Assume that there are 2 firms producing steel. There is a negative externality due to production causing pollution. Firm 1's output is q1 and firm 2's is q2. Assume that the market price for steel is Ps = 1. Now consider two scenarios :
Scenario 1 : Assume that firm 1's cost function is \({C}_1\left(\mathrm{q}_1\right)=\mathrm{q}_1^2\) and firm 2 's cost function is C2(q2, q1) = (q2 + 0.75 q1)2. In short, firm 1's production is not affected by firm 2 but firm 2's production is affected by firm 1, i.e., Firms 1's operation causes firm 2's costs to rise.
Scenario 2 : Assume that negative externality works both ways i.e. both the firms face the adverse impact. So assume C1(q1, q2) = (q1 + 0.75 q2)2 and C2(q1, q2) = (q2 + 0.75 q1)2, i.e., Firms 1's operation causes firm 2's costs to rise and vice versa.