Core allocations and the dimension of the cone of efficiency price vectors

Let (R₁,…,R_k) be an arbitrary partition of the grand coalition in an atomless exchange economy with k “large enough.” We prove that an optimal allocation x belongs to the core if and only if x cannot be improved upon by any coalition that includes at least one of the R_i's. K is “large enough” if k ? r + 1, where r is the linear dimension of the cone P of the efficiency price vectors for x. Recall that it is always true that r ? n, when n is the number of commodities in the market, and that under differentiability and interiority r = 1; thus k can be chosen to be 2 (i.e., for any coalition R, an allocation x belongs to the core of the market if and only if x is not blocked by any coalition that either contains R or contains its complement).     

On least squares estimation with a particular linear function of the dependent variable

We consider the standard linear regression model where the endogenous variable y is substituted by Ty, T being a symmetric, idempotent matrix. Comparing the mean square error (MSE) matrices we show that a ‘naive’ LS-procedure may work better than a competing estimator usually proposed in the literature and may even perform better than the LS-estimator based on untransformed data. We derive necessary and sufficient conditions for MSE-dominance and outline some ideas for testing.     

Resource egalitarianism with a dash of efficiency

We study the problem of defining inequality-averse social orderings over allocations of commodities when individuals have different preferences. We formulate a notion of egalitarianism based on the axiom that any dominance between consumption bundles should be reduced. This Dominance Aversion requirement is compatible with Consensus, a version of the Pareto principle saying that an allocation y is better than x whenever everybody finds that everyone?s bundle at y is better than at x. We characterize a family of multidimensional leximin orderings satisfying Dominance Aversion and Consensus.     

On the informational efficiency of the competitive resource allocation process

The purpose of this paper is to formalize the competitive process as a parametric process, and then prove the minimality of its message space among the message spaces for a broad class of parametric processes that includes the class of processes considered by Hurwicz (in “Studies in Resource Allocation Processes” (K. J. Arrow and L. Hurwicz, Eds.), pp. 413–423, Cambridge Univ. Press, Cambridge, 1977), Mount and Reiter (J. Econ. Theory6 (1974), 161–192), and Osana (J. Econ. Theory17 (1978), 66–78). The proof of this result turns crucially on the “asymmetry property” (which is stronger than the well-known “uniqueness property” of Hurwicz) and on an injectiveness lemma which is applicable to parametric processes.     

Zero-sum condition: A necessary and sufficient condition for a transitive voting system

A simple and quick way to ascertain whether or not any given majority voting system can always produce a transitive social preference orderings without imposing any restriction on the distribution of diverse individual preference orderings is to examine whether all individual voting (preference) vectors satisfy the Addition Rule or not. This conclusion was obtained by first reformulating the voting mechanism into that of a linear mapping from T^m defined by q = Σp_i. It was found that the subset P of T that can accommodate all possible individual preference ordering profiles and such that every sum vector q = Σ p_i of its member vectors p_i is contained in T can be expressed as P = {p: p∈ T, s(p) = 0}. It was also pointed out that this is equivalent to the requirement that all individual preference (voting) functions must satisfy the Addition Rule. Finally, Borda's Rule and Saposnik's Contributive Rule were shown to be examples of transitive voting rules which satisfy these necessary and sufficient conditions.     

On equilibrium in Hart's securities exchange model

In this paper, we use a no unbounded arbitrage condition to give a very direct proof of the existence of equilibrium in Hart's unbounded securities exchange model (J. Econ. Theory, 9 (1974), 293–311). We also examine the relationship between the no unbounded arbitrage condition and the sufficiency conditions of Hart, ibid. and Hammond (J, Econ. Theory, 31 (1983), 170–175). We present an example to show that if traders are not sufficiently risk averse, then Hammond's overlapping expectations condition is not, in general, equivalent to the no unbounded arbitrage condition or Hart's sufficiency conditions, and therefore, is not sufficient to guarantee the existence of equilibrium. We also present an example to show that it is possible for the no unbounded arbitrage condition to hold without overlapping expectations, and therefore, it is possible for equilibrium to exist without overlapping expectations.     

Oscillations in optimal growth models

This paper discusses a model of optimal growth with non-zero discount rate. Most known results concern sufficient conditions for saddle-point (SP) property of the (unique) equilibrium point Here necessary and sufficient conditions for (local) SP are found which permits one to apply bifurcation theory. In particular, the paper considers bifurcation of periodic orbits from an equilibrium point by means of the Hopf theorem, thereby generalizing a result obtained by Benhabib and Nishimura for a special case. A nonconventional theory of the trade cycle may thus be based on very conventional assumptions. Finally, certain known (SP) stability conditions are discussed and related to the main result of the paper.     

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.     

Stationary cardinal utility and optimal growth under uncertainty

In the literature on multiperiod planning under uncertainty, it is generally postulated that preferences may be represented by a von Neumann-Morgenstern utility index that is additive over time. This paper accomplishes two objectives: First, an axiomatic basis is provided for a more general class of non-additive utility indices defined over infinite consumption streams. Second, this class of utility functions is applied to extend existing results (J. Econ. Theory4 (1972), 479–513; J. Econ. Theory11 (1975), 329–339) on the nature of optimal growth under uncertainty. Of particular interest are the existence and stability of a stochastic steady state.     

The Phillips Curve and Optimal Policy in a Structural Signal Extraction Model

We study a monetary economy subject to “signal extraction” problems, and investigate within that framework the positive and normative aspects of monetary policy. As in Lucas (1972, Journal of Economic Theory,4, 103–124; 1973, American Economic Review, 63, 326–334), imperfect signal perception generates macroeconomic correlations similar to those found in the “Phillips curve” literature. Moving to normative aspects, we find that, when aggregate shocks are present, traditional nonactivist policies do not permit reaching the first best, and that an intelligent activist policy always leads to better outcomes. The specific characteristics and effectiveness of this optimal policy also depend crucially on the problem of signal extraction. Journal of Economic Literature Classification Number: E5.     

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.     

Generic 4 × 4 Two Person Games Have at Most 15 Nash Equilibria

The maximal generic number of Nash equilibria for two person games in which the two agents each have four pure strategies is shown to be 15. In contrast to Keiding (1997),Games Econ. Behav.21, 148–160, who arrives at this result by referring to the enumeration of Grünbaum and Sreedharan (1967),J. Combin. Theory2, 437–465, our argument is based on a collection of lemmas that constrain the set of equilibria. Several of these pertain to any common numberdof pure strategies for the two agents.Journal of Economic LiteratureClassification Number: C72.     

Zermelo and the Early History of Game Theory

In the modern literature on game theory there are several versions of what is known as Zermelo's theorem. It is shown that most of these modern statements of Zermelo's theorem bear only a partial relationship to what Zermelo really did. We also give a short survey and discussion of the closely related but almost unknown work by König (1927, Acta Sci. Math. Szeged, 3, 121–130) and Kálmar (1928/29, Acta Sci. Math. Szeged, 4, 65–85). Their papers extend and considerably generalize Zermelo's approach. A translation of Zermelo's paper is included in the Appendix. Journal of Economic Literature Classification Numbers: B19; C70; C72.     

On the generic nonexistence of pure strategy nash equilibria in continuous games

S. Srivastava (J. Econ. Theory 36 (1985), 26–35) states that in the space of continuous games an open and dense subset does not have a pure strategy. Nash equilibrium. The purpose of this note is twofold: first, to show that without a connectedness assumption on the strategy sets of the players this statement is false; second, to provide correct proofs of his main results.     

Experts against adverse selection: A note on the existence of equilibrium with costly appropriable information

The existence of competitive equilibrium in Laffont's (J. Econ. Theory10 (1975)) model of adverse selection with costly information is studied. The existence of an equilibrium with finite prices is demonstrated without unusual restrictions on preferences or the technology of information production. This is made possible by changing the way in which the behaviour of information producing agents is modelled, and allowing for some public information.     

Temporal risk and the nature of induced preferences

It has been known since the work of H. Markowitz (“Portfolio Selection: Efficient Diversification of Investments,” Yale Univ. Press, 1959) and J. Mossin (Amer. Econ. Rev.59 (1969), 172–174) that even an individual whose underlying preferences satisfy the von Neumann-Morgenstern axioms will not choose over delayed (i.e., “temporal”) risky prospects in a manner which can be modelled as expected utility maximizing. Since most economically important instances of risk taking (insurance, real investment, agriculture, career training) involve delayed as opposed to immediately resolved risk, the standard use of expected utility theory to model such decisions must be questioned. In this paper the technique of “generalized expected utility analysis” (M. J. Machina, Econometrica50 (1982), 277–323) and the theory of support functions (R. T. Rockafellar, “Convex Analysis,” Princeton Univ. Press, 1970) are applied to exactly model and hence determine the nature of preferences over temporal risky prospects.     

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.     

Testing for Granger non-causality in heterogeneous panels

This paper proposes a very simple test of Granger (1969) non-causality for heterogeneous panel data models. Our test statistic is based on the individual Wald statistics of Granger non causality averaged across the cross-section units. First, this statistic is shown to converge sequentially to a standard normal distribution. Second, the semi-asymptotic distribution of the average statistic is characterized for a fixed T sample. A standardized statistic based on an approximation of the moments of Wald statistics is hence proposed. Third, Monte Carlo experiments show that our standardized panel statistics have very good small sample properties, even in the presence of cross-sectional dependence.     

Hedonic prices and cost/benefit analysis

Hedonic prices have been used to evaluate the willingness to pay for attributes. We reformulate the notion of hedonic price from a composite price on housing to a unit price on traded quantities, in conformity with long run competitive equilibrium theory. This formulation was suggested (but not developed) by Rosen (J. Polit. Econ.82, No. 1 (1974), 34–35). By first characterizing an efficient allocation of consumers to space, we show that hedonic unit prices can be understood as a bid-rent function which supports the efficient allocation. This is despite the fact that the lots over which consumers bid are themselves endogenous. We show that unit hedonic prices reveal preferences in a manner different from composite hedonic expenditures.     

Overlapping expectations and Hart's conditions for equilibrium in a securities model

Hart (J. Econ. Theory9 (1974), 293–311) gave conditions for equilibrium to exist in a securities model where each agent undertakes asset transactions to maximize expected utility of wealth. These conditions rule out agents wanting to undertake unbounded balanced transactions to reach a Pareto superior allocation given their expectations. With mild extra assumptions to make agents unwilling to risk incurring unbounded losses on their portfolios, Hart's conditions become equivalent to an assumption of “overlapping expectations,” which is comparable to a much weaker form of Green's “common expectations” (Econometrica41 (1973), 1103–1124).