Index numbers are the statistical devises designed to measure the relative changes in the level of a certain phenomenon. They are known as specialized averages and are expressed in percentages. They measure the pressure of economic activities of the country. In this chapter we introduce the Index numbers, types of index numbers, and the computation methods of index numbers.
Types of Index numbers; methods of constructing index numbers; price relative of a commodity; Laspeyre’s method; Paasche’s method; Bowley’s method; Fisher’s method; Marshall’s method
The most common technique for characterizing a business or economic time series is to compute Index Numbers. An Index Number is a measure of relative change in the value added by a variable or a group of related variables over time or space. It is a statistical device for comparing the general level of magnitude of a group of distinct but related variables in two or more situations. If we want to compare the price level in 2001 with what it was in 1990, we shall have to consider a group of variables, such as the prices of rice, cloth, oil, vegetables, etc. If all these variables change in exactly the same ratio, and in the same direction, there will be no difficulty in finding out the change in the price level as a whole. But the prices of different commodities change by different ratios and in different directions. Also the prices of different commodities are expressed in different measurement units, e.g., rice and pulses are expressed in kilograms or quintals. The prices of oil, milk, etc., are expressed in liters. To avoid the difficulty in finding the average or relative changes, we use an index number that is an indicator of the change in the magnitude of the prices of different commodities as a whole. The index numbers are intended to show the average percentage of changes in the value of certain product (or products), at a specific time, place, or situation as compared to any other time, place, or situation. The study of index numbers is of great importance to the industry, the business, and to the governments for chalking out policies or fixing prices.
Index numbers are specialized averages. They are also known as economic barometers, because they reveal the state of inflation or deflation. Index numbers measure how a time series changes over time. Change is measured relative to preselected time period called the base period.
Definition: An index number is a number that measures the change in a variable over time relative to the value of the variable during a specific base period.
Many economists and statisticians have defined index numbers in their own way. Some of them are given below:
Clark and Schkade: “An index number is a percentage relative that compares economic measure, in a given period with those some measures at a fixed time period in the past.”
John I. Griffin: “An index number is a quantity which by reference to base period, shown by its variations, the changes in the magnitude over a period of time. In general, index numbers are used to measure changes over time in magnitudes which are not capable of direct measurements.”
M.M. Blair: “Index numbers are the signs and guide posts along the business highway that indicates to the business how he should drive or manage his affairs.”
A.M. Tuttle: “An index number is a single ratio (usually in percentages) which measures the combined (i.e., averaged) change of several variables between two different times, places or situations.”
Irving Fisher: “The purpose of index number is that it shall fairly represent, so far as one single figure can, the general of the many diverging ratios from which it is calculated.”
Index numbers are not only associated with the study of economic or business data, but are also used to make comparisons in the other branches of social and natural sciences. They play a key role in business planning and drafting of executive policies. Following are the characteristics of index numbers:
1. Index numbers are special type of weighted averages.
2. They are expressed in percentages, which make it feasible to compare any two or more index numbers.
3. Index numbers are of comparable nature any two timings or places or any other situation.
4. Index numbers measure changes not capable of direct measurement.
The most general and the best known use of index number is to study the changes in price over a period of time. Index numbers are very widely used in dealing with the economic and management problems.
Various uses of index numbers are:
1. To measure and compare changes: The main purpose of index numbers is to measure the relative temporal or cross-sectional changes at a point of time over some previous time.
2. To measure purchasing power of money: The purchasing power of money keeps constantly changing. The consumer’s price index helps in computing the real wages of a person. To arrive at real income of people, the income or wages for the time period are deflated by dividing them by equivalent cost of living index or wholesale price index. This helps in adjusting the wages and salaries of the employees.
3. To find trend: We use index numbers to measure the changes from time to time, which enable us to study the general trend 3/2 of the economic activity under consideration. As a measure of average change in an specified group, the index number may be used for forecasting.
4. An aid of framing policies: Economic policies like, volume of trade fixing of wholesale and retail prices are guided by index number. Fixing of wages and dearness allowance is mainly based on consumer price index.
Index numbers are the devices to measure the relative movements in variables. Which are incapable of measuring directly? The usefulness of any index number lies in the types of questions it can answer. Each index number is designed for a particular purpose and it is the purpose that determines its method of construction. Different kinds of usually constructed index numbers are:
1. Price Index Numbers: These index numbers measure changes in prices between two points of time. They measure the general change in the retail or wholesale prices of a commodity or group of commodities at current period as compared to some previous period known as reference period or base period.
2. Quantity Index Number: A quantity index number, measure the changes occurring in the quantity of goods demanded, consumed, produced, etc.
3. Value Index: A value index compares total value in some period with total value in the base period.
4. Consumer Price Index: This is a special kind of index that is constructed for the prices of only the essential items.
In the process of constructing index numbers (i.e., price index numbers) one encounters several problems, which are to be tackled and solved very carefully. In this section we shall discuss some of these problems. The following are the important problems faced in the construction of index numbers:
1. Definition of the purpose: The first and foremost objective is to clearly delineate the purpose of index number for which it is going to be constructed. All the other factors involved in the construction of index numbers mostly depend on the purpose. Choice of items, collection price quotations, selection of sources of data, and base period chiefly depend on the purpose for which the index number is required. For example, while constructing consumer price index numbers, consumer goods with retail prices will be considered, wholesale prices or exfactory prices will not be useful. Therefore it is necessary to define clearly the purpose of index number before it is constructed.
2. Selection of base period: Every index number must have a base. The base period should be recent and normal as far as possible.
According to Marris Hamburg, “It is desirable that the base period be not too far away in time from the present. The further away we move from the base period the dimmer are our recollections of economic conditions prevailing at that time. Consequently, comparisons with these remote periods tend to loose significance and to become rather tenuous in meaning.”
The base period should be clearly related also to the time period when the patterns of spending habits do not change materially. The index for base period is always taken to be 100. The base period should be fairly normal one free from fluctuations and disturbances.
3. Selection of items to be included: The items to be selected should be relevant, representative, reliable, and comparable. They should be adequate in number and importance so as to cover and reflect the overall picture and should be unaffected by violent movements. The items selected should be of standard quantity. The number to be selected will depend on the technique of collection and processing of data no. definite limit can laid for this. The larger is the number of items, the lesser will be the chances of error in the average. The list of items included in the construction of index number is called “Regimen” or “Basket.”
4. Collection of data: Data are collected on the items, which are to be included in the construction of index numbers. The choice of items totally depends on the purpose of index numbers. The sources of data should be selected with discrimination. These should be reliable and representative.
5. Choice of an average: Any measure of central tendency may be used for constructing an index number. Median and mode are erratic; hence they are not suitable for constructing index numbers. Harmonic mean is difficult to calculate, hence it is not used. Geometric mean (GM) is the best measure for measuring the relative changes and give more importance to small items and less importance to bigger items. Therefore GM is most preferred average. For the construction of general index numbers, Arithmetic mean (AM) is duly affected by extreme values and it is still widely used because of its easy computations.
6. System of weighting: In order to allow each commodity to have a reasonable influence on the index, it is advisable to use a suitable weighting system. The system of weighting depends on the purpose of index. But they ought to reflect the relative importance of the commodities in the basket in the relevant sense. The system of weighting may be either arbitrary or rational. In the case of arbitrary weighting, the statistician is free to assign weights according to his judgment and in the case of rational weighing, the statistician has some fixed criteria for assigning weights. Weights may be either implicit or explicit. Weights should be allowed to vary from period to period. Index number will give better results if weights are allowed to vary.
There are two general methods for constructing index number:
They are further be classified as follows:
1. Simple aggregative method: This is a simple method. In this method, the prices of different commodities of the current year are added and the sum is divided by the sum of the prices of those commodities in the base year and the quotient thus obtained is multiplied by 100.
Symbolically:
where 0 and 1 stand for the base period and the current period, respectively. P01=The required price index number for the current period; Σp1=The sum of the prices of commodities per unit in the current period; Σp0=The sum of the prices of commodities per unit in the base period.
Example 13.1: Calculate the index numbers from the following by simple aggregate method:
Index numbers
Item | Price in base period (in Rs.) | Price in current period (in Rs.) |
A | 5 | 7 |
B | 6 | 8 |
C | 8 | 14 |
D | 16 | 27 |
E | 30 | 28 |
F | 96 | 67 |
Solution:
Items | Price in base period (p0) | Price in current period (p1) |
A | 5 | 7 |
B | 6 | 8 |
C | 8 | 14 |
D | 16 | 27 |
E | 26 | 28 |
F | 100 | 67 |
Total | Σp0=161 | Σp1=151 |
Index number for the current period
Note: The price of every item in the current period shows that they are increased, except the item F. But the index number shows that there is a fall in the commodities. The extent of fall=100−93.7%=6.21%.
Due to presence of item F, the index number is declaring a decrease in the prices of the commodities on an average. Therefore the presence of extreme items will give misleading results. This is a demerit of this method.
Example 13.2: Calculate the index numbers from the following data by simple aggregative method, taking 1980 as base:
Commodity | Prices per unit (in Rs.) | ||||
1980 | 1981 | 1982 | 1983 | 1984 | |
A | 0.30 | 0.33 | 0.36 | 0.36 | 0.39 |
B | 0.25 | 0.24 | 0.30 | 0.32 | 0.30 |
C | 0.20 | 0.25 | 0.28 | 0.32 | 0.30 |
D | 2.00 | 2.40 | 2.50 | 2.50 | 2.60 |
Solution:
Commodity | Prices per unit (in Rs.) | ||||
1980 | 1981 | 1982 | 1983 | 1984 | |
A | 0.30 | 0.32 | 0.36 | 0.36 | 0.39 |
B | 0.25 | 0.24 | 0.30 | 0.32 | 0.30 |
C | 0.20 | 0.25 | 0.28 | 0.32 | 0.30 |
D | 2.00 | 2.40 | 2.50 | 2.50 | 2.60 |
Index number for 1981 (1980 as base) is
Index number for 1982 (1980 as base) is
Index number for 1983 (1980 as base) is
Index number for 1984 (1980 as base) is
2. Weighted aggregative method: This method is known as Laspayre’s method. In this method the base year quantities (consumption, demand, production, etc.) corresponding to the prices of the items are taken as weights. The weighted aggregative index number is constructed by using the formula:
where q0=base period quantities used as weights; p0=prices of the base year; p1=prices of the current year; =sum of products of prices of the commodities in the base period with their corresponding quantities used in the base period; =sum of products of prices of the commodities in the current period with their corresponding quantities used in the base period.
Example 13.3: Calculated weighted aggregative index number by taking 1980 as base from the following data:
Commodity | Quantity | Prices per unit (in Rs.) | ||||
1980 | 1981 | 1982 | 1983 | 1984 | ||
A | 12 units | 0.30 | 0.33 | 0.36 | 0.36 | 0.39 |
B | 10 units | 0.25 | 0.24 | 0.30 | 0.32 | 0.30 |
C | 20 units | 0.20 | 0.25 | 0.28 | 0.32 | 0.30 |
D | 1 unit | 2.00 | 2.40 | 2.50 | 2.50 | 2.60 |
Commodity | Quantity (q0) | 1980 | 1981 | 1982 | 1983 | 1984 | ||||||
p0 | p0q0 | p1 | p1q0 | p1 | p1q0 | p1 | p1q0 | p1 | p1q0 | |||
A | 12 | 0.30 | 3.60 | 0.33 | 3.96 | 0.36 | 4.32 | 0.36 | 4.32 | 0.39 | 4.68 | |
B | 10 | 0.25 | 2.50 | 0.24 | 2.40 | 0.30 | 3.00 | 0.32 | 3.20 | 0.30 | 3.00 | |
C | 20 | 0.20 | 4.00 | 0.25 | 5.00 | 0.28 | 5.60 | 0.36 | 6.40 | 0.30 | 6.00 | |
D | 1 | 2.00 | 2.00 | 2.40 | 2.40 | 2.50 | 2.50 | 2.50 | 2.50 | 2.60 | 2.60 | |
Total | 12.10 | 13.76 | 15.42 | 16.42 | 16.28 |
Weighted index numbers taking 1980 as base:
(Laspeyre’s price index numbers)
Weighted index number for 1981
Weighted index number for 1982
Weighted index number for 1983
Weighted index number for 1981
Example 13.4: Construct index numbers of price, by applying Laspeyre’s from the data:
Commodity | 1993 | 1994 | ||
Price | Quantity | Price | Quantity | |
A | 2 | 8 | 4 | 6 |
B | 5 | 10 | 6 | 5 |
C | 4 | 14 | 5 | 10 |
D | 2 | 19 | 2 | 13 |
Commodity | p0 | q0 | p1 | q1 | p0q0 | p1q0 |
A | 2 | 8 | 4 | 6 | 16 | 32 |
B | 5 | 10 | 6 | 5 | 50 | 60 |
C | 4 | 14 | 5 | 10 | 56 | 70 |
D | 2 | 19 | 2 | 13 | 38 | 38 |
Total | Σp0q0=160 | Σp1q0=200 |
3. Simple average price of relative method: Before introducing the method, we shall first explain the concept of “Price Relative.”
Price Relative of a Commodity: The price relative of a commodity in the current period with respect to base period is defined as the price of the commodity in the current period expressed as a percentage of the price in the base period.
If p0 denotes the price of commodity per unit in the base period, and
p1 denotes the price of commodity in the current period, then
4. Index number by simple average price relative method: In this method the simple average of price relatives of all the items in the data is taken as the index number.
If the AM is used as the average then the index number is
where n denotes the number of items, i.e., commodities.
If GM is used for computing price relatives, then the price index number is
Example 13.5: Compute index numbers by simple AM of price relative method.
Wheat | Cotton | Oil | |
1st year | 4 | 2 | 2 |
2nd year | 3 | 1.5 | 1.25 |
3rd year | 2.5 | 1 | 0.75 |
The price quotations are given in quantity terms converting the money prices, i.e., computing price of the commodity per quintal we get,
Price of wheat per quintal | Price of cotton per quintal | Price of oil per quintal | |
1st year | 100/4=25 | 100/2=50 | 100/2=50 |
2nd year | 100/3=33.3 | 100/1.5=66.7 | 100/1.25=80 |
3rd year | 100/2.5=40.0 | 100/1=100 | 100/0.75=133.3 |
Index numbers by simple AM of price relative method:
Year | Prices per Quintal | Price relatives | Total | ||||
Wheat | Cotton | Oil | Wheat | Cotton | Oil | ||
I | 25 | 50.0 | 50.0 | 202.4 | |||
II | 33.3 | 66.7 | 80.0 | 285 | |||
III | 40.0 | 100.0 | 133.3 | 412.3 | |||
Total | 98.3 | 216.7 | 263.3 | ||||
Average | 32.8 | 72.2 | 87.8 |
Example 13.6: From the following details, construct an index for 1982, taking 1975 as the base by the price relative method using (1) AM, (2) GM.
Commodities | Prices (1975) | Prices (1982) |
A | 10 | 13 |
B | 20 | 17 |
C | 30 | 60 |
D | 40 | 70 |
Commodities | Prices (1975) P0 | Prices (1982) P1 | Price relative (P) (P1/p0)×100 |
A | 10 | 13 | |
B | 20 | 17 | |
C | 30 | 60 | |
D | 40 | 70 | |
Total | 590 |
Commodities | Prices (1975), P0 | Prices (1982), P1 | Price relatives, P | log P |
A | 10 | 13 | 130 | 2.1139 |
B | 20 | 17 | 85 | 1.9294 |
C | 30 | 60 | 200 | 2.3010 |
D | 40 | 70 | 175 | 2.2430 |
Total | 8.5873 |
5. Weighted index number by price relative method: This is a method for computing weighted index numbers. We use value weights. The values of weights may correspond to either base period or current period or any other period. We use the formula:
i.e.,
where p0 and p1 have their usual meanings.
When GM is used, the formula is
Example 13.7: Construct index numbers from the following data for 1986 and 1987 taking prices of 1985 as base:
Commodity | Price (1985) | Price (1986) | Price (1987) |
A | 2.00 | 2.25 | 2.12 |
B | 5.00 | 8.00 | 8.00 |
C | 1.25 | 1.50 | 1.00 |
D | 20.00 | 24.00 | 21.00 |
Solution: Given that the weights of the four commodities are 1, 2, 3, 4, respectively
Commodity | Weight, | 1985 | 1986 | 1987 | ||||||
w | p0 | R | R.W. | P1 | R | R.W. | p1 | R | R.W. | |
A | 1 | 2.00 | 100 | 100 | 2.25 | 112.5 | 112.5 | 2.12 | 106 | 106 |
B | 2 | 5.00 | 100 | 300 | 1.50 | 120 | 360 | 1.00 | 80 | 240 |
C | 3 | 1.25 | 100 | 300 | 1.50 | 120 | 360 | 1.00 | 80 | 240 |
D | 4 | 20.00 | 100 | 400 | 24.00 | 120 | 480 | 21.00 | 105 | 420 |
Total | 10 | 1000 | 1272.5 | 1086 |
1. What is an Index Number? Examine the various problems involved in the construction of an index number. Discuss in brief the use of an index number.
2. Explain the use of index numbers. Describe the procedure followed in the preparation of general and cost of living index numbers.
3. What main points should be taken into consideration while constructing simple index numbers? Explain the procedure of construction of simple index numbers taking example of 10 commodities.
4. Define an “Index Number.” Distinguish between the fixed base and chain base methods of constructing the index numbers and discuss their relative methods.
5. “Index Numbers are Economic Barometers.” Explain this statement, and mention the precautions that should be taken in making in use of any published index numbers.
6. Discuss the problems of (a) Selection of base and (b) Selection of weights in the construction of index numbers.
7. What are the uses of an index number? Discuss the role of the weight in the construction of an index for the general price level.
8. Explain how “cost of living index number” are constructed? Describe briefly the problems involved and suggest their solution.
9. Laspeyre’s price index generally shows an upward trend in the price changes while Paasche’s method shows a downward trend in them elucidate the statement.
10. What is Fishers Ideal Index? Why it is called ideal? Show that it satisfies both the time reversal test as well as the factor reversal test.
11. From the following data calculate price index by using:
Commodities | Base year | Current year | ||
Quantity | Price | Quantity | Price | |
A | 20 | 4 | 30 | 6 |
B | 40 | 5 | 60 | 7 |
C | 60 | 3 | 70 | 4 |
D | 30 | 2 | 50 | 3 |
12. Calculate price index numbers for the year 1990 by using the following methods:
Ans: (a)124.699; (b) 121.769; (c) 123.234; (d) 123.225; (e) 123.323
13. Apply Fisher’s method and calculate the price index number from the following data:
Commodities | 1994 | 1995 | ||
p0 | q0 | p1 | q1 | |
A | 10 | 4 | 12 | 3 |
B | 15 | 6 | 20 | 5 |
C | 2 | 5 | 5 | 6 |
D | 4 | 4 | 4 | 4 |
14. Compute Fisher’s ideal price index number for the following data:
Commodity | 1993 | 1994 | ||
Price/unit | Expenditure | Price/unit | Expenditure | |
A | 5 | 125 | 6 | 180 |
B | 10 | 50 | 15 | 90 |
C | 2 | 30 | 3 | 60 |
D | 3 | 36 | 5 | 75 |
15. From the following data construct a price index number of the group of four commodities by using an appropriate formula:
Commodity | Base year | Current year | ||
Price/unit | Expenditure (in $) | Price/unit | Expenditure (in $) | |
A | 2 | 40 | 5 | 75 |
B | 4 | 16 | 8 | 40 |
C | 1 | 10 | 2 | 24 |
D | 5 | 25 | 10 | 60 |
16. Calculate weighted aggregative price index number taking 1972 as base from the following data:
Commodity | Quantity consumed in 1972 (Quintals) | Units | Price in base year 1972 | Price in current year 1987 |
Wheat | 4 | Per Quintal | 80 | 100 |
Rice | 1 | Per Quintal | 120 | 250 |
Gram | 1 | Per Quintal | 100 | 150 |
Pulses | 2 | Per Quintal | 200 | 300 |
17. Prepare index numbers of prices for 3 years with average price as base:
(Rate per Rs.)
Wheat (kg) | Cotton (kg) | Oil (kg) | |
1st year | 10 | 4 | 3 |
2nd year | 9 | 3.5 | 3 |
3rd year | 9 | 3 | 2.5 |
(Hint: The prices have been given in terms of quantity, convert them to money prices)
Ans: 91, 98, 110
18. Construct index number of agricultural production for 1968–69 with 1949–50 as base.
Agricultural production in India (‘000 tonnes)
Crop | 1949–50 | 1968–69 |
Rice | 25,100 | 39,761 |
Jowar | 6958 | 9809 |
Bajra | 3190 | 3802 |
Maize | 2780 | 5701 |
Ragi | 1510 | 1648 |
Small naillets | 1943 | 1804 |
Wheat | 6757 | 18,652 |
Barley | 2390 | 2424 |
19. Construct wholesale price index for the year 1968–69 with 1952–53=100
Commodity | 1952–53 | 1968–69 | ||
Production (′000 tonnes) | Price per tonne (Rs.) | Production (′000 tonnes) | Price per tonne (Rs.) | |
Rice | 23,420 | 680 | 39761 | 1300 |
Jowar | 6040 | 390 | 9809 | 560 |
Bajra | 2920 | 370 | 3802 | 611 |
Maize | 2816 | 350 | 5701 | 611 |
Wheat | 6760 | 440 | 18,652 | 1100 |
Barley | 2660 | 375 | 2424 | 625 |
Gur | 5260 | 420 | 12,003 | 900 |
Gram | 3800 | 445 | 4310 | 1100 |
(c) Marshall-Edgeworth method, and
(d) Construct Fishers ideal index method
Ans: (a) 199.6; (b) 196; (c) 198; (d) 197.8
20. Construct chain index numbers for the year 1965–66, 1966–67, and 1967–68.
Employees consumers price index (1960–61–100)
Year | Bombay | Ahmedabad | Calcutta | Delhi | Kanpur |
1965–66 | 130 | 130 | 131 | 136 | 446 |
1966–67 | 147 | 148 | 148 | 152 | 153 |
1967–68 | 162 | 168 | 163 | 172 | 174 |
Year | Rice | Milk | Coffee | Tea | Total | Average |
1961 | 81 | 77 | 119 | 55 | 332 | 83.0 |
1962 | 82 | 54 | 128 | 82 | 346 | 86.0 |
1963 | 104 | 87 | 111 | 100 | 402 | 100.5 |
1964 | 93 | 75 | 154 | 96 | 418 | 104.5 |
1965 | 60 | 43 | 165 | 88 | 356 | 89.0 |
1966 | 60 | 44 | 159 | 89 | 352 | 88.0 |
1967 | 62 | 47 | 139 | 84 | 332 | 83.0 |
Ans: 100, 107, 124.15, 102.6, 80.15, 99.95, 97.98
6. Paausche’s method: This is a method of finding weighted index number. In this method current period quantities (q1) are used as weights. Paasche’s index is also known as current year method index (or given year method index).
If p01 is the required index number for the current period then
where p01, p1 represent price per unit of commodities in the base period and current period, respectively. Paasche’s indices posses a downward bias.
Both Laspeyre’s and Paasche’s formulae stand at their sound logic and anyone cannot be out rightly rejected at the cost of the other. All the more, if the base year and given year (current year) are not much distant, both the formulae almost lead to the same index number and it makes little difference which one is chosen.
Example 13.8: Calculate the Paasche’s index number from the following data:
Commodity | Base year | Current year | ||
Quantity | Price | Quantity | Price | |
A | 12 | 10 | 15 | 12 |
B | 15 | 7 | 20 | 5 |
C | 24 | 5 | 20 | 9 |
D | 5 | 16 | 5 | 14 |
Solution: Construction of Paasche’s index
Commodity | Base year | Current year | p1q0 | p0q0 | p1q1 | p0q1 | ||
q0 | p0 | q1 | p1 | |||||
A | 12 | 10 | 15 | 12 | 144 | 120 | 180 | 150 |
B | 15 | 7 | 20 | 5 | 75 | 105 | 100 | 140 |
C | 24 | 5 | 20 | 9 | 216 | 120 | 180 | 100 |
D | 5 | 16 | 5 | 14 | 70 | 80 | 70 | 80 |
Σp1q0=505 | Σp0q0=425 | Σp1q1=530 | Σp0q1=470 |
Example 13.9: Construct index number from the data by applying Paasche’s method.
Commodity | 1995 | 1996 | ||
Price | Quantity | Price | Quantity | |
A | 2 | 8 | 4 | 6 |
B | 5 | 10 | 6 | 5 |
C | 4 | 14 | 5 | 10 |
D | 2 | 19 | 2 | 13 |
Solution: Calculation of index number (1995=100)
Commodity | p0 | q0 | p1 | q1 | p0q0 | p1q1 | p0q1 | p1q0 |
A | 2 | 8 | 4 | 6 | 16 | 24 | 12 | 32 |
B | 5 | 10 | 6 | 5 | 50 | 30 | 25 | 60 |
C | 4 | 14 | 5 | 10 | 56 | 50 | 40 | 70 |
D | 2 | 19 | 2 | 13 | 38 | 26 | 26 | 38 |
Total | 160 | 130 | 103 | 200 | ||||
Σp0q0 | Σp1q1 | Σp0q1 | Σp1q0 |
Commodity | ||
A | B | |
p0 | 1 | 1 |
q0 | 10 | 5 |
p1 | 2 | x |
q1 | 5 | 2 |
where p and q respectively stand for price and quantity and subscripts 0 and 1 stand for time period. Find x if the ratio between Laspeyre’s (L) and Paasche’s (P) index number is:
Commodity | p0 | q0 | p1 | q1 | p1q0 | p0q0 | p1q1 | p0q1 |
A | 1 | 10 | 2 | 5 | 20 | 10 | 10 | 5 |
B | 1 | 5 | X | 2 | 5x | 5 | 2x | 2 |
Total | Σp1q0=20+5x | Σp0q0=15 | Σp1q1=10+2x | Σp0q1=7 |
i.e.,
Example 13.11: Calculate Paasche’s index number for 1985 from the following data:
Commodity | Price 1975 | Quantity 1975 | Price 1985 | Quantity 1985 |
A | 4 | 50 | 10 | 40 |
B | 3 | 10 | 9 | 2 |
C | 2 | 5 | 4 | 2 |
Commodity | 1975 | 1985 | ||||||
Price, p0 | Quantity, q0 | Price, p1 | Quantity, q1 | p0q0 | p0q1 | p1q0 | p1q1 | |
A | 4 | 50 | 10 | 40 | 200 | 160 | 500 | 400 |
B | 3 | 10 | 9 | 2 | 30 | 6 | 90 | 18 |
C | 2 | 5 | 4 | 2 | 10 | 4 | 20 | 8 |
Total | 240 | 170 | 610 | 426 |
7. Dorbish and Bowley method: This method is similar to Fisher’s method, but instead of taking GM, arithmetic average is calculated.
If P01 is the required index number for the current period, then
(13.1)
where p0, p1 represents per unit of commodities in the base and current period, respectively q0, q1 represents number of units in the base period and current period, respectively. The Dorbish and Bowley index formula is the AM of Lasypeyre’s Paasche’s formula.
From Eq. (13.1) we have
Example 13.12: Calculate index number from the following data by Dorbish and Bowley method:
Commodity | Base year | Current year | ||
Price | Quantity | Price | Quantity | |
A | 6 | 50 | 10 | 56 |
B | 2 | 100 | 2 | 120 |
C | 4 | 60 | 6 | 60 |
D | 10 | 30 | 12 | 24 |
E | 8 | 40 | 12 | 36 |
Commodity | Base year | Current year | P0q0 | P0q1 | P1q0 | P1q1 | ||
p0 | q0 | p1 | q1 | |||||
A | 6 | 50 | 10 | 56 | 300 | 336 | 500 | 560 |
B | 2 | 100 | 2 | 120 | 200 | 240 | 200 | 240 |
C | 4 | 60 | 6 | 60 | 240 | 240 | 360 | 360 |
D | 10 | 30 | 12 | 24 | 300 | 240 | 360 | 288 |
E | 8 | 40 | 12 | 36 | 320 | 288 | 480 | 422 |
Total | 1360 | 1344 | 1900 | 1880 |
8. Fisher’s method: In this method two index numbers with a different set of weights are constructed and a GM is found out. Symbolically it is expressed as
where p0=price of the base year; p1=price of the current year; q0=quantity of the base; q1=quantity of the current year.
This method is also called weight formula.
9. Marshall-Edgeworth’s method: In this method the sum of base period quantities and current period quantities are used as weights. The formula for computing index numbers by this method is given as
where p0, p1 represent the price per unit of commodities in the base period and current period respectively. q0, q1 represent the number of units in the base period and current period, respectively.
The above formula can also be written as
Example 13.13: Construct index number by Marshall-Edgeworth’s method.
Commodity | 1993 | 1994 | ||
Price | Quantity | Price | Quantity | |
A | 2 | 8 | 4 | 6 |
B | 5 | 10 | 6 | 5 |
C | 4 | 14 | 5 | 10 |
D | 2 | 16 | 2 | 13 |
Solution: Construct of index number (1993–100)
Commodity | p0 | q0 | p1 | q1 | p0q0 | p1q1 | p0q1 | p1q0 |
A | 2 | 8 | 4 | 6 | 16 | 24 | 12 | 32 |
B | 5 | 10 | 6 | 5 | 50 | 30 | 25 | 60 |
C | 4 | 14 | 5 | 10 | 56 | 50 | 40 | 70 |
D | 2 | 16 | 2 | 13 | 38 | 26 | 26 | 38 |
Total | 160 Σp0q0 | 130 Σp1q1 | 103 Σp0q1 | 200 Σp1q0 |
10. Kelly’s method: In this method we compute weighted index numbers. The quantities (q) corresponding to any period can be used as weights in this method; we can also use the average quantities for two more periods as weights. Kelly’s method is also called Kelly’s fixed weight aggregative method. If p01 denotes the index number for the current period, then
where p0 and p1 have their usual meanings and q represents the quantities which are used as weights.
The greatest advantage of Kelly’s method is that the change in base year does not required to determine the new weights.
Example 13.14: Construct weighted index number by Kelly’s method from the data:
Commodity | Base year−1993 | Current year−1994 | ||
Price | Quantity | Price | Quantity | |
A | 2 | 20 | 4 | 45 |
B | 4 | 22 | 5 | 30 |
C | 6 | 30 | 8 | 40 |
D | 8 | 40 | 10 | 60 |
Commodity | Base year | Current year | p1q | p0q | |||
p0 | q0 | p1 | q1 | ||||
A | 2 | 20 | 4 | 45 | 32.5 | 130 | 65 |
B | 4 | 22 | 5 | 30 | 26.0 | 130 | 104 |
C | 6 | 30 | 8 | 40 | 35.0 | 280 | 210 |
D | 8 | 40 | 10 | 60 | 50.0 | 500 | 400 |
Total | Σp1q=1040 | Σp0q=799 |
Example 13.15: Calculate price index number for 1945 by
4. Marshall’s and Edgeworth’s method
Commodity | Price 1935 | Quantity 1935 | Price 1945 | Quantity 1945 |
A | 4 | 50 | 10 | 40 |
B | 3 | 10 | 9 | 2 |
C | 2 | 5 | 4 | 2 |
Commodity | 1935 | 1945 | p0q0 | p0q1 | p1q0 | p1q1 | ||
p0 | q0 | p1 | q1 | |||||
A | 4 | 50 | 10 | 40 | 200 | 160 | 500 | 400 |
B | 3 | 10 | 9 | 2 | 30 | 6 | 90 | 18 |
C | 2 | 5 | 4 | 2 | 10 | 4 | 20 | 8 |
Total | 240 | 170 | 610 | 426 |