Stefano Zambelli Deptartment of Economics University of Trento Trento – Italy

1. “Non-Equilibrium Dynamics: An Algorithmic Model based on Von Neumann-Sraffa-Leontief Production Schemes” Stefano Zambelli Deptartment of Economics University of Trento Trento – Italy stefano.zambelli@unitn.it 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 isrelaxed. • 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 biiunits of commodity i can be produced with ti different alternative methods. i = 1, …, n e zi = 1, …, t i

7. Technological PossibilitiesMethods of Production

8. Technological PossibilitiesMethods of Production zn = 1 zi = 5 z2 = 7 z1 = 2

9. Any ”Standard” production function can be encapsulated (approximated) in a subset of the matrix

10. 50’s Linear Programming -Samuelson – Solow The other way about – Heterogeneous production could be represented AS IF it was a ’simple’ Cobb-Douglas production function THIS ”WELL-BEHAVED” FUNCTION ”FITS” REAL METHODS

11. Methods of Production

12. Methods of Production This triple identifies a combination, z, of production methods used to produce the n commodities If the system is productive.

13. Any productive system can be re-proportioned so as to constitute another productive system The set of all possible triples constitutes a production system

14. Simple EconomySimple Productive System l- Workers l+s working units l+s consumers n commodities m production processes Letnbe a large number, say 3! E. Landau s – Producers (also Workers) 1 2 3 1 First Commodity “Products – Goods – Commodities” 2 Second Commodity 3 Third Commodity

15. Means of production necessary for the production of the quantity x11 b11 of commodity 1 1 Workers Producers 1 2 3 1 2 3

16. Means of production necessary for the production of the quantity x22 b22 of commodity 2 2 Workers Producers 1 2 3 1 2 3

17. Factors’ demand. Quantity bought for the production of commodity x33 b33 3 Workers Producers 1 2 3 1 2 3

18. REMARK: DECISION PROBLEMS CAN BE ENCAPSULATED AS DIOPHANTINE EQUATIONS – TURING MACHINES ENCODABLES 1 1 1 2 2 2 3 3 3 1 2 3

19. Exchange for Consumption Purposes Exchange for Production Purposes 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 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 are necessary – no money)

20. Non-substitution Theorem Theorem: Relative prices are independent from the production and/or demand vector.

21. Non-substitution Theorem Number of possible combinations of processes z is t1t2t3t4t5…tn w Wage-Profit Curve Wage Profit Frontier r

22. Macroeconomic Aggregates(Equilibrium values) Quantity of NNP allocated to the owners of capital Quantity of NNP allocated to the workers

23. Exchange for Consumption Purposes Exchange for Production Purposes WHAT IF? 1 1 1 2 2 2 3 3 3 1 1 1 2 2 2 3 3 3 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).

24. 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 CONDITIONS IS SIMPLER THAN THE STUDY OF OUT-OF-EQUILIBRIUM BEHAVIUOR

25. GENERAL NON-EQUILIBRIUM SPECIFICATION OF INDIVIDUAL BEHAVIOURAL FUNCTIONS COMPUTABLE ECONOMICS BEHAVIOURAL ECONOMICS EXPERIMENTAL ECONOMICS ALGORITHMIC RATIONAL AGENT A Computable Agent

26. Agents’ decisions Heterogeneity ALGORITHMIC RATIONAL AGENT A Computable Agent 1 2 3 Behavioral Experimental Computable

27. Macroeconomic Aggregates Non Equilibrium Dynamics w SHORT-RUN LONG RUN 1 r 2 3

28. Macroeconomic AggregatesOut of Equilibrium Dynamics w LOCK-IN ? 1 r 2 3

29. Macroeconomic AggregatesOut of Equilibrium Dynamics w Wage Profit Frontier 1 r 2 3

30. w GENERAL EQUILIBRIUM NEOCLASSICAL CASE Consistent with the Aggregated ”Cobb-Douglas” Wage Profit Frontier r Capital/Labor Ratio r

31. Capital Market Labor Market Macroeconomic Aggregates(Equilibrium values) • Stochastic Dynamic General Equilibrium Models . • RBC – OLG – NEW KEYNESIANS …. wWPF r Labour Supply Capital Supply Capital Demand MPK Labour Demand MPL vKWPF L

32. w GENERAL EQUILIBRIUM NOT NEOCLASSICAL CASE NOT consistent with the Aggregated ”Cobb-Douglas” Wage Profit Frontier r 60% Capital/Labor Ratio r

33. Macroeconomic Aggregates(Equilibrium values) • ARTIFICIAL ECONOMIC MODEL Labor Market Capital Market wWPF r ? ? Labor Supply Capital Supply Capital Demand MPK ? ? Labor Demand MPL vKWPF L

34. 3 commodities, 3 producers, 27 workers, 6 methods per commodity A SimulationAn example with Low Dimensional Model 1 2 3

36. HIGHLY STRUCTURED von Neumann - WolframCELLULAR AUTOMATA Workers 1 2 3 1 2 3 1 2 3 Producers “Products – Goods – Commodities” DISTINCTION BETWEEN LOCAL GLOBAL VARIABLES UNIVERSAL COMPUTABILITY MASTER DIOPHANTINE EQUATION

37. HIGHLY STRUCTURED von Neumann - WolframCELLULAR AUTOMATA Workers 1 2 3 1 2 3 1 2 3 Producers THE ALGORITHMIC COMPLEXITY AND COMPUTATIONAL COMPLEXITY OF THE CONCATENATED SYSTEM IS PROPORTIONAL TO THE COMPLEXITIES OF THE SMALLER UNIT “Products – Goods – Commodities”

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

39. IMPORTANTin 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). GENERAL EQUILIBRIUM (uniform prices – but not uniform profit rates) GENERAL EQUILIBRIUM (uniform prices, wage rates and profit rates)

40. n commodities, s producers, lworkers, m methods The wage profit frontier is independent from dimensionality 1 2 3 POSSIBLE BENCHMARK? REPRESENTATIVE SYSTEM

41. TO SUM UP • 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;

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 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 THE IMPORTANCE OF DEMAND and so on and so forth and so on and so forth …