## Overview

This lecture studies government debt in an AMSS

economy {cite}`aiyagari2002optimal` of the type described in {doc}`Optimal Taxation without State-Contingent Debt <amss>`.

economy {cite}`aiyagari2002optimal` of the type described in {doc}`Fiscal Risk and Government Debt <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.

* In a {doc}`previous lecture <amss2>`, we showed that with only two Markov states, it is possible that eventually endogenous

* In a {doc}`Fiscal Risk and Government Debt <amss2>`, we showed that with only two Markov states, it is possible that eventually endogenous

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

{doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>`.

{doc}`Fiscal Risk and Government Debt <amss2>`.

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

in {doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>` where the ergodic distribution of the par value is concentrated on one point.

in {doc}`Fiscal Risk and Government Debt <amss2>` where the ergodic distribution of the par value is concentrated on one point.

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.

We use code constructed {doc}`in a previous lecture <amss2>`.

We use code constructed {doc}`Fiscal Risk and Government Debt <amss2>`.

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

## The Economy

As in {doc}`Optimal Taxation without State-Contingent Debt <amss>` and {doc}`Optimal Taxation with State-Contingent Debt <opt_tax_recur>`,

As in {doc}`Fiscal Risk and Government Debt <amss>` and {doc}`Fiscal Risk and Government Debt <opt_tax_recur>`,

we assume that the representative agent has utility function

$$

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

The following code computes these moments; the code is recycled from {doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>`.

The following code computes these moments; the code is recycled from {doc}`Fiscal Risk and Government Debt <amss2>`.

```{code-cell} python3

def mean(x, s):

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

approximations to rates of convergence that appear in {cite}`BEGS1` and that we discuss in {doc}`a previous lecture <amss2>`.

approximations to rates of convergence that appear in {cite}`BEGS1` and that we discuss in {doc}`Fiscal Risk and Government Debt <amss2>`.

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`.

As in {doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>`, we recall that {cite}`BEGS1` used a particular

As in {doc}`Fiscal Risk and Government Debt <amss2>`, we recall that {cite}`BEGS1` used a particular

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

that we have used in previous lectures {doc}`here <amss>` and {doc}`here <amss2>`.

that we have used in previous lectures {doc}`Fiscal Risk and Government Debt <amss>` and {doc}`Fiscal Risk and Government Debt <amss2>`.

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

the ergodic distribution solves a fiscal-risk minimization problem.

As an example, notice how we used the formula for the mean of ${\mathcal B}$ in the ergodic distribution of the special AMSS economy in

{doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>`

{doc}`Fiscal Risk and Government Debt <amss2>`

* **first** we computed the ergodic distribution using a reverse-engineering construction

* **then** we verified that ${\mathcal B}$ agrees with the mean of that distribution

\hat b = \frac{ {\mathcal B}^*}{div}

$$

In the two-Markov-state AMSS economy in {doc}`Fiscal Insurance via Fluctuating Interest Rates <amss2>`,

In the two-Markov-state AMSS economy in {doc}`Fiscal Risk and Government Debt <amss2>`,

$E_t u_{c,t+1} = E u_{c,t+1}$ in the ergodic distribution and we have confirmed that

this formula very accurately describes a **constant** par value of government debt that