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StatisticsforBusiness
andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness
andEcChapter17
IndexNumbersPriceRelativesAggregatePriceIndexesComputinganAggregatePriceIndex fromPriceRelativesSomeImportantPriceIndexesDeflatingaSeriesbyPriceIndexesPriceIndexes:OtherConsiderationsQuantityIndexesChapter17
IndexNumbersPricePriceRelativesAbaseperiod
isagivenstartingpointintime.Apricerelativeexpressestheunitpriceineachperiodasapercentageoftheunitpriceinthebaseperiod.Apricerelativeshowshowthecurrentpriceperunitforagivenitemcomparestoabaseperiodpriceperunitforthesameitem.Pricerelativesarehelpfulinunderstandingandinterpretingchangingeconomicandbusinessconditionsovertime.PriceRelativesAbaseperiodiPriceRelativesTelevision11,469
23,904Newspaper$14,794$29,4121998
2008 ThepricesBescopaidfornewspaperandtelevisionadsin1998and2008areshownbelow.
Using1998asthebaseyear,computea2008price
indexfornewspaperandtelevisionadprices.Example:BescoProductsPriceRelativesTelevisionPriceRelativesTelevisionadvertisingcostincreasedatagreaterrate.NewspaperTelevisionPriceRelativesTelevisionadveAnaggregatepriceindex
isdevelopedforthespecificpurposeofmeasuringthecombinedchangeofagroupofitems.AggregatePriceIndexeswhere
Pit=unitpriceforitemiinperiodt
Pi
0=unitpriceforitemiinthebaseperiodAnunweightedaggregatepriceindexinperiodt, denotedbyIt
,isgivenbyAnaggregatepriceindexisdeLettingQi=quantityforitemi,theweightedaggregatepriceindexinperiodtisgivenbyAggregatePriceIndexeswherethesumsareoverallitemsinthegroupWithaweightedaggregateindex
eachiteminthegroupisweightedaccordingtoitsimportance,whichtypicallyisthequantityofusage.LettingQi=quantityforitemAggregatePriceIndexesWhentheweightsarebasedonperiodtusage,theindexisaPaascheindex.Whenthefixedquantityweightsaredeterminedfromthebase-yearusage,theindexiscalledaLaspeyresindex.AggregatePriceIndexesWhenthAggregatePriceIndexes Dataonenergyconsumption andexpendituresbysectorforthe cityofRockdalearegivenonthenextslide. Constructanaggregate priceindexforenergyexpendituresin2008 using1990asthebase
year.Example:CityofRockdaleAggregatePriceIndexes DataAggregatePriceIndexes Transport. 15,293 20,262 2.32 6.16 Industrial 21,287 17,832 .79 5.13 Commercial 5,416 6,015 1.97 11.32 Residential 9,473 8,804 $2.12 $10.92
Sector
1990
2008
1990
2008 Quantity(BTU)UnitPrice($/BTU)Example:CityofRockdaleAggregatePriceIndexes TranspUnweightedAggregatePriceIndexAggregatePriceIndexesExample:CityofRockdaleUnweightedAggregatePriceIndWeightedAggregateIndex(LaspeyresMethod)AggregatePriceIndexesThePaaschevaluebeinglessthantheLaspeyresindicatesusagehasincreasedfasterinthelower-pricedsectors.WeightedAggregateIndex(PaascheMethod)Example:CityofRockdaleWeightedAggregateIndex(LaspDinawantsanindexthatmeasuresthechangeintheoverallcostofherlawncare.Priceandquantitydataforherannuallawnexpensesarelistedonthenextslide.Example:AnnualCostofLawnCareDinaEversispleasedwithherlovelylawn,butsheisconcernabouttheincreasingcostofmaintainingit.Thecostincludesmowing,fertilizing,watering,andmore.AggregatePriceIndexesDinawantsanindexthatItemQuantity(Units)MowingLeafRemovalWatering(1000sgal.)FertilizingSprinklerRepair3234021UnitPrice($)2004200857.0056.001.8356.00109.0079.0071.002.7867.00128.00Example:AnnualCostofLawnCareAggregatePriceIndexesItemQuantityMowing32UnitPriceUnweightedwhere:
Pit=unitpriceforitemi
inperiodt
Pi
0=unitpriceforitemi
inthebaseperiodUnweightedaggregatepriceindexinperiodtis:A24%increaseinannuallawncareexpensesAggregatePriceIndexesUnweightedwhere:UnweightedaggWeightedaggregatepriceindexinperiodt
is:Weighted(FixedQuantity)where:Qi=quantityforitemiNotethatQidoesnothaveasecondsubscriptforatimeperiod.A36%increaseinannuallawncareexpensesAggregatePriceIndexesWeightedaggregatepriceindexWeighted(Base-PeriodQuantity)Avariable-quantityindexSpecialcaseofthefixedquantityindexLaspeyresIndexPaascheIndexMorewidelyusedthanthePaascheindexPro:Reflectscurrentusage;Con:Weightsrequirecontin-ualupdatingWeighted(PeriodtQuantity)AggregatePriceIndexesWeighted(Base-PeriodQuantityThe5-yearincreasesinunitpricerangedfromalowof17.4%forsprinklerrepairtoahighof51.9%forwater.AggregatePriceIndexesItemiMowingLeafRemovalWater(1000sgal.)FertilizingSprinklerRepairUnitPrice($)2004200857.0056.001.8356.00109.0079.0071.002.7867.00128.00Pi
0Pi
tPriceRelative(Pi
t/Pi
0)100138.6126.8151.9119.6117.4The5-yearincreasesinunitpItemPriceRelative(Pit/Pi
0)100MowingLeavesWaterFertilizeSprinklerBasePrice($)57.0056.001.8356.00109.00Qi138.6126.8151.9119.6117.4Pi
0Quantity3234021Weightwi
=
Pi
0QiWeightedPriceRelative(Pit/Pi
0)(100)wi1,824.0168.073.2112.0109.0252,806.4021,302.4011,119.0813,395.2012,796.60Total2,286.2311,419.68AggregatePriceIndexesThisvalueisthesameastheoneidentifiedbytheweightedaggregateindexcomputation.ItemPriceMowingBase57.00Qi13Itsbaseperiodis1982-1984withanindexof100.PublishedmonthlybytheU.S.BureauofLaborStatistics.Weightedaggregatepriceindexwithfixedweightsderivedfromausagesurvey.Basedon400itemsincludingfood,housing,clothing,transportation,andmedicalitems.PrimarymeasureofthecostoflivinginU.S.SomeImportantPriceIndexesConsumerPriceIndex(CPI)Itsbaseperiodis1982-1984wConsumerPriceIndex(CPI)Base1982-1984=100.0Note:For1982–1984that(96.5
+
99.6
+
103.9)/3
=
100.0SomeImportantPriceIndexesConsumerPriceIndex(CPI)NoteIsaweightedaverageofpricerelativesusingtheLaspeyresmethod.Includestheoutputofmanufacturing,agriculture,forestry,fishing,mining,gasandelectricity,andpublicutilities.Coversraw,manufactured,andprocessedgoodsateachlevelofprocessing.Usedasaleadingindicator
ofthefuturetrendofconsumerpricesandthecostofliving.MeasuresthemonthlychangesinpricesinprimarymarketsintheU.S.ProducerPriceIndex(PPI)SomeImportantPriceIndexesIsaweightedaverageofpriceAnotheraverageiscomputedfor20transportationstocks,andanotherfor15utilitystocks.TheDJIAisnotexpressedasapercentageofbase-yearprices.TheDowJonesIndustrialAverage
(DJIA)
isbasedoncommonstockpricesof30industrialfirms.IndexesdesignedtoshowpricetrendsandmovementsontheNewYorkStockExchange.DowJonesAveragesSomeImportantPriceIndexesAnotheraverageiscomputedfoDowJonesIndustrialAverage(DJIA)30CompaniesAsof04/2009SomeImportantPriceIndexesDowJonesIndustrialAverage(DeflatingaSeriesbyPriceIndexesDeflatingactualhourlywagesresultsinrealwagesorthepurchasingpower
ofwages.Inordertocorrectlyinterpretbusinessactivityovertimewhenitisexpressedindollaramounts,weshouldadjustthedatafortheprice-increaseeffect.Removingtheprice-increaseeffectfromatimeseriesiscalleddeflatingtheseries.DeflatingaSeriesbyPriceIn McNeerCleaners,with46branchlocations,hashadthetotalsalesrevenuesshownonthenext slideforthelastfiveyears.Deflatethesalesrevenue
figuresonthebasisof1982-1984constantdollars.Istheincreaseinsalesdueentirelytotheprice-increaseeffect?Example:McNeerCleanersDeflatingaSeriesbyPriceIndexes McNeerCleaners,with46braExample:McNeerCleanersDeflatingaSeriesbyPriceIndexes
200811,690215.3
200710,724207.3
20069,830201.6
20059,062195.3
20048,446188.9 YearTotalSales($1000)CPIExample:McNeerCleanersDeflaAdjustingRevenueForthePrice-IncreaseEffectAfteradjusting,revenueisstillincreasingatanaveragerateof5.0%peryear.2008(11,690/215.3)(100)=5,430+5.02007 (10,724/207.3)(100)=5,173+6.12006(9,830/201.6)(100)=4,876+5.12005(9,062/195.3)(100)=4,640+3.82004(8,446/188.9)(100)=4,471YearDeflatedSales($1000)AnnualChange(%)DeflatingaSeriesbyPriceIndexesAdjustingRevenueForthePricRealSalesRevenue($1000s)20042005200620072008Arealsalesincreaseof21.45%from2004to20084500475050005250
55004250Year44714640517354304876DeflatingaSeriesbyPriceIndexesRealSalesRevenue($1000s)20Whentheclassofitemsisverylarge,arepresentativegroup(usuallynotarandomsample)mustbeused.Thegroupofitemsintheaggregateindexmustbeperiodicallyreviewedandrevisedifitisnotrepresentativeoftheclassofitemsinmind.SelectionofItemsPriceIndexes:OtherConsiderationsWhentheclassofitemsisverForexample,aCPIwitha1945baseperiodwouldbedifficultformostindividualstounderstand,duetounfamiliaritywithconditionsin1945.SelectionofaBasePeriodThebaseperiodformostindexesisadjustedperiodicallytoamorerecentperiodoftime.Asarule,thebaseperiodshouldnotbetoofarfromthecurrentperiod.TheCPI’sbaseperiodwaschangedto1982-84in1988.PriceIndexes:OtherConsiderationsForexample,aCPIwitha1945Anincreaseinanitem’squalitymayormaynotresultinapriceincrease...andadecreaseinqualitymayormaynotresultinapricedecrease.Isaproductthathasundergoneamajorqualitychangethesameproductitwas
?Abasicassumptionofapriceindexisthatpricesovertimeareidentifiedforthesameitem.QualityChangesPriceIndexes:OtherConsiderationsAnincreaseinanitem’squaliAnindexusedtomeasurechangesinquantitylevelsovertimeiscalledaquantityindex.Aweightedaggregatequantityindexiscomputedinmuchthesamewayasaweightedaggregatepriceindex.Aquantityrelativeshowshowthecurrentquantitylevelforasingleitemcomparestoabaseperiodquantitylevelforthesameitem.QuantityIndexesAnindexusedtomeasurechangExample:ApplianceMartApplianceMartreportsthe1999and2008salesforthreemajorkitchenappliancesasshownbelow.Computequantityrelativesandusethemtodevelopaweightedaggregatequantityindexfor2008.KitchenApplianceSalesPrice(1999)DishwasherRangeRefrigeratorSales(Units)199920087205409809506101110$325$450$710QuantityIndexesExample:ApplianceMartAKitchenApplianceDishwasherRangeRefrigeratorSales(Units)199920087205409809506101110QuantityRelative(950/720)100
=
131.9(610/540)100
=
113.0(1110/980)100
=
113.3QuantityIndexesKitchenDishwasherSales(Units)Unweightedwhere:Qit=quantityforitemi
inperiodt
Qi
0=quantityforitemi
inthebaseperiodUnweightedaggregatequantityindexinperiodtis:QuantityIndexesUnweightedwhere:Qit=quanWeighted(Base-PeriodPrice)where:Pi
0=priceforitemi
inthebaseperiodWeightedaggregatequantityindexinperiodtis:QuantityIndexesWeighted(Base-PeriodPrice)whWeighted(FixedPrice)Weighted(PeriodtPrice)NotethatPidoesnothaveasecondsubscriptforatimeperiod.QuantityIndexesWeighted(FixedPrice)WeightedEndofChapter17EndofChapter17StatisticsforBusiness
andEconomicsAndersonSweeneyWilliamsSlidesbyJohnLoucksSt.Edward’sUniversityStatisticsforBusiness
andEcChapter17
IndexNumbersPriceRelativesAggregatePriceIndexesComputinganAggregatePriceIndex fromPriceRelativesSomeImportantPriceIndexesDeflatingaSeriesbyPriceIndexesPriceIndexes:OtherConsiderationsQuantityIndexesChapter17
IndexNumbersPricePriceRelativesAbaseperiod
isagivenstartingpointintime.Apricerelativeexpressestheunitpriceineachperiodasapercentageoftheunitpriceinthebaseperiod.Apricerelativeshowshowthecurrentpriceperunitforagivenitemcomparestoabaseperiodpriceperunitforthesameitem.Pricerelativesarehelpfulinunderstandingandinterpretingchangingeconomicandbusinessconditionsovertime.PriceRelativesAbaseperiodiPriceRelativesTelevision11,469
23,904Newspaper$14,794$29,4121998
2008 ThepricesBescopaidfornewspaperandtelevisionadsin1998and2008areshownbelow.
Using1998asthebaseyear,computea2008price
indexfornewspaperandtelevisionadprices.Example:BescoProductsPriceRelativesTelevisionPriceRelativesTelevisionadvertisingcostincreasedatagreaterrate.NewspaperTelevisionPriceRelativesTelevisionadveAnaggregatepriceindex
isdevelopedforthespecificpurposeofmeasuringthecombinedchangeofagroupofitems.AggregatePriceIndexeswhere
Pit=unitpriceforitemiinperiodt
Pi
0=unitpriceforitemiinthebaseperiodAnunweightedaggregatepriceindexinperiodt, denotedbyIt
,isgivenbyAnaggregatepriceindexisdeLettingQi=quantityforitemi,theweightedaggregatepriceindexinperiodtisgivenbyAggregatePriceIndexeswherethesumsareoverallitemsinthegroupWithaweightedaggregateindex
eachiteminthegroupisweightedaccordingtoitsimportance,whichtypicallyisthequantityofusage.LettingQi=quantityforitemAggregatePriceIndexesWhentheweightsarebasedonperiodtusage,theindexisaPaascheindex.Whenthefixedquantityweightsaredeterminedfromthebase-yearusage,theindexiscalledaLaspeyresindex.AggregatePriceIndexesWhenthAggregatePriceIndexes Dataonenergyconsumption andexpendituresbysectorforthe cityofRockdalearegivenonthenextslide. Constructanaggregate priceindexforenergyexpendituresin2008 using1990asthebase
year.Example:CityofRockdaleAggregatePriceIndexes DataAggregatePriceIndexes Transport. 15,293 20,262 2.32 6.16 Industrial 21,287 17,832 .79 5.13 Commercial 5,416 6,015 1.97 11.32 Residential 9,473 8,804 $2.12 $10.92
Sector
1990
2008
1990
2008 Quantity(BTU)UnitPrice($/BTU)Example:CityofRockdaleAggregatePriceIndexes TranspUnweightedAggregatePriceIndexAggregatePriceIndexesExample:CityofRockdaleUnweightedAggregatePriceIndWeightedAggregateIndex(LaspeyresMethod)AggregatePriceIndexesThePaaschevaluebeinglessthantheLaspeyresindicatesusagehasincreasedfasterinthelower-pricedsectors.WeightedAggregateIndex(PaascheMethod)Example:CityofRockdaleWeightedAggregateIndex(LaspDinawantsanindexthatmeasuresthechangeintheoverallcostofherlawncare.Priceandquantitydataforherannuallawnexpensesarelistedonthenextslide.Example:AnnualCostofLawnCareDinaEversispleasedwithherlovelylawn,butsheisconcernabouttheincreasingcostofmaintainingit.Thecostincludesmowing,fertilizing,watering,andmore.AggregatePriceIndexesDinawantsanindexthatItemQuantity(Units)MowingLeafRemovalWatering(1000sgal.)FertilizingSprinklerRepair3234021UnitPrice($)2004200857.0056.001.8356.00109.0079.0071.002.7867.00128.00Example:AnnualCostofLawnCareAggregatePriceIndexesItemQuantityMowing32UnitPriceUnweightedwhere:
Pit=unitpriceforitemi
inperiodt
Pi
0=unitpriceforitemi
inthebaseperiodUnweightedaggregatepriceindexinperiodtis:A24%increaseinannuallawncareexpensesAggregatePriceIndexesUnweightedwhere:UnweightedaggWeightedaggregatepriceindexinperiodt
is:Weighted(FixedQuantity)where:Qi=quantityforitemiNotethatQidoesnothaveasecondsubscriptforatimeperiod.A36%increaseinannuallawncareexpensesAggregatePriceIndexesWeightedaggregatepriceindexWeighted(Base-PeriodQuantity)Avariable-quantityindexSpecialcaseofthefixedquantityindexLaspeyresIndexPaascheIndexMorewidelyusedthanthePaascheindexPro:Reflectscurrentusage;Con:Weightsrequirecontin-ualupdatingWeighted(PeriodtQuantity)AggregatePriceIndexesWeighted(Base-PeriodQuantityThe5-yearincreasesinunitpricerangedfromalowof17.4%forsprinklerrepairtoahighof51.9%forwater.AggregatePriceIndexesItemiMowingLeafRemovalWater(1000sgal.)FertilizingSprinklerRepairUnitPrice($)2004200857.0056.001.8356.00109.0079.0071.002.7867.00128.00Pi
0Pi
tPriceRelative(Pi
t/Pi
0)100138.6126.8151.9119.6117.4The5-yearincreasesinunitpItemPriceRelative(Pit/Pi
0)100MowingLeavesWaterFertilizeSprinklerBasePrice($)57.0056.001.8356.00109.00Qi138.6126.8151.9119.6117.4Pi
0Quantity3234021Weightwi
=
Pi
0QiWeightedPriceRelative(Pit/Pi
0)(100)wi1,824.0168.073.2112.0109.0252,806.4021,302.4011,119.0813,395.2012,796.60Total2,286.2311,419.68AggregatePriceIndexesThisvalueisthesameastheoneidentifiedbytheweightedaggregateindexcomputation.ItemPriceMowingBase57.00Qi13Itsbaseperiodis1982-1984withanindexof100.PublishedmonthlybytheU.S.BureauofLaborStatistics.Weightedaggregatepriceindexwithfixedweightsderivedfromausagesurvey.Basedon400itemsincludingfood,housing,clothing,transportation,andmedicalitems.PrimarymeasureofthecostoflivinginU.S.SomeImportantPriceIndexesConsumerPriceIndex(CPI)Itsbaseperiodis1982-1984wConsumerPriceIndex(CPI)Base1982-1984=100.0Note:For1982–1984that(96.5
+
99.6
+
103.9)/3
=
100.0SomeImportantPriceIndexesConsumerPriceIndex(CPI)NoteIsaweightedaverageofpricerelativesusingtheLaspeyresmethod.Includestheoutputofmanufacturing,agriculture,forestry,fishing,mining,gasandelectricity,andpublicutilities.Coversraw,manufactured,andprocessedgoodsateachlevelofprocessing.Usedasaleadingindicator
ofthefuturetrendofconsumerpricesandthecostofliving.MeasuresthemonthlychangesinpricesinprimarymarketsintheU.S.ProducerPriceIndex(PPI)SomeImportantPriceIndexesIsaweightedaverageofpriceAnotheraverageiscomputedfor20transportationstocks,andanotherfor15utilitystocks.TheDJIAisnotexpressedasapercentageofbase-yearprices.TheDowJonesIndustrialAverage
(DJIA)
isbasedoncommonstockpricesof30industrialfirms.IndexesdesignedtoshowpricetrendsandmovementsontheNewYorkStockExchange.DowJonesAveragesSomeImportantPriceIndexesAnotheraverageiscomputedfoDowJonesIndustrialAverage(DJIA)30CompaniesAsof04/2009SomeImportantPriceIndexesDowJonesIndustrialAverage(DeflatingaSeriesbyPriceIndexesDeflatingactualhourlywagesresultsinrealwagesorthepurchasingpower
ofwages.Inordertocorrectlyinterpretbusinessactivityovertimewhenitisexpressedindollaramounts,weshouldadjustthedatafortheprice-increaseeffect.Removingtheprice-increaseeffectfromatimeseriesiscalleddeflatingtheseries.DeflatingaSeriesbyPriceIn McNeerCleaners,with46branchlocations,hashadthetotalsalesrevenuesshownonthenext slideforthelastfiveyears.Deflatethesalesrevenue
figuresonthebasisof1982-1984constantdollars.Istheincreaseinsalesdueentirelytotheprice-increaseeffect?Example:McNeerCleanersDeflatingaSeriesbyPriceIndexes McNeerCleaners,with46braExample:McNeerCleanersDeflatingaSeriesbyPriceIndexes
200811,690215.3
200710,724207.3
20069,830201.6
20059,062195.3
20048,446188.9 YearTotalSales($1000)CPIExample:McNeerCleanersDeflaAdjustingRevenueForthePrice-IncreaseEffectAfteradjusting,revenueisstillincreasingatanaveragerateof5.0%peryear.2008(11,690/215.3)(100)=5,430+5.02007 (10,724/207.3)(100)=5,173+6.12006(9,830/201.6)(100)=4,876+5.12005(9,062/195.3)(100)=4,640+3.82004(8,446/188.9)(100)=4,471YearDeflatedSales($1000)AnnualChange(%)DeflatingaSeriesbyPriceIndexesAdjustingRevenueForthePricRealSalesRevenue($1000s)20042005200620072008Arealsalesincreaseof21.45%from2004to20084500475050005250
55004250Year44714640517354304876DeflatingaSeriesbyPriceIndexesRealSalesRevenue($1000s)20Whentheclassofitemsisverylarge,arepresentativegroup(usuallynotarandomsample)mustbeused.Thegroupofitemsintheaggregateindexmustbeperiodicallyreviewedandrevisedifitisnotrepresentativeoftheclassofitemsinmind.SelectionofItemsPriceIndexes:OtherConsiderationsWhentheclassofitemsisverForexample,aCPIwitha1945baseperiodwo
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