Credit Card Competition and Naive Hyperbolic
Consumers
Elif Incekara-Hafalir
Tepper School of Business, Carnegie Mellon University
5000 Forbes Ave, PA 15213, USA
E-mail: [email protected]
Phone: 1-412-268-5806
April 20, 2012
Abstract
We consider a credit card market with hyperbolic consumers. We show
that naive hyperbolic borrowers might be unresponsive to interest rates when
the credit card contracts o¤er a grace period. Consequently, we demonstrate
that there might be no competition with regard to the interest rate even if the
I would like to thank Edward Green, Susanna Esteban, Isa Hafalir, Jeremy Tobacman and Neil
Wallace for their valuable comments and suggestions. I am very much indebted to Kalyan Chatterjee
for his guidance and advice.
1
consumer accepts only one card. We determine whether credit card companies
can exploit time-inconsistent consumers and gain positive expected pro…ts. We
show that, in fact, there are circumstances in which both zero and positive
expected pro…ts are possible.
Keywords: credit cards, competition, time inconsistency
1
Introduction
The credit card industry has evoked much interest in the last few decades because of
the conjunction of persistently high interest rates and what appears to be vigorous
competition among credit card providers (Ausubel, 1991). The two questions that
have been most puzzling are why interest rates remain high despite the competition
and why consumers continue to borrow at these high rates. Moreover, there is no
agreement in the literature as to whether banks/companies earn competitive pro…ts
in this market (Evans and Schmalensee, 2005). Motivated by these questions, we
present a theoretical model of credit card competition to shed light on seemingly
non-competitive interest rates and the possibility of positive pro…ts in the market
equilibrium. The main features of the model are time-inconsistent consumers and
a grace period o¤ered in credit card contracts. We therefore also contribute to the
debate as to whether biased consumers are “money pumps” or whether competition
e¤ectively eliminates the disadvantage arising from that bias.1
1
See Laibson and Yariv (2004), DellaVigna and Malmendier (2004), Eliaz and Spiegler (2006),
Gabaix and Laibson (2006), and Carlin (2009) for discussions of markets with time-inconsistent
2
We show, for some parameter values, that there are nash equilibria with noncompetitive interest rates that result in positive pro…ts. For other parameter values,
there exist zero pro…t equilibria. We observe interest rate competition among the
…rms only if the consumer accepts more than one credit card contract.
As discussed in Ausubel (1991), it seems reasonable that there are three groups of
consumers in the credit card market. The …rst group contains convenience users who
never pay interest on their cards and are hence termed "deadbeats" in the credit card
industry jargon. The second group comprises borrowers who revolve debt and pay
interest. This group searches for the lowest interest rate available. Finally, the last
group encompasses those who believe that they are not going to borrow on their cards
but end up borrowing because of commitment problems. This group does not search
for the lowest interest rate and thus is the most pro…table group for the companies.
This is because the …rst group does not generate interest revenue and the second
group consists of high-risk consumers. In this paper, we analyze the contracts o¤ered
to the last group.
In our model, there is an initial period of contracting, followed by three consumption periods.2 In the initial period, two credit card companies simultaneously o¤er
contracts that are de…ned by the interest rate and the credit limit, and the consumer
chooses contract(s). The contracts include a grace period, which we specify as a
period in which the consumer may pay his debt without interest.
consumers.
2
We use the terms "contracting period" and "initial period" interchangeably.
3
The consumer is time inconsistent and has a constant income in each period. We
model time inconsistency using the quasi-hyperbolic discount structure from Phelps
and Pollak (1968) and Laibson (1997). The consumer chooses one (or both) of the
contracts at the contracting period according to his plan for future consumption.
If there is at least one contract o¤ering enough credit (more than the amount the
consumer wants to borrow during each consumption period), the consumer chooses
only one contract because there is an in…nitesimal cost of accepting a contract.3
Otherwise, he chooses both contracts. At each period, he decides how much to
borrow and how much to pay back depending on his plan for future consumption.
The consumer does not pay interest if the previous period’s borrowing is less than his
income (grace period feature). For convenience, we consider the consumer making
decisions at di¤erent periods as di¤erent "selves" of the consumer. Owing to time
inconsistent preferences, each period self underestimates future consumptions. We
analyze only the interesting case in which the consumer’s contracting-period self
believes that he will borrow less than his income during each consumption period
and period-one self borrows more than his income.
If the consumer is not allowed to default (in other words, if the cost of default
for the consumer is prohibitively high), companies can safely o¤er high enough credit
limits and the consumer chooses only one contract. The contracting-period self does
not take the interest rate into account when choosing a contract because he believes
3
Filling out an application is such an example.
4
that he will not pay interest.4 However, the …rst-period self ends up accumulating
debt and thus pays interest. Therefore, the contracting-period self’s indi¤erence to
interest rates eliminates the interest rate competition, whereas the …rst-period self’s
high borrowing activity creates a pro…t opportunity for the chosen contract.
We also analyze the problem by allowing the consumer to default. There is an
exogenously given cost of default for the consumer. We interpret this cost as the cost
of bankruptcy proceedings and of receiving unfavorable terms in any contract in the
future after declaring bankruptcy. If this cost is high enough (the consumer’s risk of
default is low enough), we do not observe interest rate competition, and positive pro…t
equilibrium is possible similar to the no-default case. On the other hand, if the cost
is small enough, we obtain zero-pro…t equilibria. Interestingly, the existence of zeropro…t equilibrium is not necessarily because of the competition on interest rates. In
some cases, the companies o¤er only a small credit limit, which may be accompanied
by positive interest rates, preventing consumers from accumulating interest-bearing
debt.
4
Calem et al. (2006) report that convenience users are relatively indi¤erent to credit card interest
rates. Ausubel (1999) hypothesizes that consumers may systematically underestimate the extent of
their future credit card borrowing. Calem’s (2006) …nding and Ausubel’s (1999) "underestimation
hypothesis" would be consistent with our view that there exist (supposedly) convenience users who
do not pay attention to the interest rates as if placing zero probability on having to pay interest on
future credit card expenditures.
5
1.1
Related Literature
In the literature, there are experimental and empirical studies analyzing consumer
behavior in the credit card market.5 There are also a few theoretical papers that
provide alternative explanations for the sticky and high interest rates. Parlour and
Rajan (2001) construct a model in which the competing …rms cannot sustain zeropro…t equilibria if the incentive to default is high enough and if there is multiple
contracting. The predictions of our model and that of Parlour and Rajan (2001) are
completely opposite with regard to the e¤ects of the bankruptcy law changes of 2005.
The new law increased the cost of default for the consumers. According to Parlour
and Rajan’s (2001) model, this change should lower the interest rates and decrease
the positive pro…ts to a more competitive level. According to our model, however,
this change would not a¤ect the interest rates but would increase the possibility of
positive pro…ts. Simkovic (2009) also argues that Bankruptcy Abuse Prevention and
Consumer Protection Act of 2005 did not result in a lower cost of credit to consumers
and lower pro…ts for credit card companies, which is in line with the predictions of
our model.
In an important paper, DellaVigna and Malmendier (2004) analyze a …rm’s pro…tmaximizing contract design when the consumers are partially naive hyperbolic discounters. They determine that a …rm prices “leisure goods,” such as credit card…nanced consumptions, higher than the marginal cost if the …rm proposes a two-part
5
Notably, Calem and Mester (1995), Ausubel (1999), Laibson, Repetto and Tobacman (2003),
Ausubel and Shui (2004), and Calem et al. (2006).
6
tari¤. They also show that this result does not depend on the monopoly assumption,
although the pro…ts realized by the …rms are zero under perfect competition. They
provide strong evidence for the predictions of their model from a variety of markets
(e.g. health clubs, credit cards). Eliaz and Spiegler (2006) characterize the menu of
contracts that a monopolist would o¤er when facing dynamically inconsistent consumers. They also provide examples for a variety of markets, including credit cards,
for the application of their results. These two papers analyze monopoly contracts
when the consumers are dynamically inconsistent. Our interest, however, concerns
the competitive contracts when consumers are dynamically inconsistent. Moreover,
our model is speci…c to the credit card market, mainly because of the inclusion of the
default risk. It is crucial to include the default risk in a model speci…c to the credit
card market in order to contribute to the literature on the possibility of noncompetitive pro…ts.6
In another paper, Brito and Hartley (1995) show that rational individuals may
choose to pay interest on a credit card rather than to pay transaction costs on regular bank loans. We are interested in the competition between credit card companies
rather than the competition between credit card …nancing and other forms of …nancing.
In our model, we include the grace period feature of the credit card market. To
6
Ausubel (1991) …nds evidence for noncompetitive pro…ts, but Evans and Schmalensee (2005)
criticize this evidence on the basis of not adjusting for the di¤erences in risk factor. Moreover,
credit card companies argue that the interest rates are high because of the high default risks but
not because of the noncompetitive prices (Rougeau, 1996).
7
the best of our knowledge, this feature was not included in the previous credit card
competition models. If there was no grace period in the contracts, then there would
be no convenience users and these contracts would be — correctly— treated as loans.
The grace period feature enables credit card companies to earn pro…ts from consumers
who otherwise would not borrow at these high rates.
In summary, we show that some naive hyperbolic consumers are unresponsive to
interest rates because the contracting-period self places zero probability on having
to pay interest in future periods. The consumption-period self’s high expenditure
level, however, accrues interest. This discrepancy between the selves may result in
non-competitive interest rates and sometimes positive pro…ts. We demonstrate that
there are in fact circumstances in which both zero and positive expected pro…ts are
possible without competition on interest rates. The rest of the paper is organized as
follows: Section 2 presents the model. Section 3 describes the equilibria for di¤erent
circumstances. Section 4 discusses a number of extensions, and …nally, Section 5
concludes.
2
The Model
There are three periods of consumption preceded by an initial period, which is designated for contracting only.7 One good is present at each consumption period. The
7
We need at least three consumption periods to observe time-inconsistent behavior that may
result in an interest-bearing debt.
8
model contains three agents: one consumer and two companies, who compete against
each other for the credit card business of the consumer. Companies know that the
consumer is a naive hyperbolic agent.
2.1
The Consumer
Following Phelps and Pollak (1968) and Laibson (1997), we use quasi-hyperbolic
discounting to model the time-inconsistent consumer. The literature discusses two
kinds of consumers. The …rst kind is called the naive consumer, named so because he is
not aware of his time inconsistency. Speci…cally, he knows that his discounting today
is f1;
;
2
;
3
; ::g; and believes that from tomorrow on it will be f1; ;
although in reality it will be f1;
;
2
3
;
2
;
3
; ::g;
; ::g. The second kind, the sophisticated
consumer, is aware of his time inconsistency. He knows that his discounting today
is f1;
;
2
;
3
; ::g; and he correctly anticipates that it will be f1;
;
2
;
3
; ::g
from tomorrow onward.8
We analyze only the naive hyperbolic consumer with positive time discounting—
2 [0; 1] and
2 [0; 1].9 The consumer chooses contracts in the initial period and
consumes the consumption good in the following periods. There is an arbitrarily
small cost for accepting a contract. The consumer’s total utility is also a¤ected by
8
O’Donoghue and Rabin (2001) introduce a model to represent a partially naive hyperbolic
consumer who is aware of his time inconsistency but who underestimates its severity. The partially
naive hyperbolic consumer knows that his future discounting today is f1; ; 2 ; 3 ; ::g, and he
incorrectly believes that it will be f1; 0 ; 0 2 ; 0 3 ; ::g from tomorrow onward with < 0 :
9
We can get the same qualitative results for a subset of partially naive consumers, and it is
demonstrated in Section 4.3. We do not get the same results with the sophisticated hyperbolic
consumers since they correctly calculate their future debt just like time-consistent consumers.
9
whether he defaults at the last period. In each period t = 0; 1; 2, the consumer aims
to maximize Ut ; where
U0 =
2
u(c01 ) + u(c02 ) +
u(c03 ) +
U1 = u(c11 ) +
u(c12 ) +
2
U2 = u(c22 ) +
u(c23 ) +
v(d2 ):
u(c13 ) +
2
v(d0 ) ;
2
(1)
v(d1 );
We do not need to write U3 because period three is the terminal period, which
implies that the period-two plan is implemented. It is convenient to consider the
consumer who makes decisions at di¤erent periods as di¤erent "selves" of the consumer. In equation 1, ct 2 R+ represents the consumption in period
according to
the period-t self, and dt 2 f 1; 0g is the period-t self’s plan concerning defaulting,
where dt =
1 denotes a default and dt = 0 denotes no default. The function u
is strictly increasing and concave, and satis…es the standard Inada conditions. The
function v satis…es v( 1) =
C and v(0) = 0, where C > 0 is the exogenous cost of
default. Note that lower C represents riskier prospects for the companies.
The consumer chooses a trade yjt = (sj ; ntj1 ; ptj2 ; ntj2 ; ptj3 ; dt ) for each company’s
card j = 1; 2 during each period t = 0; 1; 2. Here sj 2 f0; 1g; where sj = 1 denotes
that the contract is signed with company j and sj = 0 denotes that it is rejected;
ntj 2 R+ is the new borrowing from company j at time
10
according to the period-t
self; ptj 2 R+ is the repayment to company j at time
according to the period-t self.
At each consumption period, the consumer receives an income of m.
These trades of the consumer determine the consumption plan of each period-t
self:
ct1
=m+
2
X
sj ntj1 ; t = 0; 1:
(2)
j=1
ct2
=m+
2
X
sj ntj2
ptj2 ; t = 0; 1; 2:
j=1
ct3
=m
t
1+d
2
X
sj ptj3 ; t = 0; 1; 2; 3:
j=1
The consumer chooses the trades sequentially. The consumption and default plan
of self-t does not a¤ect the plans of the subsequent selves, except for the trades
completed at time t — those for which t = .
2.2
The Companies and the Class of Contracts
For simplicity, we assume that the only source of revenue is from the interest payments.10 Company j’s pro…t is
j
= (1 + d2 ) (n1j1
p2j2 )rj + d2 n1j1 + n2j2
p2j2 :
Each credit card company j charges an interest rate of rj for loans of more than one
period, although it is not permitted to charge interest for only one-period loans (the
grace period). A credit card company loses everything lent if the consumer defaults.
10
It has been documented that over 70 percent of the credit card issuers’revenue is from revolvers
(Chakravorti, 2003).
11
Each company j 0 s strategy set consists of contracts speci…ed by a credit limit lj
and an interest rate rj 2 [0; 1].11 The consumer’s total debt cannot be greater than
his credit limit, and his repayment cannot be higher than his income or his total debt
at any period.
2.3
Strategic Interaction
The two companies make simultaneous contract o¤ers, and the consumer decides
which one to choose in the initial period; subsequently, the consumer makes two
sequential decisions as described in Section 2.1. Therefore, the only strategic game
between the companies and the consumer takes place at the initial period. We focus
on the pure strategy subgame perfect equilibria of this game.12
3
Analysis
We …rst determine how the consumer chooses among the contracts, given that the
default is not allowed. Our approach is as if there is only one interest rate a¤ecting
the consumer’s problem. We later con…rm that only one interest rate is relevant in
equilibrium, even if two di¤erent contracts are accepted. Second, we determine the
set of feasible contracts depending on the consumer’s default risk. Third, we show
11
Our results do not change as long as the upper bound for the interest rate is …nite.
Contrary to the standard subgame perfect equilibrium, a naive hyperbolic consumer has incorrect beliefs about his future decisions. This does not create a problem for the de…nition of the
equilibrium in our case, as there is no strategic game after the contracting period, just a decisionmaking problem.
12
12
the existence of equilibrium within the set of feasible contracts.
3.1
The Consumer’s Behavior with no Option to Default
The consumer has to pay all of his debt back by the last period. Therefore, the
new borrowing at the second period will never create interest revenue. On the other
hand, borrowing in the …rst period may result in interest revenue, as there are two
periods for the consumer to pay back. Therefore, we analyze only …rst-period new
borrowing.13
3.1.1
Exponential Consumer (
= 1)
We …rst analyze the exponential consumer as our benchmark case (0
1 and
= 1). The exponential consumer is time consistent, and therefore the …rst-period
borrowing according to the initial-period self (n01 ; believed amount of borrowing)
and according to the period-one self (n11 ; actual amount of borrowing) is the same
(n01 = n11 ).14 Exponential consumers can either be borrowers or convenience users.
There exists a
such that the contracting period self correctly knows that he will not
consume more than his income in any consumption period and that he will not pay
interest for
(convenience user). Therefore, he is unresponsive to interest rates.
Nevertheless, being unresponsive to interest rates does not hurt him, as he will not
13
The interest revenue will come only from the borrowing in the …rst period, even if we allow the
consumer to default. If the consumer borrows more than m at the second period, he cannot pay it
back in the third period, declares bankruptcy and does not pay interest. If he borrows less than m
in the second period, then he can pay it back in the third period and does not pay interest.
14
In general, nt = n and pt = p for all 0 t
:
13
pay interest in the future anyway, as n01 = n11 . The companies earn zero pro…t even
without competition on interest rates. If
<
; the contracting period self correctly
knows that he will consume more than his income in the following consumption period
and will pay interest (borrower). Therefore he looks for the lowest interest rate. As
a result, simple Bertrand competition drives the interest rates down to zero.
3.1.2
Naive Hyperbolic Consumer (
< 1)
A naive hyperbolic consumer is time inconsistent (
< 1) and is not aware of it.
Therefore, he underestimates his future borrowings, in contrast to the exponential
consumer; the initial-period self’s believed amount of borrowing for the …rst period,
n01 , is always less than the actual amount of borrowing, n11 .
In this section, we show that there exists a naive hyperbolic consumer (speci…ed
by
and ) with an initial-period self who does not plan to borrow more than his
income at any consumption period, but with a period-one self who ends up borrowing.
Proposition 1 For a naive hyperbolic consumer, there is a
for all
. There is also a
where
and
<
r=1 (
r=1 (
) > 0; such that n01
; such that n01
m
m < n11 for all ( ; )
):
Proof. See the Appendix.
In Figure 1, we demonstrate how a naive hyperbolic consumer’s period-one self
may end up borrowing more than his income even though his initial-period self plans
not to. The x-axis shows the
discount factor in [0; 1]: The y-axis shows the
14
hy-
perbolic discount factor in [0; 1]: The initial-period self believes that the
discount
factor does not a¤ect his future consumption plans. If the parameter values, de…ning
the consumer, are in region A1 or A2 ; the consumer believes that he will not accumulate interest-bearing debt. If the parameter values are in region C; he believes
that he will accumulate interest-bearing debt and pay interest even if the interest is
at the highest possible rate.15 On the other hand, the period-one self takes
into
account when deciding how much to borrow. For the period-one self, the interest
rate (r) and the exponential discount factor ( ) are no longer the only determinants
of borrowing; the hyperbolic discount factor ( ) plays a role as well. Therefore, the
vertical line at
r=1
separating the interest payers from convenience users at the high-
est interest rate transforms into the downward sloped line in the diagram.16 If the
consumer is in region A2 ; B1 ; or C; his period-one self accumulates interest-bearing
debt even at the highest interest rate. Note that there is a con‡ict between what
the initial-period self believes and what the period-one self ends up doing if the parameter values are in region A2 ; irrespective of the interest rate. Proposition 1 shows
the existence of consumers in this region. We analyze only the consumers in region
A2 throughout the paper. Although the initial-period self is unresponsive to interest
rates, the period-one self ends up paying interest.
Because there is an arbitrarily small cost for accepting a contract, the consumer
15
Since r will be endogenously determined, we want cuto¤s to be free of r: Therefore, we
determine the cuto¤s for r = 0 and r = 1: Later, we show that our analysis holds for all r 2 [0; 1]:
16
The …gure is only for expositional purposes, the downward sloped line does not have to be linear.
15
will choose only one contract if there is at least one company o¤ering a large enough
credit limit. Suppose the o¤ered credit limits are l1 and l2 :
if max fl1 ; l2 g
n0 > minfl1 ; l2 g, then the consumer accepts the contract with
the higher credit limit only;
if min fl1 ; l2 g
n0 , then the consumer accepts one contract randomly;17
if max fl1 ; l2 g < n0 , then the consumer accepts both contracts.
Description of the equilibrium
Recall that we consider a consumer who is in
region A2 and a context in which default is infeasible. First of all, the consumer’s
initial-period self is unresponsive to interest rates; therefore, the companies do not
have to compete on interest rates. Second, he never defaults, and therefore the
companies do not need to restrict the credit limits to prevent him from defaulting.
And third, his period-one self ends up accumulating interest-bearing debt and paying
interest. All of these features create an ideal environment for noncompetitive pro…ts,
as each company chooses the pro…t-maximizing interest rate and credit limit, just
like a monopolist.
Proposition 2 If the consumer is a naive hyperbolic discounter with ( ; ) such that
;
<
r=1 (
) and if default is not allowed, each company will o¤er a monopoly
contract (l > m; r > 0). This results in positive pro…ts.
17
See Section 4.1 for an alternative to the stated tie-breaking rule.
16
Proof. The consumer’s unresponsiveness to interest rates together with no possibility
of defaulting enables companies to o¤er monopoly contracts. The contracting-period
self of the consumer chooses one contract randomly, and the company with the chosen
contract earns a positive pro…t, as the period-one self borrows more than his income.
3.2
The Consumer’s Behavior with Defaulting
If the consumer is allowed to default in the last period, we need to take each self’s
default plan into account. The companies do not want to o¤er default-triggering
contracts; therefore, they may not be able to o¤er high enough credit limits for the
consumer to accumulate an interest-bearing debt level.
If the consumer were time consistent, it would have been enough to compare the
consumer’s gain from defaulting with the cost of defaulting only from the point of
the contracting-period self. With the time-inconsistent consumer, however, we need
to compare the consumer’s gain from defaulting with the cost of defaulting according
to each period self.
Lemma 1 Suppose that l1 and l2 are in the appropriate ranges, such that the consumer chooses only one contract. Then, the unselected contract’s interest rate does not
=
@G0
@l2
> 0;
> 0 and
@Gc
@r1
> 0;
a¤ect the gain from default. The gain from default is G0 (l1 ; l2 ) with
according to the contracting-period self, and is Gc (l1 ; r1 ) with
@Gc
@l1
@G0
@l1
according to the consumption-period selves if the selected contract is (l1 ; r1 ). More17
over, G0 (l1 ; l2 ) < Gc (l1 ; r1 ) for low enough values of l2 and for all r1 2 [0; 1]:
Proof. See the appendix.
The intuition behind this lemma is as follows. The contracting-period self believes
that he will not pay interest; therefore, the gain from defaulting is not a¤ected by
interest rates. Moreover, a marginal gain from each credit limit increase is the same,
as the credit limits always appear as a sum in the gain function. Consumption-period
selves, on the other hand, realize that they pay interest; therefore, the gain from
defaulting increases with the interest rate of the chosen contract. The credit limit
o¤ered by the contract that is not chosen does not a¤ect the gain, as the contracting
period has already passed, and the consumer has only the chosen contract on hand.
As a result, the set of credit limits that do not induce the consumer to default are
shown by the shaded area in Figure 2.18
If we further investigate gain functions from di¤erent selves’point of view, we …nd
that two di¤erent e¤ects determines the gain. One is "the credit e¤ect," and the other
is "the spending e¤ect." The credit e¤ect is the e¤ect of the potential credit limit to
default on; it is larger in the contracting period than in consumption periods, as the
total possible credit limit to default on is l1 + l2 instead of just l1 : The spending e¤ect
is the e¤ect of the expenditure level, and it is larger during the consumption periods
than the contracting period because the consumer spends more in the consumption
periods than he had planned in the contracting period. This is why Gc (li ; r) is the
18
Note that Gc (li ; r) = C would be closer to the origin for higher values of r:
18
restrictive constraint for the lower values of lj (the credit e¤ect is smaller). If the
consumer chooses both contracts, however, the credit e¤ect is the same in all periods
(the potential credit limit to default on is constant at l1 + l2 ), although the spending
e¤ect is greater in the consumption periods. Therefore, the gain from defaulting is
determined according to the consumption-period selves, namely, Gc (l1 ; l2 ; r).
Lemma 2 Suppose that l1 and l2 are in the appropriate ranges such that the consumer
chooses both contracts, and suppose that r1
Gc (l1 ; l2 ; r1 ); with
@Gc @Gc @Gc
; @l2 ; @r1
@l1
> 0 and
r2 . Then, the gain from defaulting is
@Gc (l1 ;l2 ;r1 =0)
@l1
=
@Ge (l1 ;l2 ;r1 =0)
@l2
> 0:
Proof. See the Appendix.
When the consumer has two contracts on hand, the no-default-region is determined by Gc (l1 ; l2 ; r1 = 0)
3.2.1
C. This is shown in Figure 3.19
Description of the Equilibria
We will …rst illustrate the number of contracts that the consumer will choose under
di¤erent contract o¤ers and determine whether he will pay interest on them. Afterward, we will determine the equilibria. Let us partition the Cartesian plane with
companies’credit limit o¤ers (abstracting from interest rate considerations) on the
axes into four di¤erent regions, as shown in Figure 4. If the o¤ered contracts lie in
region one, the consumer chooses only one contract and pays the agreed interest on
that contract. If the o¤ered contracts are in region two, the consumer chooses only
19
Note that the no-default region would shrink with an increase in r1 :
19
one contract but does not pay interest, because the chosen contract’s credit limit
prevents him from spending more. If the contracts are in region three, the consumer
chooses both contracts but does not pay interest (even the total credit limit o¤ered
by these contracts does not allow the consumer to accumulate interest-bearing debt).
If the contracts are in region four, the consumer chooses both contracts and pays
interest on the contract with the lower interest rate (the total credit limit will allow
him to accumulate interest-bearing debt and the …rst-period self will pay the contract
with higher interest rate within the grace period). In the following proposition, we
show the existence of equilibria (in four di¤erent regions of this graph), which are
shaped by the consumer’s risk level.
Proposition 3 The equilibria of the game depend on two cuto¤ default values, namely
C and C
with C
<C ;
If C > C ; there exists a "Region 1" equilibrium in which the companies o¤er
contracts with non-competitive interest rates and at least one company o¤ers a
high credit limit inducing the consumer to choose only one contract and accumulate interest-bearing debt, which results in positive pro…ts.
If C
C
C ; there exists a "Region 2" equilibrium, in which at least one
company o¤ers a contract with a moderate credit limit inducing the consumer to
choose only one contract but preventing him from accumulating interest-bearing
debt, which results in zero pro…ts. Moreover, there also exists a "Region 4"
equilibrium, in which the companies o¤er contracts with competitive interest
20
rates and low credit limits, inducing the consumer to choose both contracts,
which results in zero pro…ts.
If C
C ; there exists a "Region 3" equilibrium, in which the companies o¤er
contracts with possibly positive interest rates with low credit limits, inducing
the consumer to choose both contracts but preventing him from accumulating
interest-bearing debt, which results in zero pro…ts.
Proof. See the Appendix.
The intuition behind this proposition is as follows. If the cost of the default is
high enough, some of the contracts in region one are within the no-default region.
Therefore, the companies can o¤er region one contracts, which allow the consumer
to accumulate an interest-bearing level of debt from one company only. If the cost
of default is in midrange, the region one contracts are no longer feasible and the
companies cannot o¤er the most pro…table contracts. Instead, they will o¤er contracts
with moderate credit limits, which make the consumer to accept only one contract
but do not allow him to accumulate interest-bearing debt (region two contracts). As
a result, the companies make zero pro…ts, but not necessarily because of competitive
interest rates. If the cost of default is low, then the companies cannot o¤er high
enough credit limits to make the consumer to choose only one card. When the
consumer chooses to accept both contracts, interest revenue is possible only if the total
credit limit allows the consumer to accumulate interest-bearing debt. In that case,
the consumer would pay the company with the higher interest rate within the grace
21
period to minimize the interest payment. Foreseeing this, the companies compete
on interest rates giving the equilibrium in region 4. The last case is when the total
credit limit does not allow the consumer to accumulate interest-bearing debt. Since
interest revenue is not possible, the companies do not compete on interest rates but
still earn zero pro…t, which is the equilibrium in region 3.
4
Extensions
In this section, …rst, we embed the initial period contact choice decision into a search
model with no option to default. Second, we analyze the equilibria when the grace
period interest rate is endogenous. Third, we extend our analysis for quasi-hyperbolic
consumers. Last, we determine whether the consumer would have been better o¤ had
we restricted the credit limits to income.
4.1
Contract Choice with a Search Model
In this subsection, we relax the randomization-between-contracts assumption by embedding the …rst-period contract choice problem into a search model when there is
no option to default. In this new model, the credit card companies o¤er contracts
sequentially in the initial period. Each contract is equally likely to be the …rst o¤ered
contract. There is an in…nitesimal cost of inquiring about a contract. The consumer
either inquires about the …rst contract and accepts it or he inquires about the second
contract as well before deciding which one(s) to accept.
22
If the initial–period-self is planning to maintain positive debt, then he examines
the credit limit and the interest rate of the …rst o¤ered contract. If the …rst contract
either o¤ers a low credit limit (less than the consumer’s believed amount of borrowing,
n0 ) or a positive interest rate, he inquires about the second contract as well, and
decides which one(s) to accept. If the …rst contract o¤ers a high enough credit limit
and zero interest rate, then he accepts this …rst contract without inquiring about the
second one. If the contracting-self of the consumer is not planning to borrow and
pay interest, he accepts the …rst contract o¤ered as long as the …rst contract’s credit
limit is high enough. Otherwise, he inquires about the second contract as well.
If the consumer is time consistent, there exists zero pro…t equilibria because of
standard Bertrand competition. If the consumer is naive hyperbolic, however, we
show that there is a unique positive pro…t equilibrium. This follows because the
initial-period-self is unresponsive to interest rates.
Proposition 4 If the consumer is a naive hyperbolic discounter with ( ; ) such that
;
<
r=1 (
) and if default is not allowed, the monopoly contracts provide the
unique equilibirum, which results in positive expected pro…ts.
Proof. See the Appendix.
Including the default risk in this model would prohibitively complicate the analysis. Although we cannot provide the equilibria of this model with default risk, one
can argue that the default risk would decrease the pro…ts not only because of the
competition on the interest rates but also because …rms o¤er low credit limits to
23
prevent defaulting.20 With regard to the competition, another issue is whether the
second …rm would try to debias the consumer and make him sophisticated about his
future borrowing, and consequently attract him by o¤ering a lower interest rate. If
this happens, then we may expect to observe competitive interest rates with zero
pro…ts. However, Gabaix and Laibson (2006) analyzed a similar question in a general
model and showed that "educating a consumer about competitors’add-on schemes
e¤ectively teaches that consumer how to pro…tably exploit those schemes, thereby
making it impossible for the educating …rm to pro…tably attract the newly educated
consumer." In our model, educating the consumer about his self-control problem
would also teach him other ways to deal with this problem rather than making him
choose the contract with the lower interest rate, and therefore would not be a profitable strategy.21
4.2
Endogenous Introductory Interest
Our base model considers the introductory interest rate to be exogeneously set to
zero, and this corresponds to the grace period. In this subsection, we incorporate
the competition on the introductory interest rate. We specify each contract j with
a credit limit lj , an introductory interest rate rj1 ;for one-period loans, and a regular
interest rate rj2 for two-period loans. Note that interest revenue from one-period
loans is possible in this case.
20
21
As in the analysis of the original model with default risk.
For example, he can ask the credit card company to decrease his credit card limit.
24
To simplify the analysis, we only analyze the case of no default. Since the consumer’s contracting-period-self believes that he will borrow for only one period, he
is responsive to interest rates for one-period loans only, namely rj1 ; but not to interest rates for two-period loans, namely rj2 : Therefore, there will be competition on
one-period interest rates but not on two-period interest rates.
The following proposition states that interest rates for one-period loans will decrease to zero, implying a grace period, because of the competition. Moreover, there
will not be competition on the interest rates for two-period-loans.
Proposition 5 Suppose the consumer is a naive hyperbolic discounter with high
enough
and low enough
values and default is not allowed. Then in equilibrium,
the introductory interest rate is zero (for one-period loans) and the regular interest
rate is the monopoly rate (for two-period loans).
Proof. See the Appendix.
This …nding is very much in line with what we observe in the credit card market.
Companies seem to compete most …ercely on the introductory rates (teaser rates),
which are generally o¤ered at 0%, but there seems to be not much competition seems
to be not much on the regular rates. Ausubel and Shui (2005)’s …nding in their …eld
experiment is also in line with this proposition. They …nd that consumers choose a
shorter introductory period o¤er with a lower interest rate than a longer introductory
period o¤er with a higher interest rate, although ex-post borrowing behavior shows
that the longer introductory period o¤er is a better one. Ausubel and Shui (2005)
25
explains this …nding with time-inconsistent preferences.
4.3
Quasi-hyperbolic Consumer
We now show that we can get similar qualitative results for a subset of quasihyperbolic consumers as well. We analyze this case by assuming
the calculations. Note that a time-consistent consumer with
A quasi-hyperbolic consumer with
= 1 to simplify
= 1 never borrows.
< 1; however, borrows and underestimates his
borrowing.
Proposition 6 For a quasi-hyperbolic consumer with actual hyperbolic discount factor of
and a
< 1 and a believed hyperbolic discount factor of 1 > b > , there exists a
(b) such that n01
Proof. See the Appendix.
m and n11 > m for b >
and
<
r=1 (
b):
In Figure 5, we demonstrate how a partially naive hyperbolic consumer’s periodone self may end up borrowing more than his income even though his initial-period self
plans not to. The x-axis shows the believed hyperbolic discount factor b in [0; 1]: The
y-axis shows the actual hyperbolic discount factor
in [0; b]: The intuition behind
this graph is similar to that behind Figure 1. If the parameter values, de…ning the
consumer, are in region A2 ; the consumer believes that he will not accumulate debt,
but will end up borrowing more than his income. The previous proposition shows the
existence of partially naive hyperbolic consumers in this region. The same arguments
we discussed previously apply to these partially naive hyperbolic consumers as well.
26
We do not obtain the same results if the consumer is sophisticated, as demonstrated in the proof of the proposition. This is basically because the sophisticated
consumer correctly estimates how much he will end up borrowing in consumption
periods even when he is deciding in the contracting period. Therefore, he is always
responsive to interest rates if he borrows. However, we have an interesting example
if we assume that the o¤ered credit limits are the same. If the consumer is sophisticated, we expect him to choose the lower interest rate contract knowing that he is
going to pay interest. This is true for some parameter values. On the other hand,
surprisingly, for other values, the consumer chooses the contract with the higher interest rate. This is because the initial-period self of the sophisticated consumer wants
to limit his future selves’spending on credit. If he cannot do this by choosing a lower
credit limit, he can choose a higher interest rate and consequently decrease his future
selves’ credit card spending, using the contract with the higher interest rate as a
commitment device.22
4.4
Consumer Welfare
We calculate the consumer welfare according to the initial-period self. Whether he
is better o¤ with the restriction on the o¤ered credit limits depends on the utility
function and discount factors
22
and : We conclude that the welfare e¤ects of credit
We have a numerical example for each case. These are available upon request.
27
limit regulation are ambiguous. We provide an example for each case.23
Example 1 Let u(x) = x1=2 ;
= 0:61;
= 0:8; and m is income. If the consumer’s
cost of default is high enough, and if we do not restrict the credit limits, we …nd the
total utility as U00 = 2:1177m1=2 : If we restrict the credit limits to income, then the
total utility is U000 = 2:1273m1=2 : Because U00 < U000 ; the consumer would have been
better o¤ had we restricted the credit limits to income.
Example 2 Let u(x) = x1=2 ;
= 0:61;
= 0:9; and m is income. If the consumer’s
cost of default is high enough, and if we do not restrict the credit limits, we …nd the
total utility as U00 = 2:1262m1=2 : If we restrict credit limits to income, then the total
utility is U000 = 2:1258m1=2 : Because U00 > U000 ; the consumer would have been worse
o¤ had we restricted the credit limits to income.
5
Conclusion and Discussion
In our model, we focus on two aspects of a credit card contract: the interest rate and
the credit limit. Cash back and reward points (which can be modeled as transfers
to the contracting-period self of the consumer) are further aspects of credit card
contracts that were not common two decades ago. In a model with a cash back
option, we would not observe competition on interest rates, although the pro…ts
would be zero.
23
We provide the examples for each case by changing only the hyperbolic discount factor to make
it easy to interpret. This example should not mislead one to think that the equilibria we described
are speci…c to unusual discount factors.
28
Consumers’ time inconsistency and naivete, and credit card companies’ grace
period o¤ers, are the key factors that make non-competitive interest rates and positive
pro…ts possible. It is important to include a grace period when modelling the credit
card market. The credit card market is unique because of this grace period feature,
which makes it possible for consumer to treat the credit card either as a convenience
device or as borrowing medium. The grace period feature is especially important
when the consumer is naively time-inconsistent, as the contracting-period-self may
treat credit card as a convenience device, whereas consumption-period-selves may end
up using it as a borrowing medium with a high cost.
The arbitrarily small cost of applying for a credit card makes the consumer choose
only one card when the o¤ered credit limits are high enough. Once the consumer
decides to get only one contract, and if he is indi¤erent among the o¤ered contracts,
then he chooses one contract randomly. We relax this randomization-between-cards
assumption by embedding the …rst-period contract-choice problem into a search model
in Section 4.1 and show that this relaxation still gives similar results.
In our model, we do not allow contracting after the initial period. If we had done
so, the problem would have been more complicated, with additional entry deterrent
strategies to consider. For example, in our current analysis, there is an equilibrium
in which …rms o¤er monopoly contracts (with high enough credit limits) and the
consumer chooses one contract without paying attention to interest rates if the consumer’s cost of default is high enough. If there were a second period of contracting,
29
then the …rms would o¤er non-competitive contracts with credit limits just a little
bit lower than the default triggering level at the …rst period, so that no other …rm
could o¤er an additional credit limit to the consumer who is already at the margin
for defaulting.
In our model, the consumer’s cost of bankruptcy is given exogenously. With the
recent changes in the bankruptcy law of 2005, which introduced stricter bankruptcy
proceedings, such as more stringent conditions to …le under chapter 7 and a mandatory
…nancial management education program, one would expect this cost to increase for
the consumer. As a result, the consumer becomes less likely to default, and this
increases the credit card companies’pro…ts.
Because our results depend on the consumer’s incorrect beliefs regarding his future
consumption, we should obtain the qualitatively same results for more than three
periods of consumption, with appropriate cuto¤s for
and :
In summary, in this paper we show that in the presence of naive hyperbolic consumers who are unresponsive to interest rates with a grace period, it is possible to
observe high interest rates, which may result in positive expected pro…ts.
30
6
References
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American Economic Review 81(1), 50-81.
2. Ausubel, L. M., 1999. Adverse selection in the credit card market. Working
Paper. University of Maryland.
3. Ausubel, L. M., Shui, H., 2005. Time inconsistency in the credit card market.
Working Paper. University of Maryland.
4. Brito, D. L., Hartley, P.R., 1995. Consumer rationality and credit cards. Journal of Political Economy. 103(2), 400-33.
5. Calem, P. S., Mester, L. J., 1995. Consumer behavior and the stickiness of
credit-card interest rates. American Economic Review 85(5), 1327-36.
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selection in the market for credit cards: New evidence. Journal of Banking and
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7. Carlin, B. I., 2009. Strategic Price Complexity in Retail Financial Markets.
Journal of Financial Economics. 91(3), 278-287.
8. Chakravorti, S., 2003. Theory of credit card networks: a survey of the literature.
Review of Network Economics 2(2), 50-68.
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9. DellaVigna, S., Malmendier, U., 2004. Contract design and self control. The
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Time Preference: A Critical Review. Journal of Economic Literature 40(2),
351-401.
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Information Suppression in Competitive Markets. The Quarterly Journal of
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Spending: A Field Experiment. Working Paper. Carnegie Mellon University.
16. Laibson, D., 1997. Golden Eggs and Hyperbolic Discounting. The Quarterly
Journal of Economics 112(2), 443-77.
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17. Laibson, D., Repetto, A., Tobacman, J., 2003. A debt puzzle. In: Aghion, P. et.
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33
7
Appendix
It is a dominant strategy for the consumer to pay o¤
Proof of Proposition 1.
as much of his borrowing as possible within the grace period. Therefore, if nt1
then pt2 = nt1 and pt3 = nt2 ; if nt1 > m; then pt2 = m and pt3 = (nt1
m;
m) (1 + r) + nt2 :
As a result, we can write the consumer’s utility function as follows:
If nt1
m; then Unt1
m
= [u(m + nt1 ) + u(m
nt1 + nt2 ) +
2
If nt1 > m; then Unt1 >m = [u(m+nt1 )+ u(nt2 )+ 2 u(m (nt1
u(m
nt2 )]:
m) (1 + r) nt2 )]:
We will solve these two utility functions separately and show that
max Un01
m
max Un01 >m for
max Un01
m
max Un01 >m for
; irrespective of the interest rate, and
<
r=1 .
We will complete the proof by demonstrating that
1. For Un01
m,
r=1
<
:
the consumer’s maximization problem at time zero is given by:
max
0
0
n1 ; n2
s:t: n01
u(m + n01 ) + u(m
m;
n01
0; and
The Lagrangian is given by
34
n01 + n02 ) +
n02
0:
2
u(m
n02 )
(3)
u(m + n01 ) + u(m
L =
0
1 (n1
m)
2(
n01 + n02 ) +
n01 )
3(
2
n02 )
u(m
n02 )
The set of FOCs is as follows:
@L
2 0
= u0 (m + n01 )
u (m n01 + n02 )
0
@n1
@L
3 0
= 2 u0 (m n01 + n02 )
u (m n02 ) +
0
@n2
0
1 (n1
1
n01
If n01 = m; then
m) = 0;
0;
2
0 and
1
therefore n02 > 0 and
0; and
n01
m;
3
2(
n01 ) = 0; and
3
0; and
2
3(
1
+
3
2
=0
=0
(4)
(5)
n02 ) = 0
0
n02
0
= 0: If n02 = 0; (5) creates a contradiction;
= 0: Then, by (4) and (5),
u0 (2m) =
u0 (n02 ) =
35
u0 (n02 ) +
u0 (m
1
(6)
n02 )
(7)
There exists a
such that (6) and (7) both hold for
termined by u0 (2m) = u0 (n02 ) and u0 (n02 ) = u0 (m
cannot simultaneously hold for
: Hence, n01
>
where
is de-
n02 ): However, (6) and (7)
m for
:
2. For Un01 >m , the consumer’s maximization problem at time zero is given by:
max
0
0
n1 ; n2
s:t:
2
6
6
4
u(m +
2
+ u(m
n01 <
m;
n01 )
+
3
u(n02 )
(n01
m) (1 + r)
n01
0; and
n02
n02 )
0:
7
7
5
(8)
The Lagrangian is given by
L =
u(m + n01 ) + u(n02 ) +
1(
n01 + m)
2(
n01 )
36
2
u(m
3(
n01
n02 )
m (1 + r)
n02 )
The set of FOCs is as follows:
@L
3
= u0 (m + n01 )
(1 + r)u0 (m
n01 m (1 + r) n02 ) +
@n01
@L
3 0
u (m
n01 m (1 + r) n02 ) + 3 = 0
= 2 u0 (n02 )
@n02
1(
n01 + m) = 0;
1
0;
n01
2
0; and
m;
n01
If n01 = m; then
and
3
1
n01 ) = 0; and
2(
n02
0; and
2
For a given r; there exists a
n02 ) = 0
0
2
r
r
(1 + r)u0 (m
u0 (m
n02 )
(11)
1
n02 )
(12)
such that (11) and (12) both hold for
r
is determined by u0 (2m) = (1+r)u0 (m n02 ) and u0 (n02 ) = u0 (m n02 ):
However, (11) and (12) cannot simultaneously hold for
<
(10)
= 0: Then, by (9) and (10):
u0 (n02 ) =
for
=(9)0
2
= 0: If n02 = 0; (10) cannot hold; therefore n02 > 0
u0 (2m) =
where
+
0
3
0 and
3(
1
<
r:
Thus, n01 > m
r:
It is not di¢ cult to show that
r
decreases with r and
r=1
<
In a summary, the consumer borrows less than or equal to m if
37
r
<
r=0
=
.
and more
than m if
<
r:
We showed that an exponential consumer with
correctly believes that
he will borrow less than his income. A naive hyperbolic consumer with an
possesses the exact same belief (n01
exponential discount factor
m),
but his belief may not be correct, depending on his hyperbolic discount factor.
Let us analyze the naive hyperbolic consumer with
. As before, we can
write the consumer’s utility function as follows:
If nt1
m; then Unt1
m
= u(m + nt1 ) +
u(m
2
u(nt2 )+
If nt1 > m; then Unt1 >m = u(m+nt1 )+
2
nt1 + nt2 ) +
nt2 ):
u(m
m) (1 + r) nt2 )]:
u(m (nt1
We follow similar steps as before and solve these two utility functions separately,
and show that max Un11 >m
1. For Un11
m,
max Un11
m
for all ( ; ) where
and
<
r=1 (
):
the consumer’s maximization problem at time one is as follows:
u(m + n11 ) +
max
1
1
u(m
s:t: n11
0; and
n1 ; n2
m;
n11
n11 + n12 ) +
n12
2
n12 )
u(m
(13)
0:
The Lagrangian is given by
L = u(m+n11 )+
u(m n11 +n12 )+
2
u(m n12 )
38
1
1 (n1
m)
2(
n11 )
3(
n12 )
The set of FOCs ia as follows:
@L
= u0 (m + n1 )
@n11
@L
= u0 (n12 )
@n12
1
1 (n1
0;
1
n11
For
>
and
m) = 0;
u0 (m
2 0
n11
m;
n12 ) +
u (m
3
n11 ) = 0; and
2(
0; and
2
n11 + n12 )
3
1
+
2
(15)
=0
n12 ) = 0
3(
0
n12
0; and
0
= 1; the constraints are not binding, and hence
Moreover, one can show that
the constraint (n11
@n11
@
(14)
=0
< 0 and that there exists a
m) will be binding for
<
1;
2;
3
= 0:
( ) such that
( ):
2. For Un11 >m , the consumer’s maximization problem at time one is as follows:
2
6
6
max
n1 ; n1 4
1
s:t:
2
u(m + n11 ) +
+
n11 <
2
u(m
(n11
m;
n11
The Lagrangian is given by
39
3
u(n12 )
m) (1 + r)
0; and
n12
n12 )
0:
7
7
5
(16)
L = u(m + n11 ) +
n11 + m)
1(
2
u(n12 ) +
2(
n11
u(m
n11 )
m (1 + r)
n12 )
n12 )
3(
The set of FOCs is as follows:
@L
= u0 (m + n11 )
1
@n1
@L
= u0 (n12 )
1
@n2
1(
n11 + m) = 0;
1
0;
n11
For
>
2
2 0
n11
u (m
0; and
m;
n11
m (1 + r)
3(
n12 ) +
@n11
@
3
1
+
=0
2 (17)
(18)
=0
n12 ) = 0
n12
0
= 1; the constraint ( n11
; r = 1; and
n12 ) +
m (1 + r)
0
3
0; and
can also be shown that
n11
n11 ) = 0; and
2(
2
constraint ( n11
(1 + r)u0 (m
m) is binding, and it
< 0: Moreover, there exists a
m) is not binding for
<
r=1 (
r=1 (
) such that the
):24
Thus, we conclude that the initial-period self believes that he will borrow less
than or equal to m in future consumption periods, but the period-one self ends
up borrowing more than m for all ( ; ) where
24
It is possible to show that
r=1 (
)<
r
( )<
40
r=0 (
)=
and
( ) for
>
<
:
r=1 (
):
Proof of Lemma 1.
1. At the initial period, the consumer’s total utility is
Ut=0;
d= 1
= max
0
n1
u(m + n01 ) + u(m + l1 + l2
n01 ) +
2
2
u(m)
C
if he plans to default, and
Ut=0;
d=0
= 0max
n1 m; n02
u(m + n01 ) + u(m
n01 + n02 ) +
2
u(m
n02 )
if he does not plan to default.
Therefore, the gain from defaulting according to the contracting-period self is
given by G0 =
1
3
(Ut=0;
d= 1
Ut=0;
d=0 )
and
@G0
@l1
=
@G0
@l2
> 0:25
2. When the consumer reaches the …rst period with the chosen contract (which we
assume to be (l1 ; r1 ) for convenience), he realizes that his actual debt is more
than his income. Therefore, the consumer’s total utility is as follows:
Ut=1;
d= 1
= max
u(m + n11 ) +
1
u(m + l1
n1 l1
n11 ) +
2
2
u(m)
C
if he plans to default, and
Ut=1;
d=0
= 1max
n1 l1 ;
n12
u(m + n11 ) +
u(n12 ) +
2
u(m
(n11
m)(1 + r1 )
if he does not plan to default.
Therefore, the gain from defaulting planned at the …rst period is
25
All gains are adjusted according to the last-period self.
41
n12 )
G1 =
1
(Ut=1;
2
Ut=1;
d= 1
d=0 ) :
3. When the consumer reaches the second period, his total utility is
Ut=2;
d= 1
n11 ) +
= u(m + l1
u(m)
C
if he plans to default, and
Ut=2;
= max
u(n22 ) +
2
d=0
u(m
n2
(n11
m)(1 + r)
n22 )
if he does not plan to default.
Therefore, the gain from defaulting planned at the second period is G2 =
1
(Ut=2;
Ut=2;
d= 1
d=0 ) :
As a result, the gain from defaulting according to
the consumption period selves is Gc = maxfG1 (l1 ; r); G2 (l1 ; r)g with
and
@Gc
@r1
@Gc
@l1
>0
> 0:
Now let l2 = 0 and
= 1; then Ut=0;
d= 1
and Ut=1;
d= 1
are equal, and
consequently any di¤erence between G0 and G1 is due to the di¤erence between Ut=0;
d=0
and Ut=1;
d=0 :
From Proposition 1, we know that the pro…t
maximizing n1 is less than m for
Ut=1;
d=0
[u(m+n11
for
>
) u(m+n11
(
)2
>
: Therefore, Ut=0;
. Consequently G0 < G1 for
)]
< 0 such that n11 and n11
d=0
is greater than
= 1: Additionally,
@G1
@
=
represent the pro…t maximiz-
ing n1 in the case of planning and not planning default, respectively. As a
result, G0 (l1 ; l2 = 0) < G1 (l1 ; r1 )
Gc (l1 ; r1 ) for all r1 2 [0; 1]:
42
Proof of Lemma 2. The gain from defaulting according to the contracting-period
self is the same as before, namely, G0 (l1 ; l2 ). The gain from defaulting according to
the consumption-period selves, however, depends on the credit limit o¤ers of both
companies instead of only one, namely, Gc (l1 ; l2 ; r1 ) if the lower interest rate contract
is (l1 ; r1 ).26 By using the same arguments as in the previous proof, it can be shown
@Gc (l1 ;l2 ;r1 =0)
@l1
@Gc (l1 ;l2 ;r1 =0)
@l2
that G0 (l1 ; l2 ) < Gc (l1 ; l2 ; r1 );
@Gc (l1 ;l2 ;r1 )
@r1
Proof of Proposition 3.
Recall that each company o¤ers a monopoly contract
> 0; and
=
> 0.
(l ; r ) if there is no risk of default. When there is risk of default, however, the
companies may not be able to o¤er the credit limits they want without triggering
default. For convenience, from this point on, we analyze the problem from the …rst
company’s point of view. The second company’s problem is similar. Recall that the
default constraints are Gc (l1 ; r1 )
C and G0 (l1 ; l2 )
contract is chosen in equilibrium. Additionally, G0 (l1 ; l2 )
C assuming that only one
Gc (l1 ; r1 )
C for lower
values of l2 : Suppose that (l1 ; r1 ) is the …rst company’s pro…t-maximizing contract
o¤er when the other company o¤ers a zero credit limit. Let l20 be the solution to
G0 (l1 ; l2 ) = C and r20 = arg max (l20
m)r2 : Note that each company’s contract o¤er
Gc (l2 ;r2 ) C
is a¤ected by the other company’s o¤er only if G0 (l1 ; l2 ) = C.
Let C = Gc (l1 = m; r1 = 0) and C
= Gc (l1 = n0 ; r1 = 0):
26
When the consumer reaches the …rst period with two contracts in hand with di¤erent interest
rates, he will pay the contract with the higher interest rate …rst to minimize the cost of borrowing.
Moreover, he can pay all his debt back to the higher interest rate card within one period without
interest charge since his debt cannot be higher than his credit limit with this card, and it is less
than his income (recall that the reason to choose two contracts at the beginning was low credit limit
o¤ers). As a result, the higher interest rate does not enter the function representing the gain from
default.
43
Consider the case C > C ; we now demonstrate that a company o¤ers a large
enough credit limit with a positive interest rate when the other company o¤ers
a zero credit limit. If C > C ; then Gc (l ; r )
If Gc (l1 = l ; r1 = r )
C may or may not be satis…ed.
C; then the monopoly contract is feasible. Otherwise,
the company o¤ers l1 > m and r1 > 0 as the pro…t-maximizing contract with
Gc (l1 ; r1 ) = C. The argument is as follows. Suppose that l1 = m and r1 = 0:
The company can obtain a higher pro…ts by slightly increasing l1 and r1 because
the constraints C > Gc (l1 = m; r1 = 0) > G0 (l1 = m; l2 = 0) are not binding
for l1 = m and r1 = 0: Suppose that l1 = m and r1 > 0: The company can
increase its pro…t by slightly decreasing the interest rate and slightly increasing
the credit limit. Last, suppose that l1 > m and r1 = 0: The company can
increase its pro…ts by slightly decreasing the credit limit and slightly increasing
the interest rate. We now show the existence of di¤erent equilibria depending
on where G0 (l1 ; l2 )
– If l20
C starts to bind.
l1 ; then (l1 ; r1 ) and (l2 = l1 ; r2 = r1 ) is the unique symmetric
equilibrium in region 1. This is because G0 (l1 ; l2 ) < C; and each company
o¤ers the pro…t-maximizing contract without triggering default.
– If m < l20 < l1 ; then (l1 ; r1 ) and (l2 ; r2 ) is an equilibrium such that
m < l2
l20 and r2 > 0. This is because the second company cannot o¤er
more than l20 and does not have an incentive to o¤er less than m: Moreover,
the second company can o¤er a positive interest rate without triggering
44
default, as G0 (l1 ; l2 )
– If l20
C and Gc (l2 ; r1 ) < Gc (l1 ; r1 )
C:
m; then (l1 ; r1 ) and (l2 ; r2 ) is an equilibrium such that l2
and r2
l20
0: This is because the second company cannot o¤er more than
m; and consequently makes zero pro…t.
C : (l1 ; r1 ) and (l2 ; r2 ) are then an equilibrium where
Now consider C
C
0
0; and 0
l1
l1 ; r1
l20 ; r2
l2
0: The companies cannot o¤er
more than m without triggering default, and consequently there is no pro…table
deviation (region 2 or region 4). Companies make zero pro…ts with or without
competition on interest rates. In region 4, the equilibrium contracts would be
zero-interest contracts, as the consumer accepts two contracts in this region and
pays the higher interest rate within the grace period.
C ; (l1 ; r1 ) and (l2 ; r2 ) are then an equilibrium where l1 + l2
If C
r1
0; and r2
n0 ;
0: There is no pro…table deviation, as the total credit limit
to be o¤ered to the consumer without triggering default is not more than the
consumer’s income (region 3).
Proof of Proposition 4.1.
Credit card companies may o¤er low credit lim-
its for strategic reasons even though there is no possibility of default. Let ri =
arg max(n1i1
m)ri ; when the borrowing is not bounded by credit limits. Let ri =
arg max(minfn1i1 ; li g
m)ri , when the borrowing may be bounded because of the low
45
credit limit o¤er and there is only one contract in hand. Let ri
lj g
= arg max(minfn1i1 ; li +
m)ri ; when the borrowing may be bounded because of the low credit limit o¤ers
and there are two contracts in hand.
Note that o¤ering a monopoly contract provides an expected pro…t of at least
1=2(n1i1
m)ri , as this contract will be the …rst o¤ered contract with half probability
and the consumer will not inquire about a second contract (the monopoly contract
provides enough credit limit and the consumer believes that he will not maintain
positive debt).
We now show that there are no equilibria other than the monopoly contracts by
analyzing di¤erent cases.
1. li
n0 :
(a) 0
ri < r j :
i. lj
n0 and li + lj
m
Both companies make zero pro…t since the total credit limit does
not allow the consumer to accumulate interest-bearing debt; therefore
companies can pro…tably deviate by o¤ering a monopoly contract.
ii. lj
n0 and m < li + lj
Only the company with the lower interest rate earns positive pro…t;
therefore the other company can deviate by o¤ering a smaller interest
rate.
46
iii. n0 < lj
m and li < n1
m
Company i earns the expected pro…t of 1=2(li )ri which is less than the
minimum expected monopoly pro…t, 1=2(n1i1
m)ri ; therefore there
is a pro…table deviation.
iv. n0 < lj
m and n1
m
li
The company j earns zero pro…t even if it is the …rst o¤ered contract;
therefore it can deviate by o¤ering a monopoly contract.
v. m < lj
If rj = rj ; then the company i can pro…tably deviate by o¤ering a
monopoly contract since ri < ri = rj : If rj 6= rj ; then the company j
can pro…tably deviate by o¤ering a monopoly contract.
(b) 0
rj < ri : Company i makes zero pro…t. This contract is chosen only
together with the other contract. Since ri is higher, there is no interest
payment for contract i: Consequently, company i can pro…tably deviate by
o¤ering a monopoly contract.
(c) 0
ri = rj : Company i can deviate pro…tably by o¤ering a smaller interest
rate or by o¤ering a monopoly contract (when rj = 0)
2. n0 < li
m
(a) n0 < lj
Company i earns zero pro…t; therefore it can deviate by o¤ering a monopoly
47
contract.
3. m < li ; lj
The monopoly contract gives the highest pro…t. The companies cannot a¤ect
the consumer’s choice of a contract by changing interest rates but by only changing credit limit. There is no pro…table deviation from the monopoly contracts
by changing the credit limit.
Proof of Proposition 5. If companies o¤er positive interest rates for one-periodloans, then the company with the higher interest for the one-period loan can profitably deviate by o¤ering a smaller interest for the one-period-loan. Therefore, no
equilibrium contract will have a positive one-period-loan interest. Since default is not
possible and the consumer is not responsive to interest rates for two-period loans, the
companies can safely o¤er a monopoly interest rate for two-period loans.
Proof of Proposition 6.
According to both the period-zero self and period-one self, the consumer’s utility
when he reaches the second period is given by
u(m
n1 + n2 ) + bu(m
u(n2 ) + bu(m
(n1
m)r
n2 ) if n1
m; and
n2 ) if n1 > m:
Let us denote the optimal second period borrowing as n2 (n1 ; b).
48
According to the period-one self, the consumer’s utility when he reaches the …rst
period is:
u(m + n11 ) +
u(m + n11 ) +
h
h
u(m
n11 + n2 (n11 ; b)) + u(m
u(n2 (n11 ; b)) + u(m
(n11
i
n2 (n11 ; b)) if n11
m; and
i
n2 (n11 ; b)) if n11 > m:
m)(1 + r)
Let us denote the optimal …rst-period borrowing according to the period-one self
as n11 ( ; b):
According to the period-zero-self, however, the consumer’s utility when he reaches
the …rst period is:
u(m +
n01 )
h
b
+ u(m
n01
+
n2 (n01 ; b))
h
u(m + n01 ) + b u(n2 (n01 ; b)) + u(m
+ u(m
(n01
i
n2 (n01 ; b))
if n01 > m; and
i
n2 (n01 ; b)) if n01
m)(1 + r)
m:
Let us denote the optimal …rst-period borrowing according to the period-zero self
as n01 (b):
Let us consider the case when
= b (sophisticated hyperbolic consumer). If
b = 1; then n0 = n1 = 0. As b decreases, n0 = n1 increase and there is a cuto¤
1
1
1
1
such that the n01 = n11
keep b
but decrease
m for b
and n01 = n11 > m for b <
starting from b. Decreasing
: Now, let us
will not a¤ect the optimal
period-one borrowing according to the period-zero self, but will increase the optimal
period-one borrowing according to the period-one self. As
there exists a (
r=1 (
b);
) such that n01
! 0; n11 ! 1: Therefore,
m and n11 > m for
49
<
r=1 (
b) and b >
:
This means that there exist partially hyperbolic consumers with a contracting-period
self who believes he won’t borrow more than his income and with a period-one self
who ends up borrowing more than his income even at the highest interest rate of
r = 1:
50