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---

(harrison_kreps)=

In addition to what's in Anaconda, this lecture uses following libraries:

```{code-cell} ipython

---

tags: [hide-output]

---

```{code-cell} ipython3

:tags: [hide-output]

!pip install quantecon

```

Let's start with some standard imports:

```{code-cell} ipython

```{code-cell} ipython3

import numpy as np

import quantecon as qe

import scipy.linalg as la

The stationary (i.e., invariant) distributions of these two matrices can be calculated as follows:

```{code-cell} python3

```{code-cell} ipython3

qa = np.array([[1/2, 1/2], [2/3, 1/3]])

qb = np.array([[2/3, 1/3], [1/4, 3/4]])

mca = qe.MarkovChain(qa)

mcb = qe.MarkovChain(qb)

mca.stationary_distributions

```

```{code-cell} python3

```{code-cell} ipython3

mcb.stationary_distributions

```

Here's a function that can be used to compute these values

```{code-cell} python3

```{code-cell} ipython3

def price_single_beliefs(transition, dividend_payoff, β=.75):

"""

Function to Solve Single Beliefs

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

```{code-cell} python3

```{code-cell} ipython3

def price_optimistic_beliefs(transitions, dividend_payoff, β=.75,

max_iter=50000, tol=1e-16):

"""

\check p(s)

= \beta \min

\left\{

P_a(s,1) \check p(0) + P_a(s,1) ( 1 + \check p(1)) ,\;

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

P_a(s,0) \check p(0) + P_a(s,1) ( 1 + \check p(1)) ,\;

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

\right\}

```

Here's code to solve for $\check p$ using iteration

```{code-cell} python3

```{code-cell} ipython3

def price_pessimistic_beliefs(transitions, dividend_payoff, β=.75,

max_iter=50000, tol=1e-16):

"""

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

## Exercises

```{exercise-start}

:label: hk_ex1

```

First, we will obtain equilibrium price vectors with homogeneous beliefs, including when all

investors are optimistic or pessimistic.

```{code-cell} python3

```{code-cell} ipython3

qa = np.array([[1/2, 1/2], [2/3, 1/3]]) # Type a transition matrix

qb = np.array([[2/3, 1/3], [1/4, 3/4]]) # Type b transition matrix

# Optimistic investor transition matrix

We will use the price_optimistic_beliefs function to find the price under

heterogeneous beliefs.

```{code-cell} python3

```{code-cell} ipython3

opt_beliefs = price_optimistic_beliefs([qa, qb], dividendreturn)

labels = ['p_optimistic', 'p_hat_a', 'p_hat_b']