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62 changes: 31 additions & 31 deletions lectures/amss.md
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Expand Up @@ -32,7 +32,7 @@ tags: [hide-output]

## Overview

Let's start with following imports:
Let's start with the following imports:

```{code-cell} ipython
import numpy as np
Expand All @@ -46,9 +46,9 @@ from numba.experimental import jitclass

In {doc}`an earlier lecture <opt_tax_recur>`, we described a model of
optimal taxation with state-contingent debt due to
Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`.
Robert E. Lucas, Jr., and Nancy Stokey {cite}`LucasStokey1983`.

Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal` (hereafter, AMSS)
Aiyagari, Marcet, Sargent, and Seppälä {cite}`aiyagari2002optimal` (hereafter, AMSS)
studied optimal taxation in a model without state-contingent debt.

In this lecture, we
Expand All @@ -65,16 +65,16 @@ We begin with an introduction to the model.

Many but not all features of the economy are identical to those of {doc}`the Lucas-Stokey economy <opt_tax_recur>`.

Let's start with things that are identical.
We start with things that are identical.

For $t \geq 0$, a history of the state is represented by $s^t = [s_t, s_{t-1}, \ldots, s_0]$.

Government purchases $g(s)$ are an exact time-invariant function of $s$.

Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption,
Let $c_t(s^t)$, $\ell_t(s^t)$, and $n_t(s^t)$ denote consumption,
leisure, and labor supply, respectively, at history $s^t$ at time $t$.

Each period a representative household is endowed with one unit of time that can be divided between leisure
Each period, a representative household is endowed with one unit of time that can be divided between leisure
$\ell_t$ and labor $n_t$:

```{math}
Expand All @@ -93,9 +93,9 @@ c_t(s^t) + g(s_t) = n_t(s^t)

Output is not storable.

The technology pins down a pre-tax wage rate to unity for all $t, s^t$.
The technology pins down the pre-tax wage rate to unity for all $t, s^t$.

A representative household’s preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$ are ordered by
The representative household's preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0}^\infty$ are ordered by

```{math}
:label: TS_prefr_amss
Expand All @@ -106,24 +106,24 @@ A representative household’s preferences over $\{c_t(s^t), \ell_t(s^t)\}_{t=0
where

* $\pi_t(s^t)$ is a joint probability distribution over the sequence $s^t$, and
* the utility function $u$ is increasing, strictly concave, and three times continuously differentiable in both arguments.
* the utility function $u$ is increasing, strictly concave, and three times continuously differentiable in both arguments.

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

Lucas and Stokey assumed that there are complete markets in one-period Arrow securities; also see {doc}`smoothing models <smoothing>`.

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

AMSS allow the government to issue only one-period risk-free debt each period.
AMSS allows the government to issue only one-period risk-free debt each period.

Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in Robert Barro (1979) {cite}`Barro1979` than they do in Lucas and Stokey (1983) {cite}`LucasStokey1983`.

### Risk-free One-Period Debt Only

In period $t$ and history $s^t$, let

* $b_{t+1}(s^t)$ be the amount of the time $t+1$ consumption good that at time $t$, history $s^t$ the government promised to pay
* $R_t(s^t)$ be the gross interest rate on risk-free one-period debt between periods $t$ and $t+1$
* $b_{t+1}(s^t)$ be the amount of the time $t+1$ consumption good that at time $t$ and history $s^t$ the government promised to pay
* $R_t(s^t)$ be the gross interest rate on risk-free one-period debt between periods $t$ and $t+1$
* $T_t(s^t)$ be a non-negative lump-sum *transfer* to the representative household [^fn_a]

That $b_{t+1}(s^t)$ is the same for all realizations of $s_{t+1}$ captures its *risk-free* character.
Expand All @@ -138,9 +138,9 @@ The government’s budget constraint in period $t$ at history $s^t$ is
\begin{aligned}
b_t(s^{t-1})
& = \tau^n_t(s^t) n_t(s^t) - g(s_t) - T_t(s^t) +
{b_{t+1}(s^t) \over R_t(s^t )}
{b_{t+1}(s^t) \over R_t(s^t)}
\\
& \equiv z_t(s^t) + {b_{t+1}(s^t) \over R_t(s^t )},
& \equiv z_t(s^t) + {b_{t+1}(s^t) \over R_t(s^t)},
\end{aligned}
```

Expand All @@ -149,7 +149,7 @@ where $z_t(s^t)$ is the net-of-interest government surplus.
To rule out Ponzi schemes, we assume that the government is subject to a **natural debt limit** (to be discussed in a forthcoming lecture).

The consumption Euler equation for a representative household able to trade only one-period risk-free debt
with one-period gross interest rate $R_t(s^t)$ is
with a one-period gross interest rate $R_t(s^t)$ is

$$
{1 \over R_t(s^t)}
Expand All @@ -168,7 +168,7 @@ b_t(s^{t-1}) = z_t(s^t) + \beta \sum_{s^{t+1}\vert s^t} \pi_{t+1}(s^{t+1} | s
```

Components of $z_t(s^t)$ on the right side depend on $s^t$, but the left side is required to depend only
on $s^{t-1}$ .
on $s^{t-1}$.

**This is what it means for one-period government debt to be risk-free**.

Expand Down Expand Up @@ -203,7 +203,7 @@ b_t(s^{t-1})
Notice how the conditioning sets in equation {eq}`TS_gov_wo3` differ: they are $s^{t-1}$ on the left side and
$s^t$ on the right side.

Now let's
Now we

* substitute the resource constraint into the net-of-interest government surplus, and
* use the household’s first-order condition $1-\tau^n_t(s^t)= u_{\ell}(s^t) /u_c(s^t)$ to eliminate the labor tax rate
Expand All @@ -222,7 +222,7 @@ If we substitute appropriate versions of the right side of {eq}`AMSS_44_2` for
we obtain a sequence of *implementability constraints* on a Ramsey allocation in an AMSS economy.

Expression {eq}`TS_gov_wo3` at time $t=0$ and initial state $s^0$
was also an *implementability constraint* on a Ramsey allocation in a Lucas-Stokey economy:
is also an *implementability constraint* on a Ramsey allocation in a Lucas-Stokey economy:

```{math}
:label: TS_gov_wo4
Expand Down Expand Up @@ -322,7 +322,7 @@ It is helpful to apply two transformations to the Lagrangian.

Multiply constraint {eq}`AMSS_44` by $u_c(s^0)$ and the constraints {eq}`AMSS_46` by $\beta^t u_c(s^{t})$.

Then a Lagrangian for the Ramsey problem can be represented as
Then a Lagrangian for the Ramsey problem can be represented as

```{math}
:label: AMSS_lagr;a
Expand Down Expand Up @@ -380,7 +380,7 @@ and with respect to $b_t(s^t)$ as
```

If we substitute $z_t(s^t)$ from {eq}`AMSS_44_2` and its derivative
$z_c(s^t)$ into the first-order condition {eq}`AMSS_foc;a`, we find two
$z_c(s^t)$ into the first-order condition {eq}`AMSS_foc;a`, we find two
differences from the corresponding condition for the optimal allocation
in a Lucas-Stokey economy with state-contingent government debt.

Expand Down Expand Up @@ -435,7 +435,7 @@ $$
where $R_t(s^t)$ is the gross risk-free rate of interest between $t$
and $t+1$ at history $s^t$ and $T_t(s^t)$ are non-negative transfers.

Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers
Throughout this lecture, we shall set transfers to zero (for some issues about the limiting behavior of debt, this is possibly an important difference from AMSS {cite}`aiyagari2002optimal`, who restricted transfers
to be non-negative).

In this case, the household faces a sequence of budget constraints
Expand Down Expand Up @@ -502,7 +502,7 @@ The right side of equation {eq}`eqn:AMSSapp2b` expresses the time $t$ value of g
in terms of a linear combination of terms whose individual components
are measurable with respect to $s^t$.

The sum of terms on the right side of equation {eq}`eqn:AMSSapp2b` must equal
The sum of terms on the right side of equation {eq}`eqn:AMSSapp2b` must equal
$b_t(s^{t-1})$.

That implies that it has to be *measurable* with respect to $s^{t-1}$.
Expand All @@ -512,7 +512,7 @@ constraint imposed in the Lucas and Stokey model.

### Two Bellman Equations

Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries tell probabilities of moving from state $s_-$ to state $s$ in one period.
Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries give probabilities of moving from state $s_-$ to state $s$ in one period.

Let

Expand Down Expand Up @@ -639,12 +639,12 @@ means that there is no state-variable degeneracy in the AMSS model.

In the AMSS model, both $x$ and $s$ are needed to describe the state.

This property of the AMSS model transmits a twisted martingale
This property of the AMSS model transmits a twisted martingale
component to consumption, employment, and the tax rate.

### Digression on Non-negative Transfers

Throughout this lecture, we have imposed that transfers $T_t = 0$.
Throughout this lecture, we have set transfers to zero, i.e., $T_t = 0$.

AMSS {cite}`aiyagari2002optimal` instead imposed a nonnegativity
constraint $T_t\geq 0$ on transfers.
Expand All @@ -663,7 +663,7 @@ random, $V_x(x, s)$ almost surely converges to zero.
For quasi-linear preferences, the first-order condition for maximizing {eq}`eqn:AMSSapp5` subject to {eq}`eqn:AMSSapp6` with respect to $n(s)$ becomes

$$
(1-\mu(s|s_-) ) (1 - u_l(s)) + \mu(s|s_-) n(s) u_{ll}(s) =0
(1-\mu(s|s_-)) (1 - u_l(s)) + \mu(s|s_-) n(s) u_{ll}(s) =0
$$

When $\mu(s|s_-) = \beta V_x(x(s),x)$ converges to zero, in the limit
Expand Down Expand Up @@ -702,7 +702,7 @@ For convenience in matching our computer code, we have expressed
utility as a function of $n$ rather than leisure $l$.
```

We first consider a government expenditure process that we studied earlier in a lecture on
We first consider a government expenditure process that we studied earlier in a lecture on
{doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.

Government expenditures are known for sure in all periods except one.
Expand Down Expand Up @@ -846,7 +846,7 @@ If it is able to trade state-contingent debt, then at time $t=2$
* the government **sells** an Arrow security that pays off when $g_3 = g_l$
* the Ramsey planner designs these purchases and sales designed so that, regardless of whether or not there is a war at $t=3$, the government begins period $t=4$ with the *same* government debt

This pattern facilities smoothing tax rates across states.
This pattern facilitates smoothing tax rates across states.

The government without state-contingent debt cannot do this.

Expand Down Expand Up @@ -973,9 +973,9 @@ plt.show()
When the government experiences a prolonged period of peace, it is able to reduce
government debt and set persistently lower tax rates.

However, the government finances a long war by borrowing and raising taxes.
However, the government finances a long war by borrowing and raising taxes.

This results in a drift away from policies with state-contingent debt that
This results in a drift away from policies with state-contingent debt that
depends on the history of shocks.

This is even more evident in the following figure that plots the evolution of
Expand Down
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