Pr(WTP
i
* < bid
i
|
x
i
) = Pr(Y=0|
x
i
) =
F
(bid
i
|
x
i
).
(3)
Aggregating over individuals creates sample measures of
underlying population parameters. Specifically, the model is
estimated by the method of maximum likelihood, where the
likelihood function is specified as:
L =
Π
y
i
= 0
[1 –
F
(
x
i
´
δ
)]
Π
y
i
= 1
[
F
(
x
i
´
δ
)]
(4)
and the log-likelihood is
lnL =
∑
N
i
=1
{
y
i
lnF
(
x
i
´
δ
) + (1 –
y
i
) ln[1 –
F
(
x
i
´
δ
)]}
(5)
where
N
is the sample size of respondents,
y
i
takes a value of
1 if the
i
th
individual responds
Yes
to bid
i
and 0 otherwise,
(1 –
y
i
) takes a value of 1 if the
i
th
individual responds
No
to
bid
i
and 0 otherwise. A distributional assumption is necessary
for estimation. Logistic models, used here, make use of the
logistic
CDF
. The logistic has been commonly used because
the
CDF
has a closed form expression and does not require in-
tegration within the process of maximizing the log-likelihood
function. In essence, the maximum likelihood procedure fits
the weights associated with the attributes (
δ
) to most closely
match actual choices (
y
i
) by survey respondents. The intercept
and slope coefficient within the
δ
vector summarize the
WTP
response across individuals within the sample to different bid
amounts. With a large enough sample, the distribution of
WTP
is measured and through the parameters can be summarized.
From the responses to the initial
WTP
question, all the
analyst knows is whether the benefits users receive are greater
than or less than the bid amount offered to the survey respon-
dent. Using a second follow-up
CVM
question allows for im-
proved inference of economic benefits. The response to these
two questions leads to a series of
Yes
/
Yes
,
Yes
/
No
,
No
/
Yes
,
and
No
/
No
answers. These responses provide the data neces-
sary to calculate a more precise “double-bounded” estimate
of economic benefits (Hanemann
et al
., 1991). The likelihood
function is comprised of four pieces instead of two in Equa-
tions 4 and 5, but the underlying
δ
vector with the intercept
(
α
), slope (
β
), and parameters for the other attributes (
θ
s) have
the same interpretation and use.
Specifically, the model used in the Landsat valuation can
be represented as follows using the details of our application:
Pr(bid
i
|
d
ki
) =
F
(
α
+
β
l
n
(bid
i
) +
∑
K
k
=1
θ
k
d
ki
)
(6)
where
α
is the intercept,
β
is the slope parameter on the bid
amount variable,
θ
s are parameters associated with other con-
ditioning information variables (
d
s), and
F
is the logistic
CDF
.
In our application, the natural logarithm of the bid amount
is used to restrict the value of Landsat imagery from being
negative, as it is illogical that a user would take the time and
effort to download something they did not have at least some
positive value for. Specific to the logistic model:
Pr(bid
i
|
d
ki
) = exp(
α
+
β
l
n
(bid
i
) +
∑
K
k
=1
θ
k
d
ki
)
[1 + exp(
α
+
β
l
n
(bid
i
) +
∑
K
k
=1
θ
k
d
ki
)].
(7)
In the application to Landsat valuation, the conditioning
information variables
d
k
are limited to zero-one variables for
different industry sectors; these will be discussed later. In
addition, we estimate separate models for different groups
within the sample that display substantially different sets
of parameter estimates. Again, these will also be discussed
later. With a set of parameter estimates and a specific set of
attributes
d
ki
, a given bid amount can be used to calculate the
probability that a representative individual will agree to the
bid amount. Varying the bids amounts produces a probabilis-
tic demand function for the imagery: given the specifics of the
attributes of the representative individual chosen.
The probabilistic demand function Equation 7 is essential
in economic valuation. It describes
WTP
and can be used to
summarize
net
WTP
or what is also called consumer surplus.
Consumer surplus is the value of a good, i.e., Landsat imag-
ery less any cost to the user. Economic valuation of the good
hinges on relative comparisons and probabilities of choosing.
We know the number of scenes that were acquired, or con-
sumed, when the imagery was free of charge.
The following is an example where individuals respond to
charging for imagery. Using a 50-50 probability:
Pr(bid
i
|
d
ki
) = 50% =
F
(
α
+
β
l
n
(bid
i
) +
∑
K
k
=1
θ
k
d
ki
)
(8)
and choosing a set of attributes; for example, a specific user
group could be chosen or non-bid variables could be set at
their sample mean. Then, with the parameter estimates for
α
,
β
, and
θ
s, the bid amount can be solved for:
Median
WTP
= exp[(
α
+
∑
K
k
=1
θ
k
d
ki
)/
β
].
(9)
For this log-bid-amount model, the solution is the median
willingness-to-pay. If the bid amount was used linearly in the
model then the median and the mean are identical with the
symmetric logistic distribution. However, with the nonlinear
bid amount the median and mean are different. The median
WTP
amount is the price where, if charged, 50 percent of
the respondents would chose to pay, and 50 percent would
choose not to pay and would not acquire the imagery. In other
words, half the imagery that was acquired when the imagery
was free would not be acquired if the median was charged.
The probabilistic demand function converts immediately into
a typical demand function with price and quantity. Alterna-
tive quantile amounts can be similarly calculated.
The mean
WTP
is the expected value of the underlying
WTP
random variable that is observed through the answers to
the dichotomous choice question. This expected value is the
integral:
Mean
WTP
=
∫
∞
0
[1 –
F
(
WTP
* <
bid
)]
d
(
bid
)
which is:
Mean
WTP
=
∫
∞
0
[1 –
F
(
α
+
β
l
n
(
bid
) +
∑
K
k
=1
θ
k
d
k
]
d
(
bid
)
(10)
(see Hanemann, 1984). However, the integral is bounded only
if the slope coefficient (
β
), which should be negative for a
demand function, is less than −1. While the bid coefficient is
always negative in our estimated models, the estimated slope
coefficient is greater than −1. The situation of −1 <
β
< 0 is not
an unusual result in
CVM
applications. The approach then is
to choose a high-bid amount representative of the sample, and
perform the integration numerically (see Bishop and Heber-
lein, 1979; Hanemann, 1984). Specifically,
Mean
WTP
=
∫
0
highbid
[1 –
F
(
α
+
β
ln(
bid
) +
∑
K
k
=1
θ
k
d
k
)]
d
(
bid
).
(11)
We make use of the high-bid amount for the first
WTP
ques-
tion, which is $10,000
USD
, and thus our results are relatively
conservative from this aspect.
The median bid amounts, or other price-probability-and-
quantity amounts, and the mean
WTP
amounts are useful
information for interpreting behavior of individuals and con-
structing aggregate economic values. The median bid amount
is the value of imagery at which, if charged by the US Federal
government, half of the users would pay and half would forgo
650
August 2015
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
