Orders and Their Representations

An order concept, ≽(y), is introduced and interpreted as a correspondence. Some common structural properties imposed on ≽(y) are discussed. A distance function, d(x,y;g), is derived from ≽(y) and interpreted as a cardinal representation of the underlying binary relation expressed in the units of the numeraire g∈ℝ^{N}. Properties of distance functions and their superdifferential and subdifferential correspondences are treated. The chapter closes by studying the structural consequences for d(x,y;g) of different convexity axioms imposed on ≽(y).

Preferences and Production under Uncertainty

The Arrow-Savage-Debreu formalism (state space, consequence space, acts) for modelling a stochastic decision is introduced. Preferences over stochastic outcomes framed as maps (acts) from the state space to the consequence space are studied and related to nonstochastic preference structures. Distance function representations of preferences are developed and their superdifferential correspondences are shown to define subjective probability measures. Structural restrictions including uncertainty aversion, constant absolute uncertainty aversion, and constant relative uncertainty aversion are examined and related to parallel restrictions for nonstochastic preference or production structures. A model of a stochastic technology that has the nonstochastic production model as a special case is introduced, and distance function representations of it are discussed. Structural assumptions on the stochastic technology are discussed.

Decision Making and Equilibrium under Uncertainty

Rational choice under uncertainty for individuals with incomplete preferences is examined for three choice environments: the standard financial portfolio model, producer choice in the absence of financial markets, and producer choice in the presence of financial markets. Each problem is analyzed using distance functions and the zero-maximum (zero-minimum) principle. General equilibrium is analyzed using the zero-maximum (zero-minimum) principle and related to equilibrium representation in a nonstochastic setting. Choice under uncertainty for individuals with complete preferences is examined

Squiggly Economics

Three generic economic optimization problems (expenditure (cost) minimization, revenue maximization, and profit maximization) are studied using the mathematical tools developed in Chapters 2 and 3. Conjugate duality results are developed for each. The resulting dual representations (E(q;y), R(p,x), and π‎(p,q)) are shown to characterize all of the economically relevant information in, respectively, V(y), Y(x), and Gr(≽(y)). The implications of different restrictions on ≽(y) for the dual representations are examined.

Quality, Valuation, and Welfare

The analytic structure developed in the first six chapters is applied to quality-differentiated production, quality-differentiated pricing, and consumer welfare analysis. The quality-differentiated production problem is developed as a special case of the multiple-output problem for both nonstochastic and stochastic pricing regimes. The "household production" model of Gorman (1956) and Lancaster (1966) is developed in a conjugate dual framework whose solution for rational individuals obeys the zero-maximum (zero minimum) principle. The nominal concepts of compensating variation and equivalent variation are shown to have real-valued (dual) parallels in the compensating benefit and the equivalent benefit. Real, as opposed to nominal, valuation for traded and nontraded goods is treated in the benefit framework. Directional derivatives of distance functions are used to rationalize the frequently observed empirical discrepancy between willingness-to-pay and willingness-to-accept.

The (Nonstochastic) Producer Problem

The canonical profit maximization problem is surveyed. The concept of a technology set, T, is developed by interpreting ≽(y) as an order that characterizes technically feasible possibilities. Basic properties of T are discussed in terms of Y(x) and related to the generic properties of ≽(y) developed in Chapter 3. Distance functions are shown to characterize T, and the consequences of different structural restrictions routinely used in applied producer (and consumer) analysis are investigated. The canonical profit maximization problem is revisited using conjugate dual arguments and the LeChatelier Principle is discussed. The chapter closes with a brief discussion of Revealed Preference Methods in producer theory.

Differentials and Convex Analysis

Mathematical tools necessary to the argument are presented and discussed. The focus is on concepts borrowed from the convex analysis and variational analysis literatures. The chapter starts by introducing the notions of a correspondence, upper hemi-continuity, and lower hemi-continuity. Superdifferential and subdifferential correspondences for real-valued functions are then introduced, and their essential properties and their role in characterizing global optima are surveyed. Convex sets are introduced and related to functional concavity (convexity). The relationship between functional concavity (convexity), superdifferentiability (subdifferentiability), and the existence of (one-sided) directional derivatives is examined. The theory of convex conjugates and essential conjugate duality results are discussed. Topics treated include Berge's Maximum Theorem, cyclical monotonicity of superdifferential (subdifferential) correspondences, concave (convex) conjugates and biconjugates, Fenchel's Inequality, the Fenchel-Rockafellar Conjugate Duality Theorem, support functions, superlinear functions, sublinear functions, the theory of infimal convolutions and supremal convolutions, and Fenchel's Duality Theorem.

What’s Covered

Six figures that illustrate the subject matter of the book are presented. The reader is asked to take a short quiz to determine their familiarity with the figures and the underlying microeconomic concepts. Background material necessary for reading the text is discussed, and a brief précis of the remaining chapters is presented.

Equilibrium, Efficiency, and Welfare

Competitive equilibria are studied in both partial-equilibrium and general-equilibrium settings for economies characterized by consumers with incomplete preference structures. Market equilibrium determination is developed as solving a zero-maximum problem for a supremal convolution whose dual, by Fenchel's Duality Theorem, coincides with a zero-minimum for an infimal convolution that characterizes Pareto optima. The First and Second Welfare Theorems are natural consequences. The maximization of the sum of consumer surplus and producer surplus is studied in this analytic setting, and the implications of nonsmooth preference structures or technologies for equilibrium determination are discussed.

The Consumer Problem

The theory of a rational consumer characterized by an incomplete preference order is developed using distance functions and the zero-minimum (zero-maximum) principle. The essential comparative-static properties of the associated quantity-dependent and price-dependent demand structures are characterized. Utility functions are derived from distance functions for preference structures satisfying a complete ordering assumption. The Marshall-Hicks demand theory that is based on a utility-maximizing consumer is derived as a special case of rational consumer behavior. The Hicks-Allen demand decomposition is reviewed and a conjugate profit function approach to utility maximization is developed and used to discuss Revealed Preference Theory. The Chapter closes by examining the structural consequences of the independence axiom for d(x,y;g).