General conditions for the existence of maximal elements via the uncovered set

This paper disentangles the topological assumptions of classical results (e.g.,Walker, 1977 on the existence of maximal elements from rationality conditions. It is known from the social choice literature that under the standard topological conditions—with no rationality assumptions on preferences—there is an element such that the upper section of strict preference at that element is minimal in terms of set inclusion, i.e., the uncovered set is nonempty. Assuming the finite subordination property, a condition that weakens known acyclicity and convexity assumptions, each such uncovered alternative is in fact maximal. Implications are a generalization of a result of Yannelis and Prabhakar (1983) on semi-convexity, an extension of Fan’s (1961) lemma on KKM correspondences, and the existence of fixed points for subordinate convex correspondences generalizing the work of Browder (1968).     

The maximal {-1,1}-determinant of order 15

We demonstrate the D-optimality of the 15×15 −1,1-matrix discovered independently by Smith and Cohn. This matrix has determinant 25515×2¹⁴ and is unique up to Hadamard equivalence. In addition, we confirm the previously established values of the maximal determinant for all other odd orders up to 21, excluding 19, and we improve the upper bounds on the maximal determinant in orders 29, 33, and 37.     

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.     

Two reduced-form approaches to the derivation of the maximum-likelihood estimators for simultaneous-equation systems

This paper presents two new ways of deriving the maximum-likelihood estimators of subsystems of a simultaneous-equation model. The two derivations are based on an expression of the likelihood function in terms of the reduced-form parameters; and they give rise to two alternative versions of the estimating equations. These equations are compared with those that are derivable from the concentrated likelihood function of Koopmans and Hood, and the equivalence of the three sets of estimating equations is demonstrated.     

Non-parametric tests of productive efficiency with errors-in-variables

We develop a non-parametric test of productive efficiency that accounts for errors-in-variables, following the approach of Varian. [1985. Nonparametric analysis of optimizing behavior with measurement error. Journal of Econometrics 30(1/2), 445–458]. The test is based on the general Pareto–Koopmans notion of efficiency, and does not require price data. Statistical inference is based on the sampling distribution of the L_∞ norm of errors. The test statistic can be computed using a simple enumeration algorithm. The finite sample properties of the test are analyzed by means of a Monte Carlo simulation using real-world data of large EU commercial banks.     

Variable preference relations: Existence of maximal elements

We consider variable preference relations, also called reference dependent preference relations, which are typical in the study of dynamic models in economic theories. We introduce a new concept of weak consistency, a generalization of acyclicity, as an immediate regret condition for variable preferences. The main result to establish is on an existence criterion for maximal elements of a space equipped with a weakly consistent variable preference relation. It is expressed via a preference completeness condition which is equivalent to existence of aspiration points. As applications, we show that a number of results known in the recent literature on maximum principles on a space with or without topological structure can be obtained from the unifying approach of this paper. Habit formation and state functions are also discussed in the framework of variable preference relations.     

Dynamic theory of preferences: Habit formation and taste for variety

We analyze individual preferences over infinite horizon consumption choices. Our axioms provide the foundation for a recursive representation of the utility function that contains as particular cases the classical Koopmans representation (Koopmans (1960)) as well as the habit formation specification.We examine some of the consequences of our axiomatization by considering a standard consumer choice problem, and show that typically in the space of concave utility functions satisfying our axioms the consumer displays a taste for variety. The latter means that such a consumer selects optimally time variant consumption programs for any given time invariant sequence of commodities’ relative prices and for all possible sequences of market discount factors. In contrast, if a concave utility function satisfies Koopmans’ axioms the consumer does not display a taste for variety.     

Price sustainability of location assignments

Koopmans and Beckmann indicated that the assignment of indivisible plants to sites for the case of non-zero interplant flows moving at positive transportation costs was not attainable by a set of site rents or in general by a price system. In this paper, examples counter to the one in Koopmans and Beckmann's article are presented. The patterns of sites and interplant interdependencies are indicated to be crucial qualities in determining whether assignments are price sustainable. Some classical configurations are examined for price sustainability.     

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 maximal elements and equilibria in linear topological spaces

We present some mathematical theorems which are used to generalize previous results on the existence of maximal elements and of equilibrium. Our main theorem in this paper is a new existence proof for an equilibrium in an abstract economy, which is closely related to a previous result of Borglin–Keiding, and Shafer–Sonneschein, but allows for an infinite number of commodities and a countably infinite number of agents.     

Sparse estimators and the oracle property,or the return of Hodges’ estimator

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360; 2002. Variable selection for Cox's proportional hazards model and frailty model. Annals of Statistics 30, 74–99; 2004. New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. Journal of the American Statistical Association 99, 710–723] which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360]. We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity.     

The failure of Ricardian equivalence under dynastic altruism

This paper investigates the theoretical accuracy of the Barro [Barro, R.J., 1974. Are government bonds net wealth? Journal of Political Economy, 82, 1095–1117] debt neutrality proposition. We first identify a discrepancy between the transversality condition of a social planning problem and the one of altruistically linked overlapping generations. Then, this discrepancy is exploited to construct public debt policies which affect the competitive equilibrium allocation even when bequests are strictly positive in all periods: a violation of Ricardian equivalence.     

Generalized method of moments specification testing

This paper analyzes the asymptotic power properties of specification tests which are based on a finite set of moment conditions. It shows that any such test may fail against general misspecification that causes estimator inconsistency. The mutual asymptotic equivalence of maximal degree of freedom tests is shown and the form of optimal tests against specific forms of misspecification is derived. Applications to testing for exogeneity of a set of instrumental variables are presented.     

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.     

Arbitrage and equilibrium in unbounded exchange economies with satiation

In his seminal paper on arbitrage and competitive equilibrium in unbounded exchange economies, Werner (1987) proved the existence of a competitive equilibrium, under a price no-arbitrage condition, without assuming either local or global nonsatiation. Werner’s existence result contrasts sharply with classical existence results for bounded exchange economies which require, at minimum, global nonsatiation at rational allocations. Why do unbounded exchange economies admit existence without local or global nonsatiation? This question is the focus of our paper. First, we show that in unbounded exchange economies, even if some agents’ preferences are satiated, the absence of arbitrage is sufficient for the existence of competitive equilibria, as long as each agent who is satiated has a nonempty set of useful net trades– that is, as long as agents’ preferences satisfy weak nonsatiation. Second, we provide a new approach to proving existence in unbounded exchange economies. The key step in our new approach is to transform the original economy to an economy satisfying global nonsatiation such that all equilibria of the transformed economy are equilibria of the original economy. What our approach makes clear is that it is precisely the condition of weak nonsatiation – a condition considerably weaker than local or global nonsatiation – that makes possible this transformation.     

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.     

Inequalities relating maximal moments to other measures of dispersion

Let X , X ₁, ..., X_k be i.i.d. random variables, and for k ∈ N let D_k ( X ) = E ( X ₁ V ... V X _{k +1}) − EX be the k th centralized maximal moment. A sharp lower bound is given for D ₁( X ) in terms of the Lévy concentration Q_l ( X ) = sup _{x ∈ R} P ( X ∈[ x , x + l ]). This inequality, which is analogous to P. Levy's concentration-variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.     

Efficient spatial allocation in the quadratic assignment problem

Koopmans and Beckmann found that the competitive mechanism cannot sustain an assignment of indivisible plants to spatially distinct sites when intermediate product flows are required and transportation costs between sites are positive. Hartwick claims to have found valid counterexamples to this result. This paper demonstrates that while counterexamples to the Koopmans-Beckmann result do exist, Hartwick's examples are inappropriate. A correct example of a price-sustainable integer assignment is provided.     

Convexity and duality properties of a quadratic intraregional location model

This paper has two main purposes. The first one is to analyse the convexity and duality properties of a quadratic intraregional location model that has been developed for long-term indicative planning in the Stockholm region. The second one is to review the results of Koopmans and Beckmann (1957) about the inadequacy of a linear price system in sustaining an optimal assignment of plants to locations when the costs of transporting intermediary commodities are taken into consideration. At the outset a model is formulated which is a transposition of a continuous Koopmans-Beckmann model into the urban scene. It is shown that this quadratic programming model is non-convex in all practical cases of interest, due to the simple fact that transportation costs increase with distance. A modification of the model is proposed in which the centralising transportation cost criterion is traded of against a decentralising so called congestion cost which penalizes over-exploitation of urban space. It is shown that the modified model tends to be convex. In the light of these results Kuhn- Tucker theory is used to derive a set of conditions that will ensure that the optimal solution is stable relative to all potential moves by individual decision-makers. This result forms the basis for the conclusion that the failure of the price system in the Koopmans-Beckmann model is rather due to properties of the quadratic criterion function than the integral restrictions on the variables.     

Constrained inefficiency in GEI: A geometric argument

In this paper we use global analysis to study the welfare properties of general equilibrium economies with incomplete markets (GEI). Our main result is to show that constrained Pareto optimal equilibria are contained in a submanifold of the equilibrium set. This result is explicitly derived for economies with real assets and fixed aggregate resources, of which real numéraire assets are a special case. As a by product of our analysis, we propose an original global parametrization of the equilibrium set that generalizes to incomplete markets the classical one, first, proposed by Lange [Lange, O., 1942. The foundations of welfare economics. Econometrica 10, 215–228].