Consider an EOQ model with $=600,K=160,h=4,c=5,$widehat$(K)=10,$widehat$(h)=2,$ and widehat$(c)=1.$

$($a$)$ Suppose that the environmental regulations involve only a strict carbon cap. Find the cap yielding the maximum benefit from operational adjustment.

$($b$)$ Now suppose that the environmental regulations impose a cap$-$and$-$price market setting with $C=800,t=8,$ and $t^{\text{'}}=4.$ Plot the total cost $($including setup, holding, and purchasing costs as well as penalties and rewards$)$ versus the order quantity $($that can take values from the set $\{5,10,$dots,$300\}).$ Find the optimal order quantity.

$($c$)$ Compare the market setting in part $($a$)($assuming the cap is set equal to your answer$)$ tothe market setting in part $($b$).$ Which setting is desirable from a profit$-$maximizing firm'sperspective? Which setting is desirable from an environmental perspective? Why?