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Merge pull request #428 from QuantEcon/update_solow
[solow] Update editorial suggestions
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lectures/scalar_dynam.md

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@@ -235,6 +235,7 @@ Analyzing the dynamics of this model requires a different method (see below).
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Consider a fixed dynamic system consisting of set $S \subset \mathbb R$ and
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$g$ mapping $S$ to $S$.
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(scalar-dynam:steady-state)=
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### Steady states
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A **steady state** of this system is a
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(scalar-dynam:global-stability)=
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### Global stability
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A steady state $x^*$ of the dynamic system is called

lectures/solow.md

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@@ -100,17 +100,15 @@ given an exogenous initial capital stock $k_0$.
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## A graphical perspective
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To understand the dynamics of the sequence $(k_t)_{t \geq 0}$ we use a 45
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degree diagram.
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To understand the dynamics of the sequence $(k_t)_{t \geq 0}$ we use a 45-degree diagram.
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To do so, we first
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need to specify the functional form for $f$ and assign values to the parameters.
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We choose the Cobb--Douglas specification $f(k) = A k^\alpha$ and set $A=2.0$,
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$\alpha=0.3$, $s=0.3$ and $\delta=0.4$.
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The function $g$ from {eq}`solow` is then plotted, along with the 45
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degree line.
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The function $g$ from {eq}`solow` is then plotted, along with the 45-degree line.
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Let's define the constants.
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return A * s * k**alpha + (1 - delta) * k
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```
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Let's plot the 45 degree diagram of $g$.
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Let's plot the 45-degree diagram of $g$.
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```{code-cell} ipython3
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def plot45(kstar=None):
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lb = r'$g(k) = sAk^{\alpha} + (1 - \delta)k$'
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ax.plot(xgrid, g_values, lw=2, alpha=0.6, label=lb)
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ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='45')
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ax.plot(xgrid, xgrid, 'k-', lw=1, alpha=0.7, label='$45^{\circ}$')
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if kstar:
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fps = (kstar,)
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plot45()
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```
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Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45 degree line.
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Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45-degree line.
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Then we have $k_{t+1} = g(k_t) > k_t$ and capital per worker rises.
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If $g(k_t) < k_t$ then capital per worker falls.
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If $g(k_t) = k_t$, then we are at a **steady state** and $k_t$ remains constant.
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(A steady state of the model is a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) of the mapping $g$.)
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(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 $g$.)
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From the shape of the function $g$ in the figure, we see that
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there is a unique steady state in $(0, \infty)$.
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If initial capital is above this level, then the reverse is true.
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Let's plot the 45 degree diagram to show the $k^*$ in the plot.
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Let's plot the 45-degree diagram to show the $k^*$ in the plot.
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```{code-cell} ipython3
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kstar = ((s * A) / delta)**(1/(1 - alpha))
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From our graphical analysis, it appears that $(k_t)$ converges to $k^*$, regardless of initial capital
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$k_0$.
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This is a form of global stability.
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This is a form of {ref}`global stability <scalar-dynam:global-stability>`.
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The next figure shows three time paths for capital, from
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x'_t = (1-\alpha) (sA - \delta x_t)
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```
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This equation has the exact solution
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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
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$$
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x_t

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