Measuring and Testing Advertising-Induced Rotation in the Demand Curve

effect, as well as a shift-related rotation effect. Once segregated from the shift-
related rotation effects, the interaction-related rotation effects for models F - J
are Yi /( ^{c +} YiAi ) ^, Yi /( cii ⁺ Yi^ln Ai), Yi /( cii ⁺ Yi^ln Ai )^, Yi /( cii ⁺ Yid^ln A₁ ), and

n

(Y_i +Y_i lnA_i∑w_jα_ij) /(c_ii +Y_i lnA_i - w_i), respectively. Taking the Rotterdam model
j

(model I) as an example, the effect of advertising on the own-price elasticity can be
decomposed into three parts: an interaction-related rotation effect of

n

Y_i /(c_ii + Y_i lnA_i) or- Y_i /(η_iw_i), a shift-related rotation effect of∑w_jα_ij , and the
j

negative of a shift effect of α_ii .⁶ Using the above interaction-related rotation effects
to measure the advertising-induced demand curve rotation is advantageous because
it reflects the “true” rotation effects indicated by the price-advertising interaction
terms γ_i ^,s, and because of the ease of interpretation and comparison across demand
models since they are in the form of elasticities.

Additional insight can be obtained by noting that the second-order cross
partial derivatives of any particular function are unaffected by the order in which the
derivative is taken. Thus, in the simple case where quantity demanded qD is defined
to be a function of price and advertising:

(7)    q_D =D(p,A)

the following “duality relation” (Frisch 1959, p. 180) holds:

∂² D = ∂² D

∂p ∂A ∂A ∂p ’

or, in elasticity notion,

11

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