well as from chapter 19 of {cite}`Ljungqvist2012`.

In addition, we'll use ideas from linear-quadratic dynamic programming

described in [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg problems <dyn_stack>`.

described in [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg plans <dyn_stack>`.

We specify model fundamentals in ways that allow us to use

linear-quadratic discounted dynamic programming to compute an optimal government

- $\theta_t = p_{t+1} - p_t$ be the net rate of inflation between $t$ and $t+1$

- $\mu_t = m_{t+1} - m_t$ be the net rate of growth of nominal balances

The demand for real balances is governed by a perfect foresight

version of a Cagan {cite}`Cagan` demand function for real balances:

The demand for real balances is governed by a discrete time version of Sargent and Wallace's {cite}`sargent1973stability` perfect foresight version of a Cagan {cite}`Cagan` demand function for real balances:

```{math}

:label: eq_old1

related to the public's expected rate of inflation, which equals

the actual rate of inflation because there is no uncertainty here.

(When there is no uncertainty, an assumption of **rational expectations** that becomes equivalent to **perfect foresight**).

(When there is no uncertainty, an assumption of **rational expectations** becomes equivalent to **perfect foresight**).

(See {cite}`Sargent77hyper` for a rational expectations version of the model when there is uncertainty.)

({cite}`Sargent77hyper` presents a rational expectations version of the model when there is uncertainty.)

Subtracting the demand function {eq}`eq_old1` at time $t$ from the demand

function at $t+1$ gives:

An equivalence class of continuation money growth sequences $\{\mu_{t+j}\}_{j=0}^\infty$ deliver the same $\theta_t$.

We shall use this insight to help us simplify our analsis of alternative government policy problems.

We shall use this insight to help us simplify our analysis of alternative government policy problems.

That future rates of money creation influence earlier rates of inflation

makes timing protocols matter for modeling optimal government policies.

We use form {eq}`eq_old4` because we want to apply an approach described in our lecture on {doc}`Stackelberg problems <dyn_stack>`.

We use form {eq}`eq_old4` because we want to apply an approach described in our lecture on {doc}`Stackelberg plans <dyn_stack>`.

Notice that $\frac{1+\alpha}{\alpha} > 1$ is an eigenvalue of transition matrix $A$ that threatens to destabilize the state-space system.

The money demand function {eq}`eq_old1` and the utility function {eq}`eq_old5` imply that

$$

U(-\alpha \theta_t) = u_1 + u_2 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 .

U(-\alpha \theta_t) = u_0 + u_1 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 .

$$ (eq_old5a)

The``bliss level`` of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains

it is $-\frac{u_1}{u_2 \alpha}$.

## Friedman's Optimal Rate of Deflation

According to {eq}`eq_old5a`, the "bliss level" of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains it is

According to {eq}`eq_old5a`, the ``bliss level`` of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains it is

$$

Values of $V(\bar \mu)$ computed according to formula {eq}`eq:barvdef` for three different values of $\bar \mu$ will play important roles below.

* $V(\mu^{MP})$ is the value of attained by the government in a **Markov perfect equilibrium**

* $V(\mu^R_\infty)$ is the value that a continuation Ramsey planner attains at $t \rightarrow +\infty$

* We shall discover that $V(\mu^R_\infty)$ is the worst continuation value attained along a Ramsey plan

* $V(\mu^{CR})$ is the value of attained by the government in a **constrained to constant $\mu$ equilibrium**

* $V(\mu^R_\infty)$ is the limiting value attained by a continuation Ramsey planner under a Ramsey plan.

* We shall see that $V(\mu^R_\infty)$ is a worst continuation value attained along a Ramsey plan

## Structure

$\{\mu_t\}_{t=0}^\infty$ once and for all at time $0$

subject to the constraint that $\mu_t = \mu$ for all

$t \geq 0$; or

- A sequence indexed by $t =0, 1, 2, \ldots$ of separate policymakers

- A sequence of distinct policymakers indexed by $t =0, 1, 2, \ldots$

- a time $t$ policymaker chooses $\mu_t$ only and forecasts that future government decisions are unaffected by its choice.

We'll begin with the timing protocol associated with a Ramsey plan and deploy

an application of what we nickname **dynamic programming squared**.

The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a second Bellman equation.

The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a value function associated with a second Bellman equation.

Thus, our models have involved two Bellman equations:

A value $\theta$ from one Bellman equation appears as an argument of a second Bellman equation for another value $v$.

.

## A Ramsey Planner

Here we consider a Ramsey planner that chooses

$\{\mu_t, \theta_t\}_{t=0}^\infty$ to maximize {eq}`eq_old7`

subject to the law of motion {eq}`eq_old4`.

We can split this problem into two stages, as in {doc}`Stackelberg problems <dyn_stack>` and {cite}`Ljungqvist2012` Chapter 19.

We can split this problem into two stages, as in the lecture {doc}`Stackelberg plans <dyn_stack>` and {cite}`Ljungqvist2012` Chapter 19.

In the first stage, we take the initial inflation rate $\theta_0$ as given

and solve what looks like an ordinary LQ discounted dynamic programming problem.

x' = Ax + B\mu

$$

As in {doc}`Stackelberg problems <dyn_stack>`, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$.

As in the lecture {doc}`Stackelberg plans <dyn_stack>`, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$.

Guessing that $J(x) = - x'Px$ and substituting into the Bellman

equation gives rise to the algebraic matrix Riccati equation:

## Time inconsistency

As discussed in {doc}`Stackelberg problems <dyn_stack>` and {doc}`Optimal taxation with state-contingent debt <opt_tax_recur>`, a continuation Ramsey plan is not a Ramsey plan.

As discussed in {doc}`Stackelberg plans <dyn_stack>` and {doc}`Optimal taxation with state-contingent debt <opt_tax_recur>`, a continuation Ramsey plan is not a Ramsey plan.

This is a concise way of characterizing the time inconsistency of a Ramsey plan.

We now describe a model in which we restrict the Ramsey planner's choice set.

Instead of choosing a sequence of money growth rates $\vec \mu \in {\bf R}^2$, we restrict the

Instead of choosing a sequence of money growth rates $\vec \mu \in {\bf L}^2$, we restrict the

government to choose a time-invariant money growth rate $\bar \mu$.

We created this version of the model to highlight an aspect of a Ramsey plan associated with its time inconsistency, namely, the feature that optimal settings of the policy instrument vary over time.

The figure also plots the limiting value $\theta_\infty^R$ to which the promised inflation rate $\theta_t$ converges under the Ramsey plan.

In addition, the figure indicates an MPE inflation rate $\theta^{CR}$ and a bliss inflation $\theta^*$.

In addition, the figure indicates an MPE inflation rate $\theta^{MPE}$, $\theta^{CR}$, and a bliss inflation $\theta^*$.

```{code-cell} ipython3

:tags: [hide-input]

clq.V_θ(θ_inf))

```

So our claim that $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$is verified numerically.

So our claim that $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$ is verified numerically.

Since $J(\theta_\infty^R) = V^{CR}(\theta_\infty^R)$ occurs at a tangency point at which

$J(\theta)$ is increasing in $\theta$, it follows that

The horizontal dotted lines indicate values

$V(\mu_\infty^R), V(\mu^{CR}), V(\mu^{MPE}) $ of time-invariant money

growth rates $\mu_\infty^R, \mu^{CR}$ and $\mu_{MPE}$, respectfully.

growth rates $\mu_\infty^R, \mu^{CR}$ and $\mu^{MPE}$, respectfully.

Notice how $J(\theta)$ and $V^{CR}(\theta)$ are tangent and increasing at

$\theta = \theta_\infty^R$, which implies that $\theta^{CR} > \theta_\infty^R$

\begin{aligned}

\theta^{CR} & = - \frac{\alpha u_1}{\alpha^2 u_2 + c } \\

\theta^{MPE} & = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c} \\

\theta^{MPE} & = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c}

\theta^{*} & = -\frac{u_1}{u_2 \alpha}

\end{aligned}

$$

But let's see what happens when we change $c$.

```{code-cell} ipython3

The above table and figures show how

changes in $c$ alter $\theta_\infty^R$

and $\theta_0^R$ as well as $\theta^{CR}$ and $\theta^{MPE}$, but not

$\theta^*$, again in accord with formulas

$\theta^*,$ again in accord with formulas

{eq}`eq:Friedmantheta`, {eq}`eq:muRamseyconstrained`, and {eq}`eq:Markovperfectmu`.

Notice that as $c $ gets larger and larger, $\theta_\infty^R, \theta_0^R$

### Implausibility of Ramsey Plan

In settings in which governments actually choose sequentially, many economists

regard a time inconsistent plan as implausible because of the incentives to

deviate that are presented along the plan.

Many economists regard a time inconsistent plan as implausible because they question the plausibility of timing protocol in

which a plan for setting a sequence of policy variables is chosen once-and-for-all at time $0$.

A way to state this reaction is to say that a Ramsey plan is not credible because there are persistent incentives for policymakers to deviate from it.

For that reason, the Markov perfect equilibrium concept attracts many

economists.

* A Markov perfect equilibrium plan is constructed to insure that government policymakers who choose sequentially do not want to deviate from it.

* A Markov perfect equilibrium plan is constructed to insure that a sequence of government policymakers who choose sequentially do not want to deviate from it.

The *no incentive to deviate from the plan* property is what makes the Markov perfect equilibrium concept attractive.

The property of a Markov perfect equilibrium that there is *no incentive to deviate from the plan* makes it attractive.

## Comparison of Equilibrium Values