Chapter 15 deals with the basic concepts of game theory as applied to economic problems and discusses some methods of practical solution of game situations. Es wendet sich an Leser mit Kenntnissen aus einem viersemestrigen Grundstudium der Mathematik und Physik und legt seinen Schwerpunkt auf die explizite Darstellung der Lösungen. Chapters 2 and 3 together take up the leading ideas of linear programming; Chap. . It analyzes the nutritional content of a large number of common foods cooked in some agreed-upon way.
Description: ix, 525 pages : illustrations ; 22 cm Contents: Introduction -- Basic concepts of linear programming -- The valuation problem ; Market solutions -- The algebra of linear programming -- The transportation problem -- Linear-programming analysis of the firm -- Application to the firm ; Valuation and duality -- Nonlinear programming -- The statical leontif system -- The statical leontief system continued -- Dynamic aspects of linear models -- Efficient programs of capital accumulation -- Linear programming and the theory of general equilibrium -- Linear programming and welfare economics -- Elements of game theory -- Interrelations between linear programming and game. To each primal problem corresponds a dual problem, which yields additional information to the decision-maker. Then we could forget about thiamin and solve for the cheapest diet that provided adequate calories, as if we had a one-restriction problem. This book was the first to provide a wide-ranging survey of such important aspects of the topic as the interrelations between the celebrated von Neumann theory of games and linear programming, and the relationship between game theory and the traditional economic theories of duopoly and bilateral monopoly. Linear programming has been one of the most important postwar developments in economic theory, but until publication of the present volume, no text offered a comprehensive treatment of the many facets of the relationship of linear programming to traditional economic theory. Category: Mathematics Author : Clifford A. Now consider the shaded area in Fig.
Moreover, the penalties for having less than these amounts are known only for extreme cases of unbalanced diet; and there is the further point that too much of some elements, such as calories, may be as harmful as too little. Clearly there will be as many slack variables as there are factors of production. The outgoing activity is determined in the same way as in the previous iteration. We have sketched in one budget line—one that obviously is not the lowest. An intuitive economist might have arrived at this result almost immediately. The values of the elements of the third iteration are: 4. Do problems with this structure have any important place in economics? Both approaches show how economic agents consumers or producers reach optimal choices, how they do their planning or programming in order to attain maximum utility, maximum profit, minimum cost, etc.
Moreover, the quality of a diet in terms of these specifications is the mathematical sum of the qualities of its component parts, i. But remember that in dealing with problems that can be visualized the eye will put the most gigantic electronic computer to shame. Let us see how elimination works by applying it to the vitamin-calorie problem. Used textbooks do not come with supplemental materials. With the introduction of the slack variables the constraints become: a.
Modern economists will especially appreciate the treatment of the connection between linear programming and modern welfare economics and the insights that linear programming gives into the determinateness of Walrasian equilibrium. Perhaps my expectations were unrealistic. Designed primarily for economists and those interested in management economics who are not necessarily accomplished mathematicians, this text offers a clear, concise exposition of the relationship of linear programming to standard economic analysis. This latter chapter is somewhat technical and may be omitted since it adds no new economic concepts. In the columns of the five productive activities we insert the inputs of the three factors of production which are required for the production of one unit of the corresponding commodities. The consumer is conceived of as maximizing satisfaction subject to a single budgetary restraint.
Successfully used for three decades in graduate economics courses, this book stresses practical problems and specifies important concrete applications. But our minimizing procedure, within these constraints, was logically simple. Strict descriptive faithfulness is an unreasonable demand to make of any conceptualization. Überblick über die Modellierungsgrundsätze des Operations Research. Furthermore the shadow prices of the resources can be compared with their market prices and help the entrepreneur decide whether it is profitable to hire additional units of these factors. The book also offers an excellent introduction to the important Leontief theory of input-output as well as extensive treatment of the problems of dynamic linear programming. Obviously it will be the diet in the region which lies on the lowest possible efficient diet locus, and, equally obviously, the point in question is the lower left-hand corner itself.
Rather surprisingly, linear programming has turned out to be the most powerful method available for resolving the problems of general equilibrium left unsolved by Walras and his immediate successors. Connecting readers with great books since 1972. All other diets are unsatisfactory. Older books may show minor flaws. Used textbooks do not come with supplemental materials. Any other element in the second iteration table b i is found by subtracting from the corresponding original element a, in the first iteration table the product of the element of the pivot row which is in the same column as a, multiplied by the element of the incoming activity which is in the same row as a i. Thus the minimum gains become the actual gains, and the actions and returns for all participants are determinate.
In this case all we have accomplished is to substitute one mathematically intricate condition for three simple ones. If the primal problem is a maximisation problem, its dual is a minimisation problem. If x1 and x3 are increased enough, x5 will ultimately become a negative number, which it is not permitted to do since it is nonsense to consume a negative amount of any food. Since these procedures deal explicitly with the problem of planning the activities of large organizations, they are known as linear programming. When they both reach zero, x4 and x5 take on the values 100, 300 , which we earlier said were the best values. Input-output analysis was the second of the three branches of linear economics to appear. The levels of output and of unused factors in any one solution form a basis.
The rest is elementary algebra. The standard two-commodity indifference map shows that the range of relative prices at which positive amounts of both commodities will be purchased depends on the amount of curvature in the indifference contours; if the two commodities are close substitutes, the range of price ratios at which both are purchased may be. Markings do not interfere with readability. Pages and cover are clean and intact. It is the analysis of problems in which a linear function of a number of variables is to be maximized or minimized when those variables are subject to a number of restraints in the form of linear inequalities. The connection resides in the fact that the mathematical structures of linear programming and of game theory are practically identical.