Commitment savings in informal banking markets

Journal of Development Economics 107 (2014) 97–111

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Journal of Development Economics

j ourna l homepage: www.e lsev ie r .com/ locate /devec

Commitment savings in informal banking markets☆

Karna Basu 1

Dept. of Economics, Hunter College, City University of New York, United States

☆ I thank Abhijit Banerjee, Tim Besley, Saurabh BhargCortes, Greg Fischer, Xavier Gabaix, Thomas Gall, MaitreMorduch, Sendhil Mullainathan, Roger Myerson, and semSchool of Economics, Johns Hopkins University, FordhamNEUDC. This version of the paper has benefited greatly froymous referees.E-mail address: [email protected].

1 Tel.: +1 212 396 6521.

0304-3878/$ – see front matter © 2013 Elsevier B.V. All rhttp://dx.doi.org/10.1016/j.jdeveco.2013.11.006

a b s t r a c t
a r t i c l e i n f o
Article history:Received 9 June 2012Received in revised form 2 September 2013Accepted 17 November 2013Available online 5 December 2013

JEL classification:O12D03D91

Keywords:Hyperbolic discountingCommitment savingsInformal banking

I study the provision of commitment savings by informal banks to sophisticated hyperbolic discounters. Since aconsumer is subject to temptation in the period that he signs a contract, banksmight exploit his desire for instantgratification even as they help him to commit for the future. Without banking, savings decisions and welfare arenot monotonic in the degree of time-inconsistency. Consequently, commitment savings will lower welfare formoderately time-inconsistent agents. If loan contracts are enforceable, pure commitment savings will disappear.This will further lower welfare if the lender is a profit-maximizing bank, but raise welfare if the lender is awelfare-maximizing NGO. Finally, I consider the coexistence of a bank and NGO. There will be zero takeup ofNGO-provided commitment savings if there is competition from a moneylender. But the NGO's offer will raisethe agent's reservation utility, thus reducing the surplus that can be extracted by the moneylender.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

This paper addresses some questions related to informal bankingunder time-inconsistency. First, if an individual values commitmentsavings, under what conditions will such a product be offered by abank? Second, when does voluntary adoption of commitment raisethe individual's welfare? And third, what are the implications for equi-librium contracts if a welfare-minded NGO enters a region served by aprofit-minded monopolist?

Hyperbolic discounters, who in any period place an emphasis oninstant gratification, can make inefficient financial decisions. Sup-pose an individual would like to save up for a nondivisible good or in-vestment. His savings decision today depends on his future selves'willingness to continue saving. If he fears that his future selves willnot follow through, hemight abandon saving altogether. In this context,it is well understood how access to commitment devices, or contractsthat restrict future choice sets, can improve welfare. In particular, con-sider commitment savings, which I define as a contract that makes sav-ings balances illiquid until a specified date. Illiquidity, by raising futureselves' incentives to save, gives the current self a reason to save as well.

The fact that markets will respond to a demand for commitmentdoes not itself inform us about equilibrium contracts and individual

ava, Jonathan Conning, Patriciaesh Ghatak, John List, Jonathaninar participants at the LondonUniversity, Hunter College, andm the comments of two anon-

ights reserved.

welfare. I show that, depending on time preferences and the contractingenvironment, traditional commitment savings might not be offered oradopted, and that if adopted, it can lower welfare relative to autarky.In this paper, I follow O'Donoghue and Rabin (1999b) and subsequentpapers in assuming that an agent's welfare is what his lifetime utilitywould be if he were time-consistent (equivalently, it is his discountedutility from the perspective of a hypothetical “period 0”, just before heactually starts making decisions).2

The model isolates some key mechanisms through which predic-tions about contracts and welfare are made. Consider a sophisticatedquasi-hyperbolic discounter who, in any period, discounts the sum offuture utilities by a factor β b 1. His preferences are time-inconsistentsince, in any period, he places greater value on immediate consumptionthan his past selves would like him to. Much of the intuition in thispaper comes from the analysis of the strategic interaction across differ-ent incarnations of the same agent. In particular, the period 1 self makesdecisions thatmust take into account the optimal response of the period2 self. The fact that banking decisions aremade by period 1, who is him-self subject to temptation even as he tries to curb the temptation of hisfuture selves, allows us to see how markets might fail to maximizewelfare.

As a starting point, I show that, in the absence of banking, the agent'ssavings patterns and welfare are not monotonic in the degree of time-

2 This can be interpreted as, say, the preferences parents have over their children's lives.This approach is also reasonable if we are interested in thinking about how people mightvote on future changes inpolicy such as newbanking structures and new forms of contractenforcement. Given that an agent's intertemporal preferences vary over his lifetime, thiswelfare criterion does not legitimize myopia in a particular period while rejecting thesame preferences later in life.

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3 For a broader discussion, see Conning and Udry (2007).4 On the first point, see Ashraf et al. (2006) and Bryan et al. (2010). On the second, see

Morduch (1998), Banerjee et al. (2013), and Armendariz and Morduch (2010).5 Hyperbolic discounting is one of a few different ways to model problems of tempta-

tion and self-control. Other approaches include Gul and Pesendorfer (2001), Fudenberg(2006), and Banerjee and Mullainathan (2010) .

6 However, in the papers described, the “period 2” decision is discrete, unlike in thismodel.

7 Theoretical arguments are laid out in Ambec and Treich (2007) and Basu (2011). SeeGugerty (2007), Tanaka and Nguyen (2009), and Dagnelie and LeMay-Boucher (2012) forrelated evidence.

98 K. Basu / Journal of Development Economics 107 (2014) 97–111

inconsistency. Suppose he is saving for a nondivisible good in periods 1and 2, to be consumed in period 3. If he becamemore time-inconsistent(i.e. β dropped), the changes in his behavior would be driven by twoconsiderations. First, in period 1 he would wish to transfer more of thesavings burden to period 2. Second, in period 2, he would face a greatertemptation to simply consume his accumulated assets rather than con-tinue saving. At high values of β, the second considerationwould not bebinding and a drop in β would result in slower, or more imbalanced,saving. At lower values of β, the second consideration enters into play.Even though the period 1 selfwould like to save less, hewill findhimselfsavingmore than before in order to induce period 2 to continue saving.Therefore, asβ drops, the agent's period 1 savingswill fall, then rise, andultimately, when period 2 becomes sufficiently uncooperative, drop to0.

The characterization of autarky equilibrium establishes that acommitment contract is sometimes valuable. If period 2 does nothave access to period 1's deposits, he has an improved incentive tosave. At the point of adopting a contract, the hyperbolic discounterhas two objectives. He wants to improve the behavior of his futureselves, but to also limit the sacrifices required of his current self. Aprofit-maximizing bank will seek to capitalize on both objectives.

Section 6 introduces a monopolist bank. There are two possiblecases: (a) only commitment savings contracts can be offered (if thebank cannot adequately enforce repayment on loans), or (b) bothcommitment savings and loan contracts are feasible. Under a com-mitment savings contract, welfare will rise if it enables the agent tosave when he otherwise could not. However, if the agent adopts com-mitment savingwhen hewas already saving in autarky, his welfare willfall. To see why this is the case, consider the autarky outcomewhen theagent is slightly hyperbolic. In period 1, he is saving less than the wel-faremaximizing amount (but not as little as hewould like). Now, accessto commitment allows him to save even less by giving period 2 a greaterincentive to make up the balance. This serves to make savings patternsmore imbalanced than in autarky.

If the bank is able to enforce lending contracts, commitment sav-ings will no longer be offered. The bank will offer a loan instead. Thiswill cause the agent's welfare to drop relative to both autarky andcommitment savings. Borrowing is not inherently bad since it allowsthe nondivisible to be purchased while creating commitment througha fixed repayment schedule. However, pulling nondivisible consump-tion into the present creates such a large surplus for the hyperbolic dis-counter that the bank can extract high future repayment in exchangefor the instant gratification.

In Section 7, I carry out the same exercise for a welfare-maximizingNGO. The NGO, unlike the bank, will not charge fees for commitmentsavings. However, it will deny access to those agents whose welfarewould be hurt by commitment. If repayment is enforceable, it too willoffer loans instead of commitment savings, but at better terms andwith different loan sizes than the bank. In sharp contrast to the bank,theNGO achieves the first-best welfare through lending, since it can en-sure that the nondivisible is purchased while preventing over-borrowing, enforce commitment through the repayment schedule,and return surplus to the agent.

Section 8 examines equilibrium contracts when an NGO and bankcoexist. This is of interest to both practitioners and experimental re-searchers investigating nonprofit entry in areas dominated by a mo-nopolist. When both entities offer the same product, the NGO mustexpand its offers to serve those who would otherwise turn to thebank. While this erodes some of the welfare gains that an NGOcould achieve if it operated alone, it eliminates the monopoly rentsthat a bank could earn. It is also reasonable to consider the coexis-tence of a bank that can lend and an NGO that cannot, since NGOsoften lack the information and enforcement power that local money-lenders possess. In this case, the NGO's commitment savings productwill not be adopted by any agent. This is because a moneylender canalways design a loan contract that is preferable from period 1's

perspective. However, the NGO's offer improves the individual's out-side option, which reduces the amount of surplus the bank can ex-tract from him. Zero take-up of commitment savings, therefore,does not imply that it was ineffective.

Finally, Section 9 discusses the results in the context of empiricalresearch in development economics. While a number of the resultshave relevance beyond informal banking, the motivating setting forthis paper is a low-income region where people lack access to themore complicated financial instruments and contract enforcementtechnologies of industrialized nations.3 Several recent empirical papershave examined the provision and takeup of commitment savings in de-veloping countries. This paper aims to provide a theoretical comple-ment by generating predictions about the relationship between timepreferences, adoption of banking services, and welfare. The resultshave implications for the design of commitment savings contracts andallow us to put some structure on empirical hypotheses. This is perti-nent in light of concerns about market provision of commitment andambiguous welfare effects of microfinance.4

2. Related literature

Starting with Phelps and Pollack (1968) and subsequently popular-ized by Laibson (1997), several papers have studied the theoreticalproperties of hyperbolic discounting.5 Harris and Laibson (2001),Krusell and Smith (2003), and Bernheim et al. (2013) all develop tech-niques for solving consumption-savings problems. Two papers in par-ticular share some of the intuition of Section 4, which analyzes howperiod 2 incentives affect period 1 behavior6: In the context of addictivegoods, O'Donoghue and Rabin (1999a) argue that sophisticated hyper-bolic discounters are driven by two forces—a “pessimism effect” (If I ammore likely to indulge later, I might aswell indulge now) and an “incen-tive effect” (if I indulge now, I ammore likely to indulge later, so I shouldrestrain now). Diamond and Koszegi (2003) study how the option ofearly retirement affects savings decisions for hyperbolic discounters.

There is now a significant body of empirical work that points toconsumer demand for commitment. Ashraf et al. (2006) find,through a field experiment, that agents most interested in commit-ment savings display relatively greater time inconsistency and areaware of their preferences. Additional evidence on the advantages ofcommitment savings is provided by Benartzi and Thaler (2004), Bruneet al. (2013), and Dupas and Robinson (2013). There is also growing ev-idence that commitment embedded in other forms of informal bankingplays a significant role. For example, roscas (rotational savings andcredit associations) can serve as effective commitment devices.7 Ofmore direct relevance to this paper is the idea that microfinance toocan be viewed as a form of commitment. This is discussed in Banerjeeand Duflo (2011) and Bauer et al. (2012). Basu (2008) argues that si-multaneous saving and borrowing in microfinance can be rationalizedas a formof commitment. Fischer andGhatak (2010) show that a partic-ular feature of microfinance contracts – frequent repayment – allowshyperbolic discounters to access larger incentive compatible loansthan under infrequent repayment.

Finally, a number of papers study contracts between firms and time-inconsistent agents. Amador et al. (2006) and Bond and Sigurdsson(2009) look at the tradeoffs between commitment and flexibility

99K. Basu / Journal of Development Economics 107 (2014) 97–111

under uncertainty. Della Vigna and Malmendier (2004) explore theimplications of differing levels of naivete for firm pricing of leisuregoods and investment goods. Eliaz & Spiegler (2006) show howscreening contracts can be designed when the agent's degree of na-ivete is unobservable to the seller. And Heidheus and Koszegi(2010) model equilibrium credit contracts when competitive lendersface consumers of varying naivete. (Unlike in their model, the agent de-scribed below is fully sophisticated, but subject to the desire for instantgratification at the time he takes a loan). In the context of this literature,this paper can be viewed as focusing specifically on commitment sav-ings contracts, through both savings and credit, offered by profit-maximizing and welfare-maximizing banks to consumers who vary intheir levels of time-inconsistency.

3. The model: Assumptions and definitions

An agent lives for 3 periods, i ∈ {1,2,3}. He has a non-stochasticincome in each period, which is normalized to 1. His per-periodutility function, u(c), is strictly concave and twice differentiable,with u′(0) = ∞. The agent can consume a numeraire good or anondivisible good. The nondivisible good has a price of p and yieldsan instantaneous benefit of b, where 3 ≥ p ≥ 2 and b N p.8

The agent is a sophisticated quasi-hyperbolic discounter—he isaware that his future selves continue to be time-inconsistent. In any pe-riod i, his discounted utility is given by:

Ui β; c1; c2; c3ð Þ≡ u cið Þ þ βX3j¼iþ1

u cj� �

where 0 b β ≤ 1. I implicitly assume there is no other discounting. Theagent's welfare is evaluated from a hypothetical period 0:

U0 β; c1; c2; c3ð Þ≡X3j¼iþ1

u cj� �

:

In the absence of banking, the agent can allocate current wealthto consumption and savings. Let s1 denote savings in period 1 ands2 denote cumulative savings in period 2 (s2 = 1 + s1 − c2). Inthis setting, the nondivisible can only be purchased in period 3, andif s2 ≥ p − 1.

Notice that, if a time-consistent (β = 1) agent were to save for thenondivisible, the savings burden would be spread evenly across periods1 and 2, so that s1 ¼ p−1

2 and s2 = p − 1. The following assumption en-sures that the time-consistent agent prefers to save than to simply con-sume the numeraire good in each period:

U1 1;1−p−12

;1− p−12

; b� �

NU1 1;1;1;1ð Þ: ð1Þ

Depending on the nature of the banking market, the agent mighthave access to commitment savings or loans. These banking servicescan be provided by a profit-maximizing bank/moneylender or awelfare-maximizing NGO, or both. I assume that the service pro-viders have access to funds at an external interest rate of 0. Thisassumption allows us to isolate commitment motives for saving in-dependent of interest-based considerations.

4. Autarky equilibrium

I first characterize the agent's equilibrium outcome in the ab-sence of banking. This allows us to establish benchmarks againstwhich banking outcomes can be assessed. Assumption 1 ensures

8 The results of the model would be qualitatively similar if the nondivisible good werealso durable.

that, in period 0, the agent would like his future selves to save forthe nondivisible. How saving actually occurs depends on period 2'swillingness to add to period 1's savings, and on period 1's actionsbased on period 2's anticipated response. The outcome will be deter-mined by the Subgame Perfect Nash Equilibrium of a consumption-savings game played by the agent's consecutive selves.

The results below are intuitively explained in Section 4.3.

4.1. Period 1's optimal outcome

The parameter that distinguishes the exponential discounterfrom the hyperbolic discounter is β: the exponential discounter hasβ = 1, while the hyperbolic discounter has β b 1. Consider theagent with β slightly below 1. In period 1, he wishes to save for thenondivisible, but with a relatively greater savings burden on period2. Once β is small enough, period 1's optimal outcome will involveno saving at all (in this region, he is reluctant to sacrifice any imme-diate consumption for the nondivisible).

Suppose period 1 could fully control period 2's savings decisions.Since the nondivisible provides the only motive to save, he wouldchoose either s2 = s1 = 0 or, if hewished to purchase the nondivisible,s2 = p − 1 and s1 = s1

yes(β), with s1yes(β) as defined below:

syes1 βð Þ≡ arg max1≥ s1 ≥p−2

U1 β;1−s1;2þ s1−p; bð Þ: ð2Þ

This would involve setting u′(1 − s1yes) = βu′(1 + s1

yes − (p − 1)).The more time-inconsistent the agent gets, the greater the savingsburden he would like to transfer to period 2.

Period 1 prefers to save for the nondivisible only if β ≥ βmin, whereβmin is the solution to:

U1 βmin;1−syes1 βminð Þ;2þ syes1 βminð Þ−p; b� � ¼ U1 βmin;1;1;1ð Þ: ð3Þ

When β drops below βmin, the perceived benefits of a futurenondivisible are so low that they cannot justify the immediate sacrifice. 9

This gives us period 1's optimal savings decision at any β, denoteds1opt(β):

sopt1 βð Þ≡ syes1 βð Þ; if β≥βmin0; if βbβmin

:

�ð4Þ

The corresponding values of s2 are automatically determined. Whens1 = s1

yes(β), s2 = p − 1 and the nondivisible is consumed in period 3.Otherwise, no saving takes place.

The above claims are summarized in Lemma 1. All proofs are in theAppendix A.

Lemma 1. (a) βmin ∈ (0,1) exists and is uniquely determined. (b) s1opt(β)

is the optimal savings pattern. (c) For β ≥ βmin, s1opt(β) is continuous and

strictly increasing, with s1opt(βmin) N p − 2 and sopt1 1ð Þ ¼ p−1

2 .

4.2. Equilibrium

Under hyperbolic discounting, period 2 might be unwilling toimplement period 1's optimal plan. To predict actual behavior, wecan decompose the agent into three independent time-indexedagents and use backward induction to solve for equilibrium. In pe-riods 1 and 2, the agent decides how much money, si, to send to thenext period. This decision is a function of the current wealth,si − 1 + 1.

Since, by assumption, b N p, the period 3 decision is straightforward.He will consume the nondivisible if it is affordable. His consumption

9 Assume the following tie-breaking rule:when indifferent between saving andnot sav-ing, the agent saves.

10 To see when each of these regions will be non-trivial, note that one of thefollowing must be true: βmin = βmid = βmax (intersection at the discontinuity)or βmin b βmid b βmax. At βmin, U1(βmin;1 − s1

yes(βmin), 2 + s1yes(βmin) − p, b) =

U1(βmin;1,1,1). For the regions to be non-trivial, we need U2(βmin;1 − s1yes(βmin),

2 + s1yes(βmin) − p, b) b U2(βmin;1,1,1). Combining these two conditions, we get

u 1ð Þ−u 1−syes1 βminð Þð Þu 1ð Þ−u 2þsyes1 βminð Þ−pð Þð Þb1−βmin . When the RHS is low, the LHS is high (and vice versa).

So this condition is true if βmin is low enough, which will be the case if thenondivisible good is sufficiently attractive.


decision, c3aut(s2), is given by:

caut3 s2ð Þ ¼ bþ s2− p−1ð Þð Þ; if s2≥p−11þ s2; otherwise :

�ð5Þ

Period 2 observes s1 and decides how much to send to period 3. Ifhe saves less than p − 1, period 3 cannot experience the consump-tion jump from the nondivisible. Period 2's savings decision isgiven by:

saut2 ≡ arg max0≤ s2 ≤1þs1

u 1þ s1−s2ð Þ þ βu caut3 s2ð Þ� �

This determines c2aut(s1).Anticipating period 2's decision, period 1 saves the following:

saut1 ≡ arg max0≤ s1 ≤1

u 1−s1ð Þ þ βu 1þ s1−saut2 s1ð Þ� �

þ βu caut3 saut2 s1ð Þ� ��

:

ð6Þ

This specifies the equilibrium outcome.To explicitly describe equilibrium savings decisions at any β, we

need to focus on the interplay of periods 1 and 2's maximization prob-lems. In particular, period 1 must assess period 2's willingness to “topup” any savings that he receives. For any s1 ≥ p − 2, define:

syes2 ≡ arg maxs2 ≥p−1

u 1þ s1−s2ð Þ þ βu bþ 1þ s2−pð Þð Þ ð7Þ

sno2 ≡ arg maxs2

u 1þ s1−s2ð Þ þ βu 1þ s2ð Þ: ð8Þ

s2yes is period 2's optimal savings decision conditional on the

nondivisible being purchased in period 3, and s2no is period 2's opti-

mal savings decision conditional on the nondivisible not being pur-chased. Now we can define s1

min as the lowest value of s1 for whichperiod 2 is willing to save for the nondivisible:

smin1 ≡min

s1≥p−2 : U2 β;1−s1;1þ s1−syes2 ; bþ 1þ syes2 −p� �

≥U2 β;1−s1;1þ s1−sno2 ;1þ sno2� �

( ): ð9Þ

Lemma 2. (a) At any β, s1min exists. (b) At any β, s2aut ≥ p − 1 iffs1 ≥ s1

min(β). (c) s1min(β) is continuous, and is strictly decreasing inβ exceptwhen s1

min(β) = p − 2. (d) smin1 1ð Þ∈½p−2; p−1

2 Þ and limβ→0þ smin1 βð Þ ¼ ∞.

This gives us, for any β, the minimum that period 1 would have tosave to ensure that the nondivisible is purchased. As β drops, period2 gets more interested in instant gratification; hence, s1 must rise tomotivate him to save up for the nondivisible. However, we know thatas β drops, period 1's willingness to comply must also drop. We candefine s1

max as the maximum that period 1 is willing to save (underthe assumption that, if feasible, period 2 will continue to save forthe nondivisible):

smax1 βð Þ≡max s1∈ p−2;1½ � : U1 β;1−s1;2þ s1−p; bð Þ≥U1 β;1;1;1ð Þf g:

ð10Þ

If such a term does not exist, let s1max(β) = 0.

Lemma 3. (a) Given a choice between saving s1 N s1max (for the

nondivisible) and not saving, the period 1 agent strictly prefers not sav-ing. (b) For β b βmin, s1

max(β) = 0. (c) s1max(βmin) = s1opt(βmin) and for

all β b βmin, s1max(β) N s1

opt(β). (d) For β ≥ βmin, s1max(β) is continuous

and strictly increasing.Now, we have a decreasing function, s1min (period 2's minimum

requirement from period 1) and an increasing function s1max (period

1's maximum willingness). This gives us some βmid below whichthe nondivisible cannot be bought in equilibrium:

βmid ≡min β : smax1 ≥smin

1

n o: ð11Þ

Finally, let us define βmax as the lowest value of β at which period 2would agree to save for the nondivisible even if period 1 saved only hisoptimal amount:

βmax ≡min β : sopt1 ≥smin1

n o: ð12Þ

We can now describe the equilibrium outcome at any level of β.

Proposition 1. The autarky equilibrium outcome is:

saut1 ¼sopt1 ; if β∈ βmax;1½ �

smin1 ; if β∈½βmid;βmaxÞ0; if β∈ 0;βmidð Þ

saut2 ¼ p−1; if β∈ βmid;1½ �0; if β∈ 0;βmidð Þ :

�8<:

Note that, as β drops, s1aut drops, then rises, and then drops again. The

next section provides further intuition for the results above.

4.3. Graphical analysis of equilibrium

Let us first recap the functions defined so far. Refer to Fig. 1 (it iseasiest to start at β = 1 and move left). s1opt is period 1's optimallevel of saving. Clearly, when β = 1, sopt1 ¼ p−1

2 . This evenly balancesthe savings burden between periods 1 and 2, and is the welfare-maximizing outcome. As β drops, period 1 would ideally like to buythe nondivisible, but with less saving in the present and more inthe next period. Ultimately, below βmin, period 1's optimal outcomeinvolves not saving at all.

The function s1max indicates the maximum period 1 would be willing

to save if hewere assured that period 2wouldmake the additional con-tributions necessary for the nondivisible to be purchased. As we wouldexpect, s1max lies above s1

opt and drops as β drops. The function s1min indi-

cates the minimum level of period 1 savings that would make period 2willing to contribute towards the nondivisible. Again, this constraint be-comes harder to satisfy as β drops.

Now we can see that saving will occur in autarky as long ass1max ≥ s1

min, or β ≥ βmid. In this region, there is always a way to pur-chase the nondivisible so that periods 1 and 2 are left better off thanunder no saving. However, s1 will initially drop, and then rise again.The reasoning for this is the following: At high values of β (aboveβmax), period 2 is sufficiently forward looking that he is willing to saveeven if period 1 forces a disproportionate burden on him. Thus, period1 is able to save according to s1

opt. For lower levels of β, period 1 canno longer achieve his optimal. So he saves as little as possible (s1min) aslong as this gives him greater utility than not saving at all.10

This setup creates a natural need for commitment. When period 1 isunable to achieve his optimal outcome, hemight wish to change period2's incentives. In the next section, I describe commitment savings as aminimal restriction on period 2's choices, and study how this affectsequilibrium outcomes.

Fig. 1.Autarky equilibrium:Whenβ ≥ βmax, period 1's optimal is achievable.Whenβ∈[βmid;βmax), period 1 savesmore than his optimal.Whenβ b βmid, there is no saving in equilibrium.


5. The role of commitment

A commitment savings product is defined as a savings account inwhich period 1's deposits remain illiquid until period 3. While one caneasily conceive of more effective commitment contracts, it is useful tounderstand how even minimal commitment can change equilibriumoutcomes.11

To see how commitment might help, let us define a functions1minlock(β), which is the minimum that period 1 needs to save to giveperiod 2 the incentive to top up for the nondivisible (conditional on pe-riod 1 savings being inaccessible to period 2). Formally:

sminlock1 ≡min s1≥p−2 : U2 β;1−s1;1þ s1−syes2 ; bþ 1þ syes2 −p

� �≥U2 β;1−s1;1;1þ s1ð Þ

� :

ð13Þ

As defined in Eq. (7), s2yes is period 2's optimal savings decisionconditional on the nondivisible being purchased in period 3. Theconstruction of s1minlock differs from the construction of s1min in onerespect—not saving is now relatively less attractive to period 2,since he cannot consume period 1's deposits. This makes it easierfor period 1 to incentivize period 2 to save for the nondivisible.

Lemma 4. (a) s1minlock exists. (b) When period 1 savings are locked, period2 will save for the nondivisible iff s1 ≥ s1

minlock(β). (c) s1minlock(β) is con-tinuous, and is weakly decreasing in β unless s1

min(β) = p − 2. (d) sminlock1

1ð Þ∈½p−2; p−12 Þ, limβ→0þ s

minlock1 βð Þ ¼ p−1, and s1

minlock(β) b s1min(β) ex-

cept when s1min(β) = p − 2.

Since s1minlock is lower than s1

min, it will intersect s1opt and s1max at lower

values of β than βmax and βmid, respectively. As before, we can define:

βmidlock ≡min β : smax1 ≥sminlock

1

n oð14Þ

βmaxlock ≡min β : sopt1 ≥sminlock1

n o: ð15Þ

As shown in Fig. 2, commitment savings expands the set of β-valuesat which the agent is able to save for the nondivisible. For β ∈ [βmidlock,βmid), there would have been no saving in the absence of commitment.

11 This conception of commitment is also natural when the service provider has limitedability to enforce more complex contracts.

For β ∈ [βmid, βmax), commitment savings allows period 1 to save lessthan before and still purchase the nondivisible.

We can immediately see how, for β just below βmax, commitmentlowers welfare. In this region, in autarky, period 1 saves more than hewould like. But this is good from a welfare perspective, since period 2is forcing him to save an amount closer to p−1

2 . Commitment allowshim to once again save s1

opt, skewing the savings burden more heavilytowards period 2 and thus lowering welfare.

6. Profit-maximizing bank

Consider a setting where the only financial services are provided bya monopolist bank whose objective is to maximize the sum of profitsover three periods. The bank is aware of the agent's preferences andmay offer a contract in period 1. The agent will accept such a contractif it leaves him no worse off than in autarky.

The following two cases are analyzed separately: (1) lending con-tracts cannot be enforced; (2) lending contracts are feasible (i.e. repay-ment can be enforced). In case 1, the bank will offer commitmentsavings for a range of β-values. This has some interestingwelfare impli-cations: for lowβ, commitment savingswill raisewelfare, but for highβ,welfarewill drop. This shows how constraining the period 2 agent couldend up hurting the period 0 agent.

In case 2, the bankwill no longer offer commitment savings andwillinstead give period 1 a loan.Welfare will be lower than in autarky, evenif the loan enables the agent to purchase the nondivisible when earlierhe could not. This is because period 1 is willing to incur large future re-payments for greater immediate consumption.

6.1. Commitment savings

Consider the case where the bank does not have the capacity to en-force loan repayments. Then, the only possible service is commitmentsavings. Access to illiquidity can change period 2 incentives, thus raisingsurplus for the period 1 agent. This surplus can then be extracted by thebank through up-front fees. I first describe the bank's optimal contract,and then look at welfare implications.

The contract takes the following form: the monopolist sets a fee(fmon) for commitment savings. Period 1 observes fmon and chooseshow much to save (s1mon). Technically, this is equivalent to monopolistchoosing s1

mon in exchange for providing the nondivisible in period 3.The higher period 1's discounted utility from commitment savings(relative to autarky), the greater the fee the monopolist can charge.

Fig. 2. Commitment lowers the minimum period 1 savings required by period 2, from s1min to s1

minlock.


Since themonopolist cannot enforce loan contracts, the fee is charged inperiod 1. Through savings, the agent can decide how much of this feeburden to transfer to period 2).

If the bank does find it profitable to offer such a contract, it will bethe solution to the following problem:

maxf ;s1

f

s:t:U1 β;1− f−s1;2þ s1−p; bð Þ≥U1 β; caut1 ; caut2 ; caut3

� � ð16Þ

U2 β;1− f−s1;2þ s1−p; bð Þ≥U2 β;1− f−s1;1;1þ s1ð Þ: ð17Þ

Constraint (16) is a participation constraint—period 1's discountedutility must be at least as high as in autarky. Constraint (17) (equiva-lently, s1mon ≥ s1

minlock) can be interpreted as an incentive compatibilityconstraint—period 2 must actually be willing to save the remainingamount towards the nondivisible (if the agent in period 2 is not willingto save p − 1, he is best off savingnothing at all). The reason such a con-tract can improve outcomes relative to autarky is that, by making theperiod 1 savings illiquid, the temptation for period 2 to consume ratherthan save is dampened (period 2's decision is determined by s1

minlock

rather than s1min). Note that the agent is never being forced to make a

payment. The incentive to do so comes directly from the modifiedtradeoffs that emerge under illiquidity.

The monopolist will offer commitment savings only if theagent adopts it at some fmon ≥ 0.12 The actual contract dependson which constraints bind. Constraint (16) will always bind—ifit didn't, the monopolist could raise profits by raising fmon.When constraint (17) doesn't bind, the bank maximizes profitsby setting u′(1 − fmon − s1

mon) = βu′(2 + s1mon − p). By allowing

period 1 to equalize marginal utility today with the discountedmarginal utility tomorrow, period 1's surplus is maximized (andby extension, this maximizes the amount the bank can chargefor the service).

As β drops, constraint (16) leads to consumption more skewed inperiod 1's favor. But simultaneously, period 2 is more willing to notsave anything at all. When constraint (17) binds, it is no longer possibleto equalize discounted marginal utilities, so u′(1 − fmon − s1

mon) N βu′(2 + s1

mon − p). This reduces the bank's ability to extract surplus.

12 If indifferent between commitment savings and autarky, assume the agent will adoptcommitment savings as long as it changes his equilibrium actions.

Proposition 2. A monopolist bank will offer a commitment savingscontract with fmon ≥ 0 only for β ∈ [βmidlock, βmax). In any contract,Constraint 16 will bind.

Regardless of whether constraint (17) binds, period 1's discountedutility will be the same as in autarky. Welfare, however, might belower. Consider the case where there was no saving in autarky. Here,any commitment savings contract involves periods 1 and 2 voluntarilyreducing their consumption, while raising period 3 consumption.Since the welfare function weighs future periods relatively more thanperiod 1 does, this reallocation of consumption into the future wouldraise welfare. However, if the agent were already saving in autarky,commitment savings would hurt welfare. Here, the bank is able tohelp period 1 by making period 2 save more. Since period 1 has a highwillingness to transfer the savings burden to period 2, welfare goesdown.13

Proposition 3. For β ∈ (βmidlock,βmid), monopolist commitment savingsstrictly raises welfare relative to autarky. For β ∈ (βmid,βmax), monopolistcommitment savings strictly lowers welfare relative to autarky.

We see in the next section that, even if commitment savings lowerswelfare relative to autarky, lending further lowers it. Under commit-ment savings, the agent making the banking decision is giving up cur-rent consumption to fund future consumption. Since he has present-biased preferences, he is reluctant to make this trade-off, which con-strains the amount of surplus than can be extracted by the bank.

6.2. Loans

Under lending, the agent is offered a contract denoted by the vectorLmon ≡ (lmon,t2mon,t3mon). This constitutes a loan lmon ≥ 0 in period 1, withrepayment t2mon ≥ 0 and t3

mon ≥ 0 in periods 2 and 3, respectively. I as-sume that repayment is fully enforceable and that the bank can crediblycommit to not renegotiate terms.

We can easily see that the bank will always do better with a loanthan with a savings contract. Consider the profit-maximizing savingscontract and re-frame it as a loan. With a loan of p − 1 and subsequentrepayments of fmon + s1

mon and p − 1 − s1mon, period 1's discounted

utility would be strictly higher. Therefore, the bank could extract addi-tional fees in periods 2 and 3.

13 In part of this region, period 2 was already saving too much in autarky, so welfarewould drop even if commitment savings were free.

l

t

isoprofit lines

participationconstraint

0 p-1

Fig. 3. The monopolist lender's profit maximization problem when the agent's β is close to 1.

15 Here, the nondivisible good might function as a “temptation” good that further re-duces welfare. The following example illustrates this. Suppose the agent had no optionof purchasing the nondivisible good. Then, he would receive some loan contract l; t; t

� �.

Now, suppose the possibility of purchasinga nondivisible good is introduced, and continueto assume he does not do so in autarky. It is possible that the bank will now change thecontract to allow the purchase of the nondivisible, to p−1; t; t

� �. Since the indifference

constraint is satisfied in both cases, U1 β;1þ l;1−t;1−t� �

¼ U1 β; b;1−t;1−t� �

. Since


Given a loan contract L, let the agent's consumption choice in period ibe denoted ci(L). This is determined by backward induction as before.The monopolist's maximization problem is14:

maxL

−lþ t2 þ t3

s:t:sU1 β; c1 Lð Þ; c2 Lð Þ; c3 Lð Þð Þ≥U1 β; caut1 ; caut2 ; caut3

� �:

ð18Þ

The next proposition describes the profit-maximizing contract. Inany contract, it must be true that t2 = t3 ≡ t. Since the agent wouldlike to equalize marginal utilities across all future periods, a balancedrepayment plan will maximize the amount the bank can charge forthe loan. Furthermore, the agent will no longer save. He will eitherconsume the nondivisible in period 1 or not at all.

Proposition 4. (a) When lending contracts can be enforced, any agentwith β b 1 will receive a loan from amonopolist bank. (b) An agent whoreceives a loan will not save. (c) In any contract, t2

mon = t3mon ≡ tmon.

(d) Under the loan contract Lmon, either u′(c1(Lmon)) = βu′(c2(Lmon))or lmon = p − 1.

Figs. 3 and 4 illustrate two possible equilibrium contracts. Ineach case, the bank's objective is to locate itself on the highestisoprofit line subject to the participation constraint. Fig. 3 depictsan agent with β close to 1. In this case, the period managed to imple-ment his optimal savings path in autarky. Up to l = p − 1, the par-ticipation constraint is linear, which means that he is willing toaccept any loan as long as total repayment adds up to the sameamount—he will simply pass down the borrowed amount to his fu-ture selves and replicate the autarky outcome. The participationconstraint faces a jump at l = p − 1 since he prefers to consumethe nondivisible immediately than in period 3. Subsequently, theconstraint will be concave and relatively flat—any loan beyondp − 1 is relatively less attractive as he is unable to ensure balancedconsumption in periods 2 and 3. In this case, the profit-maximizingloan is p − 1.

Fig. 4 depicts an agent with β close to 0. This agent was not sav-ing in autarky. Since he is highly time-inconsistent, he is willing toaccept large repayments for small loans. His participation con-straint still has a discontinuity at l = p − 1, but the bank's profit-maximizing contract involves a smaller loan. When the bank can ex-tract a large portion of future income as repayment for a small loan,

14 Assume, if there is more than one profit-maximizing contract, the firm chooses theone with the lowest loan size.

it cannot raise profits by extending a larger loan that would enable apurchase of the nondivisible.

The agent in period 1 is willing to pay more (in terms of futureconsumption) for a loan than he would if he were time-consistent.Since the monopolist's loan contract satisfies period 1's participationconstraint while allowing him greater immediate consumption thanin autarky, his welfare drops relative to both autarky and commit-ment savings.15

Proposition 5. When a monopolist bank offers loan contracts, welfare forany agent with β b 1 is strictly lower than in autarky and strictly lowerthan under monopolist commitment savings.

7. Welfare-maximizing NGO

For the purposes of this model, an NGO is defined as a bank thatseeks to maximize the welfare of the agent, subject to a break-evenconstraint. I again consider separately the cases where it cannot en-force loan contracts and where it can. Unlike the monopolist bank,the NGO will restrict the agent's access to services to ensure thatwelfare does not drop below the autarky level. When lending is pos-sible, it will choose to lend because lending functions as amore effec-tive commitment device than commitment savings (marginalutilities are equalized across those periods in which the nondivisibleis not consumed).

7.1. Commitment savings

The NGO must decide which values of β to offer commitmentsavings to, and at what fee (fngo ≥ 0).16 An agent who accepts suchan offer must choose s1

ngo to maximize U1 subject to an incentivecompatibility constraint similar to condition (17).

First, note that if commitment savings is offered, fngo = 0. Anyhigher fee reduces consumption strictly in period 1 and weakly in

period 1 is indifferent between the two outcomes, but bN1þ l and β b 1, welfare mustbe lower. Since the nondivisible raises instantaneous utility in period 1, it simply allowsthe bank to charge more for the loan.16 Denying commitment savings is equivalent to charging an impossibly high fee.

l

tisoprofit lines

participationconstraint

0 p-1

Fig. 4. The monopolist lender's profit maximization problem when the agent's β is close to 0.


period 2. Second, commitment savings will not be offered for thosevalues of β that lie just below βmax, since commitment reduces welfarein that region. For lower values of β, commitment will be offered eitherwhen there is no saving in autarky, or when s1

min is sufficiently high thatthe NGO can generate a more equitable savings path by raising period2's incentive to save. This raises both welfare and period 1's discountedutility.

Proposition 6. The NGO will always choose fngo = 0. It will offercommitment savings for a strict subset of [βmidlock,βmax).

The NGO offers commitment savings only when it raises welfare.Unlike themonopolist bank, the it denies service to somewhowant it.17

7.2. Loans

An advantage of loan contracts relative to commitment savings isthat they do not require the NGO to know the agent's type. The NGO'soptimal lending contract is straightforward: to every agent, it willoffer a loan of p − 1 with repayments of p−1

2 in periods 2 and 3.18 Bypulling forward nondivisible consumption, period 1 gets a higherdiscounted utility than in autarky. The agent attains the first-best wel-fare since the nondivisible is purchased and marginal utilities of con-sumption are equalized across periods 2 and 3.

8. Coexistence of bank and NGO

This is a setting of particular interest in developing economies thatexperience an expansion of financial services. Suppose an NGO entersa region served by a monopolist bank. We would like to know howthis affects equilibrium contracts, and whether the NGO can eliminatethe welfare costs imposed by the bank. I consider three cases below:(a) both offer commitment savings, (b) the bank offers loans whilethe NGO offers commitment savings, and (c) both offer loans. In eachof the cases, we can solve for the Nash equilibrium of a game playedin period 1: the bank offers a contract that maximize profits given theNGO's contract, the NGO offers a contract that maximizes agent welfare

17 The NGO could expand commitment savings by setting a minimum deposit size (itwould try to get s1 as close to p−1

2 as possible, subject to the constraints imposed by autarkyutility and s1

max). Nevertheless, it would not offer commitment for β values immediatelybelow βmax.18 It is necessary to limit the loan size as agents with low βwill prefer loans larger thanp − 1.

given the bank's contract, and the agent chooses a contract (if any) thatmaximizes period 1's discounted utility.

8.1. Bank and NGO commitment savings

For any β, the bank and NGO must each choose whether to offercommitment savings, and at what fee. The monopolist's objective is tomaximize the fee while the NGO's objective is to maximize U0. Theagent chooses between the two offers and autarky, with the goal ofmaximizing U1.

Clearly, an agent who desires commitment will choose the contractwith the lowest fee. For any β where the NGO initially offered com-mitment savings, the same contract will continue to be offered, andwill be adopted. The NGO will also be forced to expand commitmentsavings to the remaining β values in [βmidlock,βmax) that it did not ini-tially serve. In these cases, the resulting welfare will be lower than inautarky. However, by undercutting the bank's fees, the NGO will en-sure a welfare improvement relative to the monopoly case.

Proposition 7. When bank and NGO commitment savings coexist, anyequilibrium will have the following property: (a) The agent will adopt acontract with f = 0 for any β ∈ [βmidlock, βmax). (b) Within this range,for any β that received commitment savings under an NGO alone, welfarewill be the same as under the NGO. For any β that did not receive commit-ment savings under an NGO alone, welfare will be strictly lower than underthe NGO and strictly higher than under the bank.

The starker results emerge below, when the monopolist is able tolend. Despite theNGO'swillingness to return surplus to the agent, amo-nopolist moneylender can always lure hyperbolic discounters awayfrom NGO commitment savings.

8.2. Monopolist moneylender and NGO commitment savings

Given that commitment savings is attractive to the period 1 agent,and given the evidence that there is some demand for it, the low real-world availability and takeup of commitment savings continues topose a puzzle. In this section, I show how the presence of a monopolistmoneylender (or bank) can, for two reasons, drive agents away fromcommitment saving.

Consider an agent who has adopted NGO commitment savings.The only advantage (over autarky) that this product offers is illiquid-ity, which leads period 2 to behave differently than he otherwisewould. But if period 2 has access to loans, he is willing to pay themoneylender to make his savings from the previous period liquid.

l

tisoprofit line

indifferenceconstraints

0 p-1

La

LB

Lb

LA

Fig. 5. The moneylender's profit maximization problem as the participation constraint is tightened (the thick curve represents an improved outside option).


The moneylender can, in effect, make all illiquid savings liquid, ren-dering commitment savings worthless. It might be possible for theNGO to overcome this problem, either by making final payments interms of a good that cannot be resold, or by making it difficult forthe moneylender to verify that the agent has illiquid assets.

However, even if the above problem is solved, the moneylendercan always offer a contract that period 1 strictly prefers to commit-ment savings. As in Section 6.2, the NGO's commitment savings con-tract can always be re-framed as a loan. This improves period 1'sdiscounted utility, thus enabling the moneylender to charge for thisservice.

Therefore, the NGO will never be able to attract agents in competi-tion with a moneylender. However, it does have the power to alter theagent's outside option, which changes the participation constraint themoneylender must satisfy. For those with β ∈ [βmidlock, βmax), an offerof commitment savings raises the minimum U1 that the agent must beleft with. Therefore, even though commitment savings will not beadopted, in equilibrium period 1's discounted utility will be higherthan if the moneylender were operating alone.

Proposition 8. (a) In equilibrium with a moneylender and NGO commit-ment savings, the agent will adopt a loan contract. For β ∉ (βmidlock,βmax),the moneylender will offer the same contract as in Proposition 4 (Lmon). Forβ ∈ (βmidlock,βmax), the moneylender's contract will be constrained by theNGO's contract, which will have f = 0. (b) Relative to NGO commitmentsavings alone, welfare will be strictly lower. Relative to monopolist lending:if the agent originally received lmon ≥ p − 1, welfarewill be strictly higher,and if the agent originally received lmon b p − 1, welfare changes areambiguous.

This demonstrates that zero take-up of commitment savings doesnot imply that it had no effect. When the NGO can alter the agent'soutside option, themoneylender is forced tomodify its loan contract.Clearly, welfare cannot be as high as in a setting with only NGO com-mitment savings. Furthermore, even relative to monopolist lending,the welfare impacts are ambiguous.

First, there are cases where welfare must rise relative to monopolistlending. Consider Fig. 5. Suppose, in the absence of an NGO, themoneylender's contract is given by La and lies at the point of tan-gency with the participation constraint. Then, it must be true thatu′(c1(La)) = βu′(c2(La)). Now, imagine the NGO's offer raises theagent's reservation utility. Since the moneylender can no longer re-main on the original isoprofit line, it must offer a new loan that sat-isfies the new participation constraint (indicated by the thickcurve). One possibility (indeed, the only possibility if there were nonondivisible), would be to relocate at the point of tangency between

an isoprofit line and the new participation constraint. The new contract,Lb, must offer a larger loan and smaller repayment than before (in orderto keep the ratio between c1′ and c2′ the same, consumption in both pe-riodsmust rise). Here, the entry of anNGOmust lead to a rise inwelfare.(Similarly, if the original monopoly loan size was at least p − 1, whichwould be true at high values of β, the entry of NGO commitment savingswould again raise welfare.)

On the other hand, because of the discontinuity in the participationconstraint, it is also possible that the improved outside optionwould re-sult in a more dramatic move: from La (a small loan) to LB (a larger loanwith larger repayment). To see this more precisely, suppose at the orig-inal participation constraint the moneylender was offering the agent La,but was virtually indifferent between that and a larger loan of p − 1 atLA. The proof of Proposition 9 shows that such a point must exist. It canthen be shown that, as the participation constraint tightens, themoney-lender strictly prefers to offer p − 1.

Here, as a result of the improved outside option, the agent gets alarger loan along with a larger repayment. If the NGO's offer generatesa sufficiently small improvement in the outside option, the agent willend up with strictly lower welfare (period 1's discounted utility underLa and LB is nearly the same, but under LB it is achieved through a largerfuture payment burden). In such a case, the NGO's refusal to offer com-mitment savings could raise welfare. However, that would not consti-tute an equilibrium: the moneylender's contract in the absence of theNGO would depress the agent's discounted utility so much that theNGO would have an incentive to offer a commitment savings contract.

8.3. Bank and NGO lending

Finally, consider the coexistence of a moneylender and a lendingNGO. The NGO would prefer to limit the loan size to p − 1 in period1, with repayments of p−1

2 in the next two periods. If β is sufficientlyhigh, this will be the equilibrium outcome. However, if β is low, themoneylender could generate profits by lending more than the wel-fare optimizing amount. In such cases, the equilibrium loan sizewill be l N p − 1, with repayments of l

2 (the NGO always has an incen-tive to drive repayment down to meet the zero-profit condition).Even in cases where the moneylender is forced to lend more thanthe optimal amount, there will be a strict welfare improvement rela-tive to the outcome with a moneylender alone since the NGO canraise welfare by driving profits down.

Proposition 9. If amoneylender and lending NGO coexist: (a) The equilib-rium contract will always satisfy a zero-profit condition and will enableimmediate purchase of the nondivisible (l ≥ p − 1 and t ¼ l

2). There is


someβ such that, ifβ∈½β;1Þ, the agent receives l = p − 1 and t ¼ l2, and if

β∈ 0;β� �

, the agent receives l N p − 1 and t ¼ l2. (b) For β∈½β;1Þ, welfare

will be the same as under a lending NGO, and for β∈ 0;β� �

, welfare will bestrictly lower than under a lendingNGO. For all β b 1,welfare will be strict-ly higher than under a monopolist lender.

9. Discussion

9.1. Implications for policy and experiments

These results put structure on the welfare losses that can ariseeven in the absence of consumer mistakes or exploitative behaviorby banks. Themodel has some implications for the design of commit-ment savings and suggests directions for further experimental study.While the mechanisms described above are stylized, they provide alink between equilibrium contracts and welfare, and suggest hetero-geneous treatment effects that can be measured in new and existingdatasets.

A testable prediction is that, in autarky, agents who display moder-ate time-inconsistency will save in a more balanced manner thanthose who are less time-inconsistent. This is because moderately time-inconsistent agents are more constrained by the next period's reluc-tance to save than mildly time-inconsistent agents are. In addition tothe growing literature on commitment savings and microcredit, therehave been several attempts to document time preferences and the de-gree of time-inconsistency.19 Similar studies in unbanked settings, com-bined with data on savings behavior, could be used to test the autarkypredictions of this model. To the extent that welfare, as defined in thisand previous papers, is measurable, it is in principle possible to alsotest the prediction that welfare does not always drop in the degree oftime-inconsistency.

This also helps us to think further about the design and takeup ofsimple financial products. While data on lending has been examinedfor potential welfare losses (which could happen through a numberof channels), less attention has been paid to the possibility that com-mitment savings too could lower welfare. This need not happenmerely through the ex post realization of shocks or misallocationacross accounts. In particular, it would be useful to identify caseswhere commitment savings helps individuals to adjust their savingspatterns to excessively disadvantage future selves, and study if thishas impacts for broader outcomes.

For intermediate values of β, commitment, rather than allowinggreater or more balanced savings, helps period 1 further indulge histaste for instant gratification. This could happen in two possible set-tings: with a profit-maximizing monopolist whose goal is to maximizeperiod 1's discounted utility and therefore meets all demand, or withan NGO that does not limit access to commitment savings (eitherbecause it is unaware of potential welfare losses, or because it mustcompetewith a profit-maximizing bank). These results on commitmentsavings lend themselves to further empirical investigation. Ashraf et al.(2006) and Dupas and Robinson (2013), for example, provide evidencethat access to commitment savings raises average savings and welfare.Since access was randomly assigned, it should be possible to examinesuch data for possible non-monotonic relationships between time pref-erences and welfare.

Furthermore, the model's examination of the interaction betweencommitment savings and lending can provide partial explanations forsome existing puzzles in the literature. Ashraf et al. (2006) ask: “Anatural question arises concerning why, if commitment products ap-pear to be demanded by consumers, the market does not alreadyprovide them.” (pg. 638) One possible answer is the following:since profit-maximizing banks can earn higher profits by offering hy-perbolic discounters credit instead of savings, an apparent demand

19 See Ameriks et al. (2007); Andersen et al. (2008); and Wang et al. (2011).

for commitment will be met through credit (which itself embedscommitment through repayment requirements) rather than by ex-plicit commitment savings. Brune et al. (2013) find that offers ofcommitment savings improve household outcomes along several di-mensions, even though there is no apparent rise in the use of thecommitment savings account. While their paper itself providessome compelling explanations, the model here suggests that thepuzzle could be further resolved by recognizing that an offer of commit-ment savings, even if it is not adopted, could have a welfare impact. Theimpacts of commitment emerge not just through participation, butthrough changes in reservation utilities that must be met by lenders.

By abstracting away from questions of default, the model oflending presented in this paper is able to generate predictions forcredit markets that are independent of contract enforceability.Lending can help hyperbolic discounters in two ways. First, it canimprove welfare by providing commitment through a balanced re-payment path. Second, it allows them to buy nondivisible goodsthat they might not have been able to save up for. However, whena monopolist provides loans, welfare drops relative to autarkysince the lender is able to feed period 1's desire for instant gratifica-tion while extracting repayment from future selves.

Loan contracts vary based on time preferences. As agents getmore hyperbolic, their demand for loans rises. A welfare-mindedNGO should be unresponsive to this, as its goal is to enable the indi-vidual to take advantage of non-convexities in consumption, not in-stant gratification. A profit-maximizing bank, however, is sensitiveto time preferences. When operating in isolation, it provides smallloans to highly time-inconsistent agents. When competing with awelfare-minded NGO, it offers the same agents loans that are largerthan optimal (since it can no longer extract large repayments fromsmall loans, it seeks to expand its profits by offering larger loansthan the NGO would like).

There remain open questions about the welfare effects ofmicrofinance, with mixed evidence (see Banerjee and Duflo, 2011;Morduch, 1998; Pitt and Khandker, 1998). Banerjee et al. (2013), inparticular, find that some heterogeneous treatment effects can ex-plain seemingly ambiguous effects of microfinance. This papermakes the complementary, and not quite novel, point that outcomesmight also vary by time preference. If loan sizes are not fixed (orfixed sufficiently high), hyperbolic discounters can find themselvesworse off than exponential discounters through over-borrowing.

The model also sheds some light on an important question, posedby Banerjee and Duflo (2011) and Karlan and Appel (2011): why ismicrocredit less popular than initially expected? Banerjee andDuflo (2011) describe how, in Hyderabad, India, despite having ac-cess to multiple sources of microcredit, more than half of their sam-ple continued to borrow from moneylenders. As they demonstrate,the rigidity of microfinance can explain some of this. In the contextof this paper, rigidity matters in a specific way: if an MFI restrictsloan sizes based on an independent welfare calculation, individualswith low βwill continue to turn to moneylenders who can offer larg-er loans, even if those loans are offered at higher rates. Again, howev-er, the presence of microcredit will affect moneylender rates, so apositive impact of microcredit might be discernable even on thosewho don't adopt it.

Finally, it is important to observe that themodel does not necessitatepaternalistic restrictions on contracts. A number of the results aboveemerge from what I argue is a realistic and necessary assumption—that in the period when an agent adopts contracts to alter future be-havior, he is time-inconsistent himself. It should be possible for someregulation to be enacted by “period 0” agents, before temptationplays a role. Just as parents restrict the set of actions available totheir children, we can make the case for contracts where individualsvoluntarily restrict the contracting ability of their future selves. Thisis most effectively done when the current self has no immediatestake in the decision.

20 Thebankwill not necessarily choose to locate at the profit-maximizing contract forβH.If it offers βH a smaller loan and repayment, this lowers available profits from βH butloosens the participation constraint that must be satisfied for βL, generating higher profitsfrom that type.21 Theprecedingdiscussionhas assumed that p and b remain constant across individuals.In cases where there is also variation in the nondivisible good being purchased, screeningmight occur across additional dimensions, even under commitment savings.


9.2. Hidden types

While this paper has focused on highlighting somenuances associat-ed with commitment savings and lending contracts under time-inconsistency, and shown how autarky equilibrium, banking equilibri-um, and welfare can sometimes diverge from intuitive predictions, apractical application of the results might depend on whether consumerpreferences are observable by the bank or NGO, as is assumed in themodel. It is arguable that, in sufficiently close-knit environmentswhere the service provider is a member of the community, the assump-tions are not entirely far-fetched. Nevertheless, it is worth discussinghow equilibrium contracts might change if consumer time preferencesare private information.

An NGO faces a particular advantage over a bank in that its optimalcontracts are less reliant on the agent's type.With commitment savings,the NGO faces a simple tradeoff. Since any contract it offers must satisfyf = 0 , it will offer commitment as long as the average welfare loss tothosewhowould be hurt by commitment is at least balanced by the av-erage gain to others. With loans, since the welfare-maximizing contractdoes not depend on β, the NGO does not need to know the agent's typeat all.

A bank, on the other hand, clearly relies on its knowledge of β to ex-tract the maximum surplus. For commitment savings, there is no possi-bility of screening—the bankmust choose a single f to maximize profits.To see how this choice depends on the distribution of preferences, wecan compare derivatives of period 1's discounted utility with respectto β (refer to Fig. 2). The following are easily derived. For β ∈ [βmid,βmax), when savings occur but period 1 is constrained by period 2:

dUaut−constrained1

dβ¼ −∂smin

1

∂β u′ c1ð Þ−βu′ c2ð Þ� �

þ u c2ð Þ þ u bð ÞN0: ð19Þ

For β ∈ [βmidlock, βmid), when savings do not occur but would undercommitment:

dUaut−nosave1

dβ¼ 2u 1ð ÞN0: ð20Þ

Under commitment savings at fee f, if period 2's incentive compati-bility constraint (see condition (17)) is not binding (s1 denotes period1's optimal savings at fee f):

dUcomm−unconstrained1

dβ¼ u c2ð Þ þ u bð ÞN0: ð21Þ

Under commitment savings at fee f, if period 2's incentive compati-bility constraint is binding:

dUcomm−constrained1

dβ¼ −∂sminlock

1

∂β u′ c1ð Þ−βu′ c2ð Þ� �

þ u c2ð Þ þ u bð ÞN0: ð22Þ

We know that, if f N 0, the agent prefers autarky over commitmentat βmidlock and βmax. Observe the following:

dUaut−constrained1

dβNdUcomm−unconstrained

1

dβNdUaut−nosave

1

dβdUaut−constrained

1

dβNdUcomm−constrained

1

dβNdUaut−nosave

1

dβ:

This tells us that an agent benefits increasingly from commitmentsavings as β drops from βmax to βmid, and then decreasingly as β con-tinues to drop from βmid to βmidlock. Therefore, as the bank raises feesfor commitment savings, its client base drops to a narrower windowaround βmid. This observation, combined with an actual distribution oftypes, allows the bank to set its profit-maximizing fee.

Under lending, the bank's optimal decision under private types issubject to more complex considerations. To illustrate this, it is con-venient to assume that the agent is, with equal probability, one oftwo possible types, βL or βH, with βL b βH. First, consider the bank'slending problem in the absence of a nondivisible good. If the bankdecides to serve only one type, it will choose βL and select theprofit-maximizing contract that satisfies the participation constraint(the more time-inconsistent the agent is, the higher are the bank'sprofits).

If the bank decides to serve both types, it will always engage inscreening. Let LH denote the bank's profit-maximizing contract at βH.The first-order condition requires that the slope of the participation

constraint,u′ 1þlHð Þ

2βHu′ 1−tHð Þ, be equal to the slope of an isoprofit line, 12. But

at this contract,u′ 1þlHð Þ

2βLu′ 1−tHð Þ N12 (the participation constraint for βL is

steeper). This means that the bank can offer a second contract with alarger loan and larger repayment that yields higher profits than LH

while being acceptable to βL.20 So, in the absence of a nondivisible, thebank will select the highest type (β∗) it chooses to serve, and will thenoffer a menu of contracts that allow full screening across all typesbelow β⁎.

Now, we can see how the outcome might change when thenondivisible is introduced. The reasoning above applies everywhere ex-cept at the discontinuity of the participation constraint. Suppose, at βH,the agent receives a contract with lH = p − 1. At this contract, it is pos-

sible that u′ 1þlHð Þ2βHu′ 1−tHð Þb

u′ 1þlHð Þ2βLu′ 1−tHð Þb

12. Therefore, theremight exist no other con-

tract that raises the lender's profits while being acceptable to βL. In thiscase, there will be no separation of types.

Finally, consider the coexistence of a bank andNGO.When both offercommitment savings, it is possible for a positive fee to survive in equilib-rium. Unlike in the case where types were publicly known, the NGO nolonger has an automatic incentive to undercut the bank's fees. Now, un-dercuttingwould have two effects: it wouldmake existing customers ofcommitment savings better off, but it would also attract new customers,some of whom might experience a welfare loss from adoption. If thebank offers loans and the NGO offers commitment savings, the outcomewill be subject to similar considerations as when the bank operatesalone, except that a higher reservation utility would have to be met.When both the bank and NGO offer loans, the equilibrium contractswill be identical to the case with public information. The NGO's incen-tive to drive any contract down to zero profits remains the same as be-fore. Since each agent is offered the contract that maximizes period 1'sdiscounted utility, he will continue to reveal his type through hischoice.21

10. Conclusion

This paper attempts to characterize equilibrium commitmentcontracts for hyperbolic discounters under different banking envi-ronments. This suggests several areas for continued research.

There is room for analyzing in greater detail the role of externalinterest rates. As interest rates rise, lending clearly becomes less at-tractive since there are non-commitment motives to save. However,the relative appeal of lending over saving will still be greater formore time-inconsistent agents. There also remains a potentially in-teresting question of how these results would change under an


infinite horizon. Such an analysis would introduce the possibility ofmultiple equilibria. Here, a bank's contract will depend not juston the agent's type but on his choice of autarky equilibrium. Final-ly, this paper makes the assumption that contracts, once signed,are exclusive and cannot be renegotiated. While this is plausiblewith a monopolist or an NGO, it is harder to justify under compe-tition, when banks could offer agents secondary loans that under-mine the benefits of commitment. This is the subject of continuingwork.

While the model presented above is a stylized representation ofmarkets for commitment savings and loans, the goal of the paperhas been to articulate the sometimes subtle mechanics at work inthe interaction between hyperbolic discounters and informalbanks.

Appendix A

Proof of Lemma 1.

(a) Period 1 will either save for nondivisible consumption or notat all. s1yes(β) is differentiable, and strictly increasing in β(since u′(0) = ∞). Note that U1(1;1 − s1

yes(1), 2 + s1yes(1) −

p, b) N U1(1;1,1,1) (Assumption 1) and limβ→0þU1ð β;1−syes1βð Þ;2þ syes1 βð Þ−p; bÞblimβ→0þU1 β;1;1;1ð Þ (since limβ→0þsyes1βð Þ≥p−2N0 ). Also note that dU1 β;1;1;1ð Þ

dβ is constant anddU1 β;1−syes1 βð Þ;2þsyes1 βð Þ−p;bð Þ

dβ ¼ u 2þ syes1 βð Þ−p� �þ u bð Þ, which is increas-

ing in β (both terms are differentiable everywhere). βmin cantherefore be uniquely determined.

(b) By construction of βmin, s1opt(β) determines the optimal savingsdecision.

(c) For β ≥ βmin, s1opt(β) = s1yes(β), which is continuous and strictly

increasing.syes1 1ð Þ ¼ p−12 . At in, c2 = 1 + s1

yes(βmin) − (p − 1) N 0(because u′(0) = ∞), so s1

opt(βmin) = s1yes(βmin) N p − 2.

Proof of Lemma 2.

(a) The following are true: for sufficiently large s1, U2(β; 1 − s1,1 + s1 − s2

yes, b + 1 + s2yes − p) N U2(β; 1 − s1, 1 + s1 − s2

no,1 + s2

no); the LHS and RHS in the previous inequality are contin-uous in s1; and by definition, s1min is bounded below at p − 2.Therefore, s1min exists.

(b) Suppose there is an s1 s.t. U2 β;1−s1;1þ s1−syes2 s1ð Þ; bþ 1þ�syes2 s1ð Þ−pÞ≤U2 β;1−s1;1þ s1−sno2 s1ð Þ;1þ sno2 s1ð Þ� �

. Then, itmust be true that s2yes attains a corner solution (p − 1), and s2

yes N

s2no. So, by concavity of u, at all s1bs1 , U2(β; 1 − s1,1 + s1 − s2

yes(s1), b + 1 + s2yes(s1) − p) bU2ð β;1−s1;1þ s1−

sno2 s1ð Þ;1þ sno2 s1ð ÞÞ≤U2 β;1−s1;1þ s1−sno2 s1ð Þ;1þ sno2 s1ð Þ� �,

which means that the period 2 agent will strictly prefer to notsave for the nondivisible.

(c) Since U2(β; 1 − s1, 1 + s1 − s2yes, b + 1 + s2

yes − p) is con-tinuous in s1, s2yes, and β; U2(β; 1 − s1, 1 + s1 − s2

no, 1 + s2no)

is continuous in s1, s2no, and β; and s2yes and s2

no are continuousin β; it must be true that s1min is continuous in β.Consider any s1 and β s.t. U2 β;1−s1;1þ s1−syes2 ð

�βÞ; bþ 1þ

syes2 β� �

−pÞ≤U2 β;1−s1;1þ s1−sno2 β� �

;1þ sno2 β� ��

. Then,s2yes attains a corner solution (p − 1) and u(1 + s2

no) b u(b).So, at allβbβ,U2 β;1−s1;1þ s1−syes2 βð Þ; bþ 1þ syes2 βð Þ−p

� �bU2

β;1−s1;1þ s1−sno2 β� �

;1þ sno2 β� ��

≤U2ð β;1−s1;1þ s1−sno2 βð Þ;1þ sno2 βð ÞÞ, which means that s1min(β) is strictly decreas-ing in β except when s1

min(β) = p − 2.(d) By assumption,smin

1 1ð Þ∈½p−2; p−12 Þ. For any s1, however large, there

exists β s.t. U2(β; 1 − s1, 1 + s1 − s2yes, b + 1 + s2

yes − p) b

U2(β; 1 − s1, 1 + s1 − s2no, 1 + s2

no). So, limβ→0þ smin1 βð Þ ¼ ∞. □

Proof of Lemma 3.

(a) This is true by definition of s1max.(b) Suppose there is an s1 s.t. U2 β;1−s1;1þ s1−syes2 s1ð Þ; bþ�

1þsyes2 s1ð Þ −pÞ≤U2 β;1−s1;1þ s1−sno2 s1ð Þ;1þ sno2 s1ð Þ� �

. Then, itmust be true that s2yes attains a corner solution (p − 1), ands2yes N s2

no. So, by concavity of u, at all s1bs1 , U2(β; 1 − s1,1 + s1 − s2

yes(s1), b + 1 + s2yes(s1) − p)bU2 β;1−s1;1þð s1−

sno2 s1ð Þ;1þ sno2 s1ð ÞÞ≤U2 β;1−s1;1þð s1−sno2 s1ð Þ;1þ sno2 s1ð ÞÞ ,which means that the period 2 agent will strictly prefer tonot save for the nondivisible.


yes − p) is con-tinuous in s1, s2yes, and β; U2(β; 1 − s1, 1 + s1 − s2

no, 1 + s2no)

is continuous in s1, s2no, and β; and s2yes and s2

no are continuousin β; it must be true that s1min is continuous in β.Consider any s1 and β s.t. U2ð β;1−s1;1þ s1−syes2 β

� �; bþ 1þ

syes2 β� �

−pÞ≤U2 β;1−s1;1þ s1−sno2 β� �

;1þ sno2 β� ��

. Then,s2yes attains a corner solution (p − 1) and u(1 + s2

no) b u(b).So, at all βbβ, U2 β;1−s1;1þ s1−syes2 ð�

βÞ; bþ 1þ syes2 βð Þ−pÞbU2 β;1−s1;1þ s1−sno2 ð�

βÞ;1þ sno2 β� �

Þ≤U2 β;1−s1;1þð s1−sno2 βð Þ;1þ sno2 βð ÞÞ, whichmeans that s1min(β) is strictly decreas-ing in β except when s1

min(β) = p − 2.(d) By assumption,smin

1 1ð Þ∈½p−2; p−12 Þ. For any s1, however large, there

exists β s.t. U2(β; 1 − s1, 1 + s1 − s2yes, b + 1 + s2

yes − p) b. So,limβ→0þsmin

1 βð Þ ¼ ∞. □

Proof of Proposition 1. By Lemmas 1–3, βmax and βmid exist, andβmax ≥ βmid ≥ βmin.

For β ∈ [βmax,1], s1opt ≥ s1

min. So, s2aut(s1opt) = p − 1. Therefore,

s1aut = s1

opt.For β ∈ [βmid, βmax), s1opt b s1

min ≤ s1max. So, s2aut(s1) = p − 1 iff

s1 ≥ s1min. Since, s1min ≤ s1

max and since U1(β; 1 − s1, p − 1, b) isstrictly decreasing in s1 for s1 ∈ [s1opt,s1max], s1

aut = s1min and

s2aut(s1min) = p − 1.

For β ∈ (0,βmid), s1min N s1

max. Therefore, in this regions1aut = s2

aut = 0. □

Proof of Lemma 4.

(a) Same argument as in Lemma 2. Also note that, for any β, s1minlock-

p − 1.(b) Suppose there is an s1 s.t. U2 β;1−s1;1þ s1−syes2 s1ð Þ; bþ 1þ�

syes2 s1ð Þ−pÞ≤U2 β;1−s1;1;1þ s1ð Þ. Then, it must be true thats2yes attains a corner solution (p − 1), and 1þ syes2 −pb1þ s1 .So, by concavity of u, at all s1bs1,U2(β; 1 − s1; 1 + s1 − s2

yes(s1);b + 1 + s2

yes(s1) − p) b U2(β; 1 − s1, 1, 1 + s1), which meansthat the period 2 agent will strictly prefer to not save for thenondivisible.


yes − p) is continu-ous in s1, s2yes, and β; U2(β; 1 − s1, 1, 1 + s1) is continuous in s1and β; s2yes is continuous in β; it must be true that s1minlock is con-tinuous in β.Suppose there is a β and s1 s.t. U2 β;1−s1;1þ s1−

�syes2 s1ð Þ; bþ

1þ syes2 s1ð Þ−pÞ≤U2 β;1−�

s1;1;1þ s1Þ. Then, it must be truethat s2yes attains a corner solution (p − 1) and bþ 1þ syes2 −p ¼bNcno3 s1ð Þ ¼ 1þ s1. Therefore, for anyβbβ,U2 β;1−ð s1;1þ s1−syes2 s1ð Þ; bþ 1þ syes2 s1ð Þ−pÞbU2 β;1−s1;1;1þ s1ð Þ, whichmeansthat s1

minlock(β) is weakly decreasing in β except whens1min(β) = p − 2.

(d) By assumption, smin1 1ð Þ∈½p−2; p−1

2 Þ. For all β, s1minlock(β) b p − 1,and for any ε N 0, there is some β s.t. s1minlock(β) N p − 1 − ε.So, limβ→0þ s

minlock1 βð Þ ¼ p−1. Finally, for any β and s1 ≥ p − 2,

U2(β; 1 − s1, 1, 1 + s1) b U2(β; 1 − s1, 1 + s1 − s2no(s1),

1 + s2no(s1)), so s1

minlock(β) b s1min(β) except when s1

min(β) =p − 2. □


Proof of Proposition 2. Themonopolist can charge a fee only if there issome s such that s ≥ s1

minlock and s ≠ s1aut, and U1(β; 1 − s, 2 + s − p,

b) ≥ U1(β;c1aut,c2aut,c3aut). This is feasible only for β ∈ [βmidlock, βmax).Suppose constraint (16) doesn't bind. Then, there must be a higher

value of fmon such that constraints (16) and (17) continue to be satisfied.

Proof of Proposition 3. For β ∈ (βmidlock,βmid), there is no saving inautarky. Since constraint (16) binds, U1(β; 1 − fmon − s1

mon, 2 +s1mon − p, b) = U1(β;1,1,1). Since 1 − fmon − s1

mon b 1 and β b 1,this equality implies that U0(β; 1 − fmon − s1

mon, 2 + s1mon − p, b)

N U0(β; 1,1,1).For β ∈ (βmid,βmax), U1(β; 1 − fmon − s1

mon, 2 + s1mon − p, b) =

U1(β; 1 − s1min, 2 + s1

min − p, b). Since u(1 − fmon − s1mon) N

u(1 − s1min) and β b 1, it must be true that U0(β; 1 − fmon − s1

mon,2 + s1

mon − p, b) b U0(β; 1 − s1min, 2 + s1

min − p, b). □

Proof of Proposition 4.

(a) Consider L ≡ p−1; p−12 ; p−1

2ð Þ. If β b 1, it follows that U1 β; c1 L� �

;�

c2 L� �

; c3 L� �ÞNU1 β; caut1 ; caut2 ; caut3

� �. Then, there exists ε N 0

such that for L ≡ p−1; p−12 −ε; p−1

2 −εð Þ,U1 β; c1 L� �

;�

c2 L� �

; c3 L� �

Þ≥ U1 β; caut1 ; caut2 ; caut3

� �. Therefore, for any agent with β N 0, there

is at least one loan contract (L) underwhich the bankmakes pos-itive profits.

(b) If the agent saves, he must save a positive amount in period 1.Suppose, under Lmon, the agent saves (s1 N 0, s2≥) but does notpurchase the nondivisible. Consider the alternative loan contractL ¼ lmon−s1; tmon

2 −s1 þ s2; t3−s2� �

. The agent's consumptionwill be identical under Lmon and L, so he is indifferent betweenthe two. Both contracts yield the same profit. But L has a strictlylower loan size, so the firm should have selected it.Suppose, under Lmon, the agent saves and purchases thenondivisible in period 3 (the same argument can be made if it ispurchased in period 2). Then, it must be true that c3(Lmon) = band c1(Lmon) + c2(Lmon) = 2 − (p − 1) + lmon − t2

mon − t3mon.

Consider the contract L ¼ p−1; 2−c1 Lmonð Þ−c2 Lmonð Þ2 ;

2−c1 Lmonð Þ−c2 Lmonð Þ2

� �.

This yields the same profit to the bank as Lmon. Since U1

β; c1 L� �

; c2 L� �

; c3 L� ��

NU1 β; c1 Lmon� �; c2 Lmon� �

; c3 Lmon� �� , Lmon

cannot be a profit-maximizing contract.(c) This follows directly from the strict concavity of u. If t2 ≠ t3, the

bank can raise its profits by offering t2 ¼ t3 ¼ t2þt32 −ε, for some

ε N 0.(d) Consider any loan contract with l N 0, l ≠ p − 1, such that

the agent's consumption choices do not involve saving. Sup-pose u′(1 + l) N βu′(1 − t). Then, there are some ε1, ε2, suchthat ε1 b ε2 and such that the agent will accept a contractlþ ε1; t þ ε2

2 ; t þ ε22ð Þ. Since the modified contract raises profits,

the original contract was not profit-maximizing. Instead, sup-pose u′(1 + l) b βu′(1 − t). Then, there are some ε1, ε2, suchthat ε1Nε2 and such that the agent will accept a contractl−ε1; t−ε2

2 ; t−ε22ð Þ. Since the modified contract raises profits, the

original contract was not profit-maximizing. □

Proof of Proposition 5. At any β b 1, U1(β; c1(Lmon), c2(Lmon),c3(Lmon)) = U1(β; c1aut,c2aut,c3aut). It must also be true that c1(Lmon) N c1-aut. Since β b 1, it follows thatU0(β; c1(Lmon), c2(Lmon), c3(Lmon)) b U0(β;c1aut,c2aut,c3aut). The same argument can be used to prove that welfare islower than under commitment savings. □

Proof of Proposition 6. (a) For any fee f, let s1ngo(f) be period 1's best pos-sible savings choice under commitment savings (subject to s1

ngo(f) ≥s1minlock). Then, s1ngo(0) ∈ [s1ngo(f), s1ngo(f) + f). So, consumption ineach period rises weakly when f drops to 0. Welfare must rise. (b)NGO commitment saving will be offered if two conditions are satisfied:

(i) Period 1 values commitment (which is the case forβ ∈ (βmidlock,βmid)),and (ii) Commitment weakly raises welfare. By construction,smin βmaxð Þbp−1

2 . Since s1min is continuous and falling in β, there

must be some β≥βmid , such that for β∈ β;βmax� �

, smin1 βð Þ≤ p−1

2 .In this region, commitment will further lower s1

min(β), thus fur-ther lowering welfare. Therefore, condition (ii) is not satisfiedin this region. □


(a) First consider β ∉ [βmidlock, βmax). In this region, there is nof ≥ 0 at which the agent will adopt commitment. Next, considerβ = βmidlock. Here, the agent can be induced to adopt commit-ment only at f = 0. Finally, consider any β ∈ (βmidlock,βmax).The NGO must offer commitment savings with f = 0. (Considerany alternate NGO strategy. Then themonopolist's best responseinvolves a positive fee, which, as shown in Proposition 6, will re-sult in lower welfare than if the NGO offered commitment forfree. Therefore the alternative NGO strategy cannot constitute abest response to the bank's best response to it).

(b) If an agent received commitment savings under anNGO alone, hereceives the same contract under coexistence. Therefore, hiswel-fare remains the same as under an NGO. If an agent did not re-ceive commitment savings under an NGO alone, his welfaremust be lower thanunder anNGO (by definition of theNGO's ob-jective) andhigher than under a bank (since in this region the feehas dropped from fmon N 0 to 0.) □


(a) The agent cannot accept a commitment savings contract in equi-librium: for any commitment savings contract, there exists a loancontract that will satisfy his participation constraint and yieldpositive profits for the bank.For β ∉ (βmidlock,βmax), commitment savings cannot change U1,so the monopolist moneylender will offer the same contract asin Proposition 4. Consider β ∈ (βmidlock,βmax). Let U1

aut and U0aut

be, respectively, period 1 discounted utility and welfare in autar-ky. LetU1(f) andU0(f) be, respectively, period 1 discounted utilityand welfare under commitment savings with fee f. And let U1(L)and U0(L) be, respectively, period 1 discounted utility and wel-fare under a loan contract L. Given f , the monopolist's offersL(f) given by:

maxL

−lþ t2 þ t3

s:t: U1 Lð Þ≥max Uaut1 ;U1 fð Þ

n o ð23Þ

Suppose f ¼ f N0. Then, there is some f ′b f such thatU1 f ′� �

NU1

L f� ��

: Since β b 1 and period 1 consumption is lower under

commitment savings than under lending, this alsomeans thatU0

f ′� �

NU0 L f� ��

. So f N 0 cannot be a best response to the

moneylender's best response to it. In equilibrium, the NGOmust offer f = 0. Since U1(0) N U1

aut, the moneylender's con-tract changes relative to Proposition 4. The agent is indifferentbetween the NGO and moneylender's contract, and acceptsthe moneylender's contract.

(b) Let the equilibrium contract be denoted L. Let the reservation

utility under coexistence be denoted U1 and reservation utilityunder autarky be denoted U1

aut. First, notice that welfare mustbe strictly lower than under NGO commitment savings: Since,

by the participation constraint, U1 is identical under L and free

commitment savings, and since c1 is higher under L than under


commitment savings, U0 must be strictly lower under L. Next,compare welfare under coexistence to welfare under monop-olist lending.

Suppose, under the moneylender, the equilibrium contract was

lmon N p − 1. This means that u c1 Lmonð Þð Þβu c2 Lmonð Þð Þ ¼ 1

2. Then, under coexistence, it

must be true that, on the new (tighter) participation constraint, there

is some lN lmon and tbtmon such that u c1 Lð Þð Þβu c2 Lð Þð Þ ¼ 1

2. At any other point on

the new participation constraint, u c1ð Þβu c2ð ÞN

12 , so it cannot be profit-

maximizing for the bank. Therefore, if lmon N p − 1, welfare risesunder coexistence.

Suppose lmon = p − 1. Now suppose lbp−1. Then,u 1þ l� �

þ 2βu

1−t� � ¼ X, where X denotes the discounted utility of the outside op-tion. Let π Xð Þ denote the lender's profits. As X changes, suppose the

firm were to continue to satisfy the participation constraint with lconstant. Then, implicitly differentiating the participation con-

straint, we get π′ Xð Þ ¼ −1βu′ 1−tð Þb0. Consider an alternate contract, L,

which satisfies l ¼ p−1 and u bð Þ þ 2βu 1−t� � ¼ X . Let π Xð Þ denote

the lender's profits from such a contract. Again, keeping l constant,

we get π′ Xð Þ ¼ −1βu′ 1−tð Þb0. Since tNt

′, π′ Xð Þbπ′ Xð Þ. Since by assumption π

U1

� �Nπ U1

� �, and Uaut

1 bU1, it must be true that π Uaut1

� �Nπ Uaut

1

� �. But

this is not possible since lmon was profit-maximizing. Therefore, if

lmon = p − 1, it must be true that l≥ lmon . Then, the new profit-maximizing point is either at tangency, in which case tbtmon , oralong l = p − 1, in which case also tbtmon.

Therefore, from the two previous paragraphs, if lmon ≥ p − 1, wel-fare under coexistence must be higher than under monopolist lending,

Now consider lmon b p − 1. If lbp−1, then it must be true that lN lmon

and tbtmon, so welfare rises relative to monopolist lending.Finally, it is possible to construct a case where welfare will drop

under coexistence. Suppose lmon b p − 1. Also consider the alternatecontract Lwith l ¼ p−1 and t such that u bð Þ þ 2βu 1−t

� � ¼ Uaut1 . Sup-

pose the moneylender is nearly indifferent between Lmon and L. Thereis some value of β at which such indifference exists: Let πL(β) be themonopolist's maximized profit from any loan conditional on thenondivisible not being purchased, and let πH(β) be the monopolist'smaximized profit from any loan large enough to allow the purchaseof the nondivisible. At sufficiently high β, when the agent isuninterested in small loans, πH(β) N πL(β). At sufficiently low β,when the agent is willing to pay nearly all future income for asmall loan, πL(β) N πH(β). Since πH(β) and πL(β) are continuous inβ, there must be some β at which the bank is indifferent between asmall loan and a large loan.

If the moneylender is nearly indifferent between lmon b p − 1

and l ¼ p−1, it can be shown that as the participation constrainttightens, the moneylender will strictly prefer to offer p − 1. Let peri-od 1's reservation utility be denoted X. As X rises, suppose the mon-eylender were to choose to remain at a point of tangency. Thetangency condition is: u(1 + l(t)) + 2βu(1 − t) ≡ X, where l(t) isgiven by the first-order condition from profit-maximization. Differ-entiating implicitly by X, we get dt

dX ¼ 1∂l∂t−2ð Þβu′ 1−tð Þ . So, as X rises, the

change in the moneylender's profits is given by: dπdX ¼ −∂l

∂tdtdX þ 2dt

dX

¼ −1βu′ 1−tð Þ. Alternatively, as X rises, suppose the moneylender were to

maintain l ¼ p−1 . Then, the change in profits is given by: dπdX

¼ −1βu′ 1−t′ð Þ. Profits drop faster at the point of tangency than at l ¼ p−

1. Therefore, if the reservation utility rises under coexistence, the

lender will switch to a contract with l ¼ p−1. Since welfare is strictlylower at L than at Lmon , if U1

∗ is sufficiently close to U1aut, welfare will

drop relative to monopolist lending. □


(a) Any equilibrium contract, L⁎, must satisfy t∗ ¼ l∗

2 and:

l� ¼ arg maxl

U1 β; bþ lþ 1−pð Þ;1− l2;1− l

2

� �: ð24Þ

If it did not satisfy the zero-profit condition t ¼ l2ð Þ, theNGO could

raise welfare by lowering t. If, conditional on satisfying the zero-profit condition, l⁎ did not satisfy condition (24), the bank couldraise profits by offering a different contract.By condition (1), l⁎must be at least p − 1. Consider the first-ordercondition of the maximization problem: u′ bþ l∗ þ 1−pð Þð Þ ¼ βu′

1−l∗

2ð Þ. Implicitly differentiating with respect to β, we get l∗′ βð Þ ¼u′ 1−l∗

2ð Þu′′ bþl∗þ1−pð Þþ1

2βu′′ 1−l∗

2ð Þb0. Since l⁎ is continuous in β, is bounded below

at p − 1, and for sufficiently low β is greater than p − 1, theremust be some βb1 as defined in the statement of the proposition.

(b) Welfare is maximized at l = p − 1 and t ¼ p−12 . Therefore, for

β∈½β;1Þ, welfare will be the same as under a lending NGO andstrictly higher than under monopolist lending.Considerβ∈ 0;β

� �. Welfare will be strictly lower than under a

lending NGO. Now we can show that it will be strictly higherthan under a monopolist lender. Under the monopolist

lender's contract, u′ c1 Lmonð Þð Þ2βu′ c2 Lmonð Þð Þ≤ 1

2 . (Strict inequality can only

occur at lmon = p − 1. Any other contract satisfies tangencybetween an isoprofit line and the participation constraint.)Under coexistence, suppose t∗ ≥ tmon. Since the individual'sreservation utility is higher than under a monopolist lender,

it must be true that l∗ N lmon. Then, u′ c1 L∗ð Þð Þ2βu′ c2 L∗ð Þð Þb

u′ c1 Lmonð Þð Þ2βu′ c2 Lmonð Þð Þ≤ 1

2. But this

means that l⁎ does not satisfy condition (24), since U1 could beraised by lowering l. Therefore, it must be true that t∗ b tmon.Under coexistence, suppose l∗ b lmon. If lmon ≤ p − 1, this is im-possible by the definition of l⁎. If lmon N p − 1, since we have

seen above that t∗ b tmon, this means that u′ c1 L∗ð Þð Þ2βu′ c2 L∗ð Þð ÞN

u′ c1 Lmonð Þð Þ2βu′ c2 Lmonð Þð Þ ¼ 1

2.

But then l⁎ does not satisfy condition (24), since U1 could beraised by raising l. Therefore, it must be true that l∗ ≥ lmon.Since t∗ b tmon and l∗ ≥ lmon, welfare is strictly higher than undera monopolist lender. □

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