476 Discrete Choice Modeling
constraint,x′d≤y, wherexis a vector of prices andyis income (or total expen-
diture). Assuming the necessary continuity and curvature conditions, a complete
set of demand equations,d∗=d
(
x,y)
results.^2 To extend the model of individ-
ual choice to observed market data, the demand system is assumed to hold at the
aggregate level, and random elements (disturbances) are introduced to account for
measurement error or optimization errors.
Since the 1960s, the availability of survey data on individual behavior has
obviated the heroic assumption underlying the aggregate utility function or
the (perhaps slightly less heroic) assumptions underlying the aggregate demand
system. That progression has evolved to the contemporary literature with the
appearance of large, detailed, high quality panel surveys, such as the Ger-
man Socio-Economic Panel Survey (GSOEP) (see Hujer and Schneider, 1989)
that we will use in this study and the British Household Panel Survey (BHPS)
(http://www.iser.essex.ac.uk/ulsc/bhps), to name only two of many. The analysis
of individual data to which the original theory applies has called for (at least) two
more detailed developments of that theory.
First, the classical theory has relatively little to say about thediscretechoices that
consumers make. Individual data detail career choices, voting preferences, travel
mode choices, discretized measures of the strength of preferences, and participa-
tion decisions of all sorts, such as labor supply behavior, whether to make a large
purchase, whether to migrate, etc. The classical, calculus based theory of decisions
made at the margins of consumption will comment on, e.g., how large a refriger-
ator a consumer will buy, but not whether they will buy a refrigerator instead of a
car (this year), or what brand of car or refrigerator they will buy.
Second, the introduction of random elements in models of choice behavior asdis-
turbancesis much less comfortable at the individual level than in market demands.
Researchers have considered more carefully the appropriate sources and form of
random variation in individual models of discrete choice.
Therandom utility modelof discrete choice provides the most general platform
for the analysis of discrete choice. The extension of the classical theory of util-
ity maximization to the choice among multiple discrete alternatives provides a
straightforward framework for analyzing discrete choice in probabilistic, statistical,
ultimately econometric, terms.
11.2.1 Discrete choice models and discrete dependent variables
Denote by “i” a consumer who is making a choice among a set ofJitchoices in
choice situationt. To put this in a context, which will help to secure the notation,
envision astated choice experimentin which individualiis offered the choice of
several,Ji 1 , brands of automobiles with differing prices and characteristics and
asked which they most prefer. In a second round of the experiment, the interviewer
changes some of the features of some of the cars, and repeats the question. Denote
byAit,1,...,Ait,Jit,Jit≥2, the set of alternatives available to the individual in
choice situationt. It will be convenient to adopt the panel data notation, in which
“t” denotes “time.” The generality of the notation allows the choice set to vary from
one individual to another, and across choice situations for the same individual. In