## Introduction

This lecture is a laboratory for experimenting with instances of competitive equilibria of an infinite-horizon pure exchange economy with

This lecture is a laboratory for experimenting with competitive equilibria of an infinite-horizon pure exchange economy with

* Markov endowments

* Complete markets in one-period Arrow state-contingent securities

* Discounted expected utility preferences of a kind often specified in macro and finance

* Discounted expected utility preferences of a kind often used in macroeconomics and finance

* Common expected utility preferences across agents

* Common beliefs across agents

* A constant relative risk aversion (CRRA) one-period utility function that implies the existence of a representative consumer whose consumption process can be plugged into a formula for the pricing kernel for one-step Arrow securities and thereby determine equilbrium prices before determing an equilibrium distribution of wealth

* A constant relative risk aversion (CRRA) one-period utility function that implies the existence of a representative consumer whose consumption process can be plugged into a formula for the pricing kernel for one-step Arrow securities and thereby determine equilbrium prices before determining an equilibrium distribution of wealth

* Diverse endowments across agents that provide motivations for reallocating goods across time and Markov states

* Diverse endowments across agents that provide motivations to reallocate across time and Markov states

We impose enough restrictions to allow us to **Bellmanize** competitive equilibrium prices and quantities

We impose restrictions that allow us to **Bellmanize** competitive equilibrium prices and quantities

We use Bellman equations to describe

In the course of presenting the model we shall describe these important ideas

* the widespread use a **resolvent operator** in this class of models

* a **resolvent operator** widely used in this class of models

* the necessity of state-by-state **borrowing limits** in infinite horizon economies

* state-by-state **borrowing limits** required in infinite horizon economies

* the absence of any required **borrowing limits** in finite horizon economies

* absence of **borrowing limits** in finite horizon economies

* a counterpart of the law of iterated expectations known as a **law of iterated values**

* a notion of **state-variable degeneracy** that prevails within a competitive equilibrium and that explains repeated appearances of resolvent operators

* a **state-variable degeneracy** that prevails within a competitive equilibrium and that explains many appearances of resolvent operators

+++

In this lecture we shall follow much of the

literatures in macroeconomics and econometrics and assume that

$\pi_t(s^t)$ is induced by a Markov process.

$\pi_t(s^t)$ is induced by a Markov process.

There are $I$ consumers named $i=1, \ldots , I$.

Consumer $i$

purchases a history-dependent consumption plan $c^i =

\{c_t^i(s^t)\}_{t=0}^\infty$ and

orders these

consumption streams by

\{c_t^i(s^t)\}_{t=0}^\infty$

Consumer $i$ orders consumption plans by

$$ U_i(c^i) =

\sum_{t=0}^\infty \sum_{s^t} \beta^t u_i[c_t^i(s^t)]

We adopt the assumption, routinely

employed in much of macroeconomics,

that consumers share probabilities $\pi_t(s^t)$ for all $t$ and $s^t$.

that consumers share probabilities $\pi_t(s^t)$ for all $t$ and $s^t$.

A **feasible allocation** satisfies

* $v^i(a,s)$ is the maximum expected discounted utility that consumer $i$ with current financial wealth $a$ can attain in state $s$.

The value function satisfies the Bellman equation

The optimal value function satisfies the Bellman equation

$$

v^i(a, s) = \max_{c, \hat a(s')} \left\{ u_i(c) + \beta \sum_{s'} v^i[\hat a(s'),s'] \pi (s' | s) \right\}

\leq y^i(s) + a

$$

and also

and also the constraints

$$

\begin{aligned}

The third condition asserts that there are zero net aggregate claims in all Markov states.

The fourth condition asserts that the economy is closed and starts off from a position in which there

The fourth condition asserts that the economy is closed and starts from a situation in which there

are zero net claims in the aggregate.

If an allocation and prices in a recursive competitive equilibrium are to be

Starting the system off with $a_0^i =0$ forall $i$ has a striking implication that we can call **state variable degeneracy**.

Starting the system with $a_0^i =0$ forall $i$ has a striking implication that we can call **state variable degeneracy**.

Here is what we mean by **state variable degeneracy**:

Thus, although there are two state variables in the value function $v^i(a,s)$, within a recursive competitive equilibrium

Notice that although there are two state variables in the value function $v^i(a,s)$, within a recursive competitive equilibrium

starting from $a_0^i = 0 \ \forall i$ at the starting Markov state $s_0$, two outcomes prevail:

* $a_0^i = 0 $ for all $i$ whenever the Markov state $s_t$ returns to $s_0$.

* Financial wealth $a$ is an exact function of the Markov state $s$.

The first finding asserts that each household recurrently visits the zero financial wealth state with which he began life.

The first finding asserts that each household recurrently visits the zero financial wealth state with which it began life.

The second finding asserts that the exogenous Markov state is all we require to track an individual within a competitive equilibrium.

+++

We are ready to dive into some Bellman equations and some Python code.

As usual, we start with Python imports

```{code-cell} ipython3

import numpy as np

import matplotlib.pyplot as plt

%matplotlib inline

```

```{code-cell} ipython3

np.set_printoptions(suppress=True)

```

### Markov asset prices primer

$$

We'll use these objects to state the following useful facts

We'll use these objects to state a useful property in asset pricing theory.

### Laws of iterated expectations and iterated values

V(d(s_{t+j})|s_t) = \sum_{s_{t+j}} d(s_{t+j}) Q_j(s_{t+j}| s_t)

$$

The law of iterated values states

The **law of iterated values** states

$$

V \left[ V (d(s_{t+j}) | s_{t+1}) \right] | s_t = V(d(s_{t+j}))| s_t

where it is understood that $ u(\alpha_k y)$ is a vector.

+++

Below we solve several fun examples with Python code.

We are ready to dive into some Python code.

As usual, we start with Python imports.

```{code-cell} ipython3

import numpy as np

import matplotlib.pyplot as plt

%matplotlib inline

```

```{code-cell} ipython3

np.set_printoptions(suppress=True)

```

First, we create a Python class to compute the objects that comprise a competitive equilibrium

with sequential trading of one-period Arrow securities.

(The reader will notice that the code is set up to handle finite-horizon economies indexed by horizon $T$.

We'll study some finite horizon economies after we look at some infinite-horizon economies.)

The reader will notice that the code is set up to handle finite-horizon economies indexed by horizon $T$.

We'll study some finite horizon economies after we look at some infinite-horizon economies.

```{code-cell} ipython3

class RecurCompetitive: