Online Appendix Time to Come Clean? Disclosure and Inspection Policies for Green Production

In this section we derive the long-run average performance measures I, R, B, and (de…ned in §3.8) assuming that the …rm employs the threshold disclosure policy described in Lemma 3 in response to the regulator’s periodic inspections. We assume that the disclosure window has an arbitrary size equal to s 2 [0; ]. Recall from §3.4 that last-minute disclosure that occurs under the threshold disclosure policy is regarded as detection. In the proofs below, U and D refer to the exponential i.i.d. random variables with means 1= and 1= representing the duration of each compliance and noncompliance episode, respectively. The most convenient unit of analysis in evaluating the performance measures under periodic inspections is an inspection cycle. An inspection cycle is the time interval that contains at most one inspection performed, beginning in the compliance state and concluding with three possible endings that leave the …rm back in the compliance state: (i) the cycle ends time units after the start when the regulator arrives for an inspection and …nds the …rm in compliance; (ii) the cycle ends when compliance is restored following last-minute disclosure (detection) at time units since the start; (iii) the cycle ends when compliance is restored following the …rm’s disclosure at the onset of noncompliance. Note that these are the only possible endings of an inspection cycle. Because of memorylessness of the exponential compliance and noncompliance durations, the start and end times of an inspection cycle mark regeneration epochs. Therefore, an inspection cycle forms a renewal (Heyman and Sobel 1982, p. 184; Tijms 2003, p. 40). Note also that an inspection cycle may end at, before, or after time units since the start, because it takes a random amount of time for compliance to be restored after detection or disclosure. Thus the length of an inspection cycle— denoted by X— is random, and it is to be distinguished from the constant inspection interval T = . Depending on whether the …rm is in compliance at the beginning of the disclosure window (at time s since the cycle start), under the threshold disclosure policy the three outcomes above are further divided into …ve cases, referred to as Case 1a, Case 1b, etc. See Figure A.1 that illustrates these cases. 1. The …rm is in compliance at time s since the cycle start. Moreover: (a) compliance lasts until or after time , at which the current inspection cycle ends with an inspection (but no detection); (b) noncompliance starts before time and at that moment the …rm discloses the state, and subsequently the current inspection cycle ends when compliance is restored. 2. The …rm is in noncompliance at time s since the cycle start. Moreover: (a) noncompliance lasts until or after time , at which detection occurs and subsequently the current inspection cycle ends when compliance is restored; (b) compliance is restored before time and it lasts until or after time , at which the current inspection cycle ends with an inspection (but no detection); (c) compliance is restored before time but it is followed by a transition to noncompliance before time which is disclosed by the …rm, and subsequently the current inspection cycle ends when compliance is restored again. Note that in all cases at most one new noncompliance episode occurs within the disclosure window because, under the threshold disclosure policy, the …rm always discloses the …rst of such occurrences (see Cases 1b and 2c in Figure A.1) and subsequently the current inspection cycle ends as soon as compliance is restored. As a result, the …ve cases described above form a complete list of categories