Abstract
In their seminal paper, Grossman, G. M., and C. Shapiro. 1984. “Informative Advertising with Differentiated Products.” The Review of Economic Studies 51: 63–81 assume that it is not profitable for a firm to deviate to the supercompetitive price of Salop, S. C. 1979. “Monopolistic Competition with outside Goods.” The Bell Journal of Economics 10: 141–56. In this note, it is shown that this assumption is violated if, roughly, each firm reaches less than half of all consumers unless it is a duopoly. This implies that most of the simulations in Grossman, G. M., and C. Shapiro. 1984. “Informative Advertising with Differentiated Products.” The Review of Economic Studies 51: 63–81 are not actually equilibria. More importantly, this implies that for their equilibrium to exist nearly all consumers must receive at least one ad. For example, with just four firms in the market, at least 96% of the consumers must receive at least one ad, and this percentage increases with the number of firms in the market.
Acknowledgments
I thank Agostino Manduchi for all his help, Fabrizio Germano and Sandro Shelegia for early conversations on the idea, and Adib Bagh for suggestions that greatly improved the proof to Proposition 1. This research was started while I was a visiting scholar at the Department of Economics and Business, Universitat Pompeu Fabra and I am grateful for their support.
Appendix A: Proof of Lemma 1
Lemma 1
If it is (not) profitable to deviate in price alone, then it is (not) profitable to deviate to in price (p) and corresponding optimal advertising reach (ϕ) given the deviation price.
Proof
Consider a deviation price p′. First, if it is profitable to deviate to p′ without adjusting advertising intensity, then it is profitable to deviate to p′ with ϕ chosen to maximize profits given the deviation to the supercompetitive price (denoted ϕ′). Consider next an unprofitable deviation to p′. To analyze this, it is convenient to use demand condition on consumers receiving an ad from the firm (and so not dependent on the firm’s ϕ), defined as
that is,
(Recall that the second argument of
As ϕ
e
maximizes profit
Thus,
□
Appendix B: Necessity of a Covered Market
Lemma
The competitive equilibrium of Grossman and Shapiro (1984) does not exist if the covered market assumption (p(ϕ, n) < v − t/2) does not hold.
Proof
In the construction of demand in Section 3.1, the last group of remaining consumers was given in Eq. (10) as
This is derived by subtracting all of the previous groups from δ (the population size). However, if the covered market condition does not hold then this last group is smaller (if zero, then the argument recurses to the penultimate group, etc.; for ease, only the case of the last group being positive is presented). Specifically, the furthest consumer with positive surplus from buying from the firm is defined by v − tx − p, that is x = (v − p)t. As this occurs in both directions from the firm, the population size is now δ[1 − 2(v − p)/t]. Subtracting the consumers in the other groups yields the final group of consumers having size
Thus, the correct demand in this case is
The profit expression, then, is
Differentiating the profit expression with respect to p and evaluating at
In contrast, if the covered market condition held, differentiating the profit expression in that case and evaluating
The difference between the two expressions is in (B.1) term
Appendix C: Existence Under Approximation
To explicitly solve their model, Grossman and Shapiro (1984) assume that the value for n is large enough such that (1 − ϕ) n−1 is effectively zero. As a result, the equilibrium price in Grossman and Shapiro (1984) is c + t/(ϕn), and equilibrium profits gross of advertising and entry costs are
Given this equilibrium price, the supercompetitive deviation price is c + t/(ϕn) − t/n = c + (1 − ϕ)t/(ϕn). To make the analysis more tractable, if the approximation (the assumption of large n) is that (1 − ϕ) n−2 is approximately zero, rather than (1 − ϕ) n−1 in Grossman and Shapiro (1984), the deviation demand (15) can be approximated as
The expression is intuitive: the firm has its market share (δ/n) plus captures all of the consumers it reaches in its two closest rivals’ markets δϕ/n. As a result, deviation profits are
Deviation profits (C.4) are greater than equilibrium profits (C.3) whenever ϕ < 1/2.
Lemma 4
Under “large n” assumption/approximation, a necessary condition for an equilibrium is that ϕ > 1/2.
The corollary is consistent with the proposition which states that if n is greater than 4 and ϕ is less than one-half, then the equilibrium does not exist. This is logical as Grossman and Shapiro (1984) interpret the approximation as assuming “large n” in their model.
References
Christou, C., and N. Vettas. 2008. “On Informative Advertising and Product Differentiation.” International Journal of Industrial Organization 26: 92–112. https://doi.org/10.1016/j.ijindorg.2006.08.004.Search in Google Scholar
Creane, A., and A. Manduchi. 2019. “Informative Advertising in Monopolistically Competitive Markets.” In MPRA Paper No. 95053.10.2139/ssrn.3290612Search in Google Scholar
Grossman, G. M., and C. Shapiro. 1984. “Informative Advertising with Differentiated Products.” The Review of Economic Studies 51: 63–81. https://doi.org/10.2307/2297705.Search in Google Scholar
Perloff, M. J., and C. S. Salop. 1985. “Equilibrium with Product Differentiation.” The Review of Economic Studies 52: 107–20. https://doi.org/10.2307/2297473.Search in Google Scholar
Salop, S. C. 1979. “Monopolistic Competition with Outside Goods.” The Bell Journal of Economics 10: 141–56. https://doi.org/10.2307/3003323.Search in Google Scholar
Tirole, J. 1988. The Theory of Industrial Organization. Cambridge: MIT Press.Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
