**Remark:** Multiple arrangements of endowments

$[w_0^1, w_0^2, w_1^1(\epsilon), w_1^2(\epsilon), \theta_0^1, \theta_0^2 ]$

associated with the same distribution of wealth $\eta$. Can you explain why?

**Hint:** Think about the portfolio indeterminacy finding above.

```{hint}

Think about the portfolio indeterminacy finding above.

```

### Modigliani-Miller theorem

We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.

Note: `scipy.stats.lognorm` expects you to pass the standard deviation

```{note}

`scipy.stats.lognorm` expects you to pass the standard deviation

first $(tH \cdot H)$ and then the exponent of the mean as a

keyword argument `scale` (`scale=np.exp(-t * H2 / 2)`).

* See the documentation [here](https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.lognorm.html#scipy.stats.lognorm).

This is peculiar, so make sure you are careful in working with the log normal distribution.

```

Here is some code that tackles these tasks

As $\theta$ is lowered, more robustness is achieved.

**Note:** The ${\sf T}$ operator is sometimes called a

```{note}

The ${\sf T}$ operator is sometimes called a

*risk-sensitivity* operator.

```

We shall apply ${\sf T}$to the special case of a linear value

We shall apply ${\sf T}$ to the special case of a linear value

function $w'(\vec r - r_f 1)$

where $\vec r - r_f 1 \sim {\mathcal N}(\mu,\Sigma)$ or

$\vec r - r_f {\bf 1} = \mu + C \epsilon$ and

past utilities and to reoptimize at date $t \geq 1$ would, if allowed, want

to deviate from a Ramsey plan.

**Note:** A modified Ramsey plan constructed under the restriction that

```{note}

A modified Ramsey plan constructed under the restriction that

$\mu_t$ must be constant over time is time consistent (see

$\check \mu$ and $\check \theta$ in the above graphs).

```

### Meaning of Time Inconsistency

For $Q$ we follow the rule in {eq}`ddp_def_ogq`.

Note:

```{note}

* The feasibility constraint is embedded into $R$ by setting $R[s, a] = -\infty$ for $a \notin A(s)$.

* Probability distributions for $(s, a)$ with $a \notin A(s)$ can be arbitrary.

```

The following code sets up these objects for us

Can you spot what features of $\tilde F$ imply this?

Hint: remember the components of $X_t$

```{hint}

Remember the components of $X_t$

```

```{code-cell} python3

# Policy function in the follower's problem

can be computed by `np.abs(fft(X))**2 / len(X)`.

Note: The NumPy function `abs` acts elementwise, and correctly handles complex numbers (by computing their modulus, which is exactly what we need).

```{note}

The NumPy function `abs` acts elementwise, and correctly handles complex numbers (by computing their modulus, which is exactly what we need).

```

A function called `periodogram` that puts all this together can be found [here](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/estspec.py).

Make sure that the boundary conditions {eq}`onefive` are satisfied.

(Note: this problem differs from the problem in the text in one important way: instead of $h > 0$ in {eq}`oneone`, $h = 0$. This has an important influence on the solution.)

```{note}

This problem differs from the problem in the text in one important way: instead of $h > 0$ in {eq}`oneone`, $h = 0$. This has an important influence on the solution.

```

```{exercise-end}

```

1. For any $f \in cb\mathbb{R}_+$, the sequence $T^k f$ converges

uniformly to $f^*$.

(Note: If you find the mathematics heavy going you can take 1--2 as given and skip to the {ref}`next section <lt_comp_eg>`)

```{note}

If you find the mathematics heavy going you can take 1--2 as given and skip to the {ref}`next section <lt_comp_eg>`

```

Recall the [Banach contraction mapping theorem](https://en.wikipedia.org/wiki/Banach_fixed-point_theorem).

is an IID process.

**Note:** A property of the state-space representation {eq}`state-space` is that in

```{note}

A property of the state-space representation {eq}`state-space` is that in

general neither $\epsilon_{1,t}$ nor $\epsilon_{2,t}$ is in

the space spanned by square-summable linear combinations of

$y_t, y_{t-1}, \ldots$.

```

In general

$\begin{bmatrix} \epsilon_{1,t} \cr \epsilon_{2t} \end{bmatrix}$

where $\hat x_t = E [x_t | y_{t-1}, y_{t-2}, \ldots ]$ and

$a_t = y_t - E[y_t |y_{t-1}, y_{t-2}, \ldots ]$.

**Note:** A key property about an *innovations representation* is that

```{note}

A key property about an *innovations representation* is that

$a_t$ is in the space spanned by square summable linear

combinations of $y_t, y_{t-1}, \ldots$.

```

For more ramifications of this property, see the lectures {doc}`Shock Non-Invertibility <hs_invertibility_example>` and

{doc}`Recursive Models of Dynamic Linear Economies <hs_recursive_models>`.