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[solow] Update editorial suggestions #428

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Apr 28, 2024
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[solow] Update editorial suggestions #428

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[solow] Update editorial suggestions
SylviaZhaooo Apr 22, 2024
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[solow] Update internal links
SylviaZhaooo Apr 22, 2024
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[Solow] Update context
SylviaZhaooo Apr 27, 2024
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2 changes: 2 additions & 0 deletions lectures/scalar_dynam.md
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Original file line number Diff line number Diff line change
Expand Up @@ -236,6 +236,7 @@ Analyzing the dynamics of this model requires a different method (see below).
Consider a fixed dynamic system consisting of set $S \subset \mathbb R$ and
$g$ mapping $S$ to $S$.

(scalar-dynam:steady-state)=
### Steady states

A **steady state** of this system is a
Expand All @@ -255,6 +256,7 @@ definition to check that



(scalar-dynam:global-stability)=
### Global stability

A steady state $x^*$ of the dynamic system is called
Expand Down
20 changes: 10 additions & 10 deletions lectures/solow.md
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Expand Up @@ -100,17 +100,15 @@ given an exogenous initial capital stock $k_0$.

## A graphical perspective

To understand the dynamics of the sequence $(k_t)_{t \geq 0}$ we use a 45
degree diagram.
To understand the dynamics of the sequence $(k_t)_{t \geq 0}$ we use a 45-degree diagram.

To do so, we first
need to specify the functional form for $f$ and assign values to the parameters.

We choose the Cobb--Douglas specification $f(k) = A k^\alpha$ and set $A=2.0$,
$\alpha=0.3$, $s=0.3$ and $\delta=0.4$.

The function $g$ from {eq}`solow` is then plotted, along with the 45
degree line.
The function $g$ from {eq}`solow` is then plotted, along with the 45-degree line.


Let's define the constants.
Expand All @@ -128,7 +126,7 @@ def g(A, s, alpha, delta, k):
return A * s * k**alpha + (1 - delta) * k
```

Let's plot the 45 degree diagram of $g$.
Let's plot the 45-degree diagram of $g$.

```{code-cell} ipython3
def plot45(kstar=None):
Expand All @@ -145,7 +143,7 @@ def plot45(kstar=None):

lb = r'$g(k) = sAk^{\alpha} + (1 - \delta)k$'
ax.plot(xgrid, g_values, lw=2, alpha=0.6, label=lb)
ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='45')
ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$')

if kstar:
fps = (kstar,)
Expand Down Expand Up @@ -175,15 +173,15 @@ def plot45(kstar=None):
plot45()
```

Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45 degree line.
Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45-degree line.

Then we have $k_{t+1} = g(k_t) > k_t$ and capital per worker rises.

If $g(k_t) < k_t$ then capital per worker falls.

If $g(k_t) = k_t$, then we are at a **steady state** and $k_t$ remains constant.

(A steady state of the model is a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) of the mapping $g$.)
(A {ref}`steady state <scalar-dynam:steady-state>` of the model is a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) of the mapping $g$.)

From the shape of the function $g$ in the figure, we see that
there is a unique steady state in $(0, \infty)$.
Expand All @@ -198,7 +196,7 @@ If initial capital is below $k^*$, then capital increases over time.

If initial capital is above this level, then the reverse is true.

Let's plot the 45 degree diagram to show the $k^*$ in the plot.
Let's plot the 45-degree diagram to show the $k^*$ in the plot.

```{code-cell} ipython3
kstar = ((s * A) / delta)**(1/(1 - alpha))
Expand All @@ -209,7 +207,7 @@ plot45(kstar)
From our graphical analysis, it appears that $(k_t)$ converges to $k^*$, regardless of initial capital
$k_0$.

This is a form of global stability.
This is a form of {ref}`global stability <scalar-dynam:global-stability>`.


The next figure shows three time paths for capital, from
Expand Down Expand Up @@ -387,6 +385,8 @@ linear differential equation
x'_t = (1-\alpha) (sA - \delta x_t)
```

To solve this equation, we need the [first ODE](https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/17%3A_Differential_Equations/17.01%3A_First_Order_Differential_Equations).
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How about we delete this sentence and then change the sentence below to

"This equation, which is a [linear ordinary differential equation](add link), has the solution"

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No problem!


This equation has the exact solution

$$
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