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.     

Efficiency First or Equity First? Two Principles and Rationality of Social Choice

This paper formalizes two different principles to socially rank allocations under efficiency-equity trade off. The efficiency-first social preference relation ranks an allocation x higher than an allocation y if and only if (i) x is Pareto-superior to y or (ii) x and y are Pareto-noncomparable and x is equity-superior to y. The equity-first social preference relation reverses the order of application of the two criteria. We show that the efficiency-first relation may have a cycle, whereas the equity-first relation is transitive. We also characterize and compare the sets of maximal elements for these social preference relations in the Edgeworth box and examine the nonemptiness of the sets. Journal of Economic Literature Classification Numbers: D71, D63, D61.     

Choice and individual welfare

We propose an abstract method of systematically assigning a “rational” ranking to non-rationalizable choice data. Our main idea is that any method of ascribing welfare to an individual as a function of choice is subjective, and depends on the economist undertaking the analysis. We provide a simple example of the type of exercise we propose. Namely, we define an individual welfare functional as a mapping from stochastic choice functions into weak orders. A stochastic choice function (or choice distribution) gives the empirical frequency of choices for any possible opportunity set (framing factors may also be incorporated into the model). We require that for any two alternatives x and y, if our individual welfare functional recommends x over y given two distinct choice distributions, then it also recommends x over y for any mixture of the two choice distributions. Together with some mild technical requirements, such an individual welfare functional must weight every opportunity set and assign a utility to each alternative x which is the sum across all opportunity sets of the weighted probability of x being chosen from the set. It therefore requires us to have a “prior view” about how important or representative a choice of x at a given situation is.     

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.     

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.     

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).     

A recursion relation for the probability of the paradox of voting

For alternatives x_i, i = 1,…, m, giving rise to m! linear preference orderings of which one is selected independently by each of N voters, a recursion relation is developed which expresses the probability that x_i is the Condorcet winner when there are N voters in terms of the probability of this event when there are N ? 1 voters. Hence the probabilities of the paradox of voting when N is odd, and of Condorcet indecision when N is even may be obtained. The relationship holds for any set of probabilities, or culture, governing the selection of the preference orderings by the voters.     

Undominated directions of tax reform

This article considers the reform of a commodity tax system. Consumers' preferences over directions of tax reform are constructed from indirect utility functions. A Wicksellian decision procedure is used to define a dominance relation on the set of directions of change; direction x dominates direction y if and only if (a) everybody prefers x to y or (b) x is the status quo and at least one person prefers x to y. A number of characterizations of undominated directions of change are provided. A related unanimity rule procedure, which does not single out the status quo for special treatment, is also considered. Particular attention is paid to the issue of whether Wicksellian reforms preserve production efficiency. Remarks on the relationship between this work, previous work in optimal taxation theory, and social choice theory are also provided.     

On relationships between numerical representations of interval orders and semiorders

The extent to which continuous numerical representations of interval orders are unique is considered. Apair of continuous, real-valued functions, <u, v>, represents an interval order, <X, >>, provided that for x, y ϵ X, x > y if and only if u(x) > v(y). Relationships which necessarily hold between any two such numerical representations are presented and a method by which one continuous representation can be derived from another is described. Similar considerations are made for special forms of continuous numerical representations of semiorders.     

Measuring Beliefs in an Experimental Lost Wallet Game

We measure beliefs in an experimental game. Player 1 may take x < 20 Dutch guilders, or leave it and let player 2 split 20 guilders between the players. We find that the higher is x (our treatment variable), the more likely is player 1 to take the x. Out of those who leave the x, many expect to get back less than x. There is no positive correlation between x and the amount y that 2 allocates to 1. However, there is positive correlation between y and 2's expectation of 1's expectation of y. Journal of Economic Literature Classification Numbers: C72, C92.     

Preference aggregation with randomized social orderings

We show that every binary and Paretian method for passing from preference profiles to lotteries over preferences is associated with a subadditive function on the set of coalitions of individuals. This function gives the power of each coalition to secure its preference for any x over any y.     

When is multidimensional screening a convex program?

A principal wishes to transact business with a multidimensional distribution of agents whose preferences are known only in the aggregate. Assuming a twist (= generalized Spence-Mirrlees single-crossing) hypothesis, quasi-linear utilities, and that agents can choose only pure strategies, we identify a structural condition on the value b(x,y) of product type y to agent type x — and on the principal?s costs c(y) — which is necessary and sufficient for reducing the profit maximization problem faced by the principal to a convex program. This is a key step toward making the principal?s problem theoretically and computationally tractable; in particular, it allows us to derive uniqueness and stability of the principal?s optimal strategy — and similarly of the strategy maximizing the expected welfare of the agents when the principal?s profitability is constrained. We call this condition non-negative cross-curvature: it is also (i) necessary and sufficient to guarantee convexity of the set of b-convex functions, (ii) invariant under reparametrization of agent and/or product types by diffeomorphisms, and (iii) a strengthening of Ma, Trudinger and Wang?s necessary and sufficient condition (A3w) for continuity of the correspondence between an exogenously prescribed distribution of agents and of products. We derive the persistence of economic effects such as the desirability for a monopoly to establish prices so high they effectively exclude a positive fraction of its potential customers, in nearly the full range of non-negatively cross-curved models.     

An extension of Fishburn's theorem on extending orders

P. C. Fishburn [J. Econom. Theory31 (1983), 000-000] has shown that one can extend any well ordering of a set to a linear ordering of the set of all non-empty subsets of that set, while satisfying two axioms denoted (GP¹) and (M¹). By applying the compactness theorem of logic, this note shows that the well ordering assumption can be waived. Definability and well ordering properties of the extension are also discussed and shown to depend on the class of basic orderings considered.     

Essential aggregation procedures on restricted domains of preferences

The restricted domains of individuals' preferences that permit the construction of Arrow social welfare functions and nonmanipulable voting procedures in which each of n voters has some power are characterized. In this context a domain is the Cartesian product of n sets of strict preference orderings. Variants of this result are obtained under the additional requirement of neutrality and in the case when alternatives are vectors whose ith components affect only the ith voter. Kalai and Muller's analogous result (J. Econ. Theory16 (1977), 457–469) concerning nondictatorial procedures is discussed and proved as a corollary to the main theorem.     

Algebra of Integrated Time Series: Evidence from Unit Root Analysis

It is argued if x_t ~ I(1) and y_t ~ I(1), then running a regression x_t on y_t would produce spurious results because e _t would generally be I(1). However, there may exist a ‘b’ such that e _t  = x _t - by _t is I(0), then running a regression x _t on y _t would not produce spurious results. This special case of two integrated time series is known in the literature as cointegration. In this particular case, x _t and y _t are said to be cointegrated. In our review of the development of the concept of cointegration, we identified that the underlying reason for this special case to arise is the proposition that if x _t  ~ I(d _x ), y _t  ~ I(d _y ), then z _t  = bx _t  + cy _t  ~ I(max(d _x ,d _y )). In this research, we offer evidence against this proposition.     

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.     

Quasitransitive social preference: why some very large coalitions have very little power

Summary. If only the strict part of social preference is required to be transitive then Independence of Irrelevant Alternatives implies that there is a coalition containing all but one individual that cannot force x to be socially ranked above y for at least half of the pairs of alternatives (x,y). Received: August 29, 1996; revised version: March 24, 1997     

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.     

A MARKET WITH A SOCIAL CONSUMPTION EXTERNALITY*

We model a spatial market in which the utility of each consumer is affected by the consumers who buy precisely the same product. The marginal contribution of consumers x's purchase on consumer y depends on |x–y|, which declines as |x–y| increases. Such modelling of preferences fits goods that signal a consumer's place in society—clothing styles, automobiles and jewellry are examples. For 2n + 1 firms we find the unique symmetric equilibrium and derive comparative statics on the optimal number of firms, the largest number of firms the market can support, and the behaviour of profits per firm as n increases.     

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.