Total pollution control cost functions for two firms are given respectively, as

TC$\_1=40+24($q$\_1)^2$

TC$\_2=40+8($q$\_2)^2$ 

where q$\_1$ and q$\_2$ are the amounts of emission reduced by the firms. With no control, each firm omits $40$ units of emission. Assuming no significant transaction costs

Explain briefly cost$-$effective allocation of control responsibility. Find how many units of emissions each firm will reduce under such an allocation