Introduction to Modern Economic Growth W (x (t) , y (t)) can be increased and thus the pair (ˆ x (t) , yˆ (t)) could not be an optimal solution Consequently, optimality requires that W (0) ≡ for all η (t) (7.8) Recall that the expression for W (0) applies for any continuously differentiable λ (t) function Clearly, not all such functions λ (·) will play the role of a costate variable Instead, as it is the case with Lagrange multipliers, the function λ (·) has to be chosen appropriately, and in this case, it must satisfy fy (t, xˆ (t) , yˆ (t)) + λ (t) gy (t, xˆ (t) , yˆ (t)) ≡ for all t ∈ [0, t1 ] (7.9) This immediately implies that Z t1 [fy (t, xˆ (t) , yˆ (t)) + λ (t) gy (t, xˆ (t) , yˆ (t))] η (t) dt = for all η (t) Since η (t) is arbitrary, this implies that xε (t, 0) is also arbitrary Thus the condition in (7.8) can hold only if the first and the third terms are also (individually) equal to zero The first term, [fx (t, xˆ (t) , yˆ (t)) + λ (t) gx (t, xˆ (t) , yˆ (t)) + λ˙ (t)], will be equal to zero for all xε (t, 0), if and only if (7.10) λ˙ (t) = − [fx (t, xˆ (t) , yˆ (t)) + λ (t) gx (t, xˆ (t) , yˆ (t))] , while the third term will be equal to zero for all values of xε (t1 , 0), if and only if λ (t1 ) = The last two steps are further elaborated in Exercise 7.1 We have therefore obtained the result that the necessary conditions for an interior continuous solution to the problem of maximizing (7.1) subject to (7.2) and (7.3) are such that there should exist a continuously differentiable function λ (·) that satisfies (7.9), (7.10) and λ (t1 ) = The condition that λ (t1 ) = is the transversality condition of continuous time optimization problems, which is naturally related to the transversality condition we encountered in the previous chapter Intuitively, this condition captures the fact that after the planning horizon, there is no value to having more x This derivation, which builds on the standard arguments of calculus of variations, has therefore established the following theorem.1 1Below we present a more rigorous proof of Theorem 7.9, which generalizes the results in Theorem 7.2 in a number of dimensions 319