From 9d05152eafbd06b193819f03379a2eb4066d659c Mon Sep 17 00:00:00 2001 From: thomassargent30 Date: Mon, 26 Sep 2022 15:03:28 -0400 Subject: [PATCH] Tom's edits of two intermediate lectures Sept 26 --- lectures/multivariate_normal.md | 55 ++++++++++++++++++++++++++---- lectures/rational_expectations.md | 56 ++++++++++++++++++++++--------- 2 files changed, 88 insertions(+), 23 deletions(-) diff --git a/lectures/multivariate_normal.md b/lectures/multivariate_normal.md index f204d6328..306b4eff1 100644 --- a/lectures/multivariate_normal.md +++ b/lectures/multivariate_normal.md @@ -1735,6 +1735,19 @@ $$ \end{aligned} $$ +We can express our finding that the probability distribution of +$x_0$ conditional on $y_0$ is ${\mathcal N}(\tilde x_0, \tilde \Sigma_0)$ by representing $x_0$ +as + +$$ + x_0 = \tilde x_0 + \zeta_0 +$$ (eq:x0rep2) + +where $\zeta_0$ is a Gaussian random vector that is orthogonal to $\tilde x_0$ and $y_0$ and that +has mean vector $0$ and conditional covariance matrix $ E [\zeta_0 \zeta_0' | y_0] = \tilde \Sigma_0$. + + + ### Step toward dynamics Now suppose that we are in a time series setting and that we have the @@ -1747,20 +1760,48 @@ $$ where $A$ is an $n \times n$ matrix and $C$ is an $n \times m$ matrix. -It follows that the probability distribution of $x_1$ conditional -on $y_0$ is +Using equation {eq}`eq:x0rep2`, we can also represent $x_1$ as $$ -x_1 | y_0 \sim {\mathcal N}(A \tilde x_0 , A \tilde \Sigma_0 A' + C C' ) +x_1 = A (\tilde x_0 + \zeta_0) + C w_1 $$ -Define +It follows that +$$ E x_1 | y_0 = A \tilde x_0 $$ -\begin{aligned} \hat x_1 & = A \tilde x_0 \cr - \Sigma_1 & = A \tilde \Sigma_0 A' + C C' -\end{aligned} + + +and that the corresponding conditional covariance matrix $E (x_1 - E x_1| y_0) (x_1 - E x_1| y_0)' \equiv \Sigma_1$ is + +$$ + \Sigma_1 = A \tilde \Sigma_0 A' + C C' +$$ + +or + $$ +\Sigma_1 = A \Sigma_0 A' - A \Sigma_0 G' (G \Sigma_0 G' + R)^{-1} G \Sigma_0 A' +$$ + +We can write the mean of $x_1$ conditional on $y_0$ as + +$$ + \hat x_1 = A \hat x_0 + A \Sigma_0 G' (G \Sigma_0 G' + R)^{-1} (y_0 - G \hat x_0) +$$ + +or + +$$ + \hat x_1 = A \hat x_0 + K_0 (y_0 - G \hat x_0) +$$ + +where + +$$ +K_0 = A \Sigma_0 G' (G \Sigma_0 G' + R)^{-1} +$$ + ### Dynamic version diff --git a/lectures/rational_expectations.md b/lectures/rational_expectations.md index c479eaae1..182b22dc8 100644 --- a/lectures/rational_expectations.md +++ b/lectures/rational_expectations.md @@ -44,9 +44,9 @@ This lecture introduces the concept of a *rational expectations equilibrium*. 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 `. @@ -79,11 +79,11 @@ 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. +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]$. @@ -93,7 +93,7 @@ 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$. +This setting allows us to speak of a representative firm that chooses to produce $y$. We want to impose that @@ -109,7 +109,7 @@ 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. @@ -177,6 +177,30 @@ 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 + +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 @@ -185,7 +209,7 @@ 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 ``` @@ -391,11 +415,11 @@ Thus, a rational expectations equilibrium equates the perceived and actual laws 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 ``` @@ -408,18 +432,18 @@ Readers accustomed to dynamic programming arguments might try to address this pr 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 @@ -431,7 +455,7 @@ Our plan of attack is to match the Euler equations of the market problem with th As we'll see, this planning problem can be solved by LQ control ({doc}`linear regulator `). -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. @@ -514,7 +538,7 @@ $H$ that the representative firm faces within a rational expectations equilibriu #### 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}