Patience in some non-additive models

Our aim is to give an axiomatization of preferences over infinite consumption streams. At first we adopt the additive case, and give a characterization of preferences which satisfy patience [Marinacci, M., 1998. An axiomatic approach to complete patience and time invariance. Journal of Economic Theory 83, 105–144] or equivalently what Diamond [Diamond, P.A., 1965. The evaluation of infinite utility streams. Econometrica 33, 170–177] named equal treatment of all generations and then, focus on stationary additive preferences. It appears that this class of functionals contains the discounting functionals axiomatized in Koopmans [Koopmans, T.C., 1972. In: McGuire, C.B., Radner, R. (Eds.), Representations of Preference Orderings Over Time. Decision and Organization, North-Holland, Amsterdam] and what is known as Banach-Mazur limit functionals. These results are extended to non-additives preferences where similar results are generalized and naive patience receives a positive treatement through the liminf criterion.     

On the impatience implications of Paretian social welfare functions

In this paper, we investigate the impatience implications resulting from the assumption of existence of a Paretian social welfare function (SWF) aggregating infinite utility streams. We show, for very general program spaces, that the set of utility streams, at which the SWF exhibits impatience, has the power of the continuum. In the context of a more special program space, which has figured prominently in the literature, we establish that this set is dense, so that even if there is a point in the program space at which the SWF does not exhibit impatience, there are points close to it at which it does. If the Paretian SWF is continuous (in the sup metric), we show that impatience is generic: the collection of points, at which the SWF does not exhibit impatience, is a closed, nowhere dense set.     

Stochastic utility theorem

This paper analyzes individual decision making. It is assumed that an individual does not have a preference relation on the set of lotteries. Instead, the primitive of choice is a choice probability that captures the likelihood of one lottery being chosen over the other. Choice probabilities have a stochastic utility representation if they can be written as a non-decreasing function of the difference in expected utilities of the lotteries. Choice probabilities admit a stochastic utility representation if and only if they are complete, strongly transitive, continuous, independent of common consequences and interchangeable. Axioms of stochastic utility are consistent with systematic violations of betweenness and a common ratio effect but not with a common consequence effect. Special cases of stochastic utility include the Fechner model of random errors, Luce choice model and a tremble model of [Harless, D., Camerer, C., 1994. The predictive utility of generalized expected utility theories. Econometrica 62, 1251–1289].     

Impatience implication of weakly Paretian orders: Existence and genericity

We study order theoretic and topological implications for impatience of weakly Paretian, representable orders on infinite utility streams. As a departure from the traditional literature, we do not make any continuity assumptions in proving the existence of impatient points. Impatience is robust in the sense that there are uncountably many impatient points. A general statement about genericity of impatience cannot be made for representable, weakly Paretian orders. This is shown by means of an example. If we assume a stronger sensitivity condition, then genericity obtains.     

No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.     

Stationary Markov equilibria on a non-compact self-justified set

The existence of stationary processes of temporary equilibria is examined in an OLG model, where there are finitely many commodities and consumers in each period, and endowments profiles and expectations profiles are subject to stochastic shocks. A state space is taken as the set of all payoff-relevant variables, and dynamics of the economy is captured as a stochastic process in the state space. In our model, however, the state space does not necessarily admit a compact-truncation consistent with the intertemporal restrictions because distributions over expectations profiles may have non-compact supports. As shown in Duffie et al. [Duffie, D., Geanakoplos, J., Mas-Colell, A., McLennan, A., 1994. Stationary Markov equilibria. Econometrica 62, 745–781), such a compact-truncation, called a self-justified set, is essential for the existence of stationary Markov equilibria. We extend their existence theorem so as to be applicable to our model.     

Equilibrium theory with unbounded consumption sets and non-ordered preferences : Part I. Non-satiation

A new condition is introduced for the existence of equilibrium for an economy where preferences need not be transitive or complete and the consumption set of each agent need not be bounded from below. The new condition allows us to extend the literature in two ways. First, the result of the paper can cover the case where the utility set for individually rational allocations may not be compact. As illustrated in Page et al. [Page Jr., F.H., Wooders, M.H., Monteiro, P.K., 2000. Inconsequential arbitrage. Journal of Mathematical Economics 34, 439–469], the no arbitrage conditions do not apply to an economy with a non-compact utility set. Second, we generalize the arbitrage-based equilibrium theory to the case of non-transitive preferences.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

Equilibria without the survival assumption

It is well known that an equilibrium in the Arrow–Debreu model may fail to exist if a very restrictive condition called the survival assumption is not satisfied. We study two approaches that allow for the relaxation of this condition. Danilov and Sotskov [Danilov, V.I., Sotskov, A.I., 1990. A generalized economic equilibrium. Journal of Mathematical Economics 19, 341–356], and Florig [Florig, M., 2001. Hierarchic competitive equilibria. Journal of Mathematical Economics 35, 515–546] developed a concept of a generalized equilibrium based on a notion of hierarchic prices. Marakulin [Marakulin, V., 1988. An equilibrium with nonstandard prices and its properties in mathematical models of economy. Discussion Paper No. 18. Institute of Mathematics, Siberian Branch of the USSR Academy of Sciences, Novosibirsk, 51 pp. (in Russian); Marakulin, V., 1990. Equilibrium with nonstandard prices in exchange economies. In: Quandt, R., Triska, D. (Eds.), Optimal Decisions in Market and Planned Economies. Westview Press, London, pp. 268–282] proposed a concept of an equilibrium with non-standard prices. In this paper, we establish the equivalence between non-standard and hierarchic equilibria. Furthermore, we show that for any specified system of dividends the set of such equilibria is generically finite. As a consequence, we have generic finiteness of Mas-Colell’s equilibria with slack, uniform dividend equilibria, and other special cases of our concept.     

Existence of an equilibrium for infinite horizon economies with and without complete information

This work proves the existence of an equilibrium for an infinite horizon economy where trade takes place sequentially over time. There exist two types of agents: the first correctly anticipates all future contingent endogenous variables with complete information as in Radner [Radner, R. (1972). Existence of equilibrium of plans, prices and price expectations in a sequence of markets. Econometrica, 289–303] and the second has exogenous expectations about the future environment as in Grandmont [Grandmont, J. M. (1977). Temporary general equilibrium theory. Econometrica, 535–572] and information based on the current and past aggregate variables including those which are private knowledge. Agents with exogenous expectations may have inconsistent optimal plans but have predictive beliefs in the context of Blackwell and Dubbins [Blackwell, D., Dubins, L. (1962). Merging of opinions with increasing information. The Annals of Mathematical Statistics, 882–886] with probability transition rules based on all observed variables. We provide examples of this framework applied to models of differential information and environments exhibiting results of market selection and convergence of an equilibrium. The existence result can be used to conclude that, by adding the continuity assumption on the probability transition rules, we obtain the existence of an equilibrium for some models of differential information and incomplete markets.     

A characterization and an impossibility of finite length anonymity for infinite generations

In the context of ranking infinite utility streams, the impartiality axiom of finite length anonymity requires the equal ranking of any two utility streams that are equal up to a finite length permutation ( Fleurbaey and Michel, 2003). We first characterize any finite length permutation as a composition of a fixed step permutation and an “almost” fixed step permutation. We then show that if a binary relation satisfies finite length anonymity, then it violates all the distributional axioms that are based on a segment-wise comparison. Examples of those axioms include the weak Pareto principle and the weak Pigou-Dalton principle.     

An adjustment process for nonconvex production economies

We prove the existence of a path of market conditions, i.e. combinations of market prices and production quantities, that links any arbitrarily chosen market condition with an equilibrium, in a general equilibrium model with possibly nonconvex production technologies based on Villar [Villar, A., 1994. Equilibrium with nonconvex production technologies. Economic Theory 4, 629–638] and Villar [Villar, A., 1999. Equilibrium and Efficiency in Production Economies, second ed., Springer Verlag, Berlin]. This existence theorem holds for any semi-algebraic version of the model and the adjustment of market conditions along the path can be given an economic interpretation as a tâtonnement process. Any such path can be approximated arbitrarily close by applying a simplicial algorithm. By restarting this algorithm in a different market condition, we may find more than one equilibrium.     

Inter-temporal preference for flexibility and risky choice

We derive an inter-temporal theory of choice, in the spirit of Kreps and Porteus [Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200], where decision makers have incomplete preferences. This can be used to model indecisiveness as well as unforeseen contingencies. The key to our approach is a time consistency condition and therefore the normative connection between ex-ante and ex-post choice. The time consistency condition enables a representation that is a straight forward extension of recursive utility with the exception that it features an inter-temporal ‘utility for flexibility’.     

A topological approach to Wilson’s impossibility theorem

We will present a topological approach to Wilson’s impossibility theorem [Wilson, R.B., 1972. Social choice theory without the Pareto principle. Journal of Economic Theory 5, 478–486] that there exists no non-null binary social choice rule which satisfies transitivity, independence of irrelevant alternatives, non-imposition and has no dictator nor inverse dictator. Our research is in line with the studies of topological approaches to discrete social choice problems initiated by [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404–415]. This paper extends the result about the Arrow impossibility theorem shown in [Tanaka, Y., 2006. A topological approach to the Arrow impossibility theorem when individual preferences are weak orders. Applied Mathematics and Computation 174, 961–981] to Wilson’s theorem.     

Implementation of voting operators

We show that a theory of implementation can be developed in the Aizerman–Aleskerov framework, capturing the main ideas regarding Nash implementation in the Arrovian case. In fact we obtain the counterparts of the results of [Maskin, E., 1977. Nash Equilibrium and Welfare Optimality. MIT, Mimeo] and [Moore, J., Repullo, R., 1990. Nash implementation. A full characterization. Econometrica 58, 1038–1100] in the new framework.     

Risk aversion in RDEU

Various notions of risk aversion can be distinguished for the class of rank-dependent expected utility (RDEU) preferences. We discuss the relationships amongst five of these, and describe simple (testable) characterizations in terms of elementary probability transformations for all but the weakest notion. The paper also provides the first complete characterization of the RDEU orderings that are risk-averse in the sense of Jewitt [Jewitt, I., 1989. Choosing between risky prospects: the characterization of comparative static results and location independent risk. Management Science 35, 60–70]. We also extend Chew et al.’s [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381] important characterization of strong risk aversion [Rothschild, M., Stiglitz, J.E., 1970. Increasing risk: I. A definition. Journal of Economic Theory 2, 225–243] by relaxing strict monotonicity and differentiability assumptions, and allowing for discontinuities in the probability transformation function. The important special case of maximin choice falls within this relaxed RDEU class. It is shown that any strongly risk-averse RDEU order is a convex combination of maximin and another RDEU order with concave utility and continuous, concave probability transformation. Our proof of the result on strong risk aversion is also simpler (as well as more general) than that of Chew et al. [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381].     

Semiparametric efficient estimation of dynamic panel data models

This paper extends the semiparametric efficient treatment of panel data models pursued by Park and Simar [Park, B.U., Simar, L., 1994. Efficient semiparametric estimation in stochastic frontier models. Journal of the American Statistical Association 89, 929–936] and Park et al. [Park, B.U., Sickles, R.C., Simar, L., 1998. Stochastic frontiers: a semiparametric approach. Journal of Econometrics 84, 273–301; Park, B.U., Sickles, R.C., Simar, L., 2003. Semiparametric efficient estimation of AR(1) panel data models. Journal of Econometrics 117, 279–309] to a dynamic panel setting. We develop a semiparametric efficient estimator under minimal assumptions when the panel model contains a lagged dependent variable. We apply this new estimator to analyze the structure of demand between city pairs for selected U.S. airlines during the period 1979 I–1992 IV.     

Risk aversion and the elasticity of substitution in general dynamic portfolio theory: Consistent planning by forward looking, expected utility maximizing investors

Using the measure of risk aversion suggested by Kihlstrom and Mirman [Kihlstrom, R., Mirman, L., 1974. Risk aversion with many commodities. Journal of Economic Theory 8, 361–388; Kihlstrom, R., Mirman, L., 1981. Constant, increasing and decreasing risk aversion with many commodities. Review of Economic Studies 48, 271–280], we propose a dynamic consumption-savings–portfolio choice model in which the consumer-investor maximizes the expected value of a non-additively separable utility function of current and future consumption. Preferences for consumption streams are CES and the elasticity of substitution can be chosen independently of the risk aversion measure. The additively separable case is a special case. Because choices are not dynamically consistent, we follow the “consistent planning” approach of Strotz [Strotz, R., 1956. Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies 23, 165–180] and also interpret our analysis from the game theoretic perspective taken by Peleg and Yaari [Peleg, B., Yaari, M., 1973. On the existence of a consistent course of action when tastes are changing. Review of Economic Studies 40, 391–401]. The equilibrium of the Lucas asset pricing model with i.i.d. consumption growth is obtained and the equity premium is shown to depend on the elasticity of substitution as well as the risk aversion measure. The nature of the dependence is examined. Our results are contrasted with those of the non-expected utility recursive approach of Epstein–Zin and Weil.     

On games with incomplete information and the Dvoretsky–Wald–Wolfowitz theorem with countable partitions

It has remained an open question as to whether the results of Milgrom–Weber [Milgrom, P.R., Weber, R.J., 1985. Distributional strategies for games with incomplete information. Mathematics of Operations Research 10, 619–632] are valid for action sets with a countably infinite number of elements without additional assumptions on the abstract measure space of information. In this paper, we give an affirmative answer to this question as a consequence of an extension of a theorem of Dvoretzky, Wald and Wolfowitz (henceforth DWW) due to Edwards [Edwards, D.A., 1987. On a theorem of Dvoretsky, Wald and Wolfowitz concerning Liapunov measures. Glasgow Mathematical Journal 29, 205–220]. We also present a direct elementary proof of the DWW theorem and its extension, one that may have an independent interest.