The bearing of duality on microeconomics

Abstract

We present some observations about links between some classical theories of microeconomics and dualities which have been used in optimization theory and in the study of firstorder Hamilton-Jacobi equations. We introduce a variant of the classical indirect utility function called the wary indirect utility function and a variant of the expenditure function. We focus the attention on the links between these functions, observing that they have better relationships with the direct functions than their classical forms and we give economic interpretations of them.Mathematics Classification Scheme 2000 91B16, 91B38, 91B02, 91B08. Keywords: conjugacy, cost function, duality, expenditure function, indirect utility function, polarity, price, subdifferential, utility function. Dedicated to the memory of Philippe MichelThe bearing of duality on a number of questions in micro-economic theory has been known for several years (see [14] .).Here we start with a conjugacy deduced from a polarity. This simple instance of duality encompasses the passage from an utility function to an indirect utility function. The observation that this passage stems from a polarity allows to deduce characterizations and some properties from general results. A similar fact occurs for production functions. The passage from an utility function to an expenditure function requires a more sophisticated duality. It appears that this duality has been used recently for the study of Hamilton-Jacobi equations ([1 . We devote some attention to a comparison between the approach used in that field and the classical scheme of mathematical economics.The main results point out that a natural variant of the inverse utility function gives rise to a simpler correspondence. Similarly, a natural variant of the expenditure function yields a function (called the wary expenditure function) which has better properties and whose characterization

Projective dualities for quasiconvex problems

“…Such a concept has been used in various contexts, from bare algebraic grounds to highly structured frameworks (see [24][25][26]31,33,34,48,52] and their references). Numerous applications have been given, from mathematical economics [8,9,12,14,16,22,34] to partial differential equations [1,5,[41][42][43] and optimization.…”

“…Such a concept has been used in various contexts, from bare algebraic grounds to highly structured frameworks (see [24][25][26]31,33,34,48,52] and their references). Numerous applications have been given, from mathematical economics [8,9,12,14,16,22,34] to partial differential equations [1,5,[41][42][43] and optimization. Many attempts have been proposed to extend the classical convex duality to the case of quasiconvex problems (see [6,11,15,18,20,21,27,32,38,39,44,49] and the surveys [8,22,31,35,36,46,[50][51][52]).…”

Glimpses upon quasiconvex analysis

“…The radiant duality is derived from a polarity (see [217], [316] for instance). In fact, setting for a subset A of X, A ∧ := {y ∈ Y : ∀x ∈ A y(x) < 1}, for all r ∈ R one has {f c∧ ≤ r} = {f < −r} ∧ since y ∈ {f c ∧ ≤ r} iff for all x ∈ {y ≥ 1} one has −f (x) ≤ r iff y(x) < 1 for all x ∈ {f < −r}, iff y ∈ {f < −r} ∧ .…”

“…As a sample of topics and references, let us mention: applications to mathematical economics ( [186,217]), applications to partial differential equations ( [1,233,235], [241], [317], [318]), asymptotic analysis ( [3], [4], [172], [213], [219]), calculus rules for subdifferentials ( [213], [238]), duality ( [90], [180]- [184], [216], [218], [227], [229]- [230]), mechanics ( [102], [107], [206]), multicriteria optimization ( [115]), numerical issues ( [149], [226], [292], [293], [305], [307]), optimality conditions, regularization ( [239]), subdifferentials ( [189], [213]), variational convergences ( [23], [240], [256], [300]). ..Also, we do not venture in the wide world of abstract convex analysis although it is rich of promises and applications ( [195], [218], [227], [257]- …”

Unilateral Analysis and Duality

“…The important case Z is a distributive lattice and Y is formed of modular or submodular functions is also included in the present example. For such examples condition (F) is not satisfied, although these conjugacies have some interest (see [70], [78], [116]- [119], [121]). …”