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Modeling Consumer Demand for a Large Set of Quality Differentiated Goods:

Estimation and Welfare Results from a Systems Approach

Roger H. von Haefen, U.S. Bureau of Labor Statistics

Daniel J. Phaneuf, North Carolina State University

George R. Parsons, University of Delaware

Working Paper 353

February 2002

 

The views expressed are those of the authors and do not necessarily reflect the policies of the U.S. Bureau of Labor Statistics or

the views of other staff members.
1

Modeling Consumer Demand for a Large Set of Quality

Differentiated Goods: Estimation and Welfare Results from a

Systems Approach*

Roger H. von Haefen

Bureau of Labor Statistics

Daniel J. Phaneuf

North Carolina State University

George R. Parsons

University of Delaware

COMMENTS WELCOME

January 15, 2002

*

We thank Kerry Smith, Matt Massey, and seminar participants at North Carolina State University and the

Environmental Protection Agency (EPA) for comments and suggestions. All remaining errors are our own. Phaneuf

acknowledges financial support from the EPA through contract #R826615010, and Parsons acknowledges financial

support from the National Oceanic and Atmospheric Administration (NOAA) through a Delaware Sea Grant. The

views expressed in this paper are the authors alone and in no way represent the views of the EPA, NOAA, or BLS.

Corresponding author: Daniel Phaneuf, Department of Agricultural and Resource Economics, NCSU, Box 8109,

Raleigh, NC 27695.
2

Modeling Consumer Demand for a Large Set of Quality Differentiated Goods: Estimation and

Welfare Results from a Systems Approach

Abstract

We develop an approach for estimating individual level preferences for a large set of quality

differentiated goods and for constructing Hicksian welfare measures for price and quality

changes within the demand system framework. Our approach uses a simulated maximum

likelihood estimation procedure for recovering estimates of the structural parameters and an

adaptive Metropolis-Hastings algorithm for constructing Hicksian consumer surplus estimates.

We illustrate our approach with a recreation data set consisting of day trips to 62 Mid-Atlantic

beaches.

JEL Classification: C15, D12, D61, Q26
3

1. Introduction

In this paper we develop a demand system approach for estimating preferences for an

arbitrarily large set of quality differentiated goods at the individual or household level. We apply

the model to an outdoor recreation data set consisting of day trips to 62 beaches in the MidAtlantic region and analyze the welfare effects of changes in beach characteristics and

availability. Interest in the value of beach recreation opportunities arises from policy makers’

need to assess the merits of beach nourishment programs and to measure the damages resulting

from acute environmental accidents that impact beach availability. The Hicksian consumer

surplus estimates reported in this paper address these issues and represent the first welfare

measures derived from a theoretically consistent demand system model that accounts for interior

and corner solutions and accommodates a large set of quality differentiated goods. 

Because of the computational difficulties associated with estimating and generating

welfare measures from demand system models, nearly all empirical strategies for modeling

consumer choice for many goods have relied on the discrete choice random utility model (RUM)

developed by McFadden [1974]. A large and growing body of empirical research has shown that

the discrete choice RUM framework is attractive for modeling extensive margin choices made on

single choice occasions, but how the framework can be modified or augmented to represent

consumer choices made over longer time horizons when realized demands are a mixture of

interior and corner solutions remains an unresolved modeling issue. At present, there are several

discrete choice RUM-based approaches for modeling consumer choices in these situations (e.g.,

Morey, Rowe, and Watson [1993] Hausman, Leonard, and McFadden [1995], Feather,

Hellerstein and Tomasi [1995], Parsons and Kealy [1995]), all of which have strengths and

weaknesses. Some common features of these RUM-based modeling strategies are the

assumptions that the time horizon of choice can be decomposed into separable choice occasions,
4

that the objects of choice on each choice occasion are quality-adjusted perfect substitutes, and,

with rare exception, that income effects are absent.1

 In part because of the restrictiveness of

these assumptions, Phaneuf, Kling, and Herriges [2000] (hereafter PKH) and Phaneuf [1999]

have recently suggested modeling consumer choice in these situations within a unified demand

system framework that consistently accounts for interior and corner solutions. Their empirical

applications, however, consider only a small number of quality differentiated goods, and thus the

relevance of the demand system framework for policy applications with many goods remained

uncertain.

We demonstrate in this paper that if preferences are additively separable the demand

system framework can be estimated and used to generate Hicksian welfare measures for

applications with many goods. Additive separability implies strong restrictions for consumer

behavior, but in our view these restrictions have close analogs to the assumptions embedded in

the discrete choice RUM models that have traditionally been used for this class of problems.

Moreover, the demand system framework has the advantage of combining the extensive and

intensive margins of consumer choice for all quality differentiated goods in a coordinated and

behaviorally consistent framework.

In addition to permitting the construction of welfare measures for a large set of quality

differentiated goods, our empirical models incorporate several innovations over existing demand

system recreation applications. Our specifications allow a subset of the parameters entering the

direct utility function to vary randomly across individuals in the population and employ a

simulated maximum likelihood estimation procedure. From an econometric perspective,

introducing random parameters is attractive because it facilitates a relatively flexible

specification for the unobserved heterogeneity without substantially expanding the number of

estimable parameters. In addition, our empirical application incorporates an approach to welfare
5

measurement suggested by von Haefen [forthcoming] that conditions on an individual’s

observed choice. In contrast to traditional approaches to welfare measurement from RUM

models (e.g. Small and Rosen [1981]), we construct welfare measures in this paper by simulating

the unobserved heterogeneity entering preferences such that our model predicts observed

behavior perfectly at baseline conditions. The structure of the model is then used to predict how

individuals respond to price, quality, and income changes. To implement our conditional

approach, we develop a sequential Monte Carlo procedure that employs an adaptive MetropolisHastings algorithm. Although conditioning on an individual’s observed choice adds more

complexity to our welfare calculation procedure, it substantially reduces the number of

simulations necessary to generate precise welfare estimates as well as the computational time

involved. 

Using an unusually rich data set consisting of 540 Delaware residents’ beach recreation

activities, our empirical application examines the demand for day trips to 62 ocean beaches in the

Mid-Atlantic region. Due to infrequent but acute oil and toxic spills, state officials in New

Jersey, Delaware, Maryland, and Virginia occasionally close beaches for health and safety

reasons. In addition, beach erosion caused by rising sea levels, development, and natural causes

has led state officials to initiate beach nourishment programs throughout the region. Since beach

recreators are among the individuals most impacted by beach closures and erosion, our welfare

scenarios can inform state officials of the potential economic losses arising from these impacts. 

The remaining structure of the paper is as follows. The next section gives a general

overview of the issues involved in demand system estimation and welfare calculation with large

choice sets. Section 3 follows with a discussion of the empirical specifications and estimation

strategies we employ in this paper, and Section 4 discusses our strategy for construction welfare

measures. Section 5 discusses the Mid-Atlantic beach recreation data set we use in our
6

application, and Section 6 summarizes our estimation results. Section 7 discusses our welfare

scenarios and results, and Section 8 concludes.

2. General Overview

In principle, there are two generic strategies for developing demand system models that

consistently account for both interior and corner solutions and can be applied to problems with

many goods. The first, referred to as the Kuhn-Tucker approach by Wales and Woodland

[1983], exploits the Kuhn-Tucker conditions that implicitly define the consumer’s optimal

consumption bundle. Alternatively, Lee and Pitt [1986] develop a demand system framework

that relies on the concepts of notional demand and virtual price functions (Neary and Roberts

[1980]). Although these approaches are dual, we focus on the Kuhn-Tucker framework in this

section and in our subsequent empirical work. Much of our general discussion that follows,

however, transfers to the dual approach in a straightforward manner. 

 As discussed in Wales in Woodland and PKH, the Kuhn-Tucker framework begins with

a specification of consumer preferences represented by a continuously differentiable, strictly

increasing, strictly quasi-concave direct utility function, U(x,Q, z,  ), where x is an Mdimension vector of consumption levels of the quality differentiated goods that are consumed in

non-negative quantities, Q denotes an M  K matrix of commodity specific quality attributes of

the goods in x (i.e., Q = 

T [ ... ] q1 qM where qk is a K 1 vector of attributes for site k), z is an

essential Hicksian composite commodity representing spending on all other goods,  is a vector

of structural parameters entering preferences, and  is a vector or matrix of unobserved

heterogeneity. Because  is interpreted as components of the utility function known to the

individual but unobserved and random from the analyst’s perspective, the structure of

preferences is consistent with McFadden’s random utility hypothesis (see McFadden [2001] for a

recent discussion). 
7

The consumer maximizes utility subject to a linear budget constraint and M nonnegativity constraints:

,

max ( , , , ) . . 0, , zU z st z y    xxQ x p xT (1)

where p is an M-dimension vector of prices, y is income, and the price of the Hicksian composite

commodity is normalized to one with no loss in generality. In addition to the constraints in (1),

the Kuhn-Tucker conditions that implicitly define the optimal solution to the consumer’s

problem can be written:

/ ( / ) , 1,..., . U x U zp j M j j      (2)

Estimation of the structural parameters entering the preference specification within the KuhnTucker framework exploits these weak inequalities. Equation (2) and an individual’s observed

choices place restrictions on the support of the distribution of the unobserved heterogeneity.

Assuming the errors representing unobserved heterogeneity are drawn from some known family

of distributions with parameter vector  , these restrictions permit recovery of estimates for 

and  within the maximum likelihood framework. 

From an econometric perspective, the Kuhn-Tucker model can be interpreted as an

endogenous regime switching model where regimes are defined as combinations of interior and

corner solutions for the M goods and determined by equation (2). When dealing with

applications involving large sets of quality differentiated goods, two related issues must be

addressed in estimation. The analyst must choose a flexible yet parsimoniously parameterized

direct utility function. This requires restricting the dimension of  to be sufficiently low.

Moreover, the analyst must specify a distribution for the unobserved heterogeneity that has an

estimable parameter vector  that is of relatively low dimension and that allows calculation of

the multiple dimensional integrals that correspond to the probabilities of observing each of the
8

M2 possible regimes. If these issues are adequately addressed, the Kuhn-Tucker framework

represents a viable approach to modeling consumer choice for many quality differentiated goods

in a systems framework. 

Welfare measurement from demand system models raises a separate and in many ways

more complicated set of issues. The Hicksian consumer surplus H CS associated with a price and

quality change from ) ( ,

0 0 p Q to ) ( ,1 1 p Q is implicitly defined as:

max ( , , , ) max ( , , , ) 0 0 1 1     

H V p Q y V p Q y CS   

, (3)

where  indexes each of the 2M separate regimes and ) ( V represents the corresponding

conditional indirect utility function. Unless preferences are quasilinear or homothetic in income,

no closed form solution exists for H CS , and iterative techniques such as numerical bisection are

required to solve for H CS . However, as discussed by PKH, procedures such as numerical

bisection require that the analyst solve the consumer’s problem at each iteration conditional on

an arbitrary set of ( p,Q, y,  ) values. PKH propose a strategy for accomplishing this task that

calculates each of the possible 2M conditional indirect utility functions and ascertains which is

the maximum. Although this strategy is computationally feasible for small M, it quickly

becomes intractable as M grows large. For example, in our subsequent empirical application

where M equals 62, the number of possible regimes is 4.61018.

An additional complication with constructing welfare estimates is that the analyst does

not observe  . This limitation suggests that the analyst cannot determine the individual’s

Hicksian consumer surplus precisely and can at best construct an estimate of the welfare

measure’s central tendency over the support of  such as its expectation, ) ( H E CS . As

described in PKH, constructing ) ( H E CS requires the use of Monte Carlo techniques that involve
9

simulating several realizations of  from its estimated distribution, solving for the H CS

conditional on each simulated  , and averaging the simulated values of H CS . Increasing the

number of simulations improves the precision of the estimate but also increases the

computational time involved. 

One final notable difficulty arises because these welfare estimates are functions of

estimates of  and  that are random variables from the analyst’s perspective. Quantifying the

implications of uncertainty about the parameters’ true values by constructing standard errors for

the welfare estimates requires replication of the entire simulation routine for several alternative

parameter estimates.

The above discussion suggests the significant computation challenges arising with

welfare estimation from demand system applications with large choice sets. For welfare

measurement to be viable from demand system models, the analyst must be able to quickly solve

for the utility the individual obtains conditional on ( p,Q, y,  ). As discussed in the introduction,

the difficulties associated with this task as well as the difficulties associated with estimating

demand system models have led researchers in the outdoor recreation literature to abandon

demand system models and instead rely on the discrete choice framework.2

 In the next section,

we develop econometrically tractable preference specifications that can be used to model the

demand for a large set of quality differentiated goods and to construct Hicksian welfare

measures.

3. Preference Specifications and Estimation Strategies

In this paper our approach to modeling consumer choice within the Kuhn-Tucker demand

system framework relies on the assumption that consumer preferences are additively separable in

each element of x and z. Although PKH employ this assumption in their empirical work, we

note that it is a strong preference restriction; it rules out a priori inferior goods and implies that
10

all goods are Hicksian substitutes (see Pollak and Wales [1992] for a discussion). We suggest,

however, that this assumption implies restrictions on consumer behavior that parallel the

implications of discrete choice RUM models. Whereas our demand system specifications

assume additive separability, the discrete choice models assume that the consumer’s time

horizon of choice can be decomposed into separable choice occasions. Additive separability in a

systems framework implies that all goods are Hicksian substitutes with non-negative Engel

curves, while the discrete choice framework assumes that on a given choice occasion all goods

are quality-adjusted perfect substitutes and, with rare exception, income effects are absent.3

Specifically, our empirical demand system specifications can be nested within the

following general structure:

( , , , ) ( , , )( ( ) )

ln ( , , ) ( ) ( )

ln ( )

j z M

j jj j

j jj

j j

Uz x z  



 

  

 

  

   







xQ sd q

sd s d

q q

T T

T

(4)

where s and j d are vectors of individual specific demographic variables and site specific

dummy variables, respectively, ( ,  ,  , ,, ) j z are estimable parameters, (   ,  ) represent

unobserved heterogeneity that varies randomly across individuals in the population, and j  is

unobserved heterogeneity that varies randomly across individuals and goods. 

Our preference structure is a close relative of the additively separable linear expenditure

system employed by PKH but differs in three important respects. First, our specification can be

interpreted as a more general specification because in the limit as all parameters approach zero,

our specification nests the linear expenditure system. Second, PKH assume that a good’s quality

attributes enter through its  parameter. This approach to introducing quality implies that weak

complementarity (i.e., /  0 U j q if  0 j x ) is not in general satisfied unless   1 (see Mäler
11

[1974] and Bradford and Hildebrand [1977] for discussions). Because our specification

introduces quality through simple repackaging parameters  () (e.g., Griliches [1964]), weak

complementarity is satisfied for all parameter values. Finally, our specification allows the

parameters for the demographic and dummy variables entering the  parameters to vary

randomly across individuals in the population. As discussed below, this feature of our model

allows us to introduce a more flexible structure for unobserved heterogeneity.

Maximizing the utility function in equation (4) with respect to p x  z y T and the nonnegativity constraints implies a set of first order conditions that, with some manipulation, can be

written:

( ) ( ) ln ln ( 1) ln( ) ( )

 (1 )ln( ( ) ) .

j z

j j z jj j

j j

j jj

p

y px

q

qx j

 

   

 

     



s d  T T

(5)

These weak inequalities, along with assumptions for the distributions of ( , , )j     , permit

estimation of the structural parameters using maximum likelihood techniques. In our application

we assume that each j  is an independent and identically distributed draw from the type I

extreme value distribution with common scale parameter  . Defining the right hand side of (5)

as ( , )    j g , the likelihood of observing a particular vector of choices x conditional on

(   ,  ) can be written:

      j

x j j l g g j ( | , ) | | [1 exp( ( , ) / ) / ] exp( exp( ( , ) / )) x  J 0      , (6)

where J is the Jacobian of transformation. The unconditional likelihood of observing x is

          l(x)  l(x | , ) f ( , )d d , (7)
12

where the integral is over the full support of ( , )    . For our application, we assume that

(   ,  ) are mean zero random draws from the normal distribution with unequal vectors of scale

parameters (   , ) and no correlations. Given these assumptions and the structure of our

conditional likelihood function, (7) has no closed form solution and cannot be evaluated using

numerical integration techniques unless the dimension of (   ,  ) is small. In this paper, we

follow common empirical practice in the discrete choice literature (e.g., Revelt and Train [1998])

and use simulation to evaluate (7). Our estimation strategy therefore falls under the rubric of

simulated maximum likelihood estimation (e.g., Gourieroux and Monfort [1996]). Although

estimation of our random parameters model is more computationally difficult than the fixed

parameter model with generalized extreme value unobserved heterogeneity employed by PKH, it

is more flexible econometrically. In addition to allowing for heteroskedasticity and correlations

in the unobserved heterogeneity across groups of sites as PKH’s specification does, our

specification also allows for heteroskedasticity across individuals.

4. Welfare Calculation

4.1 Solving the Consumer’s Problem

An essential component of constructing Hicksian welfare measures involves evaluating

the utility an individual achieves conditional on ( p,Q, y,  ). When the dimension of the choice

set is large, PKH’s strategy of analytically constructing all 2M possible conditional indirect utility

functions and determining which is the maximum is not feasible. In this paper, we pursue an

alternative, computationally tractable strategy that numerically solves the Kuhn-Tucker

conditions for the optimal consumption levels. These optimal values are then inserted into the

direct utility function to ascertain the individual’s utility conditional on ( p,Q, y,  ). 
13

Given our additive separability assumption, solving the consumer’s problem is greatly

simplified. In particular, the Kuhn-Tucker conditions take the general form:

U x U z p j j ( j ) z ( ) j ,  (8)

x j j 0,  (9)

. j j jz  y px  (10)

Note that additive separability implies that only j x and z enter the jth inequality in (8). This

structure suggests that if the analyst knew the optimal value for z, she could use (8) and (9) to

solve for each j x . Therefore, solving for the optimal value of z is equivalent to solving the

consumer’s problem. Building on this insight, we developed the following numerical bisection

algorithm to solve the consumer’s problem:

1) At iteration i, set ( ) / 2 1 1  i

h

i

l

i

a z z z . To initialize the algorithm, set 0 0

zl  and 

z y h  0 .

2) Conditional on i

a z , solve for i x using (8) and (9).

3) Use (10) and i x to construct iz

~ .

4) If i

a

i

z  z ~ , set i

a

i

l z  z & 1  ih

i

h z z . Otherwise, set 1  il

i

l z z & ia

i

h z  z .

5) Iterate until abs z z c

i

h

i ( l )  where c is arbitrarily small. 4

The ability of this algorithm to solve the consumer’s problem relies on the strict concavity of

preferences in every argument.5 By totally differentiating (8), one can show that strict concavity

implies that x / z  0 j . This inequality, in conjunction with the fact that our model has a

unique solution, ) ( ,

* * x z , suggests that i

a

i

z z ~ if * z z

i

a  and conversely, ia

i

z  z ~ if * z z

i

a .

Updating the upper and lower bounds using the criteria stipulated in step 4 and iterating solves

for ) ( ,

* * x z .

6

 Plugging these optimal solutions into (4) allows the analyst to evaluate the

consumer’s utility conditional on ( p,Q, y,  ). Nesting this algorithm for solving the consumer’s

problem within a numerical bisection routine that iteratively solves for the income compensation
14

that equates utility before and after a price and quality change will allow the analyst to construct

the Hicksian consumer surplus, H CS .

4.2 Incorporating Observed Choice

The algorithm described in the previous section permits construction of welfare measures

conditional on a set of unobserved heterogeneity values. As discussed in Section 2, a precise

estimate of the individual’s Hicksian consumer surplus is not possible because ( , , )    are not

observed. However, using simulation techniques and the distribution of the unobserved

heterogeneity, the analyst can construct estimates of H CS such as its expectation, ) ( H E CS .

The approach taken by PKH to simulating the unobserved heterogeneity employs the full

distributional support of ( , , )    . In this paper, we build on an approach suggested by von

Haefen [forthcoming] and instead simulate the unobserved heterogeneity from the region of the

unobserved heterogeneity’s support that is consistent with the individual’s observed choice. In

other words, we simulate ( , , )    such that at baseline conditions our model perfectly predicts

the observed choices we find in our data, and we use the model’s structure of substitution to

predict how individuals respond to price, quality, and income changes. This approach contrasts

with PKH’s more traditional approach that uses the structure of the model to predict both what

individuals do at baseline conditions as well as how they respond to price, quality, and income

changes. For the purposes of our application in this paper, incorporating observed choice is

appealing because, although it requires the use of more complicated simulation techniques, it

greatly reduces the number of simulations required to produce a precise estimate of ) ( H E CS as

well as the computational time involved.7

We follow von Haefen and use a sequential strategy to simulate the unobserved

heterogeneity consistent with the individual’s observed choice. Note that our objective is to
15

simulate from the distribution of ( , , )    conditional on x, ( , ,  | x)  f   . This distribution

can be decomposed as follows:

( , , | x) ( , | x) ( | , , x)    f    f    f    . (11)

Equation (11) states that the joint conditional distribution for the unobserved heterogeneity can

be decomposed into the marginal distribution for the random parameters conditional on x

multiplied by the conditional distribution for the site specific unobserved heterogeneity

conditional on x. We use an adaptive Metropolis-Hastings algorithm (Chib and Greenberg

[1995]) tailored to our problem to simulate from ( , | x)  f   . The steps of the algorithm are as

follows:

1) At iteration i, simulate a candidate vector of unobserved heterogeneity, ) ~, ~( i i

   ,

from the normal distribution with location parameters ) ( , i1 i1

   and scale

parameters (  

i i r ) where , r i r is a constant. To initialize the process, set each

element of ) ( , 0 0

   equal to zero and 0 r equal to 0.1. 

2) Construct the following statistic:

( / , / ) ( | , )

) ~, ~ / ) ( | ~ / , ~(

1 1 1 1 

 i i i i

i i i i

i

N l

N l

  

  

   

   



x

x

where N() is the probability density function for the normal distribution and l() is

defined in equation (6). If U i   where U is a uniform random draw, accept the

candidate random parameters, i.e., ) ~, ~ ( , ) ( i i i i

       . Otherwise, set

( , ) ( , ) 1 1  i i i i

      .

3) Gelman et al. [1995] argue that the Metropolis-Hastings algorithm for the normal

distribution is most efficient if the acceptance rate of candidate parameters is between

0.23 and 0.44. Accordingly, we employ the following updating rule for i r . If the

proportion of accepted candidate parameters is less than 0.3, set i i r  r  (1.1) 1 .

Otherwise, set i i r  r  (0.9) 1 .

4) Iterate.

After a burn-in period, this Monte Carlo, Markov Chain simulator generates random draws from

( , | x)  f   . 
16

After values of ( , )    are simulated, drawing from ( | , , x)  f    is far simpler. If

good j is consumed in a strictly positive quantity, the structure of our model, the simulated

random parameters, and the individual’s observed choice imply that ) ( ,

i i

j

i

j g      , where

( ) j g  is the right hand side of equation (5). Otherwise, i

j  can be simulated from the truncated

type I extreme value distribution via

  ln(ln(exp(exp(g ( ,  )/ ))U)))i i

j

i

j ,

where U again is a uniform random draw. 

4.3 Summary

Before proceeding to the data and empirical results, we summarize the key components

of our welfare measurement algorithm:

1) For simulation i, use the sequential procedure described in the previous section to

simulate ( , , )j     consistent with the individual’s observed choice. Because

simulating from ( , | x)  f   requires the use of an adaptive MetropolisHastings algorithm, discard the first T simulations.

2) For simulation i  T , use a numerical bisection routine to solve for the simulated

Hicksian consumer surplus associated with a price and quality change. 

2a) At each step of the numerical bisection routine that solves for the Hicksian

consumer surplus associated with simulation i, use the numerical bisection routine

described in Section 4.1 to solve the consumer’s problem. Inserting these optimal

values into (4) permits the analyst to determine the utility the individual achieves.

3) Average each of the simulated Hicksian consumer surplus values to construct an

estimate of ) ( H E CS , the individual’s expected Hicksian consumer surplus.

Although our algorithm for estimating welfare measures has multiple layers and numerous

details, our experience has been that it is surprisingly easy and fast to use in an applied setting.

One of the algorithm’s most appealing attributes is that each of its steps can be executed

simultaneously for every observation in the sample using vector and matrix notation. This

feature implies that coding and executing the algorithm in a matrix programming language

reduces the computational burden significantly. 
17

5. Application

We apply our demand systems framework to a random sample of Delaware residents’

recreational day trips to Mid-Atlantic ocean beaches in 1997. From a policy perspective,

understanding beach recreation demand is important for at least two reasons. Oil and toxic spills

in coastal waters often result in beach closings. Under the Oil Pollution Act of 1990 and other

Natural Resource Damage Assessment (NRDA) statutes, the public’s lost economic benefits

arising from these spills are compensable. Deacon and Kolstad [2000] discuss issues associated

with estimating these losses and argue that revealed preference recreation demand models are a

preferred method for ascertaining beach users’ resource values. Furthermore, understanding how

the characteristics of beaches such as beach width influence the demand for beach recreation can

help inform the ongoing debate over beach nourishment as a strategy for combating coastal

erosion. Beach nourishment is a technically feasible but costly way to maintain coastal areas

threatened by erosion from rising sea levels, development, and natural causes. At present,

substantial state and federal resources have been earmarked for these activities in response to a

perceived need to protect tourism and recreation related infrastructure.8

To assess the recreational values of beach amenities and to gauge residents’ willingness

to pay for beach quality improvements such as nourishment, researchers at the University of

Delaware collected data on visits by Delaware residents to 62 ocean beaches in New Jersey,

Delaware, Maryland, and Virginia during 1997. Figure 1 shows a map of the region and several

of the major beaches included in the data set. Massey [2001] discusses the data collection effort

in detail. A mail survey of 1000 randomly selected Delaware residents resulted in 540

completed responses. Although respondents provided information on the number of day trips,

overnight trips, and side trips made to each of the 62 beaches, we follow conventional empirical

practice in the recreation literature and consider only day trips in our analysis. 
18

The survey collected sufficient information to compute round trip travel costs to each of

the beaches. For every individual in the sample, a beach’s price is assumed to consist of transit

costs (valued at $0.35 per mile), beach fees, highway tolls, parking fees, a ferry toll on trips from

southern Delaware to New Jersey beaches, and the opportunity cost of travel time valued at the

individual’s average wage rate. Distances and travel times to each of the sixty-two sites from

each of 540 residents’ homes were calculated using PC Miler. The wage rate was estimated as

the individual’s annual income divided by 2040 (i.e., the typical number of hours worked in a

year). 

Household characteristic and demographic data were also collected and are used to

parameterize our utility function. Household specific variables and recreational summary

statistics are listed and described in the top part of table 1. Note that respondents took on average

9.77 trips and visited 2.77 different beaches during the season. The maximum number of beaches

visited by a respondent was nineteen, and 165 people surveyed (30% of the sample) did not visit

a beach during the season. 

In addition to the behavioral data, auxiliary information on the characteristics of the

sixty-two beaches was also gathered. Fourteen variables are used to differentiate the beaches

included in the study. These variables are listed and described in the lower part of table 1. Of

most interest for policy purposes are the indicator variables for wide and narrow beach. Of the 62

beaches, 25% are wide (i.e., greater than 200 feet in width) and 14% are narrow (less than 75 feet

in width). The impacts of the wide and narrow dummy variables on the demand for beach visits

will allow us to gauge the effects of beach width on recreation demand and to assess the welfare

implications of policies designed to alter beach width such as beach nourishment. 
19

6. Parameter Estimates

Although we estimated several demand system specifications consistent with the generic

structure described in Section 3, we present a representative set of our findings in this section.

Table 1 contains estimates for two fixed parameter models (i.e. 0      ) nested in equation

(4): a translated CES specification (column 2) resulting from the restrictions z j     j and

a second specification (column 3) resulting from the restriction 0 j  j . Table 2 contains

estimates for more general random parameter versions of these specifications that allow the

parameters in the  index to vary randomly across the population. Although the parameter

estimates in Table 1 are estimated with conventional maximum likelihood techniques, the

parameter estimates in table 2 are obtained via simulated maximum likelihood procedures. To

improve simulation efficiency we follow common empirical practice in the discrete choice

literature and employ 250 Halton draws rather than random draws in the calculation of our

simulated probabilities. Train [1999] demonstrates that for many random parameters models

Halton draws outperform random draws in terms of the number of simulations needed to achieve

an arbitrary level of precision. 

In general we find statistically significant, plausibly signed and robust coefficient

estimates for the four models. For example we find that age negatively impacts trips to all

destinations while ownership of vacation property in Delaware is positively related to increased

beach visitation. Respondents with children of all ages tend to take more trips than respondent

with no kids, although this result is not significant for all models. Students and retired indivduals

also tend to take more trips. Several site characteristics are also statistically significant

determinants of choice. Of most interest for policy purposes are the signs on the narrow and wide

beach dummy variables. We find for all four models negative and statistically significant
20

coefficient estimates for both of these variables, suggesting that respondents prefer beaches of

moderate width. This finding is consistent with Parsons et al.’s [1999] empirical findings and

suggests that individuals dislike narrow beaches with limited available recreation area and wide

beaches that require long walks to the waterfront. 

For both specifications the fixed parameter results are nested versions of the random

parameter results, allowing direct comparisons of the likelihood function values to determine the

contribution of the random parameters to the model fit. In both cases we find substantial

increases in likelihood values with the addition of random parameters, suggesting the importance

in this application of allowing for additional unobserved heterogeneity in the characterization of

preferences. This finding is further supported by noting that seven of eight and six of eight of the

parameters’ standard error estimates are statistically significant in each of the specifications,

respectively. A direct comparison of the log-likelihood values for the translated CES and the

second specification is not appropriate since the models are not nested. However, since the

models are both restricted versions of the same more general specification a likelihood

dominance criteria (Pollak and Wales [1991]) can be used to gauge their comparative fits. We

find for both the fixed and random parameter specifications that the second specification fits the

data best, suggesting on statistical grounds that our second random parameters specifications

provides the best characterization of preferences in this application. 

7. Welfare Analysis

The parameter estimates allow welfare analysis for our beach application employing the

methods described in section 4. We analyze three scenarios designed to provide different types

of valuation information for beach recreation and to provide a demonstration of the feasibility of

our welfare analysis computations. Two pertain to the recreation value lost when beaches close,
21

and the third addresses the recreational loses associated with beach erosion. The specific

scenarios analyzed include:

Closing of Rehoboth Beach 

Closing of northern Delaware beaches

Lost beach width at all Maryland, Delaware, and Virginia (MD/DE/VA) developed

beaches. 

These three scenarios have policy relevance for the mid-Atlantic region. Oil tankers enter

the Delaware Bay regularly and pass near the most frequently visited ocean beaches in the state.

The possibility of a spill and consequent closure of beaches, especially along the northernmost

beaches from Cape Henlopen State Park to the Delaware Seashore State Park (see Figure 1), is

widely recognized. For example the June 1989 Presidente Rivera oil spill off the New Jersey

coast would have resulted in beach closures had it occurred further south near the mouth of the

Delaware Bay. In our analysis the first two scenarios simulate the welfare loss that might be

associated with such a spill. We consider the closure of Rehoboth Beach, located along the

northern Delaware coast, because it is the most visited beach in the state. We also consider the

closure of all beaches from Cape Henlopen State Park to the Delaware Seashore State Park.

These are seven of the 62 beaches in our analysis and comprise the eleven northernmost miles of

Delaware’s twenty-five miles of ocean beaches. In our judgement this is the set of beaches most

likely to experience the effects of a spill. 

Our third scenario pertains to beach erosion on the MD/DE/VA beaches, a major policy

issue in the region. For more than twenty years the three states have pumped sand onto their

beaches to maintain beach width in support of recreation uses. The projects are costly and

controversial and recreation benefit analyses associated with them are rather limited and

outdated. Using our estimated model, we simulate the welfare loss associated with all developed
22

beaches in the MD/DE/VA area eroding to a width of seventy-five feet. This will affect most of

the popular beaches in the area. While a more severe erosion scenario is possible, this one seems

both plausible and within the range of our data. These losses, of course, approximate the

recreation gains associated with a nourishment project assuming that full beach migration is not a

policy option. In our scenario all the natural (park) beaches are assumed to maintain their current

width. These beaches tend to migrate inland naturally and maintain width. 

Point estimates and standard errors for the three welfare scenarios (1997 dollars per

respondent per season) are presented in table 4. Columns 2 and 3 present the translated CES and

second specification estimates, respectively. To evaluate whether our demand system models

generate qualitatively different policy inference from the discrete choice RUM-based strategies

that dominate current empirical practice, column 4 presents welfare estimates from repeated

discrete choice RUM models (e.g., Morey, Rowe, and Watson [1993]) that are generated by the

conditional welfare measurement procedure outlined in von Haefen [forthcoming].9 These

models assume: 1) the recreation season can be decomposed into 75 separable and statistically

independent choice occasions;10 2) each individual on each choice occasion makes a discrete

choice among the 62 sites and an option not to recreate, 3) preferences for a site (the outside

alternative) are a linear additive index of its price, attributes, (demographic variables) and an

i.i.d. type I extreme value random draw; and 4) a subset of the parameters entering the indexes

are independently normally distributed across the population. A table with parameter estimates

for the specifications we consider in this paper is available from the authors upon request.

For each scenario fixed and random parameter model estimates are presented together for

comparison purposes. In all cases the estimates have plausible magnitudes and standard errors

that suggest statistically significant differences from zero. Additionally, a few general patterns

emerge. Perhaps the most striking is that the inclusion of random coefficients decreases the
23

magnitude of the estimated welfare effects for all models and all scenarios. This is consistent

with some empirical findings in other applications of the random parameters framework (e.g.

Petrin [2001], Phaneuf et al. [1998]) and suggests that the additional unobserved heterogeneity

introduced by random parameter models allows for a greater degree of substitution among goods

and, as a result, smaller (absolute) economic values. Next, in spite of the statistical dominance of

the second specification over the translated CES specification, both demand system

specifications provide qualitatively similar welfare measures for both the fixed and random

parameters versions. Finally, there are some differences between estimates for the demand

system models and the repeated discrete choice models, although not for all scenarios. For the

random parameters specifications, welfare estimates for the closing of Rehoboth Beach differ

substantially across the modeling frameworks but are less dramatic for the other scenarios. 

8. Conclusion

Our general conclusion from the research presented in this paper is that the demand

system framework represents a viable strategy for modeling consumer choice and generating

Hicksian welfare measures for individual or household level applications with many qualitydifferentiated goods. Using Monte Carlo estimation and welfare construction procedures, we

present empirical results from a beach recreation application that demonstrate how this can be

accomplished. Our methodological approach and empirical findings suggest that the demand

system framework represents a genuine alternative to the RUM-based strategies that dominate

current empirical practice. Moreover, the demand system approach has the conceptual advantage

of combining the intensive and extensive margins of consumer choice in a behaviorally

consistent framework.

Numerous extensions to our approach are possible, and we discuss two in closing.

Relaxing the assumption that preferences are additively separable that is central to our modeling
24

strategy represents a significant area for future work. From a computational perspective,

relaxing additive separability without jeopardizing the demand system framework’s tractability

in estimation and welfare construction for applications with many goods represents a formidable

task. However, with continued advances in the computational performance of personal

computers, progress in this area is possible. In addition, our structure of the consumer’s problem

assumes that there are combinations of prices, site attributes, and income that would make every

individual participate in some form of beach recreation and visit each of the different sites.

Introspection suggests, however, that some individuals may not participate in beach recreation or

visit a particular site under any circumstances. Therefore, a significant modeling innovation

would be to augment our framework with a model of the individual’s decision to participate in

any form of beach recreation as well as her decision to consider visiting each of the quality

differentiated sites.
25

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Notes

1

 For a review and empirical comparison of these models, see Parsons, Jakus, and Tomasi [1999].

2

 Researchers in other fields of economics have also abandoned the demand system framework when modeling

consumer choice for many goods. For examples of this trend in the new empirical IO literature see Berry,

Levinsohn and Pakes [1995], Hendel [1999], and Nevo [2001].

3

 For a notable application with income effects, see Herriges and Kling [1999].

4

 We note that the basic numerical bisection algorithm described here can be partially modified in numerous ways

and still produce the optimal solutions. Our experience suggested that our algorithm was the fastest relative to

numerous variations of it.

5

 The assumptions of strict quasi-concavity and additive separability imply strict concavity (e.g., Pollak and Wales

[1992]).

6

 We have also compared our algorithm for constructing the individual’s utility to the procedure suggested by PKH

with a low dimensional choice set and found no difference between the two up to eight digits of numerical precision.

7

 Based on some Monte Carlo experiments with low dimensional choice sets, we found that the unconditional

approach to welfare measurement required roughly three times as many simulations to generate precise estimates

relative to the conditional approach. In terms of total computational run-time, the conditional welfare estimates took

roughly one-third the time to generate.

8

 Several studies have attempted to value beach access or beach use in general. A few have also attempted to value

beach characteristics such as beach width. These studies have tended to be contingent valuation or simple single site

travel-cost models. Examples of contingent valuation applications include McConnell [1977], Smith, Zhang, and

Palmquist [1997] and Silberman, Gerlowski, and Williams [1992]. Examples of single site travel-cost models are

Leeworthy and Wiley [1993] and McConnell [1999]. There have also been a few travel-cost random utility

maximization (RUM) models. Bockstael, Hanemann, and Kling [1987] and Bockstael, Strand, and McConnell

[1988] are examples of early work applying the RUM framework to beach recreation. More recent applications

include Parsons, Massey, and Tomasi [1999] and Environmental Economics Research Group [1998]. Finally, there

have been a few hedonic price applications treating proximity to beaches as a characteristic of coastal property

markets from which beach values have been inferred. Examples of these are Edwards and Gable [1991], Wilman

[1981], and Smith and Palmquist [1994].
31

9

 Although it is somewhat arbitrary to use the repeated discrete choice RUM model to compare our demand system

models to the alternative RUM-based models, we note that Parsons, Jakus, and Tomasi [1998] found qualitatively

similar welfare estimates across four alternative discrete choice RUM models applied to a common data set.

10 An arbitrary feature of the repeated discrete choice framework is the number of choice occasions. We specified

75 choice occasions because no one in our sample took more than 75 trips. Our experimentation with alternative

specifications of the number of choice occasions suggested that welfare estimates were largely insensitive to this

arbitrary judgement.
32

Table 1: Household and Beach Characteristics Variables

Variable Description Summarya

Household specific variables

Ln(age) Natural log of respondent age 3.82 (0.33)

Kids under 10 Respondent has kids under 10 (0/1) 0.26

Kids 10 to 16 Respondent has kids between 10 and 16 (0/1) 0.20

Vacation property in DE Respondent owns vacation home in DE (0/1) 0.03

Retired Respondent is retired (0/1) 0.24

Student Respondent is student (0/1) 0.05

Income Household annual income 49,944 (30,295)

Trips Total visits for day trips to all sites 9.77 (14.06)

Sites visited Number of beaches visited during 1997 2.70 (3.19)

Site characteristicsb

Beach length Length of beach in miles 0.62 (0.87)

Boardwalk Boardwalk with shops and attractions (0/1) 0.40

Amusements Amusement park near beach (0/1) 0.13

Private/limited access Access limited (0/1) 0.25

Park State or federal park or wildlife refuge (0/1) 0.09

Wide Beach Beach is more than 200 feet wide (0/1) 0.25

Narrow Beach Beach is less than 75 feet wide (0/1) 0.14

Atlantic City Atlantic city indicator (0/1) 0.016

Surfing Recognized as good surfing location (0/1) 0.35

High Rise Highly developed beach front (0/1) 0.24

Park within Part of the beach is a park area (0/1) 0.14

Facility Bathrooms available (0/1) 0.48

Parking Public parking available (0/1) 0.45

New Jersey Beach New Jersey beach indicator (0/1) 0.74

Price Person-specific money and time cost of travel $118c

a

Summary statistics for household variables are means (standard deviations) over the 540 individuals.

Summary statistics for site variables are means (standard deviations) over the 62 sites. 

b We thank Tony Pratt and Michael Powell of the Department of Natural Resources and Environmental

Control and Steve Hafner of Coastal Research Center at Richard Stockton College of New Jersey for their help

in compiling and verify the characteristic data. 

c

Each individual in the sample of 540 has a unique trip cost to each of the 62 sites. This statistic is the mean of

the mean of the individual trip prices. Since prices are based on distance there is substantial variability in

individual prices for the set of sites and individual prices across the sample for a given site.
33

Table 2: Kuhn-Tucker Fixed Parameter Estimatesa

#1

Translated CES #2

Log-Likelihood -7,837.40 -7,676.21

i Index Parameters

 Constant -3.8169 (-3.480) 2.2443 (1.744)

 ln(Age) -0.5822 (-2.142) -0.5356 (-2.141)

 Kids under 10 0.0593 (0.463) 0.0686 (0.596)

 Kids 10 to 16 0.1996 (1.376) 0.1960 (1.465)

 Vacation Property in DE 0.7924 (3.696) 0.9318 (4.700)

 Retired 0.3611 (1.787) 0.0566 (0.302)

 Student 0.4328 (2.140) 0.2906 (1.549)

 Park -0.0589 (-0.744) -0.0671 (-0.857)

 New Jersey Beach -1.4401 (-8.900) -1.2213 (-8.785)

Translating Parameter

  3.5132 (9.693) 4.9187 (10.954)

Simple Repackaging Quality Index Parameters

 Beach Length (Miles) 0.0726 (1.843) 0.0764 (2.034)

 Boardwalk -0.2199 (-2.670) -0.2020 (-2.589)

 Amusements 0.8568 (9.178) 0.7551 (8.650)

 Private/Limited Access -0.6099 (-5.739) -0.5750 (-5.816)

 Wide Beach -0.2286 (-2.999) -0.1901 (-2.654)

 Narrow Beach -0.4088 (-3.141) -0.3729 (-3.011)

 Atlantic City 0.7657 (7.100) 0.6994 (7.212)

 Surfing 0.1805 (2.235) 0.1554 (1.997)

 High Rise 0.1206 (1.415) 0.1331 (1.671)

 Park Within 0.5775 (6.220) 0.5256 (5.930)

 Facility -0.0347 (-0.582) -0.0275 (-0.463)

 Parking 0.4996 (5.126) 0.4667 (4.819)

Rho Parameters

 ln  -1.6185 (-6.189) -

 z ln  - -0.3063 (-3.105)

Type I Extreme Value Scale Parameter

  0.8759 (19.241) 0.8080 (22.867)

a

t-statistics based on robust standard errors reported in parentheses
34

Table 3: Kuhn-Tucker Random Parameter Estimatesa

#1

Translated CES #2

Log-Likelihood -6,828.73 -6,780.62

i Index Parameters

Mean Std. Dev. Mean Std. Dev.

 Constant -3.5091

(-8.339)

- 1.2164

(1.359)

-

 ln(Age) -0.4734

(-4.235)

0.2361

(19.663)

-0.2535

(-1.711)

0.2022

(25.258)

 Kids under 10 0.1042

(1.957)

0.0601

(1.008)

0.1431

(2.106)

0.2090

(2.894)

 Kids 10 to 16 0.0245

(0.370)

0.3576

(7.719)

-0.0404

(-0.399)

0.1203

(1.124)

 Vacation Property in DE 0.9008

(14.556)

0.2248

(3.065)

0.8610

(7.363)

0.1626

(2.505)

 Retired 0.1288

(1.018)

0.1806

(4.774)

-0.2505

(-2.520)

0.2116

(4.808)

 Student 0.4337

(4.676)

0.6611

(7.604)

0.4032

(1.259)

0.0923

(0.907)

 Park -0.0828

(-1.321)

0.3266

(5.607)

-0.0823

(-1.405)

0.3633

(6.643)

 New Jersey Beach -1.0095

(-7.871)

1.0346

(11.553)

-0.9697

(-9.915)

0.8661

(17.279)

Translating Parameter

  6.0443 (13.484) 7.4472 (15.430)

Simple Repackaging Quality Index Parameters

 Beach Length (Miles) 0.0806 (3.431) 0.0810 (3.464)

 Boardwalk -0.0527 (-0.961) -0.0516 (-0.972)

 Amusements 0.6200 (9.275) 0.5989 (9.337)

 Private/Limited Access -0.3893 (-6.685) -0.3904 (-6.853)

 Wide Beach -0.1677 (-3.279) -0.1590 (-3.112)

 Narrow Beach -0.2431 (-3.228) -0.2406 (-3.210)

 Atlantic City 0.5373 (6.084) 0.5403 (6.136)

 Surfing 0.1040 (1.974) 0.1015 (1.939)

 High Rise -0.0413 (-0.736) -0.0252 (-0.472)

 Park Within 0.3931 (5.874) 0.3859 (5.727)

 Facility -0.0119 (-0.342) -0.0101 (-0.305)

 Parking 0.2664 (4.011) 0.2620 (4.129)

Rho Parameters

 ln  -1.9391 (-7.919) -

 z ln  - -0.4294 (-4.276)

Type I Extreme Value Scale Parameter

  0.4044 (23.189) 0.4011 (26.459)

a

t-statistics based on robust standard errors in parentheses. Simulated probabilities computed using 250 Halton

draws.
35

Table 4: Mean Seasonal Welfare Estimates (1997 Dollars)a

#1 

Translated CES #2 Repeated Discrete

Choice

Closing of Rehoboth Beach

Fixed Coefficients -$81.13

(7.11)

-$93.69

(4.81)

-$83.09

(1.12)

Random Coefficientsb,c -$60.10

(4.16)

-$65.76

(3.01)

-$37.47

(1.09)

Closing of Northern Delaware Beaches

Fixed Coefficients -$160.43

(12.92)

-$171.15

(7.31)

-$195.27

(2.62)

Random Coefficientsb,c -$117.20

(8.01)

-$120.56

(4.93)

-$106.58

(2.52)

Lost Beach Width at All DE/MD/VA Developed Beaches

Fixed Coefficients -$57.68

(14.18)

-$54.44

(16.14)

-$77.78

(6.29)

Random Coefficientsb,c -$35.51

(8.71)

-$35.30

(9.58)

-$33.91

(3.82)

a

Robust standard errors based on 200 Krinsky and Robb [1986] simulations in parentheses.

b

The fixed coefficient Kuhn-Tucker welfare estimates were constructed with 25 simulations per observation. The

random coefficient Kuhn-Tucker welfare estimates were constructed with 50 simulations per observation and a

burn-in of 1000 simulations.

c

The fixed coefficient Repeated Discrete Choice estimates were constructed with 2500 simulations. The random

coefficient Repeated Discrete Choice estimates were constructed with 2500 simulations and a 1000 simulation

burn-in.
36

Reviewer’s Appendix Table: Repeated Discrete Choice Parameter Estimates

Fixed Parameters Random Parameters

Log-Likelihood -27,869.91 -27,609.78

Mean Std. Dev.

Outside Alternative Dummy 

 (OAD)

1.3704 (10.034) -2.1745 (-5.422) 4.6603 (26.409)

OAD  ln(Age) 0.7735 (22.549) 2.1512 (18.686) -

OAD  Kids under 10 0.1739 (8.759) 0.5076 (8.637) -

OAD  Kids 10 to 16 -0.3350 (-17.338) -0.9257 (-15.109) -

OAD  Vacation Property in DE -1.6901 (-53.823) -1.5286 (-6.542) 9.6724 (19.905)

OAD  Retired 0.1529 (6.058) 0.5067 (6.645) -

OAD  Student -0.2757 (-9.107) -0.8670 (-9.610) -

Park 0.4551 (3.954) 0.1848 (1.531) -

New Jersey Beach -2.0416 (-25.906) -12.243 (-11.343) 8.6077 (14.995)

Beach Length (Miles) -0.1279 (-4.075) 0.1684 (5.291) -

Boardwalk 0.2168 (1.942) 0.4474 (3.818) -

Amusements 1.4503 (37.464) 0.9940 (21.000) -

Private/Limited Access -0.7885 (-16.027) -0.8581 (-17.014) -

Wide Beach -0.8261 (-22.998) -0.6614 (-20.138) -

Narrow Beach -0.6133 (-8.369) -0.6030 (-7.898) -

Atlantic City 1.8928 (18.165) -4.4524 (-4.713) 6.2724 (9.740)

Surfing 0.8533 (23.715) 0.6530 (16.098) -

High Rise -0.1816 (-3.998) -0.1895 (-4.065) -

Park Within 1.9552 (18.312) 1.2072 (10.918) -

Facility -0.3691 (-5.757) -0.2007 (-2.826) -

Parking 0.9864 (7.844) 0.6295 (4.405) -

MU of Income/100 2.9973 (71.531) 7.3238 (33.959) -

t-statistics based on robust standard errors reported in parentheses

250 Halton draws used in estimation
37

Figure 1: Map of Application Area