Dynamic Optimization Under Uncertainty: Non-convex Feasible Set

Abstract

An editorial note in the Economic Journal (May 1930) reported the death of Frank Ramsey, and his 1928 paper was described as ‘one of the most remarkable contributions to mathematical economics ever made’. In the same issue the editor organized a symposium on increasing returns and the representative firm. This symposium seems to be a natural follow-up of a number of papers published by the Journal during 1926–8, including the well-known article of Allyn Young (1928) that is still available, and duly remembered. The problems of equilibrium of a firm under increasing returns, or more generally, of designing price-guided resource allocation processes to cope with increasing returns, has since been a topic of continuing interest. Ramsey’s contribution was enshrined as a durable piece with a resurgence of interest in intertemporal economics in the fifties. But neither John Keynes, the editor of the Economic Journal who was most appreciative of Ramsey’s talents, neither the subsequent writers on ‘growth theory’ in Cambridge, England (nor, for that matter, those in Cambridge, Massachusetts), have made any precise suggestion towards incorporating increasing returns in a Ramsey-type exercise.

[The] object [of the book] is refinement, not reconstruction; it is a study in ‘pure theory’. The motive back of its presentation is twofold. In the first place, the writer cherishes, in the face of the pragmatic, philistine tendencies of the present age, especially characteristic of the thought of our own country, the hope that careful, rigorous thinking in the field of social problems does after all have some significance for human weal and woe. In the second place, he has a feeling that the ‘practicalism’ of the times is a passing phase, even to some extent, a pose; that there is a strong undercurrent of discontent with loose and superficial thinking and a real desire, out of sheer intellectual self-respect, to reach a clearer understanding of the meaning of terms and dogmas which pass current as representing ideas. (Frank H. Knight, in Risk Uncertainty and Profit)

Research support from the National Science Foundation is gratefully acknowledged. The research of the first author was also supported by the Warshow endowment at Cornell and that of the second author by an Alfred P. Sloan Research fellowship. We would also like to thank Professors R. N. Bhattacharya, C. Clark, J. Chipman and L. Hurwicz for helpful suggestions.