rex y. du & wagner a. kamakura where did all that money go ... did all that... · allocation...

109Journal of MarketingVol. 72 (November 2008), 109–131

© 2008, American Marketing AssociationISSN: 0022-2429 (print), 1547-7185 (electronic)

Rex Y. Du & Wagner A. Kamakura

Where Did All That Money Go?Understanding How Consumers

Allocate Their Consumption BudgetAll types of consumer expenditures ultimately vie for the same pool of limited resources—the consumer’sdiscretionary income. Consequently, consumers’ spending in a particular industry can be better understood inrelation to their expenditures in others. Although marketers may believe that they are operating in distinct andunrelated industries, it is important to understand how consumers, with a given budget, make trade-offs betweenmeeting different consumption needs. For example, how much would escalating gas prices affect consumerspending on food and apparel? Which industries would gain most in terms of extra consumer spending as a resultof a tax rebate? Answers to these questions are also important from a public policy standpoint because theyprovide insights into how consumer welfare would be affected as consumers reallocate their consumption budgetin response to environmental changes. This study proposes a structural demand model to approximate thehousehold budget allocation decision, in which consumers are assumed to allocate a given budget across a fullspectrum of consumption categories to maximize an underlying utility function. The authors illustrate the modelusing Consumer Expenditure Survey data from the United States, covering 31 consumption categories over 22years. The calibrated model makes it possible to draw direct inferences about the trade-offs individual householdsmake when they face budget constraints and how their relative preferences for different consumption categoriesvary across life stages and income levels. The study also demonstrates how the proposed model can be used inpolicy simulations to quantify the potential impacts on consumption patterns due to shifts in prices or discretionaryincome.

Keywords: consumer expenditures, demand system, consumption, household budget allocation

Rex Y. Du is Assistant Professor of Marketing, Terry College of Business,University of Georgia (e-mail: [email protected]). Wagner A. Kamakurais Ford Motor Co. Professor of Global Marketing, Fuqua School of Busi-ness, Duke University (e-mail: [email protected]).

The majority of models of consumer demand in themarketing literature focus on within-category pur-chase decisions—for example, incidence, brand

choice, and quantity within a single product category (e.g.,Chiang 1991; Chintagunta 1993; Gupta 1988). Morerecently, several models have been developed to analyzechoice behavior across multiple categories using shoppingbasket data (Seetharaman et al. 2005). For example, Man-chanda, Ansari, and Gupta (1999), Russell and Petersen(2000), and Chib, Seetharaman, and Strijnev (2002) exam-ine multicategory purchase incidence decisions. Russell andKamakura (1997), Ainslie and Rossi (1998), Iyengar,Ansari, and Gupta (2003), and Singh, Hansen, and Gupta(2005) investigate multicategory brand choice decisions.Song and Chintagunta (2006) model multicategory inci-dence and brand choice decisions jointly, and Song andChintagunta (2007) allow for incidence, brand choice, andquantity decisions across multiple categories.

In contrast to research that focuses on multicategorychoice behavior, in which the budget for a particular shop-ping trip is allocated across a few selected product cate-

gories, few empirical studies in the marketing literaturehave focused on modeling how consumers allocate theirlimited discretionary income to meet different consumptionneeds, in which trade-offs must be made across a wide rangeof expenditure categories (e.g., food, apparel, recreation,transportation, medical and personal care). The empiricalstudies that examine consumer expenditures either aredescriptive in nature (e.g., Ferber 1956; Ostheimer 1958),focusing on a particular demographic group (e.g., elderly[Goldstein 1968], working wives [Bellante and Foster1984]) or a particular consumption category (e.g., food[Rogers and Green 1978], energy [Fritzsche 1981], services[Soberon-Ferrer and Dardis 1991]), or use univariate mod-els that ignore the interdependencies across consumptioncategories (e.g., Du and Kamakura 2006; Rubin, Riney, andMolina 1990; Wagner and Hanna 1983; Wilkes 1995).

Empirical studies on consumption have been far morecommon in economics than in marketing. In economics, themain issue has been the intertemporal trade-offs consumersmake when choosing between current and future consump-tion (e.g., Deaton 1992; Gourinchas and Parker 2002); here,all the consumption expenditures are typically lumped intoone aggregate account, and the real focus is on the accumu-lation of assets/debts. In the instances when economistsconsider the allocation of current consumption budget intodifferent products and services, the focus has been either ona small number of broad commodity groups, such as food,clothing, and housing (e.g., Deaton and Muellbauer 1980;

110 / Journal of Marketing, November 2008

Pollak and Wales 1978), or on a limited set of more nar-rowly defined goods, such as different types of recreation(Phaneuf, Kling, and Herriges 2000), urban transportation(Kockelman 2001), food consumed at home (Kao, Lee, andPitt 2001; Kiefer 1984), energy (Bousquet, Chakir, andLadoux 2004), and print/online newspapers (Gentzkow2007).

This relative disregard for consumption budget alloca-tion in marketing research might be due to the primaryinterest of manufacturers and retailers in influencing con-sumer choices toward their brands. However, consumerexpenditures across seemingly unrelated industries areultimately interrelated through the budget constraint andtherefore should be viewed from a systemic perspective.Marketing researchers should devote more attention tounderstanding how consumers allocate their consumptionbudget to meet all kinds of competing needs and how theresultant expenditure patterns are influenced by factors suchas income levels, inflation rates, and family life stages. Forexample, recent economic trends in the United States havefocused consumer attention on concerns such as spiralingprice increases in energy, health care, and education. Firmssuch as supermarkets that expand into mass-merchandisingand fuel centers or manufacturers that diversify into con-glomerates with multiple lines of business need to under-stand and anticipate how industry demands change as con-sumers respond to these dramatic shifts in prices byadjusting their consumption patterns while satisfying theirbudget constraints. Which industries are most affected?How do the responses differ from low- to high-incomehouseholds and from younger to older families? Similarquestions have also been raised in the public policy arenabecause policy makers need to understand how spikes in thecosts of health care, education, and energy affect con-sumers’ everyday lives or to make projections about thepotential impact on consumption of a new tax levy orrebate. To answer such questions, demand models that canapproximate individual households’ budget allocation deci-sions for any given prices across a full spectrum of con-sumption categories are needed.

Furthermore, as is often noted, tomorrow’s consumerswill have more choices, and as a result, tomorrow’s firmswill face increasingly more intense cross-industry competi-tion as consumers make trade-offs not only within productcategories but also across categories. For example, a recentstudy showed a decline in confectionery sales among teensbecause teens were using their pocket money for text mes-saging, not candy bars; similarly, after analyzing consumerfood expenditure data, Campbell Soup Company realizedthat it was competing across a variety of industries thatincluded McDonald’s and Burger King (Allen and Rigby2005). Thus, marketers need to understand better howconsumer spending in one industry may be substituted byexpenditures in others and how such cross-industry substi-tution patterns may vary across the population. To do this,again, it is necessary to model consumer budget allocations.

Finally, as households evolve through the family lifecycle, their consumption priorities and, therefore, expendi-ture pattern will change. In a society such as the UnitedStates, in which demographic composition has been going

through major shifts, the proportion of consumers in differ-ent life stages will vary substantially over time. Thus, toproject primary demand (i.e., the demand for a generalproduct category), it is necessary to consider simultane-ously population trends and differences in consumption pri-orities of different demographic groups.

In summary, we believe that modeling consumer budgetallocation should be of particular importance to marketersand public policy makers, especially in light of the dynam-ics of economic conditions (e.g., different inflation rates indifferent industries), cross-industry competition, and demo-graphic trends (e.g., aging population, more nontraditionalhouseholds). However, to develop a proper budget alloca-tion model, the following three major challenges must beaddressed.

The first challenge is that of high dimensionality. Acompressive study of consumer budget allocation shouldinclude as complete a set of consumption categories as pos-sible. This makes high dimensionality a challenge from themodeling standpoint. For example, the consumer productsand services hierarchy used by the Bureau of Labor Statis-tics (www.bls.gov/data/) to calculate the Consumer PriceIndex (CPI) comprises 11 major categories (food, alcohol,housing, apparel, transportation, medical care, recreation,education, communication, tobacco, and personal care), and34 subcategories (e.g., food at home and away from home;alcoholic beverages at home and away from home; shelter,fuels, and utilities; household furnishings and operations).At the lowest level, the CPI hierarchy consists of approxi-mately 200 mutually exclusive items. It would be intract-able to build a budget allocation model that includes 200-plus categories. This raises the issue of the appropriate levelof aggregation. At one extreme, for example, a model thatconsiders only the 11 major CPI categories could be built.The upside would be the relative ease of implementation,but the downside would be a potential loss of importantbehavioral and managerial insights. Currently availableconsumer demand models, such as the linear expendituresystem (LES) (Kao, Lee, and Pitt 2001; Pollak and Wales1969; Stone 1954), the almost ideal demand system (AIDS)(Barnett and Seck 2006; Deaton and Muellbauer 1980;Dreze, Nisol, and Vilcassim 2004), the Rotterdam model(Barten 1964; Clements and Selvanathan 1988; Theil 1965;Vilcassim 1989), or the translog model (Christensen, Jor-genson, and Lau 1975; Srinivasan and Winer 1994), wouldbe impractical to account for the interdependencies acrossmore than a few consumption categories because of thelarge number of covariance terms and/or cross-categoryinteraction effects to be estimated. This is also the case forthe existing multicategory choice models in the marketingliterature. Thus, to strike a balance between practicality andrichness, we propose a factor-analytic extension of thedemand system that Kao, Lee, and Pitt (2001) propose,making it feasible for inclusion of a set of 31 expenditurecategories, which are aligned (to the degree data are avail-able) with the CPI subcategory structure mentionedpreviously.

The second challenge is that of binding nonnegativeconstraints. A common feature of consumer expendituredata is that most households spend money in only a subset

Where Did All That Money Go? / 111

of categories. The pattern of zero consumptions varies fromhousehold to household and contains important informationabout individual preferences, making it crucial to modelthem explicitly. Unfortunately, such a requirement rules outany demand systems in which all categories are assumed tobe consumed by all households, an assumption we demon-strate subsequently to be vastly violated in householdexpenditure data. In other words, heavily censored expendi-ture data render many popular demand systems, such as theAIDS, Rotterdam, or translog, inapplicable because they areall derived from first-order conditions of constrained utilitymaximization problems, assuming the existence of interiorsolutions for all goods. As a result, they would predict posi-tive expenditures in all categories for all households, whichwould be biased and inconsistent with the actual data. Incontrast, we accommodate the binding nonnegative con-straints observed in household budget allocations byproposing a budget allocation model in which both the“whether-to-spend” and the “how-much-to-spend” deci-sions result from a common utility maximization problem,allowing for inferences about a unified preference structurefrom both zero and nonzero observations. An importantadvantage of using a common utility function for both inci-dence and quantity decisions is parsimony, which is desir-able given the high dimensionality of the demand system.

The third challenge is that of unobservable heterogene-ity. Consumer expenditure data consistently depict largevariations in the pattern of budget allocation across house-holds. Manifested in these patterns are the different con-sumption priorities of different households. Capturing theseindividual differences is important for models of budgetallocation because doing so can provide valuable insightsinto how consumers would respond differently to changesin external (e.g., price shocks, tax rebates) and internal(e.g., life stage changes) conditions. To account for hetero-geneity in preferences, existing studies of household budgetallocation have relied solely on demographic variables,such as age, income, ethnicity, and family composition.However, as the choice modeling literature has shown,demographics can often explain only a small portion ofheterogeneity in consumer preferences. In the context ofbudget allocation, unobservable heterogeneity may berevealed through the interdependencies of consumptionacross categories. This is not a trivial task given the twochallenges (i.e., high dimensionality and binding nonnega-tive constraints) we discussed previously. Conversely, as weshow in our empirical application, accounting for unobserv-able heterogeneity is beneficial in that it leads to a moreflexible demand system at the aggregate level.

In summary, we believe that it is important for mar-keters to be able to infer the underlying cross-categorytrade-offs households make in allocating their consumptionbudget, so that we can then predict how these allocationswill change in response to shifts in prices or discretionaryincome and how differences in preferences across house-holds lead to different consumption patterns. To understandhow households allocate their consumption budget across afull spectrum of expenditure categories, we use a budgetallocation model built on an approach first proposed byWales and Woodland (1983) and then extended by Kao,

Lee, and Pitt (2001) (hereinafter, the KLP model), assumingthat households allocate their consumption budget to maxi-mize a utility function that is linear in the logarithms ofquantity consumed less a constant for each category (this isalso referred to as the “Stone–Geary utility”). This budgetallocation model tackles the aforementioned three chal-lenges, leading to a demand system that (1) is feasible for alarge number of consumption categories (which is not thecase with the KLP model), (2) allows for corner solutionsobserved in censored expenditure data, and (3) results in aglobally flexible aggregate demand system for each con-sumption category by obtaining household-level estimatesof the direct utility function. Next, we briefly review theLES model and describe our more flexible extension to theKLP model. Then, we apply our proposed factor-analyticrandom coefficients budget allocation model to the Con-sumer Expenditure Survey (CEX) data from the UnitedStates, covering 31 consumption categories over a period of22 years. We conclude with a discussion and directions forfurther research.

Modeling Consumption BudgetAllocation

Our main purpose in this study is to develop a reasonable“as-if” model that approximates how individual householdsallocate their consumption budget across a comprehensiveset of expenditure categories. Because each household con-sumes only a subset of all consumption categories, ourobserved expenditure data are censored. Thus, we need ademand system that can accommodate many consumptioncategories and allow for corner solutions. As we mentionedpreviously, these requirements rule out the most populardemand systems, such as the AIDS, Rotterdam, andtranslog models, all of which (1) quickly become impracti-cal for more than a couple dozen consumption categoriesand (2) require nonzero consumption in all categories. Forthese reasons, we extend the KLP budget allocation modelto consider simultaneously whether-to-spend and how-much-to-spend decisions. Our model is distinguished fromthe KLP model in two important aspects. First, the KLPmodel is feasible only when the focus is on a small numberof consumption categories, accounting for only part of thehousehold budget (e.g., seven types of food items). How-ever, for a comprehensive analysis of household budgetallocation, the high-dimensionality challenge is inevitable.Rather than limiting the analysis to a high level of aggrega-tion or lumping a large number of distinct expenditure itemsinto an “other” category, our proposed approach affords theflexibility to cover the full spectrum of household consump-tion budget allocation at a low level of aggregation. Second,to account for a large number of consumption categorieswhile allowing for individual differences in category prefer-ences and a rich pattern of correlation in these preferencesacross categories, we extend the KLP model by imposing aflexible factor structure to the covariance matrix of the sto-chastic taste parameters that govern individual households’consumption priorities.

We assume that household h maximizes a continuouslydifferentiable, quasi-concave direct utility function G(xh)


over a set of J nonnegative quantities xh = (x1h, x2h, …, xJh),subject to a budget constraint p′xh ≤ mh, where p = (p1, p2,…, pJ)′ > 0, pi is the price of good i, and mh is household h’stotal consumption budget/discretionary income. Followingthe work of Wales and Woodland (1983) and Kao, Lee, andPitt (2001), we use the Stone–Geary utility function, whichhas the following form:

where αih > 0, (xih – βi) > 0, and J is the number of all avail-able consumption categories. The h subscript in αih impliesthat the utility function is household specific. The Kuhn–Tucker conditions for the household’s optimization problemare as follows: ∂G(xh)/∂xih – ξpi ≤ 0 for xih = 0, and ∂G(xh)/∂xih – ξpi = 0 for xih > 0, such that p′xh – mh ≤ 0 ≤ ξ, whereξ denotes the Lagrange multiplier or marginal utility perdollar.

The budget allocation problem we described impliesthat the household incrementally allocates its discretionaryincome to the consumption category that produces the high-est marginal utility per dollar,

given the current consumption levels xh, until the budget isreached, . The solution of this optimizationproblem leads to the following expenditure system, which islinear in discretionary income and prices (thus the labelLES in the literature):

where and J* is the set of consumedgoods (i.e., with positive expenditures).

Note that the demand system given by Equation 2 isdefined only for a particular consumption regime J*. If thepattern of nonzero expenditures changes, both the interceptsand the slopes of the demand system will also change. Inother words, the model implies that there is an optimal con-sumption regime for each household at each combination ofbudget and prices, and only within a particular consumptionregime are category expenditures linear functions of budgetand prices; across consumption regimes, the demand sys-tem is piecewise linear. In addition, rather than imposingarbitrary censoring mechanisms, such as the Tobit regres-sion model (Amemiya 1974), the approach based on theKuhn–Tucker condition allows for zero consumption as acorner solution to a constrained utility maximization prob-lem. It also ensures that predicted expenditures will alwaysbe nonnegative and sum to the budget.

Unlike existing budget allocation analyses, which eitherignore heterogeneity or allow for heterogeneity onlythrough demographics, we take advantage of the multivari-ate nature of the estimation problem (i.e., 31 points ofexpenditure data per household) and obtain household-specific estimates of the preference parameters (αih), which

θ α αih ihJ

jh* */= =Σ j 1

( ) *

*

21

p x p m pi ih i i ih h j j

j

J

= + −⎛

⎝⎜⎜

⎞

⎠⎟⎟

=∑β θ β , i = 11, 2, …, J*,

ΣiJ

i ih hp x m= =1

∂∂

=−

G x

x p p x ph

ih i

ih

i ih i i

( )

( ),

1 αβ

( ) ( ) ln( ),11

G x xh ih ih ii

J

= −=∑α β

1For identification purposes, in our empirical analysis, αh is setto 1 for food at home. In other words, all the preferences are rela-tive to the consumption of food at home. For identification pur-poses, we assume that βih is the same across households (i.e., βih =βi) as a result of an indeterminacy that would produce the samemarginal utility, ∂G(xh)/∂xih = αih/(xih – βih), at any given con-sumption point for an infinite pairs of αih and βih. In other words,this parametric assumption is imposed without loss of generalityand has no impact on any of our substantive findings.

means that the slope parameters (θ*ih) of the demand system

will be unique to each household as well.1 Because eachindividual household has a unique consumption regime(and, therefore, a different regime switching point) and adifferent set of demand function parameters, the impliedaggregate demand can be highly nonlinear, overcoming acommon criticism of the inflexibility of the original LESmodel.

Given the model we described, the researcher’s problemis to infer consumers’ utility function parameters (i.e., αihand βi) given the observed budget allocations and prices.The KLP model deals with variation in preferences acrosshouseholds by treating αih as stochastic; that is, αih =exp(γi + εih), where εh ~ N(0, Σ). Kao, Lee, and Pitt (2001)demonstrate that estimation through maximum likelihood isfeasible only for simple problems with few consumptioncategories because it requires the evaluation of a multivari-ate normal cumulative density function. To circumvent thisserious limitation, Kao, Lee, and Pitt propose to estimatetheir model with simulated maximum likelihood.

Although the KLP approach simplifies estimation con-siderably, the model formulation still limits the number ofconsumption categories that can be handled (e.g., onlyseven categories are considered). Specifically, the KLPmodel requires [(J – 1) × J]/2 parameters for the covariancematrix Σ. An analysis of the CEX data with 31 consumptioncategories would require the estimation of (30 × 31)/2 =465 covariance terms. In addition, the stochastic formula-tion of αih in the KLP model does not allow for household-level estimates of the taste parameter, which can beachieved through our formulation. This distinguishing char-acteristic of our model is particularly important because theability to estimate the household-specific taste parameterαih enables us to perform more realistic policy simulationsthat account for individual differences in consumption pri-orities and for a rich pattern of correlation in preferencesacross categories.

To account for unobserved heterogeneity in the tasteparameter (αih) for each category i and still have a model offeasible size, we propose a factor-analytic extension of theKLP random coefficients model by extracting the principalcomponents of the covariance matrix of the stochasticterms:

(3) αih = exp(γi + λiZh + εih),

and

(4) βi = min(xi) – exp(ηi), to ensure that xih – βi > 0 for ∀h,

where


eγ i = the geometric mean of the taste parameter αih forcategory i across the sample,

Zh = a p-dimensional vector of i.i.d. standard normalfactor scores for household h,

λi = a p-dimensional vector of factor loadings for cate-gory i, and

εih = a random disturbance normally distributed withmean zero and standard deviation σi.

Although γi and ηi provide insights into the averagepreference for category i, the product of the factor loadings(λi) and factor scores (Zh) will show how much higher orlower the (log) taste of household h is relative to the aver-age. Moreover, the factor loadings Λ = {λi} capture theessential information about how (log) tastes covary acrosscategories and households because the covariance matrixfor their distribution can be directly obtained as Λ′Λ. Thus,if two categories i and j have high loadings (λ) of the samesign on the same dimensions of the latent factors (Z),households assigning a high (low) utility to one categorywill also assign a high (low) utility to the other. In otherwords, the results from our factor-analytic model can pro-vide valuable insights into how interrelated the consump-tion categories are across consumers.

In summary, a key benefit of our proposed factor-analytic random coefficients LES model (compared withthe KLP model) is that it enables estimation of the directutility function for each household in more realistic applica-tions with a large number of consumption categories (highdimensionality), many of which may not be consumed byall households (censored data). Moreover, our proposedfactor-analytic extension to the KLP model allows not onlyfor unobserved heterogeneity in the household’s taste (αih)for each consumption category but also for a rich pattern ofcorrelation in these tastes across categories. Details aboutthe estimation of our model with simulated maximum like-lihood appear in the Appendix.

Implications of the BudgetAllocation Model

A common criticism of the Stone–Geary utility functionassumed in Equation 1 is that it does not allow for potentialcomplementarity between categories. Although thisassumption could be limiting when studying consumptionat a more micro level (e.g., pasta and pasta sauce should becomplementary because the utility derived from consumingthem together is greater than the sum of utilities derivedfrom consuming them separately), the additive separableutility assumed by the Stone–Geary function is not asrestrictive in a broader analysis of how consumers allocatetheir discretionary income, because at that point, all con-sumption categories are ultimately substitutes as they com-pete for the same budget.

Furthermore, as Gentzkow (2007, pp. 714–15) dis-cusses, separating complementarity between consumptioncategories from correlation of consumer preferences acrosscategories would require additional variables that discrimi-nate among consumption categories and/or longitudinaldata for each household. Unfortunately, because consumer

expenditure surveys (e.g., the CEX) are usually done on anannual basis, so that the analyst has only one observation onhow the household allocated its consumption budget acrossvarious expenditure categories, it is not possible to discerntrue complementarity between categories from correlationin consumer preferences across categories. Finally, giventhe high dimensionality involved in modeling households’budget allocations across a comprehensive list of expendi-ture categories, it is empirically intractable to considerpotential interaction effects among all the categories (e.g.,[30 × 31]/2 = 465 additional parameters would be needed toallow for all the potential interaction effects in the utilityfunction in our analysis). In summary, for both theoreticaland practical purposes, in a household budget allocationanalysis such as ours, a main-effect-only utility function,such as the Stone–Geary utility, should be viewed as a rea-sonable as-if model, which precludes complementaritybetween consumption categories.

By accounting for unobservable heterogeneity in prefer-ences across households, our factor-analytic random coeffi-cients LES model produces a globally flexible demand sys-tem when aggregated cross-sectionally. At the individuallevel, the additive separable Stone–Geary utility functionimplies that consumption of one category does not interactwith consumption of another category. This means thatpreferences for different consumption categories are locallyindependent (i.e., within a particular household, consump-tion of one category does not affect the marginal utility ofconsuming another category). However, in our proposedmodel, preferences for different consumption categories donot need to be independent across households, because thefactor structure embedded in αih allows tastes to be globallycorrelated, making it possible, for example, that consumerswho have a high (relative to other households) preferencefor tobacco products also have a high preference for alcoholconsumption.

The own- and cross-price elasticities implied in ourmodel for consumption categories i and j are defined at thehousehold level, respectively, as follows:

where

Note that these elasticities depend on the household’s factorscores (Zh) and the factor loadings for the particular cate-gories involved (λi).

Note also that the elasticities are defined at the house-hold level. Given that the taste parameters αih are correlatedacross consumption categories and households (accordingto the pattern reflected in the latent factors), our model

θα

α

γ λ ε

γih

ih

jhj

Ji i h ih

j

Z*

*

exp( )

exp(

= =+ +

+=∑

1

λλ εj h jhj

J

Z +=∑ )

.*

1

( )ln

ln

( )

ln

l

*5 1

1∂∂

= − +−

∂∂

x

p x

x

ih

i

ih i

ih

ih

θ β,

(6)

and

nn,*

p

p

p xjih

j j

i ih

= −θβ


allows for covariation in the consumption pattern acrosshouseholds, and the resultant aggregate demand system ismuch more flexible than the traditional (no unobservableheterogeneity) LES demand system. Despite this flexibility,the proposed model still assumes an additive separableutility function, which precludes complementarity betweencategories. To demonstrate this, using Equation 6, we canreadily derive the aggregate (population-wide) cross-elasticities (Equation 7) and decompose them into incomeand substitution effects (Equations 8 and 9, respectively)(Bohm and Haller 1987). By definition, Equation 7 equalsEquation 8 plus Equation 9.

Note that after the income effects (Equation 8), which areall negative, are accounted for, the substitution effects(Equation 9) are all positive. This implies that after the bud-get constraint is accounted for, all categories are substitutes,ruling out the possibility for complementarity. In otherwords, consumption of one category does not increase themarginal utility of any other category, and therefore theresultant demand system cannot predict that consumption of pasta will positively affect consumption of pasta sauce.Although this limitation might be critical in detailed analy-ses among a few product categories, such as within foodconsumption, it is less important in studies of householdexpenditures involving a larger number of broadly definedcommodity groups, particularly when data are availableonly at one period for each household.

In addition to price elasticities, another property ofdemand systems commonly considered in consumptionanalysis is the Engel curve, which relates consumption ineach category to the total consumption budget. If defined onquantity, the Engel curve implied by our model is linear inthe budget mh, with the slope proportional to θ*

ih andinversely proportional to price pi:

which implies that the income (or budget) elasticities are asfollows:

( ) ,*

*

101

xp

m pih iih

ih j j

j

J

= + −⎛

⎝⎜⎜

⎞

⎠⎟⎟

=∑β

θβ

(8) Income effects:

∂

∂= −

∑ln

ln

*x

p

xihIncome

h

j

ih jhθhh

ihh

j

i

i

x

p

p

x

∑∑

∂

; and

(9) Substitution effects:

ln hhSubstitution

h

j

ih jh jh

ihh

j

i

p

x

x

p

p

∑

∑∑

∂

=

−( )ln

.

*θ β

(7) Aggregate cross-elasticities:

∂

∂

∑ln

l

xihh

nn;

*

p x

p

pj

ih jh

ihh

j

i

= −∑∑θ β

Note that the Engel curve implied by our model appliesto a specific consumption regime—namely, the set of posi-tive consumption goods, J*. When the consumption budget(mh) increases, the demand regime (J*) may change, whichmeans that the intercepts and slopes of Equation 10 willalso change. More specifically, as the budget is distributedacross a larger consumption set, intercepts will continue toincrease, and slopes will continue to decrease; that is, theEngel curve becomes flatter. Consequently, when signifi-cant shifts are observed in the demand regime over a rangeof incomes, we can note only that the Engel curve is piece-wise linear. Moreover, the Engel curve is typically definedfor the same household across different income levels.Again, because we allow for heterogeneities in the tasteparameter αih, we account for the possibility that changes indiscretionary income also result in changes in the utilityfunction, so that shifts in income may result in cross-sectional changes in the Engel curve as well as movementsalong an individual Engel curve, thus producing highly non-linear Engel curves. In other words, although the traditionalLES demand system implies linear Engel curves for anindividual household at its current consumption regime, ourextension produces globally flexible aggregate demandstructures with nonlinear Engel curves, again leading to amore flexible and realistic demand system at the populationlevel than the traditional LES system. Our empirical results(which we discuss in detail subsequently) illustrate howwell our factor-analytic extension of the LES demand sys-tem produces more flexible and realistic Engel curves thanthe traditional system.

If defined on expenditure share, the Engel curve isinversely proportional to mh,

and the slope of the Engel curve is inversely proportional tothe square of mh,

Given the consumption regime J*, Equation 12 suggeststhat expenditure share can be an increasing or decreasingfunction of consumption budget, depending on the relativesizes of θ*

ih and . However, regardless of thedirection of change, the Engel curve of expenditure sharebecomes less sensitive to consumption budget as the latterincreases.

Consumption Patterns in the UnitedStates: 1982–2003

As an illustration of our proposed consumption budget allo-cation model, we apply it in an analysis of household

p pi i j jjJβ β/ *=∑ 1

( ) *

*

12 12

1∂∂

= −

⎛

⎝=∑

s

m

p

m

p

pih

h

i i

hih

j j

j

J

i i

βθ

β

β

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟ .

( ) *

*

11 11

sp x

m

p

m

p

mihi ih

h

i

hi ih

j

hj

j

J

= = + −=∑β θ β

⎛⎛

⎝⎜⎜

⎞

⎠⎟⎟

,

∂∂

=ln

ln.*

x

m

m

p xih

hih

h

i ih

θ


2We excluded three categories from the consumption budget:purchases of new and used automobiles, home rentals or equiva-lence, and home mortgages. Our rationale for excluding thesecategories was that they do not represent discretionary expendi-tures within a particular year because they typically involve installpayments that are spread over a longer horizon.

expenditures in the United States for a period of 22 years.The goal here is to obtain household-level estimates of theirdirect utility functions across a comprehensive set of con-sumption categories and use these estimates to gain insightsinto the consumption priorities across different types ofhouseholds and income levels over the 22 years covered byour data. Rather than directly relating the utility functions todemographics, we first use our factor-analytic random coef-ficients approach to obtain estimates of the utility functionfor each household in the sample, leveraging the pattern ofcovariation across consumption categories, and then weinvestigate how these estimates differ across householdtypes and income levels.

Data Description

To estimate our model, we use the CEX family extractsmade available by the National Bureau of EconomicResearch (NBER) for the 1982–2003 period (http://www.nber.org/data/ces_cbo.html). The CEX is collected fromdifferent samples each year, so that each of the 66,368households in this sample reports its consumption expendi-tures for only one year, and therefore the sample cannot betreated as a longitudinal panel. These NBER extracts fromthe CEX database contain the dollar amounts allocated byeach sample household during a one-year window across 31consumption categories.2 In defining these broad consump-tion categories, we followed the typology NBER uses.

For each of the 31 consumption categories and 22 yearsin the CEX–NBER data, we collected the relevant priceindex from the Bureau of Labor Statistics, which we nor-malized with 1982 as the base year. Figure 1 summarizesthese price indexes. Each vertical line within a bar repre-sents one year of data; note the large variation in inflationrates across categories, with the largest increases intobacco, education, and health care.

Figures 2 and 3 provide a summary description of theconsumption data from 1982 to 2003. Figure 2 shows thepercentage of households in the CEX sample reportingexpenditures in each category, with low incidence rates(less than 40% of the sample) for several categories, such ashospital and related services, airline fare, public transporta-tion, nonprescription drugs, and medical supplies. The fig-ure shows some clear trends in consumption in the twodecades under study. For example, the reporting of tobaccoand alcohol consumption declined during this period. Thereis a similar decline in incidence rate for doctors, dentists,and hospitals, which might be due to the increase in inci-dence rate for health insurance. Another category worthnoting is jewelry and watches, for which there was a sub-stantial drop in incidence rate. Most important, Figure 2shows that other demand systems, such as the popular AIDSand Rotterdam models, would be inappropriate for this typeof analysis because they assume that all households spend

on all categories, even when this assumption clearly doesnot hold, except for food at home.

Figure 3 shows how households have allocated budgetacross categories. On average, food and clothing took asmaller portion of the household budget over the years,whereas health insurance took a larger share. Although theincidence rate of tobacco usage decreased during the 22-year period, tobacco consumption took an increasing por-tion of the budget among those who still smoke. The por-tion of the budget allocated to motor and home fueldecreased substantially in the 1980s but seems to haveincreased in the last few years.

Aside from the expenditure data, we also used demo-graphic data from the CEX–NBER extracts to classify eachsample household into a specific life stage, using the typol-ogy that Du and Kamakura (2006) propose. Table 1describes the typical profile of each life stage. The last col-umn of Table 1 reports the results from our classification ofthe CEX–NBER sample into this typology. This typologyproves useful when we compare the patterns of preferencesacross households and consumption categories.

Estimation Results

To find the best compromise between parsimony and good-ness of fit, we applied our proposed budget allocationmodel to a random sample of 9526 households from theCEX–NBER extracts, varying the number of factors (Zh)from p = 3 to p = 8. From Bayesian information criterionand the interpretability of the factor solution, we chose thesix-factor solution and applied it to the entire sample of66,683 households in the CEX–NBER data. We report theparameter estimates and the goodness-of-fit measures inTable 2. The R-squares reflect our model’s overall perfor-mance in predicting budget shares for each category, includ-ing observations with zero expenditures. It is calculated asfollows: 1 – (Sum of squared prediction error/Variance ofobserved shares). At first glance, some of the R-squares inTable 2 may seem low. However, we computed theseR-squares across a large sample of households according toestimates obtained from a smaller sample of households.Note also that for low incidence categories, the dependentvariable is highly censored (i.e., with a lot of zero sharesand relatively few positive values). For these categories, theR-squares tend to be the lowest because the model mustcorrectly predict both incidence and the budget share condi-tional on incidence. The more pertinent goodness-of-fitmeasure for these categories is the hit ratio, which repre-sents the percentage of correct predictions with respect to whether a household has positive consumption in acategory.

Engel Curves by Consumption Category

We can attain a better sense of the flexibility afforded byour proposed model by comparing the cross-sectional Engelcurves implied by our model with their observed counter-parts. To obtain the cross-sectional Engel curves, we firstcomputed the expected expenditure shares for each house-hold ( ), as shown in Equation 11, according to the indi-vidual estimates of = exp(γi + λi Zh + 1⁄2σi

2) and the otherαih

sih


0

100

200

300

400

500

600

Food

at H

ome

Food

Away fr

om H

ome

Tobac

co a

nd

Smok

ing P

rodu

cts

Alcoho

lic B

ever

ages

Away fr

om H

ome

Alcoho

lic B

ever

ages

at H

ome

Appar

el

Appar

el Ser

vices

Oth

er T

han

Laun

dry a

nd D

ry C

leanin

g

Jewelr

y and

Wat

ches

Perso

nal C

are

Servic

es

Misc

ellan

eous

Perso

nal S

ervic

es

Life

Insu

ranc

e

Presc

riptio

n Dru

gs

Nonpr

escr

iption

Dru

gs

and

Med

ical S

uppli

es

Profe

ssion

al M

edica

l

Care

Servic

es

Hospit

al an

d

Relate

d Ser

vices

Health

Insu

ranc

e

0

100

200

300

400

500

600

Mot

or V

ehicl

e

Main

tena

nce

and

Repair

Mot

or F

uel

Mot

or V

ehicl

e In

sura

nce

Public

Tra

nspo

rtatio

n

Airline

Far

e

Recre

ation

Educa

tion

Charit

y

Teleph

one

Servic

es

Lodg

ing A

way fr

om H

ome

House

hold

Furnis

hings

and

Opera

tions

House

hold

Opera

tions

Electri

city

Wat

er a

nd S

ewer

and

Trash

Coll

ectio

n Ser

vices

Gas, H

eatin

g Oil,

and

Coal

FIGURE 1Price Indexes 1982–2003 (1982 = 100)

model parameters (βi), and then we averaged these expectedshares within each income decile. We report the actual andmodel-implied cross-sectional Engel curves for some of theproduct categories in Figure 4, which shows that the curvesimplied by our proposed model conform well to theirobserved counterparts. Most important, Figure 4 shows thatour factor-analytic random coefficients formulation canaccommodate monotonically increasing shares (as incomedecreases) for essential categories (e.g., electricity, tele-phone, food at home) and decreasing shares (as incomedecreases) for nonessential categories (e.g., food outside thehome, recreation, education, lodging away from home).Moreover, the model is flexible enough to capture certain

inflections in the Engel curve, as is evident for the food-at-home category (from concave to convex as incomedecreases). It also captures nonmonotonic shapes, such asan inverted U shape for motor fuel and a U shape for publictransportation, reflecting that at the lowest income deciles,more private transportation is substituted with publictransportation.

The Principal Components of Consumption

From Equations 1 and 3, we can write the log-marginal util-ity given xih as γi + λiZh + εih – ln(xih – βi). Accordingly,γi – ln(–βi) represents the average initial (when the quantityconsumed is still zero) log-marginal utility for consumption


FIGURE 2Percentage of Households Reporting Expenditures in Each Consumption Category

category i. The factor scores (Zh) for a household, weightedby the factor loadings (λi) for a consumption category, showwhether the household’s log-marginal utility is higher orlower than average for that category at a given consumptionlevel. Therefore, consumption categories that have largeloadings of the same sign for the same factor have posi-tively correlated log-marginal utilities across households.For example, because jewelry and watches as well as educa-tion have large loadings of the same sign for Factor 1,households with a higher-than-average score on this factorwill have higher-than-average log-marginal utilities for bothconsumption categories. By the same token, through thesign and magnitude of these loadings, it is possible to iden-tify sets of consumption categories that tend to have higher

(or lower) marginal utilities for the same households. Foreasier interpretation, Table 2 shows in bold the largest load-ings (in absolute value) for each consumption category.From these loadings, households with a higher-than-average score on Factor 1 would be expected to have higherlog-marginal utilities for categories, such as jewelry andwatches, education, alcohol, and recreation, suggesting thatthis factor is associated with nonessential consumption.Factor 2 is associated with smoking and drinking and couldbe labeled a “sin” factor. Factor 3 is associated with largefamily-oriented consumption needs, such as insurance,household operations, electricity, and utilities. Factor 4 isassociated with health care. Factor 5 has the highest load-ings for public transportation (which includes trains and


FIGURE 3Share of Nonzero Expenditures Allocated to Each Consumption Category

taxis), airfare, and lodging away from home and thereforecould be labeled a “travel” factor. Finally, Factor 6 has rela-tively weaker loadings than the other factors, but it hashigher loadings for categories related to the operation ofmotor vehicles.

Most important, these factors account for both hetero-geneity in the log-marginal utilities across households andthe correlation among these log-marginal utilities acrossconsumption categories. We further investigated how con-sumption preferences differ across households by perform-ing an analysis of variance (ANOVA) on the factor scoresusing three variables that describe the households: (1) lifestage, as we defined it previously; (2) income quintile, rela-tive to other households reporting expenditures in the same

year; and (3) year of data collection, which is classified intofive categories (early or late 1980s, early or late 1990s, andearly 2000s). This ANOVA, which we performed across all66,683 households for Factor 1 (nonessential consumption),showed that only the life stage × income quintiles and lifestages × year interactions (along with the main effects) werestatistically significant at the .01 level. Therefore, we reportaverages only for these two interactions in Figure 5, PanelA. Moreover, to simplify the exposition, we focus only onthe six life stages with the largest number of households,which account for more than 70% of our sample. Figure 5,Panel A, shows that average log-marginal utilities for“nonessential” consumption have increased over time and,as expected, are higher for the high-income quintiles. How-


Lif

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orce

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(%)

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ors.


FIGURE 4Actual and Estimated Engel Curves for Some of

the Consumption Categories

Notes: Engel curves implied by the proposed model are repre-sented by solid lines; Engel curves based on observed dataare represented by dots.

ever, these patterns of change are distinct for each life stage.As would be expected, the nonessential factor scores arehigher in the richest quintiles. They also tend to be lower inthe later life stages (S5) than in the earlier stages (Co/So,C1, and S1). To our surprise, these scores declined at differ-ent rates for different life stages in the 22 years covered byour data, suggesting that the marginal utility for nonessen-tial consumption (compared with food at home) decreasedover time.

Similar results for the health-related Factor 4 (see Fig-ure 5, Panel B) suggest that average log-marginal utilitiesfor the older life stages (C6 and S5) do not vary substan-tially with income, but they increase with income for theyounger life stages (C1 and S1). Figure 5, Panel B, alsoshows a trend upward in the average log-marginal utilitiesfor health care.

Consumption Priorities, Household Life Stage,and Income

In addition to investigating each factor separately, we cancapture the variation of preferences for a particular con-sumption category across households by the “preferenceshares,”

which represent the expected expenditure shares for house-hold h when there is no budget constraint. We report theaverage estimated preference shares in Table 3. For clarity,we report these averages only for the six most populated lifestages and for the richest/poorest income quintiles. Theseresults show that though there are substantial differences inpreferences between the two extreme income quintiles, thedifferences across the six main life stages are relativelyminor, particularly for the richest quintile. In general, thepoorest 20% of our sample have higher preference sharesthan the richest 20% for food at home; tobacco and smok-ing products; health insurance; telephone services; electric-ity; water and sewer and trash collection services; and gas,heating oil, and coal, suggesting that these are the moreessential consumption categories. For categories such asmotor fuel, being considered essential depends on the lifestage; the poorest 20% have higher preference shares thanthe richest 20% among some life stages (Co/So, C1, S1, andC6) but lower shares in other stages (S2 and S5), againdemonstrating the importance of accounting for unobservedheterogeneity in tastes in modeling consumption budgetallocation.

Policy Simulations

A major advantage of demand systems, such as the one wepropose here, is that they are consistent with budget-constrained utility-maximizing behavior, leading to pre-dicted expenditures that are always logically consistent (i.e.,nonnegative and sum up to the budget). Such a multicate-gory structural approach makes our proposed budget alloca-tion model valuable in anticipating consumers’ reactions toenvironmental shocks, such as price hikes or shifts in dis-

ˆˆ

ˆ

ˆ ˆ ˆ ˆ

ˆθ

α

α

γ λ σ

γih

j

J

Ze

e

e

e

ih

jh

i i h i

= =

=

+ +

∑ 1

1

22

jj j h jZ

j

J + +

=∑ ˆ ˆ ˆ,

λ σ1

21

2


FIGURE 5Average Scores by Income Quintile and Year

A: Factor 1: Nonessential Consumption

B: Factor 4: Health Care

cretionary income, under the premise that the consumers’underlying preference structures are more stable and willremain unchanged, at least in the short run. Our factor-analytic approach has the added advantages of allowing fordiversity in consumption priorities or tastes across house-holds and capturing the correlation among these priorities,leading to a more flexible demand system. In contrast, anymodel that treats each category independently (rather thanjointly) would be inapplicable because, by definition,category-specific models are not bound by the budget con-straint, and as a result, the predicted expenditures would notsum up to the household consumption budget, which is aprerequisite for any meaningful policy simulations.

To illustrate, we conducted three policy simulations ofcurrent relevancy. First, we consider the not-so-hypotheticalscenario in which prices for nonrenewable energy sources(i.e., motor fuel and gas, heating oil, and coal) increase by50%. This policy simulation exemplifies how the model canbe used to project shifts in consumer spending in responseto projected price increases in some consumption cate-gories. In the second policy simulation, we consider a sce-nario in which the federal government gives each householda $500 tax rebate specifically earmarked for stimulatingcurrent consumption. This second scenario illustrates how apolicy maker can anticipate the effects on consumer spend-ing of policies that increase or decrease discretionaryincome, such as programs to provide federal health insur-


TAB

LE

3A

vera

ge

Est

imat

ed P

refe

ren

ce S

har

es f

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or

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e S

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Ric

hes

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oo

rest

Inco

me

Qu

inti

les

Sel

ecte

d L

ife

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ge

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/So

C1

S1

C6

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Ric

hes

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oo

rest

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hes

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rest

Ric

hes

tP

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rest

Ric

hes

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rest

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rest

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nQ

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tile

Cat

ego

ries

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

Foo

d at

hom

e13

.419

.514

.722

.011

.720

.414

.120

.015

.925

.012

.519

.5F

ood

away

fro

m h

ome

6.4

5.3

6.6

5.7

6.2

5.2

5.9

4.2

6.5

5.7

5.3

4.0

Toba

cco

and

smok

ing

prod

ucts

2.3

3.8

2.3

4.0

2.3

4.2

2.3

3.6

2.7

4.4

2.2

3.4

Alc

ohol

ic b

ever

ages

at

hom

e1.

21.

21.

01.

21.

41.

31.

0.9

1.2

1.1

1.1

.8A

lcoh

olic

bev

erag

es a

way

fro

m

hom

e.8

.6.6

.51.

2.6

.6.3

.8.4

.7.3

App

arel

5.7

4.4

5.7

4.5

5.8

4.3

5.4

3.5

5.6

4.4

5.0

3.5

App

arel

ser

vice

s ot

her

than

laun

dry

and

dry

clea

ning

.8.6

.7.6

.9.6

.7.4

.7.6

.7.5

Jew

elry

and

wat

ches

.8.4

.7.4

.9.4

.7.3

.7.3

.7.3

Per

sona

l car

e se

rvic

es1.

41.

41.

41.

31.

41.

41.

41.

41.

41.

41.

41.

4M

isce

llane

ous

pers

onal

ser

vice

s1.

71.

21.

61.

21.

81.

21.

61.

11.

51.

01.

71.

1Li

fe in

sura

nce

2.9

3.0

3.0

2.9

2.7

2.9

3.0

3.2

2.8

2.9

3.1

3.3

Pre

scrip

tion

drug

s.8

1.0

.8.8

.6.9

1.1

1.8

.7.7

1.3

1.9

Non

pres

crip

tion

drug

s an

d m

edic

al s

uppl

ies

.7.7

.7.6

.6.6

.7.8

.6.6

.8.8

Pro

fess

iona

l med

ical

car

e se

rvic

es2.

32.

12.

31.

92.

11.

92.

62.

52.

21.

62.

72.

5H

ospi

tal a

nd r

elat

ed s

ervi

ces

.5.5

.5.5

.4.4

.6.8

.4.4

.6.7

Hea

lth in

sura

nce

3.9

5.6

4.0

4.8

3.5

5.6

4.9

8.4

4.0

4.9

5.5

8.8

Mot

or v

ehic

le m

aint

enan

ce a

nd

repa

ir4.

23.

34.

03.

24.

33.

13.

72.

83.

92.

63.

82.

6M

otor

fue

l5.

86.

25.

86.

45.

56.

15.

45.

65.

95.

85.

45.

3M

otor

veh

icle

insu

ranc

e4.

44.

84.

44.

74.

24.

74.

34.

94.

44.

24.

44.

7P

ublic

tra

nspo

rtat

ion

.7.6

.6.5

1.0

.7.8

.4.7

.8.7

.6A

irlin

e fa

re2.

61.

82.

31.

63.

32.

02.

61.

62.

41.

72.

81.

9R

ecre

atio

n9.

36.

98.

96.

610

.06.

88.

75.

78.

76.

28.

75.

8E

duca

tion

3.9

1.7

3.6

1.5

4.4

1.4

3.3

1.0

3.4

1.2

3.1

.8C

harit

y3.

12.

23.

01.

93.

12.

03.

42.

52.

61.

73.

82.

9Te

leph

one

serv

ices

2.9

3.4

3.0

3.4

2.8

3.4

2.9

3.5

3.0

3.5

2.9

3.5

Lodg

ing

away

fro

m h

ome

2.1

1.3

1.9

1.2

2.5

1.3

2.0

1.1

1.8

1.1

2.1

1.2

Hou

seho

ld f

urni

shin

gs a

nd

oper

atio

ns4.

42.

94.

32.

84.

62.

74.

22.

44.

02.

54.

22.

4H

ouse

hold

ope

ratio

ns2.

72.

52.

72.

22.

72.

43.

02.

82.

52.

13.

33.

0E

lect

ricity

3.9

5.1

4.1

5.2

3.5

5.1

4.1

5.7

4.1

5.3

4.2

5.7

Wat

er a

nd s

ewer

and

tra

sh

colle

ctio

n se

rvic

es1.

92.

41.

92.

31.

72.

42.

02.

81.

92.

32.

22.

9G

as,

heat

ing

oil,

and

coal

2.8

3.7

2.9

3.6

2.6

3.8

3.0

4.1

2.9

3.8

3.1

4.2


ance and funding or rebates for child care, to certain seg-ments of the population.

The third simulation is an attempt to quantify consumerwelfare losses due to the dramatic increases in prices forprescription drugs in the past 22 years, which grew by262% from 1982 to 2003 (an annual rate of 6.3%), com-pared with an inflation rate of 91% (or 3.1% per year) dur-ing the same period. Here, we consider a hypothetical sce-nario in which the costs of prescription drugs followed thegeneral inflation rate observed in the past 22 years. In thiscounterfactual simulation, we consider the income effectsof the dramatic price increases, attempting to estimate howhouseholds shifted their discretionary income away fromother consumption categories to pay for the increasing costsof prescription drugs. In this scenario, we consider allhouseholds that spent discretionary income on prescriptiondrugs in the past 22 years and compare their actual expendi-tures with those predicted by our budget allocation model,assuming that the extra discretionary income resulting fromthe lower prescription drug prices would be allocated to theother categories according to the estimated utilities for eachhousehold and consumption category. This comparisonshows how these households reduced their expenditures inall the other categories to compensate for the dramaticincreases in prescription drug prices in the past 22 years,thus providing some insights into welfare losses potentiallycaused by these price increases.

For each household in our sample, we simulate theirbudget reallocation decisions by solving the constrainedutility maximization problem, using the estimated parame-ters ( and βi for household h and category i) andobserved versus simulated prices (pi versus pi + Δpi) andbudget (mh versus mh + Δmh). The solution can be derivedthrough a five-step procedure, which we detail in theAppendix.

Policy Simulation 1: reactions to shifts in energy costs.Table 4 reports the simulated effects of a dramatic increasein oil prices, showing the percentage changes in quantityconsumed in response to a 50% increase in prices for motorfuel and gas, heating oil, and coal. As would be expected,the price increases affect the poorest quintile more dramati-cally than the richest quintile. The difference between thetwo income quintiles is the largest in the demand for motorfuel among households in the S5 stage, in which the poorest(richest) quintile reduces the quantity consumed by 43%(20%). In other words, the demand among the pooresthouseholds in the S5 stage is elastic, whereas those in therichest quintile have an inelastic demand. Table 4 alsoshows how households in different life stages would adjusttheir expenditures in other categories to compensate for theshift of discretionary income toward motor and home fuels.As would be expected, the more essential categories, suchas food at home, telephone services, electricity, and waterand sewer and trash collection, are the least affected, alongwith “addictions,” such as tobacco and alcoholic beveragesat home. The consumption categories most affected are theless essential ones, such as education (which includes booksand other educational expenses), miscellaneous personalservices, and charity. The category showing substantial

αih

3Our analysis of preferences (Table 3) across life stages andincome quintiles showed that preferences vary with income (cross-sectionally). In this policy simulation, we assumed that a relativelysmall increment of $500 in discretionary income would not causesubstantial shifts in preferences.

differences in response across life stages is jewelry andwatches, for which the highest percentage drop in demandoccurs in the richest quintile of S5 (31%), compared with adrop of only 3%–13% in the lowest quintiles, probablybecause they already spend the least in this category. A sur-prising and worrisome effect is the substantial drop in pre-scription drugs and other health care expenses, particularlyamong the older (C6 and S5) and poorer households.

The price effects we report in Table 4 show negativecross-elasticities, so that increases in fuel prices producedecreases in demand for all other categories, which happensbecause income effects dominate substitution effects, asdiscussed previously. In other words, because demand formotor and home fuel is inelastic, increases in fuel pricesleave less discretionary income to be spent elsewhere, lead-ing to a decrease in expenditures in all other categories.After we partial out these income effects (Equation 8), thesubstitution effects (Equation 9) are all indeed positive,which implies that there is no complementarity betweencategories. It might be argued that price increases in motorfuel would lead consumers to reduce their car use, whichwould lead to spending less on motor vehicle maintenanceand repair and, thus, complementarity between these twocategories (though it could also be argued that the impacton motor vehicle maintenance and repair might take longerto observe). However, after we account for income effects,these two categories become substitutes because the utilityfunction is assumed to be additive separable, and account-ing for complementarity with the CEX data is infeasible(because it is not a truly longitudinal panel).

Policy Simulation 2: reaction to a tax rebate. Table 5reports the simulated effects of a hypothetical $500 taxrebate for each household, distributed by the federal gov-ernment to boost demand and thus earmarked for consump-tion. Simulations such as this could help policy makersanticipate and quantify the differential impacts of such ashift in discretionary income on different population sectorsacross different consumption categories.3 Across the sixmain life stages, food at home receives the highest share ofthe extra $500 in discretionary income, more so among thepoorest quintile. In general, for essential categories, such ashealth insurance; telephone services; electricity; and gas,heating oil, and coal, there is a larger increase in spendingamong poorer households. After food retailing, recreationwould be one of the industries that would benefit the mostfrom the $500 tax rebate, particularly among the wealthierhouseholds. Similarly, we observe a larger increase inspending among richer households for other nonessentialcategories, such as airline fare, education, charity, andhousehold furnishings.


TAB

LE

4E

xpec

ted

Per

cen

tag

e C

han

ge

in D

eman

d in

Res

po

nse

to

a 5

0% P

rice

Incr

ease

in M

oto

r F

uel

an

d G

as,H

eati

ng

Oil,

and

Co

al

Sel

ecte

d L

ife

Sta

ge

Co

/So

C1

S1

C6

S2

S5

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Co

nsu

mp

tio

nQ

uin

tile

Qu

inti

leQ

uin

tile

Q

uin

tile

Qu

inti

le

Qu

inti

le

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Cat

ego

ries

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

Foo

d at

hom

e–2

0–2

0–4

1–4

0–1

–2–6

1F

ood

away

fro

m h

ome

–20

–12

–2–1

–2–2

–10

–2–1

Toba

cco

and

smok

ing

prod

ucts

–2–2

–31

12

–23

02

12

Alc

ohol

ic b

ever

ages

at

hom

e–2

–1–4

0–4

–5–4

–2–2

0–1

1A

lcoh

olic

bev

erag

es a

way

fro

m

hom

e–6

–9–1

–4–1

2–1

5–2

–6–1

1–1

01

–7A

ppar

el–1

00

–1–2

1–2

–2–2

–3–1

0A

ppar

el s

ervi

ces

othe

r th

an

laun

dry

and

dry

clea

ning

–4–6

–2–8

–7–8

–3–3

–3–1

4–2

–5Je

wel

ry a

nd w

atch

es–1

7–5

–12

–3–1

9–1

0–1

6–8

–13

–11

–31

–13

Per

sona

l car

e se

rvic

es–2

–1–1

–1–1

–1–2

0–2

–11

–1M

isce

llane

ous

pers

onal

ser

vice

s–1

7–2

6–1

4–1

6–2

1–1

6–1

8–2

4–2

0–2

3–1

3–2

2Li

fe in

sura

nce

–33

–3–5

02

–8–1

–12

–18

2P

resc

riptio

n dr

ugs

–9–1

8–5

–12

–3–2

2–1

3–2

6–8

–12

–11

–27

Non

pres

crip

tion

drug

s an

d m

edic

al s

uppl

ies

–1–1

–15

–2–1

–1–2

–41

0–6

Pro

fess

iona

l med

ical

car

e se

rvic

es–6

–10

–4–1

1–6

–8–1

0–1

1–5

–8–1

5–1

4H

ospi

tal a

nd r

elat

ed s

ervi

ces

–14

–19

–5–9

–11

–13

–9–1

5–1

–17

–12

–24

Hea

lth in

sura

nce

–21

–20

–41

–20

–44

–10

Mot

or v

ehic

le m

aint

enan

ce a

nd

repa

ir–7

–5–4

–5–1

0–5

–3–6

–4–4

–9–5

Mot

or f

uel

–18

–26

–18

–23

–25

–32

–17

–27

–17

–32

–20

–43

Mot

or v

ehic

le in

sura

nce

–2–1

–41

–44

–11

–42

22

Pub

lic t

rans

port

atio

n–1

0–1

2–6

–8–9

–17

–14

–15

–9–1

5–4

–12

Airl

ine

fare

–3–3

–4–2

–3–2

–30

–63

1–4

Rec

reat

ion

–6–3

–3–4

–7–4

–7–3

–2–1

–30

Edu

catio

n–2

8–3

0–3

1–2

9–2

5–3

0–2

7–2

5–3

0–2

9–3

1–2

3C

harit

y–1

4–1

3–1

1–6

–16

–10

–16

–17

–10

–3–2

1–1

4Te

leph

one

serv

ices

–2–5

–1–7

–4–1

0–2

–4–3

–12

–2–6

Lodg

ing

away

fro

m h

ome

–1–3

–4–3

–2–1

–20

–31

–2–1

Hou

seho

ld f

urni

shin

gs a

nd

oper

atio

ns–1

5–1

0–1

0–1

5–1

5–1

2–1

5–1

2–7

–11

–19

–12

Hou

seho

ld o

pera

tions

–4–5

–7–4

–2–3

–4–1

–10

–2–3

–3E

lect

ricity

–30

–30

–10

–30

–2–1

–40

Wat

er a

nd s

ewer

and

tra

sh

colle

ctio

n se

rvic

es–1

50

10

3–2

1–3

2–8

3G

as,

heat

ing

oil,

and

coal

–25

–36

–24

–39

–32

–42

–25

–34

–22

–33

–20

–34


TAB

LE

5E

xpec

ted

Allo

cati

on

of

an In

crem

enta

l $50

0 in

th

e C

on

sum

pti

on

Bu

dg

et

Sel

ecte

d L

ife

Sta

ge

Co

/So

C1

S1

C6

S2

S5

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Co

nsu

mp

tio

nQ

uin

tile

Qu

inti

leQ

uin

tile

Q

uin

tile

Qu

inti

le

Qu

inti

le

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Cat

ego

ries

($)

($)

($)

($)

($)

($)

($)

($)

($)

($)

($)

($)

Foo

d at

hom

e67

101

7211

656

110

6910

279

135

5310

3F

ood

away

fro

m h

ome

3626

2821

4528

3220

3223

3421

Toba

cco

and

smok

ing

prod

ucts

919

821

924

916

1122

610

Alc

ohol

ic b

ever

ages

at

hom

e6

65

67

76

46

55

2A

lcoh

olic

bev

erag

es a

way

fro

m

hom

e5

33

28

43

15

23

1A

ppar

el29

2229

2629

2127

1631

3022

18A

ppar

el s

ervi

ces

othe

r th

an

laun

dry

and

dry

clea

ning

44

34

65

32

35

33

Jew

elry

and

wat

ches

52

42

52

41

41

41

Per

sona

l car

e se

rvic

es7

76

66

77

77

77

8M

isce

llane

ous

pers

onal

ser

vice

s8

107

611

58

712

619

5Li

fe in

sura

nce

1712

1612

1310

1816

1310

1615

Pre

scrip

tion

drug

s3

63

32

56

143

35

13N

onpr

escr

iptio

n dr

ugs

and

med

ical

sup

plie

s3

23

23

24

33

13

4P

rofe

ssio

nal m

edic

al c

are

serv

ices

1111

1011

119

1514

118

2713

Hos

pita

l and

rel

ated

ser

vice

s2

43

32

23

52

14

3H

ealth

insu

ranc

e18

2418

2016

2524

4619

1823

51M

otor

veh

icle

mai

nten

ance

and

re

pair

2217

2117

2416

1715

1913

1312

Mot

or f

uel

2834

2636

2231

2429

3028

2023

Mot

or v

ehic

le in

sura

nce

2222

2022

2020

1923

2216

2019

Pub

lic t

rans

port

atio

n4

32

35

54

24

52

3A

irlin

e fa

re13

510

417

413

411

312

5R

ecre

atio

n49

3349

3254

3345

2644

3038

26E

duca

tion

1212

289

810

92

189

152

Cha

rity

169

148

178

2413

126

3115

Tele

phon

e se

rvic

es14

1914

1915

2212

1616

2311

19Lo

dgin

g aw

ay f

rom

hom

e11

59

413

311

48

210

3H

ouse

hold

fur

nish

ings

and

op

erat

ions

2414

2416

2212

2313

1812

2012

Hou

seho

ld o

pera

tions

1512

2212

1611

1813

1610

2518

Ele

ctric

ity19

2918

2817

2920

2919

3321

32W

ater

and

sew

er a

nd t

rash

co

llect

ion

serv

ices

910

1012

910

1014

1011

1015

Gas

, he

atin

g oi

l, an

d co

al13

1913

1912

2013

2214

2314

27

Tota

l 50

050

050

050

050

050

050

050

050

050

050

050

0


Policy Simulation 3: welfare losses due to spiralingcosts of prescription drugs. Table 6 reports the simulatedpercentage changes in household expenditure if the pricesof prescription drugs had increased at the same rate as theCPI. If that had been the case, consumers could havereduced their prescription drug expenditure by an averageof 37%, while maintaining the same level of treatment. Thesavings could then have been spent in other categories.Older, retired, and poor households (bottom income quin-tiles of the C6 and S5 life stages) would have benefitedmore, percentage-wise, than younger, working, and wealth-ier households. For example, the oldest and poorest house-holds could have increased their spending on additional lifeinsurance (3.5%), water and sewer and trash collection ser-vices (3.2%), motor vehicle insurance (3.1%), tobacco andsmoking products (3%), and motor fuel (2.6%).

Conclusions and Directions forFurther Research

The main purpose of this study was to develop a feasibledemand system that would enable us to investigate budgetallocation decisions by individual households across a com-prehensive set of consumption categories. This developmentwas motivated by the belief that marketers, research ana-lysts, and policy makers need a better understanding of howconsumers allocate their discretionary income to meet dif-ferent consumption needs, and they must be able to antici-pate how the resultant consumption patterns will change inresponse to changes in prices and budgets.

When studying consumption at this basic level, it isimportant to emulate the consumer’s resource allocationproblem. Our basic premise is that every household allo-cates its discretionary income among competing needs andwants so that when the consumption budget is exhausted,all expenditure categories offer the same marginal utilityper dollar. Because we attempt to develop a reasonable as-ifmodel to approximate the household’s basic resource allo-cation problem, when we obtain estimates of each house-hold’s direct utility function, we can simulate the house-hold’s reaction to changes in prices or income andunderstand how these changes will affect different con-sumption categories representing different industries acrossdifferent consumer segments.

By definition, a model is a simplified representation ofobserved phenomena, and therefore we made some simpli-fying assumptions in developing our proposed factor-analytic random coefficients budget allocation model. Animportant assumption to make the model parsimonious andfeasible is that of a direct utility function that is additiveseparable across consumption categories. In other words,we assume that for an individual household, expenditure inone category does not increase or decrease the marginalutility derived from consuming another category. This doesnot imply that consumption is independent across cate-gories, because all categories compete for the same budget.It also does not imply that preferences are independentacross categories and households, because our factor struc-ture accounts for possible correlations of preferencesamong the categories across households. However, the addi-

tive separable utility assumption implies that the resultantdemand system will not be able to capture potential com-plementarities among consumption categories, which couldnot be discerned from correlations in preferences because of the one-shot nature of the CEX (Gentzkow 2007). Webelieve that this is a critical limitation only in detailedanalyses that consider a limited number of potentially com-plementary product categories, such as the type of cross-category choice modeling commonly performed for con-sumer packaged goods in the marketing literature usinglongitudinal scanner panels, but this is not as critical inexpenditure analyses performed across a comprehensive setof broad consumption categories. We leave for furtherresearch the methodological challenge of extending ourmodel beyond additive separable utilities, while addressingthe issues of high dimensionality (more than a couple dozencategories), binding nonnegative constraints (householdshave no spending in many categories), unobservable hetero-geneity (unique household preferences that cannot be cap-tured by demographics), and correlation in individualhouseholds’ category preferences.

Substantively, there are several directions for extendingour work. First, the proposed framework for modelinghousehold consumption budget allocation could be adaptedfor forecasting industry sales and assessing market poten-tial. As a model of primary demand, our structural approachhas the advantage of simultaneously considering householdspending across a full spectrum of expenditure categories.Because all types of consumer expenditures ultimately viefor the same household budget, primary demand in oneindustry can be better predicted in relation to consumerexpenditures in other industries. For example, fast-foodrestaurant chains may be able to forecast sales trends betterif they can understand how consumers’ spending on foodaway from home is influenced by their spending on food at home, apparel, motor fuel, and so on. Moreover, ourapproach explicitly links household discretionary income,category price indexes, and demographics (through thehousehold-specific taste parameters) to household categoryexpenditures. This enables analysts in forecasting industrysales or assessing market potential to factor in projectionsabout household income, inflation rates, and demographictrends through an integrated framework.

Second, although developed under a different context,we believe that there is a potential to adapt our consumptionbudget allocation model for shopping-basket data analysis.As such, researchers can study many more categoriesjointly without needing to focus on a few categoriesselected a priori or lump a large number of distinct itemsinto an “other” category.

Finally, in a more realistic setting, consumers make notonly cross-category allocations but also intertemporal allo-cations (e.g., consume more today versus save more fortomorrow). In our work, we ignored the intertemporalaspect by treating the consumption budget as exogenouslydetermined. Further research could relax this assumptionand model both cross-category and intertemporal alloca-tions explicitly. However, this would require a true panelwith longitudinal information about individual households’expenditure patterns.


TAB

LE

6E

xpec

ted

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cen

tag

e C

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ge

in E

xpen

dit

ure

if t

he

Pri

ces

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gs

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ease

d a

t th

e S

ame

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e as

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I

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/So

C1

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S5

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

Ric

hes

tP

oo

rest

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nsu

mp

tio

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uin

tile

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inti

leQ

uin

tile

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uin

tile

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le

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inti

le

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Qu

inti

leQ

uin

tile

Cat

ego

ries

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

(%)

Foo

d at

hom

e.3

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1.1

.61.

6.3

.4.6

1.9

Foo

d aw

ay f

rom

hom

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1.5

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bacc

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okin

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ts.6

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2.3


AppendixEstimating the Proposed Budget AllocationModelThe problem faced by household h is to choose a consump-tion plan, xh(x1h, …, xJh ≥ 0), that maximizes the utilityfunction, , where αih > 0,(xih – βi) > 0, and J is the number of all available expendi-ture categories. Given the prices (p1, …, pJ) of unit con-sumption in each category, the household’s allocation planmust satisfy the budget constraint, mih ≤ mh, where mih represents household h’s expenditures(in dollars) in category i. The household’s optimizationproblem implies the following Kuhn–Tucker conditions:

where ξ ≥ 0 is the Lagrange multiplier. Given our parame-terization of αih = exp(γi + λiZh + εih) in Equation 3, Equa-tions A1 and A2 lead to, respectively,

(A3) γi + λiZh + εih – ln(xih – βi) – ln(pi) ≤ ln(ξ), for xih = 0,

and

(A4) γi + λiZh + εih – ln(xih – βi) – ln(pi) = ln(ξ), for xih > 0,

where Zh is a p-dimensional vector of i.i.d. standard normalfactor scores for household h and εih is a random distur-bance normally distributed with mean zero and standarddeviation σi. The parameters to be estimated, given theobserved consumptions xih (=mih/pi) and prices pi, are γi, λi,βi [= min(xi) – exp(ηi)], and σi, which we collect in the setθ. Every household in our sample has positive consumptionfor the first category, food at home (i.e., x1h > 0 for ∀h), andfor identification purposes, we set γ1, λ1, and σ1 to zero,which means that Equations A3 and A4 can be simplifiedas, respectively,

(A5) εih ≤ (τih – τ1h) – (γi + λiZh), for xih = 0,

and

(A6) εih = (τih – τ1h) – (γi + λiZh), for xih > 0,

where τih = ln(xih – βi) + ln(pi) = ln(pixih – piβi).Let ε*

ih(xh) ≡ ln(pixih – piβi) – ln(p1x1h – p1β1) – (γi +λiZh). Based on Equations A5 and A6, for a given set ofmodel parameters θ and factor scores Zh, the likelihood con-tribution of household h, Lh(Zh), or L(θ|Zh, xh, p) = L(γ2...lh,lh + 1...J, λ2...lh, lh + 1...J, σ2...lh, lh + 1...J, β1...J|Zh, X1h...Jh, p1...J), is

where xih > 0 for ∀i ∈ (2, …, lh), xih = 0 for ∀i ∈ (lh + 1,…, J), is the density function of εih, andf

ihε ( )⋅

( ) , , , ,A L Z x p f x Z p

f

h h x h hh

ih

7 L Zh h( ) = ( ) = ( )

=

θ θ

ε

| |

εεε

εεih hi

h

hihx

xf

hh

hih

* ,*

,( )⎡⎣ ⎤⎦

∂

∂× (

=∏

2

2

2

ll

l

…

…))

∞

( )

= +∫∏

-

εih h

h

x

i

J *

,l 1

( )( )

( )A

G x

x xph

ih

ih

ih ii1

∂∂

=−

≤αβ

ξ , for x = 0,ih

annd

, for x > 0ih( )( )

( )A

G x

x xph

ih

ih

ih ii2

∂∂

=−

=αβ

ξ ,,

Σ ΣiJ

i ih iJp x= ==1 1

G x xh iJ

ih ih i( ) ln( )= −=Σ 1α β

is the determinant of the (lh – 1) × (lh – 1) Jacobian of thetransformation from x2...lh,h to ...lh,h, which is a continuousone-to-one mapping, because p1x1h = (mh – Σlh

i = 2pixih).Given that εih ~ N(0, σi), we can write Equation A7 as

follows:

where ϕ and Φ are, respectively, the probability and cumu-lative density function of the standard normal distributionand

(For a detailed derivation, see Kao, Lee, and Pitt 2001, pp.210–11.)

Because Zh is unobservable, the likelihood contributionof household h, Lh, needs to be integrated over Zh ~ N(0, Ip):

We evaluate Equation A9 with simulation (Gourieroux andMonfort 2002) by replacing the multidimensional integra-tion with a summation over K Halton-sequence draws(Train 2003) of Zh from the standardized normal distribu-tion, which leads to the following:

A gradient search on the log-likelihood function L =, leads to the parameter estimates of our

demand model (i.e., γi, λi , βi, and σi). After the parametersof the demand model have been estimated, factor scores canbe estimated for each household using the same likelihoodfunction in Equation A10, except that the parameters of themodel are known and the estimation is done on the factorscores Zh. Rather than using a gradient search to estimatethe factor scores for each consumer, we use a sampling-importance-resampling procedure (Smith and Gelfand1992).

ΣhN

hL= 1 ln( )

( )*

A LK

p x p

hi

ihk

ii

i ih i ih

101 1

2

=⎛⎝⎜

⎞⎠⎟×

−(=∏σ

φεσ

βl ))−( )

⎡

⎣

⎢⎢⎢

×

=

==

∑∏

∑ i

i ih i iik

K

ih

h

hp x p

1

11

l

lβ

εΦ kk

ii

J

h

*.

σ⎛⎝⎜

⎞⎠⎟⎤

⎦

⎥⎥

= +∏

l 1

( )*

A Lp x p

hi

ih

ii

i ih i iih

91

2

∝⎛⎝⎜

⎞⎠⎟×

−( )=

=∏σφεσ

βl

1

1

l

l

h

hp x pi ih i ii

ih

i

∑∏∫

−( )

⎡

⎣

⎢⎢⎢

×⎛⎝⎜

⎞

=−∞

∞

β

εσ

Φ*

⎠⎠⎟⎤

⎦

⎥⎥× ( )

= +∏

i

J

h h

h

Z Zdl 1

φ .

∂

∂∝

−( )−

=∑ε β

β

2

2

1…

…

l h

l h

i ih i ii

i ih i

h

h

h

x

p x p

p x p

,*

,

l

iii

h ( )=∏ 1

l.

( )* ,

*

A L Zxh h

i

ih

ii

hh

h81

2

2( ) = ⎛⎝⎜

⎞⎠⎟×∂

∂=∏σ

φεσ

εl…l

22 1…l lh hh

ih

ii

J

,

*,×

⎛⎝⎜

⎞⎠⎟

= +∏ Φ

εσ

ε2*

∂

∂

ε2

2

…

…

l

l

h

h

h

hx,

*

,


Procedure for Policy Simulation

For the households in our sample, we simulate their budgetreallocation decisions by solving the constrained utilitymaximization problem, using the estimated parameters (and βi for household h and category i), observed versussimulated prices (pi versus pi + Δpi), and budget (mh versusmh + Δmh). We derive the solution in five steps.

Step 1. Calculate all categories’ marginal utilities at zeroexpenditure (i.e., mih = 0):

Step 2. Rank the categories from the lowest to the high-est by , and denote the lowest with 1′ and the high-est with J′ (i.e., 1′, 2′, …, k′, …, n′, …, J′). Note that themarginal utility for any category is always decreasing ascategory expenditure increases, and if category k′ is con-sumed, category n′ must be consumed as well, as long asn′ > k′. As such, we know that only J category consumptionregimes are possible: (1′, 2′, …, k′, …, n′, …, J′), (2′, …, k′,…, n′, …, J′), …, (k′, …, n′, …, J′), …, (n′, …, J′), or (J′).

Step 3. Assume that Category 1′ has $.01 consumption.It is fairly straightforward to calculate the expenditures forother categories (i):

∂ ˆ ( )Uih 0

∂ ( ) = ∂ =∂

=−

ˆˆ ( )

exp( ˆ )ˆ

UU m

m

m

p

ihih ih

ih

ihih

i

00

1αββ

αβ

α

ii

ih

ih i i

ih

p m p p⎛⎝⎜

⎞⎠⎟

=−

=−

1 exp( ˆ )ˆ

exp( ˆ )

ii iˆ .β

αih

which means that

Step 4. Compare with mh. If is smaller than mh,the household has a large enough budget to spend money onthe least essential category. If is greater than mh, Cate-gory 1′ will not receive any spending. Then, we go back toStep 1 and repeat Steps 2 and 3 until the first category—forexample, k′—such that the sum of is smaller than mh.We then know the categories consumed will be (k′, k +1′,…, J′ – 1, J′).

Step 5. After we know the categories consumed, solvingfor category expenditures is straightforward. Because mh =

exp – – + wehave the following:

In short, if i′ < k′, then = 0; otherwise, = exp –

( ˆ ˆ ) ˆ .m p pk h k k i i′ ′ ′ ′ ′− +β βˆ )αk h′

(αi h′mi h′mi h′

ˆ

ˆ

exp ˆ

m

m p

k h

h i ii k

J

i h

′

′ ′′ ′

′

′

=

−⎛

⎝⎜⎜

⎞

⎠⎟⎟

−

=∑ β

α ˆ

ˆ .

α

β

k hi k

J k kp

′′ ′

′ ′ ′

( )+

=∑

Σi kJ

i ip′ ′′

′ ′=ˆ ,βpk k′ ′

ˆ )β(mk h′ˆ )αk h′(αi h′Σi kJ′ ′′=

mi h′

mh

mhmh

ˆ exp( ˆ ˆ )(. ˆ )m ph i h hi

J

= − −=∑ α α β′ ′′ ′

′

′ ′11

1 101 ++=∑ pi i

i

J

′ ′′ ′

′ˆ .β

1

ˆ exp( ˆ ˆ )(. ˆ ) ˆm p pi h i h h i i′ ′ ′ ′ ′ ′ ′= − − +α α β β1 1 101 ,,

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rex y. du & wagner a. kamakura where did all that money go ... did all that... · allocation...

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