Liquidity Constraints and Precautionary Saving

Economists working with numerical solutions to the optimal consumption/saving problem under uncertainty have long known that there are quantitatively important interactions between liquidity constraints and precautionary saving behavior. This paper provides the analytical basis for those interactions. First, we explain why the introduction of a liquidity constraint increases the precautionary saving motive around levels of wealth where the constraint becomes binding. Second, we provide a rigorous basis for the oft-noted similarity between the effects of introducing uncertainty and introducing constraints, by showing that in both cases the effects spring from the concavity in the consumption function which either uncertainty or constraints can induce. We further show that consumption function concavity, once created, propagates back to consumption functions in prior periods. Finally, our most surprising result is that the introduction of additional constraints beyond the first one, or the introduction of additional risks beyond a first risk, can actually reduce the precautionary saving motive, because the new constraint or risk can hide&apos; the effects of the preexisting constraints or risks.


Introduction
A large literature has demonstrated that numerical models that take constraints and uncertainty seriously can yield different conclusions than those that characterize traditional models.For example, in heterogeneous agent New Keynesian models (e.g.Kaplan, Moll, and Violante, 2018), a major transmission mechanism for monetary and fiscal policy is the 'indirect income effect,' because in such models' equilibria many households have high marginal propensities to consume (MPC's) -a channel of minimal importance in traditional macro models, where the MPC is small.And Guerrieri and Lorenzoni (2017) and Bayer, Lütticke, Pham-Dao, and Tjaden (2019) show that such models suggest that tightened borrowing conditions and heightened income risk were important explanatory factors behind the consumption decline during the great recession.Further, Krueger, Mitman, and Perri (2016) show that numerically realistic models can match the empirical finding that the drop in consumption spending during the great recession was heavily concentrated in the middle class.
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.At least since Zeldes (1984), economists working with numerical solutions have known that liquidity constraints can strictly increase precautionary saving under very general circumstances -even for consumers with a quadratic utility function that generates no intrinsic precautionary saving motive. 1 On the other hand, simulation results have often found circumstances under which liquidity constraints and precautionary saving are substitutes rather than complements.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.
Our first main result is that introduction of a constraint at the end of period t creates 'consumption concavity' around the point where the constraint binds. 2 We further show that once consumption concavity is created (by the introduction of either uncertainty or a constraint (or in any other way)), it propagates back to periods before t. 3 Kimball (1990) showed that any utility function with a positive third derivative exhibits 'prudence,' which is the key requirement for a risk to induce precautionary saving.Our second main result is to show that prudence of the value function is (strictly) increased by (strict) 'counterclockwise concavification' of the consumption function, even when the utility function itself not exhibit prudence (e.g., in the case of quadratic utility).
Combining these results, we show that when a liquidity constraint is added to a 1 standard consumption problem, the resulting value function therefore exhibits increased 'prudence' (a greater precautionary motive)4 around the level of wealth where the constraint becomes binding. 5Constraints induce precaution because constrained agents have less flexibility in responding to shocks when the effects of the shocks cannot be spread out over time.The precautionary motive is therefore heightened by the desire (in the face of risk) to make future constraints less likely to bind. 6hen a risk is introduced into a model with constraints, the backwards propagation of concavity implies that precautionary saving is induced at any level of wealth from which there is a possibility that future constraints might bind; 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 first constraint strictly increases the precautionary saving motive in prior periods (at least at some levels of wealth), we show that the introduction of a further future constraint may actually reduce the precautionary saving motive (at some levels of wealth) by 'hiding' the effects ofthe pre-existing constraint(s).The existing constraint may be rendered irrelevant at levels of weath where the new constraint forces more saving than the precautionary motive 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 (the consumer may no longer want to borrow).Thus, it is not correct to say that the introduction of a new constraint, or of a new risk, increases the precautionary motive at all levels of wealth in all prior periods.
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.
Our most general result is to show that the precautionary saving motive is stronger at every level of wealth 7 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 in the presence of all future risks and constraints, 8 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.9 The link between constraints and consumption concavity is closely related to Carroll and Kimball (1996)'s demonstration that, within the HARA utility class, the introduction of uncertainty causes the consumption function to become strictly concave (in the absence of constraints) for all but a few knife-edge combinations of utility function and the structure of risk.Taken together, this paper and Carroll and Kimball (1996) establish rigorously the sense in which precautionary saving and liquidity constraints are substitutes: Their similar effects spring from the fact that introduction of either a constraint or a risk will 'concavify' the consumption function. 10he rest of the paper is structured as follows.To fix notation and ideas, the next section sets out our general theoretical framework.Section 3 then defines what we mean by consumption concavity and shows how consumption concavity 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.

The Setup
In this section, we present the consumer framework underlying the 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) m t , the consumer's problem can be written as where in some (but not all) of our results we consider utility functions of the HARA class with > max{−ℵc, 0}.Note that that (1) nests the case with quadratic utility (ℵ = −1).
As usual, the recursive nature of the problem makes this equivalent to the Bellman equation We define as the end-of-period value function.We can then rewrite the problem as11 Throughout, what we call 'the consumption function' is simply the mapping from wealth m t to consumption.

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.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 will allow us in the next two sections to establish how liquidity constraints affect precautionary saving, via consumption concavity and prudence.

Consumption Concavity
After defining consumption concavity ('property CC'), we show that such concavity propagates recursively; that is, once established in some period t, it also holds in periods before t under fairly general conditions.Lastly, we define what we mean by greater consumption concavity; this allows us to compare consumption functions.
We start by defining what we man when we say that the value function has property local consumption concavity (local CC).

Definition 1. (Local Consumption Concavity.)
In relation to a utility function u(c) with non-negative (u ≥ 0) and non-increasing prudence, a function V (m) has property CC (alternately, strict CC) over the interval between m 1 and m 2 , where m 2 > m 1 , if for some increasing function c(m) that satisfies concavity (alternately, strict concavity) over the interval from m 1 to m 2 .
Since (even with constraints) V (m) = u (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).12Note 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 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.
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 u > 0, u < 0, u ≥ 0 and non-increasing absolute prudence (−u /u ).Assume that no liquidity constraint applies at the end of period t and that the agent faces income risk y t+1 ∈ [y, ȳ].If V t+1 (m t+1 ) exhibits property (local) CC for all m t+1 ∈ [Ra t +y, Ra t + ȳ], then V t (m t ) exhibits property (local) CC at the level of wealth m t such that optimal consumption yields a t = m t −c t (m t ).
If also V t+1 (m t+1 ) exhibits property strict (local) CC for at least one m t+1 ∈ [Ra t + y, Ra t + ȳ], then V t (m t ) exhibits property strict (local) CC at the level of wealth m t where optimal consumption yields a t = m t − c t (m t ).
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 y t+1 , then it is also concave today.Additionally, if the the future consumption function is strictly concave for at least one realization of y t+1 , then the consumption function is strictly concave also today.
The last concept we define is when one function exhibits 'greater' concavity than another.This will allow us to compare two consumption functions and their respective concavity later.Definition 3. (Greater Consumption Concavity.)Consider two functions V (m) and V (m) that both exhibit property CC with respect to the same u(c) at a point m for some interval (m 1 , m 2 ) such that m 1 < m < m 2 .Then V (m) exhibits property 'greater CC' compared to V (m) if for all m ∈ (m 1 , m 2 ), and property 'strictly' greater CC if (2) holds as a strict inequality.
If c and ĉ exist everywhere between m 1 and m 2 , greater concavity of ĉ is equivalent to ĉ being weakly larger in absolute value than c everywhere in the range from m 1 to m 2 .The strict version of the proposition would require the inequality to hold strictly over some interval between m 1 and m 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 wealth gets lower.We call this a 'counterclockwise concavification' to capture the sense that at any specific level of wealth, one can think of the increase in the MPC at lower levels of wealth as being a counterclockwise rotation of the lower portion of the consumption function around that level of wealth.
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.3. lim υ↑m ĉ (υ) ≤ lim υ↑m c (υ) 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 wealth levels 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 ∞.
Lemma 3. (Income Risk Causes Counterclockwise Concavification.)Consider an agent who has a utility function of the HARA class (1) with u > 0, u < 0, u > 0, and decreasing absolute prudence (−u /u ).Then the consumption function in the presence of a current income risk c(m) is a counterclockwise concavification of the consumption function in the presence of no risk c(m) around ∞.
Proof.Kimball (1990) shows that positive absolute prudence − u u ensures that c(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 c (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.c(m) is therefore a counterclockwise concavification of c(m) around ∞.
Figure 1 illustrates two examples of counterclockwise concavifications: the introduction of a constraint and the introduction of a risk.In both cases, we start from the situation with no risk or constraints (solid line).The introduction of a constraint is a counterclockwise concavification around a kink point m # .Below m # , consumption is lower and the MPC is greater.The introduction of a risk also generates a counterclockwise concavification of the original consumption function, but this time around ∞.For all m < ∞, consumption is lower, the MPC is higher, and the consumption function is strictly more concave.

A Counterclockwise Concavification Increases Prudence
The section above defined a counterclockwise concavification, which describes the effects of either a constraint or a risk on current consumption.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.
Our first result relates to the effects of a counterclockwise concavification on the absolute prudence of the value function, − V (m) V (m) .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

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 line 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 ∞.
by the envelope condition.The results we derive in Lemma 4 then follow easily.Lemma 4 additionally handles cases where the consumption function is not necessarily twice differentiable.
Lemma 4. (A Counterclockwise Concavification Increases Prudence.)Consider an agent who has a utility function with u > 0, u < 0, u ≥ 0, and nonincreasing absolute prudence (−u /u ).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.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 the absolute prudence of the utility function is non-increasing, then the reduction in consumption (for some states) from the counterclockwise concavification makes agents more prudent (at those states).And third, the higher marginal propensity to consume (MPC) from the counterclockwise concavification means that any given variation in wealth 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. 13emma 4 only provides conditions for when the value function exhibits (weakly) greater prudence (that is, there is no point at which prudence is lower), 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 many values of m and strictly greater prudence only for some values of m.In Lemma 5, we provide conditions for when the value function has strictly greater prudence.See Appendix D for the proof.For prudent agents (u > 0), the value function exhibits strictly greater prudence for all levels of wealth 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 the absolute prudence of the utility 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, i.e.where ĉ (m) c (m) strictly declines at m.

Liquidity Constraints and Consumption Concavity
This section shows under which conditions liquidity constraints cause consumption concavity.The main conceptual difficulty in thinking about liquidity constraints is that the effects of introducing a new constraint depend on what constraints already exist.A constraint in some future period t + n introduces a kink in c t (m) at the m t level where the (perfect foresight) consumer knows that they will arive in period t + n with a value of m t+1 such that the constraint is on the cusp between binding and not binding.The introduction of a new constraint at the end of some other future period t + k may move or remove already existing kinks in the period-t function, potentially eliminating local consumption concavity.To get around this issue, we introduce the concept of an ordered set of relevant constraints.This allows us to add constraints recursively so that the additional constraint does not affect existing kinks.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.Corollary 1 then shows that since liquidity constraints cause consumption concavity, they also heighten prudence of the value function.

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 since we show in Theorem 1 that our results also hold with general constraints.We first define the concept of a kink point.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 in which it applies ('the constraint binds if wealth is less than ...') 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, ω t,n as the level of wealth at which constraint n stops binding or impinging on time t consumption.
A kink point corresponds to a transition from a level of wealth where a current constraint binds or a future constraint impinges, to a level of wealth 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 periods prior to t (and t itself).The set of constraints from period t to T summarizes all relevant information in period t.Further, 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 c t,n as the optimal consumption function in period t assuming that the first n constraints in T (in this chronologically backwards order) have been imposed.For example, c t,0 (m) is the consumption function in period t when no constraint (aside from the intertemporal budget constraint) has been imposed, c t,1 (m) is the consumption function in period t after the chronologically last constraint has been imposed, and so on.We define Ω t,n , V t,n , and other functions correspondingly.

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 wealth m t and earns constant income y in each period.Wealth accumulates according to m t+1 = Ra t + y.We are interested in 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 u > 0 and u < 0, faces constant income y, and is impatient (βR < 1).Assume that the agent faces a set T of N relevant constraints.Then c t,n+1 (m) is a counterclockwise concavification of c t,n (m) around ω t,n+1 .
See Appendix E for the proof.Lemma 6 shows that when we have an ordered set of constraints, T , the introduction of the next constraint in the set generates a counterclockwise concavification of the consumption function.Note that constraint n + 1 is always at a date prior to the set of the first n constraints.

Additional Constraints
In the previous section, we analyzed the case where there was 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 τ to the set of dates at which the consumer is constrained.
Call the new set of dates T 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 τ .Then note that that ĉτ,m = c τ,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 τ causes a counterclockwise concavification of the consumption function around a new kink point, ω τ,m+1 .That is, ĉτ,m+1 is a counterclockwise concavification of ĉτ,m = c τ,m .
The most interesting observation, however, is that behavior under constraints T in periods strictly before τ 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 τ will not generally correspond to kink points once the extra constraint has been added.
We present an example in Figure 2. The original T contains only a single constraint, at the end of period t + 1, inducing a kink point at ω t,1 in the consumption rule c t,1 .The expanded set of constraints, T , adds one constraint at period t + 2. T induces two kink points in the updated consumption rule ĉt,2 , at ωt,1 and ωt,2 .It is true that imposition of the new constraint causes consumption to be lower than before at every level of wealth below ωt,1 .However, this does not imply higher prudence of the value function at every m < ωt,1 .In particular, note that the original consumption function is strictly concave at m = ω t,1 , while the new consumption function is linear at ω t,1 , so prudence can be greater before than after imposition of the new constraint at this particular level of wealth.
The intuition is simple: At levels of initial wealth below ωt,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 ωt,1 , including ω t,1 .But the reason ω t,1 was a kink point in the initial situation was that it was the level of wealth where consumption would have been equal to wealth in period t + 1.Now, because of the extra savings induced by the constraint in t + 2, the larger savings induced by wealth ω t,1 implies that the period ĉt,2 Figure 2 How a future constraint can move a current kink Notes: c t,1 is the original consumption function with one constraint that induces a kink point at ω t,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: ωt,1 and ωt,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 c t,1 .
t + 1 constraint will no longer bind for a consumer who begins period t with wealth ω t,1 .
In other words, at wealth ω t,1 the extra savings induced by the new constraint moves the original constraint and prevents it from being relevant any more at the original ω t,1 .Notice, however, that all constraints that existed in T will remain relevant at some level of wealth under T even after the new constraint is imposed -they just induce kink points at different levels of wealth than before, e.g. the first constraint causes a kink at ωt,2 rather than at ω t,1 .

A More General Analysis
We now want to allow time variation in the level of income y t and in the location of the liquidity constraint (e.g. a constraint in period t might require the consumer to end period t with savings a t greater than ς).We also drop the restriction that βR < 1, allowing the consumer to desire consumption growth over time.
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 CRRA utility consumer with 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 − 1 < T − 1.The simplest case, analyzed before, was a constraint that requires the minimum level of end-of-period wealth to be a τ −1 ≥ 0.Here we generalize this to a τ −1 ≥ ς τ −1,1 where in principle we can allow borrowing by choosing ς to be a negative number.Now for constraint 1 calculate the kink points for prior periods from ) In addition, for constraint 2 recursively calculate 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).14Now 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 a τ ≥ ς τ,n+1 will be irrelevant for all m if max i [ς τ,i ] ≥ ς τ,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 ς τ,n+1 ≥ ς τ,i for any i ≤ n and for all m. 15 The fix is again simple: Counting down from i = n, find the smallest value of i for which ς τ,n+1 ≥ ς τ,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 thus establishes the following result: Theorem 1. (Liquidity Constraints Cause Counterclockwise Concavification.)Consider an agent in period t who has a utility function with u > 0, u < 0, u ≥ 0, and non-increasing absolute prudence (−u /u ).Assume that the agent faces a set T of N relevant constraints.Then c t,n+1 (m) is a counterclockwise concavification of c t,n (m) around ω t,n+1 .
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 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, c t,n+1 (m) is not necessarily strictly more concave than c t,n (m) for all m.This is where the concept of counterclockwise concavification is useful.Even though c t,n+1 (m) is not strictly more concave than c t,n (m) everywhere, it is a counterclockwise concavification and we can apply Lemma 4 and 5 to derive the consequences of imposing one more constraint on prudence.
Corollary 1. (Liquidity Constraints Increase Prudence.)Consider an agent in period t who has a utility function with u > 0, u < 0, u ≥ 0, and non-increasing absolute prudence (−u /u ).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 utility function satisfies u > 0 and m t < ω t,n+1 or if u = 0 and c t,n+1 c t,n strictly declines at m. Proof.By Theorem 1, the imposition of constraint n + 1 constitutes a counterclockwise concavification of c t,n (m).By Lemma 4 and 5, such a concavification (strictly) increases absolute prudence of the value function.
Corollary 1 states that the introduction of the next liquidity constraint increases absolute prudence of the value function.In the subsequent discussions, we consider cases where we relax the assumptions underlying Corollary 1.We first consider the case where we add an extra constraint to the set of constraints.Next, we consider the cases with time-varying deterministic income, general constraints, and no assumption on time discounting.
Finally, consider adding a new constraint to the problem and call the new set of constraints T .Suppose the new constraint applies in period τ .Then the analysis of the new situation will be like the analysis of an added constraint in the simpler case in section 4.3 if the new constraint is relevant given the constraints that apply after period τ and the new constraint does not render any of those later constraints irrelevant.If the new constraint fails either of these tests, the analysis of T can proceed from the ground up as described above.

Liquidity Constraints and Precautionary Saving
We have now established the relationship between liquidity constraints, consumption concavity, and prudence.This section takes the last step and derives the relationship between liquidity constraints and precautionary saving.First, we explain how one may go from prudence of the value function to 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.Unfortunately, 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.However, we still derive our most general results on liquidity constraints and precautionary saving in Theorem 3: the introduction of a risk results in more precautionary saving in the presence of all future risks and constraints than the case with no future risks and constraints.

Notation
We begin by defining two marginal value functions V (m) and V (m) which are convex, downward sloping, and continuous in wealth, m.We consider a risk ζ with support [ζ, ζ], and follow Kimball (1990) by defining the Compensating Precautionary Premia (CPP) as the values κ and κ such that 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 V respectively at m,16 and let κ and κ 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: ] and a given m.

κ >
] and the same given m.
Lemma 7 thus establishes that exhibiting greater prudence is equivalent to inducing a greater precautionary premium.For our purpose, it means that our results above on the absolute prudence also imply that the precautionary premium is higher, i.e. that a more prudent consumer would require a higher compensation to be indifferent about accepting the risk.17 We now take up the question of how the introduction of a risk ζ t+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 ζ t+1 will be some value ζ with support [ζ, ζ], and signify a decision rule that takes account of the presence of the immediate risk by a ∼.Thus, the perfect foresight unconstrained consumption function is c t,0 (m), the perfect foresight consumption function in the presence of the constraint is c t,1 (m), the consumption function with no constraint but with the risk is ct,0 (m) and the consumption function with both the risk and the constraint is ct,1 (m).(Corresponding notation applies to the other functions below).We now define the level of wealth such that liquidity constraint n + 1 never binds for a consumer facing the risk whose wealth is higher than that limit: Definition 7. (Wealth Limit.)ωt,n+1 is the level of wealth such that an agent who faces risk ζ t+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: How to read this limit: ω t+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. ω t+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 ω t+1,n+1 − (y + ζ) is inside the realm of feasible values of a t , in the sense of values that permit the consumer to guarantee that future levels of consumption will be within the permissible range (e.g.positive for consumers with CRRA utility).If this is not true for some level of wealth, then any constraint that binds at or below that level of wealth is irrelevant, because the restriction on wealth imposed by the risk is more stringent than the restriction imposed by the constraint.

Precautionary Saving with Liquidity Constraints
We are now in the 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 (before any choices are made in that period).
Theorem 2. (Liquidity Constraints Increase Precautionary Saving.)Consider an agent who has a utility function with u > 0, u < 0, u > 0, and nonincreasing absolute prudence (−u /u ), and who faces the risk, ζ t+1 .Assume that the agent faces a set T of N relevant constraints and n ≤ N − 1.Then and the inequality is strict if wealth is less than the level that ensures that the last constraint never binds (m t < ωt,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: We simply define ωs,n+1 as the wealth level at which the agent will arrive in the beginning of period t with wealth ωt,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, c w c t,0 ct,0 Notes: c t,0 is the consumption function with no constraint and no risk, ct,0 is the consumption function with no constraint and a risk that is realized at the beginning of period t + 1, c t,1 is the consumption function with one constraint in period t + 1 and no risk, and ct,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 c t,1 and ct,1 is always greater than the vertical distance between c t,0 and ct,0 for m < ωt,1 .
and the dashed loci reflect behavior of consumers who face the immediate risk.For levels of wealth above ω t,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, ct,1 (m t ) = ct,0 (m t ) for levels of wealth above ωt,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 ωt,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.

A More General Result?
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.Intuition might suggest 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.In this subsection, we explain why we cannot derive more general results.
To describe these results, we need to develop a last bit of notation.We define, c m 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 .Thus, relating our new notation to our previous usage, c 0 t,n = c t,n because 0 risks have been imposed.All other functions are defined correspondingly, e.g.Ω m t,n is the end-ofperiod-t value function assuming the first n constraints and m risks have been imposed.We will continue to use the notation ct,n to designate the effects of imposition of a single immediate risk (realized at the beginning of period t + 1).
Suppose now there are m future risks that will be realized between t and T .One might hope to show that, at any m, the precautionary effect of imposing all risks in the presence of all constraints would be greater than the effect of imposing all risks in the absence of any constraints: Such a hope, however, would be in vain.In fact, we will now show that even the considerably weaker condition, involving only the single risk ζ t+1 and all constraints, m), can fail to hold for some m.

An Immediate Constraint
Consider a situation in which n constraints apply in between t and T .Since c t,n−1 designates the consumption rule that will be optimal prior to imposing the period-t constraint, the consumption rule imposing all constraints will be Now define the level of wealth below which the period t constraint binds for a consumer not facing the risk as ω t,n .For values of wealth m ≥ ω t,n , analysis of the effects of the risk is identical to analysis in the previous subsection where the first n − 1 constraints were imposed.For levels of wealth m < ω t,n , we have c 1 t,n (m) = c t,n (m) = w (for the simple c ≤ w constraint; a corresponding point applies to the more sophisticated form of constraint); that is, for consumers with wealth below ω t,n , the introduction of the risk in period t + 1 has no effect on consumption in t, because for these levels of wealth the constraint at the end of t has the effect of hiding the risk from view (they were constrained before the risk was imposed and remain constrained afterwards).Thus for agents for whom inequality (10) in Theorem 2 holds strictly in the absence of the constraint at t, at levels of wealth below ω t,n , the precautionary effect of the risk is wiped out.

An Earlier Risk
Consider now the question of how the addition of a risk ζ t 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 at hand is then whether we can say that that is, does the introduction of the risk ζ t have a greater precautionary effect on consumption in the presence of the subsequent risk ζ t+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 ζ t 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.

What Can Be Said?
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 at least one strong result.
Theorem 3. (Liquidity Constraints and Risks Increase Precautionary Saving.)Consider an agent who has a utility function with u > 0, u < 0, u > 0, and nonincreasing absolute prudence (−u /u ).Then the introduction of a risk ζ t+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.
at levels of period-t wealth m such that in the absence of the new risk the consumer is not constrained in the current period (c m−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.It seems to us that a fair summary of this theorem is that in most circumstances the presence of future constraints and risks does increase the amount of 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 u > 0, u < 0, u > 0, and nonincreasing absolute prudence (−u /u ).Then the consumption function in the presence of all future risks and constraints c m t,n is a counterclockwise concavification of the consumption function with no risk and no constraints c 0 t,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.

Conclusion
The central message of this paper is that the effects of liquidity constraints and future risks on precautionary saving are very similar because the introduction of either a liquidity constraints or of a risk induce a 'counterclockwise concavification' of the consumption function.No matter how it is caused, such an increase in concavity increases prudence and makes agents save more for precautionary reasons.
In addition, we provide an explanation of the apparently contradictory results that have emerged from simulation studies, which have sometimes seemed to indicate that constraints intensify precautionary saving motives (they are complements), and sometimes have found constraints and precautionary behavior are substitutes.The insight here is that the outcome at any given m depends on whether the introduction of a constraint or risk weakens the effects of any preexisting 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 effects of any existing constraints or risks, it might weaken the precautionary saving motive at the given m.

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 (m t ) ∈ CC if V t+1 (a t + y t+1 ) ∈ CC for all realizations of y t+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(a t ) = E t [V t+1 (a t + y t+1 )] ∈ CC if V t+1 (a t + y t+1 ) ∈ CC for all realizations of y t+1 .Second, we show that property CC is preserved through the value function operator (horizontal aggregation), i.e. that V (m t ) = max s u(c t (m t −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 u ≥ 0 (ℵ ≥ −1) and non-increasing absolute prudence (ℵ / ∈ (−1, 0)).The three cases are Case I (ℵ > 0, CRRA.)We will show that concavity is preserved under vertical aggregation for c −1/ℵ to avoid clutter, but the results hold for all affine transformations, ℵc + b, with ℵ > 0. Concavity of c t+1 (a t + y t+1 ) implies that for all y t+1 ∈ [y, ȳ] if a t = pa 1 + (1 − p)a 2 with p ∈ [0, 1].Since this holds for all y t+1 , we know that 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 = 0) This implies that for ℵ ∈ (0, ∞) (CRRA) for all a t , where the inequality is strict if c t+1 is strictly concave for at least one realization of y t+1 .
Case II (a = 0, CARA).For the exponential case, property CC holds at a t if for some χ t (a t ) which is strictly concave at a t .We set b = 1 to reduce clutter, but results hold for b = 1.Consider first a case where c t+1 is linear over the range of possible values of a t + y t+1 , then which is linear in a t since the second term is a constant.Now consider a value of a t for which c t+1 (a t + y t+1 ) is strictly concave for at least one realization of y t+1 .Global weak concavity of c t+1 tells us that for every y t+1 Meanwhile, the arithmetic-geometric mean inequality states that for positive u and v, where the expression holds with equality only if v is proportional to u. Substituting in u = e −c t+1 (a 1 +y t+1 ) and v = e −c t+1 (a 2 +y t+1 ) , this means that and we can substitute for the LHS from (18), obtaining which holds with equality only when e −c t+1 (a 1 +y t+1 ) /e −c t+1 (a 2 +y t+1 ) is a constant.This will only happen if c t+1 (a 1 + y t+1 ) − c t+1 (a 2 + y t+1 ) is constant, which (given that the MPC is strictly positive everywhere) requires c t+1 (a t + y t+1 ) to be linear for y t+1 ∈ (y, ȳ).Hence, where the inequality is strict for an a t from which c t+1 is strictly concave for some realization of y t+1 .
Case III (a = −1, Quadratic).In the quadratic case, linearity of marginal utility implies that so χ t is simply the weighted sum of a set of concave functions where the weights correspond to the probabilities of the various possible outcomes for y t+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 Ω t (a t ) ∈ CC at point a t , then the first order condition implies that for some monotonically increasing χ t (a t ) that satisfies for any 0 < p < 1, and a 1 < a t < a 2 .
In addition, we know that the first order condition holds with equality such that Ω t (a t ) = u (c t (m t )) = u (χ t (a t )) which implies that a t = χ −1 t (c t ).Using this equation, we get Use the budget constraint to define t is a convex function, and ω(c t ) is the sum of a convex and a linear function, it is also a convex function satisfying so c is concave.Note that the proof of horizontal aggregation works for any utility function with u > 0 and u < 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 u by a constant corresponds to a linear transformation of c.
Strict Consumption Concavity.When V t+1 (m t+1 ) exhibits the property strict CC for at least one m t+1 ∈ [Ra t + y, Ra t + ȳ], we know that χ t (a t ) also exhibits the property strict CC from the proof of vertical aggregation.Then, equation ( 20) holds with strict inequality, and this strict inequality goes through the proof of horizontal aggregation, implying that equation ( 21) holds with strict inequality.Hence, c t (m t ) is strictly concave if c t+1 (a t + y t+1 ) is concave for all realizations of y t+1 and strictly concave for at least one realization of y t+1 .

C Proof of Lemma 4
Proof.By the envelope theorem, we know that Differentiating with respect to m yields 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 ( 22) should be interpreted as applying the left-hand and right-hand derivatives separately.(Reading (22) in this way implies that c (m − ) ≥ c (m + ); therefore V (m − ) ≤ V (m + )).Taking another derivative can run afoul of the possible discontinuity in c (m) that we will show below can arise from liquidity constraints.We therefore consider two cases: (i) c (m) exists and (ii) c (m) does not exist.
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 u (c # τ,n+1 ) = βRu (c τ +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 ) where T [n] − τ denotes the time remaining to the n'th constraint.From the assumption of decreasing marginal utility, we therefore know that . 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 )).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 ω τ,n+1 is strictly greater than the MPC above ω τ,n+1 .Thus, the consumption function c τ,n+1 constitutes a counterclockwise concavification of the consumption function c τ,n around the kink point ω τ,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 u applies to all four consumption rules, the Compensating Precautionary Premia, κ t,n and κ t,n+1 , associated with the introduction of the risk ζ t+1 must satisfy Define the amounts of precautionary saving induced by the risk ζ t+1 at an arbitrary level of wealth m in the two cases as where the mnemonic is that the first two letters of the Greek letter psi stand for precautionary saving.
We can rewrite (29) (resp.( 28)) as which implies that so the difference between precautionary saving for the consumer facing n constraints and the one facing n + 1 constraints at m + κ t,n+1 is If we can show that (32) is a positive number for all feasible levels of m satisfying m < ωt,n+1 , then we have proven Theorem 2. We know that the marginal propensity to consume is always strictly positive and that κ t,n+1 ≥ κ t,n ≥ 019 so to prove that (32) is strictly positive, we need to show one of two sufficient conditions: 1. κ t,n+1 > 0 and c t,n+1 (υ) > c t,n (υ) 2. κ t,n+1 > κ t,n Now, since u > 0, we know that κ t,n > 0 from Jensen's inequality.Hence, κ t,n+1 > 0 since κ t,n+1 ≥ κ t,n .The first integral in (32) is therefore strictly positive as long as c t,n+1 > c t,n , which is true for m < ω t,n+1 by Lemma 6. the risk, lim p↓0 sT−1,0 (m T −1 ) = a T −1,1 (m T −1 ) for every m T −1 .
To see this, start with the Euler equations for the two consumers given wealth m, u (m − a T −1,1 (m)) = u (a T −1,1 (m) + 1) (35) u (m − sT−1,0 (m)) = pu (s T −1,0 (m)) + (1 − p)u (s T −1,0 (m) + 1). (36) 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 u term on the RHS of ( 36) is finite).Thus the limit of ( 36) is (35) 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 u (m) > u (1). (37) 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 (36) is u (1 + δ).We know from concavity of the utility function that u (1 + δ) < u (1) and we know from (37) that u (m) > u (1) > u (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 u (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 i.e.Third, we know that c 1 T −1,1 (m) ≥ c T −1,0 (m) since c 1 T −1,1 (m) < c T −1,0 (m) for m ≤ ω 1 T −1,1 , lim m→∞ c 1 T −1,1 (m) − c T −1,0 (m) = 0, and that c 1 t,1 (m) is concave while c t,0 (m) is linear.Hence, c 1 T −1,1 is a counterclockwise concavification of c T −1,0 around the minimum value of wealth when the constraint will never bind and the new set is We can now add the next constraint.The consumption function now has two kink points, ω 1 T −1,1 and ω 1 T −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 The argument is similar.We still know that (strict) consumption concavity is recursively propagated and that lim m→∞ c 2 T −2,2 (m) − c T −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 Hence, we again know that c 2 T −2,2 (m) ≥ c T −2,0 (m).c 2 T −2,2 (m) is thus a counterclockwise concavification of c T −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 m) < m}.Doing this recursively and defining ωm−1 t,1 as the minimum value of wealth beyond which constraint 1 will never bind, the set of wealth levels at which Theorem 3 holds can be

Definition 4 .
(Counterclockwise Concavification.)Function ĉ(m) is a counterclockwise concavification of c(m) around m # if the following conditions hold:

Lemma 5 .
(A Counterclockwise Concavification Strictly Increases Prudence.)Consider an agent who has a utility function with u > 0, u < 0, u ≥ 0, and nonincreasing absolute prudence (−u /u ).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 satisfies u > 0 and m < m # or the utility function is quadratic (u = 0) and ĉ (m) c (m) strictly declines at m.