The many conditions under which monopolistic advertising can differ from the social optimum

Abstract

This paper takes a new approach to a classical question about the relationship of monopolistic behavior to the social optimum when advertising is admitted. By characterizing conditions in terms of consumer preferences and using an uncommon approach to comparative static analysis, we derive a general result that produces a dozen special cases of interest. We also show that a plausible preference specification is general enough to generate each of these cases. The specification is amenable to estimation and inference with common data, although empirical application is beyond the scope of this paper. Results are derived assuming that advertising follows the complementary rather than persuasive advertising paradigm where consumers have stable quasilinear preferences and the amount of advertising is seller-determined rather than offered at a unit price to consumers.

Appendix

This appendix demonstrates the possibility of each of the conditions in this paper using the specific utility specification u(q,a) = q ^α + φa ^β q + λq ^γ a assuming μ is sufficiently large and φβ(β – 1) < 0 as required for Assumptions 1–3. Our intent is merely to show that the results of each corollary can be generated by some set of feasible conditions, rather than to determine all possible conditions that generate each corollary. We first note that monopoly advertising occurs (or not) as u _qa q = τq + λγq ^γ > (≤) 1 which requires λγ > (≤) (1 – τq)q ^–γ. Advertising occurs (or not) at the social optimum as u _a   = τq + λq ^γ > (≤) 1, which requires λ > (≤) (1 – τq)q ^–γ. For Corollaries 1–12, advertising by both a monopoly and the social optimum requires min{λ, λγ} > (1 – τq)q ^–γ. For Proposition 3, advertising by a monopoly but not at the social optimum requires λ ≤ (1 – τq)q ^–γ < λγ. For Proposition 4, advertising at the social optimum without advertising by a monopoly requires λγ ≤ (1 – τq)q ^–γ < λ. For brevity of presentation, we define τ = φβa ^β-1 where sign(τ) = sign(φβ).

Except where explicitly indicated otherwise, the possibilities we demonstrate for Corollaries 1–12 assume the standard conditions 0 < α < 1, 0 < β < 1, 0 ≤ γ ≤ 1, φ ≥ 0, and that μ is sufficiently positive so that second-order conditions hold. For Corollary 1, u _qa q – u _a  < 0 if (γ – 1)λ < 0 and u _qqa q + u _qa  > 0 if λγ ² > −τq ^{1– γ}. These conditions and those for advertising by both regimes are satisfied if λ > min{0, (1 – τq)q ^–γ/γ}, among other possibilities. Corollary 2 holds if u _qa q – u _a  = 0, which requires (γ – 1)λ = 0, and u _qqa q + u _qa  > 0, which requires λγ ² > −τq ^{1– γ}, which together with conditions for advertising by both regimes are satisfied by γ = 1 and λ > min{0, q ⁻¹ – τ}, among other possibilities. Corollary 3 holds if u _qa q – u _a  > 0, which requires (γ – 1)λ > 0, and u _qq q = α(α – 1)q ^α–1 + λγ(γ – 1)q ^γ–1 a ≈ 0, which together with conditions for advertising by both regimes hold if γ > 1, α,γ → 1, and λ > min{0, (1 – τ)q ^–γ}. Compared to Corollary 3, Corollary 4 only requires confining γ to the unit interval to satisfy u _qa q – u _a  < 0, which changes the dominating constraint required for both regimes to advertise, and is satisfied by 0 < γ < 1, α,γ → 1, and λ > min{0, (1 – τ)γ ⁻¹ q ^–γ}. Corollary 5 requires u _qa q – u _a  > 0, i.e., (γ – 1)λ > 0, and that ω _qq  = θ(u _qqq q + u _qq ) + u _qq – c _qq  = α ²(α – 1)μq ^α–2 + λγ ²(γ – 1)q ^γ–2 a – c _qq is large negatively relative to u _qq  = α(α – 1)μq ^α–2 + λγ(γ – 1)q ^γ–2 a, which together with conditions for advertising by both regimes hold if γ > 1, α,γ → 1 so that u _qq /c _qq   → 0 assuming c _qq is bounded away from zero, and λ > (1 – τ)/q ^γ.

Corollary 6 includes the condition that u _qqa q + u _qa  ≪ 0 meaning that u _qqa is sufficiently negative that negative cases of θu _qqa q + u _qa (where θ is large) dominate the positive cases (where θ is small) so that E[u _qq q(θu _qqa q + u _qa )] > 0. To verify the possibility of this case, we assume that u _qq q and δ are approximately constant on θ ∈ [0,1] so that E(θu _qqa q + u _qa ) < 0 holds if ½u _qqa q + u _qa  < 0.¹³ Then Corollary 6 holds if u _qa q – u _a  > 0, i.e., (γ – 1)λ > 0, and ½u _qqa q + u _qa  < 0 if λγ(γ + 1) < −2τq ^{1– γ}. If the first condition is satisfied by γ > 1 and λ > 0, then the latter condition together with advertising by both regimes requires (1 – τq)q ^–γ < λ < −2τq ^{1– γ}/γ(γ + 1), which is possible if τq = φβa ^β-1 q > γ(γ + 1)/[γ(γ + 1) – 2] > 0. Thus, Corollary 6 holds if γ > 1, max{0, (1 – τq)q ^–γ} < λ < −2τq ^{1– γ}/γ(γ + 1), and φβa ^β-1 q > γ(γ + 1)/[γ(γ + 1) – 2], which implies that β and φ are either both positive or both negative.¹⁴ This is a stringent set of conditions and may not be possible for every set of parameter choices, but these results demonstrate that the possibility can arise.

Turning to the corollaries regarding quantity effects, the conditions of Corollaries 7 and 9 are identical to Corollaries 1 and 3, respectively, so no further demonstration is required. Further, Corollary 8 simply drops one of the requirements of Corollary 2 so trivially it applies under the conditions of Corollary 2. For Corollary 10, ω _aa  = θu _qaa q + (1 – θ)u _aa  = θφβ(β – 1)a ^β–2 q + (1 – θ)φβ(β – 1)a ^β−2 q → 0 if β → 1, u _qqa q + u _qa  > 0 if γ > 1 and λ > 0, and u _qqa q + u _qa  > 0 if λγ ² > − τq ^{1– γ}. These conditions and conditions for advertising under both regimes hold if γ > 1, β → 1, and λ > min{0,(1 – τ)/q ^γ}. For Corollary 11, γ > 1, α,γ → 1, and λ > min{0,(1 – τ)/q ^γ} imply u _qqa q + u _qa  > 0, λγ ² > − τq ^{1– γ}, and u _qa q – u _a  > 0, which satisfy the conditions for advertising by both regimes. For Corollary 12, u _qa q – u _a  < 0 can be satisfied by γ > 1 and λ < 0, which implies that conditions for both regimes to advertise require 0 > λγ > (1 – τq)q ^–γ. Then under the same illustrative approach as used for Corollary 6, ½u _qqa q + u _qa  < 0 follows from λγ(γ + 1) < −2τq ^{1– γ}, which is equivalent to λ > −2τq ^{1– γ}/γ(γ + 1). These constraints leave a non-empty set of possibilities for λ if (1 – τq) < −2τq/(γ + 1), which holds if τq > (γ + 1)/(γ – 1) > 0. Jointly, these conditions are satisfied by γ > 1, 0 > λ > (1 – τq)q ^–γ/γ, and φβa ^β-1 q = τq > (γ + 1)/(γ – 1) > 0, which implies that β and φ are either both positive or both negative.¹⁵

For Corollary 14, u _qqa  = λγ(λ – 1)q ^γ–2 = 0 if λ = 1, u _qq q → 0 if α,γ → 1, and advertising by a monopoly without advertising at the social optimum holds if (1 – τq)q ^–γ/γ < λ ≤ (1 – τq)q ^–γ, which includes λ = 1 if γ > 1 and 0 < τq < 1 (e.g., under small positive values of φ and β), which also satisfies u _qa   = τ + λγq ^γ–1 u _qqa   = λγ(γ – 1)q ^γ–2 > 0. Corollary 15 is verified similarly by relaxing the λ = 1 requirement because u _qqa q + u _qa   = τ + λγ ² q ^γ–1 > 0 if λγ ² > − τq ^{1– γ}, which is a redundant requirement if λ > 0. The possibility of conditions for Corollary 16 is verified as for Corollary 12 except that advertising by a monopoly without advertising at the social optimum with profit-maximization substitutes requires (1 – τq)q ^–γ/γ < λ < min{(1 – τq)q ^–γ, −2τq ^{1– γ}/[γ(γ + 1)]}, which requires 0 < γ < 1 rather than γ > 1, which in turn means that (1 – τq)q ^–γ/γ < −2τq ^{1– γ}/[γ(γ + 1)] requires τq < (γ + 1)/(γ – 1), i.e., either φ < 0 or β < 0.

For Corollary 17, u _qqa  = 0 if λ = 1 and u _qa   > 0 if λγ(γ – 1)q ^γ–2 > 0. Combining these conditions with the condition that advertising occurs at the social optimum but not by a monopoly requires λ > (1 – τq)q ^–γ ≥ (1 – τq)q ^–γ/γ, which applies when −1 < γ < 0. All conditions are satisfied if 0 < τq < 1 (e.g., under small positive values of φ and β) and (1 – τq)q ^–γ < 1. Corollary 18 is verified similarly by relaxing the λ = 1 requirement because u _qqa q + u _qa   = τ + λγ ² q ^γ–1 > 0 if λγ ² > − τq ^{1– γ}, which is a redundant requirement if λ > 0. The possibility of conditions for Corollary 19 is verified as for Corollary 6 except that advertising at the social optimum but not by a monopoly with profit-maximization substitutes requires (1 – τq)q ^–γ < λ < min{(1 – τq)q ^–γ/γ, −2τq ^{1– γ}/[γ(γ + 1)]}, which requires 0 < γ < 1 rather than γ > 1, which in turn means that (1 – τq)q ^–γ < −2τq ^{1– γ}/[γ(γ + 1)] is satisfied by 0 > τq > (γ + 1)/(γ – 1), i.e., where φ and β are small and have opposite signs.