You own a factory that produces shoes but also emits sulfur dioxide (SO2) into the environment. The total monetary damages caused by the emissions of SO2 from your factory are given by:
De = bq + 2cq + d
Where q is the amount of SO2 emissions from your factory and b, c, and d are positive parameters. The damage function above shows that the total damage to society increases as your SO2 emissions increase.
You have at your disposal a technology that can abate (reduce) the SO2 emissions from your factory. However, abating emissions is costly: the more you abate, the more it will cost you. The total abatement cost is given by the function:
Ra = ka + 2la + m
Where a is the amount of SO2 abated and k, l, and m are positive parameters.
Your current (unabated) level of SO2 emissions is qo. Abatement is related to emissions by the following equation:
q = qo - a
Your emissions are equal to qo when you do not abate any units of SO2. Your emissions decrease as you increase abatement of SO2.
Anum is an economist who wants to find the optimal level of emissions - the level of emissions that minimize total costs (the sum of total damage and total abatement cost).