A firm has strictly concave and increasing production function. The output q=f(z1,z2) is sold at price p>0 while w1 and w2 are the costs of the inputs.

Production activity generates also an externalities quantified by the function g(z1,z2) that is convex and increasing in the inputs levels. The government has to decide whether (1) to introduce a constraint on the level of damage k>0 such that g(z1,z2)<=k or (2) directly impose the damage effect into the social optimization problem with the following social evaluation: P(z1,z2)-ag(z1,z2) where P(z1,z2) denotes the firm profit and a>=0 is an exogenous social weight that is attributed to the environmental damage.

Are the two approaches (1),(2) the same?