THE EPISTEMOLOGY OF SIMULATION,

COMPUTATION AND DYNAMICS IN ECONOMICS

K. VELA VELUPILLAI

ALGORITHMIC SOCIAL SCIENCES RESEARCH UNIT DEPARTMENT OF ECONOMICS U N I V E R S I T Y O F T R E N T O P R E P A R E D F O R A K E Y N O T E A D D R E S S A T T H E

4TH WORLD CONGRESS OF THE SOCIETY OF SOCIAL SIMULATION

NATIONAL CHENGCHI UNIVERSITY, TAIPEI, TAIWAN

4 – 7, SEPTEMBER, 2012

ABSTRACT

We identify and characterise five frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation, in its digital mode, play crucial roles in formal modelling exercises: computable general equilibrium theory (CGE) (and its ‘extensions’: Recursive Competitive Equilibrium (RCE) & Dynamic Stochastic General Equilibrium (DSGE) theories), agent based computational economics, computational economics, classical behavioural economics (CBE, as distinct from MBE: Modern Behavioural Economics) and computable economics. These five research frontiers broach, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modes. The main claim in the paper is that there is a serious epistemological deficit in all of the approaches, but can be ‘discovered’ only in the last two, precisely because the latter are underpinned by computability and constructivity theories, in their strict mathematical senses, and the former are not. However, this claim does not imply that classical behavioural economics and computable economics are ‘complete’ from an epistemological perspective, especially from the point of view of natural or intrinsic dynamics of formal models. The epistemological deficit and the epistemological incompleteness, it is suggested, can be resolved by a theory of simulation, itself based on recognising the ‘duality’ between dynamical systems and numerical analysis, interpreted computably. Hence paying heed to Turing’s Precept: “the inadequacy of ‘reason’ unsupported by common sense.”

FOUR SCENES FROM THE THEATRE OF THE ABSURD [PACE: MARTIN ESSLIN]

"Agent-based computational methods provide the only way in which the self-regulatory capabilities of complex dynamic models can be explored so as to advance our understanding of the adaptive dynamics of actual economies."

“If you didn’t grow it [the social phenomenon], you didn’t explain its emergence.”

“We use the term “emergent” to denote stable macroscopic patterns arising from the local interaction of agents.”

“The weakness of applying GE models to policy issues is twofold: First, they provide non-constructive rather than constructive proofs of the existence of equilibrium; that is, they show that equilibria exist but do not provide techniques by which equilibria can actually be determined. Second, existence per se has no policy significance. .. They can only be employed in this way if they can be made constructive (i.e., be used to find actual equilibria). The extension of the Brouwer and Kakutani fixed point theorems in this direction is what underlies the work of Scarf.... on fixed point algorithms ..."

EPISTEMOLOGY, COMPUTATION, DYNAMICS - ECONOMICS

What is a Computation?

Kant: What is Man?

What can I know?

What must I do?

What may I hope?

Every simulation is a computation;

Every computation is a dynamic process

What is the epistemological status of a dynamic process?

What can be computed? What can be simulated? [Philosophy]

What must be done [to compute, to simulate]? [Methodology]

What may we expect [from a computation, from a simulation]? [Epistemology]

FIVE KINDS OF ECONOMICS UNDERPINNED – OSTENSIBLY -

BY ‘THEORIES’ OF COMPUTATION

I. Classical Behavioural Economics (CBE)

II. Computable General Equilibrium Theory (CGE)

III. Computational Economics (CE)

IV. Computable & Constructive Economics (CCE)

V. Agent-Based Computational Economics & Finance (ABCEF) Remark 1: I divide behavioural economics into a ‘Modern’ and a ‘Classical’ version – MBE & CBE, respectively. CBE is computably underpinned (see Velupillai-Kao).

Remark 2: CGE forms the ‘CORE’ of RCE which, in turn, is the basis for DSGE

Remark 3: Endogenous Macrodynamics is, from the point of view of my current research, part of CCE.

Remark 4: See the following references for detailed discussions, critical comparisons and ‘precise’ definitions of the above:

• Vela Velupillai & Stefano Zambelli, Computing in Economics, Ch.12, in: The Elgar Companion to Recent Economic Methodology, ed.by John B. Davis and Wade D. Hands, Edward Elgar, Cheltenham, Glos., 2011.

• Vela Velupillai & Ying-Fang Kao, Origins and Pioneers of Behavioural Economics, Interdisciplinary Journal of Economics & Business Law, Vol. 1, #. 3, 2012.

• Vela Velupillai, Stefano Zambelli & Stephen Kinsella (eds.), Computable Economics Edward Elgar International Library of Critical Writings, # 259, Edward Elgar, Cheltenham, Glos., 2011.

FIVE KINDS OF COMPUTATION THEORIES

Varieties of Constructive Mathematics and Constructive Analysis (CMCA)

Computability Theory and Varieties of Computable Analysis (CTCA)

Interval Analysis (IA)

Real Computation (RC)

Numerical Analysis (NA) See:

K. Vela Velupillai, A Computable Economist’s Perspective on Computational Complexity, ch. 4, in: Handbook of Research on Complexity, ed. by, J. Barkley Rosser, Jr., Edward Elgar, Cheltenham, Glos., 2009.

Vela Velupillai & Stefano Zambelli, Computing in Economics, Ch.12, in: The Elgar Companion to Recent Economic Methodology, ed.by John B. Davis and Wade D. Hands, Edward Elgar, Cheltenham, Glos., 2011.

EPISTEMOLOGICAL DEFICITS & INCOMPLETENESS

ECONOMICS

CBE

CCE

CGE

CE

ABCEF

MODES OF COMPUTATION

CMCA

CTCA

IA

RC

NA

Every simulation is a computation

Every mode of computation is a dynamic process

What is the dynamical system status of Numerical Analysis?

?

TOWARDS AN EPISTEMOLOGY OF COMPUTATION –

HENCE OF SIMULATION AND DYNAMICS

• “Computer science ... is not actually a science. It does not study natural objects. Neither is it, as you might think, mathematics; although it does use mathematical reasoning pretty extensively. Rather, computer science is like engineering - it is all about getting something to do something, rather than just dealing with abstractions ... . ...But this is not to say that computer science is all practical, down to earth bridge-building. Far from it. Computer science touches on a variety of deep issues. ... . It naturally encourages us to ask questions about the limits of computability, about what we can and cannot know about the world around us.”

Richard Feynman: Feynman Lectures on Computation, Addison-Wesley Publishing Company, Inc., Reading,

Mass., p. xiii

UNDECIDABILITY OF ALGORITHMIC KNOWLEDGE-

INCOMPLETENESS OF TACIT KNKOWLEDGE

“I shall reconsider human knowledge by starting from the fact that we can know more than we can tell.”

[Michael Polanyi, Tacit Knowledge, p. 4; italics in the original]

“The principle here is that you can know a lot more than you can prove! Unfortunately, it is also possible to think you know a lot more than you actually know. Hence the frequent need for proof.”

[Richard Feynman, op.cit., p. 90; italics added]

Michael Polanyi’s caveat (ibid, p.7):

“We have here examples of knowing, both of a more intellectual and more practical kind; both the ‘wissen’ and ‘können’ of the Germans, or the ‘knowing what’ and the ‘knowing how’

of Gilbert Ryle. These two aspects of knowing have a similar structure and neither is ever present without the other.”

Algorithmic Knowledge and Tacit Knowledge are capable of being encapsulated in the

formalisms of CBE & CCE – but not in CGE, CE or ABCEF. Hence: the impossibility of

discussing epistemological deficits or incompleteness in CGE,CE or ABCEF.

This is because CGE, CE & ABCEF are not subject to Turing’s Precept 1: ‘[O]f the inadequacy of ‘reason’ unsupported by common sense.’

Epistemological Deficit

Epistemological Incompleteness

FIVE KINDS OF DYNAMICAL SYSTEMS IN

ECONOMICS

Ordinary Differential Equations (Linear & Nonlinear): ODEs – Standard & Nonstandard.

Partial Differential Equations (Linear & Nonlinear): PDEs.

Stochastic Differential Equations (Linear, Nonlinear and Partial) SDEs.

Mixed Dynamic Difference-Differential Equations.

Difference Equations (Linear & Nonlinear). Remark 1:The above is a ‘descriptive’ delineation. I am able to ‘reduce’ them to two classes – perhaps even one – s.t. all the other forms can be dealt with as ‘special cases’. But this makes the discussion somewhat un-illuminating!

See:

• K. Vela Velupillai, Non-Linear Dynamics, Complexity and Randomness: Algorithmic Foundations, Journal of Economic Surveys, Vol. XXV, March, 2011.

• K. Vela Velupillai, Taming the Incomputable, Reconstructing the Nonconstructive and Deciding the Undecidable in Mathematical Economics, New Mathematics and Natural Computation, Vol. 8, #. 1, 2012

THE CANONICAL DIFFERENCE-DIFFERENTIAL

EQUATION OF ENDOGENOUS MACRODYNAMICS

All known macrodynamic models of endogenous business cycle theories, of whatever economic theoretical persuasion, can be derived, by approximations, linearizations, etc., – whether ad hoc or not – from the following canonical equation:

Remark: In general, even if analytical solutions are available, say for the special case of ODEs, via, for example the Cauchy-Peano Existence Theorem, or for the nonlinear second-order differential equation of the Rayleigh-van der Pol type by an appeal to the Poincaré-Bendixson theorem, their computational, computable and constructive statuses are highly dubious. This makes simulation, which is intrinsically also computational – but not necessarily number-theoretic – meaningless.

CAN A DYNAMICAL SYSTEM, WHOSE EQUILIBRIUM EXISTENCE IS

PROVED BY NON-FINITE MEANS BE SIMULATED BY FINITE MEANS?

Consider the Peano Existence Theorem:

However, it can be shown that ∃f(t,y), satisfying the hypotheses of the Peano existence theorem, such that there is no solution to the IVP. Why is this so? This is because the existence of a solution violates a cardinal theorem of computable calculus: the Unsovability of the Halting Problem for Turing Machines. More specifically, there are a series of nonsolvable problems by finite means, in the computable calculus, some of which have to be made solvable -- by non-finite means -- for the Peano existence theorem to be satisfied. In the case of the Peano existence theorem, the relevant non-solvable problems are:

It is undecidable (by finite means) whether, ∀a ∈ R, a or ∼a is rational. It is undecidable (by finite means) whether, ∀a∈R, a ≥ 0 or a ≤ 0.

Thus, implicit in any standard proof of the Peano existence theorem there are appeals to non-finite means to decide disjunctions.

A NONSTANDARD VERSION – I.E., APPROXIMATION – OF (1)

Consider, now, the following ODE - with α an infinitesimal - approximation of (1):

DISCOVERING DYNAMICS BY SIMULATION: ODE IN

NONSTANDARD FORM

From headed to un-headed ducks:

→

Remark 1: "Ducks are certain singular solutions of equations with a small parameter, which are studied in the theory of relaxation oscillations. These solutions were first found for the van der Pol equation, and their form resembled that of a flying duck. Duck theory is, in the authors' opinion, the most striking application of the techniques of non-standard analysis.......It was not by chance that ducks were discovered with the help of non-standard analysis and in connection with it. Zvonkin, A.K, & M.A. Shubin, (1984), Non-standard analysis and singular perturbations of ordinary differential equations, Russian Mathematical Surveys, Vol. 39, #2 Remark 2:

“[T]he conventional real number system is Dedekind complete and therefore Archimedean, which essentially means that it lacks infinitesimals; ..... .” Schechter, Eric, (1997), Handbook of Analysis and Its Foundations, Academic Press, San Diego

DISCOVERING DYNAMICS BY SIMULATION – AND

APPROXIMATIONS I: ODE IN STANDARD FORM(S)

Consider the following equation, representing a classical Keynesian nonlinear multiplier-

accelerator model of the dynamics of national income, y:

Now, there are at least six different ways to investigate solutions to this nonlinear

difference-differential equation:

In old fashioned analytical modes using:

I. Real Analysis

II. Non-Standard Analysis (as above)

III. Computable Analysis

IV. Constructive Analysis

Graphically, i.e., in terms of the geometry of dynamic behaviour, as usually

done in the qualitative theory of differential equations;

In terms of simulating, in one of the domains defined by (I) – (IV), using:

•an analogue computer

•a digital computers

)6)].....(('[)()()1()(' tytOtyty A

DISCOVERING DYNAMICS BY SIMULATION – AND

APPROXIMATIONS II: ODE IN STANDARD FORM(S)

By way of a series of economically ‘justifiable’ assumptions, (6) was ‘reduced to:

Then, ‘approximated’ by:

)8.....(0)1()( ])1([ yyyy

)7)]......(([)()1()( tytyty

For which the geometric ‘solution’, in the phase-plane is:

DISCOVERING DYNAMICS BY SIMULATION – AND

APPROXIMATIONS III: ODE IN STANDARD FORM(S)

Suppose (7) was approximated by:

)9....(01')('1''2'''3''''4)('''''24

4

tytytyCtyCtyCtyCty

Then, the ‘geometry’, obtained by digital simulation would be, depending on the choice of initial

conditions – but not quite SDIC:

The geometry of (9), using an analogue computer, can be found in: R.H. Strotz, J.C. McAnulty & J.

B.Naines, Jr., Econometrica, Vol. 21, #. 3, July, 1953

What are the lessons for The Epistemology of Simulation, Dynamics and Computation from these

three classic examples?

UNTENABLE CLAIMS ON COMPUTABILITY I:

COMPUTATIONAL ECONOMICS

"[The Johansen model] was general in that it

contained .. cost minimizing industries and utility-

maximizing household sectors....His model

employed market equilibrium assumptions in the

determination of prices. Finally, it was computable

(and applied). It produced a numerical, multi-

sectoral description of growth in Norway using

Norwegian input-output data and estimates of

household price and income elasticities derived

using Frisch's ... additive utility method.” Dixon, Peter. B & B. R. Parmenter (1996), Computable General Equilibrium Modelling for Policy Analysis and Forecasting, Chapter 1, pp. 3-85, in: Handbook of Computational Economics, Volume 1, edited by Hans M. Amman, David A. Kendrick and John Rust, North-Holland, Amsterdam.

UNTENABLE CLAIMS ON COMPUTABILITY II:

COMPUTABLE GENERAL EQUILIBRIUM THEORY

"The major result of postwar mathematical general equilibrium theory has been to demonstrate the existence of such an equilibrium by showing the applicability of mathematical fixed point theorems to economic models. ... Since applying general equilibrium models to policy issues involves computing equilibria, these fixed point theorems are important: It is essential to know that an equilibrium exists for a given model before attempting to compute that equilibrium. ........The weakness of such applications is twofold. First, they provide non-constructive rather than constructive proofs of the existence of equilibrium; that is, they show that equilibria exist but do not provide techniques by which equilibria can actually be determined. Second, existence per se has no policy significance. .... Thus, fixed point theorems are only relevant in testing the logical consistency of models prior to the models' use in comparative static policy analysis; such theorems do not provide insights as to how economic behavior will actually change when policies change. They can only be employed in this way if they can be made constructive (i.e., be used to find actual equilibria). The extension of the Brouwer and Kakutani fixed point theorems in this direction is what underlies the work of Scarf .... on fixed point algorithms ....“ Shoven, John B and John Whalley (1992, pp. 12, 20-1; italics added), Applying General Equilibrium, Cambridge University Press, Cambridge.

UNTENABLE CLAIMS ON COMPUTABILITY III: AGENT-BASED

COMPUTATION IN ECONOMICS AND FINANCE

The following two claims are made, in relation to ABCEF, in Leigh Tesfatsion & Kenneth L. Judd (ed.), 2006. "Handbook of Computational Economics" , Elsevier, Vol. 2:

• "Agent-based computational methods provide the only way in which the

self-regulatory capabilities of complex dynamic models can be explored so

as to advance our understanding of the adaptive dynamics of actual

economies.”

• “As a professor of mathematics (as well as economics), I appreciate the beauty of

classical mathematics. However, constructive mathematics is also beautiful and, in

my opinion, the right kind of mathematics for economists and other social

scientists. Constructive mathematics differs from classical mathematics in its strict

interpretation of the phrase ‘there exists’ to mean ‘one can construct’. Constructive

proofs are algorithms that can, in principle, be recast as computer programs.”

Remark 1: The first assertion is preposterous and cannot be substantiated from any serious

point of view.

Remark 2: The second – in a chapter titled, Agent-Based Computational Economics: A

Constructive Approach to Economic Theory – has nothing whatsoever to do with modelling

anything in ABCEF; indeed, it is vague, to the point of meaninglessness, in its

characterisation of ‘constructive mathematics’ – especially on the meaning of ‘construct’.

Remark 3: Quite apart from these issues, there is no awareness whatsoever that there are at

least five modes of computing – and whether one is superior to another mode, for one

purpose, and vice versa for another aim, is not clear – and at least two types of computing

devices, analogue and digital.

‘CHARACTERISING’ ABCEF

No better characterisation of the practice of agent based computational economists can be given that the one Arthur Burks gave

(Essays on Cellular Automata, University of Illinois Press, Urbana p. xviii), on a related `procedure for investigating cellular spaces':

"The investigator starts with a certain global behavior and wants to find a transition function for a cellular automaton which exhibits that behaviour. He then chooses as subgoals certain elementary behavioral functions and proceeds to define his transition function piece-meal so as to obtain these behaviors......The task of searching for a transition function which produces a specified behavior is an arduous task because there are so many possible partial transition functions to explore.”

The formal difficulties of `searching for a transition function' are provably intractable, at best; algorithmically undecidable, in general. Even when found, depending on the way the data generating process if characterised, whether the transition function -- when viewed as a finite automaton -- `halts' at the prescribed state is, again, in general, algorithmically undecidable, Correspondingly, when viewed as a dynamical system, whether the global behaviour is an attractor or is in a particular basin of attraction of the dynamical system, is algorithmically undecidable. Whether a set of initial conditions, for the transition function, can be algorithmically determined such that their halting state is the desired global behaviour, or such that the global behaviour is in the basin of attraction of the transition function as a dynamical system is decidable only for trivial sets.

Suppose we succeed in finding such a transition function -- as many agent based computational economists claim they can, and have -- and want to characterise it either in terms of computability theory, constructive mathematics or as a dynamical system. Suppose, also, that we ask questions on the feasibility of self-reproduction, self-reconstruction, evolution, computation universality, decidability of limit sets of the transition function when interpreted as a dynamical system, whether the transition function, viewed as an finite automaton, is subject to the Halting Problem, and so on. Reasonable answers to most of these question are algorithmically undecidable – whatever mathematical formalism is used.

MODESTY & HUMILITY IN THE FACE OF

EPISTEMOLOGICAL DEFICITS & INCOMPLETENESS

I. What are machines, mechanisms, computations and algorithms?

II. How interdependent are any answers to the above four sub-questions?

III. What are the limitations of mechanisms?

IV. Can a machine, encapsulating mechanisms, know its limitations.

V. Can a simulation, encapsulated within a mechanism, exhibit its own limitations?

VI. Can a Dynamical System, realised with the mechanism implementing a computation, demonstrate its own limitations

The answer(s) depend crucially on Gödel’s Incompleteness Theorems, the Turing Machine and Turing’s famous result on the Unsolvability of the Halting Problem for Turing Machines (or: The Unsolvability of Hilbert’s Tenth Problem).

KANT’S HOPES & HILBERT’S VISIONS: THE MATHEMATICS OF EPISTEMOLOGY

In terms of any mathematical formalism, validity of mathematical theorems are claimed on the basis of proof, which are, in turn, the only mechanism for expressing truth effectively – in the precise sense of recursion theory - in mathematics. Then, with Kant:

The mathematician can hope all provable mathematical statements are true;

Conversely, the mathematician can also hope that all – and only – the true

statements are provable;

And, following Hilbert’s vision, the mathematician’s task – Kant’s ‘what

must I do’ - is to build a machine to discover – Kant’s ‘what must I

know’ - valid proofs of every possible theorem in any given formal

system;

LIMITS OF EPISTEMOLOGY – HENCE OF THE EPISTEMOLOGY

OF SIMULATION, COMPUTATION AND DYNAMICS

The first two hopes were ‘derailed’ by Gödel’s Incompleteness Theorems, by the demonstration that in any reasonably strong formal system there are effectively presentable mathematical statements that are recursively undecidable – i.e., neither algorithmically provable nor unprovable. The third was ‘shaken by Church and finally demolished by Turing’, i.e., that no such machine can be ‘built’, shown in a precisely effective way.

Since, however, Gödel’s theorems were presented recursively and proved constructively – hence within the proof-as-algorithm paradigm - it must be possible to build a machine, with an effective mechanism, to check the validity of the existence of undecidable statements. This, then, it will be an instance of a mechanical verification of Gödel’s proof and, hence, a demonstration that a machine can establish the limitations of its own mechanism

Does it not mean: Man can establish the limitations of what he can know?

Only a formalism of economic theory that can make sense of the formal

limitations of what can be known is able to discourse intellignetly about

epistemological constraints of simulation, computation and dynamics