Index Numbers with Formula and Examples

Meaning and Definitions of index number

An index number is a statistical tool used to measure relative changes in a variable or group of variables in relation to time, geographical location, or other factors. It is a subset of a rate, ratio, or percentage that expresses the overall magnitude of a set of distinct but related variables in two or more situations.

Index numbers are a type of average that is used to measure relative changes in the level of a phenomenon when measuring absolute change is impossible, and the series is expressed in various types of items. It is used to compare the levels of a phenomenon over different time periods or at different locations during the same time period. Furthermore, it is used to study changes in the effects of factors that cannot be measured directly, such as general price level, which cannot be measured directly but can be measured using the index number technique.

According to Irving Fisher, an index number is “a number which measures the relative changes in some quantity or quality over time, or in different places, compared with a fixed standard.”

According to Bowley, “an index number is a simple arithmetical device for comparing the value of a group of related variables at different times or places.”

According to Laspeyres, “an index number is a number that shows how many times a certain quantity, such as price or quantity of a commodity, has changed over a given period of time in comparison with its level at some previous period, called the base period.”

Features of index number

The main highlighting features of index numbers are mentioned below–

• It is a type of average that is used to measure relative changes when absolute measurements are impossible.
• The index number only shows the first changes in things that can’t be measured directly.
• It provides an overview of the relative changes.
• The method of calculating index numbers varies from variable to variable.
• It aids in comparing the levels of a phenomenon on a specific date to those on a previous date.
• It represents a special case of averages, specifically a weighted average. Index numbers are useful in all situations.
• The same index used to determine how prices change can also determine how much is made in industry and agriculture.

Importance of index number

Index numbers are used to compare the levels of a phenomenon over different time periods or at different locations within the same time period. The following sections summarize the significance or applications of index numbers.

1. Index numbers as barometers of economic activities: Index numbers, like barometers used in physics and chemistry to measure atmospheric pressure, are used to measure the pressure of economic and business behavior. As a result, index numbers are referred to as barometers of economic activity.
2. Index numbers help study trends and tendencies of economic activities: Index numbers track changes over time, allowing us to study the overall trend and tendency of the economic activity under consideration. The index number can also be used for forecasting to measure the average change in a specific group.
3. Index numbers measure money’s purchasing power or value: A price index number can be used to calculate money’s purchasing power or value. The price index can also be used to calculate the real wage rate. Divide the money wages by the corresponding price index and multiply by 100 to get the real wage.
4. Index numbers are used for deflation: The net national product or income converted at current prices is deflated using index numbers. The deflated net national product or income represents the net national product or income at constant prices under inflationary conditions. We can also calculate real wages or income using a deflated index number.
5. Index numbers help in formulating decisions and policies: The government uses index numbers to adjust wage policies, price policies, rent control policies, and taxation policies. Index numbers guide many other economic policies, including trade volume, wholesale price setting, retail price setting, and so on.

Types of index number

Index numbers are of various types. Generally, they are categorized into the following three types.

1. Price index number: With regard to price, an index number of prices or price index is created. It is used to represent changes in the prices of a group of commodities over time. The price could be wholesale/producer or retail/consumer. There are two types of index numbers. They are:

• Wholesale price index number: The index number developed based on wholesale price is called the wholesale price index number. It is also called the producer’s index number.
• Retail price index number: If the price index number is developed based on retail price, then it is called a retail price index number. It is also called the consumer’s price index number or cost of living index number.
• Quantity index number: In relation to output, a quantity index number is calculated. It is used to track changes in the physical volume or quantity of goods (produced/ manufactured, distributed, or consumed) across various sectors of an economy over time. Quantity indexes include the index of agricultural production, the index of industrial production, the index of goods imported, and the index of goods exported.
• Value index number: A value index number is created to study the variation in the total value of a commodity or group of commodities over time. The total value, TV = P x Q, is found by multiplying the number of items by their price. Value indices include the index of retail sales, the index of export, and the index of imports.

In this article, we are concerned with price index numbers only.

Usable terminology in constructing index number

The following are some notations and terminologies used in index number (price index number):

P_o: Price of the commodities in the base year/ period.

P_i: Price of the commodities in the current year/ period.

q_o: Quantity of a commodity consumed or purchased during the base year/ period.

q₁: Quantity of a commodity consumed or purchased during the current year/ period.

P₀₁: Price index number for the current year/ period with respect to the base year/ period.

P₁₀: Price index number of base year/ period with respect to current year/ period.

W: Weight assigned to the commodities according to their relative importance in the group.

Base period: The year with respect to which comparisons are made. It is denoted by the suffix ‘0’. It is also called the reference period.

Current period: The year for which comparisons are required is called a current period. It is denoted by the suffix ‘1.’

Steps in the construction of index number

The steps for the construction of an index number can be explained below:-

1. Purpose or objective of index number: Before constructing the index number, it is critical to understand its purpose. All other steps or problems are viewed through the lens of the purpose for which the specific index number is to be created. The type of index number to be created is determined by its purpose or objective.
2. Selection of base year: The year against which the price changes in subsequent years are compared and expressed as a percentage is referred to as the base year. A normal year should be the base year (free from abnormal conditions like war, famine, flood, etc). The base can be either fixed or chained.
3. Selection of commodities: The commodities chosen are determined by the index number’s purpose and type. When constructing the index number, only representative commodities are chosen.
4. Collecting prices: Prices should be collected from a large market and from sources that are unbiased and open to the public. The choice of wholesale or retail prices is determined by the type of price index number to be created. The collected prices should be averaged.
5. Selection of average: There are several types of averages, but the choice is usually between an arithmetic mean and a geometric mean. Although the arithmetic mean is easier to understand, the geometric mean is the best measure for measuring relative changes.
6. Selection of weights: The commodities used to calculate the price index number are not all equally important. As a result, appropriate weights should be assigned to various commodities based on their relative importance in the group. Weights can be assigned to production, consumption, or distribution figures. However, the most commonly used weighting systems are quantity and value weights. Various commodities are assigned importance in quantity weights based on the amount of their quantity used, purchased, or consumed. In Value Weights, the importance of various items is assigned based on the amount of money spent on them. The weighting system can also be base period weights or current period weights. The index number’s purpose and data availability influence the selection of different weighting systems.
7. Selection of method: The goal of the study, the level of accuracy, and the availability of data all influence the process of creating a price index number. There are primarily two methods for calculating price index numbers:

(a) The simple or unweighted price index number: A simple price index number can be calculated using one of two methods:

(i) the simple aggregative method or

(ii) the simple average of price relative method.

(b) the weighted price index number:  Weighted price index numbers can be calculated using following methods:

(i) the weighted aggregative method or

(ii) the weighted average of price relative method.

These steps in the index number construction process can sometimes result in index number problems. We can’t finish all the steps needed to make the index number. As a result, these steps are also known as construction index number problems.

Limitations of Index Numbers

Index numbers are very important tools for studying economic and business activities. However, there are some limitations which are explained below:

1. Errors in sampling: Because index numbers are based on sample data, any errors during the sampling procedure creep into the index number construction.
2. Formula Error: The selection of a formula is a problem because it may introduce bias in the index number.
3. Changes: Changes in outlook, taste, and the quality of materials make it hard to make precise changes to index construction as science moves forward quickly. As a result, index numbers may not be a true measure of change.
4. Choice of the average types: Because different averages have different advantages and disadvantages, it is impossible to say that one average is completely suitable for an index number. However, arithmetic and geometric means are commonly used.
5. Errors in data collection: If data on prices, consumption, and production is not collected accurately, the index numbers will undoubtedly be misleading.
6. Limited use: Most index numbers are created with specific goals in mind. If they are used for something else, they may lead to incorrect conclusions. Index numbers prepared to learn about the economic condition of teachers, for example, cannot be used to learn about the economic condition of workers.

Types of Price index Numbers

Various price index numbers are used to track changes in the prices of goods and services over time. The following are some of the most common types:

1. Consumer Price Index (CPI): The Consumer Price Index (CPI) tracks how prices for goods and services bought by households change over time. It is often used to measure inflation. It is calculated by comparing the cost of a basket of goods and services in one time period to the cost of the same basket in another time period.
2. Producer Price Index (PPI): PPI measures how the prices that domestic producers get for their goods change on average. It is often used as an indicator of inflation in the production sector.
3. Wholesale Price Index (WPI): WPI measures the average change in prices of goods traded in wholesale markets. It is often used as an indicator of inflation in the wholesale sector.
4. Export Price Index (XPI): The XPI measures the change in the prices of goods and services sold to foreign customers. It is frequently used as a measure of international price competitiveness.
5. Import Price Index (MPI): The MPI measures how prices of goods and services bought from foreign suppliers change over time. It is often used to show how prices at home are affected by trade with other countries.
6. GDP Deflator: The GDP deflator figures out how the prices of all goods and services produced in an economy have changed on average. It is frequently used to account for inflation in nominal GDP.

Methods of CPI (Consumer Price Index)

The Consumer Price Index (CPI) measures the average change in consumer prices for a basket of goods and services over time. CPI is calculated using several methods, including:

1. Fixed-weight method: The index is calculated using a fixed basket of goods and services. The basket is chosen to represent typical consumer purchases, and the prices of the basket’s goods and services are collected at regular intervals. Because the fixed-weight method assumes that consumers buy the same basket of goods and services over time, it does not account for changes in consumer behavior or market shifts.
2. Chain-weight method: This method uses a flexible basket of goods and services that changes over time to reflect changes in consumer behavior and shifts in the market. People think the chain-weight method is more accurate than the fixed-weight method because it changes based on how consumers act.
3. Laspeyres method: This method uses a fixed basket of goods and services, but it allows for changes in the quantities purchased of each item. Most of the time, the Laspeyres method is used for goods and services whose demand stays steady over time.
4. Paasche method: This method uses a basket of goods and services that changes over time, allowing for changes in the quantities purchased of each item. The Paasche method is commonly used for goods and services with volatile demand over time.
5. Fisher’s Ideal method: This method employs a changing basket of goods and services to account for changes in both the quantity and quality of goods and services. Fisher’s Ideal method is the most accurate method of calculating CPI but is also the most complicated and time-consuming.

Popular Methods of Calculating the Price index with formula

Laspeyres and Paasche are two commonly used methods of calculating price indices.

• Laspeyre’s Price index Method: The Laspeyres index, also known as the base-weighted index, is a calculation method that uses the base period quantities as weights. This method assumes that consumers purchase the same amount of goods and services over time, regardless of price changes. The Laspeyres price index is calculated as follows:

Laspeyres Price Index = (Current Period Prices x Base Period Quantities) / (Base Period Prices x Base Period Quantities) x 100

Where:

• Current Period Prices: the prices of goods and services in the current period
• Base Period Quantities: the quantities of goods and services purchased in the base period
• Base Period Prices: the prices of goods and services in the base period

For example, if, in the base period, a consumer purchases 10 apples at Rs. 1 each and 5 oranges at Rs. 2 each, the weights for the Laspeyres index would be 10 for apples and 5 for oranges. If, in the current period, apples cost Rs. 2 each and oranges cost Rs. 3 each, the Laspeyres index would be:

Laspeyres Price Index = [(2 x 10) + (3 x 5)] / [(1 x 10) + (2 x 5)] x 100

= (20 + 15) / (10 + 10) x 100

= 135

• Paasche’s Price index Method: The Paasche index, also known as the current-weighted index, is a price index calculation method that uses current period quantities as weights. This method assumes that as prices change, consumers adjust their consumption patterns. The Paasche price index is calculated as follows:

Paasche Price Index = (Current Period Prices x Current Period Quantities) / (Base Period Prices x Current Period Quantities) x 100

Where:

• Current Period Prices: the prices of goods and services in the current period
• Current Period Quantities: the quantities of goods and services purchased in the current period
• Base Period Prices: the prices of goods and services in the base period

For example, if, in the base period, a consumer purchases 10 apples at Rs. 1 each and 5 oranges at Rs. 2 each, the weights for the Paasche index would be 10 for apples and 5 for oranges. If, in the current period, the consumer purchases 8 apples at Rs. 2 each and 7 oranges at Rs. 3 each, the Paasche index would be:

Paasche Price Index = [(2 x 8) + (3 x 7)] / [(1 x 8) + (2 x 7)] x 100

= (16 + 21) / (8 + 14) x 100

= 135.71

Both the Laspeyres and Paasche indices have advantages and disadvantages. The Laspeyres index tends to overestimate the cost of living increase, whereas the Paasche index tends to underestimate it. As a result, the Fisher index, which is the geometric mean of the Laspeyres and Paasche indices, is a widely used method.

• Fisher’s Price Index Method: Fisher’s index is a method of calculating the price index that attempts to overcome the limitations of both Laspeyres’ and Paasche’s indices by taking the geometric mean of the two. The formula for Fisher’s index is:

Fisher’s index = (√(Paasche index * Laspeyres index))

Whereas the Paasche index uses current quantities and current prices, the Laspeyres index uses base period quantities and current prices. Because it takes the geometric mean of the two indices, Fisher’s index is considered more accurate and less biased than the other two indices.

The formula considers both quantity and price changes, and it is less affected by substitution bias or changes in the composition of the basket of goods over time. Fisher’s index is a well-known way to determine inflation rates and compare how much things cost over time. It also calculates the Consumer Price Index (CPI) and other economic indicators.

Weighted price index numbers

Weighted price index numbers are used to calculate the average prices of a group of items in a way that considers each item’s importance or weight. The formula for calculating a weighted price index number is:

W_I = (Σ(P_i * W_i) / Σ(W_i)) * 100

Where,

W_I is the weighted index number,

P_i is the price of the ith item,

W_i is the weight of the ith item, and

Σ represents the sum of all items in the group.

For example, we want to calculate a weighted price index number for a basket of goods consisting of apples, bananas, and oranges. The prices and weights of the items are as follows:

Item	Price	Weight
Apples	Rs. 2.00	0.4
Bananas	Rs. 1.50	0.3
Oranges	Rs. 2.50	0.3

Using the formula above, we can calculate the weighted price index number as follows:

W_I = ((2.00 * 0.4) + (1.50 * 0.3) + (2.50 * 0.3)) / (0.4 + 0.3 + 0.3) * 100

W_I = (0.8 + 0.45 + 0.75) / 1.0 * 100

W_I = 200

So the weighted price index number for this basket of goods is 200. This means that the overall price of the basket had increased by 100% from the base period (when the index was 100).

Weighted Price Index Using Laspeyres’ Method:

P₀₁ = (∑(P_iq_o W) / ∑(P_oq_o W)) x 100

Where,

P₀₁ = Price index number for the current year/ period with respect to the base year/ period

P_i = Price of the commodity i in the current year/ period

q_o = Quantity of the commodity i consumed or purchased during the base year/ period

W = Weight assigned to the commodity i according to their relative importance in the group

P_o = Price of the commodity i in the base year/ period

Example of Laspeyres’ method:

Commodities	Base year price (Po)	Base year quantity (qo)	Current year price (Pi)	Current year quantity (q1)	Weight (W)	Po x qo	Pi x q0
Rice	20	10	24	12	2	200	288
Wheat	10	20	14	22	3	200	308
Milk	15	15	18	20	1	225	360
Total (∑)	–	–	–	–	6	625	956

Using the formula:

P₀₁ = (∑(P_iq_o W) / ∑(P_oq_o W)) x 100

P₀₁ = (288 + 308 + 360) / (200 + 200 + 225) x 100

P₀₁ = 956 / 625 x 100

P₀₁ = 153.00

Therefore, the weighted price index number using Laspeyres’ method is 153.00.

Weighted Price Index Using Paasche’s Method:

P₁₀ = (∑(P_i q₁ W) / ∑(P_o q₁ W)) x 100

Where,

P₀₁ = Price index number for the current year/ period with respect to the base year/ period

P_i = Price of the commodity i in the current year/ period

q_o = Quantity of the commodity i consumed or purchased during the base year/ period

W = Weight assigned to the commodity i according to their relative importance in the group

P_o = Price of the commodity i in the base year/ period

Example of Paasche’s method:

Commodities	Base year price (Po)	Base year quantity (qo)	Current year price (Pi)	Current year quantity (q1)	Weight (W)	Po x q1	Pi x q1
Rice	20	10	24	12	2	240	288
Wheat	10	20	14	22	3	440	308
Milk	15	15	18	20	1	300	360
Total	–	–	–	–	6	980	956

Using the formula:

P₁₀ = (∑(P_i q₁ W) / ∑(P_o q₁ W)) x 100

P₁₀ = (240 + 440 + 300) / (288 + 308 + 360) x 100

P₁₀ = 980 / 956 x 100

P₁₀ = 102.51

Therefore, the weighted price index number using Paasche’s method is 102.51.

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