360 December 1995
Journal oj Agricultural and Resource Economics
Using data elicited with the category approach to estimate average willingness-to-pay
values or a functional relationship between willingness to pay and characteristics of
respondents presents a unique estimation problem. When the data are intervals rather than
points, ordinary least squares (OLS) regression methods could be applied using the mid-
points of the intervals to represent values on the willingness-to-pay continuum. However,
Cameron and Huppert have shown that biased estimates may result. They concluded that
the application of maximum likelihood (ML) methods for “interval regression” is a more
reliable approach than OLS used on interval midpoints. Jordan and Elnagheeb also compared
results using OLS on interval midpoints and ML for interval data. Both studies adopted the
lognormal distribution as a first approximation for the valuation distribution in recognition
of the fact that valuation distributions are frequently skewed.
Given that the true willingness to pay, Yi, lies within the interval between til (the lower
bound) and till (the upper bound):
(1) Prob[ζ7 ≤ Yi < ) = Prob(ln/,7 ≤ In 1Ç < Intiu) = φ[lnZ,∙w-X1 β ∕σ]-φ[lnζ7-Xfi ∕σ],
where φ [] is the standard normal distribution function, X1is a vector of explanatory variables,
β is a parameter vector, and σ is the standard deviation of the error. If zil and ziu are the lower
and upper limits of the Zth interval, the corresponding log-likelihood function is
(2) lnZ, = ∑{ln[φ(¾)-φ(z,7)]}.
∕=ι
The maximum likelihood approach was used for this study to estimate willingness to pay
for mesoscale weather data and to determine characteristics of producers which might be
used to identify those producers particularly interested in accessing mesoscale weather data.
A lognormal distribution for willingness to pay was used, and parameter estimates were
obtained using LIMDEP’s grouped data procedure (Greene).
The independent variables included in the maximum likelihood model are defined in
table 1. Farm or ranch characteristics hypothesized to increase interest in mesoscale weather
information and thereby increase a producer’s willingness to pay include cotton, peanut or
alfalfa production, and a high level of past weather-related losses. Cotton producers might
value weather information more highly than other farmers. Soil temperature is an important
factor in the timing of cotton planting. The number of degree days before the first frost in
the fall is also an important yield and quality determinant. Peanut production is another high
input crop affected by weather conditions. Peanut producers would be expected to have a
strong interest in spraying conditions and in plant disease and insect models. Alfalfa hay is
another example of a high value crop where good drying conditions are essential at the time
ofharvest. Alfalfa is also susceptible to insects and plant diseases which are exacerbated by
weather conditions.
Gross sales is hypothesized to be positively related to willingness to pay for weather
information. The relationships between willingness to pay and total acreage and willingness
to pay and number of crops are difficult to predict. Generally, one might expect producers
with larger operations to exhibit a higher willingness to pay. However, in Oklahoma,
producers with larger acreages tend to focus on production of wheat and cattle, while the
producers with smaller acreages may grow crops, such as peanuts or vegetables, which are