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