Introduction to Modern Economic Growth for all t ∈ [0, t1 ] · For notational simplicity, in equation (7.15), we write x˙ (t) instead of xˆ (t) (= ∂ xˆ (t) /∂t) The latter notation is rather cumbersome, and we will refrain from using it as long as the context makes it clear that x˙ (t) stands for this expression Theorem 7.3 is a simplified version of the celebrated Maximum Principle of Pontryagin The more general version of this Maximum Principle will be given below For now, a couple of features are worth noting: (1) As in the usual constrained maximization problems, we find the optimal solution by looking jointly for a set of “multipliers” λ (t) and the optimal path of the control and state variables, yˆ (t) and xˆ (t) Here the multipliers are referred to as the costate variables (2) Again as with the Lagrange multipliers in the usual constrained maximization problems, the costate variable λ (t) is informative about the value of relaxing the constraint (at time t) In particular, we will see that λ (t) is the value of an infinitesimal increase in x (t) at time t (3) With this interpretation, it makes sense that λ (t1 ) = is part of the necessary conditions After the planning horizon, there is no value to having more x This is therefore the finite-horizon equivalent of the transversality condition we encountered in the previous section While Theorem 7.3 gives necessary conditions, as in regular optimization problems, these may not be sufficient First, these conditions may correspond to stationary points rather than maxima Second, they may identify a local rather than a global maximum Sufficiency is again guaranteed by imposing concavity The following theorem, first proved by Mangasarian, shows that concavity of the Hamiltonian ensures that conditions (7.13)-(7.15) are not only necessary but also sufficient for a maximum Theorem 7.4 (Mangasarian Sufficient Conditions) Consider the problem of maximizing (7.1) subject to (7.2) and (7.3), with f and g continuously differentiable Define H (t, x, y, λ) as in (7.12), and suppose that an interior continuous solution yˆ (t) ∈IntY (t) and the corresponding path of state variable xˆ (t) satisfy (7.13)-(7.15) Suppose also that given the resulting costate variable λ (t), 325