Intertemporal Substitution and Recursive Smooth Ambiguity ...

which in turn implies that on L,Hence, we deduce that:which implies that:ζ (U(l)) = V (ĉ, δ[l]) = V s t(ĉ, δ[l]) = ζ s t (U s t(l)) .U s t = ζ −1s t ◦ ζ ◦ U,(∫ ) (∫)ζ U(l)da(l) = ζ s t ζ −1s◦ ζ ◦ U(l)da(l) .tLLBy the additivity of integral formula, we haveζ −1s t◦ ζ (αx + (1 − α) y) = αζ −1s t◦ ζ(x) + (1 − α) ζ −1s t◦ ζ(y)for all x, y in the range of U and all α ∈ [0, 1]. Therefore ζ s t and ζ are identical up to positiveaffine transformations. Thus without loss of generality we can take ζ s t = ζ and U s t = U forall s t .Equations (39), (40), (46), and (47) imply that on L,(∫)φ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) = U(l) = U s t(l)C×∆(L)(∫)= φ s t u s t ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) ,C×∆(L)which in turn implies that:φ ◦ u ◦ V (c ′ , a ′ ) = U(δ[c ′ , a ′ ]) = U s t(δ[c ′ , a ′ ]) = φ s t ◦ u s t ◦ V (c ′ , a ′ ).Hence, we have φ ◦ u = φ s t ◦ u s t, which implies that:(∫) (∫)φ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) = φ s t φ −1s◦ φ ◦ u ◦ V (c ′ , a ′ )dl(c ′ , a ′ ) .tC×∆(L)C×∆(L)By the same reasoning as above, φ s t and φ are identical up to positive affine transformations.Therefore without loss of generality we can set φ s t = φ and u s t = u for all s t .Now, plugging equations (45), (46), and (47) into (44), we obtain that, on C × ∆(H +1 ),= Ŵ= WV s t(c, p +1 )( ( ∫ ∫H +1c, ζP s tφ( ∑s∈S(c, u −1 ◦ φ −1 ( ∫H +1∫P s t∫π(s)φ( ∑s∈SC×∆(L)∫π(s)u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ )46C×∆(L))u(V (c ′ , a ′ ))dh +1 (s)(c ′ , a ′ )dµ s t(π)dp +1 (h +1 ))))dµ s t(π)dp +1 (h +1 ))),