Asymptotic Linearity of Consumption Functions and Computational Efficiency

The existence of a solution to optimal savings problems has been studied by Schechtman and Escudero (1977), Chamberlain and Wilson (2000), and Li and Stachurski (2014). The recent work Ma et al. (2020) extend the Euler equation method of Li and Stachurski (2014) and show the existence and uniqueness of a solution in a general setting with Markovian shocks, capital income risk, stochastic discounting, and potentially unbounded utility functions. Ma and Toda (2021) make further extension to Ma et al. (2020) by relaxing their assumptions on utility and idiosyncratic risks. Our paper is in the spirit of Ma and Toda (2021).

Because optimal savings problems generally do not admit closed-form solutions, proving properties of the theoretical solution is often challenging. Rabault (2002) studies under what conditions borrowing constraints bind. Benhabib et al. (2015) characterize the tail behavior of the wealth distribution under iid capital and labor income shocks, which Ma et al. (2020) extend to a Markovian setting. Holm (2018) shows that with HARA preferences, tightening the liquidity constraint decreases consumption. Light (2018) shows that when the marginal utility is convex and the Markov chain has a certain monotonicity property, increasing income risk increases precautionary savings. Lehrer and Light (2018) show that with CRRA utility with risk aversion bounded above by 1, lower interest rate increases consumption. Light (2020) applies this result to prove the uniqueness of equilibrium in a certain Bewley-Aiyagari model. Stachurski and Toda (2019, 2020) show that consumption functions have linear lower bounds when the relative risk aversion is bounded, which they apply to show that wealth inherits the tail behavior of income in general equilibrium models with labor income risk only.

With HARA preferences and general income shocks, Carroll and Kimball (1996) show the concavity of consumption functions in finite horizon problems, which implies asymptotic linearity. However, under certain regularity assumptions, Toda (2020) shows that HARA is necessary for the concavity of consumption functions, implying that establishing asymptotic linearity based on concavity is possible only in very special cases. Furthermore, Ma et al. (2020) extend the concavity result of Carroll and Kimball (1996) to infinite horizon and prove asymptotic linearity of the optimal consumption function. Ma and Toda (2021) characterize the asymptotic MPCs analytically in the framework of Ma et al. (2020) specialized to CRRA utility. As discussed above, out paper separates from these studies in that we impose significantly weaker assumptions on the utility function.

In this section we introduce a general optimal savings problem that closely follows the settings in Ma et al. (2020) and Ma and Toda (2021). To avoid redundancies, we limit the discussion to the bare essentials. More details may be found in Ma and Toda (2021, Section 2.1).

Time is discrete and denoted by $t=0,1,\dots,n$, with possibly $n=\infty$. Let $a_{t}$ be the financial wealth of the agent at the beginning of period $t$. The agent chooses consumption $c_{t}\geq 0$ and saves the remaining wealth $a_{t}-c_{t}$. The period utility function is $u$ and the discount factor, gross return on wealth, and non-financial income in period $t$ are denoted by $\beta_{t},R_{t},Y_{t}$, where we normalize $\beta_{0}=1$. Thus the agent solves

	$\displaystyle\operatornamewithlimits{maximize}$	$\displaystyle\operatorname{E}_{0}\sum_{t=0}^{n}\left(\prod_{i=0}^{t}\beta_{i}\right)u(c_{t})$
	$\displaystyle\operatorname{subject~{}to}$	$\displaystyle a_{t+1}=R_{t+1}(a_{t}-c_{t})+Y_{t+1},$		(2.1a)
		$\displaystyle 0\leq c_{t}\leq a_{t},$		(2.1b)

where the initial wealth $a_{0}=a>0$ is given, (2.1a) is the budget constraint, and (2.1b) implies that the agent cannot borrow. The stochastic processes $\left\{{\beta_{t},R_{t},Y_{t}}\right\}_{t\geq 1}$ obey

$$\beta_{t}=\beta(Z_{t-1},Z_{t},\zeta_{t}),\quad R_{t}=R(Z_{t-1},Z_{t},\zeta_{t}),\quad Y_{t}=Y(Z_{t-1},Z_{t},\zeta_{t}),$$

(2.2)

where $\beta,R,Y$ are nonnegative measurable functions, $\left\{{Z_{t}}\right\}_{t\geq 0}$ is a time-homogeneous Markov chain taking values in a finite set $\mathsf{Z}=\left\{{1,\dots,Z}\right\}$ with a transition probability matrix $P$, and the innovations $\left\{{\zeta_{t}}\right\}$ are independent and identically distributed (iid) over time and could be vector-valued. To simplify the notation, we introduce the following conventions. We use a hat to denote a random variable that is realized next period, for example $Z=Z_{t}$ and $\hat{Z}=Z_{t+1}$. When no confusion arises, we write $\hat{\beta}$ for $\beta(Z,\hat{Z},\hat{\zeta})$ and define $\hat{R},\hat{Y}$ analogously. Conditional expectations are abbreviated using subscripts, for example

$$\operatorname{E}_{z}X=\operatorname{E}\left[{X}\,\middle|\,{Z=z}\right]\quad\text{and}\quad\operatorname{E}_{z,\hat{z}}X=\operatorname{E}\left[{X}\,\middle|\,{Z=z,\hat{Z}=\hat{z}}\right].$$

For $\theta\in\mathbb{R}$, we define the matrix $K(\theta)$ related to the transition probability matrix $P$, discount factor $\beta$, and return $R$ by

$$K_{z\hat{z}}(\theta)\coloneqq P_{z\hat{z}}\operatorname{E}_{z,\hat{z}}\hat{\beta}\hat{R}^{\theta}=P_{z\hat{z}}\operatorname{E}\beta(z,\hat{z},\hat{\zeta})R(z,\hat{z},\hat{\zeta})^{\theta}\in[0,\infty].$$

(2.3)

The matrix $K(\theta)$ for various values of $\theta$ appears throughout the paper. For a square matrix $A$, let $r(A)$ denote its spectral radius (largest absolute value of all eigenvalues).

Consider the following assumptions.

Assumption 1 is essentially the usual monotonicity and concavity assumptions together with a form of Inada condition.

Under the maintained assumptions, Theorem 2.2 below states that the optimal savings problem (2.1) admits a unique solution and provides a computational algorithm. To make its statement precise, we introduce further definitions. Let $\mathcal{C}$ be the space of candidate consumption functions such that $c:(0,\infty)\times\mathsf{Z}\to\mathbb{R}$ is continuous, is increasing in the first argument, $0\leq c(a,z)\leq a$ for all $a>0$ and $z\in\mathsf{Z}$, and

$$\sup_{(a,z)\in(0,\infty)\times\mathsf{Z}}\left\lvert u^{\prime}(c(a,z))-u^{\prime}(a)\right\rvert<\infty.$$

(2.4)

For $c,d\in\mathcal{C}$, define the metric

$$\rho(c,d)=\sup_{(a,z)\in(0,\infty)\times\mathsf{Z}}\left\lvert u^{\prime}(c(a,z))-u^{\prime}(d(a,z))\right\rvert.$$

(2.5)

When $u^{\prime}$ is positive, continuous, and strictly decreasing (implied by Assumption 1), it is straightforward (e.g., Proposition 4.1 of Li and Stachurski (2014)) to show that $(\mathcal{C},\rho)$ is a complete metric space.

Li and Stachurski (2014), Ma et al. (2020), and Ma and Toda (2021) show that the solution to the optimal savings problem (2.1) can be obtained as the unique fixed point of the policy iteration operator, which updates the consumption function using the Euler equation. The following lemma defines the updating rule. In what follows, long proofs are relegated to Appendix A.

When Assumptions 1, 2 hold and $c\in\mathcal{C}$, $a>0$, and $z\in\mathsf{Z}$, by Lemma 2.1 we can define a unique number $Tc(a,z)\coloneqq\xi\in[0,a]$ that solves (2.6). The following theorem establishes the existence and uniqueness of a solution to the optimal savings problem (2.1).

In this section we show that the consumption functions in the optimal savings problem (2.1) are asymptotically linear when the marginal utility is regularly varying. This is also where we depart from the setting in Ma and Toda (2021), who assume outright that the utility function exhibits constant relative risk aversion (CRRA). To state the main results, we introduce several notions. A positive measurable function $\ell:(0,\infty)\to(0,\infty)$ is slowly varying if $\ell(\lambda x)/\ell(x)\to 1$ as $x\to\infty$ for all $\lambda>0$. A positive measurable function $f:(0,\infty)\to(0,\infty)$ is regularly varying with index $\rho\in\mathbb{R}$ if $f(\lambda x)/f(x)\to\lambda^{\rho}$ as $x\to\infty$ for all $\lambda>0$. Bingham et al. (1987, Theorem 1.4.1) show that if $f$ is a positive measurable function such that $g(\lambda)=\lim_{x\to\infty}f(\lambda x)/f(x)\in(0,\infty)$ exists for $\lambda>0$ in a set of positive measure, then the limit function must be of the form $g(\lambda)=\lambda^{\rho}$ for some $\rho\in\mathbb{R}$ and $f$ is regularly varying with index $\rho$.

The assumption that marginal utility is regularly varying is related to several assumptions in the literature. Following Brock and Gale (1969) and Schechtman and Escudero (1977), we say that $u^{\prime}$ has an asymptotic exponent $-\gamma$ if $\log u^{\prime}(c)/\log c\to-\gamma$ as $c\to\infty$. We say that $u$ is asymptotically CRRA with coefficient $\gamma$ if $u$ is twice differentiable and $-cu^{\prime\prime}(c)/u^{\prime}(c)\to\gamma$ as $c\to\infty$. The following proposition clarifies the relation between these concepts.

It is clear from Proposition 2.3 that Assumption 3 is significantly weaker than CRRA (assumed in Ma and Toda, 2021) because it imposes a parametric assumption only at infinity ($c\to\infty$). Furthermore, the parameter $\gamma>0$ can be interpreted as the asymptotic relative risk aversion of the agent.

To prove our main results, we introduce a technical condition that permits us to apply the dominated convergence theorem.

Assumption 4 holds, for example, if the iid innovation $\zeta$ takes finitely many values, which is almost always the case in applied numerical works that employ discretization. Note that we allow the possibility $R=0$ with positive probability. Throughout the rest of the paper, we introduce the following conventions to simplify the notation: “1” denotes either the real number 1 or the vector $(1,\dots,1)^{\prime}\in\mathbb{R}^{Z}$ depending on the context; we interpret $0\cdot\infty=0$ and $\beta R^{1-\gamma}=(\beta R)R^{-\gamma}$, so $\beta R^{1-\gamma}=0$ whenever $\beta=0$ or $R=0$ regardless of the value of $\gamma>0$. Although the following property is an immediate implication of the above assumptions and convention, we state it as a lemma since we frequently refer to it.

Under the maintained assumptions, we can show that the consumption functions are asymptotically linear, which is our main result. We present two results, one for finite horizon ($n<\infty$) and another for infinite horizon ($n=\infty$).

A few remarks are in order. First, when the marginal utility is regularly varying, under the stated technical conditions, Theorem 2.5 shows that consumption functions in finite horizon problems are always asymptotically linear, and the asymptotic MPCs are exactly characterized as in (2.8).

Second, one might conjecture that similar statements hold in infinite horizon problems by taking the limit of both sides of (2.8), but this is not generally true. The reason is that we cannot interchange the two limits $a\to\infty$ and $n\to\infty$. To see this, consider the function $f_{n}:[0,\infty)\to\mathbb{R}$ defined by $f_{n}(a)=\max\left\{{a-n,0}\right\}$. Then clearly $\lim_{n\to\infty}f_{n}(a)=0\eqqcolon f(a)$ pointwise and $\lim_{a\to\infty}f_{n}(a)/a=1$, so

$$\lim_{n\to\infty}\lim_{a\to\infty}\frac{f_{n}(a)}{a}=1\neq 0=\lim_{a\to\infty}\frac{f(a)}{a}=\lim_{a\to\infty}\lim_{n\to\infty}\frac{f_{n}(a)}{a}.$$

This observation explains why in general we can only obtain bounds of the form (2.9).

Third, an immediate implication of Theorem 2.6 is that, when $r(K(1-\gamma))<1$, either $\liminf_{a\to\infty}c(a,z)/a=0$ for some $z\in\mathsf{Z}$, or (2.10) holds for all $z\in\mathsf{Z}$. However, the theorem does not tell which case occurs, which we investigate in the next section.

We now seek sufficient conditions for the limit (2.10) to hold. Throughout this section, we maintain Assumptions 1–4 (with $n=\infty$) and assume $r(K(1-\gamma))<1$. We first present a general result that relies on high-level assumptions, followed by applications to specific cases.

To obtain the limit (2.10), it thus suffices to verify the conditions (2.11) and (2.12). To this end, we rewrite (2.12) in a more convenient form. In what follows, assume $u$ is twice continuously differentiable and let

$$\gamma(c)\coloneqq-\frac{cu^{\prime\prime}(c)}{u^{\prime}(c)}\geq 0$$

be the local relative risk aversion coefficient.

Take any numbers $\left\{{\epsilon(z)}\right\}_{z\in\mathsf{Z}}\subset(0,1)$ and define $x=\epsilon(z)a$ and $y=\epsilon(\hat{Z})\hat{R}[1-\epsilon(z)]a$. By the mean value theorem for integrals, we obtain

	$\displaystyle\log\frac{u^{\prime}(y)}{u^{\prime}(x)}$	$\displaystyle=\int_{x}^{y}(\log u^{\prime}(c))^{\prime}\mathop{}\!\mathrm{d}c=\int_{x}^{y}\frac{u^{\prime\prime}(c)}{u^{\prime}(c)}\mathop{}\!\mathrm{d}c$
		$\displaystyle=-\int_{x}^{y}\frac{\gamma(c)}{c}\mathop{}\!\mathrm{d}c=-\int_{x}^{y}\frac{\hat{\gamma}}{c}\mathop{}\!\mathrm{d}c=-\hat{\gamma}\log\frac{y}{x},$

where $\hat{\gamma}=\gamma(\hat{c})$ for $\hat{c}=(1-\theta)x+\theta y$ with some $\theta\in(0,1)$. Taking the exponential of both sides, we obtain $u^{\prime}(y)/u^{\prime}(x)=(y/x)^{-\hat{\gamma}}$. Therefore

$$\frac{\operatorname{E}_{z}\hat{\beta}\hat{R}u^{\prime}(\epsilon(\hat{Z})\hat{R}[1-\epsilon(z)]a)}{u^{\prime}(\epsilon(z)a)}=\operatorname{E}_{z}\hat{\beta}\hat{R}\left(\frac{\epsilon(\hat{Z})}{\epsilon(z)}\hat{R}[1-\epsilon(z)]\right)^{-\hat{\gamma}}.$$

(2.13)

Thus, verifying (2.12) reduces to checking that the right-hand side of (2.13) is at most 1. Note that the right-hand side depends on $u$ and $a$ only through $\hat{\gamma}$.

We now present several sufficiency results.

In many applied works, it is often the case that the discount factor $\beta$ is constant and agents invest only in a risk-free asset. In this case we obtain the following corollary.

Under the assumptions of Corollary 2.10, Proposition 5 of Stachurski and Toda (2019) (which has been corrected as Proposition 5’ of Stachurski and Toda (2020)) establishes that consumption functions have linear lower bounds. Corollary 2.10 strengthens their result because it proves the asymptotic linearity with an exact characterization of the asymptotic MPC.

Although Proposition 2.11 does not provide an explicit threshold for the minimum income $b=\inf Y$ so that the limit (2.10) holds, it is clear from its proof that the threshold can be calculated if the utility function $u$ is explicitly given. In particular, how small $b$ can be depends on how fast the local relative risk aversion $\gamma(c)$ converges to $\gamma$.

Let $c\in\mathcal{C}$ be the solution to the optimal savings problem (2.1) and $T:\mathcal{C}\to\mathcal{C}$ be the policy iteration operator defined in Section 2.1. Theorem 2.2 of Ma et al. (2020), which also holds in the more general settings in Ma and Toda (2021) and Section 2.1, shows that $T^{k}$ is a contraction for some $k\in\mathbb{N}$. Consequently, by the contraction mapping theorem, letting $\rho$ be the marginal utility metric in (2.5), there exists a number $r\in(0,1)$ such that the approximation error can be bounded as

$$\rho(T^{kn}c_{0},c)\leq r^{n}\rho(c_{0},c)\to 0~{}\text{as}~{}n\to\infty$$

for any initial guess $c_{0}\in\mathcal{C}$. Therefore to compute $c$ efficiently, it is important to start with a good initial guess $c_{0}\in\mathcal{C}$.

If Assumptions 1–4 hold and $r(K(1-\gamma))<1$, we know from Theorem 2.6(i) that the $c(a,z)/a$ is asymptotically bounded above by $x^{*}(z)^{-1/\gamma}$, where $x$ is the unique fixed point of $F$ in (2.7). Therefore it is reasonable to use an affine function with slope $\bar{c}(z)$ as an initial guess of the consumption function $c(a,z)$. To determine its intercept, we do as follows. Let $\bar{a}(z)$ be the asset threshold for the borrowing constraint to bind (so $c(a,z)=a$ for $a\leq\bar{a}(z)$ and $c(a,z)<a$ for $a>\bar{a}(z)$). Then at $a=\bar{a}(z)$, the Euler equation (2.6) implies that

$$u^{\prime}(\bar{a}(z))=\operatorname{E}_{z}\hat{\beta}\hat{R}u^{\prime}(c(\hat{R}(\bar{a}(z)-\bar{a}(z))+\hat{Y},\hat{Z}))=\operatorname{E}_{z}\hat{\beta}\hat{R}u^{\prime}(c(\hat{Y},\hat{Z})).$$

Using the approximation $c(a,z)\approx a$ for small asset level, we obtain

$$u^{\prime}(\bar{a}(z))\approx\operatorname{E}_{z}\hat{\beta}\hat{R}u^{\prime}(\hat{Y})\iff\bar{a}(z)\approx(u^{\prime})^{-1}\left(\operatorname{E}_{z}\hat{\beta}\hat{R}u^{\prime}(\hat{Y})\right).$$

(3.1)

Therefore a reasonable initial guess based on theory is

	$\displaystyle c_{0}(a,z)$	$\displaystyle\coloneqq\min\left\{{a,\bar{c}(z)(a-\bar{a}(z))+\bar{a}(z)}\right\}$
		$\displaystyle=\min\left\{{a,\bar{c}(z)a+(1-\bar{c}(z))\bar{a}(z)}\right\},$		(3.2)

where $\bar{c}(z)=x^{*}(z)^{-1/\gamma}$ and $\bar{a}(z)$ is defined by the right-hand side of (3.1). (In (3.2), we take the minimum with $a$ to satisfy $c_{0}(a,z)\leq a$.) This $c_{0}$ trivially belongs to the candidate space $\mathcal{C}$. Furthermore, it satisfies

$$\lim_{a\to\infty}\frac{c_{0}(a,z)}{a}=x^{*}(z)^{-1/\gamma}=\bar{c}(z)=\lim_{a\to\infty}\frac{c(a,z)}{a},$$

so we can expect that $c_{0}$ approximates $c$ well.

One remaining practical issue is how to numerically compute $x^{*}(z)$, which appears in (3.2). By Theorem 2.6(i), $x^{*}$ is the unique positive solution to the equation $x=Fx$, which in principle can be computed using a nonlinear equation solver. However, doing so is not practical because $x^{*}$ tends to be a very large vector and nonlinear equation solvers tend to terminate before convergence is achieved. For instance, suppose $\beta,R$ are constant and $1\leq R<1/\beta$. Using (2.15), we obtain

$$x^{*}(z)=\bar{c}(z)^{-\gamma}=(1-(\beta R^{1-\gamma})^{1/\gamma})^{-\gamma}.$$

Suppose we fix the unit of time somehow (e.g., year), one period has time length $\Delta$, and (with a slight abuse of notation) the discount rate is $\delta>0$ and the continuously compounded risk-free rate is $r\in(0,\delta)$. Then $\beta=\mathrm{e}^{-\delta\Delta}$ and $R=\mathrm{e}^{r\Delta}$, so

$$x^{*}(z)=\left(1-\mathrm{e}^{-\frac{\Delta}{\gamma}(\delta-(1-\gamma)r)}\right)^{-\gamma}.$$

(3.3)

It is clear that (3.3) diverges to $\infty$ as $\Delta\downarrow 0$. In fact, (3.3) tends to be quite large in common settings. For instance, let the unit of time be a year and $\delta=0.04$ (4% annual discounting), $r=0.03$ (3% risk-free rate), and $\gamma=3$, which are typical values. Figure 1 plots the fixed point (3.3) in the range $\Delta\in[1/12,1]$, which implies that a period is between a month and a year. We can see that $x^{*}(z)$ tends to be quite large, which makes it impractical to numerically solve for $x^{*}(z)$.

A straightforward way to avoid this issue is to directly solve for $\bar{c}(z)=x^{*}(z)^{-1/\gamma}\in(0,1)$ instead of $x^{*}(z)$. Noting that $x^{*}$ is the fixed point of $F$ in (2.7), $\bar{c}(z)$ satisfies

$$\bar{c}(z)=\left(1+\left(\sum_{\hat{z}=1}^{Z}K_{z\hat{z}}(1-\gamma)\bar{c}(\hat{z})^{-\gamma}\right)^{1/\gamma}\right)^{-1},\quad z=1,\dots,Z.$$

(3.4)

To numerically solve the system of equations (3.4) using a nonlinear equation solver, we may use a good initial guess as follows. Conjecture that $x^{*}$ is approximately equal to $k1$ for some $k>0$ and that the right Perron vector of $K(1-\gamma)$ is close to 1. Then the equation $x=Fx$ becomes

$$k1\approx\left(1+(K(1-\gamma)k1)^{1/\gamma}\right)^{\gamma}\approx\left(1+(r(K(1-\gamma))k)^{1/\gamma}\right)^{\gamma}1.$$

Solving for $k$, we obtain

$$\bar{c}(z)\approx k^{-1/\gamma}=1-r(K(1-\gamma))^{1/\gamma}.$$

(3.5)

Since $r(K(1-\gamma))<1$ by assumption, the right-hand side of (3.5) is always in $(0,1)$. Therefore we can numerically solve the system of nonlinear equations (3.4) using the initial guess (3.5).

Given a candidate consumption function $c\in\mathcal{C}$, it is natural to update it to $Tc$ using the Euler equation (2.6). However, this is not generally feasible because $c$ is a function and thus $\mathcal{C}$ is infinite-dimensional. In practice, it is common to set up a finite grid $\mathcal{A}_{G}\coloneqq\left\{{a_{g}}\right\}_{g=1}^{G}$, where $0<a_{1}<\dots<a_{G}$ are grid points, update $c$ by solving for $\xi=c(a_{g},z)$ using (2.6) for each $(a_{g},z)\in\mathcal{A}_{G}\times\mathsf{Z}$, and interpolate (linear, spline, etc.) it if necessary.

Although this updating procedure is theoretically justified, it is not computationally efficient because it requires performing root-finding $GZ$ times for each iteration, which is computationally intensive. A straightforward way to improve the algorithm is to avoid root-finding as in the endogenous grid point method of Carroll (2006).

Instead of fixing a grid for asset, the endogenous grid method fixes a grid for saving $\mathcal{S}_{G}\coloneqq\left\{{s_{g}}\right\}_{g=1}^{G}$, where $0=s_{1}<\dots<s_{G}$, update $c$ on $\mathcal{S}_{G}\times\mathsf{Z}$, and then choose the grid points for asset endogenously based on the optimal consumption and saving. For concreteness, suppose we use linear interpolation and extrapolation. For each $n\in\mathbb{N}$, let $\mathcal{A}_{G}^{n}\coloneqq\left\{{a_{g}^{n}(z)}\right\}_{g=1}^{G}{{}_{z=1}^{Z}}$ be the endogenous grid points for asset determined in the $n$-th iteration, where $a_{g}^{n}(z)$ represents the asset grid point when saving is $s_{g}$ and exogenous state is $z$. We define $\mathcal{A}_{G}^{0}=\left\{{a_{g}^{0}(z)}\right\}_{g=1}^{G}{{}_{z=1}^{Z}}$ by $a_{g}^{0}(z)=s_{g}$ for all $z\in\mathsf{Z}$ and $g=1,\dots,G$. Furthermore, let $\mathcal{C}(\mathcal{A}_{G}^{n})$ be the set of continuous piecewise linear functions $c:(0,\infty)\times\mathsf{Z}\to\mathbb{R}$ such that for each $z\in\mathsf{Z}$, 1. $0<c(a,z)\leq a$for all $a>0$, 2. $c(a,z)=a$for $0<a\leq a_{1}^{n}(z)$, 3. $c(a,z)$is affine in $a$ on each subinterval $[a_{g}^{n}(z),a_{g+1}^{n}(z)]$ for $g=1,\dots,G-1$, and 4. $c(a,z)$is linearly extrapolated for $a>a_{G}^{n}(z)$. Policy iteration for computing the consumption functions via the endogenous grid method can be summarized as follows.