BuﬀerStockTheory Web 20210730 at 23:33:24, priorsourcecommit: 344e819 priorpubliccommit: e3cb5c2
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Abstract
This paper builds foundations for rigorous and intuitive understanding of ‘buﬀer stock’ saving
models (close cousins of Bewley (1977) models), pairing each theoretical result with quantitative
illustrations. After describing conditions under which a consumption function exists, the paper shows
that a consumer subject to idiosyncratic shocks will engage in ‘target’ saving whenever a
normalized ‘growth impatience’ condition is imposed. A related condition guarantees the
existence of an ‘expected balanced growth’ point. Together, the (provided) numerical tools
and (proven) analytical results constitute a comprehensive toolkit for understanding.
Dashboard:  https://econark.org/materials/BufferStockTheory?dashboard 
PDF:  https://llorracc.github.io/BufferStockTheory/BufferStockTheory.pdf 
Slides:  https://llorracc.github.io/BufferStockTheory/BufferStockTheorySlides.pdf 
html:  https://llorracc.github.io/BufferStockTheory 
Appendix:  https://llorracc.github.io/BufferStockTheory#Appendices 
bibtex:  https://llorracc.github.io/BufferStockTheory/LaTeX/BufferStockTheorySelf.bib 
GitHub:  https://github.com/llorracc/BufferStockTheory 
A dashboard allows users to see the consequences of alternative parametric choices in a live interactive framework; a corresponding Jupyter Notebook uses the EconARK/HARK toolkit to produce all of the paper’s ﬁgures (warning: the notebook may take several minutes to launch).
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^{1}Contact: ccarroll@jhu.edu, Department of Economics, 590 Wyman Hall, Johns Hopkins University, Baltimore, MD 21218, https://www.econ2.jhu.edu/people/ccarroll, and National Bureau of Economic Research.
In the presence of empirically realistic transitory and permanent income shocks a la Friedman (1957),^{1} 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 construct a quantitatively realistic version of such a model, spawning a subsequent literature showing that such models’ predictions can match evidence from household data reasonably well, whether or not liquidity constraints are imposed.^{2}
A related theoretical literature has derived limiting properties of inﬁnitehorizon solutions of such models, but only in cases more complex than the case with just shocks and preferences (Bewley (1977) and successors). The extra complexity has been required, in part, because standard contraction mapping theorems (beginning with Bellman (1957) and including those building on Stokey et. al. (1989)) cannot be applied when utility or marginal utility are unbounded. Many proof methods also rule out permanent shocks a la Friedman (1957), Muth (1960), and Zeldes (1989).^{3}
This paper’s ﬁrst technical contribution is to articulate conditions under which the simple problem (without complications like a consumption ﬂoor or liquidity constraints) deﬁnes a contraction mapping whose limiting value and consumption functions are nondegenerate as the horizon approaches inﬁnity. The key condition is a generalization of a condition in Ma, Stachurski, and Toda (2020), which we call the‘Finite Value of Autarky Condition’ (the other required condition, the ‘Weak Return Impatience Condition’ is unlikely to bind). Conveniently, the resulting model has analytical properties, like continuous diﬀerentiability of the consumption function, that make it easier to analyze than the standard (but more complicated) models.
The paper’s other main theoretical contribution is to identify conditions under which ‘stable’ values of the wealthtopermanentincome ratio exist, either for individual consumers (an individual consumer’s wealth can be predicted to move toward a ‘target’ ratio) or for the aggregate (the economy as a whole moves toward a ‘balanced growth’ equilibrium in which the ratio of aggregate wealth to aggregate income is constant). The requirement for stability is always that the model’s parameters must satisfy some version of a ‘Growth Impatience Condition’ where the nature of the condition depends on the quantity whose stability is required. A model that exhibits stability of this kind is what we will call a ‘buﬀer stock’ model.^{4}
Even without a formal proof of its existence, target saving has been intuitively understood to underlie central quantitative results from the heterogeneous agent macroeconomics literature; for example, the logic of target saving is central to the recent claim by Krueger, Mitman, and Perri (2016) in the Handbook of Macroeconomics that such models explain why, during the Great Recession, middleclass consumers cut their consumption more than the poor or the rich. The theory below provides the rigorous theoretical basis for this claim: Learning that the future has become more uncertain does not change the urgent imperatives of the poor (their high means they – optimally – have little room to maneuver). And, increased labor income uncertainty does not 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 wiggleroom to reduce their spending.
Analytical derivations required for the proofs provide foundations for many other results familiar from the numerical literature.^{5}
The paper proceeds in three parts.
The ﬁrst part articulates suﬃcient conditions for the problem to deﬁne a useful (nondegenerate) limiting consumption function, and explains how the model relates to those previously considered in the literature, showing that the conditions required for convergence are interestingly parallel to those required for the liquidity constrained perfect foresight model; that parallel is explored and explained. Next, the paper derives limiting properties of the consumption function as resources approach inﬁnity, and as they approach their lower bound; using these limits, the contraction mapping theorem is then proven. Last comes a proof that a corresponding model with an ‘artiﬁcial’ liquidity constraint (that is, a model that exogenously prohibits consumers from borrowing even if they could certainly repay) is a particular limiting case of the model without constraints. The analytical convenience of the unconstrained model is that it is both mathematically convenient (e.g., the consumption function is twice continuously diﬀerentiable), and arbitraily close (cf. section 2.10) to less tractable models that have heretofore been tackled with less convenient methods. For future authors, the approach here models a strategy of proving interesting propositions in this more congenial environment, and then appealing to a limiting argument to establish the analogous proposition in an explicitly constrained but more unwieldy environment.
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 useful 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 SchmittGrohé and Uribe (2003), buﬀer stock saving is an appealing method of ‘closing’ a small open economy model, because it requires no adhoc assumptions. Not even liquidity constraints.
The inﬁnite horizon solution is the (limiting) ﬁrstperiod solution to a sequence of ﬁnitehorizon problems as the horizon (the last period of life) becomes arbitrarily distant.
That is, for the value function, ﬁxing a terminal date , we are interested in the term in the sequence of value functions . We will say that the problem has a ‘nondegenerate’ inﬁnite horizon solution if, corresponding to that value function, as there is a limiting consumption function which is neither everywhere (for all ) nor everywhere.
Concretely, a consumer born periods before date solves the problem
where the utility function
 (1) 
exhibits relative risk aversion .^{6} The consumer’s initial condition is deﬁned by market resources and permanent noncapital income , which both are positive,
 (2) 
and the consumer cannot die in debt,
 (3) 
In the usual treatment, a dynamic budget constraint (DBC) incorporates several elements that jointly determine next period’s (given this period’s choices); for the detailed analysis here, it will be useful to disarticulate the steps:

where indicates the consumer’s assets at the end of period , which grow by a ﬁxed interest factor between periods, so that is the consumer’s ﬁnancial (‘bank’) balances before next period’s consumption choice;^{7} (‘market resources’) is the sum of ﬁnancial wealth and noncapital income (permanent noncapital income multiplied by a meanone iid transitory income shock factor ; transitory shocks are assumed to satisfy ). Permanent noncapital income in is equal to its previous value, multiplied by a growth factor , modiﬁed by a meanone iid shock , satisfying for (and is the degenerate case with no permanent shocks).
Following Zeldes (1989), in future periods there is a small probability that income will be zero (a ‘zeroincome event’),
 (4) 
where is an iid meanone random variable () whose distribution satisﬁes where .^{8} Call the cumulative distribution functions and (where is derived trivially from (4) and ). For quick identiﬁcation in tables and graphs, we will call this the Friedman/Muth model because it is a speciﬁc implementation of the Friedman (1957) model as interpreted by Muth (1960), needing only a calibration of the income process and a speciﬁcation of preferences (here, geometric discounting and CRRA utility) to be solvable.
The model looks more special than it is. In particular, the assumption of a positive probability of zeroincome events may seem objectionable (though it has empirical support).^{9} However, it is easy to show that a model with a nonzero minimum value of (motivated, for example, by the existence of unemployment insurance) can be redeﬁned by capitalizing the present discounted value of minimum income into current market assets,^{10} transforming that model back into this one. And no key results would change if the transitory shocks were persistent but meanreverting, 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.^{11}
This model diﬀers from Bewley’s (1977) classic formulation in several ways. The Constant Relative Risk Aversion (CRRA) utility function does not satisfy Bewley’s assumption that is well deﬁned, or that 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 are of dominant importance in micro data^{12} (permanent shocks are far more consequential for household welfare than are transitory ﬂuctuations). 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.^{13} It diﬀers from models found in Stokey et. al. (1989) because neither liquidity constraints nor bounds on utility or marginal utility are imposed.^{14} ^{15} 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 in common with this one is Ma, Stachurski, and Toda (2020), henceforth MST, who establish the 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, assume that , and that expected marginal utility of income is ﬁnite (. These assumptions are not consistent with the combination of CRRA utility and income dynamics used here, whose combined properties are key to the results.^{16}
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 ( and ) to one . Value in the last period of life is ; using (in the last line in (5) below) the fact that for our CRRA utility function, , and generically deﬁning nonbold variables as the boldface counterpart normalized by (as with ), consider the problem in the secondtolast period,

Now, in a onetime deviation from the notational convention established in the last sentence, deﬁne nonbold ‘normalized value’ not as but as , because this allows us to exploit features of the related problem,

where is a ‘growthnormalized’ return factor, and the new problem’s ﬁrst order condition is^{17}
 (5) 
Since , deﬁning from (5), we obtain
This logic induces to earlier periods; if we solve the normalized onestatevariable problem (5), we will have solutions to the original problem for any from:
The problem has a nondegenerate solution if as the horizon gets arbitrarily large the solution in the ﬁrst period of life gets arbitrarily close to a limiting :
 (6) 
that satisﬁes
 (7) 
for every
The familiar analytical solution to the perfect foresight model, obtained by setting and , allows us to deﬁne some remaining notation and terminology.
The dynamic budget constraint, strictly positive marginal utility, and the can’tdieindebt condition (3) imply an exactlyholding intertemporal budget constraint (IBC):
 (8) 
where is nonhuman wealth, and with a constant ‘human wealth’ is

In order for to be ﬁnite, we must impose the Finite Human Wealth Condition (‘FHWC’):
 (9) 
Intuitively, for human wealth to be ﬁnite, the growth rate of (noncapital) income must be smaller than the interest rate at which that income is being discounted.
Without constraints, the consumption Euler equation always holds; with ,
 (10) 
where the archaic letter ‘thorn’ represents what we will call the ‘Absolute Patience Factor,’ or APF:
 (11) 
The sense in which captures patience is that if the ‘absolute impatience condition’ (AIC) holds,^{18}
 (12) 
the consumer will choose to spend an amount too large to sustain indeﬁnitely. We call such a consumer ‘absolutely impatient.’
We next deﬁne a ‘Return Patience Factor’ (RPF) that relates absolute patience to the return factor:
 (13) 
and since consumption is growing by but discounted by :
 (14) 
from which the IBC (8) implies
 (15) 
which deﬁnes a normalized ﬁnitehorizon perfect foresight consumption function
 (16) 
where is the marginal propensity to consume (MPC) – it answers the question ‘if the consumer had an extra unit of resources, how much more would be spent.’ (’s overbar signﬁes that will be an upper bound as we modify the problem to incorporate constraints and uncertainty; analogously, is a lower bound for the MPC).
Equation (15) makes plain that for the limiting MPC to be strictly positive as goes to inﬁnity we must impose the Return Impatience Condition (RIC):
 (17) 
so that
 (18) 
The RIC thus imposes a second kind of ‘impatience:’ 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 ). A consumer who satisﬁes the RIC is ‘return impatient.’
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
 (19) 
and so in order to rule out the degenerate limiting solution we need to be ﬁnite; that is, we must impose the Finite Human Wealth Condition (9).
Because we can write a useful analytical expression for the value the consumer would achieve by spending permanent income in every period:

which (for ) asymptotes to a ﬁnite number as approaches if any of these equivalent conditions holds:

where we call ^{19} the ‘Perfect Foresight Value Of Autarky Factor’ (PFVAF), and the variants of (20) constitute alternative versions of the Perfect Foresight Finite Value of Autarky Condition, PFFVAC; they guarantee that a consumer who always spends all permanent income ‘has ﬁnite autarky value.’^{20}
If the FHWC is satisﬁed, the PFFVAC implies that the RIC is satisﬁed: Divide both sides of the second inequality in (20) by :
 (20) 
and FHWC the RHS is because (and the RHS is raised to a positive power (because )).
Likewise, if the FHWC and the GIC are both satisﬁed, PFFVAC follows:

where the last line holds because FHWC and .
The ﬁrst panel of Table 4 summarizes: The PFUnconstrained model has a nondegenerate limiting solution if we impose the RIC and FHWC (these conditions are necessary as well as suﬃcient). Imposing the PFFVAC and the FHWC implies the RIC, so PFFVAC and FHWC are jointly suﬃcient. If we impose the GIC and the FHWC, both the PFFVAC and the RIC follow, so GIC+FHWC are also suﬃcient. But there are circumstances under which the RIC and FHWC can hold while the PFFVAC fails (which we write ). For example, if , the problem is a standard ‘cakeeating’ problem with a nondegenerate solution under the RIC.
Perhaps more useful than this prose or the table, the relations of the conditions for the unconstrained perfect foresight case are presented diagrammatically in Figure 1. Each node represents a quantity considered in the foregoing analysis. The arrow associated with each inequality reﬂects the imposition of that condition. For example, one way we wrote the PFFVAC in equation (20) is , so imposition of the PFFVAC is captured by the diagonal arrow connecting and . Traversing the boundary of the diagram clockwise starting at 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 that the PFFVAC holds. Reversal of a condition will reverse the arrow’s direction; so, for example, the bottommost arrow going from to imposes ; but we can cancel the cancellation and reverse the arrow. This would allow us to traverse the diagram in a clockwise direction from to , revealing that imposition of GIC and FHWC (and, redundantly, FHWC again) let us conclude that the RIC holds because the starting point is and the endpoint is . (Consult Appendix K for a detailed exposition of diagrams of this type).
An arrowhead points to the larger of the two quantities being compared. For example, the diagonal arrow indicates that , which is one way of writing the PFFVAC, equation (20)
We next examine the perfect foresight constrained solution because it is a useful benchmark (and limit) for the unconstrained problem with uncertainty (examined next).
If a liquidity constraint requiring is ever to be relevant, it must be relevant at the lowest possible level of market resources, , deﬁned by the lower bound for entering the period, . The constraint is ‘relevant’ if it prevents the choice that would otherwise be optimal; at the constraint is relevant if the marginal utility from spending all of today’s resources , exceeds the marginal utility from doing the same thing next period, ; that is, if such choices would violate the Euler equation (5):
 (21) 
By analogy to the RPF, we therefore deﬁne a ‘growth patience factor’ (GPF) as
 (22) 
and deﬁne a ‘growth impatience condition’ (GIC)
 (23) 
which is equivalent to (21) (exponentiate both sides by ).
We now examine implications of possible conﬁgurations of the conditions.
and RIC. If the GIC fails but the RIC (17) holds, appendix A shows that, for some , an unconstrained consumer behaving according to (19) would choose for all . In this case the solution to the constrained consumer’s problem is simple: For any the constraint does not bind (and will never bind in the future); for such the constrained consumption function is identical to the unconstrained one. If the consumer were somehow^{21} to arrive at an the constraint would bind and the consumer would consume . Using the accent to designate the version of a function in the presence of constraints (and recalling that is the unconstrained perfect foresight solution):
 (24) 
GIC and RIC. More useful is the case where the return impatience and GIC conditions both hold. In this case appendix A shows that the limiting constrained consumption function is piecewise linear, with up to a ﬁrst ‘kink point’ at , and with discrete declines in the MPC at a set of kink points . As the constrained consumption function becomes arbitrarily close to the unconstrained , and the marginal propensity to consume function limits to . Similarly, the value function is nondegenerate and limits into the value function of the unconstrained consumer.
This logic holds even when the ﬁnite human wealth condition fails (), because the constraint prevents the consumer from borrowing against inﬁnite human wealth to ﬁnance inﬁnite current consumption. Under these circumstances, the consumer who starts with any amount of resources will, over time, run those resources down so that by some ﬁnite number of periods in the future the consumer will reach , and thereafter will set for eternity (which the PFFVAC says yields ﬁnite value). Using the same steps as for equation (20), value of the interim program is also ﬁnite:
So, if , value for any ﬁnite will be the sum of two ﬁnite numbers: The component due to the unconstrained consumption choice made over the ﬁnite horizon leading up to , and the ﬁnite component due to the value of consuming all thereafter.
GIC and . The most peculiar possibility occurs when the RIC fails. Under these circumstances the FHWC must also fail (Appendix A), and the constrained consumption function is nondegenerate. (See appendix Figure 8 for a numerical example). While it is true that , nevertheless the limiting constrained consumption function is strictly positive and strictly increasing in . This result interestingly reconciles the conﬂicting intuitions from the unconstrained case, where would suggest a degenerate limit of while would suggest a degenerate limit of .
Tables 3 and 4 (and appendix table 5) codify.
We now examine the case with uncertainty but without constraints, which will turn out to be a close parallel to the model with constraints but without uncertainty.
When uncertainty is introduced, the expectation of beginningofperiod bank balances can be rewritten as:
 (25) 
where Jensen’s inequality guarantees that the expectation of the inverse of the permanent shock is strictly greater than one. It will be convenient to deﬁne
 (26) 
which satisﬁes (thanks to Mr. Jensen), so we can deﬁne
 (27) 
which is useful because it allows us to write uncertaintyadjusted versions of equations and conditions in a manner exactly parallel to those for the perfect foresight case; for example, we deﬁne a normalized Growth Patience Pactor (GPFNrm):
 (28) 
and a normalized version of the Growth Impatience Condition, GICNrm:
 (29) 
which is stronger than the perfect foresight version (23) because (cf (27)).
Analogously to (20), 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:
which invites the deﬁnition of a utilitycompensated equivalent of the permanent shock,
 (30) 
which will satisfy for and nondegenerate . Deﬁning
 (31) 
we can see that will be ﬁnite as approaches if

which we call the ‘ﬁnite value of autarky condition’ (FVAC) because it guarantees that value is ﬁnite for a consumer who always consumes their (now stochastic) permanent income (and we will call the ‘Value of Autarky Factor,’ VAF).^{22} For nondegenerate , this condition is stronger (harder to satisfy in the sense of requiring lower ) than the perfect foresight version (20) because .^{23}
Figure 2, familiar from the literature, depicts the successive consumption rules that apply in the last period of life , the secondtolast period, and earlier periods under baseline parameter values listed in Table 2. (The 45 degree line is 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 . Our next purpose is to show that this appearance is not deceptive.
A precondition for the main proof is that the maximization problem (5) deﬁnes a sequence of continuously diﬀerentiable strictly increasing strictly concave^{24} functions . The straightforward but tedious proof is relegated to appendix B. For present purposes, the most important point is that the income process induces what Aiyagari (1994) dubbed a ‘natural borrowing constraint’: for all periods because a consumer who spent all available resources would arrive in period with balances of zero, and then might earn zero income over the remaining horizon, requiring the consumer to spend zero, incurring negative inﬁnite utility. To avoid this disaster, 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 in a life cycle model.^{25}
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. As the arguments below will illustrate, the analytical convenience of these features is considerable – a point that may appeal to theorists when they realize (cf. section H below) that the solution to this congenial problem is arbitraily close to the solutin to the constrained but less wieldy problem with explicit constraints.
The consumption functions depicted in Figure 2 appear to have limiting slopes as and as . This section conﬁrms that impression and derives those slopes, which will be needed in the contraction mapping proof.^{26}
Assume that a continuously diﬀerentiable concave consumption function exists in period , with an origin at , a minimal MPC , and maximal MPC . (If these will be ; for earlier periods they will exist by recursion from the following arguments.)
The MPC bound as wealth approaches inﬁnity is easy to understand: In this case, under our imposed assumption that human wealth is ﬁnite, the proportion of consumption that will be ﬁnanced out of human wealth approaches zero. In consequence, 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 useful recursive expression (used below) for the (inverse of the) limiting MPC:
 (32) 
Appendix equation (72) presents a parallel expression for the limiting maximal MPC as :
 (33) 
where is a decreasing convergent sequence if the ‘weak return patience factor’ satisﬁes:
 (34) 
a condition we dub the ‘Weak Return Impatience Condition’ (WRIC) because with it will hold more easily (for a larger set of parameter values) than the RIC ().
The essence of the argument is that as wealth approaches zero, the overriding consideration that limits consumption is the (recursive) fear of the zeroincome 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
 (35) 
because consumption starts at zero and is continuously diﬀerentiable (as argued above), is strictly concave,^{27} and always exhibits a slope between and (the formal proof is in appendix D).
As mentioned above, standard theorems in the contraction mapping literature following Stokey et. al. (1989) require utility or marginal utility to be bounded over the space of possible values of , which does not hold here because the possibility (however unlikely) of an unbroken string of zeroincome events through the end of the horizon means that utility (and marginal utility) are unbounded as . 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)), 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.^{28}
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.
Deﬁnition 1. Consider any function where is the space of continuous functions from to . Suppose with and . Then is bounded if the norm of ,
 (36) 
is ﬁnite.
For deﬁned as the set of functions in that are bounded; , , , and as examples of bounded functions; and using to indicate the function that returns zero for any argument, Boyd (1990) proves the following.
Boyd’s Weighted Contraction Mapping Theorem. Let such that^{29} ^{30}
Then deﬁnes a contraction with a unique ﬁxed point.
For our problem, take as and as , and deﬁne
Using this, we introduce the mapping ,^{31}
 (37) 
We can show that our operator satisﬁes the conditions that Boyd requires of his operator , if we impose two restrictions on parameter values. The ﬁrst is the WRIC necessary for convergence of the maximal MPC, equation (34) above. More serious is the Finite Value of Autarky condition, equation (32). (We discuss the interpretation of these restrictions in detail in section 2.11 below.) Imposing these restrictions, we are now in position to state the central theorem of the paper.
Theorem 1. is a contraction mapping if the restrictions on parameter values (34) and (32) are true (that is, if the weak return impatience condition and the ﬁnite value of autarky condition hold).
Intuitively, Boyd’s theorem shows that if you can ﬁnd a that is everywhere ﬁnite but goes to inﬁnity ‘as fast or faster’ than the function you are normalizing with , the normalized problem deﬁnes a contraction mapping. The intuition for the FVAC condition is just that, with an inﬁnite horizon, with any initial amount of bank balances , in the limit your value can always be made greater than you would get by consuming exactly the sustainable amount (say, by consuming for some small ).
The details of the proof are cumbersome, and are therefore relegated to appendix D. Given that the value function converges, appendix E.2 shows that the consumption functions converge.^{32}
This section explains why 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 zeroincome event approaches zero.
The ‘related’ problem makes two changes to the problem deﬁned above:
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 , that probability becomes arbitrarily small, so the amount of precautionary saving induced by the zeroincome events approaches zero as . But “zero” is the amount of precautionary saving that would be induced by a zeroprobability 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 H for the formal proof justifying the foregoing intuitive discussion.^{33}
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 (zeroincome events, other transitory shocks, and permanent shocks) approaches zero.
The full relationship among all the conditions is represented in Figure 3. Though the diagram looks complex, it is merely a modiﬁed version of the earlier diagram with further (mostly intermediate) inequalities inserted. (Arrows with a “because” are a new element to label relations that always hold under the model’s assumptions.) Again readers unfamiliar with such diagrams should see Appendix K) for a more detailed explanation.
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
 (38) 
If we require , the WRIC is redundant because now , so that (with and ) the RIC (and WRIC) must hold. But neither theory nor evidence demands that we assume . We can therefore approach the question of the WRIC’s relevance by asking just how low must be for the condition to be relevant. Suppose for illustration that , , and . In that case (38) reduces to
but since by assumption, the binding requirement is that
so that for example if we would need (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 , which section 2.10 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 , the consumer has no noncapital income (so that the FHWC holds) and with 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.
In the perfect foresight problem (section 2.4.2), 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 (that is, :

but since (cf. the argument below (30)), this requires ; so, given the FVAC, the RIC can fail only if human wealth is unbounded. As an illustration of the usefulness of our diagrams, note that this algebraically complicated conclusion could be easily reached diagrammatically in ﬁgure 3 by starting at the node and imposing (reversing the RIC arrow) and then traversing the diagram along any clockwise path to the PFVAF node at which point we realize that we cannot impose the FHWC because that would let us conclude .
As in the perfect foresight constrained problem, unbounded limiting human wealth () here does not lead to a degenerate limiting consumption function (ﬁnite human wealth is not a condition required for the convergence theorem). But, from equation (32) and the discussion surrounding it, an implication of is that . Thus, interestingly, in the special case (unavailable in the perfect foresight model) the presence of uncertainty both permits unlimited human wealth and at the same time prevents unlimited human wealth from resulting in inﬁnite consumption at any ﬁnite . Intutively, in the presence of uncertainty, pathological patience (which in the perfect foresight model results in a limiting consumption function of for ﬁnite ) plus unbounded human wealth (which the perfect foresight model prohibits (by assumption FHWC) because it leads to a limiting consumption function for any ﬁnite ) combine to yield a unique ﬁnite level of consumption and the MPC for any ﬁnite value of . Note the close parallel to the conclusion in the perfect foresight liquidity constrained model in the {GIC,} case. There, too, the tension between inﬁnite human wealth and pathological patience was resolved with a nondegenerate consumption function whose limiting MPC was zero.^{34}
FHWC. If the RIC and FHWC both hold, a perfect foresight solution exists (see 2.4.2 above). As the limiting consumption function and value function 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.
. The more exotic case is where FHWC does not hold; in the perfect foresight model, {RIC,} is the degenerate case with limiting . Here, since the FVAC implies that the PFFVAC holds (traverse Figure 3 clockwise from by imposing FVAC and continue to the PFVAF node), reversing the arrow connecting the and PFVAF nodes implies that under :
where the transition from the ﬁrst to the second lines is justiﬁed because So, {RIC, } implies the GIC holds. However, we are not entitled to conclude that the GICNrm holds: does not imply where . See further discussion of this illuminating case in section ??.
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.
Figures 4 and 5a,b capture the main properties of the converged consumption rule when the RIC, GICNrm, and FHWC all hold.^{35} Figure 4 shows the expected growth factors for the levels of consumption and market resources, and , for a consumer behaving according to the converged consumption rule, while Figures 5 and 6 illustrate theoretical bounds for the consumption function and the marginal propensity to consume.
Three features of behavior are captured, or suggested, by the ﬁgures. First, as the expected consumption growth factor goes to , indicated by the lower bound in Figure 4, and the marginal propensity to consume approaches (Figure 5), the same as the perfect foresight MPC. Second, as the consumption growth factor approaches (Figure 4) and the MPC approaches (Figure 5). Third, there is a value at which the expected growth rate of matches the expected growth rate of permanent income , and a diﬀerent (lower) value where the expected growth rate of consumption at is lower than . Thus, at the individual level, this model does not have a ‘balanced growth’ equilibrium in which all model variables are expected to grow at the same rate.^{36}
Deﬁne
which is the solution to an inﬁnitehorizon problem with no noncapital income (); clearly , since allowing the possibility of future noncapital income cannot reduce current consumption. Our imposition of the RIC guarantees that , 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 (19) constitutes an upper bound on consumption in the presence of uncertainty, since the introduction of uncertainty strictly decreases the level of consumption at any (Carroll and Kimball (1996)). Thus, we can write

But
so as , and the continuous diﬀerentiability and strict concavity of therefore implies

because any other ﬁxed limit would eventually lead to a level of consumption either exceeding or lower than .
Figure 5 conﬁrms these limits visually. The top plot shows the converged consumption function along with its upper and lower bounds, while the lower plot shows the marginal propensity to consume.
Next we establish the limit of the expected consumption growth factor as :
But
and

while (for convenience deﬁning ),

because ^{37} and which goes to zero as goes to inﬁnity.
Hence we have

so as cash goes to inﬁnity, consumption growth approaches its value in the perfect foresight model.
Equation (33) shows that the limiting value of is
Deﬁning as before we have
Now using the continuous diﬀerentiability of the consumption function along with L’Hôpital’s rule, we have
Figure 5 conﬁrms that the numerical solution obtains this limit for the MPC as approaches zero.
For consumption growth, as we have
where the secondtolast line follows because is positive, and the last line follows because the minimum possible realization of is so the minimum possible value of expected nextperiod consumption is positive.^{38}
Two theorems, whose proofs are sketched here and detailed in an appendix, articulate alternative (but closely related) stability criteria for the model.
One deﬁnition of a ‘stable’ point is an such that if , then . Existence of such a target turns out to require the GICNrm condition.
Theorem 2. For the nondegenerate solution to the problem deﬁned in section 2.1 when FVAC, WRIC, and GICNrm all hold, there exists a unique cashonhandtopermanentincome ratio such that
 (39) 
Moreover, is a point of ‘wealth stablity’ in the sense that

Since , the implicit equation for is
 (40) 
A traditional question in macroeconomic models is whether there is a ‘balanced growth’ equilibrium in which aggregate variables (income, consumption, market resources) all grow forever at the same rate. For the model here, we have already seen in Figure 4 that there is no single for which for an individual consumer. Nevertheless, analysis below will show that economies populated by collections of such consumers can exhibit balanced growth in the aggregate.
As an input to that analysis, we show here that if the GIC holds, the problem will have a point of what we call ‘collective stability,’ by which we mean that there is some such that, for all , , and conversely for . (‘Collective’ is meant to capture the fact that calculating the expectation of the levels of future and before dividing by is akin to examining aggregate values in a population).
The critical will be the value at which growth matches :


The only diﬀerence between (41) and (40) is the substitution of for .
We will refer to as the problem’s ExpectedBalancedGrowth State, a term motivated by the fact that an economy composed of consumers all of whom had , would exhibit balanced growth if all consumers happened to continually draw permanent and transitory shocks equal to their expected values of 1.0 forever.^{39}
Theorem 5 formally states the relevant proposition.
Theorem 3. For the nondegenerate solution to the problem deﬁned in section 2.1 when FVAC, WRIC, and GIC all hold, there exists a unique pseudosteadystate cashonhandtoincome ratio such that
 (41) 
Moreover, is a point of stability in the sense that

The proofs of the two theorems are almost completely parallel; to save space, they are relegated to Appendix M. In sum, they involve three steps:
Because the equations deﬁning target and pseudosteadystate , (40) and (41), diﬀer only by substitution of for , if there are no permanent shocks (), the conditions are identical. For many parameterizations (e.g., under the baseline parameter values used for constructing ﬁgure 4), and 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 GICNrm; now there is no target wealth level (consumption remains everywhere below the level that would keep expected constant).
The pseudosteadystate still exists because it turns oﬀ realizations of the permanent shock. But an aggregate balanced growth equilibrim can exist even when realizations of the permanent shock are implemented exactly as speciﬁed in the model. The key insight can be understood by considering the evolution of an economy that starts, at date , with the entire population at , but then evolves according to the model’s correct assumed dynamics between and . Equation (41) will still hold for this economy, so for this ﬁrst period, at least, the economy will exhibit balanced growth: the growth factor for aggregate will match the growth factor for permanent income . It is true that there will be people for whom is boosted by a small draw of . But their contribution to the aggregate variable is given by , so their reweighted by an amount that exactly undoes the boosting caused by earlier normalization.
The surprising consequence is that, if the GICholds but the GICNrm fails, it is possible to construct an aggregate economy composed of consumers all of whom have target wealth of , but in which the aggregate economy still exhibits balanced growth with a ﬁnite ratio of aggregate wealth to income. (For an example, see the software archive for the paper).
This is a good introduction to a more explicit discssion of aggregation.
A large (inﬁnite) collection of small (inﬁnitesimal) buﬀerstock consumers with identical parameter values can be thought of as a subset of the population within a single country (say, members of a given education or occupation group), or as the whole population in a small open economy with an exogenous (constant) interest rate.^{40}
Until now for convenience we have assumed inﬁnite horizons, with the implicit understanding that Poisson mortality could be handled by adjusting the eﬀective discount factor for mortality. On that basis, section 4.1 continues to omit mortality. But a reason for explicitly introducing mortality will appear at the end of section 4.2, so implications of alternative assumptions about mortality are brieﬂy examined in Section 4.3.
Formally, we assume a continuum of ex ante identical households on the unit interval, with constant total mass normalized to one and indexed by , all behaving according to the model speciﬁed above. Szeidl (2013) proves that whenever the GIC holds such a population will be characterized by invariant distributions of , , and ;^{41} designate these , , and .
The operator yields the mean of its argument in the population, as distinct from the expectations operator used above, which represents beliefs about the future.
An economist with a microeconomic dataset could calculate the average growth rate of idiosyncratic consumption in a cross section of an economy that had converged at date , and would ﬁnd
where and the last equality follows because the invariance of (Szeidl (2013)) means that . Thus, the same GIC that guaranteed the existence of an ‘individual pseudosteadystate’ value of at the microeconomic level guarantees both that there will be an invariant distribution of the population across values of the model variables and that in that invariant distribution the mean growth rates of all idiosyncratic variables are the same (see Szeidl (2013) for details).
Using boldface capital letters for aggregates, the growth factor for aggregate income is:
because of the independence assumptions we have made about and .
From the perspective of period ,
Unfortunately, the covariance term in the numerator, while generally small, will not in general be zero. This is because the realization of the permanent shock has a nonlinear eﬀect on . Matters are simpler if there are no permanent shocks; see Appendix F for a proof that in that case the growth rate of assets (and other variables) does eventually converge to the growth rate of aggregate permanent income.
One way of thinking about the problem here is that it may reﬂect the fact that, under our assumptions, permanent income does not have an ergodic distribution; its distribution of becomes forever wider over time, because our consumers never die and each immortal person is perpetually subject to symmetric shocks to their .
This is why we need to introduce mortality.
Most heterogeneous agent models incorporate a constant positive probability of death, following Blanchard (1985). In a model that mostly follows Blanchard (1985), for probabilities of death that exceed 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, which is a useful step in the direction of taming the problems caused by an unbounded distribution of . Numerical results in that paper conﬁrm the intuition that, under appropriate impatience conditions, balanced growth arises (though a formal proof remains elusive).
Even with those (numerical) results in hand, the centrality of mortality assumptions to the existence and nature of steady states requires them to be discussed brieﬂy here.
Blanchard (1985)’s model assumes the existence of an 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 several 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: If the probability of remaining alive is , then assuming that no utility accrues after death makes the eﬀective discount factor , while the enhancement to the rate of return from the annuity scheme yields an eﬀective interest rate of (recall that because of Poisson mortality, the average wealth of the two groups is identical). Combining these, the eﬀective patience factor in the new economy is unchanged from its value in the inﬁnite horizon model:
 (42) 
The only adjustments this requires to the analysis from prior parts of this paper are therefore to the few elements that involve a role for distinct from its contribution to (principally, the RIC).
The numerical ﬁnding that the covariance term above is approximately zero allows us to conclude again that the key requirement for aggregate balanced growth is presumably the GIC.
Blanchard (1985)’s innovation was useful not only for the insight it provided but also because the principal alternative, the Life Cycle model of Modigliani (1966), was computationally challenging given the thenavailable technologies. Aside from its (considerable) conceptual value, there is no need for Blanchard’s analytical solution today, when serious modeling incorporates uncertainty, constraints, and other features that rule out analytical solutions anyway.
The simplest alternative to Blanchard’s mortality is to follow Modigliani in 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 again 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 estaterelated 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. If is the probability of remaining alive, the condition changes from the plain GIC to a looser mortalityadjusted GIC:
 (43) 
With no income growth, the condition required to prohibit unbounded growth in aggregate wealth would be the condition that prevents the percapita wealth to income ratio of surviving consumers from growing faster than the rate at which mortality diminishes their collective population. With income growth, the aggregate wealthtoincome 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.
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 inconvenient) complications like liquidity constraints – have attractive analytical properties (like continuous diﬀerentiability of the consumption function, and analytical limiting MPC’s as resources approach their minimum and maximum possible values), and that (more widely used) models with liquidity constraints can be viewed as a particular limiting case of this simpler model.
The main focus of the paper, though, is on the limiting solution of the ﬁnite horizon model as the horizon extends to inﬁnity. The paper shows that the simple model has additional attractive properties: 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 Blancharidan (tighter) or Modiglianian (looser) mortality will exhibit balanced growth. Finally, the paper provides the analytical basis for a number of results about buﬀerstock saving models that are so well understood that even without analytical foundations researchers uncontroversially use them as explanations of realworld phenomena like the crosssectional pattern of consumption dynamics in the Great Recession.
The paper’s results are all easily reproducible interactively on the web or on any standard computer system. Such reproducibility reﬂects the paper’s use of the opensource EconARK toolkit, which is used to generate all of the quantitative results of the paper, and which integrally incorporates all of the analytical insights of the paper.

Under perfect foresight in the presence of a liquidity constraint requiring , this appendix taxonomizes the varieties of the limiting consumption function that arise under various parametric conditions. Results are summarized in table 5.
A consumer is ‘growth patient’ if the perfect foresight growth impatience condition fails (, ). Under the constraint does not bind at the lowest feasible value of because implies that spending everything today (setting ) produces lower marginal utility than is obtainable by reallocating a marginal unit of resources to the next period at return :^{42}

Similar logic shows that under these circumstances the constraint will never bind at for a constrained consumer with a ﬁnite horizon of periods, so for 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 () while the ﬁnite human wealth condition holds, the limiting value of this consumption function as is the degenerate function
 (44) 
(that is, consumption is zero for any level of human or nonhuman wealth).
RIC fails, FHWC fails. implies that human wealth limits to so the consumption function limits to either or depending on the relative speeds with which the MPC approaches zero and human wealth approaches .^{43}
Thus, the requirement that the consumption function be nondegenerate implies that for a consumer satisfying we must impose the RIC (and the FHWC can be shown to be a consequence of 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 from equation (19):
 (45) 
which (under these assumptions) satisﬁes .^{44} For the unconstrained consumer would choose to consume more than ; for such , the constrained consumer is obliged to choose .^{45} For any the constraint will never bind and the consumer will choose to spend the same amount as the unconstrained consumer, .
(Stachurski and Toda (2019) obtain a similar lower bound on consumption and use it to study the tail behavior of the wealth distribution.)
Imposition of the GIC reverses the inequality in (44), and thus reverses the conclusion: A consumer who starts with will desire to consume more than 1. Such a consumer will be constrained, not only in period , but perpetually thereafter.
Now deﬁne as the such that an unconstrained consumer holding would behave so as to arrive in period with (with trivially equal to 0); for example, a consumer with was on the ‘cusp’ of being constrained in period : Had been inﬁnitesimally smaller, the constraint would have been binding (because the consumer would have desired, but been unable, to enter period with negative, not zero, ). Given the GIC, the constraint certainly binds in period (and thereafter) with resources of : The consumer cannot spend more (because constrained), and will not choose to spend less (because impatient), than .
We can construct the entire ‘prehistory’ of this consumer leading up to as follows. Maintaining the assumption that the constraint has never bound in the past, must have been growing according to , so consumption periods in the past must have been
 (46) 
The PDV of consumption from until can thus be computed as

and note that the consumer’s human wealth between and (the relevant time horizon, because from onward the consumer will be constrained and unable to access post income) is
 (47) 
while the intertemporal budget constraint says
 (48) 
Deﬁning , consider the function deﬁned by linearly connecting the points for integer values of (and setting for ). This function will return, for any value of , the optimal value of for a liquidity constrained consumer with an inﬁnite horizon. The function is piecewise linear with ‘kink points’ where the slope discretely changes; for inﬁnitesimal the MPC of a consumer with assets is discretely higher than for a consumer with assets 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 (48) for the entire domain of positive real values of , we need to become arbitrarily large with . That is, we need
 (49) 
The FHWC requires , in which case the second term in (48) limits to a constant as , and (49) reduces to a requirement that
 (50) 
If the FHWC fails, matters are a bit more complex.
Given failure of FHWC, (49) requires

If RIC Holds. When the RIC holds, rearranging (51) gives
 (51) 
which with a bit of algebra^{46} can be shown to asymptote to the MPC in the perfect foresight model:^{47}
 (53) 
If RIC Fails. Consider now the case, . We can rearrange (51)as
What is happening here is that the term is increasing backward in time at rate dominated in the limit by while the term is increasing at a rate dominated by term and
Consequently, while , the limit of the ratio in (51) is zero. Thus, surprisingly, the problem has a well deﬁned solution with inﬁnite human wealth if the RIC fails. It remains true that implies a limiting MPC of zero,
 (56) 
but that limit is approached gradually, starting from a positive value, and consequently the consumption function is not the degenerate . (Figure 8 presents an example for , , , ; note that the horizontal axis is bank balances ; the part of the consumption function below the depicted points is uninteresting – – 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 . But even if the FHWC fails, the problem has a welldeﬁned 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 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.
To show that (5) deﬁnes a sequence of continuously diﬀerentiable strictly increasing concave functions , we start with a deﬁnition. We will say that a function is ‘nice’ if it satisﬁes
(Notice that an implication of niceness is that )
Assume that some is nice. Our objective is to show that this implies is also nice; this is suﬃcient to establish that is nice by induction for all because and is nice by inspection.
Now deﬁne an endofperiod value function as
 (57) 
Since there is a positive probability that will attain its minimum of zero and since , it is clear that and . So is welldeﬁned iﬀ ; it is similarly straightforward to show the other properties required for to be nice. (See Hiraguchi (2003).)
Next deﬁne as
 (58) 
which is since and are both and note that our problem’s value function deﬁned in (5) can be written as
 (59) 
is welldeﬁned if and only if . Furthermore, , , , and . It follows that the deﬁned by
 (60) 
exists and is unique, and (5) has an internal solution that satisﬁes
 (61) 
Since both and are strictly concave, both and are strictly increasing. Since both and are three times continuously diﬀerentiable, using (61) we can conclude that is continuously diﬀerentiable and
 (62) 
Similarly we can easily show that is twice continuously diﬀerentiable (as is ) (See Appendix C.) This implies that is nice, since .
First we show that is Deﬁne as . Since and
Since and are continuous and increasing, and are satisﬁed. Then for suﬃciently small . Hence we obtain a welldeﬁned equation:

This implies that the rightderivative, is welldeﬁned and

Similarly we can show that , which means exists. Since is , exists and is continuous. is diﬀerentiable because is , is and . is given by
 (63) 
Since is continuous, is also continuous.
We must show that our operator satisﬁes all of Boyd’s conditions.
Boyd’s operator maps from to A preliminary requirement is therefore that be continuous for any bounded , . This is not diﬃcult to show; see Hiraguchi (2003).
Consider condition (1). For this problem,
so implies by inspection.^{48}
Condition (2) requires that . By deﬁnition,

the solution to which is patently . Thus, condition (2) will hold if is bounded. We use the bounding function
 (64) 
for some real scalar whose value will be determined in the course of the proof. Under this deﬁnition of , is clearly bounded.
Finally, we turn to condition (3), The proof will be more compact if we deﬁne and as the consumption and assets functions^{49} associated with and and as the functions associated with ; 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,
Thus, condition (3) will certainly hold under the stronger condition
where the last line follows because by assumption.^{50}
Using and deﬁning , this condition is
which by imposing PFFVAC (equation (20), which says ) can be rewritten as:
 (65) 
But since is an arbitrary constant that we can pick, the proof thus reduces to showing that the numerator of (65) is bounded from above:

We can thus conclude that equation (65) will certainly hold for any:
 (66) 
which is a positive ﬁnite number under our assumptions.
The proof that deﬁnes a contraction mapping under the conditions (34) and (32) is now complete.
In deﬁning our operator we made the restriction . However, in the discussion of the consumption function bounds, we showed only (in (35)) that . (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 with 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 (66). Consideration of the prior two equations reveals that a suﬃcient stronger condition is
where we have used (33) for (and in the second step the reversal of the inequality occurs because we have assumed so that we are exponentiating both sides by the negative number ). To see that this is a weak condition, note that for small values of this expression can be further simpliﬁed using so that it becomes
Calling the weak return patience factor and recalling that the WRIC was , the expression on the LHS above is times the WRPF. Since we usually assume not far below 1 and parameter values such that , 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 with a unique . But since and , it must be the case that the toward which these ’s are converging is the same that was the endpoint of the contraction deﬁned by our operator . 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 deﬁned by .^{51}
Boyd’s theorem shows that deﬁnes a contraction mapping in a bounded space. We now show that also deﬁnes a contraction mapping in Euclidian space.
Calling the unique ﬁxed point of the operator , since ,
 (67) 
On the other hand, and because and are in . It follows that
 (68) 
Then we obtain
 (69) 
Since , . On the other hand, means , in other words, . Inductively one gets . This means that is a decreasing sequence, bounded below by .
Given the proof that the value functions converge, we now show the pointwise convergence of consumption functions .
Consider any convergent subsequence of converging to . By the deﬁnition of , we have
 (70) 
for any . Now letting go to inﬁnity, it follows that the left hand side converges to , and the right hand side converges to . So the limit of the preceding inequality as approaches inﬁnity implies
 (71) 
Hence, . By the uniqueness of , .
Section 4.2 asserted that in the absence of permanent shocks it is possible to prove that the growth factor for aggregate consumption approaches that for aggregate permanent income. This section establishes that result.
First deﬁne as the function that yields optimal endofperiod assets as a function of .
Suppose the population starts in period with an arbitrary value for . Then if is the invariant mean level of we can deﬁne a ‘mean MPS away from ’ function :

where the combination of the bar and the are meant to signify that this is the average value of the derivative over the interval. Since , is a constant at , if we deﬁne as the value of corresponding to , we can write

so

But since ,

and for the version of the model with no permanent shocks the GICNrm says that while the FHWC says that

This means that from any arbitrary starting value, the relative size of the covariance term shrinks to zero over time (compared to the term which is growing steadily by the factor ). Thus, .
This logic unfortunately does not go through when there are permanent shocks, because the terms are not independent of the permanent income shocks.
To see the problem clearly, deﬁne and consider a ﬁrst order Taylor expansion of around
The problem comes from the term. The concavity of the consumption function implies convexity of the function, so this term is strictly positive but we have no theory to place bounds on its size as we do for its level . We cannot rule out by theory that a positive shock to permanent income (which has a negative eﬀect on ) could have a (locally) unboundedly positive eﬀect on (as for instance if it pushes the consumer arbitrarily close to the selfimposed liquidity constraint).
For we can deﬁne and and the Euler equation (5) can be rewritten

Consider the ﬁrst conditional expectation in (5), recalling that if then . Since , is contained within bounds deﬁned by and both of which are ﬁnite numbers, implying that the whole term multiplied by goes to zero as goes to zero. As the expectation in the other term goes to (This follows from the strict concavity and diﬀerentiability of the consumption function.) It follows that the limiting satisﬁes Exponentiating by , 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 (34) for this to be a convergent sequence:
 (72) 
Since , iterating (72) backward to inﬁnity (because we are interested in the limiting consumption function) we obtain:
 (73) 
and we will therefore call the ‘limiting maximal MPC.’
The minimal MPC’s are obtained by considering the case where . If the FHWC holds, then as the proportion of current and future consumption that will be ﬁnanced out of capital approaches 1. Thus, the terms involving in (72) can be neglected, leading to a revised limiting Euler equation
 (74) 
so that is also an increasing convergent sequence, and we deﬁne
 (75) 
as the limiting (inverse) marginal MPC. If the RIC does not hold, then and so the limiting MPC is
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
 (76) 
Formally, suppose we change the description of the problem by making the following two assumptions:
and we designate the solution to this consumer’s problem . We will henceforth refer to this as the problem of the ‘restrained’ consumer (and, to avoid a common confusion, we will refer to the consumer as ‘constrained’ only in circumstances when the constraint is actually binding).Redesignate the consumption function that emerges from our original problem for a given ﬁxed as where we separate the arguments by a semicolon to distinguish between , which is a state variable, and , which is not. The proposition we wish to demonstrate is
 (77) 
We will ﬁrst examine the problem in period , then argue that the desired result propagates to earlier periods. For simplicity, suppose that the interest, growth, and timepreference factors are and there are no permanent shocks, ; the results below are easily generalized to the fullﬂedged 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 is given by (in practice, this will be ), consider the unrestrained optimization problem
 (78) 
As usual, the envelope theorem tells us that so the expected marginal value of ending period with assets can be deﬁned as

and the solution to (78) will satisfy
 (79) 
therefore answers the question “With what level of assets would the restrained consumer like to end period if the constraint did not exist?” (Note that the restrained consumer’s income process remains diﬀerent from the process for the unrestrained consumer so long as .) 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 at which the constraint makes the transition between binding and nonbinding in period .
Analogously to (79), deﬁning
 (80) 
the Euler equation for the original consumer’s problem implies
 (81) 
with solution . Now note that for any ﬁxed , . Since the LHS of (79) and (81) are identical, this means that . That is, for any ﬁxed value of such that the consumer subject to the restraint would voluntarily choose to end the period with positive assets, the level of endofperiod assets for the unrestrained consumer approaches the level for the restrained consumer as . With the same and the same , the consumers must have the same , so the consumption functions are identical in the limit.
Now consider values for which the restrained consumer is constrained. It is obvious that the baseline consumer will never choose because the ﬁrst term in (80) is , while is ﬁnite (the marginal value of endofperiod assets approaches inﬁnity as assets approach zero, but the marginal utility of consumption has a ﬁnite limit for ). The subtler question is whether it is possible to rule out strictly positive for the unrestrained consumer.
The answer is yes. Suppose, for some that the unrestrained consumer is considering ending the period with any positive amount of assets . For any such we have that . But by assumption we are considering a set of circumstances in which , and we showed earlier that . So, having assumed , we have proven that the consumer would optimally choose , which is a contradiction. A similar argument holds for .
These arguments demonstrate that for any , which is the period version of (77). But given equality of the period 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 (73) for the maximal marginal propensity to consume satisﬁes
The model is solved using an extension of the method of endogenous gridpoints (Carroll (2006)): A grid of possible values of endofperiod assets is deﬁned, and at these points, marginal endofperiod value is computed as the discounted nextperiod expected marginal utility of consumption (which the Envelope theorem says matches expected marginal value). The results are then used to identify the corresponding levels of consumption at the beginning of the period:^{52}

The dynamic budget constraint can then be used to generate the corresponding ’s:
An approximation to the consumption function could be constructed by linear interpolation between the points. But a vastly more accurate approximation can be made (for a given number of gridpoints) if the interpolation is constructed so that it also matches the marginal propensity to consume at the gridpoints. Diﬀerentiating (82) with respect to (and dropping policy function arguments for simplicity) yields a marginal propensity to have consumed at each gridpoint:

and the marginal propensity to consume at the beginning of the period is obtained from the marginal propensity to have consumed by noting that, if we deﬁne ,
which, together with the chain rule , yields the MPC from
and we call the vector of MPC’s at the gridpoints .
For any set of parameter values that satisfy the conditions required for convergence, the problem can be solved by setting the terminal consumption function to and constructing by time iteration (a method that will converge to by standard theorems). But is very far from the ﬁnal converged consumption rule ,^{53} and thus many periods of iteration will likely be required to obtain a candidate rule that even remotely resembles the converged function.
A natural alternative choice for the terminal consumption rule is the solution to the perfect foresight liquidity constrained problem, to which the model’s solution converges (under speciﬁed parametric restrictions) as all forms of uncertainty approach zero (as discussed in the main text). But a diﬃculty with this idea is that the perfect foresight liquidity constrained solution is ‘kinked:’ The slope of the consumption function changes discretely at the points . This is a practical problem because it rules out the use of derivatives of the consumption function in the approximate representation of , thereby preventing the enormous increase in eﬃciency obtainable from a higherorder approximation.
Our solution is simple: The formulae in another appendix that identify kink points on for integer values of (e.g., ) are continuous functions of ; the conclusion that is piecewise linear between the kink points does not require that the terminal consumption rule (from which time iteration proceeds) also be piecewise linear. Thus, for values we can construct a smooth function that matches the true perfect foresight liquidity constrained consumption function at the set of points corresponding to integer periods in the future, but satisﬁes the (continuous, and greater at nonkink points) consumption rule deﬁned from the appendix’s formulas by noninteger values of at other points.^{54}
This strategy generates a smooth limiting consumption function – except at the remaining kink point deﬁned by . Below this point, the solution must match because the constraint is binding. At the MPC discretely drops (that is, while ).
Such a kink point causes substantial problems for numerical solution methods (like the one we use, described below) that rely upon the smoothness of the limiting consumption function.
Our solution is to use, as the terminal consumption rule, a function that is identical to the (smooth) continuous consumption rule above some , but to replace between and with the unique polynomial function that satisﬁes the following criteria:
where is chosen judgmentally in a way calculated to generate a good compromise between smoothness of the limiting consumption function and ﬁdelity of that function to the (see the actual code for details).
We thus deﬁne the terminal function as
 (82) 
Since the precautionary motive implies that in the presence of uncertainty the optimal level of consumption is below the level that is optimal without uncertainty, and since , implicitly deﬁning (so that ), we can construct
 (83) 
which must be a number between and (since for ). This function turns out to be much better behaved (as a numerical observation; no formal proof is oﬀered) than the level of the optimal consumption rule . In particular, is well approximated by linear functions both as and as .
Diﬀerentiating with respect to and dropping consumption function arguments yields
 (84) 
which can be solved for
 (85) 
Similarly, we can solve (83) for
 (86) 
Thus, having approximated , we can recover from it the level and derivative(s) of .
This appendix explains in detail the paper’s ‘inequalities’ diagrams (Figures 1,3).
A simple illustration is presented in Figure 9, whose three nodes represent values of the absolute patience factor , the permanentincome growth factor , and the riskfree interest factor . The arrows represent imposition of the labeled inequality condition (like, the uppermost arrow, pointing from to , reﬂects imposition of the PFGICNrm condition (clicking PFGICNrm should take you to its deﬁnition; deﬁnitions of other conditions are also linked below).^{55} 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 and are all in , and that ).
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 node and impose the PFGICNrm and then the FHWC, and see that imposition of these conditions allows us to conclude that .
One could also impose directly (without imposing and ) by following the downwardsloping diagonal arrow exiting . Although alternate routes from one node to another all justify the same core conclusion (, 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,^{56} to indicate that the diagram is not commutative.^{57}
Negation of a condition is indicated by the reversal of the corresponding arrow. For example, negation of the RIC, , would be represented by moving the arrowhead from the bottom right to the top left of the line segment connecting and .
If we were to start at and then impose , that would reverse the arrow connecting and , but the node would then have no exiting arrows so no further deductions could be made. However, if we also reversed (that is, if we imposed ), that would take us to the node, and we could deduce . However, we would have to stop traversing the diagram at this point, because the arrow exiting from the node points back to our starting point, which (if valid) would lead us to the conclusion that . Thus, the reversal of the two earlier conditions (imposition of and ) requires us also to reverse the ﬁnal condition, giving us .^{ 58}
Under these conventions, Figure 1 in the main text presents a modiﬁed version of the diagram extended to incorporate the PFFVAC (reproduced here for convenient reference).
An arrowhead points to the larger of the two quantities being compared. For example, the diagonal arrow indicates that , which is an alternative way of writing the PFFVAC, (20)
This diagram can be interpreted, for example, as saying that, starting at the node, it is possible to derive the ^{59} by imposing both the PFGICNrm and the FHWC; or by imposing RIC and . Or, starting at the node, we can follow the imposition of the FHWC (twice  reversing the arrow labeled ) and then to reach the conclusion that . Algebraically,
 (87) 
which leads to the negation of both of the conditions leading into . is obtained directly as the last line in (87) and follows if we start by multipling the Return Patience Factor (RPF=) by the FHWF(=) raised to the power , which is negative since we imposed . FHWC implies FHWF so when FHWF is raised to a negative power the result is greater than one. Multiplying the RPF (which exceeds 1 because ) by another number greater than one yields a product that must be greater than one:

which is one way of writing .
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.
Figure 4 depicts the expected consumption growth factor as a strictly declining function of the cashonhand ratio. To investigate this, deﬁne
and the proposition in which we are interested is
or diﬀerentiating through the expectations operator, what we want is
 (88) 
Henceforth indicating appropriate arguments by the corresponding subscript (e.g. ), since , the portion of the LHS of equation (88) in brackets can be manipulated to yield

Now diﬀerentiate the Euler equation with respect to :

but since we can see from (89) that (88) is equivalent to

which, using (89), will be true if

which in turn will be true if both

and
The latter proposition is obviously true under our assumption . The former will be true if
The two shocks cause two kinds of variation in . Variations due to satisfy the proposition, since a higher draw of both reduces and reduces the marginal propensity to consume. However, permanent shocks have conﬂicting eﬀects. On the one hand, a higher draw of will reduce , thus increasing both and . On the other hand, the term is multiplied by , so the eﬀect of a higher could be to decrease the ﬁrst term in the covariance, leading to a negative covariance with the second term. (Analogously, a lower permanent shock can also lead a negative correlation.)
The two theorems and lemma to be proven in this appendix are:
Theorem 4. For the nondegenerate solution to the problem deﬁned in section 2.1 when FVAC, WRIC, and GICNrm all hold, there exists a unique cashonhandtopermanentincome ratio such that
 (89) 
Moreover, is a point of ‘wealth stablity’ in the sense that

Theorem 5. For the nondegenerate solution to the problem deﬁned in section 2.1 when FVAC, WRIC, and GIC all hold, there exists a unique pseudosteadystate cashonhandtoincome ratio such that
 (90) 
Moreover, is a point of stability in the sense that

The elements of the proof of theorem 4 are:
The consumption function exists because we have imposed the suﬃcient conditions (the and ; theorem 1). (Indeed, Appendix C shows that is not just continuous, but twice continuously diﬀerentiable.)
Section 2.7 shows that for all , . Since , even if takes on its minimum value of 0, , since both and are strictly positive. With and both strictly positive, the ratio inherits continuity (and, for that matter, continuous diﬀerentiability) from the consumption function.
Existence of a point where follows from:
Existence of a point where .
If RIC holds. Logic exactly parallel to that of section 3.1 leading to equation (39), but dropping the from the RHS, establishes that

where the inequality reﬂects imposition of the GICNrm (29).
If RIC fails. When the RIC fails, the fact that (see equation (32)) means that the limit of the RHS of (91) as is . In the next step of this proof, we will prove that the combination GICNrm and implies .
So we have whether the RIC holds or fails.
Existence of a point where .
Paralleling the logic for in section 3.2: the ratio of to is unbounded above as because .
Intermediate Value Theorem. If is continuous, and takes on values above and below 1, there must be at least one point at which it is equal to one.
Now deﬁne and note that

so that . Our goal is to prove that is strictly decreasing on using the fact that

Now, we show that (given our other assumptions) is decreasing (but for diﬀerent reasons) whether the RIC holds or fails.
If RIC holds. Equation (18) indicates that if the RIC holds, then . We show at the bottom of Section 2.8.1 that if the RIC holds then so that
which is negative because the GICNrm says .
If RIC fails. Under , recall that . Concavity of the consumption function means that is a decreasing function, so everywhere
which means that from (92) is guaranteed to be negative if
 (91) 
But the combination of the GICNrm holding and the RIC failing can be written:
and multiplying all three elements by gives
which satisﬁes our requirement in (91).
The elements of the proof are:
Since by assumption , our proof in M.1.1 that demonstrated existence and continuity of implies existence and continuity of .
Since by assumption , our proof in subsection M.1.1 that the ratio of to is unbounded as implies that the ratio to is unbounded as .
The limit of the expected ratio as goes to inﬁnity is most easily calculated by modifying the steps for the prior theorem explicitly:

where the last two lines are merely a restatement of the GIC (23).
The Intermediate Value Theorem says that if is continuous, and takes on values above and below 1, there must be at least one point at which it is equal to one.
Deﬁne and note that

so that . Our goal is to prove that is strictly decreasing on using the fact that

Now, we show that (given our other assumptions) is decreasing (but for diﬀerent reasons) whether the RIC holds or fails ().
If RIC holds. Equation (18) indicates that if the RIC holds, then . We show at the bottom of Section 2.8.1 that if the RIC holds then so that
which is negative because the GIC says .
If RIC fails. Under , recall that . Concavity of the consumption function means that is a decreasing function, so everywhere
which means that from (92) is guaranteed to be negative if
 (92) 
But we showed in section 2.5 that the only circumstances under which the problem has a nondegenerate solution while the RIC fails were ones where the FHWC also fails (that is, (92) holds).
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