</div>

```

# {index}`The Income Fluctuation Problem II: Stochastic Returns on Assets <single: The Income Fluctuation Problem II: Stochastic Returns on Assets>`

# {index}`The Income Fluctuation Problem IV: Stochastic Returns on Assets <single: The Income Fluctuation Problem IV: Stochastic Returns on Assets>`

```{contents} Contents

:depth: 2

## Overview

In this lecture, we continue our study of the {doc}`income fluctuation problem <ifp>`.

In this lecture, we continue our study of the income fluctuation problem described in {doc}`ifp_egm`.

While the interest rate was previously taken to be fixed, we now allow

returns on assets to be state-dependent.

Let $P$ represent the Markov matrix for the chain $\{Z_t\}_{t \geq 0}$.

Our assumptions on preferences are the same as our {doc}`previous lecture <ifp>` on the income fluctuation problem.

Our assumptions on preferences are the same as in {doc}`ifp_egm`.

As before, $\mathbb E_z \hat X$ means expectation of next period value

$\hat X$ given current value $Z = z$.

### Optimality

Let the class of candidate consumption policies $\mathscr C$ be defined

{doc}`as before <ifp>`.

Let the class of candidate consumption policies $\mathscr C$ be defined as in {doc}`ifp_egm`.

In {cite}`ma2020income` it is shown that, under the stated assumptions,

\right\}

```

(Intuition and derivation are similar to our {doc}`earlier lecture <ifp>` on

the income fluctuation problem.)

(Intuition and derivation are similar to {doc}`ifp_egm`.)

We again solve the Euler equation using time iteration, iterating with a

Coleman--Reffett operator $K$ defined to match the Euler equation

### A Time Iteration Operator

Our definition of the candidate class $\sigma \in \mathscr C$ of consumption

policies is the same as in our {doc}`earlier lecture <ifp>` on the income

fluctuation problem.

policies is the same as in {doc}`ifp_egm`.

For fixed $\sigma \in \mathscr C$ and $(a,z) \in \mathbf S$, the value

$K\sigma(a,z)$ of the function $K\sigma$ at $(a,z)$ is defined as the

### Using an Endogenous Grid

In the study of that model we found that it was possible to further

accelerate time iteration via the {doc}`endogenous grid method <cake_eating_egm>`.

accelerate time iteration via the {doc}`endogenous grid method <os_egm>`.

We will use the same method here.

Let's try to get some idea of what will happen to assets over the long run

under this consumption policy.

As with our {doc}`earlier lecture <ifp>` on the income fluctuation problem, we

As in {doc}`ifp_egm`, we

begin by producing a 45 degree diagram showing the law of motion for assets

```{code-cell} python3

```{exercise}

:label: ifpa_ex1

Let's repeat our {ref}`earlier exercise <ifp_ex2>` on the long-run

Let's repeat our {ref}`earlier exercise <ifp_egm_ex2>` on the long-run

cross sectional distribution of assets.

In that exercise, we used a relatively simple income fluctuation model.