Index Number(chapter-10)

Index Number(Chapter-10)
WORKING FORMULAE
A. Un-weighted Index Numbers (or Simple Index Numbers)

L. Simple Aggregative Method

ii. Simple Average of Price Relative Method (A.M.)

Where, Antilog of ( ) ,
Price relative

B. Weighted Index Numbers
1 Weighted Aggregative Method
a. Laspeyre's Method:

b. Paasche's Method:

c. Fisher's Ideal Methed √ × 100

d. Drobish-Bowley's Method: P01(DB) = * + × 100

e. Marshall-Edgenerth Methed: P01(ME) = * +× 100

ii. Weighted Average of Price Relative Method if Arithmetic mean is used.

(G.M.) - Antilog of ( ) if Geometric mean is used

C. Test of Cessbtency
2 Time Reversal Test:

3 Factor Reversal Test: P01

D. Base Shifting xu r t y r
D. New index of any year xu r t w sy r

E. Deflating Purchasing power of money r x Re al Wage y x
r

Real Wages Index numbers urr t y r r w s
s y rr w s

F. Cost of Living Index Numbers

i. Aggregate Expenditure Method (or Weighted Aggregate Method) Cost of Living Index

ii. Family Budget Method (or Method of Weighted Relatives) = Cost of Living Index ,

E. using weighted arithmetic mean Antilog of ( )

Antilog of ( ), using weighted geometric mean

1.(a) Solution: Chapter-10

Items Price in 1990(P0) Index Number
A 100
Exercise-10
B 80
Price in 1995(P1)
C 160 140
120
D 220 180
240
E 40 40

P0 = 600 P1 = 720

Here, × 100 = 120

P01 = × 100 =

1(b) Solution:

Commodity Price in 1984(P0) Price in 1985(P1)
250
Wheat 200 400
500
Rice 300 25
325
Pulse 400
P1 = 1500
Milk 20

Clothing 280

P0 = 1200

Here,

P01 = × 100 = × 100 = 125

2.(a) Solution:

Price in 2042 Price in 2043 Price relatives

Commodity (P0) (P1) P = × 100 Log P

A 30 40 133.33 2.1249
B 40 45 112.5 2.0512
C 70 105 150 2.1761
D6 9 150 2.1761
E5 6 120 2.0792
n=5
= 665.83 ∑ = .6 75

Here,

P01 (A.M.) = = = 133.17%

P01 (G.M.) = Antilog ( ) = Antilog ( ) = antilog (2.1215) = 132.28%
2.(b) Solution:

Price in 1982 Price in 1983 Price relatives

Items (P0) (P1) P = × 100 Log P

Wheat 40 50 125 2.0969

Oil 120 150 125 2.0969

Fruit 75 93 124 2.0934

Egg 15 24 160 2.2041
n=4
∑ = 534 = 8.4913

Here, ( By average of price relative method)

(i) P01 (A.M.) = = = 133.5% ∴ the price are increased by 33.5%

P01 (G.M.) = Antilog ( ) = Antilog ( ) = antilog(2.1228) = 132.68%
∴ the price are increased by 32.68%

3.(a) Solution:
Commodities Base Year Current Year

A P0 q0 P1 q1 16 12 32 24
B 284 6 50 25 70 35
C 5 10 7 5 56 64 70 80
D 4 14 5 16 38 76 38 26
2 19 2 13
P01(L) = =160 = 127 = 210 = 165
P01(P) =
× 100 = × 100 = 131.25% 72 90
P01(F) = √ × 100 = × 100 = 129.92% 50 60
90 120
× 100 = √ × 100 = 130.58 % 132 96

3.(b) Solution: Base Year Current Year 60 75 = 344 =366
Commodities (1980) (1982) 40 48
P0 q0 120 160 1000 2400
A 5 12 P1 q1 110 80 1200 960
B 4 10 6 15 900 1050
C 8 15 5 12 =330 =363 540 900
D 6 20 800 800
10 11 12 8
= = 6110
i) P01(L) = × 100 = × 100 = 104.24% 4440
ii) P01(P) = × 100 = × 100 = 100.83%

iii) P01(F) = √ × 100 = √ × 100 = 102.52 %

3.(c) Solution: Base Year Current Year 600 1440
(1980) (1982) 1000 800
Group P0 q0 840 980
P1 q1 480 800
1 12 50 20 120 720 720
2 10 100 12 80
3 14 60 15 70 =3640 =4740
4 16 30 18 50
5 18 40 20 40

i) P01(L) = × 100 = × 100 = 121.98

ii) P01(P) = × 100 = × 100 = 128.9

4.(a) Solution:

Items 2002 2004 p0 q0 p1 q0 P01 = × 100 = × 100 = 150
P0 q0 P0
Rice 10 12 15 120 180
Pulse 84 12 32 48
Bread 65 9 30 45

=182 =273

4.(b) 50 60 40 48
Items 48 56 42 49
18 24 15 20
A 5 10 4 12
B 8677 =116 =140 =97 =117
C 6354

H r , pr qu t ty s urr t y rs r v . S w us F s r’s t t u t pr e index
number. √

× 100

=√ × 100
= 83.6

5.(a) Solution:

Commodities Base year Current Year

Price(P0) Value( ) Price ) Value( )

A 10 100 8 96 10 12 120 80
84
B 16 96 14 98 6 7 112 30
20
C 12 36 10 40 34 48
=2
D 15 60 5 25 45 75

=292 =259 =355

i) P01(L) = × 100 = × 100 = 73.29%
ii) P01(P) = × 100 = × 100 = 72.96%

iii) P01(F) = √ × 100 = √ × 100 = 72.12 %

5.(b) Solution:

Base year Current Year

Items Price(P0) Exp.( ) Price ) Exp. ( ) 5 5 30 40
6 30 8 40 5 6 38 50
A 6 6 24 30
B 8 40 10 60 10 11 22 40
C
D 4 24 5 30 ==
124 160
2 20 4 44 t u te price

=114 =174

H r , pr qu t ty s urr t y rs r v . S w us F s r’s t

index number.

P01(F) = √ × 100 = √ × 100 = 140.34 %

6.(a) Solution: 8 12 12 18
Commodity 15 3 10 2
16 48 8 24
A 4263
B 3521 =39 =63 =30 =44
C 8246

ii) P01(F) = √ × 100 = √ × 100 = 73.3 %

i) P01(L) = × 100 = × 100 = 76.92%
P01(P) = × 100 = × 100 = 69.84%

iii) P01(F) = √ =√

P10(F) = √ =√ =1 Hence, the time reversal test is satisfied.
Now, ×√ =√
P01(F) × P10(F)= √

Again, =√
Q01(F) = √

P01(F) × Q01(F)= √ ×√ =√ == = V01

Hence, factor reversal test is satisfied.

iv) P01(L)× P10(L) = = 1

∴ L sp yr ’s x u r s t s t s y t r v rs t st.

P01(L)× Q10(L) = =

∴ L sp yr ’s x u r s t s t s y t r r v rs t st.

Again,

P01(P)× P10(P) = =1

∴ s ’s dex number does not satisfy time reversal test.

P01(P) × Q01(P)= = = V01

∴ s ’s x u r s t s t s y t r r v rs t st.

6.(b) Solution: 858 32 32 40 40
Commodity 10 6 12 50 60 60 72
647 18 21 24 28
A4 5 10 4 40 32 50 40
B5
C3 =140 =145 =174 =180
D8

P01(F) = √ × 100 = √ × 100 = 124.21 %

Q01(F) = √ × 100= √ × 100= 103.51%

Again, P01(F)= √ =√ and P10(F) = √ =√
Now, =1

P01(F) × P10(F)= √ × √ = √
Hence, the time reversal test is satisfied.

Again, =√
Q01(F) = √

P01(F) × Q01(F)= √ ×√ =√ == = V01

Hence, the factor reversal test is satisfied.

6.(c) Solution: 500 336 500 560
Commodity 200 240 200 240
360 240 360 360
A 6 50 10 56 360 240 360 288
B 2 100 2 120 480 288 480 432
C 4 60 6 60
D 10 30 12 24 =1360 =1344 =1900 =1880
E 40 12 36
=√
P01(F) = √ = √ and P10(F) = √

Q01(F) = √ =√
Now,

P01(F) × P10(F)= √ ×√ =√ =1
=√
Hence, the time reversal test is satisfied. ==

Again, ×√ 40 = V01
P01(F) × Q01(F)= √ 30
18
Hence, the factor reversal test is satisfied.
=88
6.(d) Solution:

Commodity

Food 5 485 20 25 32
2 10 3 16 24 20
Clothing 8 263 12 18 12

Housing 6 =48 =67 =64

P01(F) = √ × 100 = √ × 100 = 132.33 and
Now,
P01(F) = √ = √ and P10(F) = √ =√

Q01(F) = √ =√
Now,

P01(F) × P10(F)= √ ×√ =√ =1
==
Hence, the time reversal test is satisfied.

Again, = V01
P01(F) × Q01(F)= √ × √ = √

Hence, the factor reversal test is satisfied.

7.(a) Solution:

Group Index no.(I) Weight(W) WI
Food 150 46 6900
Fuel & Lighting 140 5 700
Clothing 125 12 1500
House rent 115 14 1610
Miscellaneous 130 13 1690

∑ =9 ∑ = 12400

Here,

Cost of living index = = = 137.78

7.(b) Solution:

Group Index no.(I) Weight(W) WI

Food 352 48 16896

Fuel & Lighting 200 10 2000

Clothing 230 8 1840

House rent 160 12 1920

Miscellaneous 190 15 2850

∑ = 93 ∑ = 25506

Here,

Cost of living index = = = 274.26

7.(c) Solution:

Group Index no.(I) Weight(W) WI
Food 225 40 9000
Clothing 128 10 1280
Electricity 150 25 3750
Housing 180 25 4500

∑= ∑ = 853

Here,

Cost of living index = = = 185.30

7(d) Solution:

Item P= × 100 W PW Log P W logP

Food 125 150 120 40 4800 2.0792 83.168
Clothing 25 40 160 12 1920 2.2041 26.4492
125 18 2250 2.0964 37.7352
Rent 80 100 120 20 2400 2.0792 41.584
Fuel 50 60 150 10 1500 2.1761 21.761
Misc. 40 60 PW=12870
= 210.6974

i) Cost of living index = = = 128.7

ii) P01 (A.M.) = = = 128.7

P01(G.M.) = antilog =antilog( ) = antilog (2.106974) = 127.93

8.(a) Solution:

Commodity 560 400
A 25 16 35 336 252
B 36 7 48 64 48
C 12 4 16 30 18
D 6 3 10 112 112
E 28 4 28
=1102 =830

Cost of living index = × 100 = × 100 = 132.77%

8.(b) Solution:

Items × 100 W PW

Food 30 42 140 4 560
150 1 150
Fuel 8 12 125 3 375
125 2 250
Clothing 16 20 120 1 120
=660 =11
House rent 12 15 =1455

Miscellaneous 25 30

n= 5

i) P01 (A.M.)= = = 132 = 132.27
ii) P01 (Weighted average)= =

8.(c) Solution:

Item P= × 100 W= PW

Food grains 3 32 38 118.75 96 11400 114
Cloth 12 5 4 80 60 4800 48
125 800 100000 1000
Electricity 40 20 25 140 1500 210000 2100
Room 3 500 700 80 1500 120000 1200
Misc. 30 50 40 PW= 446200

i) Cost of living index = × 100 = × 100 = 112.79

ii) Cost of living index = = = 112.79

8.(d) Solution: P= × 100 PW Log P W.log P

Commodity 1000 80 150 120 150 15000 2.1761 217.61
A 10 10 15 8 60 50 72 60 120 7200 2.0792 124.752
B 5 12 6 10 40 16 50 20 125 5000 2.0969 83.876
C 8 5 10 2
=200
=146 =272 =200 =27200 =426.238

i) P01(L) = × 100 = × 100 = 136
ii) P01(P) = × 100 = × 100 = 136.99

iii) P01(F) = √ × 100 = √ × 100 = 136.49

iv) P01 (A.M.) = = = 136 (Price relative)

P01(G.M.) = antilog =antilog( ) = antilog (2.13119) = 135.27

9.(a) Solution:

Items I W WI Cost of living index = = = 219.50
Food 225 52 11700
Clothing 175 8 1400 salary
Electricity 155 10 1550 Now, 5000
Housing 350 14 4900
Misc. 150 16 2400 Index no. ?
=100
= 21950 Base year: 100
Current year index: 219.50

The current salary= = Rs 10975

The required extra allowance=10975-7000= RS 3975

9.(b) Solution:

Items I W WI Cost of living index = = = 205.50
Food,drink & tobacco 225 52 11700
175 8 1400 salary
Clothing 155 10 1550 7200
Fuel & lighting 250 14 3500 Now,
150 16 2400 ?
Housing =100 Index no. = Rs 14796
Miscellaneous = 20550
Base year: 100

Current year index: 205.50

The current salary= 7

9.(c) Solution:

Items × 100 W PW Log P W log P

Food 250 360 144 45 6480 2.1584 97.128
cloth 125 210 168 20 3360 2.2253 44.506
Rent 75 90 120 10 1200 2.0792 20.792
Fuel 50 20 40 15 600 1.6020 24.03
Misc. 25 50 200 10 2000 2.3010 23.010
n= 5 =100
=13640 =209.466

i) Cost of living index = = = 136.40

∴ cost of living index has increased by 136.40-100=36.40%

Index no. salary

100 12500

136.40 ?

The required salary in 2003 = × 12500 = Rs 17050,

But the salary in 2003 is only Rs 15000, so inadequacy salary = 17050 – 15000 = Rs 2050

ii) P01 (A.M.) of price relative = = =136.40

i) P01 (G.M.) of price relative = antilog ( ) = antilog ( ) = antilog (2.09466) = 124.35

10.(a) Solution: Cost of living index = ×= = 132

Commodity
Wheat 100 8 120 9 Now,
Rice 120 10 180 15
Cloth 50 20 60 45 800 960 Index no. salary
1200 1800
1000 1200 Base year: 100 6200

=3000 =3960 Current year index: 132 ?

The current salary (1990)= = Rs 8184

11.(a) Solution:

Year I.N. I.N. with 2061 as base vaue

2059 100 80

2060 120 96

2061 125 100

2062 140 112

2063 145 116

2064 150 120

11. (b) Solution:

i) Taking Rs 4 as the base value (ii) Taking Rs ( )= Rs 5 as the base value (iii) Taking Rs 10 as the base value

Year Price/kg I.N.(1980 as base) I.N.(average of 1ST 3 years as base) I.N.(1985 as the base)

1980 4 100 80 40

1981 5 125 100 50

1982 6 150 120 60

1983 7 175 140 70

1984 7.5 187.5 150 75

1985 10 250 200 100

1986 9 225 180 90

1987 10 250 200 100

1988 11 275 220 110

12.(a) Solution:

Year Wage Index No. Real wage = Real wage index
2044 650 100 (2044 as the base)
2045 980 120
2046 1200 130 7
7

2047 1800 145

2048 2200 148

2049 2800 155 77

12.(b) Solution:

Year Wage Index No. Real wage = Real wage index
(2044 as the base)
2058 50 100
7
2059 65 104 7

2060 84 120 7
2061 98 125 7
2062 112 140

2063 124 155

2064 144 160

13.(a) Solution:

Item Weight(W) Original Price(I.N.) Increase % New index(I) WI

Food 15 100 32 132 1980

Clothing 3 100 54 154 462

Rent 4 100 47 147 588

Fuel & Light 2 100 78 178 365

Misc. 1 100 58 158 158

=25 =3544

Here, Cost of living index Salary
Cost of living index = = = 141.76 100 5000
141.76
The required salary = = Rs 7088 ?

13.(b) Solution:

Item Weight(W) Original Price(I.N.) Increase % New index(I) WI

Food 8 100 30 130 1040

Rent 2 100 57 157 314

Clothing 2.5 100 90 190 475

Fuel 1 100 75 175 175

Misc. 1.5 100 88 188 282

=15 =2286

Here, Cost of living index Salary
Cost of living index = = = 152.40 12500

100 ?

The required salary = = Rs 19050 152.40

13.(c) Solution: Let, be the % increase value in food.

Item Weight(W) Original Price(I.N.) Increase % New index(I) WI
6000+60
Food 60 100 100+
2640
Rent 16 100 65 165 3840
1680
Clothing 12 100 220 320 900
=15060+60
Fuel & Light 8 100 110 210

Misc. 4 100 125 225

=100

Here, , , cost of living index = 204.60

Now,

Cost of living index =

204.60 =

15060+60 = 20460
60 = 20460 – 15060

60 = 5400

= = 90

∴ % increase in food = 90%

The required salary = = Rs 19050

14. Solution:
Let, & be the expenditures on food and rent respectively.

Group Expenditure(W) Group index(I) WI

Food 190 190

Clothing 125 181 22625

Rent 140 140

Fuel & Lighting 100 118 11800

Miscellaneous 75 101 7575

=300+ + =42000+190 +140

=300+ + , =42000+190 +140 , Cost of living index = 160

We have, = 750 (Total Income or expenditure)

300+ + = 750
=750 – 300 –
=450 – -------------------------(i)

Again, Cost of living index =

160 =

160 =

160 × 750 = 42000 + 190 + 63000 - 140
120000 = 105000 + 50
120000 – 105000 = 5

= 15000

= = 300

Putting the value of in equation (i), we get
= 450 – 300 = 150

Hence Expenditure on food = Rs 300 and rent = Rs 150