Seminar SE 2 st. Uni Klagenfurt: und Uni Wien: Mathematische Modellbildung und Simulation Ökonometrische, systemdynamische, Input-Output Modelle sowie agent-based systems Peter Fleissner Institut für Gestaltungs- und Wirkungsforschung

websites Allgemeines ell.phphttp:// ell.php Laufende Ereignisse, Skripten, Termine MathModhttp://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 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

Einleitung Im 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 VorleistungenBruttoproduktion

Volkswirtschaftliche Gesamtrechung: Entstehung Endnachfrage Wertschöpfung Sektor n1Sektor n2Sektor n…….. = BIP=n1+n2+………= VorleistungenBruttoproduktion

Volkswirtschaftliche Gesamtrechung: Verwendung Endnachfrage = BIP=c+g+i+ex-im = Wertschöpfung Privater Konsum c Öffentl. Konsum g Investitionen i Exporte exminus Importe im Sektor n1Sektor n2Sektor n…….. = BIP=n1+n2+………= VorleistungenBruttoproduktion

Volkswirtschaftliche Gesamtrechung: Verteilung Endnachfrage Wertschöpfung Privater Konsum c Öffentl. Konsum g Investitionen i Exporte exminus Importe 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 Wertschöpfung Privater Konsum c Öffentl. Konsum g Investitionen i Exporte exminus Importe 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 ATSSectorj=1j=2j=3final dmd Y Output X i=118396, , ,134724, ,00 i=219404, , , , ,00 i=39569, , , , ,00 sum , , ,00 value added 64830, , ,00 output112200, , ,00 Direct labor Persons Vorleistungsmatrix Z = { Z ij } End- nach- frage Brutto- Produk- tion Wertschöpfung V Bruttoproduktion X

Empirical view: matrix notation [monetary units] Z = { Z ij } Y = { Y i } Endnachfrage V = { V j } Wertschöpfung VorleistungenBruttoproduktion X = { X i } X = { X j } Zeilen: Z 1 + Y = X Spalten: 1Z + 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: a ij = z ij /x j Result: Matrix A = {a ij } 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 1976 ATS i per ATS j Sectorj=1j=2j=3 i=10,160,130,02 i=20,170,340,14 i=30,090,120,18 sum1+2+30,420,590,34 value added/ output 0,580,410,66 Stand. output 1,00 l = labor/ output Persons per mill ATS 3,291,972,93 Technol. coeff matrix A = { a ij }

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 /(1c) (consumption matrix, w wages) S = i p / (1i) (surplus matrix, i investmt, p profits) Fixed capital is represented by K = i k / (1i) (capital matrix, k fixed capital) Ax + Cx + Sx = x

Extended Reproduction Labor PowerIntermed. products Capital investment Fixed Capital Intermed. goods Consumer goods

Consumption matrix C: Example Austria 1976 million ATS sectorj=1j=2j=3consum- tion c con-i= sumpti- on i= matrixi= wages(sum) output Ci=1 0, , , standari=2 0, , , dizedi=3 0, , ,

Idealized view: matrix notation [amounts, unit prices] Z = { p i a ij x j } = Y = { p i y i } = Endnachfrage V = { v j x j } = Wertschöpfung VorleistungenBruttoproduktion X = {p i x i } x…amount (Stück, Anzahl), (column) p…unit price, v…unit value added (row) Zeilen: Ax + y = x Spalten: pA + v = p Summen: pAx + vx = px X = {p j x j }

Inverse view Z = { p i a ij x j } = Y = { p i y i } = Endnachfrage V = { v j x j } = Wertschöpfung VorleistungenBruttoproduktion X = {p i x i } Zeilen: x = (I – A) -1 y Leontief-Inverse (I – A) -1 = I + A + A 2 + A Von Neumann Reihe Spalten: p = (I – A) -1 v

Excursion: Linear Programming: standard form Primal and dual problem with volumes x and prices p Primal Linear Program Maximize the Objective Function (P) P = cx subject to Ax <= b, x >= 0 Dual Linear Program Minimize the Objective Function (D) D = pb subject to pA >= c, p >= 0

Excursion: Linear Programming: standard form Primal and dual problem with amounts x and prices p Max: 2.x x 2 subject to 3x x 2 <= 24 8x x 2 <= 24 1x x 2 <= 4 x >=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 use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as a commodity Small Commodity Production Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices Competitive Production Fixed capital present

Economic Reality – a complex construction Values in use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as commodity Small Commodity Production Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices Competitive Production Fixed capital present

Values in use [observed physical units] 1/3 Output Final Demand Cons/Inv Intermediary goods A diag(x) j=1 2 3 …. i = Ax + y = x

Values in use [observed physical units] 2/3 Output Final Demand Cons/Inv/ Exp/-Imp Intermediary goods A diag(x) j=1 2 3 …. i = Ax + Cx + Sx = x Capital Investment 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?

Output x Final Demand y Intermediary flows Z other inputs minus waste (i-d) Output j=1 2 3 …. i = Z 1 + y = x 1Z +i-d =x Material flows [physical units]

Material Flows: Example Austria 1983 tonsSector123final dmdoutput sum input waste output labor

Energy Flows: Example Austria 1986 terajouleSector123final dmdoutput sum imports value added output

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 use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as commodity Small Commodity Production Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices Competitive Production Fixed 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 Ware besitzt 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 Output Final Demand Cons/Inv/ Exp/-Imp Intermediary goods Value added { Lj = lj. xj } Output L=lx wx Use of GDP Distribution of GDP wAx j=1 2 3 …. i = l,w.. row vectors

Output Final Demand Cons/Inv/ Exp/-Imp Intermediary values Value added {L j =l j.x j } Output Use of GDP Distribution of GDP j=1 2 3 …. i = wA + l = w w =l(E-A) -1 Labor Unit Values w [labor time] 2/7

Labor content: Example Austria 1976 Person- years Sector123sum final dmdoutput sum sum direct labor sum labor value of output sum

(Labor time) value of output 3/7 w = c + n = c + v + s w labor time value of output c constant capital n newly created value = socially necessary labor v variable capital (wage bill) s surplus value

Labor value is fixed in the market 4/7 w = c + n = c + v + s There 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 + v Capital available after selling: K'= c + v + s Rate of surplus (exploitation rate): e = s/v Organic composition of capital: o = v/(c + v) Classical rate of profit per production period: r = s / (c + v) = e.o Annual rate (dimensions corrected): r a = s / (c f + c z.T z + v.T v ) c f … fixed constant capital c z … circulating constant capital T z,T v … 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 use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as commodity Small Commodity Production Competitive Production Fixed capital present Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices

Final Demand Intermediary values wages profits Use of GDP Distribution of GDP j=1 2 3 …. i = p p (A+C)(1+r)=p p Prices of production p p [currency units] 1/3 r … uniform rate of profit Output

How to determine the average rate of profit? p p (A+C)(1+r) = p p -> p p [ 1/(1+r) I – A - C ] = 0 Eigenvalue-Equation for 1/(1+r) : p p (A + C) = p p Non-trivial solution for p p, if and only if determinant of the matrix MDET( I – A - C ) = 0 Hint: Solve it in Excel by its solver-function

Marxs 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 solution Iterative application of Marxs method

Solutions to the transformation problem Labor Values Marxs solution First iteration Second iteration Third iteration Bortkiewiczs solution Rate of profit Method Marxs method, iteratively applied, converges to Bortkiewiczs solution (1907)

Economic Reality – a complex construction Values in use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as a commodity Small Commodity Production Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices Competitive Production Fixed capital present

Final Demand Intermediary values wages profits Use of GDP Distribution of GDP j=1 2 3 …. i = p(A+C+S)= p Observed market prices p [currency units] different profit rates Output

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 use Goods produced family style Values in Exchange Goods and labor power as Commodities Prices of production Uniform profit rates Money as commodity Small Commodity Production Physical basis Stock market based economy Stocks as commodities Credit based economy Contemporary economic system Market prices Competitive Production Fixed 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 Commodity Production Competitive Production Physical basis 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 labor and 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] Output Final Demand Cons/Inv/ Exp/-Imp j=1 2 3 …. i = C 11 C 12 C 21 C 22 * = w 1 C 12 x 2 + w 2 C 22 x 2 L 1 0 Output S 11 <= S * A 11 A 12 A 21 A 22 * w 1 C 11 x 1 + w 2 C 21 x 1 L1L1 Total surplus value L 1 – (w 1 C 11 +w 2 C 21 )x 1 *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