Each share entitles its owner to a stream of dividends $\{d_t\}$ governed by a Markov chain defined on a state space $S \in \{0, 1\}$.

Thus, the stock is traded **ex dividend**.

The dividend obeys

\end{cases}

$$

The owner of a share at the beginning of time $t$ is entitled to the dividend paid at time $t$.

An owner of a share at the end of time $t$ and the beginning of time $t+1$ is entitled to the dividend paid at time $t+1$.

Thus, the stock is traded **ex dividend**.

The owner of the share at the beginning of time $t$ is also entitled to sell the share to another investor during time $t$.

An owner of a share at the beginning of time $t+1$ is also entitled to sell the share to another investor during time $t+1$.

Two types $h=a, b$ of investors differ only in their beliefs about a Markov transition matrix $P$ with typical element

Both types of investors are risk-neutral and both have the same fixed discount factor $\beta \in (0,1)$.

In our numerical example, we’ll set $\beta = .75$, just as Harrison and Kreps did.

In our numerical example, we’ll set $\beta = .75$, just as Harrison and Kreps {cite}`HarrKreps1978` did.

We’ll eventually study the consequences of two alternative assumptions about the number of shares $A$ relative to the resources that our two types of investors can invest in the stock.

No short sales are allowed.

This matters because it limits how pessimists can express their opinion.

This matters because it limits how pessimists can express their opinions.

* They **can** express themselves by selling their shares.

* They **cannot** express themsevles more loudly by artificially "manufacturing shares" -- that is, they cannot borrow shares from more optimistic investors and then immediately sell them.

Remember that state $1$ is the high dividend state.

* In state $0$, a type $a$ agent is more optimistic about next period's dividend than a type $b$ agent.

* In state $1$, a type $b$ agent is more optimistic about next period's dividend.

* In state $1$, a type $b$ agent is more optimistic about next period's dividend than a type $a$ agaub is.

However, the stationary distributions $\pi_a = \begin{bmatrix} .57 & .43 \end{bmatrix}$ and $\pi_b = \begin{bmatrix} .43 & .57 \end{bmatrix}$ tell us that a type $B$ person is more optimistic about the dividend process in the long run than is a type $A$ person.

However, the stationary distributions $\pi_a = \begin{bmatrix} .57 & .43 \end{bmatrix}$ and $\pi_b = \begin{bmatrix} .43 & .57 \end{bmatrix}$ tell us that a type $b$ person is more optimistic about the dividend process in the long run than is a type $a$ person.

### Information

```{raw} html

<div class="content-table"></div>

```

|$ s_t $|0|1|

|:---------------------:|:----:|:----:|

|$ p_a $|1.33|1.22|

|$ p_b $|1.45|1.91|

|$ p_o $|1.85|2.08|

|$ p_p $|1|1|

|$ \hat{p}_a $|1.85|1.69|

|$ \hat{p}_b $|1.69|2.08|

| $ s_t $ | 0 | 1 |

| :-----------: | :---: | :---: |

| $ p_a $ | 1.33 | 1.22 |

| $ p_b $ | 1.45 | 1.91 |

| $ p_o $ | 1.85 | 2.08 |

| $ p_p $ | 1 | 1 |

| $ \hat{p}_a $ | 1.85 | 1.69 |

| $ \hat{p}_b $ | 1.69 | 2.08 |

Here

$$

p_h(s) = \beta \left( P_h(s,0) (0 + p_h(0)) + P_h(s,1) ( 1 + p_h(1)) \right), \quad s = 0, 1

$$

$$ (eq:assetpricehomog)

These equations imply that the equilibrium price vector is

These equilibrium prices under homogeneous beliefs are important benchmarks for the subsequent analysis.

* $p_h(s)$ tells what investor $h$ thinks is the "fundamental value" of the asset.

* $p_h(s)$ tells what a type $h$ investor thinks is the "fundamental value" of the asset.

* Here "fundamental value" means the expected discounted present value of future dividends.

We will compare these fundamental values of the asset with equilibrium values when traders have different beliefs.

for $s=0,1$.

In the above equation, the $max$ on the right side is evidently over two prospective values of next period's payout

In the above equation, the $max$ on the right side is over the two prospective values of next period's payout

from owning the asset.

The marginal investor who prices the asset in state $s$ is of type $a$ if

purchase the asset for more than he believes its future dividends are

worth.

The reason that an investor is willing to pay more than what he believes is warranted by fundamental value of the prospective dividend stream is he expects to have the option to sell the asset later to another investor who will value the asset more highly than he will.

An investor is willing to pay more than what he believes is warranted by fundamental value of the prospective dividend stream because he expects to have the option later to sell the asset to another investor who will value the asset more highly than he will then.

* Investors of type $a$ are willing to pay the following price for the asset

* The asset changes hands whenever the state changes from $0$ to $1$ or from $1$ to $0$.

* The valuations $\hat p_a(s)$ and $\hat p_b(s)$ are displayed in the fourth and fifth rows of the table.

* Even the pessimistic investors who don't buy the asset think that it is worth more than they think future dividends are worth.

* Even pessimistic investors who don't buy the asset think that it is worth more than they think future dividends are worth.

Here's code to solve for $\bar p$, $\hat p_a$ and $\hat p_b$ using the iterative method described above

Scheinkman stresses these features of the Harrison-Kreps model:

* Compared to the homogeneous beliefs setting leading to the pricing formula, high volume occurs when the Harrison-Kreps pricing formula prevails.

* High volume occurs when the Harrison-Kreps pricing formula {eq}`hakr2` prevails.

Type $a$ investors sell the entire stock of the asset to type $b$ investors every time the state switches from $s_t =0$ to $s_t =1$.

* Type $a$ investors sell the entire stock of the asset to type $b$ investors every time the state switches from $s_t =0$ to $s_t =1$.

Type $b$ investors sell the asset to type $a$ investors every time the state switches from $s_t = 1$ to $s_t =0$.

* Type $b$ investors sell the asset to type $a$ investors every time the state switches from $s_t = 1$ to $s_t =0$.

Scheinkman takes this as a strength of the model because he observes high volume during *famous bubbles*.

* If the *supply* of the asset is increased sufficiently either physically (more "houses" are built) or artificially (ways are invented to short sell "houses"), bubbles end when the supply has grown enough to outstrip optimistic investors’ resources for purchasing the asset.

* If the *supply* of the asset is increased sufficiently either physically (more "houses" are built) or artificially (ways are invented to short sell "houses"), bubbles end when the asset supply has grown enough to outstrip optimistic investors’ resources for purchasing the asset.

* If optimistic investors finance their purchases by borrowing, tightening leverage constraints can extinguish a bubble.

Scheinkman extracts insights about the effects of financial regulations on bubbles.

```{raw} html

<div class="content-table"></div>

```

|$s_t$|0|1|

|:---------------------:|:----:|:----:|

|$p_a$|1.33|1.22|

|$p_b$|1.45|1.91|

|$p_o$|1.85|2.08|

|$p_p$|1|1|

|$\hat{p}_a$|1.85|1.69|

|$\hat{p}_b$|1.69|2.08|

| $s_t$ | 0 | 1 |

| :---------: | :---: | :---: |

| $p_a$ | 1.33 | 1.22 |

| $p_b$ | 1.45 | 1.91 |

| $p_o$ | 1.85 | 2.08 |

| $p_p$ | 1 | 1 |

| $\hat{p}_a$ | 1.85 | 1.69 |

| $\hat{p}_b$ | 1.69 | 2.08 |

You will want first to define the transition matrices and dividend payoff vector.