Timed Petri Nets” who propose using priced timed Petri nets, and Gradisar & Music on “Automated Petri-net modelling based on production management data”, who construct a timed Petri net simulator using their own MatLab-based software routines.

However, as you point out, long-run equalization of the rate of profit would have to be achieved using some kind of stock adjustment (difference equation) process rather than a Petri net so we are looking at a kind of hybrid system (though one that is still discrete and recursive). You can see why I have been grappling with this modelling issue for some time.

Gradual adjustment also means that each sector would be using a different rate of profit for the spot pricing of its outputs and this would feed in to other users of these outputs(with each represented by individual transitions) as inputs into their own production.

Finally, the presence of economies or diseconomies of scale would also influence unit costs of production! This also means, as Sraffians recognize, that the Perron-Frobenius conditions on the input-output matrix will change with shifts in the composition of output that are induced by changes in income distribution. Even a linear multi-sectoral economy is a pretty complex entity!

I suppose this would also upset John’s ‘quantum stochastics’ solution to stochatic Petri nets (which use Matrix exponentiation and the Perron-Frobenius condition). Any further suggestions on your part would be much appreciated!

]]>Or do we simplify, and use monetary units for all of the commodities…

]]>I don’t have enough knowledge of economics to make a substantive assessment of the feasibility of your idea.

That given, I did make an attempt to imagine the kind of Petri net you are describing. I didn’t get very far, but here is the thought process:

What are the places in this economic Petri net? In the Bonanno paper, production processes are described by transitions, and the places correspond to definite commodities in the network — each transition consumes a certain amount of its input commodities, and generates a certain amount of its output commodities.

Now you want to add the profit generated by the firing of each transition. So in a first approximation, that would be an additional output arc, that adds a quantity of profit into some accumulation node. These nodes would be…bank accounts of the owners of the process? That would mean using many micro-instances of a transition, say one for each factory owner. Or could it be an aggregate profit account.

But this raises the questions of modeling circulation. The profit will get reinvested in more commodities, and how would that be represented in the Petri net.

Next, you want to introduce sectoral rates of profit. Will these be places in the Petri net? Each firing of a transition will then result in a variable amount of tokens being transferred to the output account — the product of the sector rate times the input value. How is input value to be represented in the net? Are there places representing prices in the net. But then how do we accomplish multiplications in a Petri net, e.g., the value of input A is the product of the number of tokens there times the price of A. Similarly for the multiplication by the sectoral rate of profit.

This suggests that a more powerful model of computation may be needed, closer to a general dataflow graph — say something like where the firing of a transition leads to the evaluation of a formula, whose results then get deposited at the output places.

Next, I haven’t even begun to picture how the equalization of sectoral profits would be represented by the net. Can you give an references to a good computational model for this equalization process — that could serve as a point of departure for thinking about Petri-netification of this part of the process.

Anyone care to make another foray into this topic, maybe you can get further than I did here, or add some clarification.

]]>Second, I would like to point to the fact that Piero Sraffa included in the title of his book “Prelude to a Critique of Economic Theory”. This work, together with earlier complaints by Joan Robinson, launched the debates on capital theory. Amongst other things, Sraffa shows that in a multisectoral economy, with differing ratios of dead to living labour in each sector, there will be no simple monotonic functional relationship between the rate of profit and the capital intensity of chosen production techniques.

Sraffa used linear algebra and the Perron-Frobenius theorems to establish his findings, but there are different orders of temporality at play in his model: capital flows between sectors to equalize the rate of profit while economies and diseconomies of scale come into play. These processes are clearly longer-term, while changes in the distribution of income between wages and profits are of a shorter-term nature. This is exactly where a coalgebraic approach could come into play.

Here, I am not thinking so much of hybrid systems (i.e. discrete versus continuous) but more a case where you have a nesting of automatons or Petri nets within other nets to accommodate processes operating in accordance with different orders of temporality. Needless to say, this will complicate things significantly in regard to standard measures of liveliness, reachability, safety etc.

As Mirowski points out, most applications of automatons in economics have been designed to account for such things as bounded rationality in a game theory context, or in his case, for the evolution of different market forms: single-auction, double-auction etc! Instead, I want to use process algebraic techniques to remodel dynamics processes at work in the Classical economics of Ricardo and Marx.

]]>Thanks for the pointer to Bonnano’s paper. It seems he only uses Petri nets to say how certain outputs can be made from certain inputs, not to model *dynamics*, that is, how things change with time. Of course for dynamic one wants something more like a stochastic Petri net. Without dynamics one tends to focus on questions akin to the ‘reachability problem’: that is, starting with some collection of things, which other things can you eventually make? There’s been a huge amount of work on this in the Petri net literature, which I briefly sketched here.

With my background in physics, I’m drawn toward questions of dynamics.

Linear input-output models were the stock in trade of development modeling in the 1950 but have been largly “superceded” by Computerised General Equilibrium models. Many National Statistics Bureaus around the world publish I-O accounts.

I think I’ve heard of Leontief models being used to make economic predictions… but quickly reading a bit abou tthis, I get the feeling there’s no true dynamics here, just a linear map sending a vector of inputs to a vector of outputs, which could let you predict the amount of outputs if you knew the vector of inputs (if the model were right). Is that right?

There must be a huge hunger for going beyond ‘equilibrium’ models, no? I’m imagining a world where weather forecasters only had equilibrium models, and thinking about how unsatisfactory this would be.

But no one I know of has modelled Sraffa’s (1960) “Production of Commodities by Means of Commodities” using coalgebras.

I don’t know his work, either. Could you say a word about why coalgebras could be relevant?

]]>And references to the same authors in regard to automatons can be found in P. Mirowski & K. Somefun’s 1998 paper “Towards an Automata approach of Institutional (and Evolutionary) Economics” available at:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.131.7417&rep=rep1&type=pdf

But no one I know of has modelled Sraffa’s (1960) “Production of Commodities by Means of Commodities” using coalgebras. Linear input-output models were the stock in trade of development modeling in the 1950 but have been largly “superceded” by Computerised General Equilibrium models. Many National Statistics Bureaus around the world publish I-O accounts. ]]>

Hi! Your idea sounds interesting, but I don’t know enough economics to understand in detail what you’re talking about. Can you describe a tiny example of the kind of model you’re imagining?

I’d also enjoy seeing an example of one of von Neumann’s ‘input-output systems’. Presumably there are examples lurking around the web.

]]>Blake wrote:

Was Jost’s critique something along those lines?

Yes. I forget precisely which biodiversity measure was used to derive this ‘one migration per generation’ rule, but it wasn’t an effective number, and I think Jost or someone showed that if you use an effective number this rule changes to something more plausible.

(It’s not plausible that for two arbitrarily large populations of birds on separate islands, one bird flying from one island to the other per generation is enough to keep them from genetically diverging and becoming separate species!)

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