seminar se 2 st. uni klagenfurt: 814.515 und uni wien: 562.430 mathematische modellbildung und...

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+………=


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+………=


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


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


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


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





Money as commodity


Physicalbasis





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

“ “ “ “


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

“ “


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.





Money as commodity


Physicalbasis





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


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





Money as commodity



Physicalbasis




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)





Money as a commodity


Physicalbasis





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?




Money as commodity


Physicalbasis





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


Competitive Production

Physicalbasis


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


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

seminar se 2 st. uni klagenfurt: 814.515 und uni wien: 562.430 mathematische modellbildung und...

Documents