LiqConstr-last.pdftotext-nofirstline

Liquidity Constraints
and Precautionary Saving
October 15, 2020

Christopher D. Carroll1

Martin B. Holm2

Johns Hopkins University

University of Oslo

Miles S. Kimball3
University of Colorado at Boulder

Abstract
We provide the analytical explanation of the interactions between precautionary saving and
liquidity constraints. The effects of liquidity constraints and risks are similar because both stem
from the same source: a concavification of the consumption function. Since a more concave
consumption function exhibits heightened prudence, both constraints and risks strengthen
the precautionary saving motive. In addition, we explain the apparently contradictory results
that constraints and risks in some cases intensify, but in other cases weaken the precautionary
saving motive. The central insight is that the effect of introducing an additional constraint
or risk depends on whether it interacts with pre-existing constraints or risks. If it does not
interact with any pre-existing constraints or risks, it intensifies the precautionary motive. If it
does interact, it may reduce the precautionary motive in earlier periods at some levels of wealth.

Keywords

liquidity constraints, uncertainty, precautionary saving

JEL codes

C6, D91, E21
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1 Carroll: Department of Economics, Johns Hopkins University, email: ccarroll@jhu.edu
2 Holm: Department of
3 Kimball: Department of Economics, University of
Economics, University of Oslo, email: martin.b.holm@outlook.com
Colorado at Boulder, email: miles.kimball@colorado.edu

1 Introduction
A large literature has shown that numerical models that take constraints and uncertainty
seriously can yield different conclusions than those that characterize traditional models.
For example, Kaplan, Moll, and Violante (2018) show that when sufficiently many households have high marginal propensities to consume (MPC’s), a major transmission channel
of monetary policy is the ‘indirect income effect’ – a channel of minimal importance in
traditional macro models. Similarly, Guerrieri and Lorenzoni (2017) and Bayer, Lütticke,
Pham-Dao, and Tjaden (2019) show that tightened borrowing conditions and heightened
income risk can help explain the consumption decline during the great recession.
A drawback to numerical solutions is that it is often difficult to know why results
come out the way they do. A leading example is in the complex relationship between
precautionary saving behavior and liquidity constraints.1 At least since Zeldes (1984),
economists working with numerical solutions have known that liquidity constraints can
strictly increase precautionary saving under very general circumstances. On the other
hand, simulations have sometimes found circumstances under which liquidity constraints
and precautionary saving are substitutes. In an early example, Samwick (1995) showed
that unconstrained consumers with a precautionary saving motive in a retirement saving
model behave in ways qualitatively and quantitatively similar to the behavior of liquidity
constrained consumers facing no uncertainty.
This paper provides the theoretical tools to make sense of the interactions between
liquidity constraints and precautionary saving. The main theoretical innovation is to
conceptualize the effects of either constraints or risks in terms of consumption concavity.
The advantage of understanding the effects in terms of consumption concavity is that
there is a link between more consumption concavity (concavification) and prudence, and
therefore also precautionary saving (Kimball, 1990). In particular, we show that prudence
of the value function is increased by any concavification of the consumption function
regardless of its cause.
Our first main result is to show that the introduction of a constraint at the end of
period t causes consumption concavity around the point where the constraint binds.2
Furthermore, once consumption concavity is created, it propagates back to periods before
t. Carroll and Kimball (1996) showed similar results for the effects of risks on consumption
concavity. Hence, the two papers establish rigorously that both constraints and risks
create a form of consumption concavity that propagates backward.
Since prudence is heightened when the consumption function is more concave, it follows
immediately that when a liquidity constraint is added to a standard consumption problem,
the resulting value function exhibits increased prudence around the level of wealth where
the constraint becomes binding.3 Constraints induce precaution because constrained
agents have less flexibility in responding to shocks when the effects of the shocks cannot
1 For the seminal numerical examination of some of the interactions between precautionary saving and liquidity
constraints, see Deaton (1991).
2 The connection between constraints and consumption concavity has been explored in more specific settings. See e.g.
Park (2006) for CRRA utility, Seater (1997) for the case where time-discounting equals the interest rate, Nishiyama and
Kato (2012) for quadratic utility, and Holm (2018) for the case with infinitely-lived households with HARA utility.
3 A relationship between constraints and prudence has also been noted by Lee and Sawada (2007) and documented
empirically in Lee and Sawada (2010).

1

be spread out over time. The precautionary motive is heightened by the desire (in the
face of risk) to make future constraints less likely to bind.4 This can explain why such a
high percentage of households cite precautionary motives as the most important reason
for saving (Kennickell and Lusardi, 1999) even though the fraction of households who
report actually having been constrained in the past is relatively low (Jappelli, 1990).
After establishing that the introduction of a constraint increases the precautionary
saving motive, we show that the introduction of a further future constraint may actually
reduce the precautionary saving motive by ‘hiding’ the effects of pre-existing constraints or
risks. An existing constraint may be rendered irrelevant at levels of wealth where the new
constraint forces more saving than the existing constraint would induce. Identical logic
implies that uncertainty can ‘hide’ the effects of a constraint because the consumer may
save so much for precautionary reasons that the constraint becomes irrelevant. Thus, the
introduction of a new constraint or risk does not generally strengthen the precautionary
motive.
A concrete example helps clarify the intuition. A typical perfect foresight model of
consumption for a retired consumer with guaranteed income (e.g., ‘Social Security’)
implies that a legal constraint on borrowing can make the consumer run their wealth
down to zero (thereafter setting consumption equal to income). Now consider modifying
the model to incorporate the possibility of large medical expenses near the end of life (e.g.
nursing home fees; see Ameriks, Caplin, Laufer, and Van Nieuwerburgh, 2011). Under
reasonable assumptions, a consumer facing such a risk may save enough for precautionary
reasons to render the no-borrowing constraint irrelevant.
Although there is no general result for the effects of additional constraints or risks
when the consumer already faces existing constraints or risks, we can establish how the
introduction of all constraints and risks affects the precautionary saving motive. We show
that the precautionary saving motive is stronger at every level of wealth5 in the presence
of all future risks and constraints than in the case with no risks and constraints. This
is because the consumption function is concave everywhere in the presence of all future
risks and constraints,6 and since consumption concavity heightens prudence of the value
function, the precautionary saving motive is also stronger in the presence of all risks and
constraints than in the case with no risks and constraints.
Hence, we can summarize this paper as follows. The effects of liquidity constraints
and risks are similar because both stem from the same source: a concavification of
the consumption function. The effects work independently, meaning that neither risks
nor constraints are necessary to concavify the consumption function. And since a more
concave consumption function exhibits heightened prudence, both constraints and risks
strengthen the precautionary saving motive. In addition, we explain the apparently
contradictory results that constraints and risks in some cases intensify, but in other
4 To

be clear, the liquidity constraint we analyze here must be satisfied in each period (one-period bonds). This implies
that the interactions between constraints and income volatility where some households may prefer to increase (credit card)
debt today because they expect tighter credit conditions in the future are ruled out (Fulford, 2015; Druedahl and Jørgensen,
2018).
5 More precisely, we show that there is no level of wealth at which the motive is weaker, and at least some levels at
which it is strictly stronger.
6 Again, there is no level of wealth at which the consumption function becomes less concave, and at least some levels
of wealth at which it becomes strictly more concave.

2

cases weaken the precautionary saving motive. The central insight is that the effect
of introducing an additional constraint or risk depends on whether it weakens the effects
of any pre-existing constraints or risks. If it does not interact with any pre-existing
constraints or risks, it intensifies the precautionary saving motive. If it does interact, it
may weaken the precautionary saving motive at some levels of wealth.
The rest of the paper is structured as follows. To fix notation and ideas, the next section
sets out the general theoretical framework. Section 3 then defines what we mean by
consumption concavity and shows how consumption concavity propagates backward and
heightens prudence of the value function. In Section 4, we show how liquidity constraints
cause consumption concavity and thereby also prudence. And Section 5 presents our
results on the interactions between liquidity constraints and precautionary saving. The
final section concludes.

2 The Setup
In this section we present the consumer framework underlying all results. We consider a
finitely-lived consumer living from period t to T who faces some future risks and liquidity
constraints. The consumer is maximizing the time-additive present discounted value of
utility from consumption u(c). With interest and time preference factors R ∈ (0, ∞) and
β ∈ (0, ∞), and labeling consumption c, stochastic labor income y, end-of-period assets
a, liquidity constraint ς, and ‘market resources’ (the sum of current income and spendable
wealth from the past) mt , the consumer’s problem can be written as

Vt (mt )

=

max Et
c

" T −t
X

#
β k u(ct+k )

k=0

s.t.
= (mt − ct )R + yt+1
= mt − ct
≥ ςt

mt+1
at
at

As usual, the recursive nature of the problem makes this equivalent to the Bellman
equation
Vt (mt ) = max u(c) + Et [βVt+1 ((mt − c)R + yt+1 )].
c

We define Ωt (at ) = Et [βVt+1 (Rat + yt+1 )] as the end-of-period value function and rewrite
the problem as7
Vt (mt ) = max u(c) + Ωt (mt − c).
c

7 For notational simplicity we express the value function V (m) and the expected discounted value function Ω (s) as
t
t
functions simply of wealth and savings, but implicitly these functions reflect the entire information set as of time t; if,
for example, the income process is not i.i.d., then information on lagged income or income shocks could be important
in determining current optimal consumption. In the remainder of the paper the dependence of functions on the entire
information set as of time t will be unobtrusively indicated, as here, by the presence of the t subscript. For example, we
will call the policy rule in period t which indicates the optimal value of consumption ct (m). In contrast, because we assume
that the utility function is the same from period to period, the utility function has no t subscript.

3

Throughout, what we call ‘the consumption function’ is the mapping from market resources mt to consumption. In some of our results we consider utility functions of the
HARA class

α1 −1
1
α1

(α
c
+
α
)
α1 6= 0, 1
1
2
 α1 −1
−c/α
2
u(c) = −α2 e
(1)
α1 = 0


log(c + α2 )
α1 = 1
with α2 > max{−α1 c, 0}. Note that that (1) also covers the case with quadratic utility
(α1 = −1).

3 Consumption Concavity and Prudence
This section provides a set of tools necessary to prove our main results. We first define
what we mean by consumption concavity and show that consumption concavity, once
established, propagates back to prior periods. Next, we define an operation we call a
‘counterclockwise concavification’ which describes how either a liquidity constraint or
a risk affects the consumption function. The advantage of defining a counterclockwise
concavification in such general terms is that we can show that it heightens prudence
of the value function irrespective of the source of concavification. Since the relationship
between prudence and precautionary saving has already been established in the literature
(Kimball, 1990), the tools in this section allow us to establish how liquidity constraints
affect precautionary saving in the subsequent sections.

3.1 Consumption Concavity
We start by defining what we mean by consumption concavity (CC) and greater consumption concavity.
Definition 1. (Local Consumption Concavity.)
In relation to a utility function u(c) with u0 > 0, u00 < 0, and non-negative (u000 ≥ 0) and
non-increasing prudence, a function V (m) has property CC (alternately, strict CC) over
the interval between m1 and m2 , where m2 > m1 , if
V 0 (m) = u0 (c(m))
for some increasing function c(m) that satisfies concavity (alternately, strict concavity)
over the interval from m1 to m2 .
Since (even with constraints) V 0 (m) = u0 (c(m)) holds by the envelope theorem, V (m)
having property CC (alternately, strict CC) is the same as having a concave (alternately,
strictly concave) consumption function c(m).8 Note that the definition is restricted to
non-negative and non-increasing prudence. This encompasses most of the commonly used
utility functions in the economics literature (e.g. CRRA, CARA, quadratic). Also, note
that we allow for ‘non-strict’ concavity – that is, linearity – because we want to include
8 Remember that the envelope theorem depends only on being able to spend current wealth on current consumption,
so it holds whether or not there is a liquidity constraint.

4

cases such as quadratic utility in which parts of the consumption function can be linear.
Henceforth, unless otherwise noted, we will drop the cumbersome usage ‘alternately,
strict’ – the reader should assume that what we mean always applies in the two alternate
cases in parallel.
If a function has property local CC at every point, we define it as having property CC
globally.
Definition 2. (Global Consumption Concavity.)
A function V (m) has property CC in relation to a utility function u(c) with u0 > 0,
u00 < 0, and non-negative (u000 ≥ 0) and non-increasing prudence if V 0 (m) = u0 (c(m)) for
some monotonically increasing concave function c(m).
We now show that once a value function exhibits the property CC in some period t + 1,
it will also have the property CC in period t and earlier under fairly general conditions.
Lemma 1 formally provides conditions guaranteeing this recursive propagation.
Lemma 1. (Recursive Propagation of Consumption Concavity.)
Consider an agent with a HARA utility function satisfying u0 > 0, u00 < 0, u000 ≥ 0 and
non-increasing absolute prudence (−u000 /u00 ). Assume that no liquidity constraint applies
at the end of period t and that the agent faces income risk yt+1 ∈ [y, ȳ]. If Vt+1 (mt+1 )
exhibits property (local) CC for all mt+1 ∈ [Rat +y, Rat + ȳ], then Vt (mt ) exhibits property
(local) CC at the level of wealth mt such that optimal consumption yields at = mt −ct (mt ).
If also Vt+1 (mt+1 ) exhibits property strict (local) CC for at least one mt+1 ∈
[Rat + y, Rat + ȳ], then Vt (mt ) exhibits property strict (local) CC at the level of
wealth mt where optimal consumption yields at = mt − ct (mt ).
See Appendix A for the proof. The basic insight of Lemma 1 is that as long as the future
consumption function is concave for all realizations of yt+1 , then it is also concave today.
Additionally, if the the future consumption function is strictly concave for at least one
realization of yt+1 , then the consumption function is strictly concave also today.
The last circumstance we define is when a value function exhibits ‘greater’ concavity
than another. Later, this will allow us to compare two consumption functions and their
respective concavity.
Definition 3. (Greater Consumption Concavity.)
Consider two functions V (m) and V̂ (m) that both exhibit property CC with respect to the
same u(c) at a point m for some interval (m1 , m2 ) such that m1 < m < m2 . Then V̂ (m)
exhibits property ‘greater CC’ compared to V (m) if




m − m1
m2 − m
m − m1
m2 − m
ĉ(m1 ) +
ĉ(m2 ) ≥ c(m) −
c(m1 ) +
c(m2 ) (2)
ĉ(m) −
m2 − m1
m2 − m1
m2 − m1
m2 − m1
for all m ∈ (m1 , m2 ), and property ‘strictly’ greater CC if (2) holds as a strict inequality.
If c00 and ĉ00 exist everywhere between m1 and m2 , greater concavity of ĉ is equivalent to
ĉ00 being weakly larger in absolute value than c00 everywhere in the range from m1 to m2 .
The strict version of the proposition would require the inequality to hold strictly over
some interval between m1 and m2 .
5

3.2 Counterclockwise Concavification
The next concept we introduce is a ‘counterclockwise concavification,’ which describes
an operation that makes the modified consumption function more concave than in the
original situation. The idea is to think of the consumption function in the modified
situation as being a twisted version of the consumption function in the baseline situation,
where the kind of twisting allowed is a progressively larger increase in the MPC as the
level of market resources gets lower. We call this a ‘counterclockwise concavification’
to describe the sense that at any specific level of market resources, one can think of
the increase in the MPC at lower levels of market resources as being a counterclockwise
rotation of the lower portion of the consumption function around that level of resources.
Definition 4. (Counterclockwise Concavification.)
Function ĉ(m) is a counterclockwise concavification of c(m) around m# if the following
conditions hold:
1. ĉ(m) = c(m) for m ≥ m#
 0 
(m)
≥1
2. limm↑m# cĉ0 (m)
3. limµ↑m



ĉ0 (µ)
c0 (µ)

4. If limm↑m#





is weakly decreasing in m for m ≤ m#

ĉ0 (m)
c0 (m)



= 1, then limm↑m#



ĉ00 (m)
c00 (m)



>1

The limits in the definition are necessary to allow for the possibility of discrete drops
in the MPC at potential ‘kink points’ in the consumption functions. To understand
counterclockwise concavification, it is useful to derive its implied properties.
Lemma 2. (Properties of a Counterclockwise Concavification.)
If ĉ(m) is a counterclockwise concavification of c(m) around m# and c00 (m) ≤ 0 for all
m, then
1. ĉ(m) < c(m) for m < m# .
2. limµ↑m ĉ0 (µ) > limµ↑m c0 (µ) for m < m# .
3. limµ↑m ĉ00 (µ) ≤ limµ↑m c00 (µ) for m < m# .
See Appendix B for the proof. A counterclockwise concavification thus reduces consumption, increases the MPC, and makes the consumption function more concave for
all levels of market resources below the point of concavification. A prominent example
of a counterclockwise concavification is income risk. Lemma 3 shows, with a slight
abuse of notation, that a set of well-known results in the literature implies that the
introduction of a current income risk is an example of a counterclockwise concavification
of the consumption function around ∞.

6

Lemma 3. (Income Risk Causes Counterclockwise Concavification.)
Consider an agent who has a utility function of the HARA class (1) with u0 > 0, u00 < 0,
u000 > 0, and decreasing absolute prudence (−u000 /u00 ). Then the consumption function in
the presence of a current income risk c̃(m) is a counterclockwise concavification of the
consumption function in the presence of no risk c(m) around ∞.
000

Proof. Kimball (1990) shows that positive absolute prudence − uu00 ensures that c̃(m) <
c(m) for all m. Further, decreasing absolute prudence ensures that the conditions for
Corollary 1 in Carroll and Kimball (1996) are satisfied so that c̃00 (m) < 0 for all m. The
two results imply that consumption is lower, the MPC is higher, and the consumption
function is more concave everywhere in the case with risk than in the case with no risk.
c̃(m) is therefore a counterclockwise concavification of c(m) around ∞.

c

Risk

Constraint

m#

m

Figure 1 Examples of Counterclockwise Concavifications
Notes: The solid line shows the linear consumption function in the case with no constraints and no risks. The two
dashed lines show the consumption function when we introduce a constraint and a risk, respectively. The introduction of
a constraint is a counterclockwise concavification of the solid consumption function around m# , while the introduction of
a risk is a counterclockwise concavification around ∞.

Figure 1 illustrates two examples of counterclockwise concavifications: the introduction of
a constraint or a risk. In both cases, we start from the situation with no risk or constraints
(solid line). The constraint causes a counterclockwise concavification around a kink point
m# . Below m# , consumption is lower and the MPC is greater. The introduction of a risk
7

also generates a counterclockwise concavification of the original consumption function,
but this time around ∞ as described in Lemma 3.

3.3 A Counterclockwise Concavification Increases Prudence
The section above defined a counterclockwise concavification which describes the effects of
either a constraint or a risk on consumption concavity. This section shows the relationship
between consumption concavity and prudence. Our method is to compare prudence
in a baseline case where the consumption function is c(m) to prudence in a modified
situation in which the consumption function ĉ(m) is a counterclockwise concavification
of the baseline consumption function.
The first result relates to the effects of 000a counterclockwise concavification on the
(m)
.
absolute prudence of the value function, − VV 00 (m)
Lemma 4. (A Counterclockwise Concavification Increases Prudence.)
Consider an agent who has a utility function with u0 > 0, u00 < 0, u000 ≥ 0, and nonincreasing absolute prudence (−u000 /u00 ). If c(m) is concave and ĉ(m) is a counterclockwise
concavification of c(m), then the value function associated with ĉ(m) exhibits greater
absolute prudence than the value function associated with c(m) for all m.
See Appendix C for the proof. To understand the effects on prudence of a counterclockwise
concavification, note that for a twice differentiable consumption function and thrice
differentiable utility function, absolute prudence of the value function is defined as
−

V 000 (m)
u000 (c(m)) 0
c00 (m)
=
−
c
(m)
−
V 00 (m)
u00 (c(m))
c0 (m)

(3)

by the envelope condition. The results in Lemma 4 follow directly. Lemma 4 additionally
handles cases where the consumption function is not necessarily twice differentiable.
There are three channels through which a counterclockwise concavification heightens
prudence. First, the increase in consumption concavity from the counterclockwise concavification itself heightens prudence. Second, if absolute prudence of the utility function
is non-increasing, then the reduction in consumption (for some states) from the counterclockwise concavification heightens prudence (at those states). And third, the higher
marginal propensity to consume (MPC) from the counterclockwise concavification means
that any given variation in market resources results in larger variation in consumption,
increasing prudence. The channels operate separately, implying that a counterclockwise
concavification heightens prudence even if absolute prudence is zero as in the quadratic
case.9
Lemma 4 only provides conditions for when the value function exhibits greater prudence, but not strictly greater prudence. In particular, the value function associated
with ĉ(m) will in some cases (e.g., quadratic utility) have equal prudence for most m and
strictly greater prudence only for some m. In Lemma 5, we provide conditions for when
the value function has strictly greater prudence.

9 cf.

Nishiyama (2012)

8

Lemma 5. (A Counterclockwise Concavification Strictly Increases Prudence.)
Consider an agent who has a utility function with u0 > 0, u00 < 0, u000 ≥ 0, and nonincreasing absolute prudence (−u000 /u00 ). If c(m) is concave and ĉ(m) is a counterclockwise
concavification of c(m) around m# , then the value function associated with ĉ(m) exhibits
strictly greater prudence than the value function associated with c(m) if the utility function
0
satisfies u000 > 0 and m < m# or the utility function is quadratic (u000 = 0) and ĉc0 (m)
strictly
(m)
declines at m.
See Appendix D for the proof. For prudent consumers (u000 > 0), the value function
exhibits strictly greater prudence for all m where the counterclockwise concavification
affects consumption. This is because a reduction in consumption and higher marginal
propensity to consume heighten prudence if the utility function has a positive third derivative and prudence is non-increasing. If the utility function instead is quadratic, the third
derivative is zero and absolute prudence of the value function does not depend on the level
of consumption or the marginal propensity to consume. In this case, the counterclockwise
concavification
only affects prudence at the kink points in the consumption function
ĉ0 (m)
(where c0 (m) strictly declines at m).
We have now defined consumption concavity and the operation called a counterclockwise concavification. In particular, we have shown that a counterclockwise concavification
heightens prudence, which is related the precautionary saving. The next section shows
how the introduction of a liquidity constraint is a counterclockwise concavification before
we use the tools derived in this section to provide the link between liquidity constraints
and precautionary saving in Section 5.

4 Liquidity Constraints and Consumption Concavity
This section shows under which conditions liquidity constraints cause consumption concavity. The main conceptual difficulty with liquidity constraints is that the effect of
introducing a new constraint depends on already existing constraints. To get around this
issue, we introduce the concept of an ordered set of relevant constraints. This allows us
to add constraints in such a way that the next constraint does not affect behavior related
to pre-existing constraints. Our main result (Theorem 1) is that the introduction of the
next constraint from the ordered set of relevant constraints causes a counterclockwise
concavification of the consumption function. It then follows from the results in Section 3
that the introduction of the next constraint also heightens prudence of the value function.

4.1 Liquidity Constraints and Kink Points
Recall that we are working with a consumer whose horizon goes from 0 to T . We define a
liquidity constraint dated t as a constraint that requires savings at the end of period t ∈
(0, T ] to be non-negative (the assumption of non-negativity is without loss of generality
as shown in Theorem 1).
We first define what we mean by a kink point which is induced by a constraint. To
have a distinct terminology for the effects of current-period and future-period constraints,
we will use the word ‘binds’ to refer to the potential effects of a constraint in the period
9

in which it applies and will use the term ‘impinges’ to describe the effect of a future
constraint on current consumption.
Definition 5. (Kink Point.)
We define a kink point, ωt,n as the level of market resources at which constraint n stops
binding or impinging on time t consumption.
A kink point corresponds to a transition from a level of market resources where a current
constraint binds or a future constraint impinges, to a level of market resources where that
constraint no longer binds or impinges.
The timing of a constraint relative to other existing constraints matters for the effects of
the constraint. We therefore define an ordered set to keep track of the existing constraints.
Definition 6. (An Ordered Set of Relevant Constraints.)
We define T as an ordered set of dates at which a relevant constraint exists. We define
T [1] as the last period in which a constraint exists, T [2] as the date of the last period
before T [1] in which a constraint exists, and so on.
T is the set of relevant constraints, ordered from the last to the first constraint. We order
them from last to first because a constraint in period t only affects behavior prior to
period t (in addition to t itself). The set of constraints from period t to T summarizes all
relevant information in period t. Further, and as discussed below, the effect of imposing
the next constraint in T on consumption is unambiguous only if one imposes constraints
chronologically from last to first.
For any t ∈ [0, T ), we define ct,n as the optimal consumption function in period t
assuming that the first n constraints in T have been imposed. For example, ct,0 (m) is
the consumption function in period t when no constraints have been imposed, ct,1 (m) is
the consumption function in period t after the chronologically last constraint has been
imposed, and so on. Ωt,n , Vt,n , and other functions are defined correspondingly.

4.2 A Fixed Set of Constraints
We first consider an initial situation in which a consumer is solving a perfect foresight
optimization problem with a finite horizon that begins in period t and ends in period
T . The consumer begins with market resources mt and earns constant income y in each
period. Lemma 6 shows how this consumer’s behavior in period t changes from an initial
situation with n ≥ 0 constraints to a situation in which n + 1 liquidity constraints has
been imposed.
Lemma 6. (Liquidity Constraints Cause Counterclockwise Concavification.)
Consider an agent who has a utility function with u0 > 0 and u00 < 0, faces constant
income y, and is impatient (βR < 1). Assume that the agent faces a set T of N relevant
constraints. Then ct,n+1 (m) is a counterclockwise concavification of ct,n (m) around ωt,n+1
for n ≤ N − 1.
See Appendix E for the proof. When we have an ordered set of constraints, T , the
introduction of the next constraint generates a counterclockwise concavification of the
consumption function.
10

4.3 Additional Constraints
Lemma 6 analyzes the case where there is a preordained set of constraints T which were
applied sequentially in reverse chronological order. We now examine how behavior will be
modified if we add a new date τ̂ to the set of dates at which the consumer is constrained.
Call the new set of dates T̂ with N + 1 constraints (one more constraint than before),
and call the consumption rules corresponding to the new set of dates ĉt,1 through ĉt,N +1 .
Now call m the number of constraints in T at dates strictly greater than τ̂ . Then note
that that ĉτ̂,m = cτ̂,m , because at dates after the date at which the new constraint (number
m + 1) is imposed, consumption is the same as in the absence of the new constraint. Now
recall that imposition of the constraint at τ̂ causes a counterclockwise concavification
of the consumption function around a new kink point, ωτ̂,m+1 . That is, ĉτ̂,m+1 is a
counterclockwise concavification of ĉτ̂,m = cτ̂,m .
The most interesting observation, however, is that behavior under constraints T̂ in
periods strictly before τ̂ cannot be described as a counterclockwise concavification of
behavior under T . The reason is that the values of wealth at which the earlier constraints
caused kink points in the consumption functions before period τ̂ will not generally
correspond to kink points once the extra constraint has been added.
Figure 2 presents an example. The original T contains only a single constraint, at
the end of period t + 1, inducing a kink point at ωt,1 in the consumption rule ct,1 . The
expanded set of constraints T̂ adds one constraint at period t + 2. T̂ induces two kink
points in the updated consumption rule ĉt,2 , at ω̂t,1 and ω̂t,2 . It is true that imposition of
the new constraint causes consumption to be lower than before at every level of wealth
below ω̂t,1 . However, this does not imply higher prudence of the value function at every
m < ω̂t,1 . In particular, the original consumption function is strictly concave at ωt,1 ,
while the new consumption function is linear at ωt,1 , so prudence is greater before than
after imposition of the new constraint at ωt,1 .
The intuition is straightforward. At levels of initial wealth below ω̂t,1 , the consumer
had been planning to end period t + 2 with negative wealth. With the new constraint,
the old plan of ending up with negative wealth is no longer feasible and the consumer will
save more for any given level of current wealth below ω̂t,1 , including ωt,1 . But the reason
ωt,1 was a kink point in the initial situation was that it was the level of wealth where
consumption would have been equal to market resources in period t + 1. Now, because
of the extra savings induced by the constraint in t + 2, the period t + 1 constraint will no
longer bind for a consumer who begins period t with wealth ωt,1 . In other words, at wealth
ωt,1 the extra savings induced by the new constraint prevents the original constraint from
being relevant at ωt,1 .
Notice, however, that all constraints that existed in T will remain relevant at some
m under T̂ even after the new constraint is imposed - they just induce kink points at
different levels of market resources than before (in Figure 2, the first constraint causes a
kink at ω̂t,2 rather than ωt,1 ).

11

c
ĉ#
t,1

ct,1

ĉt,2

c#
t,1
ĉt,2 (ωt,1 )

ω̂t,2 ωt,1

ω̂t,1

m

Figure 2 How a future constraint can move a current kink
Notes: ct,1 is the original consumption function with one constraint that induces a kink point at ωt,1 . ĉt,2 is the modified
consumption function in where we have introduced one new constraint. The two constraints affect ĉt,2 through two kink
points: ω̂t,1 and ω̂t,2 . Since we introduced the new constraint at a later point in time than the current existing constraint,
the future constraint affects the position of the kink induced by the current constraint and the modified consumption
function ĉt,2 is not a counterclockwise concavification of ct,1 .

4.4 A More General Analysis
The preceding analyses required income to be constant, the liquidity constraints to be of
the no-borrowing type, and consumers to be impatient (βR < 1). We now relax these
requirements.
Under these more general circumstances, a constraint imposed in a given period can
render constraints in either earlier or later periods irrelevant. For example, consider a
consumer with CRRA utility and βR = 1 who earns income of 1 in each period, but who
is required to arrive at the end of period T − 2 with savings of 5. Then a constraint that
requires savings to be greater than zero at the end of period T − 3 will have no effect
because the consumer is required by the constraint in period T − 2 to end period T − 3
with savings greater than 4.
Formally, consider now imposing the first constraint, which applies in period τ < T .
The simplest case, analyzed before, was a constraint that requires the minimum level of
end-of-period wealth to be aτ ≥ 0. Here we generalize this to aτ ≥ ςτ,1 where in principle
we can allow borrowing by choosing ςτ,1 to be a negative number. Now for constraint 1
12

calculate the kink points for prior periods from
0
u0 (c#
τ,1 ) = Rβu (cτ +1,0 (Rςτ,1 + yτ +1 ))

ωτ,1 =

0 −1 0 #
) (u (cτ,1 )).
(Vτ,1

(4)
(5)

In addition, for constraint 2 recursively calculate
ς τ −1,1 = (ςτ,1 − yτ,2 + c)/R

(6)

where ς τ −1,1 is the level of wealth that constraint 1 requires the agent to end period
τ − 1 with and c is the lower bound for the value of consumption permitted by the model
(independent of constraints).10
Now assume that the first n constraints in T have been imposed, and consider imposing
constraint number n + 1, which we assume applies at the end of period τ . The first thing
to check is whether constraint number n + 1 is relevant given the already-imposed set of
constraints. This is simple: A constraint that requires aτ ≥ ςτ,n+1 will be irrelevant if
maxi∈[1,n] [ς τ,i ] ≥ ςτ,n+1 , i.e. if one of the existing constraints already implies that savings
must be greater or equal to value required by the new constraint. If the constraint is
irrelevant then the analysis proceeds simply by dropping this constraint and renumbering
the constraints in T so that the former constraint n + 2 becomes constraint n + 1, n + 3
becomes n + 2, and so on.
Now consider the other possible problem: That constraint number n + 1 imposed in
period τ will render irrelevant some of the constraints that have already been imposed.
This too is simple to check: It will be true if the proposed ςτ,n+1 ≥ ςτ,i for any i ≤ n and
for all m.11 The fix is again simple: Counting down from i = n, find the smallest value
of i for which ςτ,n+1 ≥ ςτ,i . Then we know that constraint n + 1 has rendered constraints
i through n irrelevant. The solution is to drop these constraints from T and start the
analysis over again with the modified T .
If this set of procedures is followed until the chronologically earliest relevant constraint
has been imposed, the result will be a T that contains a set of constraints that can be
analyzed as in the simpler case. In particular, proceeding from the final T [1] through
T [N ], the imposition of each successive constraint in T now causes a counterclockwise
concavification of the consumption function around successively lower values of wealth
as progressively earlier constraints are applied and the result is again a piecewise linear
and strictly concave consumption function with the number of kink points equal to the
number of constraints that are relevant at any feasible level of wealth in period t.
The preceding discussion establishes the following result:
Theorem 1. (Liquidity Constraints Cause Counterclockwise Concavification.)
Consider an agent in period t who has a utility function with u0 > 0, u00 < 0, u000 ≥ 0,
and non-increasing absolute prudence (−u000 /u00 ). Assume that the agent faces a set T of
N relevant constraints. Then ct,n+1 (m) is a counterclockwise concavification of ct,n (m)
around ωt,n+1 .
10 For example, CRRA utility is well defined only on the positive real numbers, so for a CRRA utility consumer c = 0.
In other cases, for example with exponential or quadratic cases, there is nothing to prevent consumption of −∞, so for
those models c = −∞, unless there is a desire to restrict the model to positive values of consumption, in which case the
c ≥ 0 constraint will be implemented through the use of (6).
11 If a constraint is irrelevant for the lowest m that t could enter period τ with, then it is irrelevant for all m.

13

Theorem 1 is a generalization of Lemma 6. Even if we relax the assumptions that income
is constant and the agent is impatient, the imposition of an extra (more general) constraint
increases absolute prudence of the value function as long as we are careful when we select
the set T of relevant constraints.
For an agent that only faces liquidity constraints, but no risk, the shape of the consumption function is piecewise linear. Since the consumption function is piecewise linear,
the new consumption function, ct,n+1 (m) is not necessarily strictly more concave than
ct,n (m) for all m. This is where the concept of counterclockwise concavification is useful.
Even though ct,n+1 (m) is not strictly more concave than ct,n (m) everywhere, it is a
counterclockwise concavification and we can apply Lemma 4 and 5 to show that the
introduction of the next liquidity constraint increases absolute prudence of the value
function.
Corollary 1. (Liquidity Constraints Increase Prudence.)
Consider an agent in period t who has a utility function with u0 > 0, u00 < 0, u000 ≥ 0, and
non-increasing absolute prudence (−u000 /u00 ). Assume that the agent faces a set T of N
relevant constraints. When n ≤ N − 1 constraints have been imposed, the imposition of
constraint n + 1 strictly increases absolute prudence of the agent’s value function if the
c0
strictly declines
utility function satisfies u000 > 0 and mt < ωt,n+1 or if u000 = 0 and t,n+1
c0t,n
at m.
Proof. By Theorem 1, the imposition of constraint n + 1 constitutes a counterclockwise
concavification of ct,n (m). By Lemma 4 and 5, such a concavification (strictly) increases
absolute prudence of the value function.

5 Liquidity Constraints and Precautionary Saving
The preceding sections established the relationship between liquidity constraints, consumption concavity, and prudence. This section derives the last step to understand the
relationship between liquidity constraints and precautionary saving. First, we explain
how prudence of the value function affects precautionary saving. Theorem 2 then shows
how the introduction of an additional constraint induces agents to increase precautionary
saving when they face a current risk. The results in Theorem 2 cannot be generalized
to an added risk or liquidity constraint in a later time period because it may hide or
alter the effects of current constraints or risks and thereby affect local precautionary
saving. The main conceptual issue with having both risks and constraints is that the
trick with the relevant constraints applied in Section 4 no longer applies in a setting with
risk because constraints may be relevant for some sample paths. However, we still derive
our most general result in Theorem 3: the introduction of an additional risk results in
more precautionary saving in the presence of all future risks and constraints than in the
case with no future risks and constraints.

14

5.1 Notation
We begin by defining two marginal value functions V 0 (m) and V̂ 0 (m) which are convex,
downward sloping, and continuous in wealth, m. We consider a risk ζ with support [ζ, ζ̄],
and follow Kimball (1990) by defining the Compensating Precautionary Premia (CPP)
as the values κ and κ̂ such that
V 0 (m) = E[V 0 (m + ζ + κ)]
V̂ 0 (m) = E[V̂ 0 (m + ζ + κ̂)].

(7)
(8)

The CPP can be interpreted as the additional resources an agent requires to be indifferent
between accepting the risk and not accepting the risk. The relevant part of Pratt (1964)’s
Theorem 1 as reinterpreted using Kimball (1990)’s Lemma (p. 57) can be restated as
Lemma 7. Let A(m) and Â(m) be absolute prudence of the value functions V and V̂
respectively at m,12 and let κ and κ̂ be the respective compensating precautionary premia
associated with imposition of a given risk ζ as per (7) and (8). Then the following
conditions are equivalent:
1. Â(m + ζ + κ) ≥ A(m + ζ + κ) for all ζ ∈ [ζ, ζ̄] and Â(m + ζ + κ) > A(m + ζ + κ)
for at least one [no] point ζ ∈ [ζ, ζ̄] and a given m.
2. κ̂ > [=]κ for all ζ ∈ [ζ, ζ̄] and the same given m.
Lemma 7 establishes that greater prudence is equivalent to inducing a greater precautionary premium. For our purpose, it means that our results above on absolute prudence
also imply that the precautionary premium is higher. Hence, a more prudent consumer
requires a higher compensation to be indifferent about facing the risk or not.13
We now take up the question of how the introduction of a risk ζt+1 that will be realized
at the beginning of period t + 1 affects consumption in period t in the presence and in
the absence of a subsequent constraint. To simplify the discussion, consider a consumer
for whom β = R = 1, with mean income y in period t + 1.
Assume that the realization of the risk ζt+1 will be some value ζ with support [ζ,ζ̄],
and signify a decision rule that takes account of the presence of the immediate risk by a
∼. Further, define ω̄t,n+1 as the lowest level of market resources required for the liquidity
constraint to never bind.
Definition 7. (Wealth Limit.)
ω̄ t,n+1 is the level of wealth such that an agent who faces risk ζt+1 and n + 1 constraints
saves enough to guarantee that constraint n + 1 will never bind in period t + 1. Its value
is given by:

−1
0
ω̄ t,n+1 = Ṽt,n+1
(Ω̃0t,n+1 (ωt+1,n+1 − (y + ζ)))
(9)
12 A small technicality: Absolute prudence of value functions is infinite at kink points in the consumption function, so
if both c(m) and ĉ(m) had a kink point at exactly the same m, the comparison of prudence would not yield a well-defined
answer. Under these circumstances we will say that Â(m) ≥ A(m) if the decline in the MPC is greater for ĉ(m) at m than
for c(m).
13 Note that precautionary premia are not equivalent to precautionary saving effects because precautionary premia apply
at a given level of consumption, while precautionary saving applies at a given level of wealth.

15

How to read this limit: ωt+1,n+1 is the level of wealth at which constraint n + 1 makes the
transition from binding to not binding in period t + 1. ωt+1,n+1 − (y + ζ) is the level of
wealth in period t + 1 that ensures that constraint n + 1 does not bind in period t + 1 even
with the worst possible draw, ζ.
We must be careful to check that ωt+1,n+1 − (y + ζ) is inside the set of feasible values of
at (e.g. positive for consumers with CRRA utility). If this is not true for some level of
market resources, then the constraint is irrelevant because the restriction imposed by the
risk is more stringent than the restriction imposed by the constraint.

5.2 Precautionary Saving with Liquidity Constraints
We are now in a position to analyze the relationship between precautionary saving and
liquidity constraints. Our first result regards the effect of an additional constraint on the
precautionary saving of a household facing risk at the beginning of period t + 1.
Theorem 2. (Liquidity Constraints Increase Precautionary Saving.)
Consider an agent who has a utility function with u0 > 0, u00 < 0, u000 > 0, and nonincreasing absolute prudence (−u000 /u00 ), and who faces the risk, ζt+1 . Assume that the
agent faces a set T of N relevant constraints and n ≤ N − 1. Then
ct,n+1 (m) − c̃t,n+1 (m) ≥ ct,n (m) − c̃t,n (m),

(10)

and the inequality is strict if wealth is less than the level that ensures that constraint n + 1
never binds (mt < ω̄t,n+1 ).
See Appendix F for the proof. Theorem 2 shows that the introduction of the next
constraint induces the agent to save more for precautionary reasons in response to an
immediate risk as long as there is a positive probability that the next constraint will
bind. Theorem 2 can be generalized to period s < t if there is no risk or constraint
between period s and t by defining ω̄s,n+1 as the wealth level at which the agent will
arrive in the beginning of period t with wealth ω̄t,n+1 .
To illustrate the result in Theorem 2, Figure 3 shows an example of optimal consumption rules in period t under different combinations of an immediate risk (realized
at the beginning of period t + 1) and a future constraint (applying at the end of period
t + 1). The thinner loci reflect behavior of consumers who face the future constraint,
and the dashed loci reflect behavior of consumers who face the immediate risk. For levels
of wealth above ωt,1 where the future constraint stops impinging on current behavior
for perfect foresight consumers, behavior of the constrained and unconstrained perfect
foresight consumers is the same. Similarly, c̃t,1 (mt ) = c̃t,0 (mt ) for levels of wealth above
ω̄ t,1 beyond which the probability of the future constraint binding is zero. For both
constrained and unconstrained consumers, the introduction of the risk reduces the level
of consumption (the dashed loci are below their solid counterparts). The significance of
Theorem 2 in this context is that for levels of wealth below ω̄ t,1 , the vertical distance
between the solid and the dashed loci is greater for the constrained (thin line) than for
the unconstrained (thick line) consumers because of the interaction between the liquidity
constraint and the precautionary motive.
16

c
ct,0

c̃t,0

c̃t,1
ct,1
ωt,1

ω̄t,1

m

Figure 3 Consumption Functions with and without a Constraint and a
Risk
Notes: ct,0 is the consumption function with no constraint and no risk, c̃t,0 is the consumption function with no constraint
and a risk that is realized at the beginning of period t + 1, ct,1 is the consumption function with one constraint in period
t + 1 and no risk, and c̃t,1 is the consumption function with one constraint in period t + 1 and a risk that is realized at the
beginning of period t + 1. The figure illustrates that the vertical distance between ct,1 and c̃t,1 is always greater than the
vertical distance between ct,0 and c̃t,0 for m < ω̄t,1 .

5.3 Additional Constraints or Risks?
The result in Theorem 2 is limited to the effects of an additional constraint when a
household faces income risk that is realized at the beginning of period t + 1. One might
think that this could be generalized to a proposition that precautionary saving increases
if we for example impose an immediate constraint or an earlier risk, or generally impose
multiple constraints or risks. However, it turns out that the answer is “not necessarily”
to all these possible scenarios. The insight here is that it is no longer possible to use the
trick of the relevant constraints or risks in the previous section. In a perfect-foresight
environment as in Section 4 and Theorem 2, there was a stark demarcation between
relevant and irrelevant constraints. In an environment with risk, this no longer holds
because in the presence of risk, constraints and risks may be relevant for some sample
paths. The additional constraints and risks may therefore reduce precautionary saving
for some levels of m and we cannot derive more general results on additional risks or

17

constraints. We provide two examples to illustrate this: an immediate constraint and an
earlier risk.
To describe these examples, we need a last bit of notation. Define cm
t,n as the consumption function in period t assuming that the first n constraints and the first m
risks have been imposed, counting risks, like constraints, backwards from period T . All
other functions are defined correspondingly. We will continue to use the notation c̃t,n to
designate the effects of imposition of a single immediate risk realized at the beginning of
period t + 1.

c

c0t,1

c0t,0
c1t,0

c1t,1

1
0
ωt,1
ωt,1

m

Figure 4 How an Immediate Constraint can Hide the Effect of a Future
Risk
Notes: c0t,0 is the consumption function with no constraint and no risk, c1t,0 is the consumption function with no constraint
and one future risk in t + 1, c0t,1 is the consumption function with one immediate constraint and no risk, and c1t,1 is the
consumption function with one immediate constraint and one future risk in t + 1. The figure illustrates that the future risk
1
has no effect on consumption when m < ωt−1
because the immediate constraint hides the effect of the future risk.

5.3.1 An Immediate Constraint
Consider a situation in which no constraint applies between t and T illustrated in Figure
4. Since c0t,0 designates the consumption rule that will be optimal prior to imposing
the period-t constraint, the consumption rule imposing the constraint will be c0t,1 (m) =
min[c0t,0 (m), m]. Now define the level of market resources below which the period t
18

0
0
constraint binds for a consumer not facing the risk as ωt,1
. For values of m ≥ ωt,1
,
analysis of the effects of the risk is identical to analysis in the previous subsection. For
1
where the constraint binds both in the presence and
levels of market resources m < ωt,1
the absence of the immediate risk, we have c1t,1 (m) = c0t,n (m) = m. Hence, for consumers
1
with wealth below ωt,1
, the introduction of the risk in period t + 1 has no effect on
consumption in t, because for these levels of savings at the end of t, the consumers where
constrained before the risk was imposed and remain constrained afterwards. Hence, the
immediate constraint hides the risk from view and the precautionary saving in response
to the risk is higher in the absence of the constraint than in the presence of the constraint
1
when m ≤ ωt,1
.

5.3.2 An Earlier Risk
Consider now the question of how the addition of a risk ζt that will be realized at the
beginning of period t affects the consumption function at the beginning of period t − 1, in
the absence of any constraint at the beginning of period t. The question is whether we can
say that the introduction of the risk ζt has a greater precautionary effect on consumption
in the presence of the subsequent risk ζt+1 than in its absence?
The answer again is “not necessarily.” To see why, we present an example in Appendix
G of a CRRA utility problem in which in a certain limit the introduction of a risk
produced an effect on the consumption function that is indistinguishable from the effect
of a liquidity constraint. If the risk ζt is of this liquidity-constraint-indistinguishable
form, then the logic of the previous subsection applies: For some levels of wealth, the
introduction of the risk at t can weaken the precautionary effect of any risks at t + 1 or
later.

5.4 All Risks and Constraints
It might seem that the previous subsection implies that little useful can be said about the
precautionary effects of introducing a new risk in the presence of preexisting constraints
and risks. It turns out, however, that there is one useful result about the introduction of
all risks and constraints.
Theorem 3. (Liquidity Constraints and Risks Increase Precautionary Saving.)
Consider an agent who has a utility function with u0 > 0, u00 < 0, u000 > 0, and nonincreasing absolute prudence (−u000 /u00 ). Then the introduction of a risk ζt+1 has a
greater precautionary effect on period t consumption in the presence of all future risks
and constraints than in the absence of any future risks and constraints, i.e.
m
0
1
cm−1
t,n (m) − ct,n (m) > ct,0 (m) − ct,0 (m)

(11)

at levels of period-t market resources m such that in the absence of the new risk the
consumer is not constrained in the current period (cm−1
t,n (m) > m) and in the presence of
the risk there is a positive probability that some future constraint will bind.
Appendix H presents the proof. A fair summary of this theorem is that in most circumstances the presence of future constraints and risks does increase the amount of
19

precautionary saving induced by the introduction of a given new risk. The primary
circumstance under which this should not be expected is for levels of wealth at which the
consumer was constrained even in the absence of the new risk. There is no guarantee
that the new risk will produce a sufficiently intense precautionary saving motive to
move the initially-constrained consumer off his constraint. If it does, the effect will
be precautionary, but it is possible that no effect will occur.
Our last result is part of the proof of Theorem 3, but we state it explicitly as a corollary.
Corollary 2. (Liquidity Constraints and Risk Cause Counterclockwise Concavification.)
Consider an agent who has a utility function with u0 > 0, u00 < 0, u000 > 0, and nonincreasing absolute prudence (−u000 /u00 ). Then the consumption function in the presence
of m future risks and n constraints cm
t,n is a counterclockwise concavification of the consumption function with no risk and no constraints c0t,0 .
Corollary 2 states that the consumption function in the presence of all future risks and
constraints is a counterclockwise concavification of the consumption function with no
risks or constraints. In other words, the consumption function is concave in the presence
of all future risks and constraints.

6 Conclusion
The central message of this paper is that the effects of liquidity constraints and of
future risks on precautionary saving are similar because the introduction of either a
liquidity constraints or a risk makes the consumption function more concave than the
perfect foresight consumption function. Such an increase in concavity heightens prudence,
inducing consumers at any affected level of wealth to save more for precautionary reasons.
In addition, we provide an explanation of apparently contradictory results: That
constraints in some cases intensify and in other cases weaken those motives. The insight
here is that the effect of introducing a constraint or risk depends on whether it weakens
the effect of any pre-existing constraints or risks. If the new constraint or risk does not
interact in any way with existing constraints or risks, it intensifies the precautionary
saving motive. If it ‘hides’ or moves the effect of any existing constraints or risks, it may
weaken the precautionary saving motive at some levels of market resources.

20

Appendix
A Proof of Lemma 1
Proof. First, to facilitate readability of the proof, we assume that R = β = 1 with
no loss of generality. Our goal is to prove that V (mt ) ∈ CC if Vt+1 (at + yt+1 ) ∈ CC
for all realizations of yt+1 . The proof proceeds in two steps. First, we show that
property CC is preserved through the expectation operator (vertical aggregation), i.e.
that Ω(at ) = Et [Vt+1 (at + yt+1 )] ∈ CC if Vt+1 (at + yt+1 ) ∈ CC for all realizations of
yt+1 . Second, we show that property CC is preserved through the value function operator
(horizontal aggregation), i.e. that V (mt ) = maxs u(ct (mt −s))+Ω(s) ∈ CC if Ω(s) ∈ CC.
Throughout the proof, the first order condition holds with equality since no liquidity
constraint applies at the end of period t.
Step 1: Vertical aggregation
We show that consumption concavity is preserved under vertical aggregation for three
cases of the HARA utility function with u000 ≥ 0 (α1 ≥ −1) and non-increasing absolute
prudence (α1 ∈
/ (−1, 0)). The three cases are

−1/α1

α1 ∈ (0, ∞) (CRRA)
(α1 c + b)
0
u (c) = e−c/b
(12)
α1 = 0 (CARA)


α1 c + b
α1 = −1 (Quadratic)
Case I (α1 > 0, CRRA.) We will show that concavity is preserved under vertical
aggregation for c−1/α1 to avoid clutter, but the results hold for all affine transformations,
α1 c + b, with α1 > 0. Concavity of ct+1 (at + yt+1 ) implies that
ct+1 (at + yt+1 ) ≥ pct+1 (a1 + yt+1 ) + (1 − p)ct+1 (a2 + yt+1 )

(13)

for all yt+1 ∈ [y, ȳ] if at = pa1 + (1 − p)a2 with p ∈ [0, 1]. Since this holds for all yt+1 , we
know that
io−a n h
io−α1
n h
− α1
− a1
1
Et ct+1 (at + yt+1 )
≥ Et {pct+1 (a1 + yt+1 ) + (1 − p)ct+1 (a2 + yt+1 )}
We now apply Minkowski’s inequality (see e.g. Beckenbach and Bellman, 1983, Theorem
3) which says that for u, v ≥ 0 and a scalar k < 1 (k 6= 0)



1/k
1/k
1/k
E[(u + v)k ]
≥ E[uk ]
+ E[v k ]
.
This implies that for α1 ∈ (0, ∞) (CRRA)
o−α1 n
o−α1 n
o−α1
n
− α1
− α1
− α1
1
1
1
]
≥ E[u
]
+ E[v
]
E[(u + v)
if u ≥ 0 and v ≥ 0. Thus
n h
io−α1
− 1
Et {pct+1 (a1 + yt+1 ) + (1 − p)ct+1 (a2 + yt+1 )} α1
n h
io−α1 n h
io−α1
− 1
− 1
≥ Et {pct+1 (a1 + yt+1 )} α1
+ Et {(1 − p)ct+1 (a2 + yt+1 )} α1
21

n h
io−α1
n h
io−α1
− 1
− 1
= p Et {ct+1 (a1 + yt+1 )} α1
+ (1 − p) Et {ct+1 (a2 + yt+1 )} α1
= p(Ω0 (a1 ))−α1 + (1 − p)(Ω0 (a2 ))−α1
which implies that
(Ω0 (at ))−α1 ≥ p(Ω0 (a1 ))−α1 + (1 − p)(Ω0 (a2 ))−α1
0

Thus, defining χt (at ) = {Ωt (at )}−α1 , we get
χt (at ) ≥ pχt (a1 ) + (1 − p)χt (a2 )
for all at , where the inequality is strict if ct+1 is strictly concave for at least one realization
of yt+1 .
Case II (a = 0, CARA). For the exponential case, property CC holds at at if
exp(−χt (at )/b) = Et [exp(−ct+1 (at + yt+1 )/b)]
for some χt (at ) which is strictly concave at at . We set b = 1 to reduce clutter, but results
hold for b 6= 1. Consider first a case where ct+1 is linear over the range of possible values
of at + yt+1 , then
χt (at ) = − log Et [e−ct+1 (at +yt+1 ) ]
0

= − log Et [e−(ct+1 (at +ȳ)+(yt+1 −ȳ)ct+1 ) ]
0

= ct+1 (at + ȳ) − log Et [e−(yt+1 −ȳ)ct+1 ]

(14)

which is linear in at since the second term is a constant.
Now consider a value of at for which ct+1 (at + yt+1 ) is strictly concave for at least one
realization of yt+1 . Global weak concavity of ct+1 tells us that for every yt+1
−ct+1 (at + yt+1 ) ≤ −((1 − p)ct+1 (a1 + yt+1 ) + pct+1 (a2 + yt+1 ))
Et [e−ct+1 (at +yt+1 ) ] ≤ Et [e−((1−p)ct+1 (a1 +yt+1 )+pct+1 (a2 +yt+1 )) ].

(15)

Meanwhile, the arithmetic-geometric mean inequality states that for positive u and v,
if ū = Et [u] and v̄ = Et [v], then


Et (u/ū)p (v/v̄)1−p ≤ Et [p(u/ū) + (1 − p)(v/v̄)] = 1,
implying that
Et [up v 1−p ] ≤ ūp v̄ 1−p ,
where the expression holds with equality only if v is proportional to u. Substituting in
u = e−ct+1 (a1 +yt+1 ) and v = e−ct+1 (a2 +yt+1 ) , this means that

p
1−p
Et [e−pct+1 (a1 +yt+1 )−(1−p)ct+1 (a2 +yt+1 ) ] ≤ Et [e−ct+1 (a1 +yt+1 ) ]
Et [e−ct+1 (a2 +yt+1 ) ]
and we can substitute for the LHS from (15), obtaining

p
Et [e−ct+1 (at +yt+1 ) ] ≤ Et [e−ct+1 (a1 +yt+1 ) ]
Et [e−ct+1 (a2 +yt+1 ) ]

1−p

log Et [e−ct+1 (at +yt+1 ) ] ≤ p log Et [e−ct+1 (a1 +yt+1 ) ] + (1 − p) log Et [e−ct+1 (a2 +yt+1 ) ] (16)
which holds with equality only when e−ct+1 (a1 +yt+1 ) /e−ct+1 (a2 +yt+1 ) is a constant. This will
only happen if ct+1 (a1 + yt+1 ) − ct+1 (a2 + yt+1 ) is constant, which (given that the MPC is
22

strictly positive everywhere) requires ct+1 (at + yt+1 ) to be linear for yt+1 ∈ (y, ȳ). Hence,
χt (at ) ≥ pχt (a1 ) + (1 − p)χt (a2 ).
where the inequality is strict for an at from which ct+1 is strictly concave for some
realization of yt+1 .
Case III (a = −1, Quadratic). In the quadratic case, linearity of marginal utility
implies that
u0 (χt (at )) = Et [u0 (ct+1 (at + yt+1 ))]
χt (at ) = Et [ct+1 (at + yt+1 )]
so χt is simply the weighted sum of a set of concave functions where the weights correspond
to the probabilities of the various possible outcomes for yt+1 . The sum of concave functions
is itself concave. And if additionally the consumption function is strictly concave at any
point, the weighted sum is also strictly concave.
Step 2: Horizontal aggregation:
We now proceed with horizontal aggregation, namely how concavity is preserved through
the value function operation. Assume that Ωt (at ) ∈ CC at point at , then the first order
condition implies that
Ω0t (at ) = u0 (χt (at ))
for some monotonically increasing χt (at ) that satisfies
χt (pa1 + (1 − p)a2 ) ≥ pχt (a1 ) + (1 − p)χt (a2 )

(17)

for any 0 < p < 1, and a1 < at < a2 .
In addition, we know that the first order condition holds with equality such that
0
Ωt (at ) = u0 (ct (mt )) = u0 (χt (at )) which implies that at = χ−1
t (ct ). Using this equation, we
get
χt (pa1 + (1 − p)a2 ) ≥ pχt (a1 ) + (1 − p)χt (a2 )
pa1 + (1 − p)a2 ≥ χ−1
t (pχt (a1 ) + (1 − p)χt (a2 ))
−1
−1
pχt (c1 ) + (1 − p)χt (c2 ) ≥ χ−1
t (pc1 + (1 − p)c2 )
which implies that χ−1
is a convex function.
t
Use the budget constraint to define
mt = at + ct
ω(ct ) = χ−1 (ct ) + ct
Now, since χ−1
t is a convex function, and ω(ct ) is the sum of a convex and a linear function,
it is also a convex function satisfying
pω(c1 ) + (1 − p)ω(c2 ) ≥ ω(pc1 + (1 − p)c2 )
ω −1 (pω(c1 ) + (1 − p)ω(c2 )) ≥ pc1 + (1 − p)c2
c(pm1 + (1 − p)m2 ) ≥ pc(m1 ) + (1 − p)c(m2 )
so c is concave.
23

(18)

Note that the proof of horizontal aggregation works for any utility function with u0 > 0
and u00 < 0 when R = β = 1. However, for the more general case where R or β are not
equal to one, we need the HARA property that multiplying u0 by a constant corresponds
to a linear transformation of c.
Strict Consumption Concavity. When Vt+1 (mt+1 ) exhibits the property strict CC for
at least one mt+1 ∈ [Rat + y, Rat + ȳ], we know that χt (at ) also exhibits the property
strict CC from the proof of vertical aggregation. Then, equation (17) holds with strict
inequality, and this strict inequality goes through the proof of horizontal aggregation,
implying that equation (18) holds with strict inequality. Hence, ct (mt ) is strictly concave
if ct+1 (at + yt+1 ) is concave for all realizations of yt+1 and strictly concave for at least one
realization of yt+1 .

B Proof of Lemma 2
Proof. First, condition 2 and 4 in Definition 4 imply that ĉ0 (m) > c0 (m) for m = m# − 
for a small  > 0. Condition 3 then ensures that limµ↑m ĉ0 (µ) > limµ↑m c0 (µ) holds for all
m ≤ m# −  (equivalently m < m# ). Second, condition 1 and the fact that limµ↑m ĉ0 (µ) >
limµ↑m c0 (µ) for m < m# implies that limµ↑m ĉ(µ) < limµ↑m c(µ) for m < m# . Third,
condition 3 in Definition 4 implies that
lim ĉ00 (µ) ≤ lim c00 (µ)

µ↑m

µ↑m

ĉ0 (µ)
c0 (µ)

for m < m# . Then
lim ĉ00 (µ) ≤ lim c00 (µ)

µ↑m

µ↑m

since limµ↑m ĉ0 (µ) > limµ↑m c0 (µ) for m < m# . Note that the inequality is not strict since
c00 (µ) could be 0.

C Proof of Lemma 4
Proof. By the envelope theorem, we know that
V 0 (m) = u0 (c(m))
Differentiating with respect to m yields
V 00 (m) = u00 (c(m))c0 (m)

(19)

Since c(m) is concave, it has left-hand and right-hand derivatives at every point, though
the left-hand and right-hand derivatives may not be equal. Equation (19) should be
interpreted as applying the left-hand and right-hand derivatives separately. (Reading
(19) in this way implies that c0 (m− ) ≥ c0 (m+ ); therefore V 00 (m− ) ≤ V 00 (m+ )). Taking
another derivative can run afoul of the possible discontinuity in c0 (m) that we will show
below can arise from liquidity constraints. We therefore consider two cases: (i) c00 (m)
exists and (ii) c00 (m) does not exist.
24

Case I: (c00 (m) exists.)
In the case where c00 (m) exists, we can take another derivative
V 000 (m) = u000 (c(m))[c0 (m)]2 + u00 (c(m))c00 (m)
Absolute prudence of the value function is thus defined as
V 000 (m)
u000 (c(m))[c0 (m)]2 + u00 (c(m))c00 (m)
=
−
V 00 (m)
u00 (c(m))c0 (m)
u000 (c(m)) 0
c00 (m)
V 000 (m)
= − 00
c (m) − 0
− 00
V (m)
u (c(m))
c (m)
−

(20)

From the assumption that ĉ(m) is a counterclockwise concavification of c(m), we know
000 (c(m))
from Lemma 2 that ĉ(m) ≤ c(m) and ĉ0 (m) ≥ c0 (m). Furthermore, since − uu00 (c(m))
000

000

000

(c(m))
(ĉ(m)) 0
(ĉ(m))
≥ − uu00 (c(m))
. As a result, − uu00 (ĉ(m))
ĉ (m) ≥
is non-increasing, we know that − uu00 (ĉ(m))
000

(c(m)) 0
− uu00 (c(m))
c (m).
00

(m)
The second part of the absolute prudence expression, − cc0 (m)
, is a measure of the
00

(m)
curvature of the consumption function. Since the consumption function is concave, − cc0 (m)
is a measure of the degree of concavity. Formally, if one has two functions, f (x) and g(x),
that are both increasing and concave functions, then the concave transformation g(f (x))
always has more curvature
than
f .14 A counterclockwise concavification is an example of
c00 (m)
ĉ00 (m)
such a g. Hence, − ĉ0 (m) ≥ − c0 (m) . Then

−

V̂ 000 (m)

u000 (ĉ(m)) 0
ĉ (m) −
u00 (ĉ(m))
V̂ 00 (m)
u000 (c(m)) 0
≥ − 00
c (m) −
u (c(m))
=−

ĉ00 (m)
ĉ0 (m)
c00 (m)
V 000 (m)
=
−
c0 (m)
V 00 (m)

Case II: (c00 (m) does not exist.)
Informally, if nonexistence is caused by a constraint binding at m, the effect will be a
discrete decline in the marginal propensity to consume at m, which can be thought of as
c00 (m) = −∞, implying positive infinite prudence at that point (see (20)). Formally, if
00
being a decreasing
c00 (m) does not exist, greater prudence of V̂ than V is given by V̂V 00 (m)
(m)
function of m. This is defined as
 00
 0

V̂ 00 (m)
u (ĉ(m))
ĉ (m)
≡
V 00 (m)
u00 (c(m))
c0 (m)
0

The second factor, ĉc0 (m)
, is weakly decreasing in m by the property of a counterclockwise
(m)
concavification. At any specific value of m where ĉ00 (m) does not exist because the left

14 To

see this, compute
−

d2
g(f (x))
dx2
d
g(f (x))
dx

=−

g 00 f 0
f 00
f 00
− 0 ≥− 0
g0
f
f

where the inequality holds since f 0 ≥ 0, g 0 ≥ 0, and g 00 ≤ 0.

25

and right hand values of ĉ0 are different, we say that ĉ0 is decreasing if
lim
ĉ0 (m) >
−

m →m

lim
ĉ0 (m).
+

m →m

(21)

As for the first factor, note that nonexistence of V̂ 000 (m) and/or ĉ00 (m) do not spring from
nonexistence of either u000 (c) or limm↑m ĉ0 (m) (for our purposes, when the left and right
derivatives of ĉ(m) differ at a point, the relevant derivative is the one coming from the left;
rather than carry around the cumbersome limit notation, read the following derivation as
00 (m)
applying to the left derivative). To discover whether VV̂ 00 (m)
is decreasing we differentiate

 00
(recall that the log is a monotonically decreasing transformation so the
log uu00 (ĉ(m))
(c(m))
derivative of the log of a function always has the same sign as the derivative of the
function):
d
u000 (ĉ(m)) 0
u000 (c(m)) 0
(log(u00 (ĉ(m)) − log(u00 (c(m)))) = 00
ĉ (m) − 00
c (m).
dm
u (ĉ(m))
u (c(m))
This will be negative if
u000 (ĉ(m)) 0
u000 (c(m)) 0
ĉ
(m)
≤
c (m)
u00 (ĉ(m))
u00 (c(m))
u000 (c(m)) 0
u000 (ĉ(m)) 0
ĉ (m) ≥ − 00
c (m).
⇒ − 00
u (ĉ(m))
u (c(m))

(22)

Recall from Lemma 2 that ĉ0 (m) ≥ c0 (m) and ĉ(m) ≤ c(m) so non-increasing absolute
000 (ĉ(m))
000 (c(m))
prudence of the utility function ensures that − uu00 (ĉ(m))
≥ − uu00 (c(m))
. Hence the LHS is
always greater or equal to the RHS of equation (22).

D Proof of Lemma 5
Proof. We prove each statement in Lemma 5 separately.
Case I: (u000 > 0.)
If u000 > 0, a counterclockwise concavification around m# implies that ĉ(m) < c(m) and
ĉ0 (m) > c0 (m) for all m < m# . Then
−

u000 (ĉ(m)) 0
u000 (c(m)) 0
ĉ
(m)
>
−
c (m) for m < m#
u00 (ĉ(m)
u00 (c(m))

Note that this condition is sufficient to prove Lemma 5 for the case where c00 (m) does not
exist since it then satisfies (22). In the case where c00 (m) does exist, we know that
−

ĉ00 (m)
c00 (m)
≥
−
for m < m#
ĉ0 (m)
c0 (m)

from the proof of Lemma 4. Hence,
−

V̂ 000 (m)
V̂ 00 (m)

=−

u000 (ĉ(m)) 0
ĉ00 (m)
ĉ
(m)
−
u00 (ĉ(m)
ĉ0 (m)
26

>−

u000 (c(m)) 0
c00 (m)
V 000 (m)
c
(m)
−
=
−
for m < m#
u00 (c(m))
c0 (m)
V 00 (m)

and Lemma 5 holds in the case with u000 > 0 and m < m# .
Case II: (u000 = 0.)
The quadratic case requires a different approach. Note first that the conditions in Lemma
5 hold only below the bliss point for quadratic utility. In addition, since u000 (·) = 0, strict
inequality between the prudence of V̂ and the prudence of V hold only at those points
where ĉ(·) is strictly concave.
Recall from the proof of Lemma 4 that greater prudence of V̂ (m) than V (m) occurs if
V̂ 00 (m)
is decreasing in m. In the quadratic case
V 00 (m)
u00 (ĉ(m)) ĉ0 (m)
ĉ0 (m)
V̂ 00 (m)
=
=
V 00 (m)
u00 (c(m)) c0 (m)
c0 (m)

(23)

where the second equality follows since u00 (·) is constant with quadratic utility. Thus,
0
prudence is strictly greater in the modified case only if ĉc0 (m)
strictly declines in m.
(m)

E Proof of Lemma 6
We prove Lemma 6 by induction in two steps. First, we show that all results in Lemma
6 hold when we add the first constraint. The second step is then to show that the results
hold when we go from n to n + 1 constraints.
Lemma 8. (c0t < c0t+1 )
Consider an agent who has a utility function with u0 > 0 and u00 < 0, faces constant
income, is impatient (βR < 1), and has a finite life. Then c0t < c0t+1 .
Proof. The marginal propensity to consume in period t can be obtained from the MPC
in period t + 1 from the Euler equation
u0 (ct (mt )) = βRu0 (ct+1 (R(mt − ct (mt )) + y)).
Differentiating both sides with respect to mt and omitting arguments to reduce clutter
we obtain
u00 (ct )c0t
(u00 (ct ) + βRu00 (ct+1 )c0t+1 R)c0t
c0t+1
c0t
c0t+1
c0t

= βRu00 (ct+1 )c0t+1 R(1 − c0t )
= βRu00 (ct+1 )Rc0t+1
u00 (ct ) + βRu00 (ct+1 )c0t+1 R
=
βRu00 (ct+1 )R
u00 (ct )
=
+ c0t+1
βRu00 (ct+1 )R

Since βR < 1 ensures that ct > ct+1 , we know that
u00 (ct )
u00 (ct+1 )
1
1
≥
=
>
βRu00 (ct+1 )R
βRu00 (ct+1 )R
βRR
R
27

Furthermore, we know that
R−1
R
is the MPC for an infinitely-lived agent with βR = 1. Hence,


c0t+1
u00 (ct )
1
R−1
0
=
+
c
>
+
=1
t
0
ct
βRu00 (ct+1 )R
R
R
c0t ≥

since

R−1
R

and it follows that c0t < c0t+1 .
Lemma 9. (Consumption with one Liquidity Constraint.)
Consider an agent who has a utility function with u0 > 0 and u00 < 0, faces constant
income, y, and is impatient, βR < 1. Assume that the agent faces a set T of one relevant
constraint. Then ct,1 (m) is a counterclockwise concavification of ct,0 (m) around ωt,1 .
Proof. We now prove Lemma 9 by first showing that the consumption function including
the constraint at the end of period τ is a counterclockwise concavification of the unconstrained consumption function in period τ . Next, we show how the constraint further
implies that the consumption function including the constraint is a counterclockwise
concavification of the unconstrained consumption function in periods prior to τ .
We first define τ = T [1] as the time period of the constraint. Note first that consumption is unaffected by the constraint for all periods after τ , i.e. cτ +k,1 = cτ +k,0 for any
k > 0. For period τ , we can calculate the level of consumption at which the constraint
binds by realizing that a consumer for whom the constraint binds will save nothing and
therefore arrive in the next period with no wealth. Further, the maximum amount of
consumption at which the constraint binds will satisfy the Euler equation (only points
where the constraint is strictly binding violate the Euler equation; the point on the cusp
does not). Thus, we define c#
τ,1 as the maximum level of consumption in period τ at which
the agent leaves no wealth for the next period, i.e. the constraint stops binding:
0
u0 (c#
τ,1 ) = βRu (cτ +1,0 (y))
0 −1
c#
(βRu0 (cτ +1,0 (y))) ,
τ,1 = (u )

and the level of wealth at which the constraint stops binding can be obtained from


0 −1
ωτ,1 = Vτ,1
(u0 (c#
(24)
τ,1 )).
Below this level of wealth, we have cτ,1 (m) = m so the MPC is one, while above it we
have cτ,1 (m) = cτ,0 (m) where the MPC equals the constant MPC for an unconstrained
perfect foresight optimization problem with a horizon of T − τ . Thus, cτ,1 satisfies our
definition of a counterclockwise concavification of cτ,0 around ωτ,1 .
Further, we can obtain the value of period τ − 1 consumption at which the period τ
constraint stops impinging on period τ − 1 behavior from
0 #
u0 (c#
τ −1,1 ) = βRu (cτ,1 )

and we can obtain ωτ −1,1 via the analogue to (24). Iteration generates the remaining c#
.,1
and ω.,1 values back to period t.
Now consider the behavior of a consumer in period τ − 1 with a level of wealth m <
ωτ −1,1 . This consumer knows he will be constrained and will spend all of his resources next
28

period, so at m his behavior will be identical to the behavior of a consumer whose entire
horizon ends at time τ . As shown in step I, the MPC always declines with horizon. The
MPC for this consumer is therefore strictly greater than the MPC of the unconstrained
consumer whose horizon ends at T > τ . Thus, in each period before τ +1, the consumption
function c.,1 generated by imposition of the constraint constitutes a counterclockwise
concavification of the unconstrained consumption function around the kink point ω.,1 .
We have now shown the results in Lemma 6 for n = 0. The last step is to show that
they also hold for n + 1 when they hold strictly for n. Consider imposing the n + 1’st
constraint and suppose for concreteness that it applies at the end of period τ . It will stop
binding at a level of consumption defined by
0
0
u0 (c#
τ,n+1 ) = βRu (cτ +1,n (y)) = βRu (y)

where the second equality follows because a consumer with total resources y, constant
income, and βR < 1 will be constrained. But note that by the definition of c#
τ,n , we obtain
T [n]−τ 0
u (y) < Rβu0 (y) = u0 (c#
u0 (c#
τ,n+1 )
τ,n ) = (Rβ)

where T [n] − τ denotes the time remaining to the n’th constraint. From the assumption
of decreasing marginal utility, we therefore know that
#
c#
τ,n ≥ cτ,n+1 .

This means that the constraint is relevant: The pre-existing constraint n does not force
the consumer to do so much saving in period τ that the n + 1’st constraint fails to bind.
The prior-period levels of consumption and wealth at which constraint n + 1 stops
impinging on consumption can again be calculated recursively from
0
u0 (c#
τ,n+1 ) = Rβu (cτ +1,n (y))

−1 0 #
0
ωτ,n+1 = Vτ,n
(u (cτ,n+1 )).

Furthermore, once again we can think of the constraint as terminating the horizon of a
finite-horizon consumer in an earlier period than it is terminated for the less-constrained
consumer, with the implication that the MPC below ωτ,n+1 is strictly greater than the
MPC above ωτ,n+1 . Thus, the consumption function cτ,n+1 constitutes a counterclockwise
concavification of the consumption function cτ,n around the kink point ωτ,n+1 .

F Proof of Theorem 2
Proof. Our proof proceeds by constructing the behavior of consumers facing the risk from
the behavior of the corresponding perfect foresight consumers. We consider matters from
the perspective of some level of wealth m for the perfect foresight consumers. Because the
same marginal utility function u0 applies to all four consumption rules, the Compensating
Precautionary Premia, κt,n and κt,n+1 , associated with the introduction of the risk ζt+1
must satisfy
ct,n (m) = c̃t,n (m + κt,n )
ct,n+1 (m) = c̃t,n+1 (m + κt,n+1 ).
29

(25)
(26)

Define the amounts of precautionary saving induced by the risk ζt+1 at an arbitrary level
of wealth m in the two cases as
ψt,n (m) = ct,n (m) − c̃t,n (m)
ψt,n+1 (m) = ct,n+1 (m) − c̃t,n+1 (m)

(27)
(28)

where the mnemonic is that the first two letters of the Greek letter psi stand for
precautionary saving.
We can rewrite (26) (resp. (25)) as
Z m
ct,n+1 (m) = ct,n+1 (m + κt,n+1 ) +
c0t,n+1 (µ)dµ = c̃t,n+1 (m + κt,n+1 )ψt,2 (m + κt,2 ) ≡
m+κt,n+1

which implies that
Z

m+κt,n+1

c0t,n+1 (µ)dµ,
ψt,n+1 (m + κt,n+1 ) = ct,n+1 (m + κt,n+1 ) − c̃t,n+1 (m + κt,n+1 ) =
Z m+κt,n m
c0t,n (µ)dµ
ψt,n (m + κt,n ) = ct,n (m + κt,n ) − c̃t,n (m + κt,n ) =
m

and
Z

m+κt,n+1

ψt,n (m + κt,n+1 ) = ψt,n (m + κt,n ) −

(c̃0t,n (µ) − c0t,n (µ))dµ

m+κt,n

so the difference between precautionary saving for the consumer facing n constraints and
the one facing n + 1 constraints at m + κt,n+1 is
ψt,n+1 (m + κt,n+1 ) − ψt,n (m + κt,n+1 ) =
= ψt,n+1 (m + κt,n+1 ) − ψt,n (m + κt,n ) + ψt,n (m + κt,n ) − ψt,n (m + κt,n+1 )
Z m+κt,n+1
Z m+κt,n
Z m+κt,n+1
0
0
=
ct,n+1 (µ)dµ −
ct,n (µ)dµ +
(c̃0t,n (µ) − c0t,n (µ))dµ
m
m+κt,n+1

Z
=

m

m+κt,n

(c0t,n+1 (µ) − c0t,n (µ))dµ +

m

Z

m+κt,n+1

c̃0t,n (µ)dµ

(29)

m+κt,n

If we can show that (29) is a positive number for all feasible levels of m satisfying
m < ω̄ t,n+1 , then we have proven Theorem 2. We know that the marginal propensity to
consume is always strictly positive and that κt,n+1 ≥ κt,n ≥ 015 so to prove that (29) is
strictly positive, we need to show one of two sufficient conditions:
1. κt,n+1 > 0 and c0t,n+1 (µ) > c0t,n (µ)
2. κt,n+1 > κt,n
Now, since u000 > 0, we know that κt,n > 0 from Jensen’s inequality. Hence, κt,n+1 > 0
since κt,n+1 ≥ κt,n . The first integral in (29) is therefore strictly positive as long as
c0t,n+1 > c0t,n , which is true for m < ωt,n+1 by Lemma 6.
For m ≥ ωt,n+1 , we know that c0t,n+1 = c0t,n so the first integral in (29) is always zero.
For the second integral in (29) to be strictly positive, we need to show that κt,n+1 > κt,n .
15 Since we know that liquidity constraints increase prudence (Corollary 1) and that prudence results in a positive
precautionary premium (Lemma 7).

30

First define the perfect foresight consumption functions as
=at,n+1

z}|{
c(κt,n + ζ) = ct+1,n ( st,n +y + κt,n + ζ)
c(κt,n+1 + ζ) = ct+1,n+1 (st,n+1 + y + κt,n+1 + ζ).

(30)
(31)

where at,n = at,n+1 since m ≥ ωt,n+1 . Recall also the definitions of κt,n and κt,n+1 :
u0 (ct,n ) = Et [u0 (c(κt,n + ζ))]
u0 (ct,n+1 ) = Et [u0 (c(κt,n+1 + ζ))].
Now recall that Lemma 7 tells us that if absolute prudence of u0 (c(κt,n + ζ)) is identical
to absolute prudence of u0 (c(κt,n+1 + ζ)) for every realization of ζ, then κt,n = κt,n+1 . This
is true if mt+1 ≥ ωt+1,n+1 for all possible realizations of ζ ∈ (ζ, ζ̄), i.e. that the agent is
unconstrained for all realizations of the risk. We defined this limit as mt+1 ≥ ω̄ t+1,n+1 .
We therefore know that κt,n+1 = κt,n if m ≥ ω̄ t+1,n+1 .
For all levels of wealth below this limit (m < ω̄ t+1,n+1 ), there exist realizations of ζ such
that constraint n+1 will bind in period t+1. The agent will require a higher precautionary
premia when facing constraint n + 1 in addition to the n constraints already in the set,
implying that κt,n+1 > κt,n . Equation (29) is therefore strictly positive if m < ω̄ t+1,n+1
and we have proven Theorem 2.

G Resemblance Between Precautionary Saving and a
Liquidity Constraint
This appendix repeats an illustration from appendix G of Carroll Forthcoming. (We make
no claim to novelty of this point; it is here only to aid the intuition of the reader).
In this appendix, we provide an example where the introduction of risk resembles the
introduction of a constraint. Consider the second-to-last period of life for two risk-averse
CRRA utility consumers and assume for simplicity that R = β = 1.
The first consumer is subject to a liquidity constraint cT −1 ≥ mT −1 , and earns nonstochastic income of y = 1 in period T . This consumer’s saving rule will be
(
0
if mT −1 ≤ 1
aT −1,1 (mT −1 ) =
(mT −1 − 1)/2 if mT −1 > 1.
The second consumer is not subject to a liquidity constraint, but faces a stochastic
income process,
(
0
with probability p
yT =
1
with probability (1 − p).
1−p
If we write the consumption rule for the unconstrained consumer facing the risk as
s̃T −1,0 , the key result is that in the limit as p ↓ 0, behavior of the two consumers becomes
the same. That is, defining s̃T −1,0 (m) as the optimal saving rule for the consumer facing
the risk,
lim s̃T −1,0 (mT −1 ) = aT −1,1 (mT −1 )
p↓0

31

for every mT −1 .
To see this, start with the Euler equations for the two consumers given wealth m,
u0 (m − aT −1,1 (m)) = u0 (aT −1,1 (m) + 1)
u0 (m − s̃T −1,0 (m)) = pu0 (s̃T −1,0 (m)) + (1 − p)u0 (s̃T −1,0 (m) + 1).

(32)
(33)

Consider first the case where m is large enough that the constraint does not bind for
the constrained consumer, m > 1. In this case the limit of the Euler equation for the
second consumer is identical to the Euler equation for the first consumer (because for
m > 1 savings are positive for the consumer facing the risk, implying that the limit of
the first u0 term on the RHS of (33) is finite). Thus the limit of (33) is (32) for m > 1.
Now consider the case where m < 1 so that the first consumer would be constrained.
This consumer spends her entire resources m, and by the definition of the constraint we
know that
u0 (m) > u0 (1).

(34)

Now consider the consumer facing the risk. If this consumer were to save exactly zero
and then experienced the bad shock in period T , she would have an infinite marginal
utility (the Inada condition). This cannot satisfy the Euler-equation as long as m > 0.
Therefore we know that for any p > 0 and any m > 0 the consumer will save some
positive amount. For a fixed m, hypothesize that there is some δ > 0 such that no matter
how small p became the consumer would always choose to save at least δ. But for any δ,
the limit of the RHS of (33) is u0 (1 + δ). We know from concavity of the utility function
that u0 (1 + δ) < u0 (1) and we know from (34) that u0 (m) > u0 (1) > u0 (1 + δ), so as p ↓ 0
there must always come a point at which the consumer can improve her total utility by
shifting some resources from the future to the present, i.e. by saving less. Since this
argument holds for any δ > 0 it demonstrates that as p goes to zero there is no positive
level of saving that would make the consumer better off. But saving of zero or a negative
amount is ruled out by the Inada condition at u0 (0). Hence saving must approach, but
never equal, zero as p ↓ 0.
Thus, we have shown that for m ≤ 1 and for m > 1 in the limit as p ↓ 0 the
consumer facing the risk but no constraint behaves identically to the consumer facing
the constraint but no risk. This argument can be generalized to show that for the CRRA
utility consumer, spending must always be strictly less than the sum of current wealth
and the minimum possible value of human wealth. Thus, the addition of a risk to the
problem can rule out certain levels of wealth as feasible, and can also render either future
or past constraints irrelevant, just as the imposition of a new constraint can.

H Proof of Theorem 3
Proof. To simplify notation and without loss of generality, we assume that when an
agent faces n constraints and m risks, there are one constraint and one risk for each time
period. For example, if cm
t,n faces m future risks and n future constraints, then the next
period consumption function is cm−1
t+1,n−1 (and m = n). Note that we can transform any
problem into this notation by filling in with degenerate risks and non-binding constraints.
32

However, for Theorem 3 to hold with strict inequality, we need to assume that there is
at least one relevant future risk and one relevant constraint.
We know that either the introduction of risk or a introduction of a constraint results in
a counterclockwise concavification of the original consumption function. However, this is
only true when we introduce risks in the absence of constraints (Lemma 3) and when we
introduce constraints in the absence of risk (see Theorem 1). In this proof, we therefore
need to show that the introduction of all risks and constraints is a counterclockwise
concavification of the linear case with no risks and constraints.
Here is our proof strategy. We define a set
m−1
m
1
Pt,n
= {m|cm−1
t,n (m) − ct,n (m) > ct,0 (m) − ct,0 (m)}

(35)

where Theorem 3 holds in period t when we introduce a risk at the beginning of period
t + 1. This is defined as the set where precautionary saving induced by a risk that
is realized at the beginning of period t + 1 is greater in the presence of all risks and
constraints than in the unconstrained case.
m−1
In order to show that the set Pt,n
is non-empty, we build it up recursively, starting
from period T and adding one constraint or one risk for each time period. The key to
the proof is to understand that the introduction of risks or constraints will never fully
reverse the effects of all other risks and constraints, even though they sometimes reduce
absolute prudence for some levels of wealth because risks and constraints can mask the
effects of future risks and constraints. Hence, the new consumption function must still
be a counterclockwise concavification of the consumption function with no risks and
constraints for some levels of wealth.
Since a counterclockwise concavification increases prudence by Lemma 4, and higher
prudence increases precautionary saving by Lemma 7, our required set can be redefined
as
m−1
m−1
Pt,n
= {m|cm−1
t,n (m) is a counterclockwise concavification of ct,0 (m) and ct,n (m) > m}

where we add the last condition, cm−1
t,n (m) > w to avoid the possibility that some
constraint binds such that the agent does not increase precautionary saving. In words:
m−1
Pt,n
is the set where the consumption function is a counterclockwise concavification
of ct,0 (m) and no constraint is strictly binding. We construct the set recursively for
two different cases: CARA and all other type of utility functions. We start with the
non-CARA utility functions.
0
First add the last constraint. The set PT,1
is then
0
PT,1
=∅

since we know that cT,1 (m) is a counterclockwise concavification of cT,0 (m) around ωT,1
but that the consumer is constrained below this point.
We next add the risk at the beginning of period T . To construct the new set, we note
three things. First, by Lemma 1, (strict) consumption concavity is recursively propagated
for all values of wealth where there is a positive probability that the constraint can bind,
i.e.


{mT −1 |ωT −1,1 ∈ mT −1 − c1T −1,1 (mT −1 ) + yT + ζ, mT −1 − c1T −1,1 (mT −1 ) + yT + ζ̄ }
33

has property strict CC, while it has non-strict property CC for all possible values of
mT −1 . Further, we know from Theorem 2 (rearrange equation (10)) that
c1T −1,1 (m) ≤ cT −1,1 (m) − cT −1,0 (m) + c1T −1,0 (m)
≤ cT −1,0 (m) − cT −1,0 (m) + cT −1,0 (m) = cT −1,0 (m).
Third, we know that c1T −1,1 (m)0 ≥ c0T −1,0 (m) since c1T −1,1 (m) < cT −1,0 (m) for m ≤ ωT1 −1,1 ,
limm→∞ c1T −1,1 (m) − cT −1,0 (m) = 0, and that c1t,1 (m) is concave while ct,0 (m) is linear.
Hence, c1T −1,1 is a counterclockwise concavification of cT −1,0 around the minimum value
of wealth when the constraint will never bind and the new set is


PT1 −1,1 = {mT −1 |ωT −1,1 ∈ mT −1 − c1T −1,1 (mT −1 ) + yT + ζ, mT −1 − c1T −1,1 (mT −1 ) + yT + ζ̄
∧c1T −1,1 (m) > mT −1 }.
We can now add the next constraint. The consumption function now has two kink
points, ωT1 −1,1 and ωT1 −1,2 . We know again from Lemma 1 that consumption concavity
is preserved when we add a constraint, and strict consumption concavity is preserved
for all values of wealth at which a future constraint might bind. Further, we know from
Theorem 2 that
c1T −1,2 (m) ≤ cT −1,2 (m) − cT −1,1 (m) + c1T −1,1 (m)
≤ cT −1,1 (m) − cT −1,1 (m) + cT −1,1 (m) = cT −1,1 (m) ≤ cT −1,0 (m).
Third, c1T −1,2 (m) < cT −1,0 (m), limm→∞ c1T −1,2 (m) − cT −1,0 (m) = 0, and we know that if
1
0
c1T −1,2 (m) is concave while cT −1,0 (m) is linear, then c01
T −1,2 (m) ≥ cT −1,0 (m). cT −1,2 (m)
which is a counterclockwise concavification of cT −1,0 (m) around the minimum level of
wealth at which the first constraint will never impinge on time T −1 consumption, ω̄T1 −1,1 ,
and the new set is
PT1 −1,2 = {mT −1 |mT −1 ≤ ω̄T1 −1,1 ∧ c1T −1,2 (m) > mT −1 }.
It is now time to add the next risk.
The argument is similar.
We still
know that (strict) consumption concavity is recursively propagated and that
limm→∞ c2T −2,2 (m) − cT −2,0 (m) = 0. Further, we can think of the addition of two
risks over two periods as adding one risk that is realized over two periods. Hence, the
results from Theorem 2 must hold also for the addition of multiple risks so we have
c2T −2,2 (m) ≤ cT −2,2 (m) − cT −2,1 (m) + c2T −2,1 (m)
≤ cT −2,1 (m) − cT −2,1 (m) + cT −2,1 (m) = cT −2,1 (m) ≤ cT −2,0 (m).
0
2
Hence, we again know that c02
T −2,2 (m) ≥ cT −2,0 (m). cT −2,2 (m) is thus a counterclockwise
concavification of cT −2,0 (m) around the level of wealth at minimum value of wealth when
the last constraint will never bind. The new set is therefore

PT2 −2,2 = {mT −2 |mT −2 − c2T −2,2 (mT −2 ) + yT −1 + ζT −1 ∈ PT1 −1,2 ∧ c2T −2,2 (m) < m}.
m−1
as the minimum value of wealth beyond which
Doing this recursively and defining ω̄t,1
constraint 1 will never bind, the set of wealth levels at which Theorem 3 holds can be
defined as
m−1
m−1
Pt,n
= {mt |mt ≤ ω̄t,1
∧ cm−1
t,n (m) > m}

34

In words, precautionary saving is higher if there is a positive probability that some future
constraint could bind and the consumer is not constrained today.
The last requirement is to define the set also for the CARA utility function. The
m−1
problem with CARA utility is that limm→∞ cm−1
≤ 0 where
t,n (m) − ct,0 (m) = −k
m−1
k
is some positive constant. We can therefore not use the same arguments as in the
preceding proof. However, by realizing that equation (10) in the CARA case can be
defined as
ct,n+1 (m) − c̃t,n+1 (m) − k̃ ≥ ct,n (m) − c̃t,n (m) − k̃ ≥ 0
where the last inequality follows since precautionary saving is always higher than in the
constant limit in the presence of constraints. We can therefore rearrange to get
c̃t,n+1 (m) ≤ ct,n+1 (m) − k̃ ≤ ct,n (m) − k̃ ≤ ct,0 − k̃
which implies that the arguments in the preceding section goes through also for CARA
utility with this slight modification.

35

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37