Intertemporal Substitution and Recursive Smooth Ambiguity ...

Proof of Proposition 8:When the utility function takes the homothetic form, we useProposition 7 and the definition of the pricing kernel to derive (27). Alternatively, we maywrite the pricing kernel in terms of the market return as in Epstein and Zin (1989). In acomplete market, wealth X t satisfiesX t = E t[ ∞∑s=tThat is, time t wealth is equal to the present value of the consumption stream. By Lemma6.25 in Skiadas (2009), we haveBy equation (22), we have the relation:c tV t=Thus, the consumption-wealth ratio satisfies:c tX t= c tλ tV t=ξszξtzc s].V t = λ t X t . (50)( ) −1/ρ λt.1 − β(λt1 − βEliminating λ t from equations (50) and (51) yields:V t = λ t X t = (1 − β) 1= (1 − β) 1−ρ1−ρ c1−ρt X 11−ρ−ρ1−ρ c1−ρt R 11−ρ) −1/ρλ t . (51)t (52)t (X t−1 − c t−1 ) 1where we have used equation (29) to derive the last equality. Note that the second equalityimplies thatX t= 1 ( ) 1−ρ Vt. (53)c t 1 − β c tAs a result, for unitary EIS (ρ = 1), the consumption-wealth ratio is equal to 1 − β.1−ρ ,Now, substituting equation (52) into (27) and manipulating, we derive:⎛⎞ ⎛ [ρ−γ −ρ(1−γ) 1−γ−ρ( ) −ρ 1−ρMt+1 z ct+1= β ⎜ct+1 R 11−ρ 1−ρ1−ρ⎜(E πz,t ct+1 Rt+1t+1c t⎝) ⎟ ⎜⎜⎝−ρ ⎠)−ρ1−ρ 1−ρ1−ρ 1−ρR t(cR t(cM z t+1 =β(ct+1c tt+1 R 1t+1t+1 R 1t+1Writing in terms of consumption growth and manipulating, we obtain:( ) ) 1−γ⎛ (−ρ 1−ρ⎡ ⎛(× ⎣R t⎝ β1−γ1−ρR−1t+1⎡⎝E πz,t⎣( ) ) 1−ρ 1−ρct+1R t+1c t⎞⎤⎠⎦βη−ρ(ct+1.c t]) 11−γ) −ρR t+1) 1−γ1−ρ⎞⎟⎠⎤⎞⎦⎠−(η−γ)−(η−γ)1−γ.52