To illustrate it, we describe a linear quadratic version of a model

due to Lucas and Prescott {cite}`LucasPrescott1971`.

This 1971 paper is one of a small number of research articles that ignited the *rational expectations revolution*.

That 1971 paper is one of a small number of research articles that ignited a *rational expectations revolution*.

We follow Lucas and Prescott by employing a setting that is readily "Bellmanized" (i.e., capable of being formulated in terms of dynamic programming problems).

We follow Lucas and Prescott by employing a setting that is readily "Bellmanized" (i.e., susceptible to being formulated as a dynamic programming problems.

Because we use linear quadratic setups for demand and costs, we can deploy the LQ programming techniques described in {doc}`this lecture <lqcontrol>`.

from quantecon import LQ

```

### The Big Y, Little y Trick

### The Big Y, little y Trick

This widely used method applies in contexts in which a "representative firm" or agent is a "price taker" operating within a competitive equilibrium.

This widely used method applies in contexts in which a **representative firm** or agent is a "price taker" operating within a competitive equilibrium.

The following setting justifies the concept of a representative firm.

The following setting justifies the concept of a representative firm that stands in for a large number of other firms too.

There is a uniform unit measure of identical firms named $\omega \in \Omega = [0,1]$.

All firms end up choosing to produce the same output, so that at the end of the day $ y(\omega) = y $ and $Y =y = \int_{0}^1 y(\omega) d \, \omega $.

This setting allows us to speak of a ``representative firm'' that chooses to produce $y$.

This setting allows us to speak of a representative firm that chooses to produce $y$.

We want to impose that

We begin by applying the Big $Y$, little $y$ trick in a very simple static context.

#### A Simple Static Example of the Big Y, Little y Trick

#### A Simple Static Example of the Big Y, little y Trick

Consider a static model in which a unit measure of firms produce a homogeneous good that is sold in a competitive market.

After solving for $Y$, we can compute the competitive equilibrium price $p$ from the inverse demand curve {eq}`ree_comp3d_static`.

### Related Planning Problem

Define **consumer surplus** as the area under the inverse demand curve:

$$

S_c (Y)= \int_0^Y (a_0 - a_1 s) ds = a_o Y - \frac{a_1}{2} Y^2 .

$$

Define the social cost of production as

$$ S_p (Y) = c_1 Y + \frac{c_2}{2} Y^2 $$

Consider the planning problem

$$

\max_{Y} [ S_c(Y) - S_p(Y) ]

$$

The first-order necessary condition for the planning problem is equation {eq}`staticY`.

Thus, a $Y$ that solves {eq}`staticY` is a competitive equilibrium output as well as an output that solves the planning problem.

This type of outcome provides an intellectual justification for liking a competitive equilibrium.

### Further Reading

References for this lecture include

* {cite}`Sargent1987`, chapter XIV

* {cite}`Ljungqvist2012`, chapter 7

## Defining Rational Expectations Equilibrium

## Rational Expectations Equilibrium

```{index} single: Rational Expectations Equilibrium; Definition

```

As we've seen, the firm's optimum problem induces a mapping $\Phi$ from a perceived law of motion $H$ for market-wide output to an actual law of motion $\Phi(H)$.

The mapping $\Phi$ is the composition of two operations, taking a perceived law of motion into a decision rule via {eq}`comp4`--{eq}`ree_opbe`, and a decision rule into an actual law via {eq}`ree_comp9a`.

The mapping $\Phi$ is the composition of two mappings, the first of which maps a perceived law of motion into a decision rule via {eq}`comp4`--{eq}`ree_opbe`, the second of which maps a decision rule into an actual law via {eq}`ree_comp9a`.

The $H$ component of a rational expectations equilibrium is a fixed point of $\Phi$.

## Computation of an Equilibrium

## Computing an Equilibrium

```{index} single: Rational Expectations Equilibrium; Computation

```

Unfortunately, the mapping $\Phi$ is not a contraction.

In particular, there is no guarantee that direct iterations on $\Phi$ converge [^fn_im].

Indeed, there is no guarantee that direct iterations on $\Phi$ converge [^fn_im].

Furthermore, there are examples in which these iterations diverge.

There are examples in which these iterations diverge.

Fortunately, there is another method that works here.

Fortunately, another method works here.

The method exploits a connection between equilibrium and Pareto optimality expressed in

the fundamental theorems of welfare economics (see, e.g, {cite}`MCWG1995`).

Lucas and Prescott {cite}`LucasPrescott1971` used this method to construct a rational expectations equilibrium.

The details follow.

Some details follow.

(ree_pp)=

### A Planning Problem Approach

As we'll see, this planning problem can be solved by LQ control ({doc}`linear regulator <lqcontrol>`).

The optimal quantities from the planning problem are rational expectations equilibrium quantities.

Optimal quantities from the planning problem are rational expectations equilibrium quantities.

The rational expectations equilibrium price can be obtained as a shadow price in the planning problem.

#### Structure of the Law of Motion

As you are asked to show in the exercises, the fact that the planner's

problem is an LQ problem implies an optimal policy --- and hence aggregate law

problem is an LQ control problem implies an optimal policy --- and hence aggregate law

of motion --- taking the form

```{math}