54. Competitive equilibria with Arrow securities

54.1. Introduction

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 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 determining an equilibrium distribution of wealth

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

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

We use Bellman equations to describe

  • asset prices

  • continuation wealths

  • state-by-state natural debt limits

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

  • a resolvent operator widely used in this class of models

  • state-by-state borrowing limits required in infinite 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 state-variable degeneracy that prevails within a competitive equilibrium and that explains many appearances of resolvent operators

54.2. The setting

In effect, this lecture implements a Python version of the model presented in section 9.3.3 of Ljungqvist and Sargent [LS18].

54.2.1. Preferences and endowments

In each period \(t\geq 0\), there is a realization of a stochastic event \(s_t \in {\bf S}\).

Let the history of events up and until time \(t\) be denoted \(s^t = [s_0, s_{1}, \ldots, s_{t-1}, s_t]\).

Sometimes we inadvertently reverse the recording order and denote a history as \(s^t = [s_t, s_{t-1}, \ldots, s_1, s_0]\).

The unconditional probability of observing a particular sequence of events \(s^t\) is given by a probability measure \(\pi_t(s^t)\).

For \(t > \tau\), we write the probability of observing \(s^t\) conditional on the realization of \(s^\tau\)as \(\pi_t(s^t\vert s^\tau)\).

We assume that trading occurs after observing \(s_0\), which we capture by setting \(\pi_0(s_0)=1\) for the initially given value of \(s_0\).

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.

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

Consumer \(i\) owns a stochastic endowment of one good \(y_t^i(s^t)\) that depends on the history \(s^t\).

The history \(s^t\) is publicly observable.

Consumer \(i\) purchases a history-dependent consumption plan \(c^i = \{c_t^i(s^t)\}_{t=0}^\infty\)

Consumer \(i\) orders consumption plans by

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

where \(0 < \beta < 1\).

The right side is equal to \( E_0 \sum_{t=0}^\infty \beta^t u_i(c_t^i) \), where \(E_0\) is the mathematical expectation operator, conditioned on \(s_0\).

Here \(u_i(c)\) is an increasing, twice continuously differentiable, strictly concave function of consumption \(c\geq 0\) of one good.

The utility function satisfies the Inada condition

\[ \lim_{c \downarrow 0} u'_i(c) = +\infty.\]

This condition implies that each agent chooses strictly positive consumption for every date-history pair.

Those interior solutions enable us to confine our analysis to Euler equations that hold with equality and also guarantee that natural debt limits don’t bind in economies like ours with sequential trading of Arrow securities.

We adopt the assumption, routinely employed in much of macroeconomics, that consumers share probabilities \(\pi_t(s^t)\) for all \(t\) and \(s^t\).

A feasible allocation satisfies

\[ \sum_i c_t^i(s^t) \leq \sum_i y_t^i(s^t) \]

for all \(t\) and for all \(s^t\).

54.2.2. Recursive formulation

Following descriptions in section 9.3.3 of Ljungqvist and Sargent [LS18] chapter 9, we set up a competitive equilibrium of a pure exchange economy with complete markets in one-period Arrow securities.

When endowments \(y^i(s)\) are all functions of a common Markov state \(s\), the pricing kernel takes the form \(Q(s'|s)\).

These enable us to provide a recursive formulation of a consumer’s optimization problem.

Consumer \(i\)’s state at time \(t\) is its financial wealth \(a^i_t\) and Markov state \(s_t\).

Let \(v^i(a,s)\) be the optimal value of consumer \(i\)’s problem starting from state \((a, s)\).

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

The optimal value function satisfies the Bellman equation

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

where maximization is subject to the budget constraint

\[ c + \sum_{s'} \hat a(s') Q(s' | s) \leq y^i(s) + a \]

and also the constraints

\[\begin{split} \begin{aligned} c & \geq 0, \\ - \hat a(s') & \leq \bar A^i(s'), \hskip.5cm \forall s'. \end{aligned} \end{split}\]

with the second constraint evidently being a set of state-by-state debt limits.

Note that the value function and decision rule that solve the Bellman equation implicitly depend on the pricing kernel \(Q(\cdot \vert \cdot)\) because it appears in the agent’s budget constraint.

Use the first-order conditions for the problem on the right of the Bellman equation and a Benveniste-Scheinkman formula and rearrange to get

\[ Q(s_{t+1} | s_t ) = {\beta u'_i(c_{t+1}^i) \pi(s_{t+1} | s_t) \over u'_i(c_t^i) }, \]

where it is understood that \(c_t^i = c^i(s_t)\) and \(c_{t+1}^i = c^i(s_{t+1})\).

A recursive competitive equilibrium is an initial distribution of wealth \(\vec a_0\), a set of borrowing limits \(\{\bar A^i(s)\}_{i=1}^I\), a pricing kernel \(Q(s' | s)\), sets of value functions \(\{v^i(a,s)\}_{i=1}^I\), and decision rules \(\{c^i(s), a^i(s)\}_{i=1}^I\) such that

  • The state-by-state borrowing constraints satisfy the recursion

\[ \bar A^i(s) = y^i(s) + \sum_{s'} Q(s'|s) \bar A^i(s') \]
  • For all \(i\), given \(a^i_0\), \(\bar A^i(s)\), and the pricing kernel, the value functions and decision rules solve the consumer’s problem;

  • For all realizations of \(\{s_t\}_{t=0}^\infty\), the consumption and asset portfolios \(\{\{c^i_t,\) \(\{\hat a^i_{t+1}(s')\}_{s'}\}_i\}_t\) satisfy \(\sum_i c^i_t = \sum_i y^i(s_t)\) and \(\sum_i \hat a_{t+1}^i(s') = 0\) for all \(t\) and \(s'\).

  • The initial financial wealth vector \(\vec a_0\) satisfies \(\sum_{i=1}^I a_0^i = 0 \).

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 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 consistent with the equilibrium allocation and price system that prevail in a corresponding complete markets economy with all trades occurring at time \(0\), we must impose that \(a_0^i = 0\) for \(i = 1, \ldots , I\).

That is what assures that at time \(0\) the present value of each agent’s consumption equals the present value of his endowment stream, the single budget constraint in arrangement with all trades occurring at time \(0\).

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:

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 it began life.

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

Financial wealth turns out to be redundant because it is an exact function of the Markov state for each individual.

This outcome depends critically on there being complete markets in Arrow securities.

54.2.3. Markov asset prices primer

Let’s start with a brief summary of formulas for computing asset prices in a Markov setting.

The setup assumes the following infrastructure

  • Markov states: \(s \in S = \left[\bar{s}_1, \ldots, \bar{s}_n \right]\) governed by an \(n\)-state Markov chain with transition probability

\[ P_{ij} = \Pr \left\{s_{t+1} = \bar{s}_j \mid s_t = \bar{s}_i \right\} \]
  • A collection \(k=1,\ldots, K\) of \(n \times 1\) vectors of \(K\) assets that pay off \(d^k\left(s\right)\) in state \(s\)

  • An \(n \times n\) matrix pricing kernel \(Q\) for one-period Arrow securities, where \( Q_{ij}\) = price at time \(t\) in state \(s_t = \bar s_i\) of one unit of consumption when \(s_{t+1} = \bar s_j\) at time \(t+1\):

\[ Q_{ij} = {\textrm{Price}} \left\{s_{t+1} = \bar{s}_j \mid s_t = \bar{s}_i \right\} \]
  • The price of risk-free one-period bond in state \(i\) is \(R_i^{-1} = \sum_{j}Q_{i,j}\)

  • The gross rate of return on a one-period risk-free bond Markov state \(\bar s_i\) is \(R_i = (\sum_j Q_{i,j})^{-1}\)

At this point, we’ll take the pricing kernel \(Q\) as exogenous, i.e., determined outside the model

Two examples would be

  • \( Q = \beta P \) where \(\beta \in (0,1) \)

  • \(Q = S P \) where \(S\) is an \(n \times n\) matrix of stochastic discount factors

We’ll write down implications of Markov asset pricing in a nutshell for two types of assets

  • the price in Markov state \(s\) at time \(t\) of a cum dividend stock that entitles the owner at the beginning of time \(t\) to the time \(t\) dividend and the option to sell the asset at time \(t+1\). The price evidently satisfies \(p^k(\bar s_i) = d^k(\bar s_i) + \sum_j Q_{ij} p^k(\bar s_j) \), which implies that the vector \(p^k\) satisfies \(p^k = d^k + Q p^k\) which implies the formula

\[ p^k = (I - Q)^{-1} d^k \]
  • the price in Markov state \(s\) at time \(t\) of an ex dividend stock that entitles the owner at the end of time \(t\) to the time \(t+1\) dividend and the option to sell the stock at time \(t+1\). The price is

\[ p^k = (I - Q)^{-1} Q d^k \]

Below, we describe an equilibrium model with trading of one-period Arrow securities in which the pricing kernel is endogenous.

In constructing our model, we’ll repeatedly encounter formulas that remind us of our asset pricing formulas.

54.2.4. Multi-step-forward transition probabilities and pricing kernels

The \((i,j)\) component of the \(k\)-step ahead transition probability \(P^k\) is

\[ Prob(s_{t+k} = \bar s_j | s_t = \bar s_i) = P^{k}_{i,j} \]

The \((i,j)\) component of the \(k\)-step ahead pricing kernel \(Q^k\) is

\[ Q^{(k)}(s_{t+k} = \bar s_j | s_t = \bar s_i) = Q^{k}_{i,j} \]

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

54.2.5. Laws of iterated expectations and iterated values

A law of iterated values has a mathematical structure that parallels the law of iterated expectations

We can describe its structure readily in the Markov setting of this lecture

Recall the following recursion satisfied \(j\) step ahead transition probabilites for our finite state Markov chain:

\[ P_j(s_{t+j}| s_t) = \sum_{s_{t+1}} P_{j-1}(s_{t+j}| s_{t+1}) P(s_{t+1} | s_t) \]

We can use this recursion to verify the law of iterated expectations applied to computing the conditional expectation of a random variable \(d(s_{t+j})\) conditioned on \(s_t\) via the following string of equalities

\[\begin{split} \begin{aligned} E \left[ E d(s_{t+j}) | s_{t+1} \right] | s_t & = \sum_{s_{t+1}} \left[ \sum_{s_{t+j}} d(s_{t+j}) P_{j-1}(s_{t+j}| s_{t+1} ) \right] P(s_{t+1} | s_t) \\ & = \sum_{s_{t+j}} d(s_{t+j}) \left[ \sum_{s_{t+1}} P_{j-1} ( s_{t+j} |s_{t+1}) P(s_{t+1}| s_t) \right] \\ & = \sum_{s_{t+j}} d(s_{t+j}) P_j (s_{t+j} | s_t ) \\ & = E d(s_{t+j})| s_t \end{aligned} \end{split}\]

The pricing kernel for \(j\) step ahead Arrow securities satisfies the recursion

\[ Q_j(s_{t+j}| s_t) = \sum_{s_{t+1}} Q_{j-1}(s_{t+j}| s_{t+1}) Q(s_{t+1} | s_t) \]

The time \(t\) value in Markov state \(s_t\) of a time \(t+j\) payout \(d(s_{t+j})\) is

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

The law of iterated values states

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

We verify it by pursuing the following a string of inequalities that are counterparts to those we used to verify the law of iterated expectations:

\[\begin{split} \begin{aligned} V \left[ V ( d(s_{t+j}) | s_{t+1} ) \right] | s_t & = \sum_{s_{t+1}} \left[ \sum_{s_{t+j}} d(s_{t+j}) Q_{j-1}(s_{t+j}| s_{t+1} ) \right] Q(s_{t+1} | s_t) \\ & = \sum_{s_{t+j}} d(s_{t+j}) \left[ \sum_{s_{t+1}} Q_{j-1} ( s_{t+j} |s_{t+1}) Q(s_{t+1}| s_t) \right] \\ & = \sum_{s_{t+j}} d(s_{t+j}) Q_j (s_{t+j} | s_t ) \\ & = E V(d(s_{t+j}))| s_t \end{aligned} \end{split}\]

54.3. General equilibrium model (pure exchange)

Now we are ready to do some fun calculations.

We find it interesting to think in terms of analytical inputs into and outputs from our general equilibrium theorizing.

54.3.1. Inputs

  • Markov states: \(s \in S = \left[\bar{s}_1, \ldots, \bar{s}_n \right]\) governed by an \(n\)-state Markov chain with transition probability

\[ P_{ij} = \Pr \left\{s_{t+1} = \bar{s}_j \mid s_t = \bar{s}_i \right\} \]
  • A collection of \(K \times 1\) vectors of individual \(k\) endowments: \(y^k\left(s\right), k=1,\ldots, K\)

  • An \(n \times 1\) vector of aggregate endowment: \(y\left(s\right) \equiv \sum_{k=1}^K y^k\left(s\right)\)

  • A collection of \(K \times 1\) vectors of individual \(k\) consumptions: \(c^k\left(s\right), k=1,\ldots, K\)

  • A collection of restrictions on feasible consumption allocations for \(s \in S\):

\[ c\left(s\right)= \sum_{k=1}^K c^k\left(s\right) \leq y\left(s\right) \]
  • Preferences: a common utility functional across agents \( E_0 \sum_{t=0}^\infty \beta^t u(c^k_t) \) with CRRA one-period utility function \(u\left(c\right)\) and discount factor \(\beta \in (0,1)\)

The one-period utility function is

\[ u \left(c\right) = \frac{c^{1-\gamma}}{1-\gamma} \]

so that

\[ u^\prime \left(c\right) = c^{-\gamma} \]

54.3.2. Outputs

  • An \(n \times n\) matrix pricing kernel \(Q\) for one-period Arrow securities, where \( Q_{ij}\) = price at time \(t\) in state \(s_t \bar s_i\) of one unit of consumption when \(s_{t+1} = \bar s_j\) at time \(t+1\)

  • pure exchange so that \(c\left(s\right) = y\left(s\right)\)

  • an \(K \times 1\) vector distribution of wealth vector \(\alpha\), \(\alpha_k \geq 0, \sum_{k=1}^K \alpha_k =1\)

  • A collection of \(n \times 1\) vectors of individual \(k\) consumptions: \(c^k\left(s\right), k=1,\ldots, K\)

54.3.3. Matrix \(Q\) to represent pricing kernel

For any agent \(k \in \left[1, \ldots, K\right]\), at the equilibrium allocation, the one-period Arrow securities pricing kernel satisfies

\[ Q_{ij} = \beta \left(\frac{c^k\left(\bar{s}_j\right)}{c^k\left(\bar{s}_i\right)}\right)^{-\gamma} P_{ij} \]

where \(Q\) is an \(n \times n\) matrix

This follows from agent \(k\)’s first-order necessary conditions.

But with the CRRA preferences that we have assumed, individual consumptions vary proportionately with aggregate consumption and therefore with the aggregate endowment.

  • This is a consequence of our preference specification implying that Engle curves affine in wealth and therefore satisfy conditions for Gorman aggregation

Thus,

\[ c^k \left(s\right) = \alpha_k c\left(s\right) = \alpha_k y\left(s\right) \]

for an arbitrary distribution of wealth in the form of an \(K \times 1\) vector \(\alpha\) that satisfies

\[ \alpha_k \in \left(0, 1\right), \quad \sum_{k=1}^K \alpha_k = 1 \]

This means that we can compute the pricing kernel from

\[ Q_{ij} = \beta \left(\frac{y_j}{y_i}\right)^{-\gamma} P_{ij} \]

Note that \(Q_{ij}\) is independent of vector \(\alpha\).

Thus, we have the

Key finding: We can compute competitive equilibrium prices prior to computing a distribution of wealth.

54.3.4. Values

Having computed an equilibrium pricing kernel \(Q\), we can compute several values that are required to pose or represent the solution of an individual household’s optimum problem.

We denote an \(K \times 1\) vector of state-dependent values of agents’ endowments in Markov state \(s\) as

\[\begin{split} A\left(s\right)=\left[\begin{array}{c} A^{1}\left(s\right)\\ \vdots\\ A^{K}\left(s\right) \end{array}\right], \quad s \in \left[\bar{s}_1, \ldots, \bar{s}_n\right] \end{split}\]

and an \(n \times 1\) vector of continuation endowment values for each individual \(k\) as

\[\begin{split} A^{k}=\left[\begin{array}{c} A^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ A^{k}\left(\bar{s}_{n}\right) \end{array}\right], \quad k \in \left[1, \ldots, K\right] \end{split}\]

\(A^k\) of consumer \(i\) satisfies

\[ A^k = \left[I - Q\right]^{-1} \left[ y^k\right] \]

where

\[\begin{split} y^{k}=\left[\begin{array}{c} y^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ y^{k}\left(\bar{s}_{n}\right) \end{array}\right] \equiv \begin{bmatrix} y^k_1 \cr \vdots \cr v^k_n \end{bmatrix} \end{split}\]

In a competitive equilibrium of an infinite horizon economy with sequential trading of one-period Arrow securities, \(A^k(s)\) serves as a state-by-state vector of debt limits on the quantities of one-period Arrow securities paying off in state \(s\) at time \(t+1\) that individual \(k\) can issue at time \(t\).

These are often called natural debt limits.

Evidently, they equal the maximum amount that it is feasible for individual \(i\) to repay even if he consumes zero goods forevermore.

Remark: If we have an Inada condition at zero consumption or just impose that consumption be nonnegative, then in a finite horizon economy with sequential trading of one-period Arrow securities there is no need to impose natural debt limits.

54.3.5. Continuation wealths

Continuation wealths play an important role in Bellmanizing a competitive equilibrium with sequential trading of a complete set of one-period Arrow securities.

We denote an \(K \times 1\) vector of state-dependent continuation wealths in Markov state \(s\) as

\[\begin{split} \psi\left(s\right)=\left[\begin{array}{c} \psi^{1}\left(s\right)\\ \vdots\\ \psi^{K}\left(s\right) \end{array}\right], \quad s \in \left[\bar{s}_1, \ldots, \bar{s}_n\right] \end{split}\]

and an \(n \times 1\) vector of continuation wealths for each individual \(i\) as

\[\begin{split} \psi^{k}=\left[\begin{array}{c} \psi^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ \psi^{k}\left(\bar{s}_{n}\right) \end{array}\right], \quad k \in \left[1, \ldots, K\right] \end{split}\]

Continuation wealth \(\psi^k\) of consumer \(i\) satisfies

\[ \psi^k = \left[I - Q\right]^{-1} \left[\alpha_k y - y^k\right] \]

where

\[\begin{split} y^{k}=\left[\begin{array}{c} y^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ y^{k}\left(\bar{s}_{n}\right) \end{array}\right],\quad y=\left[\begin{array}{c} y\left(\bar{s}_{1}\right)\\ \vdots\\ y\left(\bar{s}_{n}\right) \end{array}\right] \end{split}\]

Note that \(\sum_{k=1}^K \psi^k = {0}_{n \times 1}\).

Remark: At the initial state \(s_0 \in \begin{bmatrix} \bar s_1, \ldots, \bar s_n \end{bmatrix}\) the continuation wealth \(\psi^k(s_0) = 0\) for all agents \(k = 1, \ldots, K\). This indicates that the economy begins with all agents being debt-free and financial-asset-free at time \(0\), state \(s_0\).

Remark: Note that all agents’ continuation wealths recurrently return to zero when the Markov state returns to whatever value \(s_0\) it had at time \(0\).

54.3.6. Optimal portfolios

A nifty feature of the model is that optimal portfolios for a type \(k\) agent equal the continuation wealths that we have just computed.

Thus, agent \(k\)’s state-by-state purchases of Arrow securities next period depend only on next period’s Markov state and equal

\[ a_k(s) = \psi^k(s), \quad s \in \left[\bar s_1, \ldots, \bar s_n \right] \]

54.3.7. Equilibrium wealth distribution \(\alpha\)

With the initial state being a particular state \(s_0 \in \left[\bar{s}_1, \ldots, \bar{s}_n\right]\), we must have

\[ \psi^k\left(s_0\right) = 0, \quad k=1, \ldots, K \]

which means the equilibrium distribution of wealth satisfies

\[ \alpha_k = \frac{V_z y^k}{V_z y} \]

where \(V \equiv \left[I - Q\right]^{-1}\) and \(z\) is the row index corresponding to the initial state \(s_0\).

Since \(\sum_{k=1}^K V_z y^k = V_z y\), \(\sum_{k=1}^K \alpha_k = 1\).

In summary, here is the logical flow of an algorithm to compute a competitive equilibrium:

  • compute \(Q\) from the aggregate allocation and the above formula

  • compute the distribution of wealth \(\alpha\) from the formula just given

  • Using \(\alpha\) assign each consumer \(k\) the share \(\alpha_k\) of the aggregate endowment at each state

  • return to the \(\alpha\)-dependent formula for continuation wealths and compute continuation wealths

  • equate agent \(k\)’s portfolio to its continuation wealth state by state

We can also add formulas for optimal value functions in a competitive equilibrium with trades in a complete set of one-period state-contingent Arrow securities.

Call the optimal value functions \(J^k\) for consumer \(k\).

For the infinite horizon economy now under study, the formula is

\[ J^k = (I - \beta P)^{-1} u(\alpha_k y) , \quad u(c) = \frac{c^{1-\gamma}}{1-\gamma} \]

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

We are ready to dive into some Python code.

As usual, we start with Python imports.

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
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.

class RecurCompetitive:
    """
    A class that represents a recursive competitive economy
    with one-period Arrow securities.
    """

    def __init__(self,
                 s,        # state vector
                 P,        # transition matrix
                 ys,       # endowments ys = [y1, y2, .., yI]
                 γ=0.5,    # risk aversion
                 β=0.98,   # discount rate
                 T=None):  # time horizon, none if infinite

        # preference parameters
        self.γ = γ
        self.β = β

        # variables dependent on state
        self.s = s
        self.P = P
        self.ys = ys
        self.y = np.sum(ys, 1)

        # dimensions
        self.n, self.K = ys.shape

        # compute pricing kernel
        self.Q = self.pricing_kernel()
        
        # compute price of risk-free one-period bond
        self.PRF = self.price_risk_free_bond()
        
        # compute risk-free rate
        self.R = self.risk_free_rate()

        # V = [I - Q]^{-1} (infinite case)
        if T is None:
            self.T = None
            self.V = np.empty((1, n, n))
            self.V[0] = np.linalg.inv(np.eye(n) - self.Q)
        # V = [I + Q + Q^2 + ... + Q^T] (finite case)
        else:
            self.T = T
            self.V = np.empty((T+1, n, n))
            self.V[0] = np.eye(n)

            Qt = np.eye(n)
            for t in range(1, T+1):
                Qt = Qt.dot(self.Q)
                self.V[t] = self.V[t-1] + Qt

        # natural debt limit
        self.A = self.V[-1] @ ys

    def u(self, c):
        "The CRRA utility"

        return c ** (1 - self.γ) / (1 - self.γ)

    def u_prime(self, c):
        "The first derivative of CRRA utility"

        return c ** (-self.γ)

    def pricing_kernel(self):
        "Compute the pricing kernel matrix Q"

        c = self.y

        n = self.n
        Q = np.empty((n, n))

        for i in range(n):
            for j in range(n):
                ratio = self.u_prime(c[j]) / self.u_prime(c[i])
                Q[i, j] = self.β * ratio * P[i, j]

        self.Q = Q

        return Q

    def wealth_distribution(self, s0_idx):
        "Solve for wealth distribution α"

        # set initial state
        self.s0_idx = s0_idx

        # simplify notations
        n = self.n
        Q = self.Q
        y, ys = self.y, self.ys

        # row of V corresponding to s0
        Vs0 = self.V[-1, s0_idx, :]
        α = Vs0 @ self.ys / (Vs0 @ self.y)

        self.α = α

        return α

    def continuation_wealths(self):
        "Given α, compute the continuation wealths ψ"

        diff = np.empty((n, K))
        for k in range(K):
            diff[:, k] = self.α[k] * self.y - self.ys[:, k]

        ψ = self.V @ diff
        self.ψ = ψ

        return ψ

    def price_risk_free_bond(self):
        "Give Q, compute price of one-period risk free bond"
        
        PRF = np.sum(self.Q, 0)
        self.PRF = PRF
        
        return PRF
    
    def risk_free_rate(self):
        "Given Q, compute one-period gross risk-free interest rate R"

        R = np.sum(self.Q, 0)
        R = np.reciprocal(R)
        self.R = R

        return R

    def value_functionss(self):
        "Given α, compute the optimal value functions J in equilibrium"

        n, T = self.n, self.T
        β = self.β
        P = self.P

        # compute (I - βP)^(-1) in infinite case
        if T is None:
            P_seq = np.empty((1, n, n))
            P_seq[0] = np.linalg.inv(np.eye(n) - β * P)
        # and (I + βP + ... + β^T P^T) in finite case
        else:
            P_seq = np.empty((T+1, n, n))
            P_seq[0] = np.eye(n)

            Pt = np.eye(n)
            for t in range(1, T+1):
                Pt = Pt.dot(P)
                P_seq[t] = P_seq[t-1] + Pt * β ** t

        # compute the matrix [u(α_1 y), ..., u(α_K, y)]
        flow = np.empty((n, K))
        for k in range(K):
            flow[:, k] = self.u(self.α[k] * self.y)

        J = P_seq @ flow

        self.J = J

        return J

54.3.8. Example 1

Please read the preceding class for default parameter values and the following Python code for the fundamentals of the economy.

Here goes.

# dimensions
K, n = 2, 2

# states
s = np.array([0, 1])

# transition
P = np.array([[.5, .5], [.5, .5]])

# endowments
ys = np.empty((n, K))
ys[:, 0] = 1 - s       # y1
ys[:, 1] = s           # y2
ex1 = RecurCompetitive(s, P, ys)
# endowments
ex1.ys
array([[1., 0.],
       [0., 1.]])
# pricing kernal
ex1.Q
array([[0.49, 0.49],
       [0.49, 0.49]])
# Risk free rate R
ex1.R
array([1.02040816, 1.02040816])
# natural debt limit, A = [A1, A2, ..., AI]
ex1.A
array([[25.5, 24.5],
       [24.5, 25.5]])
# when the initial state is state 1
print(f'α = {ex1.wealth_distribution(s0_idx=0)}')
print(f'ψ = \n{ex1.continuation_wealths()}')
print(f'J = \n{ex1.value_functionss()}')
α = [0.51 0.49]
ψ = 
[[[ 0. -0.]
  [ 1. -1.]]]
J = 
[[[71.41428429 70.        ]
  [71.41428429 70.        ]]]
# when the initial state is state 2
print(f'α = {ex1.wealth_distribution(s0_idx=1)}')
print(f'ψ = \n{ex1.continuation_wealths()}')
print(f'J = \n{ex1.value_functionss()}')
α = [0.49 0.51]
ψ = 
[[[-1.  1.]
  [ 0. -0.]]]
J = 
[[[70.         71.41428429]
  [70.         71.41428429]]]

54.3.9. Example 2

# dimensions
K, n = 2, 2

# states
s = np.array([1, 2])

# transition
P = np.array([[.5, .5], [.5, .5]])

# endowments
ys = np.empty((n, K))
ys[:, 0] = 1.5         # y1
ys[:, 1] = s           # y2
ex2 = RecurCompetitive(s, P, ys)
# endowments

print("ys = \n", ex2.ys)

# pricing kernal
print ("Q = \n", ex2.Q)

# Risk free rate R
print("R = ", ex2.R)
ys = 
 [[1.5 1. ]
 [1.5 2. ]]
Q = 
 [[0.49       0.41412558]
 [0.57977582 0.49      ]]
R =  [0.93477529 1.10604104]
# pricing kernal
ex2.Q
array([[0.49      , 0.41412558],
       [0.57977582, 0.49      ]])
# Risk free rate R
ex2.R
array([0.93477529, 1.10604104])
# natural debt limit, A = [A1, A2, ..., AI]
ex2.A
array([[69.30941886, 66.91255848],
       [81.73318641, 79.98879094]])
# when the initial state is state 1
print(f'α = {ex2.wealth_distribution(s0_idx=0)}')
print(f'ψ = \n{ex2.continuation_wealths()}')
print(f'J = \n{ex2.value_functionss()}')
α = [0.50879763 0.49120237]
ψ = 
[[[-0.         -0.        ]
  [ 0.55057195 -0.55057195]]]
J = 
[[[122.907875   120.76397493]
  [123.32114686 121.17003803]]]
# when the initial state is state 1
print(f'α = {ex2.wealth_distribution(s0_idx=1)}')
print(f'ψ = \n{ex2.continuation_wealths()}')
print(f'J = \n{ex2.value_functionss()}')
α = [0.50539319 0.49460681]
ψ = 
[[[-0.46375886  0.46375886]
  [ 0.         -0.        ]]]
J = 
[[[122.49598809 121.18174895]
  [122.907875   121.58921679]]]

54.3.10. Example 3

# dimensions
K, n = 2, 2

# states
s = np.array([1, 2])

# transition
λ = 0.9
P = np.array([[1-λ, λ], [0, 1]])

# endowments
ys = np.empty((n, K))
ys[:, 0] = [1, 0]         # y1
ys[:, 1] = [0, 1]         # y2
ex3 = RecurCompetitive(s, P, ys)
# endowments

print("ys = ", ex3.ys)

# pricing kernel
print ("Q = ", ex3.Q)

# Risk free rate R
print("R = ", ex3.R)
ys =  [[1. 0.]
 [0. 1.]]
Q =  [[0.098 0.882]
 [0.    0.98 ]]
R =  [10.20408163  0.53705693]
# pricing kernel
ex3.Q
array([[0.098, 0.882],
       [0.   , 0.98 ]])
# natural debt limit, A = [A1, A2, ..., AI]
ex3.A
array([[ 1.10864745, 48.89135255],
       [ 0.        , 50.        ]])

Note that the natural debt limit for agent \(1\) in state \(2\) is \(0\).

# when the initial state is state 1
print(f'α = {ex3.wealth_distribution(s0_idx=0)}')
print(f'ψ = \n{ex3.continuation_wealths()}')
print(f'J = \n{ex3.value_functionss()}')
α = [0.02217295 0.97782705]
ψ = 
[[[ 0.         -0.        ]
  [ 1.10864745 -1.10864745]]]
J = 
[[[14.89058394 98.88513796]
  [14.89058394 98.88513796]]]
# when the initial state is state 1
print(f'α = {ex3.wealth_distribution(s0_idx=1)}')
print(f'ψ = \n{ex3.continuation_wealths()}')
print(f'J = \n{ex3.value_functionss()}')
α = [0. 1.]
ψ = 
[[[-1.10864745  1.10864745]
  [ 0.          0.        ]]]
J = 
[[[  0. 100.]
  [  0. 100.]]]

For the specification of the Markov chain in example 3, let’s take a look at how the equilibrium allocation changes as a function of transition probability \(\lambda\).

λ_seq = np.linspace(0, 1, 100)

# prepare containers
αs0_seq = np.empty((len(λ_seq), 2))
αs1_seq = np.empty((len(λ_seq), 2))

for i, λ in enumerate(λ_seq):
    P = np.array([[1-λ, λ], [0, 1]])
    ex3 = RecurCompetitive(s, P, ys)

    # initial state s0 = 1
    α = ex3.wealth_distribution(s0_idx=0)
    αs0_seq[i, :] = α

    # initial state s0 = 2
    α = ex3.wealth_distribution(s0_idx=1)
    αs1_seq[i, :] = α
<ipython-input-3-222e6cc36b2b>:126: RuntimeWarning: divide by zero encountered in reciprocal
  R = np.reciprocal(R)
fig, axs = plt.subplots(1, 2, figsize=(12, 4))

for i, αs_seq in enumerate([αs0_seq, αs1_seq]):
    for j in range(2):
        axs[i].plot(λ_seq, αs_seq[:, j], label=f{j+1}')
        axs[i].set_xlabel('λ')
        axs[i].set_title(f'initial state s0 = {s[i]}')
        axs[i].legend()

plt.show()
_images/ge_arrow_50_0.png

54.4. Example 4

# dimensions
K, n = 2, 3

# states
s = np.array([1, 2, 3])

# transition
λ = .9
μ = .9
δ = .05

P = np.array([[1-λ, λ, 0], [μ/2, μ, μ/2], [(1-δ)/2, (1-δ)/2, δ]])

# endowments
ys = np.empty((n, K))
ys[:, 0] = [.25, .75, .2]       # y1
ys[:, 1] = [1.25, .25, .2]      # y2
ex4 = RecurCompetitive(s, P, ys)
# endowments
print("ys = \n", ex4.ys)

# pricing kernal
print ("Q = \n", ex4.Q)

# Risk free rate R
print("R = ", ex4.R)

# natural debt limit, A = [A1, A2, ..., AI]
print("A = \n", ex4.A)

print('')

for i in range(1, 4):
    # when the initial state is state i
    print(f"when the initial state is state {i}")
    print(f'α = {ex4.wealth_distribution(s0_idx=i-1)}')
    print(f'ψ = \n{ex4.continuation_wealths()}')
    print(f'J = \n{ex4.value_functionss()}\n')
ys = 
 [[0.25 1.25]
 [0.75 0.25]
 [0.2  0.2 ]]
Q = 
 [[0.098      1.08022498 0.        ]
 [0.36007499 0.882      0.69728222]
 [0.24038317 0.29440805 0.049     ]]
R =  [1.43172499 0.44313807 1.33997564]
A = 
 [[-1.4141307  -0.45854174]
 [-1.4122483  -1.54005386]
 [-0.58434331 -0.3823659 ]]

when the initial state is state 1
α = [0.75514045 0.24485955]
ψ = 
[[[ 0.          0.        ]
  [-0.81715447  0.81715447]
  [-0.14565791  0.14565791]]]
J = 
[[[-2.65741909 -1.51322919]
  [-5.13103133 -2.92179221]
  [-2.65649938 -1.51270548]]]

when the initial state is state 2
α = [0.47835493 0.52164507]
ψ = 
[[[ 0.5183286  -0.5183286 ]
  [ 0.         -0.        ]
  [ 0.12191319 -0.12191319]]]
J = 
[[[-2.11505328 -2.20868477]
  [-4.08381377 -4.26460049]
  [-2.11432128 -2.20792037]]]

when the initial state is state 3
α = [0.60446648 0.39553352]
ψ = 
[[[ 0.28216299 -0.28216299]
  [-0.37231938  0.37231938]
  [ 0.         -0.        ]]]
J = 
[[[-2.37756442 -1.92325926]
  [-4.59067883 -3.71349163]
  [-2.37674158 -1.92259365]]]

54.5. Finite horizon economies

The Python class RecurCompetitive provided above also can be used to compute competitive equilibrium allocations and Arrow securities prices for finite horizon economies.

The setting is a finite-horizon version of the one above except that time now runs for \(T+1\) periods \(t \in {\bf T} = \{ 0, 1, \ldots, T\}\).

Consequently, we want \(T+1\) counterparts to objects described above, with one important exception: we won’t need borrowing limits because they aren’t required for a finite horizon economy in which a one-period utility function \(u(c)\) satisfies an Inada condition that sets the marginal utility of consumption at zero consumption to zero. Nonnegativity of consumption choices at all \(t \in {\bf T}\) automatically limits borrowing.

54.5.1. Continuation wealths

We denote an \(K \times 1\) vector of state-dependent continuation wealths in Markov state \(s\) at time \(t\) as

\[\begin{split} \psi_t\left(s\right)=\left[\begin{array}{c} \psi^{1}\left(s\right)\\ \vdots\\ \psi^{K}\left(s\right) \end{array}\right], \quad s \in \left[\bar{s}_1, \ldots, \bar{s}_n\right] \end{split}\]

and an \(n \times 1\) vector of continuation wealths for each individual \(k\) as

\[\begin{split} \psi_t^{k}=\left[\begin{array}{c} \psi_t^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ \psi_t^{k}\left(\bar{s}_{n}\right) \end{array}\right], \quad k \in \left[1, \ldots, K\right] \end{split}\]

Continuation wealths \(\psi^k\) of consumer \(k\) satisfy

\[ \begin{aligned} \psi_T^k & = \left[\alpha_k y - y^k\right] \cr \psi_{T-1}^k & = \left[I + Q \right] \left[\alpha_k y - y^k\right] \cr \vdots \quad & \quad \quad \quad \vdots \cr \psi_0^k & = \left[I + Q + Q^2 + \cdots + Q^T \right] \left[\alpha_k y - y^k\right] \end{aligned} \]

where

\[\begin{split} y^{k}=\left[\begin{array}{c} y^{k}\left(\bar{s}_{1}\right)\\ \vdots\\ y^{k}\left(\bar{s}_{n}\right) \end{array}\right],\quad y=\left[\begin{array}{c} y\left(\bar{s}_{1}\right)\\ \vdots\\ y\left(\bar{s}_{n}\right) \end{array}\right] \end{split}\]

Note that \(\sum_{k=1}^K \psi_t^k = {0}_{n \times 1}\) for all \(t \in {\bf T}\).

Remark: At the initial state \(s_0 \in \begin{bmatrix} \bar s_1, \ldots, \bar s_n \end{bmatrix}\), for all agents \(k = 1, \ldots, K\), continuation wealth \(\psi_0^k(s_0) = 0\). This indicates that the economy begins with all agents being debt-free and financial-asset-free at time \(0\), state \(s_0\).

Remark: Note that all agents’ continuation wealths return to zero when the Markov state returns to whatever value \(s_0\) it had at time \(0\). This will recur if the Markov chain makes the initial state \(s_0\) recurrent.

With the initial state being a particular state \(s_0 \in \left[\bar{s}_1, \ldots, \bar{s}_n\right]\), we must have

\[ \psi_0^k\left(s_0\right) = 0, \quad k=1, \ldots, K \]

which means the equilibrium distribution of wealth satisfies

\[ \alpha_k = \frac{V_z y^k}{V_z y} \]

where now in our finite-horizon economy \(V = \left[I + Q + Q^2 + \cdots + Q^T \right]\) and \(z\) is the row index corresponding to the initial state \(s_0\).

Since \(\sum_{k=1}^K V_z y^k = V_z y\), \(\sum_{k=1}^K \alpha_k = 1\).

In summary, here is the logical flow of an algorithm to compute a competitive equilibrium with Arrow securities in our finite-horizon Markov economy:

  • compute \(Q\) from the aggregate allocation and the above formula

  • compute the distribution of wealth \(\alpha\) from the formula just given

  • Using \(\alpha\) assign each consumer \(k\) the share \(\alpha_k\) of the aggregate endowment at each state

  • return to the \(\alpha\)-dependent formula for continuation wealths and compute continuation wealths

  • equate agent \(k\)’s portfolio to its continuation wealth state by state

While for the infinite horizon economy, the formula for value functions is

\[ J^k = (I - \beta P)^{-1} u(\alpha_k y) , \quad u(c) = \frac{c^{1-\gamma}}{1-\gamma} \]

for the finite horizon economy the formula is

\[ J_0^k = (I + \beta P + \cdots + \beta^T P^T) u(\alpha_k y) \]

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

54.5.2. Finite horizon example

Below we revisit the economy defined in example 1, but set the time horizon to be \(T=10\).

# dimensions
K, n = 2, 2

# states
s = np.array([0, 1])

# transition
P = np.array([[.5, .5], [.5, .5]])

# endowments
ys = np.empty((n, K))
ys[:, 0] = 1 - s       # y1
ys[:, 1] = s           # y2
ex1_finite = RecurCompetitive(s, P, ys, T=10)
# (I + Q + Q^2 + ... + Q^T)
ex1_finite.V[-1]
array([[5.48171623, 4.48171623],
       [4.48171623, 5.48171623]])
# endowments
ex1_finite.ys
array([[1., 0.],
       [0., 1.]])
# pricing kernal
ex1_finite.Q
array([[0.49, 0.49],
       [0.49, 0.49]])
# Risk free rate R
ex1_finite.R
array([1.02040816, 1.02040816])

In the finite time horizon case, ψ and J are returned as sequences.

Components are ordered from \(t=T\) to \(t=0\).

# when the initial state is state 2
print(f'α = {ex1_finite.wealth_distribution(s0_idx=0)}')
print(f'ψ = \n{ex1_finite.continuation_wealths()}\n')
print(f'J = \n{ex1_finite.value_functionss()}')
α = [0.55018351 0.44981649]
ψ = 
[[[-0.44981649  0.44981649]
  [ 0.55018351 -0.55018351]]

 [[-0.40063665  0.40063665]
  [ 0.59936335 -0.59936335]]

 [[-0.35244041  0.35244041]
  [ 0.64755959 -0.64755959]]

 [[-0.30520809  0.30520809]
  [ 0.69479191 -0.69479191]]

 [[-0.25892042  0.25892042]
  [ 0.74107958 -0.74107958]]

 [[-0.21355851  0.21355851]
  [ 0.78644149 -0.78644149]]

 [[-0.16910383  0.16910383]
  [ 0.83089617 -0.83089617]]

 [[-0.12553824  0.12553824]
  [ 0.87446176 -0.87446176]]

 [[-0.08284397  0.08284397]
  [ 0.91715603 -0.91715603]]

 [[-0.04100358  0.04100358]
  [ 0.95899642 -0.95899642]]

 [[-0.         -0.        ]
  [ 1.         -1.        ]]]

J = 
[[[ 1.48348712  1.3413672 ]
  [ 1.48348712  1.3413672 ]]

 [[ 2.9373045   2.65590706]
  [ 2.9373045   2.65590706]]

 [[ 4.36204553  3.94415611]
  [ 4.36204553  3.94415611]]

 [[ 5.75829174  5.20664019]
  [ 5.75829174  5.20664019]]

 [[ 7.12661302  6.44387459]
  [ 7.12661302  6.44387459]]

 [[ 8.46756788  7.6563643 ]
  [ 8.46756788  7.6563643 ]]

 [[ 9.78170364  8.84460421]
  [ 9.78170364  8.84460421]]

 [[11.06955669 10.00907933]
  [11.06955669 10.00907933]]

 [[12.33165268 11.15026494]
  [12.33165268 11.15026494]]

 [[13.56850674 12.26862684]
  [13.56850674 12.26862684]]

 [[14.78062373 13.3646215 ]
  [14.78062373 13.3646215 ]]]
# when the initial state is state 2
print(f'α = {ex1_finite.wealth_distribution(s0_idx=1)}')
print(f'ψ = \n{ex1_finite.continuation_wealths()}\n')
print(f'J = \n{ex1_finite.value_functionss()}')
α = [0.44981649 0.55018351]
ψ = 
[[[-0.55018351  0.55018351]
  [ 0.44981649 -0.44981649]]

 [[-0.59936335  0.59936335]
  [ 0.40063665 -0.40063665]]

 [[-0.64755959  0.64755959]
  [ 0.35244041 -0.35244041]]

 [[-0.69479191  0.69479191]
  [ 0.30520809 -0.30520809]]

 [[-0.74107958  0.74107958]
  [ 0.25892042 -0.25892042]]

 [[-0.78644149  0.78644149]
  [ 0.21355851 -0.21355851]]

 [[-0.83089617  0.83089617]
  [ 0.16910383 -0.16910383]]

 [[-0.87446176  0.87446176]
  [ 0.12553824 -0.12553824]]

 [[-0.91715603  0.91715603]
  [ 0.08284397 -0.08284397]]

 [[-0.95899642  0.95899642]
  [ 0.04100358 -0.04100358]]

 [[-1.          1.        ]
  [-0.         -0.        ]]]

J = 
[[[ 1.3413672   1.48348712]
  [ 1.3413672   1.48348712]]

 [[ 2.65590706  2.9373045 ]
  [ 2.65590706  2.9373045 ]]

 [[ 3.94415611  4.36204553]
  [ 3.94415611  4.36204553]]

 [[ 5.20664019  5.75829174]
  [ 5.20664019  5.75829174]]

 [[ 6.44387459  7.12661302]
  [ 6.44387459  7.12661302]]

 [[ 7.6563643   8.46756788]
  [ 7.6563643   8.46756788]]

 [[ 8.84460421  9.78170364]
  [ 8.84460421  9.78170364]]

 [[10.00907933 11.06955669]
  [10.00907933 11.06955669]]

 [[11.15026494 12.33165268]
  [11.15026494 12.33165268]]

 [[12.26862684 13.56850674]
  [12.26862684 13.56850674]]

 [[13.3646215  14.78062373]
  [13.3646215  14.78062373]]]

We can check the results with finite horizon converges to the ones with infinite horizon as \(T \rightarrow \infty\).

ex1_large = RecurCompetitive(s, P, ys, T=10000)
ex1_large.wealth_distribution(s0_idx=1)
array([0.49, 0.51])
ex1.V, ex1_large.V[-1]
(array([[[25.5, 24.5],
         [24.5, 25.5]]]),
 array([[25.5, 24.5],
        [24.5, 25.5]]))
ex1_large.continuation_wealths()
ex1.ψ, ex1_large.ψ[-1]
(array([[[-1.,  1.],
         [ 0., -0.]]]),
 array([[-1.,  1.],
        [ 0., -0.]]))
ex1_large.value_functionss()
ex1.J, ex1_large.J[-1]
(array([[[70.        , 71.41428429],
         [70.        , 71.41428429]]]),
 array([[70.        , 71.41428429],
        [70.        , 71.41428429]]))