December
1981
Economics Report 104
Demand for
Recreational
Facilities:
A Methodology and
Its Application to Camping
Food and Resource Economics Department
Agricultural Experiment Stations
College of Agriculture
Institute of Food and Agricultural Sciences
University of Florida, Gainesville, 32611
/
/
Kenneth
C. Gibbs
Thomas A. Jennings
~
I
I
i
ABSTRACT
Theoretical and empirical models of recreation demand were developed
and estimates for camping generated from the empirical models in Florida
state parks. Development of the theoretical model leads to criteria fa
voring the recreationist's variable daily onsite cost as the proxy price
for recreational facility use. Applicability of the models in analyzing
demand of the multipledestination recreationists was achieved by intro
duction of an adjustment factor applied to travel costs. Some new em
pirical evidence of the role of leisuretime constraints is presented
which suggests that they are of minor consequence to the recreationist's
demand for particular facilities.
Key words: Economics of outdoor recreation, recreation demand,
Floridarecreation demand.
ACKNOWLEDGEMENTS
The research reported herein was conducted at the University of Florida,
and financially supported by the Florida Agricultural Experiment Stations
through State Project No. AS01623. Manuscript preparation and secretarial
assistance were performed at Oregon State University.
The publication is based heavily upon Thomas A. Jenning's Ph.D. disser
tation, "A General Methodology for Analyzing Demand for Outdoor Recreation,
With an Application to Camping in Florida State Parks," Department of Food
and Resource Economics, University of Florida, 1975.
The authors are indebted to countless personnel of the Florida Depart
ment of Natural Resources for the data supplied, expert advice, and unhesi
tating willingness to share their information.
The authors gratefully acknowledge the review comments and helpful sug
gestions of William G. Brown, Ronald A. Oliveira, Roy Carriker, and Thomas
H. Spreen.
ABSTRACT
Theoretical and empirical models of recreation demand were developed
and estimates for camping generated from the empirical models in Florida
state parks. Development of the theoretical model leads to criteria fa
voring the recreationist's variable daily onsite cost as the proxy price
for recreational facility use. Applicability of the models in analyzing
demand of the multipledestination recreationists was achieved by intro
duction of an adjustment factor applied to travel costs. Some new em
pirical evidence of the role of leisuretime constraints is presented
which suggests that they are of minor consequence to the recreationist's
demand for particular facilities.
Key words: Economics of outdoor recreation, recreation demand,
Floridarecreation demand.
ACKNOWLEDGEMENTS
The research reported herein was conducted at the University of Florida,
and financially supported by the Florida Agricultural Experiment Stations
through State Project No. AS01623. Manuscript preparation and secretarial
assistance were performed at Oregon State University.
The publication is based heavily upon Thomas A. Jenning's Ph.D. disser
tation, "A General Methodology for Analyzing Demand for Outdoor Recreation,
With an Application to Camping in Florida State Parks," Department of Food
and Resource Economics, University of Florida, 1975.
The authors are indebted to countless personnel of the Florida Depart
ment of Natural Resources for the data supplied, expert advice, and unhesi
tating willingness to share their information.
The authors gratefully acknowledge the review comments and helpful sug
gestions of William G. Brown, Ronald A. Oliveira, Roy Carriker, and Thomas
H. Spreen.
TABLE OF CONTENTS
Page
ABSTRACT. . . . . ... . . i
ACKNOWLEDGEMENTS . . . . ... . i
LIST OF TABLES. . . . . ... .. iv
LIST OF FIGURES . . . .... .... .iv
INTRODUCTION . . . . . 1
MODEL DEVELOPMENT . . . . ... .. 3
Antecedents. . . . . ... .. 3
Direct Methods. . . . ... ... 3
Indirect Methods. . . . . 4
The General Theoretical Model. . . . 7
Choice of a Proxy for Facility Price . . 8
The Inappropriateness of Travel Cost as the
Proxy . . . . . . 9
Composite Prices, Ideal Proxies, Site
Specificity, and OnSite Cost . . .... 10
The Conceptual Model with OnSite Cost the
Choice of Proxy . . . . .13
Sample Data. . . . . ... ..... 14
THE STATISTICAL MODELS. . . . . ... 16
Definition of Variables. . ..... . 16
Choice of Equation Form. . . . ... 19
THE ESTIMATED MODELS. . . . .... 21
LengthofStay Equations . . . .... .21
FrequencyofVisits Equations .. .. ..... 24
A Note on Leisure Time Constraints .. ........ .26
INFERENCES FROM THE ESTIMATED MODELS. . . .. 28
Use as a Function of Travel Cost . . .... 28
Page
Use as a Function of OnSite Cost. . . .. 30
The Equations . . . . 30
Implications for Aggregate Demand . . .. 34
Use as a Function of Seasonal Change . ... .35
Estimation of Recreation Value . . .... .36.
Value Received from Northern Parks. . . ... 38
Value Received from Southern Parks ........ . 42
Illustration of a ValueBased Fee Policy. . .. 43
SUMMARY AND CONCLUSIONS. . .... . .47
Summary . . . . .... 47
Conclusions . . .... . . 48
Limitations . . .... . . 49
REFERENCES . ..... . . . 52
STATISTICAL APPENDIX: Mean Values of the Variables
for the Samples of Florida State Park Campers, 1973. 55
Correlation Ni'trix: Northern Parks ...... .. 57
Correlation Matrix: Southern Parks ...... .. 59
DEMAND FOR RECREATIONAL FACILITIES:
A METHODOLOGY AND ITS APPLICATION TO CAMPING
Kenneth C. Gibbs and Thomas A. Jennings
INTRODUCTION
Few would dispute the therapeutic and other intangible benefits of rec
reation. Recreation also has economic value, as attested by the fact that
people are willing to pay for it. Interest in its economic value has been
growing as policy makers and planners, especially in the public sector, re
quire more stringent justification for the increasing costs of land and other
resources that must be combined in order to provide opportunities for high
quality recreation experiences.
The research on which this study reports was designed to answer questions
such as: Is the value to users of a camping facility high enough to justify
the ever increasing costs of maintaining the parks for campers or building
new camping parks? Can a methodology be developed to accurately estimate
recreationists'response in order to suggest a more rational fee policy? The
methodology developed herein is a general methodology in the sense that it
could be applied to many types of recreational facilities whose use requires
KENNETH C. GIBBS is associate professor in the Resource Recreation Manage
ment Department, School of Forestry, Oregon State University, Corvallis. He
was formerly associate professor of food and resource economics at the University
of Florida. THOMAS A.JEININGS is currently assistant professor of economics
at the University of Louisville. He was formerly graduate research assistant
in food and resource economics at the University of Florida.
the recreationist to incur tangible cost while actually at the facility.
The specific objectives of the research were to (1) develop a conceptual
model of recreation demand; (2) estimate, on the basis of sample data, the
parameters of a statistical counterpart of the conceptual model; and (3) il
lustrate applicability of the estimated model to real problems of planning
and management. The discussion that follows is divided into three basic
sections: theory, measurement, and application, in that order. The fourth
and final section summarizes the major conclusions of this research and sug
gests some topics for future research.
MODEL DEVELOPMENT
It is of limited practical usefulness to discuss the theory of recrea
tion demand without considering the typical measurement problem arising from
insufficient variation in use fees of many recreation facilities that are
publicly provided. The facility's use fee, U (which may be zero), is all
that can be properly called the facility's own price. In order to estimate
the demand for a facility as a function of its own price, it is necessary to
know a range of prices that the consumer would be willing to pay rather than
surrender his use of the facility. It is thus difficult to estimate the effects
of a fee change directly, except on the basis of the implausible assumption
that consumers' income, prices of alternative forms of recreation, user popu
lations, etc., do not change over a several year period. Moreover, as a use
fee remains constant, the range of higher prices that would not drive the
average recreationist away from a facility is likely to be significantly above
the actual use fee if only because of general inflation of prices and consumer
incomes.
Antecedents
The literature offers two possible ways of estimating the recreationist's
response to a hypothetical fee change; the socalled "direct" and "indirect"
methods.
Direct Methods
The essence of the direct method (as described by Knetsch and Davis [17])
is to ask a sample of recreationists how much they would be willing to pay
for their use of a facility rather than do without it. The problem with this
approach is in its application. Hypothetical questions tend to elicit hypo
thetical answers, based on the respondent's own possibly prejudiced assump
tions. A respondent may overstate his willingness to pay if, for example,
he thinks he might thereby promote improvement or duplication of favored facil
ities. On the other hand, he might understate his willingness to pay if he
feared that his answer might be used to justify an increased facility use fee.
The inability to quantify such biases casts doubts on the accuracy of esti
mates based on a direct approach, even though the aim of the approach is
theoretically sound. For that reason, the authors chose an indirect approach.
Indirect Methods
Indirect methods involve the use of some surrogate, or proxy, for facility
price. More precisely, the recreationist's willingness to pay is based on
observations of costs actually incurred in order to recreate at a facility.
Hotelling [13] is credited with the original ideal of using travel cost (the
cost of overcoming distance between the facility and a series of more or less
concentric distance zones around the facility) as a proxy for the price of a
visit to the facility, although Clawson [4] provided the first application of
Hotelling's idea. Later variations and refinements on the HotellingClawson
approach include Brown et al. [2], Clawson and Knetsch [5], Burt and Brewer
[3], and Pearse [21].
Another group of writers, exemplified by Gibbs [9] and Edwards et al.
[6], divide total trip costs into travel costs, i.e., all costs incurred
while en route to and from the facility, and daily onsite cost, i.e., the
cost per day incurred while actually recreating at the facility. These two
components are then expressed as separate explanatory variables, with onsite
cost being chosen as the facility price proxy.
Gibbs' model contains a novel procedure for empirically delimiting the
5
feasible range of his equation explaining days per visit as a function of the
onsite cost. The idea is that there exist some upper limits to, or "critical"
values of, those costs above which the demand curve ceases to exist, and that
the limits would be reached at some finite quantity demanded. Specifically,
Gibbs hypothesized that (1) the critical travel cost is ceteris paribus, dir
ectly related to the recreationist's actual onsite cost; and (2) the critical
onsite cost is directly related to the actual travel cost. For the single
destination recreationist with which previous studies have been virtually ex
clusively concerned, such hypotheses seem plausible, and have been supported
by Gibb's [9] empirical evidence. For multipledestination recreationists,
the critical costs may be more strongly related to actual costs of visiting
other facilities on those people's itineraries. An easier way to obtain the
critical costs, and one that does not rest upon such hypotheses, is to find
out the minimum amount of time the recreationist would be willing to recreate
at the facility, and then to read from the demand estimate the value of the
price proxy that corresponds to that minimum amount of recreation. Examples
of this approach include Gibbs and Conner [10], and McGuire [183.
The model developed for purposes of the present study borrows more from
the GibbsEdwards type of approach than from the HotellingClawson approach,
for several reasons. First, as Clawson admitted, total trip costs represent
the price of a "total recreation experience" [5, p. 33], which includes not
only recreation at a given facility, but additional enjoyments as may accrue
frum travel and reflection on the sojourn, before and after its undertaking.
Travel cost does not measure the value of recreation at the facility per se.
Second, the HotellingClawson approach is essentially empirical, as pointed
out by Edwards et al. It relies heavily upon the tendency of large aggregates
to exhibit more uniform behavior with respect to a given influence, viz.,
average travel cost from given distance zones, than may smaller subsets of
those aggregates. By dealing with such large populations of people as exist
within a given zone, the influence of other important variables of demand may
be largely averaged out of the picture. Third, the onsite cost is a more
natural proxy for a facility's own price, which is, itself, a part of on
site cost, while travel cost is wholely offsite cost.
Finally, in contrast to the HotellingClawson focus on visits as the
quantity unit, the GibbsEdwards approach uses visitordays actually spent
at the site (per visit and per time period) as the recreation quantity var
iable. This visitorday represents a more homogeneous recreation quantity
unit than either the Clawsonian visit (which does not take into account dif
ferences among visitors' lengths of stay), or visitordays, including travel
time (which ignores obvious differences between time spent traveling and time
spent at a facility). Visitordays per time period on site of a facility
constitute a more integral measure of facility use intensity, the type of
measure of greatest relevance to recreation facility planning and management.
This visitor day is also the conventional product unit on which facility use
fees (the facility's own price) are levied.
It is not implied that the HotellingClawson approach could not be modified
to account for differences in length of stay, but only that this has not been
done by previous users of the approach. More serious problems related to the
use of travel cost as a recreation price proxy,are explained more fully in
the following section.
Whatever their differences, a major common feature of all the indirect
approaches is their implicit assumption that the price of using a recreational
facility can be reasonably well represented by the costs of certain goods and
services that are purchased in conjunction with facility use. An implied
corollary is that the recreationists would react to a change in facility use
fee in the same way as they would react to an equal change in those ancillary
costs. This assumption is subject to both theoretical and empiricaldebate,
which is singularly scarce in previous literature.
Another similarity between previous studies is the absence of suggestions
of how to deal with the multipledestination recreationist that is, the rec
reationsist who visits other sites in addition to the given facility on a
single trip. Travel cost, which explains so much of the individual recrea
tionist's demand for a singledestination facility, cannot be relied upon to
explain the demand for a facility visited on an itinerary that includes other
facilities. Since the aggregate demand for many common types of facilities
is substantially composed of multipledestination visitors, there is clearly
a need for including them in a general theory of demand for particular recrea
tion sites.
The role of leisuretime constraints on recreation demand is a topic that
has drawn considerable expression of concern, but little in the way of practi
cal suggestions of what to do about them. Some theorizing has been done, for
example by Wilson [23], who conceptualizes outdoor recreation as a consumer
produced good, and draws the arguable conclusion that the search for a facility
price proxy is irrelevant.
The General Theoretical Model
A theoretical model was developed out of the above historical context
and with regard to the basic elements of consumer demand analysis.
In general, an individual consumer's demand for a good can be expressed
as an equation relating quantity demand, Ds (measured as usage per time period),
to the good's own price, U (the facility's own price), a set of prices of rele
vant substitutes and complements, P consumer disposable income, I, and a set
of utility variables, or preferencemap shifters, T. The recreationist's usage
per time period is the product of two decisions; one of how frequently to visit
the facility, and the other of how long to stay per visit. A complete theory
of recreation demand should explain both frequency of visits, V, and length of
stay, Dv, resulting in two structural equations and an identify as follows:
(1) Dv = Dv(U, Pr' I, T)
(2) V = V (U, Pr' T)
(3) Ds = V D
Choice of a Proxy for Facility Price
In theorizing about proxy prices to represent variation in the nearly
invariant U, the logical place to seek such proxies is in the set, Pr, of
related prices. In past studies either travel cost, Et, ancillary onsite
cost, As, or both, have been used as facility price proxies. Travel cost
includes all costs incurred while en route to and from the facility, and an
cillary onsite cost includes all costs incurred while actually on site of
the facility, net of the use fee (facility's own price), U. Separating out
these presumed related prices from the set Pr gives the model:
(4) Dv = Dr(U, As, Et, Px, I, )
(5) V = V(U, A,, Et, Px I, T),
where the variable set, Px, contains all the remaining prices of Pr, after
taking out the prices, A and Et. It should be noted that U, As and Et ex
haust all the outofpocket trip costs of the recreationist.
The Inappropriateness of Travel Cost as the Proxy
As previously stated, As was chosen over Et as the source of proxy var
iation in U. The reasons are as follows: First, both U and As are onsite
costs and already expressed in the same units, viz., dollars per day on site.
Travel cost per visitor day on site (Et/D ) would be inversely dependent upon
the length of stay, which is one of the dependent variables. Even where
quantity discounts (lower daily use fees for more days per visit) may be
granted, there is no such strong reason for doubting that Dv depends more
upon As than vice versa. The U (use fee) under consideration is not an entry
fee, but a payment which is based on a standard unit of recreation time.
Travel cost, like entry fees and other purely access costs, resembles a fixed
cost of the opportunity to do any amount of recreating at the facility.
The resemblance of travel cost to a fixed cost has been strongly con
firmed by previous empirical research which has repeatedly found that, while
both the number of visits and length of stay are inversely related to As,
V is inversely related to Et (see, for example, [9, 10, and 18]). Thus,
as in the theory of production and cost, the magnitude of the fixed cost,
Et, determines the scale of the individual's recreational enterprise (view
ing recreation as partly a consumerproduced good). Other things being
equal, the typical recreationist will stay as many days per visit as are
necessary to bring his average fixed travel cost per day on site of the given
facility down to an acceptable level.
The price of travel is an important explanatory variable, even if it is
not an appropriate facility price proxy. A problem arises here, however,
with the multipledestination patron. In theory it is the marginal travel
cost of visiting the given facility to which he should be sensitive. Mar
ginal travel cost may range from Et for the singledestination visitor to
nearly zero for the recreationist who views the given facility as merely a
convenient rest stop en route to elsewhere. The portion of total travel cost
that should be counted as a marginal cost of recreating at a particular facil
ity is clearly related to the relative importance of that facility among all
other destinations of the trip. For this reason, a new technique has been
developed for the present study a travel cost adjuster.
Let r be the travel cost adjuster, Ms the subjective importance of rec
reation at the given facility, and Mo the subjective importance of other rec
reation received from the trip. Then r is hypothesized to be given by a
function such as:
Ms
(6) r M M
s Mo
Thus it is hypothesized that the relevant marginal travel cost of traveling
to a particular facility is given by rEt. A version of this hypothesis,
with tentatively suggested proxies for Ms and M is tested in a later sec
tion of this bulletin.
Composite Prices, Ideal Proxies, Site Specificity, and OnSite Cost
Certain assumptions are necessary to justify use of the ancillary on
site cost as the facility price proxy. Ancillary onsite cost is, in fact,
a composite price of a variable quantity mix of physical goods consumed.
The prices of those goods can be reasonably assumed given to the individual
recreationist, which is to say that he has no monopsony power over their de
termination. The quantities, however, are functions of the prices, consumer
income, and the parameters of consumer preferences, or utility functions.
Having decided that As is indeed a price, there remains the task of
deciding whether it is an appropriate proxy for facility price. If one price
is an ideal proxy for another, then, in technical terms, the true coefficeints
of the facility price and the proxy price must be equal. The buyer would then
be behaving as if he conceptualized the price of a day's recreation as equal
to the sum, As + U. Prices of goods that are perfect complements would meet
these criteria. A less stringent condition is that the goods be equally site
specific, though not necessarily purchased in rigidly fixed proportions. The
concept of site specificity is the crucial key to deciding the extent to which
the ancillary prices are, in fact, adequate proxies for the facility prices.
The cost of the angler's daily bait and tackle consumption could, for
example, be an ideal proxy for the facility price of a given site, but only
if there were no alternative places at which to fish. On that assumption,
a dol;larperday change in the average daily cost of bait and tackle would
change the rational consumer's quantity of site use as much as would a dollar
perday change in the daily siteuse fee. Both prices are perfectly, hence
equally, sitespecific to unique facilities.
The assumption that the ancillary inputs bought with As are perfectly
specific to a given site is generally arguable. It takes only one alterna
tive place for fishing to make the daily bait and tackle cost less than per
fectly specific to the given site. In general, the less sitespecific the cost,
the more its increase is reflected in the cost of recreating at all sites;
hence, the less the incentive to reduce use of, and/or abandon, the given
facility. The more sitespecific the cost, the greater its increase pro
vides incentive to seek out less expensive alternatives. It follows that,
other things being equal, the critical value of an onsite cost component
(U or As) would be inversely related, while the elasticity of demand with
respect to the component would be directly related to the specificity of the
component to the given facility.
One point must be clear: sitespecificity is an issue only in a compar
ative sense. If As and U are equally sitespecific, which is the same as
equally nonsitespecific, then there is no problem with the rational con
sumer whose demand will be equally elastic with respect to a given change
in the use fee and in ancillary cost. Thus, for nonunique facilities, As
may still be a good proxy for U if the market happens to be characterized by
a form of price competition designed to maintain the relative market shares
of all competing enterprises. Pure competition would give this result, al
though time would have to be allowed for expansion or contraction of indus
try capacity before a predicted equilibrium would be established. Perfect
oligopolistic price leadership would be an equally suitable and more quickly
responsive type of market situation for assuming As a good proxy for U, on
condition, of course, that the given facility is one capable of exercising
the price leadership. In either type of market structure, the key lies in
maintaining ratios between prices of all substitutable facilities. The greater
the extent to which everyone's use fees vary in tandem, the less specific is
Other onsite costs, for example food, may better reflect the cost of
subsistence, rather than recreation per se. While netting out subsistence
costs is a serious empirical problem it need not be a theoretical one. Thus,
for purposes of this theoretical discussion, A may be assumed to contain
no subsistence costs.
a given facility's price to itself.
Some degree of uniqueness or local prominence among competing facilities
is not atypical of many public outdoor recreation sites. In the cases of
facilities such as federal and state parks, their existences can generally
be attributed to the public sector's unique ability to conserve large por
tions of unusual natural environments and maintain a monopoly over them. The
public facilities may also exercise a degree of unconscious price leadership
over such establishments as, for example, commercial campgrounds, whose own
existences may be due largely to the public parks. Thus, the assumption that
the use fee, U, and ancillary cost, As, are about equally, though imperfectly,
sitespecific may be appropriate for such facilities.
The Conceptual Model, with OnSite Cost the Choice of Proxy
With the caveats stated, the theoretical model developed in the fore
going consists of the following equations:
(7) Dv = Dv(Es, rEt, I, Px, T)
(8) V = V(Es, rEt, I, Px, T)
(9) Ds = V Dv
for E < E*
The variable, E = U + A has been selected as the market equivalent facil
ity price proxy. The model is specified subject to the restriction that Es
cannot exceed the critical E*. All other variables are as previously defined.
Discussion of the remaining variables, I and the components of the sets Px
and T (in which perhaps a time constraint should be included), is contained
in a following section.
Equations (1) and (;2) are not a simultaneous system in the econometric
sense if all righthand variables can be assumed exogenous, which is deemed
reasonable in view of the data base described in a subsequent section. At
the same time, it is difficult to conceive of an independent variable belong
ing to either Equation (1) or (2) that does not also enter the other. In
this sense, estimation of the two relationships separately will provide in
formation concerning the relationship between length of stay and visiting
frequency.
Readers of Oliveira and Rausser [20] may be tempted to view Equations
(1) and (2) as reduced forms of simultaneous user demand and supply equa
tions, inasmuch as the independent variables may include some that reflect
conditions of supply (in particular, quality of supply) as well as demand.
The basic "demand" nature of the equations can be argued on two grounds:
First, that quantity supplied of tangible use potential is essentially immune
to effects of variation in any of the conceivable independent variables, and
second, that the typical state park camper encounters less than fullyoccupied
campgrounds, making usage almost totally a question of demand.
Sample Data
The theoretical model, as with any conceptual framework, needs empirical
data to test its applicability. While the main focus of this bulletin is the
development of a defensible theoretical model to describe recreationists'
demand for recreational facilities, it is appropriate to include a discussion
of the empirical counterpart.
The problem was to obtain a representative sample of data from a large
study area at reasonable cost in time and money. It was discovered that state
park registration files contain names and addresses of past campers, which
raised the possibility of a telephone survey of those campers as perhaps a
justifiable way to cover a statewide system of facilities. Camping at Florida
state parks forms the data base for applying this theoretical model. While
the results are representative of the state park system in Florida, the gen
eral implications and applicability of the model are valid for many areas.
A telephone survey was adjudged the most efficient way for purposes of this
study.
A sample of campers visiting Florida state parks between April 1972 and
July 1973 was drawn. The 36 parks with campgrounds were first stratified by
locationthose north and south of Orlando, Florida.. A representative sample
of 10 parks was chosen while maintaining a balance within important geograph
ical differences (including seaside versus inland parks). The 10 sample parks
accounted for 63 percent of total state parks registrations in fiscal year
1973. Available campsite occupancy data [8] suggested that an acceptable
sampling precision could be obtained from completed interviews with about
400 visitors divided equally between the northern and southern sample parks.
That target sample was allocated among the sample parks according to porpor
tional use received. Within each park the sample to be surveyed was distributed
between a high and a low use season, again based on proportionality. Of the
slightly over 800 names and addresses of campers obtained from the sample
park registration cards, the telephone survey produced 357 completed inter
views, or 89 percent of the 400 interviews targeted. Those completed inter
views comprise .064 percent of total 1973 registrations at the ten sample
parks and .020 percent of total 1973 registrations (including renewals) at
all state parks with camping. More details of the sampling procedure and
questionnaire can be seen in [15].
THE STATISTICAL MODELS
The statistical counterpart of the theoretical model consists of the
following equations:
(10) In Dv = In a10 + a1 In Es + al2 rEt In Es + al3 In (rEt)
+ a14 In I + a15 In (1/N) + b11 S + b12 D + b13 X
(11) In V = In a20 + a21 In Es + a22 rEt In Es + a23 In (rEt)
+ a24 In I + a25 In (1/N) + b21 S + b22 D + b23 X
(12) In Ds = In V + In Dv
for Es < E *.
Definition of Variables
The a.. and b,. are the structural parameters to be estimated. The
variables are defined as follows:
Dv = length of stay per visit by the recreation group in 12hour periods.
V = visits per month by the recreation group to a park in the universe
sampled during a given season (summer or winter).
Ds = use per month in 12hour periods (Ds = V 6 Dv).
Es = onsite cost per recreation group, in dollars per 12hour period
(Es = U + As).
rEt = adjusted roundtrip travel cost per group, where:
Et = total travel cost, including all transportation, lodging,
and other outofpocket expense from home or other point
of origin and back to origin, net of costs incurred while
actually present on site of the given facility.
r = travelcost adjuster.
I = recreationist's family income before taxes.
N = number of persons in the recreation group.
S = a seasonal dummy, where low demand season = 0 and high demand
season = 1.
D = a majordestination camper dummy, where 1 = major destination
visitor and 0 = nonmajor destination visitor to the given parks.
X = a tentcamper dummy, where 1 = tent camper and 0 = recreational
vehicle camper.
E* = the critical, or maximum, onsite cost the recreationist would
s be willing to pay per group rather than not visit the site at all.
Some of the above variables deserve more detailed explanation. Frequency
of visits is expressed on a monthly basis because of the presence of the sea
sonal dummy, S, and because the seasons as defined are of unequal length.
The case where S = 1 refers to the season in which rate of use, Ds, is above
the yearly average for the given group of parks. Table 1 explains the divi
sion of seasons. Separate estimates of the model were made for parks north
and south of Orlando.
The travelcost adjuster, r, is estimated by:
(13) r = (T1 + Dv)/T2'
where T1 is the total necessary roundtrip travel time (directly to the site
and back home, at reasonable speed), and T2 is the total time actually away
from home. Thus, time is used to measure the relative importance specified
in the theoretical travelcost adjuster, Equation (6). This admittedly ar
bitrary adjustment is deemed less arbitrary than using total unadjusted travel
cost in explaining demand for sites that serve multipledestination recrea
tionists. Actual time, T2, was determined with the questionnaire. Total ,
necessary time was estimated on the basis of map sources ([1], [8], [22]),
and the authors' own travel experiences.
Table 1.Breakdown of seasons
Area High season Low season
North...................AprilAugust (5 mo.) SeptemberMarch (7 mo.)
South...................DecemberApril (5 mo.) MayNovember (7 mo.)
SOURCE: Data from Florida Department of Natural Resources.
Personal income before taxes was the measure of income used in the model.
A better estimate of actual purchasing power would have been disposable in
come (excluding income taxes), but it was thought that a typical interviewee
would be more aware of his family's beforetax income than of how much income
taxes were paid. In view of the complexities of federal, state, and local
income tax codes, no attempt was made to derive disposable income from per
sonal income.
The seasonal shifter, S, is a catchall dummy for all seasonally induced
reactions by both sellers and buyers of Florida recreation. It is a utility
variable and also a priceofalternatives variables, since seasonal price
discrimination is a highly profitable and widely practiced art of private
enterprise, as a perusal of any major hotel's rate schedule illustrates (see,
for example, [14]). Since commercial campgrounds are a fastgrowing sideline
of nearly all the major hotel chains in Florida, the seasonal priceofalter
natives component of S can be expected to increase in the future.
The variables N, D, and X are also surrogates for utility parameters.
The tentcamper dummy, X, also may capture some of the influence of the value
of durable recreation equipment on quantity of site use. Its hypothesized
coefficient would be negative. For equally obvious reasons, the hypothesized
coefficient of D would be positive. Group size N is included primarily to
account for variation in campers' characteristics.
Choice of Equation Form
Past experience in the estimation of outdoor recreation demand suggests
that the relationships are curvilinear. The simplest types of curvilinear
equations to estimate are logarithmic forms and, in fact, logarithmic forms
2
have performed well in previous empirical studies.2
The algebraic equation form chosen is a modified constantelasticity
ofdemand (CED), or exponential, function. The basic CED function was modi
fied in order to test a version of Marshall's [19, p. 8] principle of lower
elasticity of demand for items less important in the consumer's budget; that
is, the hypothesis that the elasticity of facility use, with respect to the
facility price proxy, is inversely related to his adjusted travel cost. That
modification is as follows:
bI + d t c2 c3
(14) Dv = cE (rEt) I
which is linear in logarithms. According to Marshall's principal, dI > 0,
since bI < 0 if Es is a good proxy price and Dv is a normal (not inferior)
2According to the mathematics of the neoclassical theory of consumption
and demand, the demand of the rational consumer is homogeneous of degree zero
in prices and income. (For a relatively simple modern interpretation, see
Henderson and Quandt [11, Ch. 2].) Only a curvilinear equation could fulfil.
that condition. The modified CED equation is capable of degreezero homo
geneity, but does not force it. A strictly linear demand equation cannot
be homogeneous of degree zero.
good. By definition, the Es elasticity of Dv is given by bI + d1rEt, and
similarly for the Es elasticities of V and Ds.
An alternative model, substituting costs per person, Es/N and rEt/N,
for the pergroup costs, Es and rEt, was also estimated for each of the two
Florida subregions. The estimates are not shown in this bulletin, but are
given in [15].
It makes negligible difference to the equation estimates whether costs
per group or costs per person are specified. Where, as in the present case,
the use fee is levied on a pergroup basis, it is somewhat more convenient
to express the onsite and travel cost on the same basis. At any rate, no
more reasonable criterion is apparent.
THE ESTIMATED MODELS
Estimates of the D and V relationships are shown in Tables 2 and 3,
respectively. The variables are defined as before. Standard errors of the
coefficient estimates are shown in parentheses under the estimates. All
equations were estimated by the method of ordinary least squares.
For each estimated equation, the residuals were examined for nonsper
ical disturbances. In general, the assumption of random errors seems tenable.
There is some evidence of heteroskedasticity in estimates of the visits re
lationship, suggesting a geometrically positive correlation between the error
term and frequency of visits. There was no attempt to correct for this since
(1) the problem seemed minor among more obvious problems with the visits
equations, and (2) a vast majority of observations fall in a small range of
visits within which homoskedasticity can be reasonably assumed. Correlation
matrixes of the variables are included in the Statistical Appendix.
Lengthof stay Equations
The interaction terms, while statistically significant for the northern
parks, show up with a negative coefficient estimate for D which is con
trary to the hypothesized lower elasticity of Es demand at greater adjusted
travel cost. For southern parks the interaction term was not significant.
However, the statistical characteristics of the estimates are encouragingly
congruent with theoretically based expectations.
From a statistical standpoint, the travelcost adjuster renders notable
improvement in the Dv equation estimates. A comparison between the results
of specifying total (unadjusted) Et instead of rEt is shown in [16]. In
Table 2.Coefficient estimates of the lengthofstay (Dy) relationship: Florida state parks, 1973
constant(
Equation term In E In (rE) rEt In E
Northern parks
(15) 2.337 4.34*** .629*** .00016***
(.074) (.046) (.00004)
 Southern parks
(16) .508 .518*** .577*** .00020
(.087) (.071) (.00013)
Degree of 2
Equation In I In(/N) S D X Subject to freedom F R
Northern parks
(15) .183*** .155*** .166*** .726*** .092 E < 18.82 168 49.447 .702***
(.068) (.077) (.071 (.073) (.074) s
Southern parks
(16) .093 .069 .057 .598*** .215*** E < 21.57 171 32.600 .604***
(.066) (.096) (.086) (.091) (.085) s
Significant at 10 percent level.
** Significant at 5 percent level.
*** Significant at 1 percent level.
SOURCE: Sample data.
Table 3.Coeffcient estimates of the frequencyofvisits (V) relationship with adjusted travel cost:
Florida state parks, 1973
Constant
Equation term In Es n (rEt) rEt n E
Northern parks
(17) 1.015 .094 .335*** .00027***
(.112 (.069) (.00006)
Southern parks
(18) 1.801 .017 .084 .00001
(.069) (.057) (.00010)
Degrees of 2
Equation In I In (1/N) S D X freedom F R
Northern parks
(17) .029 .061 .387*** .459*** .023 168 6.883 .247***
(.103) (.116) (.107) (.109) (.111)
Southern parks 
(18) .048 .037 .124* .290*** .083 171 3.497 .141***
(.052) (.076) (.068) (.072) (.068)
*Signifivant at
**Significant at
***Significant at
10 percent level.
5 percent level.
1 percent level.
SOURCE: Sample data.
every case, use of adjusted travel cost results in a greater number of sta
tistically significant parameter estimates as well as a notable improvement
in the coefficient of determination.
The critical costs in the "Subject to" column of Table 2 are inferences
from the estimated lengthofstay equations. The critical cost corresponds
to that value of onsite cost obtained by substitution of the average inter
viewee's declared minimum acceptable length of stay needed to justify the
time and expense of the visit, in the manner of [10] and [18]. At the crit
ical cost the recreationist would take only his minimum acceptable length of
stay, and at higher onsite cost would not visit at all.
FrequencyofVisits Equations
It is more difficult to summon enthusiasm for the visits equation esti
mates in Table 3, although the Fvalues are high enough to suggest that the
problem may be more incompleteness than otherwise erroneous specification.
There is some evidence, however, that a part of the problem may be the need
for a better procedure to account for travel cost.
In contrast to the case with the Dv equations, the statistical charac
teristics of the V equations using adjusted travel cost are not notably dif
ferent from the result of substituting total travel cost for rEt. Indeed,
making that substitution, the coefficients of Et become significant at the
1 percent level for the southern parks, while the coefficients of rEt are
not statistically significant for those parks. (These comparisons may also
be seen in [15].) That this may not be due to random happenstance is sug
gested by both a point of theory and an empircal fact.
The theoretical point is the concept of marginal travel cost as an al
ternative cost of other desirable (and available) destinations foregone. In
this sense, the marginal travel cost of visiting a particular facility con
sists of the benefits, or value, that could have been obtained by recreating
elsewhere instead of at the given facility. Logically, this poses the seem
ingly formidable practical problem of accounting for the value of all next
mostvaluable recreational alternatives to trips within the vicinity of the
particular facilities under study.
The empirical fact is that the visitor to a southern park paid over twice
the travel cost of the visitor to a northern park, and came from over twice
as far away. This was not due solely to the length of the Florida Peninsula,
since the northern parks received a considerably higher percentage of in
state and borderstate visitors than did the southern parks. It is reasonable
to assume that the numbers and variability of alternatives considered by a
group of recreationists would, other things being equal, be directly related
to the distance (or, as suggested by the geometry of circles, the square of
the distance) between their points of origin and the particular facilities.
In other words, the greater the distance traveled, the more the travel cost
measures the influence of alternative forms of recreation foregone.
If higher total travel cost implies the consideration of greater numbers
of highervalued substitutes for use of particular facilities, then the elas
ticity of visits with respect to travel cost is perhaps not constant, but
increasing. Two alternative equation forms (the strictly linear and the power
ofe function), do give an increasing Et elasticity; however, both gave less,
rather than more, statistical confidence in the estimated influence of Et on
V. Those results, of course, by no means reject the hypothesis of increasing
Et elasticity of visiting frequency. The only conclusion to be drawn at this
point is that visiting frequency needs a better conceptualization. The
HotellingClawson method of using average costs of traveling from given dis
tance zones glosses over this problem as does any attempt to estimate a func
tion on the basis of averages instead of the highly (and only perhaps random
ly) variable observations from which the averages are computed. In statis
tical analysis, the computation of average variables for use in a regression
analysis may constitute an evasion of the hard task of improving economic
theory.
Had total usage, DS, been regressed on the independent variables speci
fied in this study's models, the numerical values of the coefficient esti
mates would necessarily have been the same as the results of combining the
Dv and V estimates through the identity Ds = V Dv. It was verified that
fairly good statistical characteristics of the Ds equation are achieved by
such an approach, as can also be seen in [6]. The decomposition of Ds into
its components, D and V, thus aids the refinement of theory by pointing out
a trouble spot. Unreliable estimates of the V relationship undermine confi
dence in statistically "reliable" estimates of the Ds relationship.
A Note on Leisure Time Constraints
The authors made two attempts to estimate the influence of a leisure
time constraint, both attempts being based on the answers of respondents to
the following question: "Why do you not camp more in these state parks?"
Is it mainly the cost of travel, etc., of recreating there, or is it mainly
the lack of time for taking such excursions?"
Of the 357 respondents in the sample, 279, or 78 percent, gave money
cost as the primary constraint of their recreation at the given type of facil
ty. Thirtyfive, or just under 10 percent, cited limited time as their more
effective constraint. The remainder could not make up their minds.
A dummy variable was used in the statistical model to distinguish campers
who cited time from those who cited cost as the contraint on their activity.
No statistical significance attached to that dummy's coefficient. A dummy
was then included to distinguish between weekend campers and all other campers.
That dummy also was not significant statistically.
The fact that such an overwhelming majority of respondents cited cost
instead of time as their constraint suggests that omission of explicit time
variables does not seriously harm empirical models of demand for facilities
such as the Florida state parks;
INFERENCES FROM THE ESTIMATED MODELS
The estimated models constitute explanations of the typical recreation
ist's behavior with respect to the independent variables. The purpose of
this section is to highlight characteristics of that behavior which may be
relevant to facility planning and management, and to illustrate how the models
can be used in formulating planning and policy criteria based on some of those
characteristics.
Use as a Function of Travel Cost
The estimates confirm the findings of previous studies that higher
travel cost, while reducing visiting frequency, increases length of stay per
visit. Moreover, the positive influence of adjusted travel cost on length
of stay prevails over the negative influence on visiting frequency, making
travel cost a net positive influence on use per time period, Ds. This can
be seen by inspection of Equations (15) through (18) (Tables 2 and 3).
In general, the form of the equation giving Ds as a function of rEt,
ceteris paribus, is:
(19) In Ds = a + b In (rEt) + c (rEt In Es),
where a is a constant resulting from holding all other variables at their
appropriate mean values, and b and c, are the sum of the estimated coeffi
cients of adjusted travel cost and interaction term, respectively, from the
V and Dv equations of each model. The elasticity of Ds with respect to rEt
is given by the derivative:
(20) (ln Ds)
(20) rIn t = b + c rEt In E.
a In (rEt)t s
For equations (16) and (18) (the model for northern parks), b = .629 .355 =
.294, and c = .00016 + .00027 = .00009. Substituting these values and the
mean value of the interaction terms into Equation (19) gives .294 + .00009
(328.99) = .590.3 Doing likewise for Equations (16) and (18) gives, in
summary:
Northern Parks:
(21) (In Ds
in (rE t = .294 + .0009 (328.99) = .590.
Southern Parks:
(22) (ln Ds)
In (rE = .493 + .00019 (391.56) = .567.
Both evaluations are positive. Thus according to the estimates, the typical
Florida State park camper makes greater use of the parks, the larger his
adjusted travel cost. He visits less frequently than the average recreation
ist, but stays so much longer per visit that his total use is greater than
average. This is consistent with the concept of travel cost as overhead.
An implication is that thetypical majordestination visitor from outofstate
3The arithmetic mean of the logarithm of a variable, X, corresponds to
the geometric mean of X, i.e.:
I N N
In (1/N) Z In X. = ( X.) 1/N
i=l i=1
The regression hyperplane passes through the arithmetic means of In V, In
D In E In (rE ), In I, and In N. The exponential form of the equations
tKus passes through the geometric means of those variables. Thus, the term
"average", as used herein for deriving predictive equations of response to
particular variables, always refers to the geometric mean of a variable
transformed into lojaritlhms. The arithmetic mean is the appropriate average
for the dummy variables only, which are not transformed into logarithms.
The interaction term is a hybrid of transformed and untransformed variables.
The regression hyperplane passes through the arithmetic mean of rE In E ,
for example, which the regression program treats as a separate variable.
uses the parks more than the typical instate camper.
Use as a Function of OnSite Cost
The variable Es, onsite cost, includes and is the proxy for the use fee,
U, and constitutes the only variable under the direct control of park manage
ment. Thus, the possibilities of influencing use by variation of U, hence
of Es, are of special interest.
The Equations
Relationships showing facility use, Ds, as a function, ceteris paribus,
of onsite costs, are derived by holding all other independent variables at
their appropriate mean values. The results of doing this for Equations (15)
and (17) and (16) and (18), are Equations (23) through (28), respectively:
Northern Parks:
(23) In Dv = 2.614 .458 In Es;
(24) In V = 1.633 + .135 In Es;
(25) In Ds = In (V Dv) = .981 .323 In Es.
Southern Parks:
(26) In Dv = 2.801 .480 In Es;
(27) In V = 1.573 .019 In Es;
(28) In Ds = In (V Dv) = 1.288 .499 In E .
The differences between the coefficients of In Es for Equations (15) through
(18) and (23) through (28) arise from the repeated presence of In Es in the
interaction term.
Onsite cost was found to be a statistically significant variable in the
equations explaining length of stay, Dv but not significant in the equations
explaining frequency of visits, V. In the V relationship, Equation (24), the
positive Es elasticity even contradicts theoretical expectation. This points
out one of the advantages of estimating the visits and lengthofstay rela
tionships separately. Anyone using the estimates is thereby apprised of the
possibility that a portion of the recreationist's total use may be unreliably
explained, viz., that portion which estimates the effects of Es upon V in the
present instance. That fact could not be known were only the Ds relationship
estimated directly, even though the coefficient estimates of the estimated
Ds equation would be the same as those of the derived Ds equation. Indeed,
it was verified that the Es elasticity estimate is significant at the I percent
level in all estimated Ds relationships, a finding which the present procedure
reveals as possibly misleading.
There are two lines of argument on the question of how to proceed with
use of the estimates from this point. On the one hand, there is a theoretical
argument for having the variable, Es, in the "visits" relationship despite
the failure to confirm its influence statistically, and since one of the goals
is to predict Ds, theoretical consistency might best be served by basing pre
dictions on Equations (25) and (28'). On the other hand, since the empirical
results fail to confirm any influence of Es over V, the more empirically minded
may be disposed to assume that onsite cost has negligible influence upon
visiting frequency, and base all predictions of the effects of Es on manipu
lation of the lengthofstay relationship. For present purposes, the predic
tions will be based on the derived Ds relationships, Equations (25) and (28).
According to the elasticity estimates contained in those equations, a 10 per
cent increase in daily onsite cost would reduce the average recreationist's
total usage by around 3.2 percent at northern parks and 5 percent at southern
parks. These predictions are also, of course, subject to the assumption of
equal site specificity of U and Es, as previously explained.
Another type of prediction that can be based on the Eselasticity is of
how the average recreationist's contribution to fee revenues may vary with
the use fee. To give an idea of the magnitudes involved, the predicted re
sults of a hypothetical 50centsperday fee increase are calculated and
shown in Table 4, for the actual sample data. Column (5) contains the pre
dicted declines in total usage per month. Thus, the predicted effect of a
50cent increase in onsite cost per 12hour period would be to reduce the
average northern park visitor's monthly use by .04 12hour periods, or .48
12hour periods per year. For the same increase in daily group onsite cost,
the average visitor to a southern park would reduce his recreation by an esti
mated .06 12hour periods per month, or .72 12hour periods per year. The
current use fee is approximately $2 per recreationist group per 12hour per
iod at all state park campgrounds; thus, the current fee burden is $4.44
(= $2 2.22 periods) per month at the sample northern parks and $4.36
(= $2 2.18 periods) per month at the sample southern parks. Raising the
fee to $2.50 would reduce use per month to an estimated 2.18 (= 2.22 .04)
periods per month at northern parks and to 2.12 (= 2.18 .06) periods per
month at southern parks. Thus the predicted fee burdens would be (2.18)
($2.50) = $5.45, and (2.12)($2.50) $5.30 per group per month at northern
and southern parks, respectively. In summary, the predicted effects of the
Table 4.Prediction of the change in usage per visitorgroup due to a 50cent increase in group daily
onsite cost
Percentage
equivalent
Arithmetic average of a 50cent Arithmetic average
Estimate onsite cost per increase in usage for sample Predicted change in
group for sample daily onsite (12hour periods usage (12 hour
($ per 12 hours) cost per month) periods per month)
(1) (2) (3) (4) (5)
North.......... 9.66 5.2 2.22 .037
South.......... 8.62 5.8 2.18 .063
SOURCE: Sample data.
50cent increase would increase the fee burden per northern recreationist
group by $1.11 per month, and per southern recreationist group by $0.94 per
month.
Implications for Aggregate Demand
These models, being explanations of the demand of the average, or "typ
ical", consumer, contain no equation explaining the total number of recrea
tionists who patronize the parks. Such an equation would have to explain
not only the behavior of the universe of actual patrons, but the behavior
of potential patrons as well. While it is beyond the scope of this study to
explain the behavior of potential patrons, a few words on the principles and
problems of relating individual to aggregate demand are appropriate.
It is reasonable to assume that some potential patrons are nonpatrons
of the state parks because.the actual onsite cost would be above its criti
cal value to them. Short of a sample survey of the population of North Amer
ica, there is no feasible way to acquire information on the individual charac
teristics of those potential patrons and their demands for use of the state
parks. In order to predict the number of new patrons that.would visit for
various hypothetical reductions of the user fee, it would be necessary to
know the numbers of potential patrons whose critical onsite costs fell within
a given range below the actual onsite costs of their use, were they to visit.
Some inferences about those numbers might be made based upon the relative
proportions of users in certain socioeconomic classes of the population in
combination with separate estimates of the model and of critical costs for
subsamples of actual patrons, stratified according to socioeconomic charac
teristics. The size of sample needed for such a procedure would be quite
large.
Use as a Function of Seasonal Change
The influence of seasonal change on facility use is measured by the
estimated coefficient of the seasonal dummy variable, S. This variable was
statistically significant in all of the estimated V relationships, and in
both of the estimated D equations for the northern park visitor. It was
not significantly different from zero for the Dv equations of the southern
park visitor.
In terms of the Dv relationships as expressed in Equations (23) and (26),
for example, the effect of seasonal change is estimated as the difference in
the constant term as S is set equal to zero (for lowdemand season) and then
equal to one (for highdemand season). For the northern parks visitor, the
resulting equations are as follows:
(29) In D = 2.504 .458 In Es (S = 0).
(30) In Dvl = 2.670 .458 In Es (S = 1).
From Equations (29) and (30), the relative difference in the estimated
average length of stay between seasons is calculated as follows:
(31) In Dvl In Dvo = 2.670 2.504 = .166,
D v/Dvo = e166
v1 vo
Dv1 = 1.18 Dvo.
Thus, the average summer visitor spends an estimated 18 percent longer per
visit than does the average winter visitor to the northern parks,
The absolute differences in seasonal length of stay given by Equations
(29) and (30) are between geometric means. To calculate the comparative
seasonal arithmetic mean lengths of stay from the equations, it is necessary
to assume that the relative seasonal differences between the arithmetic and
geometric mean lengths of stay would be approximately equal. On that assump
tion, Dvo = 5.96 12hour periods, and Dvl = 6.95 12hour periods, as compared
to the arithmetic yearly average, Dv which, for the northern sample, is 6.56
12hour periods. The difference is approximately 13 hours per visit between
the (arithmetic) average summer length of stay and the average winter length
of stay at the northern parks.
By the same method, and using the arithmetic sample averages reported
in the Statistical Appendix, estimates of seasonal variation in length of
stay, visiting frequency, and total use per visitor group were calculated
and are shown in Table 5. The much greater seasonal variation in use and its
components in northern parks is quite apparent.
Use per recreation group, Ds, is the part that the model explains of
aggregate use, which can be measured on the basis of campsite occupancy data
[8]. Assuming one campsite per visitor group, the last 3 rows of figures in
Table 5 represent what the model does not explain, namely, that an estimated
75 percent more people visited the northern parks during the summer season
than during the winter season, while twice as many visitors came to the south
ern parks during the winter season as came during the low summer season.
Estimation of Recreation Value
A measure of value known as consumer surplus can be derived from the
demand estimates. While there are several alternative concepts of consumer
surplus [11, chap. X], for practical purposes it can be defined as an area
Table 5.Estimated average and seasonal components of campground use,
Florida state park system, 1973
Use component estimate Northern parks Southern parks
Sample average length of stay per visit,
in 12hour periods (D ):
Both seasons average (D ) ............ 6.56 8.63
High season estimate (Dvl)............ 6.95 8.32
Low season estimate (Dvo )............ 5.96 8.85
Sample average visits per month (V):
Both seasons average (V)............. .370 .232
High season estimate (V1)............. .424 .246
Low season estimate (V )............ .288 .218
Sample average use per recreation group,
in 12hr. periods per month (D = VD ):
Both season average (T ) ............. 2.43 2.00
High season estimate (Dsl)............ 2.95 2.00
Low season estimate (D o) ............ 1.72 1.93
Recreation groups per month in all
park: (G):
Both seasons average (G)............. 95 91
High season estimate (G1)............. 128 129
Low season estimate (G )............. 73 64
Aggregate use, all parks, in 12hour
periods per month (GD ) :
Both seal.oni average (GDs ) .......... 230 182
High season estimate (G IDsl)......... 378 264
Low season estimate (G Dso).......... 125 124
aFigures presented are arithmetic means and equivalents.
bSeasonal estimates are not statistically different from both seasons'
average .
cEstimate of recreation groups (G) is based on the assumption that each
group ouc~cjies one .ampisite.
SOURCE: Sample data and tabulation of Florida park attendance, Tallahassee,
1972 (mimeographed).
under the pricequantity demand curve. In Figure 1, for example, the consumer
surplus is equal to the area, Es a b s, where Es* is the'camper's critical
onsite cost, Es his actual average daily onsite cost, Dsm his minimum accept
able use, and Ds actual average use. The reasoning behind this concept of
value is as follows: The camper would be willing to pay E* daily for the
first Dsm days of recreation, but, in fact, pays only Es daily for this same
amount, and, thus, earns a surplus value equal to the area E* a c E which
is the difference between the maximum money he would be willing to pay and
the actual money he pays for the minimum acceptable use. For additional days
he pays Es per day, but would be willing to pay more until he reaches equil
ibrium at Point b, where he pays for the last moment of his use exactly what
it is worth to him. Thus, the area, abc, is also a part of his surplus, or
unpaidfor, value.
With reference to Figure 1, the mathematical expression for the average
recreationist's consumer surplus, CS, is as follows:
E*
(32) CS = s f(Es) dEs.
Es
The function f(Es) is one such as Equations (25) or (28).
Value Received from Northern Parks
The average northern park visitor's use as a function, ceteris paribus,
of Es is given by Equation (25). In its untransformed version, this equation
is written as follows:
(33) Ds e981 E323
where: E* = $18.82; E = $8.50
S 5
Daily onsite costs (E )
*
E
s s a
I
0s ___ I
E I b
I I
I D s D (Es)
D Use per time period (D)
a S
s
Figure 1. Illustration of consumer surplus
The critical onsite cost corresponds to the geometric mean minimum length
of stay. Average onsite cost is also the geometric mean. The corresponding
definite integral, with its evaluation, is:
i f$18.81 .323
(34) Value per month per group = e981 Es dEs = $11.96.
$8.50
Dividing the value of Equation (34) by the geometric mean use per month (1.335)
gives an estimated value per 12hour period of $8.95 per group. Since the
mean number of visits per month is .261, this is equivalent to a value of
$45.82 per visit of the average group ($11.96 .261).
The equivalent value per person, based on the geometric mean group size
of 3.64, is $12.59 per visit. Using the number of visitors for the fiscal
year 1972 (from Florida Department of Natural Resources [83, which was an
estimated 340,700 campers to the northern sample parks, the total value pro
duced by the northern samples for campers in that year would be calculated as
approximately (340,700)($12.59), or somewhat more than $4,289,000. Assuming
the sample parks are representative of all northern parks, the estimated value
for the approximately 1,117,500 campers received from all parks north of Orlando
would be (1,117,500)($12.59), which rounds off to an estimated $14,000,000 for
the fiscal year 1972.
The effects of seasonal change are borne entirely by the seasonal dummy,
S. Thus, the effect of seasonal change on value per month is represented
entirely by varying the exponent of e in Equation (34). The exponent, .981,
results from setting S at its mean value, S = .661. When S = 0, the exponent
becomes .621; when S = 1, it is 1.16. The resulting value estimates are sum
marized in Table 6, which is derived directly from Equation (34), as just des
cribed by substitution of the appropriate value of S.
Table 6.Estimated value of camping to the recreationist group: Florida
northern state parks
Value Value per Value
Time of aar per month 12 hours per visit
(1) (2) (3) (4)
Low season (S = 0)... $ 8.34 $ 6.25 $35.01
High season (S = 1).. 14.38 10.77 55.15
Total year (S = S)... 11.96 8.95 45.86
SOURCE: Sample data.
Since the Florida state park campsite fee is levied on a groupdaily
basis, it is of interest to examine the estimated consumer surplus per day
of recreation onsite. Having used the same yearly geometric average costs
as the limits of integration in calculating both yearly and seasonal average
values, it is appropriate to use the yearly geometric mean length of stay
(= 5.122 12hour periods) for calculation of both yearly and seasonal values
per recreation day. These estimates are shown in Column (3) of Table 6. The
seasonal difference between willingness to pay might be considered by policy
makers as a basis for seasonal fee discrimination. This is not to suggest
that public enterprises should price their services at what the traffic will
bear, or even at full cost of production, which would doubtless be politically
infeasible, not to mention ideologically repugnant to many of the people. It
is something quite different to suggest that, where greater than token fees
are charged, they be made to bear some relationship to the value of services
received. On that basis, it is noted that these models give estimated differ
ences of approximately 53 percent between the value of a summer's day and a
winter's day of recreation at the northern parks.
Value Received from Southern Parks
The estimated relationship showing the southern pervisitor's use as a
function, ceteris aribus, of onsite cost is Equation (28) which, in its un
transformed version, is written:
1.228 .499
(35) Ds = e228 E
where: E* = $21.57; E = $7.60.
5 s
The'corresponding value estimate per month thus becomes the definite integral:
1.28 4* 499
(36) Value per month per group = e228 $21.57 dE = $12.93.
$7.60
Dividing the value of Equation (36) by the geometric mean use per month (1.243
12hour periods) gives an estimated value per 12hour period of $10.40 per
group. The geometric mean number of visits per month is .200; thus, the value
per visit of the average group is estimated at $64.65 ($12.93 .200).
The value received per person, based on the geometric mean group size of
3.16, is $20.45 per visit. For the estimated 213,800 campers of fiscal year
1972, the total value produced by the southern parks for campers in that year
is calculated as approximately $4,372,000. Assuming the southern sample parks
are representative of all southern Florida state parks with campgrounds, the
total value of the latter to their estimated 706,400 campers was approximately
$14,446,000 in fiscal year 1972.
For lowseason use (S = 0), the exponent of e in Equation (36) becomes
1.194; for highseason use it is 1.261. The resulting seasonal value estimates
are shown in Table 7. According to the estimates, value per group per month
ranges from $12.49 during the lowdemand, or summer, season to $13.36 during
the highdemand, or winter, season in the southern parks. Value per 12hour
period is estimated at $10.05 for low season and $10.75 for high season. That
seasonal differences in value are large in the north and small in the south
was expected in view of the comparative climatological characteristics of the
two regions. The estimated difference between the value per group per day is
also small, being around 7.5 percent higher for the southern parks.
Table 7. Estimated value of camping to the recreationist group: Southern
Florida state parks
Value Value per Value
Time of year per month 12 hours per visit
(1) (2) (3) (4)
Low season (S = 0)... $12.49 $10.05 $62.49
High season (S = 1).. 13.36 10.75 66.81
Total year (S = S)... 12.93 10.40 64.89
SOURCE: Sample data.
Illustration of a ValueBased Fee Policy
The large difference between estimated lowseason and highseason usage
and value of recreation at the northern parks raises questions about the
appropriateness of equal interseasonal fees for those parks, or, at any rate,
constitutes the most obvious example of where a significant change would be
required to make fees reflect relative interseasonal values. It may be useful
to consider an alternative policy, if only to illustrate the calculation pro
cedures involved in application of the model to such policy questions. Accord
ingly, for purposes of the following illustrative example, it is assumed that
the criteria of a new fee structure for the northern parks are: (1) the dif
ference between lowseason and highseason fees should closely reflect the
estimated difference between lowseason and highseason value received per
visitor, and (2) the new fee structure should, ceteris paribus, produce the
same total revenue per year as the present structure.
According to the seasonal vatluc estimates, the first criterion would be
met by setting the high season fee, U11, approximately 50 percent higher than
the lowseason fee, U01, i.e.:
(37) U11 = 1.5 U01.
The second criterion is that the change in revenue per year should be zero,
i.e., R0 = R1, where R0 is lowseason total revenue per year and R1 is high
season revenue per year. Expressed in terms of fees and usages, this relation
ship becomes:
(38) 7 U01Ds01 01 7 U00D0s00G0 = 5 U11DsllG11 + G U10DsloG10'
where the first numerical subscripts refer to the season (0 = lowseason; 1 =
highseason), and the second numerical subscripts refer to values before (sub
script = 0) and after subscriptt = 1) the change in fee structure. For example,
D510 is the highseason usage per visitorgroup before the change in the high
season fee. The Gij are the total numbers of visitorgroups per month; for
example, G01 is the total number of groups visiting, per month, during the
low season after the change in the lowseason fee. The U.. are the fees in
Sth
effect during the ih season under the jt fee structure. The weights, 5 and
7, are the number of months in high season and low season, respectively.
The usages per visitor, Dsij, are expressed as functions of the Uij as
follows, given the elasticity of use with respect to onsite cost which, from
Equation (25), is .323:
(E
(39) D01 Ds00 = .323 Es Eso 0) D
([Asl + Ul1 [AO+ U00])
(E EsO)
011 )10
(40) Ds1 s0 = .323 Es10 = D510
([As11 + Ull] [As10 + U10])
= .323 A0 + UO 0slo"
s10 10
Given existing knowledge and/or fair assumptions about the values of the
variables, Equations (37), (38), (39), and (40) constitute a system which can
be solved for the fee structure that meets the hypothesized policy criteria.
The "known" data for this purpose are as follows:
(1) U00 = U10 = $2 per 12hour period (the present fee is the same
for both seasons).
(2) G G1 = 19,236 visitorgroups per month; G =G 10,960
10 11 (the number of visitors per month R assued to be
independent of the fee change).
(3) As01 = As00 = As1 = A10 = As. (It is assumed that ancillary on
site cost per group is the same in both seasons, and
invariant with respect to the fee change.)
(4) DsO0 = 1.72 12hour periods per month; D = 2.96 12hour
periods per month (from the sampll~lata).
(5) E 00 = Es10: $9.66 (which follows from (1) and (3) above). Sub
stitution of these data into Equations (38), (39),
(40), and (37), respectively, yields the system:
U01(76,270 Dso0 + 144,270 Ds11) = 833.302.4.
(41)
(42) Ds01 + .058 U01 = 1.835.
(43) Ds11 + .099 U11 = 3.158.
(44) U11 1.5 U01 = 0,
which reduces to the quadratic equation:
(45) 25,010. U012 ' 595,560 OU .+ 833,;,.'.4 = 0.
The solution is as follows:
U01 = $1.50 per 12hour period
U11 = $2.25 per 12hour period
Dso0 = 1.75 12hour periods per month
Dsl1 = 2.93 12hour periods per month.
In summary, a reduction of the northern park lowseason fee to $1.50 and
an increase in the highseason fee to $2.25 per 12hour period would, other
things remaining equal, leave total yearly revenue unchanged, and approximately
equalize the estimated value received per fee dollar spent in each season. The
estimated effects on campground use (barring significant effects on numbers of
patrons) would be a decline of approximately 1 percent in high season and an
increase of approximately 2 percent in low season. In terms of absolute mag
nitudes, this would represent a decline of approximately 2,900 highseason
visitorgroupdays, and an increase of approximately 2,300 lowseason visitor
groupdays at all northern park campgrounds, for a net estimated decline in
total yearly usage of around 600 visitorgroupdays.
SUMMARY AND CONCLUSIONS
Summary
The objectives of this study were (1) to develop a general model of the
individual recreationist's demand for particular facilities; (2) to test the
appropriateness of the model by estimating parameters of the model with data
pertaining to campers; and (3) to illustrate applicability of the empirical
model to informational needs of recreation facility planning and management.
The approach taken has been to use some component of recreationists' actual
costs as a proxy for the price of a facility. On consideration of comparative
characteristics of travel and onsite cost, the daily onsite cost was selected
as the facility price proxy, while letting travel cost play a different role
as a separate explanatory variable.
The basic model consists of two structural equations explaining length
of stay and frequency of visits, along with the identity from which the use
pertimeperiod equation is derived. Applicability of the model to both single
distination and multipledestination recreationists was achieved by introduc
tion of an adjustment factor applied to travel cost. Other specific problems
dealt with include the possible role of leisuretime constraints and alterna
tive specifications of the recreationist decision unit.
The general study area consists of Florida state parks with campgrounds.
The state was divided into two areas, viz., parks north and south of Orlando,
Florida. A sample of parks was then selected to represent each area, thereby
providing two essentially different groups of facilities for separate analysis.
A sample of past campers was chosen for each group of parks, and person
ally interviewed by telephone. From the telephone survey, camper registration
cards, maps, and calendars, data were obtained on all of the variables in the
statistical model, which were then estimated by multiple linear regression.
Conclusions
The principal empirical results are estimates of the structural equa
tions explaining length of stay and frequency of visits. It makes little
difference to the estimates whether onsite and travel costs are specified
as perperson costs or pergroup costs. In general, the algebraic signs of
the statistically significant coefficient estimates conform to expectations
based on theory and previous empirical research.
The travelcost adjuster performs statistically well in the lengthof
stay equations by comparison with alternative specifications using unadjusted
travel cost. Performance in the frequencyofvisits equation is equivocal,
however, and this relationship is apparently subject to considerably greater
unspecified influences than is the lengthofstay relationship.
The question of whether a leisuretime constraint need be explicitly
considered was approached by asking the recreationists directly (the only
departure from the indirect method). Less than 10 percent of the 357 campers
queried answered that limited time primarily restricts their recreation in
the Florida state parks. It is thus tentatively hypothesized that leisure
time constraints are not a serious issue for this study area.
Examples of using the estimated models to provide information of interest
to recreation planners and policy makers included the prediction of the effect
of a general change in onsite cost on use of the two groups of parks. The
reader was advised, however, that the elasticity of use, with respect to a
general change in onsite cost, is probably less than with respect to a use
fee change unless certain conditions hold; namely, facility uniqueness and/or
vigorous price competition from private substitute facilities.
Value per visit was calculated from the derived Ds relationships, with
average magnitudes fixed for onsite cost and the seasonal shift variable.
For the northern parks, estimated value per visit during the winter season
was calculated at around $35.01 per group; during the summer season, estimated
value was around $55.12 per visit per group. For both seasons averaged, the
northern parks produced an estimated value of around $45.86 per visit per
group. For the southern parks, where seasonal differences were negligible,
the estimated value per visit was $64.89 per group.
The daily equivalents of the value estimates are of interest, since camp
site fees are a group daily levy. For the year overall, the estimated value
per 12hour recreation day in northern parks was $8.95 per group. The cor
responding estimate for a winter's day in the northern parks was $6.25; for
a summer's day, the estimated value was $10.77. Thus, the estimated value
of a summer's day is around 53 percent higher than the value of a winter's
day in the northern sample parks.
Combining the estimates of value with the estimated yearly number of visi
tors to the parks yielded an estimated value of all northern parks per year of
$14 million, while southern parks are estimated to be worth around $14.4 mil
lion per year to campers.
Limitations
A major limitation of this study is one that afflicts all previous re
search in the demand for public recreational facilities; namely, the assump
tions that must underlie the use of some ancillary cost of recreation as a
proxy for facility price. In the early .stages of 'ocy):eptualizing this project,
the authors had hoped to exploit the rare occurrence of an actual and substan
tial increase in a calmpinq fee, which was envisioned as an additional shift
variable to be estimated frvin ddta on recreationists who camTp.: both before
and after the fee increase. More evidence might thi~r ha.s be,"i :.;..led about
the relative magnitudes oF the elasticities with respNc.t to the ancillary on
site cost and the use fee. Ihis was not possible, hnw~ver. It is worth hIping
that in the future other scholars will be on the alert for such rare oppor
tunities, and exploit them while they exist.
Although it is generally accepted that. the individual c.nsumner is the
basic unit of study for consumption and demand analysis, it is clear that be
havior of the average recreationfst explains only a part of the observable
variation in aggregate demand. The absence of an equation explaining the
numbers and behavior of potential patrons thus constitutes a limit.:tiun in
sofar as predictions of aggregate demand may be sought.
No adjustment was made li the travel and onsite costs to clnopensate For
the normal subsistence components of those costs. Thus, in theory thp on
site cost is an upwardbiased facility price proxy, even if the condition of
equal site specificity is otherwise met. The problem of determoiihg the sub
sistence costs of campers is not, however, one on which any sin.ific.'ant break
through seems imminent. Another approach, which has yet to be tried, might
be the decomposition of travel and onsite costs into further separate ex
planatory variables. For example, transportation and lodging costs each may
exert different influences. Some components of ancillary onsite cost may be
better facility price proxies than others. The cost of food, which can he
eaten anywhere, is probably less appropriate than the cost of running one's
boat on a Platicular lake, for example.
Better estlmtes of the visits relationship would prbebly result f~v.r
taking into account alternative opportunities within .h.M vicinity o' the visi
tor's residence. To an extent, the availability of alternatives aTrog the way
are acrountpd for by t!i travelcost a,.juster which, in its ,., L.;. t form, how
ever, could unquestionably stand some theoretical refinement.
The ',amIple is biased in favor of recreationists who had telephones and
li;:ed number, and who .',re' at home and willing to answer the o...Mir:nnire
when alled. Also, the .i.:vey took three months to i ntrview a i'.smpri' f past
campers sipa i'ning a year's time. It may have been preferi.ble to have ;:iedulled
the survey so as to minimize the memory time of the re:pundencts Probobly the
best wa.y to minimi.e saimpi'l his.es of these types Is by means o' :n. %i 1e per.
sonal interviews. The telephone ,is, by comparison, a considerably cheaperr
dta ta yj.hring device.
While not a real limit3 o, 'fihe derivation of prpdictlcn' ir.om estimated
*:x.~~.or,,,.ial s u.!.ii'.tions is all too commonly done in a mri ne! tl ir, i'ves both
b ed Pred.i :.io; and g ns'ly maiMfi d prediction v,r'. S p'uh 4oul be the
results of drieivi ng predictions by direct substitution of hypothetical vliies
of the indrepr:ndent variables. The elasticity estimates alone c*nnfain aIpe
inrfo i)rm !..ri .i';n iorc.:t predictive lpurpo e ,e. and should be used wlenever P.os lble
in the n' hi;, ;i ...,'; '. *o' c le, .. ti. i i ra, ion o. a 'i .s :ba ed fee
policy.
VtF TI rpu !
[1] American Automobiln As socr iation. Ala 0000
ton, DC: 19N.
[2] Browni, William G., Agr Si"l4, .wij 'r!rv N. Castle. An e roiojrmicic
Evauaio o Lv ul ;OluoIn AIhin00
Sept;. 1964l.
[3] Burt, Oscar and Durwood Bir:'i, "Es timation of rH'9Va'ial "Aefits
from Outdoor [:.tra tion," %Wk.i rica 10:5 ( 1971). v. 8ll; 8V8.
[43 Clawson, Marion. 0093y, j t V. at
[53 Clawson, lidriLu ia'hi Jack L. Knetsch. truww.u3 of vhomv rtctin.'
Bal1timnore : 1he 3411 ln~ Honkino P.,
[6) Edwards, J. A., K. C. Gibbs, L. J 6WOT.l aO N N. 40t...10
1972ih~, ..,r:) 
Demandoi 'fy cl'utiii n v~,Th ik~ .*~I,?
po'vljc i~lt n f IhI~tc" Qualf i y n 1 .IItei I~: U~lt; ti. '1: An~ [YPi i cation
of~~ a I~ .~.d iqy Ou Mo We Lieq t in OW or Or 01'. ae Heai on
6'i, \701 I I , I it~IJ;i~;;j~r ~31nI ~115
[10] Gibbs, Kenneth C, nd 3, ftachard of Gieru:i, Pll'rslioq ,fro no I
Creationail iVoii urs: Vissimme i ai, Flor i.'
1A~i [ul, : 1 (duly I197S;), 0 ~~230'.
[11)3 Hendeiro., dom~ies Ni, end Vi.;harr E. Qwao~dk. HI:r~roic1I~~ A
[12) Hi cks, J. h A Revjision or Deumaind 1 hior". l'x ard: 1 h ClaruniPyn Press,
~121
[13] Hotel 1 I H rIoIc ld. L4..ttc'i cid ld '"l!e FH::no.i cs N ahli c Recradtiion:
An Ernumin iI S1ud y of t h~ ke Herietdy Ev:a 'xi on & f lecr~eati r. in the
[N1rkvnr Park,".. Wh'~1Wlultu, w;: 'U.S. Novio wri nk Sorvic~e,
1 vt
[14] Howard Johnson's Motor Lodge & Restaurant Directory i!.iited Stjlat
International. Braintree, MA: Howard Johinsons Motor Lodgs,
T97i3.
[15] Jennings, Thomas A. "A General Methodology for Analyzing Demand for
Outdoor Recreation, with an Application to Camping in Florida
State Parks." Ph.D. dissertation, Univ. of Fla., Mar. 1975.
[16] Jennings, Thomas A., and Kenneth C. Gibbs. "Some Issues Concerning
Specification and Interpretation of Outdoor Recreation Demand
Models," So. J. Agr. Econ. 6:1 (July 1974), 165169.
[17] Knetsch, Jack L. and Robert K. Davis. "Comparisons of Methods for
Recreation Evaluation," in Allen V. Kneese and Stephen C. Smith
(Eds.). Water Research. Baltimore: Johns Hopkins University
Press, 1966, pp. 125142.
[18] McGuire, John F. III. "An Application of Two Methods to Estimrtit
the Economic Value of Outdoor Recreation in the Kissimmee River
Basin." Unpublished Master of Science thesis, Univ. of Florida,
August 1972.
[19] Marshall, AlFred. Principles of Economics (8th ed.). London:
MacMillan & Co., Ltd., 1959.
[20] Oliveira, Ronald A. and Gordon C. Rausser. "Daily Fluctuations
in Campground Use: An Econometric Analysis," Am. J. Agr. Econ.
59:2 (May 1977), pp. 283293.
[21) Pearse, Peter H. "A New Approach to the Evaluation of NonPriced
Recreational Resources," Land Economics 44:1 (Feb. 1968), pp.
8799.
[22] Rand McNally. Road Atlas: United States, Canada, Mexico. Chicago;
Rand McNIally & Co.,1973.
[23] Rausser, Gordon C. and Ronald Oliveira. "An Econometric Analysis of
Wilderness Area Use," J. Amer. Statis. Assoc. 71 (1976), pp.
276285.
[24] Wilson, Robert R. "Demand Theory: Time Allocation and Outdoor Recrea
tion," So.. J.Agr. Econ. 3 (Dec. 1971), pp. 103108.
STATISTICAL AP'F'JPcDIX
Mean Values of the Variables for the
Samples of Florida State Park Camper, 1973
Northern park
Arithmetic mean
:s Southern parks
Dependent variables:
Length of stay per visit (D )
in 12hour periods ............
In Dv .........................
Visits per month (V)...........
In V ..........................
Use ,er month (Ds V.D )......
Independent variables:
Onsite cost per group (E ) per
12hour period ...............
In E .......................
Travel cost per Group (Et).....
In Et ......................
Adjusted travel cost per
group (rEt) ...................
In (rEt) ......................
Visitor income (I).............
In I ..........., ..............
Group size (N), in persons.....
In (1/N) ......................
Seasonal dummy (S).............
Destination dummy (D)..........
Tentcamper dummy (X) .........
Additional visitor characteristics:
Minimum acceptable stay per visit
(Dvm) in 12hour periods.......
In Dv ........................
Minimum roundtrip travel time
(T1) in hours.................
Total time away from home (T2)
in 12hour periods ............
Minimum 1way distance, in miles
6.562
1.634
.370
1.344
2.219
$9.661
2.140
$282.11
5.044
$153.72
4.470
$14,934.00
9.484
4.045
1.293
.661
.712
.249
8.631
1.827
.232
1.610
2.175
$8.623
2.028
$436.89
5.688
$193.12
4.850
$14,694.00
9.410
3.533
1.152
.517
.667
.339
4.427
1.269
16.844
19.548
415
5.319
1.226
35. 68
34.133
950
SOURCE: Sample data.
Variable
I
I ~  "I~~" "~ 
Correlation Matrix: Northern Parks
In(rEt)
0.3928
1.0000
rEtln Es
0.1830
0.5370
1.0000
In I
0.2243
0.2222
0.0807
1.0000
In(l/N)
0.3155
0.1575
0.0258
0.2996
1.0000
Source: Sample data.
In Es
1.0000
S
0.2717
0.1815
0.0753
0.0977
0.2732
1.0000
D
0.0547
0.1456
0.0753
0.1909
0.2377
0.0715
1.0000
X
0.0555
0.0514
0.0112
0.1154
0.0894
0.0529
0.0484
1.0000
Correlation Matrix: Southern Parks
1n(rEt)
0.3260
1.0000
rE tnE
0.4867
0.7381
1.0000
In I
0.2306
0.0115
0.0915
1.0000
ln(l/N)
0.2738
0.0623
0.1471
0.2848
1.0000
Source: Sample data.
In Es
1.0000
S
0.1101
0.1055
0.0451
0.0693
0.2612
1.0000
D
0.0348
0.1662
0.1342
0.0958
0.1655
0.1887
1.0000
X
0.0108
0.0120
0.0249
0.0949
0.0457
0.0121
0.1328
1.0000
