This lecture studies two consumers who have exactly the same

nonfinancial income process and who both conform to the linear-quadratic

permanent income of consumption smoothing model described in the

permanent income of consumption smoothing model described in this

[quantecon lecture](https://python-intro.quantecon.org/perm_income_cons.html).

The two consumers have different information about

This lecture can be regarded as an introduction to some of the **invertibility** issues that take center stage in

the analysis of **fiscal foresight** by Eric Leeper, Todd Walker, and Susan Yang {cite}`Leeper_Walker_Yang`.

## Two Representations of the **Same** Nonfinancial Income Process

## Two Representations of One Nonfinancial Income Process

Where $\beta \in (0,1)$, we study consequences of endowing a

consumer with one of the two alternative representations for the change

\sigma_a^2 = \sigma_\epsilon^2 + [ 1 + (\beta - \beta^{-1})^2 \sum_{j=0}^\infty \beta^{2j} ] = \beta^{-1} \sigma_\epsilon^2 .

$$

### Application of Kalman filter

## Application of Kalman filter

We can also obtain representation {eq}`eqn_2` from representation {eq}`eqn_1` by using

the **Kalman filter**.

We can also obtain these formulas via the classical filtering theory

described in {doc}`this lecture <classical_filtering>`.

### News Shocks and Less Informative Shocks

## News Shocks and Less Informative Shocks

Representation {eq}`eqn_1` is cast in terms of a **news shock**

$\epsilon_{t+1}$ that represents a shock to nonfinancial income

$d_a(\beta) = \frac{1 -\beta^2}{1 -\beta } = (1 + \beta)$, another

fact that will be important below.

### Representation of $\epsilon_t$ in Terms of Future $y$’s

## Representation of $\epsilon_t$ Shock in Terms of Future $y_t$

Notice that reprentation {eq}`eqn_1`, namely, $y_{t+1} - y_t = -\beta^{-1} \epsilon_t + \epsilon_{t+1}$

implies the linear difference equation

Thus, $\epsilon_t$ contains **exact** information about an

important linear combination of **future** nonfinancial income.

### Representation in Terms of $a_t$ Shocks

## Representation in Terms of $a_t$ Shocks

Next notice that representation {eq}`eqn_2`, namely, $y_{t+1} - y_t = -

\beta a_t + a_{t+1}$ implies the linear difference

E [ y_{t+1} | y^t ] = (1-\beta) \sum_{j=0}^\infty \beta^j y_{t-j}

$$

### Permanent Income Consumption-Smoothing Model

## Permanent Income Consumption-Smoothing Model

When we computed optimal consumption-saving policies for the two

representations using formulas obtained with the difference equation

All the code that we shall use below is presented in that lecture.

### Computations

## Computations

We shall use Python to form **both** of the above two state-space

representations, using the following parameter values

plt.legend()

```

### Simulating the Income Process and Two Associated Shock Processes

## Simulating Income Process and Two Associated Shock Processes

We now describe how we form a **single** $\{y_t\}_{t=0}^T$ realization

that we will use to simulate the two different decision rules associated

two consumers having **identical** incomes at each date but at each date

having **different** information about their future incomes.

### Calculating Innovations in Another Way

## Calculating Innovations in Another Way

Here we use formula {eq}`eqn_3` above to compute $a_{t+1}$ as a function

of the history

We can verify that we recover the same $\{a_t\}$ sequence

computed earlier.

### Another Invertibility Issue

## Another Invertibility Issue

This {doc}`quantecon lecture <hs_invertibility_example>` contains another example of a shock-invertibility issue that is endemic

to the LQ permanent income or consumption smoothing model.