Introduction to Modern Economic Growth is piecewise linear with first a flat portion and then an increasing portion Let ¯]), but is the space of such functions be S 00 , which is another subspace of C ([0, a not closed Nevertheless, now the second part of Theorem 6.8 applies, since starting with any nondecreasing function V (a), T V (a) will also be a piecewise linear function starting with a flat portion Therefore, the theorem implies that the unique fixed point, V (a), must have this property too The digression above used Theorem 6.8 to argue that V (a) would take a piecewise linear form In fact, in this case, this property can also be deduced directly from (6.45), since V (a) is a maximum of two functions, one of them flat and the other one linear Therefore V (a) must be piecewise linear, with first a flat portion Our next task is to determine the optimal policy using the recursive formulation of Problem A2 But the fact that V (a) is linear (and strictly increasing) after a flat portion immediately tells us that the optimal policy will take a cutoff rule, meaning that there will exist a cutoff technology level R such that all techniques above R are accepted and production starts, while those a < R are turned down and the entrepreneur continues to search This cutoff rule property follows because V (a) is strictly increasing after some level, thus if some technology a0 is accepted, all technologies with a > a0 will also be accepted Moreover, this cutoff rule must satisfy the following equation Z a¯ R = βV (a) dH (a) , (6.46) 1−β so that the individual is just indifferent between accepting the technology a = R and waiting for one more period Next we also have that since a < R are turned down, for all a < R V (a) = β Z a ¯ V (a) dH (a) = and for all a ≥ R, we have R , 1−β a 1−β 303 V (a) =