Linear Problems and Linear Algorithms

AbstractUsing predicate logic, the concept of a linear problem is formalized. The class of linear problems is huge, diverse, complex, and important. Linear and randomized linear algorithms are formalized. For each linear problem, a linear algorithm is constructed that solves the problem and a randomized linear algorithm is constructed that completely solves it, that is, for any data of the problem, the output set of the randomized linear algorithm is identical to the solution set of the problem. We obtain a single machine, referred to as the Universal (Randomized) Linear Machine, which (completely) solves every instance of every linear problem. Conversely, for each randomized linear algorithm, a linear problem is constructed that the algorithm completely solves. These constructions establish a one-to-one and onto correspondence from equivalence classes of linear problems to equivalence classes of randomized linear algorithms.Our construction of (randomized) linear algorithms to (completely) solve linear problems as well as the algorithms themselves are based on Fourier Elimination and have superexponential complexity. However, there is no evidence that the inefficiency of our methods is unavoidable relative to the difficulty of the problem

The Principle of Minimum Differentiation Reconsidered: Some New Developments in the Theory of Spatial Competition

The paper is divided into two parts: one-dimensional markets and two-dimensional markets. Also, we develop both one and two-dimensional models. Within each, we distinguish (a) bounded, (b) unbounded but finite, and (c) unbounded, infinite spaces. Among other things, we show: in one dimension, the nature of the space is not, as many investigators have thought, critical; in two dimensions, however, the very existence of equilibrium is seen to depend upon the nature of the space; the commonly-used rectangular customer density function yields results that do not generalize to any other density function; the existence of multiple equilibria in both one and two dimensions is a pervasive phenomenon in any of the spaces studied, and MD occurs only when the number of firms is restricted to two. Although the analysis and discussion are in terms of location theory and are concerned with the relationship between equilibrium configuration of firms and the transport-cost minimizing configuration, many of the results generalize to other forms of differenciation. The conditions under which the results generalize are considered in the concluding section of the paper.

The Principle of Minimum Differentiation Reconsidered: Some New Developments in the Theory of Spatial Competition

The paper is divided into two parts: one-dimensional markets and two-dimensional markets. Also, we develop both one and two-dimensional models. Within each, we distinguish (a) bounded, (b) unbounded but finite, and (c) unbounded, infinite spaces. Among other things, we show: in one dimension, the nature of the space is not, as many investigators have thought, critical; in two dimensions, however, the very existence of equilibrium is seen to depend upon the nature of the space; the commonly-used rectangular customer density function yields results that do not generalize to any other density function; the existence of multiple equilibria in both one and two dimensions is a pervasive phenomenon in any of the spaces studied, and MD occurs only when the number of firms is restricted to two. Although the analysis and discussion are in terms of location theory and are concerned with the relationship between equilibrium configuration of firms and the transport-cost minimizing configuration, many of the results generalize to other forms of differenciation. The conditions under which the results generalize are considered in the concluding section of the paper

Well-being and Affluence in the Presence of a Veblen Good

The happiness literature has established that, in the developed countries, increasing affluence has not increased well-being in recent decades. We seek an explanation for this in terms of conspicuous consumption, a phenomenon originally identified by Veblen. We develop some simple general equilibrium models that incorporate a Veblen good, among others. In all of our models, as productivity increases, the Veblen good eventually dominates the economy in the sense that, by reducing leisure, more than all the added productivity is dissipated in the production of this good. Also, in the presence of a Veblen good, productivity increases destroy social capital. Copyright � The Author(s). Journal compilation � Royal Economic Society 2009.

Myopic deterrence policies and the instability of equilibria

We develop a general equilibrium model of crime in which honest workers face an effective tax rate on earnings, consisting of an exogenous income tax, used to pay for policing, and an endogenous crime tax, given by the proportion of their disposable income that is stolen. When there are two stable equilibria, the low crime equilibrium is welfare dominant and, furthermore, welfare is a decreasing function of the income tax rate. However, as the income tax rate is reduced, the degree of stability of the low crime equilibrium diminishes, until for sufficiently low tax rates, the low crime equilibrium is unstable with respect to any small, positive perturbation in the crime rate. Thus myopic maximization of social welfare exposes the economy to random shocks in the crime rate that can destroy the preferred, low crime equilibrium. We argue that this dilemma between economizing on resources used for deterrence and ensuring the stability of the most desirable equilibrium is a fundamental trade-off in the design of many institutions intended to control antisocial behavior, including rent-seeking, corruption and theft.