chapqw1

1. Contents
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0.1 Social and Economic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Index Numbers (By: Zehiwot H) 2
1.1 Definition (Lesson 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Forms of index numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Uses of index number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 Limitations of index number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5 Types of index numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.5.1 Constructing price index numbers (Lesson 2) . . . . . . . . . . . . . . . . . . . . . . . 3
1.5.2 Classification of price index numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Deflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Base shifting and Splicing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.9 Quantity Index Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.10 Value Index Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
0.1 Social and Economic Data
Social statistics is the use of statistical measurement systems to study human behavior in a social environment.
This can be accomplished through polling a group of people, evaluating a subset of data obtained about a
group of people, or by observation and statistical analysis of a set of data that relates to people and their
behaviors.
Social statistics and indicators are about both a discrete set of domains (demography, employment, health,
education, social and family policies, welfare systems, physical security, etc...) and a specific way of looking at
these and other (economic) domains, focusing on differences in conditions among population groups and on
distributions rather than on aggregates only. Household income is an economic statistic . where as income gap
between men and women is a social statistic. The distribution of household income is (also) a social statistic.
Social scientists use social statistics for many purposes, including:
• the evaluation of the quality of services available to a group or organization,
• analyzing behaviors of groups of people in their environment and special situations,
• determining the wants of people through statistical sampling.
... Importantly, social statistics are about PEOPLE basic demographic and contextual statistics are the first
things any government collects...
− − − − − − − − −
Economic statistics is a topic in applied statistics that concerns the collection, processing, compilation, dis-
semination, and analysis of economic data. The data of concern to economic statistics may include those of an
economy of region, country, or group of countries.
Economic statistics may also refer to a subtopic of official statistics for data produced by official organizations
(e.g. national statistical services, intergovernmental organizations such as United Nations, European Union or
OECD, central banks, ministries, etc.).
Analyses within economic statistics both make use of and provide the empirical data needed in economic
research, whether descriptive or econometric. They are a key input for decision making as to economic pol-
icy. The subject includes statistical analysis of topics and problems in microeconomics, macroeconomics,
business, finance, forecasting, data quality, and policy evaluation.
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1

2. 1
Chapter
Index Numbers (By: Zehiwot H)
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Learning Objectives
By the end of this chapter you should be able to:
• represent a set of data in index number form;
• understand the role of index numbers in summarising or presenting data;
• recognise the relationship between price, quantity and expenditure index numbers;
• Understand the difference between a weighted and an unweighted index.
• splice separate index number series together;
• measure inequality using index numbers.
1.1 Definition (Lesson 1)
What is index number ?
Definition 1.1. {
An index number is commonly used descriptive measure designed to show changes in a variable or group of
related variables with respect to time, geographic location or other characteristics such as income, profession,
etc. and expressed as a percentage.
A collection of index numbers for different years, locations, etc., is sometimes called an index series. Index
numbers may be classified in terms of the variables that they are intended to measure.
In business, different groups of variables in the measurement of which index number techniques are commonly
used are price, quantity, value, and business activity. Thus, we have index of wholesale prices, consumer prices,
industrial output, value of exports, and business activity, etc.
1.2 Forms of index numbers
There are two kinds or forms of index numbers. These are Simple Index Number and Composite (aggregate)
Index Number.
Definition 1.2. {
A simple index number is a number that measures a relative change in a single variable with respect to a base.
Definition 1.3. {
A composite index number is a number that measures an average relative changes in a group of relative variables
with respect to a base.
1.3 Uses of index number
The main uses of index numbers are given below:
1. Measure fluctuations during intervals of time, group differences of geographical position of degree etc.
2. To compare the total variations in the prices of different commodities in which the unit of measurements
differs with time and price etc.
3. Measure the purchasing power of money.
4. In forecasting the future economic trends.
5. In studying difference between the comparable categories of animals, persons or items.
6. Index numbers of industrial production are used to measure the changes in the level of industrial pro-
duction in the country.
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3. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
7. Index numbers of import prices and export prices are used to measure the changes in the trade of a
country.
8. To measure seasonal variations and cyclical variations in a time series.
1.4 Limitations of index number
1. They are simply rough indications of the relative changes.
2. The choice of representative commodities may lead to fallacious conclusions as they are based on sam-
ples.
3. There may be errors in the choice of base periods or weights etc.
4. Comparisons of changes in variables over long periods are not reliable.
5. They may be useful for one purpose but not for other.
6. They are specialized types of averages and hence are subject to all those limitations with which an aver-
age suffers from.
1.5 Types of index numbers
Following types of index numbers are usually used.
Definition 1.4. {
Price index Number-measure the relative changes in prices of a commodities between two periods. Prices can
be either retail or wholesale.
Definition 1.5. {
Quantity Index Number-is considered to measure changes in the physical quantity of goods produced, con-
sumed or sold of an item or a group of items.
Definition 1.6. {
Value index-measures changes in both the price and quantities involved. A value index such as index of de-
partment store sales; needs original base year prices, original base year quantities, present year prices and the
present year quantities for its construction.
1.5.1 Constructing price index numbers (Lesson 2)
The following steps are considered for the construction of price index numbers.
1st Objective: The first and the most important steps in the construction of index numbers are to decide
the object for making the index numbers of prices. The prices may be retail or whole-sale. First of all we
decide the purpose of making the index numbers. Once the purpose is decided, then we decide about
the scope and the area or the people who are to be considered.
2nd Selection of Commodities: A list of important commodities is prepared. Those commodities are taken
into account which is commonly consumed by the consumers. There is no hard and fast rule about the
number of commodities. Only those commodities are considered on which a reasonable amount is spent.
For construction of wholesale price index numbers, about 80 commodities are taken in the list and for
retail-price index numbers about 300 commodities are considered.
Note 1.1. Sometimes the index numbers of very important commodities like wheat, rice, oil, ghee etc.
are calculated. These index numbers are based on about one dozen commodities and are called sensitive
price index numbers.
3rd Collection of Price Data: The most important and difficult step is the collection of prices. The prices
are to be taken from the field.
Note 1.2. For retail price index numbers, retail prices are needed. The prices change from place to place
and from time to time. On different shops the prices are different. In actual practice there are many
difficulties. Usually some representative shops from where the consumers mostly purchase their items
are selected and the prices are taken from those shops on daily basis and then the weekly and monthly
averages are calculated. Finally quarterly or yearly averages are calculated. Some commodities are
sold in different varieties. Rice, sugar, mangoes, etc. have different variables which are sold on different
prices. This problem is solved by assigning due weights to different varieties and then weighted average
prices is calculated. Sometimes different varieties are treated as different commodities.
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4. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
Note 1.3. For whole-sale prices, the prices are taken from the whole-sale markets, fractions depots and
the whole-sale agencies. The whole-sale prices are usually stable; therefore these prices are not taken
on daily basis. The price reporting is done on weekly or monthly basis depending upon the nature of
the commodity. The prices of some commodities are controlled by the government. These prices are
reported whenever some change takes place.
4th Selection of Base Period: The prices of the commodities in the current period are to be compared with
the prices of some period in the past. This period in the past is called the base period or the reference
period. The base period is decided statistical division of Government. This period should not be in the
remote past. The period which is economically stable and is free of disturbances and strikes is taken as
the base period.
Note 1.4. You may use fixed or chain base method. What is there difference?
Definition 1.7. In fixed base method, a particular year is generally chosen arbitrarily and the prices of
the subsequent years are expressed as relatives of the prices of the base year.
Sometimes instead of choosing a single year as the base, a period of a few years is chosen and the average
price of this period is taken as the base year’s price. The year which is selected as a base should be normal year
or in other words, the price level in this year should neither be abnormally low nor abnormally high.
The fixed base method is used by the Government in the calculation of national index numbers.
How to compute?
Pon =
Pn
P0
100 % (1.1)
Example 1.1. Find index numbers for the following data taking 1980 as base year.
Years 1980 1981 1982 1983 1984 1985 1986 1987
Price 40 50 60 70 80 100 90 110
Solution1 Apply formula 1-1 ,we have the following table
Years 1980 1981 1982 1983 1984 1985 1986 1987
Price 40 50 60 70 80 100 90 110
Index numbers =100 125 150 175 200 250 225 275
Definition 1.8. In chain base method (CBM), there is no fixed base period. The year immediately pre-
ceding the one for which price index has to be calculated; is assumed as the base year.
For the year 1994 the base year would be 1993, for 1993 it would be 1992 for 1992 it
would be 1991 and so on.
Due to this property, CBM has the following merits and demerits.
• The price relatives of a year can be compared with the price level of the immediately preceding year.
Advantage
• It is possible to include new items in an index number or to delete old times which are no more important.
Advantage
• Comparison cannot be made over a long period. Disadvantage
In Chain Base, first find link relative and then chain index number can be obtained.
Pn−1,n =
Pn
Pn−1
100 (1.2)
where
Pn−1,n =link relative of current year
Pn = price in the current year
Pn−1 = price in the preceding year
Example 1.2. Find index numbers for the following data taking 1974 as base year.
years 1974 1975 1976 1977 1978 1979
price 18 21 25 23 28 30
1Note:as we observe in the above table, the denominator is constant for calculating price index number. i.e., Po=40.
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5. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
Solution: using formula 1-2, we have,
Years Price Link relatives Chain indices
1 18 100 = 18/18*100 100
2 21 116.67 = 21/18*100 116.67= (100*116.67)/100
3 25 119.05 = 25/21*100 138.9 = (119.05*116.67)/100
4 23
5 28
6 30
5th Selection of the Suitable Average There are different averages which can be used in averaging the price
relatives or link relatives of different commodities. Geometric mean, arithmetic mean, and median.
6th Selection of Suitable Weights
Equal weighting or Unweighted Index Numbers
Note 1.5. There are two methods of constructing unweighted index numbers. These are Simple Aggregative
Method &Simple Average of Relative Method. Could you differentiate them ?
Definition 1.9. SIMPLE AGGREGATIVE METHOD (SAM) In this method, the total of the prices of commodi-
ties in a given (current) years is divided by the total of the prices of commodities in a base year and expressed
as percentage.
P0n =
P
Pn
P
P0
100 (1.3)
Definition 1.10. SIMPLE AVERAGE OF RELATIVES METHOD (SARM): In this method, we compute price
relative or link relatives of the given commodities and then use one of the averages such as arithmetic mean,
geometric mean, median etc. If we use arithmetic mean as average, then
P0n =
1
n
X
(
Pn
P0
)100 (1.4)
It is very simple and easy to apply. (A) It gives equal weight to all items. (D)
Example 1.3. The following are the prices of four different commodities for 1990 and 1991. Compute a price
index by (1) SAM and (2) SARM using both arithmetic mean and geometric mean, taking as base.
Commodity Cotton Wheat Rice Gram
Price in 1990 909 288 767 659
Price in 1991 874 305 910 573
Solution:
Commodity Price in 1990 P o Price in 1991 P n Price Relative P Log p
Cotton 909 874 1.9829
Wheat 288 305
Rice 767 910
Gram 659 573
Total 2623 2662 407.64 8.0213
The necessary calculations are given below:
• Simple Aggregate Method:
P0n =
P
Pn
P
P0
100 =
2662
2623
100 = 101.49
• Average of Price Relative Method (using arithmetic mean):
P0n =
1
n
X
(
Pn
P0
)100 = 1/4(407.64) = 101.91
• Average of Price Relative Method (using geometric mean)
P0n = antilog
P
logP
n
= 101.23
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6. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
Weighted Index Numbers
When all commodities are not of equal importance, we assign weight to each commodity relative to its im-
portance and index number computed from these weights is called weighted index numbers. There are two
methods of computing a weighted price index.
Definition 1.11. Laspeyre’s index number (L): In this index number the base year quantities are used as
weights, so it also called base year weighted index.
Pon =
P
Pnqo
P
Poqo
100 (1.5)
Adv. It requires quantity data from only the base period. This allows a more meaningful comparison over
time. The changes in index can be attributed to changes in the price.
Disadv. Doesn’t reflect changes in buying patterns overtime. Also, it may overweight goods whose prices
increase.
Definition 1.12. Paasche’s index number (P) : In this index number, the current (given) year quantities are
used as weights, so it is also called current year weighted index.
Pon =
P
Pnqn
P
Poqn
100 (1.6)
Adv. Because it uses quantities from the current period, it reflects current buying habits (patterns).
Disadv. It requires quantity data for each year, which may be difficult to obtain because different quantities are
used each year, it is impossible to attribute changes in index to changes in price alone. It tends to overweight
the goods whose prices have declined. It requires the prices to be recomputed each year.
Definition 1.13. Fisher’s ideal index number : Geometric mean of Laspeyre’s and Paasche’s index num-
bers is known as Fisher’s ideal index number. It is called ideal because it satisfies the
time reversal and factor reversal test.
Pon =
√
L × P (1.7)
Definition 1.14. Marshal-edge worth index number : In this index number, the average of the base year
and current year quantities are used as weights. This index number is proposed by two
English economists Marshal and Edge worth.
Pon =
P
Pn(qo + qn)
P
Po(qo + qn)
100 (1.8)
Definition 1.15. DORBISH-BOWLEY’S INDEX NUMBER:
Pon =
L + P
2
(1.9)
Example 1.4. Compute the weighted aggregate price index numbers for 1981 with 1980 as base year using
• Laspeyre’s Index Number
• Paashe’s Index Number
• Dorbish –Bowley’s index
• Fisher’s Ideal Index Number
• Marshal Edgeworth Index Number.
Solution:
i Laspeyre’s Index Number:
L =
456
406
100 = 112.32
ii Paashe’s Index Number :
P =
506
451
100 = 112.20
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7. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
Prices Quantities
Commodity
1980 1981 1980 1981
A 10 12 20 22
B 8 8 16 18
C 5 6 10 11
D 4 4 7 8
iv Fisher’s Ideal Index Number :
F =
√
112.32 × 112.20 = 112.26
v Marshal Edgeworth Index Number :
ME =
P
Pn(qo + qn)
P
Po(qo + qn)
100 =
456 + 506
406 + 451
100 = 112.38
Interpretation: the price of this group of items has increased 12.38% in the one year period.
1.5.2 Classification of price index numbers
We have two categories of price index. These are wholesale price index number and consumer price index
number.
Wholesale price index numbers
The wholesale price index numbers indicate the general condition of the national economy. They measure the
change in prices of products produced by different sectors of an economy. The wholesale prices of major items
manufactured or produced are included in the construction of these index numbers.
The federal bureau of statistics has been constructing and releasing wholesale price index (WPI) in USA since
1961 –1962. The list of wholesale items consists of four major groups. Food, Fuel, lighting and lubricants,
Manufactures raw material.
Consumer price index numbers
Define CPI?
Definition 1.16. Consumer price index(CPI) number : is measured the changes in the prices paid by
the consumers for purchasing a special ”basket” of goods and services during the current
year as compared to the base year.
The basket of goods and services will contain items like
1. Food
2. House Rent
3. Clothing
4. Fuel and Lighting
5. Education
6. Miscellaneous like washing, transport, newspaper etc.
Consumer price index number is also called cost of living index number or retail price
index number.
Construction of Consumer Price Index Numbers
The following steps are involved in the construction of consumer price index numbers.
a Class of People: The first step in the construction of consumer price index (CPI) is that the class of
people should be defined clearly. It should be decided whether the cost of living index number is being
prepared for the industrial workers, middle or lower class salaried people living in a particular area. It
is therefore necessary to specify the class of people and locality where they reside.
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8. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
b Family Budget Inquiry: The next step in the construction of consumer price index number is that some
families should be selected randomly these families provided information about food, clothing, house
rent, miscellaneous etc. The inquiry include questions on family size, income, the quality and quantity
consumed and the money spent on them and the weights are assigned in proportions to the expenditure
on different items.
c Price Data: The next step is to collect the data on retail prices of the selected commodities for the current
period and the base period these prices should be obtained from the shops situated in that locality for
which the index numbers are prepared.
d Selection of Commodities: The next step is the selection of the commodities to be included. We should
select those commodities which are mostly used by that class of people.
Methods of Consumer Price Index Numbers
There are two methods for the compute of consumer price index numbers.
A Aggregate expenditure method: In this method, the quantities of commodities consumed by the par-
ticular group in the base year are estimated and these figures or their proportions are used as weights.
Then the total expenditure on each commodity for each year is calculated. The price of the current year
is multiplied by the quantity or weight of the base year. These products are added. Similarly for the
base year total expenditure on each commodity is calculated by multiplying the quantity consumed by
its price in the base year. These products are also added. The total expenditure of the current year is
divided by the total expenditure of the base year and the resulting figure is multiplied by to get the
required index numbers. In this method, the current period quantities are not used as weights because
these quantities change from year to year.
Pon =
P
Pnqo
P
Poqo
100 (1.10)
Where Pn = Represent the price of the current year, Po = Represents the price of the base year and qo =
Represents the quantities consumed in the base year
B Family Budget Method: In this method, the family budgets of a large number of people are carefully
studied and the aggregate expenditure of the average family on various items is estimated. These values
are used as weights. Current year’s price are converted into price relatives on the basis of base year’s
prices and these prices relatives are multiplied by the respective values of the commodities, in the base
year. The total of these products is divided by the sum of the weights and the resulting figure is the
required index numbers.
Pon =
P
W I
P
W
I =
Pn
Po
100 (1.11)
Where W=weight
Example 1.5. Construct the consumer price index number for 88 on the basis of 87 from the following data
using: (1) Aggregate expenditure method (2) Family budget method.
commodity Quantity consumed in 1987 unit prices 1987 prices 1988
A 6 quintal quintal 315.75 316
B 6 quintal quintal 305 308
C 1 quintal quintal 416 419
D 6 quintal quintal 528 610
E 4 kg kg 12 11.5
F 1 quintal quintal 1020 1015
Solution:
commodity Quantity consumed in 1987 prices 1987 prices 1988 Poqo Pnqn
A 6 quintal 315.75 316
B 6 quintal 305 308
C 1 quintal 416 419
D 6 quintal 528 610
E 4 kg 12 11.5
F 1 quintal 1020 1015 8376.5 8884
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9. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
CPI number of 1988 by Aggregate expenditure method is:
Pon =
P
Pnqo
P
Poqo
100 =
8884
8376.5
100 = 106.06
CPI number of 1988 by Family Budget Method is:
Pon =
P
W I
P
W
=
888393.56
8376.5
== 106.06
Note 1.6. CPI number: (1) As an economic indicator. (2) As a means of adjusting income payments. (3) As a
means of preventing inflation-induced tax changes. (4) As a deflator of other economic series.
1.6 Deflation
Definition 1.17. Deflating means adjusting, correcting or reducing a value which is inflated. Hence de-
flating of the price index numbers we mean adjusting them after making allowance for
the effect of changing price levels.
• The product of birr (or any other currency) value and the purchasing power of birr is called deflated birr
value. That is:
DBV = BirrV alue ∗ P P B (1.12)
Where DBV=Deflated Birr Value and PPB=Purchasing Power of Birr
• The purchasing power of birr or any other currency is the ratio of base period price index to current
period price index.
P P B =
100
current priceindex
(1.13)
The purpose of deflated birr value is to remove the effect of price change. It is a better measure of change
than the actual value. Hence, DBVs are also called constant birr value since it maintains the price change.
Deflated birr wages called real wages. It is given by:
Real wages(or income) = money wage ∗ P P B (1.14)
• Real wage index numbers are the ratio of current period real wages to base period real wages expressed
as percentages.
Real wageindex =
Current period real wage
baseperiod real wage
∗ 100 (1.15)
1.7 Tests
Various theoretical criteria have been developed to test and evaluate the consistency of various formulas of
index numbers.
The most common test for consistency of index numbers are:
1. The unit test
2. The time reversal test
3. The factor reversal test
4. The circular test
To facilitate the application of these mathematical criteria, index numbers are expressed as ratio (proportion)
rather than percentages
1. The time reversal test: it states that ”If the time subscripts of a price (or quantity) index number formula
be interchanged, the resulting price (or quantity) index number formula should be the reciprocal of the
original formula”.
Pon × Pno = 1 (1.16)
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10. CHAPTER 1. INDEX NUMBERS (BY: ZEHIWOT H)
2. The Factor reversal test: This test states that ”If the factors prices and quantities occurring in a price (or
quantity) index number formula be interchanged so that a quantity (or price) index formula is obtained,
then the product of the index numbers should give the true value index number”.
Pon × qon = Von (1.17)
3. The circular test: This test states that ” If an index for the year ’b’ based upon the year ’a’ is P(a/b) and
for the year ’c’ based upon the year ’b’ is P(b/c) , then the circular test requires that the index for the year
’c’ based upon the year ’a’, i.e., P(a/c) should be the same as if it were compounded of these two stages”.
Pac = Pab × Pbc (1.18)
Note 1.7. All the index numbers we have considered so far do not satisfy the circular test.
1.8 Base shifting and Splicing
For a variety of reasons, it frequently becomes necessary to change the reference or base period of an index
series from one time period to another. This can be done by either re computing all period index numbers using
the new base period or an approximate index numbers obtained by dividing the index numbers corresponding
to the old base period by the index number for the new base period.
Newbaseindex =
old baseindexof current period
old baseindexof newbaseperiod
∗ 100 (1.19)
One of the application of the principle of base shifting is splicing which consists in two or more overlapping
series of index numbers to obtain a single continuous serious. Splicing two sets of price index numbers cov-
ering different periods of time is usually required when there is a major change in quantity weights. It may
also be necessary on account of a new method of calculation or the inclusion of new commodity in the index.
Furthermore, to facilitate comparison among two or more series splicing is needed.
The spliced index number may be computed as:
spliced index number
=
(current period indexno.) × (old index no. or new base period)
100
(1.20)
1.9 Quantity Index Number
Reading Assignment
1.10 Value Index Number
Reading Assignment
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