From 2cb3ac3dd58a07cd89b4770dca02d140eac63ced Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 22 Apr 2024 14:08:59 +1000 Subject: [PATCH 1/3] [solow] Update editorial suggestions --- lectures/solow.md | 20 ++++++++++---------- 1 file changed, 10 insertions(+), 10 deletions(-) diff --git a/lectures/solow.md b/lectures/solow.md index ff2efdbd..611ba100 100644 --- a/lectures/solow.md +++ b/lectures/solow.md @@ -100,8 +100,7 @@ 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. @@ -109,8 +108,7 @@ 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. @@ -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): @@ -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,) @@ -175,7 +173,7 @@ 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. @@ -183,7 +181,7 @@ 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 [steady state](https://intro.quantecon.org/scalar_dynam.html#steady-states) 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 $(0, \infty)$. @@ -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)) @@ -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 [global stability](https://intro.quantecon.org/scalar_dynam.html#global-stability). The next figure shows three time paths for capital, from @@ -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). + This equation has the exact solution $$ From 8cb45a16ee6485b77d5c4a7d8f2f1319bfe37b18 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 22 Apr 2024 17:59:57 +1000 Subject: [PATCH 2/3] [solow] Update internal links --- lectures/scalar_dynam.md | 2 ++ lectures/solow.md | 4 ++-- 2 files changed, 4 insertions(+), 2 deletions(-) diff --git a/lectures/scalar_dynam.md b/lectures/scalar_dynam.md index 864cb54d..8aee403e 100644 --- a/lectures/scalar_dynam.md +++ b/lectures/scalar_dynam.md @@ -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 @@ -255,6 +256,7 @@ definition to check that +(scalar-dynam:global-stability)= ### Global stability A steady state $x^*$ of the dynamic system is called diff --git a/lectures/solow.md b/lectures/solow.md index 611ba100..80580439 100644 --- a/lectures/solow.md +++ b/lectures/solow.md @@ -181,7 +181,7 @@ 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](https://intro.quantecon.org/scalar_dynam.html#steady-states) of the model is a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) of the mapping $g$.) +(A {ref}`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)$. @@ -207,7 +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](https://intro.quantecon.org/scalar_dynam.html#global-stability). +This is a form of {ref}`global stability `. The next figure shows three time paths for capital, from From 689cb06898708023930fa88154241bb7382a58ce Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Sat, 27 Apr 2024 14:15:50 +1000 Subject: [PATCH 3/3] [Solow] Update context --- lectures/solow.md | 4 +--- 1 file changed, 1 insertion(+), 3 deletions(-) diff --git a/lectures/solow.md b/lectures/solow.md index 80580439..484f7664 100644 --- a/lectures/solow.md +++ b/lectures/solow.md @@ -385,9 +385,7 @@ 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). - -This equation has the exact solution +This equation, which is a [linear ordinary differential equation](https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/17%3A_Differential_Equations/17.01%3A_First_Order_Differential_Equations), has the solution $$ x_t