PRICE ELASTICITY OF DEMAND VIA POINT ELASTICITY OF DEMAND (CALCULUS)

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Price elasticity of demand is the degree of responsiveness of the quantity demanded of a good to changes in the price of the good.

We have learned to solve the price elasticity of demand from one point to another, but today, we will be learning how to calculate the price elasticity of demand using calculus. 

Mathematically, the Price elasticity of demand can be calculated as:

$e=\frac{\Delta q}{\Delta p} \times \frac{p}{q}$

Where $\frac{\Delta q}{\Delta p}$ is the derivative of quantity demanded with respect to price, and p and q are price and quantity respectively.

In words, the price elasticity of demand can be defined as the ratio of price to quantity demanded multiplied by the slope of the demand function.

Example 1 

Supposed the demand function for ipad is given as $q=400-2p^2$. Calculate the price elasticity of demand when P=10 and determine whether the demand is elastic or inelastic.

Solution:

First, we need to determine the quantity demanded 

$q=400-2p^2$ 

If p=10, then 

$q=400-2(10)^2$ 

$q=400-200$ 

$q=200$ 

Recalled that elasticity of demand can be calculated as 

$E=\frac{\Delta q}{\Delta p} \times \frac{p}{q}$

To get $\frac{\Delta q}{\Delta p}$, we take the derivative of the demand function 

$Q=400-2p^2$ 

Taking the derivative 

$\frac{\Delta q}{\Delta p}=-4p$

Now let's insert it in the price elasticity demand formula

$e=\frac{\Delta q}{\Delta p} \times \frac{p}{q}$

$e=-4p \times \frac{10}{200}$

$e=-\frac{-40p}{200}$

Since p=10,

$e=\frac{-40(10)}{200}=-2$

Since the absolute value of the coefficient of price elasticity of demand (-2) is greater than 1, It follows that demand is elastic

Example 2

Daniel Knows that his demand function is $q=500-10p$, calculate the price elasticity at a price of N15. Also determine whether demand is elastic or inelastic

Solution:

$q=500-10p$

$p=15$

$q=500-10(15)$

$q=350$

The derivative of the demand function is -10, hence $\frac{\Delta q}{\Delta p}$ is -10

$e=-10 \times \frac{15}{350}$

$e=-0.428571$

As you can seen, the absolute value of the price elasticity of demand is lesser than one. Hence, demand is inelastic.

Example 3

Determine whether demand is elastic, inelastic or unit elastic given that demand function is $q=100p^{-1}$.

Solution:

$\frac{\Delta q}{\Delta p}=-100p^{-2}$

Recalled that

 $e=\frac{\Delta q}{\Delta p} \times \frac{p}{q}$

$e=-100p^{-2} \times \frac{p}{q}$

According to the question,  $q=100p^{-1}$, Hence

 $e=-100p^{-2} \times \frac{p}{100p^{-1}}$

$e=\frac{-100p^{-2+1}}{100p^{-1}}$

$e=\frac{-100p^{-1}}{100p^{-1}}=-1$

As can be observed, the absolute value of price elasticity of demand is 1, which means that demand is unit elastic.

Example 4

Company XYZ knows that its demand function is given by $q=300e^{-0.4p}$. Determine the

1. price that maximizes revenue

2. quantity that maximizes revenue

Solution:

As a rule, A firm maximizes revenue when demand is unit elastic because a rise or fall in price does not affect total revenue. 

When demand is unit elastic, the numerical value of Price elasticity of demand is -1

So, to maximize revenue, e=-1, so,

$-1=\frac{\Delta q}{\Delta p} \times \frac{p}{q}$

$\frac{\Delta q}{\Delta p}$ is the derivative of the demand function, So, taking the derivative of the demand function

$\frac{\Delta q}{\Delta p}=-120e^{-0.4p}$

If you are not familiar with differentiation of exponential function, I'd recommend you read this post.

$-1=-120e^{-0.4p} \times \frac{p}{q}$

if $q=300e^{-0.4p}$,then

$-1=-120e^{-0.4p} \times \frac{p}{300e^{-0.4p}}$

$-1=\frac{ -120e^{-0.4p} \times p}{300e^{-0.4p}}$

By cross-multiplication

$-1 \times 300e^{-0.4p}=-120e^{-0.4p} \times p$

$-300e^{-0.4p}=-120e^{-0.4p} \times p$

Divide both side by $120e^{-0.4p}$

$\frac{-300\require{cancel}\bcancel{e^{-0.4p}}}{-120\require{cancel}\bcancel{e^{-0.4p}}}=-\frac{\require{cancel}\bcancel{120e^{-0.4p}}\times p}{\require{cancel}\bcancel{120e^{-0.4p}}}$

$p=\frac{-300}{-120}$

$p=2.5$

Therefore, the firm should charge 2.5 per unit to maximize revenue.

To get the quantity that maximizes revenue, we simply insert the value of p in the demand function $q=300e^{-0.4p}$

$q=300e^{-0.4(2.5)}$

$q=300e^{-1}$

$q=10.36$

Example 5

How much should a firm sell to maximize revenue given that demand function is $q=960-5p^2$

Solution:

To maximize revenue, e must be equal to -1

 $-1=-10p  \times \frac{p}{960-5p^2}$

$-1=\frac{-10p^2}{960-5p^2}$ 

$ -960+5p^2=-10p^2$

$ -960=-10p^2-5p^2$

$ -960=-15p^2$

$p^2=64$

$p=\pm 8$

Since price cannot be negative,

$p=8$

Therefore, the firm should sell at N8 to maximize revenue 

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