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.”

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.

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.

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:

$P_{01}=\mathrm{Index}\mathrm{number}{P}_{01}\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100$

where q₀=base period quantities used as weights; p₀=prices of the base year; p₁=prices of the current year; $\sum p_0{q}_0$=sum of products of prices of the commodities in the base period with their corresponding quantities used in the base period; $\sum p_1{q}_0$=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:

Solution:

Weighted index numbers taking 1980 as base:
(Laspeyre’s price index numbers)
Weighted index number for 1981

$\begin{array}{ll}(1980\mathrm{as}\mathrm{base}\mathrm{period})\hfill & =\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100\hfill \\ \hfill & =\frac{13.76}{12.10}\times 100=113.7\hfill \end{array}$

Weighted index number for 1982

$\begin{array}{ll}(1980\mathrm{as}\mathrm{base}\mathrm{period})\hfill & =\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100\hfill \\ \hfill & =\frac{15.42}{12.10}\times 100=127.4\hfill \end{array}$

Weighted index number for 1983

$\begin{array}{ll}(1980\mathrm{as}\mathrm{base}\mathrm{period})\hfill & =\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100\hfill \\ \hfill & =\frac{16.42}{12.10}\times 100=135.7\hfill \end{array}$

Weighted index number for 1981

$\begin{array}{ll}(1980\mathrm{as}\mathrm{base}\mathrm{period})\hfill & =\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100\hfill \\ \hfill & =\frac{16.25}{12.10}\times 100=134.5\hfill \end{array}$

Example 13.4: Construct index numbers of price, by applying Laspeyre’s from the data:

Solution:
Base year is 1993

Laspeyre’s price index number

$\begin{array}{ll}(\mathrm{base}\mathrm{period}1993)\hfill & =\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100\hfill \\ \hfill & =\frac{200}{160}\times 100=125\hfill \end{array}$

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

$\mathrm{Price}\mathrm{Relative}=P=\frac{p_1}{p_0}\times 100$

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

$\mathrm{Price}\mathrm{index}\mathrm{number}=P_{01}=\frac{(\mathrm{\Sigma }{p}_1/p_0)\times 100}{n}$

where n denotes the number of items, i.e., commodities.
If GM is used for computing price relatives, then the price index number is

$P_{01}=\mathrm{Antilog}\left[\frac{\mathrm{\Sigma }\log P}{n}\right]$

Example 13.5: Compute index numbers by simple AM of price relative method.

The price quotations are given in quantity terms converting the money prices, i.e., computing price of the commodity per quintal we get,

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	$\begin{array}{cc}\left(\frac{25}{32.8}\right)& \times 100\\ & =76.2\end{array}$	$\begin{array}{cc}\left(\frac{50}{72.2}\right)& \times 100\\ & =69.3\end{array}$	$\begin{array}{cc}\left(\frac{50}{87.8}\right)& \times 100\\ & =56.9\end{array}$	202.4
II	33.3	66.7	80.0	$\begin{array}{cc}\left(\frac{33.3}{32.8}\right)& \times 100\\ & =101.5\end{array}$	$\begin{array}{cc}\left(\frac{66.7}{72.2}\right)& \times 100\\ & =92.4\end{array}$	$\begin{array}{cc}\left(\frac{80}{87.8}\right)& \times 100\\ & =91.1\end{array}$	285
III	40.0	100.0	133.3	$\begin{array}{cc}\left(\frac{40}{32.8}\right)& \times 100\\ & =138.5\end{array}$	$\begin{array}{cc}\left(\frac{100}{72.2}\right)& \times 100\\ & =138.5\end{array}$	$\begin{array}{cc}\left(\frac{133}{87.8}\right)& \times 100\\ & =151.8\end{array}$	412.3
Total	98.3	216.7	263.3
Average	32.8	72.2	87.8

We have n=3

$\mathrm{Average}\mathrm{index}\mathrm{numbers}=\frac{\mathrm{\Sigma }{P}_1/P}{n}$

$\mathrm{For}\mathrm{I}\mathrm{year}=\frac{202}{3}=67.3$

$\mathrm{For}\mathrm{II}\mathrm{year}=\frac{285}{3}=95$

$\mathrm{For}\mathrm{III}\mathrm{year}=\frac{412.3}{3}=137.4$

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.

Solution:

1. 

Commodities	Prices (1975) P0	Prices (1982) P1	Price relative (P) (P1/p0)×100
A	10	13	$(13/10)\times 100=130$
B	20	17	$(17/20)\times 100=85$
C	30	60	$(60/30)\times 100=200$
D	40	70	$(70/40)\times 100=175$
Total			590

We have n=4

$\begin{array}{ll}\mathrm{Index}\mathrm{number}\left(\mathrm{by}\mathrm{Arithmetic}\mathrm{mean}\right)\hfill & =\frac{(\mathrm{\Sigma }{p}_1/p_0)\times 100}{n}\hfill \\ \hfill & =\frac{590}{4}=147.5\hfill \end{array}$

2. Index number (by GM)

We have n=4

$\begin{array}{ll}\mathrm{Index}\mathrm{number}(\mathrm{by}\mathrm{Geometric}\mathrm{mean})\hfill & =\mathrm{Antilog}\left[\frac{\mathrm{\Sigma }\log P}{n}\right]\hfill \\ \hfill & =\mathrm{Antilog}\frac{8.5873}{4}\hfill \\ \hfill & =\mathrm{Antilog}(2.1468)=140.3\hfill \end{array}$

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:

$\mathrm{Index}\text{number}(\text{by}\text{Arithmethic}\text{mean})=P_{01}=\frac{\mathrm{\Sigma }Pw}{\mathrm{\Sigma }w}\left(P=\left(\frac{p_1}{p_0}\right)\times 100\right)$

i.e.,

$P_{01}=\frac{\mathrm{\Sigma }\frac{P_1}{P_0}\times w}{\mathrm{\Sigma }w}$

where p₀ and p₁ have their usual meanings.
When GM is used, the formula is

$P_{01}=\frac{\mathrm{\Sigma }w\log P}{\mathrm{\Sigma }w}\left(P=\left(\frac{p_1}{p_0}\right)\times 100\right)$

Example 13.7: Construct index numbers from the following data for 1986 and 1987 taking prices of 1985 as base:

Solution: Given that the weights of the four commodities are 1, 2, 3, 4, respectively

We have Σw=10

$\begin{array}{ll}\mathrm{Index}\mathrm{number}\mathrm{for}1986(1985\mathrm{as}\text{base})\hfill & =\frac{\mathrm{\Sigma }\frac{p_1}{p_0}\times w}{\mathrm{\Sigma }w}\hfill \\ \hfill & =\frac{\mathrm{\Sigma }Rw}{\mathrm{\Sigma }w}=\frac{1272.5}{10}=127.25\hfill \end{array}$

$\mathrm{Index}\mathrm{number}\mathrm{for}1987(1985\mathrm{as}\text{base})=\frac{\mathrm{\Sigma }Rw}{\mathrm{\Sigma }w}=\frac{1086}{10}=108.6$

Exercise 13.1

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:

(a) Laspeyre’s method

(b) Paasche’s method

Ans: (a) 140.38; (b) 141.140

12. Calculate price index numbers for the year 1990 by using the following methods:

(a) Laspeyre’s method

(b) Paasche’s method

(c) Bowley’s method

(d) Fisher’s method

(e) Marshall’s method

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:

Ans: 135.4

14. Compute Fisher’s ideal price index number for the following data:

Ans: 137.11

15. From the following data construct a price index number of the group of four commodities by using an appropriate formula:

Ans: 219.12

16. Calculate weighted aggregative price index number taking 1972 as base from the following data:

Ans: 148.94

17. Prepare index numbers of prices for 3 years with average price as base:
   (Rate per Rs.)

(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)

Ans: 150.4

19. Construct wholesale price index for the year 1968–69 with 1952–53=100

Using

(a) Pasche’s method

(b) Laspeyre’s method

(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)

Ans: 134.6, 149.8, 167.9

21. Construct new index numbers using chain base method.

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

$P_{01}=\frac{\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_1}\times 100$

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:

Solution: Construction of Paasche’s index

We have

$\mathrm{\Sigma }{p}_1{q}_0=505,\mathrm{\Sigma }{p}_0{q}_0=422,\mathrm{\Sigma }{p}_1{q}_1=530,\mathrm{and}\mathrm{\Sigma }{p}_0{q}_1=470$

$\begin{array}{ll}\mathrm{Paasche}’\mathrm{s}\mathrm{index}=p_{01}\hfill & =\frac{\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_1}\times 100\hfill \\ \hfill & =\frac{530}{470}\times 100=112.8\hfill \end{array}$

Example 13.9: Construct index number from the data by applying Paasche’s method.

Solution: Calculation of index number (1995=100)

We have Σp₁q₁=130, Σp₀q₁=103

$\begin{array}{ll}\text{Paasche}’\mathrm{s}\mathrm{price}\mathrm{index}\mathrm{number}\hfill & =\frac{\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_1}\times 100\hfill \\ \hfill & =\frac{130}{103}\times 100=126.21\hfill \end{array}$

Example 13.10: Given the data

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:

$\mathrm{L}:\mathrm{P}::28:27$

Solution:

$\mathrm{Laspeyre}’\mathrm{s}\mathrm{index}=P_{01}=\frac{\mathrm{\Sigma }{p}_1{q}_0}{\mathrm{\Sigma }{p}_0{q}_0}=\frac{20+5x}{15}$

$\mathrm{Paasche}’\mathrm{s}\mathrm{index}=P_{01}=\frac{\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_1}=\frac{10+2x}{7}$

It is given that

$\mathrm{L}:\mathrm{P}::28:27$

i.e.,

$\begin{array}{ll}\hfill & \frac{L}{P}=\frac{28}{27}\hfill \\ \Rightarrow \hfill & 27\mathrm{L}=28\mathrm{P}\hfill \\ \Rightarrow \hfill & 27\left[\frac{20+5x}{15}\right]=28\left[\frac{10+2x}{7}\right]\hfill \\ \Rightarrow \hfill & 27\frac{5(4\times x)}{15}=4(10+2x)\hfill \\ \Rightarrow \hfill & 9(4+x)=4(10+2x)\hfill \\ \Rightarrow \hfill & 36+9x=40+8x\hfill \\ \Rightarrow \hfill & 9x–8x=40–36\hfill \\ \therefore \hfill & x=4\hfill \end{array}$

Example 13.11: Calculate Paasche’s index number for 1985 from the following data:

Solution:

We have

$\begin{array}{l}\sum p_0{q}_0=240,\sum p_0{q}_1=170\\ \sum p_1{q}_0=610,\sum p_1{q}_1=426,\end{array}$

$\begin{array}{ll}\text{Paasche}’\mathrm{s}\mathrm{index}\mathrm{number}\hfill & =\frac{\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_1}\times 100\hfill \\ \hfill & =\frac{426}{170}\times 100=250.6\hfill \end{array}$

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

$P_{01}=\frac{\left[\frac{\sum p_1{q}_0}{\sum p_0{q}_0}+\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\right]}{2}\times 100$ (13.1)

(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

$\begin{array}{ll}{P}_{01}\hfill & =\frac{\left[\frac{\sum p_1{q}_0}{\sum p_0{q}_0}+\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\right]}{2}\times 100\hfill \\ \hfill & =\frac{\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100+\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100}{2}\hfill \\ \hfill & =\frac{\text{Laspeyre}\text{'}\mathrm{s}\mathrm{index}\mathrm{number}+\text{Paasche}\text{'}\mathrm{s}\mathrm{index}\mathrm{number}}{2}\hfill \\ \hfill & =\mathrm{Arithmetic}\mathrm{mean}\mathrm{of}\text{Laspeyr}’\mathrm{s}\mathrm{and}\text{Paasche}’\mathrm{s}\mathrm{index}\mathrm{number}\hfill \end{array}$

Example 13.12: Calculate index number from the following data by Dorbish and Bowley method:

Solution:

$\begin{array}{lll}\therefore \hfill & \sum p_0{q}_1=1360,\hfill & \sum p_0{q}_1=1900\hfill \\ \hfill & \sum p_0{q}_1=1344,\hfill & \sum p_1{q}_1=1880,\hfill \end{array}$

$\begin{array}{ll}\mathrm{Dorbish}\mathrm{and}\mathrm{Bowley}\mathrm{index}\mathrm{number}\hfill & =\frac{\left[\frac{\sum p_1{q}_0}{\sum p_0{q}_0}+\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\right]}{2}\times 100\hfill \\ \hfill & =\frac{1}{2}\left[\frac{1900}{1360}+\frac{1880}{1344}\right]\times 100\hfill \\ \hfill & =140(\text{approximately})\hfill \end{array}$

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

$\mathrm{Index}\mathrm{number}=P_{01}=\sqrt{\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times \frac{\sum p_1{q}_1}{\sum p_0{q}_1}}\times 100$

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

$P_{01}=\frac{\mathrm{\Sigma }{p}_1(q_0+q_1)}{\mathrm{\Sigma }{p}_0(q_0+q_1)}\times 100$

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

$P_{01}=\frac{\mathrm{\Sigma }{p}_1{q}_0+\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_0+\mathrm{\Sigma }{p}_0{q}_1}\times 100$

Example 13.13: Construct index number by Marshall-Edgeworth’s method.

Solution: Construct of index number (1993–100)

$\begin{array}{ll}\mathrm{Marshall}\text{Edgeworth}’\mathrm{s}\mathrm{index}\mathrm{number}\hfill & =\frac{\mathrm{\Sigma }{p}_0{q}_0+\mathrm{\Sigma }{p}_1{q}_1}{\mathrm{\Sigma }{p}_0{q}_0+\mathrm{\Sigma }{p}_0{q}_1}\times 100\hfill \\ \hfill & =\frac{200+130}{160+103}\times 100\hfill \\ \hfill & =\frac{67404.54}{24816}\times 100\hfill \\ \hfill & =125.47\hfill \end{array}$

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

$P_{01}=\frac{\mathrm{\Sigma }{p}_{1}q}{\mathrm{\Sigma }{p}_{0}q}\times 100$

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:

Solution:

Commodity	Base year		Current year		$q=\frac{q_0+q_1}{2}$	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

We have Σp₁q=1040, Σp₀q=779

$\begin{array}{ll}\text{Kelly}’s\mathrm{index}\mathrm{number}\hfill & =\frac{\mathrm{\Sigma }{p}_{1}q}{\mathrm{\Sigma }{p}_{0}q}\times 100\hfill \\ \hfill & =\frac{1040}{779}\times 100\hfill \\ \hfill & =133.5\hfill \end{array}$

Example 13.15: Calculate price index number for 1945 by

1. Laspeyre’s method

2. Paasche’s method

3. Bowley’s method

4. Marshall’s and Edgeworth’s method

5. Fisher’s method

From following data:

Solution:

$\begin{array}{cc}\sum p_0{q}_0=240,& \sum p_0{q}_1=170\\ \sum p_1{q}_0=610,& \sum p_1{q}_1=426\end{array}$