Socio-economic methodologies
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Hendrix, E. M. T. (1987). Socio-economic methodologies. Technische Universiteit Eindhoven.
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M.SC. RESEARCH ON SOCIO-ECONOMIC METHODOLOGIES AS UNIDO-INTERN
IN CLOSE COLLABORATION WITH CICA/EUT
by E. Hendrix, November, 1987
..trreek Tl:..
~ -~;t";t Eindhov~Tilburg University in November
1987.
Table of contents:
page
Preface 2
I. Introduetion . . .
3
II. A note on the L.P. methods to update input-output
tables . . .
9
III. Socio-economie interaction in the NICs, measured with socio-economie indicators ..••••.•..••••.••••.••••.••• 27 IV. Report on my activities related to the Feasibility
Studies Branch . ... 90
Preface
In February the United Nations Industrial Development Organisation
effered me the opportunity to do my M.Sc. research at their
headquaters in Vienna. This underlying report is the result of my presence within this organisation from the end of March until the end of September. With this preface I want to express my thanks to the persons without whom it would have been impossible to reach this result.
-First to my wife and children, Niki, Remi and Jodi, who managed to live relatively isolated in a strange but beautiful country.
-The people that took care of the transportation of the most necessery cammodities we needed: my father, my brother Gerlach, Leen van de Berg
and my friend Antony Winters and of course my most loyal
motorcycle, the MZ, which brought me daily from Laxenburg to the
Vienna International Centre and back and even back to Holland.
-The persons I coöperated with in my research. Specially Bert de
Bruin, my collegue from the University of Eindhoven, with whom I coöperated in the last project. At the Global and Conceptual Studies Branch Mr. Cho, Paul Wiedemann, Gerhard Margreiter and Anatoly Lyakh. -All the friends we made among and through the interns, which made our stay very pleasant, in particular Hamadou Diawara.
-The Development Co-operation Office of Eindhoven University of
Technology, Emilea van Egmon4, Paul Laperre, who took care of the
centacts with UNIDO and the practical guidance of the research project.
-And last but not least Dr. Jan van Lieshout, who coached the reseach, most of the time from a long distance, but also once very close.
24th of October
Day of the United
Na ti ons Eligius Hendrix
chapter I Introduetion
The UNIDO and my activities for this organisation during my stay as an unpaid intern in Vienna.
The United Nations Industrial Development Organisation is one of the specialised agencies of the U.N.O. system. lts task is to promate and
accelerate industrial development in developing countries. The
organisation gets its money from the contribution of some 125 member states which meet in a big conference every two years to decide upon the global strategy and policy. In between a board is meeting of 43 countries.
With the organisation chart on the next page in the hand I will discuss now how the various departments and devisions werk tagether in the direction of these targets. During my stay of six months at the headquarters in Vienna I gradually gained some insight in official and inofficial interactions and cooperations among the sections meeting various people. The target is mainly realised by the identification, selection, elsboration and finally implementation of development
projects. On the one hand this are technica! assistence projects in
the ferm of setting up training courses and institutions, organisation of workshops help in negotietien etc. and on the ether hand it are direct industrial investments. Technica! know how is available at UNIDO or originates from experts and consultants who werk for this organisation. Many people ~c,:-king for UNIDO have got an engeneering, management and/or economie background.
It eppears that the organisation chart changes regularly due to the restructuring which takes place because of the financiel crisis UNIDO is in. In the chart on the next page the names in the two first columns representing the two first departments are clear enough. The project cycle mainly takes place in the three last departments for which of course the two first ones create the necessary environment. The identification of technica! assistsnee takes in principle place in the Areas section. In cooperation with all other sections, which give economie, secterial and technical information a formuletien takes place. Targets are set and the way they have to be reached is pointed out. Financing institutions have to be found in U.N. Development
Programme (UNDP), Development banks, National funds etc. The
feasibility branch plays a very important role in the formulation of projects. Finally all information on a project: on the sector it takes place in, the government policy etc. is put together by the areas programme and can go to the appraisal stage. After the project has been approved it is implemented and monitored by the industrial operations and industrial institutions sections. It is added to the portfolio of UNIDO projects. Review and evalustion takes place. In
1986
some1693
projects were implemented or under implementation.In the field so called Senior Industrial Development Field Advisors
(SIDFA) possibly assisted by a Junior Professional Officer (JPO) can
be founn, who monitor the projects end keep contact with the
DIRECTOR-GENERAL
.
I
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DE,.UTV DEI'VTY DE"UTY DEI'UTY D!'VTY
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Ptcweu ...EXPENOITURES IN 1986
IY PROJECT COMPONENT
(In milliona of US dolltra)
AFRICA
IPF PR/SM SIS 23.0 0.3 1.4 RB IDF 2.1 2.1 TF UNFDAC Othera 1.1 0.5A SIDFA duty station • JPO duty station
IEYCMELLlS
35 EXPENOITURES 117~ TO 1g&B
(In milliona of US dolleral
20
10
I
field to visit for example a group of technical similar projects in various countries. This matrix idea of at the one hand techniques and sectors and on the other hand regions can be found through the whole of the organisation.
To get an impression of the activities related to these projects by UNIDO on the former page the activities in relation to Africa are depicted. The other regions are the Americas, Europe and the Pacific and Asia
It will be clear that not all activities of UNIDO are going through these projects. The Industrial Investment section promotes investments in industrial projects by formulating rules for joint ventures and bringing partners together for example. The system of consultations is
also directly consulted by developing countries and makes many small studies. The Transfer of Technology section does not only contain information on appropriate techniques in a field such as mechanical engeneering, but also on the very broad field of biochemie. Like all information eentres on UNIDO it has to be in contact with various academie institutions. Beside this information task it also helps institutions in developing countries directly in their negotiation with institutions in the developed world concerning the transfer of
technology.
What is left to be discussed are the activities of the studies and research section to which I was assigned during my stay in Vienna. Within this section one can find something of the matrix idea, as
there is a regional studies branche and a sectorial studies branch. The regional branch clearly makes studies concerning the economie situation in one country or region. The standardised so called
'Country Reviews' which give the information for one country are quite
popular within but alo outside of UNIDO. This branch also makes
studies for one government like a consultancy firm on the economie situation of a country. The Sectoral Branch makes studies on the methodological side of the various sectors. What is the place of the sector in the economy and which techniques are available.
The Global and Conceptual studies Branch finally was the sub-division where I was placed. In earlier times this was the place to make bigger models like world-trade models to support the global economie disussions on f.e. south-south cooperation and de-linking. Cross-country wide the economie performance of the various sectors can be compared. However as modelling is rather expensive and UNIDO is in a financial crisis this part is more and more reduced. What is left is space for smaller studies mainly to support the general policy and a forecasting model based on extrapolations of industrial data. Despite these problems the Global Studies Branch managed to publish the yearly Global Report on which the people in the Branch worked together to elaborate special questions and which contains a big statistica! part with short term forecasts for the individual countries.
As a tool in those analysis the Global Branch contains two databases. One on economie data like Gross National Product, Value added of the various sectors, capita! formation, employment etc. and one database on Input-Output tables. This last database is quite
unique and one tries to standardise all tables to the
32
sectors of the so called SITC-code. I could use both databases for my research at this branch.In this situation, in which the branch was shrinking and all people
were werking very hard at the Global Report
1987,
I entered thissectien in the end of March
1987.
There wasnota subject prepared for me to work on, so that I had to look around myself and to find in discussion with ether people of this sectien a subject. In the period to the end of September I was able to work on three different subjects which apart from the fact that I worked on them, have hardly anything in common.The first subject I started on in cons~lt with Mr.Cho the head of the Global Studies Branch and was concerning the updating of Input-Output tables. In my investigation on which methods were used to handle the I/0 tables something struck me in the formulation of an algorithm implemented in the so called Flow Matrix Analysis software that could be used for updating I/0 matrices. The formulation appeared later on to be quite unique for this purpose. The question was to do some literature study and to apply the metbod and to relate it to
ether methods. I did not only start to read literature on this
subject, but also used the software and worked out some examples with a Linear Programming package for the micro computer. In May I could already write a discussion paper in which I showed some mathematica! properties inherent to the software. Only slowly I could get more literature, as a part of it is very old. The last articles I could get in August from my own University after which I rewrote the paper in a more readible form. The secend chapter of this work consists of this paper.
At the end of May I could start werking on snother subject. In the past some research bas been done by the Global Branch on social and economie interaction: The social progress will support the economie succes and vise versa the economie progress will accelerate social
development. In this the social development is measured with social
indicators like infant mortality, education, housing etc. and the economie progress is measured with economie variables like Gross
National Product (GNP) and Manufacturing Value Added (MVA). From
fermer studies there resulted an indication that this interaction will have been strenger for the Newly Industrialising Countries (NICs), which had a more dynamic pattem of development in the sixties and seventies. The idea to do a study in the direction of the NICs was also encouraged by an artiele of A.Singh in Labor and Society, that discussed the difference in economie succes between the Latin American NICs and the Asian ones.
Perhaps we would be able to explain this difference by the the difference in the emphasis on the social part of development. The
methodological target of my study was to come to a small set of
simultaneous equations in which the economie variables are partially explained by social ones and vise versa. This would support some
statements on the importsnee of social development. Until the last week of my presence in Vienna I was werking on this subject in reading articles, collecting data, making estimates, calculating correlations and trying relations. For these purposes I used a spreadsheet and statistica! package for the P.C •• However the relations appeared to be to weak and the data to bad to try a simultaneous set of equations and at this point the results were rather disappointing. On the other hand
all the problems I met in my research and the literature on this
subject were instructive. I was encouraged to write these problems
down, before I would forget them. The third chapter of this work
describes the problems I met and is therefore in the form of half paper half report.
The third subject I worked on at UNIDO is completely represented in report form in the fourth chapter. It handles my contribution to the discussion on the methodology of feasibility studies in relation to development projects. At this point I worked together with snother Dutch intern from Eindhoven University, Bert de Bruin, which made our points of discussion much better, as we could formulate our arguments together. At the Erasmus University in Rotterdam I had studied the subject of Cost-Benefit analysis or Project Appraisal, which is in fact a part of feasibility studies. In this theory estimates of costs and benefits for the national economy are made to decide on the acceptability of a project. As well on the university as in Vienna my question was (and is ) whether there is a methodology to handle the risk aspect of a project, because every statistician will agree that if there is sn estimate, there is also a varianee of it. The range in which a benefit lays can be very big.
In the feasibility branch a lot of attention is given to a computer program which helps to make financial analysis of a project (COMFAR). We learned how to use it, applied it on a case and ,encouraged by the Dutch Ambassy in Vienna, we compared it with snother package used only internally. In the contributed discussion: project'for methodology
Global Branch there were some people who also had
in the sixties and seventies to the eest-benefit
'How do we properly measure the costs and benefits of a
the national economy'. Further we could discuss the
Chapter II
A note on the L.P. methods used to update input-output tables
The purpose of this paper is to put in a broader perspective the use of the FMA software to update I/0 tables and to show some properties that should be kept in mind whenever someone uses such algorithms. As the algorithm behind this software is not the only one that can be used for this purpose, it is useful to study the pecularities and differences with comparable methods.
page
1. INTRODUCTION: INPUT-OUTPUT ANALYSIS AND THE UPDATING •..••. 10
OF I/0 TABLES.
1.1. I/0 applied.
1.2. Adjust, update and project matrices.
2. CONVENTIONAL I/0 UPDATING METHODS ...•••••...••.... 13
3.
LINEAR PROGRAMMING METHODS ..••...•..•....••••..•...••• 174.
CHARACTER OF THE VARIO~S OBJECTIVE FUNCTIONS IN A L.PEST IMA TE . ...••••••••••••••.••••••.•...•.••••...•... 19
5. CONCLUSIONS .•.•.••••••.•••.•••••.•..•...•.••...•... 25
Chapter 1. INTRODUCTION: INPUT-OUTPUT ANALYSIS AND THE UPDATING OF I/0
T~L~.
1.1 I/0 applied.
Input-output analysis is applied as a useful tool by many
economists. It started with the werk of Leontiev in
1936.
Theclassica! werk of Chenery and Clark [3] gave a big impulse to the I/0 theory. Interindustry techniques was seen as a streng tool for
structural analysis and politica! guidance. People were rather
optimistic about the effectiveness of this instrument for policy simulation, control and optimization. But also after this first period in the sixties optimism didn't die, witness the bock of A.Kuyvenhoven
in
1978
about werk with the semi-I/0 method[9].
The link with linearprogramming with I/0 tables extended with columns representing new techniques, called activity analysis, would enable it for the national policy to choose the optima! activity and product mix.
Applications in the direction of multilevel models can be found in the research of:
The UNIDO studies devision on world scale in the seventies,
Tinbergen et. al. on the level of regional national policy in the sixties [6],
f.e. Meyboom on the level of the firm in
1986, [7],
If activity analysis would be used on all levels, we would be able to come to consistent plans in which the instruments of prices and scarce materials would make a policy on all scale possible.
Disadvantages and objections of these methods are well known. Chenery already admitted that the method would be of limited value for predictions, because of the changes in product-mix, relative prices for the inputs and availability of technological alternatives. The optimism of Tinbergen in the possibility to get insight in the mechanisms of international trade and national intersecterial and
interregional deliveries is net shared by all economists. The
applicability of activity analysis appeared to be smal!. How can the coefficients be estimated from the interindustrial deliveries and how goed do the average deliveries ferm a proxy for the marginal coefficients? The estimates of the individual coefficients appear lie in a range of some 20% around the 'real' coefficient. See [1]. However
Evans [11] showed already in
1954
that this does not lead toaccumulated problems if the total production is estimated out of the final demand.
The estimation of I/0 tables requires a big effort. The time lag of the estimation process is about
6
years for I/0 tables of a national economy. The estimated flows of goeds or money originate from varioussources~ Census, samples and national accounts, which do net form a cosistent system. There are the problems of small-scale, secend hand goeds, classification, price/value, etc,etc. For a description of such problems we can refer to Bulmer-Thomas
[1]
and Brauers[16].
Despite all these problems I/0 stays a popular as a subject of research, witness the succes of the eights international conference on I/0 organised in coöperation with the UNIDO Global and Conceptual
Studies Branch in Japan 1986. Stone reminds us [15] of the cosy first conference promoted by Jan Tinbergen and held in Driebergen 1950 in the Netherlands. In that time there were some 27 participants whereas in 1986 it contained some three hundred participants and some hundred papers were sent in.
New subjects that enter the field are those of pollution, income distribution and the extension of international trade and world models. The strengtening of the dept problems make the use of the current account eenstraint in the L.P. models more topical. The UNIDO Global Branch contributes insetting up a data-base of I/0 tables and in trying to standardise the tables with different classification of the various countries. As this organisation has connections with many
national statistical organisations and various scientific
institutions, it is an appropriate place to for the presence of a central collecting unit.
A colleetien of tables also gives the possibility to do structural analysis in camparing the tables for the various countries. Beside the difference in classification there is also the difference in the year upon which the tables are based.
1.2. Adjust, update and project matrices.
Beside the applicability of I/0 the development of I/0 tables in time has been subject to extensive research. As the estimating process of an individual table takes such a long time, there are hardly any time series of tables. A few examples of such studies are: Tilanus and later on Paelinck on a series of 13 Dutch tables, The intensive use of
three U.K. tables by the Cambridge growth project of Prof.Stone and
the efforts of Matuszewski on the Canadian tables
[3].
Mathematically seen there are three closely related problems in the estimation of I/0 tables. The first one is the adjusting or balancing of a matrix after all individual amounts are estimated. After one has finally gathered all information for the I/0 table, one becomes aware of the fact, that the primary estimates of the flows do not form a consistent system, caused by the simple fact that the information comes from different sources; the row and column sums are not right. How to change the estimates of the interindustry deliveries so that the margins are right. The second item is caused by the simple fact that one wants to have a more up-to-date estimate of the table at ones disposal. Available is an old table and information about the row and column totals which give the total production of the sectors. How to fill in the table of this year. The third aspect is then that one wants to have an estimate for the future and projects the row and column totals or also further information to the next period. Of course this idea of projecting does not essentially differ from updating. We will see the difference if some special methods are used. Especially the artiele of Leeomber describes these problems. See [13]. To come to a metbod to cope which such a problem of data creation, the Global Branch developed in coöperation with the International lnstitute for Applied Systems Analysis (IIASA) the FMA software. FMA
stands for Flow Matrix Analysis. It was not only è~veloped to update
I/0 matrices, but also to be able to estimate international trade-flow matrices of which limited information is available. The theory behind it is, that there is an iterative process in which a team of
specialists, economists is interacting with the algorithm in deciding on acceptability of outcomes and imposing constraints in trying to improve the estimate every time the algorithm has run. In this way all skills and economie insight of a team of specialists can be used and consequences of various scenarios can be obtained. It are the mathematical properties of this system that we want to study in comparison to these of ether methods.
The mathematica! translation of the above described problems is very simple. Consider the matrix with n*m cells. How to fill in these n*m cells if n+m row and column totals are known as well as an old matrix (updating and projecting} or a target matrix (adjusting}. What is left is a linear subspace with a dimension or degree of freedom of
n*m-n+m. Actually it appears to be a half-space as the flows are
usually net allowed to be negative and the degree of freedom is ene lower, as the sum of the column totals is the same as that of the row totals. To reduce this degree of freedom to get a unique estimate, some assumptions are needed.
In general the row and column totals are in the national
statistics. This can also be the case with the final demand columns or at least with the total of final demand. What is left to be additionally estimated are the interindustry deliveries of the I/0 table. Columnwise seen the coefficients, calculated as
flowij/tctal input flow sector j, represent an average technology for the whole sector j. We will regard the problem here as ene for which just the margins are known and the table itself has to be filled in completely. Let us therefore assume that also the final demand flows have to be updated in this sense.
What we will call here 'the conventional methods' assume a certain relation between the new and old table. With the aid of n+m multipliers that have to be estimated for this reletien the degree of freedom of the linear space is reduced to zero. One can try to interpret the multipliers and to find some sytemacy. These multipliers are used to project the matrix further and to make forecasts without using projections of the margins.
The secend type of methods we will call L.P.-like methods, as the
information is usually implemented by linear restrictions. The
principle is very simple; given this information fit the new matrix as goed as possible according to some loss-function (objective function) to the old or target matrix. The methods differ in the objective function or minimand which describes the difference between the new and old matrix. Actually also the methods that assume a relation have implicitely such a minimand, as we will see.
These L.P.-like methods eerreapond with the theory of Brauers [16]. that learns us that to much effort is needed to gather all information to create I/0 tables for developing countries, so that ene should concentrate on collecting of information about the most important
sectors. So the principle is first to find out what the important
sectors are, then to gather information about them and then to fill in the rest of the table with information of f.e. neighbouring countri~s
and the use of such data creation techniques. This bas the consequence that the information does net have to be in the ferm of the margins. He gives some guidelines to identify the most important sectors.
There are two major differences between the two approaches. For the methods which assume a relation, simple algorithms have been derived to estimate the multipliers. The rows and columns are iteratively adjusted until the row and column totals are right. These algorithms take much less computing time and numerical problems than the L.P. algorithms which use L.P. and quadratic programming techniques. The FMA system takes minutes computer time on a mainframe to update one
table of 32 sectors. On the ether hand the disadvantage of the
conventional methods is that the relations are lost if one wants to insert more information than the margins only. The L.P. methods are open to all additional information, as long as it has the shape of linear constraints.
It is interesting that Matuszewski [3] informs us in his artiele of 1963 about his attempts to find some functional relations with even less than n+m multipliers, whereas in his artiele of 1964 he completely switched to the L.P. methods in reporting on his succes to apply such methods to update the Canadian tables.
We will first shortly summarize the conventional methods. After this we will say something in general about the methodology of L.P.-like methods. Next we will compare some properties of the loss-function in
the FMA software with some similar functions.
Chapter 2. CONVENTIONAL I/0 UPDATING METHODS.
A well known method for this updating is the so called RAS method or biproportional method, with which the name of Prof. Stone is closely connected. He and many ether scientists did research on this updating method in deriving a lot of properties for this method and tried to interpret the multipliers and use them to make further forecasts for later I/0 tables. Bacharach wrote his dissertation on
this subject in 1970 [2]. Two matrices 0 and X are called
biproportional if there exist two veetors r and s, 80 that 0 • <r>X<s> Here <r> and <s> are diagonal matrices with the elements of vector r resp. s on the diagonal.
To explain the various methods it would be useful to introduce
8ome 8ymbols. Let us define Oij as the flows in the old
or target matrix,
Xij as the flows or estimates in the new matrix.
xi. and x.j then denote the row resp. the column totals. There are n rows and m columns.
The relation in the RAS method was already formulated by Leontiev in 1948: x1j• riOij 8j. The n+m multipliers are actually calculated with the aid of n+m column + row totals that are assumed to be known, 80 that the estimate will be unique. Important properties are that if Oij is zero then also Xij will be zero and the method tends to change
the biggest values the most. Bacharach shows
[2],
that the implicit minimization criterion ismin
rr
X .. log(X .. /0 .. )l.J l.J l.J
The normal procedure proposed by Stone. however, is to multiply
columns and rows over and over with multipliers that are decreasing in absolute value. The new matrix wi11 be certainly biproportional to the old one. The multipliers are interpreted in the following sense: ri represents a substitution effect, which is uniform for all industries, sj represents an efficiency effect, as sector j uses less or more of all inputs. If the matrix is further extrapolated with the aid of the multipliers: Xt= <r>tO<s>t, the individual coefficients form an exponential sequence.
Bacharach also compares the biproportional method with that of Friedlander (1961) [10] who proposes the folowing minimand:
2
rr
cxij- o1J>
;x1J.
If the first order conditions are written down one can find that the relation between the old and new table is:
xij • oij+r1o1j + sj oij
Actually Friedlander did not propose his method for the purpose to
updata I/0 tables. He derived it to fill in frequency tables. An
example he used was from a demographic table repreaenting the frequency of age group versus marital status. The distribution of a certain year is known, but for a later year only the frequencies of the margins are available. Of course the mathematica! problem is the same. It became known as an I/0 tool mostly by the work of Bacharach.
The method has the property that zeros are preserved but, contrary to the RAS method, the signs of the variables are not. Also with this method the multipliers can be used to create projections for the
coefficients. These projections have a linear character. The
difficulty is then to keep the flows positive. Leeomber gives in his artiele [13] the advanteges and problems of the replacement of the
2
minimand by :
tr
(Xij- Oij) /Eij where E is the matrix of standard errors attached to elements of 0. The advantage is the simple multiplier structure that can be derived and the fact that the quality of the fermer estimate is taken into account. The problem of course isthat in the application of updating I/0 there is hardly any
information on standard errors available.
The metbod of Geary (1973) [8] is very much related to that of
Friedlander. It minimizes
tr
(Xij-Oij)2 so that the first orderconditions and with that the relation between the old and new matrix is assumed to be Xij=Oij+ri+sj. The principle of this metbod goes back to a very old (1940) technology which was also used to fill in a frequency table. See [18]. Obviously the small elements are able to change more than in the ether methods.
The method of Geary also shows a kind of distance function between two matrices. A distance function d has the following properties:
If A, B and C are matrices then i} d(A,B}=O if and only if A=B ii} d(A,B)=d(B,A)
iii} d(A,C}<= d(A,B)+d(B,C)
The minimizing criterion of Friedlander violates the second property in contrast to the criterion of Geary. Geary minimizes a real distance funtion.
It is debatable whether a criterion that minimizes the change in values or a criterion which minimizes the relative change is more appropriate for the updating of I/0 tables. A distance function can be used if ene wants to compare the estimate with a realisation. This is
done in the studies of Bacharach [2], Henry [12] and Scneider
[4].
There are also techniques to make histograms of the relative errors. The character of the change from the target matrix in a information gathering process is somewhat different from the allocation of change in a process of updating of an old matrix. In the first case the change is due to administrative errors. In the secend case ene knows that changes took place in the relations of the economy and ene wants to allocate this total changes, expressed by the growth and shifts of the margins as good as possible to the individual interindustry deliveries. In a method that minimizes the absolute changes, the increase of an amount is valued the same for an original small flow as for an original big flow in the I/0 table. This difference can be seen if the methods of Friedlander and Geary are taken in consideration. We will have a look in the next chapter at the criteria in which the relative change is minimized.
As stated before, the methods mentioned above lose their relation
between X and 0 as soon as ether constraints are imposed due to the availability of new information. Bacharach [2] suggests to split the updating matrix in two parts. One part has the relation of Stone and the ether contains values that are known for sure. In this case additional information can only contain sure values and values that have to be estimated.
Fisher suggests " ••• an essentially Bayasian exercise that utilizes
expert knowledge and judgement concerning
industrial input or capital structures instead of survey statistics .••• " asinformation [14].
Though his expressions for his methodology are very beautiful, he just means that it would be easy if there existed experts which could fill in a complete target matrix. This expert should be able to fill in columnwise all coefficients of all industries. After the matrix is reformed to the flow matrix it can be used in RAS updating. This will say that the relation between the old and new matrix is not the biproportional ene. It is net clear what Fisher adds with his methodology to the existing theory. It is similar to the FMA approach to update a matrix, in which however a team of technicians end economists are interacting with the algorithm.
What do we do if also bounds on values are known or further
estimated relations among economie parameters are known. Having
arrived at this point, it is time to leave the conventional methods and go to the L.P.-like methods that use a certain criterion in minimizing a kind of distance funtion and in which all information can
1io. J 2 J
..
'
be utilised as long as it is in a linear form and consistency is taken into account.
For clarities sake we show here the scheme made by Leeomber [13],
~hich forms a summary of the various methods. Notice that the idea of Theil is very similar to the RAS method. Theil uses his information theory also in this field.
An
essay on the applicability of his method can be found in the dissertation of Bacharach [2].N.a~ rropowd "-'" furrltrr opplkorloll funrriortOI Ad f'HtJPI,r!l Norts Dr fli.'ICUSSÎOft rrlotinn~ltip proctdurr
r~~-
•"·,l'
Alm on Bachar.chx• •
tH + 0X +m
lterati~ anocatÎCifl Larpe Geary ;_ where H • 11' Of ft'Y• 1nd rolumn pr!'tJ"('rtionaldisCTepan~ chanpeo; to equ&ll) lt' ek~Mnts small elements concemed. or.
direct soluticm Friedllnder Stephan x• • t0X + 0X + oX' lrerative
r'~-
•"·l
Omar allocati!'tn ofrt> ... and column
•"•I
BacharachHenry di~~pancies
J'IO·IItl ..,;lh
elemen" !'tf ,)i. or, direct ~luti!'tn
Matuszewski Schneider · . None Linear
Ct>mputation-r':r:j-
.z,,,
proJTamminp ally heavy
"oio Bacharach
.x.l
-lterati~ ali('Qtion Sipn~ of 11X
r(sü~~)
Cambridte DAE Bacharac:bx· -
to:Uor row and column pr~ned
Omar
~AS or Waelbroec:k d iSCT"epl neies biproportional Urton. pro-r.ta ...;rh
ekrMnts obtained at ptTYioas ite-ration
Tlleil. Bacllarac:tl None lntract.bk. Theil
(
~)
lllff"ISr ...
J lof "''appro1imatinr by Jlrocedure 2
Chapter
3.
LINEAR PROGRAMMING METHODS.As long as the information is in a linear ferm it can be imposed in the L.P. systems. Examples of such an information:
bounds on certain X .. 1J
bounds on I/0 coefficients Xij/X.j eco.
etc.
ratios like VA manufacturing/GDP The basis of the constraints is formed by the m+n constraints on the row and column totals: ~ Xij .. x.j and ~ Xij .. Xi..
The methods only differ by the minimand that links the change values. As we want to concentrate on relative changes let us define:
zij = (Xij-oij)/Oij'
and z.j
=
(X.j-O.j)/O.jNotice, that
r
z1.J. ~ z .. Further z .. is not defined, if 0 .. =0. With ·J 1J 1J this the changes in Xij , for which Oij=O, are not in the objective function. Two things can be done. Xij can be in the constraints with its bounds or is left out of the formuletien which has the same effect as assuming X .. =0.
1J
Matuszewski [3] used the objective function
rr
lzijl in the sixties to make estimates of the Canadian tables. Schneider compared this metbod with the RAS metbod in his dissertation[4].
The objective min max lzijl is implemented in the FMA software at
UNIDO. The idea of this objective function is not taken out of the
air. The idea that increase (or decrease) of the flows should lay within a range that is so small as possible, can be found back in
plans on the national level in east-European countries. This
corresponds with the activity analysis of Chenery. First targets are
set on f.e. total output. Constraints are formed by flow matrices
which determine this total output and by scarce resources. Then this criterium can be used which expresses that it is impossible to change the deliveries of individual sectors very rigourously.
A third possible objective function could be
rr
z~
.• The outcome 1Jis at least comparable with optimisation process becomes one it is not the same as the metbod
these of the two mentioned above. The of quadratic programming. Remark that of Friedlander.
Bacharach [2] mentions two ways to judge the quality of an estimate of a table: 1) Look at the estimate of gross output made with the
estimated coefficients.
2) Campare the estimate and realisation of the individual coefficients.
The first methodology was also applied by Matuszewski. With known
final demand the total production was estimated of
1956
with thecoefficients of first the
1949
table and later with the updated1956
Let us first state now that, if we take also the L.P. methodology in consideration, the various methods can not be compared simply by looking at estimates made with these and relate them to reality. As extra constraints can be put on the variables, it is for an individual case always possible to make an estimate with one methad exactly the same as an estimate with snother method. For the same reasen we have to say that the succes of an estimate is due to the economie team that worked on it and not due to the methad as such. In the study of Matuszewski 5 individual coefficients are stated to be known and taken out of the updating process. The ether 251 variables are uniformly bounded by .5< Xij/X.j<2. By the way, what is left is equivalent to a transport problem with capecity restrictions. In that time this was very important, as simpler algorithms could be used. This can be compared with Tinbergen et al.
[6]
who use the same technique.The upper limit of the variables is quite arbitrary. Why was it not 2.05 for example? It is completely to the economie team to make such constraints. This is also the philosophy in the FMA approach, which is also open to this kind of restrictions.
As Leeomber [13] emphasises, the quality of the updated matrix also depends on the quality of the first estimates of information. How goed are these margins estimated? In this sense it is wrong to say that the FMA software is better than the methad of Matuszeski in camparing the difference between his estimate and realisatien-es far as we can speak
of realisation- of the Canadian table and the difference between the
estimate and realisation of the Korean table made at the Global Branch. The principle of 'garbage in - garbage out' holds.
Studies in which the estimates with two methods are compared with the realisation can be found in [4], [12] and [2]. Assumption is that only the margins are known. Matrix distance functions and histograms are used to say something on the ranking of methods. However who did formulate criteria on which methad should be preferred? Lynch [17] compares in his study a RAS update with a realisation of a U.K. table.
However, as this realisation is also only an estimate, one is just
camparing an estimate in which more statistica! information is used with one in which less is used! Such studies hardly contribute anything to the methodology of updating I/0 tables.
The three objective functions give, of course, their own properties to the estimates. They all have their own basic character that should be kept in mind by the user of one of them, like the formula-1 coureur has to keep in mind the basic characteristics of his car; is it Porsche , Alfa or Lotus. How does the methad influence the outcome? We want to gain some insight in this as far as the objective function, not the car, is concerned. What are the consequences, if such objective functions are used?
Another question closely related is the sensitivity of the outcome to imposing extra constraints or tightening constraints. How much does the extra information influence the optimum? In linear and quadratic programming the answer to this question is clearly given by the dual variables of the constraints. Surprisingly we did not encounter anything of this subject in the literature in relation to the use of L.P. in the updating of I/0 matrices. In the iterative process between an L.P. algorithm and the specialist setting bounds, this means that
there is more information that can go to the specialist than the estimate itself. A high dual price fora certain eenstraint indicates that the difference between the new table and the old table is influenced strongly by the information the eenstraint represents.
Chapter 4. CHARACTER OF THE VAR10US OBJECT1VE FUNCT10NS IN A L.P EST1MATE.
As we want to use the methodology to update 1/0 tables, let us first have a look now at the pure case in which only the old table and the margins for the new table are known. For illustrative matters we use a very aggregated version of the Korean 1975 and 1980 tables.
Beside an agricultural sector (Ag) there are two manufacturing
sectors, a construction and water supply sector (CW) and a service sector (Se). Primary input consists of only import (1m) and value
added (VA). The first manufacturing sector consistsof food and wood
processing, textiles etc (M1), whereas the secend one contains the
heavier sectors machinery, chemicals, metallurgy etc (M2). We created this table with a special aggregation program from the original tables. We do not pretend to make a real estimate. These exercises serve only as an example. Notice however the enormous growth of the output of the various sectors.
Old 1/0 1975 Ag M1 Ag 277.1 410.4 M1 171.3 1146.2 M2 174.9 474
cw
4.5 85.8 Se 80.2 468.7 VA 2209.4 1471.1 Im 69.9 888.3 Tot 2987.3 4944.5margins new I/0 1980
Ag M1
(in biljon won)
M2
cw
3.3 6.9 235.5 130.5 1440.1 628.1 116.3 11.7 525.3 247.9 1312 619.1 1645.2 108.1 5277.7 1752.3 (in biljon won)M2
cw
Se FD Tot 39.4 2250.4 2987.5 263.7 2997.2 4944.4 472.8 2087.7 5277.6 117.7 1416.2 1752.2 850.8 3855.8 6028.7 4110.5 9722.1 173.8 1045 3930.3 6028.7 13652.3 34642.8 Se FD Tot Ag 7797.2 M1 21163.4 M2 27344.4cw
9508.5 Se 27824.3 VA 37116.1 Im 16243.9 Tot 7797.2 21163.4 27344.4 9508.5 27824.3 53360 146997. There are L.P. problem. and definevarious ways to formulate the estimation problem as a One can formulate the problem with the flow variables
V i,j zij = Xij/Oij-1. and further formulate the constraints for the row and column totals:.
r
xij= x j vi andr
x.j =x. v j.i . j l . l..
However the formulation is net that simple, because L.P. keeps the variables positive and zij can in principle attain negative values. A
possible salution is to formulate: zp
1j>= Xij/Oij-1 znij>= 1 - x1j;oij zij = zpij- znij
In this way zp represents a positive change and zn a negative change. Formulated this way the number of variables and constraints is rather big. Seen from a computational point of view it is better to reduce this by translating the problem to relative variables and calculating
the flows themselves after the optimum has been reached. The
constraints then look like: ~ oijzij = x.j- o.j vj and
t Oijzij= Xi.-Oi. Vi .To reduce the number of variables, the
j
objective function itself can be expressed in zpij and znij.
In our example wich leaves out the 'value added- final demand' deliveries, there are left 2*41 variables and 13 basic constraints. Notice that also the final demand flows are updated in this case. Usually this is not done as this belengs to national statistica! data and ene is interested in the technica! coefficients. However we repeat that this is only an example.
The dual prices can directly be interpreted. It indicates the influence of one unit increase of x.j or xi. on the final objective of minimizing zpij+znij related by one of the objective functions. The first and third objective functions can just be implemented by
2
formulating min
rr
zpij+znij resp minrr
(zpij+znij) • For the minmax criterion we have to add 41 constraints zp.j+zn .. <= u and the
l. l.J
objective function becomes min u.
Let us now go to our example. It must be kept in mind however that the changes for the margins are very big and contain price changes.
The growth rates are given by:
Ag 161% M1 328% M2 418%
cw
443% Se 362% VA 282% Im 313% FD 291%We calculated the optimum of an L.P. estimation with the min max criterion with the aid of the FMA software at the mainframe. The table with an optimising set of flows is shown below:
FMA forecast in flows Ag M1 M2
cw
Se FD Tot Ag 1312.8 1944.3 17.1 37.4 186.7 4298.9 7797.2 Ml 811.6 5430.2 1218.9 708.1 1249.3 11745.0 21163.4 M2 898.3 2434.4 7453.4 3408.3 2428.2 10722.0 27344.4cw
24.4 465.6 631.1 63.5 638.7 7685.2 9508.5 Se 379.9 2220.5 2718.7 1345.2 4030.8 17129.0 27824.3 VA 4039.1 4460 6790.4 3359.4 18467 37116.1 Im 331.2 4208.4 8514.9 586.6 823.4 1779.5 16243.9 Tot 7797.2 21163.4 27344.4 9508.5 27824.3 53360 146997.However as we don't want to make a real estimate but to focus on the optimising algorithm, the relative changes stated below are more interesting:
FMA forecast in relative change ( absolute value
Ag M1 M2
cw
Se FD Tot Ag 3.7 3.7 4.2 4.4 3.7 0.9 1.6 M1 3.7 3.7 4.2 4.4 3.7 2.9 3.3 M2 4.1 4.1 4.2 4.4 4.1 4.1 4.2cw
4.4 4.4 4.4 4.4 4.4 4.4 4.4 Se 3.7 3.7 4.2 4.4 3.7 3.4 3.6 VA 0.8 2.0 4.2 4.4 3.5 2.8 Im 3.7 3.7 4.2 4.4 3.7 0.7 3.1 Tot 1.6 3.3 4.2 4.4 3.6 2.9 3.2The optimum is found at 4.4, which is also the biggest change in the margins. In general something can be said about the outcomes. Consider the optimisation problem containing n+m constraints representing the known row and column totals and the min-max criterion formulated above.
Let M be the absolute maximum of the relative changes in the margins: M= max(max z.j'max zi.). The following can be stated:
1. The optimum >= M. Prove: Assume optimum < M. lf M is realised by a column total then I z .. 1 <Mand I z .. 1 > -M for all changes in
1J 1J
this column. This will say t 0. jz .. < M t 0 ..
=
MO . • However as1 1J 1J ·J
also t Oijzij = x.j-O.j= MO.j this is impossible. The same is true
if M is realised by a row total. q.e.d.
2. If the n+m constraints are the only ones and all margins grow with M, all margins grow with the same rate, then for the optimising zij :zij= M vi,j
lt is clear that the more constraints are imposed, the bigger will be
the optimum. In our example the optimum equals M = 4.4. However if
the original matrix is poorly scaled and the growth in the various sectors differs rather, this will not be the case any more.
It is a very important character of this metbod that the outcome is net unique at all. Of course the optimum is unique, but the degree of
freedom of the location finds become M minus
linear subspace, in which the m1n1m1zer or optimum-itself, reduces only with the number of z .. which
1J
the excact number of equations in which these zij occured in first instance.
In our case it are only the equations corresponding to the row and column total of the CW sector which influence the optimum. A closely related property is, that in the case opt=M the dual price of the eenstraint with the change of M will be the only nonzero one. Actually this dual price will be exactly 1/
o.j
if M is realised by a column total. Due to the fact that the row column is equal to the column total, degeneracy occurs and only one of the active restrictions has got a non-zero dual price. In our example two constraints are really active and the minimizer is left in a linear sub-space with ten degrees of freedom less and bounded by inequalities on every variable: These are not allowed to be less than -1, because the estimated flow will be negative then; and they are not allowed to exceed M, otherwisethe optimum will be attached.
A situation in which the margins grow with exactly the same rate and the optimum location is unique, seldom occurs. In the FMA software always the same optimum will be choosen by a chosing-rule that is hidden in the automatic fixation. If automatic fixatien is used and the algorithm is run several times, it will at least give the same optimising flows. It fixes first the variables in the row and column with the biggest change, then in the row and column with the biggest grow in the margins which are left etc.
The PC, on which we calculated the L.P. problem in the formulation shown above, reached the same optimum of 4.4 for the problem and gave only a negative dual price for the CW-row. Of course the fourth column total is as restrictive, but as there is a degeneracy caused by row total = column total the algorithm has to allocate this dual price to one of the two. As long as the variables in this row and column are not touched a lot of constraints can be imposed to the model without influencing the optimum itself, but with the possibility to change the other flows. By the way, the optimising zij found by the P.C. differed apart from the fourth row and column.
The inverse of this property is found, if the objective min
rr
lzijl is used. Every eenstraint added that will change a zij influences the optimum. The optimising zij are seldom not unique. A schematic presentstion of the outcome on the P.C. can be found below.
Estimate in relative changes if the Matuszewski criterion is used
Ag M1 M2
cw
Se ~ Tot Ag 2.14 1.6 M1 13.46 0.26 3.3 M2 6.77 12.35 2.18 4.2cw
5.48 4.4 Se 5.65 3.6 VA 2.18 0.54 5.30 2.8 Im 7.48 3.1 Tot 1.6 3.3 4.2 4.4 3.6 2.9 3.2All 13 constraints gave nonzero dual prices. A comparison with the original table immediately shows the most important property of this objective function: The biggest elements are changed the most. They explain the change in absolute flows the most without costing much in
terms of the objective.
To illustrate this we can think of an I/0 matrix with the biggest elements on the diagonal, representing intrasecteral deliveries. As
the total input equals the total output for this bleek -final demand is included-, we can state:
1. For the prescribed problem the optimiser has the property zij=O if i~j and zii
=
z.i.The character of this objective is very important. Many changes will be kept zero, if no extra information is put into the system. For the first objective function it is possible to change a zij without influencing the optimum. For this function this is nearly impossible.
The FMA software uses the first loss-function in a L.P. controlled estimation process. Matusewski used the secend one to update the Canadian tables also controlled by linear constraints. Apart from the case in which the optimum is determined by the constraints itself, the outcome will be different if the same information is put into the system and the different functions are used. The inclination of the first one will be to change all flows, whereas the secend wants to allocate the grow to the biggest values.
The quadratic objective function would lead to estimates between these two extremes. Remark that this objective differs from that of Friedlander, but can be better compared with the two ethers. The main difference is that the optimisation process becomes one of quadratic programming, which causes the dual prices not to be constant over a range. It differs from a L.P. estimate of the first kind in the fact that the optimising z .. will be unique. It is in certain sense a least
1J
squares estimate with m*n parameters and m+n constraints. It differs from the secend estimate in the fact that it punishes big changes more
than small ones. We tried to make an estimate for our example, but
unfortunately the problem with 13 constraints and 41 variables (We left out the negative changes ) already gave numerical difficulties for the computerprogram used at the P.C. for quadratic programming, PC-prog
We state again, that by adding decision parameters in the form of individual weights or constraints, the outcome of the relative
automatic estimation process can be influenced, so that it gives at
least acceptable estimates. However it is useful to keep in mind the general character of the method used t
As the last point for discussion about the difference in approach we say something about the sensitivity of the change in the I/0 coefficients. As can be observed, the coefficients will change less if the first objective is used than in the case the secend is used. In our example these coefficients do not change for the CW column. In principle it is also possible to formulate the constraints and objective in terms of changes in the coefficients. Constraints on the flows can be translated to linear constraints in terms of old
coefficients, old totals and the change variables, though of course the formulation will be more abstract and difficult.
It is logical that one did not choose for this possibility in the FMA software, as this also has to tackle ether flow problems in which one does not speak of coefficients. The relation between the problems can be seen in the following identities:
(Xij/X.j)/(Oij/O.j)-1 = (Xij/Oij)/(X.j/O.j)-1 = (1+zij)/(1+z.j)-1 We can see that the coefficients only do not change, if the growth of the delivery flow is exactly the same as the growth of total intermediar input for this sector. It is discutable whether a formulation with the objective of as less possible change in the coefficients is more appropriate in an estimation process. At least it is not so difficult to change the objective function in the direction of: Let the coefficients change as few as possible. Actually this was done by Matuszewski
(3].
Let ~ij= (Xij/X.j-aij)/aij • By replacing in the FMA software the old matrix Oij by aijxij , representing the implicit flow, as the target matrix the objective becomes one of minimizing the maximum change in the coefficients, because:
~.j= l. (X.j/X .-ai .)/a.j = (X .. -ai .X j)/a .. X j • l. • J J l. l.J J • l.J •
Matuszewski used this transformation so that actually his method is not clearly represented in the table of Leeomber on page
7
nor by the table of Thomas-Bulmer [1], which suggest that the target matrix is the old matrix.Bulmer -Thomas [1] states from experience, that some flows are
changing much, but most are changing marginally over time. If an
automatic updating process is wanted, the quadratic function would be appropriate, when more than the marginal information is available. However, are the big changes then allocated to the right cells7 This is only the case if the bigger flows change the most. Here we find that one actually should know analegeus to the theory of Brauer [16] which sectors are important. The FMA idea corresponds very much with this theory. Find information about the important sectors and divide the rest of the growth without extremes.
In this last case some mathematica! problems should be kept in
mind. If we add in our example the fact that the flows of the
agricultural sector to the M1 sector has grown more than 600%, we will of course find an optimum of
6.
The bounds in wich the zij can move without influencing the optimum have widened. To get the desired result of dividing the rest of the grow, we should throw z1,2 out of the objective function. This is not the case if the other criteria are used.
5.
CONCLUSIONS.-The objective of minimizimg the maximum relative change is very useful in L.P.-like planning exercises:
1. The bottleneck sectors can be discovered in this way.
2. Degrees of freedom are left for the optimiser, so tbat secondary objectives can be taken into account.
It is not usable as such in estimating, as a necessary requirement, uniqueness, is not garanteed.
-In the FMA flow model the additional criterion to preserve uniqueness is hidden in the choice rule of of the automatic fixatien part of the algorithm. After the optimum is determined an optimizing table is fixed in a consecutive process. in which in every iteration the flows of the sector with the biggest changes are fixed in the linear space and from the sectors that are left until the degree of freedom is zero.
-In a situation in which only the boundaries are known, an algorithm like RAS is preferable, as it takes much less computer time.
-If more information is available L.P.-like methods should be used. If they are used, one should concentrate on the finding of the sectors or cells which change the most. If these are found and information about them is used, the FMA system is preferable as it changes all ether flows as less as possible, whereas the criterion of Matuszewski changes only the biggest flo~s. One should keep in mind however first to throw these individual flows for which restrictive bounds are known with certainty, out of the objective function.
-The L.P. methods cannot be compared by looking at the outcomes of the individual coefficients or flows or at the estimate of total production with the updated coefficients, as the outcome depends on the quality of the economie team involved.
-In the past no attention was paid to the use of dual prices for information in L.P. as estimating process. However if this is done, the dual prices give much less information when the FMA criterion is
