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Abstract
This paper builds foundations for rigorous and intuitive understanding of ‘buﬀer stock’ saving models (Bewley (1977)-like models with a wealth target), pairing each theoretical result with quantitative illustrations. After describing conditions under which a consumption function exists, the paper articulates stricter ‘Growth Impatience’ conditions that guarantee alternative forms of stability — either at the population level, or for individual consumers. Together, the numerical tools and analytical results constitute a comprehensive toolkit for understanding buﬀer stock models.

Keywords

Precautionary saving, buﬀer stock saving, marginal propensity to consume, permanent income hypothesis, income ﬂuctuation problem
JEL codes

D81, D91, E21

For consumers whose income is aﬀected by realistic transitory and permanent shocks (a la Friedman (1957) and Muth (1960)), only one further ingredient is required to construct a microeconomically testable model of optimal consumption: A description of preferences. Zeldes (1989) was the ﬁrst to calibrate a quantitatively plausible example, spawning a literature showing that such models’ predictions can match household life cycle data reasonably well, whether or not explicit liquidity constraints are imposed.¹

A connected literature, starting with Bewley (1977), has derived limiting properties of related inﬁnite-horizon problems – but only in models more complex than the case with just Friedman-Muth shocks and preferences, because standard contraction mapping theorems (from Bellman (1957) through Stokey et al. (1989) and beyond) cannot be applied when utility and/or marginal utility are unbounded, and many proof methods rule out Friedman-Muth permanent shocks.²

This paper’s ﬁrst contribution is to articulate conditions under which the inﬁnite-horizon Friedman-Muth(-Zeldes) problem (with permanent shocks, and without complications like a consumption ﬂoor or liquidity constraints) deﬁnes a contraction mapping whose limit is useful (neither $c = 0$ nor $c = ∞$ everywhere) as the horizon recedes. A ‘Finite Value of Autarky Condition’ is mostly suﬃcient (the ‘Weak Return Impatience Condition’³ is unlikely to bind). Because the inﬁnite-horizon solution is the limit of ﬁnite-horizon recursions, many results are also applicable to ﬁnite-horizon problems.

But the main theoretical contribution is to identify, for the inﬁnite-horizon case, conditions under which ‘stable’ points exist (the ratio of wealth to permanent income will move toward a ‘target’; alternatively, there is a ‘balanced growth’ equilibrium) either for individual consumers or for the aggregate. The requirement for stability is always that the model’s parameters satisfy a ‘Growth Impatience Condition’ whose details depend on the quantity whose stability is of interest. A model with any such stable point(s) qualiﬁes as a ‘buﬀer stock’ model. (Buﬀer stock models are neither a subset nor a superset of Bewley (1977) models – see below).⁴

Even without a formal proof of its existence, buﬀer stock saving has been intuitively understood to underlie key results in heterogeneous agent macroeconomics; for example, the logic of target saving is central to the claim by Krueger, Mitman, and Perri (2016) in the Handbook of Macroeconomics that such models explain why, during the Great Recession, middle-class consumers cut their spending more than the poor or the rich. Theory below provides the rigorous basis for this claim: Learning that the future has become more uncertain does not change the urgent imperatives of the poor (their high $u′(c)$ means they — optimally — have little room to maneuver). And, increased labor income uncertainty does not much change the behavior of the rich because it poses little risk to their consumption. Only people in the middle have both the motivation and the wiggle-room to respond to uncertainty by substantially reducing their spending. Analytical derivations required for the proofs also provide intuition for many other results familiar from the numerical literature.

The paper begins by describing suﬃcient conditions for the problem to deﬁne a sensible (nondegenerate) limiting consumption function (while explaining how the model relates to those previously considered). The conditions are interestingly parallel to those required for the liquidity constrained perfect foresight model; that parallel is explored and explained. The limiting properties of the consumption function as resources approach inﬁnity, and as they approach their lower bound, are then used to prove the contraction mapping theorem.

The next theoretical contribution is to show that a model with an ‘artiﬁcial’ liquidity constraint (it prohibits borrowing by consumers who could certainly repay) is a limiting case of the unconstrained model. The analytical appeal of the unconstrained model is that it is both mathematically convenient (the consumption function is everywhere twice continuously diﬀerentiable), and arbitrarily close (cf. Section 2.11) to less tractable models. This congenial environment makes proofs easier (if we deﬁne a proposition as holding in the inﬁnite-horizon limit when it holds as a ﬁnite horizon extends to inﬁnity).

In proving the remaining theorems, the next section examines key properties of the model. First, as cash approaches inﬁnity the expected growth rate of consumption and the marginal propensity to consume (MPC) converge to their values in the perfect foresight case. Second, as cash approaches zero the expected growth rate of consumption approaches inﬁnity, and the MPC approaches a simple analytical limit. Next, the central theorems articulate conditions under which diﬀerent measures of ‘growth impatience’ imply conclusions about points of stability (‘target’ or ‘balanced growth’ points).

The ﬁnal section elaborates the conditions under which, even with a ﬁxed aggregate interest rate that diﬀers from the time preference rate, a small open economy populated by buﬀer stock consumers has a balanced growth equilibrium in which growth rates of consumption, income, and wealth match the exogenous growth rate of 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, because it requires no ad-hoc assumptions. Not even liquidity constraints.⁵

2 The Problem

2.1 Setup

The inﬁnite horizon solution is the (limiting) ﬁrst-period solution to a sequence of ﬁnite-horizon problems as the horizon (the last period of life) becomes arbitrarily distant.

That is, for the value function, ﬁxing a terminal date $T$ , we are interested in $vT− n$ in the sequence of value functions ${v ,v ,...,v } T T−1 T −n$ . We will say that the problem has a ‘nondegenerate’ inﬁnite horizon solution if, corresponding to that $v$ , as $n ↑ ∞$ there is a limiting consumption function $c(m, p) = limn ↑∞ cT− n(m, p)$ which is neither $c = 0$ everywhere (for all $m, p$ ) nor $c = ∞$ everywhere (a ‘sensible’ solution).

exhibits relative risk aversion $ρ > 1$ .⁷ The consumer’s initial condition is deﬁned by market resources $mt$ and permanent noncapital income $pt$ , which both are positive,

In the usual exposition, a dynamic budget constraint (DBC) combines several distinct events that jointly determine next period’s $mt+1$ (given this period’s choices); for the detailed analysis here, we disarticulate and describe every separate step:⁸ $,$ ⁹

where $a t$ measures the consumer’s assets at the end of $t$ , which translate one-for-one into capital $kt+1$ at the beginning of the next period. Before the consumption choice, $kt+1$ is augmented by a ﬁxed interest factor $R = (1 + r)$ to become $bt+1$ – the consumer’s ﬁnancial (‘bank’) balances before next period’s consumption choice; $mt+1$ (‘market resources’) is the sum of ﬁnancial wealth $bt+1$ and noncapital income $pt+1ξξξt+1$ (permanent noncapital income $p t+1$ multiplied by the transitory-income-shock factor $ξξξt+1$ described below). $pt+1$ is derived from $pt$ via application of a growth factor $𝒢$ ,¹⁰ modiﬁed by a mean-one iid shock $Ψt+1$ , $𝔼t[Ψt+n ] = 1 ∀ n ≥ 1$ satisfying $Ψ ∈ [Ψ,-¯Ψ ]$ for $0 < Ψ ≤ 1 ≤ ¯Ψ < ∞ ---$ (and $Ψ = Ψ¯ = 1 ---$ is the degenerate case with no permanent shocks).

Following Zeldes (1989), in future periods $t + n ∀ n ≥ 1$ there is a small probability $℘$ that income will be zero (a ‘zero-income event’),

where $𝜃𝜃𝜃t+n$ is an iid mean-one random variable ( $𝔼t[𝜃𝜃𝜃t+n] = 1 ∀ n > 0$ ) whose distribution satisﬁes $𝜃𝜃𝜃 ∈ [𝜃𝜃𝜃,¯𝜃𝜃𝜃]$ where $0 < 𝜃𝜃𝜃 ≤ 1 ≤ ¯𝜃𝜃𝜃 < ∞ --$ .¹¹ Call the cumulative distribution functions $ℱ Ψ$ and $ℱ𝜃𝜃𝜃$ (where $ℱξξξ$ is derived trivially from (3) and $ℱ 𝜃𝜃𝜃$ ). For quick identiﬁcation in tables and graphs, we will call this the ‘Friedman/Muth’ model because it is a speciﬁc implementation of Friedman (1957)’s ideas as interpreted by Muth (1960).

The model looks more special than it is. In particular, a positive probability of zero-income events may seem objectionable (despite empirical support).¹² But a nonzero minimum value of $ξξξ$ (motivated, say, by the existence of unemployment insurance) could be handled by capitalizing the PDV of minimum income into current market assets,¹³ 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.

2.2 Comparison to Existing Literature

This model diﬀers from Bewley’s (1977) classic formulation in several ways. The CRRA utility function does not satisfy Bewley’s assumption that $u (0 )$ is well-deﬁned, or that $u′(0)$ is well-deﬁned and ﬁnite; indeed, neither the value function nor the marginal value function will be bounded. It diﬀers from Schectman and Escudero (1977) in that they impose liquidity constraints and positive minimum income. It diﬀers from both of these in that it permits permanent growth in income, and also permanent shocks to income, which a large empirical literature ﬁnds to be of dominant importance in microdata.¹⁴ It diﬀers from Deaton (1991) because liquidity constraints are absent; there are separate transitory and permanent shocks (a la Muth (1960)); and the transitory shocks here can occasionally cause income to reach zero.

It diﬀers from models found in Stokey et. al. (1989) because neither liquidity constraints nor bounds on utility or marginal utility are imposed.¹⁵ $,$ ¹⁶ Li and Stachurski (2014) show how to allow unbounded returns by using policy function iteration, but also impose constraints.

The paper with perhaps the most commonalities is Ma, Stachurski, and Toda (2020), henceforth MST, who establish existence and uniqueness of a solution to a general income ﬂuctuation problem in a Markovian setting. The most important diﬀerences are that MST impose liquidity constraints and that expected marginal utility of income is ﬁnite ( $𝔼[u′(Y )] < ∞ )$ . These assumptions are not consistent with the combination of CRRA utility and income dynamics used here, whose joint properties are key to the results.¹⁷

2.3 The Problem Can Be Normalized By Permanent Income

We establish a bit more notation by reviewing the familiar result that in such problems (CRRA utility, permanent shocks) the number of states can be reduced from two ( $m$ and $p$ ) to one $(m = m ∕p )$ . Value in the last period is $u(m ) T$ ; using (in the last line in (4)) the fact that for our CRRA utility function, $1−ρ u(xy ) = x u(y)$ , and generically deﬁning nonbold variables as the boldface counterpart normalized by $pt$ (as with $m = m ∕p$ ), consider the problem in the second-to-last period,

Now, in a one-time deviation from the notational convention established in the last sentence, deﬁne nonbold ‘normalized value’ not as $vt∕pt$ but as $vt = vt∕p1t− ρ$ , because this allows us to exploit features of the related problem,

where $ℛt+1 ≡ (R ∕𝒢t+1)$ is a ‘permanent-income-growth-normalized’ return factor, and the reformulated problem’s ﬁrst order condition is¹⁸

Since $v (m ) = u(m ) T T T$ , deﬁning $v (m ) T −1 T−1$ from (5), we obtain

This logic induces to earlier periods; if we solve the normalized one-state-variable problem (5), we will have solutions to the original problem for any $t < T$ from:

2.4 Deﬁnition of a Nondegenerate Solution

The problem has a nondegenerate solution if, as the horizon $n$ gets arbitrarily large, the solution in the ﬁrst period of life $cT− n(m)$ gets arbitrarily close to a limiting $c(m )$ :

2.5 Perfect Foresight Benchmarks

The familiar analytical solution to the perfect foresight model, obtained by setting $℘ = 0$ and $𝜃𝜃𝜃-= ¯𝜃𝜃𝜃 = Ψ--= Ψ¯ = 1$ , allows us to deﬁne remaining notation and terminology.

2.5.1 The Finite Human Wealth Condition (FHWC)

The dynamic budget constraint, strictly positive marginal utility, and the can’t-die-in-debt condition (2) imply an exactly-holding intertemporal budget constraint (IBC):

where $b$ is beginning-of-period ‘market’ balances; with $ℛ ≡ R∕𝒢$ ‘human wealth’ is

where we call $𝒢 ∕R$ the ‘Finite Human Wealth Factor’ (FHWF) because if this factor is less than one human wealth will be ﬁnite because (noncapital) income growth is smaller than the interest rate at which that income is being discounted.

2.5.2 The Return Impatience Condition (RIC)

Without constraints, the consumption Euler equation always holds; with $′ −ρ u (c) = c$ ,

where the archaic letter ‘thorn’ represents what we will call the ‘Absolute Patience Factor’ because, if the ‘absolute impatience condition’ holds,¹⁹

the consumer’s level of spending will be too large to sustain indeﬁnitely. We call such a consumer ‘absolutely impatient.’

where $κ- t$ is the marginal propensity to consume (MPC) — answering the question ‘if the consumer had an extra unit of resources, how much more spending would occur?’ (The overbar signiﬁes that $¯c$ will be an upper bound as we modify the problem to incorporate constraints and uncertainty; analogously, $κ-$ is the MPC’s lower bound).

The horizon-exponentiated term in the denominator of (13) is why, for $κ-$ to be strictly positive as $n = T − t$ goes to inﬁnity, we must impose the Return Impatience Condition:

The RIC thus implies that the consumer cannot be so pathologically patient as to wish, in the limit as the horizon approaches inﬁnity, to spend nothing today out of an increase in current wealth (the RIC rules out the degenerate limiting solution $¯c(m ) = 0$ ). We call a consumer who satisﬁes the RIC ‘return impatient.’

2.5.3 Nondegenerate PF Unconstrained Solution Requires FHWC

Given that the RIC holds, and (as before) deﬁning limiting objects by the absence of a time subscript, the limiting upper bound consumption function will be

and so in order to rule out the degenerate limiting solution $¯c(m) = ∞$ we need $h$ to be ﬁnite; that is, we must impose the Finite Human Wealth Condition (FHWC), eq. (9).

2.5.4 The Growth Impatience Condition (GIC)

2.5.5 Perfect Foresight Finite Value of Autarky Condition (PFFVAC)

Under ‘autarky,’ capital markets do not exist; the consumer has no choice but to spend permanent noncapital income $p$ in every period. Because $u(xy ) = x1 −ρu(y)$ , the value the consumer would achieve is

which (for $𝒢 > 0$ ) asymptotes to a ﬁnite number as $n = T − t$ approaches $+ ∞$ if any of these equivalent conditions holds:

where we call $ℶ$ ²⁰ the ‘Perfect Foresight Value Of Autarky Factor’ (PF-VAF), and the variants of (19) constitute alternative formulations of the Perfect Foresight Finite Value of Autarky Condition (‘PF-FVAC’); they guarantee that a perfect-foresight consumer who always spends all permanent income has ‘ﬁnite autarky value.’²¹

If the FHWC is satisﬁed, the PF-FVAC implies that the RIC is satisﬁed.²² Likewise, if the FHWC and the GIC are both satisﬁed, PF-FVAC follows:

(the last line holds because FHWC $⇒ 0 ≤ (𝒢∕R ) < 1$ and $ρ > 1 ⇒ 0 < 1 − 1∕ ρ < 1$ ).

The ﬁrst panel of Table 4 summarizes: The PF-Unconstrained model has a nondegenerate limiting solution if we impose the RIC and FHWC (these conditions are necessary as well as suﬃcient). Together the PF-FVAC and the FHWC imply the RIC. If we impose the GIC and the FHWC, both the PF-FVAC and the RIC follow, so GIC+FHWC are also suﬃcient. But there are circumstances under which the RIC and FHWC can hold while the PF-FVAC fails (‘ $¬$ PF-FVAC’). For example, if $𝒢 = 0$ , the problem is a standard ‘cake-eating’ problem with a nondegenerate solution under the RIC (when the consumer has access to capital markets).

The easiest way to grasp the relations among these conditions is by studying Figure 1. Each node represents a quantity deﬁned above. The arrow associated with each inequality imposes that condition. For example, one way we wrote the PF-FVAC in equation (19) is $ÞÞÞ < R1∕ρ𝒢1−1∕ρ$ , so imposition of the PF-FVAC is captured by the diagonal arrow connecting $ÞÞÞ$ and $R1∕ρ𝒢1−1∕ρ$ . Traversing the boundary of the diagram clockwise starting at $ÞÞÞ$ involves imposing ﬁrst the GIC then the FHWC, and the consequent arrival at the bottom right node tells us that these two conditions jointly imply the PF-FVAC. Reversal of a condition reverses the arrow’s direction; so, for example, the bottom-most arrow going to $R1∕ρ𝒢1−1∕ρ$ imposes $¬$ FHWC; but we can cancel the cancellation and reverse the arrow. This would allow us to traverse the diagram clockwise from $ÞÞÞ$ through $𝒢$ to $1∕ρ 1− 1∕ρ R 𝒢$ to $R$ , revealing that imposition of GIC and FHWC (and, redundantly, FHWC again) let us conclude that the RIC holds because the starting point is $ÞÞÞ$ and the endpoint is $R$ . (Consult Appendix I for an exposition of diagrams of this type, which are a simple application of Category Theory (Riehl (2017))).

2.5.6 PF Constrained Solution Exists Under RIC or { $¬$ RIC, GIC}

We next sketch the perfect foresight constrained solution because it deﬁnes a benchmark (and limit) for the unconstrained problem with uncertainty (our ultimate interest).

If a liquidity constraint requiring $b ≥ 0$ is ever to be relevant, it must be relevant at the lowest possible level of market resources, $mt = 1$ , deﬁned by the lower bound, $bt = 0$ (if it were relevant at any higher level of $m$ , it would certainly be relevant here, because $u′ > 0$ ). The constraint is ‘relevant’ if it prevents the choice that would otherwise be optimal; at $mt = 1$ it is relevant if the marginal utility from spending all of today’s resources $ct = mt = 1$ , exceeds the marginal utility from doing the same thing next period, $ct+1 = 1$ ; that is, if such choices would violate the Euler equation (6):

We now examine implications of possible conﬁgurations of the conditions. (Tables 3 and 4 (and the table in appendix E) codify.)

$¬$ GIC and RIC. If the GIC fails but the RIC (14) holds, Appendix E shows that, for some $0 < m < 1 #$ , an unconstrained consumer behaving according to the perfect foresight solution (16) would choose $c < m$ for all $m > m#$ . In this case the solution to the constrained consumer’s problem is simple: For any $m ≥ m#$ the constraint does not bind (and will never bind in the future); for such $m$ the constrained consumption function is identical to the unconstrained one. If the consumer were somehow²³ to arrive at an $m < m# < 1$ the constraint would bind and the consumer would consume $c = m$ . Using $`∙$ for the version of a function $∙$ in the presence of constraints (and recalling that $¯c(m )$ is the unconstrained perfect foresight solution):

GIC and RIC. When the RIC and GIC both hold, Appendix E shows that the limiting constrained consumption function is piecewise linear, with $`c(m ) = m$ up to a ﬁrst ‘kink point’ at $0 m # > 1$ , and with discrete declines in the MPC at a set of kink points $1 2 {m #, m #,...}$ . As $m ↑ ∞$ the constrained consumption function $`c(m )$ becomes arbitrarily close to the unconstrained $¯c(m )$ , and the marginal propensity to consume function $`κκκ(m ) ≡ `c′(m )$ limits to $κ-$ .²⁴ Similarly, the value function $`v (m)$ is nondegenerate and limits into the value function of the unconstrained consumer.

This logic holds even when the ﬁnite human wealth condition fails ( $¬$ FHWC), because the constraint prevents the (limiting) consumer²⁵ from borrowing against unbounded human wealth to ﬁnance unbounded current consumption. Under these circumstances, the consumer who starts with any $b > 0 t$ will, over time, run those resources down so that after some ﬁnite number of periods $τ$ the consumer will reach $bt+τ = 0$ , and thereafter will set $c = p$ for eternity (which the PF-FVAC says yields ﬁnite value). Using the same steps as for equation (19), value of the interim program is also ﬁnite:

So, even under $¬$ FHWC, the limiting consumer’s value for any ﬁnite $m$ will be the sum of two ﬁnite numbers: One due to the unconstrained choice made over the ﬁnite horizon leading up to $bt+τ = 0$ , and one reﬂecting the value of consuming $pt+τ$ thereafter.

GIC and $¬$ RIC. The most peculiar possibility occurs when the RIC fails. Under these circumstances the FHWC must also fail (Appendix E), and the constrained consumption function is nondegenerate. (See appendix Figure 8 for a numerical example). While $limm ↑∞ `κκκ(m ) = 0$ , nevertheless the limiting constrained consumption function $`c(m )$ is ﬁnite, strictly positive, and strictly increasing in $m$ . This result interestingly reconciles the conﬂicting intuitions from the unconstrained case, where $¬$ RIC would suggest a degenerate limit of $`c(m ) = 0$ while $¬$ FHWC would suggest a degenerate limit of $`c(m ) = ∞$ .

We now examine the case with uncertainty but without constraints, which turns out to be a close parallel to the model with constraints but without uncertainty.

2.6 Uncertainty-Modiﬁed Conditions

2.6.1 The Growth Impatience Condition (GIC-Mod))

When noncapital income uncertainty is introduced, the expectation of beginning-of-period bank balances $bt+1$ can be rewritten as:

where Jensen’s inequality guarantees that the expectation of the inverse of the permanent shock is greater than one. Now deﬁne

which satisﬁes $Ψ--< 1$ (thanks again to Mr. Jensen), so it is convenient to deﬁne

because this allows us to write uncertainty-modiﬁed versions of earlier equations and conditions in a manner exactly parallel to those for the perfect foresight case; for example, we deﬁne a modiﬁed Growth Patience Factor:

that is stronger than the perfect foresight version (18) because $𝒢 < 𝒢 --$ .

2.6.2 The Finite Value of Autarky Condition (FVAC)

Analogously to (19), value for a consumer who spent exactly their permanent income every period would reﬂect the product of the expectation of the (independent) future shocks to permanent income:

suggesting the deﬁnition of a utility-compensated equivalent of the permanent shock,

and (28) is the ‘ﬁnite value of autarky condition’ because it guarantees value is ﬁnite for a consumer who always consumes (now stochastic) permanent income ( $ℶ- --$ is the ‘Value of Autarky Factor’ (‘VAF’)).²⁶ For nondegenerate $Ψ$ , this condition is stronger (harder to satisfy – requiring lower $β$ ) – than the perfect foresight version (19) because $𝒢-< 𝒢 --$ .²⁷

2.7 The Baseline Numerical Solution

Figure 2, familiar from the literature, depicts the successive consumption rules that apply in the last period of life $(c (m )) T$ , the second-to-last period, and earlier periods under baseline parameter values listed in Table 2. (The 45 degree line is $cT(m ) = m$ because in the last period of life it is optimal to spend all remaining resources.)

In the ﬁgure, the consumption rules appear to converge to a nondegenerate $c(m )$ . Our next purpose is to show that this appearance is not deceptive.

2.8 Concave Consumption Function Characteristics

A precondition for the main theorem is that the maximization problem deﬁnes a sequence of continuously diﬀerentiable strictly increasing strictly concave²⁸ functions ${cT,cT− 1,...}$ . The straightforward but tedious proof is relegated to Appendix A. (Carroll and Kimball (1996) proved concavity but not continuous diﬀerentiability, which our theorem will use). For present purposes, the important point is that the income process induces what Aiyagari (1994) dubbed a ‘natural borrowing constraint’: $ct(m ) < m$ for all periods $t < T$ because a consumer who spent all available resources would arrive in period $t + 1$ with balances $bt+1$ of zero, and then might earn zero income over the remaining horizon, risking the possibility of a requirement to spend zero, yielding negative inﬁnite utility. To avoid this calamity, the consumer never spends everything. Zeldes (1989) seems to have been the ﬁrst to argue, based on his numerical results, that the natural borrowing constraint was a quantitatively plausible alternative to ‘artiﬁcial’ or ‘ad hoc’ borrowing constraints.²⁹

Strict concavity and continuous diﬀerentiability of the consumption function are key elements in many of the arguments below, but are not characteristics of models with ‘artiﬁcial’ borrowing constraints (though the analytical convenience of these features is considerable – see below).

2.9 Bounds for the Consumption Functions

The consumption functions depicted in Figure 2 appear to have limiting slopes as $m ↓ 0$ and as $m ↑ ∞$ . This section conﬁrms that impression and derives those slopes, which will be needed in the contraction mapping proof.³⁰

Assume (as justiﬁed above) that a continuously diﬀerentiable concave consumption function exists in period $t + 1,$ with an origin at $ct+1 (0 ) = 0$ , a minimal MPC $ˆκt+1 > 0$ , and maximal MPC $¯κt+1 ≤ 1$ . (If $t + 1 = T$ these will be $ˆκT = ¯κT = 1$ ; for earlier periods they will exist by recursion.)

Under our imposed assumption that human wealth is ﬁnite, the MPC bound as wealth approaches inﬁnity is easy to understand: 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. In the course of proving this, Appendix G provides a recursive expression (used below) for the (inverse of the) limiting MPC as wealth approaches $∞$ :

2.9.1 The Weak RIC Condition (WRIC)

Appendix equation (86) reﬂects the derivation of a parallel expression for the limiting maximal MPC as $m ↓ 0 t$ :

where ${ } ∞ ¯κ −T1−n n=0$ is a decreasing convergent sequence if the ‘weak return patience factor’ $℘1∕ρÞÞÞ R$ satisﬁes:

a condition we dub the ‘Weak Return Impatience Condition’ because with $℘ < 1$ it will hold more easily (for a larger set of parameter values) than the RIC ( $ÞÞÞR < 1$ ). The essence of the argument is that as wealth approaches zero, the overriding consideration that limits consumption is the (recursive) fear of the zero-income events. (That is why the probability of the zero income event $℘$ appears in the expression.)

We are now in position to observe that the optimal consumption function must satisfy

because consumption starts at zero and is continuously diﬀerentiable, is strictly concave,³¹ and always exhibits a slope between $κt$ and $¯κt$ (formal proof: Appendix C).

2.10 The Problem Is a Contraction Mapping Under WRIC and FVAC

As mentioned above, standard theorems in the contraction mapping literature before and after Stokey et. al. (1989) require utility or marginal utility to be bounded over the space of possible values of $m$ ; that requirement does not apply here because the possibility (however unlikely) of an unbroken string of zero-income events through the end of the horizon means that utility (and marginal utility) are unbounded as $m ↓ 0$ . Although a recent literature examines the existence and uniqueness of solutions to Bellman equations in the presence of ‘unbounded returns’ (see, e.g., Matkowski and Nowak (2011)), it appears that the techniques in that literature cannot be used to solve the problem here because the required conditions are violated by a problem that incorporates permanent shocks.³²

Fortunately, Boyd (1990) provided a weighted contraction mapping theorem that Alvarez and Stokey (1998) showed could be used to address the homogeneous case (of which CRRA is an example) in a deterministic framework; later, Durán (2003) showed how to extend the Boyd (1990) approach to the stochastic case.

For $𝒞 (𝒜, ℬ ) г$ deﬁned as the set of functions in $𝒞 (𝒜,ℬ )$ that are $г$ -bounded; $w$ , $x$ , $y$ , and $z$ as examples of $г$ -bounded functions; and using $0(m ) = 0$ to indicate the function that returns zero for any argument, Boyd (1990) proves the following.

Boyd’s Weighted Contraction Mapping Theorem. Let $T : 𝒞г (𝒜,ℬ ) → 𝒞 (𝒜, ℬ )$ such that³³ ^,³⁴

Using this, we introduce the mapping $𝒯 : 𝒞г(𝒜, ℬ ) → 𝒞 (𝒜, ℬ)$ .³⁵ Our operator $𝒯$ satisﬁes the conditions that Boyd requires of his operator $T$ if we impose two restrictions on parameter values.³⁶ The ﬁrst is the WRIC necessary for convergence of the maximal MPC, equation (31) above. More serious is the Finite Value of Autarky Condition, equation (28). (The interpretation of these restrictions is elaborated in Section 2.12 below.) Imposing these restrictions, we are now in position to state the central theorem of the paper.

Intuitively, Boyd’s theorem shows that if you can ﬁnd a $г$ that is everywhere ﬁnite but goes to inﬁnity ‘as fast or faster’ than the function you are normalizing with $г$ , the normalized problem deﬁnes a contraction mapping. The intuition for the FVAC condition is that, with an inﬁnite horizon, with any strictly positive initial amount of bank balances $b0$ , in the limit your value can always be made greater than you would get by consuming exactly the sustainable amount (say, by consuming $(r∕R )b0 − 𝜖$ for some arbitrarily small $𝜖 > 0$ ).

for some real scalar $η > 0$ whose value is determined in the course of the proof.³⁷ Under this deﬁnition of $г$ , ${𝒯0} (mt ) = u(¯κmt )$ is clearly $г$ -bounded.

The cumbersome details of the proof are relegated to Appendix C. Given that the value function converges, Appendix D.2 shows that the consumption functions converge.³⁸

2.11 The Liquidity Constrained Solution as a Limit

A related problem commonly considered in the literature (e.g., by Deaton (1991)), with a liquidity constraint and a positive minimum value of income, is the limit of the problem considered here as the probability $℘$ of the zero-income event approaches zero.

The essence of the argument is simple. Imposing the artiﬁcial constraint without changing $℘$ would not change behavior at all: The possibility of earning zero income over the remaining horizon already prevents the consumer from ending the period with zero assets. So, for precautionary reasons, the consumer will save something.

But 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 $℘ ↓ 0$ , that probability becomes arbitrarily small, so the amount of precautionary saving induced by the zero-income events approaches zero as $℘ ↓ 0$ . But “zero” is the amount of precautionary saving that would be induced by a zero-probability event for the impatient liquidity constrained consumer.

Another way to understand this is just to think of the liquidity constraint reﬂecting a component of the utility function that is zero whenever the consumer ends the period with (strictly) positive assets, but negative inﬁnity if the consumer ends the period with (weakly) negative assets.

See Appendix F for the formal proof justifying the foregoing intuitive discussion.³⁹

The conditions required for convergence and nondegeneracy are thus strikingly similar between the liquidity constrained perfect foresight model and the model with uncertainty but no explicit constraints: The liquidity constrained perfect foresight model is just the limiting case of the model with uncertainty as the degree of all three kinds of uncertainty (zero-income events, other transitory shocks, and permanent shocks) approaches zero.

2.12 Relations Between Parametric Restrictions

The full relationship among conditions is represented in Figure 3. Though the diagram looks complex, it is merely a modiﬁed version of the earlier simple diagram (Figure 1) with further (mostly intermediate) inequalities inserted. (Arrows with a “because” now label relations that always hold under the model’s assumptions.)⁴⁰

The ‘weakness’ of the additional condition suﬃcient for contraction beyond the FVAC, the WRIC, can be seen by asking ‘under what circumstances would the FVAC hold but the WRIC fail?’ Algebraically, the requirement is

If we require $R ≥ 1$ , the WRIC is redundant because now $β < 1 < Rρ−1$ , so that (with $ρ > 1$ and $β < 1$ ) the RIC (and WRIC) must hold. But neither theory nor evidence demand that $R ≥ 1$ . We can therefore approach the question of the WRIC’s relevance by asking just how low $R$ must be for the condition to be relevant. Suppose for illustration that $ρ = 2$ , $Ψ1-− ρ = 1.01$ , $𝒢1 −ρ = 1.01−1$ and $℘ = 0.10$ . In that case (35) reduces to

so that for example if $β = 0.96$ we would need $R < 0.096$ (that is, a perpetual riskfree rate of return of worse than -90 percent a year) in order for the WRIC to bind.

Perhaps the best way of thinking about this is to note that the space of parameter values for which the WRIC is relevant shrinks out of existence as $℘ → 0$ , which Section 2.11 showed was the precise limiting condition under which behavior becomes arbitrarily close to the liquidity constrained solution (in the absence of other risks). On the other hand, when $℘ = 1$ , the consumer has no noncapital income (so that the FHWC holds) and with $℘ = 1$ the WRIC is identical to the RIC; but the RIC is the only condition required for a solution to exist for a perfect foresight consumer with no noncapital income. Thus the WRIC forms a sort of ‘bridge’ between the liquidity constrained and the unconstrained problems as $℘$ moves from 0 to 1.

2.12.1 When the RIC Fails

In the perfect foresight problem (Section 2.5.3), the RIC was necessary for existence of a nondegenerate solution. It is surprising, therefore, that in the presence of uncertainty, the much weaker WRIC is suﬃcient for nondegeneracy (assuming that the FVAC holds). We can directly derive the features the problem must exhibit (given the FVAC) under $¬$ RIC (that is, $1∕ρ R < (Rβ) )$ :

but since $Ψ-< 1$ (cf. the argument below (26)), this requires $R∕𝒢 < 1$ ; so, given the FVAC, the RIC can fail only if human wealth is unbounded. As an illustration of the convenience of our diagrams, note that this algebraically complicated conclusion could be easily reached diagrammatically in ﬁgure 3 by starting at the $R$ node and imposing $¬RIC$ , which reverses the RIC arrow and lets us traverse the diagram along any clockwise path to the PF-VAF node at which point we realize that we cannot impose the FHWC because that would let us conclude $R > R$ .

As in the perfect foresight constrained problem, unbounded limiting human wealth ( $¬$ FHWC) here does not lead to a degenerate limiting consumption function (ﬁnite human wealth is not required for the convergence theorem). But, from equation (29) and the discussion surrounding it, an implication of $¬$ RIC is that $limm ↑∞ c′(m ) = 0$ . Thus, interestingly, in the special ${¬RIC, ¬FHWC }$ case (unavailable in the perfect foresight model) the presence of uncertainty both permits unlimited human wealth (in the $n ↑ ∞$ limit) and at the same time prevents unlimited human wealth from resulting in (limiting) inﬁnite consumption (at any ﬁnite $m$ ). Intuitively, in the presence of uncertainty, pathological patience (which in the perfect foresight model results in a limiting consumption function of $c(m ) = 0$ ) plus unbounded human wealth (which the perfect foresight model prohibits because it leads to a limiting consumption function $c(m ) = ∞$ for any ﬁnite $m$ ) combine to yield a unique ﬁnite limiting (as $n ↑ ∞$ ) level of consumption and MPC for any ﬁnite value of $m$ . Note the close parallel to the conclusion in the perfect foresight liquidity constrained model in the {GIC, $¬$ RIC} case. There, too, the tension between inﬁnite human wealth and pathological patience was resolved with a nondegenerate consumption function whose limiting MPC was zero.⁴¹

2.12.2 When the RIC Holds

FHWC. If the RIC and FHWC both hold, a perfect foresight solution exists (see 2.5.3 above). As $m ↑ ∞$ the limiting $c$ and $v$ functions become arbitrarily close to those in the perfect foresight model, because human wealth pays for a vanishingly small portion of spending. This will be the main case analyzed in detail below.

$¬$ FHWC. The more exotic case is where FHWC fails; in the perfect foresight model, {RIC, $¬$ FHWC} is the degenerate case with limiting $¯c(m ) = ∞$ . Here, the FVAC implies that the PF-FVAC holds (traverse Figure 3 clockwise from $Þ ÞÞ$ by imposing FVAC and continue to the PF-VAF node): Reversing the arrow connecting the $R$ and PF-VAF nodes implies that under $¬FHWC$ :

where the transition from the ﬁrst to the second lines is justiﬁed because $1∕ρ ¬FHWC ⇒ (R ∕𝒢) < 1$ . So, {RIC, $¬$ FHWC} implies the GIC holds. However, we are not entitled to conclude that the GIC-Mod holds: $ÞÞÞ < 𝒢$ does not imply $ÞÞÞ < Ψ-𝒢$ where $Ψ < 1 ---$ .

We have now established the principal points of comparison between the perfect foresight solutions and the solutions under uncertainty; these are codiﬁed in the remaining parts of Tables 3 and 4.

3 Analysis of the Converged Consumption Function

Figures 4-6 capture the main properties of the converged consumption rule when the RIC, GIC-Mod, and FHWC all hold.⁴²

Figure 4 shows the expected growth factors for consumption, the level of market resources, and the market resources ratio, $𝔼t[ct+1∕ct]$ and $𝔼t[mt+1 ∕mt ]$ , and $𝔼t[mt+1 ∕mt ]$ , for a consumer behaving according to the converged consumption rule, while Figures 5—6 illustrate theoretical bounds for the consumption function and the MPC.

First, as $mt ↑ ∞$ the expected consumption growth factor goes to $ÞÞÞ$ , indicated by the lower bound in Figure 4, and the marginal propensity to consume approaches $κ-= (1 − ÞÞÞR )$ (see Figure 5) — the same as the perfect foresight MPC. Second, as $mt$ approaches zero the consumption growth factor approaches $∞$ (Figure 4) and the MPC approaches $1∕ρ ¯κ = (1 − ℘ ÞÞÞR)$ (Figure 5). Third, there is a value of the market resources ratio $mt = mˇ$ at which the expected growth rate of the level of market resources $m$ matches the expected growth rate of permanent income $𝒢$ , and a diﬀerent (larger) target ratio $mˆ$ where $𝔼 [mt+1∕mt ] = 1$ (and the expected growth rate of consumption is lower than $𝒢$ ). Thus, at the individual level, this model does not have a single $m$ at which $p, m,$ and $c$ all are expected to grow at the same rate.

3.1 Limits as $m$ Approaches Inﬁnity

which is the solution to an inﬁnite-horizon problem with no noncapital income ( $ξξξt+n = 0 ∀ n ≥ 1$ ); clearly $c(m ) < c(m )$ , since allowing the possibility of future noncapital income cannot reduce current consumption. Our imposition of the RIC guarantees that $κ-> 0$ , so this solution satisﬁes our deﬁnition of nondegeneracy, and because this solution is always available it deﬁnes a lower bound on both the consumption and value functions.

Assuming the FHWC holds, the inﬁnite horizon perfect foresight solution (16) constitutes an upper bound on consumption in the presence of uncertainty, since the introduction of uncertainty strictly decreases the level of consumption at any $m$ (Carroll and Kimball (1996)). Thus,

so as $m ↑ ∞, c(m )∕c(m ) → 1$ , and the continuous diﬀerentiability and strict concavity of $c(m )$ therefore implies

because any other ﬁxed limit would eventually lead to a level of consumption either exceeding $¯c(m )$ or lower than $c(m )$ . Figure 5 illustrates these limits by plotting the numerical solution.

Next we establish the limit of the expected consumption growth factor as $mt ↑ ∞$ :

because $lim a′(m ) = ÞÞÞ mt↑∞ R$ ⁴³ and $¯ ¯ 𝒢t+1ξξξt+1 ∕mt ≤ (𝒢 Ψ 𝜃𝜃𝜃∕(1 − ℘))∕mt$ which goes to zero as $mt$ goes to inﬁnity.

so as cash goes to inﬁnity, consumption growth approaches its value $ÞÞÞ$ in the perfect foresight model.

3.2 Limits as m Approaches Zero

Now using the continuous diﬀerentiability of the consumption function along with L’Hôpital’s rule, we have

Figure 5 visually conﬁrms that the numerical solution obtains this limit for the MPC as $m$ approaches zero.

where the second-to-last line follows because $[( c(ℛ a(m ))) ] limmt ↓0 𝔼t ---t+κ1¯mt--t- 𝒢t+1$ is positive, and the last line follows because the minimum possible realization of $𝜃𝜃𝜃 t+1$ is $𝜃𝜃𝜃 > 0$ so the minimum possible value of expected next-period consumption is positive.⁴⁴

3.3 Unique ‘Stable’ Points

Theorems whose substance is described here (and whose details are in an appendix) articulate alternative (but closely related) stability criteria.

3.3.1 ‘Individual Target Wealth’ $ˆm$

One kind of ‘stable’ point is a ‘target’ value $ˆm$ such that if $m = mˆ t$ , then $𝔼 [m ] = m t t+1 t$ . Existence of such a target turns out to require the GIC-Mod condition.

Since $mt+1 = (mt − c(mt ))ℛt+1 + ξξξt+1$ , the implicit equation for $mˆ$ is

3.3.2 Individual Balanced-Growth ‘pseudo steady state’ $ˇm$

Our least restrictive deﬁnition of ‘stability’ derives from a traditional question in macro models: whether there is a ‘balanced growth’ equilibrium in which aggregate variables (income, consumption, market resources) all grow forever by the same factor $𝒢$ . For our model, Figure 4 showed that there is no single $m$ for which $𝔼 [m ]∕m = 𝔼 [c ]∕c = 𝒢 t t+1 t t t+1 t$ for an individual consumer. Nevertheless, the next section will show that economies populated by heterogeneous collections of such consumers can exhibit balanced growth in the aggregate, and in the cross-section of households.

As an input to that analysis, we show here that if the GIC holds, the problem will exhibit a balanced-growth ‘pseudo-steady-state’ point, by which we mean that there is some $ˇm$ such that, for all

$mt > mˇ$ , $𝔼t [mt+1 ∕mt ] < 𝒢$ , and conversely if $mt < ˇm$ then $𝔼t [mt+1 ∕mt ] > 𝒢$ .

The only diﬀerence between (41) and (40) is the substitution of $ℛ$ for $¯ ℛ$ .⁴⁵ $,$ ⁴⁶

The proofs of these theorems are intuitive, and almost completely parallel; to save space, they are relegated to Appendix H. They involve three steps:

3.4 Example With Balanced-Growth $mˇ$ But No Target $ˆm$

Because the equations deﬁning target and pseudo-steady-state $m$ , (40) and (41), diﬀer only by substitution of $ℛ$ for $ℛ¯ = ℛ 𝔼 [Ψ −1]$ , if there are no permanent shocks ( $Ψ ≡ 1$ ), the conditions are identical. For many parameterizations (e.g., under the baseline parameter values used for constructing ﬁgure 4), $ˆm$ and $mˇ$ will not diﬀer much.

An illuminating exception is exhibited in Figure 7, which modiﬁes the baseline parameter values by quadrupling the variance of the permanent shocks, enough to cause failure of the GIC-Mod; now there is no target wealth level $mˆ$ . But the pseudo-steady-state still exists because it turns oﬀ realizations of the permanent shock. It is tempting to conclude that the reason target $ˆm$ does not exist is that the increase in the size of the shocks induces a precautionary motive that increases the consumer’s eﬀective patience. But that interpretation is not correct when the FHWC holds, 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 $ℛ$ to $¯ℛ > ℛ$ which reﬂects the fact that in the presence of uncertainty the expectation of the inverse of the growth factor increases: $𝒢-> 𝒢$ . That is, in the limit as $m ↑ ∞$ the increase in eﬀective impatience reﬂected in $ÞÞÞ𝒢-< ÞÞÞ𝒢$ is entirely due to the certainty-equivalence growth adjustment, 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 GIC, which ignores the magnitude of permanent shocks, not the GIC-Mod.⁴⁷

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 $mt = mˇ$ , but then evolves according to the model’s assumed dynamics between $t$ and $t + 1$ . Equation (41) will still hold, so for this ﬁrst period, at least, the economy will exhibit balanced growth: the growth factor for aggregate $m$ will match the growth factor for permanent income $𝒢$ . It is true that there will be people for whom $bt+1 = atR∕(𝒢 Ψt+1)$ is boosted by a small draw of $Ψt+1$ . But their contribution to the level of the aggregate variable is given by $bt+1 = bt+1Ψt+1$ , so their $b t+1$ 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 $Mˇ$ (this is not numerically the same as the individual pseudo-steady-state ratio $ˇm$ because the problem’s nonlinearities have consequences when aggregated).⁴⁸

4 Invariant Relationships

Assume a continuum of ex ante identical buﬀer-stock households on the unit interval, with constant total mass normalized to one and indexed by $i ∈ [0,1 ]$ .

Szeidl (2013) proved that such a population will be characterized by invariant distributions of $m$ , $c$ , and $a$ under the condition⁴⁹

which is stronger than our GIC ( $ÞÞÞ 𝒢 < 1$ ), but weaker than our GIC-Mod ( $ÞÞÞ 𝒢 < 1$ ).⁵⁰

Harmenberg (2021) substitutes a clever change of probability-measure into Szeidl’s proof, with the implication that if the GIC holds, invariant permanent-income-weighted distributions exist. This allows him to prove a conjecture from an earlier draft of this paper (Carroll (2019, Submitted)) that under the GIC, aggregate consumption grows at the same rate $𝒢$ as aggregate noncapital income in the long run (with the corollary that aggregate assets and market resources grow at that same rate). Harmenberg (2021) shows that his reformulation of the problem can reduce costs of calculation enormously.⁵¹ In conﬁrmation, this notebook ﬁnds that the Harmenberg method reduces the simulation size required for a given degree of accuracy by roughly a factor of 100 (!) under the baseline parameter values deﬁned above.

The remainder of this section brieﬂy draws out some implications of these points.

4.1 Individual Balanced Growth of Income, Consumption, and Wealth

Deﬁne $𝕄 [∙ ]$ to yield the mean of its argument in the population (as distinct from the expectations operator $𝔼[∙]$ which represents beliefs about the future). Using boldface capitals for aggregates, the growth factor for aggregate noncapital income is:

Consider an economy that satisﬁes the Szeidl impatience condition (44) 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 rate of consumption for each individual household, and take the average:

Because this economy is Szeidl-converged, distributions of $ct$ and $ct+1$ will be identical, so that the second term in (45) disappears; hence, mean cross-sectional growth rates 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 $m$ . This fact is one way of interpreting Harmenberg’s deﬁnition of the density of the permanent-income-invariant distribution of $m$ ; call this density $p (m )$ .⁵² Call $ct(m )$ the total amount of consumption at date $t$ by persons with market resources $m$ , and note that in the invariant economy this is given by the converged consumption function $c(m )$ multiplied by the amount of permanent income accruing to such people $p(m )pt$ . Since $p(m )$ is invariant and aggregate permanent income grows according to $Pt+1 = 𝒢Pt$ , for any $m$ :

4.2 Aggregate Balanced Growth and Idiosyncratic Covariances

Harmenberg shows that the covariance between the individual consumption ratio $c$ and the idiosyncratic component of permanent income $p$ 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 $𝔠t+n ≡ 𝕄 [ct+n]$ . Normalizing period- $t$ aggregate permanent income to $Pt = 1$ , total consumption at $t + 1$ and $t + 2$ are

In a Szeidl-invariant economy, $𝔠t+2 = 𝔠t+1$ , 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 $𝔠$ terms have constant growth factors $Ωcov$ and $Ω 𝔠$ ,⁵³ an equation corresponding to (48) will hold in $t + n$ :

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

4.3 Implications for Microfoundations

Thus we have microeconomic propositions, for both growth rates and for covariances of observable variables,⁵⁴ 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 attempt direct measurement of ‘permanent income’; Carroll, Slacalek, Tokuoka, and White (2017), for example, show that their assumptions about the magnitude of permanent shocks (and mortality; see below) yield a simulated distribution of permanent income that roughly matches answers in the U.S. Survey of Consumer Finances (‘SCF’) to a question designed to elicit a direct measure of respondents’ permanent income. They use those results to calibrate a model to match empirical facts about the distribution of permanent income and wealth, showing that the model also does ﬁts empirical facts about the marginal propensity to consume. The quantitative credibility of the argument depends on the model’s match to the distribution of permanent income inequality, which would not be possible in a model without a nondegenerate steady-state distribution of 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.

4.4 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, Slacalek, Tokuoka, 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 $β$ and probability of remaining alive (not dead) is $ℒ$ , then the assumption that no utility accrues after death makes the eﬀective discount factor $β- = βℒ$ while the enhancement to the rate of return from the annuity scheme yields an eﬀective interest factor $R¯= R∕ℒ$ (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 $βR¯$ 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 $ÞÞÞ$ (principally, the RIC, which becomes $Þ ¯ ÞÞ ∕R$ ).

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, 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 $ℒ$ is the probability of remaining alive, the condition changes from the plain GIC to a looser mortality-adjusted GIC:

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.

5 Conclusions

Numerical solutions to optimal consumption problems, in both life cycle and inﬁnite horizon contexts, have become standard tools since the ﬁrst reasonably realistic models were constructed in the late 1980s. One contribution of this paper is to show that ﬁnite horizon (‘life cycle’) versions of the simplest such models, with assumptions about income shocks (transitory and permanent) dating back to Friedman (1957) and standard speciﬁcations of preferences — and without plausible (but computationally and mathematically inconvenient) complications like liquidity constraints — have attractive properties (like continuous diﬀerentiability of the consumption function, and analytical limiting MPC’s as resources approach their minimum and maximum possible values).

The main focus of the paper, though, is on the limiting solution of the ﬁnite horizon model as the time horizon approaches inﬁnity. This simple model has other appealing features: A ‘Finite Value of Autarky’ condition guarantees convergence of the consumption function, under the mild additional requirement of a ‘Weak Return Impatience Condition’ that will never bind for plausible parameterizations, but provides intuition for the bridge between this model and models with explicit liquidity constraints. The paper also provides a roadmap for the model’s relationships to the perfect foresight model without and with constraints. The constrained perfect foresight model provides an upper bound to the consumption function (and value function) for the model with uncertainty, which explains why the conditions for the model to have a nondegenerate solution closely parallel those required for the perfect foresight constrained model to have a nondegenerate solution.

The main use of inﬁnite horizon versions of such models is in heterogeneous-agent macroeconomics. The paper articulates intuitive ‘Growth Impatience Conditions’ under which populations of such agents, with Blanchardian (tighter) or Modiglianian (looser) mortality will exhibit balanced growth. Finally, the paper provides the analytical basis for many results about buﬀer-stock saving models that are so well understood that even without analytical foundations researchers uncontroversially use them as explanations of real-world phenomena like the cross-sectional pattern of consumption dynamics in the Great Recession.

A $c$ Functions Exist, are Concave, and Diﬀerentible

To show that (5) deﬁnes a sequence of continuously diﬀerentiable strictly increasing concave functions ${cT,cT− 1,...,cT−k}$ , we start with a deﬁnition. We will say that a function $n(z)$ is ‘nice’ if it satisﬁes

Assume that some $vt+1$ is nice. Our objective is to show that this implies $vt$ is also nice; this is suﬃcient to establish that $vt−n$ is nice by induction for all $n > 0$ because $vT (m ) = u (m)$ and $u(m ) = m1 −ρ∕(1 − ρ)$ is nice by inspection.

Since there is a positive probability that $ξξξt+1$ will attain its minimum of zero and since $ℛt+1 > 0$ , it is clear that $lima ↓0𝔳t(a) = − ∞$ and $′ lima ↓0𝔳t(a) = ∞$ . So $𝔳t(a)$ is well-deﬁned iﬀ $a > 0$ ; it is similarly straightforward to show the other properties required for $𝔳t(a)$ to be nice. (See Hiraguchi (2003).)

which is $C3$ since $𝔳t$ and $u$ are both $C3$ , and note that our problem’s value function deﬁned in (5) can be written as

$vt$ is well-deﬁned if and only if $0 < c < m$ . Furthermore, $limc↓0vt(m, c) = limc ↑m vt(m, c) = − ∞$ , $∂2vt(m2,c)< 0 ∂c$ , $limc ↓0 ∂vt(m,c)= + ∞ ∂c$ , and $limc↑m ∂vt(m,c)= − ∞ ∂c$ . It follows that the $ct(m )$ deﬁned by

Since both $u$ and $𝔳t$ are strictly concave, both $ct(m )$ and $at(m ) = m − ct(m )$ are strictly increasing. Since both $u$ and $𝔳t$ are three times continuously diﬀerentiable, using (57) we can conclude that $c(m ) t$ is continuously diﬀerentiable and

Similarly we can easily show that $ct(m )$ is twice continuously diﬀerentiable (as is $at(m )$ ) (See Appendix B.) This implies that $vt(m )$ is nice, since $vt(m ) = u (ct(m )) + 𝔳t(at(m ))$ .

B $ct(m )$ is Twice Continuously Diﬀerentiable

First we show that $ct(m )$ is $C1$ . Deﬁne $y$ as $y ≡ m + dm$ . Since $u′(c (y)) − u′(c(m )) = 𝔳′(a(y)) − 𝔳′(a(m )) t t t t t t$ and $at(y)−-at(m) = 1 − ct(y)−ct(m) dm dm$ ,

Since $ct$ and $at$ are continuous and increasing, $u′(c (y))−u′(c(m)) lim ---tct(y)−ct(mt)---< 0 dm →+0$ and $𝔳′(at(y))−𝔳′(at(m)) dmlim→+0-t-at(y)−att(m)--- < 0$ are satisﬁed. Then $u′(ct(y))− u′(ct(m)) 𝔳′(at(y))−𝔳′(at(m )) ---ct(y)−-ct(m)---+ t-at(y)−att(m-)---< 0$ for suﬃciently small $dm$ . Hence we obtain a well-deﬁned equation:

Similarly we can show that $c′+(m ) = c′−(m ) t t$ , which means $c′(m ) t$ exists. Since $𝔳t$ is $C3$ , $′ ct(m )$ exists and is continuous. $′ ct(m )$ is diﬀerentiable because $′′ 𝔳t$ is $1 C$ , $ct(m)$ is $1 C$ and $′′ ′′ u (ct(m )) + 𝔳t (at(m )) < 0$ . $′′ ct(m )$ is given by

C $𝒯$ Is a Contraction Mapping

Boyd’s operator $T$ maps from $𝒞г(𝒜, ℬ )$ to $𝒞 (𝒜,ℬ )$ . A preliminary requirement is therefore that ${ 𝒯z}$ be continuous for any $г −$ bounded $z$ , ${𝒯z } ∈ 𝒞 (ℝ++, ℝ)$ . This is not diﬃcult to show; see Hiraguchi (2003).

so $x(∙) ≤ y(∙)$ implies ${𝒯x }(mt) ≤ { 𝒯y}(mt )$ by inspection.⁵⁵

the solution to which is patently $u(¯κmt )$ . Thus, condition (2) will hold if $1−ρ (¯κmt )$ is $г$ -bounded, which it is if we use the bounding function

Finally, we turn to condition (3), ${𝒯(z + ζг )}(mt) ≤ {𝒯z }(mt ) + ζα г (mt )$ . The proof will be more compact if we deﬁne $˘c$ and $˘a$ as the consumption and assets functions⁵⁶ associated with $𝒯z$ and $ˆc$ and $ˆa$ as the functions associated with $𝒯(z + ζг )$ ; using this notation, condition (3) can be rewritten

Now note that if we force the $⌣$ consumer to consume the amount that is optimal for the $∧$ consumer, value for the $⌣$ consumer must decline (at least weakly). That is,

Using $1−ρ г (m ) = η + m$ and deﬁning $ˆat = ˆa(mt )$ , this condition is

which by imposing PF-FVAC (equation (19), which says $ℶ < 1$ ) can be rewritten as:

But since $η$ is an arbitrary constant that we can pick, the proof thus reduces to showing that the numerator of (61) is bounded from above:

The proof that $𝒯$ deﬁnes a contraction mapping under the conditions (31) and (28) is now complete.

C.1 $𝒯$ and $v$

In deﬁning our operator $𝒯$ we made the restriction $κmt ≤ ct ≤ ¯κmt$ . However, in the discussion of the consumption function bounds, we showed only (in (32)) that $κ-mt ≤ ct(mt ) ≤ ¯κtmt t$ . (The diﬀerence is in the presence or absence of time subscripts on the MPC’s.) We have therefore not proven (yet) that the sequence of value functions (5) deﬁnes a contraction mapping.

Fortunately, the proof of that proposition is identical to the proof above, except that we must replace $¯κ$ with $¯κT− 1$ and the WRIC must be replaced by a slightly stronger (but still quite weak) condition. The place where these conditions have force is in the step at (62). Consideration of the prior two equations reveals that a suﬃcient stronger condition is

where we have used (30) for $¯κ T− 1$ (and in the second step the reversal of the inequality occurs because we have assumed $ρ > 1$ so that we are exponentiating both sides by the negative number $1 − ρ$ ). To see that this is a weak condition, note that for small values of $℘$ this expression can be further simpliﬁed using $− 1 (1 + ℘1∕ρÞÞÞR) ≈ 1 − ℘1 ∕ρÞÞÞR$ so that it becomes

Calling the weak return patience factor $ÞÞÞ℘ = ℘1 ∕ρÞÞÞ R R$ and recalling that the WRIC was $℘ ÞÞÞ R < 1$ , the expression on the LHS above is $−ρ βÞÞÞR$ times the WRPF. Since we usually assume $β$ not far below 1 and parameter values such that $ÞÞÞR ≈ 1$ , this condition is clearly not very diﬀerent from the WRIC.

The upshot is that under these slightly stronger conditions the value functions for the original problem deﬁne a contraction mapping in $г −$ bounded space with a unique $v(m )$ . But since $limn →∞ κT−n = κ-$ and $limn →∞ ¯κT−n = ¯κ$ , it must be the case that the $v(m )$ toward which these $vT−n$ ’s are converging is the same $v(m )$ that was the endpoint of the contraction deﬁned by our operator $𝒯$ . Thus, under our slightly stronger (but still quite weak) conditions, not only do the value functions deﬁned by (5) converge, they converge to the same unique $v$ deﬁned by $𝒯$ .⁵⁸

D Convergence in Euclidian Space

D.1 Convergence of $vt$

Boyd’s theorem shows that $𝒯$ deﬁnes a contraction mapping in an $г$ -bounded space. We now show that $𝒯$ also deﬁnes a contraction mapping in Euclidian space.

Calling $∗ v$ the unique ﬁxed point of the operator $𝒯$ , since $∗ ∗ v (m ) = 𝒯v (m )$ ,

On the other hand, $vT − v∗ ∈ 𝒞г (𝒜,ℬ )$ and $κ = ∥vT − v∗∥г < ∞$ because $vT$ and $v ∗$ are in $𝒞г (𝒜,ℬ )$ . It follows that

Since $m1−ρ- vT (m ) = 1−ρ$ , $(¯κm)1−ρ vT−1(m ) ≤ 1−ρ < vT(m )$ . On the other hand, $vT −1 ≤ vT$ means $𝒯vT −1 ≤ 𝒯vT$ , in other words, $vT−2(m ) ≤ vT−1(m )$ . Inductively one gets $vT −n(m ) ≥ vT−n− 1(m )$ . This means that ${vT −n+1(m )}∞n=1$ is a decreasing sequence, bounded below by $v∗$ .

D.2 Convergence of $ct$

Given the proof that the value functions converge, we now show the pointwise convergence of consumption functions $∞ {cT −n+1(m )}n=1$ .

Consider any convergent subsequence ${ } cT−n(i)(m )$ of ${cT− n+1(m )} ∞n=1$ converging to $c∗$ . By the deﬁnition of $c (m ) T−n$ , we have

for any $cT−n(i) ∈ [κm, ¯κm ]$ . Now letting $n(i)$ go to inﬁnity, it follows that the left hand side converges to $u(c∗) + β 𝔼t[𝒢1t−ρv (m )]$ , and the right hand side converges to $u(c ) + β 𝔼 [𝒢1−ρv (m )] T− n(i) t t$ . So the limit of the preceding inequality as $n (i)$ approaches inﬁnity implies

Hence, ${ } c∗ ∈ argmax u (cT−n(i)) + β 𝔼t[𝒢1− ρv(m )] cT−n(i)∈[κm,¯κm] t+1$ . By the uniqueness of $c(m )$ , $c∗ = c(m )$ .

E Perfect Foresight Liquidity Constrained Solution

Under perfect foresight in the presence of a liquidity constraint requiring $b ≥ 0$ , this appendix taxonomizes the varieties of the limiting consumption function $`c(m )$ that arise under various parametric conditions.

E.1 If GIC Fails

A consumer is ‘growth patient’ if the perfect foresight growth impatience condition fails ( $¬$ GIC, $1 < ÞÞÞ∕𝒢$ ). Under $¬$ GIC the constraint does not bind at the lowest feasible value of $mt = 1$ because $1 < (Rβ )1∕ρ∕𝒢$ implies that spending everything today (setting $ct = mt = 1$ ) produces lower marginal utility than is obtainable by reallocating a marginal unit of resources to the next period at return $R$ :⁵⁹

Similar logic shows that under these circumstances the constraint will never bind at $m = 1$ for a constrained consumer with a ﬁnite horizon of $n$ periods, so for $m ≥ 1$ such a consumer’s consumption function will be the same as for the unconstrained case examined in the main text.

RIC fails, FHWC holds. If the RIC fails ( $1 < ÞÞÞR$ ) while the ﬁnite human wealth condition holds, the limiting value of this consumption function as $n ↑ ∞$ is the degenerate function

RIC fails, FHWC fails. $¬$ FHWC implies that human wealth limits to $h = ∞$ so the consumption function limits to either $`cT −n(m ) = 0$ or $`cT−n(m ) = ∞$ depending on the relative speeds with which the MPC approaches zero and human wealth approaches $∞$ .⁶⁰

Thus, the requirement that the consumption function be nondegenerate implies that for a consumer satisfying $¬$ GIC we must impose the RIC (and the FHWC can be shown to be a consequence of $¬$ GIC and RIC). In this case, the consumer’s optimal behavior is easy to describe. We can calculate the point at which the unconstrained consumer would choose $c = m$ from Equation (16):

which (under these assumptions) satisﬁes $0 < m# < 1$ .⁶¹ For $m < m#$ the unconstrained consumer would choose to consume more than $m$ ; for such $m$ , the constrained consumer is obliged to choose $`c(m ) = m$ .⁶² For any $m > m#$ the constraint will never bind and the consumer will choose to spend the same amount as the unconstrained consumer, $¯c(m )$ .

(Stachurski and Toda (2019) obtain a similar lower bound on consumption and use it to study the tail behavior of the wealth distribution.)

E.2 If GIC Holds

Imposition of the GIC reverses the inequality in (68), and thus reverses the conclusion: A consumer who starts with $m = 1 t$ will desire to consume more than 1. Such a consumer will be constrained, not only in period $t$ , but perpetually thereafter.

Now deﬁne $bn#$ as the $bt$ such that an unconstrained consumer holding $bt = bn#$ would behave so as to arrive in period $t + n$ with $bt+n = 0$ (with $b0#$ trivially equal to 0); for example, a consumer with $bt−1 = b1 #$ was on the ‘cusp’ of being constrained in period $t − 1$ : Had $bt−1$ been inﬁnitesimally smaller, the constraint would have been binding (because the consumer would have desired, but been unable, to enter period $t$ with negative, not zero, $b$ ). Given the GIC, the constraint certainly binds in period $t$ (and thereafter) with resources of $mt = m0# = 1 + b0# = 1$ : The consumer cannot spend more (because constrained), and will not choose to spend less (because impatient), than $0 ct = c# = 1$ .

We can construct the entire ‘prehistory’ of this consumer leading up to $t$ as follows. Maintaining the assumption that the constraint has never bound in the past, $c$ must have been growing according to $ÞÞÞ𝒢$ , so consumption $n$ periods in the past must have been

and note that the consumer’s human wealth between $t − n$ and $t$ (the relevant time horizon, because from $t$ onward the consumer will be constrained and unable to access post- $t$ income) is

Deﬁning $n n m # = b# + 1$ , consider the function $`c(m )$ deﬁned by linearly connecting the points ${mn#, cn# }$ for integer values of $n ≥ 0$ (and setting $`c(m ) = m$ for $m < 1$ ). This function will return, for any value of $m$ , the optimal value of $c$ for a liquidity constrained consumer with an inﬁnite horizon. The function is piecewise linear with ‘kink points’ where the slope discretely changes; for inﬁnitesimal $𝜖$ the MPC of a consumer with assets $n m = m # − 𝜖$ is discretely higher than for a consumer with assets $n m = m # + 𝜖$ because the latter consumer will spread a marginal dollar over more periods before exhausting it.

In order for a unique consumption function to be deﬁned by this sequence (72) for the entire domain of positive real values of $b$ , we need $n b#$ to become arbitrarily large with $n$ . That is, we need

E.2.1 If FHWC Holds

The FHWC requires $ℛ − 1 < 1$ , in which case the second term in (72) limits to a constant as $n ↑ ∞$ , and (73) reduces to a requirement that

E.2.2 If FHWC Fails

which with a bit of algebra⁶³ can be shown to asymptote to the MPC in the perfect foresight model:⁶⁴

If RIC Fails. Consider now the $¬$ RIC case, $ÞÞÞ > 1 R$ . We can rearrange (75)as

What is happening here is that the $cn#$ term is increasing backward in time at rate dominated in the limit by $𝒢 ∕ÞÞÞ$ while the $b#$ term is increasing at a rate dominated by $𝒢 ∕R$ term and

Consequently, while $limn ↑∞ bn# = ∞$ , the limit of the ratio $cn#∕bn#$ in (75) is zero. Thus, surprisingly, the problem has a well deﬁned solution with inﬁnite human wealth if the RIC fails. It remains true that $¬$ RIC implies a limiting MPC of zero,

but that limit is approached gradually, starting from a positive value, and consequently the consumption function is not the degenerate $`c(m ) = 0$ . (Figure 8 presents an example for $ρ = 2$ , $R = 0.98$ , $β = 1.00$ , $𝒢 = 0.99$ ; note that the horizontal axis is bank balances $b = m − 1$ ; the part of the consumption function below the depicted points is uninteresting — $c = m$ — so not worth plotting).

We can summarize as follows. Given that the GIC holds, the interesting question is whether the FHWC holds. If so, the RIC automatically holds, and the solution limits into the solution to the unconstrained problem as $m ↑ ∞$ . But even if the FHWC fails, the problem has a well-deﬁned and nondegenerate solution, whether or not the RIC holds.

Although these results were derived for the perfect foresight case, we know from work elsewhere in this paper and in other places that the perfect foresight case is an upper bound for the case with uncertainty. If the upper bound of the MPC in the perfect foresight case is zero, it is not possible for the upper bound in the model with uncertainty to be greater than zero, because for any $κ > 0$ the level of consumption in the model with uncertainty would eventually exceed the level of consumption in the absence of uncertainty.

Ma and Toda (2020) characterize the limits of the MPC in a more general framework that allows for capital and labor income risks in a Markovian setting with liquidity constraints, and ﬁnd that in that much more general framework the limiting MPC is also zero.

F The Perfect Foresight Liquidity Constrained Solution as a Limit

Formally, suppose we change the description of the problem by making the following two assumptions:

Redesignate the consumption function that emerges from our original problem for a given ﬁxed $℘$ as $ct(m; ℘ )$ where we separate the arguments by a semicolon to distinguish between $m$ , which is a state variable, and $℘$ , which is not. The proposition we wish to demonstrate is

We will ﬁrst examine the problem in period $T − 1$ , then argue that the desired result propagates to earlier periods. For simplicity, suppose that the interest, growth, and time-preference factors are $β = R = 𝒢 = 1$ , and there are no permanent shocks, $Ψ = 1$ ; the results below are easily generalized to the full-ﬂedged version of the problem.

The solution to the restrained consumer’s optimization problem can be obtained as follows. Assuming that the consumer’s behavior in period $T$ is given by $cT(m )$ (in practice, this will be $cT (m ) = m$ ), consider the unrestrained optimization problem

As usual, the envelope theorem tells us that $v′T (m ) = u′(cT(m ))$ so the expected marginal value of ending period $T − 1$ with assets $a$ can be deﬁned as

$`a ∗ (m ) T−1$ therefore answers the question “With what level of assets would the restrained consumer like to end period $T − 1$ if the constraint $cT−1 ≤ mT −1$ did not exist?” (Note that the restrained consumer’s income process remains diﬀerent from the process for the unrestrained consumer so long as $℘ > 0$ .) The restrained consumer’s actual asset position will be

reﬂecting the inability of the restrained consumer to spend more than current resources, and note (as pointed out by Deaton (1991)) that

is the cusp value of $m$ at which the constraint makes the transition between binding and non-binding in period $T − 1$ .

with solution $a∗ (m; ℘) T−1$ . Now note that for any ﬁxed $a > 0$ , $lim℘ ↓0 𝔳′ (a;℘ ) = `𝔳 ′ (a) T−1 T−1$ . Since the LHS of (83) and (85) are identical, this means that $lim a∗ (m; ℘ ) = `a ∗ (m ) ℘↓0 T−1 T−1$ . That is, for any ﬁxed value of $1 m > m #$ such that the consumer subject to the restraint would voluntarily choose to end the period with positive assets, the level of end-of-period assets for the unrestrained consumer approaches the level for the restrained consumer as $℘ ↓ 0$ . With the same $a$ and the same $m$ , the consumers must have the same $c$ , so the consumption functions are identical in the limit.

Now consider values $1 m ≤ m #$ for which the restrained consumer is constrained. It is obvious that the baseline consumer will never choose $a ≤ 0$ because the ﬁrst term in (84) is $lima ↓0 ℘a− ρ = ∞$ , while $lima ↓0(m − a)−ρ$ is ﬁnite (the marginal value of end-of-period assets approaches inﬁnity as assets approach zero, but the marginal utility of consumption has a ﬁnite limit for $m > 0$ ). The subtler question is whether it is possible to rule out strictly positive $a$ for the unrestrained consumer.

The answer is yes. Suppose, for some $m < m1#$ , that the unrestrained consumer is considering ending the period with any positive amount of assets $a = δ > 0$ . For any such $δ$ we have that $lim 𝔳′ (a;℘ ) = `𝔳′ (a) ℘↓0 T− 1 T−1$ . But by assumption we are considering a set of circumstances in which $∗ `aT −1(m ) < 0$ , and we showed earlier that $∗ ∗ lim ℘↓0aT−1(m; ℘ ) = `a T−1(m )$ . So, having assumed $a = δ > 0$ , we have proven that the consumer would optimally choose $a < 0$ , which is a contradiction. A similar argument holds for $m = m1#$ .

These arguments demonstrate that for any $m > 0$ , $lim c (m; ℘) = `c (m ) ℘↓0 T−1 T− 1$ which is the period $T − 1$ version of (81). But given equality of the period $T − 1$ consumption functions, backwards recursion of the same arguments demonstrates that the limiting consumption functions in previous periods are also identical to the constrained function.

Note ﬁnally that another intuitive conﬁrmation of the equivalence between the two problems is that our formula (87) for the maximal marginal propensity to consume satisﬁes

G The Limiting MPC’s

For $mt > 0$ we can deﬁne $et(mt) = ct(mt)∕mt$ and $at(mt ) = mt − ct(mt)$ and the Euler equation (6) can be rewritten

Consider the ﬁrst conditional expectation in (6), recalling that if $ξξξt+1 > 0$ then $ξξξt+1 ≡ 𝜃𝜃𝜃t+1∕(1 − ℘)$ . Since $limm ↓0at(m ) = 0$ , $𝔼t[(et+1(mt+1 )mt+1𝒢t+1)− ρ | ξξξt+1 > 0 ]$ is contained within bounds deﬁned by $(e (𝜃𝜃𝜃∕(1 − ℘ ))𝒢 Ψ 𝜃𝜃𝜃∕(1 − ℘))−ρ t+1 -- ----$ and $¯ ¯¯ −ρ (et+1(𝜃𝜃𝜃∕(1 − ℘))𝒢Ψ 𝜃𝜃𝜃 ∕(1 − ℘ ))$ both of which are ﬁnite numbers, implying that the whole term multiplied by $(1 − ℘)$ goes to zero as $ρ m t$ goes to zero. As $mt ↓ 0$ the expectation in the other term goes to $¯κ−tρ+1(1 − ¯κt)− ρ$ . (This follows from the strict concavity and diﬀerentiability of the consumption function.) It follows that the limiting $¯κt$ satisﬁes $−ρ 1− ρ −ρ −ρ ¯κt = β ℘R κ¯t+1 (1 − ¯κt)$ . Exponentiating by $ρ$ , we can conclude that

which yields a useful recursive formula for the maximal marginal propensity to consume:

As noted in the main text, we need the WRIC (31) for this to be a convergent sequence:

Since $¯κT = 1$ , iterating (86) backward to inﬁnity (because we are interested in the limiting consumption function) we obtain:

The minimal MPC’s are obtained by considering the case where $mt ↑ ∞$ . If the FHWC holds, then as $mt ↑ ∞$ the proportion of current and future consumption that will be ﬁnanced out of capital approaches 1. Thus, the terms involving $ξξξt+1$ in (86) can be neglected, leading to a revised limiting Euler equation

so that $({κ−1 })∞ -T− n n=0$ is also an increasing convergent sequence, and we deﬁne

as the limiting (inverse) marginal MPC. If the RIC does not hold, then $limn →∞ κ-−T1−n = ∞$ and so the limiting MPC is $κ-= 0$ .

For the purpose of constructing the limiting perfect foresight consumption function, it is useful further to note that the PDV of consumption is given by

H Unique and Stable Target and Steady State Points

H.1 Proof of Theorem 4

H.2 Existence and Continuity of $𝔼t[mt+1 ∕mt ]$

The consumption function exists because we have imposed suﬃcient conditions (the $WRIC$ and $FVAC$ ; Theorem 1).

Section 2.8 shows that for all $t$ , $at− 1 = mt−1 − ct−1 > 0$ . Since $mt = at−1ℛt + ξξξt$ , even if $ξξξ t$ takes on its minimum value of 0, $a ℛ > 0 t−1 t$ , since both $a t−1$ and $ℛ t$ are strictly positive. With $mt$ and $mt+1$ both strictly positive, the ratio $𝔼t[mt+1 ∕mt ]$ inherits continuity (and, for that matter, continuous diﬀerentiability) from the consumption function.

H.3 Existence of a point where $𝔼t[mt+1∕mt ] = 1$ .

H.3.1 Existence of $m$ where $𝔼t[mt+1 ∕mt] < 1$

If RIC holds. Logic exactly parallel to that of Section 3.1 leading to equation (38), but dropping the $𝒢t+1$ from the RHS, establishes that

If RIC fails. When the RIC fails, the fact that $′ limm ↑∞ c(m ) = 0$ (see equation (29)) means that the limit of the RHS of (93) as $m ↑ ∞$ is $ℛ¯ = 𝔼t [ℛt+1 ]$ . In the next step of this proof, we will prove that the combination GIC-Mod and $¬$ RIC implies $ℛ¯ < 1$ .

H.3.2 Existence of $m$ > 1 where $𝔼 [m ∕m ] > 1 t t+1 t$

Paralleling the logic for $c$ in Section 3.2: the ratio of $𝔼t[mt+1 ]$ to $mt$ is unbounded above as $mt ↓ 0$ because $limmt ↓0𝔼t [mt+1 ] > 0$ .

Intermediate Value Theorem. If $𝔼t[mt+1 ∕mt]$ is continuous, and takes on values above and below 1, there must be at least one point at which it is equal to one.

H.3.3 $𝔼t[mt+1] − mt$ is monotonically decreasing.

so that $ζζζ(ˆm ) = 0$ . Our goal is to prove that $ζζζ(∙)$ is strictly decreasing on $(0,∞ )$ using the fact that

Now, we show that (given our other assumptions) $′ ζζζ (m )$ is decreasing (but for diﬀerent reasons) whether the RIC holds or fails.

If RIC holds. Equation (15) indicates that if the RIC holds, then $κ-> 0$ . We show at the bottom of Section 2.9.1 that if the RIC holds then $0 < κ-< c′(mt) < 1$ so that

If RIC fails. Under $¬$ RIC, recall that $limm ↑∞ c′(m ) = 0$ . Concavity of the consumption function means that $c′$ is a decreasing function, so everywhere

H.4 Proof of Theorem 5

H.4.1 Existence and Continuity of the Ratio

Since by assumption $¯ 0 < Ψ--≤ Ψt+1 ≤ Ψ < ∞$ , our proof in H.2 that demonstrated existence and continuity of $𝔼t[mt+1 ∕mt]$ implies existence and continuity of $𝔼t[Ψt+1mt+1 ∕mt ]$ .

H.4.2 Existence of a stable point

Since by assumption $0 < Ψ--≤ Ψt+1 ≤ ¯Ψ < ∞$ , our proof in Subsection H.2 that the ratio of $𝔼t[mt+1 ]$ to $mt$ is unbounded as $mt ↓ 0$ implies that the ratio $𝔼t [Ψt+1mt+1 ]$ to $mt$ is unbounded as $mt ↓ 0$ .

The limit of the expected ratio as $mt$ goes to inﬁnity is most easily calculated by modifying the steps for the prior theorem explicitly:

The Intermediate Value Theorem says that if $𝔼t[Ψt+1mt+1 ∕mt ]$ is continuous, and takes on values above and below 1, there must be at least one point at which it is equal to one.

H.4.3 $𝔼t[Ψt+1mt+1 ] − mt$ is monotonically decreasing.

so that $ζζζ(ˆm ) = 0$ . Our goal is to prove that $ζζζ(∙)$ is strictly decreasing on $(0,∞ )$ using the fact that

Now, we show that (given our other assumptions) $′ ζζζ (m )$ is decreasing (but for diﬀerent reasons) whether the RIC holds or fails ( $¬$ RIC).

If RIC holds. Equation (15) indicates that if the RIC holds, then $κ-> 0$ . We show at the bottom of Section 2.9.1 that if the RIC holds then $0 < κ-< c′(mt) < 1$ so that

I Relational Diagrams for the Inequality Conditions

This appendix explains in detail the paper’s ‘inequalities’ diagrams (Figures 1, 3).

I.1 The Unconstrained Perfect Foresight Model

A simple illustration is presented in Figure 9, whose three nodes represent values of the absolute patience factor $ÞÞÞ$ , the permanent-income growth factor $𝒢$ , and the riskfree interest factor $R$ . The arrows represent imposition of the labeled inequality condition (like, the uppermost arrow, pointing from $ÞÞÞ$ to $𝒢$ , reﬂects imposition of the GIC condition (clicking GIC should take you to its deﬁnition; deﬁnitions of other conditions are also linked below)).⁶⁵ Annotations inside parenthetical expressions containing $≡$ are there to make the diagram readable for someone who may not immediately remember terms and deﬁnitions from the main text. (Such a reader might also want to be reminded that $R, β,$ and $Γ$ are all in $ℝ++$ , and that $ρ > 1$ ).

Navigation of the diagram is simple: Start at any node, and deduce a chain of inequalities by following any arrow that exits that node, and any arrows that exit from successive nodes. Traversal must stop upon arrival at a node with no exiting arrows. So, for example, we can start at the $ÞÞÞ$ node and impose the GIC and then the FHWC, and see that imposition of these conditions allows us to conclude that $ÞÞÞ < R$ .

One could also impose $ÞÞÞ < R$ directly (without imposing $GIC$ and $FHWC$ ) by following the downward-sloping diagonal arrow exiting $ÞÞÞ$ . Although alternate routes from one node to another all justify the same core conclusion ( $ÞÞÞ < R$ , in this case), $⁄=$ symbol in the center is meant to convey that these routes are not identical in other respects. This notational convention is used in category theory diagrams,⁶⁶ to indicate that the diagram is not commutative.⁶⁷

Negation of a condition is indicated by the reversal of the corresponding arrow. For example, negation of the RIC, $¬RIC ≡ ÞÞÞ > R$ , would be represented by moving the arrowhead from the bottom right to the top left of the line segment connecting $ÞÞÞ$ and $R$ .

If we were to start at $R$ and then impose $¬FHWC$ , that would reverse the arrow connecting $R$ and $𝒢$ , but the $𝒢$ node would then have no exiting arrows so no further deductions could be made. However, if we also reversed $GIC$ (that is, if we imposed $¬GIC$ ), that would take us to the $ÞÞÞ$ node, and we could deduce $R > ÞÞÞ$ . However, we would have to stop traversing the diagram at this point, because the arrow exiting from the $ÞÞÞ$ node points back to our starting point, which (if valid) would lead us to the conclusion that $R > R$ . Thus, the reversal of the two earlier conditions (imposition of $¬FHWC$ and $¬GIC$ ) requires us also to reverse the ﬁnal condition, giving us $¬RIC$ .⁶⁸

Under these conventions, Figure 1 in the main text presents a modiﬁed version of the diagram extended to incorporate the PF-FVAC (reproduced here for convenient reference).

This diagram can be interpreted, for example, as saying that, starting at the $ÞÞÞ$ node, it is possible to derive the $PF -FVAC$ ⁶⁹ by imposing both the GIC and the FHWC; or by imposing RIC and $¬$ FHWC. Or, starting at the $𝒢$ node, we can follow the imposition of the FHWC (twice — reversing the arrow labeled $¬FHWC$ ) and then $¬RIC$ to reach the conclusion that $ÞÞÞ < 𝒢$ . Algebraically,

which leads to the negation of both of the conditions leading into $ÞÞÞ$ . $¬$ GIC is obtained directly as the last line in (96) and $¬$ PF-FVAC follows if we start by multiplying the Return Patience Factor (RPF= $ÞÞÞ ∕R$ ) by the FHWF (= $𝒢∕R$ ) raised to the power $1∕ ρ − 1$ , which is negative since we imposed $ρ > 1$ . FHWC implies FHWF $< 1$ so when FHWF is raised to a negative power the result is greater than one. Multiplying the RPF (which exceeds 1 because $¬$ RIC) by another number greater than one yields a product that must be greater than one:

The complexity of this algebraic calculation illustrates the usefulness of the diagram, in which one merely needs to follow arrows to reach the same result.

After the warmup of constructing these conditions for the perfect foresight case, we can represent the relationships between all the conditions in both the perfect foresight case and the case with uncertainty as shown in Figure 3 in the paper (reproduced here).

Finally, the next diagram substitutes the values of the various objects in the diagram under the baseline parameter values and veriﬁes that all of the asserted inequality conditions hold true.

J Apparent Balanced Growth in $𝔠$ and $cov (c,p )$

Section 4.2 demonstrates some propositions under the assumption that, when an economy satisﬁes the GIC, there will be constant growth factors $Ω 𝔠$ and $Ωcov$ respectively for $𝔠$ (the average value of the consumption ratio) and $cov(c,p )$ . In the case of a Szeidl-invariant economy, the main text shows that these are $Ω = 1 𝔠$ and $Ω = 𝒢 cov$ . If the economy is Harmenberg- but not Szeidl-invariant, no proof is oﬀered that these growth factors will be constant.

J.1 $log c$ and $log (cov(c,p))$ Grow Linearly

Figures 13 and 14 plot the results of simulations of an economy that satisﬁes Harmenberg- but not Szeidl-invariance with a population of 4 million agents over the last 1000 periods (of a 2000 period simulation).⁷⁰ The ﬁrst ﬁgure shows that $log𝔠$ increases apparently linearly. The second ﬁgure shows that $log (− cov(c,p ))$ also increases apparently linearly. (These results are produced by the notebook ApndxBalancedGrowthcNrmAndCov.ipynb).

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Theoretical Foundations of Buﬀer Stock Saving

1 Introduction