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

where we have made use of (9)-(11) in the last row of the matrix. Solving (12) we
obtain for ∂a∕∂p_c (we omit the expressions of the other effects being of no further
interest here):

where the denominator D is given by

D = p²a[UccUww - Uc²w]+ pc²[UaaUww - Ua²w]+ p²w[UaaUcc - Ua²c]
+2p_ap_c[U_aw U_cw - U_acU_ww] + 2p_ap_w [U_ac U_cw - U_cc U_aw]
+2pcpw [UacUaw ^- UaaUcw]

and is greater than zero by strict quasi-concavity and the resulting second-order con-
dition of the consumer’s expenditure minimization problem.

We now look at the restricted Marshallian demand a M(p_a,p_c,p_w,c,y) which by
definition is the solution of max U(a, c, w) sub ject to
a,c,w

paa + pcc + pww = y                         (14)

and c ≤ C. The Lagrange function is then given by L(a,c,w,μ,ν) = U(a,c,w) +
μ [У — paa — pcc — p_w w] + ν [C — c]. Assuming that the constraint c ≤ C holds with
equality, the first-order necessary conditions for the utility maximum are given by

Differentiating (15), (16) and (14) with respect to c we obtain:

Solving this system we obtain for ∂a∕∂c (again omitting the other expressions) we

where the denominator D' = p²_υlU_aa + p²_aU_ww - 2p_ap_wU_aw is negative by strict quasi-
concavity.

Finally, comparing (13) and (17) we obtain

∂aH _ ∂aM μD'
∂p_c    ∂c D

establishing (7).

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