Borrowing and Saving: with or without Self-Control?∗

Alex Groves†

11/28/11‡

Abstract

I study the Gul and Pesendorfer (2001)(2005) model of temptation and self-control and the

βδ discounting model in a stochastic income setting. I prove that the optimal savings plan

for the GP model consists of a mandatory minimum deposit. Using the mandatory minimum

deposit framework I show that for a given distribution of income the strength of temptation can

be decreased and the discount factor increased in the GP model so that it is indistinguishable

from the βδ model. However the two can be quantitatively differentiated if the discount factor

for the individual is known, or by using one distribution to calibrate both models and the second

for prediction. It is because in the GP model commitment reduces the cost of self-control even

when it is not binding that the two models are distinguishable. Differentiating the two models

is of practical interest for designing loan and savings plans, and will improve the understanding

of the empirical relevance of self-control. Further, I characterize the optimal borrowing plan

under the GP model as consisting of a debt limit and mandatory payments. Finally, I show

that mandatory deposits are strictly preferred to liquidity constraints because they allow for

greater flexibility and are more effective in reducing the cost of self-control.

∗I would like to thank Philipp Sadowski for the many helpful conversations and input. I am grateful tomy adviser Rachel Kranton, and my committee: Curtis Taylor, Atila Abdulkadiroglu, and Dan Ariely, fortheir help and support. I’d also like to thank the Duke University theory lunch for useful comments. Allerrors are my own.†Email: alexander.groves@duke.edu‡For the most recent version of this paper please visit my website:

http://econ.duke.edu/people/groves/research

1

1 Introduction

There is a large amount of evidence that at times people have a preference for commit-

ment devices (Thaler and Benartzi (2004), Loewenstein and Prelec (1992), Beshears et al.

(2011)1). This suggests that popular financial tools and organizations that require people

to commit to some course of action and reduce their flexibility may exist in part because

of the preference for commitment. Defined contribution 401(k)s and IRAs, pensions, saving

up clubs, purchasing via layaway, and rotating savings and credit associations (ROSCAS)

all contain predefined payments. Mortgages, car financing, credit cards, and microfinance

loans all require regular payments, and a debt limit. While there may be many reasons that

people opt into these devices it is the act of willingly reducing their flexibility that I am

interested in.

Why do people want to reduce their flexibility? One explanation is that temptation

is only present for choices that affect immediate consumption, and that by committing an

individual is trying to control her temptation2. There is a large body of evidence in cognitive

neuroscience for an intrinsic difference between consumption now and consumption in the

future, which supports this explanation (McClure (2004) for example, and see Luhmann

(2009) for a survey of the neuroscience literature)3. When making the decision about whether

or not to save an individual is explicitly weighing immediate consumption against future

consumption. Because it is precisely during these types of decisions that temptation plays

a role it is necessary to take temptation into account when modeling saving and borrowing

choices. While there are many different models of temptation two of the most commonly used

are the Gul and Pesendorfer (2001)(2005) (henceforth GP) model of temptation and costly

self-control and the βδ quasi-hyperbolic discounting model, originally introduced by Phelps

and Pollak (1968). Given that these two models seem to interpret temptation differently

is there any observable difference between the two models, and if so, which one fits data

better? This paper outlines a method to differentiate the two models experimentally using

1For surveys of the literature see Bryan et al. (2010) and Frederick et al. (2002).2For experimental evidence for this explanation see Houser et al. (2010).3See Loewenstein (1996) for a discussion of the consequences of visceral urges on behavior in general.

2

commitment and saving decisions in a stochastic income setting and also illustrates when

in this context the two models will be indistinguishable. I use stochastic income instead

of stochastic preferences because the former is reproducible in a laboratory setting, and is

easier to observe in, and estimate from real world data.

The βδ model explains the difference between consumption today and in the future via

changes in impatience. The model is a specialized version of the time-inconsistent pref-

erences of Strotz (1955) that was introduced by Phelps and Pollak (1968) as a model of

intergenerational altruism, and re-tailored by Laibson (1997) as a model of intrapersonal

dynamic conflict. In the βδ model discounting is quasi-hyperbolic: the individual discounts

at a greater rate between today and tomorrow than she does between any other two days

in the future. Because she knows that her future selves will discount in the same way, she

disagrees with them about how resources should be allocated. It is this disagreement be-

tween the selves that leads to a preference for commitment. The βδ model is prolific in the

applied literature because it is easy to estimate the level of a person’s time-inconsistency

and then correlate this with the variable of interest4. Inconsistent preferences do complicate

the analysis of the model because either each self must be treated as a separate agent with

preferences that do not fully align with those of all the other agents (Krusell and Smith Jr

(2003)), or if actions are found recursively there can be existence problems (Caplin and

Leahy (2006), Harris and Laibson (2001)).

The GP model reproduces many of the same observations that the Strotz (1955) model

can explain but with preferences that are time-consistent instead of inconsistent. Time-

consistent preferences result in straightforward welfare analysis and a tractable mathemat-

ical model. The GP model explains the visceral component of consumption today through

temptation and costly self-control. For instance, if an individual receives all of her paycheck

today she may be tempted to spend it all at once, and she will have to control herself in

order to avoid spending it all: having to control oneself is unpleasant, or undesirable. Before

she receives her paycheck she knows that she will be tempted and therefore will have to

4See Meier and Sprenger (2010) for a discussion of the estimation of time-inconsistency and its relationto credit card debt.

3

exercise self-control, so she chooses to have part of her paycheck automatically deposited

into an illiquid asset in order to avoid exerting this effort. In this model it is the desire

to avoid temptation and costly self-control that creates a preference for commitment. The

combination of temptation with costly self-control results in consistent preferences.

In a stochastic income setting I prove that the optimal savings plan for the GP model

consists of mandatory minimum deposits that are committed to before income is realized. A

mandatory minimum deposit is designed to balance flexibility and commitment. In the GP

framework the cost of self-control is defined by the most tempting element of the current

menu of options, which in the consumption-saving setting corresponds to the individual’s

option of consuming her entire income. Therefore when designing a savings program to

reduce the cost of self-control only the minimum amount she has committed to save matters.

Any other fees or mechanisms that are put in place to try and control future choices are

either ineffective or get in the way. The result is a mandatory minimum deposit that is

committed to before income is realized. This optimal plan is qualitatively the same as that

found for the βδ model by Amador et al. (2006) in a similar setting.

Using the mandatory minimum deposit framework I am able to precisely describe situ-

ations in which the two models are indistinguishable, because they both can describe the

same revealed preferences over the level of commitment and savings, and others in which they

cannot and predict different outcomes. The additional cost of self-control in the GP model is

what differentiates the two models. Commitment benefits an individual with GP preferences

even when it is not binding because it restricts the most tempting option, thereby reducing

the amount the individual must control herself. However, if the researcher uses only one

distribution of income the strength of the individual’s temptation can be decreased and her

discount factor increased so that the GP model matches any preferences that the βδ model

can describe. But if the researcher determines the individual’s discount factor (experimen-

tally or otherwise), thereby fixing one of the independent variables, the two models will no

longer agree when stochastic income is non-degenerate. When the discount factor is known

the strength of temptation in the GP model can be adjusted so that the βδ and GP models

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agree on either savings decisions or on commitment decisions, but not both. If the models

describe the same savings decisions then the GP model will predict a higher level of desired

commitment than the βδ. Again, this occurs because in the GP model commitment reduces

the cost of self-control even when it is not binding, while in the βδ model commitment has

no benefits when it is not binding. In order to make both agree instead on the level of

commitment the strength of temptation must be decreased in the GP model, and so it will

predict a larger savings decision than the βδ model.

Alternatively, the two models can be calibrated with one distribution of income, and if

they are used to make predictions using another they may no longer agree on both com-

mitment and savings decisions. This is because by calibrating both models for a particular

distribution the strength of the individual’s temptation and her discount factor have been

pinned down, so when using these calibrated models in another setting they can no longer

be adjusted so that the two models agree. For instance, if the models are calibrated using

deterministic income and then used with a non-degenerate stochastic income they will agree

on savings decisions, but the GP model will predict a higher level of commitment. If in-

stead the models are used with another degenerate distribution they will continue to agree

on both saving and commitment decisions. These results mean that the time-inconsistent

βδ model and the time-consistent GP model can be differentiated empirically. This will

be useful for policy design, for example it will allow for more accurate estimation of the

effects of offering different types of retirement or borrowing plans and subsequent changes

to those plans. It will also help researchers better understand whether costly self-control is

empirically relevant.

This paper also finds interesting results when the individual is borrowing rather than

saving. The GP model predicts an optimal borrowing plan in a stochastic income setting

that is very similar to the optimal savings plan because it relies on regular mandatory

minimum payments that are defined before income is realized. Additionally the borrowing

plan has a debt limit. Mandatory payments and the debt limit help the individual reduce

her cost of self-control. This is an explanation of why regular payment schedules are a fixture

5

of most microfinance loan plans.

This paper contributes to the literature comparing the βδ model, and more generally

the Strotz (1955) model, to the GP model. Dekel and Lipman (2011) relate a random

preference Strotz model and a random preference GP. They find that if both agree on the

choices from menus then the random GP individual is more temptation-averse than the

random Strotz individual: they are more willing to commit. This is in line with what I

find except that instead of random preferences I use stochastic income. Stochastic income

is attractive because it can be measured empirically and reproduced experimentally while

the estimation of stochastic preferences requires large amounts of preference data. Gul and

Pesendorfer (2005) axiomatize the Strotz model and show that any preferences over a finite

choice set that are able to be described by a Strotz representation can be described by a

GP representation as well. This comparison is similar to what I find in this paper except

that they are focusing on a finite choice set for the Strotz model, while in my model the

choice set is continuous. Additionally they use a version of their 2001 model introduced by

Krusell et al. (2010) (henceforth KKS) in which temptation depends not only on current

consumption but on all future consumption as well. To make this model agree with the

βδ model temptation must be overwhelming, and therefore there is no cost of self-control

because the individual always succumbs. This makes it impossible to test the implications of

costly self-control as I do in this paper. Gul and Pesendorfer (2004) directly compares the βδ

and GP models in a simple market with liquid and illiquid assets available simultaneously,

and in an intergenerational transfer environment. While these settings differ from what I

analyze here, they do note that it is precisely the additional cost of self-control in the GP

model that drives any difference in choices between it and the βδ model.

The literature on saving and time-inconsistency has used the βδ model to explain under-

saving and the desirability of commitment (Phelps and Pollak (1968), Laibson (1997)).

Amador et al. (2006) is the only other paper to study optimal savings plans in the pres-

ence of temptation. They use the βδ model and the KKS version of the GP model. Their

paper considers stochastic preferences, which are different from stochastic income except

6

when utility is exponential, in which case the two are equivalent. This suggests that for

stochastic income in general the optimal savings plan for the βδ model will be a mandatory

minimum deposit as well.

The only other paper that I am aware of that studies borrowing and temptation is Fischer

and Ghatak (2010)5. In their paper using the βδ model they show that an individual would

be willing to borrow more if she can commit to a regular repayment schedule than she would

if she could not commit. They offer this as a reason why most microfinance institutions

require regular predefined payments after a loan is taken out.

The remainder of the paper is organized as follows: Section 2 introduces the GP model

in the stochastic income consumption-savings setting. Section 3 finds the optimal savings

plan. Section 4 uses this finding to compare the GP and βδ models. Section 5 studies

the optimal borrowing plan for the GP model and shows that a similar comparison can be

made in this situation between the GP and βδ models. I then extend the GP model to look

at other applications: the difference between mandatory minimum deposits and liquidity

constraints in Section 6; and borrowing with a simple interest rate in Section 7. The final

section concludes. Any proof that does not appear in the text has been relegated to the

appendix.

2 The model

This section introduces the model specific to saving. The model will be adjusted to encom-

pass borrowing in section 5.

The model in this paper consists of three periods: 0, 1, 2. In period 0 the individual

makes a choice about commitment and consumes nothing. In period 1 income, y1, is realized

and she decides how much to save, s, and consume, c1. Finally in period 2 income, y2, is

realized, the individual receives her savings plus interest, and consumes c2.

There is a bank which exists for two purposes. The first is to be the entity with whom

5Basu (2008) studies a general equilibrium setting using the βδ model in which there is a profit maximizingbank that offers loans and a welfare maximizing microfinance bank that offers savings.

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the individual can sign binding contracts. The second is to offer an interest rate, r, for the

individual’s savings that is large enough so that non-zero savings is optimal for some level

of income. The bank has no profit or costs, and makes no decisions.

The person cannot borrow or consume a negative amount, so the lower bound on the

distribution of income must be greater than or equal to zero. Income is stochastic and

the pdfs of the distributions, f(y1) and f(y2), are continuous, independent, not necessarily

identical, and known to both the individual and the bank. Realized incomes, y1 and y2,

are the individual’s private information. Therefore the available savings mechanisms are a

function of either the individual’s period 1 savings decision, s(s), or of her reported income,

s(y). These mechanisms define a mandatory deposit that the individual is contractually

obligated to make given either her savings decision, s, or her reported income, y. A portion

of this mandatory deposit could be kept by the bank as a fee. s(·) can be discontinuous

and/or nonlinear. Let S denote the space of feasible savings mechanisms.

The individual has GP preferences represented by commitment utility, u(c), and temp-

tation utility, v(c). Working backwards, the period 2 decision problem for the individual

is

maxc2∈B2

[u(c2) + v(c2)]− maxc2∈B2

[v(c2)] . (1)

Where B2 ≡ {c2 ∈ R+|c2 ≤ y2 + (1 + r)(s + s(s))}, if the mandatory deposit is a function

of s. Denoting as c∗2 and c∗2 the values of c2 and c2 that solve the maximization problems

above, the individual’s period 1 decision problem is

maxc1,s∈B1

[u(c1, s) + v(c1, s)]− maxc1,s∈B1

[v(c1, s)] + δE [u(c∗2) + v(c∗2)− v(c∗2)] . (2)

Where B1 ≡ {(c1, s) ∈ R2+|c1 +s ≤ y1− s(s)}. Finally, the individual’s period 0 commitment

decision consists of choosing a feasible savings mechanism that maximizes her period 0

expected utility given how she will make her period 1 and 2 consumption choices.

8

maxs(s)∈S

E [u(c∗1, s∗) + v(c∗1, s

∗)− v(c∗1, s∗1) + δE [u(c∗2) + v(c∗2)− v(c∗2)]] .

From equations (1) and (2) one can see that the individual’s actual choice in periods 1 and

2 maximizes the sum of the commitment utility and the temptation utility u(c, s) + v(c, s)6.

The net cost of self-control for any choice (c, s) is[max(c,s) v(c, s)− v(c, s)

]. I refer to

max(c,s) v(c, s) as the cost of self-control. Both u(c, s) and v(c, s) depend only on current

consumption, not on future consumption. I am ruling out preferences such as those defined

by the KKS version of the GP model so that I can study the effects of the cost of self-control.

u(·) is continuous, strictly concave, and increasing: u′(c) > 0, u′′(c) < 0. Temptation

utility v(·) is continuous and increasing. It can be convex, in which case v′(c) > 0, v′′(c) ≥ 0.

If it is concave then v′(c) > 0 , v′′(c) < 0. Assuming that v(c) is convex is akin to having

a convex cost function in producer problems and results in a convex optimization problem.

The possibility of a concave v(c) is necessary in order to compare the GP and βδ models

and the exact form it takes will be discussed in Section 4. Finally, u(c) + v(c) is concave

and increasing⇒ ‖u′′(c)‖ > v′′(c). This means that the budget constraints will always bind.

Also, because v(c) is continuous and increasing the individual is most tempted by the option

to consume her entire available wealth this period. Therefore in period 2 the individual is

not going to have to control herself because she will consume her entire period 2 wealth: her

net cost of self-control will be zero. This reduces the problem to a two period problem where

the individual chooses how much to save in period 1 and how to commit in period 0:

Period 1:

maxs≤y1−s(s)

[u(y1 − s− s(s)) + v(y1 − s− s(s))]

− maxs≤y1−s(s)

[v(y1 − s− s(s))] + δEu(y2 + (1 + r)(s+ s(s)))

Period 0:

6For period 2 there is no s.

9

maxs(s)∈S

E [u(y1 − s∗ − s(s∗)) + v(y1 − s∗ − s(s∗))

− v(y1 − s∗ − s(s∗)) + δEu(y2 + (1 + r)(s∗ + s(s∗)))]

3 Saving

This section analyzes the GP model in the stochastic income setting presented above and

the resulting optimal savings plan. To focus on savings I assume that the individual cannot

borrow, so consumption must be greater than or equal to zero and the distribution of income

has a lower bound of at least zero.

If the distribution for income is degenerate (income is deterministic), then period 1 utility

without commitment is

U(s) = u(y1 − s) + v(y1 − s)− v(y1) + δu(y2 + (1 + r)s).

In this setting the individual can reduce her net cost of self-control to zero by committing

to a savings device s(s) = s = s∗, where s∗ is her optimal savings with no temptation. This

would allow her to achieve her first-best commitment utility level

U(s∗) = maxsu(y1 − s) + δu(y2 + (1 + r)s).

When income is stochastic (non-degenerate) the ideal savings mechanism would be one

that induces the individual to truthfully reveal her income so that it can assign a mandatory

deposit that nullifies the cost of self-control. Note that the most tempting report is the one

that allows her to save the minimum amount so that she can consume the maximum amount:

s = miny s(y). Outside of the minimum mandatory deposit, s, any functional form of s(y)

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does not effect her period 1 savings decision, s, because the mechanism is not a function of

s. Therefore if the savings mechanism is to be anything other than a mandatory minimum

deposit it must be a function of s, not of reported income. The remaining mechanisms can

be split into two categories: those that levy fees on the individual, and those that return

all deposits the following period with interest7. Both of these categories of mechanisms will

rely on mandatory deposits and/or mandatory fees because this is the only way to affect the

cost of self-control. As soon as a deposit or fee is optional the cost of self-control is defined

by the upper limit of the budget constraint.

1 2 3 4 5 6s

1

2

3

4

5

6

Savings

s`HsLs+s`HsLs

s`

(a) Decreasing s(s)

1 2 3 4 5 6s

1

2

3

4

5

6

Savings

s`HsLs+s`ss

`

(b) Increasing discontinuous s(s)

Figure 1: Saving mechanisms that are a function of period 1 savings, s.

Ignoring fees for a moment, can a savings mechanism that depends on s do any better

than a mandatory minimum deposit? That is, can s(s) 6= constant? The most tempting

level of period 1 savings is the one that allows her to save, in total, the minimum amount

so that she can consume the maximum amount: s = mins(s + s(s)). s is effectively the

mandatory minimum deposit. If the mandatory deposit is weakly decreasing in s then any

level of savings that can be obtained with s(s) is obtainable with a mandatory minimum

deposit of s, and both mechanisms will be equivalent in terms of the cost of self-control. For

example, in figure 1(a) the savings mechanism is s(s) = 1/(s2 + 0.2) − 1/(y2 + 0.2). This

function decreases from s(0) ≈ 5 to zero when s = y = 6, the upper limit of possible income

7Remember this model only has two periods for this mechanism to work in, if it had more periods wecould include an additional category, which I will do in Section 6.

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in this example. The effective mandatory minimum deposit for this savings mechanism is

s ≈ 1.8, and net savings (s+ s(s)) can obtain any amount between 1.8 and 6.

If instead the mandatory deposit is weakly increasing in s some levels of savings will be

unobtainable, in addition to those below s. An example of such a savings mechanism is

shown in figure 1(b) where s(s) is a piece-wise function such that

s(s) =

2 0 ≤ s ≤ 3

3 3 < s ≤ y

where the upper limit of income, y, is 6. In this example the individual is unable to obtain a

net savings level between 5 and 6. Restricting savings options above the minimum mandatory

deposit forces the individual to either save more or less than she would prefer. Forcing her

to save more than she would otherwise choose increases her net cost of self-control by more

than it increases her commitment utility. Forcing her to save less than she would choose

decreases her net cost of self-control by less than the increase in her commitment utility. A

mandatory minimum deposit with the same minimum, s, does not restrict the individual’s

options in this way and is therefore strictly preferred.

Introducing a fee does not reduce the individual’s cost of self-control any more than a

mandatory deposit would, and has the additional detriment of reducing her consumption in

the final period. While a fee that is a function of s may induce the individual to save more

than a similar mandatory deposit this does not benefit her because her fee induced choice is

even further from the choice that optimally balances her commitment utility and her net cost

of self-control. Therefore a fee mechanism that is a function of s is strictly less preferred to a

mandatory deposit that is a function of s, which is less preferred to a mandatory minimum

deposit.

Theorem 1. The optimal savings plan consists of a mandatory minimum deposit, s, that is

a function of the distribution for income. This deposit is returned to the saver the following

period with interest.

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If v(c) is convex or linear then s is unique8. If v(c) is concave and f(y) is either uniform

or monotonically decreasing then it is unique as well. Simulations show that it is also unique

for a variety of other continuous single mode distributions9.

When period 1 realized income is very low the mandatory minimum deposit is overly

restrictive and forces the individual to save more than she would like. In a middle range of

income the mandatory minimum deposit reduces her net cost of self-control both through

the reduction of her cost of self-control and by forcing her to save more than she is tempted

to. For high levels of income the mandatory minimum deposit provides a benefit by reducing

her cost of self-control, but it does not reduce her temptation because she chooses to save

more than the mandatory minimum deposit.

In this way the mandatory minimum deposit, s, creates the following piece-wise period

zero expected utility function:

EUGP =

y�

y

(u(y − s) + v(y − s)− v(y − s) + δE [u(y2 + (1 + r)s)]) f(y)dy

︸ ︷︷ ︸Decreased cost of self-control

+

y�

y

(u(y − s) + δE [u(y2 + (1 + r)s)]) f(y)dy

︸ ︷︷ ︸Decreased temptation

(3)

+

y�

y

(u(y − s) + δE [u(y2 + (1 + r)s)]) f(y)dy

︸ ︷︷ ︸Reduced flexibility

Where y is defined by the first order condition for savings with temptation, and y is defined

by the first order condition for savings without temptation:

8When v(c) is convex or linear this result extends to any number of time periods.9If you would like a copy of the code that shows this please contact me.

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y : −u′(y − s)− v′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s) = 0 (4)

y : −u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s) = 0. (5)

The optimal level of commitment, s, solves the first order condition of EU :

∂EUGP

∂s=

y�

yGP

v′(y − s)f(y)dy

︸ ︷︷ ︸> 0

⇒ Reducing the cost of self-control

+

yGP�

yGP

[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸> 0

⇒ Decreasing temptation

(6)

+

yGP�

y

[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸< 0

⇒ Restricting choice in a negative way

= 0

As equations (4) and (5) show, the individual will choose to save less when she is tempted

than when she is not, so y > y. Equation (6) illustrates that s only affects temptation when

income is between y and y, hence the reduction in temptation. Above y the mandatory

minimum deposit reduces the cost of self-control by restricting the maximum possible con-

sumption, but it has no effect on temptation in this range of income because s is not binding.

At low levels of income, below y, the individual would ideally save less than s, but cannot,

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so the mandatory minimum deposit reduces utility in this range. The optimal s balances

these three forces.

Because the individual is not permitted to borrow s cannot be greater than y. y > 0

is reasonable if the individual is entering the setting with some savings, or if she receives a

nonzero income with certainty.

In Sections 4 and 5 I assume that there is a unique optimal mandatory minimum deposit.

4 GP and βδ

Amador et al. (2006) show that the optimal savings plan for the βδ model in a setting

with stochastic preferences consists of mandatory minimum deposits. Because stochastic

preferences are identical to stochastic income when utility is exponential, this implies that

the optimal savings plan for the βδ model in general for the stochastic income setting may

be the same. Given that the GP and βδ models result in qualitatively the same policy

prediction is there a difference between the GP and the βδ models when using mandatory

minimum deposits?

Theorem 2 illustrates how for a given distribution a GP model can be calibrated so that

it replicates any βδ model’s choices of commitment level and period 1 savings.10

Theorem 2. Given a distribution for income there is a unique (up to positive affine trans-

formations) functional form of the GP model that replicates the βδ model in the three period

setting:

u(c) = w(c)

v(c) = γw(c),

where δGP(1+γ)

= βδβδ, and γ ∈[0, 1−β

β

].

10Theorem 2 is similar to Theorem 6 in Gul and Pesendorfer (2005), except that they use the KKS versionof the GP model and a finite choice set.

15

The full proof of Theorem 2 is in the appendix, but I now give a sketch of the proof

to provide the intuition behind the result. The form of the GP model above gives us two

variables to modulate so that it will fit the βδ model for a given distribution: δGP and γ.

For the period 1 savings to be the same in both models the ratios of the period 1 and 2

marginal utilities must equal each other:

u′(y − s) + v′(y − s)δGPu′(y + (1 + r)s)

=w′(y − s)

βδβδw′(y + (1 + r)s). (7)

When u(c) = w(c) and v(c) = γw(c) then equation (7) becomes the condition in Theorem 2:

δGP(1 + γ)

= βδβδ.

If the commitment decisions are to be the same then the period 0 first order conditions must

equal zero for the same value of s. The FOCs for the two models are:

∂EUGP

∂s=

y�

yGP

γw′(y − s)f(y)dy

︸ ︷︷ ︸>0

+

yGP�

yGP

[−w′(y − s) + δGP (1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yGP�

y

[−w′(y − s) + δGP (1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

= 0 (8)

for the GP model (where u(c) = w(c) and v(c) = γw(c) have been substituted in), and

16

∂EUβδ

∂s=

yβδ�

yβδ

[−w′(y − s) + δβδ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yβδ�

y

[−w′(y − s) + δβδ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

= 0 (9)

for the βδ model. When γ = 0, ∂EUGP∂s

< ∂EUβδ∂s

, and when γ = (1 − β)/β, ∂EUGP∂s

≥ ∂EUβδ∂s

(these are equal given deterministic income). It can be shown that ∂(∂EUGP/∂s

)/∂γ > 0,

which means that γ can be modulated between 0 and (1 − β)/β until the two marginal

expected utilities are equal. It is this ability to calibrate both δGP and γ that allows the GP

model to exactly reproduce the choices of the βδ model. As soon as one of these variables

is pinned down the two models may no longer agree with each other. If the individual’s

discount factor is determined (experimentally or otherwise) then δGP = δβδ = δ.

Theorem 3. Given a value for the discount factor, δ, in the stochastic income environment

a GP model can be calibrated to replicate the βδ model either in period 0 or in period 1, but

not both. When both models agree on

1. period 1 savings, s, then the GP model will predict a higher level of commitment, s,

than the βδ model.

2. period 0 commitment, s, then the GP model will predict a higher level of optional

savings, s, than the βδ model.

If both models are to agree on savings decisions in period 1 then γ = (1 − β)/β. Assume

for a moment that they agree on the commitment decision s as well: so yGP = yβδ, and

the second and third integrals in (8) are equal to (9). The additional positive integral in

equation (8) results in a higher level of commitment for the GP model. Therefore sGP >

17

sβδ, a contradiction. It is the first integral in equation (8) that captures the difference

between the GP and βδ models: costly self-control. Commitment benefits both models by

reducing temptation (the second integral in equation (8) and the first in equation (9)) and

is detrimental in both models because of the reduction in flexibility when income is low (the

third integral in equation (8) and the second in equation (9)). But the only the GP model

benefits from the reduction in costly self-control due to commitment.

If instead the two models to agree on commitment decisions then γ is reduced until sGP =

sβδ (remember, this works because ∂(∂EUGP/∂s

)/∂γ > 0 and when γ = 0, ∂EUGP

∂s< ∂EUβδ

∂s,

while when γ = (1− β)/β, ∂EUGP∂s≥ ∂EUβδ

∂s). Now γ < (1− β)/β, so

∂UGP

∂s=

(1 + γ)w(y1 − s)δEw(y2 + (1 + r)s)

<w(y1 − s)

βδEw(y2 + (1 + r)s)=∂Uβδ

∂s, ∀s.

Therefore the period 1 savings decisions predicted by the GP model will be greater than

those of the βδ model. At first this may seem counter-intuitive. Initially one may think

that because the GP model predicts a higher level of commitment when the two models

agree on period 1 savings, forcing the models agree on the level of commitment would shift

the period 1 savings predicted by the GP model down to a lower level than that of the βδ

model. However, the cost of self-control does not play a part in the period 1 savings decision,

only temptation has any relevance (when s is not binding). Because the discount factors in

both models are the same, in order for two to agree on the level of commitment γ must be

decreased, which reduces the strength of temptation in the GP model. This reduction in the

salience of temptation leads to larger savings decisions in period 1.

Corollary 4. If the GP model is calibrated so that it matches the βδ model when income

is deterministic, when both models are used to make predictions with a non-degenerate dis-

tribution for income they will both agree on savings but the GP model will predict a higher

desired level of commitment.

When the two models are calibrated using deterministic incomes δGP = δβδ and γ =

(1 − β)/β. Then moving to a non-degenerate distribution for income the second and third

18

integrals in equation (8) would equal equation (9) if sGP = sβδ. Even under the new distribu-

tion the two models still agree on period 1 savings decisions because δGP(1+γ)

= βδβδ still holds

(so yGP = yβδ). However, equation (8) has the additional positive first integral, therefore the

two equations cannot be equal and the two models will not agree on the level of commitment.

The additional positive integral in equation (8) results in a higher level of commitment for

the GP model: sGP > sβδ, a contradiction. Again it is the additional benefit to the GP

model of the reduction in costly self-control that drives this result.

5 Borrowing, GP and βδ

In this section I focus on borrowing, abstracting away from savings. The optimal borrowing

plan for the GP model in the three period setting works in a very similar manner to the

optimal savings plan, except a debt limit plays the part of the mandatory minimum deposit

in decreasing the cost of self-control. In equation (10) b is the debt limit, b is the amount

borrowed when the debt limit is not binding, and q is the interest rate. The limit will not

bind for higher income draws because the individual will not desire to borrow much in that

situation, though it will bind for lower income draws. In this scenario borrowing has two

costs, interest and the cost of self-control.

19

EUGP =

y�

y

(u(y + b) + v(y + b)− v(y + b) + δE [u(y2 − (1 + q)b)]

)f(y)dy

︸ ︷︷ ︸Decreased cost of self-control

+

y�

y

(u(y + b) + δE

[u(y2 − (1 + q)b)

])f(y)dy

︸ ︷︷ ︸Decreased temptation

(10)

+

y�

y

(u(y + b) + δE

[u(y2 − (1 + q)b)

])f(y)dy

︸ ︷︷ ︸Reduced flexibility

The reasoning behind the optimal borrowing mechanism is parallel to that of the optimal

savings mechanism. Like the optimal savings plan, the debt limit cannot be a function of

reported income, because only the lowest possible debt limit, b = miny∗ b(y∗), will have any

effect on the cost of self-control.

Theorem 5. The optimal borrowing plan for the GP model consists of a debt limit, b, that

is a function of the distribution of income.

If v(c) is convex or linear then b is unique. If v(c) is concave and f(y) is either uniform

or monotonically decreasing then it is unique as well.

Theorem 2 applies here as well. We can again determine the discount rate of the indi-

vidual experimentally and compare the GP and βδ models’ behavior using a non-degenerate

distribution for income.

Theorem 6. Given a value for the discount factor, δ, in the stochastic income environment

a GP model can be calibrated to replicate the βδ model either in period 0 or in period 1, but

not both. When both models agree on

• period 1 borrowing, b, then the GP model will predict a higher level of commitment (a

lower debt limit, b) than the βδ model.

20

• period 0 commitment, b, then the GP model will predict a lower level of optional bor-

rowing, b, than the βδ model.

The derivatives for period 0 expected utility with respect to the debt limit are:

∂EUGP

∂b= −

y�

yGP

γw′(y + b)f(y)dy

︸ ︷︷ ︸<0

+

yGP�

yGP

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸<0

+

yGP�

y

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸>0

(11)

for the GP model and:

∂EUβδ

∂b=

yβδ�

yβδ

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸<0

+

yβδ�

y

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸>0

(12)

for the βδ model. Note that an increase in the debt limit leads to an increase in the cost

of self-control (the first integral in equation (11)), increased temptation (second integral in

equation (11), and the first in equation (12)), and increased flexibility (the final integrals in

both equations).

If γ = (1 − β)/β then the two models agree on period 1 borrowing and yGP = yβδ (if

bGP = bβδ). But if yGP = yβδ and bGP = bβδ, then ∂EUGP/∂b < ∂EUβδ/∂b. Therefore the

two debt limits cannot be equal: it must be that bGP < bβδ.

Alternatively we could increase γ until both models agree on the debt limit. This works

21

because ∂(∂EUGP/∂b

)/∂γ < 0, and when γ = 0, ∂EUGP/∂b > ∂EUβδ/∂b, and when

γ = (1 − β)/β, ∂EUGP/∂b ≤ ∂EUβδ/∂b; therefore there is γ ∈ [0, (1 − β)/β] for which the

two marginal expected utilities will be equal. Because γ > (1− β)/β

∂UGP

∂b=

(1 + γ)w(y1 + b)

δEw(y2 − (1 + q)b)>

w(y1 + b)

βδEw(y2 − (1 + q)b)=∂Uβδ

∂b, ∀b.

So the level of period 1 borrowing will be less in the GP model than in the βδ model.

6 Saving, more time periods

In this section I focus on the GP model in an extended setting with an additional time pe-

riod. This allows for the comparison of liquidity constraints (an IRA or CD with pre-defined

date of maturity) and mandatory deposits (automatic paycheck deduction). I find that an

individual would strictly prefer mandatory deposits to liquidity constraints. This is surpris-

ing given that most existing savings devices have a liquidity constraint as one of their main

components. For instance, as just noted, IRAs and CDs, and additionally pensions, 401(k)

plans, and life insurance to name a few. However, often times these liquidity constraints are

coupled with mandatory deposits in the form of automatic paycheck deductions11. Manda-

tory minimum deposits are preferred to liquidity constraints because they are more flexible:

any liquidity constraint can be exactly reproduced by an appropriately designed series of

mandatory minimum deposits, but the reverse is not true. This could potentially explain

the anomaly presented in Noor (2007)

11Though I have not thoroughly examined them I suspect that most other models of temptation and time-inconsistency (the βδ model, the multiple-selves model in Fudenberg and Levine (2006), the Strotz model)would have a similar preference for mandatory deposits over liquidity constraints because there is nothingin these models that matures at any certain future date.

22

Self-control problems have been put forward as an explanation for the apparent undersaving

in the U.S. The earlier noted models imply that undersavers do not need added incentives to

participate in saving schemes such as 401(k) and IRAs which provide a means to commit to

saving for retirement. Yet such saving schemes have substantial tax benefits associated with

them: all contributions are tax deductible. Furthermore, participation in these schemes is

closely related to the tax benefits. For instance, IRA contributions fell by 62% when the

Tax Reform Act of 1986 excluded higher-income groups from tax benefits [Venti and Wise

(1987), Poterba, Venti, Wise (2001)]. The fall in participation took place although there was

no change in the commitment aspect of IRAs (early withdrawal penalties). This suggests

that the appeal of such saving vehicles is primarily the tax benefits, not their commitment

value [Akerlof, Gale, Hall (1998)]Perhaps the liquidity constraints imposed by the 401(k)s and IRAs are not flexible

enough, and so when the incentives are reduced they become much less attractive. If in-

stead the savings plan consisted of the more flexible mandatory minimum deposits without

liquidity constraints maybe this decline would not have been so precipitous.

The liquidity constraints examined are absolute in the sense that if the person decides to

save an amount st in the current period she will not have access to αt ∈ [0, 1] of the savings

again until the predefined date of maturity. Additionally in this section I assume that v(c)

is convex for convenience. A four period model in which the savings matures in period three

(period 0 is the date in which the level and type of commitment is decided upon) when there

is a necessary expenditure x, takes the following form:

EU(α1) =E [u(y1 − s1) + v(y1 − s1)− v(y1)

+ δE [u(y2 + (1 + r)(1− α1)s1 − s2)

+ v(y2 + (1 + r)(1− α1)s1 − s2)− v(y2 + (1 + r)(1− α1)s1)]

+ δ2Eu(y3 − x+ (1 + r)2α1s1 + (1 + r)s2)]]

The parallel four period model with mandatory deposits instead is:

23

EU(s1, s2) =E [u(y1 − s1) + v(y1 − s1)− v(y1 − s1)

+ δE [u(y2 + (1 + r)s1 − s2)

+ v(y2 + (1 + r)s1 − s2)− v(y2 + (1 + r)s1 − s2))

+ δ2Eu(y3 − x+ (1 + r)s2)]]

The mandatory deposits are assumed to be set at the optimal level for the analysis in this

subsection. Because v(c) is convex these exist and are unique for any continuous distribution.

The individual must commit to one or the other of these two saving regimes. As they are

modeled above the two savings plans are draconian versions of an IRA and an automatic

paycheck deduction savings plan respectively.

Liquidity constraints do not dictate the level of savings as in the mandatory minimum

deposit regime, hence some temptation and self-control costs are present: v(y1 − s1) −

v(y1) in the first period, and v(y2 + (1 + r)(1 − α1)s1 − s2) − v(y2 + (1 + r)(1 − α1)s1)

in the second. However, liquidity constraints do damp the self-control cost by tightening

the available budget constraint. On the other hand, as savings becomes less liquid the

individual has less access to the interest income and, naturally, to her savings. Mandatory

minimum deposits allow her more flexibility in the second period. Because there are still self-

control costs with liquidity constraints the person will prefer mandatory deposits to liquidity

constraints.

Theorem 7. A person with costly self-control will strictly prefer optimal mandatory deposits

to optimal liquidity constraints.

Any liquidity constraint αt can be recreated with a mandatory deposit of st+1 = αtst+rst,

where st is the amount saved in time t, and r is the interest rate. Mandatory minimum

deposits are more effective in reducing the cost of self-control if the individual is expected to

save part of her realized income (that is, if st > (1+r)st−1) because this additional reduction

in the her self-control cost cannot be achieved with liquidity constraints.

24

In addition, during the first period a mandatory deposit can be defined and thereby

reduce the cost of self-control, whereas there cannot be a liquidity constraint. Therefore

even if somehow the optimal mandatory deposit level for each subsequent period is identical

to the liquidity constraints this first period will make the individual strictly prefer the optimal

mandatory deposits to liquidity constraints.

Liquidity constraints and mandatory payments will be identical whenever it is possible

that income will be zero, but may not be otherwise. If we do not allow for borrowing then if

zero income has positive probability the most a particular mandatory minimum payment can

be is the amount of savings plus interest income from the previous period, st = (1 + r)st−1.

This and any amount less than this can be replicated with a liquidity constraint. Once

minimum income is greater than zero then the mandatory payment can be larger than

savings plus interest from the previous period and thus liquidity constraints and mandatory

payments may not be equivalent. As mentioned in the proof for Theorem 7 no matter what

in the first period liquidity constraints and mandatory payments would be different, because

no liquidity constraint can exist in the first period.

Theorem 7 suggests an experiment that tests people’s preference for mandatory payments

over liquidity constraints. If the results are consistent with Theorem 7 then many current

saving mechanisms (IRAs, 401(k)s and so on) should remove their focus on long term liquidity

constraints. If the data reveal that people instead prefer liquidity constraints over mandatory

payments then a more elaborate model will be necessary. This could include a cognitive cost

from thinking about financial decisions often (see Ergin and Sarver (2010), and Conlisk

(1988)).

7 Borrowing, extended model

This section focuses on borrowing and regular repayment. With the additional period, and

a simple interest rate (instead of compounding) the optimal mandatory borrowing plan has

25

two elements: a debt limit as before, and regular mandatory payments12.

The individual’s expected utility would take the following form in a four period model

with a simple interest rate and a loan that is taken out in period 1 and repaid in period 3:

E[U ] = E[u(y1 + b− x) + v(y1 + b− x)− v(y1 + b− x)

+ δE [u(y2 − a) + v(y2 − a)− v(y2 − a) + δE [u(y3 − (1 + q)b+ a)]]]

A debt limit, b, and mandatory payment, a, that are identical to the desired level of borrowing

and repayment would be ideal. The individual cannot rely on her own truthful report of

income, and so the best that she can do is to tie both the mandatory repayment and budget

constraint to expected wealth.

Theorem 8. The optimal borrowing plan will consist of predefined payments made each

period and a debt limit, both of which are functions of the income distributions.

Loans in this setting have two types of costs: interest, and the self-control required to

avoid borrowing up the the debt limit plus the self-control necessary to make oneself repay

in the second period. Mandatory payments prevent the individual from delaying repayment

of the loan until later periods, which increases her overall expected utility from a loan of any

size. This means that instituting mandatory payments that are optimized with respect to

the distribution for income actually increases the size of the loan the individual would like

to take out.

Theorem 9. The amount the person would like to borrow is greater when there are optimized

mandatory payments than when payments are completely flexible.

Theorem 9 is analogous to the main result in Fischer and Ghatak (2010), although here

the effect on welfare is not ambiguous: welfare strictly increases because mandatory payments

allow the individual to move closer to her first-best, full commitment borrowing level. This

12If interest is compounding then the payments can be rolled into the debt limit since compounding interestis like taking out a new loan each period.

26

is one explanation of why most microfinance institutions require regular fixed payments that

begin almost immediately after the loan is taken out13.

8 Conclusion

The most important result of this paper is that a mandatory minimum deposit savings plan in

a stochastic income setting can be used to differentiate between the GP and βδ models. If the

individual’s discount factor is known and if both models agree on period 1 savings decisions

then the GP model will predict a higher level of desired commitment than the βδ model. If

instead the two models agree on the level of commitment the GP model will predict larger

savings decisions in period 1. The test is simple and experimentally feasible: determine the

individual’s discount factor, then given a particular non-degenerate distribution for income

observe her savings choice. Fit the βδ model to this data (this will simultaneously fit the

GP model presented in Section 3). Finally, using the same distribution for income observe

the level of commitment that the individual prefers. It is the last step that will differentiate

between the two models because commitment has the additional benefit of reducing the cost

of self-control for the GP model. Such an experiment could observe commitment decisions

first and then savings decisions are compared after to differentiate the two.

In short, my model provides an empirical criterion to decide whether the GP or the βδ

model is more accurate in the stochastic income consumption-saving setting. The answer

to this question bears onto the empirical relevance of the cost of self-control. The answer

is important for welfare analysis, and is also of immediate practical interest, for example in

the design of microfinance loans or retirement plans.

13The vast majority of microfinance loans have an interest rate that is simple instead of compounding.

27

9 Appendix

9.1 Saving

On the way to proving Theorem 1 first I prove two lemmas that will be useful.

Lemma 10. The individual is strictly better off if any fee that is assessed in one period is

payed back with interest in the following period.

Proof. Say there is some mandatory deposit that is a function of the amount saved, s(s), and

that the fee takes a fraction of this deposit, 1− θ(s), and gives it to the bank. This fee can

also be a function of s. Also assume without loss of generality that arg mins(s + s(s)) = 0.

The expected utility of the person at time 0 when the fee is repaid is:

E[U ] =

y�

y

[u(y − s− s(s)) + v(y − s− s(s))− v(y − s(0))

+ δEu(y2 + (1 + r)(s+ θ(s)s(s)))] f(y)dy (13)

+

y�

y

[u(y − s(0)) + δEu(y2 + (1 + r)θ(s)s(0))] f(y)dy

Create a variable σ that modulates the magnitude of θ(s), so increasing σ increases the

magnitude of θ(s) for every s. Take the derivative of E[U ] with respect to σ. After using

the Envelop condition and canceling out the derivatives of the limits of the integrals:

∂E[U ]

∂σ=

y�

y

δ(1 + r)s(s)∂θ(s)

∂σEu′(y2 + (1 + r)θ(s)(s+ s(s)))f(y)dy

+

y�

y

δ(1 + r)s(0)∂θ(s)

∂σEu′(y2 + (1 + r)θ(s)s(0))f(y)dy. (14)

28

Which is positive. Therefore 1− θ(s) = 0, ∀s.

Lemma 11. The functional form of the any fee, s(s), where ∂s/∂s < 0, does not affect the

person’s utility when s > arg mins(s+ s(s)).

Proof. Assume without loss of generality that arg mins(s + s(s)) = 0. The expected utility

of the person at time t is:

E[U ] =

y�

y

[u(y − s− s(s)) + v(y − s− s(s))− v(y − s(0))

+ δEu(y2 + (1 + r)(s+ s(s)))] f(y)dy

y�

y

[u(y − s(0) + δEu(y2 + (1 + r)s(0)] f(y)dy

Pin down s(0) and create a variable φ that modulates the magnitude of s(s). Take the

derivative of E[U ] with respect to φ (I substituted c1 and c2 in for the arguments in the

equation above to shorten notation):

∂E[U ]

∂φ=

y�

y

∂s(s)

∂φ[−u′(c1)− v′(c1) + δ(1 + r)Eu′(c2)] f(y)dy

=0

From the first order conditions for saving the above equation is always zero as long as saving

some nonzero amount is optimal. When saving is zero then s(s) = s(0), which we’ve pinned

down so that ∂s(0)/∂φ = 0, and so again, the equation above is zero.

Theorem. 1 The optimal savings plan consists of a minimum mandatory deposit that is a

function of expected wealth. This deposit is returned to the saver the following period with

interest.

29

Proof. Step 1: show that a mandatory minimum deposit that is a function of expected wealth

exists.

Step 2: prove that any optimal savings mechanism consists of a minimum mandatory

deposit and that this deposit is returned to the person the following period.

Step 1. Expected utility is a piecewise function that is split at the point that the manda-

tory deposit binds. Optimal optional savings, s(y1), is increasing with increasing realized

period 1 income, so there exists a y below which the mandatory minimum deposit, s, will

bind. Additionally, when s is binding the net cost of self-control is zero, and so there is only

commitment utility, u(c1), in this range. This means that there is another level of income,

y, at which s is binding for just the commitment utility. y ≥ y because s is increasing with

y1.

U =

u(y1 − s) + v(y1 − s)− v(y1 − s) + δEu(y2 + (1 + r)s) when y < y1

u(y1 − s) + δEu(y2 + (1 + r)s) when y > y1 ≥ y

u(y1 − s) + δEu(y2 + (1 + r)s) when y > y1

If the first derivative with respect to s crosses zero while decreasing then an optimal s

necessarily exists. If it crosses more than once while decreasing than several optimal solutions

may exist.

30

∂EU∂s

=

y�

y

v′(y − s)f(y)dy

︸ ︷︷ ︸>0

+

y�

y

[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

(15)

+

y�

y

[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

y and y are increasing with s, so when s = 0 the second and third terms in equation (15)

are zero, so ∂EU/∂s > 0 when s = 0.

The second derivative with respect to s is:

∂2EU∂s2

= −y�

y

v′′(y − s)f(y)dy

+

y�

y

[u′′(y − s) + δ(1 + r)2Eu′′(y2 + (1 + r)s)

]f(y)dy (16)

When v(·) is convex ∂2EU/∂s2 < 0. Therefore when v(·) is convex there is a unique s given

any continuous distribution for income.

If v(·) is concave then the first integral in equation (16) is no longer negative, but

∂2EU/∂s2 can still be shown to be negative for some types of distributions. First rewrite

equation (15) so that s is in the limits of the first two integrals:

31

∂EU∂s

=

y−s�

y−s

v′(y)f(y + s)dy

︸ ︷︷ ︸>0

+

y−s�

y−s

[−u′(y) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y + s)dy

︸ ︷︷ ︸>0

+

y�

y

[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

Then take the derivative with respect to s:

∂EU∂s

= −v′(y − s)f(y)−(∂y

∂s− 1

)v′(y − s)f(y) +

y−s�

y−s

v′(y)f ′(y + s)dy

+

(∂y

∂s− 1

)[−u′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)]

+

y−s�

y−s

δ(1 + r)2Eu′′(y2 + (1 + r)s)f(y + s)dy

+

y−s�

y−s

[−u′(y) + δ(1 + r)2Eu′(y2 + (1 + r)s)

]f ′(y + s)dy

+

y�

y

[u′′(y − s) + δ(1 + r)2Eu′′(y2 + (1 + r)s)

]f(y)dy

This can be simplified slightly by realizing that it contains the first order condition for period

1 saving with temptation: −u′(y − s)− v′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s) = 0.

32

∂EU∂s

= −v′(y − s)f(y) +

y−s�

y−s

v′(y)f ′(y + s)dy

+

y−s�

y−s

δ(1 + r)2Eu′′(y2 + (1 + r)s)f(y + s)dy

+

y�

y

[−u′(y − s) + δ(1 + r)2Eu′(y2 + (1 + r)s)

]f ′(y)dy

+

y�

y

[u′′(y − s) + δ(1 + r)2Eu′′(y2 + (1 + r)s)

]f(y)dy

As you can see, all of the remaining terms are negative when f(y) is constant or decreasing

monotonically.

Step 2. Lemma 10 proves that any mandatory deposit will be returned with interest

in the following period. Equation (14) above shows that a minimum mandatory deposit is

desirable to the individual. Lemma 11 proves that when a saving mechanism is dependent

on a report of income (or level of s chosen), and is decreasing in the report (or s) then the

functional form of a mechanism, other than the minimum mandatory deposit, does not affect

the amount saved by the individual or her utility, and therefore is immaterial. If on the other

hand the saving mechanism is increasing in the report of income (or s) then this will limit

the options of savings the individual has. This may reduce the realized period 1 utility, and

will reduce the period 0 expected utility. To illustrate, imagine between realized period 1

incomes of y and y you force the individual to save below her period 1 optimal savings level,

at s. And between y and...y you force her to save more than she’d like: s. Her period 0

marginal expected utility between these ranges of income is going to be:

33

� ...y

y[−u′(y − s)− v′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

+� yy

[−u′(y − s)− v′(y − s) + δ(1 + r)Eu′(y2 + (1 + r)s)] f(y)dy

The first integral will be negative, because she is forced to save too much, and the second

is positive, because she is forced to save too little. Therefore her period 0 expected utility

would benefit from allowing her to save more than s when she would like to in period 1,

and less than s when she so desires in period 1. Therefore a savings mechanism with an

increasing mandatory deposit is not beneficial to her in any way.

9.2 GP and βδ

Theorem. 2 Given a distribution for income there is a unique (up to positive affine transfor-

mations) functional form of the GP model that replicates the βδ model, which is as follows:

u(c) = w(c)

v(c) = γw(c),

where δGP(1+γ)

= βδβδ, and γ ∈[0, 1−β

β

].

Proof. • Step 1: show that there is a unique form of the GP model (when temptation is

instantaneous) that can replicate the time one behavior of any βδ model.

• Step 2: show that this functional form of the GP model can be calibrated so that it

also replicates the time zero behavior of a βδ model.

Step 1. The βδ person makes her savings decisions based on her marginal utility:

w′(y1 − sβδ)βδβδEw′(y2 + (1 + r)sβδ)

= (1 + r). (17)

The GP person decides on her savings analogously:

34

u′(y1 − sGP ) + v′(y1 − sGP )

δGPEu′(y2 + (1 + r)sGP )= (1 + r). (18)

These two equations need to equal each other in order for the GP person and the βδ person

to agree on all of their savings decisions in time 1 (sβδ(y) = sGP (y) = s(y)).

This conversion needs to take a time separable form given the formulation of the GP

model,

u(c) + v(c)− v(B) = g(w(c)).

From the first order conditions in period 1 (substituting in c1 = y1−s and B2 = y2 +(1+r)s

to make notation more concise):

∂g(w(c1))

∂s+ δGP

∂Eg(w(B2)

∂s= −g′(w(c1))w′(c1) + δGP (1 + r)Eg′(w(B2))w′(B2)

= −u′(c1)− v′(c1) + δGP (1 + r)Eu′(B2).

Therefore g′(w(c1))w′(c1) = u′(c1) + v′(c1), and g′(w(B2))w′(B2) = u′(B2). Also equation

(18) must be equal to to equation (17) in order for period 1 optional savings decisions to

coincide between the two models.

g′(w(c1))w′(c1)

δGPEg′(w(B2))w′(B2)=

w′(c1)

βδβδEw′(B2)

⇒ g′(w(c1))

Eg′(w(B2))=

δGPβδβδ

. (19)

So g′(w(c)) needs to be a constant because (19) must hold for all y, and we know that

g′(w(c1))w′(c1) = u′(c1) + v′(c1) and g′(w(B2))w′(B2) = u′(B2). In order to satisfy these

conditions define

35

u(c) = w(c)

v(c) = γw(c)

and so g(w(c)) = w(c) + γw(c) − γw(B) = u(c) + v(c) − v(B). Using this in equation (19)

shows that δGP/(1+γ) = βδβδ must hold in order for period 1 saving to be the same between

the models.

Step 2: time zero behavior. The FOCs for the two models are:

∂EUGP

∂s=

y�

y

γw′(y − s)f(y)dy

︸ ︷︷ ︸>0

+

y�

yGP

[−w′(y − s) + βδβδ(1 + γ)(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yGP�

y

[−w′(y − s) + βδβδ(1 + γ)(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

(20)

for the GP model (where u(c) = w(c) and v(c) = γw(c), and δGP = βδβδ(1 + γ) have been

substituted in), and

∂EUβδ

∂s=

y�

yβδ

[−w′(y − s) + δβδ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yβδ�

y

[−w′(y − s) + δβδ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

(21)

for the βδ model. At γ = 0 ∂EUGP∂s

< ∂EUβδ∂s

, and at γ = (1 − β)/β ∂EUGP∂s

≥ ∂EUβδ∂s

(these

36

are equal given deterministic income). If ∂(∂EUGP/∂s

)/∂γ > 0, then γ can be modulated

between 0 and (1− β)/β until that the two marginal expected utilities are equal.

Taking the derivative of ∂EUGP/∂s with respect to γ and keeping s constant gives us

the following:

∂(∂EUGP/∂s

)∂γ

= −∂yGP∂γ

γw′(yGP − s)f(yGP ) +

y�

yGP

w′(y − s)f(y)dy

+∂yGP∂γ

[w′(yGP − s) + βδβδ(1 + γ)(1 + r)Ew′(y2 + (1 + r)s)] f(yGP )

+

yGP�

y

βδβδ(1 + r)Ew′(y2 + (1 + r)s)f(y)dy

=

y�

yGP

w′(y − s)f(y)dy +

yGP�

y

βδβδ(1 + r)Ew′(y2 + (1 + r)s)f(y)dy

+∂yGP∂γ

f(yGP ) [−(1 + γ)w′(yGP − s) + βδβδ(1 + γ)Ew′(y2 + (1 + r)s)]︸ ︷︷ ︸=0

.

The two remaining integrals are both positive. Because δGP = βδβδ(1 + γ) both models

agree on period 1 saving, and γ can be adjusted so that both models agree on the level of

commitment as well.

Theorem. 3 Given a value for the discount factor, δ, in the stochastic income environment

a GP model can be calibrated to replicate the βδ model either in period 0 or in period 1, but

not both. When both models agree on

• period 1 savings, s, then the GP model will predict a higher level of commitment, s,

than the βδ model.

• period 0 commitment, s, then the GP model will predict a higher level of optional

savings, s, than the βδ model.

Proof. Assume that δGP = δβδ = δ.

37

First assume both agree in period 1, therefore from Theorem 2 we know that γ = (1 −

β)/β.

The derivative of period 0 expected utility with respect to s for the GP model is:

∂EUGP

∂s=

y�

yGP

γw′(y − s)f(y)dy

︸ ︷︷ ︸>0

+

yGP�

yGP

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yGP�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

,

and for the βδ:

∂EUβδ

∂s=

yβδ�

yβδ

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yβδ�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

.

Assuming that both models agree on period 1 behavior can sGP = sβδ? Say they do,

this means that yGP = yβδ because if both models save identically in period 1 and if both

models have the same s then the level of income for which s is just binding will be the same

for both. If they agree on s then naturally the FOC for both models are equal to zero.

∂EUGP

∂s=

∂EUβδ

∂s= 0

38

⇒y�

y

1− ββ

w′(y − s)f(y)dy +

y�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy (22)

>

y�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy.

Notice that the first integral on the left side of equation (22) is always strictly positive, and

when sGP = sβδ then the second integral on the left side and the integral on the right side

are equal to each other, hence the strict inequality. Because ∂EUGP/∂s only crosses zero

once, in order to make the two equations equal to each other in (22) sGP will need to be

increased. Therefore, when δGP = δβδ and the two models agree in period 1 behavior then

sGP > sβδ.

Now assume both models agree in period 0 behavior: sGP = sβδ = s.

Given that we are assuming that s is optimal for both models, then:

∂EUGP

∂s=

∂EUβδ

∂s= 0

⇒y�

yGP

γw′(y − s)f(y)dy +

yGP�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy (23)

=

yβδ�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy.

If both models agreed on period 1 behavior then yGP = yβδ, but this would mean that the

second integral on the left side of equation (23) and the integral on the right side would be

equal, and given that the first integral on the left side is strictly positive this would break

39

the equality. Further dissecting the left side of equation (23) gives us

∂EUGP

∂s=

y�

yGP

γw′(y − s)f(y)dy

︸ ︷︷ ︸>0

+

yGP�

yGP

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸>0

+

yGP�

y

[−w′(y − s) + δ(1 + r)Ew′(y2 + (1 + r)s)] f(y)dy

︸ ︷︷ ︸<0

.

The first integral is the gain from reducing the cost of self-control, the second integral is the

gain from reducing over consumption due to temptation, and the final integral is the cost

from constraining the option to save less when income is low. We know from Theorem 2

that ∂(∂EUGP/∂s

)/∂γ > 0. Going back to equation (23), in order to maintain the equality

when sGP = sβδ = s we need to decrease γ, namely decrease the strength of temptation and

the cost of self-control. Revisiting the FOCs from period 1:

GP:(1 + γ)w′(y1 − s)Ew′(y2 + (1 + r)s)

= δ(1 + r)

βδ :w′(y1 − s)

βEw′(y2 + (1 + r)s)= δ(1 + r).

We know that 1/(1 + γ) < β now, so therefore optional savings, s, in the GP model will be

greater than that of the βδ model.

Corollary. 4 If the GP model is calibrated so that it matches the βδ model when income

is deterministic, when both models are used to make predictions with a non-degenerate dis-

tribution for income they will both agree on savings but the GP model will predict a higher

40

desired level of commitment.

Proof. When both models are calibrated using deterministic income δGP = δβδ and γ =

(1 − β)/β. When used for prediction using a non-degenerate distribution of income both

models will continue to agree on period 1 savings decisions because δGP/(1 + γ) = βδβδ,

and equation (22) will hold. Therefore the GP model will predict a higher desired level of

commitment than the βδ model.

9.3 Liquidity constraints

Theorem. 7 A person with costly self-control will strictly prefer optimal mandatory deposits

to optimal liquidity constraints.

Any liquidity constraint αt can be recreated with a mandatory deposit of st+1 = αtst+rst,

where st is the amount saved in time t, and r is the interest rate.

The largest a liquidity constraint can be is αt = 1: under liquidity constraints the

minimum cost of self-control is v(yt+1). It is feasible for a mandatory minimum deposit to

require the individual to save her entire income, so the minimum cost of self-control is v(0).

Therefore mandatory minimum deposits have the ability to reduce the cost of self-control

by more than can liquidity constraints.

In this model of liquidity constraints, once a portion of savings is constrained it is not

available until the date of maturity. In a model with T > 4 periods it can be the case that

the wealth accrued through mandatory minimum deposits is greater than that available each

period to consume given liquidity constraints. In this case liquidity constraints cause over

consumption in period T and under consumption in some date t < T .

Even if neither of the two cases above occur, during the first period a mandatory deposit

can be defined and thereby reduce the cost of self-control, whereas there cannot be a liquidity

constraint. Therefore the individual will strictly prefer the optimal mandatory deposits to

liquidity constraints.

41

9.4 Borrowing

Theorem. 5 The optimal borrowing plan consists of a debt limit, b, that is a function of

that is a function of the distribution for income.

Proof. This proof’s reasoning exactly parallels that of Theorem 1.

Theorem. 6 Given a value for the discount factor, δ, in the stochastic income environment

a GP model can be calibrated to replicate the βδ model either in period 0 or in period 1, but

not both. When both models agree on

• period 1 borrowing, b, then the GP model will predict a higher level of commitment (a

lower debt limit, b) than the βδ model.

• period 0 commitment, b, then the GP model will predict a lower level of optional bor-

rowing, b, than the βδ model.

Proof. The text gives the majority of the proof, all that is left is to show that ∂(∂EUGP/∂b

)/∂γ <

0.

∂EUGP

∂b= −

y�

yGP

γw′(y + b)f(y)dy

︸ ︷︷ ︸<0

+

yGP�

yGP

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸<0

+

yGP�

y

[w′(y + b)− δ(1 + q)Ew′(y2 − (1 + q)b)

]f(y)dy

︸ ︷︷ ︸>0

(24)

Take the derivative of equation (24) with respect to γ:

∂2EUGP

∂b∂γ= −

y�

yGP

w′(y + b)f(y)dy < 0.

Similar to Theorem 2 the derivatives of the limits of the integrals are multiplied by the period

1 first order condition for borrowing which are zero.

42

Theorem. 8 The optimal borrowing plan will consist of predefined payments made each

period and a debt limit, both of which are functions of per period expected wealth.

Proof. Proving that neither the debt limit or the mandatory payments can be functions of

reported current wealth is the same as above as well.

Step 1: Show that mandatory payments and a debt limit optimized with respect to the

distributions for income exist.

Step 2: Show that both a debt limit and mandatory payments can exist simultaneously.

Step 1.

E[U ] = E[u(y1 + b− x) + v(y1 + b− x)− v(y1 + b− x)

+ δE[u(y2 − a) + v(y2 − a)− v(y2 − a) + δ2E [u(y3 − (1 + q)b+ a)]

]]a is the mandatory payment in time t. If a loan of size b was taken out in the first period, and

there are no defaults, then the sum of the payments made by the person in the subsequent

periods naturally must be equal to (1 + q)b. In addition, as above, there will be a level

y = y(a) such that when income is above y the mandatory payment is not binding, and

when it is below it is binding. The result is a piece-wise expected utility function, whose

derivative with respect to a takes the following form:

∂EU∂a

=

y�

y

v′(y − a)f(y)dy −y�

y

(u′(y − a)− δE [u′(y − (1 + q)b+ a)]) f(y)dy (25)

When a = 0 then the mandatory payment is never binding, y = y, the first term above is

positive and the second is zero. Given that both terms in equation (25) are continuous at

least one value of a > 0 exists that will maximize expected utility.

An optimal debt limit exists:

43

∂E[U ]

∂b= −

y�

y

v′(y + b)f(y)dy +

y�

y

(u′(y + b)− δ2(1 + q)E

[u′(y − (1 + q)b+ a)

])f(y)dy

(26)

Because both integrals are continuous an optimal b will exist.

Step 2. By the implicit function theorem ∂b/∂a > 0, and ∂a/∂b < 0, so there does exist

a stable pair b, a2, ..., aT that maximizes expected utility.

Theorem. 9 The amount the person would like to borrow is greater when there are optimized

mandatory payments than when payments are completely flexible.

Proof. Assuming four periods for simplicity, the period 1 expected utility will take the fol-

lowing form:

EU = u(y1 + b) + v(y1 + b)− v′(y + b)

+δ

y�

y

[u(y − a) + v(y − a)− v(y − a) + δ2Eu(y3 − (1 + q)b+ a)

]f(y)dy

+δ

y�

y

[u(y − a) + δ2Eu(y3 − (1 + q)b+ a)

]f(y)dy

Since we are interested in the effect of mandatory payments on borrowing the debt limit is

assumed not to bind.

a is not a function of b, so neither is y. Using the envelope condition the first order

condition for borrowing is as follows:

44

∂E[U ]

∂b= u′(y + b) + v′(y + b)− v′(y + b)

−δ2(1 + q)

y�

y

E[u′(y − (1 + q)b+ a)

]f(y)dy

−δ2(1 + q)

y�

y

E[u′(y − (1 + q)b+ a)

]f(y)dy

Using the implicit function theorem (note that the derivatives of the limits of the two integrals

cancel out):

∂b

∂a=

� yyE[u′′(y − b+ a)

]f(y)dy

∂2E[U ]∂b2

> 0

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