Kooreman, P.

English

Research Papers in Economics

ELSEVIER  Economics Letters 48 (1995)  29-32 
economics 
letters 
Individual  discounting  and the purchase  of durables  with 
random lifetimes 
Peter  Kooreman 
Wageningen Agricultural  University,  P.O.B. 8060, Wageningen, 6700 DA, Netherlands 
Received 24 June  1994; accepted 12 July 1994 
Abstract 
A  number  of  papers  have  suggested  that  the  returns  consumers  require  on  energy  efficiency  investments 
are  much  higher  than  the  capital market  rate  of return.  The  earlier literature has  typically assumed  that  the 
lifetime of the energy-using durables is fixed. I  show that if risk-neutral consumers  anticipate that the lifetime is 
random, ignoring the randomness results in an upward bias of estimated discount rates. The bias may be as large 
as 35%. 
JEL classification:  D12 
1.  Introduction 
A  number of papers have suggested that the returns consumers require on energy efficiency 
investments are much higher than the capital market rate of return. A  widely cited study is the 
article  by  Hausman  (1979)  on  the  purchase  and  utilization  of  air-conditioners.  Hausman 
estimated  an  average  annual  discount  rate  of 26.4%,  considerably  higher  than  the  capital 
market  rate  of  return.  In  a  recent  paper,  Loewenstein  and  Thaler  (1989)  discussed  two 
explanations  that  might  be  offered:  information  barriers,  in  particular  with  respect  to  the 
operating costs of durables,  and liquidity constraints. 
The  present  paper  supplements  these  explanations.  When  comparing  the  discounted 
purchase  and  utilization costs  of low-efficiency and  high-efficiency versions  of durables,  the 
existing literature has typically assumed that the lifetime of the durables is fixed; examples are 
Hausman (1979),  Gately (1980),  Dubin and McFadden (1984),  and Ruderman et al.  (1986). 
In this paper I  show that if risk-neutral consumers anticipate that the lifetime is random, the 
assumption of a  deterministic lifetime results in an upward bias of estimated discount rates. 
The bias may be as large as 35%. 
0165-1765/95/$09.50  (~ 1995  Elsevier Science B.V.  All rights reserved 
SSDI  0165-1765(94)00584-2 30  P. Kooreman  /  Economics Letters  48 (1995) 29-32 
2.  Framework  and  main  proposition 
Consider a low-efficiency version (i =  L) and a high-efficiency version (i =  H) of a consumer 
durable good. Let z  i be the operating costs per time period and let Ri be the purchase price of 
type i,  i = L,H.  I  assume % >  ~H  and R L <  R H.  L  and H  generate identical services and  are 
identical in all other respects.  Consumers are risk-neutral, i.e. the choice between L  and H  is 
made on the basis  of expected cost minimization, at a  given service demand. The per-period 
subjective discount rate is  denoted by r. 
The lifetime t  of both types is  a  random variable with distribution  function f(.).  As  in the 
literature referred to above,  time is discrete.  In case of failure the durable is replaced by an 
identical one. Operating costs are paid at the end of each period. Let 0 <  s~ < s 2 < s 3 <  ...  be 
the  random  points  in  time  at  which  the  durable  fails  and  is  replaced.  Since  s 1,  (s z -Sl), 
(s 3 -s2) ....  are independent drawings from f(.),  the expected discounted purchase costs  of 
type i  over an infinite horizon can be written as 
E[Ri(1 +  (1 + r) -s' + (1 + r) -~2 +  ---)1 
=  E[R,(1 +  (1 + r) -s' +  (1 + r)-S'.(1 +  r)  ~s2  .3,) +  ...)] 
=E  R,E  (l+r)-'*  =R,E  [E(l+r)-']  k  R~ 
,=o  ,=0  -  1-  E(l+r)-''  (1) 
Consider the sum of expected discounted purchase  and operating costs: 
R,  +z,  (2) 
C~(r) =  1 -  E(1 +  r) -t  r  " 
The reservation  value  of the discount rate is defined as the value of r  that solves the equation 
CL(r  ) =  Cn(r  ).  A  consumer will purchase a  high-efficiency version if and only if his subjective 
discount rate does not exceed the reservation discount rate. 
Using (2),  Ce(r  ) =  C,(r)  can be rewritten as 
1-E(l+r)-'  R u-R  L 
h(r) -  -  -  B  .  (3) 
r  T L  --  T H 
B  is the payback period, i.e. the period of time before the additional investment for type H  is 
recovered from its lower operating costs. 
Lemma.  Let  0 <  B  <  Et.  Then for  any  lifetime  distribution  f(. )  there  exists  a  unique  positive 
value  of r  that solves  Eq.  (3). 
Proof.  Consider the function 
g(r) =  E(1 +  r)-'=  ~,  P(t = k). (1 + r) -~ , 
k=l 
(4) 
with P. Kooreman  /  Economics Letters  48 (1995) 29-32  31 
oc 
g'(r) =  -  ~'~  P(t=k).k(1  +r)  -k-1  . 
k=l 
(5) 
Since  limr__,0 g'(r)=-Et,  it  follows  from  l'H6pital's  rule  that  limr__,  0  h(r)= Et.  Thus  a 
consumer  with  r =  0  will  purchase  type  H  if  and  only  if  the  expected  lifetime  exceeds  the 
payback  period. We  therefore  assume 0 <  B  <  Et. 
The  existence  and  uniqueness  of  a  positive  solution  to  Eq.  (3)  follows  from  lim,__,  0 
h(r) = Et >  B, limr__,= h(r) = 0 <  B, and h'(r) <  0 for all r >  0. To prove that h'(r) <  0, note that 
h'(r)={-rg'(r)-l+g(r)}/r  2.  Thus,  it  suffices  to  show  that  ~b(r)=--rg'(r)-I  +g(r)<0. 
Now 
c~(r) =  ~  P(t = k). qJ(r,k)  ,  (6) 
k=l 
with 
~O(r,k) = rk(1 +  r) -k-' -  1 +  (1 +  r) -k 
=  (1 +  r)-k-l[rk--  (1 +  r) k+' +  (1 +  r)] 
=  (1 +  r)-~-l[1  +  (k +  1)r-  (1 +  r) k+~] 
=(l+r)  -k-1  l+(k+l)r-  ~=0  ~  k+lj  r~ 
r  k+l 
(7) 
Thus  qJ(r,k) <  0  for all  k =  1,2,..  and  r >  0.  As  a  consequence,  ~b(r) <  0  and h'(r) <  0.  [] 
The main result can  now be formulated  as follows. 
Proposition.  Let r v  denote the reservation  discount  rate when lifetimes are fixed and equal to  T, 
i.e.  r v  solves  hv(r ) =  {1-  (1 +  r)-r}/r  = B.  Let r s  denote the reservation  discount rate  when 
lifetime t follows a distribution function f(.) with Et =  T, i.e.  r s solves hs(r  ) =  {1 -  E(1 +  r)-'}/ 
r=  B.  Then  rv>~r s. 
Proof.  Using Jensen's  inequality  we  have  0 <  (1 +  r) -Et <~ E(1 +  r)-' <  1.  As  a  consequence, 
hv(r  )/> hs(r ), for all r >  0. Since hF(r  ) and hs(r  ) are monotonically decreasing (see proof of the 
lemma),  it follows that  r v I> r s.  [] 
Note  that  no  restrictions  on f(-)  are  imposed other  than Et =  T. 32  P.  Kooreman  /  Economics  Letters 48 (1995) 29-32 
3.  Implications 
Consider  a  population  of consumers who  are  homogeneous except  with  respect  to  their 
subjective  discount  rates.  Suppose  that  q%  are  observed  to  purchase  the  low-efficiency 
version. Assuming deterministic lifetimes, one would conclude that q %  of the consumers have 
a  subjective discount rate exceeding r v.  Assuming random lifetimes, one would conclude that 
q%  of  the  consumers  have  a  subjective  discount  rate  only  exceeding  r s  (~<rF).  Thus  the 
assumption of a deterministic lifetime would result in an upward bias of the estimated average 
discount rate.  The intuitive explanation is that the possible benefit of a  late failure does not 
offset the possible loss incurred at  an early failure because  'late'  is discounted more heavily 
than 'early'. 
Hausman's (1979) widely cited estimate of 26.4%  was determined by solving r  with a fixed 
lifetime of 9.94  years))  Assuming a  hazard  rate  that  increases  linearly such that  Et = 9.94, 
Hausman's parameter estimates imply an annual discount rate  of 23.5%.  Assuming that the 
hazard rate is constant over time, so that t follows the geometric distribution, with Et = 9.94, 
the implied discount rate is  19.2%,  much closer to the interest rate for personal loans at the 
end of the  1970s.  2 
References 
Dubin,  J.  and  D.  McFadden,  1984,  An  econometric  analysis  of  residential  electric  appliance  holding  and 
consumption, Econometrica 52,  345-362. 
Gately, D.,  1980,  Individual discount rates and the purchase and utilization of energy-using durables: Comment, 
The Bell Journal of Economics 11,  373-374. 
Hausman, J.,  1979,  Individual discount rates and the purchase and utilization of energy-using durables, The Bell 
Journal of Economics 10, 33-54. 
Loewenstein, G.  and R.H.  Thaler,  1989,  Anomalies; intertemporal choice, Journal of Economic Perspectives 3, 
181-193. 
Ruderman,  H.,  M.  Levine  and  J.  McMahon,  1986,  Energy-efficiency choice  in  the  purchase  of  residential 
appliances, in: Kempton, Willett and M. Neiman, eds., Energy Efficiency: Perspectives on individual behavior, 
American Council for an Energy Efficient Economy, Washington, DC. 
9.94 was the estimated mean lifetime of air-conditioners. Thus the calculation of the discount rate was based on 
discounted costs evaluated at the expected lifetime rather than on expected discounted costs. 
2 In case  of a  constant hazard,  E(1 + r) -t = (1 +  Tr) -1,  which  allows r s  to  be  solved from  (3)  analytically as 
r s = B  ~  -  T-I. In the case of a  linearly increasing hazard rate,  the rate of increase is chosen such that Et =  T. 
Next r s is solved from (3) numerically, with E(1 + r)-' being evaluated numerically at each trial value of r s. 

An econometric analysis of residential electric appliance holding and consumption,

Individual discounting and the purchase of durables with random lifetimes.

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