Vind, Karl (1933–2004)
Abstract
Karl Vind was born in a small provincial town in Denmark on 3 April 1933. His mother died when he was only a few years old, and his father was often absent from home, so Karl and his two brothers were taken care of by relatives. The family reports early interest and skill in economics and mathematics.
Keywords
Control theory Cores Exchange equilibrium Expected utility hypothesis Fairness General equilibrium Lyapunov theorem Mathematical economics Optimal control Statistics Utility representations Vind, K Walrasian equilibriaJEL Classifications
B31Karl Vind was born in a small provincial town in Denmark on 3 April 1933. His mother died when he was only a few years old, and his father was often absent from home, so Karl and his two brothers were taken care of by relatives. The family reports early interest and skill in economics and mathematics.
Karl Vind finished his school years in 1951. He studied economics at the University of Copenhagen, and attended lectures in mathematics by Werner Fenchel (known for his contributions to the theory of convexity). He graduated in 1958, and after finishing military service was employed at the Faculty of Social Sciences; he also had a position as scientific researcher at what was later known as the Institute of Economics. His future scientific orientation was formed in the years 1962–3, which he spent as a Rockefeller Fellow at the University of California (Berkeley), where he was inspired by the highly fertile research environment around Gerard Debreu. He returned to Berkeley as visiting associate professor from 1964 to 1966. While at Berkeley Karl Vind was married in 1962 to Anni (Mortensen); they had sons named Lars and Jacob and adopted a daughter named Dorthe.
After his return to Copenhagen in 1966 Karl Vind obtained a position as professor in economics, later changed to a chair in mathematical economics, which he held until his retirement in April 2003 at the age of 70. He spent several long periods in Berkeley as well as at the Center for Operations Research and Analysis (CORE) in Louvain. After retirement, he retained his office at the institute and participated in its everyday activities until his death in July 2004 after a short illness.
Karl Vind’s published research covers many fields – international trade theory, control theory, general equilibrium, game theory, the theory of choice under uncertainty – so that his publication record is consistent with his own interpretation of mathematical economics as ‘the derivative of economic theory’ – what is mathematical economics today becomes economic theory tomorrow. However, most of the topics he studied engaged him over long periods; some of the results appearing in the later years were at least partially obtained in the early years of his career.
In the years after graduating, Karl Vind had been interested in mathematical statistics and control theory, something which is witnessed by his work on optimal control with jumps in the state variables (1967). He never returned to control theory, but it usefully inspired him to apply Lyapunov’s theorem, which at that time was known to control theorists but not to researchers in general equilibrium. Vind demonstrated that it could be used to show the equivalence of core and Walrasian equilibria in large economies (1964). This was a major breakthrough at the time, achieved independently by Aumann (1964). The approach using Lyapunov’s theorem was innovative and offered a new approach to modelling large economies.
Also from this period is the short piece on the core of an exchange economy (1965), which pioneers the extension of the results obtained for economies with infinitely many agents to economies with a finite number of agents. Vind’s result does not go all the way to establishing a connection between core and equilibrium, something which was achieved only several years later. This later development might perhaps have been simpler and faster if researchers had followed Vind’s early approach; he had the bad luck of being ahead of his time.
In the following years, Vind’s published research dealt with extensions of the general equilibrium model in several directions. An example is the paper written with David Schmeidler (1972) on fair net trades, proposing a new approach to the concept of fairness as well as an elegant formalism. In much of his work from this period Karl Vind was concerned with the structural properties of exchanges. His concept of an exchange equilibrium was intended to capture the essential properties of trade in markets. He was, however, not quite satisfied with the initial formulations of the exchange equilibrium, which were never published. After several reformulations the concept appeared in 1983 as ‘equilibrium with coordination’.
In the late 1960s Karl Vind started on another project, dealing with utility representations of preferences, which remained at the draft stage until it was finally published as a monograph in 2003. His work on the socalled mean groupoids was inspired by the need to extend the general equilibrium model to include time and uncertainty, which at that time seemed to call for specific functional forms of utility representations. Vind realized that there was a common structure behind utility representations over time and under uncertainty, related to the operation of taking mixtures of consumption programmes, and this led Vind’s theory of mean groupoids. The work had already taken shape as a draft for a research monograph around 1970, but its final publication was considerably delayed, partly for practical reasons and partly due to the emergence of new results from Vind himself and others. Some of these had to do with the extension of the expected utility hypothesis to preferences that are not necessarily complete, one of Karl Vind’s later research projects.
As a researcher, Karl Vind remained active after retirement as an organizer of and participant in scientific meetings and seminars. His influence on the mathematical economics profession goes beyond his published work, since he took great pleasure in following the work of other researchers, in particular young ones, who received valuable suggestions from a person genuinely interested in their work. Due to this aspect of his scientific activity, he has had a lasting influence on the development of the field.
See Also
Selected Works

1964. Edgeworthallocations in an exchange economy with many traders. International Economic Review 5: 165–177.

1965. A theorem on the core of an economy. Review of Economic Studies 32: 47–48.

1967. Control systems with jumps in the state variables. Econometrica 35: 273–277.

1972. (With D. Schmeidler.) Fair net trade. Econometrica 40: 637–642.

1983. Equilibrium with coordination. Journal of Mathematical Economics 12: 275–285.

2003. Independence, additivity, uncertainty. Berlin: Springer.
Bibliography
 Aumann, R. 1964. Markets with a continuum of traders. Econometrica 32: 265–290.Google Scholar