On the Nature of Elasticity Function: An Investigation and a Kernel Estimation

Abstract

In economics, we know the law of demand: a higher price will lead to a lower quantity demanded. The question is to know how much lower the quantity demanded will be. Similarly, the law of supply shows that a higher price will lead to a higher quantity supplied. Another question is to know how much higher. To find answers to these questions which are critically important in the real world, we need the concept of elasticity. Elasticity is an economics concept that measures the responsiveness of one variable to changes in another variable. Elasticity is a function $e\left(x\right)$ that can be built from an arbitrary function $y=g\left(x\right)$ . Elasticity at a certain point is usually calculated as $e\left(x\right)=\left(dy/dx\right)\left(x/y\right)$ . Elasticity can be expressed in many forms. An interesting form, from an economic point of view, is the ratio between the derivative of the logarithm of the distribution function with respect to the logarithm of the point $x$ , which is developed in this article. The aim of this article is to study the direction of variation of this elasticity function and to construct a nonparametric estimator because the estimators that have been constructed so far are parametric estimators and admit many deficiencies in practice. And finally, we study the strong consistency of the said estimator. A numerical study was carried out to verify the adequacy of the theory.