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[harrison_kreps] Update lecture #763

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112 changes: 58 additions & 54 deletions lectures/harrison_kreps.md
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Original file line number Diff line number Diff line change
Expand Up @@ -3,12 +3,10 @@ jupytext:
text_representation:
extension: .md
format_name: myst
format_version: 0.13
jupytext_version: 1.17.1
kernelspec:
name: python3
display_name: Python 3 (ipykernel)
display_name: Python 3
language: python
name: python3
---

(harrison_kreps)=
Expand All @@ -31,9 +29,10 @@ kernelspec:

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

```{code-cell} ipython3
:tags: [hide-output]
```{code-cell} ipython
---
tags: [hide-output]
---
!pip install quantecon
```

Expand All @@ -52,10 +51,9 @@ The model features

Let's start with some standard imports:

```{code-cell} ipython3
import numpy as np
```{code-cell} ipython
import quantecon as qe
import scipy.linalg as la
import jax.numpy as jnp
```

### References
Expand All @@ -73,7 +71,7 @@ The Harrison-Kreps model illustrates the following notion of a bubble that attra

> *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
Expand Down Expand Up @@ -133,8 +131,8 @@ But in state $1$, a type $a$ investor is more pessimistic about next period's
The stationary (i.e., invariant) distributions of these two matrices can be calculated as follows:

```{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]])
qa = jnp.array([[1/2, 1/2], [2/3, 1/3]])
qb = jnp.array([[2/3, 1/3], [1/4, 3/4]])
mca = qe.MarkovChain(qa)
mcb = qe.MarkovChain(qb)
mca.stationary_distributions
Expand All @@ -150,7 +148,7 @@ The stationary distribution of $P_b$ is approximately $\pi_b = \begin{bmatrix} .

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 $t+1$.

Expand All @@ -167,23 +165,23 @@ Case 1 is the case studied in Harrison and Kreps.

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 themsevles 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 $a$ agnet 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.

Expand All @@ -195,7 +193,7 @@ This price function is endogenous and to be determined below.

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.

Expand All @@ -208,7 +206,7 @@ assumptions about beliefs:
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).
Expand Down Expand Up @@ -242,7 +240,7 @@ The row corresponding to $p_p$ would apply if neither type of investor has enoug

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.

Expand Down Expand Up @@ -277,15 +275,17 @@ def price_single_beliefs(transition, dividend_payoff, β=.75):
Function to Solve Single Beliefs
"""
# First compute inverse piece
imbq_inv = la.inv(np.eye(transition.shape[0]) - β * transition)
imbq_inv = jnp.linalg.inv(
jnp.eye(transition.shape[0]) - β * transition
)
# Next compute prices
prices = β * imbq_inv @ transition @ dividend_payoff
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.
Expand All @@ -294,7 +294,7 @@ These equilibrium prices under homogeneous beliefs are important benchmarks for
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.
Expand Down Expand Up @@ -334,7 +334,7 @@ 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))
$$
**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
Expand Down Expand Up @@ -407,31 +407,31 @@ def price_optimistic_beliefs(transitions, dividend_payoff, β=.75,
Function to Solve Optimistic Beliefs
"""
# We will guess an initial price vector of [0, 0]
p_new = np.array([[0], [0]])
p_old = np.array([[10.], [10.]])
p_new = jnp.array([[0], [0]])
p_old = jnp.array([[10.], [10.]])
# We know this is a contraction mapping, so we can iterate to conv
for i in range(max_iter):
p_old = p_new
p_new = β * np.max([q @ p_old
+ q @ dividend_payoff for q in transitions],
axis=0)
p_new = β * jnp.max(jnp.stack([q @ p_old
+ q @ dividend_payoff for q in transitions]),
1)
# If we succeed in converging, break out of for loop
if np.max(np.sqrt((p_new - p_old)**2)) < tol:
if jnp.max(jnp.sqrt((p_new - p_old)**2)) < tol:
break
ptwiddle = β * np.min([q @ p_old
+ q @ dividend_payoff for q in transitions],
ptwiddle = β * jnp.min(jnp.stack([q @ p_old
+ q @ dividend_payoff for q in transitions]),
axis=0)
phat_a = np.array([p_new[0], ptwiddle[1]])
phat_b = np.array([ptwiddle[0], p_new[1]])
phat_a = jnp.array([p_new[0], ptwiddle[1]])
phat_b = jnp.array([ptwiddle[0], p_new[1]])
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.
Expand All @@ -445,8 +445,8 @@ Instead of equation {eq}`hakr2`, the equilibrium price satisfies
\check p(s)
= \beta \min
\left\{
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))
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))
\right\}
```
Expand Down Expand Up @@ -475,24 +475,24 @@ def price_pessimistic_beliefs(transitions, dividend_payoff, β=.75,
Function to Solve Pessimistic Beliefs
"""
# We will guess an initial price vector of [0, 0]
p_new = np.array([[0], [0]])
p_old = np.array([[10.], [10.]])
p_new = jnp.array([[0], [0]])
p_old = jnp.array([[10.], [10.]])
# We know this is a contraction mapping, so we can iterate to conv
for i in range(max_iter):
p_old = p_new
p_new = β * np.min([q @ p_old
+ q @ dividend_payoff for q in transitions],
axis=0)
p_new = β * jnp.min(jnp.stack([q @ p_old
+ q @ dividend_payoff for q in transitions]),
axis=0)
# If we succeed in converging, break out of for loop
if np.max(np.sqrt((p_new - p_old)**2)) < tol:
if jnp.max(jnp.sqrt((p_new - p_old)**2)) < tol:
break
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.
Expand All @@ -513,6 +513,8 @@ Scheinkman extracts insights about the effects of financial regulations on bubbl
He emphasizes how limiting short sales and limiting leverage have opposite effects.
## Exercises
```{exercise-start}
:label: hk_ex1
```
Expand Down Expand Up @@ -570,22 +572,24 @@ First, we will obtain equilibrium price vectors with homogeneous beliefs, includ
investors are optimistic or pessimistic.
```{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
qa = jnp.array([[1/2, 1/2], [2/3, 1/3]]) # Type a transition matrix
qb = jnp.array([[2/3, 1/3], [1/4, 3/4]]) # Type b transition matrix
# Optimistic investor transition matrix
qopt = np.array([[1/2, 1/2], [1/4, 3/4]])
qopt = jnp.array([[1/2, 1/2], [1/4, 3/4]])
# Pessimistic investor transition matrix
qpess = np.array([[2/3, 1/3], [2/3, 1/3]])
qpess = jnp.array([[2/3, 1/3], [2/3, 1/3]])
dividendreturn = np.array([[0], [1]])
dividendreturn = jnp.array([[0], [1]])
transitions = [qa, qb, qopt, qpess]
labels = ['p_a', 'p_b', 'p_optimistic', 'p_pessimistic']
for transition, label in zip(transitions, labels):
print(label)
print("=" * 20)
s0, s1 = np.round(price_single_beliefs(transition, dividendreturn), 2)
s0, s1 = jnp.round(
price_single_beliefs(transition, dividendreturn), 2
)
print(f"State 0: {s0}")
print(f"State 1: {s1}")
print("-" * 20)
Expand All @@ -601,7 +605,7 @@ labels = ['p_optimistic', 'p_hat_a', 'p_hat_b']
for p, label in zip(opt_beliefs, labels):
print(label)
print("=" * 20)
s0, s1 = np.round(p, 2)
s0, s1 = jnp.round(p, 2)
print(f"State 0: {s0}")
print(f"State 1: {s1}")
print("-" * 20)
Expand All @@ -613,4 +617,4 @@ with **permanently optimistic** investors - this is due to the marginal investor
```{solution-end}
```
[^f1]: By assuming that both types of agents always have "deep enough pockets" to purchase all of the asset, the model takes wealth dynamics off the table. The Harrison-Kreps model generates high trading volume when the state changes either from 0 to 1 or from 1 to 0.
[^f1]: By assuming that both types of agents always have "deep enough pockets" to purchase all of the asset, the model takes wealth dynamics off the table. The Harrison-Kreps model generates high trading volume when the state changes either from 0 to 1 or from 1 to 0.
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