Relationship of the arrow-pratt coefficients with the tangent to the utility function

Abstract

The article discusses a geometric approach for determining some properties of the utility function using the Arrow-Pratt coefficient. In particular, the geometric properties of the Arrow-Pratt coefficient are disclosed, it turned out that the Arrow-Pratt coefficient determines the distance from the origin to the straight line parallel to the tangent line at the point where the Arrow-Pratt coefficient is calculated. Therefore, an increase and decrease in the value of this coefficient determines the convexity or concavity of the utility function. With an increase in the value of the Arrow-Pratt coefficient, the tangent line moves away from the origin. This process describes the convexity of the utility function as a function of location from the origin. There are two concepts: the expected utility model and the utility function. Our approach is that due to a tangent line and a straight line parallel to the tangent with a free term, associated with the variance of variables and the Arrow-Pratt coefficient, we determine the behavior of the utility function: monotonicity, convexity, etc. The location of these lines relative to the origin gives us twenty-four different options, which are divided into four groups, depending on the signs of the first and second derivatives of the utility function at the point where the tangent line passes.