Erio Castagnoli

Generalized Convexity and Fractional Programming with Economic Applications

I. Generalized Convexity.- to generalized convexity.- Structural developments of concavity properties.- Projectively-convex models in economics.- Convex directional derivatives in optimization.- Differentiable (? , ?)-concave functions.- On the bicriteria maximization problem.- II. Fractional Programming.- Fractional programming - some recent results.- Recent results in disjunctive linear fractional programming.- An interval-type algorithm for generalized fractional programming.- A modified Kelleys cutting plane algorithm for some special nonconvex problems.- Equivalence and parametric analysis in linear fractional programming.- Linear fractional and bicriteria linear fractional programs.- III. Duality and Conjugation.- Generalized conjugation and related topics.- On strongly convex and paraconvex dualities.- Generalized convexity and fractional optimization.- Duality in multiobjective fractional programming.- An approach to Lagrangian duality in vector optimization.- Rubinstein Duality Scheme for Vector Optimization.- IV. Applications of Generalized Convexity in Management Science and Economics.- Generalized convexity in economics: some examples.- Log-Convexity and Global Portfolio Immunization.- Improved analysis of the generalized convexity of a function in portfolio theory.- On some fractional programming models occurring in minimum-risk problems.- Quasi convex lower level problem and applications in two level optimization.- Problems of convex analysis in economic dynamical models.- Recent bounds in coding using programming techniques.- Logical aspects concerning Shephards axioms of production theory.- Contributing Authors.

Insurance Premia Consistent with the Market

We consider insurance prices in presence of an incomplete and competitive market. We show that if the insurance price system is internal, sublinear, and consistent with the market, then insurance prices are the maxima of their expected payments with respect to a family of risk neutral probabilities. We also show that under a simple additional assumption it is possible to decompose the obtained price in net premium plus safety loading.

About Derivatives of some Generalized Concave Functions

Abstract The definitions of (α, λ)-concave and (G, h)-concave functions, introduced in a previous work, are reconsidered. The first one is simplified by symmetrizing λ. A theorem characterizing (α, λ)-concave differentiable functions is given. Moreover it is shown that it is possible to define a new directional derivative so that the usual properties of classical concave functions still hold. A similar approach is also developed for (G, h)-concave functions: the new directional derivative can be expressed in two equivalent ways and can also be used to define a sort of directional derivative for quasi concave functions intended as a limiting case of (G, h)-concave functions.

A Note On Inequality Measures And The Pigou-Dalton Principle Of Transfers

The question whether a given function is a suitable measure of income inequality is mostly answered in the following way: it is obvious that the proposed function satisfies some properties that are desiderable in the context. In our opinion, the only way is to derive inequality measures axiomatically. Our aim is therefore to characterize inequality measures with a minimal number of “essential” and “natural” properties. There are many properties that either seem to be “natural” or are satisfied by functions being introduced as inequality measures: the principle of transfers, homogeneity of degree zero, symmetry, decomposability, extensibility, etc.. For a discussion of these properties see C.E. Bonferroni (1930, 1940), G.S. Fei and Y.C.H. Fields (1978), S.C. Kolm (1976 a,b) W. Eichhorn and W. Gehrig (1982), W. Gehrig (1983, 1984). In this paper we will focus on the Pigou-Dalton principle of transfers without any further requirement. The Pigou-Dalton criterion has central role in the theory of inequality measurement and is at the heart of several well-known results. The intuitive appeal of the principle of transfers is further supported by the results of A.B. Atkinson (1970), P. Dasgupta, A. Sen and D. Starret (1973) M. Rothschild and J.E. Stigliz (1970), which relate it to the well-known Lorenz criterion and to the social welfare approach to inequality. As a basis for inequality comparisons, however, its scope is severely limited (e. g. when the distributions are defined over populations of different sizes, or having different means). Transfer sensitivity has been seen as a means of strengthening the Pigou- Dalton principle by ensuring that a higher weigth in the inequality assessment is attached to transfers taking place in the lower tail of the distribution. A transfer sensitivity requirement of this type has been discussed by many authors: A.B.

Restricting independence to convex cones

Abstract Several nonexpected utility models restrict the independence axiom to particular families of convex cones. The present paper considers this possibility from an abstract point of view. A vector space is endowed with a complete and archimedean preorder satisfying independence on each convex cone of an arbitrary family covering the entire space. Necessary and sufficient conditions are given for the representability of the preorder by a functional which is linear on each cone of the family.

Generalized connectedness for families of Arcs

In this paper the topological property of are wise connectedness is generalized both for sets and functions: a family of “modified secants” is defined, not referring to particular kinds of functions, quadratic, logarithmic, exponential, etc., but fixing the extreme configurations of the generalized means. Such a theory is developed by proving properties which generalize convexity and quasi-convexity in a topological rather than a purely algebraic framework. A new separation theorem is given and a theory of the alternative is stated for the new functions introduced in order to unify the convex and quasi-convex results.

Towards a unified type of concavity

Abstract Starting from the pioneering definition of concavity, many extensions have been suggested in the literature, some of them based on the first and second order approximations. They are mainly linked to practical problems: see, for instance, the case of utility and production functions, risk analysis and, in mathematical programming problems, the ones related to fractional and geometric programming. In our opinion it is very worth looking for the basic idea underlying these properties and a tool may be represented by generalized means as in Ben-Tal (1977). In that paper most of the attention has been given to analyze the properties of (G, h)- functions. Our aim is devoted to look for a unifying tool so that most of the known definitions can be considered as a continuous development from the classical case to the quasi concave one. In this effort we propose a new definition showing that it preserves the most important properties gradually going to more general cases.

Expected Utility without Utility: A Model of Portfolio Selection

We present a model for portfolio selection which is based on an alternative interpretation of the expected utility model. According to this interpretation, the von Neumann-Morgenstern utility function is read as the cumulative distribution function of a stochastic benchmark and lotteries are ranked by their probability to outperform the benchmark. Under different assumptions about the type of benchmark used, we show that the induced behavior is not consistent with mean-variance analysis and tends to select the efficient portfolio with the highest expected rate of return.

Direct and indirect duality for dominance relations

This paper studies the general structure of representation theorems when a dominance relation on a given space X is induced by a partial order on another (vector) space Y. Dual characterizations of the dominance relation are of two kinds: “direct,” which are based on the dominance ordering on X, and “indirect,” which are derived from the partial ordering on Y. Indirect representations are easier to obtain but less prone to meaningful interpretations, while direct characterizations have a closer relationship to the set on which dominance is defined. We show how and when direct dual characterizations can be obtained by the indirect representations.

On the completeness of a constrained market

We show how the well-known Farkas Lemma, commonly used to characterise absence of arbitrages in perfect markets, is also exploitable to ascertain the completeness of a market with total short sales constraints. The generalisation of this lemma to convex cones also allows to characterise the completeness of a market with general conic constraints on investment strategies. Such results can be also applied to tell whether it is possible to hedge perfectly any risky position with a given set of tools.