A global non-linear extension of the Le Chatelier-Samuelson Principle for linear Leontief models

The extension is concerned with non-linear mappings A(·) which map the space of n-vectors of total gross outputs into the space of n-vectors of final demands. Roughly speaking, it is shown that, if throughout the non-negative orthant the Jacobian matrix of A(·) is of the Leontief type and satisfies the Hawkins-Simon condition, and if A(·) satisfies a certain other condition, then propositions of a global nature entirely analogous to the Le Chatelier-Samuelson Principle in both the weak and strong forms for linear Leontief systems are valid for A(·).     

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.     

A note on the underestimation and overestimation in stochastic input-output models

Here we show that (i) the Leontief inverse (I – A)^-1 is underestimated when the elements of A, the fixed coefficient input–output matrix, are biproportionally stochastic, and (ii) the output vector in a non-linear input–output system is overestimated when the final demand vector is stochastic.     

The expected probability of Condorcet's paradox

Let P(n,p) be the probability that there is a Condorcet winner on three alternatives for n (odd) voters. The vector p defines the probability that a randomly selected voter will have any of the six linear rankings on three alternatives as his or her preference ranking. It is shown that if all p vectors are equally likely, the expected value of P(n,p) is given by 15(n+3)²/[16(n+2)(n+4)].     

A proposed convention for measuring the state of the art of products or processes

During 1980, The Futures Group performed a study under contract to the National Science Foundation to devise a convention for describing—in quantitative terms—technological state of the art of essential any technology. In designing this convention we hope that different analysts, working independently, would be able to arrive at a similar conclusion about the state of the art of a technology under study. With such a measure at hand, it would be possible, for example, to evaluate the effectiveness of investment in R&D by evaluating the improvement in the technological state of the art of a given technology, per unit investment.The form of the equation chosen for the state-of-the-art convention was SOA = K₁(P₁/P'₁ + K₂(P₂/P'₂) ? K_n(P_n/P'_n) where SOA = state of the art, K_n = the relative weight associated with each parameter describing the technology, P_n = the value of the particular parameter useful in describing the state of the art, and P'_n = a reference value of the parameter. Various approaches to the selection of the parameters and their weights were described. Of particular interest is a statistical approach which assumes that the state of the art, over time, is an S-shaped curve. In this instance it is possible to compute the value of the weights through an iterative numerical technique.The convention was applied to two technologies: computers and antibiotics, and state-of-the-art measures were developed for each. In the case of antibiotics, selected organism/antibiotic pairs were analyzed to identify the state of the art of a particular antibiotic as applied to the control of a particular microorganism, over time.     

An efficiency theorem for incompletely known preferences

There are n agents who have von Neumann-Morgenstern utility functions on a finite set of alternatives A. Each agent i's utility function is known to lie in the nonempty, convex, relatively open set Ui. Suppose L is a lottery on A that is undominated, meaning that there is no other lottery that is guaranteed to Pareto dominate L no matter what the true utility functions are. Then, there exist utility functions ui∈Ui for which L is Pareto efficient. This result includes the ordinal efficiency welfare theorem as a special case.     

People,skills, and resources: an interaction model for world population growth

A model is set up that yields the equation followed by world population (P), past and present: P = A/(D - t)^M, where t is time and A, D, and M are constants. Cumulated historical population estimates confirm this hyperbolical pattern first noticed by Cailleux, Meyer, and Foerster. Technological indices should grow according to the same pattern; partial confirmation of this part of the model is presented. The growth acceleration crisis that we are now facing may have had a counterpart around 4500 B.C., during the neolithic agricultural revolution. World-population projections up to 2000 A.D. are discussed in the light of the interaction model, with and without resource limitations.     

Tops-only domains

In this paper we consider the standard voting model with a finite set of alternatives A and n voters and address the following question: what are the characteristics of domains \({\mathcal D}\) that induce the property that every strategy-proof social choice function \({f: {\mathcal D}^n \rightarrow A}\) satisfying unanimity, has the tops-only property? We first impose a minimal richness condition which ensures that for every alternative a, there exists an admissible ordering where a is maximal. We identify conditions on \({\mathcal D}\) that are sufficient for strategy-proofness and unanimity to imply tops onlyness in the general case of n voters and in the special case, n = 2. We provide an algorithm for constructing tops-only domains from connected graphs with elements of A as nodes. We provide several applications of our results. Finally, we relax the minimal richness assumption and partially extend our results.     

Sufficient conditions for D-stability

Sufficient conditions for an n by n matrix to be D-stable are surveyed. Use is made of some transformations under which the D-stables are invariant and relations among the conditions are given. The verifiability of the thirteen conditions cited is also discussed. The lack of an effective characterization of D-stability motivates the discussion.     

The information matrix test: Simplified calculation via a score test interpretation

In Chesher (1982) I show that the Information Matrix test introduced by White (1982) is a score test for parameter constancy. In this letter I show that this result leads to a simple computational procedure for calculating the Information Matrix test. The procedure involves computing, for a sample of n observations, n times the R² from the least squares regression of a column of ones on a matrix whose elements are functions of 1st and 2nd derivatives of the log density function.     

Algebraic aggregation theory

An aggregation procedure merges a list of objects into a representative object. This paper considers the problem of aggregating n rows in an n-by-m matrix into a summary row, where every entry is an element in an algebraic field. It focuses on consistent aggregators, which require each entry in the summary row to depend only on its column entries in the matrix and to be the same as the column entry if the column in constant. Consistent aggregators are related to additive, linear and projective aggregators.     

How to gamble against all odds

We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the A-martingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too.We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains rA, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered. Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).     

The network structure of the United States input-output table

Economists triangulate input-output tables in order to find a predominant direction in the interindustrial movement of goods and services. Typically, industries, such as Mining, which are far removed from Final Demand, are situated at one end of the input-output table after rearrangement. At the other end are industries that produce for Final Demand, such as Food and Automobiles. Triangulation yields a linear ordering of industries. Much greater detail than this on the complex directionalities of interindustrial flows is presented here. The maximum flow minimum cut algorithm of network flow theory is employed to identify production and consumption complexes. These are relatively self-contained blocks of industries. In each block, there is one industry — the node of the complex — whose total production (consumption) is greater than the amount produced (consumed) by all members of the collection for (from) non-members. Many production complexes have as their nodes primary industries — ones that would be situated near one end of the table after triangulation — and as their other members Final Demand-oriented industries — ones that would be located near the other end. If there arem industries in an input-output table, ann-member production complex is composed of one nodal industry with total output,P, and (n?1) other industries, which together as then-member complex sell less thanP to the (m?n) other industries. 200 production and 42 consumption complexes of the most highly detailed version of the 1967 U.S. input-output table available are presented.     

Bilateral trade in a monetized pure exchange economy

A model is constructed having n traders and m commodities, one of which is money and must be used in every trade. Exchange rates are flexible. If bilateral exchange takes place whenever utility increasing trade is possible, the allocation matrix converges to the Pareto locus.     

Mixed Equilibrium in a Downsian Model with a Favored Candidate

This paper examines competition in the standard one-dimensional Downsian model of two-candidate elections, but where one candidate (A) enjoys an advantage over the other candidate (D). Voters' preferences are Euclidean, but any voter will vote for candidate A over candidate D unless D is closer to her ideal point by some fixed distance δ. The location of the median voter's ideal point is uncertain, and its distribution is commonly known by both candidates. The candidates simultaneously choose locations to maximize the probability of victory. Pure strategy equilibria often fail to exist in this model, except under special conditions about δ and the distribution of the median ideal point. We solve for the essentially unique symmetric mixed equilibrium with no-gaps, show that candidate A adopts more moderate policies than candidate D, and obtain some comparative statics results about the probability of victory and the expected distance between the two candidates' policies. We find that both players' equilibrium strategies converge to the expected median voter as A's advantage shrinks to 0. Journal of Economic Literature Classification Numbers: C72, D72.     

Characterization of certain choice functions

We characterize rationality of certain choice functions C by C(A) ∩ C(B) ? C(A ∪ B) ? C(A) ∪ C(B). We also characterize choice functions coming from preference relations which are partial orders and some other types of relations. Furthermore, we also characterize the admissible set of a binary relation by the fact it is invariant under certain changes in the relation.     

A progress report on Kelly's majority conjectures

LetC(m, n) be the proportion of n-voter profiles on m alternatives that have a majority winner. Jerry Kelly conjectured that C(m, n) > C(m + 1, n) for m ? 3 and n = 3 and n ? 5, and C(m, n) > C(m, n + 2) for m ? 3 and n ? 3. We prove these for special cases.     

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).