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The Income Fluctuation Problem and the Evolution of Wealth

We analyze the household savings problem in a general setting where returns on assets, non-financial income and impatience are all state dependent and fluctuate over time. All three processes can be serially correlated and mutually dependent. Rewards can be bounded or unbounded and wealth can be arbitrarily large. Extending classic results from an earlier literature, we determine conditions under which (a) solutions exist, are unique and are globally computable, and (b) the resulting wealth dynamics are stationary, ergodic and geometric mixing. We show how these results can be used to extend recent studies of the wealth distribution. Our conditions have natural economic interpretations in terms of asymptotic growth rates for discounting and return on savings.


Introduction
It has been observed that, in the US and several other large economies, the wealth distribution is heavy tailed and wealth inequality has risen sharply over the last few decades. 2 This matters not only for its direct impact on taxation and redistribution 1 This paper has benefitted from discussion with many colleagues. We particularly thank Fedor Iskhakov, Larry Liu and Chung Tran for valuable feedback and suggestions. The second author gratefully acknowledges financial support from ARC grant FT160100423. Email addresses: qingyin.ma@anu.edu.au, john.stachurski@anu.edu.au, atoda@ucsd.edu.
2 For example, in a study based on capital income data, Saez and Zucman (2016) find that, in the case of the US, the share of total household wealth held by the top 0.1% increased from 7 percent to 22 percent between 1978 and 2012. For a discussion of the heavy-tailed property of the wealth distribution, see Pareto (1896), Davies and Shorrocks (2000), Benhabib and Bisin (2018), Vermeulen (2018) or references therein.
policies, but also for potential flow-on effects for productivity growth, business cycles and fiscal policy, as well as for the political environment that shapes these and other economic outcomes. 3 At present, our understanding of these phenomena is hampered by the fact that standard tools of analysis-such as those used for heterogeneous agent models-are not well adapted to studying the wealth distribution as it stands. For example, while we have sound understanding of the household problem when returns on savings and rates of time discount are constant (see, e.g., Schechtman (1976), Schechtman and Escudero (1977), Deaton and Laroque (1992), Carroll (1997), or Açıkgöz (2018)), our knowledge is far more limited in settings where these values are stochastic. This is problematic, since injecting such features into the household problem is essential for accurately representing the joint distribution of income and wealth (e.g., Benhabib, Bisin, and Zhu (2015), Benhabib, Bisin, and Luo (2017), Stachurski and Toda (2019)). 4 Moreover, models with time-varying discount rates and returns on assets are at the forefront of recent quantitative analysis of wealth and inequality. 5 While it might be hoped that the analysis of the income fluctuation problem (or household consumption and savings problem) changes little when we shift from constant to state dependent asset returns and rates of time discount, this turns out not to be the case. Effectively modeling these features and the way they map to the wealth distribution requires significant advances in our understanding of choice and stochastic dynamics in the setting of optimal savings.
3 One analysis of the two-way interactions between inequality and political decision making can be found in Acemoglu and Robinson (2002). Glaeser, Scheinkman, and Shleifer (2003) show how inequality can alter economic and social outcomes through subversion of institutions. The same study contains references on linkages between inequality and growth. Regarding fiscal policy, Brinca, Holter, Krusell, and Malafry (2016) find strong correlations between wealth inequality and the magnitude of fiscal multipliers, while Bhandari, Evans, Golosov, and Sargent (2018) study the connection between fiscal-monetary policy, business cycles and inequality. Ahn, Kaplan, Moll, Winberry, and Wolf (2018) discuss the impact of distributional properties on macroeconomic aggregates. 4 Also related is the recent experimental study of Epper, Fehr, Fehr-Duda, Kreiner, Lassen, Leth-Petersen, and Rasmussen (2018), which finds a strong positive connection between dispersion in subjective rates of time discounting across the population and realized dispersion in the wealth distribution. This in turn is consistent with earlier empirical studies such as Lawrance (1991). 5 For a recent quantitative study see, for example, Hubmer, Krusell, and Smith (2018), where returns on savings and discount rates are both state dependent (as is labor income). Kaymak, Leung, and Poschke (2018) find that asset return heterogeneity is required to match the upper tail of the wealth distribution.
One difficulty is that state-dependent discounting takes us beyond the bounds of dynamic programming theory. This matters little if there exists some constantβ < 1 such that the discount process {β t } satisfies β t ≤β for all t with probability one, since, in this case, a standard contraction mapping argument can still be applied (see, e.g., Miao (2006) or Cao (2018)). However, recent quantitative studies extend beyond such settings. For example, AR(1) specifications are increasingly common, in which case the support of β t is unbounded above at every point in time. 6 Even if discretization is employed, the outcome β t ≥ 1 can occur with positive probability when the approximation is sufficiently fine. Moreover, such outcomes are not inconsistent with empirical and experimental evidence, at least for some households in some states of the world. 7 Do there exist conditions on {β t } that allow for β t ≥ 1 in some states and yet imply existence of optimal polices and practical computational techniques?
Another source of complexity for the income fluctuation problem in the general setting considered here is that the set of possible values for household assets is typically unbounded above. For example, when returns on assets are stochastic, a sufficiently long sequence of favorable returns can compound one another to project a household to arbitrarily high levels of wealth. This model feature is desirable: We wish to analyze these kinds of outcomes rather than rule them out. Indeed, Benhabib, Bisin, and Zhu (2015) and other related studies argue convincingly that such outcomes are a key causal mechanism behind the heavy tail of the current distribution of wealth. 8 However, if we accept this logic, then stationarity and ergodicity of the wealth process-which are fundamental both for estimation and for simulation-based numerical methods-must now be established in a setting where the wealth distribution has unbounded support. In such a scenario, what conditions on preferences and financial and labor income are necessary for these properties to hold? 6 See, for example, Hills and Nakata (2018), Hubmer, Krusell, and Smith (2018) or Schorfheide, Song, and Yaron (2018). 7 See, for example, Loewenstein and Prelec (1991) and Loewenstein and Sicherman (1991). 8 One related study is Benhabib, Bisin, and Zhu (2011), who show that capital income risk is the driving force of the heavy-tail properties of the stationary wealth distribution. In Blanchard-Yaari style economies, Toda (2014), Toda and Walsh (2015) and Benhabib, Bisin, and Zhu (2016) show that idiosyncratic investment risk generates a double Pareto stationary wealth distribution. Gabaix, Lasry, Lions, and Moll (2016) point out that a positive correlation of returns with wealth ("scale dependence") in addition to persistent heterogeneity in returns ("type dependence") can well explain the speed of changes in the tail inequality observed in the data.
A final and related example of the need for deeper analysis is that, if we wish to understand the upper tail of the wealth distribution, we should avoid unnecessarily truncating the upper tail of the set of possible asset values. While truncation is convenient because finite or compact state spaces are easier to handle computationally, we can attain greater accuracy in modeling the wealth distribution if truncation at the upper tail can be replaced locally by a parameterized savings function, such as a linear function (Gouin-Bonenfant and Toda, 2018). However, any such approximation must be justified by theory. What conditions can be imposed on primitives to generate such properties while still maintaining realistic assumptions for asset returns and non-financial income?
In this paper we address all of these questions, along with other key properties of the income fluctuation problem, such as continuity and monotonicity of the optimal consumption policy. Our setting admits capital income risk, labor earnings shocks and time-varying discount rates, driven by a combination of iid innovations and an exogenous Markov chain {Z t }. The supports of the innovations can be unbounded, so we admit practical innovation sequences such as normal and lognormal. As a whole, this environment allows for a range of realistic features, such as stochastic volatility in returns on asset holdings, or correlation in the shocks impacting asset returns and non-financial income. The utility function can be unbounded both above and below, with no specific structure imposed beyond differentiability, concavity and the usual slope (Inada) conditions. 9 To begin, when considering optimality in the household problem, we require a condition on the state dependent discount process {β t } that generalizes the classical condition β < 1 from the constant case and, for reasons discussed above, permits β t > 1 with positive probability. To this end, we introduce the restriction 10 9 While the assumption that the exogenous state process {Z t } is a (finite state) Markov chain might appear restrictive, it fits most practical settings and avoids a host of technical issues that tend to obscure the key ideas. We treat the case of general state space in a separate paper, showing that, once these technical issues are taken care of, the same core ideas still apply. Moreover, the innovation shocks are not restricted to be discrete, and the same is true of course for assets and consumption. 10 Here and below we set β 0 ≡ 1, so n t=1 β t = n t=0 β t .

Condition
(1) clearly generalizes the classical condition β < 1 for the constant discount case. In the stochastic case, ln G β can be understood as the asymptotic growth rate of the probability weighted average discount factor. Indeed, if B t := E n t=1 β t is the average t-period discount factor, then, from the definition of G β and some straightforward analysis, we obtain ln(B t+1 /B t ) → ln G β , so the condition G β < 1 implies that the asymptotic growth rate of the average t-period discount factor is negative, drifting down from its initial condition β 0 ≡ 1 at the rate ln G β . This does not, of course, preclude the possibility that β t > 1 at any given t.
We show that condition (1) is in fact a necessary condition in those settings where the classical condition is necessary for finite lifetime values. In this sense it cannot be further weakened for the income fluctuation problem apart from special cases. At the same time, it admits the use of convenient specifications such as the discretized AR(1) process from Hubmer, Krusell, and Smith (2018). We also prove that G β can be represented as the spectral radius of a positive matrix, and hence can be computed by numerical linear algebra (as discussed below).
We also generalize the standard condition βR < 1, where R is the gross interest rate in the constant case, which is used to ensure stability of the asset path and finiteness of lifetime valuations, as well as existence of stationary Markov policies (see, e.g., Deaton and Laroque (1992), Chamberlain and Wilson (2000) or Li and Stachurski (2014)). Analogous to (1), we introduce the generalized condition Here {R t } is a stochastic capital income process. Analogous to the case of G β , the value ln G βR can be understood as the asymptotic growth rate of average gross payoff on assets, discounted to present value.
We show that, when Conditions (1)-(2) hold and non-financial income satisfies two moment conditions, then a unique optimal consumption policy exists. We also show that the policy can be computed by successive approximations and analyze its properties, such as monotonicity and asymptotic linearity. This asymptotic linearity can be used to successfully model wealth inequality by accurately representing asset path dynamics for very high wealth households (Benhabib, Bisin, and Zhu, 2015;Gouin-Bonenfant and Toda, 2018).
One important feature of Conditions (1)-(2) is that they take into account the autocorrelation structure of preference shocks and asset returns. For example, if these processes depend only on iid innovations, then (1) reduces to Eβ t < 1 and (2) reduces to Eβ t R t < 1. But returns on assets are typically not iid, since both mean returns and volatility are, in general, time varying, and preference shocks are typically modeled as correlated (see, e.g., Hubmer, Krusell, and Smith (2018) or Schorfheide, Song, and Yaron (2018)). This dependence must be and is accounted for in (2), since long upswings in {β t } and {R t } can lead to explosive paths for valuations and assets.
Next we study asymptotic stability, stationarity and ergodicity of wealth. Such properties are essential to existence of stationary equilibria in heterogeneous agent models (e.g., Huggett (1993), Aiyagari (1994) or Cao (2018)), as well as standard estimation, calibration and simulation techniques that connect time series averages with crosssectional moments. 11 These properties require an additional restriction, placed on the asymptotic growth rate of mean returns. Analogous to (1) and (2), this is defined as We show that if G R is sufficiently restricted and a degree of social mobility is present, then there exists a unique stationary distribution for the state process, the distributional path of the state process under the optimal path converges globally to the stationary distribution, and the stationary distribution is ergodic. We also show that, under some mild additional conditions, the rate of convergence of marginal distributions to the stationary distribution is geometric, and that a version of the Central Limit Theorem is valid. The latter provides a rate of convergence for time series averages and asymptotic distributions for standard estimators.
Our study is related to Benhabib, Bisin, and Zhu (2015), who prove the existence of a heavy-tailed wealth distribution in an infinite horizon heterogeneous agent economy with capital income risk. In the process, they show that households facing a stochastic return on savings possess a unique optimal consumption policy characterized by the (boundary constraint-contingent) Euler equation, and that a unique and unbounded stationary distribution exists for wealth under this consumption policy. They assume isoelastic utility, constant discounting, and mutually independent, iid returns and labor income processes, both supported on bounded closed intervals with strictly positive lower bounds. We relax all of these assumptions. Apart from allowing more general utility and state dependent discounting, this permits such realistic features for 11 A well-known example of a computational technique that uses ergodicity can be found in Krusell and Smith (1998). On the estimation side see, for example, Hansen and West (2002).
household income as positive correlations between labor earnings and wealth returns (an extension that was suggested by Benhabib, Bisin, and Zhu (2015)), or time varying volatility in returns. 12 Another related paper is Chamberlain and Wilson (2000), which studies an income fluctuation problem with stochastic income and asset returns and obtains many significant results on asymptotic properties of consumption. Their study imposes relatively few restrictions on the wealth return and labor income processes. Our paper extends their work by allowing for random discounting, as well as dropping their boundedness restriction on the utility, which prevents their work from being used in many standard settings such as constant relative risk aversion. We also develop a set of new results on stability and ergodicity, as well as asymptotic normality of the wealth process.
Our optimality theory draws on techniques found in Li and Stachurski (2014), who show that the time iteration operator is a contraction mapping with respect to a metric that evaluates consumption differences in terms of marginal utility, while assuming a constant discount factor and constant rate of return on assets. 13 We show that these ideas extend to a setting where both returns and discount rates are stochastic and time varying. Our results on dynamics under the optimal policy have no counterparts in Li and Stachurski (2014).
In a similar vein, our work is related to several other papers that treat the standard income fluctuation problems with constant rates of return on assets and constant discount rates, such as Rabault (2002), Carroll (2004) and Kuhn (2013). While Carroll (2004) constructs a weighted supremum norm contraction and works with the Bellman operator, the other two papers focus on time iteration. In particular, Rabault (2002) exploits the monotonicity structure, while Kuhn (2013) applies a version of the Tarski fixed point theorem. Our techniques for studying optimality are close to those in Li and Stachurski (2014), as discussed above.
The rest of this paper is structured as follows. Section 2 formulates the problem and establishes optimality results. Sufficient conditions for the existence and uniqueness of optimal policies are discussed. Section 3 focuses on stochastic stability. Section 4 discusses our key conditions and how they can be checked. Section 5 provides a set of 12 Empirical motivation for these kinds of extensions can be found in numerous studies, including Guvenen and Smith (2014) and Fagereng, Guiso, Malacrino, and Pistaferri (2016a,b). 13 Coleman (1990) introduced the time iteration operator as a constructive method for solving stochastic growth models. It has since been used in Datta, Mirman, and Reffett (2002), Morand and Reffett (2003) and many other studies.
applications and Section 6 concludes. All proofs are deferred to the appendix. Code that generates our figures can be found at https://github.com/jstac/ifp_public.

The Income Fluctuation Problem and Optimality Results
This section formulates the income fluctuation problem we consider, establishes the existence, uniqueness and computability of a solution, and derives its properties.
2.1. Problem Statement. We consider a general income fluctuation problem, where a household chooses a consumption-asset path {(c t , a t )} to solve where u is the utility function, {β t } t≥0 is discount factor process with β 0 = 1, {R t } t≥1 is the gross rate of return on wealth, and {Y t } t≥1 is non-financial income. These stochastic processes obey where β, R and Y are measurable nonnegative functions and {Z t } t≥0 is an irreducible time-homogeneous Z-valued Markov chain taking values in finite set Z. Let P (z,ẑ) be the probability of transitioning from z toẑ in one step. The innovation processes {ε t }, {ζ t } and {η t } are iid independent and their supports can be continuous and vector-valued.
The next period value of a random variable X is typically denotedX. Expectation without a subscript refers to the stationary process, where Z 0 is drawn from its (necessarily unique) stationary distribution.
Example 2.1. The optimization problem stated above includes the problem faced by households in Cao (2018), which in turn builds on Krusell and Smith (1998). In that model, randomness in the discount factor process is driven by idiosyncratic preference shocks {i t }, while returns on assets fluctuate due to an aggregate shock process {s t } impacting productivity. Non-financial income is affected by both the idiosyncratic shocks and the aggregate shocks. His scenario fits our framework when Z t = (s t , i t ). The innovation vectors in (5) are not required.
2.2. Key Conditions. Our conditions for optimality are listed below. In what follows, G β is the asymptotic growth rate of the discount process as defined in (1).
Assumption 2.1 is a natural extension of the standard condition β < 1 from the constant discount case. If β t ≡ β for all t, then G β = β, as follows immediately from the definition. It is weaker than the obvious sufficient condition β t ≤β with probability one for some constantβ < 1, since in such a setting we have G β ≤β < 1.
In fact it cannot be significantly weakened, as the theorem shows.
Theorem 2.1 (Necessity of the discount condition). Let β t and u(Y t ) be positive with probability one for all t and all initial states z in Z. If, in this setting, we have G β ≥ 1, then the objective in (4) is infinite at every initial state (a, z).
The positivity assumed here may or may not hold in applications, but Theorem 2.1 shows that special conditions will have to be imposed on preferences if Assumption 2.1 fails. Put differently, allowing G β ≥ 1 is tantamount to allowing β ≥ 1 in the case when the discount rate is constant.
Next, we need to ensure that the present discounted value of wealth does not grow too quickly, which requires a joint restriction on asset returns and discounting. When {R t } and {β t } are constant at values R and β, the standard restriction from the existing literature is βR < 1. A generalization using G βR as defined in (2) is Assumption 2.2. The discount factor and return processes satisfy G βR < 1.
Finally, we impose routine technical restrictions on non-financial income. The second restriction is needed to exploit first order conditions.
Next we provide one example where Assumptions 2.1-2.3 are easily verified. More complex examples are deferred to Sections 4 and 5.
Example 2.2. Suppose, as in Benhabib et al. (2015), that there is a constant discount factor β < 1, utility is CRRA with γ ≥ 1, {R t } and {Y t } are iid, mutually independent, supported on bounded closed intervals of strictly positive real numbers, and, moreover, Assumptions 2.1-2.3 are all satisfied in this case. To see this, observe that G β = β in the constant discount case, so G β < 1 is immediate. The inequalities in (7) imply that βER t < 1, which in turn gives G βR < 1, since G βR = βER t in the iid setting of Benhabib et al. (2015). Thus, Assumptions 2.1-2.2 hold. Assumption 2.3 also holds because Y t is restricted to a compact subset of the positive reals.

Optimality: Definitions and Fundamental Properties.
To consider optimality, we temporarily assume that a 0 > 0 and set the asset space to (0, ∞). 14 The state space for the state process where {a t } is the asset path generated by (c, (a, z)). In the Appendix we show that V c is well-defined on S 0 . A feasible policy c * is called optimal if V c ≤ V c * on S 0 for any feasible policy c. A feasible policy is said to satisfy the first order optimality condition if for all (a, z) ∈ S 0 , and equality holds when c(a, z) < a. Noting that u is decreasing, the first order optimality condition can be compactly stated as for all (a, z) ∈ S 0 . A feasible policy is said to satisfy the transversality condition if, for all (a, z) Theorem 2.2 (Sufficiency of first order and transversality conditions). If Assumptions 2.1-2.3 hold, then every feasible policy satisfying the first order and transversality conditions is an optimal policy.
2.4. Existence and Computability of Optimal Consumption. Let C be the space of continuous functions c : S 0 → R such that c is increasing in the first argument, To compare two consumption policies, we pair C with the distance which evaluates the maximal difference in terms of marginal utility. While elements of C are not generally bounded, ρ is a valid metric on C . In particular, ρ is finite , and the last two terms are finite by (12). In Appendix B, we show that (C , ρ) is a complete metric space. The following proposition shows that, for any policy in C , the first order optimality condition (10) implies the transversality condition.
If c ∈ C and the first order optimality condition (10) holds for all (a, z) ∈ S 0 , then c satisfies the transversality condition. In particular, c is an optimal policy.
Inspired by Proposition 2.1, we aim to characterize the optimal policy as the fixed point of the time iteration operator T defined as follows: for fixed c ∈ C and (a, z) ∈ S 0 , the value of the image T c at (a, z) is defined as the ξ ∈ (0, a] that solves where ψ c is the function on The following theorem shows that the time iteration operator is an n-step contraction mapping on a complete metric space of candidate policies and its fixed point is the unique optimal policy. Theorem 2.3 (Existence, uniqueness and computability of optimal policies). If Assumptions 2.1-2.3 hold, then there exists an n in N such that T n is a contraction mapping on (C , ρ). In particular, (1) T has a unique fixed point c * ∈ C .
(2) The fixed point c * is the unique optimal policy in C . ( Part (3) shows that, under our conditions, the familiar time iteration algorithm is globally convergent, provided one starts with some policy in the candidate class C .
2.5. Properties of Optimal Consumption. In this section we study the properties of the optimal consumption function obtained in Theorem 2.3. Assumptions 2.1-2.3 are held to be true throughout. The following two propositions show the monotonicity of the consumption function, which is intuitive.
Proposition 2.2 (Monotonicity with respect to wealth). The optimal consumption and savings functions c * (a, z) and i * (a, z) := a − c * (a, z) are increasing in a.
Proposition 2.3 (Monotonicity with respect to income). If {Y 1t } and {Y 2t } are two income processes satisfying Y 1t ≤ Y 2t for all t and c * 1 and c * 2 are the corresponding optimal consumption functions, then c * 1 ≤ c * 2 pointwise on S 0 .
Under further assumptions we can show that the optimal policy is concave and asymptotically linear with respect to the wealth level.
Proposition 2.4 (Concavity and asymptotic linearity of consumption function). If for each z ∈ Z and c ∈ C that is concave in its first argument, Remark 2.1. Condition (17) imposes some concavity structure on utility. It holds for the constant relative risk aversion (CRRA) utility function as shown in Appendix B.
Proposition 2.4 states that c * (a, z) ≈ α(z)a + b(z) for some function b(z) when a is large. This provides justification for linearly extrapolating the policy functions when computing them at high wealth levels.
Together, parts (1) and (2) of Proposition 2.4 imply the linear lower bound c * (a, z) ≥ α(z)a, although they do not provide a concrete number for α(z). The following proposition establishes an explicit linear lower bound.
Proposition 2.5 (Linear lower bound on consumption). If there exists a nonnegative constants such that The second inequality in (19) restricts marginal utility derived from transferring wealth to the next period and then consuming versus consuming wealth today. The values can be clarified once primitives are specified, as the next example illustrates.
Example 2.3. Suppose that utility is CRRA, as in (18). If we now takē ands < 1, then the conditions of Proposition 2.5 hold. In particular, the second inequality in (19) holds, as follows directly from the definition ofs and u (x) = x −γ .
In the case of Benhabib et al. (2015), where the discount rate is constant and returns are iid, the expression in (20) reduces tos := (βER 1−γ t ) 1/γ . The requirements < 1 then reduces to βER 1−γ t < 1, which is one of their assumptions (see Example 2.2).

Stationarity and Ergodicity of Wealth
This section focuses on stationarity and ergodicity of wealth under the unique optimal policy c * obtained in Theorem 2.3. So that this policy exists, Assumptions 2.1-2.3 are always taken to be valid. We extend c * to S by setting c * (0, z) = 0 for all z ∈ Z and consider dynamics of (a t , Z t ) on S := R + × Z, the law of motion for which is 15 We adopt the convention 0 · ∞ = 0, so condition (19)  Let Q be the joint stochastic kernel of (a t , Z t ) on S. See Appendix A for this and other definitions of dynamic properties used below.
To obtain existence of a stationary distribution we need to restrict the asymptotic growth rate for asset returns G R defined in (3).
Below is one straightforward example of a setting where this holds, with more complex applications deferred to Sections 4-5.
Example 3.1. Assumption 3.1 holds in the setting of Benhabib et al. (2015). As shown in Example 2.3, withs := (βER 1−γ t ) 1/γ and the assumptions of Benhabib et al. (2015) in force, the conditions of (19) hold. Moreover, in their iid setting we have G R = ER t , sos G R < 1 reduces to (βER 1−γ t ) 1/γ ER t < 1. This is one of their conditions, as discussed in Example 2.2.
By Proposition 2.5, the values in Assumption 3.1 is an upper bound on the rate of savings. G R is an asymptotic growth rate for each unit of savings invested. If the product of these is less than one, then probability mass contained in the wealth distribution will not drift to +∞, which allows us to obtain the following result. 16 Theorem 3.1 (Existence of a stationary distribution). If Assumption 3.1 holds, then Q admits at least one stationary distribution on S.
The stationarity in Theorem 3.1 is required to establish existence of recursive equilibria in traditional heterogeneous agent models with idiosyncratic risk such as Huggett (1993) or Aiyagari (1994), as well as for models with aggregate risk, such as Krusell and Smith (1998). For a discussion of the latter case see Cao (2018).
While Assumption 3.1 implies existence of a stationary distribution, it is not in general sufficient for uniqueness or asymptotic stability, as well as closely related sample path properties such as ergodicity. For these additional properties to hold, we must impose sufficient mixing. In doing so, we consider the following two cases: 16 Assumption 3.1 is weaker than any restriction implying wealth is bounded from above-a common device for compactifying the state space and thereby obtaining a stationary distribution. Indeed, under many specifications of {Y t } and {R t } that fall within our framework, wealth of a given household can and will, over an infinite horizon, exceed any finite bound with probability one. See, for example, Benhabib, Bisin, and Zhu (2015), Proposition 6.
for all Borel sets A ⊂ R + and all z in Z.
Assumption 3.2. There exists az in Z such that P (z,z) > 0. Moreover, with y ≥ 0 defined as the greatest lower bound of the support of {Y t }, either • (Y1) holds and P{Y t = y | Z t =z} > 0, or • (Y2) holds and there exists a δ > y such that f (· |z) > 0 on (y , δ).
Assumption 3.2 requires that there is a positive probability of receiving low labor income at some relatively persistent state of the worldz. This is a mixing condition that enforces social mobility. The reason is that {Z t } is already assumed to be irreducible, soz is eventually visited by each household. For any such household, there is a positive probability of low labor income over a long period. Wealth then declines. In other words, currently rich households or dynasties will not be rich forever. This guarantees sufficient social mobility between rich and poor, generating an ergodic wealth distribution.
To state our uniqueness and stability results, let Q t be the t-step stochastic kernel, let · T V be total variation norm and let V (a, z) := a + m V , where m V is a constant to be defined in the proof. For any integrable real-valued function h on S, let , where E indicates, as usual, expectation under stationarity.
Theorem 3.2 (Uniqueness, stability, ergodicity and mixing). If Assumptions 3.1 and 3.2 hold, then (1) the stationary distribution ψ * of Q is unique and there exist constants λ < 1 and M < ∞ such that, for all (a, z) ∈ S, (2) For all z in Z and real-valued functions h on S such that E|h(a t , Z t )| < ∞, Part 1 of Theorem 3.2 states that the stationary distribution ψ * is unique and asymptotically attracting at a geometric rate. Part 2 states that the state process is ergodic, and hence long-run sample moments for individual households coincide with cross-sectional moments. The notion of mixing discussed in Part 3 is defined in the appendix. It states that social mobility holds asymptotically and mixing occurs at a geometric rate, although the rate may be arbitrarily slow. This mixing is enough to provide a Central Limit Theorem for the state process, which is the second claim in Part 3.

Testing the Growth Conditions
The three key conditions in the paper are the restrictions on the growth rates G β , G βR and G R , with the first two required for optimality and the last need for stationarity (see Assumptions 2.1, 2.2 and 3.1 respectively). In this section we explore the restrictions implied by these conditions. We begin with the following result, which yields a straightforward method for computing these growth rates.
Lemma 4.1 (Long-run growth rates and spectral radii). Let ϕ t = ϕ(Z t , ξ t ), where ϕ is a nonnegative measurable function and {ξ t } is an iid sequence with marginal distribution π. In this setting we have and r(L ϕ ) is the spectral radius of the matrix defined by The matrix L ϕ is expressed as a function on Z × Z in (24) but can be represented in traditional matrix notation by enumerating Z. 17 17 Specifically, if Z := {z 1 , . . . , z N }, then L ϕ = P D ϕ where P is, as before, the transition matrix for the exogenous state, and D ϕ := diag (E z1 ϕ, . . . , E z N ϕ) when E z ϕ := E z ϕ(z,ξ). In what follows, D β , D R and D βR are defined analogously to D ϕ .
What factors determine the long-run average growth rates embedded in our assumptions, such as G β or G R ? Lemma 4.1 tells us how to compute these values for a given specification of dynamics, but how should we understand them intuitively and what factors determine their size? To address these questions, let us consider an AR(1) discount factor process, which has been adopted in several recent quantitative studies (see, e.g., Hubmer, Krusell, and Smith (2018) or Hills and Nakata (2018)). In particular, suppose that the state process follows a discretized version of and β t = Z t . (The discretization implies that β t is always positive.) To simplify interpretation, the process (25) is structured so that the stationary distribution of {Z t } is N (µ, σ 2 ). We use Rouwenhorst (1995)'s method to discretize {Z t } and then calculate G β using Lemma 4.1, studying how G β is affected by the parameters in (25).
Since β t = Z t for all t, the structure of (25) implies that µ is the long-run unconditional mean of {β t }. It can therefore be set to standard calibrated value for the discount factor, such as 0.99 from Krusell and Smith (1998). What we wish to understand is how the remaining parameters ρ and σ affect the value of G β . While no closed form expression is available in this case, Figure 1 sheds some light by providing a contour plot of G β over a set of (ρ, σ) pairs. The figure shows that G β grows with both the persistence term ρ and volatility term σ. In particular, the condition G β < 1 fails when the persistence and volatility of the discount factor process are sufficiently high. This is because G β is the limit of (E n t=1 β t ) 1/n and, for positive random variables, sequence of large outcomes have a strong compounding effect on their product. High volatility and high persistence reinforce this effect.
This discussion has focused on G β but similar intuition applies to both G R and G βR . If β t and R t are both increasing functions of the state process, then these asymptotic growth rates also increase with greater persistence and volatility in the state process, as well as higher unconditional mean. The next section further illustrates these points.

Application: Stochastic Volatility and Mean Persistence
We showed in Examples 2.2, 2.3 and 3.1 that, in the setting of Benhabib et al. (2015), where the discount factor is constant and returns and labor income are iid,  (1) discounting of Y t is finite or Y t has a positive density, say, then the conditions of Theorem 3.2 also hold and the stationary solution is ergodic, geometrically mixing and its time series averages are asymptotically normal.
While some of the results just stated are new, we can obtain more important results by relaxing the iid restrictions on financial and non-financial returns. For example, we can allow for both mean persistence and time varying volatility in returns on assets, the importance of which for wealth dynamics has been highlighted in Fagereng et al. (2016a) and other recent studies. (We do not, however, intend to reproduce or extend the important results on Pareto tails of the stationary wealth distribution obtained by Benhabib et al. (2015), which are beyond the scope of this study.) To illustrate, we continue in the setting of Benhabib et al. (2015) but now shift returns on assets to where {ζ t } is iid and standard normal and {µ t } and {σ t } are finite-state Markov chains, discretized from Innovations are iid and standard normal. Using the data in Fagereng et al. (2016b) on Norwegian financial returns over 1993-2003, we estimate these AR(1) models to obtainμ = 0.0281, ρ µ = 0.5722, δ µ = 0.0067,σ = −3.2556, ρ σ = 0.2895 and δ σ = 0.1896. Based on this calibration, the stationary mean and standard deviation of {R t } are around 1.03 and 4%, respectively.
To distinguish the effects of stochastic volatility and mean persistence, we consider two subsidiary models. The first reduces {µ t } to its stationary meanμ, while the second reduces {σ t } to its stationary meanσ := eσ +δ 2 σ /2(1−ρ 2 σ ) . In summary, We set β = 0.95 and γ = 1.5. To test the stability properties of Model I, we set K = 5 and explore a neighborhood of the calibrated (ρ σ , δ σ ) values, while in Model II, we set M = 5 and do likewise for (ρ µ , δ µ ) pairs. In each scenario, other parameters are fixed to the benchmark. The results are shown in Figures 2 and 3.
In part (a) of each figure, we see that G βR is increasing in the persistence and volatility parameters of the state process. The intuition behind this feature was explained in Section 4 for the case of G β and is similar here. (Note that G βR = βG R in the present case, since β t ≡ β is a constant, so G βR has the same shape as G R in terms of contours.) The dots in the figures show that G βR < 1 at the estimated parameter values.
Part (b) of each figure shows the set of parameters under which the model is globally stable and ergodic. The stability threshold is the boundary of the set of parameter pairs that produce max{G βR ,s,sG R } < 1, wheres is given by (20). For such pairs, Assumptions 2.2 and 3.1 both hold, so the conditions of Theorems 3.1-3.2 are satisfied.
(We are continuing to suppose that Y t is finite or has a positive density, so that Assumption 3.2 holds. Assumptions 2.1 and 2.3 are always valid in the current setting). Observe that the estimated parameter values (dot points) lie inside the stable set.

Conclusion
We studied an updated version of the income fluctuation problem, the "common ancestor" of modern macroeconomic theory (Ljungqvist and Sargent (2012)   Our work leaves several open questions. For example, in the context of an infinite horizon heterogeneous agent economy, Benhabib, Bisin, and Zhu (2015) prove that capital income risk drives the heavy upper tail observed in the distribution of wealth. While capital income risk is iid in that study, the theory presented above allows researchers to model households facing a richer stochastic environment, with stochastic volatility in returns on savings and time-varying means. If capital income risk generates inequality, then perhaps time-varying risk can explain changes in inequality. Exploring such ideas requires embedding our household problem in a larger model and is left for future research.

Appendix A. Preliminaries
Given a topological space T, let B(T) be the Borel σ-algebra and P(T) be the probability measures on B(T). A stochastic kernel Π on T is a map Π : is a probability measure on B(T) for each x ∈ T. For all t ∈ N, x, y ∈ T and B ∈ B(T), we define Π 1 := Π and Π t (x, B) := Π t−1 (y, B)Π(x, dy). Furthermore, for all µ ∈ P(T), let (µΠ t )(B) : is continuous on T whenever h is bounded and continuous on T. We call ψ ∈ P(T) stationary for Π if ψΠ = ψ.
A sequence {µ n } ⊂ P(T) is called tight, if, for all ε > 0, there exists a compact K ⊂ T such that µ n (T\K) ≤ ε for all n. A stochastic kernel Π is called bounded in probability if the sequence {Q t (x, ·)} t≥0 is tight for all x ∈ T. Given µ ∈ P(T), we define the total variation norm µ T V := sup g:|g|≤1 g dµ . Given any measurable map V : T → [1, ∞), we say that Π is V -geometrically mixing if there exist constants M < ∞ and λ < 1 such that, for all x ∈ T and t ≥ 0, the corresponding Markov process Below we use (Ω, F , P) to denote a fixed probability space on which all random variables are defined. E is expectations with respect to P. The state process {Z t } and the innovation processes {ε t }, {ζ t } and {η t } introduced in (5) live on this space. In what follows, {Z t } is a stationary version of the chain, where Z 0 is drawn from its unique stationary distribution-henceforth denoted π Z . The marginal distributions of the innovations are denoted by π ε , π ζ and π η respectively. We let {F t } be the natural filtration generated by {Z t } and the three innovation processes. P z conditions on Z 0 = z and E z is expectation under P z .
We first prove Lemma 4.1, since its implications will be used immediately below. In the proof, we consider the matrix L ϕ as a linear operator on R Z and identify vectors in R Z with real-valued functions on Z.
Proof of Lemma 4.1. A proof by induction confirms that, for any function h ∈ R Z , where L n ϕ is the n-th composition of the operator L ϕ with itself (or, equivalently, the n-th power of the matrix L ϕ ). The positivity of L ϕ and Theorem 9.1 of Krasnosel'skii et al. (2012) imply that r(L ϕ ) = lim n→∞ L n ϕ h 1/n when · is any norm on R Z and h is everywhere positive on Z. With h ≡ 1 and f = E|f(Z 0 )|, this becomes where the second equality is due to (27) and h = 1 and the third is by the law of iterated expectations.
Lemma A.1. Let {ϕ t } and G ϕ be as defined in Lemma 4.1. If G ϕ < 1, then there exists an N in N and a δ < 1 such that max z∈Z E z n t=1 ϕ t < δ n whenever n ≥ N .
Proof. Recalling from the proof of Lemma 4.1 that r(L ϕ ) = lim n→∞ L n ϕ h 1/n when · is any norm on R Z and h is everywhere positive on Z, we can again take h ≡ 1 but now switch to f = max z∈Z |f (z)|, so that (28) Since r(L ϕ ) = G ϕ and G ϕ < 1, the claim in Lemma A.1 now follows.

Appendix B. Proof of Section 2 Results
Proof of Theorem 2.1. Pick any a ≥ 0 and z ∈ Z. Since c t = Y t for all t is dominated by a feasible consumption path, monotonicity of u and the law of iterated expectations give where h(Z t ) := E Zt u(Y ) and the Monotone Convergence Theorem has been employed to pass the expectation through the sum. In view of (27) and β 0 = 1, we then have By the assumed almost sure positivity of β t and the irreducibility of P , the matrix L β is irreducible. Hence, by the Perron-Frobenius Theorem, we can choose an everywhere positive eigenfunction e such that L β e = r(L β )e. By the everywhere positivity of u(Y t ), the function h is everywhere positive on Z, and hence we can choose α > 0 such that e α := αe is less than h pointwise on Z. We then have By lemma 4.1 we know that r(L β ) ≥ 1, and since α and e are positive, this expression is infinite. Returning to (30), we see that the value function is infinite at our arbitrarily chosen pair (a, z).
For the rest of this section we suppose that Assumptions 2.1-2.3 hold.
are finite, as are the constants M 3 = max z∈Z E z Y and M 4 = max z∈Z E z u (Y ).
Proof. That M 1 and M 2 are finite follows directly from Lemma A.1, with ϕ t = β t and ϕ t = β t R t respectively. Regarding M 3 , Assumption 2.3 states that EY < ∞. By the Law of Iterated Expectations, we can write this as z∈Z E z Y π Z (z) < ∞. As {Z t } is irreducible, we know that π Z is positive everywhere on Z. Hence, M 3 < ∞ must hold. The proof of M 4 < ∞ is similar.

Then the Monotone Convergence Theorem and the Markov property imply that
By Lemma B.1, we now have, for all (a, z) ∈ S 0 , Applying Lemma B.1 again gives M (a, z) < ∞, as was to be shown.
Proof. By the assumptions on the utility function, there exists a constant B ∈ R + such that u(c) ≤ c + B, and hence V c (a, z) ≤ E a,z The last term is finite by Lemma A.1.
Proof of Thoerem 2.2. The proof is a long but relatively straightforward extension of Theorem 1 of Benhabib et al. (2015) and thus omitted. A full proof is available from the authors upon request.
Proof. The proof is a straightforward extension of Proposition 4.1 of Li and Stachurski (2014) and thus omitted. A full proof is available from the authors upon request.
Proof of Proposition 2.1. Let c be a policy in C satisfying (10). To show that any asset path generated by c satisfies the transversality condition (11), observe that, by condition (12), we have Regarding the first term on the right hand side of (33), fix A > 0 and observe that with probability one, whereã t is the maximal path defined in (31). We then have By Lemma B.1, we have and the last expression converges to zero as t → ∞ by Lemma A.1. The second term in (34) also converges to zero by Lemma B.2. Hence E a,z t i=0 β i u (a t )a t → 0 as t → ∞, which, combined with (33) and another application of Lemma B.2, gives our desired result. Proposition B.3. For all c ∈ C and (a, z) ∈ S 0 , there exists a unique ξ ∈ (0, a] that solves (14).
Proof. Fix c ∈ C and (a, z) ∈ S 0 . Because c ∈ C , the map ξ → ψ c (ξ, a, z) is increasing. Since ξ → u (ξ) is strictly decreasing, the equation (14) can have at most one solution. Hence uniqueness holds.

For part (a), it suffices to show that
is continuous on (0, a]. To this end, fix ξ ∈ (0, a] and ξ n → ξ. By (32) we havê The last term is integrable, as follows easily from Lemma B.1. Hence the dominated convergence theorem applies. From this fact and the continuity of c, we obtain g(ξ n ) → g(ξ). Hence, ξ → ψ c (ξ, a, z) is continuous.
Part (b) clearly holds, since u (ξ) → ∞ as ξ → 0 and ξ → ψ c (ξ, a, z) is increasing and always finite (since it is continuous as shown in the previous paragraph). Part (c) is also trivial (just set ξ = a).
Proposition B.4. We have T c ∈ C for all c ∈ C .
Proof. Fix c ∈ C and let g (ξ, a, z) : Step 1. We show that T c is continuous. To apply a standard fixed point parametric continuity result such as Theorem B.1.4 of Stachurski (2009), we first show that ψ c is jointly continuous on the set G defined in (15). This will be true if g is jointly continuous on G. For any {(ξ n , a n , z n )} and (ξ, a, z) in G with (ξ n , a n , z n ) → (ξ, a, z), we need to show that g(ξ n , a n , z n ) → g(ξ, a, z). To that end, we define whereβ := β(Ẑ,ε),R := R(Ẑ,ζ) andŶ := Y (Ẑ,η) as defined in (5). Then h 1 and h 2 are continuous in (ξ, a,Ẑ) by the continuity of c and nonnegative by (35).
This implies that The function g is then continuous, since the above inequality is equivalent to the statement lim inf n→∞ g(ξ n , a n , z n ) ≥ g(ξ, a, z) ≥ lim sup n→∞ g(ξ n , a n , z n ). Hence, ψ c is continuous on G, as was to be shown. Moreover, since ξ → ψ c (ξ, a, z) takes values Step 2. We show that T c is increasing in a. Suppose that for some z ∈ Z and a 1 , a 2 ∈ (0, ∞) with a 1 < a 2 , we have ξ 1 := T c(a 1 , z) > T c(a 2 , z) =: ξ 2 . Since c is increasing in a by assumption, ψ c is increasing in ξ and decreasing in a. Then u (ξ 1 ) < u (ξ 2 ) = ψ c (ξ 2 , a 2 , z) ≤ ψ c (ξ 1 , a 1 , z) = u (ξ 1 ). This is a contradiction.
Step 3. We have shown in Proposition B.3 that T c(a, z) ∈ (0, a] for all (a, z) ∈ S 0 .
Step 4. We show that u for all (a, z) ∈ S 0 . The right hand side is easily shown to be finite via Lemma B.1.
Next, we aim to prove Theorem 2.3. Recall H defined in the proof of Proposition B.2. Given h ∈ H , letT h be the function mapping (a, z) ∈ S 0 into the κ that solves The next lemma implies thatT is a well-defined self-map on H , as well as topologically conjugate to T under the bijection H : C → H defined by Hc := u • c. Proof. Pick any c ∈ C and (a, z) ∈ S 0 . Let ξ := T c(a, z), then ξ solves We need to show that HT c andT Hc evaluate to the same number at (a, z). In other words, we need to show that u (ξ) is the solution to But this is immediate from (37). Hence, we have shown thatT H = HT on C . Since H : C → H is a bijection, we haveT = HT H −1 . Since in addition T : C → C by Proposition B.4, we haveT : H → H . This concludes the proof.
Proof. Let h 1 , h 2 be functions in H with h 1 ≤ h 2 . Suppose to the contrary that there exists (a, z) ∈ S 0 such that κ 1 :=T h 1 (a, z) >T h 2 (a, z) =: κ 2 . Since functions in H are decreasing in the first argument, we have This is a contradiction. Hence,T is order preserving.
Lemma B.5. There exists an n ∈ N and θ < 1 such thatT n is a contraction mapping of modulus θ on (H , d ∞ ).
Proof. SinceT is order preserving and H is closed under the addition of nonnegative constants, based on Blackwell (1965), it remains to verify the existence of n ∈ N and θ < 1 such thatT n (h + γ) ≤T n h + θγ for all h ∈ H and γ ≥ 0. By Lemma A.1 and Assumption 2.2, it suffices to show that for all k ∈ N and (a, z) ∈ S 0 , we havẽ Fix h ∈ H , γ ≥ 0, and let h γ (a, z) := h(a, z) + γ. By the definition ofT , we havẽ Here, the first inequality is elementary and the second is due to the fact that h ≤ h γ andT is order preserving. Hence,T (h+γ)(a, z) ≤T h(a, z)+γE z β 1 R 1 and (38) holds for k = 1. Suppose (38) holds for arbitrary k. It remains to show that it holds for k + 1. For z ∈ Z, define f (z) := γE z β 1 R 1 · · · β k R k . By the induction hypothesis, the monotonicity ofT and the Markov property, Hence, (38) is verified by induction. This concludes the proof.
Proof of Theorem 2.3. Let n and θ be as in Lemma B.5. In view of Propositions 2.1, B.2 and B.4, to show that T n is a contraction and verify claims (1) Hence, T n is a contraction and claims (1)-(3) are verified.
Our next goal is to prove Proposition 2.2. To begin with, we define Lemma B.6. C is a closed subset of C , and T c ∈ C for all c ∈ C .
Proof. To see that C is closed, for a given sequence {c n } in C and c ∈ C with ρ(c n , c) → 0, we need to show that c ∈ C . This obviously holds since a → a−c n (a, z) is increasing for all n, and, in addition, ρ(c n , c) → 0 implies that c n (a, z) → c(a, z) for all (a, z) ∈ S 0 .
Fix c ∈ C . We now show that ξ := T c ∈ C . Since ξ ∈ C by Proposition B.4, it remains to show that a → a − ξ(a, z) is increasing. Suppose the claim is false, then there exist z ∈ Z and a 1 , a 2 ∈ (0, ∞) such that a 1 < a 2 and a 1 −ξ(a 1 , z) > a 2 −ξ(a 2 , z).
Proof of Proposition 2.2. Since T is a self-map on the closed subset C by Lemma B.6 and T c * = c * , we have c * ∈ C by Theorem 2.3. Hence, the stated claims hold.
Proof of Proposition 2.3. Let T j be the time iteration operator for the income process j established in Proposition B.4. It suffices to show T 1 c ≤ T 2 c for all c ∈ C . To see this, note that by Lemma B.4, we have T j c 1 ≤ T j c 2 whenever c 1 ≤ c 2 . Therefore if T 1 c ≤ T 2 c for all c ∈ C , we obtain T 1 c 1 ≤ T 1 c 2 ≤ T 2 c 2 . Iterating this starting from any c ∈ C , by Theorem 2.3, it follows that c * 1 = lim n→∞ (T 1 ) n c ≤ lim n→∞ (T 2 ) n c = c * 2 , completing the proof.
To show that T 1 c ≤ T 2 c for any c ∈ C , take any (a, z) ∈ S 0 and define ξ j = (T j c)(a, z). To show ξ 1 ≤ ξ 2 , suppose on the contrary that ξ 1 > ξ 2 . Since c is increasing in a and u < 0 (hence u is decreasing), it follows from the definition of the time iteration operator in (14)-(16), Y 1 ≤ Y 2 , u < 0 and the monotonicity of c ∈ C that which is a contradiction.
To prove Proposition 2.4, we need several lemmas.
Lemma B.7. For all c ∈ C , there exists a thresholdā c (z) such that T c(a, z) = a if and only if a ≤ā c (z). In particular, there exists a thresholdā(z) such that c * (a, z) = a if and only if a ≤ā(z).
Proof. Recall that, for all c ∈ C , ξ(a, z) := T c(a, z) solves For each z ∈ Z and c ∈ C , definē Let a ≤ā c (z). We claim that ξ(a, z) = a. Suppose to the contrary that ξ(a, z) < a. Then (u • ξ)(a, z) > u (a). In view of (39), we have From this we get a >ā c (z), which is a contradiction. Hence, ξ(a, z) = a.
The second claim follows immediately from the first claim and the fact that c * is the unique fixed point of T in C .
Consider a subset C 1 defined by C 1 := {c ∈ C : a → c(a, z) is concave for all z ∈ Z}.
Lemma B.8. C 1 is a closed subset of C , and, T c ∈ C 1 for all c ∈ C 1 .
Proof. The first claim is immediate because limits of concave functions are concave.
To prove the second claim, fix c ∈ C 1 . We have T c ∈ C since T is a self-map on C . It remains to show that a → ξ(a, z) := T c(a, z) is concave for all z ∈ Z. Given z ∈ Z, Lemma B.7 implies that ξ(a, z) = a for a ≤ā c (z) and that ξ(a, z) < a for a >ā c (z). Since in addition a → ξ(a, z) is continuous and increasing, to show the concavity of ξ with respect to a, it suffices to show that a → ξ(a, z) is concave on (ā c (z), ∞).
Proof of Proposition 2.4. By Theorem 2.3, T : C → C is a contraction mapping with unique fixed point c * . Since C 1 is a closed subset of C and T : C 1 → C 1 by Lemma B.8, we know that c * ∈ C 1 . The first claim is verified. Regarding the second claim, note that c * ∈ C 1 implies that a → c * (a, z) is increasing and concave for all z ∈ Z. Hence, a → c * (a, z)/a is a decreasing function for all z ∈ Z. Since 0 ≤ c * (a, z) ≤ a for all (a, z) ∈ S 0 , α(z) := lim a→∞ c * (a, z)/a is well-defined and α(z) ∈ [0, 1].
To prove Proposition 2.5, lets be as in (19) and define Lemma B.9. C 2 is a closed subset of C , and T c ∈ C 2 for all c ∈ C 2 .
Proof. To see that C 2 is closed, for a given sequence {c n } in C 2 and c ∈ C with ρ(c n , c) → 0, we need to verify that c ∈ C 2 . This obviously holds since c n (a, z)/a ≥ 1 −s for all n and (a, z) ∈ S 0 , and, on the other hand, ρ(c n , c) → 0 implies that c n (a, z) → c(a, z) for all (a, z) ∈ S 0 .
We next show that T is a self-map on C 2 . Fix c ∈ C 2 . We have T c ∈ C since T is a self-map on C . It remains to show that ξ := T c satisfies ξ(a, z) ≥ (1 −s)a for all (a, z) ∈ S 0 . Suppose ξ(a, z) < (1 −s)a for some (a, z) ∈ S 0 . Then Since u ((1 −s)a) > u (a) and c ∈ C 2 , this implies that which contradicts (19) since ((1 −s)a, z) ∈ S 0 . Hence, ξ(a, z)/a ≥ 1 −s for all (a, z) ∈ S 0 and we conclude that T c ∈ C 2 .
Proof of Proposition 2.5. We have shown in Theorem 2.3 that T is a contraction mapping on the complete metric space (C , ρ), with unique fixed point c * . Since in addition C 2 is a closed subset of C and T C 2 ⊂ C 2 by Lemma B.9, we know that c * ∈ C 2 . The stated claim is verified.

Appendix C. Proof of Section 3 Results
As before, Assumptions 2.1-2.3 are in force. Notice that Assumption 2.2, Assumption 3.1 and Lemma A.1 imply existence of an n in N such that Lemma C.1. For all (a, z) ∈ S, we have sup t≥0 E a,z a t < ∞.
Proof. Since c * (0, z) = 0, Proposition 2.5 implies that c * (a, z) ≥ (1 −s)a for all (a, z) ∈ S. For all t ≥ 1, we have t = kn + j in general, where the integers k ≥ 0 and j ∈ {0, 1, . . . , n − 1}. Using these facts and (4), we have: with probability one. Taking expectations of the above while noting that M 0 := max 1≤ ≤n, z∈Z E z t=1 R t < ∞ by Assumption 3.1 and Lemma A.1, we have or all (a, z) ∈ S and t ≥ 0. Here we have used M 3 in Lemma B.1 and the Markov property. Hence, sup t≥0 E a,z a t < ∞ for all (a, z) ∈ S, as was claimed.
A function w * : S → R + is called norm-like if all its sublevel sets (i.e., sets of the form {x ∈ S : w(x) ≤ b}, b ∈ R + ) are precompact in S (i.e., any sequence in a given sublevel set has a subsequence that converges to a point of S).
Proof of Theorem 3.1. Based on Lemma D.5.3 of Meyn and Tweedie (2009), a stochastic kernel Q is bounded in probability if and only if for all x ∈ S, there exists a norm-like function w * x : S → R + such that the (Q, x)-Markov process {X t } t≥0 satisfies lim sup t→∞ E x [w * x (X t )] < ∞. Fix (a, z) ∈ S. Since Z is finite, P is bounded in probability. Hence, there exists a norm-like function w : Z → R + such that lim sup t→∞ E z w(Z t ) < ∞. Then w * : S → R + defined by w * (a 0 , Z 0 ) := a 0 + w(Z 0 ) is a norm-like function on S. The stochastic kernel Q is then bounded in probability since Lemma C.1 implies that lim sup t→∞ E a,z w * (a t , Z t ) ≤ sup t≥0 E a,z a t + lim sup t→∞ E z w(Z t ) < ∞. Regarding existence of stationary distribution, since P is Feller (due to the finiteness of Z), whenever z n → z, the product measure satisfies Since in addition c * is continuous, a simple application of the generalized Fatou's lemma of Feinberg, Kasyanov, and Zadoianchuk (2014) (Theorem 1.1) shows that the stochastic kernel Q is Feller. Moreover, since Q is bounded in probability, based on the Krylov-Bogolubov theorem (see, e.g., Meyn and Tweedie (2009) That is, for all (a, z) ∈ S, we have P a,z (∪ t≥0 {c t = a t }) > 0.
Recall that R m paired with its Euclidean topology is a second countable topological space (i.e., its topology has a countable base). Since R + and Z are respectively Borel subsets of R and R m paired with the relative topologies, they are also second countable. Hence, S := R + × Z satisfies B(S) = B(R + ) ⊗ B(Z) (see, e.g., page 149, Theorem 4.44 of Aliprantis and Border (2006)). Recall (22). With slight abuse of notation, in proofs, we use f to denote the density of {Y t } in both cases (Y1) and (Y2) and write dy = ν(dy), where ν is the related measure. Specifically, ν is the Lebesgue measure when (Y2) holds. Moreover, Let ϑ be the counting measure.
Recallz ∈ Z and the greatest lower bound y ≥ 0 of the support of {Y t } given by Assumption 3.2. Letp := P (z,z). Thenp > 0 by Assumption 3.2.
Note that the set (t) := {c t < a t } ∪ (∪ t i=0 {Z i =z}) can be written as Assumption 3.2 then implies that, for all t ≥ 0, Let n and θ be defined by (43) and let t = kn + 1. Similar to the proof of Lemma B.7, we can show that, with probabilityp t > 0, for some constant M ∈ R + . Since θ ∈ (0, 1) and (u • c * )(a,z) > 0, Lemma B.1 implies that there exists N ∈ N such that As a result, we have (u • c * )(a,z) < (u • c * )(a,z) with probabilityp N n+1 > 0. This is a contradiction. Hence the stated claim is verified.
Note that, by Assumption 3.2, f (y | z )P (z, z ) > 0 whenever (y , z ) ∈ D. Since in addition ϕ (45) and (46) imply that Therefore, we have shown that any measurable subset with positive ϕ measure can be reached in finite time with positive probability, i.e., {(a t , Z t )} is ϕ-irreducible. Based on Proposition 4.2.2 of Meyn and Tweedie (2009), there exists a maximal probability measure ψ on B(S) such that {(a t , Z t )} is ψ-irreducible.
Proof. Suppose (Y1) holds andā(z) < y . Then, by Lemma B.7, for all t ∈ N, Hence, for all a ∈ (ā(z), ∞), c 0 < a 0 by Lemma B.7, and, for all t ∈ N, where the last equality follows from (21a), which implies that a t ≥ Y t ≥ y with probability one. This is contradicted with Lemma C.3.
Proof. By the definition of strong aperiodicity, we need to show that there exists a v 1 -small set D 1 with v 1 (D 1 ) > 0, i.e., there exists a nontrivial measure v 1 on B(S) and a subset D 1 ∈ B(S) such that v 1 (D 1 ) > 0 and inf (a,z)∈D 1 Q ((a, z), A) ≥ v 1 (A) for all A ∈ B(S).
Lemma C.7. Let h : S → R + be an integrable map such that a → h(a, z) is decreasing for all z ∈ Z. Then, for all t ∈ N and z ∈ Z, the map a → (a, z, t) := h(a , z )Q t ((a, z), d(a , z )) is decreasing.
Since a → h(a, z) is decreasing, and by Proposition 2.2 and (21a), the optimal asset accumulation path a t+1 is increasing in a t with probability one, we know that a → h(a , z )F (da | a, z, z ) is decreasing for all z ∈ Z. Thus, a → (a, z, 1) is decreasing. The claim holds for t = 1. Suppose this claim holds for arbitrary t, it remains to show that it holds for t + 1. Note that (a, z, t + 1) = h(a , z )Q t ((a , z ), d(a , z ))Q((a, z), d(a , z )) = (a , z , t)Q((a, z), d(a , z )).
Since a → (a , z , t) is decreasing for all z ∈ Z, based on the induction argument, a → (a, z, t + 1) is decreasing. The stated claim then follows.
Since a → m(z )1{a ≤ā(z ), z =z} is decreasing for all z ∈ Z, by Lemma C.7, Note that v N is a nontrivial measure on B(S) since (49) implies that v N (S) > 0. Furthermore, since (a, z) is chosen arbitrarily, the above inequality implies that (50) holds. We have shown that B is a v N -small set, and hence a petite set. Since finite union of petite sets is petite for ψ-irreducible chains (see, e.g., Proposition 5.5.5 of Meyn and Tweedie (2009)), the set [0, d] × Z must also be petite.
Proof. Since c * (a, z) ≥ (1 −s)a by Proposition 2.5 and M 0 := max z∈Z E zR < ∞ by Assumption 3.1 and Lemma A.1, by Lemma B.1 and the Markov property, E a,z a n ≤s n E z R n · · · R 1 a + n t=1s n−t E z R n · · · R t+1 Y t ≤ γa + n t=1s n−t E z Y t E Zt R t+1 · · · R n ≤ γa + n t=1s n−t M n−t 0 M 3 .
Proof of Theorem 3.2. Claim (1) can be proved by applying Theorem 19.1.3 (or a combination of Proposition 5.4.5 and Theorem 15.0.1) of Meyn and Tweedie (2009). The required conditions in those theorems have been established by Lemmas C.4, C.6, C.8 and C.9 above. Regarding claim (2), Lemmas C.8 and C.9 imply that E a,z V (a n , Z n )−V (a, z) ≤ −n+b1{(a, z) ∈ B} for all (a, z) ∈ S, where B := [0, d]×Z is petite. Since in addition {(a t , Z t )} is ψ-irreducible by Lemma C.4, Theorem 19.1.2 of Meyn and Tweedie (2009) implies that {(a t , Z t )} is a positive Harris chain. Claim (2) then follows from Theorem 17.1.7 of Meyn and Tweedie (2009).
To verify claim (3), since we have shown that Φ := {(a t , Z t )} is positive Harris with stationary distribution ψ * , based on Theorem 16.1.5 and Theorem 17.5.4 of Meyn and Tweedie (2009), it suffices to show that Q is V -uniformly ergodic. Let Φ n be the n-skeleton of Φ (see page 62 of Meyn and Tweedie (2009)). Then Φ n is ψ-irreducible and aperiodic by Proposition 5.4.5 of Meyn and Tweedie (2009). Theorem 16.0.1 of Meyn and Tweedie (2009) and Lemmas C.8-C.9 then imply that Φ n is V -uniformly ergodic, and, there exists N ∈ N such that |||Q nN − 1 ⊗ ψ * ||| V < 1, where µ V := sup g:|g|≤V | g dµ| for µ ∈ P(S) and, for all t ∈ N, To show that Q is V -uniformly ergodic, by Theorem 16.0.1 of Meyn and Tweedie (2009), it remains to verify: |||Q t − 1 ⊗ ψ * ||| V < ∞ for t ≤ nN . This obviously holds since, by the proof of Lemma C.9, there exist L 0 , L 1 ∈ R such that, for all t ∈ N, |f (a , z )|Q t ((a, z), d(a , z )) V (a, z) Hence, Q is V -uniformly ergodic and claim (3) follows. The proof is now complete.