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.

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

### Related planning problem

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

* {cite}`Sargent1987`, chapter XIV

* {cite}`Ljungqvist2012`, chapter 7

## Rational Expectations Equilibrium

## Rational expectations equilibrium

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

```

We formulate a rational expectations equilibrium in terms of a fixed point of an operator that maps beliefs into optimal beliefs.

(ree_ce)=

### Competitive Equilibrium with Adjustment Costs

### Competitive equilibrium with adjustment costs

```{index} single: Rational Expectations Equilibrium; Competitive Equilbrium (w. Adjustment Costs)

```

* $Y_t = \int_0^1 y_t(\omega) d \omega = y_t$ is the market-wide level of output

(ree_fp)=

#### The Firm's Problem

#### The firm's problem

Each firm is a price taker.

We turn to this problem now.

#### Prices and Aggregate Output

#### Prices and aggregate output

In view of {eq}`ree_comp3d`, the firm's incentive to forecast the market price translates into an incentive to forecast aggregate output $Y_t$.

That justifies firms in regarding their forecasts of aggregate output as being unaffected by their own output decisions.

#### Representative Firm's Beliefs

#### Representative firm's beliefs

We suppose the firm believes that market-wide output $Y_t$ follows the law of motion

The **belief function** $H$ is an equilibrium object, and hence remains to be determined.

#### Optimal Behavior Given Beliefs

Because of this, at this stage $Y_{t+1}$ only means the perceived output in the next period, $Y^e_{t+1}$.

#### Optimal behavior given beliefs

For now, let's fix a particular belief $H$ in {eq}`ree_hlom` and investigate the firm's response to it.

Evidently $v$ and $h$ both depend on $H$.

#### Characterization with First-Order Necessary Conditions

#### Characterization with first-order necessary conditions

In what follows it will be helpful to have a second characterization of $h$, based on first-order conditions.

v_y(y,Y) = a_0 - a_1 Y + \gamma (y' - y)

$$

and equivalently, $v_y(y', H(Y)) = a_0 - a_1 H(Y) +\gamma (y'' - y')$

Substituting this equation into {eq}`comp5` gives the **Euler equation**

```{math}

:label: ree_comp7

-\gamma (y_{t+1} - y_t) + \beta [a_0 - a_1 Y_{t+1} + \gamma (y_{t+2} - y_{t+1} )] =0

-\gamma (y_{t+1} - y_t) + \beta [a_0 - a_1 H(Y_t) + \gamma (y_{t+2} - y_{t+1} )] =0

```

The firm optimally sets an output path that satisfies {eq}`ree_comp7`, taking {eq}`ree_hlom` as given, and subject to

Note that solving the Bellman equation {eq}`comp4` for $v$ and then $h$ in {eq}`ree_opbe` yields

a decision rule that automatically imposes both the Euler equation {eq}`ree_comp7` and the transversality condition.

#### The Actual Law of Motion for Output

#### The actual law of motion for output

As we've seen, a given belief translates into a particular decision rule $h$.

Thus, when firms believe that the law of motion for market-wide output is {eq}`ree_hlom`, their optimizing behavior makes the actual law of motion be {eq}`ree_comp9a`.

(ree_def)=

### Definition of Rational Expectations Equilibrium

### Definition of rational expectations equilibrium

```{prf:definition}

A **rational expectations equilibrium** or **recursive competitive equilibrium** of the model with adjustment costs is a decision rule $h$ and an aggregate law of motion $H$ such that

1. Given belief $H$, the map $h$ is the firm's optimal policy function.

1. The law of motion $H$ satisfies $H(Y)= h(Y,Y)$ for all

$Y$.

```

Thus, a rational expectations equilibrium equates the perceived and actual laws of motion {eq}`ree_hlom` and {eq}`ree_comp9a`.

#### Fixed Point Characterization

#### Fixed point characterization

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 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 $\Phi$.

## Computing an Equilibrium

## Computing an equilibrium

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

```

Now let's compute a rational expectations equilibrium.

### Failure of Contractivity

### Failure of contractivity

Readers accustomed to dynamic programming arguments might try to address this problem by choosing some guess $H_0$ for the aggregate law of motion and then iterating with $\Phi$.

There are examples in which these iterations diverge.

To see this intuitively, consider Blackwell's sufficient condition and suppose there are two beliefs with $H_a(Y) > H_b(Y)$ for every $Y$.

By the Euler equation {eq}`ree_comp7`, the actual law of motion $Y_{t+1} = h(Y_t, Y_t)$ decreases as $H$ increases, so $\Phi$ reverses the ordering of beliefs.

Hence the monotonicity required by Blackwell's condition fails.

Fortunately, another method works here.

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

Some details follow.

(ree_pp)=

### A Planning Problem Approach

### A planning problem approach

```{index} single: Rational Expectations Equilibrium; Planning Problem Approach

```

subject to an initial condition for $Y_0$.

### Solution of Planning Problem

### Solution of planning problem

Evaluating the integral in {eq}`comp10` yields the quadratic form $a_0

Y_t - a_1 Y_t^2 / 2$.

\beta a_0 + \gamma Y_t - [\beta a_1 + \gamma (1+ \beta)]Y_{t+1} + \gamma \beta Y_{t+2} =0

```

### Key Insight

### Key insight

Return to equation {eq}`ree_comp7` and set $y_t = Y_t$ for all $t$.

A small amount of algebra will convince you that when $y_t=Y_t$, equations {eq}`comp16` and {eq}`ree_comp7` are identical.

In a rational expectations equilibrium the perceived and actual laws of motion agree, so $H(Y_t) = Y_{t+1}$.

Imposing this alongside $y_t = Y_t$, a small amount of algebra will convince you that equations {eq}`comp16` and {eq}`ree_comp7` are identical.

Thus, the Euler equation for the planning problem matches the second-order difference equation

that we derived by

The optimal policy function for the planning problem is the aggregate law of motion

$H$ that the representative firm faces within a rational expectations equilibrium.

#### Structure of the Law of Motion

#### 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 control problem implies an optimal policy --- and hence aggregate law

:class: dropdown

```

To map a problem into a [discounted optimal linear control

problem](https://python.quantecon.org/lqcontrol.html), we need to define

To map a problem into a {doc}`discounted optimal linear control problem <lqcontrol>`, we need to define

- state vector $x_t$ and control vector $u_t$

- matrices $A, B, Q, R$ that define preferences and the law of

= n 96.949 + (1 - n 0.046) Y_t

$$

For the case of a unit measure of firms,

$$

\begin{aligned}

\int_0^1 y_{t+1}(\omega)\, d\omega

&= h_0 + h_1 \int_0^1 y_{t}(\omega)\, d\omega + h_2 Y_t \\

Y_{t+1} &= h_0 + h_1 Y_t + h_2 Y_t \\

Y_{t+1} &= 96.949 + (1 - 0.046) Y_t

\end{aligned}

$$

```{solution-end}

```