IJPAM: Volume 40, No. 1 (2007)

RELATIVE DERIVATIVES AND THE ECONOMIC SCIENCE

Gregory L. Light
Providence College
Providence, Rhode Island, 02918, USA
e-mail: glight@providence.edu


Abstract.``Relative derivative'', $(a/b)(dy/dx)$, is a generalization of the ordinary derivative for $a=b=1$, and that of ``elasticity'' in economics for $a=x$ and $b=y$; the role of $(a,b)$ is twofold: scaling and removing units. In a previous publication, this author applied relative derivatives to Taylor series in n variables and the fundamental theorem of calculus. Nature does not come with unit labels; rather, it is characterized by proportionalities, which are precisely what relative derivatives reveal - the geometric invariance of the Ricci scalar curvature $R$ for example. The economic science has a pronounced feature, viz., the indeterminacy of the units of variables. This paper shows how relative derivatives, by revealing proportionalities, streamline the mathematical logic of economics and integrate all the building blocks therein, and at the same time change the mode of analysis from qualitative to quantitative. To the extent that many fields share the same interests as in economics of (post) optimization and equilibrium analysis yet also the same problem of unit specifications, this paper provides an illustration of how relative derivatives can be applied to fruitful theoretical derivations by four fundamental examples in economics.

Received: August 1, 2007

AMS Subject Classification: 26A24, 90C31, 37N40, 91B02, 91B62

Key Words and Phrases: quantitative parametric perturbation, comparative statics, sensitivity analysis, scale invariance, generalized elasticities

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 40
Issue: 1