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## Über dieses Buch

In recent years, the usual optimization techniques, which have proved so useful in microeconomic theory, have been extended to incorporate more powerful topological and differential methods, and these methods have led to new results on the qualitative behavior of general economic and political systems. These developments have necessarily resulted in an increase in the degree of formalism in the publications in the academic journals. This formalism can often deter graduate students. The progression of ideas presented in this book will familiarize the student with the geometric concepts underlying these topological methods, and, as a result, make mathematical economics, general equilibrium theory, and social choice theory more accessible.

## Inhaltsverzeichnis

### 1. Sets, Relations, and Preferences

Abstract
Chapter 1 introduces elementary set theory and the notation to be used throughout the book. We also define the notions of a binary relation, of a function, as well as the axioms of a group and field. Finally we discuss the idea of an individual and social preference relation, and mention some of the concepts of social choice and welfare economics.
Norman Schofield

### 2. Linear Spaces and Transformations

Abstract
Chapter 2 surveys material on linear or vector spaces, and introduces the idea of a linear transformation between vector spaces, and shows how a linear transformation can be represented by a matrix. We also prove the dimension theorem, which shows how a linear transformation can be represented in a simple form, given by its kernel and image. We also introduce the notion of an eigenvalue and eigenvector of a matrix.
Norman Schofield

### 3. Topology and Convex Optimisation

Abstract
Chapter 3 covers Topology and convex optimization. In the Chap. 2 we introduced the notion of the scalar product of two vectors in ℜ n . More generally if a scalar product is defined on some space, then this permits the definition of a norm, or length, associated with a vector, and this in turn allows us to define the distance between two vectors. A distance function or metric may be defined on a space, X, even when X admits no norm. More general than the notion of a metric is that of a topology. This notion allows us to define the idea of continuity of a function as well as analogous ideas for a correspondence. We then introduce three powerful theorems, the Brouwer Fixed Point Theorem for a function, Michael’s Selection Theorem, and the Browder Fixed Point Theorem for a correspondence.
Norman Schofield

### 4. Differential Calculus and Smooth Optimisation

Abstract
In this chapter we develop the ideas of the differential calculus. Under certain conditions a continuous function f:ℜ n →ℜ m can be approximated at each point x in ℜ n by a linear function df(x):ℜ n →ℜ m , known as the differential of f at x. In the same way the differential df may be approximated by a bilinear map d 2 f(x). When all differentials are continuous then f is called smooth. For a smooth function f, Taylor’s Theorem gives a relationship between the differentials at a point x and the value of f in a neighbourhood of a point. This in turn allows us to characterise maximum points of the function by features of the first and second differential. For a real-valued function whose preference correspondence is convex we can virtually identify critical points (where df(x)=0) with the maxima of the function. We use calculus to derive important results in economic theory, namely conditions for existence of a price equilibrium for an economy, and the Welfare Theorem for an exchange economy.
Norman Schofield

### 5. Singularity Theory and General Equilibrium

Abstract
In Chap. 5 we introduce the fundamental result in singularity theory, that the set of singularity points of a smooth preference profile almost always has a particular geometric structure. We then go on to use this result to discuss the Debreu-Smale Theorem on the generic existence of regular economies. Section 5.4 uses an example of Scarf (Int. Econ. Rev. 1:157–172, 1960) to illustrate the idea of an excess demand function for an exchange economy. The example provides a general way to analyse a smooth adjustment process leading to a Walrasian equilibrium. Sections 5.5 and 5.6 introduce the more abstract topological ideas of structural stability and chaos in dynamical systems.
Norman Schofield

### 6. Topology and Social Choice

Abstract
In this chapter we apply earlier results to the study of social choice and modelling elections. In Chap. 3 we showed the Nakamura Theorem. that a social choice could be guaranteed as long as the dimension of the space did not exceed k(σ)=2. We now consider what can happen in dimension above k(σ)=1. We then go on to consider “probabilistic” social choice, where there is some uncertainty over voters’ preferences, by constructing an empirical model of the 2008 U.S. presidential election.
Norman Schofield

### 7. Review Exercises

Abstract
This section gives review exercises for each of the chapters.
Norman Schofield

### Backmatter

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