Index Numbers
Introduction
In our daily lives, we constantly encounter statements about changes over time. We hear that prices are rising, industrial production has increased, or the stock market is booming. But how are these complex changes measured? A single commodity like salt might have a stable price, while the price of petrol may have doubled. How do we measure the overall change in the "general price level"?
Direct comparison of absolute figures is often not feasible or meaningful. Comparing the total value of industrial output this year with last year can be misleading if prices have changed. We need a tool that can summarise the average change in a group of related variables over time. This tool is the index number. Index numbers are one of the most widely used statistical tools in economics and business. They act as economic barometers, helping us to understand and quantify the dynamics of an economy.
What Is An Index Number
An index number is a specialised statistical measure designed to show the average change in a variable or a group of related variables over time, with respect to a base. It is a technique for measuring the relative change in a variable when direct measurement is not possible or meaningful.
Key concepts associated with an index number are:
- Base Period/Year: This is the reference period against which all comparisons are made. The index number for the base period is always taken as 100. The base period should be a normal period, free from any major economic disturbances like wars, famines, or depressions.
- Current Period/Year: This is the period for which the change is being measured.
In essence, an index number is a percentage. For instance, if the Consumer Price Index (CPI) for a given year is 125 with a base year of 2012=100, it means that the general price level of the basket of goods and services consumed by an average household has increased by 25% in that year compared to 2012.
While most commonly used for measuring price changes, index numbers can also be constructed for changes in quantity (e.g., agricultural production) or value (e.g., value of exports).
Construction Of An Index Number
The construction of an index number involves several steps, from selecting the items and base year to choosing an appropriate formula. There are two primary methods for constructing price index numbers: the Aggregative Method and the Method of Averaging Relatives.
The Aggregative Method
In this method, the aggregate (sum) of prices of a group of commodities in the current year is compared with the aggregate of their prices in the base year.
1. Simple Aggregative Method
This is the simplest method where we just sum up the prices of all selected commodities in the current and base years.
Formula: $ P_{01} = \frac{\sum p_1}{\sum p_0} \times 100 $
where $p_1$ = price in the current year, and $p_0$ = price in the base year.
Limitation: This method has a major flaw. It is heavily influenced by commodities with high absolute prices, regardless of their importance in consumption. Also, it is affected by the units of measurement (e.g., price per kg vs. price per quintal).
2. Weighted Aggregative Method
To overcome the limitation of the simple aggregative method, we assign weights to each commodity according to its relative importance. For a price index, the weight used is the quantity ($q$) consumed or produced. Several methods have been proposed by different statisticians.
a) Laspeyres' Price Index: Proposed by Laspeyres in 1871, this method uses the quantities of the base year ($q_0$) as weights. The logic is that the consumption pattern of the base year should be used to see how much the cost of that specific basket of goods has changed.
Formula: $ P_{01}(L) = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 $
b) Paasche's Price Index: Proposed by Paasche in 1874, this method uses the quantities of the current year ($q_1$) as weights. It measures the change in cost of the current year's consumption basket.
Formula: $ P_{01}(P) = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 $
c) Fisher's Ideal Price Index: Irving Fisher described this index as "ideal" because it avoids the biases associated with Laspeyres' (which tends to have an upward bias) and Paasche's (which tends to have a downward bias) methods. It is the geometric mean of the Laspeyres' and Paasche's indices.
Formula: $ P_{01}(F) = \sqrt{P_{01}(L) \times P_{01}(P)} = \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 $
Example 1. From the following data, calculate Laspeyres', Paasche's, and Fisher's Price Index Numbers for the year 2022 with 2015 as the base year.
| Commodity | 2015 (Base Year) | 2022 (Current Year) | ||
|---|---|---|---|---|
| Price ($p_0$) | Quantity ($q_0$) | Price ($p_1$) | Quantity ($q_1$) | |
| A | 10 | 50 | 12 | 60 |
| B | 20 | 25 | 25 | 30 |
| C | 5 | 100 | 8 | 120 |
Answer:
First, we need to calculate the necessary product columns.
| Comm. | $p_0$ | $q_0$ | $p_1$ | $q_1$ | $p_1 q_0$ | $p_0 q_0$ | $p_1 q_1$ | $p_0 q_1$ |
|---|---|---|---|---|---|---|---|---|
| A | 10 | 50 | 12 | 60 | (12x50)=600 | (10x50)=500 | (12x60)=720 | (10x60)=600 |
| B | 20 | 25 | 25 | 30 | (25x25)=625 | (20x25)=500 | (25x30)=750 | (20x30)=600 |
| C | 5 | 100 | 8 | 120 | (8x100)=800 | (5x100)=500 | (8x120)=960 | (5x120)=600 |
| Total | $\sum p_1 q_0=2025$ | $\sum p_0 q_0=1500$ | $\sum p_1 q_1=2430$ | $\sum p_0 q_1=1800$ |
1. Laspeyres' Index ($P_{01}(L)$):
$ P_{01}(L) = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 = \frac{2025}{1500} \times 100 = 1.35 \times 100 = 135 $
2. Paasche's Index ($P_{01}(P)$):
$ P_{01}(P) = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 = \frac{2430}{1800} \times 100 = 1.35 \times 100 = 135 $
3. Fisher's Index ($P_{01}(F)$):
$ P_{01}(F) = \sqrt{135 \times 135} = \sqrt{18225} = 135 $
Conclusion: The price level in 2022 was 35% higher than in 2015. (In this specific case, all three indices gave the same result, which is unusual).
Method Of Averaging Relatives
In this method, we first calculate a price relative for each commodity. A price relative is the price of a single commodity in the current year expressed as a percentage of its price in the base year.
$ \text{Price Relative} = \frac{p_1}{p_0} \times 100 $
Then, we take an average (simple or weighted) of these individual relatives to get the overall index number. The weighted average of price relatives, when weighted by the base year values ($p_0q_0$), gives the same result as Laspeyres' method.
Some Important Index Numbers
Index numbers are widely used in India to track various aspects of the economy. Some of the most important ones are:
Consumer Price Index (CPI) or Cost of Living Index
The Consumer Price Index (CPI) measures the average change over time in the prices paid by urban or rural consumers for a market basket of consumer goods and services. It is a measure of the cost of living.
Construction: It is generally constructed using a weighted average of price relatives, where the weights are derived from family budget surveys that determine the proportion of income spent on different items (e.g., food, housing, clothing, fuel).
Uses:
- It is used to calculate the inflation rate.
- It is used by the government to grant Dearness Allowance (DA) to its employees to compensate them for the rising cost of living.
- It is used to adjust wages and salaries in private contracts.
In India, the National Statistical Office (NSO) releases CPI for different consumer groups (e.g., Industrial Workers, Agricultural Labourers) and a combined CPI for the whole country.
Wholesale Price Index (WPI)
The Wholesale Price Index (WPI) measures the average change in the prices of goods sold in wholesale markets. It tracks prices at the first point of bulk sale in the domestic market. For a long time, WPI was the main headline measure of inflation in India.
Index Of Industrial Production (IIP)
The IIP is a quantity index. It measures the changes in the level of production in the industrial sector, including mining, manufacturing, and electricity. It gives a snapshot of the industrial growth in the economy.
Human Development Index (HDI)
The HDI is a composite index published by the United Nations Development Programme (UNDP). It is not an economic index but a measure of overall human development, combining statistics on life expectancy, education, and per capita income.
Sensex
The Sensex (S&P BSE SENSEX) is a stock market index for the Bombay Stock Exchange (BSE). It is a value-weighted index composed of 30 of the largest and most actively-traded stocks on the BSE. It is considered a barometer of the health of the Indian stock market and the broader economy.
Issues In The Construction Of An Index Number
Constructing a reliable index number is a complex task that involves several practical and conceptual challenges. The usefulness of an index depends heavily on the choices made during its construction.
- Purpose of the Index: The very first step is to be clear about the purpose. An index to measure the cost of living for industrial workers will have a different basket of goods and weights than one designed to measure wholesale prices.
- Selection of the Base Period: The base period should be recent and economically stable. A period of crisis like a war or a drought is unsuitable. Periodically, the base year needs to be revised to reflect changing economic structures.
- Selection of Items: It is impossible to include all items. A representative sample of items must be chosen. The items should be popular, stable in quality, and representative of the tastes and habits of the people.
- Selection of Prices: Price data can be tricky. Should it be wholesale or retail? From which market should it be collected? A consistent method of price collection is crucial.
- Selection of Weights: Assigning appropriate weights is the most critical step. Should we use base year quantities (Laspeyres) or current year quantities (Paasche)? Using the wrong weights can lead to a biased index.
- Choice of Formula: As seen, different formulas (Laspeyres, Paasche, Fisher) can give different results. The choice depends on the purpose, availability of data, and the desired properties of the index.
Index Number In Economics
Index numbers are indispensable tools in modern economics for a variety of purposes:
- Measuring Inflation: CPI and WPI are used to measure price inflation, which is a key variable for monetary policy decisions by the Reserve Bank of India (RBI).
- Economic Barometers: Indices like IIP and Sensex act as barometers that indicate the overall health and direction of the economy.
- Policy Formulation: The government uses various indices to formulate policies regarding taxation, wages (DA), trade, and agriculture.
- Deflating: Index numbers are used to adjust nominal values for price changes to arrive at real values. For example, to find the "real wage," we adjust the money wage for the cost of living.
$ \text{Real Wage} = \frac{\text{Money Wage}}{\text{Consumer Price Index}} \times 100 $
This tells us about the actual purchasing power of a person's income. - Forecasting: By analysing the trends in various index numbers over time, economists can make forecasts about future economic activity.
Conclusion
Index numbers are powerful statistical devices that simplify complexity. They enable us to express the average change in a large number of diverse items as a single, easily understood figure. From measuring the cost of living for a household to gauging the health of the national economy, their applications are vast and varied.
While their construction involves several challenges and requires careful judgment, their utility is undeniable. By providing a standardised metric for comparison over time, index numbers serve as a vital tool for economists, policymakers, businesses, and the general public to understand, analyse, and navigate the ever-changing economic landscape.