A characterization of quasi-concave functions

This paper provides a simple and short proof that the following two properties are equivalent under ▽f(x) ≠ 0: (A) ▽f(x) h = 0 implies h^T▽²f(x) h ? 0 for any x; (B) f(x¹) ? f(x⁰) implies f(tx¹ + (1?t)x⁰) ?f(x⁰) for 0? t ? 1.     

On the existence of equilibria in economies with infinitely many commodities and without ordered preferences

The present paper deals with the existence of equilibria in economies whose commodity space is L_∞(M, M, μ) and where the agents' preferences need not be complete or transitive. Applying a fixed point theorem of Browder, an equilibrium existence theorem for abstract economies (generalized qualitative games) is proven where each agent's choice set is contained in an arbitrary topological vector space. With the help of this theorem the existence of Walrasian general equilibrium for a suitably specified economic model is obtained. The final result is a generalization of T. F. Bewley's (J. Econ. Theory4 (1972), 514–540) equilibrium existence theorem to the case of non-ordered preferences.     

Characterization of certain choice functions

We characterize rationality of certain choice functions C by C(A) ∩ C(B) ? C(A ∪ B) ? C(A) ∪ C(B). We also characterize choice functions coming from preference relations which are partial orders and some other types of relations. Furthermore, we also characterize the admissible set of a binary relation by the fact it is invariant under certain changes in the relation.     

Asymptotic distribution of the sup-Wald statistic under specification errors

Consider a simple structural break model where y_t=α₁+β₁f(x_t)+u_t for t≤k₀ and y_t=α₂+β₂f(x_t)+u_t for t>k₀. The timing of break and the structural parameters are unknown. Suppose the true functional form of the regressor f(·) is misspecified as g(·). We do not place too many restrictions on the functional forms of f(·) and g(·). A frequently encountered example in economics is that the true model is measured in level, but we estimate a log-linear model, i.e. when f(x_t)=x_t and g(x_t)=log(x_t) For any f(·) and g(·), we derive a nonstandard limiting null distribution of the sup-Wald test statistic under some very general regularity conditions. Monte Carlo simulations support our findings.     

Mean-dispersion preferences and constant absolute uncertainty aversion

We axiomatize, in an Anscombe–Aumann framework, the class of preferences that admit a representation of the form V(f)=μ−ρ(d)$V(f)=\mu -\rho (d)$, where μ is the mean utility of the act f with respect to a given probability, d   is the vector of state-by-state utility deviations from the mean, and ρ(d)$\rho (d)$ is a measure of (aversion to) dispersion that corresponds to an uncertainty premium. The key feature of these mean-dispersion   preferences is that they exhibit constant absolute uncertainty aversion. This class includes many well-known models of preferences from the literature on ambiguity. We show what properties of the dispersion function ρ(⋅)$\rho (\cdot )$ correspond to known models, to probabilistic sophistication, and to some new notions of uncertainty aversion.     

Aggregation and the existence of a social utility function

In a well-known paper Gorman (Econometrica21 (1953)) established that the necessary and sufficient condition for the existence of an aggregate, or social, utility function, independent of the distribution of income, is that all individuals' income consumption paths be parallel straight lines. Recently Chipman (J. Econ. Theory8 (1974)), building on the paper of Hurwicz and Uzawa (in “Preference Utility and Demand”) has shown that if the distribution of income is proportional and individual preferences are homothetic, aggregate consumption behavior obeys the necessary integrability conditions. It is shown here that the consistency of aggregate behavior can be derived from more general conditions than the ones used by Chipman and Gorman. Examples of demand systems from which aggregate behavior implies a social utility function are provided. It is then shown that if individual demand functions are linear in income—a form employed by both Gorman and Chipman—it is not necessary that the distribution of income be fixed.     

On incentive structures of a socialist economy

This paper considers an incentive structure B_T as an alternative to the incentive structure B_D recently suggested by Domar. Both of these incentive structures induce managers in a planned economy to provide the socially desirable output level while allowing them to set prices. In two aspects B_T is better than B_D. First, B_D works only if demand is elastic at the optimal output level while B_T works whether demand is elastic or inelastic at that level. Second, even when demand is elastic at the optimal level, there are circumstances for which output converges to the optimal level faster under B_T than under B_D.     

Recursive preferences and balanced growth

We study a class of utility functions that are defined recursively by an aggregator W(x,y) where u_t=W(c_t,u_t+1). In single-agent economies it is known that a sufficient condition for the existence of a balanced growth path is that utility should be homogenous of degree γ. In the context of a multi-agent economy we show that this restriction implies that either a balanced growth equilibrium fails to exist or all agents have the same constant discount factor. We suggest a generalization of recursive preferences wherein the intertemporal utility function is time dependent. Within this class we establish that there may exist a balanced growth equilibrium even if agents are different.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

Market equilibrium with heterogeneous recursive-utility-maximizing agents

Summary The paper by C. Ma [1] contains several errors. First, statement and proof of Theorem 2.1 on the existence of intertemporal recursive utility function as a unique solution to the Koopmans equation must be amended. Several additional technical conditions concerning the consumption domain, measurability of certainty equivalent and utility process need to be assumed for the validity of the theorem. Second, the assumptions for Theorem 3.1 need to be amended to include the Feller's condition that, for any bounded continuous functionf C(S × ⁿ ₊), (f(S_t+1, )¦s_t =s) is bounded and continuous in (s, ). In addition, for Theorem 3.1, the pricep, the endowmente and the dividend rate as functions of the state variables S are assumed to be continuous.The Feller's condition for Theorem 3.1 is to ensure the value function to be well-defined. This condition needs to be assumed even for the expected additive utility functions (See Lucas [2]). It is noticed that, under this condition, the right hand side of equation (3.5) in [1] defines a bounded continuous function ins and. The proof of Theorem 3.1 remains valid with this remark in place.A correct version of Theorem 2.1 in [1] is stated and proved in this corrigendum. Ozaki and Streufert [3] is the first to cast doubt on the validity of this theorem. They point out correctly that additional conditions to ensure the measurability of the utility process need to be assumed. This condition is identified as conditionCE ₄ below. In addition, I notice that, the consumption space is not suitably defined in [1], especially when a unbounded consumption set is assumed. In contrast to what claimed in [3], I show that the uniformly bounded consumption setX and stationary information structure are not necessary for the validity of Theorem 2.1.I would like to thank Hiroyuki Ozaki and Peter Streufert for pointing out correctly some mistakes made in the original article. Comments and suggestions from an anonymous referee are gratefully appreciated. Financially support from SSHRC of Canada is acknowledged.     

A class of multiattribute utility functions

A function u(z) is a utility function if u′(z) > 0. It is called risk averse if we also have u′′(z) < 0. Some authors, however, require that u ⁽ⁱ⁾(z) > 0 if i is odd and u ⁽ⁱ⁾(z) < 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function. A stronger condition, similar to the one in case of a univariate utility function, requires that, in addition, all partial derivatives of total order m should be positive if m is odd and negative if m is even. In this paper, we present a class of functions in analytic form such that each of them satisfies this stronger condition. We also give sharp lower and upper bounds for E[u(X ₁,... , X _s )] under moment information with respect to the joint probability distribution of the random variables X ₁,... , X _s assumed to be discrete and representing wealths. Partially supported by OTKA grants F-046309 and T-047340 in Hungary.     

Transitive measurable utility

This paper continues a study of theories of preferences under risk that do not use the independence axiom of the von Neumann-Morgenstern theory. Unlike its predecessor, it assumes that preferences are transitive. The effects of transitivity are noted in two representations of preferences. The first, which also uses continuity and dominance axioms, involves a function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. Although u might be nonlinear, it has other features of a von Neumann-Morgenstern linear utility function. The second representation has linear functions u and w on P, with w strictly positive except perhaps at preference-extreme measures—where it might vanish, such that u(p) w(q) > u(q) w(p) if and only if p is preferred to q. A symmetry axiom along with the axioms for the first representation are necessary and sufficient for the second representation.     

Rising marginal cost and concavity of the production function

Suppose a production function, f, is continuous, quasi-concave and weakly monotone on the non-negative orthant of Euclidean n-space. Let c(·, ·) be the associated cost function. Then it is shown that f is concave if and only if for each w, c(w, ·) is convex.     

Justifiable preferences

We characterize preferences over acts that can be represented by a utility function and a multiple-prior, such that an act f is preferred to act g if there is a prior under which the expected utility induced by f is higher than that induced by g. These preferences are referred to as justifiable preferences. We further introduce a generalized model of ambiguity that involves a collection of multiple-priors, namely, multiple multiple-priors and incorporate Bewley?s Knightian model in justifiability: f is preferred to g if, according to at least one set of priors, f is unanimously preferred to g.     

On the existence of lindahl equilibria in economies with a measure space of non-transitive consumers

We present a generalization of Roberts' theorem on the existence of Lindahl equilibria in economies with a measure space of agents. The principal contribution of this paper is methodological. We show that by formulating the problem in what we consider to be its natural infinite dimensional setting, the basic structure of Debreu's proof in the theory of value can be applied. Our proof makes use of functional analysis and relies, in particular, on Artstein's characterization of weak sequential convergence in L₁(μ).     

A global non-linear extension of the Le Chatelier-Samuelson Principle for linear Leontief models

The extension is concerned with non-linear mappings A(·) which map the space of n-vectors of total gross outputs into the space of n-vectors of final demands. Roughly speaking, it is shown that, if throughout the non-negative orthant the Jacobian matrix of A(·) is of the Leontief type and satisfies the Hawkins-Simon condition, and if A(·) satisfies a certain other condition, then propositions of a global nature entirely analogous to the Le Chatelier-Samuelson Principle in both the weak and strong forms for linear Leontief systems are valid for A(·).     

Minimal conditions for upper-hemicontinuity of the choice correspondence

We denote by ?(M, r) the set of all maximal elements in the constraint set M with respect to a (partial) order r and find the coarsest topologies on the domain of ? such that ? is u.h.c.     

Numerical representation of intransitive preferences on a countable set

Let > be a preference relation on a countable set X. We prove that if > is acyclic (that is, has irreflexive transitive closure), then there exists a mapping u of X into $\text{R}$ such that x > y entails u(x)>u(y). We also give a simple proof of a representation theorem of Fishburn when > is an interval order.     

On some global properties of Gorman class demand systems

This letter explores some global properties of the Gorman class of demand functions. We find that generalizing the homothetic preferences case to non-homotheticity gives rise to an unexpected global problem. In particular, interaction of the bounded budget share condition with either Slutsky symmetry or with homogeneity reduces the set of admissible Gorman class demand functions to but one — the polar form.