“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 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. 2 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. 3 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. 4 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(non) convergence towards equilibrium; • the emergence of monetary-financial magnitudes; • price dynamics; • the effects of technological innovations 5 Technological Possibilities Methods of Production bii units of commodity i can be produced with ti different alternative methods. ( zi , :, i ) : i = 1, …, n e a zi i1 zi i2 zi in a a zi ii L b (2, :, i ) zi = 1, …, t i ai11 ai12 2 2 a a i1 i2 Φ( :, :, i ) ti t ai1 ai 2i zi i ain1 ain2 ainti L1i bii1 2 2 Li bii ti ti Li bii 6 Technological Possibilities Methods of Production a1n1 2 a Φ(:, :, n) n1 tn an1 1 1 ai1 ai 2 ain1 2 ai1 ai22 ain2 Φ( :, :, i ) ti ti t 1 1i a in a1 a1 aai1 1 ai 2L b 21 22 2n 2 2 2 a a a 2n Φ( :, :, 2) 21 22 t2 t2 t2 a21 a22 a2 n 1 1 a11 a12 a11n 2 2 2 a a a 1n Φ( :, :, 1) 11 12 t1 t1 t1 a11 a12 a1n 1 L11 b11 L12 b112 L1t1 b11t1 2 2 2 L Lt22 22 2 22 b t2 b22 a1n 2 a1nn L1n 2 an22 ann L2n tn antn2 ann L1i bii1 L2i bii2 Ltii biiti Ltnn 1 bnn 2 bnn tn bnn 7 Technological Possibilities z1 2 Methods of Production z 2 7 z 5 i z n 1 a1n1 2 a Φ(:, :, n) n1 tn an1 1 1 ai1 ai 2 ain1 2 ai1 ai22 ain2 Φ( :, :, i ) ti ti t 1 1i a in a1 a1 aai1 1 ai 2L b 21 22 2n 2 2 2 a a a 2n Φ( :, :, 2) 21 22 t2 t2 t2 a21 a22 a2 n 1 1 a11 a12 a11n 2 2 2 a a a 1n Φ( :, :, 1) 11 12 t1 t1 t1 a11 a12 a1n 1 L11 b11 L12 b112 L1t1 b1t1 2 2 2 L t2 2 L 22 2 22 b t2 b22 zn = 1 a1n 2 a1nn L1n 2 an22 ann L2n tn antn2 ann L1i bii1 L2i bii2 Ltii biiti Ltnn 1 bnn 2 bnn tn bnn zi = 5 z2 = 7 z1 = 2 8 Any ”Standard” production function can be encapsulated Φ( :, :, i ) (approximated) in a subset of the matrix bii1 bii2 biif bii ai11 ai12 0 2 2 ai1 ai 2 0 Φ( :, :, i ) f f a a i2 0 i1 a ti a ti a ti i2 in i1 0 bii1 0 bii2 0 biif Ltii biiti aiz2i 1 i2 a 2 i2 a bii1 bii2 bii3 ai32 biif aif2 bii ai11 ai21 ai31 ai1f aiz1i 9 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 bii1 bii2 biif bii ai11 ai12 0 2 2 ai1 ai 2 0 Φ( :, :, i ) f f a a i2 0 i1 a ti a ti a ti i2 in i1 a ) a a ) a a 1 i1 1 1 i2 bii 2 i1 2 1 i2 bii f i1 f 1 i2 bii F (a , a ) a 1 i1 1 i2 2 i1 2 i2 f i1 f i2 F (a , a F (a , a 0 bii1 0 bii2 0 biif Ltii biiti aiz2i 1 i2 a 2 i2 a bii1 bii2 bii3 ai32 biif aif2 bii ai11 ai21 ai31 ai1f aiz1i THIS ”WELL-BEHAVED” FUNCTION 10 ”FITS” REAL METHODS Φ Φ:, :,1, Φ:, :, 2,, Φ:, :, i ,, Φ:, :, n Methods of Production z1 a 11 ( z1 ,1 : n,1) ( z ,1 : n,2) a z2 2 21 Az zn ( z n ,1 : n, n) a k 1 a 1n z2 z2 a 22 a 2 n zn a nn z1 z1 a 12 z ( z1 , n 2,1) b111 0 ( z , n 2,2) 0 b z2 2 22 Bz ( z n , n 2, n) 0 0 0 zn b nn 11 Φ Φ:, :,1, Φ:, :, 2,, Φ:, :, i ,, Φ:, :, n Methods of Production ( z1 , n 1,1) L1z1 ( z , n 1,2) z2 L2 2 z L zn ( zn , n 1, n) Ln z z z (B , A , L ) This triple identifies a combination, z, of production methods used to produce the n commodities If e B z A z 0 the system is productive. 12 E B , A ,L z z z z Any productive system can be re-proportioned so as to constitute another productive system x11 0 X 0 0 0 x22 0 0 0 0 0 0 xnn z 0 0 z z z ( XB , XA , XL ) The set of all possible triples constitutes a production system E B ,A ,L z z z 13 Simple Economy Simple Productive System l- Workers l +s working units l +s consumers n commodities m production processes Let n be a large number, say 3! E. Landau 1 2 3 s – Producers (also Workers) “Products – Goods – Commodities” 1 First Commodity 2 Second Commodity 3 Third Commodity 14 Means of production necessary for the production of the quantity x11 b11 of commodity 1 1 x11L1z1 Workers 1 1 2 3 x11a11z1 Producers x11a12z1 x11a13z1 x11L1z1 x11b11z1 2 3 15 Means of production necessary for the production of the quantity x22 b22 of commodity 2 2 x22 Lz22 Workers 1 2 3 Producers 1 2 3 z2 x22a21 z2 x22a22 z2 x22a23 x22 Lz22 x22b22z2 16 Factors’ demand. Quantity bought for the production of commodity x33 b33 3 x33 L3z3 Workers 1 2 3 Producers 1 2 3 z3 x33a31 z3 x33a32 z3 x33a33 x33 L3z3 x33b33z3 17 x22 Lz22 z1 11 1 x L x33 L3z3 z z z ( XB , XA , XL ) REMARK: DECISION PROBLEMS 1 2 CAN BE ENCAPSULATED AS DIOPHANTINE EQUATIONS – TURING MACHINES ENCODABLES 3 1 x11a11z1 x11a12z1 x11a13z1 x11L1z1 2 z2 x22a21 z2 x22a22 z2 x22a23 x22 Lz22 x22b22z2 3 z3 x33a31 z3 x33a32 z3 x33a33 x33 L3z 3 x33b33z 3 x11b11z1 18 Exchange for Production Purposes 1 2 3 Exchange for Consumption Purposes 1 2 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 19 necessary – no money) Non-substitution Theorem Theorem: Relative prices are independent from the production and/or demand vector. 20 Non-substitution Theorem Number of possible combinations of processes z is t1t2t3t4t5…tn w w (r , η) z 1 ηB A (1 r ) L z z 1 z wz ( 6 ) w w z (1) z ( 4) w z ( 2) w z ( 3) Wage-Profit Curve wz (5) Wage Profit Frontier r 21 Macroeconomic Aggregates (Equilibrium values) XA z p(1 r) XLz w XB z p z GNP vY (r , η) e' XB p (r , η) z z z vYNNP (r , η) e' X(B z A z )p z (r , η) vK z (r , η) e' XA z p z (r , η) vC (r , η) e' X(B A )p(r , η) z z z z vYNNP e' XAp z (r , η)r e' XLwz (r , η) vΠ z (r , η) vW z (r , η) Quantity of NNP allocated to the owners of capital Quantity of NNP allocated22 to the workers Exchange for Production Purposes Exchange for Consumption Purposes WHAT IF? 1 2 3 1 2 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 emergence23of credit-debt ... i.e., clearing contracts). 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 24 GENERAL NON-EQUILIBRIUM SPECIFICATION OF INDIVIDUAL BEHAVIOURAL FUNCTIONS BEHAVIOURAL ECONOMICS COMPUTABLE ECONOMICS EXPERIMENTAL ECONOMICS ALGORITHMIC RATIONAL AGENT A Computable Agent 25 Agents’ decisions Heterogeneity ALGORITHMIC RATIONAL AGENT A Computable Agent 1 2 3 Behavioral Experimental Computable 26 Macroeconomic Aggregates Non Equilibrium Dynamics w SHORT-RUN LONG RUN 1 2 3 r 27 Macroeconomic Aggregates Out of Equilibrium Dynamics w LOCK-IN ? 1 2 3 r 28 Macroeconomic Aggregates Out of Equilibrium Dynamics w wz ( 6 ) w w 1 2 3 w z (1) z ( 4) z ( 2) w z ( 3) Wage Profit Frontier wz (5) r 29 w GENERAL EQUILIBRIUM wz ( 6 ) NEOCLASSICAL CASE Consistent with the Aggregated ”Cobb-Douglas” w z (1) wz (5) Wage Profit Frontier r vKWz PF e' XW PFL Capital/Labor Ratio r 30 Macroeconomic Aggregates (Equilibrium values) Stochastic Dynamic General Equilibrium Models . RBC – OLG – NEW KEYNESIANS …. Capital Market Labor Market wWPF r Capital Supply Capital Demand MPK vKWPF Labour Supply Labour Demand MPL L 31 w GENERAL EQUILIBRIUM wz ( 6 ) NOT NEOCLASSICAL CASE NOT consistent with the Aggregated ”Cobb-Douglas” w z (1) wz (5) Wage Profit Frontier r vKWz PF e' XW PFL 60% Capital/Labor Ratio r 32 Macroeconomic Aggregates (Equilibrium values) ARTIFICIAL ECONOMIC MODEL Labor Market Capital Market r ? ? wWPF Capital Supply Capital Demand MPK vKWPF ? Labor Supply ? Labor Demand MPL L 33 3 commodities, 3 producers, 27 workers, 6 methods per commodity A Simulation An example with Low Dimensional Model Frontier 4.5 2 z 2 4 4 3.5 1 2 3 Wage Rates 3 2.5 2 1.5 2 z 5 4 1 5 z 2 1 5 z 2 4 0.5 0 0 0.2 0.4 0.6 0.8 Profit Rates 1 1.2 1.4 34 Capital/Labor Ratio at the Frontier 8 7 Capital/Labor Ratio 6 5 4 3 2 1 0 0 0.2 0.4 0.6 0.8 Profit Rates 1 1.2 1.4 35 HIGHLY STRUCTURED von Neumann - Wolfram CELLULAR AUTOMATA Workers 1 2 3 1 2 3 1 2 3 Producers – Goods – Commodities” DISTINCTION“Products BETWEEN LOCAL GLOBAL VARIABLES UNIVERSAL COMPUTABILITY MASTER DIOPHANTINE EQUATION 36 HIGHLY STRUCTURED von Neumann - Wolfram CELLULAR AUTOMATA Workers 1 2 3 1 2 3 1 2 3 Producers “Products – Goods COMPLEXITY – Commodities” THE ALGORITHMIC AND COMPUTATIONAL COMPLEXITY OF THE CONCATENATED SYSTEM IS PROPORTIONAL TO THE COMPLEXITIES OF THE SMALLER UNIT 37 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 38 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). GENERAL EQUILIBRIUM (uniform prices – but not uniform profit rates) X A p (1 r ) X L w X B p * z* * * * z* * * z* * GENERAL EQUILIBRIUM (uniform prices, wage rates and profit rates) X A p (1 r ) X L w X B p* * z* * * * z* * * z* 39 n commodities, s producers, l workers, m methods The wage profit frontier is independent from dimensionality Frontier 4.5 4 3.5 1 2 3 Wage Rates 3 2 z 2 4 2.5 2 1.5 POSSIBLE BENCHMARK? 0.5 0 REPRESENTATIVE 0 SYSTEM 2 z 5 4 1 0.2 0.4 5 z 2 1 0.6 0.8 Profit Rates 5 z 2 4 1 1.2 1.4 40 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; 41 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 42 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 43 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 … 44
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