Liquidity Constraints and Precautionary Saving

______________________________________________________________________________________

Abstract
We provide the analytical explanation of strong interactions between precautionary saving and liquidity constraints that are regularly observed in numerical solutions to consumption/saving models. The effects of constraints and of uncertainty spring from the same cause: concavification of the consumption function, which can be induced either by constraints or by uncertainty. Concavification propagates back to consumption functions in prior periods. But, surprisingly, once a linear consumption function has been concavified by the presence of either risks or constraints, the introduction of additional concavifiers in a given period can 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

1 Introduction

Numerical solutions have now supplanted analytical methods for modeling consumption/saving choices, because analytical solutions are not available for realistic descriptions of utility, uncertainty, and constraints.

A large literature in both micro and macroeconomics has demonstrated that numerical models that take constraints and uncertainty seriously can yield quite different conclusions than those obtainable for traditional models. For example, in heterogeneous agent New Keynesian models (e.g. Kaplan, Moll, and Violante, 2018), a major transmission mechanism for monetary policy is the indirect income effect because a substantial share of households have high marginal propensities to consume – a channel that is of minimal importance in perfect foresight unconstrained models. And Guerrieri and Lorenzoni (2017) and Bayer, Lütticke, Pham-Dao, and Tjaden (2019) show that tightened borrowing capacity and heightened income risk may be 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.

But 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. These tools provide a rigorous theoretical foundation that can be used to clarify the reasons for the numerical literature’s apparently contrasting findings.

For example, one of the paper’s main results is a proof that when a liquidity constraint is added to a standard consumption problem, the resulting value function exhibits increased ‘prudence’ (a greater precautionary motive) around the level of wealth where the constraint becomes binding.² 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. We show that the precautionary motive is heightened by the desire (in the face of risk) to make future constraints less likely to bind.³

At a deeper level, we show that the effect of a constraint on prudence is an example of a general theoretical result: Prudence is induced by concavity of the consumption function. Since a constraint creates consumption concavity around the point where the constraint binds,⁴ adding a constraint necessarily boosts prudence around that point.⁵ We show that this concavity-boosts-prudence result holds for any utility function with non-negative third derivative; “prudence” in the utility function as in Kimball (1990) is not necessary, because prudence is created by consumption concavity.

These results connect closely 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 structure of risk. Taken together, this paper and Carroll and Kimball (1996) can be seen as establishing rigorously the sense in which precautionary saving and liquidity constraints are substitutes.⁶ To illustrate this point, we provide an example of a specific kind of uncertainty that (under CRRA utility, in the limit) induces a consumption function that is point-wise identical to the consumption function that would be induced by the addition of a liquidity constraint.

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 the period in which the concavity has been introduced.⁷ Precautionary saving is induced by the possibility that 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).

Our final theoretical contribution is to show that the introduction of further liquidity constraints beyond the first one may actually reduce precautionary saving at some levels of wealth by ‘hiding’ the effects of the pre-existing constraint(s); they are no longer relevant because the liquidity 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. For example, a typical perfect foresight model of retirement consumption for a consumer with Social Security (guaranteed pension) income implies that a legal constraint on borrowing against benefits will cause the consumer to run assets down to zero, and thereafter set consumption equal to income. Now consider adding 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 analysis proceeds in five steps. We present our general theoretical framework in the next section. We then show that consumption concavity increases prudence (Section 3); that concavity, once created, propagates to previous periods (Section 4); that constraints cause consumption concavity (Section 5); and when additional constraints or risks increase the precautionary saving motive (Section 6). The final section concludes.

2 The Setup

Consider a consumer who faces some future risks but is not subject to any current or future 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 $y$ , and gross wealth (inclusive of period-t labor income) $wt$ , the consumer’s problem can be written as

with $b>max{− ac,0}$ . Note that that (1) nests the case with quadratic utility ( $a=−1$ ).

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

as the end-of-period value function where $st = wt − ct$ is the portion of period t resources saved. We can then rewrite the problem as⁸

3 Consumption Concavity and Prudence

Our ultimate goal is to understand the relationship between liquidity constraints and precautionary saving. In this section we describe the relationship between consumption concavity and prudence; Kimball (1990) shows that prudence induces precautionary saving, and below we that consumption concavity is induced by either liquidity constraints or precautionary saving.

Our analysis of consumption concavity and prudence is couched in general terms and therefore applies whether the source of concavity is liquidity constraints or something else (e.g., uncertainty).⁹

3.1 Definitions

Our approach shows that the crucial question is whether the value function exhibits a property we call consumption concavity (CC). So we define property CC first, and then we define a counterclockwise concavification which captures a specific class of transformations of a consumption function that make the modified function globally “more” concave.

Since (even with constraints) $′ ′ V (w) = u (c(w))$ holds by the envelope theorem, $V(w)$ having property CC (alternately, strict CC) is the same as having a concave (alternately, strictly concave) consumption function $c(w)$ .¹⁰ 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 encompass 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 CC at every point, we define it as having property CC globally.

We now show how consumption concavity affects the prudence of the value function. To compare two consumption functions and their respective concavity, we need to define when one function exhibits ‘greater’ concavity than another.

If $′′ c$ and $′′ ˆc$ exist everywhere between $w1$ and $w2$ , property CC is equivalent to $′′ ˆc$ being weakly larger in absolute value than $′′ c$ everywhere in the range from $w1$ to $w2$ . The strict version of the proposition would require the inequality to hold strictly over some interval between $w1$ and $w2$ .

The next concept we introduce is ‘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 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 A 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. 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 $# w$ . Below $# w$ , 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 $w< ∞$ , consumption is lower, the MPC is higher, and the consumption function is strictly more concave.

3.2 Consumption Concavity and Prudence

The section above established all the tools necessary to show the relationship between consumption concavity and prudence. Our method in this section is to compare prudence in a baseline case where the consumption function is $c(w )$ to prudence in a modified situation in which the consumption function $ˆc(w )$ 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.

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 we are about to derive in Theorem 1 then follow easily. Theorem 1 itself handles cases where the consumption function is not necessarily twice differentiable.

See Appendix B 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 (in 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.¹¹

Theorem 1 only provides conditions for when the value function exhibits greater prudence, but not strictly greater prudence. In particular, the value function associated with $ˆc(w )$ will in some cases exhibit equal prudence for many values of $w$ and strictly greater prudence only for some values of $w$ . In Corollary 1, we provide conditions for when the value function exhibits strictly greater prudence.

See Appendix C 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 $ˆc′(w) c′(w)$ strictly declines at $w$ .

4 Recursive Propagation of Consumption Concavity

Section 3 defined conditions under which consumption concavity heightens prudence, by comparing value functions and consumption functions at a specific point in time. In this section, we provide conditions guaranteeing that if the consumption function is concave in period $t+1$ , it will be concave in period $t$ and earlier, whatever the source of that concavity may be.

See Appendix D for the proof. Theorem 2 presents conditions to ensure that the consumption function is concave today if the consumption function is concave in the future. The basic insight 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.

5 Liquidity Constraints and Consumption Concavity

We now move on to the sources of consumption concavity. In our setting, there are two sources of consumption concavity: risk and constraints. The properties of consumption under risk have already been derived in Carroll and Kimball (1996). We therefore restrict our attention to showing how liquidity constraints make the consumption function concave. Once the relationship between liquidity constraints and consumption concavity is established, we use the results on consumption concavity and prudence to show under which conditions liquidity constraints heighten prudence.

5.1 Notation

Throughout this paper, we are working with a finite horizon household 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; we show in Theorem 5 that our results also hold with general constraints.

The timing of a constraint relative to other existing constraints matters for the effects of the constraint. We therefore need to 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 periods prior to $t$ . The set of constraints from period $t$ to $T$ summarizes all relevant information in period $t$ . Further, the effect of imposing one extra constraint 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, $ct,0(w)$ is the consumption function in period $t$ when no constraint (aside from the intertemporal budget constraint) has been imposed, $ct,1(w)$ is the consumption function in period $t$ after the chronologically last constraint has been imposed, and so on. We define $Ω ,V t,n t,n$ , and other functions correspondingly.

To have a distinct terminology for the effects of current-period and future-period constraints, we will restrict the use of the term ‘binds’ 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. We can now define the concept of a kink point.

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.

5.2 Perfect Foresight Consumption with Liquidity Constraint

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 $wt$ and earns constant income $y$ in each period. Wealth accumulates according to $wt+1 = Rst + y$ . 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.

See Appendix E for the proof. Theorem 3 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. From the proof of Theorem 3, we also know the shape of the perfect foresight consumption function with liquidity constraints:

Since the consumption function is piecewise linear, the new consumption function, $ct,n+1(w)$ is not necessarily strictly more concave than $ct,n(w )$ for all $w$ . This is where the concept of counterclockwise concavification is useful. Even though $c (w ) t,n+1$ is not strictly more concave than $ct,n(w )$ everywhere, it is a counterclockwise concavification and we can apply Theorem 1 to derive the consequences of imposing one more constraint on prudence.

Proof.By Theorem 3, the imposition of constraint $n + 1$ constitutes a counterclockwise concavification of $c (w ) t,n$ . By Theorem 1 and Corollary 1, such a concavification strictly increases absolute prudence of the value function for the cases in Corollary 1. □

Theorem 4 is the main result in the current section: 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 Theorem 4. We first consider the case where we add an extra constraint to the set of relevant constraints. Next, we consider the cases with time-varying deterministic income, general constraints, and no assumption on time discounting.

5.3 Increasing the Number of Constraints

In the previous section, we analyzed a 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 $ˆτ$ to the set of dates at which the consumer is constrained.

Call the new set of dates $ˆ𝒯$ with $N + 1$ constraints (one more constraint than before), and call the consumption rules corresponding to the new set of dates $ˆct,1$ through $ˆc t,N+1$ . Now call $m$ the number of constraints in $𝒯$ at dates strictly greater than $ˆτ$ . Then note that that $ˆcˆτ,m = cˆτ,m$ , 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 $𝒯ˆ$ 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.

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, $ˆ 𝒯$ , adds one constraint at period $t + 2$ . $ˆ 𝒯$ induces two kink points in the updated consumption rule $ˆct,2$ , at $ωˆt,1$ 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 $ˆω t,1$ . However, this does not imply higher prudence of the value function at every $w<ˆωt,1$ . In particular, note that the original consumption function is strictly concave at $w=ωt,1$ , while the new consumption function is linear at $ωt,1$ , 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 $ˆω t,1$ , 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 wealth in period $t + 1$ . Now, because of the extra savings induced by the constraint in $t+2$ , the larger savings induced by wealth $ωt,1$ implies that 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 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 $ˆ𝒯$ 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 $ˆωt,1$ rather than at $ωt,1$ .

5.4 A More General Analysis

We now want to allow time variation in the level of income $yt$ 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 $st$ 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 $sτ−1 ≥ 0$ . Here we generalize this to $sτ−1 ≥ ςτ−1,1$ 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

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 $sτ≥ςτ,n+1$ will be irrelevant for all $w$ if $maxi[ςτ,i] ≥ ςτ,n+1$ , i.e. if one of the existing constraints already implies that savings must be greater or equal to value required by the new constraint. If the constraint is irrelevant then the analysis proceeds simply by dropping this constraint and renumbering the constraints in $𝒯$ so that the former constraint $n + 2$ becomes constraint $n + 1$ , $n + 3$ becomes $n + 2$ , and so on.

Now consider the other possible problem: That constraint number $n + 1$ imposed in period $τ$ will render irrelevant some of the constraints that have already been imposed. This too is simple to check: It will be true if the proposed $ς ≥ ς τ,n+1 τ,i$ for any $i ≤ n$ and for all $w$ .¹³ 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 $t$ .

Theorem 5 is a generalization of Theorem 4. 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.

Finally, consider adding a new constraint to the problem and call the new set of constraints $ˆ𝒯$ . Suppose the new constraint applies in period $ˆτ$ . Then the analysis of the new situation will be like the analysis of an added constraint in the simpler case in section 5.3 if the new constraint is relevant given the constraints that apply after period $ˆτ$ 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 $ˆ𝒯$ can proceed from the ground up as described above.

6 Liquidity Constraints and Precautionary Saving

In the three previous sections, we have derived the relationships between liquidity constraints, consumption concavity, and prudence. It is now time to be explicit about the last step: the relationship between liquidity constraints and precautionary saving. We first explain the relationship between the precautionary premium and absolute prudence. We then use this result to show how the introduction of an additional constraint induces agents to increase precautionary saving when they face a current risk. Next, we explain why the result cannot be generalized to an added risk or liquidity constraint in a later time period. We end this section by showing our most general result on liquidity constraints and precautionary saving: The introduction of a risk has a greater precautionary effect in the presence of all future risks and constraints than in the absence of any future risks or constraints.

6.1 Notation

We begin by defining two marginal value functions $V ′(w )$ and $ˆV ′(w )$ which are convex, downward sloping, and continuous in wealth, $w$ . 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

Lemma 2 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.¹⁵

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 $∼$ . Thus, the perfect foresight unconstrained consumption function is $ct,0(w )$ , the perfect foresight consumption function in the presence of the constraint is $ct,1(w)$ , the consumption function with no constraint but with the risk is $˜ct,0(w )$ and the consumption function with both the risk and the constraint is $˜ct,1(w )$ . (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:

We must be careful to check that $ωt+1,n+1 − (y + ζ)$ is inside the realm of feasible values of $st$ , 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.

6.2 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).

See Appendix F for the proof. Theorem 6 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 6 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 $¯ω t,n+1$ .

To illustrate the result in Theorem 6, 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 $ω t,1$ 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, $˜ct,1(wt ) = ˜ct,0(wt )$ for levels of wealth above $ω¯t,1$ beyond which the probability of the future constraint binding is zero. For both constrained and unconstrained consumers, the introduction of the risk reduces the level of consumption (the dashed loci are below their solid counterparts). The significance of Theorem 6 in this context is that for levels of wealth below $ω¯t,1$ , the vertical distance between the solid and the dashed loci is greater for the constrained (thin line) than for the unconstrained (thick line) consumers, because of the interaction between the liquidity constraint and the precautionary motive.

6.3 A More General Result?

The result in Theorem 6 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, $cm t,n$ , 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 $T$ . Thus, relating our new notation to our previous usage, $c0t,n = ct,n$ because 0 risks have been imposed. All other functions are defined correspondingly, e.g. $Ωmt,n$ is the end-of-period- $t$ 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 $w$ , 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, $01 0 1 ct,n(w)−ct,n(w) ≥ ct,0(w ) − ct,0(w ),$ can fail to hold for some $w$ .

6.3.1 An Immediate Constraint

Consider a situation in which $n$ constraints apply in between $t$ and $T$ . Since $ct,n−1$ designates the consumption rule that will be optimal prior to imposing the period- $t$ constraint, the consumption rule imposing all constraints will be

6.3.2 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 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.

6.4 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.

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.

7 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 $w$ 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 $w$ .

References

Ameriks, John, Andrew Caplin, Steven Laufer, and Stijn Van Nieuwerburgh (2011): “The Joy Of Giving Or Assisted Living? Using Strategic Surveys To Separate Public Care Aversion From Bequest Motives,” The Journal of Finance, 66(2), 519–561.

Bayer, Christian, Ralph Lütticke, Lien Pham-Dao, and Volker Tjaden (2019): “Precautionary savings, illiquid assets, and the aggregate consequences of shocks to household income risk,” Econometrica, 87(1), 255–290.

Beckenbach, Edwin F, and Richard Bellman (1983): Inequalities, vol. 30. Springer, reprint edn.

Carroll, Christopher D., and Miles S. Kimball (1996): “On the Concavity of the Consumption Function,” Econometrica, 64(4), 981–992, http://econ.jhu.edu/people/ccarroll/concavity.pdf.

Deaton, Angus S. (1991): “Saving and Liquidity Constraints,” Econometrica, 59, 1221–1248, http://www.jstor.org/stable/2938366.

Druedahl, Jeppe, and Casper Nordal Jørgensen (2018): “Precautionary borrowing and the credit card debt puzzle,” Quantitative Economics, 9(2), 785–823.

Fernandez-Corugedo, Emilio (2000): “Soft Liquidity Constraints and Precautionary Savings,” Manuscript, Bank of England.

Fulford, Scott L (2015): “How important is variability in consumer credit limits?,” Journal of Monetary Economics, 72, 42–63.

Guerrieri, Veronica, and Guido Lorenzoni (2017): “Credit crises, precautionary savings, and the liquidity trap,” The Quarterly Journal of Economics, 132(3), 1427–1467.

Holm, Martin Blomhoff (2018): “Consumption with liquidity constraints: An analytical characterization,” Economics Letters, 167, 40–42.

Jappelli, Tullio (1990): “Who is Credit Constrained in the U.S. Economy?,” Quarterly Journal of Economics, 105(1), 219–34.

Kaplan, Greg, Benjamin Moll, and Giovanni L. Violante (2018): “Monetary Policy According to HANK,” American Economic Review, 108(3), 697–743.

Kennickell, Arthur B., and Annamaria Lusardi (1999): “Assessing the Importance of the Precautionary Saving Motive: Evidence from the 1995 SCF,” Manuscript, Board of Governors of the Federal Reserve System.

Kimball, Miles S. (1990): “Precautionary Saving in the Small and in the Large,” Econometrica, 58, 53–73.

Krueger, Dirk, Kurt Mitman, and Fabrizio Perri (2016): “Macroeconomics and Household Heterogeneity,” Handbook of Macroeconomics, 2, 843–921.

Lee, Jeong-Joon, and Yasuyuki Sawada (2007): “The degree of precautionary saving: A reexamination,” Economics Letters, 96(2), 196–201.

(2010): “Precautionary saving under liquidity constraints: Evidence from rural Pakistan,” Journal of Development Economics, 91(1), 77–86.

Mendelson, Haim, and Yakov Amihud (1982): “Optimal Consumption Policy Under Uncertain Income,” Management Science, 28(6), 683–697.

Nishiyama, Shin-Ichi, and Ryo Kato (2012): “On the Concavity of the Consumption Function with a Quadratic Utility under Liquidity Constraints,” Theoretical Economics Letters, 2(05), 566.

Park, Myung-Ho (2006): “An analytical solution to the inverse consumption function with liquidity constraints,” Economics Letters, 92(3), 389–394.

Pratt, John W. (1964): “Risk Aversion in the Small and in the Large,” Econometrica, 32, 122–136.

Samwick, Andrew A. (1995): “The Limited Offset Between Pension Wealth and Other Private Wealth: Implications of Buffer-Stock Saving,” Manuscript, Department of Economics, Dartmouth College.

Seater, John J (1997): “An optimal control solution to the liquidity constraint problem,” Economics Letters, 54(2), 127–134.

Zeldes, Stephen P. (1984): “Optimal Consumption with Stochastic Income,” Ph.D. thesis, Massachusetts Institute of Technology.

A Proof of Lemma 1

Proof.First, condition 2 and 4 in Definition 4 imply that $ˆc′(w ) > c′(w )$ for $w = w# − 𝜖$ for a small $𝜖> 0$ . Condition 3 then ensures that $lim υ↑w ˆc′(υ) > lim υ↑w c′(υ)$ holds for all $# w≤w − 𝜖$ (equivalently $# w < w$ ). Second, condition 1 and the fact that $′ ′ limυ↑wˆc(υ)>lim υ↑w c (υ)$ for $# w < w$ implies that $limυ↑w ˆc(υ) < lim υ↑w c(υ )$ for $w<w#$ . Third, condition 3 in Definition 4 implies that

′′ ′′ ˆc′(υ ) liυm↑w ˆc (υ) ≤ lυi↑mw c (υ)c′(υ-)

for $w<w#$ . Then

′′ ′′ lυi↑mw ˆc (υ) ≤ liυm↑w c (υ)

since $limυ↑wˆc′(υ ) > lim υ↑w c′(υ)$ for $w < w#$ . Note that the inequality is not strict since $c′′(υ)$ could be 0. □

B Proof of Theorem 1

Proof. By the envelope theorem, we know that

V ′(w ) = u′(c(w ))

Differentiating with respect to $w$ yields

V ′′(w) = u ′′(c(w ))c′(w )

(15)

Since $c(w)$ 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 (15) should be interpreted as applying the left-hand and right-hand derivatives separately. (Reading (15) in this way implies that $c′(w−)≥c′(w+ )$ ; therefore $V ′′(w− ) ≤ V′′(w+ )$ ). Taking another derivative can run afoul of the possible discontinuity in $c′(w)$ that we will show below can arise from liquidity constraints. We therefore consider two cases: (i) $c′′(w )$ exists and (ii) $c′′(w)$ does not exist.

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

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

Absolute prudence of the value function is thus defined as

From the assumption that $ˆc(w )$ is a counterclockwise concavification of $c(w )$ , we know from Lemma 1 that $ˆc(w) ≤ c(w )$ and $ˆc′(w) ≥ c′(w)$ . Furthermore, since $u′′′(c(w)) −u′′(c(w))$ is non-increasing, we know that $u′′′(ˆc(w)) u′′′(c(w)) − u′′(ˆc(w)) ≥ − u′′(c(w))$ . As a result, $−u′′′(ˆc(w))′′ˆc′(w) ≥ − u′′′′′(c(w))c′(w) u(ˆc(w)) u(c(w ))$ .

The second part of the absolute prudence expression, $c′′(w ) − c′(w)-$ , is a measure of the curvature of the consumption function. Since the consumption function is concave, $− c′′(w-) c′(w)$ 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 $f$ .¹⁶ A counterclockwise concavification is an example of such a $g$ . Hence, $′′ ′′ − ˆcˆc(′(ww))-≥ − cc′((ww))-$ . Then

Case II: ( $′′ c(w )$ does not exist.)
Informally, if nonexistence is caused by a constraint binding at $w$ , the effect will be a discrete decline in the marginal propensity to consume at $w$ , which can be thought of as $c′′(w)=−∞$ , implying positive infinite prudence at that point (see (16)). Formally, if $c′′(w )$ does not exist, greater prudence of $ˆV$ than $V$ is given by $Vˆ′′(w) V-′′(w)$ being a decreasing function of $w$ . This is defined as

ˆ ′′ ( ′′ ) ( ′ ) V--(w)-≡ u--(ˆc(w-)) ˆc(w-) V ′′(w) u ′′(c(w )) c′(w )

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

lim ˆc′(w) > lim ˆc′(w). (17 ) w−→w w+→w

As for the first factor, note that nonexistence of $ˆV ′′′(w )$ and/or $ˆc′′(w )$ do not spring from nonexistence of either $u ′′′(c)$ or $limw ↑w ˆc′(w )$ (for our purposes, when the left and right derivatives of $ˆc(w )$ 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 $ˆ′′ VV-′′(w(w-))$ is decreasing we differentiate $(u′′(ˆc(w))) logu′′(c(w))$ (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

Recall from Lemma 1 that $ˆc′(w) ≥ c′(w)$ and $ˆc(w) ≤ c(w )$ so non-increasing absolute prudence of the utility function ensures that $− u′′′′′(ˆc(w)) ≥ − u′′′′′(c(w)) u(ˆc(w )) u(c(w ))$ . Hence the LHS is always greater or equal to the RHS of equation (18). □

C Proof of Corollary 1

Proof.We prove each statement in Corollary 1 separately.

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

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

Note that this condition is sufficient to prove Corollary 1 for the case where $c′′(w )$ does not exist since it then satisfies (18). In the case where $c′′(w)$ does exist, we know that

ˆc′′(w ) c′′(w) − -′----≥ − -′----for w < w# cˆ(w) c (w )

from the proof of Theorem 1. Hence,

and Corollary 1 holds in the case with $u′′′ > 0$ and $w < w#$ .

Case II: ( $u′′′= 0$ .)
The quadratic case requires a different approach. Note first that the conditions in Corollary 1 hold only below the bliss point for quadratic utility. In addition, since $u′′′(⋅) = 0$ , strict inequality between the prudence of $ˆ V$ and the prudence of $V$ hold only at those points where $ˆc(⋅)$ is strictly concave.

Recall from the proof of Theorem 1 that greater prudence of $ˆV(w )$ than $V (w )$ occurs if $ˆV′′(w) V′′(w)$ is decreasing in $w$ . In the quadratic case

Vˆ′′(w-) u′′(ˆc(w-))ˆc′(w-) ˆc′(w) V ′′(w ) = u′′(c(w ))c′(w ) = c′(w)

(19)

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

D Proof of Theorem 2

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 $V (wt) ∈ CC$ if $Vt+1(st + yt+1) ∈ CC$ 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 $Ω(st)= 𝔼t[Vt+1 (st + yt+1)] ∈ CC$ if $Vt+1(st + yt+1) ∈ CC$ for all realizations of $yt+1$ . Second, we show that property CC is preserved through the value function operator (horizontal aggregation), i.e. that $V (wt) = maxs u (ct(wt − s)) + Ω(s) ∈ CC$ 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 $′′′ u ≥ 0$ ( $a ≥ − 1$ ) and non-increasing absolute prudence ( $a∕∈(−1,0)$ ). The three cases are

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

(20)

Case I ( $a>0$ , CRRA.) We will show that concavity is preserved under vertical aggregation for $−1∕a c$ to avoid clutter, but the results hold for all affine transformations, $ac + b$ , with $a>0$ . Concavity of $ct+1(st + yt+1)$ implies that

ct+1(st + yt+1) ≥ pct+1(s1 + yt+1 ) + (1 − p)ct+1(s2 + yt+1)

(21)

for all $yt+1∈[y,y¯]$ if $st = ps1 + (1 − p)s2$ with $p ∈ [0,1]$ . Since this holds for all $yt+1$ , we know that

{[ − 1]} −a { [ − 1]} −a 𝔼tct+1(st+ yt+1) a ≥ 𝔼t {pct+1(s1 + yt+1) + (1 − p)ct+1(s2 + yt+1)} a

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 ] .

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

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

if $u≥0$ and $v ≥ 0$ . Thus

which implies that

(Ω′(st))−a ≥ p (Ω ′(s1))−a + (1 − p)(Ω′(s2))−a

Thus, defining $′ χt(st) = {Ωt(st)} −a$ , we get

for all $st$ , where the inequality is strict if $ct+1$ is strictly concave for at least one realization of $yt+1$ .

Case II ( $a=0$ , CARA). For the exponential case, property CC holds at $st$ if

exp (− χt(st)∕b) = 𝔼t[exp(− ct+1(st + yt+1)∕b)]

for some $χt(st)$ which is strictly concave at $st$ . 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 $st+yt+1$ , then

−ct+1(st+yt+1) χt(st) = − log 𝔼t[e ] ′ = − log 𝔼t[e−(ct+1(st+¯y)+(yt+1− ¯y)ct+1)] ′ = ct+1(st + ¯y) − log 𝔼t[e−(yt+1− ¯y)ct+1] (22 )

which is linear in $st$ since the second term is a constant.

Now consider a value of $st$ for which $ct+1(st + yt+1)$ is strictly concave for at least one realization of $y t+1$ . Global weak concavity of $c t+1$ tells us that for every $y t+1$

−ct+1(st + yt+1 ) ≤ − ((1 − p )ct+1(s1 + yt+1 ) + pct+1(s2 + yt+1)) − ct+1(st+yt+1) −((1−p)ct+1(s1+yt+1)+pct+1(s2+yt+1)) 𝔼t[e ] ≤ 𝔼t[e ]. (23 )

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] ≤ ¯upv¯1− p,

where the expression holds with equality only if $v$ is proportional to $u$ . Substituting in $−ct+1(s1+yt+1) u=e$ and $− ct+1(s2+yt+1) v = e$ , this means that

{ }p { }1− p 𝔼t[e−pct+1(s1+yt+1)− (1−p)ct+1(s2+yt+1)] ≤ 𝔼t[e−ct+1(s1+yt+1)] 𝔼t[e−ct+1(s2+yt+1)]

and we can substitute for the LHS from (23), obtaining

−c (s+y ) { −c (s+y )}p { −c (s+y )}1− p 𝔼t[e t+1 t t+1] ≤ 𝔼t[e t+1 1 t+1] 𝔼t[e t+1 2 t+1 ] log𝔼[e−ct+1(st+yt+1)] ≤ plog𝔼 [e−ct+1(s1+yt+1)] + (1 − p)log 𝔼 [e−ct+1(s2+yt+1)] (24 ) t t t

which holds with equality only when $e−ct+1(s1+yt+1)∕e−ct+1(s2+yt+1)$ is a constant. This will only happen if $ct+1(s1 + yt+1 ) − ct+1(s2 + yt+1)$ is constant, which (given that the MPC is strictly positive everywhere) requires $ct+1(st + yt+1)$ to be linear for $yt+1 ∈ (y, ¯y)$ . Hence,

χt(st) ≥ p χt(s1) + (1 − p)χt(s2).

where the inequality is strict for an $st$ 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(st)) = 𝔼t[u (ct+1(st + yt+1 ))] χt(st) = 𝔼t[ct+1(st + 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 $Ω (s) ∈ CC t t$ at point $s t$ , then the first order condition implies that

Ω ′t(st) = u′(χt(st))

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

for any $0<p < 1$ , and $s1 < st < s2$ .

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

which implies that $−1 χt$ 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

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 $Vt+1(wt+1)$ exhibits the property strict consumption concavity for at least one $wt+1 ∈ [Rst + y, Rst + ¯y]$ , we know that $χt(st)$ also exhibits the property strict consumption concavity from the proof of vertical aggregation. Then, equation (25) holds with strict inequality, and this strict inequality goes through the proof of horizontal aggregation, implying that equation (26) holds with strict inequality. Hence, $ct(wt)$ is strictly concave if $ct+1(st + yt+1)$ is concave for all realizations of $yt+1$ and strictly concave for at least one realization of $yt+1$ . □

E Proof of Theorem 3

We prove Theorem 3 by induction in two steps. First, we show that all results in Theorem 3 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(wt)) = βRu (ct+1(R(wt − ct(wt)) + y)).

Differentiating both sides with respect to $wt$ 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 c′ u′′(c ) + βRu ′′(c )c′ R -t+1 = ----t----------t+1--t+1-- c′t βRu ′′(ct+1)R c′t+1 u′′(ct) ′ --′- = ----′′-------+ ct+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

Furthermore, we know that

′ R-−-1- ct ≥ R

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

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

and it follows that $′ ′ ct < ct+1$ . □

Proof.We now prove Lemma 4 by first showing that the consumption function including the constraint at the end of period $τ$ is a counterclockwise concavification of the unconstrained consumption function in period $τ$ . Next, we show how the constraint further implies that the consumption function including the constraint is a counterclockwise concavification of the unconstrained consumption function in periods prior to $τ$ .

We first define $τ = 𝒯 [1]$ as the time period of the constraint. Note first that consumption is unaffected by the constraint for all periods after $τ$ , i.e. $cτ+k,1 = cτ+k,0$ for any $k > 0$ . For period $τ$ , we can calculate the level of consumption at which the constraint binds by realizing that a consumer for whom the constraint binds will save nothing and therefore arrive in the next period with no wealth. Further, the maximum amount of consumption at which the constraint binds will satisfy the Euler equation (only points where the constraint is strictly binding violate the Euler equation; the point on the cusp does not). Thus, we define $c# τ,1$ as the maximum level of consumption in period $τ$ at which the agent leaves no wealth for the next period, i.e. the constraint stops binding:

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

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

ω = (V ′ )−1(u ′(c# )). (27 ) τ,1 τ,1 τ,1

Below this level of wealth, we have $cτ,1(w ) = w$ so the MPC is one, while above it we have $cτ,1(w)=cτ,0(w )$ 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 $c τ,0$ around $ω τ,1$ .

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# ) = βRu ′(c# ) τ−1,1 τ,1

and we can obtain $ωτ− 1,1$ via the analogue to (27). 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 $w<ωτ−1,1$ . This consumer knows he will be constrained and will spend all of his resources next period, so at $w$ 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 Theorem 3 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

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 $ωτ,n+1$ . Thus, the consumption function $cτ,n+1$ constitutes a counterclockwise concavification of the consumption function $cτ,n$ around the kink point $ωτ,n+1$ .

F Proof of Theorem 6

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 $w$ for the perfect foresight consumers. Because the same marginal utility function $u ′$ applies to all four consumption rules, the Compensating Precautionary Premia, $κt,n$ and $κt,n+1$ , associated with the introduction of the risk $ζt+1$ must satisfy

ct,n(w) = ˜ct,n(w + κt,n) (28 ) ct,n+1(w) = ˜ct,n+1(w + κt,n+1 ). (29 )

Define the amounts of precautionary saving induced by the risk $ζ t+1$ at an arbitrary level of wealth $w$ in the two cases as

ψt,n(w) = ct,n(w) − ˜ct,n(w ) (30 ) ψt,n+1(w) = ct,n+1(w) − ˜ct,n+1(w ) (31 )

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

∫ w ′ ct,n+1(w) = ct,n+1(w + κt,n+1) + ct,n+1(υ)d υ = ˜ct,n+1(w + κt,n+1) w+κt,n+1

which implies that

∫ w+κt,n+1 ψt,n+1(w+κt,n+1) = ct,n+1(w + κt,n+1) − ˜ct,n+1(w + κt,n+1) = c′ (υ)dυ, w t,n+1 ∫ w+κt,n ψt,n(w+κt,n) = ct,n(w + κt,n) − ˜ct,n(w + κt,n) = c′t,n (υ )dυ w

and

∫ w+κt,n+1 ′ ′ ψt,n(w + κt,n+1) = ψt,n(w + κt,n) − (˜ct,n(υ) − ct,n(υ))dυ w+κt,n

so the difference between precautionary saving for the consumer facing $n$ constraints and the one facing $n+1$ constraints at $w + κt,n+1$ is

If we can show that (32) is a positive number for all feasible levels of $w$ satisfying $w<¯ωt,n+1$ , then we have proven Theorem 6. We know that the marginal propensity to consume is always strictly positive and that $κt,n+1≥κt,n≥0$ ¹⁷ so to prove that (32) is strictly positive, we need to show one of two sufficient conditions:

$κt,n+1>0$ and $′ ′ ct,n+1(υ ) > ct,n (υ )$
$κt,n+1>κt,n$

Now, since $u′′′>0$ , we know that $κt,n > 0$ from Jensen’s inequality. Hence, $κt,n+1 > 0$ since $κt,n+1≥κt,n$ . The first integral in (32) is therefore strictly positive as long as $c′t,n+1 > c′t,n$ , which is true for $w < ω t,n+1$ by Theorem 3.

For $w≥ωt,n+1$ , we know that $′ ′ ct,n+1 = ct,n$ so the first integral in (32) is always zero. For the second integral in (32) to be strictly positive, we need to show that $κt,n+1>κt,n$ .

First define the perfect foresight consumption functions as

=st,n+1 c(κ + ζ) = c ( ◜s◞◟◝ +y + κ + ζ) (33 ) t,n t+1,n t,n t,n c(κt,n+1 + ζ) = ct+1,n+1(st,n+1 + y + κt,n+1 + ζ). (34 )

where $st,n=st,n+1$ since $w ≥ ωt,n+1$ . 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 2 tells us that if absolute prudence of $u′(c(κ + ζ)) t,n$ 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 $wt+1≥ωt+1,n+1$ 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 $wt+1 ≥ ¯ωt+1,n+1$ . We therefore know that $κt,n+1 = κt,n$ if $w ≥ ω¯t+1,n+1$ .

For all levels of wealth below this limit ( $w < ¯ω t+1,n+1$ ), 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 (32) is therefore strictly positive if $w < ¯ωt+1,n+1$ and we have proven Theorem 6. □

G Resemblance Between Precautionary Saving and a Liquidity Constraint

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 $c ≥ w T −1 T− 1$ , 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 $˜sT− 1,0,$ the key result is that in the limit as $p ↓ 0$ , behavior of the two consumers becomes the same. That is, defining $˜s (w ) T−1,0$ 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 $w$ ,

Consider first the case where $w$ is large enough that the constraint does not bind for the constrained consumer, $w > 1$ . 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 $w>1$ 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 $w>1$ .

Now consider the case where $w < 1$ so that the first consumer would be constrained. This consumer spends her entire resources $w$ , 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 $w>0$ . Therefore we know that for any $p > 0$ and any $w > 0$ the consumer will save some positive amount. For a fixed $w$ , 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 $′′ ′ u(w)>u(1) > u (1 + δ)$ , 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 $w ≤ 1$ and for $w > 1$ 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 7

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 $cmt,n$ faces $m$ future risks and $n$ future constraints, then the next period consumption function is $m −1 ct+1,n− 1$ (and $m = n$ ). Note that we can transform any problem into this notation by filling in with degenerate risks and non-binding constraints. However, for Theorem 7 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 (see Carroll and Kimball, 1996) and when we introduce constraints in the absence of risk (see Theorem 4). 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

where Theorem 7 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 $m −1 𝒫t,n$ is non-empty, we build it up recursively, starting from period $T$ and adding one constraint or one risk for each time period. The key to the proof is to understand that the introduction of risks or constraints will never fully reverse the effects of all other risks and constraints, even though they sometimes reduce absolute prudence for some levels of wealth because risks and constraints can mask the effects of future risks and constraints. Hence, the new consumption function must still be a counterclockwise concavification of the consumption function with no risks and constraints for some levels of wealth.

Since a counterclockwise concavification increases prudence by Theorem 1, and higher prudence increases precautionary saving by Lemma 2, our required set can be redefined as

where we add the last condition, $m −1 ct,n (w) > w$ to avoid the possibility that some constraint binds such that the agent does not increase precautionary saving. In words: $𝒫mt,−n1$ is the set where the consumption function is a counterclockwise concavification of $ct,0(w )$ 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 $𝒫0T,1$ is then

𝒫0T,1 = ∅

since we know that $cT,1(w )$ is a counterclockwise concavification of $cT,0(w)$ 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 Theorem 2, (strict) consumption concavity is recursively propagated for all values of wealth where there is a positive probability that the constraint can bind, i.e.

[ 1 1 ¯] {wT−1|ωT−1,1 ∈ wT −1 − cT−1,1(wT −1) + yT + ζ,wT −1 − cT−1,1(wT −1) + yT + ζ }

has property strict CC, while it has non-strict property $CC$ for all possible values of $wT− 1$ . Further, we know from Theorem 6 (rearrange equation (10)) that

Third, we know that $1 ′ ′ cT−1,1(w ) ≥ cT−1,0(w)$ since $1 cT−1,1(w ) < cT−1,0(w )$ for $1 w ≤ ωT− 1,1$ , $limw→∞c1T−1,1(w ) − cT−1,0(w ) = 0$ , and that $c1t,1(w)$ is concave while $ct,0(w )$ 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 Theorem 2 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 6 that

Third, $c1(w ) < c (w ) T−1,2 T−1,0$ , $lim c1 (w ) − c (w ) = 0 w→ ∞ T−1,2 T−1,0$ , and we know that if $1 cT−1,2(w)$ is concave while $cT−1,0(w )$ is linear, then $′1 ′ cT−1,2(w) ≥ cT−1,0(w)$ . $1 cT−1,2(w )$ which is a counterclockwise concavification of $cT− 1,0(w)$ 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

1 1 1 𝒫T −1,2 = {wT −1|wT− 1 ≤ ¯ωT−1,1 ∧ cT− 1,2(w) > wT −1}.

It is now time to add the next risk. The argument is similar. We still know that (strict) consumption concavity is recursively propagated and that $limw → ∞ c2T−2,2(w ) − cT−2,0(w ) = 0$ . 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 6 must hold also for the addition of multiple risks so we have

Hence, we again know that $′2 ′ cT−2,2(w ) ≥ cT−2,0(w )$ . $2 cT−2,2(w )$ is thus a counterclockwise concavification of $cT− 2,0(w)$ around the level of wealth at minimum value of wealth when the last constraint will never bind. The new set is therefore

𝒫2={wT −2|wT− 2 − c2 (wT −2) + yT −1 + ζT−1 ∈ 𝒫1 ∧ c2 (w ) < w }. T−2,2 T −2,2 T− 1,2 T −2,2

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 7 holds can be defined as

m−1 m− 1 m −1 𝒫t,n = {wt |wt ≤ ω¯t,1 ∧ ct,n (w) > w }

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 $m −1 m −1 limw → ∞ ct,n (w) − ct,0(w) = − k ≤ 0$ 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 (10) 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. □