Estimating willingness to pay by risk ad
Applied Economics, 2013, 45, 37–46
Estimating willingness to pay by risk
adjustment mechanism
Joo Heon Parka,* and Douglas L. MacLachlanb
a
Department of Economics, Dongduk Women’s University,
23-1 Wolgok-dong, Sungbuk-Gu, Seoul 136-714, Korea
b
Michael G. Foster School of Business, University of Washington, Seattle,
WA 98195-3226, USA
Measuring consumers’ Willingness To Pay (WTP) without considering the
level of uncertainty in valuation and the consequent risk premiums will
result in estimates that are biased toward lower values. This research
proposes a model and method for correctly assessing WTP in cases
involving valuation uncertainty. The new method, called Risk Adjustment
Mechanism (RAM), is presented theoretically and demonstrated empirically. It is shown that the RAM outperforms the traditional method for
assessing WTP, especially in a context of a nonmarket good such as a
totally new product.
Keywords: purchase decisions; willingness to pay; contigent valuation
method; adjustment mechanism
JEL Classification: D12; D81; M31
I. Introduction
Traditional consumer theory presumes that a consumer makes a purchasing decision by comparing his
or her own subjective value of a good (or service) with
its objective value. The objective value is the posted
price, while the subjective value is usually measured
by the so-called Willingness To Pay (WTP) that is
defined as the maximum dollar amount a consumer
is willing to pay for the good. If the posted price is
below a consumer’s WTP, the purchase would be
made; otherwise the purchase would be abandoned.
We call this the Traditional Decision Mechanism
(TDM).
A problem from the researcher’s perspective is that
a consumer never reveals his or her own WTP
directly, so that the WTP is not observable, whereas
the price is directly observed in a market.
However, consumers reveal their WTP indirectly
through transactions, so that their WTP can be
estimated by analysing their observed transaction
behaviours. While the TDM provides a theoretical
basis for inferring the WTP from purchase behaviours, it makes sense only when there is no uncertainty in consumers valuing the good of interest. If
consumers are uncertain over the value they place on
the good and do not know their WTP exactly as a
single value, they are unable to compare their WTP
with the price. Therefore, the TDM should not be
employed as a basis for inferring WTP for a good
when uncertainties are involved in valuation of
the good.
A question for us is how consumers make decisions
to purchase a good or not if they are ignorant of their
WTP for the good as a single value because of
uncertainties. As an answer, we propose a new
purchasing decision mechanism called a Risk
Adjustment Mechanism (RAM) that can be applied
*Corresponding author. E-mail: [email protected]
Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online ! 2013 Taylor & Francis
http://www.informaworld.com
DOI: 10.1080/00036846.2011.568404
37
38
under valuation uncertainty. A key characteristic of
RAM is that consumers are presumed to have in their
minds a statistical distribution of their WTP instead
of a single value. Furthermore, typically risk averting
consumers are expected to purchase a good if the
mean of their WTP is greater than the good’s price by
at least their risk premium. This is in contrast to
TDM in which consumers are expected to purchase a
good if their WTP is simply greater than the price. If
consumers are assumed to have aversion to risk, the
WTP of consumers using RAM who have decided to
purchase must be sufficiently greater than the price,
while it is theoretically enough to purchase for
consumers using TDM that the WTP be just greater
than the price even by a single penny. Therefore, from
a RAM point of view, the empirically assessed TDM
underestimates WTP by structure, since the TDM
devalues the WTP of a risk averting consumer at least
by his or her positive risk premium.
The notion that consumers are uncertain about
valuation is not new. Uncertainty in valuation has
been addressed by many researchers in a number of
different ways since Ready et al. (1995) raised the
issue explicitly in a context of Contingent Valuation
Method (CVM).
In fact, there are many well-known uncertain
factors that can and do influence a consumer’s
valuation. Wang (1997) argued that uncertainty
might exist in a commodity or service itself,1 in an
unpredictable market situation, in socio-demographic
variables, and in people’s uncertain preferences,
among others. Recognizing the existence of such
uncertainties, various studies have been made that
incorporate the uncertainties into the valuation
process.
Van Kooten et al. (2001) employed fuzzy set theory
to represent the underlying vagueness of preferences.
Hanemann and Kristro¨m (1995) and Wang (1997)
suggested adding a random variable to the utility
function. But they do not take the consumer’s
behaviour into account, in contrast with our proposed RAM where the risk premium reflecting the
consumer’s behaviour against uncertainties plays an
explicit role in the purchasing decision.
Consumers react to uncertainties in different ways,
leading to different market results. Mauro and
Maffioletti (2004) recognized consumers’ reaction to
uncertainties in the face of an incentive compatible
J. H. Park and D. L. MacLachlan
mechanism through experiments, and suggested that
ambiguity may be a relevant factor that affects
market prices and allocations. And a purchasing
decision mechanism like RAM that reflects consumers’ means of dealing with uncertainties can also
explain more efficiently the market results of a new
product with embedded uncertainties. For example,
we may also explain through the RAM the behaviour
difference between early adopters and slow movers
for new products.
Studies on how uncertainties affect the purchasing
decisions have been usually made in a hypothetical
context such as the CVM because the consumers’
attempts to resolve uncertainties are not observed
directly in real markets. We cannot deduce the
consumers’ uncertainty resolution behaviours from
their purchasing decisions.
However, valuation based on a CVM is often
questioned because of the so-called hypothetical bias
caused by difference in respondents’ behaviour
between a hypothetical situation and a real purchase
situation (Paradiso and Trisorio, 2001). Champ et al.
(1997), Johannesson et al. (1998) and Alberini et al.
(2003) used a calibration process with follow-up
questions regarding the certainty of the stated valuation in a CVM context in order to reduce the
hypothetical bias; for example, ‘very uncertain’, ‘not
likely’, ‘very sure’ and so on. But these assessments of
certainty could be as wrong as the valuation reports
because the assessments are also asked in the same
hypothetical situation as the valuation. Contrary to
these approaches, the RAM does not have the bias
problems caused by double sources of uncertainty
because it incorporates uncertainties into the model
instead of trying to handle uncertainties outside the
model. And furthermore, the RAM can explain and
reduce the hypothetical bias because it contains a risk
premium variable that should be measured in the
same situation where a product is evaluated.
Ariely et al. (2003) and Hanley et al. (2009) posited
that consumers probably have some range of acceptable values rather than specific WTP value for a
good. Following their thinking, we might assume that
consumers have in their heads a distribution of WTP
values rather than a single value. This assumption has
been used explicitly or implicitly by many researchers
such as Cameron and Quiggin (1994),2 Carson et al.
(1994) and Wang (1997).
1
One would face uncertainty over how much gains could be obtained from a nonmarket good that has never before existed.
Such uncertainty would not be removed until the good is actually consumed. For example, the value of improved environment
is not realized until it is actually provided and consumed.
2
Cameron and Quiggin (1994) said ‘Respondents seem not to hold in their heads a single immutable ‘true’ point valuation . . . .
At best, they may hold a distribution of values-amounts they would be willing to pay with some associated probability
density. This might be interpreted as ‘uncertainty’. Whenever they are asked to produce value for the resource, they make a
draw from this distribution and use it as a basis for their response to the current discrete-choice CV question’.
Estimating willingness to pay by risk adjustment mechanism
Wang (1997) and Chang et al. (2007) among others
asked a polychotomous choice WTP question with an
implicit assumption that the certainty of a respondent’s answer depends upon the difference between
the WTP distribution and the bid price. Especially,
Wang (1997) allows the trichotomous response
options of ‘yes’, ‘no’ or ‘don’t know (DK)’. In his
model, there is a vagueness band over which a
respondent has an option of ‘DK’ instead of making
a clear purchasing decision. It should be noted that
‘DK’ option is never allowed by structure in a real
purchase situation. The true outcomes made by
individuals in a real situation are only either ‘yes’ or
‘no’. The RAM has only dichotomous options of ‘yes’
or ‘no’, which is consistent with real market
transactions.
In the next section, the RAM is formally introduced along with the TDM. In Section III, the
estimation issue is addressed. An empirical comparison of WTP estimates by TDM and RAM is made in
Section IV to see how seriously the TDM underestimates the WTP. Some concluding remarks follow in
the last section.
II. Risk Adjustment Mechanism
A consumer decides to purchase a good (z) if his
or her utility can be increased by purchasing the
good, i.e.
!
"
Ii ¼ 1 if U p, yi " pðzÞ þ w&i ðzÞ ' Uð p, yi Þ
ð1Þ
Ii ¼ 0 otherwise
where U(.) is an indirect utility function; yi denotes
the ith consumer’s income; p is the price vector of all
other goods except good z; p(z) the price of good z;
wi&(z) the ith consumer’s WTP for the good z and Ii is
a binary response variable taking one for purchasing
and zero for not purchasing good z.
Since the WTP for a good is the maximum amount
of money that a consumer is willing to give up in
exchange for the good without decreasing the initial
utility level, the utility change caused by purchasing a
good is exactly the same as that caused by changing
the consumer’s income by the difference between
the price and the WTP for the good. Therefore the
left-hand side of the inequality in Equation 1 is the
utility level after purchase that the ith consumer could
attain by purchasing good z with its price p(z) paid
while the right-hand side is the utility level before
purchase. To sum up, a consumer purchases a good if
39
the utility after purchase is greater than the utility
before purchase.
Since the indirect utility function is monotonically
increasing in income y, the purchasing decision
mechanism (Equation 1) may be converted simply
as follows:
Ii ¼ 1 if w&i ðzÞ ' pðzÞ
Ii ¼ 0 otherwise
ð2Þ
The mechanism (2) means that a consumer purchases a good if the WTP is greater than the price,
which is usually presumed the purchasing mechanism
by traditional consumer theory and is called the
TDM in this article.
It is worth noting that the TDM works correctly
only if there is no uncertainty related with the
purchase. The TDM can be applied only when a
consumer knows his or her own WTP exactly as a
single value with certainty, and then can compare the
WTP with the price as suggested by Equation 2.
However, the decision mechanism is not as simple as
in Equation 2 if uncertainties are involved. Suppose
uncertainties stem from the benefits expected from
consuming a good z, the WTP is also uncertain and
should be treated as random because it totally
depends upon the benefits. Since the benefits of a
good are actually realized by consuming the good, a
consumer cannot know his or her WTP for the good
precisely until he or she consumes it. Therefore,
a consumer can only guess the WTP at the purchase
time when actual consumption is not yet realized. In
this article, the WTP realized after consumption is
called the ex post WTP and the WTP guessed before
consumption is called the ex ante WTP.
There is usually a difference between the ex post
and ex ante WTP’s, which depends upon the degree
of uncertainty. Hanley et al. (2009) argued that this
kind of uncertainty is reduced when a person has
more experience, i.e. the uncertainty might be
inversely related with experience with the good. One
would expect little difference between the two WTP’s
for standard and frequently used goods, because
consumers have already considerable experiences
with the amount of benefit that they can expect
from those goods. Contrarily, one would expect a
large difference between ex post and ex ante WTP’s
for the goods such as totally new products or
nonmarket goods that have never been traded in a
market before. Consumers are ignorant of how much
benefit they can obtain from such goods until actual
consumption.3
3
Although the ex post WTP is a more accurate indicator of the true value of a good and it would be desirable to know that
value when doing a cost benefit analysis and developing public policy, only the ex ante WTP is typically known or knowable
at that decision point.
J. H. Park and D. L. MacLachlan
40
At the purchase time, the ex post WTP is random
and is assumed to be distributed around the ex ante
WTP as
wi ðzÞ ¼ w&i ðzÞ þ ui ,
E ½ui ) ¼ 0,
Var½ui ) ¼ !u2
ð3Þ
where ui is an error term reflecting uncertainties
associated with consumption; wi(z) the ex post WTP;
wi&(z) the ex ante WTP. Note that the consumers are
posited to guess their own ex ante WTP as the mean
value of the ex post WTP distribution even at the time
of purchase. And the consumer is exposed to uncertainty to a degree of the variance of the error
term (!u2 ).
Now consider how a consumer would make a
purchasing decision when facing uncertainty and
being ignorant of the ex post WTP. The consumer
should make a purchase if the expected utility is
greater than the utility without purchase but otherwise he or she should not. This can be expressed as
Ii ¼ 1
Ii ¼ 0
if Ewi ðzÞ ½Uðp, yi " pðzÞ þ wi ðzÞÞ) ' Uðp, yi Þ
otherwise
ð4Þ
A consumer tends to avoid uncertainties if possible.
A consumer is willing to pay an incremental amount
of money if he or she is guaranteed to have a certain
offer instead of an uncertain offer. The amount of
money a consumer is willing to pay for a certain offer
of ensuring the mean value of an uncertain offer
instead of the uncertain offer is called the risk
premium of the uncertain offer. And the risk
premium of a new good z can be defined as Ri (z)
satisfying the following equation:
!
"
U p, yi " pðzÞ þ w&i ðzÞ " Ri ðzÞ
¼ Ewi ðzÞ ½Uðp, yi " pðzÞ þ wi ðzÞÞ)
ð5Þ
Therefore, combining Equations 5 and 4 and using
the fact that the indirect utility function U(.) is a
monotonically increasing function, yields the following purchasing decision mechanism:
Ii ¼ 1
if w&i ðzÞ " pðzÞ ' Ri ðzÞ
Ii ¼ 0
otherwise
theoretically by TDM, but the purchase should not
be always made by RAM. The RAM considers
consumers who decide to purchase the good as
having a much higher WTP than the price while the
TDM regards them as having just higher WTP than
the price. And thus the TDM would yield an
underestimation of WTP when there are
uncertainties.
III. Estimation
The RAM presented as in Equation 6 implies that
consumers take their own ex ante WTP together with
the price and risk premium into account to make a
purchase decision. The ex ante WTP is assumed to be
linearly determined as
w&i ðzÞ ¼ x0i " þ "i ,
"i * Nð0, !"2 Þ
ð7Þ
where xi is a vector of covariates, " the vector of
coefficients and "i the error term. Now the error terms
in Equation 7 should be distinguished from those in
Equation 3. The error term in Equation 3 denotes a
part of ex post WTP that a consumer cannot grasp at
the purchasing stage and is the source of uncertainty.
But the error term "i in Equation 7 denotes a part of
the ex ante WTP that a researcher cannot explain
with a vector of covariates even though a consumer
knows it completely. So the researcher regards the
ex ante WTP as being statistically distributed even
though a consumer knows it as a single value.
The log-likelihood functions implied by TDM and
RAM are presented as in Equations 8 and 9,
respectively.
TDM:
%
#
$&
n '
X
pðzÞ " x0i "
Ii ln 1 " !"
ln L ¼
!"
i¼1
#
$(
pðzÞ " x0i "
þ ð1 " Ii Þ ln !"
!"
ð6Þ
ð8Þ
RAM:
which we call the RAM. A consumer purchases the
good z if his or her ex ante WTP exceeds the price at
least by the risk premium; the consumer should not
purchase otherwise. This is a salient point differentiating the RAM from the TDM. Note that if there is
no uncertainty and the risk premium is zero, then
the RAM coincides with the TDM exactly. Actually,
the TDM need not distinguish the ex ante WTP
from the ex post WTP since it assumes no uncertainty
at all. If the WTP is greater than the price even by a
miniscule value, the purchase should be made
n '
X
%
#
$&
pðzÞ " x0i " þ Ri ðzÞ
Ii ln 1 " !"
!"
i¼1
#
$(
0
pðzÞ " xi " þ Ri ðzÞ
þ ð1 " Ii Þ ln !"
!"
ln L ¼
ð9Þ
where !S(+) is the cumulative distribution function of
the standardized normal distribution.
The TDM can only identify the coefficients (") of
the model (7) up to a scale transformation. Since
a common price is given to all the consumers,
Estimating willingness to pay by risk adjustment mechanism
the inequality condition of TDM can be converted as
follows:
"0 þ "1 xi1 þ + + + þ "1k xik þ "i ' pðzÞ
, ð"0 " pðzÞÞ þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
, #0 þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
ð10Þ
where #0 ¼ "0 " pz . Therefore the coefficients of "
can be identified only up to a scale transformation
because
#0 þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
, $ð#0 þ "1 xi1 þ + + + þ "1k xik þ "i Þ ' 0
for $ 4 0
ð11Þ
The TDM cannot predict the value of WTP, but
only the probability of the inequality condition of
TDM being satisfied.
In contrast, the RAM can identify all coefficients
fully because different risk premiums are taken by
each consumer, even when a common price is given to
them. The inequality condition of RAM being
multiplied by a positive number, all coefficients can
be identified because the multiplication number ($)
can be estimated as the coefficient of the risk
premium (Ri(z)) as
"0 þ "1 xi1 þ + ++ þ "1k xik þ "i ' pðzÞ þ Ri ðzÞ
, ð"0 " pðzÞÞ " Ri ðzÞ þ "1 xi1 þ ++ + þ "1k xik þ "i ' 0
, #0 " Ri ðzÞ þ "1 xi1 þ ++ + þ "1k xik þ "i ' 0
, $ð#0 " Ri ðzÞ þ "1 xi1 þ + ++ þ "1k xik þ "i Þ ' 0 for $ 40
ð12Þ
However, the above identification of RAM is
possible only when the risk premium is observable.
Yet in reality this does not occur. The risk premium
associated with the good z is not actually observed.
We need to assume a model of risk premium and then
estimate it. It is already proven that there exists a
particular relationship like Equation 13 between the
risk premium and the attitude toward the risk if the
expected return of a risky offer is relatively very small
(Malinvaud, 1985).
Ri ¼ "
! 2 U00 ð yÞ ! 2 ri
,
¼
2U0 ð yÞ
2
ri ¼ "
U00 ð yÞ
U0 ð yÞ
ð13Þ
where Ri is the risk premium for a risky offer with its
variance ! 2, and ri the Arrow–Pratt absolute Risk
Aversion Index (ARAI) reflecting a person’s attitude
toward risk. A respondent with a higher ARAI has a
higher risk premium. Now if a Constant Absolute
4
41
Risk Aversion (CARA) utility function (Equation 14)
is assumed for simplicity, then the ARAI is fixed at ri
whatever the consumer’s income (y) level is
1
CARA utility function: Uð yÞ ¼ " e"ri y
ri
ð14Þ
Assuming that each consumer has a CARA utility
function, the log-likelihood of RAM (Equation 9)
can be transformed as
%
#
$&
n '
X
pðzÞ " x0i " þ !u2 ðri =2Þ
Ii ln 1 " !"
!"
i¼1
#
$(
0
pðzÞ " xi " þ !u2 ðri =2Þ
þ ð1 " Ii Þ ln !"
!"
ð15Þ
ln L ¼
The risk premium having been replaced by !u2 ri =2,
the RAM turns out not to be identifiable because a
parameter !u2 is added. So we need an additional
assumption to make the model parameters identified.
For simplicity, we assume that the degree of uncertainty a researcher faces is exactly the same as that
amount a consumer faces, i.e. !u2 ¼ !"2 . Then the
log-likelihood of Equation 15 becomes identifiable
because the new parameter !u2 is out of the estimation
process.
The log-likelihood functions of Equation 15 are,
however, based upon market transactions so that
they cannot be applied to the estimation of WTP for a
nonmarket good such as a totally new product. By
definition no transaction of a nonmarket good has
been made and no information pertaining to the
market such as the price is available to be observed.
Hence a different method is needed to estimate the
WTP for a nonmarket product.
The CVM has been used extensively for estimating
the value of nonmarket goods. See for example,
Cummings et al. (1986) and Carson and Mitchell
(1994).4 The CVM has also been employed as a pretest-market evaluation method in the marketing
arena (Cameron and James, 1987; Wertenbroch and
Skiera, 2002).
The CVM uses a sample survey asking respondents
to value a nonmarket good of interest in a hypothetical situation where no actual transaction is required.
And we are free to give each respondent different
prices in CVM because the respondents are supposed
to answer ‘yes, I’ll buy it’ or ‘no, I won’t buy it’ with
the price given to them. Then the log-likelihood
functions of TDM and RAM with CARA
According to Carson and Mitchell (1994), up to 1994 there had been about 1600 CVM studies and papers from over 40
countries on many topics including transportation, sanitation, health, the arts and education as well as the environment.
J. H. Park and D. L. MacLachlan
42
assumption are somewhat changed as follows:
risk premium involved in the purchasing decision,
we do not need the CARA assumption but just
employ the likelihood including the risk premium
itself; otherwise, we should make an assumption like
CARA in order to get an estimable likelihood
function.
TDM:
n '
X
%
#
$&
pi ðzÞ " x0i "
Ii ln 1 " !"
!"
i¼1
#
$(
pi ðzÞ " x0i "
þð1 " Ii Þ ln !"
!"
ln L ¼
ð16Þ
RAM:
%
#
$&
n '
X
pi ðzÞ " x0i " þ !u2 ðri =2Þ
Ii ln 1 " !"
ln L ¼
!"
i¼1
#
$(
pi ðzÞ " x0i " þ !u2 ðri =2Þ
þ ð1 " Ii Þ ln !"
!"
ð17Þ
It should be noted that there is no parameter
identification concern for both TDM and RAM in
Equations 16 and 17 because ! " can be identified as
the inverse of the coefficient of pi(z). All the coefficients are fully identified for both TDM and RAM
even without the assumption of !u2 ¼ !"2 .
The last problem that must be solved to make the
log-likelihood function workable is that an individual
consumer’s ARAI (ri) cannot be observed directly.
But if we make the assumption of CARA utility, we
can infer the ARAI indirectly by observing how the
consumer behaves when he or she faces uncertainty
and can avoid it by paying some risk premium. If we
can observe how much uncertainty is involved and
how much risk premium the consumer pays, then we
can infer the consumer’s ARAI indirectly by the
relationship (13) under the CARA assumption.
Once an inferred ARAI is available, maximization
of the log-likelihood of RAM (Equation 17) yields
the estimates of all parameters in RAM.
Now it should be noted that the reason for
assuming CARA utility function is that the risk
premium in Equation 9 cannot be observed directly.
With CARA utility function, we can present the risk
premium as a (reduced ) function of the ARAI that
can be subsequently inferred from observation of
another risk averting behaviour. However, the
CARA assumption is not necessary to connect the
risk premium with the ARAI. Actually, the risk
premium is a function of the ARAI with any utility
function. But we need the CARA utility function in
order to get a reduced form like Equation 13 with
which the risk premium is substituted so that an
estimable likelihood function such as Equation 17
could be obtained. Therefore, if we can observe the
5
IV. An Empirical Study
In this section, the RAM is compared to TDM with a
real CVM data. The CVM data were collected to
obtain information about consumers’ WTP for a new
customized cell phone service. In this case, a Korean
mobile telecommunication company attempts to
determine the best rate program for each customer
by analysing actual data. The company calculates due
amounts for the past 3 months under all rate
programmes available to find the lowest due
amount and its associated rate programme and then
informs customers of this information. This is a
totally new service that has never been offered before.
Customers are thereby not sure how much benefit
they can actually obtain from the service.
Each respondent was asked his or her WTP for the
new service in different formats – open-ended or
close-ended.5 However, the data set used here is only
the close-ended format data set because both TDM
and RAM are designed to be applicable only to a
dichotomized reaction of such as ‘yes’ or ‘no’ in the
close-ended format. For the close-ended format
survey, potential respondents were divided into
eight groups to which a bid price was chosen and
assigned from eight different predetermined bid
prices that are distributed from 2000 Won to
9000 Won.
The survey was fielded around Seoul and its
suburbs in South Korea by six trained interviewers
for 2 weeks during spring of 2008. We conducted 400
interviews with the close-ended CV formats. Of them,
we succeeded in getting 347 responses, resulting in a
response rate of 86.8%. Deleting questionnaires for
which there were no responses to key questions, we
obtained 249 final responses ready to be used in
estimation. Table 1 presents the percentages of ‘yes’
votes for each bid price. The higher percentage of
‘yes’ votes occurs with the lower bid prices, which is
consistent with the law of demand.
This CV survey is different from others in that,
along with the question regarding the WTP for the
new product, a respondent was also asked his or her
The open-ended format asks how much a respondent is willing to pay for the product while the close-ended format asks
whether the respondent is willing to pay a bid price for the product.
Estimating willingness to pay by risk adjustment mechanism
Table 1. Percentage of ‘Yes’ votes by bid prices
Bid price (Won)
Sample size
2000
3000
4000
5000
6000
7000
8000
9000
Total
30
29
30
34
36
28
30
32
249
Table 2. Frequencies of the consumers’ risk premium for the
game and absolute risk aversion index
Percentage of
‘Yes’ votes
60.00
65.52
60.00
38.24
38.89
28.57
30.00
40.63
WTP for a gamble from which the ARAI of each
respondent could be conjectured. Each respondent
was asked to play a gamble in which tossing a coin,
a player is paid 15 000 Won if the head is upside, or
5000 Won otherwise. The expected value of the
gamble is 10 000 Won. The respondent is asked how
much he or she was willing to pay for playing the
gamble, in an auction manner. A respondent was
asked initially whether he or she were willing to pay
6000 Won or not for the gamble. If the answer was
‘no’ for the initial question, the auction stopped and
the WTP for the game was regarded as 5000 Won
that was the guaranteed minimum amount to pay
back to the player. However, if the answer was ‘yes’,
the auction continued by asking the same question
but the price raised by 1000 Won, i.e. whether they
were willing to pay 7000 Won or not for the game.
Subsequently, if the answer was ‘no’, the WTP was
regarded as 6000 Won. This process was continued
until the respondent gave an answer of ‘no’. But the
maximum price given to the respondents was set at
10 000 Won because a risk averting respondent is
believed to be willing to pay no more than the
expected value of the game.
The risk premium of each respondent can be
obtained by subtracting the expected return of the
game, 10 000 Won from the WTP of each respondent
inferred through the series of question items. For
example, if a respondent’s inferred WTP for the game
is 7000 Won, then his or her risk premium of the
game is calculated to be 3000 Won. Then with the
CARA assumption, the ARAI can be also calculated
by using the relationship (13). Since the variance of
the returns of the game is 25 million Won, the ARAI
of each respondent is calculated as
ri ¼
2Ri ð gÞ
25 000 000
43
ð18Þ
where Ri(g) is the risk premium of ith respondent.
The risk premiums and the calculated ARAI’s
are presented with their frequencies in Table 2.
Risk premium
ARAI
Frequencies
5000
4000
3000
2000
1000
0
Average or total
0.00040
0.00032
0.00024
0.00016
0.00008
0.00000
0.00020
103
57
45
28
10
6
249
The ARAI’s obtained from this gamble experiment
are used in the log-likelihood of RAM (Equation 17).
The ex ante WTP for the service is assumed to be a
linear function as
w&i ¼ "0 þ "1 EDUi þ "2 AGEi þ "3 GENDERi
þ "4 ln INCi þ "i
ð19Þ
where the covariates are explained in Table 3.
The coefficient estimates of the WTP model
(Equation 19) and the estimated mean value of the
WTP’s of the respondents in the sample are reported
in Table 4. The left column reports the TDM
estimation results and the right column reports the
RAM. It is disappointing to learn that most of
coefficient estimates are not statistically significant
except that of education level. Furthermore, it should
be recognized that the WTP model of Equation 19
itself is also somewhat lacking as a model predicting
how much a potential consumer is willing to pay for
the service. However, the main purpose of this
empirical study is not estimating the coefficients but
demonstrating how much lower the TDM would
estimate the WTP, and how important the risk
premium is when estimating the WTP by using
purchasing behaviour data. Even with a perfect
model, an estimation process without considering
the risk premium could not predict the WTP as
precisely as would an estimation process that does so,
if a consumer takes the risk premium into account
when making a purchasing decision. What is important is not the RAM’s predictive power itself but its
relative prediction superiority over TDM.
An initial basis of comparison can be made in
terms of the distributions of the WTP obtained from
TDM and RAM. Figure 1 presents the distribution of
WTP estimates by the two mechanisms and the risk
premiums. As expected, the mean of WTP estimates
is much higher for the RAM than for the TDM. The
difference reflects the risk premium the consumers are
willing to pay for avoiding the uncertainties associated with the new cell phone service.
J. H. Park and D. L. MacLachlan
44
Table 3. Definition and sample statistics of variables
Variable
Definition
Format in survey
Format in use
Mean
SD
EDU
Education (unit: years of schooling)
Continuous
1: 10
2: 11.5
3: 13.5
4: 15.5
14.00
2.03
AGE
GENDER
Age
Gender
21.24
0.50
Monthly household
total income after tax
deduction (Unit: 10 000 Won)
Continuous
Category
1: male
0: female
Continuous
1: 200
2: 250
3: 350
4: 450
5: 550
6: 650
7: 750
8: 850
9: 950
10: 1000
31.98
0.50
INC
Category
1: 59
2: 10–12
3: 13–14
4: 15–16
5: 417
Continuous
Category
1: male
0: female
Category
1: 5200
2: 201–300
3: 301–400
4: 401–500
5: 501–600
6: 601–700
7: 701–800
8: 801–900
9: 901–1000
10: 41001
482.81
216.19
Table 4. Comparison of CVM/RAM with CVM/TDM
CVM/TDM
Independent variables
WTP function
Constant
EDU
AGE
GENDER
ln(INC)
!u
!"
Mean of WTP
Median of WTP
log likelihood
Number of observations
CVM/RAM
Estimates
SE
Estimates
SE
9524.38
"809.45&
6.29
1770.04
860.87
9241.63
370.19
30.75
1298.95
1455.12
7599.02
4511
4343
"0.6484
249
2126.83
1804
15 042.12
"658.11&
13.15
415.01
837.66
6765.98
6756.93
11 552
11 474
"0.6145
249
8581.47
318.46
31.15
1177.57
1308.59
1123.86
1725.80
1372
Note: &Denotes significance at the 5% level.
We may compare the two mechanisms in terms of
binary prediction accuracies. The real binary answers
(yes or no) given by respondents can be compared
with the predicted binary answers made by TDM and
RAM to see which mechanism predicts the real
answer more accurately. According to each mechanism (Equation 2) and (Equation 6), the predicted
answers of RAM are made by comparing the
estimated WTP with the differently assigned bid
price plus estimated risk premium while those of
TDM are made by comparing the estimated WTP
with the differently assigned bid price only. There are
four possible combinations of prediction and realization as presented in Table 5.
We find that the RAM makes a more accurate
binary prediction than the TDM. The correct prediction ratio of RAM is 65.06% while that of TDM is
60.24%. Even though we cannot make a rigorous
assertion, this provides support for the RAM being
the purchasing decision mechanism for a new product
Estimating willingness to pay by risk adjustment mechanism
Fig. 1.
Distribution of WTP and risk premium
Table 5. Prediction accuracies of TDM and RAM
Realized answer
Yes
Predicted answer TDM Yes
No
RAM Yes
No
21.69%
23.29%
24.10%
20.88%
No
45
Brier score
16.47% 60.24%
38.55%
14.06% 65.06%
40.96%
having considerable performance uncertainty.
Companies that are about to release a new product
can use the RAM to make better demand predictions
for each possible price.
V. Concluding Remarks
As Kalish (1985) notes, normally there is uncertainty
with regard to a product’s experience attributes. This
is especially true for nonmarket goods such as new
products. So the actual value of the product to the
consumer is not known for sure. A risk averse
consumer will pay less for a risky choice in comparison with buying a product with a sure expected
value. One expects the risk premium to result in lower
demand at any price, the larger the degree of
valuation uncertainty.
Much literature dealing with uncertainty in consumer purchasing decisions assumes a distribution of
WTP and ad hoc purchasing rules. An example is
Wang (1997) where a distribution for WTP and two
arbitrary thresholds are assumed. However, no
explanation is given regarding the location of the
thresholds. In our approach, the thresholds can be
determined by the risk premium.
Future research should examine the dynamics of
how consumers adjust their risk premium over time.
We conjecture that the risk premium for a really new
product will decline over time as uncertainty
diminishes with experience and knowledge.
Marketers would be expected to change price levels
accordingly.
References
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Johannesson, M., Liljas, B. and Johansson, P.-O. (1998)
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1359–64.
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Preference uncertainty in non-market valuation: a
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contingent valuation surveys: a random valuation
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Estimating willingness to pay by risk
adjustment mechanism
Joo Heon Parka,* and Douglas L. MacLachlanb
a
Department of Economics, Dongduk Women’s University,
23-1 Wolgok-dong, Sungbuk-Gu, Seoul 136-714, Korea
b
Michael G. Foster School of Business, University of Washington, Seattle,
WA 98195-3226, USA
Measuring consumers’ Willingness To Pay (WTP) without considering the
level of uncertainty in valuation and the consequent risk premiums will
result in estimates that are biased toward lower values. This research
proposes a model and method for correctly assessing WTP in cases
involving valuation uncertainty. The new method, called Risk Adjustment
Mechanism (RAM), is presented theoretically and demonstrated empirically. It is shown that the RAM outperforms the traditional method for
assessing WTP, especially in a context of a nonmarket good such as a
totally new product.
Keywords: purchase decisions; willingness to pay; contigent valuation
method; adjustment mechanism
JEL Classification: D12; D81; M31
I. Introduction
Traditional consumer theory presumes that a consumer makes a purchasing decision by comparing his
or her own subjective value of a good (or service) with
its objective value. The objective value is the posted
price, while the subjective value is usually measured
by the so-called Willingness To Pay (WTP) that is
defined as the maximum dollar amount a consumer
is willing to pay for the good. If the posted price is
below a consumer’s WTP, the purchase would be
made; otherwise the purchase would be abandoned.
We call this the Traditional Decision Mechanism
(TDM).
A problem from the researcher’s perspective is that
a consumer never reveals his or her own WTP
directly, so that the WTP is not observable, whereas
the price is directly observed in a market.
However, consumers reveal their WTP indirectly
through transactions, so that their WTP can be
estimated by analysing their observed transaction
behaviours. While the TDM provides a theoretical
basis for inferring the WTP from purchase behaviours, it makes sense only when there is no uncertainty in consumers valuing the good of interest. If
consumers are uncertain over the value they place on
the good and do not know their WTP exactly as a
single value, they are unable to compare their WTP
with the price. Therefore, the TDM should not be
employed as a basis for inferring WTP for a good
when uncertainties are involved in valuation of
the good.
A question for us is how consumers make decisions
to purchase a good or not if they are ignorant of their
WTP for the good as a single value because of
uncertainties. As an answer, we propose a new
purchasing decision mechanism called a Risk
Adjustment Mechanism (RAM) that can be applied
*Corresponding author. E-mail: [email protected]
Applied Economics ISSN 0003–6846 print/ISSN 1466–4283 online ! 2013 Taylor & Francis
http://www.informaworld.com
DOI: 10.1080/00036846.2011.568404
37
38
under valuation uncertainty. A key characteristic of
RAM is that consumers are presumed to have in their
minds a statistical distribution of their WTP instead
of a single value. Furthermore, typically risk averting
consumers are expected to purchase a good if the
mean of their WTP is greater than the good’s price by
at least their risk premium. This is in contrast to
TDM in which consumers are expected to purchase a
good if their WTP is simply greater than the price. If
consumers are assumed to have aversion to risk, the
WTP of consumers using RAM who have decided to
purchase must be sufficiently greater than the price,
while it is theoretically enough to purchase for
consumers using TDM that the WTP be just greater
than the price even by a single penny. Therefore, from
a RAM point of view, the empirically assessed TDM
underestimates WTP by structure, since the TDM
devalues the WTP of a risk averting consumer at least
by his or her positive risk premium.
The notion that consumers are uncertain about
valuation is not new. Uncertainty in valuation has
been addressed by many researchers in a number of
different ways since Ready et al. (1995) raised the
issue explicitly in a context of Contingent Valuation
Method (CVM).
In fact, there are many well-known uncertain
factors that can and do influence a consumer’s
valuation. Wang (1997) argued that uncertainty
might exist in a commodity or service itself,1 in an
unpredictable market situation, in socio-demographic
variables, and in people’s uncertain preferences,
among others. Recognizing the existence of such
uncertainties, various studies have been made that
incorporate the uncertainties into the valuation
process.
Van Kooten et al. (2001) employed fuzzy set theory
to represent the underlying vagueness of preferences.
Hanemann and Kristro¨m (1995) and Wang (1997)
suggested adding a random variable to the utility
function. But they do not take the consumer’s
behaviour into account, in contrast with our proposed RAM where the risk premium reflecting the
consumer’s behaviour against uncertainties plays an
explicit role in the purchasing decision.
Consumers react to uncertainties in different ways,
leading to different market results. Mauro and
Maffioletti (2004) recognized consumers’ reaction to
uncertainties in the face of an incentive compatible
J. H. Park and D. L. MacLachlan
mechanism through experiments, and suggested that
ambiguity may be a relevant factor that affects
market prices and allocations. And a purchasing
decision mechanism like RAM that reflects consumers’ means of dealing with uncertainties can also
explain more efficiently the market results of a new
product with embedded uncertainties. For example,
we may also explain through the RAM the behaviour
difference between early adopters and slow movers
for new products.
Studies on how uncertainties affect the purchasing
decisions have been usually made in a hypothetical
context such as the CVM because the consumers’
attempts to resolve uncertainties are not observed
directly in real markets. We cannot deduce the
consumers’ uncertainty resolution behaviours from
their purchasing decisions.
However, valuation based on a CVM is often
questioned because of the so-called hypothetical bias
caused by difference in respondents’ behaviour
between a hypothetical situation and a real purchase
situation (Paradiso and Trisorio, 2001). Champ et al.
(1997), Johannesson et al. (1998) and Alberini et al.
(2003) used a calibration process with follow-up
questions regarding the certainty of the stated valuation in a CVM context in order to reduce the
hypothetical bias; for example, ‘very uncertain’, ‘not
likely’, ‘very sure’ and so on. But these assessments of
certainty could be as wrong as the valuation reports
because the assessments are also asked in the same
hypothetical situation as the valuation. Contrary to
these approaches, the RAM does not have the bias
problems caused by double sources of uncertainty
because it incorporates uncertainties into the model
instead of trying to handle uncertainties outside the
model. And furthermore, the RAM can explain and
reduce the hypothetical bias because it contains a risk
premium variable that should be measured in the
same situation where a product is evaluated.
Ariely et al. (2003) and Hanley et al. (2009) posited
that consumers probably have some range of acceptable values rather than specific WTP value for a
good. Following their thinking, we might assume that
consumers have in their heads a distribution of WTP
values rather than a single value. This assumption has
been used explicitly or implicitly by many researchers
such as Cameron and Quiggin (1994),2 Carson et al.
(1994) and Wang (1997).
1
One would face uncertainty over how much gains could be obtained from a nonmarket good that has never before existed.
Such uncertainty would not be removed until the good is actually consumed. For example, the value of improved environment
is not realized until it is actually provided and consumed.
2
Cameron and Quiggin (1994) said ‘Respondents seem not to hold in their heads a single immutable ‘true’ point valuation . . . .
At best, they may hold a distribution of values-amounts they would be willing to pay with some associated probability
density. This might be interpreted as ‘uncertainty’. Whenever they are asked to produce value for the resource, they make a
draw from this distribution and use it as a basis for their response to the current discrete-choice CV question’.
Estimating willingness to pay by risk adjustment mechanism
Wang (1997) and Chang et al. (2007) among others
asked a polychotomous choice WTP question with an
implicit assumption that the certainty of a respondent’s answer depends upon the difference between
the WTP distribution and the bid price. Especially,
Wang (1997) allows the trichotomous response
options of ‘yes’, ‘no’ or ‘don’t know (DK)’. In his
model, there is a vagueness band over which a
respondent has an option of ‘DK’ instead of making
a clear purchasing decision. It should be noted that
‘DK’ option is never allowed by structure in a real
purchase situation. The true outcomes made by
individuals in a real situation are only either ‘yes’ or
‘no’. The RAM has only dichotomous options of ‘yes’
or ‘no’, which is consistent with real market
transactions.
In the next section, the RAM is formally introduced along with the TDM. In Section III, the
estimation issue is addressed. An empirical comparison of WTP estimates by TDM and RAM is made in
Section IV to see how seriously the TDM underestimates the WTP. Some concluding remarks follow in
the last section.
II. Risk Adjustment Mechanism
A consumer decides to purchase a good (z) if his
or her utility can be increased by purchasing the
good, i.e.
!
"
Ii ¼ 1 if U p, yi " pðzÞ þ w&i ðzÞ ' Uð p, yi Þ
ð1Þ
Ii ¼ 0 otherwise
where U(.) is an indirect utility function; yi denotes
the ith consumer’s income; p is the price vector of all
other goods except good z; p(z) the price of good z;
wi&(z) the ith consumer’s WTP for the good z and Ii is
a binary response variable taking one for purchasing
and zero for not purchasing good z.
Since the WTP for a good is the maximum amount
of money that a consumer is willing to give up in
exchange for the good without decreasing the initial
utility level, the utility change caused by purchasing a
good is exactly the same as that caused by changing
the consumer’s income by the difference between
the price and the WTP for the good. Therefore the
left-hand side of the inequality in Equation 1 is the
utility level after purchase that the ith consumer could
attain by purchasing good z with its price p(z) paid
while the right-hand side is the utility level before
purchase. To sum up, a consumer purchases a good if
39
the utility after purchase is greater than the utility
before purchase.
Since the indirect utility function is monotonically
increasing in income y, the purchasing decision
mechanism (Equation 1) may be converted simply
as follows:
Ii ¼ 1 if w&i ðzÞ ' pðzÞ
Ii ¼ 0 otherwise
ð2Þ
The mechanism (2) means that a consumer purchases a good if the WTP is greater than the price,
which is usually presumed the purchasing mechanism
by traditional consumer theory and is called the
TDM in this article.
It is worth noting that the TDM works correctly
only if there is no uncertainty related with the
purchase. The TDM can be applied only when a
consumer knows his or her own WTP exactly as a
single value with certainty, and then can compare the
WTP with the price as suggested by Equation 2.
However, the decision mechanism is not as simple as
in Equation 2 if uncertainties are involved. Suppose
uncertainties stem from the benefits expected from
consuming a good z, the WTP is also uncertain and
should be treated as random because it totally
depends upon the benefits. Since the benefits of a
good are actually realized by consuming the good, a
consumer cannot know his or her WTP for the good
precisely until he or she consumes it. Therefore,
a consumer can only guess the WTP at the purchase
time when actual consumption is not yet realized. In
this article, the WTP realized after consumption is
called the ex post WTP and the WTP guessed before
consumption is called the ex ante WTP.
There is usually a difference between the ex post
and ex ante WTP’s, which depends upon the degree
of uncertainty. Hanley et al. (2009) argued that this
kind of uncertainty is reduced when a person has
more experience, i.e. the uncertainty might be
inversely related with experience with the good. One
would expect little difference between the two WTP’s
for standard and frequently used goods, because
consumers have already considerable experiences
with the amount of benefit that they can expect
from those goods. Contrarily, one would expect a
large difference between ex post and ex ante WTP’s
for the goods such as totally new products or
nonmarket goods that have never been traded in a
market before. Consumers are ignorant of how much
benefit they can obtain from such goods until actual
consumption.3
3
Although the ex post WTP is a more accurate indicator of the true value of a good and it would be desirable to know that
value when doing a cost benefit analysis and developing public policy, only the ex ante WTP is typically known or knowable
at that decision point.
J. H. Park and D. L. MacLachlan
40
At the purchase time, the ex post WTP is random
and is assumed to be distributed around the ex ante
WTP as
wi ðzÞ ¼ w&i ðzÞ þ ui ,
E ½ui ) ¼ 0,
Var½ui ) ¼ !u2
ð3Þ
where ui is an error term reflecting uncertainties
associated with consumption; wi(z) the ex post WTP;
wi&(z) the ex ante WTP. Note that the consumers are
posited to guess their own ex ante WTP as the mean
value of the ex post WTP distribution even at the time
of purchase. And the consumer is exposed to uncertainty to a degree of the variance of the error
term (!u2 ).
Now consider how a consumer would make a
purchasing decision when facing uncertainty and
being ignorant of the ex post WTP. The consumer
should make a purchase if the expected utility is
greater than the utility without purchase but otherwise he or she should not. This can be expressed as
Ii ¼ 1
Ii ¼ 0
if Ewi ðzÞ ½Uðp, yi " pðzÞ þ wi ðzÞÞ) ' Uðp, yi Þ
otherwise
ð4Þ
A consumer tends to avoid uncertainties if possible.
A consumer is willing to pay an incremental amount
of money if he or she is guaranteed to have a certain
offer instead of an uncertain offer. The amount of
money a consumer is willing to pay for a certain offer
of ensuring the mean value of an uncertain offer
instead of the uncertain offer is called the risk
premium of the uncertain offer. And the risk
premium of a new good z can be defined as Ri (z)
satisfying the following equation:
!
"
U p, yi " pðzÞ þ w&i ðzÞ " Ri ðzÞ
¼ Ewi ðzÞ ½Uðp, yi " pðzÞ þ wi ðzÞÞ)
ð5Þ
Therefore, combining Equations 5 and 4 and using
the fact that the indirect utility function U(.) is a
monotonically increasing function, yields the following purchasing decision mechanism:
Ii ¼ 1
if w&i ðzÞ " pðzÞ ' Ri ðzÞ
Ii ¼ 0
otherwise
theoretically by TDM, but the purchase should not
be always made by RAM. The RAM considers
consumers who decide to purchase the good as
having a much higher WTP than the price while the
TDM regards them as having just higher WTP than
the price. And thus the TDM would yield an
underestimation of WTP when there are
uncertainties.
III. Estimation
The RAM presented as in Equation 6 implies that
consumers take their own ex ante WTP together with
the price and risk premium into account to make a
purchase decision. The ex ante WTP is assumed to be
linearly determined as
w&i ðzÞ ¼ x0i " þ "i ,
"i * Nð0, !"2 Þ
ð7Þ
where xi is a vector of covariates, " the vector of
coefficients and "i the error term. Now the error terms
in Equation 7 should be distinguished from those in
Equation 3. The error term in Equation 3 denotes a
part of ex post WTP that a consumer cannot grasp at
the purchasing stage and is the source of uncertainty.
But the error term "i in Equation 7 denotes a part of
the ex ante WTP that a researcher cannot explain
with a vector of covariates even though a consumer
knows it completely. So the researcher regards the
ex ante WTP as being statistically distributed even
though a consumer knows it as a single value.
The log-likelihood functions implied by TDM and
RAM are presented as in Equations 8 and 9,
respectively.
TDM:
%
#
$&
n '
X
pðzÞ " x0i "
Ii ln 1 " !"
ln L ¼
!"
i¼1
#
$(
pðzÞ " x0i "
þ ð1 " Ii Þ ln !"
!"
ð6Þ
ð8Þ
RAM:
which we call the RAM. A consumer purchases the
good z if his or her ex ante WTP exceeds the price at
least by the risk premium; the consumer should not
purchase otherwise. This is a salient point differentiating the RAM from the TDM. Note that if there is
no uncertainty and the risk premium is zero, then
the RAM coincides with the TDM exactly. Actually,
the TDM need not distinguish the ex ante WTP
from the ex post WTP since it assumes no uncertainty
at all. If the WTP is greater than the price even by a
miniscule value, the purchase should be made
n '
X
%
#
$&
pðzÞ " x0i " þ Ri ðzÞ
Ii ln 1 " !"
!"
i¼1
#
$(
0
pðzÞ " xi " þ Ri ðzÞ
þ ð1 " Ii Þ ln !"
!"
ln L ¼
ð9Þ
where !S(+) is the cumulative distribution function of
the standardized normal distribution.
The TDM can only identify the coefficients (") of
the model (7) up to a scale transformation. Since
a common price is given to all the consumers,
Estimating willingness to pay by risk adjustment mechanism
the inequality condition of TDM can be converted as
follows:
"0 þ "1 xi1 þ + + + þ "1k xik þ "i ' pðzÞ
, ð"0 " pðzÞÞ þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
, #0 þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
ð10Þ
where #0 ¼ "0 " pz . Therefore the coefficients of "
can be identified only up to a scale transformation
because
#0 þ "1 xi1 þ + + + þ "1k xik þ "i ' 0
, $ð#0 þ "1 xi1 þ + + + þ "1k xik þ "i Þ ' 0
for $ 4 0
ð11Þ
The TDM cannot predict the value of WTP, but
only the probability of the inequality condition of
TDM being satisfied.
In contrast, the RAM can identify all coefficients
fully because different risk premiums are taken by
each consumer, even when a common price is given to
them. The inequality condition of RAM being
multiplied by a positive number, all coefficients can
be identified because the multiplication number ($)
can be estimated as the coefficient of the risk
premium (Ri(z)) as
"0 þ "1 xi1 þ + ++ þ "1k xik þ "i ' pðzÞ þ Ri ðzÞ
, ð"0 " pðzÞÞ " Ri ðzÞ þ "1 xi1 þ ++ + þ "1k xik þ "i ' 0
, #0 " Ri ðzÞ þ "1 xi1 þ ++ + þ "1k xik þ "i ' 0
, $ð#0 " Ri ðzÞ þ "1 xi1 þ + ++ þ "1k xik þ "i Þ ' 0 for $ 40
ð12Þ
However, the above identification of RAM is
possible only when the risk premium is observable.
Yet in reality this does not occur. The risk premium
associated with the good z is not actually observed.
We need to assume a model of risk premium and then
estimate it. It is already proven that there exists a
particular relationship like Equation 13 between the
risk premium and the attitude toward the risk if the
expected return of a risky offer is relatively very small
(Malinvaud, 1985).
Ri ¼ "
! 2 U00 ð yÞ ! 2 ri
,
¼
2U0 ð yÞ
2
ri ¼ "
U00 ð yÞ
U0 ð yÞ
ð13Þ
where Ri is the risk premium for a risky offer with its
variance ! 2, and ri the Arrow–Pratt absolute Risk
Aversion Index (ARAI) reflecting a person’s attitude
toward risk. A respondent with a higher ARAI has a
higher risk premium. Now if a Constant Absolute
4
41
Risk Aversion (CARA) utility function (Equation 14)
is assumed for simplicity, then the ARAI is fixed at ri
whatever the consumer’s income (y) level is
1
CARA utility function: Uð yÞ ¼ " e"ri y
ri
ð14Þ
Assuming that each consumer has a CARA utility
function, the log-likelihood of RAM (Equation 9)
can be transformed as
%
#
$&
n '
X
pðzÞ " x0i " þ !u2 ðri =2Þ
Ii ln 1 " !"
!"
i¼1
#
$(
0
pðzÞ " xi " þ !u2 ðri =2Þ
þ ð1 " Ii Þ ln !"
!"
ð15Þ
ln L ¼
The risk premium having been replaced by !u2 ri =2,
the RAM turns out not to be identifiable because a
parameter !u2 is added. So we need an additional
assumption to make the model parameters identified.
For simplicity, we assume that the degree of uncertainty a researcher faces is exactly the same as that
amount a consumer faces, i.e. !u2 ¼ !"2 . Then the
log-likelihood of Equation 15 becomes identifiable
because the new parameter !u2 is out of the estimation
process.
The log-likelihood functions of Equation 15 are,
however, based upon market transactions so that
they cannot be applied to the estimation of WTP for a
nonmarket good such as a totally new product. By
definition no transaction of a nonmarket good has
been made and no information pertaining to the
market such as the price is available to be observed.
Hence a different method is needed to estimate the
WTP for a nonmarket product.
The CVM has been used extensively for estimating
the value of nonmarket goods. See for example,
Cummings et al. (1986) and Carson and Mitchell
(1994).4 The CVM has also been employed as a pretest-market evaluation method in the marketing
arena (Cameron and James, 1987; Wertenbroch and
Skiera, 2002).
The CVM uses a sample survey asking respondents
to value a nonmarket good of interest in a hypothetical situation where no actual transaction is required.
And we are free to give each respondent different
prices in CVM because the respondents are supposed
to answer ‘yes, I’ll buy it’ or ‘no, I won’t buy it’ with
the price given to them. Then the log-likelihood
functions of TDM and RAM with CARA
According to Carson and Mitchell (1994), up to 1994 there had been about 1600 CVM studies and papers from over 40
countries on many topics including transportation, sanitation, health, the arts and education as well as the environment.
J. H. Park and D. L. MacLachlan
42
assumption are somewhat changed as follows:
risk premium involved in the purchasing decision,
we do not need the CARA assumption but just
employ the likelihood including the risk premium
itself; otherwise, we should make an assumption like
CARA in order to get an estimable likelihood
function.
TDM:
n '
X
%
#
$&
pi ðzÞ " x0i "
Ii ln 1 " !"
!"
i¼1
#
$(
pi ðzÞ " x0i "
þð1 " Ii Þ ln !"
!"
ln L ¼
ð16Þ
RAM:
%
#
$&
n '
X
pi ðzÞ " x0i " þ !u2 ðri =2Þ
Ii ln 1 " !"
ln L ¼
!"
i¼1
#
$(
pi ðzÞ " x0i " þ !u2 ðri =2Þ
þ ð1 " Ii Þ ln !"
!"
ð17Þ
It should be noted that there is no parameter
identification concern for both TDM and RAM in
Equations 16 and 17 because ! " can be identified as
the inverse of the coefficient of pi(z). All the coefficients are fully identified for both TDM and RAM
even without the assumption of !u2 ¼ !"2 .
The last problem that must be solved to make the
log-likelihood function workable is that an individual
consumer’s ARAI (ri) cannot be observed directly.
But if we make the assumption of CARA utility, we
can infer the ARAI indirectly by observing how the
consumer behaves when he or she faces uncertainty
and can avoid it by paying some risk premium. If we
can observe how much uncertainty is involved and
how much risk premium the consumer pays, then we
can infer the consumer’s ARAI indirectly by the
relationship (13) under the CARA assumption.
Once an inferred ARAI is available, maximization
of the log-likelihood of RAM (Equation 17) yields
the estimates of all parameters in RAM.
Now it should be noted that the reason for
assuming CARA utility function is that the risk
premium in Equation 9 cannot be observed directly.
With CARA utility function, we can present the risk
premium as a (reduced ) function of the ARAI that
can be subsequently inferred from observation of
another risk averting behaviour. However, the
CARA assumption is not necessary to connect the
risk premium with the ARAI. Actually, the risk
premium is a function of the ARAI with any utility
function. But we need the CARA utility function in
order to get a reduced form like Equation 13 with
which the risk premium is substituted so that an
estimable likelihood function such as Equation 17
could be obtained. Therefore, if we can observe the
5
IV. An Empirical Study
In this section, the RAM is compared to TDM with a
real CVM data. The CVM data were collected to
obtain information about consumers’ WTP for a new
customized cell phone service. In this case, a Korean
mobile telecommunication company attempts to
determine the best rate program for each customer
by analysing actual data. The company calculates due
amounts for the past 3 months under all rate
programmes available to find the lowest due
amount and its associated rate programme and then
informs customers of this information. This is a
totally new service that has never been offered before.
Customers are thereby not sure how much benefit
they can actually obtain from the service.
Each respondent was asked his or her WTP for the
new service in different formats – open-ended or
close-ended.5 However, the data set used here is only
the close-ended format data set because both TDM
and RAM are designed to be applicable only to a
dichotomized reaction of such as ‘yes’ or ‘no’ in the
close-ended format. For the close-ended format
survey, potential respondents were divided into
eight groups to which a bid price was chosen and
assigned from eight different predetermined bid
prices that are distributed from 2000 Won to
9000 Won.
The survey was fielded around Seoul and its
suburbs in South Korea by six trained interviewers
for 2 weeks during spring of 2008. We conducted 400
interviews with the close-ended CV formats. Of them,
we succeeded in getting 347 responses, resulting in a
response rate of 86.8%. Deleting questionnaires for
which there were no responses to key questions, we
obtained 249 final responses ready to be used in
estimation. Table 1 presents the percentages of ‘yes’
votes for each bid price. The higher percentage of
‘yes’ votes occurs with the lower bid prices, which is
consistent with the law of demand.
This CV survey is different from others in that,
along with the question regarding the WTP for the
new product, a respondent was also asked his or her
The open-ended format asks how much a respondent is willing to pay for the product while the close-ended format asks
whether the respondent is willing to pay a bid price for the product.
Estimating willingness to pay by risk adjustment mechanism
Table 1. Percentage of ‘Yes’ votes by bid prices
Bid price (Won)
Sample size
2000
3000
4000
5000
6000
7000
8000
9000
Total
30
29
30
34
36
28
30
32
249
Table 2. Frequencies of the consumers’ risk premium for the
game and absolute risk aversion index
Percentage of
‘Yes’ votes
60.00
65.52
60.00
38.24
38.89
28.57
30.00
40.63
WTP for a gamble from which the ARAI of each
respondent could be conjectured. Each respondent
was asked to play a gamble in which tossing a coin,
a player is paid 15 000 Won if the head is upside, or
5000 Won otherwise. The expected value of the
gamble is 10 000 Won. The respondent is asked how
much he or she was willing to pay for playing the
gamble, in an auction manner. A respondent was
asked initially whether he or she were willing to pay
6000 Won or not for the gamble. If the answer was
‘no’ for the initial question, the auction stopped and
the WTP for the game was regarded as 5000 Won
that was the guaranteed minimum amount to pay
back to the player. However, if the answer was ‘yes’,
the auction continued by asking the same question
but the price raised by 1000 Won, i.e. whether they
were willing to pay 7000 Won or not for the game.
Subsequently, if the answer was ‘no’, the WTP was
regarded as 6000 Won. This process was continued
until the respondent gave an answer of ‘no’. But the
maximum price given to the respondents was set at
10 000 Won because a risk averting respondent is
believed to be willing to pay no more than the
expected value of the game.
The risk premium of each respondent can be
obtained by subtracting the expected return of the
game, 10 000 Won from the WTP of each respondent
inferred through the series of question items. For
example, if a respondent’s inferred WTP for the game
is 7000 Won, then his or her risk premium of the
game is calculated to be 3000 Won. Then with the
CARA assumption, the ARAI can be also calculated
by using the relationship (13). Since the variance of
the returns of the game is 25 million Won, the ARAI
of each respondent is calculated as
ri ¼
2Ri ð gÞ
25 000 000
43
ð18Þ
where Ri(g) is the risk premium of ith respondent.
The risk premiums and the calculated ARAI’s
are presented with their frequencies in Table 2.
Risk premium
ARAI
Frequencies
5000
4000
3000
2000
1000
0
Average or total
0.00040
0.00032
0.00024
0.00016
0.00008
0.00000
0.00020
103
57
45
28
10
6
249
The ARAI’s obtained from this gamble experiment
are used in the log-likelihood of RAM (Equation 17).
The ex ante WTP for the service is assumed to be a
linear function as
w&i ¼ "0 þ "1 EDUi þ "2 AGEi þ "3 GENDERi
þ "4 ln INCi þ "i
ð19Þ
where the covariates are explained in Table 3.
The coefficient estimates of the WTP model
(Equation 19) and the estimated mean value of the
WTP’s of the respondents in the sample are reported
in Table 4. The left column reports the TDM
estimation results and the right column reports the
RAM. It is disappointing to learn that most of
coefficient estimates are not statistically significant
except that of education level. Furthermore, it should
be recognized that the WTP model of Equation 19
itself is also somewhat lacking as a model predicting
how much a potential consumer is willing to pay for
the service. However, the main purpose of this
empirical study is not estimating the coefficients but
demonstrating how much lower the TDM would
estimate the WTP, and how important the risk
premium is when estimating the WTP by using
purchasing behaviour data. Even with a perfect
model, an estimation process without considering
the risk premium could not predict the WTP as
precisely as would an estimation process that does so,
if a consumer takes the risk premium into account
when making a purchasing decision. What is important is not the RAM’s predictive power itself but its
relative prediction superiority over TDM.
An initial basis of comparison can be made in
terms of the distributions of the WTP obtained from
TDM and RAM. Figure 1 presents the distribution of
WTP estimates by the two mechanisms and the risk
premiums. As expected, the mean of WTP estimates
is much higher for the RAM than for the TDM. The
difference reflects the risk premium the consumers are
willing to pay for avoiding the uncertainties associated with the new cell phone service.
J. H. Park and D. L. MacLachlan
44
Table 3. Definition and sample statistics of variables
Variable
Definition
Format in survey
Format in use
Mean
SD
EDU
Education (unit: years of schooling)
Continuous
1: 10
2: 11.5
3: 13.5
4: 15.5
14.00
2.03
AGE
GENDER
Age
Gender
21.24
0.50
Monthly household
total income after tax
deduction (Unit: 10 000 Won)
Continuous
Category
1: male
0: female
Continuous
1: 200
2: 250
3: 350
4: 450
5: 550
6: 650
7: 750
8: 850
9: 950
10: 1000
31.98
0.50
INC
Category
1: 59
2: 10–12
3: 13–14
4: 15–16
5: 417
Continuous
Category
1: male
0: female
Category
1: 5200
2: 201–300
3: 301–400
4: 401–500
5: 501–600
6: 601–700
7: 701–800
8: 801–900
9: 901–1000
10: 41001
482.81
216.19
Table 4. Comparison of CVM/RAM with CVM/TDM
CVM/TDM
Independent variables
WTP function
Constant
EDU
AGE
GENDER
ln(INC)
!u
!"
Mean of WTP
Median of WTP
log likelihood
Number of observations
CVM/RAM
Estimates
SE
Estimates
SE
9524.38
"809.45&
6.29
1770.04
860.87
9241.63
370.19
30.75
1298.95
1455.12
7599.02
4511
4343
"0.6484
249
2126.83
1804
15 042.12
"658.11&
13.15
415.01
837.66
6765.98
6756.93
11 552
11 474
"0.6145
249
8581.47
318.46
31.15
1177.57
1308.59
1123.86
1725.80
1372
Note: &Denotes significance at the 5% level.
We may compare the two mechanisms in terms of
binary prediction accuracies. The real binary answers
(yes or no) given by respondents can be compared
with the predicted binary answers made by TDM and
RAM to see which mechanism predicts the real
answer more accurately. According to each mechanism (Equation 2) and (Equation 6), the predicted
answers of RAM are made by comparing the
estimated WTP with the differently assigned bid
price plus estimated risk premium while those of
TDM are made by comparing the estimated WTP
with the differently assigned bid price only. There are
four possible combinations of prediction and realization as presented in Table 5.
We find that the RAM makes a more accurate
binary prediction than the TDM. The correct prediction ratio of RAM is 65.06% while that of TDM is
60.24%. Even though we cannot make a rigorous
assertion, this provides support for the RAM being
the purchasing decision mechanism for a new product
Estimating willingness to pay by risk adjustment mechanism
Fig. 1.
Distribution of WTP and risk premium
Table 5. Prediction accuracies of TDM and RAM
Realized answer
Yes
Predicted answer TDM Yes
No
RAM Yes
No
21.69%
23.29%
24.10%
20.88%
No
45
Brier score
16.47% 60.24%
38.55%
14.06% 65.06%
40.96%
having considerable performance uncertainty.
Companies that are about to release a new product
can use the RAM to make better demand predictions
for each possible price.
V. Concluding Remarks
As Kalish (1985) notes, normally there is uncertainty
with regard to a product’s experience attributes. This
is especially true for nonmarket goods such as new
products. So the actual value of the product to the
consumer is not known for sure. A risk averse
consumer will pay less for a risky choice in comparison with buying a product with a sure expected
value. One expects the risk premium to result in lower
demand at any price, the larger the degree of
valuation uncertainty.
Much literature dealing with uncertainty in consumer purchasing decisions assumes a distribution of
WTP and ad hoc purchasing rules. An example is
Wang (1997) where a distribution for WTP and two
arbitrary thresholds are assumed. However, no
explanation is given regarding the location of the
thresholds. In our approach, the thresholds can be
determined by the risk premium.
Future research should examine the dynamics of
how consumers adjust their risk premium over time.
We conjecture that the risk premium for a really new
product will decline over time as uncertainty
diminishes with experience and knowledge.
Marketers would be expected to change price levels
accordingly.
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