Supplementary Appendix for ‘ Collective Dynamic Choice : The Necessity of Time Inconsistency ’

Proof of the Paper’s Theorem 4: As in the proof of Theorem 1 in the paper, we apply a theorem by Koopmans (1960, Section 14). First note that by the fact that each ui is nondecreasing and increasing in some dimension on [0, 1] and V satisfies unanimity, his postulates 2 and 5 are satisfied. Next, his postulate 1 follows from continuity (in fact, differentiability) of V under the metric d(C,C ′) = supt ‖ct − ct‖. Finally, time consistency implies his postulates 3, 3’, and 4. Thus, there exists 0 < δ < 1 and a continuous u such that V [δ1, u1; . . . ; δn, un](C) = ∑ t δ u(ct) for all C. By unanimity, it follows that u is nondecreasing and is increasing in some arguments. Assume now that it is twice differentiable and that δi 6= δj for all i 6= j. Without loss of generality, let us normalize u so that u(0) = 0 and u(1) = 1 (where the inputs are now vectors).