ons methodology working paper series no 3 a note on

ONSMethodologyWorkingPaperSeriesNo3

AnoteondistributionsusedwhencalculatingestimatesofconsumptionoffixedcapitalCraigMcLarenandChrisStapenhurstMarch2015

A note on distributions used when calculating estimates of capital consumption, March 2015

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0 Summary

Thisisatechnicalreferencepaperwhichdescribesdistributionsusedwhencalculatingestimatesofcapitalconsumption.WefocusondistributionsusedthatcanbeusedwithinthePerpetualInventoryMethod.Inpractice,thechoiceofretirementdistributionsforcapitalconsumptioncalculationscanbesomewhatarbitrary,sothispaperdescribespropertiesofdistributionstohelpunderstandtherelativemeritsofdifferentapproachesusedinthatcontext.

1 Introduction

ThePerpetualInventoryMethod(PIM)isawellestablishedmethodtoestimatecapitalstocksandconsumptionwithinNationalAccounts.WithintheOfficeforNationalStatistics(ONS),outputfromthePIMispublishedwithintheCapitalStocksandConsumptionofFixedCapitalrelease(ONS,2014).ForadetailedoverviewofthePIM,andrelatedconceptswithinthewidercontextoftheNationalAccountsframework,seeOECD(2009).

Describedbriefly,thePIMsumsflowsofinvestmentsoverpreviousperiods,subtractingretiredcapitalandcorrectingforreductionsinproductivity(value)togivethestock(value)ofproductivecapital.Incorporatingappropriatepriceprofilegivesthenetcapitalstock.Finally,therateofreturnandholdinggainsandlossesareusedtoderivethevalueofcapitalservices.

Thispaperisconcernedwiththeestimationofcapitalstocksfromhistoricalinvestmentsdataandassumptionsaboutassetretirementanddegradation.Forthisthefollowingarerequired:

Aclassificationofcapitalinputsintoreasonablyhomogenoustypes,i.e.exhibitingsimilarcharacteristicsintermsofdegradationandretirement

andthenforeachclassofassets:

Atimeseriesofinvestmentsineachclassofcapital(forourpurposesweassumepriceshavebeendeflatedappropriately)

Anage‐priceorage‐efficiencyprofile(collectively,age‐profiles)forasingleassetineachclassofcapital

Aretirementdistributionforeachclassofcapital

Oncethesearegiven,foreachassetclasstheage‐profileforasingleassetiscombinedwitharetirementdistributiontogiveanage‐profileforacohortofassets.Thisisthenappliedtotheseriesofinvestmentsofthatclass;finallywesumacrossallclassestogivethecapitalstock.

Theaimofthispaperistocatalogueanddescribesomeofthemoreprevalentfunctionalformsusedforage‐profilesandretirementdistributionsintoasinglereference,whilenotingthatsomeformsmaybeusedforbothpurposes.Wethengoontodescribehowthesemaybecombinedtogiveanage‐profileforacohortofassets.Inpractice,thechoiceforage‐profilesandretirementdistributionscanbesomewhatarbitraryasitcandependontheassetclassunderconsiderationandhistoricalconvention,sowemakenorecommendationsonthepracticaluseofeachdistribution.OECD(2009,p42)makessomepracticalrecommendationsregardingthelinearandgeometricforms.


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2 Age‐efficiency, age‐price and depreciation profiles

TheOECDCapitalManual(OECD,2009)definesage‐efficiency,age‐priceanddepreciationprofilesasfollows:

Age‐efficiencyprofile:Thisdescribesanasset’sproductivecapacityoveritsservicelifeandisrepresentedasanindex.Theindexissettoequaloneforanewassetandbecomeszerowhentheassethasreachedtheendofitsservicelife.Thedeclineinproductivecapacityisaresultofwearandtearand/orobsolescenceoftheasset.

Age‐priceprofile:Thisisanindexofthepriceofacapitalgoodwithregardtoitsage.Theage‐priceprofileshowsthelossinvalueofacapitalgoodasitages,orthepatternofrelativepricesfordifferentvintagesofthesame(homogenous)capitalgood.Theage‐priceprofilecomparesidenticalcapitalgoodsofdifferentageatthesamepointintime

Depreciationprofile:Valuelossofanassetduetoaging,expressedaspercentageofthevalueofanewasset

Age‐efficiencyandage‐priceprofilesareanalogousinmanyrespects,forthisreasonwemayoftenrefertothemcollectivelyas‘age‐profiles’.ThechoiceofwhichisusedinthePIMisdeterminedbywhatwewishtocalculate:theage‐priceprofileisusedforcalculatingnetcapitalstockanddepreciation,whereastheefficiencyprofilegivesproductivecapitalstock.Thesedefinitionsrefertoasingleasset;althoughitisalsoappropriatetorefertotheage‐profileofacohortofassets.Therelationshipbetweenthatofasingleassetandthatofacohortorassetsisconsideredlaterinthisnote.

Wedefineanage‐profilemoreformallyasa(deterministic)function

: →

satisfying:

i) 0 1ii) 0∀ ∈

Inmostcasesthefollowingalsoholds:

iii) lim → 0

Weinterpret astheprice(orefficiency)oftheassetattime relativetoitsprice(orefficiency)attime0.Theconditionscanbeinterpretedassayingthatthepriceoftheassetisnormalisedtoequal1atthebeginningofitslife,thatthepriceoftheassetisalwayspositive,andthatthepriceoftheassettendstowardszerowithtime(thismaynotholdinsomespecialcasesforassetssuchasland).Notethattheage‐profileitselfdoesnotnecessaryrequirethatthepriceeverdoesreachzero;thismaybedealtwithbychoosingsomethresholdvalueafterwhichthevalueoftheassetisreducedtozero.Forinstance,if isanage‐profileandthereexistsnopositiverealnumber suchthat

0;let beourthresholdvaluethenwecanredefine sothat 0forallwhere istheinverseof .


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Alternatively,manytexts,includingOECD(2009),defineage‐profilesasasequencesatisfyinganalogousproperties.Thisisentirelysufficientsinceweonlyeverwishtocalculateefficiencyofvalueatdiscreteintervals,forinstanceannuallyorquarterly.Weusecontinuousfunctions1withinthispaperonlybecausemanyofthemhaveformsusedalsoinretirementdistributionssowemaybesavedpresentingthesameformtwice,thoughwenotethatsomeofthefunctionalformsdescribedbelowaretraditionallydefinedonlyinthediscretecase.Movingbetweendiscreteandcontinuousformsisusuallytrivial:ifasequence canbedescribedbyaclosedform ,wemaysometimesevaluate atnon‐integervaluestogiveacontinuousfunction(forexamplethegeometricform);converselyifacontinuousfunction isgivenwemaydefineasequence by .

Wealsonotethatthepriceandefficiencyprofilesarerelatedbutbynomeansidentical.Thepriceprofilecanbederivedfromtheefficiencyprofileandviceversaiftherateofreturnandassetpriceinflationareknown.TheproceduresfordoingsoforsingleassetsandcohortsareillustratedinOECD(2009)Chapter3andAnnexDrespectively,andinMcLellan(2004,p12‐16).Forthispurposeage‐profilesaredefinedassequences.Let

betheage‐priceprofilesequence, theage‐efficiencyprofile,Tthelifespanoftheasset(whichmaybeinfinite),πtherateofassetpriceinflationandρthediscountrate,thenOECD(2009,p228)gives:

∑ 11

∑ 11

(1)

Theprocesscanbereversedtoderivetheage‐efficiencyprofilefromtheage‐priceprofileonOECD(2009,p229):

1 11 1

(2)

Similarly,thedepreciationprofileisdeterminedbytheage‐priceprofilebythefollowingformula:

1 (3)

whereonceagain isasequencedescribingapriceprofileand isthedepreciationprofile.Acontinuousanalogueisgivenby:

′ (4)

Intheliteratureitisnormaltofirstestimatetheage‐priceorage‐efficiencyprofileandthenderivethedepreciationprofile,thoughinsomecasesitmaybeappropriatetoestimatedepreciationdirectly(seeBlades,1998a).

3 Retirement distributions

Retirementisthe‘actofputtinganassetoutofservicebecauseithasreachedtheendofitsservicelife’,duetoeitherwearandtearorobsolescence,orperhapsduetolegal

1‘Continuous’inthesenseofafunctionwithdomain ,opposedtothemathematicalsenseofafunctionsatisfyinglim →


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requirement(seeOECD,2009).Intermsofage‐efficiencyprofiles,thisisthesameassayingthattheproductivecapacityofanassetiszero.Werecognisethatingeneralnotallassetsofthesametypehaveidenticallifespansandthatthelifespanofanassetisusuallystochastic.Thismeansweareabletodescribethelifespansofacertainclassofassetswithaprobabilitydistribution.

Everyprobabilitydistributionmaybecharacterisedbythefollowingstatisticalfunctions.TheseparticularformsaregivenastheymaybeusefulinconceptualisingcapitalretirementandinunderstandingtheimpactswithinthePIM.

ProbabilityDensityFunction(PDF):Let : → 0,∞ beafunctionsatisfying0∀ ∈ and 1,then isaprobabilitydensityfunction

(PDF).Wecanthinkof asvalueswhichavariatemaytakeand astheprobabilitythatthevariatetakesthevalueinaninfinitesimallysmallintervalaround .Inthecaseofretirementdistributionswecaninterpret astheageofanassetand astheprobabilitythatanassetretiresatage .Itisconventionthatforacontinuousdistribution , 0∀ ∈ onthebasisthatsincemaytakeuncountablymanyvalues,itsprobabilityofobtaininganyone

particularvalueiszero;ratherwesaythat istheprobabilitythat takesavaluebetween and .

CumulativeDistributionFunction(CDF):Let : → 0,1 beafunctionsatisfying(i)lim → 0andlim → 1;(ii) ismonotonicallydecreasing,i.e. ⇒ ;(iii) iscontinuousfromtheright,i.e.lim →

∀ ∈ .Then isacumulativedistributionfunction(CDF).Foranygivendistribution,thePDFandCDFarerelatedasfollows:

(5)

Thus istheprobabilitythatthevariatetakesavaluelessthan ,orinourcase,theproportionofassetswithlifespanlessthan .

Thesurvivalfunction(alsoreferredtoasareliabilityfunction):ifavariatehasCDF ,thesurvivalfunctionisgivenby

1 (6)

Itshowswhatproportionoftheoriginalmembersofthegroupofassetsarestillinserviceateachpointduringthelifetimeofthelongest‐livedmemberofthegroup(Blades,1983).Inorderwords,theprobabilityofsurvivingatleasttotimepoint .Inpractice,bankruptcyandscrappagecanbedealtwithseparately.

Thehazardfunction:ifavariatehasPDF andsurvivalfunction ,then

thehazardfunctionisdefinedastheratiooftheprobabilitydensityfunctiontothesurvivalfunction:

(7)

Inthestatisticalliteraturethisisalsoreferredtoastheconditionalfailuredensityfunction,intensityfunction,theage‐specificfailurerate,instantaneousfailurerate,ortheforceofmortalitybecauseitreflectstheprobabilitythatthe


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eventoccursatagiventime,conditionalontheassethavingsurvivedupuntilthattime.

ForapplicationofthePIM,onlythePDFandthesurvivalfunctionofaretirementdistributionarenecessarythoughinthefollowingsectionwegivethehazardfunctionforcompleteness.

Manyofthedistributionsbelowtakestrictlypositivevaluesalongthewholerealline,whichistosaythatforanyage thereisanon‐zeroprobabilitythatanassetmaylivetothatage,thoughitmaybesmall.Forpracticalpurposesitisusefultoassumeeveryassetclasshasafinitemaximumlifespan ,andthatassetslivesdonottakenegativevalues.ThismaybedonebycarryingoutthePIMcalculationsonlybetween0and ,whichisequivalenttocuttingoffthetailsoftheretirementdistribution:if istheformoftheretirementdistributionofchoice,thenthedistributionusedinpracticeis

, (8)

(where istheindicatorfunction)whichhastheproperty 1,introducingasmalldownwardbiasintheresult.Alternativelywemaytruncatethedistributionatthepointofuse.Thisisequivalenttocuttingoffthetailsofthedistributionasbeforeanddistributingthelostareabetweenthepointsoftruncation.AdistributionwithPDF andCDF maybetruncatedusingthefollowingformula:

,

0 (9)

whichpreserves 1.

4 Examples of functional forms used within the PIM

Wenowlistsomeofthefunctionalformscommonlyusedforage‐profilesandretirementdistributions.Someformsareusedforbothage‐profilesandretirementdistributionsandinthesecasesthefunctionusedfortheage‐profilewillcorrespondtothesurvivalfunctionofthecorrespondingretirementdistribution.Noticethatage‐profilesandsurvivalfunctionsaresimilarinthatbothareinitiallyequalto1andtendtowards0.Forthisreasonwedonotplottheexampleage‐profilesinadditiontosurvivalfunctions.Itisimportanttorecognisethatage‐profilesarenotthesameassurvivalfunctions:theage‐profileshowsdeterministicallyhowthevalue(orproductivity)ofanassetdeclinesovertimewhereasthesurvivalfunctiongivestheprobabilityoftheassetsurvivingtoagivenageandsaysnothingaboutitsvalue(orproductivity)pathingettingthere.Anyassetclassmayhaveage‐profilesofoneformandretirementdistributionsofanotherdifferentform.Moreover,noteveryage‐profilewillbeabletoberepresentedasasurvivalfunctionsinceage‐profilesmaybeincreasingonpartoftheirdomain(seeBlades,1998a).Intheexamplesbelow, referstothemeanassetlifelengthinthecontextofaretirementdistribution,buthaslessnaturalinterpretationinthecontextofage‐profileswhereitreferstomeanvalueorefficiency.Withregardtoage‐profiles,themaximumassetlifelength, isofgreaterinterest.


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Name Simultaneousexit/Onehossshay

Background Consideredasaretirementdistribution,thisreflects thecasewhereallassetsliveforexactlythesamelengthoftime, ,atwhichpointtheyretire.Asanage‐profile,thisdescribesthecasewhereanassetretainsitsfullproductivecapacityorvalueuntiltheendofitslife, ,whenitsproductivityorvaluedropinstantaneouslytozero.

Theoriginoftheterm‘onehossshay’isdescribedhere:http://stats.oecd.org/glossary/detail.asp?ID=1904

Reading Referencesinclude:

OECD(2009,p114) OECD(1997,p6)

Formula Theprobabilitydensityfunctionisgivenby

∞when

0when

Thesurvivalfunctionandage‐profilearegivenby

1

where0 .

andthehazardfunctionisgivenby

∞when

0when0

Themeanageofretirementis .

Example Thesimultaneousexitfunctioncanbeapproximatedbyeitherusingthelineardistributionandchoosing → ,orconsideringanormaldistributionwherethevarianceparameterapproacheszero.Theareaunderthedensitycurvewillstillequal1and istheaveragelifelength.

Density Survival Hazard

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Name Delayedlinearfunction

Background Retirementdistribution:(referredtoastheuniformdistribution)Assetsareequallylikelytoretireatanyagebetweentheminimumage (whichmaybezero)andthemaximumage .

Age‐profile:reinterpreting asthetimeatwhichthevalueoftheassetbeginstofall,thevalueorefficiencyofanassetdeclinesbythesameamounteachperiod.

Setting 0reducestothelinearcase.FunctionsfortheuniformdistributionareavailableinthebasepackageofR(www.r‐project.org).


OECD(2009,p114) OECD(1997,p2) NIST/SEMATECH(2012)withspecificreference

http://www.itl.nist.gov/div898/handbook/eda/section3/eda3662.htm

Formula Let bethemaximumlifelengthand theminimumlifelengththenthePDFisgivenby:

1

where .

Thesurvivalfunctionandage‐profilearegivenby

where0 1


1

where0 1.

Themeanageofretirementisgivenby .

Example Examplewhere istheaveragelifelength,andthereisadelayof whichimplies, 2 ,and 2 .Thisgives 1/ 2 , 1 / 2 ,and

1/ 2 .


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Name Geometricdistribution

Background Thegeometricdistributionisadiscretefunction.FunctionsforthegeometricdistributionareavailableinthebasepackageorR(www.r‐project.org).

Efficiency/valuefallsbyaconstantproportioneachperiod.Lifespanisinfinite;inapplicationsathresholdischosenwhentheefficiency/valueistakentobezero.


http://mathworld.wolfram.com/GeometricDistribution.html OECD(2009p93,96‐97) Biorn(1998,p618)

Formula Thegeneralformulafortheprobabilitydensityfunctionisgivenby

1

where0 1, 1 ,andthemeanis 1 / .

Thedistributionfunctionisdefinedas

1 1

Thesurvivalfunctionisthengivenby

1


1

Supposewewishourthresholdvalueatwhichweretiretheassettobe ,thenwecanselectavalueof suchthattheassetretiresatexactlyage :

1

⇒ 1

Example Geometricdistributionwith: 0.5 andwhere istheaveragelifelength.Noticethatforthisvalueof ,thedensityandsurvivalcurvescoincideandthehazardfunctionisconstantat1.

Notes Blades(1998a,p7)notesthatthegeometricdepreciationiscalculatedbyapplyingaconstantrateofdeclinetotheinitialvalueoftheassetandthatdouble‐decliningbalancedepreciationissimilartogeometricdepreciationexceptthateachyear’sdepreciationissetattwicethatofthepreviousyear’sdepreciatedvalue.Essentially,thiscanbecapturedwithaparameterchangeinthestandardgeometricdistribution,e.g.thechoiceof .Foradetaileddiscussionondouble‐decliningbalancealsoseeStatisticsCanada(2007p13and14).ThisisalsodiscussedinOECD(2009,p97).SNA2008p418paragraph20.22alsocoversgeometricaspects.

Setting givesthedoubledecliningbalance,noticethatlim → exp 2 .


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Name Hyperbolicform(alsoreferredtoasbeta‐decay)

Background Thehyperbolicformhasbeenhistoricallyusedforthedevelopmentofarelevantage‐efficiencyfunction.Itdescribesassetswhichdecaygraduallyatfirstandmorequicklywithage.WithinOECD(2009),theconceptofage‐efficiencyisconsideredtobeequivalenttotheformusedforthesurvivalfunction.Botharefunctionswhichgiveasurvivalorfullefficiencyattime0,andthenovertimereducetozero.Ifweacceptthiscomparison,thenequivalentprobabilitydensityfunctionsandhazardfunctionscanbederivedbasedonahyperbolicformforthesurvivalfunction.Thisisgivenbelow.

NotethatmoregeneralstatisticalfunctionsforthehyperbolicdistributionareavailablethroughtheRpackageGeneralizedHyperbolic:

(http://cran.univ‐lyon1.fr/web/packages/GeneralizedHyperbolic/index.html)


OECD(2009,p92) http://www.federalreserve.gov/releases/g17/CapitalStockDocLatest.pdfp9 Blades(1998,p2)

Formula Let bethemaximumservicelifeand assumethatthesurvivalfunctionisofthehyperbolicform

where0 and0 1.

Thisimplies(as 1 )thatthecumulativedensityfunctionisgivenby

1

Derivingtheprobabilitydensityfunctionbydifferentiating meanswecanderive

1

Sothehazardfunctionisgivenby

1

Notethatif 0thenformulaisequivalenttothelineardistribution.

Example Hyperbolicformwith: 0.5, 10 andwhere istheaveragelifelength.


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Name Weibulldistribution

Background TheWeibulldistributionisnamedafterWaloddiWeibull whoformulateditin1951todescribethefatiguelifeofsteel.ItisusedbyanumberofcountriesasaninputintothePIMcalculations.

Retirementdistribution:TheWeibullfunctiontakesallpositiverealvaluesandshouldbetruncatedfor and 0.

Age‐profile:theshapeparameter,γ,maybechosentoreflectwhetherthevalueofefficiencyoftheassetrisesorfallsintheearlyperiodofitslife,andhowlongthisperiodlasts.Setting 1reducestotheexponentialdistributionwhichisthecontinuousanalogueofthegeometricdistribution.StatisticsCanada(2007p17‐18)usestheWeibullfunctiontoempiricallyestimateage‐priceprofiles.

FunctionsfortheWeibulldistributionareavailableinthebasepackageorR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.


Meinen,G.et.al(1998,p14) OECD(2009,p115)whichdiscussesrangesforWeibullparametersbasedonStatistics

Netherlandsresearch. NIST/SEMATECH(2012)

http://www.itl.nist.gov/div898/handbook/eda/section3/eda3668.htm StartisticsCanada(2007,p17‐18)

Formula Fora3‐parameterWeibulldistribution,theprobabilitydensityfunctionisgivenby

/

where =shapeparameter, =scaleparameterand =locationparameter.When 1and 0thisisthestandardWeibulldistribution.Thesurvivalfunctionisgivenby

where 0, 0,andthehazardfunctionisgivenby

where 0, 0.

Themeanisgivenby 1 1 ,whereΓisthegammafunction.

Example Weibulldistributionwith: 1, 1, andwhere istheaveragelifelength.

Weibulldistributionwith: 2.5, 1, .


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Name Sumofthedigits

Background Blades(1998a,p7)givesthefollowingdescription:

Sumofthedigitsisamethodofdepreciatinganassetbyanamountwhichdeclineslinearlyoverthelifeoftheassetsuchthatthedepreciationaccumulatedoverthelifetimeoftheassetsequalsitsinitialvalue.Foryeart,theamountofdepreciationisobtainedbymultiplyingtheinitialvalueoftheassetby

1 /

Thedenominatoristhe“sumofthedigits”ofL,i.e.,15+14+...+1=120.Thefirstyear’sdepreciationisthereforecalculatedas15/120oftheinitialassetvalue,thesecondyear’sdepreciationis14/120oftheinitialvalueandsoon.


Blades(1998a,p7) StatisticsCanada(2007,p13‐14)

Formula Noticeintheabovedescriptionthatdepreciationisgivenasanabsoluteamountinagivenyear,notasarateofdeclineofvalueasgivenabove.Wederivethecorrespondingage‐priceprofileasfollows.Ifthevolumeofdepreciationoccurringattime isgivenby ,thenthetotalvolumeof

depreciationbytime willbegivenby

2 11

21

1

21

11

22 1

1

Thusthevalueattime willbe

12 1

1

Example Thesumofthedigitsfunctionhasthefollowinggraph:

Sum of Digits

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Name Generalconvex/concavecurves

Background Aconvexfunction : → satisfies

1 1

forany , ∈ and0 1,whichistosaythatthecurveisbowedtowardstheorigin;converselyaconcavefunctionisbowedawayfromtheorigin:

1 1


Biorn(1998,p619‐621)Formula Let betheasset’slifespan,then

1

1

where , 0givesthedegreeofcurvaturefor and respectively. isstrictlyconvexfor 1andstrictlyconcavefor 1.Thereverseistruefor ,whichisstrictlyconcavefor 1andstrictlyconvexfor 1.Both and reducetothelinearcasefor , 1.

Example Thefollowingillustrateshowthefunction maybeeitherconvexorconcavedependingonchoiceofparameter(thefirstandsecondgraphs),andhowthetwofunctionscomparewiththesamechoiceofparameter(thesecondandthirdgraphs)

Notes Althoughboth and maybeeitherconcave orconvex,theyarenotequivalentfor , 1.Inotherwords,afunctionofform cannotbealternativelypresentedasafunctionofform .Otherformsfromstatisticaldistributionscouldbeusedtogenerateconvexorconcavecurves.

g(x;s=1/2) g(x;s=2) f(x;s=2)

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Name Logistic

Background Usedforage‐profiles,valueorefficiencyfallsslowlyatfirst,thenveryquicklyinitsmidlife,slowingdownagainneartheendoflife.Thepointatwhichtheassetdepreciatesofdeterioratequicklyisdeterminedbythelocationparameter,andthedifferencesbetweentheratesofdeclineacrosstheasset’slifearegivenbyashapeparameter.

Aswithsomeoftheotherage‐profilefunctionsdescribedhere,thelogisticfunctionneverreaches0ontherealline,moreoverthereisnovalueforwhichthelogisticfunctionequals1.Thuswehavetoimposetheserestrictionsartificially.


Meinenet.al(1998,p29)Formula Let bethelocationparameterand bethescaleparameter,thelogisticfunctionisgivenby

exp

1 exp

Noticethat foranyvalueof ,thuswerequirethat0 ,where istheasset

lifelength.Adefaultoptionmightbetoset .Theshapeparametercanthenbechosentoensurethat 0 1and 0.

Example Thefollowinggraphsillustratethelogisticfunctionwithvaryinglocationsandshapeparameters. isthemaximumlifelength.

m=T/2 s=T/10 m=2T/3, s=T/20 m=T/2, s=T/5

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Name Normaldistribution

Background Thenormaldistributionisoftenusedtodescriberetirements.Itisspecifiedbyameanageofretirement,µ,andthestandarddistribution, ,ofretirements.Thenormaldistributionhastheusefulpropertythatapproximately95%ofretirementsoccurwithintwostandarddeviationsofthemeanandapproximately99%ofretirementsoccurwithinthreestandarddeviationsofthemean.Sincethereisnoimpliedmaximumorminimumlifelength,itisnecessarytousetruncationinapplications.FunctionsforthenormaldistributionareavailableinthebasepackageofR(www.r‐project.org).


OECD(2009,p118) NIST/SEMATECH(2012)

http://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htmFormula Thegeneralformulafortheprobabilitydensityfunctionisgivenby

/ 2

√2

1∅

where isthestandarddeviationand isthemean.When 1and 0thisisthestandardnormaldistribution.Thefollowingfunctionsareforthestandardform.

Thecumulativedistributionfunctionisdefinedas

Φ1

√2/2


1 Φ


,∅Φ

Example Normaldistributionwith: , 1 where istheaveragelifelength.

Normaldistributionwith: , 2.5.


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Name Truncatednormaldistribution

Background FunctionsforthenormaldistributionareavailablethroughtheRpackagemsm

(http://cran.univ‐lyon1.fr/web/packages/msm/index.html)

Reading SeeNormaldistributioninformation.

Formula Thegeneralformulafortheprobabilitydensityfunctionisgivenby

1∅

Φ Φ

where =scaleparameter, =locationparameter, =lowerlimit,and =upperlimit.Φ isthecumulativedistributionfunctionforthenormaldistribution,and∅ isdefinedasperthenormaldistribution.

Thecumulativedistributionfunctionforatruncatednormalisthendefinedas

ΦΦ Φ

Φ Φ


1 Φ


,Φ

Example Normaldistributionwith: , 1, withcut‐offsofupperlimitof3,andlowerlimitof‐3timestheaveragelifelength,where istheaveragelifelength.

Normaldistributionwith: , 2.5,withcut‐offsofupperlimitof3,andlowerlimitof‐3timestheaveragelifelength,where istheaveragelifelength.


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Name Log‐normaldistribution

Background Usedasaretirementdistribution.DefineX tobearandomvariablewithanormaldistribution.ThenY=exp(X)willhavealog‐normaldistribution.Similarly,ifYhasalog‐normaldistribution,thenX=log(Y)willhaveanormaldistribution.Thelog‐normaldistributiononlytakespositiverealvalues.Aswiththenormaldistribution,itisnecessarytotruncatethelog‐normaldistributionatsomepre‐defined and ,unlesstheminimumlifelengthiszero.

Functionsforthelog‐normaldistributionareavailableinthebasepackageofR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.



http://www.itl.nist.gov/div898/handbook/eda/section3/eda3669.htmFormula Fora3‐parameterlog‐normaldistribution,the probabilitydensityfunctionisgivenby

exp / / 2

√2

where ; , 0.Also =shapeparameter, =scaleparameterand =locationparameter.When 1and 0thisisthestandardlog‐normaldistribution.Thefollowingfunctionsareforthestandardform.

Thesurvivalfunctionisgivenby

1 Φln

where 0, 0,andΦisthecumulativedistributionfunctionofthenormaldistribution.Thehazardfunctionisgivenby

,

1∅

ln

Φln

where 0, 0,and∅istheprobabilitydensityfunctionofthenormaldistribution,andΦisthecumulativedistributionofthenormaldistribution.

Themeanisgivenby exp μ

Otherparameterisationscanbeused,e.g.aswithinOECD(2009,p118).

Example Log‐normaldistributionwith: 1, 1, where istheaveragelifelength.


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Name Gammadistribution

Background Thegammadistributiontakesallpositiverealvalues,soaswiththelog‐normaldistribution,itisnecessarytotruncateit andat ifthisisdifferentfromzero.FunctionsforthegammadistributionareavailableinthebasepackageofR(www.r‐project.org),orthroughtheRpackageFAdist(http://cran.r‐project.org/web/packages/FAdist/)whichincludesa3‐parameterversion.



http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htmFormula Fora3‐parametergammadistribution,theprobabilitydensityfunctionisgivenby

Γ γ

where ; , 0.Also =shapeparameter, =scaleparameter, =locationparameter,and

Γ

When 1and 0thisisthestandardgammadistribution.Thefollowingfunctionsareforthestandardform.

Thesurvivalfunctionisgivenby

1ΓΓ

where 0, 0,andΓ istheincompletegammafunctiondefinedas

Γ


Γ γ Γ

where 0, 0.

Themeanisgivenby .

Example Gammadistributionwith: 2, 1, where isthemedianlifelength.

Notes Schmalwasser,O.andSchidlowski,M.(2006,p9) describestheGammadistributionusedwithinGermanyforthemortality(density)functionbutwithadifferentparameterisation.Referringtotheirpapertheparametersare: , μ, .


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Name Winfreycurves(symmetric)

Background ThisisnamedafterRobleyWinfrey,whowasaresearchengineerattheIowaEngineeringExperimentationStationduringthe1930s.Winfreycollectedinformationondatesofinstallationandretirementof176groupsofindustrialassetsandcalculated18“type”curvesthatgavegoodapproximationstotheirobservedretirementpatterns.Therewere7symmetrical,6leftskewedand5rightskewed(seeWinfrey,1935;OECD,2009;andBlades,1998b).


Winfrey(1935).Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised

OECD(2009,p115) Blades(1998b).Link:http://www.oecd.org/std/nationalaccounts/2662425.pdf Hypergeometricfunction:http://mathworld.wolfram.com/HypergeometricFunction.html

Formula TheformulaisgiveninWinfrey(1935)whileBlades (1998b)alsoprovidesanexcellentsummary.FortheWinfreysymmetricalcurves,theequationderivedbyWinfreyis

1

where istheordinatetothefrequencycurveatage (originatthemeanage), istheordinatetothefrequencycurveatitsmode,and ,and areparameterswhichdeterminethekurtosisandtheskewnessofeachcurve.Blade(1998b)givesexamplesforthechoiceofparameters.

Theexistingliteraturedoesnotgiveaformaldefinitionintermsofstatisticaldistributions,oranequivalentsurvivalorhazardfunction.However,afterinvestigationswecandemonstratethatthehypergeometricfunctionisactuallyequivalenttotheexpressionabovefortheWinfreycurves.Forexample,usingaspecificcaseandnotationofthegeneralisedhypergeometricfunction,thisgives

, ; ; 1

Sowithanappropriatechoiceofparameters,wemaketheconjecturethatthewellknownWinfreycurvesforthesymmetriccaseareactuallyequivalenttoahypergeometricfunctionwiththefollowingparameters

, ; ; 1

Sodefiningthisasourprobabilitydensityfunctionmeansthatthesurvivalandhazardfunctionwillhavetheform

1 , ; ; , ; ; andthen .

Example Examplewith 10, 3.7, 11.911 whichisequivalenttoS2inOECD(2009p117),whereistheaveragelifelength.


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Name Winfreycurves(asymmetric)

Background SeeWinfrey(symmetric)fordetails.Winfrey(1935)givesdetailson18“type”curvesthatgavegoodapproximationstoobservedretirementpatterns.Therewere7symmetrical,6leftskewedand5rightskewed(seeWinfrey,1935;OECD,2009;andBlades,1998b).


Winfrey(1935).Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised

OECD(2009,p115) Blades(1998b).Link:http://www.oecd.org/std/nationalaccounts/2662425.pdf Hypergeometricfunction:http://mathworld.wolfram.com/HypergeometricFunction.html

Formula DetailedformulafortheasymmetriccurvesaregiveninWinfrey(1935),whileBlades(1998b)alsoprovidesanexcellentsummaryanddiscussion.Whilesimilartothesymmetricalcurves,theasymmetricformulasaremuchmorecomplicated.Forexample,

1 1 1 1

where istheordinatetothefrequencycurveatage (originatthemeanage), , aretheordinatetothefrequencycurveatitsmean,and , , , , , , , , , areparameterswhichdeterminethekurtosisandtheskewnessofeachcurve.ExamplesforchoiceofparametersaregiveninWinfrey(1935,p98).Inpracticeitseemsthatthe signischangedinanarbitraryway.

Again,theexistingliteraturedoesnotgiveaformaldefinitionoftheWinfreycurvesintermsofstatisticaldistributions,ortheassociatedsurvivalorhazardfunctions.Basedontheapproachforthesymmetriccase,theasymmetriccurvescanberewritteninaformwhichincludescombinationsofthehypergeometricdistribution.However,thecombinationofatleastfourcurveswillleadtoacomplicatedsolution.Forexample,

, ; ; , ; ;

, ; ; , ; ;

Fromthis,thesurvivalandhazardfunctionscanbederivedthroughstandardmethods(e.g.integration)althoughtheywillbecomplicatedandnotreproducedhere.

Example DensityexampleswithdifferentparameterchoicesforL2(red),L3(blue),R3(black)andR4(green).Averagelifelength isalongthex‐axiswherewillbedifferentforeachdistribution.

Notes WhileWinfrey(1935)hasderivedacomplicatedformfortheasymmetriccurves,furtherworkcouldconsiderwherethiscouldbesimplified.Forexample,ratherthantheconvolutionoffourhypergeometrictermsperhapstheuseofjustoneorthemultiplicationoftwohypergeometricfunctionswithappropriateparameterchoicewouldprovideareasonablealternative.

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5 Examples with cohorts

Onceanage‐profileforasingleandaretirementdistributionhavebeenfoundforaclassofassets,weareabletoderivetheage‐profileforacohortofassets.OECD(2009)definesacohortas:

Cohortofassets:Setofassetsofthesamekindandsameage,butnotnecessarilythesamelifespan.

Theage‐profileofacohortofassetsdescribestheproductivityorvalueofthewholecohortatanygivenpointoftime.FollowingthenotationusedinOECD(2009),let denotetheage‐profileofasingleassetand betheage‐profileofthecohort;each isgivenbytakingtheaverageofthe ofalltheindividualassetssurvivinginperiod ,weightedbytheretirementdistribution.Thismaybedonebydefiningadifferentage‐profileforeachretirementageinthedistribution,orbyusingasingleage‐profileforthehighestageintheretirementdistribution.Weexaminethesetwoapproachesinmoredetailbelow.Inbothcasesweneedtospecify:

Amaximumassetlife Afunctionalformfortheage‐profile Aretirementdistribution

Theretirementdistributionmayormaynotbetruncated.

Method1:seeOECD(2009,p118‐121).Foreachlife0 wedefineauniqueage‐profile(noticethatalltheage‐profilesabovearegivenasfunctionslifespan ),

, ,i.e.where , denotestheefficiencyorvalueofan yearoldassetwithlifespan.Thentheefficiencyorvalueofthecohortinyear isgivenbytheaverageof , ,for0 ,weightedbytheprobabilityofanassethavinglifespan ,givenby :

, ∙ (10)

wherewestarttheindexat because , 0forall .

Weillustratetheprocessinthefollowingtable.

year 0 1 2 ⋯ 1 ret.dist.

1 , 1 , 0 1

2 , 1 , , 0 2

⋮ ⋮ ⋮ ⋮ ⋱ ⋮

1, 1 , , ⋯ , 0

1

, 1 , , ⋯ , ,0

, , ,

⋯, 0

Weconjecturethatacontinuousanalogueofthismaybegivenby


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, ∙ (11)

where , isacontinuousage‐profilewith denotingageand ,retirementage.

Method2:seeOECD(2009,p121).Wedefineasingleage‐profileforanassetwithretirementage ,callit .Thistimetheage‐profileforanassetretiringatage

takesthefirst 1termsof andiszerothereafter.Asabove,thetermsofthecohortprofilearetheweightedaverageoftheindividualprofiles.Tocomparethiswiththepreviousmethodweconstructatableanalogoustotheoneabove,thoughthisisnotstrictlynecessary.

year 0 1 2 ⋯ n‐1 n ret.dist.

1 1 1

2 1 2

⋮ ⋮ ⋮ ⋮ ⋱ ⋮

n‐1 1 ⋯ 1

n 1 ⋯0

⋯ 0

Weseeinthelastlinethattheformulasimplifiesto

(12)

Infactthelastterm,∑ ,isadiscreteestimateoftheareaunderthePDFboundedby and ,whichisgivenpreciselybythesurvivalfunction, .Thispaperdoesnotprescribetheuseofoneortheother,howeverwedo,asformethod1postulatethatacontinuousanaloguemaybegivenby

(13)

Wenowpresentanumberofgraphsshowingafewdifferentcombinationsofage‐profileandretirementdistributions.Ineachcaseweassumeminimum,maximumandmeanassetlivesof2,16and9.5yearsrespectively,inlinewiththeexamplein(OECD2009,p118‐121)andillustratetheresultsofbothmethods,whilealsoreplicatingtheexamplewithinOECD(2009).Inallcases,graphaillustratesthecohortprofileresultingfrommethod1(inred)andaselectionofindividualassetprofiles(inblack);graphbcomparesthemethod1andmethod2cohortprofiles(redandbluerespectively)andtheindividualassetprofilecorrespondingtothemaximumlifelength(inblack);graphscanddshowthePDF(usedinmethod1)andsurvivalfunction(usedinmethod2)respectively.


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Linearage‐profile,normalretirement

Fortheage‐profileweassumedecayisnotdelayed,intheretirementdistributionweuseameanof9.5yearsandastandarddeviationof2.021inlinewith(OECD,2009p118‐121).Wenotethatthisgivesrisetoastrictlyconcavecohortprofileundermethod1andalogisticshapedcohortprofileundermethod2.

Geometricage‐profile,normalretirement

Anotherpopular,andindeedtheonlyotherprescribedwithinESA10 forthechoiceofage‐profileisthegeometricprofilewith 0.02 ⇒ 0.22.Thistimeboththecohortprofilesarebroadlygeometricinshapeandmoreconcavethanthepreviouscase.

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Linearage‐profile,Weibullretirement

Thistimewevarytheretirementdistribution,usingaWeibull distributionwithshape,scaleandlocationparameters: 17, 20,and 10respectively.Theoutcomeissimilartousingthenormaldistribution.

Linearage‐profile,normalretirement(specialcase)

Thisexampleisidenticaltothefirstexceptthatwehavereducedthestandarddeviationto0.1.Thisillustratesapotentialissuewiththemethodologysincethemethod1cohortprofilenowdropsalmostimmediatelytozero,lowerthananyoftheindividualprofiles.ThisoccursbecausethePDFis(closeto)zeroeverywhereoutsideasmallintervalaroundthemeanvalue,9.5,whichliesbetweentwointegers.TheconsequenceisthatthePDFiszeroforallthe(integer)values 1 , … , ,(seetableabove),usedinthecalculation.Ifthemeanhappenstofallnearaninteger,theoppositeoccursandthecohortprofileovershootsanyoftheindividualprofiles.ThisproblemisliabletooccurwhereverthePDFhasasteepgradientandmayberesolvedbyreducingthesizeoftheincrementsatwhichthecalculationiscarriedoutorusingthecontinuousversionoftheformula.

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Whatisclearfromthecohortprofilesaboveisthattheseoftenexhibitsimilarshapestothesingleassetprofiles,andweaskwhetheritmightbemorefeasibletoestimatecohortprofilesdirectly,ratherthanestimatingandcombiningseparateassetprofilesandretirementdistributions.Besidesbeingsimplertoimplement,thiswouldavoidanotherproblem:ifaretirementdistributionhasasmallstandarddeviation(orthelengthofperiodisgreat)andameanneartheminimumlifelength,itispossiblethatthecohortprofilewillbegreaterthan1evenwheresingleassetprofilesaremonotonicallydecreasing.Thisimpliesthattheefficiencyofacohortofassetscanexceedthatofeventhemostefficientassetinthecohort,whichisclearlyunrealistic,andarisesfromthediscretenatureofthecalculation.Similarlyitispossiblethatthereversecanhappenandtheefficiencyofthecohortislowerthaneventheleastefficientassetinthecohort.

6 Conclusion

Thispaperhasdescribedarangeofdistributionswhichareusedinpracticeforcalculatingestimatesofcapitalconsumption.Pointstohighlightinclude:

Inpractice,thechoiceofretirementdistributioncanbesomewhatarbitrary.Therearesomeconventionsandrecommendedpracticesbutthisdoesdependontheassetclassunderconsideration.

Priceandefficiencyprofilessharemanycommoncharacteristics.Theycannotbedefinedindependently,butonecanbederivedfromtheother.

Itisoftennecessarytotruncateretirementdistributionsinpracticalapplications.

Itmaybefeasibletoestimatetheage‐profileofawholecohortinonestep,ratherthancombiningasingleassetprofileandaretirementdistribution.

Usingasteepretirementdistributionatalowresolution(e.g.largeincrementsoftime)canproduceunreasonableresultsincalculatingcohortprofiles.

Furtherworkcouldexploretheimpactofthechoiceofretirementdistributionandsingleassetprofileondifferentcohortprofiles,andfromapracticalperspectiveofwhattypesofdistributionsareappropriatefordifferentassets,seeforinstance(Meinenet.al,1998p28‐34).

7 References and further reading

Biorn,E.(1998),Survivalandefficiencycurvesforcapitalandthetime‐age‐profileofvintageprices,EmpiricalEconomics,23,p611to633.Partialpdflink:http://link.springer.com/article/10.1007%2FBF01205997

Blades,D.(1983),Servicelivesoffixedassets,OECDWorkingpaper,March1983.Link:http://www.oecd.org/economy/productivityandlongtermgrowth/1915571.pdf


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Blades,D.(1998a),MeasuringDepreciation,SecondmeetingoftheCanberraGrouponCapitalStockstatistics.Link:http://www.oecd.org/std/na/2662743.pdf

Blades,D.(1998b),MortalityFunctionsforEstimatingCapitalStocks,SecondmeetingoftheCanberraGrouponCapitalStockstatistics.Linkhttp://www.oecd.org/std/na/2662425.pdf

Dey‐ChowdhuryS.(2008),Perpetualinventorymethod:methodsexplained,EconomicandLabourMarketReview,Vol2,No.9,September2008,OfficeforNationalStatistics.

McLellan,N.(2004),MeasuringProductivityusingtheIndexNumberApproach:AnIntroduction.NewZealandTreasury,Workingpaper04/05,June2004.Link:http://www.treasury.govt.nz/publications/research‐policy/wp/2004/04‐05/twp04‐05.pdf

Meinen,G.,Verbiest,P.,anddeWolf,P.(1998)PerpetualInventoryMethod:Servicelives,discardpattersanddepreciationmethods,OECD.Link:http://www.oecd.org/std/na/2552337.pdf

NIST/SEMATECH(2012)e‐HandbookofStatisticalMethods,Link:http://www.itl.nist.gov/div898/handbook/

OECD(1997)MortalityandSurvivalFunctions,CapitalStockConferenceMarch1997,Link:http://www.oecd.org/std/nationalaccounts/2666812.pdf

OECD(2009)MeasuringCapital:OECDManual,Secondedition.Link:http://www.oecd.org/dataoecd/16/16/43734711.pdf,Chapter13.1andalsoAnnexA.

OfficeforNationalStatistics(2014)CapitalStocksandConsumptionofFixedCapitalpublication,link:http://www.ons.gov.uk/ons/rel/cap‐stock/capital‐stock‐‐capital‐consumption/capital‐stocks‐and‐consumption‐of‐fixed‐capital‐‐2013/index.html

Oulton,N.,andSrinivason,S.(2003)Capitalstocks,capitalservices,anddepreciation:anintegratedframework,BankofEngland.

StatisticsCanada(2007)EconomicDepreciationandRetirementofCanadianAssets:AComprehensiveEmpiricalStudy,Link:http://www.statcan.gc.ca/pub/15‐549‐x/15‐549‐x2007001‐eng.pdf

Schmalwasser,O.andSchidlowski,M.(2006)MeasuringCapitalStockinGermany.Link:https://www.destatis.de/EN/Publications/Specialized/Nationalaccounts/MeasuringCapitalStockWista1106.pdf


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UnitedNations,SystemofNationalAccounts(SNA2008).Link:http://unstats.un.org/unsd/nationalaccount/sna2008.asp

West,P.andClifton‐Fearnside,A.(1999).ImprovingtheNon‐financialBalanceSheetsandCapitalStocksEstimates,OfficeforNationalStatistics.

Winfrey,R.(1935).StatisticalAnalysisofIndustrialPropertyRetirements,EngineeringExperimentStationBulletin125,Revised.Link:http://www.scribd.com/doc/34898535/Statistical‐Analysis‐of‐Industrial‐Property‐Retirements‐Engineering‐Experiment‐Station‐Bulletin‐125‐Revised

8 Acknowledgements

TheauthorswouldliketothankErikBiørn,SteveDrew,WesleyHarrisandJimO’Donoghuefortheirfeedbackandcommentsinpreparingthisarticle.Ofcourse,anyerrorsandomissionsareourresponsibility.

9 Disclaimer

Theviewsinthispaperdonotnecessarilyreflecttheofficialviews,endorsedorcurrentlyusedmethodsoftheOfficeforNationalStatistics.

ons methodology working paper series no 3 a note on

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