Knowledge that Transforms

In this article Gomory's method of solution of integer linear programming problems is described briefly (with an example of the method of solution). The bulk of the paper is devoted to a discussion of the dual prices and their relationship to the marginal yields of scarce indivisible resources and their efficient allocation. IT HAS been known for some time that a method of solution of the general linear programming problem in which the variables are required to take integer values would also permit the solution of a considerable variety of other problems many of which are not obviously related to it.1 For example, Markowitz and Manne [13] have shown that the difficult concave (nonlinear) programming problem (e.g., a cost minimization problem in which the total cost function is shaped like a hill) can, at least in principle, be approximated as an integer program which permits the determination of a global, and not just a local minimum. Nonconvex feasible regions can also, at least in principle, be handled by integer programming. Among the economic problems which are related to integer programming are the travelling salesman problem and problems in which fixed (inescapable) costs are present. A surprisingly wide range of problems including diophantine problems and the four color map problem2 can be given an integer programming formulation. Some of these applications will be described in greater detail in section five of this paper. Recently one of the authors of this article developed a method, which he calls the method of integer forms (MIF), for solving integer programming problems. In the next section the method of solution will be described in some detail. No proof that the algorithm arrives at the optimal integer solution in a finite number of steps will be described since it is rather lengthy and is being published elsewhere (see Gomory [6] and [7]. For an alternative approach see Land and Doig [12]).

Frisch's work in all its phases exemplifies the interplay of economic theory, empirical analysis, and statistical method which peculiarly characterizes econometrics. Starting with his earliest papers, theoretical considerations have essentially the role of interpreting data. Repeatedly and in many different contexts, the need for a model, as we would say today, to enable us to learn from observations is stressed. This is particularly true when the exigencies of imperfections in data and theory lead us to statistical analysis and therewith implicitly to the admission of stochastic elements. From his empirical work, Frisch was led to examine the subtleties in the interpretation of statistical relations and the methods of statistical analysis appropriate to theoretical understanding. Such a program has inevitably meant the extensive use of mathematics, which involves two interrelated dangers: cutting the lines of communication with economists who lack mathematical training, and a tendency to value mathematical technique over economically meaningful results. In Frisch's work, the sterile Byzantinism that might be implied by these dangers is completely avoided. At all points, there is an open-minded receptivity to economic ideas derived from all sources, whether or not expressed mathematically, and the focus of all research is the underlying economic issue, not the mathematics used. This does not, however, mean any reluctance to use difficult mathematics when it is necessary to the solution. At all times, the economic problem is the master; the necessary mathematics is neither complicated for reasons of elegance and generality nor skimped for reasons of popularity. The enormous volume and variety of Frisch's work is indicated by the attached bibliography (which has been prepared by Professor Trygve Haavelmo and the University Institute of Economics, University of Oslo). This excludes (with few exceptions) his mimeographed memoranda and lectures, some of which are of the greatest importance. In reviewing his work I have been forced to be selective in a way which undoubtedly reflects my own interests and accidents of reading. No attempt has been made to discuss

I SHALL PROVE two theorems using a new method in the problem of stability of equilibrium based upon the second method of Liapounov [4, p. 256ff.]. The novelty of method lies in the selection of the function V(p) whose decrease with time leads to the equilibrium position.2 This is the price weighted sum of the positive excess demands. I shall first prove the existence and stability3 in the large of the set of equilibrium points in the case of cross-elasticities which are nonnegative. The set of equilibrium points is compact and convex, and if the gross substitution matrix is indecomposable at equilibrium, the equilibrium is unique. When a numeraire is not present, it is possible to proceed beyond the limitation of nonnegative cross-elasticities to consider cases where certain weighted sums of the partial derivatives of excess demands with respect to prices are positive. This appears to be a natural generalization. Although the second theorem is primarily of local interest, one hardly need apologize for that. Global stability is not to be expected in general. This type of study was initiated by Walras [8, p. 170) and given its present formulation by Samuelson [5, p. 269]. I shall not elaborate on its limitations. Suffice it to say that, strictly interpreted, the groping for equilibrium which