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Let y ∈ intΓ (x). 1 in [3], y ∈ intΓ (xn ) for n large enough. Therefore V (xn ) ≥ −δ(xn , y) + V (y) which implies that V ∗ ≥ −δ(x, y) + V (y) Since V is upper semicontinuous, V (x) ≥ V ∗ . Let us assume that V (x) > V ∗ . Let y ∈ Γ (x), y = τ (x). Let yλ = λy + (1 − λ)τ (x). 12), V (x) = −δ(x, τ (x)) + V (τ (x)). 15) and the strict concavity of −δ(x, ·) that follows from H8, we obtain V ∗ ≥ −δ(x, yλ ) + V (yλ ) > −λ(δ(x, y) + V (y)) + (1 − λ)(−δ(x, τ (x)) + V (τ (x))) 1. Optimal Growth Without Discounting 15 = −λ(δ(x, y) + V (y)) + (1 − λ)V (x) > V ∗ hence a contradiction.

The proof is obvious since the problem is {max u(x) : x ∈ Π(x0 )} with u upper semi-continuous and Π(x0 ) compact. We say that a function ϕ from X into R ∪ {−∞} which satisfies ϕ(x) > −∞ if x = 0, ϕ(0) = −∞, is continuous in the generalized sense if (i) it is continuous at any point x = 0 and (ii) if a sequence {xn } of points in X\{0} converges to 0 then ϕ(xn ) → −∞. 2. Assume H1-H2-H3bis-H4. (i) The Value function V is upper semi-continuous. (ii) ∀x0 ∈ X ∀x ∈ Π(x0 ), lim supT β t V (xt ) ≤ 0. (iii) For every x0 ∈ X, and for every x ∈ Π (x0 ), we have limt→+∞ β t V (xt ) = 0.

The reader can check that the initial problem is equivalent to: +∞ max β t V (xt , xt+1 ) t=0 under the constraint: ∀t ≥ 0, xt+1 ∈ Γ (xt ), xt ∈ R2+ and x0 is given in R2+ . 1 Bounded from Below Utility We consider a model where the technology can exhibit a non zero maximal rate of growth. In the first section we assume that the absolute value of the utility is bounded by an affine function. This excludes Cobb-Douglas utility functions with negative elasticities or logarithm. We devote the next section to the case where the utility function can take the value −∞.

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