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How to Calculate Elasticity ?


Solved questions, 
1- Tom purchased 10 units of a commodity when its price was $5 per unit. He purchased 12 units when its price falls to $4 per unit what is the price elasticity of demand at that price?
Solution:

According to the question;
P1 = 5,     Q1 = 10         Initial price and quantity
P2 = 4,    Q2 = 12         New price and quantity,
ΔP = P2 - P1,         ΔQ = Q2 - Q1,
      = 4 - 5                      = 12 -10
      = -1                           = 2
As we know that
`\ep=-\frac{ΔQ}{ΔP}×frac{P}{Q}` ......... (i)

substituting vatues in the above equation,
`\=-\frac{2}{-1}×frac{5}{10}`
`\=2×0.5`
`\= 1`

2- If demand for a good rises from 100 units to 120 units as a result of 10 percent fall in its price, what will be the price elasticity of demand?

solution:
According to the question,
% change in price = -10 Given,
% change in demand 
`\=\frac{Q2-Q1}{Q1}×100`
                                     `\=\frac{120-100}{100}×100`
                                     `\=\20`
As we know that
`\ep=\frac{%quantity}{%price}`
    `\=\frac{20}{-10}`
      = 2       (Minus sign is ignored)

3- Certain quantity of a commodity is purchased when its price
is $10 per unit. Quantity demanded increases by 50 percentage in response to a fall in the price by $2 per unit. Find elasticity of demand.

Solution:
According to question;
%ΔQ = 50            (Given)
Initial price P1 = 10
New price P2 = 8        (There is a fall in price by 2 per unit)
%ΔP `\=\frac{P2-P1}{P1}×100`
               `\=\frac{8-10}{10}×100`   (a Fall in price by 2 per unit)
              `\=-20`
`\ep=\frac{%quantity}{%price}`
    `\=\frac{50}{-20}`
      `\=2.5`             (Minus sign is ignored)

4- A consumer buys 80 units of a good at $4 per unit. With a fall in price he buys 100 units. If ep = (-)1, find the new price.

Solution:
According to the question;
Initial quantity Q1 = 80
New quantity   Q2 = 100
Initial price P1 = 4
New price   P2 = ?
                    ep = -1
Let P2 = x
ΔQ = Q2 - Q1
       = 100 - 80
       = 20
ΔP  = P2 - P1
       = x - 4
As we know that
`\ep=-\frac{ΔQ}{ΔP}×frac{P}{Q}` ......... (i)
Substituting the values in the above equation,
Or, `\ep=-\frac{20}{x-4}×frac{4}{80}`
Or, `\1=-\frac{20}{x-4}×frac{1}{20}`  
Or, `\1=-\frac{1}{x-4}`
Or, `\x-4=-1`
Or, `\x=-1+4`
Hence `\x=3` 
Hence Q2 = $3

5- A consumer buys 40 units of a good when its price is $5 per unit. Given the elasticity (-)1.5, how much will he buy at $4 per unit price?

Solution:
Given ep = 1.5
Initial price P1 = 5
New price    P2 = 4
Initial Qnt.  Q1 = 40 units,
New Qnt.     Q2 = ?

Let Q2 = x
ΔQ = X - 40
       = X - 40
ΔP  = P2 - P1
       = 4 - 5
       = -1
As we know that
`\ep=-\frac{ΔQ}{ΔP}×frac{P}{Q}` ......... (i)
Substituting the values in the above equation,
Or, `\1.5=-\frac{x-40}{-1}×frac{5}{40}`
Or, `\1.5=-\frac{x-40}{-1}×frac{1}{8}`
Or, `\1.5=\frac{-x+40}{-8}`
Or, `\-12=\-x+40`
Or, `\-12-40=\-x`
Or, `\-52=\-x`
Or, `\52=\x`
Hence, P2 = 52

6- Given the demand function Qd = 10 - 2p and supply function Qs = - 2 + p, find price elasticity at the equilibrium price.

Solution:
Qd = Qs ........(i)
Substituting demand and supply functions for Qd and Qs;
    10 - 2p = -2 + p
Or, 10 + 2 = p + 2p
Or, 12 = 3p
Or, 12/3 = p
Hence P = 4

Substituting 4 in demand for p;
Qd = 10 - 2×4
Qd = 2

At equilibrium price 4 quantity demanded is equal to 2 units
Hence P1 = 4
            Q1 = 2
Given the demand function Qd = 10 - 2p,
Differentiating demand function with respect to P,

`\frac{dQd}{dP}=frac{d(10-2P)}{dP}`
`\frac{dQd}{dP}\=\-2`
`\ep=-\frac{ΔQ}{ΔP}×frac{P1}{Q1}`
Substituting dQd/dP for ΔQ/ΔP and 4 for P1 and 2 for Q1
`\ep=-\(-2)×frac{4}{2}`
`\ep=4`

Q.no.7- A consumer's demand curve for `X` is given by the equation `P=100-\sqrt{Q}`. What is the point elasticity of demand when the price of `X` is 60?

Solution:
Given the question;
`P=100-\sqrtQ`
`Or,\sqrtQ=100-P............(i)`

Applying derivative,
`\frac{d\sqrtQ}{dQ}=\frac{d(100)}{dP}-\frac{d(P)}{dP}`
`Or,\frac{d\sqrtQ}{dQ}×\frac{dQ}{dP}=0-1`
`Or,\frac{1}{2\sqrtQ}×\frac{dQ}{dP}=-1`
`\therefore\frac{dQ}{dP}=-2\sqrtQ`

substituting `100-P` for `\sqrtQ` in the above equation, we get,
`\frac{dQ}{dP}=-2(100-P)`
Given the price `P=60`, we get;
`\frac{dQ}{dP}=-2(100-60)`
`Or,\frac{dQ}{dP}=-2(40)`
`Or,\frac{dQ}{dP}=-80`

 Calculating quantity demanded,
`\sqrtQ=100-P`
`\sqrtQ=100-60`
`\sqrtQ=40`

Squaring both sides we get;
`(\sqrt{Q})^2=(40)^2`
`Q=1600`

Now, `e_P=-\frac{△Q}{△P}×\frac{P}{Q}`
Plotting the required values in the equation we get;
`e_p=-(-80)×\frac{60}{1600}`
`\therefore\e_p=3`

Q.no.8- A monopoly sells 30 units of output when the price is $12 and 40 units when the price is $10. If its demand schedule is linear, what is the specific form of the demand function? Use this function to predict the quantity sold when the price is $8.

Solution:
Let the two points `P1=12\&\Q1=30` and `P2=10\&\Q2=40`  on a line.
Applying the concept of the equation of a straight line, we get the following form of the demand function.
`Q-Q1=\frac{Q2-Q1}{P2-P1}\(P-P1)`
`Or,\Q-30=\frac{40-30}{10-12}\(P-12)`
`Or,\Q-30=\frac{10}{-2}\(P-12)`
`Or,\Q-30=-5(P-12)`
`Or,\Q-30=-5P+60`
`Or,\Q=-5P+60+30`
`Or,\Q=90-5P.............(i)` 

Thus, equation `(i)` is the specific form of the demand function.
When the price is $8, quantity demand will be as calculated below,

`Q=90-5P.............(i)`
`Or,\Q=90-5(8)`   
`Or,\Q=90-40` 
`\therefore\Q=50` 

The quantity demanded is equal to 50 units. If it will be the equilibrium quantity, the supply will also be 50 units.

Q.no.9- If the quantity demanded of a commodity reduces from 20 units to 10 units in response to an increase in price from $10 to $20, what is the coefficient of price elasticity?

Solution:
Given;
`P1=10,\P2=20` and `Q1=20,\Q2=10`
`△P=P2-P1` and `△Q=Q2-Q1`
`Or,\△P=20-10` and `Or,\△Q=10-20`
`\therefore△P=10` and `\therefore△Q=-10`
As we know that the coefficient of elasticity of demand is calculated by using the formula as follows.
`e_p=-\frac{△Q}{△P}×\frac{P}{Q}`
Plotting the required values in the equation, we get;
`e_p=-\frac{-10}{10}×\frac{10}{20}`
`Or,\e_p=\frac{1}{2}`
`\therefore\e_p=0.5`



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