“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
ηB  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
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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 …
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