Time: The Maximum Principle

Abstract As in the case of uncertainty, the study of optimization over time requires no new general principles. The variables to be chosen will pertain to different dates, but we can always stack them together in one large vector x, and the general problem remains one of maximizing the value of a function F(x) subject to a vector inequality constraint G(x) ::; c. At the time when the decision is taken, the knowledge of future tastes and technology may be very imperfect. But this simply requires us to capture the uncertainties and attitudes to risk in the functions F and G. As time unfolds, there may be opportunities to rethink the current decision and revise the plan. But this merely requires us to recognize such future revisions in our current decision. Such a consideration may lead us to take more flexible decisions now so as to allow later choices in the light of better knowledge. But it may also mean making commitments now to foreclose certain future avenues that tomorrow’s preferences would tempt us into, when today’s preferences dictate otherwise; here today’s decision involves playing a game of strategy against one’s own future self. Once all such considerations are incorporated in the objective function and the constraints, the formal theory of the previous chapters continues to apply.