Introduction to Modern Economic Growth Combining and rearranging these, we have u (f (k) + (1 − δ) k − s (k)) − u (f (k) + (1 − δ) k − s (k0 )) ≥ β [V (s (k0 )) − V (s (k))] ≥ u (f (k0 ) + (1 − δ) k0 − s (k)) −u (f (k0 ) + (1 − δ) k0 − s (k0 )) Or denoting z ≡ f (k) + (1 − δ) k and x ≡ s (k) and similarly for z and x0 , we have (6.36) u (z − x0 ) − u (z − x) ≤ u (z − x0 ) − u (z − x) But clearly, (z − x0 ) − (z − x) = (z − x0 ) − (z − x) , which combined with the fact that z > z (since k0 > k) and x > x0 by hypothesis, and that u is strictly concave and increasing implies u (z − x0 ) − u (z − x) > u (z − x0 ) − u (z − x) , contradicting (6.36) This establishes that s (k) must be nondecreasing everywhere Ô In addition, Assumption (the Inada conditions) imply that savings and consumption levels have to be interior, thus Theorem 6.6 applies and immediately establishes: Proposition 6.2 Given Assumptions 1, and 3’, the value function V (k) defined above is differentiable Consequently, from Theorem 6.10, we can look at the Euler equations The Euler equation from (6.35) takes the simple form: u0 (c) = βV (s) where s denotes the next date’s capital stock Applying the envelope condition, we have V (k) = [f (k) + (1 − δ)] u0 (c) Consequently, we have the familiar condition (6.37) u0 (c (t)) = β [f (k (t + 1)) + (1 − δ)] u0 (c (t + 1)) 294