BufferStockTheory.tex
Theoretical Foundations of
Buffer Stock Saving
August 4, 2017
Christopher D. Carroll^{1} 
_____________________________________________________________________________________
Abstract
“Bufferstock” models of saving are now common in the consumption literature. This paper builds
theoretical foundations for rigorous understanding of the main features of such models, including the
existence of a target wealth ratio and the proposition that aggregate consumption growth equals
aggregate income growth in a small open economy populated by buffer stock savers.
Precautionary saving, buffer stock saving, marginal propensity to consume, permanent income hypothesis
D81, D91, E21
PDF:  http://www.econ2.jhu.edu/people/ccarroll/papers/BufferStockTheory.pdf 
Slides:  http://www.econ2.jhu.edu/people/ccarroll/papers/BufferStockTheorySlides.pdf 
Web:  http://www.econ2.jhu.edu/people/ccarroll/papers/BufferStockTheory/ 
Github:  http://github.com/llorracc/BufferStockTheoryLatest 
(Contains software for solving and simulating the model) 
^{}
^{}
^{1}Contact: ccarroll@jhu.edu, Department of Economics, 440 Mergenthaler Hall, Johns Hopkins University, Baltimore, MD 21218, http://www.econ2.jhu.edu/people/ccarroll, and National Bureau of Economic Research.
Spurred by the success of Modigliani and Brumberg’s (1954) Life Cycle model and Friedman’s (1957) Permanent Income Hypothesis, a vast literature in the 1960s and 1970s formalized the idea that household spending can be modeled as reflecting optimal intertemporal choice. Famous papers by Schectman and Escudero (1977) and Bewley (1977) capped this literature, providing key building blocks for the ascendancy of dynamic stochastic optimizing models in economics.
Given this pedigree, it is surprising that the nowstandard method for analyzing such problems, contraction mapping theory, has not yet established some basic properties of the solution to the benchmark consumption problem with unbounded (e.g. constant relative risk aversion) utility, uncertainty about permanent and transitory income a la Friedman (1957), and no liquidity constraints (nor has any other method established such results). The gap exists because (except in a few special cases) standard theorems from the contraction mapping literature (including those in Stokey et. al. (1989) and up through the recent work of Matkowski and Nowak (2011)) cannot be used for this problem (for reasons explained below).
This paper fills that gap, deriving the conditions that must be satisfied for this standard problem to have a nondegenerate solution.
The reader could be forgiven for not having noticed a gap. A large literature solving precisely such problems has emerged following Zeldes (1989), fueled by advances in numerical solution methods. But numerical solutions are a ‘black box’: They make it possible to use a model without really understanding it. Indeed, without foundational theory, it can be difficult even to be sure that a computational solution is correct, given the notorious difficulty of writing errorfree computer code. Furthermore, without theoretical underpinnings, the analyst often has little intuition for how results might change with changes in the structure (or even the calibration) of the model.
For example, numerical solutions typically imply the existence of a target level of nonhuman wealth (‘cash’ for short) such that if cash exceeds the target, the consumer will spend freely and cash will fall (in expectation), while if cash is below the target the consumer will save and cash will rise. Carroll (1992; 1997) showed that target saving behavior arises under plausible parameter values for both infinite and finite horizon models. Gourinchas and Parker (2002) estimate the Carroll (1997) model using household data and conclude that for the mean household the bufferstock phase of life lasts 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. But none of these papers provides a rigorous delineation of the circumstances under which target saving will emerge or an analytical explanation for why such behavior is optimal.
This paper provides the analytical foundations for target saving and many other results that have become familiar from the numerical literature. All theoretical conclusions are paired with numerically computed illustrations (using software available on the author’s website), providing an integrated framework for understanding bufferstock saving.
The paper proceeds in three parts.
The first part specifies the conditions required for the problem to define a unique limiting consumption function. The conditions turn out to strongly resemble those required for the liquidity constrained perfect foresight model to have a solution; that parallel is explored and explained. Next, some limiting properties are derived for the consumption function as cash approaches infinity and as it approaches its lower bound, and the theorem asserting that the problem defines a contraction mapping is proven. 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 recently proven 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 shows that 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 aggregate consumption tends toward the (exogenous) growth rate of aggregate permanent income. A similar proposition holds at the level of individual households.
The consumer solves an optimization problem from the current period until the end of life at defined by the objective
where is a constant relative risk aversion utility function with .^{3} ^{4} The consumer’s initial condition is defined by market resources (what Deaton (1991) calls ‘cashonhand’) and permanent noncapital income . (This will henceforth be called a ‘Friedman/Buffer Stock’ (FBS) income process because its definition corresponds reasonably well to the descriptions in Friedman (1957) and because such a process has been widely used in the numerical buffer stock saving literature.)In the usual treatment, a dynamic budget constraint (DBC) simultaneously incorporates all of the elements that determine next period’s 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 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;^{5} (‘market resources’ or ‘money’) is the sum of financial wealth and noncapital income (permanent noncapital income multiplied by a meanone iid transitory income shock factor ; from the perspective of period , all future transitory shocks are assumed to satisfy ). Permanent noncapital income in period is equal to its previous value, multiplied by a growth factor , modified by a meanone iid shock , satisfying for where is the degenerate case with no permanent shocks.^{6} (Hereafter for brevity we occasionally drop time subscripts, e.g. signifies .)Following Carroll (1992), assume that in future periods there is a small probability that income will be zero (a ‘zeroincome event’),
 (3) 
where is an iid meanone random variable () that has a distribution satisfying where (degenerately ). Call the cumulative distribution functions and (and is derived trivially from (3) and ). Permanent income and cash start out strictly positive, and , and the consumer cannot die in debt,
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 (motivated, for example, by the existence of unemployment insurance) can be redefined by capitalizing the PDV of minimum income into current market assets,^{7} 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 and clarifies 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 is well defined, or that 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, and also permanent shocks to income, which a large empirical literature finds are to be quantitatively important in micro data (MaCurdy (1982); Abowd and Card (1989); Carroll and Samwick (1997); Jappelli and Pistaferri (2000); Storesletten, Telmer, and Yaron (2004); Blundell, Low, and Preston (2008)) and which the theory since Friedman (1957) suggests are far more consequential for household welfare than are transitory fluctuations. (The incorporation of permanent shocks also rules out application of the tools of Matkowski and Nowak (2011) and the extensive literature cited therein). 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.^{8} Finally, it differs from models found in Stokey et. al. (1989) because neither liquidity constraints nor bounds on utility or marginal utility are imposed.^{9}
The number of relevant state variables can be reduced from two ( and ) to one as follows. Defining nonbold variables as the boldface counterpart normalized by (as with ), assume that value in the last period of life is , and consider the problem in the secondtolast period,
Now consider the related problem
where is a ‘growthnormalized’ return factor, and the problem’s first order condition is
Since , defining from (??) for , (5) reduces to
This logic induces to all earlier periods, so that if we solve the normalized onestatevariable problem specified in (??) we will have solutions to the original problem for any from:^{10}
We say that a consumption problem has a nondegenerate solution if it defines a unique limiting consumption function whose optimal satisfies
for every (‘Degenerate’ limits will be cases where the limiting consumption function is either or .)
The analytical solution to the perfect foresight specialization of the model, obtained by setting and , provides a useful reference point and defines some remaining notation.
The dynamic budget constraint, strictly positive marginal utility, and the can’tdieindebt condition (4) imply an exactlyholding intertemporal budget constraint (IBC)
where is ‘human wealth,’ the discounted value of noncapital income, and with a constant , human wealth will be (9) makes plain that in order for to be finite, we must impose the Finite Human Wealth Condition (‘FHWC’) 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.
In the absence of a liquidity constraint, the consumption Euler equation holds in every period; with , this says
where the Old English letter ‘thorn’ represents what we will call the ‘absolute patience factor’ .^{11} The sense in which captures patience is that if the ‘absolute impatience condition’ (AIC) holds, the consumer will choose to spend an amount too large to sustain (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:
so that from which the IBC (8) implies which defines a normalized finitehorizon perfect foresight consumption function where 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.’ Equation (14) makes plain that for the limiting MPC to be strictly positive as goes to infinity we must impose the condition so thatEquation (16) 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. This is the condition that rules out the degenerate limiting solution . Henceforth (16) will be called the ‘return impatience condition’ or RIC, and a consumer who satisfies the condition is called ‘return impatient.’
Given that the RIC holds, and defining limiting objects by the absence of a time subscript (e.g., ), the limiting consumption function will be
and we now see that in order to rule out the degenerate limiting solution we need to be finite so we must impose the finite human wealth condition (10).A final useful point is that since the perfect foresight growth factor for consumption is , using yields the following expression for value:
which asymptotes to a finite value as approaches if ; with a bit of algebra, this requirement can be shown to be equivalent to the RIC.^{12} Thus, the same conditions that guarantee a nondegenerate limiting consumption function also guarantee a nondegenerate limiting value function.If the liquidity constraint is ever to be relevant, it must be relevant at the lowest possible level of market resources, , which obtains for a consumer who enters period with . 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 (6):
By analogy to the return patience factor, we therefore define a ‘perfect foresight growth patience factor’ as
and define a ‘perfect foresight growth impatience condition’ (PFGIC) which is equivalent to (19) (exponentiate both sides by ).If the RIC and the FHWC hold, appendix A shows that an unconstrained consumer behaving according to (18) would choose for all for some . The solution to the constrained consumer’s problem in this case is simple: For any 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 to arrive at an the constraint would bind and the consumer would have to consume (though such values of are of questionable relevance because they could only be obtained by entering the period with which the constraint rules out). We use the accent to designate the limiting constrained consumption function:
 (22) 
More useful is the case where the PFGIC and the RIC 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 successively increasing kink points . As the constrained consumption function approaches 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. Surprisingly, this logic holds even when the finite human wealth condition fails (denoted " class="math" > " class="oalign" >). 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 will run those resources down over time so that by some finite number of periods in the future the consumer will reach , and thereafter will set for eternity, a policy that will yield value of
The most peculiar possibility occurs when the RIC fails. Remarkably, the appendix shows that although under these circumstances the FHWC must also fail, the constrained consumption function is nondegenerate even in this case. 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 " class="math" > " class="oalign" > would suggest a dengenerate limit of while " class="math" > " class="oalign" > would suggest a degenerate limit of .
Tables 2 and 3 (and appendix table 4) codify the key points to help the reader keep them straight (and to facilitate upcoming comparisons with the surprisingly parallel results in the presence of uncertainty but the absence of liquidity constraints (also tabulated for comparison)).
When uncertainty is introduced, the expectation of can be rewritten as:
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 because this permits us to write expressions like the RHS of (25) compactly as, e.g., ^{13} We refer to this as the ‘return compensated’ permanent shock, because it compensates for the effect of uncertainty on the expected growthnormalized return (in the sense implicitly defined in (25)). Note that Jensen’s inequality implies that for nondegenerate (since by assumption).Using this definition, we can transparently generalize the PFGIC (21) by defining a ‘compensated growth factor’
and a compensated growth patience factor and a straightforward derivation using some of the results below yields the conclusion that which implies that if we wish to prevent from heading to infinity (that is, if we want to be guaranteed to be expected to fall for some large enough value of ) we must impose a generalized version of (21) which we call simply the ‘growth impatience condition’ (GIC):^{14} which is stronger than the perfect foresight version (21) because .A consumer who spent his permanent income every period would have value
which invites the definition of a utilitycompensated equivalent of the permanent shock, which will satisfy for and nondegenerate (and for the preferred (though not required) case of ); defining we can see that will be finite as approaches if which we call the ‘finite value of autarky’ condition (FVAC) because it is the value obtained by always consuming permanent income, and which for nondegenerate is stronger (harder to satisfy in the sense of requiring lower ) than the perfect foresight version (23) because .
Figure 1 depicts the successive consumption rules that apply in the last period of life (), the secondtolast period, and various earlier periods under the baseline parameter values listed in Table 1. (The 45 degree line is labelled as because in the last period of life it is optimal to spend all remaining resources.)
Description  Parameter  Value  Source
 
Permanent Income Growth Factor  1.03  PSID: Carroll (1992)
 
Interest Factor  1.04  Conventional  
Time Preference Factor   0.96  Conventional
 
Coefficient of Relative Risk Aversion  2  Conventional
 
Probability of Zero Income  0.005  PSID: Carroll (1992)
 
Std Dev of Log Permanent Shock  0.1  PSID: Carroll (1992)
 
Std Dev of Log Transitory Shock  0.1  PSID: Carroll (1992)
 
Approximate  
Calculated  
Description  Symbol and Formula  Value  
Finite Human Wealth Measure  0.990  
PF Finite Value of Autarky Measure  0.932  
Growth Compensated Permanent Shock  0.990  
UncertaintyAdjusted Growth  1.020  
Utility Compensated Permanent Shock  0.990  
Utility Compensated Growth  1.020  
Absolute Patience Factor  0.999  
Return Patience Factor  0.961  
PF Growth Patience Factor  0.970  
Growth Patience Factor  0.980  
Finite Value of Autarky Measure  0.941  
In the figure, the consumption rules appear to converge as the horizon recedes (below we show that this appearance is not deceptive); we call the limiting infinitehorizon consumption rule
A precondition for the main proof is that the maximization problem (??) defines a sequence of continuously differentiable strictly increasing strictly concave^{15} functions .^{16} The proof of this precondition is straightforward but tedious, and so is relegated to appendix B. For present purposes, the most important point is the following intuition: for all periods because if the consumer spent all available resources, he would arrive in period with balances of zero, then might earn zero noncapital income for the rest of his life (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)).)^{17}
The consumption functions depicted in Figure 1 appear to have limiting slopes as and as . This section confirms that impression and derives those slopes, which also turn out to be useful in the contraction mapping proof.
Assume (as discussed above) 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.)
For we can define and and the Euler equation (6) can be rewritten
Consider the first conditional expectation in (32), 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 We can conclude that
which yields a useful recursive formula for the maximal marginal propensity to consume:Then is a decreasing convergent sequence if
a condition that 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 ().Since , iterating (34) backward to infinity (because we are interested in the limiting consumption function) we obtain:
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 (31) can be neglected, leading to a revised limiting Euler equation
and we know from L’Hôpital’s rule that , and so a further limit of the Euler equation is and the same sequence of derivations used above yields the conclusion that if the RIC holds, then a recursive formula for the minimal marginal propensity to consume is given by so that is also an increasing convergent sequence, with being the ‘limiting minimal MPC.’ If the RIC does not hold, then and so the limiting MPC isWe are now in position to observe that the optimal consumption function must satisfy
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 and (the formal proof is provided in appendix D).
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 , because the possibility (however unlikely) of an unbroken string of zeroincome events for the remainder of life means that as approaches zero 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 Matkowski and Nowak (2011) for a recent contribution), 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.^{18}
Fortunately, Boyd (1990) provides a weighted contraction mapping theorem that can be used. To use Boyd’s theorem we need
Definition 1. Consider any function where is the space of continuous functions from to . Suppose with and . Then is bounded if the norm of ,
 (39) 
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^{19} ^{20}
Then defines a contraction with a unique fixed point.For our problem, take as and as , and define
Using this, we introduce the mapping ,^{21}
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 restriction is the WRIC necessary for convergence of the maximal MPC, equation (35) above. A more serious restriction is the utilitycompensated Finite Value of Autarky condition, equation (29). (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.
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.
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
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
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 (41) will satisfytherefore 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 (42), defining
the Euler equation for the original consumer’s problem implies with solution . Now note that for any fixed , . Since the LHS of (42) and (44) 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 (43) 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 (41). 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 (36) for the maximal marginal propensity to consume satisfies
which makes sense because the marginal propensity to consume for a constrained restrained consumer is 1 by our definitions of ‘constrained’ and ‘restrained.’
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 to exist. We thus begin our discussion by asking what features the problem must exhibit (given the FVAC) if the RIC fails (that is, :
but since and (because we have assumed ), this can hold only if ; 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 (37), an implication of " class="math" > " class="oalign" > is that . 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 level of consumption for any finite value of . Note the close parallel to the conclusion in the perfect foresight liquidity constrained model in the PFGIC, " class="math" > " class="oalign" > 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
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 , the WRIC is indeed redundant because now , so that 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 (46) 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. 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 , 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.
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 " class="math" > " class="oalign" > is depicted in Figure 2. The consumption function is shown along with the locus that identifies the ‘sustainable’ level of spending at which 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 software archive for details), the consumption function never reaches the locus; indeed, when the RIC holds but the GIC does not, the consumption function’s limiting slope is shallower than that of the sustainable consumption locus ,^{22} so the gap between the two actually increases with in the limit. That is, although a nondegenerate consumption function exists, a target level of does not (or, rather, the target is ), because no matter how wealthy a consumer becomes, he will always spend less than the amount that would keep stable (in expectation).
For the reader’s convenience, Tables 2 and 3 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)
 
The
growth
factor
for
permanent
income
must
be
smaller
than
the
discounting
factor
,
for
human
wealth
to
be
finite. 
The
model’s
risks
are
meanpreserving
spreads,
so
the
PDV
of
future
income
is
unchanged
by
their
introduction. 
Absolute Impatience Condition (AIC)
 
The
unconstrained
consumer
is
sufficiently
impatient
that
the
level
of
consumption
will
be
declining
over
time: 
If
wealth
is
large
enough,
the
expectation
of
consumption
next
period
will
be
smaller
than
this
period’s
consumption: 
Return Impatience Conditions
 
Return Impatience Condition (RIC)  Weak RIC (WRIC)

The
growth
factor
for
consumption
must
be
smaller
than
the
discounting
factor
,
so
that
the
PDV
of
current
and
future
consumption
will
be
finite: 
If
the
probability
of
the
zeroincome
event
is
then
income
is
always
zero
and
the
condition
becomes
identical
to
the
RIC.
Otherwise,
weaker. 
Growth Impatience Conditions
 
PFGIC  GIC

Guarantees
that
for
an
unconstrained
consumer,
the
ratio
of
consumption
to
permanent
income
will
fall
over
time.
For
a
constrained
consumer,
guarantees
the
constraint
will
eventually
be
binding. 
By
Jensen’s
inequality,
stronger
than
the
PFGIC.
Ensures
consumers
will
not
expect
to
accumulate
unboundedly. 
Finite Value of Autarky Conditions
 
PFFVAC  FVAC

equivalently  
The
discounted
utility
of
constrained
consumers
who
spend
their
permanent
income
each
period
should
be
finite. 
By
Jensen’s
inequality,
stronger
than
the
PFFVAC
because
for
and
nondegenerate
,
. 
Model  Conditions  Comments

PF Unconstrained  RIC, FHWC  RIC ; FHWC 
RIC prevents  
FHWC prevents  
PF Constrained  PFGIC  If RIC, 
If " class="math" > " class="oalign" >,  
Buffer Stock Model  FVAC, WRIC  FHWC 
" class="math" > " class="oalign" >+RIC  
" class="math" > " class="oalign" >+" class="math" > " class="oalign" >  
GIC guarantees finite target wealth ratio  
FVAC is stronger than PFFVAC  
WRIC is weaker than RIC  
For feasible , limiting consumption function defines unique value of satisfying . RIC, FHWC are necessary as well as sufficient. Solution also exists for " class="math" > " class="oalign" > and RIC, but is identical to the unconstrained model’s solution for feasible .
Figures 3 and 4a,b capture the main properties of the converged consumption rule when the RIC, GIC, and FHWC all hold.^{23} Figure 3 shows the expected consumption growth factor 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 the expected consumption growth factor goes to , indicated by the lower bound in Figure 3, and the marginal propensity to consume approaches (Figure 4), the same as the perfect foresight MPC.^{24} Second, as the consumption growth factor approaches (Figure 3) and the MPC approaches (Figure 4). Third (Figure 3), there is a target cashonhandtoincome ratio such that if then , and (as indicated by the arrows of motion on the curve), the model’s dynamics are ‘stable’ around the target in the sense that if then cashonhand will rise (in expectation), while if , it will fall (in expectation). Fourth (Figure 3), at the target , 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 . 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.^{25}
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.^{26}Assuming the FHWC holds, the infinite horizon perfect foresight solution (18) 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 .
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 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 :
But
and

while
because ^{27} 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.
This argument applies equally well to the problem of the restrained consumer, because as approaches infinity the constraint becomes irrelevant (assuming the FHWC holds).
Now consider the limits of behavior as gets arbitrarily small.
Equation (36) 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 4 confirms that the numerical solution method obtains this limit for the MPC as approaches zero.
For consumption growth, 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.^{28}Define the target cashonhandtoincome ratio as the value of such that
 (47) 
where the accent is meant to invoke the fact that this is the value that other ’s ‘point to.’
We prove existence by arguing that is continuous on , 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 .
The limit as goes to infinity is
where the last line is guaranteed by our imposition of the GIC (28).Stability means that in a local neighborhood of , values of above will result in a smaller ratio of than at . That is, if then . This will be true if
at . But which will be negative if its numerator is negative. Define as the expectation of the numerator,
 (48) 
The target level of market resources satisfies
At the target, equation (48) is
Substituting for the first term in this expression using (49) gives
where the step introducing the inequality imposes the fact that which is an implication of the concavity of the consumption function.We have now proven that some target must exist, and that at any such the solution is stable. Nothing so far, however, rules out the possibility that there will be multiple values of that satisfy the definition (47) 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., . The argument just completed implies that since is continuously differentiable there must exist some small such that for . (Continuous differentiability of follows from the continuous differentiability of .)
Now assume there exists a second value of satisfying the definition of a target, . Since is continuous, it must be approaching 1 from below as , since by the intermediate value theorem it could not have gone above 1 between and without passing through 1, and by the definition of it cannot have passed through 1 before reaching . But saying that is approaching 1 from below as implies that
at . However, we just showed above that, under our assumption that the GIC holds, precisely the opposite of equation (50) must hold for any 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 , 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 at particular values of . But we can use the fact that the restrained model is the limit of the baseline model as 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 . That is, if a consumer starting with will save nothing, , then the target level of in the restrained model will be 1; if a consumer with 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 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 then
and since and it is clear that cov which implies that the entire term added to in (51) 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
and the proposition in which we are interested is or differentiating through the expectations operator, what we want isHenceforth indicating appropriate arguments by the corresponding subscript (e.g. ), since , the portion of the LHS of equation (52) in brackets can be manipulated to yield
Now differentiate the Euler equation with respect to :
but since we can see from (54) that (52) is equivalent to which, using (53), will be true if which in turn will be true if both andThe 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.)
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. We will continue to take the aggregate interest rate as exogenous and constant. It is also possible, and only slightly more difficult, to solve for the steadystate of a closedeconomy version of the model where the interest rate is endogenous.
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.^{29}
Szeidl (2012) proves that such a population will be characterized by an invariant distribution of that induces invariant distributions for and ; designate these , , and .^{30}
Szeidl’s proof, however, does not yield any sense of how quickly convergence occurs, which in principle depends on all of the parameters of the model as well as the initial conditions. To build intuition, Figure 5 supplies an example in which a population begins with a particularly simple distribution that is far from the invariant one:
which would characterize a population in which all assets had been wiped out immediately before the receipt of period 1’s noncapital income.^{31}The figure plots the distributions of (for technical reasons, this is slightly better than plotting ) at the ends of 1, 4, 10, and 40 periods.^{32}
The figure illustrates the fact that, under these parameter values, convergence to the invariant distribution has largely been accomplished within 10 periods. By 40 periods, the distribution is indistinguishable from the invariant distribution.
It is useful to define the operator which yields the mean value of its argument in the population, as distinct from the expectations operator which represents beliefs about the future.
An economist with a microeconomic dataset could calculate the average growth rate of idiosyncratic consumption, and would find
where and the last equality follows because the invariance of means that .^{33}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.^{34} 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:
because of the independence assumptions we have made about and .Aggregate assets are:
where designates the mean level of permanent income across all individuals, and we are assuming that was distributed according to the invariant distribution with a mean value of Since permanent income grows at mean rate while the distribution of is invariant, if we normalize to one we will similarly have for any periodUnfortunately, Szeidl (2012)’s proof of the invariance of does not yield the information about how the crosssectional covariance between and evolves required to show that the covariance term grows by a factor smaller than ; 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 prevents proof in the general case with both transitory and permanent shocks.
A wide range of simulation experiments confirms that the role of that covariance term is more an irritating theoretical curiosum than an important practical consideration. An example is given in Figure 6, which plots for the economy whose converging CDFs were depicted in Figure 5. After the 40 periods of simulation that generated CDFs plotted in 5, we conduct an experiment designed to flush out the role of the annoying covariance term: We reset the level of permanent income to be identical for all consumers (‘the revolution’):
and we redistribute cash among consumers in such a way as to leave each consumer with the same value of that they would have had in the absence of the revolution. This leaves us with the same distribution of as before the revolution, but no covariance between and .
The effect on aggregate consumption growth of even such an extreme revolution in covariance is small, and dissipates immediately (no effect is visible after the period of revolution itself). This experiment is representative of many that suggest that the practical effects of timevariaton in the covariance between and are negligible.
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 cash will tend.
Another contribution is provision a set of tools for numerical solution and simulation (available on the author’s web page) that confirm and illustrate the theoretical propositions. These programs demonstrate how the incorporation of the paper’s theoretical results can make numerical solution algorithms more efficient and simpler. A goal of the paper has been to make these tools accessible and easy to use while incorporating the full rigor of the theoretical results in the structure.
This appendix taxonomizes the characteristics of the limiting consumption function under perfect foresight in the presence of a liquidity constraint requiring under various conditions. Results are summarized in table 4.
Name  Condition  Outcome/Comments  
" class="math" > " class="oalign" >  Constraint never binds for  
RIC  FHWC holds ()  
for  
" class="math" > " class="oalign" >  is degenerate  
PFGIC  Constraint binds in finite time for any  
RIC  FHWC may or may not hold  
" class="math" > " class="oalign" >  " class="math" > " class="oalign" >  
A consumer is ‘growth patient’ if the perfect foresight growth impatience condition fails ( " class="math" > " class="oalign" >, ). Under " class="math" > " class="oalign" > 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 :^{35}
Similar logic shows that under these circumstances the constraint will never bind for an unconstrained consumer with a finite horizon of 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 () while the finite human wealth condition holds, the limiting value of this consumption function as is the degenerate function
If the RIC fails and the FHWC fails, 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 .^{36}
Thus, the requirement that the consumption function be nondegenerate implies that for a consumer satisfying " class="math" > " class="oalign" > we must impose the RIC (and the FHWC can be shown to be a consequence of " class="math" > " class="oalign" > 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 (18):
which (under these assumptions) satisfies .^{37} For the unconstrained consumer would choose to consume more than ; for such , the constrained consumer is obliged to choose .^{38} For any the constraint will never bind and the consumer will choose to spend the same amount as the unconstrained consumer, .Imposition of the PFGIC reverses the inequality in (55)(57), 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 PFGIC, 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
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 while the intertemporal budget constraint says from which we can solve for the such that the consumer with would unconstrainedly plan (in period ) to arrive in period with :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, because 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 (65) for the entire domain of positive real values of , we need to become arbitrarily large with . That is, we need
The FHWC requires , in which case the second term in (65) limits to a constant as , and (66) reduces to a requirement that
Given the PFGIC , this will hold iff the RIC holds, . But given that the FHWC holds, the PFGIC is stronger (harder to satisfy) than the RIC; thus, FHWC and the PFGIC together imply the RIC, and so a welldefined solution exists. Furthermore, in the limit as approaches infinity, the difference between the limiting constrained consumption function and the unconstrained consumption function becomes vanishingly small, because as the date at which the constraint binds becomes arbitrarily distant, the effect of that constraint on current behavior shrinks to nothing. That is,
If the FHWC fails, matters are a bit more complex. Given failure of FHWC, (66) requires
If RIC Holds. When the RIC holds, rearranging (68) gives
and for this to be true we need which is merely the RIC again. So the problem has a solution if the RIC holds. Indeed, we can even calculate the limiting MPC from which with a few lines of algebra can be shown to asymptote to the MPC in the perfect foresight model:^{39}If RIC Fails. Consider now the " class="math" > " class="oalign" > case, . In this case the constant multiplying in (68) will be positive if
which is merely the PFGIC which we are maintaining. So the first term’s limit is . The combined limit will be if the term involving goes to faster than the term involving goes to ; that is, if which merely confirms the starting assumption that the RIC fails. Thus, surprisingly, the problem has a well defined solution with infinite human wealth if the RIC fails. It remains true that " class="math" > " class="oalign" > implies a limiting MPC of zero, but that limit is approached gradually, starting from a positive value, and consequently the consumption function is not the degenerate . (Figure 7 presents an example for , , , ).
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 . But even if the FHWC fails, the problem has a welldefined solution, whether or not the RIC holds.
To show that (??) 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
 (72) 
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
 (73) 
which is since and are both and note that our problem’s value function defined in (??) can be written as
is welldefined if and only if . Furthermore, , , , and . It follows that the defined by
exists and is unique, and (??) has an internal solution that satisfies
 (76) 
Since both and are strictly concave, both and are strictly increasing. Since both and are three times continuously differentiable, using we can conclude that is continuously differentiable and
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
 (78) 
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.^{40}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
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^{41} 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 Using and defining , this condition is which by imposing the PFFVAC (23) can be rewritten as:But since is an arbitrary constant that we can pick, the proof thus reduces to showing that the numerator of (80) is bounded from above:
We can thus conclude that equation (80) will certainly hold for any:
which is a positive finite number under our assumptions.The proof that defines a contraction mapping under the conditions (35) and (29) is now complete.
In defining our operator we made the restriction . However, in the discussion of the consumption function bounds, we showed only (in (38)) 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 (??) defines a contraction mapping.
Fortunately, the proof of that proposition is identical to the proof in 2.9, except that we must replace with and the WRIC must be replaced by a stronger condition. The place where these conditions have force is in the step at (81). Consideration of the prior two equations reveals that a sufficient stronger condition is
where we have used (34) for . For small values of this expression can be further simplified using so that it becomes which for small values of is plainly easy to satisfy.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 (??) converge, they converge to the same unique defined by .^{42}
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.
Since ,
 (83) 
On the other hand, and because and are in . It follows that
 (84) 
Then we obtain
 (85) 
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 .
We start by showing that
 (86) 
is uniquely determined. We show this by contradiction. Suppose there exist and that both attain the supremum for some , with mean . satisfies
 (87) 
where and . is concave for concave . Since the space of continuous and concave functions is closed, is also concave and satisfies
 (88) 
On the other hand, Then one gets
 (89) 
Since is a feasible choice for , the LHS of this equation cannot be a maximum, which contradicts the definition.
Using uniqueness of we can now show
 (90) 
Suppose this does not hold for some . In this case, has a subsequence that satisfies and . Now define . because . Because and there exist satisfying and . It follows that and the convergence is uniform on . (Uniform convergence is obtained from Dini’s theorem.^{43} ) Hence for any , there exists an such that
for all . It follows that if we define
 (91) 
then satisfies
 (92) 
On the other hand, there exists an such that
 (93) 
because is uniformly continuous on . and
 (94) 
This implies
 (95) 
From (92) and (95), we obtain and this implies . This implies that is not uniquely determined, which is a contradiction.
Thus, the consumption functions must converge.
The text 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 with an arbitrary value for . Then if is the invariant mean level of we can define a ‘mean MPS away from ’ function
and since , is a constant at we can write soBut since ,

and for the version of the model with no permanent shocks the GIC says that 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 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 an unboundedly positive effect on (as for instance if it pushes the consumer arbitrarily close to the selfimposed liquidity constraint).
The model is solved using an extension of the method of endogenous gridpoints (Carroll (2016)): A grid of possible values of endofperiod assets is defined (aVec in the software), 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:^{44}
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 (96) 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 ,^{45} 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 appendix A 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.^{46}
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^{47}
 (98) 
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
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
which can be solved forSimilarly, we can solve (99) for
Thus, having approximated , we can recover from it the level and derivative(s) of .^{48}
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