We begin with an introduction to the model.

## Competitive Equilibrium with Distorting Taxes

## 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>`.

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

### Risk-free one-period debt only

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

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

### Comparison with Lucas-Stokey Economy

### 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 $t$.

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.

### Ramsey Problem Without State-contingent Debt

### Ramsey problem without state-contingent debt

After we have substituted the resource constraint into the utility function, we can express the Ramsey problem as being to choose an allocation that solves

given $b_0(s^{-1})$.

#### Lagrangian Formulation

#### Lagrangian formulation

Let $\gamma_0(s^0)$ be a non-negative Lagrange multiplier on constraint {eq}`AMSS_44`.

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

### Some Calculations

### Some calculations

It is helpful to apply two transformations to the Lagrangian.

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

using techniques like those in our lectures on {doc}`dynamic Stackelberg models <dyn_stack>` and {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.

## Recursive Version of AMSS Model

## Recursive version of AMSS model

We now describe a recursive formulation of the AMSS economy.

$0$ Ramsey planner and for time $t \geq 1$, history $s^t$

continuation Ramsey planners.

### Recasting State Variables

### Recasting state variables

In the AMSS setting, the government faces a sequence of budget constraints

for $t \geq 1$.

### Measurability Constraints

### Measurability constraints

Write equation {eq}`eqn:AMSSapp2` as

Equations {eq}`eqn:AMSSapp2b` are the *measurability constraints* that the AMSS model adds to the single time $0$ implementation

constraint imposed in the Lucas and Stokey model.

### Two Bellman Equations

### 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.

u_{c,0} b_0 = u_{c,0} (n_0-g_0) - u_{l,0} n_0 + x_0

```

### Martingale Supercedes State-Variable Degeneracy

### Martingale supercedes state-variable degeneracy

Let $\mu(s|s_-) \Pi(s|s_-)$ be a Lagrange multiplier on the constraint {eq}`eqn:AMSSapp6`

for state $s$.

```{exercise-end}

```

### Absence of State Variable Degeneracy

### Absence of state variable degeneracy

Along a Ramsey plan, the state variable $x_t = x_t(s^t, b_0)$

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

This property of the AMSS model transmits a twisted martingale

component to consumption, employment, and the tax rate.

### Digression on Non-negative Transfers

### Digression on non-negative transfers

Throughout this lecture, we have imposed that transfers $T_t = 0$.

We now turn to some examples.

### Anticipated One-Period War

### Anticipated one-period war

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.

* 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

#### Perpetual war alert

History dependence occurs more dramatically in a case in which the government

perpetually faces the prospect of war.