Travel cost methods article

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Parsons G.R. (2013) Travel Cost Methods. In: Shogren, J.F., (ed.) Encyclopedia of Energy, Natural
Resource, and Environmental Economics, volume 3, pp. 349-358 Amsterdam: Elsevier.
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Travel Cost Methods
GR Parsons, University of Delaware, Newark, DE, USA
ã 2013 Elsevier Inc. All rights reserved.

Introduction
Economists have been concerned with measuring the economic
value of recreational uses of the environment for over 50 years.
This has been largely motivated by benefit–cost analyses of
environmental policies and damage assessments where legal
rules call for valuation in circumstances where some harm has
been done to the environment. Benefit–cost analysis under the
Clean Water Act (USA) is a good example of the former wherein
values due to improved water quality, such as better fishing and
swimming, are needed to judge the regulatory impact of water
quality standards. Damage assessment under the Oil Pollution
Act (USA) is a good example of the latter where analysts seek
values of lost recreational uses, such as beach closures or lower
hunting quality, to establish the size of compensable payments.
Economists have used both revealed and stated preference
methods to estimate recreational use values. The travel cost
model (TCM) is the primary revealed preference method used
in this context. It has been in existence since Harold Hotelling
suggested the method in his now famous letter to the director
of the National Park Service in 1949. By most accounts, it is
one of the success stories in valuing environmental goods. It
enjoys broad application in policy settings, and the scholarly
literature addressing the theory and empirical method has
grown considerably in the past three decades.
The basic insight underlying the TCM is that an individual’s
‘price’ for recreation at a site, such as hiking in a park or fishing
at a lake, is his or her trip cost (including out-of-pocket travel
and time cost) of reaching that site. Viewed in this way, individuals reveal their willingness to pay for recreational uses of
the environment in the number of trips they make and/or the
sites they choose to visit. The actual measurement of these
values entails some form of demand estimate for recreation
trips and, in turn, measurement of consumer surplus.
This article provides an introduction to the TCM and how it
is used to value recreation sites and the attributes of recreation
sites. It is intended to be comprehensive but will not develop
the theory or econometrics underlying each model or address
all of the methodological inquiries pertaining to the validity
and robustness of the model. It is instead a statement of the
current practice in practical terms.
It is common to classify TCMs into three groups: seasonal
demand, site choice, and Kuhn–Tucker (KT). The lines between these models have blurred somewhat with the advance
of research, but this classification still helps in organizing the
material.
The earliest TCMs, dating back to the 1960s, are seasonal
demand models that work much like a demand curve for
any consumer good. Trip cost is treated as the ‘price’ of the
good and the number of trips taken over a season is treated
as the ‘quantity demanded.’ The simple observation that
the closer one lives to a site (lower price), the more trips
one takes (higher quantity demanded) is taken as a simple
downward-sloping demand curve. To estimate such a curve,

Encyclopedia of Energy, Natural Resource and Environmental Economics

one gathers cross-sectional data on the number of trips taken
by people living at different distances from the site. Then, by
regressing the number of trips on the measured trip cost, a
demand relationship is revealed, from which conventional
measures of surplus may be derived. Seasonal demand
models have proved to be useful for valuing the opening and
closure of a site and computing per trip values for use in
transfer studies. It is especially useful if policies are focused
on a single site or on only a few sites that serve as substitutes for
one another.
Site choice models, however, are now the most commonly
used form of the TCM. These were introduced in the mid1980s and were motivated by a need for models that allow
for valuation of changes in site quality (e.g., improved water
quality at lakes) and to consider recreation demand where the
number of sites is large. These models were built during the
rapid expansion of random utility theory in the 1980s and
1990s that eventually led to Daniel McFadden’s Nobel Prize
for work in this area. His Nobel Prize lecture actually included
an application using a TCM.
Site choice models consider an individual’s choice of visiting one site from among many possible sites on a given
choice occasion. While approaches exist for accommodating
seasonal demand in site choice models, at their heart, they are
based on the choice of a single site during a single choice
occasion. Which site a person visits is assumed to be a function
of the attributes of the sites (size, quality, access, etc.) and the
trip cost of reaching the site. Individuals reveal their relative
values of site attributes in the sites they choose. Because trip
cost is one of the attributes, an individual’s choice reveals the
relative values for the attributes of sites in money terms. To
estimate a site choice model, one gathers data on the actual site
choices made by individuals. Then, usually using some form of
multinomial logit in the context of a random utility model
where trip cost and site attributes serve as arguments in the
utility function, a probabilistic choice model is estimated. As
these parameters are hypothesized to come from individual
utility functions, they readily accommodate welfare analysis.
Site choice models have been useful for valuing quality
changes in site attributes, closure of one or more sites in a
region, and addition of new sites. Given the proliferation of
software that can accommodate discrete choice random utility
models and their ability to address many policy issues in a
defensible and easy-to-understand way, site choice models
have come to dominate the TCM literature.
The final TCM is the KT model. Application of this model
came into practice in the early 2000s. KT models are seasonal,
but they are set in a probabilistic framework that shares many
of the properties of site choice models. In a sense, KT models
bring together the strengths of seasonal and site choice models
in a unified model. However, it has proven to be computationally more cumbersome than the simpler seasonal demand and
site choice models. While its rise has been slow, its use will no
doubt increase in the years ahead.

http://dx.doi.org/10.1016/B978-0-12-375067-9.00002-4

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Like any area of economic research, there are numerous
theoretical and empirical issues surrounding application of
the TCM. These include, among other things, measuring trip
cost (especially the time cost component), dealing with
multiple-purpose trips, incorporating time interdependence,
treatment of congestion, and defining sites and choice sets.
Each of these issues is briefly addressed following a presentation of the three models outlined earlier.
Finally, an area of increasing interest in TCM research is
the use of state preference data in combination with revealed
preference travel cost data, for example, data on how people
report they would change trips if water quality were improved
in response to a hypothetical survey question on this matter. This
enables an analyst to explore unobserved, yet policy-relevant,
changes to recreation sites. Given the more than adequate
coverage of this topic elsewhere in this volume in connection
with several valuation models, it is only briefly addressed in
this article.

Seasonal Demand Models
Introduction and Theory
The TCM in its ‘seasonal demand’ form is the traditional variety. It considers demand for use of a site over an entire season.
It treats trips to a site as the ‘quantity demanded’ and trip cost
as the ‘price.’ This gives a conventional demand function
rn ¼ f ðpn , psn , zn Þ

[1]

where rn is the number of trips taken by individual n to the site
during the season, pn is the price or trip cost for individual n to
reach the site (travel and time cost), psn is the vector of trip cost
substitute sites, and zn is the vector of individual characteristics
believed to influence the number of trips taken in a season
(e.g., age and income). This form may be readily derived from
conventional utility theory or household production theory. It
is usually done using separate budget and time constraints to
explicitly show the opportunity cost of time in trip cost.
Consumer surplus for access to the site for a season is
csn ¼

ð p∗n
p0n

f ðpn , psn , zn Þdpn

[2]

where p0n is the current trip cost to the site and p∗
n is the ‘choke
price’ – the trip cost at which demand for trips goes to zero for
individual n. If the site were lost, csn is the loss in welfare –
sometimes called ‘access value.’ Analysts sometimes report
mean per trip values that take the form csn/rn. Exact Hicksian
measures of surplus may also be derived as usual.

Estimation
In estimation, an analyst gathers cross-sectional data on a
sample of individuals for a given season: number of trips
taken to the site, trip cost, and other relevant demand shifters.
Then, via the spatial variation in trip cost (people living at
different distances from the site have different ‘prices’ for
trips), one estimates an equation like eqn [1].
The choice of functional form for estimation has been the
subject of inquiry since the methodology was first developed.

Earlier functional forms were continuous – linear, log-linear,
log-log, etc. Most modern forms are from the family of count
data models – Poisson, negative binomial, zero-inflated,
hurdle, and so on. Count data models are designed for analyses
with a nonnegative integer-dependent variable and are quite
versatile for handling truncation, large number of zero trips in
the data, and preference heterogeneity. This has made them
popular for seasonal demand function estimation. A Poisson
model is the simplest form. An individual’s probability of
making y trips to a site in a given season is
y

exp ðmn Þmn
where y ¼ 0, 1, 2, . . . ;
y!


mn ¼ exp bp pn þ bps psn þ bz zn

pr ðrn ¼ yÞ ¼

where

[3]
mn is the expected number of trips taken by a person n. It serves
as the ‘demand’ expression for eqn [1], and the Poisson model
puts it in probabilistic form. The parameters in eqn [3] are
estimated by maximum likelihood where each person’s probability of taking the number of trips actually taken is used as an
entry in the likelihood function. Seasonal consumer surplus
^ , where
for a person n in the Poisson form is ^csn ¼ ^r n =  b
p
‘hatted’ values denote estimates. Per trip consumer surplus is
^ .
a simple constant ^csn =^r n ¼ 1=  b
p
An undesirable feature of the Poisson model is an implicit
constraint that the mean and variance of rn are equal. If upon
testing the data fail to support this assumption, it is common
to use a negative binomial model to relax this constraint. As
testing usually shows that equality of mean and variance does
not hold, the negative binomial form is frequently used.

On-Site and Off-Site Samples
One of the more important decisions an analyst makes when
estimating a seasonal demand model is whether to gather data
on-site or off-site. Often, the number of visitors to a particular
recreation site is a small fraction of the general population. If
so, sampling the general population may require a large number of contacts to form a reasonable sample size of recreational
users. On-site data have the advantage that every individual
‘intercepted’ has taken at least one recreation trip. In this way,
gathering data on-site is often cost-effective. However, there are
at least two disadvantages of on-site data: endogenous stratification and truncation. As individuals taking more trips over a
season are more likely to be drawn for inclusion in the sample,
there is oversampling in direct proportion to the number of
trips one takes over a season (e.g., a person taking two trips is
twice as likely to be sampled as a person taking one trip).
Estimation that fails to account for this effect will give biased
parameter estimates for the general population. At the same
time, the analyst never observes individuals taking zero trips in
the season, so there is no direct observation at the ‘choke price’
on the demand function, which is important in the computation of consumer surplus. Both endogenous stratification and
truncation are easily corrected econometrically. One can show
in the Poisson form that simply using y  1 instead of y in
estimation in eqn [3] corrects for both effects. The correction
is somewhat more complicated in more complex forms such as
negative binomials, but it is possible there as well.

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Off-site sampling has the advantage that nonparticipants
are observed. This presumably makes for more accurate estimation of the choke price and hence estimation of consumer
surplus. In some models, nonparticipants get special treatment,
wherein the analyst estimates a two-part model: First, the decision to take a recreation trip or not (the participation decision)
and second, the number of trips to take (the frequency decision). These models are also members of the count data family
of models and are known as hurdle and zero-inflated Poisson
models. The model uses, more or less, a simple bivariate choice
model for the participation decision and a Poisson model of
one form or another for the frequency decision. If one believes
that participation and frequency are governed by different decision processes, these models are beneficial. A zero-inflated
count model applied in our context has the form
prðrn ¼ 0Þ ¼ ’n ðpn , psn , zn Þ þ ð1  ’n ðpn , psn , zn ÞÞexpðmn Þ
y

prðrn ¼ yÞ ¼ ð1  ’n ðpn , psn , zn ÞÞ

expðmn Þmn
y!

[4]

The first equation is the probability of observing a person
take zero trips in the season, and the second equation is the
probability of observing a person taking one or more trips in
the season where mn is the same as shown in eqn [3] and ’n is a
simple bivariate logit model. The first term in the first equation
models the participation decision, the probability of a person
being someone who engages in the type of recreation under
study at all. The second term in the first equation captures
those who engage in the type of recreation under study but
happen to not make a trip in the current season, the probability
of being a participant but taking no trips. The second equation
is the frequency decision for those taking at least one trip, the
probability of taking rn trips in the season given that that
person is a participant. Again, estimation is by maximum
likelihood wherein probabilities from eqn [4] are loaded for
each observation according to actual choices. Seasonal and
per trip consumer surplus in a zero-inflated model have the
forms ð1  ’
^ n Þ^
mn = bp and 1/bp, where the ‘weighting’ term,
^ n Þ, accounts for participation.
ð1  ’

Valuing Quality Changes and Multiple Site Models
Although the strength of seasonal demand models is not in the
valuation of site attributes, such as water quality or acres of
open space, there are empirical approaches using the model for
this purpose. The most popular approach uses contingent
behavior response data in combination with the trip data. For
example, in addition to asking respondents to report total
number of trips over a season, one also asks them to report
the total number of trips they would have taken ‘if the expected
catch rate of fish at the site had been ½ the current rate’ or ‘if
the width of the beach had been twice its current width.’ Then,
using the newly created contingent trip count, a second TCM
under the new hypothetical conditions is estimated. The area
between this new demand curve and the original – the difference in consumer surplus estimates in the two conditions – is
an estimate for the value of the attribute change. These demand
models are typically estimated simultaneously and often with
logical parameter and error term restrictions. This approach
accepts the condition of ‘weak complementarity’ in the

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behavior model – that a person receives utility from the attributes of a site only by visiting the site.
It is also possible to observe demand function shifts using
actual trips to several different sites that vary in attribute quality. ‘Pooling’ or ‘stacking’ multiple sites in this way allows the
analyst to enter site attributes in eqn [1], estimate parameters
for site attributes, and then use that estimate to calculate welfare as described in the contingent behavior case above. These
cross-sectional models have variously been called varying
parameter models, pooled models, and stacked multiple site
models. The basic problem underlying these models is that
they fail to integrate site choice across the set of sites under
consideration in a meaningful way. For this reason, these
models have largely fallen out of favor, although applications
do still appear from time to time in the published literature.
The final form of the seasonal demand model to consider is a
multiple-site model in which a system of count demand equations allowing for substitution across sites and utility theoretic
restrictions is developed. There are a number of applications
along these lines, but this has largely been confined to settings
with a few sites and has focused on access values instead of
valuing site attributes. For the most part, these models have
given way to site choice random utility models and KT models
as a way of integrating many sites into the decision model and to
conduct welfare analysis for changes in site attributes. The KT
model is really the state-of-the-art demand system model and is
presented in the section ‘Kuhn–Tucker Models.’

Site Choice Models
Introduction
The most commonly used TCM in the literature today is a
model of recreation site choice based on the random utility
theory. Known as the random utility maximization (RUM)
model, it has proven to be quite versatile for measuring access
value (e.g., opening or closure of one or more recreation sites)
and quality changes at one or more sites (e.g., improved water
quality, wider beaches, and increased bag rate for hunting). Its
appeal hinges on its ability to handle many sites and substitution among sites in a plausible and easily estimable way. The
behavioral basis underlying the model is also easily understood and intuitive making it all the more attractive for policy
analysis and damage assessment.

Theory
The time frame in a site choice model is a single choice occasion, usually a day, in which an individual makes one recreation trip. The individual is assumed to face a set of S possible
sites for a trip. Each site i (i ¼ 1, 2, . . ., S) is assumed to give
individual n (n ¼ 1, 2, . . ., N) some utility Uin on a given choice
occasion. The utilities are assumed to be a function of the trip
cost of reaching the site and attributes of the site, such as
natural amenities, water quality, size, and access. As before,
trip cost includes travel and time cost.
Letting pin be trip cost, qi and q~i be vectors of site attributes
that may or may not share some of the same terms, and ~zi be a
vector of individual characteristics, site utility for person n at
site i is

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Valuation Tools | Travel Cost Methods
Uin ¼ btc pin þ bq qi þ bqz q~i ~zn þ ein

[5]

Typically, site utility is linear as shown or some form with a
nonlinear transformation of characteristics (such as the log of
beach width) that maintains linearity. The coefficient on trip
cost is the marginal utility of income as it describes how site
utility changes with a decrease in income if a trip is taken. Site
utility often includes site-specific constants in the vector qi
capturing ‘average’ differences across sites missed by the vector
of attributes. The vector ~zn is interacted with q~i to capture
individual heterogeneity. For example, ‘boat ramp’ as a site
attribute in q~i might be interacted with ‘boat ownership’ as an
individual characteristic in ~zn , if one believes that boat ramps
only matter to people who own a boat. Individual characteristics cannot be entered alone because they are invariant across
sites. Individual characteristics may, however, be interacted
with a site-specific constant if there is a compelling reason
that an individual characteristic affects a person’s proclivity to
visit one site over another. The error term, ein, captures site
attributes and individual characteristics that influence site
choice but are unobserved by the analyst.
On any given choice occasion, an individual is assumed to
choose the site with the highest site utility giving trip utility of
the form
Vn ¼ max ðU1n ;U2n ; :::: ;USn Þ

[6]

Trip utility, Vn, is the basis for welfare analysis in the RUM
model. It is used to value loss or gain of sites (access values)
and changes in site attributes. For example, consider an oil spill
that closes sites 1 and 2. Trip utility with the closures becomes
Vnclosure ¼ max ðU3n ;U4n ; :::: ;USn Þ

[7]

where sites 1 and 2 have been dropped from the choice set. Trip
utility declines from Vn to Vnclosure.
A similar expression can be generated for a change in site
quality at one or more sites. Suppose the water quality at sites 2
and 3 is improved through some regulation. If so, trip utility
for person n becomes


*
*
;U3n
;U4n ; ::: ;USn
Vnclean ¼ max U1n ;U2n
U*2n

[8]

U*3n

and
denote the now higher utility due to the
where
improved quality. In this case, trip utility increases from Vn to
Vnclean. In both cases, the change in utility is monetized by
dividing the change by the coefficient on trip cost bp, which
is our marginal utility of income, in eqn [5]. This gives the
following compensating variation (also equivalent variation)
measures for changes in trip utility


wclosure
¼ Vnclosure  Vn =  bp and
n


wclean
¼ Vnclean  Vn =  bp
n

[9]

These are changes in welfare on a per trip per person basis.

Estimation
Because the error term, ein, on each site utility is unknown to
researchers, the choice is treated as the outcome of a stochastic
process in estimation. By assuming some explicit distribution
for the error terms in eqn [5], each person’s probability of

visiting a site can be expressed in some form. The simplest is
to assume that the error terms are independently and identically distributed (iid) type 1 extreme value random variables.
This gives a closed form expression, a multinomial logit, for the
choice probabilities. Each person’s probability of choosing any
site k from the set of S sites in the multinomial logit from is


exp btc pkn þ bq qk þ bqz q~k ~zn


prn ðkÞ ¼ P
~ zn
i2S exp btc pin þ bq qi þ bqz qi ~

[10]

The parameters are estimated using data on actual site
choices and maximum likelihood with the logit probabilities
in eqn [10] – so a person’s entry into the likelihood function is
the probability of visiting the site actually chosen on a given
choice occasion. Because researchers proceed as though
choices are the outcome of a stochastic process, trip utility in
eqns [6]–[8] is also stochastic. Expected trip utility is used as an
estimate of Vn in empirical work. Using the assumption of iid
type 1 extreme value distributions for the error terms gives each
individual’s expected trip utility as
EðVn Þ ¼ Efmax
( ðU1n ;U2n ; . . . ;USn Þg
)
S
X
expðbp pin þ bq qi þ bqz q~i ~zn Þ þ C
¼ ln

[11]

i¼1

i ¼ 1, where C is some unknown additive constant. It is a
manifestation that the absolute level of utility is unmeasurable,
and as it is shared and constant across all expected trip utilities,
it is of no practical relevance in welfare analysis. E(Vn) is often
referred to as the ‘log-sum’ and is the empirical form of Vn used
in welfare analysis. The steps in such an analysis are straightforward: estimate the parameters of site utility, use the parameters to construct expected trip utilities with and without some
resource change using eqn [11], and finally compute per trip
losses per person substituting E (Vn) for Vn in eqn [9] and with
the estimates of bp used to monetize the change in expected
utility (note that C discussed above drops out when one differences the equations). In some cases, rarely, however, researchers will consider site utilities that are nonlinear in trip
cost, allowing for nonconstant marginal utility of income and
empirical forms of eqn [9] that are not a closed form. In this
case, welfare is calculated using numerical methods.
One of the major drawbacks of the multinomial logit model
is the restrictive way in which substitution occurs. Since site
substitution is the pathway through which welfare effects are
captured, it is important to handle it in as realistic a way as
possible. The multinomial logit model assumes that the closure
or decline in quality at one or more sites leads to a proportional
increase in the visitation to all other sites – their shares remain in
fixed proportion. This property, known as the independence of
irrelevant alternatives, is usually unrealistic. For this reason,
economists have turned almost entirely to alternative forms
that allow for more realistic patterns of substitution. This is
achieved, at least in a stochastic sense, by allowing for correlated
error terms across the sites in eqn [5]. There are two common
methods that allow for such correlation: nested and mixed logit.
These forms dominate the travel cost random utility model
literature. Both are generalizations of the multinomial logit
model outlined above and follow the same steps outlined there.

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The nested model has been used since the introduction of
travel cost RUM models. Nested models place sites that share
unobserved characteristics in a common ‘nest’ under the assumption that they serve as better substitutes for one another
than sites outside the nest. It also renders a closed form choice
probability similar to the conditional logit model, but it includes a new parameter for each nest that captures the degree of
substitution among the sites within that nest. Researchers often
nest sites by proximity (e.g., grouping sites in the same region
together), resource type (e.g., grouping lakes and rivers in
separate nests), and purpose (e.g., grouping trout and bass
fishing trips separately). The option of purpose actually
expands the choice model to include than just site choice.
The mixed logit model (or random parameters logit) is
a more flexible approach for enriching the patterns of substitution. It allows for more, and overlapping, substitution structures
and can also be easily configured to mimic what a nested logit
model does. The mixed logit model induces correlation among
the error terms by allowing parameters on the site attributes to
have random components or dispersion terms. The random
components then become a part of the error term in site utility,
and this, in effect, causes correlation among site utilities. Consider an example: A site utility includes a dummy variable for
‘state park’ to distinguish community-run beaches from statelevel park beaches. As state parks are likely to share unobserved
attributes, one might estimate the model with a random parameter on ‘state park’. If so, the estimated dispersion term, the
variance, on ‘state park’ would cause all the sites sharing the
attribute to be correlated as their shared error component
would move in concert. The larger the dispersion, the greater
the degree of correlation and hence substitutability among the
state park sites. This, in turn, is implicitly captured in the welfare
analysis.
Because models are estimable with a large number of random terms, the possible patterns of correlation are almost
endless, making mixed logit an obvious choice for the recreation applications with welfare analysis. Unlike the multinomial
and nested logit models, the mixed logit model does not yield a
closed-form choice probability. Instead, it uses simulated probability methods to solve for choice probabilities that present
themselves as integrals and yields estimates for the mean and
dispersion of designated parameters. In some circumstances,
the disturbance term is used to interpret the degree of taste
heterogeneity in the sample. In this case, the greater the dispersion, the greater the unobserved heterogeneity in the sample.
Welfare analysis with mixed logit uses a log-sum term exactly
like eqn [11], but it requires the use of a simulated log-sum as
some or all of the parameters in the equation vary by some
known distribution. Due to its flexibility and now widespread
presence in standard econometric packages, mixed logit has
become extremely popular in travel cost random utility model
applications.
Finally, like the seasonal demand model, the major decision of on-site versus off-site data affects the econometrics used
to estimate the model. Again, on-site sampling may be a costeffective way of obtaining trip data but the data must be
adjusted to account for the relative amount of time spent
sampling at each site; otherwise, the choice probabilities will
reflect in part (perhaps in large part) the relative extent of
sampling at each site instead of the relative preferences for

353

each site. To correct for on-site sampling bias in a meaningful
way, one needs to know, or at least should be able to estimate,
the proportion of trips to each site in the population. This
population proportion can be used to adjust or weigh the
sample choice probabilities in estimation. An alternative solution is to design a sampling strategy that applies equal sampling pressure across all sites, independent of their popularity.
Off-site data are ‘cleaner’ in the sense that no such adjustment
is needed.

Seasonal Forms
Site choice models, at their core, are occasion-based, centering
on an individual trip. Oftentimes, analysts are interested either
in the seasonal implications of a policy change or in resource
changes that may engender changes in the number of trips
taken (e.g., fewer fishing trips if catch rates decline). The site
choice model described earlier disallows taking fewer or more
trips over a season or no trip on a single choice occasion as the
model is conditioned on taking a trip.
There are essentially two methods used to modify the basic
choice model to make it seasonal and incorporate the possibility of adjusting the number of trips taken over a season: a
repeated discrete choice model with a no-trip alternative and a
linked seasonal demand model.
The repeated choice model simply adds a no-trip utility to
the individual’s choice. This typically takes the form
U0n ¼ d0 þ dz zn þ e0n

[12]

where zn is a vector of individual characteristics believed to
influence whether or not a person takes a trip on a given choice
occasion (zn usually different from ~zn ). This might include age,
family composition, years engaged in recreation, and so on. Each
person now has S þ 1 choices on each choice occasion: visiting
one of the S sites or taking no trip. The model is made ‘seasonal’
by simply repeating it for every choice occasion in the season,
where the choice probabilities now include no-trip as one of the
alternatives. The log-sum becomes an expected choice occasion
utility instead of expected trip utility with the form
max ðU0n ,U1n , .. ., USn Þg
EðVn Þ ¼ Ef(
)
S
X
¼ ln exp ðdz zn Þ þ
exp ðbp pin þ bq qi þ bqz q~i ~zn Þ
þ Cn

i¼1

[13]

Per trip welfare changes are calculated as before (see eqn [9] and
discussion following eqn [11]) but become per choice occasion
per person. Seasonal estimates of welfare change are simply per
choice occasion values multiplied by the number of choice occasions in a season, Wn ¼ M  wnco, where wnco denotes the per
choice occasion value and M the number of choice occasions.
Usually, an analyst will have data on trips over an entire
season without knowing the specific date for each trip. If so, if a
person took Tn trips, on M choice occasions, each of the Tn trips
would enter the likelihood function as the probability of taking a trip and each of the M  Tn no-trips would enter as the
probability of taking no trip. This expands the data set considerably in estimation. In nested logit, the S sites are usually
placed in a nest separate from the no-trip choice. In mixed
logit, no-trip utility usually includes its own alternative-specific

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constant as shown in eqn [12] and will be treated as a random
parameter. It is also desirable to allow for correlation of utilities across choice occasions in repeated models in estimation.
The alternative approach for introducing a seasonal dimension into a site choice model is a pseudo–seasonal demand
model that ‘links’ the number of trips taken over a season with
the expected trip utility from the site choice model. The linked
model has the form


Tn ¼ f EðVn Þ= bp , zn

[14]

where Tn is the number of trips taken by a person n over the
season, E(Vn)/bp is the expected trip utility estimated in a site
choice model (eqn [11]) divided by the estimated coefficient
on trip cost from the same model, and zn is a vector of individual characteristics. E(Vn)/bp is the expected value of a
recreation trip for person n on any given choice occasion.
One would expect the number of trips a person takes to
increase with the expected value of a trip. In this way, the
model can be used to predict how the number of trips over a
season changes with changes in site characteristics or the loss/
addition of sites in the choice set. For example, the expansion
of designated open space at one or more recreation sites would
increase predicted E(Vn) from the site choice model, which, in
turn, would increase the number of predicted trips in a linked
model, thereby picking up seasonal adjustments in the number
of trips taken. The linked model is typically estimated using a
count data model, either stepwise or simultaneously with the
site choice model. Seasonal changes in welfare in Poisson or
negative binomial form can be shown to be DT^n =^g for person
n, where DT^n is the change in trips due the resource change
and ^g is the parameter estimate on expected trip utility in the
linked model.
The model is admittedly ad hoc in the sense that it is not
built from a consistent utility theoretic framework at the site
choice and trip demand levels. Nevertheless, it has proved to be
quite versatile and is usually easy to estimate. The repeated
choice model can be written in a linked form as f(.) ¼ M 
(1  pr(no-trip)), where M is the number of choice occasions
in a season and pr(no-trip) is the probability of taking no trip
in the repeated choice model. In this way, the two models,
while ostensibly different, can be seen as simply different
functional forms for the seasonal component of a site choice
model.

Kuhn–Tucker Models
The KT model is the most recent of the travel cost models. The
KT model in some ways brings together the best attributes of
the seasonal demand and site choice models by modeling site
choice and total trips over a season in a utility-consistent way.
Although introduced into the recreation demand literature
over 10 years ago, it has not seen particularly wide use, especially when compared with the site choice models mentioned
in the previous section. This seems to be largely due to the
complexity and computation difficulties one often encounters
when estimating a KT model. Nevertheless, the KT model is at
the cutting edge of travel cost demand modeling and is likely to
see increased use in due time.

In the KT model, individuals are assumed to maximize a
seasonal direct utility function subject to a usual budget constraint. To ease notation, let us assume one site for now. An
individual’s choice is defined by
max fuðr, a; q, z, e, gÞg s:t: pr þ a ¼ y, r  0
r, a

[15]

where r is the number of trips taken to the site, a is a numeraire
good with price one, q is a vector of site attributes, z is a vector
of individual characteristics, e is an error term, and g is a
parameter vector to be estimated. The use of the subscript n
denoting individuals has been suppressed. In the budget
constraint, there are p trip costs and y income. The KT firstorder conditions for utility maximization then are
@uðr, y  pr; q, z, e, gÞ=@r
 tc; r  0;
@uðr, y  pr; q, z, e, gÞ=@a
r½@uðr, y  pr; q, z, e, gÞ=@r  tc@uðr, y  pr; q, z, e, gÞ=@a ¼ 0
[16]
These are the usual complementary slackness conditions
that allow for both corner (zero trips) and interior (nonzero
trips) solutions. The trick to making the KT model operational
is to select a form of the utility function that allows one to
rewrite the conditions in eqn [16] as
e  gðr, p, y, q, z, gÞ; r > 0;

r½e  gðr, p, y, q, z, gÞ ¼ 0 [17]

Equation [17] makes the model empirical because it allows the
analyst to write realizations from a data set (number of trips
taken by respondents) in probabilistic terms that can be entered into a maximum likelihood function for estimation. For a
given individual n, if rn ¼ 0, then en < gn, whereas if rn > 0,
en ¼ gn. So, if £ has some known distribution, an explicit form
for the probabilities for each observation can be used in estimation. Some applications, for example, have used iid type 1
extreme value error terms, such as those used in the multinomial logit. The estimated parameters ^g are then used to
construct the fitted direct utility functions for each individual,
which, in turn, may be used to estimate exact measures of
surplus for some hypothetical changes in site attributes. The
analog to the expected trip utility in eqn [7] for the RUM model
is the maximum indirect seasonal utility corresponding to the
problem in eqn [15] or


vn ¼ max v1n ðp, y, q, z, e, gÞ1 , v0n ðp, y, q, z, e, gÞ

[18]

where v1n is the indirect seasonal utility conditioned on taking
trips and v0n is the indirect seasonal utility of not taking a trip.
Eqn [18] simultaneously defines whether or not a person is a
participant and, if so, how many trips are taken over the entire
season. In welfare analysis for a change in site attributes, compensating variation is just the value of DWn that solves


max v1n ðp,
 y, x, z, e, gÞ1 , v0n ðp, y, x, z, e, gÞ

¼ max v1n ðp, y  DWn , x*, z, e, gÞ1 , v0n ðp, y  DWn , x*, z, e, gÞ
[19]
As the error term e is random, the welfare change DWn is
also random. Also, because each element in eqn [19] is itself a
maximum value function, there is no closed form solution to
DWn like the log-sum. It must be solved numerically using

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repeated draws on the assumed distribution for the error term.
Importantly, the outcome of this process allows each individual in the sample to adjust visits to the site and their frequency.
When more than one site is included in the model, eqn [18]
includes 2S available combinations of sites that might be visited
over the season (including no trip), and each combination has
an optimal number of trips for its included sites. In this way,
attribute changes and site closures can lead to adjustments in
the sites visited and the number of trips taken to each site.
As noted at the outset, due to the computation complexity
of the KT model, it has not been used as widely as expected.
Also, to make it operational, the form of the utility functions
used has been rather restrictive. Still, unlike the seasonal
demand and site choice models, the KT model provides utility
theoretic consistency between site choice and trip frequency
and allows for substitution among sites in the traditional
(through cross-price terms) and stochastic (through error
term correlation) ways. The model has also been applied in
settings with a large number of sites and with numerous site
characteristics.

Issues and Complications
This section addresses several perennial topics that complicate
estimation and, in some cases, interpretation and use of the
results in TCMs. These include multiple-purpose and overnight
trips, measuring time cost, intertemporal substitution, choice
set formation, and congestion.

Multiple-Purpose and Overnight Trips
Sometimes the purposes of a trip will extend beyond recreation
at the site. For example, a person may visit family and friends,
or go shopping, or even visit more than one site on a recreation
outing. In these instances, the trip is producing (i.e., the trip
cost is buying) more than single-site recreation, and it is no
longer clear whether the simple travel cost paradigm applies.
For this reason, researchers often confine their analysis to
day trips where multiple purposes are less likely to occur. This
is sometimes done by either focusing on trips made within a
day’s drive from a person’s home (assuming that these will
largely be day trips) or by identifying day trips through a survey
question. In some cases in the survey, the analyst identifies
single-purpose day trips or day trips where recreation is the
primary purpose. Sometimes, ‘other purposes’ are handled as
an attribute of a site. For example, nearby shopping may be
variable in a beach choice model. This, in effect, expands the
nature of the recreation experience.
Expanding the model to overnight trips is problematic
for a number of reasons. There are more costs to estimate
(e.g., lodging at all sites). Length of stay can vary significantly
over the sample (e.g., some people stay one night, others for
2 weeks.) The relevant choice set is likely to be considerably
larger. For example, for a household in the United States, the
set of substitutes for a week-long beach vacation may include
all beaches in the United States and even beyond. Also, if
people use long trips as a ‘getaway,’ nearby sites with low trip
cost may be undesirable. Greater trip cost then, at least over
some range, would be viewed as a positive attribute,

355

complicating ‘price’ in the simple TCM. Finally, many overnight trips will be multiple-purpose/multiple-site excursions
wherein the individual transits from one site to the next obviously straining the TCM paradigm.
In cases where individuals visit multiple sites on a single
trip, one of the more promising approaches is to redefine a site
such that there are ‘single-site’ sites and ‘multiple-site’ sites and
then proceed with the logic of the TCM. Trip cost would be
recalculated for a site with multiple sites by accounting for the
costs of visiting all the sites on one trip. In a site choice model,
one can think of this as a portfolio choice problem wherein
each person chooses a portfolio of sites on any given trip.
Characteristics of the portfolio would simply be alternativespecific constants for each site in the portfolio.

Measuring Travel and Time Costs
Trip cost is measured as the sum of travel and time cost plus
any other expenses necessary to make the recreation trip possible. Travel cost includes fuel and depreciation of the owner’s
vehicle. In some instances, analysts will ignore the depreciation
costs as inconsequential. In either case, travel cost is typically
measured using round-trip distance from home to site times
some standard cost per mile of operating a vehicle. Distance
from a person’s home to the site is usually calculated using a
standard over-road software such as PC Miler. It should be
noted that this has to be done to all sites in a persons’ choice
set. An alternative is to use an individual’s reported trip cost
from a survey question. This has the advantage of being the
‘perceived’ cost but the disadvantage of measurement/reporting error in the survey and the complication of usually having
it reported only for the site actually visited by the respondent.
Measuring the time cost component is a thorny issue. In a
world where everyone has a continuous labor-leisure budget
constraint, the wage rate is an ideal value of a person’s opportunity cost of time for a recreation trip. But, many (most?)
individuals simply do not fit this prototype. If a person is a
retired person, a student, a homemaker, an unemployed person, or is paid a fixed annual salary to work 40 h per week,
there is no clear forgone wage and the opportunity cost of time
is not so obvious. There are essentially two ways economists
have handled this issue: using a ‘wage-analogy’ or inferring the
value of time directly in the recreation choice.
The ‘wage-analogy’ is ad hoc but is the most common. One
simply divides a person’s annual income by the number of
hours worked in a year (usually 2000) and uses this as a ‘wage.’
As people are not on the continuous labor-leisure budget
constraint described earlier, this estimate is only loosely tied
to theory. In the final calculation, analysts typically use onethird of this calculated wage as the estimated value of time.
There is evidence from a number of sources that this is a
reasonable adjustment. The mode choice literature in transportation studies, for example, supports this adjustment. Another
reason given for using less than the full wage is that the trip to a
recreation site itself may be of value. A nice ride through the
country side, for example, may be a desirable part of the trip.
For these reasons, albeit highly imperfect, the tradition of onethird of the wage continues to be used in applied work.
Inferring the value of time directly in a travel cost model is
done by entering out-of-pocket travel cost and time separately

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in the model. In the random utility model in eqn [5], bppin is
replaced with btctcin þ btmtmin, where tcin is the out-of-pocket
travel cost measured in money terms and tmin is the simple
round-trip time. In this way, the researcher is not explicitly
placing an ex ante value on time. Instead, time is being
accounted for in the analysis without prior explicit restriction
and can be valued implicitly as btm/btc, the relative contribution
of travel cost and time to a person’s utility. Heterogeneity in the
value of time can be accounted for in the usual way by interacting tmin with attributes of individuals one believes may govern
differences (e.g., income or not working full-time). Another
strategy along these same lines is to identify in advance people
who work for an hourly wage and have a flexible work schedule
and then value their time directly with the wage while treating
only those not working for an hourly wage, as previously
described. The chief drawback of inferring time value within
the analysis is that the tcin and tmin are often highly correlated
as both are determined in part by one’s distance from a site.
Despite some other creative efforts to value time, the two
methods outlined continue to dominate applications in the
literature. Neither is particularly satisfying, but there has been
no compelling reason to abandon either in favor of a new
approach.

Intertemporal Substitution
The bulk of the literature and most of the active research on
TCMs ignore any dynamic aspect of decision making, yet it is
hard to deny that it is important. Dynamic elements allow
people to substitute sites over time and allow experiences
early in the season or perhaps in the last season (such as a
good catch rate of fish) to affect the choice of site and number
of trips in the current season. Individuals may even base current site choices on expectations about future trips.
The repeated site choice model is, in principle, set up for
just such an analysis as it considers an individual’s trip choice
day by day over a season. Nevertheless, few applications consider interdependence in this framework. The typical analysis
treats each trip choice as independent of the previous and
upcoming choices and takes no account of temporal characteristics (such as weather and day of the week). There are two
good reasons for this. First, the data are more difficult to
collect. To gather trip data by date of trip usually requires a
diary to be maintained by the respondents for the recall to be
accurate. This means repeating survey administration
(perhaps, monthly throughout the season) or continual reminders to complete a diary sent early in the season. This
increases the cost of the survey and leads to sample attrition.
Second, there is an inherent endogeneity in trip choice over time.
Unobserved factors affecting trips in period t are no doubt present in periods t  1 and t þ 1. If so, this feedback needs to be
dealt with by purging the explanatory variables of any historical
or future content (most notably, the lag of past trips to the site
used as explanatory variables) of their endogeneity. The instrumental variables needed to make this possible have been elusive.
There have been a few efforts to build time interdependence
into site choice models. As just noted, one way is to use a
measure of past trips to a site as an explanatory variable in
a TCM. For example, some have considered a simple dummy
variable for whether or not a person has visited the site in the

previous season or the current season as an explanatory variable in site utility in eqn [5]. A positive coefficient would imply
‘habit formation’ and a negative coefficient, ‘variety seeking.’
Other types of time-interdependent variables have or might
include time since previous trip to a site, quality of experience
on previous visits to a site, or known upcoming visitation
plans. Research along these lines has been limited to a few
exploratory studies and has largely ignored the issue of
endogeneity; as such, it has not become a standard method
in the literature. Most analyses continue to use data gathered
without date-specific information and simply allow correlated
errors over the season. Interestingly, seasonal demand models
and the KT model implicitly estimate diminishing marginal
utility of trips to individual sites within a season and, at least in
this way, if it exists, implicitly capture the extent of habitforming versus variety-seeking behavior in the sample.
A fully dynamic model where choices over the season are
the result of solving a dynamic programming problem has
been estimated, but given the computational difficulty, this
has only been possible for a single-site discrete choice model
(‘go – don’t go’ each day of the season).
Finally, there have been efforts that combine contingent
behavior data with trip choice data to infer intertemporal
effects. For example, simply asking people what they might
do if a site had been closed or a catch rate of fish at a site had
been lower and then allowing for trip and site choice (substitution) in response to be in different time periods within or
across seasons provides a data set with some time interdependence. These types of data have been exploited by allowing the
response information to serve as alternative sites, but have
fallen short because of not explicitly accounting for the dynamics inherent in the choice.
The evidence to date based on a small collection of studies
and simple common sense is that accounting for intertemporal
substitution and dynamics makes for better models of behavior and is likely to have a large impact on measures of welfare.

Choice Set Formation
When forming the relevant choice set for a multiple-site TCM
study, the usual approach is to begin with the sites of policy
significance and then expand it to include a reasonable set of
substitute sites without reaching a number so large that estimation is infeasible. It is often driven by arbitrary political or
geographic boundaries and, in some instances, leads to highly
aggregated sites. In some cases, counties or even regions larger
than counties can serve as individual sites. Most recent studies
have used less aggregated sites, such as individual lakes, rivers,
parks, or beaches. As a general rule, the more homogeneous
the sites, the less the error faced in aggregation. In practice, it is
best to err on the side of less aggregate sites.
The model being used can also be a factor in choice set size.
Seasonal demand and KT models are nearly always estimated
with fewer than ten or so sites, often with as few as three or
four. It has, however, been shown that KT models can be
estimated with a significantly larger number of sites. RUMbased site choice models are usually used when the number
of relevant substitutes gets large. In some cases, this can be in
the hundreds or even thousands. With certain restrictions applied to the model, it is possible to estimate a TCM using

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randomly drawn alternatives as a proxy for the full choice set,
in effect, allowing for extremely large choice sets.
Most applications use choice sets defined by the analyst, but
there is an ongoing debate about constructing and using choice
sets defined by the respondents; that is, sites people are aware
of and consider in making a choice. The difficulty with using
choice sets formed by individuals is that the choice process that
individuals use to form the narrow set of considered sites is
part and parcel of the process of choosing the best site. Put
differently, sites falling outside the set of considered sites are
simply those with low utility and should be included in the
analysis. If, on the other hand, individuals are unaware of sites,
there is a reasonable argument for dropping them from the
choice set in estimation.

Congestion
Congestion readily comes to mind when ones thinks of recreation sites. Given the growth in population and income and
the decline in transit cost over time, recreation sites naturally
see more use. In some cases, congestion can become a major
policy issue. While the theory of incorporating congestion into
TCMs is well understood, as is the idea of an efficient or
optimal congestion at a site, it is difficult to incorporate its
effect empirically, and there have been only a handful of
studies that have attempted to do so. The difficulty is captured
in a famous Yogi Berra quote about a favorite restaurant of his:
“Nobody goes there anymore, it’s too crowded.” Observing
many people at a site signals its desirability and hence high
probability of visitation. At the same time, it may have gotten
so popular that visitation is actually somewhat lower than it
would have otherwise been but for congestion. How does one
tease out the latter effect? An obvious start is to put some
measure of congestion on the right-hand side of a seasonal
demand or site choice model. But almost any measure one
considers is correlated with excluded unobservables that influence individual demand – the same factors that effect individual demand (make a site desirable) also effect congestion.
There is an inherent endogeneity problem, and sorting out
the partial effect of congestion, without some cleaver instruments, is no easy task.
There are a few instances in the site choice random utility
literature where instrumental variables have been successfully
introduced to identify congestion effects. The instruments are
still somewhat dubious (e.g., weather conditions, day of the
week), but they have passed statistical tests. An alternative
strategy, also appearing in the literature, is the use of some
form of contingent behavior, combining stated and revealed
preference data, where the stated component constitutes
a response to some hypothetically introduced level of congestion. The effects of accounting for congestion in both
the revealed preference/instrumental variable approach and
the contingent behavior approach show that accounting
for the effects of congestion are important to welfare analysis.

Conclusions
The TCM has been in use for over 50 years. It has grown in
sophistication and use along with the growth in applied

357

microeconometrics and applied welfare economics. It is the
mainstay of nonmarket valuation for recreational uses of the
environment.
This article has presented the three primary forms of
the model in use today: seasonal demand, site choice, and
KT. The seasonal demand models are the traditional forms
and are best used in applications where there is a single or
a few sites of interest, substitution outside this set is of limited
relevance, and the focus is on access values. In some
circumstances, cases can be made for using the model to
value site attributes. However, the preferred model when the
number of sites in question is large and/or quality changes at
the sites are of interest is the site choice model using random
utility theory. The site choice RUM model is the dominant
model in the literature due to its flexibility and relative ease
in application. The third model, the KT model, brings together
the best of the seasonal and site choice models in a theoretically consistent way and may very well be the model of the
future. However, because of its computational complexity, it
has not been used widely.
The TCM certainly has its share of issues and complications.
The major issues are essentially the same set confronting the
model since its inception: measuring the value of time, dealing
with multiple-purpose and overnight trips, accounting for
intertemporal substitution, and forming the relevant choice
set for estimation. Despite its flaws and blemishes, research
and application of the TCM appear to be robust and poised for
still more growth.

See also: Allocation Tools: Environmental Cost-Benefit Analysis;
Media: Water Pollution from Industrial Sources; Valuation Tools:
Benefit Transfer; Contingent Valuation Method; Hedonics.

Further Reading
Bockstael NE and Hanemann WM (1987) Time and the recreation demand model.
American Journal of Agricultural Economics 69: 293–302.
Bockstael NE, Hanemann WM, and Strand IE (1986) Measuring the benefits of water
quality improvements using recreation demand models. Report to the U. S.
Environmental Protection Agency, Cooperative Agreement CR-811043-01-0.
College Park, MD: University of Maryland.
Creel MD and Loomis JB (1990) Theoretical and empirical advantages of truncated
count estimators for analysis of deer hunting in California. American Journal of
Agricultural Economics 72(2): 434–441.
Habb TC and McConnell KE (2002) Valuing Environmental and Natural Resources.
Cheltenham: Edward Elgar ch. 7 and 8.
Herriges JA and Kling CL (eds.) (1999) Valuing recreation and the environment:
Revealed preference methods in theory and practice. Cheltenham: Edward Elgar.
Herriges JA and Kling CL (eds.) (2008) Revealed Preference Approaches to
Environmental Valuation, vols. 1 and 2. Burlington, VT: Ashgate Publishing
Company (Part IV).
Landry CE and Liu H (2009) A semi-parametric estimator for revealed and stated
preference data – An application to recreational beach visitation. Journal of
Environmental Economics and Management 57: 205–218.
McFadden D (2001) Economic choices. American Economic Review 91(3): 351–378.
Mendelsohn R, Hof J, Peterson G, and Johnson R (1992) Measuring recreation values
with multiple destination trips. American Journal of Agricultural Economics
74: 926–933.
Morey E, Rowe RD, and Watson M (1993) A nested-logit model of Atlantic Salmon
Fishing. American Journal of Agricultural Economics 75: 578–592.
Parsons GR (2003) The travel cost model. In: Champ PA, Boyle KJ, and Brown TC (eds.)
A Primer for Nonmarket Valuation. Boston, MA: Kluwer Academic Publishers ch. 9.

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Phaneuf DJ, Kling CL, and Herriges JA (2000) Estimation and welfare calculations
in a generalized corner solution model with and application to recreation demand.
The Review of Economics and Statistics 82: 83–92.
Phaneuf DJ and Smith VK (2005) Recreation demand models. In: Maler KG and Vincent JR
(eds.) Handbook of Environmental Economics, vol. 2. North Holland: Elsevier.

Provencher B and Bishop RC (1997) An estimable dynamic model of recreation
behavior with an application to great lakes fishing. Journal of Environmental
Economics and Management 33: 107–127.
Shonkweiler JS and Shaw WD (1996) Hurdle count-data models in recreation demand
analysis. Journal of Agricultural and Resource Economics 21(2): 210–219.

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