This book provides an introduction to the application of group theory, a mathematical
tool, to examine the structure of economic models. Even for models that can
be analyzed by mathematical methods now standard in economics, group theory
provides better intuition. Complex conditions that characterize the equilibrium path
of a dynamicmodel can be expressed as a simple conservation law. Reflecting recent
developments and a broadening interest among economists, this new enhanced
edition has two parts. The first part reproduces the chapters from the 1998 edition
of Symmetry and Economic Invariance and the second consists of four chapters
(Chapters 8 – 11) and a mathematical appendix (Chapter 12). The appendix
supplements a survey of mathematical techniques in Chapter 1.
We first met symmetry in the philosophy of Pythagoras, who used it to attribute
mythical properties to some numbers and geometric figures. In contrast, the
twentieth-century physicists under the influence of analytical philosophy used it to
expunge classical physics of concepts that have no observable consequences! Sophus
Lie showed how continuous groups can be used to determine the integrability
of differential equations while his contemporary Felix Klein used discrete groups
to codify Euclidean and non-Euclidean geometries. Today mathematicians relate
group theory to differential topologies while simultaneously developing geometric
methods that provide visual interpretation of analytical results. We have adopted in
Part , whenever possible, the geometric approach so as to relate our discussion to
diagrammatic analysis in economics; the interpretations we provide are consistent
with the positivist traditions now prevalent in our science.
Chapter 1 provides a heuristic introduction to group theory. Basic mathematical
techniques are introduced and the use of group theory to determine the symmetric
invariance of economic models is explained.
Chapter 2 uses group theory to analyze the controversy overwhether productivity
growth can be decomposed into that arising from technical progress and that from
scale economies. We argue, following Sato (1981), that certain types of technical
progress are indistinguishable from scale economies generated by particular types
of production functions. In the process we develop additional analytical concepts in
group theory.
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