A Tractable Model of Buffer Stock Saving

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Abstract

In the spirit of Merton (1969) and Samuelson (1969), we present an analytically tractable model of the effects of nonfinancial risk on intertemporal choice. Our simple framework can be adopted in contexts where modelers have, until now, chosen not to incorporate serious nonfinancial risk because available methods did not readily yield transparent insights. Our model produces an intuitive formula for target assets, and we show how to analyze transition dynamics using a familiar Ramsey-style phase diagram. Despite its starkness, we argue that our model captures the key implications of nonfinancial risk for intertemporal choice.

Keywords: risk, uncertainty, precautionary saving, buffer stock saving
JEL codes: C61, D11, E24

¹Carroll: ccarroll@jhu.edu, Department of Economics, Johns Hopkins University, Baltimore Maryland 21218, USA; and National Bureau of Economic Research. http://econ.jhu.edu/people/ccarroll ²Toche: ptoche@cityu.edu.hk, Department of Economics and Finance, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, HK.

1 Introduction

The Merton-Samuelson model of portfolio choice is the foundation for the vast literature analyzing financial risk,² not because it offers conclusions that cannot be obtained from other frameworks,³ but because it is easy to use and its key insights emerge in a way that is natural, transparent, and intuitive — in a word, the Merton-Samuelson model is tractable.

Unfortunately, nonfinancial risk⁴ (which is much more important than financial risk for most households)⁵ has proven more difficult to analyze. Of course, a large and impressive numerical literature has carefully computed the theoretical effects of realistically calibrated nonfinancial risks in a variety of contexts.⁷ But because a formidable investment in human capital is required to learn the computational methods necessary to solve such models, much of the economic literature (and much graduate-level instruction) has dodged the question of how nonfinancial risk influences choices, by assuming perfect insurance markets or perfect foresight or risk neutrality or quadratic utility or Constant Absolute Risk Aversion, or by attempting only to match aggregate risks (which are orders of magnitude smaller than idiosyncratic risks). These approaches rob the question of its essence, either by assuming that markets transform nonfinancial risk into financial risk or by making implausible assumptions in order to reach the implausible conclusion that decisions are largely or entirely unaffected by such risk.⁸

Our paper’s contribution is to offer a tractable model that captures the key qualitative features (and many quantitative features) of realistic models of the optimal response to nonfinancial risk, but without the customary technical difficulties. The model is a natural extension of the no-risk perfect foresight framework. Its solution is characterized by simple, intuitive equations and we show how the model’s results can be analyzed using a phase diagram analysis that will be familiar to every student of the canonical Ramsey growth model taught in graduate school.

The trick that yields tractability is to distill all nonreturn risk into a stark and simple possibility: A one-time uninsurable permanent loss in nonfinancial income. For an individual’s decision problem, this may be directly interpreted as the risk of a permanent transition into unemployment (or disability, or retirement). Our view is that the consumer’s response to this single, tractable risk can capture most of the substantive essence (that is, the qualitative and quantitative implications) of the results obtained in the numerical literature under more realistic but more complex assumptions about income dynamics.

The same framework could be interpreted to apply in other contexts as well. For instance, the risk faced by a country whose exports are dominated by a commodity whose price might collapse permanently (e.g., oil exporters, if cold fusion had worked).⁹

The optimal response to such a risk is to aim to accumulate a buffer stock of precautionary assets, as a form of “self-insurance.” The existing literature has found the numerical value of the target under specific parametric assumptions, but has struggled to present an intuitive picture of the determinants of that target. In contrast, we are able to derive an analytical formula for the target level of wealth, and show transparently how the precautionary motive interacts with the other saving motives that have been well understood since Irving Fisher (1930)’s work: The income, substitution, and human wealth effects.

The literature’s principal alternative to numerical methods for analysis of precautionary behavior has been to attempt to approximate the nonlinear part of the logarithmic consumption Euler equation (the part that drives precautionary saving and the target level of assets). The Euler equation approach, however, has foundered because the higher-level (beyond first-order) components of the Euler equation are endogenous in a way that has proven difficult to master. Thanks to our model’s tractability, we are able to derive a simple expression that shows how the familiar perfect-foresight consumption Euler equation is modified by the presence of a one-shot risk. Specifically, our equation shows that the effect of the risk on consumption growth is related to the probability of the risky event, to its magnitude, to the consumer’s degree of risk aversion, and to the consumer’s wealth. We obtain an exact analytical expression (not a log-linearized one) for the combined value of the higher-order terms at the target. With this expression in hand, the intuition comes into clear focus, and the problems that have bedeviled the literature can be plainly articulated and understood.

Our hope is that tractability of this kind will eventually allow a model like ours to become the standard reference point to which more specialized models can be compared in much of economics, replacing the perfect foresight, certainty equivalent, or perfect markets models that are currently so widely used because of their own tractability. A further ambition is that even the specialist literatures like heterogeneous-agents macroeconomics may find ours a useful ‘toy model’ with which to exposit more clearly some of the subtle points that authors in those literatures find difficult to communicate simply by appealing to results from numerical simulations.¹⁰

2 The Decision Problem

For concreteness, we analyze the problem of an individual consumer facing a labor income risk. Other interpretations (like the ones mentioned in the introduction) are left for future work or other authors. We couch the problem in discrete time, but in most cases we provide the logarithmic approximations that will correspond to the exact solution to the corresponding problem in continuous time.¹¹

The aggregate wage rate, $Wt,$ grows by a constant factor $G$ from the current time period to the next, reflecting exogenous productivity growth:

The interest rate is exogenous and constant (the economy is small and open); the interest factor is denoted $R$ .¹² Define $mmm$ as market resources (financial wealth plus current income), $aaa$ as end-of-period assets after all actions have been accomplished (specifically, after the consumption decision), and $bbb$ as bank balances before receipt of labor income. Individuals are subject to a dynamic budget constraint (DBC) that can be decomposed into the following elements:

2.1 The Unemployed Consumer’s Problem

There is no way out of unemployment; once an individual becomes unemployed, that individual remains unemployed forever, $ξt = 0=⇒ ξt+1 = 0 ∀ t$ . Consumers have a CRRA utility function $u(∙ ) = ∙1-ρ∕ (1 - ρ)$ , with $ρ > 1$ , and they discount future utility geometrically by $β$ per period. We show below that the simplicity of the unemployed consumer’s behavior is what makes employed consumer’s problem tractable. The solution to the unemployed consumer’s optimization problem is simply:¹⁴

2.1.1 Parameter Restrictions

Table 1 summarizes our notation and should serve as a useful guide to the reader. We follow the terminology in Carroll (2011), where a detailed discussion of the concepts is provided. The marginal propensity to consume out of total wealth $κu$ is related to the ‘return patience factor’ $ÞÞÞ R$ as follows¹⁵

The MPC for the problem without risk is strictly below the MPC for the problem with risk (Carroll and Kimball, 1996). We impose a ‘return impatience condition’ (RIC),

For short, we will sometimes say that a consumer is ‘return impatient’ (or, ‘the RIC holds’) if $ÞÞÞ R < 1$ or if $þr < 0$ or if $u κ > 0$ , all three conditions being equivalent.¹⁷ A consumer who is return impatient is someone who will be spending enough to make the ratio of consumption to total wealth decline over time.

The return patience factor can be compared to the ‘absolute patience factor’

2.2 The Employed Consumer’s Problem

The consumer’s preferences are the same in the employment and unemployment states; only exposure to risk differs.

2.2.1 A Human-Wealth-Preserving Spread in Unemployment Risk

A consumer who is employed in the current period has $ξt = 1$ ; if this person is still employed next period ( $ξt+ 1 = 1$ ), market resources will be:

However, there is no guarantee that the consumer will remain employed: Employed consumers face a constant risk $℧$ of becoming unemployed. It is convenient to define $//℧ ≡ 1 - ℧$ , the complementary probability that a consumer does not become unemployed. We assume that $ℓ$ grows by a factor $-1 //℧$ every period,

because under this assumption, for a consumer who remains employed, labor income will grow by factor $Γ = G ∕//℧$ , so that the expected labor income growth factor for employed consumers is the same $G$ as in the no-risk perfect foresight case:

2.2.2 First Order Optimality Condition

The usual steps lead to the standard consumption Euler equation. Using $i ∈ {e,u }$ to stand for the two possible states,

Henceforth nonbold variables will be used to represent the bold equivalent divided by the level of permanent labor income for an employed consumer, e.g. $cet = cccet∕(Wt ℓt)$ ; thus we can rewrite the consumption Euler equation as:

2.2.3 Analysis and Intuition of Consumption Growth Path

It will be useful now to define a ‘growth patience factor’ $1∕ρ ÞÞÞΓ = (R β) ∕Γ$ which is the factor by which the consumption-income ratio $ce$ would grow in the absence of labor income risk. With this notation, (14) can be written as:

To understand (15), it is useful to consider an approximation. Define $( ) cet+1-cut+1 ∇t +1 ≡ cu t+1$ , the proportion by which consumption next period would drop in the event of a transition into unemployment; we refer to this loosely as the size of the ‘consumption risk.’ Define $ω$ , the ‘excess prudence’ factor, as $ω = (ρ - 1)∕2$ .¹⁸ Applying a Taylor approximation to (15) (see appendix A) yields:

The approximations in (16) or (17) capture the essence of equation (15). As a consequence of missing insurance markets, consumption growth depends on the employment outcome;¹⁹ consumption if employed next period $ce t+1$ is greater than consumption if unemployed next period $u ct+1$ , so that $∇t +1$ is positive. In the limit case, as unemployment risk $℧$ vanishes, so does consumption risk $∇$ ,²⁰ and thus $ce ∕cet t+1$ approaches $ÞÞÞΓ$ . Equation (16) thus shows that the presence of unemployment risk boosts consumption growth by an amount proportional to the probability of becoming unemployed $℧$ multiplied by a factor that is increasing in the amount of consumption risk $∇$ . In the logarithmic case, equation (17) shows that the precautionary boost to consumption growth is directly proportional to the size of the consumption risk.

The effect of risk on saving is transparent. For a given value of $met$ , risk has no effect on the PDV of future labor income and human wealth, but the larger is $℧$ , the faster consumption growth must be, as equation (16) shows. For consumption growth to be faster while keeping the PDV constant, the level of current $e c$ must be lower. Thus, the introduction of a risk of unemployment $℧$ induces a (precautionary) increase in saving.

In the (persuasive) case where $ρ > 1$ , (16) implies that a consumer with a higher degree of prudence (larger $ρ$ and therefore larger $ω$ ) will anticipate greater consumption growth. This reflects the greater precautionary saving motive induced by a higher degree of prudence.

To compute the steady state of the model, we must find the $Δce = 0 t+ 1$ and $e Δm t+1 = 0$ loci. Consider a consumer who is unemployed in period $t + 1$ . Dividing both sides of (4) by $W ℓ t+1 t+1$ yields $mu = bu = (me - ce)R t+1 t+1 t t$ (where $R ≡ R ∕Γ$ ). Substituting $e e ct+1 = ct$ and $u u u ct+1 = κ m t+1$ into (15) yields:

We now turn to parameter restrictions necessary to guarantee a positive steady state.

2.2.4 Parameter Restrictions (Continued)

Consider equation (18). We know that $met > cet > 0$ because, with CRRA preferences, zero consumption carries an infinite penalty, implying that a consumer facing the risk of perpetual unemployment will never borrow. Since we have assumed $u κ > 0$ (the RIC), steady-state consumption is positive only if $Π$ is positive; so we impose the condition

In the limit as $℧$ approaches zero, (19) therefore reduces to a requirement that the growth patience factor $ÞÞÞ Γ$ be less than one,

Using $γ ≡ lo gΓ$ , we similarly define the corresponding ‘growth patience rate’

2.2.5 Why Increased Unemployment Risk Increases Growth Impatience

Under the maintained assumption that the RIC holds, the (generalized) GIC in (19) slackens (becomes easier to satisfy) as unemployment risk rises because, with relative risk aversion $ρ > 1$ , an increase in $℧$ reduces the right-hand side of (19). This occurs for two reasons. First, an increase in $℧$ is like a reduction in the future downweighting factor (that is, a decrease in patience), conditional on the consumer remaining employed.²³ Second, an increase in $℧$ slackens the GIC because our mean-preserving-spread assumption requires that labor productivity growth be adjusted so that the value of human wealth is independent of $℧$ – see (12). The higher $℧$ is, the faster growth is conditional on remaining employed. As income growth (conditional on employment) increases, the continuously-employed (lucky) consumer is effectively more ‘impatient’ in the sense of desiring consumption growth below employment-conditional income growth.²⁴

The fact that the GIC is easier to satisfy as $℧$ increases means that if the PF-GIC (20) is satisfied, then (19) must be satisfied.

2.2.6 The Target Level of $me$

We first characterize the steady state. Setting $e Δc t+ 1 = 0$ and $e Δm t+ 1 = 0$ yields, respectively

The steady-state levels of $e m$ and $e c$ are the values for which both (24) and (23) hold. This system of two equations in two unknowns can be solved explicitly (see the appendix). For illustration, consider the special case of logarithmic utility ( $ρ = 1$ ). The appendix shows that an approximation of the target level of market resources is

This expression encapsulates several of the key intuitions of the model. The human wealth effect of labor income growth (conditional upon remaining employed) is captured by the first $γ$ term in the denominator; for any calibration for which the denominator is positive, increasing $γ$ reduces the target level of wealth. This reflects the fact that a consumer who anticipates being richer in the future will choose to save less in the present, and the result of lower saving is smaller wealth. The human wealth effect of interest rates is correspondingly captured by the $- r$ term, which goes in the opposite direction to the effect of income growth, because an increase in the rate at which future labor income is discounted constitutes a reduction in human wealth. An increase in the rate at which future utility is discounted, $ϑ$ , reduces the target wealth level. Finally, a reduction in unemployment risk raises $(γ + ϑ - r)∕℧$ and therefore reduces the target wealth level.²⁵ $,$ ²⁶

Note that the different effects interact with each other, in the sense that the strength of, say, the human wealth effect of interest rates will vary depending on the values of the other parameters.

The assumption of log utility is implausible; empirical estimates from structural estimation exercises (e.g. Gourinchas and Parker (2002), Cagetti (2003), or the subsequent literature) typically find estimates considerably in excess of $ρ = 1$ , and evidence from Barsky, Juster, Kimball, and Shapiro (1997) suggests that values of 5 or higher are not implausible. Another special case helps to illuminate how results change for $ρ > 1$ . The appendix shows that, in the special case where $ϑ = r$ , the target level of wealth is:

In the $ω > 0$ case, the interaction effects between parameter values are particularly intense for the $2 (γ∕℧ )$ term that multiplies $ω$ ; this implies, e.g., that a given increase in unemployment risk can have a greater effect on the target level of wealth for a consumer who is more prudent.

2.2.7 The Phase Diagram

Figure 1 presents the phase diagram of system (23)-(24) under our baseline parameter values. Since the employed consumer never borrows, market resources never fall below the value of current labor income, which is the value selected for the origin of the diagram.²⁷ An intuitive interpretation is that the $Δmet = 0$ locus characterized by (24) shows how much consumption $e ct$ would be required to leave resources $e m t$ unchanged so that $met = met$ .²⁸ Thus, any point below the $e Δm t = 0$ line would have consumption below the break-even amount, implying that wealth would rise. Conversely for points above $e Δm t = 0$ . This is the logic behind the horizontal arrows of motion in the diagram: Above $Δmet = 0$ the arrows point leftward, below $Δmet = 0$ the arrows point rightward.

The intuitive interpretation of the $Δcet = 0$ locus characterized by (23) is more subtle. Recall that expected consumption growth depends on the amount by which consumption would fall if the unemployment state were realized. At a given level of resources, the farther actual consumption (if employed) is below the break-even (sustainable) amount, the smaller the $cet∕cut$ ratio is, and therefore the smaller consumption growth is. Points below the $e Δc t = 0$ locus are associated with negative values of $Δcet$ . This is the logic behind the vertical arrows of motion in the diagram: Above $e Δc t = 0$ the arrows point upward, below $e Δc t = 0$ the arrows point downward.

2.2.8 The Consumption Function

Figure 2 shows the optimal consumption function $c(m )$ for an employed consumer (dropping the $e$ superscript to reduce clutter). This is of course the stable arm of the phase diagram. Also plotted are the 45 degree line along which $ct = mt$ and

The consumption function $c (m )$ is concave: The marginal propensity to consume $κκκ(m ) ≡ dc (m )∕dm$ is higher at low levels of $m$ because the intensity of the precautionary motive increases as resources $m$ decline.³¹ The MPC is higher at lower levels of $m$ because the relaxation in the intensity of the precautionary motive induced by a small increase in $m$ (Kimball, 1990) is relatively larger for a consumer who starts with less than for a consumer who starts with more resources (Carroll and Kimball, 1996).

To see this important point, consider a counterfactual. Suppose the consumer were to spend all his resources in period $t$ , i.e. $c = m t t$ . In this situation, if the consumer were to become unemployed in the next period, he would then be left with resources $mu = (m - c )R = 0 t+1 t t$ , which would induce consumption $u u u ct+1 = κ m t+1 = 0$ , yielding negative infinite utility. A rational, optimizing consumer will always avoid such an eventuality, no matter how small its likelihood may be. Thus the consumer never spends all available resources.³² This implication is illustrated in figure 2 by the fact that consumption function always remains below the 45 degree line.

2.2.9 Expected Consumption Growth Is Downward Sloping in $me$

Figure 3 illustrates some of the key points in a different way. It depicts the growth rate of consumption $cccet+1∕cccet$ as a function of $met$ .

Figure 3 illustrates the result that consumption growth is equal to what it would be in the absence of risk, plus a precautionary term; for algebraic verification, multiply both sides of (15) by $Γ$ to obtain

2.2.10 Summing Up the Intuition

We are finally in position to get an intuitive understanding of how the model works and why a target wealth ratio exists. On the one hand, consumers are growth-impatient: It cannot be optimal for them to let wealth become arbitrarily large in relation to income. On the other hand, consumers have a precautionary motive that intensifies as the level of wealth falls. The two effects work in opposite directions. As resources fall, the precautionary motive becomes stronger, eventually offsetting the impatience motive. The point at which prudence becomes exactly large enough to match impatience defines the target wealth-to-income ratio.

It is instructive to work through a couple of comparative dynamics exercises. In doing so, we assume that all changes to the parameters are exogenous, unexpected, and permanent. Figure 4 depicts the effects of increasing the interest rate to $ˋr > r$ . The no-risk consumption growth locus shifts up to the higher value $-1 þˋr ≈ ρ (ˋr - ϑ )$ , inducing a corresponding increase in the expected consumption growth locus. Since the expected growth rate of labor income remains unchanged, the new target level of resources $mˋˇe$ is higher. Thus, an increase in the interest rate raises the target level of wealth, an intuitive result that carries over to more elaborate models of buffer-stock saving with more realistic assumptions about the income process (Carroll (2011)).

The next exercise is an increase in the risk of unemployment $℧.$ The principal effect we are interested in is the upward shift in the expected consumption growth locus to $Δˋccc t+1$ . If the household starts at the original target level of resources $ˇm$ , the size of the upward shift at that point is captured by the arrow orginating at ${mˇ, γ }$ .

In the absence of other consequences of the rise in $℧$ , the effect on the target level of $m$ would be unambiguously positive. However, recall our adjustment to the growth rate conditional upon employment, (12); this induces the shift in the income growth locus to $ˋγ$ which has an offsetting effect on the target $m$ ratio. Under our benchmark parameter values, the target value of $m$ is higher than before the increase in risk even after accounting for the effect of higher $γ$ , but in principle it is possible for the $γ$ effect to dominate the direct effect. Note, however, that even if the target value of $m$ is lower, it is possible that the saving rate will be higher; this is possible because the faster rate of $γ$ makes a given saving rate translate into a lower ratio of wealth to income. In any case, our view is that most useful calibrations of the model are those for which an increase in uncertainty results in either an increase in the saving rate or an increase in the target ratio of resources to permanent income. This is partly because our intent is to use the model to illustate the general features of precautionary behavior, including the qualitative effects of an increase in the magnitude of transitory shocks, which unambiguously increase both target $m$ and saving rates.

2.2.11 Death to the Log-Linearized Consumption Euler Equation!

Our simple model may help explain why the attempt to estimate preference parameters like the degree of relative risk aversion or the time preference rate using consumption Euler equations has been so signally unsuccessful (Carroll (2001)). On the one hand, as illustrated in figures 3 and 4, the steady state growth rate of consumption, for impatient consumers, is equal to the steady-state growth rate of income,

To illustrate further the workings of the model, consider an increase in the growth rate of income. On the one hand, the right-hand side of (32) rises. But, lower wealth raises consumption risk, so that the new target level of $mˇ$ must be lower, and this raises the left-hand side of (32). In equilibrium, both sides of the expression rise by the same amount.

The fact that consumption growth equals income growth in the steady-state poses major problems for empirical attempts to estimate the Euler equation. To see why, suppose we had a collection of countries indexed by $i$ , identical in all respects except that they have different interest rates $ri$ . In the spirit of Hall (1988), one might be tempted to estimate an equation of the form

2.2.12 Dynamics Following An Increase in Patience

We now consider a final experiment: Figure 6 depicts the effect on consumption of a decrease in the rate of time preference (the change is exogenous, unexpected, permanent), starting from a steady-state position. A decrease in the discount rate (an increase in patience) causes an immediate drop in the level of consumption; successive points in time are reflected in the series of dots in the diagram. The new consumption path (or consumption function) starts from a lower consumption level and has a higher consumption growth than before the decrease in $ϑ$ .³⁴

Consumption eventually approaches the new, higher equilibrium target level. This higher level of consumption is financed, in the long run, by the higher interest income provided by the higher target level of wealth.

Note again, however, that equilibrium steady-state consumption growth is still equal to the growth rate of income (this follows from the fact that there is a steady-state level for the ratio of consumption to income). The higher target level of the wealth-to-income ratio is precisely enough to reduce the precautionary term by an amount that exactly offsets the effect of the rise in $-1 - ρ ϑ$ .

Figures 8 and 9 depict the time paths of consumption, market wealth, and the marginal propensity to consume following the decrease in $ϑ$ . The dots are spread out evenly over time to give a sense of the rate at which the model adjusts toward the steady state.

3 Conclusions

Despite its simplicity, the core logic of the model analyzed above emerges in almost every detail (after much more work) under more realistic assumptions about risk that allow for transitory shocks, permanent shocks, and unemployment in a form that is calibrated to match a large literature exploring the details of the household income process (Carroll (2011)).

We hope that the simplicity of our framework will encourage its use as a building block for analyzing questions that have so far been resistant to a transparent treatment of the role of nonreturn risk. For example, Carroll and Jeanne (2009) construct a fully articulated model of international capital mobility for a small open economy using the model analyzed here as the core element. We can envision a variety of other direct purposes the model could serve, including applications to topical questions such as the effects of risk in a search model of unemployment.

A Taylor Approximation for Consumption Growth

B The Exact Formula for $mˇ$

The steady-state value of $e m$ will be where both (23) and (24) hold. To simplify the algebra, define $ζ ≡ R κuΠ$ so that $R κuΠ = ζ Γ$ . Then:

We now show that the RIC and GIC ensure that the denominator of the fraction in (34) is positive:

However, note that $℧$ also affects $Γ$ ; thus, the first inequality above does not necessarily imply that the denominator is decreasing as $℧$ moves from $0$ to $1$ .

C An Approximation for $ˇm$

The formula also provides insight about how the human wealth effect works in equilibrium. All else equal, the human wealth effect is captured by the $(γ - r)$ term in the denominator of (40), and it is obvious that a larger value of $γ$ will result in a smaller target value for $m$ . But it is also clear that the size of the human wealth effect will depend on the magnitude of the patience and prudence contributions to the denominator, and that those terms can easily dominate the human wealth effect.

For (40) to make sense, we need the denominator of the fraction to be a positive number; defining

The same set of derivations imply that we can replace the denominator in (40) with the negative of the RHS of (42), yielding a more compact expression for the target level of resources,

We are now in position to discuss (40), understanding that the impatience conditions guarantee that its numerator is a positive number.

Two specializations of the formula are particularly useful. The first is the case where $ρ = 1$ (logarithmic utility). In this case,

The other useful case to consider is where $r = ϑ$ but $ρ > 1$ . In this case,

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