71. Competitive Equilibria with Arrow Securities#

71.2.1. Preferences and endowments#

In each period $t\ge 0$, a stochastic event $s_t\in \mathbf{S}$ is realized.

Let the history of events up until time $t$ be denoted $s^t=[s_0,s_1,\dots ,s_{t-1},s_t]$.

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

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 ${\pi }_t(s^t|s^{\tau })$.

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 macroeconomics and econometrics and assume that ${\pi }_t(s^t)$ is induced by a Markov process.

There are $K$ consumers named $k=1,\dots ,K$.

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

The history $s^t$ is publicly observable.

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

Consumer $k$ orders consumption plans by

where $0<\beta <1$.

The right side is equal to $E_0\sum _{t=0}^{\mathrm{\infty }}{\beta }^t{u}_k(c_t^k)$, where $E_0$ is the mathematical expectation operator, conditioned on $s_0$.

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

The utility function of person $k$ satisfies the Inada condition

This condition implies that each agent chooses strictly positive consumption for every date-history pair $(t,s^t)$.

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

for all $t$ and for all $s^t$.

71.5.1. Exogenous Pricing Kernel#

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 $\beta \in (0,1)$

• $Q=SP$ 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 $p^h({\overline{s}}_i)=d^h({\overline{s}}_i)+\sum _j{Q}_{ij}{p}^h({\overline{s}}_j)$, which implies that the vector $p^h$ satisfies $p^h=d^h+Q{p}^h$ which implies the formula

• 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

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.

71.5.2. Multi-Step-Forward Transition Probabilities and Pricing Kernels#

The $(i,j)$ component of the $\ell$-step ahead transition probability $P^{\ell }$ is

The $(i,j)$ component of the $\ell$-step ahead pricing kernel $Q^{\ell }$ is

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

71.5.3. Laws of Iterated Expectations and Iterated Values#

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

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

Recall the following recursion satisfied by $j$ step ahead transition probabilites for our finite state Markov chain:

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

The pricing kernel for $j$ step ahead Arrow securities satisfies the recursion

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

The law of iterated values states

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:

71.6.1. Inputs#

• Markov states: $s\in S=[{\overline{s}}_1,\dots ,{\overline{s}}_n]$ governed by an $n$-state Markov chain with transition probability

• A collection of $K\times 1$ vectors of individual $k$ endowments: $y^k\left(s\right),k=1,\dots ,K$

• 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: $c^k\left(s\right),k=1,\dots ,K$

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

• Preferences: a common utility functional across agents $E_0\sum _{t=0}^{\mathrm{\infty }}{\beta }^{t}u(c_t^k)$ with CRRA one-period utility function $u\left(c\right)$ and discount factor $\beta \in (0,1)$

The one-period utility function is

so that

71.6.2. Outputs#

• An $n\times n$ matrix pricing kernel $Q$ for one-period Arrow securities, where $Q_{ij}$ = price at time $t$ in state $s_t={\overline{s}}_i$ of one unit of consumption when $s_{t+1}={\overline{s}}_j$ at time $t+1$

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

• a $K\times 1$ vector distribution of wealth vector $\alpha$, ${\alpha }_k\ge 0,\sum _{k=1}^K{\alpha }_k=1$

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

71.6.3. $Q$ is the Pricing Kernel#

For any agent $k\in [1,\dots ,K]$, at the equilibrium allocation, the one-period Arrow securities pricing kernel satisfies

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 are affine in wealth and therefore satisfy conditions for Gorman aggregation

Thus,

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

This means that we can compute the pricing kernel from

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

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

71.6.4. 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

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

$A^k$ of consumer $k$ satisfies

where

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 $k$ 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. See the section on a Finite Horizon Economy below.

71.6.5. Continuation Wealth#

Continuation wealth plays 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

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

Continuation wealth ${\psi }^k$ of consumer $k$ satisfies

where

Note that $\sum _{k=1}^K{\psi }^k=0_{n\times 1}$.

Remark: At the initial state $s_0\in \left[\begin{array}{c}{\overline{s}}_1,\dots ,{\overline{s}}_n\end{array}\right]$, the continuation wealth ${\psi }^k(s_0)=0$ for all agents $k=1,\dots ,K$. 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 $0$.

71.6.6. Optimal Portfolios#

A nifty feature of the model is that an optimal portfolio of a type $k$ agent equals the continuation wealth that we 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

71.6.7. Equilibrium Wealth Distribution $\alpha$#

With the initial state being a particular state $s_0\in [{\overline{s}}_1,\dots ,{\overline{s}}_n]$, we must have

which means the equilibrium distribution of wealth satisfies

where $V\equiv {[I-Q]}^{-1}$ 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 formula (71.1)

• compute the distribution of wealth $\alpha$ from the formula (71.4)

• Using $\alpha$ assign each consumer $k$ the share ${\alpha }_k$ of the aggregate endowment at each state

• return to the $\alpha$-dependent formula (71.2) and compute continuation wealths

• via formula (71.3) 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

where it is understood that $u({\alpha }_{k}y)$ is a vector.

71.7.1. Continuation Wealths#

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

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

Continuation wealths ${\psi }^k$ of consumer $k$ satisfy

where

Note that $\sum _{k=1}^K{\psi }_t^k=0_{n\times 1}$ for all $t\in \mathbf{T}$.

Remark: At the initial state $s_0\in \left[\begin{array}{c}{\overline{s}}_1,\dots ,{\overline{s}}_n\end{array}\right]$, for all agents $k=1,\dots ,K$, 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 $s_0\in [{\overline{s}}_1,\dots ,{\overline{s}}_n]$, we must have

which means the equilibrium distribution of wealth satisfies

where now in our finite-horizon economy

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 formula (71.1)

• compute the distribution of wealth $\alpha$ from formulas (71.6) and (71.7)

• using $\alpha$, assign each consumer $k$ the share ${\alpha }_k$ of the aggregate endowment at each state

• return to the $\alpha$-dependent formula (71.5) 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

for the finite horizon economy the formula is

where it is understood that $u({\alpha }_{k}y)$ is a vector.

71.9.1. 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]]]

71.9.2. 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      ]])

Note that the pricing kernal in example economies 1 and 2 differ.

This comes from differences in the aggregate endowments in state 1 and 2 in example 1.

ex2.β * ex2.u_prime(3.5) / ex2.u_prime(2.5) * ex2.P[0,1]

0.4141255848169731

ex2.β * ex2.u_prime(2.5) / ex2.u_prime(3.5) * ex2.P[1,0]

0.5797758187437624

# 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]]]

71.9.3. 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, 0.99, 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

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'α{j+1}')
        axs[i].set_xlabel('λ')
        axs[i].set_title(f'initial state s0 = {s[i]}')
        axs[i].legend()

plt.show()

71.9.4. Example 4#

# dimensions
K, n = 2, 3

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

# transition
λ = .9
μ = .9
δ = .05

# prosperous, moderate, and recession states
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)