We'll start this lecture with a quick review of deterministic (i.e., non-random)

first-order and second-order linear difference equations.

## Linear difference equations

## Linear Difference Equations

We'll use the *backward shift* or *lag* operator $L$.

The algebra of lag and forward shift operators can simplify representing and solving linear difference equations.

### First order

### First Order

We want to solve a linear first-order scalar difference equation.

The distributed lead in $u$ in {eq}`equn_5` need not

converge when $|\lambda| < 1$.

### Second order

### Second Order

Now consider the second order difference equation

sequence $c \lambda^{-t}$ where $c$ is an arbitrary positive

constant.

## Some Python code

## Some Python Code

We’ll construct examples that illustrate {eq}`equation_3`.

- it happens that in this example future $m$’s are always less

than the current $m$

## Alternative code

## Alternative Code

We could also have run the simulation using the quantecon

**LinearStateSpace** code.

plt.show()

```

### Special case

### Special Case

To simplify our presentation in ways that will let focus on an important

idea, in the above second-order difference equation {eq}`equation_6` that governs

Please keep these formulas in mind as we investigate an alternative

route to and interpretation of our formula for $F$.

## Another perspective

## Another Perspective

Above, we imposed stability or non-explosiveness on the solution of the key difference equation {eq}`equation_1`

in Cagan's model by solving the unstable root of the characteristic polynomial forward.

This is the unique **stabilizing value** of $p_0$ expressed as a function of

$m_0$.

### Refining the formula

### Refining the Formula

We can get an even more convenient formula for $p_0$ that is cast

in terms of components of $Q$ instead of components of

Q_1 = \begin{bmatrix} Q_{11} \\ Q_{21} \end{bmatrix}.

$$

### Some remarks about feedback

### Remarks about Feedback

We have expressed {eq}`equation_8` in what superficially appears to be a form in

which $y_{t+1}$ feeds back on $y_t$, even though what we

the log money supply actually does feed back on the log of the price

level.

## Log money supply feeds back on log price level

## Log money Supply Feeds Back on Log Price Level

An arrangement of eigenvalues that split around unity, with one being

below unity and another being greater than unity, sometimes prevails when there is *feedback* from the log price level to the log

magic_p0(1, δ=0.2)

```

## Big $P$, little $p$ interpretation

## Big $P$, Little $p$ Interpretation

It is helpful to view our solutions of difference equations having feedback from the price level or inflation to money or the rate of money

creation in terms of the Big $K$, little $k$ idea discussed in {doc}`Rational Expectations Models <rational_expectations>`.

F_check[0] + F_check[1] * F_star, F_star

```

## Fun with SymPy code

## Fun with SymPy

This section is a gift for readers who have made it this far.