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Topic-15

Topic-15. Index Numbers. Business Index: An Important Statistical Tool

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Topic-15

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  1. Topic-15 Index Numbers

  2. Business Index: An Important Statistical Tool Business Index: Many business index numbers are published, reviewed, and studied on a daily basis. Those index numbers, as another important statistical tools, provide a wide range of information to the society (as a whole) and all business organizations. Classical examples include: - Consumer Price Index; - Dow Jones Industrial Average; - unemployment Rate index; - ..................., * An Index Number: a statistic number (e.g., a ratio) that measures the overall trend or a relative change in an economic indicator (such as: average product price, production quantity, stock value, etc.) from one time period to another. * Why Convert Data into Indexes? - A convenient way of expressing a change in a heterogeneous group of items (e.g., consumer price index). - An easy approach to evaluate the trend in a series of composed of exceptionally large numbers (retail sales index).

  3. Simple Index Numbers- Measuring One Variable * Simple Index: when an index number is designed to measuring the relative change (a trend) in one variable only - it is called “Simple Index Numbers”. - A simple index number is essentially a ratio of two values of the variable (of interest) expressed as a percentage, such as: (1) Average Hourly Wage in Manufacturing Index: In 1990, $10.50, in 1997, it is increased to $15.80: AHWM = (15.8/10.5)x100 = 150.5% (2) Median Home Sales Price Index: In 1994, $150,000, in 1998, it is increased to $180,000: MHSP = (180,000/150,000)x100 = 120% (Examples) * Base Period: each index number needs a “base period” to be compared with (i.e., setting the value of base period as 100%, or other values such as: 50, 100 ). Different indexes have different base period, and those base periods may be updated upon well agreed among corresponding business or social organizations.

  4. Major Business Index Types * There are 4 major types (Price, Quantity, Value, and Special-Purpose) of Business Index: 1. Price Index: measuring the changes in prices for a selected period over the based period. Typical examples include: - Consumer Price Index: measures the trend in retail prices. - Producer Price Index: measures the trend in the price of “primary market” (industrial goods & non-retail market). 2. Quantity Index: measuring the changes in quantity consumed (or purchased, produced, etc.) for a selected period over the based period. Typical examples include: - Federal Reserve Board Indexes of Quantity Output: measures the trend in overall manufactured goods. - Annual Automobiles Production Index 3. Value Index: measuring the changes in the value of one or more items from base period to the given period. Examples include: - National Football Superbowl Advertising Value Index. - Annual Construction Value Index

  5. 4. Special-Purpose Indexes: are designed to reflect the overall economic activity over certain periods of time (by combining and weighing a very dislike group of items, such as: prices, inflation, unemployment rate, production level, bank-debit ratio, education performance, health care cost, etc.) - often referred as “leading economic indicators” by different governmental agencies and business organizations. Typical examples include: - Federal Government Leading Economic Indicators: include indexes about stock price, orders for durable goods, building permits issued, etc. - Forbes Economic Index: measures the manufacturing production, department store sales, average inventory turnover ratios, etc. - International Trade balance Index: measures the trend of international trade among different regions/nations with their changes in the trade balances (surplus/deficit) over the given period (published by International Trade Organization).

  6. How to Construct Index Numbers? 1. Simple Price Index (P): P = (Pt/Po)•100 where Po - price in base period, and Pt - price in selected period. Note: Two or more periods may be used as the base period. (Examples) The index computed by above simple method, however, may be misleading - due to the different importance of items to be averaged or different units of measurement used in the computation. Thus, in practice, index numbers are more often calculated through weighted-average method. Two common weighing methods for weighted indexes are: 2. “Laspeyres Weighted Price Index”: use base-period weights: P = (Sptqo/Spoqo)•100 where (qo) is the quantity consumed in the base period (i.e., used as the weights). “Laspeyres Weighted Quantity Index”: use base-period weights: Q = (Spo qt/Spoqo)•100 Where (qt) is the quantity consumed in the current period (and base period price (po) is used as the weights).

  7. 3. “Paasche Weighted Price Index”: use the present-period weights: P = (Sptqt/Spoqt)•100 Where (qt) is the quantity consumed in the current period (i.e., used as the weights). (Examples) The advantage of “Paasche” method (compared with Laspeyres method) is that the “consumption pattern” is always updated to current, but the huge disadvantage is that the consumption patterns for all items related are now needed to be collected every year to updated all indexes. So, the “Lasperes” method is more practical and commonly used. 4. Value Index: considering the changes in both price and quantity together from the base period to the given period: V = (Sptqt/Spoqo)•100 (Example) 5. Special-Purpose Index: as general economic indicators, all special -purposeindexes have their own methods of calculation and need a great amount of related data, and some kind of subjective judgements in determining various weights. (Examples)

  8. Consumer Price Index (CPI) * CPI is designed to measure the price changes of a selected “basket” of goods & services from base period to given period. * Due to different consumption patterns, there are two CPIs (published by Labor Dept.), one is for all urban consumers (80% population) & another is urban wage earners (32% population). * The above Governmental CPIs are determined through: - Over 400 items are carefully selected (based on the feedbacks of customer survey), - Their prices are examined monthly from 21,000 retail stores, 60,000 housing units in 91 urban areas, and - Published since 1921 with a frequently updated base period (current base: 1982-84). * There is a group of CPIs developed for different regions/cities, markets/categories of goods, for different purposes. * CPIs have been used in many social and business applications: - To determine “real income” (after considering inflation), - To determine the “deflating sales” (considering cost changes), - To determine “actual” purchasing power of dollars, and - To calculate the “cost-of-living” adjustments.

  9. Using CPI as A Yardstick 1. Determine “Real Income”: after considering inflation: Also called as “deflated income”(or “constant dollars”), then the CPI is called as “deflator”. 2. Determine “Deflating Sales”: to show the trend of “real” sales after considering price changes (Sales-in-Constant-Dollars): PPI (Producer Price Index) is often used in above computation.

  10. 3. Determine “actual” Purchasing Power of Dollar: another way to measure the impact of price changes to consumers’ income. 4. Calculate Cost-of-Living Adjustments: the CPIs have been used in determining: wage increase (for unionized workers), social security payment rise (for retirees), welfare, alimony, pensionbenefits, and child support payment adjustment, and other living expense related items.

  11. Summary * Shifting the Base Period : when two or more series of index numbers to be compared but do not have the base period, those indexes cannot be compared directly. Thus, their “original” base periods then must be shifted to a common period which will be used as the new “base” period: (Examples) * There are hundreds of business and industrial indexes published on daily, weekly, monthly, quarterly, semi-annually, and annually basis for different business organizations, industrial associations, and various governmental (Federal, State, and Local) agencies, as important economic indicators. Many indexes (e.g., most industrial indexes) are constructed uniquely and computed based on specific formulations. Specialized knowledge is needed in calculating those unique business and industrial indexes.

  12. Example-1 • Mr. Wagner owns stock in three companies. Shown below is the price per share at the end of 1991 and 1998 for the three stocks and the quantities he owned in 1991 and 1998. • Compute a simple price index for each stock. Use 1991 as the base year (1991=100). • The simple price indexes are: (2/1)(100)=200; (4/5)(100) = 80; and (6/6) (100) = 100 • Compute a simple index for the number of shares owned for each stock. Use 1991 as the base year. • The shares indexes are: (50/30) (100) = 166.67; • (30/15) (100) = 200; and (20/40) (100) =50.

  13. Example-2 • From the information given in Example 1, perform the following operations: • Compute a simple aggregate price index for the three stocks. (12/12) = 1 • Compute the price index using the Laspeyres method. • P= [2(30) +4(15)+6(40)]/[1(30)+5(15)+6(40)](100) = [360/345](100) = 104.35 • Compute the price index using the Paasche method. • P = [2(50)+4(30)+6(20)]/[1(50)+5(30)+6(20)](100=[340/320](100)=106.25 • Develop a value index. • V=[2(50)+4(30)+6(20)]/[1(30)+5(15)+6(40)](100) = (360/345)(100) = 98.55

  14. Exercises A. 1. The annual incomes in 2000 for a few selected companies are: 2. The average hourly earnings of production workers fro selected periods are given below:

  15. a) Using 1991 as the base period and 100 as the base value, determine the indexes for 1999and for the preliminary 2000 data. Interpret the index. b) Use the average of 1991, 1992 and 1993 as the base and determine indexes for 1999 and the preliminary 2000 data using 100 as the base value. Interpret the index. c)What is the index for the preliminary 2000 data using 1995 as the base? B. An index of clothing prices for 2002 based in 1995 is to be constructed. The clothing items considered are shoes and dresses. The information for prices and quantities for both years is given below. Use 1995 as the base period and 100 as the base value.

  16. Determine the simple average of the price indexes. • Determine the aggregate price indexes for the two years. • Determine Laspeyres’ price index. • Determine the Paasche price index. • Determine Fisher’s ideal index. C.The take-home pay of Jon Greene and the CPI for 1995 and 2000 are: • What was Jon’s real income in 1995? • What was his real income in 2000? • Interpret your findings.

  17. Table 15-1: Prices of Benson AUTOMATIC Stapler, Model 3, Converted to Indexes. Using three different base periods Table 17-2: Computation of Index for Food Price 1998, 1995 = 100.

  18. The simple index for bread is 115.6, found by using formula (17-1) Simple Average of the price relatives: [17-2]

  19. Simple Aggregate Index [17-3] Laspeyres’ Price Index: [17-4]

  20. Table 17-3 Computation of Laspeyres and Paasche Indexes of Food Price, 1995 = 100 The weighted price index for 1998 is 108.8, found by

  21. PAASCHE’S Price Index: [17-5] The Paasche index is 109.4, found by Fisher’s ideal index: [17-6]

  22. Fisher’s ideal index= =109.1 Value Index: [17-7]

  23. Example Suppose the prices and quantities sold at the Waleska Department Store for various items of apparel for May 1982 and May 1995 are: What is the index of value for May 1995 using May 1982 as the base period?

  24. Table 17-6: Construction of Value Index for 1995 (1982=100)

  25. Table 17-7: Data for the Computation if the Index of General Business Activity Department store sales: ($44/$20) 100 X 0.40 = 88.0 Employment: ($125/100) 100 X 0.30 = 37.5 Freight car loadings: (18/50) 100 X 0.10 = 3.6 Exports: (700/500) 100 X 0.20 = 28/157.10 The index of general business activity for June 1995 is 157.1. Interpreting, business activity increased 57.1 percent from the base period (arbitrarily selected as 1986) to June 1995.

  26. Table 17-8: Computation if real Income for 1982-84 and Present Year

  27. Table 17-9: Computations for Deflating Sales

  28. Suppose the Consumer Price Index this month is 200.0 (1982-84 = 100). What is the purchasing power of the dollar? • Using Formula (17-8), it is 50 cents, found by: • Purchasing power of dollar = $1/200.0 (100) = $0.50 • Table 17-10: Computing the purchasing Power of the Dollar *Source: U.S. Department of Labor, Monthly Labor Review, January 1995, p.102.

  29. Table 17-11: Trend Consumer Prices to 1995 (1982-84 = 100) *Source: U.S. Department of Labor, Monthly Labor Review, November 1994, p.102.

  30. A problem arises, however, when two or more series being compared do not have the same base period. *Source: Federal Reserve Bulletin, July 1988, p.A25; and p.A24; and June 1993, PA27 The calculations for the 1993 American Stock Exchange price index using 1985 = 100 are: 418.54/229.10 X 100 = 182.7 The complete set of indexes using 1985 = 100 is:

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