Exercise 5.13. (1) Suppose that utility of individual i given by P t=0 t ui xi

Exercise 5.13. (1) Suppose that utility of individual i given by P∞ t=0 βt ui ¡ xi (t) ¢ , where xi (t) ∈ X ⊂ RK + , ui is continuous, X is compact, and β < 1. Show that the hypothesis that for any x, x0 ∈ Xi with ui (x) > ui (x0 ), there exists T¯ such that ui (x [T]) > ui (x0 [T]) for all T ≥ T¯ in Theorem 5.7 is satisfied. (2) Suppose that the production structure is given by a neoclassical production function, where the production vector at time t is only a function of inputs at time t and capital stock chosen at time t − 1, and that higher capital so contributes to greater production and there is free disposal. Then show that the second hypothesis in Theorem 5.7 that for each y ∈ Y f , there exists T˜ such that y [T] ∈ Y f for all T ≥ T˜ is satisfied.