Introduction to Modern Economic Growth which denotes the action of the agent at time t, which is either to produce with the current technique he has discovered, at , or to choose q (t) =“search” and spend that period searching for or researching a new technique Let Pt be the set of functions from At into at ∪ {search}, and P ∞ the set of infinite sequences of such functions The most general way of expressing the problem of the individual would be as follows Let E be the expectations operator Then the individual’s problem is max ∞ {q(t)}t=0 ∈P E ∞ ∞ X β t c (t) t=0 subject to c (t) = if q (t) =“search” and c (t) = a0 if q (t) = a0 Naturally, written in this way, the problem looks complicated, even daunting In fact, the point of writing it in this way is to show that in certain classes of models, while the sequence problem will be complicated, the dynamic programming formulation will be quite tractable To demonstrate this, we now write this optimization problem recursively using dynamic programming techniques First, it is clear that we can discard all of the techniques that the individual has sampled except the one with the highest value Therefore, we can simply denote the value of the agent when the technique he has just sampled is a ∈ [0, a ¯] by V (a) Moreover, let us suppose that once the individual starts producing at some technique a0 , he will continue to so forever, i.e., he will not go back to searching again This is a natural conjecture, since the problem is stationary If the individual is willing to accept production at technique a0 rather than searching more at time t, he would also so at time t + 1, etc (see Exercise 6.17) In that case, if the individual accepts production at some technique a0 at date t, he will consume c (s) = a0 for all s ≥ t, thus obtain a value function of the form V accept (a0 ) = a0 1−β Therefore, we can write (6.43) © ª V (a0 ) = max V accept (a0 ) , V ẵ ắ a = max , βV , 1−β 301