Merge pull request #376 from QuantEcon/Consumption_Smoothing_Improvement

[cons_smooth] Update consumption smoothing lecture

lectures/cons_smooth.md

@@ -17,9 +17,9 @@ kernelspec:
 ## Overview
 
 
-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 original  Keynesian consumption function  described in this quantecon lecture {doc}`geometric series <geom_series>`  missed.
+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 original  Keynesian consumption function  described in this QuantEcon lecture {doc}`geometric series <geom_series>`  missed.
 
-In this lecture, we'll study what is often  called the "consumption-smoothing model"  using  matrix multiplication and matrix inversion, the same tools that we used in this quantecon lecture {doc}`present values <pv>`. 
+In this lecture, we'll study what is often  called the "consumption-smoothing model"  using  matrix multiplication and matrix inversion, the same tools that we used in this QuantEcon lecture {doc}`present values <pv>`. 
 
 Formulas presented in  {doc}`present value formulas<pv>` are at the core of the consumption smoothing model because we shall use them to define a consumer's "human wealth".
 
@@ -29,7 +29,7 @@ and that standard asset-pricing formulas could be applied to compute a person's
 ''non-financial wealth'' that capitalizes the  earnings stream.  
 
 ```{note}
-As we'll see in this quantecon lecture  {doc}`equalizing difference model <equalizing_difference>`,
+As we'll see in this QuantEcon lecture  {doc}`equalizing difference model <equalizing_difference>`,
 Milton Friedman had used this idea  in his PhD thesis at Columbia University, 
 eventually published as {cite}`kuznets1939incomes` and {cite}`friedman1954incomes`.
 ```
@@ -58,21 +58,14 @@ The consumer faces a gross interest rate of $R >1$ that is constant over time, a
 
 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 non-financial incomes $y_t$
-
- * $a = \{a_t\}_{t=0}^{T+1}$ be a sequence of financial wealth
-
- * $c = \{c_t\}_{t=0}^T$ be a sequence of non-negative consumption rates
-
- * $R \geq 1$ be a fixed gross one period rate of return on financial assets
-
- * $\beta \in (0,1)$ be a fixed discount factor
-
+ * $T \geq 2$  be a positive integer that constitutes a time-horizon. 
+ * $y = \{y_t\}_{t=0}^T$ be an exogenous  sequence of non-negative non-financial incomes $y_t$. 
+ * $a = \{a_t\}_{t=0}^{T+1}$ be a sequence of financial wealth.  
+ * $c = \{c_t\}_{t=0}^T$ be a sequence of non-negative consumption rates. 
+ * $R \geq 1$ be a fixed gross one period rate of return on financial assets. 
+ * $\beta \in (0,1)$ be a fixed discount factor.  
  * $a_0$ be a given initial level of financial assets
-
- * $a_{T+1} \geq 0$  be a terminal condition on final assets
+ * $a_{T+1} \geq 0$  be a terminal condition on final assets. 
 
 The sequence of financial wealth $a$ is to be determined by the model.
 
@@ -83,7 +76,7 @@ We require it to satisfy  two  **boundary conditions**:
 
 The **terminal condition** $a_{T+1} \geq 0$ requires that the consumer not leave the model in debt.
 
-(We'll soon see that a utility maximizing consumer won't **want** to die leaving positive assets, so she'll arrange her affairs to make
+(We'll soon 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   sequences $(y, c, a)$
@@ -94,7 +87,7 @@ $$ (eq:a_t)
 
 Equations {eq}`eq:a_t` constitute  $T+1$ such budget constraints, one for each $t=0, 1, \ldots, T$. 
 
-Given a sequence $y$ of non-financial incomes, a large  set of **pairs** $(a, c)$ of (financial wealth, consumption) sequences  satisfy the sequence of budget constraints {eq}`eq:a_t`. 
+Given a sequence $y$ of non-financial incomes, a large  set of pairs $(a, c)$ of (financial wealth, consumption) sequences  satisfy the sequence of budget constraints {eq}`eq:a_t`. 
 
 Our model has the following logical flow.
 
@@ -116,7 +109,7 @@ with the exogenous non-financial income stream $y$, the initial financial  asset
 
 In general, there are **many** budget feasible consumption paths $c$.
 
-Among all budget-feasible consumption paths, which one **should** a consumer want?
+Among all budget-feasible consumption paths, which one should a consumer want?
 
 
 To answer this question, we shall eventually evaluate alternative budget feasible consumption paths $c$ using the following utility functional or **welfare criterion**:
@@ -131,7 +124,7 @@ where $g_1 > 0, g_2 > 0$.
 
 When $\beta R \approx 1$, 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.  
 
-Indeed, we shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`),  criterion {eq}`welfare` assigns higher welfare to **smoother** consumption paths.
+Indeed, we shall see that when $\beta R = 1$ (a condition assumed by Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`),  criterion {eq}`welfare` assigns higher welfare to smoother consumption paths.
 
 By **smoother** we mean as close as possible to being constant over time.  
 
@@ -147,11 +140,11 @@ We create a Python **namedtuple** to store these parameters with default values.
 ConsumptionSmoothing = namedtuple("ConsumptionSmoothing", 
                         ["R", "g1", "g2", "β_seq", "T"])
 
-def creat_cs_model(R=1.05, g1=1, g2=1/2, T=65):
+def create_consumption_smoothing_model(R=1.05, g1=1, g2=1/2, T=65):
     β = 1/R
     β_seq = np.array([β**i for i in range(T+1)])
-    return ConsumptionSmoothing(R=1.05, g1=1, g2=1/2, 
-                                β_seq=β_seq, T=65)
+    return ConsumptionSmoothing(R, g1, g2, 
+                                β_seq, T)
 ```
 
 
@@ -167,7 +160,7 @@ $$
 
 Human or non-financial wealth  at time $0$ is evidently just the present value of the consumer's non-financial income stream $y$. 
 
-Formally it very much resembles the asset price that we computed in this quantecon lecture {doc}`present values <pv>`.
+Formally it very much resembles the asset price that we computed in this QuantEcon lecture {doc}`present values <pv>`.
 
 Indeed, this is why Milton Friedman called it "human capital". 
 
@@ -185,7 +178,7 @@ $$ (eq:budget_intertemp)
 
 Equation {eq}`eq:budget_intertemp`  says that the present value of the consumption stream equals the sum of finanical and non-financial (or human) wealth.
 
-Robert Hall {cite}`Hall1978` showed that when $\beta R = 1$, a condition Milton Friedman had also  assumed, it is "optimal" for a consumer to **smooth consumption** by setting 
+Robert Hall {cite}`Hall1978` showed that when $\beta R = 1$, a condition Milton Friedman had also  assumed, it is "optimal" for a consumer to smooth consumption by setting 
 
 $$ 
 c_t = c_0 \quad t =0, 1, \ldots, T
@@ -223,7 +216,7 @@ $$
 Compute an  time $0$   consumption $c_0 $ :
 
 $$
-c_t = c_0 = \left( \frac{1 - R^{-1}}{1 - R^{-(T+1)}} \right) (a_0 + \sum_{t=0}^T R^t y_t ) , \quad t = 0, 1, \ldots, T
+c_t = c_0 = \left( \frac{1 - R^{-1}}{1 - R^{-(T+1)}} \right) (a_0 + \sum_{t=0}^T R^{-t} y_t ) , \quad t = 0, 1, \ldots, T
 $$
 
 ### Step 3
@@ -305,7 +298,7 @@ a0 = -2     # such as "student debt"
 # non-financial Income process
 y_seq = np.concatenate([np.ones(46), np.zeros(20)])
 
-cs_model = creat_cs_model()
+cs_model = create_consumption_smoothing_model()
 c_seq, a_seq, h0 = compute_optimal(cs_model, a0, y_seq)
 
 print('check a_T+1=0:', 
@@ -399,7 +392,7 @@ plot_cs(cs_model, a0, y_seq_neg)
 
 #### Experiment 2: permanent wage gain/loss
 
-Now we assume a **permanent**  increase in income of $W$ in year 21 of the $y$-sequence.
+Now we assume a permanent  increase in income of $W$ in year 21 of the $y$-sequence.
 
 Again we can study positive and negative cases
 
@@ -514,7 +507,7 @@ Let's compute that function.
 We require
 
 $$
-\sum_{t=0}^T \left[ \xi_1 \phi^t - \xi_0 \right] = 0
+\sum_{t=0}^T R^{-t}\left[ \xi_1 \phi^t - \xi_0 \right] = 0
 $$
 
 which implies that
@@ -543,7 +536,7 @@ where $v$ is a budget-feasible variation.
 Given $R$, we thus have a two parameter class of budget feasible variations $v$ that we can use
 to compute alternative consumption paths, then evaluate their welfare.
 
-Now let's compute and plot consumption path variations variations
+Now let's compute and plot consumption path variations
 
 ```{code-cell} ipython3
 def compute_variation(model, ξ1, ϕ, a0, y_seq, verbose=1):
@@ -663,7 +656,7 @@ plt.show()
 ## Wrapping up the consumption-smoothing model
 
 The consumption-smoothing model of Milton Friedman {cite}`Friedman1956` and Robert Hall {cite}`Hall1978`) is a cornerstone of modern macro that has important ramifications for the size of the Keynesian  "fiscal policy multiplier" described briefly in
-quantecon lecture {doc}`geometric series <geom_series>`.  
+QuantEcon lecture {doc}`geometric series <geom_series>`.  
 
 In particular,  it  **lowers** the government expenditure  multiplier relative to  one implied by
 the original Keynesian consumption function presented in {doc}`geometric series <geom_series>`.