Leonid Vitaliyevich Kantorovich – Autobiography
was born in Petersburg (Leningrad) on 19th January 1912. My father, Vitalij Kantorovich,
died in 1922 and it was my mother, Paulina (Saks), who brought me up. Some of
the first events of my childhood were the February and the October Revolutions
of 1917, and a one-year trip to Byelorussia during the Civil War.
My first interest in sciences and the first displays of self-dependent thinking manifested themselves about 1920. On entering the Mathematical Department of the Leningrad University in 1926, I was mainly interested in sciences (but also in political economy and modern history, thanks to the most vivid lectures of academician E. Tarle).
At the University, I attended lectures and worked in seminars of V.I. Smirnov, G.M. Fichtengolz, B.N. Delaunay; my University friends were I.P. Natanson, S.L. Sobolev, S.G. Michlin, D.K. and V.N. Faddeevs.
My scientific activities started in my second university year covering the rather more abstract fields of mathematics. I think my most significant research in those days was that connected with analytical operations on sets and on projective sets (1929-30) where I solved some N.N. Lusin problems. I reported these results to the First All-Union Mathematical Congress in Kharkov (1930).
My participation in the work of the Congress was an important episode in my life; here I met such outstanding Soviet mathematicians as S.N. Bernstein, P.S. Alexandrov, A.N. Kolmogorov, A.O. Gelfond, et al, and some foreign guests, among whom were J. Hadamard, P. Montel, W. Blaschke.
The Petersburg mathematical school combined theoretical and applied research. On graduating from the university in 1930, simultaneously with my teaching activities at the higher school educational institutions, I started my research in applied problems. The ever expanding industrialization of the country created the appropriate atmosphere for such developments. It was precisely at that time such works of mine, A New Method of Approximate Conformal Mapping , and The New Variational Method were published. This research was completed in Approximate Methods of Higher Analysis, a book that I wrote with V.I. Krylov (1936). By that time I was a full professor confirmed in this rank in 1934, and in 1935, when the system of academic degrees was restored in USSR, I received my doctoral degree. At that time I worked at the Leningrad University and in the Institute of Industrial Construction Engineering.
The Thirties was a time of intensive development of functional analysis which became one of the fundamental parts of modern mathematics.
My own efforts in this field were concentrated mainly in a new direction. It was the systematical study of functional spaces with an ordering defined for some of pairs of elements. This theory of partially-ordered spaces turned out to be very fruitful and was being developed at approximately the same time in the USA, Japan and the Netherlands. On this subject I contacted J. von Neumann, G. Birkhoff, A.W. Tucker, M. Frechet and other mathematicians whom I met at the Moscow Topological Congress (1935). One of my memoires on functional equations was published as a result of the invitation extended to me by T. Carleman in Acta Mathematica. Functional Analysis in Semiordered Spaces, the first complete book of our contributions in this field, was published in 1950 by my colleagues, B.Z. Vulikh and A.G. Pinsker, and myself.
In those days, my theoretical and applied research had nothing in common. But later, especially in the postwar period, I succeeded in linking them and showing broad possibilities for using the ideas of functional analysis in Numerical Mathematics. This I proved in my paper, the very title of which, Functional Analysis and Applied Mathematics, seemed, at that time, paradoxical. In 1949, the work was awarded the State Prize and later was included in the book, Functional Analysis in Normed Spaces, written with G.P. Akilov (1959) .
The Thirties was also important for me as I began my first economics. The very starting point was rather accidental. In 1938, as professor of the university, I acted as a consultant for the Laboratory of the Plywood Trust in a very special extreme problem. Economically, it was a problem of distributing some initial raw materials in order to maximize equipment productivity under certain restrictions. Mathematically, it was a problem of maximizing a linear function on a convex polytope. The well-known general recommendation of calculus to compare the function values in the polytope vertices lost its force since the vertices number was enormous even in very simple problems.
But this accidental problem turned out to be very typical. I found many different economic problems with the same mathematical form: work distribution for equipment, the best use of sowing area, rational material cutting, use of complex resources, distribution of transport flows.* This was reason enough to find an efficient method of solving the problem. The method was found under influence of ideas of functional analysis as I named the "method of resolving multipliers".
In 1939, the Leningrad University Press printed my booklet called The Mathematical Method of Production Planning and Organization which was devoted to the formulation of the basic economic problems, their mathematical form, a sketch of the solution method, and the first discussion of its economic sense. In essence, it contained the main ideas of the theories and algorithms of linear programming. The work remained unknown for many years to Western scholars. Later, Tjalling Koopmans, George Dantzing, et al, found these results and, moreover, in their own way. But their contributions remained unknown to me until the middle of the 50s.
I recognized the broad horizons offered by this work at an early stage. It could be carried forward in three directions:
1) The further development of methods of solving these extremal problems and their generalization; their application to separate classes of problems;
2) A mathematical generalization of these problems such as, non-linear problems, problems in functional spaces, the application of these methods to extremal problems of mathematics, mechanics and technical sciences;
3) The spreading of the method of description and analysis from separate economic problems to general economic systems with their application to planning problems on the level of an industry, a region, the whole national economy as well as the analysis of the structure of economic indices.
Some activity took place in the first two directions (the results were published partly immediately, partly after the war), but the third one lured me most. I hope that the reasons were clarified enough in my Nobel lecture.
The studies were interrupted by the war. During the war, I worked as Professor of the Higher School for Naval Engineers. But even then I found time to continue my deliberations in the realm of economics. It was then that I wrote the first version of my book. Having returned to Leningrad in 1944, I worked at the University and at the Mathematical Institute of the USSR Academy of Sciences, heading the Department of Approximate Methods. At that time, I became interested in computation problems, with some results in the automation of programming and in computer construction.
My economics studies progressed as well. I particularly wish to mention the work done in 1948-1950 at the Leningrad Carriage-Building Works by geometrist V.A. Zalgaller under my guidance. Here the optimal use of steel sheets was calculated by linear programming methods and saved material. Our book of 1951 summarized our experience and gave a systematic explanation of our algorithms including the combination of linear programming with the idea of dynamic programming (independently of R. Bellman).
In the middle of the 50s, the interest in the improvement of economic control in the USSR increased significantly, and conditions for studies in the use of mathematical methods and computers for general problems of economics and planning became more favourable. At that time, I made a series of reports and publications and prepared the above-mentioned book for publication. It appeared in 1959 under the title, The Best Use of Economic Resources, and contained a broad exposition of the optimal approach to such central problems of economics as planning, pricing, rent valuations, stock efficiency, "hozraschet" problems and decentralization of decisions. Precisely at that time, I contacted foreign scholars in this field. As a particular result, thanks to the initiative of Tjalling Koopmans, my 1939 booklet was published in Management Science, and, somewhat later, the 1959 book was translated as well.
Some of the Soviet economists met the new methods guardedly. Together with the book, I must mention the special Conference on Mathematical Methods in Economics and Planning held by the Academy of Science. The participants of the conference were some prominent Soviet mathematicians and economists. The conference approved the new scientific direction. But this time we had obtained some positive experience of its applications.
The field attracted a number of young talented scientists, and the preparation of such hybrid specialists (mathematician-economist) began in Leningrad, Moscow, and some other cities. It is worth noting that in the newly-organized Siberian Branch of the Academy of Sciences, conditions for new scientific directions were especially favourable. A special laboratory on the application of mathematics in economics headed by Nemchinov V.S. and me was created. Its main body belonged to the Leningrad and Moscow schools. In Akademgorodok it was integrated into Institute of Mathernatics as a department.
I was elected Corresponding-Member of the Academy in 1958 and came to Novosibirsk in 1960. Out of my group in Novosibirsk, a number of talented mathematicians and economists emerged.
In spite of continual discussions and some critique, the scientific direction gained recognition more and more by both the scientific community and governmental bodies. The token of this recognition was the Lenin Prize which I was awarded in 1965.
Now I head the Research Laboratory at the Institute of National Economy Control, Moscow, where high-ranking executives are introduced to new methods of control and management. I act as consultant to various governmental bodies.
I was married in 1938. My wife, Natalie, is a physician. We have two adult children (d. and s.), both working in mathematical economy.
* A. Tolstoy had stated this problem before me (1930). He gave an approximate method of its solution. Later, the same problem was stated by F. Hitchcock.
From Nobel Lectures, Economics 1969-1980, Editor Assar Lindbeck, World Scientific Publishing Co., Singapore, 1992
This autobiography/biography was written at the time of the award and later published in the book series Les Prix Nobel/Nobel Lectures. The information is sometimes updated with an addendum submitted by the Laureate. To cite this document, always state the source as shown above.
Leonid Kantorovich died on April 7, 1986.
Leonid Vitaliyevich Kantorovich – Prize Lecture
Lecture to the memory of Alfred Nobel, December 11, 1975
Mathematics in Economics: Achievements, Difficulties, Perspectives
I am deeply excited by that high honour which fell on my lot and I am happy for the opportunity to appear here as a participant of this honorable series of lectures.
In our time mathematics has penetrated into economics so solidly, widely and variously, and the chosen theme is connected with such a variety of facts and problems that it brings us to cite the words of Kozma Prutkov which are very popular in our country: "One can not embrace the unembraceable". The appropriateness of this wise sentence is not diminished by the fact that the great thinker is only a pen-name.
So, I want to restrict my theme to the topics which are nearer to me, mainly to optimization models and their use in the control of the economy for the purpose of the best use of resources for gaining the best results. I shall touch mainly on the problems and experience of a planned economy, especially of the Soviet economy. Certainly even within these limits I will succeed in considering only a few problems.
Specific peculiarities of the problems considered
Before discussing methods and results I think it will be useful to talk about the specific peculiarities of our problems. These are distinctive for the Soviet economy and many of them appeared already in the years just after the October Revolution. Then for the first time in history all main means of production passed into the possession of the people and there arose the need for the centralized and unified control of the economy of the vast country. This need appeared in very complicated and social conditions and met with some specific peculiarities. The following problems are related both to the economic theory and to the practice of planning and control.
1) First of all, the main purpose of economic theory was altered. There appeared a necessity to shift from study and observation of economic processes and from isolated policy measures to systematic control of the economy, to the common and united planning being based on the common aims and covering a long time horizon. This planning must be so detailed as to include specific tasks to individual enterprises for specific periods and to that common consistency of the whole this giant set of decisions was guaranteed.
It is clear that a planning problem of such scale did appear for the first time, so its solution could not be based on the existing experience and economic theory.
2) Economic science must yield not only conclusions on general economic problems concerning the national economy as a whole but also serve as the basis for solutions concerning single enterprises and projects. So it needs the proper information and methodology to provide decisions that are in accordance with general goals and interests of the national economy. Finally, it must contribute not only general qualitative recommendations but also concrete quantitative and sufficiently precise accounting methods which could provide the objective choice of economic decisions.
3) Together with material flows and funds in capitalist economies there are also studied and directly observed such important economic indices as prices, rents, interest rate in their static and dynamic properties. The indices mentioned serve as the background for all economic calculations, for aggregation, for the construction of the synthetic indices. It became clear that a, consistently planned economy cannot do without indices characterizing the analogous aspects. They could not be observed here and were given as normatives. The problem of their calculation was however not restricted only by technical aspects of calculation and statistics. It is important that in the new conditions similar indices received a quite different sense and significance, and some problems as to their nature, role and structure arose. For example, it was unclear and open to discussion whether a land rent should exist in a society where land is in the possession of the people or whether such an index as the interest rate has a right to exist.
4) The previous problems are displayed in one more peculiarity of the planned economy. Obviously the economy of such scale and complexity cannot be quite centralized 'up to the least nail' and a valuable part of decisions should be retained for the lower levels of the control system.
The decisions of different control levels and from different places must be linked by material balance relations and should follow the main object of the economy.
The problem is to construct a system of information, accounting, economic indices and stimuli which permit local decision-making organs to valuate the advantage of their decisions from the point of view of the whole economy. In other words to make profitable for them the decisions profitable for the system, give a possibility to check the validity of the work of local organs activity also from the point of view of the whole economy.
5) New problems of control of the economy and new methods put forward the question of the most efficient structural forms of control organization.
Some changes of these forms have taken place both owing to the tendency to perfection of the control system as well as to changes in the economy itself, connected with the increase of its scale, the increasing complexity of connections and with new problems and conditions. The problem of the most efficient structure of a planning system has also a scientific aspect, but its solution is not well advanced.
6) Some complex problems of economic control were generated by the contemporary development of the economy, by the so-called scientific-technical revolution. I mean the problems of prediction and control in conditions of large shifts in the weights of different branches, of the rapid changes in production and technology, national economy. The problems of estimating technical innovations and the general effect of technical progress. The problems of ecology connected with the deep changes of the natural environment under the influence of human activity, the prospects of exhausting the natural resources. The prediction of social changes and their influence on the economy. The changes in presence of contemporary computational technique, means of communication, managerial devices and so on.
Most of these problems arise also in countries with capitalist economy but in socialist economy they have their own difficulties and peculiarities.
There existed neither experience nor sufficient theoretical foundation for the solving of these hard problems.
The economic theory of Karl Marx became the methodological background of the new created Soviet economic science and of the new control system. A number of its important and fundamental statements on general economical situations turned out to be applicable immediately to a socialist economy. However a practical use of Marx' ideas needed serious theoretical research. There was no practical economic experience under the new conditions.
These problems were being solved practically by governmental bodies and economic executives. They were being solved under the first years of the state in difficult conditions of the Civil war, Devastation and postwar reconstruction. Nevertheless the problem of building up an effective economic mechanism was resolved. I have no possibilities to describe it in detail but I just wish to point out that the system of planning organs was created on the initiative of the founder of our state V. Lenin and simultaneously on the same initiative a system of economic accounting (hozraschet) was introduced which gave a certain financial form of balance and control of separate economic activities.
An evidence of significant efficiency of this mechanism lies in the great improvements of the economy, successful solution of the industrialization problem, of the economic problems of state defence before and during the Second World War, the postwar reconstruction and further development.
The system of planning and economic organs was, improved and altered in connection with new problems. The generalization of this experience built a stock in anticipation of economic theory of the planned socialist economy.
At the same time in our country the necessity of further improvements of the control mechanism, some defects in the use of resources, incomplete realization of the potential advantages of the planned economy were pointed out repeatedly. It was obvious that such improvements should be based on new ideas and new means. This led to the natural idea to introduce and use quantitative mathematical methods.
The new methods
The first attempts to use mathematics in the Soviet economic researches were made in the 20-ies. Let me name the well-known demand models of E. Slutsky and A. Konjus, the first growth models of G. Feldman, the 'chesstable' balance analysis done in the Central Statistical Department, which was later developed both mathematically and economically using the data of the US economy by W. Leontiev. The attempt of L. Jushkov to determine rate of investment efficiency received a profound continuation researches of V. Novojilov. The above mentioned researches had common features with the mathematical direction in Western economic science which developed at the same time and was presented in the works of R. Harrod, E. Domar, F. Ramsey, A. Wald, J. von Neumann, J. Hicks et al.
Here I would like to talk mainly about optimization models which appeared in our country in the late 30-ies (and later independently in USA) and which were in a certain sense the most suitable means to treat the problems I have mentioned.
The optimizing approach is here a matter of prime importance. The treatment of the economy as a single system, to be controlled toward a consistent goal, allowed the efficient systematization of enormous information material, its deep analysis for valid decision-making. It is interesting that many inferences remain valid even in cases when this consistent goal could not be formulated, either for the reason that it was not quite clear or for the reason that it was made up of multiple goals, each of which to be taken into account.
For the present the multi-product linear optimizing models seems to be mostly used. I suppose that now it is spread in economic science not less than for instance Lagrange equations of motion in mechanics.
I see no need to describe in detail this well-known model which is based on the description of an economy as a set of main kinds of production (or activities, - the term of professor T. Koopmans), each characterized by use and production of goods and resources. It is well-known that the choice of optimal program i.e. of the set of intensities of these activities under some resource and plan restriction gives us a problem to maximize a linear function of many variables satisfying some linear restrictions.
This reduction has been described too many times so that it can be treated as well-known. It is more important to show those of its properties which determine its wide and various use. I can name the following ones:
a) Universality and flexibility. The model structure permits various forms of its application, it can describe very different real situations for extremely different branches of economy and levels of its control. It is possible to consider a series of models where necessary conditions and restrictions are introduced step-by-step while the needed descriptive precision is not reached.
In more complicated cases when the linearity hypothesis significantly contradicts the problem specifics and we must take into account non-linear inputs and outputs, indivisible decisions and non-deterministic information. Here the linear model becomes a good 'elementary block' and the take-of point for generalizations.
b) Simplicity. In spite of its universality and good precision the linear model is very elementary in its means which are mainly those of linear algebra, so even people with very modest mathematical training can understand and master it. The last is very important for a creative and non-routine use of the analytical means which are given by the model.
c) Efficient computability. The urgency of solving extremal linear problems implied an elaboration of special, very efficient methods worked out both in USSR (method of successive improvements, method of resolving multipliers) and in USA (well-known simplex-method of G. Dantzig), and a detailed theory of these methods. An algorithmic structure of the methods has allowed later to write corresponding computer codes and nowadays modern variants of the methods on modern computers can rapidly resolve problems with hundreds and thousands of restrictions, with tens and hundreds of thousands of variables.
d) Qualitative analysis, indices. Together with the optimal planning solution the model gives valuable devices of qualitative analysis of concrete tasks and of the whole problem. This possibility is given by a system of indices for activities and limiting factors which is found simultaneously with the optimal solution and is in accordance with it. Professor T. Koopmans named them 'shadow prices', my term was 'resolving multipliers' since they were used as an auxiliary device for optimal solution finding like Lagrange multipliers. However shortly after their economic meaning and importance for analysis were realized, and they have been named in economic treatment objectively-determined valuations (Russian version gives an obreviation 'o.o.o.'). They have the sense of value indices of goods and factor equivalence, intrinsically determined for a given problem, and showing how the goods and factors can be exchanged in fluctuations of extremal state. Thus these valuations give an objective way of calculating accounting prices and other economic indices and a way of analysing of their structure.
e) Concordance of the means with the problems. Though separate firms and even government bodies in states of capitalist economy successfully used these methods their spirit corresponds closer to the problems of socialist economy. Evidence of their efficiency is in their successful application to a number of concrete problems of economic science and operations research. They have such large-scale applications as the long-term planning of some branches of Soviet economy, territorial allocation of agricultural production. Now we are discussing problems of model complexes including the model of long-term planning of the whole national economy. These problems are investigated in special large research institutes - Central economic-mathematical institute in Moscow (headed by academician N. Fedorenko) and Institute of economic science and industry organization in Novosibirsk (headed by academician A. Aganbegjan).
It is necessary to point out as well the current position of optimal planning and mathematical methods in theoretical investigations of Soviet economic science. The linear model has proved a good means of simplest logical description for problems of planning control and economic analysis. It has contributed to significant advancements in pricing problems. For instance, it has given justification of accounting basic funds in production prices, principles of accounting the use of natural resources. It is given also a quantitative approach to reflecting the time factor in investments. Note that a model describing a simple economic index has sometimes a rather sophistical mathematical form (We can mention here as an example a model for the use of a stock of equipment from which the structure of amortization payments was derived).
A problem that needs to be pointed out especially is that of decentralized decisions. The investigation of a two-level model complex leads us to the conclusion that in principle the decentralization of decisions with observance of the total object of the complex is possible by the means of a correct construction of objects in local models. We must point out here a brilliant mathematical formalism of the idea of decomposition given by G. Dantzig and Ph. Wolfe. The value of their paper of 1960 is far from the limits of the algorithm and its mathematical foundation. It gave rise to a lot of active discussions and various treatments in the whole world and particularly in our country.
Together with input-output analysis and optimization models as a result of the activity of a large community of scientists the economic theory and practice was provided with such analytical tools as statistics and stochastic programming, optimal control, simulation methods, demand analysis, social economic science and so on.
Summing up we say that as result of about 15 years of intensive development and spreading of the mentioned methods we have some significant results.
The level of development and especially that of applications may cause however a feeling of dissatisfaction. The solving of many problems was not completed. Many applications are episodical, they don't became regular and are not united into a system. In the most complicated and perspective problems, such as those of national planning, have up till now effective and generally acceptable forms for the realization not been found. The attitude to these methods like to many other innovations went sometimes from scepticism and resistance through enthusiasm and exaggerated hopes to some disappointment and dissatisfaction.
Certainly we can say that the results are not too bad for such a short period of time that has passed. We can refer to the longer periods of widespreading of many technical innovations or to physics and mechanics where some theoretical models are not realized in spite of two-hundreds-year experience. However we prefer to mention some concrete problems to clarify the main difficulties and their causes and to outline some ways to overcome them. Difficulties arise both from the specific features of the object under investigation and from defects in the researches and their practical realization.
The economic matter is a difficult object for a formal description in view of its complexity and pecularity. The models emphasize only a few of its aspects and take into account the real economic situation very roughly and approximately, so as a rule it is difficult to estimate the correctness of the descriptions and inferences.
So in spite of the above mentioned universality of the model and its generalizations a routine approach is often non-efficient. A work on each serious model and its practical application demands hard research elaboration with joined efforts of economists, mathematicians and specialists in the concrete field, but even in successful cases the widespreading of the model needs several years, especially for testing and improving of practical instructions.
It is especially important to test the influence of the difference between the model and reality on the obtained result and to correct the result or the model itself. This part of work is not often observed.
The hard thing in a model realization is to receive and often to construct necessary data which in many cases have considerable errors and sometimes are completely absent, since none needed them previously. Difficulties of principle lie in the future prediction data and in the estimation of industry development variants.
The computation of optimal solution has its difficulties as well. In spite of the presence of efficient algorithms and codes practical linear programs are not too simple since they are very large. The difficulties grow significantly when the linear model is modified by any of its generalizations.
It was mentioned that theoretically in the linear model there is a perfect accordance and harmony of the optimal solution and the estimating indices and stimuli based on o.o.o. However real decisions and the work of local bodies are evaluated not by the theoretical indices but by actual prices and methods of assessment which are not so simple to replace. Even if one branch or region adopts its proper indices the disharmonies will appear on boundaries with its neighbours. Moreover various parts of the economic system are described by mathematical models with difficult success and they have not always distinct quantitative characteristics. Thus the industrial production is described better than demand and consuming preferences. At the same time in a wide statement of the plan optimization problem it is natural to tend not only to least possible use of resources but as well to a structure of production which is optimal for consumers. This condition complicates the correct choice of objective function.
Certainly the situation is not hopeless. For instance, one can use an idea of extremal state (i.e. of a state which cannot be improved all-round, 'efficient decision' of A. Wald), which is pithy enough. Then one can make a compromise of a few criteria or be less rigorous and solve the industrial part of the problem by optimization methods and the consuming one by the traditional expert methods. One can try to use econometrics, - too many 'can' mean that the problem is very far from solution.
In planning the idea of decentralization must be connected with routines of linking plans of rather autonomous parts of the whole system. Here one can use a conditional separation of the system by means of fixing values of flows and parameters transmitted from one part to another. One can use an idea of sequential recomputation of the parameters, which was successfully developed by many authors for the scheme of Dantzig-Wolfe and for aggregative linear models.
A solution of newly appearing economic problems, and in particular those connected with the scientific-technical revolution often cannot be based on existing methods but needs new ideas and approaches. Such one is the problem of the protection of nature. The problem of economic valuation of technical innovations efficiency and rates of their spreading cannot be solved only by the long-term estimation of direct outcomes and results without accounting peculiarities of new industrial technology, its total contribution to technical progress.
The accounting methods based on mathematical models, the use of computers for computations and information data processing make up only one part of the control mechanism, another part is the control structure. Thus success of the control depends on to what extent and how there is guaranteed in the system the possibility of personal interest in correct and complete information, in proper realization of decisions achieved. The construction of such interest and of checking systems is not an easy job either.
Moreover, in order to achieve a real spread of the new methods it is necessary that they be studied and mastered by the people who are employed in planning and economic science. It is necessary to reorganize this system, to overcome some psychological barrier, to shift from many-years-used routines to new ones.
For this purpose we have an educational system which serves to acquaint the planning administration up to top level with the new methods. The accounting reorganization usually is combined with the introduction of computerbased information systems. It is clear that such a recognization of methods and consciousness is difficult and time-spending.
In spite of mentioned difficulties I am looking optimistically on the prospects of wide spread of mathematical methods, especially those of optimization, in economic science and in all-level economic control. It can give us a significant improvement of planning activity, better use of resources, increment of national income and living standards.
The difficulties of modelling and data creation can be overcome like similar difficulties were overcome in the natural and technical sciences. My hope is based on the more and more intensive steam of research for new methods and algorithms in this field, on the fact of appearance of new theoretical approaches and problem statements, on a series of concrete studies of general and special problems concerning separate economic branches, on the fact that a whole army of talented young researchers work now in this field. A significant progress is now being made in the development of computer hard- and software and their mastering.
The mathematicians, economists and practical managers have achieved a better mutual understanding.
The favourable conditions for the work in this field were given by well-known important statements on control methods and their improvements which were made in last years by our authorities.
From Nobel Lectures, Economics 1969-1980, Editor Assar Lindbeck, World Scientific Publishing Co., Singapore, 1992
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