This lecture describes several linear-quadratic versions of a model that Guillermo Calvo {cite}`Calvo1978` used to illustrate the **time inconsistency** of optimal government

plans.

Like Chang {cite}`chang1998credible`, we use the models as a laboratory in which to explore consequences of timing protocols for government decision making.

Like Chang {cite}`chang1998credible`, we use these models as laboratories in which to explore consequences of timing protocols for government decision making.

The models focus attention on intertemporal tradeoffs between

linear-quadratic discounted dynamic programming to compute an optimal government

plan under each of our timing protocols.

In addition to what's in Anaconda, this lecture will need the following libraries:

A sister lecture {doc}`calvo_machine_learn` studies some of the same models but does not use dynamic programming.

Instead it uses a **machine learning** approach that does not explicitly recognize the recursive structure structure of the Ramsey problem that Chang {cite}`chang1998credible` saw and that we exploit in this lecture.

In addition to what's in Anaconda, this lecture will use the following libraries:

```{code-cell} ipython3

:tags: [hide-output]

from IPython.display import display, Math

```

## Model components

## Model Components

There is no uncertainty.

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

## 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

To deduce this recommendation, Milton Friedman assumed that the taxes that government must impose in order to acquire money at rate $\mu_t$ do not distort economic decisions.

- for example, the government imposes lump sum taxes that distort no decisions by private agents

- for example, perhaps the government can impose lump sum taxes that distort no decisions by private agents

## Calvo's perturbation of optimal deflation rate

## Calvo's Distortion of Friedman's optimal Deflation Rate

The starting point of Calvo {cite}`Calvo1978` and Chang {cite}`chang1998credible`

is that such lump sum taxes are not available.

decisions will make $\vec \mu$ endogenous, i.e., a theoretical *output* instead of an *input*.

## Intertemporal structure

## Intertemporal Structure

Criterion function {eq}`eq_old7` and the constraint system {eq}`eq_old4` exhibit the following

structure:

We'll also study outcomes under other timing protocols.

## Four timing protocols

## Four Timing Protocols

We consider four models of government policy making that differ in

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

## A Ramsey planner

## A Ramsey Planner

Here we consider a Ramsey planner that chooses

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

\theta_0 = \theta_0^R = - \frac{P_{21}}{P_{22}}

$$

### Representation of Ramsey plan

## Representation of Ramsey Plan

The preceding calculations indicate that we can represent a Ramsey plan

$\vec \mu$ recursively with the following system created in the spirit of Chang {cite}`chang1998credible`:

equation {eq}`eq_old3`.

### Digression on timeless perspective

## Digression on Timeless Perspective

As our subsequent calculations will verify, $ \vec \mu^R, \vec \theta^R, \vec v^R $ are each monotone sequences that are bounded below and converge from above to limiting values.

- the timing protocol and/or

- assumptions about how government decision makers think their decisions affect the representative agent's beliefs about future government decisions

## Constrained-to-constant-growth-rate Ramsey plan

## Constrained-to-Constant-Growth-Rate Ramsey Plan

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

government in order eventually to highlight the time-variation of

$\mu_t$ that is a telltale sign of a Ramsey plan's **time inconsistency**.

## Markov perfect governments

## Markov Perfect Governments

We now describe yet another timing protocol.

* we equate $\mu_t = \mu$ only *after* we have computed a time $t$ government's first-order condition for $\mu_t$.

(compute_lq)=

## Outcomes under three timing protocols

## Outcomes under Three Timing Protocols

We want to compare outcome sequences $\{ \theta_t,\mu_t \}$ under three timing protocols associated with

In some subsequent calculations, we'll use our Python code to study how gaps between

these outcome vary depending on parameters such as the cost parameter $c$ and the discount factor $\beta$.

### Ramsey planner's value function

### Ramsey Planner's Value Function

The next code plots the Ramsey Planner's value function $J(\theta)$ as well as the value function

of a constrained Ramsey planner who must choose a constant

We leave it to the reader to explore consequences of other constellations of parameter values.

### Time inconsistency of Ramsey plan

### Time Inconsistency of Ramsey Plan

The variation over time in $\vec \mu$ chosen by the Ramsey planner

is a symptom of time inconsistency.

A constrained-to-constant-$\mu$ Ramsey plan is time consistent by construction. So is a Markov perfect plan.

```

### Implausibility of Ramsey plan

### 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

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

### Ramsey plan strikes back

### Ramsey Plan Strikes Back

Research by Abreu {cite}`Abreu`, Chari and Kehoe {cite}`chari1990sustainable`

{cite}`stokey1989reputation`, and Stokey {cite}`Stokey1991` discovered conditions under which a Ramsey plan can be rescued from the complaint that it is not credible.

We turn to such theories of *sustainable plans* next.

## A fourth model of government decision making

## A Fourth Model of Government Decision Making

In this model

- at each $t$, the government chooses $\mu_t$ to maximize

a continuation discounted utility.

### Government decisions

### Government Decisions

$\vec \mu$ is chosen by a sequence of government

decision makers, one for each $t \geq 0$.

expect an associated $\theta_0^A$ for $t+1$. Here $\vec \mu^A = \{\mu_j^A \}_{j=0}^\infty$ is

an alternative government plan to be described below.

### Temptation to deviate from plan

### Temptation to Deviate from Plan

The government's one-period return function $s(\theta,\mu)$

described in equation {eq}`eq_old6` above has the property that for all

current payoff, it is only because it forecasts adverse consequences that

its setting of $\mu_t$ would bring for continuation government payoffs via alterations in the private sector's expectations.

## Sustainable or credible plan

## Sustainable or Credible Plan

We call a plan $\vec \mu$ **sustainable** or **credible** if at

each $t \geq 0$ the government chooses to confirm private

The key is an object called a **self-enforcing** plan.

### Abreu's self-enforcing plan

### Abreu's Self-Enforcing Plan

A plan $\vec \mu^A$ (here the superscipt $A$ is for Abreu) is said to be **self-enforcing** if

We shall use a construction featured in Abreu ({cite}`Abreu`) to construct a

self-enforcing plan with low time $0$ value.

### Abreu's carrot-stick plan

### Abreu's Carrot-Stick Plan

Abreu ({cite}`Abreu`) invented a way to create a self-enforcing plan with a low

initial value.

For an appropriate $T_A$, this plan can be verified to be self-enforcing and therefore credible.

### Example of self-enforcing plan

### Example of Self-Enforcing Plan

The following example implements an Abreu stick-and-carrot plan.

check_ramsey(clq)

```

### Recursive representation of a sustainable plan

### Recursive Representation of a Sustainable Plan

We can represent a sustainable plan recursively by taking the

continuation value $v_t$ as a state variable.

expectations by setting $\mu_t$ equal to the recommended value

$\hat \mu_t$, or whether it disappoints those expectations.

## Whose plan is it?

## Whose Plan is It?

A credible government plan $\vec \mu$ plays multiple roles.

An argument in favor of the *simply confirm* interpretation is gathered from staring at the key inequality {eq}`eq_old100a` that defines a credible policy.

## Comparison of equilibrium values

## Comparison of Equilibrium Values

We have computed plans for

- a better plan -- possibly one that attains values associated with

Ramsey plan -- that is not self-enforcing.

## Note on dynamic programming squared

## Note on Dynamic Programming Squared

The theory deployed in this lecture is an application of what we nickname **dynamic programming squared**.