Measuring and Testing Advertising-Induced Rotation in the Demand Curve

Appendix B

With price-advertising interaction terms included in equations (2) - (6), the
corresponding own demand elasticities (in absolute values) are

η_i^LM (=- (c_ii +γ_iA_i)( p_i Iq_i) ), η_i^SL(=-(c_ii+γ_ilnA_i)Iq_i), η_i^DL

(=-(cii+γilnAi)), ηi^RM (=-(cii+γidlnAi)Iwi), and ηi^AIDS

(=- ((c_ii +γ_i ln A_i)Iw_i - 1)), respectively. The ∂lnηI∂lnA term in table 1 is

derived as follows:

(B1)

∂ lnη_i^LM

∂ ln A_i

∂ηLM
_iLM

η_i^LM∂lnA_i

p_i

LM
η_i^LM

p_iq_i

- (c_ii + γ_iA_i)p_i

-c_ii∂q_i

q_i²∂lnA_i

- c_ii ∂ ln q_i₊q_i ∂ ln A_i

γ_i(q_i∂A_i /∂lnA_i - A_i∂q_i /∂lnA_i)

qi²

YiAiqi (^d^ln Ai I^d^lnA₁^-d^ln q_i I^d^lnAi)

^qi²

= YiAi ^- (cii ⁺ YiAi )^αii
cii ⁺ YiA

(B2)

∂ lnη_i^SL ∂ ln[-(c_ii +γ_i ln A_i)] ∂lnq_i

∂ln A_i

∂ ln A_i

∂[-(c_ii +γ_ilnA_i)]
(c_ii +γ_ilnA)∂lnA_i

-α_ii

(B3)

∂ lnη_i^DL

∂ ln A_i

(B4)

∂ lnη_i^RM

∂ ln A_i

γ_i

c_ii +γ_ilnA_i

∂ηi^DLη_i^DL ∂ ln A_i

∂η_i^RMη_i^RM∂lnA_i

-1

RM
η_i^RM

-α_ii,

-1

c_ii +γ_i lnA_i

-1

RM
^ηi^RM

∂(-c_ii) γ_i∂lnA_i

∂ ln A_i ∂ ln A_i

c_ii∂w_i

w_i² ∂ ln A_i

^Yⁱ and

c_ii +Y_ilnA_i

γ_i(w_i -dlnA_i∂w_iI∂lnA_i)

^wi²

^-cii^d^lnwi ₊ Yi(1^-d^lnA^d^lnwi I^d^lnA-)

w_i ∂ ln A_i

w_i

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