Spillovers in Attribute Advertising

Steven Schmeiser

doi:10.1515/roms-2014-0020

Article

Spillovers in Attribute Advertising

Steven Schmeiser

Published/Copyright: April 26, 2016

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From the journal Review of Marketing Science Volume 14 Issue 1

Abstract

When firms advertise a vertically differentiated product attribute, they increase the weight consumers put on that attribute. This creates a spillover effect as one firm’s advertising can increase demand for other firms that also have the attribute. I develop a market share attraction model and find that as spillover increases, firms advertise less. Profit can be increasing or decreasing in spillover depending on the particular industry environment. Spillover lends a “public good” quality to advertising and firms free-ride in equilibrium. An advertising ban has an ambiguous effect on the profits of firms with the attribute, but always increases profits of firms without the attribute.

Keywords: advertising; spillover; vertical differentiation

Acknowledgments

This paper benefited greatly from helpful comments by the editor and anonymous referees.

Appendix

A Existence of Equilibrium

Consider a general attraction model

m(s)=f(s)f(s)+g(s)

where f(s) denotes a firm’s attraction as a function of the level of its own advertising s and g(s) denotes the sum of all other firms’ attractions which may depend on the original firm’s advertising. It is assumed that f′(⋅)>0 and g′(⋅)≥0. Then

∂m∂s=(1−m)f′(s)−mg′(s)f(s)+g(s).

Note that even in the general case, the marginal benefit of advertising is decreasing in market share. This is therefore a feature of all attraction models and not specific to the case of linear attraction functions.

The second derivative is given by

∂2m∂s2=−2∂m∂s[f′(s)+g′(s)f(s)+g(s)]+(1−m)f″(s)−mg″(s)(f(s)+g(s))2.

For a firm advertising in equilibrium, ∂m/∂m∂s∂s is positive (the marginal benefit is equated to the strictly positive marginal cost of advertising c), making the first-term negative. The second term could be positive or negative depending on the market share of firm i and the second derivatives of f(⋅) and g(⋅). If f(⋅) and g(⋅) are linear then the second term is zero, and the second derivative of market share is guaranteed to be negative. This ensures that second-order condition is satisfied for firms with an interior solution and that a Nash equilibrium exists.

The problem of existence of Nash equilibrium for attraction function models is not entirely due to the spillover effect. Gruca and Sudharshan (1991) have pointed out the non-existence of equilibrium for the logit model in advertising settings. Mesak and Means (1998) show that if f(⋅) is concave, then an equilibrium does exist and they propose several attraction models with concave f(⋅). Here, these modifications do not work because of the spillover effect: g″(⋅) enters the second term with the opposite sign as f″(⋅). Depending on the amount of spillover and specific market shares, (1−m)f″(s)−mg″(s) could be positive.

B Proofs

B.1 Proposition 4

First, I show that if the total level of advertising increases (decreases) in equilibrium, then the each firm’s individual level also increases (decreases). Let s* denote the total level of advertising. By assumption, a small change in a parameter does not result in the set of advertisers changing. If s* increases, then it must be the case that at least one firm i of the L advertisers increased its advertising. All else equal, this increases the market share of firm i. But in equilibrium, all advertisers have the same market share, so firm i’s increase must result in increases by all remaining advertisers. Therefore, I can do comparative statics on the aggregate level of advertising s*, and individual comparative statics must have the same sign.

The aggregate amount of advertising in equilibrium can be implicitly defined by adding up the L first-order conditions of the advertising firms:

[3]L−xL+s∗(1+γ(M−1))xN+s∗(1+γ(M−1))(1+γ(M−1))=Lc(xN+s∗(1+γ(M−1)))

where x_N is the sum of all firms’ x attributes and x_L is the sum of the advertising firms’ x attributes. I then use the implicit function theorem to do comparative statics on s* with respect to the exogenous variable of interest.

∂s∗∂c=−((1+γ(M−1))+xN)3(1+γ(M−1))(c((1+γ(M−1))s∗+xN)2+(1+γ(M−1))(xN−xL))

The numerator is clearly positive, and the denominator is positive as xN−xL>0. Therefore ∂si∗/∂si∗∂c<0∂c<0.

∂s∗∂γ=−(M−1)(s∗(s∗(1+γ(M−1))2(1+cs∗)+2(1+γ(M−1))(1+cs∗)xN+cxN2)+xNxL)(1+γ(M−1))(c((1+γ(M−1))s∗+xN)2+(1+γ(M−1))(xN−xL))

Both the numerator and denominator are positive, therefore ∂si∗/∂si∗∂γ<0∂γ<0.

B.2 Proposition 5

Following Theorem 1 in Caputo (1996), the envelope results for the model are given by the following. Suppose firm i has yi=1 and advertises in equilibrium. Then

∂mi∗∂c=∑j≠i|∂mi∗∂sjs=s∗∂sj∗∂c+|∂mi∗∂cs=s∗=γ−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c≷0

and

∂πi∗∂c=∑j≠i|∂πi∗∂sjs=s∗∂sj∗∂c+|∂πi∗∂cs=s∗=γ−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c−si∗≷0

The sign of the second line is indeterminate, as γ−mi∗(1+γ(M−1)) could be positive or negative. Similar steps show how the profit of an advertiser changes with γ.

∂mi∗∂γ=∂πi∗∂γ=γ−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c+s∗(1−mi∗(M−1))−si∗xN+s∗(1+γ(M−1))≷0

Now suppose firm i has yi=1 but does not advertise in equilibrium. Then

∂mi∗∂c=∂πi∗∂c=γ−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c≷0mi∗∂γ=∂πi∗∂γ=γ−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c+s∗(1−mi∗(M−1))−si∗xN+s∗(1+γ(M−1))≷0

Suppose firm i has yi=0. Then

∂mi∗∂c=∂πi∗∂c=−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c>0∂mi∗∂γ=∂πi∗∂γ=−mi∗(1+γ(M−1))xN+s∗(1+γ(M−1))∑j≠i∂sj∗∂c−s∗mi∗(M−1)xN+s∗(1+γ(M−1))≷0

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Published Online: 2016-4-26

Published in Print: 2016-6-1

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Articles in the same Issue

https://doi.org/10.1515/roms-2014-0020

Keywords for this article

advertising; spillover; vertical differentiation