Incorporating Zero Values in the Economic Valuation of Environmental Program Benefits Benjamin Reiser Department of Statistics, University of Haifa, Haifa 31905, Israel and Mordechai Shechter Department of Economics and Natural Resource & Environmental Research Center University of Haifa, Haifa 31905, Israel Send correspondence to: Prof. Benjamin Reiser Department of Statistics University of Haifa Haifa 31905, Israel E-mail: [email protected]
Transcript
Incorporating Zero Values in the Economic Valuation ofEnvironmental Program Benefits
Benjamin ReiserDepartment of Statistics, University of Haifa, Haifa 31905, Israel
and
Mordechai ShechterDepartment of Economics and Natural Resource & Environmental Research Center
University of Haifa, Haifa 31905, Israel
Send correspondence to:Prof. Benjamin ReiserDepartment of StatisticsUniversity of HaifaHaifa 31905, IsraelE-mail: [email protected]
Incorporating Zero Values in the Economic Valuation ofEnvironmental Program Benefits
Benjamin Reiser and Mordechai Shechter∗
Abstract
The contingent valuation method estimates individuals’ willingness to pay (WTP) for
non-market environmental assets via preferences elicited by either open-ended or
dichotomous choice questions. Traditional analysis of such data has tended to ignore
zero WTP values, or treat then in a unsatisfactory manner. Recently, spike models,
which explicitly allow for and incorporate zero responses have been suggested. The
paper extends the spike model approach to allow for explanatory covariates, and
shows how standard computer software can be used to carry out the computations.
In addition, the paper develops estimates of mean or median willingness to pay as a
function of these covariates.
Key words: Contingent valuation, logistic regression, willingness to pay, spikemodel.
∗ We thank Sagi Nevo and Natalia Zaitsev for valuable and dedicated research assistance.
1. Introduction
Economic valuation of societal benefits from environmental improvements and the their
associated costs are today essential informational inputs in environmental policy making. The
attributes of environmental quality, a public good, require the adoption of different approaches
from those customarily employed in studies of private (market) goods, in carrying out such
valuations. Two approaches for the valuation of public goods have been used in this context.
The first employs so-called “indirect” valuation methods, which infer an implicit value for the
public good from observable market prices of private goods1. The second approach - the so-
called “contingent valuation method” (CVM) - is viewed, on the other hand, as a “direct”
approach, because it uses a questionnaire to elicit from surveyed respondents in a direct
fashion their own valuations of posited changes in the quantity (or quality) of the non-market
public good.
A typical CVM scenario involves inquiry into the amount of money the individual
would be willing to pay for a change in government policy concerning, for example, pollution
control, scenic area regulation, or the supply of environmental amenities. The “consumers”
(i.e., the respondents) in this “contingent” market scenario are typically provided with a
detailed description of the good being evaluated, and are then asked questions concerning their
willingness to pay (WTP) - i.e., their subjective “price” - for the good under study, or in
utility-theoretic terms, what change in income would leave the respondent’s utility level
unchanged. In addition, questions are posed about the demographic and socioeconomic
characteristics of the respondent (age, sex, education, income, etc.), as proxies for variations
in individual preferences.
The Contingent Valuation Method (CVM) has become an important tool for estimating
willingness to pay (WTP) for non-market environmental goods. Preferences are typically
elicited by either open-ended questions which inquire after the individual’s WTP, or
referendum (closed-ended) dichotomous choice questions. In either case, WTP (over some
population of interest) is generally assumed to be a random variable from some continuous
distribution. Popular choices for this distribution include the normal, lognormal, logistic, log-
logistic distributions. Empirical open-ended data (e.g., Langford and Bateman, 1993) indicates
1 For example, Shechter (1991) posited a relationship between a public good, air quality and two private goods, residentialhousing and medical services. Changes in air quality levels are expected to shift the demand schedules for these marketgoods: Air quality affects housing prices as well as the demand for preventive and medical care that are associated with theeffect of air pollution on health. From the extent of these shifts, implicit prices of the public good in question can beinferred.
that the distribution of WTP is generally skewed to the right with a substantial lumping at zero
WTP.
Typically, these zero values are either ignored (as in carrying out a normal or logistic
analysis) or excluded to allow a log transformation. Recently (McFadden, 1994; Kriström,
1995; Johansson et al., 1994) there has been interest in models which explicitly allow for a
positive probability of zero WTP in the analysis of both open-ended or dichotomous choice
CVM data. Kristrom refers to such a model as a spike model. McFadden (1994) considers a
rather complex distributional model for WTP which depends on real discretionary income (for
which information is often unavailable) and an “income elasticity” parameter. He discusses
both open-ended and referendum data. Ignoring these complicating features he essentially
suggests for the distribution of WTP a mixture of a point mass at zero with some continuous
distribution such as lognormal, gamma or weibull or alternatively censoring a continuous
distribution at zero. He also allows for consumer characteristics (covariates), but in his mixture
models does not allow the possibility of a zero WTP to depend on any covariates. This
mixture model is similar to bioassay models constructed to take natural mortality into account
(e.g., Collett, 1991). Kristrom (1995) and Johansson et al. (1992) only consider the
dichotomous choice framework but add a second question in order to determine whether the
individual is willing to contribute at all to the project. Their underlying model is based on
censoring but can readily be extended to the mixture model. In addition, their papers do not
take covariate information into account.
The mixture model implies that the population of interest can be considered to be
composed of two sub-populations. One sub-population is simply not willing to pay at all for
the good in question, while the other sub-population is willing to pay and has a continuous
WTP distribution. We find this to be more appealing than the model based on censoring and
note that McFadden found it to be empirically preferable for the data he studied.
*
In Section 2 we discuss the mixture model as applied to open-ended data. We show
that covariate effects on the probability of zero WTP can be readily introduced and that the
likelihood function factors in such a way that standard software can be used to estimate the
model. In Section 3 we examine the mixture model for the closed-ended case when the
additional question suggested by Kristrom is used. Again, factorization of the likelihood
function permits the use of standard methods to estimate the parameters and covariate
information can be readily introduced. Section 4 provides numerical examples of the methods
described in Sections 2 and 3, using data from a study on nonuse value of Mt. Carmel National
Park in Israel. Finally, Section 5 presents a concluding discussion.
2. Open-Ended Data
2.1 No covariate information
First consider the case where no covariate information on the individuals is available. Let p
denote the probability that an individual chosen at random has WTP=0 and let F(x), x > 0
represent the continuous cumulative distribution function (cdf) for the sub-population which is
willing to pay. Then the cdf for an open-ended response w is
P WTP ww
p wp p F w w
( ),,
( ) ( ) ,< =
<=
+ − >
0 00
1 0(1)
For an observed random sample of n individuals let δi = 1(0) if the ith individual’s observed
WTP is zero (wi > 0). Consequently the likelihood function can be written as proportional to
( ) ( )[ ] ( )p p f w p p f wii
ni
i i ii
ni
w i
δ δ δ δ=
− −= >
∏ − = −∏ ∏1
1 1
1 01 1 ( ) (2)
where f is obtained as the derivative of F and wi o>
∏ represents the product taken over all
individuals with observed WTP > 0. Typically f (and F) will depend on unknown parameters
which will need to be estimated.
A reasonable choice for F would be
( )F zz= −
Φ log µ
σ(3)
with ( )Φ t et u= ∫− ∞
−12
22
π du and z o> , i.e. a lognormal distribution. This would reflect the
right skewness discussed in the introduction. Other skewed distributions such as weibull or
gamma are also reasonable candidates.2 Conventional choice of the lognormal has the
advantage of allowing the use of standard computer programs and familiarity to economists.
The likelihood function (2) breaks up into two separate pieces
2 McFadden (1994) suggests use of the highest likelihood or the Akaike information criteria inorder to choose the “best” parametric model. Whether these criteria can effectively distinguishbetween various skewed models without exceedingly large amount of data is questionable andneeds to be further studied.
( )i
ni ip p
=−∏ −
1
11δ δ (4a)
and
( )∏>wi
if w0
(4b)
which can be maximized separately to provide maximum likelihood estimates (MLEs) of the
unknown parameters.
From (4a) we obtain the MLE
$pn
i=∑ δ
i.e. the observed percentage of zero WTPs in the sample. Maximizing (4b) while assuming (3)
results in the usual MLEs for µ and σ based solely on the positive WTPs.
One is frequently interested in the mean or median of WTP for the population under
study. From (1) we have that
Mean = ( ) ( )1 − ∫∞p wf w dwo (5a)
and
Median = F p
p
pp
− −−
<
>
1121
12
12
0
,
,
(5b)
which for the lognormal case (3) results in
Mean = ( )12
2− +p e
µ σ (6a)
and
Median
p
pp
p
=+
−
−
<
≥
−exp ,
,
µ σφ 1121
12
012
(6b)
Note that ( )Φ − 1 p * denotes the p* percentile point of the standard normal distribution.
Estimates of the Mean and Median can be obtained by substituting the MLEs of the unknown
parameters in the above formulae.
Distributions other than the lognormal can be used in the above simply by substituting
the appropriate F and f into the above formulae. It is more convenient to use distributions such
as the gamma or weibull for which programs to compute the MLEs are readily available.
2.2 Covariate information
In many situations covariate information (Age, Sex, etc.) is available for each individual. This
covariate information may effect both the parameters in the distribution F and p (that is the
probability of zero WTP may be influenced by age, sex, etc.). In order to keep the notation
simple, however, let us assume that there is only one covariate and denote by xi the covariate
value for individual i. Both p and F (or f) can now change with the individual i as xi changes.
Introducing the subscript i on p, F (or f) in order to denote the effect results in (4) becoming
( )p pi i i ii
n δ δ1 1
1−∏ −
=(7a)
and
( )wi o
i if w>
∏ (7b)
each of which can be optimized separately.
First consider (7a). We need to model pi as a function of xi in order to proceed. A
convenient formulation would be to assume that
log logit p pp xi i
i o i= −
= +1 1α α (8)
An alternative formulation would be to use the probit transformation. The use of (8) in (7a)
results in the usual logistic regression analysis where the dependent variable for each individual
is simply 1 or 0 according to whether her WTP is zero or greater. The likelihood factorization
in (7) shows that this analysis has no connection with the actual nonzero WTP values.
Standard computations provide estimates of α o, α1 , and the pi.
Turning to (7b), let us first consider the lognormal situation. Following the usual
conventions we assume that σ is constant while µ varies in a linear manner with xi,
i.e., µ β βi ix= +0 1(9)
The use of (9) together with (3) in (7b) results in just the likelihood function for a lognormal
linear regression model considering only the individuals who provide a positive WTP.
Consequently standard regression computations in the log scale provide estimates of β0, β1, σ
and the µi. The Mean and Median are now a function of the covariate xi and thus in (6) p and µ
need to be replaced by pi and µi respectively. Estimates can be obtained as before.
Extending (8) and (9) to more than one explanatory variable is trivial. It is important
to note that the “α‘s” and “β‘s” in (8) and (9) respectively will in general not be the same.
Furthermore, in the case of many potential explanatory variables (covariates) those which are
important for estimating p may not be important for estimating µ. Due to the likelihood
factorization, model building techniques can be used to pick the important covariates in (8) and
(9) completely independently. Although gamma or weibull regression models are not as well
known as the (log)normal model, software to carry these out does exist and consequently they
provide an alternative to (3).
3. Closed-Ended Data
3.1. No covariate information
Consider now a dichotomous choice situation with an additional question, as suggested by
Kristrom (1995). The first question asks whether or not the individual is at all prepared to
contribute to the project. If the response is positive the second question asks whether the
individual is willing to contribute A, where A is one of the possible bids in the study. For
individual i, let Si = 1(0) if the response to the first question is yes (no) and set Ti = 1(0) if the
response to the bid Ai is yes (no). We further assume that the underlying probability
distribution according to which the individual responds to the question is described by (1) with
w being replaced by the bid Ai presented to the i-th individual. (Si ,Ti ) can take on the values:
a) (1,1)
b) (1,0)
c) (0,0)
where in c) the second 0 is implied since for Si = 0 there is no follow-up question. For the i-th
individual the likelihood function corresponding to these three possibilities is
a) (1-p)(1-F(Ai))
b) (1-p) F(Ai)
c) p
resulting in the overall likelihood:
( ) ( ) ( )[ ] ( ) ( )( )[ ]p p F A p F ASi
S Ti
S T
i
ni i i i i1 1
11 1 1− −
=− − −
∏ ( )
= ( ) ( )[ ] ( )[ ]{ }p p F A F AS S
i
ni
S Ti
S T
i
ni i i i i i1
1
1
11 1−
=
−
=−
∏ −∏( ) ( )
= ( ){ } ( )[ ]( ) ( )[ ]p p F A F AS S
i
ni
Ti
T
i Si i i i
i
( )
,
1
1
1
11 1−
=
−
=−∏ −
∏ (10)
where i Si, =
∏1
denotes taking the product over all individuals with Si = 1.
The likelihood function (10) breaks up into two separate parts
( )( )p pSii
n Si1
11−
=∏ − (11a)
and
( )[ ] ( )[ ]i Si
iTi
iTiF A F A
, =
−∏ −1
1 1 (11b)
From (11a) the MLE of p is
p
ii
nn S
n^ =
− ∑=1 (12)
i.e. the observed percentage of those unwilling to participate at all. F will depend on unknown
parameters which require estimation. Various choices of F are possible.
Assuming a lognormal distribution results in
( ) ( ) ( )F AA
Aii
o i i= −
= + =Φ Φ Φlog
logµ
σβ β η1 (13)
with β µ σ βσ
η β βo i o iA= − = = +/ , , log1 11 . Use of (12) together with (11b) results in the
usual probit model in a log scale.
Assuming a log-logistic distribution results in
( ) ( )( )
( )( )F A
AAi
o i
o i
i
i=
++ +
=+
exp logexp log
expexp
β ββ β
ηη
1
11 1(14)
which combined with (11b) results in the usual logit model in a log scale. Consequently
standard probit and logit regression results apply. Computer software for these is readily
available. Note that following (11b) the computations are carried out only on the individuals
who are prepared to participate ( )Si =1 .
For the computation of Mean (Median) WTP, the general result given in (5) applies
and for F lognormal (6) holds. The estimates for the unknown parameters in (6) will be
obtained from (12) and the probit analysis described above. For F log-logistic, the
corresponding population Means and Medians are obtained from (5) to be
Mean = ( ) ( )
( )1
11
1 11
− −>
p oπ β ββ π β
βexp ( /sin /
, (15a)
and
Median = exp
log( ),
,
1 2 12
012
0
1
− −
<
≥
ββ
p
p(15b)
where for β1 1< , the population mean does not exist. Estimates for the population Mean and
Median are obtained from (15) by substituting for the unknown parameters their MLEs
obtained from (12) and the logit regression analysis described above.
3.2 Covariate Information
Again, for the sake of simplicity of notation, we assume one covariate taking the value x i for
individual i and use p and Fi i (or f i ) to denote the effect of the individual on p and F (or fi ).
Consequently the likelihood (11) can now be written as the product of
p piS
iS
i
ni i( ) ( )1
11−
=−∏ (16a)
and
( )[ ] ( )[ ]i Si
i iTi
i iTiF A F A
, =
−∏ −1
1 1 (16b)
each of which can be optimized separately.
In order to proceed we need to assume a model for p i as a function of x i . As in
Section 2.2 we typically assume that
log it p xi o i= +α α1 (17)
Alternatively a probit model can be used. Thus a standard logit or probit analysis based on all
the individuals according to their response to the first question can be used to estimate the p i
as a function of x i .
The analysis of (16b) is very similar to that of (11b) as discussed in Sect. 3.1. We
generalize ηi to include x i , i.e.
η β β βi i iA x= + +0 1 2log (18)
and proceed as above with either a probit or logit regression using a log scale for the bids.
This regression only includes individuals who expressed a willingness to contribute in the first
question.
For computation of Mean (Median) WTP (5) applies after replacing p, f and F
with p f and Fi i i, , respectively. As in the open-ended case the Mean (Median) now depends
on the covariate x i . In the lognormal case we thus obtain a generalized version of (6) to be
( )Mean p ei i i= − +12 2µ σ / (19a)
where as before β σ1 1= , but now β β µσ0 2+ = −x i i
and
Median
p
pp
pi
i i
i
=+
−−
<
≥
−exp ,
,
µ σφ 11
21
12
012
(19b)
For the logistic assumption on Fi we obtain a generalized version of (15) to be
( ) ( )[ ]( )Mean p
xi i
i= −− +
10 2 1
1 1
π β β ββ π β
exp /
sin /, β1 1> (20a)
and
Medianp x p
pi
i i i
i=
− − +
<
≥exp
log( ) ( ) ,
,
1 2
0
1212
0 2
1
β ββ (20b)
The modeling described above for one covariate can readily be extended to k
covariates, say x x xi i ki1 2, , ..., by replacing α1x i by j
kj jix
=∑
1a in (17) and β2x i by
j
kj jix
=+∑
11β
in (18). Estimates of the Mean (Median) are obtained by substituting in (20) the estimates of
the unknown parameters. It should be again emphasized that the covariates important for
estimating p i need not be identical to those important for estimating ηi .
3.3 Question Order
The order of the two questions described in Sec. 3.1 can be reversed. The first question would
be whether the individual is willing to contribute a specified bid A and if the response is
negative the second question asks whether the individual is prepared to contribute at all. An
analysis parallel to that described in Sec. 3.1 results in a likelihood identical to (10).
Consequently the methodology described above can also be used for the reordered questions.
However, there is one logical difference between the two situations. Previously the bid
presented to the individual could not influence his response to the question on his willingness
to contribute at all; but now, due to the reversal of the order, it may. Consequently in this case
the bid A i needs to be considered as a potential explanatory variable for p i .
4. An Application
As an example of the analysis of spike models we examine the data of a CVM survey discussed
by Shechter et al. (1997). The paper analyzes real donations in the context of natural resource
damage valuation, as an explicit expression of use and nonuse value, and compares them with
CVM-derived valuations. The donations were raised in the wake of severe damage suffered by
a natural resource in Israel. In the summer of 1989 large areas of Mt. Carmel National Park
were destroyed by fire. The park is located in the Carmel mountain range in northern Israel. It
contains one of the few natural forests in the country and a wildlife reserve, whose major
function is the replenishment and reintroduction into a natural habitat of threatened species to
the region. Recreational activities are mainly restricted to hiking and picnicking. Significant
portions of the park's natural forest stand as well as the wildlife reserve were consumed by the
fire. For the subsequent three years, portions of the burned area were unusable or had limited
public access. While the fire was still raging, in anticipation of the need to finance active
intervention to rehabilitate damaged areas, a national media campaign was launched to solicit
donations. The "Carmel Fund" was established as the depository for all the money collected.
The fund has been managed by the Ministry of the Environment to finance ongoing research,
rehabilitation and preservation activities in the aftermath of the fire.
A follow-up CVM survey was carried out in 1993. The respondents were explicitly
queried about their willingness to donate in order to preserve the park. The survey attempted to
reproduce as much as possible the conditions of the fund raising campaign in order to specifically
test for the existence and magnitude a nonuse value component in WTP. Information on various
demographic and socio-economic characteristics (explanatory variables) was also obtained for
each respondent.
Two sub-populations were sampled in the follow-up study: a donor group which
consisted of people who had actually pledged donations in 1989 and a general population (GP)
“control” group. We will only discuss the GP sample in this paper. This sample was randomly
selected from the telephone directory. Respondents in this sample were divided into two
groups: 200 respondents were presented with an open-ended version of WTP questions
(denoted OE), and a second group of 500 respondents were given dichotomous choice
(closed-ended) WTP questions (denoted DCH).3 The respondents who were given a
dichotomous choice question and replied negatively to the presented bid were further asked
how much they were willing to contribute. In order to exemplify the methodology discussed in
Sec. 3 we ignored the actual amount an individual in the DCH group is willing to contribute,
and simply took any positive (zero) amount as showing a willingness (unwillingness) to
contribute.
The Appendix lists only those explanatory variables found to be useful in our modeling
below. Excepting Bid, the other variables in the Appendix are taken as categorical and are
3 A detailed discussion of the survey is given by Shechter et al. (1996).
defined for modeling purposes as dummy variables. For each explanatory variable the number
of dummies is one less than the number of classes. For example, Income is defined by two
dummies, the first of which takes on the values 1 or 0 depending on whether the individual is in
the lower than average national income or not. The second dummy is similarly defined for the
average income group. The higher than average group is defined by both dummies taking the
value of zero, thus the estimated coefficient for each dummy must be interpreted relative to the
higher income group which serves as a baseline.
4.1 The OE sample
Following Sec. 2, we use a logistic regression model for the P(WTP = 0) and a lognormal
linear regression for the positive WTPs. A backwards procedure was used to reduce the
number of explanatory covariates. Tables 1 and 2 present the covariates which were found to
be significant. Due to the method used for constructing dummy variables for each categorical
variable the coefficient associated with the highest level of each categorical variable is
automatically set to zero and consequently is not listed in these tables.
- Tables 1 and 2 -
Table 1 shows that for individuals having a positive WTP, users have a higher WTP
than non-users. Examining Table 2 we see that the probability of not contributing at all
increases with increasing age, decreases with increasing income and is smaller for white collar
workers when compared to blue collar workers. For this data set the covariates important for
the P(WTP=0) are completely different from those important for the positive WTP regression.
Means and Medians can be computed as functions of the covariates as in Section 2.2.
According to the combined spike model (Table 3), white collar users belonging to the
highest income and lowest age group will on average have the highest WTP, while blue collar
non-users belonging to the highest age and lowest income group have on average the lowest
WTP. Table 3 presents these two extreme cases, along with a number of other cases with
different combinations of levels of the covariates. As expected, due to skewness the estimated
Medians are smaller than their corresponding Means. Note that due to the strong covariate
effects, examining the overall mean (or median) may be misleading.
- Table 3 -
4.2 The DCH sample
For the dichotomous choice data we assume that the response to a bid for those willing to
donate, can be modeled by a logistic model in terms of log bid4 (as in (14) and (18)) while the
probability of not being willing to donate at all can be modeled by a logistic model (as in (17)).
As in Sec. 4.2 a backwards procedure was used to reduce the number of covariates. Tables 4
and 5 present the final models. The two questions presented to the individual are as discussed
in Sec. 3.3 in reverse order and thus log(bid) is considered as an explanatory covariate for p i .
- Tables 4 and 5 -
In the DCH sample, none of the socioeconomic variables examined effect the WTP of
those individuals who are prepared to participate. We suspect that the anchoring effect of the
Bid variable overshadows that of the other variables. However, the P(WTP=0) is effected by
user, age and income as well as the size of the bid. Nonusers have a higher probability of not
contributing than users. The probability of not contributing to all decreases with increasing
income. The age effect is not monotonic but the highest age group has a higher probability of
not contributing than the lowest. It should be noted that P(WTP=0) increases with bid. We
postulate that high bids seemingly discourage people from contributing anything, although this
requires further corroboration. If this is the case, one ought to seriously consider the option of
presenting a bid only after the individual has indicated a willingness to pay something at all.
Finally, Table 6 presents predictions of Means and Medians for some combinations of
covariate values computed according to the spike model. The largest WTP is obtained by the
high bid, high income, low age, user group, while the smallest WTP corresponds to the low
bid, low income, high age, nonuser group.
- Table 6-
4.3 Comparison of the OE and DCH results
There are both similarities and differences between the OE and DCH results. If we look at
only those individuals willing to contribute the OE sample indicates a user effect with the
model estimates of Table 1 resulting in a mean WTP (for only those willing to contribute some
positive amount) of NIS 74 for users and NIS 38 for nonusers respectively, while the model
estimates of Table 4 result in an overall mean WTP of 128. The P(WTP=0) is, for the two
samples, effected in the same general direction by age and income, but differs in the effect of
bid, user and occupation, with bid and user only effecting the OE sample and occupation only
the DCH sample. It is important to note, however, that since the DCH and OE groups are
4 Log bid is used rather than bid to account for the skewness noted in the Introduction.
random sub-samples of the same population, we would expect to see similar results. The
observed differences again raises the question as to which approach provides a better
indication of the individuals’ WTP (see, e.g., Mitchell and Carson, 1989).
5. Discussion
In the analysis of WTP data the possibility of zeros is often ignored even though there is
empirical evidence that this frequently occurs. Recently spike models have been introduced to
take this zero lumping into account. We have shown how spike models, when considered as a
mixture of a point mass at zero with some continuous distribution, can be extended to take
explanatory variables into account.
For both open-ended and closed-ended (with a secondary question) data the likelihood
function breaks up in such a way that computations are carried out separately for individuals
willing to contribute and those not willing. Standard computer programs can then be used to
carry out the computations and the results can be recombined to provide estimates of Mean or
Median WTP which do not ignore the zero value.
The methodology can be applied to various choices of distributions (F) for those
willing to pay and of models for the probability of not being willing to pay at all (p). Our
examples have used conventional choices but others are certainly possible. The question of
how to pick the best choice remains an open issue, and deserves additional study. Another
problem of interest which we are currently examining is the development of good confidence
intervals for the mean (median) WTP.
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Appendix
List of Explanatory Variables
Age: Respondent’s age group, 1 (youngest) to 5 (oldest).
Bid: Bid amount in NIS (New Israeli Shekel) for the dichotomous choice.
Occup.: Occupation 1 = white collar, 2 = blue collar, 3 = other.
Income: Household’s income level, 3 = higher than (national) average, 2 = about
average, 1 = lower than average.
User: User (of park) = 1, passive-user = 2.
Symbols
δ delta
π pi
σ sigma
φ phi
∫ integral sign
∞ infinity
Π product (capital pi)
∑ summation (capital sigma)
µ mu
α alpha
β beta
η eta
Table 1: OE Sample: Regression Model for positive WTP
Parameter Coefficient Estimate p-value
Intercept 3.15
User
1 0.66 0.001
σ2 0.996
Table 2: OE Sample: Logistic Regression Model for P(WTP=0)
Parameter Coefficient Estimate p-value
Intercept -0.44
Age 0.045
1 -1.88.
2 -1.63
3 -1.55
4 -1.13
Occup 0.049
1 -1.13
2 0.62
Income 0.021
1 1.24
2 0.11
Table 3: OE Sample: Spike Model Predictions
Estimates of
Users Income Occup Age µi pi Meani Mediani
2 1 2 5 3.14 0.81 7.4 0
1 3 1 1 3.81 0.03 72.0 47.0
1 2 2 1 3.81 0.17 61.8 35.1
1 2 2 2 3.81 0.21 59.0 32.6
1 2 2 3 3.81 0.22 58.0 31.7
1 2 2 4 3.81 0.30 52.0 25.8
1 2 2 5 3.81 0.57 31.9 0
2 2 2 3 3.15 0.22 29.9 16.3
1 1 2 3 3.81 0.47 39.7 9.9
1 3 2 3 3.81 0.20 59.4 32.9
Table 4: DCH Sample: Logistic regression model for ηi
Parameter Coefficient Estimate p-value
Intercept -9.73 0.001
Logbid 2.17 0.001
Table 5: DCH Sample: Logistic Regression Model for pi
