1 Gran-in-Aid for Scienific Research(S) Real Esae Markes, Financial Crisis, and Economic Growh : An Inegraed Economic Approach Working Paper Series No.4 A Concepual Framework for Commercial Propery Price Indexes Erwin Diewer Chihiro Shimizu December, 2013 HIT-REFINED PROJECT Insiue of Economic Research, Hiosubashi Universiy Naka 2-1, Kuniachi-ciy, Tokyo , JAPAN Tel: hp://www.ier.hi-u.ac.jp/ifn/
2 1 A Concepual Framework for Commercial Propery Price Indexes Erwin Diewer and Chihiro Shimizu 1 Ocober 8, 2013 School of Economics, The Universiy of Briish Columbia, Vancouver, Canada, V6T 1Z1. Absrac The paper sudies he problems associaed wih he consrucion of price indexes for commercial properies ha could be used in he Sysem of Naional Accouns. Propery price indexes are required for he socks of commercial properies in he Balance Shees of he counry and relaed price indexes for he land and srucure componens of a commercial propery are required in he Income Accouns of he counry if he Mulifacor Produciviy of he Commercial Propery Indusry is calculaed as par of he Sysem of Naional accouns. The paper suggess a varian of he capializaion of he Ne Operaing Income approach o he consrucion of propery price indexes and uses he one hoss shay or ligh bulb model of depreciaion as a model of depreciaion for he srucure componen of a commercial propery. Key Words Commercial propery price indexes, Ne Operaing Income, discouned cash flow, Sysem of Naional Accouns, Balance Shees, mehods of depreciaion, land and srucure prices. Journal of Economic Lieraure Classificaion Numbers C2, C23, C43, D12, E31, R21. W. Erwin Diewer: School of Economics, Universiy of Briish Columbia, Vancouver B.C., Canada, V6T 1Z1 and he School of Economics, Universiy of New Souh Wales, Sydney, Ausralia ( and Chihiro Shimizu, Reiaku Universiy, Kashiwa, Chiba, , Japan and he School of Economics, Universiy of Briish Columbia, ( The auhors hank David Fenwick, Jan de Haan François Des Rosiers, Alice Nakamura, Alicia Rambaldi and Iqbal Syed for valuable commens.
3 2 1. Inroducion Many of he propery price bubbles experienced during he 20h cenury were riggered by seep increases and sharp decreases in commercial propery prices. 2 Given his, here is a need o consruc commercial propery price indexes bu exacly how should hese prices be measured? Since commercial propery is highly heerogeneous compared o housing and he number of ransacions is also much lower, i is exremely difficul o capure rends in his marke. In addiion, many counries have been experiencing large invesmens in commercial properies and in counries where he marke has maured, depreciaion and invesmens in improvemens and renovaions represens a subsanial fracion of naional oupu. Bu clear measuremen mehods for he reamen of hese expendiures in he Sysem of Naional Accouns are lacking. Given his, one may say ha he economic value of commercial propery in paricular is one of he indicaors ha is mos difficul o measure on a day-o-day basis and ha saisical developmen relaed o his is one of he fields ha has perhaps lagged he furhes behind. Indexes based on ransacion prices for commercial properies have begun o appear in recen years, especially in he U.S. However, in many cases, hese indexes are based on propery appraisal prices. Bu appraisal prices need o be based on a firm mehodology. Thus in his paper, we will briefly review possible appraisal mehodologies and hen develop in more deail wha we hink is he mos promising approach. In summary: he purpose of his paper is o presen a framework for he organizaion of commercial propery daa ino a form which would be suiable for naional income accouning purposes and for he consrucion of propery price indexes. Basically, he paper looks a he problems associaed wih he consrucion of oupu and inpu price indexes for periodic flow inpus as well as for he fixed asses used in he commercial propery indusry. Thus secion 2 below looks a he problems associaed wih he consrucion of oupu indexes while secion 3 sudies he problems associaed wih he consrucion of variable inpu indexes such as inermediae inpu and labour indexes. Secions 4 and 5 presen mehodologies for consrucing boh sock and flow indexes for he fixed inpus used by commercial propery firms, namely srucures and he land ha he srucures si on. Secion 6 concludes. I should be noed ha he problems associaed wih consrucing price indexes for commercial properies are more complex when he prices are required for naional income accouning purposes as opposed for oher purposes ha simply require he deerminaion of he join value of commercial propery srucures and he associaed land. 3 For naional income accouning purposes, he value of a commercial propery needs o be decomposed ino land and srucure componens, a problem ha has been largely overlooked in he academic lieraure on commercial propery price indexes. Mos of he difficul problems associaed wih he consrucion of consan qualiy price 2 See Schreyer (2009b) for maerial on real esae bubbles. 3 Commercial propery valuaions may be required for morgage applicaions and for he analysis of propery bubbles where only he join price of he srucures and he associaed land maers.
4 3 indexes for he commercial propery secor are associaed wih he land and srucure inpus used by he secor. Basically, hree disinc mehods have been used o consruc commercial propery price indexes for he fixed asses of a propery firm: Mehods based on commercial propery sales ransacions over ime. Varians of his mehod include he repea sales mehodology and hedonic regression mehods. 4 Mehods based on observing published share prices for propery socks or REITs (Real Esae Invesmen Truss) in order o esimae he asse value of he propery. 5 Mehods based on capializing he Ne Operaing Income (NOI) of a building rus. 6 The major problem wih he ransacions based mehods for deermining he consan qualiy prices for commercial properies is ha sales of commercial properies are very sparse. Thus hedonic and repea sales mehods, which can work well for consrucing consan qualiy price indexes for residenial properies, may no work well in he commercial propery conex due o he infrequency of sales of hese properies. Sock marke based mehods for deermining he prices of commercial properies are subjec o hree problems: The sock marke valuaion of a commercial propery REIT ha has a consan porfolio of properies does no provide a consan qualiy price for he fixed asses in ha porfolio due o he depreciaion of he srucures in he porfolio. For naional income accouning purposes, a decomposiion of he asse value of a REIT ino is srucure and land componens is required and he sock marke valuaion canno provide his breakdown. Sock marke valuaions will generally be oo volaile for saisical agency purposes. The sock marke price of a REIT will ypically over reac o daily news abou macroeconomic evens. 7 4 Appraisers frequenly base heir esimaes for he asse value of a commercial propery on sales of similar properies during he same ime period: Real esae valuaion is founded primarily on he use of comparable sales informaion. Andrew Baum and Neil Crosby (2008; 17). This valuaion mehod can be considered as a ype of informal hedonic regression model. For a fairly comprehensive review of he hedonic regression mehodology used in he valuaion of residenial properies, see de Haan and Diewer (2011). For comparisons of he hedonic regression and he repea sales mehodologies, see Diewer, Nakamura and Nakamura (2009), Diewer (2010) (2011), Shimizu, Nishimura and Waanabe (2010) and Deng, McMillen and Sing (2013). See Devaney and Diaz (2010) for an applicaion of he hedonic regression mehodology o he commercial propery conex. 5 Papers ha use his mehodology include Fisher, Gelner and Webb (1994), Gelner (1997), Gelner, Pollakowski, Horrigan and Case (2010), Bokhari and Gelner (2012) and Shimizu, Diewer, Nishimura and Waanabe (2013). 6 Mehods of his ype are discussed in some deail in Baum and Crosby (2008; 27) and are labelled as discouned cash flow models or capializaion of income approaches. Our paricular varian of his class of mehods o be explained laer can be regarded as a discouned cash flow model.
5 4 Thus in our view, he mos useful mehod for obaining consan qualiy prices for commercial properies is he capializaion of ne operaing income mehod. We will presen a varian of his mehod in some deail in secions 4 and 5 below. Our analysis will give pracical advice on how saisical agencies could consruc boh sock and flow price indexes for commercial srucures and he land ha he srucures si upon. A paricular problem wih exising mehodologies for he capializaion of he NOI of a paricular propery in order o obain esimaes of he curren asse value of he propery is ha hey do no deal adequaely wih he problems raised by he depreciaion of he srucure over ime. These difficulies were discussed by Dixon, Crosby and Law (1999) and Crosby, Devaney and Law (2012) and heir papers conain exensive reviews of he depreciaion lieraure as i applies o commercial properies. However, our suggesed soluion o hese problems is quie differen han he soluions presened in hese publicaions The Consrucion of Oupu Price Indexes We consider he problems associaed wih consrucing quarerly inpu and oupu price indexes for a paricular commercial propery in a single locaion. We assume ha he propery has N sources of revenue and he quarer price for produc n = 1,2,...,N is p n and he corresponding quaniy sold during he quarer is q n. Before proceeding furher, we need o discuss he exac meaning of he microeconomic prices and quaniies if here are muliple ransacions for say commodiy n wihin quarer. In his case, i is naural o inerpre q n as he oal amoun of commodiy n sold wihin quarer. In order o conserve he value of ransacions, i is necessary ha p n be defined as a uni value 9 ; i.e., p n mus be equal o he value of ransacions for commodiy n during quarer divided by he oal quaniy ransaced, q n. Thus define he revenue of he commercial propery in quarer as: (1) R n=1 N p n q n p q where p (p 1,, p N ) is he quarer oupu price vecor, q (q 1,, q N ) is he quarer oupu quaniy vecor and p q denoes he inner produc of hese wo vecors. The oupus produced by an office or reail building will ypically consis primarily of he renal or leasing of individual unis of floor space. The oal floor space leased or rened will generally be well below he oal floor space of he building since some space will be 7 However, an advanage of he share price mehod over he NOI mehod is ha he former mehod may predic urning poins in he propery marke in advance of he laer mehod since sock marke valuaions are forward looking, whereas Ne Operaing Income gives a snapsho of he propery s curren (or slighly backward looking) income generaing abiliies. 8 Our mehod relies on a depreciaion model discussed in he naional income accouning lieraure. 9 The early index number heoriss Walsh (1901; 96), Fisher (1922; 318) and Davies (1924; 96) all suggesed uni values as he prices ha should be insered ino an index number formula. This advice is followed in he Consumer Price Index Manual: Theory and Pracice wih he proviso ha he uni value be a narrowly defined one; see he ILO (2004; 356).
6 5 aken up by hallways, uiliy rooms, careaker and managerial offices. 10 When measuring oupus, rened space is wha couns bu in secion 5 when valuing he cos of he services provided by he basic building srucure, i is oal floor space ha maers. In addiion o leased office and reail space, he building may make addiional revenues from rening parking spaces and oher miscellaneous sources of revenues. 11 If he revenue generaing propery leases (for commercial space in he building) are monhly or quarerly, here is no problem in deermining quarerly revenues. However, if he leases exend longer han a quarer and here is a fixed paymen a he beginning of he lease for he use of leased space ha covers he enire leasing period, hen his oal lease paymen has o be amorized ino impued quarerly paymens over he life of he lease. There are various commercial accouning mehods for accomplishing his amorizaion and he price saisician may have no choice bu o use whaever amorizaion mehod was used in he quarerly saemens of he propery owner. However, if deailed informaion on he building leases is available, hen one hoss shay amorizaion is a useful mehod ha can be recommended. This mehod will be explained in secion 4 below in he conex of amorizing he cos of a building ha lass L quarers afer i is consruced. In he presen oupu price conex, he lengh of life L is now inerpreed as he lengh of he lease. 12 The index number problem can be explained in he following manner. Consider he propery s revenue raio going from say quarer 0 o 1, R 1 /R 0. Index numbers aemp o decompose a value raio for he wo periods under consideraion ino a price change componen P imes a quaniy change componen Q. Thus we look for wo funcions of 4N variables, P(p 0,p 1,q 0,q 1 ) and Q(p 0,p 1,q 0,q 1 ) such ha: 13 (2) p 1 q 1 /p 0 q 0 = P(p 0,p 1,q 0,q 1 )Q(p 0,p 1,q 0,q 1 ). I can be seen ha if he price index funcion P(p 0,p 1,q 0,q 1 ) has been deermined, hen he quaniy index Q(p 0,p 1,q 0,q 1 ) can be residually deermined using equaion (2). If he funcional form for P(p 0,p 1,q 0,q 1 ) is known, hen we can use (2) o deermine he period 0 and 1 aggregae price levels, P 0 and P 1 respecively, and he period 0 and 1 aggregae quaniy (or volume) levels, Q 0 and Q 1 respecively, as follows: 10 However, an indusrial building is ypically leased o a single business and in his case, oal floor space would coincide wih rened floor space. 11 For example, a building may derive revenues from a ransmission ower on he op of he building. 12 The amorizaion mehod is explained below by equaions (13)-(16). If he lease paymen is made a he beginning of quarer and he lengh of he lease is L quarers, hen P 0 in equaion (13) can be inerpreed as he oal lease paymen, f is he quarer impued revenue for he firm, i is he expeced quarerly inflaion rae for he renal of space of he ype under consideraion, r is he firm s opporuniy cos of capial a he beginning of period and (1+i ) s f is he impued revenue for he space under consideraion in quarer +s for s= 1,2,...,L1. 13 If N = 1, hen we define P(p 0 1,p 1 1,q 0 1,q 1 1 ) p 1 1 /p 0 1 and Q(p 0 1,p 1 1,q 0 1,q 1 1 ) q 1 1 /q 0 1, he single price raio and he single quaniy raio respecively. In he case of a general N > 1, we hink of P(p 0 1,p 1 1,q 0 1,q 1 1 ) as being a weighed average of he price raios p 1 1 /p 0 1, p 1 2 /p 0 2,..., p 1 N /p 0 N. Thus we inerpre P(p 0 1,p 1 1,q 0 1,q 1 1 ) as an aggregae price raio, P 1 /P 0, where P is he aggregae price level for period for = 0,1.
7 6 (3) P 0 1; P 1 P(p 0,p 1,q 0,q 1 ); Q 0 p 0 q 0 ; Q 1 p 1 q 1 /P(p 0,p 1,q 0,q 1 ). Thus once he funcional form for he price index P(p 0,p 1,q 0,q 1 ) is deermined (and deailed price and quaniy daa are available for he wo quarers), aggregae oupu price and quaniy levels for he wo quarers under consideraion can be deermined using he definiions in (3). There are four main approaches o he deerminaion of he funcional form for a price index P(p 0,p 1,q 0,q 1 ) ha compares he prices (and associaed quaniies) beween wo periods: Fixed baske and averages of fixed baske approaches; The es or axiomaic approach; The sochasic approach and The economic approach. These four approaches are explained in deail elsewhere. 14 There are wo main funcional forms for he price index ha are used by saisical agencies as ideal arge indexes: he Fisher ideal price index P F and he Törnqvis-Theil price index P T. Before defining hese wo indexes, i is useful o define some oher indexes ha are frequenly used by saisical agencies. One of he earlies approaches o defining a price index he fixed baske approach. In his approach, we are given a represenaive baske of commodiies ha is defined by he posiive quaniy vecor q. Given he price vecors for periods 0 and 1, p 0 and p 1 respecively, we can calculae he cos of purchasing his same baske in he wo periods, p 0 q and p 1 q. Then he raio of hese coss is a very reasonable indicaor of pure price change over he wo periods under consideraion, provided ha he baske vecor q is represenaive. Thus define he Lowe (1823) price index, P Lo, as follows: (4) P Lo (p 0,p 1,q) p 1 q/p 0 q. As ime passed, economiss and price saisicians demanded a bi more precision wih respec o he specificaion of he baske vecor q. There are wo naural choices for he reference baske: he period 0 commodiy vecor q 0 or he period 1 commodiy vecor q 1. These wo choices lead o he Laspeyres (1871) price index P L defined by (5) and he Paasche (1874) price index P P defined by (6): 15 (5) P L (p 0,p 1,q 0,q 1 ) p 1 q 0 /p 0 q 0 = n=1 N s n 0 (p n 1 /p n 0 ) ; (6) P P (p 0,p 1,q 0,q 1 ) p 1 q 1 /p 0 q 1 = [ n=1 N s n 1 (p n 1 /p n 0 ) 1 ] 1 14 See Diewer (2012) or he Consumer Price Index Manual, ILO/IMF/OECD/UNECE/Eurosa/The World Bank (2004; ). For breviy, in he fuure, we will refer o he CPI Manual as ILO (2004). 15 Noe ha P L (p 0,p 1,q 0,q 1 ) does no acually depend on q 1 and P P (p 0,p 1,q 0,q 1 ) does no acually depend on q 0.
8 7 where he period expendiure share on commodiy n, s n, is defined as p n q n /p q for n = 1,,N and = 0,1. Thus he Laspeyres price index P L can be wrien as a base period expendiure share weighed average of he N price raios (or price relaives), p n 1 /p n The las equaion in (6) shows ha he Paasche price index P P can be wrien as a period 1 (or curren period) expendiure share weighed harmonic average of he N price raios. 17 The problem wih hese index number formulae is ha hey are equally plausible bu in general, hey will give differen answers. This suggess ha if we require a single esimae for he price change beween he wo periods, hen we should ake some sor of evenly weighed average of he wo indexes as our final esimae of price change beween periods 0 and 1. An examples of a symmeric average is he geomeric mean, which leads o he Fisher (1922) ideal index, P F, defined as (7) P F (p 0,p 1,q 0,q 1 ) [P L (p 0,p 1,q 0,q 1 ) P P (p 0,p 1,q 0,q 1 )] 1/2. Anoher useful funcional form for he arge price index is he Törnqvis-Theil index P T. Theil (1967; 137) provided a srong jusificaion for he use of his index from he perspecive of he sochasic or descripive saisics approach o index number heory. Theil s measure of overall logarihmic price change is defined as follows: (8) lnp T (p 0,p 1,q 0,q 1 ) n=1 N (1/2)(s n 0 +s n 1 )ln(p n 1 /p n 0 ). Exponeniaing he righ hand side of (8) provides a formula for lnp T (p 0,p 1,q 0,q 1 ). 18 The Fisher index P F receives a srong jusificaion as a arge index from he perspecives of he baske, axiomaic and economic approaches o index number heory while he Törnqvis-Theil index P T receives a srong jusificaion from he perspecives of he axiomaic, sochasic and economic approaches o index number heory. 19 Thus choosing beween hese wo alernaive indexes is difficul bu forunaely, using ime series daa, hese wo index formulae give very similar resuls and so ypically, i will no maer which of hese wo indexes is chosen. 20 The above maerial deal wih aggregaing revenue informaion on muliple sources of revenue from he same building. This informaion can be aggregaed over muliple buildings in he same locaion and in he same indusry o form an indusry oupu price index using he wo sage aggregaion procedures ha are commonly used in he 16 This resul is due o Walsh (1901; 428 and 539). 17 This expendiure share and price raio represenaion of he Paasche index is described by Walsh (1901; 428) and derived explicily by Fisher (1911; 365). 18 The U.S. Bureau of Labor Saisics uses his index number formula as is arge Consumer Price Index a higher levels of aggregaion. 19 See he ILO (2004). 20 Diewer (1978; 888) showed ha P T (p 0,p 1,q 0,q 1 ) approximaes P F o he second order around an equal price and quaniy poin. However, Diewer s resuls assumed ha all prices and quaniies were posiive.
9 8 consrucion of a Producer Price Index. 21 Typically, commercial buildings in a paricular locaion are classified ino hree groups: Offices; Reail sales and Indusrial (or facories). Buildings could be furher subdivided according o he ype of consrucion and oher characerisics. We conclude his secion wih a discussion of he problems raised by vacancies; i.e., for some quarers, some pars of he building may be emporarily vacan and hus for hese componens n of he building s revenue sream for quarer, i may be he case ha p n equals 0. These zero componens can cause he Törnqvis-Theil index P T o be ill-defined since he log of 0 is infinie. 22 The soluion o his problem is o noe ha ypically, he individual oupu quaniies q n are consan 23 and so we have q n 0 = q n 1 for n = 1,2,...,N, irrespecive of wheher all unis are rened for he wo periods or no. Thus under hese condiions, he Laspeyres, Paasche and Fisher oupu price indexes reduce o he Lowe index, P Lo (p 0,p 1,q) p 1 q/p 0 q where q is he common quaniy vecor for he wo periods under consideraion. Under hese condiions, leing P be eiher P L, P P or P F, we find using definiions (3) ha he price and quaniy levels for quarers 0 and 1 become: (9) P 0 1; P 1 P(p 0,p 1,q 0,q 1 ) = R 1 /R 0 = p 1 q 1 /p 0 q 0 ; Q 0 p 0 q 0 ; Q 1 p 0 q 0 = Q 0. Thus he price index P(p 0,p 1,q 0,q 1 ) collapses o he revenue raio, R 1 /R 0 if we use he Laspeyres, Paasche or Fisher index number formula. Hence our soluion o he vacancy problem is simple: use he Fisher formula P F and no he index P T in order o form a commercial propery oupu price index. I should be noed ha our suggesed naional income accouning reamen of commercial propery revenues differs somewha from he reamen used by commercial propery appraisers. Appraisers impue a normal ren for vacan suies in he building and add i o acual ren. They hen reduce his impued oal ren by one minus a normal vacancy rae and hey use his adjused ren as heir esimae of Ne Operaing Revenues for he building in he period under consideraion. 24 However, his ype of impuaion would ypically no be made for naional accouning purposes where revenues are aken o be acual period operaing revenues for he producion uni, no impued revenues. I can be seen ha, ypically, appraised operaing income will be smooher han acual operaing revenues, since vacancies are no uniform over ime. 21 See he IMF/ILO/OECD/UNECE/Eurosa/The World Bank (2004). 22 The second alernaive formulae for he Laspeyres and Paasche indexes defined by (5) and (6) can also become ill-defined if some prices are zero. 23 If renovaions occur while he uni is vacan, hen here will be a qualiy adjusmen problem. 24 See Baum and Crosby (2008; 67).
10 9 We urn our aenion o he consrucion of (non capial) inpu indexes for a commercial propery. 3. The Consrucion of Variable Inpu Price Indexes In his secion, we focus on nondurable variable inpus ha have well defined price and quaniy componens and ha are used by he commercial propery firm in quarers = 0,1. We assume ha here are M such variable inpus. 25 Denoe he quarer uni value price of inpu m by p Im and he corresponding oal quaniy purchased during he quarer by q Im for m = 1,2,...,M. 26 Examples of hese nondurable inpus include he following: Inpus used o hea he building such as fuel oil, coal and naural gas; Elecriciy inpus; Telecommunicaion inpus; Cleaning supplies; Janiorial, mainenance and repair inpus; Securiy and careaker services and Managerial and legal services inpus. In he sandard Sysem of Naional Accouns and in Mulifacor Produciviy Accouns, 27 hese nondurable inpus are furher classified as inermediae inpus or as labour inpus. The las hree classes of inpus lised above could be lised as labour inpus if he ype of service rendered is provided by an employee of he commercial propery firm. Alernaively, hese hree ypes of inpu could be classified as inermediae inpus if he ype of service is conraced ou. The consrucion of an inpu price index for he M classes of inpu proceeds in a manner ha is enirely analogous o our discussion of an oupu price index in he previous secion. Thus he propery s quarer variable inpu cos C is defined as follows: (10) C m=1 M p Im q Im p I q I where p I (p I1,, p IM ) denoes he quarer variable inpu price vecor, q I (q I1,, q IM ) is he corresponding quarer variable inpu quaniy vecor and p I q I denoes he inner produc of hese wo vecors. Consider he propery s variable cos raio going from say quarer 0 o 1, C 1 /C 0. As in he previous secion, we decompose his value raio for he wo quarers under consideraion ino an inpu price change componen P I imes an inpu quaniy change componen Q I. 25 Using naional income accouning erminology, he variables inpus in his secion refer o inermediae inpus and labour inpus. 26 We have added he subscrip I o he prices and quaniies in order o disinguish inpu prices and quaniies for he oupu prices and quaniies considered in he previous secion. 27 For explanaions on how Mulifacor Produciviy Accouns can be consruced, see Jorgenson and Griliches (1967), Chrisensen and Jorgenson (1973), Diewer (1980) (1992) and Schreyer (2001) (2009a).
11 10 Thus we look for wo funcions of 4M variables, P I (p 0,p 1,q 0,q 1 ) and Q I (p 0,p 1,q 0,q 1 ) such ha: (11) p I 1 q I 1 /p I 0 q I 0 = P I (p I 0,p I 1,q I 0,q I 1 )Q I (p I 0,p I 1,q I 0,q I 1 ). I can be seen ha if he price index funcion P I (p I 0,p I 1,q I 0,q I 1 ) has been deermined, hen he quaniy index Q I (p I 0,p I 1,q I 0,q I 1 ) can be residually deermined using equaion (11). If he funcional form for P I (p I 0,p I 1,q I 0,q I 1 ) is known, hen we can use (11) o deermine he quarer 0 and 1 aggregae inpu price levels, P I 0 and P I 1 respecively, and he quarer 0 and 1 aggregae inpu quaniy (or volume) levels, Q I 0 and Q i 1 respecively, as follows: (12) P I 0 1; P I 1 P I (p I 0,p I 1,q I 0,q I 1 ); Q I 0 p I 0 q I 0 ; Q I 1 p I 1 q I 1 /P I (p I 0,p I 1,q I 0,q I 1 ). As in he previous secion, he four major approaches o index number heory ha have been considered in he lieraure o dae sugges ha he Fisher or Törnqvis-Theil funcional forms defined earlier by (7) and (8) would be good choices for P I (p I 0,p I 1,q I 0,q I 1 ). The problem of zero purchases of a paricular inpu during one of he wo periods under consideraion needs o be addressed. 28 We canno use he soluion o his problem ha was used in he previous secion because quaniies of variable inpus are generally no consan across periods. To explain our suggesed soluion o he problem of zero purchases of an inpu during one period, we suppose q m 0 unis of a paricular inpu are purchased in quarer 0 a price p m 0 bu no unis of he inpu are purchased in quarer 1. Thus i is clear ha we can se q m 1 equal o 0 bu if we se p m 1 equal o zero as well, our preferred Fisher and Törnqvis-Theil inpu indexes can generae anomalous resuls. In order o obain sable inpu indexes over ime, i is bes o impue a posiive price for he missing price, p m 1. There are a leas hree possible choices for his impuaion exercise: 29 Carry forward he price of he previous period; i.e., se p 1 m equal o he observed price p 0 m for he produc m in quarer 0. Collec a price for he same produc in quarer 1. Thus if a quaniy of a paricular ype of cleaning fluid was purchased by he firm in quarer 0 bu no purchased in period 1, a price for he same produc is colleced for period 1. Assume ha he rae of price change for produc m going from quarer 0 o 1 is he same as he rae of change of an available price index for a similar produc or class of producs. Thus suppose he level of a saisical agency price index for cleaning fluids is P 0 CF in quarer 0 and P 1 CF in quarer 1, hen he impued price for produc m in quarer 1, p 1 m, is se equal o p 0 m [P 1 CF /P 0 CF ]. 28 If he inpu m is no purchased in boh quarers 0 and 1, hen his inpu can simply be omied in he lis of inpus and normal index number heory is applied o he remaining commodiies. 29 Addiional impuaion mehods are considered in Diewer (1980; ) and in Feensra and Diewer (2001).
12 11 The firs mehod is no recommended if he inflaion rae for produc m is high or very variable. 30 The second mehod is a preferred mehod bu i may be very cosly o obain he missing price quoe from he markeplace. 31 The hird mehod will generally be saisfacory bu of course, price indexes by produc caegory mus be available. The above discussion has focussed on nondurable variable inpus for which (a leas in principle) i is possible o obain period by period uni value prices and he corresponding period by period quaniies purchased for he commercial propery under consideraion. However, here are hree addiional periodic inpu coss for which here are values bu no obvious breakdown ino price and quaniy componens. These hree classes of value only nondurable variable inpu coss are as follows: Quarerly propery ax paymens, say C PT for period ; Quarerly business income ax paymens, say C IT for period and Quarerly propery insurance paymens, say C PI for period. These coss need o be decomposed ino price and quaniy componens so ha he real oupu and inpu of he Commercial Propery secor can be compued for naional income accouning purposes. We will defer he problems associaed wih hese decomposiions unil we have sudied he problems associaed wih cosing ou he conribuion of he building or srucure and of he land ha he srucure sis on. Thus in he following secion, we sudy how sock and flow price indexes for he building can be consruced and hen in he nex secion, we show how sock and flow price indexes for he plo area can be consruced The Consrucion of Sock and Flow Prices for he Commercial Building In his secion, we address he problem of pricing he services he building provides in each period. This is a flow of services price bu a sock price is also required; i.e., we require an esimae of he marke value for he srucure a he beginning of each accouning period and a decomposiion of his value ino a price and quaniy componen. 33 Unforunaely, consrucing price indexes for a durable inpu is a much more difficul ask han consrucing prices for oupus and variable inpus because impuaions are required in order o consruc he sock and flow prices for a durable 30 In periods of high or moderae inflaion, he carry forward mehod of impuaion will end o undersae inpu cos inflaion over he periods where he inpu is no purchased bu hen he index will jump up when he inpu is finally repurchased. 31 We are essenially assuming ha he price saisician has access o building level daa on revenues and coss, where he coss are lised by ransacion. In many cases, only quarerly accouning daa will be available and coss will generally be decomposed only by a few produc caegories and while values by caegory migh be available, average prices by caegory may no be available. 32 There is one addiional cos caegory associaed wih a commercial propery ha needs o be aken ino accoun and his caegory is period by period capial expendiures, C CE, on he propery. We will deal wih hese expendiures in he following secion. 33 This informaion is required for he consrucion of he Balance Shees for he commercial propery secor of he economy. The value of he flow of srucure services (and he decomposiion of his value flow ino price and quaniy componens) is also required in he Sysem of Naional Accouns in order o consruc esimaes of he Mulifacor Produciviy growh of he commercial propery secor.
13 12 inpu. The problem is ha a marke price for he building can only be observed a he ime he building is consruced when i is possible o deermine he consrucion cos. 34 Allocaing his consrucion cos across he useful lifeime of he building is a fundamenal problem of accouning. There is no universally acceped single soluion o his ineremporal cos allocaion problem. In his secion, we will sugges a possible mehod for solving his problem bu he reader should keep in mind ha oher mehods may be equally appropriae. Capial heory 35 suggess ha he value of a durable inpu (or asse) a a poin in ime is equal o is expeced discouned sream of flow reurns. We will formalize his suggesion in (13) below. Le P 0 be he asse value of a newly consruced building a he beginning of quarer and suppose ha he expeced life of he srucure is L quarers. Suppose also ha he cash flow ha can be aribued o he services provided by he new srucure in quarer is f 0. Now underake a menal experimen and suppose ha he same srucure is n quarers older a he beginning of quarer (where 1 n L1) and is asse value is P n, where P n < P 0 since an older asse has a smaller number of periods lef in is useful life and hence will be worh less han a brand new asse. Suppose ha he cash flow ha can be aribued o he services provided by he n quarers old srucure in period is f n where f n f 0 for n = 1,2,...,L1. 36 Finally, suppose ha he firm ha owns he building faces he nominal quarerly ineres rae (cos of financial capial) r a he beginning of period 37 and ha value of building srucure services is expeced o rise a he quarerly inflaion rae i. Then capial heory suggess ha he value of he new srucure a he beginning of quarer, P 0, should be equal o he following sum of discouned expeced cash flows: 38 (13) P 0 = f 0 + f 1 + ( ) 2 f ( ) L1 f L1 where he quarer discoun facor is defined as follows: (14) (1+i )/(1+r ). 34 One migh hink ha a marke price could be observed when a building is sold bu ypically, he sale price includes no only he srucure bu he land ha he srucure sis on and hence an impuaion mus be made in order o decompose he sale price ino srucure and land componens. 35 Walras (1954) (firs ediion published in 1874) was one of he earlies economiss o sae ha capial socks are demanded because of he fuure flow of services ha hey render. For background maerial on capial heory and producion heory, see Jorgenson (1963) (1989), Chrisensen and Jorgenson (1969), Diewer (1980) (2005), Hulen (1990), Diewer and Lawrence (2000) and Schreyer (2001) (2009a). 36 For n L, f n = 0 since he srucure is abandoned or demolished afer delivering srucure services for L quarers. 37 Concepually, r is equal o a gross rae of reurn ha is high enough o cover business income axes and pay invesors a rae of reurn ha will induce hem o inves in he building. Thus business income axes are folded ino his rae of reurn. 38 See Diewer (2005; ) for addiional maerial on his model. The quarer cash flow for a building ha is n quarers old, f n, can be inerpreed as a quarer impued renal price for he building of he ype under consideraion ha is n quarers old.
14 13 Under he same se of assumpions, capial heory suggess ha he value of a similar srucure ha is n quarers old a he beginning of period, P n, should be equal o he following sum of discouned expeced cash flows: (15) P n = f n + f n+1 + ( ) 2 f n ( ) L1n f L1 ; n = 1,2,...,L1. In order o value he flow of srucure services in he SNA for he commercial propery under consideraion, we require esimaes for he f n for each quarer and for all ages n of he srucure for n = 0,1,2,...,L1. In order o consruc balance shee esimaes in he SNA for he asses of he commercial propery secor, we require esimaes for he P n for each quarer and for all ages n of he srucure for n = 0,1,2,...,L1. Obviously, his is a raher dauning ask. In order o consruc a pracical mehod for esimaing hese prices, we need o make addiional assumpions. In order o simplify he above general model of asse valuaion, we will make a furher assumpion; namely, ha he flow of building services by age of he building is consan over he useful life of he building. 39 This assumpion seems o be a reasonable one: basically he services provided by he srucure are he same quarer by quarer over he useful life of he asse. 40 This assumpion means ha all of he quarer flow prices for buildings of he ype under consideraion by age n, f n, are equal o he quarer flow of services of a new building f 0. Thus we have: (16) f f 0 = f 1 =... = f L1. This is known as he ligh bulb or one hoss shay model of depreciaion. 41 Upon subsiuing equaions (16) ino (13), we obain he following relaionship beween he sock and flow prices for a new building a he beginning of quarer : (17) P 0 = f [1 + + ( ) ( ) L1 ] = f [1 ( ) L ]/[1 ] where he second equaion follows if [1+i ]/[1+r ] < 1. Thus if he price saisician has esimaes for he quarer building services inflaion rae i, he quarer cos of capial r and an esimae for he per square meer new building consrucion cos a he 39 We hope o implemen a more general depreciaion model in fuure work. 40 As Dixon, Crosby and Law (1999) noed, basically hree depreciaion models have been considered for commercial srucures: declining balance (or geomeric) depreciaion, sraigh line depreciaion or one hoss shay (or ligh bulb) depreciaion. The problem wih he firs wo depreciaion models is ha he srucure services provided by he building will approach zero as he building nears he end of is life. Bu even near he end, he srucure is sill providing basic floor space services ha are much he same as he services provided in previous periods. Thus we hink ha he one hoss shay model of depreciaion is more realisic for a single srucure han he compeing geomeric and sraigh line depreciaion models. 41 I is due o Böhm-Bawerk (1891). I is described in more deail by Hulen (1990; 124) and Diewer (2005; ). The name comes from a poem by Oliver Wendell Holmes: The Deacon s Maserpiece or he Wonderful One-Hoss Shay : A Logical Sory.
15 14 beginning of period, P 0, 42 equaion (17) can be used o deermine f, he per square meer value of building services for quarer. Now subsiue equaions (16) ino (15) and we obain he following expression for he beginning of period asse price P n of a building of he same ype ha is n quarers old a he sar of period : (18) P n = f [1 + + ( ) ( ) L(n+1) ] ; n = 1,2,...,L1 = f [1 ( ) Ln ]/[1 ] where he second equaion follows if < Thus if an esimae for he consrucion cos of a new building of he ype under consideraion, P 0, is available along wih esimaes for he expeced life of he building L, he cos of capial r and he expeced building inflaion rae i, hen a complee se of flow prices for srucure services, f n = f, and asse prices by age n of he building P n can be consruced. We now suppose ha we are considering a specific commercial propery in quarers and +1 and he problem is o consruc prices and quaniies of building services and esimaes of he sock value (and price and quaniy) of he building for quarers and +1. Suppose ha P 0 represens he period price of a square meer of a newly consruced building a he beginning of quarer of he ype of building under consideraion. Le Q B be he quarer floor area of he building in square meers. We suppose ha he building floor area remains consan from period o period; i.e., (19) Q B = Q B ; = 0,1,2,...,L1. We assume ha he price saisician has esimaes for he lengh of life L for he building, for he firm s cos of capial a he beginning of quarer, r, for expeced building consrucion cos inflaion, i, and for he new building cos per meer squared a he beginning of period, P 0, for quarers and +1. The quarer and +1 esimaed prices of building services are equal o f and f +1 defined by (20): 44 (20) f P 0 [1 ]/[1( ) L ] ; f +1 P 0 +1 [1 +1 ]/[1( +1 ) L ]. The corresponding quarers and +1 building services quaniies, Q B and Q +1 B, are defined by (19); i.e., he quaniies are fixed a he number of square meers of floor space, Q B. Thus f, f +1, Q B and Q +1 B defined by (19) and (20) are he prices and quaniies ha can be insered ino an index number formula ha aggregaes all building inpus ino an 42 Noe ha we have made an assumpion here ha he value of he new building is equal o is consrucion cos. This assumpion is likely o be approximaely rue bu i will generally be only an approximaion. 43 Cross secional depreciaion for a building of age n a he beginning of quarer is defined as D n P n P n+1 for n = 0,1,2,...,L1. Using he firs equaion in (18), i can be seen ha D n = f ( ) L(n+1) for n = 0,1,2,...,L1. Thus if < 1, cross secional depreciaion increases as he age of he asse increases. However, if r = i = 0, hen = 1 and D n = f for n = 0,1,2,...,L1; i.e., cross secional depreciaion is consan under hese condiions and one hoss shay depreciaion collapses down o sraigh line depreciaion. 44 We are assuming ha [1+i ]/[1+r ] < 1 for periods and +1. Noe ha if r = r +1 and i = i +1, hen = +1 and f +1 /f = P +1 0 /P 0. Thus if he cos of capial and he expeced srucure services inflaion rae are consan from period o period, he renal price for srucure space f will be proporional o P 0, he new consrucion price index for quarer.
16 15 overall inpu price index for he services of he building under consideraion. This is he ype of inpu index ha would be used in sudies of he Mulifacor Produciviy of he commercial propery under consideraion. Consrucing suiable prices for beginning of quarer and +1 asse values of he building ha can be used in an index number formula is a more difficul ask. The esimaed asse value of he building a he beginning of quarer is P n Q B where P n is defined by (18) and Q B is defined by (19). The esimaed asse value of he building a he beginning of quarer +1 is P n+1 +1 Q B +1 where Q B +1 is defined by (19) and P n+1 +1 is defined as: (21) P n+1 +1 = f +1 [ ( +1 ) ( +1 ) L(n+2) ] ; n = 1,2,...,L1 = f +1 [1 ( +1 ) L(n+1) ]/[1 +1 ] where he second equaion in (21) follows if +1 < 1. A firs glance, i migh appear ha one could simply use he asse prices P n and P +1 n+1 defined by (18) and (21) and he quaniies Q B and Q +1 B as suiable prices and quaniies ha could be used in an asse price index number formula. However, his is no a concepually correc procedure because he qualiy of he building is no being held consan as we go from period o period +1. The period +1 building is no he same as a period building since he older building has one less period lef in is expeced life 45 and hus P +1 n+1 /P n does no represen a consan qualiy asse price index for he building under consideraion. Forunaely, here is a way o overcome he above lack of comparabiliy problem. For each quarer, we can conver he building quaniy Q B ino an equivalen number of unis of a new building. Thus define he quarer qualiy adjused quaniy of building unis, Q * B, as follows: (22) Q B * Q B [P n /P 0 ] = Q B [1 ( ) Ln ]/[1 ( ) L ] where he second equaion in (22) follows from he second equaions in (17) and (18). The corresponding quarer +1 qualiy adjused quaniy of building unis, Q B +1*, is defined as follows: (23) Q B +1* Q B +1 [P n+1 +1 /P 0 +1 ] = Q B +1 [1 ( +1 ) L(n+1) ]/[1 ( +1 ) L ]. Thus when consrucing a consan qualiy asse price index for he commercial propery under consideraion going from quarer o +1, he new asse building prices P 0 and P 0 +1 can be used in he index number formula along wih he qualiy adjused building quaniies, Q B * and Q B +1* defined above by (22) and (23) We did no have his problem when consrucing a price index for building services since he one hoss shay model of depreciaion explicily assumes ha he flow of building services is consan over all ages of he building (unil i is reired or demolished). 46 If = +1, our suggesed qualiy adjusmen procedure can be jusified using Hicks Aggregaion Theorem: Thus we have demonsraed mahemaically he very imporan principle, used exensively in he ex, ha if he prices of a group of goods change in he same proporion, ha group of goods behaves
17 16 There are limiaions of our suggesed price consrucion models: The esimae for he expeced age of he building L will be subjec o some uncerainy; Similarly, here will be uncerainy abou he esimaed cos of capial r and expeced building inflaion rae i ; The assumpion ha depreciaion is of he one hoss shay ype may no be correc; The new building consrucion price index (in level form) P 0 may no be appropriae for he paricular building under consideraion and The assumpion ha new building value is equal o new building cos may also be only an approximaion. However, i should be noed ha naional income accounans sruggle wih he same se of problems when hey consruc naional balance shee esimaes for oher componens of he capial sock. There is an alernaive way of looking a our suggesed mehod for finding flow of services prices and asse values for he building on he commercial propery: our mehod can simply be regarded as a mehod for amorizing he iniial consrucion cos of he building. Since amorizaion models are somewha arbirary, he mehod we have suggesed seems o be a reasonable mehod. 47 In secion 2 above, we menioned how he one hoss shay algebra could be used o impue quarerly revenues for say office space in a building ha is leased ou o various enans on long erm leases. Now suppose ha he enire building is leased ou o a single enan who in urn subleases he space o oher enans or simply uses he building o generae revenues. The srucure (and land) services valuaion problem for he firm ha has leased he enire building is now differen from a firm ha owns he land (he laer case will be considered in he following secion). If he enan pays a quarerly ren o he firm ha owns he building, his renal paymen can be considered an inermediae inpu cos for he leasing firm in he producion accouns for he commercial propery secor. This quarerly renal paymen will appear as a revenue iem for he firm ha owns he building. This is reasonably sraighforward. However, he problem here is do we regard he quaniy of he inermediae inpu (combined srucure and land services) for he leasing enan o be consan over he life of he lease? If so, here is no problem in decomposing he inermediae inpu paymen ino price and quaniy componens over he life of he jus as if i were a single commodiy. J.R. Hicks (1946; ). I is like ha will be approximaely consan going from quarer o quarer since r and i are unlikely o change much going from quarer o quarer. If is consan over ime, i can be seen ha he enire schedule of asse prices by age n, P n, moves in sric proporion o he movemens in P 0 over ime. 47 If commercial properies could be sold as building only sales, hen i would be possible o check on he validiy of alernaive depreciaion models using informaion on sales of commercial properies. However, ypically sales of properies include he land ha sis under he srucure so informaion on he value of he used srucure will be conaminaed by he land value of he propery.
18 17 lease. If he qualiy of he srucure service changes over ime (due o changes in mainenance or o improvemens in ameniies), hen here is a problem in qualiy adjusing he quaniy of service over ime. There is no easy answer o his qualiy adjusmen problem, and so in pracice, i may be necessary o assume ha his qualiy is consan over he life of he lease. However, his is no he end of our measuremen problems. Suppose he enan does no pay quarerly ren over he life of he lease bu raher pays a lump sum for he services of he srucure over he lifeime of he lease a he beginning of he leasing period. In order o value he inpu conribuion o he single enan firm of he propery over he life of he lease, i is reasonable o use he one hoss shay model explained above o value he long erm lease and decompose i ino period by period inpu coss. The value of he lease now covers he combined services of he srucure and he land ha i sis on for he duraion of he lease. 48 However, noe ha following he resuling impued renal prices for he space over ime will no give us a reasonable index of curren commercial propery rens since he sequence of impued renals is enirely deermined by he iniial lease paymen, he duraion of he lease and assumpions abou ineres raes and expeced inflaion raes made a he beginning of he lease. 49 We can now sugges a possible reamen of capial expendiures on a commercial propery. The same model of depreciaion ha we have suggesed for he original srucure can be applied o each capial expendiure. All ha is required is an esimae for he lengh of life for each period s capial expendiure (his will generally be shorer han he lengh of life L used for he main srucure) and an appropriae deflaor for hese expendiures. The resuling algebra will be messy 50 bu concepually, here do no seem 48 If he lease paymen is made a he beginning of period and he lengh of he lease is L quarers, hen P 0 in equaion (13) can be inerpreed as he oal lease paymen, f is he period impued inpu cos for he srucure (wih he land), i is he expeced quarerly inflaion rae for he renal of space of he ype under consideraion, r is he (single) enan s opporuniy cos of capial a he beginning of period and (1+i ) s f is he impued period +s inpu cos for he space under consideraion for s= 1,2,...,L1. Noe ha he iniial lease paymen P 0 is now reaed as an invesmen cos by he enan and hence is amorized over ime. This lease paymen P 0 is also reaed as a deferred revenue asse by he firm ha owns he building and he owning firm has o amorize his asse over he lengh of he lease (possibly using a differen mehod of amorizaion). Accouning for commercial propery ransacions in he Sysem of Naional Accouns is no a rivial maer! 49 Since long erm leasing ransacions are ypically more prevalen han ourigh sales of commercial properies, i migh be hough ha a hedonic regression approach o obaining consan qualiy price indexes for commercial properies would be more successful han a hedonic regression approach on sales. However, hese long erm lease ransacions are sill relaively sparse (since he leases are long erm!) and so i will be difficul o find a parsimonious se of characerisics for he leases and he underlying properies ha will lead o successful hedonic regressions ha have high explanaory power. The advanages of he NOI approach o consrucing indexes are: (i) he daa are available every quarer for he buildings under consideraion and (ii) all of he characerisics of he propery are held (approximaely) consan from period o period, excep for building depreciaion. Thus he NOI approach has all of he advanages of he repea sales mehod for consrucing house price indexes (in erms of holding characerisics o be relaively consan) bu has he addiional advanage of providing informaion every quarer. 50 Wih one hoss shay depreciaion, here will be a separae consrucion expendiures asse for each ime period when a capial expendiure ook place and hus as ime marches on, here will be more and more consrucion asses o depreciae and accoun for.
19 18 o be any problems in addiion o he problems ha have been menioned for our analysis presened above for he underlying srucure. Now ha we have developed price and quaniy esimaes for srucures, we can reurn o he problem of obaining price and quaniy decomposiions for insurance paymens and propery axes ha fall on srucures. We address he insurance problem firs. The main componen of propery insurance is he proecion of he srucure from damage due o fire and oher causes. An insurance policy on a commercial propery could provide a paymen in case of he complee desrucion of he srucure which could eiher (i) cover he cos of complee replacemen of he srucure wih a new srucure of he same ype or (ii) cover he value of he depreciaed srucure. Le C PI be he period value of propery insurance paymens. Then for case (i), an appropriae period price of insurance is p PI C PI /Q B and he corresponding quaniy of insurance services is Q B, he number of square meers of floor space for he srucure in period. For case (ii), he appropriae period price of insurance is p PI * C PI /Q B * and he corresponding quaniy of insurance services is Q B *, he qualiy adjused number of square meers of floor space defined above by (22). In boh cases, he insurance price is viewed as he per square meer cos of proecion of he srucure from damage. An implici assumpion for boh price reamens is ha he risks of damage remain consan over ime. 51 The period propery ax paymens, C PT, can usually be decomposed ino wo addiive componens: C PT = C ST + C LT where C ST + C LT are he period ax paymens ha can be aribued o he srucure and C LT are he paymens ha can be aribued o he plo of land ha he srucure is locaed upon. The ax paymens on he srucure are almos always assessed on he basis of he curren (depreciaed) value of he srucure. Thus he appropriae price-quaniy decomposiion of he value of srucure ax paymens C ST is o define he period price p ST and he corresponding quaniy q ST as follows: (24) p ST C ST /Q B * ; q ST Q B * where Q B * is he qualiy adjused number of square meers of floor space defined above by (22). An appropriae price-quaniy decomposiion of he value of land ax paymens C LT is defined as follows: (25) p LT C LT /Q L ; q LT Q L where Q L is he size of he propery s land area in square meers (which remains consan over he periods ). Wha services do propery axes provide o he firm? These paymens allow he firm o be in exisence and o obain some specific services from he governmens ha levy he axes, such as access o a publically provided road nework. 51 This assumpion is only a useful approximaion o realiy. Unforunaely, obaining consan qualiy prices for propery insurance under condiions of changing risk facors is exremely complex; see Diewer (1993; ) (1995).
20 19 We will no decompose business income axes ino price and quaniy componens. Insead, we assume ha business income axes are refleced in he beginning of he period cos of financial capial, r ; i.e., his rae is approximaely equal o he afer business ax rae of reurn ha invesors require plus he prevailing rae of business income axaion. We now urn our aenion o he final se of problems ha are associaed wih he deerminaion of sock and flow prices for he land plo. 5. The Consrucion of Sock and Flow Prices for he Land Area of he Commercial Propery The saring poin of our analysis in his secion is he definiion of he Ne Operaing Income (NOI ) of he commercial propery under consideraion for period : (26) NOI R C C PI C PT ; = 0,1,2,...,L1 where R is he revenue generaed by he propery in period, C is he period variable cos for he propery, C PI is he period cos of propery insurance and C PT denoes period propery ax paymens. 52 I should be noed ha we are basically following he propery assessmen lieraure in defining NOI. 53 Our firs ask is o deermine he period rens ha can be aribued o he land (and of course, he locaion of he propery). We assume ha land rens in period, LR, are equal o he period ne operaing income less our impued flow of building services, which is equal o he building flow of services price per m 2, f, defined in (20) imes he building floor space, Q B : (27) LR NOI f Q B. Once oal land rens LR have been deermined by (27), he corresponding consan qualiy land services price, p L, can be defined by dividing land rens by he area of he propery, Q L, which will be consan over : (28) p L LR /Q L. Thus he appropriae decomposiion of land rens LR ino price and quaniy componens, is given by (28); i.e., LR = p L Q L. Finally, if an index of srucure and land service flows for he propery is required, hen one should choose an appropriae bilaeral index number formula for prices, such as he Fisher price index P F, and hen calculae he aggregae 52 See (1), (10), (24) and (25) for decomposiions of R, C, C ST and C LT ino price and quaniy componens where C PT = C ST + C LT. 53 There is some variaion across auhors in he reamen of propery axes and depreciaion: some auhors include hese iems as par of NOI and some do no.