Introduction to Modern Economic Growth Since by definition of the admissible pairs (x (t) , y (t)) and (ˆ x (t) , yˆ (t)), we have · xˆ (t) = g (t, xˆ (t) , yˆ (t)) and x˙ (t) = g (t, x (t) , y (t)), (7.20) implies that W (x (t) , y (t)) ≤ W (ˆ x (t) , yˆ (t)) for any admissible pair (x (t) , y (t)), establishing the first part of the theorem If M is strictly concave in x, then the inequality in (7.17) is strict, and therefore the same argument establishes W (x (t) , y (t)) < W (ˆ x (t) , yˆ (t)), and no other xˆ (t) could achieve the same value, establishing the second part Ô Theorems 7.4 and 7.5 play an important role in the applications of optimal control They ensure that a pair (ˆ x (t) , yˆ (t)) that satisfies the necessary conditions specified in Theorem 7.3 and the sufficiency conditions in either Theorem 7.4 or Theorem 7.5 is indeed an optimal solution This is important, since without Theorem 7.4 and Theorem 7.5, Theorem 7.3 does not tell us that there exists an interior continuous solution, thus an admissible pair that satisfies the conditions of Theorem 7.3 may not constitute an optimal solution Unfortunately, however, both Theorem 7.4 and Theorem 7.5 are not straightforward to check since neither concavity nor convexity of the g (·) function would guarantee the concavity of the Hamiltonian unless we know something about the sign of the costate variable λ (t) Nevertheless, in many economically interesting situations, we can ascertain that the costate variable λ (t) is everywhere positive For example, a sufficient (but not necessary) condition for this would be fx (t, xˆ (t) , yˆ (t) , λ (t)) > (see Exercise 7.9) Below we will see that λ (t) is related to the value of relaxing the constraint on the maximization problems, which also gives us another way of ascertaining that it is positive (or negative depending on the problem) Once we know that λ (t) is positive, checking Theorem 7.4 is straightforward, especially when f and g are concave functions 7.2.2 Generalizations The above theorems can be immediately generalized to the case in which the state variable and the controls are vectors rather than scalars, and also to the case in which there are other constraints The constrained case requires constraint qualification conditions as in the standard finite-dimensional optimization case (see, e.g., Simon and Blume, 1994) These are slightly more messy 328