<p>Consider the dynamic system <span class="math-container">$Y_{t+1} = f(Y_t)$</span>. In order to investigate the stability locally at <span class="math-container">$Y^\ast = f(Y^\ast)$</span> you can take a Taylor expansion of <span class="math-container">$Y_{t+1}$</span> around <span class="math-container">$Y^\ast$</span>.
<span class="math-container">$$
\begin{align*}
&amp;Y_{t+1} \approx f(Y^\ast) + f'(Y^\ast)(Y_{t} - Y^\ast),\\
\to &amp;Y_{t+1} \approx Y^\ast + f'(Y^\ast) (Y_{t} - Y^\ast),\\
\to &amp;Y_{t+1} - Y^\ast \approx f'(Y^\ast) (Y_t - Y^\ast)
\end{align*}
$$</span></p>
<p><a href="https://econwpa.ub.uni-muenchen.de/econ-wp/mac/papers/0409/0409011.pdf" rel="nofollow noreferrer">These lecture notes from Oded Galor</a> seem quite complete.
So:
<span class="math-container">$$
(Y_t - Y^\ast) = (f'(Y^\ast))^t(Y_1 - Y^\ast) 
$$</span>
Now if <span class="math-container">$|f'(Y^\ast)| &lt; 1$</span> then this converges to zero. So <span class="math-container">$Y_t$</span> converges back to <span class="math-container">$Y^\ast$</span> after a (small) local deviation.</p>

CC BY-SA 4.0
2021-11-14T08:36:48.073Z
2021-11-14T08:36:48.073Z
tdm
<p>Consider a variable <span class="math-container">$Y_t$</span> that evolves over time according to the fixed relation <span class="math-container">$Y_{t+1} = f(Y_t)$</span>. Clearly, fixed points of <span class="math-container">$f$</span> are absorbing states of the process. To assess <em>stability</em>, we may now allow for small errors, i.e. <span class="math-container">$Y_{t+1} = f(Y_t) + \epsilon_t$</span> (I don't think it matters much exactly how <span class="math-container">$\epsilon_t$</span> is distributed, but we should assume it has zero mean). In that case, I believe that the stable points will depend on the slope of <span class="math-container">$f$</span>. In particular, we need to ask whether the derivative of <span class="math-container">$f$</span> is greater or less than one (at various points).</p>
<p><strong>Question</strong>: Can anyone refer me to texts that clarify the mathematics of such processes?</p>
<p>Comment 1: I know this kind of thing crops up in a bunch of areas of economics, e.g. in macroeconomics (think of the Solow model), growth theory (think of the S shape in 'Poor Economics'), and evolutionary game theory. References from these would be fine, but ideally I am looking for a relatively abstract treatment.</p>
<p>Comment 2: It may be helpful if I provide an example of what I am after. Let <span class="math-container">$f(Y) = Y^2$</span> and suppose that the initial value <span class="math-container">$Y_0$</span> is in <span class="math-container">$[0, 1]$</span>. Then <span class="math-container">$Y_t \in [0, 1]$</span> for all <span class="math-container">$t &gt; 0$</span>. Moreover, the function has fixed points at <span class="math-container">$Y = 0$</span> and <span class="math-container">$Y = 1$</span>; so if <span class="math-container">$Y_t$</span> somehow hits these points, it will get stuck there. Considering now the `perturbed' process, let's suppose that <span class="math-container">$Y_{t+1} = Y_t^2 + \epsilon_t$</span>. Computing the derivative, <span class="math-container">$f'(Y) = 2Y$</span> and so the derivatives are <span class="math-container">$f'(0) = 0 &lt; 1$</span> and <span class="math-container">$f'(1) = 2 &gt; 1$</span>. This suggests that, in some sense (to be clarified I hope!), the fixed point at <span class="math-container">$Y = 0$</span> is the stable one and where the process should generally end up.</p>

CC BY-SA 4.0
2021-11-11T20:21:33.877Z
Sounds like you want to study dynamic systems? This is a branch of mathematics.
Giskard
CC BY-SA 4.0
2021-11-11T20:23:03.437Z
Yes it is a maths question, but with plenty of economics applications. I guess I could post on Math SE if nobody has any ideas (though I'll wait a bit to see; I believe that this theory should be well known in various areas of econ).
afreelunch
CC BY-SA 4.0
2021-11-11T21:22:52.677Z
You misunderstand, I was answering your question "*Can anyone refer me to texts that clarify the mathematics of such processes?*" You can find a description of the theory and some textbooks here: [https://en.wikipedia.org/wiki/Dynamical_systems_theory](https://en.wikipedia.org/wiki/Dynamical_systems_theory)
Giskard
CC BY-SA 4.0
2021-11-11T21:38:21.063Z
The actual criterion for discrete-time stability is as you suspect: $|f'(Y^*)| < 1$.
Giskard
CC BY-SA 4.0
2021-11-12T11:42:26.377Z
I think what you're after is a contraction mapping. If $f$ is from $\mathbb{R}$ to $\mathbb{R}$ then $\sup |f'(x)| < 1$ guarantees this as Giskard said.
tdm
CC BY-SA 4.0
2021-11-12T14:05:59.387Z
Thanks, this is helpful. Though I also want to understand the dynamics of functions that have more than one fixed point.
afreelunch
CC BY-SA 4.0
2021-11-14T08:43:24.400Z
Usually, one uses different criteria for what it means to be stable for a deterministic dynamical system.
Michael Greinecker
CC BY-SA 4.0
2021-11-14T14:41:35.403Z
@MichaelGreinecker Thanks; if you could point to these criteria, I would be very grateful. (I think the key phrase may be "Lyapunov stability"?)
afreelunch
<markov-chain>
Stable outcomes of a dynamic process
afreelunch
# Stable outcomes of a dynamic process

(Asked by: username_0 on 2021-11-11)

Consider a variable $Y\_t$ that evolves over time according to the fixed relation $Y\_{t+1} = f(Y\_t)$. Clearly, fixed points of $f$ are absorbing states of the process. To assess *stability*, we may now allow for small errors, i.e. $Y\_{t+1} = f(Y\_t) + \epsilon\_t$ (I don't think it matters much exactly how $\epsilon\_t$ is distributed, but we should assume it has zero mean). In that case, I believe that the stable points will depend on the slope of $f$. In particular, we need to ask whether the derivative of $f$ is greater or less than one (at various points).

**Question**: Can anyone refer me to texts that clarify the mathematics of such processes?

Comment 1: I know this kind of thing crops up in a bunch of areas of economics, e.g. in macroeconomics (think of the Solow model), growth theory (think of the S shape in 'Poor Economics'), and evolutionary game theory. References from these would be fine, but ideally I am looking for a relatively abstract treatment.

Comment 2: It may be helpful if I provide an example of what I am after. Let $f(Y) = Y^2$ and suppose that the initial value $Y\_0$ is in $[0, 1]$. Then $Y\_t \in [0, 1]$ for all $t > 0$. Moreover, the function has fixed points at $Y = 0$ and $Y = 1$; so if $Y\_t$ somehow hits these points, it will get stuck there. Considering now the `perturbed' process, let's suppose that $Y\_{t+1} = Y\_t^2 + \epsilon\_t$. Computing the derivative, $f'(Y) = 2Y$ and so the derivatives are $f'(0) = 0 < 1$ and $f'(1) = 2 > 1$. This suggests that, in some sense (to be clarified I hope!), the fixed point at $Y = 0$ is the stable one and where the process should generally end up.

**Tags:** markov-chain

> **Comments:**
> **username_1**: Sounds like you want to study dynamic systems? This is a branch of mathematics.  
> **username_0**: Yes it is a maths question, but with plenty of economics applications. I guess I could post on Math SE if nobody has any ideas (though I'll wait a bit to see; I believe that this theory should be well known in various areas of econ).  
> **username_1**: You misunderstand, I was answering your question "*Can anyone refer me to texts that clarify the mathematics of such processes?*" You can find a description of the theory and some textbooks here: [https://en.wikipedia.org/wiki/Dynamical_systems_theory](https://en.wikipedia.org/wiki/Dynamical_systems_theory)  
> **username_1**: The actual criterion for discrete-time stability is as you suspect: $|f'(Y^*)| < 1$.  
> **username_2**: I think what you're after is a contraction mapping. If $f$ is from $\mathbb{R}$ to $\mathbb{R}$ then $\sup |f'(x)| < 1$ guarantees this as username_1 said.  
> **username_0**: Thanks, this is helpful. Though I also want to understand the dynamics of functions that have more than one fixed point.  
> **username_3**: Usually, one uses different criteria for what it means to be stable for a deterministic dynamical system.  
> **username_0**: @MichaelGreinecker Thanks; if you could point to these criteria, I would be very grateful. (I think the key phrase may be "Lyapunov stability"?)  

---

## 1 answers

### Answer by username_2 on 2021-11-14. Score: 3 | Accepted answer

Consider the dynamic system $Y\_{t+1} = f(Y\_t)$. In order to investigate the stability locally at $Y^\ast = f(Y^\ast)$ you can take a Taylor expansion of $Y\_{t+1}$ around $Y^\ast$.
$$
\begin{align\*}
&Y\_{t+1} \approx f(Y^\ast) + f'(Y^\ast)(Y\_{t} - Y^\ast),\\
\to &Y\_{t+1} \approx Y^\ast + f'(Y^\ast) (Y\_{t} - Y^\ast),\\
\to &Y\_{t+1} - Y^\ast \approx f'(Y^\ast) (Y\_t - Y^\ast)
\end{align\*}
$$

[These lecture notes from Oded Galor](https://econwpa.ub.uni-muenchen.de/econ-wp/mac/papers/0409/0409011.pdf) seem quite complete.
So:
$$
(Y\_t - Y^\ast) = (f'(Y^\ast))^t(Y\_1 - Y^\ast)
$$
Now if $|f'(Y^\ast)| < 1$ then this converges to zero. So $Y\_t$ converges back to $Y^\ast$ after a (small) local deviation.


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