extension: .md

format_name: myst

format_version: 0.13

jupytext_version: 1.14.1

jupytext_version: 1.16.2

kernelspec:

display_name: Python 3 (ipykernel)

language: python

Our discussion and simulations draw on [high quality lectures](https://comp-econ.org/CEF_2013/downloads/Complex%20Econ%20Systems%20Lecture%20II.pdf) by [Cars Hommes](https://www.uva.nl/en/profile/h/o/c.h.hommes/c.h.hommes.html).

+++

We will use the following imports.

plt.show()

```

## The model

Let's return to our discussion of a hypothetical soybean market, where price is determined by supply and demand.

($D(p_t)$ is the quantity demanded in some fixed unit, such as thousands of tons.)

Because the crop of soybeans for time $t$ is planted at $t-1$, supply of soybeans at time $t$ depends on *expected* prices at time $t$, which we denote $p^t_{t-1}$.

Because the crop of soybeans for time $t$ is planted at $t-1$, supply of soybeans at time $t$ depends on *expected* prices at time $t$, which we denote $p^e_t$.

We suppose that supply is nonlinear in expected prices, and takes the form

$$

S(p^t_{t-1}) = \tanh(\lambda(p^t_{t-1} - c)) + d

S(p^e_t) = \tanh(\lambda(p^e_t - c)) + d

$$

where $\lambda$ is a positive constant, $c, d$ are nonnegative constants and $\tanh$ is a type of [hyperbolic function](https://en.wikipedia.org/wiki/Hyperbolic_functions).

Market equilibrium requires that supply equals demand, or

$$

a - b p_t = S(p^t_{t-1})

a - b p_t = S(p^e_t)

$$

Rewriting in terms of $p_t$ gives

$$

p_t = - \frac{1}{b} [S(p^t_{t-1}) - a]

p_t = - \frac{1}{b} [S(p^e_t) - a]

$$

Finally, to complete the model, we need to describe how price expectations are formed.

```{math}

:label: p_et

p^t_{t-1} = f(p_{t-1}, p_{t-2})

p^e_t = f(p_{t-1}, p_{t-2})

```

where $f$ is some function.

In other words,

$$ p_{t-1}^t = p_{t-1} $$

$$ p_t^e = p_{t-1} $$

Using {eq}`price_t`, we then have

```{math}

:label: pe_adaptive

p_{t-1}^t = \alpha p_{t-1} + (1-\alpha) p^t_{t-2}

p_t^e = \alpha p_{t-1} + (1-\alpha) p^e_{t-1}

\qquad (0 \leq \alpha \leq 1)

```

Another way to write this is

```{math}

:label: pe_adaptive_2

p_{t-1}^t = p^t_{t-2} + \alpha (p_{t-1} - p_{t-2}^t)

p_t^e = p^e_{t-1} + \alpha (p_{t-1} - p_{t-1}^e)

```

This equation helps to show that expectations shift

Using {eq}`pe_adaptive`, we obtain the dynamics

$$

p_t = - \frac{1}{b} [ S(\alpha p_{t-1} + (1-\alpha) p^t_{t-2}) - a]

p_t = - \frac{1}{b} [ S(\alpha p_{t-1} + (1-\alpha) p^e_{t-1}) - a]

$$

Let's call the function with prices starting at $p_0 = 5$.

```{code-cell} ipython3

ts_price_plot_adaptive(m, 5, ts_length=30)

```

```{math}

:label: pe_blae

p_{t-1}^t = \alpha p_{t-1} + (1-\alpha) p_{t-2}

p_t^e = \alpha p_{t-1} + (1-\alpha) p_{t-2}

```

```{code-cell} ipython3

```