Index Number

 Index Number     

Introduction 

We seen in measures of central tendency the data can be reduced to a single figure by calculating an average and two series can be compared by their averages. But the data are homogeneous then the average is meaningful. (Data is homogeneous means data in same type). If the two series of the price of commodity for two years. It is clear that we cannot compare the cost of living for two years by using simple average of the price of the commodities. For that type of problem we need type of average is called Index number. Index number firstly defined or developed to study the effect of price change on the cost of living. But now days the theory of index number is extended to the field of wholesale price, industrial production, agricultural production etc. Index number is like barometers to measure the change in change in economics activities. 

 An index may be defined as a "specialized average designed to measure the change in the level of a phenomenon in Reported like time and location."  This single value which  describe the change in a phenomenon with respect to time etc.  For example when we say that the cost of living index is 125 in 2000 as compared to 2002, that means there is a 25% rise the cost of living in that year. 

a)  Number is a specialized average: -

The average calculate from the figure of the same like marks, height. Is called simple average. Index number is not that type of average. The commodities are measure in different units like sugar in Kg, milk in Litters, a simple average of that prices is meaningless. Therefore the average defined in different that represents the series of that type. Index number is specialized average.

b) Index number measure the change in level of a phenomenon.:-

 Index number gives information about whole they does not say anything about it's components.  If the cost of living index is 117 in 1998 as compared to 1994, is means that there is 17% total expenditure increase of a family. It doesn't mean that the price of components is increasing 17% .  It give information about total change in expenditure of family. 

c)  Number also measure the effect of change over period of time. :-

An index number measure change in variable over period of time. That mean it measures the change the value of variable form considered base year. 

The problems involved in construction of index number

1. The purpose of index number:-  we know there different types of index numbers such as wholesale price index number, industrial production etc. And each index number gives information about particular economics activities. Hence the index number defined on purpose of study. 

2. Selection of base year:- We know the index number measure change with reference to time. An index number express the value of current year in percentage with respect to typical year. 

Following point used to select base year. 

I. The base year is normal:- the base year normal mean the year selected for comparison is normal one. (that is the selected base year is not affect by any irregular variation or any unusual situations like famines, earthquake, etc. 

II. The base year is not be too distant:-  The economic activities are change over time hence the base year is distant past the results may be affected by cumulative effect.

3. Selection methods for base year :- There are two method fixed  base year method  and chain base year method.  

I. Fixed base year method:- the comparison is made with always with same base year which is fixed. 

II. The chain bases method :- the comparison is made always with previous year. Thin we get series of figure which gives information about activity. The chain bases method is superior then fixed base method.

4. Selection of items.:-  the selection of items is based on the purpose of that index number. The item selected for one index number is irrelevant to the other index number. e.g. we constructing cost of living index number for working class then we doesn't include the costly items. Like car's expenses, laptops. And if we selecting the item the quality of items must be decided. If the quality of items changes then the index number is not reliable. 

5. Price quotation:  the problems is the fix the price of selected commodities. Because we know the price are different in different cities and the price is different in different shop's in that city. For that we select the price form reasonable shop and taking it average and considering this price. In that also we decided which price select wholesale price or retailer price.   Again there is problems to give quotation price on two ways. As first money price and second is qualify price. In the first method we give preference two price in rupees per unit commodity. And in second method gives preference to quantity of the commodity. 

6. Choice of an average:- an index number is  essentially an one the average of special kind of average and such average can be chosen. In this method generally only arithmetic mean and geometric mean are used. But the we select one of them but geometric mean is theoretically superior than arithmetic mean but is not easy to calculate. Then we use arithmetic mean is mostly preferred and used. 

7. Selection of weight:- weights are more important of item in whole phenomenon. In all items are not equally important. e.g. sugar is not important as rice. There  are two types to assigned  in two ways I. Quantity weights and ll. Value weight

 I. Quantity weights:- the quality may be used as  actually weights. But they are not always appreciate. And another difficulty connected with the quantity weights that in many problem the quantity not measure in same unit. 

 II. Value of weight:- when the units are measured in different then it is convenient to use value weight. The value weight of commodity is the ratio of expenditure on that item to the total expenditure multiplied by 100. There p is price of commodity, q is the quantity consumed for the base year, they defined as (p+q)/(total expenditure)x100. It is easy than quantity weights are assigned I this manner the sum of all the weight will be equal to 100

 8. Selection of formula:- there are many formula available to calculate an index number. And the choice depends upon the purpose and nature of data. . 

Construction of Index Number

we see the problems of construction of index number there are many problem to construct the index number,  therefore for that different problems we used the different formula to construct the index number.

For construction of index number the formulae divided into following two parts as:

1. Unweighted index Number  2. Weight Index Number

And each part divided into two parts.

i) Aggregative and ii)Average of price relatives.

First method is Un-weighted Index Number

i. Aggregative Method: it is simplest method and is defined as

P₀₁ =  Index number for current year 1 as base year 0.

 = Sum of the Price in current year.

 = Sum of Price in Base year.

It is simplest method but have two major drawbacks.

i. it does not consider the weights of items. (i.e. it consider only price of items.)

ii. it not consider unit of measurement.

 

For better understanding see following example

Example 1. Calculate simple aggregative index number by taking 1980 as base year.

Commodity	Price in year 1980	Price in year 1984
Sugar	38	40
Rice	35	39
Clothes	102	110
Rent	98	105

Solution:-

For constructing index number

Commodity	Price in year 1980	Price in year 1984
Sugar	38	40
Rice	35	39
Clothes	102	110
Rent	98	105
total	273	294

Aggregative Index Number.

Where p₁ = price of commodity in the current year

 p₀ = price of commodity in  base year.

2. Average of price relative:  This method also simple as aggressive method there is small change in formula. Firstly calculating price relative for each commodity and price relative means the ration of price in current year to the price in base year. In percentage. Is given as

Where p₁ = price of commodity in the current year

 p₀ = price of commodity in  base year.

Average of price relative method is superior than aggregative method . because it consider the ratios of the prices are calculated. Here only consider the price unit of measurement is not affected. it has drawback that the weight.

 

Example 1. From the following data compute price index number by using  average of price relative using arithmetic mean.

Commodity	Price in year 1980	Price in year 1984
Sugar	38	40
Rice	35	39
Clothes	102	110
Rent	98	105
total	273	294

Solution:

For constructing index number

Commodity	Price in year 1980	Price in year 1984	Price Relative
Sugar	38	40	105.2632
Rice	35	39	111.4286
Clothes	102	110	107.8431
Rent	98	105	107.1429
total			431.6777

Index number using Average of relative price  is 107.9194

 Weighted Index Number

 Weighted index Number is the second type of construction of index number. in that point we see the first type as un-weighted index number. and they are divide into parts aggregative and average price relative index number.

Let's see the second type as weighted index number is divided into parts as 

1. Aggregative index number: In this type there are three method to construct an Index Number are as,

i. Laspeyre's Index Number

ii. Paasche's Index Number

iii. Fisher's Index Number

now we see one by one 

i. Laspeyre's Index Number: All three methods are belongs to weighted index number then in this method we consider the one quantity as weight form current year and base year. (i.e. one of the year is considered as weight) 

In laspeyre's index number the base year quantity is taken weights. (i.e. in laspeyre's index number the weights are considered as base year quantity)

Is formulated as P01=((∑p₁q₀)/( (∑p₀q₀)) x 100                                                                                                   Where p₁= Current Year Price,  q₀= Base Year Quantity, p₀=Current Year Price

ii. Paasche's Index Number:  In this method we consider the one quantity as weight form current year and base year. (i.e. one of the year is considered as weight) In Passche's Index Number the Current year  quantity is taken weights . (i.e. in Passche's index number the weights are considered as Current year quantity)

Is formulated as P01=((∑p₁q₁)/( (∑p₀q₁)) x 100      

these methods are used for construction of index number so we compare the both index number to finding we method is chose for construction of index number.

1.     In paasche's index nymber we see the current year quantity is used as weight so for constructing paasche's index number the actual quantities of every year are used. hence paasche's index number is required the actual quantities of every year, for calculating the paasche's index number we collect quantity of each item of the current year.   but in the laspeyre's index number required the base year quantity, there  the laspeyre's index number formula preferred.

2.     In the Paasche's index number the denominator  ∑p₀q_{1 is computes every year, hence it is required to calculated again and again. but in laspeyre's index number the denominator is} ∑p₀q_{0 both quantities are same for every year. so again we preferred}  laspeyre's index number formula.

3.   Paasche's index number has downward and laspeyre's index number has upward bias.                                if the price go up the quantities consumed are reduced and we see the both formulae the laspeyre's index number use the old year quantity as base year, and the paasche's index number use the new year quantities of current year. hence laspeyre's index number  is slightly larger than paasche's index number.

form above we say that the Laspeyre's index number preferred.  

iii. Fisher's Index Number: 

is formulated as  P₀₁ =√[((∑p₁q₀)/( (∑p₀q₀)) X((∑p₁q₁)/( (∑p₀q₁))] X 100

from this formula we say that the fisher's index number is the geometric mean of Paasche's and Laspeyre;s index number.

as Fisher = √(P X L)

therefore the Fisher's Index Number is superior than above two index number.

we discus the example of this index numbers on next page.           

 Examples:

1. Aggregative index number: In this type there are three method to construct an Index Number are as,

i. Laspeyre's Index Number

ii. Paasche's Index Number

iii. Fisher's Index Number

now we see one  Examples of Weighted Index Number

For Aggregative method in this type three important method to constructing index number.

i.                    Laspeyre’s index number is P₀₁={(∑p₁q₀)/ (∑p₀q₀)} x 100

ii.                  Paasche’s  index number is P₀₁={(∑p₁q₁)/ (∑p₀q₁)} x 100

iii.                fisher’s index number is geometric mean of  Laspeyre’s And Paasche’s index number

Fisher’s index number  = √( Laspeyre’s x Paasche’s )

Example 1. Construct Index Number using Laspeyre’s index number.

Commodity	Base year		Current year
	Price	Quantity	Price	Quantity
A	7	402	9	410
B	9	268	15	278
C	12	298	14	277
D	15	315	12	258
E	8	254	10	304
H	2	125	4	127

Solution:-

The formula for calculating Laspeyre’s index number is P₀₁={(∑p₁q₀)/ (∑p₀q₀)} x 100

Firstly we calculate ∑p₁q₀ and ∑p₀q₀

Commodity	P0	q0	P1	q1	p1q0	p0q0
A	7	402	9	410	3618	2814
B	9	268	15	278	4020	2412
C	12	298	14	277	4172	3576
D	15	315	12	258	3780	4725
E	8	254	10	304	2540	2032
H	2	125	4	127	500	250
Total					18630	15809

∑p₁q₀ =18630, ∑p₀q₀ = 15809

P₀₁={(18630)/ (15809)} x 100

      =1.178443 x 100

      =117.8443

Laspeyre’s index number is 117.8443

Example 2. Construct Index Number using Laspeyre’s, Paasche’s and fisher’s  index number.

Commodity	Base year		Current year
	Price	Quantity	Price	Quantity
A	9	399	14	410
B	15	257	17	278
C	12	126	18	134
D	10	78	7	84
E	6	158	8	145
H	4	104	9	158

Solution: For calculating firstly we prepare table

Commodity	P0	q0	P1	q1	p1q0	p0q0	p1q1	P0q1
A	9	399	14	410	5586	3591	5740	3690
B	15	257	17	278	4369	3855	4726	4170
C	12	126	18	134	2268	1512	2412	1608
D	10	78	7	84	546	780	588	840
E	6	158	8	145	1264	948	1160	870
H	4	104	9	158	936	416	1422	632
Total					14969	11102	16048	11810

Here  ∑ p₁q₀ = 14696, ∑ p₀q₀ = 11102, ∑ p₁q₁ = 16048, ∑ p₀q₁ = 11810

Now we calculate one by one index number as

i.Laspeyre’s index number is P₀₁={(∑p₁q₀)/ (∑p₀q₀)} x 100

P₀₁={(14696)/ (11102)} x 100

    = 1.3483 x 100

Laspeyre’s index number   = 134.83

ii. Paasche’s  index number is P₀₁={(∑p₁q₁)/ (∑p₀q₁)} x 100

            P₀₁={(16048)/ (11810)} x 100           

               = 1.3588 x 100

         Paasche’s  index number     = 135.88

iii. fisher’s index number is geometric mean of  Laspeyre’s And Paasche’s index number.

fisher’s index number  = √(134.83 x 135.88)  = 135.3572

Laspeyre’s index number is 134.83 ,  Paasche’s  index number  is 135.88

 Average of Price Relative:

In weighted  index number two formula are used to calculating the index number they are aggregative price index number and average of price relative.

if the all the weights  of an items are given then we use weighted average price relative index number is formulated as: it is a weighted Average of price relative.

           P₀₁₌(∑PW)/ (∑W)

where P ={p₁/ p₀} x 100 and W = the Weight of an Item.

here the value weight or percentage of the total expenditure then calculated  the sum of all the weights will be equal to 100.

if the index number of group are given then we use the formula as: 

  P₀₁₌(∑IW)/ (∑W)

where I = Index number of group. and w = weight of the items.

Example. 1. compute the index number for the following data.

Items	Percentage of total expenditure	Price in 1950	price in 1957
A	24	157	168
B	16	18	17
C	34	38	47
D	26	44	47

Solution: here we see that data given that the Percentage of total expenditure so we can used average of relative index number for that index number use the following formula

  P₀₁₌(∑PW)/ (∑W)

where P = {p₁/ p₀} x 100 and W = the Weight of an Item.

firstly we calculates P for that constructing table as

Items	Price in 1950	price in 1957	Price Relative  P = {p1/ p0} x 100	W	PW
A	157	168	107.0064	24	2568.153
B	18	17	94.44444	16	1511.111
C	38	47	123.6842	34	4205.263
D	44	47	106.8182	26	2777.273
Total				100	11061.8

  P₀₁₌ (∑PW)/ (∑W)  = 11061.8 / 100 = 110.6180

Example 2. construct the cost of living index number for the following group. calculate cost of living index number using weighted Arithmetic  mean.

Groups	Index Number	Weight
food	248	7
rent	165	2
cloths	249	1
fuel	187	1

 

Solution: Cost of living index number is calculated using weighted mean.

  P₀₁₌(∑IW)/ (∑W)

where I = Index number of group. and w = weight of the items.

Groups	Index Number  I	Weight   W	IW
food	248	6	1488
rent	165	2	330
cloths	249	1	249
fuel	187	1	187
Total		10	2254

P₀₁₌  2254 / 10  = 225.4

Cost of living index number is 225.4

 The cost of living index measure the effect of change in the price on the cost of living of particular class of society in  the particular  place in particular period. e.g. the cost of living index number for the year 1970 is 125 with reference year 1967 for a working class. that means the worker in a city pay 25% more in 1970 as compared to 1967.

Quantity and Value Index Number

1. Quantity Index Number:- 
Quantity Index Number: 
A Quantity Index Number is a statistical method used to tracking the change in the quantity od item over time, the quantity index number is used in economic analysis and it is calculates by comparing the current year quantity of item to its base year quantity of an item.
Quantity index number is calculated using formula they are depended upon the importance of different items in that index. there are various types of Quantity Index Numbers such as Laspeyre's index, Paasche's index and Fisher's index number and each index number has its own formula and methodology. the selection of index number is depends on the purpose and requirements of the analysis. 
We know the Price index number, in which we are interested in the price. In Price index number measure the change in level of price. Now we are interested in quantity of item, the quantity index number measure the change in level of quantity. We see many formula for price index number in which we can interchange the price by quantity we get Quantity Index Number.
They are, 

i.     Laspeyre’s index number is Q₀₁={(∑q₁p₀)/ (∑q₀p₀)} x 100

ii.     Paasche’s  index number is Q₀₁={(∑q₁p₁)/ (∑q₀p₁)} x 100

iii.    fisher’s index number is Q₀₁ =√[((∑q₁p₀)/ (∑q₀p₀)) X((∑q₁p₁)/ (∑q₀p₁))] X 100

The Quantity Index Number play an important role in various real life application. following are the examples.

1. Economic Analysis: Quantity index number are widely used in economic analysis to measure the change in production. it provide the overall performance and trend over the time. these measurements helps the businesses and researchers to understand the economic growth,  and productivity.

2. Market Research: Quantity index number is used in market research  to monitor the changes in consumers demand for specific product, or items, form this understand the consumers preference towards the any products, this information od analysis is used in companies to take decision regarding the product development, marketing. 

 


2. Value index number :- 

A value index number is also known as price index number. a value index number is a statistical measure used to track the change in value or price of specific item. or product over time. this measure is used to compare the value of variable in current year to the value of the item in base ( previous or reference) year.

Another type of index number is value index number. The value index number are based on both price and quantity of an item in base and current year.
It is defined as the ratio of total expenditure of current year to total expenditure of base year in percentage
Value index number is =  (∑p₁q₁)/ (∑p₀q_{0)  x 100}

3. Cost of living index:

 The cost of living index measure the effect of change in the price on the cost of living of particular class of society in  the particular  place in particular period. e.g. the cost of living index number for the year 1970 is 125 with reference year 1967 for a working class. that means the worker in a city pay 25% more in 1970 as compared to 1967.