Figure 4. The final 
ratio plotted as a function of 
under the constraint 
, for (a) 
(b) 
(c) 
and (d) 
.
The constant is chosen as 
so that three curves are shown for each each value of 
. Each curve terminates at 
. Panel (b) is projected from Fig. 3(a). Note that for all 
values, the most efficient allocation of resources is attained for the intermediate 
. In particular, for 
the final 
ratio is attained at 
, 
and 
as shown in panel (b). Also note the emergence of maxima in panels (b) and (c).