Com Put Ability and Economic Planning

8/2/2019 Com Put Ability and Economic Planning

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Computability and Economic

PlanningGreg Michaelson

School of Mathematical & Computer Sciences

Heriot-Watt University


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Overview

computability results misused in otherdomains e.g. philosophy: Nagel & Newman (1959) on

Gdel results motivation often ideological

in economics longstanding debates

between free market and rational planning planning presented as demonstrable

failure in pre-1989 Soviet-style economies


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Overview

arguments that planning inherentlyinfeasible

Marciszewski (2002)

planning is hypercomputational;

Murphy (2006)

planning undermined by Cantor diagonalisation;

Nove (1983) planning is computationally intractable


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Planning

planning formulated as input-output model

Sraffa (1960)

Aapa+Bapb+...+Kapk= Apa

Abpa+Bbpb+...+Kbpk= Bpb...

Akpa+Bkpb+...+Kkpk= Kpawhere:

I= total annual quantity of commodity I

Ij= quantity of commodityjused to produce commodity I

pj= unknown value of commodityj


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Planning

reformulate as:

MP= QP

where:

M= matrix of IjP= vector of PjQ= diagonal matrix of I

solve:

(M-Q)P= 0 for P

Aapa+Bapb+...+Kapk= ApaAbpa+Bbpb+...+Kbpk= Bpb

...

Ak

pa

+Bk

pb

+...+Kk

pk

= Kpa

where:I= annual quantity of

commodity I

Ij= quantity of commodityjused to producecommodity I

pj= unknown value ofcommodityj


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Is planning hypercomputational?

Marciszewski argues that: Among the simplifications made in the [Club of Rome] Report

there was the total omitting of the factors of scientific researchand technological invention ... Obviously, such factors cannot begrasped in central economic planning. Even if the Laplaceandemon revealed what is to be going on in the heads of futurediscoverers, the unimaginable complexity of each brainseparately and still greater of their world-wide interactions wouldunavoidably hamper any computer-based predictions.

On the other hand, an intuitive understanding as expressed inaxiom-like maxims, e.g. the more economic freedom, the moreeconomic information may prove more reliable and more useful

than the results of algorithmic procedures. Do suchunderstandings result from some hypercomputational processesin our brains? This is an open question.


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Is planning hypercomputational?

reminiscent of Nagel & Newmans

response to Gdels Godels results

couldnt be mechanized

but human brain can construct these results

so brain is of greater computational powerthan machine

Ammon (1993) provided a mechanisation


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Is planning computational?

Murphy argues that:

list of prices Pis infinite

must take into account all possible future

commodities

list of prices Pis uncountably infinite

diagonalise over prices


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Are prices uncountably infinite?

diagonalisation

lay out prices in table

construct unique new price which differs fromprice in row I at column J


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Are prices uncountably infinite?

diagonalisation applies to irrational numbers

prices are of fixed precision

at worst rational

diagonalisation inapplicable

prices are of commodities

number of possible distinct commodities is countablyinfinite

enumerate sequences of atoms in distinctcommodities

if infinite, prices are countablyinfinite


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Is there an infinite number ofprices?

an infinite number of potential commodities hasan infinite number of prices

but it doesnt matter at any given moment:

a finite number of producers from a finite range/quantity of actual input

commodities produce a finite range/quantity of possible output

commodities with a finite number of prices

so even if the system is infinite, instances arevery sparse


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Is planning tractable?

solving (M-Q)P = 0 for N prices using Gaussianelimination is O(N3) poor algorithm

can approach O(N2) each commodity only requires a small proportion of other commodities

(S) heuristic iteration of 10-12 cycles

for 10,000,000 prices S*10*1014 * C(ost of base ops) current processors are 3*109 instructions per sec there are 60*60*24 = 86*103 secs in a day

need S*C*1015*/(3*109*86*103) = S*C*4 days on oneprocessor

Google has 250,000 processors...


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Acknowledgements

joint work with Paul Cockshott (University ofGlasgow) and Allin Cottrell (Wake-ForrestUniversity)

more details in: A. Cottrell, P. Cockshott and G. J. Michaelson, `Iseconomic planning hyper-computational? Theargument from Cantor diagonalisation', InternationalJournal of Unconventional Computing, to appear,

2009 P.Cockshott, A. Cottrell, G. Michaelson, I. Wright and

V. Yakovenko, Classical Econophysics, Routledge,May 2009


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References

Nagel, E. and J. R. Newman (1959) Gdels Proof, Routledge andKegan Paul.

Ammon, K. (1993) An automatic proof of Gdels incompletenesstheorem, Artificial Intelligence, 61(2): 291306.

W. Marciszewski. (2002). Hypercomputational vs. Computational

Complexity A Challenge for Methodology of the Social Sciences.Free Market and Computational Complexity. Essays inCommemoration of Friedrich Hayek (1899-1992). Series: Studies inLogic, Grammar and Rhetoric, 5:18.

Alex Nove. (1983). The Economics of Feasible Socialism. George

Allen and Unwin, London. R.P. Murphy. (2006). Cantors Diagonal Argument: an Extension tothe Socialist Calculation Debate. Quarterly Journal of AustrianEconomics, 9(2):3..11.

Com Put Ability and Economic Planning

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