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 {cite}`Barro1979` than they do in {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` than they do in Lucas and Stokey (1983) {cite}`LucasStokey1983`.

### Risk-free One-Period Debt Only

\begin{aligned}

b_t(s^{t-1})

& = \tau^n_t(s^t) n_t(s^t) - g_t(s_t) - T_t(s^t) +

& = \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 )}

\\

& \equiv z(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}

```

where $z(s^t)$ is the net-of-interest government surplus.

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).

```{math}

:label: TS_gov_wo2

b_t(s^{t-1}) = z(s^t) + \beta \sum_{s^{t+1}\vert s^t} \pi_{t+1}(s^{t+1} | s^t)

b_t(s^{t-1}) = z_t(s^t) + \beta \sum_{s^{t+1}\vert s^t} \pi_{t+1}(s^{t+1} | s^t)

{ u_c(s^{t+1}) \over u_c(s^{t}) } \; b_{t+1}(s^t)

```

Components of $z(s^t)$ on the right side depend on $s^t$, but the left side is required to depend only

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

on $s^{t-1}$ .

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

Therefore, the sum on the right side of equation {eq}`TS_gov_wo2` also has to depend only on $s^{t-1}$.

Therefore, the right side of equation {eq}`TS_gov_wo2` also has to depend only on $s^{t-1}$.

This requirement will give rise to **measurability constraints** on the Ramsey allocation to be discussed soon.

particular realization $s_{t}$) we get

$$

b_t(s^{t-1}) = z(s^t) + \sum_{s^{t+1}\vert s^t} \beta \pi_{t+1}(s^{t+1} | s^t)

b_t(s^{t-1}) = z_t(s^t) + \sum_{s^{t+1}\vert s^t} \beta \pi_{t+1}(s^{t+1} | s^t)

{ u_c(s^{t+1}) \over u_c(s^{t}) }

\, \left[z(s^{t+1}) + {b_{t+2}(s^{t+1}) \over R_{t+1}(s^{t+1})}\right]

\, \left[z_{t+1}(s^{t+1}) + {b_{t+2}(s^{t+1}) \over R_{t+1}(s^{t+1})}\right]

$$

After making similar repeated substitutions for all future occurrences of

government indebtedness, and by invoking the natural debt limit, we

government indebtedness, and by invoking a natural debt limit, we

arrive at:

```{math}

\begin{aligned}

b_t(s^{t-1})

&= \sum_{j=0}^\infty \sum_{s^{t+j} | s^t} \beta^j \pi_{t+j}(s^{t+j} | s^t)

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j})

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z_{t+j}(s^{t+j})

\end{aligned}

```

* 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

so that we can express the net-of-interest government surplus $z(s^t)$ as

so that we can express the net-of-interest government surplus $z_t(s^t)$ as

```{math}

:label: AMSS_44_2

z(s^t)

= \left[1 - {u_{\ell}(s^t) \over u_c(s^t)}\right] \left[c_t(s^t)+g_t(s_t)\right]

-g_t(s_t) - T_t(s^t)\,.

z_t(s^t)

= \left[1 - {u_{\ell}(s^t) \over u_c(s^t)}\right] \left[c_t(s^t)+g(s_t)\right]

-g(s_t) - T_t(s^t)\,.

```

If we substitute appropriate versions of the right side of {eq}`AMSS_44_2` for $z(s^{t+j})$ into equation {eq}`TS_gov_wo3`,

If we substitute appropriate versions of the right side of {eq}`AMSS_44_2` for $z_{t+j}(s^{t+j})$ into equation {eq}`TS_gov_wo3`,

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 $s^0$

:label: TS_gov_wo4

b_0(s^{-1}) = \mathbb E_0 \sum_{j=0}^\infty \beta^j

{ u_c(s^{j}) \over u_c(s^{0}) } \;z(s^{j})

{ u_c(s^{j}) \over u_c(s^{0}) } \;z_j(s^{j})

```

Indeed, it was the *only* implementability constraint there.

:label: TS_gov_wo4a

b_t(s^{t-1}) = \mathbb E_t \sum_{j=0}^\infty \beta^j

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j})

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z_{t+j}(s^{t+j})

```

Equation {eq}`TS_gov_wo4a` must hold for each $s^t$ for each $t \geq 1$.

$$

\max_{\{c_t(s^t),b_{t+1}(s^t)\}}

\mathbb E_0 \sum_{t=0}^\infty \beta^t

u\left(c_t(s^t),1-c_t(s^t)-g_t(s_t)\right)

u\left(c_t(s^t),1-c_t(s^t)-g(s_t)\right)

$$

where the maximization is subject to

:label: AMSS_44

\mathbb E_{0} \sum_{j=0}^\infty \beta^j

{ u_c(s^{j}) \over u_c(s^{0}) } \;z(s^{j}) \geq b_0(s^{-1})

{ u_c(s^{j}) \over u_c(s^{0}) } \;z_j(s^{j}) \geq b_0(s^{-1})

```

and

\mathbb E_{t} \sum_{j=0}^\infty \beta^j

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;

z(s^{t+j}) = b_t(s^{t-1})

z_{t+j}(s^{t+j}) = b_t(s^{t-1})

\quad \forall \, t, s^t

```

&\;\geq\; (\leq)\;\, 0 \quad \text{if the constraint binds in the following direction }

\\

& \mathbb E_{t} \sum_{j=0}^\infty \beta^j

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z(s^{t+j}) \;\geq \;(\leq)\;\, b_t(s^{t-1})

{ u_c(s^{t+j}) \over u_c(s^{t}) } \;z_{t+j}(s^{t+j}) \;\geq \;(\leq)\;\, b_t(s^{t-1})

\end{aligned}

$$

\begin{aligned}

J &= \mathbb E_{0} \sum_{t=0}^\infty \beta^t

\biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\right)\\

\biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g(s_t)\right)\\

& \qquad + \gamma_t(s^t) \Bigl[ \mathbb E_{t} \sum_{j=0}^\infty \beta^j

u_c(s^{t+j}) \,z(s^{t+j}) - u_c(s^{t}) \,b_t(s^{t-1}) \biggr\}

u_c(s^{t+j}) \,z_{t+j}(s^{t+j}) - u_c(s^{t}) \,b_t(s^{t-1}) \biggr\}

\\

&= \mathbb E_{0} \sum_{t=0}^\infty \beta^t

\biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g_t(s_t)\right)

\biggl\{ u\left(c_t(s^t), 1-c_t(s^t)-g(s_t)\right)

\\

& \qquad + \Psi_t(s^t)\, u_c(s^{t}) \,z(s^{t}) -

& \qquad + \Psi_t(s^t)\, u_c(s^{t}) \,z_t(s^{t}) -

\gamma_t(s^t)\, u_c(s^{t}) \, b_t(s^{t-1}) \biggr\}

\end{aligned}

```

\begin{aligned}

u_c(s^t)-u_{\ell}(s^t) &+ \Psi_t(s^t)\left\{ \left[

u_{cc}(s^t) - u_{c\ell}(s^{t})\right]z(s^{t}) +

u_{cc}(s^t) - u_{c\ell}(s^{t})\right]z_t(s^{t}) +

u_{c}(s^{t})\,z_c(s^{t}) \right\}

\\

& \hspace{35mm} - \gamma_t(s^t)\left[

\mathbb E_{t} \left[\gamma_{t+1}(s^{t+1})\,u_c(s^{t+1})\right] = 0

```

If we substitute $z(s^t)$ from {eq}`AMSS_44_2` and its derivative

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

differences from the corresponding condition for the optimal allocation

in a Lucas-Stokey economy with state-contingent government debt.

* This term reflects the constraint that

beginning-of-period government indebtedness must be the same across all

realizations of next period’s state, a constraint that would not be present if

government debt could be state contingent.

government debt could be state-contingent.

1. The Lagrange multiplier $\Psi_t(s^t)$ in the first-order condition

{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.

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 makes a possibly

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

\beta V_x(x(s),s) = \mu(s|s_-)

```

Applying the envelope theorem to Bellman equation {eq}`eqn:AMSSapp5` gives

Applying an envelope theorem to Bellman equation {eq}`eqn:AMSSapp5` gives

```{math}

:label: eqn:AMSSapp8

:label: eqn:AMSSapp9

V_x(x_-, s_-) = \sum_{s} \left( \Pi(s|s_-) {\frac{u_c(s)}{\sum_{\tilde s}

\Pi(\tilde s| s_-) u_c(\tilde s)}} \right) V_x(x(s), s)

\Pi(\tilde s| s_-) u_c(\tilde s)}} \right) V_x(x, s)

```

Equation {eq}`eqn:AMSSapp9` states that $V_x(x, s)$ is a *risk-adjusted martingale*.

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

the Lagrange multiplier on the Lucas-Stokey implementability constraint

* time invariance of $V_x(x,s)$ is the source of a key

feature of the Lucas-Stokey model, namely, **state variable degeneracy** in which $x_t$ is an exact time-invariant function of $s_t$)

feature of the Lucas-Stokey model, namely, **state variable degeneracy** in which $x_t$ is an exact time-invariant function of $s_t$.

That $V_x(x,s)$ varies over time according to a twisted martingale

means that there is no state-variable degeneracy in the AMSS model.

### Anticipated One-Period War

In our lecture on {doc}`optimal taxation with state contingent debt <opt_tax_recur>`

In our lecture on {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`

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

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

utility as a function of $n$ rather than leisure $l$.

```

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

{doc}`optimal taxation with state contingent debt <opt_tax_recur>`.

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.

```{literalinclude} _static/lecture_specific/opt_tax_recur/crra_utility.py

```

The following figure plots the Ramsey plan under both complete and incomplete

The following figure plots Ramsey plans under complete and incomplete

markets for both possible realizations of the state at time $t=3$.

Optimal policies when the government has access to state contingent debt are

represented by black lines, while the optimal policies when there is only a risk-free bond are in red.

Ramsey outcomes and policies when the government has access to state-contingent debt are

represented by black lines and by red lines when there is only a risk-free bond.

Paths with circles are histories in which there is peace, while those with

triangle denote war.

```{code-cell} python3

# Initialize μgrid for value function iteration

μ_grid = np.linspace(-0.7, 0.01, 200)

μ_grid = np.linspace(-0.7, 0.01, 300)

time_example = CRRAutility()

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 $g_3 = g_l$

* these purchases are designed in such a way that regardless of whether or not there is a war at $t=3$, the government will begin period $t=4$ with the *same* government debt

* 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

This pattern facilities smoothing tax rates across states.

The government without state contingent debt cannot do this.

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 $t=3$.

It responds to this constraint by smoothing tax rates across time.

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**.

A **low** risk-free rate of return on government debt between time $2$ and time $3$ allows the government to enter period $3$ with **lower** government debt than it entered period $2$.

To finance a war it raises taxes and issues more debt.

To finance a war at time $3$ it raises taxes and issues more debt to carry into perpetual peace that begins in period $4$.

To service the additional debt burden, it raises taxes in all future periods.

The absence of state contingent debt leads to an important difference in the

The absence of state-contingent debt leads to an important difference in the

optimal tax policy.

When the Ramsey planner has access to state contingent debt, the optimal tax

When the Ramsey planner has access to state-contingent debt, the optimal tax

policy is history independent

* the tax rate is a function of the current level of government spending only,

given the Lagrange multiplier on the implementability constraint

Without state contingent debt, the optimal tax rate is history dependent.

Without state-contingent debt, the optimal tax rate is history dependent.

* A war at time $t=3$ causes a permanent increase in the tax rate.

* A war at time $t=3$ causes a permanent **increase** in the tax rate.

* Peace at time $t=3$ causes a permanent **reduction** in the tax rate.

#### Perpetual War Alert

model with state-contingent debt.

The figure below plots optimal tax policies for both the economy with

state contingent debt (circles) and the economy with only a risk-free bond

state-contingent debt (circles) and the economy with only a risk-free bond

(triangles).

```{code-cell} python3

```

When the government experiences a prolonged period of peace, it is able to reduce

government debt and set permanently lower tax rates.

government debt and set persistently lower tax rates.

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

the two policies over 200 periods.

This outcome reflects the presence of a force for **precautionary saving** that the incomplete markets structure imparts to the Ramsey plan.

In {doc}`this subsequent lecture <amss2>` and {doc}`this subsequent lecture <amss3>`, some ultimate consequences of that force are explored.

```{code-cell} python3

T = 200 # Set T to 200 periods

sim_seq_long = log_sequential.simulate(0.5, 0, T)