Sufficient conditions in free final time optimal control problems. A comment

By an application of sufficient conditions, assume that an optimal pair (χ_τ(·), u_τ(·)) and an adjoint function p_τ(·) were found in the control problem in question with the final time τ fixed but arbitrary. Then a sufficient condition for one of these pairs, say , to be optimal in the corresponding free final time problem is that the Hamiltonian, with (χ_τ(t), u_τ(τ?), p_τ(τ), τ) inserted, is nonnegative (nonpositive) to the left (right) of $\text{τ}{}^1$.     

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.     

Topological upper limits of mixed Nash equilibria

Our purpose in this article is to prove that given any integer n ≥ 2 and any non-empty compact Polish spaces S ₁, ..., S _n , if for any u ∈ C( S ₁ × ... × S _n , R) ⁿ , we denote by MNE(u) the set of mixed Nash equilibria of (S ₁, ..., S _n , u), then MNE(u) is a non-empty compact subset of P(S ₁) × ... × P(S _n ) and if u ^k → u in C(S ₁ × ... × S _n , R) ⁿ as k → ∞, then lim sup _{k → ∞} MNE (u ^k ) MNE(u). The author would like to thank the referee for offering critical comments on this paper.     

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.     

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.     

A new test for normality

Burr (1942) type XII distribution $\text{?(u)=kc u}{}^{\mathrm{c?1}}\text{(1+u}{}^{\mathrm{c}}\text{)}{}^{\mathrm{-(k+1)}}\text{u?0, k > 0, c > 0}$ is considered. Particular values of k and c give β₁ ? 0 and β₂ ? 3. Using this fact tests for normality of observations and regression disturbances are constructed.u.1. Introduction     

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.     

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.     

Tiebout equilibria in local public good economies with spillovers

This paper analyzes the effects of spillovers on the equilibrium population distribution across jurisdictions in a local public good economy with free mobility. Spillovers are parametrized by a matrix [α_ij] where α_ij ∈ [0, 1]. When spillovers are symmetric and close to 0 or 1 (pure local public goods and pure public goods), all equilibrium jurisdiction structures are symmetric. However, any population distribution can be sustained in equilibrium for some value of the spillover parameter α. In the class of utility functions with additive externalities, we identify the unique family of utility functions for which equilibria are symmetric except for an isolated value of α. This is a class of utility functions which are linear in the public good and a power function of the private good, u(c, γ) = − A(1 − c)^β + γ. With this specification of utility, we show that an increase in α results in a more fragmented equilibrium population distribution, and that when spillovers are asymmetric and large, a jurisdiction which is more centrally located (i.e. benefits more from the public goods provided in other jurisdictions) has a larger population in equilibrium.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Greater prudence and greater downside risk aversion

Variation in the degree of downside risk aversion across decision makers has implications for efficient risk sharing. However, except for small differences in risk preferences, there is no index, analogous to the Arrow-Pratt index of risk aversion, that depends only on local properties of the utility function and indicates the degree of aversion to downside risk. A measure that does depend only on local properties of the utility function u, the index of prudence p=−u^?/u^″, is related to downside risk aversion, which is indicated by a positive value for u^?. Although we show that the degree of prudence is not an accurate indicator of the degree of downside risk aversion, we nonetheless demonstrate that a uniform increase in prudence accompanied by a uniform increase (decrease) in risk aversion is sufficient to indicate greater downside risk aversion, provided prudence is greater (less) than three times the degree of risk aversion.     

Min, Max, and Sum

This paper provides characterization theorems for preferences that can be represented by U(x₁, …, x_n)=min{x_k}, U(x₁, …, x_n)=max{x_k}, U(x₁, …, x_n)=∑ u(x_k), or combinations of these functionals. The main assumption is partial separability, where changing a common component of two vectors does not reverse strict preferences, but may turn strict preferences into indifference. We discuss applications of our results to social choice. Journal of Economic Literature Classification Numbers: C0, D1, D6.     

Public-good valuation and intra-family allocation

I derive values of marginal changes in a public good for two-person households, measured alternatively by household member i’s willingness to pay (WTP) for the good on behalf of the household, WTP _i (H), or by the sum of individual WTP values across family members, WTP(C). Households are assumed to allocate their resources in efficient Nash bargains over private and common household goods. WTP _i (H) is then defined by trade-offs between the public good and the household good, and WTP(C) by trade-offs of between the public good and private goods. WTP _i (H) is found to be higher (lower) than WTP(C) when member i has a relatively high (low) marginal valuation of the public good, but tends on average to equal WTP(C). As a consequence, individuals tend to represent households correctly on average when questioned about the household’s WTP for a public good, even when they are purely selfish and answer truthfully. Adding all members’ WTP answers on behalf of the household then leads to double counting. Pure and paternalistic altruism (the latter attached to consumption of the public good) move each member’s WTP on behalf of the household closer to the true sum of individual WTP, but only paternalistic altruism raises this sum.      

More pessimism than greediness: a characterization of monotone risk aversion in the rank-dependent expected utility model

Summary. This paper studies monotone risk aversion, the aversion to monotone, mean-preserving increase in risk (Quiggin [21]), in the Rank Dependent Expected Utility (RDEU) model. This model replaces expected utility by another functional, characterized by two functions, a utility function u in conjunction with a probability-perception function f. Monotone mean-preserving increases in risk are closely related to the notion of comparative dispersion introduced by Bickel and Lehmann [3,4] in Non-parametric Statistics. We present a characterization of the pairs (u,f) of monotone risk averse decision makers, based on an index of greediness G _u of the utility function u and an index of pessimism P _f of the probability perception function f: the decision maker is monotone risk averse if and only if . The index of greediness (non-concavity) of u is the supremum of taken over . The index of pessimism of f is the infimum of taken over 0 < v < 1. Thus, , with G _u = 1 iff u is concave. If then , i.e., f is majorized by the identity function. Since P _f = 1 for Expected Utility maximizers, forces u to be concave in this case; thus, the characterization of risk aversion as is a direct generalization from EU to RDEU. A novel element is that concavity of u is not necessary. In fact, u must be concave only if P _f = 1.Received: 10 April 2001, Revised: 18 November 2003, JEL Classification Numbers: D81. Correspondence to: Michéle CohenAlain Chateauneuf, Michéle Cohen, Isaac Meilijson: We are most grateful to Mark Machina, Peter Wakker and two anonymous referees for very helpful suggestions and comments.     

Budget preferences and direct utility

This paper extends a result of Sakai, who presented conditions for indirect preferences from which a utility function can be deduced if demand is single valued. It will be shown that—adding a “partial Lipschitz condition”—Sakai's theorem can be extended to multi-valued demand. Our result follows from the extension of a theorem of Hurwicz and Richter, who have shown that, under certain hypotheses on demand correspondences, an upper semicontinuous utility function f exists, so that the set of utility maximal elements μ_f(B) is contained in the choice set h(B) for every budget B. By our partial Lipschitz condition h(B) ?- μ_f(B) also follows.     

Von Neumann-Morgenstern stable sets, discounting, and Nash bargaining

We establish a link between von Neumann-Morgenstern stable set and the Nash solution in a general n-player utility set. The stable set-solution is defined with respect to a dominance relation: payoff vector u dominates v if one player prefers u even with one period delay. We show that a stable set exists and, if the utility set has a smooth surface, any stable set converges to the Nash bargaining solution when the length of the period goes to zero.     

The network of weighted majority rules and its geometric realizations

In a previous work the first author considered the network G(n,p) of weighted majority rules (WMR) for n decision makers whose competencies are given by their probabilities p=(p₁,…,p_n) of making a correct decision. On this paper we consider chains of decision profiles, which must occur in G(n,p) in a fixed order, and show that they can be mapped onto straight lines in a low-dimensional geometric realization. The minimal number of directions which must used to separate all edges is given as the chromatic number of a certain incidence graph. We also define degenerate networks in which several nodes coalesce.     

Further evidence on asymptotic tests for homogeneity and symmetry in large demand systems

The asymptotically equivalent Lagrange multiplier, likelihood ratio and Wald tests are compared when testing for homogeneity and symmetry. The need for size correction becomes apparent as does the superior performance of the LM test under H₀.     

Saddle points of Hamiltonian systems in convex Lagrange problems having a nonzero discount rate

Problems are studied in which an integral of the form ∫₀^+∞L(k(t),k(t))e^?ptdt is minimized over a class of arcs k: [0, +∞) → Rⁿ. It is assumed that L is a convex function on Rⁿ × Rⁿ and that the discount rate ? is positive. Optimality conditions are expressed in terms of a perturbed Hamiltonian differential system involving a Hamiltonian function H(k, q) which is concave in k and convex in q, but not necessarily differentiable. Conditions are given ensuring that, for ? sufficiently small, the system has a stationary point, in a neighborhood of which one has classical “saddle point” behavior. The optimal arcs of interest then correspond to the solutions of the system which tend to the stationary point as t → +∞. These results are motivated by questions in theoretical economics and extend previous work of the author for the case ? = 0. The case ? < 0 is also covered in part.     

Optimal control problems from second-order difference equations

We examine solutions x?l_∞(Rⁿ) of the equations $\text{F}{}_{\mathrm{t}}\text{(}\text{x}{}_{\mathrm{t?1}}\text{,}\text{x}{}_{\mathrm{t}}\text{,}\text{x}{}_{\mathrm{t+1}}\text{) = 0}$ and derive conditions for the existence of objective functions G_t so that $\text{x}$ solves max_x?l∞ΣG_t(x_t, x_t+1). We specialize the conditions to time autonomous equations and apply them to a competitive industry in temporary equilibrium. The objective function in the example is of the cost-benefit form: consumer surplus minus opportunity cost of labor minus industry-wide cost of investment.