Tom's edit or ree lecture

Author: thomassargent30

lectures/rational_expectations.md

@@ -39,18 +39,19 @@ tags: [hide-output]
 
 ## Overview
 
-This lecture introduces the concept of *rational expectations equilibrium*.
+This lecture introduces the concept of a *rational expectations equilibrium*.
 
-To illustrate it, we describe a linear quadratic version of a famous and important model
+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 kicked off the *rational expectations revolution*.
+This 1971 paper is one of a small number of research articles that ignited the *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).
 
-Because we use linear quadratic setups for demand and costs, we can adapt the LQ programming techniques described in {doc}`this lecture <lqcontrol>`.
+Because we use linear quadratic setups for demand and costs, we can deploy the LQ programming techniques described in {doc}`this lecture <lqcontrol>`.
 
-We will learn about how a representative agent's problem differs from a planner's, and how a planning problem can be used to compute rational expectations quantities.
+We will learn about how a representative agent's problem differs from a planner's, and how a planning problem can be used to compute  quantities and prices in a rational expectations
+equilibrium.
 
 We will also learn about how a rational expectations equilibrium can be characterized as a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_%28mathematics%29) of a mapping from a *perceived law of motion* to an *actual law of motion*.
 
@@ -82,10 +83,22 @@ from quantecon import LQ
 
 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.
+
+There is a uniform unit measure  of identical firms named  $\omega \in \Omega = [0,1]$.
+
+The output of firm $\omega$ is $y(\omega)$.  
+
+The output of all firms is $Y = \int_{0}^1 y(\omega) d \, \omega $.
+
+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 $y$.
+
 We want to impose that
 
-* The representative firm or individual takes *aggregate* $Y$ as given when it chooses individual $y$, but $\ldots$.
-* At the end of the day, $Y = y$, so that the representative firm is indeed representative.
+* The representative firm or individual firm takes *aggregate* $Y$ as given when it chooses individual $y(\omega)$, but $\ldots$.
+* At the end of the day, $Y = y(\omega) = y$, so that the representative firm is indeed representative.
 
 The Big $Y$, little $y$ trick accomplishes these two goals by
 
@@ -98,9 +111,9 @@ We begin by applying the  Big $Y$, little $y$ trick in a very simple static cont
 
 #### A Simple Static Example of the Big Y, Little y Trick
 
-Consider a static model in which a collection of $n$ firms produce a homogeneous good that is sold in a competitive market.
+Consider a static model in which a unit measure of  firms produce a homogeneous good that is sold in a competitive market.
 
-Each of these $n$ firms sells output $y$.
+Each of these firms ends up producing and selling output $y (\omega) = y$.
 
 The price $p$ of the good lies on an inverse demand curve
 
@@ -113,7 +126,9 @@ p = a_0 - a_1 Y
 where
 
 * $a_i > 0$ for $i = 0, 1$
-* $Y = n y$ is the market-wide level of output
+* $Y = \int_0^1 y(\omega) d \omega$ is the market-wide level of output
+
+For convenience, we'll often just write $y$ instead of $y(\omega)$ when we are describing the choice problem of an individual firm $\omega \in \Omega$.
 
 Each firm has a total cost function
 
@@ -149,13 +164,13 @@ The first-order condition for problem {eq}`max_problem_static` is
 a_0 - a_1 Y - c_1 - c_2 y = 0
 ```
 
-At this point, *but not before*, we substitute $Y = ny$ into {eq}`BigYsimpleFONC`
+At this point, *but not before*, we substitute $Y = y$ into {eq}`BigYsimpleFONC`
 to obtain the following linear equation
 
 ```{math}
 :label: staticY
 
-a_0 - c_1 - (a_1 + n^{-1} c_2) Y = 0
+a_0 - c_1 - (a_1 +  c_2) Y = 0
 ```
 
 to be solved for the competitive equilibrium market-wide output $Y$.
@@ -175,13 +190,13 @@ References for this lecture include
 ```{index} single: Rational Expectations Equilibrium; Definition
 ```
 
-Our first illustration of a rational expectations equilibrium involves a market with $n$ firms, each of which seeks to maximize the discounted present value of profits in the face of adjustment costs.
+Our first illustration of a rational expectations equilibrium involves a market with a unit measure of identical  firms, each of which seeks to maximize the discounted present value of profits in the face of adjustment costs.
 
 The adjustment costs induce the firms to make gradual adjustments, which in turn requires consideration of future prices.
 
 Individual firms understand that, via the inverse demand curve, the price is determined by the amounts supplied by other firms.
 
-Hence each firm wants to  forecast future total industry supplies.
+Hence each firm wants to  forecast future total industry output.
 
 In our context, a forecast is generated by a belief about the law of motion for the aggregate state.
 
@@ -198,7 +213,7 @@ We formulate a rational expectations equilibrium in terms of a fixed point of an
 
 To illustrate, consider a collection of $n$ firms producing a homogeneous good that is sold in a competitive market.
 
-Each of these $n$ firms sell output $y_t$.
+Each firm sell output $y_t(\omega) = y_t$.
 
 The price $p_t$ of the good lies on the inverse demand curve
 
@@ -211,7 +226,7 @@ p_t = a_0 - a_1 Y_t
 where
 
 * $a_i > 0$ for $i = 0, 1$
-* $Y_t = n y_t$ is the market-wide level of output
+* $Y_t = \int_0^1 y_t(\omega) d \omega = y_t$ is the market-wide level of output
 
 (ree_fp)=
 #### The Firm's Problem
@@ -256,11 +271,11 @@ In view of {eq}`ree_comp3d`, the firm's incentive to forecast the market price t
 
 Aggregate output depends on the choices of other firms.
 
-We assume that $n$ is such a large number  that the output of any single firm has a negligible effect on aggregate output.
+The output $y_t(\omega)$ of a single firm $\omega$ has a negligible effect on aggregate output $\int_0^1 y_t(\omega) d \omega$.
 
 That justifies firms in regarding their forecasts of aggregate output as being unaffected by their own output decisions.
 
-#### The Firm's Beliefs
+#### Representative Firm's Beliefs
 
 We suppose the firm believes that market-wide output $Y_t$ follows the law of motion
 
@@ -301,13 +316,13 @@ where
 ```{math}
 :label: ree_opbe
 
-h(y, Y) := \argmax_{y'}
+h(y, Y) := \textrm{argmax}_{y'}
 \left\{ a_0 y - a_1 y Y - \frac{ \gamma (y' - y)^2}{2}   + \beta v(y', H(Y))\right\}
 ```
 
 Evidently $v$ and $h$ both depend on $H$.
 
-#### A First-Order Characterization
+#### 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.
 
@@ -342,7 +357,7 @@ The firm optimally sets  an output path that satisfies {eq}`ree_comp7`, taking {
 
 This last condition is called the *transversality condition*, and acts as a first-order necessary condition "at infinity".
 
-The firm's decision rule solves the difference equation {eq}`ree_comp7` subject to the given initial condition $y_0$ and the transversality condition.
+A representative  firm's decision rule solves the difference equation {eq}`ree_comp7` subject to the given initial condition $y_0$ and the transversality condition.
 
 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.
@@ -351,12 +366,12 @@ a decision rule that automatically imposes both the Euler equation {eq}`ree_comp
 
 As we've seen, a given belief translates into a particular decision rule $h$.
 
-Recalling that $Y_t = ny_t$, the *actual law of motion* for market-wide output is then
+Recalling that in equilbrium  $Y_t = y_t$, the *actual law of motion* for market-wide output is then
 
 ```{math}
 :label: ree_comp9a
 
-Y_{t+1} = n h(Y_t/n, Y_t)
+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`.
@@ -367,7 +382,7 @@ Thus, when firms believe that the law of motion for market-wide output is {eq}`r
 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 $H(Y)= nh(Y/n,Y)$ for all
+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`.
@@ -385,7 +400,7 @@ The $H$ component of a rational expectations equilibrium is a fixed point of $\P
 ```{index} single: Rational Expectations Equilibrium; Computation
 ```
 
-Now let's consider the problem of computing the rational expectations equilibrium.
+Now let's compute a  rational expectations equilibrium.
 
 ### Failure of Contractivity
 
@@ -420,7 +435,6 @@ The optimal quantities from the planning problem are rational expectations equil
 
 The rational expectations equilibrium price can be obtained as a shadow price in the planning problem.
 
-For convenience, in this section, we set $n=1$.
 
 We first compute a sum of  consumer and producer surplus at time $t$
 
@@ -441,7 +455,7 @@ $$
 
 subject to an initial condition for $Y_0$.
 
-### Solution of the Planning Problem
+### Solution of  Planning Problem
 
 Evaluating the integral in {eq}`comp10` yields the quadratic form $a_0
 Y_t - a_1 Y_t^2 / 2$.
@@ -478,20 +492,17 @@ equation
 \beta a_0 + \gamma Y_t - [\beta a_1 + \gamma (1+ \beta)]Y_{t+1} + \gamma \beta Y_{t+2} =0
 ```
 
-### The Key Insight
+### Key Insight
 
 Return to equation {eq}`ree_comp7` and set $y_t = Y_t$ for all $t$.
 
-(Recall that for this section we've set $n=1$ to simplify the
-calculations)
-
 A small amount of algebra will convince you that when $y_t=Y_t$, 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
 
 1. finding the Euler equation of the representative firm and
-1. substituting into it the expression $Y_t = n y_t$ that "makes the representative firm be representative".
+1. substituting into it the expression $Y_t =  y_t$ that "makes the representative firm be representative".
 
 If it is appropriate to apply the same terminal conditions for these two difference equations, which it is, then we have verified that a solution of the planning problem is also a rational expectations equilibrium quantity sequence.
 
@@ -550,7 +561,7 @@ $$
 
 Express the solution of the firm's problem in the form {eq}`ree_ex5` and give the values for each $h_j$.
 
-If there were $n$ identical competitive firms all behaving according to {eq}`ree_ex5`, what would {eq}`ree_ex5`  imply for the *actual* law of motion {eq}`ree_hlom` for market supply.
+If there were a unit measure of  identical competitive firms all behaving according to {eq}`ree_ex5`, what would {eq}`ree_ex5`  imply for the *actual* law of motion {eq}`ree_hlom` for market supply.
 
 (ree_ex2)=
 ### Exercise 2