(c) Assume an infinite horizon, continuous time and certainty. Furthermore, assume an additively separable utility function in consumption C and pollution stock S so that, U(C,S) = u(C) + v(S), where uc > 0; Ucc < 0; vs < 0; Vss > 0. Note that the first derivative, and the second subscript denotes the second derivative. The evolution of the pollution stock S over time is a function of consumption, decay rate of the stock of pollution S and abatement through the function, g(S) with gs > 0 and gss < 0. Time subscripts are ignored for ease of notation. Show that in the case of a stock pollutant, the marginal utility of consumption should equal the present value of disutility associated with the pollution stock. Interpret the condition. (d) Suppose there is a single firm whose profits are determined by: B(E) = 10E 1 E2 where E denotes emissions and the expected damages associated with E is given by: ED(E)=√²+(1-x) where 0 < < ẞ. Suppose that the true level of damages is actually equal to Ꮎ D(E) - E2. Page 4 of 5 ES30031 Show that under uncertainty the policymaker is indifferent between using a Pigouvian tax or a standard to control pollution. Illustrate your answer with a diagram. Section 1 Answer all questions. Show all your workings. (a) Suppose there are two firms 1 and 2, whose abatement costs are given by c₁(e₁) and c₂ (e₂), where e denotes emissions and subscripts denote the firm. We assume that c{(e;) < 0; c{' (e;) > 0 for i = 1,2 and for any level of emission e we have c₁'(e) # c₂'(e). Furthermore, assume the two firms make different contributions towards pollution concentration in a nearby river captured by the transfer coefficients ε₁ and 2 such that for any level of emission e we have 2(e) +2 The regulator does not know the resulting C₁'(e) Τι environmental damages. Using an analytical approach explain carefully how the regulator may limit the concentration of pollution using (i) a Pigouvian tax scheme and (ii) uniform emissions standards. Discuss the cost-effectiveness of both approaches to control pollution. [200 marks] (b) "Whether the regulator sells or gives away tradeable emission permits free of charge, the quantities of emissions produced by firms are the same." Assume that there are n identical profit-maximising firms where profit for each firm is given by π(e) with л'(e) > 0; π"(e) < 0 and e denotes emissions. Individual emissions summed over all firms gives Ę which generates environmental damages D(E). Show that the regulator achieves the optimal level of total pollution through a tradeable emission permit scheme, where the permits are distributed according to the following cases: Case (i) the firm purchases all permits; Case (ii) the firm receives all permits free; and Page 3 of 5 ES30031 Case (iii) the firm purchases a portion of its permits and receives the remainder free of charge.