1. $\{X_t\}$ is a finite Markov chain with state space $S$ and

transition probabilities

$$

P(x, y) := \mathbb P \{ X_{t+1} = y \,|\, X_t = x \}

\qquad (x, y \in S)

$$

1. $g$ is a given function on $S$ taking nonnegative values

You can think of

* a single *tree* that each period yields *fruit* that equals the total amount of consumption available to the economy

* a competitive market in *shares* in the tree that entitles their owners to corresponding shares of the *dividend* stream, i.e., the *fruit* stream, yielded by the tree

* a representative consumer who in a competitive equilibrium

* a representative consumer who in a competitive equilibrium

* consumes the economy's entire endowment each period

* owns 100 percent of the shares in the tree

v = tree_price(ap)

ax.plot(states, v, lw=2, alpha=0.6, label=rf"$\gamma = {γ}$")

ax.set_title('Price-divdend ratio as a function of the state')

ax.set_title('Price-dividend ratio as a function of the state')

ax.set_ylabel("price-dividend ratio")

ax.set_xlabel("state")

ax.legend(loc='upper right')

```{exercise}

:label: ma_ex3

Let's consider finite horizon call options, which are more common than

Let's consider finite horizon call options, which are more common than

infinite horizon ones.

Finite horizon options obey functional equations closely related to {eq}`FEoption0`.

Not surprisingly, options with larger $k$ are worth more.

This is because an owner has a longer horizon over which

This is because an owner has a longer horizon over which

the option can be exercised.

```{solution-end}

```

```