Tom's May 18 edits of two Cass-Koopmans lectures

Author: thomassargent30

lectures/cass_koopmans_1.md

@@ -18,20 +18,20 @@ kernelspec:
 </div>
 ```
 
-# Cass-Koopmans Planning Problem
+# Cass-Koopmans Model
 
 ```{contents} Contents
 :depth: 2
 ```
 
 ## Overview
 
-This lecture and lecture {doc}`Cass-Koopmans Competitive Equilibrium <cass_koopmans_2>` describe a model that Tjalling Koopmans {cite}`Koopmans`
+This lecture and {doc}`Cass-Koopmans Competitive Equilibrium <cass_koopmans_2>` describe a model that Tjalling Koopmans {cite}`Koopmans`
 and David Cass {cite}`Cass` used to analyze optimal growth.
 
 The model can be viewed as an extension of the model of Robert Solow
 described in [an earlier lecture](https://lectures.quantecon.org/py/python_oop.html)
-but adapted to make the saving rate the outcome of an optimal choice.
+but adapted to make the saving rate be a choice.
 
 (Solow assumed a constant saving rate determined outside the model.)
 
@@ -43,6 +43,17 @@ organized as a **competitive equilibrium**.
 
 This lecture is devoted to the planned economy version.
 
+In the planned economy, there are
+
+- no prices
+- no budget constraints
+
+Instead there is a dictator that tells people
+
+- what to produce
+- what to invest in physical capital
+- who is to consume what  and when
+
 The lecture uses important ideas including
 
 - A min-max problem for solving a planning problem.
@@ -66,22 +77,77 @@ import numpy as np
 
 Time is discrete and takes values $t = 0, 1 , \ldots, T$ where $T$ is  finite.
 
-(We'll study a limiting case in which  $T = + \infty$ before concluding).
+(We'll eventually study a limiting case in which  $T = + \infty$)
 
 A single good can either be consumed or invested in physical capital.
 
 The consumption good is not durable and depreciates completely if not
 consumed immediately.
 
-The capital good is durable but depreciates some each period.
+The capital good is durable but depreciates.
 
-We let $C_t$ be a nondurable consumption good at time $t$.
+We let $C_t$ be the total consumption of a nondurable  consumption good at time $t$.
 
 Let $K_t$ be the stock of physical capital at time $t$.
 
 Let $\vec{C}$ = $\{C_0,\dots, C_T\}$ and
 $\vec{K}$ = $\{K_0,\dots,K_{T+1}\}$.
 
+### Digression: an Aggregation Theory
+
+We use a concept of a representative consumer to be thought of as follows.
+
+There is a unit mass of identical consumers.
+
+For $\omega \in [0,1]$, consumption of consumer  is $c(\omega)$.
+
+Aggregate consumption is 
+
+$$ 
+C = \int_0^1 c(\omega) d \omega
+$$
+
+Consider the a welfare problem of choosing an allocation $\{c(\omega)\}$ across consumers to maximize
+
+$$
+ \int_0^1 u(c(\omega)) d \omega
+$$ 
+
+where $u(\cdot)$ is a concave utility function with $u' >0, u'' < 0$ and  maximization is subject to 
+
+$$ 
+C = \int_0^1 c(\omega) d \omega .
+$$ (eq:feas200)
+
+Form a Lagrangian $L = \int_0^1 u(c(\omega)) d \omega + \lambda [C - \int_0^1 c(\omega) d \omega ] $.
+
+Differentiate under the integral signs with respect to each $\omega$ to  obtain the first-order
+necessary condtions 
+
+$$
+u'(c(\omega)) = \lambda. 
+$$ 
+
+This condition implies that $c(\omega)$ equals a constant $c$ that is independent 
+of $\omega$.  
+
+To find $c$, use the  feasibility constraint {eq}`eq:feas200` to conclude that 
+
+$$ 
+c(\omega) = c = C.
+$$
+
+This line of argument indicates the special *aggregation theory* that lies beneath outcomes in which a representative consumer 
+consumes amount $C$.
+
+It appears often in aggregate economics. 
+
+We shall use it in this lecture and in {doc}`Cass-Koopmans Competitive Equilibrium <cass_koopmans_2>`.
+
+
+#### An  Economy
+
+
 A representative household is endowed with one unit of labor at each
 $t$ and likes the consumption good at each $t$.
 
@@ -153,7 +219,7 @@ $$
 \mathcal{L}(\vec{C} ,\vec{K} ,\vec{\mu} ) =
 \sum_{t=0}^T \beta^t\left\{ u(C_t)+ \mu_t
 \left(F(K_t,1) + (1-\delta) K_t- C_t - K_{t+1} \right)\right\}
-$$
+$$ (eq:Lagrangian201)
 
 and then pose the following min-max problem:
 
@@ -226,7 +292,7 @@ $$
 
 ### First-order necessary conditions
 
-We now compute **first order necessary conditions** for extremization of the Lagrangian:
+We now compute **first-order necessary conditions** for extremization of the Lagrangian {eq}`eq:Lagrangian201`:
 
 ```{math}
 :label: constraint1
@@ -255,7 +321,7 @@ K_{T+1}: \qquad -\mu_T \leq 0, \ \leq 0 \text{ if } K_{T+1}=0; \ =0 \text{ if }
 In computing  {eq}`constraint3` we recognize that $K_t$ appears
 in both the time  $t$ and time $t-1$ feasibility constraints.
 
-{eq}`constraint4` comes from differentiating with respect
+Restrictions {eq}`constraint4` come from differentiating with respect
 to $K_{T+1}$ and applying the following **Karush-Kuhn-Tucker condition** (KKT)
 (see [Karush-Kuhn-Tucker conditions](https://en.wikipedia.org/wiki/Karush-Kuhn-Tucker_conditions)):
 
@@ -745,7 +811,7 @@ def plot_saving_rate(pp, c0, k0, T_arr, k_ter=0, k_ss=None, s_ss=None):
 plot_saving_rate(pp, 0.3, k_ss/3, [250, 150, 75, 50], k_ss=k_ss)
 ```
 
-## A Limiting Economy
+## A Limiting Infinite Horizon Economy
 
 We want to set $T = +\infty$.
 
@@ -816,12 +882,11 @@ plot_saving_rate(pp, 0.3, k_ss*1.5, [130], k_ter=k_ss, k_ss=k_ss, s_ss=s_ss)
 In {doc}`Cass-Koopmans Competitive Equilibrium <cass_koopmans_2>`,  we study a decentralized version of an economy with exactly the same
 technology and preference structure as deployed here.
 
-In that lecture, we replace the  planner of this lecture with Adam Smith's **invisible hand**
+In that lecture, we replace the  planner of this lecture with Adam Smith's **invisible hand**.
 
-In place of quantity choices made by the planner, there are market prices somewhat produced by
-the invisible hand.
+In place of quantity choices made by the planner, there are market prices that are set by a mechanism outside the model, a so-called invisible hand.
 
-Market prices must adjust to reconcile distinct decisions that are made independently
+Equilibrium market prices must reconcile distinct decisions that are made independently
 by a representative household and a representative firm.
 
 The relationship between a command economy like the one studied in this lecture and a market economy like that

lectures/cass_koopmans_2.md

@@ -28,39 +28,39 @@ kernelspec:
 
 This lecture continues our analysis in this  lecture
 {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` about the  model that Tjalling Koopmans {cite}`Koopmans`
-and David Cass {cite}`Cass` used to study optimal growth.
+and David Cass {cite}`Cass` used to study optimal capital accumulation.
 
 This lecture illustrates what is, in fact, a
 more general connection between a **planned economy** and an economy
-organized as a **competitive equilibrium**.
+organized as a competitive equilibrium or a **market economy**.
 
 The earlier  lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>` studied a planning problem and used ideas including
 
-- A min-max problem for solving the planning problem.
+- A Lagrangian formulation of the  planning problem that leads to a system of difference equations.
 - A **shooting algorithm** for solving difference equations subject
   to initial and terminal conditions.
 - A **turnpike** property that describes optimal paths for
   long-but-finite horizon economies.
 
 The present lecture uses  additional  ideas including
 
-- Hicks-Arrow prices named after John R. Hicks and Kenneth Arrow.
-- A connection between some Lagrange multipliers in the min-max
+- Hicks-Arrow prices, named after John R. Hicks and Kenneth Arrow.
+- A connection between some Lagrange multipliers from the planning 
   problem and the Hicks-Arrow prices.
 - A **Big** $K$ **, little** $k$ trick widely used in
   macroeconomic dynamics.
     * We shall encounter this trick in [this lecture](https://python.quantecon.org/rational_expectations.html)
       and also in [this lecture](https://python-advanced.quantecon.org/dyn_stack.html).
 - A non-stochastic version of a theory of the **term structure of
   interest rates**.
-- An intimate connection between the cases for the optimality of two
-  competing visions of good ways to organize an economy, namely:
+- An intimate connection between  two
+   ways to organize an economy, namely:
     * **socialism** in which a central planner commands the
       allocation of resources, and
-    * **capitalism** (also known as **a  market economy**) in
+    * **competitive markets** in
       which competitive equilibrium **prices** induce individual
       consumers and producers to choose a socially optimal allocation
-      as an unintended consequence of their selfish
+      as  unintended consequences of their selfish
       decisions
 
 Let's start with some standard imports:
@@ -134,7 +134,7 @@ where $\delta \in (0,1)$ is a depreciation rate of capital.
 In this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>`, we studied a problem in which a planner chooses an allocation $\{\vec{C},\vec{K}\}$ to
 maximize {eq}`utility-functional` subject to {eq}`allocation`.
 
-The allocation that solves the planning problem plays an important role in a competitive equilibrium as we shall see below.
+The allocation that solves the planning problem reappears in a competitive equilibrium, as we shall see below.
 
 ## Competitive Equilibrium
 
@@ -145,14 +145,16 @@ technology and preference structure as the planned economy studied in this lectu
 
 But now there is no planner.
 
-Market prices adjust to reconcile distinct decisions that are made
+There are (unit masses of) price taking consumers and firms.
+
+Market prices are set to reconcile distinct decisions that are made
 separately by a representative household and a representative firm.
 
 There is a representative consumer who has the same preferences over
-consumption plans as did the consumer in the planned economy.
+consumption plans as did a consumer in the planned economy.
 
-Instead of being told what to consume and save by a planner, the
-household chooses for itself subject to a budget constraint
+Instead of being told what to consume and save by a planner, a
+consumer (also known as a *household*) chooses for itself subject to a budget constraint
 
 - At each time $t$, the household receives wages and rentals
   of capital from a firm -- these comprise its **income** at
@@ -175,8 +177,7 @@ household chooses for itself subject to a budget constraint
 - The representative household and the representative firm are both
   **price takers** who believe that prices are not affected by their choices
 
-**Note:** We can think of there being a large number
-$M$ of identical representative consumers and $M$
+**Note:** We can think of there being  unit measures of identical representative consumers and 
 identical representative firms.
 
 ## Market Structure
@@ -195,25 +196,28 @@ all other dates $t=1, 2, \ldots, T$.
 
 There are  sequences of prices
 $\{w_t,\eta_t\}_{t=0}^T= \{\vec{w}, \vec{\eta} \}$
-where $w_t$ is a wage or rental rate for labor at time $t$ and
-$\eta_t$ is a rental rate for capital at time $t$.
+where 
+
+- $w_t$ is a wage or rental rate for labor at time $t$
 
-In addition there is are intertemporal prices that work as follows.
+- $\eta_t$ is a rental rate for capital at time $t$
 
-Let $q^0_t$ be the price of a good at date $t$ relative
+In addition there is a vector $\{q_t^0\}$ of  intertemporal prices where  
+
+- $q^0_t$ is the price of a good at date $t$ relative
 to a good at date $0$.
 
 We call $\{q^0_t\}_{t=0}^T$  a vector of **Hicks-Arrow prices**,
 named after the 1972 economics Nobel prize winners.
 
-Evidently,
+Units of $q_t^0$ could be 
 
 $$
-q^0_t=\frac{\text{number of time 0 goods}}{\text{number of time t goods}}
+\frac{\text{number of time 0 goods}}{\text{number of time t goods}}
 $$
 
-Because $q^0_t$ is a **relative price**, the units in terms of
-which prices are quoted are arbitrary -- we are free to normalize them.
+But because $q^0_t$ is a **relative price**, the units in terms of
+which prices are quoted are arbitrary, we are free to re-normalize them.
 
 ## Firm Problem
 
@@ -296,7 +300,7 @@ $\tilde k_t$ and $\tilde n_t$.
 
 If $\frac{\partial F}{\partial \tilde k_t}> \eta_t$, then the
 firm makes positive profits on each additional unit of
-$\tilde k_t$, so it will want to make $\tilde k_t$
+$\tilde k_t$, so it would want to make $\tilde k_t$
 arbitrarily large.
 
 But setting $\tilde k_t = + \infty$ is not physically feasible,
@@ -307,7 +311,7 @@ A similar argument applies if
 $\frac{\partial F}{\partial \tilde n_t}> w_t$.
 
 If $\frac{\partial \tilde k_t}{\partial \tilde k_t}< \eta_t$,
-the firm will set $\tilde k_t$ to zero, something that is not feasible.
+the firm would want to set  $\tilde k_t$ to zero, which is not feasible.
 
 It is convenient to define
 $\vec{w} =\{w_0, \dots,w_T\}$and $\vec{\eta}= \{\eta_0, \dots, \eta_T\}$.
@@ -520,7 +524,7 @@ k_t: \quad -\lambda q_t^0 \left[(1-\delta)+\eta_t \right]+\lambda q^0_{t-1}=0 \q
 k_{T+1}: \quad -\lambda q_0^{T+1} \leq 0, \ \leq 0 \text{ if } k_{T+1}=0; \ =0 \text{ if } k_{T+1}>0
 ```
 
-Now we plug in our guesses of prices and embark on some algebra in the hope of derived all first-order necessary conditions
+Now we plug in our guesses of prices and embark on some algebra in the hope of recovering all first-order necessary conditions
 {eq}`constraint1`-{eq}`constraint4` for the planning problem from this lecture {doc}`Cass-Koopmans Planning Model <cass_koopmans_1>`.
 
 Combining {eq}`cond1` and {eq}`eq-price`, we get:
@@ -563,7 +567,7 @@ $$
 \sum_{t=0}^T \beta^t \mu_{t} \left(C_t+ (K_{t+1} -(1-\delta)K_t)-f(K_t)+K_t f'(K_t)-f'(K_t)K_t\right) \leq 0
 $$
 
-which simplifies
+which simplifies to
 
 $$
 \sum_{t=0}^T  \beta^t \mu_{t} \left(C_t +K_{t+1} -(1-\delta)K_t - F(K_t,1)\right) \leq 0
@@ -616,7 +620,7 @@ $$
 
 which is exactly {eq}`eq-pr4`.
 
-So at our guess for the equilibrium price system, the allocation
+Thus, at our guess for the equilibrium price system, the allocation
 that solves the planning problem also solves the problem faced by a firm
 within a competitive equilibrium.
 
@@ -963,6 +967,3 @@ Now we plot when $t_0=20$
 plot_yield_curves(pp, 20, 0.3, k_ss/3, T_arr)
 ```
 
-We aim to have more to say about the term structure of interest rates
-in a planned lecture on the topic.
-