Tobacco and Alcohol: Complements or Substitutes? - A Statistical Guinea Pig Approach

Appendix

A Equivalence of Measures of Complementarity

In this appendix we show that the cross price effect of increasing the price of tobacco
(alcohol) on the Hicksian demand for alcohol (tobacco) has always the opposite sign
of the effect resulting from increasing the consumption of tobacco (alcohol) on the
Marshiallian demand for alcohol (tobacco).

To see this, we write the consumer’s direct utility as U (a, c, w), where we denote by
a, c, and w the amounts of consumed alcohol, tobacco and a compound good consisting
of all other goods, respectively. For simplicity, any subscripts i and t denoting specific
individuals and periods are skipped. The corresponding prices are p_a ,p_c, and p_w .
Hicksian demand for alcohol is written as aH(p_a,p_c,p_w, U), for some fixed utility level
U. Accordingly, the restricted Marshallian demand for alcohol, if the consumption of
tobacco c is given, is denoted by a M(p_a,p_c,p_w,c,y) where y is income. We now state
the following result:

Proposition: If U is strictly quasi-concave, and both the Marshallian and the
Hicksian demand is characterized by interior solutions in a, c, and w, then

. ∂^aH(⁽Pa^,Pc^,Pw^,U )1       ∙ ∂^aM(⁽Pa^,Pc^,Pw ^,С,У Л              ∕_7λ

sign                      = - sign                        .            (7)

∂pc        J            |_          ∂c

Proof: By definition aH(Pa,Pc,Pw,U) is the solution of min {paa + pcc + Pww}
a,c,w

subject to

U(a, c, w) = U.                                  (8)

The first-order necessary conditions for the expenditure minimum are given by

Ua ( a,c,w )  =  A ¹ Pa ≡ μpa                          (9)

Uc(a,c,w)  =  A ¹ Pc ≡ μpc                         (10)

Uw (a,c,w)  =  A ¹ Pw ≡ μpw                      (11)

where A is the Langrange multiplier with respect to (8) and μ = A ¹. In order to obtain
∂a∕∂p_c we differentiate the equation system (9)-(11) and (8) totally with respect to p_c

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