A Tractable Model of Buffer Stock Saving

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Abstract
We present a tractable model of the effects of nonfinancial risk on intertemporal choice. Our purpose is to provide a simple framework that can be adopted in fields like representative-agent macroeconomics, corporate finance, or political economy, where most modelers have chosen not to incorporate serious nonfinancial risk because available methods were too complex to yield transparent insights. Our model produces an intuitive analytical formula for target assets, and we show how to analyze transition dynamics using a familiar Ramsey-style phase diagram. Despite its starkness, our model captures most of 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. ²Toche: p@toche.net, Department of Economics and Finance, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, HK.

1 Introduction

The Merton (1969)-Samuelson (1969) model of portfolio choice is the foundation for the vast literature analyzing financial risk, not because it provides insights that are unavailable in any other framework, but because those insights are packaged in a form that is tractable, transparent, and easy to use. These qualities make the Merton-Samuelson model the natural starting point (though often not the finishing point) for analyzing any problem where rate-of-return risk is the only kind of risk worth worrying about.

Unfortunately, nonfinancial risks (such as unemployment risk for a consumer) have proven much more difficult to analyze. Of course, there is a large and sophisticated literature that carefully examines the theoretical effects of realistically calibrated nonfinancial risks. But much of the economic literature, and much graduate-level instruction, dodge 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 calibrating models to match aggregate risks which are orders of magnitude smaller than idiosyncratic risks. These assumptions rob the question of its essence, either by assuming that markets transform nonfinancial risk into financial risk or by making implausible assumptions that yield the conclusion that decisions are largely or entirely unaffected by such risk.²

Often, nonreturn risk is avoided not because economists judge it to be unimportant, but because they have a perception that a fully realistic treatment would entail too much additional complexity. The specialized literature on precautionary saving and heterogeneous-agents macroeconomic models has reinforced that perception by showing just how much effort can be required to properly analyze behavior in the presence of empirically plausible specifications of risk.

This paper offers a compromise. We present a tractable model that captures the key qualitative features of models that incorporate a serious treatment of nonreturn risk. Our model is a natural extension of the benchmark perfect foresight framework, and we show how to analyze the model using a phase diagram that will look familiar to every economist because of its close kinship to the Ramsey model of economic growth universally taught in graduate school.

Our model’s tractability springs from our distillation of all nonreturn risk into a stark and simple possibility: The decisionmaker might experience an uninsurable one-time permanent reduction in the flow of nonfinancial income. When that decisionmaker is an employed household, this can be interpreted as an exogenous and permanent transition into unemployment (or disability, or retirement). A similar risk is faced by a country whose exports are dominated by a commodity whose price might collapse (e.g., oil exporters, if cold fusion had worked). The model could even be interpreted as applying to the behavior of a firm controlled by a risk-neutral manager, so long as the collapse of a line of business could have the effect of reducing the firm’s collateral value and therefore increasing its cost of external finance a la Bernanke, Gertler, and Gilchrist (1996).³

The optimal response to this risk is to aim to accumulate a buffer stock of precautionary assets, as a form of “self-insurance.” The existing literature has employed cumbersome numerical solution and simulation methods to explore the determinants of the target stock of wealth under alternative assumptions. 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 other approach (besides numerical solutions) to analyzing precautionary behavior has been the examination of a generalized approximation to the consumption Euler equation that incorporates nonlinear (higher-order) terms. The influences determining the magnitude of the higher-order terms, especially for a consumer away from the target level of assets, have mostly been treated as an impenetrable mystery. We derive a simple expression that shows how the familiar perfect-foresight consumption Euler equation is modified in an intuitive way by our one-shot risk; whether or not the consumer’s assets are at the target, the effect of the risk on consumption growth is related to the probability of the bad event, its magnitude, the degree of risk aversion, and the consumer’s wealth position. At the target, we are able to obtain an exact analytical expression for the combined value of the higher-order terms.

Our chief ambition is to persuade nonspecialist modelers that incorporating a serious treatment of nonreturn risk is not as hard as they think. (Specialists are already aware of how difficult the problem can be; but they may be surprised at how simple it can be, when stripped down to its essence). The treatment of risk may need to be stylized (as ours is) to preserve tractability, but incorporating a stylized treatment of risk is much better than ignoring it altogether.⁴

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.⁵

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 $R$ . 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, $ξ = 0= ⇒ ξ = 0 ∀ t t t+1$ . Consumers have a CRRA utility function $1-ρ u(∙) = ∙ ∕(1 - ρ)$ , with $ρ > 1$ , and discount future utility geometrically by $β$ per period. The solution to the unemployed consumer’s optimization problem is simply:⁷

is the ‘return patience factor’ (see Carroll (2004) for a detailed discussion).⁹ We will show below that the simplicity of the unemployed consumer’s behavior (in particular, the closed-form consumption function (5)) is what makes the problem of the employed consumer tractable (given our assumption that the employed consumer faces only a single kind of risk).

The $κu$ for the problem without risk is strictly below the MPC for the problem with risk (Carroll and Kimball, 1996). We impose what Carroll (2004) calls the ‘return impatience condition’ (RIC),

For short, we will sometimes say that a consumer is ‘return impatient’ (or, ‘the RIC holds’) if $ÞÞÞR < 1$ or if $þr < 0$ or if $u κ > 0$ , all three conditions being equivalent.¹¹ 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 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’ $ÞÞÞ Γ = (Rβ)1∕ρ∕Γ$ 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 $( ce -cu ) ∇t+1 ≡ t+1cu--t+1- t+1$ , 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 $ω = (ρ - 1)∕2$ .¹² Applying a Taylor approximation to (15) (see appendix A) yields:

Consumption growth depends on the employment outcome because insurance markets are missing (by assumption);¹³ consumption if employed next period $cet+1$ is greater than consumption if unemployed $cut+1$ , so that $∇t+1$ is positive. Recall that $cet+1∕cet$ approaches $ÞÞÞΓ$ as the risk vanishes. Thus equation (16) shows that risk boosts consumption growth for the employed consumer 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.

For any given $me t$ , an increase in risk does not change the PDV of future labor income, so that the human wealth term in the intertemporal budget constraint is not affected by an increase in $℧$ . 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 $ce$ 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 perform a phase-diagram analysis of this model, we must find the $e Δc t+1 = 0$ and $e Δm t+1 = 0$ loci. Consider a consumer who is unemployed in period $t + 1$ . Dividing both sides of (4) by $Wt+1 ℓt+1$ yields $u u e e m t+1 = bt+1 = (m t - ct)R$ , where the shorthand $R ≡ R∕ Γ$ has been used.

We know that $met+1 - cet+1 > 0$ because a consumer facing the risk of perpetual unemployment will never borrow. Since the RIC imposes $κu > 0$ , (18) implies that steady-state consumption is positive only if $Π$ is positive. From the definition of $Π$ above, we need the condition

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

2.2.4 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.¹⁵ 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.¹⁶

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.

2.2.5 The Target Level of $me$

We first characterize the steady state. Consider first $e Δc t+1 = 0$ . Imposing the RIC and the GIC, we substitute $e e ct+1 = ct+1$ into equation (18):

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 income growth 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. (Less human wealth results in lower consumption and therefore higher target wealth). An increase in the rate at which future happiness 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.¹⁷ $,$ ¹⁸

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) regularly 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 $(γ ∕℧ )2$ term that multiplies $ω$ ; this implies, e.g., that a given increase in unemployment risk (say, from 5 percent to 10 percent) can have a much more powerful effect on the target level of wealth for a consumer who is more prudent.

2.2.6 The Phase Diagram

Figure 1 presents the phase diagram of system (23)-(24) under our baseline parameter values.¹⁹ An intuitive interpretation is that the $e Δm t+1 = 0$ locus characterized by (24) shows how much consumption $e ct$ would be required to leave resources $e m t$ unchanged so that $met+1 = met$ .²⁰ Thus, any point below the $Δme = 0 t+1$ line would have consumption below the break-even amount, implying that wealth would rise. Conversely for points above $e Δm t+1 = 0$ . This is the logic behind the horizontal arrows of motion in the diagram: Above $Δmet+1 = 0$ the arrows point leftward, below $Δmet+1 = 0$ the arrows point rightward.

The intuitive interpretation of the $Δce = 0 t+1$ 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 $cet+1∕cut+1$ ratio is, and therefore the smaller consumption growth is. Points below the $Δce = 0 t+1$ locus are associated with negative values of $e Δc t+1$ . This is the logic behind the vertical arrows of motion in the diagram: Above $e Δc t+1 = 0$ the arrows point upward, below $e Δc t+1 = 0$ the arrows point downward.

2.2.7 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.²³ 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).

This important point is clearest as $m$ approaches zero. Consider a counterfactual. Suppose the consumer were to spend all his resources in period $t$ , i.e. $ct = mt$ . In this situation, if the consumer were to become unemployed in the next period, he would then be left with resources $mut+1 = (mt - ct)R = 0$ , which would induce consumption $cut+1 = κumut+1 = 0$ , 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.²⁴

This implication is illustrated in figure 2 by the fact that consumption function always remains below the 45 degree line.

2.2.8 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 $ccce ∕ccce t+1 t$ as a function of $me t$ . Since $℧ ≥ 0$ , the no-risk GIC for this model implies:

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

2.2.9 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 $`r > r$ . The no-risk consumption growth locus shifts up to the higher value $` -1 þr ≈ ρ (`r - ϑ )$ , 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 $m`ˇe$ 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 (2004)).

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 $Δ`ccct+1$ . 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 $`γ$ 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.

2.2.10 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 (33) rises. But, lower wealth raises consumption risk, so that the new target level of $ˇm$ must be lower, and this raises the left-hand side of (33). 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

2.2.11 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 $ϑ$ .²⁶

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 $- ρ-1ϑ$ .

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 as analyzed above is reflected 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 (2004)).

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 $me$ will be where both (23) and (24) hold. To simplify the algebra, define $u ζ ≡ R κ Π$ so that $u Rκ Π = ζΓ$ . Then:

We now show that the RIC and GIC ensure that the denominator of the fraction in (35) 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 (41), 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 (41) 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 (41) with the negative of the RHS of (43), yielding a more compact expression for the target level of resources,

We are now in position to discuss (41), 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,

References

BARSKY, ROBERT B., F. THOMAS JUSTER, MILES S. KIMBALL, AND MATTHEW D. SHAPIRO (1997): “Preference Parameters and Behavioral Heterogeneity: An Experimental Approach in the Health and Retirement Survey,” Quarterly Journal of Economics, CXII(2), 537–580.

BERK, JONATHAN, RICHARD STANTON, AND JOSEF ZECHNER (2009): “Human Capital, Bankruptcy, and Capital Structure,” Manuscript, Hass School of Business, Berkeley.

BERNANKE, BENJAMIN, MARK GERTLER, AND SIMON GILCHRIST (1996): “The Financial Accellerator and the Flight to Quality,” Review of Economics and Statistics, 78, 1–15.

CAGETTI, MARCO (2003): “Wealth Accumulation Over the Life Cycle and Precautionary Savings,” Journal of Business and Economic Statistics, 21(3), 339–353.

CARROLL, CHRISTOPHER D. (2001): “Death to the Log-Linearized Consumption Euler Equation! (And Very Poor Health to the Second-Order Approximation),” Advances in Macroeconomics, 1(1), Article 6, Available at http://econ.jhu.edu/people/ccarroll/death.pdf.

__________ (2004): “Theoretical Foundations of Buffer Stock Saving,” NBER Working Paper No. 10867 (Status: Revise and Resubmit, Review of Economic Studies), Latest version available at http://econ.jhu.edu/people/ccarroll/BufferStockTheory.pdf.

CARROLL, CHRISTOPHER D., AND OLIVIER JEANNE (2009): “A Tractable Model of Precautionary Reserves, Net Foreign Assets, or Sovereign Wealth Funds,” NBER Working Paper Number 15228.

CARROLL, CHRISTOPHER D., AND MILES S. KIMBALL (1996): “On the Concavity of the Consumption Function,” Econometrica, 64(4), 981–992, Available at http://econ.jhu.edu/people/ccarroll/concavity.pdf.

FISHER, IRVING (1930): The Theory of Interest. MacMillan, New York.

FRIEDMAN, MILTON A. (1957): A Theory of the Consumption Function. Princeton University Press.

GOURINCHAS, PIERRE-OLIVIER, AND JONATHAN PARKER (2002): “Consumption Over the Life Cycle,” Econometrica, 70(1), 47–89.

HALL, ROBERT E. (1988): “Intertemporal Substitution in Consumption,” Journal of Political Economy, XCVI, 339–357, Available at http://www.stanford.edu/~rehall/Intertemporal-JPE-April-1988.pdf.

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

MERTON, ROBERT C. (1969): “Lifetime Portfolio Selection under Uncertainty: The Continuous Time Case,” Review of Economics and Statistics, 50, 247–257.

PARKER, JONATHAN A., AND BRUCE PRESTON (2005): “Precautionary Saving and Consumption Fluctuations,” American Economic Review, 95(4), 1119–1143.

SAMUELSON, PAUL A. (1969): “Lifetime Portfolio Selection By Dynamic Stochastic Programming,” Review of Economics and Statistics, 51, 239–46.

TOCHE, PATRICK (2005): “A Tractable Model of Precautionary Saving in Continuous Time,” Economics Letters, 87(2), 267–272, Available at http://ideas.repec.org/a/eee/ecolet/v87y2005i2p267-272.html.