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[cons_smooth] Update consumption smoothing lecture #376

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[cons_smooth] Update consumption smoothing lecture #376

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57 changes: 25 additions & 32 deletions lectures/cons_smooth.md
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Expand Up @@ -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".

Expand All @@ -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`.
```
Expand Down Expand Up @@ -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.

Expand All @@ -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
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@longye-tian perhaps we can use a note admonition rather than brackets

```{note}
<text>
```

$a_{T+1} = 0$.)

The consumer faces a sequence of budget constraints that constrains sequences $(y, c, a)$
Expand All @@ -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.

Expand All @@ -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**:
Expand All @@ -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.

Expand All @@ -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):
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@mmcky mmcky Feb 19, 2024 •
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thanks @longye-tian.

The build is failing and I suspect the calling location needs to be updated to this new function name as well.

NameError: name 'creat_cs_model' is not defined

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Thank you @mmcky !

I just changed the function name at the calling location. :)

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thanks @longye-tian -- you will see the preview is now generating. 👍

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So glad to contribute !!

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)
```


Expand All @@ -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".

Expand All @@ -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

$$
c_t = c_0 \quad t =0, 1, \ldots, T
Expand Down Expand Up @@ -223,7 +216,7 @@ $$
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
Expand Down Expand Up @@ -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:',
Expand Down Expand Up @@ -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

Expand Down Expand Up @@ -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
Expand Down Expand Up @@ -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):
Expand Down Expand Up @@ -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>`.
Expand Down
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