Consider again the case of one firm as in exercise 9.5 , but now assume that the firm can observe its parameter ν.

(a) Determine the firm’s efficient choice of l for each value ν_L and ν_H.

(b) Determine the optimal tax rate for this case.

Data from exercise 9.5

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 probability 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.

Example 9.2

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Suppose there are two firms with abatement cost functions C(1)=2/2 and emissions e,=(e-v), where , is the maximum (unregulated) emission level, and v and determine the effectiveness of abatement activities 1 and 2, respectively. Denote =+ as the aggregate unregulated emission level, and let ambient pollution be given by A=n (e+e)=n (nh-2), where represents the aggregate uncertainty. We consider two damage functions, given by D(4)=& and D(-4)=5.4/2, which represent constant and increasing marginal damages, respectively. The random variables (1,12.) are jointly distributed with independence as a special case, and we have dd/den and dejl-vj For the case of constant marginal damage we have D'(4)=5. According to Eq. (9.8), the first-order condition with constant marginal damage is cl, = SEV (nv) = 8v, +SCOV (n. v,), (9.10)