diff --git a/lectures/solow.md b/lectures/solow.md index 72c7a8f8..ff2efdbd 100644 --- a/lectures/solow.md +++ b/lectures/solow.md @@ -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 @@ -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 @@ -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): @@ -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)) @@ -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 @@ -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, @@ -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