Index Numbers — Study Notes

1. Introduction

Economic magnitudes such as the price level, the cost of living, industrial production and agricultural output change from period to period, but they are made up of many components that move differently. It is impossible to judge at a glance whether prices in general have risen or fallen, or by how much. An index number solves this problem by summarising the changes in a group of related variables into a single figure. An index number is thus a statistical device that measures the relative change in the level of a variable or a group of variables with reference to a base period.

The most familiar index numbers are the Consumer Price Index, which measures changes in the retail prices of goods and services purchased by a typical consumer, and the Wholesale Price Index, which measures changes in the prices of goods at the wholesale level. Index numbers are also constructed for industrial production, agricultural output, imports, exports, share prices and many other economic series.

In this chapter we study the meaning and uses of index numbers, the problems faced in their construction, the methods of constructing price and quantity index numbers - including the Laspeyres, Paasche and Fisher methods - the concept of weighted and unweighted indices, and the Consumer Price Index and Wholesale Price Index of India.

2. Meaning and Uses of Index Numbers

An index number is a number that shows the average relative change in the price, quantity or value of a group of commodities or variables between one period (the current period) and another (the base period). It is expressed as a percentage, with the base period generally taken as 100. A price index of 120 means prices have risen by 20 percent over the base period; a value below 100 means a fall.

The uses of index numbers are:

1. Measurement of price changes: They measure the general price level and changes in the cost of living.
2. Formulation of economic policy: The government uses price and production indices for policy decisions on inflation, wages, dearness allowance and planning.
3. Adjustment of money values: Index numbers are used to deflate money values into real values, e.g. real wages or real national income, removing the effect of price changes.
4. Forecasting: Indices of business activity, production and share prices help in forecasting the state of the economy.
5. Comparison over time and space: They allow comparison of prices, production and incomes across different periods and regions.
6. Industrial and commercial decisions: Business firms use index numbers to adjust wages, rents and contract prices.

3. Problems in the Construction of Index Numbers

The construction of index numbers involves several conceptual and practical difficulties:

1. Purpose of the index: The index must be constructed for a clear purpose, since the choice of items, weights and methods depends on the objective.
2. Selection of items: Only representative commodities should be selected; the number and nature of items depend on the purpose.
3. Choice of base period: The base period should be a normal year - free from abnormal conditions such as war, famine or inflation - and should be recent.
4. Sources of data: Price data must be obtained from reliable and comparable sources.
5. Selection of weights: Items are not of equal importance; appropriate weights must be given to each item.
6. Choice of the method: The appropriate formula - simple or weighted, aggregative or average of price relatives - must be chosen.

4. Types of Index Numbers

Index numbers may be classified as:

1. Price index numbers: They measure the relative change in the prices of commodities over time. Examples are the Consumer Price Index, the Wholesale Price Index and the Producer Price Index.
2. Quantity index numbers: They measure the relative change in the quantity of goods produced, consumed or traded, e.g. index of industrial production, index of agricultural output.
3. Value index numbers: They measure the relative change in the total money value (price x quantity) of goods, e.g. the value of exports.

5. Methods of Constructing Index Numbers

Index numbers are constructed by two broad groups of methods:

A. Simple (unweighted) methods

1. Simple aggregative method: The sum of the prices of all items in the current year is divided by the sum of their prices in the base year and multiplied by 100:

$$P_{01} = \frac{\sum P_1}{\sum P_0} \times 100$$

where P0 is the base year price and P1 the current year price. This method gives equal importance to all items, ignoring their relative importance.

2. Simple average of price relatives method: The price relative of each item is first computed as (P1/P0) x 100, and then the arithmetic mean of these relatives is taken:

$$P_{01} = \frac{1}{N} \sum \frac{P_1}{P_0} \times 100$$

B. Weighted methods

1. Weighted aggregative method: The prices are weighted by quantities or values. The two most common formulas are: - Laspeyres' index: Base year quantities are used as weights.

   $$P_{01} = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100$$

• Paasche's index: Current year quantities are used as weights.

  $$P_{01} = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100$$

• Fisher's ideal index: It is the geometric mean of the Laspeyres and Paasche indices and satisfies both the time reversal and factor reversal tests:

  $$P_{01} = \sqrt{\frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times \frac{\sum P_1 Q_1}{\sum P_0 Q_1}} \times 100$$

2. Weighted average of price relatives method: The price relatives are weighted by the values of the commodities in the base year.

6. Tests of an Index Number

Two important tests are applied to check the adequacy of an index number formula:

Time reversal test: The index for period 1 with period 0 as base, multiplied by the index for period 0 with period 1 as base, should equal 1 (ignoring the factor 100). That is, P01 x P10 = 1. Fisher's ideal index satisfies this test.

Factor reversal test: The product of the price index and the quantity index should equal the value index, i.e. the ratio of the total value in the current period to the total value in the base period. Fisher's ideal index satisfies this test as well.

7. Consumer Price Index and Wholesale Price Index

Consumer Price Index (CPI): The CPI measures the changes in the prices of goods and services consumed by a particular group of consumers, e.g. industrial workers, agricultural labourers or urban consumers. It is constructed using the weighted aggregative or the weighted average of price relatives method, with the consumption pattern of the target group providing the weights. The CPI is used to measure the cost of living, to adjust wages, dearness allowance and pensions, and to measure the purchasing power of money. In India, CPIs are published for industrial workers (CPI-IW), agricultural labourers (CPI-AL) and urban consumers, and a combined CPI is used by the Reserve Bank for inflation targeting.

Wholesale Price Index (WPI): The WPI measures the changes in the prices of goods at the wholesale level - that is, prices at which transactions are done in bulk between producers, wholesalers and retailers. In India the WPI is constructed for a large basket of commodities grouped into primary articles, fuel and power, and manufactured products. The WPI is used to measure the level of wholesale prices and is computed by the Office of the Economic Adviser, Ministry of Commerce and Industry.

The limitations of index numbers are that they are not exact, they use representative items that may not match every user's consumption basket, the base period may become outdated, and changes in the quality of goods are difficult to incorporate.

Quick Revision Tables

Mind Map

Important Diagrams (SVG)

Diagram 1: Price Index Comparison - Base and Current Year

Diagram 2: Laspeyres, Paasche and Fisher Indices

Common Mistakes

1. Confusing the base year with the current year; the base year has index 100 and should be a normal, recent year.
2. Using the simple aggregative method when items have very different importance; weights must be used to give proper importance to each item.
3. Mixing up the Laspeyres and Paasche weights; Laspeyres uses base year quantities Q0, Paasche uses current year quantities Q1.
4. Forgetting that the simple average of price relatives uses (P1/P0) x 100 for each item before averaging.
5. Confusing the CPI with the WPI; the CPI measures retail prices for a consumer group at the retail level, while the WPI measures wholesale prices of commodities.
6. Believing that index numbers are exact measures; they are approximate and depend on the choice of items, weights, base year and method.
7. Applying an outdated base year; the base period should be recent and normal, otherwise the index misrepresents current changes.

Exam Tips

1. Define an index number and explain its uses in economics.
2. List the problems faced in constructing index numbers.
3. Distinguish price, quantity and value index numbers.
4. State the formulas for the simple aggregative and simple average of price relatives methods.
5. Write the Laspeyres, Paasche and Fisher formulas and state which weights each uses.
6. Explain the time reversal and factor reversal tests and why Fisher's index is called ideal.
7. Explain the meaning and uses of the Consumer Price Index and the Wholesale Price Index in India.

Conclusion

This chapter explained index numbers, the statistical devices that measure relative changes in prices, quantities and values over time. We studied their importance in measuring the cost of living, guiding government policy, deflating money values and forecasting economic activity. We examined the problems of constructing index numbers - the selection of items, the choice of a normal and recent base year, the use of weights, and the choice of method. We learned the simple aggregative and simple average of price relatives methods, the weighted Laspeyres and Paasche indices, and Fisher's ideal index, which is the geometric mean of the two and satisfies the time reversal and factor reversal tests. Finally, we studied the Consumer Price Index and the Wholesale Price Index of India and their uses. These statistical tools complete our preparation for applied economic analysis, and the next chapter ties all the tools together in the use of statistical tools in economic decision making.