Yakar Kannai

Concavifiability and constructions of concave utility functions

Abstract Concavifiable convex preference orderings are characterized and minimally concave utilities are constructed, using three different approaches. One involves the intersection of arbitrary lines with the three indifference surfaces, another involves conditions on the normals of two indifference surfaces and is related to the super-gradient map of a possible concave utility. In the third approach it is assumed that the ordering is induced by a twice-differentiable utility and Perrors integral of a certain expression formed from the derivatives is used. A possible economic interpretation of minimally concave utilities is suggested, and it is shown that one cannot select concave utilities so that they depend continuously on the ordering.

The core and balancedness

Publisher Summary Of all solution concepts of cooperative games, the core is probably the easiest to understand. It is the set of all feasible outcomes (payoffs) that no player (participant) or group of participants (coalition) can improve upon by acting for themselves. Once an agreement in the core has been reached, no individual and no group could gain by regrouping. In a free market, outcomes should be in the core; economic activities should be advantageous to all parties involved. For many games, feasible outcomes that cannot be improved upon may not exist.. In such cases one possibility is to ask that no group could gain much by recontracting. It is as if communications and coalition formations are costly. The minimum size of the set of feasible outcomes required for non-emptiness of the core is given by the so-called balancedness condition. The sets containing outcomes upon which nobody could improve by much are called “ɛ-cores.” The chapter discusses the theory of cores in the case of transferable utility (TU) game, the theory of cores of games with non-transferable utility (NTU), and some economic applications of the theory.

Wave kernels related to second-order operators

The wave kernel for a class of second-order subelliptic operators is explicitly computed. This class contains degenerate elliptic and hypo-elliptic operators (such as the Heisenberg Laplacian and the Gru šin operator). Three approaches are used to compute the kernels and to determine their behavior near the singular set. The formulas are applied to study propagation of the singularities. The results are expressed in terms of the real values of a complex function extending the Carnot-Caratheodory distance, and the geodesics of the associated sub-Riemannian geometry play a crucial role in the analysis.

A characterization of monotone individual demand functions

Abstract Monotone demand functions of consumers with concave utility functions are characterized in terms of geometric properties of the indifference surfaces (or, alternatively, in terms of least concave utility representations). Relations to other results (by Mitjuschin and Polterovitch) are discussed.

One-point feedback approach to distributed linear systems

Abstract The disturbance attenuation problem is studied for linear (distributed) plants described by partial differential operators and by using one lumped- (i.e. acting at a single point) feedback loop. A result emphasizing the trade-off between stability and disturbance attenuation is proven. Also, the effect of the feedback loop location on disturbance attenuation is pointed out. Finally, a synthesis procedure is derived, valid also for plants having bounded uncertainty in the gain. The procedure is illustrated by two significant examples in a one-dimensional, bounded domain.

Non-standard concave utility functions

Abstract Every reasonable concave preference ordering possesses a concave utility function assuming values in a suitable non-standard extension of the reals. Even if a real-valued concave utility function does exist, this function is not least concave if non-standard utilities are allowed, unless a certain finiteness (or piecewise linearity) condition holds.

Inferior Goods, Giffen Goods, and Shochu

According to a well-known result by W. Hildenbrand [6], if all consumers possess the same demand function and the density of the expenditure distribution is decreasing, than the average income effect term is non-negative even if inferior goods are present, so that the aggregate demand must be monotone. We show that if the expenditure density is uni-modal and a certain relation between the income density and individual demand is satisfied, than the average income effect term is negative and Giffen goods are not ruled out. We show that the lowest-grade rice-based Japanese spirit (shochu) satisfies this condition. The data suggest that this commodity might be a Giffen good.

Violation of the Law of Demand

Following the classic work of Mitjuschin, Polterovich and Milleron, necessary and sufficient as well as sufficient conditions have been developed for when the multicommodity Law of Demand holds. We show when the widely cited Mitjuschin and Polterovich sufficient condition also becomes necessary. Using this result, violation regions for the very popular Modified Bergson (or hyperbolic absolute risk aversion) class of utility functions are fully characterized in terms of preference parameters. For a natural extension of the constant elasticity of substitution member of the Modified Bergson family that is neither homothetic nor quasihomothetic, we create the first simple, explicit example of which we are aware that (i) fully characterizes violation regions in both the preference parameter and commodity spaces and (ii) analyzes the range of relative income and price changes within which violations occur.

When is individual demand concavifiable

Abstract We describe results and open problems concerning the relation between the (observed) demand and the underlying (not directly observable) preference relation, both in the smooth and in the general convex categories. We focus on the relevance of concavifiability to demand.

Arbitrarily low sensitivity (ALS) in linear distributed systems using pointwise linear feedback

The sensitivity problem is defined for feedback systems with plants described by linear partial differential operators having constant coefficients, in a bounded one-dimensional domain. there are also finitely many observation points and finitely many lumped feedback loops, and a finite number of disturbance inputs. The sensitivity problem is studied in detail for the heat equation, and comments are made about the linearized damped beam equation and the damped wave equation. It is shown that it is possible to reduce arbitrarily the sensitivity over any temporal frequency interval uniformly in the space domain (except for the undamped wave equation, where a limitation in the frequency interval is induced by the plant). This reduction may require a high-gain feedback around the points where the disturbances appear. >