The presentation is presented for providing a solution to different simple mathematical problems in the income elasticity of demand. The presentation includes all the measurement methods of income elasticity of demand that are covered in the simplest way. The PDF file is available at https://knoweco11.blogspot.com/2020/05/microeconomics-note-bbs-income-elasticity.html

1. Solution to Mathematical
Questions of Income
Elasticity of Demand

2. Formula used in measure of income elasticity of demand
• Percentage Method:
𝑒 𝑌=
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒
• Proportion Method:
𝑒 𝑌 =
∆𝑄
∆𝑌
.
𝑌
𝑄
• Arc Method:
𝑒 𝑦=
∆𝑄
∆𝑌
.
𝑌1+𝑌2
𝑄1+𝑄2
Tilak Mahara

3. 1. Suppose that the initial income of a person is Rs.2000 and quantity
demanded for the commodity by him is 20 units. When his income
increases to Rs.3000, quantity demanded by him also increases to 40 units.
Find out the income elasticity of demand.
Solution:
Here, 𝑄1 = 20 units, 𝑄2 = 40 So, ∆Q = (40-20) units = 20 units
𝑌1 = Rs.2000, 𝑌2=Rs. 3000 So ,∆Y = Rs. (3000-2000) =Rs.1000
Now,
𝑒 𝑌=
∆𝑄
∆𝑌
.
𝑌
𝑄
=
20
1000
.
2000
20
= 2
Thus 1% increase in income leads to a rise of 2% in quantity demanded.
Tilak Mahara

4. 2. Given the demand schedule and compute;
• Income elasticity of demand at movement from B to D by percentage method.
• Income elasticity of demand at movement from B to D by proportion method.
• Income elasticity of demand at movement from B to D by arc method.
Solution
Here given
Measure of income elasticity of demand at movement from B to D by percentage
method
𝑄1 = 200 units, 𝑄2 = 400 So, ∆Q = (400-200) units = 200 units &
Tilak Mahara
Combination A B C D
Income 20,000 40,000 60,000 80,000
Demand 100 200 300 400

5. So %∆Q=
200
200
*100= 100%
𝑌1 = Rs.40,000, 𝑌2=Rs. 80,000 So ,∆Y = Rs. (80,000-
40,000) =Rs.40,000 &
%∆Y=
40,000
40,000
*100= 100%
Now
𝑒 𝑌=
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒
=
100%
100%
= 1
Tilak Mahara

6. Again
Measure of income elasticity of demand at movement from B to
D by proportion method.
𝑒 𝑌 =
∆𝑄
∆𝑌
.
𝑌
𝑄
=
200
40,000
.
40,000
200
= 1
And
Measure of income elasticity of demand at movement from B to
D by arc method.
𝑒 𝑦=
∆𝑄
∆𝑌
.
𝑌1+𝑌2
𝑄1+𝑄2
=
200
40,000
.
40,000+80,000
200+400
= 1
Tilak Mahara

7. 3. Suppose income increase by 25 percent and, as a result, the quantity of
a particular brand of automobile demanded (holding the price for this
particular automobile constant) increases by 40 percent.
Solution
Here given
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦= 40%
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒= 25%
Now
𝑒 𝑌=
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒
=
40
25
= 1.6
The income elasticity of income greater than one means that this brand of
automobile is a luxury good.
Tilak Mahara

8. 4. If demand increased by 50 percent due to an increase in income by 75
percent, calculate the income elasticity of demand.
Solution
Here given,
Percentage change in demand= 50 %
Percentage change in income= 75 %
And ss we know,
𝐸 𝑌 =
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑖𝑛 𝑖𝑛𝑐𝑜𝑚𝑒
=
50%
75%
= 0.666
Therefore coefficient of income elasticity of demand is 0.66.
Tilak Mahara

9. 5. Suppose demand for cars in Kathmandu as a function of income is given
by the following equation: Q= 20,000 + 5M
Where Q is quantity demanded, M is per capita level f income in rupees.
Find out income elasticity of demand when per capita annual income in
Kathmandu is Rs. 15,000.
Solution:
Here
Income demand function= Q= 20,000 + 5M
Per capita income (M) = Rs. 15,000 So Q= 20,000+5.15, 000= 95,000
Now,
𝑑𝑄
𝑑𝑀
=
𝑑(20,000 + 5M)
𝑑𝑀
= 5
By formula,
𝐸 𝑌=
∆𝑄
∆𝑌
×
𝑌
𝑄
= 5.
15,000
95,000
= 0.8
Therefore coefficient of income elasticity of demand is 0.80.
Tilak Mahara

10. 6. Consider the given table and calculate income
elasticity of demand at point B, C and D.
Solution:
To solve the problem of elasticity under point method we
need to draw the graph of given schedule and after
drawing the curve we can measure the coefficient of
elasticity at particular point by using needed formula.
Tilak Mahara
Combinations A B C D E
Income (Rs.) 2,000 3,000 4,000 5,000 6,000
Quantity (Units) 100 150 200 250 300

11. Tilak Mahara
Now,
Income elasticity at point B=
𝑂𝑀
𝑂𝑀
=
150
150
= 1
Income elasticity at point C=
𝑂𝑁
𝑂𝑁
=
200
200
= 1
Income elasticity at point D=
𝑂𝑅
𝑂𝑅
=
250
250
=1

12. 7. Consider the following table and compute income elasticity of
demand at point B and D.
Solution:
The graphical representation of given schedule is shown as;
Tilak Mahara
Points A B C D E
Income (Rs.) 4,000 6,000 8,000 10,000 12,000
Demand (Units) 50 100 120 140 150

13. In the graph our income demand curve is non-linear. To measure income
elasticity of demand at a particular point on the non-linear curve we have to
draw a straight line in such a way that the line is tangent to that point where
we have to measure elasticity and then we can apply the as usual process of
calculation.
In this case, we have to measure income elasticity of demand at pint B and
D. Here straight line LL1 and OT are drawn so as to tangent to the point D
and B respectively.
Now,
Income elasticity at point B on the demand curve =
𝑂𝑅
𝑂𝑅
=
100
100
=1
Income elasticity at point D on the demand curve =
𝐿𝑆
𝑂𝑆
=
140−70
140
= 0.5
Tilak Mahara

14. 8. The demand for a product has income elasticity of 3 and if there is a 10
% increase in the consumer’s income then compute the change in demand.
Solution:
Here,
Coefficient of income elasticity (𝐸 𝑌)= 3
Percentage increase in income= 10%
Percentage increase in demand=?
As we know
𝐸 𝑌=
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑐𝑜𝑚𝑒
Or 3=
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑
10%
Therefore, percentage change in quantity demand of the product= 3*10= 30
percent.
Tilak Mahara

15. 9. A business firm’s demand function is gives as Q= 60-60P + 2Y. Where P is
price and Rs. 1, Q is quantity and Y is income and Rs. 40. Suppose price is
increases by Rs. 2 and income increases by Rs. 80, then find the quantity
demand and also income elasticity of demand.
Solution:
Here given demand function is; Q= 60-60P + 2Y
Initial price (𝑃1) = Rs. 1 and Income (𝑌1) = Rs. 40
So initial quantity demanded (𝑄1) = 60- 60*1 + 2*40= 80 units
Now price is increases by Rs. 2 means new price (𝑃2) = 1+2= Rs. 3and
Income is increased by Rs. 80 means new income(𝑌2) = Rs. 40+80= Rs. 120 then
New quantity demand(𝑄2) = 60-60*3 + 2*120= 120 units
Thus new quantity demand with reflection of change in price and income both is
120 units.
Tilak Mahara

16. For income elasticity of demand,
Here,
(𝑌1) = Rs. 40,(𝑌2) = Rs. 120, (𝑄1)= 80 units and (𝑄2)=60-60*1+ 2*120=
240 (while measuring income elasticity we have to assume price remains
constant so increased price is ignored).
So
𝐸 𝑌=
∆𝑄
∆𝑌
×
𝑌
𝑄
=
240−80
120−40
×
40
80
=
160
80
×
40
80
=1
𝑜𝑟
𝐸 𝑌=
∆𝑄
∆𝑌
×
𝑌
𝑄
= 2.
40
80
= 1 {where,
𝑑𝑄
𝑑𝑌
=
𝑑(60−60P + 2Y)
𝑑𝑌
= 2}
Tilak Mahara

17. 10. The quantity demand for a commodity increases from 100 to 120 units
with decline in price from Rs. 12 to Rs.8. On the other hand, when income
increased by 5 percent, the commodities’ demand in increased from 100 to
110 units. What is the income elasticity of the commodity?
Solution:
Here given
Percentage increase in income= 5 % and
Percentage increase in demand=
110−100
100
*100 = 10 %
Now
𝐸 𝑌=
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑
𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑐𝑜𝑚𝑒
=
10
5%
= 2
Therefore, income elasticity is 2.
Tilak Mahara

18. 11. Last year Thomas purchased 6 pairs of shoes when his income was Rs.
40,000. This year, his income is Rs. 35,000 and he purchased 5 pair of
shoes. All else unchanged, what do you say about income elasticity.
Solution:
Initial income (𝑌1) = 𝑅𝑠. 40,000, Current income (𝑌2)= Rs. 35,000
So change in income (∆𝑌) = (𝑌2 − 𝑌1) = Rs. 35,000-40,000= Rs. -5000
Initial demand (𝑄1) =6 units, Current demand (𝑄2) =5 units
So change in demand (∆𝑄) = (𝑄2 − 𝑄1) =5-6 = -1 units
Now
Coefficient of income elasticity (𝐸 𝑌) =
∆𝑄
∆𝑌
×
𝑌
𝑄
=
−1
−5000
×
40,000
6
= 1.33
Here coefficient of income elasticity is 1.33, which shows shoes are
luxurious good for Thomas.
Tilak Mahara

19. 12. Suppose that the initial income of a person is Rs.2000 and quantity
demanded for the commodity by him is 20 units. When his income
increases to Rs.3000, quantity demanded by him also increases to 40 units.
Find out the income elasticity of demand.
Solution:
Here, 𝑄1 = 20 units, 𝑄2 = 40 So, ∆Q = (40-20) units = 20 units
𝑌1 = Rs.2000, 𝑌2=Rs. 3000 So, ∆Y = Rs. (3000-2000) =Rs.1000
Now,
𝑒 𝑌=
∆𝑄
∆𝑌
.
𝑌
𝑄
=
20
1000
.
2000
20
= 2
Thus 1% increase in income leads to a rise of 2% in quantity demanded.
Tilak Mahara

20. 13. If income demand function is given as; Q= 10,000 + 5Y. Find income
elasticity of demand by point method at income Rs. 5000.
Solution:
Here given
Income demand function as; Q= 10,000 + 5Y and at income Rs. 5000, Q=
10,000 +5.5,000= 35000 units
The point income elasticity of demand can be measured as;
𝑒 𝑌 =
𝑑𝑄
𝑑𝑌
.
𝑌
𝑄
= 5.
5,000
35,000
=
25,000
35,000
= 0.71
Here coefficient of income elasticity is less than one indicating the goods
as necessity good.
Tilak Mahara

21. 14. Suppose income increase by 25 percent and, as a result, the quantity of a
particular brand of automobile demanded (holding the price for this
particular automobile constant) increases by 40 percent.
Solution:
Here given
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦= 40%
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒= 25%
Now
𝑒 𝑌=
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑚𝑎𝑛𝑑𝑒𝑑 𝑜𝑓 𝑎 𝑐𝑜𝑚𝑚𝑜𝑑𝑖𝑡𝑦
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑡𝑒 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝐶𝑜𝑛𝑠𝑢𝑚𝑒𝑟′ 𝑠 𝐼𝑛𝑐𝑜𝑚𝑒
=
40
25
= 1.6
The income elasticity of income greater than one means that this brand of
automobile is a luxury good.
Tilak Mahara

22. Thank You
Tilak Mahara