A System of Categorized Normal Distribution
Population Model will perhaps bring a Revolution

in an Economic Science

Masaichi HIRAYAMAThe Summary of High Points＊＊＊＊＊＊＊ Preface ＊＊＊＊＊＊＊（1）． Period from 1930 to 1980， the author calls it a policybeloving

age for an Economic Science．
（2）． What are the reasons why such up‑side‑down‑concept， tting a cart before a horsePut‑

has reigned more t；han half a century

in Economics as well as in Political World？
（3）． A research．to find a proper an'?right model for the economic

science is entirely different from a research to solve a problem
of a limited field of an economic policy．

（4）． Hypothesis of a Rational or of a Theoretical Economics is a
Fallacy．

（s）． We all know the Central Limit Theory which prohibits
Categorization of any kind， and which automatically prohibits to
use Subjective Economic Value System in the ：operational system．
How can we call such Macro Econometrics as an Economic Science ？
（6）． The Limits Theory of Categorization．
（7）． Theory of Half Normal．
一 1 一徳山大学論叢 10周年記念論文集（第16号）（8）． Elasticity Coefficients．

（9）． From one to Several Categorized Normal Population Distribu‑

tion of Subjective Value System V．S． Many Categorized Normal
Population Distributions， the lntegral Parts of the Corresponding

Subjective Value System （Categorized Objective Values of Goods
and Services）．

（10）． Total System of Japanese lnternal Economy based on the

Economic Marginal Theorem．
（11）． Linkaged Parametric Simulation System， Consumer Economics
Example No．1．

（12）． Summary Table for Elasticity Coefficients by lncome Strata
of Japanese Household in 1959．

（13）． A Linkaged Parametric Simulation System．
（14． A Separate Prodticer Producing Several Products．
a5）． Example No．2． Butter Consumption Estimate Simulation for 1960．
（16）． Example No．3． Calculation Table for Basis of Linkaged Para‑

metric S；．mulation for 10％ lncrease in Butter Production （1200
tons） for Upper 10 Enterprises in the 35th Year of Showa．
（17）． Exarnple No．4． Calculation 'Table for Basic Linkaged Para‑

metric Simulation of Oligopoly lndustry， 1977．

＊＊＊＊＊＊＊ Preface ＊＊＊＊＊＊＊
This paper is written in answer to Late Professor Harold Hotel‑
ling's paper， entitled，

lmpact of R． A． Fisher

that was given me

by him personally at the Tokyo lmperial Hotel， where and when
I．S．S． session was held there．

Present paper is rather brief one， but it is the concluding‑ 2 一paper of my past 16 years experimental and research study．
Between times， however 1 have written several numerical tables，
books and many essays on the related topics．
1． Period from 1930 to 1980， the author calls it a policybeloving

age for an Economic Science．
Modern Statistics has been right to have a normal distribution
belonging to the same family for its population． model． This isbecause， so far as the economic date are subjected to an ex‑

perimental design of some sort， through various methods ofrandomization processes， the researcher can bring the data to thepoint at which he can apply an analysis of variance．

However， the economic data that are not subject to an
experimental design， Economists have been quite odd at them，and have been unable to find a correct and right distributionalmodel for them．
Under pressure of the emergency situation 1930s， economists aswell as politicians all over the countries could not help but adopted

Keynsian Theorem，

lnflation had better been adomitted rather

than letting the rate of unemployMent increasing freely．
Without exception， the followers of the Keyns：，an Theorem， thepolicy‑loving economists and econometricians have not only wasted
their energies but also have let peoples of the world away fromapproaching the true story of the economic science． lt seems tome that the policy‑loving economists and econometoriians havebeen beating along the bushes fruitlessly past half a century．
Since a scientific procedure must proceed the policy to attain an

aim． Econometric science basing on a hypothesis of a theoretical
economics or a simultanious equations that has been stinged with‑ 3一

徳山大学論叢 10周年記念論文集（第16号）
a strOng doubt， won'．t last long．

2． What are the reasons why such up‑side‑down‑concept，ting a cart before a horsePut‑

has reigned more than half a century

in Economics as well as in Political World？
As th e author wishes to write as briefly as possible， 一he canmention three reasons as followes ：lst reason， what he can mention is， of course， the pressure of thebusiness depression of 1930s， that necessitated an urgent financial

action on the part of the Governments'to save the National
Economies．
2nd reason has been the fame and authority of J． M． Keynes．
3rd reason has been that the researchers have failed to find aproper and right population modei for the economic science， buthave mentioned some mathematical equations based on a prevailingcommon sense in the mathematical world．
3． A research to find a proper and right model for the economic
science is entirely different from a research．． to solve a problem
of a limited f．ield of an economic policy．

So far as the author knows， neither economist， nor mathe‑
matician or statistician has ever attempted to find a proper andright model or hypothesis bn which the economi'c science can bebuilt． ln order to do so， the following conditions have to be met
by a single model or a single hypothesis．

（1） The model or the hypothesis must be inclusive of the
Marginal Theorem on an operational state．
（2） The model must overcome the difficulties．of non‑experimental

state of economic data．

（3） Reproductive property of the parametric information must
一一 4 一be maintained．

（4） Additive property of the parameters must be maintained
throughout the operational system．
（5） The system should be operational and simulational at thesame time．
（6） Values of the paramaters in the operation must correspondactual da'ti｛ realized in the ec；ono｛'！ttic． worlcl．

（7） Dynamic concept （income concept） retained in the economic
data must be treated very carefully in such a way that the effect‑

iveness of the dynamic information can be ret：ained in the mostsuitable and stable condition．

These very rigid 7 conditions are given for the purpose offinding an effective hypothesis or a population model to deal withthe economic science． Judging from the informational science， acertain familyっf a Population of Normal Distributional Model wm
probably do the best work．

4． Hypothesis of a Rational or of a Theoretical Economics is
a Fallacy．

The Theoretical Economics based on an Objective Economic
Value System without Categolization， the Economic Subjective
Value System does not have or can not have a full informationthat we need to start to ana！v． se with．

Without Categolization of the Economic Subjects， we cannot put

the information of the Marginal Theorem in the economic oper‑
ational model in identifiable way． The proof of this fact is that the

Mathematical Model or Simultaneous Equations based on the
Theoretical Economics must always bripg a very troublesome
problem， if not a questionable，‑ 5 一

ldentification Problem， since， if

徳山大学論叢 10周年記念論文集（第16号）

our assumption iS based not on the Theoretical Economics， but onsome Axiomatic Assumption such as

the Marginal Theorem

we should not be so nervous as to rely on such an，

ldentification

Problem
Defective points of Theoretical Economics may be summed up
as follows ：（a） Among others， a hypothesis of Theoretjcal Economics lacks

information about Marginal Theorem．
fo） Thus it lacks an Axiomatic Hypothesis．

（c） Theoretical Economics lacks entirely information about thesystem of Categorization． The lacks of the concep． t of thecategolization in the social science is said to have no strategic

information or behavior patterns of the economic subjects．

（d） Central Limit Theory upon which a normal distribution of
random error of the simultaneous equations based， has nothing
to do with Gaussian Normal Laws of Error which is ristricted
to apply a primar．v distribution， while Central Limit Theory
can on！y be applied to．the Secondary Distribution， and thatmeans， that the theory furnishes us with a consolidated infor‑
mation， but prohibits Categorization of any kind．

5． We all know the Central Limit Theory， which prohibits
Categorization of any kind， and which automatically prohibits

to use Subjective Economic Value System in the operational
system． How can we call such Macrp Econometrics as an Eco‑nomic Science？

The Marginal Theorem furnishes us with a full information
for the Subjective EconQmic Value System， while the system of

the Macro Econometorics based on a system of simultaneous
一一 6 一equations furnishes us but with a consolidated total objective

economic value information （Market Price times Quantity） andno Categしrization． The system does not furnishes us with any

information corresponding to the actual date realized in pro‑duction as well as in consumption．
The best we can expect the reult of， or the simmulation of theMacro Economics are， a General price level， a General pro‑duction level， a General stock level， a Total export and importlevel， a General unemployment level with certain probability

allowances
Economic Value System consists of two Systems， Subiective
Value and Objective Value， Subjective Economic Value System

supplies a full information about the Marginal Theorem of
Economics which consist of Elasticity Coefficients and Engel
Coet'ficients， while Objective Economic Va1ue System supplies
only the market price times the quantity that can be derived

from the information furnished through Engel Coefficients．
Therefore， this fact has proved that the Subjective Economic
Value System embraces the Obje'ctive Value System completely．

After a long research study， the author has found that aright and proper hypothetical population model for the economic

science in order to consolidate the foregoing 7 conditions andproblems discussed in Sections， 3， 4， ・and 5， is the marginal

theorem and a System of a Categorized Normal Population
Model for the Economic Science．
6． The Limits Theory of Categorization．
What the limits theory of Categolization in the social sciencecorresponding to the Normal Distribution condition is equal to
一一一 7 一

徳山大学論叢 10周年記念論文集（第16号）

what the Lirnits Theory of a mathematical function in thephysical science to calculus． Thus， the concept of categori‑zation in economics has made it possible to use distributional
theory in economics． The theory runs as follows：：K Pearson made it known his famous coefficient of variation
as a categorization coefficie'nt in the philosophical Jour，nal of1896 a・ ・，v．（・・effi・i・nt・f va・iati・n）1

Ls・Theref・re・iti・nece‑ssary for us to find a deliverance of the coefficient of variatlon．
100

Since

is constant， as m is unity， and therefore， the Limits

m

．

． ds

・fCat・g・rlzati・n ls dx‑O
sis a decreasing function as a Categorization progresses，．thev alue of s becomes less than any preasigned value e， andinthel ??狽窒?高?case， one sample one Category maybehad．
In that case the value of s becomes O．

7． Theory of Half Normal．
The second study in this paper deals with the theory of Half

Normal． It sounds somewhat strange， however， it is quite
effective approach in establishing parametric relationships among
three factors， changes in price level， changes in consumption
leve1， and changes in income level． In Micrp＿economics state， aconsumer tends to allocate his in．come in such a way that themarginal utility that he can realize from each category of goods

tendto be equal． How can we be possible to treat Macro‑economics‑state so th at they may be able to behave themselvesむto allocate their income in such a way that・the marginal utility
that they can realize from each category of goods tend to be equal〜

The only possibility is the assumption of a normal distribution‑8 一

of the same income strata． This is the very important condition
which makes it pQssible to treat a macro‑eco：nomics‑state unit
as a micro‑economics‑state unit， in the apPlication of an econo‑

mic theory originated in micro‑economics‑state1，
When we came to find the elasticity coefficients in distribution
theory， we realized that the distribution of an income strata isnormal and therefore， symmetric on both sides （of course， wedeliveratel．y took it so）， however， the distribution of the upper

half and the lower half of expenditure values are not symmetrica1．

Therefore， 至n deriving the elasticity coeffcients， the best
combinat1on can be had by letting the lower half of the income
distribution corresponds to the lower half of the expenditure valueof a category， and the upPer half of the income distribution

corresponds to the upPer half of the expenditure value of acategory． That is，ヘユ

勇・7刑σ、）％・邑がN1（Vl）
C●V。

t C．▽．▽
ユ当・百刑σ、）〃・当〆1（vi）
X

C。V．h

‑

C。V・Vん（lmeans the lowest point of the category th means the highest
point of the category）

8． Elasticity Coefficients
Let v stands for value and P for price or y＝一P where y stands
for income．

We have two formula for the elastcity coefficient．
（i） Point formula．
η・一；≡多｝／多1（ii） Arc formula．
Tl 一Vo ／ Vl ＋To

opp＝wh，一p'，／M，＋pl

一9 一一

徳山大学論叢（iii）

10周年記念言命∫文集 （第16．号）

Expanding （i） in distribution theory ：^1・一1｝：：：gg｛1一：；ll‑i3｛｝i一：：：：：gi．IOI：o'．D：i．i；：8'：'，'／一：'；：：

t Rapge of a％ D．．．．po（Po）
Range of a％∠）c．v．vo（7／o）・φo

Range of a％1）c．v．Po（Po）・？ノo

Range of a％ D．．．．vo（voPo）
Range of a％ D c．．．po（Poz，o）

Range of 99％ Dc．v．vo（50）Range of 9990
（iv）

・・・…We have thefor this value

D c．v．po（50）Expanding （ii） in distribution theory ：η。一者Range of％ D…o（？ノ。）^71・＋7'・

SRange of a％ De．v．po（Po） ／ Pi＋Po
Range of a％Dc．v．vo（z／o）・（φ■十φo）＝＝

垂狽狽?e o f a％一'D・c．v．po（Po）．（vi＋'vo）

Range of a％ Dc．v．vo（PQVo十7ioPi十Po7'i‑Po7，i）R／Jaftilg e of a％ D．．．．po（Povo十PoTi）

Range of a％ Dc．v．vo（Pbvo十Povi）
Range of a％ De．v．po（PoTo十loVi）
Range of a％ Dc．．．vo（Pi十uo‑Po7i）

Rit［iig6‑6irin e of a％DTET． III1561．．（一Pr，vo＋Povi）

Range of a％ D e．．．vo（Pivo‑Povi） ・・・…Generally
RiEtfil15'6r‑ie o f a％D，．．．p5（iti5i 5‑V5iJT5‑ovo＋poTi） negrigible．

Range of a％ De．．．vo（Povo十Povi）
Range．of a％ D．．．．po（Povo十PoTi）．d Range of 99％ De．．．．o（50） ・・．…We have a table for
一：：RI1iitil16‑6irgge o f g g％ D，．．．． s o） this v alueThus we have the Elasticity Coefficient for the value of the'items against the price and the Elasticity Coeffcient for thePrice against the value of the items．
Range of 99％ Dc．v．vo（50）？p＝wt． e of ggli；D：i；iSgNso5

一 10 一一

tableη・一三呂鴇湯÷塞．lill∴●●認￡識♂table fo「Thus we have the elasticity coefficients for the value of theitems against the price and for the price against the value ofthe items．

Theorem of Marginal Decreasing Utility may be absorbed intoa system of an Elasticity Coefficient in a Categorized Distri‑butional Theorem． Thus the concept of the marginal decreasing
utility theorem may correspond to a deliverance in calculus， ora tangent in a trigonometric function．

9． From one to Several Categorized Normal Population
Distributions of SubJ'ecitive Value System'V． S． many Cate‑gorized Normal Population Distributions， the lntegral Parts of

the Corresponding Objective Value System （Categorized
Objectve Values of Goods and Services）．

The Limits Theory of Categorization makes it possible for us
to write the Distributional Populations of the Subjective Value
System as well as the Objective Value System as follows ：Coresponding Objective Value

I． Consumption Economy System（Consumption Distribution）（Theory of Half‑Normal）（Categorized Gbods andServices）
ガユ

よNI
2

Σ：D，．v．i、σ・）ZD

c．v．iLガ

4N2

ΣD，．。．了、σ・）

2

V
レヨ

Σ0，．v．了，σ・）

D

（vi）

者N・

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c．viL
上N3
2

XD

c．v．iL

一11一

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2

（vi）；ZD
c．v． 一i u

工1V32（vi）；Σoc。v．乞u（T，）

徳山大学論．叢 10周年記念論文集（第16号）

Corresponding Objective Value

11． Production Economy System（Theory of Half‑Norma！）（Production Distribution）

Sales Price Sqles Price
Distribution Distribution：1］Dc．v．k （‑k）

ZDc．v．KL（iK）； ZDc．v．i〈u（liE7K）（Theory‑gLf ii！ggt1）Lg1：g！guH alf一一Norma1）

III． Local and Central Govern‑

Sales Price Sales Price
Distribution Distribution

ment Finance and Econo‑mic Mixed Economy（1） Categorized Resales In‑（1） Categorized lncome
Distribution Population

come Distribution Population

Model

Model

Number of Categorized In‑

Number of Categorized ln‑

come Distributions Population

come Distributions Popu‑

Models in Operational Formlation MQdels in Operational

Formt（2） Categorized Produc‑（2） Categorized Goods andtion Distribution

Service Expense Distribution

Population Model

Population Model

Number of Categorized Pro‑

Number of Catego rized Goods，

duction Distribution Popula‑Services， and Expense Dis‑

tion Model in Operational

tribution Popuiation Model

Formin Operational Form

This is an

Explanatory Graph 1that is to show A Total

System of Japanese lnternal Economy based on the Economic
Marginal Theorem． lt is shown by the Equilibrium Distribution

Coefficient under Marginal Theorem into Three Sectors，Consumption Economic
Sector， Pro．duction Economic Sector．
and Public or Local as well as Central Government Economic
Sector．
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徳山大学論叢 ．10凋年記念論文集（第16号）

This is the first time to reveal the technical structural const‑

ructions of the coefficient of equilibriuM distribution for a
Consumer with Stratified iMultiple Strata．
op ，一 ， × 一31；，． liY￠i ＝＝rTi
Σηゴi×YAnd also to reveal the technical structural construction of
the coefficient of equilibrium distribution for a Producer．
tpBk × 一CV‑k

X￠ ic ＝＝ 一一一一一＝劣k＝rp B ic ×x

Explanations of the Mixed Economics，of the Public Sectors
are postponed to the next essay．

There are three categories of parameter appearing on aStrategic Simulation．
a． The Example of this． ： Coefficient of Variation， Elasti‑

city Coefficient， lncome Distribution， Production Distribu‑
tion， Cosumer Value Distribdtion， Sqles Price Distribution．

b． Those parameters which belong to the purely Objective
Value System．

The examples ： Consume'r Engel Coefficient， Market Pricex Quantity， Producer Engel Function．

c． Those parameters which belong to the consolidated
Subjective Economic Value Systern． Examples ： The Consumer
Equilibrium Distribution Coefficie，nts， the Producer Equilibrium

Distribution Coefficients．

Regarding Distribution Coefticients， we may be able to‑14一

categorize these parameters as ：a． Distributional Category belonging to Elasticity Coef‑

ficient or Parameters belong to the marginal theorem of
decreasing utility． ．

b． Functional Category belong to Engel Coefficient or Para‑
meters belonging to the equilibrium of their marginality．
11． Linkaged Parametric Simulation， Example， No． 1．

The followings are the results of the decomposition of the
Income Distribution in Japan in 1959 by means of FACOiM 270‑
20／30．

STRATIFIED MULTIPLE NORMAL POPULATIONSXBRI 30．5 COEFI 32．0 CONSI 1174133． O
XBAR2 49．5 COEF2 25．0 CONS2 515268．7
XBAR3 95．0 COEF3 26．9 CONS3 243613．1
ERROR O． 0240

DATA

STRATUM2

STRATUM1

STRATUM3

SUM

ERROR

22947．0

0．58250．0496｝10598．9397．922456．8

92．1

145895．0155565．5

4497．0327．7160390．2444733．0407808．6

27185．5

1002．9435997．1

0．0098528284．0407808．6

88199．3

2641．3498649．2

0．0280312562．0155565．5

153932．6

5986．1315484．2

0．0046261342．0

23667．9217948．6

31270．4272886．9

0．0220132255．0

24．2

23078．2

1G1139．6

124242．1

0．03020．02640．0

9．6

101118．9

101128．5

1172897．1

515248．7243579．0

1931725．2

96055．0

From the foregoing computation， we have the income model
under 2，000，000 Yen a year as follows：
ユ

き ア

Σ0
32

ユら

ヨらフ

（30．5）十Σ∠） （49．5）十Σ∠） （95．0）
25

27

For the income group over 2，000，000 yen a year， the estima‑tion of the normal population parameters are made by means‑15一曽網︵毯︒︶ミq囚︵寸剛．・一q惑奥q臼︵卜二野︵8・・8㌫q囚一祠︵℃§㌫q囚

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ひま一εコ2財3＝80白話ξもε霊あ㊤ε8唱一言．切︒嘱﹂o︒08δ335欄︒罎8りう旧︒欄葛

@16 一

一一

10周年記念論文集（第16号）

徳．山大学論叢

of the table specially constructed for the purpose of finding

the distribution parameters of Oligopoly， the explanation ofwhich will be made in section （Part 3）． The results are：35209 5039 1092

＝D （350．． 9） ＋ ZD （750． 0） ＋ ZD （1500． 0）19 19 25

13． Linkaged Parametric Simulation System．

Alinkaged Parametric Simulation Systern is a system inwhich all the parameters in a linkage under the equilibrium

distribution coefficient system have additive nature， andtherefore， interchangeably quite freely， ．namely， decrease ofincrease in their values， prices， and quantities of the basis of

the deviations with equal probability densities for the rest of
the others．

As has been stated before， elasticity coefficients coupledwith income distributions and corresponding value distributionsof the categories of goods consumed in the family， not only
enable us to find an equal marginal utility relationships among

the categories of goods in the household consumption in themacro‑economics state， but also permit us to make parametriccomputation by means of simple arithmetic method．

The following conditions are given in an example of a

Linkaged Parametric Simulation applied to three income
strata．

The conditions：
Income Level ・・・・・・・・・・・…．・・・・・・… 8％ up．
Cereals一・・一・一・一・・一・一・・，・一・・一一 15％ up．

Other food ・・・・・・・・・・・・・・・・・・・・・・・・… 12％ up．

Housing・・・・・・・・・・・・・・・・・・・・…一・・・・・… 5％ up．

一17一徳山大学論叢

10周年記念論文集（第16号）

Clothing ・・・・・・・・・・・・・・・・・・・・・・・・・・・… Stays the same．

others・… 一・… 一・一・・・・… 一・一・… 一… 8％ up．
Saving・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・… No change．

Elasticity CoeMcients and Values of Seven Categories
vii

v童

v

ノ

ノ

ﾅPユ

η，P2

η，P3

η，P4、

η，Ps

η，P6

η，P7

】

ηv2

ηv3

ηv4

ηッ5

ηv6

ηv7

ηP1
ﾅコ1

ηP2
？v2

ηP3
ηv3

ηP4

ηP5

ηP6・

ηP7

ηび4

ηひ5

ηv6

ηv7 '

1

V2

V3

v4

V5

V6

v7

ﾅ'

Lower Income Strata

iv

ド

'

UpPer Income Strata

血rpZ q， v

五︐‑

7 Categories

Central Value foraoth Upper andkower Income Strata

The basic allocation ratio on equal marginal utility in macro‑economics state for the lower half of the income strata may
be had：？7 vi 'Ti ＋ ？7 v2'Zt2 ＋ ？7 v3'Zt3 ＋ ？7 v 4'V．i ＋ 77 vi） '7is ＋ ？7 v6 '7e） ＋ 77 v7'Ztv

The basic allocation ratio on equal marginal utility in macro‑economics state for the upper half of the income strata may
be had：rp 'v1'Ztl＋V 'v

o 'V2 ＋ op 'v3'la'N3 ＋？ 'v4'Ztl＋V 'v i） 'Vor ＋ op 'v6T6 ＋ op 'v7'Tr

COMPUTATION SHEET FOR 1 INCOME STRATA＼＼、、Income Strata

Lower Half

Upper Half

Ihcome Strata

Income Strata
％

％

1帰

Cereals

1．709×3806

6504

17，3373。851×3806

14657

31，845

Other Food

1．925×8070

15573

41，5101．025×8070

8272

17，973

Housing．899x2365

2621

6，986

2．049x2365

4846

10，529

Fuel＆Light

1．155×1228

1418

3，780

1．709x1228

2099

4，561

Clothing．879x3008

2644

7，048

1．419×3008

4286

9，273

Mlscellaneous．879×8053

7079

18，8691．246×8053

10034

21，801

Saving．839×1999

1677

4，470

．925×1999

1849

4，017

7Categorles

ηPf×η'

η'P，×謬'

ηP，び'

、

37516

46025

For the lst income strata， 8％ increase in income brings
2，033 yen increase per month while increase in prices of the一エ8一items brings deficit of 571， 968， 118， 120， 644， 2421 （yen）， if

people don'twant to cut the consumption of items of which
prices rise． The balances of 2421 yen一一一一230 yen or 338 yen will

be deflated according to the basic allocation percentages．
COMPUTATION SHEET FOR II INCOME 5C TRATA
一

Lower Half

Income Strata

Income Strata

ηP，×

ゴ

％

Other Food

．Upper Half％

7Categories
Cereals

Income Strata''

ﾅρf×か'

ηPゴび'

ﾅP，

'

3．537×4349

15382

32，173．844×10203

8611

18，011

25，389．962x2892

2782

5，819

3，818

6219

13，008．

3075

6，432

9628

20，1382108

4409

2．489×4349

10864

18，0481．371×10203

13988

23，238

Hou ing

L545×2892

15283

Fuel＆Light

1．458×1576

2298

C且othing

1．037×4224

4380

7，276

1．088x1576．728x4224

Miscellaneous，927×11408

10575

17，568．844×11408

Saving．927x3029

2807

4，663

．696×3029

47805

60195

COMPUTATION SHEET FOR III INCOME STRATA@

Lower Half

Income Strata

Income Strata

Upper Half

Income Strata％％

7Categories

ηPゴ×か'

Cereals

3．811×4837

18434

21，7706．702×4937

32418

24，424

Other Food

1．480x15274

22606

26，6972．960×1527

45211

34，062

HQusjng

1．038×4426

1．284×4426

5683

4，282

1．909×2237

4594
4270

5，425

Fuel＆Light

5，043

2．433×2237

5443

4，101

Clothing．757x10805

8179

9，659

1．284×10805

13874

10，453

Miscellaneous．961×21814

20963

24，7571．038×21814

22643

17，059

Saving．845×6661

5629

6，648

1．120×6661

7460

5，620

ηPゴ

η'

r

o

×ひゴ

'

ﾅP8び∫

84675 1 ' 1 13av32

For the second income
strata． we have incorne increase of 3300'
yen and the price increase by 3029 yen． The balance of 197 yen

is to be allocated over seven categories according to the basic

allocation percentage．
For the third income strata， we have inome increase of 6333
yen and the price increase by 4957 yen． The balance， 1376 yen
is to be allocated over seven categories of it：ems according to‑19一

10周年記念論文集（第16号）

徳山大学論叢the basic allocation percentage．
ALLOCATION TABLE・．． lncome

1 lncome Strata

x．．． Strata

7 Categorie，S'x'・一×

Fuel ＆ Light

Clothing
Miscelleneous

Saving（一）

Lower Half
（＋）

36

70

S6

41

T0

36
85
16

Lower Half

（＋）

124

18

III lncome Strata

Upper Half

W14359Housing1

Other Food

II lncome Strata

Upper Half

3
5
1
6
3
ρ0
31
9一
14
・ 09

（一）

7
1ハ7
O
7．・
37
ρ0
09F
U1
27
‑凸

×
Cereals

Lower Half

（＋）

322

367

469

75
69

59

133

144

341

235

91

77

Scores of Oligopoly Producers Producing Categorized．
Product．

lst Assumption ： A Categorized Normal Distribution Model

Each member of the Oligopoly Producers is characterizedby：（1） Production Occupancy Percentages （Production Engel
Coefficient）〈2） The Rank of the members of the Oligopoly．

（3） The Number of the members of the Oligopoly．

A greatcare is， however， to be taken t，o formulate the． We use a production or a market share in the density
model． ln so doing
we havemodel t'一〇 find the distribution
parameter

three assumptions：（i） A producer producing a commodity， his production
density forms one half normal distribution in macro‑economics‑
state．

〈ii） His present production or market share is supposed to

一20一

（＋）

300

14． A Separate Producer Producing Several Products．

have the outcome of the four factors：

Upper Half

56

Initial Conditions十Non‑economic Conditions＋Economic Con‑

ditions十Random Error（iii） Assuming an ideal case what the market or production

shares each producer is solely the outcome of random error．

With these three assumptions， can we find any sensible
density formula to detect the producer's behaviour？
The following is the formula I have found：lie一一一

奄奄?一（i‑i）

Dc．v．k（一liil）

P， ＝2．〉一xx‑ N（i）

The formula tells us immediately what the distribution
parameter （in c．v．） is， as soon as you find the percentage ofthe producer's share his rank， and the number of his
camerades competing in the market． Amount of his annual
production is also ahother parameter （in this case， a central
parameter， i）．

．ln the computation of the mathematical table for the above

formula， a specially constructed probability integral tables
with the central parameter， T＝50， and the distribution para‑
meter in c． v． from 2 to 100． Thus the formula becomes：
50

50‑

（i‑1）

N
De．'v．k（50）P， ＝：2．〉

50

50‑

N

の

The table has been constructed in the range of c． v． 2 to c． v・一21一

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10周忌記念論文：集（第16写）

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徳山大学論叢

10周年記念論文集（第16号）

100 and N＝ 11．

15． Example No． '2，

ButterConsumPtion Estimate Simulationfor 1960．

Ref． Summary Table for CoefficientS of Elasticity by lncome
Strata of Japanese Households in 1960．

Butter Consumption Estimates in 1960．

Equilibrium Distribution Coefficients for Other‑foods＼＼＿璽・S・・a・・
恥・L ヘへDis． C・e翫＼
Other Foods Dis‑
tribution Coef．

lst lncome Stratq

2nd lncome Strata

boWer

Lower

Upper

Upper

3rd lncome Strata

Lower

Upper
．111996

Half

Half

Half

Half

Half

． 159343．375536．227136．291689．173549

Number of Linkaged Parameters：
Coefficient of Elasticity 7×6 ＝42

Engel Coefficients （7×6）÷2＝ 一g15‑

Parameter of the mathematical means entered in Engel
Coefficients amounting to 21．

16． Example NO． 3． Calculation Table for Basic Linkaged
Parametric Simulation for 10％ Increase of Butter Produc‑tion （1200 tons） for．Upper 10 Enterprises in the 35th Yearof Showa．

Explanation： lnside Data of the blacked line enclosure are
the data subject t o the Strategic Economic Policies of Each
Member of the Oligopoly P roducers． The rest of the out‑sideof the enclosure are calculated on the basis of the system inorder to have Linkaged Parametric System （basing on a Coef‑
ficient of Equilibrium） has to be established．
一 24 一一HalfColumn：︵

ラー234567

Ranks of the member of Olig'opoly Producers．

ラ一

Distribution of Production．

︵

ラSupply Price Distribution Half Normal （Lower Half）．

一

Supply price Distribution Half Normal （Upper Half）．

ラ︵

Coefficients of Elasticity， Upper Hal￡ラ
一

Coefficients of Elasticity， Lower Hal￡ラ︵

Elimination of Price difference among the member pro一

ducers．

（8） Lower Half of the Supply Coefficient．
（9） Upper Half of the Supply Coefficient．
（10） Coefficients of the Equilibution， Distribution， Lower Half．

（11） Coefficients of the Equilibrium Distribution， Upper Half．

（12） Distribution of the arnount of increased production， Lower
Half．

（13） Distributio．n of the amount of increased production， Upper
Half．

（14） Decreases in Sa1es Prices， Lower Half．
（ls） Decreases in Sales Prices， Upper Half．
（16） Average decrease in Sales prices．

（17） Production Distribution， New Equilibrium Point．
as） Sales Prices Distribution， Lower Half．
（lg） Sales Prices Distribution， Upper Hal￡皿．

Calculation on Micro Economic System to find a New
Equilibrum Point of 10％ Increase of Butter Production（1，200 tons） 10 Enterprises in 1960．

Number of Linkaged Parameters to find ，a New Equilibrium
Point of 10％ Increase of Butter Production．
一25一

徳山大学論叢

10周年記念論文集（第16号）

Coefficients of Elasticity 2×10＝＝20

Engel Coefficients （2×10）÷2＝＝10

Total Parameters 20＋10＝ 30
17． Example NO． 4． Calculation Table for Basic Linkaged
Parametric Simulation of Oligopoly lndustry ， 1977．
Number of Linkaged Parametric Simulation
皿． Oligopoly Industry．
Number of Linkaged Parameters to find a Strategic Simu‑
lation for the purpose of determing Production， Sales， andPrices．

Coefficients of Elasticities 2×11＝22
11Engel Coefficients （2×11）÷2＝'lit／

Explanation：ラ︵

Column：Ranks of the member of Oligopoly Producers．

ラ一

Initials of the Makers．
ラ一

Production Occupancy， that is， equal to Engel Coefficients．

ラ︵

Production Distributions．

ラ︵

Sales Price Distributions， Lower Half．

ラ︵Sales Price Distributions， Upper Half．

ラ︵

Elasticity Coefficients， Lower Half．

ラ︵

Elasticity Coefficients， Upper Half．

ラ︵

Elimination of Price differences among the member pro一一

ducers．

aO） Lower Half of the Supply Coefficients．

aO Upper Half of the Supply Coefficients．
aZ Coefficients of the Equilibrium Distribution， Lower Half．
一26一

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徳山大学論叢、

10周年記念論文集（第16号）

a3） Coefficients of the Equilibrium Distribution， Upper Half．
ae Effects of Pri'ce Policies Comparison．

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