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Seminar SE 2 st.Uni Klagenfurt: 814.515 und Uni Wien: 562.430
Mathematische Modellbildung und Simulation
Ökonometrische, systemdynamische, Input-Output
Modelle sowie agent-based systems
Peter FleissnerInstitut für Gestaltungs- und Wirkungsforschung
websites
Allgemeines
• http://www.iff.ac.at/socec/lehre/lehre_aktuell.php
Laufende Ereignisse, Skripten, Termine
• http://cartoon.iguw.tuwien.ac.at/zope/lvas/MathMod
Termine• Vorbespr: Donnerstag; 3. März 2005, 17 Uhr • 1. Block: Montag, 7. März (9 -17 Uhr)• 2. Block: Donnerstag, 17. März (9 Uhr)
– Vortrag ANYLOGIC TEAM (Ort: Kontaktraum der TU, Gusshausstrasse 25-29, 6. Stock)
• 3. Block: Montag, 4. April (9 -17 Uhr)• 4. Block: Montag, 11. April (9 -17 Uhr)• 5. Block: Montag, 2. Mai (9 -17 Uhr)
– Ab 15:30 Uhr: Vortrag DI Klug, Seibersdorf
• Die ganztägigen Termine finden im Seminarraum 187-2 bzw. im Computerlabor des IGW statt
Ausblick
Teil 5 (Montag, 2. Mai, ganzer Tag)• Agent-based modelling
• Praktische Beispiele, Vortrag DI Klug ab 15:30 Uhr
Teil 4
Montag, 11. April, ganzer Tag
• Volkswirtschaftliche Gesamtrechung (Einführung)• Grundzüge der Input-Output-Analyse,• Mehrebenenökonomie • Anwendungen auf volkswirtschaftliche Modelle,
Stoffstromrechnung
EinleitungIm 1. Teil des Seminars haben wir uns mit einer
spezifischen Widerspiegelungsform der Welt als • mathematisches Konstrukt
– Systeme nichtlineare Differential/Differenzengleichungen
und seiner Vergegenständlichung als • Computerprogramm
– Stella/Ithink/Dynamo/Vensim…beschäftigt. Leitgedanke: Die Welt als System von miteinander
wechselwirkenden Bestands- und Flussgrößen Damit sind beliebige komplexe dynamische Vorgänge
gut beschreibbar (vom Ort wird meist abstrahiert). Die Variablen haben eine vorgegebene Qualität und zu
jedem Zeitpunkt einen quantitativen Wert.
Einleitung
Diese Sicht erlaubt die Modellierung von Strukturen, die sich aus Bilanz- und Verhaltensgleichungen zusammensetzen:
Im Teil 2 des Seminars haben wir die Konstruktion von Verhaltensgleichungen mittels Regressionsanalyse studiert.In folgenden Teil 3 werden spezielle Bilanzgleichungen im Bereich der Volkswirtschaft analysiert.
Grundelemente der
Volkswirtschaftlichen Gesamtrechung
Volkswirtschaftliche Gesamtrechung: Grundschema
Endnachfrage
Wertschöpfung
Vorleistungen Bruttoproduktion
Volkswirtschaftliche Gesamtrechung: Entstehung
Endnachfrage
Wertschöpfung
Sek
tor
n1
Sek
tor
n2
Sek
tor
n…
…..
= BIP=n1+n2+………=
Vorleistungen Bruttoproduktion
Volkswirtschaftliche Gesamtrechung: Verwendung
Endnachfrage
= BIP=c+g+i+ex-im =Wertschöpfung
Priv
ater
Kon
sum
c
Öffe
ntl.
Kon
sum
g
Inv
estit
ione
n i
Exp
orte
ex
min
us Im
port
e im
Sek
tor
n1
Sek
tor
n2
Sek
tor
n…
…..
= BIP=n1+n2+………=
Vorleistungen Bruttoproduktion
Volkswirtschaftliche Gesamtrechung: Verteilung
Endnachfrage
WertschöpfungP
rivat
er K
onsu
m c
Öffe
ntl.
Kon
sum
g
Inv
estit
ione
n i
Exp
orte
ex
min
us Im
port
e im
Löhne v
Unv. Gewinne pr
Abschreibungen d
Eink Selbständiger s
Ind Steuern min Sub
= BIP=c+g+i+ex-im =
= BIP=v+pr+s+ind+d =
= BIP=n1+n2+………=
BruttoproduktionVorleistungen
National Economic Accounting: Input-Output Scheme
Endnachfrage
WertschöpfungP
rivat
er K
onsu
m c
Öffe
ntl.
Kon
sum
g
Inv
estit
ione
n i
Exp
orte
ex
min
us Im
port
e im
Löhne v
Unv. Gewinne pr
Abschreibungen d
Eink Selbständiger s
Ind Steuern min Sub
= BIP=c+g+i+ex-im =
= BIP=v+pr+s+ind+d =
= BIP=n1+n2+………=
BruttoproduktionVorleistungen Bruttoproduktion
Current prices: Example Austria 1976
million ATS
Sector j=1 j=2 j=3 final dmd Y
Output
X
i=1 18396,73 77305,34 11773,13 4724,80 112200,00
i=2 19404,07 210142,46 75713,31 307308,15 612568,00
i=3 9569,20 72819,19 99498,56 361828,05 543715,00
sum 1+2+3 47370,00 360267,00 186985,00
value added
64830,00 252301,00 356730,00
output 112200,00 612568,00 543715,00
Direct labor
Persons 369610 1207657 1594369
VorleistungsmatrixZ = { Zij }
End-nach-frage
Brutto-Produk-
tion
Wertschöpfung V
Bruttoproduktion X‘
Empirical view: matrix notation [monetary units]
Z = { Zij } Y = { Yi }
Endnachfrage
V = { Vj }
Wertschöpfung
Vorleistungen Bruttoproduktion
X = { Xi }
X‘ = { Xj }
Zeilen: Z 1 + Y = X Spalten: 1’Z + V = X’Symbols in caps!!
How can we characterize the I-O system?
Try to find invariants which will increase the understanding of the economy and allow also for comparisons -> standardize the figures
Easy procedure: divide each figure of the intermediary table by the corresponding output of the sector. Be aware of the units of measurement!
The figures of one column are divided by the same numbers:
aij = zij/xj
Result: Matrix A = {aij } of technical coefficients: input needed for the production of one unit of output
(in this case in monetary units, e.g. Euro or ATS)
Standardized I-O: Example Austria 1976ATSi per
ATSj
Sector j=1 j=2 j=3
i=1 0,16 0,13 0,02
i=2 0,17 0,34 0,14
i=3 0,09 0,12 0,18
sum 1+2+3 0,42 0,59 0,34
value added/
output
0,58 0,41 0,66
Stand.
output
1,00 1,00 1,00
l = labor/
output
Persons per mill ATS
3,29 1,97 2,93
Technol. coeffmatrix A = { aij}
Simplified two-class model of extended reproduction
There are only two loops in the system, • accumulation of capital and • reproduction of labor power• No explicit public sector, closed economy• All capital investment goes to capitalists• All consumption goes to laborers
Final demand approximated by two matrices • C = c w‘ /(1‘c) (consumption matrix, w wages)• S = i p‘ / (1‘i) (surplus matrix, i investmt, p profits)Fixed capital is represented by • K = i k‘ / (1‘i) (capital matrix, k fixed capital)
Ax + Cx + Sx = x
Extended Reproduction
Labor PowerIntermed. products
Capital investment
Fixed Capital Intermed. goods
Intermed. goods Consumer goods
Consumption matrix C: Example Austria 1976
million ATS
sector j=1 j=2 j=3 consum-
tion c
con- i=1 1027 9859 13733 24619
sumpti-on
i=2 6536 62710 87353 156598
matrix i=3 7254 69601 96952 173807
wages (sum) 14817 142169 198038
output 112200 612568 543715
C i=1 0,009158 0,0160939 0,0252573
standar i=2 0,058250 0,1023715 0,1606593
dized i=3 0,064651 0,1136215 0,1783146
Idealized view: matrix notation [amounts, unit prices]
Z = { pi aij xj } =
Y = { piyi } =
Endnachfrage
V = { vj xj } =
Wertschöpfung
Vorleistungen Bruttoproduktion
X = {pixi}
x…amount (Stück, Anzahl), (column)p…unit price, v…unit value added (row)
Zeilen: Ax + y = x Spalten: pA + v = pSummen: pAx + vx = px
X‘ = {pjxj}
^xA
^p yp
^
^
xv
Inverse view
Z = { pi aij xj } =
Y = { piyi } =
Endnachfrage
V = { vj xj } =
Wertschöpfung
Vorleistungen Bruttoproduktion
X = {pixi}
Zeilen: x = (I – A)-1y Leontief-Inverse (I – A)-1=
I + A + A2 + A3 +..Von Neumann ReiheSpalten: p = (I – A)-1
v
^xA
^p yp
^
^
xv
Excursion: Linear Programming: standard formPrimal and dual problem with volumes x and prices p
Primal Linear Program
Maximize the Objective Function (P)P = c‘x subject toAx <= b, x >= 0
Dual Linear Program
Minimize the Objective Function (D)D = p‘b subject to p‘A >= c‘, p >= 0
Excursion: Linear Programming: standard formPrimal and dual problem with amounts x and prices p
Max: 2.x1 + 1.x2
subject to3x1 + 8 x2 <= 248x1 + 3 x2 <= 241x1 + 1 x2 <= 4x >=o
Multi-level Economics
Multilevel Economics
1. How to look at the economy?
2. Appearance and Essence
3. A multilevel perspective
4. Three ways to understand “productivity”
5. Labor values and price systems
6. Transformation of values into prices
7. How to handle services?
Looking through the surface
• General rule– From empirical findings to abstractions– From appearance to essence– and back
• Application of the rule– From observed market prices to labor values
and use-values– and back
Economic Reality – a complex construction
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as a commodity
Small CommodityProduction
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Competitive ProductionFixed capital present
Economic Reality – a complex construction
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as commodity
Small CommodityProduction
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Competitive ProductionFixed capital present
Values in use [observed physical units] 1/3
Ou
tpu
tFinal Demand
Cons/Inv
Intermediarygoods
A diag(x)
j=1 2 3 ….
i=123....
Ax + y = x
“ “ “ “
Values in use [observed physical units] 2/3
Ou
tpu
tFinal DemandCons/Inv/Exp/-Imp
Intermediarygoods
A diag(x)
j=1 2 3 ….
i=123....
Ax + Cx + Sx = x
CapitalInvestment
S diag(x)
Con-sumption
C diag(x)
=“ ““ “
(A+C+S)x= Tx = x
“ “
Values in use [observed physical units] 3/3
Two physical ways how to make goods commeasurable:
• By mass/weight• By energy content/dissipated heat/entropy
If we have commeasurable variables, we can add up the columns of the table, not only the rows
Problems: • How to handle services, waste?• What are the balance equations?
Ou
tpu
t xFinal
Demandy
Intermediaryflows
Z
other inputs minus waste
(i-d)
Output
j=1 2 3 ….
i=123....
Z 1 + y = x 1’Z +i-d =x’
Material flows[physical units]
Material Flows: Example Austria 1983
tons Sector 1 2 3 final dmd output
1 30545305 18172255 12045635 39127282 99890477
2 2065667 7593987 5033740 5525759 20219153
3 0 0 0 0 0
sum 1+2+3 32610973 25766242 17079375
input 71757963 7389707 0
- waste -4478459 -12936796 -17079375
output 99890477 20219153 0
labor 369610 1207657 1594369
Energy Flows: Example Austria 1986
terajoule Sector 1 2 3 final dmd output
1 181725 122719 43311 116725 464480
2 53924 181721 62689 190779 489113
3 5084 1906 912 1788 9690
sum 1+2+3 240734 306346 106912
imports 89982 400735 37899
„value added“
133764 -217967 -135121
output 464480 489113 9690
But there is a third way also. It is leading to the societal sphere…..
Human labor is THE activity which makes human beings different from animals
Excursion: what is the difference between labor and other human activities?
Human labor and other human activities
Labor is a special human activity which could in principle be performed by someone else. The person performing the activity is replaceable.
This is in contrast to human activities which are related to the individual personality and individual experience =>
Labor bears an internal and an external aspect for the individual worker
Labor does exist formally and informally
While formal labor is remunerated by wages (like in the case of employees) or compensated by income via the market (like in the case of self-employed),
informal labor is not directly compensated financially (e.g. the work of wives in households or social work during leisure time).
The contemporary economic system does not deal with informal labor, nevertheless a comparison between formal and informal labor might be interesting.
Economic Reality – a complex construction
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as commodity
Small CommodityProduction
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Competitive ProductionFixed capital present
Fragestellung„Die positive Fragestellung, die sich NUR mit Hilfe der Arbeitswerttheorie beantworten lässt, geht von der Tatsache aus, dass die wachsenden Wirtschaften laufend einen größeren Output als Input erstellen, d.h. der Ertrag übersteigt die Kosten. Ertrag und Kosten können prinzipiell in drei Dimensionen definiert werden, in Mengengrößen, in monetären (Preis-) Größen und Arbeitswertgrößen.Die Mengenrechnung hat nur sehr eingeschränkte Verwendbarkeit, da sie bei den vielen heterogenen Gütern zu keiner einheitlichen Ertrags- bzw. Kostenziffer führt. Anders ist dies bei der monetären und Arbeitswertrechnung.Während jedoch die monetären Größen keine exakte Widerspiegelung der Menge darstellen, sondern sich bei inflationären Entwicklungen davon lösen, bestehen zwischen Gütermengen und den in ihnen steckenden Arbeitsmengen genaue technische Beziehungen. Es liegt somit nahe, zu untersuchen, wie mit einem gegebenen Arbeitsquantum als Input eine wachsende Gütermenge hergestellt werden kann und welche Relation zwischen Arbeitsmenge und Gütermenge besteht. Um den Output (Brutto- oder Nettoproduktion einer Volkswirtschaft) und den Input als Ausdruck von Arbeitsmengen begreifen zu können, ist es notwendig, die Produktionsmittel, also das sogenannte Kapital, auf Arbeit zurückzuführen."
(H.G.Zinn, Arbeitswertheorie, Westberlin, 1972, S.10)
Die Warebesitzt • Gebrauchswert (value in use, Nutzen)• Tauschwert (value in exchange)
Wertgröße = durchschnittliche gesellschaftlich notwendige Arbeitszeit (individuellem Arbeitszeitaufwand)
Mehrwert = Gesellschaftliche Arbeitszeit, die über jene Arbeitszeit hinausgeht, die für die einfache Reproduktion notwendig ist
Technischer Fortschritt reduziert den Stückwert = durchschnittl. Arbeitszeitaufwand pro Stück
Labor Values w[labor time] 1/7
Ou
tpu
tFinal DemandCons/Inv/Exp/-Imp
Intermediarygoods
Value added { Lj = lj .
xj }
Output
L=lx
wx
Use of GDP
Distribution of GDP
wAx
j=1 2 3 ….
i=123....
l,w.. row vectors
Ou
tpu
t Final DemandCons/Inv/Exp/-Imp
Intermediaryvalues
Value added{Lj=lj.xj}
Output
Use of GDP
Distribution of GDP
j=1 2 3 ….
i=123....
wA + l = w w=l(E-A)-1
Labor Unit Values w[labor time] 2/7
Labor content: Example Austria 1976
Person-years
Sector 1 2 3 sum
1+2+3
final dmd output
1 99568 418396 63719 581683 25572 607255
2 94423 1022581 368431 1485435 1495402 2980836
3 43655 332202 453913 829770 1650663 2480432
sum 1+2+3 237646 1773179 886063 sum 2896888
direct labor
369610 1207657 1594369 sum 3171636
labor
value of output
607255 2980836 2480432 sum 6068524
(Labor time) value of output 3/7
w = c + n = c + v + s
w labor time value of outputc constant capitaln newly created value = socially necessary laborv variable capital (wage bill)s surplus value
Labor value is fixed in the market 4/7
w = c + n = c + v + sThere is a difference between
• individual value • market value (average value)
Market • Engine to compare individual performance
with the performance of society• Rewards the efficient• Punishes the less productive• Creates a tendency towards more efficiency
Labor values• shrinking with technical/organis. improvements
Various indicators 5/7
Capital advanced: K = c + vCapital available after selling: K'= c + v + sRate of surplus (exploitation rate): e = s/vOrganic composition of capital: o = v/(c + v)Classical rate of profit per production period:
r = s / (c + v) = e.oAnnual rate (dimensions corrected):
ra = s / (cf + cz.Tz + v.Tv)cf … fixed constant capital
cz … circulating constant capital
Tz,Tv … turnover periods
Classical indicators in matrix terms 6/7
Capital advanced: K = w(A+C)x
Capital available after selling: K'=wAx+L
Rate of surplus: e = (L - wCx)/wCx
Organic composition of capital:
o= wCx/w(A+C)x
Classical rate of profit per production period:
r = (L - wCx)/w(A+C)x = e.o
Implicit assumption 7/7
Producers are fully compensated for their effort they have put into the production of goods
Economic Reality – a complex construction
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as commodity
Small CommodityProduction
Competitive ProductionFixed capital present
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Final Demand
Intermediaryvalues
wages
profits
Use of GDP
Distribution of GDP
j=1 2 3 ….
i=123....
pp(A+C)(1+r)=pp
Prices of production pp [currency units] 1/3
r… uniform rate of profit Output
Ou
tpu
t
How to determine the average rate of profit?
pp(A+C)(1+r) = pp -> pp[ 1/(1+r) I – A - C ] = 0
Eigenvalue-Equation for 1/(1+r) :
pp (A + C) = pp
Non-trivial solution for pp, if and only if determinant of the matrix
MDET( I – A - C ) = 0
Hint: Solve it in Excel by its solver-function
Marx’s solution of the transformation of labor values into prices of production
• Basis: costs of production In terms of labor values of commodities
• Profit added Share of total value added proportionally to costs of production
• Result“prices of production”
• Intrinsic problem input prices are different from output prices
• Proposed solutionIterative application of Marx’s method
Solutions to the transformation problem
Labor Values
Marx’s solution
• First iteration
• Second iteration
• Third iteration
Bortkiewicz’s solution
1.140
1.035
1.030
1.025
1.025
1.140
Rate of profit Method
Marx’s method, iteratively applied, converges to Bortkiewicz’s solution (1907)
Economic Reality – a complex construction
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as a commodity
Small CommodityProduction
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Competitive ProductionFixed capital present
Final Demand
Intermediaryvalues
wages
profits
Use of GDP
Distribution of GDP
j=1 2 3 ….
i=123....
p(A+C+S)= p
Observed market prices p [currency units]
different profit rates
Output
Ou
tpu
t
Conclusions on price systems
• While A and C may remain constant, • Changes in the price system go together with
changes of the surplus matrix S • Changes of S must leave the row sums of
Sdiag(x) invariant (no change in the surplus product, but change in the distribution of profits)
• In mathematical terms this can be done by post-multiplication of S by an appropriately chosen Markov-matrix
• The changes result in a change of sectorial profits resp. surplus values, and sectorial rates of profit resp. rates of surplus
How to treat services on the labor value level?
Values in useGoods produced “family style”
Values in ExchangeGoods and labor power as Commodities
Prices of productionUniform profit rates
Money as commodity
Small CommodityProduction
Physicalbasis
Stock market based economy Stocks as commodities
Credit based economy
Contemporary economic system Marketprices
Competitive ProductionFixed capital present
Does every wage-earner create surplus value?
w = c + n = c + v + s
Here comes the story of an agrarian society consisting only of a farmers.
What will be the effect to the surplus if
a. They bring an additional farmer into their society?
b. They bring a shaman into their society?
Three kinds of the productivity of labor
Productivity (1) Production of use-values
Productivity (2)Production of values in exchange
Productivity (3)Profitable Production
Small CommodityProduction
Competitive Production
Physicalbasis
Credit based economy
Contemporary economic system
Stock market based economy
Does every worker create additional value? My answer is: “no”
• Value is a concept to make qualitative different things comparable
• The common background is labor power embodied in physical goods
• If there is no physical/material good, there is no surplus, and there is no value nor surplus value
• Accumulation is based on surplus and surplus value, which is the basis for profits and the profit rate
• This statement is equivalent to: human beings cannot live without a material environment and without exchange of material flows (metabolism) with nature
There is value creating laborand value consuming labor
• Value creating labor results in things, which can be stored, resold and accumulated
• Value consuming labor results in pure use-values, which are useful, but cannot be stored, neither resold nor accumulated
• A typical sector of value consuming labor is the service sector
• All the value consuming sectors have zeros in the rows of the S (surplus) matrix = their output cannot be invested
• But, their output can have important indirect effects (e.g. increasing the productivity of labor)
Examples and borderline cases of
value consuming production Services
Education
Entertainment (opera ticket)
Material production
The story of the coffee mugs
Customer specific production
Energy transformation
Software production
Labor Values w[labor time]
Ou
tpu
tFinal DemandCons/Inv/Exp/-Imp
j=1 2 3 ….
i=123....
C11 C12
C21 C22
*
=“ ““ “ “ “
w1C12x2+ w2C22x2
L1 0
Output
S11 <= S12
0 0
*
A11 A12
A21 A22
*
w1C11x1
+w2C21x1
L1
Total surplus valueL1 – (w1C11+w2C21)x1
*Note: All matrices have to be premultiplied by diag(w) and postmultiplied by diag(x)
Possible effects of an expansion of value consuming sectors
Effects of first order (ceteris paribus)
• Decrease of the average rate of profit
• Reduction of the rate of economic growth
• Example: Outsourcing and GDP (“vertical” and “horizontal” growth, labor productivity change)
Possible secondary effects
• Services may increase their own productivity of labor and the productivity of other sectors by improving technology and management techniques
• This may compensate for the effects of first order
Danke für Ihre Aufmerksamkeit!
Nächster Termin: 2. Mai, 9:00 Uhr
Ab 16:30 Uhr
Vortrag von Herrn DI Klug
über agentenbasierte Systeme
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