Seneca Cliff

Esset aliquod inbecillitates nostrae solacium rerumque nostrarum si tam tarde perirent cuncta quam fiunt: nunc incrementa lente exeunt, festinatur in damnum.

Seneca: Epistulae Morales, Liber XIV, XCI

Sarebbe una consolazione per la nostra debolezza  e per i nostri beni se tutto andasse in rovina con la stessa lentezza con cui si produce e, invece, l'incremento è graduale, la rovina precipitosa.  

It would be some consolation for the feebleness of our selves and our works, if all things should perish as slowly as they come into being; but as it is, increases are of sluggish growth, but the way to ruin is rapid.
The Seneca Effect
Neverending growth based on finite physical resources is not possible: this obvious concept has been the object of extensive investigation in the book The Limits to Growth. An interesting qualitative aspect of the simulations reported and analyzed in the book, is that many quantities exhibit a characteristic trend of growth followed by decline, with the steepnes of the two being unequal, the latter often being steeper than the former: the so called Seneca Effect.
An example of a simple model describing a symmetric growth-decline pattern was provided by Hubbert and it was found to be quantitatively accurate in the description of some phenomena among which crude oil production in the 48 US lower states. It is of some interest to create similarly simple models capable of exhibiting the asymmettric behaviour exposed by the complex simulation model employed in the book The Limits to Growth. Such a simple model was introduced not too long ago by prof. Bardi and can be used to experience some interesting scenarios depicting the evolution of economic growth in our energy (and material) thirsty society. Bardi's model is defined by the following differential equations:
\begin{eqnarray} \frac{d P}{d t}  & = & k_2 E P - l_3 P\\ \frac{d E}{d t}  & = & k_1 E R - l_2 E - k_2 E P\\  \frac{d R}{d t}  & = & - l_1 E R - k_1 E R\end{eqnarray}
which link the dynamics of three reservoirs or stocks:
  • $R$: the set of (non renewable) resources foraging the economy;
  • $E$: the economy (or capital) transforming the resources and creating pollution;
  • $P$: pollution.
Wording the model:
Now, can we say in words what is that generates the Seneca cliff? Yes, we can. It goes like this: first, consider that the effect of pollution is to drain economic capital. Secondly, consider that the pollution stock grows by feeding on the economy stock - so it has to wait for the economy to have grown before it can grow itself. It is this delay that causes an increase in the rate of energy draining from the economy as the process goes on. Since the size of the economy stock determines the production rate, we see also that parameter going down rapidly after the peak. This is the essence of the Seneca effect.
Ugo Bardi
The model is controlled by five parameters: two of them ($k_1$, $k_2$) define the direct couplings between stocks while the remaining one ($l_1$, $l_2$, $l_3$, where $l$ stands for loss) describe the flows going to outer space.
Following the detailed description and graphical presentation provided by Bardi the model can be implemented as a Vensim model (and run through Vensim PLE, the free evaluation and educational version of the simulation software). The attached file seneca.mdl is a sample Vensim implementation.
The Seneca app
A simple Android app enables the curious reader to easily change the future, albeit the simulated one, by changing the parameters controlling the simulation. The app visualizes the following quantities
  • $R$: resources;
  • $E$: economy;
  • $P$: pollution;
  • $-dR$: growth, the amount of extracted resources at a given time sustaining economic growth.  

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seneca.mdl3.61 KB