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Persuasive Advertising in a Vertically Differentiated Competitive Marketplace

Yuanfang Lin EMAIL logo and Chakravarthi Narasimhan

Abstract

Despite the widely acknowledged existence in practice, the theoretical literature on persuasive advertising is generally vague about exactly how such advertising could affect consumer preferences, except for the general assumption that persuasive advertising affects consumer willingness to pay or simply “shifts demand.” This paper proposes a theoretical framework for characterizing different ways that persuasive advertising may affect consumer utility in a vertically differentiated marketplace. Firstly, persuasive advertising could simply raise consumers’ reservation price for the product category. Secondly, persuasive advertising could enhance consumers’ perception about the product quality offered by the advertising firm. Thirdly, persuasive advertising could increase consumers’ willingness to pay for quality increment. Preliminary evidences from lab studies are presented to support the existences of the proposed effects. Using a game-theoretic approach, we study two firms’ decision in the adoption of persuasive advertising of a particular effect and the associated price competition. Findings from the theoretical model analyses indicate that factors influencing a firm’s decision in persuasive advertising include consumer heterogeneity, degree of product differentiation, the effectiveness and the cost of such advertising. In a vertically differentiated competitive marketplace, persuasive adverting is a more desirable strategic tool for firms of higher-quality products to further establish a competitive advantage.


Corresponding author: Yuanfang Lin, Gordon S. Lang School of Business and Economics, University of Guelph, Guelph, ON, Canada, E-mail:

Appendix 1

Proof of Proposition 1

We use backward induction by deriving the Nash Equilibrium in pricing between the two firms, conditional upon each firm’s decision on whether or not to invest in persuasive advertising to increase consumer reservation value (V). There are four subgames as each firm can choose to advertising or not. Returning to the first stage, a firm’s equilibrium in advertising decision is derived by comparing the payoffs across the different subgames.

For subgame 1 (where neither firm chooses persuasive advertising), Equation (1) represents the utility of a consumer i from buying product of Firm j if she has marginal willingness to pay for incremental quality θi. It is straightforward to derive the equilibrium prices of the two firms to be p1 =(Δ + δ)/3, p2 = 2(Δ + δ)/3. And the equilibrium profits for the two firms are π1 = (Δ + δ)2/9Δ, π2 = (2Δ − δ)2/9Δ.

For subgame 2 (only lower-quality Firm 1 invests in persuasive advertising a1 on V), we now use Equation (2) to derive the marginal consumer who is indifferent between buying from Firm 1 and from Firm 2. Notice that (V + a1) canceled out when equating Ui1 to Ui2 of Equation (2). It is straightforward to show that the indifference consumer, equilibrium prices and profits are the same as in the above subgame 1. In order to maximize profit in this subgame π1 = (Δ + δ)2/9Δ − ca12/2, Firm 1’s optimal advertising decision is at the corner solution of a1 = 0.

The derivations for Subgame 3 (only higher-quality Firm 2 invests in persuasive advertising a2 on V) and subgame 4 (both firms invest in persuasive advertising aj on V) follow the same process as in Subgame 2, with Firm j’s optimal advertising decision being aj = 0. This proves the first part of Proposition 1. The proof for the second part of Proposition 1 follows similar process after replacing Equation (2) with Equation (5). This completes the proof of Proposition 1. ▪

The Proofs for the remaining Propositions are similar to the approach used in the above proof for Proposition 1. Therefore we present the proofs in a more concise format to simplify the exposition.

Appendix 2

Proof of Proposition 2

Again we start by deriving the pricing equilibrium and profits from each of the four subgames conditional upon whether a firm decides to invest aj to enhance consumers’ quality perception on product qj, j = 1, 2. Subgame 1 is exactly the same as in the above proof for Proposition 1. For subgame 2 (only lower-quality Firm 1 invests in persuasive advertising a1 to enhance quality perception of q1), from Equation (3) we can derive the pricing equilibrium and Firm 1’s gross profit to be (Δ − a1 + δ)2/9(Δ − a1) which is positive as long as a1 < Δ. Within the subgame Firm 1 decides optimal level of advertising (a1) by maximizing π1q = (Δ − a1 + δ)2/9(Δ − a1) − ca12/2. It can be shown that π1q is a monotone decreasing function of a1, which implies that Firm 1’s optimal advertising decision is at the corner solution of a1 = 0. The derivations for Subgame 3 (only higher-quality Firm 2 invests in persuasive advertising a2 on q2) and subgame 4 (both firms invest in persuasive advertising aj on qj) follow the same process with the result of Firm 2’s optimal advertising decision being, a2>0 and π2(Subgame3)π2(Subgame1)>0. This completes the proof of Proposition 2. ▪

Appendix 3

Proof of Proposition 3

Again we start by deriving the pricing equilibrium and profits from each of the four subgames conditional upon whether a firm decides to invest aj to enhance consumers’ willingness to pay for incremental quality (θ). Subgame 1 is exactly the same as in the above proof for Proposition 1. The derivation for subgame 2 (only lower-quality Firm 1 invests in persuasive advertising a1) is similar as in the proof of Proposition 2 which results in the corner solution of a1 = 0.

For subgame 3 (only higher-quality Firm 2 invests in persuasive advertising a2), from Equation (4) we can identify the marginal consumer who is indifferent between buying from Firm 1 and from Firm 2. In particular, the marginal consumer is found to have willingness to pay for incremental quality θi =( p2 − p1)/Δ − a2∈[0, 1]. The two firms simultaneously set prices by maximizing the respective profits of

(A1)π1θ=p1(p2p1Δa2)
(A2)π2θ=p2(1p2p1Δ+a2)12ca22

The equilibrium prices are derived by solving the simultaneous equations of ∂π/∂pj = 0, j = 1, 2 and check the negative semi-definiteness of the second-order Hessian matrix. This yields p1 =( Δ(1 − a2) + δ)/3, p2 =( 2(Δ + δ) + Δa2)/3. Substituting the price expressions into Equation (A2), Firm 2’s optimal advertising level can be derived by solving ∂π2θ/∂a2 = 0. This results in a2 = 2(2Δ − δ)/(9c − 2Δ). Substituting a2 into the above price expressions and further into (A1) and (A2), the two firms’ equilibrium profits from subgame 3 can be written as:

(A3)π1θ=(3c(Δ+δ)2Δ2)2Δ(9c2Δ)2
(A4)π2θ=c(2Δδ)2Δ(9c2Δ)

Note that a2, π2θ > 0 if Δ < 9c/2. Otherwise a2 = 0 and the results are back to those in subgames 1 and 2.

The derivation of subgame 4 (both firms invest in aj) follow the same process as above and the two firms’ advertising decisions within the subgame are a1′ = 0 (corner solution) and a2′ = 2(2Δ − δ)/(9c − 2Δ) with profits as reported in Equations (A3) and (A4). Given Firm 1 does not invest in persuasive advertising, Firm 2 will have unilateral incentive of advertising if π2(Subgame 3) − π2(Subgame 1) = c(2Δ − δ)2/Δ(9c − 2Δ) − (2Δ − δ)2/9Δ > 0. Solving the inequality results in the parameter condition as reported in Proposition 3. This completes the proof of Proposition 3. ▪

Appendix 4

Proof of Proposition 4

It is straightforward to show that when V = 0, consumers with θ∈[0, p1/q1] will not buy from either Firm. Price competition without any type of persuasive advertising results in p1 =( δq1 − q12 + q1q2)/(4q2 − q1), p2 = (2q22 + 2δq2 − 2q1q2)/(4q2 − q1), and the two firms’ profits are π1N = q1q2(Δ + δ)2/Δ(4q2 − q1)2, π2N = (δq1 + 2q2(Δ − δ))2/Δq1(4q2 − q1)2.

To simplify exposition, we skip the proofs for Part 1) and Part 2) of Proposition 4 as they follow the exact same process as in that of Proposition 1 and Proposition 2. For Part 3) of Proposition 4, the payoffs for subgame 2 and 3 (where only one firm invests in persuasive advertising) can be derived similarly as those in Proposition 3. Below we present details of derivation for subgame 4 where both firms invest in persuasive advertising (aj, j = 1, 2) aiming to increase consumers’ willingness to pay for incremental quality (θ).

From Equation (4) with V = 0, it can be derived that consumers with θ∈[0, θ0] will not buy from either Firm, where θ0 = p1/q1 − a2 − a1, and the marginal consumer with θ1 =( p2 − p1)/Δ − a2 − a1 is indifferent between buying from Firm 1 and from Firm 2. The two firms simultaneously set prices by maximizing the respective profits of

(A5)π1=p1((p2p1Δa2a1)(p1q1a2a1))12ca12
(A6)π2=(p2δ)(1(p2p1Δa2a1))12ca22

The equilibrium prices are derived by solving the simultaneous equations of ∂πj/∂pj = 0, j = 1, 2 and check the negative semi-definiteness of the second-order Hessian matrix. This yield p1 =( Δ(1 − a2 − a1) + δ + 2)/3, p2 =( 2(Δ + δ) + Δ(a2 + a1) + 1)/3. Substituting the price expressions into Equations (A5) and (A6), the two firms’ equilibrium advertising levels can be derived by solving the simultaneous equations of ∂πj/∂aj = 0, j = 1, 2 and check the negative semi-definiteness of the second-order Hessian matrix. This results in a1 = 2(3c(1 − Δ − δ) + 2Δ2)/3c(9c − 4Δ), and a2 = 6c(3c(1 + 2Δ − δ) − 4Δ2)/3c(9c − 4Δ). Substituting aj(j = 1, 2) into the above price expressions and further into (A5) and (A6), the two firms’ equilibrium profits from subgame 4 can be written as:

(A7)π1=(9c2Δ)(2Δ23cΔ+3c3cδ)2729c3Δ648c2Δ2+144cΔ3
(A8)π2=(9c2Δ)(2Δ26cΔ3c+3cδ)2729c3Δ648c2Δ2+144cΔ3

Further algebra can show that there exists parameter conditions on c where π1 − π1(a1 = 0, a2) > 0 and π2 − π2(a1, a2 = 0) > 0. This completes the proof of Proposition 4. ▪

Appendix 5

Proof of Proposition 5

To simplify exposition, we skip the proofs for Part 1) and Part 2) of Proposition 5 as they follow the exact same process as in that of Proposition 1 and Proposition 2. For Part 3) of Proposition 5, we follow the same process as in that of Proposition 3 by providing detailed derivation for subgame 3 (only higher-quality Firm 2 invests in persuasive advertising a2). For those α proportion of consumers reached by Firm 2’s persuasive ad, marginal consumer can be identified as having θr =( p2 − p1)/Δ − a2. For the remaining 1-α proportion of consumers who are not reached by Firm 2’s persuasive ad, marginal consumer can be identified as having θu = p2 − p1/Δ which is the same as in subgame 1 (where neither firm advertises). The two firms then simultaneously set prices by maximizing the respective profits of

(A9)π1r=p1(α(p2p1Δa2)+(1α)(p2p1Δ))
(A10)π2r=p2(α(1p2p1Δ+a2)+((1α))(1p2p1Δ))12ca22

The equilibrium prices are derived by solving the simultaneous equations of ∂π/∂pj = 0, j = 1, 2 and check the negative semi-definiteness of the second-order Hessian matrix. This yields p1 =( Δ + δ − Δαa2)/3, p2 =( 2(Δ + δ) + Δαa2)/3. Substituting the price expressions into Equation (A10), Firm 2’s optimal advertising level can be derived by solving ∂π2r/∂a2 = 0. This results in a2 = 2α(2Δ − δ)/(9c − 2Δα2). Substituting a2 into the above price expressions and further into (A9) and (A10), the two firms’ equilibrium profits from subgame 3 can be written as:

(A11)π1r=(3c(Δ+δ)2Δ2α2)2Δ(9c2Δα2)2
(A12)π2r=c(2Δδ)2Δ(9c2Δα2)

Note that a2, π2r > 0 if Δ < 9c/2α2. Otherwise a2 = 0 and the results are back to those in subgames 1 and 2.

The derivation of subgame 4 (both firms invest in aj) follow the same process as above and the two firms’ advertising decisions within the subgame are a1′ = 0 (corner solution) and a2′ = 2α(2Δ − δ)/(9c − 2Δα2) with profits as reported in Equations (A11) and (A12). Given Firm 1 does not invest in persuasive advertising advertising, Firm 2 will have unilateral incentive of advertising if π2(Subgame 3) − π2(Subgame 1) = c(2Δ − δ)2/Δ(9c − 2Δα2) − (2Δ − δ)2/9Δ > 0. Solving the inequality results in the parameter condition as reported in Proposition 5. This completes the proof. ▪

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Received: 2019-12-11
Accepted: 2020-05-17
Published Online: 2020-09-23

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