## 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).

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*,

*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:

Therefore, a consumer buys no report if and only if her type (*θ*) is on the interval

## Claim 2:

*A consumer buys both reports if and only if her type*,

*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:

Therefore, a consumer buys both reports if and only if her type (*θ*) is on the interval

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,
*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,

## Scenario 1:

Suppose
^{[11]}

Given prices *p*
_{1} and *p*
_{2}, a consumer type, *θ*′, is indifferent between buying both products and product 2 only iff
*θ*″, is indifferent between buying product 2 only and no product iff
*θ*) on the interval

Because the demand for the composite good is

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

Firm 1 maximizes the following profit:

Firm 2 maximizes the following profit:

The candidate equilibrium prices are:

Now we want to check whether the candidate equilibrium prices satisfy the supposition. Let’s consider the first inequality of the supposition,

Let’s consider the second inequality of the supposition,

The supposition also states

The above two inequalities lead to

## Scenario 2:

Suppose

Given prices *p*
_{1} and *p*
_{2}, a consumer type, *θ*′, is indifferent between buying both products and product 2 only iff
*θ*″, is indifferent between buying product 2 only and product 1 only iff
*θ*‴, is indifferent between buying product 1 only and no product iff

Because
*θ*) on the interval

Because the demand for the composite good is

The demand for product 1 is:

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

Firm 2 maximizes the following profit:

Firm 1 maximizes the following profit:

The best response of firm 2 to *p*
_{1}:

The best response of firm 1 to *p*
_{2}:

The candidate equilibrium prices are:

Now we want to check whether the candidate prices satisfy the supposition:

Let’s consider the second inequality of the supposition,

It is clear that condition (B3) is stronger than condition (B2). We show that the candidate equilibrium derived above is indeed an equilibrium iff

We can also confirm that information products are substitutes in this scenario from

## Scenario 3:

Suppose

Given prices *p*
_{1} and *p*
_{2}, a consumer type, *θ*′, is indifferent between buying both products and product 1 only iff
*θ*″, is indifferent between buying product 1 only and no product iff

Because
*θ*) on the interval [

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

Firm 1 maximizes the following profit:

Firm 2 maximizes the following profit:

The candidate equilibrium prices are:

Now we want to check whether the candidate prices satisfy the supposition. Let’s consider the first inequality of the supposition,

## Scenario 4:

Suppose

## Scenario 5:

Suppose

Given prices *p*
_{1} and *p*
_{2}, a consumer type, *θ*′, is indifferent between buying both products and no product iff

Consumer types (*θ*) on the interval

Because the demand for the composite good is

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

Firm 1 maximizes the following profit:

Firm 2 maximizes the following profit:

The best response of firm 1 to *p*
_{2}:

The best response of firm 2 to *p*
_{1}:

The candidate equilibrium prices are:

Now we want to check whether the candidate prices satisfy the supposition. Let’s consider the first inequality of the supposition,

By substituting

Hence, we show that this scenario will lead to an equilibrium iff

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**Received:**2021-01-21

**Accepted:**2021-06-03

**Published Online:**2021-07-09

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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