Intertemporal Substitution and Recursive Smooth Ambiguity ...

Define v = ψ ◦ ū −1 ◦ u, where ū is defined in (13). 21Using Axiom A6, we immediatelyobtain part (iii) of the theorem. Plugging this definition of v in (36) yields:( ∫ ( ) )∑V s t(ĉ, g +1 ) = ξ v ◦ u −1 ◦ ū g +1 (s)π(s) dµ s t(π)P s ts∈S( ∫ ( ∑∫) )= ξ v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) ,P s ts∈SC×Mwhere the second equality follows from (13).When restricting V s t to the domain M, we obtain:(∫)V s t(ĉ, m) = ξ ◦ v ◦ u −1 u(V (c ′ , m ′ ))dm(c ′ , m ′ )C×M(∫)= V (ĉ, m) = ζ u(V (c ′ , m ′ ))dm(c ′ , m ′ ) ,C×Mfor all m ∈ M, where the last equality follows from (33). Therefore, we have ξ ◦ v ◦ u −1 = ζ,implying u −1 ◦ ζ −1 ◦ ξ = v −1 .Define:y ≡ V s t(ĉ, g +1 ) = ξ( ∫P s tUsing equation (35), we obtain:v ◦ u −1 ( ∑s∈S∫) )π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) .C×MV s t(c, g +1 ) = Ŵ (c, y) = Ŵ ( c, ζ ◦ u(u −1 ◦ ζ −1 (y)) ) = W ( c, u −1 ◦ ζ −1 (y) )(( ∫ ( ∑∫) ))= W c, u −1 ◦ ζ −1 ◦ ξ v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π)P s ts∈SC×M( ( ∫ ( ∑∫) ))= W c, v −1 v ◦ u −1 π(s) u(V (c ′ , m ′ ))dg +1 (s)(c ′ , m ′ ) dµ s t(π) ,P s ts∈SC×Mwhere the third equality follows from the definition of W in Appendix A1.For any g ∈ G, for each s ∈ S, and each (c ′ , g ′ ) in the support of g(s) ∈ ∆(C × G),there exists a risk equivalent (c ′ , m ′ ) ∈ C × M such that (c ′ , m ′ ) ∼ s t ,s (c ′ , g ′ ). Let g +1 bea one-step-ahead act such that g +1 (s)(L ′ ) = g(s)(L) holds for all pairs L ⊂ C × G andL ′ ⊂ C × M where L ′ consists of all risk equivalents (c ′ , m ′ ) of corresponding elements(c ′ , g ′ ) in L. By construction, g +1 (s) and g (s) are stochastically equivalent. By Axiom A521 Note that ū is increasing on M when M is ordered by first-order stochastic dominance. So its inverseexists.35