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22 changes: 11 additions & 11 deletions lectures/solow.md
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Expand Up @@ -13,14 +13,14 @@ kernelspec:
# The Solow-Swan Growth Model

In this lecture we review a famous model due
to [Robert Solow (1925--2014)](https://en.wikipedia.org/wiki/Robert_Solow) and [Trevor Swan (1918--1989)](https://en.wikipedia.org/wiki/Trevor_Swan).
to [Robert Solow (1925--2023)](https://en.wikipedia.org/wiki/Robert_Solow) and [Trevor Swan (1918--1989)](https://en.wikipedia.org/wiki/Trevor_Swan).

The model is used to study growth over the long run.

Although the model is simple, it contains some interesting lessons.


We will use the following imports
We will use the following imports.

```{code-cell} ipython3
import matplotlib.pyplot as plt
Expand Down Expand Up @@ -59,7 +59,7 @@ Production functions with this property include
* the **CES** function $F(K, L) = \left\{ a K^\rho + b L^\rho \right\}^{1/\rho}$
with $a, b, \rho > 0$.

We assume a closed economy, so domestic investment equals aggregate domestic
We assume a closed economy, so aggregate domestic investment equals aggregate domestic
saving.

The saving rate is a constant $s$ satisfying $0 \leq s \leq 1$, so that aggregate
Expand Down Expand Up @@ -121,14 +121,14 @@ x0 = 0.25
xmin, xmax = 0, 3
```

Now, we define the function $g$
Now, we define the function $g$.

```{code-cell} ipython3
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 Down Expand Up @@ -198,7 +198,7 @@ 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 Down Expand Up @@ -259,15 +259,15 @@ def simulate_ts(x0_values, ts_length):
simulate_ts(x0, ts_length)
```

As expected, the time paths in the figure both converge to this value.
As expected, the time paths in the figure all converge to $k^*$.

## Growth in continuous time

In this section we investigate a continuous time version of the Solow--Swan
In this section, we investigate a continuous time version of the Solow--Swan
growth model.

We will see how the smoothing provided by continuous time can
simplify analysis.
simplify our analysis.


Recall that the discrete time dynamics for capital are
Expand All @@ -291,7 +291,7 @@ Taking the time step to zero gives the continuous time limit
```

Our aim is to learn about the evolution of $k_t$ over time,
given initial stock $k_0$.
given an initial stock $k_0$.

A **steady state** for {eq}`solowc` is a value $k^*$
at which capital is unchanging, meaning $k'_t = 0$ or, equivalently,
Expand All @@ -308,7 +308,7 @@ the next figure, maintaining the parameterization we used
above.

Writing $k'_t = g(k_t)$ with $g(k) =
s Ak^\alpha - \delta k$, values of $k$ with $g(k) > 0$ imply that $k'_t > 0$, so
s Ak^\alpha - \delta k$, values of $k$ with $g(k) > 0$ imply $k'_t > 0$, so
capital is increasing.

When $g(k) < 0$, the opposite occurs. Once again, high marginal returns to
Expand Down