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61 changes: 39 additions & 22 deletions lectures/rational_expectations.md
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Expand Up @@ -77,7 +77,7 @@ We'll also use the LQ class from `QuantEcon.py`.
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.

Expand Down Expand Up @@ -107,7 +107,7 @@ Please watch for how this strategy is applied as the lecture unfolds.

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.

Expand Down Expand Up @@ -175,7 +175,7 @@ to be solved for the competitive equilibrium market-wide output $Y$.

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:

Expand Down Expand Up @@ -207,7 +207,7 @@ References for this lecture include
* {cite}`Sargent1987`, chapter XIV
* {cite}`Ljungqvist2012`, chapter 7

## Rational Expectations Equilibrium
## Rational expectations equilibrium

```{index} single: Rational Expectations Equilibrium; Definition
```
Expand All @@ -228,7 +228,7 @@ law of motion generated by production choices induced by this belief.
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)
```
Expand All @@ -251,7 +251,7 @@ where
* $Y_t = \int_0^1 y_t(\omega) d \omega = y_t$ is the market-wide level of output

(ree_fp)=
#### The Firm's Problem
#### The firm's problem

Each firm is a price taker.

Expand Down Expand Up @@ -287,7 +287,7 @@ This includes ones that the firm cares about but does not control like $p_t$.

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 $Y_t$.

Expand All @@ -297,7 +297,7 @@ The output $y_t(\omega)$ of a single firm $\omega$ has a negligible effect on ag

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

Expand All @@ -311,7 +311,9 @@ where $Y_0$ is a known initial condition.

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.

Expand Down Expand Up @@ -344,7 +346,7 @@ h(y, Y) := \textrm{argmax}_{y'}

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.

Expand All @@ -364,12 +366,14 @@ $$
v_y(y,Y) = a_0 - a_1 Y + \gamma (y' - y)
$$

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
Expand All @@ -384,7 +388,7 @@ A representative firm's decision rule solves the difference equation {eq}`ree_c
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 $h$.

Expand All @@ -399,32 +403,34 @@ Y_{t+1} = h(Y_t, Y_t)
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 $\Phi$.

Expand All @@ -434,6 +440,10 @@ Indeed, there is no guarantee that direct iterations on $\Phi$ converge [^fn_im]

There are examples in which these iterations diverge.

To see this intuively from Blackwell's sufficient condition, let us assume there are two beliefs $H_a(Y) > H_b(Y)$ for any $Y$.

Then by Euler equation {eq}`ree_comp7`, the optimal $y_{t+1} = h(Y_t, Y_t)$ decreases as $H$ increases, which indicates the monotoncity required in the Blackwell's condition is not satisfied.

Fortunately, another method works here.

The method exploits a connection between equilibrium and Pareto optimality expressed in
Expand All @@ -444,7 +454,7 @@ Lucas and Prescott {cite}`LucasPrescott1971` used this method to construct a rat
Some details follow.

(ree_pp)=
### A Planning Problem Approach
### A planning problem approach

```{index} single: Rational Expectations Equilibrium; Planning Problem Approach
```
Expand Down Expand Up @@ -477,7 +487,7 @@ $$

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 $a_0
Y_t - a_1 Y_t^2 / 2$.
Expand Down Expand Up @@ -514,7 +524,7 @@ equation
\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 $t$.

Expand All @@ -533,7 +543,7 @@ It follows that for this example we can compute equilibrium quantities by formin
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
Expand Down Expand Up @@ -590,8 +600,7 @@ If there were a unit measure of identical competitive firms all behaving accord
: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
Expand Down Expand Up @@ -700,6 +709,14 @@ Y_{t+1}
= n 96.949 + (1 - n 0.046) Y_t
$$

For the case of a unit measure of firms,
$$
\begin{aligned}
\int_0^1 y_{t+1}(\omega)\, d\omega &= h_0 + h_1 \int_0^1 y_{t}(\omega)\, dω + h_2 Y_t \\
Y_{t+1} &= h_0 + h_1 Y_t + h_2 Y_t \\
Y_{t+1} &= 96.949 + (1 - 0.046) Y_t
\end{aligned}
$$
```{solution-end}
```

Expand Down
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