When firms are deciding in the short-run on how much output to produce in a given period they implicitly or explicitly equate marginal cost to marginal benefits.  In the short-run different types of taxation schemes can change the optimum production levels while other forms of taxation do not change the output decisions of firms.  In the long-run when all inputs are flexible, Lump Sum Taxes and Proportional Taxes on Profits  reduces the number of firms in the market, but Lump-sum taxes on profits and a proportional tax on profits do not affect the firms output decision in the short-run.  A Tax on Output or an Input Tax on a variable input does reduce the output that the representative firm decides to produce in the short-run as well as increasing the likelihood of a firm to exit this industry.  The mathematical proof for these statements are outlined using partial derivatives and multivariable calculus in conjunction with the microeconomic theory of profit maximizing firms.