Statistics For Economics Chapter 8 Index Numbers

A.

weighted Index

A.

 base year

A.

small

A.

retail prices

A.

food

A.

wholesale price index

C.

Food

B.

 Retail prices

B.

indicates changes in the general price level

A.

Consumer price index

We need index number for measuring changes in the magnitude of a group of related variables.

It is essential to have different CPIs for different categories of consumers as the cost of living of different categories of consumers differ.

A consumer price index for industrial workers measures general inflation.

A price index measures changes in prices between two points of time whereas a quantity index measures changes in the volume of goods produce.

The change in any price is not reflected in price index number. Price index only shows the changes in the price of goods which are included in the construction of a particular index number.

Yes, the CPI number for urban non-manual employees represents the changes in the cost of living of the President of India, because he is also non-manual employee.

Wheat, Rice, Toothpaste, pulses, clothing, petroleum, Housing, means of entertainment such as T.V., transistor, Radio, telephone, mobile, vehicle, books, stationery.

Suppose the percentage of weight given to food = x

In this case percentage of weight given to other items would be = (100 – x)

Hence,

Hence, percentage of the total weight given to food = 66.66

Annual income in base year = Rs. 4000. Annual income required in the base
year to maintain the same standard

Present annual salary = Rs.6000.

The increase in annual salary = 16,000 - 6,000 = Rs.10,000

Desirable properties the base year :

1. As far as possible, the base year should be a normal year i.e. it should be the one without ups and downs.

2. Extreme values should not be selected as base period.

3. The period should also not belong to too far in the past.

The table shows that the growth performance of the broad industrial categories differ. The general index represents the average performances of these categories.There is about 45% rise in general index. There has been a comparatively lower performance of mining and quarrying. It is about 25%. The comparatively lower performance of mining and quarrying has not been able to push down the general index due to high increase in manufacturing index and due to maximum weightage given to this sector.

Item	Expenses in Percentage	Price in Rs. in 1995	Price in Rs. in 2004 (P1)		WR
Food	35	1400	1500	107.14	3750
Fuel	10	200	250	125.00	1250
Clothing	20	500	750	150.00	3000
Rent	15	200	300	150.00	2250
Misce	20	250	400	160.00	3200
	ΣW = 100				ΣWR = 13450

Cost of living Index

The students are advised that they should enquire their parents about the daily expenditure. Quantities bought and prices paid per unit of daily purchases of their family for two weeks. They will come to know that the rising prices of goods have effected that family very much.

Calculation of inflation rate

(i) Industrial Workers :

(ii) Non-manual employees :

(iii) Agricultural Labourers :

(iv) Whole Sale Price Index (WPI) :

(b) Comments on the relative values Index Number :

1. C.P.I. of industrial workers increased in two initial years, but there was decline in its increase rate.

2. These were many ups and downs in the CPI of non-manual employees.

3. There were many ups and downs in the CPI and agricultural labourers.

(c) They are comparable.

An index number is a statistical measure designed to show change in a variable or group of related variables with respect to time, geographic location or other characteristics.

In the simple index number all items of the series are treated as of equal importance. In the weighted index, weights are accorded to different items depending on their relative importance.

The base year should be the year of economic stability. It should not to be too distant from the current year.

Price index number is that number measures and permits comparison of the prices of certain goods.

1. Consumer price index or cost of living index number.

2. Wholesale price index number.

In India, the consumer price index are mainly constructed for the following consumer groups:

1. Industrial workers.

2. Urban-non manual employees.

3. Agricultural labourers.

The other name of the consumer price index is price deflater of income.

Inflation refers to increase in prices and deflation refers to decrease in prices.

1. Index numbers are expressed in percentage.

2. They are a specialised types of averages.

Consumer price index measures the average changes in retail prices at which the consumers buy goods and services in the market at a given price.

The wholesale price index is an indicator of change in general price level.

1993-1994

Inflation Rate:

where

(i) X_t refers WPI for the th week.

(ii) X_t-1 refers to WPI for the (t-1) week.

Consumer price Index (CPI).

Purchasing power of money

Purchasing power of one rupee

o maintain the same standard of living

a man’s salary should be Rs. 15780

so the salary should be raised Rs. 15780 This salary should be added by Rs. 5780.

quivalent of a rupee =   
It means that Rel in Jan., 2005 was worth 19 paise in 1982. In other worths 19 paise in 1982 is worth Rs. 1 in 2005. There is 526 times rise in p

An index number can be defined as a special type of average which measures changes in the value of such variables as prices of commodities. Industrial production, cost of living etc.

It is the actual difference in prices relative to the original price. For example the prices of Potato were Rs. 8/- and Rs. 10/- per kg. respectively in August. and October. In this case the relative change in prices will be as follows:

Relative change in prices is more important than actual differences in prices.

The base period is the reference period with which the current period is compared. If we measure the prices in Dec. 2001 relative to those in October 2001, then October is called the base year.

It reflects the importance of the variables to be composed.

Some important index numbers in use are as follows:

1. Consumer Price Index (CPI).

2. The Wholesale Price Index (WPI)

3. Index of Industrial Production.

(i) An increase in the general price level.

(ii) An increase in the real output.

(i) Mining, (ii) Manufacturing, (iii) Electricity.

1. The Industrial workers.

2. The Urban non-manual workers.

3. The agricultural labourers.

Widely used index numbers are wholesale price index number, consumer price index, index for industrial production, agricultural production index and sensex.

1. Items selected should be relevant to the purpose of the index.

2. The total number of commodities selected for the index should be neither too small nor too large.

In Laspeyre’s method, the weights are the quantities of the base year.

In Paasche’s method, the weights are the quantities of current year.

Laspeyre’s method is widely used as it in based on fixed weights of the last year.

The other name of the consumer price index is the cost of living index.

It means that the general price-level has rises by 89.1 percent during the year.

The index number of industrial production measures changes in the level of industrial production comprising many industries.

The rise in sensex reflects the good health of the economy in general.

The weightages assigned to mining, construction and light are 10.47,79.36 and 10.17 respectively.

Industries are grouped into three categories : (i) Mining, (ii) Manufacturing and, (iii) Electricity for constructing index of industrial production in India.

Following are main methods of constructing index number:

(i) Simple Aggregative method (ii) Simple average of price relative method (tit) Weighted Aggregative method and (iv) Weighted average of price relative method.

Sensex stands for Bombay Stock Exchange Sensitive Index with 1978-79 as base.

Sensex consists of 30 stocks.

Producer price index number measures price changes from the producers, perspective.

Producers price index number will replace wholesale price index very soon.

Some mathematical tools used in Economics are equation of a line, slope of line and slope of a curve.

An equation means equating the value of two sides. An equation has two sides with equality sign. For example : a = b + c in between.

Equality sign in an equation indicates that the value of two sides of an equation is equal.

Functional equation is that equation which shows that a dependent variable is the function of an independent variable.

A functional equation is expressed as

Y = f(X)

The functional function given in the question indicates that the value of Y depends on the value of X.

An independent variable is that variable which is not dependent on any other values. Its value is free.

A dependent variable is that variable whose value depends on the value of an independent value/s.

Here, X is a dependent variable and Y is an independent variable.

A curve is a graphical representation of a functional equation or a functional tabular data.

Types of curve: (i) Convex curve, (ii) Concave curve and (iii) Straight line curve.

A convex curve is that upward sloping curve which is bent towards the X-axis or it is that downward sloping curve which is bent towards the origin.

A concave curve is that upward sloping curve which is bent away from the X-axis (i.e., towards Y-axis) or that downward sloping curve which is bent towards away from the origin.

The curve is a straight line when it is neither upward sloping and nor downward sloping.

Given two variables — X and Y₁, X variable is represented on the X-axis and Y on the Y-axis. In this case slop is the ratio of change in Y and change in X between two points of curve. The method of calculation is as follows:

In this case slop is zero.

In this case slope = ∞

The property of the slope of a straight line is that it remains unchanged.

The slope goes on decreasing as we move along the curve downwards.

The slope increases as we move along the curve upwards.

The slope of downward sloping concave curve increases as we move along the curve downwards.

It decreases it we move along the curve upwards.

Inflation refers to a situation of rise in the general price-level in a country over a fairly long time. In other words a consistent rise in the wholesale price index over a time implies a situation of inflaction.

Often, measures inflation the wholesale price index.

Inflation causes erosion of purchasing power of the people if their money income emains constaint.

Relative change in the price index from reek to week is known as the rate of inflation. a equation

Rate of inflation =

A₁ = Whole Price index of week 1.

A₂ = Whole price index of week 2.

Rate of inflation imples the increase or decrease in the pace of inflation or increase/ drease in the speed of inflation.

No, I do not agree with him. It is not fall in the price level. It is only a fall in the speed at which prices tend to rise.

Primary ettect of the inflation is decrease in the value of money.

(i) In first case it will be as follows

y = a + b.

(ii) In second case it will be as follows

y = bx

(iii) In third case it will be as follows

y = a + bx

Here, a will be negative.

y = a – bx

Index number is a statistical measure showing percentage or relative changes in variables like prices, production, exports etc.

According to Spiegel:An index number is a statistical measure designed to show changes in a variable or group of related variables with respect to time, geographic location or other characteristics.

Utility of Index Numbers:

1. Index Numbers are helpful to policy makers.

2. Index Numbers simplify the facts.

3. Comparative study become easy.

4. Index number help us to study the general trend.

5. They are helpful in determination of real change in income.

In the construction of Index Numbers, there are some practical difficulties and theoretical limitations.

1. Index numbers are not fully true. They simply indicate arithmetical tendency of the temporal change in the variable.

2. Index numbers do not help in international comparison.

3. Index numbers are prepared with certain specific objectives.

4. It is difficult to collect retail prices so index numbers based on wholesale prices may be misleading.

Basic consideration underlying the choice of base period:
1. The base period should be a normal period. Abnormal periods like period of war or floods etc. should not be selected as a base year.

2. The base period should be neither too short nor too long.

3. It should not be the period for which actual data are not available.

4. It should not be too far back in the past.

Simple Aggregative Price Index:A simple Aggregative Price Index is defined as

Which is the sum of prices of commodities in the current period ‘I’ expressed as the sum of prices in the base year ‘0’.

Limitations:
1. It has limited applicability.

2. No weightage is given to the relative importance of items.

3. Index is influenced by the items with the large unit prices.

Uses of the Wholesale Price Index Numbers:
1. The time series of wholesale price index numbers can be used to forecast future prices.

2. With the help of wholesale price index number, we can estimate the future demand and supply situations.

3. It is used to measure the rate of inflation

4. It can be used to eliminate the effect of change in price on aggregates such as national income, capital formation etc.

The purpose of constructing an index number of prices is to measure the relative changes in the price of a group of commodities and the purpose of constructing index number of quantities is to measure the relative changes in the quantities of different commodities consumed or produced to obtain “quantity index”

“Weighted” Index of Prices : In order to allow adequate importance to different commodities in a composite index, we assign suitable weights to them. The weighted index number is simply the weighted arithmetic mean of price relatives defined as

Where the weights W₁ W₂..... ..... W₁₁ are such that ΣW₁ = 1

The index numbers reveal the following trend with respect to the area under cultivation for grain and non-foodgrains.

Over a period of time i.e. for the last 7 years area under cultivation for food grains is showing a downward/ decreasing trend and that of nonfood grains is showing an upward trend.

Broad Industrial groupings and their weights:

Many problems are faced while constructing an index number. Two of them are as follows:

1. Selection of base year : The first problem which arises while preparing index number is the selection of base year. Which year should be selected as a base? While selecting a base year, we should remember that it should be a normal period. It should not be either too short or too long. It should be free from all sorts of abnormalities or irregular fluctuations.

2. Selection of items : We should also be careful in selecting the items to be included in

Following are the usefulness of consumer price index:

1. It is helpful in wage negotiation.

2. It is helpful in the formation of income policy, price policy, taxation and general economic policy formulation.

3. It is used in calculating the purchasing power of money.

4. With the help of CPI, we can calculate the real wages.

Following are the uses of the wholesale price index numbers:

1. The wholesale price index (WPI) is used to eliminate the effect of changes in prices on aggregate such as national income, capital formation etc.

2. With the help of WPI, we can measure the rate of inflation. The following formula is used for calculating inflation rate.

Inflation rate
=

Equal weightage is assigned to all the items included in the index number. There are two methods of calculating index number (i) simple aggregative method and (ii) Simple average of price relatives.

1. Simple aggregative method : This is the simplest method of calculating index numbers. In this method, total of the current year prices for the various commodities is divided by the total of the base year and the quotient multiplied by 100. Symbolically.

Where P₁ and P₀ indicate the price of the commodity in the current and base periods respectively.

2. Simple Average of price relatives : In this method, first of all, price relatives are calculated. A price relative is the price for the current period expressed as the percentage of the price of the base period. Symbolically,

Differences between Laspeyre’s method and Paasche’s method:

Bases of difference: 1. Base 2. Formula Laspeyre’s method:This method uses the base period quantities as weights.

2. Following formula is used in this method

Paasche’s Method:This method uses the current period quantities as weights.

2. Following formula is used in this method

Sensex is the short from of Bombay Stock Exchange Sensitive Index with 1978-79 as base. The value of sensex is with reference to this period. It is the benchmark of index for the India stock market. It consists of 30 stocks represented by 13 sectors of the economy. If sensex rises, it indicates that the market is doing well and investors expect better earnings on their investments. It indicates a growing confidence of investors in the basic health of the economy.

There are many types index number. Some of the important ones are as follows:

1. Consumer price index number having three types:(i) CPI for industrial workers (ii) CPI for urban non-manual employees and (iii) CPI for agricultural labourers.

2. Wholesale price index.

3. Industrial production index.

4. Index number for agricultural production.

5. Sensex.
6. Human development Index.

Human development index (HDD is the quality of life index prepared by UNDP. It takes into consideration the three most basic human capabilities - (i) Longevity of life (ii) Knowledge and (iii) Quality of life. Symbolically.

Following are the limitations of the simple aggregative method:

1. Here the items are treated equally. The weightage is assigned to any item according to its importance. The costlier goods get high weightage naturally. For example, the price of gold is very high. So, it gets more weightage.

2. Items with large weights also get more weightage than the items with low weights.

Method of constructing simple index numbers:

(a) Simple aggregative method

(b) Simple average of price relative method

(ii) Methods of constructing weighted index numbers. 

(a) Weighted aggregative method

(b) Weighted average of price relatives method

Features of CPI. Following are main features of CPI.

1. In India three CPI’s are constructed:

(i) CPI for industrial workers (ii) CPI for urban non-manual employees and (iii) CPI for agricultural labourers.

2. The base year for CPI for industrial workers, CPI for urban non-manual employees and CPI for agricultural labourers are 1982,1984-85 and 1986-87, respectively.

3. They are routinely calculated every month to analyse the impact of changes in the retail price on the cost of living of these three broad categories of consumers.

4. The CPI for industrial workers and agricultural labourers are published by Labour Bureau Shimla.

5. The Central Statistical Organization publishes the CPI for urban non-manual employees.

6. The items shown in the table are taken while calculating CPI. for industrial workers. From the table we come to know that food has the largest weight.

CPI for industrial workers

Importance of Index Numbers :

Index Numbers have a great importance in statistics. They are used in the various fields of economics such as production, consumption, price, etc. With the help of statistics, economic conditions can be studied and forecast can be made. They are called the “Economic Barometer”. Their importance can be judged with the following facts.

1. Calculation of price levels:Index Numbers help in calculating the price levels of various items.

2. Helpful in taking decisions:Index Numbers are very helpful to Government in taking important decisions such as granting dearness allowances to the employees.

3. Helpful in comparative study:They also help us in making comparative study of the changes occurring in the economy.

4. Helpful to insurance companies and Banking Institution:Index Numbers help the insurance companies and banking institutions in determining of premium-rates and interests respectively.

5. Helpful in assessing the development of an economy:With the help of statistics, the development of an economy can be assessed.

Following are the main problems in the construction of index numbers.

1. The purpose of the Index:Before going to construct an index number, it is very essential to define the purpose and nature of its line.

There is no all-purpose index. Every index is of limited and particular use. For example, the cost of living for workers in an industrial town will have different requirements compared with the one for agriculture workers.

2. Base year:Another problem in the construction of Index Numbers is the choice of a proper base year. The base year should be a normal one. It should be free from abnormalities like wars, earthquakes, famines, booms etc. Moreover it should not be too distant in the past.

3. Selection of items:The items should be selected in such a manner that they are representative of the taste, habits and customs of the people for whom the index is meant

4. Price Quotations:The next problem is to obtain price quotations for the commodities selected. We know that prices of many commodities vary from place to place and even from shop to shop in the same market. A decision must also be made as to whether the wholesale price or retail price are required. The choice would depend upon the purposes of Index.

5. Choice of average:Next comes the choice of average. Mostly choice is to be made between arithmetic mean and geometric mean. Though geometric mean is the best in theory yet arithmetic mean is more popularly used while constructing Index Numbers.

6. Selection of appropriate weight:All the items included for construction of Index Numbers are not of equal importance. So they should be given weightage according to their importance.

There are two broad methods of constructing Index Numbers:

(i) The unweighted Method and

(ii) The weighted Method

Where the unweighted method is used, the index numbers are called the unweighted index numbers. Similarly, if the weighted method is used, the index numbers that we obtain are called the weighted index numbers.

Unweighted and weighted index numbers are of two kinds:(i) The aggregative:and

(ii) The average of price relative

These methods have been illustrated by the following chart:

Let us now consider these methods:

I. Unweighted Index Number

(a) Simple Aggregate method:This is the simplest method of constructing Index numbers. The formula used is as follows:

Where P₀₁ is the Current year Index Number:

ΣP₁= total of current year prices for different commodities

ΣP_o = total of base year prices for different commodities.

Steps : (i) Total the current year prices for different commodities to get ΣP₁

(ii) Total the base year prices for these different commodities to get ΣP₀

(iii) Divide P₁ and P₂ and multiply the result with 100.

Limitations:
1. No weight is given to the relative importance of items.

2. Index is influenced by the items with the large unit prices.

(b) Simple Average of Price Relative Method : In this method, first of all, price relatives are calculated. A price relative is the price for the current period expressed as a percentage of the period of the base period symbolically p₁/p₀ x 100. Index Numbers by this method is the arithmetic mean or median or geometric mean of these calculated price relatives.

II. Weighted Index Numbers : In this method, appropriate weights are assigned to various commodities to reflect their relative importance in the group.

(a) Weighted Aggregative Method : Weights are assigned to the various items included in the index. There are various methods of assigning weights and consequently a large number of formulae have been given by different persons like, Laspeyre’s Method, Paasche’s Methods, Dorbish and Bowley’s Method. Fisher’s Method etc. Simple types of index number do not indicate the significance of commodities. But in this method, weights are assigned as per the relative significance.

(b) Weighted Average of Price Relatives Method under this Methods : Weighted sum of the price relatives is divided by the sum total of the weights. Here, goods are given weights according to their quality.

(i ) Price relatives of the current year is calculated

(ii) The weights are assigned i.e. PoQo

(iii) Then they are multiplied i.e.

(iv) The results obtained from multiplication are added i.e. [ΣPV]

(v) This is divided by total value weights Pol =

Unweighted Index of Price =