> *A component of an asset price can be interpreted as a bubble when all investors agree that the current price of the asset exceeds what they believe the asset's underlying dividend stream justifies*.

## Structure of the Model

## Structure of the model

The model simplifies things by ignoring alterations in the distribution of wealth

among investors who have hard-wired different beliefs about the fundamentals that determine

Thus, a type $a$ investor is more pessimistic on average.

### Ownership Rights

### Ownership rights

An owner of the asset at the end of time $t$ is entitled to the dividend at time $t+1$ and also has the right to sell the asset at time $t+1$.

In case 2, both types of investors always hold at least some of the asset.

### Short Sales Prohibited

### Short sales prohibited

No short sales are allowed.

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.

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

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

### Optimism and Pessimism

### Optimism and pessimism

The above specifications of the perceived transition matrices $P_a$ and $P_b$, taken directly from Harrison and Kreps, build in stochastically alternating temporary optimism and pessimism.

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 than a type $a$ agaub is.

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

When investors choose whether to purchase or sell the asset at $t$, they also know $s_t$.

## Solving the Model

## Solving the model

Now let's turn to solving the model.

1. There are two types of agents differentiated only by their beliefs. Each type of agent has sufficient resources to purchase all of the asset (Harrison and Kreps's setting).

1. There are two types of agents with different beliefs, but because of limited wealth and/or limited leverage, both types of investors hold the asset each period.

### Summary Table

### Summary table

The following table gives a summary of the findings obtained in the remainder of the lecture

(in an exercise you will be asked to recreate the table and also reinterpret parts of it).

The row corresponding to $p_p$ would also apply if both types have enough resources to buy the entire stock of the asset but short sales are also possible so that temporarily pessimistic investors price the asset.

### Single Belief Prices

### Single belief prices

We’ll start by pricing the asset under homogeneous beliefs.

return prices

```

#### Single Belief Prices as Benchmarks

#### Single belief prices as benchmarks

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

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

### Pricing under Heterogeneous Beliefs

### Pricing under heterogeneous beliefs

There are several cases to consider.

The marginal investor is of type $b$ if

$$

P_a(s,1) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) <

P_b(s,1) \bar p(0) + P_b(s,1) ( 1 + \bar p(1))

P_a(s,0) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) <

P_b(s,0) \bar p(0) + P_b(s,1) ( 1 + \bar p(1))

$$

**Thus the marginal investor is the (temporarily) optimistic type**.

*Thus the marginal investor is the (temporarily) optimistic type*.

Equation {eq}`hakr2` is a functional equation that, like a Bellman equation, can be solved by

return p_new, phat_a, phat_b

```

### Insufficient Funds

### Insufficient funds

Outcomes differ when the more optimistic type of investor has insufficient wealth --- or insufficient ability to borrow enough --- to hold the entire stock of the asset.

return p_new

```

### Further Interpretation

### Further interpretation

Jose Scheinkman {cite}`Scheinkman2014` interprets the Harrison-Kreps model as a model of a bubble --- a situation in which an asset price exceeds what every investor thinks is merited by his or her beliefs about the value of the asset's underlying dividend stream.

He emphasizes how limiting short sales and limiting leverage have opposite effects.

## Exercises

```{exercise-start}

:label: hk_ex1

```