Competitive Advertising and Pricing

Hwang, Kim, and Boleslavsky study how firms in an oligopoly simultaneously choose prices and advertising strategies, where advertising is modeled as the choice of how much product information to disclose to consumers. The paper extends the canonical Perloff-Salop (1985) random-utility discrete-choice framework — in which n firms engage in Bertrand competition for a consumer whose value for each product is independently drawn from a common distribution F — by endogenizing the information environment: each firm may choose any mean-preserving contraction (MPC) of F as its advertising strategy, with no structural restriction on feasible content. This full flexibility, drawn from the information design literature, allows each firm to choose the consumer’s effective value distribution, ranging from full information (choosing F itself) to complete concealment (a degenerate distribution at the mean). The model is silent on advertising costs, which are assumed to be zero throughout.

The central result is that intense competition forces firms to provide precise product information. Formally, the full information equilibrium — in which every firm chooses F — exists in the advertising game (the subgame in which prices are fixed symmetrically) if and only if F^(n-1) is convex over its support. Because F^(n-1) represents the distribution of the consumer’s best outside option, convexity means the consumer likely faces an attractive alternative, incentivizing each firm to maximize the chance of offering the highest possible value. Crucially, this convexity condition is guaranteed to hold when n is sufficiently large, regardless of the shape of F, because the power function x^(n-1) becomes more convex as n rises. This establishes that under sufficiently intense competition, full information disclosure is the unique symmetric equilibrium.

The general equilibrium advertising strategy G* — which governs cases where full information is not an equilibrium — satisfies two necessary and sufficient conditions: (i) (G*)^(n-1) is convex over the support of G*, and (ii) for almost all values in the support, G* either coincides with F (where the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (where the firm is locally risk-neutral and has no incentive to alter its distribution). The paper proves existence and uniqueness of G* for any F satisfying the stated regularity conditions (density positive, continuously differentiable, bounded, with finitely many peaks). When F has log-concave density, a unique symmetric pure-price equilibrium (p*, G*) exists in the full game.

The paper demonstrates that strategic advertising has ambiguous implications for prices and consumer welfare. Strategic advertising necessarily reduces social surplus through information loss, since consumers select suboptimal products with positive probability when G* differs from F. However, it compresses the support of the value distribution relative to F, which — by a new result (Proposition 3) — tends to lower the equilibrium price. Offsetting this, strategic advertising also redistributes marginal consumers in ways that may raise or lower the price. In the duopoly case with power distributions F(v) = v^alpha on [0,1], strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) (approximately 0.7071), and raises consumer surplus if and only if alpha > 0.7928.

The paper examines three extensions: (1) a binding consumer outside option, (2) multi-unit (k-out-of-n) demand, and (3) asymmetric firms with two types. In all three cases, full information cannot be a strict equilibrium for any finite n under the relevant structural condition, yet the equilibrium distribution G* converges pointwise to F as n tends to infinity, preserving the paper’s core asymptotic insight.

Q: What is the main research question? A: The paper asks how much product information firms will voluntarily disclose when they compete both on price and advertising content in an oligopoly. Unlike the monopoly literature, the oligopoly context creates strategic interdependencies — each firm’s optimal disclosure depends on rivals’ disclosure choices — that the paper characterizes fully.

Q: How is advertising modeled, and why use mean-preserving contractions? A: Each firm’s advertising strategy is modeled as a choice of any mean-preserving contraction (MPC) of the true value distribution F. An MPC preserves the expected value but reduces dispersion, capturing the idea that a firm can selectively conceal information (moving toward a degenerate distribution) but cannot fabricate value dispersion beyond what F allows. Because consumers are risk-neutral and buy based on expected values net of prices, this MPC formulation captures full flexibility in information design without loss of generality.

Q: What is the precise necessary and sufficient condition for the full information equilibrium in the advertising game? A: The full information equilibrium — in which every firm chooses F — exists if and only if F^(n-1) is convex over its support [v, v̄]. The “only if” direction follows from Lemma 1: in any equilibrium, (G*)^(n-1) must be convex, so if F^(n-1) is not convex, F is not an equilibrium. The “if” direction follows because a convex F^(n-1) makes each firm locally risk-loving, so no MPC of F yields a higher payoff than F itself.

Q: Why does sufficiently intense competition force full information disclosure? A: For any distribution F with positive, continuously differentiable, bounded density f with bounded derivative f’, the second derivative of F^(n-1) satisfies F(v)^(n-1)’’ >= (n-1)F(v)^(n-3)[(n-2)epsilon^2 - M], where epsilon = min f(v)^2 > 0 and M = max |f’(v)| < infinity. This expression is strictly positive for n sufficiently large, so F^(n-1) is convex and the full information equilibrium exists. Economically, with many competitors each firm wins the consumer only when it offers the highest possible value, so providing full information is optimal.

Q: What are the two necessary and sufficient properties characterizing the general equilibrium advertising strategy G?* A: First (Lemma 1), (G*)^(n-1) must be convex over the support of G* — this prevents any firm from profitably concentrating mass to reduce dispersion. Second (Lemma 2), for almost all values in the support, either G* = F locally (the MPC constraint binds, preventing further dispersion) or (G*)^(n-1) is locally linear (the firm is locally risk-neutral and indifferent over distributions with the same local mean). Theorem 1 proves these two conditions are both necessary and sufficient, and that G* is unique for any F satisfying the stated regularity conditions.

Q: What structure does G take when F^(n-1) has strictly quasi-concave density?* A: By Corollary 2(1), there exists a cutoff v* in [v, v̄] such that G*(v) = F(v) for v <= v* (full information below the cutoff) and (G*)^(n-1) is linear above v*. As n increases, v* rises, meaning the region of full disclosure expands, and G* increases in convex order — so consumers receive strictly more information. One immediate implication is that consumer surplus strictly increases in n: consumers benefit both from more options and from more accurate information about each product.

Q: What happens when F^(n-1) is concave? A: By Corollary 3, when F^(n-1) is concave, (G*)^(n-1) is linear over the entire support, with lower bound v. In the illustrative Example 1 (truncated exponential with n=2), this yields G* = U[0, 2*mu_F] — a uniform distribution on an interval whose upper bound is twice the mean of F.

Q: Does strategic advertising raise or lower equilibrium prices, and consumer surplus? A: Both effects are ambiguous and depend on the shape of F. Strategic advertising compresses the support of the value distribution (since G* is an MPC of F), which by Proposition 3(1) tends to lower equilibrium prices. But it also reshapes the distribution of marginal consumers, which may raise or lower prices. In the power distribution example (n=2, F(v) = v^alpha on [0,1]), strategic advertising lowers the market price if and only if alpha > 1/sqrt(2) ≈ 0.7071, and raises consumer surplus if and only if alpha > 0.7928. Thus even with deadweight loss from information suppression, consumers can be better off under strategic advertising than under forced full disclosure.

Q: What does Proposition 3 contribute about equilibrium prices in the Perloff-Salop model? A: Proposition 3 delivers two results about how the distribution of marginal consumers (integral (F^(n-1))’ dF) determines equilibrium prices. First, the measure of marginal consumers decreases if F is proportionally stretched over a larger support, confirming that longer support raises equilibrium prices. Second — presented as novel — among all distributions with support in [v, v̄], the power distribution F(v) = ((v-v)/(v̄-v))^(2/n) minimizes the measure of marginal consumers, corresponding to the maximum equilibrium price. The key property is that marginal consumers are uniformly distributed under this power distribution, and any deviation from uniformity allows a “flattening” adjustment that increases the measure of marginal consumers and lowers the price.

Q: Under what condition does the full game (price plus advertising) have a unique symmetric pure-price equilibrium? A: Theorem 2 states that log-concavity of the density f is sufficient for existence and uniqueness of a symmetric pure-price equilibrium (p*, G*) as characterized in Theorems 1 and 2. Log-concavity ensures that the equilibrium distribution G* has a convex-linear structure (as in Corollary 2), which preserves log-concavity of each firm’s profit function even under compound deviations (simultaneous changes to both price and advertising strategy), making the first-order conditions sufficient for global optimality.

Q: Can strategic advertising create or destroy pure-price equilibria relative to the Perloff-Salop benchmark? A: Yes, both directions are possible. When F^(n-1) is convex (so G* = F), equilibrium existence in the Perloff-Salop (PS) model is necessary but not sufficient for existence in the full model, because compound deviations (changing both price and advertising) may be profitable even when pure price deviations are not. Conversely, when G* differs from F, the changed distribution of marginal consumers can sustain an equilibrium in the full model even when none exists in PS. Appendix E of the paper provides a specific example of the latter phenomenon.

Q: What happens with a binding consumer outside option? A: Proposition 4 shows that a full information equilibrium never exists in the advertising game when the consumer has a binding outside option (p* in (v, v̄)). The firm’s value function acquires a discrete jump at p* due to the indicator 1_{v >= p*}, making it optimal to pool mass around p* rather than disclose fully. Nevertheless, Proposition 5 proves that G* converges pointwise to F as n tends to infinity, because the jump of size F(p*)^(n-1) vanishes exponentially fast as n grows.

Q: Does the full information result survive multi-unit demand? A: No. Proposition 6 shows that with k > 1 units demanded (out of n products), the full information equilibrium never exists for any finite n or F. The reason is that phi’(v; F) — the firm’s marginal value of offering value v — is zero at v̄ when k > 1, so the firm can profitably pool values near the top of the support. However, Proposition 7 shows that G* converges pointwise to F as n tends to infinity (with k fixed), preserving the asymptotic full information result.

Q: What happens with asymmetric firms differing in their value distribution supports? A: Proposition 8 shows a sharp dichotomy. If both firm types share the same upper bound of their value supports (v̄_1 = v̄_2), the full information equilibrium exists whenever both F_1^(n1-1) and F_2^(n2-1) are convex. If the supports have different upper bounds (v̄_1 < v̄_2), the full information equilibrium never exists regardless of n_1 and n_2, because type-2 firms face a downward kink in their winning probability at v̄_1 and always have an incentive to pool mass there. The authors conjecture that G*_1 and G*_2 still converge to F_1 and F_2 asymptotically but do not prove this due to technical complexity.

Q: How does this paper relate to Ivanov (2013)? A: Ivanov (2013) also uses the Perloff-Salop framework and shows that full information is an equilibrium when n is sufficiently large, but restricts advertising to rotation-ordered strategies (in the sense of Johnson and Myatt, 2006). The present paper imposes no structural restriction and strengthens Ivanov’s result by: (a) providing a necessary and sufficient condition for the full information equilibrium (not just a sufficient condition for large n); (b) fully characterizing G* when full information is not an equilibrium; and (c) demonstrating robustness across multiple model variants.

Q: What policy implication does the ambiguity result carry? A: The paper warns against assuming that mandating full information disclosure is unambiguously consumer-beneficial. While strategic advertising creates deadweight loss through information suppression, it can simultaneously compress support and alter the marginal consumer distribution in ways that lower equilibrium prices significantly. The power distribution example (alpha > 0.7928) shows consumers can be strictly better off under strategic advertising than under forced full disclosure. This ambiguity is a cautionary tale for disclosure regulation.

Mean-Preserving Contraction (MPC): A distribution G_i is an MPC of F if it has the same mean as F but less dispersion (in the sense of second-order stochastic dominance). In the paper, each firm’s feasible advertising strategies are exactly the set MPC(F) — this captures all informationally feasible disclosures without structural restriction on content.

Advertising Game: A restricted subgame of the full market game in which firms choose their advertising strategies G_i taking the symmetric price as given. An equilibrium in the advertising game is a necessary condition for equilibrium in the full game. The advertising game’s equilibrium uniquely pins down G* independently of the price level (under the baseline model without binding outside option).

Full Information Equilibrium: An equilibrium of the advertising game in which every firm chooses the true underlying distribution F as its advertising strategy. This corresponds to complete, unobstructed product disclosure. The paper’s central result is that this equilibrium exists if and only if F^(n-1) is convex over its support.

Convexity of F^(n-1): The key distributional condition governing advertising equilibria. F^(n-1) is the distribution of the consumer’s best alternative among (n-1) rivals’ products. Convexity of F^(n-1) means its density is increasing, signaling a likely attractive outside option, which makes each firm risk-loving and induces full disclosure. This convexity is guaranteed for n sufficiently large.

Locally Linear (G)^(n-1):* A region of the equilibrium distribution where (G*)^(n-1) has constant slope, making the firm locally risk-neutral. Over such a region, the firm is indifferent among all distributions with the same local mean, and the equilibrium G* need not coincide with F — it is only required to be an MPC of F on that interval. This alternating structure (coinciding with F on strictly convex regions; linear elsewhere) fully characterizes G*.

Marginal Consumers: In the Perloff-Salop pricing formula, the equilibrium price p* = (1/n) / integral [(G*(v)^(n-1))’ dG*(v)]. The integrand (G*(v)^(n-1))’ * g*(v) is the density of consumers who are indifferent between a given firm’s product and their best alternative at value v. A larger measure of marginal consumers implies lower equilibrium prices through greater competitive pressure.

Compound Deviation: In the full game, a deviation by a firm that changes both its price p_i and its advertising strategy G_i simultaneously, rather than varying only one dimension. The possibility of compound deviations is what distinguishes equilibrium existence conditions in the full model from those in the standard Perloff-Salop model, even when G* = F.