Modify example 9.2 by assuming there is only one firm. Assume that both and 1 = can take two values: L ,

Modify example 9.2 by assuming there is only one firm. Assume that both η and ν₁=ν can take two values: ν_L, ν_H, and η_L, η_H. Denote p_LL=Pr(η=η_L, ν=ν_L) as the probablity that both parameters take their low values, and similarly define p_LH, p_HL, and p_HH. Define abatement costs by C(l)=c·l²/2 and damages by D(A)=δ⋅A²/2, where A=η(e^max−νl) is the ambient pollution level.

(a) Calculate the optimal level for abatement input and the optimal ambient emission tax rate.

(b) Consider the following special cases and interpret: (i) p_LH=p_HL=0, (ii) p_LL=p_HH=0, and (iii) Pr(η
,ν)=Pr(η)×Pr(ν)=p_η•p_ν. 

Example 9.2

Example 9.2 (Numerical/Analytical) Suppose there are two firms with abatement cost functions C(1)=cl/2 and emissions e-(e-vlj), where e; is the maximum (unregulated) emission level, and and determine the effectiveness of abatement 1/1 activities 1 and 12, respectively. Denote E=+ as the aggregate unregulated emission level, and let ambient pollution be given by A=n (e+)=n (E-v-22), where n represents the aggregate uncertainty. We consider two damage functions, given by D(4)=8.4 and D(4)=84/2, which represent constant and increasing marginal damages, respectively. The random variables (,2,n) are jointly distributed with independence as a special case, and we have d./de-n and defalvj. For the case of constant marginal damage we have D'(A)=8. According to Eq. (9.8), the first-order condition with constant marginal damage is cl = SEV (nv) = nv, +SCOV (n. v), (9.10) where = EV (1) and V = EV(v) are the means of the random variables, and the constant marginal damage is deterministic. From this expression, it is apparent that if n and v; are positively correlated, so that a large positive shock to ambient pollution is likely to accompany a high