Shrouded Attributes and
Information Suppression in
Competitive Markets
Xavier Gabaix
MIT and NBER
David Laibson
Harvard University and NBER
Current Draft: March 4, 2004∗
∗
For useful suggestions we thank Simon Anderson, Roland Bénabou, Douglas
Bernheim, Andrew Caplin, Victor Chernozhukov, Avinash Dixit, Edward Glaeser,
Penny Goldberg, Robert Hall, Sergei Izmalkov, Julie Mortimer, Barry Nalebuff,
Aviv Nevo, Ariel Pakes, Nancy Rose, José Scheinkman, Andrei Shleifer, David
Thesmar, Wei Xiong and seminar participants at Berkeley, Columbia, Harvard,
MIT, NBER, New York University, Princeton, Stanford, the University of Virginia,
the 2003 European Econometric Society meeting, and the 2004 American Economic
Association meeting. Steve Joyce provided great research assistance. We acknowledge financial support from the NSF (SES-0099025). Gabaix thanks the Russell
Sage Foundation for their hospitality during the year 2002-3. Xavier Gabaix:
MIT, 50 Memorial Drive, Cambridge, MA 02142, [email protected]. David Laibson: Harvard University, Department of Economics, Cambridge, MA, 02138, [email protected].
1
Abstract: We propose an alternative way of thinking about
search costs. Standard models of search costs describe markets
in which consumers want to access information but can only do
so at a cost. In standard search models, firms will compete by reducing search costs (if they can) and advertising low prices. We
study markets in which firms’ equilibrium incentives turn out to
be reversed. We derive conditions in which consumers overlook
information in equilibrium because firms suppress the information
and not because information is fundamentally hard to transmit.
Our analysis explains why informational shrouding/suppression
flourishes even in highly competitive markets.
JEL classification: D00, D80, L00.
Keywords: add-on, aftermarket, bounded rationality, behavioral
economics, behavioral industrial organization, information suppression, loss leader, myopia, profit center, shrouded attributes.
2
1
Introduction
When choosing a good or service, some consumers do not take full account of
shrouded product attributes, including maintenance costs, prices for necessary
add-ons, or hidden fees.1 Bank accounts have lots of shrouded attributes
like fees and surcharges. A bank may prominently advertise “free checking,”
but the advertising won’t highlight the $30 fee for a bounced check, the $15
surcharge for a checkbook, or a $2 penalty per check for writing more than
five checks in a month. Even if all consumers know that such fees exist, many
consumers may not know the particular value of these fees when they choose
a bank account.
Unfortunately, bank accounts are not uniquely opaque. Many products
have shrouded attributes that some customers fail to observe or fail to incorporate in their decision when they initially choose a product. Hotels do
not catalog their long distance phone fees when a potential customer inquires
about the price of a room. Mail-order retailers segregrate shipping and handling surcharges, typically specifying them only at the end of a phone/web
transaction. Rental car companies do not highlight their gas refill charges,
which are only found in the fine print in the contract the rentor is handed
when she drives off with her rental car. Printer manufacturers rarely advertise the operating costs of owning a printer, and only a tiny fraction of printer
buyers claim that they knew the ink price per page when they bought their
printer.2 Most mutual funds do not advertise management fees and most
individual investors do not know the fees that they are paying.3
Even when consumers understand that shrouded costs exist, they may
not take these costs fully into account when deciding what good to buy.
Firms will exploit the existence of shrouded attributes. If add-ons – like
printer ink cartridges – are shrouded, then firms will price these add-ons at
supra-competitive prices (Lal and Matutes 1989 and Borenstein et al.1995a).
Because of competitive pressures, the presence of shrouded attributes
does not raise profits in general, instead reallocating profits across different
segments of a business. Firms compete to attract customers by choosing loss
leader prices for the non-shrouded base good. Printer manufacturers sell
1
See Caplin and Leahy (2003) for a discussion of the economics of situations where
consumers are not aware of some future events.
2
Hall (2001) conducts a survey in which only 3% of printer buyers claim that they knew
the ink price per page at the time that they bought their printer.
3
Alexander et al (1998), Barber et al (2002).
3
printers below marginal cost so they can capture consumers to whom they
will sell above marginal cost printer cartridges. Banks attract depositors
with giveaways (“open an account and receive a free DVD player”4 ) and
then charge hefty fees for banking services like check books, ATM usage,
money orders, and overdrafts. In general, this shift in pricing will generate
inefficiencies, both because of distorted purchases of the underpriced base
good and because of distortions in long-run demand for the overpriced addons.
In this paper, we show that competition and advertising will not generally
pull down prices for add-ons. One might conjecture that firms will compete
by advertising (and committing to) low add-on prices, thereby forcing other
firms to follow suit. However, we show that firms will lose market share
(and/or profitability) by advertising low add-on prices. In equilibrium, sophisticated consumers prefer to give their business to a firm with high add-on
prices, because the sophisticated consumers end up with an implicit subsidy
from the unsophisticated customers of that firm. Staying in an underpriced
hotel room is a great idea as long as you avoid the marked up phone calls,
room service, and mini-bar items on which the hotel makes its profits. The
sophisticated consumer gets the benefit of the loss-leader hotel room and
avoids the high cost of the add-ons. Many markets work the same way.
For example, a sophisticate gets the benefit of a $25 gift for opening a bank
account and avoids paying the banking fees that snare naive consumers.
Sophisticated consumers would rather pool with the naive consumers at
firms with overpriced add-ons instead of switching to firms with marginal
cost pricing of both the base good and the add-on. This preference for
pooling with the naive consumers reflects the fact that in equilibrium the sophisticated consumers are cross-subsidized by the naives (cf Della Vigna and
Malmendier 2003, 2004).5 Because of these cross-subsidies, firms will often
not be able to profitably attract consumers by educating naive consumers
about shrouded attributes and advertising low add-on prices.
Hence, supra-competitive pricing of the add-on will persist in markets
with high levels of competition and free advertising. This prediction distinguishes our model from standard search models, which imply that firms
have an incentive to disseminate information about their products, and only
4
Citibank is currently advertising this offer.
DellaVigna and Malmendier study cross-subsidies that arise from self-control problems. Naive consumers who don’t recognize their self-control problems cross-subsidize
sophisticated consumers who do.
5
4
choose not to do so because such dissemination is costly. We identify conditions under which firms will choose not to advertise and hence not to compete
on add-on pricing, even when advertising is free. Shrouding is a more profitable strategy. This explains why industries with nearly costless marginal
information dissemination still shroud their add-on prices,6 contradicting the
traditional search cost framework.
The rest of this paper formalizes our claims. Section 2 defines a shrouded
attribute and presents a simple equilibrium analysis of a market with discrete
demand for add-ons. Section 3 discusses the general case with continuous
demand. Section 4 concludes.
1.1
Literature review
Our model of shrouded attributes is related to a literature that studies products that have an “aftermarket” or “add-ons.”7 Ink cartridges are a good
example of an add-on market. If a consumer buys a deskjet printer, he will
need to purchase replacement ink cartridges from time to time. The original
producer of the printer typically has some market power in the aftermarket. For example, many printers only accept ink cartridges that contain a
print-head patented by the printer manufacturer (Hall, 2000).
Many authors have developed models that explain why add-ons have relatively high mark-ups. For example, Lal and Matutes (1994) point out
that consumers have difficulty observing add-on prices because of information/search costs. Borenstein et al. (1995a, 1995b, 2000) note that base
good producers sometimes can’t commit to add-on prices. When consumers
can’t observe add-on mark-ups or when firms can’t commit to add-on prices,
producers will set add-on prices above marginal cost. In the current paper,
we analyze an environment in which high markups persist even when firms
can costlessly reveal and commit to add-on prices (e.g., in our setting printer
manufacturers can freely and credibly preannounce the price of their printer
cartridges).
High add-on markups have also been explained as the result of price discrimination (Klein 1995, Ellison 2002). In Ellison’s model, firms use add-ons
6
Finding the operating cost of a printer (e.g., price of ink per page) on the web site of
a printer manufacturer is like looking for a needle in a haystack (cf Hall (2000)).
7
Lal and Matutes 1989, 1994, Salop 1993, Borenstein et al 1995a, 1995b, 2000, Shapiro
1995, Klein 1995, Hall 2000, Ellison and Ellison 2001, Ayres and Nalebuff 2002, Ellison
2002, Hortacsu and Syverson 2003.
5
to generate price discrimination of rational consumers. By contrast, we assume that consumer myopia plays an important role in firms’ pricing decisions
for add-ons. Unlike Ellison, our add-on effects survive in an environment in
which firms do not have market power.
Our paper is related to the literature on search (e.g., Stigler 1961, Diamond 1972, Salop and Stiglitz 1977, Hortacsu and Syverson 2004). Like
search models, our model predicts that consumers will not always purchase
the most competitively priced good. However, our model also generates some
contrast with the predictions of traditional search models. For example, in
a search model, easy dissemination of information – e.g., free advertising –
eliminates equilibrium market power and mark-ups. However, in our equilibrium, these effects will vanish since firms prefer to shroud information about
the add-on market and competition does not lead this shrouding to unravel.
Our model generates voluntary information suppression in equilibrium, an
effect that won’t arise in search equilibria.
2
Shrouded Attributes: Definitions and an
Example of a Market Equilibrium
We analyze consumers who sometimes make purchase decisions without incorporating the effects of future contingencies, like the need to pay an overdraft fee at a bank. A shrouded attribute is any contingency that is not
fully incorporated into the initial purchase decision. We introduce a formal
framework below. At this stage we present some examples for motivation.
Shrouded attributes may include surcharges, fees, accessories, options,
fraud, or any other hard-to-anticipate feature of the ongoing relationship
between a consumer and a firm. We simplify this list by dividing it into
two mutually exclusive categories: (unavoidable) surcharges and (avoidable)
add-ons.
Hidden surcharges are shrouded attributes that a consumer cannot avoid
once she buys a product. Surcharges are quite common in the rental car
industry. Rental car companies typically charge “airport fees,” “concession
recoupment fees,” and “vehicle license fees.” These fees are paid to the rental
car company and then remitted to third parties. Rental car companies could
incorporate these charges into their base prices, but instead they choose to
isolate these costs and include them as surcharges tacked on at the end of
6
the transaction.
While unavoidable surcharges are quite common, most shrouded attributes
are complementary goods that consumers have the option to avoid. For example, hotel guests can avoid paying telephone charges if they instead use
a public phone or a cell phone. Likewise, hotel guests can avoid paying
for a room service meal by finding a local restaurant. Both hotel phone
use and hotel room service complement a hotel stay. Such complementary
(voluntary) goods are often referred to as add-ons.8
This paper introduces a framework that unifies these examples, including
both unavoidable surcharges and avoidable add-ons. To generalize this discussion we distinguish between a “base good” and the shrouded attribute.
In the immediately preceding examples, the base goods are rental cars and
hotel rooms and the shrouded attributes are surcharges and hotel services.
We assume that consumers may pick a base good – e.g., hotel – without observing shrouded attributes. For example, a consumer could simply
compare prices of closely located four star hotels on a web travel site (e.g.,
Orbitz), without observing the hotels’ telephone pricing schedule (Ayres and
Nalebuff 2002). Indeed, we don’t know anybody who has bothered to inquire
about a hotel’s phone charges at the time they reserved a room. Hence, we
consider shrouded attributes to be the leading case in many market transactions.
When the shrouded attribute is a surcharge, our model implies that such
surcharges will be aggressively adopted by firms and limited only by legal
constraints and fairness norms. When the shrouded attribute is an add-on,
our model implies that add-ons will be priced with high mark-ups relative to
the mark-ups for base goods. We find that even if firms have access to free
advertising, firms will often not use this advertising in equilibrium and the
high mark-ups on the add-ons will persist.
2.1
A simple example with discrete demand
The body of this paper presents a general model of shrouded attributes. To
develop intuition, we first analyze a stripped down example that illustrates
many of the key points. To motivate this example, consider a bank that sells
two kinds of services. For a price p a consumer can open an account. If the
account holder subsequently violates the minimum balance of the account,
8
See Ellison (2003) for another approach to the analysis of add-ons.
7
she pays a fee pb. In our language, being “allowed” to violate the minimum
balance is an “add-on” service with price pb. Without loss of generality, we
assume that the cost to the bank of having the consumer violate the minimum
balance is zero.
We assume that violating the minimum balance requirement generates
extra liquidity with value V to the shareholder. To keep things simple,
assume that the consumer can raise this liquidity either by violating the
minimum balance or by exerting effort e > 0 to cut back earlier purchases.
We assume V > e, implying that if pb is high enough the consumer should
exert effort e to generate the extra liquidity. To sum up, the consumer’s
payoff table is given by:
Cut back early spending V − e
Violate minimum balance V − pb
Do neither
0
In this example, we assume that the add-on service comes only in a single
discrete increment. So the minimum balance violation is a zero-one outcome.
We consider the general case of continuous add-on quantities in section 3
below.
We assume that sophisticated consumers anticipate the fee pb and optimally decide which of the three rows to pick above. Because we assume
V > e, the sophisticated consumer will choose one of the first two options no
matter what value pb takes:
max {V − e, V − pb} ≥ V − e > 0.
We assume that naive consumers do not accurately anticipate the fee
pb. Naive consumers may completely overlook the aftermarket or they may
mistakenly believe that pb < e. Under either assumption, naive consumers
will not cut back their early spending. So when the naive consumer needs
the extra liquidity, the naive consumer must choose between foregoing payoff
V or paying fee pb.
Given this simple discrete structure, sophisticates will buy the add-on iff
V − pb ≥ V − e. Naives will buy the add-on iff V − pb ≥ 0.
We can now characterize the symmetric equilibrium in this market.9 Let
D(xi ) represent the probability that a consumer opens an account at bank
9
See Appendix A for a discussion of the existence of a symmetric equilibrium.
8
i, assuming that xi is the average anticipated net surplus from opening an
account at bank i less the average anticipated net surplus from opening an
account at the best alternative bank. The demand function is probabilistic
either because of trembles or because of idiosyncratic consumer preferences.10
For a sophisticated consumer,
xi = [−pi + max {V − pbi , V − e}] − [−p∗ + max {V − pb∗ , V − e}] .
Throughout the paper we use starred variables to represent the (uniform)
prices set by other firms.11 For a naive consumer,
x = −pi + p∗ ,
since the naive consumer neglects the aftermarket when deciding where to
open her bank account. D is weakly increasing, bounded below by zero, and
bounded above by one.
Let α represent the share of sophisticated consumers in the marketplace.
The following proposition shows that firms will choose high mark-ups in the
add-on market. Indeed, when the share of sophisticated consumers is small
(α < 1− Ve ), firms will choose mark-ups that are so high that the sophisticated
consumers substitute out of the add-on market.
Proposition 1 Call
e
.
V
(1)
D (0)
.
D0 (0)
(2)
α† = 1 −
and
µ=
If α < α† , equilibrium prices are
p = − (1 − α) V + µ
pb = V
(3)
(4)
p = −e + µ
pb = e
(5)
(6)
and only naive agents consume the add-on.
If α ≥ α† , prices are
10
See Appendix B for a microfoundation of the demand function. Also see McFadden
(1981) and Anderson et al (1992).
11
We analyze symmetric equilibria. We give sufficient conditinos for the existence of
such equilibria in Appendix A.
9
and all agents consume the add-on.
Proof.
Step 1. We observe that if (p, pb) is a best response, then pb ∈ {e, V }.
To see this, consider first the case pb < e. A firm can increase pb by a small
positive increment, decrease p, and not change the demand of sophisticated
consumers (the total price they face does not change), but increase strictly
the demand of naive consumers.
Consider next the case e < pb < V . The firm can earn more by charging
pb = V , as this will not change the demand of naive consumers. For pb > e,
the sophisticated consumer would rather substitute away from the add-on.
Step 2. For pb ∈ {e, V }, the profit function reduces to
Π (p, pb = e) = (p + e) D (p∗ − p)
Π (p, pb = V ) = (p + (1 − α) V ) D (p∗ − p)
(7)
(8)
Demand is independent of the aftermarket, since sophisticates get no utility
from it and naives do not anticipate it.
Equations 7 and 8 imply that the problem is recursive. First, the firm
picks the maximum of e and (1 − α) V . The firm will choose pb = V iff
e < (1 − α) V,
i.e., iff α ≤ α† . Second, the firm picks p:
max (p + max {e, (1 − α) V }) D (p∗ − p) .
p
As in equilibrium p = p∗ , the first order condition is:
p + max {e, (1 − α) V } =
D0 (0)
= µ.
D (0)
¤
The proposition shows that in equilibrium firms will choose high markups in the add-on market. Firms set pb∗ ≥ e, though the marginal cost of
producing the add-on is 0. These mark-ups are particularly high when the
fraction of sophisticated consumers is small. When α < 1 − Ve , the mark-up
is V. High mark-ups for the add-on are offset by low or negative mark-ups
on the base good. This is easiest to see when the market is approximately
10
competitive (i.e., the demand curve is highly elastic, and hence µ is close
to zero). In a relatively competitive market with small µ, the base good
is always a loss leader with a negative mark-up: p∗ ≈ − (1 − α) V < 0 or
p∗ ≈ −e < 0. This explains why in a host of seemingly competitive markets
(printers, hotels, car rentals) the price of the base good is typically set below
marginal cost of the base good while the price of the shrouded add-ons are
set well above marginal cost of the add-ons (printer cartridges, local phone
calls, gas charges12 ).
The model implies that the shrouded attribute will be the “profit-center”
and the base good will be the “loss leader.” The shrouded attribute market
becomes the profit-center because at least some consumers don’t anticipate
the shrouded add-on market and won’t respond to a price cut in the shrouded
attribute market.
We now evaluate the robustness of this result. Previous authors have
conjectured that advertising would drive down after-market prices. Shapiro
(1995) describes the inefficiency caused by high mark-ups in the aftermarket
and then observes that competition and advertising should drive them away.
Furthermore, manufacturers in a competitive equipment market have incentives to avoid even this inefficiency by providing
information to consumers. A manufacturer could capture profits
by raising its [base-good] prices above market levels (i.e., closer
to cost), lowering its aftermarket prices below market levels (i.e.,
closer to cost), and informing buyers that its overall systems price
is at or below market. In this fashion, the manufacturer could
eliminate some or all of the deadweight loss, attract consumers
by offering a lower total cost of ownership, and still capture as
profits some of the eliminated deadweight loss. In other words,
and unlike traditional monopoly power, the manufacturers have
a direct incentive to eliminate even the small inefficiency caused
by poor consumer information (1995, p. 495).
We will show that this intuition will sometimes fail to apply. In general,
high mark-ups in the aftermarket will not go away as a result of competition
or advertising. We will also describe cases in which advertising does eliminate
high mark-ups in the add-on market.
12
See Ellison (2002) for more examples.
11
2.2
The curse of education: A No-Advertising Result
In this subsection, we show that firms will not cut prices in the aftermarket,
even if the firms can freely advertise the resulting efficiency gains to consumers. Naturally, our findings also extend to the case in which advertising
is costly.13
We assume that advertising makes more consumers anticipate the aftermarket, which will lead these previously naive consumers to care about
inefficiencies in aftermarket prices. For example, such advertising might
report, “When you choose a printer, remember to take into account the aftermarket for printer cartridges. Our cartridge prices are low relative to our
competitors.” More formally, advertising increases the proportion of sophisticated consumers from α to α0 ∈ (α, 1]. We assume that all sophisticated
consumers anticipate the aftermarket and take into account all aftermarket
prices.
We look for a Nash equilibrium, defined as a set of prices and advertising
decisions that are all simultaneously chosen. If no firm advertises, then α is
low. If at least one firm advertises, then α rises to α0 . We take the prices
from the previous proposition and call them the Shrouded Market Prices.
Proposition 2 If α < α† , the Shrouded Market Prices support an equilibrium in which firms choose not to advertise.
Proof: If the firm advertises (and thereby deviates from the no-advertising
equilibrium), the firm’s optimal value pb must still be in {e, V }. Its profit, as
in (7) and (8), will be
Π (b
p = e) = max (p + e) D (p∗ − p)
p
Π (b
p = V ) = max (p + (1 − α0 ) V ) D (p∗ − p) .
p
(9)
(10)
If the firm does not advertise, its profit will be
Π0 = max (p + (1 − α) V ) D (p∗ − p) ,
p
(11)
as implied by the conjectured no-advertising equilibrium. Given that e and
(1 − α0 ) V are each less than (1 − α) V , the deviation is not profitable and
the no-advertising equilibrium survives.¤
13
See Anderson and Renault (2003) for a complementary analysis of the information
revealed by advertising, and Bagwell (2002) for a review of the literature on advertising.
12
At first glance these results seem highly counter-intuitive. Why doesn’t
Shapiro’s argument about advertising apply? Shapiro conjectured that firms
will compete by advertising low add-on prices, thereby making naive consumers more sophisticated and attracting consumers who as a result of the
advertising appreciate the benefit of being able to buy efficiently priced addons. However, the current proposition shows that this business-stealing
effect is offset by a pooling effect. In equilibrium, sophisticated consumers
would rather pool with the naive consumers at firms with high add-on prices
than defect to firms with marginal cost pricing of both the base good and
the add-on.
To gain intuition for this effect consider the case in which the firm has no
market power, so µ = 0. If a sophisticated consumer gives her business to a
firm with Shrouded Market Prices, the sophisticated consumer’s surplus will
be,
sophisticated surplus =
=
>
=
−p + V − e
(1 − α) V + V − e
(1 − α) V + V − (1 − α) V
V.
By contrast, if the sophisticated consumer gives her business to a firm with
zero mark-ups on both the base good and the add-on, the sophisticated consumers surplus will only be V. So the sophisticated consumer is strictly better
off pooling at the firm with Shrouded Market Prices (and high aftermarket
mark-ups), then deviating to the firm with marginal cost pricing.
This preference for pooling reflects the fact that the sophisticated consumers are being cross-subsidized by the naive consumers. The sophisticated
consumers benefit from the “free gifts” and can avoid the high fees.
Because of this effect, no firm has an incentive to cut their add-on prices,
raise their base good prices, and advertise these efficiency-enhancing changes.
Making more consumers sophisticated won’t attract consumers and won’t
raise profits. Hence, monopoly pricing of the add-on will persist in markets
with high levels of competition and free advertising.
Note that the current proposition characterizes equilibrium for the case
α ≤ α† . Our no-advertising result will not change if we consider the case α >
α† . Recall that this latter case generates no inefficiency, since all consumers
purchase the add-on in equilibrium. Without inefficiency, no firm will strictly
benefit from advertising.
13
These “no-advertising” results describe equilibria of games in which all
actions are chosen simultaneously, including advertising. We can change
the timing of the advertising decision without affecting the “no-advertising”
results. Assume that advertising is chosen first and that all firms pick their
prices after observing the advertising decision. To eliminate indeterminacy,
also assume that advertising has a vanishingly small cost. Then we find
that advertising will not arise in equilibrium, since the firms that advertise
make lower profits then the firms that don’t. Advertising won’t change the
market-wide level of profits (once all firms adjust their prices in response to
the advertising), and it won’t change the gross profit per firm. Advertising
will only reduce the profits of the firms that choose to advertise.
This section’s toy model is non-robust in one important sense. In this
illustrative model, the sophisticated agents use a perfectly elastic substitute
for the add-on service. In many important settings, substitutes are imperfectly elastic, a feature that we incorporate in the general model of continuous
add-on consumption in section 3.
2.2.1
Another impediment to unshrouding
Until now we have ignored the consumer entry decision, since we have assumed that all consumers must buy a base good. We now endogenize the
consumer entry decision and find that this reinforces our no-advertising result.
When some consumers naively believe that aftermarket expenses are minimal, these consumers may buy the base good when they should avoid the
market altogether. Think of a consumer who buys a $50 deskjet printer
without realizing that the lifetime operation costs of such a printer typically
add up to hundreds of dollars of ink cartridges. Firms that compete by unshrouding high aftermarket prices, will drive some of these naive consumers
out of the base good market. Hence, naive entry decisions on the part of
consumers are another force that discourages firms from using advertising to
unshroud the aftermarket.14
To illustrate this, we adopt a random utility framework (see Appendix
B). The perceived value of the good sold by firm i to sophisticated consumer
a is
via = V − p − min (e, pb) + σεia ,
14
Ellison (2002) anticipates this effect in the discussion of his model with sophisticated
consumers.
14
with εia a random variable with values in [−1, 1].
the good sold by firm i to naive consumer a is
The perceived value of
via = V − p + σεia .
Note that naive consumers overlook the add-on market, implying that their
perceptions omit the term min (e, pb).
We call R the reservation value. A consumer first picks the good i with
the highest via , and buys that good iff via ≥ R.
The following illustrative proposition shows that introducing a binding
reservation value reinforces our earlier “no-advertising” prediction.
Proposition 3 Assume that a fraction γ of naive consumers have a high
reservation utility R > V + σ. If α ≥ α† and γ = 0, the Shrouded Market
Prices support an equilibrium in which firms are indifferent between advertising and not advertising. If α ≥ α† and γ > 0, then all firms strictly prefer
not to advertise.
Condition R > V + σ implies that, even if the price of the base-good
and the add-ons were 0, the consumer would still prefer not to consume the
good.15
Proof. Consider a firm that charges prices p and pb and advertises. The other
firms charge the pre-deviation price p∗ = −e + µ, and pb∗ = e. We first derive
the optimal prices p and pb of the advertising firm. By the same reasoning as
in the proof of Proposition 1, the optimal pb belongs to {e, V }.
Case 1. We first consider pb = e. The profit per consumer, p + pb, has to
be non-negative for the deviation to be profitable. But the high-reservation
utility consumers will not take the good, as V − (p + pb) + σ < R. So the
total number M of potential consumers becomes less than 1, as a fraction
(α0 − α) (1 − γ) drops out of the market. This fraction is equal to the mass
of newly educated consumers, α0 − α, times the probability 1 − γ that those
consumers have high reservation utility. So:
M = 1 − (α0 − α) (1 − γ) < 1.
The profit is:
max Π (p, pb = e) = M max (p + e) D (p∗ − p)
p
p
= M max (p + e) D (µ − (p + e)) .
p
15
With marginal costs c and b
c, this condition would become: R > V + σ − c − min (e, b
c).
15
The maximum is reached at p + e = µ , and we get profit
max Π (p, pb = e) = MµD (0) < µD (0) ,
p
which implies that firm profits fall below the pre-deviation profit µD (0), as
M < 1.
Case 2. We now consider the case pb = V . A fraction (1 − α0 ) of consumers
will buy the add-on, and a fraction m ∈ [M, 1] will buy the base good,
implying that firm profit will be
Π (b
p = V ) = max (mp + (1 − α0 ) V ) D (p∗ − p) .
p
Simple algebra shows that
1 − α0
< 1 − α,
M
which means that the fraction of consumers who are naive buyers is lower
than it was before. We get:
µ
¶
(1 − α0 )
Π (b
p = V ) = m max p +
V D (p∗ − p)
p
m
µ
¶
(1 − α0 )
V D (p∗ − p)
< m max p +
p
M
< 1 max (p + (1 − α) V ) D (p∗ − p) .
p
Because maxp (p + (1 − α) V ) D (p∗ − p) was realizable before the deviation,
it is weakly less than the pre-deviation profit µD (0). We conclude:
Π (b
p = V ) < µD (0)
So the post-deviation profit is less than the pre-deviation profit, µD (0). The
deviation is not profitable.
2.3
Welfare
In this subsection we consider the welfare consequences of naiveté.
16
Proposition 4 Suppose that the reservation utility is such that all consumers would buy the good, even if they were aware of the cost of the add-on
(γ = 0). Relative to the case of sophisticated consumers (α = 1), the equilibrium social welfare loss is:
£
¤
Λ = αe for α ∈ 0, α†
= 0 for α ∈ (α† , 1)
For α ≤ α† , sophisticated consumers are V − e units better off than naive
consumers. For α > α† , all consumers are equally well.
Proof. The social welfare loss is proportional (with factor e) to the
fraction of agents who exert
costly
effort when a socially costless substitute
£
¤
†
is available. When α ∈ 0, α , all of the sophisticated agents exert effort e,
so the deadweight loss is αe. When α > α† , no agents exert costly effort,
since the firms price the add-on at p = e.
Consumption of the add-on is socially efficient since the add-on is produced at zero social cost. In the case with inefficiency (α < α† ), the welfare
losses increase as the fraction of sophisticates increase, since sophisticates are
the source of socially inefficient underconsumption of the add-on. In equilibrium, all naive consumers purchase the add-on, hence there are no social
deadweight losses when there are only naive consumers (i.e., α = 0).16
However, social losses will arise if shrouded attributes generate distorted
consumer decisions to buy the base good. Assume that a fraction γ (1 − α)
of consumers shouldn’t buy the good. This corresponds to a social loss of
R −V , where R is the consumer’s reservation utility, and V is the consumer’s
actual utility. The welfare accounting is given by the following corollary.
Corollary 5 Suppose that a fraction γ of naive consumers satisfy R > V
(with σ = 0). Those consumers would not buy the good if they were aware of
the cost of the add-on. Then, the equilibrium social welfare loss is
£
¤
Λ = αe + (1 − α) γ (R − V ) for α ∈ 0, α†
= (1 − α) γ (R − V ) for α ∈ (α† , 1).
The total deadweight loss is decomposed into losses arising from price distortions, αe, plus losses arising from purchases of goods that are not socially
beneficial, (1 − α) γ (R − V ). The second term falls linearly with fraction of
sophisticated consumers (α).
16
This conclusion is a special property of the toy model and does not generalize to the
continuous demand model of the next section.
17
3
General case of shrouded attributes with
continuous demand
In this section, we generalize the arguments above to the case of continuous
demand for the add-on. While most of our results do not change, we show
that the no-advertising result only applies when sophisticates have relatively
inexpensive substitutes for the add-ons. For example, a sophisticated hotel
visitor will bring her cell phone, providing a low-cost substitute for hotel
phone service. When sophisticates do not have a relatively inexpensive
substitute technology, advertising will arise.
As before, each firm sells a base good with a single shrouded attribute.
Each firm sets a price pi for the base good. Each firm also sets a price
pbi associated with the shrouded attribute. If the shrouded attribute is a
surcharge, then pbi is the value of the surcharge. If the shrouded attribute is
an add-on, then pbi is the per-unit price of the add-on.
Total utility is decomposed into two parts: the value of owning the base
good assuming that the shrouded attribute did not exist and the incremental value of the shrouded attribute. Let uai represent agent a’s gross value
of firm i’s base good, not including the value associated with the shrouded
attribute. Let u
b (b
eai , qbai ) represent the gross utility flow associated with
the after-market, reflecting both the quantity of the add-on that is consumed, qbai , and any costly efforts, ebai , to substitute away from the add-on.
So u
b1 (b
eai , qbai ) ≤ 0 u
b2 (b
eai , qbai ) ≥ 0, and u
b12 (b
eai , qbai ) ≤ 0. Note that substitution is only possible when the shrouded attribute is at least partially
avoidable. In the surcharge case, ebai = 0 and qbai = 1.
Summing up all of the terms, the net value of buying the base-good can
be written
uai − pi + u
b (b
e , qb ) − pbi qbai
| {z }
| ai ai
{z
} .
base good
shrouded attribute
3.1
Sophisticated consumers
Sophisticated consumers take into account the existence of the add-on, so
they choose eb optimally. For example, a sophisticated consumer will avoid
hotel phone charges by bringing a cell phone. A sophisticated bank customer
will avoid late payment fees by mailing mortgage payments early. A sophisticated printer owner will avoid buying expensive ink cartridges by printing
18
large jobs on the office machine (instead of printing at home).
We adopt the following timing assumptions for the sophisticated consumer. The sophisticated consumer first observes all prices at all firms,
then picks a base good i, then picks a level of substitution effort ebai , and
finally picks a quantity of add-ons qbai . Formally, the complete maximization
problem for the sophisticated consumer can be written,
½
¾
b (b
eai , qbai ) − pbi qbai .
max max uai − pi + u
i
eeia ,e
qia
We introduce notation that enables us to compactly summarize the choices
S
of the sophisticated consumer. Let b
eSai (b
p) and qbai
(b
p) represent the equilibrium choices of the sophisticated consumer who buys good i,
¡
¢
S
S
uai − pi + u
b ebSai (b
p), qbai
(b
p) − pbi qbai
(b
p)
b (b
eai , qbai ) − pbi qbai .
= max uai − pi + u
eeia ,e
qia
Without loss of generality, we assume that Ea uai = 0. Then the average
utility of sophisticated consumers at firm i is
¡ S S¢
S
b b
eai , qbai − pbi qbai
.
uS (pi , pbi ) ≡ −pi + u
3.2
Naive consumers
Naive consumers do not (fully) take account of the effect of the shrouded attribute, and they may choose b
e suboptimally. For example, a naive consumer
will not recognize the true (high) price of printing on her inkjet printer17 and
will therefore fail to appropriately reduce her use of the printer. Eventually,
the naive consumer may learn the true costs of the add-on, but these learning
effects are not modeled in this paper to preserve tractability.
We make the following timing assumptions. The naive consumer first
observes only base-good prices, then picks a base good i, and then picks some
default level of substitution effort b
eai . Later, the add-on price pbi is revealed
to the naive consumer and she finally picks an add-on quantity, qbai .
17
At the moment, black and white text costs between 2 cents and 15 cents per page,
depending on the deskjet printer. Color text costs a bit more than black and white text
A photographic image costs an order of magnitude more. Printing 10 pages per day at
10 cents a page costs $1460 over four years.
19
The complete maximization problem for the naive consumer can be decomposed into three parts. First the naive consumer picks a base good:
©
ª
max uai − pi + constant .
i
The constant in this equation captures the naive consumer’s expectations
about the aftermarket. Note that this constant does not depend on the true
value of pbi . This last assumption could be relaxed as we discuss below.
The naive consumer now chooses a substitution level ebai . We assume ebN
ai
S
does not depend on pbi and that b
eN
≤
e
b
.
To
motivate
this
setup,
we
assume
ai
ai
that naive agents choose their substitution level under the false belief that
the add-on price is zero. We could weaken this assumption by making ebN
ai
depend on pbi with some sensitivity, but with insufficient sensitivity relative
to the optimal level of responsiveness.18 It is notationally easier to consider
the extreme case of no dependence between b
eN
bi , which is the approach
ai and p
we take in this paper. Alternatively, we could assume that naive agents
underestimate how much of the add-on they will end up consuming. As
an example of the last mechanism, consider a consumer who knows the true
price of hotel phone calls, but mistakenly believes that she will not need to
use the hotel phone.
Finally, the naive consumer chooses qbai , the continuous quantity of addon consumption. We assume that this choice is made after the substitution
effort level ebN
bi , is evenai has already been fixed and after the add-on price, p
tually revealed.
¡ N
¢
b b
eai , qbai − pbi qbai .
max u
qeai
We introduce notation that enables us to summarize the choices of the
N
naive consumer. Let qbai
(b
p) represent the equilibrium choices of the naive
consumer who buys good i.
¡ N
¢
N
b b
eai , qbai − pbi qbai
(b
p) ≡ arg max u
qbai
3.3
qeai
The firm’s problem
To analyze the firm’s optimization problem, we need to know how firm level
demand responds to the average perceived value of buying the firm’s base
18
For example, the naive consumer could simply underestimate the true price of the
add-on by a factor θ.
20
good. Let D(x) represent the demand of a firm that offers an average
perceived surplus x units greater than the average perceived surplus provided
by its competitors (Appendix B presents a standard microfoundation for D).
We assume that a fraction α of consumers are sophisticated and a fraction
1 − α of consumers are naive. Finally, we assume that the marginal cost
of manufacturing the base-good (e.g., printer) is c, and the marginal cost of
manufacturing the add-on (printer cartridge) is b
c.
Drawing all of our assumptions together, the firm’s profits will be,
¡
¢ ¡
¢
Π = α p − c + (b
p−b
c) qbS (b
p) D uS (p, pb) − uS (p∗ , pb∗ )
¡
¢
+ (1 − α) p − c + (b
p−b
c) qbN (b
p) D (−p + p∗ ) .
(12)
The first line represents profit deriving from sophisticated consumers and
the second line represents profit deriving from naive consumers.
3.4
Characterization of Equilibrium
We characterize equilibria when a symmetric equilibrium exists. Existence is
guaranteed for sufficiently high levels of idiosyncratic variation in consumer
preferences (see Appendix A).
Definition 6 The average demand for the add-on is:
b
q (b
p) := αb
qS (b
p) + (1 − α) qbN (b
p) .
Proposition 7 If a symmetric equilibrium exists, it satisfies:
i
¤
p) £
d h
αb
q S (b
q (b
p) =
p)
(b
p−b
c) b
p − c + (b
p−b
c) qbS (b
db
p
µ
p − c + (b
p−b
c) b
q (b
p) = µ.
(13)
(14)
(15)
where µ is the inverse elasticity of the demand for the good,
µ = D(0)/D0 (0).
(16)
Corollary 8 If all consumers are naive (α = 0), the add-on has a price
corresponding to the case of full monopoly:
pb − b
c
1
= ,
pb
η
p) /b
q N (b
p) is the price elasticity of the naive demand for
where η = −b
pqbN0 (b
the shrouded attribute.
21
Corollary 9 If all consumers are sophisticated (α = 1), the add-on is priced
at marginal cost:
pb − b
c = 0
p − c = µ.
Corollary 10 If the sophisticated and naive consumers have the same demand function qb (b
p) for the add-on, then:
pb − b
c
1−α
=
pb
η
(17)
where η = −b
pqb0 (b
p) /b
q (b
p) is the price elasticity of the demand for the shrouded
attribute.
Corollary 11 In the case of (unavoidable) surcharges, the surcharge pb is set
to its upper bound and
p − c = µ − (b
p−b
c) .
∂ S
Proof for the Proposition. We observe ∂p
u (p, pb) = −1, and the en∂ S
S
q (b
p). We take the first order condition
velope theorem gives ∂ peu (p, pb) = −b
∗ ∗
in (12) at (p, pb) = (p , pb ).
¡
¢
¡
¢ ¡
¢
∂Π
= αD uS (p, pb) − uS (p∗ , pb∗ ) − α p − c + (b
p−b
c) qbS (b
p) D0 uS (p, pb) − uS (p∗ , pb∗ )
∂p
¡
¢
p−b
c) qbN (b
p) D0 (−p + p∗ )
+ (1 − α) D (−p + p∗ ) − (1 − α) p − c + (b
¢
¡
p) D0 (0)
= αD (0) − α p − c + (b
p−b
c) qbS (b
¡
¢
+ (1 − α) D (0) − (1 − α) p − c + (b
p−b
c) qbN (b
p) D0 (0)
³
´
= D (0) − p − c + (b
p−b
c) b
q (b
p) D0 (0)
0 =
so that the unit profit is:
q (b
p) =
p − c + (b
p−b
c) b
22
D (0)
= µ.
D0 (0)
Optimization of the add-on’s price gives:
¤ ¡
¢
∂Π
d £
p) D uS (p, pb) − uS (p∗ , pb∗ )
=α
(b
p−b
c) qbS (b
∂ pb
db
p
¤ ∂uS (p, pb) 0 ¡ S
¢
£
D u (p, pb) − uS (p∗ , pb∗ )
p)
+α p − c + (b
p−b
c) qbS (b
∂ pb
¤
d £
(b
p−b
c) qbN (b
p) D (−p + p∗ )
+ (1 − α)
db
p
¤
£
¤
£
d
p) D (0) − α p − c + (b
p−b
c) qbS (b
p) qbS (b
p) D0 (0)
(b
p−b
c) qbS (b
= α
db
p
¤
d £
(b
p−b
c) qbN (b
p) D (0)
+ (1 − α)
db
p
i
£
¤
d h
(b
p−b
c) b
q (b
p) D (0) − α p − c + (b
p−b
c) qbS (b
p) qbS (b
p) D0 (0)
=
db
p
0 =
which gives (15). For sufficient conditions on D, see Appendix B.
The proofs of the corollaries are immediate. ¤
If all consumers are sophisticated (α = 1), the market equilibrium reflects marginal cost pricing of the add-on pb − b
c = 0 (Corollary 9), replicating
standard economic efficiency results. By contrast, if no consumers are sophisticated (α = 0), we get monopoly pricing of the add-on (Corollary 8).
Firms exploit the invisibility of the add-on market and charge their captured
customers full monopoly prices for add-ons.
In the intermediate cases (0 < α < 1), the add-on will have supracompetitive mark-ups and these markups are offset by low or negative markups on the base good. This is easiest to see when the market is approximately
competitive (i.e., the demand curve is highly elastic, and hence µ is close to
zero). Equation (15) decomposes profits per-customer into two terms: profits on the base good and profits on the add-on. Equation (15) implies that
in a relatively competitive market (i.e., µ small), the base good is always a
loss leader with a negative mark-up: p − c ≈ − (b
p−b
c) b
q (b
p) < 0.
Equation (17) enables an analyst to use data on markups and demand
elasticities to impute the fraction of sophisticated consumers, α.
µ
¶
pb − b
c
α=1−η
.
pb
Markets with high markups and high add-on elasticities are likely to have a
low level of consumer sophistication. Intuitively, competitive pressure would
23
normally push down add-on prices in markets with elastic demand curves,
unless some consumers do not anticipate the aftermarket when they pick
their base good.
In practice a good is likely to have many aftermarket components with
different degrees of shroudedness. In Appendix C we present the generalization of our results to this multi-attribute case. Once again, high add-on
mark-ups are generated by high levels of shroudedness.
3.5
The curse of education: A no-advertising result
In this subsection, we show that firms may not cut prices in the aftermarket
(raising prices in the base good market), even if the firms can advertise the
resulting efficiency gains to naive consumers. As before, we assume that
advertising is free and that advertising leads previously naive consumers to
become aware of inefficiencies in aftermarket prices. In the current subsection, we assume that advertising increases the proportion of sophisticated
consumers from α to 1. We assume that sophisticated consumers anticipate
the aftermarket and take into account all aftermarket prices.
We look for a Nash equilibrium, defined as a set of prices and advertising decisions, which are all simultaneously chosen. If no firm advertises, then α is unchanged. If at least one firm advertises, then α rises
to 1. We take the prices from the previous proposition (7) and call them
the Shrouded
Market Prices.
Let u
bS (b
p) represent aftermarket utility, so
¡
¢
S
S
S
u
b (b
p) ≡ u
b eb (b
p) , qb (b
p) .
Proposition 12 The Shrouded Market Prices support an equilibrium in which
firms choose not to advertise iff
¡
¢ £ S
¤ £ S
¤
(1 − α) (b
p−b
c) qbN (b
p) − qbS (b
p) > u
b (b
c) − b
cqbS (b
c) − u
b (b
p) − b
cqbS (b
p) .
Proof. Without loss of generality, we take b
c = c = 0 to simplify exposition.19
∗
∗
In this proof, we call p and pb the equilibrium values from Proposition 7.
We start from the situation where no firm advertises. Suppose that a firm
advertises, and sets new prices p and pb. Its profit is
¢
¡
Π = (p + pbqb) D uS (p, pb) − uS (p∗ , pb∗ )
¡ S
¢
= xD u
b (b
p) − x − uS (p∗ , pb∗ ) ,
19
To go back to the general case, replace pb by pb − b
c, p by p − c, and for θ =
N aive, Sophisticate, u
bθ (b
p) by u
bθ (b
p) − b
c qbθ (b
p).
24
where x = p + pbqb is the total profit per customer. Maximizing Π (x, pb) over
pb we get pb = 0, so the highest profit the firm can get after deviating is:
Π = max xD (−x + x∗ )
x
(18)
with
x∗ = u
bS (0) − uS (p∗ , pb∗ ) .
As the pre-deviation profit is µ, the firm doesn’t want to deviate iff
Π < µ.
(19)
Given µ = maxy yD (−y + µ) and Eq. (18), and the fact that maxy yD (−y + z)
is increasing in z, (19) is equivalent to x∗ < µ. In other terms, x∗ is the price
offered by the competitor firms to Sophisticates after the deviation, and the
deviation is unprofitable iff this price is less than µ, which is the pre-deviation,
or “normal,” level of profits. To find the sign of x∗ − µ, we calculate:
By (15):
bS (0) − uS (p∗ , pb∗ ) − µ
x∗ − µ = u
bS (b
p∗ ) + p∗ + pb∗ qbS (b
p∗ ) − µ.
= u
bS (0) − u
h
i
x∗ − µ = u
bS (0) − u
bS (b
p∗ ) + p∗ + pb∗ qbS (b
p∗ ) − p∗ + pb∗b
q (b
p∗ )
¤
£ S ∗
= u
bS (0) − u
bS (b
p∗ ) + pb∗ qbS (b
p∗ ) − pb∗ αb
p ) + (1 − α) qbN (b
p∗ )
q (b
¢
¡
= u
bS (0) − u
bS (b
p∗ ) − (1 − α) pb∗ qbN (b
p∗ ) − qbS (b
p∗ ) .
As the firm doesn’t want to deviate iff x∗ −µ < 0, the Proposition is proven.¤
This no-advertising result contains a boundary condition that determines
when advertising will appear. Intuitively, advertising – and resulting segmentation – does not arise when the distortion from the monopoly pricing
in the add-on market is less costly to sophisticates than the benefits of loss
leader pricing in the base good market. Hence, advertising does not appear because a firm can’t pull sophisticated consumers away from firms that
cater to naive consumers. Sophisticates receive cross-subsides from the naive
consumers.
Formally, advertising does not arise when the cross-subsidies to sophisticates are larger than the distortions arising from pricing that deviates from
25
marginal cost. To see this worked out in a transparent example, consider
the case in which µ = 0, so firms have no market power. Now consider the
equilibrium payoff of a sophisticated consumer in the no-advertising equilibrium:
−p + u
bS (b
p) − pbqbS (b
p) .
Compare this to the equilibrium payoff of a sophisticated consumer if the
consumer has access to a firm with marginal cost pricing:
−c + u
bS (b
c) − b
cqbS (b
c) .
To further simplify exposition assume (without loss of generality) that c =
b
c = 0. Then the sophisticated payoff with marginal cost pricing reduces to:
u
bS (0) .
The no-advertising equilibrium is robust if the payoff with marginal cost
pricing is less than the payoff in the no-advertising equilibrium:
bS (b
p) − pbqbS (b
p) .
u
bS (0) < −p + u
Substituting equation (15) implies that this inequality can be reexpressed
q (b
p) + u
bS (b
p) − pbqbS (b
p) ,
u
bS (0) < pbb
q (b
p) = αb
q S (b
p) + (1 − α) qbN (b
p) (cf equation 13). Rearranging yields
where b
¢
¡
p) − qbS (b
p) > u
bS (0) − u
bS (b
p) ,
(1 − α) pb qbN (b
which is equivalent to the condition in Proposition 12 when c = b
c = 0.
Formally, advertising arises when the cross-subsidies to sophisticates (the
left hand side of the inequality) are smaller than the distortions arising from
pricing that deviates from marginal cost (the right hand side).
Until now, this subsection has focussed on the case of shrouded add-ons.
When the shrouded attribute is instead a surcharge, two types of equilibria
exist: one with advertising and indeterminate use of surcharges (holding the
“total” price fixed) and one with maximal surcharges and no advertising. If
advertising has a vanishingly small cost then the second type of equilibrium
is selected. Hence, the model predicts that maximal surcharges will always
arise when advertising is nearly costless.
26
Finally, our earlier conclusions about the effects of endogenous entry (subsection 2.2.1) apply to the general model discussed in this section. Again
endogenizing the consumer’s entry decision reinforces our no-advertising results. The benefits to producers of advertising the aftermarket are further reduced when advertising reveals aftermarket costs that drive some consumers
out of the base good market altogether.
3.6
Welfare losses from shrouded attributes
We have already seen that if all consumers are sophisticated (α = 1), then
add-ons are priced at marginal cost. When some consumers are naive, addons are priced above marginal costs, producing classic welfare losses associated with underconsumption of the add-on. We call this loss ΛH (H for
“Harberger’s triangle”).
Deadweight losses may also arise from excess consumer purchases in the
base-good market. Some naive consumers buy the base good because they
have not thought about the (high) price of the add-on. Had they considered
the add-on price, they might not have purchased anything at all. In the
worst case, the welfare loss is the full marginal cost of the base good and
the (purchased) add-on. We call this loss ΛShouldn’t have , as it arises from
consumers who shouldn’t have entered the base good market in the first
place.
For small distortions, it is well known that Harberger’s triangles ΛH are
of second order magnitude. However, the losses ΛShouldn’t have are first order
and are likely to be big in practice. For instance, if some consumers in the
printer market should not have bought their $50 printer, then the social
welfare loss may be as high as the full printer price. Such results are an
additional reason why naiveté is very important for welfare. The first-order
entry loss, ΛShouldn’t have , is more important than classical second-order losses,
ΛH . Appendix D develops these points and shows that ΛShouldn’t have can
attain its upper bound of the sum of the marginal cost of the base good and
the marginal cost of the (purchased) add-ons.20
20
See Caplin and Leahy (2003) for a thoughtful discussion of the economics of situations
where consumers are not aware of some future events.
27
4
Conclusion
We have analyzed an environment in which consumers fail to anticipate
“shrouded attributes” of goods, like the high cost of replacement printer
cartridges. Following Lal and Matutes (1989) and Borenstein et al. (1995a),
we show that firms will price shrouded add-ons at supra-competitive prices.
We identify settings in which competition and advertising fail to fix these
distortions. Even when advertising is free, firms will not use it, thereby suppressing the competitive pressure that advertising normally produces. In
equilibrium, sophisticated consumers buy the loss-leader base good and partially substitute away from add-on consumption. Sophisticated consumers
take advantage of subsidized hotel room prices and avoid high hotel service
charges (e.g., phone, food, mini-bar, etc). So sophisticated consumers end
up with a cross-subsidy from their naive counterparts. Sophisticated consumers would rather pool with naive consumers than defect to firms with
marginal cost pricing. Advertising low mark-ups and educating consumers
about the aftermarket won’t (profitably) increase a firm’s business.
We show that improvements in information technology may have surprisingly little impact on competitiveness. Low information dissemination costs
do not necessarily make markets competitive. Our findings stand in stark
contrast to standard economic intuitions. In classical discrete choice models
with commodity products and nearly homogeneous consumers (e.g., Anderson et al 1999) markups vanish as search costs fall to zero. In such models,
falling search costs force all producers to unshroud aftermarket prices and
set mark-ups close to zero. In our framework mark-ups robustly stay high,
since firms prefer not to compete by disseminating information about the
aftermarket even when such information dissemination is nearly costless.
Our model provides an alternative way of thinking about search costs.
Standard models of search costs describe situations in which consumers want
to access information but can only do so at a cost. In such standard models,
firms will compete by reducing search costs and advertising low prices. Our
analysis studies markets in which information plays a very different role. We
have argued that consumers do not anticipate shrouded attributes because
firms go out of their way to suppress this information and not because information is fundamentally hard to transmit. We believe that information
suppression – even costly information suppression – plays a prominent role
in most market equilibria. Economic theory needs to explain why shrouding
flourishes in highly competitive markets.
28
5
Appendix A: Existence of the symmetric
equilibrium
We now provide sufficient conditions for existence of a symmetric equilibrium.
Proposition 13 If α = 0, then a symmetric equilibrium exists.
Proof. We just set:
¢
¡
p) D (−p + p∗ )
Π = p + pbqbN (b
p), and the price is unique:
There is a unique maximum arg max pbqbN (b
p = µ − pbqbN (b
p) .
Proposition 14 Given demand functions u
bS (b
p), q N (b
p), and value of α,
assume that
d ³ b ´
pbq (b
p) − αb
qS (b
p)
φ (b
p) =
db
p
p1 ) < 0 (this ensure that pb1 is a profit
has a unique root pb1 , and that φ0 (b
maximizing price). Parametrize the demand function by
Dσ (x) = D1 (x/σ)
There is a σ ∗ such that for all σ ≥ σ ∗ , there exists a unique symmetrical equilibrium (p, pb) of the game maximizing profit (12). It is described by
Proposition 7.
Proof: We will begin with a purely mathematical Lemma.
Lemma 15 Suppose that n is a non-zero integer, I a compact of Rn , and
a > 0, and a continuous function H : I × [0, a] → Rn . Suppose that there is a
unique X0 in the interior of I such that H (X0 , 0) = 0, that H and ∂X H are
continuous in a neighborhood of (X0 , 0), and that ∂X H (X0 , 0) is invertible.
Then, there is an ε∗ > 0 such that for all ε ∈ [0, ε∗ ], there is a unique solution
X (ε) of H (X (ε) , ε) = 0.
29
This slight extension of the Implicit Function Theorem states that for ε
small enough, there is a unique solution of H (X, ε), if this is the case for
ε = 0.
Proof. The Implicit Function Theorem guaranties the existence of a set of
solutions X (ε) such that H (X (ε) , ε) = 0. We just have to show that for
ε small enough, there is indeed just one solution to H (·, ε) = 0. Suppose
0
the opposite. We would have a series εn → 0 and Xn 6= Xn such that
H (Xn , εn ) = H (Xn0 , εn ) = 0.
As Xn ∈ I compact, Xn → X0 . Otherwise, one would be extract a
subseries Xnk converging to some Y 6= X0 , and one would have H (Y, 0),
contradicting the hypothesis that X0 is the unique zero of H (·, 0). So Xn →
0
X0 , and likewise Xn → X0 .¡
°
0 ¢
0 °
We observe that un = Xn − Xn / °Xn − Xn ° is in a compact, so one
can extract a subseries ni such that uni → v for some kvk = 1, so v 6= 0.
Because:
¡ 0
¢
H (Xni , εni ) − H Xni , εni
°
°
→ ∂X H (X0 , 0) · v
0=
°Xni − Xn0 °
i
we have ∂X H (X0 , 0) · v = 0, which contradicts the fact that ∂X H (X0 , 0) is
invertible. This concludes the proof.
We are now ready to prove the Proposition. We set ε = 1/σ and p0 = p/σ,
0
Π = Π/σ, and without loss of generality c = b
c = 0. We call D1 = D to
simplify notations. With those notations, the profit function (12) is written:
¡
¢ ¡
¢
pqbS (b
p) D −p + εb
uS (b
p) + p∗ − εb
uS (b
p∗ )
Π0 (p, pb, p∗ , pb∗ , ε) = α p + εb
¡
¢
+ (1 − α) p + εb
pqbN (b
p) D (−p + p∗ )
This makes clear that a small ε means that the add-on is “small” for the
determination of the overall demands and profits. The strategy of the proof
is to show that for ε close to 0 , there is a unique equilibrium value of p, and
pb. The difficulty is that
Π0 (p, pb, p∗ , pb∗ , ε = 0) = pD (−p + p∗ )
so at ε = 0, Π0 is independent of pb. To circumvent this degeneracy at ε = 0,
we define the function F : R3 → R2 :
¶
µ
∂1 Π0 (p∗ , pb∗ , p∗ , pb∗ , ε)
∗ ∗
F (p , pb , ε) =
∂2 Π0 (p∗ , pb∗ , p∗ , pb∗ , ε) /ε
30
We take F because the f.o.c. for (p∗ , pb∗ ) to be an equilibrium is F (p∗ , pb∗ , ε) =
0, and the division by ε in the second component of F removes the degeneracy at ε = 0. Indeed, to calculate the second component at ε = 0, we observe
that as ∂2 Π0 (p∗ , pb∗ , p∗ , pb∗ , ε = 0) = 0, then:
lim ∂2 Π0 (p∗ , pb∗ , p∗ , pb∗ , ε) /ε = ∂2 (∂5 Π0 (p∗ , pb∗ , p∗ , pb∗ , 0))
ε→0
with
£ S
¤
q (b
p) D (−p + p∗ ) + αD0 (−p + p∗ ) u
b (b
p) − u
bS (b
p∗ )
∂5 Π0 (p, pb, p∗ , pb∗ , 0) = pbb
As the envelope’s theorem gives ∂peu
bS (b
p) = −b
qS (b
p), we get:
∗
∗
F (p , pb , ε = 0) =
So F (p∗ , pb∗ , ε = 0) = 0 iff:
µ
d
de
p∗
¶
−b
p∗ D0 (0) + D (0)
³
´
pb∗b
q (b
p∗ ) D (0) − αp∗ qbS (b
p∗ ) D0 (0)
p∗ = D (0) /D0 (0)
d ³ ∗b ∗ ´
pb q (b
p ) = αb
q S (b
p∗ )
db
p∗
This ensures a unique zero P ∗ (0) = (p∗ , pb∗ ) for ε = 0. By continuity, there
is also a unique zero P ∗ (ε) of F (P ∗ , ε) for ε close enough to 0.
We have now constructed our local equilibria P ∗ (ε). We want to show
that they are global equilibria, at least for ε close enough to 0. To do this,
we define, for X = (x, x
b) ∈ R2
¶
µ
∂1 Π0 (x + p∗ (ε) , x
b + pb∗ (ε) , p∗ (ε) , pb∗ (ε) , ε)
G (X, ε) =
b + pb∗ (ε) , p∗ (ε) , pb∗ (ε) , ε) /ε
∂2 Π0 (x + p∗ (ε) , x
G is gives a condition for a P (ε) to be a global equilibrium:
Π0 (p, pb, p∗ (ε) , pb∗ (ε) , ε) has a unique maximum for (p∗ , pb) = (p∗ (ε) , pb∗ (ε))
(20)
⇔ The only solution of G (X, ε) = 0 is X = (0, 0) .
In other terms, we have a global equilibrium iff the only zero of G (X, ε) is
X = (0, 0). As G (0, ε) = 0 by construction of G, Lemma 15 shows that this
31
will be true for ε small enough if GX (X = 0, 0), a 2 × 2 matrix, is invertible.
To calculate it, we observe that:
¶
µ
d
(pD (p − p∗ ))
dp
³
´
G (X, 0) =
d
b
p
b
q
(b
p
)
D (p − p∗ ) − αpb
qS (b
p) D0 (p − p∗ )
de
p
evaluated at (p∗ , pb∗ ) = (p∗ (0) , pb∗ (0)) and (p, pb) = (p∗ (0) + x, pb∗ (0) + x
b).
So,
GX (X = 0, ε = 0) =
Ã
d2
dp2
(pD (p∗ − p))|p=p∗
B
D (0) dedp∗
h
d
de
p∗
0
³
´
i
pb∗b
q (b
p∗ ) − αb
qS (b
p∗ )
where the specific value of B doesn’t matter. As G11 and G22 are < 0,
GX (X = 0, 0) is invertible. By applying Lemma 15, we conclude that there
is an ε∗ such that for ε ∈ [0, ε∗ ], (20) holds. Given σ = 1/ε, our Proposition
holds for σ ∗ = 1/ε∗ .¤
Intuitively, a firm might want to deviate from the conjectured symmetric
equilibrium (with monopoly pricing in the aftermarket) and instead charge
efficient add-on prices so it would attract the rational consumers, and capture
their associated surplus. But such a firm would lose the market of naive
consumers. This creates a drop in profit proportional to (1 − α) µ. So if µ is
large enough, a firm will not want to deviate from the symmetric equilibrium.
6
Appendix B: On the demand function D (x)
We define D (x) as the demand of a firm that offers an average perceived
surplus x units greater than the average perceived surplus provided by its
competitors. We examine here the microfoundations for D(x).
Random utility theory offers its best-known foundation (see Anderson et
al. 1992 for an excellent review). We can suppose that good i gives agent a
a decision utility equal to:
Uia = vi − pi + σ i εia
where εia are i.i.d. random variables. We normalize the mass of consumers
to 1. If the agent has reservation utility R for the good, the demand function
is:
µ
µ
¶¶
Di = P Uia ≥ max max Uja , R
j6=i
32
!
We will be looking for symmetrical equilibria where a firm posts quality v
and prices p, while the other firms post the same quality v ∗ and price p∗ ,
and all firms have the same σ. In those cases, one can set S = v − p and
S ∗ = v ∗ − p∗ the net average surplus from the good, and the demand for firm
1 is:
µ
µ
¶¶
∗
∗
D (S, S , R) = P S + σε1 ≥ max S + σ max εj , R
j=2...n
This is expressed:
∗
D (S, S , R) =
Z
∞
f (ε) dεF
(R−S)/σ
n−1
µ
¶
S − S∗
+ ε dε
σ
(21)
where f and F are respectively ε’s density and cumulative distribution functions.
A useful simplification arises when the consumer must buy one of the n
goods. Formally, this corresponds to a reservation surplus R = −∞. If this
is true, one can express
D (S, S ∗ ) = D (S − S ∗ )
for a function D (x) that has just one argument:
Z ∞
´
³x
+ ε dε
f (ε) dεF n−1
D (x) =
σ
−∞
(22)
(23)
The demand depends only on the difference S − S ∗ between the surplus S
offered by the firms, and the surplus S ∗ offered by its competitors.
If marginal costs are c and profits (p − c) D, a symmetrical equilibrium
will satisfy:
p∗ = arg max (p − c) D (v − p, v − p∗ , R)
p
∗
∗
which implies, with x = p − c,
−x∗ ∂1 D (x∗ , x∗ , R − v) + D (x∗ , x∗ , R − v + c) = 0
so that the equilibrium markup satisfies:
x∗ = p − c = µ (x∗ , R − v + c) :=
D (x∗ , x∗ , R − v + c)
∂1 D (x∗ , x∗ , R − v + c)
(24)
The value of the equilibrium mark up µ does not depend on x∗ and R in
two classes of cases
33
Proposition 16 There is a symmetrical equilibrium µ is:
µ=
D (0)
D0 (0)
and does not depend on x∗ and R, if (I) the global maximum of xD (µ − x)
is reached at x = µ, and either (II.i) R = −∞, or (II.ii)
v − c − µ + σε ≥ R
(25)
where ε is the lower bound of the support of ε.
Proof. First we consider R = −∞. If there is a symmetrical equilibrium, it
satisfy p−c = µ by equation (24). In terms, µ is an equilibrium iff xD (µ − x)
reaches its unique maximum at x = µ.
If (25) we have. For p = c + µ, and the equilibrium surplus is
Uia = v − c − µ + σεia ≥ v − µ − c + σε ≥ R
i.e. the surplus is always (weakly) greater than the reservation surplus. So
the profit function
Π (p, p∗ , R) := (p − c) D (v − p, v − p∗ , R)
satisfies
Π (c + µ, c + µ, R) = µD (0)
(26)
So for any p,
Π (p, c + µ, R) ≤ Π (p, c + µ, −∞) because Π is nonincreasing in R
≤ Π (c + µ, c + µ, −∞) by condition (I)
= Π (c + µ, c + µ, R) because of (26)
This means that p = c + µ, is the best response to p∗ = c + µ, so that
p = c + µ is an equilibrium.
Proposition 16 says that, for a low enough reservation surplus R, the
equilibrium markup µ does not depend on the marginal cost c and the valuation v for the good. For simplicity, in the body of the paper we express our
results in the sufficient cases of Proposition 16. It would be straightforward
to extend them to general reservation surplus R, but that would increase the
notational burden.
Condition (I) of Proposition 16 means that p − c = µ is an equilibrium.
The literature provides sufficient conditions for it.
34
Proposition 17 Condition (I) of Proposition 16 is satisfied if ln f is concave.
Proof: We use Caplin and Nalebuff (1991), Theorem 2 and Proposition
7. ¤
For concrete calculation, it is useful to have some closed forms. One
can get closed forms in at least 3 cases. If ε follows a Gumbel distribution
F (ε) = exp (−e−ε ), it is well known (e.g., Anderson et al. 1992) that
ex/σ
ex/σ + n − 1
n
µ =
σ
n−1
D (x) =
For the exponential distribution f (ε) = e−ε 1ε>0 , integration yields:
¡
¢n i
ex/σ h
D (x) =
1 − 1x≥0 1 − e−x/σ
n
µ = σ
In the case where ε is uniformly distributed in [−1, 1], one has:
µ=
7
2σ
.
n
Appendix C: Extension of Proposition 7 to
several shrouded attributes
We generalize the case of K shrouded attributes, indexed by k = 1...K. We
suppose that the decision utility for a consumer of a is
P
uk (b
eaik , qbaik ) − pbik qbaik ]
Uai = −p + uai − pi +
k Aak [b
where Aak ∈ [0, 1] is the amount of attention that consumer a devotes to the
attribute k. If Aak = 1, the consumer pays full attention to the good. If
Aak = 0, the consumer pays no attention to the good. In principle Aak can
take intermediate values.
Call Pi = (pi , pi1 , ..., pik ) firm i’s (K + 1) − dimensional vector made of
its price for the base good and the K add-ons. After maximization on the
preventive action ebaik , the indirect utility from the good is:
X
Ua (Pi ) = −p + max
Aak [b
uk (b
eaik , qbaik ) − pbik qbaik ]
(e
eaik )k=1...K
k
35
Consumer a chooses the good that gives him the highest prospective maxi Ua (Pi ).
If the other firms post prices P ∗ , a firm that posts prices P will have
profit:
"Ã
!
#
X
Π=E
p−c+
(b
pk − b
ck ) qbk (a) · D (Ua (P ) − Ua (P ∗ ))
(27)
k
We take the expectation on the various types of attentions Ak . For instance,21 in the case K = 1, we get expression (12).
In this setup with many attributes, we can generalize Proposition 7.
Proposition 18 If a symmetric equilibrium exists, it satisfies
∀k,
d
[(b
pk − b
ck ) E [b
qk (a, pbk )]] =
db
pk
"
E p−c+
where
X
k
#
(b
pk − b
ck ) qbk (a)
(28)
!#
Ã
"
X
1
(b
pk0 − b
ck0 ) qbk0 (a)
E Aka qbk (a) p − c +
µ
0
k
(29)
= µ.
µ = D(0)/D0 (0).
Corollary 19 If all consumers have identical demand for add-on k, and Aka
is independent of the attention paid to the other add-ons, then:
1 − E [Ak ]
ck
pbk − b
=
pbk
ηk
(30)
pk qbk0 (b
pk ) /b
qk (b
pk ) is the price elasticity of the demand for the
where η k = −b
add-on, and E [Ak ] is the average amount of attention paid to it. In particular, if consumers all overlook the add-on (Ak = 0), it is priced at monopoly
pricing, and if all consumers pay full attention to the add-on (Ak = 1), then
it is priced at marginal cost.
21
If the distribution of types is that Ak = 1 with probability αk , and Ak = 0 with
probability 1 − αk , and the draws are independent, the profit function is:
!
"
#Ã
Y
X
X
1−Ak
Ak
αk (1 − αk )
(b
pk − b
ck ) qbk (Ak ) ·D (Ua (P ) − Ua (P ∗ ))
p−c+
Π=
(A1 ,...Ak )∈{0,1}K
k
k
36
Corollary 20 In the case of (unavoidable) surcharges, the surcharge pbk is
set to its upper bound.
Condition (15) means that the overall profit remains µ, independently of
the structure of the add-ons. Hence add-ons do not create additional unit
ck ) qbk (a). We have fixed
profits, they just reallocate the profit centers (b
pk − b
the market size at 1. If add-on can change the market size, as is likely in
practice, then they would increase total profits, if not the profit on each unit.
Condition (14) is most easy to interpret in the special cases of the Corollary 19.
Corollary 19 predicts that, for the same good, the markup (actually, the
Lerner index) (b
pk − b
ck ) /b
pk of the add-ons could vary from add-on to add-on.
In fact, by observing the mark-up and the elasticity, one can in theory infer
the attention paid by consumers for the add-on. This gives the scope for a
market-based inference of the amount of attention E [Aka ].
∂
Proof. As before, we ∂p
u (P ) = −1, and ∂∂pek u (P ) = −Aka pbk . We take the
(K + 1) first order conditions in (27) at P = P ∗ . We start with the first
component, the price of the base good p.
∂Π
∂p
= E[D (ua (P ) − ua (P ∗ ))
Ã
!
X
+ p−c+
(b
pk − b
ck ) qbk (a) D0 (ua (P ) − ua (P ∗ ))]
0 =
"
Ãk
= E D (0) + p − c +
X
k
!
(b
pk − b
ck ) qbk (a) D0 (0)
so the average unit profit is:
#
"
X
D (0)
= µ.
(b
pk − b
ck ) qbk (a) = 0
E p−c+
D
(0)
k
which proves (29).
37
#
Optimization of the k-th add-on’s price gives:
0 =
∂Π
∂ pbk
d
= E[
[(b
pk − b
ck ) qbk (a)] D (0)
db
pk
!
Ã
X
(b
pk0 − b
ck0 ) qbk0 (a) Aka qbk (a) D0 (0)]
− p−c+
k0
which gives:
"
Ã
!#
∙
¸
X
d
D0 (0)
E
E Aka qbk (a) p − c +
[(b
pk − b
ck ) qbk (a)] =
(b
pk0 − b
ck0 ) qbk0 (a)
db
pk
D (0)
k0
which proves (28).
To prove the first corollary, we observe that Aka and qbk are independent
on qbk0 (a), then (28) reads:
"
#
X
1
d
E [Aka ] qbk E p − c +
[(b
pk − b
ck ) E [b
qk (b
pk )]] =
(b
pk − b
ck ) qbk (a)
db
pk
µ
k
= E [Aka ] qbk
by (29). This gives (30).
The proof of 20 is immediate.
8
Appendix D: Welfare Analysis.
This Appendix proves the welfare claims in subsection 3.6.
If consumer a buys the good in a symmetric equilibrium, he will get
surplus:
Sa (p, pb) = ua − p + u
b (b
ea , qba ) − pbqba
where ua = maxi uai . Call SaE the surplus he expects, perhaps naively. The
consumers buys the good iff SaE ≥ 0. We say Ba = 1 if the consumer buys
the good, and Ba = 0 otherwise.
In equilibrium, each firm has profits of p − c + (b
p−b
c) qba . So, if consumer
a buys the good, the transaction yields a total social surplus equal to the
consumer’s surplus and the firms’s profit
Va (p, pb) = ua − c + u
b (b
ea , qba ) − b
cqba .
38
(31)
The surplus from consumer a is, in general
Wa (p, pb) = Ba Va (p, pb) .
We define the welfare loss as the difference between the welfare under
marginal cost pricing, and the welfare under actual pricing.
c) − Wa (p, pb)] .
Λ = E [Wa (c, b
If all consumers should buy the good and all consumers do buy the good,
then welfare losses are just,
Λ = E [Va (c, b
c) − Va (p, pb)] .
This is the classic oligopolistic distortion.
For the general case in which some consumers should not buy the good
and/or some consumers do not buy the good the total welfare loss can be
expressed as,
Λ = ΛH + ΛShould have + ΛShouldn’t have,
where,
£
¤
ΛH = E 1SaE (p,ep)≥0, SaE (c,ec)≥0 (Va (c, b
c) − Va (p, pb))
is the “Harberger’s triangle” loss due to marginal consumption distortions of
the add-on;
£
¤
ΛShould have = E 1SaE (p,ep)<0, SaE (c,ec)≥0 Va (c, b
c)
is the loss due to consumers that haven’t bought the good at prices (p, pb),
but would have bought the good if it had been priced at marginal cost (c, b
c);
and
£
¤
ΛShouldn’t have = −E 1SaE (p,ep)≥0, SaE (c,ec)<0 Va (p, pb)
is the welfare loss due to naive consumers who bought the good but shouldn’t
have done so. They bought it because they mistakenly expected a high
surplus, whereas the true surplus is less than 0, even under marginal cost
pricing.
The following bound gives us a rough quantification of ΛShouldn’t have .
Proposition 21 Assume free disposal, i.e. ua + u
ba ≥ 0. Then
£
¤
cqba (b
p))
0 ≤ ΛShouldn’t have ≤ E 1SaE (p,ep)≥0, SaE (c,ec)<0 (c + b
39
(32)
i.e. the loss due to naive consumers buying a good they shouldn’t have bought,
ΛShouldn’t have , is bounded above by the amount of their purchase, valued at
marginal cost, multiplied by the mass of those naive consumers who shouldn’t
have purchased the good.
Proof. Immediate as
−Va (p, pb) = c + b
cqba − (ua + u
b (b
ea , qba )) .
It is easy to get conditions under which bound 32 is reached,22 so it is the
least upper bound for ΛShouldn’t have .
As sketched above, one expects ΛH and ΛShould have to be second order
losses, while ΛShouldn’t have is first order. We illustrate this tendency in the
following proposition.
Proposition 22 Consider the case
´
³
b
c) , µ > 0.
η = max (b
p−b
c) q (b
Let η → 0+ . We have
¡ ¢
ΛH = O η 2
¡ ¢
ΛShould have = O η 2 .
If (i) at marginal cost pricing, we have a non-zero density of naive consumers
just indifferent between buying or not buying, and (ii) η tends to 0 in such a
way that
b
q (b
c) (b
p−b
c) − µ ∼ k 0 µ
for a positive constant k 0 , then there is a constant k such that
ΛShouldn’t have = kη.
In particular,
ΛShouldn’t have À max (ΛH , ΛShould have ) .
22
For instance, take a competitive market, with u = 0, u
b = 1, b
c = 0, c = 1, pb = 1,p = 1,
and assume further it is entirely populated by naive consumers, with mass 1. The naive
consumer expects a price pb = 0, so she makes the purchase iff her valuation for the whole
experience is V ≥ 1. However, his true surplus is V − 2, so the loss is
ΛShouldn’t
One has ΛShouldn’t
have
have
=c+b
cqb = 1 if V = 1.
40
=2−V
Proof. The first two statements are classic results. They derive from the fact
that the welfare optimum is reached for η = 0, and the fact that the welfare
function is C 2 . To prove the last statement, recall that naive consumers have
a subjective surplus
SaE (p) = ua − p + u
bE .
where ua is a random variable with density f (ua ). Pivotal naive consumers
are those such that SaE (c) = 0, i.e. ua = c − u
bE . For them, equation (31)
shows that the loss is
¡ ¡ N N¢
¢
c) = L := u
bE − u
b b
e , qb − b
cqbN ,
−Va (c, b
which is the gap between the expected and realized surplus from the add-on.
So as η → 0
£
¤
ΛShouldn’t have = −E 1SaE (p,ep)≥0, SaE (c,ec)<0 Va (p, pb)
¤
£
∼ LE 1SaE (p,ep)≥0, SaE (c,ec)<0
¢
¡
bE
= LP p − u
bE ≤ ua < c − u
¡
¢
∼ Lf c − u
bE (c − p)
¡
¢
= Lf c − u
bE k 0 µ
and the result holds with k = Lf (c − u
be ) k 0 .
So the largest deadweight losses come from naive consumers who are lured
into the market because they see the low price for the base good but not the
high price of the add-ons. This is an additional reason why naiveté matters
for welfare analysis.
9
References
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Alexander, Gordon, Jonathan D. Jones, & Peter J. Nigro (1998). “Mutual
fund shareholders: Characteristics, investor knowledge, and sources of
information.” Financial Services Review, 7, 301-16.
Anderson, Simon, André de Palma and Jacques-Francois Thisse, Discrete
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