WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 109Revista de Análisis Económico, Vol. 16, Nº 2, pp. 109-135 (Diciembre 2001)
VIVIANA FERNANDEZ*
Universidad de Chile
WHAT DRIVES REPLACEMENT OF DURABLE GOODS
AT THE MICRO LEVEL?
* Department of Industrial Engineering (DII). Email: vfernand@dii.uchile.cl. The author wishes to
thank two anonymous referees for their extensive and insightful comments. Pablo Rodríguez
provided able research assistance. Funding was provided by an institutional grant from the Hewlett
Foundation to the Center for Applied Economics (CEA) at DII. All remaining errors are the
author’s.
Abstract
Technological innovations have contributed over the years to an increas-
ing stock of durable goods -those products that are not immediately con-
sumed but provide a stream of services over a long period of time.
Indeed, virtually every household in a modern economy owns a refrig-
erator, a personal computer and an automobile. Given the inter-tempo-
ral nature of replacement decisions, the existing literature has resorted
to the technique of dynamic programming, and most recently to the theory
of stochastic processes.
This article focuses on micro replacement decisions. We survey some
representative models of the recent literature, and discuss their empiri-
cal testability. In addition, we study replacement of home appliances in
the United States, and construct a test statistic that leads to conclude
that replacement decisions might be correlated across appliances. Fi-
nally, we enrich our analysis by developing a theoretical model in which
replacement decisions are interdependent.
110 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
I. Introduction
Technological innovations have contributed over the years to an increasing
stock of durable goods –products that are not immediately consumed but provide
a stream of services over a long period of time. Indeed, virtually every household
in the United States, and to a great extent in the rest of the world, owns or has
access to a microwave oven, a cloth washer, a computer, among many other durable
goods (see Figures 1A and 1B). Despite the rich theoretical body of knowledge
existing in different fields (e.g., economics and operations research) to analyze
durable goods purchases, only in the past few years have applied researchers
succeeded in identifying the forces behind replacement of durable goods.
Given that time plays an essential role in replacement decisions, dynamic
programming has naturally arisen as an adequate mathematical tool to tackle the
replacement problem (e.g., Beckmann, 1968; Kamien and Schwartz, 1971;
Bertsekas, 1976; Sargent, 1987). The most recent literature has also resorted to
the theory of stochastic processes, and characterized the physical decay of a du-
rable good as a Markov process in either discrete time or continuous time (e.g.,
Rust, 1985, 1987; Ye, 1990; Dixit and Pindyck, 1994; Mauer and Ott, 1995; Huei
Yeh, 1997).
Source: U.S. Department of Energy.
FIGURE 1A
PENETRATION OF HOME APPLIANCES
8
35
74
5961
43
14
66
34
50
8
71
3335
75 77
83
0
10
20
30
40
50
60
70
80
90
M
ic
ro
w
aiv
e
O
ve
n
D
ish
w
as
he
r
W
at
er
be
d
H
ea
te
rs
Cl
ot
h
W
as
he
r
Cl
ot
h 
dr
ye
r
Fr
ee
ze
rs
Appliance
Pe
rc
en
t o
f H
ou
se
ho
ld
s
1978
1987
1997
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 111
However, even though enormous progress has been made on the theoretical
ground, most empirical studies of acquisition and replacement of durable goods
do not derive from a consumer or firm’s optimization process. Instead, they present
ad-hoc statistical models developed from the techniques of discrete choice and
duration analysis. Exceptions, among others, are the work of Dubin and McFadden
(1984), Rust (1987), Caballero and Engle (1991, 1999), Cooper, Haltiwanger and
Power (1999), Lai, Leung, Tao and Wang (2000), and Martin (2001).
The aim of this article is threefold. First, to present some micro replacement
models – which are somehow representative of the most recent literature, and dis-
cuss their empirical testability. In addition, to briefly refer to the state of the art
of demand for durable goods modeling in the field of macroeconomics. Second,
to study replacement of home appliances in the United States – particularly, re-
frigerators and water heaters, and test whether replacement decisions might be
correlated across appliances. Finally, to develop a richer framework where house-
holds do not replace a durable good in isolation, but where they consider replac-
ing several items simultaneously.
The main contributions of this article are the following. First, to give an
overview of how the micro and macroeconomists have tackled the replacement
problem. Second, to present an empirical application where the interdependence
of replacement decisions is tested. Finally, to obtain an empirical testable model
for multiple replacement decisions. The author of this paper is not aware of any
study of similar characteristics.
FIGURE 1B
NUMBER OF PCs BY ANNUAL HOUSEHOLD INCOME IN THE UNITED STATES, 1997
Source: U.S. Department of Energy.
8
15
33
51
2 2 4
14
0
10
20
30
40
50
60
Less than
$ 10,000
$ 10,000-
$ 24,999
$ 25,000-
$ 49,999
$ 50,000 or
more
Number of PCs
Pe
rc
en
t o
f H
ou
se
ho
ld
s
1 PC
2 or more PCs
112 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
This article is organized as follows. Section II goes through micro models by
Rust (1985, 1987), Ye (1990), and Mauer and Ott (1995). Section III briefly dis-
cusses new developments in the demand for durable goods in macroeconomics
and other fields. Finally, Section IV focuses on replacement of home appliances.
Specifically, Section 4.1 studies replacement of refrigerators and water heaters,
using data from the U.S. Residential Energy Consumption Survey (RECS). Sec-
tion 4.2 develops a test statistic for interdependence of replacement decisions, and
applies it to the estimation results of Section 4.1. Section 4.3 develops a theoreti-
cal model for multiple replacement decisions. Finally, the main conclusions are
presented.
II. Replacement Decisions in a Dynamic Context
The models surveyed in this section deal with replacement decisions under
uncertainty in an infinite-time horizon. In particular, we go through some recent
papers where the replacement problem has been tackled by resorting to the theory
of stochastic processes and stochastic calculus: Rust (1985, 1987), Ye (1990), and
Mauer and Ott (1995). Rust studies the existence of a stationary equilibrium in a
market for a durable asset, which physically deteriorates according to a discrete-
time Markov process. A similar set-up, but in continuous time, is postulated by
Ye. Mauer and Ott in turn show that uncertainty about the arrival of technological
innovation may lead to a significant decrease in replacement investment.
We should point out at the outset that the set of models presented in this
section is by no means exhaustive. However, these models share some interesting
features. In particular, all of them determine the optimal replacement time by the
stopping time technique, and assume that equipment physical decay can be char-
acterized as a Markovian process. The concept of stopping time will be particu-
larly useful to the empirical application presented in Section IV. A more complete
overview of micro and macro models developed to analyze the demand for du-
rable goods is presented in Section III.
2.1 Some micro replacement models
Early attempts to analyze the replacement problem have modeled the optimal
time until preventive maintenance of a machine must be performed. For example,
Beckmann (1968) focuses on a firm that each period decides to either spend on
preventive maintenance that will make its current equipment “as good as new;” or
to wait until failure occurs, and purchase new equipment. Another paper along
the same lines is Kamien and Schwartz (1971).
The most recent literature introduces randomness in the replacement problem
by assuming that the physical condition of the piece of equipment deteriorates
according to some continuous or discrete stochastic process. For example, Rust
(1985, 1987) assumes that a durable good can be described as a discrete Markov
process, whose physical condition at time t is described by a nonnegative real
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 113
number.1 Continuous Markov processes are analyzed by Ye (1990) and Mauer
and Ott (1995), among others.
Specifically, Rust (1985) studies the existence of a stationary equilibrium in
a market for a stochastically deteriorating durable asset. His model tracks the
trading process of a durable good from its production in the primary market
through its sequence of owners in the secondary market, until it is scrapped.
Each durable good is represented as a discrete time Markov process whose
state at time t is described by zt, a nonnegative real number. The level of zt
indicates the degree of physical deterioration of the piece of equipment. The
consumer holds at most one durable per period over an infinite time horizon,
and chooses an optimal durable selection and replacement policy to maximize
the expected utility of owning an infinite sequence of durable goods.
At the beginning of each period, the consumer faces two choices: either to
continue with his/her current piece of equipment or to scrap it and replace it.
Under a set of assumptions, Rust shows that the optimal replacement policy
takes the form: “replace if zt is greater or equal than z*, an optimal stopping
barrier; do not replace otherwise”. Furthermore, the author shows the existence
and uniqueness of a stationary equilibrium, in which the distribution of asset
lifetimes is the first time passage distribution to the optimal stopping barrier z*.
Equilibrium rental rates and durable prices are shown to embody the functional
form of population distribution of preferences and technological features of
durable goods.
In latter work, Rust (1987) develops a statistical specification for a model
of bus engines replacement, similar in nature to that briefly described above.
Based upon data for the time-period December 1974-May 1985, the author tests
whether the decisions on bus engine replacement of the “Madison Metropolitan
Bus Company” can be described by an optimal stopping rule.
Rust’s base model assumes that the state variable, zt, is the accumulated
mileage since last replacement on the bus engine at time t, and expected per
period operating costs are given by an increasing, differentiable function of
zt, c(zt, ?1), where ?1 is some parameter. Each month, the following discrete
decision is faced: (i) perform maintenance on the current bus engine and incur
operating costs c(zt, ?1), or (ii) scrap the old bus engine for P , install a new (or
rebuilt) bus engine at cost P , and incur operating costs c(0, ?1). If it denote the
replacement decision at t, it=0 (keep), it=1 (replace), then the stochastic process
governing {it, zt} is the solution for the following regenerative optimal stopping
problem2:
V z E u z f zt
j t
j j t
j t
? ? ?( ) sup ( , , )=
?
??
?
??
?
=
?
?? 1 (1)
where the agent’s utility function, u, is given by:
114 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
u z i
c z if i
P P c if it t
t t
t
( , , ) ( , )( ( , ))?
?
?1
1
1
0
0 1
=
? =
? ? + =
??? (2)
and ? is an infinite sequence of decision rules given by ?={ft, ft+1,...}, where
each ft specifies the replacement decision at time t as a function of the entire
history of the stochastic process it = ft (zt, it–1, zt–1, it–2, zt–2,...}. The expectation in
(1) is taken with respect to the controlled stochastic process {zt}, whose probabil-
ity distribution is defined from ? and the transition probability p (zt+1|zt, it, ?2):
p z z
z z
z
if i z z
if i z
otherwise
t t
t t
t
t t t
t t( , , )
( ))
)
,
,+
+
+
+
+=
??
??
??
= ?
= ?1
2 2 1
2 2 1
1
1
0
0
1 0 i  
exp(
exp(   
            
t 2?
? ?
? ? (3)
Expression (3) states that, if the decision is to keep, then accumulated mile-
age zt+1 is drawn from the exponential distribution, 1–exp{?2(zt+1–zt)}. If the
decision it is to replace, zt regenerates to state 0, and then zt+1 represents a draw
from the exponential distribution 1–exp(?2(zt+1 – 0)).
The function V?(zt) is the value function and is the unique solution for
Bellman’s equation given by:
V z max i EV zt i C z t tt t? ?? ?( ) , , ) ( , ))( )= +?  (u(z  it t1 (4)
where C(zt)={0, 1}, and EV z V y p dy zt t? ? ?( , ) ( ) ( , , ) i  i  t t 2= ??0 .
The solution for this maximization problem is given by the optimal stationary
Markov replacement policy ?=(f, f, ...), where f is given by:
i f z
if z
if z
replace
do not replacet t
t
t
= =
???
?
<
( , , ) ( , )( , )? ?
? ? ?
? ? ?1 2
1 2
1 2
1
0
   
       
   
(5)
and ? (?1, ?2) represents a threshold value of mileage or the optimal stopping
barrier.
Rust’s data refute the assumption of monthly mileage data being exponen-
tially distributed. Therefore, he considers a class of more general dynamic dis-
crete choice models, which do not necessarily have a closed-form solution for the
agent’s stochastic control problem. In order to estimate such models, via maxi-
mum likelihood, Rust resorts to a “nested fixed point” algorithm. He concludes
that this algorithm can be a practical, efficient, and a numerically stable method,
and that the data seem consistent with his regenerative optimal stopping model.
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 115
Similar in nature to Rust’s (1985, 1987), Ye’s (1990) replacement model
assumes that the instantaneous maintenance and operation cost increases stochas-
tically with physical deterioration. One appealing feature of Ye’s set-up is that it
gives rise to a parsimonious structural model that can be fitted to real data. We
illustrate this point in Section 4.1 by presenting replacement models of refrigera-
tors and electric water heaters.
Ye assumes that in every instant of time the consumer or firm must decide
either to continue paying a rising maintenance and operation cost for the deterio-
rating piece of equipment; or, to sell it in the secondary market, and pay a fixed
cost to purchase a new piece of equipment with a guaranteed low initial mainte-
nance and operation cost. The objective function in this model is the expected
total discounted cost of maintenance and operation as well as of purchasing. As
in Rust’s work, the optimal replacement rule is defined as a stopping barrier.
The instantaneous maintenance and operating cost is represented by xt. This
may also be indicative of the state of the equipment. In particular, as in Rust’s
set-up, a higher xt indicates a more physically deteriorated piece of equipment.
The evolution of xt is described by an arithmetic Brownian motion with constant
drift, b, and instantaneous volatility, ?, where b>0 and ??0:
dxt = b dt + ? dWt (6)
where dWt represents an increment of a standard Wiener process.
The expected discounted total cost of obtaining the required service from this
piece of equipment is given by:
K x E e x ds x xrs s( ) = =[ ]???0 0 (7)
where xs evolves according to (6), r is the discount rate, and x0 represents the
state of the piece of equipment at time zero, which does not necessarily equal that
of a new one, x*.
The installation cost of new equipment is a fixed amount, ˜C, and the scrap
value of the previous equipment is zero. When x reaches x, an upper barrier,
replacement takes place and the following condition is satisfied:
K( x)  = ˜C  + K(x*) (8)
That is, the total cost right before replacement, K( x), equals the total cost after
replacement, K(x*), plus the cost of installing a new piece of equipment, ˜C.
The function K(x) is assumed bounded to avoid the problem of explosive
behavior:
lim K xx?? < ?( ) (9)
Ye shows that the solution of K(x) is:
116 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
K x e
e e
C x x
r
x
r
b
r
x
x x
( ) ˜
*
*
=
?
?
?
?
??
?
??
+ +
?
? ? 2 (10)
where ? is the positive root of the characteristic equation (1/2)?2p2+bp– r=0.
The  optimal  upper  barrier  is  unique,  and  can  be  found  from  the  condition
K x' ( ) := 0
1 + ? (r ˜C +x*– x) = exp(? (x*– x)) (11)
In addition, Ye shows that the total operation cost, K(x), at the optimal upper
barrier, is increasing in ˜C, x*, and b, and decreasing in r. The derivatives of x
with respect to ˜C, a, b, and x*, and r are, however, generally ambiguous.
An issue that is tackled neither by Rust or Ye is the existence of technologi-
cal change. This can be an important determinant of replacement for some du-
rable goods, such as personal computers3 and automobiles. Based upon a theoreti-
cal framework similar in nature to Rust and Ye’s, Mauer and Ott (1995) analyze
equipment replacement in the presence of uncertainty about the arrival of techno-
logical innovation. Their base model considers a firm that operates a machine that
produces a fixed level of output for a given maintenance and operation cost. The
before-tax cost, Ct, evolves according to a geometric Brownian motion:
dCt = ?Ctdt + ˜?CtdWt (12)
where ? and ˜?  are the instantaneous drift and the volatility rate, respectively,
with ?>0 and ˜?  ?0, and dWt represents the increment of a standard Wiener
process. All equipment has the same initial maintenance and operation cost CN>0,
which subsequently evolves according to (12).
The net purchase price of new equipment is P(1–?), where ? ? [0,1) denotes
the investment tax credit, and P is the purchase price of a new piece of equip-
ment. For tax purposes, it is assumed that a piece of equipment depreciates expo-
nentially over time at the rate ? ?0. Thus, if the piece of equipment was pur-
chased at time zero, its remaining book value at time t is given by P(1–?)e–?t.
For convenience, the elapsed time is modeled as a function of Ct. As a proxy for
t, the authors use the expected first passage time from CN to Ct, E(t). For a geo-
metric Brownian motion, this is given by:
E t C
C
t
N
( )
˜ /
ln=
?
?
??
?
??
1
22? ?
(13)
Using this result, the tax book value can be approximated by P(1–?)(Ct/CN) –?/Z,
with Z/?– ˜?2 /2>0. Then the depreciation tax shield of the machine over the time
interval [t, t+dt] equals:
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 117
?? ?
?
P C
C
dtt
N
Z( )1? ???
?
??
?
(14)
where ? ? [0,1) is the corporate tax rate.
At some unknown level of Ct, C , the firm discontinues operation of the piece
of equipment, and replace it with a stochastically equivalent one. If V(C) repre-
sents the discounted expected value of the after-tax cost from the optimal replace-
ment policy, then:
V C min E e C P C
C
dt C CC
rt
t
t
N
Z( ) ( ) ( )= ? ? ? ???
?
?? =
?
?
???
?
?
???
?
??
??
?
??
??
?
?
?
?0 01 1? ?? ?
?
(15)
where Ct evolves according to equation (12), r denotes the discount rate4, and Co
is the state of the piece of equipment at time zero.
For given values of the parameters r, ?, ˜?2 , ?, ?, CN and P, and a functional
specification of the relation between salvage value and cost, a solution of C  can
be obtained (see Mauer and Ott for details). On the other hand, the expected
replacement cycle, T  (years), can be computed, once C  is determined:
T C C C
C
N
N
=
?
?
?
?
?
?
??
?
??
?
?
???
?
?
???
?
ln( ) ln( )
˜ /
˜
( ˜ / )
˜
? ?
?
? ?
?
?
2
2
2 2
1 2
2 2 2
1
2
(16)
This expression yields the mean time it takes for Ct to reach C , conditional
on having started at CN.
Mauer and Ott carry out some sensitivity analysis on the optimal replacement
policy. They specify the relation between salvage value and cost as S(Ct)=? Ct?1,
with ?>0. They find, for example, that an increase in the volatility of equipment
cost ˜? , makes the firm replace less often.5 Hence, C , T  and V(CN) increase.
Similarly, for an increase in the price of new equipment, P.
Mauer and Ott also explore the effect on the firm’s replacement decisions of
uncertainty about a technological change that lowers the initial maintenance and
operation cost of new equipment, CN. It is assumed that the technological break-
through follows a Poisson process. Although the firm will not replace the existing
equipment right after the innovation takes place, the value function V(.) will change
because the firm rationally anticipates that at the next replacement CN will be
lower.
118 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
The authors carry out some simulations to measure the impact of changes in
CN and the parameter of the Poisson process ? on the expected replacement cycle.
They find that, for a technological breakthrough that lowers the initial mainte-
nance and operating cost by 10 percent, a jump of ? from 0 to 0.5 increases the
expected replacement cycle by over 40 percent. Intuitively, the firm keeps the
deteriorating equipment longer as ? increases, hoping that technological uncer-
tainty about a reduction in initial operating and maintenance cost will be rapidly
resolved.
III. Other Studies on the Demand for Durable Goods
Given that durable goods provide a stream of services over time, the litera-
ture has typically modeled the demand for these goods in the context of an inter-
temporal utility maximization process. An example of such an approach is Parks
(1974). The author develops a continuous-time model in which at t=0 an indi-
vidual must choose a consumption and investment path for his/her indefinite fu-
ture, given a certain income stream and known price paths for a perishable con-
sumer good and for a durable good.6 Other references of this sort of models are
Malcomson (1975) and van Hilten (1991) – an extension of Malcomson’s work.
The most recent literature has postulated that the demand for durable goods can
be modeled as a dynamic programming problem, in which changes in the stock
of capital take place at optimal stopping times, as previously discussed.
In the field of macroeconomics, the demand for durable goods has been the
focus of several articles. In particular, one topic that has been extensively studied
is the impact of transaction costs on the frequency of purchase.7 Early studies
postulated that changes in the aggregate stock of capital could be well explained
by the stock adjustment model. Specifically, this assumes the existence of qua-
dratic adjustment costs, so that the change in the stock of capital in period t is a
fraction ? of the gap between the current and the desired stock. Bar-Ilan and
Blinder (1992) point out that this approach has two flaws. First, empirical studies
have come up with estimated values of ? that seem improbably low to represent
speed of adjustment. Second, the assumption that marginal adjustment costs are
zero at zero and increasing thereafter seems, in general, hard to believe.
In the last few years the literature has moved away from the stock adjustment
model and emphasized the lumpiness of expenditure on durable goods (e.g., Bar-
Ilan and Blinder, 1992; Caballero and Engel, 1991, 1999; Caballero, 1994; Coo-
per, Haltiwanger, and Power, 1999). In particular, individuals and firms seem to
update their durable stock infrequently, and when they do so, their expenditures
are large.
One possible explanation is that, durable goods are replaced according to an
(S, s) policy (see, for example, Caballero and Engel, 1991, 1999). That is, when
the durable stock depreciates to some lower bound, s, a purchase is made to
increase the durable stock to S, the upper bound. If the stock remains above s, it
is optimal to do nothing. And, therefore, agents’ behavior is strictly inertial within
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 119
the band (S, s). In the limit as transaction costs vanish, the levels of s and S
should coincide. An alternative, but somehow similar, approach is presented by
Martin (2001). Based upon Grossman and Laroque (1990)’s set-up, the author
develops a model where agents employ an optimal stopping rule for the purchase
of durable goods defined by two boundaries, and an optimal returning point.
The fields of marketing and economic psychology have also contributed to
the analysis of the acquisition of consumer durable goods. This literature has
focused primarily on information search and decision making, planning of pur-
chases and the acquisition sequence of durable goods, post-purchase behavior
(disposition), and dissatisfaction and complaint behavior. Progress has been also
made on determining what factors affect replacement decisions.
In particular, several studies have looked at the importance of demographic
and lifestyles variables, perceived obsolescence, styling and fashion, prices, envi-
ronmental awareness, and uncertainty, among other variables, on the likelihood of
replacement. See, for example, Hoffer and Reilly, 1984; Bayus, 1988, 1991;
Antonides, 1990; Bayus and Gupta, 1992; Cripps and Meyer, 1994; Marrel,
Davidsson and Garling, 1995.
IV. An Empirical Application of Replacement of Home Appliances
This section is divided into three parts. In section 4.1 we estimate replace-
ment models for refrigerators and water heaters using a sample of U.S. house-
holds from the Residential Energy Consumption Survey (RECS). Based upon our
estimation results of Section 4.1, in Section 4.2 we test the dependence of re-
placement decisions across appliances. Finally, in section 4.3 we develop a more
general model where households consider the replacement of more than one ap-
pliance at a time.
4.1 A replacement model
Fernandez (2000) shows that the probability density function (p.d.f.) of the
first passage time for the stochastic process xt, T, is given by:
g (t b,  ,  x,  x ) =  x x
2 t
(x x bt)
2 tT
*
3
* 2
*?
? ? ?
? ? ? ?
?
??
?
??exp 2 t 0 (17)
where x  is given by the implicit function H( x , b, ?2)?1+?(r ˜C + x*– x) –
exp[?(x*– x)] = 0, 8 is the positive root of the characteristic equation (1/2)?2p2 +
bp – r = 0, and the parameters b and ?2 come from the dynamics of xt, dxt = bdt +
?dWt, as described in Section 2.1.
In order to calibrate our model, we take a sample from the Residential En-
ergy Consumption Survey (RECS). The RECS is a statistical survey of the U.S.
120 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
Department of Energy that collects energy-related data for occupied primary hous-
ing units in the 50 states and the District of Columbia. Conducted triennially
since 1978, it provides information on the use of energy in residential housing
units in the United States. This information includes the physical characteristics
of the housing units, the appliances utilized, including space heating and cooling
equipment, demographic characteristics of the household, the types of fuels used,
and other information that relates to energy use. The RECS also provides energy
consumption and expenditures data for natural gas, electricity, fuel oil, liquefied
petroleum gas (LPG), and kerosene.
Data for the RECS are obtained from three different sources: on-site 30-minute
personal interviews conducted in the housing unit; telephone interviews with the
rental agents of those rented housing units that have any of their energy use in-
cluded in their rent; and, mail questionnaires mailed to the housing units’ energy
suppliers asking them to provide the units’ actual energy consumption. Our sample
was taken from the RECS 1990, which contains approximately 5,100 households,
out of which 3,398 are homeowners.
One shortcoming of the RECS is that it does not provide information on
replacement times. It only records current equipment ages in intervals. For ex-
ample, category 01 = equipment is less than 2 years old; category 02 = equipment
is between 2 and 4 years old, etc. Therefore, the model parameters cannot be
estimated directly from the p.d.f. of replacement times. Instead, the p.d.f. of equip-
ment age, U, must be used. It can be shown that an asymptotic approximation for
the p.d.f. of U can be obtained from the renewal theorem (see Fernández, 2000,
for the technical details):
f u b x x expU , , ,
)*?
? ? ?
( ) =
?
? ?
?
??
?
?? ?
?
?
??
?
??
? ? ?
?
??
?
??
?
?
??
?
?
??
b
x x
x x bu
u
(x x )b
 
(x x bu
u*
* * *
? ?2 2
(18)
u  0
where ?(.) represents the cumulative distribution function of a standard normal,
and x is the solution for H( x, b, ?2) = 0.
Characteristics of household ‘i’ are incorporated into the model through
( x–x*)i/?i:
(x x )* i?
? i
 = exp(?'zi) (19)
where ? is a vector of parameters, and zi represents a vector of household char-
acteristics. This functional form ensures the non-negativity of ( x–x*)i/?i. For sim-
plicity, the ratio bi/?i is assumed to be constant across households, and equal to
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 121
b/?. Under these extra assumptions, an asymptotic approximation to the likeli-
hood function of Ui, the age of current equipment of household ‘i’, is given by:
f (u | bU ii ˜ , ?, zi) =
˜ ) (
˜
˜ ) (
˜
b exp( ' ' ) bu
u
b exp( ' ' ) bu
u
i
i i
i
i
i i
i
?
?
?
??
?
?? ? ( ) ? ?
?
??
?
??
?
?
??
?
?
???
? ? ?z z z z? ?exp exp exp2 , (20)
where ˜b? b/?.
For a sample of n independent observations, the likelihood function of equip-
ment age is given by8:
f U U U1 2 n, , ...,  (u1, u2, ..., un| ˜b , ?, z1, ..., zn) =
˜ ) (
˜
˜ ) (
˜
b exp( ' ' ) bu
u
b exp( ' ' ) bu
u
i
i i
i
i
i i
ii 1
n
?
?
?
??
?
?? ? ( ) ? ?
?
??
?
??
?
?
??
?
?
??=? ?
? ? ?z ? ?exp exp expz z z
? ?
2 (21)
Estimates of ( x –x*)i, bi, ?i for household ‘i’ can be obtained from equation
(22), once the estimates for ˜b  and ? become available:
1 + ?i(r ˜Ci +(x*– x)i) = e i?  (x x)
*
i? i=1, 2, ..., n (22)
where ?i represents the positive root of the characteristic equation (1/2) ?i2 p2 + bi p –
r = 0.
Our application deals with replacement of refrigerators and water heaters. We
first estimate separate replacement models for each appliance, and then test whether
replacement decisions are correlated. Figure 2 illustrates how replacement sales
have become a sizeable share of total annual shipments of refrigerators and elec-
tric water heaters. Indeed, this holds for all consumer durable goods with high
market penetrations.
Based on the annual average operation costs of new refrigerators from the
RECS, and on the “Consumer Reports” (December 1992), we estimated the aver-
age price of a new refrigerator in 1990 to be $ 1,355. Our estimate of the annual
operating and maintenance costs of a new refrigerator is $ 105, which corresponds
with the annual operation cost of equipment aged two years or less reported in
the RECS. Our estimate of the average price of a new electric water heater in
1990 is $ 662, and it is based on information provided in the RECS 1990 and the
“National Construction Estimator” (1990). From the RECS 1990 its annual oper-
ating and maintenance cost is estimated to be $ 241.
122 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
The regressors in the replacement model of refrigerators are a constant, the
age of the head of the household (per 10 years), monthly income (per $ 10,000),
a dummy variable that takes on the value of 1 if the household lives in an urban
area and 0 otherwise, the size of the refrigerator (cubic feet), family size (number
of members), and a dummy variable that takes on the value of 1 if the household
has a poor credit rating and 0 otherwise. Households are classified as having a
poor credit rating in case they have received aid in terms of food stamps, unem-
ployment benefits or income from AFDC (Aid to Families with Dependent Chil-
dren) during the 12 months prior to the conduction of the survey.
Table 1 presents the estimation results for the refrigerator data. The exog-
enous variables that are statistically significant at the 5 percent level are the age
of the head of the household and the size of the refrigerator. In particular, the
older the head of the household, the less likely he/she will replace his/her piece
of equipment. It is possible that older people have higher discount rates or, alter-
natively, that their preferences may change more slowly. By contrast, a greater
refrigerator size accelerates replacement. This may be due to the fact that size is
highly correlated with operating costs, after controlling for income, family size,
and electricity rate, among others factors, as Table 2 shows. Although income and
Source: Own elaboration based upon the distribution of lifetime equipment cali-
brated with the RECS data, and data on annual shipments of appliances
from the Statistical Abstract of the United States, various issues.
FIGURE 2
ESTIMATED ANNUAL REPLACEMENT UNITS AS A PERCENTAGE
OF TOTAL ANNUAL SHIPMENTS
0
10
20
30
40
50
60
70
80
90
100
19
46
19
50
19
54
19
58
19
62
19
66
19
70
19
74
19
78
19
82
19
86
19
90
19
94
Year
Pe
rc
en
t
Refrigerators Electric water heaters
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 123
family size are not statistically significant at the conventional levels, they have
the expected sign. That is, as income and family size increase, the gap between
x and x* shrinks making replacement more likely.
The estimation results for water heaters are presented in Table 3. The regres-
sors are in this case a constant, age of the head of the household, monthly in-
come, a dummy variable for those households that live in an urban area, a dummy
variable for those households for which natural gas is available in their neighbor-
hood, the tank size of the water heater (gallons), family size, and a dummy vari-
able for those households with a poor credit rating. As we see, the regressors
statistically significant at the 5 and 10 percent levels are the age of the head of
the household, natural gas availability, tank size, and the poor credit rating dummy.
As before, replacement is less likely as the head of the household becomes
older. Natural gas availability and a poor credit rating have the same effect. In
particular, natural gas availability might delay replacement because of differen-
tials in operation costs between natural gas and electric powered equipment. In-
deed, those households that would like to reduce operation costs by switching
from electric to natural gas equipment cannot do it when natural gas is not avail-
able in their neighborhoods. Consequently, replacement of electric equipment
becomes less likely. A poor credit rating delaying replacement is self-evident.
Like in the case of refrigerators, a larger equipment size makes replacement more
likely because of its high and positive correlation with operation costs.
TABLE 1
REPLACEMENT MODEL FOR REFRIGERATORS
Regressor Parameter Standard Asymptotic
estimate error t-statistic
Constant 2.529 0.145 17.444*
Age head of household (per 10 years) 0.099 0.013 7.679*
Monthly income (per $10,000) – 0.006 0.010 – 0.538
Urban area dummy (=1 if yes) 0.046 0.040 1.149
Family size (number of members) – 0.010 0.015 – 0.703
Refrigerator size (cubic3) – 0.040 0.006 7.218*
Poor credit rating dummy (=1 if yes) – 0.067 0.082 – 0.819
Standardized drift, b/? 0.624 0.032 19.375*
Log of likelihood function at convergence = –3,612
Number of observations = 2,440
*: Statistically significant at 5% level for H0: ?=0 against H1: ??0.
124 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
TABLE 2
REFRIGERATOR MONTHLY OPERATING COST MODELED AS A LINEAR
FUNCTION OF EXOGENOUS REGRESSORS
Regressor Parameter Standard t-statistic
estimate error
Constant – 48.145 7.567 – 6.362*
Monthly income (per $ 10,000) 1.847 0.679  2.721*
Urban area dummy (=1 if yes) 7.831 2.707  2.893*
Family size (number of members) – 0.634 0.831 – 0.764
Refrigerator size (feet3) 5.053 0.348  14.541*
Average electricity rate ($/kwh) 0.947 0.056  16.906*
R2 = 0.194, Adjusted R2 = 0.193
Number of observations = 2,440
*: Statistically significant at 5% level for H0: ?=0 against H1: ??0.
TABLE 3
REPLACEMENT MODEL FOR ELECTRIC WATER HEATERS
Regressor Parameter Standard Asymptotic
estimate error t-statistic
Constant 1.249 0.179 6.939 *
Age head of household (per 10 years) 0.137 0.018 7.449 *
Monthly income (per $ 10,000) – 0.069 0.140 – 0.491
Urban area dummy (=1 if yes) – 0.064 0.059  – 1.086
Natural gas availability (=1 if yes) 0.219 0.057  3.830 *
Tank size (gallons) – 0.004 0.002  – 1.757**
Family size (number of members) 0.146 0.090 1.617
Poor credit rating dummy (=1 if yes) 0.039 0.021  1.866**
Standardized drift, b/? 0.516 0.025 20.615 *
Log of likelihood function at convergence = –2,612.9
Number of observations = 1,057
* : Statistically significant at 5% level for H0: ?=0 against H1: ??0.
** : Statistically significant at 10% level for H0: ?=0 against H1: ??0.
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 125
Table 4 presents estimates for the difference between the threshold operation
cost x and the operation cost of new equipment x*, equipment lifetime, the drift
and the standard deviation of the arithmetic Brownian process, b and ?, respec-
tively, and for the total expected discounted costs for both appliances. Our esti-
mate of the expected lifetime of a refrigerator is approximately 16.5 years. If we
start up with new equipment, the expected total discounted cost amounts to
$ 4,271.87. We should point out that our lifetime estimate is fairly close to that
of the industry: an average lifetime of 16 years with a range of 10 (low) –20
(high) years9. For water heaters, our estimates of the expected lifetime and ex-
pected total cost are 13.7 years and $ 5,526.6, respectively. Like for refrigerators,
our estimate of equipment lifetime is quite close to that given by the industry in
1992: 14 years with a range of 10 (low) –18 (high) years.
TABLE 4
ESTIMATES OF x–x*, b, ?, EXPECTED EQUIPMENT LIFETIME
AND TOTAL DISCOUNTED COST
Mean Standard deviation
Estimates
Refrigerator Water Refrigerator Water
 heater heater
x–x* ($) 243.2 139.9 42.5 26.1
b ($) 16.3 11.3 6.5 4.7
? ($) 26.1 21.8 10.4 9.2
Expected lifetime (years) 16.5 13.7 4.3 3.4
Total discounted cost ($) 4,271.6 5,539.3 596.8 309.6
The overall fit for both models is quite good, as Table 5 shows. The percent
error for all age categories of refrigerators is below 10 percent, being the greatest
for equipment that are less than 2 years old, and between 5 and 9 years old. For
water heaters in turn, the greatest percent error is below 5 percent in absolute
value. Finally, Table 6 shows the impact of marginal changes in the value of the
regressors on the probability of replacement over time. For example, a cubic-foot
increase in refrigerator size leads to an increase of 9.93 per cent in the probability
of replacement within 20 years. The overall probability of replacement is very
small within the first 9 years, as we might have expected.
126 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
4.2 Are replacement decisions independent?
In the previous section we modeled household decisions to replace a given
set of appliances independently. However, it may be the case that such decisions
are indeed correlated. Theoretically, the demand for durable goods is derived from
a utility function that depends on the services these goods provide over time.
Therefore, it would not be surprising to observe some degree of either substitu-
tion or complementarity in replacement decisions of different durable goods.
TABLE 5
FITTED AND ACTUAL FREQUENCY FOR EACH AGE CATEGORY
Fitted Actual Percent error
Age Category
Refrigerator Water Refrigerator Water Refrigerator Water
heater heater heater
Less than 2 years old 0.124 0.156 0.136 0.153  8.8 – 1.9
2-4 years old 0.185 0.228 0.187 0.227  1.1 – 0.4
5-9 years old 0.289 0.289 0.270 0.299 – 7.0 3.3
10-19 years old 0.311 0.242 0.318 0.234  2.2 – 4.7
Over 20 years old 0.090 0.084 0.088 0.086 – 2.3 2.3
TABLE 6
IMPACT ON THE PROBABILITY OF REPLACEMENT DUE TO MARGINAL
CHANGES IN THE REGRESSORS
Time Period (years)
Regressor 1-3 7-9 1-20
Refrigerator Water Refrigerator Water Refrigerator Water
heater heater heater
Age of head of household – 5.61e-4 – 0.065 – 0.033 – 0.062 – 0.256 – 0.366
(per 10 years)
Monthly income 1.07e-6 0.003 0.002 0.031 0.016 0.285
(per $10,000)
Equipment size (*) 2.17e-4 1.90e-4 0.013 0.002 0.099 0.011
Probability of replacement 4.4e-4 0.011 0.088 0.131 0.730 0.796
Notes: Marginal impacts are evaluated at sample means. (*): Equipment size is measured in feet for
refrigerators and in gallons for water heaters.
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 127
In order to test the hypothesis of independent replacement decisions for an
individual household, we constructed a set of generalized residuals (Gourieroux
and Monfort, 1987) based on the difference between observed and expected elapsed
duration.
Equation (18) is derived from the fact that an asymptotic approximation of
the p.d.f. of elapsed duration U, is given by:
f u S uU T( )
( )
=
µ
u 0 (23)
where ST(u) is the survival function of completed duration or equipment lifetime,
and µ is the expected lifetime (see, for example, Lancaster, 1990, pages 91-93).
From (23), the kth moment around the origin of elapsed duration (current
equipment age) is given by:
m E u w S w dw
k
w
S w
w
g w
k
dwk
k k T k T k T
' ( ) ( ) ( ) ( )( )= = = +
?
??
?
?? + +
?
?
+
?
? ?
0 0
1
0
1
1 1µ µ µ
(24)
= 
1
1 1
1
0
1
( ) ( )
'
( )k w g w dw k
k
T
k
+
=
+
+
?
+?µ
µ
µ
 
by integration by parts. The term µk+1' is the (k+1)th moment around the origin
of completed duration t, where µ1=E(t)? µ.
From equation (24), and from the fact that for our model the moment gener-
ating function of completed duration is MT(t) = exp x x (b b 2t
*
2 2?
? ?
?
??
?
????
?2 ) , the
expectation of elapsed duration is given by:
E U x x b x x b
x x b
x x b
b
( ) ' ( ) / ( ) /( ) /
( )* *
*
*
= =
? + ?
?
=
+ ?µ
µ
? ?2
2 3 2 2 2
22 2 2
(25)
given that it can be shown that the variance and expected value of completed
duration are (x x
b
*
3
? )?2
 and x x
b
*
?
, respectively.
In order to test the independence of replacement decisions, we utilized a score
test of the form:
?
? ?
? ?
?n
i i
i
n
i i
i
n
d
=
?
??
?
??
? ??=
=
?
?
ˆ ˆ
( ˆ ˆ )
( )
1 2
1
2
1 2
2
1
2 1 (26)
128 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
where ?ˆ1i  and ?ˆ2i  represent the estimates of the generalized residuals of appli-
ance 1 (refrigerator) and 2 (water heater) for household i, respectively. This test
is asymptotically distributed as chi-square with 1 degree of freedom (see Gourieroux
and Monfort for more details).
Each residual is computed as the difference between the observed equipment
age and its expected value, evaluated at the parameter estimates of Section 4.1:
ˆ
ˆ
ˆ
ˆ
?
?
mi i
i i i
i
u
b
b
= ?
+ ?
?
2
22
(x x )*
m=1, 2; i=1, 2,.., n (27)
In order to compute ?ˆ1i  and ?ˆ2i , we had to approximate equipment age. As
described earlier, our data set only provides equipment age in intervals. There-
fore, for the first category, “less than 2 years old”, we took u = 1 year. That is,
the mid-point between 0 and 2. Similarly, for the categories “2-4 years old”, u = 3
years; “5-9 years old”, u = 7; “10-19 years old”, u = 14.5 years. The fifth age
category, however, gathers all equipment aged 20 years or older. In this case, we
had to estimate at which age the survival probability becomes negligible for each
appliance. Based upon our estimation results of Section 4.1, we found that at age
60 the probability of survival of a water heater equals 0.055 percent (5.5e-4),
whereas at age 55 the probability of survival of a refrigerator is 0.09 percent
(9e-4). Therefore, for the last age category, we took u = 40 for water heaters, and
u = 37.5 for refrigerators.
Only those households that own both appliances are considered in our com-
putations. We found positive correlation among residuals of refrigerators and water
heaters: 11.6 percent, and ?n = 20.8, being the 95-percent critical value for a ?2
(1) equal to 3.84. That is, we reject at the 95-percent confidence level the null
hypothesis of independence of the residuals of the replacement models of refrig-
erators and water heaters. In addition, the positive correlation between the residu-
als indicates that unobservable factors that either accelerate or delay replacement
of one appliance will also affect the other in the same direction.
4.3 Estimating replacement decisions simultaneously:
Minimum and maximum stopping times
Given the evidence of the previous section, we now model replacement deci-
sions of a set of appliances simultaneously rather than each one in isolation. In
order to do so, we resort to the concept of stopping time. Although this has been
already used in previous sections of this paper, we now provide a formal defini-
tion.
Let t1, t2, ..., be a sequence of independent random variables. An integer-
valued random variable T is said to be a stopping time for the sequence t1, t2, ...,
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 129
if the event {T = n} is independent of tn+1, tn+2, for all n = 1, 2, ... This means that
we observe the tn’s in sequential order and N denotes the number observed before
stopping. If T = n, then we have stopped after observing t1, t2, ..., tn and before
observing tn+1, tn+2, ...(see Ross, 1996, page 104).
Let us now consider two independent stopping times, T1 and T2, with corre-
sponding cumulative density functions FT1  and FT2 , and density functions fT1
and fT2 . For example, let us think of two home appliances whose times of either
technical failure or obsolescence are independent of one another. This is the as-
sumption in Section 4.1. But, how do we reconcile the assumption of independent
stopping times with the evidence in Section 4.2? One way to go about it is by
thinking that, although stopping times are independent, households replace their
appliances jointly.
For instance, a household might wait and replace its obsolete microwave oven
until the cutting-edge technology of refrigerators becomes available at the market
place.10 Or, alternatively, the household might decide to replace its microwave
oven and refrigerator at once, as soon as the technology of the former falls behind
the new trends.
If that is so, then we can find the minimum and maximum bounds for the
replacement time of both appliances. Let Z1= max(T1, T2) and Z2= min (T1, T2).
The probability density function of Z1 and Z2 are given by:
F z P max T T z
P T z T z
P T z P T z
Z1 1 2
1 2
1 2
( ) ( ( , ) )
          ( , )
          ( ) ( )
= ?
= ? ?
= ? ?
by independence of T1 and T2.
Then:
F z F z F zZ T T1 1 2( ) ( ) ( )= (28)
From (28) it follows that the probability density of Z1 is:
f z f z F z F z f zZ T T T T1 1 2 1 2( ) ( ) ( ) ( ) ( )= + z  0 (29)
Similarly, for the minimum:
F z P min T T zZ 2 1 2( ) ( ( , ) )= ?
= 1–P(min(T1, T2)>z)
= 1–P(T1>z, T2>z)
130 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
= 1– P(T1>z) P(T2>z) by independence of T1 and T2.
= 1– (1– F zT1 ( ) )(1– F zT2 ( ))
= F zT1 ( )  + F zT2 ( ) – F zT1 ( ) F zT2 ( )
But from (28) F z F z F zZ T T1 1 2( ) ( ) ( )= , then:
F z F z F z F zZ T T Z2 1 2 1( ) ( ) ( ) ( )= + ? (30)
Therefore, the density function of the minimum is given by:
f z f z f z f zZ T T Z2 1 2 1( ) ( ) ( ) ( )= + ? z  0 (31)
One way to model joint replacement decisions is by assuming that replace-
ment will take place somewhere between the minimum and the maximum stop-
ping times (Figure 3). Therefore, we can define a new random variable W=Z2–Z1
that denotes the time elapsed between the minimum and the maximum stopping
times. Intuitively, households might replace both appliances right after anyone of
them either becomes obsolete or breaks down. Or, alternatively, they might as
well wait until both appliances render inadequate to their needs. The exact time
at which households will replace both appliances is therefore random, and will be
located somewhere between Z1 and Z2.
FIGURE 3
JOINT REPLACEMENT DECISIONS
Now, in order to determine the distribution function of the W, we make use
of convolutions:
F w Z Z w f z z dz dzW Z Z
w Z
( ) Pr( ) ( , )= ? ? = ???
?
??
+?
??2 1
0
1 2 2
0
11 2
1
Z1 = min (T1, T2) Z2 = max (T1, T2)
Replacement of appliances 1
and 2
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 131
? f w d
dw
F w d
dw
f z z dz dzW W Z Z
w z
( ) ( ) ( , )= = ???
?
??
+?
?? 1 2
1
0
1 2 2
0
1
= f z w z dzZ Z1 2
0
1 1 1
?
? +( , ) by Leibnitz’s rule
Now, given that both Z1 and Z2 are stopping times, Z2 is independent of Z1.
Therefore, the distribution function of W boils down to:
f w f z f w z dzW Z Z( ) ( ) ( )= +
?
? 1 2
0
1 1 1 w  0 (32)
where fW(w) is the convolution of f zZ1 1( )  and f wZ 2 ( ).
It remains to characterize fW(w) based upon the distribution functions of our
model of Section 4.1. According to equation (17), T1 and T2 are distributed as:
f t b expT j j j
j1
2( , , , )?
? ? ?
x x
x x
2 t
(x x b t )
2 tj j
* j j
*
j
2
j j
*
j j
2
j
=
? ? ? ?
?
??
?
??j
tj  0, j = 1, 2. (33)
with cumulative distribution functions:
  
F t b x x expT j j j j j
j j1
1
2
2( , , , )
)
*
*
?
? ? ?
= ?
? ?
?
??
?
?? +
?
?
??
?
??
? ? ?
?
??
?
??? ?
x x b t
t
(x x )b
 
(x x b t
t
j j
*
j j j j
*
j j j j j
jj j
tj ? 0, j = 1, 2. (34)
where ?(.) represents the cumulative distribution function of a standard normal,
and x j  is given by the implicit function H( xj , bj, ? j2 )?1+?j(r ˜Cj+ x j* – xj )–
exp[?j( x j* – xj )] = 0, ?j is the positive root of the characteristic equation (1/2)? j2
p2+bjp–r = 0, and the parameters bj and ? j
2
 come from the dynamics of xjt, dxjt = bj
dt + ?j dWjt.
From (29), the distribution function of the maximum is given by:
  
f z expz1 2 1
2( ) =
? ? ? ?
?
??
?
??
?
???
?
???
x x
2 z
(x x b z)
2 z
1 1
*
1 1
*
1
2
? ? ?1
132 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
  
*
)*
 
x x b z (x x )b
 
(x x b z
z
2 2
*
2 2 2
*
2 2 2 21 2
2
2?
? ?
?
??
?
?? +
?
?
??
?
??
? ? ?
?
??
?
??
?
??
?
??? ?? ? ?2 2z exp
  
+ ?
? ?
?
??
?
?? +
?
?
??
?
??
? ? ?
?
??
?
??
?
??
?
?? 
x x b z (x x )b
 
(x x b z
z
1 1
*
1 1 1
*
1 1 1 11 2
1
2? ?? ? ?1 1z
exp
*)
  
* 
x x
2 z
(x x b z)
2 z
2 2
*
2 2
*
2
2
? ? ? ?
?
??
?
??
?
???
?
???? ? ?2 2 22
exp z  0 (35)
In turn, from (31), the distribution function of the minimum is given by:
  
f z exp expZ 2 2 1
2 2 2
2( ) =
? ? ? ?
?
??
?
??
?
???
?
??? +
? ? ? ?
?
??
?
??
?
???
?
???
x x
2 z
(x x b z)
2 z
x x
2 z
(x x b z)
2 z
1 1
*
1 1
*
1
2
2 2
*
2 2
*
2
2
? ? ? ? ? ?1 2
– f zZ1 ( ) z  0 (36)
where f zZ1 ( )  is given by (35).
Now suppose we have a cross section of n independent pairs of stopping
times for n households. Then the likelihood function of the sample is given by:
f w f z f w z dzW i
i
n
Z Z i
i
n
i i i
( ) ( ) ( )
=
?
=
? ??= +???
?
??1 0 1 1 11 1 2 wi ? 0, i = 1, 2, ...n
? ln( ( )) ln ( ) ( )f w f z f w z dzW i
i
n
Z Z i
i
n
i i i
=
?
=
? ??= +???
?
??1 0 1 1 11 1 2 (37)
where wi= Z2i –Z1i, Z2i = max(T1i, T2i) and Z1i=min(T1i, T2i).
How do we go about approximating f z f w z dzZ Z ii i1 2
0
1 1 1
?
? +( ) ( ) ? We know, from
our estimation results of Sections 4.1 and 4.2, that the probability mass for a
stopping time greater than some constant M, large enough (say M = 60), goes to
zero. Therefore the above improper integral can be suitably truncated:
f z f w z dz f z f w z dzZ Z i Z
M
Z i ii i i i1 2 1 2
0
1 1 1
0
1 1
?
? ?+ ? +( ) ( ) ( ) ( )
WHAT DRIVES REPLACEMENT OF DURABLE GOODS AT THE MICRO LEVEL? 133
Therefore the log likelihood function of the sample becomes:
ln( ( )) ln ( ) ( )f w f z f w z dzW i
i
n
Z
M
Z i
i
n
i i i
= =
? ??? +???
?
??1 0 1 1 11 1 2 (38)
In turn the integral f z f w z dzZ
M
i Z i i ii i1 2
0
1 1 1? +( ) ( )  can be approximated by some nu-
meric method, such as the trapezoidal rule:11
f z f w z dz z y y y y yZ
M
Z i m mi i1 2
0
1 1 1
1
0 1 2 22
2 2 2? + ? + + + + +( )?( ) ( ) ...? (39)
where ?z M
m1
= , y g z f z f w zZ Z ii i= ? +( ) ( ) ( )1 1 11 2 , and g(z1k)=g(k?z1), k=0, 1, 2, ..., m.
As in Section 4.1, the parameters of the distributions of Z1i and Z2i, i = 1,
2, ..., n, may be modeled as functions of household characteristics and appliances
features.
V. Summary and Conclusions
In this paper we focused on micro replacement decisions. We surveyed some
representative models of the recent literature, and discuss their empirical testabil-
ity. In particular, we went through papers where the replacement problem has
been tackled by the theory of stochastic processes and stochastic calculus: Rust
(1985, 1987), Ye (1990), and Mauer and Ott (1995). In addition, we briefly re-
ferred to the state of the art of demand for durable goods modeling in macroeco-
nomics and other fields.
The core of this paper is the empirical results for replacement of home appli-
ances in the United States, and the theoretical model of multiple replacement
decisions. Based upon individual replacement models for electric water heaters
and refrigerators, we concluded that demographics and appliance features might
either accelerate or delay replacement. In addition, we constructed a test statistic
that led us to conclude that replacement decisions might be correlated across
appliances. Based upon this evidence, we enriched our model by allowing house-
holds to replace a set of appliances simultaneously rather than each one in isola-
tion. Although the estimation process of this extension may be computationally
intensive, it is still tractable.
134 REVISTA DE ANALISIS ECONOMICO, VOL. 16, Nº 2
Notes
1 Kristensen (1994) presents a survey of Markov decision programming techniques applied to the
animal replacement problem.
2 Once the bus engine is replaced, the system regenerates to state xt = 0.3 The number of personal computers (PCs) in U.S. households has risen from zero in 1976, when
the first 200 Apple I PCs were manufactured, to nearly 43 million in 1997, when 35 percent of all
U.S. households had at least one PC (source: U.S. Department of Energy).
4 More specifically, Mauer and Ott consider a real options set-up, in which the risk of operation cost
dW can be spanned by traded financial assets. And, therefore, r is a riskless rate. In Ye’s model
no point is made as to whether r is a riskless rate or not. So one can simply assume that agents
are risk neutral.
5 In the spirit of the real options literature, an increase in volatility makes the option of waiting more
valuable.
6 For an example in discrete time, see Roberts (1978).
7 Explicit transaction costs, such as large commissions, or implicit transaction costs, such as the
search for information on performance characteristics and prices of heterogeneous durable goods.
8 When estimating the likelihood function we took account of the discreteness of the data. That is
to say, that it is necessary to compute the probability of falling into each category.
9 Source: “A Portrait of the U.S. Appliance Industry 1992”, Appliance, September 1992, Dana Chase
Publications.
10 This might be also the case if the durable goods present some degree of substitution. For example,
the replacement of a stereo system might not be as urgent given that audio CDs can be played on
the personal computer. On the other hand, a household’s postponing the purchase of new durable
goods might be indicative of borrowing constraints.
11 The area of the first trapezoid is 1/2(y0+y1)?z1, the area of the second trapezoid is 1/2(y1+y2)?z1,
etc. up to the area of the nth trapezoid, which is 1/2(yn–1+yn)?z1.
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