Understanding Index Numbers: Definition, Characteristics, Types, and Applications, Study notes of Statistics

Download Understanding Index Numbers: Definition, Characteristics, Types, and Applications and more Study notes Statistics in PDF only on Docsity! Index Numbers Dr. ViVek Tyagi STaTiSTicS DeparTmenT n.a.S. college, meeruT Index Numbers • What is Index Number ? • Definitions used for all Index Numbers • Why Convert Data to indices ? • Types of Index Number • Some Important Price indices Characteristics Of Index Numbers • Index numbers are specialised averages. • Index numbers measure the change in the level of a phenomenon. • Index numbers measure the effect of changes over a period of time. • Index numbers are Economic Barometers. • Index numbers are sign and guide posts of Business. • Index number are the ratio of the current value to a base value. Terms used for all Index Numbers  Current period :- The period for which you wish to find the Index Number.  Base period :- The period with which you wish to compare prices of the current period.  Price :- Price of the commodity or items you want to compare.  Quantity :- Quantity of the commodity or items you want to compare. • An index is a convenient way to express a change in a diverse group of items. • Converting data to indices also makes it easier to assess the trend in a series composed of exceptionally large numbers. my ree Index Number Construction Aggregative Method Relative Method [-—— Simple Average Weighted Average of Relatives of Relatives Simple Aggregative Weighted Aggregative Formula Formula | | | | | Laspeyres’ Paasche’s Edgeworth- Fisher's Formula Formula Marshall's “Ideal” Formula Formula Price Index  Price Relative :- The price relative of an item is defined as: Where: pt = price in current period po = price in base period  Price Relative Index provides a ratio that indicates the change in price of an item from one period to another. o t p p RelativePrice Simple Price Index  Simple Price Index is a common method of expressing this change as a percentage: Where: pt = price in current period po = price in base period 100 Index Price Simple  o t p p WHAT IS THE INCREASE EACH YEAR? We could choose Year 1 as the base year. Year Price Calculation Index 1 2.00 (2.00 * 100/ 2) 100 2 2.20 (2.20 * 100/ 2) 110 3 2.40 (2.40 * 100/ 2) 120 4 2.90 (2.90 * 100/ 2) 145 Unweighted indices Simple Aggregate Price Index :- In most cases we are interested in the prices of a “basket of goods”, and not just one item. We therefore need an aggregate index. 1. Add up column of prices 2. Use Item Price Yr0 Price Yr1 Price Yr2 A 1.00 1.10 1.15 B 2.00 2.30 2.35 C 5.00 5.60 5.70 8.00 9.00 9.20 100 112.5 115 100   o t P P 100 8 9  Interpretation • A price index of 113 would indicate an increase of 13% relative to the base year. • A price index of 75 would indicate a decrease of 25% relative to the base year. Average of Relative Prices Where: k = number of items pt = price in current period po = price in base period k p p k o t          100 indexes price simple theof sum Prices Relative of Average Example 1. Find PR’s for each item. 2. Add up columns. 3. Find the average Item Price Yr0 Price Yr1 Price Yr2 A 1.00 1.10 1.15 B 2.00 2.30 2.35 C 5.00 5.60 5.70 PR Yr0 PR Yr1 PR Yr2 100 110 115 100 115 117.5 100 112 114 300 337 346.5 100 112.33 115.5 The indices we have discussed either dealt with a single item or assumed that all items are of equal importance. This is obviously not true! We need an index which can deal with a “basket of goods” and take account of the relative importance of the items in the basket. Where: qo = quantity bought in base period pt = price in current period po = price in base period 100 Index sLaspeyres’ 00 0     qp qpt Laspeyres’s Index 1. Multiply each price by the quantity in Year 0. 2. Add up each column. 3. Use the given formula Item Quant Yr0 Price Yr0 Price Yr1 Price Yr2 A 50 1.00 1.10 1.15 B 20 2.00 2.30 2.35 C 5 5.00 5.60 5.70 PoQo P1Qo P2Qo 50 55 57.5 40 46 47 25 28 28.5 115 129 133 100 112.2 115.7     100 00 0    QP QPt Disadvantage with Laspeyres • Laspeyres’s Index assumes that the same amount of each item is bought every year. • If I bought 35 kg of oranges in base year, the index assumes I bought the same amount every year, when in reality if the price went up, one might buy less. Paasche’s Index • Note the structure of this. • We need to find Sum of P1Q1 and Sum of P0Q1 • Use Formula Item Quant Yr0 Quant Yr1 Quant Yr2 Price Yr0 Price Yr1 Price Yr2 A 50 55 60 1.00 1.10 1.15 B 20 21 23 2.00 2.30 2.35 C 5 5 4 5.00 5.60 5.70 P1Q1 PoQ1 60.5 55 48.3 42 28 25 136.8 122 P2Q2 PoQ2 69 60 54.05 46 22.8 20 145.85 126     100 0    t tt QP QP 100 122 8.136 100 126 85.145 112.1311 115.754 Laspeyres versus Paasche Index Lasperyres’s Index  The Laspeyres Index measures the ratio of expenditures on base year quantities in the current year to expenditures on those quantities in the base year.  The Laspeyres’s Index is usually larger than the Paasche’s Index. Paasche’s Index  The Paasche index measures the ratio of expenditures on current year quantities in the current year to expenditures on those quantities in the base year.  With the Paasche index it is difficult to make year-to-year comparisons since every year a new set of weights is used. Laspeyres versus Paasche Index Lasperyres’s Index  Since the Laspeyres index uses base period weights, it may overestimate the rise in the cost of living, because people may have reduced their consumption of items that have become costly.  Laspeyres’s Index tends to overweight goods whose prices have increased. Paasche’s Index  Since the Paasche index uses current period weights, it may underestimate the rise in the cost of living, because people may have increased their consumption of items.  Paasche’s Index, on the other hand, tends to overweight goods whose prices have gone down. Quantity indices • An index that measures changes in quantity levels over time is called a quantity index. • Probably the best known quantity index is the Index of Industrial Production. • A weighted aggregate quantity index is computed in much the same way as a weighted aggregate price index. • A weighted aggregate quantity index for period t is given by 100_ 0    t tt wQ wQ IndexQuantity Value Index • A value index measures changes in both the price and quantities involved. • A value index, such as the index of department store sales, needs the original base-year prices, the original base-year quantities, the present-year prices, and the present year quantities for its construction. • Its formula is given as: 100 year base the in value Total year current the in value Total Index Value Where: qt = quantity bought in current period qo = quantity bought in base period pt = price in current period po = price in base period 100 0     qp qp Index Value 0 tt Consumer Price Index (CPI) is given by: 100 CPI 0     wp wp t CPI Uses - Formulas 100 CPI 1 Money OfPower Purchasing  100 CPI IncomeMoney Income Real Price indices: Other Considerations • Selection of Items  When the class of items is very large, a representative group (usually not a random sample) must be used.  The group of items in the aggregate index must be periodically reviewed and revised if it is not representative of the class of items in mind. • Selection of a Base Period  As a rule, the base period should not be too far from the current period.  The base period for most indices is adjusted periodically to a more recent period of time.