A model of obfuscation with
Heterogeneous Firms ∗
Samir Mamadehussene†
May 28, 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, at which point they buy 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. I find empirical evidence that supports the model’s
predictions.
∗
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 on 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 support obfuscation being collectively rational emerge from the economics literature that connects search costs and equilibrium profits.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 article, I
1
For example, Diamond [1971], Wolinsky [1986] and Anderson and Renault [1999].
2
propose a model of obfuscation with heterogeneous firms, with two main goals: i) to provide
foundations for obfuscation to be individually rational; ii) to understand the incentives of
obfuscation for different types of firms.
In the model proposed in this article, 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 assumption that search is with recall and the first search is for free.
As consumers follow an exogenous search order, this implies that all consumers know the
price of 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.
When a firm obfuscates, it can only prevent consumers from learning its price. To see
this, suppose there are 3 firms in 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
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 article, I refer to the ith firm on 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
consumers learn which prices. α2 consumers will not have time to learn the price of firm
2, so they will only observe the price of firm 1. α3 consumers will have enough time to
learn the price of firm 2, but not enough to learn the price of firm 3. Finally, (1 − α2 − α3 )
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 that will learn its price. This is not surprising because firm
3 is the last firm on the search order. Hence, firm 3 could never prevent consumers from
searching another store, as 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 that will learn its
price. Notice that the share of consumers that 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 that will learn all prices and
increase the number of consumers that 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 a Bertrand equilibrium, making zero profits. If firm 2 obfuscates,
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, as 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
4
price of firm 1.
When firm 2 obfuscates it softens the 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 that it is competing for. 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 in the
same direction as Arbatskaya [2007]. The intuition is straightforward. When a consumer
enters firm i he already knows the prices of all firms ranked above i on 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 opposite is not true. Firm i − 1 can sell even if it has a price higher than firm i,
because some consumers will not have enough time to learn the price of firm i.
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 on 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 that 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 power to firms 1 and 2. Consumers that 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.
5
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 that will learn all prices.
However, because firm 2 charges higher prices than 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 pay, on average, higher prices. As the time that consumers spend
searching is determined by firms’ obfuscation levels, this means that the market responds
to increase 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 induced 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 that 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 the complex pricing scheme. Casual observation suggests that, even though firms change
prices frequently, they tend to stick to their pricing schemes for long periods of time.
Another extension allows for exogenous market 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.
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 that learn all prices.
6
I find that, essentially, the equilibrium does not change, as long as the market frictions are
not too large.
In order to test the prediction that firms ranked higher on 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 follows 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 its prices: i) firms may list 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.
Regarding the search order that consumers use, I rely on the psychology literature
(Sutherland and Sylvester [2000]) that argues that consumers will visit first the stores they
are more aware of. Hence, I interpret the exogenous search order in the model as a ranking
of firms by their 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 both in the theoretical and
empirical literature (Carlin [2009]; Ellison and Wolitzky [2012]; Muir et al. [2013]). This
article provides a new link between brand awareness and obfuscation.
5
A list of all ISPs operating in Canada can be found at www.canadianisp.com.
7
Related Literature
This article 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 (eg. different fees depending on what contingency arises). Boundedly rational
consumers only compare 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 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 affects 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] 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 article 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 that are able to understand which firm has
the lowest price. The remaining consumers purchase from a firm at random. He finds that
8
firms that list higher prices are the ones that will have incentives to obfuscate. My model
also shares this prediction. However, whereas in Carlin’s equilibrium firms choose either full
transparency or full obfuscation, 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 article
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 on their prominence level. Consumers that 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 difference between my model
and Wilson’s 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 their 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 with reverse obfuscation strategies. In equilibrium, the firm that obfuscates makes a lower profit than the transparent firm. Hence, even though obfuscation is
individually rational, each firm strictly prefers the equilibrium in which it is not obfuscating.
9
Ellison and Wolitzky [2012] formalize a model 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 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 understanding the various obfuscation strategies for
different types of firms. Second, the insight for why obfuscation is individually rational is
very different in both models. Whereas in Ellison and Wolitzky firms obfuscate to prevent
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 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
6
I assume that the first search is for free to avoid philosophical questions regarding the behavior of a
consumer that does not have enough time to learn any price. The results of the model would not change if,
instead, all consumers were endowed with a positive amount of time to search (eg. time budget drawn from
the uniform distribution on [θ, 1 + θ]) and no firm could obfuscate more than θ. In this case, firm 1 would
10
Firm heterogeneity comes from their ranking in the search order. Throughout this article,
I refer to the ith firm on 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 by Fi and πi the equilibrium price distribution and profit of firm i. All
omitted proofs are in the appendix.
Proposition 1 For any obfuscation level that firm 2 chooses, α ∈ [0, 1], there exists a unique
equilibrium in the pricing stage characterized by
F1 (p) =




0




if p < αv
v
1 − αp






1
if αv ≤ p < v
F2 (p) =
if p ≥ v




0




1−






1
if p < αv
α v−p
1−α p
if αv ≤ p < v
if p ≥ v
π1 = αv
π2 = α(1 − α)v
Any consumer that learns the price of firm 2, also knows the price of firm 1. Hence, firm
2 will only be able to sell to some consumers 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 firm
2, as 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
choose the maximum obfuscation level, θ, and the remaining firms would choose the same obfuscation as
described in the model, as long as θ is large enough.
11
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
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 that will buy at that price are
those that 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 soften market competition. The downside is that, when firm 2
obfuscates, it reduces the share of consumers that it is competing for. 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
12
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 that 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
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 that 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 over 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,
13
the other firm is indifferent between all obfuscation strategies. Hence, there exists a fully
transparent equilibrium.
Proposition 7 For any obfuscation level that firms 2 and 3 choose, α and β, equilibrium
profits are characterized by
π1 = αv



 2βα(1−α) v
β +α(1−α)
π2 =


(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)
Inspecting the profits of firm 3, I find that, in equilibrium, firm 3 will never obfuscate
more than firm 2. In fact, when firm 3 obfuscates more than firm 2, i.e. α < β, its profits
are 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 example illustrates
why I compare α∗ and
β∗
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
fraction of 0.4 will not have enough time to learn the price of firm 2. Hence, firm 2 prevents
40% of consumers that start searching it from learning its price. Only the consumers that
were successful at learning the price of firm 2 (a mass 0.6 of consumers) will start searching
firm 3. Given the choice of 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 that start searching it
14
from learning its price. Even though α > β, firm 3 is effectively obfuscating more than firm
2.
β
1−α
is a measure of the obfuscation of firm 3 that is comparable to α, as it quantifies the
proportion of consumers that start searching the price of firm 3 but are not successful.
Proposition 8 states that firms ranked higher on the search order obfuscate more. As it
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
that the obfuscating firm can compete for.
When firm 2 obfuscates, it provides firm 1 with direct market power. I say direct market
power in the sense that consumers that 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 only with indirect market power. Consumers that 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 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 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 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 firm 2 and (1 − α − β) consumers
will observe all prices. Hence, when either firm obfuscates, it reduces the share of consumers
that will learn all prices. Whereas the market of firm 2 includes both consumers that only
learn prices of firms 1 and 2 and consumers that learn all prices, the market of firm 3 consists
solely of consumers that 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) >
15
R
0
v
[1 − F1 (p)][1 − F3 (p)]pdF2 (p)
As firm 3 extracts a higher surplus from consumers that 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




F1 (p) = 1 − 2(1−α)α v if P̂ < p ≤ v
β +α(1−α) p






1
if p > v
F2 (p) =




0







1 −



1−







1
F3 (x) =




0




if p ≤ P
αβ
v
β 2 +α(1−α) p
if P < p ≤ P̂
α v−p
β p
if P̂ < p ≤ v
1−α
−
1−α−β






1
where P ≡
if p > v
if p ≤ P
(1−α)αβ
v
[β 2 +α(1−α)](1−α−β) x
αβ
v
β 2 +α(1−α)
if P < p ≤ P̂
if p > P̂
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 that 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
16
price, as some consumers do not have time to search and only learn its price. This leads to
the result that firms ranked higher on 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 that have enough time to search firms in 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 that has enough time to search firm 3 will never go back and purchase
from firm 1.
A consumer that 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 12 Firm 2 will choose the same obfuscation level in both the two-firm and the
three-firm equilibria
Since 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
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.
17
result of this section is presented in proposition 13.
Proposition 13 When the market size increases from 2 to 3 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 of 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 14 When the market size increases from 2 to 3 firms, consumers that have
search time lower than α∗ + β ∗ pay higher prices, while the remaining consumers pay lower
prices.
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 that 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 that 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.
18
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 15 Zero obfuscation is a weakly dominant strategy for all firms
The result in Proposition 15 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 obfuscation levels. In fact, while 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.
Proposition 16 analyzes firms equilibrium profits.
Proposition 16 In equlibrium, 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
19
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. Since all consumers that 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.
Since there is a positive probability that firm n has the lowest price in the market, it will
want to maximize the number of consumers that learn its price. But if firm n makes zero
obfuscation, all consumers that 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 true that all consumers
that 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 that will purchase from the lowest-price firm,
a la Bertrand. The equilibrium involves both firms charging marginal cost and making zero
profits. This contradicts that firm n makes profits.
Exogenous market 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
20
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 17 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 5
minutes, firms will make their prices more complex, so that consumers spend an additional
5 minutes in order to understand its 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.
4
Empirical Findings
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 on how each variable
was computed.
Obfuscation Scores
A challenging task when studying obfuscation is to measure the complexity of firms’ pricing
schemes. I follows 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
8
Number of Google searches between September 14, 2014 and September 12, 2015.
21
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, while others list it in the fine print.
Charging a lower price for the first few months makes it harder to compare different
pricing schedules. Consider a consumer that 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
afterwards. 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 of them, charge
different prices in the first few months is not straightforward. The calculations needed to do
such comparison can also increase due to the fact that the length of the discount period may
be different. In fact, while some firms offer a lower rate for the first 6 months, others only
do it for 3 (or even only 1) months. I construct a score that reflects whether a firm engages
in this obfuscation device. I let OD equal to 1 if a firm offers a different price in the first few
months of the contract and equal to 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 to 0 if a firm does not charge an activation/installation fee, equal
to 1 if it charges a fee but lists it in the main table of prices and equal to 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
22
There is significant dispersion in the complexity of pricing schemes. Figure 3 presents
the distribution of firms across obfuscation scores. There are 4 possible levels of obfuscation.
Only 2 firms present the highest possible level of obfuscation (which involves charging a
lower price for the first few months and listing an activation fee in the small print). All the
remaining obfuscation levels have significant share of firms adopting it.
The obfuscation range is very wide. While 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
4, 5 and 6 present the pricing schemes for two firms. Figure 4 details the available packages
and prices for Bell Canada, the largest firm that operates in the market. It offers a wide
range of packages. Each package has a regular price and a 3-month promo price. After
selecting a particular package, the consumer is directed to the full offer details presented in
Figure 5. 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. Figure 6 details
the available packages and prices for Coextro, a much smaller ISP. This firm only offers 2
packages and it does not charge activation or installation fees. Moreover, the monthly price
is flat.
Brand Awareness Scores
Measuring firms’ 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 their brand awareness. In fact, a firm 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 firms’ brand awareness. Google trends provides information
on the number of Google searches for each firm. Even though data on the magnitude of
23
Google searches is not available for each firm, 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 more well known and have a positive score. For this reason,
I construct another brand awareness score, AB , as a binary variable described below.
AB =



0 if A = 0


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 2 presents the summary statistics for the obfuscation, brand awareness and price
variables.
24
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
11.86
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 2: Summary statistics
4.1
Empirical Analysis
Relationship brand awareness - obfuscation
I start by running the most simple OLS regression to analyze the effect of firms’ brand
awareness on their obfuscation level. I do this for each of the obfuscation and brand awareness
variables defined in the previous section.
Obf uscationi = α + βBrandAwarenessi + i
where Obf uscation ∈ {OD , OF , OT } and BrandAwareness ∈ {A, AB }
The results are presented in table 3. 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 on 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
4 groups. The first group consists of all 65 firms with a brand awareness score of zero.
The remaining 24 firms are evenly split among the remaining 3 groups, in increasing brand
25
OD
A
OD
0.009***
(0.002)
AB
Observations
R-squared
OF
OF
OT
0.022***
(0.006)
0.162**
(0.067)
OT
0.031***
(0.007)
0.724***
(0.165)
0.887***
(0.174)
89
0.145
89
89
89
0.063
0.116
0.181
Standard errors in parentheses
*** p < 0.01, ** p < 0.05, * p < 0.1
89
0.199
89
0.230
Table 3: The impact of Brand Awareness on Firms’ Obfuscation
awareness. So, for example, group 4 constitutes of the 8 ISPs with 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
Relationship brand awareness - price
I run an OLS regression to analyze the effect of firms’ brand awareness on their prices. I do
this for each of the brand awareness and price variables defined in the previous section.
26
P ricei = α + βBrandAwarenessi + i
where P rice ∈ {P12 , V P12 , P36 , V P36 } and BrandAwareness ∈ {A, AB }
The results are presented in table 4.
P12
P12
A 2.3**
(1.2)
AB
69.7**
(30.9)
Observations
R-squared
89
0.044
V P12
V P12
2.2**
(1.0)
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
89
0.055 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 4: The impact of Brand Awareness on Firms’ Price
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, more well-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 use that obfuscation device will have significantly lower
prices.
Another obfuscation device that firms use is to charge an activation fee. The price of
firms with higher brand awareness scores could be higher simply because they engage in such
obfuscation device. To control for this possible issue, I also run the regression of variable
27
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 more well-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 more well-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 practice is more frequently
adopted by more well-known firms.
I propose a model under which consumers search stores in order of their brand awareness.
The model predicts that more well-known firms will obfuscate more and list higher prices,
just like the empirical findings suggest. The model has important welfare implications. By
28
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 increase, consumers end up paying, on
average, higher prices.
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 practice. This is challenging to
implement in practice. 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 such as: the main features, amount and costs of credit,
the Annual Percentage Rate (APR), the number and frequency of payments requested from
the credit provider, as well as a note on important legal aspects. This allows [consumers] to
compare the offers of different credit providers and select the credit offer that works best for
[them].”
In a similar attempt to reduce the level of obfuscation, the U.K. Competition and Markets
Authority is enforcing supermarkets to display unit prices, for ease of comparison of ”the
29
costs of different products, regardless of their respective sizes”.
In the model presented in this paper, firms that are higher on the search order make
more profits. This could lead to competition for higher positions in the list. I interpret the
search order as a ranking of firms’ brand awareness. Hence, competition for a higher ranking
on the list is simply an advertisement game.
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32
Appendix A - Omitted Proofs
Proof of Proposition 1. Since 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, since 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.
Since 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 that 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, since it always sells to consumers that 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. Since firms
must be indifferent between all prices in the support of their cdf, and since v is the upper
bound of both cdfs, it follows that firm 2 makes positive profits when it charges price v.
Since 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 that 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
When firm 1 charges a price x < v, its profits are
33
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
Since there are no mass points at the lower bound, firm two will sell to all consumers that
learn its price, when it charges the lower bound. Since the mass of consumers that 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.
v
x
α (1 − α)v
=1−
1−α
x
α v−x
≤1−
1−α x
F1 (x) = 1 − α
= F2 (x)
where the inequality follows from the fact that the lower bound of F1 and F2 is αv
34
Proof of Proposition 3.
v
p
v
≤ 1 − α0
p
00
F1α (p) = 1 − α00
0
= F1α (p)
Proof of Proposition 4. Since firm 2 chooses α, we must have that
α∗ = arg max π2
α
where π2 is described in Proposition 1. The first order condition yields the result.
Proof of Proposition 5. For p ≥ v and p ≤ αv, the result is trivial. For αv < p < v
F1 (p) = 1 − α
v
p
v−p
=1−α−α
p
"
#
α v−p
= (1 − α) 1 −
1−α p
= (1 − α)F2 (p)
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 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 still an equilibrium, because as long as one firm is playing zero obfuscation,
35
the other firm will make zero profits regardless of its obfuscation strategy.
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, since 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 1 − β[1 − F2 (P 1 )]P 1 − (1 − α − β)[1 − F2 (P 1 )][1 − F3 (P 1 )]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
π1 (P ) = P
36
π2 (P ) − π2 (P 2 ) = (1 − α)P − β[1 − F1 (P 2 )]P 2 − (1 − α − β)[1 − F1 (P 2 )][1 − F3 (P 2 )]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.
Since 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
37
Moreover, it must be that the cdf of firm 1 has the highest upper bound. Indeed, if some
firm charged a price higher than the upper bound of firm 1, it would not make sales, since
all consumers know the price of firm 1. Furthermore, it must be that the upper bound of
firm 1 is v, since if it was lower than v, firm 1 would prefer to increase its price, since 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. Since the upper bound of the cdf of firm 1 is v
and F2 (v) = F3 (v) = 1, it follows that π1 = αv.
Since v is in the support of the cdf of firm 2,
π2 = π2 (v) = β[1 − F1 (v)]v =
Since P =
αβ
v
β 2 +α(1−α)
βα(1−α)
v
β 2 +α(1−α)
is in the support of the cdf of firm 3, and since 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. Since 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 ∗ =
38
√
2−√ 2
.
2 2
Proof of Proposition 9. First notice that π3 = (1 − α − β)
It follows from Proposition 7 that
R
0
R
v
0
[1 − F1 (p)][1 − F2 (p)]pdF3 (p).
v
[1 − F1 (p)][1 − F2 (p)]pdF3 (p) =
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
β2
(1 − α)α
v
+ 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
h
i
π1 (P̂ ) − π1 (x) = α(P̂ − x) + β [1 − F2 (P̂ )]P̂ − [1 − F2 (x)]x − (1 − α − β)[1 − F2 (x)][1 − F3 (x)]x
i
= α(P̂ − x) − [1 − F2 (x)]x − (1 − α − β)[1 − F2 (x)][1 − F3 (x)]x
i
≥ β(P̂ − x) − [1 − F2 (x)]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−
αβ
v
β 2 + α(1 − α) x
= F2 (x)
"
#
1−α
αβ
v
≤
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 (since no consumer would learn its
price), and hence would prefer a lower β.
Proof of Proposition 12. 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)
Since firm 2 chooses α, it must be that α∗ = arg max π2 . The FOC yields that α∗ = 12 , which
α
is the same obfuscation level that firm 2 chooses in the two-firm equilibrium, as shown in
Proposition 4
Proof of Proposition 13. 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 spend α∗ + β ∗ time. The average
42
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 the three-firm equilibrium is
(1−α∗ )α∗ (α∗ +2β ∗ )
v
(β ∗ )2 +α∗ (1−α∗ )
Since firm 3 chooses β, it must be that β ∗ = arg max π3 . It then follows from Proposition
∗
7 that β =
√
2−√ 2
.
2 2
β
From Proposition 4, it follows that α∗ = 21 . Replacing the equilibrium
obfuscation levels in the profit functions yields 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 14. Let’s start by analyzing the expected price for consumers that 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 that 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
Two-firm equilibrium: xdF12F (x) = 1 + ln α1∗ α∗ v
h
i
R
∗ 2
∗ (1−α∗ ) α∗ (1−α∗ )
Three-firm equilibrium: xdF13F (x) = 1 + ln (β )α∗+α
v
(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 that 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.
Two-firm equilibrium:
(α∗ )2
= 2α∗ v − ln α1∗ 1−α
∗v
R
{f12F (x)[1 − F22F (x)] + f22F (x)[1 − F12F (x)]}xdx
43
Three-firm equilibrium:
=
α∗ β ∗
(β ∗ )2 +α∗ (1−α∗ )
ln
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 that 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. Since 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
h
i
R 3F
β∗
β∗
α∗ β ∗ v
3F
3F
3F
{f2 (x)[1 − F3 (x)] + f3 (x)[1 − F2 (x)]}xdx = (β ∗ )2 +α∗ (1−α∗ ) 2 + 1−α∗ −β ∗ ln 1−α∗
Plugging in the values of α∗ and β ∗ from the proof of Proposition 13, we find that these
consumers pay lower prices under the three-firm equilibrium.
Proof of Proposition 15. Fix a profile of prices p ≡ (p1 , ..., pn ) and obfuscation levels α ≡
(α2 , ..., αn ). The profit of firm i is
n
o
n−1
P
αk+1 1 pi = min{p1 , ..., pk } pi +
πi (p, α) =
k=i
1−
n
P
j=2
!
n
o
αj 1 pi = min{p1 , ..., pn } pi
It follows that
∂πi (p,α)
∂αi
n
o
= −1 pi = min{p1 , ..., pn } pi ≤ 0
Proof of Proposition 17. 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
Appendix B - Additional Figures
33%
2%
39%
OT
OT
OT
OT
=0
=1
=2
=3
26%
Figure 3: Distribution of firms over obfuscation scores
45
Figure 4: Bell Canada pricing scheme - part 1
46
Figure 5: Bell Canada pricing scheme - part 2
47
Figure 6: Coextro pricing scheme
48