Page 1 Index Numbers Studying this chapter should enable you to: • understand the meaning of the term index number; ? become familiar with the use of some widely used index numbers; ? calculate an index number; ? appreciate its limitations. 1. INTRODUCTION You have learnt in the previous chapters how summary measures can be obtained from a mass of data. Now you will learn how to obtain summary measures of change in a group of related variables. Rabi goes to the market after a long gap. He finds that the prices of most commodities have changed. Some items have become costlier, while others have become cheaper. On his return from the market, he tells his father about the change in price of the each and every item, he bought. It is bewildering to both. The industrial sector consists of many subsectors. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The changes are not uniform. Description of the individual rates of change will be difficult to understand. Can a single figure summarise these changes? Look at the following cases: Case 1 An industrial worker was earning a salary of Rs 1,000 in 1982. Today, he CHAPTER Page 2 Index Numbers Studying this chapter should enable you to: • understand the meaning of the term index number; ? become familiar with the use of some widely used index numbers; ? calculate an index number; ? appreciate its limitations. 1. INTRODUCTION You have learnt in the previous chapters how summary measures can be obtained from a mass of data. Now you will learn how to obtain summary measures of change in a group of related variables. Rabi goes to the market after a long gap. He finds that the prices of most commodities have changed. Some items have become costlier, while others have become cheaper. On his return from the market, he tells his father about the change in price of the each and every item, he bought. It is bewildering to both. The industrial sector consists of many subsectors. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The changes are not uniform. Description of the individual rates of change will be difficult to understand. Can a single figure summarise these changes? Look at the following cases: Case 1 An industrial worker was earning a salary of Rs 1,000 in 1982. Today, he CHAPTER 108 STATISTICS FOR ECONOMICS earns Rs 12,000. Can his standard of living be said to have risen 12 times during this period? By how much should his salary be raised so that he is as well off as before? Case 2 You must be reading about the sensex in the newspapers. The sensex crossing 8000 points is, indeed, greeted with euphoria. When, sensex dipped 600 points recently, it eroded investors’ wealth by Rs 1,53,690 crores. What exactly is sensex? Case 3 The government says inflation rate will not accelerate due to the rise in the price of petroleum products. How does one measure inflation? These are a sample of questions you confront in your daily life. A study of the index number helps in analysing these questions. 2. WHAT IS AN INDEX NUMBER An index number is a statistical device for measuring changes in the magnitude of a group of related variables. It represents the general trend of diverging ratios, from which it is calculated. It is a measure of the average change in a group of related variables over two different situations. The comparison may be between like categories such as persons, schools, hospitals etc. An index number also measures changes in the value of the variables such as prices of specified list of commodities, volume of production in different sectors of an industry, production of various agricultural crops, cost of living etc. Conventionally, index numbers are expressed in terms of percentage. Of the two periods, the period with which the comparison is to be made, is known as the base period. The value in the base period is given the index number 100. If you want to know how much the price has changed in 2005 from the level in 1990, then 1990 becomes the base. The index number of any period is in proportion with it. Thus an index number of 250 indicates that the value is two and half times that of the base period. Price index numbers measure and permit comparison of the prices of certain goods. Quantity index numbers measure the changes in the physical volume of production, construction or employment. Though price index numbers are more widely used, a production index is also an important indicator of the level of the output in the economy. Page 3 Index Numbers Studying this chapter should enable you to: • understand the meaning of the term index number; ? become familiar with the use of some widely used index numbers; ? calculate an index number; ? appreciate its limitations. 1. INTRODUCTION You have learnt in the previous chapters how summary measures can be obtained from a mass of data. Now you will learn how to obtain summary measures of change in a group of related variables. Rabi goes to the market after a long gap. He finds that the prices of most commodities have changed. Some items have become costlier, while others have become cheaper. On his return from the market, he tells his father about the change in price of the each and every item, he bought. It is bewildering to both. The industrial sector consists of many subsectors. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The changes are not uniform. Description of the individual rates of change will be difficult to understand. Can a single figure summarise these changes? Look at the following cases: Case 1 An industrial worker was earning a salary of Rs 1,000 in 1982. Today, he CHAPTER 108 STATISTICS FOR ECONOMICS earns Rs 12,000. Can his standard of living be said to have risen 12 times during this period? By how much should his salary be raised so that he is as well off as before? Case 2 You must be reading about the sensex in the newspapers. The sensex crossing 8000 points is, indeed, greeted with euphoria. When, sensex dipped 600 points recently, it eroded investors’ wealth by Rs 1,53,690 crores. What exactly is sensex? Case 3 The government says inflation rate will not accelerate due to the rise in the price of petroleum products. How does one measure inflation? These are a sample of questions you confront in your daily life. A study of the index number helps in analysing these questions. 2. WHAT IS AN INDEX NUMBER An index number is a statistical device for measuring changes in the magnitude of a group of related variables. It represents the general trend of diverging ratios, from which it is calculated. It is a measure of the average change in a group of related variables over two different situations. The comparison may be between like categories such as persons, schools, hospitals etc. An index number also measures changes in the value of the variables such as prices of specified list of commodities, volume of production in different sectors of an industry, production of various agricultural crops, cost of living etc. Conventionally, index numbers are expressed in terms of percentage. Of the two periods, the period with which the comparison is to be made, is known as the base period. The value in the base period is given the index number 100. If you want to know how much the price has changed in 2005 from the level in 1990, then 1990 becomes the base. The index number of any period is in proportion with it. Thus an index number of 250 indicates that the value is two and half times that of the base period. Price index numbers measure and permit comparison of the prices of certain goods. Quantity index numbers measure the changes in the physical volume of production, construction or employment. Though price index numbers are more widely used, a production index is also an important indicator of the level of the output in the economy. INDEX NUMBERS 109 3. CONSTRUCTION OF AN INDEX NUMBER In the following sections, the principles of constructing an index number will be illustrated through price index numbers. Let us look at the following example: Example 1 Calculation of simple aggregative price index TABLE 8.1 Commodity Base Current Percentage period period change price (Rs) price (Rs) A 2 4 100 B5 6 20 C4 5 25 D2 3 50 As you observe in this example, the percentage changes are different for every commodity. If the percentage changes were the same for all four items, a single measure would have been sufficient to describe the change. However, the percentage changes differ and reporting the percentage change for every item will be confusing. It happens when the number of commodities is large, which is common in any real market situation. A price index represents these changes by a single numerical measure. There are two methods of constructing an index number. It can be computed by the aggregative method and by the method of averaging relatives. The Aggregative Method The formula for a simple aggregative price index is P P P 01 1 0 100 =¥ S S Where P 1 and P 0 indicate the price of the commodity in the current period and base period respectively. Using the data from example 1, the simple aggregative price index is P 01 46 5 3 2 542 100 138 5 = ++ + ++ + ¥= . Here, price is said to have risen by 38.5 percent. Do you know that such an index is of limited use? The reason is that the units of measurement of prices of various commodities are not the same. It is unweighted, because the relative importance of the items has not been properly reflected. The items are treated as having equal importance or weight. But what happens in reality? In reality the items purchased differ in order of importance. Food items occupy a large proportion of our expenditure. In that case an equal rise in the price of an item with large weight and that of an item with low weight will have different implications for the overall change in the price index. The formula for a weighted aggregative price index is P Pq Pq 01 11 01 100 =¥ S S An index number becomes a weighted index when the relative Page 4 Index Numbers Studying this chapter should enable you to: • understand the meaning of the term index number; ? become familiar with the use of some widely used index numbers; ? calculate an index number; ? appreciate its limitations. 1. INTRODUCTION You have learnt in the previous chapters how summary measures can be obtained from a mass of data. Now you will learn how to obtain summary measures of change in a group of related variables. Rabi goes to the market after a long gap. He finds that the prices of most commodities have changed. Some items have become costlier, while others have become cheaper. On his return from the market, he tells his father about the change in price of the each and every item, he bought. It is bewildering to both. The industrial sector consists of many subsectors. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The changes are not uniform. Description of the individual rates of change will be difficult to understand. Can a single figure summarise these changes? Look at the following cases: Case 1 An industrial worker was earning a salary of Rs 1,000 in 1982. Today, he CHAPTER 108 STATISTICS FOR ECONOMICS earns Rs 12,000. Can his standard of living be said to have risen 12 times during this period? By how much should his salary be raised so that he is as well off as before? Case 2 You must be reading about the sensex in the newspapers. The sensex crossing 8000 points is, indeed, greeted with euphoria. When, sensex dipped 600 points recently, it eroded investors’ wealth by Rs 1,53,690 crores. What exactly is sensex? Case 3 The government says inflation rate will not accelerate due to the rise in the price of petroleum products. How does one measure inflation? These are a sample of questions you confront in your daily life. A study of the index number helps in analysing these questions. 2. WHAT IS AN INDEX NUMBER An index number is a statistical device for measuring changes in the magnitude of a group of related variables. It represents the general trend of diverging ratios, from which it is calculated. It is a measure of the average change in a group of related variables over two different situations. The comparison may be between like categories such as persons, schools, hospitals etc. An index number also measures changes in the value of the variables such as prices of specified list of commodities, volume of production in different sectors of an industry, production of various agricultural crops, cost of living etc. Conventionally, index numbers are expressed in terms of percentage. Of the two periods, the period with which the comparison is to be made, is known as the base period. The value in the base period is given the index number 100. If you want to know how much the price has changed in 2005 from the level in 1990, then 1990 becomes the base. The index number of any period is in proportion with it. Thus an index number of 250 indicates that the value is two and half times that of the base period. Price index numbers measure and permit comparison of the prices of certain goods. Quantity index numbers measure the changes in the physical volume of production, construction or employment. Though price index numbers are more widely used, a production index is also an important indicator of the level of the output in the economy. INDEX NUMBERS 109 3. CONSTRUCTION OF AN INDEX NUMBER In the following sections, the principles of constructing an index number will be illustrated through price index numbers. Let us look at the following example: Example 1 Calculation of simple aggregative price index TABLE 8.1 Commodity Base Current Percentage period period change price (Rs) price (Rs) A 2 4 100 B5 6 20 C4 5 25 D2 3 50 As you observe in this example, the percentage changes are different for every commodity. If the percentage changes were the same for all four items, a single measure would have been sufficient to describe the change. However, the percentage changes differ and reporting the percentage change for every item will be confusing. It happens when the number of commodities is large, which is common in any real market situation. A price index represents these changes by a single numerical measure. There are two methods of constructing an index number. It can be computed by the aggregative method and by the method of averaging relatives. The Aggregative Method The formula for a simple aggregative price index is P P P 01 1 0 100 =¥ S S Where P 1 and P 0 indicate the price of the commodity in the current period and base period respectively. Using the data from example 1, the simple aggregative price index is P 01 46 5 3 2 542 100 138 5 = ++ + ++ + ¥= . Here, price is said to have risen by 38.5 percent. Do you know that such an index is of limited use? The reason is that the units of measurement of prices of various commodities are not the same. It is unweighted, because the relative importance of the items has not been properly reflected. The items are treated as having equal importance or weight. But what happens in reality? In reality the items purchased differ in order of importance. Food items occupy a large proportion of our expenditure. In that case an equal rise in the price of an item with large weight and that of an item with low weight will have different implications for the overall change in the price index. The formula for a weighted aggregative price index is P Pq Pq 01 11 01 100 =¥ S S An index number becomes a weighted index when the relative 110 STATISTICS FOR ECONOMICS importance of items is taken care of. Here weights are quantity weights. To construct a weighted aggregative index, a well specified basket of commodities is taken and its worth each year is calculated. It thus measures the changing value of a fixed aggregate of goods. Since the total value changes with a fixed basket, the change is due to price change. Various methods of calculating a weighted aggregative index use different baskets with respect to time. Example 2 Calculation of weighted aggregative price index TABLE 8.2 Base period Current period Commodity Price Quantity Price Quality P 0 q 0 p 1 q 1 A21045 B512610 C420515 D215310 P Pq Pq 01 11 01 100 =¥ S S = ¥+ ¥+ ¥ + ¥ ¥+ ¥+ ¥ + ¥ ¥ 4 106 12 5 203 15 2 10 5 12 4202 15 100 =¥ = 257 190 100 135 3 . This method uses the base period quantities as weights. A weighted aggregative price index using base period quantities as weights, is also known as Laspeyre’s price index. It provides an explanation to the question that if the expenditure on base period basket of commodities was Rs 100, how much should be the expenditure in the current period on the same basket of commodities? As you can see here, the value of base period quantities has risen by 35.3 per cent due to price rise. Using base period quantities as weights, the price is said to have risen by 35.3 percent. Since the current period quantities differ from the base period quantities, the index number using current period weights gives a different value of the index number. P Pq Pq 01 11 01 100 =¥ S S = ¥ + ¥+ ¥+ ¥ ¥ + ¥+ ¥+ ¥ ¥ 45 6 105 15 3 10 2 5 510 4 15 215 100 =¥ = 185 140 100 132 1 . It uses the current period quantities as weights. A weighted aggregative price index using current period quantities as weights is known as Paasche’s price index. It helps in answering the question that, if the Page 5 Index Numbers Studying this chapter should enable you to: • understand the meaning of the term index number; ? become familiar with the use of some widely used index numbers; ? calculate an index number; ? appreciate its limitations. 1. INTRODUCTION You have learnt in the previous chapters how summary measures can be obtained from a mass of data. Now you will learn how to obtain summary measures of change in a group of related variables. Rabi goes to the market after a long gap. He finds that the prices of most commodities have changed. Some items have become costlier, while others have become cheaper. On his return from the market, he tells his father about the change in price of the each and every item, he bought. It is bewildering to both. The industrial sector consists of many subsectors. Each of them is changing. The output of some subsectors are rising, while it is falling in some subsectors. The changes are not uniform. Description of the individual rates of change will be difficult to understand. Can a single figure summarise these changes? Look at the following cases: Case 1 An industrial worker was earning a salary of Rs 1,000 in 1982. Today, he CHAPTER 108 STATISTICS FOR ECONOMICS earns Rs 12,000. Can his standard of living be said to have risen 12 times during this period? By how much should his salary be raised so that he is as well off as before? Case 2 You must be reading about the sensex in the newspapers. The sensex crossing 8000 points is, indeed, greeted with euphoria. When, sensex dipped 600 points recently, it eroded investors’ wealth by Rs 1,53,690 crores. What exactly is sensex? Case 3 The government says inflation rate will not accelerate due to the rise in the price of petroleum products. How does one measure inflation? These are a sample of questions you confront in your daily life. A study of the index number helps in analysing these questions. 2. WHAT IS AN INDEX NUMBER An index number is a statistical device for measuring changes in the magnitude of a group of related variables. It represents the general trend of diverging ratios, from which it is calculated. It is a measure of the average change in a group of related variables over two different situations. The comparison may be between like categories such as persons, schools, hospitals etc. An index number also measures changes in the value of the variables such as prices of specified list of commodities, volume of production in different sectors of an industry, production of various agricultural crops, cost of living etc. Conventionally, index numbers are expressed in terms of percentage. Of the two periods, the period with which the comparison is to be made, is known as the base period. The value in the base period is given the index number 100. If you want to know how much the price has changed in 2005 from the level in 1990, then 1990 becomes the base. The index number of any period is in proportion with it. Thus an index number of 250 indicates that the value is two and half times that of the base period. Price index numbers measure and permit comparison of the prices of certain goods. Quantity index numbers measure the changes in the physical volume of production, construction or employment. Though price index numbers are more widely used, a production index is also an important indicator of the level of the output in the economy. INDEX NUMBERS 109 3. CONSTRUCTION OF AN INDEX NUMBER In the following sections, the principles of constructing an index number will be illustrated through price index numbers. Let us look at the following example: Example 1 Calculation of simple aggregative price index TABLE 8.1 Commodity Base Current Percentage period period change price (Rs) price (Rs) A 2 4 100 B5 6 20 C4 5 25 D2 3 50 As you observe in this example, the percentage changes are different for every commodity. If the percentage changes were the same for all four items, a single measure would have been sufficient to describe the change. However, the percentage changes differ and reporting the percentage change for every item will be confusing. It happens when the number of commodities is large, which is common in any real market situation. A price index represents these changes by a single numerical measure. There are two methods of constructing an index number. It can be computed by the aggregative method and by the method of averaging relatives. The Aggregative Method The formula for a simple aggregative price index is P P P 01 1 0 100 =¥ S S Where P 1 and P 0 indicate the price of the commodity in the current period and base period respectively. Using the data from example 1, the simple aggregative price index is P 01 46 5 3 2 542 100 138 5 = ++ + ++ + ¥= . Here, price is said to have risen by 38.5 percent. Do you know that such an index is of limited use? The reason is that the units of measurement of prices of various commodities are not the same. It is unweighted, because the relative importance of the items has not been properly reflected. The items are treated as having equal importance or weight. But what happens in reality? In reality the items purchased differ in order of importance. Food items occupy a large proportion of our expenditure. In that case an equal rise in the price of an item with large weight and that of an item with low weight will have different implications for the overall change in the price index. The formula for a weighted aggregative price index is P Pq Pq 01 11 01 100 =¥ S S An index number becomes a weighted index when the relative 110 STATISTICS FOR ECONOMICS importance of items is taken care of. Here weights are quantity weights. To construct a weighted aggregative index, a well specified basket of commodities is taken and its worth each year is calculated. It thus measures the changing value of a fixed aggregate of goods. Since the total value changes with a fixed basket, the change is due to price change. Various methods of calculating a weighted aggregative index use different baskets with respect to time. Example 2 Calculation of weighted aggregative price index TABLE 8.2 Base period Current period Commodity Price Quantity Price Quality P 0 q 0 p 1 q 1 A21045 B512610 C420515 D215310 P Pq Pq 01 11 01 100 =¥ S S = ¥+ ¥+ ¥ + ¥ ¥+ ¥+ ¥ + ¥ ¥ 4 106 12 5 203 15 2 10 5 12 4202 15 100 =¥ = 257 190 100 135 3 . This method uses the base period quantities as weights. A weighted aggregative price index using base period quantities as weights, is also known as Laspeyre’s price index. It provides an explanation to the question that if the expenditure on base period basket of commodities was Rs 100, how much should be the expenditure in the current period on the same basket of commodities? As you can see here, the value of base period quantities has risen by 35.3 per cent due to price rise. Using base period quantities as weights, the price is said to have risen by 35.3 percent. Since the current period quantities differ from the base period quantities, the index number using current period weights gives a different value of the index number. P Pq Pq 01 11 01 100 =¥ S S = ¥ + ¥+ ¥+ ¥ ¥ + ¥+ ¥+ ¥ ¥ 45 6 105 15 3 10 2 5 510 4 15 215 100 =¥ = 185 140 100 132 1 . It uses the current period quantities as weights. A weighted aggregative price index using current period quantities as weights is known as Paasche’s price index. It helps in answering the question that, if the INDEX NUMBERS 111 the current period basket of commodities was consumed in the base period and if we were spending Rs 100 on it, how much should be the expenditure in current period on the same basket of commodities. A Paasche’s price index of 132.1 is interpreted as a price rise of 32.1 percent. Using current period weights, the price is said to have risen by 32.1 per cent. Method of Averaging relatives When there is only one commodity, the price index is the ratio of the price of the commodity in the current period to that in the base period, usually expressed in percentage terms. The method of averaging relatives takes the average of these relatives when there are many commodities. The price index number using price relatives is defined as P n p p 01 1 0 1 100 =¥ S where P 1 and P o indicate the price of the ith commodity in the current period and base period respectively. The ratio (P 1 /P 0 ) × 100 is also referred to as price relative of the commodity. n stands for the number of commodities. In the current example P 01 1 4 4 2 6 5 5 4 3 2 100 149 =+++ Ê Ë Á ˆ ¯ ˜ ¥= Thus the prices of the commodities have risen by 49 percent. The weighted index of price relatives is the weighted arithmetic mean of price relatives defined as P W P P W 01 1 0 100 = ¥ Ê Ë Á ˆ ¯ ˜ S S where W = Weight. In a weighted price relative index weights may be determined by the proportion or percentage of expenditure on them in total expenditure during the base period. It can also refer to the current period depending on the formula used. These are, essentially, the value shares of different commodities in the total expenditure. In general the base period weight is preferred to the current period weight. It is because calculating the weight every year is inconvenient. It also refers to the changing values of different baskets. They are strictly not comparable. Example 3 shows the type of information one needs for calculating weighted price index. Example 3 Calculation of weighted price relatives index TABLE 8.3 Commodity Base Current Price Weight year year price relative in % price (in Rs) (in Rs.) A 2 4 200 40 B 5 6 120 30 C 4 5 125 20 D 2 3 150 10Read More

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### Index Numbers (Part - 2)

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### Methods of Construction

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### Weighted Index Numbers

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