2.1 Discrete Choice Models
Random utility theory provides a suitable framework for our analysis, as it pre-
dicts choices by comparing the utility associated with distinct retrofitting alter-
natives. Each household faces a choice set C with K elements. The utility Uij
of household i for alternative j ∈ C comprises a deterministic and a stochastic
component:
(1)
Uij = Vij + eij,
with Vij = αj + Xijβ as representative utility, determined by the alternative
specific constant αj and the matrix Xij , which captures alternative-specific at-
tributes (e.g. costs) as well as characteristics of the household (e.g. income). The
portion of utility that is unobservable to the researcher is represented by eij.
Household i chooses alternative j if and only if Uij >Uik for all k = j, with
j, k ∈C. The probability Pi(j) of selecting j from the set of alternatives is thus
dependent on eij and is equal to:
(2)
Pi (j ) = Pr (Vij + eij > Vik + eik )
= Pr (eik - eij < Vij - Vik) , ∀k = j∙
Assuming the error terms to be identically and independently (iid) distributed
as Gumbel (or Type I extreme value), the resulting probability model is logistic,
giving rise to the well-known conditional logit model (see e.g. Ben-Akiva and
Lerman 1985), with choice probabilities equal to:
eVij
(3) Pi(j) =
eVik
k
One drawback of this model is its imposition of the independence of irrelevant
alternatives (IIA) assumption, requiring that when one alternative is removed
from the choice set C , the choice probabilities of the remaining alternatives rise
by the same proportion. This assumption is, in particular, violated when the error