Intertemporal Substitution and Recursive Smooth Ambiguity ...

D Appendix: Proofs for Section 4.4Proof of Proposition 3: If v ◦ u −1 is concave, it is straightforward to check that {≽ s t}is risk averse in the first stage. We now prove the reverse direction. Pick any l 1 , l 2 ∈ L andλ ∈ [0, 1]. By Theorem 3, we havewhereV (c, λl 1 + (1 − λ)l 2 ) = W ( c, v −1 ( λv ◦ u −1 (V ∗ (l 1 )) + (1 − λ)v ◦ u −1 (V ∗ (l 2 )) )) ,∫V ∗ (l) =C×∆(L)u(V (c ′ , a ′ ))dl(c ′ , a ′ )is a mixture linear function on L. Also we have(∫))V (c, δ[λl 1 ⊕ (1 − λ)l 2 ]) = W(c, v −1 v ◦ u −1 (λV ∗ (l 1 ) + (1 − λ)V ∗ (l 2 )) .LFrom the definition of risk aversion in the first stage, we have (c, δ[λl 1 ⊕ (1 − λ)l 2 ]) ≽ s t(c, λl 1 + (1 − λ)l 2 ). Thus,λv ◦ u −1 (V ∗ (l 1 )) + (1 − λ)v ◦ u −1 (V ∗ (l 2 )) ≤ v ◦ u −1 (λV ∗ (l 1 ) + (1 − λ)V ∗ (l 2 )).We may vary l 1 and l 2 to cover the whole domain of v ◦ u −1 . The above inequality impliesthat v ◦ u −1 is concave.Proof of Proposition 4: Suppose {≽ i s} is more risk averse than {≽ j t s} in the first stage.tBy definition they rank deterministic consumption streams in the same way and rank lotteriesin the second stage in the same way. So there exist representations such that V i = V j ,W i = W j and u i = u j .Since v i (V i (·)) and v j (V j (·)) are ordinally equivalent over C ∞ , there is a monotonetransformation Ψ such that v i = Ψ ◦ v j . The rest is to show that Ψ is concave. Let y, y ′ , y ′′ ∈C ∞ be such that (c, δ[δ[y]]) ∼ j s(c, λδ[δ[y ′ ]]+(1−λ)δ[δ[y ′′ ]]). This is possible due to continuitytof preference ordering. Thus, we have v j (V j (y)) = λv j (V j (y ′ )) + (1 − λ)v j (V j (y ′′ )). Since≽ i sis more risk averse than ≽ j t sin the first stage, we have (c, δ[δ[y]]) ≽ i t s(c, λδ[δ[y ′ ]] + (1 −tλ)δ[δ[y ′′ ]]), which implies v i (V i (y)) ≥ λv i (V i (y ′ )) + (1 − λ)v i (V i (y ′′ )).Since v i (V i (·)) = v i ((V j (·))) = Ψ(v j (V j (·))), we obtainΨ(u j (V j (y))) ≥ λΨ(u j (V j (y ′ ))) + (1 − λ)Ψ(u j (V j (y ′′ ))).One can choose y, y ′ , y ′′ so as to cover the whole range of u j ◦ V j . Thus, Ψ is concave. Theproof of the other direction of the proposition is routine.49