In addition to what’s in Anaconda, this lecture uses following libraries:
!pip install quantecon
This lecture describes a version of a model of Harrison and Kreps [Harrison and Kreps, 1978].
The model determines the price of a dividend-yielding asset that is traded by two types of self-interested investors.
The model features
Let’s start with some standard imports:
import numpy as np
import quantecon as qe
import scipy.linalg as la
Prior to reading the following, you might like to review our lectures on
Economists differ in how they define a bubble.
The Harrison-Kreps model illustrates the following notion of a bubble that attracts many economists:
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.
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 asset payouts.
There is a fixed number $ A $ of shares of an asset.
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\} $.
The dividend obeys
$$ d_t = \begin{cases} 0 & \text{ if } s_t = 0 \\ 1 & \text{ if } s_t = 1 \end{cases} $$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.
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
$$ P(i,j) = \mathbb P\{s_{t+1} = j \mid s_t = i\} $$Investors of type $ a $ believe the transition matrix
$$ P_a = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{2}{3} & \frac{1}{3} \end{bmatrix} $$Investors of type $ b $ think the transition matrix is
$$ P_b = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{4} & \frac{3}{4} \end{bmatrix} $$Thus, in state $ 0 $, a type $ a $ investor is more optimistic about next period’s dividend than is investor $ b $.
But in state $ 1 $, a type $ a $ investor is more pessimistic about next period’s dividend than is investor $ b $.
The stationary (i.e., invariant) distributions of these two matrices can be calculated as follows:
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
mcb.stationary_distributions
The stationary distribution of $ P_a $ is approximately $ \pi_a = \begin{bmatrix} .57 & .43 \end{bmatrix} $.
The stationary distribution of $ P_b $ is approximately $ \pi_b = \begin{bmatrix} .43 & .57 \end{bmatrix} $.
Thus, a type $ a $ investor is more pessimistic on average.
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 $.
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 [Harrison and Kreps, 1978] 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.
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.
No short sales are allowed.
This matters because it limits how pessimists can express their opinions.
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.
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.
Investors know a price function mapping the state $ s_t $ at $ t $ into the equilibrium price $ p(s_t) $ that prevails in that state.
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 $.
Now let’s turn to solving the model.
We’ll determine equilibrium prices under a particular specification of beliefs and constraints on trading selected from one of the specifications described above.
We shall compare equilibrium price functions under the following alternative assumptions about beliefs:
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 table reports implications of Harrison and Kreps’s specifications of $ P_a, P_b, \beta $.
| $ 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
We’ll explain these values and how they are calculated one row at a time.
The row corresponding to $ p_o $ applies when both types of investor have enough resources to purchase the entire stock of the asset and strict short sales constraints prevail so that temporarily optimistic investors always price the asset.
The row corresponding to $ p_p $ would apply if neither type of investor has enough resources to purchase the entire stock of the asset and both types must hold the asset.
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.
We’ll start by pricing the asset under homogeneous beliefs.
(This is the case treated in the lecture on asset pricing with finite Markov states)
Suppose that there is only one type of investor, either of type $ a $ or $ b $, and that this investor always “prices the asset”.
Let $ p_h = \begin{bmatrix} p_h(0) \cr p_h(1) \end{bmatrix} $ be the equilibrium price vector when all investors are of type $ h $.
The price today equals the expected discounted value of tomorrow’s dividend and tomorrow’s price of the asset:
$$ p_h(s) = \beta \left( P_h(s,0) (0 + p_h(0)) + P_h(s,1) ( 1 + p_h(1)) \right), \quad s = 0, 1 \tag{76.1} $$
These equations imply that the equilibrium price vector is
$$ \begin{bmatrix} p_h(0) \cr p_h(1) \end{bmatrix} = \beta [I - \beta P_h]^{-1} P_h \begin{bmatrix} 0 \cr 1 \end{bmatrix} \tag{76.2} $$
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
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)
# Next compute prices
prices = β * imbq_inv @ transition @ dividend_payoff
return prices
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.
There are several cases to consider.
The first is when both types of agents have sufficient wealth to purchase all of the asset themselves.
In this case, the marginal investor who prices the asset is the more optimistic type so that the equilibrium price $ \bar p $ satisfies Harrison and Kreps’s key equation:
$$ \bar p(s) = \beta \max \left\{ P_a(s,0) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) ,\; P_b(s,0) \bar p(0) + P_b(s,1) ( 1 + \bar p(1)) \right\} \tag{76.3} $$
for $ s=0,1 $.
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
$$ P_a(s,0) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) > P_b(s,0) \bar p(0) + P_b(s,1) ( 1 + \bar p(1)) $$The marginal investor is of type $ b $ if
$$ P_a(s,1) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) < P_b(s,1) \bar p(0) + P_b(s,1) ( 1 + \bar p(1)) $$Thus the marginal investor is the (temporarily) optimistic type.
Equation (76.3) is a functional equation that, like a Bellman equation, can be solved by
$$ \bar p^{j+1}(s) = \beta \max \left\{ P_a(s,0) \bar p^j(0) + P_a(s,1) ( 1 + \bar p^j(1)) ,\; P_b(s,0) \bar p^j(0) + P_b(s,1) ( 1 + \bar p^j(1)) \right\} \tag{76.4} $$
for $ s=0,1 $.
The third row of the table labeled $ p_o $ reports equilibrium prices that solve the functional equation when $ \beta = .75 $.
Here the type that is optimistic about $ s_{t+1} $ prices the asset in state $ s_t $.
It is instructive to compare these prices with the equilibrium prices for the homogeneous belief economies that solve under beliefs $ P_a $ and $ P_b $ reported in the rows labeled $ p_a $ and $ p_b $, respectively.
Equilibrium prices $ p_o $ in the heterogeneous beliefs economy evidently exceed what any prospective investor regards as the fundamental value of the asset in each possible state.
Nevertheless, the economy recurrently visits a state that makes each investor want to purchase the asset for more than he believes its future dividends are worth.
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.
Evidently, $ \hat p_a(1) < \bar p(1) $ and $ \hat p_b(0) < \bar p(0) $.
Investors of type $ a $ want to sell the asset in state $ 1 $ while investors of type $ b $ want to sell it in state $ 0 $.
Here’s code to solve for $ \bar p $, $ \hat p_a $ and $ \hat p_b $ using the iterative method described above
def price_optimistic_beliefs(transitions, dividend_payoff, β=.75,
max_iter=50000, tol=1e-16):
"""
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.]])
# 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],
1)
# If we succeed in converging, break out of for loop
if np.max(np.sqrt((p_new - p_old)**2)) < tol:
break
ptwiddle = β * np.min([q @ p_old
+ q @ dividend_payoff for q in transitions],
1)
phat_a = np.array([p_new[0], ptwiddle[1]])
phat_b = np.array([ptwiddle[0], p_new[1]])
return p_new, phat_a, phat_b
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.
In this case, the asset price must adjust to attract pessimistic investors.
Instead of equation (76.3), the equilibrium price satisfies
$$ \check p(s) = \beta \min \left\{ P_a(s,1) \check p(0) + P_a(s,1) ( 1 + \check p(1)) ,\; P_b(s,1) \check p(0) + P_b(s,1) ( 1 + \check p(1)) \right\} \tag{76.5} $$
and the marginal investor who prices the asset is always the one that values it less highly than does the other type.
Now the marginal investor is always the (temporarily) pessimistic type.
Notice from the sixth row of that the pessimistic price $ p_o $ is lower than the homogeneous belief prices $ p_a $ and $ p_b $ in both states.
When pessimistic investors price the asset according to (76.5), optimistic investors think that the asset is underpriced.
If they could, optimistic investors would willingly borrow at a one-period risk-free gross interest rate $ \beta^{-1} $ to purchase more of the asset.
Implicit constraints on leverage prohibit them from doing so.
When optimistic investors price the asset as in equation (76.3), pessimistic investors think that the asset is overpriced and would like to sell the asset short.
Constraints on short sales prevent that.
Here’s code to solve for $ \check p $ using iteration
def price_pessimistic_beliefs(transitions, dividend_payoff, β=.75,
max_iter=50000, tol=1e-16):
"""
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.]])
# 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],
1)
# If we succeed in converging, break out of for loop
if np.max(np.sqrt((p_new - p_old)**2)) < tol:
break
return p_new
Jose Scheinkman [Scheinkman, 2014] 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.
Scheinkman stresses these features of the Harrison-Kreps model:
Scheinkman takes this as a strength of the model because he observes high volume during famous bubbles.
Scheinkman extracts insights about the effects of financial regulations on bubbles.
He emphasizes how limiting short sales and limiting leverage have opposite effects.
This exercise invites you to recreate the summary table using the functions we have built above.
| $ 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.
In addition, below we’ll add an interpretation of the row corresponding to $ p_o $ by inventing two additional types of agents, one of whom is permanently optimistic, the other who is permanently pessimistic.
We construct subjective transition probability matrices for our permanently optimistic and permanently pessimistic investors as follows.
The permanently optimistic investors(i.e., the investor with the most optimistic beliefs in each state) believes the transition matrix
$$ P_o = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{3}{4} \end{bmatrix} $$The permanently pessimistic investor believes the transition matrix
$$ P_p = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{2}{3} & \frac{1}{3} \end{bmatrix} $$We’ll use these transition matrices when we present our solution of exercise 1 below.
First, we will obtain equilibrium price vectors with homogeneous beliefs, including when all investors are optimistic or pessimistic.
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
qopt = np.array([[1/2, 1/2], [1/4, 3/4]])
# Pessimistic investor transition matrix
qpess = np.array([[2/3, 1/3], [2/3, 1/3]])
dividendreturn = np.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)
print(f"State 0: {s0}")
print(f"State 1: {s1}")
print("-" * 20)
We will use the price_optimistic_beliefs function to find the price under heterogeneous beliefs.
opt_beliefs = price_optimistic_beliefs([qa, qb], dividendreturn)
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)
print(f"State 0: {s0}")
print(f"State 1: {s1}")
print("-" * 20)
Notice that the equilibrium price with heterogeneous beliefs is equal to the price under single beliefs with permanently optimistic investors - this is due to the marginal investor in the heterogeneous beliefs equilibrium always being the type who is temporarily optimistic.
[1] 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.