Introduction to Modern Economic Growth where (6.44) V = Z a ¯ V (a) dH (a) is the continuation value of not producing at the available techniques The expression in (6.43) follows from the fact that the individual will choose whichever option, starting production or continuing to search, gives him higher utility That the value of continuing to search is given by (6.44) follows by definition At the next date, the individual will have value V (a) as given by (6.43) when he draws a from the distribution H (a), and thus integrating over this expression gives V The integral is written as a Lebesgue integral, since we have not assumed that H (a) has a continuous density a slight digression* It is also useful to note that we can directly apply the techniques developed in Section 6.3 to the current problem For this, combine the two previous equations and write (6.45) ½ a0 ,β V (a ) = max 1−β = T V (a0 ) , Z a ¯ ¾ V (a) dH (a) , where the second line defines the mapping T Now (6.45) is in a form to which we can apply the above theorems Blackwell’s sufficiency theorem (Theorem 6.9) applies directly and implies that T is a contraction since it is monotonic and satisfies discounting Next, let V ∈ C ([0, a ¯]), i.e., the set of real-valued continuous (hence bounded) functions defined over the set [0, a ¯], which is a complete metric space with the sup norm Then the Contraction Mapping Theorem, Theorem 6.7, immediately implies that a unique value function V (a) exists in this space Thus the dynamic programming formulation of the sequential search problem immediately leads to the existence of an optimal solution (and thus optimal strategies, which will be characterized below) Moreover, Theorem 6.8 also applies by taking S to be the space of nondecreasing continuous functions over [0, a¯], which is a closed subspace of C ([0, a¯]) Therefore, V (a) is nondecreasing In fact, using Theorem 6.8 we could also prove that V (a) 302