Theoretical
Tests for Index Numbers:
According to Dr. Irvin Fisher, a good
index number is required to satisfy the theoretical tests given below:
(a)
Time
Reversal Test: This test may be stated
as follows:
“If
the time subscripts of a price (or quantity) index number formula be
interchanged, the resulting price (or quantity) index formula should be
reciprocal of the original formula”
or P_{on} × P_{no}
= 1
The Fisher’s
and MarshallEdgeworth’s formulae satisfy the ‘Time Reversal Test’.
(i)
Lets take Lespeyre’s formula and interchange the time
subscripts:
Therefore,
Lespeyre’s index does not satisfy the tests.
(ii)
Now take Paasche’s formula and by interchanging the time
subscripts, we get:
The
above calculation shows that the Paasche’s formula does not satisfy the time
reversal test.
(iii)
The Fisher’s index formula is given below:
On
interchanging the time subscripts, we get
Now,
by multiplying the resulting formula to the original formula, we get:
It
means that Fisher’s index number satisfies the time reversal test.
(iv)
The time reversal test for MarshallEdgeworth index is as
follows:
It
means that the time reversal test for MarshallEdgeworth index is satisfied.
(b)
Factor
Reversal Test: This test may be stated
as follows:
“If
the factors Ps and Qs occurring in a price (or quantity) index formula be
interchanged so that a quantity (or price) index formula is obtained, then the
product of the two index numbers should give the true value index number”
(i)
Lets take the Lespeyre’s formula and interchange the
factors and then multiply the same with the resulting formula:
Therefore,
the factor reversal test on Lespeyre’s formula is failed and it is concluded
that Lespeyre’s index number is not a Value Index Number.
(ii)
Now take Paasche’s formula and interchange the factors
and then multiply both the formulae:
The
above test shows that Paasche’s index number does not satisfy the factor
reversal test and it means that Paasche’s index formula is not a value index
formula.
(iii)
For Fisher’s index:
Therefore,
it is concluded that factor reversal test is also satisfied in case of
Fisher’s index. Since the
Fisher’s index satisfies the time reversal test and factor reversal test, it
is called Fisher’s Ideal Number.
(iv)
The factor reversal test is not satisfied for MarshallEdgeworth
index, as:
Ideal Index Number:
The Fisher’s index number is also known ideal index number in this sense that
other index number, i.e., Lespeyre’s and Paasche’s either overstate or
understate expenditure index. The
Lespeyre’s index number overstates the index and Paasche’s index number
understates the index. The
‘true’ index lies somewhere between Lespeyre’s and Paasche’s indices,
and Fisher has suggested that it is equivalent to the GM of the two, i.e.:
Therefore, Fisher’s index numbers are
ideal in the sense that they are the only ones that correctly predict the
expenditure index.
Wholesale
Price Index:
The whole price index number is designed
to measure changes in the goods and services produced in different sectors of
the economy and traded in wholesale markets.
These goods and services even include electricity, gas, petrol,
telecommunication, etc. These
indices are constructed by weighted aggregative method with quantities produced
or sold as weights.
Consumer
Price Index (CPI):
1. This index is also known as the ‘Cost of Living Index’ or ‘Retail Price Index’. It is designed to measure changes in the cost of living.
2. By cost of living, the cost of goods and services of daily use purchased by a particular class of people in a city or town is meant. These goods and services are known as ‘market basket’ consists of food, house rent, apparel, energy, education, health and miscellaneous items.
3.
CPI is essentially a weighted aggregative price index.
The prices used are the coverage retail prices paid by the consumers for
purchase of goods and services. The
weights are proportion of expenditure on different goods and services.
Construction of Consumer Price
Index:
Following steps are involved in the
construction of CPI:
(a) Scope: As the first step the scope of index number is determined, e.g., the industrial workers, middle class, salaried individuals, lowincome earners, etc. It is, therefore, necessary to specify the class of people and the locality where they reside. The class or group of people considered should, as far as possible, be homogenous with regard to their incomes and consumption patterns.
(b) Family Budget Inquiry and Allocation of Weights: The second step is to conduct a family budget inquiry so as to ascertain the proportions of expenditure on different items and to assign weights to various items. The information regarding the nature, quality and quantities of commodities consumed should be analysed and weights assigned in proportion to the expenditure on different items.
(c) Price Data: The prices used in the construction of consumer price index are the retail prices.
(d)
Methods of Construction:
Two methods are used for the construction of CPI.
The index numbers under both methods are the same.
These methods are:
(i)
Aggregative Expenditure Method:
In this method, the quantities consumed in the base year are taken as weights.
It is thus the base year weighted index number given by Lespeyre’s
formula:
(ii)
Family Budget Method:
This method is the weighted average of relatives.
The amounts of expenditure incurred by families on various items in the
base period are used as weights. This
method is known as ‘Family Budget Method’ because the amounts of money spent
by the families are obtained from a family inquiry:
Where
and W = P_{o}Q_{o
}
Example:
Commodity 
Qty. Consumed 
Unit
of Price 
Price 

2001 
2005 

Wheat 
250 kgs 
Rs. per 10 kgs bag 
120 
150 
Rice 
100 kgs 
Rs. per 40 kgs bori (bag) 
800 
1120 
Sugar 
18 kgs 
Rs. per 40 kgs bori (bag) 
440 
1200 
Milk 
57 kgs 
Rs. per kg 
18 
32 
Tea 
14 kgs 
Rs. per 100 kgs bag 
17800 
18000 
Salt 
1 kg 
Rs. per kg 
1 
5 
Compute consumer price index number for
the year 2005 taking 2001 as base year using:
(a) Aggregative Expenditure Method, and
(b)
Family Budget Method.
Solution:
(a)
Aggregative Expenditure Method:

Quantity Consumed 
Q_{o } 
Prices 
Aggregate Expenditure 

P_{o } 
P_{n } 
P_{o}Q_{o } 
P_{n}Q_{o } 

Wheat 
250 kgs 
25 bags 
120 
150 
3000 
3750 
Rice 
100 kgs 
2.5 boris (bags) 
800 
1120 
2000 
2800 
Sugar 
18 kgs 
0.45 bori (bag) 
440 
1200 
198 
540 
Milk 
57 kgs 
57 kgs 
18 
32 
1026 
1824 
Tea 
14 kgs 
0.14 bag 
17800 
18000 
2492 
2520 
Salt 
1 kg 
1 kg 
1 
5 
1 
5 





8717 
11439 
(b)
Family Budget Method:

Qty. Consumed 
Q_{o } 
P_{o } 
P_{n } 
V 
W_{ } 
WV 
Wheat 
250
kgs 
25
bags 
120 
150 

3000 
375000 
Rice 
100
kgs 
2.5
boris (bags) 
800 
1120 

2000 
280000 
Sugar 
18
kgs 
0.45
bori (bag) 
440 
1200 

198 
54000.54 
Milk 
57
kgs 
57
kgs 
18 
32 

1026 
182402.28 
Tea 
14
kgs 
0.14
bag 
17800 
18000 

2492 
251991.04 
Salt 
1
kg 
1
kg 
1 
5 

1 
500 






8717 
1143893.86 
Uses
of Consumer Price Index:
(a)
Purchasing Power of Money:
The purchasing power of a rupee is the reciprocal of CPI. It expresses the purchasing power of rupee in a current time
period relative to the base period:
(b)
Deflation
of Per Capita Income: The effect of
changing prices on per capita income may be removed by deflating the income
expressed in current money by CPI to produce a measure expressed in terms of
deflated (real) money. This
relationship is:
The deflated
(or real) per capital income is expressed in terms of the price level at the
time of the base period of CPI.
Example:
For the following CPI, calculate the
purchasing power of rupee for each year:
Year 
CPI 
2000 
100 
2001 
103.54 
2002 
106.75 
2003 
111.63 
2004 
121.98 
Solution:
Year 
CPI 
Purchasing
Power of Rupee 
2000 
100 
1.0000 
2001 
103.54 
0.9658 
2002 
106.75 
0.9368 
2003 
111.63 
0.8958 
2004 
121.98 
0.8198 
Example:
Deflate the Per Capita Income (PCI) by
the consumer price index given in the following table, with base year 2000:
Year  CPI 
PCI (In
US$) 
2000 
100 
526 
2001 
103.54 
501 
2002 
106.75 
503 
2003 
111.63 
579 
2004 
121.98 
657 
Solution:
Deflated Per Capita Income:
Year 
CPI 
Per Capita Income 

Current 
Real 

2000 
100 
526 

2001 
103.54 
501 

2002 
106.75 
503 

2003 
111.63 
579 

2004 
121.98 
657 

Construction
of Wholesale and Consumer Price Indices in Pakistan:
Three price indices are prepared in
Pakistan by Federal Bureau of Statistics:
(a) Wholesale Price Index,
(b) Consumer Price Index, and
(c)
Sensitive Price Indicator.
(a) Wholesale Price Index: This index is based on 91 commodities comprising 690 specifications. Lespeyre’s method is used and the average wholesale prices for the current year are used.
(b) Consumer Price Index (CPI): This index is based on around 500 items of daily use grouped in 9 groups. Prices are collected from 25 major cities of the country. Number of markets varied from 1 to around 15 depending upon the size of the town. The average prices collected from the markets of a town represent the average retail price of that town. Lespeyre’s method is used for construction of CPI.
(c)
Sensitive Price Indicator:
This index measures changes in the retail prices of around 50 essential items of
daily use of lowincome earners’ group.
Splicing
Index Numbers:
1. A series of index numbers may be discontinued because of obsolete commodities included in it or because of change in weights of these commodities.
2. If a new series of index numbers is constructed with changed commodities or changed weights, the two series (old and new) are not comparable. The old and new series must therefore be adjusted so that the two series are comparable.
3.
To adjust the new series, new index numbers are multiplied by the ratio
of the old to the new index in the period of discontinuation:
4.
Likewise, to adjust the old series, old index numbers are multiplied by
the ratio of the new to old index:
The above
procedures are known as ‘Splicing Index Numbers’.
Example:
In the data given below, 2001 is the
year of discontinuation of the old series.
Construct a continuous series by splicing:
(a) Old series, and
(b)
New series.
Year 
Index (Old
Series) 
Index (New
Series) 
1995 
99.8 

1996 
96.7 

1997 
95.3 

1998 
111.9 

1999 
134.6 

2000 
159.8 

2001 
173.2 
96.7 
2002 

100.0 
2003 

100.9 
2004 

109.1 
2005 

111.0 
Solution:
(a) Old Series: To splice the old series, multiply old indices by 0.5583, i.e., .
(b)
New Series:
To splice the new series, multiply new indices by 1.7911, i.e.,
.
Year 
Index (Old) 
Index (New) 
Spliced Index (Old) 
Spliced Index (New) 
1995 
99.8 

55.7 
99.8 
1996 
96.7 

54.0 
96.7 
1997 
95.3 

53.2 
95.3 
1998 
111.9 

62.5 
111.9 
1999 
134.6 

75.1 
134.6 
2000 
159.8 

89.2 
159.8 
2001 
173.2 
96.7 
96.7 
173.2 
2002 

100.0 
100.0 
179.1 
2003 

100.9 
100.9 
180.7 
2004 

109.1 
109.1 
195.4 
2005 

111.0 
111.0 
198.8 