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, Appendix Proofs of Propositions Proposition 1. Under Axioms, vol.2

.. .. , T. , and .. , p T ?1 ) ? (x, t + 1; p 0

,. .. , T. , and .. ,

, Under Axiom 3, temporal weights in (1) satisfy: (i) ?(t +

, T ? 1, and (ii) ?(s, s + 1) = ?(s, s + 2) = ... = ?(s, T ), ?s = 0, p.1

, 2) and ?(1) = ?(2). Likewise, indifference between dates t = 2, vol.1

?. , ? p t?1 ?

, The equality holds for any {p 0

, = ?(s, s + 2) = ... = ?(s, T )

. Proof, The indifference conditions (x, 0) ? (y, t), (x, 0) ? (z, s) and (z, s) ? (y , t) express as: u(x) = (1 ? ? 0 ? p? 1 ? ... ? p? s ?