The Rocky Mashed Potato Factory produces output at costs

$($marginal costs

$),$ where

is the quantity of mashed potatoes produced, in tons. In addition, $2$ units of emissions are produced for each ton of mashed potatoes

$.$ Pollution damage is $$2$ for each unit of emissions, which leads the government to charge $$2$ per unit of emissions as a Pigovian fee. The firm's output sells competitively for