model.

In contrast to that lecture, this one describes an instance of a model authored by Bisin, Clementi, and Gottardi {cite}`BCG_2018`

in which financial markets are **incomplete**.

Instead of being able to trade equities and a full set of one-period

It is useful to watch how outcomes differ in the two settings.

- there is a unique stochastic discount factor that prices all assets

- consumers’ portfolio choices are indeterminate

- while **individual** firms' financial structures are indeterminate, thus conforming to part of a Modigliani-Miller theorem,

{cite}`Modigliani_Miller_1958`, the **aggregate** of all firms' financial structures **is** determinate.

A more subtle version of a `Big K, little k` features in the BCG incomplete markets environment here.

We use it to convey the heart of what BCG call a **rational conjectures** equilibrium in which conjectures are about

equilibrium pricing functions in regions of the state space that an average consumer or firm does not visit in equilibrium.

Note that the absence of complete markets means that we can compute competitive equilibrium prices and allocations by first solving

Instead, we compute an equilibrium by solving a system of simultaneous inequalities.

the model, we follow BCG in assuming that there are unit measures of

- consumers of type $i=1$

sits at the red dot in the above graph.

This contrasts sharply with the *unqualified* Modigliani-Miller theorem

There the **market’s** financial structure was indeterminate.

optimal taxation with state-contingent debt due to

Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`.

Let's start with things that are identical.

The government imposes a flat rate tax $\tau_t(s^t)$ on labor income at time $t$, history $s^t$.

It is at this point that AMSS {cite}`aiyagari2002optimal` modify the Lucas and Stokey economy.

{eq}`AMSS_foc;a` may change over time in response to realizations of the state,

while the multiplier $\Phi$ in the Lucas-Stokey economy is time-invariant.

on optimal taxation with state-contingent debt sequential allocation implementation:

To analyze the AMSS model, we find it useful to adopt a recursive formulation

becomes a function of the history $s^t$ and initial

government debt $b_0$.

found that

* a counterpart to $V_x(x,s)$ is time-invariant and equal to

we studied how the government manages uncertainty in a simple setting.

As in that lecture, we assume the one-period utility function

We consider the same government expenditure process studied in the lecture on

Government expenditures are known for sure in all periods except one.

This utility function is implemented in the following class.

perpetually faces the prospect of war.

This case was studied in the final example of the lecture on

There, each period the government faces a constant probability, $0.5$, of war.

This lecture extends our investigations of how optimal policies for levying a flat-rate tax on labor income and issuing government debt depend

on whether there are complete markets for debt.

This is because the implementability restriction that a competitive equilibrium with a distorting tax imposes on allocations in the Lucas-Stokey model is just one among a set of

implementability conditions imposed in the AMSS model.

can be traded.

Differences between the Ramsey allocations in the two models indicate that at least some of the measurability constraints of the AMSS model of

Another way to say this is that differences between the Ramsey allocations of the two models indicate that some of the measurability constraints of the

AMSS model are violated at the Ramsey allocation of the Lucas-Stokey model.

we assume that the representative agent has utility function

Here are several classes that do most of the work for us.

want to borrow the same value $\bar b$ next period for all states and all dates.

We accomplish this by using various equations for the Lucas-Stokey {cite}`LucasStokey1983` model

We use the following steps.

This lecture studies government debt in an AMSS

We study the behavior of government debt as time $t \rightarrow + \infty$.

We study an {cite}`aiyagari2002optimal` economy with three Markov states driving government expenditures.

interest rate fluctuations support complete markets allocations and Ramsey outcomes.

* The presence of three states prevents the full spanning that eventually prevails in the two-state example featured in

The lack of full spanning means that the ergodic distribution of the par value of government debt is nontrivial, in contrast to the situation

Nevertheless, {cite}`BEGS1` (BEGS) establish for general settings that include ours, the Ramsey

planner steers government assets to a level that comes

**as close as possible** to providing full spanning in a precise a sense defined by

BEGS that we describe below.

**Warning:** Key equations in {cite}`BEGS1` section III.D carry typos that we correct below.

we assume that the representative agent has utility function

We'll want first and second moments of some key random variables below.

```{code-cell} python3

The long simulation apparently indicates eventual convergence to an ergodic distribution.

It takes about 1000 periods to reach the ergodic distribution -- an outcome that is forecast by

We discard the first 2000 observations of the simulation and construct the histogram of

the part value of government debt.

We begin by computing objects required by the theory of section III.i

of {cite}`BEGS1`.

notation to represent what we can regard as a generalization of the AMSS model.

We introduce some of the {cite}`BEGS1` notation so that readers can quickly relate notation that appears in their key formulas to the notation

BEGS work with objects $B_t, {\mathcal B}_t, {\mathcal R}_t, {\mathcal X}_t$ that are related to notation that we used in

earlier lectures by

If these assumptions are correct, then each new observation $X_t, X_{t+1},\ldots$ can provide additional information about the time-invariant features, allowing us to learn from as data arrive.

For this reason, we will focus in what follows on processes that are *stationary* --- or become so after a transformation

(arma_defs)=