Theoretical Foundations of Buffer Stock Saving

Introduction¶

The precautionary motive to save springs from the fact that extra resources improve a consumer’s ability to buffer spending against shocks. A consumer who, in the absence of shocks, would be impatient enough to plan to spend down their resources, will (when shocks are present) experience an intensifying precautionary motive as their buffering capacity shrinks. The result of this competition between impatience and ‘prudence’ (Kimball, 1990) has been described, starting with (Deaton, 1991), as ‘buffer stock saving,’ with a ‘target’ defined^[1] as the point where the precautionary motive to accumulate becomes exactly strong enough to counter the impatience motive to decumulate.

The logic of buffer stock saving underpins key findings in heterogeneous-agent (HA) macroeconomics. For example, it can explain why, during the Great Recession, middle-class consumers cut spending more than the poor or the rich (Krueger, Mitman, and Perri, 2016). Buffer stock saving also can explain why consumption growth tracks income growth over much of the life cycle,^[2] rather than being determined solely by preferences and interest rates as Irving (Fisher, 1930) had proposed.

Buffer stock saving models are neither a subset nor a superset of the closely-related class of (Bewley, 1977) models (or, more generically, ‘income fluctuation’ problems (Schectman, 1976); we will use the terms ‘Bewley model’ and ‘income fluctuation problem’ interchangeably). That is, not all Bewley models with fluctuating income exhibit buffer stock saving, and not all models that exhibit buffer stock saving satisfy the mathematical assumptions that guarantee boundedness of marginal marginal utility imposed by Schectman or Bewley (and inherited by almost all of the subsequent literature through to the recent contributions of (Ma, Stachurski, and Toda, 2020; Ma, Stachurski, and Toda, 2022a)).

The purpose of this paper is to provide a comprehensive statement and explanation of the conditions under which buffer stock saving arises in a class of problems broader in important respects than Bewley models. Specifically, we consider the problem of an agent who is subject to a Friedman-Muth(-Zeldes) income process incorporating permanent shocks to noncapital income (Friedman, 1957; Muth, 1960; Zeldes, 1989) in addition to the transitory shocks traditionally examined in the income fluctuation literature,^[3] and who does not face an ‘artificial’ borrowing constraint (a constraint that prohibits borrowing even when the loan could certainly be repaid).

In the course of proving our main theoretical results, we define a variety of alternative measures of ‘patience’ – an intuitive term that nonetheless has had multiple interpretations in the literature. Different measures of impatience guarantee two kinds of theoretical results: The existence of a nondegenerate limiting consumption function, and the existence of a buffer stock ‘target’.

Patience Requirements for Nondegeneracy¶

We will define the ‘limiting’ consumption function as the limit of the sequence of consumption rules constructed by iterating backward from a terminal period $T$, and we will say that this limiting function is ‘nondegenerate’ if it exists, is real-valued, and is strictly positive for every reachable circumstance the consumer could be in.

When an artificial borrowing constraint is imposed and noncapital income is stationary – it is subject to no permanent shocks and exhibits no long-term growth – our problem coincides with a standard ‘income fluctuation problem.’ But our new proof methods in this paper’s first main contribution, allows us also to solve models where permanent income is unbounded above and below and there is no artificial constraint.

As noted by (Szeidl, 2013), the impatience condition $(\Rfree \DiscFac) < 1$ that is commonly imposed in Bewley models to guarantee existence and stability of the stochastic distributions is in general not necessary nor sufficient for ensuring the existence of a non-degenerate limiting solution.

For instance, in the unconstrained perfect foresight version of our problem, one type of degeneracy arises if permanent income perpetually grows faster than the rate at which it is discounted: With no limiting upper bound to the PDV of future income and no borrowing constraint, there is no upper bound to limiting consumption. Imposition of a ‘Finite Human Wealth’ condition ($\PermGroFac < \Rfree$), where the growth factor of permanent income ($\PermGroFac$) is strictly dominated by the discount factor ($\Rfree$), is required to prevent this. Another type of degeneracy arises if preferences fail the ‘return impatience’ condition: $\APFacRaw/\Rfree < 1$, where $\CRRA$ is the coefficient of relative risk aversion. Without return impatience, the limiting consumption function is zero everywhere. Thus, both of these conditions are necessary for the existence of a nondegenerate limiting consumption function.

Intuition would suggest that, by activating the precautionary saving motive, introduction of the Friedman-Muth stochastic income process might necessitate a stronger degree of impatience to avoid degeneracy. In fact, we show that the required impatience condition is weaker, for reasons that follow from the lack of an artificial borrowing constraint and the presence of a ‘natural’ borrowing constraint,^[4] (which has the additional effect of eliminating the need to impose the Finite Human Wealth condition). We further demonstrate that the artificial constraint emerges as the limit, as a certain parameter goes to zero, of the natural constraint. This provides an intuitive conceptual bridge between the two.

In the existing literature going all the way back to (Fisher, 1930),^[5] time preference ‘$\DiscFac$’ and patience have often been treated as synonymous.

However, we show that, in the presence of nonstationary permanent income, the corresponding generalized mathematical steps yield a ‘finite value of autarky’ condition that incorporates both $\DiscFac$ and characteristics of the income growth process. The fact that the generalized condition involves terms other than $\DiscFac$ undermines the temptation to identify ‘patience’ solely with the pure time preference factor. We therefore propose that henceforth the literature should deprecate use of the term ‘patience’ unadorned with any adjective identifying precisely which kind of patience is under consideration.

One such kind of patience is absolute patience, $\APFacRaw$, which is the rate at which the consumer is willing to move consumption forward without any precautionary saving motive (under perfect foresight). Importantly, we show that for a consumer to have a non-degenerate value function in the limit as the planning horizon recedes, absolute patience cannot exceed both the market return factor $\Rfree$ and the expected income growth factor $\PermGroFac.$ (Growth impatience must hold if return impatience fails and vice-versa.) When both growth impatience and return impatience fail, the limiting consumption function is either $\cFunc=0$ or $\cFunc=\infty$.

Patience Requirements and the Existence of Buffer Stock Targets¶

Once we have established the existence of a non-degenerate solution, the second (and more important) main result of the paper is to identify conditions under which buffer stock ‘targets’ exist, for individual consumers or in the aggregate.

The appropriate definition of a buffer stock ‘target’ turns out to depend on whether we are interested in the microeconomic behavior of individual consumers, or the aggregate behavior of the entire population of consumers. The requirement for the existence of an individual target is ‘strong growth impatience,’ ($\Ex[\APFacRaw/\PermGroFacRnd] < 1$) which prevents the ratio of a household’s market resources $(\mLvl)$ to permanent income $(\pLvl)$ (‘normalized market resources’ $\mNrm = \mLvl/\pLvl$) from growing without bound. Specifically, strong growth impatience guarantees that at some large-enough value $\mNrm = \acute{\mNrm}$ it must be the case that the expectation of next period’s $\mNrm$ is less than this period’s: (if $\mNrm_{t} > \acute{m}$, then $\Ex_{t}[\mNrm_{t+1}] < \mNrm_{t}$). This turns out to guarantee that normalized market resources eventually revert back toward a target.

A weaker requirement, ‘growth impatience,’ ensures the existence of an aggregate buffer stock target even when individual target ratios are unbounded. Growth impatience requires the ratio of absolute patience to the expected growth factor of permanent income to be less than one: $\APFacRaw/\PermGroFac < 1$.

As (Harmenberg, 2021a) points out, a stationary distribution of market resources, weighted by permanent income still exists under growth impatience. The trick to understanding how there can be an aggregate target even when there is no individual target is to realize that one reason that $\mLvl/\pLvl$ can grow is that individuals can have negative shocks to $\pLvl$. But the people whose ratio grows because their $\pLvl$ shrinks by definition account for a smaller portion of the level of aggregate permanent income. That is, even as their $\mNrm$ rises they become smaller contributors to the aggregate economy. ^[6]

Thus in the aggregate, even with a fixed interest rate that differs from the time preference rate, a small open economy populated by buffer stock consumers has a balanced growth path in which growth rates of consumption, income, and market resources match the exogenous growth rate of aggregate permanent income (equivalent, here, to productivity growth). In the terms of (Schmitt-Grohé and Uribe, 2003), buffer stock saving is an appealing method of ‘closing’ a small open economy model, because it requires no ad-hoc assumptions. Not even borrowing constraints.

Relationship to Literature¶

Although the elements of buffer stock saving behavior were informally articulated by (Friedman, 1957), the term was introduced to the literature by (Deaton, 1991) to describe the behavior of liquidity-constrained impatient consumers with transitory income shocks.^[7] (Carroll, 1992) showed (numerically) that buffer stock saving could arise even in the absence of borrowing constraints, and defined the individual buffer stock ‘target’ as the point where a measure of normalized resources is expected to stay the same.

Traditional Bellman approaches to showing existence rely on assumptions that guarantee the boundedness of utility and marginal utility (Stokey, Lucas, and Prescott, 1989).^[8] The results by (Ma, Stachurski, and Toda, 2020; Ma, Stachurski, and Toda, 2022a) are the most general we are aware of that tackle income fluctuation problems, and can be specialized to show existence in a model one step away from our normalized model with a stochastic rate of return and stochastic effective discount factor. The discrepant step is that they impose an artificial constraint and positive minimum value of income; this bounds utility from below (it can never be lower than the marginal utility of consumption) and thus cannot be applied here.

Our approach to constructing the weighted-norm space of value functions uses results on unbounded dynamic programming by (Boyd, 1990).^[9] Our approach differs from previous approaches in its use of limiting marginal propensities to consume to construct per-period bounds on the Bellman operator. Moreover, our patience restrictions are grounded in intuitive economic ideas (rather than abstract mathematical assumptions) that arise naturally in the presence of permanent income uncertainty and growth. To the best of our knowledge, these economic mechanisms have not been explored elsewhere.

Our discussion of aggregate results builds on (Szeidl, 2013) and (Harmenberg, 2021a) who give results on the existence and convergence of stationary wealth distributions that apply to the model presented here. While their conditions for stationarity relate to growth impatience and strong growth impatience, our objective is to establish existence of stable buffer stock targets, which is relatively easily tested empirically, rather than to establish stationarity of distributions, which is much harder to imagine testing with empirical data.

(defun my-tex-command-expand-options () (let ((opts (TeX-command-expand-options))) (replace-regexp-in-string " -interaction=nonstopmode" “” opts)))

(setq TeX-expand-list-builtin (mapcar (lambda (item) (if (equal (car item) “mode”) ’(“mode” my-tex-command-expand-options) item)) TeX-expand-list-builtin))

Theoretical Foundations¶

This section formalizes the consumer income fluctuation problem and proves the existence of a limiting non-degenerate solution. In doing so, we also introduce our consumer patience conditions and use them to derive the consumer’s MPCs. The MPCs are formulae, for any period $t$ earlier than the terminal period $T$, for the maximum and minimum MPCs as wealth approaches zero and infinity. If the environment is that of an infinite-horizon ‘income fluctuation problem,’ our formulae yield the limiting upper and lower bounds of the limiting non-degenerate solution.

We first state the finite horizon problem and then define the limiting solution as the limit of finite horizon solutions as the terminal period becomes arbitrarily distant. This way, the economic intuition of limiting consumer behaviour can be directly linked to consumer behaviour in life-cycle models (see (Gourinchas and Parker, 2002) for an instance where buffer stock saving is discussed in the context of a life-cycle model). Nonetheless for the class of problems we consider, a non-degenerate limiting solution, if it exists, is mathematically equivalent ((Bertsekas, 2012), Ch. 1.) to a stationary solution to an infinite stochastic sequence problem commonly used in the literature (for example, (Ma, Stachurski, and Toda, 2020)).

Setup¶

We start by stating the consumer problem with permanent income growth in levels and then normalize by permanent income. The normalized problem then becomes the subject of our formal results in the paper.

Our time index $t$ can take on values in $\{T,T-1,T-2,\dots \}$. We assume that our consumer has a Constant Relative Risk Aversion (CRRA) per-period utility function, $\uFunc(c)=\frac{c^{1-\CRRA}}{1-\CRRA}$, where $\CRRA>1$. The term $\DiscFac$ is the (strictly positive) discount factor. In each period $t$, the consumer faces independently and identically distributed (iid) income shocks, with the permanent shock given by $\permShk_{t} \in \Reals_{++}$ and the transitory shock by $\tranShkAll_{t} \in \Reals_{+}$.^[10]

In each $t$, the finite horizon value function for the problem in levels will be denoted by $\vFuncLvl_{t}$, with $\vFuncLvl_{t}\colon \Reals_{++}^{2}\rightarrow \Reals$. Value, $\vFunc_{t}(\mLvl_{t}, \permLvl_{t})$, depends on two strictly positive state variables: ‘market resources’ $\mLvl_{t}$ and permanent income $\permLvl_{t}$. After the terminal period, we assume the consumer cannot die in debt:

Letting $\vFuncLvl_{T+1} = 0$, it follows that the value function for the terminal period satisfies $\vFuncLvl_{T} = \uFunc(\mNrm_{T})$. For $t<T$, the finite-horizon value functions are recursively defined by:

where i) $\cLvl_{t}$ is the level of consumption at time $t$, ii) $\Ex_{t}$ is the expectation operator over the shocks $\permShk_{t+1}$ and $\tranShkAll_{t+1}$

, and iii) $\mLvl_{t+1}$ is determined from this period’s $\mLvl_{t}$ and choice of $\cLvl_{t}$ as follows:^[11]

The consumer’s assets at the end of $t$, $\aLvl_{t}$, translate one-for-one into capital $\kLvl_{t+1}$ at the beginning of the next period. In turn, $\kLvl_{t+1}$ is augmented by a fixed interest factor $\Rfree$ to become the consumer’s financial (‘bank’) balances $\bLvl_{t+1} = \Rfree \kLvl_{t+1}$. ‘Market resources,’ $\mLvl_{t+1}$, are the sum of financial wealth $\Rfree \kLvl_{t+1}$ and noncapital income $\yLvl_{t+1}=\permLvl_{t+1}\tranShkAll_{t+1}$ (permanent noncapital income $\permLvl_{t+1}$ multiplied by the transitory shock $\tranShkAll_{t+1}$). Permanent noncapital income $\permLvl_{t+1}$ is derived from $\permLvl_{t}$ by application of a growth factor $\PermGroFac$,^[12] modified by the permanent income shock $\permShk_{t+1}$,^[13] and the resulting idiosyncratic growth factor for permanent income is written as $\Rnd{\PermGroFac}_{t+1}$.

Letting $n$ denote the planning horizon, the finite-horizon problems furnish a sequence of value functions $\{\vFuncLvl_{T},\vFuncLvl_{T-1},\ldots,\vFuncLvl_{T-n}\}$ and associated consumption functions $\{\cFuncLvl_{T},\cFuncLvl_{T-1},\ldots,\cFuncLvl_{T-n}\}$. The limiting consumption function, denoted by $\usual{\cFuncLvl}(\mLvl,\permLvl) = \lim\limits_{n \rightarrow \infty} \cFuncLvl_{T-n}(\mLvl,\permLvl)$, will be called a ‘non-degenerate limiting solution’ if neither $\usual{\cFuncLvl}=0$ everywhere (for all $(\mLvl,\permLvl)$) nor $\usual{\cFuncLvl}=\infty$ everywhere.

Before turning to the normalized problem, we present the income process and its implications for the consumer problem. The following assumption defines the income process.

Following (Zeldes, 1989), the income process incorporates a small probability $\pZero$ that income will be zero (a ‘zero-income event’). At date $T-1$, the (strictly positive) probability $q$ of zero income in period $T$ will prevent the consumer from spending all resources, because saving nothing would mean arriving in the following period with zero bank balances and thus facing the possibility of being required to consume 0, which would yield utility of $-\infty$. This logic holds recursively from $T-1$ back, so the consumer will never spend everything, giving rise to what (Aiyagari, 1994) dubbed a ‘natural borrowing constraint.’^[14] (Thus, the upper-bound constraint on consumption in the problem (2) will not bind.)

The model looks more special than it is. In particular, a positive probability of zero-income events may seem objectionable (despite empirical support). However, a nonzero minimum value of $\tranShkAll$ (motivated, say, by the existence of unemployment insurance) could be handled by capitalizing the present discounted value (PDV) of minimum income into current market assets,^[15] and transforming that model back into this one. And no key results would change if the transitory shocks were persistent but mean-reverting (instead of iid). Also, the assumption of a positive point mass for the worst realization of the transitory shock is inessential, but simplifies the proofs and is a powerful aid to intuition.

Normalized Problem¶

Let nonbold variables be the boldface counterpart normalized by $\permLvl_{t}$, allowing us to reduce the number of states from two ($\mLvl$ and $\permLvl$) to one $(\mNrm = \mLvl/\permLvl)$. Now, in a one-time deviation from the notational convention established in the last sentence, define nonbold ‘normalized value’ not as $\vLvl_{t}/\permLvl_{t}$ but as $\vNrm_{t} = \vLvl_{t}/\permLvl_{t}^{1-\CRRA}$, because this allows us to write nonbold $\vFunc_{t}$, with $\vFunc_{t}\colon \Reals_{++}\rightarrow \Reals$, to denote the ‘normalized value function’:

where $\RNrmByGRnd_{t+1}:= (\Rfree/\PermGroFacRnd_{t+1})$ is a ‘permanent-income-growth-normalized’ return factor.^[16] (Appendix Paragraph explains how the solution to the original problem in levels can be recovered from the normalized problem.)

The time $t$ normalized consumption policy function for the finite-horizon problem, $\cFunc_{t}$, is defined by:

The normalized problem’s first order condition becomes:

Since our main results pertain to the normalized problem, we define the limiting non-degenerate solution to the normalized problem formally.

We use $\TMap$ to denote the stationary Bellman operator for the normalized problem. To define $\TMap$, let $\RNrmByGRnd := \Rfree/\PermGroFacRnd$ and let $\TMap$ denote the mapping $\vFunc_{t+1} \mapsto \vFunc_{t}$ given by Problem (5):

The mapping $\mNrm\mapsto (0, \mNrm)$ defines the feasibility correspondence. To define $\TMap$, we excluded the boundary of the feasible values that consumption can take (0 and $\mNrm$) to ensure the maximand above is real-valued for all feasible values of consumption. It is straightforward to show (using the Bellman Principle of Optimality) that a finite valued solution, $\vFunc$, to the functional equation $\TMap\vFunc = \vFunc$ defines a limiting non-degenerate solution. However, because the feasibility correspondence does not include the boundary of feasible consumption, existing dynamic programming arguments cannot be used to show that such a solution (a fixed point to $\TMap$) exists.

Dynamic Programming Challenges¶

Standard dynamic programming (Stachurski, 2022) works by showing that $\TMap$ is a well-defined contraction map on a Banach space, which would allow us to conclude that the sequence of value functions given by Problem (5) converges to a fixed point of $\TMap$, a non-degenerate solution. At first, we must contend with the fact that both $\uFunc$ and $\vFunc$ are unbounded below. We resolve unboundedness by constructing a weighted-norm (see below). Setting aside unboundedness, the natural liquidity constraint introduces a more pernicious challenge related to continuity: $\TMap$ will not a be well defined self-map on a vector space of continuous functions. In particular, we cannot assert $\TMap$ maps continuous functions to continuous functions since the feasiblility correspondence $\mNrm\mapsto (0, \mNrm)$ is not compact-valued.

If we reintroduce the boundary points 0 and $\mNrm$ to the feasibility correspondence, the operator $\TMap$ will be able to map upper semicontinuous functions to upper semicontinuous functions (Lemma 1, (Jaśkiewicz and Nowak, 2011)). However, $\vFunc$ must now be defined on $\Reals_{+}$ and take on values in $\mathbb{R}_{+}\cup\{-\infty\}$ and spaces of such functions will not be a vector space. The approach taken by (Ma, Stachurski, and Toda, 2022b) is to impose an artificial liquidity constraint, which yields a real-valued continuation value, even if $\cNrm= \mNrm$, and forces the value function to be bounded below as a function of end-of-period assets. This allows (Ma, Stachurski, and Toda, 2022b) to define a functional operator operator within which the feasibiltiy correspondence is the compact interval $[0, \mNrm]$. Without an artificial constraint, no such strategy is possible.^[17]

A related approach, which uses Euler operators is used by (Ma, Stachurski, and Toda, 2020). While (Ma, Stachurski, and Toda, 2020) also assume an artificial liquidity constraint to bound the marginal utility of consumption, it is useful to consider how the structure of our model relates to theirs once the artificial liquidity constraint is imposed.

Notwithstanding Remark Remark 2, there are important economic consequences relating consumer patience to buffer stock saving due to the fact that in our problem $\RNrmByGRnd_{t+1}=\Rfree/\Rnd{\PermGroFac}_{t+1}$ is tightly tied to the ‘normalized stochastic discount factor,’ $\DiscFac \Rnd{\PermGroFac}_{t+1}^{1-\CRRA}$; these will become apparent as we proceed.

Consumer Patience Conditions¶

In order to have a central reference point for them, we now collect conditions relating consumer discounting and patience to the rate of return and income growth that underpin results in the remainder of the paper. Assumptions Assumption 2 - Assumption 4 (finite value of autarky, return impatience and weak return impatience) will be used to prove the existence of limiting solutions in Section Paragraph, and Assumptions Assumption 6 - Assumption 7 (growth impatience and strong growth impatience) are required for existence of alternative definitions of a stable target buffer stock in Section Paragraph.

We start by generalizing the standard $\DiscFac<1$ condition to our setting with permanent income growth and uncertainty.^[18] The updated condition requires that the expected net discounted value of utility from consumption is finite under our definition of ‘autarky’ – where consumption is always equal to permanent income. A finite value of autarky helps guarantee that as the horizon extends, discounted value remains finite along any consumption path the consumer might choose. (See Appendix Recovering the Non-Normalized Problem).

We now turn to consumer patience and start with ‘absolute (im)patience.’ We will say that an unconstrained perfect foresight consumer exhibits absolute impatience if they optimally choose to spend so much today that their consumption must decline in the future. The growth factor for consumption implied by the Euler equation of a perfect foresight model is $\cLvl_{t+1}/\cLvl_{t} = {(\Rfree\DiscFac)}^{1/\CRRA}$,^[19] which motivates our definition of an ‘absolute patience factor’ whose centrality (to everything that is to come later) justifies assigning to it a special symbol; we have settled on the archaic letter ‘thorn’:

We will say that (in the perfect foresight problem) ‘an absolutely impatient’ consumer is one for whom $\APFac < 1$; that is an absolutely impatient consumer prefers to consume more today than tomorrow (and vice versa for an ‘absolutely patient’ consumer, whose consumption will grow over time):

A consumer who is absolutely impatient, $\APFac<1$, satisfies the standard impatience condition commonly used in the income fluctuation literature, $\DiscFac\Rfree<1$, which guarantees the existence of a stable asset distribution when there is no permanent income growth. However, as pointed out by (Szeidl, 2013) and (Ma, Stachurski, and Toda, 2022a), $\DiscFac\Rfree<1$ is not necessary for an infinite-horizon solution.

Recall now our earlier requirement that the limiting consumption function $\cFunc(\mNrm)$ in our model must be ‘sensible.’ We will show below that for the perfect foresight unconstrained problem this requires

Return impatience can be best understood as the tension between the income effect of capital income and the substitution effect. As we show below in Section Paragraph, in the perfect foresight model, it is straightforward to derive the MPC out of overall (human plus nonhuman) wealth that would result in next period’s wealth being identical to the current period’s wealth. The answer turns out to be an MPC (‘$\MPC$’) of $\MPC=(1-\APFac/\Rfree)$. The interesting point here is that $\MPC$ depends both on our absolute patience factor $\APFac$ and on the return factor. This is the manifestation in this context of the interaction of the income effect (higher wealth yields higher interest income if $\Rfree>1$) and the substitution effect (which we have already captured with $\APFac$).

Next, consider the weaker condition of a consumer whose absolute patience factor is suitably adjusted to take account of the probability of zero income is less than the market return.

This condition is ‘weak’ (relative to the plain return impatience) because the probability of the zero income events $\pZero$ is strictly less than 1. The role of $\pZero$ in this equation is related to the fact that a consumer with zero end-of-period assets today has a probability $\pZero$ of having no income and no assets to finance consumption (and $\mNrm_{t+1}=0$ would yield negative infinite utility). In the case with no artificial constraint, our main results below, in Section Paragraph, show weak return impatience and finite value of autarky are sufficient to guarantee a sensible (non-degenerate) solution.

Weak return impatience cannot be relaxed further without an artificial liquidity constraint. Even though $\pZero^{1/\CRRA} \RPFac\rightarrow 0$ as $\pZero\rightarrow 0$ the weak return impatience condition does not approach irrelevance as the possibility of the zero income event approaches zero. Instead, we show below in Section Paragraph that the limiting consumption function with a natural constraint approaches the solution to a model with an artificial constraint.

Now that we have finished discussing the requirements for a non-degenerate solution, we turn to assumptions required for stability.

We will call the ratio of the $\APFac$ to the expected growth factor for permanent income $\PermGroFac = \Ex[\PermGroFac \permShk])$ the :

as exhibiting ‘growth impatience:’

We speak of a consumer whose absolute patience factor is less than the expected growth factor for their permanent income $\PermGroFac = \Ex[\PermGroFac \permShk])$ as exhibiting ‘growth impatience:’

A final useful definition is ‘strong growth impatience’

which holds for a consumer for whom the expectation of the ratio of the absolute patience factor to the (stochastic) growth factor of permanent income is less than one,

(The difference between growth impatience and strong growth impatience is that the first is the ratio of an expectation to an expectation, while the latter is the expectation of the ratio. With non-degenerate mean-one stochastic shocks to permanent income, the expectation of the ratio is strictly larger than the ratio of the expectations).

Since $\CRRA>1$, note that strong growth impatience is weaker than the impatience condition $\DiscFac \Rnd{\PermGroFac}_{t+1}^{1-\CRRA}\RNrmByGRnd<1$ used by (Ma, Stachurski, and Toda, 2020) to guarantee stability. Moreover, while neither growth impatience nor return impatience will by themselves be required for the existence of a limiting solution, the finite value of autarky condition stops individuals from becoming both growth and return patient.

We discuss further intuition for the consumer patience conditions below when they are used in the main results.

The relationship between the conditions and their implications for consumption behaviour will also be be discussed in detail in Section Figure 3.

Perfect Foresight Benchmarks¶

To understand the economic implications of the patience conditions, we begin with the perfect foresight case.

Below, when we say we assume perfect foresight, what we mean mathematically is:

Throughout this sub-section, we assume Assumption Assumption 8 remains in force.

Under perfect foresight, finite value of autarky reduces to a ‘perfect foresight finite value of autarky’ condition:

Perfect Foresight without Liquidity Constraints¶

Consider the familiar analytical solution to the perfect foresight model without liquidity constraints. In this case, the consumption Euler Equation always holds as an equality; with $\uP(\cLvl)=\cLvl^{-\CRRA}$ and $\uFunc^{\prime}(\cLvl_{t})=\Rfree\DiscFac\uFunc^{\prime}(\cLvl_{t+1})$, we have:

Recalling $\RNrmByG = \Rfree/\PermGroFac$, ‘human wealth’, is the present discounted value of income:

For human wealth to have finite value, we must have:

If $\RNrmByG^{-1}$ is less than one, human wealth will be finite in the limit as $T \rightarrow \infty$ because (noncapital) income growth is smaller than the interest rate at which that income is being discounted.

Under these conditions we can define a normalized finite-horizon perfect foresight consumption function (see Appendix Paragraph for details) as follows:

where $\MPCmin_{t}$ is the marginal propensity to consume (MPC) and satisfies:

Let $\MPCmin = \lim\limits_{n\rightarrow\infty}\MPCmin_{T-n}$. For $\Min{\MPC}$ to be strictly positive, we must impose return impatience. The limiting consumption function then becomes:

where, under return impatience, the limiting MPC becomes:

In order to rule out the degenerate limiting solution in which $\bar{\cFunc}(\mNrm) = \infty$, we also require (in the limit as the horizon extends to infinity) that human wealth remain bounded (that is, we require ‘finite human wealth’). Thus, while return impatience prevents a consumer from saving everything in the limit, ‘finite limiting human wealth’ prevents infinite borrowing (against infinite human wealth) in the limit.

The following two results consider the normalized problem without liquidity constraints and with perfect foresight income (Assumption Assumption 8).

The claim implies that if we impose finite limiting human wealth, then growth impatience is sufficient for nondegeneracy since finite value of autarky and return impatience follow. However, there are circumstances under which return impatience and finite limiting human wealth can hold while the finite value of autarky fails. For example, if $\PermGroFac=0$, the problem is a standard ‘cake-eating’ problem with a non-degenerate solution under return impatience.

Perfect Foresight with Liquidity Constraints¶

Our ultimate interest is in the unconstrained problem with uncertainty. Here, we show that the perfect foresight constrained solution defines a useful limit for the unconstrained problem with uncertainty.

Consider that if a liquidity constraint requiring $\aNrm_{t} \geq 0$ binds at any $\mNrm_{t}$, it must bind at the lowest possible level of $\mNrm_{t}$, $\mNrm_{t}=1$, defined by the lower bound of having arrived into the period with $\bNrm_{t}=0$ (if the constraint were binding at any higher $\mNrm_{t}$, it would certainly be binding here, because $\uFunc^{\prime\prime}<0$ and $\cFunc^{\prime}>0$). At $\mNrm_{t}=1$ the constraint binds if the marginal utility from spending all of today’s resources $c_{t}=m_{t}=1$, exceeds the marginal utility from doing the same thing next period, $\cNrm_{t+1}=1$; that is, if such choices would violate the Euler equation, Equation  (7), yielding

which is just a restatement of growth impatience. So, the constraint is relevant if and only if growth impatience holds.

For the following result, consider the normalized perfect foresight problem with a liquidity constraint (that is, assume $\cNrm_{t}\leq \mNrm_{t}$ for each $t$.)

The proof for the result follows from the discussion in Section Paragraph, which outlines the cases under which perfect foresight liquidity constraint solutions are non-degenerate.

Importantly, if return impatience fails ($\Rfree\leq \APFac$) and growth impatience holds ($\APFac<\PermGroFac$), then finite human wealth also fails $(\Rfree \leq \PermGroFac)$. Despite the unboundedness of human wealth as the horizon extends arbitrarily, for any finite horizon the relevant liquidity constraint prevents borrowing. Similarly, when uncertainty is present, the natural borrowing constraint plays an analogous role in permitting a finite limiting solution with unbounded limiting human wealth – we discuss the various parametric cases in Section Figure 3.

Main Results for Problem with Uncertainty¶

We are now ready to return to our primary interest, the model with permanent and transitory income shocks. Throughout this section, we assume the Friedman-Muth income process (Assumption Assumption 1 holds) and examine the normalized problem, Problem (5).

Limiting MPCs¶

We first establish results regarding the shape of the consumption function.^[20]

Next, we note that the ratio of optimal consumption to market resources ($\cNrm/\mNrm$) is bounded by the minimal and maximal marginal propensities to consume (MPCs). Recall that the MPCs answer the question ‘if the consumer had an extra unit of resources, how much more spending would occur?’, The minimal and maximal MPCs are the limits of the MPC as $\mNrm \rightarrow \infty$ and $\mNrm \rightarrow 0$, which we denote by $\MPCmin_{t}$ and $\MPCmax_{t}$ respectively. Since the consumer spends everything in the terminal period, $\MPCmin_{T}=1$ and $\MPCmax_{T}=1$. Furthermore, Proposition Proposition 3 will imply:^[21]

We define:

as the ‘limiting minimal and maximal MPCs’. The following result verifies that the consumption share is bounded each period by the minimal and maximal MPCs, that the consumption function is asymptotically linear and that the MPCs converge to the limiting MPCs as the terminal period recedes.^[22]

The MPC bound as market resources approach infinity is easy to understand. Recall that $\bar{\cFunc}$ from the perfect foresight case will be an upper bound in the problem with uncertainty; analogously, $\MPCmin$ becomes the MPC’s lower bound. As the proportion of consumption that will be financed out of human wealth approaches zero, the proportional difference between the solution to the model with uncertainty and the perfect foresight model shrinks to zero.

To understand the maximal limiting MPC, the essence of the argument is that as market resources approach zero, the overriding consideration that limits consumption is the (recursive) fear of the zero-income events — this is why the probability of the zero income event $\pZero$ appears in the expression for the maximal MPC. Weak return impatience is too weak to guarantee a lower bound on the share of consumption to market resources; it merely prevents the upper bound on the share of consumption to market resources from approaching zero. Weak return impatience thereby prevents a situation where everyone consumes an arbitrarily small share of current market resources as the terminal period recedes. This insight plays a key role in the proof for the existence of a non-degenerate solution in what follows.

Existence of Limiting Non-degenerate Solution¶

Let $\mathcal{C}(\Reals_{++},\Reals)$ be the space of continuous functions from $\Reals_{++}$ to $\Reals$. To address the challenges posed by unbounded state-spaces, Boyd ((1990)) provided a weighted contraction mapping theorem. Our strategy is to use this approach to first show that while the stationary operator $\TMap$ may be undefined on a suitable Banach space (recall Remark Remark 1), operators defining each period’s problem (which we define below) will be contractions on a space of continuous functions with a finite weighted norm. We then show the sequence of finite horizon value functions given by Problem (5) generates a Cauchy sequence; since the weighted norm space is complete, the sequence of value functions converges to a non-degenerate solution in $\mathcal{C}(\Reals_{++},\Reals)$.

We define the weighting function as

where $\zeta \in \Reals_{++}$ is a constant derived from the model primitives and the upper and lower bound on the consumption share (see Claim Remark 5 in Appendix Paragraph for the parametrization of $\zeta$).

Next, for any lower bound $\MPCminInf$ and upper-bound $\MPCmaxInf$ on the share of consumption to market resources, define the ‘MPC bounded Bellman operator’ $\TMap^{\MPCminInf, \MPCmaxInf}$, with $\TMap^{\MPCminInf, \MPCmaxInf}:\mathcal{C}_{\boundFunc }\left( \Reals_{++},\Reals\right) \rightarrow \mathcal{C}_{\boundFunc }\left( \Reals_{++},\Reals\right)$, as:

The value functions defined by Problem (5) will satisfy $\vFunc_{t} = \TMap^{\MPCmin_{t}, \MPCmax_{t}}\vFunc_{t+1}$ for each period $t$, since consumption shares are bounded by the minimal and maximal MPCs (Lemma Lemma 1 and Equation (24)). We now show the operator $\TMap^{\MPCminInf, \MPCmaxInf}$ is a contraction on $\mathcal{C}_{\boundFunc }\left( \Reals_{++},\Reals\right)$ for a suitably narrow interval $[\MPCminInf, \MPCmaxInf]$.

The theorem says eventually the maximal MPCs will be small enough such that the Bellman operators generating the sequence of finite horizon value functions given by (5) are contraction maps.

We can now relate the sequence of contraction maps to the limiting solution defined in Section Paragraph.

The proof above shows that the sequence of value functions produced by the iteration of the per-period Bellman operators $\TMap^{\MPCmin_{T-n}, \MPCmax_{T-n}}$ will be a Cauchy sequence converging to the limiting solution. Due to weak return impatience, the upper bound on the per-period consumption converges to a strictly positive share of market resources, preventing consumption from converging to zero.

Finite value of autarky is the second assumption required to show existence of limiting solutions and guarantees the value is finite (in levels) for a consumer who spent exactly their permanent income every period (see Section Paragraph). The intuition for the finite value of autarky condition is that, with an infinite-horizon, with any strictly positive initial amount of bank balances $\bNrm_{0}$, in the limit your value can always be made greater than you would get by consuming exactly the sustainable amount (say, by consuming $(\rfree/\Rfree)\bNrm_{0}-\epsilon$ for some arbitrarily small $\epsilon>0$).

Finally, we verify that the converged non-degenerate consumption functions satisfies the same marginal propensities to consume the per-period consumption functions.

The Liquidity Constrained Solution as a Limit¶

Recall the common assumption (Deaton, 1991; Aiyagari, 1994; Li and Stachurski, 2014; Ma, Stachurski, and Toda, 2020) of a strictly positive minimum value of income and a non-trivial artificial liquidity constraint, namely $\aNrm_{t}\geq 0$. We will refer to the set-up from Section Paragraph, with Assumption Theorem 2 modified so $\pZero=0$ as the “liquidity constrained problem.” Let $\cFunc_{t}(\bullet;\pZero)$ be the consumption function for a problem where Assumption Assumption 1 holds for a given fixed $\pZero$, with $\pZero>0$. Moreover, let $\cnstr{\cFunc}_{t}$ be the limiting consumption function for the liquidity constrained problem (note that the liquidity constraint $\cNrm_{t}\leq \mNrm_{t}$, or $\aNrm_{t}\geq 0$, becomes relevant only when $\pZero= 0$). The discussion in Appendix Paragraph shows how an finite-horizon solution to the liquidity constrained problem, $\cnstr{\cFunc}_{t}$ , is the limit of the problems as the probability $\pZero$ of the zero-income event approaches zero.

Intuitively, if we impose the artificial constraint without changing $\pZero$ and maintain $\pZero>0$, it would not affect behavior. This is because the possibility of earning zero income over the remaining horizon already prevents the consumer from ending the period with zero assets. For precautionary reasons, the consumer will save something. However, the extent to which the consumer feels the need to make this precautionary provision depends on the probability that it will turn out to matter. As $\pZero \rightarrow 0$, the precautionary saving induced by the zero-income events approaches zero, and “zero” is the amount of precautionary saving that would be induced by a zero-probability event by the impatient liquidity constrained consumer. See Appendix Paragraph for the formal proof.

Individual Buffer Stock Stability¶

In this section we analyse two notions of stability which will be useful for studying either an individual or a population of individuals who behave according to the converged consumption rule. Consider an individual who at time $t$ holds normalized market resources $\mNrm_{t}$, and market resources in levels $\mLvl_{t}$, and follows the converged decision function $\cFunc$. The time-$t$ consumption for the consumer will be $\cNrm_{t} = \cFunc(\mNrm_{t})$ and normalized market resources in time $t+1$ will be a random variable $\mNrm_{t+1} = \RNrmByGRnd_{t+1}(\mNrm_{t} - \cFunc(\mNrm_{t})) + \tranShkAll_{t+1}$.^[23]

Our first notion of stability concerns the existence of a unique ‘buffer stock target’ for the individual; we are interested in whether the current level of normalized market resources is above or below a ‘target’ level such that the magnitude of the precautionary motive (which causes a consumer to save) exactly balances the impatience motive (which makes them want to dissave). At the individual target, expected normalized market resources in the next period, conditioned on current normalized market resources, will be the same as in the current period. The intensifying strength of the precautionary motive with decreasing market resources can ensure stability of the target. Below the target, the urgency to save due to the precautionary motive leads to an expected rise in market resources. Conversely, above the target, impatience prevails, leading to an expected reduction of market resources.

Our second, weaker, notion of stability gives conditions for the invidiual such that an aggregate balanced growth path exists. To motivate this notion, consider Figure Figure 1 which shows the expected growth factors for consumption, the level of market resources, and normalized market resources, $\Ex_{t}[\cLvl_{t+1}/\cLvl_{t}]$, $\Ex_{t}[\mLvl_{t+1}/\mLvl_{t}]$, and $\Ex_{t}[\mNrm_{t+1}/\mNrm_{t}]$.^[24] Begin by noting how the figure shows that as $\mNrm_{t} \rightarrow \infty$ the expected consumption growth factor goes to ${\APFac}$, indicated by the lower bound in Figure Figure 1. Moreover, as $\mNrm_{t}$ approaches zero, the consumption growth factor approaches $\infty$. (Proposition Proposition 4 in Appendix Proposition 4 establishes the asymptotic growth factors formally.)

Next, consider the implications of Figure Figure 1 for individual stability. The figure shows a buffer stock target for normalized market resources, $\mNrm_{t} = \mBalLvl$, at which the expected growth factor of the level of market resources $\mLvl$ matches the expected growth factor of permanent income $\PermGroFac$. A distinct and larger target ratio, $\mTrgNrm$, also exists. At this ratio, $\Ex_{t}[\mNrm_{t+1}/\mNrm_{t}]=1$, and the expected growth factor of consumption is less than $\PermGroFac$. Importantly, conditioned on an individual’s time $t$ state, this model does not have a single $\mNrm$ at which $\permLvl$, $\mLvl$ and $\cLvl$ are all expected to grow at the same rate. Yet, when we aggregate across individuals, balanced growth paths can exist. Importantly, balanced growth paths can exist even if a buffer stock target, where $\Ex_{t}[\mNrm_{t+1}/\mNrm_{t}]=1$, does not exist. What we require for aggregate stability is the weaker notion of a ‘pseudo-target’ target, namely that there is some ${\mBalLvl}$ such that if $\mNrm_{t}>{\mBalLvl}$, then $\Ex_{t}[\mLvl_{t+1}/\mLvl_{t}] < \PermGroFac$.

Existence of Target and Pseudo-Target¶

For both results below, consider the problem defined in Section Paragraph. The first stabiltiy result guarantees the existence of a buffer stock target $\mTrgNrm$ such that if $\mNrm_{t}=\mTrgNrm$, then $\Ex_{t}[\mNrm_{t+1}]=\mNrm_{t}$. Existence of such a target requires strong growth impatience.

Since $\mNrm_{t+1}= (\mNrm_{t}-\cFunc(\mNrm_{t}))\RNrmByGRnd_{t+1} +\tranShkAll_{t+1}$, the implicit equation for $\mTrgNrm$ becomes:

The second, and less restrictive, definition of a target derives from a traditional aggregate question in macro models: whether or not there is a ‘balanced growth’ path in which aggregate variables (income, consumption, market resources) all grow by the same factor $\PermGroFac$. In particular, if growth impatience holds, the problem will exhibit a ‘pseudo-target’, by which we mean that there is some ${\mBalLvl}$ such that if $\mNrm_{t}>{\mBalLvl}$, then $\Ex_{t}[\mLvl_{t+1}/\mLvl_{t}] < \PermGroFac$.

Conversely if $\mNrm_{t}<{\mBalLvl}$ then $\Ex_{t}[\mLvl_{t+1}/\mLvl_{t}] > \PermGroFac$. The pseudo-target $\mBalLvl$ will be such that $\mLvl$ growth matches $\PermGroFac$, allowing us to write the implicit equation for $\mBalLvl$ as follows:

The only difference between (38) and (37) is the substitution of $\RNrmByG$ for $\bar{\RNrmByGRnd}$.^[25]$^{,}$

Under the weaker growth impatience condition, we can verify the existence of this pseudo-target, $\mBalLvl$.

Example With Balanced-Growth $\mBalLvl$ But No Target $\mTrgNrm$¶

Because the equations defining the buffer stock target and pseudo-target, (37) and (38), differ only by substitution of $\RNrmByG$ for $\bar{\RNrmByGRnd}=\RNrmByG \Ex[\permShk^{-1}]$, if there are no permanent shocks ($\permShk \equiv 1$), the conditions are identical. For many parameterizations (e.g., under the baseline parameter values used for constructing figure Figure 1), $\mTrgNrm$ and $\mBalLvl$ will not differ much.

An illuminating exception is exhibited in Figure Figure 2, which modifies the baseline parameter values by quadrupling the variance of the permanent shocks, enough to cause failure of strong growth impatience; now there is no target level of market resources $\mTrgNrm$. Nonetheless, the pseudo-target still exists because it turns off realizations of the permanent shock. It is tempting to conclude that the reason target $\mTrgNrm$ does not exist is that the increase in the size of the shocks induces a precautionary motive that increases the consumer’s effective patience. The interpretation is not correct because as market resources approach infinity, precautionary saving against noncapital income risk becomes negligible (as the proportion of consumption financed out of such income approaches zero).

The correct explanation is more prosaic: The increase in uncertainty boosts the expected uncertainty-modified rate of return factor from $\RNrmByG$ to $\bar{\RNrmByGRnd}>\RNrmByG$ which reflects the fact that in the presence of uncertainty the expectation of the inverse of the growth factor increases: $\PermGroFacAdj < \PermGroFac$. That is, in the limit as $\mNrm \rightarrow \infty$ the increase in effective impatience reflected in $\GPFacMod > \GPFacRaw$ is entirely due to the elevation of the expected normalized return factor under uncertainty, not to a (limiting) change in precaution. In fact, the next section will show that an aggregate balanced growth equilibrium will exist even when realizations of the permanent shock are not turned off: The required condition for aggregate balanced growth is the regular growth impatience, which ignores the magnitude of permanent shocks.^[26]

Before we get to the formal arguments, the key insight can be understood by considering an economy that starts, at date $t$, with the entire population at $\mNrm_{t}=\mBalLvl$, but then evolves according to the model’s assumed dynamics between $t$ and $t+1$. Equation (38) will still hold, so for this first period, at least, the economy will exhibit balanced growth: the growth factor for aggregate $\MLvl$ will match the growth factor for permanent income $\PermGroFac$. It is true that there will be people for whom the financial balances ratio, $\bNrm_{t+1}$, where $\bNrm_{t+1} = \aNrm_{t}\Rfree/(\PermGroFac\permShk_{t+1})$, is boosted by a small draw of $\permShk_{t+1}$. However, their contribution to the level of the aggregate variable is given by $\bLvl_{t+1}=\bNrm_{t+1}\pLvl_{t}\PermGroFac\permShk_{t+1}$, so their $\bNrm_{t+1}$ is reweighted by an amount that exactly unwinds that divisor-boosting. This means that it is possible for the consumption-to-permanent-income ratio for every consumer to be small enough that their market resources ratio is expected to rise, and yet for the economy as a whole to exhibit a balanced growth equilibrium with a finite aggregate balanced growth steady state $\BalGroFac{\MNrm}$ (this is not numerically the same as the individual pseudo-target ratio $\mBalLvl$ because the problem’s nonlinearities have consequences when aggregated).^[27]

Aggregate Invariant Relationships¶

In this section, we move from characterizing the individual decision rule to properties of a distribution of individuals following the converged non-degenerate consumption rule $\cFunc$. Assume a continuum of ex ante identical buffer-stock households, with constant total mass normalized to one and indexed by $i$. Szeidl ((2013)) proved that such a population, following the consumption rule $\cFunc$, will be characterized by invariant distributions of $\mNrm$, $\cNrm$, and $\aNrm$ under the log growth impatience condition:^[28]

which is stronger than our growth impatience ($\GPFacRaw < 1$), but weaker than our strong growth impatience ($\GPFacMod < 1$).^[29]

(Harmenberg, 2021a) substitutes a clever change of probability-measure into Szeidl’s proof, with the implication that under growth impatience, invariant permanent-income-weighted distributions of $\mNrm$ and $\cNrm$ exist (see Section Paragraph in the Appendix). In particular, let $\CDF_{\mNrm_{t},\permLvl_{t}}$ be the joint CDF of normalized market resources and permanent income at time $t$.^[30] The permanent-income-weighted CDF of $\mNrm_{t}$, $\Harm{\CDF}_{\mNrm_{t}}$, will be:

Simply put, the permanent-income-weighted CDF shows how the total ‘mass’ of permanent income is distributed along normalized market resources. The change of variables allows (Harmenberg, 2021a) to prove a conjecture from an earlier draft of this paper ((Carroll, 2019, Submitted)) that under growth impatience, aggregate consumption grows at the same rate $\PermGroFac$ as aggregate noncapital income in the long run (with the corollary that aggregate assets and market resources grow at that same rate).

(Harmenberg, 2021a) also shows how the reformulation can reduce costs of calculation by over a factor of 100.^[31] The remainder of this section draws out the implications of these points for aggregate balanced growth factors.

Aggregate Balanced Growth of Income, Consumption, and Wealth¶

Define $\Mean$ to yield the expected value operator with respect to the empirical distribution of a variable across the population (as distinct from the operator $\Ex$ which represents beliefs about the future for a given individual).^[32] Using boldface capitals for aggregates, the growth factor for aggregate noncapital income becomes:

because of the independence assumptions we have made about the shocks $\tranShkAll$ and $\permShk$.

Consider an economy that satisfies the Szeidl impatience condition (41) and has existed for long enough by date $t$ that we can consider it as Szeidl-converged. In such an economy a microeconomist with a population-representative panel dataset could calculate the growth factor of consumption for each individual household, and take the average:

Because this economy is Szeidl-converged, distributions of $\cNrm_{t}$ and $\cNrm_{t+1}$ will be identical, so that the second term in (44) disappears; hence, mean cross-sectional growth factors of consumption and permanent income are the same:

In a Harmenberg-invariant economy (and therefore also any Szeidl-invariant economy), a similar proposition holds in the cross-section as a direct implication of the fact that a constant proportion of total permanent income is accounted for by the successive sets of consumers with any particular $\mNrm$ (recall Equation (42)). This fact is one way of interpreting Harmenberg’s definition of the density of the permanent-income-weighted invariant distribution of $\mNrm$; call this density $\Harm{f}$. To understand $\Harm{f}$, we can see how total aggregate market resources held by people with given $\mNrm$ will be:

By implication of Theorem Theorem 4, $\MLvl_{t}$ grows at a rate $\PermGroFac$. We will now use this property of $\Harm{f}$ to show that aggregate consumption also grows at rate $\PermGroFac$. Call $\CLvl_{t}(\mNrm)$ the total amount of consumption at date $t$ by persons with market resources $\mNrm$, and note that in the invariant economy this is given by the converged consumption function $\cFunc(\mNrm)$ multiplied by the amount of permanent income accruing to such people $\Harm{f}(\mNrm)\PermLvlAgg_{t}$. Since $\Harm{f}(\mNrm)$ is invariant and aggregate permanent income grows according to $\PermLvlAgg_{t+1} = \PermGroFac \PermLvlAgg_{t}$, for any $\mNrm$, the following characterizes the growth of total consumption:

Aggregate Balanced Growth and Idiosyncratic Covariances¶

Harmenberg shows that the covariance between the individual consumption ratio $\cNrm$ and the idiosyncratic component of permanent income $\permLvl$ does not shrink to zero; thus, covariances are another potential measurement for construction of microfoundations.

Consider a date-$t$ Harmenberg-converged economy, and define the mean value of the consumption ratio as $\cNrmAvg_{t+n} \equiv \Mean[\cNrm_{t+n}]$. Normalizing period-$t$ aggregate permanent income to $\PermLvlAgg_{t}=1$, total consumption at $t+1$ and $t+2$ are

$$ \begin{aligned}

\CLvl_{t+1} & = \Mean[\cNrm_{t+1}\permLvl_{t+1}] = \cNrmAvg_{t+1}\PermGroFac^{1} + \cov_{t+1}(\cNrm_{t+1}, \permLvl_{t+1})
\\  \CLvl_{t+2} & = \Mean[\cNrm_{t+2}\permLvl_{t+2}] = \cNrmAvg_{t+2}\PermGroFac^{2} + \cov_{t+2}(\cNrm_{t+2}, \permLvl_{t+2})

\end{aligned}$$and Harmenberg’s proof that$\CLvl_{t+2}-\PermGroFac \CLvl_{t+1}=0$ allows us to obtain:

$$\begin{aligned}

\left(\cNrmAvg_{t+2} - \cNrmAvg_{t+1}\right)\PermGroFac^{2} & = \PermGroFac \cov_{t+1} - \cov_{t+2} .

\end{aligned}$$

In a Szeidl-invariant economy, $\cNrmAvg_{t+2} = \cNrmAvg_{t+1}$, so the economy exhibits balanced growth in the covariance:

The more interesting case is when the economy is Harmenberg- but not Szeidl-invariant. In that case, if the $\cov$ and the $\cNrmAvg$ terms have constant growth factors $\GroFac_{\cov}$ and $\GroFac_{\cNrmAvg}$,^[33] an equation corresponding to Paragraph will hold in $t+n$:

so for the LHS and RHS to grow at the same rates we need

This is intuitive: In the Szeidl-invariant economy, it just reproduces our result above that the covariance exhibits balanced growth because $\GroFac_{\cNrmAvg}=1$. The revised result just says that in the Harmenberg case where the mean value $\cNrmAvg$ of the consumption ratio $\cNrm$ can grow, the covariance must rise in proportion to any ongoing expansion of $\cNrmAvg$ (as well as in proportion to the growth in $\permLvl$).

Implications for Microfoundations¶

Thus we have microeconomic propositions, for both growth factors and for covariances of observable variables,^[34] that can be tested in either cross-section or panel microdata to judge (and calibrate) the microfoundations that should hold for any macroeconomic analysis that requires balanced growth for its conclusions.

At first blush, these points are reassuring; one of the most persuasive arguments for the agenda of building microfoundations of macroeconomics is that newly available ‘big data’ allow us to measure cross-sectional covariances with great precision, so that we can use microeconomic natural experiments to disentangle questions that are hopelessly entangled in aggregate time-series data. Knowing that such covariances ought to be a stable feature of a stably growing economy is therefore encouraging.

But this discussion also highlights an uncomfortable point: In the model as specified, permanent income does not have a limiting distribution; it becomes ever more dispersed as the economy with infinite-horizon consumers continues to exist indefinitely.

A few microeconomic data sources permit direct measurement of ‘permanent income’; among the best (in data quality and span) is data from the Norwegian national registry, which has a long span of well-measured data for millions of Norwegians. Recent work by (Crawley, Holm, and Tretvoll, 2022) demonstrates that these data point strongly to the presence of a component of income shocks that is either truly permanent, or so extremely highly serially correlated as to be indistinguishable from permanent shocks. Using IRS tax data, (DeBacker, Heim, Panousi, Ramnath, and Vidangos, 2013) similarly find a large permanent (or very nearly permanent) component to income shocks. In quite a different exercise (Carroll, Sla{}calek, To{}ku{}o{}ka, and White, 2017) show that their calibration of the magnitude of permanent shocks (and mortality; see below) yield a simulated distribution of permanent income that matches answers in the U.S. Survey of Consumer Finances (‘SCF’) to a question designed to elicit a direct measure of respondents’ permanent income.

For macroeconomists who want to build microfoundations by comparing the microeconomic implications of their models to micro data (directly – not in ratios to difficult-to-meaure ‘permanent income’), it would be something of a challenge to determine how to construct empirical-data-comparable simulated results from a model with no limiting distribution of permanent income.

Death can solve this problem.

Mortality Yields Invariance¶

Most heterogeneous-agent models incorporate a constant positive probability of death, following (Blanchard, 1985) and (Yaari, 1965). In the Blanchardian model, if the probability of death exceeds a threshold that depends on the size of the permanent shocks, (Carroll, Sla{}calek, To{}ku{}o{}ka, and White, 2017) show that the limiting distribution of permanent income has a finite variance. (Blanchard, 1985) assumes a universal annuitization scheme in which estates of dying consumers are redistributed to survivors in proportion to survivors’ wealth, giving the recipients a higher effective rate of return. This treatment has considerable analytical advantages, most notably that the effect of mortality on the time preference factor is the exact inverse of its effect on the (effective) interest factor. That is, if the ‘pure’ time preference factor is $\DiscFacRaw$ and probability of remaining alive (not dead) is $\Alive$, then the assumption that no utility accrues after death makes the effective discount factor $\DiscFacLiv=\DiscFacRaw\Alive$ while the enhancement to the rate of return from the annuity scheme yields an effective interest factor $\RfreeEff=\Rfree/\Alive$ (recall that because of white-noise mortality, the average wealth of the two groups is identical). Combining these, the effective patience factor in the new economy $\DiscFacLiv \RfreeEff$ is unchanged from its value in the infinite-horizon model:

The only adjustments this requires to the analysis above are therefore to the few elements that involve a role for the interest factor distinct from its contribution to $\APFac$ (principally, the RIC, which becomes $\APFac/\RfreeEff$).

(Blanchard, 1985)’s innovation was valuable not only for the insight it provided but also because when he wrote, the principal alternative, the Life Cycle model of (Modigliani, 1966), was computationally challenging given then-available technologies. Despite its (considerable) conceptual value, Blanchard’s analytical solution is now rarely used because essentially all modern modeling incorporates uncertainty, constraints, and other features that rule out analytical solutions anyway.

The simplest alternative to Blanchard is to follow Modigliani in constructing a realistic description of income over the life cycle and assuming that any wealth remaining at death occurs accidentally (not implausible, given the robust finding that for the great majority of households, bequests amount to less than 2 percent of lifetime earnings, (Hendricks, 2001; Hendricks, 2016)).

Even if bequests are accidental, a macroeconomic model must make some assumption about how they are disposed of: As windfalls to heirs, estate tax proceeds, etc. We again consider the simplest choice, because it represents something of a polar alternative to Blanchard. Without a bequest motive, there are no behavioral effects of a 100 percent estate tax; we assume such a tax is imposed and that the revenues are effectively thrown in the ocean: The estate-related wealth effectively vanishes from the economy.

The chief appeal of this approach is the simplicity of the change it makes in the condition required for the economy to exhibit a balanced growth equilibrium (for consumers without a life cycle income profile). If $\Alive$ is the probability of remaining alive, the condition changes from the plain growth impatience to a looser mortality-adjusted version of growth impatience:

With no income growth, what is required to prohibit unbounded growth in aggregate wealth is the condition that prevents the per-capita wealth-to-permanent-income ratio of surviving consumers from growing faster than the rate at which mortality diminishes their collective population. With income growth, the aggregate wealth-to-income ratio will head to infinity only if a cohort of consumers is patient enough to make the desired rate of growth of wealth fast enough to counteract combined erosive forces of mortality and productivity.

Consumer Patience and Limiting Consumption¶

Having established our formal results, we are ready to describe how the various patience conditions determine the characteristics of the limiting consumption function. To fix ideas, we start with a quantitative example using the familiar benchmark case where return impatience, growth impatience and finite human wealth all hold, shown by Figure Figure 3. The figure depicts the successive consumption rules that apply in the last period of life $(\cFunc_{T})$, the second-to-last period, and earlier periods under parameter values listed in Table Table 2. (The 45 degree line is $\cFunc_{T}(\mNrm) = m$ because in the last period of life it is optimal to spend all remaining resources.)

Under the same parameter values, Figures Figure 4–Figure 5 capture the theoretical bounds and MPCs of the converged consumption rule. In Figure Figure 4, as $\mNrm$ rises, the marginal propensity to consume approaches $\MPCmin=(1-\RPFac)$ as $m \rightarrow \infty$, the same as the perfect foresight MPC. Moreover, as $\mNrm$ approaches zero, the MPC approaches $\MPCmax=(1-\pZero^{1/\CRRA}\RPFac)$.

While in the presence of a constraint neither return impatience nor growth impatience is individually necessary for nondegeneracy of $\cFunc(m)$, a key conclusion of this section is that if both return impatience and growth impatience fail, the consumption function will be degenerate (limiting either to $\cFunc(m)=0$ or $c(\mNrm)=\infty$ as the horizon recedes). So, for a useful solution, at least one of these conditions must hold.^[35] The case with growth impatience but return patience is particularly surprising, because it is not immediately clear what prevents our earlier conclusion (in other circumstances) that return patience leads $\cFunc(m)$ to asymptote to zero. The trick is to note that if return patience holds, $\Rfree < \APFac$, while failure of growth impatience means $\APFac < \PermGroFac$; together these inequalities tell us that $\Rfree < \PermGroFac$ so (limiting) human wealth is infinite.^[36] But, if at any $\mNrm$ human wealth is unbounded, what prevents $\cFunc$ from asymptoting to $\cFunc(m)=\infty$ as the horizon gets arbitrarily long? This is where the natural borrowing constraint comes in. We will show that growth impatience is sufficient, at any fixed $\mNrm$, to guarantee an upper bound to $\cFunc(m)$. The insight is best understood by first abstracting from uncertainty and studying the perfect foresight case (with and without constraints).

Model with Perfect Foresight¶

Claims Property 1-Property 2 established the relationship between the finite value of autarky, return impatience and growth impatience in the context of a model with uncertainty. The easiest way to grasp the relations among these conditions is by studying Figure Figure 6. Each node represents a quantity defined above. The arrow associated with each inequality imposes the condition, which is defined by the originating quantity being smaller than the arriving quantity. For example, one way we wrote the

finite value of autarky (under perfect foresight) in Equation (15) is $\APFac < \Rfree^{1/\CRRA} \PermGroFac^{1-1/\CRRA}$, so imposition of

finite value of autarky is captured by the diagonal arrow connecting $\APFac$ and $\Rfree^{1/\CRRA}\PermGroFac^{1-1/\CRRA}$. Traversing the boundary of the diagram clockwise starting at $\APFac$ involves imposing first growth impatience ($\APFac < \PermGroFac$) then