update

Author: longye-tian

lectures/calvo_abreu.md

@@ -55,7 +55,7 @@ our    timing protocol.
 
 
 
-## Model Components
+## Model components
 
 We'll start with a brief review of the setup.  
 
@@ -133,7 +133,7 @@ makes  timing protocols matter for modeling optimal government policies.
 Quantecon lecture {doc}`calvo` used this  insight to  simplify   analysis of alternative  government policy problems.
 
 
-## Another  Timing Protocol
+## Another timing protocol
 
 The Quantecon lecture {doc}`calvo`  considered three  models of government policy making that  differ in
 
@@ -182,7 +182,7 @@ In this version of our model
 - at each $t$, the government chooses $\mu_t$ to maximize
   a continuation discounted utility.
 
-### Government Decisions
+### Government decisions
 
 $\vec \mu$ is chosen by a sequence of government
 decision makers, one for each $t \geq 0$.
@@ -214,7 +214,7 @@ for each $t \geq 0$:
   expect an associated $\theta_0^A$ for $t+1$. Here $\vec \mu^A = \{\mu_j^A \}_{j=0}^\infty$ is
   an alternative government plan to be described below.
 
-### Temptation to Deviate from Plan
+### Temptation to deviate from plan
 
 The government's one-period return function $s(\theta,\mu)$
 described in equation {eq}`eq_old6` in quantecon lecture {cite}`Calvo1978`  has the property that for all
@@ -240,7 +240,7 @@ If the government at $t$ is to resist the temptation to raise its
 current payoff, it is only because it forecasts adverse  consequences that
 its setting of $\mu_t$ would bring for continuation  government payoffs via  alterations  in the private sector's expectations.
 
-## Sustainable or Credible Plan
+## Sustainable or credible plan
 
 We call a plan $\vec \mu$ **sustainable** or **credible** if at
 each $t \geq 0$ the government chooses to confirm private
@@ -294,7 +294,7 @@ import matplotlib.pyplot as plt
 import pandas as pd
 ```
 
-### Abreu's Self-Enforcing Plan
+### Abreu's self-enforcing plan
 
 A plan $\vec \mu^A$ (here the superscipt $A$ is for Abreu) is said to be **self-enforcing** if
 
@@ -359,7 +359,7 @@ agents' expectation.
 We shall use a construction featured in Abreu ({cite}`Abreu`) to construct a
 self-enforcing plan with low time $0$ value.
 
-### Abreu's Carrot-Stick Plan
+### Abreu's carrot-stick plan
 
 {cite}`Abreu` invented a way to create a self-enforcing plan with a low
 initial value.
@@ -518,7 +518,7 @@ Let's create an instance of ChangLQ with the following parameters:
 clq = ChangLQ(β=0.85, c=2)
 ```
 
-### Example of Self-Enforcing Plan
+### Example of self-enforcing plan
 
 The following example implements an Abreu stick-and-carrot plan.
 
@@ -634,7 +634,7 @@ def check_ramsey(clq, T=1000):
 check_ramsey(clq)
 ```
 
-### Recursive Representation of a Sustainable Plan
+### Recursive representation of a sustainable plan
 
 We can represent a sustainable plan recursively by taking the
 continuation value $v_t$ as a state variable.
@@ -665,7 +665,7 @@ depends on whether the government at $t$ confirms the representative agent's
 expectations by setting $\mu_t$ equal to the recommended value
 $\hat \mu_t$, or whether it disappoints those expectations.
 
-## Whose  Plan is It?
+## Whose plan is it?
 
 A credible government plan $\vec \mu$ plays multiple roles.
 

lectures/calvo_machine_learn.md

@@ -72,7 +72,7 @@ We pose  some of  those questions at the end of this lecture and  answer them  b
 Human intelligence, not the ``artificial intelligence`` deployed in our machine learning approach, is a key input into choosing which regressions to run. 
 
 
-## The Model
+## The model
 
 We study a   linear-quadratic version of a model that Guillermo Calvo {cite}`Calvo1978` used to illustrate the **time inconsistency** of optimal government plans.
 
@@ -93,7 +93,7 @@ The model combines ideas from  papers by Cagan {cite}`Cagan`, {cite}`sargent1973
 
 
 
-## Model Components
+## Model components
 
 There is no uncertainty.
 
@@ -239,7 +239,7 @@ $$
 
 
 
-## Parameters and Variables
+## Parameters and variables
 
 
 **Parameters:**  
@@ -265,7 +265,7 @@ $$
 
 
 
-### Basic Objects
+### Basic objects
 
 To prepare the way for our calculations, we'll remind ourselves of the  mathematical objects
 in play.
@@ -321,7 +321,7 @@ An optimal  government plan under this timing protocol is an example of what is
 Notice that while the government is in effect choosing a bivariate **time series** $(\vec mu, \vec \theta)$, the government's problem is **static** in the sense that it chooses treats that time-series as a single object to be chosen at a single point in time. 
 
 
-## Approximation and Truncation parameter $T$
+## Approximation and truncation parameter $T$
 
 We anticipate that under a Ramsey plan the sequences  $\{\theta_t\}$ and $\{\mu_t\}$  both converge to stationary values. 
 
@@ -392,7 +392,7 @@ $$
 
 where $\tilde \theta_t, \ t = 0, 1, \ldots , T-1$ satisfies formula (1).
 
-## A Gradient Descent Algorithm
+## A gradient descent algorithm
 
 We now describe  code that  maximizes the criterion function {eq}`eq:Ramseyvalue` subject to equations {eq}`eq:inflation101` by choice of the truncated vector  $\tilde \mu$.
 
@@ -689,7 +689,7 @@ compute_V(clq.μ_series, β=0.85, c=2)
 
  
 
-### Restricting  $\mu_t = \bar \mu$ for all $t$
+### Restricting $\mu_t = \bar \mu$ for all $t$
 
 We take  a brief detour to solve a restricted version of  the Ramsey problem defined above.
 
@@ -729,7 +729,7 @@ V_CR
 compute_V(jnp.array([clq.μ_CR]), β=0.85, c=2)
 ```
 
-## A More Structured ML Algorithm
+## A more structured ML algorithm
 
 By thinking  about the mathematical structure of the Ramsey problem and using some linear algebra, we can simplify the problem that we hand over to a ``machine learning`` algorithm. 
 
@@ -1063,7 +1063,7 @@ the  limit $\bar \mu$ of  $\mu_t$ as $t \rightarrow +\infty$.
 This pattern reflects how formula {eq}`eq_grad_old3`  makes $\theta_t$ be a weighted average of future $\mu_t$'s.
 
 
-## Continuation Values
+## Continuation values
 
 For subsquent analysis, it will be useful to  compute a sequence $\{v_t\}_{t=0}^T$ of  what we'll call ``continuation values`` along a Ramsey plan.
 
@@ -1163,7 +1163,7 @@ time-less perspective." A more descriptive phrase is "the value of the worst con
 ```
 
 
-## Adding Some Human Intelligence 
+## Adding some human intelligence 
 
 We have used our machine learning algorithms to compute a Ramsey plan.
 
@@ -1351,7 +1351,7 @@ Evidently, continuation values $v_t > V^{CR}$ for $t=0, 1, 2$ while $v_t < V^{CR
 
 
 
-## What has Machine Learning Taught Us?
+## What has machine learning taught us?
 
 
 Our regressions tells us that along the Ramsey outcome $\vec \mu^R, \vec \theta^R$, the linear function