How to measure the average rate of change?

Authors: Aleksandr Alekseev, Mikhail Sokolov

Abstract: This paper contributes to the theory of average rate of change (ARC) measurement. The contribution is twofold. First, it relates ARC measurement to intertemporal choice. We show that an ARC of a variable can be identified with a discount rate which makes an economic agent indifferent between the initial and final temporal states of the variable. Furthermore, there is a one-to-one correspondence between ARC measures and one-parameter families of time preferences indexed by a discount rate. Second, we employ an axiomatic approach to generalize the conventional ARC measures (such as the difference quotient and the continuously compounded growth rate) in several directions: to variables with arbitrary connected domains, to not necessarily time-shift invariant dependence on dates, to sets of time points other than an interval, to a benchmark-based evaluation. The generalized ARC measures turn out to correspond to the existing time preference models such as the discounted utility and the relative discounting model of Ok and Masatlioglu [Ok, E.A., Masatlioglu, Y., 2007. A theory of (relative) discounting. J. Econ. Theory 137(1), 214–245].