recreation demand choices and revealed ...home.uchicago.edu/sabina/economic inquiry.pdfleisure time...

RECREATION DEMAND CHOICES AND REVEALED VALUES OFLEISURE TIME

DOUGLAS M. LARSON and SABINA L. SHAIKH*

Anapproach to jointly estimating an endogenous marginal value of time function anda recreation budget share equation is developed. Thespecifhation requirements for themarginal value of time function, to ensure consistency with the matnlained hypothesis oftwo binding constraints on choice, are articulated. The estimating model consistent withthese requirements that nests several conventional treatments of the marginal value oftime is highly significant. The general endogenous marginal value of time modeloutperforms other approaches and .shows a more complex relationship between themarginal value of time and the individual's wage ihan has been u.sed in previous work.{JEL J22, Q26)

I, INTRODUCTION

At least since the work of Becker (1965), ithas been recognized widely that time may playan important role in eonsumer demand. Oneof the areas where this may be most significantempirically is in the valuation of natural resourceamenities through their associated recreationdemands. A principal reason is that recreationis a time-intensive commodity, so that the valueofthetimcspcnt in recreation would be expectedto play a large role in its overall value. The poten-tial bias to consumer's surplus estimates of theneteconomic value of recreation from excludingtime "prices" from the demand model has beenrecognized from the outset, for example byKnetsch (1963) and Clawson (1959).

To incorporate time into recreation demandmodels, researchers have generally adopted thepractice suggested by McConnell (1975) of defin-ing the '*fuir priceof recreation as the money costof a trip plus its monetized time cost, though theuse offuil" budgets alsocalled for by the Beckerapproach is less widespread. Bockstael et al.

*We thank Paul Mason and participants in the 74thAnnual WEA International Conference in San Diegofor helpful comments, though any errors are our own.Commcnls by an anonymous reviewer improved the articleconsiderably.

Lar.son: Professor, Department of Agricultural andResource Efonomius, and Member of the GianniniFoundation of Agricultural Economics, University ofCalifornia. Davis. CA 95616, Phone l-5?(l-752-3586.Fax 1-530-752-3586. E-mail dmlarsDn(a)ucdavis.edu

Shaikh: Senior Research Economist. RCF Economic andFinancial Consulting. 333 N, Michigan Avenue. SuiteS04. Chicago. IL 60601. Phone 1-312-431-1540. Fax1-312-431-1170, E-mail sabinaCMiuchicago.edu

(1987) pointed out quite clearly that both fullbudgets and full prices are required in thedemand function if the recreationist is jointlychoosing labor supply with recreation at anexogenous marginal wage, but that the structureof demand was unclear if the individual was notmaking such marginal labor supply choices (i.e.,if he or she was working fixed hours). Subse-quently, Larson and Shaikh (2001) showedthat the full prices full budgets specification issufficient to satisfy the constraints on demandparameters that arise because choice is madesubject lo two binding constraints, regardlessof the individual's position in the labor market.They also pointed out that models that use fullprices but only money income (in effect ignoringthe role of time as a resource eonstraint) cannotbe consistent with the requirements of choicesubject to two constraints.

Although the outlines of how time shouldenter the structure of demand are becomingclear, a major unresolved issue is how to deter-mine its value in practice. Several studies havefollowed the basic logic of Becker's early work,assuming that the opportunity cost of recrea-tion time (i.e., the value of other time forgone infavor of recreation) is an exogenous parameter,such as the average wage rate or. more com-monly, some fraction thereof This fraction iseither chosen arbitrarily or estimated as part ofthe recreation demand model, as in Cesario(1976), McConnell and Strand (1981), andSmith etal. (1983).

Such assumptions may be erroneous for avariety of reasons. Assuming the average wageis the appropriate opportunity cost of time

Economic Inquiry(ISSN 0095-2583)Vol. 42, No, 2, April 2004, 264-278

264

DOI: IO.lO93/ei/cbhO59C.1 Western Economic Association International

LARSON & SHAIKH: REVEALED LEISURE TIME VALUES 265

presumes that the individual faces no con-straints on hours worked, derives no utilityor disutility from work, and has a linearwage function, as has been noted by Chiswick(1967), Bockstael et al. (1987), and Smith et al.(1983). This is unlikely to be true for manypeople. Assuming some constant fraction ofthe wage is the opportunity cost of time, espe-cially an arbitrarily chosen fraction, also seemslikely to be incorrect for most people. For all ofthese reasons, an individual's average wagedoes not necessarily reveal anything aboutthe shadow value of discretionary leisuretime, either as an upper or lower bound.

In a recent advance. Feather and Shaw(2000) adapted the Heckman (1974) labor sup-ply model to include information supplied byrecreationists about their labor supply status.in particular whether they felt under- or over-employed relative to their desired number ofhours. This information is used to identify sub-groups in thedata for which the shadow value ofleisure time is less than the wage or more than thewage, which helps in the estimation of both awage function and a shadow value of leisuretime function. This approach produces indivi-dual-specific estimates of the opportunity costof leisure time, based on their demographicsand labor market decisions, which are used ina subsequent recreation demand analysis.

The Feather-Shaw work is a substantialimprovement in assessing the shadow value oftime by allowing it to diverge from the indiv-idual's wage rate in predictable ways. Likemost of the other literature, the Feather-Shaw marginal value of time is determinedoutside the recreation demand model: that is.it is identified in an auxiliary relationship(in their case, the individual's labor supplychoice) and is used as a predetermined valuein the recreation demand model. The marginalvalue oftime is not. then, responsive to changesin the parameters of the recreation choiceproblem, nor is it necessarily utility-consistentwith the recreation demand function.

This article takes a different approach toincorporating the value oftime into recreationdemand models by estimating a latent, endo-genous marginal value oftime function jointlywith recreation demand. In this approach, theconsumer's observed choices, made in responseto the constraints and prices he or she faces,reveal the marginal value of time consistentwith those choices. The marginal value oftime function is internally consistent with the

demand model in that its functional formis consistent with the homogeneity and othercurvature properties required for models ofconsumer choice subject to two binding con-straints (in this case, a money constraint and atime constraint).

The article makes three contributions to therecreation demand literature. First, it is thefirst to estimate an endogenous marginalvalue of time jointly with observable behavior(in this case, recreation demands) in a utility-consistent framework. To this end, the curva-ture properties required for the marginal valueoftime function to be consistent with consumerchoice subject to two constraints (on time andon money) are identified, and an empiricalspecification consistent with these propertiesis developed. Second, the marginal value oftime function is specified so that it nests theother empirical approaches used in the recrea-tion demand literature, which allows for testsof which approaches best explain the data. Inparticular, it permits (but does not require) aconnection between the marginal value oftimeand the individual's wage.

Finally, the accompanying demand modelthat embeds the marginal value of time func-tion is, to our knowledge, the first applicationof the popular almost ideal demand system(AIDS) model of Deaton and Muellbauer(1980) to the case of choice subject to two bind-ing constraints. We show how the functionalform of this demand model can be derived bytracing throtigh the dual relationships betweenthe two expenditure functions and the indirectutiHty function that hold for the two-constraintchoice model.

The empirical setting for the demand-marginal value of time model is Californiawhale watching, an increasingly popular acti-vity that takes place along Western shoresduring the months of Deeember-Aprii. Boatsin ports along the California coast offer daytrips of varying lengths that afford an oppor-tunity for close-up viewing of the gray whalemigration between the Bering Sea ofl' Alaskaand the Gulf of California in Mexico. In addi-tion, one can often glimpse the migration frommany points on headlands along the coast.

II. BACKGROUND: THE TWO-CONSTRAINTRECREATION CHOICE MODEL

The two-constraint model used to moti-vate the analysis has its origins in the work

266 ECONOMIC INOUIRY

of Beeker (1965) and DeSerpa (1971) and wasused to analyze reereation ehoiees by Bockstaelet al. (1987). It can be interpreted, with minormodifieations, for either the ease where laborsupply is predetermined relative to recreationchoices or joint with them. The basic presenta-tion is for predetermined labor supply, beeausemany reereation choices are made conditionalon (not jointly with) the individual's laborsupply decision. The modifieations needed toincorporate joint labor supply are then brieflynoted.

When labor supply is predetermined, reerea-tion demands are part ofthe second stage ofa two-stage budgeting process, which allocatesthe money and time resulting from the first-stage labor supply deeision to all consump-tion activities. The money-budget M andtime-budget T applicable to the recreationdemand choice are therefore fixed.

Let u{x,s) be the individual's direct utilityfunction, with x an /i-vector o^ consumptiongoods with corresponding money-prices pand time-prices t. Second-stage recreationehoiees are made subjeet to strictly bindingconstraints on money-budget A/ —px andtime-budget 7'=tx.

Attention is contlned to the case of twostrictly binding constraints, which is not veryrestrictive for analyzing outdoor reereation,a time-intensive good whose time requirementshave long been reeognized to have an oppor-tunity eost. For time to have a value or oppor-tunity cost, the constraint must be strictlybinding. Additionally, nonsatiation is asufficient condition for the money constrainttobind.OneeouId introduee cases in which oneor both constraints go slack, though neither isvery compelling for the analysis of reereationdemand.' The existence of at least one numer-aire good with respect to eaeh constraintis sufficient for both constraints to bindcontinuously." The individual's utility isalso influeneed by an exogenous vector of

1. I f the time constraint goes slack, the money value oftioK" is zero and the standard single-constraint consumerproblem results. If the money constraint goes slack, thevalue of time goes infinite and a standard consumer choiceproblem of allocating lime results. Morey (1984) providesan example of the latter choice problem applied to outdoorrecreation.

2. Examples are goods, such as walks on the beach(requiring time but no money), or making charitable con-tributions (requiring money but no—or liUle—time).LaFrance and Hanemann (1989) discuss the constructionof numeraire goods for incomplete demand systems.

individual and site eharaeteristies s, whiehmay include one or more unpriced qualityvariables.

The primal version of the problem leads tothe indirect utihty funetion K(p, t ,s. A/. T),defined as

(1) r (p , t , s . A/, 7")

= maxM(x,sX

[M — px}

The solution to (I) yields the Marshalliandemands .vXp, t.s, M, 7*), i=\.,...n, alongwith the shadow values ^(p, t, s. A/, 7 ) and|a(p, t, s, M. T) that are of particular interest."*They are the marginal utilities of the money-budget and time-budget, respectively, whoseratio.'' \ilX= f- is the marginal moneyvalue of time, as noted by McConnell (1975).Beeause \i and X are funetions ofthe observablearguments (p. t, s, M. T), their ratio is termedthe Marshallian value of time, denoted byp(p,t ,s ,A/T).

The presence of an additional binding (time)constraint implies additional strueture on theconsumer choice problem, as there are two ver-sions of Roy's identity for this model. From theenvelope theorem applied to (1), Vp — —Xx,,Vii^ -H-Vy, VM^ X, and K/'^| i , so that for allgoods in the estimated incomplete demandsystem one can write

(2)

These two Roy's identities are a source ofparameter restrictions in the empirieal demandsystem and prove useful for speeification andidentification of the marginal value of leisuretime from demand system coefficients. Larsonand Shaikh (2001) show that even whenp = p(p, t, s., M.T) is an endogenous functionof all parameters, these restrictions are satisfiedby systems of demands that are functions of"full" priees Pj + p- tj, for / ^ 1 n. and

3. To consider labor supply choice joint with goodsdemands, dctinean element of x, say .t|, as hours worked.with a money price pi = w. where w is the wage, and it - 1(Bockstael etal, 1987).

4. Subscripts that are parameters represent partialderivatives; for example, yi = d


"full" budgets M + pT,

(3) .v / (p , t , s ,M,n

A result they derive, that is useful later, is thatbecause the'full-budget'Marshallian demandsare homogeneous of degree zero in (t, T), therelationship between the time and money-budget slopes of (3) is

(4)

Equation (3) can be viewed as a structuralform lor the empirical recreation demand func-tion that satisfies the two-constraint choicerequirements. By specifying a form forp(p,t,s. M. T) that is also consistent with thetwo-constraint choice requirements., one candirectly estimate the endogenous marginalvalue of time function jointly with the param-eters of demand. However, the propertiesofp(p, t,s, A/, T) have not yet been articulated.Knowledge of these properties is neededto ensure that empirical specifications ofp(p.t.s, A^,r) are internally consistent withMarshallian demands and that both are utility-theoretic.

To develop the properties of the marginalvalue of time, it is necessary to consider the dualstructure ofthe two-constraint model. Becausechoice is subject lo two constraints, the modelhas two dual expenditure functions, one mini-mizing money expenditure subject to utilityand time constraints, the other minimizingtime expenditure subject to utility and moneyconstraints. The dual money expenditure func-tion t?{p,t,s, T,u) is

(5) e(p. t. s. T. u) = min px + (j) • [H - M(X. S)]

-\-\\f • {T - h - t\}

where (t) = Cwisthemarginalmoneycostofutilityand, in this problem, - y ^ -e7-(p, t.s, 7", H)

5. For demand systems, the requirements aresomewhat more stringent. The full-budget term mustenter muliiplicativcly and have the same budget slopeacross all demands, and the (Marshallian) cross-fullprice effects must be symmeiric (Larson and Shaikh 2001).

is the marginal value of additional time.'' Thesolution to (5) yields a set of money-com-pensated Hicksian demands, .Y, '(p. t. s. 7".»).that vary money income to keep utility con-stant given fixed time-budget T.

The dual time expenditure function isdefined as

(6) ^(p, t. s, M, M) = min tx -|- (p • [w - M(X, S)]

+ Q{T-h-tx}

and in this problem cp ^ ^,, is the marginal timecost of utility and - 1 / 9 ^ - 1 1^M{ p. t. s, M. tt) isthe marginal value of additional time.^ Thesolution to (6) is the set of time-compensatedHicksian demands, .vf (p, t .s. A/, w), that varytime expenditure to maintain constant utilitygiven fixed money-budget M.

Summarizing the different marginal value oftime relationships in terms of the slopes of theobjective functions instead of the shadowvalues, the marginal value of time is

(7) p(p,t .s,M,7') ^ -e r (p , t , s ,7 ' ,M) .

This equivalence ofthe dual expressions for themarginal value of time at the optimum is usefulfor analyzing the comparative statics of themarginal value of time function estimated aspart ofthe recreation demand model.^

These dual expenditure functions havebeen defined and partially described for two-constraint models by Smith (1986) but havenot (to our knowledge) been used to analyzethe comparative statics of the marginal valueof time, nor for specifying utility-theoreticempirical two-constraint demand functions.These topics are taken up in the next twosections.

6, This can be veriHed by noting tlic identity relatingthe indirect utility function in (I) to the money expendi-ture function in (5), K(p, t.s,f[p,l.s, T,u],T) = u. PartialdilTerenliation with respect to Tyields K

7. Here the identity is K(p, t. s. M. ^[p, t, s. M. u\) = u.and partial differentiation with respect to M and rearran-ging yields the result.

S. In equation (7), each expression for the marginalvalue of time has a difTerent set of arguments becauseeach objective function holds a dilTerent set of paratn-cters constant. At the optimum, of course, they all areequal.

268 ECONOMIC INQUIRY

III. COMPARATIVE STATICS OF THEMARGINAL VALUE OF TIME

What, in fact, should one expect to be the"correct" signs on the derivatives ofp(p, t,s, A/, r ) ? This section develops com-parative statics of the marginal value of timefunction and shows that under the full prices,full budgets specification of demands, it mustsatisfy analogs to Roy's identity: the ratios ofprice to budget slopes of the Marshallianmargitial value oftime ftinction equal the tiega-tive of corresponding quantity consumed.

By letting utihty vary in equation (7), the twoidentities

(8) p(p , t , s ,A/ , r )

and

(9) p(p,t ,s ,M,

To evaluate (12), note the identity relatingmoney-compensated demands to Marshalliandemands,

.xf{p, t,s, T, u) - X;(p, t s, e[p, t.s, r , MJ, r ) .

Differentiating both sides with respect to T. thetime-budget slope of money compensateddemand can be written as

(13)

Substituting (4) and (7) simplifies (13) to

(14) dxf{pA,s,T

Xs,M,T)/dM

=0.

result. These are useful in evaluating the com-parative statics ofthe marginal value oftime.

Relating Money-Price and Money-BudgetSlopes of the Marginal Value of Time

Consider first the money-budget and priceslopes ofthe marginal value oftime. Differen-tiating (8) with respect to M and pjj= ],...n,one obtains

(10)

and

(11)

The cross-partial derivative er^ of themoney expenditure function with respect tomoney-price p,- and time-budget T is also, byYoung's theorem, the time-budget slope ofmoney-compensated Hicksian demand, that is.

(12)

This interesting result shows that with afull-prices/full-budgcts Marshallian demandspecification, the time-budget slope of Ihemoney-compensated Hicksian demand isalways zero at given (H, M , T) points. Becauseit is a local result rather than a global resultabout the demand function, though, themoney-compensated Hicksian demands arcgenerally functions of T.

Because of (14), the derivative f/ (p, t,s, T, u) is always zero in (11). Taking the ratioof (11) to (10), a fundamental result emerges.

(15)

Equation (15) is striking for the structure itreveals about the endogenous marginal valueoftime function. The ratio of its money-price tomoney-budget slopes is the same as for theindirect utility function and equals the negativeof corresponding quantity consumed, viaRoy's identity.

9. This can be seen by noting ihat ihe equalityej{p,t.s. T.u)= - p ( p , t,s, .W, T) holds only at a point.Liniess ulility is varied to keep money-budget constant(as in equation [8]). But the money-compensated Hicksiandemand holds utility constant by definition.


Relating Time-Price and Time-BudgetSlopes of the Marginal Value of Time

Parallel results hold for the time-budgetarguments. By differentiating (9) with respectto time-budget Tand time-prices tjj= ! , . . .« ,

(16)

identity relating time-price and time-budgetslopes.

(17)

where in (17) arguments of the functions com-mon to (16) are suppressed. The cross-partialderivative ^^ {pXs,M,u)^dxJ{p,Us,M,u)/()M is the m'oney-budget slope of the time-compensated demand for.v, again by Young'stheorem. To evaluate this term, note theidentity relating time-compensated demand toMarshallian demand,

.\/(p, t, s, M, u) = Xj{p, t, s, M, ^[p, t, s, M,u]).

Differentiating with respect to M,

Again substituting from (4) and (7), this timeinto (18),

xJ(19) dxJ{p,t,s,M,u)/dM

• dxjip, t, s, M, T)/dM]•[- i /p(p, t ,s ,A/ , r ) ] ,

which parallels the result in (14). As with themoney-compensated Hicksian demands, thetime-compensated Hicksian demands havezero money-budget slopes at given (;/, M, T)points.

As before, substituting (19) into (17) andtaking the ratio of (17) to (16) yields the

Xs, M,T)/dtj],t,s, M,T)/dT]

(20)

Equations (15) and (20) are the two principalresults useful in specifying the marginal valueoftime function. When the marginal value ofleisure time is endogenous, a demand modelwith full prices and full budgets is sufficient(not necessary) to satisfy the two-binding con-straint hypothesis. Given this demand specifi-cation, the marginal value of time satisfiesanalogs to the two Roy's identities given in(2): the ratio ofthe price to budget slopes withineach constraint is the negative of the corre-sponding quantity consumed.

Implications for the Structure of theMarginal Value of Time Function

To identify a specification for p(p,t,s, M, 7")consistent with the requirements of equations(15) and (20). it is useful to note that incompletedemand systems are estimated in practice.By introducing A*^ and NT as numerairegoods such that p^^ = \. /A/ = 0, PT = O, andtT=U the budget constraints can be expressedas M = p\ = p'\' -j- NM and 7"=tx = t'x' +Nr with N/^f and A'y representing expendituresofmoneyandtimeoutside the A/-good empiricaldemand system of interest, which explains x'.

Given this, a general structural specificationfor the marginal value oftime that satisfies (15)and (20) is™

(21) p(p,t,s,A/,7-;x')-/(A/-p'x',r-t 'x',s).

That is, given the parameters of the problemand the individual's optimal choices .v, the"revealed" value of time is a function of thediscretionary money and time after expendi-tures are allocated to goods in the empiricalincomplete demand system time and of theother shifters s.

10. This can be easily verified by noting thatp i } f _ ^ f ) { , ) p { p , , .

{fM). so equation (15) holds. Similar resultshold for (and T, so ihal equation (20) also is satisfied.


Given the development of the marginalvaltie of time function, equations (3) and (21)are collected together to form an M + 1 goodsystem of structural equations in the endogen-ous variables x' and p,

(3')x'(p,t,s,A/,7-;p)-x'(p-^p-t,s,A/-t-pr),

i=\,...n

and

(21') p(p,t,s,A/,r;x') =

where the definitions of the numeraire goodsN^, = M- pV and NT=T- tV are used tosimplify notation in (21). Because p is notobserved, equation (21') cannot be estimateddirectly as part of the econometric system.However, it can be substituted directly into(3). yielding the Af-good implicit system

(22)

Equation (22) can be estimated given theempirical marginal value of time functionQ—f(Nxf, Nj-.s) from (21) and suitable choicesof functional forms for the goods x in theincomplete demand system.

IV. AN EMPIRICAL TWO-CONSTRAINTDEMAND MODEL

This section develops the first empiricalapplication of a two-constraint version ofthe AIDS model of Deaton and Muellbauer(1980). This demand model has an endogen-ous marginal value of time consistent withthe two-constraint requirements. The implicitdemand system with endogenous marginalvalue of time described in (22) is considerablymore general than recreation demand modelsestimated in practice: in fact, single-equationmodels are more the norm. After describingthe general form of a two-constraint demandmodel, then a specific form for the marginalvalue of time is adopted and a single-equation version of the model is estimatedempirically.

Beginning with any one of the optimizedchoice functions v(), e(), or ^(). one canderive the others from the dual structure

of the optimization problem. It is most con-venient to develop the dual structure in termsof the money cost function, in a mannersimilar to the development of the standardAIDS model. Let the money expenditurefunction be

is the (money-compensated) Hicksian value oftime function, =/?,+ p"(p,t,s,7') •are full prices formed using this value of timefunction.

The indirect utility function K(p,t,s, A/, T)dual to e(p,t,s. T,u) is'^

(24) V{p,t,s,M,T)

where/Jy^ = Pj + p(/>,t,s, Af, T) • tj and A/^=M-l- p(p, t, s, M,T)T are full prices andfull budget formed using the empirical(Marshallian) value of time functionp(p, t, s. A/, 7"). It can be verified in (24) that' ^ r /^M~p(P- t s , A/, 7"), consistent with thedefinition of p(p, t,s, M,T) in (7). The timeexpenditure function corresponding to (23)

11. It is straightforward to show that -CT —p*'{p, t.s, r.«) for this model, as required. Though Tappears throughout (25) because it is an argument ofpy. homogeneity of degree zero of money expenditurefunction in (t.T) ensures that all terms involving Op'^^tOTsum to zero.

12. That ('(p. t. s. r.«) and F(p.t.s.W.r) are M-uinverses can be verified by substituting (24) into (23) andnoting that p'"(p. t.s, V, F[p. t.s. A/. r])= p(p.l.s. A/,r).When terms cancel, what remains isdp. l.s. 7', )''[p. t.s. M, T]) - M. Conversely, using (23jin (24) and noting that p(p.t.s.f[p.t.s, 7",((].7") =p *'(p, t, s, T, u), the result is Kp. t. s. e[p, t. s, T,«], T) = u.


and (24) is

(25) ^(p,

Marshallian demand funetion is

where p^( ) = p^(p, t,s, M, a) is the time-compensated value of time funetion and/j j =p, + p^(p, t,s, M, u) • tj are full prices formedusing this value of time function. The time ex-penditure funetion is a V- 7" inverse with (24)and an M-T inverse with respeet to (23),and it can be verified that p^(p, t, s. A/, w) =-l/^A/(P' ts , A/, w), again eonsistent with (7).

The Implied Share Equations

DifTerentiating (25) with respeet to themoney-priee p, yields the Hicksian money-compensated demand .vf^(p, t, s, T,«) —()c(p. t, s. T. u)/dpi. Because /J, is embedded inthe full price variable/jf' — Pi + p'^^(p,t,s,/•..»)-ti, the chain rule applies and the derivativecan be written as t?t'(p,t,s. 7", w)

From (25), it ean be seen that

and d\og{pf)/dpi = \/pf. Combining these,the Hicksian money-compensated demand is

because p'^'(p, t, s, T. V[p, t, s, M, T]) - p(p, t,s,M,T) and, therefore, pf^ becomes/jf ande{pA.s.T,u)+p''\p,ts,T,u)T is M^. Theterm PI is the budget deflator.'^ Multiplyingboth sides by pf /M^, the Marshallian shares

are

(26) log(,v,)

whieh take a form identical to shares in thesingle-constraint AIDS model, aside fromthe fact that pf = pj + p(p, t, s, M, T) • tj, arefull priees and A/''= A/+p(p,t ,s , A/, T ) - Tis full budget.

One ean also derive the same system fromthe time expenditure funetion (25), differentiat-ing with respeet to tj and using the enveloperesult that .v, ^ c>^(p,t,s, M,u)/i)ti. Not sur-prisingly, no new information results becausethe two expenditure funetions are jointlydependent. The shares in (26) arc full sharesbeeause they are defined in terms of full prieespf and full budget A/''.

The form of the share model in (26) isessentially a generalization of the AIDSshare model from single constraint choice.The arguments ofthe share equations are fullpriees and full budgets, and the conversion fac-tor between time and money is the marginalvalue of time function that must satisfy therestrictions on its price and budget slopes in(15) and (20).

The Empirical Value of Time Eunction

Because so much ofthe reereation demandliterature focuses on the connection betweenthe marginal value of leisure time and thewage, it is natural to include the individual'swage as an argument of the vector of shifters s.The speeifieation should be flexible enough.

By substituting the indirect utility function in(24) for the utility term everywhere, the

13. It is defined as / > / - [oo H-Ijocf logl/jf),2;yT . log(pf) log(/)f) -f-\ l


though, to allow for different effects of thewage on the marginal value of time for thosewho make a marginal labor supply decision(e.g., by working flexible hours) and thosewho don't (e.g., work fixed hours). Other indi-vidual characteristics may also be important.For example, it seems likely that the individ-ual's age is a determinant of the marginal valueof leisure time. The individual's avidity for theparticular recreation activity could also play arole.

With these observations and equation (21)in mind, the empirical marginal value oftime is

(27) [So

where f is a dummy variable equaling 1 if theindividual works fixed hours and 0 otherwise.This specification recognizes that the individ-ual's average wage, as well as the amounts ofthe numeraire goods Nxi and Nj-, are likely toatTect the marginal value of time in the fullprices and full budgets; but the effects can bedifferent for those working fixed versus flexiblehours. One would expect that wage plays less ofa role for those working fixed hours (i.e., 63 — 0and 5|, 52?^0) and more of a role for thoseworking llexible hours and ofTering discretion-ary labor supply (i.e.. 617^0 and Q], 62 —0). Ifall individuals report their marginal wages cor-rectly, one might further suppose that 63 — 1.

In addition to Wage, two other shifters areincluded. Age and Avidity. Avidity is an indexranging from I (lowest) to 4 (highest) based onresponses to several categorieal variables prob-ing the individual's strength of opinion aboutthe importance of preserving marine mammalpopulations.

The specification in (27) encompasses severalapproaches used for treating time in the empiri-cal recreation demand literature. One is theapproach suggested by MeConnell and Strand(1981), which estimates the constant fraction ofthe wage rate that the marginal value oftime isassumed to represent. When 61—62 —(pi =<P2 ~ 0 and 9: — 1. the value of time for indivi-duals working flexible hours {F= 0) is

and is the natural log of the constant wagefraction. More generally, of course, thefraction of the wage could vary with individualcharacteristics, which is accommodated by theless restrictive condition 63 = I, in which easethe marginal value of time for those workingflexible hours is

Wage.

One could also estimate the value of time forthose with fixed hours as a fraction of the wage,as (27) indicates, though this is less compellingbecause presumably there is less of a connec-tion between the marginal value oftime and thewage for these people.

Another common approach is to assumethat the marginal value ofleisure time is a frac-tion of the wage, with 1/4 to 1/2 often used inpractice by reference to the value of time savedin transportation studies (e.g., Cesario 1976).This type of assumption is also easily evaluatedas the additional restriction on (28) that9() —ln(A'), where k is the value to be tested.To test whether time is valued at zero, onecan replace exp(9o) in (28) with a constant icand test whether K — 0.

V. AN APPLICATION TO CALIFORNIAWHALE WATCHING

(28) o) • Wage

The data used to illustrate the model arefrom on-site intercepts of whale watchers atfour sites in California during the winter1991-92. The survey instrument was pretestedusing individuals who had gone whale watch-ing in the previous year. It collected informa-tion on trips taken so far that season, expectedfuture trips, travel time, travel costs, whetherthe trip was their primary destination, and soon. Also collected was information includ-ing actual contributions to marine mammalgroups; time spent reading, watching, or think-ing about wildlife and whales; as well aspurchases of whale-related merchandise. Last,demographic information including workstatus, wage rates, and income was asked.The survey was presented in booklet form.

In total. 1.402 visitor surveys were handedout, and 1,003 were returned, for an overallresponse rate of 71.3%. The response rate

14. We thank an anonymous reviewer for pointing outthis simpler test.

LARSON & SHAIKH; REVEALED LEISURE TIME VALUES 273

was reasonably similar across the four loca-tions, varying from a low of 65.2% for inter-cepts at Point Loma (San Diego) to a high of80.3% for intercepts at Point Reyes. On-siterefusals were not a problem. For e.xample. atPoint Reyes, only 10 people of roughly 600contacted (about 1.6%) refused to take a surveypacket.

The empirical demand system focuses onestimating the budget share for whale watchingtrips but recognizes there are related actionsindividuals can take with respect to whaleand marine mammal populations. These arethe expenditure ofone's time or money to con-tribute to the maintenanee or enhancement ofgray whale populations. The relative prices ofthese other actions differ across the popuiationand therefore may affect the frequency of whalewatching trips taken, just as substitute pricesaffect consumption of other goods. Augment-ing the single-equation incomplete demand/share equation for trips arc the two numerairegoods A' and A', the residual expenditures oftime and money from their respective budgetsafter accounting for trips.

The own (full) price of whalewatching trips(A) is the money cost of travel to the site andwhile on site, plus the time costs of travel (hourstraveled and spent on site) valued at the (endo-genous) marginal value of time.''' Thus the fullpriceof.vis/?f -t- p{Nm.Nr.s) • t\, where/?] isthe money cost ofthe trip and/1 is the time costofthe trip, or the trip length. Monetary dona-tions have a money price because charitablecontributions are tax-deductible at both thefederal and state level, so the marginal priceof a dollar donation is less than a dollar andvaries across individuals according to house-hold income. These donations do not have asignificant time-price, so the full price of dona-tions is just the money cost of donating; that is,P2 — T. where T is the marginal tax rate forthe individual. Both California and federaltaxes were included in figuring T for each indi-vidual, based on their total household incomecategory.

Time donations (.v ) have a time-price i > 1,measured by the total number of hoursrequired to deliver an hour of time to the volun-teer organization. Total time may be higher

15. Time on site is taken to be exogenous because allirips covered in this analysis are day trips and roughly halfof all whale watching trips represented are boat trips offixed duration.

because of transactions time costs from drivingback and forth, and so on. Because we don'thave information on how this time-price willdiffer across individuals in our sample, we takethetime-price?i= 1 for all. There may also be amoney cost p^ of time donations, which wouldrepresent the money costs of driving to andfrom the site where volunteering occurs andother transactions costs of donating time.We have no information on this variablefrom our sampie but suspect it is small, thusit is taken to be zero. Thus price pj, = 0, andthe full price of time donations is p^ —^(NM,NT,S), which varies across the samplebased on differences in individual charac-teristics.

In addition to the time- and money-prices, itisexpected that the individual's whale watchingsuccess will inlluence both trips demand andpotentially the willingness to make donationsof time and money. The success variable (r) isthe individual's ex ante expectation of whalesightings for the trip when they were contacted.Money-budget (M) is the household incomebefore taxes, and the time-budget (T) is theamount of nonworking time measured asthe number of weekend and paid vacationdays. Table 1 provides some descriptive infor-mation on these variables used in the estima-tion model.

Results

The empirical model consists ofthe demandshare for whale watching trips (equation [26])with the marginal value of time function{equation [27]) embedded in the full pricesand budgets ofthe share equation. It was esti-mated using nonlinear least squares in SASversion 6.12. Stone's price index (Stone 1954)was used for the deflator.

Estimation results for the full model aregiven in Table 2. The model is highly significantoverall, and most coefficients individually aresignificant at the a = 0.05 level. The share coef-ficients indicate that demand is price inelastic(with mean own-full price elasticity of -0.31)and income elastic (with mean full income elas-ticity of 0.66). Expected sightings have a sig-niftcant positive effect on trips share, and timedonations are a substitute for trips, whereasmoney donations are complementary withtrips. There are significant site-specific effectsat the two northernmost sites. Point Reyes andHalf Moon Bay.


TABLE 1Some Characteristics of the Whale Watcher Sample (n ^ 393)

Variable

Household incomePaid vacation

Time-budget

Wage

AgeAvidityE.xpected sightings

Whale watching tripsMoney-price of trips

Time-price of trips

Money-price of donalionsTime-price of donations

Money-price of volunteeringTime-price of volunteering

Units

S/year

days/year

hours/year$/hoiiryears

—no./trip

number$/trip

hrs/trip

—

hrs/S

—

Mean

65,178

136,771

2536.63.266.542.20

109.31

3.05

0.69001

Minimum

5.000 10

6,672

1

171.5O.II

0.465

1.1

0.640

0

1

Maximum

150.000

1807.272

100•m3.755024

m -1770.84

0

1

TABLE 2Parameter Estimates for the Full Trips Share-Value of Time Model

Variable Parameter Estimate Asymptotic Student's t

Trips share equationConstantPoint Reyes dummy

Hair Moon dummyMonterey dummy

Own full priceTime donations full price

Money donations full price£[sigh tings]Full budget

Marginal value of time function

ConstantMoney numeraire* 10Time numeraire* 10ln( Wage)Constant

Money numeraireTime numeraireln( Wage)AgeAvidity

N 393LogZ, -807.04

Log /.. (constants only) -949.15

P«ai

Yii

Yl2

Yl3

YL-

P.

8o8,828,Oo

e,e.83

9i

<f>2

5.1214-0.1176

-0.59970.11860.7255

-0.46630.1312

0.0476-0.3626

-42.4001

0.00680.0070

-0.2022-3.4330

-0.01421.04610.8866

-0,7297-0.0194

-1.84-2.91

0,7310.50-5.08

2.162.55

-3.52

-3.151.153.50

-0 .56

-0.27

-2 .46

2.16

2.76

1.32

"Trips share is scaled up by a factor of 1.000.


TABLE 3A Comparison of Some "Standard" Methods of Valuing Time with the Endogenous

Marginal Value of Time Model

Hypothesis Parameter Restrictions Test Statistic df

Age & avidity do notaffect the marginalvalue of time

Time is valued at difTerentwage fractions for fixedand tlex hours

Time is valued at same wagefraction for fixed and flex

Time is valued at 1/3 wage

Time is valued at zero"

S1-S2 —

81-82-08] =: 02 = 0

9.90

32.21

33.361

81-62-083-63=180-60

81-62=061-82=083-6 , -18o = 6,, = ln(0.33)

54.64

23.30

5.99

12.59

14.06

15.51

3.84

"Estimated with one free constant such that p = K.

Within the value of time function, it is clearthat the parameter vectors are different forthose who work fixed versus flexible hours.For those who work ftxed hours, the time con-straint was the most important determinant ofthe marginal value of time. As one wouldexpect, wage is not a significant determinantofthe marginal value of time for these individ-uals. In contrast, for those working flexiblehours, the most important factors in the mar-ginal value of time are the wage rate and themoney-budget constraint. Age has a significantnegative effect on the marginal value of time,controlling for budget and other effects, andavidity also has a negative sign but is notstatistically significant.

Table 3 presents the results of hypothesistests on the treatment of the marginal valueof time. The first hypothesis test is whetherpersonal characteristics (in our case. Age andAvidity) make a signiftcant difference to anindividual's marginal value of time. TheX" statistic for the restrictions that coefficientsof these variables are zero is 9.90, exceeding thecritical x^05' which is 5.99. The hypothesisthat personal characteristics are irrelevant istherefore rejected.

The rest of Table 3 explores the implicationsof using various conventional treatments ofthe value of time. These are nested withinthe endogenous value of time model and canbe tested as parameter t estrictions on the more

general model. If the parameter restrictionscorresponding to alternative conventionaltreatments are not rejected, one can say thatthere is no additional explanatory power fromthe more general model. If. on the otherhand, the parameter restrictions are rejected,one can conclude that the more generalendogenous marginal value of time model ispreferred.

From Table 3, one can see that assuming themarginal value of time is a fraction ofthe wageis rejected., because the x^ test statistics for fourdifferent variants of this assumption exceedthe a — 5"A> critical values. The most generalassumption, that those working fixed and flex-ible hours have a difTerent fraction, with bothestimated in the model, is rejected, as is theadditional restriction that the fraction be thesatTie for both groups. Similarly, using twocommon assumptions in the literature (thattime is valued at 1/3 the wage and at zero)are also rejected. These results indicate thatthe endogenous marginal value of timemodel does a significantly better job of explain-ing trips shares than do the conventionaltreatments.

It is interesting to ask what relationship themodel estimates ofthe marginal value of timehave with the individual's reported wagecategory. Table 2 indicates a strong link tothe wage for individuals working flexiblehours, as one would expect. The parameter


estimate on ln( Wage), O ., is the elasticity of themarginal value of time with respeet to wage,which is 0.886. This means that for every 1%increase in the wage, the marginal value of timefor individuals working flexible hours increasesby 0.89%. This is a partial clTect, however, thatdoesn't account for other changes in moneyincome and discretionary time-budget thatare induced by changes in hours worked.

Figure 1 graphs the model estimates of meanmarginal value of time by wage category andlabor class, along with error bars representingthe 95% confidence intervals. The mean esti-mates are conditional sample means and takethe estimated parameters as given, so the eon-fidence intervals reported refleet the effeets ofvariations in individual eharaeteristies on themean value of time estimates. The numbersused in Figure I are also presented in Table 4.

Figure 1 illustrates two interesting findingsof the empirieal model. First, marginal values

FIGURE 1Values of Time by Wage Group

of time for both labor elasses {fixed and fiexiblehours worked) diverge signifieantly from thewage at moderate to high wages (aboveabout $17.5O/h). There is a modest increasein mean marginal value of time with wagefor flexible hours workers but no diseernibletrend for fixed hours workers (Table 4). Over-all, the relationship of the marginal value oftime to the wage is more complex than mostmodels in the literature allow, with the mar-ginal value of time exceeding the wage forlow wage rates and less than the wage forhigh wage rates. That is, the ratio of the mar-ginal value of time to the wage deereases withincreasing wage.

The second interesting finding is that there isnot a statistically significant differenee betweenthe marginal values of time for fixed versusflexible hours workers. This is due in part,no doubt, to the fact that when the data aredisaggregated into many subsets, the confi-denee intervals widen beeause the number ofobservations used for each statistic decreases.It may also refleet the fact that schedules arenot strictly fixed for any workers in the longrun, so that when a serious imbalance betweenthe wage and the marginal value of time arises,an adjustment of hours worked oecurs to bringthem more in line.

The key point is the flexibihty of theendogenous marginal value of time approachto better represent the complex relationship ofthe marginal value of time to individual's eon-straints and opportunities. Figure 1 illustrates

TABLE 4Estimated Marginal Values of Time by Wage Category and Flexibility of Work Time

Wage Category ($/hr)

<4

4-6

6-1010-1515-2020-2525-3030-5050-100>iOO

Cjrand total

Flexible Hours

Time Value (S/hr)

1.83(1.17)"7.02 (0.95)7.91 (1.14)9.95(1.14)

11.07(1.10)12.53(2.21)11.25(1.94)15.88(1.53)16.23(1.78)18.81 (2.43)12.08(0.58)

Number

22

17

302916132015

9153

Fixed Hours

Time Value (S/hr)

10.07 (6.40)6.26(1.50)7.24 (0.80)7.47 (0.58)

19.26(7.54)8.85 (0.82)9.70 (0.78)

13.72(2.99)11.89(1.94)9.57(3.94)

10.75(1,34)

Number

31026

5840

3629

28

73

240

All

Time Value (S/hr)

6.77 (4.06)6.39(1.24)7.50 (0.65)8.32 (0.56)

15.81 (4.40)9.98 (0.90)

10.18(0.80)14.62(1.85)14.85(1.41)16.50(2.32)11.27(0.85)

Number

5

12

438869

5242

48

2212

393

"Standard errors of the means in parentheses.


this flexibility of the model to represent ele-ments of two approaches to the marginalvalue of time in the literature: the conventionalassumption ofa constant fraction of the wage,and the approach of inferring a constant traveltirne value for all individuals in the sample(e.g., Hausman et al. 1995). The patterns oftime values are revealed by the data on choices,prices, and constraints, rather than beingimposed a priori.

VI. CONCLUSIONS, LIMITATIONS. ANDEXTENSIONS

This article has developed an approach toestimate an endogenous marginal value of timefunction jointly with recreation demands.It identifies the structural requirements thatendogenous marginal values of time tnustsatisfy to be consistent with the hypothesis ofchoice subject to two binding constraints. Ananalog to Roy's identities for the two-constraint model holds for the endogenousmarginal value of tirne function, wherein theratios of time-price slopes to time-budget slopeof the marginal value of time function equalthe negative of corresponding quantities con-sumed, which also equal the ratios of money-price slopes to money-budget slope. Wepropose a functional form for the endogenousmarginal value of time that nests the other mainempirical specifications in the literature, andestimate this function jointly with the recrea-tion demand share in a two-constraint versionof the AIDS. The approach is attractive econ-otnetrically because joint estimation of theparameters of the marginal value of time anddetnand means that full information about themodel is used in its estimation. The parametersofthe tnarginal value of time function are thosewhich best ftt the data, in providing the closestmatch with recreationists' actual choices givenrelative prices and constraints on time andmoney they face.

The model is applied to data on gray whalewatching along the California coast. The sharemodel is specified using full prices and budgets,with the endogenous marginal value of timefunction as the tenns of trade between timeand money. Based on the econometric results,joint estimation ofthe marginal value of timeand the demand (share) model seems superiorto conventional treatments of this importantvariable in the literature. Allowing for differ-ences in the structure of demand between

individuals working ftxed versus flexiblehours, the demand share and value of timefunctions are highly significant. The marginalvalue of time appears to have a more complexrelationship with the wage than is allowed for inmost conventional treatments in the literature.

It is encouraging that very little additionalinformation is required to implement a recrea-tion demand model fully consistent withthe two-constraint requirements. Principally,information on time-budgets is needed beyondwhat is routinely collected. When labor supplyis predetermined relative to recreation choice,the relevant time constraint is nonwork time,and this can be obtained easily through surveysthat ask about the respondent's work schedule:how many hours worked per week and howmany weeks worked per year. The nonworktime is then the difference between total timein the period of analysis and the time spentworking. When, alternatively, labor supply isjoint with recreation choice, the relevant time-budget is simply all time during the period ofanalysis. This does not vary across respon-dents, but the monetized value of the time-budget does, because the marginal value oftime varies. The other important time variablesin the two-constraint model, time spent intravel and on site, are commonly collected inrecreation demand surveys because of thewidespread recognition of the importance oftime-prices.

An area for further work is to incorporatethe individual's labor supply decision into theframework of this article, as another source ofinformation about the individual's choicesthat help reveal the marginal value of time.Another possible direction for further workis to allow for the presence of multiple con-straints on use of nonwork time. We have pre-sumed a single time constraint, but in realitymost people face multiple constraints on theirability to get free for recreational activities.Development of empirical models of prefer-ences that generate both demand and the mar-ginal value of time from the same expenditureor indirect utility functions is a natural gener-alization to pursue. Hopefully the identifica-tion of structural requirements for shadowvalues herein will assist with and motivatesuch work. In a sense., it allows researchersto widen the optimization space for findingworkable empirical models that better fitobserved choices while maintaining consis-tency with theory.


REFERENCES

Becker. G. "ATbeory ofthe Allocation of Time." EeonomicJournal, 75(3), 1965. 493-517.

Boekstael, N. E., I. E. Strand, and W. M. Hanemann."Time and the Recreation Demand Model."American Journal of Agricultural Economics. 69(2),1987. 293-302.

Cesario, F. J. "Value of Time in Recreation BenefitStudies." Lttnd Economics, 52(1). 1976, 32-41.

Chiswick. B. R. "The Economic Value of Time and theWage Rale: Comment," Western Economic Journal.Mareh 1967,294-95.

Clawson. M. "Meihods of Measuring the Demand for andValue of Oiildoor Recreation," Resources for theFuture. Reprint No. 10. Washington. DC. 1959.

Deaton. A., and J. J. Muelibauer. "An Almost IdealDemand System." American Eeonomic Review,70(3). June 1980.312-26.

DeSerpa. A, C. "A Theory of the Economics of Time."Economic Journal. 81(4), 1971. 828-46,

Feather, Peter M., and W. Douglass Shaw, "The Demandlor Leisure Time in the Presence of Constrained WorkHours." Economic Int/uiry. 38(4). 2000. 651-66,

Hausman. J.. G. Leonard, and D. McFadden. "A Utility-Consistent Combined Discrete Choice and CountData Model: Assessing Recreational Use LossesDue to Natural Resource D-dmage." Journal of PublicEconomics. 56(1). \995. \ 30.

Heckman. J. J. "Shadow Prices, Market Wages, and LaborSupply," Econometrica. 42(4), 1974. 679-94.

Knetsch. J, L. "Outdoor Recreation Demands and Bene-fits." Land Economics. 39(4). 1963, 387-96.

LaFrance. J. T.. and W, M, Hanemann. "The DualStrueture of Incomplete Demand Systems." AmericanJournal of Agricultural Economics, 71(2). 1989.262-74.

Larson. D. M.. and S. L. Shaikh, "Empirical SpeeificationRequirements forTwo-Conslraint Modelsof Recrea-tion Demand." American Journal of AgriculturalEconomics. 83(2). 2001, 428-40.

MeConnell, K. E. "Some Problems in Estimating theDemand for Outdoor Recreation." American Journalof Agrkvltural Economics. 57(2). 1975. 330-34.

MeConnell. K. E,. and I, E, Strand, "Measuring the Costof Time in Recreation Demand Analysis: AnApplication to Sportfisbing." American Journal ofAgricultural Economies, biO). 1981. 153-56.

Morey. E. R. "The Choice of Ski Areas: Estimation ofa Generalized CES Preference Ordering withCharacteristics." Review of Economics and Statistics,66(4), 1984. 584-90,

Smith, T, P. "A Comparative Statics Analysis of the TwoConstraint Case." appendix 4.1 in Measuring theBenefits of Water Quality Improvements UsingRecreation Demand Models, edited by N. E. Boek-stael, W. M. Hanemann, and I. E. Strand. Reportto the Environmental Protection Agency. Washing-ton D.C.. 1986.

Smith. V,K..W.H. Desvousges.and M. P. McGivney, "TheOpportunityCostofTravelTimein Recreation DemandMotids." Land Economies. 590). [9HX 259-n.

Stone, J. R. N. "Linear Expenditure Systems and DemandAnalysis: An Application to the Pattern of BritishPost-War Demand." Economic Journal, 64(3). 1954.511-27.

recreation demand choices and revealed ...home.uchicago.edu/sabina/economic inquiry.pdfleisure time...

Documents