You signed in with another tab or window. Reload to refresh your session.You signed out in another tab or window. Reload to refresh your session.You switched accounts on another tab or window. Reload to refresh your session.Dismiss alert
Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45degree line.
176
+
Suppose, at some $k_t$, the value $g(k_t)$ lies strictly above the 45-degree line.
179
177
180
178
Then we have $k_{t+1} = g(k_t) > k_t$ and capital per worker rises.
181
179
182
180
If $g(k_t) < k_t$ then capital per worker falls.
183
181
184
182
If $g(k_t) = k_t$, then we are at a **steady state** and $k_t$ remains constant.
185
183
186
-
(A steady state of the model is a [fixed point](https://en.wikipedia.org/wiki/Fixed_point_(mathematics)) of the mapping $g$.)
184
+
(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 $g$.)
187
185
188
186
From the shape of the function $g$ in the figure, we see that
189
187
there is a unique steady state in $(0, \infty)$.
@@ -198,7 +196,7 @@ If initial capital is below $k^*$, then capital increases over time.
198
196
199
197
If initial capital is above this level, then the reverse is true.
200
198
201
-
Let's plot the 45degree diagram to show the $k^*$ in the plot.
199
+
Let's plot the 45-degree diagram to show the $k^*$ in the plot.
202
200
203
201
```{code-cell} ipython3
204
202
kstar = ((s * A) / delta)**(1/(1 - alpha))
@@ -209,7 +207,7 @@ plot45(kstar)
209
207
From our graphical analysis, it appears that $(k_t)$ converges to $k^*$, regardless of initial capital
210
208
$k_0$.
211
209
212
-
This is a form of global stability.
210
+
This is a form of {ref}`global stability <scalar-dynam:global-stability>`.
213
211
214
212
215
213
The next figure shows three time paths for capital, from
@@ -387,7 +385,7 @@ linear differential equation
387
385
x'_t = (1-\alpha) (sA - \delta x_t)
388
386
```
389
387
390
-
This equationhas the exact solution
388
+
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
0 commit comments