Complementarity of Information Products

Andrew T. Ching; Ignatius Horstmann; Hyunwoo Lim

doi:10.1515/roms-2021-0004

Article

Complementarity of Information Products

Andrew T. Ching , Ignatius Horstmann and Hyunwoo Lim

Published/Copyright: July 9, 2021

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From the journal Review of Marketing Science Volume 19 Issue 1

Abstract

In “Marketing Information: A Competitive Analysis,” Sarvary, M., and P. M. Parker. 1997. “Marketing Information: A Competitive Analysis.” Marketing Science 16 (1): 24–38 (S&P) argue that in part of the parameter space that they considered, a reduction in the price of one information product can lead to an increase in demand for another information product, i.e. information products can be gross complements. This result is surprising and has potentially important marketing implications. We show that S&P obtain this complementarity result by implicitly making the following internally inconsistent assumptions: (i) after purchasing information products, consumers update their beliefs using a Bayesian updating rule that assumes they have a diffuse initial prior (i.e. their initial prior variance is ∞ before receiving any information); (ii) if consumers choose not to purchase any information product, it is assumed that their initial prior variance is 1 (implied by the utility function specification). This internal inconsistency leads to the possibility that when information products are uncorrelated and their variances are close to 1, marginal utility is increasing in the number of products purchased, and hence information products can be complements in their model. We show that if we remove this internal inconsistency, in the parameter space considered by S&P, information products cannot be complements because the marginal utility of information products will be diminishing. We also show that, in parts of the parameter space not considered by S&P, it is possible that information products are complements; this space of parameters requires consumer’s initial prior to be relatively precise and information products to be highly correlated (either positively or negatively).

Keywords: Bayesian decision making; information complements; information substitutes; pricing

JEL Classification: D11; D81; D83; L15; L86; M31

Corresponding author: Andrew T. Ching, Carey Business School, Johns Hopkins University, Baltimore, MD, USA, E-mail: andrew.ching@jhu.edu

We thank the editor, Ernan Haruvy, and four anonymous reviewers for their constructive comments and suggestions, which helped us to significantly improve this paper. We also thank Preyas Desai, Liang Guo, Susumu Imai, Xi Li, Pinar Yildirim, Juanjuan Zhang, Yi Zhu and participants at Marketing Science Conference for their helpful comments. An early version of this paper was circulated under the title “Can Information Products be Complements?”

Appendix A

We explain how we derive the intervals in the example discussed in Subsection 2.2.

Claim 1:

A consumer buys no report if and only if her type, θ ∈ ( 0 , 1 6 ) .

Proof. Note that a consumer buys no report if and only if her consumer surplus of buying no report is (i) greater than buying one report, and (ii) greater than buying both reports. We can formulate the necessary and sufficient condition for buying no report as CS (no report) > CS (both reports) & CS (no report) > CS (one report), and derive the interval for the corresponding consumer type as follows:

C S ( no report ) > C S ( both reports ) & C S ( no report ) > C S ( one report )

⇔ C S ( no report ) − C S ( both reports ) > 0 & C S ( no report ) − C S ( one report ) > 0

⇔ 0 − 0.6 ⋅ θ − 0.1 > 0 & 0 − 0.2 ⋅ θ − 0.05 > 0 from Equations 3 a , 3 b & 3 c

⇔ 0.1 − 0.6 ⋅ θ > 0 & 0.05 − 0.2 ⋅ θ > 0

⇔ θ < 1 6 & θ < 1 4

⇔ θ < 1 6 .

Therefore, a consumer buys no report if and only if her type (θ) is on the interval ( 0 , 1 6 ) . □

Claim 2:

A consumer buys both reports if and only if her type, θ ∈ [ 1 6 , 1 ) .

Proof. Note that a consumer buys both reports if and only if her consumer surplus of buying both reports is (i) greater than buying one report, and (ii) greater than buying no report. We can formulate the necessary and sufficient condition for buying both reports as CS (both reports) ≥ CS (one report) & CS (both report) ≥ CS (no report), and derive the interval for the corresponding consumer type as follows:

C S ( both reports ) ≥ C S ( one report ) & C S ( both reports ) ≥ C S ( no report )

⇔ C S ( both reports ) − C S ( one report ) ≥ 0 & C S ( both reports ) − C S ( no report ) ≥ 0

⇔ 0.6 ⋅ θ − 0.1 − 0.2 ⋅ θ − 0.05 ≥ 0 & 0.6 ⋅ θ − 0.1 − 0 ≥ 0 from Eqs. 3 a , 3 b & 3 c

⇔ 0.4 ⋅ θ − 0.05 ≥ 0 & 0.6 ⋅ θ − 0.1 ≥ 0

⇔ θ ≥ 1 8 & θ ≥ 1 6

⇔ θ ≥ 1 6 .

Therefore, a consumer buys both reports if and only if her type (θ) is on the interval [ 1 6 , 1 ) . □

Appendix B

Here, we formally show how to derive pure strategy equilibrium prices in our model. The logic of the proof is similar to S&P and Tirole (1988, p. 96–97). In the model, each consumer faces four different choices: (i) buying both products, (ii) product 1 only, (iii) product 2 only, and (iv) no product, and chooses one of them to maximize her consumer surplus. Note that we can simply regard buying both products as buying a high quality composite product. Let’s define product i’s quality, s i : = σ 0 2 − σ i ˜ 2 and the composite product’s quality, S : = σ 0 2 − Σ ˜ 2 . Note that S > s i for i = 1, 2. We will discuss the following five possible scenarios, where we characterize consumer demand schedules by the relationships among “quality per dollar” of products, S p 1 + p 2 , s 1 p 1 , and s 2 p 2 . Then, we will show, out of five scenarios, only scenarios 2 and 5 lead to an equilibrium. The equilibrium in scenario 2 corresponds to information products being substitutes; the equilibrium in scenario 5 corresponds to information products being complements. Without loss of generality, we assume σ 1 2 > σ 2 2 (i.e. s 2 > s 1 ) in what follows.

Scenario 1:

Suppose s 2 p 2 > s 1 p 1 ≥ S p 1 + p 2 . In this scenario, buying product 1 only is “dominated” by buying product 2 only, i.e. no consumer will buy product 1 only.^[11]

Given prices p ₁ and p ₂, a consumer type, θ′, is indifferent between buying both products and product 2 only iff θ ′ S − p 1 − p 2 = θ ′ s 2 − p 2 . A consumer type, θ″, is indifferent between buying product 2 only and no product iff θ ″ s 2 − p 2 = 0 . From these conditions, it is straightforward to show θ ′ = p 1 S − s 2 and θ ″ = p 2 s 2 . Because S > s 2 , consumer types (θ) on the interval [ θ ′ , 1 ) buy both products, those on the interval [ θ ″ , θ ′ ) buy product 2 only, and those on the interval ( 0 , θ ″ ) buy no product.

Because the demand for the composite good is D 1,2 = ( 1 − θ ′ ) , the demand for product 1 is D 1 = D 1,2 + D 1 _ o n l y = ( 1 − θ ′ ) + 0 = 1 − θ ′ . The demand for product 2 is D 2 = D 1,2 + D 2 _ o n l y = ( 1 − θ ′ ) + ( θ ′ − θ ″ ) = 1 − θ ″ .

The demand functions for products 1 and 2 can be re-written as:

D 1 = 1 − p 1 S − s 2 , D 2 = 1 − p 2 s 2 .

Firm 1 maximizes the following profit: π 1 = p 1 ⋅ ( 1 − p 1 S − s 2 ) = p 1 − p 1 2 S − s 2 .

Firm 2 maximizes the following profit: π 2 = p 2 ⋅ ( 1 − p 2 s 2 ) = p 2 − p 2 2 s 2 .

The candidate equilibrium prices are:

p 1 * = S − s 2 2 ∵ ∂ π 1 ∂ p 1 = 1 − 2 p 1 S − s 2 ,

p 2 * = s 2 2 ∵ ∂ π 2 ∂ p 2 = 1 − 2 p 2 s 2 .

Now we want to check whether the candidate equilibrium prices satisfy the supposition. Let’s consider the first inequality of the supposition, s 2 p 2 > s 1 p 1 . By substituting p 1 * and p 2 * into it, we have

2 s 2 s 2 > 2 s 1 S − s 2

⇔ 2 > 2 · s 1 S − s 2

⇔ S − s 2 − s 1 S − s 2 > 0

(B1) ⇔ S > s 1 + s 2 .

Let’s consider the second inequality of the supposition, s 1 p 1 ≥ S p 1 + p 2 ⇔ s 1 ≥ p 1 p 1 + p 2 · S .

The supposition also states s 2 p 2 > S p 1 + p 2 ⇔ s 2 > p 2 p 1 + p 2 · S .

The above two inequalities lead to s 1 + s 2 > S , which contradicts Condition (B1), S > s 1 + s 2 . This shows that these candidate equilibrium prices contradict the supposition. Therefore, this scenario does not lead to an equilibrium.

Scenario 2:

Suppose s 1 p 1 ≥ s 2 p 2 ≥ S p 1 + p 2 . In this scenario, no product is dominated.

Given prices p ₁ and p ₂, a consumer type, θ′, is indifferent between buying both products and product 2 only iff θ ′ S − p 1 − p 2 = θ ′ s 2 − p 2 . A consumer type, θ″, is indifferent between buying product 2 only and product 1 only iff θ ″ s 2 − p 2 = θ ″ s 1 − p 1 . A consumer type, θ‴, is indifferent between buying product 1 only and no product iff θ ′ ′ ′ s 1 − p 1 = 0 . From these conditions, it is straightforward to show θ ′ = p 1 S − s 2 , θ ″ = p 2 − p 1 s 2 − s 1 and θ ′ ′ ′ = p 1 s 1 , and θ ′ > θ ′′ > θ ′ ′ ′ .

Because S > s 2 > s 1 , consumer types (θ) on the interval [ θ ′ , 1 ) buy both products, those on the interval [ θ ″ , θ ′ ) buy product 2 only, those on the interval [ θ ′ ′ ′ , θ ′′ ) buy product 1 only, and those on the interval 0 , θ ' ' ' buy no product.

Because the demand for the composite good is D 1,2 = ( 1 − θ ′ ) , the demand for product 2 is:

D 2 = D 1,2 + D 2 _ o n l y = ( 1 − θ ′ ) + ( θ ′ − θ ″ ) = 1 − θ ″ .

The demand for product 1 is:

D 1 = D 1,2 + D 1 _ o n l y = 1 − θ ′ + θ ′′ − θ ′ ′ ′ .

Therefore, the demand for product 1 and product 2 under this scenario can be re-written as:

D 2 = 1 − p 2 − p 1 s 2 − s 1 , D 1 = 1 − p 1 S − s 2 + p 2 − p 1 s 2 − s 1 − p 1 s 1 .

Firm 2 maximizes the following profit: π 2 = p 2 1 − p 2 − p 1 s 2 − s 1 = p 2 s 2 − s 1 + p 1 s 2 − s 1 − p 2 2 1 s 2 − s 1 .

Firm 1 maximizes the following profit:

π 1 = p 1 1 − p 1 S − s 2 + p 2 − p 1 s 2 − s 1 − p 1 s 1 = p 1 s 2 − s 1 + p 2 s 2 − s 1 − p 1 2 s 2 − s 1 s 1 + S − s 2 s 1 + S − s 2 s 2 − s 1 S − s 2 s 2 − s 1 s 1 .

The best response of firm 2 to p ₁:

p 2 ( p 1 ) = s 2 − s 1 + p 1 2 , ∵ ∂ π 2 ∂ p 2 = s 2 − s 1 + p 1 − 2 p 2 s 2 − s 1 .

The best response of firm 1 to p ₂:

p 1 ( p 2 ) = 1 2 ⋅ ( s 2 − s 1 + p 2 ) ( S − s 2 ) s 1 ( s 2 − s 1 ) s 1 + ( S − s 2 ) s 1 + ( S − s 2 ) ( s 2 − s 1 ) , ∵ ∂ π 1 ∂ p 1 = s 2 − s 1 + p 2 s 2 − s 1 − 2 p 1 ( s 2 − s 1 ) s 1 + ( S − s 2 ) s 1 + ( S − s 2 ) ( s 2 − s 1 ) ( S − s 2 ) ( s 2 − s 1 ) s 1 .

The candidate equilibrium prices are:

p 2 * = 3 ( S − s 2 ) ( s 2 − s 1 ) s 1 + 2 ( s 2 − s 1 ) 2 ( S − s 2 + s 1 ) 4 ( s 2 − s 1 ) s 1 + 4 ( S − s 2 ) s 1 + 3 ( S − s 2 ) ( s 2 − s 1 ) ,

p 1 * = 3 ( S − s 2 ) ( s 2 − s 1 ) s 1 4 ( s 2 − s 1 ) s 1 + 4 ( S − s 2 ) s 1 + 3 ( S − s 2 ) ( s 2 − s 1 ) .

Now we want to check whether the candidate prices satisfy the supposition: s 1 p 1 ≥ s 2 p 2 ≥ S p 1 + p 2 . Let’s consider the first inequality of the supposition, s 1 p 1 ≥ s 2 p 2 . By substituting p 1 * and p 2 * into it, we obtain

s 1 p 1 * ≥ s 2 p 2 *

⇔ s 1 p 2 * ≥ s 2 p 1 *

⇔ s 1 p 2 * − s 2 p 1 * ≥ 0

⇔ 3 ( s 1 − s 2 ) ( S − s 2 ) ( s 2 − s 1 ) s 1 + 2 s 1 ( s 2 − s 1 ) 2 ( S − s 2 + s 1 ) ≥ 0

⇔ ( s 2 − s 1 ) 2 ( − 3 ( S − s 2 ) s 1 + 2 s 1 ( S − s 2 + s 1 ) ) ≥ 0

⇔ s 1 ( s 2 − s 1 ) 2 ( − 3 S + 3 s 2 + 2 S − 2 s 2 + 2 s 1 ) ≥ 0

⇔ s 1 ( s 2 − s 1 ) 2 ( 2 s 1 + s 2 − S ) ≥ 0

(B2) ⇔ 2 s 1 + s 2 ≥ S

Let’s consider the second inequality of the supposition, s 2 p 2 ≥ S p 1 + p 2 . By substituting p 1 * and p 2 * into it, we obtain

s 2 p 2 * ≥ S p 1 * + p 2 *

⇔ s 2 ( p 1 * + p 2 * ) − S p 2 * ≥ 0

⇔ s 2 p 1 * + ( s 2 − S ) p 2 * ≥ 0

⇔ s 2 3 S − s 2 s 2 − s 1 s 1 + s 2 − S ( 3 S − s 2 s 2 − s 1 s 1 + 2 s 2 − s 1 2 S − s 2 + s 1 ) ≥ 0

⇔ 3 2 s 2 − S S − s 2 s 2 − s 1 s 1 + 2 s 2 − S s 2 − s 1 2 S − s 2 + s 1 ≥ 0

⇔ ( s 2 − s 1 ) ( S − s 2 ) ( 3 s 1 ( 2 s 2 − S ) − 2 ( s 2 − s 1 ) ( S − s 2 + s 1 ) ) ≥ 0

⇔ s 2 − s 1 S − s 2 − 3 s 1 − 2 s 2 + 2 s 1 S + 6 s 1 s 2 + 2 s 2 − s 1 2 ≥ 0

⇔ 2 s 1 2 + 2 s 2 2 + 2 s 1 s 2 s 1 + 2 s 2 ≥ S

(B3) ⇔ 2 s 1 + s 2 − 3 s 1 s 2 s 1 + 2 s 2 ≥ S

It is clear that condition (B3) is stronger than condition (B2). We show that the candidate equilibrium derived above is indeed an equilibrium iff s 2 p 2 * ≥ S p 1 * + p 2 * is satisfied.

We can also confirm that information products are substitutes in this scenario from ∂ D 2 ∂ p 1 = 1 s 2 − s 1 > 0 and ∂ D 1 ∂ p 2 = 1 s 2 − s 1 > 0 .

Scenario 3:

Suppose s 1 p 1 > S p 1 + p 2 ≥ s 2 p 2 . In this scenario, buying product 2 only is “dominated” by the composite product, i.e. no consumer will buy product 2 only.

Given prices p ₁ and p ₂, a consumer type, θ′, is indifferent between buying both products and product 1 only iff θ ′ S − p 1 − p 2 = θ ′ s 1 − p 1 . A consumer type, θ″, is indifferent between buying product 1 only and no product iff θ ″ s 1 − p 1 = 0 . From the conditions, it is straightforward to show θ ′ = p 2 S − s 1 and θ ″ = p 1 s 1 .

Because S > s 1 , consumer types (θ) on the interval [ θ ′ , 1 ) buy both products, those on the interval [ θ ″ , θ ′ ) buy product 1 only, and those on the interval ( 0 , θ ″ ) buy no product. Because the demand for the composite good is D 1,2 = ( 1 − θ ′ ) , the demand for product 1 is D 1 = D 1,2 + D 1 _ o n l y = ( 1 − θ ′ ) + ( θ ′ − θ ″ ) = 1 − θ ″ . The demand for product 2 is D 2 = D 1,2 + D 2 _ o n l y = ( 1 − θ ′ ) + 0 = ( 1 − θ ′ ) .

The demand functions for products 1 and 2 can be re-written as:

D 1 = 1 − p 1 s 1 , D 2 = 1 − p 2 S − s 1 .

Firm 1 maximizes the following profit:

π 1 = p 1 ⋅ ( 1 − p 1 s 1 ) = p 1 − p 1 2 s 1 .

Firm 2 maximizes the following profit:

π 2 = p 2 ⋅ ( 1 − p 2 S − s 1 ) = p 2 − p 2 2 S − s 1 .

The candidate equilibrium prices are:

p 1 * = s 1 2 ∵ ∂ π 1 ∂ p 1 = 1 − 2 p 1 s 1 ,

p 2 * = S − s 1 2 ∵ ∂ π 2 ∂ p 2 = 1 − 2 p 2 S − s 1 .

Now we want to check whether the candidate prices satisfy the supposition. Let’s consider the first inequality of the supposition, s 1 p 1 > S p 1 + p 2 . By substituting p 1 * and p 2 * into it, we obtain s 1 p 1 * > S p 1 * + p 2 * ⇔ s 1 s 1 2 > S s 1 + S − s 1 2 ⇔ 2 > 2 . This shows that these candidate equilibrium prices contradict the supposition. Therefore, this scenario does not lead to an equilibrium.

Scenario 4:

Suppose s 2 p 2 > S p 1 + p 2 ≥ s 1 p 1 . A similar argument as Scenario 3 holds for this scenario. Therefore, this scenario does not lead to an equilibrium.

Scenario 5:

Suppose S p 1 + p 2 > s 1 p 1 and S p 1 + p 2 > s 2 p 2 . In this scenario, both buying product 2 only and buying product 1 only are “dominated” by the composite product, i.e. no consumer will buy product 1 only or product 2 only.

Given prices p ₁ and p ₂, a consumer type, θ′, is indifferent between buying both products and no product iff θ ′ S − p 1 − p 2 = 0 . From the condition, it is straightforward to show θ ′ = p 1 + p 2 S .

Consumer types (θ) on the interval [ θ ′ , 1 ) buy both products, those on the interval [ 0 , θ ′ ) buy no product.

Because the demand for the composite good is D 1,2 = ( 1 − θ ′ ) , the demand for product 1 is D 1 = D 1,2 + D 1 _ o n l y = ( 1 − θ ′ ) + 0 = 1 − θ ′ . The demand for product 2 is D 2 = D 1,2 + D 2 _ o n l y = ( 1 − θ ′ ) + 0 = 1 − θ ′ .

The demand functions for products 1 and 2 can be re-written as:

D 1 = 1 − p 1 + p 2 S , D 2 = 1 − p 1 + p 2 S .

Firm 1 maximizes the following profit: π 1 = p 1 ⋅ 1 − p 1 + p 2 S = p 1 · S − p 1 − p 2 S .

Firm 2 maximizes the following profit: π 2 = p 2 ⋅ 1 − p 1 + p 2 S = p 2 · S − p 1 − p 2 S .

The best response of firm 1 to p ₂:

p 1 ( p 2 ) = S − p 2 2 , ∵ ∂ π 1 ∂ p 1 = − 2 p 1 + S − p 2 S .

The best response of firm 2 to p ₁:

p 2 ( p 1 ) = S − p 1 2 , ∵ ∂ π 2 ∂ p 2 = − 2 p 2 + S − p 1 S .

The candidate equilibrium prices are:

p 1 * = p 2 * = S 3 .

Now we want to check whether the candidate prices satisfy the supposition. Let’s consider the first inequality of the supposition, S p 1 + p 2 > s 1 p 1 . By substituting p 1 * and p 2 * into it, we obtain

S p 1 * + p 2 * > s 1 p 1 *

⇔ S 2 S 3 > s 1 S 3

⇔ 3 2 > 3 s 1 S

⇔ S > 2 s 1 .

By substituting p 1 * and p 2 * into the second inequality of the supposition, S p 1 + p 2 > s 2 p 2 , we obtain

S p 1 * + p 2 * > s 2 p 2 *

⇔ S 2 S 3 > s 2 S 3

⇔ 3 2 > 3 s 2 S

⇔ S > 2 s 2 .

Hence, we show that this scenario will lead to an equilibrium iff S > 2 s 1 and S > 2 s 2 hold. We also confirm that information products are complements in this scenario from ∂ D 2 ∂ p 1 = − 1 S < 0 and ∂ D 1 ∂ p 2 = − 1 S < 0 .

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Received: 2021-01-21

Accepted: 2021-06-03

Published Online: 2021-07-09

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Articles in the same Issue

https://doi.org/10.1515/roms-2021-0004

Keywords for this article

Bayesian decision making; information complements; information substitutes; pricing