BuﬀerStockTheory.tex, September 13, 2019
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Abstract
This paper builds theoretical foundations for rigorous understanding of ‘buﬀer stock’ saving
models, and pairs each theoretical result with a quantitative exploration. After articulating conditions
under which the consumption function converges, the paper shows that the ‘target’ saving behavior
that deﬁnes buﬀer stock models arises only under conditions strictly stronger than those that
guarantee convergence. It also shows that average consumption growth equals average income growth
in a small open economy populated by buﬀer stock savers. Together, the (provided) numerical tools
and the analytical results constitute a comprehensive toolkit for understanding buﬀer stock models.
Precautionary saving, buﬀer stock saving, marginal propensity to consume, permanent income hypothesis
D81, D91, E21
PDF:  http://econ.jhu.edu/people/ccarroll/papers/BufferStockTheory.pdf 
Slides:  http://econ.jhu.edu/people/ccarroll/papers/BufferStockTheorySlides.pdf 
Web:  http://econ.jhu.edu/people/ccarroll/papers/BufferStockTheory/ 
GitHub:  http://github.com/llorracc/BufferStockTheory 
(In GitHub repo, see /Code for tools for solving and simulating the model) 
CLICK HERE for an interactive Jupyter Notebook that uses the EconARK/HARK toolkit (Christopher D. Carroll, Alexander M. Kaufman, Jacqueline L. Kazil, Nathan M. Palmer, and Matthew N. White (2018)) to produce all of the paper’s ﬁgures (warning: it 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, http://econ.jhu.edu/people/ccarroll, and National Bureau of Economic Research.
A benchmark consumption/saving problem is the case with unbounded (constant relative risk aversion) utility, uncertainty about permanent and transitory income a la Friedman (1957), and no liquidity constraints. While a large literature has explored related models (see the summary below, and references throughout), that literature has not established some key properties of this benchmark case. It turns out that many of the models for which results have been established can be seen as special cases of this model; for example, the case with liquidity constraints is a particular limit of the model here; or, results that hold with both transitory and permanent shocks also hold if the permanent shocks are turned oﬀ.
The literature gap exists partly because standard theorems from the contraction mapping literature (beginning with those in Bellman (1957) and including those of Stokey et. al. (1989) cannot be directly applied for this problem. (See the end of section 2.1 for details). It is unclear whether newer methods such as those of Matkowski and Nowak (2011)) could be used, or how diﬃcult it would be to do so; but in any case this particular problem does not seem to have been tackled by those methods or any others.
The reader could be forgiven for not having noticed a gap. A large literature using numerical models to solve precisely such problems has emerged following Zeldes (1989). But without theoretical underpinnings, the ‘black box’ character of numerical solutions makes it diﬃcult to build intuition for how results might change with changes in the structure or calibration of the model. Indeed, without such theory, it can be diﬃcult even to check whether a computational solution is correct.
For example, numerical solutions typically imply the existence of a target level of nonhuman wealth (‘cash’ for short). Carroll (1992; 1997) showed numerically that target saving behavior arises under plausible parameter values for both inﬁnite and ﬁnite horizon models. Gourinchas and Parker (2002) estimate that for the mean household, buﬀer stock behavior characterizes behavior from age 25 until around age 4045; using the same model with diﬀerent data Cagetti (2003) ﬁnds target saving behavior into the 50s for the median household. Such target saving plays a key role in understanding the main results of the recent heterogeneous agent macroeconomics literature, including, for example, the insight in Krueger, Mitman, and Perri (2016) that explains why, during the Great Recession, middleclass consumers cut their consumption more than the poor or the rich. The ‘wealthy handtomouth’ in Violante, Kaplan, and Weidner (2014) are people with plenty of illiquid assets but with liquid assets low relative to their target levels. Despite the centrality of the mechanism, none of these papers provides a characterization of the circumstances under which target saving will emerge.
The paper’s main technical contributions are to articulate the (surprisingly loose) conditions under which the problem deﬁnes a contraction mapping with a nondegenerate consumption function, and conditions under which the resulting consumption function a implies expstence of a ‘target’ wealthtopermanentincome ratio. (This is the sense in which the paper studies the class of ‘buﬀer stock’ saving models.) The key condition required for target saving is that the consumer’s preferences need to satisfy a “Growth Impatience Condition” which relates preferences to the growth rate of income.
The paper also provides analytical foundations for other results that have become familiar from the numerical literature. All theoretical conclusions are paired with numerically computed illustrations (using opensource toolkit available from the EconARK project). All of the insights of this paper are instantiated in the toolkit, which algorithmically ﬂags parametric choices under which a problem fails to deﬁne a contraction mapping, under which a target level of wealth does not exist, or under which the solution is otherwise degenerate.
The paper proceeds in three parts.
The ﬁrst part articulates the conditions required for the problem to deﬁne a unique nondegenerate limiting consumption function, and discusses the relation of the paper’s model to models previously considered in the literature. The required conditions turn out to be interestingly parallel to those required for the liquidity constrained perfect foresight model; that parallel is explored and explained. Next, the paper derives some limiting properties of the consumption function as cash approaches inﬁnity and as it approaches its lower bound, and the theorem is proven explaining when the problem deﬁnes a contraction mapping. Finally, a related class of commonlyused models (exempliﬁed by Deaton (1991)) is shown to constitute a particular limit of this paper’s more general model.
The next section examines ﬁve 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. Third, if the consumer is suﬃciently ‘impatient’ (in a particular sense), a unique target cashtopermanentincome ratio will exist. Fourth, at the target cash ratio, the expected growth rate of consumption is slightly less than the expected growth rate of permanent noncapital income. Finally, the expected growth rate of consumption is declining in the level of cash. The ﬁrst four propositions are proven under general assumptions about parameter values; the last is shown to hold if there are no transitory shocks, but may fail in extreme cases if there are both transitory and permanent shocks.
Szeidl (2012) has shown that such an economy will be characterized by stable invariant distributions for the consumption ratio, the wealth ratio, and other variables.^{2} Using Szeidl’s result, the ﬁnal section discusses conditions under which, even with a ﬁxed aggregate interest rate that diﬀers from the time preference rate, an economy populated by buﬀer stock consumers converges to a balanced growth equilibrium in which the growth rate of consumption tends toward the (exogenous) growth rate of permanent income.
The consumer solves an optimization problem from period t until the end of life at T deﬁned by the objective
 (1) 
where u(∙) = ∙^{1−ρ}∕(1 −ρ) is a constant relative risk aversion utility function with ρ > 1.^{3} ^{,}^{4} The consumer’s initial condition is deﬁned by market resources m_{t} (Deaton (1991) called it ‘cashonhand’) and permanent noncapital income p_{t}.
In the usual treatment, a dynamic budget constraint (DBC) simultaneously incorporates all of the elements that determine next period’s m given this period’s choices; but for the detailed analysis here, it will be useful to disarticulate the steps so that individual ingredients can be separately examined:

where a_{t} indicates the consumer’s assets at the end of period t, which grow by a ﬁxed interest factor R = (1+r) between periods,^{5} so that b_{t+1} is the consumer’s ﬁnancial (‘bank’) balances before next period’s consumption choice;^{6} m_{t+1} (‘market resources’ or ‘money’) is the sum of ﬁnancial wealth b_{t+1} and noncapital income p_{t+1}ξ_{t+1} (permanent noncapital income p_{t+1} multiplied by a meanone iid transitory income shock factor ξ_{t+1}; from the perspective of period t, future transitory shocks are assumed to satisfy 𝔼 _{t}[ξ_{t+n}] = 1 ∀ n ≥ 1). Permanent noncapital income in period t + 1 is equal to its previous value, multiplied by a growth factor Γ, modiﬁed by a meanone iid shock ψ_{t+1}, 𝔼 _{t}[ψ_{t+n}] = 1 ∀ n ≥ 1 satisfying ψ ∈ [ψ,] for 0 < ψ ≤ 1 ≤ < ∞ where ψ = = 1 is the degenerate case with no permanent shocks.^{7} (Hereafter for brevity we occasionally drop time subscripts, e.g. 𝔼[ψ^{−ρ}] signiﬁes 𝔼 _{t}[ψ_{t+1}^{−ρ}].)
In future periods t + n ∀ n ≥ 1 there is a small probability ℘ that income will be zero (a ‘zeroincome event’),
 (2) 
where 𝜃_{t+n} is an iid meanone random variable (𝔼 _{t}[𝜃_{t+n}] = 1 ∀ n > 0) that has a distribution satisfying 𝜃 ∈ [𝜃,] where 0 < 𝜃 ≤ 1 ≤ < ∞ (degenerately 𝜃 = = 1). (See Rabault (2002) and Li and Stachurski (2014) for analyses of cases where the shock processes have unbounded support). Call the cumulative distribution functions ℱ_{ψ} and ℱ_{𝜃} (and ℱ_{ξ} is derived trivially from (2) and ℱ_{𝜃}). Permanent income and cash start out strictly positive, {p_{t},m_{t}}∈ (0,∞), and as usual the consumer cannot die in debt, so that
 (3) 
The model looks more special than it is. In particular, the assumption of a positive probability of zeroincome events may seem objectionable. However, it is easy to show that a model with a nonzero minimum value of ξ (motivated, for example, by the existence of unemployment insurance) can be redeﬁned by capitalizing the PDV of minimum income into current market assets,^{8} analytically transforming that model back into the model analyzed here. Also, the assumption of a positive point mass (as opposed to positive density) for the worst realization of the transitory shock is inessential, but simpliﬁes the proofs and is a powerful aid to intuition.
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, so 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 quantitatively important in micro data^{9} and which since Friedman (1957) have been understood to be 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.^{10} Finally, it diﬀers from models found in Stokey et. al. (1989) because neither liquidity constraints nor bounds on utility or marginal utility are imposed.^{11} Alvarez and Stokey (1998) relaxed the bounds on the return function, but they address only the deterministic case.
The incorporation of permanent shocks rules out application of the tools of Matkowski and Nowak (2011), who followed and corrected an error in the fundamental work on the local contraction mapping method developed in RincónZapatero and RodríguezPalmero (2003). Martinsda Rocha and Vailakis (2010) provides another correction to RincónZapatero and RodríguezPalmero (2003), and provides conditions that are easier to verify than those of Martinsda Rocha and Vailakis (2010) in many applications, but again only addresses the deterministic case.
We establish a bit more notation by reviewing the standard result that in problems of this class (CRRA utility, permanent shocks) the number of relevant state variables can be reduced from two (m and p) to one (m = m∕p) as follows. Deﬁning nonbold variables as the boldface counterpart normalized by p_{t} (as with m just above), assume that value in the last period of life is u(m_{T }), and consider the problem in the secondtolast period,

Now, in a onetime notational deviation, deﬁne nonbold ‘normalized value’ as v_{t} = v_{t}∕p_{t}^{1−ρ}, and consider the related problem

where ℛ_{t+1} ≡ (R∕Γ_{t+1}) is a ‘growthnormalized’ return factor, and the problem’s ﬁrst order condition is
 (4) 
Since v_{T }(m_{T }) = u(m_{T }), deﬁning v_{T−1}(m_{T−1}) from (4) for t = T − 1, (4) reduces to
This logic induces to all earlier periods, so that if we solve the normalized onestatevariable problem speciﬁed in (4) we will have solutions to the original problem for any t < T from:
We say that this problem has a nondegenerate solution if it deﬁnes a unique limiting consumption function whose optimal c satisﬁes
 (5) 
for every 0 < m < ∞. (‘Degenerate’ limits will be cases where the limiting consumption function is either c(m) = 0 or c(m) = ∞.)
Articulating the wellknown analytical solution to the perfect foresight specialization of the model, obtained by setting ℘ = 0 and 𝜃 = = ψ = = 1, allows us to deﬁne some remaining notation and terminology, and to deﬁne a convenient reference point.
The dynamic budget constraint, strictly positive marginal utility, and the can’tdieindebt condition (3) imply an exactlyholding intertemporal budget constraint (IBC)
 (6) 
where b is nonhuman wealth and h_{t} is ‘human wealth,’ and with a constant ℛ≡R∕Γ,

(7) makes plain that in order for h ≡ lim _{n→∞}h_{T−n} to be ﬁnite, we must impose the Finite Human Wealth Condition (‘FHWC’)
 (7) 
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.
The consumption Euler equation holds in every period; with u′(c) = c^{−ρ}, this says
 (8) 
where the Old English letter ‘thorn’ represents what we will call the ‘absolute patience factor’ (Rβ)^{1∕ρ}.^{12} The sense in which Þ captures patience is that if the ‘absolute impatience condition’ (AIC) holds,
 (9) 
the consumer will choose to spend an amount too large to sustain indeﬁnitely (the level of consumption must fall over time). We say that such a consumer is ‘absolutely impatient’ (this is the key condition in Bewley (1977)).
We next deﬁne a ‘return patience factor’ that relates absolute patience to the return factor:
 (10) 
and note that since consumption is growing by Þ but discounted by R:
 (11) 
which deﬁnes a normalized ﬁnitehorizon perfect foresight consumption function
 (12) 
where κ_{t} is the marginal propensity to consume (MPC) because it answers the question ‘if the consumer had an extra unit of wealth, how much more would he spend.’ (The overbar on c reﬂects the fact that this will be an upper bound as we modify the problem to incorporate constraints and uncertainty; analogously, the underbar for κ indicates that it is a lower bound). Equation (11) makes plain that for the limiting MPC to be strictly positive as n = T −t goes to inﬁnity we must impose the condition
 (13) 
so that
 (14) 
Equation (13) 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; that is, the condition rules out the degenerate limiting solution (m) = 0. Because the return patience factor ÞR is the absolute patience factor divided by the return, we call equation (13) the ‘return impatience condition’ or RIC; we will say that a consumer who satisﬁes the condition is ‘return impatient.’
Given that the RIC holds, and deﬁning limiting objects by the absence of a time subscript (e.g., (m) = lim _{n↑∞} _{T−n}(m)), the limiting consumption function will be
 (15) 
and we now see that in order to rule out the degenerate limiting solution (m_{t}) = ∞ we need h to be ﬁnite so we must impose the ﬁnite human wealth condition (7).
A ﬁnal useful point is that since the perfect foresight growth factor for consumption is Þ, using u(xy) = x^{1−ρ}u(y) yields an analytical expression for value:
 (16) 
which asymptotes to a ﬁnite value as n = T −t approaches +∞ if βÞ^{1−ρ} < 1 (related to a condition in Alvarez and Stokey (1998)); with a bit of algebra, this requirement can be shown to be equivalent to the RIC.^{13} Thus, the same conditions that guarantee a nondegenerate limiting consumption function also guarantee a nondegenerate limiting value function (this will not be true in the version of the model that incorporates uncertainty).
If a liquidity constraint requiring b ≥ 0 is ever to be relevant, it must be relevant at the lowest possible level of market resources, m_{t} = 1, which obtains for a consumer who enters period t with b_{t} = 0. The constraint is ‘relevant’ if it prevents the choice that would otherwise be optimal; at m_{t} = 1 the constraint is relevant if the marginal utility from spending all of today’s resources c_{t} = m_{t} = 1, exceeds the marginal utility from doing the same thing next period, c_{t+1} = 1; that is, if such choices would violate the Euler equation (4):
 (17) 
By analogy to the return patience factor, we therefore deﬁne a ‘perfect foresight growth patience factor’ as
 (18) 
and deﬁne a ‘perfect foresight growth impatience condition’ (PFGIC)
 (19) 
which is equivalent to (17) (exponentiate both sides by 1∕ρ).
If the RIC and the FHWC hold, appendix A shows that, for some 0 < m_{#} < 1, an unconstrained consumer behaving according to (15) would choose c < m for all m > m_{#}. The solution to the constrained consumer’s problem in this case is simple: For any m ≥m_{#} the constraint does not bind (and will never bind in the future) and so the constrained consumption function is identical to the unconstrained one. In principle, if the consumer were somehow^{14} to arrive at an m < m_{#} < 1 the constraint would bind and the consumer would have to consume c = m. We use the ∘ accent to designate the limiting constrained consumption function:
 (20) 
More useful is the case where the perfect foresight growth and return impatience conditions both hold. In this case appendix A shows that the limiting constrained consumption function is piecewise linear, with c(m) = m up to a ﬁrst ‘kink point’ at m_{#}^{1} > 1, and with discrete declines in the MPC at successively increasing kink points {m_{#}^{1},m_{ #}^{2},...}. As m ↑∞ the constrained consumption function c(m) becomes arbitrarily close to the unconstrained (m), and the marginal propensity to consume function (m) ≡c^{′}(m) limits to κ. Similarly, the value function v(m) is nondegenerate and limits into the value function of the unconstrained consumer. Surprisingly, this logic holds even when the ﬁnite human wealth condition fails (denoted ). A solution exists 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 b_{t} > 1 will run those resources down over time so that by some ﬁnite number of periods n in the future the consumer will reach b_{t+n} = 0, and thereafter will set c = m = 1 for eternity, a policy that will (using (16)) yield value of

which will be ﬁnite whenever

which we call the Perfect Foresight Finite Value of Autarky Condition, PFFVAC, because it guarantees that a consumer who always spends all his permanent income will have ﬁnite value (the consumer has ‘ﬁnite autarky value’). Note that the version of the PFFVAC in (21) implies the PFGIC ÞΓ < 1 whenever R < Γ holds. So, if , value for any ﬁnite m will be the sum of two ﬁnite numbers: The component due to the unconstrained consumption choice made over the ﬁnite horizon leading up to b_{t+n} = 0, and the ﬁnite component due to the value of consuming all income thereafter. The consumer’s value function is therefore nondegenerate.
The most peculiar possibility occurs when the RIC fails. The appendix shows that under these circumstances the FHWC must also fail, and the constrained consumption function is nondegenerate. (See Figure 6 for a numerical example). While it is true that lim _{m↑∞}(m) = 0, nevertheless the limiting constrained consumption function c(m) is strictly positive and strictly increasing in m. This result interestingly reconciles the conﬂicting intuitions from the unconstrained case, where would suggest a dengenerate limit of c(m) = 0 while would suggest a degenerate limit of c(m) = ∞.
Tables 3 and 4 (and appendix table 5) codify the key points to help the reader keep them straight (and to facilitate upcoming comparisons with the results in the presence of uncertainty but the absence of liquidity constraints (also tabulated for comparison)). The model without constraints but with uncertainty will turn out to be a close parallel to the model with constraints but without uncertainty.
When uncertainty is introduced, the expectation of b_{t+1} can be rewritten as:
 (21) 
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 the object
We can now transparently generalize the PFGIC (19) by deﬁning a ‘compensated growth factor’
 (22) 
and a compensated growth patience factor
 (23) 
and a straightforward derivation using some of the results below yields the conclusion that
 (24) 
which is stronger than the perfect foresight version (19) because Γ < Γ.
Analogously to (16), a consumer who spent his permanent income every period would have value

which we call the ‘ﬁnite value of autarky’ condition (FVAC) because it is the value obtained by always consuming permanent income. For nondegenerate ψ, this condition is stronger (harder to satisfy in the sense of requiring lower β) than the perfect foresight version (??) because Γ < Γ.
Figure 1 depicts the successive consumption rules that apply in the last period of life (c_{T }(m)), the secondtolast period, and various earlier periods under the baseline parameter values listed in Table 2. (The 45 degree line is labelled as c_{T }(m) = m because in the last period of life it is optimal to spend all remaining resources.)
Calibrated Parameters  
Description  Parameter  Value  Source
 
Permanent Income Growth Factor  Γ  1.03  PSID: Carroll (1992)
 
Interest Factor  R  1.04  Conventional  
Time Preference Factor  β  0.96  Conventional
 
Coeﬃcient of Relative Risk Aversion  ρ  2  Conventional
 
Probability of Zero Income  ℘  0.005  PSID: Carroll (1992)
 
Std Dev of Log Permanent Shock  σ_{ψ}  0.1  PSID: Carroll (1992)
 
Std Dev of Log Transitory Shock  σ_{𝜃}  0.1  PSID: Carroll (1992)
 
Approximate  
Calculated  
Description  Symbol and Formula  Value  
Finite Human Wealth Measure  ℛ^{−1}  ≡  Γ∕R  0.990 
PF Finite Value of Autarky Measure  ℶ  ≡  βΓ^{1−ρ}  0.932 
Growth Compensated Permanent Shock  ψ  ≡  (𝔼[ψ−1])^{−1}  0.990 
UncertaintyAdjusted Growth  Γ  ≡  Γψ  1.020 
Utility Compensated Permanent Shock  ψ  ≡  (𝔼 _{t}[ψ^{1−ρ}])^{1∕(1−ρ)}  0.990 
Utility Compensated Growth  Γ  ≡  Γψ  1.020 
Absolute Patience Factor  Þ_{ }  ≡  (Rβ)^{1∕ρ}  0.999 
Return Patience Factor  ÞR  ≡  Þ∕R  0.961 
PF Growth Patience Factor  ÞΓ  ≡  Þ∕Γ  0.970 
Growth Patience Factor  ÞΓ  ≡  Þ∕Γ  0.980 
Finite Value of Autarky Measure  ℶ  ≡  βΓ^{1−ρ}ψ^{1−ρ}  0.941 
In the ﬁgure, the consumption rules appear to converge as the horizon recedes (our purpose is to show that this appearance is not deceptive); we call the limiting inﬁnitehorizon consumption rule
 (25) 
A precondition for the main proof is that the maximization problem (4) deﬁnes a sequence of continuously diﬀerentiable strictly increasing strictly concave^{17} functions {c_{T }, c_{T−1},...}.^{18} The straightforward but tedious proof of this precondition is relegated to appendix B. For present purposes, the most important point is the following intuition: c_{t}(m) < m for all periods t < T because a consumer who spent all available resources would arrive in period t + 1 with balances b_{t+1} of zero, and then might earn zero noncapital income over the remaining horizon (an unbroken series of zeroincome events is unlikely but possible). In such a case, the budget constraint and the can’tdieindebt condition mean that the consumer would be forced to spend zero, incurring negative inﬁnite utility. To avoid this disaster, the consumer never spends everything. (This is an example of the ‘natural borrowing constraint’ induced by a precautionary motive (Zeldes (1989)).)^{19}
The consumption functions depicted in Figure 1 appear to have limiting slopes as m ↓ 0 and as m ↑∞. This section conﬁrms that impression and derives those slopes, which also turn out to be useful in the contraction mapping proof. In a recent paper, Benhabib, Bisin, and Zhu (2015) show that the consumption function becomes linear as wealth approaches inﬁnity in a model with capital income risk and liquidity constraints; it seems clear that their results would generalize to the limits derived here if capital income risk were added to the model. See also Ma, Stachurski, and Toda (2018) for an analysis of the stability of models with capital income risk.
Assume (as discussed above) that a continuously diﬀerentiable concave consumption function exists in period t + 1, with an origin at c_{t+1}(0) = 0, a minimal MPC κ_{t+1} > 0, and maximal MPC _{t+1} ≤ 1. (If t + 1 = T these will be κ_{T } = _{T } = 1; 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 about ﬁnite human wealth, the proportion of consumption that will be ﬁnanced out of human wealth approaches zero. The consequence is that 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 point, appendix F provides a useful recursive expression for the (inverse of the) limiting MPC:
 (26) 
It turns out that there is a parallel expression for the limiting maximal MPC as m ↓ 0: appendix equation (27) shows that, as m_{t} ↑∞,
 (27) 
Then _{n=0}^{∞} is a decreasing convergent sequence if
 (28) 
a condition that we dub the ‘Weak Return Impatience Condition’ (WRIC) because with ℘ < 1 it will hold more easily (for a larger set of parameter values) than the RIC (ÞR < 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 consideration is the reason 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
 (29) 
because consumption starts at zero and is continuously diﬀerentiable (as argued above), is strictly concave (Carroll and Kimball (1996)), and always exhibits a slope between κ_{t} and _{t} (the formal proof is provided in appendix D).
These limits are useful at least in the sense that they can be hardwired into a solution algorithm for the model, which has the potential to make the solution more eﬃcient (cf. Carroll, Chipeniuk, Tokuoka, and Wu (2020)). Alternatively, they can provide a useful check on the accuracy of a solution algorithm that does not impose them directly.
To prove that the consumption rules converge, we need to show that the problem deﬁnes a contraction mapping. This cannot be proven using the standard theorems in, say, Stokey et. al. (1989), which require marginal utility to be bounded over the space of possible values of m, because the possibility (however unlikely) of an unbroken string of zeroincome events for the remainder of life means that as m approaches zero c must approach zero (see the discussion in 2.7); thus, marginal utility is unbounded. 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 involves permanent shocks.^{20}
Fortunately, Boyd (1990) provided a weighted contraction mapping theorem that Alvarez and Stokey (1998) showed how to use to to address the homogeneous case (of which CRRA formulation is an example) in a deterministic framework, and Durán (2003) showed how to extend Boyd (1990) approach to the stochastic case.
Deﬁnition 1. Consider any function ∙∈𝒞(𝒜,ℬ) where 𝒞(𝒜,ℬ) is the space of continuous functions from 𝒜 to ℬ. Suppose г ∈𝒞(𝒜,ℬ) with ℬ ⊆ℝ and г > 0. Then ∙ is гbounded if the гnorm of ∙,
 (30) 
is ﬁnite.
For 𝒞_{г} deﬁned as the set of functions in 𝒞(𝒜,ℬ) that are гbounded; w, x, y, and z as examples of гbounded 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 : 𝒞_{г} →𝒞 such that^{21} ^{,}^{22}
For our problem, take 𝒜 as ℝ_{++} and ℬ as ℝ, and deﬁne
Using this, we introduce the mapping 𝒯 : 𝒞_{г} →𝒞,^{23}
 (31) 
We can show that our operator 𝒯 satisﬁes the conditions that Boyd requires of his operator T, if we impose two restrictions on parameter values. The ﬁrst restriction is the WRIC necessary for convergence of the maximal MPC, equation (28) above. A more serious restriction is the utilitycompensated Finite Value of Autarky condition, equation (25). (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. 𝒯 is a contraction mapping if the restrictions on parameter values (28) and (25) are true.
The proof is cumbersome, and therefore relegated to appendix D. Given that the value function converges, appendix D.3 shows that the consumption functions converge.
This section shows that a related problem commonly considered in the literature (e.g., with a simpler income process, 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 essence of the argument is easy to state. As noted above, there is a ﬁnite possibility of earning zero income over the remainder of the horizon, which prevents the consumer from ending the current period with zero assets because with some ﬁnite probability the consumer would be forced to consume zero, which would be inﬁnitely painful.
But 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 approaches zero. But zero precautionary saving is the amount of saving that a liquidity constrained consumer with perfect foresight would choose.
Another way to think about this is just to think of the liquidity constraint as being imposed by specifying a component of the utility function that is zero whenever the consumer ends the period with (weakly) positive assets, but negative inﬁnity if the consumer ended the period with (strictly) negative assets.
See appendix G for the formal proof justifying the foregoing intuitive discussion.
In the perfect foresight unconstrained problem (section 2.4.2), the RIC was required for existence of a nondegenerate solution. It is surprising, therefore, that in the presence of uncertainty, the RIC is neither necessary nor suﬃcient for a nondegenerate solution. We thus begin our discussion by asking what features the problem must exhibit (given the FVAC) if the RIC fails (that is, R < (Rβ)^{1∕ρ}):

but since ψ < 1 and 0 < 1 − 1∕ρ < 1 (because we have assumed ρ > 1), this can hold only if R∕Γ < 1; that is, given the FVAC, the RIC can fail only if human wealth is unbounded. Unbounded human wealth is permitted here, as in the perfect foresight liquidity constrained problem. But, from equation (26), an implication of is that lim _{ m↑∞}c^{′}(m) = 0. Thus, interestingly, the presence of uncertainty both permits unlimited human wealth and at the same time prevents that unlimited wealth from resulting in inﬁnite consumption. That is, in the presence of uncertainty, pathological patience (which in the perfect foresight model with ﬁnite wealth results in consumption of zero) plus inﬁnite human wealth (which the perfect foresight model rules out because it leads to inﬁnite consumption) combine here to yield a unique ﬁnite limiting MPC for any ﬁnite value of m. Note the close parallel to the conclusion in the perfect foresight liquidity constrained model in the {PFGIC,} case (for detailed analysis of this case see the appendix). There, too, the tension between inﬁnite human wealth and pathological patience was resolved with a nondegenerate consumption function whose limiting MPC was zero.
The ‘weakness’ of the additional requirement for contraction, the weak RIC, can be seen by asking ‘under what circumstances would the FVAC hold but the WRIC fail?’ Algebraically, the requirement is
 (32) 
If there were no conceivable parameter values that could satisfy both of these inequalities, the WRIC would have no force; it would be redundant. And if we require R ≥ 1, the WRIC is indeed redundant because now β < 1 < R^{ρ−1}, so that the RIC (and WRIC) must hold.
But neither theory nor evidence demands that we assume 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, ψ^{1−ρ} = 1.01, Γ^{1−ρ} = 1.01^{−1} and ℘ = 0.10. In that case (32) reduces to

but since β < 1 by assumption, the binding requirement is that

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. Thus, the relevance of the WRIC is indeed “Weak.”
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 2.10 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 ℘ = 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.
If both the GIC and the RIC hold, the arguments above establish that the limiting consumption function asymptotes to the consumption function for the perfect foresight unconstrained function. The more interesting case is where the GIC fails. A solution that satisﬁes the combination FVAC and is depicted in Figure 2. The consumption function is shown along with the 𝔼 _{t}[Δm_{t+1}] = 0 locus that identiﬁes the ‘sustainable’ level of spending at which m is expected to remain unchanged. The diagram suggests a fact that is conﬁrmed by deeper analysis: Under the depicted conﬁguration of parameter values (see the code for details), the consumption function never reaches the 𝔼 _{t}[Δm_{t+1}] = 0 locus; indeed, when the RIC holds but the GIC does not, the consumption function’s limiting slope (1 −Þ∕R) is shallower than that of the sustainable consumption locus (1−Γ∕R),^{24} so the gap between the two actually increases with m in the limit. That is, although a nondegenerate consumption function exists, a target level of m does not (or, rather, the target is m = ∞), because no matter how wealthy a consumer becomes, he will always spend less than the amount that would keep m stable (in expectation).
For the reader’s convenience, Tables 3 and 4 present a summary of the connections between the various conditions in the presence and the absence of uncertainty.
Perfect Foresight Versions  Uncertainty Versions  
Finite Human Wealth Condition (FHWC)
 
Γ∕R < 1  Γ∕R < 1  

 
Absolute Impatience Condition (AIC)
 
Þ < 1  Þ < 1  

 
c_{t+1} < c_{t}  lim _{mt→∞}𝔼 _{t}[c_{t+1}] < c_{t}  
Return Impatience Conditions
 
Return Impatience Condition (RIC)  Weak RIC (WRIC)
 
Þ∕R < 1  ℘^{1∕ρ}Þ∕R < 1  

 
c^{′}(m) = 1 −Þ∕R < 1  c^{′}(m) < 1 −℘^{1∕ρ}Þ∕R < 1  
Growth Impatience Conditions
 
PFGIC  GIC
 
Þ∕Γ < 1  Þ𝔼[ψ^{−1}]∕Γ < 1  



lim _{mt→∞}𝔼 _{t}[m_{t+1}∕m_{t}] = ÞΓ  
Finite Value of Autarky Conditions
 
PFFVAC  FVAC
 
βΓ^{1−ρ} < 1  βΓ^{1−ρ} 𝔼[ψ^{1−ρ}] < 1  
equivalently Þ∕Γ < (R∕Γ)^{1∕ρ}  



Model  Conditions  Comments

PF Unconstrained  RIC, FHWC^{∘}  RIC ⇒v(m)< ∞; FHWC ⇒ 0 < v(m) 
RIC prevents (m) = 0  
FHWC prevents (m) = ∞  
PF Constrained  PFGIC^{∗}  If RIC, lim _{ m→∞}c(m) = (m), lim _{m→∞}(m) = κ 
If , lim _{ m→∞}(m) = 0  
Buﬀer Stock Model  FVAC, WRIC  FHWC ⇒ lim _{m→∞}c(m) = (m), lim _{m→∞}(m) = κ 
+RIC ⇒ lim _{ m→∞}(m) = κ  
+ ⇒ lim _{ m→∞}(m) = 0  
GIC guarantees ﬁnite target wealth ratio  
FVAC is stronger than PFFVAC  
WRIC is weaker than RIC  
^{‡}For feasible m, the limiting consumption function deﬁnes the unique value of c satisfying 0 < c < ∞. ^{∘}RIC, FHWC are necessary as well as suﬃcient. ^{∗}Solution also exists for and RIC, but is identical to the unconstrained model’s solution for feasible m ≥ 1.
Figures 3 and 4a,b capture the main properties of the converged consumption rule when the RIC, GIC, and FHWC all hold.^{25} Figure 3 shows the expected consumption growth factor 𝔼 _{t}[c_{t+1}∕c_{t}] for a consumer behaving according to the converged consumption rule, while Figures 4a,b illustrate theoretical bounds for the consumption function and the marginal propensity to consume.
Five features of behavior are captured, or suggested, by the ﬁgures. First, as m_{t} ↑∞ the expected consumption growth factor goes to Þ, indicated by the lower bound in Figure 3, and the marginal propensity to consume approaches κ = (1 −ÞR) (Figure 4), the same as the perfect foresight MPC.^{26} Second, as m_{t} ↓ 0 the consumption growth factor approaches ∞ (Figure 3) and the MPC approaches = (1 −℘^{1∕ρ}Þ R) (Figure 4). Third (Figure 3), there is a target cashonhandtoincome ratio m such that if m_{t} = m then 𝔼 _{t}[m_{t+1}] = m_{t}, and (as indicated by the arrows of motion on the 𝔼 _{t}[c_{t+1}∕c_{t}] curve), the model’s dynamics are ‘stable’ around the target in the sense that if m_{t} < m then cashonhand will rise (in expectation), while if m_{t} > m, it will fall (in expectation). Fourth (Figure 3), at the target m, the expected rate of growth of consumption is slightly less than the expected growth rate of permanent noncapital income. The ﬁnal proposition suggested by Figure 3 is that the expected consumption growth factor is declining in the level of the cashonhand ratio m_{t}. This turns out to be true in the absence of permanent shocks, but in extreme cases it can be false if permanent shocks are present.^{27}
Deﬁne

which is the solution to an inﬁnitehorizon problem with no noncapital income (ξ_{t+n} = 0 ∀ n ≥ 1); clearly c(m) < c(m), since allowing the possibility of future noncapital income cannot reduce current consumption.^{28}
Assuming the FHWC holds, the inﬁnite horizon perfect foresight solution (15) constitutes an upper bound on consumption in the presence of uncertainty, since Carroll and Kimball (1996) show that the introduction of uncertainty strictly decreases the level of consumption at any m.
Thus, we can write

But

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 (m) or lower than c(m).
Figure 4 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 m_{t} ↑∞:

But

while
because lim _{mt↑∞}a^{′}(m) = Þ R^{29} and Γ_{t+1}ξ_{t+1}∕m_{t} ≤ (Γ∕(1 −℘))∕m_{t} which goes to zero as m_{t} goes to inﬁnity.Hence we have

so as cash goes to inﬁnity, consumption growth approaches its value Þ in the perfect foresight model.
This argument applies equally well to the problem of the restrained consumer, because as m approaches inﬁnity the constraint becomes irrelevant (assuming the FHWC holds).
Now consider the limits of behavior as m_{t} gets arbitrarily small.
Equation (72) shows that the limiting value of is

Deﬁning e(m) = c(m)∕m as before we have

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

Figure 4 conﬁrms that the numerical solution method obtains this limit for the MPC as m approaches zero.
For consumption growth, as m ↓ 0 we have
Deﬁne the target cashonhandtoincome ratio m as the value of m such that
 (33) 
where the ∨ accent is meant to invoke the fact that this is the value that other m’s ‘point to.’
We prove existence by arguing that 𝔼 _{t}[m_{t+1}∕m_{t}] is continuous on m_{t} > 0, and takes on values both above and below 1, so that it must equal 1 somewhere by the intermediate value theorem.
Speciﬁcally, the same logic used in section 3.2 shows that lim _{mt↓0} 𝔼 _{t}[m_{t+1}∕m_{t}] = ∞.
The limit as m_{t} goes to inﬁnity is
Stability means that in a local neighborhood of m, values of m_{t} above m will result in a smaller ratio of 𝔼 _{t}[m_{t+1}∕m_{t}] than at m. That is, if m_{t} > m then 𝔼 _{t}[m_{t+1}∕m_{t}] < 1. This will be true if

at m_{t} = m. But
 (34) 
The target level of market resources is the m such that if m_{t} = m then 𝔼 _{t}[m_{t+1}] = m.

At the target, equation (34) is

Substituting for the ﬁrst term in this expression using (35) gives
We have now proven that some target m must exist, and that at any such m the solution is stable. Nothing so far, however, rules out the possibility that there will be multiple values of m that satisfy the deﬁnition (33) of a target.
Multiple targets can be ruled out as follows. Suppose there exist multiple targets; these can be arranged in ascending order and indexed by an integer superscript, so that the target with the smallest value is, e.g., m^{1}. The argument just completed implies that since 𝔼 _{ t}[m_{t+1}∕m_{t}] is continuously diﬀerentiable there must exist some small 𝜖 such that 𝔼 _{t}[m_{t+1}∕m_{t}] < 1 for m_{t} = m^{1} + 𝜖. (Continuous diﬀerentiability of 𝔼 _{ t}[m_{t+1}∕m_{t}] follows from the continuous diﬀerentiability of c(m_{t}).)
Now assume there exists a second value of m satisfying the deﬁnition of a target, m^{2}. Since 𝔼 _{t}[m_{t+1}∕m_{t}] is continuous, it must be approaching 1 from below as m_{t} →m^{2}, since by the intermediate value theorem it could not have gone above 1 between m^{1} + 𝜖 and m^{2} without passing through 1, and by the deﬁnition of m^{2} it cannot have passed through 1 before reaching m^{2}. But saying that 𝔼 _{ t}[m_{t+1}∕m_{t}] is approaching 1 from below as m_{t} →m^{2} implies that
 (35) 
at m_{t} = m^{2}. However, we just showed above that, under our assumption that the GIC holds, precisely the opposite of equation (35) must hold for any m that satisﬁes the deﬁnition of a target. Thus, assuming the existence of more than one target implies a contradiction.
The foregoing arguments rely on the continuous diﬀerentiability of c(m), so the arguments do not directly go through for the restrained consumer’s problem in which the existence of liquidity constraints can lead to discrete changes in the slope c^{′}(m) at particular values of m. But we can use the fact that the restrained model is the limit of the baseline model as ℘ ↓ 0 to conclude that there is likely a unique target cash level even in the restrained model.
If consumers are suﬃciently impatient, the limiting target level in the restrained model will be m = 𝔼 _{t}[ξ_{t+1}] = 1. That is, if a consumer starting with m = 1 will save nothing, a(1) = 0, then the target level of m in the restrained model will be 1; if a consumer with m = 1 would choose to save something, then the target level of cashonhand will be greater than the expected level of income.
In Figure 3 the intersection of the target cashonhand ratio locus at m with the expected consumption growth curve lies below the intersection with the horizontal line representing the growth rate of expected permanent income. This can be proven as follows.
Strict concavity of the consumption function implies that if 𝔼 _{t}[m_{t+1}] = m = m_{t} then

and since m_{t+1} = (R∕Γ_{t+1})a(m) + ξ_{t+1} and a(m) > 0 it is clear that cov_{t}(Γ_{t+1},m_{t+1}) < 0 which implies that the entire term added to Γ in (36) is negative, as required.
Figure 3 depicts the expected consumption growth factor as a strictly declining function of the cashonhand ratio. To investigate this, deﬁne

or diﬀerentiating through the expectations operator, what we want is
 (36) 
Henceforth indicating appropriate arguments by the corresponding subscript (e.g. c_{t+1}^{′} ≡ c^{′}(m_{ t+1})), since Γ_{t+1}ℛ_{t+1} = R, the portion of the LHS of equation (36) in brackets can be manipulated to yield

Now diﬀerentiate the Euler equation with respect to m_{t}:

but since _{t+1} > 0 we can see from (37) that (36) is equivalent to

which, using (37), will be true if

which in turn will be true if both

and
The latter proposition is obviously true under our assumption ρ > 1. The former will be true if
The two shocks cause two kinds of variation in m_{t+1}. Variations due to ξ_{t+1} satisfy the proposition, since a higher draw of ξ both reduces c_{t+1}^{−ρ−1} and reduces the marginal propensity to consume. However, permanent shocks have conﬂicting eﬀects. On the one hand, a higher draw of ψ_{t+1} will reduce m_{t+1}, thus increasing both c_{t+1}^{−ρ−1} and c_{ t+1}^{′}. On the other hand, the c_{t+1}^{−ρ−1} term is multiplied by Γψ_{ t+1}, so the eﬀect of a higher ψ_{t+1} could be to decrease the ﬁrst term in the covariance, leading to a negative covariance with the second term. (Analogously, a lower permanent shock ψ_{t+1} can also lead a negative correlation.)
The software archive associated with this paper presents an example in which this perverse eﬀect dominates. However, extreme assumptions were required (in particular, a very small probability of the zeroincome shock) and the region in which _{t+1}^{′} > 0 was tiny. In practice, for plausible parametric choices, 𝔼 _{t}[_{t+1}^{′}] < 0 should generally hold.
This section examines the behavior of large collections of buﬀerstock consumers with identical parameter values. Such a collection can be thought of as either 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.^{31}
Formally, we assume a continuum of ex ante identical households on the unit interval, with constant total mass normalized to one and indexed by i ∈ [0, 1], all behaving according to the model speciﬁed above.^{32}
Szeidl (2012) proves that such a population will be characterized by an invariant distribution of m that induces invariant distributions for c and a; designate these ℱ^{m}, ℱ^{a}, and ℱ^{c}.^{33}
It is useful to deﬁne the operator 𝕄 which yields the mean value of its argument in the population, as distinct from the expectations operator 𝔼 which represents beliefs about the future.
An economist with a microeconomic dataset could calculate the average growth rate of idiosyncratic consumption, and would ﬁnd
Thus, in a population that has reached its invariant distribution, the growth rate of idiosyncratic log consumption matches the growth rate of idiosyncratic log permanent income.
Attanasio and Weber (1995) point out that concavity of the consumption function (or other nonlinearities) can imply that it is quantitatively important to distinguish between the growth rate of average consumption and the average growth rate of consumption.^{35} We have just examined the average growth rate; we now examine the growth rate of the average.
Using capital letters for aggregate variables, the growth factor for aggregate income is given by:
Aggregate assets are:
where P_{t} designates the mean level of permanent income across all individuals, and we are assuming that a_{t,i} was distributed according to the invariant distribution with a mean value of A. Since permanent income grows at mean rate Γ while the distribution of a is invariant, if we normalize P_{t} to one we will similarly have for any period n ≥ 1

Unfortunately, Szeidl (2012)’s proof of the invariance of ℱ^{a} does not yield the information about how the crosssectional covariance between a and p evolves required to show that the covariance term grows by a factor smaller than Γ; if that were true, its relative size would shrink to zero over time. (A proof that the covariance shrinks fast enough would mean that the term could be neglected).
The desired result can be proven if there are no permanent shocks; see appendix E for that proof, along with a discussion of the characteristics of a covariance term that are an obstacle to proof in the general case with both transitory and permanent shocks.
This paper provides theoretical foundations for many characteristics of buﬀer stock saving models that have heretofore been observed in simulations but not proven. Perhaps the most important such proposition is the existence of a target cashtopermanentincome ratio toward which actual resources will move. The intuition provided by the existence of such a target can be a powerful aid to understanding a host of numerical results.
Another contribution is integration of the paper’s results with an 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.
This appendix taxonomizes the characteristics of the limiting consumption function c(m) under perfect foresight in the presence of a liquidity constraint requiring b ≥ 0 under various conditions. Results are summarized in table 5.
Name  Condition  Outcome/Comments  
1 <  Þ∕Γ  Constraint never binds for m ≥ 1  
RIC  Þ∕R  < 1  FHWC holds (R > Γ)  
c(m) = (m) for m ≥ 1  
1 <  Þ∕R  c(m) is degenerate  
PFGIC  Þ∕Γ  < 1  Constraint binds in ﬁnite time for any m  
RIC  Þ∕R  < 1  FHWC may or may not hold  
lim _{m↑∞}(m) −c(m) = 0  
lim _{m↑∞}(m) = κ  
1 <  Þ∕R  
lim _{m↑∞}(m) = 0  
A consumer is ‘growth patient’ if the perfect foresight growth impatience condition fails (, 1 < Þ∕Γ). Under the constraint does not bind at the lowest feasible value of m_{t} = 1 because 1 < (Rβ)^{1∕ρ}∕Γ implies that spending everything today (setting c_{t} = m_{t} = 1) produces lower marginal utility than is obtainable by reallocating a marginal unit of resources to the next period at return R:^{36}
 (37) 
Similar logic shows that under these circumstances the constraint will never bind for a constrained consumer with a ﬁnite horizon of n periods, so such a consumer’s consumption function will be the same as for the unconstrained case examined in the main text.
If the RIC fails (1 < ÞR) while the ﬁnite human wealth condition holds, the limiting value of this consumption function as n ↑∞ is the degenerate function
 (38) 
If the RIC fails and the FHWC fails, human wealth limits to h = ∞ so the consumption function limits to either c_{T−n}(m) = 0 or c_{T−n}(m) = ∞ depending on the relative speeds with which the MPC approaches zero and human wealth approaches ∞.^{37}
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 c = m from (15):
 (39) 
which (under these assumptions) satisﬁes 0 < m_{#} < 1.^{38} 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.^{39} For any m > m_{#} the constraint will never bind and the consumer will choose to spend the same amount as the unconstrained consumer, (m).
Imposition of the PFGIC reverses the inequality in (37), and thus reverses the conclusion: A consumer who starts with m_{t} = 1 will desire to consume more than 1. Such a consumer will be constrained, not only in period t, but perpetually thereafter.
Now deﬁne b_{#}^{n} as the b_{ t} such that an unconstrained consumer holding b_{t} = b_{#}^{n} would behave so as to arrive in period t + n with b_{t+n} = 0 (with b_{#}^{0} trivially equal to 0); for example, a consumer with b_{t−1} = b_{#}^{1} was on the ‘cusp’ of being constrained in period t − 1: Had b_{t−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 PFGIC, the constraint certainly binds in period t (and thereafter) with resources of m_{t} = m_{#}^{0} = 1 + b_{ #}^{0} = 1: The consumer cannot spend more (because constrained), and will not choose to spend less (because impatient), than c_{t} = c_{#}^{0} = 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 ÞΓ, so consumption n periods in the past must have been
 (40) 
The PDV of consumption from t −n until t can thus be computed as

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 postt income) is
 (41) 
while the intertemporal budget constraint says
 (42) 
Deﬁning m_{#}^{n} = b_{ #}^{n} + 1, consider the function c(m) deﬁned by linearly connecting the points {m_{#}^{n},c_{ #}^{n}} 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, because for inﬁnitesimal 𝜖 the MPC of a consumer with assets m = m_{#}^{n} −𝜖 is discretely higher than for a consumer with assets m = m_{ #}^{n} + 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 (42) for the entire domain of positive real values of b, we need b_{#}^{n} to become arbitrarily large with n. That is, we need
 (43) 
The FHWC requires ℛ^{−1} < 1, in which case the second term in (42) limits to a constant as n ↑∞, and (43) reduces to a requirement that
 (44) 
If the FHWC fails, matters are a bit more complex. Given failure of FHWC, (43) requires

If RIC Holds. When the RIC holds, rearranging (45) gives
 (45) 
which with a few lines of algebra can be shown to asymptote to the MPC in the perfect foresight model:^{40}
 (46) 
If RIC Fails. Consider now the case, Þ R > 1. In this case the constant multiplying ℛ^{−n} in (45) will be positive if
 (47) 
but that limit is approached gradually, starting from a positive value, and consequently the consumption function is not the degenerate c(m) = 0. (Figure 6 presents an example for ρ = 2, R = 0.98, β = 0.99, Γ = 1.0).
We can summarize as follows. Given that the PFGIC 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 welldeﬁned solution, whether or not the RIC holds.
To show that (4) deﬁnes a sequence of continuously diﬀerentiable strictly increasing concave functions {c_{T }, c_{T−1},..., c_{T−k}}, we start with a deﬁnition. We will say that a function n(z) is ‘nice’ if it satisﬁes
(Notice that an implication of niceness is that lim _{z↓0}n^{′}(z) = ∞.)
Assume that some v_{t+1} is nice. Our objective is to show that this implies v_{t} is also nice; this is suﬃcient to establish that v_{t−n} is nice by induction for all n > 0 because v_{T }(m) = u(m) and u(m) = m^{1−ρ}∕(1 −ρ) is nice by inspection.
Now deﬁne an endofperiod value function 𝔳_{t}(a) as
 (48) 
Since there is a positive probability that ξ_{t+1} will attain its minimum of zero and since ℛ_{t+1} > 0, it is clear that lim _{a↓0}𝔳_{t}(a) = −∞ and lim _{a↓0}𝔳_{t}^{′}(a) = ∞. So 𝔳_{ t}(a) is welldeﬁned iﬀ a > 0; it is similarly straightforward to show the other properties required for 𝔳_{t}(a) to be nice. (See Hiraguchi (2003).)
Next deﬁne v_{t}(m,c) as
 (49) 
which is C^{3} since 𝔳_{ t} and u are both C^{3}, and note that our problem’s value function deﬁned in (4) can be written as
 (50) 
v_{t} is welldeﬁned if and only if 0 < c < m. Furthermore, lim _{c↓0}v_{t}(m,c) = lim _{c↑m}v_{t}(m,c) = −∞, < 0, lim _{c↓0} = +∞, and lim _{c↑m} = −∞. It follows that the c_{t}(m) deﬁned by
 (51) 
exists and is unique, and (4) has an internal solution that satisﬁes
 (52) 
Since both u and 𝔳_{t} are strictly concave, both c_{t}(m) and a_{t}(m) = m − c_{t}(m) are strictly increasing. Since both u and 𝔳_{t} are three times continuously diﬀerentiable, using we can conclude that c_{t}(m) is continuously diﬀerentiable and
 (53) 
Similarly we can easily show that c_{t}(m) is twice continuously diﬀerentiable (as is a_{t}(m)) (See Appendix C.) This implies that v_{t}(m) is nice, since v_{t}(m) = u(c_{t}(m)) + 𝔳_{t}(a_{t}(m)).
First we show that c_{t}(m) is C^{1}. Deﬁne y as y ≡m+dm. Since u^{′}−u^{′} = 𝔳_{t}^{′}(a_{ t}(y))−𝔳_{t}^{′}(a_{ t}(m)) and = 1 −,

This implies that the rightderivative, c_{t}^{′+}(m) is welldeﬁned and

Similarly we can show that c_{t}^{′+}(m) = c_{ t}^{′−}(m), which means c_{ t}^{′}(m) exists. Since 𝔳_{ t} is C^{3}, c_{t}^{′}(m) exists and is continuous. c_{ t}^{′}(m) is diﬀerentiable because 𝔳_{ t}^{′′} is C^{1}, c_{ t}(m) is C^{1} and u^{′′}(c_{ t}(m)) + 𝔳_{t}^{′′} < 0. c_{t}^{′′}(m) is given by
 (54) 
Since 𝔳_{t}^{′′}(a_{ t}(m)) is continuous, c_{t}^{′′}(m) is also continuous.
We must show that our operator 𝒯 satisﬁes all of Boyd’s conditions.
Boyd’s operator T maps from 𝒞_{г}(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).
Consider condition 1). For this problem,
so x(∙) ≤ y(∙) implies {𝒯x}(m_{t}) ≤ {𝒯y}(m_{t}) by inspection.^{41}Condition 2) requires that {𝒯0}∈𝒞_{г}. By deﬁnition,

the solution to which is patently u(m_{t}). Thus, condition 2) will hold if (m_{t})^{1−ρ} is гbounded. We use the bounding function
 (55) 
for some real scalar η > 0 whose value will be determined in the course of the proof. Under this deﬁnition of г, {𝒯0}(m_{t}) = u(m_{t}) is clearly гbounded.
Finally, we turn to condition 3), {𝒯(z + ζг)}(m_{t}) ≤{𝒯z}(m_{t}) + ζαг(m_{t}). The proof will be more compact if we deﬁne c and a as the consumption and assets functions^{42} associated with 𝒯z and c and a 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 г(m) = η + m^{1−ρ} and deﬁning a_{ t} = a(m_{t}), this condition is
 (56) 
But since η is an arbitrary constant that we can pick, the proof thus reduces to showing that the numerator of (56) is bounded from above:

We can thus conclude that equation (56) will certainly hold for any:
 (57) 
which is a positive ﬁnite number under our assumptions.
The proof that 𝒯 deﬁnes a contraction mapping under the conditions (28) and (25) is now complete.
In deﬁning our operator 𝒯 we made the restriction κm_{t} ≤c_{t} ≤m_{t}. However, in the discussion of the consumption function bounds, we showed only (in (29)) that κ_{t}m_{t} ≤ c_{t}(m_{t}) ≤ _{t}m_{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 (4) deﬁnes a contraction mapping.
Fortunately, the proof of that proposition is identical to the proof above, except that we must replace 57). Consideration of the prior two equations reveals that a suﬃcient stronger condition is
with _{T−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 (Calling the weak return impatience factor ÞR^{℘} = ℘^{1∕ρ}Þ R and recalling that the WRIC was ÞR^{℘} < 1, the expression on the LHS above is βÞ R^{−ρ} times the WRIF. Since we usually assume β not far below 1 and parameter values such that ÞR ≈ 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 with a unique v(m). But since lim _{n→∞}κ_{T−n} = κ and lim _{n→∞} _{T−n} = , it must be the case that the v(m) toward which these v_{T−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 (4) converge, they converge to the same unique v deﬁned by 𝒯.^{44}
Boyd’s theorem shows that 𝒯 deﬁnes a contraction mapping in a гbounded space. We now show that 𝒯 also deﬁnes a contraction mapping in Euclidian space.
Since v^{∗}(m) = 𝒯v^{∗}(m),
 (58) 
On the other hand, v_{T } − v^{∗} ∈𝒞_{г} and κ = _{г} < ∞ because v_{ T } and v^{∗} are in 𝒞_{г}. It follows that
 (59) 
Then we obtain
 (60) 
Since v_{T }(m) = , v_{T−1}(m) ≤ < v_{T }(m). On the other hand, v_{T−1} ≤ v_{T } means 𝒯v_{T−1} ≤𝒯v_{T }, in other words, v_{T−2}(m) ≤ v_{T−1}(m). Inductively one gets v_{T−n}(m) ≥ v_{T−n−1}(m). This means that _{n=1}^{∞} is a decreasing sequence, bounded below by v^{∗}.
Given the proof that the value functions converge, we now show the pointwise convergence of consumption functions _{n=1}^{∞}.
We start by showing that
 (61) 
is uniquely determined. We show this by contradiction. Suppose there exist c_{1} and c_{2} that both attain the supremum for some m, with mean c = (c_{1} + c_{2})∕2. c_{i} satisﬁes
 (62) 
where m_{t+1}(m,c_{i}) = (m −c_{i})ℛ_{t+1} + ξ_{t+1} and i = 1, 2. 𝒯v is concave for concave 𝔳. Since the space of continuous and concave functions is closed, 𝔳 is also concave and satisﬁes
 (63) 
On the other hand, < u(c). Then one gets
 (64) 
Since c is a feasible choice for c_{i}, the LHS of this equation cannot be a maximum, which contradicts the deﬁnition.
Using uniqueness of c(m) we can now show
 (65) 
Suppose this does not hold for some m = m^{∗}. In this case, _{ n=1}^{∞} has a subsequence _{ i=1}^{∞} that satisﬁes lim _{ i→∞}c_{T−n(i)}(m^{∗}) = c^{∗} and c^{∗}≠c(m^{∗}). Now deﬁne c_{T−n+1}^{∗} = c_{ T−n+1}(m^{∗}). c^{∗} > 0 because lim _{ i→∞}v_{T−n(i)+1}(m^{∗}) ≤ lim _{ i→∞}u(c_{T−n(i)}^{∗}). Because a(m^{∗}) > 0 and ψ ∈ [ψ,] there exist {m_{ +}^{∗}, _{ +}^{∗}} satisfying 0 < m_{ +}^{∗} < _{ +}^{∗} and m_{T−n+1}(m^{∗},c_{ T−n+1}^{∗}) ∈. It follows that lim _{n→∞}v_{T−n+1}(m) = v(m) and the convergence is uniform on m ∈. (Uniform convergence is obtained from Dini’s theorem.^{45} ) Hence for any δ > 0, there exists an n_{1} such that
 (66) 
then v_{T−n}(m^{∗}) satisﬁes
 (67) 
On the other hand, there exists an i_{1} ∈ℕ such that
 (68) 
because v is uniformly continuous on [m_{+}^{∗}, _{ +}^{∗}]. lim _{ i→∞} = 0 and
 (69) 
This implies
 (70) 
From (67) and (70), we obtain lim _{i→∞}v_{T−n(i)}(m^{∗}) = w(m^{∗},c^{∗}) and this implies w(m^{∗},c^{∗}) = v(m^{∗}). This implies that c(m) is not uniquely determined, which is a contradiction.
Thus, the consumption functions must converge.
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.
Suppose the population starts in period t with an arbitrary value for cov_{t}(a_{t+1,i},p_{t+1,i}). Then if m is the invariant mean level of m we can deﬁne a ‘mean MPS away from m’ function

and since ψ_{t+1,i} = 1, ℛ_{t+1,i} is a constant at ℛ we can write

so

But since R^{−1}(℘Rβ)^{1∕ρ} < a(m) < Þ R,

and for the version of the model with no permanent shocks the GIC says that Þ < Γ, which implies

This means that from any arbitrary starting value, the relative size of the covariance term shrinks to zero over time (compared to the AΓ^{n} term which is growing steadily by the factor Γ). Thus, lim _{n→∞}A_{t+n+1}∕A_{t+n} = Γ.
This logic unfortunately does not go through when there are permanent shocks, because the ℛ_{t+1,i} terms are not independent of the permanent income shocks.
To see the problem clearly, deﬁne ℛ = 𝕄 and consider a ﬁrst order Taylor expansion of a(m_{t+1,i}) around m_{t+1,i} = ℛa_{t,i} + 1,
The problem comes from the a^{′} term. The concavity of the consumption function implies convexity of the a function, so this term is strictly positive but we have no theory to place bounds on its size as we do for its level a. We cannot rule out by theory that a positive shock to permanent income (which has a negative eﬀect on m_{t+1,i}) could have an unboundedly positive eﬀect on a^{′} (as for instance if it pushes the consumer arbitrarily close to the selfimposed liquidity constraint).
For m_{t} > 0 we can deﬁne e_{t}(m_{t}) = c_{t}(m_{t})∕m_{t} and a_{t}(m_{t}) = m_{t} − c_{t}(m_{t}) and the Euler equation (4) can be rewritten

Consider the ﬁrst conditional expectation in (71), recalling that if ξ_{t+1} > 0 then ξ_{t+1} ≡𝜃_{t+1}∕(1 −℘). Since lim _{m↓0}a_{t}(m) = 0, 𝔼 _{t}[(e_{t+1}(m_{t+1})m_{t+1}Γ_{t+1})^{−ρ}  ξ_{ t+1} > 0] is contained within bounds deﬁned by (e_{t+1}(𝜃∕(1 −℘))Γψ𝜃∕(1 −℘))^{−ρ} and (e_{ t+1}(∕(1 −℘))Γ∕(1 −℘))^{−ρ} both of which are ﬁnite numbers, implying that the whole term multiplied by (1 −℘) goes to zero as m_{t}^{ρ} goes to zero. As m_{ t} ↓ 0 the expectation in the other term goes to _{t+1}^{−ρ}(1 − _{ t})^{−ρ}. (This follows from the strict concavity and diﬀerentiability of the consumption function.) It follows that the limiting _{t} satisﬁes _{t}^{−ρ} = β℘R^{1−ρ} _{ t+1}^{−ρ}(1 − _{ t})^{−ρ}. 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 (28) for this to be a convergent sequence:
 (71) 
Since = 1, iterating (71) backward to inﬁnity (because we are interested in the limiting consumption function) we obtain:
_{T }
 (72) 
and we will therefore call
the ‘limiting maximal MPC.’The minimal MPC’s are obtained by considering the case where m_{t} ↑∞. If the FHWC holds, then as m_{t} ↑∞ the proportion of current and future consumption that will be ﬁnanced out of capital approaches 1. Thus, the terms involving ξ_{t+1} in (71) can be neglected, leading to a revised limiting Euler equation
 (73) 
so that {κ_{T−n}^{−1}}_{ n=0}^{∞} is also an increasing convergent sequence, and we deﬁne
κ^{−1} ≡  lim _{ n↑∞}κ_{T−n}^{−1}  (74) 
as the limiting (inverse) marginal MPC. If the RIC does not hold, then lim _{n→∞}κ_{T−n}^{−1} = ∞ 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
which, combined with the intertemporal budget constraint, yields the usual formula for the perfect foresight consumption function:
c_{t}  = (b_{t} + h_{t})κ_{t}  (75) 
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 c_{t}(m). 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 c_{t}(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 timepreference factors are β = R = Γ = 1, and there are no permanent shocks, ψ = 1; 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 T is given by c_{T }(m) (in practice, this will be c_{T }(m) = m), consider the unrestrained optimization problem
 (76) 
As usual, the envelope theorem tells us that v_{T }^{′}(m) = u′(c_{ T }(m)) so the expected marginal value of ending period T − 1 with assets a can be deﬁned as

and the solution to (76) will satisfy
 (77) 
a_{T−1}^{∗}(m) therefore answers the question “With what level of assets would the restrained consumer like to end period T − 1 if the constraint c_{T−1} ≤m_{T−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 nonbinding in period T − 1.
Analogously to (77), deﬁning
 (78) 
the Euler equation for the original consumer’s problem implies
 (79) 
with solution a_{T−1}^{∗}(m; ℘). Now note that for any ﬁxed a > 0, lim _{ ℘↓0}𝔳_{T−1}^{′}(a; ℘) = 𝔳_{ T−1}^{′}(a). Since the LHS of (77) and (79) are identical, this means that lim _{℘↓0}a_{T−1}^{∗}(m; ℘) = a_{ T−1}^{∗}(m). That is, for any ﬁxed value of m > m_{#}^{1} 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 ℘ ↓ 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 m ≤m_{#}^{1} for which the restrained consumer is constrained. It is obvious that the baseline consumer will never choose a ≤ 0 because the ﬁrst term in (78) is lim _{a↓0}℘a^{−ρ} = ∞, while lim _{ a↓0}(m −a)^{−ρ} 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 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 < m_{#}^{1}, 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 _{℘↓0}𝔳_{T−1}^{′}(a; ℘) = 𝔳_{ T−1}^{′}(a). But by assumption we are considering a set of circumstances in which a_{T−1}^{∗}(m) < 0, and we showed earlier that lim _{℘↓0}a_{T−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 = m_{#}^{1}.
These arguments demonstrate that for any m > 0, lim _{℘↓0}c_{T−1}(m; ℘) = c_{T−1}(m) which is the period T − 1 version of (76). 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 (72) 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 a is deﬁned (aVec in the software), and at these points, marginal endofperiodt 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:^{46}

The dynamic budget constraint can then be used to generate the corresponding m’s:
An approximation to the consumption function could be constructed by linear interpolation between the {m,c} 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 (80) with respect to a (and dropping policy function arguments for simplicity) yields a marginal propensity to have consumed 𝔠^{a} 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 𝔪(a) = 𝔠(a) −a,
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 c_{T }(m) = m and constructing {c_{T−1}, c_{T−2},...} by time iteration (a method that will converge to c(m) by standard theorems). But c_{T }(m) = m is very far from the ﬁnal converged consumption rule c(m),^{47} 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 {m_{#}^{1},m_{ #}^{2},...}. This is a practical problem because it rules out the use of derivatives of the consumption function in the approximate representation of c(m), thereby preventing the enormous increase in eﬃciency obtainable from a higherorder approximation.
Our solution is simple: The formulae in appendix A that identify kink points on c(m) for integer values of n (e.g., c_{#}^{n} = Þ Γ^{−n}) are continuous functions of n; the conclusion that c(m) 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 n ≥ 0 we can construct a smooth function c(m) 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 n at other points.^{48}
This strategy generates a smooth limiting consumption function – except at the remaining kink point deﬁned by {m_{#}^{0},c_{ #}^{0}}. Below this point, the solution must match c(m) = m because the constraint is binding. At m = m_{#}^{0} the MPC discretely drops (that is, lim _{m↑m#0}c^{′}(m) = 1 while lim _{ m↓m#0}c^{′}(m) = κ_{ #}^{0} < 1).
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 c(m) above some n ≥n, but to replace c(m) between m_{#}^{0} and m_{ #}^{n} with the unique polynomial function c(m) that satisﬁes the following criteria:
where n is chosen judgmentally in a way calculated to generate a good compromise between smoothness of the limiting consumption function c(m) and ﬁdelity of that function to the c(m) (see the actual code for details).
We thus deﬁne the terminal function as
 (80) 
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 c(m) ≥c(m), implicitly deﬁning m = e^{μ} (so that μ = log m), we can construct
 (81) 
which must be a number between −∞ and +∞ (since 0 < c_{t}(m) < c(m) for m > 0). 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 c_{t}(m). In particular, χ_{t}(μ) is well approximated by linear functions both as m ↓ 0 and as m ↑∞.
Diﬀerentiating with respect to μ and dropping consumption function arguments yields
 (82) 
which can be solved for
 (83) 
Similarly, we can solve (81) for
 (84) 
Thus, having approximated χ_{t}, we can recover from it the level and derivative(s) of c_{t}.
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