The name is absent



l3 = θlι + (1


ʌ

— θ)l2 is feasible as the choice set of possible loadings is unbounded.


[T (m)](X3 )


max { kl - X32 + δ ʃ


∕m(c(X3, l) + v + u)f1(u)du g(v)dv
-∞


c 00
δ /
⅛c-


У m(c(X3,l)+ r + v + u)f2 (u)du g(v)dv j∙


Xc
kl3 - X32 + δ

J-00


[   m(c(X3, l3) + v + u)f1(u)du g(v)dv

— —o


c 00
δ /
⅛c-


/00

m(c(X3,13 ) + r + v + u)f2(u)du g(v)dv


k [θl^χ + (1 - θ)⅛] - [θX1 + (1 - θ)X2]2

-sr       λ , -sr f ∙.     ,→ zʌ. , -sr f ∙.


/■Xc

+δ

— — o


ec(X1,l1) (1 +c(X2,l2) y∞ m (θ [c(X1,ι^1 ) + υ + U +(1 - θ) [c(X2 ,l^2) + v + u]) f1(u)du g(v)dv


C 00
+δ /

Xc


θc(X1∕1b (1 — ff)c(X2,t2)


J'   m (θ [c(X1,l^1) + r + v + u] +(1 - θ) [c(X2,l^2) + r + v + u]) f2(u)dug(v)dv


The second line uses the fact that l3 is feasible and hence the value under the optimum by definition

ʌ

has to be at least as high. The third line uses the definition of X3 and l3. Using Proposition 1 in the

above equation, namely that c(X1, l1) = c(X2, l2) = c we get (the second line utilizes the fact that
both m and — x2 are concave functions).

[T(m)](X3)   > k [θl^1 + (1 - θ)⅛] - [θX1 + (1 - θ)X2]2

^ ,J- .              .   ,       i-.


Xc
+δ

— — о


Ω ~/ V 7.∖  /1 Ω∖^δV. 7.∖  . „

У m (θ [c(X1, /1 ) + V + u] +(1 - θ) [c(X2, Z2) + V + u]) f1(u)du g(v)dv

C 00
+δ /

Xc


ʌ ʌ

θc(X1,l 1) — (1 — ff)c(X2,l 2)


J' m (θ [c(X1,l^1) + r + v + u] +(1 - θ) [c(X2,l^2) + r + v + u]) f2(u)du g(v)dv


12 - (1 - θ)X22


ʌ                          ʌ               I

> θklι + (1 - θ)k^2 - θX

+δ              [θm ^c(X1 ,l^1)+ v + u) +(1 - θ)m (c(X2,Z^2 ) + v + u)] f1(u)du g(v)dv

+δ J'    J' [θm (c(X1,l^1)+ r + v + u) +(1 - θ)m (c(X2 ,l^2)+ r + v + u)] f2 (u)dug(v)dv

θ fkl^1 - X12 + δ [      [ m fc(X1,l"1 ) + v + u) f1 (u)du g(v)dv

—00  ——∞

+δ J^    J^   m (c(X1,l^1)+ r + v + u) f2(u)dug(v)dv]

+(1 - θ) kl2


- X2 + δ [      [ m (c(X2,l^2) + v + u) f1(u)du g(v)dv

— —     — —

+δX    J^   m (c(X2,l^2) + r + v + u) f2(u)dug(v)dv]

θ[T(m)](X1) + (1 - θ)[T(m)](X2)

The last two lines are simple rearrangements and definition of the value function. We hence know
that the unique attractor, the value function V(X) is concave.

28



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