samuelson and davidson on ergodicity

© 2014 Dirk Ehn

ts Samuelson and Davidson on

ergodicity: a reformula>on

Dirk Ehnts, Free University Berlin Miguel Carrión Álvarez, Banco Santander Madrid

12th International Post-Keynesian Conference (UMKC)

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ts

Samuelson and Davidson on ergodicity: a reformula>on

September 26th 2014 Samuelson and Davidson on ergodicity: a reformula>on 2

Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the “ergodic hypothesis” by analogy to the use of this term in statistical mechanics.

Paul Samuelson, 1969

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Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the “ergodic hypothesis” by analogy to the use of this term in statistical mechanics.

Furthermore in an article published in 1969 Samuelson argued that the „ergodic hypothesis [axiom]“ is a necessary foundation if economics is a hard science [Samuelson, 1969, p. 184].

Paul Samuelson, 1969

Paul Davidson, 2006

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ts

Ergodicity in a nutshell: If you throw six dice at one point in >me and the average is the same as throwing a die six >mes in a row, then the ensemble average and the 'me average are iden>cal – the system is ergodic. In [non-‐]ergodic system, you can[not] derive the probabili>es of a single future event just by looking at the past historical development of a single instance of the system.

14. April 2014 Neuere Ansätze der Geldtheorie 4


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„by analogy to the use of this term in sta>s>cal mechanics“: Hence, to the mythical ques>on of whether the ergodic hypothesis jus>fies sta>s>cal mechanics, the answer is worse than no: it is not the right ques>on. Indeed, the ques>on cannot be anymore whether nature strictly obeys the demands of ergodicity; rather, the ques>on ought to ask how good an idealiza>on the theory really is. Emch and Liu (2001, §9.5)



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Economics: ergodicity as shibboleth? hfp://www.merriam-‐webster.com/dic>onary/shibboleth



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“[I]t is possible, but not necessarily assumed, that an ergodic state for P [the probability distribution] will emerge in the limit as T [time] goes to infinity.” Samuelson (1965, 43)

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“Finally, there was an even more interesting third assumption implicit and explicit in the classical mind. It was a belief in unique long-run equilibrium independent of initial conditions. I shall call it the “ergodic hypothesis” by analogy to the use of this term in statistical mechanics. Remember that the classical economists were fatalists (a synonym for “believers in equilibrium”!) . Harriet Martineau, who made fairy tales out of economics (…), believed that if the state redivided income each morning, by night the rich would again be sleeping in their comfortable beds and the poor under the bridges. (I think she thought this a cogent argument against egalitarian taxes.)” Samuelson (1968, 11-12)

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“Now, Paul Samuelson, aged 20 a hundred years later, was not Harriet Martineau or even David Ricardo; but as an equilibrium theorist he naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theorists hoped not to introduce hysteresis phenomena into our model, as the Bible does when it says “We pass this way only once“ and, in so saying, takes the subject out of the realm of science into the realm of genuine history. Specifically, we did not build into the Walrasian system the Christian names of particular individuals, because we thought that the general distribution of income between social classes, not being critically sensitive to initial conditions, would emerge in a determinate way from our equilibrium analysis.”

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In 1968, MIT Professor and later Nobel Prize winner Paul Samuelson wrote that in their quest to provide a hard scientific basis for the economics discipline modern economists must believe in a ‚unique long run equilibrium [i.e., an inevitable outcome for the economy] independent of the initial conditions’. Davidson (1996, 65)

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Indeed, Samuelson (1969, p. 184) has made the acceptance of the „ergodic hypothesis“ the sine qua non of the scientific method in economics. [Samuelson (1969, p. 184) indicated that he used the term ergodic „by analogy to the use of this term in [19th century] statistical mechanics“ in order to remove economics from the „realm of genuine history,“ and keep it in the „realm of science.“] Davidson (2006,11)

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“Now, Paul Samuelson, aged 20 a hundred years later, was not Harriet Martineau or even David Ricardo; but as an equilibrium theorist he naturally tended to think of models in which things settle down to a unique position independently of initial conditions. Technically speaking, we theorists hoped not to introduce hysteresis phenomena into our model, as the Bible does when it says “We pass this way only once“ and, in so saying, takes the subject out of the realm of science into the realm of genuine history.“ … ‘having once been a jackass’ ... “What Classical and Neoclassical Monetary Theory Really was” (1968), Paul Samuelson (1915-2009)

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Uncertainty and ergodicity Davidson has often stressed (fundamental/true) uncertainty in his writings. However, there are ergodic systems which are completely deterministic but do not allow predictions to be made. These chaotic systems are part of the real world! While terms like "ergodic", "stationary", "conservative", "predictable" or "deterministic" have natural language meanings, they are also overloaded from dynamical systems theory and practice where their meanings are most definitely not coterminous.

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Uncertainty and ergodicity We believe that a different term should be used to describe the ‘non-predictability of the future’ that is compatible with un-/certain but also compatible with non-/ergodic.

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Stochastic (and non-stochastic): 1660s, "pertaining to conjecture," from Greek stokhastikos "able to guess, conjecturing," from stokhazesthai "to guess, aim at, conjecture," from stokhos "a guess, aim, target, mark," literally "pointed stick set up for archers to shoot at," from PIE *stogh-, variant of root *stegh- "to stick, prick; pointed" (see sting (v.)). The sense of "randomly determined" is from 1934, from German stochastik (1917). Online Etymology Dictionary

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“Speaking of randomness in the ordinary sense of this word, we mean those phenomena in which we do not find regularities allowing us to predict their behavior. Generally speaking, there are no reasons to assume that random in this sense phenomena are subject to some probabilistic laws. Hence, it is necessary to distinguish between randomness in this broad sense and stochastic randomness (which is the subject of probability theory).”

Kolmogorov (1983, first paragraph)

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Davidson (2007, 102): ‚In a wider sense, however, ergodicity means the presumption of a preprogrammed stable, conservative system where the past, present, and future reality are predetermined whether the system is stochastic or not.’

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Davidson (2007, 102): ‚In a wider sense, however, ergodicity means the presumption of a preprogrammed stable, conservative system where the past, present, and future reality are predetermined whether the system is stochastic or not.’

“When I use a word,” Humpty Dumpty said in rather a scornful tone, “it means just what I choose it to mean — neither more nor less.” “The question is,” said Alice, “whether you can make words mean so many different things.” “The question is,” said Humpty Dumpty, “which is to be master—that's all.” Lewis Carroll (1797: Through the looking glass)

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Keynes (1921): probability of future events not enough to understand today’s prices! Demand for insurance (CDS, …) depends on ‘beauty contest’ (Keynes, GT) and this situation is one of reflexivity (Soros). AIG mattered, underwrote CDS market and got prices wrong when ‘specially large demand’ arose! In our terms: markets are described by non-stochastic processes!

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Book-making deals with non-stochastic randomness (or uncertainty, etc.) – to ‘protect the book’ is of utmost importance! Even in the case of fundamental uncertainty in the sense of Keynes and Davidson, it still makes sense to ‘protect the book’. Failures on a systemic level will be solved on a systemic level, hence individuals do not prepare for them! (citi’s Chuck Prince: “as long as the music is playing, you’ve got to get up and dance”)

samuelson and davidson on ergodicity

Economy & Finance