Continuity properties of the private core

Let be a sequence of differential information economies, converging to a limit differential information economy (written as ). Denote by the set of all ε-private core allocations, ε ≥ 0 (for ε=0 we get the private core of Yannelis (1991), denoted by ). Under appropriate conditions, we prove the following stability results

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.     

Remarks on the application of statistical games to technological development forecasting

This work presents the probability of determining a quantitative forecast of technological development S(t) defined by a set m of parameters S⁽¹⁾(t),S⁽²⁾(t),…,S^(m)(t), based on statistical game theory. Assuming that the coordinates S⁽ⁱ⁾(t) (i = 1, 2,…,m) of a forecasted vector S(t) are stochastic processes with given probabilistic characteristics, a formula of a function forecasting the value of a coordinate S⁽ⁱ⁾(t) of this vector can be obtained. This formula permits to determine a vector of forecasts _τT(x) of technological development S(t) at a given moment t = τ+T.     

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.     

Anonymous sequential games: Existence and characterization of equilibria

Summary In this paper we consider Anonymous Sequential Games with Aggregate Uncertainty. We prove existence of equilibrium when there is a general state space representing aggregate uncertainty. When the economy is stationary and the underlying process governing aggregate uncertainty Markov, we provide Markov representations of the equilibria.Table of notation Agents' characteristics space ( ) - A Action space of each agent (aA) - Y Y = x A - Aggregate distribution on agents' characteristics - (X) Space of probability measures onX - C(X) Space of continuous functions onX - _X Family of Borel sets ofX - State space of aggregate uncertainty ( ) - x _t=1 aggregate uncertainty for the infinite game - = (₁,₂,...,_t,...) - ^{t t} (₁, ₂,..., _t) - L₁(^t,C ×A),v _t Normed space of measurable functions from ^t toC( x A) - 8o(^t,( x A)) Space of measurable functions from ^tto( x A) - X^t X^t= x _s=1 ^t X - _X ^t Borel field onX ^t - v Distribution on - v_t Marginal distribution of v on ^t - v(^t)((¦^t)) Conditional distribution on given ^t - v^t(^s)(v_t(¦^s)) Conditional distribution on ^t given ^s (wheres) - _t Periodt distributional strategy - Distributional strategy for all periods =(₁,₂,...,_t,...) - _t Transition process for agents' types - ( _t,^t,y)(P _t+1(, _t , ^t ,y)) Transition function associated with _t - u _t Utility function - V _t (, a, , ^t) Value function for each collection (, a, , ^t ) - W _t (, , ^t ) Value function given optimal action a - C() Consistency correspondence. Distributions consistent with and characteristics transition functions - B() Best response correspondence (which also satisfy consistency) - E Set of equilibrium distributional strategies - x _t=1 ( ^t , (x A)) - S Expanded state space for Markov construction - (, a, ) Value function for Markov construction - P( _t ^* , _t y)(P(, _t ^* , _t , _y )) Invariant characteristics transition function for Markov game We wish to acknowledge very helpful conversations with C. d'Aspremont, B. Lipman, A. McLennan and J-F. Mertens. The financial support of the SSHRCC and the ARC at Queen's University is gratefully acknowledged. This paper was begun while the first author visited CORE. The financial support of CORE and the excellent research environment is gratefully acknowledged. The usual disclaimer applies.     

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.     

A modified logit model for time series with an application to the pricing behaviour of manufacturing firms in Australia

The modified logit model (Amemiya and Nold, 1975) is generalised to the case where the error term is autocorrelated. The asymptotic distribution (as n →∞ and T →∞) of a feasible GLS estimator of β is derived. Tests of linear restrictions on β and the significance of ρ are presented. The results of the applied work suggest that the factors which explain the pricing behaviour of manufacturing firms, as reported in the tendency survey conducted by the Australian Chamber of Commerce and Industry and the Westpac Banking Corporation, include historical inflation rates of up to 7 quarters and capacity utilisation. First version received: March 2001/Final version received: July 2002 RID="*" ID="*"  The first draft of this paper was written while the author was on study leave at the Department of Econometrics, University of Sydney, Australia.     

The closedness of the free-disposal hull of a production set

Summary Assume thatL is a topological vector lattice andY is a closed subset ofL ₊ ×R ^N, whereR ^N denotes theN-dimensional Euclidean space. It is shown that the setY–L ₊ ×R ₊ ^N is closed ifY has appropriate monotonicity properties. The result is applicable to the case ofL equal toL with the Mackey topology, (L ,L ¹).     

Balanced Contributions Axiom in the Solution of Cooperative Games

In a gamevin characteristic function form, suppose the Banzhaf value ψ is used to pay a coalitionSalready formed. Then coalitionSno longer receivesv(S); instead it receivesR^ψ(S) = ∑_{i ∈ S}ψ_i(v_s), wherev_Sdenotes the subgame of coalitionS. Surprisingly, the Shapley value of this new game Sh(N, R^ψ) is equal to the Banzhaf value ofv. In this paper we establish a similar result for all values satisfying balanced contributions axiom. Additionally, we introduce player's weights to obtain the corresponding result in the nonsymmetric case.Journal of Economic LiteratureClassification Number: C71     

Structural interactions in spatial panels

Until recently, considerable effort has been devoted to the estimation of panel data regression models without adequate attention being paid to the drivers of interaction amongst cross-section and spatial units. We discuss some new methodologies in this emerging area and demonstrate their use in measurement and inferences on cross-section and spatial interactions. Specifically, we highlight the important distinction between spatial dependence driven by unobserved common factors and those based on a spatial weights matrix. We argue that purely factor-driven models of spatial dependence may be inadequate because of their connection with the exchangeability assumption. The three methods considered are appropriate for different asymptotic settings; estimation under structural constraints when N is fixed and T → ∞, whilst the methods based on GMM and common correlated effects are appropriate when T ≫ N → ∞. Limitations and potential enhancements of the existing methods are discussed, and several directions for new research are highlighted.     

Equilibria and Pareto optima of markets with adverse selection

Summary This paper examines the efficiency properties of competitive equilibrium in an economy with adverse selection. The agents (firms and households) in this economy exchange contracts, which specify all the relevant aspects of their interaction. Markets are assumed to be complete, in the sense that all possible contracts can, in principle, be traded. Since prices are specified as part of the contract, they cannot be used as free parameters to equate supply and demand in the market for the contract. Instead, equilibrium is achieved by adjusting the probability of trade. If the contract space is sufficiently rich, it can be shown that rationing will not be observed in equilibrium. A further refinement of equilibrium is proposed, restricting agents' beliefs about contracts that are not traded in equilibrium. Incentive-efficient and constrained incentive-efficient allocations are defined to be solutions to appropriately specified mechanism design problems. Constrained incentive efficiency is an artificial construction, obtained by adding the constraint that all contracts yield the same rate of return to firms. Using this notion, analogues of the fundamental theorems of welfare economics can be proved: all refined equilibria are constrained incentive-efficient and all constrained incentive-efficient allocations satisfying some additional conditions can be decentralized as refined equilibria. A constrained incentive-efficient equilibrium is typically not incentive-efficient, however. The source of the inefficiency is the equilibrium condition that forces all firms to earn the same rate of return on each contract.Notation ={ ₁,..., _k } set of outcomes - : ₊ generic contract or lottery - A = () ; - A_o A{, where denotes the null contract or no trade - S={1,...,¦S¦} set of seller types - L(s) number of type-s sellers - M number of buyers - u: × S seller's utility function, which can be extended toA× S by puttingu(, s) ; - v. × S buyer's utility function, which can be extended toA × S by puttingv(, s) ; - f:A ₀ ×S ₊ allocation of sellers - g:A ₀ ×S ₊ allocation of buyers - A ₊ sellers' trading function - :A ×S ₊ buyers' trading function This paper has had a long gestation period, during which I have been influenced by helpful conversations with many persons, by their work, or both. Among those who deserve special mention are Martin Hellwig, Roger Myerson, Edward Prescott, Robert Townsend and Yves Younés. Earlier versions were presented to the NBER/CEME Conference on Decentralization at the University of Toronto and the NBER Conference on General Equilibrium at Brown University. I would like to thank John Geanakoplos, Walter Heller, Andreu Mas Colell, Michael Peters, Michel Poitevin, Lloyd Shapley, John Wooders, Nicholas Yannelis and an anonymous referee for their helpful comments and especially Robert Rosenthal for his careful reading of two drafts. The financial support of the National Science Foundation under Grant No. 912202 is gratefully acknowledged.     

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.     

A note on the fundamental theorem of exact aggregation

If individual i's demands for a commodity are a function of prices, p, income M_i and a vector of attributes A_i, then aggregate demand is

This paper derives the necessary and sufficient conditions of f_i, F and a system of functions g_k(M₁,…,M_N,A₁,…,A_N) symmetric in the M's and A's such that F can be written in the form F(p, g, (M₁,…,M_n,A₁,…,A_N),…, g_n(M₁,…,M_n,A₁,… ,A_n)) for all values of its arguments.     

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.     

The (S,s) Policy is an Optimal Trading Strategy in a Class of Commodity Price Speculation Problems

This paper introduces a model of commodity price speculation and proves that the optimal trading strategy is of the (S,s) form when a no expected loss condition holds. A strong form of this condition is that the retail price charged to consumers at time t exceeds the expected wholesale price of the commodity at time t+1, i.e. , where β ∈(0,1) is the speculator’s discount factor. We are extremely grateful to Herbert Scarf for pointing out an important error in a previous draft of this paper and for suggesting the key argument in a revised proof that fixed the problem. We also benefited from helpful feedback from an anonymous referee, William Brainard, Zvi Eckstein, participants of seminars at Yale, the Operations Research Center at MIT, and the Econometric Society Winter School at the Indian Statistical Institute, New Delhi.     

Some results on “an income fluctuation problem”

A consumer at each period, given the income available, y, has to decide how much to consume and save. If he consumes c ? 0 units he gets u(c) units of satisfaction or utility, and if x = y ? c ? 0 is the amount saved then the available income in the next period is rx + ω^k, where ω^k is a random variable, and r is an interest factor that is assumed to be known with certainty. Infinite time horizon problems are considered, and it is shown that if 0 < δr < 1, where 0 < δ < 1 is a discount factor, then the limiting policy is optimal. Questions about the behavior of the stock level, such as boundness, are considered, and an example is given that shows that the stock level might converge almost surely to infinity. Finally an economic explanation is given.     

Existence of Sparsely Supported Correlated Equilibria

We show that every N-player K ₁ × ... × K _N game possesses a correlated equilibrium with at least zero entries. In particular, the largest N-player K × ... × K games with unique fully supported correlated equilibrium are two-player games. We thank an anonymous referee for most useful comments. The first author acknowledges financial support from Spanish Ministry of Science and Technology, grant SEJ2004-03619, and in form of a Ramón y Cajal fellowship. The second author acknowledges support by the PASCAL Network of Excellence under EC grant no.506778, as well as from Spanish Ministry of Science and Technology and FEDER, grant BMF2003-03324. Both authors also acknowledge financial support from BBVA grant “Aprender a jugar.”     

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 $\text{τ}{}^1$.     

The Costs of Implementing the Majority Principle: The Golden Voting Rule

In a context of constitutional choice of a voting rule, this paper presents an economic analysis of scoring rules that identifies the golden voting rule under the impartial culture assumption. This golden rule depends on the weights β and (1−β) assigned to two types of costs: the cost of majority decisiveness (‘tyranny’) and the cost of the ‘erosion’ in the majority principle. Our first main result establishes that in voting contexts where the number of voters n is typically considerably larger than the number of candidates k, the golden voting rule is the inverse plurality rule for almost any positive β. Irrespective of n and k, the golden voting rule is the inverse plurality rule if β ≥ 1/2 .. This hitherto almost unnoticed rule outperforms any other scoring rule in eliminating majority decisiveness. The golden voting rule is, however, the plurality rule, the most widely used voting rule that does not allow even the slightest ‘erosion’ in the majority principle, when β=0. Our second main result establishes that for sufficiently “small size” voting bodies, the set of potential golden rules consists at most of just three rules: the plurality rule, the Borda rule and the inverse plurality rule. On the one hand, this finding provides a new rationalization to the central role the former two rules play in practice and in the voting theory literature. On the other hand, it provides further support to the inverse plurality rule; not only that it is the golden rule in voting contexts, it also belongs, together with the plurality rule and the Borda method of counts, to the “exclusive” set of potential golden voting rules in small committees. We are indebted to Jim Buchanan, Amichai Glazer, Noa Nitzan, Ken Shepsle, and an anonymous referee for their useful comments.     

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