Introduction to Modern Economic Growth continuous From the definition (7.31), at all t where yˆ (t) is continuous, V (t, x) will also be differentiable in t Moreover, an envelope theorem type argument also implies that when yˆ (t) is continuous, V (t, x) should also be differentiable in x (though the exact conditions to ensure differentiability in x are somewhat involved) Second, (7.37) is a partial differential equation, since it features the derivative of V with respect to both time and the state variable x Third, this partial differential equation also has a similarity to the Euler equation derived in the context of discrete time dynamic programming In particular, the simplest Euler equation (6.22) required the current gain from increasing the control variable to be equal to the discounted loss of value The current equation has a similar interpretation, with the first term corresponding to the current gain and the last term to the potential discounted loss of value The second term results from the fact that the maximized value can also change over time Since in Theorem 7.9 there is no boundary condition similar to x (t1 ) = x1 , we may expect that there should be a transversality condition similar to the condition that λ (t1 ) = in Theorem 7.1 One might be tempted to impose a transversality condition of the form (7.38) lim λ (t) = 0, t→∞ which would be generalizing the condition that λ (t1 ) = in Theorem 7.1 But this is not in general the case We will see an example where this does not apply soon A milder transversality condition of the form (7.39) lim H (t, x, y, λ) = t→∞ always applies, but is not easy to check Stronger transversality conditions apply when we put more structure on the problem We will discuss these issues in Section 7.4 below Before presenting these results, there are immediate generalizations of the sufficiency theorems to this case Theorem 7.11 (Mangasarian Sufficient Conditions for Infinite Horizon) Consider the problem of maximizing (7.28) subject to (7.29) and (7.30), with f and g continuously differentiable Define H (t, x, y, λ) as in (7.12), and suppose that a piecewise continuous solution yˆ (t) and the corresponding path of state variable 336