Measuring and Testing Advertising-Induced Rotation in the Demand Curve

Appendix A

When price-advertising interaction terms are not present in equations (2) - (6), the
own demand elasticities (in absolute values) are η_i^LM (=- c_ii(p_i /q_i)),

η (= - Ci / qi ), ηΓ (= - Ci ), η (= - c∙ / ), and n (= - (c« / w -1) ),
respectively. The ∂lnη / ∂ln A term in table 1 is derived as follows:

budget share, ∂w_i /∂ln A_i is derived beforehand. Note that:

∂w_i    w_i∂lnw_i   w_i(∂ln p_i + ∂lnq_i - ∂lnY)

∂ ln A_i    ∂ ln A_i               ∂ ln A_i

Under the assumption of fixed price, we have:

n  nn

w_i (∂ ln q_i - ∂ ln∑pjqj)          ∂∑pjqj           ∑p_j∂q_j

----~ =-----------------j-------= w_i (α_ii--^j------) = w_i (α_ii - —--------).
∂lnA_i              ∂lnA_i                ^{i ii}    Y∂lnA_i       ^{i ii}    Y∂lnA_i

nn

Since ∑ p_j∂q_j is identical to∑ p_jq_j∂lnq_j , (A5) leads to:
jj

(A6)    ^dwi = w (α_i - ∑ w α_i ), or ^d ln wⁱ = (α_ii - ∑ wα ■ ).

_∂lnAi       i ii _j j ij         _∂lnAi        ii _j j ij

It follows that:

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