfrac{\text{d}q}{\text{d}p}=-0.4*200e^{-0.4p}=-80e^{-0.4p}$

Therefore:

$E = -\dfrac{20}{200e^{-8}}*-80e^{-0.4*20}\\=\dfrac{1}{10e^{-20}}*80e^{-20}\\\\E=8$

(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.