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Browsing by Author "Hurwicz, Leonid"

Now showing 1 - 11 of 11
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    A note on the Lagrangian saddle-points
    (University of Minnesota, 1958-09) Hurwicz, Leonid; Uzawa, Hirofumi
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    Construction of Outcome Functions Guaranteeing Existence and Pareto Optimality of Nash Equilibria
    (Center for Economic Research, Department of Economics, University of Minnesota, 1976-08) Hurwicz, Leonid; Schmeidler, David
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    Implicit Functions and Diffeomorphisms without C1
    (Center for Economic Research, Department of Economics, University of Minnesota, 1995-03) Hurwicz, Leonid; Richter, Marcel K.
    We prove implicit and inverse function theorems for non-C1 functions, and characterize non-C1 diffeomorphisms.
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    Incentive Structures Maximizing Residual Gain Under Incomplete Information
    (Center for Economic Research, Department of Economics, University of Minnesota, 1977-04) Hurwicz, Leonid; Shapiro, Leonard
    Two agents are involved in our model. The first agent is to announce a schedule of rewards (or, equivalently, charges) which is a function of the amount produced by the second agent. Then the second agent will decide, using utility maximization, how much to produce. Knowing only the formof the second agent's utility and production functions--not the exact values of their parameters--the first agent seeks to choose a schedule which maximizes the minimum (over all possible utility and productivity parameter values) of a quantity related to his residual gain (residual gain being that part of output remaining after rewards have been paid out). We show that in a broad class of cases the only such maximum is a schedule which takes one-half of production. It should be noted that this result is valid even when schedules are allowed to have certain kinks and/or discontinuities, so that such discontinuities and kinks do not yield any special incentive properties in our model. This problem is motivated by situations in which the first agent may be thought of as the government and the residual gain (revenue from taxation) is to be used for a paramount national or social objective, e.g., defense to ensure national survival; in this case the second agent represents the country's labor force to be rewarded so as to stimulate a degree of effort maximizing the residual available for national defense. Another possible interpretation is with first agent as a landlord, the second as sharecropper, with value added as the "product" and the problem, seen from the landlord's point of view, being that of maximizing his share of value added.
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    The Informational Efficiency of Finite Price Mechanisms
    (Center for Economic Research, Department of Economics, University of Minnesota, 2002-02) Hurwicz, Leonid; Marschak, Thomas
    This paper obtains finite counterparts of previous results that showed the informational efficiency of the Walrasian mechanism among all mechanisms yielding Pareto-optimal individually rational trades in an exchange economy while using a continuum of possible messages. Such mechanisms lack realism, since it is not possible to transmit or announce all points of a continuum, and it generally takes infinite time to find an equilibrium message, among all the messages in a continuum. Accordingly, the paper studies approximations of the continuum Walrasian mechanism, in which the number of messages is finite. It applies general results from a companion paper, which considered finite approximations of continuum mechanisms in general organizations, with exchange economies as a particular example. For classic exchange economies, we compare the continuum Walrasian mechanism with alternative continuum mechanisms that also find a Pareto-optimal and individually rational allocation. There are many of them, and some of them, like the continuum Direct Revelation mechanism, do not use prices at all. A finite approximation to a continuum mechanism will have an error. Its overall error for a given class of economies is the worst distance (over all members of the class) between the continuum mechanism's final allocation and the approximation's final allocation. We measure a finite mechanism's cost by the number of its (equilibrium) messages. We consider exchange economies in which traders' utility functions are quasi-linear and strictly concave. We find that the overall error of a sufficiently fine finite approximation of the Walrasian mechanism is arbitrarily close to the overall error of a not more costly approximation of an alternative continuum mechanism that has the same number of message variables. The former overall error is smaller than the latter if the alternative continuum mechanism has a larger number of message variables. A continuum Direct Revelation mechanism is an example of an alternative mechanism with a larger number of message variables than the Walrasian mechanism. Thus the informational superiority of the Walrasian mechanism emerges again when we approximate it and take the finite number of messages as our cost measure.
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    An Integrability Condition, with Applications to Utility Theory and Thermodynamics
    (Center for Economic Research, Department of Economics, University of Minnesota, 1977-02) Hurwicz, Leonid; Richter, Marcel K.
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    Optimization and Lagrange Multipliers: Non-C1 Constraints and "Minimal" Constraint Qualifications
    (Center for Economic Research, Department of Economics, University of Minnesota, 1995-03) Hurwicz, Leonid; Richter, Marcel K.
    When do Lagrange multipliers exist at constrained maxima? In this paper we establish: a) Existence of multipliers, replacing C1 smoothness of equality constraint functions by differentiability (for Jacobian constraint qualifications) or, for both equalities and inequalities, by the existence of partial derivatives (for path-type constraint qualifications). This unifies the treatment of equality and inequality constraints. b) A notion of "minimal" Jacobian constraint qualifications. We give new Jacobian qualifications and prove they are minimal over certain classes of constraint functions. c) A path-type constraint qualification, weaker than previous constraint qualifications, that is necessary and sufficient for existence of multipliers. (It only assumes existence of partial derivatives.) A survey of earlier results, beginning with Lagrange's own multipliers for equality constraints is contained in the last section. Among others, it notes contributions and formulations by Weierstrass; Bolza; Bliss; Caratheodory; Karush; Kuhn and Tucker; Arrow, Hurwicz, and Uzawa; Mangasarian and Fromovitz; and Gould and Tolle.
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    The Second Welfare Theorem of Classical Welfare Economics
    (Center for Economic Research, Department of Economics, University of Minnesota, 2001-08) Hurwicz, Leonid; Richter, Marcel K.
    We extend the Second Fundamental Theorem of Welfare Economics in several directions. For pure exchange economies, we drop all insatiability requirements on preferences. For economies with production, we use a concept of directional optimality to provide necessary and sufficient conditions for a given allocation to be competitive. This enables us to show, for example, that not all consumers need to be locally nonsatiated, if the economy is "connected." (An example due to Stanley Reiter shows that such extra conditions are unavoidable.) We use weak assumptions on feasibility sets, allowing, but not requiring, short sales and a very general form of disposability. We do not require that preferences be reflexive, transitive, total, or negatively transitive; and we replace full continuity of preferences by a semicontinuity condition for strict preferences. This provides decentralization results extending some of Arrow's original results [1], as well as those in Arrow and Hahn [2, Theorem 4, pp. 93-94] Debreu [6, Theorem 6.4, p. 95], [4, p. 281], and elsewhere.
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    Ville Axioms and Consumer Theory
    (Center for Economic Research, Department of Economics, University of Minnesota, 1977-02) Hurwicz, Leonid; Richter, Marcel K.