The Generalized Multiple Discrete-Continuous Extreme Value (GMDCEV) Model:
Allowing for Non-Additively Separable and Flexible Utility Forms
Chandra R. Bhat
The University of Texas at Austin
Department of Civil, Architectural & Environmental Engineering
1 University Station C1761, Austin, Texas 78712
Tel: 512-471-4535, Fax: 512-475-8744
Email: bhat@mail.utexas.edu
and
Abdul Rawoof Pinjari
University of South Florida
Department of Civil & Environmental Engineering
4202 E. Fowler Ave., ENB 118, Tampa, Florida 33620
Tel: 813-974-9671, Fax: 813-974-2957
Email: apinjari@usf.edu
ABSTRACT
Many consumer choice situations are characterized by the simultaneous demand for
multiple alternatives that are imperfect substitutes for one another. A simple and
parsimonious Multiple Discrete-Continuous Extreme Value (MDCEV) econometric
approach to handle such multiple discreteness was formulated by Bhat (2005) within the
broader Kuhn-Tucker (KT) multiple discrete-continuous economic consumer demand
model of Wales and Woodland (1983). However, the MDCEV model, and other extant
multiple discrete-continuous models, use an additively-separable utility function, with the
assumption that the marginal utility of one good is independent of the consumption of
another good. The implication is that the marginal rates of substitution between any pair of
goods is dependent only on the quantities of the two goods in the pair, and independent of
the quantity of other goods, thus substantially reducing the ability of the utility function to
accommodate rich and flexible substitution patterns. Also, the specification of a quasiconcave and increasing utility function with respect to the consumption of goods in extant
models, when coupled with the additive utility across goods, immediately implies that
goods cannot be inferior and cannot be complements. In this paper, we develop a closedform model formulation for multiple discrete-continuous choices that allows a non-additive
utility structure, and accommodates rich substitution structures and complementarity
effects. As importantly, the utility functional form proposed here remains within the class
of flexible forms, while also retaining global theoretical consistency properties (unlike the
Translog and related flexible quadratic functional forms). The result is also clarity in the
interpretation of the model parameters. The proposed Generalized Multiple DiscreteContinuous Extreme Value (GMDCEV) model is easy to estimate, and should represent a
substantially more accurate representation of the underlying behavioral process for many
multiple discrete-continuous choice situations.
Keywords: Discrete-continuous system, Multiple discreteness, Kuhn-Tucker demand
systems, Mixed discrete choice, Random Utility Maximization, Non-additively separable
utility form.
1. INTRODUCTION
Several consumer demand choices related to travel and other decisions are characterized by
a discrete dimension as well as a continuous dimension. Examples of such choice situations
include vehicle type holdings and usage, appliance choice and energy consumption,
housing tenure (rent or purchase) and square footage, brand choice and quantity, and
activity type choice and duration of time investment of participation. There is a substantial
amount of literature now that underscores the importance of considering the interrelatedness in the discrete and continuous dimensions of choice to obtain unbiased
statistical estimates of model parameters. Two broad model structures may be identified in
the literature to handle such discrete-continuous choice situations. The first structure
(sometimes referred to as the “reduced-form” structure) typically has a separate equation
for the discrete choice and another separate equation for the continuous choice, with
jointness introduced through the statistical correlation in the random stochastic components
of each equation. This first structure has seen extensive use and has proved useful to handle
many empirical situations, but it is not based off an underlying (and unifying) theoretical
economic model. The second structure to discrete-continuous choice modeling, and the one
of interest in this paper, originates from microeconomic consumer utility maximization
theory. While much work in the context of consumer utility maximization has been focused
on the case of a single discrete-continuous (SDC) choice situation (where the choice
involves the selection of one of many alternatives and the continuous dimension associated
with the chosen alternative), there has been increasing interest and research recently in the
multiple discrete-continuous choice (MDC) situation (where the choice situation involves
1
the selection of one or more alternatives, along with a continuous quantity dimension
associated with the consumed alternatives).
1.1. MDC Context and Utility Framework
Multiple discrete-continuous (MDC) choice situations are quite ubiquitous in consumer
decision-making, and constitute a generalization of the more classical single discretecontinuous choice situation. Examples of MDC contexts include the participation decision
of individuals in different types of activities over the course of a day and the duration in the
chosen activity types (see Bhat, 2005), and household holdings of multiple vehicle
body/fuel types and the annual vehicle miles of travel on each vehicle (Bhat et al., 2009,
Ahn et al., 2008).
There are several differences between the single discrete choice (SDC) and MDC
utility frameworks, primarily originating in the functional form of the utility function. MDC
models typically assume imperfect substitution among alternatives based on a more general
utility function than the SDC case, which assumes perfect substitution among alternatives.
But, at a basic level, the choice process faced by the consumer in both the SDC and MDC
situations is formulated as follows:
K
Max U ( x) subject to
p
k 1
k
x k  E , x k  0,
(1)
where U(x) is the utility function corresponding to a consumption vector x, p k is the unit
price of good k, and E is the total expenditure. Note that the vector x may or may not
include an outside composite (or numeraire) good that is always consumed and has a unit
price equal to unity.
2
The functional form of the utility function U(x) determines the characteristics of,
and the solution for, the constrained utility maximization formulation of Equation (1). More
importantly, the functional form determines whether the formulation corresponds to an
SDC or an MDC model. For instance, Hanemann (1984) considers the “perfect substitutes”
case when he writes the utility function U(x) as follows (Hanemann considers an outside
good, which we will assume to be good 1):
K
U ( x )  U * (  k x k , x1 ),
(2)
k 2
where U * is a bivariate utility function and  k (k = 2, …, K) represents the quality index
specific to each inside good k. This functional form assures that, in addition to the outside
good which is consumed, exactly one inside good (k = 2, 3, …, K) is also consumed.
Hanemann (1984) refers to this as the “extreme corner solution”.1 As is typical in SDC
analysis, rather than deriving the consumption function based on solving the constrained
maximization problem of the direct utility function in Equation (2), Hanemann assumes a
functional form for the indirect utility function, introduces random stochasticity into the
formulation, and then derives expressions for the probabilities of the discrete and
continuous choices. Chiang (1991) and Chintagunta (1993) extend Hanemann’s SDC
formulation to include the possibility of no inside goods being selected for consumption by
including a “reservation price”. If the quality-adjusted prices of all the inside goods exceed
the reservation price, no inside goods are selected, but if the quality adjusted prices of one
1
Of course, the utility function in Equation (2) is easily modified for the case when there is no outside good,
and only one of the inside goods is consumed. In this case, the utility function becomes U ( x ) 
K

k
xk .
k 2
This, of course, corresponds to the case of the traditional discrete choice model, since all the expenditure is on
the chosen good and so the continuous component drops out.
3
or more of the inside goods are below the reservation price, an inside good is selected based
on Hanemann’s framework. The demand functions for the continuous components of
choice are obtained using Roy’s identity (Roy, 1947).
As indicated above, SDC analysis is usually undertaken using an indirect utility
approach, based on the argument that it is usually difficult and, often intractable, to adopt a
direct utility approach for estimating parameters and obtaining analytic expressions for
demand functions. However, as clearly articulated by Bunch (2009), the direct utility
approach has the advantage of being closely tied to an underlying behavioral theory, so that
interpretation of parameters in the context of consumer preferences is clear and
straightforward. Further, the direct utility approach also provides insights into identification
issues. Of course, when one moves to the MDC models, the indirect utility approach all but
falls apart because multiple inside goods can be selected for consumption and nonnegativity of the consumption vector must be guaranteed (see Wales and Woodland, 1983).
Thus, in addition to conceptual and behavioral advantages, it has been the norm to examine
MDC situations using the direct utility approach, especially because, through clever
stochastic term distribution assumptions, one can obtain a closed form for the probability of
the consumption patterns of goods.
Earlier direct utility-based MDC models have their origins in Hanemann’s (1978)
and Wales and Woodland’s (1983) Kuhn-Tucker or KT first-order conditions approach for
constrained random utility maximization. This approach assumes the utility function U(x)
to be random (from the analyst’s perspective) over the population, and then derives the
consumption vector for the random utility specification subject to the linear budget
constraint by using the KT conditions for constrained optimization. All earlier MDC
4
models (with the exception of a recent study by Vasquez-Lavin and Hanemann, 2008) have
adopted, as in the SDC case, an additively separable utility function, which assumes that
the marginal utility of one good is independent of the consumption of another good. This
assumption has at least two important implications. First, the marginal rate of substitution
between any pair of goods is dependent only on the quantities of the two goods in the pair,
and independent of the quantity of other goods. As indicated by Pollak and Wales (1992),
this has consequences on the preferences directly. For example, let there be three food
items: milk (x1), cornflakes (x2), and raisin bran (x3). Consider an individual who tends to
have milk and cornflakes, or milk and raisin bran, but not milk alone. Such an individual
may prefer the triplet [20,1,20] over [10,10,20], but may also prefer [10,10,1] over [20,1,1].
This violates additive utility, because, if the individual prefers [20,1,20] over [10,10,20],
s/he must prefer [20,1,x3] over [10,10,x3] according to additive utility. Overall, the
additively separable assumption substantially reduces the ability of the utility function to
accommodate rich and flexible substitution patterns. Second, the specification of a quasiconcave and increasing utility function with respect to the consumption of goods, along
with additive utility across goods, immediately implies that goods cannot be inferior and
cannot be complements (i.e., they must be strict substitutes; see Deaton and Muellbauer,
1980, page 139).
1.2. Paper Objectives and Structure
The objective of this paper is to extend Bhat’s (2008) multiple discrete-continuous extreme
value (MDCEV) model to relax the assumption of an additively separable utility function.
In doing so, we propose a particular utility functional form that remains within the class of
5
flexible forms, while also retaining global theoretical consistency properties. The form also
allows clarity in the interpretation of parameters and helps understand identification issues.
In addition, we show that, with a specific way of introducing stochasticity, we are able to
retain the closed form and simple nature of the MDCEV formulation. The structure of the
Jacobian in the likelihood function is also relatively simple because of the way stochasticity
is introduced. The non-additively separable utility-based MDCEV (i.e. the Generalized
MDCEV or GMDCEV) model can be compared with the basic MDCEV model using wellestablished likelihood ratio tests to examine the hypothesis of additively separable utility.
The rest of this paper is structured as follows. The next section formulates a
functional form for the non-additive utility specification that enables the isolation of the
role of different parameters in the specification. This section also identifies empirical
identification considerations in estimating the parameters in the utility specification.
Section 3 discusses the stochastic form of the utility specification, the resulting general
structure for the probability expressions, and associated identification considerations. For
presentation ease, Sections 2 and 3 consider the case of the absence of an outside good. In
Section 4, we extend the discussions of the earlier sections to the case when an outside
good is present. Section 5 provides an empirical demonstration of the GMDCEV model
proposed in this paper. The final section concludes the paper.
2. FUNCTIONAL FORM OF UTILITY SPECIFICATION
The starting point for our utility functional form is Bhat (2008), who proposes a linear BoxCox version of the constant elasticity of substitution (CES) direct utility function for MDC
models:
6
K
U ( x)  
k 1
k

 k  xk 

 k   1  1
 k   k 


(3)

where U(x) is a strictly quasi-concave, strictly increasing, and continuously differentiable
function with respect to the consumption quantity (Kx1)-vector x ( xk  0 for all k), and
 k ,  k and  k are parameters associated with good k. The function in Equation (3) is a
valid utility function if  k  0 ,  k  0 , and  k  1 for all k. For presentation ease, we
assume temporarily that there is no outside good, so that corner solutions (i.e., zero
consumptions) are allowed for all the goods k (this assumption is being made only to
streamline the presentation and should not be construed as limiting in any way; in fact, as
we will show later, the econometrics become much easier when there is an outside good).
The possibility of corner solutions implies that the term  k , which is a translation
parameter, should be greater than zero for all k.2 The reader will note that there is an
assumption of additive separability of preferences in the utility form of Equation (3).
Bhat’s utility form clarifies the role of the various parameters  k ,  k and  k , and
explicitly indicates the inter-relationships between these parameters that relate to theoretical
and empirical identification issues.3 In particular,  k represents the baseline marginal
utility, or the marginal utility at the point of zero consumption. Thus, a higher baseline  k
2
As illustrated in Kim et al. (2002) and Bhat (2005), the presence of the translation parameters makes the
indifference curves strike the consumption axes at an angle (rather than being asymptotic to the consumption
axes), thus allowing corner solutions.
3
The utility formulation of Equation (3) also imposes the untestable, but intuitive, condition of weak
complementarity (see Mäler, 1974), which implies that the consumer receives no utility from a non-essential
good’s attributes if s/he does not consume it (i.e., a good and its quality attributes are weak complements, or
Uk = 0 if xk = 0, where Uk is the sub-utility function for the kth good). The reader is referred to Hanemann
(1984), von Haefen (2004), and Herriges et al. (2004) for a detailed discussion of the advantages of using the
weak complementarity assumption.
7
implies less likelihood of a corner solution for good k.  k is the vehicle to introduce corner
solutions for good k (that is, zero consumption for good k). But it also serves the role of a
satiation parameter that determines the rate at which the marginal utility of good k
decreases with consumption of the good; the higher the value of  k , the less is the satiation
effect in the consumption of x k . The entire range of satiation effects from immediate and
full satiation (flat line) to linear satiation (constant marginal utility) can be accommodated
by different values of  k . Finally, the express role of  k is to capture satiation effects.
When  k  1 for all k, this represents the case of absence of satiation effects or,
equivalently, the case of constant marginal utility. The utility function in Equation (3) in
such a situation collapses to

k
xk , which represents the perfect substitutes case as
k
proposed by Deaton and Muellbauer (1980) and applied in Hanemann (1984), Chiang
(1991), Chintagunta (1993), and Arora et al. (1998), among others. Intuitively, when there
is no satiation, the consumer will invest all expenditures on the single good with the highest
price-normalized marginal (and constant) utility  k / pk . This is the case of single
discreteness. As  k moves downward from the value of 1, the satiation effect for good k
increases. When  k  0  k , the utility function collapses to the following linear
expenditure system (LES) form:
K
x

U ( x )    k k ln  k  1
k 1
k

(4)
The LES form above is empirically indistinguishable or similar to the functional form used
in the environmental economics literature (von Haefen et al., 2004, Mohn and Hanemann,
8
2005, von Haefen and Phaneuf, 2005, von Haefen, 2003, Phaneuf et al., 2000, Phaneuf and
Herriges, 2000, and Herriges et al., 2004).
Overall,  k , in addition to allowing corner solutions, controls satiation by
translating consumption quantity, while  k
controls satiation by exponentiating
consumption quantity. Clearly, both these effects operate in different ways, and different
combinations of their values lead to different satiation profiles. However, empirically
speaking, it is very difficult to disentangle the two effects separately, which leads to serious
empirical identification problems and estimation breakdowns when one attempts to
estimate both  k and  k parameters for each good. Thus earlier studies have either
constrained  k to zero for all goods (technically, assumed  k  0  k ) and estimated the
 k parameters, or constrained  k to 1 for all goods and estimated the  k parameters. This
is discussed in detail by Bhat (2008), who suggests testing both these normalizations and
selecting the model with the best fit.
Vasquez-Lavin and Hanemann (2008) extended Bhat’s additively separable linear
Box-Cox form and presented a quadratic version of it, as below:
K
U ( x)  
k 1

 k
 
 k
k
m

 x



 m  x m 
1 K


k
 1  1  
  1  1  k   km

2 m 1
 m   m 

  k
 
 




(5)
where  k  0 ,  k  0 , and  k  1 for all k. The new interaction parameters  km allow
quadratic effects (when k = m) as well as allow the marginal utility of good k to be
dependent on the level of consumption of other goods (note that  km =  mk for all k and m).
Of course, if  km  0 for all k and m, the utility function collapses to Bhat’s linear Box-Cox
9
form. If  k  0  k , the function collapses to the well-known direct basic translog utility
function (see Christensen et al., 1975), and if  k  1 k , we obtain the quadratic utility
function used by Wales and Woodland (1983). The quadratic form of Equation (5) is a
flexible functional form that has enough parameters to provide a second-order
approximation to any true unknown direct twice-differentiable utility functional form. It
also is a non-additive functional form. However, the flexibility is also a limitation, since the
function can provide nonsensical results and be theoretically inconsistent for some
combinations of the parameters and consumption bundles, an issue that has not received
much attention in the literature (but see Sauer et al., 2006). In fact, it is not possible to
achieve global consistency (over all consumption bundles) in terms of the strictly
increasing and quasi-concave nature of the utility function using the translog form. In the
next section, we extend Vasquez-Lavin and Hanemann’s discussion to clarify the role of
parameters, identify issues of theoretical consistency and restrictions that need to be
maintained, present identification considerations, and recommend a flexible form similar to
the translog but that is easier to estimate and reduces global inconsistency problems
associated with the translog.
2.1 Role of Parameters in Non-Additively Separable Utility Specification
2.1.1. Role of  k
The marginal utility of consumption with respect to good k can be written from Equation
(5) as:
10

U ( x )  x k
   1
x k
k

 k 1
m

K
 
 m  x m 






1

1


 k  km



m

1

 
m
m




(6)
The difference between the above expression and the corresponding one in Bhat’s (2008)
linear Box-Cox additively separable case is the presence of the second term in parenthesis,
which includes the consumptions of other goods. Thus, the formulation is not additively
separable, but one in which the marginal utility of a good is dependent on the consumption
amounts of other goods. The marginal utility at zero consumption of good k collapses to:
U ( x )
x k
xk  0
m

 
 m  x m 
  k   k   km
 1  1 

 m   m 
m k
 


(7)
From above, it is clear that  k is no more the baseline (marginal) utility of good k at the
point of zero consumption of good k. Rather, it should be viewed as the baseline (marginal)
utility of good k at the point at which no good has yet been “consumed”; that is, when
xm  0  m (no consumption decision has yet been made). This also indicates that, if prices
of all goods are the same, then the good with the highest value of  k will definitely see
some positive consumption.4
Another important point to note from Equation (6) is that for the utility function to
be strictly increasing, the following condition should be satisfied for all possible values of
the consumption vector x:
4
If there is price variation across goods, one needs to take the derivative of the utility function with respect to
expenditures (ek = pk xk ) on the goods. Then ∂U / ∂ek = ψk / pk at the point of no consumption of any goods, and
the good with the highest price-normalized marginal utility ψk / pk will definitely see some positive
consumption.
11
m

K
 
 m  x m 






1

1

   0 for all k.
 k  km



m

1

 
m
m




(8)
This is in addition to the condition in the linear case where  k  0  k . The condition
above is needed because we are considering the case of economic goods. In addition, a
sufficiency condition for maintaining the decreasing marginal utility (or strict quasiconcavity) of the utility function is that the left side of Equation (8) be a non-increasing
function of xk. We return to these conditions later in the paper.
2.1.2. Role of  k
As in the linear case, the  k parameter allows for corner solutions. In particular, the  k
terms shift the position of the point at which the indifference curves are asymptotic to the
axes from (0,0,0…,0) to ( 1 ,   2 ,   3 , ...,   K ) , so that the indifference curves strike the
positive orthant with a finite slope. This, combined with the consumption point
corresponding to the location where the budget line is tangential to the indifference curve,
results in the possibility of zero consumption of good k. In addition to allowing corner
solutions, the  k terms also serve as satiation parameters. In general, the higher the value of
 k , the less is the satiation effect in the consumption of x k . However, unlike the linear
case,  k affects satiation for good k in two ways. The first effect is through the first linear
term on the right side of Equation (5), and the second is through the second term on the
right side of Equation (5) that generates quadratic effects. The overall effect depends on the
sign and magnitude of the parameter  kk in the second term. If this term is negative, and
12
particularly for high values of  k , we can get an inappropriate parabolic shape for the
contribution of alternative k to overall utility within the range of x k . In particular, beyond a
certain point of consumption of alternative k, there is negative marginal utility. This is
because of the violation of the condition in Equation (8). An illustration is provided in
Figure 1, which plots the utility contribution of alternative k for  k  1 ,  k  0 ,
 kk  0.02 ,  km  0  m  k , and different values of  k (  k = 1, 10, and 30). As can be
observed, for the  k value of 30, we get a profile that peaks at about 110 units, and violates
the requirement that the utility function be strictly increasing. On the other hand, if  kk is
positive and quite high in magnitude, it is possible that, for high  k values, there is in fact
an increase in the marginal utility effect at low values of x k (essentially a violation of the
strictly quasi-concave assumption of the utility function). This is because the left side of
Equation (8) becomes an increasing function of x k at low x k values. Figure 2 illustrates
such a case for  k  1 ,  k  0 ,  kk  0.2 ,  km  0  m  k , and different values of  k
(  k = 1, 10, and 30). For  k = 10, one can discern the increasing marginal utility until about
6.5 units after which the shape becomes one of decreasing marginal utility. The increasing
marginal utility at low values is particularly pronounced for  k = 30, which continues until
a value of 40 units before starting to decrease in marginal utility. We will return to these
issues in Section 2.2.
The translation parameters  m of other goods also have an impact on the utility
contribution of good k, through the influence on the baseline (marginal) utility of good k
13
(see Equation (7)). Specifically, for a given value of x m , the baseline (marginal) utility for
good k increases as  m increases for positive  km values and decreases as  m increases for
negative  km values.
2.1.3. Role of  k
The express role of  k is to reduce the marginal utility with increasing consumption of
good k; that is, it represents a satiation parameter. However, as in the case of the  k effect
on consumption of good k, there are two effects of the  k parameter – one through the first
linear term on the right side of Equation (5) and the second through the quadratic effect
caused by the combination of the first and second terms on the right side of Equation (5).
The overall  k effect depends on the sign and magnitude of the parameter  kk in the
second term. If this term is negative, and particularly for values of  k close to 1, we can get
a “nonsensical” parabolic shape for the utility contribution of alternative k within the usual
possible range of x k . An illustration is provided in Figure 3, which plots the utility
contribution of alternative k for  k  1,  k  1 ,  kk  0.03 ,  km  0  m  k , and different
values of  k . As can be observed, at the  k value of 0.6, we get a profile peaks at about
150 units and decreases thereafter, violating the requirement that the utility function be
strictly increasing. On the other hand, if  kk is positive and quite high in magnitude, it is
possible that, for high  k values, there is in fact an increase in the marginal utility effect at
some low values of x k . Figure 4 illustrates such a case for  k  1,  k  1 ,  kk  0.2 ,
14
 km  0  m  k , and different values of  k . The non-conforming utility profile is obvious
for the  k value of 0.8.
The  m parameters for other goods also impact the baseline (marginal) utility of
good k (see Equation (7)). For a given value of x m , the baseline (marginal) utility for good
k decreases as  m falls down from 1 for positive  km values and increases as  m falls
down from 1 for negative  km values.
2.2. Empirical Identification Issues Associated with Utility Form
The total number of parameters in the flexible utility functional form of Equation (5) rises
rapidly with the number of alternatives, especially in the  km terms (k = 1, 2, …, K; m = 1,
2, …, K). There are also identification issues that arise with the utility form. As in the linear
case, empirically speaking, it is next to impossible to disentangle the effects of the  k and
 k parameters for each good separately (see Bhat, 2008). Thus one has to impose some
constraints on the  k and  k parameters. While many combinations of constraints are
possible, the easiest approaches are to constrain  k to zero for all goods (technically,
assume  k  0  k ) and estimate the  k parameters (the  profile), or constrain  k to 1
for all goods and estimate the  k parameters (the  profile).
In the case of the non-additively separable utility function, there is an additional
empirical identification issue in both the  profile case and the  profile case. This is
because the  kk parameters in the quadratic utility functional form essentially also serve as
15
“satiation” parameters by providing appropriate curvature to the utility function. However,
empirically speaking, it is difficult to disentangle the  kk effects from the  k effects (for
the  profile) and from the  k effects (for the  profile) as long as the  kk effects do not
become that negative as to bring on a parabolic shape at even low to moderate consumption
levels (this latter case would anyway be inappropriate to represent the utility function). In
fact, a utility profile based on a combination of  kk and  k values for the  profile case
can be closely approximated by a utility function based solely on  k values with  kk  0 .
Similarly, a utility profile based on a combination of  kk and  k values for the  profile
case can be closely approximated by a utility function based solely on  k values with
 kk  0 . This is illustrated in Figures 5 for the  profile, with  k  1 ,  k  0  k , and
 km  0  m  k . The figure shows that alternative k’s contribution to utility based on a
certain combination of  k and  kk values can be closely replicated by other combination
values of  k and  kk . In particular, the utility profiles corresponding to combinations of  k
and  kk values can be replicated very closely by a profile that corresponds to  k  1 and
some specific  kk value, or by a profile that corresponds to  kk  0 and some specific  k
value. Thus, in Figure 5, the utility profiles corresponding to  k  7.45 and  kk  0 , and
 k  1 and  kk  1.3 , are able to closely replicate all the other utility profiles. A similar
situation may be observed from Figure 6 for the  profile, where the utility profiles of
different combinations of  kk and  k values can be approximated closely by the profile
corresponding to  k  0.442 and  kk  0 , and  k  0 and  kk  1.38 .
16
The discussion above suggests that, without loss of empirical generality, one can
normalize  k  1 (and estimate  kk ) or set  kk  0 (and estimate  k ) for each good k in the
 profile case. In the  profile case, one can normalize  k  0 (and estimate  kk ) or set
 kk  0 (and estimate  k ) for each good k in the utility function. We propose to set  kk  0
for each good, since this immediately removes the possibility of a parabolic shape for the
utility contribution of good k. At the same time, we immediately ensure that the marginal
utility is strictly decreasing over the entire range of consumption values of the good k.
These are important theoretical considerations that we are able to maintain globally without
any loss in functional form flexibility. In fact, the functional form proposed in this paper
remains within the class of flexible forms, while also retaining global theoretical
consistency properties (unlike the translog and related flexible quadratic functional forms).
The result is also clarity in the interpretation of the  k and  k parameters, which now have
the same interpretation as satiation parameters corresponding to good k as in the linear
utility function case of Bhat (2008). Besides, the baseline marginal utility of good k now
remains unchanged with the consumption of good k, which is intuitive.
There is still, however, one remaining issue, which is that the baseline marginal
utility of all goods should be positive (  k  0 , k = 1, 2, …, K). The only way this condition
will hold globally is if  km  0 for all k and m (see Equation (7)). The condition  km  0
implies that the goods k and m are complements (since the consumption of good m would
increase the baseline marginal utility of good k and therefore consumption of good k).
However, we would also like to allow rich substitution patterns in the utilities of goods by
17
allowing  km  0 for some pairs of goods. As we discuss later, our methodology
accommodates this, while also recognizing the constraint  k  0 (k = 1, 2, …, K) during
estimation and ensuring that it holds in the range of consumptions observed in the data.
To summarize, we propose the following general formulation for the non-additively
separable utility specification:
K
U ( x)  
k 1

 k
 
 k
k
m

 x



 m  x m 
1


k
 1  1  
  1  1  k   km

2 mk
 m   m 

  k
 
 




(9)
In the above function, the analyst will need to set  k  0  k and estimate the  profile, or
set  k  1 and estimate the  profile. The  profile takes the following form:
K

x
 
x
 
1
U ( x )     k ln  k  1  k   km  m ln  m  1  ,

2 m k
k 1
k
 
m
  

(10)
and the  profile takes the following form:




K
 1


x k  1k  1  k  1   km x m  1m  1   .
U ( x )   
2 mk  m
k 1   k


(11)
In actual application, it would behoove the analyst to estimate models based on both the
estimable profiles above, and choose the one that provides a better statistical fit. In the rest
of this paper, we will use the general form in Equation (9) for ease in presentation.
18
3. THE ECONOMETRIC MODEL
3.1. Optimal Expenditure Allocations
The consumer maximizes utility U(x) as provided by Equation (9) subject to the budget
constraint that
K
K
k 1
k 1
 pk xk   ek  E , where pk is the unit price of good k, ek is the
expenditure on good k, and E is total expenditure across all goods. The analyst can solve for
the optimal expenditure allocations by forming the Lagrangian and applying the KuhnTucker (KT) conditions. The Lagrangian function for the problem is:
K

L  U ( x )     ek  E  ,
 k 1

(12)
where  is the Lagrangian multiplier associated with the expenditure constraint (that is, it
can be viewed as the marginal utility of total expenditure or income). The KT first-order
conditions for the optimal expenditure allocations (the ek* values) are given by:

 k  ek*

 1
pk  pk  k


 k  ek*

 1
pk  pk  k

 k 1
   0, if ek*  0 , k = 1, 2, …, K
(13)
 k 1
   0, if ek*  0 , k = 1, 2, …, K,
where  k is the baseline utility of good k as given in Equation (7).
3.2. Introducing Stochasticity
To complete the econometric model, the analyst needs to introduce stochasticity. This is an
important component of the model formulation, since how one decides to do so can make
the difference between analytically tractable and intractable models. In Bhat’s additively
19
separable form of the utility function (and in other restricted versions of this formulation),
the baseline marginal utility term  k collapses to  k , and stochasticity has been introduced
through a multiplicative random element as follows:
 k   ( zk ,  k )   ( zk )  e ,
k
(14)
where zk is a set of attributes characterizing alternative k and the decision maker, and  k is
an extreme value error term that captures idiosyncratic (unobserved) characteristics that
impact the baseline utility for good j. The exponential form for the introduction of the
random term guarantees the positivity of the baseline marginal utility as long as  ( z k )  0 .
To ensure this latter condition,  ( z k ) is further parameterized as exp(  zk ) , which then
leads to the following form for the baseline random utility associated with good k:
 k   ( zk ,  k )  exp(  zk   k ) .
(15)
Vasquez-Lavin and Hanemann indicate that, in the non-additively separable utility
form, introducing stochasticity in the multiplicative exponential form as in the additively
separable case above does not help in any way in simplifying the first-order conditions for
optimization in the non-additively separable utility form. They instead write  k in a linear
form as follows:
 k   ( zk ,  k )   zk   k .
(16)
The problem with this, however, is that it allows negative values for  k , which is
theoretically inappropriate. The derivation of the probability expressions for any given
consumption pattern from the first order conditions in Equation (13), after introducing the
form for  k from the above equation in  k , is also relatively messy and difficult. Further,
20
Vasquez-Lavin and Hanemann indicate that there is no closed form for the resulting
probability expressions.
In this paper, we propose a method to introduce stochasticity that is a logical
extension of Bhat’s additively separable MDCEV formulation. The approach also helps in
developing an analytically tractable form for the probability of any consumption pattern, as
we discuss in Section 3.3. In particular, we include stochasticity multiplicatively in the
baseline marginal utility  k of the non-additively separable formulation. Thus, we write:
k   k  e ,
(17)
k
and replace  k with  k in Equation (13). Essentially, we are introducing stochasticity
through the baseline (marginal) utility for each alternative just as in the additively separable
case. However, there is a very important interpretational difference in the way we are
introducing stochasticity in the case here and in the additively separable (AS) case.
Specifically, in the AS case, the stochasticity in the baseline marginal utility (Equation
(15)) can be viewed as a consequence of a random utility specification as follows:
k




 e

U ( x )   k exp(  z k   k )  k  1  1
k k
  k pk



(18)

According to this view, any stochasticity in the KT conditions originates from the analyst’s
inability to observe all factors relevant to the consumer’s utility formation. There is no
stochasticity originating from random individual errors in the process of maximizing utility
(subject to the expenditure constraint). An alternative perspective, equally applicable (and
empirically indistinguishable from the random utility perspective) in the AS case, is that the
analyst observes all factors relevant to utility formation. This corresponds to the assumption
21
of deterministic utility. However, consumers can make random errors in the optimization
process, which gets manifested in the form of stochasticity in the baseline marginal utility
of the KT conditions. Both the above interpretations are valid for the AS case. In the nonadditively separable case, one cannot derive a random utility specification that is consistent
with the stochastic specification of the baseline marginal utility in Equation (17) and so we
cannot use the first interpretation above. Thus, we use the second interpretation for the
source of stochasticity for the non-additively separable case.5
3.3. Stochastic Model Form
To develop the stochastic model form, we start by replacing  k with k   k exp(  k ) in the
KT conditions of Equation (13). The resulting first order conditions are:

 k exp(  k )  ek*

 1
pk
 pk  k


 k exp(  k )  ek*

 1
pk
 pk  k

5
 k 1
   0, if ek*  0 , k = 1, 2, …, K
(19)
 k 1
   0, if ek*  0 , k = 1, 2, …, K,
Note that a utility function of the form:
U ( x) 
K

k 1

  k

 k


k
m


 x




 m  x m
1 K



k
 1  1  k    km
 1  1   exp(  k ) 



2



m 1
 k
 
 
m  m







leads to a baseline marginal utility form as follows:
m

 
 m  x m 

1 K
1 


  k   km

1

1
exp(

)

 km m



k


2 m 1
 m   m 
2 m k
m
 



m

 x



m



1

1
exp(

)


m


  m


xk  0

The expression above is a function of all the error terms, and does not collapse to  k exp(  k ). This creates
U ( x )
x k
problems in developing the probability expression for the consumption vector. As we show in the next
section, the analytic tractability of our model hinges critically on writing the baseline marginal utility as
 k exp(  k ), so that the KT conditions simplify when a logarithm operator is applied.
22
K
The optimal demand satisfies the conditions above plus the budget constraint

k 1
ek*  E .
The budget constraint implies that only K-1 of the ek* values need to be estimated, since the
quantity consumed of any one good is automatically determined from the quantity
consumed of all the other goods. To accommodate this constraint, designate activity
purpose 1 as a purpose to which the individual allocates some non-zero amount of
consumption (note that the individual should consume at least one good, given the strictly
increasing utility specification assumed and given that E > 0). For the first good, the KT
condition may then be written as:

 exp(  1 )  e1*

 1
 1
p1
  1 p1

1 1
(20)
Substituting for  from above into Equation (18) for the other activity purposes (k = 2, …,
K), and taking logarithms, we can rewrite the KT conditions as:
Vk   k  V1   1 if ek*  0 (k = 2, 3, …, K)
Vk   k  V1   1 if ek*  0 (k = 2, 3, …, K), where
(21)
 ek*

Vk  ln(  k)  ln pk  ( k  1) ln 
 1 (k = 1, 2, 3, …, K).
  k pk

In our estimations, one needs to solve the KT conditions above. In doing so, note that, as
indicated earlier, we should ensure  k  0 for each good k. This is recognized in the
logarithmic transformation of  k appearing above. We ensure that  k  0 for each good k
in our estimation by using a constrained maximum likelihood procedure, in which we
explicitly impose this condition. At the same time, we also require that  k  0 . This latter
23
condition can be ensured by writing  k  exp(  zk ) . Also, since only differences in the Vk
from V1 matters in the KT conditions, a constant cannot be identified in the term for one of
the K alternatives. Similarly, individual-specific variables are introduced in the Vk ’s for
(K-1) alternatives, with the remaining alternative serving as the base.
In the case that a  profile is estimated, the  k values need to be greater than zero,
which can be maintained by reparameterizing  k as exp(  k ) . Additionally, the translation
parameters can be allowed to vary across individuals by writing  k   k wk , where wk is a
vector of individual characteristics for the kth alternative, and  k is a corresponding vector
of parameters. In the case when an  profile is estimated, the  k values need to be
bounded from above at the value of 1.
To enforce these conditions,  k can be
parameterized as [1  exp(  k )] , with  k being the parameter that is estimated. Further, to
allow the satiation parameters (i.e., the  k values) to vary across individuals, Bhat (2005)
writes  k   k yk , where y k is a vector of individual characteristics impacting satiation for
the kth alternative, and  k is a corresponding vector of parameters.
3.4. Probability Expressions
Let the joint probability density function of the  k terms be f(  1 ,  2 , …,  K ). Then, the
probability that the individual allocates expenditure to the first M of the K goods is:
24

P( e1* , e2* , e3* , ..., eM* , 0, 0, ..., 0)  | J |

V1 VM 1 1 V1 VM  2 1

1    M 1  

V1 VK 1 1 V1 VK 1

 M  2  

 K 1  

 K  
f ( 1 , V1  V2   1 , V1  V3   1 , ..., V1  VM   1 ,  M 1 ,  M  2 , ...,  K 1 ,  K )
(22)
d K d K 1 ...d M  2 d M 1d 1 ,
where J is the Jacobian whose elements are given by (see Bhat, 2005):
J ih 
[V1  Vi 1  1 ] [V1  Vi 1 ]

; i, h = 1, 2, …, M – 1.
eh* 1
eh* 1
(23)
Following Bhat (2005, 2008), consider an extreme value distribution for  k and assume
that  k is independent of  k ,  k , and  k (k = 1, 2, …, K). The  k ’s are also assumed to be
independently distributed across alternatives with a scale parameter of  (  can be
normalized to one if there is no variation in unit prices across goods). In this case, the
probability expression collapses to the following form:


P e1* , e2* , e3* , ..., eM* , 0, 0, ..., 0  abs J M
1
 M 1
 M
  eVi / 
 i1
M
 K

   eVk /  
  k 1



 ( M  1)!,



(24)
where the elements of the Jacobian JM are given by:
J ih 
[V1  Vi 1   1 ] [V1  Vi 1 ]

, i  1,2,...M  1; h  1,2,...M  1
eh*1
eh*1
(25)
To write these Jacobian elements, define zih  1 if i  h , and zih  0 if i  h. Also, define
the following:
 k 1
 e

k   k  1
 pk  k

for k  1, 2, ..., K .
(26)
25
Then, the elements of the Jacobian can be derived to be:6
J ih 

  
 1   i 1 
 1  1 
h 1 1,h 1

  zih 
 1  zih  i 1,h 1   1 1,i 1  

ph 1   1
 i 1  p1  i 1  e1   1 p1 
 ei 1   i 1 pi 1 
(27)
Unfortunately, there is no concise form for the determinant of the Jacobian for M > 1
(unlike the case of the additively separable case, where Bhat derived a simple form for any
value of M). In the case when M = 1 (i.e., only one alternative is chosen) for all individuals,
there are no satiation effects (  k = 1 for all k),  km  0  k , m (k  m) , and the Jacobian
term drops out (that is, the continuous component drops out, because all expenditure is
allocated to good 1). Then, the model in Equation (24) collapses to the standard MNL
model.
The parameters in the non-additively separable or generalized MDCEV model
(GMDCEV) may be estimated in a straightforward way using the maximum likelihood
inference approach. Extensions to include covariance and heteroscedasticity among the
error terms (  k ) do not pose any particular problems, and may be accomplished in the same
way as for the MDCEV model (see Bhat, 2008).
4. THE MODEL WITH AN OUTSIDE GOOD
Thus far, the discussion has assumed that there is no outside numeraire good (i.e., no
essential Hicksian composite good). If an outside good is present, the econometrics
simplify relative to the case without an outside good. To see this, label the outside good as
the first good which now has a unit price of one (i.e., p1  1) . This good, being an outside
6
The derivation is rather straightforward, but requires some cumbersome differentiation. Interested readers
may obtain the derivation from the authors.
26
good, has no interaction term effects with the inside goods; i.e., 1m  0m (k  1) . The
utility functional form of Equation (9) now needs to be modified as follows:
U ( x) 
1
1
( x
1

K
 1) 1  
1
k 2

 k
 
 k
k
m

 x



 m  x m 
1


k
 1  1   (28)
  1  1  k   km

2 m k
 m   m 

  k
 
 




In the above formula, we need  1  0 , while  k  0 for k > 1. Also, we need x1   1  0 .
The magnitude of  1 may be interpreted as the required lower bound (or a “subsistence
value”) for consumption of the outside good. As in the “no-outside good” case, the analyst
will generally not be able to estimate both  k and  k for the outside and inside goods. The
analyst will have to use either an  -profile or a  -profile. One can go through the same
procedure as earlier by writing the KT conditions and introducing stochasticity. For
identification, we impose the condition that  1  1 . The final expression for probability in
this outside good case and the GMDCEV model is the same as in Equation (24) with the
following modifications to the V k terms:
 ek*

Vk  ln(  k)  ln pk  ( k  1) ln 
 1 ( k  2) ; V1  (1  1) ln( e1*   1 ) .
  k pk

(29)
The Jacobian elements in this case simplify relative to Equation (27), with
1m  0m (k  1) . The elements now are given as follows:
J ih 

  1  1 
 1   i 1 

  zih 
 1  zih  i 1,h 1   

ph 1 
 i 1   e1   1 
 ei 1   i 1 pi 1 
h 1 
(30)
27
5. EMPIRICAL DEMONSTRATION
5.1. The Context
Transportation expenses account for nearly 20 percent of total household expenses and 1215 percent of total household income. It is little surprise, therefore, that the study of
transportation expenditures has been of much interest in recent years (Gicheva et al., 2007;
Cooper, 2005; Hughes et al., 2006; Thakuriah and Liao 2005, 2006, Choo et al., 2007,
Sanchez et al., 2006). Several of these studies examine the factors that influence total
household transportation expenditures and/or examine transportation expenditures in
relation to expenditures on other commodities and services (such as in relation to housing,
telecommunications, groceries, and eating out). But there has been relatively little research
on identifying the many disaggregate-level components of transportation expenditures -rather all transportation expenditures are usually lumped into a single category.
In the current paper, we demonstrate the use of the GMDCEV model for an
empirical case of household transportation expenditures in six disaggregate categories: (1)
Vehicle purchase, (2) Gasoline and motor oil (termed as gasoline in the rest of the
document), (3) Vehicle insurance, (4) Vehicle operation and maintenance (labeled as
vehicle maintenance from hereon), (5) Air travel, and (6) Public transportation. In addition,
we consider all other household expenditures in a single “outside good” category that
lumps all non-transportation expenditures, so that total transportation expenditure is
endogenously determined. Households expend some positive amount on the “outside good”
category, while expenditures are zero for one or more transportation categories for some
households.
28
Data for the analysis is drawn from the 2002 Consumer Expenditure (CEX) Survey,
which is a national level survey conducted by the US Census Bureau for the Bureau of
Labor Statistics (BLS, 2003). This survey has been carried out regularly since 1980 and is
designed to collect information on incomes and expenditures/buying habits of consumers in
the United States. In addition, information on individual and household socio-economic,
demographic, employment, and vehicle characteristics is also collected. Details of the data
and sample extraction process for the current analysis are available in Ferdous et al.,
(2010). A non-additively separable utility form is adopted to accommodate rich substitution
patterns as well as allowing complementarity among the transportation expenditure
categories. Since annual household income is considered exogenous in the current analysis,
we use the proportion of annual household income spent in each of the six transportation
categories and the “outside” non-transportation category as the dependent variables in the
analysis.
The final sample for analysis includes 4100 households with the information
identified above.
About one-quarter of the sample reports expenditures on vehicle
purchase. About 94% of the sample incurs expenditures on gasoline, and about 90% of the
sample indicates vehicle maintenance expenses. About 80 percent of the sample has
vehicle-insurance related expenses, suggesting that a sizeable number of households
operate motor vehicles with no insurance or have insurance costs paid for them (possibly by
an employer or self-employed business). About one-third of the sample reports spending
money on public transportation and air travel. All together, expenditures on transportationrelated items account for about 15 percent of household income, a figure that is quite
consistent with reported national figures. Of these 4100 households, a random sample of
29
3600 households was used for model estimation and the remaining sample of 500
households was held for validation.
5.2 Model Specification
The MDCEV and GMDCEV models were estimated using the GAUSS matrix
programming language. The analysis first estimated the best MDCEV model based on
intuitive and statistical significance considerations, and then explored alternative
specifications for the interaction parameters in the GMDCEV model. The  profile of
Equation (10) is used in all specifications, because this  profile consistently provided
better results than the  profile. Also, the  1 value for the outside good is set to zero for
convenience and stability in convergence.
The results of the final MDCEV and GMDCEV specifications are provided in Table
1. In this paper, the emphasis is on demonstrating the application of the GMDCEV model,
and not expressly on contributing to a substantive understanding of expenditure patterns or
to policy analysis related to expenditures. However, the empirical results are intuitive. Also,
while there are differences in the estimated coefficients between the MDCEV and
GMDCEV models, the general pattern and direction of variable effects are similar. The
alternative specific constants in the baseline utility for all the transportation categories are
negative, indicating the generally higher baseline utility of the “outside” non-transportation
good category relative to each transportation category (this is a reflection of the higher
expenditure on the outside good than on the transportation categories). As the number of
workers increases, so does the proportion of income allocated to all vehicle-related
30
transportation expenses, presumably to support the transportation needs of multi-worker
households. In the context of income, the middle income group (30-70K annual income)
spends a lower proportion of its income on gasoline relative to the low income group. This
result indicates that transportation expenditures constitute a major share of expenditures for
the low income group. Higher income groups also tend to spend a lower proportion of their
resources on gasoline, most likely due to a travel saturation effect combined with high
income. As one would expect, the proportion of vehicle insurance expenses decreases as
income rises, while the proportion on new vehicle purchases and air travel increases as
income rises. Multicar households tend to allocate a greater proportion of their income to
all transportation categories, except on public transportation (the effect of multicar
households on public transportation is negative in the MDCEV model, and positive but
statistically insignificant in the GMDCEV model). Finally, non-caucasians, those residing
in urban areas, and those living in the Northeast and West regions of the United States
spend a higher proportion on public transportation than Caucasians, those residing in nonurban areas, and those living in the South and Midwest regions of the US.
The satiation parameters, as estimated by the  k parameters, indicate statistically
significant satiation. The interaction parameters indicate a significant complementary effect
in vehicle purchase and gasoline expenditures, and in vehicle purchase and vehicle
maintenance expenditures. Also, as expected, there are complementary effects in the
expenditures on gasoline, vehicle insurance, and vehicle maintenance, as well as between
air travel and public transportation expenditures. This last complementary effect perhaps
reflects the use of public transportation to get to/from the airport and the use of public
31
transportation at the non-home end. On the other hand, there are particularly sensitive
substitution effects in gasoline and air/public transportation expenditures, and vehicle
insurance and air/public transportation expenditures. Such complementary and rich
substitution effects are not possible within the additively-separable utility formulation of
the MDCEV model framework, and require the non-additive utility formulation of the
GMDCEV framework proposed here.
5.3. Model Evaluation
In this section, we compare the model performance of the GMDCEV and the MDCEV
models both in the estimation sample of 3600 households as well as a validation sample of
500 households.
In terms of model fit in the estimation data, the log-likelihood value at convergence
of the GMDCEV model is -36522, while that of the MDCEV model is -37045. A likelihood
ratio test between these two models returns a value of 1046, which is larger than the chisquared statistic value with 10 degrees of freedom at any reasonable level of significance,
indicating the substantially superior fit of the GMDCEV model compared to the MDCEV
model. Of course, both the MDCEV and GMDCEV models at convergence provide a much
better data fit than the naïve MDCEV model with only the alternative-specific constant
terms and the translation parameters (with the effect of all explanatory variables assumed to
be zero), which has a log-likelihood value of -37692.
To further compare the performance of the MDCEV and GMDCEV models, we
computed an out-of-sample log-likelihood function (OSLLF) using the validation sample of
500 observations for (a) the naïve MDCEV model with only constants and translation
32
parameters model, (b) the MDCEV model with independent variables, and (c) the
GMDCEV model. The OSLLF is computed by plugging in the out-of-sample (i.e.,
validation) observations into the log-likelihood function, while retaining the estimated
parameters from the estimation sample. As indicated by Norwood et al. (2001), the model
with the highest value of OSLLF is the preferred one, since it is most likely to generate the
set of out-of-sample observations. Table 2 reports the OSLLF values for the entire
validation sample (of 500 households) as well as for different socio-demographic segments
within the sample. As can be observed from the first row, the OSLLF for the MDCEV and
GMDCEV models are higher than for the naïve MDCEV model, and the OSLLF for the
GMDCEV model is higher than the MDCEV model. The OSLLF for the MDCEV model
is, in general, higher than the OSLLF for the naïve MDCEV model for all sociodemographic segments, except for the “number of workers in household = 0” and “number
of vehicles > 2” segments. Finally, the OSLLF for the GMDCEV model is consistently
higher than those of both the MDCEV and naïve MDCEV models for all sociodemographic segments.
In summary, the data fit of the GMDCEV model is superior to that of the MDCEV
model in both estimation and validation samples.
6. CONCLUSIONS
Classical discrete and discrete-continuous models deal with situations where only one
alternative is chosen from a set of mutually exclusive alternatives. Such models assume
that the alternatives are perfectly substitutable for each other. On the other hand, many
consumer choice situations are characterized by the simultaneous demand for multiple
33
alternatives that are imperfect substitutes or even complements for one another. The
traditional MDCEV model developed by Bhat adopts a additively-separable utility form
that assumes that the marginal utility of a good is independent of the consumption amounts
of other goods. It also is not able to allow complementarity among goods. This paper
develops a closed-form model formulation that allows a non-additive utility structure and
complementarity effects. As importantly, the utility functional form proposed here remains
within the class of flexible forms, while also retaining global theoretical consistency
properties (unlike the Translog and related flexible quadratic functional forms). The result
is also clarity in the interpretation of the model parameters.
The proposed GMDCEV model should have several applications. In the current
paper, we demonstrate the application of the formulation to the empirical case of household
transportation expenditures in six disaggregate categories: (1) Vehicle purchase, (2)
Gasoline and motor oil, (3) Vehicle insurance, (4) Vehicle operation and maintenance, (5)
Air travel, and (6) Public transportation. In addition, we consider other household
expenditures in a single “outside good” category that lumps all non-transportation
expenditures, so that total transportation expenditure is endogenously determined.
Households expend some positive amount on the “outside good” category, while
expenditures are zero for one or more transportation categories for some households. Data
for the analysis is drawn from the 2002 Consumer Expenditure (CEX) Survey, which is a
national level survey conducted by the US Census Bureau for the Bureau of Labor Statistics
(BLS, 2003). The results indicate statistically significant complementary and substitution
effects in the utilities of selected pairs of transportation categories, and indicates the
34
substantially superior data fit of the GMDCEV model relative to the MDCEV model. The
GMDCEV model performed better in a validation sample as well.
35
REFERENCES
Ahn J, Jeong G, Kim Y. 2008. A forecast of household ownership and use of alternative
fuel vehicles: A multiple discrete-continuous choice approach. Energy Economics
30(5): 2091-2104.
Arora N, Allenby GM, Ginter JL. 1998. A hierarchical Bayes model of primary and
secondary demand. Marketing Science 17: 29-44.
Bhat CR. 2005. A multiple discrete-continuous extreme value model: formulation and
application to discretionary time-use decisions. Transportation Research Part B 39(8):
679-707.
Bhat CR. 2008. The multiple discrete-continuous extreme value (MDCEV) model: role of
utility function parameters, identification considerations, and model extensions.
Transportation Research Part B 42(3): 274-303.
Bhat CR, Sen S, Eluru N. 2009. The impact of demographics, built environment attributes,
vehicle characteristics, and gasoline prices on household vehicle holdings and use.
Transportation Research Part B 43(1): 1-18.
Bunch DS. 2009. Theory-based functional forms for analysis of disaggregated scanner
panel data. Technical working paper, Graduate School of Management, University of
California-Davis, Davis CA.
Bureau of Labor Statistics (BLS). 2003. 2002 Consumer expenditure interview survey
public use microdata documentation. U.S. Department of Labor Bureau of Labor
Statistics, Washington, D.C., http://www.bls.gov/cex/csxmicrodoc.htm#2002.
Chiang J. 1991. A simultaneous approach to whether to buy, what to buy, and how much to
buy. Marketing Science 4: 297-314.
Chintagunta PK. 1993. Investigating purchase incidence, brand choice and purchase
quantity decisions of households. Marketing Science 12: 194-208.
Christensen L, Jorgenson D, Lawrence L. 1975. Transcendental logarithmic utility
functions. The American Economic Review 65: 367-83.
Choo S, Lee T, Mokhtarian PL. 2007. Do transportation and communications tend to be
substitutes, complements, or neither? U.S. consumer expenditures perspective, 19842002. Transportation Research Record 2010: 121-132.
36
Cooper M. 2005. The impact of rising prices on household gasoline expenditures.
Consumer Federation of America, www.consumerfed.org/.
Deaton A, Muellbauer J. 1980. Economics and Consumer Behavior. Cambridge University
Press: Cambridge.
Ferdous N, Pinjari AR, Bhat CR, Pendyala RM. 2010. A comprehensive analysis of
household transportation expenditures relative to other goods and services: an
application to United States consumer expenditure data. Transportation, forthcoming.
Gicheva D, Hastings J, Villas-Boas S. 2007. Revisiting the income effect: gasoline prices
and grocery purchases. Working Paper No. 13614, National Bureau of Economic
Research, Cambridge, MA.
Hanemann WM. 1978. A methodological and empirical study of the recreation benefits
from water quality improvement. Ph.D. dissertation, Department of Economics,
Harvard University.
Hanemann WM. 1984. The discrete/continuous model of consumer demand. Econometrica
52: 541-561.
Herriges JA, Kling CL, Phaneuf DJ. 2004. What’s the use? Welfare estimates from
revealed preference models when weak complementarity does not hold. Journal of
Environmental Economics and Management 47: 55-70.
Hughes JE, Knittel CR, Sperling D. 2006. Evidence of a shift in the short-run price
elasticity of gasoline demand. The Energy Journal, 29(1), 93-114.
Kim J, Allenby GM, Rossi PE. 2002. Modeling consumer demand for variety. Marketing
Science 21: 229-250.
Mäler K-G. 1974. Environmental Economics: A Theoretical Inquiry. The Johns Hopkins
University Press for Resources for the Future, Baltimore, MD.
Mohn C, Hanemann M. 2005. Caught in a corner: using the Kuhn-Tucker conditions to
value Montana sportfishing. In Explorations in Environmental and Natural Resource
Economics: Essays in Honor of Gardner M. Brown, Jr., Halvorsen R, Layton DF (eds).
Ch 10, pp. 188-207, Edward Elgar Publishing, Inc: Northampton, MA.
37
Norwood B, Ferrier P, Lusk J. 2001. Model selection criteria using likelihood functions and
out-of-sample performance. Paper presented at the NCR-134 Conference on Applied
Commodity Price Analysis, Forecasting, and Market Risk Management, St. Louis,
Missouri, April 23-24.
Phaneuf DJ, Herriges JA. 2000. Choice set definition issues in a Kuhn-Tucker model of
recreation demand. Marine Resource Economics 14: 343-355.
Phaneuf DJ, Kling CL, Herriges JA. 2000. Estimation and welfare calculations in a
generalized corner solution model with an application to recreation demand. The
Review of Economics and Statistics 82(1): 83-92.
Pollak R, Wales T. 1992. Demand System Specification and Estimation. Oxford University
Press: New York.
Roy R. 1947. La distribution du revenu entre les divers biens. Econometrica 15: 205-225.
Sanchez TW, Makarewicz C, Hasa PM, Dawkins CJ. 2006. Transportation costs, inequities,
and trade-offs. Presented at the 85th Annual Meeting of the Transportation Research
Board, Washington, D.C., January.
Sauer J, Frohberg K, Hockmann H. 2006. Stochastic efficiency measurement: The curse of
theoretical consistency. Journal of Applied Economics 9: 139-165.
Thakuriah P, Liao Y. 2005. An analysis of variations in vehicle-ownership expenditures.
Transportation Research Record 1926: 1-9.
Thakuriah P, Liao Y. 2006. Transportation expenditures and ability to pay: evidence from
consumer expenditure survey. Transportation Research Record 1985: 257-265.
Vasquez Lavin F, Hanemann M. 2008. Functional forms in discrete/continuous choice
models with general corner solution. Department of Agricultural & Resource
Economics, University of California Berkeley. CUDARE Working Paper 1078.
von Haefen RH. 2003. Incorporating observed choice into the construction of welfare
measures from random utility models. Journal of Environmental Economics &
Management 45(2): 145-165.
von Haefen RH. 2004. Empirical strategies for incorporating weak complementarity into
continuous demand system models. Working paper, Department of Agricultural and
Resource Economics, University of Arizona.
38
von Haefen RH, Phaneuf DJ. 2005. Kuhn-Tucker demand system approaches to nonmarket valuation. In Applications of Simulation Methods in Environmental and
Resource Economics, Scarpa R, Alberini AA (eds). Springer: The Netherlands.
von Haefen RH, Phaneuf DJ, Parsons GR. 2004. Estimation and welfare analysis with large
demand systems. Journal of Business and Economic Statistics 22(2): 194-205.
Wales TJ, Woodland AD. 1983. Estimation of consumer demand systems with binding
non-negativity constraints. Journal of Econometrics 21(3): 263-85.
39
LIST OF FIGURES
Figure 1. Effect of  k , due to Negative  kk , on Good k’s Subutility Function Profile
Figure 2. Effect of  k , due to Positive  kk , on Good k’s Subutility Function Profile
Figure 3. Effect of  k , due to Negative  kk , on Good k’s Subutility Function Profile
Figure 4. Effect of  k , due to Positive  kk , on Good k’s Subutility Function Profile
Figure 5. Alternative Subutility Profiles (for Good k) with Different  kk and  k Values
Figure 6. Alternative Subutility Profiles (for Good k) with Different  kk and  k Values
LIST OF TABLES
Table 1. Model Estimation Results
Table 2. Predictive Likelihood-based Measures of Fit in the Validation Sample
40
Figure 1. Effect of  k , due to Negative  kk , on Good k’s Subutility Function Profile
Figure 2. Effect of  k , due to Positive  kk , on Good k’s Subutility Function Profile
41
Figure 3. Effect of  k , due to Negative  kk , on Good k’s Subutility Function Profile
Figure 4. Effect of  k , due to Positive  kk , on Good k’s Subutility Function Profile
42
Figure 5. Alternative Subutility Profiles (for Good k) with Different  kk and  k Values
Figure 6. Alternative Subutility Profiles (for Good k) with Different  kk and  k Values
43
Table 1. Model Estimation Results
MDCEV
t-stat
Parameter
t-stat
-7.127
-67.76
-8.143
-18.27
Gasoline/oil
-2.523
-35.02
-2.919
-33.56
Vehicle insurance
Vehicle maintenance
-3.975
-3.446
-63.50
-52.87
-4.475
-4.230
-28.22
-30.29
Air travel
-6.144
-65.24
-5.196
-41.96
Public transportation
-5.820
-39.30
-4.699
-48.43
0.182
4.39
0.189
3.53
0.209
0.081
0.192
5.70
2.82
6.79
0.254
0.104
0.281
5.74
2.54
6.14
Annual HH income 30-70K
Vehicle purchase
Gasoline
Air travel
0.809
-0.284
0.756
7.88
-3.43
7.95
1.404
-0.300
0.281
4.10
-3.03
5.01
Annual HH income >70K
Vehicle purchase
Gasoline
Vehicle insurance
Air travel
0.807
-0.792
-0.337
1.188
6.18
-6.06
-3.81
9.14
1.461
-0.881
-0.332
0.536
4.03
-5.22
-2.81
5.44
0.304
10.58
0.379
11.38
Gasoline
0.305
9.21
0.387
14.11
Vehicle insurance
Vehicle maintenance
0.275
0.269
10.89
11.00
0.348
0.364
12.03
13.65
Air travel
Public transportation
0.073
-0.122
2.04
-3.24
0.084
0.010
4.79
1.19
Non-Caucasian HH – Public transportation
0.417
5.71
0.084
2.02
Urban location – Public transportation
0.491
4.02
0.085
2.84
North East Region – Public transportation
0.722
9.64
0.182
3.80
Western Region – Public transportation
0.590
8.50
0.131
3.48
Baseline Utility Parameters
Baseline Constants
Vehicle purchase
Number of workers in household
Vehicle purchase
Gasoline
Vehicle Insurance
Vehicle Maintenance
Number of vehicles in household
Vehicle purchase
Parameter
GMDCEV
44
Table 1 (Continued.) Model Estimation Results
MDCEV
Translation (  k ) Parameters
Vehicle purchase
Gasoline
GMDCEV
Parameter
t-stat
Parameter
20.889
0.196
11.10
10.58
21.607
0.166
10.67
8.81
Vehicle insurance
Vehicle maintenance
0.613
0.284
17.48
16.98
0.579
0.269
15.78
16.12
Air travel
Public transportation
0.677
0.237
13.81
17.70
0.548
0.187
13.44
16.81
Interaction Parameters (  km parameters)
Vehicle purchase and gasoline
Vehicle purchase and vehicle insurance
Vehicle purchase and vehicle maintenance
Vehicle purchase and air travel
Vehicle purchase and public transportation
Gasoline and vehicle insurance
t-stat
-
-
1.355x10-3
0.323x10-3
2.053x10-2
3.22
2.11
4.09
Gasoline and vehicle maintenance
Gasoline and air travel
Gasoline and public transportation
-
-
5.167x10-2
-5.216x10-3
-8.205x10-3
6.33
-3.95
-4.91
Vehicle insurance and vehicle maintenance
Vehicle insurance and air travel
Vehicle insurance and public transportation
Vehicle maintenance and air travel
Vehicle maintenance and public transportation
Air travel and public transportation
-
-
3.248x10-3
-1.409x10-3
-2.488x10-3
6.607x10-3
2.16
-4.14
-5.27
10.78
Log-likelihood with only constants and
translation parameters
-37692
-37692
Log-likelihood at convergence
-37045
-36522
45
Table 2. Predictive Likelihood-based Measures of Fit in the Validation Sample
Sample details
Full validation sample
Number of workers in HH
0
1
2
>2
Household income ($)
< 30K/annum
30K-70K/annum
>70K/annum
Number of vehicles
0
1
2
>2
Race
Non-Caucasian
Caucasian
Residential location
Urban
Rural
Number of
observations
OSLLF for MDCEV
OSLLF for MDCEV
OSLLF for GMDCEV
500
-5630.58
-5575.26
-5475.03
14
109
240
137
-147.24
-1160.73
-2696.74
-1625.86
-147.99
-1139.70
-2667.63
-1619.94
-145.57
-1119.53
-2608.45
-1601.48
10
168
322
-103.25
-1865.47
-3661.86
-100.62
-1862.07
-3612.57
-100.40
-1835.87
-3538.75
13
94
179
214
-149.76
-1045.80
-1881.21
-2553.81
-141.42
-1016.61
-1852.94
-2564.29
-140.18
-1004.42
-1819.24
-2511.19
453
47
-5097.93
-532.64
-5047.85
-527.42
-4953.62
-521.41
470
30
-5272.37
-358.21
-5217.54
-357.72
-5125.17
-349.86
46