A simple framework for the axiomatization of expon

Authors: Nina Anchugina

Publish Date: 2016/07/25

Volume: 82, Issue: 2, Pages: 185-210

PDF Link

Abstract

The main goal of this paper is to investigate which normative requirements or axioms lead to exponential and quasihyperbolic forms of discounting Exponential discounting has a wellestablished axiomatic foundation originally developed by Koopmans Econometrica 282287–309 1960 1972 and Koopmans et al Econometrica 321/282–100 1964 with subsequent contributions by several other authors including Bleichrodt et al J Math Psychol 526341–347 2008 The papers by Hayashi J Econ Theory 1122343–352 2003 and Olea and Strzalecki Q J Econ 12931449–1499 2014 axiomatize quasihyperbolic discounting The main contribution of this paper is to provide an alternative foundation for exponential and quasihyperbolic discounting with simple transparent axioms and relatively straightforward proofs Using techniques by Fishburn The foundations of expected utility Reidel Publishing Co Dordrecht 1982 and Harvey Manag Sci 3291123–1139 1986 we show that Anscombe and Aumann’s Ann Math Stat 341199–205 1963 version of Subjective Expected Utility theory can be readily adapted to axiomatize the aforementioned types of discounting in both finite and infinite horizon settingsI would like to thank my supervisor Matthew Ryan for thoughtful advice and support I am grateful to Arkadii Slinko and Simon Grant for helpful discussions I would also like to thank the Australian National University for their hospitality while working on the paper Helpful comments from seminar participants at Australian National University The University of Auckland and participants at Auckland University of Technology Mathematical Sciences Symposium 2014 Centre for Mathematical Social Sciences Summer Workshop 2014 the joint conferences “Logic Game theory and Social Choice 8” and “The 8th PanPacific Conference on Game Theory” 2015 Academia Sinica are also gratefully acknowledged Financial support from the University of Auckland is gratefully acknowledged Finally I would like to thank the anonymous reviewer for their effort and the constructive comments providedStep 1 Applying Theorem 1 of Fishburn 1982 to the mixture set X it follows from Axioms I1 I3 I4 that there exists a mixture linear utility function u preserving the order on X unique up to positive affine transformations Normalize u so that ux 0=0 Note that by nontriviality ux 0 is in the interior of the nondegenerate interval uXConvert streams into their utility vectors by replacing the outcomes in each period by their utility values Define the following order v 1 v 2 ldots succcurlyeq u 1 u 2 ldots Leftrightarrow there exist mathbfx mathbfy in Xinfty such that mathbfx succcurlyeq mathbfy and ux t=v t and uy t=u t for every t This order is unambiguously defined because of the monotonicity assumption ie if x i sim xprime i then x 1 ldots x i ldots sim x 1 ldots xprime i ldots Step 2 Normalize U so that U0 0 ldots =Umathbf0=0 Since 0 is in the interior of uX and since Umathbfv lambda mathbf0=lambda Umathbfv for any mathbfvin mathbb Rinfty and for every lambda in 0 1 we can assume that U is defined on mathbb Rinfty Mixture linearity of U implies standard linearity of U on mathbb Rinfty To prove this we need to show that Uk mathbfv= kUmathbfv for any k and Umathbfv+mathbfu=Umathbfv+Umathbfu for any mathbfu mathbfv in mathbb Rinfty

Keywords: