Intertemporal Substitution and Recursive Smooth Ambiguity ...

In complete markets, the following Euler equation holds:E t[Mzt+1 R t+1]= 1.Substituting the preceding pricing kernel into this Euler equation, we obtain:1 = E t⎧⎪ ⎨⎪ ⎩(β(ct+1c t⎡ ⎛(× ⎣R t⎝ βNoting that E t = E µt E πz,t) ) 1−γ⎛ ⎡(−ρ 1−ρR t+1⎝E πz,t⎣ β( ) ) 1−ρ 1−ρct+1R t+1c t⎞⎤⎠⎦η−ρ.(ct+1c t) ) 1−γ⎤−ρ 1−ρR t+1⎦and using the definition of R t , we obtain:⎡ ⎛(( ) ) ⎞⎤1−ρ 1−ρ⎣R t⎝ ct+1β R t+1⎠⎦c t1−ρ= 1.⎞⎠⎫−(η−γ)1−γ ⎪⎬⎪⎭Thus,⎛(( ) ) ⎞1−ρ 1−ρR t⎝ ct+1β R t+1⎠ = 1,c tso that we can write the pricing kernel as equation (28).53