Index Numbers II

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 Pon × Pno = 1

The Fisher’s and Marshall-Edgeworth’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 Marshall-Edgeworth index is as follows:

It means that the time reversal test for Marshall-Edgeworth 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 Marshall-Edgeworth 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, low-income 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 = PoQo

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

Qo

Prices

Aggregate

Expenditure

Po

Pn

PoQo

PnQo

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

Qo

Po

Pn

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 low-income 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

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