A model of obfuscation with
Heterogeneous Firms ∗
Samir Mamadehussene†
June 11, 2016
Abstract
I propose a model under which obfuscation is individually rational,
even though, by obfuscating, a firm can only keep consumers away
from its store. Consumers search stores sequentially and they follow an
exogenous search order. Consumers search until they run out of time
and purchase from the firm with the lowest price, among the ones they
searched. Firm heterogeneity comes from their ranking in the search
order. The model predicts that the higher the ranking of a firm, the
more it will obfuscate and the higher its price. Empirical evidence from
Internet Service Providers in Canada supports the model’s predictions.
JEL Classification: D43, D83, L13, L80
Keywords: Obfuscation, Brand Awareness, Search Order
∗
I thank Jeff Ely, Wojciech Olszewski and Rob Porter for helpful comments and suggestions. All errors are my own.
†
Northwestern University, Department of Economics, 312 Arthur Anderson Hall, 2001
Sheridan Road, Evanston, IL 60208, [email protected]
1
1
Introduction
“Figuring out the monthly phone bill has become the consumer’s equivalent of deciphering hieroglyphics, with baffling new fees creating a
thicket of items at the bottom of the bill. Some providers [...] have
even started adding a ’paper bill’ fee to pay for the bill itself.”
– Ken Belson, The New York Times
Shopping for the lowest price has become a painful activity for consumers,
with firms adopting convoluted price schemes that are difficult to compare.
These confusing prices are observed in a variety of markets. In the US telecommunication industry, ”the four major carriers offer a total of nearly 700 combinations of smartphone plans” (The Wall Street Journal, July 31, 2013). In the
banking industry, Citigroup was fined more than $700 million in July 2015,
”partly for misleading customers when it came to fees and using confusing
language to make it more likely that customers would buy services they didn’t
need” (Washington Post, July 22, 2015). In the energy industry, ”companies have admitted giving dire service and baffling customers with confusing
bills designed to hide the true cost of heating and lighting out homes” (Daily
Mail, March 31, 2014). Confusing prices are also observed in the supermarket industry. The U.K. Competition and Markets Authority reported that it
has ”discovered supermarket prices and promotions that have the potential to
confuse or mislead consumers” (The Telegraph, July 16, 2015).
Arguments that obfuscation is collectively rational emerge from the economics literature that finds that equilibrium prices are increasing in consumers’
search costs.1 However, as Ellison and Wolitzky [2012] point out, arguments
based on collective rationality bring up a natural critique: why collude on
obfuscation rather than just colluding directly on price?
The literature on obfuscation has mostly focused on homogeneous firms.
In this paper, I propose a model of obfuscation with heterogeneous firms, with
1
For example, Diamond [1971], Wolinsky [1986] and Anderson and Renault [1999].
2
two main goals: i) to provide foundations for obfuscation to be individually
rational; ii) to analyze the obfuscation strategies of different types of firms.
In the model proposed in this paper, consumers have a time budget that
they allocate to search. Consumers have different time budgets and, for simplicity, I assume that the time that each consumer allocates to search is drawn
from the uniform distribution in the unit interval. I define obfuscation in a similar way as Wilson [2010] and Ellison and Wolitzky [2012]. Each firm chooses
the length of time that is required for a consumer to learn its price.2 This time
is not observable to consumers until after they have visited the firm. Similarly
to Arbatskaya [2007], consumers search stores sequentially and they follow the
same exogenous search order. The main difference between the model presented here and Arbatskaya’s is that, whereas in her model search costs are
exogenous, in my model firms can choose their search costs by engaging in
obfuscation.
I make the standard assumptions that search is with recall and that the
first search is for free. As consumers follow an exogenous search order, this
implies that all consumers know the price offered by firm 1.3 In that sense,
the obfuscation of firm 1 is meaningless. In a first stage, firms choose their
obfuscation levels, i.e., the amount of time it will take for a consumer to
learn their price. After that, obfuscation levels become common knowledge.
In a second stage, firms set prices simultaneously. Consumers search stores
sequentially, following the same search order, until they run out of time. At
that point, they purchase the product from the firm with the lowest price,
among the ones they were able to search, provided that the price is not higher
than their valuation, v.
In this model, when a firm obfuscates, it can only prevent consumers from
learning its price. By engaging in obfuscation, firms cannot stop consumers
from searching another store. To see this, suppose there are three firms in
2
Other interpretations would also work. For example, consumers could be homogeneous
in their time budgets and differ instead in how quickly they understand the price complexity
chosen by firms.
3
Throughout this paper, I refer to the ith firm in the exogenous search order as firm i.
3
Share of Consumers
Prices observed
α2
α3
1-α2 -α3
{p1 }
{p1 , p2 }
{p1 , p2 , p3 }
Table 1: Prices observed by consumers
the market, and let αi denote the obfuscation level of firm i. Notice that the
choice of obfuscation levels, in the first stage, will determine which consumers
learn which prices. A mass α2 of consumers (those that have time budgets
lower than α2 ) will not have time to learn the price of firm 2, so they will
only observe the price of firm 1. A mass α3 of consumers (those that have
time budgets between α2 and α2 + α3 ) will have enough time to learn the price
of firm 2, but not enough time to learn the price of firm 3. Finally, a mass
(1 − α2 − α3 ) of consumers will have enough time to learn all prices. Table
1 summarizes the prices that consumers observe. It is clear that, as firm 3
increases its obfuscation level (α3 ), it is only reducing the share of consumers
who will learn its price. This is not surprising because firm 3 is the last firm in
the search order. Hence, firm 3 could never prevent consumers from searching
another store, because there is no other store for them to search. However,
it is also the case that, by obfuscating, firm 2 can only reduce the share of
consumers who will learn its price. Notice that the share of consumers who
learn prices of firms 1 and 2 but do not have time to learn the price of firm 3
depends exclusively on the obfuscation level of firm 3. By obfuscating, firm 2
can only reduce the number of consumers who will learn all prices and increase
the number of consumers who will learn only the price of firm 1.
Even though, by obfuscating, firms can only keep consumers away from
their stores, firms still want to engage in obfuscation because it softens market
competition. The argument is better understood when we consider a duopoly.
If firm 2 does not obfuscate, all consumers will be able to learn the price of
both firms. In the second stage of the game, when firms choose prices, they
will play the Bertrand equilibrium, making zero profits. If firm 2 obfuscates,
4
it prevents some consumers from learning its price. Those consumers will buy
from firm 1. The market power that firm 1 has over these consumers allows
it to charge a price higher than marginal cost and make profits. Firm 2 also
benefits, because now it can also charge a price higher than marginal cost and
still sell to some consumers if it has a price lower than the price of firm 1.
When firm 2 obfuscates, it softens market competition at the expense of
losing some consumers. By obfuscating, firm 2 decreases the slice of the pie it
can obtain, but improves the quality of the pie for which it is competing (in the
sense of reduced competition). If firm 2 obfuscates too much, no consumer will
learn its price, and firm 1 will operate as a monopolist. In this case, the quality
of the pie is the highest (in the sense that the market has no competition), but
firm 2 does not get any of it. If firm 2 does not obfuscate, it will compete with
firm 1 in a Bertrand equilibrium. Firm 2 can obtain the entire pie, but the
quality of the pie is the lowest. In equilibrium, firm 2 will choose an interior
level of obfuscation.
The model predicts that the higher the ranking of a firm, the more it will
obfuscate and the higher its price. The result that firms higher on the list
charge higher prices is similar to Arbatskaya [2007]. The intuition is straightforward. When a consumer enters firm i, he already knows the prices of all
firms ranked above i in the search order. In order for firm i to make sales, it
has to set a price lower than the price of the firms above it.
The intuition behind the result that firms higher on the list obfuscate more
is as follows. When a firm obfuscates, it benefits from creating incentives for
firms ranked above it in the search order to charge higher prices. The cost is
that the firm prevents some consumers from learning its price. Firms higher
on the list derive both a higher benefit and a lower cost from obfuscation.
Regarding the benefit, the higher the firm is on the list, the bigger the
impact that its obfuscation will have on the price of the firms above it. When
firm 2 obfuscates, it gives direct market power to firm 1. I say direct market
power because consumers who do not have time to learn the price of firm 2
will buy from firm 1. When firm 3 obfuscates, it gives only indirect market
5
power to firms 1 and 2. Consumers who do not have enough time to learn the
price of firm 3 will buy from the firm with the lowest price among firms 1 and
2.
To see that the cost of obfuscation is lower for firms higher on the list, notice
that, when either firm 2 or 3 obfuscates, it reduces the share of consumers who
will learn all prices. However, because firm 2 charges higher prices than does
firm 3, it has a lower shot at selling to these consumers.4 Hence, the revenue
loss incurred by firm 2 is lower than that of firm 3.
The model also predicts that increasing the number of firms will lead consumers to spend more time searching and to pay, on average, higher prices.
Because the time that consumers spend searching is determined by firms’ obfuscation levels, this means that the market responds to increased competition
with higher prices and less transparency. This result is in the same direction
as findings in Spiegler [2006], Carlin [2009] and Chioveanu and Zhou [2013].
In Section 3, I consider some extensions to the model. First, I analyze the
equilibrium when firms cannot commit to their obfuscation level. In this case,
firms choose obfuscation and prices simultaneously, and zero obfuscation is a
weakly dominant strategy for all firms. Firms only engage in obfuscation to
provide incentives for other firms to charge higher prices. When prices and
obfuscation levels are chosen simultaneously, a firm cannot use its obfuscation
to induce other firms to charge higher prices. This result highlights the fact
that the timing of the model is key for the results.
The assumption that firms are committed to their obfuscation level before
choosing prices is connected to what we observe in reality, as prices tend to be
more flexible than obfuscation. Indeed, choosing an obfuscation level implies
designing a pricing scheme that is complex enough so that a consumer who
wishes to understand it will take the amount of time that the firm desires. On
the other hand, firms can change prices merely by changing the numbers in a
pricing scheme. Casual observation suggests that, even though firms change
4
As I show in Section 2, firm 2 charges higher prices than firm 3 in the sense of First
Order Stochastic Dominance. Firm 2 may have a lower price than firm 3, and may sell to
consumers who learn all prices.
6
prices frequently, they tend to stick to their pricing schemes for long periods
of time.
Another extension allows for exogenous search frictions, i.e., even if the
firm does not obfuscate, there is some minimum amount of time required
for a consumer to learn its price. I find that any change in the exogenous
component of search costs is completely offset by firms through obfuscation
and the equilibrium prices and profits do not change, as long as the exogenous
component of search costs is not too large.
A final extension allows for consumers to be impatient, i.e., there is a
maximum amount of time that they will spend on a firm, before moving on
to the next firm. In the original model, consumers cannot move to the next
firm until they have spent the time required to learn the price of the current
firm. I find that, qualitatively, the results do not change when consumers are
impatient, i.e., firms ranked higher in the search order will still obfuscate more
and charge higher prices.
In order to test the prediction that firms ranked higher in the search order
obfuscate more and charge higher prices, I examine a novel dataset on Internet
Service Providers (ISPs) in Ontario, Canada. There are 89 ISPs operating in
that region.5 The data includes the price schedule for each ISP. There are two
challenging aspects to test the model’s prediction: i) how to measure firms’
obfuscation; ii) how to measure the ranking of firms in the search order that
consumers use.
In order to measure obfuscation, I follow the approach from Muir et al.
[2013] and focus on observable criteria on pricing schedules. I find two major
devices that firms use that increase the complexity of their prices: i) firms
may list a promotional price, i.e., a lower price for the first few months; ii)
firms may charge an activation/installation fee. Among firms that charge an
activation fee, some of them list it near the price, whereas others list it in the
fine print. I construct measures of obfuscation that depend on the number of
these devices that firms include in their price schemes.
5
A list of all ISPs operating in Canada can be found at www.canadianisp.com.
7
Regarding the search order that consumers use, I rely on the psychology
literature (Sutherland and Sylvester [2000]) that argues that consumers will
first visit the stores of which they are more aware. Hence, I interpret the
exogenous search order in the model as a ranking of firms by brand awareness.
I measure brand awareness by the number of Google searches in the recent
past.
The findings are consistent with the model’s predictions. Firms that have
higher levels of brand awareness obfuscate more and charge higher prices.
The result that higher prices are associated with higher levels of obfuscation
is established in both the theoretical and empirical literature (Carlin [2009];
Ellison and Wolitzky [2012]; Muir et al. [2013]). This paper provides a new
link between brand awareness and obfuscation.
Related Literature
This paper contributes to a growing theoretical literature that formalizes how
firms can benefit from confusing consumers. Ellison [2005] and Gabaix and
Laibson [2006] study models of add-on pricing. Spiegler [2006] analyzes how
firms price multiple dimensions of a product (e.g,. different fees depending
on what contingency arises). Boundedly rational consumers compare only a
subset of the dimensions of the product, even though the price they will pay
may come from any dimension. Firms set prices lower in some dimensions,
to attract consumers, and higher in other dimensions, to profit from the consumers they were able to attract. Salant and Rubinstein [2008], Eliaz and
Spiegler [2011], Piccione and Spiegler [2012], Chioveanu and Zhou [2013] and
Spiegler [2014] formalize models where firms can influence consumers’ ability
to compare market alternatives through their choice of price frame.
Empirical literature also suggests that obfuscation raises firms’ profits. Ellison and Ellison [2009] analyze how internet price search engines affect competition. They find that, for some products, the easy price search makes
demand tremendously price-sensitive. Retailers engage in obfuscation which
leads to much less price sensitivity on some other products. Brown et al. [2010]
8
study the revenue effect of varying the level and disclosure of shipping charges.
They conclude that disclosure affects revenues and increasing shipping charges
boosts revenues when these charges are hidden. Chetty et al. [2009] study the
effect of salience on behavioral response to commodity taxation. They show
that posting tax-inclusive price tags reduces demand.
A recent body of literature interprets obfuscation as actions that firms
take to increase consumers’ search costs. The model presented in this paper is
more closely related to Carlin [2009], Wilson [2010] and Ellison and Wolitzky
[2012]. In Carlin’s model, the level of price complexity determines the fraction
of consumers who are able to understand which firm has the lowest price. The
remaining consumers purchase from a firm at random. He finds that firms that
list higher prices are the ones that have incentives to obfuscate. My model also
shares this prediction. However, whereas firms choose either full transparency
or full obfuscation in Carlin’s equilibrium, in my model the optimal level of
obfuscation is interior. As Ellison and Wolitzky [2012] point out, Carlin does
make obfuscation individually rational and not just collectively rational, but
this is done in a straightforward manner so that the paper can focus on other
things: the probability that a consumer will be able to understand which firm
has the lowest price is a function of all obfuscation levels, so an increase in
obfuscation by any one firm leads to exactly the same outcome as would a
smaller, coordinated increase by all firms.
Gu and Wenzel [2014] extend the framework developed in Carlin [2009] to
allow for firm heterogeneity based on their prominence level. Consumers who
are not able to understand which firm has the lowest price are more likely to
buy from the prominent firm. This assumption guarantees that the prominent
firm benefits more from confusing consumers. Whereas, in Gu and Wenzel
[2014], a firm’s level of prominence affects purchasing decisions even after the
agents observe prices, in my model prominence only impacts the consumers’
search order.
Wilson [2010] is closer to my model, as firms obfuscate to soften competition and provide incentives for other firms to list higher prices. The key
9
difference between my model and Wilson’s, besides the fact that my model
allows for firm heterogeneity, is that he assumes that consumers can observe
firms’ obfuscation levels before deciding which firm(s) to search. The insight is
that some consumers will refrain from searching a firm that obfuscates, which
allows the other firm to charge a higher price. However, Wilson’s model features an awkward prediction that firms that obfuscate charge lower prices,
i.e., firms that charge higher prices are fully transparent, whereas firms that
charge lower prices go out of their way to make the price hard to find. Wilson analyzes a duopoly and finds two equilibria: one in which firm 1 is fully
transparent and firm 2 fully obfuscates and another one where the obfuscation strategies are reversed. In equilibrium, the firm that obfuscates makes a
lower profit than does the transparent firm. Hence, even though obfuscation
is individually rational, each firm strictly prefers the equilibrium in which it
is not obfuscating.
Ellison and Wolitzky [2012] formalize models where, by obfuscating, a firm
increases consumers’ expected marginal cost of search. They propose two
models to achieve this goal. One model involves search costs that are convex
in shopping time. A second model assumes that there is common uncertainty
about how much time is required to learn a firm’s price in the absence of
obfuscation. My model differs from theirs in two regards. First, I allow for
firm heterogeneity with the purpose of analyzing the obfuscation strategies of
different types of firms. Second, the insight for why obfuscation is individually
rational is very different. Whereas, in Ellison and Wolitzky, firms obfuscate
to stop consumers from searching another store, in my model firms obfuscate
to prevent consumers from searching its own store.
2
Model
A finite number of firms compete to supply a homogeneous product. Firms
face the same marginal cost, normalized to zero. Firms choose both prices
and the amount of time it takes for a consumer to learn the price (henceforth,
10
obfuscation level). The timing is as follows. In a first stage, firms choose obfuscation levels simultaneously. After that, obfuscation levels become common
knowledge. In a second stage, firms choose prices simultaneously.
There is a unit mass of consumers, each demanding at most one unit of
the product if their valuation of v > 0 is not exceeded. Each consumer has a
time budget that he allocates to search. Consumers’ time budgets are drawn
from the uniform distribution on [0, 1]. Consumers search stores sequentially
and they follow the same exogenous search order. Consumers search until
they run out of time. Search is with recall and, as is standard in the search
literature, the first search is for free. This implies that, when they enter the
market, all consumers know the price of firm 1, which renders its obfuscation
meaningless.6
Firm heterogeneity comes from their ranking in the search order. Throughout this paper, I refer to the ith firm in the search order as firm i. The model
becomes less tractable as the number of firms increases. Fortunately, analyzing
the two-firm and three-firm equilibria is sufficient to derive the main insights
of the model.
Two-firm equilibrium
As the obfuscation level of firm 1 is meaningless, the only relevant obfuscation
level is that of firm 2, denoted by α. Proposition 1 characterizes the equilibrium
price distributions and profits. I denote the equilibrium price distribution
and profit of firm i as Fi and πi , respectively. All omitted proofs are in the
appendix.
6
I assume that the first search is for free to avoid philosophical questions regarding the
behavior of a consumer who does not have enough time to learn any price. The results of
the model would not change if, instead, time budgets were drawn from a distribution whose
support was bounded away from zero (eg. time budget drawn from the uniform distribution
on [θ, 1 + θ]) and obfuscation levels higher than θ were not feasible. In this case, firm 1
would choose the maximum obfuscation level, θ, and the remaining firms would choose the
same obfuscation levels as described in the model, as long as θ is large enough.
11
Proposition 1 For any obfuscation level that firm 2 chooses, α ∈ [0, 1], there
exists a unique equilibrium in the pricing stage characterized by



0


F1 (p) = 1 − α vp



1
F2 (p) =



0


if p < αv
if αv ≤ p < v
if p ≥ v
if p < αv
1
1−α



1
−
α v
1−α p
if αv ≤ p < v
if p ≥ v
π1 = αv
π2 = α(1 − α)v
Any consumer who learns the price of firm 2 also knows the price of firm
1. Hence, firm 2 will only be able to sell if its price is lower than that of firm
1. The reverse is not true. Firm 1 will sell some of its product even if it has a
price higher than that of firm 2, because some consumers will not have enough
time to learn the price of firm 2. Hence, firm 2 has higher incentives to charge
lower prices. This leads to the following result.
Proposition 2 F1 first-order stochastically dominates (FOSD ) F2
When firm 2 obfuscates, some consumers will never learn its price. Those
consumers will purchase from firm 1. Because firm 1 will sell to those consumers regardless of the price it charges (as long as it charges a price lower
than v), it will have an incentive to charge a higher price, to extract more
revenue from those consumers.
Proposition 3 Let F1α denote the equilibrium price distribution of firm 1 when
the obfuscation level of firm 2 is α.
00
α00 > α0 =⇒ F1α FOSD F1α
0
12
Proposition 3 summarizes the key insight behind why firms want to obfuscate. The more firm 2 obfuscates, the higher the equilibrium price of firm 1. In
particular, as Proposition 1 shows, the price distribution of firm 1 has a mass
point at v. When firm 1 charges v, it extracts all surplus from consumers. The
only consumers who will buy at that price are those who do not have time to
learn the price of firm 2. Firm 1 has monopoly power over those consumers.
Firm 1 will charge v with probability α. The higher the obfuscation level of
firm 2, the higher the monopoly power of firm 1, and the more likely firm 1
will charge the monopoly price.
By obfuscating, firm 2 softens market competition. The downside is that,
when firm 2 obfuscates, it reduces the share of consumers for which it is competing. If firm 2 obfuscates too much (α = 1), firm 1 becomes a monopolist.
The market has no competition, but firm 2 is not able to make profits because no consumer has time to learn its price. When firm 2 obfuscates too
little (α = 0), the market is perfectly competitive and both firms compete in
a Bertrand equilibrium, charging marginal cost. All consumers will learn the
price of firm 2, but the firm cannot extract any surplus from them. In equilibrium, firm 2 will choose an interior obfuscation level. I denote the equilibrium
obfuscation level of firm 2 by α∗ .
Proposition 4 α∗ =
1
2
The two-firm equilibrium has interesting properties. Firm 2 will obfuscate
in such a way that half of the consumers will not have time to learn its price.
These consumers will buy from firm 1. The remaining half of consumers will
learn both prices and will purchase from the firm with the lowest price. Firm
2 will have the lowest price with probability
R
0
v
[1 − F1 (x)]f2 (x)dx =
1+α
2
Consumers who have more time to search pay, on average, lower prices.
Another interesting feature of the equilibrium is summarized in Proposition 5
Proposition 5 F1 = αv + (1 − α)F2
13
Firm 1 will either choose the monopoly price (with probability α) or draw
a price from the same distribution as firm 2.
Because the obfuscation of firm 1 is not relevant (as consumers learn its
price for free), we cannot use the results from the two-firm equilibrium to
understand how the incentives to obfuscate change with a firm’s ranking in
the search order. To address this, I analyze the three-firm equilibrium.
Three-firm equilibrium
I denote by α and β the obfuscation levels of firms 2 and 3, respectively.
Proposition 6 Zero obfuscation is a weakly dominated strategy for firms 2 and
3. Nevertheless, there exists a fully transparent equilibrium (α∗ = β ∗ = 0).
When a firm does not obfuscate, all consumers who are able to learn the
price of the firm above it in the search order will also learn its price. This
implies that, in the pricing stage, the firm is competing against the firm immediately above it for the same consumers, a la Bertrand. Both firms will
charge marginal cost and make no profits. Therefore, zero obfuscation is a
weakly dominated strategy.
However, if one firm is playing zero obfuscation, the firm immediately above
it will make zero profits, regardless of its obfuscation level. Therefore, when
one firm is not obfuscating, the other firm is indifferent between all obfuscation
strategies. Hence, there exists a fully transparent equilibrium.
Proposition 7 For any obfuscation levels that firms 2 and 3 choose, α and
β, equilibrium profits are characterized by
π1 = αv

 βα(1−α) v
2
π2 = β +α(1−α)
(1 − α)αv
π3 =
if α > β
if α ≤ β

 βα(1−α−β) v
if α > β
(1 − α − β)αv
if α ≤ β
β 2 +α(1−α)
14
Inspecting the profits of firm 3, I find that, in equilibrium, α ≥ β. In
fact, when α < β, the profit of firm 3 is decreasing in its obfuscation level, β.
Proposition 8 summarizes the obfuscation levels in equilibrium.
Proposition 8 In any equilibrium that does not involve weakly dominated
strategies
α∗ >
β∗
1−α∗
>0
In particular, Proposition 8 implies that α∗ > β ∗ > 0. The following
β∗
∗
∗
example illustrates why I compare α∗ and 1−α
∗ (instead of α and β ). Suppose
that α = 0.4 and β = 0.3. Because all consumers already know the price of
firm 1, they all start searching firm 2. A mass 0.4 of consumers will not
have enough time to learn the price of firm 2. Hence, firm 2 prevents 40% of
consumers who start searching it from learning its price. Only the consumers
who are successful at learning the price of firm 2 (a mass 0.6 of consumers)
will start searching firm 3. Given the obfuscation level of firm 3, a mass 0.3
of consumers will not be able to learn its price. Hence, firm 3 prevents 50% of
consumers who start searching it from learning its price. Even though α > β,
β
is a measure of the
firm 3 is effectively obfuscating more than firm 2. 1−α
obfuscation of firm 3 that is comparable to α, as it quantifies the proportion
of consumers who start searching the price of firm 3 but are not successful.
Proposition 8 states that firms ranked higher in the search order obfuscate
more. As was clear in the analysis of the two-firm equilibrium, obfuscation has
two effects. On one hand, it softens competition, by providing firms higher on
the list with market power, which leads them to charge higher prices. On the
other hand, it reduces the share of consumers for which the obfuscating firm
can compete.
When firm 2 obfuscates, it provides firm 1 with direct market power. I say
direct market power in the sense that consumers who do not have time to learn
the price of firm 2 will buy from firm 1, even if firm 1 charges the monopoly
price. When firm 3 obfuscates, it provides firms 1 and 2 with only indirect
market power. Consumers who do not have time to learn the price of firm 3
will buy from the firm with the lowest price, among firms 1 and 2. Firms 1 and
15
2 have some market power over these consumers, in the sense that they can sell
to them even if they charge a higher price than that of firm 3. However, firms
1 and 2 still have to compete over these consumers. Hence, the obfuscation of
firm 2 is more effective in increasing prices than is the obfuscation of firm 3.
Moreover, firm 3 suffers more from the downside of obfuscation. To see
that, notice that, when firms 2 and 3 choose obfuscation levels α and β, α
consumers will only observe the price of firm 1, β consumers will only observe
the price of firms 1 and 2 and (1 − α − β) consumers will observe all prices.
Hence, when either firm obfuscates, it reduces the share of consumers who
will learn all prices. Whereas the market of firm 2 includes both consumers
who only learn prices of firms 1 and 2 and consumers who learn all prices, the
market of firm 3 consists solely of consumers who learn all prices. Therefore,
firm 3 is able to charge lower prices and extract a higher surplus from those
consumers, as stated in Proposition 9.
Proposition 9
R
0
v
[1 − F1 (p)][1 − F2 (p)]pdF3 (p) >
R
0
v
[1 − F1 (p)][1 − F3 (p)]pdF2 (p)
As firm 3 extracts a higher surplus from consumers who learn all prices,
it suffers more from reducing the share of those consumers by engaging in
obfuscation.
Summing up, firm 3 faces a larger downside than firm 2 from engaging in
obfuscation. Moreover, regarding the effect that obfuscation has on softening
market competition by inducing other firms to charge higher prices, the obfuscation level of firm 3 is less powerful than that of firm 2. It follows that firm
3 will obfuscate less than firm 2.
Proposition 10 characterizes the equilibrium price distributions.
Proposition 10 In an equilibrium that does not involve weakly dominated
strategies,



0
if p ≤ P̂


v
F1 (p) = 1 − β 2(1−α)α
if P̂ < p ≤ v
+α(1−α) p



1
if p > v
16



0




αβ
v
1 −
β 2 +α(1−α) p
F2 (p) =
α+β


− αβ vp

β



1
F3 (p) =



0


if P < p ≤ P̂
if P̂ < p ≤ v
if p > v
if p ≤ P
1−α
1−α−β



1
where P ≡
if p ≤ P
−
(1−α)αβ
v
[β 2 +α(1−α)](1−α−β) p
if P < p ≤ P̂
if p > P̂
αβ
v
β 2 +α(1−α)
and P̂ ≡
α(1−α)
v
β 2 +α(1−α)
The lower the ranking of a firm in the search order, the more competition
it faces. In fact, any consumer who has enough time to learn the price of firm
3 also knows the prices of firms 1 and 2. If firm 3 wants to make sales, it must
have a lower price than both firms 1 and 2. Firm 1 is on the opposite side.
This firm will make sales even if it has the highest price, as some consumers
only learn its price. This leads to the result that firms ranked higher in the
search order charge higher prices. Proposition 11 states the result, analogous
to Proposition 2 in the two-firm equilibrium.
Proposition 11 F1 FOSD F2 FOSD F3
The equilibrium features many interesting properties. Firms higher on the
list obfuscate more and charge higher prices. Consumers who have enough time
to search firms on the bottom of the list will pay lower prices. In particular, as
shown in Proposition 10, the upper bound on the price distribution of firm 3
is the lower bound on the price distribution of firm 1. Hence, a consumer who
has enough time to search firm 3 will never go back and purchase from firm
1. Proposition 12 is analogous to Proposition 5 in the two-firm equilibrium.
Firm 2 either draws a price from the same price distribution as firm 1 or a
price from the same price distribution as firm 3.
Proposition 12 F2 =
β
F
1−α 1
+
1−α−β
F3
1−α
17
A consumer who has a time budget between α and α + β will search only
firms 1 and 2. After searching firm 2, he may go back to firm 1 to purchase the
product, in case firm 1 has a lower price. This feature - consumers returning
to a previously searched store before exhausting all stores - is not present in
standard models of sequential search. However, it is consistent with empirical
evidence on search, such as Kogut [1990] and De los Santos et al. [2012].7
Comparative statics
In this section, I analyze how the equilibrium changes when another firm
enters, by comparing the two-firm and three-firm equilibria.
Proposition 13 Firm 2 will choose the same obfuscation level in both the
two-firm and the three-firm equilibria.
As the equilibrium obfuscation level of firm 2 is the same in both equilibria,
I abuse notation and keep using α∗ to denote the equilibrium obfuscation level
of firm 2. The main result of this section is presented in Proposition 14.
Proposition 14 When the market size increases from two to three firms, consumers, on average, spend more time searching and pay higher prices.
Increasing the number of firms induces more obfuscation (measured by the
time consumers spend searching). Firms reply to higher competition with less
transparency. This result is in the same direction as findings in Spiegler [2006],
Carlin [2009], and Chioveanu and Zhou [2013].
I also find that more competition leads to consumers paying, on average,
higher prices. However, competition has different effects on different types of
consumers, depending on their time budgets.
Proposition 15 When the market size increases from two to three firms, consumers who have search time lower than α∗ + β ∗ pay higher prices, whereas
the remaining consumers pay lower prices.
7
Ellison and Wolitzky [2012] analyze an extension to their model in which obfuscation
is costly. In that version, their model also presents this feature.
18
Even though competition leads, on average, to higher prices and higher
obfuscation levels, the welfare implications of increasing the number of firms
are not straightforward, because consumers who have higher time budgets will
pay lower prices. It is well documented in the literature (Marvel [1976]; Masson
and Wu [1974]; Phlips [1989]) that consumers who allocate more time to search
(i.e., consumers with low search costs) tend to be the poorest. It follows that,
even though competition has, on average, negative effects on consumers, it has
positive effects on the welfare of poor consumers.
3
Extensions
Simultaneous choice of price and obfuscation
In this section, I analyze the equilibrium in a setting under which firms choose
prices and obfuscation levels simultaneously. I consider a general model with
n firms and denote by αi the obfuscation level of firm i.
Proposition 16 Zero obfuscation is a weakly dominant strategy for all firms
The result in Proposition 16 is intuitive. When firms choose prices and
obfuscation levels simultaneously, a firm’s obfuscation level has no impact on
other firms’ prices. Hence, there is no longer an upside of obfuscating. When
a firm obfuscates it simply prevents some consumers from learning its own
price.
Whereas zero obfuscation was a weakly dominated strategy when firms
were able to commit to their obfuscation level, it is weakly dominant when
firms cannot. The timing of the model is of central importance.
Prices are usually more flexible than are obfuscation levels. In fact, whereas
changing obfuscation levels implies designing a new pricing schedule that has
the complexity that the firm desires, changing prices only involves modifying
the numbers in the existing price schedule. Casual observation of markets
that are associated with high levels of obfuscation suggests that, indeed, firms
change prices more often than they change pricing schemes.
19
Proposition 17 In equilibrium, all firms make zero profits.
Proof. I will start by showing that firm i makes positive profits if and only if
firm i+1 makes positive profits. Let Fi denote the cdf from which firm i draws
prices, in equilibrium. Suppose that firm i makes positive profits. It follows
that the lower bound of Fi is higher than the marginal cost. Firm i + 1 can
then make profits by choosing no obfuscation and a price slightly lower than
the lower bound of Fi . Now suppose that firm i + 1 makes positive profits. It
follows that the lower bound of Fi+1 is higher than the marginal cost. Firm i
can then make profits by choosing a price slightly lower than the lower bound
of Fi+1 .
It follows that, in equilibrium, either all firms make positive profits, or no
firm does.
Suppose, by contradiction, that all firms make positive profits. In particular, firm n makes positive profits. Because all consumers who learn the price
of firm n also learn the prices of all remaining firms, it follows that, with some
probability, firm n is the lowest-price firm in the market. This in turn implies
that firm n would strictly prefer to play zero obfuscation. Indeed, when firm n
obfuscates, it is only keeping some consumers from learning its price. Because
there is a positive probability that firm n has the lowest price in the market,
it will want to maximize the number of consumers who learn its price.
But if firm n does not obfuscate, all consumers who learn the price of firm
n − 1 also learn the price of firm n. Moreover, given the search order that
consumers use, it is also the case that all consumers who learn the price of
firm n also know the price of firm n − 1. Firms n and n − 1 are competing over
the same group of consumers who will purchase from the lowest-price firm, a
la Bertrand. The equilibrium involves both of these firms charging marginal
cost and making zero profits. This contradicts that firm n makes profits.
20
Exogenous search frictions
In this extension, I relax the assumption that, in the absence of obfuscation,
consumers can learn the price of any firm without spending any time searching.
I denote by τ the amount of time it takes for a consumer to learn the price of
a firm that does not obfuscate. When a firm chooses obfuscation level α, the
total amount of time needed to learn that firm’s price is τ + α.
Let αi∗ , Fi∗ , and πi∗ denote the equilibrium obfuscation level, price distribution, and profits of firm i when there are no exogenous market frictions
(τ = 0), and let α̂i∗ , F̂i∗ , and π̂i∗ be defined analogously in the presence of
exogenous market frictions.
Proposition 18 If τ ≤ min{αi∗ }, then α̂i∗ = αi∗ − τ , F̂i∗ = Fi∗ , and π̂i∗ = πi∗ .
i
Any reduction in the exogenous component of search costs is completely
offset by firms, through obfuscation. If a search engine reduces the time it
takes to learn a firm’s price by five minutes, firms will make their prices more
complex, so that consumers spend an additional five minutes in order to understand a firm’s price schedule. Equilibrium prices and profits do not change.
This result is equivalent to findings in Ellison and Wolitzky [2012]. As they
argue, this provides a formalization of the observation in Ellison and Ellison
[2009] that improvements in search technology need not make search more
efficient.
Impatient Consumers
In this extension, I allow for consumers to move to the next firm before learning
the price of the firm they are currently searching. In particular, consumers
will not spend more than θ amount of time in any firm.
Proposition 19 Let αθ and βθ denote the equilibrium obfuscation levels of
firms 2 and 3, respectively, in the three-firm equilibrium.
If θ ≤ 0.25 then αθ = βθ = θ
If θ > 0.25 then αθ > βθ > 0. Moreover, if αθ < θ then αθ >
21
βθ
1−αθ
If consumers are too impatient, both firms will obfuscate in such a way
that the amount of time it takes to learn the price of each is exactly the
amount of time consumers would spend searching before moving on to the
next firm. When they are not so impatient, the equilibrium still features firm
2 obfuscating more than firm 3, in the sense that αθ > βθ . As discussed in
Section 2, a more accurate analysis of which firm is obfuscating more involves
βθ
. However, if αθ = θ, firm 2 is obfuscating as much
comparing αθ and 1−α
θ
βθ
as it can. So, even if αθ < 1−α
, we can still argue that firm 2 is obfuscating
θ
more than firm 3, in the sense that it is feasible for firm 3, but not for firm 2,
to obfuscate more.
4
Empirical Findings
4.1
Data
The dataset consists of price schedules for the 89 Internet Service Providers
(ISPs) that offer residential services in Ontario (Canada), as well as the number
of Google searches for each ISP, in the past 12 months.8 I use the dataset to
construct obfuscation scores, brand awareness scores and prices for each ISP.
I present a detailed description of how each variable was computed.
Obfuscation Scores
A challenging task when studying obfuscation is to measure the complexity
of firms’ pricing schemes. I follow the approach of Muir et al. [2013] and
focus on observable criteria in pricing schedules. I find two major devices
that firms use that increase the complexity of their prices: i) firms may list
a lower (promotional) price for the first few months; ii) firms may charge an
activation/installation fee. Among firms that charge an activation fee, some
of them list it near the price, whereas others list it in the fine print.
8
Number of Google searches between September 14, 2014 and September 12, 2015. Price
schedules are from September, 2015.
22
Charging a lower price for the first few months makes it harder to compare
different pricing schedules. Consider a consumer who is faced with a firm that
charges a flat rate for all months, and another that charges a lower price for
the first few months and a higher price afterward. In order for the consumer
to make the correct decision, he has to know how long he plans to use the
service. Even then, it still requires some calculation to figure out which firm
has the lowest price. Even comparing prices from two firms that both charge
promotional prices in the first few months is not straightforward. The calculations needed to do such a comparison can also be complicated due to the fact
that the length of the discount period may be different. In fact, whereas some
firms offer a lower rate for the first six months, others do so for only three or
even one month. I construct a score that reflects whether a firm engages in
this obfuscation device. I let OD equal 1 if a firm offers a different price in the
first few months of the contract and equal 0 otherwise.
The existence of an activation fee clearly makes it much harder for a consumer to compare prices. Indeed, when a consumer is faced with two firms
that charge different prices and the firm that charges the lowest price also
charges an activation fee, he needs to perform some calculations in order to
understand what is best for him. Some firms make it even more time consuming by hiding those fees in the fine print. In those cases, consumers have to
spend time reading through all the fine print even before starting to compare
which firm has the lowest price. I let OF equal 0 if a firm does not charge an
activation/installation fee; 1 if it charges a fee but lists it in the main table of
prices; and 2 if it charges a fee and lists it in the fine print.
Besides the obfuscation scores already mentioned, OD and OF , I construct
another obfuscation score, OT , that captures the total obfuscation from the
two devices. I define OT ≡ OD + OF .
There is significant dispersion in the complexity of pricing schemes. Table
2 presents the distribution of firms across obfuscation scores. There are four
possible levels of obfuscation. Only two firms present the highest possible level
of obfuscation (which involves charging a lower price for the first few months
23
Obfuscation Level
Firm share
0
1
2
3
33%
39%
26%
2%
Table 2: Firm share by obfuscation level
and listing an activation fee in the small print). All the remaining obfuscation
levels are used by a substantial share of firms.
The obfuscation range is very wide. Whereas some firms adopt pricing
schemes that are immediately understandable upon seeing them, other firms
present such convoluted pricing schemes that one has to take some time to
fully understand all conditions and fees. Figures B.1, B.2 and B.3 present
the pricing schemes for two firms. Figure B.1 details the available packages
and prices for Bell Canada, the largest firm operating in the market. It offers
a wide range of packages. Each package has a regular price and a threemonth promotional price. After selecting a particular package, the consumer
is directed to the full offer details, presented in Figure B.2. It is only at
this point, after already having spent some time choosing which plan is best,
that the consumer becomes aware of a nearly $50 activation fee. A footnote
then states that, even though the table suggests that Bell Install is included,
conditions do apply. In order to find those conditions, the consumer must
follow another hyperlink. For comparison, Figure B.3 details the available
packages and prices for Coextro, a much smaller ISP. This firm offers only two
packages and does not charge activation or installation fees. Moreover, the
monthly price is flat (i.e., no promotional price).
Brand Awareness Scores
Measuring brand awareness is a challenging task. Data on sales is not available
for the majority of firms. Even if the availability of data was not an issue,
firms’ revenue may not be a good indicator of brand awareness. In fact, a firm
24
could be very well known and, at the same time, charge a high price and sell
only to a small fraction of consumers. I resort to Google trends to measure
brand awareness. Google trends provides information on the number of Google
searches for each firm. Even though data on the magnitude of Google searches
is not available, it provides the ratio of the number of searches between different
firms. The firm with the highest number of Google searches is attributed a
score of 100. A firm that, for example, has half of that number of searches will
get a score of 50. I construct a brand awareness score A by using the output
from Google trends.
There are 89 ISPs that operate in Ontario, Canada. Most of them are
relatively unknown. In fact, 65 of the firms are so rarely searched on Google
that their Google trends score is 0. The remaining 24 firms are better known
and have a positive score. For this reason, I construct another brand awareness
score, AB , as a binary variable described below.

0 if A = 0
AB =
1 if A > 0
Prices
I measure the total price for two different time horizons: 12 and 36 months.
I denote them by P12 and P36 , respectively. I also measure variable prices
for those time horizons, i.e., prices that do not include installation/activation
fees. I denote those variables by V P12 and V P36 . All prices are for an internet
speed of 10 Mbps. I choose that speed because it is a very standard speed that
most ISPs offer. For ISPs that do not offer that speed, I select the package
with the speed closest to 10 Mbps. The standard deviation in ISPs speeds is
only 2.33 Mbps.
Table 3 presents the summary statistics for obfuscation level, brand awareness and price variables.
25
OD
OF
OT
A
AB
P12
V P12
P36
V P36
Average
0.09
0.89
0.98
3.33
0.27
560
512
1,598
1,549
St. Dev.
0.29
0.76
0.82
11.9
0.44
132
113
351
339
Maximum
1
2
3
100
1
1,275
1,125
3,525
3,375
Minimum
0
0
0
0
0
347
347
1,042
1,042
Table 3: Summary statistics
4.2
Empirical Analysis
Relationship between brand awareness and obfuscation
I start by running an OLS regression to analyze the effect of brand awareness
on firms’ obfuscation levels. I do this for each of the obfuscation and brand
awareness variables defined in the previous section.
Oi = α + βBAi + i
where O ∈ {OD , OF , OT } and BA ∈ {A, AB }
The results are presented in Table 4. I find a positive relationship between
firm brand awareness and obfuscation. Not only is this result significant, it
is also robust to the various measures of obfuscation and brand awareness.
This result is consistent with the predictions of the model, under which firms
that are ranked higher in the search order obfuscate more. Indeed, the interpretation of the search order is that consumers search in order of the firms
that pop up first in their minds. Brand awareness reflects the search order
of consumers. This link between brand awareness and obfuscation is a new
empirical finding.
Figure 1 plots average obfuscation level by firm brand awareness. I split
firms into four groups. The first group consists of all 65 firms with a brand
26
OD
OD
OF
OF
A 0.009***
0.022***
(0.002)
(0.006)
AB
0.162**
(0.067)
N
R2
89
0.145
OT
OT
0.031***
(0.007)
0.724***
(0.165)
0.887***
(0.174)
89
89
89
89
0.063
0.116
0.181
0.199
Standard errors in parentheses
*** p < 0.01, ** p < 0.05, * p < 0.1
89
0.230
Table 4: The impact of Brand Awareness on Firms’ Obfuscation
awareness score of zero. The remaining 24 firms are evenly split among the
remaining three groups, listed in order of increasing brand awareness. So, for
example, Group 4 constitutes the eight ISPs with the highest brand awareness
scores.
Average Obfuscation
2.5
2.3
2
1.5
1.5
1.1
1
0.7
0.5
Group 1 Group 2 Group 3 Group 4
Figure 1: Obfuscation by firm brand awareness
27
Relationship between brand awareness and price
I run an OLS regression to analyze the effect of brand awareness on firms’
prices. I do this for each of the brand awareness and price variables defined in
the previous section.
Pi = α + βBAi + i
where P ∈ {P12 , V P12 , P36 , V P36 } and BA ∈ {A, AB }
P12
A
2.3**
(1.2)
AB
N
R2
P12
V P12
2.2**
(1.0)
69.7**
(30.9)
89
0.044
V P12
89
0.055
P36
P36
8.4***
(3.0)
67.6**
(26.3)
V P36
V P36
8.3***
(2.9)
239.5***
(80.8)
89
89
89
89
0.055 0.071 0.080
0.092
Standard errors in parentheses
*** p < 0.01, ** p < 0.05, * p < 0.1
237.3***
(77.7)
89
0.084
89
0.097
Table 5: The impact of Brand Awareness on Firms’ Price
The results are presented in Table 5. I find that brand awareness scores are
a significant predictor of firms’ prices. When I consider prices for a 12-month
period, all results are significant at the 5% level. However, when I consider
a longer horizon of 36 months, the results become even more significant. For
prices for a 36-month period, all results are significant at the 1% level.
The results become more significant when we consider a longer horizon
because, as detailed previously, better-known firms tend to obfuscate more.
One obfuscation device that firms use is to charge a lower price in the first
months of the contract. Hence, when we consider a short horizon, firms that
28
use that obfuscation device will have significantly lower prices.
Another obfuscation device is to charge an activation fee. Firms with
higher brand awareness scores could have higher prices simply because they
engage in this obfuscation device. To control for this possible issue, I also run
the regression of variable prices on brand awareness. Variable prices consist
only on monthly rates, and do not include activation fees. I find that brand
awareness is also a significant predictor of variable prices. Hence, not only do
better-known firms have higher prices because they are more likely to charge
an activation fee, they also have higher monthly rates.
Figure 2 plots average prices by firms’ brand awareness. I consider only
two brand awareness groups, using their AB score. As the figure shows, prices
of better-known firms are about 14% higher than prices of less-known firms.
Average Monthly Price
60
51
49
50
48
47
45
43
41
41
40
30
P12
V P12
AB = 0
P36
V P36
AB = 1
Figure 2: Price by firm brand awareness
5
Discussion
In many markets, firms are adopting confusing price schemes in order to increase consumers’ search costs. Empirical findings in this paper show that this
29
practice is more frequently adopted by better-known firms.
I propose a model under which consumers search stores following an exogenous search order. The model predicts that firms higher on the list will
obfuscate more and charge higher prices, just as empirical findings suggest.
Firms that are higher in the search order make more profits. This could lead
to competition for higher positions on the list. I interpret the search order as
a ranking of brand awareness. Hence, competition for a higher ranking on the
list is simply an advertisement game.
The model has important welfare implications. By adopting convoluted
price schedules, firms reduce consumers’ surplus. The reduction in consumers’
welfare comes from two sources. First, consumers have to incur higher search
costs when searching for a product. Second, equilibrium prices are higher than
they would be if firms were not adopting complex prices.
Competition alone is not enough to restore consumers’ welfare. In fact,
firms respond to competition with less transparency, i.e., increasing the number
of firms leads to even more complex prices. Besides, when the number of firms
increases, consumers end up paying higher prices on average.
In order to bring consumer surplus back to its original level (when firms
were not obfuscating), it is necessary to stop firms from engaging in such practices. This is challenging to execute. Regulation of some markets has taken
steps to, at least, limit the amount of obfuscation that a firm can implement.
The European Parliament and the Council of the European Union have
regulated the way that a creditor advertises credit agreements. In particular,
Article 4 of Directive 2008/48/EC of 23 April 2008, specifies ”standard information to be included in advertising”. It states that ”the standard information
shall specify in a clear, concise and prominent way by means of a representative example: a) the borrowing rate [...] b) the total amount of credit c) the
annual percentage rate of charge d) the duration of the credit agreement”. As
the European Commission explains, the Standard European Consumer Credit
form ”is designed to give [consumers] the best overview of the terms and conditions of any credit contract [they] consider and includes key details [...] that
30
allows [them] to compare the offers of different credit providers and select the
credit offer that works best.”
In a similar attempt to reduce the level of obfuscation, the U.K. Competition and Markets Authority is directing supermarkets to display unit prices,
for ease of comparison of ”the costs of different products, regardless of their
respective sizes”.
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Appendix (for online publication)
A
Omitted Proofs
Proof of Proposition 1. As there are only two firms, the cdfs of both firms
must have the same upper and lower bound.
There can be no mass points at prices lower than v. If one firm, call it i,
played price x < v with positive probability, the other firm, call it j, would
never play prices in the interval (x, x + ) for small enough, as it would be
better to play a price slightly lower than x. But then firm i would prefer to
play x + than x, which is a contradiction.
As there are no mass points at prices lower than v, the upper bound of the
cdfs must be v. If the upper bound was y < v, firm 1 would prefer to play v
instead of y. Indeed, in both cases firm 1 would only sell to consumers who
do not have time to learn the price of firm 2. But by charging v, firm 1 can
extract more surplus from those consumers.
Only one firm can play v with positive probability. If both firms played v with
positive probability, each firm would prefer to play a slightly lower price.
Firm 1 makes profits in equilibrium, because it always sells to consumers who
do not have time to learn the price of firm 2. It follows that the lower bound
on prices is higher than the marginal cost, zero. This in turn implies that firm
2 also makes positive profits. As firms must be indifferent between all prices
in the support of their cdf, and because v is the upper bound of both cdfs, it
follows that firm 2 makes positive profits when it charges price v. As firm 2
only sells when it has a price lower than firm 1, it follows that firm 1 must
play price v with positive probability. This in turn implies that the cdf of firm
2 has no mass points.
The mass of consumers who do not have time to learn the price of firm 2 is α.
It than follows that the profit of firm 1, when charging price v, is
π1 (v) = αv
34
When firm 1 charges a price x < v, its profits are
π1 (x) = αx + (1 − α)[1 − F2 (x)]x
Indifference of firm 1 yields
F2 (x) = 1 −
α v−x
1−α x
It follows that the lower bound of F2 , which is the same as the lower bound of
F1 is αv
As there are no mass points at the lower bound, firm two will sell to all
consumers who learn its price, when it charges the lower bound. As the mass
of consumers who have enough time to learn the price of firm 2 is (1 − α), it
follows that the profit of firm 2, when charging the lower bound, is
π2 (αv) = (1 − α)αv
The profit of firm 2 when charging price x is
π2 (x) = (1 − α)[1 − F1 (x)]x
Indifference of firm 2 yields
F1 (x) = 1 − α xv
Proof of Proposition 2. F1 (p) = 1 − α vp ≤
1
1−α
−
α v
1−α p
= F2 (p)
0
00
Proof of Proposition 3. F1α (p) = 1 − α00 vp ≤ 1 − α0 vp = F1α (p)
Proof of Proposition 4. As firm 2 chooses α, α∗ = arg max π2
α
The first order condition yields the result.
Proof of Proposition 5. Follows from Proposition 1
Proof of Proposition 6. When α = 0, all consumers learn both the price of
firm 1 and the price of firm 2. Hence, firms 1 and 2 compete over the same
consumers, a la Bertrand, making zero profits. Any strictly positive level of
obfuscation weakly dominates α = 0. Indeed, when a firm chooses α > 0 and
35
price no lower than 0, it cannot do worse than zero profits. An analogous
argument implies that β = 0 is also weakly dominated.
α∗ = β ∗ = 0 is an equilibrium because when one firm is playing zero obfuscation, the other firm will make zero profits regardless of its obfuscation.
Proof of Proposition 7. I start with the case α < β. I will show that, in this
scenario, all three cdfs will have the same lower bound. Notice that at least
two firms must have the same lower bound (otherwise, the firm with the lowest
lower bound would prefer to play higher prices, because by doing so it would
sell to the same number of consumers at a higher price). I denote by P i the
lower bound of the cdf of firm i and P the minimum of the three lower bounds.
I will show, by contradiction, that no cdf can have a lower bound higher than
P.
Case 1: P 1 > P
Notice that
π1 (P 1 ) = αP 1 + β[1 − F2 (P 1 )]P 1 + (1 − α − β)[1 − F2 (P 1 )][1 − F3 (P 1 )]P 1
π1 (P ) = P
π2 (P 1 ) = βP 1 + (1 − α − β)[1 − F3 (P 1 )]P 1
π2 (P ) = (1 − α)P
π1 (P ) − π1 (P 1 ) > P − αP − βP 1 − (1 − α − β)[1 − F3 (P 1 )]P 1
= π2 (P ) − π2 (P 1 )
=0
This contradicts that P 1 is the lower bound of the support of the cdf of firm
1.
Case 2: P 2 > P
π2 (P 2 ) = β[1 − F1 (P 2 )]P 2 + (1 − α − β)[1 − F1 (P 2 )][1 − F3 (P 2 )]P 2
π2 (P ) = (1 − α)P
π1 (P 2 ) = αP 2 + βP 2 + (1 − α − β)[1 − F3 (P 2 )]P 2
36
π1 (P ) = P
π2 (P ) − π2 (P 2 ) > P − αP 2 − βP 2 − (1 − α − β)[1 − F3 (P 2 )]P 2
= π1 (P ) − π1 (P 2 )
=0
This contradicts that P 2 is the lower bound of the support of the cdf of firm
2.
Case 3: P 3 > P
π3 (P 3 ) = (1 − α − β)[1 − F1 (P 3 )][1 − F2 (P 3 )]P 3
π3 (P ) = (1 − α − β)P
π2 (P 3 ) = (1 − α)[1 − F1 (P 3 )]P 3
π2 (P ) = (1 − α)P
π3 (P ) − π3 (P 3 ) = (1 − α − β)P − (1 − α − β)[1 − F1 (P 3 )][1 − F2 (P 3 )]P 3
> (1 − α − β)P − (1 − α − β)[1 − F1 (P 3 )]P 3
1 − α − β
=
(1 − α)P − (1 − α)[1 − F1 (P 3 )]P 3
1−α
1−α−β
[π2 (P ) − π2 (P 3 )]
=
1−α
=0
This contradicts that P 3 is the lower bound of the support of the cdf of firm
3.
As all the firms have the same lower bound, we can compute profits as
π1 = π1 (P ) = P
π2 = π2 (P ) = (1 − α)P
π3 = π3 (P ) = (1 − α − β)P
Moreover, it must be that the cdf of firm 1 has the highest upper bound.
37
Indeed, if some firm charged a price higher than the upper bound of firm
1, it would not make sales, because all consumers know the price of firm 1.
Furthermore, it must be that the upper bound of firm 1 is v, as if it was lower
than v, firm 1 would prefer to increase its price, because that would imply
selling to the same consumers at a higher price. Hence
π1 = π1 (v) = αv
We can conclude that π1 (P ) = π1 (v) ⇐⇒ P = αv
Replacing in the profit function we get the result:
π1 = αv
π2 = (1 − α)αv
π3 = (1 − α − β)αv
I will now analyze the case α ≥ β
For this, I use the results in Proposition 10. As the upper bound of the cdf of
firm 1 is v and F2 (v) = F3 (v) = 1, it follows that π1 = αv.
As v is in the support of the cdf of firm 2,
π2 = π2 (v) = β[1 − F1 (v)]v =
βα(1−α)
v
β 2 +α(1−α)
αβ
As P = β 2 +α(1−α)
v is in the support of the cdf of firm 3, and because F1 (P ) =
F2 (P ) = 0, it follows that
π3 = π3 (P ) = (1 − α − β)P =
βα(1−α−β)
v
β 2 +α(1−α)
Proof of Proposition 8. As firm 2 chooses α and firm 3 chooses β, it must be
that
α∗ = arg max π2
α
β ∗ = arg max π3
β
If α∗ < β ∗ , the profit of firm 3 would be (1 − α∗ − β ∗ )α∗ v, so firm 3 would
want to decrease its obfuscation level. Hence, it must be that α∗ ≥ β ∗ .
The First Order conditions yield α∗ = 21 , β ∗ =
Proof of Proposition 9. First notice that
38
√
2−√ 2
.
2 2
π3 = (1 − α − β)
R
0
v
[1 − F1 (p)][1 − F2 (p)]pdF3 (p).
It follows from Proposition 7 that
R
v
[1 − F1 (p)][1 − F2 (p)]pdF3 (p) =
0
αβ
v
β 2 +α(1−α)
Using the result from Proposition 10
R
0
v
[1 − F1 (p)][1 − F3 (p)]pdF2 (p) =
αβ
v
β 2 +α(1−α)
h
1−
β
[ln(1
1−α−β
i
− α) − ln(β)]
Notice that α + β < 1, hence ln(1 − α) − ln(β) > 0.
Proof of Proposition 10. I will show that the above constitutes an equilibrium.
I will show that each firm is indifferent between all prices in the support of its
cdf and it weakly prefers those prices than any price that is not on the support
of its cdf.
Firm 2 For x ∈ (P̂ , v)
π2 (x) = β[1 − F1 (x)]x + (1 − α − β)[1 − F1 (x)][1 − F3 (x)]x
=β
(1 − α)α
v
β 2 + α(1 − α)
For x ∈ (P , P̂ )
π2 (x) = β[1 − F1 (x)]x + (1 − α − β)[1 − F1 (x)][1 − F3 (x)]x
= βx + (1 − α − β)[−
=
β
(1 − α)αβ
v
+ 2
]x
1 − α − β [β + α(1 − α)](1 − α − β) x
(1 − α)αβ
v
+ α(1 − α)
β2
Firm 1
For x ∈ (P̂ , v)
39
π1 (x) = αx + β[1 − F2 (x)]x + (1 − α − β)[1 − F2 (x)][1 − F3 (x)]x
α
= αx + β (v − x)
β
= αv
For x ≤ P̂
First notice that
[1 − F2 (P̂ )]P̂ = [1 − F2 (x)]x. Hence,
π1 (P̂ ) − π1 (x) = α(P̂ − x) − (1 − α − β)[1 − F2 (x)][1 − F3 (x)]x
≥ β(P̂ − x) − (1 − α − β)[1 − F3 (x)]x
= π2 (P̂ ) − π2 (x)
=0
Firm 3
For x ∈ (P , P̂ )
π3 (x) = (1 − α − β)[1 − F1 (x)][1 − F2 (x)]x
=
(1 − α − β)αβ
v
β 2 + α(1 − α)
For x ≥ P̂
40
π3 (P̂ ) − π3 (x) = (1 − α − β)[1 − F2 (P̂ )]P̂ − (1 − α − β)[1 − F1 (x)][1 − F2 (x)]x
≥ (1 − α − β)[1 − F2 (P̂ )]P̂ − (1 − α − β)[1 − F2 (x)]x
i
1−α−β h
β [1 − F2 (P̂ )]P̂ − [1 − F2 (x)]x
=
β
#
"
i
1−α−β h
≥
β [1 − F2 (P̂ )]P̂ − [1 − F2 (x)]x + α(P̂ − x)
β
=
1−α−β
[π1 (P̂ ) − π1 (x)]
β
=0
Proof of Proposition 11. First let’s consider the region (P , P̂ )
F1 (x) = 0
≤1−
β2
αβ
v
+ α(1 − α) x
= F2 (x)
"
#
αβ
v
1−α
1− 2
≤
1−α−β
β + α(1 − α) x
= F3 (x)
For the region (P̂ , v)
41
F3 (x) = 1
≥1−
αv−x
β x
= F2 (x)
α(1−α)
α v − β 2 +α(1−α) v
≥1−
β
x
αβ
v
=1− 2
β + α(1 − α) x
α(1 − α) v
≥1− 2
β + α(1 − α) x
= F1 (x)
where the last inequality follows from the fact that α + β < 1. This is true
because if α + β ≥ 1, firm 3 would make zero profits in equilibrium (as no
consumer would learn its price), and hence would prefer a lower β.
Proof of Proposition 12. Follows directly from Proposition 10.
Proof of Proposition 13. From Propositions 7 and 8, it follows that the profits
of firm 2 in the three-firm equilibrium are
π2 =
βα(1−α)
v.
β 2 +α(1−α)
As firm 2 chooses α, it must be that α∗ = arg max π2 . The FOC yields that
α
α∗ = 21 , which is the same obfuscation level that firm 2 chooses in the two-firm
equilibrium, as shown in Proposition 4
Proof of Proposition 14. In the two-firm equilibrium, consumers either search
only the first store (which they do without incurring any time), or search both
stores and spend α∗ time. Consumers will only search both stores if they
have enough time. Hence, the average time a consumer spends searching is
(1 − α∗ )α∗
In the three-firm equilibrium, consumers either search only one store, or they
search the first two stores (spending α∗ time) or they search all stores and
42
spend α∗ + β ∗ time. The average time a consumer spends searching is β ∗ α∗ +
(1 − α∗ − β ∗ )(α∗ + β ∗ ) = (1 − α∗ )α∗ + (1 − α∗ − β ∗ )β ∗
It follows from Proposition 1 that the total firm profit in the two-firm equilibrium is α∗ (2 − α∗ )v. It follows from Proposition 7 that the total firm profit in
∗ )α∗ (α∗ +2β ∗ )
v
the three-firm equilibrium is (1−α
(β ∗ )2 +α∗ (1−α∗ )
As firm 3 chooses β, it must be that β ∗ = arg max π3 . It then follows from
∗
β
√
2−√ 2
.
2 2
Proposition 7 that β =
From Proposition 4, it follows that α∗ =
1
. Replacing the equilibrium obfuscation levels in the profit functions yields
2
that the total firm profit is greater in the three-firm equilibrium. This then
implies that, in the three-firm equilibrium, the average price consumers pay is
higher.
Proof of Proposition 15. Let’s start by analyzing the expected price for consumers who have search time lower than α∗ . I denote by Fi2F and Fi3F the
equilibrium cdf for firm i in the two-firm and three-firm equilibrium, respectively. Consumers who have search time lower than α∗ only learn the price of
firm 1, so they will pay the expected price of firm 1.
h
i ∗
R
1
2F
Two-firm equilibrium: xdF1 (x) = 1 + ln α∗ α v
h
i
R
∗ 2
∗ (1−α∗ ) α∗ (1−α∗ )
v
Three-firm equilibrium: xdF13F (x) = 1 + ln (β )α∗+α
∗
(1−α )
(β ∗ )2 +α∗ (1−α∗ )
The two expressions are equal when β ∗ = 1 − α∗ . Moreover, the second
expression is decreasing in β ∗ . Because β ∗ < 1 − α∗ (otherwise no consumer
would learn the price of firm 3 and firm 3 would rather choose a lower β), it
follows that the average price for consumers with search time lower than α∗ is
greater under the three-firm equilibrium.
I will now analyze the average price for consumers who have search time in
(α∗ , α∗ + β ∗ )
These consumers have enough time to learn the price of firms 1 and 2, but not
to learn the price of firm 3. They will pay the expected value of the minimum
price between firms 1 and 2.
R
Two-firm equilibrium: {f12F (x)[1 − F22F (x)] + f22F (x)[1 − F12F (x)]}xdx
43
= 2α∗ v − ln
1
α∗
(α∗ )2
1−α∗
v
Three-firm equilibrium:
=
∗
α∗ β ∗
ln 1−α
(β ∗ )2 +α∗ (1−α∗ )
β∗
v+
R
{f13F (x)[1 − F23F (x)] + f23F (x)[1 − F13F (x)]}xdx
"
#
(α∗ )2 (1−α∗ )v
β ∗ [(β ∗ )2 +α∗ (1−α∗ )]
2(β ∗ )2 v
α∗ (1−α∗ )−ln
(β ∗ )2 +α∗ (1−α∗ )
α∗ (1−α∗ )
The argument is the same as above. The expressions are equal when β ∗ =
1 − α∗ and the second expression is decreasing in β ∗ . It follows that these
consumers will pay higher prices under the three-firm equilibrium.
Lastly, I analyze the average price for consumers who have search time higher
than α∗ + β ∗ . These consumers have enough time to search all stores. For the
two-firm equilibrium, this just means that they search the two existing stores,
so they pay the same as consumers with search time in (α∗ , α∗ +β ∗ ), which was
analyzed before. For the three-firm equilibrium, these consumers will search
all three stores. As the price of firm 3 is always lower than the price of firm
1 (from Proposition 10), it follows that the average price these consumers will
pay is
R
{f23F (x)[1 − F33F (x)] + f33F (x)[1 − F23F (x)]}xdx =
α∗ β ∗ v
(β ∗ )2 +α∗ (1−α∗ )
h
2+
β∗
β∗
ln 1−α
∗
1−α∗ −β ∗
i
Plugging in the values of α∗ and β ∗ from the proof of Proposition 14, we find
that these consumers pay lower prices under the three-firm equilibrium.
Proof of Proposition 16. Fix a profile of prices p ≡ (p1 , ..., pn ) and obfuscation
levels α ≡ (α2 , ..., αn ). The profit of firm i is
πi (p, α) =
n−1
P
n
o
αk+1 1 pi = min{p1 , ..., pk } pi +
k=i
1−
n
P
j=2
!
n
o
αj 1 pi = min{p1 , ..., pn } pi
It follows that
n
o
∂πi (p,α)
=
−
1
p
=
min{p
,
...,
p
}
pi ≤ 0
i
1
n
∂αi
Proof of Proposition 18. When firm i chooses obfuscation level α̂i , the time it
takes for a consumer to learn its price is τ + α̂i . We can interpret the game as if
firms are choosing the total amount of time it takes for consumers to learn its
price. The game is the same as in the absence of exogenous market frictions,
with the additional restriction that no firm can choose an obfuscation level
lower than τ . By assumption, this additional restriction is not binding.
44
Proof of Proposition 19. Using the same argument as in the proof of Proposition 8 we find that, in any equilibrium, αθ ≥ βθ . Hence, using the result from
Proposition 7
βθ α(1 − α)v
αθ = arg max 2
α≤θ β + α(1 − α)
θ
βθ = arg max
β≤θ
βαθ (1 − αθ − β)v
β 2 + αθ (1 − αθ )
The solution is αθ = min{θ, 21 } and βθ = min{θ,
√
αθ − αθ }
For θ ≤ 0.25, αθ = βθ = θ
√
For θ > 0.25, βθ = αθ − αθ < αθ
When αθ < θ it must be that αθ = 21 . Hence, βθ =
βθ
αθ > 1−α
θ
45
√
2−√ 2
,
2 2
which implies that
B
Additional Figures
Figure B.1: Bell Canada pricing scheme - part 1
46
Figure B.2: Bell Canada pricing scheme - part 2
47
Figure B.3: Coextro pricing scheme
48