The Optimal pricing of Pollution When Enforcement is Costly

Stranlund, Chavez, Villena (2008)

1 Section 4: The robustness of the optimality of full compliance

Can a regulator desing a tax/enforcement policy with a non-constant expected marginal penalty that leads to a lower expected enforcement costs than inducing full compliance with a constant expected marginal penalty?

Denote a tax enforcment policy for fimr i as p_{i}, consisting of a tax and an expected penalty function, under which the firm optimally chooses its emissions q(p_{i}, x_{i}, ε_{i}) and emissions report r(p_{i}, x_{i}, ε_{i}). The regulator cannot observe these, but as it knows the firm´ s decision problem, it can form conditional expectations of these variables. Let \overline{q}(p_{i}) = E [q(p_{i}, x_{i}, ε_{i})] and \overline{r}(p_{i}) = E [r(p_{i}, x_{i}, ε_{i})] .

The authors consider thre policy types that induce the same expected emissions from the firm. First: policy of Porposition 1: t_{i}^{c} = π_{i}^{c}ϕ_{i}. The superscript c indicates that the tax and the detection probability are chosen to induce full compliance, given the unit penalty ϕ_{i}.In short, this policy is denoted p_{i}^{c} and is composed of [t_{i}^{c}, π_{i}^{c}ϕ_{i} (q_{i} − r_{i})] . the regulator expectation of the firm´ s emissions under p_{i}^{c} is \overline{q}(p_{i}^{c}). Second policy: p_{i}^{q} = [t_{i}^{q}, π_{i} (q_{i}, r_{i}) f_{i} (q_{i} − r_{i})] . This is a policy with a non-constant unit penalty function and a detection probability based on actual emissions. These are not observed but estimated by observing a correlate. Policy three, instead, has the regulator calculating the expected emissions given the tax and the enforcement policy, when such a correlate does not exist. This is policy p\overline{q} ; i.

To simplify their analysis they assume that the functional forms of π_{i} and f_{i} are the same under p_{i}^{q} and p\overline{q} ; i and that π_{i} and f′_{i} are lienar functions. The latter implies that E [π (q(p_{i}, x_{i}, ε_{i}), r(p_{i}, x_{i}, ε_{i}))] = π [\overline{q}(p_{i}), \overline{r}(p_{i})] and E [f′ (q(p_{i}, x_{i}, ε_{i}) − r(p_{i}, x_{i}, ε_{i}))] = f′ [\overline{q}(p_{i}) − \overline{r}(p_{i})] .

What the author do in this section is to compare the detection probability under p_{i}^{c} to the expected detection probabilities under p_{i}^{q} and p\overline{q} ; i that induce the same expected emissions from the firm. Clearly, the detection probabilities can be adjusted by adjusting the marginal penalties. To avoid this problem the authors assume that ϕ_{i} = f′\overline{q}(p_{i}) − \overline{r}(p_{i}) for both p_{i}^{q} and p\overline{q} ; i.

Proposition 1 Suppose that the regulator whishes to induce a fixed level of expected emissions from a firm. Relative to inducing the firm’s compliance with a constant expected marginal penalty, a regulator can reduce its expected monitoring costs of the firm if and only if the probability of detection is a strictly increasing function of the firm´ s actual emissions and the regulator implements an enforcement strategy that it expects will result in the firm´ s noncompliance.

Proof. Appendix 2 in the paper

Since inducing full compliance with a constant marginal expected penalty does not incur in sanctioning costs, implementing an enforcement strategy that results in noncomplaince can only reduce total expected enforcement costs if significantly reduces detection probabilities. Since this can only happen if the detection probabilities are increasing in the firms ´ actual emissions, the followinc Corollary follows inmediately.

Corollary 2 Any tax/enforcement policy that the regulator expects will result in firms ´ noncompliance and that featires a combination of a strcitly convex penalty function, the firms ´ emissions report, and the regulator´ s expectations of the firms emissions will produce higher expected enforcement costs than a policy that motivates full compliance with constant expected marginal penalties.