Introduction to Modern Economic Growth Theorem 7.7 (Mangasarian Sufficient Conditions) Consider the problem of maximizing (7.21) subject to (7.22) and (7.23), with f and g continuously differentiable Define H (t, x, y, λ) as in (7.24), and suppose that an interior continuous solution y ˆ (t) ∈IntY (t) and the corresponding path of state variable x ˆ (t) satisfy (7.25)-(7.27) Suppose also that for the resulting costate variable λ (t), H (t, x, y, λ) ˆ (t) and the corresponding x ˆ (t) is jointly concave in (x, y) for all t ∈ [0, t1 ], then y achieves a global maximum of (7.21) Moreover, if H (t, x, y, λ) is strictly jointly concave, then the pair (ˆ x (t) , y ˆ (t)) achieves the unique global maximum of (7.21) Theorem 7.8 (Arrow Sufficient Conditions) Consider the problem of maximizing (7.21) subject to (7.22) and (7.23), with f and g continuously differ- entiable Define H (t, x, y, λ) as in (7.24), and suppose that an interior continuous solution y ˆ (t) ∈IntY (t) and the corresponding path of state variable x ˆ (t) satisfy (7.25)-(7.27) Suppose also that for the resulting costate variable λ (t), define ˆ, λ) If M (t, x, λ) is concave in x for all t ∈ [0, t1 ], then M (t, x, λ) ≡ H (t, x, y y ˆ (t) and the corresponding x ˆ (t) achieve a global maximum of (7.21) Moreover, if M (t, x, λ) is strictly concave in x, then the pair (ˆ x (t) , y ˆ (t)) achieves the unique global maximum of (7.21) The proofs of both of these Theorems are similar to that of Theorem 7.5 and are left to the reader 7.2.3 Limitations The limitations of what we have done so far are obvious First, we have assumed that a continuous and interior solution to the optimal control problem exists Second, and equally important for our purposes, we have so far looked at the finite horizon case, whereas analysis of growth models requires us to solve infinite horizon problems To deal with both of these issues, we need to look at the more modern theory of optimal control This is done in the next section 7.3 Infinite-Horizon Optimal Control The results presented so far are most useful in developing an intuition for how dynamic optimization in continuous time works While a number of problems in economics require finite-horizon optimal control, most economic problems–including 330