Equality between those demands and supply gives a *dynamic* model in which money supply

and price level *sequences* are simultaneously determined by a set of simultaneous linear equations.

These equations take the form of what are often called vector linear **difference equations**.

These equations take the form of what is often called vector linear **difference equations**.

In this lecture, we'll roll up our sleeves and solve those equations in two different ways.

* perverse dynamics under rational expectations in which the system converges to the higher stationary inflation tax rate

* a peculiar comparative stationary-state outcome connected with that stationary inflation rate: it asserts that inflation can be *reduced* by running *higher* government deficits, i.e., by raising more resources by printing money.

The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.

The same qualitative outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.

These outcomes set the stage for the analysis to be presented in this lecture {doc}`laffer_adaptive` that studies a nonlinear version of the present model; it assumes a version of "adaptive expectations" instead of rational expectations.

That lecture will show that

* replacing rational expectations with adaptive expectations leaves the two stationary inflation rates unchanged, but that $\ldots$

* it reverse the pervese dynamics by making the *lower* stationary inflation rate the one to which the system typically converges

* it reverses the perverse dynamics by making the *lower* stationary inflation rate the one to which the system typically converges

* a more plausible comparative dynamic outcome emerges in which now inflation can be *reduced* by running *lower* government deficits

This outcome will be used to justify a selection of a stationary inflation rate that underlies the analysis of unpleasant monetarist arithmetic to be studies in this lecture {doc}`unpleasant`.

This outcome will be used to justify a selection of a stationary inflation rate that underlies the analysis of unpleasant monetarist arithmetic to be studied in this lecture {doc}`unpleasant`.

We'll use theses tools from linear algebra:

We'll use these tools from linear algebra:

* matrix multiplication

* matrix inversion

print(f'R_l, g_l = {msm.R_l:.4f}, {g2:.4f}')

```

Now let's compute the maximum steady-state amount of seigniorage that could be gathered by printing money and the state state rate of return on money that attains it.

Now let's compute the maximum steady-state amount of seigniorage that could be gathered by printing money and the state-state rate of return on money that attains it.

## Two computation strategies

```{note}

The same qualitive outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.

The same qualitative outcomes prevail in this lecture {doc}`money_inflation_nonlinear` that studies a nonlinear version of the model in this lecture.

```