<p>The economic essence of saddle-path stability, is that it requires <em>conscious behavior and decision making from the part of economic agents</em>, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, economic systems are unstable.</p>

<p>Contrast this with the basic Solow growth model: there, fixing arbitrarily the savings rate (i.e. depriving ourselves from making a decision on it period per period) makes the system properly stable. But once we want to optimize with respect to the savings rate, we find out that there is a single path that leads to long-term equilibrium.</p>

<p>Now, perusing the pages of a book like Barro and Sala-i-Martin's <a href="https://mitpress.mit.edu/books/economic-growth" rel="nofollow">"Economic Growth"</a>, which contains many-many extensions of the basic RCK growth model, one will <em>not</em> find a model with intertemporally optimizing agents that is also mathematically stable.</p>

<p><strong>ADDEDNDUM</strong><br>
Elaborating, in response to a thoughtful comment by the OP, my approach to the issue of "stability" of an economy is the following: The fact that economies tend to "correct course" is <em>not</em> an indication, let alone proof, that they posses some inherent stabilization mechanism <em>independent</em> of the purposeful action of the economic agents.<br>
If this was the case, then indeed, we should have models that possessed the mathematical property of stability (I remind again that "saddle-path stable" although it uses the word "stable", is a mathematically unstable system). </p>

<p>On the other hand, economies do not plunge into chaos every day. The solution to the puzzle from my point of view is not that we "continuously fine tune the economy" -such dreams and visions have been tried and squashed (irrespective of what ideological/philosophical (dis) agreements one may have with socialism, the factual lesson to be learned from the failure of centrally planned economies is that an economy is such a complex system that no one can "plan and then fine-tune it", especially in our era).<br>
My answer is that economies, after being hit by external shocks or internal institutional failures or technology upheavals, or.., tend to return to a dynamic path that tends to some fixed point by the aggregate consequences of decentralized purposeful action of economic agents that try not to fix the economy as a whole, but their own economic present and future. Economies survive not "because they inherently do", not because the government is "steering the boat with a certain and effective hand", but because individuals try hard every day to survive.  </p>

<p>And the RCK models reflect just that: if one performs comparative statics on it, assuming an external shock that suddenly puts the economy out of the saddle-path, the prediction of the model as to what will happen <em>in order</em> for the economy to return to the (new) saddle path towards the (new) long-run equilibrium, is more or less what we observe in reality. But this happens not because the government makes it so, but because individual economic agents realize that they must do so in order not to eventually self-destruct.</p>

<p>I will indulge myself and I will link to a little exposition I crafted a year ago for a class of graduate students, to show them how the simplest RCK model can help them think about the economic crisis (and subsequent depression) in Greece -which was sudden enough to be described as a shock. The exposition can be found <a href="https://alecospapadopoulos.wordpress.com/2015/10/31/ramsey-on-the-greek-crisis-and-depression/" rel="nofollow">here</a>.</p>

CC BY-SA 3.0
2016-10-03T07:02:11.720Z
Your description of the models is fair, and in particular, much more precise than most textbooks.  Unfortunately, this fair interpretation of R-C-K neoclassical models also shows that they are not  realistic. The prediction of these models is that we continuously fine tune our whole economy in order to avoid disaster, which, barring extremely precise and immediate action, will fall into a bottomless pit far from optimality. This is not realistic. Even under strange policy circumstances, economies can show surprising stability. In summary : your summary is realistic, the models are not.
tomatosoup
CC BY-SA 3.0
2016-10-03T09:08:10.187Z
This second comment is on the thoughtful addendum of Alecos Papadopoulos. Again, I agree with the realistic facts invoked in the addendum. There is no continuous fine tuning, and individuals attempt to survive, and the economy as a whole does. The crucial difference in attitude is that I imagine a model in which the stabilizing effect of the actions of individuals is modeled as an aggregate effect, which (apparently, realistically, often) stabilizes the economy. In my eyes, this should be part of the theory of economics. It should be included in a good model, which then is stable (often) .
tomatosoup
CC BY-SA 3.0
2016-10-03T09:10:05.870Z
More technically, I wish to describe a model that would result (!) from "performing comparitive statics" (see Addendum), without having to perform this further operation.
tomatosoup
CC BY-SA 3.0
2016-10-01T17:11:54.107Z
2016-10-03T08:48:09.853Z
2016-10-03T08:48:09.853Z
Alecos Papadopoulos
<p>An economy described by the Ramsey-Cass-Koopmans model obeys a mathematically stable 1D dynamical system.</p>
<p>Here's the key point that I think will resolve the confusion: The Hamiltonian dynamical system for the consumer’s utility maximization (which is a 2D dynamical system with an unstable &quot;saddle-path&quot;) does not describe the dynamics of the economy. Rather, it arises merely as an intermediate step in solving the consumer’s utility maximization problem.</p>
<p>Using the notation from <a href="https://en.wikipedia.org/wiki/Ramsey%E2%80%93Cass%E2%80%93Koopmans_model" rel="nofollow noreferrer">Wikipedia</a>, the dynamics of per capita capital are given by:
<span class="math-container">$$\dot k = f(k) - (n+\delta)k - c$$</span></p>
<p>By maximizing utility, the consumer's consumption at time <span class="math-container">$t$</span> is a function of the amount of capital at time <span class="math-container">$t$</span>. Let <span class="math-container">$\tilde c(k)$</span> denote the utility-maximizing choice of <span class="math-container">$c$</span>, given <span class="math-container">$k$</span>. Mathematically, <span class="math-container">$\tilde c(k)$</span> is given by the saddle path (i.e. the trajectory of the Hamiltonian dynamical system for the optimization problem that converges to the saddle point, see <a href="https://en.wikipedia.org/wiki/Hamiltonian_(control_theory)#Example:_Ramsey%E2%80%93Cass%E2%80%93Koopmans_model" rel="nofollow noreferrer">Wikipedia</a>). More specifically, if <span class="math-container">$(c_s(t),k_s(t))$</span> is the &quot;saddle-path&quot; trajectory of the Hamiltonian dynamical system, then <span class="math-container">$\tilde c(k) = c_s(k_s^{-1}(k))$</span> where <span class="math-container">$k_s^{-1}(k)$</span> is the inverse function of <span class="math-container">$k_s(t)$</span>.</p>
<p>Then, the economy obeys
<span class="math-container">$$\dot k = f(k) - (n+\delta)k - \tilde c(k)$$</span>
which is a 1D dynamical system with a stable fixed point.</p>

CC BY-SA 4.0
2024-06-29T06:53:15.153Z
2024-06-29T06:53:15.153Z
C.M.O.B.
<p>The Ramsey-Cass-Koopmans neoclassical model of growth is saddle path stable. In other words, it is stable upon perturbation from the steady state on a one-dimensional line, but is unstable towards a generic perturbation in the two-dimensional space of capital/labor ratio and consumption.</p>

<p>My question is whether there are (realistic, standard) neoclassical growth models that are stable upon perturbation (beyond the simplest Solow type models) ?</p>

<macroeconomics><economic-growth><dynamic-optimization>
Are there stable improvements of the Ramsey-Cass-Koopmans model?
tomatosoup
# Are there stable improvements of the Ramsey-Cass-Koopmans model?

(Asked by: username_0 on 2016-03-19)

The Ramsey-Cass-Koopmans neoclassical model of growth is saddle path stable. In other words, it is stable upon perturbation from the steady state on a one-dimensional line, but is unstable towards a generic perturbation in the two-dimensional space of capital/labor ratio and consumption.

My question is whether there are (realistic, standard) neoclassical growth models that are stable upon perturbation (beyond the simplest Solow type models) ?

**Tags:** macroeconomics, economic-growth, dynamic-optimization


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## 2 answers

### Answer by username_1 on 2016-10-01. Score: 1

The economic essence of saddle-path stability, is that it requires *conscious behavior and decision making from the part of economic agents*, in order for the system to remain on the saddle path. Simply put, we cannot do "whatever we like" and still expect the system to settle down. But that is what makes it interesting (and true to reality): on their own, economic systems are unstable.

Contrast this with the basic Solow growth model: there, fixing arbitrarily the savings rate (i.e. depriving ourselves from making a decision on it period per period) makes the system properly stable. But once we want to optimize with respect to the savings rate, we find out that there is a single path that leads to long-term equilibrium.

Now, perusing the pages of a book like Barro and Sala-i-Martin's ["Economic Growth"](https://mitpress.mit.edu/books/economic-growth), which contains many-many extensions of the basic RCK growth model, one will *not* find a model with intertemporally optimizing agents that is also mathematically stable.

**ADDEDNDUM**  
Elaborating, in response to a thoughtful comment by the OP, my approach to the issue of "stability" of an economy is the following: The fact that economies tend to "correct course" is *not* an indication, let alone proof, that they posses some inherent stabilization mechanism *independent* of the purposeful action of the economic agents.  
If this was the case, then indeed, we should have models that possessed the mathematical property of stability (I remind again that "saddle-path stable" although it uses the word "stable", is a mathematically unstable system).

On the other hand, economies do not plunge into chaos every day. The solution to the puzzle from my point of view is not that we "continuously fine tune the economy" -such dreams and visions have been tried and squashed (irrespective of what ideological/philosophical (dis) agreements one may have with socialism, the factual lesson to be learned from the failure of centrally planned economies is that an economy is such a complex system that no one can "plan and then fine-tune it", especially in our era).  
My answer is that economies, after being hit by external shocks or internal institutional failures or technology upheavals, or.., tend to return to a dynamic path that tends to some fixed point by the aggregate consequences of decentralized purposeful action of economic agents that try not to fix the economy as a whole, but their own economic present and future. Economies survive not "because they inherently do", not because the government is "steering the boat with a certain and effective hand", but because individuals try hard every day to survive.

And the RCK models reflect just that: if one performs comparative statics on it, assuming an external shock that suddenly puts the economy out of the saddle-path, the prediction of the model as to what will happen *in order* for the economy to return to the (new) saddle path towards the (new) long-run equilibrium, is more or less what we observe in reality. But this happens not because the government makes it so, but because individual economic agents realize that they must do so in order not to eventually self-destruct.

I will indulge myself and I will link to a little exposition I crafted a year ago for a class of graduate students, to show them how the simplest RCK model can help them think about the economic crisis (and subsequent depression) in Greece -which was sudden enough to be described as a shock. The exposition can be found [here](https://alecospapadopoulos.wordpress.com/2015/10/31/ramsey-on-the-greek-crisis-and-depression/).

> **Comments:**
> **username_0**: Your description of the models is fair, and in particular, much more precise than most textbooks.  Unfortunately, this fair interpretation of R-C-K neoclassical models also shows that they are not  realistic. The prediction of these models is that we continuously fine tune our whole economy in order to avoid disaster, which, barring extremely precise and immediate action, will fall into a bottomless pit far from optimality. This is not realistic. Even under strange policy circumstances, economies can show surprising stability. In summary : your summary is realistic, the models are not.  
> **username_0**: This second comment is on the thoughtful addendum of username_1. Again, I agree with the realistic facts invoked in the addendum. There is no continuous fine tuning, and individuals attempt to survive, and the economy as a whole does. The crucial difference in attitude is that I imagine a model in which the stabilizing effect of the actions of individuals is modeled as an aggregate effect, which (apparently, realistically, often) stabilizes the economy. In my eyes, this should be part of the theory of economics. It should be included in a good model, which then is stable (often) .  
> **username_0**: More technically, I wish to describe a model that would result (!) from "performing comparitive statics" (see Addendum), without having to perform this further operation.  

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### Answer by username_2 on 2024-06-29. Score: 0

An economy described by the Ramsey-Cass-Koopmans model obeys a mathematically stable 1D dynamical system.

Here's the key point that I think will resolve the confusion: The Hamiltonian dynamical system for the consumer’s utility maximization (which is a 2D dynamical system with an unstable "saddle-path") does not describe the dynamics of the economy. Rather, it arises merely as an intermediate step in solving the consumer’s utility maximization problem.

Using the notation from [Wikipedia](https://en.wikipedia.org/wiki/Ramsey%E2%80%93Cass%E2%80%93Koopmans_model), the dynamics of per capita capital are given by:
$$\dot k = f(k) - (n+\delta)k - c$$

By maximizing utility, the consumer's consumption at time $t$ is a function of the amount of capital at time $t$. Let $\tilde c(k)$ denote the utility-maximizing choice of $c$, given $k$. Mathematically, $\tilde c(k)$ is given by the saddle path (i.e. the trajectory of the Hamiltonian dynamical system for the optimization problem that converges to the saddle point, see [Wikipedia](https://en.wikipedia.org/wiki/Hamiltonian_(control_theory)#Example:_Ramsey%E2%80%93Cass%E2%80%93Koopmans_model)). More specifically, if $(c\_s(t),k\_s(t))$ is the "saddle-path" trajectory of the Hamiltonian dynamical system, then $\tilde c(k) = c\_s(k\_s^{-1}(k))$ where $k\_s^{-1}(k)$ is the inverse function of $k\_s(t)$.

Then, the economy obeys
$$\dot k = f(k) - (n+\delta)k - \tilde c(k)$$
which is a 1D dynamical system with a stable fixed point.


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