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23DURABLES AND USER COSTS

Introduction23.1 When a durable good (other than housing) is

purchased by a consumer, national consumer priceindices (CPIs) attribute all that expenditure to the periodof purchase, even though the use of the good extendsbeyond the period of purchase.1 By definition, a durablegood delivers services longer than the period underconsideration.2 The System of National Accounts 1993defines a durable good as follows:

In the case of goods, the distinction between acquisitionand use is analytically important. It underlies the dis-tinction between durable and non-durable goods exten-sively used in economic analysis. In fact, the distinctionbetween durable and non-durable goods is not based onphysical durability as such. Instead, the distinction isbased on whether the goods can be used once only forpurposes of production or consumption or whether theycan be used repeatedly, or continuously. For example,coal is a highly durable good in a physical sense, but it canbe burnt only once. A durable good is therefore defined asone which may be used repeatedly or continuously over aperiod of more than a year, assuming a normal or averagerate of physical usage. A consumer durable is a good thatmay be used for purposes of consumption repeatedly orcontinuously over a period of a year or more (Commissionof the European Communities et al. (1993, p. 208)).

This chapter is mainly concerned with the problemsinvolved in pricing durable goods according to theabove definition.3 Durability is more than the fact that agood can physically persist for more than a year (this istrue of most goods): a durable good is distinguishedfrom a non-durable good by its ability to deliver usefulservices to a consumer through repeated use over anextended period of time.23.2 Since the benefits of using the consumer dur-

able extend over more than one period, it may not beappropriate to charge the entire purchase cost of the

durable to the initial period of purchase. If this point ofview is taken, then the initial purchase cost must bedistributed somehow over the useful life of the asset.This is a fundamental problem of accounting.4

Charles R. Hulten (1990, pp. 120–121) explains theconsequences for accountants of the durability of apurchase as follows:

Durability means that a capital good is productive fortwo or more time periods, and this, in turn, implies that adistinction must be made between the value of using orrenting capital in any year and the value of owning thecapital asset. This distinction would not necessarily leadto a measurement problem if the capital services used inany given year were paid for in that year; that is, if allcapital were rented. In this case, transactions in the rentalmarket would fix the price and quantity of capital in eachtime period, much as data on the price and quantity oflabor services are derived from labor market transactions.But, unfortunately, much capital is utilized by its ownerand the transfer of capital services between owner anduser results in an implicit rent typically not observed bythe statistician. Market data are thus inadequate forthe task of directly estimating the price and quantity ofcapital services, and this has led to the development ofindirect procedures for inferring the quantity of capital,like the perpetual inventory method, or to the acceptanceof flawed measures, like book value.

23.3 There are three main methods for dealing withthe durability problem:

� ignore the problem of distributing the initial cost ofthe durable over the useful life of the good and allo-cate the entire charge to the period of purchase. Thisis known as the acquisitions approach, and is theapproach currently used by CPI statisticians for alldurables, with the exception of housing;

� the rental equivalence or leasing equivalence approach.In this approach, a period price is imputed for thedurable which is equal to the rental price or leasingprice of an equivalent consumer durable for the sameperiod of time;

1 This treatment of the purchases of durable goods dates back to AlfredMarshall (1898, pp. 594–595) at least:

We have noticed also that though the benefits which a man derives fromliving in his own house are commonly reckoned as part of his real income,and estimated at the net rental value of his house; the same plan is notfollowed with regard to the benefits which he derives from the use of hisfurniture and clothes. It is best here to follow the common practice, andnot count as part of the national income or dividend anything that is notcommonly counted as part of the income of the individual.

2 An alternative definition of a durable good is that the good deliversservices to its purchaser for a period exceeding three years: ‘‘TheBureau of Economic Analysis defines consumer durables as thosedurables that have an average life of at least 3 years’’ (Katz (1983, p.422)).3 In paragraphs 23.136 to 23.145 there is a brief discussion aboutaccounting for the purchase, consumption and inventory holdings ofnon-durable goods.

4According to Stephen Gilman (1939) and David Solomons (1961):

The third convention is that of the annual accounting period. It is thisconvention which is responsible for most of the difficult accounting pro-blems. Without this convention, accounting would be a simple matter ofrecording completed and fully realized transactions: an act of primitivesimplicity (Gilman (1939, p. 26)).All the problems of income measurement are the result of our desire to

attribute income to arbitrarily determined short periods of time. Every-thing comes right in the end; but by then it is too late to matter (Solomons(1961, p. 378)).

Note that these authors do not mention the additional complicationsthat arise from the fact that future revenues and costs must be dis-counted to yield values that are equivalent to present dollars.

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� the user cost approach. In this approach, the initialpurchase cost of the durable is decomposed into twoparts: one part which reflects an estimated cost ofusing the services of the durable for the period, andanother part which is regarded as an investment thatmust earn some exogenous rate of return.

These three approaches will be discussed more fully inthe following three sections.

23.4 The above three approaches to the treatment ofdurable purchases can be applied to the purchase ofany durable commodity. Historically, it turns out thatthe rental equivalence and user cost approaches haveonly been applied to owner-occupied housing. In otherwords, the acquisitions approach to the purchase of con-sumer durables has been universally used by statisticalagencies, with the exception of owner-occupied housing.A possible reason for this is tradition; Marshall set thestandard, and statisticians have followed his example forthe past century. Another possible reason is that unlessthe durable good has a very long useful life, it usuallywill not make a great deal of difference in the long runwhether the acquisitions approach or one of the twoalternative approaches is used. This fact is demonstratedin paragraphs 23.39 to 23.42.

23.5 a major component of the user cost approachto valuing the services of owner-occupied housing is thedepreciation component. In paragraphs 23.43 to 23.68, ageneral model of depreciation for a consumer durable ispresented and then specialized to the three most com-mon models of depreciation that are in use. Thesemodels assume that homogeneous units of the durableare produced in each period so that information on theprices of the various vintages of the durable at any pointin time can be used to determine the pattern of depre-ciation. However, many durables (like housing) arecustom produced and thus the methods for determiningthe form of depreciation explained in paragraphs 23.43to 23.68 are not applicable. The special problems causedby these uniquely produced consumer durables areconsidered in paragraphs 23.69 to 23.78.

23.6 Subsequent sections treat some of the specialproblems involved in implementing the user cost andrental equivalence methods for valuing the servicesprovided by owner-occupied housing. Paragraphs 23.79to 23.93 present a derivation for the user cost of owner-occupied housing and various approximations to it.Paragraphs 23.94 to 23.120 consider some of the coststhat are tied to home-ownership, while paragraphs23.121 to 23.133 consider how a landlord’s costs mightdiffer from a home-owner’s costs. This material is rele-vant if the rental equivalence approach to valuing theservices of owner-occupied housing is used: care mustbe taken to remove some costs that are embedded inmarket rents that home-owners do not face.

23.7 Following Marshall, statistical agencies haveused alternatives to the acquisitions approach whendealing with owner-occupied housing. In addition to therental equivalence approach (which is the usualapproach by statistical agencies) and the user cost

approach, a fourth approach has been used: the pay-ments approach,5 which is a type of cash flow approach.It is explained in paragraphs 23.134 and 23.135.

23.8 Paragraphs 23.136 to 23.145 outline some ofthe problems involved in implementing the three mainapproaches for pricing owner-occupied housing.

The acquisitions approach23.9 The net acquisitions approach to the treatment

of owner-occupied housing is described by CharlesGoodhart (2001, p. F350) as follows:

The first is the net acquisition approach, which is thechange in the price of newly purchased owner-occupieddwellings, weighted by the net purchases of the referencepopulation. This is an asset based measure, and thereforecomes close to my preferred measure of inflation as achange in the value of money, though the change in theprice of the stock of existing houses rather than just of netpurchases would in some respects be even better. It is,moreover, consistent with the treatment of other dur-ables. A few countries, e.g., Australia and New Zealand,have used it, and it is, I understand, the main contenderfor use in the Euro-area Harmonized Index of ConsumerPrices (HICP), which currently excludes any measureof the purchase price of (new) housing, though it doesinclude minor repairs and maintenance by home-owners,as well as all expenditures by tenants.

23.10 The weights for the net acquisitions approachare the net purchases of the household sector of housesfrom other institutional sectors in the base period. Notethat, in principle, purchases of second-hand dwellingsfrom other sectors are relevant here; for example, a localgovernmentmay sell rental dwellings to owner-occupiers.Typically, however, newly built houses form amajor partof these types of transactions. Thus the long-term pricerelative for this category of expenditure will be primarilythe price of (new) houses (quality adjusted) in the currentperiod relative to the price of new houses in the baseperiod.6 If the net acquisitions approach is applied toother consumer durables, it is extremely easy to imple-ment: the purchase of a durable is treated in the same wayas a non-durable or service purchase is treated.

23.11 One additional implication of the net acqui-sitions approach is that major renovations and additionsto owner-occupied dwelling units could also be con-sidered as being in scope for this approach. In practice,these costs typically are not covered in a standard CPI.The treatment of renovations and additions is con-sidered in more detail in paragraphs 23.107 to 23.117.

23.12 Traditionally, the net acquisitions approachalso includes transfer costs relating to the buying andselling of second-hand houses as expenditures that are inscope for an acquisitions type CPI. These costs are

5This is the term used by Goodhart (2001, pp. F350–F351).

6 This price index may or may not include the price of the land onwhich the new dwelling unit is situated. Thus a new house price con-struction index would typically not include the land cost. The acqui-sitions approach concentrates on the purchases by households ofgoods and services that are provided by suppliers from outside thehousehold sector. If the land on which a new house is situated waspreviously owned by the household sector, then presumably the cost ofthis land would be excluded from an acquisitions type new house priceindex.

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mainly the costs of using a real estate agent’s services andasset transfer taxes. These transfer costs are further dis-cussed in paragraphs 23.100, 23.101 and 23.118 to 23.120.23.13 The major advantage of the acquisitions

approach is that it treats durable and non-durable pur-chases in a completely symmetric manner, and thus nospecial procedures have to be developed by a statisticalagency to deal with durable goods. As will be seen later,the major disadvantage of this approach is that theexpenditures associated with this approach will tend tounderstate the corresponding expenditures on durablesthat are implied by the rental equivalence and user costapproaches.23.14 Some differences between the acquisitions

approach and the other approaches are:

� If rental or leasing markets for the durable existand the durable has a long useful life, then theexpenditure weights implied by the rental equivalenceor user cost approaches will typically be much largerthan the corresponding expenditure weights impliedby the acquisitions approach; see paragraphs 23.34 to23.42.

� If the base year corresponds to a boom year (or aslump year) for the durable, then the base periodexpenditure weights may be too large or too small.Put another way, the aggregate expenditures thatcorrespond to the acquisitions approach are likely tobe more volatile than the expenditures for the aggre-gate that are implied by the rental equivalence or usercost approaches.

� In making comparisons of consumption across coun-tries where the proportion of owning versus rentingor leasing the durable varies greatly,7 the use ofthe acquisitions approachmay lead tomisleading cross-country comparisons. The reason for this is thatcapital costs are excluded in the net acquisitionsapproach, whereas they are explicitly or implicitlyincluded in the other two approaches.

23.15 More fundamentally, whether the acquisitionsapproach is the right one or not depends on the overallpurpose of the index number. If the purpose is to mea-sure the price of current period consumption services,then the acquisitions approach can only be regardedas an approximation to a more appropriate approach(which would be either the rental equivalence or usercost approach). If the purpose of the index is to measuremonetary (or non-imputed) expenditures by householdsduring the period, then the acquisitions approach ispreferable.

The rental equivalence approach23.16 The rental equivalence approach simply values

the services yielded by the use of a consumer durablegood for a period by the corresponding market rental

value for the same durable for the same period of time(if such a rental value exists). This is the approach takenin the System of National Accounts 1993 (SNA 1993) forowner-occupied housing:

As well-organized markets for rented housing exist inmost countries, the output of own-account housing ser-vices can be valued using the prices of the same kinds ofservices sold on the market with the general valuationrules adopted for goods and services produced on ownaccount. In other words, the output of housing servicesproduced by owner-occupiers is valued at the estimatedrental that a tenant would pay for the same accom-modation, taking into account factors such as location,neighbourhood amenities, etc. as well as the size andquality of the dwelling itself (Commission of the Euro-pean Communities et al. (1993, p. 134)).

23.17 The SNA 1993 nevertheless follows Marshall(1898, p. 595) and does not extend the rental equivalenceapproach to consumer durables other than housing.This seemingly inconsistent treatment of durables isexplained in the SNA 1993 as follows:

The production of housing services for their own finalconsumptionbyowner-occupiers has always been includedwithin the production boundary in national accounts,although it constitutes an exception to the generalexclusion of own-account service production. The ratio ofowner-occupied to rented dwellings can vary significantlybetween countries and even over short periods of timewithin a single country, so that both international andintertemporal comparisons of the production and con-sumption of housing services could be distorted if noimputation were made for the value of own-accountservices (Commission of the European Communities et al.(1993, p. 126)).

23.18 Eurostat’s (2001) Handbook on Price andVolume Measures in National Accounts also recommendsthe rental equivalence approach for the treatment of thedwelling services for owner-occupied housing: ‘‘Theoutput of dwelling services of owner-occupiers at cur-rent prices is in many countries estimated by linking theactual rents paid by those renting similar propertiesin the rented sector to those of owner-occupiers. Thisallows the imputation of a notional rent for the serviceowner-occupiers receive from their property’’ (Eurostat(2001, p. 99)).

23.19 The United States statistical agencies, theBureau of Labor Statistics and the Bureau of EconomicAnalysis, both use the rental equivalence approach tovalue the services of owner-occupied housing. Arnold J.Katz (1983, p. 411) describes the Bureau of EconomicAnalysis (BEA) procedures as follows:

Basically, BEA measures the gross rent (space rent) ofowner-occupied housing from data on the rent paid forsimilar housing with the same market value. To get theservice value that is added to GNP (gross housing pro-duct), the value of intermediate goods and servicesincluded in this figure (e. g., expenditures for repair andmaintenance, insurance, condominium fees, and closingcosts) are subtracted from the space rent. To obtain a netreturn (net rental income), depreciation, taxes, and netinterest are subtracted from, and subsidies added to, theservice value.

23.20 There are some problems with the abovetreatment of housing and they are discussed in later

7According to Hoffmann and Kurz (2002, pp. 3–4), about 60 per centof German households live in rented dwellings, whereas only about 20per cent of Spaniards rent their dwellings.

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sections, after the user cost approach to durables hasbeen discussed.8

23.21 To summarize the above material, it can beseen that the rental equivalence approach to the treat-ment of durables is conceptually simple: impute a cur-rent period rental or leasing price for a comparableproduct as the price for the purchase of a unit of aconsumer durable. For existing stocks of used consumerdurables, the rental equivalence approach would entailfinding rental prices for comparable used units.9 Todate, as noted above, statistical agencies have not usedthe rental equivalence approach to the treatment ofdurables, with the single exception of owner-occupiedhousing. Note, however, that in order to implement therental equivalence approach, it is necessary for therelevant rental or leasing markets to exist. Often this willnot be the case, particularly when it is recognized thatvintage specific rental prices are required for all vintagesof the durable held by households.10

The user cost approach23.22 The user cost approach to the treatment of

durable goods is in some ways very simple: it calculatesthe cost of purchasing the durable at the beginning ofthe period, using the services of the durable during theperiod and then netting off from these costs the benefitthat could be obtained by selling the durable at the endof the period. Several details of this procedure are,however, somewhat controversial. These involve the useof opportunity costs, which are usually imputed costs,the treatment of interest, and the treatment of capitalgains or holding gains.

23.23 Another complication with the user costapproach is that it involves making distinctions betweencurrent period (flow) purchases within the period underconsideration and the holdings of physical stocks of thedurable at the beginning and the end of the accounting

period. Up to this point in the manual, all prices andquantity purchases have been thought of as taking placeat a single point in time, say the middle of the periodunder consideration, and consumption has been thoughtof as taking place within the period as well. Thus, therehas been no need to consider the behaviour (andvaluation) of stocks of consumer durables that house-holds may have at their disposal. The rather complexproblems involved in accounting for stocks and flowsare unfamiliar to most price statisticians.

23.24 To determine the net cost of using the durablegood during, say, period 0, assume that one unit of thedurable good is purchased at the beginning of period 0at the price P0. The ‘‘used’’ or ‘‘second-hand’’ durablegood can be sold at the end of period 0 at the price P1S. Itmight seem that a reasonable net cost for the use of oneunit of the consumer durable during period 0 is its initialpurchase price P0 less its end of period 0 ‘‘scrap value’’P1S. However, money received at the end of the period isnot as valuable as money received at the beginning ofthe period. Thus, in order to convert the end of periodvalue into its beginning of period equivalent value, it isnecessary to discount the term P1S by the term 1+r0,where r0 is the beginning of period 0 nominal interestrate that the consumer faces. Hence the period 0 usercost u0 for the consumer durable11 is defined as:

u0 � P 0 � P1S(1+r0)

(23:1)

23.25 There is another way to view the user costformula (23.1): the consumer purchases the durable atthe beginning of period 0 at the price P0 and chargeshimself or herself the rental price u0. The remainder ofthe purchase price, I 0, defined as

I0 � P 0 � u0 (23:2)

can be regarded as an investment, which is to yield theappropriate opportunity cost of capital r0 that the con-sumer faces. At the end of period 0, this rate of returncould be realized provided that I 0, r0 and the sellingprice of the durable at the end of the period P1S satisfythe following equation:

I0(1+r0)=P1S (23:3)

Given P1S and r0, equation (23.3) determines I 0, which in

turn, given P0, determines the user cost u0 via equa-tion (23.2).12

23.26 It should be noted that some price statisticiansobject to the user cost concept as a valid pricing conceptfor a CPI:

A suitable price concept for a CPI ought to reflect onlya ratio of exchange of money for other things, not a ratioat which money in one form or time period can be tradedfor money in another form or time period. The ratio atwhich money today can be traded for money tomorrow

8To anticipate the later results: the main problem is that the rentalequivalence approach to valuing the services of owner-occupiedhousing may give a higher valuation for these services than the usercost approach.9Another method for determining rental price equivalents for stocks ofconsumer durables is to ask households what they think their durableswould rent for. This approach is used by the United States Bureau ofLabor Statistics in order to determine expenditure weights for owner-occupied housing; i.e., home-owners are asked to estimate what theirhouse would rent for if it were rented to a third party; see the Bureau ofLabor Statistics (1983). Lebow and Rudd (2003, p. 169) note that theseestimates of imputed rents in the United States based on a consumerexpenditure survey differ considerably from the corresponding Bureauof Economic Analysis estimates for imputed rents, which are derivedby applying a rent-to-value ratio for rented properties to the owner-occupied stock of housing. Lebow and Rudd feel that the expendituresurvey estimates may be less reliable than the ratio of rent-to-valuemethod because of the relatively small size of the consumer expendi-ture survey plus the difficulties households may have in recalling orestimating expenditures.10 If the form of depreciation is of the ‘‘one hoss shay’’ or light bulbtype, then the rental price for the durable will be the same for allvintages, and hence a detailed knowledge of market rentals by vintagewill not be required. The light bulb model of depreciation dates back toBohm-Bawerk (1891, p. 342). For more recent material on this model,see paragraphs 23.62 to 23.68, or Hulten (1990) or Diewert (2003b).

11 This approach to the derivation of a user cost formula was used byDiewert (1974b), who in turn based it on an approach attributable toHicks (1946, p. 326).12 This derivation for the user cost of a consumer durable was alsomade by Diewert (1974b, p. 504).


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by paying an interest rate or by enjoying actual orexpected holding gains on an appreciating asset has nopart in a measure of the current purchasing power ofmoney (Reinsdorf (2003)).

User costs are not like the prices of non-durables orservices because the user cost concept involves pricingthe durable at two points in time rather than at a singlepoint in time.13 Because the user cost concept involvesprices at two points in time, money received or paid outat the first point in time is more valuable than moneypaid out or received at the second point in time, sointerest rates creep into the user cost formula. Further-more, because the user cost concept involves prices attwo points in time, expected prices can be involved if theuser cost is calculated at the beginning of the periodunder consideration instead of at the end. With all thesecomplications, it is no wonder that many price statisti-cians would like to avoid using user costs as a pricingconcept. However, even for price statisticians whowould prefer to use the rental equivalence approach tothe treatment of durables over the user cost approach,there is some justification for considering the user costapproach in some detail, since this approach givesinsights into the economic determinants of the rentalor leasing price of a durable. As is seen in paragraphs23.121 to 23.133, the user cost for a house can differsubstantially for a landlord compared to an owner. Thusadjustments should be made to market rents for dwell-ing units, if they are to be used as imputations forowner-occupied rents.23.27 The user cost formula (23.1) can be put into a

more familiar form if the period 0 economic depreciationrate d and the period 0 ex post asset inflation rate i0 aredefined. Define d by:

(1�d) � P1S=P1 (23:4)

where P1S is the price of a used asset at the end of period 0and P1 is the price of a new asset at the end of period 0.The period 0 inflation rate for the new asset, i0, is definedby:

1+i 0 � P1S=P1 (23:5)

Eliminating P1 from equations (23.4) and (23.5) leads tothe following formula for the end of period 0 used assetprice:

P1S=(1�d)(1+i 0)P 0 (23:6)

Substitution of equation (23.6) into equation (23.1)yields the following expression for the period 0 usercost u0:

u0=[(1+r0)�(1�d)(1+i 0)]P 0

1+r0(23:7)

Note that r0�i0 can be interpreted as a period 0 realinterest rate and d(1+i 0) can be interpreted as aninflation-adjusted depreciation rate.

23.28 The user cost u0 is expressed in terms of pricesthat are discounted to the beginning of period 0. It isalso possible to express the user cost in terms of pricesthat are ‘‘discounted’’ to the end of period 0.14 Definethe end of period 0 user cost p0 as:15

p 0 � (1+r0)u0=[r0�i 0+d(1+i 0)]P 0 (23:8)

where the last equation follows using equation (23.7). Ifthe real interest rate r0* is defined as the nominal interestrate r0 less the asset inflation rate i0, and the small termdi0 is neglected, then the end of period user cost definedby equation (23.8) reduces to:

p0=(r0*+d)P 0 (23:9)

23.29 Abstracting from transactions costs andinflation, it can be seen that the end of period user costdefined by equation (23.9) is an approximate rental cost;i.e., the rental cost for the use of a consumer (or pro-ducer) durable good should equal the (real) opportunitycost of the capital tied up, r0*P0, plus the decline invalue of the asset over the period, dP0. Formulae(23.8) and (23.9) thus cast some light on the economicdeterminants of rental or leasing prices for consumerdurables.

23.30 If the simplified user cost formula defined byequation (23.9) is used, then forming a price index forthe user costs of a durable good is not very much moredifficult than forming a price index for the purchaseprice of the durable good, P0. The price statisticianneeds only to:

� make a reasonable assumption as to what an appro-priate monthly or quarterly real interest rate r0*should be;

13Woolford suggested that interest should be excluded from an idealprice index that measured inflation. In his view, interest is not a con-temporaneous price; i.e., an interest rate necessarily refers to two pointsin time: a beginning point when the capital is loaned and an endingpoint when the capital loaned must be repaid. Thus, if attention isrestricted to a domain of definition that consists of only contem-poraneous prices, interest rates are excluded. Woolford (1999, p. 535)noted that his ideal inflation measure ‘‘would be contemporary innature, capturing only the current trend in prices associated withtransactions in goods and services. It would exclude interest rates onthe grounds that they are intertemporal prices, representing the relativeprice of consuming today rather than in the future.’’

14 The beginning of the period user cost u0 discounts all monetary costsand benefits into their dollar equivalent at the beginning of period 0,whereas p0 discounts (or appreciates) all monetary costs and benefitsinto their dollar equivalent at the end of period 0. This leaves openhow flow transactions that take place within the period should betreated. Following the conventions used in financial accounting sug-gests that flow transactions taking place within the accounting periodshould be regarded as taking place at the end of the accounting period.Following this convention, end of period user costs should be used bythe price statistician.15 Christensen and Jorgenson (1969) derived a user cost formula similarto equation (23.7) in a different way, using a continuous time optimi-zation model. If the inflation rate i equals 0, then the user cost formula(23.7) reduces to that derived byWalras (1954, p. 269; first edition 1874).This zero inflation rate user cost formula was also derived by theindustrial engineer A. Hamilton Church (1901, pp. 907–908), whoperhaps drew on the work of Ewing Matheson (1910, p. 169, firstpublished in 1884): ‘‘In the case of a factory where the occupancy isassured for a term of years, and the rent is a first charge on profits, therate of interest, to be an appropriate rate, should, so far as it applies tothe buildings, be equal (including the depreciation rate) to the rentalwhich a landlord who owned but did not occupy a factory would let itfor.’’ Additional derivations of user cost formulae in discrete time havebeen made by Katz (1983, pp. 408–409) and Diewert (2003b).


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� make an assumption as to what a reasonable monthlyor quarterly depreciation rate d should be;16

� collect purchase prices P0 for the durable;

� make an estimate of the total stock of the durablewhich was held by the reference population during thebase period for quantities. In order to construct asuperlative index, estimates of the stock held will haveto be made for each period.

23.31 If it is thought necessary to implement themore complicated user cost formula (23.8) in place ofthe simpler formula (23.9), then the situation is morecomplicated. As it stands, the end of period user costformula (23.8) is an ex post user cost: the asset inflationrate i0 cannot be calculated until the end of period 0has been reached. Formula (23.8) can be converted intoan ex ante user cost formula if i0 is interpreted as ananticipated asset inflation rate. The resulting formulashould approximate a market rental rate for the assetunder inflationary conditions.17

23.32 Note that in the user cost approach to thetreatment of consumer durables, the entire user costformula (23.8) or (23.9) is the period 0 price. Thus, in thetime series context, it is not necessary to deflate eachcomponent of the formula separately; the period 0 pricep0:[r0�i0+d(1+i0)]P0 is compared to the correspond-ing period 1 price, p1 � [r1�i1+d(1+i1)]P1 and so on.

23.33 In principle, depreciation rates can be esti-mated using information on the selling prices of usedunits of the durable good. This methodology will beexplained in more detail in paragraphs 23.43 to 23.68.Before this is done, however, it will be useful to use thematerial in this section to explain what the relationshipbetween the user cost and acquisition approaches to thetreatment of durables is likely to be. This topic is dis-cussed in the following section.

The relationship betweenuser costs and acquisition costs

23.34 In this section, the user cost approach to thetreatment of consumer durables is compared to theacquisitions approach. Obviously, in the short run, thevalue flows associated with each approach could be verydifferent. For example, if real interest rates, r0�i0, arevery high and the economy is in a severe recession ordepression, then purchases of new consumer durables,Q0 say, could be very low and even approach zero forvery long-lived assets, such as houses. In contrast, using

the user cost approach zero, existing stocks of consumerdurables would be carried over from previous periodsand priced out at the appropriate user costs, andthe resulting consumption value flow could be quitelarge. Thus, in the short run, the monetary values ofconsumption under the two approaches could be vastlydifferent. Hence, in what follows, a (hypothetical)longer-run comparison is considered where real interestrates are held constant.18

23.35 Suppose that in period 0, the reference popu-lation of households purchases q0 units of a consumerdurable at the purchase price P0. Then the period 0 valueof consumption from the viewpoint of the acquisitionsapproach is:

V0A � P 0q0 (23:10)

23.36 Recall that the end of period user cost forone new unit of the asset purchased at the beginning ofperiod 0 is p0 defined by equation (23.8). In order tosimplify the analysis, declining balance depreciation isassumed; i.e., at the beginning of period 0, a one-period-old asset is worth (1�d)P0; a two-period-oldasset is worth (1�d)2P0; a t-period-old asset is worth(1�d)tP0; etc. Under these hypotheses, the corre-sponding end of period 0 user cost for a new assetpurchased at the beginning of period 0 is p0; the end ofperiod 0 user cost for a one-period-old asset at thebeginning of period 0 is (1�d)p0; the correspond-ing user cost for a two-period-old asset at the beginningof period 0 is (1�d)2p0; the corresponding user cost fora t-period-old asset at the beginning of period 0 is(1�d)tp0; etc.19 The final simplifying assumption is thathousehold purchases of the consumer durable havebeen growing at the geometric rate g into the indefinitepast. This means that if household purchases of thedurable were q0 in period 0, then in the previous periodthe households purchased q0/(1+g) new units; twoperiods ago, they purchased q0/(1+g)2 new units; tperiods ago, they purchased q0/(1+g)t new units; etc.Putting all of these assumptions together, it can be seenthat the period 0 value of consumption from the view-point of the user cost approach is:

V0U � p0q0+

(1�d)p0q01+g

+(1�d)2p0q0

(1+g)2+ . . . (23:11)

=(1+g)p0q0

g+dsumming the infinite series

=(1+g)[r0�i 0+d(1+i 0)]P 0q0

g+dusing equation

(23:8) (23:12)

23.37 Equation (23.12) can be simplified by lettingthe asset inflation rate i0 be 0 (or by replacing r0�i0 bythe real interest rate r0* and by ignoring the small termdi0). Under these conditions, the ratio of the user cost

16 The geometric model for depreciation, explained in more detail inparagraphs 23.43 to 23.68, requires only a single monthly or quarterlydepreciation rate. Other models of depreciation may require the esti-mation of a sequence of vintage depreciation rates. If the estimatedannual geometric depreciation rate is da, then the correspondingmonthly geometric depreciation rate d can be obtained by solvingthe equation (1�d)12=1�da. Similarly, if the estimated annualreal interest rate is ra*, then the corresponding monthly real interest rater* can be obtained by solving the equation (1+r*)12=1+ra*.17 Since landlords must set their rent at the beginning of the period (andin fact, they usually set their rent for an extended period of time), if theuser cost approach is used to model the economic determinants ofmarket rental rates, then the asset inflation rate i0 should be interpretedas an expected inflation rate rather than an ex post actual inflation rate.

18The following material is based on Diewert (2002c).19 For many consumer durables, the ‘‘one hoss shay’’ assumption fordepreciation may be more realistic than the declining balance model;see paragraphs 23.43 to 23.68, or Hulten (1990) or Diewert andLawrence (2000).


424

flow of consumption (23.12) to the acquisitions measureof consumption in period 0, (23.10), is:

V0U

V0A

=(1+g)(r0*+d)

g+d(23:13)

23.38 Using formula (23.13), it can be seen that if1+g>0 and d+g>0, then V0

U=V0A will be greater than

unity if

r0*>g(1�d)1+g

(23:14)

a condition that will usually be satisfied.20 Thus undernormal conditions and over a longer time horizon,household expenditures on consumer durables using theuser cost approach will tend to exceed the correspondingmoney outlays on new purchases of the consumer dura-ble. The difference between the two approaches will tendto grow as the life of the asset increases (i.e., as thedepreciation rate d decreases).23.39 To get a rough idea of the possible magnitude

of the value ratio for the two approaches, V0U=V

0A, equa-

tion (23.13) is evaluated for a ‘‘housing’’ example usingannual data where the depreciation rate is 2 per cent (i.e.,d=0.02), the real interest rate is 4 per cent (i.e., r0*=0.04)and the growth rate for the production of new houses is 1per cent (i.e., g=0.01). In this base case, the ratio of usercost expenditures on housing to the purchases of newhousing in the same period, V0

U=V0A, is 2.02. If the

depreciation rate is increased to 3 per cent, then V0U=V

0A

decreases to 1.77; if the depreciation rate is reduced to 1per cent, then V0

U=V0A increases to 2.53. Again looking at

the base case, if the real interest rate is increased to 5 percent, then V0

U=V0A increases to 2.36, while if the real

interest rate is reduced to 3 per cent, then V0U=V

0A

decreases to 1.68. Finally, if the growth rate for newhouses is increased to 2 per cent, thenV0

U=V0A decreases to

1.53, while if the growth rate is reduced to 0, then V0U=V

0A

increases to 3.00. Thus an acquisitions approach tohousing in the CPI is likely to give about half the expen-diture weight that a user cost approach would give.23.40 For shorter-lived assets, the difference between

the acquisitions approach and the user cost approachwill not be so large, indicating that the acquisitionsapproach is approximately ‘‘correct’’ as a measure ofconsumption services.21

23.41 The following is a list of some of the problemsand difficulties that might arise in implementing a usercost approach to purchases of a consumer durable:22

� It is difficult to determine what the relevant nominalinterest rate r0 is for each household. If a consumerhas to borrow to finance the cost of a durable goodpurchase, then this interest rate will typically be muchhigher than the safe rate of return that would be theappropriate opportunity cost rate of return for aconsumer who had no need to borrow funds tofinance the purchase.23 It may be necessary to simplyuse a benchmark interest rate that would be deter-mined by either the government, a national statisticalagency or an accounting standards board.

� It will generally be difficult to determine what the rele-vant depreciation rate is for the consumer durable.24

� Ex post user costs based on formula (23.8) will betoo volatile to be acceptable to users25 (owing to thevolatility of the asset inflation rate i0), and hence an exante user cost concept will have to be used. This cre-ates difficulties in that different national statisticalagencies will generally make different assumptions anduse different methods to construct forecasted struc-tures and land inflation rates. Hence, the resulting exante user costs of the durable may not be comparableacross countries.26

� The user cost formula (23.8) must be generalized toaccommodate various taxes that may be associated

20Note that if the real interest rate r0 equals g, the real rate of growthin purchases of the durable, then from the ratio (23.13),V0U=V

0A=(1+g) and the acquisitions approach will be more or less

equivalent to the user cost approach over the long run.21 The simplified user cost approach can be used for other consumerdurables as well. In formula (23.13), let r0*=0.04, g=0.01 and d=0.15. Under these conditions, V0

U=V0A=1:20; i.e., for a declining bal-

ance depreciation rate of 15 per cent, the user cost approach leads toan estimated value of consumption that is 20 per cent higher than theacquisitions approach under the conditions specified. Thus for con-sumer durable depreciation rates that are lower than 15 per cent, itcould be useful for the statistical agency to produce user costs for thesegoods and for the national accounts division to produce the corre-sponding consumption flows as ‘‘analytic series’’. It should be notedthat this extends the present national accounts treatment of housing toother long-lived consumer durables. Note also that this revised treat-ment of consumption in the national accounts would tend to make richcountries richer, since poorer countries hold fewer long-lived consumerdurables on a per capita basis.

22 For additional material on difficulties with the user cost approach,see Diewert (1980, pp. 475–479) and Katz (1983, pp. 415–422).23Katz (1983, pp. 415–416) comments on the difficulties involved indetermining the appropriate rate of interest to use:

There are numerous alternatives: a rate on financial borrowings, onsavings, and a weighted average of the two; a rate on nonfinancialinvestments. e.g. residential housing, perhaps adjusted for capital gains;and the consumer’s subjective rate of time preference. Furthermore, thereis some controversy about whether it should be the maximum observedrate, the average observed rate, or the rate of return earned on invest-ments that have the same degree of risk and liquidity as the durableswhose services are being valued.

24 It is not necessary to assume declining balance depreciation in theuser cost approach: any pattern of depreciation can be accommodated,including ‘‘one hoss shay’’ depreciation, where the durable yields aconstant stream of services over time until it is scrapped. See Diewertand Lawrence (2000) for some empirical examples for Canada, usingdifferent assumptions about the form of depreciation. For references tothe depreciation literature and for empirical methods for estimatingdepreciation rates, see Hulten and Wykoff (1981a; 1981b; 1996) andJorgenson (1996).25Goodhart (2001, p. F351) comments on the practical difficulties ofusing ex post user costs for housing, as follows:

An even more theoretical user cost approach is to measure the cost fore-gone by living in an owner-occupied property as compared with selling it atthe beginning of the period and repurchasing it at the end . . .But this givesthe absurd result that as house prices rise, so the opportunity cost falls;indeed the more virulent the inflation of housing asset prices, the morenegative would this measure become. Although it has some academicaficionados, this flies in the face of common sense; I am glad to say that nocountry has adopted this method.

As will be seen later, Iceland has in fact adopted a simplified user costframework.26 For additional material on the difficulties involved in constructing exante user costs, see Diewert (1980, pp. 475–486) and Katz (1983,pp. 419–420). For empirical comparisons of different user cost for-mulae, see Harper, Berndt andWood (1989) and Diewert and Lawrence(2000).


425

with the purchase of a durable or with the continuinguse of the durable.27

23.42 Some of the problems associated with esti-mating depreciation rates will be discussed in the nextsection.

Alternative models of depreciation

A general model of depreciation for(unchanging) consumer durables

23.43 In this subsection, a ‘‘general’’ model ofdepreciation for durable goods that appear on the mar-ket each period without undergoing quality change willbe presented. In three subsequent subsections, this gen-eral model will be specialized to the three most commonmodels of depreciation that appear in the literature. Inparagraphs 23.69 to 23.78 below, the additional pro-blems that occur when the durable is built as a uniquegood will be discussed.

23.44 The main tool that can be used to identifydepreciation rates for a durable good is the (cross-sectional) sequence of vintage asset prices that units ofthe good sell for on the second-hand market at any pointof time.28

23.45 Some notation is required. Let P0 be the priceof a newly produced unit of the durable good at thebeginning of period 0 (this is the same notation as wasused earlier). Let Ptv be the second-hand market price atthe beginning of period t of a unit of the durable goodthat is v periods old.29 Let d0v be the period 0 deprecia-tion rate for a unit of the durable good that is v periodsold at the beginning of period 0. These depreciationrates can be defined recursively, starting with the period0 depreciation rate for a brand new unit, d00, using theperiod 0 vintage asset prices P0v as follows:

1�d00=P01=P 0 (23:15)

Once d00 has been defined by equation (23.15), the per-iod 0 cross-sectional depreciation rate for a unitof the durable good that is one period old at the begin-ning of period 0, d01, can be defined using the followingequation:

(1�d01)(1�d00)=P

02=P

0 (23:16)

Note that P02 is the beginning of period 0 asset price of aunit of the durable good that is two periods old, and it iscompared to the price of a brand new unit of the dur-able, P0 (which is equal to P00 using the vintage goodnotation).

23.46 Given that the period 0 cross-sectionaldepreciation rates for units of the durable that are 0, 1,2, . . . , v�1 periods old at the beginning of period 0 aredefined (these are the depreciation rates d00, d

01, d

02, . . . ,

d0v�1), then the period 0 cross-sectional depreciation ratefor units of the durable that are v periods old at thebeginning of period 0 can be defined using the followingequation:

(1�d0v) . . . (1�d01)(1�d00)=P

0v+1=P

0 (23:17)

23.47 It should be clear how the sequence of period 0vintage asset prices P0v can be converted into a sequenceof period 0 vintage depreciation rates. It should also beclear that the sequence of equations (23.15)–(23.17) canbe repeated using the vintage asset price data pertainingto the beginning of period t, Ptv, in order to obtain asequence of period t vintage depreciation rates, dtv. In theliterature, it is usually assumed that the sequence ofvintage depreciation rates, dtv, is independent of theperiod t so that:

dtv=dv for all periods t and all vintages v (23:18)

23.48 The above material shows how the sequenceof vintage or used durable goods prices at a point in timecan be used in order to estimate depreciation rates. Thistype of methodology, with a few extra modifications toaccount for differing ages of retirement, was pioneeredby Beidelman (1973; 1976) and Hulten and Wykoff(1981a; 1981b; 1996).30

23.49 Recall the user cost formula for a new unitof the durable good under consideration defined byequation (23.1). The same approach can be used inorder to define a sequence of period 0 user costs forall vintages v of the durable. Thus suppose that P1av+1 isthe anticipated end of period 0 price of a unit of thedurable good that is v periods old at the beginning ofperiod 0, and let r0 be the consumer’s opportunity costof capital. Then the discounted to the beginning ofperiod 0 user cost of a unit of the durable good that is vperiods old at the beginning of period 0, u0v , is defined asfollows:

u0v � P0v�P1av+1=(1+r0) v=0, 1, 2, . . . (23:19)

23.50 It is now necessary to specify how the end ofperiod 0 anticipated vintage asset prices P1av are relatedto their counterpart beginning of period 0 vintage asset

27 For example, property taxes are associated with the use of housingservices and hence should be included in the user cost formula; seeparagraphs 23.100 and 23.101. As Katz (1983, p. 418) noted, taxationissues also have an impact on the choice of the interest rate: ‘‘Shouldthe rate of return be a before or after tax rate?’’ From the viewpoint ofa household that is not borrowing to finance the purchase of thedurable, an after tax rate of return seems appropriate; but from thepoint of view of a leasing firm, a before tax rate of return seemsappropriate. This difference helps to explain why rental equivalenceprices for the durable might be higher than user cost prices.28Another information source that could be used to identify depre-ciation rates for the durable good is the sequence of vintage rental orleasing prices that might exist for some consumer durables. In theclosely related capital measurement literature, the general frameworkfor an internally consistent treatment of capital services and capitalstocks in a set of vintage accounts was set out by Jorgenson (1989) andHulten (1990, pp. 127–129; 1996, pp. 152–160).29Using this notation for vintages, it can be seen that the vintage v=0price at the beginning of period t=0, P00, is equal to the price of anew unit of the good, P0. If these second-hand vintage prices dependon how intensively the durable good has been used in previous periods,then it will be necessary to further classify the durable good not onlyby its vintage v, but also according to the intensity of its use. In thiscase, think of the sequence of vintage asset prices, P0v , as correspondingto the prevailing market prices of the various vintages of the good atthe beginning of period 0 for assets that have been used at ‘‘average’’intensities.

30 See also Jorgenson (1996) for a review of the empirical literature onthe estimation of depreciation rates.


426

prices P0v . The assumption that is made now is that theentire sequence of vintage asset prices at the end ofperiod 0 is equal to the corresponding sequence of assetprices at the beginning of period 0 times a generalanticipated period 0 inflation rate factor (1+i0), where i0

is the anticipated period 0 (general) asset inflation rate.Thus it is assumed that

P1av =(1+i 0)P0v v=0, 1, 2, . . . (23:20)

Substituting equations (23.20) and (23.15)–(23.18) intoequation (23.19) leads to the following beginning ofperiod 0 sequence of vintage user costs:31

u0v=(1�dv�1)(1�dv�2) . . . (1�d0)� [(1+r0)�(1+i 0)(1�dv)]P 0=(1+r0)

=(1�dv�1)(1�dv�2) . . . (1�d0)� [r0�i 0+dv(1+i 0)]P 0=(1+r0) v=0, 1, 2, . . .

(23:21)

23.51 Note that if v=0, then the u00 defined byequation (23.21) agrees with the user cost formula for anew purchase of the durable u0 that was derived earlierin equation (23.7).23.52 The sequence of vintage user costs u0v defined

by equation (23.21) are expressed in terms of prices thatare discounted to the beginning of period 0. However, aswas done in paragraphs 23.22 to 23.33, it is also possibleto express the user costs in terms of prices that are‘‘discounted’’ to the end of period 0. Thus, define thesequence of vintage end of period 0 user cost, p0v , asfollows:

p0v � (1+r0)u0v=(1�dv�1)(1�dv�2) . . . (1�d0)� [r0�i 0+dv(1+i 0)]P 0 v=0, 1, 2, . . . (23:22)

23.53 If the real interest rate r0* is defined as thenominal interest rate r0 less the asset inflation rate i0,and the small terms dv i

0 are neglected in equation (23.22),then the sequence of end of period user costs defined byequation (23.22) reduces to:

p0v=(1�dv�1)(1�dv�2) . . . (1�d0)[r0*+dv]P 0

v=0, 1, 2, . . . (23:23)

Thus, if the price statistician has estimates for thevintage depreciation rates dv and the real interest rater0*, and is able to collect a sample of prices for newunits of the durable good P0, then the sequence ofvintage user costs defined by equation (23.23) can becalculated. To complete the model, the price statisticianshould gather information on the stocks held by thehousehold sector of each vintage of the durable good.Then normal index number theory can be applied tothese p and Q values, with the set of p being vintageuser costs and the set of Q being the vintage stockspertaining to each period. For some worked examplesof this methodology under various assumptions aboutdepreciation rates and the calculation of expected asset

inflation rates, see Diewert and Lawrence (2000) andDiewert (2003c).32

23.54 In the following three subsections, the generalmethodology described above is specialized by makingadditional assumptions about the form of the vintagedepreciation rates dv.

Geometric or declining balancedepreciation

23.55 The declining balance method of depreciationdates back to Matheson (1910, p. 55) at least.33 Interms of the algebra presented in the previous subsection,the method is very simple: all the cross-sectional vintagedepreciation rates d0v defined by equations (23.15)–(23.17) are assumed to be equal to the same rate d, whered is a positive number less than one; i.e., for all timeperiods t and all vintages v, it is assumed that

dtv=d v=0, 1, 2, . . . (23:24)

Substitution of equation (23.24) into equation (23.22)leads to the following formula for the sequence of period0 vintage user costs:

p0v=(1�d)v[(1+r0)�(1+i 0)(1�d)]P 0 v=0, 1, 2, . . .

=(1�d)vp00 (23:25)

23.56 The second set of equations in (23.25) saysthat all the vintage user costs are proportional to the usercost for a new asset. This proportionality means that itis not necessary to use an index number formula toaggregate over vintages to form a durable servicesaggregate. To see this, it is useful to calculate theaggregate value of services yielded by all vintages ofthe consumer durable at the beginning of period 0. Letq�v be the quantity of the durable purchased by thehousehold sector v periods ago for v=1, 2, . . . and let q0

be the new purchases of the durable during period 0.The beginning of period 0 price for these vintages of agev will be p0v , defined by equation (23.25) above. Thus theaggregate services of all vintages of the good, includingthose purchased in period 0, will have the followingvalue, S0:

S0=p00q0+p01q

�1+p02q�2+ . . .

=p00q0+(1�d)p00q�1+(1�d)2p00q�2+ . . .

using equation (23:25)

=p00[q0+(1�d)q�1+(1�d)2q�2+ . . . ]

=p00Q0 (23:26)

31When v=0, define d�1:1; i.e., the terms in front of the squarebrackets on the right-hand side of equation (23.21) are set equal to 1.

32Additional examples and discussion can be found in two recentOECD manuals on productivity measurement and the measurement ofcapital; see OECD (2001a; 2001b).33A case for attributing the method to Walras (1954, pp. 268–269)could be made, but he did not lay out all of the details. Matheson(1910, p. 91) used the term ‘‘diminishing value’’ to describe themethod. Hotelling (1925, p. 350) used the term ‘‘the reducing balancemethod’’, while Canning (1929, p. 276) used the term ‘‘the decliningbalance formula’’.


427

where the period 0 aggregate (quality-adjusted) quantityof durable services consumed in period 0, Q0, is definedas

Q0 � q0+(1�d)q�1+(1�d)2q�2+ . . . (23:27)

23.57 Thus the period 0 services quantity aggregateQ0 is equal to new purchases of the durable in period 0,q0, plus one minus the depreciation rate d times thepurchases of the durable in the previous period, q�1, plusthe square of one minus the depreciation rate times thepurchases of the durable two periods ago, q�2, and so on.The service price that can be applied to this quantityaggregate is p00, and the imputed rental price or user costfor a new unit of the durable purchased in period 0.

23.58 If the depreciation rate d and the purchasesof the durable in prior periods are known, then theaggregate service quantity Q0 can readily be calculatedusing equation (23.27). Then, using equation (23.26), itcan be seen that the value of the services of the durable(over all vintages), St, decomposes into the price term p00times the quantity term Q0. Hence, it is not necessary touse an index number formula to aggregate over vintagesusing this depreciation model.

Straight line depreciation23.59 Another very common model of depreciation

is the straight line model.34 In this model, a most prob-able length of life for the durable is somehow deter-mined, say L periods, so that after being used for Lperiods, the durable is scrapped. In the straight linedepreciation model, it is assumed that the period 0 cross-sectional vintage asset prices P0v follow the followingpattern of linear decline relative to the period 0 price ofa new asset P0:

P0vP 0

=L�vL

for v=0, 1, 2, . . . ,L�1 (23:28)

For v=L, L+1, . . . , it is assumed that P0v=0. Nowsubstitute equations (23.20) and (23.28) into the begin-ning of the period user cost formula (23.19) in order toobtain the following sequence of period 0 vintage usercosts for the durable:

u0v=P0v�(1+i 0)P0v+1=(1+r

0) for v=0, 1, 2, . . . ,L�1

=(L�v)P 0

L� (1+i

0)(L�v�1)P 0

(1+r0)L

=P 0

1+r0*(L�v)r0*L

+1

L

� �(23:29)

where the asset-specific real interest rate for period 0, r0*,is defined by

1+r0* � 1+r0

1+i 0(23:30)

23.60 The user costs for units of the durable goodthat are older than L periods are zero; i.e., u0v � 0 for

v�L. Looking at the terms in square brackets on theright-hand side of equation (23.29), it can be seen thatthe first term is a real interest opportunity cost forholding and using the unit of the durable that is v periodsold (and this imputed interest cost declines as the dur-able good ages), and the second term is a depreciationterm that is equal to the constant rate 1/L.

23.61 In this model of depreciation, it is necessary tokeep track of household purchases of the durable for Lperiods and weight up each vintage quantity q�v of thesepurchases by the corresponding vintage user cost u0v ,defined by equation (23.29), or the end of period vintageuser costs p0v , defined as (1+r0)u0v , could be used.35

‘‘One hoss shay’’ or lightbulb depreciation

23.62 The final model of depreciation that is incommon use is the ‘‘light bulb’’ or ‘‘one hoss shay’’ modelof depreciation.36 In this model, the durable deliversthe same services for each vintage: a chair is a chair,no matter what its age (until it falls to pieces and isscrapped). Thus this model also requires an estimateof the most probable life L of the consumer durable.37

In the ‘‘one hoss shay’’ model, it is assumed that thesequence of vintage beginning of the period user costsu0v , defined by the first line of equation (23.29), is con-stant for all vintages younger than the asset lifetime L;i.e., it is assumed that

u0=u0v=P0v�(1+i0)P0v+1=(1+r

0) for v=0,1,2, . . . ,L�1=P0v�gP0v+1 (23:31)

where the discount factor g is defined as

g � 1+i 0

1+r0=

1

1+r0*(23:32)

and the asset-specific real interest rate r0* was defined byequation (23.30). Now the second equation in (23.31)can be used to express the vintage v asset price P0vin terms of the common user cost u0 and the vintage v+1asset price, P0v+1, so that

P0v=u0+gP0v+1 (23:33)

23.63 Now start out using equation (23.33) withv=0, then substitute out P01 using equation (23.33) withv=1, then substitute out P02 using equation (23.33)with v=2, etc. The process finally ends after L such

34This model of depreciation dates back to the late 1800s; seeMatheson(1910, p. 55), Garcke and Fells (1893, p. 98) or Canning (1929, pp.265–266).

35A worked example using this model of depreciation can be found inDiewert (2003b).36 This model can be traced back to Bohm-Bawerk (1891, p. 342). Fora more comprehensive exposition, see Hulten (1990, p. 124) or Diewert(2003b).37 The assumption of a single life L for a durable can be relaxed using amethodology attributable to Charles R. Hulten:

We have thus far taken the date of retirement T to be the same for allassets in a given cohort (all assets put in place in a given year). However,there is no reason for this to be true, and the theory is readily extended toallow for different retirement dates. A given cohort can be broken intocomponents, or subcohorts, according to date of retirement and a sepa-rate T assigned to each. Each subcohort can then be characterized by itsown efficiency sequence, which depends among other things on the sub-cohort’s useful life Ti (Hulten (1990, p. 125)).


428

substitutions, when P0L is reached and, of course, P0Lequals zero. The following equation is obtained:

P 0=u0+gu0+g2u0+ . . .+gL�1u0

=u0[1+g+g2+ . . .+gL�1]

=u01

1�g�gL

1�g

� �provided that g< 1

=u01�gL1�g

� �: (23:34)

23.64 Now use the last equation in (23.34) in orderto solve for the constant over vintages (beginning ofthe period) user cost for this model, u0, in terms of theperiod 0 price for a new unit of the durable, P0, and thediscount factor g defined by equation (23.32):

u0=1�g1�gL

� �P 0: (23:35)

23.65 The end of period 0 user cost, p0, is, as usual,equal to the beginning of period 0 user cost, u0, times thenominal interest rate factor, 1+r0:

p0 � (1+r0)u0 (23:36)

23.66 The aggregate services of all vintages of thegood, including those purchased in period 0, will havethe following value, S0:

S0=p00q0+p01q

�1+p02q�2+ . . .

=p0[q0+q�1+q�2+ . . .+q�(L�1)]

=p0Q0 (23:37)

where the period 0 aggregate (quality-adjusted) quantityof durable services consumed in period 0, Q0, is definedas follows for this ‘‘one hoss shay’’ depreciation model:

Q0 � q0+q�1+q�2+ . . .+q�(L�1) (23:38)

23.67 Thus in this model of depreciation, the vintagequantity aggregate is the simple sum of household pur-chases over the last L periods. As was the case with thegeometric depreciation model, the ‘‘one hoss shay’’model does not require index number aggregation overvintages: there is a constant service price p0, and theassociated period 0 quantity Q0 is a weighted sum ofpast purchases for the geometric model and a simplesum over the purchases of the last L periods for thelight bulb model.38

23.68 How can the different models of depreciationbe distinguished empirically? In principle, informationon the prices of second-hand durable goods may be usedin order to decide which model of depreciation best fitsthe empirical facts. In practice, this is difficult to do, andhence different statistical agencies may make differentassumptions about the ‘‘correct’’ pattern of depreciation(which generates the ‘‘correct’’ pattern of vintage usercosts), based on whatever information they have at theirdisposal.

Unique durable goods and theuser cost approach

23.69 In the previous sections, it was assumed that anewly produced unit of the durable good remained thesame from period to period. This means that the variousvintages of the durable good repeat themselves goingfrom period to period. Hence, a particular vintage of thegood in the current period can be compared with thesame vintage in the next period. In particular, considerthe period 0 user cost of a new unit of a durable good,p00, defined by equation (23.8). For convenience, theformula is repeated here:

p00=[(1+r0)�(1+i 0)(1�d0)]P 0=[r0�i 0+d0(1+i 0)]P 0

(23:39)

23.70 Recall that P0 is the beginning of period 0purchase price for the durable, r0 is the nominal oppor-tunity cost of capital that the household faces in period 0,i0 is the anticipated period 0 inflation rate for the durablegood, and d0 is the one-period depreciation rate for a newunit of the durable good. In previous sections, it wasassumed that the period 0 user cost p00 for a new unit ofthe durable could be compared with the correspondingperiod 1 user cost p10 for a new unit of the durable pur-chased in period 1. This period 1 user cost can be definedas follows:

p10=[(1+r1)�(1+i1)(1�d0)]P1=[r1�i1+d0(1+i1)]P1

(23:40)

23.71 However, many durable goods are producedas one of a kind models. For example, a new house mayhave many features that are specific to that particularhouse. An exact duplicate of it is unlikely to be built inthe following period. Thus, if the user cost for the houseis constructed for period 0 using formula (23.39), wherethe new house price P0 plays a key role, then since therewill not necessarily be a comparable new house price forthe same type of unit in period 1, it will not be possibleto construct the period 1 user cost for a house of thesame type, p10 defined by equation (23.40), because thecomparable new house price P1 will not be available.

23.72 Recall the notation that was introduced inparagraphs 23.43 to 23.54 above, where Ptv was thesecond-hand market price at the beginning of period t ofa unit of a durable good that is v periods old. Define dvto be the depreciation rate for a unit of the durable goodthat is v periods old at the beginning of the period underconsideration. Using this notation, the user cost of thehouse (which is now one period old) for period 1, p11, canbe defined as follows:

p11 � (1+r1)P11�(1+i1)(1�d1)P11 (23:41)

where P11 is the beginning of period 1 price for the house

that is now one period old, r1 is the nominal opportunitycost of capital that the household faces in period 1, i1 isthe anticipated period 1 inflation rate for the durablegood and d1 is the one-period depreciation rate for ahouse that is one period old. For a unique durable good,there is no beginning of period 1 price for a new unit of

38Thus equation (23.38) is the quantity aggregate counterpart toequation (23.27).


429

the durable, P1, but it is natural to impute this price asthe potentially observable market price for the useddurable, P11, divided by one minus the period 0 depre-ciation rate, d0; i.e., define an imputed period 1 price fora new unit of the unique durable as follows:

P1 � P11(1�d0)

(23:42)

23.73 If equation (23.42) is solved for P11 and thesolution is substituted into the user cost defined byequation (23.41), then the following expression isobtained for p11, the period 1 user cost of a one-period-oldunique consumer durable:

p11 � (1�d0)[(1+r1)�(1+i1)(1�d1)]P1 (23:43)

23.74 If it is further assumed that the unique con-sumer durable follows the geometric model of depre-ciation, then

d=d0=d1 (23:44)

Substituting equation (23.44) into equations (23.43) and(23.40) leads to the following relationship between theimputed rental cost in period 1 of a new unit of the con-sumer durable, p10, and the period 1 user cost of the one-period-old consumer durable, p11:

p01=p11

(1�d) (23:45)

23.75 Thus, in order to obtain an imputed rentalprice for the unique consumer durable for period 1, p10,that is comparable to the period 0 rental price for a newunit of the consumer durable, p00, it is necessary to make aquality adjustment to the period 1 rental price for theone-period-old durable, p11, by dividing this latter priceby one minus the one-period geometric depreciation rate,d. This observation has implications for the qualityadjustment of observed market rents of houses. With-out this type of quality adjustment, observed dwellingunit rents will have a downward bias, since the observedrents do not adjust for the gradual lowering of thequality of the unit as a result of depreciation of the unit.39

23.76 Note that in order to obtain an imputedpurchase price for the unique consumer durable forperiod 1, P1, that is comparable to the period 0 purchaseprice for a new unit of the consumer durable, P0, it isnecessary to make a quality adjustment to the period 1used asset price for the one-period-old durable, P11, bydividing this latter price by one minus the period 0depreciation rate, d0; recall equation (23.42) above.40

23.77 This section concludes with some observationson the difficulties for economic measurement that occurwhen attempting to determine depreciation rates empiri-cally for unique assets. Consider again equation (23.42),which allows the potentially observable market price ofthe unique asset at the beginning of period 1, P11, to beexpressed as being equal to (1�d0)P1, where P1 is a hypo-thetical period 1 price for a new unit of the unique asset.If it is assumed that this hypothetical period 1 new assetprice is equal to the period 0 to 1 inflation rate factor(1+i0) times the observable period 0 asset price P0, thenthe following relationship between the two observableasset prices is obtained:

P11=(1�d0)(1+i 0)P 0 (23:46)

Thus the potentially observable period 1 used asset priceP11 is equal to the period 0 new asset price P0 times theproduct of two factors: (1�d0), a quality adjustmentfactor that takes into account the effects of ageing on theunique asset; and (1+i0), a period-to-period pure pricechange factor holding quality constant. The problemwith unique assets is that cross-sectional information onused asset prices at any point in time is no longeravailable to make it possible to sort out the separateeffects of these two factors. Thus there is a fundamentalidentification problem with unique assets; without extrainformation or assumptions, it will be impossible todistinguish the separate effects of asset deterioration andasset inflation.41 In practice, this identification problemis solved by making somewhat arbitrary assumptionsabout the form of depreciation that the asset is expectedto experience.42

23.78 Housing is the primary example of a uniqueasset. However, in addition to the problems outlined inthis section, there are other major problems associatedwith this particular form of unique asset. These pro-blems are discussed in the following sections.

The user cost of owner-occupiedhousing

23.79 Owner-occupied housing is typically anexample of a unique consumer durable, so the materialon the quality adjustment of both stock and rental pricesdeveloped in the previous section applies to this com-modity. Owner-occupied housing is, however, also an

39There is an exception to this general observation: if housing depre-ciation is of the ‘‘one hoss shay’’ type, then there is no need to quality-adjust observed rents for the same unit over time. However, ‘‘one hossshay’’ depreciation is empirically unlikely in the housing market sincerenters are generally willing to pay a rent premium for a new unit overan older unit of the same type. For empirical evidence of this agepremium, see Malpezzi, Ozanne and Thibodeau (1987), and Hoffmannand Kurz (2002, p. 19).40 This type of quality adjustment to the asset prices for unique con-sumer durables will always be necessary; i.e., there is no exception tothis rule, as was the case for ‘‘one hoss shay’’ depreciation in thecontext of quality-adjusting rental prices.

41 Special cases of this fundamental identification problem have beennoted in the context of various econometric housing models, by MartinJ. Bailey, Richard F. Muth and Hugh O. Nourse (1963, p. 936): ‘‘Forsome purposes one might want to adjust the price index for deprecia-tion. Unfortunately, a depreciation adjustment cannot be readily esti-mated along with the price index using our regression method. . . . Inapplying our method, therefore, additional information would beneeded in order to adjust the price index for depreciation.’’42 For example, if the unique asset is a painting by a master, then thedepreciation rate can be assumed to be very close to zero. As anotherexample, a reasonable guess at the likely length of life L of the uniqueasset could be made, and then the ‘‘one hoss shay’’ or straight linedepreciation models could be implemented. Alternatively, the length oflife L could be converted into an equivalent geometric depreciationrate d using the conversion rule d=n/L, where n is a number between 1and 2.


430

example of a composite good; i.e., two distinct com-modities are bundled together and sold (or rented) at asingle price. The two distinct commodities are:

– the structure;

– the land on which the structure is situated.

23.80 To model this situation, consider a particularnewly constructed dwelling unit that is purchased at thebeginning of period 0. Suppose that the purchase priceis V0. This value can be regarded as the sum of a costof producing the structure, say P0SQ

0S, where Q

0S is the

number of square metres of floor space in the structureand P0S is the beginning of period 0 price of constructionper square metre, and the cost of the land, say P0LQ

0L,

where Q0L is the number of square metres of the ground

on which the structure is situated and the associatedland, and P0L is the beginning of period 0 price of theland per square metre.43 Thus at the beginning of period0, the value of the dwelling unit, V0, is defined as follows:

V0=P0SQ0S+P

0LQ

0L (23:47)

23.81 Suppose that the anticipated price of a unit ofa new structure at the beginning of period 1 is P1aS andthat the anticipated price of a unit of land at thebeginning of period 1 is P1aL . Define the period 0 antici-pated inflation rates for new structures and land, i 0S and i

0L,

respectively, as follows:

1+i 0S �P1aSP0S

(23:48)

1+i 0L �P1aLP0L

(23:49)

23.82 Let d0 be the period 0 depreciation rate for thestructure. Then the anticipated beginning of period 1value for the structure and the associated land isequal to

V1a=P1aS (1�d0)Q0S+P

1aL Q

0L (23:50)

Note the presence of the depreciation term (1�d0) onthe right-hand side of equation (23.50). Should thisterm be associated with the expected beginning ofperiod 1 price for a new unit of structure P1aS or withthe structure quantity term Q0

S? On the principle thatlike should be compared with like for prices, it seemspreferable to associate (1�d0) with the quantity termQ0S. This is consistent with the treatment of unique

assets that was suggested in the previous section; i.e.,the initial quantity of structure Q0

S should be qualityadjusted downwards to the amount (1�d0) Q0

S at thebeginning of period 1.23.83 Now calculate the cost (including the imputed

opportunity cost of capital r0) of buying the dwellingunit at the beginning of period 0 and (hypothetically)selling it at the end of period 0. The following end ofperiod 0 user cost or imputed rental cost R0 for thedwelling unit is obtained using equations (23.47)–(23.50):

R0 � V0(1+r0)�V1a

=[P0SQ0S+P

0LQ

0L](1+r

0)�[P1aS (1�d0)Q0S+P

1aL Q

0L]

=[P0SQ0S+P

0LQ

0L](1+r

0)

�[P0S(1+i 0S )(1�d0)Q0S+P

0L(1+i

0L)Q

0L]

=p0SQ0S+p

0LQ

0L (23:51)

where separate period 0 user costs of structure and land,p0S and p

0L, are defined as follows:

p0S � [(1+r0)�(1+i 0S )(1�d0)]P0S=[r0�i 0S+d0(1+i 0S )]P

0S (23:52)

p0L � [(1+r0)�(1+i 0L)]P0L=[r0�i 0L]P0L (23:53)

23.84 Note that the above algebra indicates some ofthe major determinants of market rents for rentedproperties. The user cost formulae defined by equations(23.52) and (23.53) can be further simplified if the sameapproximations that were made in paragraphs 23.22 to23.33 are made here (recall equation (23.9); i.e., assumethat the terms r0�i 0S and r0�i 0L can be approximated bya real interest rate r0* and neglect the small term d0times i 0S in equation (23.52). Then the user costs definedby equations (23.52) and (23.53) reduce to:

p0S � [r0*+d0]P0S (23:54)

p0L � r0*P0L (23:55)

23.85 Thus the imputed rent for an owner-occupieddwelling unit is made up of three main costs:

� the real opportunity cost of the financial capital tied upin the structure;

� the real opportunity cost of the financial capital tiedup in the land;

� the depreciation cost of the structure.

23.86 The above simplified approach to the user costof housing can be even further simplified by assumingthat the ratio of the quantity of land to structure is fixed,and so the aggregate user cost of housing is equal to[r0*+d]P0H , where PH is a quality-adjusted housing priceindex that is based on all properties sold in the countryto households during the period under considerationand d is a geometric depreciation rate that applies tothe composite of household structure and land. Thisextremely simplified approach is used in Iceland;see Gudnason (2003, pp. 28–29).44 A variant of this

43 If the dwelling unit is part of a multiple unit structure, then the landassociated with it will be the appropriate share of the total land space.

44 The real interest rate that is used is approximately 4 per cent peryear and the combined depreciation rate for land and structures isassumed to equal 1.25 per cent per year. The depreciation rate forstructures alone is estimated to be 1.5 per cent per year. Propertytaxes are accounted for separately in the Icelandic CPI. Housingprice information is provided by the State Evaluation Board, basedon property sales data of both new and old housing. The StateEvaluation Board also estimates the value of the housing stock andland in Iceland, using a hedonic regression model based on propertysales data. The value of each household’s dwelling is collected in thehousehold budget survey.


431

approach is used by the United States Bureau of Eco-nomic Analysis: Lebow and Rudd (2003, p. 168) notethat the United States national accounts imputation forthe services of owner-occupied housing is obtained byapplying rent-to-value ratios for tenant-occupied hous-ing to the stock of owner-occupied housing. The rent-to-value ratio can be regarded as an estimate of theapplicable real interest rate plus the depreciation rate.

23.87 Returning to the period 0 imputed rental costmodel for a new structure defined by equations (23.47)–(23.53), now calculate the cost (including the imputedopportunity cost of capital r1) of buying the used dwell-ing unit at the beginning of period 1 and (hypothetically)selling it at the end of period 1. Thus at the beginning ofperiod 1, the value of the depreciated dwelling unit is V1

defined as follows:

V1=P1S(1�d0)Q0S+P

1LQ

0L (23:56)

where P1S is the beginning of period 1 constructionprice for building a new dwelling unit of the sametype and P1L is the beginning of period 1 price of landfor the dwelling unit. Note that equation (23.56) is anend of period 0 ex post or actual value of the dwellingunit, whereas the similar expression (23.50) defineda beginning of period 0 ex ante or anticipated value ofthe dwelling unit.

23.88 Suppose that the anticipated price of a unit ofa new structure at the beginning of period 2 is P2aS andthat the anticipated price of a unit of land at thebeginning of period 2 is P2aL . Define the period 1 antici-pated inflation rates for new structures and land, i1S and i

1L,

respectively, as follows:

1+i1S � P2aS =P1S (23:57)

1+i1L � P2aL =P1L (23:58)

23.89 Let d1 be the period 1 depreciation rate forthe structure. Then the anticipated beginning of period2 value for the structure and the associated land isequal to

V2a=P2aS (1�d0)(1�d1)Q0S+P

2aL Q

0L (23:59)

23.90 The following end of period 1 user cost orimputed rental cost, R1

1, for a one-period-old dwellingunit is obtained using equations (23.56)–(23.59):

R11 � V1(1+r1)�V2a

=[P1S(1�d0)Q0S+P

1LQ

0L](1+r

1)

�[P2aS (1�d0)(1�d1)Q0S+P

2aL Q

0L]

=[P1S(1�d0)Q0S+P

1LQ

0L](1+r

1)

�[P1S(1+i1S)(1�d0)(1�d1)Q0S+P

1L(1+i

1L)Q

0L]

=p1S1(1�d0)Q0S+p

1LQ

0L (23:60)

where the period 1 user costs of one-period-old structuresand land, p1S1 and p


p1S1 � [(1+r1)�(1+i1S)(1�d1)]P1S=[r1�i1S+d1(1+i1S)]P

1S (23:61)

p1L � [(1+r1)�(1+i1L)]P1L=[r1�i1L]P1L (23:62)

23.91 Comparing the period 0 user cost of land,p0L, defined by equation (23.53) with the period 1 usercost of land, p1L, defined by equation (23.62), it can beseen that these user costs have exactly the same formand hence are comparable. However, comparing theperiod 0 user cost for a new structure, p0S, defined byequation (23.52) with the period 1 user cost for a one-period-old structure, p1S, defined by equation (23.61),it can be seen that these user costs are not quite com-parable unless the period 0 depreciation rate d0 isequal to the period 1 depreciation rate d1. If decliningbalance depreciation for structures is assumed, thend0=d1=d, where d is the common depreciation rateacross all periods. Under this assumption, p1S is com-parable to the period 0 user cost for a new unit ofstructures p0S. Even under the assumption of geometricdepreciation, it can be seen that the period 1 imputedrent for a one-period-old dwelling unit, R1

1, defined byequation (23.60) is not comparable to the corre-sponding period 0 imputed rent for a new dwellingunit, R0, defined by equation (23.51). The imputed rentR1 that would be comparable to R0 can be defined asfollows:

R1 � p1SQ0S+p

1LQ

0L=R

11+p

1SdQ

0S (23:63)

where the period 1 user cost of structures, p1S, is definedby the right-hand side of equation (23.61), with d1 equalto the common depreciation rate d and the period 1 usercost of land, p1L, defined by equation (23.62). Equation(23.63) has the following implication for the qualityadjustment of the price of a rented property: if R0 is theobserved rent of the unit in period 0 and R1

1 is theobserved rent for the same dwelling unit in period 1,then the observed rent R1

1 is too low compared to R0 andso the period 1 observed rent should be quality adjustedupwards by the period 1 rental price for structures, p1S,times the amount of physical depreciation, dQ0

S, in thestructure that occurred in the previous period. This isthe same point that was made in paragraphs 23.69 to23.78, but in this section the complications arising fromthe fact that housing services are a mixture of structureand land services are taken into account.

23.92 It is evident that the main drivers for the usercosts of structures and land are a price index for newdwelling construction, PtS, and a price index for resi-dential land, PtL. Most statistical agencies have a con-stant quality price index for new residential structures,since this index is required in the national accounts inorder to deflate investment expenditures on residentialstructures. This index could be used as an approxima-tion to PtS.

45 The national accounts also require an

45 This index may only be an approximation since it covers the con-struction of rented properties, as well as owner-occupied dwellings.


432

imputation for the services of owner-occupied housingand thus the constant quality price component of thisimputation may be suitable for CPI purposes.46 If thenational accounts division also computes quarterly realbalance sheets for the economy, then a price index forresidential land may be available to the prices division.Even if this is the case, there will be problems in pro-ducing this price index for land on a timely basis andat a monthly frequency.47 A further possible source ofinformation on land prices may be found in land titleregistry offices and in the records of real estate firms.The associated information on transactions involvingthe same property can be used in a hedonic regressionframework; see for example, Malpezzi, Ozanne andThibodeau (1987).48

23.93 There are many other difficulties associatedwith measuring the price and quantity of owner-occupiedhousing services. The following section discusses someof the problems involved in modelling the costs ofcertain expenditures that are tied to the ownership of ahome.

The treatment of costs that aretied to owner-occupied housing23.94 There are many costs that are quite directly

tied to home-ownership. It is not always clear, however,how these costs can be decomposed into price andquantity components. Several of these cost componentsare listed below, and some ways of forming their asso-ciated prices are suggested.

The treatment of mortgage interest costs23.95 The derivation of the user cost or expected

rental price that an owner of a home should charge forthe use of the dwelling unit for one period implicitlyassumes that the owner has no mortgage interest costs,so the interest rate r0 refers to the owner’s opportunitycost of equity capital. In this section, the case where theowner has a mortgage on the property is considered.

23.96 Recall the notation in the previous sectionwhere the user cost or imputed rental cost, R0, for anequity-financed dwelling unit is obtained; see equation(23.51). Suppose now that the property purchase ispartly financed by a mortgage of M0 dollars at thebeginning of period 0. Let f 0 be the fraction of thebeginning of period 0 market value of the property thatis financed by the mortgage so that

M0=f 0V0=f 0[P0SQ0S+P

0LQ

0L] (23:64)

23.97 Let the one-period nominal mortgage interestrate be r0M . The owner’s period 0 benefits of owning thedwelling unit remain the same as in the previous sectionand are equal to V1a, defined by equation (23.50).However, the period 0 costs are now made up of anexplicit mortgage interest cost equal to M0(1+r0M) plusan imputed equity cost equal to (1�f 0)V 0(1+r0). Thusthe new imputed rent for using the property duringperiod 0 is now

R0� (1�f 0)V0(1+r0)+M0(1+r0M)�V1a

=(1�f 0)[P0SQ0S+P

0LQ

0L](1+r

0)

+f 0[P0SQ0S+P

0LQ

0L](1+r

0M)�[P1aS (1�d0)Q0

S+P1aL Q

0L]

=p0*S Q0S+p

0*L Q

0L (23:65)

where the new mortgage interest adjusted period 0 usercosts of structures and land, p0*S and p0*L , are defined asfollows:

p0*S � [(1+r0)(1�f 0)+(1+r0M)f0�(1+i 0S )(1�d0)]P0S

=[r0(1�f 0)+r0Mf 0�i 0S+d0(1+i 0S )]P0S

=[(r0�i 0S )(1�f 0)+(r0M�i 0S )f 0+d0(1+i 0S )]P0S

(23:66)

p0*L � [(1+r0)(1�f 0)+(1+r0M)f0�(1+i 0L)]P0L

=[r0(1�f 0)+r0Mf 0�i 0L]P0L=[(r0�i 0L)(1�f 0)+(r0M�i 0L)f 0]P0L (23:67)

23.98 Comparing the new user costs for structuresand land defined by equations (23.66) and (23.67) withthe corresponding equity-financed user costs defined byequations (23.52) and (23.53), it can be seen that the oldequity opportunity cost of capital r0 is now replaced by aweighted average of this equity opportunity cost andthe mortgage interest rate, r 0(1�f 0)+r0Mf 0, where f

0 isthe fraction of the beginning of period 0 value of thedwelling unit that is financed by the mortgage.

23.99 Central bankers often object to the inclusionof mortgage interest in a CPI. Examination of the lastequation in (23.66) and in (23.67) nevertheless showsthat the nominal mortgage interest rate r0M has an off-setting benefit resulting from anticipated price inflation inthe price of structures, i 0S in equation (23.66), and in theprice of land, i 0L in equation (23.67), so, as usual, whatcounts in these user cost formulae are real interest costsrather than nominal ones.

46However, the national accounts imputation for the services ofowner-occupied housing will only be produced on a quarterly basis,and so some additional work will be required to produce a pricedeflator on a monthly basis. Also, even though the SNA 1993recommends that the imputation for the services of owner-occupiedhousing be based on the rental equivalent method, it may be the casethat the imputation covers only the imputed depreciation on thestructure part of owner-occupied housing. As was pointed out above,there are two other important additional components that should alsobe included in owner-occupied housing services; namely, the imputedreal interest on the structures and the land on which the structure issituated. These latter two components of imputed expenditures arelikely to be considerably larger than the depreciation component.47Another source of information on the value of residential land maybe available from the local property tax authorities, particularly ifproperties are assessed at market values.48Many hedonic regression studies use the logarithm of a transactionprice as the dependent variable. This specification of the hedonic modelis usually not consistent with the additive nature of the structure andland components of a property, and the multiplicative nature of thedepreciation adjustment, as appearing in equations (23.47) and (23.56)which defined the value of a specific property in successive periods.


433

The treatment of property taxes23.100 Recall the user costs of structures and land

defined by equations (23.52) and (23.53). It is nowsupposed that the owner of the housing unit must paythe property taxes T0

S and T0L for the use of the structure

and land, respectively, during period 0.49 Define theperiod 0 structures tax rate t0S and land tax rate t0L, asfollows:

t0S �T0S

P0SQ0S

(23:68)

t0L �T0L

P0LQ0L

(23:69)

23.101 The new imputed rent for using the propertyduring period 0, R0, including the property tax costs, isdefined as follows:

R0� V0(1+r0)+T0S+T

0S�V1a

=[P0SQ0S+P

0LQ

0L](1+r

0)+t0SP0SQ

0S+t0LP

0LQ

0L

�[P0S(1+i 0S )(1�d0)Q0S+P

0L(1+i

0L)Q

0L]

=p0SQ0S+p

0LQ

0L (23:70)

where separate period 0 tax-adjusted user costs of struc-tures and land, p0S and p


p0S� [(1+r0)�(1+i 0S )(1�d0)+t0S]P0S

=[r0�i 0S+d0(1+i 0S )+t0S]P0S (23:71)

p0L� [(1+r0)�(1+i 0L)+t0L]P0L

=[r0�i 0L+t0L]P0L (23:72)

Thus the property tax rates, t0S and t0L, defined byequations (23.68) and (23.69), enter the user costs ofstructures and land, p0S and p0L, defined by equations(23.71) and (23.72), in a simple additive manner; i.e.,these terms are additive to the previous depreciation andreal interest rate terms.50

The treatment of property insurance23.102 At first glance, it would seem that property

insurance could be treated in the same manner as thetreatment of property taxes in the previous subsection.Thus, let C0

S be the cost of insuring the structure at thebeginning of period 0 and define the period 0 structurespremium rate g0S as follows:

g0S �C0S

P0SQ0S

(23:73)

23.103 The new imputed rent for using the propertyduring period 0, R0, including property tax and insur-ance costs, is defined as follows:

R0 � V0(1+r0)+T 0S+T

0S+C

0S�V1a=p0SQ

0S+p

0LQ

0L

(23:74)

where separate period 0 tax and insurance adjusted usercosts of structures and land, p0S and p0L, are defined asfollows:

p0S � [(1+r0)�(1+i 0S )(1�d0)+t0S+g0S]

P0S=[r0�i 0S+d0(1+i 0S )+t0S+g0S]P0S (23:75)

p0L � [(1+r0)�(1+i 0L)+t0L]P0L=[r0�i 0L+t0L]P

0L

(23:76)

23.104 Thus the insurance premium rate g0S appearsin the user cost of structures, p0S, defined by equation(23.75), in an additive manner, analogous to the additiveproperty tax rate term.51 If it is desired to have a sepa-rate CPI price component for insurance, then the cor-responding period 0 and 1 prices can be defined as g0SP

0S

and g1SP1S, respectively, while the corresponding period 0

and 1 expenditures can be defined as g0SP0SQ

0S and

g1SP1S(1�d)Q0

S, respectively.52 Of course, if this separate

treatment is implemented, then these terms have to bedropped from the corresponding user costs of structures.

23.105 The above treatment of property taxationand insurance assumes that the property taxes and thepremium payments are made at the end of the periodunder consideration; see equation (23.74) above. Whilethis may be an acceptable approximation for the pay-ment of property taxes, it is not acceptable for thepayment of insurance premiums: the premium must bepaid at the beginning of the period of protection ratherthan at the end. When this complication is taken intoaccount, the user cost of structures becomes

p0S � [(1+r0)�(1+i 0S )(1�d0)+t0S+g0S(1+r0)]P0S

=[r0�i 0S+d0(1+i 0S )+t0S+g0S(1+r0)]P0S (23:77)

23.106 There are some additional problems asso-ciated with the modelling of property insurance:

� The above user cost derivations assume that the riskof property damage remains constant from period toperiod. If the risk of damage changes, then an argu-ment can be made for quality adjustment of the pre-mium to hold constant the risk so that like can becompared with like.

� The gross premium approach to insurance is taken inthe above treatment; i.e., it is assumed that dwel-ling owners pay premiums for property protection

49 If there is no breakdown of the property taxes into structure andland components, then just impute the overall tax into structure andland components based on the beginning of the period values of bothcomponents.50 If the price statistician uses the national accounts imputation for thevalue of owner-occupied housing services, care should be taken toensure that the value of property taxes is included in this imputation.

51 This treatment of property insurance dates back to Walras (1954,pp. 268–269).52 Similarly, if it is desired to have a separate CPI price component forproperty taxes on structures, then the corresponding period 0 and 1prices can be defined as t0SP

0S and t1SP

1S , respectively, while the corre-

sponding period 0 and 1 expenditures can be defined as t0SP0SQ

0S and

t1SP1S(1�d)Q0

S , respectively.


434

services, no matter whether they have a claim or not. Inthe net premium approach, payments to settle claimsare subtracted from the gross premium payments.

� The property protection may not be complete; i.e., theinsurance policy may have various limitations on thetype of claim that is allowed and there may be adeductible or damage threshold, below which noclaim is allowed. If the deductible changes from periodto period, then the price statistician is faced with arather complex quality adjustment problem.

Thus it can be seen that there are many difficult prob-lems that remain to be resolved in this area.

The treatment of maintenance andrenovation expenditures23.107 Another problem associated with home-

ownership is the treatment of maintenance expenditures,major repair expenditures and expenditures associatedwith renovations or additions.23.108 Empirical evidence suggests that the normal

decline in a structure resulting from the effects of ageingand use can be offset by maintenance and renovationexpenditures. How exactly should these expenditures betreated in the context of modelling the costs and benefitsof home-ownership?23.109 A common approach in the national accounts

literature is to treat major renovation and repair expen-ditures as capital formation, and smaller routine main-tenance and repair expenditures as current expenditures.If this approach is followed in the CPI context, thenthese smaller routine maintenance expenditures can betreated in the same manner as other non-durable goodsand services. The major renovation and repair expen-ditures do not enter the CPI in the period that they aremade, but are capitalized and added to expenditures onnew structures for the period under consideration, sothat period 0 investment in structures in constantdollars, I0S say,53 would include both types of expendi-tures. Let Q0

S and Q1S be the stocks (in constant quality

units) of owner-occupied structures in the referencepopulation at the beginning of period 0 and 1, respec-tively. Then if the geometric model of depreciation isused, so that the constant period-to-period depreciationrate d is applicable, then the beginning of period 1 stockof owner-occupied structures Q1

S is related to thebeginning of period 0 stock of structures Q0

S and theperiod 0 investment in structures I0S, according to thefollowing equation:

Q1S=(1�d)Q0

S+I0S (23:78)

23.110 Thus, if declining balance depreciation isassumed for structures, then the treatment of majorrepair and renovation expenditures does not pose majorconceptual problems using a conventional capital accu-mulation model: it is only necessary to have an estimate

for the monthly or quarterly depreciation rate d, astarting value for the stock of owner-occupied structuresfor some period, information on new purchases of resi-dential housing structures by the household sector,information on expenditures by owners on major repairsand renovations, and a construction price index for newresidential structures. With this information on a timelybasis, up-to-date CPI weights for the stock of owner-occupied structures could be constructed.54

23.111 This section concludes by looking at howmajor repair and renovation expenditures could betreated in a repeat sales regression model that uses trans-actions data on the sale of the same housing unit in twoor more periods. In order to minimize notational com-plexities, consider a highly simplified situation wheredata on the sale of N houses of a relatively homogeneoustype for two consecutive periods are available. Supposethat these sale prices are V0

n for period 0 and V1n for

period 1, for n=1, 2, . . . , N. Suppose that a price indexfor structures of this type of property in period 0, P0S,and a corresponding price index for land in period 0,P0L, have been constructed.55 The price statistician’sproblem is to use the data on the matched sales for thetwo periods in order to construct estimates of these twoindices for period 1; i.e., the problem is to construct P1Sand P1L.

23.112 The period 0 dwelling unit values for the Nproperties can be decomposed into the structure andland components as follows:

V0n=V

0Sn+V

0Ln=anP0SQ

0Sn+bnP

0LQ

0Ln; n=1, 2, . . . ,N

(23:79)

where V0Sn and V

0Ln are the estimated period 0 values of

the structure and land of property n in period 0, P0S andP0L are the (known) price index values for structures andland for all properties of this type in period 0, and Q0

Snand Q0

Ln are (known) estimates of the quantity of struc-tures and land for property n. The numbers an and bn areproperty n quality adjustment factors that convert theproperty standardized values of structures and land,P0SQ

0Sn and P

0LQ

0Sn, respectively, into the period 0 actual

market values, V0Sn and V0

Ln, respectively; i.e., if esti-mates of the period 0 market values of the structures andland for property n are available, then an and bn can bedefined as follows:

an �V0Sn

P0SQ0Sn

; bn �V0Ln

P0LQ0Ln

; n=1, . . . ,N (23:80)

23.113 Suppose that information on the dollaramount of major repairs and renovations made toproperty n during period 0, VR0

n, are also available foreach property n in the sample of properties. Then theperiod 1 value for property n, V1

n , should be approxi-mately equal to

53Let VI0S be the nominal value of investment in new owner-occupiedstructures in period 0, plus the value of major renovation expendituresmade during period 0. Then the constant dollar quantity of investmentcould be defined as I0S � VI0S=P0

S , where P0S is the period 0 construction

price index for new structures.

54 The practical problems involved in obtaining all this information ona timely basis are, however, not trivial.55 If these period 0 indices are not available, then set P0

S and P0L equal

to 1.


435

V1n=anP1S(1�d)Q0

Sn+VR0n+bnP

1LQ

0Ln; n=1, . . . ,N

(23:81)

where d is the geometric depreciation rate for structures.All the variables on the right-hand side of equation(23.81) are assumed to be known, with the exception ofthe period 1 price index values for structures and land,P1S and P1L, respectively, and the one-period geometricdepreciation rate, d. If the number of observations N isgreater than three, then it would appear that these threeparameters, P1S, P

1L and d, could be estimated by a linear

regression using the N equations in (23.81) as estimatingequations. It turns out, however, that this is not quitecorrect. The problem is that the parameters P1S and(1�d) appear in equation (23.81) in a multiplicativefashion so that while the product of these two terms willbe identified by the regression, the individual termscannot be uniquely identified. This is just a reappearanceof the same problem that was discussed in paragraphs23.69 to 23.78 in relation to unique consumer durables:the separate effects of ageing of the asset (depreciationor capital consumption) and price appreciation overtime cannot be separately identified using just marketdata.56

23.114 There are three possible solutions to thisidentification problem:

� use an external estimate of the depreciation rate d;� use an external construction price index P1S instead ofestimating it as a parameter in equations (23.81);

� abandon the repeat sales approach and use a hedonicregression approach instead.

23.115 What would a hedonic regression model looklike, taking into account the approximate additivity ofthe value of the housing structure and the value of theland on which the structure is situated? If the renova-tions problem is ignored and geometric depreciation ofthe structure is assumed, then the value of a housingunit n in period t that is v periods old, Vtn, should beapproximately equal to the depreciated value of thestructure plus the value of the land plus an error term;i.e., the following relationship should hold approxi-mately:

Vtn=PtS(1�d)

vQSn+PtLQL+u

tn (23:82)

where d is the one-period geometric depreciation rate,QSn is the number of square metres of floor space of theoriginal structure for housing unit n, QL is the numberof square metres of land on which the housing structure

is situated, and utn is an error term. PtS is the beginning ofperiod t price level for structures of this type and PtL isthe corresponding price of land for this class of housingunits. As long as there is more than one vintage ofstructure in the sample (i.e., more than one v), then theparameters PtS, P

tL and d can be identified by running a

non-linear regression model using equation (23.82).Why can the price levels be identified in the presenthedonic regression model, whereas they could not beidentified in the repeat sales model? The answer is thatthe hedonic model (23.82) does not assume property-specific quality adjustment factors for each housing unit;instead, all the housing units in the sample are assumedto be of comparable quality once prices are adjusted forthe age of the unit and the quantity (in square metres) oforiginal structure and the quantity of land.

23.116 Unfortunately, many housing structures thatmay have started off as identical structures do notremain the same over time, because of differing stan-dards of maintenance, as well as major renovationsand additions to some of the structures. To model thisphenomenon, let Rtn be real maintenance, repair andrenovation expenditures on housing unit n during periodt, and suppose that these real expenditures depreciate atthe geometric rate dR. It is reasonable to assume thatthese expenditures add to the value of the housing unit,and so equation (23.82) should be replaced by the fol-lowing equation:

Vtn=PtS(1�d)

vQSn+PtR[R

tn+(1�dR)Rt�1n

+(1�dR)2Rt�2n + � � �+(1�dR)vRt�vn ]+PtLQL+utn

(23:83)

where PtR is the period t price level for real maintenance,repair and renovation expenditures on this class ofhousing units. If information on these real renovationand repair expenditures, Rtn,R

t�1n ,Rt�2n , . . . ,Rt�vn , is

available for each housing unit in the sample of housingunits that sold in period t, then the parameters PtS, P

tL,

PtR, d and dR can be identified by running a non-linearregression model using equation (23.83).57 A majorpractical problem with implementing a hedonic regres-sion model along the above lines is that, usually, accu-rate data on renovation and repair expenditures on aparticular dwelling unit between the construction of theinitial housing unit and the present period are notavailable. Without accurate data on repairs and reno-vations, it will be impossible to obtain accurate esti-mates of the unknown parameters in the hedonicregression model.

23.117 A final practical problem with the abovehedonic regression model will be mentioned. Theoreti-cally, ‘‘normal’’ maintenance expenditures could beincluded in the renovation expenditure terms, Rtn, inequation (23.83). If this is done, then including normalmaintenance expenditures in Rtn will have the effect ofincreasing the estimated depreciation rates d and dR.Thus different statistical agencies, with different criteria

56Recall equation (23.46). This fundamental identification problemwas recognized by Bailey, Muth and Nourse (1963, p. 936) in theoriginal repeat sales housing article, but it was ignored by them andsubsequent users of the repeat sales methodology. Another problemwith the housing hedonic regression literature is that, usually, thelogarithm of the purchase price is taken as the dependent variable inthe regression. While this specification has some advantages, it doesnot recognize properly the additive nature of the structure and landcomponents of the housing property. A final problem with the tradi-tional hedonic housing literature is that, usually, separate price indicesfor land and structures are not estimated. It is important to allow forseparate price indices for these two components since, usually, theprice of land is more volatile and tends to increase faster than the priceof structures over long periods of time.

57Alternatively, if price levels are available for PtS and PtR from con-struction price indices, then these parameters do not have to be esti-mated.


436

for deciding where to draw the line between ‘‘normal’’maintenance and ‘‘major’’ repair and renovations, willproduce different estimated depreciation rates.

The treatment of the transactions costsof home purchase23.118 Another cost of home-ownership needs to be

discussed. Normally, when a family purchases a dwell-ing unit, they have to pay certain fees and costs, whichcan include:

� the commissions of real estate agents who help thefamily find the ‘‘right’’ property;

� various transactions taxes that governments canimpose on the sale of the property;

� various legal fees that might be associated with thetransfer of title for the property.

23.119 Should the above fees be considered asexpenditure in the period of purchase, or should theysimply be regarded as part of the purchase price of theproperty and hence be depreciated over time in a man-ner analogous to the treatment of structures in thenational accounts?23.120 An argument can be made for either treat-

ment. From the viewpoint of the opportunity costtreatment of purchases of durable goods, the relevantprice of the dwelling unit in the periods following thepurchase of the property is the after tax and transactionsfees value of the property. This viewpoint suggests thatthe transactions costs of the purchaser should be countedas expenses in the period of purchase. From the view-point of a landlord who has just purchased a dwellingunit for rental purposes, however, it would not be sen-sible to charge the tenant the full cost of these transac-tions fees in the first month of rent. The landlord wouldtend to capitalize these costs and recover them graduallyover the time period that the landlord expects to own theproperty. Thus, either treatment could be justified andthe statistical agency will have to decide which treatmentis most convenient from its particular perspective.

User costs for landlordsversus owners23.121 In the previous section, the various costs

associated with home-ownership were discussed. Bothhome-owners and landlords face these costs. Thus, theywill be reflected in market rents, and this must be kept inmind if the imputed rent approach is used to value theservices of owner-occupied housing. If some or all ofthese associated costs of owner-occupied housing arecovered elsewhere in the CPI (e.g., home insurancecould be separately covered), then the value of imputedrents for owner-occupied housing must be reduced bythe amount of these expenditures covered elsewhere.23.122 In addition to the costs of home-ownership

covered in the previous section, landlords face a numberof additional costs compared to the home-owner. Theseadditional costs will be reflected in market rents. Thus, ifmarket rents are used to impute the services provided bythe ownership of a dwelling unit, then these extra costs

should also be removed from the market rents that areused for imputation purposes, since they will not berelevant for owner-occupiers. These additional landlord-specific costs are discussed in paragraphs 23.123 to23.133.

Damage costs23.123 Tenants do not have the same incentive to

take care of a rented property compared to an ownedproperty, so depreciation costs for a rented property arelikely to exceed depreciation rates for comparable ownedproperties. Usually, landlords demand damage deposits,but often these deposits are not sufficient to cover thecosts of the actual damages that some tenants inflict.

Non-payment of rent and vacancy costs23.124 At times, tenants run into financial difficul-

ties and are unable to pay landlords the rent that isowned. Usually, eviction is a long-drawn-out process,and so landlords can lose several months of rent beforea non-paying tenant finally leaves. The landlord alsoincurs extra costs compared to a home-owner when arented property remains vacant because of lack ofdemand.58 These extra costs will be reflected in marketrents but should not be reflected in the user costs ofowner-occupied housing.

Billing and maintenance costs23.125 A landlord may have to rent office space and

employ staff to send out monthly bills to tenants, andemploy staff to respond to requests for maintenance. Ahome-owner who gives his or her time in order to pro-vide maintenance services59 offers this time at his or herafter income tax wage rate, which may be lower than thebefore income tax wage rate that a landlord must pay hisor her employees. The net effect of these factors leads tohigher market rents compared to the correspondingowner-occupied user cost.

The opportunity cost of capital23.126 The home-owner’s after tax opportunity cost

of capital that appeared in the various user cost for-mulae considered earlier in this chapter will typically belower than the landlord’s before tax opportunity cost ofcapital. Put another way, the landlord has an extraincome tax cost compared to the home-owner. In addi-tion, the landlord may face a higher risk premium forthe use of capital because of the risks of damage andnon-payment of rent. Care must be taken that theseadditional landlord costs are not counted twice.

58 The demand for rented properties can vary substantially over thebusiness cycle, and this can lead to depressed rents or very high rentscompared to the user costs of home-ownership. Thus imputed rentsbased on market rents of similar properties can differ substantiallyfrom the corresponding user costs of owner-occupied housing over thebusiness cycle.59 Typically, these imputed maintenance costs will not appear in theCPI, but if the user cost of an owned dwelling unit is to be comparablewith the market rent of a similar property, these imputed labour costsshould be included.


437

The supply of additional services forrented properties

23.127 Often, properties that are to let will containsome major consumer durables that home-owners haveto provide themselves, such as refrigerators, stoves,washing-machines, driers and air conditioning units. Inaddition, landlords may pay for electricity or fuel insome rented apartments. Thus, to make the marketrental comparable to an owner-occupied imputed rent,the market rental should be adjusted downwards toaccount for the above factors (which will appear else-where in the expenditures of owner-occupiers).

23.128 The factors listed above will tend to makeobserved market rental prices higher than the corre-sponding user cost for an owner-occupier of a propertyof the same quality. Thus, if the imputed rentalapproach is used to value the services of owner-occupiedhousing, then these market-based rents should beadjusted downward to account for the above factors.

23.129 Although all the above factors will tend tolead to an upward bias if unadjusted market rental ratesare used to impute the services of owner-occupiedhousing, there is another factor not discussed thus farthat could lead to a large downward bias. That factor isrent controls.

23.130 Under normal conditions, the acquisitionsapproach to the treatment of owner-occupied housingwill give rise to the smallest expenditures, the user costapproach will give rise to the next highest level ofexpenditures, and the use of imputed market rentals willgive the largest level of expenditures for owner-occupiedhousing. For the first two approaches, a main driver ofthe price of owner-occupied housing is the price ofnew housing construction. For the user cost approach,another main driver is the price of land. For the imputedrent approach, the main driver of the price of owner-occupied housing is the rental price index.

23.131 The above discussion is far from beingcomplete and definitive, but it does illustrate that it isnot completely straightforward to impute market rentalrates to owner-occupied dwelling units. Care must betaken to ensure that the ‘‘correct’’ expenditure weightsare constructed.

23.132 As can be seen from the material above, thetreatment of owner-occupied housing presents specialdifficulties. Astin (1999, p. 5) discussed some of thedifficulties that the European Union encountered intrying to find the ‘‘best’’ approach to use in its Har-monized Index of Consumer Prices (HICP), as follows:

A special coverage problem concerns owner-occupiedhousing. This has always been one of the most difficultsectors to deal with in CPIs.

Strictly, the price of housing should not be included ina CPI because it is classified as capital. On the otherhand, the national accounts classifies imputed rents ofowner-occupiers as part of consumers’ expenditure. Thisis a reasonable thing to do if the aim is to measure thevolume of consumption of the capital resource of hous-ing. But that is not what a CPI is measuring.

Some countries, following the compensation indexconcept, would prefer to have mortgage interest includedin the HICP. This approach could indeed be defended for

a compensation index, because there is no doubt that themonthly mortgage payment is an important element inthe budget of many households: a rise in the interest rateacts in exactly the same way as a price increase from thepoint of view of the individual household. But this is notacceptable for a wider inflation index.

So, after many hours of debate, the Working Partycame to the conclusion that there were just two options.The first was to simply exclude owner-occupied housingfrom the HICP. One could at least argue that this was aform of harmonization, although it is worrying that thereare such large differences between Member States in thepercentages of the population which own or rent theirdwellings. Exclusion also falls in line with the interna-tional guideline issued 10 years ago by the ILO. Fur-thermore, it would be possible to supplement the HICPwith a separate house price index, which could be used byanalysts as part of a battery of inflation indicators.

The second option was to include owner-occupiedhousing on the basis of acquisition costs, essentiallytreating them like any other durable. Most secondhandhousing would be excluded: in practice the index wouldinclude new houses plus a small volume of housing newto the household sector (sales from the company or gov-ernment sectors to the household sector).

The main problem here is practical: several countriesdo not have new house price indices and their construc-tion could be difficult and costly. A Task Force is atpresent examining these matters. Final recommendationsare due at the end of 1999.

Because of the complexities involved in modelling thetreatment of owner-occupied housing, final recommen-dations have still not emerged for the HICP.

23.133 A fourth approach to the treatment ofhousing will be studied in the following section. Sincethis approach has only been applied to owner-occupieddwellings, it is not as ‘‘universal’’ as the other threeapproaches.60

The payments approach23.134 A fourth possible approach to the treatment

of owner-occupied housing, the payments approach, isdescribed by Charles Goodhart (2001, pp. F350–F351)as follows:

The second main approach is the payments approach,measuring actual cash outflows, on down payments,mortgage repayments and mortgage interest, or somesubset of the above. This approach always, however,includes mortgage interest payments. This, though com-mon, is analytically unsound. First, the procedure is notcarried out consistently across purchases. Other goodsbought on the basis of credit, e.g., credit card credit, areusually not treated as more expensive on that account(though they have been in New Zealand). Second, thetreatment of interest flows is not consistent across per-sons. If a borrower is worse off in some sense wheninterest rates rise, then equivalently a lender owning aninterest bearing asset is better off; why measure one andnot the other? If I sell an interest earning asset, say a

60 The acquisitions, user cost and rental equivalence approaches can beapplied to any consumer durable, but to apply the rental equivalenceapproach, appropriate rental or leasing markets for the durable mustexist.


438

money market mutual fund holding, to buy a house, whyam I treated differently to someone who borrows on a(variable rate) mortgage? Third, should not the questionof the price of any purchase be assessed separately fromthe issue of how that might be financed? Imports,inventories and all business purchases tend to be pur-chased in part on credit. Should we regard importsas more expensive, when the cost of trade credit rises?Money, moreover, is fungible. As we know from calcu-lations of mortgage equity withdrawal, the loan may besecured on the house but used to pay for furniture. Wheninterest rates rise, is the furniture thereby more expensive?Moreover, the actual cash out-payments totally ignorechanges in the ongoing value of the house whether bydepreciation, or capital loss/gain, which will often dwarfthe cash flow. Despite its problems, such a cash paymentapproach was used in the United Kingdom until 1994 andstill is in Ireland.

23.135 Thus, the payments approach to owner-occupied housing is a kind of cash flow approach to thecosts of operating an owner-occupied dwelling. Possibleobjections to this approach are that it ignores the oppor-tunity costs of holding the equity in the owner-occupieddwelling, it ignores depreciation, and it uses nominalinterest rates without any offset for inflation. If thepayments approach is adjusted for these imputed costs,however, then the result is a rather complicated usercost approach to the treatment of housing. Nevertheless,as was mentioned in Chapter 10, under some condi-tions, the payments approach to the treatment of owner-occupied housing may be a reasonable compromise. Ingeneral, the payments approach will tend to lead tomuch smaller monthly expenditures on owner-occupiedhousing than the other three main approaches, exceptduring periods of high inflation, when the nominalmortgage rate term becomes very large without anyoffsetting item for inflation.61

Alternative approaches for pricingowner-occupied housing23.136 For consumer durables that have long useful

lives, the usual acquisitions approach will not be ade-quate for CPI users who desire prices that measure theservice flows that consumer durables generate. This isparticularly true for owner-occupied housing. Henceit will be useful to many users if, in addition to theacquisitions approach, the statistical agency implementsa variant of either the rental equivalence approach orthe user cost approach for long-lived consumer durablesand for owner-occupied housing in particular. Users canthen decide which approach best suits their purposes.Any one of the three main approaches could be chosenas the approach that would be used in the ‘‘headline’’CPI. The other two approaches could be made availableto users as analytic tables.

23.137 We conclude this chapter by outlining someof the problems involved in implementing the three mainapproaches to the measurement of price change forowner-occupied housing.

The acquisitions approach23.138 In order to implement the acquisitions

approach, a constant quality price index for the sales ofnew residential housing units will be required.

The rental equivalence approach

Option 1: Using home-owners’ estimatesof rents

23.139 In this option, home-owners would be sur-veyed and asked to estimate a rental price for theirhousing unit. Problems with this approach are:

� Home-owners may not be able to provide very accu-rate estimates for the rental value of their dwellingunit.

� The statistical agency should make an adjustment tothese estimated rents over time in order to take intoaccount the effects of depreciation, which causes thequality of the unit to decline slowly over time (unlessthis effect is offset by renovation and repair expendi-tures).62

� Care must be taken to determine exactly what extraservices are included in the home-owner’s estimatedrent; i.e., does the rent include insurance, electricityand fuel or the use of various consumer durables inaddition to the structure? If so, these extra servicesshould be stripped out of the rent, since they arecovered elsewhere in the CPI.63

Option 2: Using a hedonic regression model ofthe rental market to impute rents

23.140 In this option, the statistical agency wouldcollect data on rented properties and their character-istics, and then use this information to construct ahedonic regression model for the housing rental mar-ket.64 This model would then be used to impute pricesfor owner-occupied properties. Problems with thisapproach are:

� It is information intensive; in addition to requiringinformation on the rents and characteristics of rentedproperties, information on the characteristics ofowner-occupied properties would also be required.

� The characteristics of the owner-occupied populationcould be quite different from the characteristics of therental population. In particular, if the rental marketfor housing is subject to rent controls, this approach isnot recommended.

61 If there is high inflation, then the statistical agency using the pay-ments approach may want to consider adjusting nominal mortgageinterest rates for the inflation component, as is done in paragraphs23.95 to 23.99. For additional material on the payments approach, seeChapter 10.

62Recall paragraphs 23.79 to 23.93.63 It could be argued that these extra services that might be included inthe rent are mainly a weighting issue; i.e., that the trend in the home-owner’s estimated rent would be a reasonably accurate estimate of thetrend in the rents after adjusting for the extra services included in therent.64 See Hoffmann and Kurz (2002) for an example of such a model.


439

� Hedonic regression models suffer from a lack ofreproducibility in that different researchers will havedifferent characteristics in the model and use differentfunctional forms.

� From the previous discussions, it is seen that marketrents can be considerably higher than the opportunitycosts of home-owners. Hence, using market rents toimpute rents for owner-occupiers may lead to rentsthat are too high.65 On the other hand, if there arerent controls or a temporary glut of rented properties,then market rents could be too low compared to theopportunity costs of home-owners.

� There is some evidence that depreciation is somewhatdifferent for rental units compared to owner-occupiedhousing units.66 If this is so, then the imputationprocedure will be somewhat incorrect. However, allstudies that estimate depreciation for owner-occupiedhousing suffer from biases resulting from the inade-quate treatment of land and the lack of informationon repair, renovation and maintenance expendituresover the life of the dwelling unit. It is not certain thatdepreciation for rental units is significantly differentfrom that for owner-occupied units.

The user cost approach23.141 It is first necessary to decide whether an ex

ante or ex post user cost of housing is to be calculated. Itseems that the ex ante approach is the more useful forCPI purposes; these are the prices that should appear ineconomic models of consumer choice. Moreover, the expost approach will lead to user costs that fluctuate toomuch to suit the needs of most users. Of course, theproblem with the ex ante approach is that it will bedifficult to estimate anticipated inflation rates for houseprices.

Option 3: The rent-to-value approach23.142 In this option, the statistical agency collects

information on market rents paid for a sample of rentedproperties, but it also collects information on the salesprice of these rented properties when they are sold. Usingthese two pieces of information, the statistical agency canform an estimated rent-to-value ratio for rented proper-ties of various types. It can be seen that this rent-to-valueratio represents an estimate of all the terms that go intoan ex ante user cost formula, except the asset price of theproperty; i.e., the rent-to-value ratio for a particularproperty can be regarded as an estimate of the interestrate less anticipated housing inflation plus the deprecia-tion rate plus the other miscellaneous rates discussedin paragraphs 23.94 to 23.120, such as insurance and

property tax rates. Under the assumption that these ratesremain reasonably constant over the short run, changesin user costs are equal to changes in the price of owner-occupied housing. Thus, this approach can be imple-mented if a constant quality price index for the stockvalue of owner-occupied housing can be developed. Itmay be decided to approximate the comprehensive priceindex for owner-occupied housing by a new housingprice index. If this is done, the approach essentiallyreduces to the acquisitions approach, except that theweights will generally be larger using this user costapproach than those obtained using the acquisitionsapproach.67 Problems with this approach include:

� It will require a considerable amount of resources toconstruct a constant quality price index for the stockof owner-occupied housing units. If a hedonic regres-sion model is used, there are problems associated withthe reproducibility of the results.

� Rent-to-value ratios can change considerably overtime. Hence it will be necessary to keep collectinginformation on rents and selling prices of rentedproperties.

� As was noted in paragraphs 23.121 and 23.122, theuser cost structure of rented properties can be quitedifferent from the corresponding user cost structure ofowner-occupied properties. Hence, the use of rent-to-value ratios can give misleading results.68

Option 4: The simplified user cost approach23.143 This approach is similar to that of Option 3

above, but instead of using the rent-to-value ratio toestimate the sum of the various rates in the user costformula, direct estimates are made of these rates. If thesimplified Icelandic user cost approach discussed inparagraphs 23.79 to 23.93 is used, all that is required is aconstant quality owner-occupied housing price index, anestimated real interest rate, and an estimated compositedepreciation rate on the structure and land together.Problems with this approach are:

� As was the case with Option 3 above, it will require aconsiderable amount of resources to construct aconstant quality price index for the stock of owner-occupied housing units. If a hedonic regression modelis used, there are problems associated with thereproducibility of the results.

� It is not known with any degree of certainty what theappropriate real interest rate should be.

� Similarly, it is difficult to determine what the ‘‘cor-rect’’ depreciation rate should be.69

65Again, it could argued that this is a mainly a weighting issue; i.e.,that the trend in market rents would be a reasonably accurate estimatefor the trend in home-owners’ opportunity costs.66According to Stephen Malpezzi, Larry Ozanne and Thomas G.Thibodeau (1987, p. 382): ‘‘The average depreciation rate for rentalproperty is remarkably constant, ranging from 0.58 per cent to 0.60 percent over the 25 year period. Depreciation rates for owner-occupiedunits show more variation than the estimated rates for renter occupiedunits. The average depreciation rate for owner-occupied housing ran-ges from 0.9 per cent in year 1 to 0.28 per cent in year 20.’’

67Recall the discussion in paragraphs 23.34 to 23.42.68 This is primarily a weighting issue, however, so the trend in theconstant quality stock of owner-occupied housing price index shouldbe an adequate approximation to the trend in owner-occupied usercosts.69 Because of the lack of information on repairs and renovations,estimated housing depreciation rates vary widely:

One striking feature with the results of all three approaches used in theseand related studies is their variability: estimates range from about a halfper cent per year to two and a half per cent (Malpezzi, Ozanne andThibodeau (1987, pp. 373–375).


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This problem is complicated by the fact that, over time,the price of land tends to increase faster than the price ofbuilding a residential structure, so the land price com-ponent of an owner-occupied housing unit will tend toincrease in importance, which in turn will tend todecrease the composite depreciation rate.

Option 5: A national accounting approach23.144 This approach makes use of the fact that the

national accounts division of the statistical agency willusually collect data on investment in residential hous-ing, as well as on repair and renovation expenditures onhousing. In addition, many statistical agencies will alsoconstruct estimates for the stock of residential dwellingunits, so that estimates for the structures’ depreciationrates are available. Finally, if the statistical agency alsoconstructs a national balance sheet, then estimates forthe value of residential land will also be available. Thus,all the basic ingredients that are necessary to constructstocks for residential structures and the associatedland stocks are available. If, in addition, assumptionsabout the appropriate nominal interest rate and aboutexpected prices for structures and land are made,70 thenaggregate user costs of residential structures and resi-dential land can be constructed. The proportion ofthese stocks that is rented can be deducted, and esti-mates for the user costs and corresponding values forowner-occupied residential land and structures can bemade. Of course, it would be almost impossible to do

all this on a current basis, but all the above computa-tions can be done for a base period in order to obtainappropriate weights for owner-occupied structures andland. Then, it can be seen that the main drivers for themonthly user costs are the price of a new structureand the price of residential land. Hence, if timelymonthly indicators for these two prices can be devel-oped, the entire procedure is feasible. Problems withthis approach include:

� As was the case with Option 4 above, it will be difficultto determine what the ‘‘correct’’ depreciation ratesand real interest rates are.71

� It will be difficult to construct a monthly price ofresidential land index.

� It may be difficult to convert the residential housinginvestment price deflator from a quarterly to amonthly basis.

23.145 All the above five options have their advan-tages and disadvantages; there does not appear to be aclear ‘‘winning’’ option. Thus, each statistical agencywill have to decide whether they have the resources toimplement any of these five options in addition to theusual acquisitions approach to the treatment of owner-occupied housing. From the viewpoint of the cost ofliving approach to the CPI, any one of the five optionswould be an adequate approximation to the idealtreatment from the perspective of measuring the flow ofconsumption services in each period.72

70Alternatively, an appropriate real interest rate can be assumed.

71As usual, however, it can be argued that errors in estimating theseparameters will mainly affect the weights used in the price index.72 For consumer durables that do not change in quality over time,Option 5 will probably suffice.


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DURABLES AND USER COSTS - International Labour … · DURABLES AND USER COSTS 23 Introduction 23.1...

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