the (latent) demand for one good - say alcohol - ait as a function of the (latent)
consumption of tobacco c*t and common explanatory variables xit as well as alcohol-
specific ones zait, and vice versa. Time and regional effects, including those due to
temporal and regional price variation, are accounted for by including sets of dummy
variables in the vector xit .
ait = γacit + βa xit + δ'a zait + εait (1)
cU = γcaU + βc xit + δC zcit + εcit (2)
This structural model that explains the demand for one good by the consumption
level of the other stays in line with micro-economic theory if the individual consumers’
optimization problem is subject to a fixed consumption constraint concerning the lat-
ter good. This exactly holds for an experimental situation. In this representation
the coefficient γa measures what would happen to the (latent) consumption of alcohol
if the (latent) consumption of tobacco were exogenously reduced by one unit; i.e. γa
represents the derivative of the Marshallian demand for alcohol with respect to the
restricted consumption of tobacco.9 This analogously applies to γc. In this study we
use these coefficients as a measure of complementarity in consumption which exactly
answers the question which often is relevant when thinking about side-effects of drug
related regulation: “Imagine the regulator could manage to reduce smoking by a cer-
tain amount, how would this affect the consumption of alcohol?” As the equations
(1) and (2) describe (hypothetical) experiments they can be both interpreted in terms
of a causal relationship and the “autonomy requirement” (cf. Wooldridge 2002: 209)
is satisfied, even though both structural equations explain the behavior of the same
economic unit.
Obviously, the coefficients γ originate from restricted Marshallian demand and do
not coincide with cross-price derivatives of Hicksian demand functions. However, it is
shown in Appendix A that for any regular utility function our proposed measures of
complementarity γa and γc necessarily show the opposite sign than the corresponding
cross-price derivatives of Hicksian demand functions do, given restricted Marshallian
demand is evaluated at the unrestricted consumer’s optimum.10 Therefore, if the
qualitative question of whether alcohol and tobacco are consumed as complements or
substitutes is addressed, our empirical approach that mimics an experimental study -
9If feedback-effects are taken into account, one might think of (1 - γaγc)-1γa as the more appro-
priate measure. For model stability, the condition 1 - γaγc > 0 needs to be satisfied.
10 This condition holds as we analyze survey data collected from a situation with consumption
restrictions not yet in place.