The name is absent

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).

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

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]

- 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.

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