\n",
" \n",
" \n",
" \n",
"

"
]
},
{
"cell_type": "markdown",
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"source": [
"# Asset Pricing III: Incomplete Markets\n",
"\n",
"\n",
""
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Contents\n",
"\n",
"- [Asset Pricing III: Incomplete Markets](#Asset-Pricing-III:--Incomplete-Markets) \n",
" - [Overview](#Overview) \n",
" - [Structure of the Model](#Structure-of-the-Model) \n",
" - [Solving the Model](#Solving-the-Model) \n",
" - [Exercises](#Exercises) \n",
" - [Solutions](#Solutions) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Overview\n",
"\n",
"This lecture describes a version of a model of Harrison and Kreps [[HK78]](https://julia.quantecon.org/../zreferences.html#harrkreps1978).\n",
"\n",
"The model determines the price of a dividend-yielding asset that is traded by two types of self-interested investors.\n",
"\n",
"The model features\n",
"\n",
"- heterogeneous beliefs \n",
"- incomplete markets \n",
"- short sales constraints, and possibly $ \\ldots $ \n",
"- (leverage) limits on an investor’s ability to borrow in order to finance purchases of a risky asset "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### References\n",
"\n",
"Prior to reading the following you might like to review our lectures on\n",
"\n",
"- [Markov chains](https://julia.quantecon.org/../tools_and_techniques/finite_markov.html) \n",
"- [Asset pricing with finite state space](https://julia.quantecon.org/markov_asset.html) "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Bubbles\n",
"\n",
"Economists differ in how they define a *bubble*.\n",
"\n",
"The Harrison-Kreps model illustrates the following notion of a bubble that attracts many economists:\n",
"\n",
"> *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.*"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Setup"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"hide-output": true
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"outputs": [],
"source": [
"using InstantiateFromURL\n",
"# optionally add arguments to force installation: instantiate = true, precompile = true\n",
"github_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.8.0\")"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"hide-output": false
},
"outputs": [],
"source": [
"using LinearAlgebra, Statistics"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Structure of the Model\n",
"\n",
"The model simplifies by ignoring alterations in the distribution of wealth\n",
"among investors having different beliefs about the fundamentals that determine\n",
"asset payouts.\n",
"\n",
"There is a fixed number $ A $ of shares of an asset.\n",
"\n",
"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\\} $.\n",
"\n",
"The dividend obeys\n",
"\n",
"$$\n",
"d_t =\n",
"\\begin{cases}\n",
" 0 & \\text{ if } s_t = 0 \\\\\n",
" 1 & \\text{ if } s_t = 1\n",
"\\end{cases}\n",
"$$\n",
"\n",
"The owner of a share at the beginning of time $ t $ is entitled to the dividend paid at time $ t $.\n",
"\n",
"The owner of the share at the beginning of time $ t $ is also entitled to sell the share to another investor during time $ t $.\n",
"\n",
"Two types $ h=a, b $ of investors differ only in their beliefs about a Markov transition matrix $ P $ with typical element\n",
"\n",
"$$\n",
"P(i,j) = \\mathbb P\\{s_{t+1} = j \\mid s_t = i\\}\n",
"$$\n",
"\n",
"Investors of type $ a $ believe the transition matrix\n",
"\n",
"$$\n",
"P_a =\n",
" \\begin{bmatrix}\n",
" \\frac{1}{2} & \\frac{1}{2} \\\\\n",
" \\frac{2}{3} & \\frac{1}{3}\n",
" \\end{bmatrix}\n",
"$$\n",
"\n",
"Investors of type $ b $ think the transition matrix is\n",
"\n",
"$$\n",
"P_b =\n",
" \\begin{bmatrix}\n",
" \\frac{2}{3} & \\frac{1}{3} \\\\\n",
" \\frac{1}{4} & \\frac{3}{4}\n",
" \\end{bmatrix}\n",
"$$\n",
"\n",
"The stationary (i.e., invariant) distributions of these two matrices can be calculated as follows:"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"hide-output": false
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"outputs": [
{
"data": {
"text/plain": [
"1-element Array{Array{Float64,1},1}:\n",
" [0.5714285714285715, 0.4285714285714286]"
]
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"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
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"source": [
"using QuantEcon\n",
"\n",
"qa = [1/2 1/2; 2/3 1/3]\n",
"qb = [2/3 1/3; 1/4 3/4]\n",
"mcA = MarkovChain(qa)\n",
"mcB = MarkovChain(qb)\n",
"stA = stationary_distributions(mcA)"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"hide-output": false
},
"outputs": [
{
"data": {
"text/plain": [
"1-element Array{Array{Float64,1},1}:\n",
" [0.42857142857142855, 0.5714285714285714]"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"stB = stationary_distributions(mcB)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The stationary distribution of $ P_a $ is approximately $ \\pi_A = \\begin{bmatrix} .57 & .43 \\end{bmatrix} $.\n",
"\n",
"The stationary distribution of $ P_b $ is approximately $ \\pi_B = \\begin{bmatrix} .43 & .57 \\end{bmatrix} $."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Ownership Rights\n",
"\n",
"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 $.\n",
"\n",
"Both types of investors are risk-neutral and both have the same fixed discount factor $ \\beta \\in (0,1) $.\n",
"\n",
"In our numerical example, we’ll set $ \\beta = .75 $, just as Harrison and Kreps did.\n",
"\n",
"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.\n",
"\n",
"1. Both types of investors have enough resources (either wealth or the capacity to borrow) so that they can purchase the entire available stock of the asset **[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."
]
}
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