Separable utility functions

We present a necessary and sufficient condition for the norm properness of separable utility functions. The condition is illustrated with a variety of examples. The condition and the examples indicate that norm uniformly proper separable utility functions are much “closer” to linear utility functions than previously suspected.We also take this opportunity and present in a systematic and simplified manner the basic properties of separable utility functions that are scattered in a fragmented way throughout the literature.     

Nonparametric Bayes inference for concave distribution functions

Bayesian inference for concave distribution functions is investigated. This is made by transforming a mixture of Dirichlet processes on the space of distribution functions to the space of concave distribution functions. We give a method for sampling from the posterior distribution using a Pólya urn scheme in combination with a Markov chain Monte Carlo algorithm. The methods are extended to estimation of concave distribution functions for incompletely observed data.     

On intermediate utility functions

Intermediate utility functions have been recently introduced by Persson and Tabellini (2000, Political Economics: Explaining Economics Policy, MIT Press, Cambridge) and ensure the existence of a majority voting equilibrium at multi-dimensional voting. Here, we show that monotone transforms of these utility functions also are sufficient for the existence of a Condorcet winner.     

Proper preferences and quasi-concave utility functions

It is shown that preferences which are continuous, convex and uniformly proper [Mas-Colell (1983)] on the positive cone of a Banach lattice can be represented by a quasi-concave utility function which is defined on a larger domain with non-empty interior. This utility function may be chosen to be either upper or lower semi-continuous on its domain, and continuous at each point of the positive cone. Conversely, any preference relation on the positive cone which is monotone and arises from such a utility function is shown to satisfy a condition which is slightly weaker than uniform properness but which (in the presence of appropriate compactness assumptions) is sufficient to establish the existence of quasi-equilibria. An example is presented to illuminate the role played by the uniformity requirement.     

On continuous utility functions derived from demand functions

In the theory of revealed preference and in the approach to integrability theory of Hurwicz and Uzawa certain conditions are proposed implying the existence of a utility function generating the given demand function. This article presents a hypothesis which, under supposition of some well-known axioms of those models, is necessary and sufficient for the existence of a continuous utility function. This hypothesis implies the existence of a utility function u with the property that all of the boundary points of the set $\text{{x|u(x)≧α}}$ for every $\text{α?}\text{R}$ are lower boundary points, being fundamental for the continuity of the utility function.     

Lipschitz continuous policy functions for strongly concave optimization problems

We prove that the policy function, obtained by optimizing a discounted infinite sum of stationary return functions, is Lipschitz continuous when the instantaneous function is strongly concave. Moreover, by using the notion of α-concavity, we provide an estimate of the Lipschitz constant which turns out to be a decreasing function of the discount factor.     

Concepts of similarity for utility functions

A generalization of the compact-open topology is defined for a space of utility functions with different choice sets. The space is a complete separable metric space. A continuous representation theorem of Levin (1983) gives a homeomorphism between this space and the space of preference relations, topologized by closed convergence. A map into the space is measurable with regard to the Borel algebra iff the choice set correspondence is measurable and the utility function is measurable on its graph. A coarser topology (that of subgraph convergence) is also studied. This topology is coarser than the compact-open topology when the utility functions are defined on the same choice set. However the demand correspondence is still upper-hemicontinuous. A homeomorphism is given between this space and the space of preference relations, with the latter given a certain topology coarser than closed convergence.     

Risk and prices in separable utility functions

Many decision problems involve more than one attribute. Separable multi-attribute utility functions are commonly used to model preferences in such situations. We consider the case in which one attribute can be identified as money. The price at which non-monetary attributes may be substituted by money, the relation of this price to a decision-maker's wealth, and the implications on attitudes toward risk are examined for additively and multiplicatively separable multi-attribute utility functions. In particular, it is shown that additive separability, price independent of wealth and monetary risk-aversion are mutually inconsistent.     

Myopic utility functions on sequential economies

In Brown and Lewis (1981) continuity in the Mackey topology of (l^∞, l¹) is related to myopic (or impatient) economic behavior. They also show that finer (locally convex) topologies admit continuous non-myopic utility functions. In that work the space of bounded sequences, l^∞, is interpreted as all time sequences of bounded consumption plans. In Brown (1981) the analysis is extended to study the theory of interest on related sequence spaces.This note applies our simple technique for ‘computing’ Mackey continuity of real-valued functions defined on l^∞. Our first result is motivated by Bewley's (1972, app. II) theorem, but extends it in several important ways (on sequential economies). First, Beweley's examples (specialized to the sequential setting) are all ‘temporally separable’,that is, consumption in one time period does not affect indifference sets in another. We give new explicit examples of Cobb-Douglass-like utility functions and show that the ‘obvious’ infinite-dimensional Cobb-Douglass functions are non-myopic. Known equilibrium theory [from Bewley (1972), but pre-dating him in the sequential case] applies to these new examples. Second, we remove the assumptions of concavity and monotony from the proof of continuity.Our second result shows that some of the ‘stationary’ utility functions studied by Koopmans, Diamond and Williamson (1964) are also myopic in the sense of Brown and Lewis. In general their work is based on the finer uniform topology.Finally, we show how to transform our technique so that it applies to Brown's more general sequential economies. A change of variables transfers our examples to these spaces.     

Differentiated oligopolistic markets with concave cost functions via Ky Fan inequalities

A Nash–Cournot model for oligopolistic markets with concave cost functions and a differentiated commodity is analyzed. Equilibrium states are characterized through Ky Fan inequalities. Relying on the minimization of a suitable merit function, a general algorithmic scheme for solving them is provided. Two concrete algorithms are therefore designed that converge under suitable convexity and monotonicity assumptions. The results of some numerical tests on randomly generated markets are also reported.     

Decision analysis using targets instead of utility functions

A common precept of decision analysis under uncertainty is the choice of an action which maximizes the expected value of a utility function. Savage's (1954) axioms for subjective expected utility provide a normative foundation for this principle of choice. This paper shows that the same set of axioms implies that one should select an action which maximizes the probability of meeting an uncertain target. This suggests a new perspective and an alternate target-based language for decision analysis. We explore the implications and the advantages of this target-based approach for both individual and group decision-making.     

Universal Pareto dominance and welfare for plausible utility functions

We study Pareto efficiency in a setting that involves two kinds of uncertainty: Uncertainty over the possible outcomes is modeled using lotteries whereas uncertainty over the agents’ preferences over lotteries is modeled using sets of plausible utility functions. A lottery is universally Pareto undominated if there is no other lottery that Pareto dominates it for all plausible utility functions. We show that, under fairly general conditions, a lottery is universally Pareto undominated iff it is Pareto efficient for some vector of plausible utility functions, which in turn is equivalent to affine welfare maximization for this vector. In contrast to previous work on linear utility functions, we use the significantly more general framework of skew-symmetric bilinear (SSB) utility functions as introduced by Fishburn (1982). Our main theorem generalizes a theorem by Carroll (2010) and implies the ordinal efficiency welfare theorem. We discuss three natural classes of plausible utility functions, which lead to three notions of ordinal efficiency, including stochastic dominance efficiency, and conclude with a detailed investigation of the geometric and computational properties of these notions.     

Ordering utility functions based on mean-seeking behavior1

We consider utility function partial orderings to predict comparative portfolio features with two risky assets. No utility ordering can predict comparative holdings of the riskier asset for any reasonable definition of the latter. We thus reinterpret results of Arrow-Pratt and Ross as predicting comparative mean-seeking behavior. We also present a stronger utility ordering which predicts comparative portfolio means with no joint distribution restrictions. Thus we present a progression of contexts, with successively more relaxed distributional restrictions and hence successively stronger restrictions on utility function pairs, in which comparative mean-seeking behavior (not comparative risk avoidance) is predictable.     

Advantageous redistribution with three smooth CES utility functions

We present a parametric example of three-country advantageous redistribution with two Cobb–Douglas utility functions and one CES utility function for which the elasticity of substitution is 1/2. This paper indicates that the possibility of advantageous redistribution strongly depends on the three countries’ taste patterns, endowment distributions, and the elasticity of substitution. In particular, we will show with specific examples that greater difference between the donor and recipient’s taste patterns and a lower elasticity of substitution can increase the chance of advantageous redistribution.