The Income Fluctuation Problem and the Evolution of Wealth

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.\footnote{For example, in a study based on capital income data, saez2016wealth 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 Pareto1896LaCourbe, davies2000distribution, BenhabibBisin2018, vermeulen2018fat or references therein.} This matters not only for its direct impact on taxation and redistribution 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.\footnote{One analysis of the two-way interactions between inequality and political decision making can be found in acemoglu2002political. glaeser2003injustice 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, brinca2016fiscal find strong correlations between wealth inequality and the magnitude of fiscal multipliers, while bhandari2018inequality study the connection between fiscal-monetary policy, business cycles and inequality. ahn2018inequality discuss the impact of distributional properties on macroeconomic aggregates.}

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., schechtman1976income, SchechtmanEscudero1977, deaton1992behaviour, carroll1997buffer, or accikgoz2018existence), 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., benhabib2015wealth, BenhabibBisinLuo2017, stachurski2019impossibility).\footnote{Also related is the recent experimental study of epper2018time, 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 lawrance1991poverty.} Moreover, models with time-varying discount rates and returns on assets are at the forefront of recent quantitative analysis of wealth and inequality.\footnote{For a recent quantitative study see, for example, hubmer2018comprehensive, where returns on savings and discount rates are both state dependent (as is labor income). kaymak2018accounting find that asset return heterogeneity is required to match the upper tail of the wealth distribution. }

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.

One difficulty is that state-dependent discounting takes us beyond the bounds of traditional dynamic programming theory. This matters little if there exists some constant $\bar \beta < 1$ such that the discount process $\{\beta_t\}$ satisfies $\beta_t \leq \bar \beta$ for all $t$ with probability one, since, in this case, a standard contraction mapping argument can still be applied (see, e.g., miao2006competitive or cao2020recursive). However, recent quantitative studies extend beyond such settings. For example, AR(1) specifications are increasingly common, in which case the support of $\beta_t$ is unbounded above at every point in time.\footnote{See, for example, hills2018fiscal, hubmer2018comprehensive or schorfheide2018identifying.} Even if discretization is employed, the outcome $\beta_t \geq 1$ 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.\footnote{See, for example, loewenstein1991negative and loewenstein1991workers.} Do there exist conditions on $\{\beta_t\}$ that allow for $\beta_t \geq 1$ 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, benhabib2015wealth and other related studies argue convincingly that such outcomes are a key causal mechanism behind the heavy tail of the current distribution of wealth.\footnote{One related study is benhabib2011distribution, 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, Toda2014JET, TodaWalsh2015JPE and benhabib2016distribution show that idiosyncratic investment risk generates a double Pareto stationary wealth distribution. gabaix2016dynamics 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.} 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?

A final and related example of the need for deeper analysis is as follows: To understand the upper tail of the wealth distribution, we must avoid unnecessarily truncating the upper tail of the set of possible asset values in quantitative work. 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 \citep*{gouin2018pareto}. 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 {\sc 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.\footnote{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. Moreover, the innovation shocks are not restricted to be discrete, and the same is true for assets and consumption.}

To begin, when considering optimality in the household problem, we require a condition on the state dependent discount process $\{\beta_t\}$ that generalizes the classical condition $\beta < 1$ from the constant case and, for reasons discussed above, permits $\beta_t > 1$ with positive probability. To this end, we introduce the restriction\footnote{Here and below we set $\beta_0 \equiv 1$, so $\prod_{t=1}^n \beta_t = \prod_{t=0}^n \beta_t$.}

Condition (ref) clearly generalizes the classical condition $\beta < 1$ for the constant discount case. In the stochastic case, $\ln G_\beta$ can be understood as the asymptotic growth rate of the probability weighted average discount factor. Indeed, if $B_n := \mathbbm E \prod_{t=1}^n \beta_t$ is the average $n$-period discount factor, then, from the definition of $G_\beta$ and some straightforward analysis, we obtain $\ln (B_{n+1}/B_n) \to \ln G_\beta$, so the condition $G_\beta < 1$ implies that the asymptotic growth rate of the average $n$-period discount factor is negative, drifting down from its initial condition $\beta_0 \equiv 1$ at the rate $\ln G_\beta$. This does not, of course, preclude the possibility that $\beta_t > 1$ at any given $t$.

We show that condition (ref) 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 hubmer2018comprehensive. In addition, we prove that $G_\beta$ can be represented as the spectral radius of a nonnegative matrix, and hence can be computed by numerical linear algebra (as discussed below).

We also generalize the standard condition $\beta R < 1$, 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., deaton1992behaviour, chamberlain2000optimal or li2014solving). Analogous to (ref), we introduce the generalized condition

Here $\{R_t\}$ is a stochastic capital income process. Analogous to the case of $G_\beta$, the value $\ln G_{\beta R}$ can be understood as the asymptotic growth rate of average gross payoff on assets, discounted to present value.

We show that, when Conditions (ref)--(ref) hold and non-financial income satisfies two moment conditions, 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 \citep*{gouin2018pareto}.

One important feature of Conditions (ref)--(ref) is that they take into account the autocorrelation structure of preference shocks and asset returns. For example, if these processes depend only on {\sc iid} innovations, then (ref) reduces to $\mathbbm E \beta_t < 1$ and (ref) reduces to $\mathbbm E \beta_t R_t < 1$. But returns on assets are typically not {\sc iid}, since both mean returns and volatility are, in general, time varying, and preference shocks are typically modeled as correlated (see, e.g., hubmer2018comprehensive or schorfheide2018identifying). This dependence must be and is accounted for in (ref), since long upswings in $\{\beta_t\}$ 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., huggett1993risk, aiyagari1994uninsured or cao2020recursive), as well as standard estimation, calibration and simulation techniques that connect time series averages with cross-sectional moments.\footnote{A well-known example of a computational technique that uses ergodicity can be found in krusell1998income. On the estimation side see, for example, hansen2002generalized.} These properties require an additional restriction, placed on the asymptotic growth rate of mean returns. Analogous to (ref) and (ref), 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. Finally, under some mild additional conditions, we prove that the stationary distribution of assets is Pareto tailed, consistent with the data.

Our study is related to benhabib2015wealth, 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, {\sc 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 household income as positive correlations between labor earnings and wealth returns (an extension that was suggested by benhabib2015wealth), or time varying volatility in returns.\footnote{Empirical motivation for these kinds of extensions can be found in numerous studies, including guvenen2014inferring and fagereng2016heterogeneityNBER, fagereng2016heterogeneityAERPP.}

Another related paper is chamberlain2000optimal, 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 li2014solving, 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.\footnote{Coleman1990 introduced the time iteration operator as a constructive method for solving stochastic growth models. It has since been used in DattaMirmanReffett2002, MorandReffett2003 and many other studies.} 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 li2014solving.

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 rabault2002borrowing, carroll2004theoretical and kuhn2013recursive. While carroll2004theoretical constructs a weighted supremum norm contraction and works with the Bellman operator, the other two papers focus on time iteration. In particular, rabault2002borrowing exploits the monotonicity structure, while kuhn2013recursive applies a version of the Tarski fixed point theorem. Our techniques for studying optimality are close to those in li2014solving, as discussed above.\footnote{Our paper is also related to cao2017persistent, who study wealth inequality in a continuous-time framework with heterogeneous returns following a two-state Markov chain. While we do not pursue the connection here, the generality of our setup, including a persistent shock structure to wealth returns, might permit a study of the continuous-time limit that yields the tail results of cao2017persistent in a general framework.}

The rest of this paper is structured as follows. Section (ref) formulates the problem and establishes optimality results. Sufficient conditions for the existence and uniqueness of optimal policies are discussed. Section (ref) focuses on stochastic stability. Section (ref) discusses our key conditions and how they can be checked. Section (ref) provides a set of applications and Section (ref) concludes. All proofs are deferred to the appendix. Code that generates our figures can be found at \url{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.

Problem Statement

We consider a general income fluctuation problem, where a household chooses a consumption-asset path $\{(c_t, a_t)\}$ to solve

Here $u$ is the utility function, $\{\beta_t\}_{t \geq 0}$ is discount factor process with $\beta_0=1$, $\{R_t\}_{t \geq 1}$ is the gross rate of return on wealth, and $\{Y_t \}_{t \geq 1}$ is non-financial income. These stochastic processes obey

where $\beta$, $R$ and $Y$ are measurable nonnegative functions and $\{Z_t\}_{t \geq 0}$ is an irreducible time-homogeneous $\mathsf Z$-valued Markov chain taking values in finite set $\mathsf Z$. Let $P(z, \hat z)$ be the probability of transitioning from $z$ to $\hat z$ in one step. The innovation processes $\{\varepsilon_t\}$, $\{\zeta_t\}$ and $\{\eta_t\}$ are {\sc iid} independent and their supports can be continuous and vector-valued.

The function $u$ maps $\mathbbm R_+$ to $\{ - \infty \} \cup \mathbbm R$, is twice differentiable on $(0, \infty)$, satisfies $u' > 0$ and $u'' < 0$ everywhere on $(0, \infty)$, and that $u'(c) \to \infty$ as $c \to 0$ and $u'(c) < 1$ as $c \to \infty$. We define

The next period value of a random variable $X$ is typically denoted $\hat{X}$. Expectation without a subscript refers to the stationary process, where $Z_0$ is drawn from its (necessarily unique) stationary distribution.

Key Conditions

Our conditions for optimality are listed below. In what follows, $G_\beta$ is the asymptotic growth rate of the discount process as defined in (ref).

Assumption (ref) is a natural extension of the standard condition $\beta < 1$ from the constant discount case. If $\beta_t \equiv \beta$ for all $t$, then $G_\beta = \beta$, as follows immediately from the definition. It is weaker than the obvious sufficient condition $\beta_t \leq \bar \beta$ with probability one for some constant $\bar \beta < 1$, since in such a setting we have $G_\beta \leq \bar \beta < 1$. In fact it cannot be significantly weakened, as the proposition shows.

The positivity assumed here may or may not hold in applications, but Proposition (ref) shows that special conditions will have to be imposed on preferences if Assumption (ref) fails. Put differently, allowing $G_\beta \geq 1$ is tantamount to allowing $\beta \geq 1$ 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 $\{ \beta_t\}$ are constant at values $R$ and $\beta$, the standard restriction from the existing literature is $\beta R < 1$. A generalization using $G_{\beta R}$ as defined in (ref) is

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 (ref)--(ref) are easily verified. More complex examples are deferred to Sections (ref) and (ref).

Optimality: Definitions and Fundamental Properties

To consider optimality, we temporarily assume that $a_0>0$ and set the asset space to $(0, \infty)$.\footnote{Assumption (ref) combined with $u'(0)= \infty$ implies that $\mathbbm P \{ Y_t > 0 \}=1$ for all $t \geq 1$. Hence, $\mathbbm P \{ a_t > 0 \} = 1$ for all $t \geq 1$ and excluding zero from the asset space makes no difference to optimality.} The state space for $\{(a_t, Z_t) \}_{t \geq 0}$ is then $\mathsf S_0:= (0, \infty) \times \mathsf Z$. A feasible policy is a Borel measurable function $c \colon \mathsf S_0 \to \mathbbm R$ with $0 \leq c(a,z) \leq a$ for all $(a,z) \in \mathsf S_0$. A feasible policy $c$ and initial condition $(a,z) \in \mathsf S_0$ generate an asset path $\{ a_t\}_{t \geq 0}$ via (ref) when $c_t = c (a_t, Z_t)$ and $(a_0, Z_0) = (a,z)$. The lifetime value of policy $c$ is

where $\{ a_t\}$ is the asset path generated by $(c,(a,z))$. In the Appendix we show that $V_c$ is well-defined on $\mathsf S_0$. A feasible policy $c^*$ is called optimal if $V_c \leq V_{c^*}$ on $\mathsf S_0$ for any feasible policy $c$. A feasible policy is said to satisfy the first order optimality condition if

for all $(a,z) \in \mathsf S_0$, 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 $(a,z) \in \mathsf S_0$. A feasible policy is said to satisfy the transversality condition if, for all $(a, z) \in \mathsf S_0$,

Existence and Computability of Optimal Consumption

Let $\mathscr C$ be the space of continuous functions $c \colon \mathsf S_0 \to \mathbbm R$ such that $c$ is increasing in the first argument, $0 < c(a,z) \leq a$ for all $(a,z) \in \mathsf S_0$, and

To compare two consumption policies, we pair $\mathscr C$ with the distance

which evaluates the maximal difference in terms of marginal utility. While elements of $\mathscr C$ are not generally bounded, $\rho$ is a valid metric on $\mathscr C$. In particular, $\rho$ is finite on $\mathscr C$ since $\rho(c,d) \leq \left\| u' \circ c - u' \right\| + \left\| u' \circ d - u' \right\|$, and the last two terms are finite by (ref). In Appendix (ref), we show that $(\mathscr C, \rho)$ is a complete metric space. The following proposition shows that, for any policy in $\mathscr C$, the first order optimality condition (ref) implies the transversality condition.

We aim to characterize the optimal policy as the fixed point of the time iteration operator $T$ defined as follows: for fixed $c \in \mathscr C$ and $(a,z) \in \mathsf S_0$, the value of the image $Tc$ at $(a,z)$ is defined as the $\xi \in (0,a]$ that solves

where $\psi_c$ is the function on

defined by

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.

Part ((ref)) shows that, under our conditions, the familiar time iteration algorithm is globally convergent, provided one starts with some policy in the candidate class $\mathscr C$.

Properties of Optimal Consumption

In this section we study the properties of the optimal consumption function obtained in Theorem (ref). Assumptions (ref)--(ref) are held to be true throughout. The following two propositions show the monotonicity of the consumption function, which is intuitive.

Under further assumptions we can show that the optimal policy is concave and asymptotically linear with respect to the wealth level.

Proposition (ref) states that $c^*(a, z) \approx \alpha(z) a + b(z)$ 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 (ref) imply the linear lower bound $c^*(a,z) \geq \alpha(z)a$, although they do not provide a concrete number for $\alpha(z)$. The following proposition establishes an explicit linear lower bound.

The second inequality in (ref) restricts marginal utility derived from transferring wealth to the next period and then consuming versus consuming wealth today. The value $\bar s$ can be clarified once primitives are specified, as the next example illustrates.

Stationarity, Ergodicity, and Tail Behavior

This section focuses on stationarity, ergodicity and tail behavior of wealth under the unique optimal policy $c^*$ obtained in Theorem (ref). So that this policy exists, Assumptions (ref)--(ref) are always taken to be valid. We extend $c^*$ to $\mathsf S$ by setting $c^* (0,z) = 0 $ for all $z \in \mathsf Z$ and consider dynamics of $(a_t, Z_t)$ on $\mathsf S := \mathbbm R_+ \times \mathsf Z$, the law of motion for which is

Let $Q$ be the joint stochastic kernel of $(a_t, Z_t)$ on $\mathsf S$. See Appendix (ref) for this and related definitions.

Stationarity

To obtain existence of a stationary distribution we need to restrict the asymptotic growth rate for asset returns $G_R$ defined in (ref).

Below is one straightforward example of a setting where this holds, with more complex applications deferred to Sections (ref)--(ref).

By Proposition (ref), the value $\bar s$ in Assumption (ref) 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 $+\infty$, which allows us to obtain the following result.\footnote{Assumption (ref) 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, benhabib2015wealth, Proposition 6.}

Stationarity of the form obtained in Theorem (ref) is required to establish existence of stationary recursive equilibria in heterogeneous agent models with idiosyncratic risk, such as huggett1993risk or aiyagari1994uninsured.\footnote{For models with aggregate shocks, such as krusell1998income, a fully specified recursive equilibrium requires that households take the wealth distribution as one component of the state in their savings problem, and that stationarity holds for the entire joint distribution (defined over a product space encompassing both the wealth distribution and the exogenous state process). These problems fall outside the scope of Theorem (ref), since $\{Z_t\}$ is finite-valued. For a careful treatment of stationary recursive equilibrium in Krusell--Smith type models, see cao2020recursive.}

Ergodicity

While Assumption (ref) implies existence of a stationary distribution, it is not in general sufficient for uniqueness or stability. For these additional properties to hold, we must impose sufficient mixing. In doing so, we consider the following two cases:

Condition (Y2) means that there exists a function $f$ from $\mathbbm R_+ \times \mathsf Z$ to $\mathbbm R_+$ such that

for all Borel sets $A \subset \mathbbm R_+$ and all $z$ in $\mathsf Z$.

Assumption (ref) requires that there is a positive probability of receiving low labor income at some relatively persistent state of the world $\bar{z}$. This is a mixing condition that enforces social mobility. The reason is that $\{Z_t\}$ is already assumed to be irreducible, so $\bar z$ 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 ergodicity.

To state our uniqueness and stability results, let $Q^t$ be the $t$-step stochastic kernel, let $\| \cdot \|_{TV}$ 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 $\mathsf S$, let

and

where, here and in the theorem below, $\mathbbm E$ indicates expectation under stationarity.

Part 1 of Theorem (ref) states that the stationary distribution $\psi_\infty$ 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.

Tail Behavior

Having established the stationarity and ergodicity of wealth, we now study the tail behavior of the wealth distribution. We show that the wealth distribution is either bounded or (unbounded and) heavy-tailed under mild conditions. To prove this result we introduce the following assumption.

To state our result on tail behavior, we introduce the following notation. For any nonnegative function $A(z,\hat{z},\hat{\zeta})$, define the $\mathsf Z \times \mathsf Z$ matrix-valued function $M_A$ by

Elements of $M_A(s)$ are conditional moment generating functions of $\log A$. In the statement below, $\odot$ denotes the Hadamard (entry-wise) product, and $r(\cdot)$ returns the spectral radius of a matrix. Also $a_\infty$ is a random variable with distribution $\psi_\infty(\cdot , \mathsf Z)$.

Condition (ref) implies that for any $\varepsilon>0$, there exists a constant $C(\varepsilon)>0$ such that

for large enough $a$, so the upper tail of the wealth distribution is at least Pareto.

Testing the Growth Conditions

The three key conditions in the paper are the restrictions on the growth rates $G_\beta$, $G_{\beta R}$ and $G_R$, with the first two required for optimality and the last for stationarity (see Assumptions (ref), (ref) and (ref) 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.

The matrix $L_\varphi$ is expressed as a function on $\mathsf Z \times \mathsf Z$ in (ref) but can be represented in traditional matrix notation by enumerating $\mathsf Z$.\footnote{Specifically, if $\mathsf Z := \{z_1, \dots, z_N \}$, then $L_\varphi = P D_\varphi$ where $P$ is, as before, the transition matrix for the exogenous state, and $D_\varphi := \operatorname{diag} \left( \mathbbm E_{z_1} \varphi, \dots, \mathbbm E_{z_N} \varphi \right)$ when $\mathbbm E_{z} \varphi := \mathbbm E_z \varphi (z, \hat \xi) $. In what follows, $D_\beta$, $D_R$ and $D_{\beta R}$ are defined analogously to $D_\varphi$.}

What factors determine the long-run average growth rates embedded in our assumptions, such as $G_\beta$ or $G_R$? Lemma (ref) 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., hubmer2018comprehensive or hills2018fiscal). In particular, suppose that the state process follows a discretized version of

and $\beta_t = Z_t$. (The discretization implies that $\beta_t$ is always positive.) To simplify interpretation, the process (ref) is structured so that the stationary distribution of $\{Z_t\}$ is $N(\mu, \sigma^2)$. We use rouwenhorst1995's method to discretize $\{Z_t\}$ and then calculate $G_\beta$ using Lemma (ref), studying how $G_\beta$ is affected by the parameters in (ref).

Since $\beta_t = Z_t$ for all $t$, the structure of (ref) implies that $\mu$ is the long-run unconditional mean of $\{\beta_t\}$. It can therefore be set to standard calibrated value for the discount factor, such as $0.99$ from krusell1998income. What we wish to understand is how the remaining parameters $\rho$ and $\sigma$ affect the value of $G_\beta$. While no closed form expression is available in this case, Figure (ref) sheds some light by providing a contour plot of $G_\beta$ over a set of $(\rho, \sigma)$ pairs. The figure shows that $G_\beta$ grows with both the persistence term $\rho$ and volatility term $\sigma$. In particular, the condition $G_\beta < 1$ fails when the persistence and volatility of the discount factor process are sufficiently high. This is because $G_\beta$ is the limit of $\left(\mathbbm E \prod_{t=1}^n \beta_t \right)^{1/n}$ 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 $G_\beta$ but similar intuition applies to both $G_R$ and $G_{\beta R}$. If $\beta_t$ 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 (ref), (ref) and (ref) that, in the setting of benhabib2015wealth, where the discount factor is constant and returns and labor income are {\sc iid}, Assumptions (ref)--(ref) and Assumption (ref) are all satisfied. Hence, by Theorems (ref) and (ref), the household optimization problem has a unique optimal policy and the wealth process under this policy has a stationary solution. If, in addition, the support of $Y_t$ is finite or $Y_t$ has a positive density, say, then the conditions of Theorem (ref) also hold and the stationary solution is ergodic, geometrically mixing and its time series averages are asymptotically normal.

Let us now bring the model closer to the data by relaxing the {\sc iid} restrictions on financial and non-financial returns, introducing both mean persistence and time varying volatility in returns on assets.\footnote{The importance of these features for wealth dynamics was highlighted in fagereng2016heterogeneityNBER.} In particular, we set

where $\{ \zeta_t\}$ is {\sc iid} and standard normal and $\{\mu_t\}$ and $\{\sigma_t\}$ are finite-state Markov chains, discretized from

Innovations are {\sc iid} and standard normal. Using the data in fagereng2016heterogeneityAERPP on Norwegian financial returns over 1993--2003, we estimate these AR(1) models to obtain $\bar{\mu} = 0.0281$, $\rho_\mu = 0.5722$, $\delta_\mu = 0.0067$, $\bar{\sigma}=-3.2556$, $\rho_\sigma=0.2895$ and $\delta_\sigma=0.1896$. 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 $\{ \mu_t\}$ to its stationary mean $\bar{\mu}$, while the second reduces $\{ \sigma_t \}$ to its stationary mean $\tilde{\sigma} := \mathrm{e}^{\bar{\sigma} + \delta_\sigma^2/ 2(1 - \rho_\sigma^2)}$. In summary,

We set $\beta=0.95$ and $\gamma=1.5$. To test the stability properties of Model \expandafter\@slowromancap\romannumeral 1@, we explore a neighborhood of the calibrated $(\rho_\sigma, \delta_\sigma)$ values, while in Model \expandafter\@slowromancap\romannumeral 2@, we do likewise for $(\rho_\mu, \delta_\mu)$ pairs. In each scenario, other parameters are fixed to the benchmark. The results are shown in Figures (ref) and (ref).

In part (a) of each figure, we see that $G_{\beta R}$ is increasing in the persistence and volatility parameters of the state process. The intuition behind this feature was explained in Section (ref) for the case of $G_\beta$ and is similar here. (Note that $G_{\beta R} = \beta G_R$ in the present case, since $\beta_t \equiv \beta$ is a constant, so $G_{\beta R}$ has the same shape as $G_R$ in terms of contours.) The dots in the figures show that $G_{\beta R} < 1$ 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 $\max \{ G_{\beta R}, \bar s, \bar s G_R \} < 1$, where $\bar s$ is given by (ref). For such pairs, Assumptions (ref) and (ref) both hold, so the conditions of Theorems (ref)--(ref) are satisfied. (We are continuing to suppose that $Y_t$ is finite or has a positive density, so that Assumption (ref) holds. Assumptions (ref) and (ref) 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 (ljungqvist2012recursive, p. 3.) Working in a setting where returns on financial assets, non-financial income and impatience are all state dependent and fluctuate over time, we obtained conditions under which the household savings problem has a unique solution that can be computed by successive approximations and the wealth process under the optimal savings policy has a unique stationary distribution with Pareto right tail. We also obtained conditions under which wealth is ergodic and exhibits geometric mixing and asymptotic normality. We investigated the nature of our conditions and provided methods for testing them in applications. While our work was motivated by the desire to better understand the joint distribution of income and wealth, the income fluctuation problem also has applications in asset pricing, life-cycle choice, fiscal policy, monetary policy, optimal taxation, and social security. The ideas contained in this paper should be helpful for those fields after suitable modifications or extensions.