Liquidity Constraints and Precautionary Saving

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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

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

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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 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 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

As usual, the recursive nature of the problem makes this equivalent to the Bellman equation

We deﬁne $Ω (a) = 𝔼 [βV (Ra + y )] t t t t+1 t t+1$ as the end-of-period value function and rewrite the problem as⁷

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

with $α2 > max {− α1c,0}$ . 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.

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 $c(m )$ .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 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.

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.

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.

If $c′′$ and $ˆc′′$ exist everywhere between $m1$ and $m2$ , greater concavity of $ˆc$ is equivalent to $ˆc′′$ being weakly larger in absolute value than $c′′$ 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 $m2$ .

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.

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.

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 $∞$ .

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 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 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 $ˆc(m )$ is a counterclockwise concaviﬁcation of the baseline consumption function.

The ﬁrst result relates to the eﬀects of a counterclockwise concaviﬁcation on the absolute prudence of the value function, $V′′′(m ) − V-′′(m-)$ .

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

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 $ˆc(m )$ 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.

See Appendix D for the proof. For prudent consumers ( $u′′′ > 0$ ), 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 (where $′ ˆc′(m) c(m)$ 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 in which it applies and will use the term ‘impinges’ to describe the eﬀect of a future constraint on current 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.

$𝒯$ 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. $Ω ,V t,n t,n$ , 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 $m t$ 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.

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.

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 $ˆτ$ to the set of dates at which the consumer is constrained. Call the new set of dates $𝒯ˆ$ with $N + 1$ constraints (one more constraint than before), and call the consumption rules corresponding to the new set of dates $ˆct,1$ through $ˆct,N+1$ . Now call $m$ the number of constraints in $𝒯$ at dates strictly greater than $ˆτ$ . Then note that that $ˆcτˆ,m = cˆτ,m$ , 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 $ˆτ$ causes a counterclockwise concaviﬁcation of the consumption function around a new kink point, $ωˆτ,m+1$ . That is, $ˆcˆτ,m+1$ is a counterclockwise concaviﬁcation of $ˆcˆτ,m = cˆτ,m$ . The most interesting observation, however, is that behavior under constraints $ˆ𝒯$ in periods strictly before $ˆτ$ 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 $ˆτ$ 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 $𝒯ˆ$ adds one constraint at period $t + 2$ . $𝒯ˆ$ induces two kink points in the updated consumption rule $ˆct,2$ , at $ˆωt,1$ and $ˆωt,2$ . It is true that imposition of the new constraint causes consumption to be lower than before at every level of wealth below $ˆωt,1$ . However, this does not imply higher prudence of the value function at every $m < ˆωt,1$ . 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 $ˆωt,1$ , 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 $ˆωt,1$ , 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 $ˆ 𝒯$ 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 $ˆωt,2$ rather than $ωt,1$ ).

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 $1$ calculate the kink points for prior periods from

where $ςτ−1,1$ 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 $max [ς ] ≥ ς i∈[1,n]-τ,i τ,n+1$ , 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 $ςτ,n+1 ≥ ςτ,i$ 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 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.

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.

5.1 Notation

We begin by deﬁning two marginal value functions $V ′(m )$ and $ˆV ′(m )$ 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 $ˆκ$ 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 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 [ $ζ-$ , $¯ ζ$ ], and signify a decision rule that takes account of the presence of the immediate risk by a $∼$ . Further, deﬁne $¯ωt,n+1$ as the lowest level of market resources required for the liquidity constraint to never bind.

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 $t + 1$ .

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 $¯ωs,n+1$ as the wealth level at which the agent will arrive in the beginning of period $t$ with wealth $ω¯ t,n+1$ .

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, $˜ct,1(mt ) = ˜ct,0(mt )$ 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.

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 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 $˜ct,n$ to designate the eﬀects of imposition of a single immediate risk realized at the beginning of period $t + 1$ .

5.3.1 An Immediate Constraint

Consider a situation in which no constraint applies between $t$ and $T$ illustrated in Figure 4. Since $c0 t,0$ 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 $t$ constraint binds for a consumer not facing the risk as $ω0 . t,1$ For values of $m ≥ ω0 t,1$ , analysis of the eﬀects of the risk is identical to analysis in the previous subsection. For levels of market resources $1 m < ω t,1$ where the constraint binds both in the presence and the absence of the immediate risk, we have $c1t,1(m ) = c0t,n(m ) = m$ . Hence, for consumers with wealth below $ω1t,1$ , 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 when $1 m ≤ ω t,1$ .

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.

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 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 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.

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 $u ′′′ ≥ 0$ ( $α1 ≥ − 1$ ) and non-increasing absolute prudence ( $α ∕∈ (− 1,0) 1$ ). The three cases are

( −1∕α |{ (α1c + b) 1 α1 ∈ (0,∞ ) (CRRA ) u ′(c) = e−c∕b α = 0 (CARA ) |( 1 α1c + b α1 = − 1 (Quadratic)

(9)

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, $α1c + b$ , 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)

(10)

for all $yt+1 ∈ [y, ¯y]$ if $at = pa1 + (1 − p)a2$ with $p ∈ [0,1]$ . Since this holds for all $yt+1$ , we know that

{ [ ]}−a { [ 1]} −α1 𝔼t ct+1(at + yt+1)− 1a ≥ 𝔼t {pct+1 (a1 + yt+1) + (1 − p)ct+1(a2 + yt+1)}−α1

(11)

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 $k < 1 (k ⁄= 0 )$

{ k }1∕k { k}1∕k { k}1∕k 𝔼 [(u + v) ] ≥ 𝔼 [u ] + 𝔼 [v ] .

(12)

This implies that for $α1 ∈ (0,∞ )$ (CRRA)

{ -1 }− α1 { -1 }−α1 { -1 } −α1 𝔼[(u + v)−α1] ≥ 𝔼 [u −α1] + 𝔼[v−α1]

(13)

if $u ≥ 0$ and $v ≥ 0$ . Thus

which implies that

(Ω ′(at))−α1 ≥ p(Ω ′(a1))−α1 + (1 − p)(Ω ′(a2))− α1

(14)

Thus, deﬁning $χ (a ) = {Ω′(a )}−α1 t t t t$ , we get

for all $at$ , where the inequality is strict if $ct+1$ 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 $at$ 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 $b ⁄= 1$ . Consider ﬁrst a case where $ct+1$ is linear over the range of possible values of $at + yt+1$ , then

−ct+1(at+yt+1) χt(at) = − log 𝔼t[e ] −(ct+1(at+¯y)+(yt+1− ¯y)c′t+1) = − log 𝔼t[e ] = c (a + ¯y) − log 𝔼 [e −(yt+1− ¯y)c′t+1] t+1 t t

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 $y t+1$

− ct+1 (at + yt+1) ≤ − ((1 − p)ct+1(a1 + yt+1 ) + pct+1(a2 + yt+1 )) 𝔼 [e− ct+1(at+yt+1)] ≤ 𝔼 [e−((1−p)ct+1(a1+yt+1)+pct+1(a2+yt+1))]. t t

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

[ ] 𝔼t (u∕¯u )p(v∕ ¯v)1− p ≤ 𝔼t[p(u∕¯u) + (1 − p)(v∕¯v)] = 1,

implying that

𝔼t[upv1 −p] ≤ u¯p ¯v1−p,

where the expression holds with equality only if $v$ is proportional to $u$ . Substituting in $−ct+1(a1+yt+1) u = e$ and $−ct+1(a2+yt+1) v = e$ , this means that

−pct+1(a1+yt+1)− (1−p)ct+1(a2+yt+1) { −ct+1(a1+yt+1) }p{ − ct+1(a2+yt+1)}1−p 𝔼t[e ] ≤ 𝔼t[e ] 𝔼t[e ]

and we can substitute for the LHS from (15), obtaining

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

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 strictly positive everywhere) requires $c (a + y ) t+1 t t+1$ to be linear for $yt+1 ∈ (y, ¯y)$ . Hence,

χ (a ) ≥ pχ (a ) + (1 − p )χ (a ). t t t 1 t 2

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

′ ′ u (χt(at)) = 𝔼t[u (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

Ω ′t(at) = u ′(χt(at))

(15)

for some monotonically increasing $χt(at)$ that satisﬁes

χt(pa1 + (1 − p)a2) ≥ pχt(a1) + (1 − p)χt(a2)

(16)

for any $0 < p < 1$ , and $a1 < at < a2$ .

In addition, we know that the ﬁrst order condition holds with equality such that $′ ′ ′ Ω t(at) = u (ct(mt )) = u(χt(at))$ which implies that $−1 at = χt (ct)$ . Using this equation, we get

which implies that $χ −t1$ is a convex function.

Use the budget constraint to deﬁne

Now, since $−1 χt$ 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(p ω(c1) + (1 − p)ω (c2)) ≥ pc1 + (1 − p)c2 c(pm + (1 − p )m ) ≥ pc(m ) + (1 − p )c(m ) 1 2 1 2

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 $Vt+1(mt+1 )$ exhibits the property strict CC for at least one $mt+1 ∈ [Rat + y,Rat + ¯y ] --$ , we know that $χt(at)$ also exhibits the property strict CC from the proof of vertical aggregation. Then, equation (16) holds with strict inequality, and this strict inequality goes through the proof of horizontal aggregation, implying that equation (17) 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 $cˆ′(m ) > c′(m )$ for $m = m# − 𝜖$ for a small $𝜖 > 0$ . Condition 3 then ensures that $lim ˆc′(μ ) > lim c′(μ) μ↑m μ↑m$ holds for all $# m ≤ m − 𝜖$ (equivalently $# m < m$ ). Second, condition 1 and the fact that $lim μ↑m ˆc′(μ) > limμ↑m c′(μ )$ for $m < m#$ implies that $lim μ↑m ˆc(μ ) < lim μ↑m c(μ)$ for $m < m#$ . Third, condition 3 in Deﬁnition 4 implies that

ˆc′(μ) lim ˆc′′(μ ) ≤ lim c′′(μ)---- μ↑m μ↑m c′(μ)

for $m < m#$ . Then

lim ˆc′′(μ) ≤ lim c′′(μ ) μ↑m μ↑m

since $lim μ↑m ˆc′(μ) > limμ↑m c′(μ )$ for $m < m#$ . Note that the inequality is not strict since $c′′(μ)$ could be 0. □

C Proof of Lemma 4

Proof. By the envelope theorem, we know that

V ′(m ) = u′(c(m ))

(17)

Diﬀerentiating with respect to $m$ yields

V ′′(m ) = u ′′(c(m ))c′(m )

(18)

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 (18) should be interpreted as applying the left-hand and right-hand derivatives separately. (Reading (18) 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.

Case I: ( $c′′(m )$ exists.)
In the case where $c′′(m )$ exists, we can take another derivative

V ′′′(m ) = u′′′(c(m ))[c′(m )]2 + u′′(c(m ))c′′(m )

(19)

Absolute prudence of the value function is thus deﬁned as

V′′′(m-) u′′′(c(m-))[c′(m-)]2-+-u′′(c(m-))c′′(m-)- − V ′′(m ) = − u ′′(c(m ))c′(m) ′′′ ′′′ ′′ − V--(m-)= − u-(c(m-))c′(m ) − c-(m-) V ′′(m ) u′′(c(m )) c′(m )

From the assumption that $ˆc(m )$ is a counterclockwise concaviﬁcation of $c(m )$ , we know from Lemma 2 that $ˆc(m ) ≤ c(m )$ and $ˆc′(m ) ≥ c′(m )$ . Furthermore, since $u′′′(c(m)) − u′′(c(m))$ is non-increasing, we know that $u′′′(cˆ(m)) u′′′(c(m)) − u′′(ˆc(m)) ≥ − u′′(c(m))$ . As a result, $− u′′′′(′ˆc(m-))ˆc′(m ) ≥ − u′′′′′(c(m))c′(m ) u (ˆc(m )) u (c(m))$ .

The second part of the absolute prudence expression, $c′′(m ) − c′(m-)$ , is a measure of the curvature of the consumption function. Since the consumption function is concave, $− c′′′(m-) c (m )$ 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 such a $g$ . Hence, $− ˆc′′(m)≥ − c′′(m-) cˆ′(m ) c′(m)$ . Then

Case II: ( $′′ c (m )$ 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 $c′′(m ) = − ∞$ , implying positive inﬁnite prudence at that point (see (20)). Formally, if $′′ c (m )$ does not exist, greater prudence of $ˆ V$ than $V$ is given by $ˆ′′ VV′′((mm))$ being a decreasing function of $m$ . This is deﬁned as

( ) ( ) ˆV-′′(m-)- u-′′(ˆc(m-)) ˆc′(m-) V ′′(m ) ≡ u ′′(c(m )) c′(m )

(21)

The second factor, $ˆc′(m ) c′(m-)$ , is weakly decreasing in $m$ by the property of a counterclockwise concaviﬁcation. At any speciﬁc value of $m$ where $ˆc′′(m )$ does not exist because the left and right hand values of $′ ˆc$ are diﬀerent, we say that $′ ˆc$ is decreasing if

lim cˆ′(m ) > lim ˆc′(m ). m −→m m+→m

(22)

As for the ﬁrst factor, note that nonexistence of $ˆV ′′′(m )$ and/or $ˆc′′(m )$ do not spring from nonexistence of either $u ′′′(c)$ or $limm ↑m ˆc′(m )$ (for our purposes, when the left and right derivatives of $ˆc(m )$ 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 applying to the left derivative). To discover whether $ˆV′′(m) V′′(m)$ is decreasing we diﬀerentiate $( ′′ ) log uu′(′(ˆcc(m(m-))))$ (recall that the log is a monotonically decreasing transformation so the derivative of the log of a function always has the same sign as the derivative of the function):

This will be negative if

′′′ ′′′ u-(ˆc(m-))ˆc′(m ) ≤ u--(c(m-))c′(m ) u′′(ˆc(m )) u′′(c(m)) u′′′(ˆc(m )) u′′′(c(m)) ⇒ − -′′-------ˆc′(m ) ≥ − --′′------c′(m ). u (ˆc(m )) u (c(m ))

Recall from Lemma 2 that $ˆc′(m ) ≥ c′(m )$ and $ˆc(m ) ≤ c(m )$ so non-increasing absolute prudence of the utility function ensures that $− u′′′(ˆc(m))≥ − u′′′(c(m)) u′′(ˆc(m )) u′′(c(m))$ . Hence the LHS is always greater or equal to the RHS of equation (23). □

D Proof of Lemma 5

Proof. We prove each statement in Lemma 5 separately.

Case I: ( $u′′′ > 0$ .)
If $u′′′ > 0$ , a counterclockwise concaviﬁcation around $m#$ implies that $ˆc(m ) < c(m )$ and $′ ′ ˆc (m) > c (m )$ for all $# m < m$ . Then

u′′′(ˆc(m))-′ u′′′(c(m-))-′ # − u′′(ˆc(m )cˆ(m ) > − u′′(c(m ))c(m ) for m < m

(23)

Note that this condition is suﬃcient to prove Lemma 5 for the case where $c′′(m )$ does not exist since it then satisﬁes (23). In the case where $c′′(m )$ does exist, we know that

′′ ′′ − ˆc-(m-)≥ − c-(m)-for m < m# ˆc′(m ) c′(m )

(24)

from the proof of Lemma 4. Hence,

and Lemma 5 holds in the case with $u′′′ > 0$ and $m < m#$ .

Case II: ( $′′′ u = 0$ .)
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 $u′′′(⋅) = 0$ , strict inequality between the prudence of $ˆV$ and the prudence of $V$ hold only at those points where $ˆc(⋅)$ is strictly concave.

Recall from the proof of Lemma 4 that greater prudence of $Vˆ(m )$ than $V (m )$ occurs if $ˆV′′′(′m) V (m)$ is decreasing in $m$ . In the quadratic case

ˆV ′′(m ) u′′(ˆc(m ))ˆc′(m ) ˆc′(m ) --′′----= -′′-------′----= -′---- V (m ) u (c(m ))c (m ) c(m )

(25)

where the second equality follows since $u′′(⋅)$ is constant with quadratic utility. Thus, prudence is strictly greater in the modiﬁed case only if $ˆc′(m) c′(m)$ strictly declines in $m$ . □

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.

Proof. The marginal propensity to consume in period $t$ can be obtained from the MPC in period $t + 1$ from the Euler equation

u ′(ct(mt)) = βRu ′(ct+1(R (mt − ct(mt )) + y)).

Diﬀerentiating both sides with respect to $mt$ and omitting arguments to reduce clutter we obtain

′′ ′ ′′ ′ ′ u (ct)ct = βRu (ct+1)ct+1R (1 − ct) (u ′′(ct) + βRu ′′(ct+1 )c′t+1R )c′t = βRu ′′(ct+1)Rc ′t+1 ′ ′′ ′′ ′ ct+1 = u-(ct)-+-βRu--(ct+1)ct+1R-- c′t βRu ′′(ct+1 )R c′ u′′(ct) -t+′1 = ----′′--------+ c′t+1 ct βRu (ct+1 )R

Since $βR < 1$ ensures that $ct > ct+1$ , we know that

u′′(ct) u′′(ct+1) 1 1 -----′′------- ≥ -----′′------- = ----- > -- βRu (ct+1)R βRu (ct+1)R βRR R

(26)

Furthermore, we know that

c′t ≥ R-−-1- R

(27)

since $R−1- R$ is the MPC for an inﬁnitely-lived agent with $βR = 1$ . Hence,

( ) c′t+1- ----u′′(ct)--- ′ 1- R--−-1 c′ = βRu ′′(ct+1)R + ct > R + R = 1 t

(28)

and it follows that $c′t < c′t+1$ . □

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. $c = c τ+k,1 τ+k,0$ 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 $# cτ,1$ 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:

′ # ′ u (cτ,1) = βRu (cτ+1,0(y )) c# = (u ′)−1 (βRu ′(cτ+1,0(y ))) , τ,1

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

ω = (V ′ )−1(u′(c# )). τ,1 τ,1 τ,1

(29)

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

u′(c#τ−1,1) = βRu ′(c#τ,1)

and we can obtain $ω τ−1,1$ via the analogue to (29). 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 $m < ω τ−1,1$ . This consumer knows he will be constrained and will spend all of his resources next 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

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 $ωτ,n+1$ 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 $ω τ,n+1$ .

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 $κt,n+1$ , associated with the introduction of the risk $ζt+1$ must satisfy

c (m ) = ˜c (m + κ ) t,n t,n t,n ct,n+1(m ) = ˜ct,n+1 (m + κt,n+1).

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 ) − ˜ct,n(m ) ψt,n+1 (m) = ct,n+1 (m ) − ˜ct,n+1(m )

(30)

where the mnemonic is that the ﬁrst two letters of the Greek letter psi stand for precautionary saving.

We can rewrite the last (resp. the ﬁrst) equation of (30) as

∫ m ct,n+1(m ) = ct,n+1(m + κt,n+1) + c′ (μ)dμ = ˜ct,n+1 (m + κt,n+1 )ψt,2(m + κt,2) ≡ m+ κt,n+1 t,n+1

which implies that

∫ m+κt,n+1 ψ (m + κ ) = c (m + κ ) − ˜c (m + κ ) = c′ (μ)dμ, t,n+1 t,n+1 t,n+1 t,n+1 t,n+1 t,n+1 m t,n+1 ∫ m+ κt,n ψt,n (m + κt,n ) = ct,n (m + κt,n ) − ˜ct,n(m + κt,n) = c′t,n (μ )dμ m

and

∫ m+κt,n+1 ψ (m + κ ) = ψ (m + κ ) − (˜c′ (μ) − c′ (μ))dμ t,n t,n+1 t,n t,n m+κt,n t,n t,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

ψ (m + κ ) − ψ (m + κ ) = t,n+1 t,n+1 t,n t,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) ∫ m+κt,n+1 ∫ m+ κt,n ∫ m+ κt,n+1 = c′t,n+1(μ)dμ − c′t,n (μ )dμ + (˜c′t,n(μ ) − c′t,n (μ ))dμ m m m+ κt,n ∫ m+κt,n+1 ∫ m+κt,n+1 = (c′t,n+1(μ) − c′t,n(μ))dμ + ˜c′t,n(μ )dμ m m+κt,n

If we can show that (31) 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 $κt,n+1 ≥ κt,n ≥ 0$ ¹⁵ so to prove that (31) is strictly positive, we need to show one of two suﬃcient conditions:

$κt,n+1 > 0$ and $c′t,n+1(μ) > c′t,n(μ )$
$κ > κ t,n+1 t,n$

Now, since $u′′′ > 0$ , we know that $κ > 0 t,n$ from Jensen’s inequality. Hence, $κt,n+1 > 0$ since $κt,n+1 ≥ κt,n$ . The ﬁrst integral in (31) is therefore strictly positive as long as $c′t,n+1 > c′t,n$ , which is true for $m < ωt,n+1$ by Lemma 6.

For $m ≥ ωt,n+1$ , we know that $c′ = c′ t,n+1 t,n$ so the ﬁrst integral in (31) is always zero. For the second integral in (31) to be strictly positive, we need to show that $κt,n+1 > κt,n$ .

First deﬁne the perfect foresight consumption functions as

=at,n+1 c(κt,n + ζ) = ct+1,n(◜◞s◟t,n◝ +y + κt,n + ζ) c(κt,n+1 + ζ) = ct+1,n+1(st,n+1 + y + κt,n+1 + ζ).

where $at,n = at,n+1$ since $m ≥ ωt,n+1$ . Recall also the deﬁnitions of $κt,n$ and $κt,n+1$ :

′ ′ u (ct,n ) = 𝔼t [u (c(κt,n + ζ))] u′(ct,n+1 ) = 𝔼t [u ′(c(κt,n+1 + ζ))].

Now recall that Lemma 7 tells us that if absolute prudence of $′ u (c(κt,n + ζ ))$ is identical to absolute prudence of $u′(c(κt,n+1 + ζ))$ 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 $ζ ∈ (ζ, ¯ζ) --$ , i.e. that the agent is unconstrained for all realizations of the risk. We deﬁned this limit as $mt+1 ≥ ¯ωt+1,n+1$ . We therefore know that $κt,n+1 = κt,n$ if $m ≥ ¯ωt+1,n+1$ .

For all levels of wealth below this limit ( $m < ω¯t+1,n+1$ ), 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 (31) is therefore strictly positive if $m < ¯ωt+1,n+1$ 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 non-stochastic income of $y = 1$ in period $T$ . This consumer’s saving rule will be

The second consumer is not subject to a liquidity constraint, but faces a stochastic income process,

If we write the consumption rule for the unconstrained consumer facing the risk as $˜s , T− 1,0$ the key result is that in the limit as $p ↓ 0$ , behavior of the two consumers becomes the same. That is, deﬁning $˜sT−1,0(m )$ as the optimal saving rule for the consumer facing the risk,

To see this, start with the Euler equations for the two consumers given wealth $m$ ,

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 the second line of (31) is ﬁnite). Thus the limit of the second equation in (31) is the ﬁrst equation in (31) 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

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 the second line of (31) is $u ′(1 + δ)$ . We know from concavity of the utility function that $u′(1 + δ) < u′(1)$ and we know from (31) 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 period. For example, if $m ct,n$ faces $m$ future risks and $n$ future constraints, then the next period consumption function is $cmt+−11,n−1$ (and $m = n$ ). Note that we can transform any problem into this notation by ﬁlling in with degenerate risks and non-binding constraints. 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

𝒫mt,n−1= {m |cmt,−n1(m ) − cmt,n(m) > ct,0(m ) − c1t,0(m)}

(32)

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.

In order to show that the set $𝒫mt,−n1$ 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

where we add the last condition, $m− 1 ct,n (m ) > w$ to avoid the possibility that some constraint binds such that the agent does not increase precautionary saving. In words: $𝒫m −1 t,n$ 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.

First add the last constraint. The set $𝒫0 T,1$ is then 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 + ¯ζ }

(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 (7)) that

Third, we know that $1 ′ ′ cT− 1,1(m ) ≥ cT−1,0(m)$ since $1 cT−1,1(m ) < cT− 1,0(m )$ for $1 m ≤ ω T−1,1$ , $1 limm → ∞ cT−1,1(m ) − cT−1,0(m ) = 0$ , and that $1 ct,1(m )$ 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

We can now add the next constraint. The consumption function now has two kink points, $ω1 T− 1,1$ and $ω1 T−1,2$ . 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

Third, $c1T −1,2(m ) < cT−1,0(m )$ , $limm →∞ c1T −1,2(m ) − cT −1,0(m ) = 0$ , and we know that if $c1 (m ) T −1,2$ is concave while $c (m ) T −1,0$ is linear, then $′1 ′ cT− 1,2(m ) ≥ cT−1,0(m )$ . $1 cT−1,2(m )$ 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, $¯ω1T− 1,1$ , and the new set is

𝒫1T−1,2 = {mT −1|mT −1 ≤ ¯ω1T− 1,1 ∧ c1T− 1,2(m ) > mT −1}.

(34)

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 $2 limm → ∞ cT−2,2(m ) − cT−2,0(m ) = 0$ . 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

Hence, we again know that $′2 ′ cT−2,2(m ) ≥ cT−2,0(m )$ . $2 cT−2,2(m)$ 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

𝒫2T −2,2 = {mT − 2|mT − 2 − c2T− 2,2(mT −2) + yT−1 + ζT−1 ∈ 𝒫1T −1,2 ∧ c2T−2,2(m ) < m }.

(35)

Doing this recursively and deﬁning $m−1 ¯ω t,1$ 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 deﬁned as

𝒫mt,−n 1= {mt|mt ≤ ¯ωmt,1−1∧ cmt,n−1(m ) > m }

(36)

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 problem with CARA utility is that $lim cm−1(m ) − c (m ) = − km −1 ≤ 0 m→ ∞ t,n t,0$ where $m− 1 k$ is some positive constant. We can therefore not use the same arguments as in the preceding proof. However, by realizing that equation (7) in the CARA case can be deﬁned as

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

which implies that the arguments in the preceding section goes through also for CARA utility with this slight modiﬁcation. □

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