A Simple Model of Technolof^ical Research 153 



product is reached. The optimal rate at which this research is to be 

 conducted can be determined in the following way. Define 



V{T) = -^yiT)^'£cx' i2E{Ms)\-f;^a' cp)-^a' c^[Ay(tXy(t)] 

 t=T+l s=T+\ t^j t=T 



(8.23) 



where Tis the first point in time at which a? = 1 is optimal, [note 

 that>'(0= >'(0)]. From Ton, A7 will proceed at a rate satisfying 

 8.22. The problem in the present case is to increase 7 (0)to 7(71 = 

 rq(l) over the period[0,r]while maximizing the present value of 

 the system. Formally: maximize ViO) with respect to Tand A7 (t) 

 (r=0, 1,2, ...)in 



T-l T-l 



V(0) = V(T)-^(x'c^[Ay(t),y{t)] +X)a'.v(0) (8.24a) 

 t=0 t=0 



subject to 



^l(l) = ^ - 7[t(0) +1; A7(0]. (8.24b) 



To obtain the solution to this problem, maximize 8.24a subject 

 to 8.24b for each value of Tand choose as optimal Tthe value that 

 will maximize the value of the system. 



To maximize 8.24a subject to 8.24b, write the Lagrangian 



T-l T-l 



L=V{T)-J2oi'c2[Ayit),yit)] + ui{rc^(l)-y{0) -J^ ^yiO\ (8.25) 

 r=0 ^=0 



The condition for optimal A7(r), 0<t<T, is 



