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The Nobel Series: Kantorovich & Koopmans

Leonid Kantorovich and Tjalling C Koopmans made major contributions to the field of economic planning by way of developing the theory of the optimal allocation of resources — relevant not only for socialist countries but also free-market economies

The Nobel Series: Kantorovich & Koopmans
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Kantorovich and Koopmans were jointly awarded the Nobel Prize in Economic Sciences in 1975 for their work on the theory of the optimal allocation of resources. As we know, this is a question that is important in normative economics and answers the question 'what should be…', rather than 'what is…'. Hence, both the awardees sought to answer questions such as:

What goods should be produced

What methods of production should be used

How much of current production should be consumed, and how much is reserved to create new resources for future production and consumption.

Works of Kantorovich

Kantorovich took admission in the mathematics programme at Leningrad University in 1926. He graduated in 1930 and did work on mainly theoretical mathematical problems. He got his doctorate from the same University in 1935. Only after graduating did he turn his attention to applications. Kantorovich's first brush with economics was accidental. In 1938, he worked as a consultant to the plywood industry where the problem was one of distributing raw materials in order to maximise capital productivity subject to certain constraints. Kantorovich found that the problem in the plywood industry was typical in many different economic arenas such as transport, agriculture etc. In 1939, the Leningrad University Press printed Kantorovich's booklet titled, 'The Mathematical Method of Production Planning and Organization', which was nothing but a compilation of the main ideas of linear programming. More

generally, in this book, Kantorovich showed that all problems of economic allocation can be seen as maximising a function subject to constraints. As we know, John Hicks in Britain and Paul Samuelson in the United States were reaching the same conclusion at around the same time.

As the Nobel website informs us, Kantorovich's 1939 booklet referred to above spoke of the meaning and significance of efficient use of resources in individual enterprises. Kantorovich's best book is perhaps, 'The Best Use of Economic Resources', published in 1959, where he developed some of the points made in his earlier 1939 book. In this book, he discussed the connection between the allocation of resources and the price system, both at a certain point in time and in a growing economy using linear programming. He showed how the possibility of decentralising decisions in a planned economy is dependent on the existence of a rational price system, including a uniform accounting interest rate to form a foundation for investment decisions. Further, even centrally planned economies have to be concerned with using prices to allocate resources. As Kantorovich mentions in his Nobel speech:

"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 decentralisation of decisions. 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."

Besides receiving the Nobel Prize, Kantorovich was awarded the Soviet Government's Lenin Prize in 1965 and the Order of Lenin in 1967. From 1944 to 1960, Kantorovich was a professor at the University of Leningrad. In 1960 he became director of mathematical economic methods at the Siberian Division of the Soviet Academy of Sciences. In 1971, he was appointed laboratory chief of the Institute of National Economic Management in Moscow.

Works of Koopmans

Koopmans studied mathematics at Utrecht University in the Netherlands in the late 1920s and later switched to physics. Only in 1933 when he came in touch with the Nobel laureate Jan Tinbergen, did he move to Amsterdam to study mathematical economics and econometrics under him. He got his doctorate from Leiden University in 1936. His thesis was titled, 'Linear regression analysis of economic time series'. Koopmans moved to the US in 1940, and joined the Cowles Commission for Research in Economics in Chicago (affiliated to University of Chicago) and became Director of this Commission in 1948. When this Commission fell out with the University of Chicago, Koopmans moved it to Yale University in 1955.

Koopmans' broad area of work was the optimal allocation of resources. In the United States, Koopmans became a statistician with the Combined Shipping Adjustment Board in Washington where he tried to solve the practical problem of how to reorganise shipping to minimise transportation costs. Basically, the purpose was to organise the most optimal routing of empty ships. Since the variables included thousands of merchant ships, millions of tons of cargo, and hundreds of ports, he solved it using linear programming tools. The technique he developed to do so was called "activity analysis" and is now called linear programming. His first write-up of the analysis is in a 1942 memorandum. His techniques were very similar to those used by Kantorovich, whose work he discovered only much later.

Koopmans was also like Kantorovich in generalising his approach from one sector of the economy to the economy as a whole. Koopmans showed the conditions required for economy-wide efficiency in allocating resources. He also, again like Kantorovich, used his activity analysis techniques to derive efficient criteria for allocating between the present and the future.

Koopmans is also well known for his work titled, 'Analysis of Production as an Efficient Combination of Activities', where he developed his activity analysis. Here he proposed new ways of interpreting the relationship between inputs and outputs of a production process, which is then used to arrive at a relationship between efficiency and a system of calculation prices. This led to a connection between the normative allocation theory and the general equilibrium theory. During the sixties, Koopmans studied the problem of finding criteria for an optimum growth rate for an economy.

Kantorovich, Koopmans and public policy

As we have seen above, Koopmans was influenced a great deal by Jan Tinbergen, a fellow Dutch and the first Nobel Prize winner in Economic Sciences. Tinbergen's work in the shipping industry, as well as his statistical and econometric modelling, were some specific areas which Koopmans developed further in his work. For example, Koopmans' formal mathematical models in transportation were inspired by Tinbergen. Specifically, the problem of determining the shipping plan that minimises the total cost, given a preassigned pattern of availability of supplies and demands, is known as Koopmans transportation problem. This was Koopmans first posing of a linear programming problem, that is, the maximisation of a linear function of several variables, subject to a series of linear inequality constraints. Koopmans had earlier done a lengthy study of the relationship between freight rates and the construction of oil tankers. The work was published as a monograph titled 'Tanker Freight Rates and Tankship Building' by the Netherlands Economic Institute in 1939. There is a clear foreshadowing in the monograph of Koopmans's subsequent interest in the construction of optimal transportation routes.

Kantorovich made a name for original contributions in mathematics and economics. He effortlessly introduced mathematical concepts in economics and is credited with being one of the founders of linear programming. His major mathematical discovery was a concept called 'K-spaces', which he used to develop linear programming.

A well-known policy application of Kantorovich linear programming work is in the approach to central planning. Kantorovich wanted to maximise a linear output function subject to linear production constraints. He proposed the problem in his 1939 book 'Mathematical Method of Production Planning and Organization' and in his 1959 book, 'The Best Uses of Economic Resources.' He argued that knowing the price of inputs and outputs is important even when the prices are hidden, as they are in planned economies, a concept which came to be known as shadow prices.

Kantorovich is also well known for the Monge-Kantorovich transportation problem. This problem originated in three influential papers — the first one written by G. Monge in 1781, the second and third were due to Kantorovich. The Monge-Kantorovich theory has several applications in various areas of sciences including economics, optic (e.g., the reflector problem), meteorology, oceanography, kinetic theory, partial differential equations, (PDEs) etc. The problem was first posed by Monge in 1781 as: 'How do you best move given piles of sand to fill up given holes of the same total volume'. About 150 years later, Kantorovich simplified the problem a great deal with tools of linear programming.

Conclusion

Both Leonid Kantorovich and Tjalling Koopmans developed the theory of the optimum allocation of resources, which analysed how available productive resources can be optimally used in the production of goods and services. One direct relevance of this work was in the area of economic planning, not only in the socialist countries but also in market economies of the western countries.

It is interesting to note that both Kantorovich and Koopmans were involved in applying linear programming to many policy issues. In particular, their applications to the transportation problem is well known as we saw above.

The writer is an IAS officer, working as Principal Resident Commissioner, Government of West Bengal. Views expressed are personal

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