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59 changes: 30 additions & 29 deletions lectures/amss.md
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Expand Up @@ -63,9 +63,9 @@ We begin with an introduction to the model.

## Competitive Equilibrium with Distorting Taxes

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

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

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

Expand All @@ -74,7 +74,7 @@ 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,
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
In 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 @@ -93,7 +93,7 @@ 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 a pre-tax wage rate equal 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

Expand All @@ -110,13 +110,14 @@ where

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>`.
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.

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`.
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`.
This is in contrast to the behavior they exhibit in Lucas and Stokey (1983) {cite}`LucasStokey1983`.

### Risk-free One-Period Debt Only

Expand Down Expand Up @@ -168,7 +169,7 @@ 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 +204,7 @@ 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 let us

* 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 Down Expand Up @@ -246,13 +247,15 @@ Equation {eq}`TS_gov_wo4a` must hold for each $s^t$ for each $t \geq 1$.

### Comparison with Lucas-Stokey Economy

The expression on the right side of {eq}`TS_gov_wo4a` in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government net-of-interest surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $t$.
The expression on the right side of {eq}`TS_gov_wo4a` in the Lucas-Stokey (1983) economy would equal the present value of a continuation stream of government net-of-interest surpluses.
This present value would be evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $t$.

In the Lucas-Stokey economy, that present value is measurable with respect to $s^t$.

In the AMSS economy, the restriction that government debt be risk-free imposes that that same present value must be measurable with respect to $s^{t-1}$.
In the AMSS economy, the restriction that government debt be risk-free imposes that the same present value must be measurable with respect to $s^{t-1}$.

In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each $(t, s^t)$ what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to the **marketable subspace** of the AMSS model.
In the language used in the literature on incomplete markets models, the AMSS model requires the following.
At each $(t, s^t)$, what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to the **marketable subspace** of the AMSS model.

### Ramsey Problem Without State-contingent Debt

Expand Down Expand Up @@ -312,7 +315,7 @@ A negative multiplier $\gamma_t(s^t)<0$ means that if we could
relax constraint {eq}`AMSS_46`, we would like to *increase* the beginning-of-period
indebtedness for that particular realization of history $s^t$.

That would let us reduce the beginning-of-period indebtedness for some other history [^fn_b].
This would allow us to reduce the beginning-of-period indebtedness for some other history [^fn_b].

These features flow from the fact that the government cannot use state-contingent debt and therefore cannot allocate its indebtedness efficiently across future states.

Expand All @@ -322,7 +325,7 @@ 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 @@ -435,8 +438,8 @@ $$
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
to be non-negative).
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 @@ -512,7 +515,7 @@ 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 @@ -544,8 +547,8 @@ for each $s \in {\cal S}$:
```

A continuation Ramsey planner at $t \geq 1$ takes
$(x_{t-1}, s_{t-1}) = (x_-, s_-)$ as given and before
$s$ is realized chooses
$(x_{t-1}, s_{t-1}) = (x_-, s_-)$ as given.
Before $s$ is realized, the planner chooses
$(n_t(s_t), x_t(s_t)) = (n(s), x(s))$ for $s \in {\cal S}$.

The **Ramsey planner** takes $(b_0, s_0)$ as given and chooses $(n_0, x_0)$.
Expand Down Expand Up @@ -669,9 +672,8 @@ $$
When $\mu(s|s_-) = \beta V_x(x(s),x)$ converges to zero, in the limit
$u_l(s)= 1 =u_c(s)$, so that $\tau(x(s),s) =0$.

Thus, in the limit, if $g_t$ is perpetually random, the government
accumulates sufficient assets to finance all expenditures from earnings on those
assets, returning any excess revenues to the household as non-negative lump-sum transfers.
Thus, in the limit, if $g_t$ is perpetually random, the government accumulates sufficient assets to finance all expenditures from earnings on those assets.
It returns any excess revenues to the household as non-negative lump-sum transfers.

### Code

Expand Down Expand Up @@ -705,7 +707,7 @@ 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
{doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.

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

* For $t<3$ or $t > 3$ we assume that $g_t = g_l = 0.1$.
* At $t = 3$ a war occurs with probability 0.5.
Expand Down Expand Up @@ -844,13 +846,13 @@ If it is able to trade state-contingent debt, then at time $t=2$

* the government **purchases** an Arrow security that pays off when $g_3 = g_h$
* 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
* the Ramsey planner designs these purchases and sales 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.

Instead, it must enter time $t=3$ with the same level of debt falling due whether there is peace or war at $t=3$.
Instead, the government must enter time $t=3$ with the same level of debt falling due, regardless of whether there is peace or war at $t=3$.

The risk-free rate between time $2$ and time $3$ is unusually **low** because at time $2$ consumption at time $3$ is expected to be unusually **low**.

Expand All @@ -877,7 +879,7 @@ Without state-contingent debt, the optimal tax rate is history dependent.
#### Perpetual War Alert

History dependence occurs more dramatically in a case in which the government
perpetually faces the prospect of war.
perpetually faces the possibility of war.

This case was studied in the final example of the lecture on
{doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.
Expand Down Expand Up @@ -973,7 +975,7 @@ 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, when faced with a long war, the government finances it by borrowing and raising taxes.

This results in a drift away from policies with state-contingent debt that
depends on the history of shocks.
Expand Down Expand Up @@ -1003,8 +1005,7 @@ titles = ['Consumption', 'Labor Supply', 'Government Debt',

fig, axes = plt.subplots(3, 2, figsize=(14, 10))

for ax, title, ls, amss in zip(axes.flatten(), titles, sim_ls, \
sim_amss):
for ax, title, ls, amss in zip(axes.flatten(), titles, sim_ls, sim_amss):
ax.plot(ls, '-k', amss, '-.b', alpha=0.5)
ax.set(title=title)
ax.grid()
Expand Down
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