This lecture describes a version of a model of Harrison and Kreps [HK78]
The model determines the price of a dividend-yielding asset that is traded by two types of self-interested investors
The model features
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
using InstantiateFromURL
github_project("QuantEcon/quantecon-notebooks-julia", version = "0.2.0")
Activated /home/qebuild/repos/lecture-source-jl/_build/website/jupyter/Project.toml Info quantecon-notebooks-julia 0.1.0 activated, 0.2.0 requested
using LinearAlgebra, Statistics
The model simplifies by ignoring alterations in the distribution of wealth among investors having 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
$$ d_t = \begin{cases} 0 & \text{ if } s_t = 0 \\ 1 & \text{ if } s_t = 1 \end{cases} $$The owner of a share at the beginning of time $ t $ is entitled to the dividend paid at time $ t $
The owner of the share at the beginning of time $ t $ is also entitled to sell the share to another investor during time $ t $
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
$$ P_b = \begin{bmatrix} \frac{2}{3} & \frac{1}{3} \\ \frac{1}{4} & \frac{3}{4} \end{bmatrix} $$The stationary (i.e., invariant) distributions of these two matrices can be calculated as follows:
using QuantEcon
qa = [1/2 1/2; 2/3 1/3]
qb = [2/3 1/3; 1/4 3/4]
mcA = MarkovChain(qa)
mcB = MarkovChain(qb)
stA = stationary_distributions(mcA)
1-element Array{Array{Float64,1},1}: [0.5714285714285715, 0.4285714285714286]
stB = stationary_distributions(mcB)
1-element Array{Array{Float64,1},1}: [0.42857142857142855, 0.5714285714285714]
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} $
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 did
We’ll eventually study the consequences of two different 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 investor always hold at least some of the asset
No short sales are allowed
This matters because it limits pessimists from expressing 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
Transition matrices for the temporarily optimistic and pessimistic investors are constructed as follows
Temporarily optimistic investors (i.e., the investor with the most optimistic beliefs in each state) believe the transition matrix
$$ P_o = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{3}{4} \end{bmatrix} $$Temporarily pessimistic believe the transition matrix
$$ P_p = \begin{bmatrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{4} & \frac{3}{4} \end{bmatrix} $$We’ll return to these matrices and their significance in the exercise
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
This amounts to determining equilibrium prices under the different possible specifications of beliefs and constraints listed above
In particular, we 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 (you will be asked to recreate the table in an exercise)
It records 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
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:
$$ p_h(s) = \beta \left( P_h(s,0) (0 + p_h(0)) + P_h(s,1) ( 1 + p_h(1)) \right), \quad s = 0, 1 $$These equations imply that the equilibrium price vector is
$$ \begin{bmatrix} p_h(0) \cr p_h(1) \end{bmatrix} = \beta [I - \beta P_h]^{-1} P_h \begin{bmatrix} 0 \cr 1 \end{bmatrix} \tag{1} $$
The first two rows of of the table report $ p_a(s) $ and $ p_b(s) $
Here’s a function that can be used to compute these values
using LinearAlgebra
function price_single_beliefs(transition, dividend_payoff;
β=.75)
# First compute inverse piece
imbq_inv = inv(I - β * transition)
# Next compute prices
prices = β * ((imbq_inv * transition) * dividend_payoff)
return prices
end
price_single_beliefs (generic function with 1 method)
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 agent 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:
$$ \bar p(s) = \beta \max \left\{ P_a(s,0) \bar p(0) + P_a(s,1) ( 1 + \bar p(1)) ,\; P_b(s,0) \bar p(0) + P_b(s,1) ( 1 + \bar p(1)) \right\} \tag{2} $$
for $ s=0,1 $
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 (2) is a functional equation that, like a Bellman equation, can be solved by
$$ \bar p^{j+1}(s) = \beta \max \left\{ P_a(s,0) \bar p^j(0) + P_a(s,1) ( 1 + \bar p^j(1)) ,\; P_b(s,0) \bar p^j(0) + P_b(s,1) ( 1 + \bar p^j(1)) \right\} \tag{3} $$
for $ s=0,1 $
The third row of the table 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 $
Equilibrium prices $ \bar p $ in the heterogeneous beliefs economy 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
The reason is that he expects to have the option to sell the asset later to another investor who will value the asset more highly than he will
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
function price_optimistic_beliefs(transitions,
dividend_payoff;
β=.75, max_iter=50000,
tol=1e-16)
# We will guess an initial price vector of [0, 0]
p_new = [0,0]
p_old = [10.0,10.0]
# We know this is a contraction mapping, so we can iterate to conv
for i ∈ 1:max_iter
p_old = p_new
temp = [maximum((q * p_old) + (q * dividend_payoff))
for q in transitions]
p_new = β * temp
# If we succed in converging, break out of for loop
if maximum(sqrt, ((p_new - p_old).^2)) < 1e-12
break
end
end
temp=[minimum((q * p_old) + (q * dividend_payoff)) for q in transitions]
ptwiddle = β * temp
phat_a = [p_new[1], ptwiddle[2]]
phat_b = [ptwiddle[1], p_new[2]]
return p_new, phat_a, phat_b
end
price_optimistic_beliefs (generic function with 1 method)
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 (2), the equilibrium price satisfies
$$ \check p(s) = \beta \min \left\{ P_a(s,1) \check p(0) + P_a(s,1) ( 1 + \check p(1)) ,\; P_b(s,1) \check p(0) + P_b(s,1) ( 1 + \check p(1)) \right\} \tag{4} $$
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 $ \underline p $ is lower than the homogeneous belief prices $ p_a $ and $ p_b $ in both states
When pessimistic investors price the asset according to (4), optimistic investors think that the asset is underpriced
If they could, optimistic investors would willingly borrow at the one-period 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 (2), 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
function price_pessimistic_beliefs(transitions,
dividend_payoff;
β=.75, max_iter=50000,
tol=1e-16)
# We will guess an initial price vector of [0, 0]
p_new = [0, 0]
p_old = [10.0, 10.0]
# We know this is a contraction mapping, so we can iterate to conv
for i ∈ 1:max_iter
p_old = p_new
temp=[minimum((q * p_old) + (q* dividend_payoff)) for q in transitions]
p_new = β * temp
# If we succed in converging, break out of for loop
if maximum(sqrt, ((p_new - p_old).^2)) < 1e-12
break
end
end
return p_new
end
price_pessimistic_beliefs (generic function with 1 method)
[Sch14] 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 the asset’s underlying dividend stream
Scheinkman stresses these features of the Harrison-Kreps model:
Type $ a $ investors sell the entire stock of the asset to type $ b $ investors every time the state switches from $ s_t =0 $ to $ s_t =1 $
Type $ b $ investors sell the asset to type $ a $ investors every time the state switches from $ s_t = 1 $ to $ s_t =0 $
Scheinkman takes this as a strength of the model because he observes high volume during famous bubbles
Scheinkman extracts insights about effects of financial regulations on bubbles
He emphasizes how limiting short sales and limiting leverage have opposite effects
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 first need to define the transition matrices and dividend payoff vector
First we will obtain equilibrium price vectors with homogeneous beliefs, including when all investors are optimistic or pessimistic
qa = [1/2 1/2; 2/3 1/3] # Type a transition matrix
qb = [2/3 1/3; 1/4 3/4] # Type b transition matrix
qopt = [1/2 1/2; 1/4 3/4] # Optimistic investor transition matrix
qpess = [2/3 1/3; 2/3 1/3] # Pessimistic investor transition matrix
dividendreturn = [0; 1]
transitions = [qa, qb, qopt, qpess]
labels = ["p_a", "p_b", "p_optimistic", "p_pessimistic"]
for (transition, label) in zip(transitions, labels)
println(label)
println(repeat("=", 20))
s0, s1 = round.(price_single_beliefs(transition, dividendreturn), digits=2)
println("State 0: $s0")
println("State 1: $s1")
println(repeat("-", 20))
end
p_a ==================== State 0: 1.33 State 1: 1.22 -------------------- p_b ==================== State 0: 1.45 State 1: 1.91 -------------------- p_optimistic ==================== State 0: 1.85 State 1: 2.08 -------------------- p_pessimistic ==================== State 0: 1.0 State 1: 1.0 --------------------
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) ∈ zip(opt_beliefs, labels)
println(label)
println(repeat("=", 20))
s0, s1 = round.(p, digits = 2)
println("State 0: $s0")
println("State 1: $s1")
println(repeat("-", 20))
end
p_optimistic ==================== State 0: 1.85 State 1: 2.08 -------------------- p_hat_a ==================== State 0: 1.85 State 1: 1.69 -------------------- p_hat_b ==================== State 0: 1.69 State 1: 2.08 --------------------
Notice that the equilibrium price with heterogeneous beliefs is equal to the price under single beliefs with optimistic investors - this is due to the marginal investor being the temporarily optimistic type
Footnotes
[1] By assuming that both types of agent 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.