CSIS Discussion Paper No.157

Measuring the Services of Durables and Owner Occupied
Housing
W. Erwin Diewert and Chihiro Shimizu∗
Center for Sparial Information Science,
The University of Tokyo.
January 6, 2019
Abstract
This paper provides an update to the chapter on the treatment of durables in the Consumer Price Index Manual (2004)[76]. The most important durable good is housing
services, which typically account for approximately 25% of total consumption. A large
fraction of housing services is the services of Owner Occupied Housing (OOH). The main
approaches to measuring the services of OOH are (i) the acquisitions approach; (ii) the
rental equivalence approach and (iii) the user cost approach. Two other approaches are
sometimes used: (iv) the opportunity cost approach and (v) the payments approach.
The purpose of this paper is to present the main approaches to the treatment of OOH
and to discuss the beneﬁts and costs of the alternative approaches. The paper also discusses the problems associated with forming imputations for the services of “ordinary”
consumer durable goods.
Journal of Economic Literature Classiﬁcation Numbers
C23, C43, C81, D12, E31.
C43, C82, E01.
Key Words
Durable goods; Consumer Price Index; Owner Occupied Housing; hedonic regression
models; rental equivalence approach; user cost approach, acquisitions approach, opportunity cost approach.

∗ W. Erwin Diewert:

School of Economics, University of British Columbia, Vancouver B.C., Canada,
V6T 1Z1 and the School of Economics, University of New South Wales, Sydney, Australia (email: erwin.diewert@ubc.ca) and Chihiro Shimizu, Nihon University, Setagaya, Tokyo, 154-8513, Japan, (email:shimizu.chihiro@nihon-u.ac.jp). The authors gratefully acknowledges the ﬁnancial support of the SSHRC
of Canada and Nomura Foundation of Japan.

11 Introduction
When a durable good (other than housing) is purchased by a consumer, national Consumer
Price Indexes typically attribute all of that expenditure to the period of purchase, even though
the use of the good extends beyond the period of purchase.*1 This is known as the acquisitions
approach to the treatment of consumer durables in the context of determining a pricing concept
for the CPI. However, if one takes a cost of living approach to the Consumer Price Index, thenit may be more appropriate to take the cost of using the services of the durable good during
the period under consideration as the pricing concept. There are two broad methods for
estimating this imputed cost for using the services of a durable good during a period:• If rental or leasing markets for a comparable consumer durable exist, then this market
rental price could be used as an estimate for the cost of using the durable during theperiod. This method is known as the rental equivalence approach.
• If used or second hand markets for the durable exist, then the imputed cost of purchasinga durable good at the beginning of the period and selling it at the end could be computed
and this net cost could be used as a estimate for the cost of using the durable during
the period. This method is known as the user cost approach.
The major advantages of the acquisitions approach to the treatment of consumer durablesare:• It is conceptually simple and entirely similar to the treatment of nondurables and services and• No complex imputations are required.
The major disadvantage of the acquisitions approach compared to the other two approaches
is that the acquisitions approach is not likely to reﬂect accurately the consumption servicesof consumer durables in any period. Thus suppose that real interest rates in a country arevery high due to a macroeconomic crisis. Under these conditions, purchases of automobiles
and houses and other long lived consumer durables may drop dramatically, perhaps to zero.
However, the actual consumption of automobile and housing services of the country’s population will not fall to zero under these circumstances: households will still be consuming theservices of their existing stocks of motor vehicles and houses. Thus for at least some purposes,
rather than taking the cost of purchasing a consumer durable as the pricing concept, it will
be more useful to take the cost of using the services of the durable good during the period
under consideration as the pricing concept.
The above paragraphs provide a brief overview of the three major approaches to the treatmentof consumer durables. In the remainder of this introduction, we explore these approaches ina bit more detail and give the reader an outline of the detailed discussion that will follow insubsequent sections.
Since the beneﬁts of using the consumer durable extend over more than one period, it does notseem to be appropriate to charge the entire purchase cost of the durable to the initial period*1 This treatment of the purchases of durable goods dates back to Alfred Marshall (1898; 594-595)[86] atleast: “We have noticed also that though the beneﬁts which a man derives from living in his own houseare commonly reckoned as part of his real income, and estimated at the net rental value of his house;
the same plan is not followed with regard to the beneﬁts which he derives from the use of his furnitureand clothes. It is best here to follow the common practice, and not count as part of the national income
or dividend anything that is not commonly counted as part of the income of the individual.”2

of purchase. If this point of view is taken, then the initial purchase cost must be distributed
somehow over the useful life of the asset. This is the fundamental problem of accounting.*2
Hulten (1990)[71] explains the accounting problems that arise from the purchase of a durable
good as follows:“Durability means that a capital good is productive for two or more time periods, andthis, in turn, implies that a distinction must be made between the value of using or
renting capital in any year and the value of owning the capital asset. This distinction
would not necessarily lead to a measurement problem if the capital services used in any
given year were paid for in that year; that is, if all capital were rented. In this case,transactions in the rental market would ﬁx the price and quantity of capital in eachtime period, much as data on the price and quantity of labor services are derived fromlabor market transactions. But, unfortunately, much capital is utilized by its ownerand the transfer of capital services between owner and user results in an implicit rent
typically not observed by the statistician. Market data are thus inadequate for thetask of directly estimating the price and quantity of capital services, and this has led
to the development of indirect procedures for inferring the quantity of capital, like theperpetual inventory method, or to the acceptance of ﬂawed measures, like book value.”
Charles R. Hulten (1990; 120-121)[71].
Thus the treatment of durable goods is more complicated than the treatment of nondurable
goods and services due to the simple fact that the period of time that a durable is used by
the consumer extends beyond the period of purchase. For nondurables and services, the pricestatistician’s measurement problems are conceptually simpler: prices for the same commodity
need only be collected in each period and compared. However, for a durable good, the periodsof payment and use do not coincide and so complex imputation problems arise if the goal of
the price statistician is to measure and compare the price of using the services of the durable
in two time periods.
As mentioned above, there are 3 main methods for dealing with the durability problem:• Ignore the problem of distributing the initial cost of the durable over the useful life of
the good and allocate the entire charge to the period of purchase. As noted above, thisis known as the acquisitions approach and it is the present approach used by Consumer
Price Index statisticians for all durables with the exception of housing.
• The rental equivalence approach. In this approach, a period price is imputed for the
durable which is equal to the rental price or leasing price of an equivalent consumer
durable for the same period of time.
• The user cost approach. In this approach, the initial purchase cost of the durable is decomposed into two parts: one part which reﬂects an estimated cost of using the servicesof the durable for the period and another part, which is regarded as an investment,
which must earn some exogenous rate of return.
*2 “The third convention is that of the annual accounting period. It is this convention which is responsible

for most of the diﬃcult accounting problems. Without this convention, accounting would be a simple
matter of recording completed and fully realized transactions: an act of primitive simplicity.” Stephen
Gilman (1939; 26)[57]. “All the problems of income measurement are the result of our desire to attributeincome to arbitrarily determined short periods of time. Everything comes right in the end; but by thenit is too late to matter.” David Solomons (1961; 378)[103]. Note that these authors do not mention the
additional complications that are due to the fact that future revenues and costs must be discounted toyield values that are equivalent to present dollars. For more recent papers on the fundamental problemof accounting, see Cairns (2013)[12] and Diewert and Fox (2016)[18].

3These three major approaches will be discussed more fully in sections 2, 3 and 4 below.*3 Thereis fourth approach that has not been applied but seems conceptually attractive that will be
discussed in section 5: the opportunity cost approach. This approach takes the maximum of therental equivalence and user cost as the price for the use of the services of a consumer durable
over a period of time. Finally, there is a ﬁfth approach to the treatment of consumer durables
that has only been used in the context of pricing owner occupied housing and that is thepayments approach.*4 This is a kind of cash ﬂow approach, which is not entirely satisfactory.
It will be discussed in section 17 after we have discussed the other approaches in more detail.
The main three approaches to the treatment of durable purchases can be applied to the purchase of any durable commodity. However, historically, it turns out that the rental equivalence
and user cost approaches have only been applied to owner occupied housing. In other words,
the acquisitions approach to the purchase of consumer durables has been universally usedby statistical agencies, with the exception of owner occupied housing. A possible reason forthis is tradition; i.e., Marshall (1898)[86] set the standard and statisticians have followed his
example for the past century. However, another possible reason is that unless the durable
good has a very long useful life, it usually will not make a great deal of diﬀerence in the longrun whether the acquisitions approach or one of the two alternative approaches is used. This
point is discussed in more detail in section 10 below.
A major component of the user cost approach to valuing the services of Owner Occupied
Housing (OOH) is the depreciation component. In section 6, a general model of depreciation
for a consumer durable is presented and then it is specialized in sections 7-9 to the three mostcommon models of depreciation that are widely used.
The general model presented in section 6 assumes that homogeneous units of the durable areproduced in each period so that information on the prices of the various vintages of the durable
at any point in time can be used to determine the pattern of depreciation. However, many
durables (like housing) are custom produced and thus the methods for determining the formof depreciation explained in section 6 are not immediately applicable. The special problems
caused by uniquely produced consumer durables are considered in section 11.
The remainder of this paper looks at the particular problems associated with measuring theservices of housing. Sections 12-14 show how information on the sales of dwelling units canbe used to decompose the sales price into land and structure components. This information isrequired for the country’s national balance sheet accounts. The decomposition into land andstructure components is also required for the construction of rental prices and user costs.*5
Section 12 looks at land and structure decompositions for the sale of detached housing units
while section 13 does the same for the sales of condominium units. Hedonic regression models
are explained in sections 12 and 13 that are basically supply side models while section 14
looks at a demand side hedonic regression model for the sales of detached houses. Section 15
considers hedonic regression models for rents. Section 16 looks at the factors that inﬂuencerents. This section also explains why the amount that an owned dwelling unit could rent foris in general diﬀerent from the user cost that could be used to price the services of the unit
*3 t should be noted that in principle, the user cost and rental equivalence approaches should be muchthe same: the owner of a rental property needs to construct a user cost for the current period (using
its opportunity cost of capital as the interest rate that appears in the user cost formula) so that theresulting user cost can be used as the rental price that will just allow the owner to make the target rateof return on the property investment. In practice, the exact equality does not hold due to various market
imperfections which will be discussed later.
*4 This is the term used by Goodhart (2001; F350-F351)[58].
*5 Depreciation applies to the structure part of property value but not to the land part.

4

to an owner. This section brings up important issues that pertain to the measurement of
the services of OOH. Thus section 16 revisits issues surrounding the use of either the rental
equivalence or user cost approaches to the valuation of Owner Occupied Housing.
As mentioned early, section 17 explains the payments approach while section 18 concludes.

2 The Acquisitions Approach
The net acquisitions approach to the treatment of owner occupied housing is described by
Goodhart as follows:“The ﬁrst is the net acquisition approach, which is the change in the price of newly
purchased owner occupied dwellings, weighted by the net purchases of the reference
population. This is an asset based measure, and therefore comes close to my preferredmeasure of inﬂation as a change in the value of money, though the change in the priceof the stock of existing houses rather than just of net purchases would in some respects
be even better. It is, moreover, consistent with the treatment of other durables. A few
countries, e.g., Australia and New Zealand, have used it, and it is, I understand, themain contender for use in the Euro-area Harmonized Index of Consumer Prices (HICP),
which currently excludes any measure of the purchase price of (new) housing, though it
does include minor repairs and maintenance by home owners, as well as all expendituresby tenants.” Charles Goodhart (2001; F350)[58].
Thus the weights for the net acquisitions approach are the net purchases of the household sectorof houses from other institutional sectors in the base period. Note that in principle, purchasesof second-hand dwellings from other sectors are relevant here; e.g., a local government may sellrental dwellings to owner occupiers. However, typically, newly built houses form a major partof these types of transactions. Thus the long term price relative for this category of expenditure
will be primarily the price of (new) houses (quality adjusted) in the current period relative tothe price of new houses in the base period.*6 If this approach is applied to other consumer
durables, it is extremely easy to implement: the purchase of a durable is treated in the same
way as a nondurable or service purchase is treated.
One additional implication of the net acquisition approach is that major renovations andadditions to owner occupied dwelling units could also be considered as being in scope for thisapproach. In practice, major renovations to a house are treated as investment expendituresand not covered as part of a consumer price index. Normal maintenance expenditures on adwelling unit are usually treated in a separate category in the CPI.
Traditionally, the net acquisitions approach also includes transfer costs relating to the buying
and selling of second hand houses as expenditures that are in scope for an acquisitions typeconsumer price index. These costs are mainly the costs of using a real estate agent’s services
and asset transfer taxes. These costs can be measured but the question arises as to what is theappropriate deﬂator for these costs. An overall property price index is probably a satisfactory*6 This price index may or may not include the price of the land that the new dwelling unit sits on; e.g., a

new house price construction index would typically not include the land cost. The acquisitions approachconcentrates on the purchases by households of goods and services that are provided by suppliers fromoutside the household sector. Thus if the land on which a new house sits was previously owned by thehousehold sector, then presumably, the cost of this land would be excluded from an acquisitions typenew house price index. In this case, the price index that corresponds to the acquisitions approach isbasically a new house price index (excluding land) or a modiﬁcation of a construction cost index where
the modiﬁcation takes into account builder’s margins.

5

deﬂator.*7
The major advantage of the acquisitions approach is that it treats durable and nondurable
purchases in a completely symmetric manner and thus no special procedures have to be developed by a statistical agency to deal with durable goods.*8 As will be seen in section 5
below, the major disadvantage of this approach is that the expenditures associated with thisapproach will tend to understate the corresponding expenditures on durables that are impliedby the rental equivalence and user cost approaches.
Some diﬀerences between the acquisitions approach and the other approaches are:• If rental or leasing markets for the durable exist and the durable has a long useful life,
then as mentioned above, the expenditure weights implied by the rental equivalence oruser cost approaches will typically be much larger than the corresponding expenditure
weights implied by the acquisitions approach; see section 16 below.
• If the base year corresponds to a boom year (or a slump year) for the durable, thenthe base period expenditure weights may be too large or too small. Put another way,
the aggregate expenditures that correspond to the acquisitions approach are likely to
be more volatile than the expenditures for the aggregate that are implied by the rental
equivalence or user cost approaches.*9• In making comparisons of consumption across countries where the proportion of owning versus renting or leasing the durable varies greatly,*10 the use of the acquisitions
approach may lead to misleading cross country comparisons. The reason for this is that
opportunity costs of capital are excluded in the net acquisitions approach whereas they
are explicitly or implicitly included in the other two approaches.
More fundamentally, whether the acquisitions approach is the right one or not depends on theoverall purpose of the index number. If the purpose is to measure the price of current period
consumption services, then the acquisitions approach can only be regarded as an approximation
to a more appropriate approach (which would be either the rental equivalence or user cost
approach). If the purpose of the index is to measure monetary (or nonimputed) expendituresby households during the period, then the acquisitions approach might be preferable (provided
the land component of property value is in scope), since the rental equivalence and user cost
approaches necessarily involve imputations.*11*7 See the discussion in section 16 below on transfer costs.
*8 The acquisitions approach is straightforward and simple for most durable goods but not for housing, if

the land component of property value is regarded as out of scope. Properties are sold with a single pricethat includes both the land and structure components of housing and so if the land part of property
value is regarded as out of scope for the index, then there is a problem in decomposing property value
into land and structure components. This decomposition problem can be avoided if information on theconstruction costs for building a new housing unit are available. In this case, the construction cost index(including builder’s markups) can serve as the price index for newly constructed dwelling units.
*9 Hill, Steurer and Waltl (2017; 6)[68] summarize the problem of variable weights as follows: “Hence theexpenditure weights on OOH under the acquisitions approach can ﬂuctuate very signiﬁcantly over the
housing cycle. If the weights are updated regularly this may have a destabilizing eﬀect on the CPI. If
the weights are not updated regularly, then the treatment of OOH may be highly sensitive to the choiceof reference year.”*10 From Hoﬀmann and Kurz (2002; 3-4)[69], about 60% of German households lived in rented dwellings
whereas only about 11% of Spaniards rented their dwellings in 1999.
*11 Fenwick (2009)[49] (2012)[50] laid out the case for the use of the acquisitions approach as a usefulmeasure of general inﬂation. He also argued for the construction of multiple consumer price indexes tosuit diﬀerent purposes.

6The details of the acquisitions approach (as applied to OOH) are discussed in great detailin Eurostat (2017)[48].*12 Eurostat is considering the use of the acquisitions approach forthe treatment of OOH in its Harmonized Index of Consumer Prices (HICP) but at this date,no decision has been ﬁnalized. At present, OOH is simply omitted in the HICP. Eurostat is
considering the use of the acquisitions approach for OOH because at ﬁrst sight, it seems thatno imputations have to be made in order to implement it. The HICP was created as an index
of consumer prices that used actual transactions prices without the use of any imputations.*13
As such, it was thought to be particularly useful for monitoring inﬂation by central banks.
However, the sale of a newly constructed dwelling unit typically includes a land component
which Eurostat wishes to exclude but existing methods for excluding the land component
involve imputations.*14

3 The Rental Equivalence Approach
The rental equivalence approach simply values the services yielded by the use of a consumer
durable good for a period by the corresponding market rental value for the same durable forthe same period of time (if such a rental value exists). This is the approach taken in the
System of National Accounts: 1993 for owner occupied housing:“As well-organized markets for rented housing exist in most countries, the output of
own-account housing services can be valued using the prices of the same kinds of services
sold on the market with the general valuation rules adopted for goods and services
produced on own account. In other words, the output of housing services produced byowner-occupiers is valued at the estimated rental that a tenant would pay for the sameaccommodation, taking into account factors such as location, neighbourhood amenities,
etc. as well as the size and quality of the dwelling itself.” Eurostat, IMF, OECD, UN
and World Bank (1993; 134)[44].
However, the System of National Accounts: 1993 follows Marshall (1898; 595)[86] and does
not extend the rental equivalence approach to consumer durables other than housing. This
seemingly inconsistent treatment of durables is explained in the SNA 1993 as follows:“The production of housing services for their own ﬁnal consumption by owner-occupiers
has always been included within the production boundary in national accounts, although
it constitutes an exception to the general exclusion of own-account service production.
The ratio of owner-occupied to rented dwellings can vary signiﬁcantly between countries
and even over short periods of time within a single country, so that both international
and intertemporal comparisons of the production and consumption of housing services
could be distorted if no imputation were made for the value of own-account services.”
Eurostat, IMF, OECD, UN and World Bank (1993; 126)[44].
Eurostat’s (2001)[45] Handbook on Price and Volume Measures in National Accounts also
recommends the rental equivalence approach for the treatment of the dwelling services for*12 This very useful publication also discusses the main methods for the treatment of OOH and it also

covers the methods used to construct residential property price indexes. The latter topic is also coveredin Eurostat (2013)[47].
*13 However, with the passage of time, it became apparent that some imputations for changes in the qualityof consumer goods and services had to be made. Thus the current HICP is not completely free from
imputations. See Astin (1999)[1] for the methodological foundations of the HICP.
*14 The use of a construction cost index also involves an imputation (but it is a reasonably straightforward
one).

7

owner occupied housing:“The output of dwelling services of owner occupiers at current prices is in many countries
estimated by linking the actual rents paid by those renting similar properties in therented sector to those of owner occupiers. This allows the imputation of a notional rent
for the service owner occupiers receive from their property.” Eurostat (2001; 99)[45].
To summarize the above material, it can be seen that the rental equivalence approach to thetreatment of durables is conceptually simple: impute a current period rental or leasing pricefor a comparable dwelling unit as the price for the services of an owned dwelling unit.*15
But where will the statistical agency ﬁnd the relevant rental data to price the services of OOH?
There are at least three possible methods:• Ask home owners what they think the market rent for their dwelling unit is;*16• Undertake a survey of owners of rental properties or managers of rental properties andask what rents they charge for their rental properties by type of property or• Use one of the above two methods to get a rent to value ratio for various types ofproperty for a benchmark period and then link these ratios to indexes of purchaseprices for the various types of property.*17
There are some disadvantages associated with the use of the rental equivalence approach tothe valuation of OOH services:• Homeowners may not be able to provide very accurate estimates for the rental value of
their dwelling unit.
• On the other hand, if the statistical agency tries to match the characteristics of an owned
dwelling unit with a comparable unit that is rented in order to obtain the imputed
rent for the owned unit, there may be diﬃculties in ﬁnding such comparable units.
Furthermore, even if a comparable unit is found, the rent for the comparable unit may
not be an appropriate opportunity cost for valuing the services of the owned unit.*18• The statistical agency should make an adjustment to these estimated rents over timein order to take into account the eﬀects of depreciation, which causes the quality of theunit to slowly decline over time (unless this eﬀect is completely oﬀset by renovation andrepair expenditures).*19• Care must be taken to determine exactly what extra services are included in the homeowner’s estimated rent; i.e., does the rent include insurance, electricity and fuel or theuse of various consumer durables in addition to the structure? If so, these extra services
should be stripped out of the rent, if they are covered elsewhere in the consumer priceindex.*20*15 As will be seen in section 16 below, the situation is not quite as simple as indicated above.
*16 This approach is used by the Bureau of Labor Statistics (1983)[10] in order to determine expenditure

weights for owner occupied housing; i.e., homeowners are asked to estimate what their house would rent
for if it were rented to a third party.
*17 Lebow and Rudd (2003; 169)[83] note that the US Bureau of Economic Analysis applies a benchmark
rent to value ratio for rented properties to the value of the owner occupied stock of housing. It can beseen that this approach is essentially a simpliﬁed user cost method where all of the key variables in theuser cost formula (to be discussed later) are held constant except the asset price of the property.
*18 We will return to this point after we have discussed the opportunity cost approach to the valuation of
OOH services.
*19 This issue will be discussed in more detail in section 16 below.
*20 However, it could be argued that these extra services that might be included in the rent are mainly aweighting issue; i.e., it could be argued that the trend in the homeowner’s estimated rent would be areasonably accurate estimate of the trend in the rents after adjusting for the extra services included in8In order to overcome the ﬁrst diﬃculty listed above, statistical agencies, including the Japanese
government, are currently collecting housing rent data from property management companies
or owners who rent out their dwelling units; i.e., Japan uses the second method to value theservices of OOH. However, the characteristics of the owner occupied population of dwelling
units are generally quite diﬀerent from the characteristics of the rental population.*21 Thus
typically, it is diﬃcult to ﬁnd rental units that are comparable to owned dwelling units. The
use of hedonic regression techniques can mitigate this lack of matching problem. Moreover theuse of hedonic regressions can deal with the depreciation or quality decline problem mentioned
above. We will discuss hedonic regression techniques later in this paper in sections 12-15.
In addition to the above possible biases in using the rental equivalence approach to the valuation of the services of OOH, there are diﬀerences between contract rent and market rent.
Contract rent refers to the rent paid by a renter who has a long term rental contract withthe owner of the dwelling unit and market rent is the rent paid by the renter in the ﬁrst
period after a rental contract has been negotiated. In a normal economy which is experiencing
moderate or low general inﬂation, typically market rent will be higher than contract rent.
However, if there are rent controls or a temporary glut of rental units, then market rent could
be lower than contract rent. In any case, it can be seen that if we value the services of anowner occupied dwelling at its current opportunity cost on the rental market, we should beusing market rent rather than contract rent.
The rents used to estimate the cost of rented dwellings in the Japanese CPI is the aggregateof rents paid for rental accommodation. These rents include a combination of newly signed
rental contracts and rollover contracts for existing tenants. It is appropriate to use both typesof contract to measure the actual cost of rental housing (but of course, these rents should be
quality adjusted for depreciation and other changes in quality). But it is not appropriate to use
both types of contract to impute rents for owner occupied housing: only market rents should
be used. It is known that price adjustments are basically not made for rollover contracts (i.e.
renewed leases). As a result, it is to be expected that new contract rents determined freely by
the market will diverge considerably from rollover contract rents.*22
Genesove (2003)[56], based on a study using individual data from the American Housing Survey and survey research, analyzed the stickiness of rents by dividing them into new contracts
and rollover contracts.
In Japan, Shimizu, Nishimura and Watanabe (2010b)[99] and Shimizu and Watanabe(2011)[101] used data from a housing listing magazine and a property management company
to measure the extent of housing rent stickiness in the country and analyzed the micro
structure of rental adjustments.
In the following section, we provide an introduction to user cost theory for a non-housing
durable good. In subsequent sections, we will deal with the problems associated with measuring depreciation and the aggregation of user costs over diﬀerent ages of the same good. And
later yet, we will look at the additional diﬃculties that are associated with the formation of
the rent.

*21 For example, according to the 2013 Japanese Housing and Land Survey, the average ﬂoor space (size)of owner occupied housing in Tokyo was 110.64 square meters for single family houses and 82.71 square
meters for rental housing, a diﬀerence of over 30 square meters. For condominiums, an even greater
discrepancy exists: the average ﬂoor space is 65.73 square meters for owner-occupied housing and 37.64
square meters for rental housing. Moreover, in addition to the diﬀerence in ﬂoor space between rented
and owned units, the quality of the owned units tends to be higher than the rented units and thesequality diﬀerences need to be taken into account.
*22 On this point, see also Lewis and Restieaux (2015)[84].

9user costs for housing.

4 The User Cost Approach for Pricing the Services of a
Non-Housing Durable Good
The user cost approach to the treatment of durable goods is in some ways very simple: it
calculates the cost of purchasing the durable at the beginning of the period, using the servicesof the durable during the period and then netting oﬀ from these costs the beneﬁt that could beobtained by selling the durable good at the end of the period. However, there are several detailsof this procedure that are somewhat controversial. These details involve the use of opportunity
costs, which are usually imputed costs, the treatment of interest and the treatment of capitalgains or holding gains.
Another complication with the user cost approach is that it involves making distinctions between current period (ﬂow) purchases within the period under consideration and the holdingsof physical stocks of the durable at the beginning and the end of the accounting period. Typically, when constructing a consumer price index, we think of all quantity purchases as taking
place at a single point in time, say the middle of the period under consideration, at the (unitvalue) average prices for the period. In constructing user costs, prices at the beginning andend of an accounting period play an important role.
To determine the net cost of using a durable good during say period 0, it is assumed that one
unit of the durable good is purchased at the beginning of period 0 at the price P 0 . The “used”
or “second-hand” durable good can be sold at the end of period 0 at the price PS1 .*23 It mightseem that a reasonable net cost for the use of one unit of the consumer durable during period
0 is its initial purchase price P 0 less its end of period 0 “scrap value”, PS1 . However, money
received at the end of the period is not as valuable as money that is received at the beginningof the period. Thus in order to convert the end of period value into its beginning of the period
equivalent value, it is necessary to discount the term PS1 by the term 1 + r0 where r0 is the
beginning of period 0 nominal interest rate that the consumer faces. Hence the period 0 user
cost u0 for the consumer durable*24 is deﬁned asu0 ≡ P 0 − PS1 /(1 + r0 ).

(1)

There is another way to view the user cost formula (1): the consumer purchases the durable
at the beginning of period 0 at the price P 0 and charges himself or herself the rental price u0 .
The remainder of the purchase price, I 0 , deﬁned as
I 0 ≡ P 0 − u0(2)

can be regarded as an investment, which is to yield the appropriate opportunity cost of capitalr0 that the consumer faces. At the end of period 0, this rate of return could be realized provided
that I 0 , r0 and the selling price of the durable at the end of the period PS1 satisfy the following
equation:I 0 (1 + r0 ) = PS1 .
(3)*23 Note that this approach to pricing the services of a durable good assumes the existence of second handmarkets for units of the durable that have aged. This assumption may not be satisﬁed for many consumer
durables including unique assets such as dwelling units and works of art, which are not bought and soldevery period. We will deal with this situation later in section 12.
*24 This approach to the derivation of a user cost formula was used by Diewert (1974)[22] who in turn based
it on an approach due to Hicks (1946; 326)[65].

10Given PS1 and r0 , (3) determines I 0 , which in turn, given P 0 , determines the user cost u0 via(2).*25
Thus user costs are not like the prices of nondurables or services because the user cost concept
involves pricing the durable at two points in time rather than at a single point in time. Because
the user cost concept involves prices at two points in time, money received or paid out at theﬁrst point in time is more valuable than money paid out or received at the second point intime and so interest rates creep into the user cost formula. Furthermore, because the user
cost concept involves prices at two points in time, expected prices can be involved if the user
cost is calculated at the beginning of the period under consideration instead of at the end.
With all of these complications, it is no wonder that many price statisticians would like to
avoid using user costs as a pricing concept. However, even for price statisticians who would
prefer to use the rental equivalence approach to the treatment of durables over the user cost
approach, there is some justiﬁcation for considering the user cost approach in some detail,since this approach gives insights into the economic determinants of the rental or leasing priceof a durable.
The user cost formula (1) can be put into a more familiar form if the period 0 economic
depreciation rate δ and the period 0 ex post asset inﬂation rate i0 are deﬁned. Deﬁne δ by:(1 − δ) ≡ PS1 /P 1

(4)

where PS1 is the price of a one period old used asset at the end of period 0 and P 1 is the priceof a new asset at the end of period 0. Typically, if a new asset and a one period older asset are
sold at the same time, then the new asset will be worth more than the used asset and hence δ
will be a positive number between 0 and 1. The period 0 inﬂation rate for the new asset, i0 ,is deﬁned by:
1 + i0 ≡ P 1 /P 0 .
(5)Eliminating P 1 from equations (4) and (5) leads to the following formula for the end of period
0 used asset price:
PS1 = (1 − δ)(1 + i0 )P 0 .
(6)
Substitution of (6) into (1) yields the following expression for the period 0 user cost u0 :u0 = [(1 + r0 ) − (1 − δ)(1 + i0 )]P 0 /(1 + r0 ).

(7)

Note that r0 − i0 can be interpreted as a period 0 real interest rate and δ(1 + i0 ) can beinterpreted as an inﬂation adjusted depreciation rate.
The user cost u0 is expressed in terms of prices that are discounted to the beginning of period
0. However, it is also possible to express the user cost in terms of prices that are “antidiscounted” or appreciated to the end of period 0.*26 Thus deﬁne the end of period 0 user

*25 This derivation for the user cost of a consumer durable was also made by Diewert (1974; 504)[22].
*26 Thus the beginning of the period user cost u0 discounts all monetary costs and beneﬁts into their dollar
equivalent at the beginning of period 0 whereas p0 discounts (or appreciates) all monetary costs andbeneﬁts into their dollar equivalent at the end of period 0. This leaves open how ﬂow transactions thattake place within the period should be treated. Following the conventions used in ﬁnancial accounting
suggests that ﬂow transactions taking place within the accounting period be regarded as taking place atthe end of the accounting period and hence following this convention, end of period user costs should be
used by the price statistician; see Peasnell (1981)[91].

11cost p0 as:*27

p0 ≡ (1 + r0 )u0 = [(1 + r0 ) − (1 − δ)(1 + i0 )]P 0(8)where the last equation follows using (7). If the real interest rate r0∗ is deﬁned as the nominalinterest rate r0 less the asset inﬂation rate i0 and the small term δi0 is neglected, then theend of the period user cost deﬁned by (8) reduces to:p0 = (r0∗ + δ)P 0 .

(9)Abstracting from transactions costs and inﬂation, it can be seen that the end of the perioduser cost deﬁned by (9) is an approximate rental cost; i.e., the rental cost for the use of aconsumer (or producer) durable good should equal the (real) opportunity cost of the capitaltied up, r0∗ P 0 , plus the decline in value of the asset over the period, δP 0 . Formulae (8) and(9) thus cast some light on what are the economic determinants of rental or leasing prices for
consumer durables.
If the simpliﬁed user cost formula deﬁned by (9) above is used, then at ﬁrst glance, forming a
price index for the user cost of a durable good is not very much more diﬃcult than forming a
price index for the purchase price of the durable good, P 0 . The price statistician needs onlyto:
• Make a reasonable assumption as to what an appropriate monthly or quarterly realinterest rate r0∗ should be;• Make an assumption as to what a reasonable monthly or quarterly depreciation rate δshould be;*28• Collect purchase prices P 0 for the durable and use formula (9) to calculate the simpliﬁeduser cost.*29
If it is thought necessary to implement the more complicated user cost formula (8) in placeof the simpler formula (9), then the situation is more complicated. As it stands, the end of
the period user cost formula (8) is an ex post (or after the fact) user cost: the asset inﬂation
rate i0 cannot be calculated until the end of period 0 has been reached. Formula (8) can*27 Christensen and Jorgenson (1969)[14] derived a user cost formula similar to (7) in a diﬀerent way using

a continuous time optimization model. If the inﬂation rate i equals 0, then the user cost formula (7)reduces to that derived by Walras (1954; 269)[107] (ﬁrst edition 1874). This zero inﬂation rate user
cost formula was also derived by the industrial engineer A. Hamilton Church (1901; 907-908)[15], who
perhaps drew on the work of Matheson: “In the case of a factory where the occupancy is assured for aterm of years, and the rent is a ﬁrst charge on proﬁts, the rate of interest, to be an appropriate rate,should, so far as it applies to the buildings, be equal (including the depreciation rate) to the rental
which a landlord who owned but did not occupy a factory would let it for.” Ewing Matheson (1910;169)[87], ﬁrst published in 1884. Additional derivations of user cost formulae in discrete time have been
made by Katz (1983; 408-409)[80] and Diewert (2005a)[25]. Hall and Jorgenson (1967)[62] introduced
tax considerations into user cost formulae.
*28 The geometric model for depreciation to be explained in more detail in section 6 below requires only asingle monthly or quarterly depreciation rate. Other models of depreciation may require the estimationof a sequence of vintage depreciation rates. If the estimated annual geometric depreciation rate is δa ,then the corresponding monthly geometric depreciation rate δ can be obtained by solving the equation(1 − δ)12 = 1 − δa . Similarly, if the estimated annual real interest rate is ra∗ , then the corresponding
monthly real interest rate r ∗ can be obtained by solving the equation (1 + r ∗ )12 = 1 + ra∗ .
*29 Iceland uses a variant of the simpliﬁed user cost formula (9) to estimate the services of OOH with areal interest rate approximately equal to 4% and depreciation rate of 1.25%. The depreciation rate isrelatively low because it is applied to the entire property value and not to just the structure portion ofproperty value; see Gudnason and Jonsdottir (2011)[60]. Eurostat (2005)[46] also uses a simpliﬁed user
cost formula. Additional simpliﬁed user cost formulae have been developed by Hill, Steurer and Waltl(2017)[68] and many others.

12be converted into an ex ante (or before the fact) user cost formula if i0 is interpreted as an
anticipated asset inﬂation rate. The resulting formula should approximate a market rental
rate for the durable good.*30
Note that in the user cost approach to the treatment of consumer durables, the entire user cost
formula (8) or (9) is the period 0 price. Thus in the time series context, it is not necessary to
deﬂate each component of the formula separately; the period 0 price p0 ≡ [r0 −i0 +δ(1+i0 )]P 0
is compared to the corresponding period 1 price, p1 ≡ [r1 − i1 + δ(1 + i1 )]P 1 and so on.
In principle, depreciation rates can be estimated using information on the selling prices ofused units of the durable good.*31 However, for housing, the situation is more complex, as
will be explained later.
We conclude this introductory section by noting some practical problems that statistical agencies will face when calculating user costs for durable goods:*32• It is diﬃcult to determine what the relevant nominal interest rate r0 is for each household. If a consumer has to borrow to ﬁnance the cost of a durable good purchase,then this interest rate will typically be much higher than the safe rate of return thatwould be the appropriate opportunity cost rate of return for a consumer who had noneed to borrow funds to ﬁnance the purchase.*33 It may be necessary to simply use abenchmark interest rate that would be determined by either the government, a national
statistical agency or an accounting standards board.*34• It will generally be diﬃcult to determine what the relevant depreciation rate is for the
consumer durable.*35*30 Since landlords must set their rent at the beginning of the period (and in fact, they usually set their rent

for an extended period of time), if the user cost approach is used to model the economic determinantsof market rental rates, then the asset inﬂation rate i0 should be interpreted as an expected inﬂation rate
rather than an after the fact actual inﬂation rate. This use of ex ante prices in this price measurement
context should be contrasted with the preference of national accountants to use actual or ex post prices
in the system of national accounts.
*31 For housing, the situation is more complex because typically, a dwelling unit is a unique good; its location
is a price determining characteristic and each housing unit has a unique location and thus is a uniquegood. It also changes its character over time due to renovations and depreciation of the structure. Thus
the treatment of housing is much more diﬃcult than the treatment of other durable goods. Note thatthe deﬁnitions (4) and (5) of the depreciation rate δ and the asset inﬂation rate i0 implicitly assumed
that prices for a new asset and a one period old asset were available in both periods 0 and 1. This
assumption is not satisﬁed for a unique asset.
*32 For additional material on diﬃculties with the user cost approach, see Diewert (1980; 475-479)[23] andKatz (1983; 415-422)[80].
*33 Katz (1983; 415-416)[80] comments on the diﬃculties involved in determining the appropriate rate ofinterest to use: “There are numerous alternatives: a rate on ﬁnancial borrowings, on savings, and aweighted average of the two; a rate on nonﬁnancial investments. e.g., residential housing, perhaps
adjusted for capital gains; and the consumer’s subjective rate of time preference. Furthermore, there issome controversy about whether it should be the maximum observed rate, the average observed rate, or
the rate of return earned on investments that have the same degree of risk and liquidity as the durables
whose services are being valued.”*34 One way for choosing the nominal interest rate for period t, r t , is to set it equal to (1 + r ∗ )(1 + ρt ) − 1
where ρt is a consumer price inﬂation rate for period t and r∗ is a reference real interest rate. The
Australian Bureau of Statistics has used this method for determining r t with r ∗ ≡ 0.04; i.e., a 4 percent
real interest rate was chosen. Other methods for determining the appropriate interest rate that should be
inserted into user cost formula are discussed by Harper, Berndt and Wood (1989)[63], Schreyer (2001)[94]and Hill, Steurer and Waltl (2017)[68].
*35 We will discuss geometric or declining balance depreciation and one hoss shay depreciation below. For
references to the depreciation literature and for empirical methods for estimating depreciation rates,
see Hulten and Wykoﬀ (1981a)[73] (1981b)[74] (1996)[75], Beidelman (1973)[6] (1976)[7], Jorgenson(1996)[79] and Diewert and Lawrence (2000)[32].

13• Ex post user costs based on formula (8) may be too volatile to be acceptable to users*36(due to the volatility of the ex post asset inﬂation rate i0 ) and hence an ex ante user
cost concept may have to be used. For most durable goods, the asset inﬂation rates are
smaller than the reference nominal interest rate so that subtracting an ex post asset
inﬂation rate from the sum of the nominal interest rate plus the asset depreciation rate
will usually lead to reasonably stable positive user costs. However, for durable goods
with very low depreciation rates, like a housing structure or like land (which has a zerodepreciation rate), the resulting ex post user costs may turn out to be negative for someperiods. This means that the resulting negative user costs are not useful approximations
to rental prices for these long lived durable goods. This creates diﬃculties in that
diﬀerent national statistical agencies will generally make diﬀerent assumptions and use
diﬀerent methods in order to construct forecast inﬂation rates for structures and land
and hence the resulting ex ante user costs of the durable may not be comparable acrosscountries.*37• The user cost formula (8) should be generalized to accommodate various taxes that
may be associated with the purchase of a durable or with the continuing use of the
durable.*38
Some of the problems associated with estimating depreciation rates will be discussed in section
6 below.

5 The Opportunity Cost Approach
The opportunity cost approach to the valuation of the services of a consumer durable during
a time period is very easy to describe: the opportunity cost valuation is simply the maximumof the foregone rental or leasing price for the services of the durable during a period of time
and the corresponding user cost for the durable. It is easy to see that when a household has*36 Goodhart (2001; F351)[58] commented on the practical diﬃculties of using ex post user costs for housing

as follows: “An even more theoretical user cost approach is to measure the cost foregone by living in anowner occupied property as compared with selling it at the beginning of the period and repurchasing itat the end ... But this gives the absurd result that as house prices rise, so the opportunity cost falls;
indeed the more virulent the inﬂation of housing asset prices, the more negative would this measure
become. Although it has some academic aﬁcionados, this ﬂies in the face of common sense; I am gladto say that no country has adopted this method.” As noted above, Iceland and Eurostat have in fact
adopted a simpliﬁed user cost framework which seems to work well enough. Moreover, the user cost
concept is used widely in production function and productivity studies and by national statisticians whoconstruct multifactor productivity accounts for their countries.
*37 For additional material on the diﬃculties involved in constructing ex ante user costs, see Diewert (1980;
475-486)[23] and Katz (1983; 419-420)[80]. For empirical comparisons of diﬀerent user cost formulae, see
Harper, Berndt and Wood (1989)[63], Diewert and Lawrence (2000)[32] and Diewert and Fox (2018)[31].
In the latter paper, the authors calculated Jorgensonian (ex post) user costs for US land used in residential
housing and found that negative user costs occurred. Diewert and Fox then replaced the ex post capitalgains term in the user cost for land with the long term inﬂation rate for land over the previous rolling
window of 25 years and this substitution led to positive user costs for land that were relatively smooth.
Hill, Steurer and Waltl (2017)[68] also recommend the use of long run asset inﬂation rates to avoid chain
drift in housing indexes based on user costs.
*38 For example, property taxes are associated with the use of housing services and hence should be includedin the user cost formula; see section 16 below. As Katz (1983; 418)[80] noted, taxation issues also impact
the choice of the interest rate: “Should the rate of return be a before or after tax rate?” From theviewpoint of a household that is not borrowing to ﬁnance the purchase of the durable, an after tax
rate of return seems appropriate but from the point of a leasing ﬁrm, a before tax rate of return seems
appropriate. This diﬀerence helps to explain why rental equivalence prices for the durable might behigher than user cost prices; see also section 16 below.

14

a consumer durable in its possession, the household forgoes the money that one could earnby renting out the services of the durable good for the period of time under consideration.
(Such rental markets may not exist, in which case, this opportunity cost is 0). Thus therental equivalent (at current market rates) is one opportunity cost that the household incurs
by continuing to own and use the services of the durable during the period. However, there isanother opportunity cost that is applicable to using the services of the durable good during
the period under consideration. By using the services of the durable good, the householdalso forgoes a ﬁnancial opportunity cost. Thus the durable good could be sold on the second
hand market at the beginning of the period at the price P 0 . This amount of money could be
invested in some ﬁnancial instrument that earns the one period rate of return of r0 . Thus at
the end of the period, the household would have accumulated P 0 (1 + r0 ) dollars as a resultof selling the consumer durable at the beginning of the period. Now suppose at the end of
the period, the household buys back the consumer durable that it sold at the beginning of
the period. The value of the durable good at the end of the period will be (1 + i0 )(1 − δ 0 )P 0
where i0 is the asset appreciation rate over period 0 and δ 0 is the depreciation rate for the
durable good. Thus the net opportunity cost of using the services of the durable for period 0
from the ﬁnancial perspective is P 0 (1 + r0 ) − (1 + i0 )(1 − δ 0 )P 0 which is exactly the end of
period user cost for the durable good that was derived earlier; see equation (8) above. A true
opportunity cost for using the services of a durable good should equal the maximum of the
beneﬁts that are foregone by not using these services. Thus the opportunity cost approach topricing the services of a consumer durable is equivalent to taking the maximum of the rent
and user cost that the durable could generate over the period under consideration.*39

6 A General Model of Depreciation for Consumer Durables
In this section, a “general” model of depreciation for durable goods that appear on the market
each period without undergoing quality change will be presented. In the following three sections, this general model will be specialized to the three most common models of depreciationthat appear in the literature. In section 11 below, the additional problems that occur when
the durable is a unique good (or when second hand markets do not exist) will be discussed.
The main tool that can be used to identify depreciation rates for a durable good is the crosssectional sequence of asset prices classiﬁed by their age that units of the good sell for on
the second hand market at any point of time.*40 Thus in order to apply this method of
measurement, it is necessary that such second hand markets exist.
Some notation is required. Let P0t be the price of a newly produced unit of the durable goodat the beginning of period t. Let Pvt be the second hand market price at the beginning of
period t of a unit of the durable good that is v periods old.*41 The beginning of period t cross*39 The opportunity cost approach to pricing the services of Owner Occupied Housing was ﬁrst proposed by

Diewert (2008)[27]. It was further developed by Diewert and Nakamura (2011)[33] and Diewert, Nakamura and Nakamura (2009)[34]. To our knowledge, there have been only two studies that implemented
the opportunity cost approach to the valuation of the services of OOH; see Shimizu, Diewert, Nishimura
and Watanabe (2012)[97] and Aten (2018)[2].
*40 Another information source that could be used to identify depreciation rates for the durable good is thesequence of vintage rental or leasing prices that might exist for some consumer durables. In the closely
related capital measurement literature, the general framework for an internally consistent treatment of
capital services and capital stocks in a set of vintage accounts was set out by Jorgenson (1989)[78] andHulten (1990; 127-129)[71] (1996; 152-160)[72].
*41 If these second hand vintage prices depend on how intensively the durable good has been used in previous
periods, then it will be necessary to further classify the durable good not only by its vintage v but also15

sectional depreciation rate for a brand new unit of the durable good, δ0t , is deﬁned as follows:
1 − δ0t ≡ P1t /P0t .

(10)

Once δ0t has been deﬁned by (10), the period t cross sectional depreciation rate for a unit ofthe durable good that is one period old at the beginning of period t, δ1t , can be deﬁned using
the following equation:(1 − δ1t )(1 − δ0t ) ≡ P2t /P0t .
(11)
Note that P2t is the beginning of period t asset price of a unit of the durable good that is 2
periods old and it is compared to the price of a brand new unit of the durable, P0t .
Given that the period t cross sectional depreciation rates for units of the durable that are
0, 1, 2, . . . , v − 1 periods old at the beginning of period 0 are deﬁned (these are the depreciationt), then the period t cross sectional depreciation rate for units of therates δ0t , δ1t , δ2t , . . . , δv−1
durable that are v periods old at the beginning of period t, δvt , can be deﬁned using thefollowing equation:tt(1 − δvt )(1 − δv−1
) · · · (1 − δ1t )(1 − δ0t ) ≡ Pv+1
/P0t .

(12)

Thus it is clear how the sequence of period 0 vintage asset prices Pvt can be converted into asequence of period t vintage depreciation rates, δvt . In the depreciation literature, it is usuallyassumed that the sequence of vintage depreciation rates, δvt , is independent of the period t sothat:for all periods t and all ages v.
(13)δvt = δv
The above material shows how the sequence of vintage or used durable goods prices at apoint in time can be used in order to estimate depreciation rates. This method for estimating
depreciation rates using data on second hand assets, with a few extra modiﬁcations to accountfor diﬀering ages of retirement, was pioneered by Beidelman (1973)[6] (1976)[7] and Hulten
and Wykoﬀ (1981a)[73] (1981b)[74] (1996)[75].*42
Recall the user cost formula for a new unit of the durable good under consideration which was
deﬁned by (1) above. The same approach can be used in order to deﬁne a sequence of period
1ais the anticipated end
0 user costs for all vintages v of the durable. Thus suppose that Pv+1of period 0 price of a unit of the durable good that is v periods old at the beginning of period
0 and let r0 be the consumer’s opportunity cost of capital for period 0. Then the discounted
to the beginning of period 0 user cost of a unit of the durable good that is v periods old at
the beginning of period 0, u0v , is deﬁned as follows:1au0v ≡ Pv0 − Pv+1/(1 + r0 );v = 0, 1, 2, . . .

(14)

It is now necessary to specify how the end of period 0 anticipated vintage asset prices Pv1a are
related to their counterpart beginning of period 0 vintage asset prices Pv0 . The assumptionthat is made now is that the entire sequence of vintage asset prices at the end of period 0
is equal to the corresponding sequence of asset prices at the beginning of period 0 times aaccording to the intensity of its use. In this case, think of the sequence of vintage asset prices Pvt ascorresponding to the prevailing market prices of the various vintages of the good at the beginning of
period t for assets that have been used at “average” intensities.
*42 See also Jorgenson (1996)[79] for a review of the empirical literature on the estimation of depreciation
rates.

16

general anticipated period 0 inﬂation rate factor, (1 + i0 ), where i0 is the anticipated period 0(general) asset inﬂation rate. Thus it is assumed that*43
Pv1a = (1 + i0 )Pv0 ;v = 0, 1, 2, . . .

(15)

Substituting (15) and (10)-(13) into (14) leads to the following beginning of period 0 sequenceof vintage user costs:*44
u0v = (1 − δv−1 )(1 − δv−2 ) · · · (1 − δ0 )[(1 + r0 ) − (1 − δv )(1 + i0 )]P00 /(1 + r0 )= (1 − δv−1 )(1 − δv−2 ) · · · (1 − δ0 )[r0 − i0 + δv (1 + i0 )]P00 /(1 + r0 );v = 1, 2, . . .

(16)

If v = 0, then u00 ≡ [r0 − i0 + δ0 (1 + i0 )]P00 /(1 + r0 ) and this agrees with the user cost formulafor a new purchase of the durable u0 that was derived earlier in (7) (with our changes innotation; i.e., P 0 is now called P00 ).

The sequence of vintage user costs u0v deﬁned by (16) are expressed in terms of prices that are
discounted to the beginning of period 0. However, as was done in section 4 above, it is also
possible to express the user costs in terms of prices that are “anti-discounted” to the end of
period 0. Thus deﬁne the sequence of vintage end of period 0 user cost p0v as follows:
p0v ≡ (1 + r0 )u0v = (1 − δv−1 )(1 − δv−2 ) · · · (1 − δ0 )[r0 − i0 + δv (1 + i0 )]P00 ; v = 1, 2, . . . (17)with p00 deﬁned as follows:
p00 ≡ (1 + r0 )u00 = [r0 − i0 + δv (1 + i0 )]P00 .

(18)

Thus if the price statistician has estimates for the vintage depreciation rates δv and the realinterest rate r0∗ and is able to collect a sample of prices for new units of the durable good P00 ,then the sequence of vintage user costs deﬁned by (17) can be calculated. To complete themodel, the price statistician should gather information on the stocks held by the household
sector of each vintage of the durable good and then normal index number theory can beapplied to these p’s and q’s, with the p’s being vintage user costs and the q’s being the vintage
stocks pertaining to each period. For some worked examples of this methodology under various
assumptions about depreciation rates and the calculation of expected asset inﬂation rates, see
Diewert and Lawrence (2000)[32] and Diewert (2005a)[25].*45
In the following three sections, the general methodology described above is specialized bymaking additional assumptions about the form of the vintage depreciation rates δv .*46

*43 More generally, we assume that assumptions (15) hold for subsequent periods t as well; i.e., it is assumed
that Pvt+1a = (1 + it )Pvt for v = 0, 1, 2, . . . and t = 0, 1, 2, . . . where Pvt+1a is the anticipated price ofa unit of the durable good that is v periods old at the end of period t, it is a period t expected asset

inﬂation rate for all ages of the durable and Pvt is the second hand market price for a unit of the durable
good that is v periods old at the beginning of period t.
*44 When v = 0, deﬁne δ−1 ≡ 1; i.e., the terms in front of the square brackets on the right hand side of (16)are set equal to 1.
*45 Additional examples and discussion can be found in two recent OECD Manuals on productivity measurement and the measurement of capital; see Schreyer (2001)[94] (2009)[95].
*46 In the case of one hoss shay depreciation, assumptions are made about the sequence of user costs, ut ,v
as the age v varies.

17

7 Geometric or Declining Balance Depreciation
The declining balance method of depreciation dates back to Matheson (1910; 55)[87] at least.*47
In terms of the algebra presented in the previous section, the method is very simple: all of thecross sectional vintage depreciation rates δvt deﬁned by (10)-(12) are assumed to be equal to
the same rate δ, where δ is a positive number less than one; i.e., for all time periods t and all
vintages v, it is assumed thatδvt = δ;v = 0, 1, 2, . . . .

(19)

Substitution of (19) into (17) leads to the following formula for the sequence of end of period
0 vintage user costs:
p0v = (1 − δ)v [r0 − i0 + δ(1 + i0 )]P 0 ;v = 0, 1, 2, . . .

= (1 − δ)v p00(20)

where the second equation follows using deﬁnition (18). The second set of equations in (20)says that all of the vintage user costs are proportional to the user cost for a new asset. This
proportionality means that it is not necessary to use an index number formula to aggregate
over vintages to form a durable services aggregate. To see this, it is useful to calculate theaggregate value of services yielded by all vintages of the consumer durable at the beginningof period 0. Let q −v be the quantity of the new durable purchased by the household sector v
periods ago for v = 1, 2, . . . and let q 0 be the new purchases of the durable during period 0.
The beginning of period 0 user cost for the holdings of the durable of age v will be p0v deﬁnedby (20) above. Thus the aggregate value of services over all vintages of the good, including
those purchased in period 0, will have the value v 0 deﬁned as follows:v 0 = p00 q 0 + p01 q −1 + p02 q −2 + · · ·= p00 q 0 + (1 − δ)p00 q −1 + (1 − δ)2 p00 q −2 + · · ·using (20)= p00 [q 0 + (1 − δ)q −1 + (1 − δ)2 q −2 + · · · ]= p00 Q0(21)

where the period 0 aggregate (quality adjusted) quantity of durable services consumed inperiod 0, Q0 , is deﬁned as
Q0 ≡ q 0 + (1 − δ)q −1 + (1 − δ)2 q −2 + · · · .

(22)

Thus the period 0 services quantity aggregate Q0 is equal to new purchases of the durable inperiod 0, q 0 , plus one minus the depreciation rate δ times the purchases of the durable in theprevious period, q −1 , plus the square of one minus the depreciation rate times the purchasesof the durable two periods ago, q −2 , and so on. The service price that can be applied to thisquantity aggregate is p00 , the imputed rental price or user cost for a new unit of the durable
purchased in period 0.
*47 A case for attributing the method to Walras (1954; 268-269)[107] could be made but he did not lay out

all of the details. Matheson (1910; 91)[87] used the term “diminishing value” to describe the method.
Hotelling (1925; 350)[70] used the term “the reducing balance method” while Canning (1929; 276)[13]
used the term the “declining balance formula”. For a modern exposition of the geometric model ofdepreciation, see Jorgenson (1989)[78].

18The above result greatly simpliﬁes the valuation of consumer durables. Normally, the pricestatistician would have to keep track of all new purchases of the durable good by the reference
population by period, calculate the relevant user costs p0v and ptv for periods 0 and t, andapply the relevant index number formula (Laspeyres, Paasche, Fisher or whatever formula isbeing used in the CPI) to these age speciﬁc prices and quantities for periods 0 and t. But
because under assumptions (13), (15) and (19), all vintage user costs vary in a proportionalmanner over time,*48 so any reasonable index number formula will ﬁnd that the price index
going from period 0 to t is equal to pt0 /p00 , the ratio of user costs for a new unit of the durable
good. Moreover the corresponding aggregate quantity index will be equal to Qt /Q0 , where Q0
is deﬁned by (22) and Qt is deﬁned byQt ≡ q t + (1 − δ)q t−1 + (1 − δ)2 q t−2 + · · ·= q t + (1 − δ)Qt−1 .

(23)Note that the second equation simpliﬁes the calculation of the period t aggregate service ﬂow(in real terms) over all vintages of the consumer durable: the period t aggregate ﬂow, Qt , isequal to period t new purchases of the durable, q t , plus (1 − δ) times the aggregate ﬂow ofservices in the previous period, Qt−1 .
If the depreciation rate δ and the purchases of the durable in prior periods are known, then theaggregate service quantity Q0 can readily be calculated using (22). Then using (21), it can beseen that the period 0 value of the services of the durable (over all vintages), v 0 , decomposes
into the price term p00 times the quantity term Q0 . Hence, it is not necessary to use an index
number formula to aggregate over vintages using this depreciation model.
The stock of consumer durables held by the household sector of a country should appear in
the balance sheets of the country. Using the geometric model of depreciation, it is very easy
to calculate the nominal and real value of the stock of consumer durables held by households.
At time t, the stocks held by the household sector for the particular type of consumer durable
under consideration are q t , q t−1 , q t−2 , . . . and the corresponding asset prices by age of asset
are P0t , P1t , P2t , . . . . Assumptions (12), (13) and (19) imply that these period t asset prices
satisfy the following equations:
Pvt = (1 − δ)v P0t ;v = 1, 2, . . . .

(24)Equations (24) can be used to deﬁne period t aggregate asset value for the stocks held byhouseholds for the durable good over all ages of the durable good, V t :V t ≡ P0t q t + P1t q t−1 + P2t q t−2 + P3t q t−3 + · · ·= P0t [q t + (1 − δ)1 q t−1 + (1 − δ)2 q t−2 + · · · ]= P0t Qt

using (24)(25)

where Qt is deﬁned by (23). Thus Qt serves as a measure of the real capital stock of theconsumer durable at the end of period t and it also serves as a measure of the real consumption
services provided by this capital stock during period t.
*48 Equations (20) for period t are the following ones:

ptv = (1 − δ)v pt0 for v = 1, 2, . . . and so the entire
sequence of user costs by age of asset vary in a proportional manner over time under our assumptions.
Thus an aggregate period t price for the entire group of assets of varying ages is pt0 and the corresponding
aggregate quantity will be Qt deﬁned by (23). This is an application of Hicks’ (1946; 312-313)[65]Aggregation Theorem: “Thus we have demonstrated mathematically the very important principle, used
extensively in the text, that if the prices of a group of goods change in the same proportion, that groupof goods behaves just as if it were a single commodity.”

19

The above algebra explains why the geometric model of depreciation is used so widely in production function studies and in the measurement of Total Factor Productivity or Multifactor
Productivity in the production accounts of countries: it is very simple to work with!*49

8 Straight Line Depreciation
Another very common model of depreciation is the straight line model.*50 In this model, amost probable length of life for the durable is somehow determined, say L periods, so that after
being used for L periods, the durable is scrapped. In the straight line depreciation model, it
is assumed that the period 0 cross sectional vintage asset prices Pv0 decline in a linear fashion
relative to the period 0 price of a new asset P00 :Pv0 /P00 = [L − v]/Lfor v = 0, 1, 2, . . . , L − 1.

(26)

For v = L, L + 1, . . ., it is assumed that Pv0 = 0. Now use deﬁnitions (14) and (17) along withassumptions (15) in order to obtain the following sequence of end of period 0 vintage user
costs for a unit of the durable good of age v at the beginning of period 0:0p0v = Pv0 (1 + r0 ) − (1 + i0 )Pv+1

for v = 0, 1, 2, . . . , L − 1= [1/L][(L − v)(1 + r0 ) − (L − v − 1)(1 + i0 )]P00
0

0

−1

= [(r − i )(L − v)L0−1+ (1 + i )L

]P00 .

using assumptions (26)(27)

The user costs for units of the durable good that are older than L periods are zero; i.e.,p0v ≡ 0 for v ≥ L. Looking at the terms in square brackets on the right hand side of (27),
it can be seen that the ﬁrst term (r0 − i0 )(L − v)P00 /L is a real interest opportunity costfor holding and using the unit of the durable that is v periods old (and this imputed realinterest cost declines as the durable good ages; i.e., as the age v increases) and the second
term (1 + i0 )(1/L)P00 is an inﬂation adjusted depreciation term that is equal to the constantstraight line depreciation rate 1/L times the adjustment factor for asset inﬂation over theperiod, (1 + i0 ), times the price of a new unit of the durable good P00 . Note that in periodt, the corresponding end of period user cost for a unit of the durable good that is v periods
old is ptv ≡ [(rt − it )(L − v)L−1 + (1 + it )L−1 ]P0t for v = 0, 1, 2, . . . , L − 1. Thus in both
periods 0 and t, the sequences of end of period user costs by age, {p0v } and {ptv } for v =0, 1, 2, . . . , L − 1, are proportional to the price of a new unit of the durable for periods 0
and t, P00 and P0t respectively*51 but if r0 and/or i0 change to a diﬀerent rt or it , then thefactors of proportionality will change as we go from period 0 to t and so we cannot apply
Hicks’ Aggregation Theorem in this case. Thus in the case of changing nominal interest rates
r and/or changing expected or actual asset price inﬂation rates, it , we cannot assume thatthe overall inﬂation rate between periods 0 and t for all ages of the durable good is equal to
P0t /P00 as was the case with the geometric model of depreciation. Thus for the straight line
model of depreciation, it is necessary to keep track of household purchases of the durable for L*49 See Jorgenson (1989)[78] who popularized the use of the geometric model of depreciation in production

function and Total Factor Productivity studies.

*50 This model of depreciation dates back to the late 1800’s; see Matheson (1910; 55)[87], Garcke and Fells(1893; 98)[54] or Canning (1929; 265-266)[13].

*51 Thus as the price of a new unit of the durable good changes over time, the value of depreciation will

also change in line with the change in the price of the new unit. Thus economic depreciation as we havedeﬁned it is diﬀerent from historical cost accounting depreciation which does not adjust depreciation
allowances for changes in the levels of asset prices over time.

20

periods and weight up each vintage quantity q −v of these purchases by the corresponding endof period user costs vintage user cost p0v deﬁned by (27) for period 0 and a similar calculation ofhousehold holdings of the durable good by age for period t along with the period t counterparts
to the period 0 user costs deﬁned by (27) will be necessary. Once these vectors of prices andquantities have been calculated for both periods, then normal index number theory can beapplied to get the overall price index for the household holdings of the durable good andthis index can be used to deﬂate the user cost aggregate values to get an appropriate volume
index.*52 Thus the straight line model of depreciation is considerably more complicated to
implement than the geometric model of depreciation explained in the previous section.*53

9 One Hoss Shay or Light Bulb Depreciation
The ﬁnal model of depreciation that is in common use is the “light bulb” or one hoss shaymodel of depreciation.*54 In this model, the durable delivers the same services for each vintage:a chair is a chair, no matter what its age is (until it falls to pieces and is scrapped). Thus thismodel also requires an estimate of the most probable life L of the consumer durable.*55 In
this model, it is assumed that the sequence of vintage beginning of the period user costs u0v
deﬁned by (14) and (15) is constant for all vintages younger than the asset lifetime L; i.e., it
is assumed that0u0v ≡ Pv0 − (1 + i0 )Pv+1/(1 + r0 ) = u0 ;v = 0, 1, 2, . . . , L − 1(28)where u0 > 0 is a constant. Equations (28) can be rewritten in the following form:0u0 = Pv0 − γPv+1;

v = 0, 1, 2, . . . , L − 1(29)*52 Diewert and Lawrence (2000)[32] noted this problem with the straight line model of depreciation; i.e.,that in general, an index number formula should be used to aggregate over the diﬀerent ages of the asset
in order to obtain an aggregate of the capital services of the diﬀerent vintages of the asset.
*53 However, if one is willing to assume that the reference interest rate for period t, r t , and the expectedasset inﬂation rate over all ages of the asset, it , both remain constant, then all reasonable index number
formula will estimate the overall rate of user cost inﬂation between periods 0 and t as the new goodprice ratio, P0t /P00 . However, the assumption that r t and it remain constant over time is only a roughapproximation to reality. Note that in order to calculate real and nominal consumption of the durable(over all ages of the durable), it will be necessary to use the vintage user costs deﬁned by (27) for aconstant r and i to weight up past purchases of the durable good. Thus deﬁne the constants αv ≡[(r − i)(L − v)L−1 + (1 + i)L−1 ] for v = 0, 1, 2, . . . , L − 1 and αv ≡ 0 for v ≥ L. Then the periodt nominal value of durable services is deﬁned as v t ≡ pt0 q t + pt1 q t−1 + pt2 q t−2 + · · · + ptL−1 q t−L+1 =α0 P0t q t + α1 P0t q t−1 + α2 P0t q t−2 + · · · + αL−1 P0t q t−L+1 = P0t Qt where Qt is the real value or volume ofdurable services deﬁned as Qt ≡ α0 q t + α1 q t−1 + α2 q t−2 + · · · + αL−1 q t−L+1 . Deﬁne βv ≡ (L − v)/Lfor v = 0, 1, 2, . . . , L − 1. The period t asset value of consumer holdings of the durable good is deﬁned ast
V t ≡ P0t q t + P1t q t−1 + P2t q t−2 + · · · + PL−1
q t−L+1 = P0t [β0 q t + β1 q t−1 + β2 q t−2 + · · · + βL−1 q t−L+1 ] =tt∗
P0 Q where we have used assumptions (26) applied to period t and the real value of durable stocks heldby households at the end of period t is deﬁned as Qt∗ ≡ β0 q t + β1 q t−1 + β2 q t−2 + · · · + βL−1 q t−L+1 .
The decomposition of V t into P0t Qt∗ does not require the assumption of constant rt and it .
*54 This model can be traced back to Böhm-Bawerk (1891; 342)[8]. For a more comprehensive exposition,
see Hulten (1990; 124)[71] or Diewert (2005a)[25].
*55 The assumption of a single life L for a durable can be relaxed using a methodology due to Hulten: “Wehave thus far taken the date of retirement T to be the same for all assets 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 readilyextended to allow for diﬀerent retirement dates. A given cohort can be broken into components, or
subcohorts, according to date of retirement and a separate T assigned to each. Each subcohort can thenbe characterized by its own eﬃciency sequence, which depends among other things on the subcohort’s
useful life Ti .” Charles R. Hulten (1990; 125)[71].

21where the discount factor γ is deﬁned asγ ≡ (1 + i0 )/(1 + r0 ) = 1/(1 + r0∗ ).

(30)

The interest rate r0∗ can be regarded as an asset speciﬁc real interest rate; i.e., 1 + r0∗ ≡(1 + r0 )/(1 + i0 ) so that one plus the nominal interest rate r0 is deﬂated by one plus the
expected asset price inﬂation rate, i0 . Note that equations (29) can be rewritten as follows:0;
Pv0 = u0 + γPv+1

v = 0, 1, 2, . . . , L − 1.

(31)

Use equation (31) with v = 0 to express P00 in terms of u0 and P10 . Now use (31) with v = 1
to express P20 in terms of u0 and P10 and then substitute out P10 using the previous expressionthat expressed P10 in terms of P00 and u0 . Continue this substitution process until ﬁnallyit ends after L such substitutions when PL0 is reached and of course, PL0 equals zero. The
following equation is obtained:
P00 = u0 + γu0 + γ 2 u0 + · · · + γ L−1 u0= u0 [1 + γ + γ 2 + · · · + γ L−1 ]= {u0 /(1 − γ)} − {u0 γ L /(1 − γ)}provided that γ < 1*56

= u0 (1 − γ L )/(1 − γ).

(32)

Now use the last equation in (32) in order to solve for the constant over vintages (beginningof the period) user cost for this model, u0 , in terms of the period 0 price for a new unit of the
durable, P00 , and the discount factor γ deﬁned by (31):u0 = (1 − γ)P00 /(1 − γ L ) = u0v ;v = 0, 1, 2, . . . , L − 1.

(33)

The sequence of end of period 0 user cost, p0v , is as usual, equal to the corresponding beginningof the period 0 user cost, u0v , times the period 0 nominal interest rate factor, 1 + r0 :p0v ≡ (1 + r0 )u0v = [1 + r0 ][1 − γ 0 ][1 − (γ 0 )L ]−1 P00 = p00 ;v = 0, 1, 2, . . . , L − 1(34)and p0v = 0 for v = L, L + 1, . . . and γ 0 ≡ (1 + i0 )/(1 + r0 ).
The aggregate services of all vintages of the good for period 0, including those purchased inperiod 0, will have the following value, v 0 :v 0 = p00 q 0 + p01 q −1 + p02 q −2 + · · · + p0L−1 q −(L−1)= p00 [q 0 + q −1 + q −2 + · · · + q −(L−1) ]= p00 Q0(35)

where the period 0 aggregate (quality adjusted) quantity of durable services consumed in period0, Q0 , is deﬁned as follows for this depreciation model:
Q0 ≡ q 0 + q −1 + q −2 + · · · + q −(L−1) .

(36)

Thus in this model of depreciation, the service quantity aggregate is the simple sum of household purchases over the last L periods.*57 As was the case with the geometric depreciation*56 If γ ≥ 1, then use the second equation in (32) to express u0 in terms of P 0 and the various powers of γ.
0*57 In the national income accounting literature, this measure is sometimes called the gross capital stock.

22

model, the one hoss shay model does not require index number aggregation over vintages whencalculating aggregate services from all vintages of the durable: there is a constant service pricep00 for all assets that are less than L periods old and the associated period 0 quantity Q0 is
the simple sum deﬁned by (36) over the purchases of the last L periods for the one hoss shaymodel.*58
The ﬁrst two models of depreciation considered in sections 6 and 7 made assumptions about
the pattern of depreciation rates for durables of diﬀerent ages. The model in this section madeassumptions about the pattern of user costs for durable goods of diﬀerent ages. For a moregeneral model of depreciation that allows for an arbitrary pattern of user costs by age of asset,
see Diewert and Wei (2017)[42].
How can the diﬀerent models of depreciation be distinguished empirically? For durable goods
that do not change in quality over time, there are three possible methods for determining thesequence of vintage depreciation rates:*59• By making a rough estimate of the average length of life L for the durable good andthen by assuming a depreciation model that seems most appropriate.*60• By using cross sectional information on used durable prices at a single point in time
and then using equations (10)-(12) above to determine the corresponding sequence of
vintage depreciation rates.*61• By using cross sectional information on the rental or leasing prices of the durable as
a function of the age of the durable and then equations (17) and (18), along withinformation on the appropriate nominal interest rate r0 and expected durables inﬂation
rate i0 along with information on the price of a new unit of the durable good P 0 canbe used to determine the corresponding sequence of vintage depreciation rates.

10 The Relationship Between User Costs and Acquisition Costs
In this section, the user cost approach to the treatment of consumer durables will be comparedto the acquisitions approach. Obviously, in the short run, the value ﬂows associated with eachapproach could be very diﬀerent. For example, if real interest rates, r0 − i0 , are very high andthe economy is in a severe recession or depression, then purchases of new consumer durables,q 0 say, could be very low and even approach 0 for very long lived assets, like houses. On
the other hand, using the user cost approach, existing stocks of consumer durables would
be carried over from previous periods and priced out at the appropriate user costs and theresulting consumption value ﬂow could be quite large. Thus in the short run, the monetary
values of consumption under the two approaches could be vastly diﬀerent. Hence, in what
follows, a (hypothetical) longer run comparison is considered where real interest rates are held
*58 Using equations (31),

it can be shown that Pv0 = u0 [1 + (γ 0 ) + (γ 0 )2 + · · · + (γ 0 )L−1−v ] for v =0, 1, 2, . . . , L − 1 where γ 0 ≡ (1 + i0 )/(1 + r 0 ) and Pv0 = 0 for v ≥ L. Thus the period 0 value of the∑
0 −v . The corresponding asset prices for period t are equal tostock of consumer durables is L−1
v=0 Pv qtttt2
Pv = u [1 + (γ ) + (γ ) + · · · + (γ t )L−1−v ] for v = 0, 1, 2, . . . , L − 1 where ut ≡ [1 − (γ t )]P0t /[1 − (γ t )L ],γ t ≡ (1 + it )/(1 + r t ) and Pvt = 0 for v ≥ L. The period t value of the stock of consumer durables∑L−1 t t−v. An index number formula will have to be used to form aggregate price and quantity
is v=0Pv q
indexes for the stocks of consumer durables using the one hoss shay model of depreciation.
*59 These three classes of methods were noted in Malpezzi, Ozanne and Thibodeau (1987; 373-375)[85] inthe housing context.
*60 A length of life L is can be converted into an equivalent geometric depreciation rate δ by setting δ equal
to a number between 1/L and 2/L.
*61 This method will be pursued in sections 12 and 13 for housing depreciation rates.

23

constant.*62
Suppose that in period 0, the reference population of households purchased q 0 units of aconsumer durable at the purchase price P 0 . Then the period 0 value of consumption from theviewpoint of the acquisitions approach is:
VA0 ≡ P 0 q 0 .

(37)

Recall that the end of period user cost for one new unit of the asset purchased at the beginningof period 0 was p0 deﬁned by (8) above. In order to simplify the analysis, the geometric modelof depreciation is assumed; i.e., at the beginning of period 0, a one period old asset is worth(1 − δ)P 0 , a two period old asset is worth (1 − δ)2 P 0 , . . . , a t period old asset is worth(1 − δ)t P 0 , etc. Under these hypotheses, the corresponding end of period 0 user cost for a new
asset purchased at the beginning of period 0 is p0 ; the end of period 0 user cost for a one period
old asset at the beginning of period 0 is (1 − δ)p0 ; the corresponding user cost for a two period
old asset at the beginning of period 0 is (1 − δ)2 p0 ; . . . ; the corresponding user cost for a t
period old asset at the beginning of period 0 is (1−δ)t p0 ; etc. The ﬁnal simplifying assumption
is that household purchases of the consumer durable have been growing at the geometric rate
g into the indeﬁnite past. This means that if household purchases of the durable were q 0 inperiod 0, then in the previous period they purchased q 0 /(1 + g) new units, two periods ago,
they purchased q 0 /(1 + g)2 new units, . . . , t periods ago, they purchased q 0 /(1 + g)t newunits, etc. Putting all of these assumptions together, it can be seen that the period 0 value ofconsumption services from the viewpoint of the user cost approach is:
VU0 ≡ p0 q 0 + [(1 − δ)p0 q 0 /(1 + g)] + [(1 − δ)2 p0 q 0 /(1 + g)2 ] + · · ·= (1 + g)p0 q 0 /(g + δ)

summing the inﬁnite series0= (1 + g)[(1 + r ) − (1 − δ)(1 + i0 )]P 0 q 0 /(g + δ)using (8).

(38)

Equation (38) can be simpliﬁed by letting the asset inﬂation rate i0 be 0 ( or by replacingr0 − i0 by the real interest rate r0∗ and by ignoring the small term δi0 ) and under these
conditions, the ratio of the user cost ﬂow of consumption (38) to the acquisitions measure of
consumption in period 0, (37) is:
VU0 /VA0 = (1 + g)(r0∗ + δ)/(g + δ).

(39)

Using formula (39), it can be seen that if 1 + g > 0 and δ + g > 0, then VU0 /VA0 will begreater than unity if r0∗ > g(1 − δ)/(1 + g), a condition that will usually be satisﬁed. Thus
under normal conditions and over a longer time horizon, household expenditures on consumer
durables using the user cost approach will tend to exceed the corresponding expenditures on newpurchases of the consumer durable. Since the value of consumption services using the rental
equivalence approach will tend to approximate the value of consumption services using theuser cost approach, it can be seen that the acquisitions approach to household expenditures
will tend to understate the value of consumption services estimated by the user cost and rental
equivalence approaches. The diﬀerence between the user cost and acquisitions approach willtend to grow as the depreciation rate δ decreases.
To get a rough idea of the possible magnitude of the value ratio for the two approaches, VU0 /VA0 ,equation (39) is evaluated for a “housing” example using annual data where the depreciation
rate is 2% (i.e., δ = .02), the real interest rate is 3% (i.e., r0∗ = .03) and the growth rate
*62 The following material is based on Diewert (2002)[24].

24for the production of new houses is 1% (i.e., g = .01). In this base case, the ratio of user
cost expenditures on housing to the purchases of new housing in the same period, VU0 /VA0 , is1.68. If the depreciation rate is decreased to 1%, then VU0 /VA0 increases to 2.02. If the realinterest rate is decreased to 2% (with δ = .02 and g = .01) then VU0 /VA0 decreases to 1.35
while if the real interest rate is increased to 4%, then VU0 /VA0 increases to 2.02. Thus anacquisitions approach to housing in the CPI is likely to give a substantially smaller weight tohousing services than a user cost approach would give.
However, or shorter lived consumer durables like clothing, the diﬀerence between the acquisitions approach and the user cost approach will not be so large and hence, the acquisitions
approach can be justiﬁed as being an approximately “correct” as a measure of consumption
services for these high depreciation rate durable goods.*63

11 Calculating User Costs for Unique Durable Goods
Calculating rental prices or user costs for durable goods that are unique so that second handmarkets for this type of good are either very thin or nonexistent will in general be impossible.
Examples of such goods are paintings and unique jewels.*64 It should be noted that dwelling
units are also examples of unique goods in that the location of each dwelling unit is uniqueand a house at a certain location does not remain the same over time due to renovations anddepreciation of the structure. However, as we shall see in subsequent sections, the measurement
situation is not so dire with respect to measuring housing service as it is for measuring valuable
services.
As was mentioned above, it is impossible to measure the services of a unique good that nevertrades. If the good trades sporadically, it is possible to make estimates of the service ﬂows
generated by the good between sales of the good in an ex post fashion. We will indicate how
this can be done below. The resulting estimates will not be very accurate but some kind of
estimate is probably better than no estimate at all.
Suppose that a valuable is purchased at the beginning of period 0 at the price P 0 and it is soldat the beginning of period T at the price P T . It is assumed that both asset prices are observed
and there is an average one period nominal interest rate r that provides an opportunity costof borrowing or lending for the owner of the asset over the T periods. An average geometricasset inﬂation rate i for the asset over the T periods is deﬁned as follows:
1 + i ≡ [P T /P 0 ]1/T .

(40)

We assume that the purchase price of the asset, P 0 , is set equal to the discounted imputed
ﬂow of services that the asset generates for its owner plus the discounted selling price of the*63 Let r 0∗ = .03, g = .01 and δ = .2.

Under these assumptions, using (39), we ﬁnd that VU0 /VA0 = 1.11;
i.e., using a geometric depreciation rate of 20%, the user cost approach leads to an estimated value ofconsumption that is 11% higher than the acquisitions approach under the conditions speciﬁed. Thus
the acquisitions approach for consumer durables with high depreciation rates is probably satisfactory.
However, for longer lived durables such as houses, automobiles and household furnishings, it would be
useful for a national statistical agency to produce user costs for these goods and for the national accounts
division to produce the corresponding consumption ﬂows as “analytic series”. This would extend thepresent national accounts treatment of housing to other long lived consumer durables. Note also that
this revised treatment of consumption in the national accounts would tend to make rich countries richer,since poorer countries hold fewer long lived consumer durables on a per capita basis.
*64 In the international System of National Accounts, these unique goods are listed as valuables.

25

asset at the beginning of period T ; i.e., we assume that the following equation holds:
P 0 = u0 + γu1 + γ 2 u2 + · · · + γ T −1 uT −1 + (1 + r)−T P T(41)

where ut is the constant quality consumption value for the durable’s services in period t*65
for t = 0, 1, . . . , T − 1 and the discount factor γ is deﬁned as follows:γ ≡ (1 + i)/(1 + r).

(42)

Make the further assumption that the quality of the service rendered by the unique durable
good is constant over the T periods so that:ut = u 0 ;t = 1, 2, . . . , T − 1.

(43)

Substitution of assumptions (43) into (41) enables one to solve for u0 :u0 = [P 0 − P T (1 + r)−T ]/[1 + γ + γ 2 + · · · + γ T −1 ].

(44)

Once u0 has been calculated, the sequence of imputed rental prices for the unique asset forperiods 0, 1, . . . , T −1 is u0 , (1+i)u0 , (1+i)2 u0 , . . . , (1+i)T −1 u0 . The corresponding quantities
are all equal to 1. Note that these computations can only be done once P T is known; i.e.,these user cost valuations cannot be made until period T occurs.
The above analysis assumes that P 0 > P T /(1 + r)T which ensures that u0 > 0. If P 0 =P T /(1 + r)T , then u0 = 0 and the services of the valuable for the T periods are provided tothe owner at no (imputed) charge! If P 0 < P T /(1 + r)T , then u0 < 0 and the services ofthe valuable for the T periods are provided to the owner for no charge and in addition, thevaluable provides a source of income to the owner. The total beneﬁt to the owner in terms ofdollars at the beginning of period 0 is P T /(1 + r)T − P 0 . The income beneﬁt to the owner interms of dollars at the end of period T − 1 is P T − (1 + r)T P 0 .
For some unique assets, the quality of the service ﬂow from using the services of the durable
may decline over time. For example, the service ﬂow from a custom built automobile or custom
built horse trailer may decline over time due to the aging of the asset. The above model canbe modiﬁed to take into account this complication but it is necessary to assume an exogenous
service ﬂow quality diminution rate δ where 0 < δ < 1. Thus in place of the constant relative
quality assumption (43), the following assumption is made:ut = (1 − δ)t u0 ;t = 1, 2, . . . , T − 1.

(45)

Assumptions (41) and (42) still hold. Now substitute assumptions (45) into (41) in order toobtain the following equation:
P 0 = u0 + γ(1 − δ)u0 + γ 2 (1 − δ)2 u0 + · · · + γ T −1 (1 − δ)T −1 u0 + (1 + r)−T P T .

(46)

Deﬁne the constant φ as follows:φ ≡ γ(1 − δ) = (1 + i)(1 − δ)/(1 + r).

(47)

There is a new deﬁnition for i as a constant quality asset inﬂation rate over the T periods
between period 0 and period T :1 + i = [P T /P 0 ]1/T /(1 − δ).
*65 The nominal imputed user cost for period t is (1 + i)t ut .

26

(48)

Thus the quality adjusted asset inﬂation rate is now adjusted upwards by dividing the oldasset inﬂation rate by 1 − δ. Using deﬁnition (47), equation (46) can be rewritten as follows:
P 0 = u0 [1 + φ + φ2 + · · · + φT −1 ] + (1 + r)−T P T .

(49)

Thus u0 can be determined from equation (49) as follows:*66
u0 = [P 0 − P T (1 + r)−T ]/[1 + φ + φ2 + · · · + φT −1 ].

(50)

Once u0 has been calculated, the sequence of imputed rental prices for the unique asset forperiods 0, 1, . . . , T −1 is u0 , (1+i)(1−δ)u0 , (1+i)2 (1−δ)−2 u0 , . . . , (1+i)T −1 (1−δ)−T +1 u0 . The
corresponding sequence of constant quality quantities is 1, (1−δ)−1 , (1−δ)−2 , . . . , (1−δ)−T +1 .
Again, note that these computations can only be done once P T is known; i.e., these user cost
valuations cannot be made in real time.
The above models for measuring the services of a unique durable good are subject to many
criticisms but perhaps these models can serve as starting points for more realistic models. In
any case, having an imperfect model for measuring the services of a unique durable good isbetter than having no model at all.
In the remaining sections of this paper, the focus will be on the special problems that areassociated with both measuring the value of the housing stock as well as on valuing theservices of Owner Occupied Housing (OOH).

12 Decomposing Residential Property Prices into Land andStructure Components
In this section, the problems associated with the construction of constant quality residential
property price indexes will be studied. In this section, we will look at the construction of
constant quality indexes for the stock of residential housing units; in subsequent sections, wewill look at the problems associated with pricing the services of a residential dwelling unit.
There are two diﬃcult measurement problems associated with the construction of a constantquality house price index:• A dwelling unit is a unique consumer durable good; i.e., the location of a housing unit isa price determining characteristic of the unit and each house or apartment has a uniquelocation.
• There are two main components of a dwelling unit: (i) the size of the structure (measured
in square meters of ﬂoor space) and (ii) the size of the land plot that the structure sitson (also measured in square meters). However, the purchase price of a dwelling unit isfor the entire property and thus the decomposition of property price into its two main
components will involve imputations.
The ﬁrst problem area listed above might not be a problem if the same dwelling unit soldat market prices at a frequent rate so that the location would be held constant and it would
seem that the usual matched model methodology that is used in constructing price indexes
could be applied. But houses do not transact all that frequently; typically, a house is held
*66 If P 0 − P T (1 + r)−T

< 0, then as before, u0 becomes negative (and u1 , . . . , uT −1 become negative aswell) and again, the services of the unique durable are free of charge and −(1 + i)t−1 (1 − δ)t−1 ut =−(1 + i)t−1 u0 > 0 becomes an addition to household income for period t.

27

for 10-20 years by the same owner before it is resold. Moreover, the structure is not constantover time; depreciation of the structure occurs over time and owners renovate and replaceaging components of the structure. For example, the rooﬁng materials for many dwellings are
replaced every 20 or 30 years. Thus depreciation and renovation constantly change the qualityof the structure.
The second problem area is associated with the diﬃculty of decomposing the transaction pricefor a housing unit into separate components representing the structure value and the land
value; i.e., the single property price is for both components of the housing unit but for many
purposes, we require separate valuations for the two components. The international System
of National Accounts, requires separate valuations for the land and structure components ofresidential housing in the National Balance Sheets of the country. Many countries constructestimates for the Total Factor Productivity or Multifactor Productivity of the various sectors
in the economy and the methodology used to construct these estimates requires separate priceand quantity information on both structures and the land that the structures sit on. In thissection, we will indicate a possible method that can be used to accomplish this decompositionof property value into constant quality land and structure components.
The builder’s model for valuing a detached dwelling unit postulates that the value of theproperty is the sum of two components: the value of the land which the structure sits on plus
the value of the structure. This model can be justiﬁed in two situations:• A household purchases a residential land plot with no structure on it (or if there are
structures on the land plot, they are immediately demolished).*67• A household purchases a land plot and immediately builds a new dwelling unit on theproperty.
In the ﬁrst case, it is clear that the property value is equal to the land value. In the second
case, The total cost of the property after the structure is completed will be equal to the ﬂoor
space area of the structure, say S square meters, times the building cost per square meter βtduring period t, plus the cost of the land, which will be equal to the cost per square meter αttimes the area of the land site, say L square meters. Now think of a sample of properties ofthe same general type in the same general location, which have prices or values Vtn in period t(where t = 1, . . . , T ) and structure ﬂoor space areas Stn and land areas Ltn for n = 1, . . . , N (t)where N (t) is the number of observations in period t. Assume that these prices are equal to thesum of the land and structure costs plus error terms εtn which we assume are independently
normally distributed with zero means and constant variances. This leads to the following
hedonic regression model for period t where the αt and βt are the parameters to be estimatedin the regression:*68
Vtn = αt Ltn + βt Stn + εtn ;t = 1, . . . , T ; n = 1, . . . , N (t).

(51)

The hedonic regression model deﬁned by (51) applies to new structures and to purchasesof vacant residential lots in the neighbourhood under consideration where Stn = 0. Note
*67 The cost of the demolition should be added to the purchase price for the land to get the overall land

price for the land plot.

*68 Other papers that have suggested hedonic regression models that lead to additive decompositions ofproperty values into land and structure components include Clapp (1980; 257-258)[16], Bostic, Longhofer
and Redfearn (2007; 184)[9], Francke and Vos (2004)[53], Diewert (2008; 19-22)[27] (2010)[28], Francke(2008; 167)[52], Koev and Santos Silva (2008)[81], Rambaldi, McAllister, Collins and Fletcher (2010)[92],Diewert, Haan and Hendriks (2011)[39] (2015)[40], Eurostat (2013)[47], Diewert and Shimizu (2015)[35](2016)[36] (2017a)[37], Burnett-Issacs, Huang and Diewert (2016)[11] and Diewert, Huang and BurnettIssacs (2017)[41].

28that there are some strong simplifying assumptions built into the model deﬁned by (51): (i)the period t land price αt (per m2 ) is assumed to be constant across all properties in theneighbourhood under consideration and (ii) the construction cost (per m2 ) is also assumed tobe constant across all housing units built in the neighbourhood during period t. The above
model applies to raw land purchases and the purchases of new dwelling units during period
t in the neighbourhood under consideration. It is likely that a model that is similar to (51)applies to sales of older structures as well. Older structures will be worth less than newer
structures due to the depreciation of the structure. Assuming that we have information on
the age of the structure n at time t, say A(t, n), and assuming a geometric (or decliningbalance) depreciation model, a more realistic hedonic regression model than that deﬁned by(51) above is the following basic builder’s model:
Vtn = αt Ltn + βt (1 − δ)A(t,n) Stn + εtn ;t = 1, . . . , T ; n = 1, . . . , N (t)(52)

where the parameter δ reﬂects the net geometric depreciation rate as the structure ages one
additional period. Thus if the age of the structure is measured in years, we would expect anannual net depreciation rate to be around 1 to 3 percent per year.*69 Note that (52) is now
a nonlinear regression model whereas (51) was a simple linear regression model. The period t
constant quality price of land will be the estimated coeﬃcient for the parameter αt and theprice of a unit of a newly built structure for period t will be the estimate for βt . The periodt quantity of land for property n is Ltn and the period t quantity of structure for propertyn, expressed in equivalent units of a new structure, is (1 − δ)A(t,n) Stn where Stn is the ﬂoor
space area of the structure for property n in period t.
Note that the above model can be viewed as a supply side model as opposed to a demand sidemodel.*70 Basically, we are assuming a valuation of a housing structures that is equal to thecost per unit ﬂoor space area of a new unit times the ﬂoor space area times an adjustment forstructure depreciation. The corresponding land value of the property is determined residuallyas total property value minus the imputed value of structures quality adjusted for the ageof the structure. This assumption is justiﬁed for the case of newly built houses and sales ofvacant lots but it is less well justiﬁed for sales of properties with older structures where ademand side model may be more relevant.
There is a major practical problem with the hedonic regression model deﬁned by (52): The
multicollinearity problem. Experience has shown that it is usually not possible to estimate
sensible land and structure prices in a hedonic regression like that deﬁned by (52) due tothe multicollinearity between lot size and structure size.*71 Thus in order to deal with themulticollinearity problem, the parameter βt in (52) is replaced by pSt , an exogenous periodt construction cost price for houses in the area under consideration.*72 The exogenous construction price index may be an oﬃcial construction price index estimated by the national*69 This estimate of depreciation is regarded as a net depreciation rate because it is equal to a “true” grossstructure depreciation rate less an average renovations appreciation rate. Since typically information onrenovations and major repairs to a structure is not available, the age variable will only pick up averagegross depreciation less average real renovation expenditures.
*70 We will pursue a demand side model in Section 14 below.
*71 See Schwann (1998)[96] and Diewert, de Haan and Hendriks (2011)[39] (2015)[40] on the multicollinearity
problem.
*72 This formulation follows that of Diewert (2010)[28], Diewert, Haan and Hendriks (2011)[39] (2015)[40],Eurostat (2013)[47], Diewert and Shimizu (2015)[35] (2016)[36] (2017a)[37], Burnett-Issacs, Huang andDiewert (2016)[11] and Diewert, Huang and Burnett-Issacs (2017)[41]. These authors assume that property value is the sum of land and structure components but movements in the price of structures areproportional to an exogenous structure price index. Note that the index pSt should be a levels price thatgives the period t cost of building one square meter of structure.

29

statistical agency or a relevant commercially available residential construction price index.
Thus the new model that replaces (52) is the following nonlinear hedonic regression model:
Vtn = αt Ltn + pSt (1 − δ)A(t,n) Stn + εtn ;t = 1, . . . , T ; n = 1, . . . , N (t).

(53)

This model has T land price parameters (the αt ) and one (net) geometric depreciation rateδ. Note that the replacement of the βt by the exogenous construction price level, pSt , meansthat we have saved T degrees of freedom as well as eliminated the multicollinearity problem.
In order to allow for a ﬁner structure of local land prices, the sales data may be further
classiﬁed into a ﬁner classiﬁcation of locations. For example, the initial regression (53) may
be applied to say city wide sales of residential properties. Suppose that the postal code of eachsale is also available and there are J postal codes. Then one can introduce the following postal
code dummy variables, DP C,tn,j , into the hedonic regression (53). These J dummy variables
are deﬁned as follows: for t = 1, . . . , T ; n = 1, . . . , N (t); j = 1, . . . , J:
DP C,tn,j ≡ 1 if observation n in period t is in Postal Code j;≡ 0 if observation n in period t is not in Postal Code j.

(54)

We now modify the model deﬁned by (53) to allow the level of land prices to diﬀer across the
J postal codes. The new nonlinear regression model is the following one:)(∑J
A(t,n)
ωD
Stn + εtn ; t = 1, . . . , T ; n = 1, . . . , N (t).
Vtn = αt
j=1 j P C,tn,j Ltn + pSt (1 − δ)(55)Comparing the models deﬁned by equations (53) and (55), it can be seen that we have added
an additional J neighbourhood relative land value parameters, ω1 , . . . , ωJ , to the model deﬁnedby (53). However, looking at (55), it can be seen that the T land time parameters (the αt )and the J location parameters (the ωj ) cannot all be identiﬁed. Thus it is necessary to impose
at least one identifying normalization on these parameters. The following normalization is aconvenient one:*73
ω1 ≡ 1.
(56)
Thus Model 2 is deﬁned by equations (55) and (56) has J − 1 additional parameters comparedto Model 1 deﬁned by (53). Note that if we initially set all of the ωj equal to unity, Model
2 collapses down to Model 1. It is useful to make use of this fact in running a sequence of
nonlinear hedonic regressions. The models that are proposed in this section are nested so thatthe ﬁnal parameter estimates from a previous model can be used as starting parameter values
in the next model’s nonlinear regression.*74*73 Equivalently, one could make the normalization α1 = 1 and not normalize the ωj . The resulting estimated

αt for t = 2, 3, . . . , T can then be interpreted as a constant quality land price index for the entire region
relative to period 1 where α1 ≡ 1. In this section, we are drawing heavily on Diewert, Huang and
Burnett-Isaacs (2017)[41] and using the normalization used in that paper.
*74 In order to obtain sensible parameter estimates in our ﬁnal (quite complex) nonlinear regression model,
it is absolutely necessary to follow our procedure of sequentially estimating gradually more complex
models, using the ﬁnal coeﬃcients from the previous model as starting values for the next model. The
models that are being described in this section were implemented in Diewert, Huang and Burnett-Issacs(2017)[41] where the econometric software Shazam was used to perform the nonlinear regressions; seeWhite (2004)[108].

30

In the next model, some nonlinearities in the pricing of the land area for each property are
introduced. The land plot areas in a typical sample of properties can vary 5 or 10 fold.*75 Up
to this point, we have assumed that land plots in the same neighbourhood sell at a constantprice per square meter of lot area. However, it is likely that there is some nonlinearity in thispricing schedule; for example, it is likely that large lots sell at a per m2 price that is well
below the per m2 price of medium sized lots. In order to capture this nonlinearity, divide up
the total number of observations into K groups of observations based on their lot size. The
Group 1 properties have lot size less than L1 m2 , the Group 2 properties Ltn have lot sizes
which satisfy the inequalities L1 ≤ Ltn < L2 ; the Group 3 properties Ltn have lot sizes which
satisfy the inequalities L2 ≤ Ltn < L3 ; . . . ; the Group K properties Ltn have lot sizes which
satisfy the inequalities LK−1 ≤ Ltn . The break points L1 < L2 < · · · < LK−1 should bechosen so that the sample probability that any property in the sample will fall into any oneof the groups is approximately equal. For each observation n in period t, the K land dummy
variables, DL,tn,k , for k = 1, . . . , K are deﬁned as follows:
DL,tn,k ≡ 1 if observation tn has land area that belongs to group k;≡ 0 if observation tn has land area that does not belong to group k.

(57)

These dummy variables are used in the deﬁnition of the following piecewise linear function of
Ltn , fL (Ltn ), deﬁned as follows:fL (Ltn ) ≡ DL,tn,1 λ1 Ltn + DL,tn,2 [λ1 L1 + λ2 (Ltn − L1 )]+ DL,tn,3 [λ1 L1 + λ2 (L2 − L1 ) + λ3 (Ltn − L2 )]+ · · · + DL,tn,K [λ1 L1 + λ2 (L2 − L1 ) + · · · + λK (Ltn − LK−1 )](58)

where the λk are unknown parameters. The function fL (Ltn ) deﬁnes a relative valuation
function for the land area of a house as a function of the plot area, Ltn . The new nonlinear
regression model is the following one:(∑)
J
A(t,n)
Vtn = αtωD
Stn + εtn ; t = 1, . . . , T ; n = 1, . . . , N (t).
j=1 j P C,tn,j fL (Ltn ) + pSt (1 − δ)(59)Comparing the models deﬁned by equations (55) and (59), it can be seen that we have added
an additional K land plot size parameters, λ1 , . . . , λK , to the model deﬁned by (55). However,looking at (59), it can be seen that the T land time parameters (the αt ), the J postal code
parameters (the ωj ) and the K land plot size parameters (the λk ) cannot all be identiﬁed.
Thus the following identiﬁcation normalizations on the parameters for Model 3 deﬁned by (59)and (60) are imposed:ω1 ≡ 1; λ1 ≡ 1.
(60)
Note that if all of the λk are set equal to unity, Model 3 collapses down to Model 2. Typically,*75 This brings up an important point that has not been mentioned up to now.

Panel data on the selling
prices of properties and on the characteristics of the properties are subject to tremendous variationsin the ratio of the say highest price property to the lowest price property, to the largest lot size tothe smallest lot size, to the largest ﬂoor space area to the smallest ﬂoor space area and so on. The
observations that appear in the tales of the distribution of prices and in the distributions of property
characteristics are inevitably sparse and subject to measurement error. Thus in order to obtain sensible
estimates in running these hedonic regressions, it is typically necessary to delete the observations thatare in the tales of these distributions.

31the log likelihood for Model 3 will be considerably higher than for Model 2.*76 Land prices
as functions of lot size do not always decline monotonically but for very large land plots, themarginal price of an extra square foot of land is typically quite low.
The next model is similar to Model 3 except that now the marginal price of adding an extra
amount of structure is allowed to vary as the size of the structure increases. It is likelythat the quality of the structure increases as the size of the structure increases. In order tocapture this nonlinearity, divide up the sample observations into M groups of observations
based on their structure size. The Group 1 properties have structures with ﬂoor space area Stnless than S1 m2 , the Group 2 properties have structure areas Stn satisfying the inequalities
S1 ≤ Stn < S2 , . . . , the Group M − 1 properties have structure areas Stn satisfying theinequalities SM −2 ≤ Stn < SM −1 , and the Group M properties have structure areas Stnsatisfying the inequalities SM −1 ≤ Stn where the M − 1 break points satisfy the inequalities
S1 < S2 < · · · < SM −1 . Again, the break points should be chosen so that the sample
probability that any property in the sample will fall into any one of the groups is approximately
equal. For each observation n in period t, we deﬁne the M structure dummy variables, DS,tn,m ,for m = 1, . . . , M as follows:
DS,tn,m ≡ 1 if observation tn has structure area that belongs to structure group m;≡ 0 if observation tn has structure area that does not belong to group m.

(61)

These dummy variables are used in the deﬁnition of the following piecewise linear function of
Stn , gS (Stn ), deﬁned as follows:gS (Stn ) ≡ DS,tn,1 µ1 Stn + DS,tn,2 [µ1 S1 + µ2 (Stn − S1 )]+ DS,tn,3 [µ1 S1 + µ2 (S2 − S1 ) + µ3 (Stn − S2 )]+ DS,tn,4 [µ1 S1 + µ2 (S2 − S1 ) + µ3 (S3 − S2 ) + µ4 (Stn − S3 )] + · · ·+ DS,tn,M [µ1 S1 + µ2 (S2 − S1 ) + µ3 (S3 − S2 ) + · · · + µM (Stn − SM −1 )].

(62)

where the µm are unknown parameters. The function gS (Stn ) deﬁnes a relative valuation
function for the structure area of a house as a function of the structure area.
The new nonlinear regression model is the following Model 4:(∑)
J
A(t,n)
Vtn =αtgS (Stn ) + εtn ;j=1 ωj DP C,tn,j fL (Ltn ) + pSt (1 − δ)t = 1, . . . , T ; n = 1, . . . , N (t).

(63)

Comparing the models deﬁned by equations (59) and (63), it can be seen that an additional
M structure ﬂoor space parameters, µ1 , . . . , µM , have been added to the model deﬁned by(59).*77 Again, we add the normalizations (60) in order to identify all of the parameters in*76 For the example in Diewert, Huang and Burnett-Isaacs (2017)[41] where the models described in thissection were estimated, the log likelihood increased by 1762 log likelihood points and the R2 jumped

from 0.7662 for Model 2 to 0.8283 for Model 3 for the addition of 6 new λk parameters.
*77 At this stage of the sequential estimation procedure, it is usually not necessary to impose a normalizationon the parameters µ1 -µM . This lack of a normalization means that the scale of the exogenous structure
price levels pSt is allowed to change; i.e., essentially, allowance is now made to quality adjust theexogenous index to a certain extent. However, if the resulting estimated structure values turn out
to be unreasonably large or small, then it will be necessary to set one of the µm to equal 1.

32the model. Note that if all of the µm are set equal to unity, Model 4 collapses down to Model
3. Typically, the log likelihood for Model 4 will be considerably higher than for Model 3.*78
At this stage, it is often the case that an acceptable model has been estimated. How canthe estimated parameters from the ﬁnal model be used in order to form price and quantity
indexes?
The sequence of price levels for the land component of residential property sales is deﬁned tobe α1 , α2 , . . . , αT and the corresponding sequence of price levels for the structure component ofresidential property sales in the T periods is deﬁned to be the exogenous sequence of indexes,
pS1 , pS2 , . . . , pST . The land and structure values of properties transacted in period t, VLt and
VSt , are deﬁned by using the estimated land and structure additive components of transacted∑J
properties in period t, αt ( j=1 ωj DP C,tn,j )fL (Ltn ) and pSt (1 − δ)A(t,n) gS (Stn ) respectively,
and summing over properties that were sold in period t:(∑)∑
JVLt ≡ n∈N (t) αtωDfL (Ltn );t = 1, . . . , T ;(64)jPC,tn,jj=1∑
VSt ≡ n∈N (t) pSt (1 − δ)A(t,n) gS (Stn );t = 1, . . . , T.
(65)
Using the prices α1 , α2 , . . . , αT and the corresponding estimated land values, VL1 , . . . , VLT andthe prices pS1 , pS2 , . . . , pST and the corresponding estimated structure values, VS1 , . . . , VST ,one can just apply normal index number theory using these data to construct Laspeyres,
Paasche, Fisher or whatever index formula is being used by the statistical agency in order
to construct constant quality price and quantity overall property indexes for the sales ofresidential properties in the area under consideration for the T periods.
However, constant quality land and structure price indexes for sales of Owner Occupied Residential houses is not what is needed for most purposes; what is required are constant quality
price and quantity indexes for the stock of residential houses. In order to accomplish thistask, it is necessary to have a census of the housing stock in the country which would include
information on the characteristics that are used in the hedonic regression model that is deﬁned by (63). The information that is required in order to estimate (63) is information on thefollowing variables:• The selling price of the residential properties (Ptn );• The age of the structure on the property (Atn );• The area of the land plot (Ltn );• The ﬂoor space area of the structure (Stn );• The neighbourhood of the property (or the postal code) and• An exogenous structure price index which provides the construction cost of a new structure per meter squared or per square foot (pSt ).
If a national housing Census has information on the above property characteristics (excluding
the information on selling prices Ptn and on the exogenous structure price index pSt )*79 thenit will be possible to insert the characteristics of each residential dwelling unit into the righthand side of (63) and then using appropriate modiﬁcations of deﬁnitions (64) and (65), it*78 For the example in Diewert, Huang and Burnett-Isaacs (2017)[41], the log likelihood increased by 935
log likelihood points and the R2 jumped from 0.8283 for Model 3 to 0.8520 for Model 4 for the additionof 5 new µM parameters.
*79 Every country will have a national residential construction deﬂator because this deﬂator is required inorder to form estimates of real investment in residential structures. However, this national deﬂator may
not be entirely appropriate for the type of buildings in a particular neighbourhood.

33

will be possible to obtain estimates for the land and structure value for each dwelling unit in
the area covered by the regression. If there is no national housing census information or therequired characteristics are not included in the census, then it will be very diﬃcult to form
estimates for the value of residential land.
Additional information on house and property characteristics will lead to more accurate landand structure decompositions of property value. Examples of useful additional structure pricedetermining characteristics are: (i) the number of bathrooms; (ii) the number of bedrooms;(iii) the type of construction material; (iv) the number of stories; etc. Examples of usefuladditional land price determining characteristics are: (i) the distance to the nearest subway
station; (ii) the distance to the city core; (iii) the quality of neighbourhood schools; (iv) the
existence of various neighbourhood amenities; etc. For examples of how these characteristics
can be integrated into the builder’s model, see Diewert, de Haan and Hendriks (2011)[39](2015)[40], Eurostat (2013)[47] (2017)[48], Diewert and Shimizu (2015)[35] and Diewert, Huang
and Burnett-Isaacs (2017)[41].*80
The estimates for the geometric depreciation rate generated by the application of the builder’smodel are useful for national income accountants because they facilitate the accurate estimation of structure depreciation, which is required for the national accounts. However, thedepreciation estimates that are generated by the builder’s model are wear and tear depreciation estimates that apply to structures that continue in existence over the sample period. The
estimated depreciation rate measures (net) depreciation*81 of a structure that has survived
from its birth to the period of its sale. However, there is another form of structure depreciationthat the estimated depreciation rate misses; namely the loss of residual structure value that
results from the early demolition of the structure. This problem was noticed and addressed byHulten and Wykoﬀ (1981a; 377-379)[73] (1981b)[74] (1996)[75]. Wear and tear depreciation isoften called deterioration depreciation and demolition or early retirement depreciation is sometimes called obsolescence depreciation.*82 Methods for estimating this form of depreciation
have been proposed by Hulten and Wykoﬀ as mentioned above and by Diewert and Shimizu(2017a; 512-516)[37]. Both methods require information on the distribution of the ages of retirement for the asset class. The Hulten and Wykoﬀ method absorbs demolition depreciation
into the wear and tear depreciation rate whereas the Diewert and Shimizu method uses thewear and tear depreciation rate that is generated by sales of surviving buildings but adds a
separate depreciation rate that is due to early demolition of the structures in the asset class.
Both methods require information on the age of structures when they are demolished.*83
The above paragraph simply warns the reader that wear and tear depreciation*84 for surviving*80 It is also possible to estimate more general models of depreciation using the builder’s model; see Diewert

and Shimizu (2017a)[37] and Diewert, Huang and Burnett-Isaacs (2017)[41].

*81 It is a net estimate since renovation and replacement investments in the building tend to extend the lifeof the building or augment its value. Thus the gross wear and tear depreciation rate for the structure
will tend to be larger than the estimated net depreciation rate.
*82 Crosby, Devaney and Law (2012; 230)[20] distinguish the two types of depreciation and in addition, they
provide a comprehensive survey of the depreciation literature as it applies to commercial properties.
*83 The Hulten and Wykoﬀ method estimates the age of retirement in a somewhat arbitrary fashion whereas
the Diewert and Shimizu method relies on mortality distributions on the age of buildings at the timethey are demolished. Over long periods of time and using country wide data, the two methods shouldbe equivalent. However, the Diewert and Shimizu method should give more accurate results at the ﬁrm
and regional levels since their method is consistent with the hedonic estimation of structure depreciation
rates as explained in this section.
*84 What has been labeled as wear and tear depreciation could be better described as anticipated amortization of the structure rather than wear and tear depreciation. Once a structure is built, it becomes a ﬁxed
asset which cannot be transferred to alternative uses (like a truck or machine). Thus amortization of the34

buildings is not the entire depreciation story: there is also a loss of asset value that resultsfrom the early retirement of a building that needs to be taken into account when constructing
national income accounting estimates of depreciation.
There is one additional complication that needs to be taken into account when running ahedonic regression on the sales of houses; i.e., what happens when the sales information for
an additional period becomes available? The simplest way of dealing with this problem dates
back to Court (1939)[17]. His method works as follows: set T = 2 and run a hedonic regressionthat has a time dummy variable in it. In the context of the hedonic regression model deﬁnedby (63), estimates for the price of land for periods 1 and 2 would be obtained, say α11 andα21 . The price index for land for periods 1 and 2 is deﬁned as PL1 = 1 and PL2 = α21 /α11 . Now
run a new hedonic regression using (63) for t = 2, 3 and obtain new estimates for the priceof land in periods 2 and 3, say α22 and α32 . The price index for land in period 3 is deﬁned as
PL3 = PL2 (α32 /α22 ); i.e., we update the price index value for period 2, PL2 , by the rate of change
in land prices going from period 2 to 3, (α32 /α22 ). Thus the previously estimated index is
updated each period as new information becomes available. This adjacent period time dummy
model has the advantage that it does not revise the previously estimated indexes as the new
information becomes available.*85
The above method does not always work well in the context of estimating property priceindexes due to the sparseness of sales in a neighbourhood and the multiplicity of parametersthat are required to adequately control for diﬀerences in housing characteristics. Thus Shimizu,
Nishimura and Watanabe (2010a; 797)[98] suggested extending the number of periods from
2 to a longer window of T consecutive periods, leading to the rolling window time dummyhedonic regression model. Thus for the model deﬁned by (63), the land price parameters thatare estimated by the ﬁrst regression using the data for periods 1 to T are α11 , α21 , . . . , αT1 andthe corresponding land price indexes for periods 1 to t are PLt ≡ αt1 /α11 for t = 1, . . . , T . The
second hedonic regression uses the data for periods 2, 3, . . . , T, T + 1 and the estimated land
price parameters are α22 , α32 , . . . , αT2 , αT2 +1 . The price index for land in period T + 1 is deﬁned
as PLT +1 = PLT (αT2 +1 /αT2 ); i.e., the price index for period T , PLt , is updated by the rate ofchange in land prices going from period T to T + 1, αT2 +1 /αT2 .
There are two additional issues that need to be addressed when using a rolling window time
dummy hedonic regression model:• How long should the window length be? A longer window length will probably lead tomore stable estimates for the unknown parameters in the hedonic regression. A shorter
window length will allow for taste changes to take place more quickly. A window lengthof one year plus one period will allow for seasonal eﬀects. At this stage of our knowledge,
it is diﬃcult to give deﬁnitive advice on the length of the window.
cost of the structure should be proportional to the cash ﬂows or to the service ﬂows of utility that thebuilding generates over its expected lifetime. However, technical progress, obsolescence or unanticipatedmarket developments can cause the building to be demolished before it is fully amortized. See Diewert
and Fox (2016)[29] for a more complete discussion of the ﬁxity problem.
*85 The two period time dummy variable hedonic regression (and its extension to many periods) was ﬁrst
considered explicitly by Court (1939; 109-111)[17] as his hedonic suggestion number two. Court usedadjacent period time dummy hedonic regressions as links in a longer chain of comparisons extending
from 1920 to 1939 for US automobiles: “The net regressions on time shown above are in eﬀect price link
relatives for cars of constant speciﬁcations. By joining these together, a continuous index is secured.” If
the two periods being compared are consecutive years, Griliches (1971; 7)[59] coined the term “adjacentyear regression” to describe this method for updating the index as new information becomes available.
Diewert (2005b)[26] looked at the axiomatic properties of adjacent year time dummy hedonic regressions.

35• When a new window is computed, how should the index results from the new window
be linked to the previous index values? The same issue applies when a multilateralmethod is used in the time series context. Ivancic, Diewert and Fox (2011)[77] alongwith Shimizu, Nishimura and Watanabe (2010a)[98] and Shimizu, Takatsuji, Ono andNishimura (2010)[100] suggested that the movement of the indexes for the last two
periods in the new window be linked to the last index value generated by the previous
window. However Krsinich (2016)[82] suggested that the movement of the indexes
generated by the new window over the entire new window period be linked to theprevious window index value for the second period in the previous window. Krsinich
called this a window splice as opposed to the movement splice explained above. De
Haan (2015; 27)[21] suggested that perhaps the linking period should be in the middleof the old window which the Australian Bureau of Statistics (2016; 12)[3] termed a half
splice. Ivancic, Diewert and Fox (2011; 33)[77] suggested that the average of all possible
links of the new window to the old window be used and they called this a mean splicemethod for linking the results of the new window to the previous window.*86 Again,
there is no consensus at this time on which linking method is “best”. However, it islikely that all of these linking methods will generate much the same results.
It can be seen that estimating price indexes for houses (or detached dwelling units) is not astraightforward task, particularly if one wants separate constant quality indexes for the landand structure components of property value.*87 In the following section, it will be seen that
it is even more complicated to obtain separate indexes for the land and structure components
for condominium sales.

13 Decomposing Condominium Sales Prices into Land andStructure Components
A starting point for applying the builder’s model to condominium sales is the hedonic regression model deﬁned by equations (53) in the previous section.*88 For convenience, equations(53) are repeated below as equations (66):
Vtn = αt Ltn + pSt (1 − δ)A(t,n) Stn + εtn ;t = 1, . . . , T ; n = 1, . . . , N (t)(66)

where Vtn is the selling price of a condominium property in a neighbourhood in period t, αt
is the price of the land that the structure sits on (per m2 ), Ltn is the land area that can be
attributed to the condo unit, pSt is an exogenous period t construction cost for the type of
condo under consideration (per m2 ), δ is the one period wear and tear geometric depreciation
rate for the structure, Atn = A(t, n) is the age of the structure in periods, Stn is the ﬂoor
space of unit n that is sold in period t (in m2 ) and εtn is an error term.
A problem with the above model is that it is not appropriate to allocate the entire land valueof the condominium property to any particular unit that is sold in period t. Thus each condo
unit in the building should be allocated a share of the total land value of the property. The
problem is: how exactly should this imputed land share be calculated? There are two simple*86 For the details on how the mean splice method works, see Diewert and Fox (2017)[30].
*87 For additional hedonic regression models for detached houses, see Verbrugge (2008)[106], Garner andVerbrugge (2011)[55], Eurostat (2013)[47] (2017)[48], Hill (2013)[66], Hill, Scholz, Shimizu and Steurer(2018)[67], Rambaldi and Fletcher (2014)[93] and Silver (2018)[102].
*88 The analysis in this section follows that of Diewert and Shimizu (2016)[36].

36methods for constructing an appropriate land share: (i) Use the unit’s share of ﬂoor space tototal structure ﬂoor space or (ii) simply use 1/N as the share where N is the total numberof units in the building. Thus deﬁne the following two land share imputations for unit n inperiod t:
LStn ≡ (Stn /T Stn )T Ltn ; LN tn ≡ (1/Ntn )T Ltn ;t = 1, . . . , T ; n = 1, . . . , N (t)(67)

where Stn is the ﬂoor space area of unit n which is sold in period t, T Stn is the total building
ﬂoor space area, T Ltn is the total land area of the building and Ntn is the total number ofunits in the building for unit n sold in period t. The ﬁrst method of land share imputation
is used by the Japanese land tax authorities. The second method of imputation implicitly
assumes that each unit can enjoy the use of the entire land area and so an equal share of landfor each unit seems “fair”.
There is a problem with the deﬁnition of LStn in (67): the ﬂoor space “share” of unit n,
Stn /T Stn if summed over all units in the building would be less than 1 because the privately
held ﬂoor space of each unit in the building does not account for shared building ﬂoor spaces
such as halls, elevators, storage spaces, furnace rooms and other “public” ﬂoor spaces, which
are included in total building ﬂoor space, T Stn . Thus the “share” Stn /T Stn must be adjusted
upward by some percentage to account for these shared building facilities.*89 In what follows,
it is assumed that this adjustment has been made to Stn (so that Stn is now interpreted as
adjusted condo ﬂoor space area).
In order to obtain sensible decompositions of the condominium selling price into land andstructure components, it may be necessary to assume a structure value and focus on thedeterminants of land value at the initial stages of the sequential estimation procedure. Thus
following Diewert and Shimizu (2016)[36], assume that the imputed structure value for unit nin period t, VStn , is deﬁned as follows:
VStn ≡ pSt (1 − δ)A(t,n) Stn ;t = 1, . . . , T ; n = 1, . . . , N (t)(68)

where δ is an assumed geometric depreciation rate.*90 Once the imputed value of the structure
has been deﬁned by (68), the imputed land value for condo n in period t, VLtn , is deﬁned bysubtracting the imputed structure value from the total value of the condo unit, which is Vtn :VLtn ≡ Vtn − VStn ;t = 1, . . . , T ; n = 1, . . . , N (t).

(69)

In the hedonic regressions which follow immediately, the imputed value of land for the condominium unit, VLtn , is used as the dependent variable in a hedonic regression. The following
regressions explain variations in these imputed land values in terms of the property characteristics.
*89 Diewert and Shimizu (2016; 303)[36] constructed estimates of Tokyo total building private ﬂoor space

to total building ﬂoor space for each observation nt as Ntn Stn /T Stn , where Ntn is the number of units
in the building which contained condo sale n in period t, Stn is the private ﬂoor space of the soldunit and T Stn is the total ﬂoor space of the building. The sample wide average of these ratios was
0.899. Thus the ﬁrst imputation method in deﬁnitions (67) was changed from LStn ≡ (Stn /T Stn )T Ltn
to LStn ≡ (1/0.899)(Stn /T Stn )T Ltn = (1.1)(Stn /T Stn )T Ltn . Burnett-Issacs, Huang and Diewert(2016)[11] estimated a similar condo model and consulted with construction experts and determinedthat on average, the ratio of total space to private space for Ottawa condominium apartments wasapproximately 1.33. Thus they changed LStn ≡ (Stn /T Stn )T Ltn to LStn ≡ (1.33)(Stn /T Stn )T Ltn .
*90 Diewert and Shimizu (2016)[36] assumed δ = 0.03 and Burnett-Isaacs, Huang and Diewert (2016)[11]
assumed δ = 0.02 where the age variable Atn is measured in years. Later, δ will be estimated.

37

Suppose that the postal code of each sale is also available and there are J postal codes.
Then one can introduce the following postal code dummy variables, DP C,tn,j , as explanatory
variables into a hedonic regression. Deﬁne these J dummy variables using deﬁnitions (54) inthe previous section and estimate the following hedonic regression which is a land counterpart
to the hedonic regression deﬁned by (55) in the previous section: are deﬁned as follows:(∑)
J
VLtn = αtωDt = 1, . . . , T ; n = 1, . . . , N (t).
(70)
j=1 j P C,tn,j LStn + εtn ;Note that the imputed value of land, VLtn deﬁned by (69), replaces total property value Vtn
which was the dependent variable in (55).*91
It is likely that the height of the building (number of stories) increases the value of the land plot
supporting the building, all else equal. Thus deﬁne the number of stories dummy variables,
DN S,tn,s , as follows: t = 1, . . . , T ; n = 1, . . . , N (t); s = 1, . . . , N S:
DN S,tn,s ≡ 1 if observation n in period t is in a building with s stories ;≡ 0 if observation n in period t is not in building with s stories.

(71)

The new nonlinear regression model is the following one:
(∑
) (∑
)
J
NS
VLtn = αtωDχD
j=1 j P C,tn,js=1 s N S,tn,s LStn + εtn ; t = 1, . . . , T ; n = 1, . . . , N (t).
(72)
Comparing the models deﬁned by equations (70) and (72), it can be seen that an additional
N S building height parameters, χ1 , . . . , χN S , have been added to the model deﬁned by (70).*92
As usual, the models deﬁned by (70) and (72) are nested so that the ﬁnishing parameter values
from the nonlinear regression (70) can be used as starting values for (72) along with the starting
values χ1 = χ2 = · · · = χN S = 1.
The higher up a unit is, the better is the view on average and so it could be expected that theprice of the unit increases as its height increases. The quality of the structure probably does
not increase as the height of the unit increases so it seems reasonable to impute the heightpremium as an adjustment to the land price component of the unit.
It is possible to introduce the height of the unit (the H variable) as a categorical variable(like the number of stories N S in the last hedonic regression model). However, both Diewert
and Shimizu (2016)[36] (hereafter DS) and Burnett-Isaacs, Huang and Diewert (2016)[11](hereafter BHD) found that this dummy variable approach could be replaced by using H
as a continuous variable with little change in the ﬁt of the model. Thus the new nonlinear
regression model is the following one where t = 1, . . . , T ; n = 1, . . . , N (t):
)
) (∑
(∑
NS
J
χ
D
(1 + γ(Htn − 3))LStn + εtn ;(73)ωDVLtn = αtsNS,tn,sjPC,tn,js=1
j=1
where Htn is the height of the sold unit n in period t (measured in number of stories fromground level) and γ is a height of the unit parameter to be estimated.*93 The above model
assumes that the lowest height for the units sold in the sample was Htn = 3. Thus for all the*91 As usual, we need a normalization on the parameters such as α1= 1 in order to identify all of theremaining parameters, α2 , . . . , αT , ω1 , . . . , ωJ . Note that this regression uses the ﬁrst method of land
imputation deﬁned by (67). Later, the second method will also be considered.
*92 Again normalizations like α1 ≡ 1; χ1 ≡ 1 are required in order to identify the remaining parameters. If
all χs = 1, then the model deﬁned by (72) collapses down to the model deﬁned by (70).
*93 Normalizations like α1 ≡ 1; χ1 ≡ 1 need to be imposed in order to identify the remaining parameters.

38

observations that correspond to the sold unit being located on the third ﬂoor of the building,the new parameter γ in (73) will not aﬀect the predicted value in the regression. However, forheights of the sold units that were greater than 3, the regression implies that the land value
will increase by γ for each story that is above 3.*94
As was mentioned earlier, there are two simple methods for imputing the share of the building’s
total land area to the sold unit. Up until now, we have used the ﬁrst method of imputation
deﬁned by (67) which set the share of total land imputed to unit n in period t, LStn , equalto (Stn /T Stn )T Ltn whereas the second method set LN tn equal to (1/Ntn )T Ltn . In the next
model, the land imputation for unit n in period t is set equal to a weighted average of the two
imputation methods and the best ﬁtting weight, λ, is estimated. Thus deﬁne:
Ltn (λ) = [λ(Stn /T Stn ) + (1 − λ)(1/Ntn )]T Ltn ;t = 1, . . . , T ; n = 1, . . . , N (t).

(74)

The new nonlinear regression model is the following one where t = 1, . . . , T ; n = 1, . . . , N (t)and Ltn (λ) is deﬁned by (74).*95 :
)
) (∑
(∑
NS
J
χ
D
(75)
ω
D
VLtn = αts=1 s N S,tn,s (1 + γ(Htn − 3))Ltn (λ) + εtn .
j=1 j P C,tn,jConditional on the land area of the building, one would expect the sold unit’s land imputation
value to increase as the number of units in the building increases. Thus one could use thetotal number of units in the building, Ntn , as a quality adjustment variable for the imputed
land value of a condo unit. DS introduced this variable as a continuous variable. The smallest
number of units in the buildings in their sample was 11. Thus they introduced the term
1 + κ(Ntn − 11) as an explanatory term in the nonlinear regression. The new parameter κis the percentage increase in the unit’s imputed value of land as the number of units in thebuilding grows by one unit. The new nonlinear regression model is the following one wheret = 1, . . . , T ; n = 1, . . . , N (t) and Ltn (λ) is deﬁned by (74):
)
) (∑
(∑
NS
J
χ
D
(1+γ(Htn −3))(1+κ(Ntn −11))Ltn (λ)+εtn .
ωDVLtn = αtsNS,tn,sjPC,tn,js=1
j=1(76)
where Ltn (λ) is deﬁned by (74).
The next explanatory variable to be introduced into the hedonic regression model is one which
is not obvious but turned out to be very signiﬁcant in the regressions run by DS and BHD.
The footprint of a building is the area of the land that directly supports the structure. An
approximation to the footprint land for unit n in period t is the total structure area T Stn
divided by the total number of stories in the structure T Htn . If footprint land is subtracted
from the total land area, T Ltn , the resulting variable is excess land,*96 ELtn , deﬁned asfollows:
ELtn ≡ T Ltn − (T Stn /T Htn );t = 1, . . . , T ; n = 1, . . . , N (t).
(77)
In the Tokyo data used by DS, excess land ranged from 47 m2 to 2912 m2 . Now group thesample observations into M categories, depending on the amount of excess land that pertained*94 The studies that have implemented this model found that the estimated γ was in the 2-4% range. Thus

the imputed land value of a unit increases by 2 to 4% for each story above the threshold level of 3.

*95 For the DS Tokyo condo data, the estimated λ turned out to be λ∗ = 0.3636 (t = 9.84) so that the very

simple land imputation method that just divided the total land plot size by the number of units in thebuilding got a higher weight (0.6364) than the weight for the ﬂoor space allocation method (0.3636). For
the Ottawa condo data, the estimated λ turned out to be λ∗ = 0.2525 (t = 12.10).
*96 This is land that is usable for purposes other than the direct support of the structure on the land plot.

39

to each observation. Group 1 consists of observations tn where ELtn is less than some number
EL1 ; Group 2: observations such that EL1 ≤ ELtn < EL2 ; . . . ; Group M : ELM −1 ≤ ELtn .
The break points, EL1 , EL2 , . . . , ELM −1 should be chosen so that the number of observations
in each group is approximately equal. Deﬁne the excess land dummy variables, DEL,tn,m , asfollows for t = 1, . . . , T ; n = 1, . . . , N (t); m = 1, . . . , M :DEL,tn,m ≡ 1 if observation n in period t is in excess land group m;≡ 0 if observation n in period t is not in excess land group m.

(78)

The new regression model is the following one:
(∑
) (∑
) (∑
)
J
NS
M
VLtn = αt
ω
D
χ
D
µ
D
j=1 j P C,tn,js=1 s N S,tn,s
m=1 m EL,tn,m ×(1 + γ(Htn − 3))(1 + κ(Ntn − 11))Ltn (λ) + εtn ; t = 1, . . . , T ; n = 1, . . . , N (t). (79)
Not all of the parameters in (79) can be identiﬁed so the following normalizations on theparameters in (79) are imposed:
α1 ≡ 1; χ1 ≡ 1; µ1 ≡ 1.

(80)

Introducing the excess land dummy variables led to huge jumps in the log likelihoods for thehedonic regressions run by DS and BHS: 1020 for DS and 2652 for BHS.*97 Both studies
found that the estimated µm were positive but their magnitudes decreased monotonically as
the excess land variable increased.
There are three additional explanatory variables that were used by DS that may aﬀect theprice of land. Deﬁne T W as the walking time in minutes to the nearest subway station; T T
as the subway running time in minutes to the Central Tokyo station from the nearest stationand the SOUTH dummy variable is set equal to 1 if the sold condo unit faces south and0 otherwise. Let DS,tn,2 equal the SOUTH dummy variable for sale n in period t. Deﬁne
DS,tn,2 = 1 − DS,tn,1 . In the Tokyo data set used by DS, T W ranged from 1 to 19 minutes
while T T ranged from 12 to 48 minutes. These new variables are inserted into the previous
nonlinear regression model (79) in the following manner for t = 1, . . . , T ; n = 1, . . . , N (t):
)
) (∑
) (∑
(∑
M
NS
J
µ
D
χ
D
ω
D
VLtn = αtm=1 m EL,tn,m ×
s=1 s N S,tn,s
j=1 j P C,tn,j(φ1 DS,tn,1 + φ2 DS,tn,2 )(1 + γ(Htn − 3))(1 + κ(Ntn − 11))×(1 + η(T Wtn − 1))(1 + θ(T Ttn − 12))Ltn (λ) + εtn ;(81)

where Ltn (λ) is deﬁned by (74). Not all of the parameters in (81) can be identiﬁed so thefollowing normalizations (82) are imposed on the parameters in (81):α1 ≡ 1; χ1 ≡ 1; µ1 ≡ 1; φ1 ≡ 1.

(82)

Using the DS Tokyo data, the R2 for this model turned out to be 0.6308 and the log likelihood
increased by 406 points over the log likelihood of the previous model deﬁned by (79) for the*97 Recall the hedonic regression model deﬁned by (59) in the previous section which introduced linear

splines on the valuation of the land area of a stand alone housing unit. This introduction also greatly
increased the log likelihood of the regression. In the present context, the excess land dummy variables
take the place of the linear spline functions in (59).

40

addition of 3 new parameters. The estimated parameters had the expected signs and hadreasonable magnitudes.
At this point, DS concluded that the imputed land value for each condominium in their sample
was predicted reasonably well by the hedonic regression deﬁned by (81) and (82). Thus in thefollowing regression, they switched from using the imputed land value VLtn deﬁned by (69)as the dependent variable in the regressions to using the actual selling price of the property,
Vtn . They used the speciﬁcation for the land component of the property that that is deﬁnedby (81) and (82) but they also added the structure term pSt (1 − δ)A(t,n) Stn to account forthe structure component of the value of the condo unit. Note that the annual depreciation
rate δ is now estimated by the new hedonic regression model, rather than assuming that itwas equal to 3%. Thus the number of unknown parameters in the new model increased by
1. They used the estimated values for the coeﬃcients in (81) as starting values in this new
nonlinear regression.*98
Using their Tokyo data, DS found that the R2 for this new model was 0.8190 and the estimateddepreciation rate was δ ∗ = 0.0367 (t = 27.1). Note that the R2 is satisfactory; i.e., the newmodel explains a substantial fraction of the variation in condo prices.
DS and BHD introduced some additional explanatory variables as quality adjusting variables
for the imputed value of structures. DS introduced the number of bedrooms and the type ofbuilding as quality adjusters for the value of the structure. BHD introduced the number ofbedrooms, the number of bathrooms, the presence of balconies, the use of natural gas as theheating fuel and whether there was commercial space in the building as additional variables
that could determine the value of the structure. These variables were signiﬁcant explanatory
variables but the overall R2 for the ﬁnal hedonic regression did not increase by a large amountwith the addition of these variables to the regression. The details may be found in Diewert
and Shimizu (2016)[36] and Burnett-Isaacs, Huang and Diewert (2016)[11].
Once the ﬁnal hedonic regression has been run, the sequence of land prices is given by
α1 , α2 , . . . , αT and the sequence of condo structure prices is given by the exogenous structure price indexes, pS1 , pS2 , . . . , pST . To obtain overall property price indexes for sales ofcondos, form the following counterparts to equations (64) and (65) in the previous section to
obtain an estimate of period t condo land value, VLt , and estimated period t structure value,
VSt , for t = 1, . . . , T :
)
) (∑
) (∑
(∑
∑
M
NS
J
µ
D
χ
D
ω
D
VLt ≡ n∈N (t) αt
m=1 m EL,tn,m ×
s=1 s N S,tn,s
j=1 j P C,tn,j(φ1 DS,tn,1 + φ2 DS,tn,2 )(1 + γ(Htn − 3))(1 + κ(Ntn − 11))×(1 + η(T Wtn − 1))(1 + θ(T Ttn − 12))Ltn (λ);∑
VSt ≡ n∈N (t) pSt (1 − δ)A(t,n) Stn .

(83)
(84)

Using the prices α1 , α2 , . . . , αT and the corresponding estimated land values, VL1 , . . . , VLT andthe prices pS1 , pS2 , . . . , pST and the corresponding estimated structure values, VS1 , . . . , VST ,one can again apply normal index number theory using these data to construct Laspeyres,
Paasche, Fisher or whatever index formula is being used by the statistical agency in order*98 Attempting to estimate the parameters in (83) without good starting values for the nonlinear regression

will not lead to sensible parameter estimates. Thus it is necessary to obtain good starting values for (83)by estimating the rather long sequence of regressions explained above, starting with a very simple modeland gradually introducing additional explanatory variables. Each regression in the sequence contains
the previous one as a special case so that the ﬁnal estimates of one regression can be used as starting
values for the subsequent one.

41

to construct constant quality price and quantity overall property indexes for the sales ofcondominium units in the area under consideration for the T periods.
In summary: the builder’s model can be modiﬁed to apply to the sales of condominium
units and reasonable decompositions of property value into land and structure components
can be obtained. However, the nonlinear regressions that are required in order to implement
the model end up being rather complex. In addition, information on more characteristics ofthe condominium properties needs to be collected in order to implement the models. The
information that is required in order to estimate the ﬁnal model and calculate (83) and (84)is as follows:• The selling prices of the condominium properties in the sample (Ptn );• The age of the structure on the property (Atn );• The total area of the land plot (T Ltn );• The ﬂoor space area of the condo unit (Stn );• The total ﬂoor space area of the entire building (T Stn );• The neighbourhood of the property (or the postal code);• An exogenous structure price index which provides the construction cost of a new structure per meter squared or per square foot (pSt );• The number of stories of the building (N Stn );• The height of the sold unit (the number of stories from ground level) (Htn );• The number of units in the building (Ntn );• The walking time in minutes to the nearest subway station (T Wtn ) and• The subway running time in minutes to the city center from the nearest station (T Ttn ).
The last two variables are not essential (and are not relevant in small towns and cities). Other
non-essential variables which could be useful are the number of bedrooms, the number ofbathrooms, the existence of balconies, the type of construction, the number of parking spaces
and so on.
The hedonic regression models that were considered in the last two sections are essentially
modiﬁed supply side models. In the following section, demand side hedonic regressions are
considered.

14 Demand Side Property Price Hedonic Regressions
A way of rationalizing the traditional log price time dummy hedonic regression model for
properties with varying amounts of land area L and constant quality structure area S ∗ isthat the utility that these properties yield to consumers is proportional to the Cobb-Douglas
utility function Lα S ∗β where α and β are positive parameters (which do not necessarily sum
to one).*99 Initially, assume that the constant quality structure area S ∗ is equal to the ﬂoor
space area of the structure, S, times an age adjustment, (1 − δ)A , where A is the age of thestructure in years and δ is a positive depreciation rate that is less than 1. Thus S ∗ is related
to S as follows:S ∗ ≡ S(1 − δ)A .
(85)
In any given time period t, assume that the sale price of transacted property n, Vtn , with the∗amount of land Ltn and the amount of quality adjusted structure Stnis equal to the following*99 The early analysis in this section follows that of McMillen (2003; 289-290)[88], Shimizu, Nishimura and

Watanabe (2010a; 795)[98] and Diewert, Huang and Burnett-Isaacs (2017)[41]. McMillen assumed thatα + β = 1. We follow Shimizu, Nishimura and Watanabe in allowing α and β to be unrestricted.

42expression:∗β
Vtn = pt Lα
tn StnA(t,n) β= pt Lα]tn [Stn (1 − δ)using (85)ββA(t,n)= pt Lαtn Stn (1 − δ)β A(t,n)= pt Lαtn Stn φ(86)

where A(t, n) = Atn is the age of house n sold in period t, pt can be interpreted as a period t
property price index and the constant φ is deﬁned as follows:φ ≡ (1 − δ)β .

(87)

Thus if Vtn is deﬂated by the period t property price index pt , the real value or utility utn of∗the property with characteristics Ltn and Stn
is obtained:∗βVtn /pt = Lα
tn Stn ≡ utn .

(88)∗
Thus utn ≡ qt is the aggregate real value of the property with characteristics Ltn and Stn.

Deﬁne ρt as the logarithm of pt and γ as the logarithm of φ; i.e.,ρt ≡ ln pt ; γ ≡ ln φ.

(89)

After taking logarithms of both sides of the ﬁrst equation in (88), using deﬁnitions (85) and(89) and adding error terms, the following system of estimating equations is obtained:*100
ln Vtn = ρt + α ln Ltn + β ln Stn + γAtn + εtn ;t = 1, . . . , T ; n = 1, . . . , N (t)(90)

where the εtn are independently distributed error terms with 0 means and constant variances.
It can be seen that (90) is a traditional log price time dummy hedonic regression model with aminimal number of characteristics. The unknown parameters in (90) are the constant quality
log property prices, ρ1 , . . . , ρT , and the taste parameters α, β and the transformed depreciation
rate γ. Once these parameters have been determined, the geometric depreciation rate δ whichappears in equations (86) can be recovered from the regression parameter estimates as follows:δ ≡ 1 − eγ/β .

(91)

We now explain how the hedonic pricing model deﬁned by (86) can be manipulated to provide a decomposition of property value in period t into land and quality adjusted structurecomponents.
Once estimates for α, β and δ have been obtained, deﬁne period t value of a property with∗characteristics Ltn and Stn
is given by the following period t property valuation function by the∗β∗right hand side of (86); i.e., deﬁne V (pt , Ltn , Stn) ≡ pt Lα
tn Stn . In empirical applications of thehedonic regression model deﬁned by (90), it will often happen that estimates for α and β aresuch that α+β is less than 1.*101 This means that a property in a given period that has double
the land and quality adjusted structure than another property will sell for less than double
the price of the smaller property. This follows from the fact that the Cobb-Douglas hedonic*100 Log price hedonic regressions for property prices date back to Bailey, Muth and Nourse (1963)[4].
*101 See for example the estimated model in Diewert, Huang and Burnett-Isaacs (2017)[41].

43utility function, u(L, S ∗ ) ≡ Lα S ∗β , exhibits diminishing returns to scale when α + β < 1; i.e.,
we have:u(λL, λS ∗ ) = λα+β u(L, S ∗ )(92)
for all λ > 0. This behavior is roughly consistent with our builder’s Models 5-7 where there
was a tendency for property prices to increase less than proportionally as L and S ∗ increased.
The marginal prices of land and constant quality structure in period t for a property with characteristics L and S ∗ , πL (pt , L, S ∗ ) and πS ∗ (pt , L, S ∗ ), are deﬁned by partially diﬀerentiating
the property valuation function with respect to L and S ∗ respectively:∗∂V (pt , Ltn , Stn)∗β∗≡ pt αLαtn Stn /Ltn = αV (pt , Ltn , Stn )/Ltn ;∂L∗∂V (pt , Ltn , Stn
)
∗β
∗
∗
∗
∗
πS ∗ (pt , Ltn , Stn)≡≡ pt βLαtn Stn /Stn = βV (pt , Ltn , Stn )/Stn .
∗∂S∗
πL (pt , Ltn , Stn
)≡

(93)
(94)

Multiply the marginal price of land by the amount of land in the property and add to thisvalue of land the product of the marginal price of constant quality structure by the amountof constant quality structure on the property in order to obtain the following identity:∗∗∗∗(α + β)V (pt , Ltn , Stn) = πL (pt , Ltn , Stn)Ltn + πS ∗ (pt , Ltn , Stn
)Stn
.

(95)

If α + β is less than one, then using marginal prices to value the land and constant quality
structure in a property will lead to a property valuation that is less than its selling price.
Thus to make the land and structure components of property value add up to property value,
divide the marginal prices deﬁned by (93) and (94) by α + β in order to obtain the following∗
adjusted prices of land and structures for property n sold in period t, ptL (pt , Ltn , Stn) and∗ptS ∗ (pt , Ltn , Stn ):∗∗∗ptL (pt , Ltn , Stn) ≡ πL (pt , Ltn , Stn)/(α + β) = α(α + β)−1 V (pt , Ltn , Stn)/Ltn ;

(96)

∗
∗
∗
∗
ptS ∗ (pt , Ltn , Stn) ≡ πS ∗ (pt , Ltn , Stn)/(α + β) = β(α + β)−1 V (pt , Ltn , Stn
)/Stn
.

(97)

The above material outlines a theoretical framework that can generate a decomposition ofproperty value into land and structure components using the results of a traditional log pricetime dummy hedonic regression model. To complete the analysis, it is necessary to ﬁll in thedetails of how the individual property land and structure prices that are generated by themodel can be aggregated into period t overall land and structure price indexes.
Run the hedonic regression model deﬁned by (90). Deﬁne the constant quality property priceindex pt for period t as follows:pt ≡ exp(ρt );t = 1, . . . , T.

(98)

Deﬁne the geometric depreciation rate δ by (91). Once δ has been deﬁned, the amount of∗quality adjusted structure for property n in period t, Stn
is deﬁned as follows:∗ln(Stn) ≡ ln(Stn ) + Atn ln(1 − δ);t = 1, . . . , T ; n = 1, . . . , N (t).

(99)∗
Now that pt , Ltn , Stn, α and β have all been deﬁned, we use these data in order to deﬁne the∗predicted prices for property n sold in period t, Vtn
:
∗
∗ β
Vtn
≡ pt (Ltn )α (Stn) ;t = 1, . . . , T ; n = 1, . . . , N (t).
44

(100)

Use equations (96) and (97) in order to deﬁne constant quality land and structure prices for
sold property n in period t, ptnL and ptnS ∗ , as follows:∗
ptnL ≡ α(α + β)−1 Vtn/Ltn ;−1

ptnS ∗ ≡ β(α + β)

∗
∗
Vtn
/Stn
;

t = 1, . . . , T ; n = 1, . . . , N (t);(101)

t = 1, . . . , T ; n = 1, . . . , N (t).

(102)

Finally, unit value constant quality land and structure prices for all properties sold in periodt, ptL and ptS ∗ , are deﬁned as follows:/∑∑N (t)N (t)ptL ≡ n=1 ptnL Ltn
t = 1, . . . , T ;(103)
n=1 Ltn/∑N (t)∑N (t) ∗∗ptS ∗ ≡ n=1 ptnS ∗ Stn
t = 1, . . . , T.
(104)
n=1 Stn
The period t land and structure prices that are deﬁned by (103) and (104) are reasonablesummary statistic prices for land and structures sold in period t that are generated by the log
price time dummy hedonic regression model deﬁned by (90).
The time dummy log price hedonic regression model deﬁned by (90) will generate very diﬀerent
constant quality land and structure subindexes when compared to the corresponding indexes
estimated by the builder’s model. To see this, suppose the same house n sold in period t and∗∗sold again in the following period t + 1. The period t data for this house are Vtn, Ltn and Stn∗∗∗
while the period t + 1 data are Vt+1n, Lt+1n = Ltn and St+1n= (1 − δ)Stn. Use deﬁnitions(101) and (102) for this house for periods t and t + 1 and calculate the following land andstructure inﬂation rates for this house going from period t to period t + 1:∗∗∗∗
pt+1nL /ptnL = [α(α + β)−1 Vt+1n/Ltn ]/[α(α + β)−1 Vtn/Ltn ] = Vt+1n
/Vtn
;

(105)

∗
∗
∗
∗
∗
∗
pt+1nS ∗ /ptnS ∗ = [β(α + β)−1 Vt+1n/(1 − δ)Stn]/[β(α + β)−1 Vtn/Stn] = (1 − δ)−1 (Vt+1n
/Vtn
).
(106)

Thus (one plus) the imputed land inﬂation rate, pt+1nL /ptnL , will equal (one plus) the growth∗∗/Vtn
in property value, Vt+1n, and (one plus) the imputed constant quality structure inﬂation∗∗
rate, pt+1nS ∗ /ptnS ∗ , will equal (1 − δ)−1 (Vt+1n/Vtn). Hence if δ is small, then the land andstructure inﬂation rates will be almost identical and approximately equal to (one plus) thegrowth rate for overall property value. Thus the constant quality price indexes for land andstructures will move in an almost proportional manner. In most countries, the price of land
will grow much more rapidly than the price of structures so the hedonic regression model
deﬁned by (90) is not suitable for ﬁnding usable land price indexes for residential housing.
However, the hedonic regression model deﬁned by (90) (and its generalizations) can generatevery reasonable overall constant quality property price indexes, provided that the model generates a plausible estimate for the structure depreciation rate. To see why this result mightoccur, a highly simpliﬁed comparison of a builder’s model and the log price traditional hedonic
regression model studied in this section will be undertaken below.
Consider the valuation of a representative property in periods 1 and 2 using both the builders
model and the traditional hedonic regression model explained in this section. In period 1, thequantity of land and constant quality structure is L1 and S1∗ with total property value equal
to V1 . In period 2, the quantity of land and constant quality structure is L2 = (1 + gL )L1 andS2∗ = (1 + gS )S1∗ with total property value equal to V2 . The Lt and St∗ are known and hence
the growth rates gL and gS are also known. Using the property valuation function deﬁned45

by (100), the two properties have the following value decompositions where p1 and p2 are theconstant quality property price levels for periods 1 and 2:∗βV1 = p1 Lα
1 S1 ;(107)∗βV2 = p2 Lα
2 S2= p1 (1 + ρ)[L1 (1 + gL )]α [S1∗ (1 + gS )]βαwhere 1 + ρ = p2 /p1

β= V1 (1 + ρ)(1 + gL ) (1 + gS )≈ V1 (1 + ρ)[α(1 + gL ) + β(1 + gS )](108)

where the last approximate equality follows if α + β = 1 and the geometric mean (1 + gL )α (1 +gS )β is approximated by the corresponding arithmetic mean, α(1 + gL ) + β(1 + gS ).
Now use the builder’s model to value the same properties. Let pL1 and pL2 be the pricelevels for land in periods 1 and 2 and let pS1 and pS2 be the constant quality price levels
for structures in periods 1 and 2. The builder’s model imputes the following values for theproperties in the two periods:
V1 = pL1 L1 + pS1 S1∗ ;(109)

V2 = pL2 L2 + pS2 S2∗= pL1 (1 + ρL )(1 + gL )L1 + pS1 (1 + ρS )(1 + gS )S1∗

(110)

where the land and structure constant quality price indexes are deﬁned as 1 + ρL = pL2 /pL1
and 1 + ρS = pS2 /pS1 . Deﬁne the land and structure share of property value in period 1 assL1 ≡ pL1 L1 /V1 and sS1 ≡ pS1 S1∗ /V1 respectively. The Laspeyres quantity and Paasche priceindexes for properties, QL and PP , are deﬁned as follows:
QL ≡ sL1 (L2 /L1 ) + sS1 (S2∗ /S1∗ )= sL1 (1 + gL ) + sS1 (1 + gS );(111)

PP ≡ [V2 /V1 ]/QL
= [V2 /V1 ]/[sL1 (1 + gL ) + sS1 (1 + gS )](112)

where the last equality follows using (111). Using (108), we have the following approximate
expression for 1 + ρ, which is the property price index generated by the traditional hedonic
regression model:
1 + ρ ≈ [V2 /V1 ]/[α(1 + gL ) + β(1 + gS )].
(113)
Comparing (112) to (113), it can be seen that the Paasche property price index that is generated by the builder’s model, PP , will be approximately equal to the property price index 1 + ρthat is generated by a traditional log price time dummy hedonic regression model provided
that α is approximately equal to the land share sL1 and β is approximately equal to structure
share sS1 .*102 Since the hedonic utility function for the traditional model is Cobb Douglas,
this approximate equality is likely to hold. Thus the traditional model is likely to generate
approximately the same overall property price indexes as would be generated by the builder’smodel.*103*102 To obtain this approximation result, it is also necessary that the depreciation rate that is estimated by

the log price time dummy model be reasonable.

*103 For examples of studies where it was found that this approximate equality held, see Diewert (2010;21)[28], Diewert and Shimizu (2015; 1692)[35] and Diewert, Huang and Burnett-Isaacs (2017; 32)[41].

46The approximation result in the previous paragraph opens up another possible method for
obtaining aggregate land values for residential housing. There are residential property priceindexes for many countries that are based on traditional hedonic regression models. Consider
such a country that also conducts periodic censuses of housing where owners of residential
dwelling units are asked to value their properties. Let the estimated value of housing inperiods 1 and t be V1 and Vt . Suppose the aggregate housing price index levels for thesetwo periods are p1 and pt . Using these data, one can form aggregate volume estimates forresidential housing as q1 ≡ V1 /p1 and qt ≡ Vt /pt . From the country’s system of national
accounts, it should be possible to obtain estimates for the aggregate price and quantity orvolume of residential structures which we denote by pS1 and qS1 for period 1 and pSt and qSt
for period t. With these data in hand, aggregate Laspeyres, Paasche and Fisher (1922)[51]
price and quantity indexes for residential land can be formed using (p1 , pS1 ) and (pt , pSt ) as
period 1 and t price vectors and using (q1 , −qS1 ) and (qt , −qSt ) as period 1 and t quantity
vectors. The resulting land prices (pL1 , pLt ) and volumes (qL1 , qLt ) would ﬁll a gap in the
System of National Accounts for the country.
For data series on residential property prices for either the sales of properties or the stockof properties, see the European Central Bank (2018)[43] (which lists 228 series for European
countries) and the Bank for International Settlements (2018)[5], which lists long series for 18
advanced economies. For additional information on alternative approaches for the measurement of residential property price indexes for sales of properties and for making estimates
for the stock of residential properties, see Statistics Portugal (2009)[104], Eurostat (2013)[47](2017)[48], Hill (2013)[66], Hill, Scholz, Shimizu and Steurer (2018)[67] and Silver (2018)[102].

15 Price Indexes for Rental Housing
At ﬁrst sight, it would seem that the construction of price indexes for rental housing would
be fairly straightforward since typically, rents are paid to owners every month. Thus all that
seems to be necessary is to collect information on rents paid (from either the tenants or fromthe owners), say Rtn and Rt+1n for rental unit n in periods t and t + 1, form the price ratios,
Rt+1n /Rtn , and take a suitable average of these ratios to form a rent index. However, theproblem is that the quality of the rental unit does not in general remain constant going fromone period to the next due to depreciation of the structure and possible renovations andimprovements to the structure. Thus the structure is a unique good in general.
Three procedures for dealing with the above problem will be outlined in this section. The ﬁrst
procedure assumes that the builder’s model has been run on sales of dwelling units that could
be rented and so asset prices, PLtn and PStn *104 can be assigned to the land and structureareas, Ltn and Stn , that can be imputed for rental dwelling n in period t. The rental price
Rtn is approximated by the sum of its (end of period) user cost components for land andstructures, pLtn and pStn respectively. The geometric model of depreciation for structures isused and the one period depreciation rate is 0 < δ < 1. The depreciation rate for land is 0.
The age of the structure for rental unit n in period t is A(t, n) periods. Setting the rental price*104 PStn is the price of a square meter of new structure of the type used by rental unit n at the beginningof period t.

47

of unit n in period t and t + 1 to the corresponding user costs leads to the following equations:
Rtn = pLtn Ltn + pStn (1 − δ)A(t,n)−1 Stn ;n = 1, . . . , N= [rt − iLt ]PLtn Ltn + [rt − iSt + (1 + iSt )δ]PStn (1 − δ)A(t,n)−1 Stn ;Rt+1n = pLt+1n Ltn + pSt+1n (1 − δ)A(t,n) Stn ;(114)

n = 1, . . . , N= [rt+1 − iLt+1 ]PLt+1n Ltn + [rt+1 − iSt+1 + (1 + iSt+1 )δ]PSt+1n (1 − δ)A(t,n) Stn(115)
where rt is the period t opportunity cost of capital for the owner of the rental unit and iLt andiSt are the land and structure price inﬂation rates that the owner expects at the beginningof period t. Note that the land and structure areas for unit n, Ltn and Stn , do not changeover time since by hypothesis, we are collecting rent information for the same units over time.
It is well known in the housing literature that user costs for dwelling units are much morevolatile than the corresponding rents for the same units.*105 Thus in order for the rents Rtn
and Rt+1n to approximate their corresponding user costs on the right hand sides of (114) and(115), it is necessary to use a nominal smoothed values for the nominal interest rates rt andparticularly for the expected asset inﬂation rates, iLt and iSt .*106 Note that the quantity of∗
constant quality structure for property n in periods t and t + 1 are Stn≡ (1 − δ)A(t,n)−1 Stn∗≡ (1 − δ)A(t,n) Stn ; i.e., the imputed constant quality amount of structure constantand St+1n
quality quantity declines as time increases. The corresponding constant quality amount ofland rent, Ltn , remains constant over all periods when the dwelling unit is rented. To form a
constant quality overall price index for rents, calculate Laspeyres, Paasche or Fisher indexes
where the price data for periods t and t + 1 are the vectors [pLt1 , . . . , pLtN ; pSt1 , . . . , pStN ]and [pLt+11 , . . . , pLt+1N ; pSt+11 , . . . , pSt+1N ] and the quantity data for periods t and t + 1 are
the vectors [Lt1 , . . . , LtN ; (1 − δ)A(t,1)−1 St1 , . . . , (1 − δ)A(t,N )−1 StN ] and [Lt1 , . . . , LtN ; (1 −δ)A(t,1) St1 , . . . , (1 − δ)A(t,N ) StN ].
If estimates for the price of land for the rented units, PLtn , are not available, then with some
additional simplifying assumptions, it is possible to turn equations (114) into a system of
estimating equations. Thus assume that the price of land and the price of a new structure
per m2 is constant across the N rented properties in each period so that PLtn = PLt andPStn = PSt for t = 1, . . . , T . Further assume that the new structure price level for period t
is known so that PSt is a given exogenous variable. Then equations (114) simplify into thefollowing nonlinear regression model:
Rtn = pLt Ltn + µt PSt (1 − δ)A(t,n)−1 Stn ;n = 1, . . . , N ; t = 1, . . . , T(116)

where the user cost of land in period t, pLt ≡ [rt −iLt ]PLt , is a parameter which is estimated andµt ≡ [rt − iSt + (1 + iSt )δ]*107 is also a parameter which is estimated for t = 1, . . . , T . The de*105 On

this point, see Genesove (2003)[56], Verbrugge (2008)[106], Shimizu, Nishimura and Watanabe(2010b)[99], Diewert and Nakamura (2011)[33], Garner and Verbrugge (2011)[55] and Suzuki, Asami
and Shimizu (2018)[105].
*106 The expected land inﬂation rate iLt should be an average of land price inﬂation over the past 15 to 25
years to reﬂect the long holding periods that investors have for rental properties and the high transactions
costs of buying and selling properties. Diewert and Fox (2018)[31] used a rolling window annualized 25
year inﬂation rate for land for the 25 years prior to period t to generate very smooth estimates for the
expected land inﬂation rate in their user costs for land in the US.
*107 µt is also known as a capitalization rate; i.e., it is the ratio of the rental price of the structure to its
capital value.

48

preciation parameter δ is also estimated.*108 The period t price and quantity vectors generated
by this model are [pL1 , . . . , pLt ; µ1 PS1 , . . . , µt PSt ] and [Lt1 , . . . , LtN ; (1−δ)A(t,1)−1 St1 , . . . , (1−δ)A(t,N )−1 StN ] and normal index number theory can be applied to these vectors. Of course,
this simple model can be generalized along the same lines as was done in sections 12 and 13
above for the basic builder’s model. If the geometric model of depreciation for the structurecomponent of the rental unit is changed to another model of depreciation such as one hoss
shay depreciation, then the estimating equations must be modiﬁed to suit the alternative depreciation model. Finally, a rolling window approach to this model can be implemented which
will allow for gradually changing parameters over time.
The second method for dealing with the quality adjustment problems for rents due to theaging of the structure does not require as much information and can be implemented withguesses on the magnitude of a few key parameters. Recall that the market rents for rented
unit n in periods t and t + 1 were deﬁne above by (114) and (115). Deﬁne a constant quality∗
rent for unit n in period t + 1, Rt+1n, as follows:∗
Rt+1n≡ pLt+1n Ltn + pSt+1n (1 − δ)A(t,n)−1 Stn ;n = 1, . . . , N= Rt+1n [1 + δ(1 − δ)−1 sSt+1n ](117)

where Rt+1n is the period t + 1 market rent for unit n deﬁned by (115) and sSt+1n is thefollowing share of structures in the market rent for unit n in period t + 1:
sSt+1n ≡ pSt+1n (1 − δ)A(t,n) Stn /Rt+1n ;n = 1, . . . , N.

(118)

∗
/Rtn is a constant quality rent index for unit n for period t + 1 for n = 1, . . . , N .
Thus Rt+1n
This index can be calculated if the market rents for both periods, Rtn and Rt+1n , are known
along with the geometric depreciation rate δ and the imputed share of structures in market
rent for unit n in period t + 1, sSt+1n deﬁned by (118). Thus if market rents are known andthe statistician makes educated guesses on the magnitudes of the geometric depreciation rate δ∗and on sSt+1n , then Rt+1n
deﬁned by (117) can be calculated as can 1 + δ(1 − δ)−1 sSt+1n . To
form a constant quality price index for rents, calculate the Laspeyres, Paasche or Fisher
indexes where the price data for periods t and t + 1 are the vectors [Rt1 , . . . , RtN ] and∗∗, . . . , RtN] and the quantity data for periods t and t + 1 are the vectors [1, . . . , 1] and[Rt1[{1 + δ(1 − δ)−1 sSt+11 }−1 , . . . , [{1 + δ(1 − δ)−1 sSt+1N }−1 ]. This adjustment to rents for the
aging of the units will increase the rental price index for period t + 1 and decrease the corresponding quantity index for period t + 1 as compared to an index which just assumed that
there was no aging bias.
The third method for dealing with the quality adjustment problems for rents due to the agingof the structure is to run a hedonic regression with the logarithm of rents as the dependent
variable. Thus recall the demand side hedonic regression for property prices that was describedby equations (86)-(91) above. Using these equations, replace the period t selling price forproperty n, Vtn , by the observed rent for unit n in period t, Rtn , and reinterpret the constantquality price for property sales in period t, pt , as the period t constant quality price level forrents for the dwelling units in scope. With these changes, the rent counterparts to equations*108 If multicollinearity becomes a problem, it may be necessary to set µt = µ or assume that that the µt are

slowly trending over time.

49

(86) are the following equations:A(t,n) β
Vtn = pt Lα]
tn [Stn (1 − δ)n = 1, . . . , N ; t = 1, . . . , T

β A(t,n)= pt Lαtn Stn φ(119)

where A(t, n) = Atn , Ltn and Stn are the age, land plot area and ﬂoor space area of rental
unit n and the constant φ is deﬁned as (1 − δ)β . Deﬁne ρt as the logarithm of pt and γ as thelogarithm of φ. Take logarithms of both sides of (119) and add error terms in order to obtainthe following system of estimating equations:ln Rtn = ρt + α ln Ltn + β ln Stn + γAtn + εtn ;t = 1, . . . , T ; n = 1, . . . , N.

(120)

Once the unknown parameters in the linear regression model (120) have been determined,
the geometric depreciation rate δ which appears in equations (119) can be recovered from theregression parameter estimates as δ ≡ 1 − eγ/β . The sequence of constant quality rent levels
can be recovered as pt ≡ exp[ρt ] for t = 1, . . . , T .
The estimated depreciation rate δ could equal 0. In this case, renters do not experience any
reduction in the quality of the rented structure as the structure ages. This corresponds to one
hoss shay or light bulb depreciation. It this case were to occur, it would imply that the aging
bias adjustments made in the above two models are not warranted and the estimating equations
for those two models would need to be changed to reﬂect the one hoss shay depreciation of
the structures. However, the empirical evidence is that depreciation rates are positive.*109
Other explanatory variables could be added to the basic log price time dummy hedonic regression model.*110 The explanatory variables that were used in sections 12 and 13 could also
be added to the present model deﬁned by equations (120).

16 Valuing the Services of OOH: User Cost versus Rental
Equivalence
In this section, various factors that cause the user cost of an owned dwelling unit to diﬀer
from a rental price for a comparable property will be examined.*111 In addition, other factorsthat aﬀect user costs in general will be discussed.
*109 “The average [annual] depreciation rate for rental property is remarkably constant, ranging from 0.58%

to 0.60% over the 25 year period. Depreciation rates for owner occupied units show more variation thanthe estimated rates for renter occupied units. The average depreciation rate for owner occupied housing
ranges from 0.9% in year 1 to 0.28% in year 20.” Stephen Malpezzi, Larry Ozanne and Thomas G.
Thibodeau (1987; 382)[85]. Note that these depreciation rates are underestimates for the “true” rates
since demolition depreciation is not taken into account using this methodology. Put another way, thegeometric model of depreciation may not be the “right” model of depreciation for rental housing.
*110 For example, see Malpezzi, Ozanne and Thibodeau (1987)[85], Crone, Nakamura and Voith (2000)[18](2011)[19], Verbrugge (2008)[106], Shimizu, Nishimura and Watanabe (2010a)[98] and Garner and Verbrugge (2011)[55].
*111 Our discussion here is similar to that of Hill, Steurer and Waltl (2017; 7)[68]: “The services a householdobtains from renting a dwelling are not the same as the services obtained by owner-occupying.” They
consider some additional factors that can cause rents to diﬀer from user costs. They also assert thatsince OOH services are derived from both the structure and land, it follows that there is no need to try
and separate land from structure in the rental house price index. However, depreciation aﬀects only the
structure part of rents and if one attempts to adjust a market rent for this aging factor, it is necessary
to apply the depreciation adjustment to only the structure part of rents.

50• Utilities such as electricity, water and natural gas may be included in the rent for adwelling unit that is similar to an owned unit. The net beneﬁt of renting an owned unit
should exclude these costs since these expenditures are covered in other categories of a
Consumer Price Index.
• When calculating the beneﬁt to the owner of a dwelling unit of renting the unit, there
is a problem of determining what is the correct market rental opportunity cost. It
turns out that all rents paid in say period t for comparable units to an owned unit
can be classiﬁed into 3 categories: (i) the rental agreement is not being renegotiatedduring this period; (ii) the rental agreement is renegotiated during this period with thesame tenants and (iii) the rental agreement is a new one with new tenants. Typically,
there are no escalations of rents for continuing tenants during the leasehold period andoften, renegotiated rents with continuing tenants are also sticky; i.e., there is not muchchange in these renegotiated rents.*112 For purposes of measuring the net beneﬁt to an
owner of renting an owned unit, category (iii) rents should be used as the appropriatecomparable market rent.*113• Property taxes will be included in market rents and they should also be included in anowner’s user cost. However, if property tax payments are treated as a separate category
in the CPI, then property taxes should be deducted from the comparable market rents
to avoid double counting of these tax expenditures.
• Normal maintenance expenditures on the structure will be part of market rents. These
expenditures should be deducted from the comparable market rents since these expenditures by home owners should already be included in other expenditure categories in
the CPI. Again, it is necessary to avoid double counting these expenditures. Landlords
may also have considerable overhead expenses that are associated with the managementof rental properties. These expenses can perhaps be grouped together with maintenanceexpenditures.
• The structure depreciation rate for rented dwelling units will probably be higher than
the rate for comparable owned dwelling units, since owners are likely to take better careof their property and will avoid property damage. This expected diﬀerence in the valueof depreciation should be deducted from the market rent that is applied to a comparable
owned home.
• The owners of rental properties need to charge a small premium to the rents that theyreceive from rented units in order to cover the loss of rental income due to vacancies.
This vacancy premium does not apply to the user cost of an owned unit and thus thecomparable market rent for an owned unit should be adjusted downward to account forthis vacancy factor.
• Insurance payments are included in market rents. However, in the CPI, insurance
payments are included in another category so the imputed insurance premiums should
be deducted from the market rent that is applied to a comparable owned home.
• The opportunity cost of capital for a landlord and for an owner may be diﬀerent. In
particular, the owner of a house may be risk averse and have a very low opportunity
cost of capital. A landlord who rents properties to tenants will have to include a risk
*112 On the stickiness of rents, see Shimizu, Nishimura and Watanabe (2010b)[99], Lewis and Restieaux (2015;72-75)[84] Suzuki, Asami and Shimizu (2018)[105] and Hill, Steurer and Waltl (2017; 9)[68]. Lewis andRestieaux label their three categories as (i) Occupied Let, (ii) Renewal and (iii) New Let. Their category(i) is a stock measure that includes all occupied rental units while their categories (ii) and (iii) match up
with categories (ii) and (iii) in the text above. Rents in categories (ii) and (iii) may be subject to rent
controls which means that rents in these categories do not reﬂect current opportunity costs.
*113 However, when constructing a rental price index for renters, rents for all 3 categories should be used.

51

premium in his or her cost of capital to account for possible downturns in the rental
market.
Consider the user cost formula for a dwelling unit that was deﬁned by the right hand side of(114) in section 15. Using the same notation for property n, deﬁne period t property value
for property n as:
Vtn ≡ PLtn Ltn + PStn (1 − δ)A(t,n)−1 Stn ;n = 1, . . . , N.

(121)

Deﬁne the period t, property n share of property land value as sLtn ≡ PLtn Ltn /Vtn and theshare for constant quality structure as sStn ≡ PStn (1 − δ)A(t,n)−1 Stn /Vtn for n = 1, . . . , N .
The using (114) and the above deﬁnitions the ratio of user cost to property value (or the
implied rent capitalization ratio) can be written as follows:
Rtn /Vtn = [rt − iLt ]sLtn + [rt − iSt + (1 + iSt )δ]sStn ;n = 1, . . . , N.

(122)

Recall that rt is a smoothed longer term opportunity cost of capital for period t, iLt is the longterm expected land price inﬂation rate, iSt is a long term expected structure price inﬂation
rate and δ is the geometric structure depreciation rate. The rent to capital value ratio deﬁnedby (122) or capitalization rate*114 does not take into account the complications that werediscussed above. Thus it is necessary to modify (122) to take into account these complications.
Deﬁne vt as the period t rate of expected loss of rental income due to vacancies (as a fraction of
period t capital value), deﬁne mtn as expected period t maintenance and overhead expenditures
for property n divided by the corresponding period t structure value,*115 deﬁne the land tax
rate τLtn as the ratio of land taxes paid by the owners of property n in period t to the imputed
land value PLtn Ltn and the structure tax rate τStn as the ratio of structure property taxes paidin period t for property n to imputed structure value, PStn (1 − δ)A(t,n)−1 Stn . Finally deﬁne
πtn as the ratio of insurance payments made in period t by property n to imputed structure
value, PStn (1 − δ)A(t,n)−1 Stn . Using the above discussion on complications to the standard
user cost model, it can be seen that a more meaningful rent to value ratio decomposition forproperty n in period t is given by the following modiﬁcation of (122) for n = 1, . . . , N :Rtn /Vtn = [rt − iLt + vt + τLtn ]sLtn + [rt − iSt + (1 + iSt )δ + vt + τStn + mtn + πtn ]sStn . (123)
If property tax payments are not a separate category in the CPI, then the appropriate user
cost for an owner of property n in period t, Utn , as a fraction of property value, Vtn , is equal
to the following expression:
Utn /Vtn = [rt − iLt + τLtn ]sLtn + [rt − iSt + (1 + iSt )δ + τStn ]sStn(124)

Note that the terms vt , mtn and πtn have been dropped from (124). Thus the diﬀerences
between (123) and (124) are equal to the following expressions for n = 1, . . . , N :Rtn /Vtn − Utn /Vtn = vt + [mtn + πtn ]sStn .

(125)*114 Crone, Nakamura and Voith (2000)[18] used hedonic techniques to estimate both a rent index and aselling price index for housing in the U.S. They also suggested that capitalization rates (i.e., the ratio of
the market rent of a housing property to its selling price) can be applied to an index of housing selling
prices in order to obtain an imputed rent index for OOH. As will be shown below, capitalization rates
are functions of many variables, some of which can change considerably over time. Also it will be seenthat capitalization rates for rented houses are not exactly appropriate as estimators for capitalization
rates for owned houses.
*115 Older structures will probably have higher mtn ratios.

52Thus looking at (125), it can be seen that simply applying the rent of a comparable rented
dwelling unit to an owned unit will overstate the appropriate user cost that should be applied
to the owned unit. However, the above computations did not take into account the likelihoodthat the depreciation rate for a rental property is greater than the corresponding depreciation
rate for a similar owned property. Thus let δO be the depreciation rate for an owned property
and suppose that 0 < δO < δ where δ is the depreciation rate for a comparable rented property.
Rewriting (123) in absolute form rather than in ratio form leads to the following expression
for the user cost value of rented property n in period t:
Rtn = [rt − iLt + vt + τLtn ]PLtn Ltn+ [rt − iSt + (1 + iSt )δ + vt + τStn + mtn + πtn ]PStn (1 − δ)A(t,n)−1 Stn .

(126)

Taking into account that the depreciation rate is diﬀerent, the corresponding user cost of asimilar owned property n in period t is the following one:
Utn = [rt − iLt + τLtn ]PLtn Ltn + [rt − iSt + (1 + iSt )δO + τStn ]PStn (1 − δO )A(t,n)−1 Stn . (127)
If δO is considerably smaller than δ, then PStn (1 − δO )A(t,n)−1 Stn will be considerably larger
than PStn (1 − δ)A(t,n)−1 Stn and thus in this case, it is likely that Utn will be larger than Rtn
for older properties. Thus the rental equivalence imputation for the services of a comparable
owned unit could be considerably smaller than the corresponding imputed long run user costfor the owned unit.*116
The user cost formulae deﬁned by (124)-(127) look rather complicated and they require information that may not be available to the statistician. Thus additional assumptions may haveto be made which allow approximate user costs for owned dwelling units to be calculated.
In situations where equivalent rental prices are not available, this may be the only feasible
method to value the services of OOH. For example, the European Union issued the following regulation in 2005 that gives guidance in forming estimates of the services of OOH whenequivalent rental prices are not available:“Under the user-cost method, the output of dwelling services is the sum of intermediate
consumption, consumption of ﬁxed capital (CFC), other taxes less subsidies on production and net operating surplus (NOS). For owner occupied dwellings, no labour input
is recorded for work done by the owners (1). Experience suggests that CFC and NOS
are the two largest items, each representing 30 to 40 % of output.
CFC should be calculated based on a perpetual inventory model (PIM) or other approved methods. A separate estimate for the owner-occupied residential buildings
should be available.
The net operating surplus should be measured by applying a constant real annual rateof return of 2.5% to the net value of the stock of owner-occupied dwellings at current
prices (replacement costs). The real rate of return of 2.5% is applied to the value of thestock at current prices since the increase in current value of dwellings is already taken
account of in the PIM. The same rate of return should be applied to the value of theland at current prices on which the owner-occupied dwellings are located.
The value of land at current prices may be diﬃcult to observe annually. Ratios of land
value to the value of buildings in diﬀerent strata may be derived from an analysis of thecomposition of the costs of new houses and associated land.” Eurostat (2005)[46].
*116 The algebra will be diﬀerent for diﬀerent models of depreciation but the same conclusion will follow.

53

To value the services of OOH in Iceland, the highly simpliﬁed user cost formula Ut = (rt∗ +δ)Pt
was used where Ut is the period t property user cost, rt∗ is a real interest rate (varied between
3.6% and 4.3%), δ is a property depreciation rate (set equal to 1.25%) and Pt is a period t
constant quality property price index.*117
The Oﬃce for National Statistics in the UK used the user cost formula Ut = (r + m + δ − i)Pt
to value the services of OOH where r is a rate of return which includes a risk premium, δ isa depreciation rate, m is the maintenance rate, i is the expected capital appreciation rate of
the unit and Pt is a period t property price index.*118
Returning back to the user cost formulae deﬁned by (126) and (127), there is another factor
which will tend to make the user cost valuation of the services of an owned dwelling unit muchbigger than the corresponding actual rental price: households who rent tend to be poorer thanhouseholds who own. Thus renters simply cannot aﬀord to rent high end housing units. Highend dwelling units that do rent will tend to rent for prices that are much less than their longrun user costs.*119 Thus in advanced countries, the rent to property value ratio for the more
expensive properties tends to be about one half the rent to property value ratio for the least
expensive properties.*120 Thus it is likely that the widespread use of the rental equivalence
approach to the valuation of the services of owner occupied housing results in a measures ofthe value of housing services which give much lower valuations than valuations based on longrun user costs.
There is one additional troublesome issue that has not been discussed thus far and that is
the issue of what to do with transfer costs. Transfer costs are the costs associated with thepurchase of a dwelling unit. These costs include transactions taxes, legal fees and real estateagent fees. These costs can be substantial. Thus when a household purchases a dwelling unit,
the ﬁnal cost of the purchase should include all of the associated transfer costs. According
to user cost theory, the appropriate valuation of the property at the end of the period shouldbe the value of the sale of the house after transfer costs. This viewpoint suggests that thetransactions costs of the purchaser should be immediately expensed in the period of purchase.
However, from the viewpoint of a landlord who has just purchased a dwelling unit for rental
purposes, it would not be sensible to charge the tenant the full cost of these transaction feesin the ﬁrst month of rent. The landlord would tend to capitalize these costs and recover them
gradually over the time period that the landlord expects to own the property. Thus takethe capitalized transfer costs that are charged to property n in period t and divide by total
property value Vtn to obtain the imputed property transfer cost ratio, λtn . The new rental
cost formula for rented unit n in period t, the counterpart to (126), becomes the following
formula:Rtn = [rt − iLt + vt + τLtn + λtn ]PLtn Ltn+ [rt − iSt + (1 + iSt )δ + vt + τStn + mtn + πtn + λtn ]PStn (1 − δ)A(t,n)−1 Stn .
*117 See Gudnason and Jonsdottir (2011;(128)148)[60]. Note that as in the case of Iceland, the depreciation
rate is applied to total property value and not to just the structure value. This may be an acceptableapproximation if the shares of land and structure in total property value remain roughly constant overtime.
*118 See Lewis and Restieaux (2015; 156)[84]. We have changed their notation to match up with our notation.
*119 Often high end houses that are not being used by their owners are rented out at prices that are far below
their user costs just so someone will be in the house to maintain it and deter theft and vandalism.
*120 See Heston and Nakamura (2011)[64]. Hill, Steurer and Waltl (2017; 8)[68] ﬁnd similar results forAustralia and Aten (2018)[2] ﬁnds similar results for the US. Shimizu, Diewert, Nishimura and Watanabe(2012)[97] found that user cost valuations for OOH in Tokyo were about 1.7 times as big as the equivalent
rent estimates.

54

From the viewpoint of an owner of a newly purchased dwelling unit, the owner does not
actually sell the unit in the next period; the owner holds on to the dwelling unit for average
periods that range from 10 to 20 years. Thus it is probably best to regard the transfer costs
as a ﬁxed cost that should be amortized over the expected holding period before the dwelling
unit is sold again. If this amortisation is appropriate, then the new user cost formula that isthe counterpart to (127) is the following formula which should be used to value the servicesof the owned unit if it is not rented out to tenants:
Utn = [rt − iLt + τLtn + λtn ]PLtn Ltn+ [rt − iSt + (1 + iSt )δO + τStn + λtn ]PStn (1 − δO )A(t,n)−1 Stn .

(129)

The above discussion indicates that it is not a straightforward matter to determine the conceptually correct rental equivalent price to value the services of an owned dwelling unit.*121

17 The Payments Approach
The ﬁfth possible approach to the treatment of owner occupied housing in a CPI, the payments
approach, is described by Goodhart as follows:“The second main approach is the payments approach, measuring actual cash outﬂows,
on down payments, mortgage repayments and mortgage interest, or some subset of theabove. ... Despite its problems, such a cash payment approach was used in the United
Kingdom until 1994 and still is in Ireland.” Charles Goodhart (2001; F350-F351)[?].
Thus the payments approach to owner occupied housing is a kind of a cash ﬂow approach tothe costs of operating an owner occupied dwelling. It consists mainly of mortgage interest
and principle payments along with property taxes. Imputations for capital gains, for the costof capital tied up in house equity and depreciation are ignored in this approach. This leads
to the following objections to this approach; i.e., it ignores the opportunity costs of holding
the equity in the owner occupied dwelling, it ignores depreciation and it uses nominal interest
rates without any oﬀset for inﬂation in the price of land and the structure. In general, thepayments approach will tend to lead to much smaller monthly expenditures on owner occupied
housing than the other 4 main approaches, except during periods of high inﬂation, when thenominal mortgage rate term may become very large without any oﬀsetting item for inﬂation.
One reason for implementing this approach is that it may be useful for indexing the pensions of
homeowners; i.e., as the cash costs of home ownership increase, it may be popular to increase
pensions to oﬀset these costs.*122 This line of argument has some validity but in recent years,
perhaps it is less compelling in many countries due to the ability of homeowners to draw ontheir equity with reverse mortgages and to postpone paying property taxes until the property
is sold. However, a cash ﬂow or payments approach to the valuation of the costs of homeownership may be useful for some users.*123*121 For a more comprehensive decomposition of the user cost formula for an owned dwelling unit with amortgage on the unit, see Diewert, Nakamura and Nakamura (2009)[34] and Diewert and Nakamura(2011)[33].
*122 Thus the UK still uses the payments approach to value OOH in its Retail Prices Index.
*123 Fenwick (2009)[?] (2012)[?] has argued strongly that statistical agencies responsible for consumer priceindexes should produce a range of indexes that suit diﬀerent purposes. Thus the payments approachto OOH could be produced by statistical agencies that provide multiple consumer price indexes to suitdiﬀerent purposes. However, the payments approach cannot serve as a reliable guide for pricing theservices of OOH.

55

In the following paragraphs, we outline in more detail under what circumstances it may be
useful to use the payments approach in the CPI.*124
The payments approach is more consistent with the traditional approach to CPI construction,
which is a carry-over from a time when the CPI was mostly used as a compensation tool. It
also has much to commend it from the point of view of public acceptability as changes in thecost of servicing the interest on loans that have been taken out to purchase owner-occupier
housing, following changes in interest rates, are reﬂected in the index. Home owners who facereal increases in costs from a rate rise, for instance, would expect to see this reﬂected in theircost of living as measured by an oﬃcial CPI. A user cost approach would reﬂect this increasein costs but a rental equivalence approach may not reﬂect the magnitude of the increase ininterest rates. It is counter-intuitive to home owners that under the “rental equivalence”
approach owner occupier housing costs can increase when house prices are falling and interest
rates are stable and more people are looking to live in rent accommodation thereby increasing
rents. The latter happened in the UK following the ﬁnancial crisis of 2008.*125 “Mortgage
interest” is more likely to be understood by the general public than “rental equivalence” and,unlike the latter, the index will directly reﬂect changes in house prices and interest rates.
From a technical viewpoint the payments approach measures the following direct expendituresby owners of property: mortgage interest payments; repayment of capital; large repairs (associated with depreciation from wear and tear as properties get older). There are no imputed
prices. Index compilers typically take every eﬀort to avoid the latter in other expenditurecategories in a CPI.
As with other approaches to the measurement of owner-occupier housing costs there are someproblems associated with the payments approach. A CPI should only relate to consumption
items and should exclude cash disbursements or expenditures which are in the nature of savings
or “investments”. The acquisition of a house can represent a substantial capital asset – a point
which emerges when comparing the position of owner-occupiers with tenants who rent. It cantherefore be argued that the capital element of mortgage repayments should be regarded as aninvestment or saving rather than consumption expenditure and should therefore be excluded
from the index. But the counter-argument is that the capital gains of owning a house which
appreciates in value, is of limited relevance when people have to bear such costs from current
income. The question also arises over whether the weight and price indicator should be net ofany tax allowances for mortgage interest payments. It is in accordance with the principle that a
CPI should be based on the amounts actually paid, that the weight and price indicator shouldboth be based on payments after tax relief. The biggest practical barrier to the adoption of
the payments approach is that it requires a large volume of data, which may or may not be
available to the compiler. The calculation of the price indicator for interest charges involves
two components: the rate of interest and the average amount of mortgage debt outstanding.
To calculate the average outstanding debt at any one point in time can be problematic as
it consists of a large number of individual debts, some from mortgages taken out recently
and others from mortgages taken out some time ago at historic prices and with some of thedebt paid back. Consistent with the ﬁxed basket approach, average payments are calculatedwith respect to a ﬁxed stock of new and existing mortgages (of various ages) equivalent tothose existing in the base period. Finally, it can be noted that the inclusion of owner-occupier
housing costs using the payments approach can increase the volatility of the index in periods
where interest rates frequently change.
*124 he following four paragraphs are due to David Fenwick.
*125 The user cost approach could also generate an increase in OOH under these circumstances:rates of property inﬂation could fall, which would lead to an increase in user costs.

56

expectedTo conclude this discussion of the possible uses of the payments approach, we note that theapproach chosen should align as closely as possible with the conceptual basis which bestsatisﬁes the principal purpose of the CPI, subject to adequate data being available to make areliable calculation. This applies, of course, to all expenditures covered in a CPI.
At this point, it is useful to review the three ways which can be used to measure consumption
expenditures. The following quotation from the Oﬃce for National Statistics (2010; 6)[90]:“Consumption expenditure can be measured in three ways which it is important todistinguish. These ways are:Acquisition means that the total value of all goods and services delivered during a given
period is taken into account, whether or not they were wholly paid for during the period.
Use means that the total value of all goods and services consumed during a given period
is taken into account.
Payment means that the total payments made for goods and services during a given
period is taken into account, whether or not they were delivered.
For practical purposes, these three concepts cannot be distinguished in the case of nondurable items bought for cash, and they do not need to be distinguished for many
durable items bought for cash. The distinction is, however, important for purchases
ﬁnanced by some form of credit, notably major durable goods, which are acquired at a
certain point of time, used over a considerable number of years, and paid for, at least
partly, some time after they were acquired, possibly in a series of installments. Housing
costs paid by owner-occupiers are an obvious example.”
In what follows, we will look at the problems associated with the three methods of valuation
in a number of speciﬁc cases.
Case 1 : The payment period coincides with the acquisition period. Let P1 be the acquisition
price for such a unit of a durable good in period 1. Then the acquisition price in period 1 isobviously P1 , the payments price is also P1 and the period 1 user cost price is p1 and its exact
form depends on the model of depreciation that is applicable for this particular durable good.
In other words, there are no problems in sorting out the three methods of valuation in thiscase.
Case 2 : The initial payment period coincides with the acquisition period but payments forthe purchase of the durable continue on for subsequent periods. Suppose that payments must
be made for T periods and the sequence of monetary payments is π1 , π2 , . . . , πT . Suppose
also that the sequence of expected one period ﬁnancial opportunity costs of capital for thepurchasing household is r1 , r2 , . . . , rT −1 . Then the discounted stream of payments, P1 , is theperiod 1 (expected) cost of purchasing the good where P1 is deﬁned as follows:
P1 ≡ π1 + (1 + r1 )−1 π2 + (1 + r1 )−1 (1 + r2 )−1 π3 + · · · + (1 + r1 )−1 (1 + r2 )−1 · · · (1 + rT −1 )−1 πT .
(130)
In this case, the acquisitions price for the durable good in period 1 is deﬁned to be P1 , thepayments price is π1 and the user cost will be determined using the appropriate depreciation
model where P1 is taken to be the beginning of the period price for the durable good. In
a subsequent period t ≤ T , the acquisitions price for the used durable good will be 0, thepayments price will be πt and the period t user cost value vt will be determined using theappropriate depreciation model for this type of durable good. If the useful life of the durable
good happens to equal T and if the period t payment is equal to the corresponding period tuser cost valuation vt for t = 1, 2, . . . , T , then obviously, the period t user cost valuation vt
57

will be equal to the observable period t payment πt .*126
There are problems associated with the computation of the P1 deﬁned by (130); i.e., in order
to compute P1 when the durable good is purchased during period 1, the sequence of future
payments πt has to be known and guesses will have to be made on the magnitudes of thesequence of expected nominal interest rates rt . However, the important point to be made here∑Tis that P1 deﬁned by (130) will be less than the simple sum of the πt , t=1 πt , provided thatthe nominal interest rates rt are positive.
Case 3 : The full payment for the good (or service) is made in period 1 but the services of thecommodity are not delivered until period t. Let the period 1 payment be π1 as usual. Thus thesequence of payments associated with the purchase of the commodity under consideration is
π1 for period 1 and 0 for all subsequent periods. The acquisition of the commodity does nottake place until period t but the appropriate acquisition price Pt is not the period 1 payment,
π1 , but the following escalated period 1 price:Pt ≡ (1 + r1 )(1 + r2 ) · · · (1 + rt−1 )π1 .

(131)

The logic behind this valuation is the following one. During period 1 when the product was
paid for, the payment could have been used to pay down debt (at the interest rate r1 ) or thepayment could have been used to invest in an asset that earned the rate of return r1 . Thus
after one period, the opportunity cost of the investment in the pre-purchased product has
grown to π1 (1 + r1 ), after 2 periods, the opportunity cost has grown to π1 (1 + r1 )(1 + r2 ),. . . ,and by period t when the good or service is acquired, the opportunity cost has grown toπ1 (1 + r1 )(1 + r2 ) · · · (1 + rt−1 ), which is (131). The important point to be made here isthat Pt deﬁned by (131) will be greater than the period 1 prepayment, π1 , provided thatthe nominal interest rates rt are positive. Since the product has not been acquired by thehousehold for periods 1, 2, . . . , t − 1, the corresponding user cost valuations, v1 , v2 , . . . , vt−1
should be set equal to 0. However, when period t is reached, “normal” user costs can becalculated for durable goods using the Pt deﬁned by (131) as the beginning of period t priceof the durable, assuming that the form of depreciation is known.
Prepayment for services or durable goods is widespread; e.g., trip and hotel reservations madein advance and paid for in advance are service examples and prepayment for condominium
units that are under construction is a durable good example.
Case 4 : The good or service is acquired in period 1 but is not paid for until period 2. In thiscase, the sequence of payments is 0, π2 , 0, . . . , 0. The commodity is acquired in period 1 andthe appropriate period 1 acquisition price is P1 deﬁned as follows:
P1 ≡ (1 + r1 )−1 π2 .

(132)

The justiﬁcation for this acquisition price runs as follows: The purchasing household lays
aside the amount of money P1 to buy the product in period 1. This money is invested andearns the one period rate of return r1 . Thus when period 2 comes along, the household has
P1 (1 + r1 ) = π2 which is just enough money to complete the purchase in period 2. Thus P1 isan appropriate period 1 acquisitions price. If the commodity is a durable good, then assuming*126 The period t user cost valuation vt for a unit of the durable good that is t periods old can be converted

into an equivalent amount of a new unit of a durable good if the geometric or one hoss shay model ofdepreciation is applicable for the durable good under consideration. Otherwise, units of the durable goodof diﬀerent ages at the same point in time need to be aggregated using an index number formula.

58

that the form of depreciation is known, P1 deﬁned by (132) can be used as the beginning of
period 1 price for the period 1 user cost and the entire sequence of user costs can be calculated.
This form of pricing is used as a way of oﬀering lower prices for a wide variety of products.
A particular application of this model to a service is the use of credit cards to purchaseconsumption items. A household that pays its balance owing on time can avoid interest
charges and thus can postpone payment for its household purchases for up to one month inmany cases.*127
If interest rates are very low, then statistical agencies may well ﬁnd it is not worth taking
into account the above reﬁnements. However, if nominal interest rates are high, it may benecessary to make some of the above adjustments.

18 Summary and Conclusion
It is clear that constructing constant quality price indexes for consumer durables is not as conceptually simple as constructing price indexes for nondurables and services where the matchedmodel approach can guide index construction. The fundamental problem of accounting arises
when constructing a price index for the services of a durable good: imputations will have to bemade in order to decompose the initial purchase cost into period by period components overthe life time of the durable good. The method of imputation will involve assumptions which
may not be accepted by all interested parties. In spite of this diﬃculty, it will be useful for statistical agencies to construct analytical series for the services of long lived consumer durables
that can be made available to the public. This will meet the needs of diﬀerent users.*128
When constructing property price indexes based on sales of properties, there is another factor
that reinforces the argument for multiple price indexes: when transactions are sparse, property
indexes based on the sparse data can be very volatile. Thus for some purposes, it may be useful
to construct a smoothed index (that is revised for a certain number of months) in addition toa volatile real time index.*129
For non-housing consumer durables, at present, statistical agencies produce consumer price indexes based on the acquisitions approach. This type of index is useful for measuring consumer
price inﬂation based on market transactions, with minimal imputations (except for possible
quality change). In addition to this standard index, statistical agencies should produce supplementary indexes based on the user cost approach in order to more accurately measure theﬂow of services generated by stocks of consumer durables.*130
The valuation of the services of housing is very diﬃcult due to the fact that housing services
are unique: the location of each dwelling unit is unique and the location aﬀects the land pricecomponent of the property and thus aﬀects rents and user costs. Moreover, the structurecomponent of housing does not remain constant over time due to depreciation of the structure*127 However, a household that does not pay oﬀ its balance owing in a timely fashion will ﬁnd itself in Case

3 above.

*128 Hill, Steurer and Waltl (2017)[68], using Australian data, found substantial diﬀerences using the threemain approaches to the valuation of OOH. This emphasizes the need for statistical agencies to produce
estimates for all three approaches if possible.
*129 See Rambaldi and Fletcher (2014)[93] on various smoothing methods that could be used. Diewert andShimizu (2017b)[38] suggested a very simple method which worked well in their empirical application.
*130 The rental equivalence approach could be used for durables that are rented or leased but typically, mostconsumer durables are not rented. Depreciation rates will in most cases be based on educated guesses.
Durable stock estimates can be made once depreciation rates have been determined. The current valueof household stocks of consumer durables should also be constructed and added to household balance
sheets.

59

and to renovation expenditures. Various methods that can deal with these diﬃculties (to somedegree at least) were explained in sections 12-16. The details of the methods are too complex
to summarize here but the suggested methods based on various hedonic regression models
have been applied and oﬀer possible ways forward.
For Owner Occupied Housing, the three main approaches should be implemented. There are
two possible versions for the acquisitions approach: (i) construct a price index for the purchaseof new dwelling units in an inclusive basis, including the price of land and (ii) exclude land
cost from the purchase cost. The latter index should be well approximated by a construction
cost index (with appropriate margins added for developer margins). The inclusive index will
be useful for new house buyers, who have to pay for the land plot as well as the new structure.
A rental equivalence price index for the services of OOH should also be constructed. For
many countries, such an (implicit) index is already available as part of the national accounts
valuation for the services of OOH.*131 A user cost index for the services of OOH should also
be constructed since the user cost valuation for the services of a high end dwelling unit will
typically be much greater than the corresponding price that the unit could rent for.*132 If therental equivalent rent and user cost for an owned unit are constructed and are of the same
quality, then applying the opportunity cost approach to the valuation of the services of theowned unit is appropriate.
For rented housing, the measurement problems are perhaps not so severe; monthly or weekly
rents can be observed for the same rental unit and so it would seem that the usual matchedmodel methodology could be applied in this situation. However, an index based on the matchedmodel methodology and normal index number theory will generally have an upward bias dueto the neglect of depreciation or a lowering of quality due to the aging of the structure. In
order to deal with this bias, it will in general require a hedonic regression approach with age
as one of the explanatory variables.
We will conclude by noting some speciﬁc recommendations that emerge from the paper:• There are three main approaches for the treatment of consumer durables in a CPI: theacquisitions approach, the rental equivalence approach and the user cost approach.
• The acquisitions approach is suitable (for most purposes) for durable goods with arelatively short expected useful life.
• The acquisitions approach is particularly useful for central bankers who want consumer
inﬂation indexes that are largely free from imputations.
• The acquisitions approach provides an index for purchases of a durable good and thisindex is a required input into the construction of a user cost index.
• The remaining two approaches are useful for measuring the ﬂow of services yielded byconsumer durables over their useful lives.
• At present, only the ﬂow of services for OOH is estimated by national statistical agencies (using the rental equivalence or user cost approaches) because this information isrequired for the international System of National Accounts.
• The acquisitions approach will substantially understate the value of the service ﬂow
from consumer durables that have relatively long lives. Hence at least one of the rental
equivalence or user cost approaches should be implemented by statistical agencies for*131 However, the equivalent rents should be based on new contract rents if possible in order to provide acurrent opportunity cost for using the services of an owned dwelling unit; recall the discussion on thispoint in section 16.
*132 Recall the evidence on this point in Heston and Nakamura (2011)[64].

60durables with long lives.*133 Examples of long lived durables are automobiles andhousehold furnishings.
• The rental equivalence approach to the valuation of the services provided by consumer
durables is the preferred method of valuation when rental or leasing markets for theclass of durables exist, because, in principle, no imputations are required to implement
this method.*134• However, when rental markets for the durable good under consideration are thin or do
not exist, then the user cost approach should be used to value the services of the durable
good.
• The user cost approach requires the construction of a price index for new acquisitionsof the durable. It also requires a model of depreciation and assumptions about theopportunity cost of capital and about expected asset inﬂation rates. Thus the user cost
approach necessarily involves imputations.
• In order to avoid unnecessary volatility in the user costs, long run expected asset inﬂation rates should be used in the user cost formula.*135• Rental markets for high end dwelling units are generally nonexistent or very thin andhence, it may not be possible to use the rental equivalence approach for high end OOH.
Even if some rental information on high end housing units is available, usually theserents are far below the corresponding user costs.
• The “true” opportunity cost for using the services of a consumer durable is the maximumof its rental price (if it exists) and its user cost. Thus the use of the rental equivalence
approach to value the services of a high end housing unit will understate the “true”
service ﬂow by a substantial amount.*136• In order to construct national balance sheets and to measure national multifactor productivity, it is necessary to decompose the selling prices of dwelling units into structure
and land components. This can be done for both detached housing and condominium
units using hedonic regression techniques; see sections 12 and 13 above. This decomposition is also required in order to construct accurate user costs for housing units since
depreciation applies to the structure but not to the land component of the property.
• When constructing price indexes for rental housing, statistical agencies need to makean adjustment to observed rents for the same unit for depreciation of the structure andpossible improvements to the structure.
• When using observed rents to measure the service ﬂow for comparable owned properties,
statistical agencies should use new contract rents to evaluate the service ﬂow for theowned units since rents for continuing tenants may be sticky and not reﬂect current
opportunity costs.
• When constructing user costs for OOH, statistical agencies need to avoid double count*133 If the acquisitions approach is used in the headline CPI, the alternative approaches can be published as

experimental or supplementary series.

*134 However, for housing, the “comparable” rental property may not be exactly the same as the owned

unit. Moreover, the observed rents may include insurance services and the services of some utilities andpossibly furniture. It will be diﬃcult to extract these costs from the observed rent.
*135 The long run asset inﬂation rate over the past 20 or 25 years or the long run rate of inﬂation in housing
rents could be used to predict future asset inﬂation rates. Many other prediction methods could be
used; see for example Verbrugge (2008)[106]. However, the focus should be on predicting long run asset
inﬂation rather than period to period inﬂation.
*136 Long run user costs and rents will tend to be approximately equal to each other for lower end housing
units since this type of housing unit will be built by property developers who provide rental housing andthey need to set rents that are approximately equal to their long run user costs. However, short run
dynamics can cause user costs and rents to diverge even for lower end housing units.

61

ing of some housing related costs that may appear elsewhere in the CPI such as insurance costs. Similar double counting problems may arise with housing rents, which may
include the services of some utilities or furniture and of course, the housing rent will
include insurance costs. In principle, these associated costs should be deducted fromthe observed rent and placed in the appropriate classiﬁcation of the CPI. In practice,
this is a diﬃcult imputation problem.
• A variant of the acquisitions approach is sometimes applied to OOH. This variant excludes the land component of the purchase of a new house. Thus this variant reduces
to a construction cost index for housing with some allowance made for builders’ proﬁt
margins. This variant generates valuations for OOH that are far below the comparable
rental equivalent and user cost valuations. It is diﬃcult to justify the use of this variant
in a CPI.*137
Which of the three main methods for valuing the purchase of a consumer durable should
be used for indexing pensions or indexing salaries for consumer inﬂation? This is a diﬃcult
question to answer. If we start out with the idea that we want a national consumer priceindex, then if there were no durable goods, a national acquisitions price index would be thetarget index. But it is not clear that this is the “correct” price index once we recognize theexistence of consumer durables: an acquisitions index does not recognize the imputed costsof previously purchased consumer durable goods. Thus in order to deal with this diﬃculty,
we need to move to a rental equivalence index or a user cost index if rental markets are thin.
But if a national index based on say the rental equivalence approach were used to determine
pension payments for veterans or retired civil servants or for employees in an industry, theresulting payments do not take into account that diﬀerent households have diﬀerent holdingsof consumer durables (housing in particular) and they do not need to be compensated fortheir consumption of existing holdings. There are additional complications that need to beaddressed:• If the goal is to maintain the purchasing power of a certain group of households (such as
retirees or veterans), then an appropriate index needs to be constructed for the relevantgroup.
• The relevant group may live in diﬀerent regions of the country and so in principle,
separate indexes need to be constructed for each region by group.
• The index may be a plutocratic one (where well oﬀ members of the group get a higher
weight in the index) or a democratic one (where each individual gets an equal weightin the index).
The resolution of these diﬃculties is not available at present.

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