The second term on the right hand side of equation (10) can be rewritten
as (since ∑i α'i(Dτ) = 0)
v O ( D ) ( —___1__~ V αi ( Dτ ) D
1
ηi ( Xi )
2-α i( τ) ∖(x(Xi) ηM(Dτ)J Dτ^ O (Dτ) '
=Dj ∑
i
d ln αi
d ln Dτ
1
ηi ( Xi )
пм ( Dτ )
Dτ
1
ηi ( Xi )
12
— ■" J αi( Dτ )
The last equation follows from d ln αi∕d ln Dτ = d ln xi/d ln Dτ — 1 and equa-
tion (8) which implies d ln αi∕d ln Dτ = ηM(Dτ)[1 /ηi(xi) — 1 /ηM(Dτ)].
Multiplying equation (10) by ηM(Dτ) proves Lemma 1.
6.2 Proof of Proposition 1
For 0 ≤ t ≤ s the forward stochastic discount factor is defined by Φt,s =
Φ0,s/Φ0,t. Because of the martingale property Φ0,t = E(Φ0,s∣Dt), the elas-
ticity of the (forward) stochastic discount factor Φt,s with respect to the
dividend Dt is given by
∂ ln Φt,s = ∂ ln Φo,s
∂ ln Dt ∂ ln Ds
e(∂^ Φ ts
∂ ln Ds
Df)) = — ηΦ D + E ( ηφ D φts ∣ Dt),
since ∂ ln Ds = 1 d φ,D ≡ d l∏φ0,s
since ∂ ln Dt = 1 and ηs ≡ ∂ ln Ds ■
Differentiating the logarithm of equation (4) with respect to ln Dt yields
after some manipulation
S,D
ηt ,
1+
ʃ∞ exp (—rf (s — t)) E (dsΦt,s (—ηφD + E (ηsφ,DΦt>s∣ Dt)) ∣ Dt) ds
St
The forward stochastic discount factor defines an equivalent martingale mea-
sure P , i.e. P (A)= A Φt,sdP , where P is the physical or objective proba-
bility measure. Hence the previous equation yields
∞
S,D
ηt =1 -J
t
covP ( Ds,ηφ ,D IDt ) ds
exp (rf (s — t)) St ,
where covP(.) is the covariance under the equivalent martingale measure P7.
Thus, if ηφ,D is constant, then ηS,D ≡ ∂П Dt = 1. Declining aggregate RRA,
d∂Djr- < 0, implies covP(Ds,ηφ,D∣Dt) < 0 and, hence, ηS,D > 1. Increasing
aggregate RRA, 'DDD, > 0, implies ηS,D < 1. ■
21
More intriguing information
1. Public-private sector pay differentials in a devolved Scotland2. MATHEMATICS AS AN EXACT AND PRECISE LANGUAGE OF NATURE
3. Endogenous Heterogeneity in Strategic Models: Symmetry-breaking via Strategic Substitutes and Nonconcavities
4. The name is absent
5. Family, social security and social insurance: General remarks and the present discussion in Germany as a case study
6. Langfristige Wachstumsaussichten der ukrainischen Wirtschaft : Potenziale und Barrieren
7. Influence of Mucilage Viscosity On The Globule Structure And Stability Of Certain Starch Emulsions
8. Epistemology and conceptual resources for the development of learning technologies
9. Forecasting Financial Crises and Contagion in Asia using Dynamic Factor Analysis
10. The name is absent