diff --git a/lectures/_config.yml b/lectures/_config.yml
index c60ac8dc0..4b97f1551 100644
--- a/lectures/_config.yml
+++ b/lectures/_config.yml
@@ -120,6 +120,9 @@ sphinx:
schelling: https://intro.quantecon.org/schelling.html
scalar_dynam: https://intro.quantecon.org/scalar_dynam.html
complex_and_trig: https://intro.quantecon.org/complex_and_trig.html
+ # sphinx-proof
+ proof_minimal_theme: true
+ # sphinx-tojupyter
tojupyter_static_file_path: ["source/_static", "_static"]
tojupyter_target_html: true
tojupyter_urlpath: "https://python.quantecon.org/"
diff --git a/lectures/_static/quant-econ.bib b/lectures/_static/quant-econ.bib
index 0fd46856d..5c5f15b80 100644
--- a/lectures/_static/quant-econ.bib
+++ b/lectures/_static/quant-econ.bib
@@ -3,6 +3,62 @@
Note: Extended Information (like abstracts, doi, url's etc.) can be found in quant-econ-extendedinfo.bib file in _static/
###
+
+@article{harsanyi1968games,
+ title={Games with Incomplete Information Played by ``{B}ayesian'' Players, {I}--{III} Part {II}. {B}ayesian Equilibrium Points},
+ author={Harsanyi, John C.},
+ journal={Management Science},
+ volume={14},
+ number={5},
+ pages={320--334},
+ year={1968},
+ publisher={INFORMS}
+}
+
+@article{harsanyi1968games3,
+ title={Games with Incomplete Information Played by ``{B}ayesian'' Players, {I}--{III} Part {III}. {T}he Basic Probability Distribution of the Game},
+ author={Harsanyi, John C.},
+ journal={Management Science},
+ volume={14},
+ number={7},
+ pages={486--502},
+ year={1968},
+ publisher={INFORMS}
+}
+
+@article{harsanyi1967games,
+ title={Games with Incomplete Information Played by ``{B}ayesian'' Players, {I}--{III} Part {I}. {T}he Basic Model},
+ author={Harsanyi, John C.},
+ journal={Management Science},
+ volume={14},
+ number={3},
+ pages={159--182},
+ year={1967},
+ publisher={INFORMS}
+}
+
+@article{miller1977risk,
+ title={Risk, uncertainty, and divergence of opinion},
+ author={Miller, Edward M},
+ journal={The Journal of finance},
+ volume={32},
+ number={4},
+ pages={1151--1168},
+ year={1977},
+ publisher={JSTOR}
+}
+
+@article{jeffreys1946invariant,
+ title={An invariant form for the prior probability in estimation problems},
+ author={Jeffreys, Harold},
+ journal={Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences},
+ volume={186},
+ number={1007},
+ pages={453--461},
+ year={1946},
+ publisher={The Royal Society London}
+}
+
@article{blume2018case,
title={A case for incomplete markets},
author={Blume, Lawrence E and Cogley, Timothy and Easley, David A and Sargent, Thomas J and Tsyrennikov, Viktor},
@@ -1378,6 +1434,16 @@ @article{HarrKreps1978
pages = {323-336}
}
+@article{Morris1996,
+ author = {Stephen Morris},
+ title = {Speculative Investor Behavior and Learning},
+ journal = {The Quarterly Journal of Economics},
+ year = {1996},
+ volume = {111},
+ number = {4},
+ pages = {1111-1133}
+}
+
@article{pal2013,
title = {Fitted value function iteration with probability one contractions},
author = {P{\'a}l, Jen{\H{o}} and Stachurski, John},
diff --git a/lectures/_toc.yml b/lectures/_toc.yml
index 5886601da..1474e4679 100644
--- a/lectures/_toc.yml
+++ b/lectures/_toc.yml
@@ -112,6 +112,7 @@ parts:
- file: markov_asset
- file: ge_arrow
- file: harrison_kreps
+ - file: morris_learn
- caption: Data and Empirics
numbered: true
chapters:
diff --git a/lectures/harrison_kreps.md b/lectures/harrison_kreps.md
index 631f12f22..03cc6d269 100644
--- a/lectures/harrison_kreps.md
+++ b/lectures/harrison_kreps.md
@@ -3,10 +3,12 @@ jupytext:
text_representation:
extension: .md
format_name: myst
+ format_version: 0.13
+ jupytext_version: 1.17.1
kernelspec:
- display_name: Python 3
- language: python
name: python3
+ display_name: Python 3 (ipykernel)
+ language: python
---
(harrison_kreps)=
@@ -29,10 +31,9 @@ kernelspec:
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
```
@@ -51,7 +52,7 @@ The model features
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
@@ -131,7 +132,7 @@ 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} 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)
@@ -139,7 +140,7 @@ mcb = qe.MarkovChain(qb)
mca.stationary_distributions
```
-```{code-cell} python3
+```{code-cell} ipython3
mcb.stationary_distributions
```
@@ -270,7 +271,7 @@ The first two rows of the table report $p_a(s)$ and $p_b(s)$.
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
@@ -399,7 +400,7 @@ Investors of type $a$ want to sell the asset in state $1$ while investors of typ
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):
"""
@@ -444,8 +445,8 @@ Instead of equation {eq}`hakr2`, the equilibrium price satisfies
\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\}
```
@@ -467,7 +468,7 @@ Constraints on short sales prevent that.
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):
"""
@@ -512,8 +513,6 @@ 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
```
@@ -570,7 +569,7 @@ We'll use these transition matrices when we present our solution of exercise 1 b
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
@@ -595,7 +594,7 @@ for transition, label in zip(transitions, labels):
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']
diff --git a/lectures/morris_learn.md b/lectures/morris_learn.md
new file mode 100644
index 000000000..b41d81da4
--- /dev/null
+++ b/lectures/morris_learn.md
@@ -0,0 +1,698 @@
+---
+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)
+ language: python
+---
+
+(morris_learn)=
+```{raw} jupyter
+
+```
+
+# Speculative Behavior with Bayesian Learning
+
+```{index} single: Models; Morris Learning
+```
+
+```{contents} Contents
+:depth: 2
+```
+
+## Overview
+
+This lecture describes how {cite:t}`Morris1996` extended the Harrison–Kreps model {cite}`HarrKreps1978` of speculative asset pricing.
+
+Like Harrison and Kreps's model, Morris's model determines the price of a dividend-yielding asset that is traded by risk-neutral investors who have heterogeneous beliefs.
+
+The Harrison-Kreps model assumes that the traders have dogmatic, hard-wired beliefs about the asset's dividend stream.
+
+Morris replaced Harrison and Kreps's traders with hard-wired beliefs about the dividend stream with traders who use Bayes' Law to update their beliefs about prospective dividends as new dividend data arrive.
+
+```{note}
+Morris's traders don't use data on past prices of the asset to update their beliefs about the
+dividend process.
+```
+
+Key features of the environment in Morris's model include:
+
+* All traders share a set of statistical models for prospective dividends
+* A single parameter indexes the set of statistical models
+* All traders observe the same dividend history
+* All traders use Bayes' Law to update beliefs
+* Traders have different initial *prior distributions* over the parameter
+* Traders' *posterior distributions* over the parameter eventually merge
+* Before their posterior distributions merge, traders disagree about the predictive density over prospective dividends
+ * therefore they disagree about the value of the asset
+
+Just as in the hard-wired beliefs model of Harrison and Kreps, those differences of opinion induce investors to engage in *speculative behavior* in the following sense:
+
+ * sometimes they are willing to pay more for the asset than what they think is its "fundamental" value, i.e., the expected discounted value of its prospective dividend stream
+
+
+Prior to reading this lecture, you might want to review the following quantecon lectures:
+
+* {doc}`Harrison-Kreps model `
+* {doc}`Likelihood ratio processes `
+* {doc}`Bayesian versus frequentist statistics `
+
+Let's start with some standard imports:
+
+```{code-cell} ipython3
+import numpy as np
+import matplotlib.pyplot as plt
+```
+
+
+
+## Structure of the model
+
+There is a fixed supply of shares of an asset.
+
+Each share entitles its owner to a stream of *binary* i.i.d. dividends $\{d_t\}$ where
+
+$$
+d_{t+1} \in \{0,1\}
+$$
+
+The dividend at time $t$ equals $1$ with unknown probability $\theta \in (0,1)$ and equals $0$ with probability $1-\theta$.
+
+Unlike {cite}`HarrKreps1978` where traders have hard-wired beliefs about a Markov transition matrix, in Morris's model:
+
+* The true dividend probability $\theta$ is unknown
+* Traders have *prior beliefs* about $\theta$
+* Traders observe dividend realizations and update beliefs via Bayes' Law
+
+There is a finite set $\mathcal{I}$ of *risk-neutral* traders.
+
+All traders have the same discount factor $\beta \in (0,1)$.
+
+* You can think of $\beta$ as being related to a net risk-free interest rate $r$ by $\beta = 1/(1+r)$.
+
+Owning the asset at the end of period $t$ entitles the owner to dividends at time $t+1, t+2, \ldots$.
+
+Because the dividend process is i.i.d., trader $i$ thinks that the fundamental value of the asset is the capitalized value of the dividend stream, namely, $\sum_{j=1}^\infty \beta^j \hat \theta_i
+= \frac{\hat \theta_i}{r}$, where $\hat \theta_i$ is the mean of the trader's posterior distribution over $\theta$.
+
+### Possible trades
+
+Traders buy and sell the risky asset in competitive markets each period $t = 0, 1, 2, \ldots$ after dividends are paid.
+
+As in Harrison-Kreps:
+
+* The asset is traded *ex dividend*
+* An owner of a share at the end of time $t$ is entitled to the dividend at time $t+1$
+* An owner of a share at the end of period $t$ also has the right to sell the share at time $t+1$ after having received the dividend at time $t+1$.
+
+*Short sales are prohibited*.
+
+This matters because it limits how pessimists can express their opinions:
+
+* They *can* express themselves by selling their shares
+* They *cannot* express themselves more emphatically by borrowing shares and immediately selling them
+
+All traders have sufficient wealth to purchase the risky asset.
+
+## Information and beliefs
+
+At time $t \geq 1$, all traders observe $(d_1, d_2, \ldots, d_t)$.
+
+All traders update their subjective distribution over $\theta$ by applying Bayes' rule.
+
+Traders have *heterogeneous priors* over the unknown dividend probability $\theta$.
+
+This heterogeneity in priors produces heterogeneous posterior beliefs.
+
+## Source of heterogeneous priors
+
+Imputing different statistical models to agents inside a model is controversial.
+
+Many game theorists and rational expectations applied economists think it is a bad idea.
+
+While these economists often construct models in which agents have different *information*, they prefer to assume that all of the agents inside their model always share the same statistical model -- i.e., the same joint probability distribution over the random process being modeled.
+
+For a statistician or an economic theorist, a statistical model is a joint probability distribution that is characterized by a known parameter vector.
+
+When working with a *set* of statistical models swept out by parameters, say $\theta$ in a known set $\Theta$, economic theorists
+reduce the set of models to a single model by imputing to all agents inside the model the same prior probability distribution over $\theta$.
+
+
+```{note}
+A set of statistical models that has a particular geometric structure is called a [manifold](https://en.wikipedia.org/wiki/Manifold) of statistical models. Morris endows traders with a shared manifold of statistical models.
+```
+
+Proceeding in this way adheres to the *Harsanyi Common Priors Doctrine*.
+
+
+
+{cite}`harsanyi1967games`, {cite}`harsanyi1968games`, {cite}`harsanyi1968games3` argued that if two rational agents have
+the same information and the same reasoning capabilities, they will have the same joint probability distribution over outcomes of interest.
+
+Harsanyi interpreted disagreements about prospective outcomes as arising from differences in agents' information sets, not differences in their statistical models.
+
+
+Evidently, {cite}`HarrKreps1978` departed from the Harsanyi common statistical model assumption when they hard-wired dogmatic disparate beliefs.
+
+{cite:t}`Morris1996` abandons the Harsanyi doctrine less completely than Harrison and Kreps had.
+
+ * Morris does assume that agents share the same set of statistical models, but $\ldots$
+ * Morris assumes that they have different initial prior distributions over the parameter that indexes the models
+
+Morris's agents express their initial ignorance about the parameter differently -- they have different priors.
+
+
+Morris defends his assumption by alluding to the apparent ''mispricing'' of initial public offerings presented by {cite}`miller1977risk`.
+
+Miller described a situation in which agents have access to little or no data about a new enterprise.
+
+Morris wanted his traders to be open to changing their opinions as information about the parameter arrives.
+
+Knowledgeable statisticians have been known to disagree about an appropriate prior.
+
+For example, Morris described *different* respectable ways to express ''maximal ignorance'' about the parameter of a Bernoulli distribution
+
+ * a uniform distribution on $[0, 1]$
+ * a Jeffreys prior {cite}`jeffreys1946invariant` that is invariant to reparameterization; in the present situation, the Jeffreys prior takes the form of a Beta distribution with parameters $.5, .5$
+
+Is one of these priors more ''rational'' than the other?
+
+Morris thinks not.
+
+
+
+
+## Beta priors
+
+For tractability, assume trader $i$ has a Beta prior over the dividend probability
+
+$$
+\theta \sim \text{Beta}(a_i, b_i)
+$$
+
+where $a_i, b_i > 0$ are the prior parameters.
+
+```{note}
+The Beta distribution also appears in these quantecon lectures {doc}`divergence_measures`, {doc}`likelihood_ratio_process`, {doc}`odu`.
+```
+
+Suppose trader $i$ observes a history of $t$ periods in which a total of $s$ dividends are paid
+(i.e., $s$ successes with dividend and $t-s$ failures without dividend).
+
+By Bayes' rule, the posterior density over $\theta$ is:
+
+$$
+\pi_i(\theta \mid s, t) = \frac{\theta^s (1-\theta)^{t-s} \pi_i(\theta)}{\int_0^1 \theta^s (1-\theta)^{t-s} \pi_i(\theta) d\theta}
+$$
+
+where $\pi_i(\theta)$ is trader $i$'s prior density.
+
+```{note}
+The Beta distribution is the conjugate prior for the Binomial likelihood. This means that when the prior is $\text{Beta}(a_i, b_i)$ and we observe $s$ successes in $t$ trials, the posterior is $\text{Beta}(a_i+s, b_i+t-s)$.
+```
+
+The posterior mean (or expected dividend probability) is:
+
+$$
+\mu_i(s,t) = \int_0^1 \theta \pi_i(\theta \mid s, t) d\theta
+= \mathbb{E}[\text{Beta}(a_i+s, b_i+t-s)] = \frac{a_i + s}{a_i + b_i + t}
+$$
+
+Morris refers to $\mu_i(s,t)$ as trader $i$'s **fundamental valuation** of the asset after history $(s,t)$.
+
+This is the probability trader $i$ assigns to receiving a dividend next period.
+
+It embeds trader $i$'s updated belief about $\theta$.
+
+## Market prices with learning
+
+Fundamental valuations equal expected present values of dividends that our heterogeneous traders
+attach to the option of holding the asset *forever*.
+
+The equilibrium price process is determined by the condition that the asset is held at time $t$ by the trader who attaches the highest valuation to the asset at time $t$.
+
+An owner of the asset has the option to sell it after receiving that period's dividend.
+
+Traders take that into account.
+
+That opens the possibility that a trader will be willing to pay more for the asset than that trader's fundamental valuation.
+
+```{prf:definition} Most Optimistic Valuation
+:label: most_optimistic_valuation
+
+After history $(s,t)$, the *most optimistic fundamental valuation* is:
+
+$$
+\mu^*(s,t) = \max_{i \in \mathcal{I}} \mu_i(s,t)
+$$
+```
+
+```{prf:definition} Equilibrium Asset Price
+:label: equilibrium_asset_price
+
+Write $\tilde{p}(s,t,r)$ for the competitive equilibrium price of the risky asset (in current dollars) after history $(s,t)$ when the interest rate is $r$.
+
+The equilibrium price satisfies:
+
+$$
+\tilde{p}(s,t,r) = \frac{1}{1+r} \Bigl[ \mu^*(s,t) \{1 + \tilde{p}(s+1,t+1,r)\}
++ (1 - \mu^*(s,t)) \tilde{p}(s,t+1,r) \Bigr]
+$$
+```
+
+The equilibrium price equals the highest expected discounted return among all traders from holding the asset to the next period.
+
+```{prf:definition} Normalized Price
+:label: normalized_price
+
+Define the normalized price as:
+
+$$
+p(s,t,r) = r \tilde{p}(s,t,r)
+$$
+
+Since the current "dollar" price of the riskless asset is $1/r$, this represents the price of the risky asset in terms of the riskless asset.
+```
+
+Substituting the preceding formula into the equilibrium condition gives:
+
+$$
+p(s,t,r) = \frac{r}{1+r} \mu^*(s,t) + \frac{1}{1+r}
+\Bigl[ \mu^*(s,t) p(s+1,t+1,r) + (1 - \mu^*(s,t)) p(s,t+1,r) \Bigr]
+$$
+
+or equivalently:
+
+$$
+p(s,t,r) = \mu^*(s,t) + \frac{r}{1+r}
+\Bigl[ \mu^*(s,t) p(s+1,t+1,r) + (1 - \mu^*(s,t)) p(s,t+1,r) - \mu^*(s,t) \Bigr]
+$$
+
+A price function that satisfies the equilibrium condition can be computed recursively.
+
+Set $p^0(s,t,r) = 0$ for all $(s,t,r)$, and define $p^{n+1}(s,t,r)$ by:
+
+$$
+p^{n+1}(s,t,r) = \frac{r}{1+r} \mu^*(s,t) + \frac{1}{1+r}
+\Bigl[ \mu^*(s,t) p^n(s+1,t+1,r) + (1 - \mu^*(s,t)) p^n(s,t+1,r) \Bigr]
+$$
+
+The sequence $\{p^n(s,t,r)\}$ converges to the equilibrium price $p(s,t,r)$.
+
+```{prf:definition} Speculative Premium
+:label: speculative_premium
+
+When the identity of the most optimistic trader can switch with future dividend realizations, the market price exceeds *every* trader's fundamental valuation.
+
+In normalized units:
+
+$$
+p(s,t,r) > \mu_i(s,t) \quad \text{for all } i \in \mathcal{I}
+$$
+
+Define the **speculative premium** as:
+
+$$
+p(s,t,r) - \mu^*(s,t) > 0
+$$
+```
+
+
+## Two Traders
+
+We now focus on an example with two traders with Beta priors with parameters $(a_1,b_1)$ and $(a_2,b_2)$.
+
+```{prf:definition} Rate Dominance (Beta Priors)
+:label: rate_dominance_beta
+
+Trader 1 **rate-dominates** trader 2 if:
+
+$$
+a_1 \geq a_2 \quad \text{and} \quad b_1 \leq b_2
+$$
+```
+
+```{prf:theorem} Global Optimist (Two Traders)
+:label: two_trader_optimist
+
+For two traders with Beta priors:
+
+1. If trader 1 rate-dominates trader 2, then trader 1 is a **global optimist**: $\mu_1(s,t) \geq \mu_2(s,t)$ for all histories $(s,t)$
+2. In this case where $p(s,t,r) = \mu_1(s,t)$ for all $(s,t,r)$, there is *no speculative premium*.
+```
+
+When neither trader rate-dominates the other, the identity of the most optimistic trader can switch as dividends accrue.
+
+Along a history in which perpetual switching occurs, the price of the asset strictly exceeds both traders' fundamental valuations so long as traders continue to disagree:
+
+$$
+p(s,t,r) > \max\{\mu_1(s,t), \mu_2(s,t)\}
+$$
+
+Thus, along such a history, there is a persistent speculative premium.
+
+### Implementation
+
+For computational tractability, let's work with a finite horizon $T$ and solve by backward induction.
+
+```{note}
+On page 1122, {cite:t}`Morris1996` provides an argument that the limit as $T\rightarrow + \infty$ of such finite-horizon economies provides a useful
+selection algorithm that excludes additional equilibria that involve a Ponzi-scheme price component that Morris dismisses as fragile.
+```
+
+Following {prf:ref}`equilibrium_asset_price`, we use the discount factor parameterization $\beta = 1/(1+r)$ and compute dollar prices $\tilde{p}(s,t)$ via:
+
+$$
+\tilde{p}(s,t) = \beta \max_{i\in\{1,2\}} \Bigl[ \mu_i(s,t) \{1 + \tilde{p}(s+1,t+1)\} + (1-\mu_i(s,t)) \tilde{p}(s,t+1) \Bigr]
+$$
+
+We set the terminal price $\tilde{p}(s,T)$ equal to the perpetuity value under the most optimistic belief.
+
+```{code-cell} ipython3
+def posterior_mean(a, b, s, t):
+ """
+ Compute posterior mean μ_i(s,t) for Beta(a, b) prior.
+ """
+ return (a + s) / (a + b + t)
+
+def perpetuity_value(a, b, s, t, β=.75):
+ """
+ Compute perpetuity value (β/(1-β)) * μ_i(s,t).
+ """
+ return (β / (1 - β)) * posterior_mean(a, b, s, t)
+
+def price_learning_two_agents(prior1, prior2, β=.75, T=200):
+ """
+ Compute \tilde p(s,t) for two Beta-prior traders via backward induction.
+ """
+ a1, b1 = prior1
+ a2, b2 = prior2
+ price_array = np.zeros((T+1, T+1))
+
+ # Terminal condition: set to perpetuity value under max belief
+ for s in range(T+1):
+ perp1 = perpetuity_value(a1, b1, s, T, β)
+ perp2 = perpetuity_value(a2, b2, s, T, β)
+ price_array[s, T] = max(perp1, perp2)
+
+ # Backward induction
+ for t in range(T-1, -1, -1):
+ for s in range(t, -1, -1):
+ μ1 = posterior_mean(a1, b1, s, t)
+ μ2 = posterior_mean(a2, b2, s, t)
+
+ # One-step continuation values under each trader's beliefs
+ cont1 = μ1 * (1.0 + price_array[s+1, t+1]) \
+ + (1.0 - μ1) * price_array[s, t+1]
+ cont2 = μ2 * (1.0 + price_array[s+1, t+1]) \
+ + (1.0 - μ2) * price_array[s, t+1]
+ price_array[s, t] = β * max(cont1, cont2)
+
+ def μ1_fun(s, t):
+ return posterior_mean(a1, b1, s, t)
+ def μ2_fun(s, t):
+ return posterior_mean(a2, b2, s, t)
+
+ return price_array, μ1_fun, μ2_fun
+```
+
+(hk_go)=
+### Case A: global optimist (no premium)
+
+Pick priors with rate dominance, e.g., trader 1: $\text{Beta}(a_1,b_1)=(2,1)$ and trader 2: $(a_2,b_2)=(1,2)$.
+
+Trader 1 is the global optimist, so the normalized price equals trader 1's fundamental valuation: $p(s,t,r) = \mu_1(s,t)$.
+
+```{code-cell} ipython3
+β = 0.75
+price_go, μ1_go, μ2_go = price_learning_two_agents(
+ (2,1), (1,2), β=β, T=200)
+
+perpetuity_1 = (β / (1 - β)) * μ1_go(0, 0)
+perpetuity_2 = (β / (1 - β)) * μ2_go(0, 0)
+
+print("Price at (0, 0) =", price_go[0,0])
+print("Valuation of trader 1 at (0, 0) =", perpetuity_1)
+print("Valuation of trader 2 at (0, 0) =", perpetuity_2)
+```
+
+The price equals trader 1's perpetuity value.
+
+### Case B: perpetual switching (positive premium)
+
+Now assume trader 1 has $\text{Beta}(1,1)$, trader 2 has $\text{Beta}(1/2,1/2)$.
+
+These produce crossing posteriors, so there is no global optimist and the price exceeds both fundamentals early on.
+
+```{code-cell} ipython3
+price_ps, μ1_ps, μ2_ps = price_learning_two_agents(
+ (1,1), (0.5,0.5), β=β, T=200)
+
+price_00 = price_ps[0,0]
+μ1_00 = μ1_ps(0,0)
+μ2_00 = μ2_ps(0,0)
+
+perpetuity_1 = (β / (1 - β)) * μ1_ps(0, 0)
+perpetuity_2 = (β / (1 - β)) * μ2_ps(0, 0)
+
+print("Price at (0, 0) =", np.round(price_00, 6))
+print("Valuation of trader 1 at (0, 0) =", perpetuity_1)
+print("Valuation of trader 2 at (0, 0) =", perpetuity_2)
+```
+
+The resulting premium reflects the option value of reselling to whichever trader becomes temporarily more optimistic as dividends arrive sequentially.
+
+Within this setting, we can reproduce two key figures reported in {cite:t}`Morris1996`
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: Normalized price against interest rate
+ name: fig-normalized-price-interest
+---
+def normalized_price_two_agents(prior1, prior2, r, T=250):
+ """Return p(s,t,r) = r \tilde p(s,t,r) for two traders."""
+ β = 1.0 / (1.0 + r)
+ price_array, *_ = price_learning_two_agents(prior1, prior2, β=β, T=T)
+ return r * price_array
+
+# Figure I: p*(0,0,r) as a function of r
+r_grid = np.linspace(1e-3, 5.0, 200)
+priors = ((1,1), (0.5,0.5))
+p00 = np.array([normalized_price_two_agents(
+ priors[0], priors[1], r, T=300)[0,0]
+ for r in r_grid])
+
+fig, ax = plt.subplots()
+ax.plot(r_grid, p00, lw=2)
+ax.set_xlabel(r'$r$')
+ax.set_ylabel(r'$p^*(0,0,r)$')
+ax.axhline(0.5, color='C1', linestyle='--')
+plt.show()
+```
+
+In the first figure, notice that:
+
+- The resale option pushes the normalized price $p^*(0,0,r)$ above fundamentals $(0.5)$ for any finite $r$.
+
+- As $r$ increases ($\beta$ decreases), the option value fades and $p^*(0,0,r) \to 0.5$.
+
+- At $r = 0.05$ the premium is about $8–9\%$, consistent with Morris (1996, Section IV).
+
+
+```{code-cell} ipython3
+---
+mystnb:
+ figure:
+ caption: Normalized price against time
+ name: fig-normalized-price-time
+---
+# Figure II: p*(t/2,t,0.05) as a function of t
+r = 0.05
+T = 60
+p_mat = normalized_price_two_agents(priors[0], priors[1], r, T=T)
+t_vals = np.arange(0, 54, 2)
+s_vals = t_vals // 2
+y = np.array([p_mat[s, t] for s, t in zip(s_vals, t_vals)])
+
+fig, ax = plt.subplots()
+ax.plot(t_vals, y, lw=2)
+ax.set_xlabel(r'$t$')
+ax.set_ylabel(r'$p^*(t/2,t,0.05)$')
+ax.axhline(0.5, color='C1', linestyle='--')
+plt.show()
+
+p0 = p_mat[0,0]
+μ0 = 0.5
+print("Initial normalized premium at r=0.05 (%):",
+ np.round(100 * (p0 / μ0 - 1.0), 2))
+```
+
+In the second figure, notice that:
+
+- Along the symmetric path $s = t/2$, both traders’ fundamental valuations equal $0.5$ at every $t$, yet the price starts above $0.5$ and declines toward $0.5$ as learning reduces disagreement and the resale option loses value.
+
+
+### General N–trader extension
+
+The same recursion extends to any finite set of Beta priors $\{(a_i,b_i)\}_{i=1}^N$ by taking a max over $i$ each period.
+
+```{code-cell} ipython3
+def price_learning(priors, β=0.75, T=200):
+ """
+ N-trader version with heterogeneous Beta priors.
+ """
+ price_array = np.zeros((T+1, T+1))
+
+ def perp_i(i, s, t):
+ a, b = priors[i]
+ return perpetuity_value(a, b, s, t, β)
+
+ # Terminal condition
+ for s in range(T+1):
+ price_array[s, T] = max(
+ perp_i(i, s, T) for i in range(len(priors)))
+
+ # Backward induction
+ for t in range(T-1, -1, -1):
+ for s in range(t, -1, -1):
+ conts = []
+ for (a, b) in priors:
+ μ = posterior_mean(a, b, s, t)
+ conts.append(μ *
+ (1.0 + price_array[s+1, t+1])
+ + (1.0 - μ) * price_array[s, t+1])
+ price_array[s, t] = β * max(conts)
+
+ return price_array
+
+β = 0.75
+priors = [(1,1), (0.5,0.5), (3,2)]
+price_N = price_learning(priors, β=β, T=150)
+
+# Compute valuations for each trader at (0,0)
+μ_vals = [posterior_mean(a, b, 0, 0) for a, b in priors]
+perp_vals = [(β / (1 - β)) * μ for μ in μ_vals]
+
+print("Three-trader example at (s,t)=(0,0):")
+print(f"Price at (0,0) = {np.round(price_N[0,0], 6)}")
+print(f"\nTrader valuations:")
+for i, (μ, perp) in enumerate(zip(μ_vals, perp_vals), 1):
+ print(f" Trader {i} = {np.round(perp, 6)}")
+```
+
+Note that the asset price is above all traders' valuations.
+
+Morris tells us that no rate dominance exists in this case.
+
+Let's verify this using the code below
+
+```{code-cell} ipython3
+dominant = None
+for i in range(len(priors)):
+ is_dom = all(
+ priors[i][0] >= priors[j][0] and priors[i][1] <= priors[j][1]
+ for j in range(len(priors)) if i != j)
+ if is_dom:
+ dominant = i
+ break
+
+if dominant is not None:
+ print(f"\nTrader {dominant+1} is the global optimist (rate-dominant)")
+else:
+ print(f"\nNo global optimist and speculative premium exists")
+```
+
+Indeed, there is no global optimist and a speculative premium exists.
+
+## Concluding remarks
+
+{cite:t}`Morris1996` uses his model to interpret a ''hot issue'' anomaly described by {cite}`miller1977risk` according to which opening market prices of initial public offerings seem higher than values prices that emerge later.
+
+
+
+## Exercise
+
+```{exercise-start}
+:label: hk_ex3
+```
+
+Morris {cite}`Morris1996` provides a sharp characterization of when speculative bubbles arise.
+
+The key condition is that there is no *global optimist*.
+
+In this exercise, you will verify this condition for the following sets of traders with Beta priors:
+
+1. Trader 1: $\text{Beta}(2,1)$, Trader 2: $\text{Beta}(1,2)$
+2. Trader 1: $\text{Beta}(1,1)$, Trader 2: $\text{Beta}(1/2,1/2)$
+3. Trader 1: $\text{Beta}(3,1)$, Trader 2: $\text{Beta}(2,1)$, Trader 3: $\text{Beta}(1,2)$
+4. Trader 1: $\text{Beta}(1,1)$, Trader 2: $\text{Beta}(1/2,1/2)$, Trader 3: $\text{Beta}(3/2,3/2)$
+
+```{exercise-end}
+```
+
+```{solution-start} hk_ex3
+:class: dropdown
+```
+
+Here is one solution:
+
+```{code-cell} ipython3
+def check_rate_dominance(priors):
+ """
+ Check if any trader rate-dominates all others.
+ """
+ N = len(priors)
+
+ for i in range(N):
+ a_i, b_i = priors[i]
+ is_dominant = True
+
+ for j in range(N):
+ if i == j:
+ continue
+ a_j, b_j = priors[j]
+
+ # Check rate dominance condition
+ if not (a_i >= a_j and b_i <= b_j):
+ is_dominant = False
+ break
+
+ if is_dominant:
+ return i
+
+ return None
+
+# Test cases
+test_cases = [
+ ([(2, 1), (1, 2)], "Global optimist exists"),
+ ([(1, 1), (0.5, 0.5)], "Perpetual switching"),
+ ([(3, 1), (2, 1), (1, 2)], "Three traders with dominant"),
+ ([(1, 1), (0.5, 0.5), (1.5, 1.5)], "Three traders, no dominant")
+]
+
+for priors, description in test_cases:
+ dominant = check_rate_dominance(priors)
+
+ print(f"\n{description}")
+ print(f"Priors: {priors}")
+ print("=="*8)
+ if dominant is not None:
+ print(f"Trader {dominant+1} is the global optimist (rate-dominant)")
+ else:
+ print(f"No global optimist exists")
+ print("=="*8 + "\n")
+```
+
+```{solution-end}
+```
+
+