# ELASTICITY OF DEMAND (CALCULUS)

The elasticity of demand is the measurement of change in quantity demanded to changes in another variable.

There are three types of elasticity of demand, namely; Price elasticity of demand, Cross-price elasticity of demand, Income elasticity of demand.

Price elasticity of demand is calculated as a percentage change in quantity demanded divided by a change in the price. This is illustrated below:

$PED=\frac{\%∆Q}{\%P}$

However, using some basic calculus, PED can represent this way:

$PED=\frac{\%∆Q}{\%∆P}=\frac{dQ}{dP}\times\frac{P}{Q}$

Where $\frac{dQ}{dP}$ is the partial derivative of quantity demanded in respect to price.

Cross-price elasticities of demand are calculated as a percentage change in quantity demanded of good $a$ divided by percentage change in the price of good $b$

$XED=\frac{\%∆Q_a}{\%∆P_b}$

With some basic calculus, this can be represented like this:

$XED=\frac{\%∆Q_a}{\%∆P_b}=\frac{dQ_a}{dP_b}\times\frac{P_b}{Q_a}$

Where $\frac{dQ_a}{dP_b}$ is the partial derivative of Quantity demanded of goods $a$ in respect to price of $y$

Income elasticity of demand, on the other hand, measures the responsiveness of quantity demanded to changes in price.

$YED=\frac{\%∆Q}{\%∆Y}$

As with other elasticities, this can be defined in this order

$YED=\frac{\%∆Q}{\%Y}=\frac{dQ}{dY}\times\frac{Y}{Q}$

Where $\frac{dQ}{dY}$ is the partial derivative of quantity demanded in respect to consumer income.

Using the fact, let's solve these examples

#### Example 1

Assuming the demand function of rice for Daniel is given as $Q=700-2P+0.02Y$. Find the price and income elasticity of demand when $P=25$ and $Y=5000$.

Solution:

$Q=700-2P+0.02Y$

By substitution.

$Q=700-2(25)+0.02(5000)$

$Q=750$

$PED=\frac{dQ}{dP}\times\frac{P}{Q}$

From the demand function, $\frac{dQ}{dP}=-2$. Accorsingly

$PED=-2(\frac{25}{750})$

$PED=\frac{-50}{750}$

$PED=-0.067$

In economics, we treat elasticity of demand as an absolute number, therefore

$PED=0.067$

Now, let's solve for income elasticity of demand.

$YED=\frac{dQ}{dY}\times\frac{Y}{Q}$

From the demand function, $\frac{dQ}{dY}=0.02$

By substitution

$YED=0.02(\frac{5000}{750})$

$YED=\frac{100}{750}$

$YED=0.13$

#### Example 2

Given the demand function of notebook $Q_n=200-2P_n-4P_b+0.1Y$. Where $P_b$ is the price of pen and $Y$ is Joshua's income. If $P_n=10$ and $P_b=5$ and $Y=2000$. Determine

1. The price elasticity of demand

2. If notebook and pen are complementary or substitute goods.

3.  The kind of goods book is to Joshua

Solution:

$Q_n=200-2Pn-4P_b+0.1Y$

$Q_n=200-2(10)-4(5)+0.1(2000)$

$Qn=360$

Let's determine the price elasticity of demand

$PED=\frac{dQ_n}{dP_n}\times\frac{P_n}{Q_n}$

From the demand function, $\frac{dQ_n}{dP_n}=-2$.

By substitution

$PED=-2(\frac{10}{360})$

$PED=\frac{-20}{360}$

$PED=-0.056$

Economist treat PED as an absolute term

$PED=0.056$

Therefore, demand is said to be relatively elastic.

To determine if the goods is a complementary or substitute good. We would solve for cross-price elasticity of demand

$XED=\frac{dQ_n}{dP_b}\times\frac{P_b}{Q_n}$

Frome the equation $\frac{dQ_n}{dQ_b}=-4$

$XED=-4(\frac{5}{360})$

$XED=\frac{-20}{360}$

$XED=-0.056$

As can be seen, the numerical value of the cross-price elasticity of demand is negative. Therefore, in this case, notebook and pen are complementary goods

Having determined price and cross-price elasticity of demand. Our next task is to determine the type of goods the notebook is to Joshua.

To achieve this, we would solve the income elasticity of demand.

$YED=\frac{dQ_n}{dY}\times\frac{Y}{Q_n}$

From the demand function, $\frac{dQ_n}{dY}=0.1$.

By substitution

$YED=0.1(\frac{2000}{360})$

$YED=\frac{200}{360}$

$YED=0.56$

As you can see, the numerical value of income elasticity of demand is positive, this means that notebooks are normal goods for Joshua

More specifically, we can say a notebook is a necessity good for Joshua.

Remember that necessity goods are a type of normal goods

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#### Example 3

Given the demand function of e-book is $Q_a=100-P_a+0.75P_b-0.25P_c+0.0075Y$ where $P_a$ is Price of e-book, $P_b$ is Price of printed book and $P_c$ is price of e-book reader and $Y$ is the income of a consumer. And given also that $P_a=10$, $P_b=20$, $P_c=40$, $y=10,000$

1. Determine how e-book and printed book are related

2. Also Detemrmine how e-book and e-book reader are related.

Solution

$Q_a=100-P_a+0.75P_b-0.25P_c+0.0075Y$

By substitution

$Q_a=100-10+0.75(20)-0.25(40)+0.0075(10,000)$

$Q_a=170$

We can determine how e-book and printed book are related via cross-price elasticity of demand

$XED=\frac{dQ_a}{dP_b}\times\frac{P_b}{Q_a}$

From the equation $\frac{dQ_a}{dP_b}=0.75$

By substitution

$XED=0.75(\frac{20}{170})$

$XED=\frac{15}{170}$

$XED=0.088$

The cross-price elasticity of demand is positive. Therefore, we say e-book and printed books are substitutes goods.

Having determined that e-book and printed books are substitutes. We now turn our attention to determining the relationship between e-book and e-book readers.

To achieve this, we also solve for the cross-price elasticity of demand

$XED=\frac{dQ_a}{dP_c}\times\frac{P_c}{Q_a}$

From the equation $\frac{dQ_a}{dP_b}=-0.25$

By substitution

$XED=-0.25(\frac{40}{170})$

$XED=\frac{-10}{170}$

$XED=-0.0588$

The cross-price elasticity of demand is negative. Therefore, we say e-book and printed books are complementary goods.

#### Example 4

A demand function of a certain good is given by $Q = —20P + 0.04Y + 47 + 3P_y$

where Q and P denote the quantity and price of the good, Y is income, T is taste and $P_y$ is the price of a related good.

1. Calculate Q when P = 8, Y = 1000, T = 15 and $P_y$ = 30.

2. Is the related good substitutable or complementary? Give a reason for your answer.

3. Find the value of P when Q = 235, Y = 8000, T = 30 and $P_y= 25$.

Solution:

1. To calculate quantity, we simply insert the value of the unknown.

$Q=-20(8)+0.04(1000)+4(15)+3(30)$

$Q=-160+40+60+90$

$Q=30$.

2. A good will be complementary if its cross-price elasticity of demand is negative. If a good has a positive cross-price elasticity of demand is positive.

$XED=\frac{∆Q}{∆P_y}×\frac{P_y}{Q}$

From the equation, $\frac{∆Q}{∆P}=3$.

$XED=3 × \frac{8}{30}$

$XED=0.8$

The good is a complimentary good because its numerical value is positive.

3. when Q = 235, Y = 8000, T = 30 and $P_y= 25$.

$235=-20P+0.04(8000)+4(30)+3(25)$

$235=-20p+515$

$20p=280$

$p=14$

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