The Sensitivity of Travel CostEstimates of Recreation Demand

to the Functional Form andDefinition of Origin Zones

Ronald J. Sutherland

The travel-cost method of estimating a recreation demand function requires specify-ing the functional form of the first-stage demand curve and defining the width of theconcentric origin zones. A Monte Carlo approach is used to determine the sensitivity ofdemand and valuation estimates to alternative choices about these two issues. Demandand valuation estimates are shown to be sensitive to the definition of the origin zone andto the use of a semilog versus a double log first-stage demand curve. The proper choiceor origin zones is unclear, but a semilog form is more appropriate than a double log form.

Since Clawson's paper in 1959, the travelcost method (TCM) of obtaining a recreationdemand curve has been used frequently toestimate demand and value of a recreationsite. Despite widespread acceptance and theofficial sanction of the Water ResourcesCouncil, the TCM has been subject tonumerous criticisms, for example, the timebias and the identification problem. The im-plication of these criticisms is that the TCMis not sufficiently rigorous and comprehen-sive to produce reliable demand and varia-tion estimates.

The focus of this study is on two specifica-tion choices required by the TCM which mayinfluence estimates of the demand curve andconsumers' surplus. The issues investigatedhere are (1) the functional form of the first-stage demand curve; and (2) the width of theconcentric origin zones. The objective is todetermine the sensitivity of travel cost de-mand and valuation estimates to various as-

Ronald J. Sutherland is currently with the Los AlamosNational Laboratory. The research for this paper wasconducted while the author was with the EnvironmentalProtection Agency, Corvallis Environmental ResearchLaboratory.

sumptions concerning these points. Themethod of analysis is to apply the TCM toseveral sites under various assumptions andto contrast the results.

Most empirical demand curves in the eco-nomics literature are specified in double logform, perhaps because the coefficients maybe interpreted as elasticities.1 In the recrea-tion literature the semilog specification ismost prevalent, although linear functionshave been used.2 The relative merit of thesemilog and double log specification of thefirst-stage demand curve and the sensitivityof the valuation estimates to the choice ofthese two functional forms are considered inthis study.

In the TCM, visit rates from various ori-gins are regressed against correspondingtravel costs. Since the pioneering work of

In their literature surveys on the demand for money,Laidler and Goldfeld present empirical evidence infavor of a double log specification.

2 Linear demand curves were used by Burt and Brewerand by Cicchetti, Fisher and Smith because this specifi-cation is required by some properties of their models.

87

Western Journal of Agricultural Economics

Clawson and Knetsch, origins have been de-fined by a series of concentric rings aroundthe recreation site. For instance, if recrea-tionists travel a maximum of 200 miles andrings are defined every 20 miles, then thereare 10 origin zones and 10 observations forthe experience demand schedule. Similarly,if a ring is defined every 10 miles, there willbe 20 travel zones and 20 observations forestimating the visit rate schedule. As an al-ternative to a system of concentric rings,each population centroid may be construedas a separate origin, and the number of ob-servations is therefore determined by thenumber of such centroids. The sensitivity ofthe demand and valuation estimates to thedefinition of the origin zone is also examinedin this study.

An Overview of a RegionalTravel Cost Model

The simulation methodology used in thisstudy requires the capability to estimate atravel cost demand curve for a large numberof recreation sites. In two recent papers(1981, 1982), I present a regional recreationdemand and benefits model, which is de-signed to estimate demand and valuation foreach of 179 centroids. The model uses theTCM to analyze camping, fishing, boatingand swimming in the Pacific Northwest. Theessence of the approach is that householdrecreation surveys are used to estimatevisitor days emanating from each origin zone,and a gravity model is used to estimate visitsby origin to each recreation centroid in theregion. The estimates of visit rates by originand the corresponding travel costs are usedto construct travel cost demand and valuationestimates.

Only a brief overview of the model ispresented here. A gravity model is specifiedin the form

(1) T = p Aj FijAj F.'

j iJ

where

Tij = number of activity days produced atorigin i and attracted to destination j

Pi = number of activity days produced ati

Aj = number of activity days attracted tothe jth recreation centroid

Fij = a calibration term reflecting spatialimpedance for interchange ij.

The gravity model, as specified in (1), is adistribution model; that is, it distributes agiven number of trips according to relativeattractiveness and relative effect of spatialimpedance of each origin and destinationcentroid. As employed in the regional recrea-tion model, the gravity model estimates thenumber of visits to each of 179 recreationdestinations from each of 144 origins in thesystem. Total visitor days weighted by corre-sponding populations is the visit rate, whichis the critical input in the TCM.

Travel costs are estimated as the round tripcost per person per mile times the minimumdriving distance.3 Population centroids de-fine trip origin points and at least one popula-tion centroid is defined for each county. Rec-reation centroids are the center of recreationactivity in the county, and, where appropri-ate, multiple recreation centroids are definedfor each county.4 Each origin and destinationcentroid was identified and a highway net-work was constructed showing various possi-ble routes from the origins to the destina-tions. The 'minimum distance routes wereestimated via a computer program, and thesedistances form an impedance matrix. This

3Travel costs are defined here as pecuniary costs and noallowance is made for the time value of travel. Usingdata from a recent household recreation survey in theNorthwest, I show -in a paper in progress - thatrecreationists do not consider time spent for recreationtravel as a cost.

4 There are 119 counties in the three states of Washing-ton, Oregon and Idaho, but 179 recreation centroidsand 132 internal population centroids and 12 externalzones. A list of the population and recreation centroidsis given in Sutherland (1981).

88

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Recreation Demand Sensitivity

matrix is used to estimate travel costs and tocalibrate the gravity model.

Although the regional model considersfour activities, this study will consider onlyboating. Estimates of the number of boatingtrips by origin were obtained from threehousehold recreation surveys taken in 1976by the state parks authorities in Washington,Oregon and Idaho. These surveys also in-clude information on travel distances by ac-tivity. Distance data, along with the corre-sponding number of trips, were used to esti-mate a relative trip length frequency dis-tribution for boating. The Fij values in (1) areobtained by substituting the impedance(travel distance) values into the trip lengthfrequency distribution. The Fij values can beinterpreted as the probability that a boaterwill travel the distance from origin i to desti-nation j.

The boating attractiveness of each recrea-tion centroid is hypothesized to be a multipli-cative function of the number of boat rampsand the accessibility of the centroid. Boatramps are a proxy for a combination of factors- for example, water quality, acres of water- which determine the attractiveness of aboating site. Accessibility (RAj), or total po-tential demand, is estimated as the sum ofboating visitor days from each populationcentroid in the region weighted by the prob-ability that a boater will drive to the particu-lar centroid. That is, the accessibility of thejth centroid is

(2)132

RAj = E FijPi,i

where 132 internal origins have been definedfor Washington, Oregon and Idaho. Recrea-tion accessibility varies positively with thenearness of population centers and with thenumber of boating days produced by theseorigins.

U.S. Forest Service data on visitor daysand boat ramps (BRj) by ranger district wereused with accessibility estimates of (2) to

estimate an attractions model. 5 The regres-sion results are

0.60 1.74,(3) Aj = 0.001BRj RAj

(4.33) (4.79)

R2 = 0.52, n = 37

where the t values (in parentheses) indicatethat each coefficient is highly significant.Boat ramp and accessibility data for eachrecreation centroid in the region were sub-stituted into (3) to estimate the relative at-tractions.

Travel cost demand schedules are es-timated by first regressing visit rates by ori-gin (Vi) against travel cost (Ci) per person pervisitor day (5.52 cents per mile).6 In generalform, the experience or first-stage demandequation is

(4)

where the hat means predicted and i refers tothe origin zone. Equation (4) is estimated foreach recreation centroid in the region withordinary least squares and is used to analyzethe issues of functional form and definition oforigin zone. Multiplying (4) by the popula-tion of origin i and summing gives

(5) T Ti = NiVi,i i

which estimates total visits as a function ofthe estimated visit rate of each origin zone

5These data are unpublished and are part of the Recrea-tion Information Management (RIM) system of the U.S.Forest Service. The data were obtained from the re-gional office in Portland and reflect ranger districts inWashington and Oregon.

6 According to the 1979 edition of Principles and Stan-dards, of the Water Resources Council, the cost pervehicle mile for a standard vehicle was 8.4 cents in1976. Using the U.S. Department of Commerce gasand oil price deflator, this figure was adjusted to 9.66cents per mile in 1979. Next, 9.66 cents was doubled toadjust for round trip costs and divided by the averagenumber of persons per vehicle (3.5) to obtain 5.52cents.

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Vi = f(Ci),

Western Journal of Agricultural Economics

and its corresponding population. Totalquantity demanded can also be estimatedexogenously by site attendance data, whichin this study is the sum of the appropriatecolumn vector in the Tij matrix produced bythe gravity model. Although the regressionestimate of (4) estimates visit rates with anaverage error of zero, total visits are notnecessarily estimated correctly, that is, NiVi¥ NiVi.

The effect of various hypothetical prices ontotal quantity demanded is estimated via

(6) Ti =/ Nif(Ci + AP).

The quantities obtained from (6) and thecorresponding price increments are used toestimate a recreation site demand curve. In-stead of using regression analysis to estimatea sign demand curve, a fourth degree polyno-mial is fit to every five consecutive price-quantity observations and Bode's Rule isused to measure the area under the polyno-mial. 7 The integral under this polynomialestimates consumers' surplus.

The site visit rate data obtained here viathe gravity model would not correspondidentically to data obtained from attendancesurveys. However, the gravity model is cali-brated so that the trip length frequency dis-tribution formed from the trip interchangematrix (Tij) corresponds closely to the dis-tribution estimated from household surveydata. The travel cost estimates presentedhere should, on the average, be representa-tive of those obtained using site attendancedata or household survey data. For purposesof a simulation analysis, the visit rate andtravel cost data may be considered exact.

Sensitivity of Travel CostEstimates to Various Assumptions

Since the 179 recreation centroids and 4activities included in the regional model are

7Bode's Rule is given in Davis and Rabinowitz (p.30), inAbramowitz and Stegun (p. 886), and discussed inSutherland (1981).

90

more than sufficient for this analysis, I arbit-rarily consider the demand for boating at 20Washington recreation centroids, numbered17.0 to 26.0 (column 1 in the accompanyingtables). These centroids include those inKing County, which contains Seattle and isheavily populated, as well as sparselypopulated counties east of the CascadeMountains. By including both urban and rur-al counties in the sample, the travel costestimates reflect a diversity of realistic condi-tions. The rationale for sampling a relativelylarge number of centroids (20) is that certainadverse consequences may be observed onlyoccasionally, and a large sample increases thelikelihood of such a result. Also, results basedon a single site may reflect a special case,inconsistent with results obtained over awide range of experience.

The sensitivity of travel cost estimates toeach of the two issues being considered de-pends upon the assumption made on theother issue. The interdependence of theseissues precludes analyzing them individually.The functional form of the first stage demandcurve is considered first by focusing on thesemilog form and the double log form. Travelcost estimates will then be presented usingvarious size origin zones. The results areshown to be sensitive to the definition oforigin zone, and this sensitivity in turn de-pends on the functional form.

Functional Form of theFirst-Stage Demand Curve

The proper form of a recreation demandcurve has been studied by Ziemer et al. andby Smith. The studies are similar in that onlyone site was considered and a statistical anal-ysis, namely a Box-Cox transformation, wasused to statistically estimate the most appro-priate functional form. Smith rejected thelinear form because it provided a poorer fit ofthe data than the double log and semilogform. However, Smith also concluded thateven though the latter two forms fit the dataand provided reasonable results, each formmust be considered inappropriate. Ziemer et

July 1982

Recreation Demand Sensitivity

al. used the Box-Cox transformation proce-dure and concluded that a semilog form isappropriate and a linear form is inappropri-ate, and further that consumers' surplus esti-mates are highly sensitive to the choice offunctional form.

In considering the various functionalforms, double log and semilog (logarithm ofthe dependent variable) are candidates, butthe linear form need not be considered.Ziemer et al. and Smith provide evidenceagainst the linear form, and scatter plots ofseveral sites indicate a distinct curvilinearrelationship. The evidence against the appro-priateness of the linear form is persuasive,and in this study we consider the double logand semilog functional form.

Analyzing these two forms determines ifthe results are sensitive to the choice offunctional form, and if so, which of the twoforms seems most appropriate. Four criteriaare suggested which may be useful in iden-tifying the most appropriate form. The coeffi-cients of determination are a relevant but notdecisive indicator, particularly if estimatedover several sites. Secondly, estimates ofconsumers' surplus per trip should be some-what stable across sites and should be similarto those reported elsewhere in the literature.Thirdly, the first-stage demand curve shouldestimate closely known quantity demandedat a zero price when P = 0 is used inequation (6). Finally, goodness of fit andconsumers' surplus estimates should be in-sensitive to other computational decisions,particularly if the decisions are made arbit-rarily. These properties are not asserted tobe rigorous statistical criteria that will neces-sarily determine the unambiguous superiori-ty of one functional form. Since previousstudies have not been able to resolve thisissue on statistical or theoretical grounds, it isappropriate to employ a Monte Carlo analy-sis, where a demand curve for several sites isestimated with each functional form and theresults are compared.

First-stage demand curves for boating,equation (4), are estimated for 20 centroidsusing both double log and semilog forms,

where the logarithm is taken of the depen-dent variable. These estimates are based onpopulation centroids as origin zones andquantity demanded estimated exogeneously.The results are presented in Table 1. Thecoefficients of determination, columns (4)and (7), indicate that each form fits the datareasonably well, but the semilog model hasmore explanatory power in 19 of the 20 cases.The semilog surplus per day estimates aremore stable than the corresponding doublelog estimates. Dwyer, Kelley and Bowes re-view several empirical studies of recreationbehavior, but only a few of these studies dealspecifically with boating. If we presume thatother water-based activities have a valuecomparable to boating or that boating is typi-cal of outdoor recreation in general, we mayconjecture on the basis of Dwyer et al. thatvalue per day estimates below $1 or above$10 are outside the range of most existingstudies. The double log estimate of surplusper day of $68.95 for centroid 24.3 is clearlyuntenable, and the double log surplus perday estimates of $10.85 and $11.05 appearsuspiciously high.

A few of the surplus per day estimates,such as those for centroids 18.0 are highlysensitive to this choice. This result exposesthe inadequacy of analyzing the issue of func-tional form by considering only one site. Theresults in Table 1 do not establish that eitherform is correct, but the consistently lowerexplanatory power of the double log form andthe wide variation in surplus per day esti-mates cast some doubt about the appropri-ateness of this form.

Size of Origin Zone

When the travel cost method was pre-sented by Clawson and by Clawson andKnetsch, origins were aggregated into zonesdefined by a series of concentric circles. Tomy knowledge there has been no seriousanalysis of the appropriate size of these originzones nor of the sensitivity of the results to

91

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92

Recreation Demand Sensitivity

various size zones.8 The above results useeach population centroid in the region as apotential origin zone. Evidence on the sen-sitivity of travel cost demand and valuation tothe definition of the origin zone is obtainedby comparing the above results to those ob-tained using 10-mile and 20-mile originzones. Consider two systems of concentriccircles, one at 10-mile and one at 20-mileintervals from the recreation centroid. Originzones are now defined as the area betweeneach ring; visit rates, as total trips from eachzone per 1,000 population of the zone. Thetravel cost from each zone is the weightedaverage travel cost of all centroids within thezone where the weights are the number oftrips per centroid.

Travel cost valuation estimates using asemilog form and 10- and 20-mile originzones are presented in Table 2. Comparingthe results using a 10-mile zone with those ofa 20-mile origin shows similar estimates forseveral sites but quite dissimilar estimates forothers. The most important result in Table 2is the instability of consumers' surplus pertrip estimates using both 10- and 20-mileorigin zones. Columns (4) and (7) indicatethat surplus per trip estimates range fromjust over $1 to above $20. An estimationprocedure which occasionally gives unstableresults cannot be relied upon when analyzinga single site.

The estimates in Table 2 are comparable tothe semilog results in Table 1, with the dif-ference being that population centroids areused as origins in the analysis reported inTable 1. A comparison of the results on thesetwo tables indicates that aggregating popula-tion centroids into concentric zones increasesconsumers' surplus by an average of over $1per trip. Furthermore, consumer's surplusestimates on Table 1 appear uncorrelated

SBrown and Newas have argued that observations shouldbe based on individuals, rather than aggregations ofpeople. Since they use site attendance data, visit ratesreflect the frequency of participation of recreators andof the participation rate of the entire population, whichmay also decline with distance from the site.

with those on Table 2. Estimates of totalsurplus for centroids 17.1 and 17.2 are over$1 million lower when population centroidsare aggregated into zones. However, theaggregation process increases the surplus es-timates per trip for centroids 22.0 and 22.2by over 300 percent. The surplus per tripestimates for these two recreation centroidsexceed $20, and the coefficients of determi-nation cast doubt on the reliability of theseestimates. The results for these two centroidsmay be regarded as outliers and thereforedismissed, but it is significant that aggregat-ing population centroids into zones producesoutliers while use of population centroids asorigins did not.

The conclusion that travel cost valuationestimates are sensitive to the definition of theorigin zone is significant, and raises the ques-tion of which definition is most appropriate.The average of the coefficients of determi-nation favor the use of population centroidsas origin zones, but the differences in R2

values between models do not provide suffi-cient evidence to resolve this issue. The twoextreme estimates (centroid 22.0 and 22.2)obtained from the 10- and 20-mile originzone equations raise a question about ag-gregating, but are not compelling evidenceagainst it. A third potential indicator of theproper model is the ability of the statisticalestimate of equation (6) to estimate knownquantity demanded at a zero price.

Table 3 depicts the assumed known quan-tities demanded, column (2), and endogen-ous estimates of this variable using 10- and20-mile origin zones and recreation centroidsas origin zones. This table also depicts quan-tity estimates using semilog and double logfunctional forms. Regardless of the choice ofthe origin zone, the semilog form predictstotal quantity demanded more accuratelythan the double log form. This result is fur-ther evidence in favor of the semilog formover the double log form. An additional re-sult is that aggregating population centroidsinto either 10- or 20-mile zones substantiallyimproves the ability of the model to predicttotal use at a zero price. Although aggregat-

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Western Journal of Agricultural Economics

TABLE 2. Semilog Valuation Estimates Using 10-Mile and 20-Mile Origin Zones.

10 Mile Origin Zones 20 Mile Origin Zones

Recreation Consumers' Consumers'Centroid Surplus Surplus Surplus SurplusNumber R2 (in $000) Per Trip R2 (in $000) Per Trip

(1) (2) (3) (4) (5) (6) (7)

17.0 0.675 $1,394 $4.01 0.787 $1,415 $4.0717.1 0.764 3,463 3.62 0.749 3,606 3.7717.2 0.768 3,775 3.45 0.868 2,830 2.5918.0 0.935 111 1.77 0.977 91 1.4419.0 0.631 82 2.47 0.668 81 2.4519.1 0.547 444 5.10 0.470 587 6.7320.0 0.619 394 3.75 0.751 337 3.2121.0 0.824 767 4.44 0.827 882 5.1122.0 0.258 117 20.52 0.216 217 38.1222.1 0.620 60 5.70 0.716 44 4.2122.2 0.170 361 24.87 0.474 313 21.5823.0 0.882 118 2.82 0.948 114 2.7323.1 0.895 120 2.62 0.928 109 2.3923.2 0.815 890 4.11 0.913 868 4.0124.0 0.916 9 1.36 0.919 7 0.9724.1 0.903 53 4.31 0.903 44 3.5624.2 0.835 29 2.02 0.877 23 1.6124.2 0.830 139 5.97 0.806 106 4.5625.0 0.840 98 2.35 0.915 75 1.7926.0 0.699 102 4.80 0.698 80 3.76

Mean 0.720 $626 $5.50 0.771 $591 $5.93

*These estimates are based on a $1 price increment in equation (6), and quantity demand estimated exogene-ously.

ing populations into zones improves the pre-dictive ability of the model in this sense, thequantity estimates for several centroids stillcontain substantial errors.

The result -that aggregating populationcentroids into concentric zones does not im-prove the R2 values as we may expect, butdoes improve the estimates of total quantitydemanded - is easily explained. Visit ratesdiminish with distance from the site, but thenumber of population centroids increaseswith distance from the site. When populationcentroids are used as origins, there are manyobservations of low visit rates which are closeto the regression line. The very few originzones which have high visit rates and accountfor most of the total visits have relativelylittle influence on the regression line. The

94

visit rates of the close origin zones are oftenestimated with large residuals. Aggregationresults in a large number of good fittingobservations being combined into a few ob-servations and hence reduces their influenceon R2.

Aggregation decreases the total number ofobservations and thereby increases the rela-tive weight of the close origins in determin-ing the regression line. The error in estimat-ing these visit rates thereby decreases; withit, the error in estimating total visits. The"solution" to the visit estimation problem isnot obtained by aggregating because ag-gregating from a 10-mile origin zone to a 20-mile origin zone actually decreases the relia-bility of predicting total visits, see Table 3,columns (4) and (5). Indeed, total visits could

July 1982

Recreation Demand Sensitivity

TABLE 3. Estimates of Quantity Demanded by Centroid Using Semilog and Double LogForms and Various Definitions of Origin Zones (in Thousands of Visitor Days).

Semilog Results Double Log Results

Ten Twenty Ten TwentyRecreation Exogeneous Mile Mile Mile MileCentroid Quantity Recreation Origin Origin Recreation Origin OriginNumber Demanded Centroids Zone Zone Centroids Zone Zone

(1) (2) (3) (4) (5) (6) (7) (8)17.0 347 534 352 313 3,751 492 54017.1 956 1,270 912 829 17,657 1,357 1,38117.2 1,092 1,354 1,009 833 111,807 1,656 1,33318.0 63 70 47 34 572 56 4419.0 33 36 32 27 122 45 5919.1 87 134 97 87 475 134 15520.0 105 135 114 98 882 261 24021.0 113 244 148 145 800 243 33722.0 6 4 18 24 15 34 5122.1 10 8 18 14 130 39 4822.2 15 9 43 40 102 101 11323.0 42 57 37 31 389 48 5123.1 45 59 39 33 1,279 55 5023.2 217 303 206 185 10,549 345 32524.0 7 5 7 4 52 6 424.1 12 9 26 21 1,223 20 1824.2 14 10 16 13 84 15 1224.3 23 19 61 43 7,081 68 4325.0 42 49 30 24 91 37 3426.0 21 17 32 26 1,015 106 116

Mean 166 216 171 141 8,234 256 248

NOTE: The quantity estimates in columns (3) through (8) are obtained by letting AP =squares estimate of equation (6).

be predicted exactly if populations were ofconstant size across origins. 9

The composite influence of a double logspecification and 10- and 20-mile origin zoneson valuation estimates is depicted in Table 4.The coefficients of determination, columns(2) and (5), are lower for a double log modelthan for a semilog model (Table 2) when

9 The estimated residuals in predicting visit rates neces-sarily sum to zero, i.e.,

S(Vi-Vi) = S(Ti/N -Ti/Ni) = 0.

If the population of each origin is identical, NY(T, -Ti)= 0, visits (Ti) are also predicted exactly. Bowes andLoomis (1980) show that specifying populations insquare root form results in a first-stage demand curvewhich necessarily predicts total visits identically.

0 in the appropriate least

population centroids are aggregated into 10-or 20-mile zones. Furthermore, most of thesurplus per day estimates are higher than onecould reasonably expect, and they are veryunstable across sites. Overall, the use of adouble log form together with 10- or 20-mileconcentric origin zones results in very unten-able valuation estimates. This result also fol-lows when we consider the double log esti-mates of total use at a zero price. As seen inTable 3, aggregating population centroids in-to 10- or 20-mile origin zones improved thepredictability of the model in terms of totaluse. However the double log model predictstotal use with a larger error than a semilogmodel, regardless of the choice of originzone. Overall, the double log estimates aremuch more sensitive to the definition of theorigin zone than are semilog estimates.

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Western Journal of Agricultural Economics

TABLE 4. Double Log Valuation Estimates Using 10-Mile and 20-Mile Origin Zones.

10 Mile Origin Zones 20 Mile Origin Zones

Recreation Consumers' Consumers'Centroid Surplus Surplus Surplus SurplusNumber R2 (in $000) Per Trip R2 (in $000) Per Trip

(1) (2) (3) (4) (5) (6) (7)

17.0 0.556 $8,810 $25.36 0.596 $18,546 $53.3917.1 0.599 34,694 36.29 0.588 38,203 39.9617.2 0.608 36,984 33.84 0.673 19,616 17.9518.0 0.803 405 6.44 0.804 770 12.2519.0 0.493 500 15.10 0.454 1,631 49.3819.1 0.536 2,312 26.54 0.434 5,044 57.9020.0 0.535 8,386 79.72 0.555 7,732 73.5121.0 0.723 7,171 41.52 0.668 15,291 88.5422.0 0.366 1,050 184.24 0.283 2,114 371.1022.1 0.491 906 87.75 0.469 1,658 158.7222.2 0.175 4,326 298.29 0.399 5,345 368.5423.0 0.775 902 21.52 0.878 1,445 34.4623.1 0.844 926 20.31 0.903 1,150 25.2523.2 0.736 9,984 46.10 0.855 9,324 43.0624.0 0.797 15 2.04 0.776 15 2.1524.1 0.813 180 14.69 0.792 227 18.4924.2 0.782 62 4.26 0.822 60 4.1524.3 0.711 238 10.23 0.670 230 9.9225.0 0.764 505 12.05 0.772 561 13.3926.0 0.461 3,242 153.28 0.459' 4,083 193.05

Mean 0.627 $6,080 $55.93 0.643 $6,653 $81.76

Conclusions and Implications

This study presents travel cost demandand valuation estimates for boating in 20recreation centroids in Washington. The ob-jective of the analysis is to determine thesensitivity :f the results to the functionalform of the first-stage demand curve and tothe definition of the origin zone.

The preference of most recreation analystsfor a semilog specification of the first-stagedemand function over a double log is con-firmed by the results of this study. In termsof goodness of fit, stability of results acrosssites, accuracy of predicting quantity de-manded at a zero price, and a priorireasonableness of results, this specification isclearly superior to the double log.

Some recreation analysts, such as Com-mon, have used a double log specificationwith satisfactory results. However, Commonand others have tried alternative specifica-

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tions for only one site. For some centroids,consumers' surplus estimates are insensitiveto the specification, but this result is a specialcase which may be observed in a sample ofsize one. A particularly serious problem withthe double log specification is that on occa-sion it produces unrealistic results. Thesource of this problem is unclear and cannotbe determined from the regression estimatesof the experience demand schedule. Thecause of these occasional untenable resultsmay, to a lesser extent, affect the apparentlytenable results; hence, these estimatesshould also be considered suspect.

This analysis of boating at 20 recreationcentroids reflects a small sample of the 179centroids and four recreation activities con-sidered in my regional model. Recreationexperience demand curves were estimatedfor each centroid and for each activity usingboth a double log and semilog specification.

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Recreation Demand Sensitivity

The results are similar to those reportedhere. Some estimates of consumers' surplusare sensitive to the choice of functional form;some, insensitive. About five percent of theresults using a double log specification wereunreasonable.

The most disconcerting result of this studyis that valuation estimates are sensitive to thedefinition of the origin zone. When eachpopulation centroid is construed as a separatezone, the explanatory power of the model ishigher on the average than when centroidsare aggregated into 10- or 20-mile zones.Furthermore, aggregating centroids resultsin a substantial loss of degrees of freedom,which ceteris paribus is undesirable and inthis case causes the results to become unsta-ble. However, aggregating population cen-troids into origin zones improves the accura-cy by which total use is predicted at a zeroprice.

Most travel cost studies have been basedon an aggregation of population centroidsinto concentric zones. The choice of a 10-mile versus a 20-mile system of concentriccircles affects the results, but there is a great-er disparity between using zones and usingpopulation centroids as origins. A conse-quence of using each centroid as an origin isthat a large proportion of the centroids ac-count for a small proportion of the trips. Inrough numbers, about 96 percent of the cen-troids account for only 10 to 15 percent of thetrips. The experience demand curve is there-fore influenced disproportionately by cen-troids which account for very few trips.There is some justification for using eachpopulation centroid as an origin zone and foraggregating centroids into concentric zones.The best choice is unclear. Since travel costvaluation estimates are sensitive to the defin-ition of the origin zone, this is an importanttopic for future work.

References

Abramowitz, Milton and Irene Stegun, Handbook ofMathematical Functions, Dover Publications Inc.,1965.

Brown, William B. and Farid W. Newas, "Impact ofAggregation on Estimation of Outdoor Recreation De-mand Functions." American Journal of AgriculturalEconomics, 55(1973):246-49.

Bowes, Michael D. and John B. Loomis, "A Note on theUse of Travel Cost Models with Unequal Zonal Popu-lations." Land Economics, 56(1980):465-70.

Burt, O. D. and D. Brewer, "Evaluation of Net SocialBenefits from Outdoor Recreation." Econometrica,39(1971):813-27.

Cicchetti, C. J., A. C. Fisher and V. K. Smith, "AnEconometric Evaluation of a Generalized ConsumerSurplus Measure: The Mineral King Controversy,"Econometrica 44(1976):1259-76.

Clawson, Marion, "Methods of Measuring the Demandfor and Value of Outdoor Recreation," Resources forthe Future, Inc., Reprint #10, Washington, D.C.,1959.

Clawson, Marion, and Jack Knetsch, Economics of Out-door Recreation, Baltimore: The Johns Hopkins Uni-versity Press, Baltimore 1966.

Common, M. S., "A Note on the Use of the ClawsonMethod for the Evaluation of Recreation Site Bene-fits." Regional Studies 7(1973):401-406.

Davis, Phillip J. and Phillip Rabinowitz, Numerical In-tegration, Blaisdell Publishing Co., 1967.

Dwyer, John F., John R. Kelley and Michael D. Bowes,Improved Procedures for Valuation of the Contribu-tion of Recreation to National Economic Develop-ment, University of Illinois, Water Resources Center,September 1977.

Goldfeld, Stephen M. "The Demand for Money Re-visited." Brookings Papers on Economic Activity3(1973):577-646.

Laidler, David E. W., The Demandfor Money: Theoriesand Evidence New York: Dun-Donnelly, Second Edi-tion, 1977.

Smith, V. Kerry, "Travel Cost Demand Models forWilderness Recreation: A Problem for Non-NestedHypotheses," Land Economics, 51(1975):103-11.

Sutherland, Ronald J., A Regional Recreation BenefitsModel, U.S. Enviromental Protection Agency, Cor-vallis Environmental Research Laboratory, 1981.

Sutherland, Ronald J., "A Regional Approach to Es-timating Recreation Benefits of Improved WaterQuality." Journal of Economics and EnvironmentalManagement, forthcoming.

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Sutherland

Western Journal of Agricultural Economics

Water Resources Council, "Procedures for Evaluation ofNational Economic Development (NED) Benefits andCosts in Water Resources Planning (Level C)," Feder-al Register, (December, 1979):72,950-72,965.

Ziemer, Rod, F., Wesly N. Musser and R. Carter Hill."Recreation Demand Equations: Functional Formand Consumer Surplus." American Journal of Ag-ricultural Economics, 62(1980):136-41.

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