Functional Forms in Discrete/Continuous Choice Models With General Corner Solution

Transcription

1 Functional Fors in Discrete/Continuous Choice Models With General Corner Solution Felipe Vásquez Lavín Departent of Agricultural and Resource Econoics, University of California, Bereley Michael Haneann Departent of Agricultural and Resource Econoics, University of California, Bereley Deceber 30, 2008 Abstract In this paper we present a new utility odel that serves as the basis for odeling discrete/continuous consuer choices with a general corner solution. The new odel involves a ore exible representation of preferences than what has been used in the previous literature and, unlie ost of this literature, it is not additively separable. This functional for can handle richer substitution patterns such as copleentarity as well as substitution aong goods. We focus in part on the Quadratic Box-Cox utility function and exaine its properties fro both theoretical and epirical perspectives. We identify the signi cance of the various paraeters of the utility function, and deonstrate an estiation strategy that can be applied to deand systes involving both a sall and large nuber of coodities. Introduction Consuer behavior generally involves two types of decisions. On one hand, people decide which goods to purchase or not purchase (the goods actually purchased are typically only a sall subset of all goods available in the aret); on the other, people decide what quantity (how any units) to purchase of the coodities they have chosen to acquire. In principle, researchers would lie to explain these two decisions using a uni ed utility odel. Researchers explain the rst decision using a discrete choice odel such as the ultinoial logit, the nested logit odel (McFadden, 979), and recently the ixed logit odel (Train, 998,

2 2003). For the second decision, one can use a set of deand equations with either a continuous or discrete distribution depending on how uch one is concerned about the integer nature of the quantities consued. Soe attepts to lined these two decisions include the applications by Bocstael et al. (987), Hausann et al. (995), Parson and Kealy (995) and Feather et al. (995). They use a two-step estiation procedure that in the rst step estiates a repeated logit odel and steing fro this step, proxy variables for price and quality of the good are built and incorporated as exogenous variables in the estiation of a deand function that explains the total deand. Unfortunately, their procedure fails to integrate both decisions in a uni ed odel of consuer utility axiization. A uni ed odel of consuer choice should account for other iportant characteristics of consuer behavior. First, consuer deand is a ected by other attributes of the goods besides prices and incoe. We use the generic ter of quality to represent these other atributes. Mareting applications have shown that brand selection and the quantity purchased of goods such as coputers or soft drins depend on the objective and subjective attributes of the di erent brands (Hendell, 999; Chan, 2002; Dubé, 2004). In the context of outdoor recreation, environental quality at di erent sites is an iportant variable explaining site visitation behavior (Herriges and Kling, 999; Phaneuf and Sith, 2004). Quality has been incorporated in deand odels through a transforation of either the deand functions or the utility function (Bocstael et al., 984). However, incorporating quality into the deand functions directly is not always recoended, especially with large deand systes, because the integrability of these systes is coplex and there ay soeties not be a closed for solution for the iplied indirect utility function. By contrast, the transforation of the utility function provides a clear relationship between preferences and the resulting deand functions with quality attributes. A second and very iportant feature of individual consuer choice behavior is the prevalence of corner solutions, wherein consuers are observed not to purchase any quantity of certain coodities. This stands in contrast to an interior solution, where the consuer consues soe positive quantity of every available coodity. It turns out that a corner solution has a relatively siple analytical structure if the consuer purchases a positive quantity of at ost one or two coodities (including the nueraire good); Haneann (984) referred to this as an extree corner solution. In a general corner solution, however, that condition does not There are at least three ways to incorporate quality into utility functions. First, each good in the utility function can be ultiplied by its own quality index (scaling). Second, the quality index could be added to each good of the utility function (translating). Finally, the product of the quality index and the quantity can be incorporated into the utility function through the nueraire good (cross repacaging approach). All of these alternatives di er in ters of the iplicit relationship between welfare and quality, deand and quality and epirical tractability. A coprehensive analysis of the can be found in Haneann, (982; 984) and Bocstael et al. (984). 2

3 hold: the consuer purchases positive quantities of ore than two coodities but not of all coodities. Analyzing deand functions for general corner solutions turns out to be rather coplex. Since the wor by Haneann (978, 984), Wales and Woodland (983) and Bocstael, Haneann and Strand (984), the approach used to forulate a lielihood function for axiu lielihood estiation of a deand syste with a general corner solution has been based on the Kuhn-Tucer (KT) conditions for the solution to the consuer s utility axiization proble. The KT approach starts with the forulation of a utility function whose arguents include the level of consuption of each coodity over a period of tie and also the quality or other attributes of soe or all of the coodities, possibly represented in the for of a sub function that serves as overall index of quality for each relevant coodity. The axiization of this utility function subject to the budget and nonnegativity constraints generates the rst order conditions (Karush-Kuhn-Tucer conditions) governing whether a positive or zero quantity of each good is consued and, if the forer, how large a quantity. Adding a rando ter to the utility function aes it possible to generate probability stateents for the consuption bundle observed vector which serve as the building blocs for axiu lielihood estiation of the paraeters of the utility function. Any stochastic speci cation can be used for the error ter. However, the choice of this speci cation deterines the tractability of the lielihood function and therefore the nuber of coodities that can be handled in the estiation. For instance, using a ultivariate noral distribution for the error ter aes it hard to apply the odel to a deand syste with ore than a sall nuber of coodities because of the di culty of evaluating high-diensional ultivariate noral probability integrals; the integrals have to be calculated nuerically aing the estiation process very slow. In contrast, using an extree value distribution for the error ter produces a siple closed for solution for the probabilities in the lielihood function. An additional coplication arises fro the fact that one often wants to use the estiated utility odel to predict coodity deands under di erent scenarios (price-attribute cobinations) fro those observed in the data. Moreover, one ay wish to calculate welfare easures Hicsian copensating or equivalent variations for changes in prices or attributes. The coplication here is that, in both of these cases, one need to predict the new general corner solution that will be chosen with the new cobination of prices and attributes. This is coplex because, if there are M coodities (including the nueraire) there are 2 M alternative possible solutions to the consuer s utility axiization (including an interior solution and all possible corner solutions). Evaluating all of these alternatives in a brute-force deterination of the utility-axiizing optiu is coputationally burdensoe for large M. 2 2 Lee and Pitt (986;987) subsequently extended this approach to Kuhn-Tucer lie conditions that apply to price-derivatives of the indirect utility function. But this does not reduce the diensionality of the proble, 3

4 In the last few years there has been renewed interest in the KT odel triggered by three ajor developents. First, it has proved feasible to apply the KT approach to deand functions with a signi cant nuber of coodities based on a clever use of the extree value distribution (which provides a closed for representation of the lielihood function) and faster coputers that perit estiation in a reasonable tie (Von Haefen et al. 2004; Bhat, 2007). Second, the developent of siulation techniques allows researchers to avoid the probles associated with nuerical evaluation when integrals in the lielihood function lac a closed for representation. Any lielihood function lacing a closed for solution can in principle be approxiated using siulation. Furtherore, siulation introduces greater exibility in the rando structure of the odel and facilitates the calculation of welfare easures. This exibility can be exploited when researchers have icro level data since the rando paraeter odels can be used to identify individual heterogeneity and to capture several patterns of substitution (see for exaple, Train 2003; Revelt and Train, 999). This is also true in the context of Bayesian estiation of the discrete/continuous odel such as Ki et al. (2002). Finally, the use of an additively separable utility function has provided a siple way to calculate welfare easures without requiring an explicit coparison of all possible solutions to the utility axiization proble. Von Haefen and Phaneuf (2004) and Von Haefen et al. (2004) developed a ethodology to estiate welfare easures in a KT fraewor with a large deand syste and an additively separable utility function. This approach has draatically increased the nuber of alternatives that can be handled with this fraewor fro 3 or 4 (Wales et al., 978; Lee and Pitt, 986; Phaneuf et al., 2000) to 2 or 4 (Von Haefen et al. 2004), and even ore than 50 (Von Hae en et al., 2004; Mohn and Haneann, 2005). Bhat (2007) shows that the KT odel with an additively separable utility function and an extree value distribution for the error ter is the natural extension of the traditional discrete choice odel suggested by McFadden alost four decades ago. Therefore, the rando paraeter KT odel is the natural extension of the ixed logit odel (or rando paraeter logit odel) suggested aong others by Train and McFadden (998), Train (998), and Train (2003). An additively separable utility function is unfortunately a restrictive functional for since it reduces the exibility of the utility function in ters of substitution patterns. Nevertheless, the di culty of nding an appropriate lielihood function and of calculating welfare easures with a nonadditively separable utility function has ept researcher fro eploying ore general odels. In this paper we explore the estiation of a generalized KT odel with a nonadditively separable utility function. Depending on the value of the paraeters, this utility function can tae alost any particular functional for used in the literature such as the translog, linear, both in estiating the lielihood function and in predicting deands or calculating welfare easures for new price-attribute cobinations. 4

5 and quadratic utility functions. We discuss estiation strategies and welfare calculation for both sall and large deand systes. In our application we estiate both additively and nonadditively separable utility functions and copare welfare easures derived fro the. The next section develops the general KT odel and presents a discussion of functional fors used in the literature with ephasis on Bhat s suggestion. Section 3 shows the generalization to a nonadditively separable utility function, its stochastic forulation, and the lielihood function. Section 4 presents an application of the odel to a large deand syste. 2 Kuhn-Tucer fraewor In this odel consuers have a continuously di erentiable, strictly increasing, and strictly quasiconcave utility function (Haneann 978, 984; Wales and Woodland, 984; Von Haefen et al., 2004) denoted by U = U(x; q;z; ; "); where x is a M-diensional vector of consuption levels, q is a Mx atrix of attributes for the vector of coodities and z is a Hicsian good. is a vector of paraeters and " is a vector of unobserved coponents. Given a vector of prices (p) for the coodities and a level of incoe (y) for the individual, the utility axiization proble is The rst order Kuhn-Tucer conditions are ax x; z U(x; q;z; ; ") s.t. p0 x + z = y; x 0: p = ; p = 0 = ; :::; M x 0: Assuing an additive error ter in these equations, they can be rewritten as " g (x; q; p; ) (2) x (" g (x; q; p; )) = 0 (3) x 0 j = ; :::; M; where g (:) is a function that contains only the deterinistic coponent of the FOC. Let ^x n represent the observed cobination of zero and positive consuption of goods for individual n; that is ^x n = (x ; :::x ; 0; :::; 0) ; then the probability of observing an individual consuing 5

6 only the rst goods is f (^x n ) = Z gm Z g+ ::: f " (g ; :::g ; " + ; :::; " M ) jj j d" + :::d" M ; (4) where f " (:) is the joint density function of the error ters and J is the Jacobian of the transforation. For the goods that are consued we now by equation (3) that " = g (x; q; p; ) ; however for the rest of the goods that are not consued we only now that " g (x; q; p; ) by equation (2). Given a distribution function for the error ter and a functional for for the utility function, it is possible in principle to construct the lielihood function. However, whether or not the lielihood function is coputationally tractable, depends on how we represent the interaction aong coodities in the utility function. This interaction is odeled through the choice of a functional for or an error structure. The siplest case uses a functional for and an error structure that does not allow any interaction i.e., substitution aong the alternatives coodities. Using a distribution function that allows dependence aong error ters creates a proble for the lielihood function because for any distributions there is not a closed for solution for the integrals in equation (4) and this severely liits the nuber of goods that can be handled in the odel. Siilarly, ore sophisticated functional fors increase the di culties in the de nition of the lielihood function and the estiation process when there are any coodities. For instance, if we assue the error ters have a joint noral distribution with zero ean and not a diagonal covariance atrix, then the integrals in the expression f (^x n ) (equation 4) do not have a closed for and have to be calculated nuerically. Other error structures, such as a GEV distribution aes the calculation of the density function coplex as the nuber of alternatives increases. These facts see to explain why the literature has shown cobinations of a few nuber of goods with a ultivariate noral distribution or a large nuber of goods but with an extree value distribution. For exaple, in this last case the probability of observing an individual consuing only the rst goods is given by f (^x n ) = jj j MY e ( g (x;q;p;)=) =u Ix >0 e ( e ( g (x;q;p;)=) ) ; whose closed for solution enables us to estiate the odel with a large nuber of goods. Previous applications of the KT odel used a noral distribution and a quadratic utility function but with only 4 goods (Woodland et al., 983). The odel has been also estiated using a generalized extree value distribution, an additively separable utility function and four goods by Phaneuf et al. (999). More recently, there have been soe applications using an extree 6

7 value distribution, an additively separable utility function and a larger nuber of goods (Von Haefen et al., 2004; Von Haefen and Phaneuf, 2004; Mohn and Haneann, 2005). Calculation of welfare easures requires solving a constrained optiization proble. This fact has constrained the application of KT odels with large deand systes to utility functions that are additively separable. The copensating variation (CV) for a change fro (p 0 ; q 0 ) to (p ; q ) is calculated iplicitly using the indirect utility function as V (p ; q ; y CV; ; ") = V (p 0 ; q 0 ; y; ; "): (5) This iplicit de nition of CV does not have a closed for solution, therefore researchers use a atheatical algorith to nd the value of CV that satis es the equality condition. For the new vector (p ; q ) we need to nd the optial consuption pattern that axiizes utility. For a sall nuber of alternatives Phaneuf et al. (2000) copare all the 2 M possible consuptions patterns and nd the alternative that provides the higher level of utility. Given this consuption pattern and the new utility level they use a nuerical bisection routine to nd the value of CV in (5). Both procedures perfor well if the diension of the proble is sall, however for large deand syste the coparison of all consuption patterns is not possible and the bisection routine could also be very slow in nd the value of CV. Von Haefen, Phaneuf and Parson (2004) suggest an algorith to nd the optial consuption pattern using the properties of the additively separable utility function and the nonnegativity condition for the nueraire good. The level of z is the only inforation needed to deterine the consuption of all other goods. After the optial quantities have been found, they also use the nuerical bisection approach to nd the CV. Von Haefen (2004) suggests a ore e cient approach to nd the CV which uses the expenditure function instead of the indirect utility function. In this case the optial quantities are obtained with a siilar routine as in Von Haefen et al. but nding the optial quantities that iniize the total expenditure. The welfare easure is calculated as CV = y e(p ; q ; U 0 ; ; "); (6) where the initial level of utility U 0 is obtained evaluating the utility function at the initial values of the explanatory variables using the paraeters given in the estiation process. 2. Previous functional fors in the KT odel One of the ost coonly used functional fors for the utility function is U(x; Q; z; ; ") = j ln ( x + ) + z ; 7

8 with ln = 0 s+" ; ln = 0 q ; = exp( ); ln = and ln = : x is the level of consuption of good ; s is a vector of individual characteristics, q is a vector of attributes of the good and z = y p 0 x: " is an extree value error ter with scale paraeter and is a translating paraeter. For this functional for the KT conditions are given p ( x + ) y p 0 x pj = 0 0 s+ ln ( p ) + ln ( x + ) + ( ) ln y p 0 x = g : This functional for is used by Von Haefen et al. (2004) and Herriges et al. (999). A slightly di erent utility function has been suggested by Phaneuf et al. (2000) and Mohn and Haneann (2005), given by U(x; Q; z; ; ") = ln (x + ) + ln z and whose FOC are " ln (p ) + ln (x + ) ln y p 0 x 0 s: Bhat generalizes these functional fors using a Box-Cox transforation of the quantities consued in the odel, the utility function is U = = x + where the Box-Cox transforation is applied over x = (x = + ) ; i.e., we have that (x )( ) = ((x ) ) = : ; and are paraeters to be estiated. Consistency conditions require 0 and : Unlie previous functional fors there is not an outside good in this forulation. This odel allows corner solutions given the presence of the paraeter : If = the odel reduces to the extree corner solution discussed by Haneann (984) and Chiang and Lee (992) with utility function equal to U = P M = x : When = 0 the utility function is U = P M = ln x + which is siilar to functional fors described above if we de ne =, = and =. 8

9 Bhat suggests the following interpretation for the paraeters; is the baseline arginal = x +, therefore, if x = = : The paraeters and are satiation paraeters where the forer shifts the position of the point at which the indi erence curve hits the positive orthant allowing corner solutions while the latter de nes the satiation level of a good. As in the other cases the stochastic part of the odel is included in the de nition of (s ; " ) = (s )e " = exp( 0 s + " ). Following Bhat s the axiization proble is ax x U = = exp( 0 x s + " ) + subject to whose FOC are given by p x = e = E = = exp( 0 s + " ) p e + p 0 this is satis ed with equality when e > 0 and with inequality when e = 0: Individuals are constrained to consue at least one good, therefore the arginal utility of incoe can be de ned as = exp(0 s +" ) e p p + using good without loss of generality. Replacing it into the FOC we obtain exp( 0 s + " ) p this can be reduced to e + p exp( 0 z + " ) p e + 0; (7) p where V + " V + " V = 0 e s + " ln p + ( ) ln + p With this set up the individual s contribution to the lielihood function is with g i = V l i = jj j R g MR " = " M = g + R " + = V i + " : The eleents of the Jacobian are f (" ; g 2 ; :::; g ; " + ; :::; " M ) d" + :::d" M d" J ih [V V i + " ; i; h = 2; :::; : h 9

10 Additionally, and cannot be identi ed separately so we have to noralize the odel with respect to one of these paraeters, then V is either V = 0 s + " ln p + ( ) ln e p + or 0 s + " ln p ln e p + : Using an extree value distribution for the error ters, the individual s contribution to the lielihood becoes l i = jjj " R= KQ " = i=2 V V i + " M Q s=k+ V V s + " " d" where denotes the standard extree value density and denotes the cuulative distribution. The Jacobian reduces to KQ jjj = c i i= K X c i= i! with c i = i e i + ip i and after solving the integral in the lielihood function Bhat s solution is l i = KQ K c i i= K X c i= i! Q K e v i= i= P M = ev = (K )! This elegant forulation collapses to the siple conditional logit odel forulation when K=, e i.e., l i = v i P = K : = ev = 3 A non-additively separable utility function In this section we present a Quadratic Box-Cox functional for, which is the natural extension of the additively separable linear Box-Cox functional for developed by Bhat (2007). We present this non-additively separable utility function, discuss the interpretation of the paraeters in the odel, and copare our ndings with Bhat s results. The utility function is U = = + 2 = = x + x x + + ; Again, x is the quantity of good and ; and are paraeters to be estiated in the odel with the sae assuptions as before, i.e., 0 and : This utility function includes as particular cases ost of the other utility functions used in 0

11 the KT approach. For instance, if = 0 for all ; this utility function becoes the linear Box-Cox utility function, with the properties discussed above. If! 0 the utility becoes the translog function U = KX = x ln KX = = x x ln + ln + ; and if = we obtain the quadratic utility function used by Wales and Woodland (983). U = KX = x + 2 KX x x = = Now we analyze the role of the paraeters ; ; and in the odel using this utility function. 3. Interpretation of paraeters 3.. Marginal utility: Taing a derivative with respect to x produces the arginal utility of consuption at x = 0 we x = + = x + x = + x =0 x + then the arginal utility depends on the coe cients ;,, and consuption of the other goods. In this case and the level 2 U + x + ; 6= and the e ect depends on the sign of : Siilarly to Bhat s linear Box-Cox additively separable utility function the contribution to the utility is zero when x = 0, since x + = 0; then all the interactions including x are eliinated. But, unlie his forulation, the arginal utility in this general utility function includes the e ect of consuption of other goods on the desirability of x ; for instance, if the good already consued is copleentary with x ; ( > 0) then it is ore liely that the x will be consued. is the baseline utility only if

12 all other goods are not consued, in other words represents the desirability of x before any consuption decision has been ade Corner solution and satiation paraeter: : For our analysis we consider only two goods and assue the following values for the paraeters = 2 = ; = 2 = 0:5; 2 =, = 22 = 0:0; 2 = 0:00: The indi erence curves for three di erent values of gaa are given in gure..5 Indiference curves for different values of Gaa. gaa =.25 Consuption good gaa = gaa= Consuption of good Figure As in the linear Box-Cox additively separable functional for, shifts the position of the intersection between the indi erence curve and the x-axis, allowing for corner solutions since the budget line could be tangent to the indi erence curve at the point where this curve intersects the axis. The indi erence curves becoe steeper as the value of increases which can be interpreted as a stronger preference for good. consuption of good when! 0 is given by x U = ln ln x + x ln x2 ln + 2 The contribution to the utility function of x2 ln + 2 2

13 since li!0 x x + = ln + Figure 2 plots the utility contribution for values of equal to, 5, and Effect of different values of gaa on the utility contribution of consuption 50 gaa =00 40 gaa =0 Utility gaa = Consuption of good Figure 2 According to gure 2, the paraeter also de nes the satiation points in the odel since it deterines the arginal utility of consuption and the level where we have a negative contribution to utility Satiation paraeter: Again, interpretation of this paraeter is siilar to the linear Box-Cox results. deterines how fast the arginal utility decreases with increasing consuption of good : for Exaple if = 8 there is no satiation, the utility function is U = = x + 2 = = x x With Bhat s utility function only the rst ter in the equation exists. Therefore, unlie the siple case we do not have perfect substitute goods since there are soe substitution given by the second ter of the equation. Figure 3 shows that the higher the alpha the steeper the slope of the function representing the utility contribution of consuption. In other words, a higher alpha iplies slower satiation in consuption. 3

14 8 Effect of different values of alfa on utility contribution of consuption 7 alfa= Utility 4 alfa= alfa= Consuption of good Figure Lielihood function With the general utility function the calculation of the Jacobian and the construction of the lielihood function becoe ore coplicated. The axiization proble is ax U = = + 2 = = x + x x + + subject to p x = e = E = = For convenience ost of the literature has de ned the error ter inside either by a ultiplicative approach as = e 0 s e " (Bhat, 2007; Von Haefen et al. 2004; Haneann, 984) or an additive approach = 0 s + " (Wales et al. 983; Haneann, 984) and the assuptions on " are either a extree value distribution or a noral distribution. As we showed above in the siple case the assuption = e 0 s e " is convenient since the FOC are reduced to an expression easily sipli ed by applying the logarith operator (see equation 7) and using it directly in the lielihood function. However, in our new forulation this ultiplicative assuption does not help us siplify the odel. 4

15 There are two solutions for this proble. First, we could follow the approach by Wales and Woodland and de ne = 0 s + " : Second, we could eep the de nition of = e 0 s e " but de ne an outside good which will siplify the calculation of the FOC. We show details of each alternative below Linear stochastic structure In the rst case the FOC can be expressed as 0 0 s + " p 0 s + " p e + + p 2 e + p = 2 = p p e + p e + p e + p e + p with equality if e > 0 and inequality if e = 0: The last coponent in this expression is the arginal utility of incoe : Lets de ne a = e p + and a = e p + then the equation is 0 " p a " p a + 0 z p a + 2p a a 2p = = e p + 0 " p a " p a + V V = " p p a a " + p a V e 0 z + a p p p a V this area is given by p " < p V V + p a " a a p a " < a V a V + a a " = V V + " Z V V +" e " e e " d" = e e (V The lielihood function follows a siilar patterns as before, l i = jj j " R= KQ " = i=2 V V i + " M Q s=k+ V +" ) V V s + " " d" Unfortunately there is no siple solution for the Jacobian that can be generalized to any con- 5

16 suption pattern. Furtherore, the integral in the lielihood function does not have a closed for solution. Nevertheless, the integral can be calculated nuerically or by siulation and, since there is only a single integral in the lielihood function, this is not burdensoe. Because it is possible to include rando paraeters in the odel the additional e ort to calculate this integral is not signi cant. For exaple, if we include a rando paraeter structure such that n = b + ; where b is the ean e ect and a deviation with respect to this ean, then the lielihood function is Z L() = " R= KQ " = i=2 e (V V i +" ) e e (V which can be calculated using siulation. V i +" ) MQ e e (V s=k+ V s +" ) e " e e " d" f()d Outside good The alternative way to overcoe the di culty in the de nition of the lielihood function is to assue the existence of an outside good which does not have an error ter. In that case the utility function will be U = U + z Where U is the Quadratic Box-Cox utility function. With this assuption the FOC are sipler since we can tae advantage of the FOC for z to de ne the arginal utility of = z = 0 ) z = and we can replace this value in the FOC for other goods. e 0 exp(0 s + " ) + p p + MP e 2 + p =! e + p z and " 0 s + ln p 2 e X M! e + p + + p z = p ; which has the sae lielihood function as the siple case except we do not have any integrals in its de nition because the outside good does not have an error ter. In the siple case of the linear Box-Cox utility function, Bhat gives two arguents against the outside good forulation. 6

17 First, it is arbitrary to assue one good does not have an error ter while all the others are stochastic variables. Second, in his siple linear forulation the lielihood function does not reduce to the siple discrete choice odel when J =. Fro our perspective these arguents ay be less iportant than the advantage of having a siple lielihood function which facilitates the estiation of this ore coplex odel. 3.3 Welfare easures With this new utility function we face an additional proble in calculating welfare easures. In this case it is no longer possible to use Von Haefen et al. (2004) algorith to obtain the optial quantities after a change in the vector (p; q) : The optial quantities depend on all other goods since fro the F OC we cannot separate z fro the vector of quantities x. We found that the use of a constrained optiization routine solve this proble. Using this routine we iniize the total expenditure needed to reach the initial level of utility U 0 at the new price and quality conditions. To understand this proble we repeat the relevant part of Von Haefen et al. algorith for our estiation and explain where we use the constrained optiization ethod.. In the rst step Von Haefen et al. siulate the unobserved heterogeneity of the odel that coes fro the rando paraeter assuption and fro the traditional error coponent. The process starts taing rando draws fro the distribution of t, and accepting or rejecting these siulated paraeters using a Metropolis-Hasting algorith. Then it siulates the error ters conditional on the value of the paraeters t given in the rst step: Since we have not used rando paraeters yet no siulation is needed for t ; (this can be easily incorporated in the odel later), so we only need a siulation for the error ters " t fro the conditional distribution f(" t j t ; x t ): Following von Haefen et al., for each eleent of " t we set " tj = g tj (:) if the j-th good is consued and " tj = ln( ln(exp( exp( g jt (:))) tj )) where tj is a unifor rando draw if the j-th good is not consued. 2. For each set of the siulated heterogeneity given in () we need to copute the iniu level of expenditure e(p ; q ; U 0 ; ; ") required to reach the initial utility level U 0 : Von Haefen et al. nd the optial quantities that iniize the total expenditure using the following subroutine: (a) at iteration i the level of z is set at za i = z i l + zu i =2; where z 0 l = 0 and zu 0 = y: (b) with this suggested value for z solve the F OC derived fro an expenditure iniization given j(x j j p j ; x j 0 8 j and obtain the new quantities x i : 7

18 (c) given results in (b) nd the new value for z equal to z i = y P p x (d) if z i > z i a set z i l = z i a and z i u = z i u : And if z i < z i a set z i l = zi l and z i u = z i a: (e) iterate until abs(z i l z i u) < c for c arbitrarily sall. 3. The total expenditure given at the last iteration of this subroutine is replaced in equation (6) to obtain the welfare easure. Finally the average of the siulated welfare easures is used as an estiate of the E(CV ). The second part of this algorith is only appropriate if the utility function is additively separable. In our case the F OC conditions do not allow us to separate the de nitions of z fro the other goods. To nd the optial bundle of goods we use a constrained optiization routine that nds the values of x and z that iniize the total expenditure subject to the nonnegativity constraints and the initial level of utility. With the value of x and z we calculate the new level of expenditure needed to reach U 0 and replace it into equation (6) to obtain the corresponding welfare easure. 4 Application Our application uses a data containing a panel of 063 Alasan anglers who too 685 shing trips in the suer of 986. We have inforation on the nuber of trips taen by each angler in each of the 22 wees of the season and on the characteristics of each of these trips including date, duration, destination zone and type of sh targeted. For each individual n; we have inforation on his travel cost (T C nj ) to each of the 29 sites included in the saple. This T C was calculated as the round trip travel cost fro the origin zone to the corresponding site i. We also have inforation on the quality of shing (Q jt ) at each site on each decision occasion. This is based on detailed sport shing advisories published each wee by the Alasa Departent of Fish & Gae (ADFG) describing shing conditions at sites around the state. They de ne the shing in qualitative ters using adjectives and descriptors, rather than predicting a speci c catch rate per hour of e ort. Accordingly, we view this as an ordinal easure of shing quality. Based on the advisories, our variable is coded as an eight-level indicator of shing quality, starting fro ( no sh are available ) to 8 ( excellent shing ). The sh advisories are speci c to di erent types of sh. These species are classi ed into 3 acrospecies (Salon, Saltwater, Freshwater) and a no target alternative. For each acrospecies there are a nuber of subspecies. In the acrospecies salon there are four subspecies; King, Red, Silver and Pin. Freshwater has ve subspecies; Rainbow Trout, Dolly Varden, Lae Trout, Grayling, and others. Saltwater has only three subspecies, Halibut, Razor Clas, and others. 8

19 For each subspecies there are several sites at which shing is available. Our de nition of a "site" therefore is a cobination of acrospecies/subspecies/site. There are 8 potential goods, an individual chooses to visit a subset of these cobinations. Although there are 29 destination sites in the saple not all of the are available for each subspecies. The axiu nuber of sites available for each subspecies is presented in table in the appendix. Two other variables describing the sites include a easureent of harvest output in previous year (LogHarv85 ) and the variable Developed which taes the value when there is soe degree of developent lie boats and tourist facilities in site i and 0 otherwise. There are nine variables describing individuals. These variables are Sill, Leisure, Avlong, Own, Site focus, boatown, trophy, release, cabin and crowding. Sill is an index of the individual s experience in sport shing which ranges fro for a novice to 4 for an expert angler. Leisure easures the aount of leisure tie available for an angler. Avlong is the average length of all shing trips taen by the individual in Alasa over the suer. Own is a duy variable that taes the value if the individual owns a cabin, a boat or an RV and 0 otherwise. Site focus is a duy variable taing the value if the choice of a site was ore iportant to an individual than the choice of a target species. Boatown taes the value if the individual owns a boat and 0 otherwise. Trophy taes the value if the individual prefers trophy sport shing and 0 otherwise. Release taes the value if the individual practices catch and release shing and the value 0 otherwise. Cabin is a duy variable that taes the value when the individual owns a cabin at site i and 0 otherwise. Crowding is a easure of subjective crowding conditions at site i and it was coputed as the product of an individual s crowding tolerance (positive if the individual lies crowded places and negative otherwise) and a easure of crowding conditions at the site i, that ranges fro 0 for not crowded to 2 for very crowded. This is based on inforation provided by ADFG. Note crowding and cabin varies for both the individual and the site. Given the size of the choice set, we estiate the both the additively separable utility function and a translog utility function including an outside good. The translog utility function is U(x; Q; z; ; ") = j j ln j x j i ij ln ( i x i + ) ln j x j + + ln (z) ; j with ln j = 0 s+" j ; ln j = 0 q j ; ln = and ln = ; the vector s include the variables varying across individuals while q includes the quality variables 9

20 varying across sites. The interaction paraeters ij can be represented in the atrix B = M M M2 MM which is a syetric and negative sei de nite atrix of paraeters. The KT conditions are given by j j x j + j + i j ij j x j + ln ( ix i + )! " j = ln p j j x j + j y p 0 x X M ij ln ( i x i + ) whose Jacobian j i = p j j A j i ; p j (y p 0 x) = 0 ln j ln y p 0 x 0 s + P M! jp j i ij ln ( i x i + ) jj 2 j (y p 0 x) A j j x j + + p j A j y p 0 ; x P j p M! i i ij ln ( i x i + ) j i (y p 0 x) ji + p i A j ( i x i + ) A j y p 0 ; x and A j = p j j x j + j (y p 0 x) P M i ij ln ( i x i + ) : To assure the negativity and syetry conditions for the atrix B we ae the following adjustent (Wiley et al., 973) B = CC 0! ; where 2 C = 6 4 a 0 0 a 2 a a + a 2 + ::: + a (M ) a 22 + a 23 + ::: + a 2(M ) and we de ne the paraeters a ij as a function of observable variables (D ij ) coon to alternatives i and j; that is, a ij = 0 D ij ; In this application we use a duy variable to indicate the two alternatives belong to the sae sh subspecies. This is intended to reduce the nuber of paraeters that we need to estiate when we have a large deand syste, but it is not necessary with a sall deand 20

21 syste. 4. Data and Results Table 2 present the results of the KT odel with the additively separable utility function while table 3 presents results of the translog utility function. All the covariates varying aong individuals are included in the set s and enter into the utility function through = exp 0 s + ". Results show that ve out of nine paraeters are signi cant in the rst equation. The other four coe cients on own, site focus, release and crowding are not signi cant. Crowding is at the edge of being signi cant. These paraeters suggest an increase in the index of quality or in the level of harvest aes that site ore attractive to anglers. Both coe cients are signi cant. In the translog utility function the qualitative results are siilar to the previous case, except that now release becoes signi cant and Harvest is not signi cant and has an unexpected sign. The 3 duy variables (D to D3) represent the 2 di erent subspecies and the no target alternative. In other words, D J = if the two goods belong to the sae sh group and 0 otherwise. All these coe cients are signi cant. Other variables describing siultaneously the two goods under consideration could also be incorporated, for exaple we could use another duy variable to indicate the two goods belong to the sae acrospecies. Finally, Table 4 presents welfare easures for several scenarios. For exaple, closing site (Gulana River) for all salon species (King and Red in this case), for all freshwater species (Rainbow Trout and Grayling) and for all subspecies. Additionally we include two scenarios for closing one of the ost iportant shing sites in Alasa; the Kenai River. The rst scenario closes the Kenai River for all species (the four types of salon in this case) and the second scenario closes the river only for ing salon. The welfare loss of closing the Kenai River for ing salon is signi cantly greater than closing the Gulana River for this subspecies. This shows the relevance of the Kenai River in the recreational shing activities in Alasa due to better shing qualities of the river. The translog forulation produces saller welfare easures than the welfare easures of the traditional utility function which can be explained by the substitution patterns allowed in the translog utility function. Even though the agnitudes of the di erence in welfare easures are not large, the results show that aing the e ort to estiate ore exible functional for could help us enrich the substitution patterns of the odel and to correctly asses welfare iplications of di erent policies. 2

22 5 Conclusions In this paper we analyze and estiate a ore exible utility function for the Kuhn-Tucer approach to ultiple/discrete continuous choice odels. Unlie previous literature our utility function is not additively separable and it can be applied to estiate paraeters of a large deand syste. We estiated this odel considering 80 coodities which, to our nowledge, is the largest aount of goods that have been used in this approach. We thin we have overcoe the proble of welfare calculation using a sequential quadratic prograing ethod to nd the optial quantities needed to perfor welfare analysis. The paper contributes to odel the ultiple discrete continuous choice odel, adding greater exibility to include richer substitution patterns. References [] Bhat, C. (2007) "The Multiple Discrete-Continuous Extree Value (MDCEV) Model: Role of Utility Function Paraeters, Identi cation Considerations, and Model Extension" Forthcoing Transportation Research. [2] Bocstael, N., M. Haneann and I. Strand (984). "Measuring the Bene ts of Water Quality Iproveents using Recreation Deand Models" US Environental Protection Agency Washington D.C [3] Bocstael, N., M. Haneann and C. Kling (987) "Estiating the Value of Water Quality Iproveents in a Recreational Deand Fraewor" Water Resources Research 23(5): [4] Chan, T. (2002) Estiating a Continuous Choice Model with an Application to Deand for Soft Drins, unpublished (R&R at Rand). [5] Chiang J. and L. Lee (992) Discrete Continuous Models of Consuer Deand with Binding Nonnegativity Constraints" Journal of Econoetrics 54: 79:93. [6] Dubé, Jean-Pierre (2004). Multiple Discreteness and Product Di erentiation: Strategy and Deand for Carbonated Soft Drins, Mareting Science 23():66-8. [7] Haneann, M. (978) A Methodological and Epirical Study of the Recreation Bene ts fro Water Quality Iproveents. Ph. D. Dissertation, Departent of Econoics, Harvard University. 22

23 [8] Haneann, M. (982) "Quality and Deand Analysis" In New Directions in Econoetric Modeling and Forecasting in US agriculture. Edited by Gordon C. Rausser. New Yor Elsevier/North Holland. p.p [9] Haneann, M. (984). "Discrete/Continuous Models of Consuer Deand" Econoetrica 52(3): [0] Hausan, J., G. Leonard and D. McFadden (995) " A utility Consistent, Cobined Discrete Choice and Count Data Model Assessing Recreational Use Losses Due to Natural Resource Daage" Journal of Public Econoics 56:-30. [] Hendel, Igal (999). Estiating Multiple-Discrete Choice Models: An Application to Coputerization Returns, Review of Econoic Studies, 66(999): [2] Herriges, J. and C. Kling (999). Valuing Recreation and the Environent. Edward Elgar. [3] Herriges, J, C. Kling and D. Phaneuf (999). "Corner Solutions Models of Recreation Deand: A Coparison of Copeting Fraewors: in Valuing Recreation and the Environent, Herriges J and C. Kling ed Edward Elgar. [4] Ki, J., G. Allenby and P. Rossi (2002) "Modeling Consuer Deand for Variety" Mareting Science 2(3): [5] Lee, L. and M. Pitt (986). "Microeconoetric Deand Syste with Binding Nonnegativity Constraint: The Dual Approach" Econoetrica 54(5): [6] Lee, L. and M. Pitt (987). "Microeconoetric Models of Rationating, Iperfect Marets and Nonegativity Constraints" Journal of Econoetrics 3:89-0. [7] McFadden, D. (978). "Modeling the Choice of Residential Location" In Spatial Interaction Theory and Planning Models ed. by A. Karlquist, L. Lundquist, F. Snicars and J.L. Weibull. Asterda: North Holland. [8] McFadden, D. (979). "Quantitative Methods for Analyzing Travel Behavior of Individuals: Soe Recent Developents." Behavioral Travel Modeling. David Hensher and P. Stopher, eds. pp London: Croo Hel,979. [9] Mohn, Craig and M. Haneann (2005) "Caught in a Corner Solution: Using the Kuhn- Tucer Conditions to Value Montana Sport shing" Woring paper, University of California. [20] Parson, G., P. and M. Kealy (995). "A Deand Theory for Nuber of Trips in a Rando Utility Model of Recreation" Journal of Environental Econoics and Manageent 29:

24 [2] Parson, G., P. Jaus and T. Toasi (999). "A Coparison of Welfare Estiates fro Four Models for Lining Seasonal Recreational Trips to Multinoial Logit Models of Site Choice" Journal of Environental Econoics and Manageent 38, [22] Phaneuf, D. and K. Sith (2004) "Recreation Deand Models" Prepared for Handboo of Environental Econoics. K. Mäler and J Vincent, Editors. [23] Phaneuf, D., C. Kling and J.Herriges (999). "Estiation and Welfare Calculations in a Generalized Corner Solution Model with an Application to Recreation Deand" Review of Econoics and Statistics 82(): [24] Revel, D. and K. Train (999). "Mixed Logit with Repeated Choices: Households Choices of Appliance E ciency Level" The Review of Econoics and Statistics 80(4): [25] Train, K. (998). "Recreation Deand Models with Taste Di erences Over People" Land Econoics 74(2): [26] Train K. (2003). Discrete Choice Model with Siulation Cabridge University Press. [27] Von Haefen, R. (2004) "Epirical Strategies for Incorporating Wee Copleentarity into Continuous Deand Syste Models" Woring paper. [28] Von Haefen, R. and D. Phaneuf (2004) "Kuhn Tucer Deand Syste Approaches to Non- Maret Valuation" prepared for Applications of Siulation Methods in Environental and Resource Econoics, Anna Alberini and Riccardo Scarpa, editors. [29] Von Haefen, R., D. Phaneuf and G. Parson (2004). "Estiation and Welfare Analysis with Large Deand Syste" Journal of Business & Econoic Statistics 22(2): [30] Wales T. and A. Woodland (983). "Estiation of Consuer Deand Systes With Binding Nonnegative Constraints" Journal of Econoetrics 2: [3] Wiley, D., W. Schidt and W. Brable (973) "Studies of a Class Covariance Structure Models" Journal of the Aerican Statistical Association 68:

25 Table : Maxiu nuber of sites available for each subspecies Species Nuber of sites Species Nuber of sites King 2 Lae Trout 7 Red 7 Arctic Grayling Silver 24 Other Freshwater 9 Pin Halibut 7 Rainbow Trout 9 Razor Cla 2 Dolly Varden 7 Other Saltwater 7 No target 29 total 8 Table 2. ML estiates of KT odel variable estiates s.e. Est./s.e. Scale µ Constante δ Sill δ Leisure δ Avlong δ own δ site focus δ boatown δ trophy δ release δ cabin δ Crowding δ theta θ quality γ harvest γ Devp γ rho ρ Mean log-lielihood

26 Table 3. ML estiates of Translog KT odel variable estiates s.e. Est./s.e. Scale µ Constante δ Sill δ Leisure δ Avlong δ own δ site focus δ boatown δ trophy δ release δ cabin δ Crowding δ theta θ quality γ harvest γ Devp γ D D D D D D D D D D D D D Mean log-lielihood Table 4. Mean WTP KT odel Separable utility function Translog ean Total Per choice occasion ean Total Per choice occasion Gulana river all Species Gulana river all Salon Gulana river King salon Gulana river Red salon Gulana river All FW Gulana River Rainbow trout Gulana River Grayling Kenai River all species Kenai River King