from mpl_toolkits.mplot3d import Axes3D

```

## Key Formulas

## Key formulas

To start, let $c$ be a real number that lies strictly between

$-1$ and $1$.

We want to evaluate geometric series of two types -- infinite and finite.

### Infinite Geometric Series

### Infinite geometric series

The first type of geometric that interests us is the infinite series

that if $c \in (-1,1)$, then the outcome is the

equation $1 = 1$.

### Finite Geometric Series

### Finite geometric series

The second series that interests us is the finite geometric series

The geometric series formula {eq}`infinite` is at the heart of the classic model of the money creation process -- one that leads us to the celebrated

**money multiplier**.

### A Simple Model

### A simple model

There is a set of banks named $i = 0, 1, 2, \ldots$.

\sum_{i=0}^\infty (1-r)^i D_0 = \frac{D_0}{1 - (1-r)} = \frac{D_0}{r}

```

### Money Multiplier

### Money multiplier

The **money multiplier** is a number that tells the multiplicative

factor by which an exogenous injection of cash into bank $0$ leads

equal demand** (e.g., prices and interest rates are frozen)

- national income is entirely determined by aggregate demand

### Static Version

### Static version

An elementary Keynesian model of national income determination consists

of three equations that describe aggregate demand for $y$ and its

of the multiplier as the outcome of a dynamic process that we describe

next.

### Dynamic Version

### Dynamic version

We arrive at a dynamic version by interpreting the nonnegative integer

$t$ as indexing time and changing our specification of the

So if someone has a claim on $x$ dollars at time $t+j$, it

is worth $x R^{-j}$ dollars at time $t$ (e.g., today).

### Application to Asset Pricing

### Application to asset pricing

A **lease** requires a payments stream of $x_t$ dollars at

times $t = 0, 1, 2, \ldots$ where

Similarly, $\frac{\partial p_0}{\partial g}>0$ as long as $r>g$, $r>0$ and $g>0$ and $x_0$ is positive, so $\frac{\partial p_0}{\partial g}$

will always be positive.

## Back to the Keynesian Multiplier

## Back to the Keynesian multiplier

We will now go back to the case of the Keynesian multiplier and plot the

time path of $y_t$, given that consumption is a constant fraction