Adjust styling

Author: xuanguang-li

lectures/rational_expectations.md

@@ -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.
 
@@ -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.
 
@@ -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:
 
@@ -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
 ```
@@ -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)
 ```
@@ -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.
 
@@ -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 $Y_t$.
 
@@ -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
 
@@ -313,7 +313,7 @@ The **belief function** $H$ is an equilibrium object, and hence remains to be de
 
 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
+#### 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.
 
@@ -346,7 +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.
 
@@ -388,7 +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 $h$.
 
@@ -403,32 +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 $\Phi$.
 
@@ -452,7 +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
 ```
@@ -485,7 +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 $a_0
 Y_t - a_1 Y_t^2 / 2$.
@@ -522,7 +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 $t$.
 
@@ -541,7 +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
@@ -598,8 +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