Mathematical Economics

The Dynamic Programming approach for a family of optimal investment models with vintage capital is here developed. The problem falls into the class of infinite horizon optimal control problems of PDE's with age structure that have been studied in various papers (see e.g. , ) either in cases when explicit solutions can be found or using Maximum Principle techniques.

This paper proposes a multi-prize ''reverse'' nested lottery contest model, which can be viewed as the ''mirror image'' of the conventional nested lottery contest of Clark and Riis (1996a). The reverse-lottery contest model determines winners by selecting losers based on contestants' one-shot effort through a hypothetical sequence of lotteries. We provide a microfoundation for the reverse-lottery contest from a perspective of (simultaneous) noisy performance ranking and establish that the model is underpinned by a unique performance evaluation rule. We further demonstrate that the noisy-ranking model can be interpreted intuitively as a ''worst-shot'' contest, in which contestants' performances are evaluated based on their most severe mistakes. The reverse-lottery contest model thus depicts a great variety of widely observed competitive activities of this nature. A handy closed-form solution for a symmetric equilibrium of the reverse-lottery contest is obtained. We show that the winner-take-all principle continues to hold in reverse-lottery contests. Moreover, we find that a reverse-lottery contest elicits more effort than a conventional lottery contest whenever the prizes available to contestants are relatively scarce.

We study a general class of pure exchange economies that have multiple equilibria. This class generalizes an example presented by Shapley and Shubik. For such economies, we find easily verified conditions that determine whether there are multiple equilibria. We also provide simple methods for constructing economies in which arbitrary pre-specified sets of prices are equilibria. These economies have strong comparative statics properties, since prices at interior competitive equilibrium depend on the parameters of utility but not on the endowment quantities. We believe that this easily manipulated special case is a valuable addition to the class of simple general equilibrium economies that can be used as testing grounds in economic theory.

In this paper we argue that indeterminacy of equilibrium is a possibility inherent in economies with a double infinity of agents and goods, large-square economies. We develop a framework that is quite different from the overlapping generations one and that is amenable to analysis by means of differential calculus in linear spaces. The commodity space is a separable, inlinitedimensional Hilbert space, and each of a continuum of consumers is described by means of an individual excess demand function detined on an open set of prices. In this setting we prove an analog of the Sonnenschein-MantelLDebreu theorem. Using this result, we show that the set of economies whose equilibrium sets contain manifolds of arbitrary dimension is non-empty and open in the appropriate topology. In contrast, we also show that, if the space of consumers is sufficiently small, then local uniqueness is a generic property of economies. The concept of sufftciently small has a simple mathematical formulation (the derivatives of individual excess demands form a uniformly integrable family) and an equally simple economic interpretation (the variation across consumers is not too great).

Economic methodologists most often study the relations between models and reality
while focusing on the issues of the model’s epistemic relevance in terms of its relation
to the ‘real world’ and representing the real world in a model. We complement the
discussion by bringing the model’s constructive mechanisms or self-implementing
technologies in play. By this, we mean the elements of the economic model that are
aimed at ‘implementing’ it by envisaging the ways to change the reality in order to
bring it more in line with the model. We are thus concerned mainly not with the ways to
change the model to ‘fit’ the reality, but rather with the model’s own armature that is
supposed to transform the world along theoretical lines. The case we study is Arrow–
Debreu–McKenzie general equilibrium model. In particular, we show the following:
gradient methods and stability could be regarded as constructive mechanisms of
general equilibrium modeling in the context of market socialism debates; the obsession
of general equilibrium theorists with these concepts can be partly explained by the fact
that they hoped not to be faithful to reality, but rather to adjust it to fit the theoretical
model; mechanism design theory initiated by the stability theorist Leonid Hurwicz
could be seen as a successor of this position. We conclude by showing the relevance of
this analysis for epistemic culture of much of contemporary economics and hence,
claim that it is an important complement to the traditional philosophy of economic
modeling.

It has never been accomplished to describe our behavior mathematically. Due to the fact that human behavior is highly erratic even the understanding of its causes are still sketchy. Assuming that we are all equal in our regulation of thought and behavior there are simply too many differences and partially inconsistencies, the attempts stopped in its onset. For having defined the five major groups of mankind, the sociopath, the artist, the median, the, and the psychopath［1］each group is related to each other but his decision making is hardwired differently and thou probably more easy to grasp than sticking to the consistency of all appearance for the regulation of behavior is quite similar in the five groups but different in its limits. Using the more formalistic tools of mathematics, this can open the review of the equations in verifying or falsifying the predictions on future behavior in an individual, at least after defining the group affiliation. Therefore a self-test has been established［2］to predetermine the group. In a hybrid-species ［3］, the eight main neuro-receptors in each group to have two optional origins. Measuring each by its own dominating patterns not only the amalgamation in each group can be defined but also the native patterns of the non-hybrid ancestry ［4］. Not only the variance of possible combination can distribute to the limits of brain-equations but also the time-axis of our memory, being rather different, illuminating the highly different decision making among offspring of hunters and farmers. A phenomenon probably explaining the variance in processing memory by peripheral distributed groups of ADHD and autism for ADHD memorizes in combining data with importance and such is been given an emotional response to the recall, while autism is mainly been given the exact time reference stored in a continuous frame of time-preference. The latter therefore have problems to distinguish between important and not important and the former lacks a passing timeframe, mirroring the primary form of acquiring resource, farming or hunting. With the boundaries set the graphs of the equation on resource-projection looks highly different by only changing its limits. Having not only focussed on the designation of human groups but also in behavioral shifts over time (from social to non-social) and the exact denomination of similar behavior, some rather simple equation could be defined to predict and consequently proof the predicate. Not only the proper use of words is necessary but also awareness that non-social individuals will often not answer truthfully. A topic that also can be mathematically predicted. The outcome will revolutionize our perception of mankind and ourselves, dawning more than one academic discipline, probably enabling us to go virtual and back again.