A Simple Model of Technological Research 151 



The system now has two control variables, n and A7 . The new 

 possibility of modifying the variance will not affect the optimal 

 rate of experimentation. Here, too, optimal number of trials is « = 

 lifq(l)<-i- J and zero otherwise (see equation 8.18). 



Two cases can now be distinguished. The first, assume that ci(l) 

 < y/r and search takes place with n= \. The problem of optimal 

 variance modification can be stated formally as (let t = 0): 



I t =1 s=l t=0 t=0 ) 



(8.20) 



where the index /emphasizes thatA.v, A7 , and 7 may differ 

 from one stage (time period) to another. 



Optimization of equation 8.20 is done by solving a set of simul- 

 taneous equations whose solution will determine Ajit) (/= 0, 1, 

 2, . . .). It will be convenient to look at the equation for / = 0. 

 Rewriting 8.20, and isolating the expressions which are functions 

 of A7 (0): 



max MO) (1^+2.) _|;,.,^^^^(,),^( J (8.2I) 



The first term in equation 8.21 is obtained by substituting in 

 the second term in equation 8.20 



t-\ 



E\Ay{t)\ = 7(0= 7(0) +X;^t(5) 



The condition for optimal A7 (0) is 



dC2 _ 1 + 2r Y> t dC2 (8.22) 



aA7(0) ~ r2 ~^^ dy(t) 



The parameter 7 is nondecreasing in time; therefore, by 8.19 

 both the left hand-side and the second term on the right of 8.22 



