Stefano Zambelli Deptartment of Economics University of Trento Trento – Italy

“Non-Equilibrium Dynamics: An Algorithmic Model based on Von Neumann-Sraffa-Leontief

Production Schemes”

Stefano ZambelliDeptartment of Economics

University of Trento

Trento – Italy

[email protected]

28.05.2009

2

Modern macroeconomics:– the economic system is in a perpetual state of

general economic equilibrium (postulate)

– The aggregate dynamics is explained by the existence of (real or monetary) shocks that require a revision of the agents’ (inter)temporal decisions - Stochastic Dynamic General Equilibrium Models

– These low dimensional Stochastic Dynamic General Equilibrium Models are also the ‘benchmark’ models for the cases in which out of equilibrium behaviours are considered.

3

In this work an attempt is made to design:– a dynamic system– where the postulate of perpetual general economic

equilibrium is relaxed. – an algorithmic model in which interactions between agents

and regions is constructed using the theoretical toolbox of coupled dynamical systems.

4

To be more specific the algorithmic model is based on the tradition set by:– von Neumann’s growth model, – by the Keynes-Stone’s conceptual work on

national accounting;– Simon’s work on behevioural economics,

decision making; – Vellupillai’s computable economics.

5

The final aim is – to use the model as a type of virtual laboratory

in which to implement analytical conceptual experiments aimed to study: • the convergence towards equilibrium; • the emergence of monetary-financial

magnitudes; • price dynamics; • the effects of technological innovations

(non)

6

Technological PossibilitiesMethods of Production

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bii units of commodity i can be produced with ti different alternative methods.

i = 1, …, n e zi = 1, …, t i ):,,2( i

7


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)1:,:,(

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Φ

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Φ z1 = 2

z2 = 7

zi = 5

zn = 1

1

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9

Any ”Standard” production function can be encapsulated

(approximated) in a subset of the matrix

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):,:,( iΦ

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10

50’s Linear Programming - Samuelson – Solow

izia 2

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The other way about – Heterogeneous production could be represented AS IF it was a ’simple’ Cobb-Douglas production function

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THIS ”WELL-BEHAVED” FUNCTION”FITS” REAL METHODS

11

Methods of Production ni :,:,,,:,:,,,2:,:,,1:,:, ΦΦΦΦΦ

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12

Methods of Production ni :,:,,,:,:,,,2:,:,,1:,:, ΦΦΦΦΦ

nzn

z

z

n L

L

L

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nz

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2

1

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1

),1,(

)2,1,(

)1,1,(

zL

),,( zzz LABThis triple identifies a combination, z, of production methodsused to produce the n commodities

If 0ABe zz the system is productive.

13

nnx

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x

000

000

000

000

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X

zzzz LABE ,,

),,( zzz XLXAXB

Any productive system can be re-proportioned so as to constitute another productive system

The set of all possible triplesconstitutes a production system

zzzz LABE ,,

14

Simple EconomySimple Productive System

1 2 3 s – Producers (also Workers)

l- Workers

“Products – Goods – Commodities”1

2

3

First Commodity

Second Commodity

Third Commodity

l +s working units

l +s consumers

n commodities

m production processes

Let n be a large number, say 3! E. Landau

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Means of production necessary for the production of the

quantity x11 b11 of commodity 1

111111111111131112111111zzzzz bxLxaxaxax

1 2 3 Producers

Workers

1

2

3

1

1111zLx

16


1 2 3 Producers

Workers

1

2

3

2222zLx

2

Means of production necessary for the production of the

quantity x22 b22 of commodity 2

17

Factors’ demand. Quantity bought for the production of

commodity x33 b33


1 2 3 Producers

Workers

1

2

3

3333zLx

3

18


1111zLx



1

2

3

1 2 31 2 3

2222zLx

1 2 3

3333zLx

),,( zzz XLXAXBREMARK: DECISION PROBLEMSCAN BE ENCAPSULATED AS DIOPHANTINE EQUATIONS –TURING MACHINES ENCODABLES

19

1 2 31 2 31 2 3 1 2 31 2 31 2 3

Exchange for ProductionPurposes

Exchange for ConsumptionPurposes

GENERAL EQUILIBRIUM Walrasian or Marshallian

THE VALUE OF THE QUANTITIES SOLD BY THE INDIVIDUAL AGENTS IS EQUAL TO THE VALUE BOUGHT BY THEM (no credit-debt contracts arenecessary – no money)

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Non-substitution Theorem

Theorem: Relative prices are independent from the production and/or demand vector.

21

Non-substitution Theorem

zzz

z

L)(ABηη 1

1

1),(

rrw

)3(zw)2(zw

w

r

Wage-Profit Curve

Number of possible combinations of processes z is t1t2t3t4t5…tn

)1(zw

)6(zw

)5(zw

)4(zw

Wage Profit Frontier

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Macroeconomic Aggregates(Equilibrium values)

),(),(),('),(' ηηηXLeηXApe zzzzz rvWrvΠrwrrvYNNP

pXBXLpXA zzz wr)1(

),()('),( ηpABXeη zzz rrvC

Quantity of NNP allocatedto the owners of capital

Quantity of NNP allocatedto the workers

),('),( ηpXBeη zzz rrvYGNP

),('),( ηpXAeη zzz rrvK

),()('),( ηpABXeη zzzz rrvYNNP

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1 2 31 2 31 2 3 1 2 31 2 31 2 3

Exchange for ProductionPurposes

Exchange for ConsumptionPurposes

GENERAL NON-EQUILIBRIUM THE VALUE OF THE REAL QUANTITIES SOLD BY THE INDIVIDUAL AGENTS IS NOT EQUAL TO THE VALUE BOUGHT BY THEM (They are by definition equal – but with the emergence of credit-debt ... i.e., clearing contracts).

WHAT IF?

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GENERAL NON-EQUILIBRIUM

• Bilateral trade (non uniform prices – exchange prices are

NOT equal to equilibrium natural prices)• Purchasing power unbalances. For most agents the values of the

real quantities sold is not equal to the values of the real quantities bought.

• Money as Debt-Credit relations. The individuals write bilateral contracts (the sellers of real commodities sell them in exchange of I Owe You contracts IOU – as clearing devices)

THE STUDY OF EQULIBRIUM CONDITIONSIS SIMPLER THAN THE STUDY OF OUT-OF-EQUILIBRIUM BEHAVIUOR

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GENERAL NON-EQUILIBRIUM SPECIFICATION OF INDIVIDUAL BEHAVIOURAL FUNCTIONS

BEHAVIOURAL ECONOMICS

EXPERIMENTAL ECONOMICS

COMPUTABLE ECONOMICS

ALGORITHMICRATIONAL

AGENTA Computable Agent

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Agents’ decisions

1 2 3

ALGORITHMICRATIONAL

AGENTA Computable Agent

Heterogeneity

Experimental

Behavioral

Computable

27

Macroeconomic Aggregates Non Equilibrium Dynamics

w

r2

3

1

SHORT-RUN

LONG RUN

28

Macroeconomic AggregatesOut of Equilibrium Dynamics

w

r2

3

1

LOCK-IN ?

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Macroeconomic AggregatesOut of Equilibrium Dynamics

)3(zw)2(zw

w

r

)1(zw

)6(zw

)5(zw

)4(zw


2

3

1

30

w

r

)1(zw

)6(zw

)5(zw

Capital/Labor Ratio

LXe

z

WPF

WPFvK

'

r


NEOCLASSICAL CASEConsistent with theAggregated ”Cobb-Douglas”

GENERAL EQUILIBRIUM

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Stochastic Dynamic General Equilibrium Models .

RBC – OLG – NEW KEYNESIANS ….

Capital Market Labor Market

LvKWPF

r wWPF

Labour DemandMPL

Labour SupplyCapital Supply

Capital DemandMPK

32

w

r

)1(zw

)6(zw

)5(zw

Capital/Labor Ratio

LXe

z

WPF

WPFvK

'

r


NOT NEOCLASSICAL CASENOT consistent with the Aggregated ”Cobb-Douglas”

GENERAL EQUILIBRIUM

60%

33


ARTIFICIAL ECONOMIC MODEL

LvKWPF

r wWPF

Labor DemandMPL

Labor SupplyCapital Supply

Capital DemandMPK

?

?

?

?

Capital Market Labor Market

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A SimulationAn example with Low Dimensional Model

3 commodities, 3 producers, 27 workers, 6 methods per commodity

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5Frontier

Profit Rates

Wag

e R

ates

1 2 3

4

2

2

z

4

5

2

z

4

2

5

z

1

2

5

z

35

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1

2

3

4

5

6

7

8Capital/Labor Ratio at the Frontier

Profit Rates

Cap

ital/L

abor

Rat

io

36

HIGHLY STRUCTURED von Neumann - Wolfram

CELLULAR AUTOMATA

1 2 3 Producers

Workers

“Products – Goods – Commodities”

1 2 3 1 2 3

UNIVERSAL COMPUTABILITY

MASTER DIOPHANTINE EQUATION

DISTINCTION BETWEEN LOCAL GLOBAL VARIABLES

37

HIGHLY STRUCTURED von Neumann - Wolfram

CELLULAR AUTOMATA

1 2 3 Producers

Workers

“Products – Goods – Commodities”

1 2 3 1 2 3

THE ALGORITHMIC COMPLEXITY AND COMPUTATIONAL COMPLEXITY OF THE CONCATENATED SYSTEM IS PROPORTIONAL TO THE COMPLEXITIES OF THE SMALLER UNIT

38

Notions of Equilibrium

• Uniform prices• Desired-planned exchanges equal actual exchanges

(ex-ante=ex-post)• Supply equals demand • IOUs=0 (ΔIOUs=0)• … and so on

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GENERAL EQUILIBRIUM

(uniform prices – but not uniform profit rates) ******* ***

)1( pBXwLXrpAX zzz

GENERAL EQUILIBRIUM

(uniform prices, wage rates and profit rates)

IMPORTANT in equilibrium the dimensionality is not important and the ”aggregate” system is simply a ’multiple’ (ω) of the (equilibrium) subsystems (or a linear combination of them).

******* ***

)1( pBXLXpAX zzz wr

40

n commodities, s producers, l workers, m methods

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.5

1

1.5

2

2.5

3

3.5

4

4.5Frontier

Profit Rates

Wag

e R

ates

1 2 3

4

2

2

z

4

5

2

z

4

2

5

z

1

2

5

z

The wage profit frontier is independent from dimensionality

POSSIBLE BENCHMARK?REPRESENTATIVESYSTEM

41

• THERE ARE NO STOCHASTIC ELEMENTS IN THE ALGORITHMIC MODEL

• THE DIMENSION OF THE MODEL IS PARAMETRIC (it functions well also with a high number of agents and regions)

• IT GENERATES ALL THE STANDARD NATIONAL ACCOUNTING DATA

• ALL THE ECONOMIC AND ALGORITHMIC CHECKS (CONTROLS) GIVE CONSISTENT RESULTS BOTH AT THE MICRO AS WELL AS AT THE MACRO LEVEL (for example accounting - double-book keeping – NO ERRORS AND OMISSIONS)

• All the different ARAs’ algorithms for the determination of the expected sales, future prices and buying and selling decisions function well;

TO SUM UP

42

WORK IN PROGRESS

• INITIAL VALUE PROBLEM – INITIAL CONDITIONS

• AFTER SOME ITERATIONS SOME PRODUCERS STOP PRODUCING BECAUSE EXPECTED REVENUES ARE LOWER THAN EXPECTED COSTS

• COORDINATION PROBLEM?• CORRIDOR?• THE WORKERs’ MOBILITY HAS NOT YET

BEEN INTRODUCED

43

SOME EXAMPLES OF RESEARCH QUESTIONS

• Can the system function without the introduction of institutions such as Central Bank and Government?

• Will the system(s) converge towards a uniform equilibrium? Or are we facing a PASTA-ULAM-FERMI problem?

• What are the determinants of the equilibrium?– Demand?– Policy?– None of the above

• WHAT IS THE RELATION BETWEEN MONETARY MAGNITUDES AND REAL MAGNITUDES?

• What is the relation between the ”real” interest rate and the ”monetary” interest rate?

• Effects of technological innovations

44

THE STUDY OF OUT-OF-EQUILIBRIUM BEHAVIUOR IS NECESSARY FOR THE UNDERSTANDING OF:

THE IMPORTANCE OF DEMAND

THE EFFECTS OF MONEY AND FINANCIAL MAGNITUDES ON REAL VARIABLES

THE IMPLEMENTATION OF NEW METHODS OF PRODUCTION: NEW PRODUCTION TECHNIQUES

THE EQUILIBRIUM IN THE LABOUR MARKET AND INDIVIDUAL WELFARES

and so on and so forth and so on and so forth …

Documents

Stefano Zambelli Deptartment of Economics University of Trento Trento – Italy