A monopolist produces a good x and pollution as a

Question

A monopolist produces a good x and pollution as a by-product. The production of x units of the good generates x gross emission units (i.e. E_G = x). Assume the existence of an abatement technology that can reduce emissions by an arbitrary amount 0 ≤ ER_x (≤ E_G). Hence, net emissions are given by E = E_G - ER_x.

Environmental damages are assume to be given by D(E) = 5E². The marginal private cost function and the inverse demand function are assumed to be given by MPC(x) = 10 and P(x) = 110 - 5x. Marginal abatement cost by application of the abatement technology are given by MAC(ER_x) = 5ER_x.

• Determine the socially optimal gross and net emission level E_G^** and E^**
• Determine the unregulated equilibrium emission level, E_M
• To induce the socially optimal emission level E^**, the government considers charging an emission tax. Determine the appropriate tax level t_M and the corresponding gross emission and abatement level of monopolist.