6 Appendix
6.1 Proof of Lemma 1
Investor i derives her optimal portfolio of date τ -claims from
maxE[ui(xi)] s.t. E[xiφ(Dτ)] = w0τ.
w0τ is the investor’s endowment reserved for buying claims on Dτ . φ(Dτ )is
the stochastic discount factor, i.e. φ(Dτ) = Φ0,τ exp(-rf τ). The FOC for
xi is (λi denotes the Lagrange-multiplier of the budget constraint)
u'i(Xi) = λiφ(Dτ) ; ∀ Dτ.
Differentiate the log of this equation with respect to ln Dτ . This yields
d ln x
ni (Xi) = ПМ (Dτ) ; ∀ Dτ (8)
d ln Dτ
Since d ln xi/d ln Dτ = (dxi∕dDτ)/αi (Dτ) and ]∑i dxi∕dDτ = 1, aggregating
equation (8) across all investors yields
1 = αi ( Dτ )
ПМ ( Dτ ) ʌ ni (Xi )
(9)
Differentiate equation (9) with respect to Dτ . This yields
nM( Dτ ) = V ni( xi ) dxi α ( D ) -V 1 α ( D ) (10)
[ПМ(Dτ)]2 b [ni(Xi)]2 dDτα(T) ≥-η ηi(Xi)αi(t). (0)
The first term on the right hand side of equation (10) can be rewritten using
(8) as
ni ( Xi ) d ln Xi
[ni(Xi)]2 dln Dτ
[αi(Dτ)]2
1 v ni ( Xi ) ∣^ D D }MM ( Dτ ) ^∣ 2
nM ( Dτ ) V ni (Xi ) I- i( τ ) ni (Xi ) -
20
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