CD Appendix AC Index Numbers

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1 CD Appedix AC Idex Numbers I Chapter 20, we preseted a variety of techiques for aalyzig ad forecastig time series. This appedix is devoted to the simpler task of developig descriptive measuremets of the chages i a time series. These measuremets are called idex umbers. As is the case with the descriptive statistics itroduced i Chapter 4 ad used throughout this book, idex umbers allow the statistics practitioer to summarize a large body of data with a sigle umber. For example, shoppers who observe the price of goods ad services chagig frequetly have difficulty assessig what is happeig to overall prices. They might kow that the price of T-boe steaks has icreased, that baby-sitters charge more, ad that retal ad housig prices seem to have icreased; but at the same time, persoal computers have become cheaper, airlie deregulatio has lowered the price of may flights, ad the cost of certai types of food staples does ot seem to have icreased sigificatly. By lookig at the Cosumer Price Idex, they ca get a immediate geeral idea as to what has happeed to overall prices. (I Chapter 2 we briefly itroduced the Cosumer Price Idex ad showed how it is used.) The Cosumer Price Idex (CPI) measures what has happeed to the prices of hudreds of cosumer goods ad services. I , the CPI was set to equal 00. By 990, it had rise to 30.7, idicatig that prices had geerally icreased by 30.7% over the precedig 7 years. By 2006, the CPI was Aother popular idex is the Dow Joes Idustrial Average, which measures the average daily closig prices of 30 large corporatios listed o the New York Stock Exchage. As such, it is perceived by may people (ad especially by the media that report it o a daily ad sometimes hourly basis) as beig a good idicator ot oly of the curret status of the stock market but of the ecoomic well-beig of the coutry as a whole. Although we ca develop a idex to measure chages i ay time series, i practice most idices measure chages i prices. Our otatio will reflect this fact.

2 Simple ad Aggregate Idex Numbers We start with the most basic type of idex umber, the simple idex. Simple Idex A simple idex is a ratio of the price of a commodity at time t divided by its value at some base period. We ca express the ratio as a percetage by multiplyig by 00. That is, where 0 I I P P P 00 Idex i the curret period Price i the curret period P 0 Price i the base period Example AC. Costruct the idex of the per capita persoal icome i the Uited States for 990, 995, 2000, ad 2004, usig 990 as the base year. Year Per Capita Aual Icome 990 $9, , , ,907 Source: Statistical Abstract of the Uited States, Table 657 Solutio For each year, we compute 0 I P P 00 2

3 The results are as follows. Year Idex Number of Per Capita Earigs 990 9, , , , , , , ,500 These umbers ca be used, for example, by a labor uio to compare its members' wage icreases sice 990 with those of the average America. Example AC.2 illustrates this type of compariso. Example AC.2 Suppose the members of a uio were paid the mea weekly wages listed below. Compute the idex of earigs for 990, 995, ad 997 (usig 985 as the base year) for this uio ad compare it with the idex computed i Example. Year Mea Weekly Earigs for a Labor Uio 990 $ Solutio The idex of earigs for this uio is as follows. 3

4 Year Idex Number of Earigs I compariso to the wages represeted by the idex umbers i Example AC., this uio's wages grew more quickly tha the per capita persoal icome. The simple idex umber measures the price chages of oly oe item. Frequetly though, we would like to measure price chages for a group of commodities. The simple aggregate idex umber performs this fuctio. Simple Aggregate Idex A simple aggregate idex is the ratio of the sum of the prices of several commodities i the curret period to the sum i the base period, multiplied by 00. where P i i P i 0i I P i 00 Price of item i i the curret period Price of item i i the base period P 0i Example AC.3 Costruct a simple aggregate idex of the prices of meat, chicke, ad fish items show i the accompayig table. 4

5 Price per Poud Item July 2006 July 2007 Beef $5.50 $5.92 Veal Pork Chicke Fish Solutio P i i P i 0i I Thus, the oe-year icrease i the total prices is 8.3%. The figure calculated i Example AC.3 caot be iterpreted as the price icrease for this part of our diets, however. For example, if the average perso's diet cosisted mostly of chicke, the average costs might actually have decreased. Thus, the simple aggregate idex must be modified if we do ot cosume equal quatities of each item. I the weighted aggregate idex, each item is weighted by its relative importace. Weighted Aggregate Idex where Note that P Q i i i P Q i 0i 0i I 00 Weight assiged to item i i the curret period Q i Weight assiged to item i i the base period Q 0i i Qi ad Q i 0i 5

6 Example AC.4 I July 2006, a family's weekly diet cosisted of 6 pouds of fish, 2 pouds of beef, ad 2 pouds of veal. Oe year later, because of the cost icreases i these products, the family's diet chaged so that each week they cosumed 4 pouds of chicke ad poud of each other item. Assumig that the prices are those listed i the table i Example AC.3, calculate the weighted aggregate idex for 2007, usig 2006 as the base period. Solutio The weight assiged to each food item i each year is the percetage of the etire diet supplied by that item. The relevat weights are show i the followig table, July 2006 July 2007 Item Price Quatity Weightig Price Quatity Weightig Beef $ $ Veal Pork Chicke Fish Totals The weighted aggregate idex is P Q i i i P Q i 0i 0i I (.25) (.25) (.25) (.500) (.25) 5.50(.2) (.2) (0) (0) + 5.3(.6) Thus, there has bee a decrease of 3.9% i this part of the family's food budget. The evidet problem with this calculatio, however, is that it does ot reflect the price chages that have take place. 6

7 Oe way of correctig the weighted aggregate idex is to substitute the weights i the base period for the weights i the curret period. The result is the Laspeyres idex. Laspeyres Idex L P Q i i 0i P Q i 0i 0i 00 By keepig the weights the same, the Laspeyres idex more accurately measures the true price chages, as Example AC.5 illustrates. Example AC.5 Calculate the Laspeyres idex for the family described i Example AC.4. Solutio L P Q i i 0i P Q i 0i 0i (.2) (.2) (0) (0) (.6) (.2) 7.48(.2) 4.80(0) 3.62(0) 5.3(.6) This umber tells us that, if the 2007 diet had bee the same as the 2006 diet, this part of the family budget would show a cost icrease of 5.4%. Of course, sice i 2007 the family o loger eats 6 pouds of fish, 2 pouds of beef, ad 2 pouds of veal weekly, this figure does ot accurately reflect the situatio, Aother way of measurig the price icrease is to use the curret year's weightig. Whe this is doe, the result is the Paasche idex, 7

8 Paasche Idex P Q i i i P Q i 0i i P 00 Example AC.6 Calculate the Paasche idex for the family i Exercise AC.4. Solutio P Q i i i P Q i 0i i P (.25) (.25) (.25) (.500) (.25) (.25) 7.48(.25) 4.80(.25) 3.62(.500) 5.3(.25) Usig the 2003 quatities, the aggregate price icrease is 4.5%. We ow have three differet idices based o the same data. The weighted aggregate idex (from Example W4) is 86.. The Laspeyres idex (from Example AC.5) is L 5.4. The Paasche idex (from I Example W6) is P Which idex should we use to describe the price chage? To aswer this questio, we must first address aother questio: what do we wat to measure? Do we wat to fid out what actually happeed to this family's food budget betwee 2006 ad 2007? If so, we use I 86.; the cost decreased by 3.9%. Do we wat to kow what happeed to the prices, assumig that the 2006 diet remaied uchaged? The we use L 5.4; prices icreased by 5.4%. Do we wat to kow what happeed to the prices, assumig that the 20037diet was i effect i 2006? The we use P 04.5; prices icreased by oly 4.5%. Thus, our choice of idex depeds o what we wat to measure. 8

9 Govermets i the Uited States, Caada, ad other coutries have chose to use the Laspeyres idex to measure how prices chage mothly. The Cosumer Price Idex is a example of a Laspeyres idex, ad it is without doubt the most importat measure of iflatio i may coutries. It is importat for maagers ad ecoomists to uderstad this descriptive measuremet. Cosumer Price Idex The Cosumer Price Idex works with a basket of some 300 goods ad services i the Uited States (ad a similar umber i other coutries), icludig such diverse items as food, housig, clothig, trasportatio, health, ad recreatio. Prices for each item i this basket are computed o a mothly basis, by city, state, ad regio, ad the CPI is computed from these prices. The basket is defied for the "typical" or "average" middle-icome family, ad the set of items ad their weights are revised periodically (every 0 years i the Uited States ad every 7 years i Caada). There are, of course, a umber of problems associated with the costructio ad the cotiuig validity of a idex such as the CPI. I costructig the idex, the followig steps must be performed:. Select the appropriate items for the basket. 2. Select the appropriate weights. 3. Select a appropriate base period. For example, startig the idex durig a year of low prices or i the midst of a recessio will cause a later ecoomic recovery to appear to be imposig substatial icreases o the price idex. To maitai the validity of the CPI over a umber of years, the idex's sposors must deal with several problems. Has the typical family chaged from the curretly used moderate-icome urba couple? How much distortio occurs durig the years betwee official revisios of the CPI (at which time ew cosumptio patters are icorporated ito the ew idex)? Have defiitioal chages occurred with 9

10 respect to terms such as "food away from home"? How importat are qualitative chages, ad does the revised CPI capture these differeces? For example, computers are more powerful today tha last year ad last year's computers were more powerful tha those produced the previous year. Ad yet, the price has decreased. The CPI, despite ever really beig iteded to serve as the official measure of iflatio, has come to be iterpreted i this way by the geeral public. Pesio-pla paymets, old-age security, ad some labor cotracts are automatically liked to the CPI ad automatically idexed (so it is claimed) to the level of iflatio. However, if yours is ot the "typical family," price chages may affect you quite differetly from the way suggested by the CPI. Usig the Cosumer Price Idex to Deflate Prices Despite its flaws, the Cosumer Price Idex is used i umerous applicatios. Oe applicatio ivolves adjustig prices by removig the effect of iflatio. To do this, we first idetify the CPI from 980 to 997 (see Table AC.). The base year of the idex is the average of We ca use this table to deflate the aual values of prices or wages. This removes the effect of iflatio, makig comparisos more realistic. For example, suppose a worker eared $6.50/hour i 985 ad $9.25/hour i 997. To determie whether his or her purchasig power really icreased, we deflate both figures by dividig each by the CPI for the correspodig year ad the multiplyig by 00. Thus, we have the followig figures: DEFLATED YEAR WAGE CPI WAGE 985 $ $

11 The deflated wages are ow beig measured i dollars. I dollars, the worker eared less i 997 tha he or she did i 985. Aother way of makig such comparisos is by dividig the 997 wage by the 997 CPI ad the multiplyig by the 985 CPI: This figure represets the worker's wages measured i 985 dollars. Because i 985 he or she actually eared $6.50, we ca see that our earlier coclusio remais uchaged. Table AC. Uited States Cosumer Price Idex, Year CPI Year CPI Source: Statistical Abstract of the Uited States, 2006, Table 706.

12 Example AC.7 The gross domestic product (GDP) is ofte used as a measure of the ecoomic growth of a coutry. The aual GDP of the Uited States for the years is show i the accompayig table. Year GDP ($billios) 995 7, , , , , , , , , ,735.0 Source: Statistical Abstract of the Uited States, 2006, Table 650. Use the CPI i Table AC. to deflate these figures to dollars. Solutio To covert the GDP to (sometimes referred to as "costat ") dollars, we divide the GDP by its associated CPI ad the multiply by the CPI i (which is 00). The results follow. GDP ($billios) GDP ($billios) Year Curret Dollars CPI Costat Dollars 995 7, , , ,

13 999 9, , , , , , As you ca see, the GNP i curret dollars gives the impressio that the ecoomy has grow rapidly i the period , whereas whe measured i costat dollars the growth is quite modest. Our coclusio would be the same if we used some year other tha as our basis. EXERCISES AC. Takig 2000 as the base year, compute the simple idex for the price of atural gas i Natural Gas Price Year (per,000 cubic feet) 998 $ Source: Statistical Abstract of the Uited States, 2006, Table 880. AC.2 Takig 989 as the base year, calculate the simple idex for the price of a short to of coal i Year Coal Price (per short to) 3

14 989 $ Source: Statistical Abstract of the Uited States, 998. AC.3 Repeat Exercise AC., takig 993 as the base year. AC.4 A cake recipe calls for the igrediets below. Compute a simple aggregate idex, takig 990 as the base year. Price Igrediet Butter (per poud $.27 $.87 Sugar (per poud) Flour (per poud) Eggs (per doze) AC.5 The hotel idustry is very iterested i uderstadig how tourists sped moey. I order to measure the price chages i three importat compoets of a tourist's budget, a statisticia computed the average cost of a hotel room (oe ight), a meal, ad a car retal loe day) i 990 ad i 998. The results of these computatios are show i the accompayig table. Compute a simple aggregate idex, takig 990 as the base year Cost Compoet Hotel (oe ight) $75 $80 Meal 2 6 Car retal (oe day)

15 AC.6 Refer to Exercise AC.4. Suppose the chef at Chez Gerard's has decided to make his ow improvemets (addig more butter) to the cake recipe he used i 990. The old ad ew quatities of igrediets are listed i the accompayig table. Compute a weighted aggregate idex to measure the price icreases Igrediet Price Quatity Price Quatity Butter $.27/lb 6 oz $.87/lb 20 oz Sugar.65/Ib 8 oz.95/lb 80 oz Flour.49/lb 8 oz.89/lb 8 oz Eggs.52/doze 3eggs.85/doze 3eggs AC.7 Refer to Exercise AC.5. Suppose that i 990 the average tourist stayed i the hotel for 6 days, ate 8 meals at the hotel, ad reted a car for 2 days. I 998, the average tourist stayed for 4 days, ate 6 meals at the hotel, ad reted a car for 3 days. Calculate a weighted aggregate idex. AC.8 Compute the Laspeyres idex for Exercise AC.7. AC.9 Compute the Paasche idex for Exercise AC.7. AC.0 For Exercise AC.6, calculate the Laspeyres idex to measure the chage i the cost of a cake's igrediets. AC. For Exercise AC.6, calculate the Paasche idex to measure the chage i the cost of a cake's igrediets. AC.2 What is beig measured by each of the idices computed i Exercises AC.6, AC.0, ad AC.? AC.3 People cotiually look for ivestmets i periods of high ad uexpected iflatio, buyig gold, silver, ad eve platium, as a hedge agaist iflatio becomes attractive to some ivestors. Actual prices for these three precious metals are listed below. Compute the simple idex for each metal for the years 992, 994, ad 997, takig 989 as the base year. 5

16 Gold Price Silver Price Platium Price Year (per fie ouce) (per fie ouce) (per troy ouce) 989 $383 $5.50 $ Source: Statistical Abstract of the Uited States: 998. AC.4 For Exercise AC.3 calculate the aggregate idex for all three metals for 997, takig 989 as the base year. AC.5 Suppose a ivestor i 989 owed 00 ouces of gold,,000 ouces of silver, ad 50 ouces of platium. By 997, she had 500 ouces of gold,,500 ouces of silver, ad 00 ouces of platium. Costruct a weighted aggregate idex to measure how her ivestmet's value chaged betwee 989 ad 997. AC.6 For Exercise AC.5, compute the Laspeyres idex. AC.7 For Exercise AC.5, compute the Paasche idex. AC.8 The aual price of a barrel of oil for the years is show below. Calculate a simple idex showig the price icreases for 990, 993, ad 997. Use 989 as the base year. Crude Petroleum Price Year (per barrel) 989 $

17 Source: Statistical Abstract of the Uited States: 998. AC.9 Usig Table AC., deflate the aual prices of crude petroleum i Exercise AC.8 so that they are measured i costat dollars. Exercises AC.20 through AC.24 are based o the followig problem. A gasolie service statio determied the price ad the umber of uits sold per day of its four most popular items (after gasolie). These data were recorded for the years 985, 990, ad 995, as show below. Takig 985 as the base year, calculate the simple aggregate idex for 990 ad Item Price Quatity Price Quatity Price Quatity Oil (quart) $ $.85 5 Tire Atifreeze (quart) Battery AC.2 Compute the weighted aggregate idex for 990 ad 995, takig 985 as the base year. AC.22 Compute the Laspeyres idex for 990 ad 995. AC.23 Compute the Paasche idex for 990 ad 995. AC.24 Deflate (to costat dollars) each of the idices i Exercises AC.20 through AC.23, usig the CPI i Table 20.. What coclusios ca you reach about these results? AC.25 Is real estate a good ivestmet? I particular, will the price of a house keep up with iflatio? To aswer this questio, the media sales price of ew privately owed, oe-family houses was recorded for 7

18 These data are show ext. Compute the prices i costat dollars, usig the CPI i Table AC.. What coclusios do these results lead to? Media Sales Price of New Year Oe-family Houses ($000) Source: Statistical Abstract of the Uited States, 998. AC.26 The media sales price of existig oe-family homes is listed ext. Repeat Exercise AC.25 usig these data. Media Sales Price of Existig Year Oe-family Houses ($000) Source: Statistical Abstract of the Uited States, 998. AC.27 The table that follows lists the per capita gross atioal product. Covert these figures to costat dollars. What do these values tell you? Year Per Capita GNP 989 2,984 8

19 990 22, , , , , , , ,6 Source: Bureau of Ecoomic Aalysis, Natioal Icome ad Products Accouts of the Uited States, July

STRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans

STRAND: FINANCE. Unit 3 Loans and Mortgages TEXT. Contents. Section. 3.1 Annual Percentage Rate (APR) 3.2 APR for Repayment of Loans CMM Subject Support Strad: FINANCE Uit 3 Loas ad Mortgages: Text m e p STRAND: FINANCE Uit 3 Loas ad Mortgages TEXT Cotets Sectio 3.1 Aual Percetage Rate (APR) 3.2 APR for Repaymet of Loas 3.3 Credit Purchases