Symmetry and Economic Invariance: An Introduction

Samuelson (1947, p.3) argued that seemingly diverse fields in economics possessed formal similarities and that the same inequalities and theorems appeared again and again in these theories. He recognized that each field involved interdependent unknowns determined by presumably efficacious equilibrium conditions but argued that there exists identically meaningful theorems in these fields, each derived by essentially analogous methods. Consider microeconomic theory. The interaction between consumers and firms is studied using two major analytical techniques - optimization and equilibrium (Varian (1984, pp.l-3)). The characterization of the optimum behavior requires specification of the actions that the unit can take, the constraints to such actions and the objective function that the unit has. In examining the equilibrium of the model, we are considering whether the actions of all units are compatible. Equilibrium is modeled as the solution of a set of equations.

Economists are interested in returns to scale for three reasons. First, the equilibrium of an industry is dependent on the nature of its technology. Next, growth theorists, while attributing most of the growth in per capita output to factors other than the increase in capital intensity, are not able to agree whether productivity increases should be modelled as arising from technological change or scale effects. Finally, econometricians recognize the problems in identifying the two sources of productivity in any empirical study.

In the last chapter we examined the question whether the increases in efficiency of inputs due to technical progress can be explained by economies of scale. Economies of scale can be considered as resulting in a transformation of the production function which in turn leads to a renumbering of the isoquants. The condition for technical progress to have the same effect is that it should also map one isoquant to another. This was shown in Figure 2.5 by constructing virtual expansion paths based on the coefficients of and in the operator U in equation (10); there we indicated, without deriving, that the production function holothetic to Hicks neutral technical progress is a homothetic production function and that the one for biased technical progress has to be non-homothetic. In this chapter, we will discuss the derivation of production functions that are holothetic to given technical progress functions and show the types of technological changes that are holothetic to some well-known production functions.

Microeconomic theory recognizes that the allocation of resources in economy is driven forces of demand and supply. Supply is determined by technology and the cost of inputs. In previous chapters, we discussed how group theoretic method allow us to examine the effects of technological change on supply. In this chapter, we look how the traditional results of demand theory can be reinterpreted using group theoretic methods.

The wide use of duality in economics arises from the theoretical insights that it provides and from its usefulness for empirical work. The economic implications of the optimization problem and the restrictions that have to be imposed on functional relations to enable econometric estimation become more transparent in the dual than in the original formulation.

The theory of index numbers has a long and distinguished history. The quantity theory of money asserts that the value of money, which in itself is a function of the general level of prices, varies with its supply. If the change in the money supply is followed by a proportionate change in all prices, then the measurement of the changes in the price level will not constitute any problem. Scholars like Edgeworth and Bowley were quick to recognize that the difficulties in quantifying fluctuations in purchasing power of money arose from the absence of such proportionality (Allen (1975, p.2)). As Frisch (1936, p.1) noted:

In dynamic analysis, we examine the movements of a system over time. From the infancy of their science, economists were interested in the microeconomic and macroeconomic adjustment processes. As for long-term trends, classical economists assumed that economy would tend towards a stationary state. Modern growth theory, as developed in the second half of this century, indicated the possibility of a non-stationary equilibrium and provided strong impetus to appropriate and adopt for study of economic dynamics the mathematical tools developed in physical sciences.