1. society
2. welfare

Naivete-Based Discrimination∗
Paul Heidhues
ESMT
Botond K˝oszegi
Central European University
April 20, 2015
Abstract
We initiate the study of naivete-based discrimination, the practice of conditioning offers on
external information about a consumer’s naivete. We identify a broad class of situations in which
such discrimination lowers social welfare under a weak condition. In our primary example, a
credit market with time-inconsistent borrowers, firms lend more than socially optimal to increase
profits from naive borrowers’ unexpected eagerness to put off repayment. Information about
consumer naivete leads firms to (inefficiently) differentiate the extent of overlending according to
naivete as well as to (also inefficiently) raise total lending—so that naivete-based discrimination
always lowers welfare. Because the overlending distortion is the same whether or not firms
observe beliefs, information about a consumer’s beliefs has at most distributional implications,
while information about naivete given beliefs always strictly decreases total welfare. We show
that naivete-based discrimination follows a closely related logic in many other applications,
including models of bank accounts and mobile phones, where the distortion from exploiting
naivete falls homogenously on naive and sophisticated consumers. We also point out important
settings in which the distortion differs across types, and identify the effect of information about
naivete in some of those cases.
Keywords: sophistication, naivete, first-degree price discrimination, third-degree price discrimination
∗
We thank Dan Benjamin, Fabian Herweg, Michael Grubb, Takeshi Murooka, Rani Spiegler, and numerous seminar
and conference audiences for helpful comments. Financial support from the European Research Council (Starting
Grant #313341) is also gratefully acknowledged. Part of the research on this project was carried out while Heidhues
was visiting the Institute for Advanced Studies at CEU, whose hospitality he gratefully acknowledges.
1
1
Introduction
What is the welfare effect of firms knowing more about consumers? The classical approach to
answering this old question in economics, in the literatures on (first- and third-degree) price discrimination and privacy, typically presumes that the information firms acquire about consumers
pertains to preferences. Then, the welfare effect of information is in general ambiguous, but as
a notable extreme case, perfect information always maximizes social welfare given the number
of firms in the market. A growing literature in behavioral economics, however, documents that
many consumers are naive about key fees or product features, raising the possibility that some
of the information firms acquire about consumers pertains to naivete—and hence firms engage in
naivete-based discrimination. This possibility is especially relevant since most models in behavioral
industrial organization imply that firms have incentives to differentially target profitable naive consumers, and the more and more detailed information they collect about consumers is likely useful
in predicting who is prone to be naive.
In this paper, we initiate the study of naivete-based discrimination by identifying its welfare
effects in a number of key canonical environments. We delineate a broad class of settings in which
the welfare effect is under weak conditions unambiguously negative. In our primary example, a
credit market with time-inconsistent borrowers, improving lenders’ information about borrower
naivete always lowers welfare. Hence, while perfect preference-based discrimination leads to the
best possible outcome, perfect naivete-based discrimination leads to the worst possible outcome.
Although the underlying type of naivete may be different, we demonstrate that naivete-based
discrimination follows a closely related logic in any setting in which the distortion from exploiting
naivete falls equally on naive and sophisticated consumers. This class of settings includes variants
of many types of naivete that have been identified in the literature for specific markets, including
bank accounts, mobile phones, and hotels. We also characterize the effect of information about
naivete for some important settings outside this class.
Section 2 presents our model of the credit market, which is a simplified Hotelling-type variant
of the model in Heidhues and K˝
oszegi (2010). Two lenders interact over three periods, 0, 1, and 2,
with consumers whose utility from borrowing satisfies prudence. In period 0, each lender makes an
1
offer consisting of a loan amount, interest rate, and discount—e.g., cash back or other credit-card
perk—to consumers. If a consumer takes a loan, she decides in period 1 how much of it to repay
then, and how much to put off (at the chosen firm’s interest rate) to period 2. Borrowers have
time-inconsistent preferences derived from hyperbolic discounting. Capturing the notion that at
the time of accepting the credit offer both consumption and repayment are things of the future,
we assume that self 0 puts the same weight on the utility from borrowing and the disutilities of
repayment in periods 1 and 2. Self 1, in contrast, downweights the disutility in period 2 relative to
that in period 1 by a factor β ≤ 1. Following much of the literature on time inconsistency, we take
the long-run perspective and equate welfare with self 0’s preferences. Self 0 believes that she will
discount period 2 by a factor βˆ ≥ β, so that βˆ represents her beliefs about β.
In Section 3.1, we identify the properties of equilibrium when βˆ is known to firms, and there
ˆ sophisticated (β = β)
ˆ and naive (β < β).
ˆ Similarly to the
are two possible β types for each β,
logic of Heidhues and K˝
oszegi (2010), firms choose the interest rate so that borrowers expect to
repay their loans in period 1 but naive consumers just put off repayment to period 2, maximizing
the unexpected interest naive consumers pay per dollar of borrowing. And in order to increase the
total amount of unexpected interest collected, in equilibrium lenders extend more credit than is
socially optimal.
To understand naivete-based discrimination, we compare social welfare when firms face a single
pool of consumers to that when firms are able to sort consumers into two pools with different
shares of naive individuals. Firms respond by raising inefficient lending to consumers more likely to
be naive and lowering inefficient lending to consumers more likely to be sophisticated. In Section
3.2, we show that for two reasons the former effect always dominates the latter effect, so that
social welfare always decreases. First, because the utility function is concave, it is inefficient to
lend different amounts to different consumers. Second, total lending, which was inefficiently high
to start with, increases. Intuitively, for a prudent consumer the risk of which pool she will be
allocated to—and therefore how much she will borrow—increases the expected marginal utility of
consumption, leading to a greater average willingness to borrow.
In Section 3.3, we show that the negative effect of naivete-based discrimination is only reinforced
2
when there are multiple naive types who differ in their time inconsistency β. Then, information
about the distribution of naive types allows a firm to reoptimize its interest rate and collect more
in unexpected interest from naive consumers per dollar of borrowing. This increases the firm’s
incentive to overlend, lowering welfare.
In Section 4, we consider an extension of our model in which both β and βˆ are unknown to
firms, and firms can offer contract menus to induce self-selection among consumers with different
beliefs. This exercise goes beyond the common practice in the classical literature not to combine
third-degree price discrimination with screening. For this model, we assume that in addition to
interest, a firm can introduce fixed penalties. We show that in equilibrium all consumers borrow
the same amount and pay the same interest as when firms know beliefs—although the distribution
of discounts may be different. While the construction is intricate, for a rough intuition suppose that
there are two belief types, βˆ1 and βˆ2 > βˆ1 , and denote their perceived utilities from the equilibrium
contracts when firms know beliefs by u
ˆ1 and u
ˆ2 , respectively. If u
ˆ2 ≥ u
ˆ1 , then the same contracts are
incentive compatible and hence constitute an equilibrium even when firms do not know beliefs: an
optimistic consumer does not want to take the lower-utility contract 1, and because a pessimistic
consumer is worried she will pay too much in interest or penalties, she does not want to take
contract 2. If u
ˆ2 < u
ˆ1 , in contrast, the contracts that firms offer when they know beliefs are not
incentive compatible: the optimistic consumer—unworried about interest and penalties—prefers
the higher-utility contract 1. The equilibrium pair of contracts, then, redistributes the discount
toward the more optimistic type until the two types’ perceived utilities are equalized. Although
pessimistic consumers are then more profitable than optimistic consumers, if a firm attempted to
attract a greater number of pessimistic consumers by improving on their contract, it would also
attract a greater number of optimistic consumers, making the deviation unprofitable.
The above characterization has several implications for the welfare consequences of lender information. Since both with and without information the level of borrowing and hence the distortion is
the same as when firms know beliefs, our results for the latter case immediately imply that lender
information still (weakly) lowers welfare. For the same reason, total welfare does not depend on
what firms know about beliefs. Rather, it is specifically information about the likelihood that a
3
consumer with given beliefs is naive—that is, the likelihood that her future behavior will deviate from what she expects—that lowers welfare. Accordingly, for welfare to decrease it is neither
necessary, nor sufficient for firms to receive information about consumers’ time inconsistency β.
In Section 5, we move beyond the credit market and offer a partial characterization of how
information about consumer naivete affects welfare in different economic environments, restricting attention to our basic setting with known beliefs. We argue that naivete-based discrimination
tends to lower welfare whenever the exploitation of naivete generates the same, “homogenous” distortion on the naive and sophisticated sides of the market. To make our argument, we introduce
a reduced-form model in which firms choose—in addition to a transparent “up-front price”—an
“additional price” a that naive but not sophisticated consumers unexpectedly pay, and that generates a welfare loss k(a) on each consumer. We show that if an arguably weak condition on k(·) is
satisfied, information about consumer naivete lowers welfare. Our credit-market model simplifies
to this reduced-form model, with the additional price being the unexpected interest paid by naive
consumers, and the welfare cost being the cost of overlending. Although the source of distortion
and the type of naivete may be different, many other settings have the same reduced-form structure. We formally outline one such setting, and discuss others. In our alternative model, naive
consumers underestimate their demand for an additional service such as overdraft protection or
roaming, firms exploit this mistake by charging a high price for the service, and this induces all
consumers to undertake socially inefficient ex-ante steps to avoid the service.
The welfare effect of naivete-based discrimination, however, is different when the exploitation
of naivete generates distortions that are heterogeneous across naive and sophisticated consumers.
If the welfare cost arises only on the sophisticated side of the market—as for instance when all
consumers expect to pay costly attention to avoid overdrafting, but only sophisticated consumers
do—then perfect information about consumers maximizes welfare. Intuitively, if a firm knows that
a consumer is sophisticated and hence cannot be exploited, then it does not charge an overdraft
fee, so no distortion arises. And if the firm knows that the consumer is naive, then it can charge
a high overdraft fee without triggering inefficient avoidance. Nevertheless, we show that if the
share of naive consumers is small, then a small amount of information about naivete tends to
4
lower social welfare. And if the welfare cost arises only on the naive side of the market—as for
instance when collecting the additional price distorts naive consumers’ behavior—then naivetebased discrimination has no effect on welfare. Intuitively, since naive consumers do not realize the
distortion when contracting, firms do not care about it either, so that—independently of the share
of naive consumers—firms maximize the profits from the additional price ex post.
In Section 6, we relate our paper to the empirical and theoretical literatures on markets with
naive consumers and on price discrimination and privacy. While direct evidence that firms condition offers on information about naivete is limited, we argue based on theoretical and empirical
considerations that this is or will soon be a major issue—yet to our knowledge no paper has systematically addressed how it affects welfare. Also recognizing that firms may want to discriminate
between consumers of different sophistication, some authors have began to analyze how firms can
design contract menus to induce self-selection of consumers according to their naivete (Eliaz and
Spiegler 2006, 2008, Heidhues and K˝
oszegi 2010, for example). In contrast to this “second-degree
naivete-based discrimination,” we ask how firms’ access to outside information about consumer
naivete affects welfare.
While our paper covers many of the relevant cases, it is just a first step in understanding the
welfare effects of naivete-based discrimination. In Section 7, we point out some important aspects
of markets with naive consumers missing from our framework. Most crucially, the different types of
distortions generated by the exploitation of naive consumers—those that fall on both sides, on the
sophisticated side, and on the naive side—interact, and we do not have a general handle on how these
interactions depend on information about naivete. In addition, the naivete-based discrimination
we analyze is likely to occur simultaneously with classical preference-based discrimination, and the
interaction of these considerations requires further research.
2
A Model of Information in the Credit Market
In this section, we introduce our model of a credit market with sophisticated and partially naive
time-inconsistent consumers. Our model is a variant of that in Heidhues and K˝oszegi (2010), which
in turn builds on previous work by DellaVigna and Malmendier (2004) and Eliaz and Spiegler
5
(2006). While the logic of the equilibrium contracts signed by consumers is similar to that in these
previous papers, we move beyond the literature in asking how information about consumers affects
welfare, and in fully characterizing equilibrium in a setting where firms observe neither beliefs nor
time inconsistency.
We use a Hotelling-type model of imperfect competition. Two risk-neutral profit-maximizing
lenders are located at the endpoints of the unit interval representing consumer tastes, and borrowers
are uniformly distributed on the interval. Borrowers have an outside option with perceived and
actual utility 0. If a borrower located at y ∈ [0, 1] chooses lender n ∈ {0, 1}, she incurs a disutility
(or “transportation cost”) of t|y − n|, where t is the product-differentiation parameter.
The lenders interact with consumers over three periods. In period 0, firm n makes an offer
consisting of a loan amount bn , interest rate rn , and discount dn to consumers. The discount
could, for instance, capture airline miles, cash back, or other credit-card perks. If a consumer takes
firm n’s loan, then in period 1 she chooses an amount q ∈ [0, bn ] to repay in that period, leaving
(bn − q)(1 + rn ) to be repaid in period 2.1 Firms’ fixed cost of serving a consumer is zero, and they
acquire funds at zero interest.
Borrowers have time-inconsistent preferences derived from hyperbolic discounting a la Laibson
(1997) and O’Donoghue and Rabin (1999). Self 0 has utility u(bn )−q−(bn −q)(1+rn )+dn −t|y−n|,
where u(·) is the gross utility from funds. We assume that self 0 does not discount the cost
of repayment relative to the utility from consumption because at the time of accepting a credit
offer, typically both are things of the future.2 Furthermore, by assuming that the disutility from
repayment is linear, we isolate the overlending distortion generated by the exploitation of consumer
naivete,3 and avoid a setting in which non-linear pricing of repayment schedules is optimal, a
consideration almost never studied in the context of third-degree price discrimination. We suppose
that u(·) is differentiable, u(0) = 0, u0 (b) > 0 and u00 (b) < 0 for all b ≥ 0, limb→∞ u0 (b) = 0, and
1
Consistent with the observation that most credit and store cards offer grace periods, we assume that no interest
accrues between periods 0 and 1; our results would be identical if interest also accrued before period 1.
2
Our model also assumes that dn is paid at the ex-post stage (entering utility outside u(·)) rather than at the
ex-ante stage (where it would enter inside u(·)). In Section 3.1, we provide an endogenous reason for this specification.
3
If the cost of repayment is non-linear, then additional welfare costs arise because welfare depends not only on
the total repayment amount, but also on how that amount is distributed between periods 1 and 2. Although we
have not been able to derive a full characterization of the welfare effects of information in this case, in Section 5.3 we
discuss how our results may be modified.
6
u0 (0) ≥ 1. We also assume that consumers are prudent (u000 (b) ≥ 0 for all b ≥ 0), which is generally
considered to be a weak assumption on a consumption-utility function.
In contrast to self 0, self 1 discounts payments in period 2 relative to period 1 by a factor β ≤ 1,
choosing q to minimize q + β(bn − q)(1 + rn ). To ensure an interior solution to a firm’s lending
problem, we assume that β > 1/2.4 Following much of the literature on time inconsistency, we
take the long-run perspective and equate welfare with self 0’s utility.5 Nevertheless, our results
that firms overlend and do so more with information suggest that naivete-based discrimination also
lowers the welfare of selves 1 and 2—the selves who have to repay the oversized loans.6 Given our
welfare measure, the efficient level of borrowing, be , satisfies u0 (be ) = 1.
Following O’Donoghue and Rabin (2001), we assume that in period 0 a consumer has point
ˆ n−
beliefs βˆ about her future β; that is, she believes that self 1 will choose q to minimize q + β(b
q)(1 + rn ). A consumer chooses a contract or the outside option to maximize her perceived utility,
given her prediction about her own future behavior. In general, we think of both βˆ and β as
heterogeneous and potentially unobservable to firms, but in our basic model we assume that βˆ is
ˆ there are two types: sophisticated—who
known to firms. Within a set of consumers with beliefs β,
ˆ 7 The share of
are correct about their future preferences—and naive—who have a given β < β.
naive consumers is α, and consumers’ tastes y are independent of their naivete.
Our specification of firms’ initial knowledge about consumers, whereby they know a consumer’s
4
If β < 1/2, then a borrower is willing to pay more than a 100% interest to delay repayment. Our analysis implies
that if all consumers are naive, a lender can then give a discount equal to the loan amount and still run a profit, so
that borrowing more seems both costless to the consumer and profitable to the lender. This implies that a lender
can always increase profits by increasing the loan amount.
5
Earlier work that takes the same long-run approach to welfare includes Gruber and K˝
oszegi (2004), DellaVigna
and Malmendier (2004), O’Donoghue and Rabin (2006), and Galperti (forthcoming). As we explain in Heidhues
and K˝
oszegi (2010), although we simplify things by considering a three-period model, in reality time inconsistency
seems to be mostly about very immediate gratification that plays out over many short periods. Hence, arguments
by O’Donoghue and Rabin (2006) in favor of a long-run perspective apply: in deciding how to weight any particular
week of a person’s life relative to future weeks, it is reasonable to snub that single week’s self—who prefers to greatly
downweight the future—in favor of the many earlier selves—who prefer more equal weighting. This is especially so
since the utility of consumption from borrowing on a credit card is distributed over many weeks. In addition, the
models in Bernheim and Rangel (2004a, 2004b) can be interpreted as saying that a taste for immediate gratification
is often a mistake not reflecting true welfare.
6
At the same time, analyzing the utilities of selves 1 and 2 would require us to specify more precisely when the
utility u(bn ) is realized, something we prefer to remain agnostic about.
7
Our presentation abuses notation somewhat in that β first refers to a generic short-term discount factor, and
here—as well as in what follows—it denotes naive consumers’ specific discount factor.
7
beliefs βˆ but not her time inconsistency β, differs from most previous specifications in the literature
(Eliaz and Spiegler 2006, 2008, for instance), where firms know ex-post preferences but not beliefs.
As shown by previous work, in the latter case firms’ optimal response is often to induce selfselection among consumers with different beliefs. But because consumers with the same beliefs and
preferences choose from any contract menu in the same way, it is impossible to induce self-selection
among them. Hence, our approach of assuming known beliefs is consistent with the common practice
in the literature of not combining third-degree price discrimination with screening. In addition,
since our main question is the effect of discrimination based on external information rather than
self-selection, focusing on an unscreenable pool seems most relevant. Consistent with this view, in
Section 4 we consider a model of the credit market in which both βˆ and β are unknown to firms,
and show that because consumers self-select according to their beliefs, the welfare implications of
naivete-based discrimination are exactly the same as when βˆ is known to firms.
Throughout, we solve for symmetric pure-strategy Nash equilibria of the contract-offer game
played between firms, assuming that firms correctly predict consumers’ behavior. To ensure that
the market is covered—i.e., all consumers purchase in equilibrium and therefore firms effectively
compete with each other—we impose that u(be ) − be > 3t/2. We define social welfare as the sum
of firms’ profits and the population-weighted sum of naive and sophisticated consumers’ utilities.
Our main interest is the welfare effect of naivete-based discrimination, which we think of as using
information about α to design credit contracts. Without discrimination, firms believe α = αns .
With discrimination, firms get—and can condition offers on—a common binary signal that changes
their beliefs to either α = αn > αns or α = αs < αns , but provides no information about consumers’
tastes y. In our preferred interpretation, firms get signals that are independent across consumers—
sorting the population into two pools with different shares of naive consumers—although the welfare
effects are the same if firms receive a single signal on the aggregate share of naive consumers. A
theoretically relevant extreme case is perfect naivete-based discrimination, where αn = 1 and
αs = 0. Note that we have an environment with a type of two-part tariff—by adjusting d, a firm
can extract any surplus it generates for a consumer—and in such a setting classical perfect price
discrimination maximizes welfare.
8
3
Lender Information Lowers Welfare
In this section, we analyze the welfare effect of naivete-based discrimination.
3.1
Characterizing the Equilibrium Contract
To characterize the effect of information about α, we first solve for the properties of equilibrium
for a given α. We break down firm n’s problem into two parts. First, given any u
ˆn , the firm solves
for the profit-maximizing contract that gives consumers a perceived utility gross of transportation
costs of u
ˆn . Second, the firm chooses u
ˆn .
Part 1. Notice that the highest interest rate at which naive consumers are willing to put off
repayment to period 2 satisfies β(1 + rn ) = 1, or rn = (1 − β)/β. If this is the interest rate, then
consumers—who all have beliefs βˆ > β—predict in period 0 that they will repay the entire loan bn
in period 1, and sophisticated consumers will actually do so. We continue assuming for a moment
that this interest rate—which maximizes the unanticipated interest paid by consumers—is optimal,
and argue below that it indeed is.
Given that the interest rate is set at rn = (1 − β)/β, a profit-maximizing contract offering
perceived utility u
ˆn must solve
1−β
max α bn +
· bn + (1 − α)bn −dn − bn
bn ,dn
β
{z
}
|
actual repayment
subject to u(bn ) − bn +dn = u
ˆn .
|{z}
anticipated repayment
While the firm’s profit includes the interest collected from naive consumers, since consumers believe
that they will repay early, the same interest does not appear in the constraint. Solving the constraint
for dn and plugging into the maximand yields the reduced problem
max u(bn ) − bn + α ·
bn
| {z } |
surplus
1−β
· bn −ˆ
un .
β
{z
}
(1)
unanticipated interest
The firm maximizes the sum of the social surplus from lending—the utility from funds net of the
cost of funds—and unanticipated interest—the difference between the total payment consumers
9
will make and how much they think they will make. Equation (1) implies that the equilibrium level
of borrowing, b∗ , satisfies u0 (b∗ ) = 1 − α(1 − β)/β independently of u
ˆn .
The logic of our derivation makes clear that it is not optimal for the firm to induce sophisticated
consumers to also pay interest on the loan. If the firm did so, consumers would anticipate paying
interest, negatively impacting the constraint. In particular, the unanticipated interest in the firm’s
profit in Expression (1) would disappear, lowering profits for any u
ˆn .
Part 2. If firms 0 and 1 offer perceived gross utilities u
ˆ0 and u
ˆ1 , respectively, and both firms
have positive demand, then the consumer who is is indifferent between them has taste y satisfying
u
ˆ1 − ty = u
ˆ2 − t(1 − y). This implies that firm 0’s demand is y = (ˆ
u1 − u
ˆ2 + t)/(2t), and that
∂y/∂dn = 1/(2t). Using this to solve for the equilibrium discount gives:
Lemma 1. The contract (b∗ , r∗ , d∗ ) firms offer in a symmetric pure-strategy equilibrium satisfies
u0 (b∗ ) = 1 − α(1 − β)/β, r∗ = (1 − β)/β, and d∗ = αr∗ b∗ − t. Sophisticated consumers’ utility (and
all consumers’ perceived utility) is increasing in α, while social welfare is decreasing in α. The
firms’ equilibrium profits are t/2 for any α.
From the perspective of efficiency and the welfare effect of information, the key implication
of Lemma 1 is that firms overlend (b∗ > be ). Intuitively, in order to increase the profits from
the unanticipated payments naive consumers make, firms induce consumers to borrow more than
is socially optimal. Although beyond the efficient level a consumer’s willingness to pay for a $1
increase in consumption is less than the cost of funds ($1), since naive consumers will also pay
unanticipated interest, extending the extra credit is profitable for the firm.
Although naive consumers are more profitable than sophisticated consumers, Lemma 1 also says
that an increase in the share of naive consumers leaves firms’ equilibrium profits unchanged. The
reason is that an increase in ex-post profits leads firms—much like a common cost reduction does
in other Hotelling-type settings—to compete more aggressively for consumers, exactly offsetting
the extra profits.
To conclude this section, we note some potentially important implications of our analysis for
how firms want to pay the discount d. Since u0 (b∗ ) < 1, the consumer derives greater utility from—
and hence firm n prefers to disburse—dn at the ex-post stage (entering utility outside u(·)) rather
10
than at the ex-ante stage (where it would enter inside u(·)). In addition, to avoid lowering the
unexpected interest it receives, firm n wants to pay dn in a way that does not lower the borrower’s
interest-bearing balance due. These insights both provide an endogenous reason for our (exogenous)
specification of how firms pay the discount, and are consistent with many forms of real-life creditcard perks. Most credit-card perks are available only some time after purchase, and hence cannot
be used to augment the purchase itself.8 In addition, possibly as an attempt to avoid decreasing
the amount due, many credit-card companies dole out perks in goods (e.g., airline miles) that
consumers may value less than the cash equivalent. And even when the perk is in the form of cash
back, issuers use a variety of incentives to encourage borrowers to use it for purchases rather than
debt repayment.
3.2
The Welfare Effect of Lender Information
We now turn to our main result in this section:
Proposition 1. For any αns , αn , αs , naivete-based discrimination strictly lowers welfare.
In response to information, firms increase the extent of overlending to the pool with more naive
consumers—thereby lowering social welfare—and lower the extent of overlending to the pool with
more sophisticated consumers—thereby raising social welfare. But for two reasons the former effect
outweighs the latter effect, so information always lowers total welfare. The intuition is in two parts.
First, because u(·) is concave, an increase in lending to a consumer hurts social welfare more than
an equal decrease in lending to a consumer raises social welfare. Second, the total amount of
lending—too high to begin with—increases. In equilibrium, the marginal utility of consumption
equals the marginal cost of funds net of naive consumers’ unanticipated interest payments. For
a prudent consumer, the risk of which pool she will be allocated to—and therefore how much she
will borrow—increases the expected marginal utility of consumption relative to that with average
consumption. Because the same risk does not change the expected marginal cost of lending a dollar,
total lending increases.9
8
9
An exception to this regularity is free rental-car insurance.
If the cost of repayment is non-linear, then risk also affects the expected marginal cost of lending. Because
11
Although our main interest is in the effect of seller information on the total welfare of naive
and sophisticated consumers, it is worth considering briefly their individual welfare.10 We do so
for perfect information:
Proposition 2 (Effect of Perfect Information on Individual Consumer Types). For any u(·), αns , β,
perfect naivete-based discrimination (αs = 0, αn = 1) strictly lowers the welfare of sophisticated
consumers. For any u(·) and αns , if β is sufficiently close to 1/2, then perfect naivete-based
discrimination strictly lowers the welfare of naive consumers.
We begin with the simpler case of sophisticated consumers. In a competitive equilibrium the
ex-post profits firms make from naive consumers are redistributed to all consumers ex ante, so
that—similarly to the logic of Gabaix and Laibson (2006) and the literature following it—without
price discrimination naive consumers in effect cross-subsidize sophisticated ones. Since perfect
discrimination eliminates the cross-subsidy, it hurts sophisticated consumers.
The effect of perfect information on naive consumers’ welfare is more complicated. On the one
hand, since perfect information eliminates the cross-subsidy from naive to sophisticated consumers,
it benefits naive consumers. On the other hand, perfect information leads firms to increase inefficient
lending, hurting naive consumers. The net effect is in general ambiguous, and the latter effect
dominates if and only if lending is sufficiently sensitive to information about naivete. This is the
case if β is sufficiently close to 1/2.
The above cross-subsidy from naive to sophisticated consumers is a major theme in the existing
literature, leading researchers to conclude that the main effect of many policies is redistributional
in nature (Armstrong and Vickers 2012, for instance). Proposition 2 says that lender information
about consumers can hurt both naive and sophisticated consumers, so that the welfare effect of a
policy banning the use of information is not merely redistributional.
repayment is in that case split over multiple periods, however, one would expect the curvature of the consumption
benefit to matter more than the curvature of the repayment cost, so that prudence still tends to imply that information
increases lending. We confirm this intuition in Section 5.3.
10
Recall that by Lemma 1, naivete-based discrimination does not affect firms’ profits in equilibrium.
12
3.3
Multiple Naive Types
In this section, we allow for multiple naive types, showing that this only reinforces the negative
effect of lender information on social welfare.
We continue to assume that firms know borrowers’ beliefs, and focus on a pool with the same
ˆ Within this pool, the actual time inconsistency β takes one of I values, β1 through βI ,
beliefs β.
ˆ The fraction of consumers with time inconsistency βi or less is αi . To
where β1 < · · · < βI = β.
avoid uninteresting indifferences, we assume that the values ρi ≡ αi (1 − βi )/βi are all distinct. We
denote the vector (α1 , . . . , αI−1 ) of proportions describing the population by α.
The values ρi play a central role in the structure and implications of equilibrium. If a firm
chooses the interest rate ri ≡ (1 − βi )/βi for i < I, then borrowers believe that they will repay in
period 1, yet a share αi will repay in period 2, generating ρi in unexpected interest payments per
dollar lent. Then, defining i∗ ≡ arg maxi<I ρi , in a Nash equilibrium the firm sets interest rate ri∗ :
this maximizes unexpected interest and hence the firm’s profits while leaving consumers’ perceived
utility from the contract unaltered. This leads to a simple characterization of equilibrium:
Lemma 2. In a symmetric pure-strategy equilibrium, firms lend an amount satisfying u0 (b∗ ) =
1 − ρi∗ and set interest rate r∗ = (1 − βi∗ )/βi∗ .
Now we are ready to consider the effect of lender information. Suppose that lenders receive
information about α, sorting consumers into two groups with α1 and α2 , respectively. We define
the values ρ1i and ρ2i from α1 and α2 analogously to ρi above, and assume that these values are also
all distinct.
Proposition 3. For any α, α1 , α2 , naivete-based discrimination lowers total social welfare. If
αi1∗ 6= αi2∗ , i∗ 6= arg maxi ρ1i , or i∗ 6= arg maxi ρ2i , then naivete-based discrimination strictly lowers
total social welfare.
Part of the reason that information about consumers lowers welfare is the same as in our basic
model with a single naive type: even holding the interest rate fixed, if firms receive information
as to the proportion of consumers sufficiently naive to pay this interest rate, they lend different
amounts to the two pools and also increase total lending, lowering social welfare. In addition,
13
information might allow firms to collect more unanticipated interest per dollar lent by choosing
their interest rate better. If so, this further increases firms’ incentive to overlend and leads to an
additional reduction in welfare.
4
Unobservable Beliefs
In this section, we consider the effect of naivete-based discrimination in a generalization of our
model in which firms do not know consumers’ beliefs. This exercise goes beyond the common
practice in classical economics, where almost no study combines third-degree price discrimination
with screening.11 We show that our main result that seller information lowers welfare survives, and
also discuss exactly what kind of information lowers welfare.
Formally, there are I possible consumer beliefs, βˆ1 through βˆI , where βˆ1 < βˆ2 < · · · < βˆI . For
each belief type βˆi , there are two possible actual levels of β, β = βˆi and β = βi < βˆi .12 We denote
the share of belief type βˆi by si , and the share of naive consumers among those with belief type βˆi
by αi . We assume that |αi − αj | < 1 for any i, j; this rules out the likely unrealistic scenario in
which a firm cannot distinguish two belief types in a pool, yet knows that one belief type is certain
to be sophisticated and the other belief type is certain to be naive.
For this version of our model, we posit a somewhat different contract space than above: we
assume that in addition to specifying bn , rn , dn , firm n can impose capped penalties or charges
(e.g., late or over-the-limit fees) of a specific type. Namely, the firm sets ∆n , pn satisfying ∆n ≥ 0,
0 ≤ pn ≤ p, with self 1 deciding—on top of how much of her loan to repay in period 1—whether
to pay an extra ∆n in period 1 or an extra ∆n + pn in period 2. As we discuss in more detail in
Section 4.1, the role of these penalties is to ensure that a consumer with a lower βˆ does not take a
ˆ but for many parameter values our main result
contract intended for a consumer with a higher β,
holds even without the penalties.
We look for symmetric pure-strategy equilibria in the contract-offer game played between firms,
11
The only exception we are aware of is Bergemann, Brooks and Morris (2015). Studying the robustness of their
results on third-degree price discrimination, they consider the implications of second-degree price discrimination
when there is a small amount of non-linearity in consumer preferences and hence the returns to second-degree price
discrimination are small.
12
At the cost of some extra notation, it is possible to extend our results to more naive types per βˆi .
14
assuming that each firm can offer any finite contract menu. For tractability, we assume that a firm
can break consumers’ indifference between contracts at will: it can assign each belief type to the
contract it prefers among the ones that maximize the type’s perceived utility.
4.1
Characterization of Equilibrium
In this subsection, we characterize the equilibrium of our model for given si , αi . While we think
of this construction as a major contribution of our paper, readers not interested in the theoretical
details can jump to Section 4.2, where we identify the implications of our characterization for the
welfare impact of naivete-based discrimination.
We start by noting that when beliefs are observable, the possibility of imposing a penalty p
does not change equilibrium social welfare.13 The penalty has a convenient flexibility property: for
any β, a firm can choose ∆ such that a consumer with time inconsistency β is indifferent between
ˆ the firm can choose ∆ such that naive but
paying and not paying a penalty of p. Hence, for any β,
not sophisticated consumers pay a penalty of p, thereby collecting an unexpected payment of p from
naive consumers. Since lending more does not increase the penalty firms can collect, the penalty
does not affect the amount firms lend in equilibrium, and hence does not affect total welfare.14
We illustrate the logic of our results for unobservable beliefs by discussing why the equilibrium
we find is an equilibrium; Proposition 4 below establishes that no other equilibrium exists. Our
construction hinges on different types’ equilibrium perceived utilities gross of transportation costs.
We define u
ˆi as type βˆi ’s equilibrium perceived utility when beliefs are observable, and ui as type
βˆi ’s equilibrium perceived utility when beliefs are unobservable, thinking of u
ˆ and u as functions
from {1, . . . , I} to R. For any set of belief types I, we let the average perceived utilities of those
types be u
ˆI and uI in the two cases.
First, we make an important observation: u is increasing. Notice that if a belief type βˆi expects
to repay a given loan early and/or not pay penalties, then so does a belief type βˆj > βˆi . This
implies that type βˆj expects to receive at least as much utility out of any contract as does type βˆi .
13
See the proof of Proposition 4 for a formal argument.
Nevertheless, the possibility of imposing penalties does change the distribution of welfare by increasing the transfer from naive to sophisticated consumers. Furthermore, in equilibrium ∆ is indeterminate, but this indeterminacy
is irrelevant because a higher ∆ transfers one-to-one into a higher discount.
14
15
Hence, ui > uj is incompatible with equilibrium, since type βˆj would prefer to deviate and take
the contract type βˆi is taking.
Now we argue that if u
ˆi is weakly increasing, then the contracts firms offer when they know
beliefs are incentive compatible and hence constitute an equilibrium even when firms do not know
beliefs. If type βˆi takes the contract intended for type βˆj < βˆi , she expects to behave the same
way as type βˆj expects to behave—to repay the loan in period 1 and pay no penalty. Hence, the
two types have the same perceived utility from the contract intended for type βˆj . But since type
βˆi ’s perceived utility from the contract intended for her is higher, she prefers not to deviate in this
direction.
Conversely, type βˆi is tempted to take a contract intended for type βˆj > βˆi , since if she can
avoid interest and penalties, she obtains higher utility. But due to the above flexibility property, a
firm can design the penalty in the contract intended for type βˆj such that type βˆi expects to pay
it while type βˆj does not, making the contract unattractive to type βˆi .
The more difficult case arises when u
ˆi > u
ˆi+1 for some i. Then, the contracts consumers get
when beliefs are observable violate the constraint that ui+1 ≥ ui —thereby violating type βˆi+1 ’s
incentive-compatibility constraint—and hence cannot (all) be chosen in equilibrium. Through
changing the transfers d, however, a firm can “iron out” decreasing parts of u
ˆi to create an increasing ui and thereby eliminate more optimistic types’ incentive to take contracts intended for
more pessimistic types. If there are two types and u
ˆ1 > u
ˆ2 , for instance, the firm can lower the
discount to βˆ1 and increase the discount to βˆ2 to equalize the perceived utilities of the two types,
while holding the average markup constant. Then, type βˆ2 no longer prefers to take the contract
intended for type βˆ1 .
As it happens, this set of contracts constitutes a symmetric equilibrium. First, because it
maximizes consumers’ average perceived utility given the average markup, a firm cannot do better
by trying to attract the two types in equal numbers. Second, although type βˆ1 is more profitable
than type βˆ2 , a firm cannot disproportionately attract these profitable consumers: because type βˆ2
derives at least as much perceived utility from any contract as does type βˆ1 , any contract the firm
offers to attract more βˆ1 types also attracts more βˆ2 types.
16
An interesting aspect of equilibrium when ironing is necessary is that some contracts are subsidized and hence may generate negative profits for a firm. Continuing with the two-type example,
if u
ˆ1 > u
ˆ2 and t is sufficiently small—that is, competition is sufficiently intense—then in the symmetric equilibrium firms lose money on type βˆ2 . The question arises why firms voluntarily offer
money-losing contracts. In particular, why does a firm not pull its unprofitable contract? If it did
so, its type-βˆ2 consumers would take its other contract and generate higher losses for the firm.
To construct a symmetric equilibrium with more than two types, we define the notion of ironing
we need in general:
Definition 1. The function u is an admissible ironing of u
ˆ if (i) it is weakly increasing; and
(ii) for any maximal set {i, . . . , i0 } on which u takes the same value, u{i,...,i0 } = u
ˆ{i,...,i0 } , and
u
ˆ{i,...,i00 } ≥ u
ˆ{i00 +1,...,i0 } whenever i ≤ i00 < i0 .
Intuitively, an admissible ironing u is an ironing of u
ˆ that keeps the average markup the same for
all sets of consumers, and in as much as markups between consumers differ, a firm cannot separate
on-average more profitable consumers from less profitable ones. This has three implications. First,
on any strictly increasing part of u, ui = u
ˆi . Second, on any flat part, the average of u is the same
as the average of u
ˆ. Third, within any flat part, the higher types are not more profitable on average
than the lower types. If higher types were more profitable, then a firm would prefer to separate
and disproportionately attract them with a contract featuring slightly better terms and a penalty
that puts off lower types. It turns out that there is exactly one such ironing:
Lemma 3. For any u
ˆ, there is exactly one admissible ironing.
Now we are ready to state our main proposition:
Proposition 4 (Separation According to Beliefs). For any si , αi , if p is sufficiently large, then
any symmetric pure-strategy equilibrium is fully separating between belief types, with the borrowed
amount and interest rate equal to those in the observable case for each belief type βˆi , and the
discounts chosen such that u is the admissible ironing of u
ˆ.
While the penalties p—which, as we have explained, prevent more pessimistic types from taking
a contract intended for a more optimistic type—are somewhat special in ensuring the generality
17
of Proposition 4, it is worth emphasizing that for many parameter combinations, the proposition
holds even in an environment without penalties. First, this is the case if u
ˆi is decreasing. Then,
in the admissible ironing ui is constant, so a more pessimistic type does not prefer a contract
intended for a more optimistic type. Second, the same holds if u
ˆi is increasing and consumers are
only slightly naive, i.e., βi+1 > βˆi for all i. Then, if type βˆi takes the contract intended for type
βˆj > βˆi , she expects to repay late and pay costly interest, making the contract unattractive. Third,
by continuity, the addition of penalties is unnecessary if consumers are only slightly naive and the
redistributions required in the admissible ironing are sufficiently small.
4.2
The Welfare Effect of Naivete-Based Discrimination
Given the characterization of equilibrium in the previous subsection, we proceed to identify its
implications for the welfare effect of lender information about consumers. We begin with perfect
information, which reveals a consumer’s β and βˆ perfectly to firms. We assume that p is sufficiently
high for Proposition 4 to hold for firms’ initial information. Then, Proposition 4 implies that the
unobservability of beliefs does not affect the equilibrium distortion faced by a consumer with a
given βˆi . By Proposition 1, starting from a situation with observable beliefs, perfect information
strictly lowers welfare. Hence:
Corollary 1. For any si , αi , perfect naivete-based discrimination strictly lowers welfare.
Now we turn to imperfect information. As in our basic model, we assume that with naivetebased discrimination firms sort consumers into two pools. We denote the shares of belief type βˆi
in the two pools by s1i and s2i , respectively, and the shares of naive consumers among those with
belief type βˆi by αi1 and αi2 , respectively. For each h = 1, 2, we assume that—similarly to initial
beliefs—|αih − αjh | < 1 for any i, j, and that p is sufficiently high for Proposition 4 to hold both
with and without information about consumers. Then, once again Proposition 4 implies that the
unobservability of beliefs does not affect the equilibrium distortion faced by a consumer with a
given βˆi . Hence, both with and without price discrimination the distortion is as in the observable
case. Using Proposition 1, this yields:
Corollary 2. For any s1i , s2i , αi1 , αi2 , naivete-based discrimination lowers total social welfare.
18
Our results also allow us to characterize exactly what kind of information lowers social welfare.
Information about beliefs does not affect welfare:
Corollary 3. For any s1i , s2i , if αi1 = αi2 for all i, then naivete-based discrimination does not affect
social welfare.
Again, since total welfare does not depend on whether firms know beliefs, it also does not depend
on how much firms know about beliefs. Of course, our equilibrium characterization implies that
information about beliefs can affect the distribution of welfare across consumers with different
beliefs. On the other hand, any information about the likelihood that a consumer with given
beliefs is naive lowers welfare:
Corollary 4. If αi1 6= αi2 for any i, then lender information strictly lowers social welfare.
Our results complement in an interesting way the small literature on screening with potentially
naive consumers (Eliaz and Spiegler 2006, 2008, for instance), which assumes that when contracting
a firm knows consumers’ ex-post behavior but not their beliefs. While this is the natural way to set
up a screening problem, Corollaries 3 and 4 imply that when it comes to third-degree naivete-based
discrimination, it is information about ex-post behavior given beliefs that matters, so that the
unobservability of ex-post behavior is crucial.
To conclude our discussion of the welfare effects of information, we show that for naivete-based
discrimination to lower welfare, it is neither necessary nor sufficient for firms to learn about a
consumer’s time inconsistency β.
Example 1. Suppose there are 30 consumers, I = 2, and βˆ1 = β2 < βˆ2 . There are 10 consumers
with βˆ1 , and they are all sophisticated. There are also 10 consumers each of type (βˆ2 , β2 ) and
(βˆ2 , βˆ2 ). Information sorts consumers into two groups, in which the numbers of each type are 4,6,5
and 6,4,5, respectively.
The above piece of information does not reveal anything about a consumer’s time inconsistency,
since in both groups the share of consumers with β = β2 is 2/3. Yet by Corollary 4, the information
strictly lowers welfare. Intuitively, although firms do not receive information about consumers’ time
inconsistency, they do receive information about the proportion of naive consumers among those
19
of belief type βˆ2 . Once consumers sort according to beliefs, this information leads to a reduction
in welfare. This implies that for total welfare to decrease, it is not necessary for firms to receive
information about time inconsistency.
Conversely, note that since in our model β and βˆ can be correlated, information about βˆ can
also provide information about β. Yet by Corollary 3, such information does not affect welfare.
Hence, for total welfare to decrease, it is not sufficient for firms to receive information about time
inconsistency.
5
Homogenous versus Heterogeneous Distortions
In this section, we move beyond the credit market and offer a partial characterization of how
naivete-based discrimination affects welfare in different economic environments. We show that the
welfare implications depend on which sides of the market—both the naive and sophisticated sides,
the sophisticated side, or the naive side—the distortion from exploiting consumer naivete falls on.
Throughout the section, we assume that firms know consumers’ beliefs. As we have emphasized,
given that assuming unknown beliefs would call for analyzing screening menus, this approach is
consistent with the standard practice in the literature of not combining third-degree price discrimination with second-degree price discrimination. Furthermore, while we have not confirmed this in
a screening setting for other applications, our credit-market model suggests that allowing for beliefs
to be unobservable is less crucial for understanding third-degree price discrimination than allowing
for naivete given beliefs to be unobservable.
5.1
A Reduced-Form Model and Other Applications
We first show that naivete-based discrimination tends to lower total welfare in the class of environments in which the distortion from exploiting naive consumers falls equally on the naive
and sophisticated sides of the market. To do so, we introduce a simple reduced-form model of
naivete-based discrimination with such “homogenous distortions” that (we will argue) fits many
applications, including our credit-market framework.
20
Two risk-neutral profit-maximizing firms located at the endpoints of the unit interval simultaneously make offers consisting of an “anticipated price” fn ∈ R and an “additional price” an ∈ [0, amax ]
to a population of consumers uniformly distributed on the interval. Each consumer is naive with
probability α and sophisticated with probability 1 − α independently of her location, and is interested in buying at most one unit of one product. A naive consumer does not take the additional
prices into account when making purchase decisions, acting as if the total price of product n was
fn ; but if she buys firm n’s product, she ends up paying an as well. A sophisticated consumer
anticipates the additional price and takes costless steps to avoid paying it. Hence, if a consumer’s
value for the product is v, she makes purchase decisions under the expectation that her utility
from product n will be v − fn − t|y − n|, but if she is naive, her actual utility will instead be
v − fn − an − t|y − n|. We assume that v is sufficiently high for all consumers to purchase in
equilibrium for every α ∈ [0, 1].
Crucially, we posit that firm n’s cost of serving any consumer is c+k(an ), where the “exploitation
cost” function k(an ) satisfies k(a) = 0 for a ≤ a and is three times differentiable for a > a, with
k 0 (a) = 0, k 00 (a) > 0 for a > a, and k 0 (amax ) > 1. This is a situation with homogenous distortions
because the welfare cost k(a) is the same for all consumers, independently of whether they are
naive or sophisticated.
The equilibrium additional price and the welfare effect of information are easy to characterize
in this reduced-form model:
Lemma 4. Given α, a Nash equilibrium has additional price a(α) = (k 0 )−1 (α). If k 0 (a)/k 00 (a) is
strictly increasing in a over the interval [a(0), a(1)], then for any αns , αn , αs , seller information
about consumer naivete strictly lowers social welfare.
In response to information, firms increase the additional price to the pool with more naive
consumers—thereby increasing the exploitation distortion and lowering social welfare—and lower
the additional price to the pool with more sophisticated consumers—thereby lowering the exploitation distortion and raising social welfare. Because an increase in a pre-existing distortion is more
costly than an identical decrease is beneficial, the net effect is often negative. Qualifying this tendency is that information may have asymmetric effects on the additional prices chosen for the two
21
pools, which could mitigate or exacerbate the adverse effect of seller information. In particular, if
the additional price decreases for the sophisticated pool sufficiently more than it increases for the
naive pool, then the welfare effect of information is positive. This would, however, require that the
marginal exploitation cost increases much faster for increases in a than it decreases for decreases
in a. The condition in Lemma 4 that k 0 (a)/k 00 (a) is increasing in the relevant range, a property
called decreasing absolute convexity, rules this out.
As the above discussion suggests, decreasing absolute convexity is an arguably weak condition.
Since k 0 (a)/k 00 (a) is constant if k(·) is exponential, the condition says roughly that the exploitation
cost is increasing slower than exponential. For instance, any convex power function—a common
functional form for cost functions in economics—satisfies decreasing absolute convexity.15 If decreasing absolute convexity is not satisfied, then the curvature of the cost function increases very
fast, so that at higher levels the additional price becomes relatively very unresponsive to incentives.
As an example, consider a proportional tax τ imposed on the additional price. A simple derivation
shows that ∂a(α)/∂τ |τ =0 = −k 0 (a(α))/k 00 (a(α)).16 If k 0 (a)/k 00 (a) is decreasing, therefore, then the
absolute responsiveness of firms’ profits from the additional price to a small tax is decreasing in
the proportion of naive consumers, so that the tax elasticity of firms’ profits from the additional
price is drastically decreasing. Although ultimately an empirical question that needs to be settled,
our guess is that this is unlikely to be the case in most applications.
Given that decreasing absolute convexity appears to be a weak condition, Lemma 4 says that
information about consumer naivete is likely to lower welfare in the class of situations in which
the distortion from exploiting naive consumers is homogenous. In fact, our proof of Proposition
1—which establishes that in the credit market naivete-based discrimination is welfare-decreasing—
proceeds by showing that our credit-market model simplifies to the reduced-form model above,
and the resulting k(·) satisfies decreasing absolute convexity. In the credit market, the additional
price arises from the interest payments naive consumers fail to anticipate, the exploitation cost k(·)
derives from overborrowing, and prudence implies that k(·) satisfies decreasing absolute convexity.
For power cost functions k(a) = kar , strict convexity requires r > 1. Hence, k0 (a)/k00 (a) = a/(r − 1) is strictly
increasing in a. An example of a function that does not satisfy decreasing absolute convexity is k(a) = (exp(a) − 1)2 .
16
In equilibrium, k0 (a(α)) = α(1 − τ ). Differentiating with respect to τ gives k00 (a(α))∂a(α)/∂τ = −α. Solving
for ∂a(α)/∂τ and using that at τ = 0 we have k0 (a(α)) = α yields the formula.
15
22
To illustrate that our reduced-form model applies to other economically important settings,
we consider an alternative application. In this application, naive consumers underestimate their
demand for an additional service, firms exploit this mistake by charging a high price for the service,
and this induces consumers to undertake socially inefficient ex-ante steps to avoid the service. This
mechanism is explicitly discussed by Gabaix and Laibson (2006) in the context of hotels and by
Armstrong and Vickers (2012) in the context of bank accounts. One important difference is that
in our setting naive and sophisticated have the same initial beliefs regarding their demand for the
service, so they both undertake costly avoidance.
Formally, we suppose that firms sell a basic product—e.g., a bank account or mobile-phone
contract—with cost c and an additional service—e.g., overdraft protection or roaming—with cost
zero. Firm n chooses f˜n for the basic service and a
˜n for the additional service, and consumers
observe all prices. Each consumer is interested in contracting with at most one firm, and can
only purchase the additional service from the firm from which she bought the basic service. A
sophisticated consumer needs the additional service with probability θs − t, where θs is the baseline probability and t is the avoidance effort—such as arranging for sufficient funds in the banking
example or buying a phone card in the mobile-phone example—undertaken by the consumer. Similarly, a naive consumer needs the additional service with probability θn − t, where θn > θs . Both
naive and sophisticated consumers initially believe that they will need the additional service with
baseline probability θs . The cost of avoidance is κ(t) = φtγ , with φ > 0 and γ > 1. To ensure that
consumers choose an interior avoidance level, we furthermore suppose that (γ − 1)/γ < θs /θn .
In this case, seller information about consumer naivete once again lowers welfare:
Proposition 5 (Welfare with Avoidance Costs). For any αns , αs , αn , naivete-based discrimination
strictly lowers total welfare.
The proof of Proposition 5 maps our avoidance-cost application to the reduced-form model
above, and shows that in the reduced form k(·) satisfies decreasing absolute convexity. In this
mapping, the part of add-on expenses that naive consumers fail to anticipate becomes the additional
price, and the avoidance cost given consumers’ optimal response to the add-on price becomes the
exploitation cost. That it is consumers rather than firms who pay the exploitation cost makes no
23
difference, since in equilibrium the cost is transferred to firms through lower prices.
As we show in our working paper (Heidhues and K˝oszegi 2014), other applications in which
previous research has argued consumer naivete plays a role also fit our reduced-form framework. For
instance, firms might exacerbate naive consumers’ unexpected spending on a profitable product by
oversupplying a complementary good (e.g., alcohol and glitter in casinos). And firms might increase
naive consumers’ unexpected willingness to pay for an upgrade to a base product by distorting the
quality of the base product downwards (e.g., computer programs, vacation packages).
Beyond identifying a class of settings in which naivete-based discrimination lowers welfare under weak conditions, however, the reduced-form model also raises the question of when positing
homogenous distortions is a good assumption, and when it is not. It is easy to see that in any situation in which the distortion from exploiting naive consumers arises from the ex-ante considerations
of firms and consumers, the distortion is homogenous. Since naive and sophisticated consumers
have the same ex-ante beliefs and preferences and hence make the same ex-ante choices, any distortion that is generated at the ex-ante stage must apply equally to them. For any such distortion,
information about consumer naivete tends to lower welfare.
But distortions generated by ex-post behavior—in particular, the choices that lead naive but
not sophisticated consumers to pay an additional price—could be different for the two types of
consumers. In the next subsection, we discuss the effects of information for such distortions.
5.2
Heterogeneous Distortions
We now analyze the welfare implications of information about consumer naivete when the exploitation cost is not the same on the naive and sophisticated sides of the market. We consider two
extreme cases: when the welfare cost arises only on the sophisticated side, and when the welfare
cost arises only on the naive side. An analysis of the important case in which the welfare cost arises
on both sides but is different is beyond the scope of this paper.
Sophisticated Side. We first suppose that the exploitation cost k(·) arises only on the sophisticated side. This would be the case, for instance, if—as in Grubb’s (forthcoming) model of the
mobile-phone market—both naive and sophisticated consumers expect to pay costly attention to
24
make sure they avoid an add-on, but only sophisticated consumers do so in the end. Consistent
with this example, we assume that it is the consumer who has to pay the cost k(·).17 Then:
Proposition 6. For sophisticated-side distortions, perfect naivete-based discrimination (αs =
0, αn = 1) maximizes social welfare. If for a ∈ (a(0), a(1)) the derivative of k 0 (a)/k 00 (a) is positive and bounded away from zero, then there is an α∗ such that if αns , αn , αs < α∗ , naivete-based
discrimination strictly lowers welfare.
Perfect information, which typically minimizes welfare for homogenous distortions, maximizes
welfare for sophisticated-side distortions. Intuitively, if a firm knows that a consumer is sophisticated—
and hence she anticipates any additional price and dislikes the associated cost—it imposes no additional price, so that no distortion arises. And if a firm knows that a consumer is naive, it can
exploit the consumer without triggering an associated exploitation cost.
The contrast between our model’s avoidance costs—which both naive and sophisticated consumers pay—and the Grubb (forthcoming) model’s avoidance costs—which only sophisticated consumers pay—highlights that determining whether an exploitation cost is homogenous can be tricky.
Economically, if avoidance steps are taken before naive consumers are hit with the surprise they
did not anticipate, then the cost is an ex-ante, homogenous exploitation cost. If avoidance steps
are taken later, then the cost is ex-post, and may be heterogeneous.
Although perfect information always maximizes social welfare, so long as k 0 (a)/k 00 (a) is increasing at a rate bounded away from zero—which, as we have argued above, appears to be a weak
condition—a sufficiently small amount of information lowers welfare if the share of naive consumers
is sufficiently small. Intuitively, because the majority of consumers in both pools is sophisticated,
the majority of consumers has to bear an exploitation cost. Hence, since it leads firms to charge
different additional prices to the two pools and k(·) is convex, naivete-based discrimination lowers
17
Unlike with homogenous distortions, it now matters whether the consumer or the firm pays k(·). If the firm pays,
the cost enters profits as a term −(1 − α)k(a). But since all consumers make choices as if they were sophisticated, if
the consumer pays the cost enters consumers’ utility as a term −k(a). Hence, the cost does not transfer one-to-one
between the consumer and the firm.
In some cases, the assumption that the firm pays (some of) the exploitation cost is plausible. For instance,
if the additional price arises from a rebate all consumers expect to return but only sophisticated consumers do,
administrative costs associated with this have to be borne by the firm. Nevertheless, these examples seem less
important in practice than examples in which consumers pay the cost.
25
welfare. For instance, our proof of Proposition 6 implies that if k(·) is quadratic, then naivete-based
discrimination lowers welfare if αns , αn , αs < 1/3.
Naive side. We now suppose that the distortion from exploitation arises only on the naive side.
We distinguish two cases according to who pays the exploitation cost. In some situations, such as
when the failure to anticipate the additional price leads to undersaving or another distortion in a
consumer’s choices, naive consumers pay the cost. In other situations, such as when collecting the
additional price is associated with administrative or default costs, it is firms who pay the cost.
Proposition 7. For naive-side distortions, be they paid by consumers or firms, naivete-based discrimination does not affect welfare.
If naive consumers pay the exploitation cost, the cost affects neither the firm’s margin nor (since
she does not anticipate it) the consumer’s willingness to accept a contract, so that the cost per se
does not affect the firm’s considerations. As a result, for any α the firm charges a = amax —simply
maximizing the additional price it can extract from naive consumers—and information about α
has no effect on welfare. If the firm pays the exploitation cost, then both the benefit and the
cost of raising the additional price arises only for naive consumers in the firm’s pool. As a result,
the optimal additional price is again independent of α, and information about α has no effect on
welfare.
5.3
Heterogeneous Distortions in Credit Markets: Non-Linear Repayment Costs
We return briefly to the credit market to discuss a source of heterogeneous distortions. Our basic
model assumes that repayment costs are linear, thereby isolating the overborrowing distortion
that results from exploiting naive consumers. Now we suppose in contrast that repayment costs
are non-linear. Then, ex-post distortions arise: holding the total repayment amount constant,
welfare is maximized when a consumer repays the same amount in periods 1 and 2, yet with the
firm’s optimal interest rate, typically neither consumer does so. Furthermore, because naive and
sophisticated consumers repay differently, the distortions are heterogeneous. Although we have not
been able to fully characterize the effect of information about naivete in this case, the analysis in
Section 5.2 indicates that naive-side distortions do not qualify the effect of information, whereas
26
with sophisticated-side distortions information could be beneficial, so that—factoring in its negative
effect on the overlending distortion—the net effect of information may be ambiguous.18
Our intuition suggests, however, that the sophisticated-side repayment distortion may in reality
be economically small or non-existent because sophisticated consumers can arrange for alternative
sources of funds—from their own budgets or from other sources of credit—that naive consumers
either do not have available or procrastinate in using. To capture this possibility in a simple way,
we assume that in period 1 sophisticated consumers have access to external funds at zero interest.
The instantaneous disutility from repaying an amount q is c(q) = φq γ with φ > 0, γ > 1, where the
power-function form is to keep our analysis tractable. Also for tractability, we keep the quasi-linear
form and assume that the discount d is paid in period 3 and the consumer’s utility from it is linear.
Hence, when taking the loan (bn , rn , dn ) and repaying amounts q1 and q2 in periods 1 and 2, self
0’s utility is u(bn ) − c(q1 ) − c(q2 ) + dn . For any rn > 0, a sophisticated consumer repays the entire
loan to the firm in period 1, but uses alternative sources of funds to set q1 = q2 = bn /2. A naive
consumer believes in period 0 that she will do the same, but—having no alternative funds—her self
1 will choose q1 to minimize c(q1 ) + βc((1 + rn )(bn − q1 )). We still assume that the surplus from
lending is sufficiently high relative to t for the market to be covered for any α.
In this model, our results extend for a broad set of circumstances:
Proposition 8 (Welfare with Non-Linear Repayment Costs). Suppose u000 (b) > c000 (b/2)/4 for all
b. Then, for any αns , αn , αs , naivete-based discrimination strictly raises total lending and strictly
lowers social welfare.
The logic of Proposition 8 is similar to that of our main result. Because firms can collect
unexpected interest from naive consumers, they have an incentive to overlend. Information leads
firms to overlend by different amounts to different consumers, lowering welfare for two reasons.
First, because u(·) is concave and c(·) is convex, an increase in lending to a consumer hurts social
welfare more than an equal decrease in lending to a consumer raises social welfare. Second, total
lending—too high to begin with—increases so long as u000 (b) > c000 (b/2)/4. As in our basic model,
18
Of course, extrapolating from the insights of Section 5.2 must be taken with caution, since the different types of
distortions could interact.
27
the risk of which pool she will be allocated to—and therefore how much she will borrow—increases
a prudent consumer’s expected marginal utility of consumption. Because c(·) is convex, however,
unlike in our basic model the same risk also increases the consumer’s expected marginal cost of
consumption. But because the cost of repayment is distributed over multiple periods, the shape of
u(·) is more important in determining the amount of consumption than the shape of c(·). Hence,
borrowing is likely to increase in many situations.19
Beyond overlending—an effect that hurts both types of consumers—naive consumers are hurt
because they allocate repayment between periods 1 and 2 in an ex-ante suboptimal way. In addition, the two types of distortions interact: as lending increases, the ex-ante cost of repaying in a
suboptimal way increases, and hence the increase in overlending increases the additional distortion
naive consumers face.
6
Related Literature
6.1
Empirical Background
One of the key assumptions of our model is that naive consumers incur unexpected charges. This
assumption is made in different forms in many papers in behavioral industrial organization, and is
consistent with empirical findings from a number of industries. For instance, Stango and Zinman
(2009) find that consumers incur many avoidable fees, and the Office of Fair Trading (2008) reports
that most consumers who use overdraft protection do so unexpectedly. Evidence by Agarwal,
Driscoll, Gabaix and Laibson (2008) indicates that many credit-card consumers seem to not know
or forget about various fees issuers impose. Ausubel (1991) documents that consumers receiving
credit-card solicitations overrespond to the introductory (“teaser”) interest rate relative to the postintroductory rate, suggesting that they end up borrowing or revolving debt more than they intended
19
For instance, although our model simplifies things by considering a three-period model, in a more realistic,
long-horizon model both u(·) and c(·) would be derived from the consumer’s instantaneous utility function over
consumption. If consumption is approximately in steady state, therefore, we would have u000 (b) ≈ c000 (b/2), so that
the condition of the proposition would hold. Furthermore, given the observation that the shape of u(·) matters
more than the shape of c(·) because repayment is divided over multiple periods, intuition suggests that the condition
required would be much weaker if repayment was divided into more periods, and in the limit with many periods the
condition would reduce back to u000 (b) > 0, as in our basic model.
28
or expected. And regulators are worried about the “bill shock” many mobile-phone consumers face
when they unknowingly run up charges (Federal Communication Commission 2010).
The other central assumption in our model is that firms acquire and use information about
consumer naivete for designing offers. Although not conclusive, some direct evidence is consistent with this assumption. Gurun, Matvos and Seru (2013) document that lenders targeted less
sophisticated populations with ads for expensive mortgages. Schoar and Ru (2014) find that the
offers credit-card companies send to less educated borrowers feature more back-loaded payments,
including low introductory interest rates but high late fees, penalty interest rates and over-the-limit
fees.
Beyond the direct evidence, there are economic reasons to believe that naivete-based discrimination is or will soon be pervasive. Since in the settings we study naive consumers are more
profitable than sophisticated consumers with the same beliefs and ex-ante preferences, firms have
a strong incentive to obtain outside information about consumers’ naivete. In addition, researchers
have documented several simple correlates of the tendency to make financial mistakes,20 making it
likely that firms also have access to some (perhaps different) information regarding naivete. This
is especially so given recent technological advances in collecting and processing information about
consumers. As a simple example, the complexity of the words a person uses in an email message
may well be correlated with naivete, and Google allows firms to condition offers on this information.
Given that information about naivete is profitable and seems to be obtainable based on simple and
intuitive measures, it is likely that firms are or will soon be able to use such information in targeting
offers. Indeed, researchers such as Bar-Gill and Warren (2008, pages 23-25) take it for granted that
firms are already doing so.
20
For instance, Agarwal, Driscoll, Gabaix and Laibson (2007) find an age pattern in the amount of financial
mistakes individuals make, Calvet, Campbell and Sodini (2007) report that consumers with lower levels of education
or income make more investing mistakes, and Stango and Zinman (2011) document that it is possible to predict,
based on two simple hypothetical questions on the Survey of Consumer Finances, the consumers who buy the most
overpriced loans.
29
6.2
Related Theory
In asking how outside information about consumers affects economic outcomes, our paper is related
to the literature on first- and third-degree price discrimination, as well as to the literature on privacy.
To our knowledge, however, no paper has considered the question we address in this paper: how
information about whether a consumer is likely to be naive affects welfare.
The existing classical literature overwhelmingly assumes that the consumer type about which
firms may acquire information concerns preferences, not naivete. The starkest contrast is for perfect
price discrimination: while in classical settings perfect discrimination always maximizes welfare
given the number of firms in the market (Stole 2007), in our model of the credit market perfect
discrimination always minimizes welfare.
The welfare effect of third-degree preference-based price discrimination is in general ambiguous.
Building on a large literature, Aguirre, Cowan and Vickers (2010) analyze monopolistic third-degree
price discrimination and establish how the overall welfare effect depends on the interplay between
the misallocation effect first introduced by Pigou (1920) and the output effect originally discussed
by Robinson (1933).21 Bergemann et al. (2015) show that third-degree price discrimination can
generate any combination of producer profit and consumer surplus such that producer profit is at
least as high as without information, consumer surplus is non-negative, and total surplus is at most
as high as with efficient trade.
The literature on privacy often finds that it is socially beneficial for firms to know more about
consumers or employees. Stigler (1980) argues that the protection of personal information leads
firms to substitute other, less efficient, forms of information acquisition or screening, and Posner
(1981) contends that privacy protection creates asymmetric information that impedes the functioning of markets. Varian (1996) reasons that it is in both a consumer’s and a firm’s best interest to
know which product the consumer would like—this lowers search costs for the consumer—although
the consumer would not like the firm to know how much she likes the product. Hoffmann, Inderst
and Ottaviani (2013) consider a model in which each product has two dimensions, and firms—
21
Stole (2007) highlights that the same basic logic determines the welfare effects in a homogenous-good Cournot
model, while additional effects are relevant in differentiated price-competition models.
30
having obtained information about the consumer’s preferences—reveal a consumer’s utility in the
dimension in which her utility is higher. Although such “selective disclosure” is biased, it still
provides useful information, and hence raises social welfare unless a number of market frictions are
present at the same time.22,23
In considering how firms respond to the presence of naive consumers, our paper also belongs
to the growing literature on behavioral industrial organization. Our reduced-form model can capture many forms of naivete discussed in this literature.24 In contrast to our paper, the existing
behavioral industrial organization literature takes firms’ information regarding consumer naivete as
given, and asks how firms respond to this knowledge. In particular, several authors have analyzed
how firms may screen consumers with different degrees of naivete (Eliaz and Spiegler 2006, 2008).
In contrast to this “second-degree naivete-based discrimination,” we ask how firms’ access to outside information about consumer naivete affects welfare. Furthermore, while most of the existing
literature assumes that when contracting a firm knows consumers’ ex-post behavior but not their
beliefs, for the bulk of the paper we assume that a firm knows beliefs but not ex-post behavior, and
study the effect of firms’ information about ex-post behavior. In our credit-market model, we allow
for both beliefs and ex-post behavior to be unknown to firms, and show that from the perspective
of third-degree discrimination it is indeed information about ex-post behavior given beliefs that
22
Taylor (2004) and Calzolari and Pavan (2006) consider dynamic pricing games in which a consumer makes two
purchases in sequence from two monopolists, and her first purchase reveals information about her willingness to
pay for the second product. The authors ask whether disclosing information about the first purchase to the second
monopolist benefits sellers or consumers. Unlike in our paper, the focus is on how the potential for disclosure affects
the first interaction, not on how any information revealed affects welfare in the second interaction. Similarly, Acquisti
and Varian (2005) consider a single monopolist selling two products in a row, and analyze optimal pricing strategies
when consumers can use anonymizing technologies in the first purchase. Taylor (2004) and Acquisti and Varian
(2005) also point out that if consumers do not anticipate how firms will use information on early purchases, firms
take advantage of this naivete, to the detriment of consumers and possibly social welfare.
23
Hermalin and Katz (2006) take an entirely different perspective on the privacy debate. They show that releasing
information about an individual before contracting can create a kind of reclassification risk that risk-averse individuals
dislike. Hence, it is optimal to forbid the release of private information, even if consumers have no taste for privacy
per se. Hermalin and Katz’s argument applies best to settings where insurance is a major consideration, which does
not seem to be the case for most of the applications we consider.
24
See Spiegler (2011) for an introduction to and overview of the behavioral-industrial-organization literature.
Researchers consider two distinct kinds of naive consumers, those who understand contracts but mispredict their
own future behavior, and those who misunderstand contract terms or product features. For models incorporating
the former type of naivete, see for instance DellaVigna and Malmendier (2004) Eliaz and Spiegler (2006), Eliaz and
Spiegler (2008), Grubb (2009), Heidhues and K˝
oszegi (2010), and Laibson and Yariv (2007). For models using the
latter type of naivete, see among others Armstrong and Vickers (2012), Gabaix and Laibson (2006), Heidhues, K˝
oszegi
and Murooka (2014), Heidhues, K˝
oszegi and Murooka (2012), Piccione and Spiegler (2012), and Spiegler (2006).
31
matters.
Like we do in Section 4, Galperti (forthcoming) studies the screening of consumers according to
beliefs under time inconsistency. But because he largely assumes that all consumers are sophisticated and the problem he studies—how to provide a flexible incentive to save under uncertainty—is
quite different from ours—how to lend to a mix of sophisticated and partially naive consumers—the
screening contracts he derives have no obvious relationship to ours. Nevertheless, some of the considerations are related. For instance, similarly to how in our setting more optimistic types derive
greater utility from any contract than do more pessimistic types, in his setting less time-inconsistent
types derive more benefit from savings incentives than do more time-inconsistent types. To reduce
the resulting information rent, the subsidized saving for more time-inconsistent types is capped.
7
Conclusion
The main message of our paper is that the welfare consequences of firms acquiring information about
preferences and firms acquiring information about naivete are qualitatively different. One important
implication of this message is that empirical research on the nature of information acquired and
used by firms is crucial to determine the effects of first- and third-degree price discrimination.
Our analysis also calls for further theoretical research on a number of economically important
realities in markets with naive consumers that our framework ignores. As an immediate example,
all of our analysis ignores possible distortions arising from participation decisions—that consumers
respond to the “wrong” prices when deciding whether to buy. In Heidhues and K˝oszegi (2015), we
argue that the participation distortion in markets with naive consumers can be massive. How seller
information affects participation distortions is an important topic. It is clear that this effect depends
on consumers’ valuation relative to firms’ cost (and, with consumer heterogeneity in valuations,
on the demand curve). To illustrate this fact, consider the perfect-competition limit of our model
(t → 0), and suppose that firms’ fixed cost of serving a given consumer is positive. Then, since
consumers perceive the product to be cheaper than it is, there may be overparticipation in the
market. Information about consumer naivete can make overparticipation better or worse. For
instance, it may be the case that without information firms cannot break even and hence do not
32
serve any consumer, but information allows firms to serve the more naive pool, lowering welfare.
In contrast, it may also be the case that without information consumers are served, but with
information the sophisticated pool is no longer profitable to serve, increasing welfare.
As we have emphasized, we designed our framework to isolate the question of how seller information about naivete affects welfare. Sellers, however, have other concerns related to consumer
heterogeneity as well. In most settings, for instance, the information sellers can observe about
consumers pertains not just to naivete, but also to preferences. In addition, it is likely to be
optimal for a seller not just to rely on outside information to discriminate between consumers,
but also to take advantage of self-selection according to preferences or naivete. How third-degree
naivete-based discrimination interacts with other forms of discrimination is an important topic for
future research. Even within our framework, the different types of distortions generated by the
exploitation of naive consumers—those that fall on both sides, on the sophisticated side, and on
the naive side—interact, and we do not have a general handle on how these interactions depend on
information about naivete.
Finally, our paper does not analyze potential policy responses to price discrimination. For
instance, a commonly advocated solution to privacy concerns is to require firms to obtain a consumer’s consent before using her private information. In on-going work, we investigate whether
this policy helps in our framework, and find that it does not: because a naive consumer does not
understand that the firm will use information to exploit her, she agrees to giving her information
too easily. In addition, a regime with required consent may transfer even more money from naive
to sophisticated consumers than does a regime without required consent.
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Appendix: Proofs
Proof of Lemma 1. We have already established in the text that r∗ = (1 − β)/β and that b∗ is
implicitly defined through u0 (b∗ ) = 1 − α(1 − β)/β. In any interior symmetric equilibrium—that is
a symmetric equilibrium in which all consumers sign a contract— d∗ must solve
u
ˆ1 (d) − u
ˆ2 (d∗ ) + t
∗ ∗
max (α r b − d)
.
d
2t
36
(2)
Using the fact that y = 1/2 and ∂y/∂dn = 1/(2t) in an interior symmetric equilibrium, the first-oder
condition is
1
1
− + (αr∗ b∗ − d∗ ) = 0.
2 2t
Rewriting gives that at an interior equilibrium d∗ = αr∗ b∗ − t. In such a candidate equilibrium, a
sophisticated consumers located at y has equilibrium utility is
1−β ∗
b − t − min{y, 1 − y}t,
β
i
h
b
and the right-hand side is strictly
which increases in α because b∗ = arg maxb u(b) − b + α 1−β
β
u(b∗ ) − b∗ + α
increasing in α for any given b > 0.
Next, we verify that for any α, all sophisticated borrowers prefer borrowing to their outside
option (which implies that naive consumers also accept the contract offer as they believe to be
sophisticated at that stage). Since sophisticated consumers utility is increasing in α, this condition
holds if a sophisticated consumer located at 1/2 prefers signing the contract offer when α = 0, i.e.
if u(be ) − be − (3/2)t > 0, which holds by assumption.
To rule out a non-interior symmetric equilibrium, suppose such an equilibrium exists and let
0
y(d∗ ) < 1/2 be the marginal consumer who buys from firm 1 in such a candidate equilibrium, and
note that in this case
0
0
u(b∗ ) − b∗ + d∗ − ty(d∗ ) = 0.
0
0
Solving for y(d∗ ), in such an equilibrium d∗ must solve
u(b∗ ) − b∗ + d
∗ ∗
,
max (α r b − d)
d
t
(3)
giving rise to the first order condition:
1
0
0
−y(d∗ ) + α r∗ b∗ − d∗
=0
t
0
Using that y(d∗ ) < 1/2 and comparing the first order condition to that of the interior symmetric
0
equilibrium implies that d∗ > d∗ ; but since the consumer located at 1/2 already prefers buying
0
0
at a discount d∗ , she must strictly prefer buying at d∗ , contradicting the fact that y(d∗ ) < 1/2.
Hence there is a unique symmetric equilibrium.
37
Note that firms earn t in expectation per contract they sell in the unique symmetric equilibrium,
and have a market share of 1/2. Thus, equilibrium profits are t/2. Finally, social welfare gross of
transportation costs is equal to u(b∗ ) − b∗ , which decreases in α since for α = 0 one has b∗ = be ,
and b∗ is strictly increasing in α.
Proof of Proposition 1. We first prove that our model simplifies to the reduced-form model
of Section 5.1. Let a = (1 − β)b/β. Let x = β/(1 − β) and note that b = xa, d = αa − t, and
x − xu0 (xa) = α, so that xa ≥ be . Using this, the equilibrium welfare of a sophisticated consumer
gross of transportation cost is u(b) − b + d = u(xa) − xa + αa.
Now we define k(a) = 0 for xa < be and k(a) = (u(be ) − be ) − (u(xa) − xa) for xa ≥ be , and
v = u(be ) − be . Then, the above implies that in equilibrium k 0 (a) = α, sophisticated consumers’
utility gross of transportation costs is
u(b) − b + αa = u(be ) − be + [(u(b) − b) − (u(be ) − be )] + αa
= v − [(u(xa) − xa) − (u(be ) − be )] + αa
= v − k(a) + αa,
and naive consumers’ utility gross of transportation costs is a lower, exactly as in our reduced-form
model (where our application satisfies c = 0).
To complete the proof, we show that in this application, k satisfies the condition of Lemma 4.
We have
k 0 (a)
x(1 − u0 (xa))
=
.
00
k (a)
−x2 u00 (xa)
The derivative of the left-hand side with respect to a is strictly positive if [−x2 u00 (xa)]2 +x3 u000 (xa)[1−
u0 (xa)] > 0, and hence a sufficient condition is that u000 (b) ≥ 0 for all b ≥ 0.
Proof of Proposition 2. In the proof of Lemma 1, we established that a sophisticated consumer’s
equilibrium utility is
u(b∗ ) − b∗ + α
1−β ∗
b − t − min{y, 1 − y}t.
β
Using the fact (established in the text) that only naive consumers incur the interest b∗ (1 − β)/β,
38
their equilibrium utility is
u(b∗ ) − b∗ − (1 − α)
1−β ∗
b − t − min{y, 1 − y}t,
β
∗
where b∗ is implicitly defined through u0 (b∗ ) = 1 − α 1−β
β . Hence, limβ→ 1 b = ∞. Thus, when the
2
firms have perfect information the equilibrium utility of naive consumers goes to negative infinity
as β approaches 1/2. For a given u and αns , the amount borrowed by consumers is finite because
limβ→ 1 b∗ = (u0 )−1 (1 − α). Substituting this into a naive consumer’s equilibrium utility above
2
implies that this is also bounded from below as β approaches 1/2, and thus strictly greater than
when firms have perfect information. Continuity of a naive consumer’s equilibrium utility in β
implies that for a β sufficiently close to 1/2, perfect information hurts naive consumers.
We now establish that the sophisticated consumers’ equilibrium utility is strictly increasing in
α, and hence that perfect information about consumers (implying αs = 0) lowers sophisticated
consumer’s equilibrium utility. In the text, we showed that the equilibrium amount borrowed b∗
maximizes
α·
1−β
· bn + u(bn ) − bn .
β
Since for a given bn the above expression is strictly increasing in α, it must also increase when the
optimal b∗ is chosen. This directly implies that a sophisticated consumer’s equilibrium utility
u(b∗ ) − b∗ + α
1−β ∗
b − t − min{y, 1 − y}t
β
is strictly increasing in the share of naive consumers α.
Proof of Lemma 2. We break firm n’s problem down into deciding what perceived utility level
gross of transportation costs u
ˆn it want to offer to consumers, and solve for the optimal contract
(bn , rn , dn ) that provides this utility level. In the main text, we already established that the firm
chooses an interest rate ri = (1 − βi )/βi . Using this fact and the notation introduced in the main
text, the firm’s problem is equivalent to
max
bn ,dn ,ρn
bn + ρn bn − dn − bn
subject to u(bn ) − bn + dn ≥ u
ˆn
ρn ∈ {ρ1 , . . . , ρI−1 }.
39
Using that the participation constraint must hold with equality to substitute dn in the maximand,
gives
max
bn ,dn ,ρn
ρn bn + u(bn ) − bn − u
ˆn
subject to ρn ∈ {ρ1 , . . . , ρI−1 }.
Because the maximand is increasing in ρn , the optimal contract has ρn = max{ρ1 , . . . , ρI−1 }, and
thus r∗ = (1 − βi∗ )/βi∗ where i∗ = arg maxj ρj . Differentiating establishes that at the optimal
contract u0 (b∗ ) = 1 − ρi∗ .
Proof of Proposition 3. Note that since in a symmetric equilibrium all consumers borrow, the
change in total welfare in each pool l ∈ {1, 2} depends only on the amount of overborrowing, and
the welfare cost of overborrowing is k(bln ) = (u(be ) − be ) − (u(bln ) − bln ) per consumer in pool l. We
think of the firms as responding to information in two steps. First, they respond having to hold
interest rate fixed at r∗ , thus in each pool l ∈ {1, 2} maximizing
max αil∗ r∗ bln + u(bln ) − bln − u
ˆn .
bln ,dln
Second, firms are allowed to optimally adjust their interest rate (and readjust all other contract
terms) in each pool.
Because in this case, all consumers with a type i ≤ i∗ delay repaying while the others do not, the
welfare effects of the first step are equivalent to those of a two-type model in which the share αil∗ is
naive in pool l and has value β = βi∗ . Thus, Proposition 1 implies that the first step above strictly
lowers total welfare. In the second step, the firm adjusts bn in each pool l so that u0 (bln ) = 1 − ρl
for ρl = arg maxj ρlj rather than u0 (bln ) = 1 − αil∗ r∗ . This implies that the marginal utility of
borrowing weakly decreases in both pools. Therefore, the second step further increases the cost of
overborrowing, and thus further decreases total welfare.
Proof of Lemma 3. We begin by establishing that an admissible ironing exists. Our proof is
constructive.
Step 1: Start with the lowest belief type βˆ1 , and define a preliminary pool I = {1} interpreted
as a preliminary pool 1 consisting of the lowest type only.
40
Step 2: Consider the highest preliminary pool, I = {i, . . . , i0 }. Compare the perceived utility
of player i0 + 1 in the observable case to that of the preliminary pool. If u
ˆi0 +1 > uI , we introduce
a new highest preliminary pool I + 1 = {i0 + 1}. Otherwise, reduce the perceived utility of belief
types {i, . . . , i0 } by an equal amount and raise that of player i + 1 in such a way that keeps the
P
Pi+1
u
ˆi constant until either (a) uI = ui0 +1 or (b) uI = uI−1 . In case (a) define the
ui = i+1
i
i
new highest preliminary pool as I new = {i, . . . , i0 + 1} with the new corresponding perceived utility
level. In case (b) define the new highest preliminary pool as (I − 1)new = (I − 1) ∪ I. Repeated
Step 2 until no higher belief type is left.
Note that since we have a finite number of belief types, the algorithm converges in a finite
number of steps. Furthermore, in Step 2 we change perceived utilities until u is increasing, so
that the algorithm converges to a u satisfying condition (i) of an admissible ironing. In Step 2,
P
P
we enforce that in every preliminary pool i+1
ui = i+1
u
ˆi , and hence our algorithm converges
i
i
to a u for which any maximal set I that receives the same utility must satisfy uI = u
ˆI . Finally,
in Step 2, we add belief types {i0 + 1, . . . , i000 } to an existing preliminary pool {i, . . . , i0 } only in
case u
ˆ{i,...,i0 } ≥ u
ˆ{i0 ,...,i000 } . Hence, because by construction the preliminary pool {i, . . . , i0 } satisfies
u
ˆ{i,...,i00 } ≥ u
ˆ{i00 ,...,i0 } whenever i ≤ i00 ≤ i0 − 1, and the preliminary pool {i0 + 1, . . . , i000 } also
satisfies the corresponding property, the resulting preliminary pool {i0 , . . . , i000 } must also satisfies
this property. Together, these facts imply that the algorithm converges to an admissible ironing.
To prove that the admissible ironing is unique, it is useful to note that for a maximal set
{i, . . . , i0 } of an admissible ironing, condition (ii) implies that for any i ≤ i00 ≤ i0 ,
00
i
X
00
sj uj ≤
j=i
i
X
sj u
ˆj ,
j=i
and that the above holds with equality if i00 = i0 .
We will now argue by contradiction that there is a unique admissible ironing. Suppose otherwise, i.e. there exists two admissible ironings u and u0 . Clearly, condition (ii) implies that the
maximal sets over which belief types get the same perceived utility must differ across these ironings.
Consider the lowest set for which the belief types differ, and wlog let the maximal set {i, . . . , i0 }
of the admissible ironing u be contained in the maximal set {i, . . . , ˜i} of the admissible ironing u0 .
41
Condition (ii) of an admissible ironing implies that
0
i
X
0
sj uj =
j=i
and that
˜i
X
i
X
sj u
ˆj ,
j=i
0
sj uj ≤
j=i0 +1
i
X
sj u
ˆj .
(4)
j=i
Condition (i) of an admissible ironing implies that u is increasing, thus
P˜i
Pi0
j=i0 +1 sj uj
j=i sj uj
< P˜
.
Pi0
i
j=i sj
j=i0 +1 sj
The last inequality together with Inequality 4, implies that
P˜i
P i0
ˆj
j=i0 +1 sj u
j=i sj uj
=u
ˆ{i0 +1,...,˜i} ,
< P˜
u
ˆ{i,...,i0 } = u{i,...,i0 } = Pi0
i
j=i sj
j=i0 +1 sj
and hence that the set {i, . . . , ˜i} of the admissible ironing u0 violates condition (ii) of an admissible
ironing, a contradiction.
Proof of Proposition 4. The proof proceeds in five steps. The first step makes a preliminary
observation about the equilibrium with observable belief types that is helpful in solving for the
equilibrium with unobservable ones. Step (ii) establishes that the maximal perceived-utility difference between belief types in the unobservable case is less than the maximal difference in the
observable case. This allows us to establish that the firm can always use the penalty in the unobservable belief case to deter lower belief types from deviating and choosing contracts designed for
higher belief types. Building on this, Step (iii) establishes that every belief type borrows the same
amount and pays the same interest rate as in the observable belief case; hence the perceived surplus
is maximized also in the unobservable belief case. Step (iv) uses these facts to establish that any
equilibrium is an admissible ironing that maximizes perceived surplus. Step (v) verifies that such a
perceived-surplus maximizing admissible ironing is indeed a symmetric equilibrium. Finally, since
the admissible ironing is unique by Lemma 3, Steps (iii) to (v) directly imply the proposition.
Denote the highest perceived utility of any type in the observable case by u
ˆmax = maxi {ˆ
ui },
and denote the lowest by u
ˆmin = mini {ˆ
ui }.
42
Step (i): If |αi − αj | < 1 for all i, j ∈ {1, . . . , I}, then there exists a p¯ such that for all p¯ > p¯,
u
ˆmax (¯
p) − u
ˆmin (¯
p) < p¯. We begin by establishing the analogue of Lemma 1 for observable case in
which the firm also chooses ∆i , pi ; are derivation of the equilibrium contract is almost identical and
hence we only sketch the steps that do not differ.
We think of firm n as optimally providing a given level of perceived utility gross of transportation
costs u
ˆn to belief type i, and then selecting the optimal perceived utility. The firm must choose
the maximal interest rate at which naive consumers are willing to delay repayment, i.e. as before
ri∗ = (1 − βi )/βi . Furthermore, it designs ∆i , pi such that naive consumers unexpectedly and incur
the fine pi but sophisticated consumers do not because then consumers do not anticipate incurring
the fine, and hence it does not affect which contract they select. But since consumers incur the
fine only unexpectedly, the firm selects the maximal fine possible pi = p¯. Now naive consumers
incur the fine only if ∆i ≥ βi (∆i + p¯), while sophisticated consumers expect not to delay only if
βi
∆i ≤ βˆi (∆i + p¯). Hence an optimal ∆∗i ∈ [ 1−β
,
i
βˆi
].
1−βˆi
Fix any such optimal ∆∗i . Because consumers
do not anticipate paying interest or the fine, their perceived utility gross of transportation costs is
u
ˆn = u(bi ) − bi − ∆∗i + di . Using this constraint, we can rewrite the firms maximization problem as
1 − βi
max αi ·
· bi + p¯ +u(bi ) − bi − ∆∗i − u
ˆn .
b
βi
{z
}
|
unanticipated payment
i
Thus, b∗i is implicitly defined through u0i (b∗i ) = 1 − αi 1−β
ˆn and ∆∗i .
βi independently of u
Analogous to the proof of Lemma 1 , the optimal discount in a symmetric equilibrium in which
all consumers select a contract must solve
max [αi (ri∗ b∗i + p¯) + ∆∗i − d]
d
u
ˆ1 (d) − u
ˆ2 (d∗ ) + t
.
2t
Imposing that y = 1/2 and ∂y/∂dn = 1/(2t) in a symmetric equilibrium on the first-oder condition
and rewriting gives that in an interior equilibrium d∗ − ∆∗i = αi (ri∗ b∗i + p¯) − t. Therefore, in an
symmetric interior equilibrium a consumer’s perceived utility gross of transportation costs is
1 − βi ∗
∗
∗
b + p¯ − t.
u
ˆi (¯
p) = u(bi ) − bi + αi
βi i
By the exact same argument as in the proof of Lemma 1 this utility is increasing in αi , and hence
43
that all consumers prefer selecting a contract to not buying if a sophisticated consumer located at
1/2 does for α = 0, which in turn our parameter restriction on the transportation costs ensures.
Because dˆ
ui (¯
p)/d¯
p = αi , the condition |αi − αj | < 1 ensures that for sufficiently high p¯ the
difference between any pair u
ˆi (¯
p) − u
ˆj (¯
p) < p¯ and thus also that u
ˆmax (¯
p) − u
ˆmin (¯
p) < p¯.
Step (ii): The equilibrium perceived utility levels with unobservable belief types satisfy u
ˆmax (¯
p) ≥
ui ≥ u
ˆmin (¯
p) for all i. We first show that ui ≤ u
ˆmax (¯
p). Suppose otherwise, i.e. that ui > u
ˆmax (¯
p)
for some i. Consider all types i that receive the highest perceived utility, and denote their set by
˜ Note that a higher belief type receives a weakly greater perceived utility from any contract as it
I.
believes to be (weakly) less likely to incur interest or pi , and hence ui must be weakly increasing
in i. Thus, if i ∈ I˜ so are all higher belief types. Let i be the lowest type that receives maximal
perceived utility uI˜.
˜ for
Observe that the firm must earn strictly less than t from any consumer type i in the pool I;
if this was not the case, then the firm’s contract in the unobservable case (bi , ri , di , ∆i , pi ) would
give a consumer of type i strictly higher utility (and hence increase the firm’s demand), and at the
same time earn a weakly greater profit margin than in the observable case, contradicting the fact
that the contract in the observable case is profit-maximizing.
We now identify a profitable deviation. First, we replace any contract designed for belief type i
in which the sophisticated type consumer either incurs the fine or delays repayment with one that
also gives ui , imposes the maximal fine p0i = p¯ but in which a belief type i expects neither to incurs
the fine nor the penalty but all lower belief types anticipate paying the fine, and a naive consumer
of type i delays repayment and incurs the fine. In particular, in case the sophisticated type i incurs
the penalty, we replace the contract with one in which p0i = p¯ and


βˆi +βi
 βˆi +βˆi−1

2
2
,
∆i = p¯ · max
 1 − βˆi +βˆi−1 1 − βˆi +βi 
2
2
and in case belief type i expects to pay interest we replace the contract with one in which ri0 = ri∗ .
To see that we can do so without inducing types to select a contract other than that designed
for them, consider changing one contract at the time. Observe that since belief type i receives the
same utility ui from the new contract, our tie-breaking assumption ensures that the firm can induce
44
belief type i to select the contract designed for her. Second, note that a higher belief types also
do not pay interest nor incurs the fine when selecting the contract designed for belief type i, and
hence also receive a perceived utility ui gross of transportation costs when selecting that contract.
But since in any candidate equilibrium u is weakly increasing, they do not have a incentive to
choose a contract designed for a lower type. Note that a lower type now anticipates incurring the
maximal fine when selecting the contract designed for consumer i, and anticipates to pay weakly
more interest than belief type i. We now argue that she will not get a higher utility from i’s contract
after the change. First, suppose i expected to pay the fine before the contract change; then after
the contract change, the discount di was reduced to reflect the fact that i does not anticipate to
incur the fine anymore, and hence for all lower types who still do anticipate incurring the fine,
the contract designed for i becomes strictly less attractive. Second, suppose i anticipated to incur
interest; then after the contract change to hold ui fixed the discount was reduced by the amount
of interest that i now does not anticipate paying anymore; hence the contract does not become
more attractive to lower belief types and since they (weakly) preferred their own contract over that
designed for type i prior to the contract change, they still do so after the contract change. Our
tie-breaking assumption thus ensures that no lower belief type will switch to the contract designed
for type i.
Take these new contracts as given. Now consider the class of deviations in which firm 1 reduces
di for all belief types i ≥ i by , so that following the deviation it offers these types a perceived
utility u0i = ui − , where we restrict ourselves to deviations in which < ui − ui−1 and where we
define ui−1 = 0 in case all types get the highest-utility level in the candidate equilibrium. First,
observe that when firm 1 deviates no belief type i has an incentive to choose a contract from firm
1 that was not designed for her. Since u0i is weakly increasing, no belief type wants to choose a
contract designed for a lower belief type. Furthermore, because the contracts for all types i ≥ i
now give strictly lower perceived utility to anyone accepting them, the fact that we started in a
candidate equilibrium implies that no belief type i < i wants to choose a contract designed for
˜ Finally, since the perceived utility of all contracts designed for i ≥ i
belief types that belong to I.
change by the same amount, no consumer i ∈ I˜ has an incentive to switch to a different contract
45
in this set, and hence our tie-breaking assumption ensures that all consumers choose the contract
designed for them.
Using the fact that ∂yi /∂di = 1/(2t) as long as all consumers keep choosing the contract
designed for them and denoting the average per-consumer profit of the firm in pool i by πi , the
marginal profit of changing di for all i ≥ i by the same amount is



 

I
I
I
I
X
X
X
X
1 ∂yi
1
1
1 
1




si − +
πi = −
si +
si πi <
si
− + t = 0,
2 ∂di
2
2t
2 2t
i=i
i=i
i=i
i=i
where we have used that in a symmetric equilibrium firm n has market share 1/2 and the inequality
follows from the fact that the average per-consumer profit for all consumer belief types i ≥ i is
less than t. Thus, firm 1’s profits increase if it slightly lowers di for all i ≥ i by the same , a
contradiction. We conclude that ui ≤ u
ˆmax (¯
p) .
We next show that ui ≥ u
ˆmin (¯
p) for all i. Suppose otherwise, i.e. ui < u
ˆmin (¯
p) for some type i.
Since ui is weakly increasing, there exists a set of belief types {1, · · · , i} that gets perceived utility
ui < u
ˆmin (¯
p). Consider firm 1 deviating and replacing any contract designed for a belief type i ≤ i
with a contract in which in which b0i = b∗i , ri0 = ri∗ and


βˆi +βi

 βˆi +βˆi−1
2
2
,
,
∆0i = p¯ · max
 1 − βˆi +βˆi−1 1 − βˆi +βi 
2
2
p0i = p¯, and d0i is chosen such that the perceived utility of belief type i is ui (b0i , ri0 , d0i , ∆0i , p0i )) = ui +,
where < min{ˆ
umin (¯
p) − ui , ui+1 − ui }. First, note that now belief-type has an incentive to select a
contract other than that designed for herself, and hence our tie-breaking rule ensures that all belief
βˆi
], and hence
1−βˆi
optimal ∆0i , however,
βi
,
types choose the contract designed for themselves. Second, note that ∆0i ∈ [ 1−β
i
b0i , ri0 , ∆i , p0i are optimal choices in the observable case. For this particular
the firm must be offering a lower discount d0i than in the observable case, because the consumers
receive a lower utility gross of transportation costs then in the observable case. Thus, for all these
belief types i ≤ i the per consumer profit πi is strictly greater than t. But then the marginal profit
of slightly increasing the di for all consumers in the pool is
 

  
i
i
i
i
X
X
X
X
1 ∂yi
1
1
1
1
si − +
πi = −   + 
si πi  >   − + t = 0,
2 ∂di
2
2t
2 2t
i=1
i=1
i=1
46
i=1
(5)
a contradiction. We conclude that u
ˆmax (¯
p) ≥ ui ≥ u
ˆmin (¯
p) for all i.
Step (iii): For any p¯ > u
ˆmax (¯
p)−ˆ
umin (¯
p), each belief type i’s contract has bi = b∗i , ri = ri∗ , pi = p¯,
βi
and ∆i ∈ [ 1−β
,
i
βˆi
]
1−βˆi
. In other words, each belief type receives a contract that can differ from an
optimal contract with observable beliefs only in the discount. Suppose not. Consider the candidate
equilibrium ui , and note that ui is (weakly) increasing. Now consider firm 1 deviating and offering
contracts to each belief type i such that b0i = b∗i , ri0 = ri∗ are set as in the observable case,


βˆi +βi

 βˆi +βˆi−1
2
2
,
∆0i = p¯ · max
 1 − βˆi +βˆi−1 1 − βˆi +βi 
2
2
p0i = p¯, and d0i is chosen such that the perceived utility of belief type i is ui (b0i , ri0 , d0i , ∆0i , p0i )) = ui .
First, observe that since ui is weakly increasing and no contract contains a repayment option that
yields higher long-run utility than ui , no belief type i has an incentive to select a contract designed
for a lower type. Second, since each belief type realizes that it will incur the penalty p¯ when
choosing the contract designed for a higher type, no belief type will want to do so, because the
difference in perceived utilities between any belief types i, ˆi is less than u
ˆmax (¯
p) − u
ˆmin (¯
p) < p¯.
Thus, our tie-breaking rule ensures that each belief type still chooses the contract designed for
herself, and firm 1’s market share does not change. Firm 1’s profits from each belief type, however,
increases; hence this is a profitable deviation, completing the contradiction.
Step (iv): A symmetric equilibrium must be a perceived-surplus maximizing admissible ironing.
We already established that bi , ri , pi , ∆i are chosen such that they are optimal in the observable
case for all belief types i, and that ui is weakly increasing. Hence, a symmetric equilibrium must
maximize the perceived surplus.
We next argue that for any maximal set {i, . . . , i0 } on which u takes the same value, u{i,...,i0 } =
u
ˆ{i,...,i0 } . Suppose not. Since for all belief types i, only the discount di can differ from a profitmaximizing contract for the observable case, we can think of social planner who chooses to divide
the given perceived surplus among borrowers and firms, imposing that firms are treated equally.
Now if u{i,...,i0 } > u
ˆ{i,...,i0 } then consumers get a higher perceived utility level than in the observable
case, and hence the firms’ profits must be lower. Hence, the firms earn less than t on average across
all belief types in the set {i, . . . , i0 }. Now consider a firm that lowers all di for all belief types in
47
the set; as long as all belief types still choose the contract designed for them, this change has a
positive marginal profit at the symmetric equilibrium. Furthermore, when setting


βˆi +βi
 βˆi +βˆi−1

2
2
∆0i = p¯ · max
,
 1 − βˆi +βˆi−1 1 − βˆi +βi 
2
2
and p0i = p¯, no consumer has an incentive to chose a contract designed for a higher type. And as
long as the reduction in di is small enough such that the perceived utility u{i,...,i0 } > ui−1 , no type
has an incentive to choose a lower type’s contract. Thus, in this case there is a profitable deviation.
Analogously, if u{i,...,i0 } < u
ˆ{i,...,i0 } the firms earn more than t on average across all belief types
in the maximal set {i, . . . , i0 } when choosing bi , ri , as in the observable case and setting pi = p
and ∆i = ∆0i . Now consider increasing the discount di for all consumers in the set {i, . . . , i0 } in
such a way that u{i,...,i0 } < ui+1 ; observe that the marginal profit of doing so at the symmetric
equilibrium is positive. We conclude that a necessary condition for a symmetric equilibrium is that
u{i,...,i0 } = u
ˆ{i,...,i0 } .
To show that a symmetric equilibrium must be an admissible ironing, we are left to argue
that for any maximal set for which u{i,...,i0 } = u
ˆ{i,...,i0 } , u
ˆ{i,...,i00 } ≥ u
ˆ{i00 +1,...,i0 } for all i ≤ i00 < i0 .
Suppose otherwise, i.e., there exists some i00 ∈ {i, · · · , i0 − 1} such that u
ˆ{i,...,i00 } < u
ˆ{i00 +1,...,i0 } . Note
that if both firms would choose di ’s that induce the perceived utility level u
ˆ{i00 +1,...,i0 } for the set
{i00 + 1, . . . , i0 } instead, firms would earn on average t per consumer. Hence, they must earn more
than t per consumer from belief types in the set {i00 + 1, . . . , i0 }. Now consider firm 1 deviating
and offering a slightly higher discount di to all consumers in the set {i00 + 1, . . . , i0 }; no lower type
will choose this contract and if the increase is small enough so that u
ˆ{i00 +1,...,i0 } < ui0 +1 after the
increase, no higher belief type will choose a contract designed for belief types in {i00 + 1, . . . , i0 }.
But since the per-consumer profit in this pool is greater than t, such a deviation raises profits of
firm 1 by a calculation analogous to that in Equation 5, a contradiction.
Step (v): An admissible ironing is a symmetric equilibrium.
Consider possible deviations by
firm 1. Note that any deviation induces a profile u0 of perceived utilities, and that this profile must
be weakly increasing since for any contract, a higher belief type gets a higher utility than a lower
one. We will enlarge firm 1’s set of possible deviations by allowing it to observe belief types (and
48
hence condition its contract offer on the observable type) with the restriction that any profile of
perceived utilities it offers must be weakly increasing. (Intuitively, this amount to ignoring the
constraint that a lower belief type may want to choose a contract designed for a higher belief type.)
Clearly, this is a larger class of feasible deviations, and as we will establish that there is no profitable
deviation in this larger class, there is also no profitable deviation in the original game.
When belief types are observable, it follows from the proof for the observable case that the firm
will offer a contract in which (bi , ri , ∆i , pi ) are chosen such that they are optimal in the observable
case. For each belief type fix these parameters in following, which amounts to selecting an optimal
βi
∆i ∈ [ 1−β
,
i
βˆi
].
1−βˆi
Hence, from now on we only need to consider deviations in which the firm
changes di for some belief type.
Now consider any maximal set {i, . . . , i0 } in which the admissible ironing induces a constant
perceived utility level. We will first argue that it is optimal for firm 1 to induce the same utility
level in any best response over such a set. Denote the profit the firm 1 earns from selling a contract
to belief type i by πi (di ). Hence, the profits from selling to the set {i, . . . , i0 } of belief types is
0
i
X
sj πj (dj )yj (dj ),
j=i
where yj (dj ) denotes the belief type j’s marginal consumer willing to buy from firm 1. Hence the
change in the profits from selling to belief types {i, . . . , i0 } for any small change (∆di , . . . , ∆di0 )
that does not violate the monotonicity of u0 is
i0
i0
j=i
j=i
X
1 X
sj πj (dj )∆dj −
sj yj (dj )∆dj ,
2t
(6)
where we use the facts that ∂πj (dj )/∂dj = −1 and ∂yj (dj )/∂dj = 1/(2t).
Since in an admissible ironing, u
ˆ{i,...,i00 } ≥ u
ˆ{i00 +1,...,i0 } for all i ≤ i00 < i0 , the firm earns higher
average per-consumer profits from selling to belief types {i, . . . , i00 } than from selling to belief types
{i00 + 1, . . . , i0 } for any weakly increasing u0 ; by definition, this is true when firm one offers the
same perceived utility level to all consumers and charges on average t as it does in the observable
beliefs’ case. Since utility is transferable, it thus also holds for any other amount the firm charges.
And when the perceived utility is increasing, firm 1 offers greater discounts to higher than to lower
49
types. Hence, the average per-consumer profits satisfy
Pi00
j=i sj πj (dj )
Pi00
j=i sj
P i0
j=i00 +1 sj πj (dj )
,
Pi0
s
00
j
j=i +1
≥
(7)
with the inequality being strict in case the perceived utility level is not constant.
Now suppose for the sake of contradiction that there exists some i00 ∈ {i, . . . , i0 − 1} for which
u0i00 < u0i00 +1 . Then for it to be unprofitable to lower the discount for all belief types in the set
{i00 + 1, . . . , i0 }, it must be that
0
0
j=i +1
j=i +1
i
i
X
1 X
sj πj (dj ) −
sj yj (dj ) ≥ 0,
2t
00
00
(8)
while for it to be unprofitable for to raise the discount for all belief types in the set {i, . . . , i00 } it
must be that
i00
i00
j=i
j=i
X
1 X
sj πj (dj ) −
sj yj (dj ) ≤ 0.
2t
Dividing Equation 8 by
Pi0
j=i00 +1 sj
and Equation 9 by
Pi00
j=i sj ,
(9)
and using that Equation 7 holds
with a strict inequality, this implies that
P i0
j=i00 +1 sj yj (dj )
Pi0
j=i00 +1 sj
Pi0
<
j=i sj yj (dj )
.
Pi0
j=i sj
But because firm 2 offers all belief types the same perceived utility while firm 1 offers strictly
higher utility to the belief types in the set {i00 + 1, . . . , i0 } than to those in {i, . . . , i00 }, the indifferent
consumer in the former set is located at a higher point in the unit interval than in the latter one,
contradicting the last inequality. We conclude that firm 1 offers all consumers that get the same
perceived utility in the admissible ironing, the same perceived utility in an optimal deviation.
To conclude the proof of this step, we show that for any maximal set {i, . . . , i0 } on which uj is
constant, firm 1’s best response is to set u0j = uj . To do so, we will furthermore relax the constraint
that u0 is increasing by allowing any response such that if uj is constant on a set {i, . . . , i0 }, then
so is u0j ; that is, we do not require that u0j be increasing between pools. We show that in this larger
class of possible strategies, the best response is to offer the symmetric equilibrium contract. To
see this, note that for each belief type ∂yi /∂di = 1/(2t) in case yi is interior and zero otherwise.
50
Furthermore, as both the discount and the transportation cost are additively separable and firm
2 offers the same perceived utility to all consumers in the pool, we are only considering choices of
discounts that induce the same yi = y for all belief types in the maximal set. Clearly, y = 0 reduces
profits and whenever y = 1 the firm does not want to increase the discount to the consumers in
this set. Think of firm 1 as choosing a location for the indifferent consumers y, and denote the
corresponding discount to belief type j as dj (y). Firm 1’ problem is thus to select y to maximize
0
i
X
sj πj (dj (y)) y,
j=i
giving rise to the first-order condition
0
−
i
X
j=i
0
sj (2t)y +
i
X
sj πj (dj (y)) = 0.
j=i
Since in a surplus-maximizing admissible ironing y = 1/2 and
Pi0
j=i sj πj (dj (y))
= t, it satisfies
the first order condition. Because furthermore πj (dj (y)) is decreasing in y, the left-hand side is
decreasing in y and hence the discounts offered in an admissible ironing are the unique ones to
solve firm 1’s problem. We conclude that playing the symmetric equilibrium is the unique best
βi
reply. Hence, up to the possible selection of an optimal ∆i ∈ [ 1−β
,
i
βˆi
]
1−βˆi
for each belief type and
the induced necessary change in the discount to hold ui constant, the best reply is unique.
To conclude, we note that there exists a unique symmetric equilibrium up to inconsequential
βi
variation in ∆i ∈ [ 1−β
,
i
βˆi
]
1−βˆi
and the corresponding changes in di to hold ∆i − di fixed, and that
this equilibrium satisfies the properties stated in the proposition. We argued in Step (iii) that for each
belief type i the borrowed amount, the interest rate and the penalty are set as in the observable case,
and that ∆i is selected among those that are optimal when belief types are observable. Hence, a
symmetric equilibrium is fully separating between belief types, with the borrowed amount, fines, and
interest rates as in the observable case. Furthermore, for any given optimal selection of ∆1 , . . . , ∆I ,
the discounts are chosen such u is the admissible ironing of u
ˆ by Step (iv). Conversely, by Step (v),
an admissible ironing in which (bi , ri , pi , ∆i ) are chosen such that they are optimal with observable
beliefs is a symmetric equilibrium.
51
Proof of Lemma 4. We think of firm n as determining the optimal prices fn , an for any given
level of perceived utility gross of transportation cost u
ˆn that it wants to offer to consumers, and
then selecting the optimal u
ˆn . The firm, thus, solves
α(fn + an ) + (1 − α)fn − k(an ) − c
max
(fn ,an )
v − fn ≥ u
ˆn .
s.t.
Hence, independently of u
ˆn , at the optimum k 0 (a(α)) = α or a(α) = (k 0 )−1 (α).
If firms 0 and 1 offer gross perceived utilities u
ˆ0 and u
ˆ1 , respectively, and v that is high enough so
that all consumers select one of the offered contracts, then there exists an indifferent consumer with
taste y satisfying u
ˆ0 − ty = u
ˆ1 − t(1 − y). This implies that firm 0’s demand is y = (ˆ
u0 − u
ˆ1 + t)/(2t),
and that ∂y/∂f0 = −1/(2t). Hence, the equilibrium up-front price must solve
max
f
u
ˆ1 − u
ˆ2 + t
[f + a(α) − k(a(α)) − c] .
2t
(10)
Solving the above establishes that f (α) = c + t − [a(α) − k(a(α))].
Since v is high enough for consumer to buy in a symmetric for any α, they do so when α = 0
and hence v > c + t/2 so that it is efficient for all consumers to buy from the closest firm. Hence,
the social welfare loss of the symmetric equilibrium for any share of naive consumers α is simply
DW L(α) = k(a(α)). Seller information about consumer naivete strictly lowers social welfare if
DW L(αns ) < pn DW L(αn ) + ps DW L(αs ),
where pn is the probability of signal αn and ps that of αs . Hence, αns = pn αn + ps αs . Therefore, a
sufficient condition for welfare to strictly decrease when the firms have access to the signal is that
DW L(α) is strictly convex at every α ∈ (0, 1). The first derivative equals
DW L0 (α) = k 0 (a(α))a0 (α) =
k 0 (a(α))
> 0,
k 00 (a(α))
where the second equality follows from totally differentiating the equilibrium condition k 0 (a(α)) =
α. Hence,
d k 0 (a(α))
1
DW L (α) =
> 0,
00
00
da k (a(α)) k (a(α))
00
52
where the inequality follows from the fact that k 0 (a)/k 00 (a) is strictly increasing in a over the interval
[a(0), a(1)].
Proof of Proposition 5. Again, we begin by proving that the model simplifies to the reducedform model introduced in Section 5.1. Denote the implicit solution to κ0 (t) = a
˜ by t∗ (˜
a). If a
consumer selects the contract f˜, a
˜, her avoidance effort equals min{θs , t∗ (˜
a)}. From now one, we
will ignore the constraint that consumers will not spend a higher avoidance level than θs , and
verify that this constraint is indeed satisfied at our solution ex post. We transform the model into
our reduced-form setup by defining the anticipated price as the overall payment of a sophisticated
consumer, the additional price as the unanticipated additional payment by a naive consumer, and
the exploitation cost as the inefficient consumer avoidance costs: f = f˜+(θs −t∗ (˜
a))˜
a, a = (θn −θs )˜
a,
and k(a) = κ(t∗ (a/(θn − θs )).
Hence, the optimal way to offer a given perceived utility level u
ˆ gross of transportation cost is
for the firm to solve
max α(f + a) + (1 − α)f
f,a
subject to
v − f − k(a) = u
ˆ.
Substituting the constraint in the maximization problem gives the exact same problem as in the
proof of Lemma 4. In an interior symmetric pure-strategy equilibrium, thus, a(α) = (k 0 )−1 (α) and
f (α) = c + t − [a(α) − k(a(α))]. Note that in a symmetric pure-strategy equilibrium in which all
consumers buy, the welfare loss of information depends only on the exploitation cost. Furthermore,
solving for the equilibrium avoidance costs in this case, one can show that k 0 (a)/k 00 (a) = (γ − 1)a,
which is increasing in a. Lemma 4, hence, implies the result.
It remains to verify that t∗ (˜
a) < θs . Since the firm chooses the maximal additional price a if all
consumers are naive, it suffice to verify that at a = (k 0 )−1 (1) the inequality holds. Using that in
this case
0
1 = k 0 (a) = κ0 (t∗ (˜
a)) t∗ (˜
a)
1
κ0 (t∗ (˜
a))
1
t∗ (˜
a)
= 00 ∗
=
,
θn − θ s
κ (t (˜
a)) θn − θs
(γ − 1)(θn − θs )
one has that t∗ (˜
a) < θs if θn < θs γ/(γ
˙
− 1), which holds by assumption.
53
Proof of Proposition 6. We again think of a firm as first solving for the optimal contract given
the perceived utility u
ˆ it wants to give consumers, and then choosing u
ˆ. The equilibrium distortion
is determined entirely in the first step.
The firm’s problem is
max α(f + a) + (1 − α)f
f,a
subject to
v − f − k(a) = u
ˆ.
In addition, note that if firms choose a = a(α) in equilibrium, then the welfare loss relative to
first-best is (1 − α)k(a(α)).
If α = 0, then it is obvious that in a solution to the above problem, k(a) = 0; otherwise, the
firm could decrease k(a) to zero and increase f correspondingly, increasing profits. And if α = 1,
then the welfare loss is zero. This completes the proof that perfect information maximizes social
welfare.
For α > 0, solving the constraint for f and plugging into the maximand gives the first-order
condition k 0 (a(α)) = α for the equilibrium a(α). We will show that if k 000 (a)/k 00 (a)2 is bounded
from above, then there is an α∗ such that the function (1 − α)k(a(α)) is convex on [0, α∗ ]; this
implies that for αns , αs , αn < α∗ , information lowers welfare.
The first derivative is
−k(a(α)) + (1 − α)k 0 (a(α))a0 (α) = −k(a(α)) + (1 − α)αa0 (α).
The second derivative is
(1 − 3α)a0 (α) + (1 − α)αa00 (α).
Differentiating the first-order condition k 0 (a(α)) = α totally with respect to α gives
a0 (α) =
1
k 000 (a(α)) 0
00
,
and
therefore
a
(α)
=
−
· a (α).
k 00 (a(α))
k 00 (a(α))2
Hence, the second derivative can be rewritten as
k 0 (a(α))k 000 (a(α))
a0 (α) (1 − 3α) − (1 − α)
,
k 00 (a(α))2
54
which is positive for sufficiently small α since a0 (α) > 0 and because the fact that k 0 (a)/k 00 (a) is
strictly increasing and bounded away from zero implies that
k 0 (a(α))k 000 (a(α))
<1
k 00 (a(α))2
and is bounded away from 1.
Proof of Proposition 7. Proceeding similarly as in the previous proposition, we solve for a firm’s
optimal contract that gives perceived gross utility u
ˆ to a consumer. If the consumer bears the
exploitation cost, then the problem is
max α(f + a) + (1 − α)f
f,a
subject to v − f = u
ˆ.
Since increasing a increases profits without affecting the constraint, the firm chooses a = amax for
any α > 0. This implies that information has no effect on welfare.
If the firm bears the cost, then the problem is
max α(f + a) + (1 − α)f − αk(a)
f,a
subject to v − f = u
ˆ,
so that the firm chooses k 0 (a) = 1 for any α > 0, and again information has no effect on welfare.
Proof of Proposition 8. Given b, we can solve the firm’s revenue maximization problem taking
into account the consumers’ behavior. Naive consumers who accepted a contract solve
min φq1γ + βφ[(1 + r)(b − q1 )]γ .
q1
Solving this for q1 yields
"
q1 =
γ
1
#
β γ−1 (1 + r) γ−1
1
γ
b
1 + β γ−1 (1 + r) γ−1
{z
}
|
≡η(r)
For any given b, the firm thus chooses r to maximize its interest income from naive consumers
delaying repayment, i.e.
max b(1 − η(r))(1 + r)
r
55
Denote the optimal revenue the firm earns by zb, where we know z > 1.
Now for a given α, a firm’s problem choosing to offer a perceived utility level gross of transportation costs of u
ˆn solves
max(1 − α)b + αzb − b − d
b,d
s.t. u(b) − 2c(b/2) + d = u
ˆn
Solving for d in the constraint and substituting into the maximand gives
max u(b) + α(z − 1)b − 2c(b/2),
b
yielding the first-order condition
u0 (b∗ ) = c0 (b∗ /2) − α(z − 1).
(11)
This implies that there is overlending in equilibrium to sophisticated types. Note that the marginal
cost of repaying of a naive consumer equals
γ−1
η(r)γφ(η(r)b)
γ−1
+(1−η(r))γφ [(1 − η(r))b]
γ−1
= γφb
b γ−1
{η(r) + (1 − η(r)) } > γφ
= c0
2
γ
γ
b
,
2
where the inequality follows from the fact that the term in curly brackets is minimized at η = 1/2
and then equal to (1/2)γ−1 , but η(r) < 1/2 and hence the term is greater (1/2)γ−1 . Thus, the
marginal cost of repaying is greater for naive consumers than for sophisticated ones, and hence
they also overborrow.
Differentiating Expression 11 totally with respect to α and solving for the derivative of b∗ with
respect to α gives
b∗ 0 =
z−1
.
− u00 (b(α))
c00 (b(α)/2)
This derivative is strictly increasing in α since u000 (b) > c000 (b/2)/4 for all b. Since b∗ is therefore
convex in α, lending increases with information about α. Because information yields different
individuals to borrow different amounts while increasing total lending, it thus lowers total social
welfare.
56