<blockquote>
<p>The assertion that <span class="math-container">$k^{\ast}$</span> is the fixed point follows from the intermediate value theorem (as <span class="math-container">$g(k)$</span> is continuous over its domain and <span class="math-container">$g(k') \neq g(k'')$</span> for <span class="math-container">$k' \neq k''$</span>). The uniqueness stems from the fact that <span class="math-container">$g(k)$</span> is strictly increasing</p>
</blockquote>
<p>Your approach about the proof of existence and uniqueness is substantially correct, but it   needs some specifications, in particular about the <strong>function to which the intermediate value theorem is applied</strong> and the <strong>use of the Inada conditions</strong>.</p>
<p><strong>Existence of a fixed point</strong></p>
<p>It must be specified to which function we apply the intermediate value theorem.</p>
<p>The existence of a <em>steady state equilibrium</em> means that there exists a constant  <span class="math-container">$k^*$</span> such that
<span class="math-container">$$k(t+1)= g(k(t))= s f(k(t)) +(1- \delta) k(t) = k(t)\tag {1}.$$</span></p>
<p>Therefore, a steady state value <span class="math-container">$k^*$</span> must satisfy, for <span class="math-container">$k\neq 0$$^1$</span>:</p>
<p><span class="math-container">$$\frac {f(k)}{k}= \frac {\delta}{s} \tag {2}$$</span></p>
<p>We can apply the intermediate value theorem to the function <span class="math-container">$\frac {f(k)}{k}$</span> above.</p>
<p>Here, the <strong>Inada conditions</strong> are crucial. In fact, applying the Inada conditions and the De l'Hôpital rule, we have:</p>
<p><span class="math-container">$$\lim_{k\rightarrow 0} {\frac {f(k)}{k}}=  \lim_{k\rightarrow 0}{\frac {f'(k)}{1}}= +\infty \tag {3}$$</span></p>
<p><span class="math-container">$$\lim_{k\rightarrow +\infty} {\frac {f(k)}{k}}=  \lim_{k\rightarrow 0}{\frac {f'(k)}{1}}= 0,\tag {4}$$</span></p>
<p>where the second equalities of equations <span class="math-container">$(3)$</span> and <span class="math-container">$(4)$</span> are the Inada conditions.</p>
<p>The <strong>intermediate value theorem</strong> says that a continuous function on an interval <span class="math-container">$I$</span> takes all the values between its <span class="math-container">$\inf$</span> and its <span class="math-container">$\sup$</span>.<span class="math-container">$^2$</span></p>
<p>Here, <span class="math-container">$\inf {\frac {f(k)}{k}}=0$</span> and <span class="math-container">$\sup {\frac {f(k)}{k}}=+\infty$</span>, so that <span class="math-container">$\frac {f(k)}{k}$</span> takes all values in <span class="math-container">$\mathbb{R}^+$</span>, <strong>including  <span class="math-container">$\frac {\delta}{s}$</span></strong></p>
<p>Therefore, a steady state equilibrium exists.</p>
<p><span class="math-container">$$***$$</span></p>
<p>However, I can’t understand why you said:</p>
<blockquote>
<p>and <span class="math-container">$g(k') \neq g(k'')$</span> for <span class="math-container">$k' \neq k''$</span>.</p>
</blockquote>
<p>This means just that the function <span class="math-container">$g$</span> is injective, what we already know as it is strictly increasing.</p>
<p>But what has this to do with existence and intermediate value theorem? The intermediate value theorem doesn’t require  the function to be injective, and the proof above will hold even if <span class="math-container">$g$</span> wouldn't be injective.</p>
<p>Consider the usual graph, in which we have a not injective <span class="math-container">$g$</span>: this evidently does not affect the existence of a steady state, but only its unicity:</p>
<p><a href="https://i.sstatic.net/ZVb8Z.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZVb8Z.png" alt="enter image description here" /></a>
<span class="math-container">$$\;$$</span></p>
<p><strong>Uniqueness</strong></p>
<p>Your statement concerning uniqueness is correct, we in particular  observe that, because of the strict monotonicity of <span class="math-container">$g(k)$</span> (and <span class="math-container">$g''&lt;0$</span>), for <span class="math-container">$k&lt;k^* $</span> <span class="math-container">$g(k)&gt;k$</span>, and for <span class="math-container">$k&gt;k^*$</span> <span class="math-container">$g(k)&gt;k$</span>.
Therefore, the only point in which <span class="math-container">$g(k)=k$</span> is <span class="math-container">$k^*$</span>.</p>
<p><strong>Stability</strong></p>
<p>A qualitative argument to establish stability is to consider that if <span class="math-container">$k(t)$</span> is above <span class="math-container">$k^*$</span>, it decreases, and if it is below <span class="math-container">$k^*$</span> it increases, and this is related to diminishing returns of capital.</p>
<p>A formal, analitycal proof of stability, however, requires theorems concerning the stability of discrete dynamical systems, much more complicated.</p>
<hr />
<p><span class="math-container">$^1$</span> <sup> Of course, there is a trivial fixed point, for <span class="math-container">$k=0$</span>, but we discard it as of no economic interest, and moreover unstable.
</sup></p>
<p><span class="math-container">$^2$</span><sup>  I specify this version of the intermediate value theorem, with <span class="math-container">$\inf$</span> and <span class="math-container">$\sup$</span>, because often we find a different version, where the values of the function at the extreme of a closed interval <span class="math-container">$[a,b]$</span> are specified, but the two versions  are equivalent.</sup></p>

CC BY-SA 4.0
2024-03-02T18:14:16.873Z
Hi! You seem to be proving that the equilibrium exists and is unique. Where do we see global stability?
Giskard
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2024-03-02T18:18:26.853Z
@Giskard, if you read the question, it is about existence and uniqueness, stability is in the title only. There is no question in the text about stability. Maybe the OP asks for proofs of existence and uniqueness as a prerequisite for studyng stability? I don't know.
BakerStreet
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2024-03-02T18:20:06.343Z
Hmm, what is this then? "*I attempt to prove global stability as follows:*"
Giskard
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2024-03-02T18:22:31.873Z
I agree that the OP seems to be confusing existence and uniqueness with global stability in his attempt, but they indicate both in the title and in the body that they are after global stability.
Giskard
CC BY-SA 4.0
2024-03-02T18:24:26.007Z
It is a mistake.  The OP says, I try to prove prove stability', but actually goes on talking about existence and uniqueness, and asking about them, as you can see also in the comments. .The reference to intermediate value theorem is cleraly related to existence, no internmediate value is involved in proofs of stability.
BakerStreet
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2024-03-02T18:25:25.633Z
In the comments the OP makes explicit that they is asking about existence and uniqueness. Otherwise the whole question makes no sense.
BakerStreet
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2024-03-02T18:26:56.783Z
Can you please link to the comment where they do that? I only see [this one](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95821_57806) where they write (emphasis by me) "**global stability** comes from the diminishing returns of capital."
Giskard
CC BY-SA 4.0
2024-03-02T18:28:23.820Z
The OPs proof and goal are not compatible, it is true. But it is quite surprising that you see in their soul that hey are mistaken when they state the objective, not when they try to prove it.
Giskard
CC BY-SA 4.0
2024-03-02T18:28:53.393Z
I refer to this comment: "What I was trying to say is that *the existence* of a fixed point is guaranteed by the intermediate value theorem. The * uniqueness* of ∗ is guaranteed by the fact that () is strictly increasing, and the global stability comes from the diminishing returns of capital. Are these statements correct?"
BakerStreet
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2024-03-02T18:29:34.767Z
I quoted the same comment. Do you not see the part on global stability in it...?
Giskard
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2024-03-02T18:30:11.917Z
You write "*Otherwise the whole question makes no sense.*" Why would a question asking how to prove global stability in this model make no sense?
Giskard
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2024-03-02T18:32:30.413Z
The part about stability is in this comment only and not in the text of the question. In the question stability is only formally, and erroneusly, present, but then the question is only about existence and uniqueness. All the OP says in is 'proof' has nothing to do with stability.
BakerStreet
CC BY-SA 4.0
2024-03-02T18:33:43.790Z
Yeah, they fail to prove stability. How do you know that was not their goal though? They write several times that is what they were trying to do.
Giskard
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2024-03-02T18:34:38.377Z
Nothing in the text of the question is related to stability, for this reason if it were about stability would make no sense.
BakerStreet
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2024-03-02T18:35:04.567Z
Nothing in the proof is related to it. They are asking if it is correct or not though, so...
Giskard
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2024-03-02T18:37:06.430Z
The OP says it in the notes, and it is clear that is talking about the *usual* arguments of the proofs of existence and uniqueness, that they probably studied in a course.
BakerStreet
CC BY-SA 4.0
2024-03-02T18:38:16.993Z
Do you teach? If yes, maybe you notice that students frequently use "usual" arguments for things where they do not really work? Left OP a [message](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95840_57806), maybe they will clarifiy.
Giskard
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2024-03-02T18:41:22.473Z
@Giskard, yes, I have a long experience of teaching, and this is my interpretation of the question. If you don't agree, there is no problem.
BakerStreet
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2024-03-02T18:41:24.930Z
You have a deep trust in others' understanding of this proof and what it shows. I like that!
Giskard
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2024-03-02T19:12:23.407Z
Hey, why did you add a section on stability 20 minutes after my last comment since you are 100% sure that is not what the question is about, and if it was about that it would be meaningless
Giskard
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2024-03-02T19:20:51.440Z
**Comments have been [moved to chat](https://chat.stackexchange.com/rooms/151800/discussion-on-answer-by-bakerstreet-proving-the-global-stability-in-the-solow-sw); please do not continue the discussion here.** Before posting a comment below this one, please review the [purposes of comments](/help/privileges/comment). Comments that do not request clarification or suggest improvements usually belong as an [answer](/help/how-to-answer), on [meta], or in [chat]. Comments continuing discussion may be removed.
1muflon1
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2024-03-02T19:34:12.150Z
@1muflon1 I'm pretty sure my comments suggested an improvement, the answer got a new section after I posted them. Since I am not credited in it, maybe you should bring back some of my comments?
Giskard
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2024-03-02T19:42:29.143Z
@Giskard I added the part about stability just to avoid misunderstandings, after your comments, I dind't write it before because I thought that it was not really in the question, and also because it wasn't a very interesting part (unless one studies the theory of stability of discrete dynamical systems),  rather obvious from the graph of the Solow model. Much more 'substantial' , and useful, are the proofs of existence and uniqueness.
BakerStreet
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2024-03-02T19:56:44.830Z
@giskard  I brought back the comments, please continue the discussion in the chat
1muflon1
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2024-03-02T20:33:37.467Z
I made the question more clear. Sorry for the confusion
Maximilian
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2024-03-02T21:08:35.123Z
@BakerStreet Care to guess which one of us had the right interpretation? https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95851_57806
Giskard
CC BY-SA 4.0
2024-03-02T21:14:15.517Z
"*wasn't a very interesting part (unless one studies the theory of stability of discrete dynamical systems)*" is a funny comment on a question asking about global stability in a discrete dynamical system.
Giskard
CC BY-SA 4.0
2024-03-02T21:15:36.773Z
@Giskard The question in the mind of the OP was about stability, but before Maxilian edited the question it was impossible to undertsand it clearly,  Maximilian has completely changed the question, incorporating also  aspect of my answer, that clearly answered to their questions about existence,  as you can see in the comments.  The question was not clear, that's all.
BakerStreet
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2024-03-02T21:16:49.680Z
"*it was impossible to undertsand it clearly*" and yet I managed to make the correct guess. "*The question was not clear, that's all.*" Indeed. And yet you were so certain. Perhaps we are all Watsons in some way every now and then, and that is okay?
Giskard
CC BY-SA 4.0
2024-03-02T21:21:02.810Z
@Giskard, you don't even try to understand what one is saying. The part of stability, without studying dynamical systems, reduces to write arrows on the graph, and establish the sign of the growth of $k(t)$, which Maximilian does also with formulas. But this is a partial and intuitive explanation, not a real sufficient proof based on the theory of dynamical systems.
BakerStreet
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2024-03-02T21:23:08.990Z
@Giskard, I was just reading the actual text, and what the OP said in the comments, that confirmed the questions about existence and uniqueness . We don't agree, that's all.
BakerStreet
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2024-03-02T21:23:41.133Z
That is not all, sir, but I have no further arguments. Good night to you!
Giskard
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2024-03-02T17:44:21.927Z
2024-03-02T19:01:17.170Z
2024-03-02T19:01:17.170Z
BakerStreet
<p>A unique fixed point may not be stable. For an example, check the dynamic system
<span class="math-container">$$
x_{t+1} = 2x_t,
$$</span>
where the unique fixed point is <span class="math-container">$x^* = 0$</span>, but this is not stable, in fact, starting from any point other than 0 you will get a trajectory that diverges from the fixed point. To prove global stability, you need to show that no trajectories diverge from the fixed point, show that their distance is not increasing.
<br><br>
<a href="https://en.wikipedia.org/wiki/Stability_theory#Overview_in_dynamical_systems" rel="nofollow noreferrer">https://en.wikipedia.org/wiki/Stability_theory#Overview_in_dynamical_systems</a>
<br><br>
The same equations can be used, but this is what you need to prove and should be working towards.</p>

CC BY-SA 4.0
2024-03-02T06:50:42.080Z
2024-03-02T07:00:50.067Z
2024-03-02T07:00:50.067Z
Giskard
<p>Consider a simplified version of the Solow-Swan model in discrete time, where technology is normalized to one and the population size is constant.</p>
<p>The Solow equation is given by
<span class="math-container">\begin{equation}
    k_{t+1} = s f(k_t) + (1-\delta) k_t
\end{equation}</span>
where <span class="math-container">$k_t \equiv \frac{K_t}{L}$</span> represents the capital per worker, <span class="math-container">$s \in (0,1)$</span> is the saving rate, <span class="math-container">$f(k)$</span> is the production function in intensive form, and <span class="math-container">$\delta \in (0,1)$</span> is the depreciation rate of physical capital.</p>
<p>I attempt to prove global stability as follows:</p>
<p>First, a steady state <span class="math-container">$k^{\ast}$</span> is a fixed point of the law of motion of capital: <span class="math-container">$k^{\ast} = g(k^{\ast})$</span>, and it is about:
<span class="math-container">\begin{equation}
\frac{k^{\ast}}{f(k^{\ast})} = \frac{s}{\delta}
\end{equation}</span></p>
<p>The existence of a steady state <span class="math-container">$k^{\ast}$</span> satisfying the equation above is guaranteed by the intermediate value theorem since <span class="math-container">$g(k)$</span> is continuous over its domain and defined over a closed interval, say <span class="math-container">$[a,b]$</span>, with <span class="math-container">$g(a) \ne g(b)$</span>. The uniqueness (discarding the trivial steady state, <span class="math-container">$k^{\ast}=0$</span>) stems from the fact that <span class="math-container">$g(k)$</span> is strictly increasing (yet bounded above due to the Inada conditions, i.e., <span class="math-container">$f'(\infty) = 0$</span>).</p>
<p>Now, I prove global stability as follows:</p>
<p>The function <span class="math-container">$g(k) = s f(k) + (1-\delta) k_t$</span> satisfies: <span class="math-container">$i) g'(k) &gt; 0$</span> and <span class="math-container">$ii) g''(k) &lt; 0$</span>. Consequently, <span class="math-container">$g(k) &gt; k$</span> if and only if <span class="math-container">$k &lt; k^{\ast}$</span>. Similarly, <span class="math-container">$g(k) &lt; k$</span> if and only if <span class="math-container">$k &gt; k^{\ast}$</span>. The global stability is due to the diminishing marginal return of capital. To see this, let <span class="math-container">$\gamma$</span> denote the growth of <span class="math-container">$k$</span>:</p>
<p><span class="math-container">\begin{equation}
\gamma = \frac{k_{t+1} -k_t}{k_t} = s \frac{f(k_t)}{k_t} - \delta = \frac{s}{z(k_{t})} - \delta
\end{equation}</span>
Here, <span class="math-container">$z(k_{t}) = \frac{k_t}{f(k_t)}$</span> is strictly increasing in <span class="math-container">$k_t$</span>, and <span class="math-container">$z(k^{\ast}) = s/\delta$</span>. Trivially, the growth rate of capital per worker at <span class="math-container">$k^{\ast}$</span> is zero. It is evident that if <span class="math-container">$k' &gt; k^{\ast}$</span>, negative growth in <span class="math-container">$k$</span> occurs until capital per worker converges to <span class="math-container">$k^{\ast}$</span>. Similarly, if <span class="math-container">$k' &lt; k^{\ast}$</span>, positive growth in <span class="math-container">$k$</span> occurs until it converges to <span class="math-container">$k^{\ast}$</span>.</p>
<p>Is the proof presented in the following manner sufficiently rigorous, or do you believe there is room for improvement?</p>

CC BY-SA 4.0
2024-03-02T06:54:15.537Z
You seem to start from the assumption that $k^*$ is a fixed point, which you use to prove stuff about $g(k)$'s and $k$'s relation, which you then use to prove that $k^*$ is a fixed point. This is circular logic. Perhaps I misunderstand what you are doing here, because $k'$ and $k''$ are not defined.
Giskard
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2024-03-02T09:49:06.310Z
What I was trying to say is that the existence of a fixed point is guaranteed by the intermediate value theorem. The uniqueness of $k^{\ast}$ is guaranteed by the fact that $g(k)$ is strictly increasing, and the global stability comes from the diminishing returns of capital. Are these statements correct?
Maximilian
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2024-03-02T18:13:12.337Z
I don't think they are precise enough to be correct, e.g.; I don't see you using the definition of global stability anywhere, you are just saying that it "comes".
Giskard
CC BY-SA 4.0
2024-03-02T18:36:46.103Z
Hi! Can you please clarify for me and the other person who answered, if you in fact want to prove that the steady state is globally stable, or if you just want to show that it exists and is unique?
Giskard
CC BY-SA 4.0
2024-03-02T20:02:52.333Z
Thanks for flagging the global stability aspect. Apologies for any confusion caused by the lack of clarity. The main question is about global stability. I provided a qualitative explanation, saying that if $k$ exceeds $k^{\ast}$, then $k$ diminishes, albeit at a decreasing rate, ultimately converging towards $k^{\ast}$. Similarly, if $k$ is below $k^{\ast}$, it increases towards $k^{\ast}$. This stability emerges due to diminishing marginal returns of capital. By the way, let me clarify the question above
Maximilian
CC BY-SA 4.0
2024-03-02T21:12:28.277Z
Hi Maximilian! Thank you for the clarification. Please do not edit your question in substantive ways, SE is a poor fit for this sort of iterative processes, because the answers become out of sync with the questions. On your new proof: the sign of the change of $k$ does not prove convergence. For an example see Xeno's/Achilles and the tortoise paradox. The good news is that you do not have to prove convergence, that is asymptotic stability. You can use the signs to argue that the distance does not increase, and that should be fine for global stability.
Giskard
CC BY-SA 4.0
2024-03-02T21:38:51.083Z
@Maximilian Just a little observation. The use of the intermediate value theorem is not completely correct, that $g(a)\neq g(b)$ is not sufficient, you sholud show that there are  $a$ and $b$ such that $g(a)<\delta/s$ and $g(b)>\delta/s$. It is better to use the version of the theorem with inf and sup and use the Inada conditions.
BakerStreet
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2024-03-02T21:40:35.747Z
@BakerStreet thanks for the suggestion.
Maximilian
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2024-03-02T21:55:58.720Z
@Maximilian As for the proof of stability, your argument is correct and well done, but it is not a sufficient rigorous mathematical proof. To prove stability we need to prove that every sequence $k (t)$ converges to $k^*$. But to prove, for example, that if $k(t)<k^*$, $k_{t+1}>k(t)$ doesn't ensure stability: $k(t+1)$ can go far beyond $k^*$ and then jump again on the left of $k^*$, and the sequence can oscillate around $k^*$, going further and further away from $k^*$. For a rigorous proof see Acemoglu, Modern Economic Growth.
BakerStreet
CC BY-SA 4.0
2024-03-03T08:55:55.730Z
@BakerStreet Good point on jumping oscillations in discrete time systems. I was wrong about signs being enough to show not asymptotic stability here. (See how easy it is to admit when one made a mistake?)
Giskard
CC BY-SA 4.0
2024-03-03T11:15:02.360Z
I went through Acemoglu's book, but I think it's not very intuitive, and maybe it's not a good starting point for studying the mathematics behind dynamic systems. Do you have any recommendations for comprehensive mathematical books that address these topics but are suitable for economics students?
Maximilian
CC BY-SA 4.0
2024-03-03T11:33:03.060Z
A classical reference for economic dynamics is Gandolfo, *Economic dynamics*, where you find also difference equations, that is discrete time equations. As for a matematical book about dynamics systems in general, they are often advanced, but there is a very interesting book, not advanced, no particular mathematics is required, but it is not addressed to economists, so there is a lot of subjects that are not very important for economics, the book is Strogatz, *Non linear Dynamics and Chaos*.
BakerStreet
CC BY-SA 4.0
2024-03-03T14:19:00.067Z
@Maximilian It would be useful to add in your question, in a note at the end  or as you want, a previous version of your question, before you edited it. It was really different. 
After having received two answers, you  changed radically your question, eliminating previous questions and incorporating also parts of the answers: without reference to the previous version, the two answers  and the whole thread become nonsensical for future readers. In the present form the thread is not a good suitable thread for Stack Exchange.
BakerStreet
CC BY-SA 4.0
2024-03-03T14:19:39.027Z
This is of course not a criticism to you, just a way to not leave on a Stack Exchange a thread disordered and incomprehensible.
BakerStreet
CC BY-SA 4.0
2024-03-03T18:59:03.487Z
@BakerStreet I'd love to repost the initial question but it was a bit messy, and I'm uncertain about accurately recalling how I phrased it initially. While I did integrate a bit of your answer, it aligned with my pre-existing thoughts. I genuinely appreciated your explanation regarding the concept of a fixed point and the comment on stability.
Maximilian
CC BY-SA 4.0
2024-03-03T19:06:40.133Z
@Maximilian Ok, don't worry, I said this just to make clear to possible readers the logic of this thread. Thank you for your appreciation, I'm happy having been helpful.
BakerStreet
CC BY-SA 4.0
2024-03-03T20:49:28.993Z
@BakerStreet, if you don't mind, I'd like to inquire about the final matter. Since I'm sure you know a lot about dynamic systems, I checked Gandolfo's book (1997), and it seems very well-written with plenty of examples. I'll study it, for sure. But, to integrate, what about a rigorous reference for economics students that covers topics like difference/differential equations and key theorems, such as the existence and uniqueness theorem, which appear to be overlooked in Gandolfo's book? Any recommendations?
Maximilian
CC BY-SA 4.0
2024-03-03T21:52:36.897Z
@Maximilian  I agree with you, Gandolfo is a good book, but it lacks theory.  A beautiful book, not advanced, not very theorical and complicated, absolutely readable because it deals with  also mathematical prerequisites,  as the linear algebra that is needed, full of example is Braun, *Differential Equations and Their Applications*, Springer, Fourth Edition: there is the existence and unicity theorem, with proof. It is not a book addressed to economists, but it may  be worth  looking  at a book less ‘dedicated’, to get a wider idea of the subject. But it is very useful also for economics.
BakerStreet
CC BY-SA 4.0
2024-03-03T21:53:17.167Z
As for a book addressed to economists, but that deals  with theorems also, as theorems of existence and unicity, and both discrete and continuous dynamics , you can look at a  beautiful book of mathematics for economists, De La Fuente, *Mathematical Methods  for Economists*, Cambridge University Press, 2000,  more ‘theorical’  than other books for economists.  There are theorems also about discrete dynamic systems.
BakerStreet
<macroeconomics><solow><steady-state>
Proving the Global stability in the Solow Swan Model
Maximilian
# Proving the Global stability in the Solow Swan Model

(Asked by: username_0 on 2024-03-01)

Consider a simplified version of the Solow-Swan model in discrete time, where technology is normalized to one and the population size is constant.

The Solow equation is given by
\begin{equation}
k\_{t+1} = s f(k\_t) + (1-\delta) k\_t
\end{equation}
where $k\_t \equiv \frac{K\_t}{L}$ represents the capital per worker, $s \in (0,1)$ is the saving rate, $f(k)$ is the production function in intensive form, and $\delta \in (0,1)$ is the depreciation rate of physical capital.

I attempt to prove global stability as follows:

First, a steady state $k^{\ast}$ is a fixed point of the law of motion of capital: $k^{\ast} = g(k^{\ast})$, and it is about:
\begin{equation}
\frac{k^{\ast}}{f(k^{\ast})} = \frac{s}{\delta}
\end{equation}

The existence of a steady state $k^{\ast}$ satisfying the equation above is guaranteed by the intermediate value theorem since $g(k)$ is continuous over its domain and defined over a closed interval, say $[a,b]$, with $g(a) \ne g(b)$. The uniqueness (discarding the trivial steady state, $k^{\ast}=0$) stems from the fact that $g(k)$ is strictly increasing (yet bounded above due to the Inada conditions, i.e., $f'(\infty) = 0$).

Now, I prove global stability as follows:

The function $g(k) = s f(k) + (1-\delta) k\_t$ satisfies: $i) g'(k) > 0$ and $ii) g''(k) < 0$. Consequently, $g(k) > k$ if and only if $k < k^{\ast}$. Similarly, $g(k) < k$ if and only if $k > k^{\ast}$. The global stability is due to the diminishing marginal return of capital. To see this, let $\gamma$ denote the growth of $k$:

\begin{equation}
\gamma = \frac{k\_{t+1} -k\_t}{k\_t} = s \frac{f(k\_t)}{k\_t} - \delta = \frac{s}{z(k\_{t})} - \delta
\end{equation}
Here, $z(k\_{t}) = \frac{k\_t}{f(k\_t)}$ is strictly increasing in $k\_t$, and $z(k^{\ast}) = s/\delta$. Trivially, the growth rate of capital per worker at $k^{\ast}$ is zero. It is evident that if $k' > k^{\ast}$, negative growth in $k$ occurs until capital per worker converges to $k^{\ast}$. Similarly, if $k' < k^{\ast}$, positive growth in $k$ occurs until it converges to $k^{\ast}$.

Is the proof presented in the following manner sufficiently rigorous, or do you believe there is room for improvement?

**Tags:** macroeconomics, solow, steady-state
> **Comments:**
> **username_1**: You seem to start from the assumption that $k^*$ is a fixed point, which you use to prove stuff about $g(k)$'s and $k$'s relation, which you then use to prove that $k^*$ is a fixed point. This is circular logic. Perhaps I misunderstand what you are doing here, because $k'$ and $k''$ are not defined.  
> **username_0**: What I was trying to say is that the existence of a fixed point is guaranteed by the intermediate value theorem. The uniqueness of $k^{\ast}$ is guaranteed by the fact that $g(k)$ is strictly increasing, and the global stability comes from the diminishing returns of capital. Are these statements correct?  
> **username_1**: I don't think they are precise enough to be correct, e.g.; I don't see you using the definition of global stability anywhere, you are just saying that it "comes".  
> **username_1**: Hi! Can you please clarify for me and the other person who answered, if you in fact want to prove that the steady state is globally stable, or if you just want to show that it exists and is unique?  
> **username_0**: Thanks for flagging the global stability aspect. Apologies for any confusion caused by the lack of clarity. The main question is about global stability. I provided a qualitative explanation, saying that if $k$ exceeds $k^{\ast}$, then $k$ diminishes, albeit at a decreasing rate, ultimately converging towards $k^{\ast}$. Similarly, if $k$ is below $k^{\ast}$, it increases towards $k^{\ast}$. This stability emerges due to diminishing marginal returns of capital. By the way, let me clarify the question above  
> **username_1**: Hi Maximilian! Thank you for the clarification. Please do not edit your question in substantive ways, SE is a poor fit for this sort of iterative processes, because the answers become out of sync with the questions. On your new proof: the sign of the change of $k$ does not prove convergence. For an example see Xeno's/Achilles and the tortoise paradox. The good news is that you do not have to prove convergence, that is asymptotic stability. You can use the signs to argue that the distance does not increase, and that should be fine for global stability.  
> **username_2**: @Maximilian Just a little observation. The use of the intermediate value theorem is not completely correct, that $g(a)\neq g(b)$ is not sufficient, you sholud show that there are  $a$ and $b$ such that $g(a)<\delta/s$ and $g(b)>\delta/s$. It is better to use the version of the theorem with inf and sup and use the Inada conditions.  
> **username_0**: @BakerStreet thanks for the suggestion.  
> **username_2**: @Maximilian As for the proof of stability, your argument is correct and well done, but it is not a sufficient rigorous mathematical proof. To prove stability we need to prove that every sequence $k (t)$ converges to $k^*$. But to prove, for example, that if $k(t)<k^*$, $k_{t+1}>k(t)$ doesn't ensure stability: $k(t+1)$ can go far beyond $k^*$ and then jump again on the left of $k^*$, and the sequence can oscillate around $k^*$, going further and further away from $k^*$. For a rigorous proof see Acemoglu, Modern Economic Growth.  
> **username_1**: @BakerStreet Good point on jumping oscillations in discrete time systems. I was wrong about signs being enough to show not asymptotic stability here. (See how easy it is to admit when one made a mistake?)  
> **username_0**: I went through Acemoglu's book, but I think it's not very intuitive, and maybe it's not a good starting point for studying the mathematics behind dynamic systems. Do you have any recommendations for comprehensive mathematical books that address these topics but are suitable for economics students?  
> **username_2**: A classical reference for economic dynamics is Gandolfo, *Economic dynamics*, where you find also difference equations, that is discrete time equations. As for a matematical book about dynamics systems in general, they are often advanced, but there is a very interesting book, not advanced, no particular mathematics is required, but it is not addressed to economists, so there is a lot of subjects that are not very important for economics, the book is Strogatz, *Non linear Dynamics and Chaos*.  
> **username_2**: @Maximilian It would be useful to add in your question, in a note at the end  or as you want, a previous version of your question, before you edited it. It was really different.  After having received two answers, you  changed radically your question, eliminating previous questions and incorporating also parts of the answers: without reference to the previous version, the two answers  and the whole thread become nonsensical for future readers. In the present form the thread is not a good suitable thread for Stack Exchange.  
> **username_2**: This is of course not a criticism to you, just a way to not leave on a Stack Exchange a thread disordered and incomprehensible.  
> **username_0**: @BakerStreet I'd love to repost the initial question but it was a bit messy, and I'm uncertain about accurately recalling how I phrased it initially. While I did integrate a bit of your answer, it aligned with my pre-existing thoughts. I genuinely appreciated your explanation regarding the concept of a fixed point and the comment on stability.  
> **username_2**: @Maximilian Ok, don't worry, I said this just to make clear to possible readers the logic of this thread. Thank you for your appreciation, I'm happy having been helpful.  
> **username_0**: @BakerStreet, if you don't mind, I'd like to inquire about the final matter. Since I'm sure you know a lot about dynamic systems, I checked Gandolfo's book (1997), and it seems very well-written with plenty of examples. I'll study it, for sure. But, to integrate, what about a rigorous reference for economics students that covers topics like difference/differential equations and key theorems, such as the existence and uniqueness theorem, which appear to be overlooked in Gandolfo's book? Any recommendations?  
> **username_2**: @Maximilian  I agree with you, Gandolfo is a good book, but it lacks theory.  A beautiful book, not advanced, not very theorical and complicated, absolutely readable because it deals with  also mathematical prerequisites,  as the linear algebra that is needed, full of example is Braun, *Differential Equations and Their Applications*, Springer, Fourth Edition: there is the existence and unicity theorem, with proof. It is not a book addressed to economists, but it may  be worth  looking  at a book less ‘dedicated’, to get a wider idea of the subject. But it is very useful also for economics.  
> **username_2**: As for a book addressed to economists, but that deals  with theorems also, as theorems of existence and unicity, and both discrete and continuous dynamics , you can look at a  beautiful book of mathematics for economists, De La Fuente, *Mathematical Methods  for Economists*, Cambridge University Press, 2000,  more ‘theorical’  than other books for economists.  There are theorems also about discrete dynamic systems.  

---

## 2 answers

### Answer by username_2 on 2024-03-02. Score: 3

> The assertion that $k^{\ast}$ is the fixed point follows from the intermediate value theorem (as $g(k)$ is continuous over its domain and $g(k') \neq g(k'')$ for $k' \neq k''$). The uniqueness stems from the fact that $g(k)$ is strictly increasing

Your approach about the proof of existence and uniqueness is substantially correct, but it needs some specifications, in particular about the **function to which the intermediate value theorem is applied** and the **use of the Inada conditions**.

**Existence of a fixed point**

It must be specified to which function we apply the intermediate value theorem.

The existence of a *steady state equilibrium* means that there exists a constant $k^\*$ such that
$$k(t+1)= g(k(t))= s f(k(t)) +(1- \delta) k(t) = k(t)\tag {1}.$$

Therefore, a steady state value $k^\*$ must satisfy, for $k\neq 0$$^1$:

$$\frac {f(k)}{k}= \frac {\delta}{s} \tag {2}$$

We can apply the intermediate value theorem to the function $\frac {f(k)}{k}$ above.

Here, the **Inada conditions** are crucial. In fact, applying the Inada conditions and the De l'Hôpital rule, we have:

$$\lim\_{k\rightarrow 0} {\frac {f(k)}{k}}= \lim\_{k\rightarrow 0}{\frac {f'(k)}{1}}= +\infty \tag {3}$$

$$\lim\_{k\rightarrow +\infty} {\frac {f(k)}{k}}= \lim\_{k\rightarrow 0}{\frac {f'(k)}{1}}= 0,\tag {4}$$

where the second equalities of equations $(3)$ and $(4)$ are the Inada conditions.

The **intermediate value theorem** says that a continuous function on an interval $I$ takes all the values between its $\inf$ and its $\sup$.$^2$

Here, $\inf {\frac {f(k)}{k}}=0$ and $\sup {\frac {f(k)}{k}}=+\infty$, so that $\frac {f(k)}{k}$ takes all values in $\mathbb{R}^+$, **including $\frac {\delta}{s}$**

Therefore, a steady state equilibrium exists.

$$\*\*\*$$

However, I can’t understand why you said:

> and $g(k') \neq g(k'')$ for $k' \neq k''$.

This means just that the function $g$ is injective, what we already know as it is strictly increasing.

But what has this to do with existence and intermediate value theorem? The intermediate value theorem doesn’t require the function to be injective, and the proof above will hold even if $g$ wouldn't be injective.

Consider the usual graph, in which we have a not injective $g$: this evidently does not affect the existence of a steady state, but only its unicity:

[![enter image description here](https://i.sstatic.net/ZVb8Z.png)](https://i.sstatic.net/ZVb8Z.png)
$$\;$$

**Uniqueness**

Your statement concerning uniqueness is correct, we in particular observe that, because of the strict monotonicity of $g(k)$ (and $g''<0$), for $k $g(k)>k$, and for $k>k^\*$ $g(k)>k$.
Therefore, the only point in which $g(k)=k$ is $k^\*$.

**Stability**

A qualitative argument to establish stability is to consider that if $k(t)$ is above $k^\*$, it decreases, and if it is below $k^\*$ it increases, and this is related to diminishing returns of capital.

A formal, analitycal proof of stability, however, requires theorems concerning the stability of discrete dynamical systems, much more complicated.

---

$^1$  Of course, there is a trivial fixed point, for $k=0$, but we discard it as of no economic interest, and moreover unstable.

$^2$ I specify this version of the intermediate value theorem, with $\inf$ and $\sup$, because often we find a different version, where the values of the function at the extreme of a closed interval $[a,b]$ are specified, but the two versions are equivalent.

> **Comments:**
> **username_1**: Hi! You seem to be proving that the equilibrium exists and is unique. Where do we see global stability?  
> **username_2**: @Giskard, if you read the question, it is about existence and uniqueness, stability is in the title only. There is no question in the text about stability. Maybe the OP asks for proofs of existence and uniqueness as a prerequisite for studyng stability? I don't know.  
> **username_1**: Hmm, what is this then? "*I attempt to prove global stability as follows:*"  
> **username_1**: I agree that the OP seems to be confusing existence and uniqueness with global stability in his attempt, but they indicate both in the title and in the body that they are after global stability.  
> **username_2**: It is a mistake.  The OP says, I try to prove prove stability', but actually goes on talking about existence and uniqueness, and asking about them, as you can see also in the comments. .The reference to intermediate value theorem is cleraly related to existence, no internmediate value is involved in proofs of stability.  
> **username_2**: In the comments the OP makes explicit that they is asking about existence and uniqueness. Otherwise the whole question makes no sense.  
> **username_1**: Can you please link to the comment where they do that? I only see [this one](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95821_57806) where they write (emphasis by me) "**global stability** comes from the diminishing returns of capital."  
> **username_1**: The OPs proof and goal are not compatible, it is true. But it is quite surprising that you see in their soul that hey are mistaken when they state the objective, not when they try to prove it.  
> **username_2**: I refer to this comment: "What I was trying to say is that *the existence* of a fixed point is guaranteed by the intermediate value theorem. The * uniqueness* of ∗ is guaranteed by the fact that () is strictly increasing, and the global stability comes from the diminishing returns of capital. Are these statements correct?"  
> **username_1**: I quoted the same comment. Do you not see the part on global stability in it...?  
> **username_1**: You write "*Otherwise the whole question makes no sense.*" Why would a question asking how to prove global stability in this model make no sense?  
> **username_2**: The part about stability is in this comment only and not in the text of the question. In the question stability is only formally, and erroneusly, present, but then the question is only about existence and uniqueness. All the OP says in is 'proof' has nothing to do with stability.  
> **username_1**: Yeah, they fail to prove stability. How do you know that was not their goal though? They write several times that is what they were trying to do.  
> **username_2**: Nothing in the text of the question is related to stability, for this reason if it were about stability would make no sense.  
> **username_1**: Nothing in the proof is related to it. They are asking if it is correct or not though, so...  
> **username_2**: The OP says it in the notes, and it is clear that is talking about the *usual* arguments of the proofs of existence and uniqueness, that they probably studied in a course.  
> **username_1**: Do you teach? If yes, maybe you notice that students frequently use "usual" arguments for things where they do not really work? Left OP a [message](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95840_57806), maybe they will clarifiy.  
> **username_2**: @Giskard, yes, I have a long experience of teaching, and this is my interpretation of the question. If you don't agree, there is no problem.  
> **username_1**: You have a deep trust in others' understanding of this proof and what it shows. I like that!  
> **username_1**: Hey, why did you add a section on stability 20 minutes after my last comment since you are 100% sure that is not what the question is about, and if it was about that it would be meaningless  
> **username_3**: **Comments have been [moved to chat](https://chat.stackexchange.com/rooms/151800/discussion-on-answer-by-bakerstreet-proving-the-global-stability-in-the-solow-sw); please do not continue the discussion here.** Before posting a comment below this one, please review the [purposes of comments](/help/privileges/comment). Comments that do not request clarification or suggest improvements usually belong as an [answer](/help/how-to-answer), on [meta], or in [chat]. Comments continuing discussion may be removed.  
> **username_1**: @1muflon1 I'm pretty sure my comments suggested an improvement, the answer got a new section after I posted them. Since I am not credited in it, maybe you should bring back some of my comments?  
> **username_2**: @Giskard I added the part about stability just to avoid misunderstandings, after your comments, I dind't write it before because I thought that it was not really in the question, and also because it wasn't a very interesting part (unless one studies the theory of stability of discrete dynamical systems),  rather obvious from the graph of the Solow model. Much more 'substantial' , and useful, are the proofs of existence and uniqueness.  
> **username_3**: @giskard  I brought back the comments, please continue the discussion in the chat  
> **username_0**: I made the question more clear. Sorry for the confusion  
> **username_1**: @BakerStreet Care to guess which one of us had the right interpretation? https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95851_57806  
> **username_1**: "*wasn't a very interesting part (unless one studies the theory of stability of discrete dynamical systems)*" is a funny comment on a question asking about global stability in a discrete dynamical system.  
> **username_2**: @Giskard The question in the mind of the OP was about stability, but before Maxilian edited the question it was impossible to undertsand it clearly,  Maximilian has completely changed the question, incorporating also  aspect of my answer, that clearly answered to their questions about existence,  as you can see in the comments.  The question was not clear, that's all.  
> **username_1**: "*it was impossible to undertsand it clearly*" and yet I managed to make the correct guess. "*The question was not clear, that's all.*" Indeed. And yet you were so certain. Perhaps we are all Watsons in some way every now and then, and that is okay?  
> **username_2**: @Giskard, you don't even try to understand what one is saying. The part of stability, without studying dynamical systems, reduces to write arrows on the graph, and establish the sign of the growth of $k(t)$, which Maximilian does also with formulas. But this is a partial and intuitive explanation, not a real sufficient proof based on the theory of dynamical systems.  
> **username_2**: @Giskard, I was just reading the actual text, and what the OP said in the comments, that confirmed the questions about existence and uniqueness . We don't agree, that's all.  
> **username_1**: That is not all, sir, but I have no further arguments. Good night to you!  

---

### Answer by username_1 on 2024-03-02. Score: 2

A unique fixed point may not be stable. For an example, check the dynamic system
$$
x\_{t+1} = 2x\_t,
$$
where the unique fixed point is $x^\* = 0$, but this is not stable, in fact, starting from any point other than 0 you will get a trajectory that diverges from the fixed point. To prove global stability, you need to show that no trajectories diverge from the fixed point, show that their distance is not increasing.
  
  
<https://en.wikipedia.org/wiki/Stability_theory#Overview_in_dynamical_systems>
  
  
The same equations can be used, but this is what you need to prove and should be working towards.


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