Curvature and the elasticity of substitution: What is the link?
Relation between curvature and the elasticity of substitution is the old question important for economic theory. Opinions of economists concerning presence or absence of a link between these two concepts radically diverge. Also now there is a steady trend of the use of the Arrow-Pratt coefficient of relative risk aversion and the coefficient of relative prudence as characteristics of utility functions and production functions even in non-stochastic models, and these two coefficients are also commonly interpreted as measures of curvature. The purpose of the paper is to contribute to clarification of the links between all these concepts. We suggest a simple unifying approach based on the notions of prototype functions and osculating curves. In framework of this approach we easily derive the classic geometric curvature and show the relations between the Arrow-Pratt coefficient, the prudence coefficient, the elasticity and the elasticity of substitution. As an example, demonstrating the role of such relations in economic models, we study a simple macroeconomic model with a non-homothetic production function.