Tom's edits of consumption smoothing lecture, June 29

Author: thomassargent30

lectures/cons_smooth.md

@@ -23,7 +23,7 @@ It will take a while for a "present value" or asset price explicilty to appear i
 
 In this lecture, we'll study a famous model of the "consumption function" that Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`)  proposed to fit some empirical data patterns that the simple Keynesian model described in this quantecon lecture {doc}`geometric series <geom_series>` had missed.
 
-The key insight of Friedman and Hall was that today's consumption ought not to depend just on today's income: it should also depend on a person's anticipations of her **future** incomes at various dates.  
+The key insight of Friedman and Hall was that today's consumption ought not to depend just on today's non-financial income: it should also depend on a person's anticipations of her **future** non-financial incomes at various dates.  
 
 In this lecture, we'll study what is sometimes called the "consumption-smoothing model"  using only linear algebra, in particular  matrix multiplication and matrix inversion.
 
@@ -39,16 +39,19 @@ from collections import namedtuple
 
 +++ {"user_expressions": []}
 
-Our model describes the behavior of a consumer who lives from time $t=0, 1, \ldots, T$, receives an income stream $\{y_t\}_{t=0}^T$, 
-and chooses a consumption stream $\{c_t\}_{t=0}^T$.
+Our model describes the behavior of a consumer who lives from time $t=0, 1, \ldots, T$, receives a stream $\{y_t\}_{t=0}^T$ of non-financial income and chooses a consumption stream $\{c_t\}_{t=0}^T$.
+
+We usually think of the non-financial income stream as coming from the person's salary from supplying labor.  
+
+The model  takes that non-financial income stream as an input, regarding it as "exogenous" in the sense of not being determined by the model. 
 
 The consumer faces a gross interest rate of $R >1$ that is constant over time, at which she is free to borrow or lend, up to some limits that we'll describe below.
 
 To set up the model, let 
 
  * $T \geq 2$  be a positive integer that constitutes a time-horizon
 
- * $y = \{y_t\}_{t=0}^T$ be an exogenous  sequence of non-negative financial incomes $y_t$
+ * $y = \{y_t\}_{t=0}^T$ be an exogenous  sequence of non-negative non-financial incomes $y_t$
 
  * $a = \{a_t\}_{t=0}^{T+1}$ be a sequence of financial wealth
 
@@ -62,31 +65,40 @@ To set up the model, let
 
  * $a_{T+1} \geq 0$  be a terminal condition on final assets
 
+While the sequence of financial wealth $a$ is to be determined by the model, it must satisfy  two  **boundary conditions** that require it to be equal to $a_0$ at time $0$ and $a_{T+1}$ at time $T+1$.
+
+The **terminal condition** $a_{T+1} \geq 0$ requires that the consumer not die leaving debts.
+
+(We'll see that a utility maximizing consumer won't **want** to die leaving positive assets, so she'll arrange her affairs to make
+$a_{T+1} = 0.)
+
 The consumer faces a sequence of budget constraints that  constrains the triple of sequences $y, c, a$
 
 $$
 a_{t+1} = R (a_t+ y_t - c_t), \quad t =0, 1, \ldots T
 $$ (eq:a_t)
 
-Notice that there are $T+1$ such budget constraints, one for each $t=0, 1, \ldots, T$.  
+Notice that there are $T+1$ such budget constraints, one for each $t=0, 1, \ldots, T$. 
+
+Given a sequence $y$ of non-financial income, there is a big set of **pairs** $(a, c)$ of (financial wealth, consumption) sequences that satisfy the sequence of budget constraints {eq}`eq:a_t`. 
 
 Our model has the following logical flow.
 
- * start with an exogenous income sequence $y$, an initial financial wealth $a_0$, and 
+ * start with an exogenous non-financial income sequence $y$, an initial financial wealth $a_0$, and 
  a candidate consumption path $c$.
  
  * use the system of equations {eq}`eq:a_t` for $t=0, \ldots, T$ to compute a path $a$ of financial wealth
  
  * verify that $a_{T+1}$ satisfies the terminal wealth constraint $a_{T+1} \geq 0$. 
     
-     * If it does, declare that the candidate path is budget feasible. 
+     * If it does, declare that the candidate path is **budget feasible**. 
  
      * if the candidate consumption path is not budget feasible, propose a path with less consumption sometimes and start over
      
 Below, we'll describe how to execute these steps using linear algebra -- matrix inversion and multiplication.
 
 The above procedure seems like a sensible way to find "budget-feasible" consumption paths $c$, i.e., paths that are consistent
-with the exogenous income stream $y$, the initial financial  asset level $a_0$, and the terminal asset level $a_{T+1}$.
+with the exogenous non-financial income stream $y$, the initial financial  asset level $a_0$, and the terminal asset level $a_{T+1}$.
 
 In general, there will be many budget feasible consumption paths $c$.
 
@@ -103,10 +115,14 @@ W = \sum_{t=0}^T \beta^t (g_1 c_t - \frac{g_2}{2} c_t^2 )
 
 where $g_1 > 0, g_2 > 0$.  
 
-We shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`), this criterion assigns higher welfare to **smoother** consumption paths.
+The fact that the utility function $g_1 c_t - \frac{g_2}{2} c_t^2$ has diminishing marginal utility imparts a preference for consumption that is very smooth when $\beta R \approx 1$.  
+
+Indeed, we shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`), this criterion assigns higher welfare to **smoother** consumption paths.
 
 By **smoother** we mean as close as possible to being constant over time.  
 
+The preference for smooth consumption paths that is built into the model gives it the  name "consumption smoothing model".
+
 Let's dive in and do some calculations that will help us understand how the model works. 
 
 Here we use default parameters $R = 1.05$, $g_1 = 1$, $g_2 = 1/2$, and $T = 65$. 
@@ -137,7 +153,7 @@ h_0 \equiv \sum_{t=0}^T R^{-t} y_t = \begin{bmatrix} 1 & R^{-1} & \cdots & R^{-T
 \begin{bmatrix} y_0 \cr y_1  \cr \vdots \cr y_T \end{bmatrix}
 $$
 
-Human or non-financial wealth is evidently just the present value at time $0$ of the consumer's income stream $y$. 
+Human or non-financial wealth is evidently just the present value at time $0$ of the consumer's non-financial income stream $y$. 
 
 Notice that formally it very much resembles the asset price that we computed in this quantecon lecture {doc}`present values <pv>`.
 
@@ -169,14 +185,14 @@ $$
 In this case, we can use the intertemporal budget constraint to write 
 
 $$
-c_t  = \left(\sum_{t=0}^T R^{-t}\right)^{-1} (a_0 + h_0), \quad t= 0, 1, \ldots, T.
+c_t = c_0  = \left(\sum_{t=0}^T R^{-t}\right)^{-1} (a_0 + h_0), \quad t= 0, 1, \ldots, T.
 $$ (eq:conssmoothing)
 
 Equation {eq}`eq:conssmoothing` is the consumption-smoothing model in a nutshell.
 
 +++ {"user_expressions": []}
 
-## Permanent income model of consumption 
+## Mechanics of Consumption smoothing model  
 
 As promised, we'll provide step by step instructions on how to use linear algebra, readily implemented
 in Python, to compute all the objects in play in  the consumption-smoothing model.
@@ -194,15 +210,17 @@ $$
 
 ### Step 2
 
-Compute
+Compute the optimal level of  consumption $c_0 $ 
 
 $$
-c_0 = \left( \frac{1 - R^{-1}}{1 - R^{-(T+1)}} \right) (a_0 + \sum_{t=0}^T R^t y_t )
+c_t = c_0 = \left( \frac{1 - R^{-1}}{1 - R^{-(T+1)}} \right) (a_0 + \sum_{t=0}^T R^t y_t ) , \quad t = 0, 1, \ldots, T
 $$
 
 ### Step 3
 
-Formulate the system of difference equations as follows (we'll say more about the mechanics of using linear algebra to solve such difference equations later in the last part of this lecture):
+In this step, we use the system of equations {eq}`eq:a_t` for $t=0, \ldots, T$ to compute a path $a$ of financial wealth.
+
+To do this, we translated that system of difference equations into a single matrix equation as follows (we'll say more about the mechanics of using linear algebra to solve such difference equations later in the last part of this lecture):
 
 $$
 \begin{bmatrix} 
@@ -232,6 +250,9 @@ $$
 a_{T+1} = 0.
 $$
 
+We have built into the our calculations that the consumer leaves life with exactly zero assets, just barely satisfying the
+terminal condition that $a_{T+1} \geq 0$.  
+
 Let's verify this with our Python code.
 
 First we implement this model in `compute_optimal`
@@ -260,13 +281,13 @@ def compute_optimal(model, a0, y_seq):
 
 We use an example where the consumer inherits $a_0<0$ (which can be interpreted as a student debt).
 
-The income process $\{y_t\}_{t=0}^{T}$ is constant and positive up to $t=45$ and then becomes zero afterward.
+The non-financial process $\{y_t\}_{t=0}^{T}$ is constant and positive up to $t=45$ and then becomes zero afterward.
 
 ```{code-cell} ipython3
 # Financial wealth
 a0 = -2     # such as "student debt"
 
-# Income process
+# non-financial Income process
 y_seq = np.concatenate([np.ones(46), np.zeros(20)])
 
 cs_model = creat_cs_model()
@@ -276,15 +297,15 @@ print('check a_T+1=0:',
       np.abs(a_seq[-1] - 0) <= 1e-8)
 ```
 
-The visualization shows the path of income, consumption, and financial assets.
+The visualization shows the path of non-financial income, consumption, and financial assets.
 
 ```{code-cell} ipython3
 # Sequence Length
 T = cs_model.T
 
-plt.plot(range(T+1), y_seq, label='income')
+plt.plot(range(T+1), y_seq, label='non-financial income')
 plt.plot(range(T+1), c_seq, label='consumption')
-plt.plot(range(T+2), a_seq, label='asset')
+plt.plot(range(T+2), a_seq, label='financial wealth')
 plt.plot(range(T+2), np.zeros(T+2), '--')
 
 plt.legend()
@@ -327,6 +348,10 @@ $$
 \sum_{t=0}^T R^{-t} v_t = 0
 $$
 
+This equation says that the **present value** of admissible variations must be zero.
+
+(So once again, we encounter our formula for the present value of an "asset".)
+
 Here we'll compute a two-parameter class of admissible variations
 of the form