Self-reporting and Market Structure

Many regulators utilize self-reporting, that is, wrongdoers reporting their own crimes to the authority, to enforce regulations in a variety of market contexts. This paper studies the eﬀectiveness of self-reporting within the context of an oligopoly. We identify two important consequences of implementing self- reporting (relative to no-reporting) for a welfare-maximizing regulator. First, if the regulator can control only the audit probability and ﬁne, then whether compliance rises or falls upon implementing self- reporting depends on the level of competition. Second, if the regulator can also control the market size, then the welfare-maximizing policy entails self-reporting but with more competition and lower compliance than under no-reporting.


INTRODUCTION
Self-reporting is the reporting of harmful or non-compliant behaviour by the wrongdoer to the enforcement authority. Many regulators utilize self-reporting to enforce their regulations (Innes 2000). The US Environmental Protection Agency (EPA) and the UK Environment Agency, for example, encourage firms to self-report environmental 'crimes' such as spills of oil or of untreated sewerage. Similarly, the US Department of Agriculture (USDA) and the Food and Drug Administration (FDA) have recently adopted self-reporting to regulate firms for compliance with food safety standards. 1 The essence of self-reporting is that offenders are incentivized to self-report violations in exchange for weaker sanctions, whereas those who do not self-report face stricter sanctions if they are caught. Accordingly, self-reporting is beneficial to a regulator because it need not audit those who confess to the crime, thereby saving on auditing costs. Indeed, formal analysis of self-reporting (Malik 1993;Kaplow and Shavell 1994) has shown that it can implement a given level of compliance at a lower cost than enforcement without self-reporting ('no-reporting'). 2 These qualities make self-reporting an attractive policy in an era of smaller budgets for regulators.
Previous research that studies the efficacy of self-reporting (Malik 1993;Kaplow and Shavell 1994;Innes 1999Innes , 2001) assumes a large number of (atomistic) agents or pricetaking firms. That is, this literature implicitly assumes that regulators are monitoring firms that operate in perfectly competitive industries. We assert that this is not realistic because most regulation occurs in imperfectly competitive markets. For example, the EPA and the FDA regulate, respectively, the oligopolistic energy and pharmaceutical industries, and the USDA regulates an agricultural industry that is less than perfectly competitive. 3 Despite this, little is known about how self-reporting interacts with market structure-especially whether the effectiveness or impact of self-reporting varies with market structure.
The goal of this paper is to study the effectiveness of self-reporting under nonperfectly-competitive markets-that is, monopolistically competitive and oligopolistic markets. The questions that we wish to address are as follows. First, how does the optimal self-reporting policy vary by industry structure? Second, under what market conditions will self-reporting yield a higher level of compliance? Finally, if a planner is unconstrained and can choose both the level of enforcement and the size of the market, will implementing self-reporting give rise to more or less competitive markets?
To study these questions we develop a 'Cournot-style' model in which oligopolistic firms generate a negative externality (e.g. environmental pollution) during production. Firms can reduce this harm by investing in abatement. However, since abatement is costly, in the absence of any regulation, firms do not abate. To incentivize abatement, firms are audited by a regulator who can choose either a self-reporting regime or a noreporting regime to fine firms for causing harm. Enforcement, through auditing firms with a given probability, is costly, and these costs may be either fixed or variable in nature. Under a fixed cost structure, enforcement cost does not vary with firm size, whereas under a variable cost structure it does.
Analysing this framework yields three important results concerning the value of regulating via self-reporting relative to no-reporting.
First, by utilizing self-reporting, a regulator can introduce welfare-enhancing regulations in markets where regulation would otherwise be inefficient to implement. To elaborate, if the regulations that are needed to correct a market failure (such as an external harm) are too costly, then it may be more efficient to permit the external harm rather than impose even costlier regulation. In these situations the laissez-faire policy of 'no regulation' can be optimal in a second-best sense, even though regulating the harm would be welfare-maximizing if regulation were costless (i.e. the first-best policy). Framed in our context, in the absence of self-reporting, there exists a threshold level of competition above which regulation becomes so costly that the regulator prefers the laissez-faire outcome over no-reporting. But if the regulator implements self-reporting, then for any level of competition, we show that the regulator prefers regulation (through self-reporting) to the laissez-faire policy that would be (second-best) optimal under noreporting. Thus by utilizing self-reporting, it is always optimal to correct the market failure, whereas under no-reporting it may not always be optimal to do so.
Second, if a regulator is constrained in that it cannot choose the level of competition, self-reporting need not yield a higher level of compliance (relative to no-reporting) even though it is always welfare-enhancing. Specifically, if enforcement costs are fixed with respect to firm size, then the socially optimal audit probability and level of compliance are higher (lower) under self-reporting than under no-reporting when the market is sufficiently competitive (concentrated). If, however, enforcement costs vary with firm size, then this result is reversed: the optimal audit probability and level of compliance are higher (lower) under self-reporting than under no-reporting when the market is sufficiently competitive (concentrated). Thus whether or not implementing self-reporting yields a higher level of compliance, relative to regulation through no-reporting, depends on the level of competition. Importantly, the nature of this effect is mediated by the structure of enforcement costs: fixed or variable.
Third, if the regulator is unconstrained in that it can choose both the level of enforcement and the number of firms, then the regulator always chooses to favour more competition and a lower level of compliance, relative to a no-reporting regime. Thus selfreporting allows for a larger, more competitive, market with larger consumer surplus, but at the expense of lower compliance and greater harm. This result, importantly, implies that there should be more partnership and joint enforcement between competition (antitrust) authorities, which determine market concentration levels, and other regulators such as the EPA.
It is insightful to relate these findings to the broader literature on self-reporting. The main benefit to self-reporting is that the regulator can save on enforcement costs Malik 1993). Innes (1999Innes ( , 2001) also identifies two further advantages to self-reporting. First, if firms can engage in clean-up activities, then under self-reporting firms always engage in clean-up, whereas under no-reporting firms cleanup only when they are caught. Since clean-up is welfare-improving, self-reporting improves welfare for this additional reason. Second, if firms can invest in costly detection avoidance, then under self-reporting there is less avoidance. Since avoidance is wasteful, self-reporting enhances welfare. While these studies agree that self-reporting is welfare-improving, only Innes (1999) 4 recognizes the possibility that implementing self-reporting can cause the level of compliance to fall. 5 Further, to date there has been no analysis of the exact conditions under which this will occur. Indeed, as Toffel and Short (2011) note in their recent review of the self-reporting literature: '[a]lthough this scholarship identifies some important dynamics that underlie self-reporting ... [its] connection to improving compliance or reducing harm is unclear'. By introducing market structure into this framework, we show that in the context of market regulation, this outcome is determined by the level of competition and other market characteristics.
Besides the literature on self-reporting, this paper contributes to the small but recently growing literature on the relationship between market structure and various public and private enforcement mechanisms. Dechenaux and Samuel (2019) find that whether a regulator prefers announced or surprise inspections (from a compliance maximization standpoint) depends on whether or not the market is sufficiently concentrated. In the context of private enforcement mechanisms, Daughety and Reinganum (2006) study the effectiveness of liability rules in various market contexts, and find that whether strict liability is preferred to negligence also depends on market competition. Our paper contributes to this literature by characterizing the welfaremaximizing policy; this has not so far been addressed, perhaps due to its complexity.
The rest of this paper is organized as follows. Section I sets up the basic model as well as the market equilibrium. Section II studies the welfare-maximization problem under self-reporting and no-reporting for a constrained regulator that cannot choose the level of competition. Section III conducts the same analysis for an unconstrained regulator, and Section IV concludes. All proofs are provided in the Appendix.

I. THE MODEL
Consider a market with N≥1 oligopolistic firms that each produce q i units of a product. The total market quantity is Q ¼ ∑ N i¼1 q i . The cost of producing each unit is c, and there are no fixed costs of producing q i . Firms sell products to consumers with quasilinear utility function Uðq,q 0 Þ¼q 0 þ uðqÞ, where good 0 is the numeraire, with p 0 ¼ 1. We assume that U has the Bowley form Maximizing this utility function with respect to a standard budget constraint yields the linear inverse demand P ¼ β À γQ: occurs only with probability ð1 À a i Þ. Abatement, however, costs kða i Þ per unit where we assume that kða i Þ¼ka 2 i =2. Since abatement is costly, and the harm does not affect a firm's profits, a firm will not choose to abate unless there is some regulation. That is, the laissez-faire level of abatement is zero.
To incentivize abatement, a welfare-maximizing regulator may choose to implement either a self-reporting or a no-reporting regulatory regime, where z 2 {NR,SR} denotes the no-reporting and self-reporting regimes, respectively. In the NR regime, each firm is audited with probability q NR , and when harm has occurred (with probability 1 À a i )itis fined F NR ∈ ½0, F per unit, where F is the maximal feasible fine. Thus in the NR regime, a firm's profit is In a self-reporting regime (SR), if harm occurs, then the firm self-reports the occurrence of harm with probability τ i ∈ ½0, 1, in which case it is fined F SR ∈ ½0, F.I n keeping with Kaplow and Shavell (1994), the firm is audited with probability q SR when it does not make a report (or reports no harm), and is fined at the same rate F NR that applies to unreported harm in the NR regime. 6 Thus a firm's profit in the SR regime is The timing of this game is as follows.
1. Stage 1. The regulator chooses fq NR ,F NR g in the no-reporting regime, and fq SR , F SR g in the self-reporting regime. 2. Stage 2. Firms choose a and q. 3. Stage 3. Harm is realized or not. 4. Stage 4. In the SR regime, if harm occurs (with probability 1 À a i ), then the firm chooses whether or not to self-report it. 5. Stage 5. The regulator audits with probability q NR in the no-reporting regime, and with probability q SR in the self-reporting regime when it does not receive a report.
Using backwards induction (and subgame perfection), we first solve the model in the case of the SR regime. In stage 4 (taking quantities and abatement levels as given), firms choose τ to maximize profits. The derivative of equation (2) with respect to τ i is q SR ð1 À a i Þ F NR Àð1 À a i ÞF SR : have not yet introduced the regulator's welfare-maximization problem, we find it convenient to note here that as long as auditing costs are increasing in the audit probability, the regulator sets q SR F NR ¼ F SR . Choosing q SR F NR > F SR cannot be optimal because then q SR can be lowered (up to the point of equality) while also improving welfare. Also, q SR F NR < F SR cannot be optimal as then firms would never self-report and the equilibrium would be identical to the NR regime. Thus q SR F NR ¼ F SR is optimal, so firms always self-report when harm occurs. 8 Thus equation (2) The first-order condition with respect to a i yields the profit-maximizing level of abatement in the SR regime: For now we assume that the solution to a Ã is interior (i.e. F SR =k < 1), but in Assumption 1(c) below we ensure that this condition is always met.
Substituting the value for a Ã into the profit function yields Maximizing this expression with respect to the firms' quantity yields the symmetric Cournot-Nash equilibrium. This equilibrium is characterized in the following lemma.
Lemma 1. Denote a firm's full marginal cost in each regime by At a symmetric Nash equilibrium, a firm's quantity, profits and abatement are Note that the two regimes affect the equilibrium quantity only through m z . Since fines in the SR regime are chosen such that F SR ¼ q SR F NR , the algebraic expression of the full marginal cost m is identical in both regimes, for given {ρ,F}. Accordingly, although Lemma 1 specifies the expression for q z at the optimal policy under the SR and NR regimes, the expressions for a, q and are identical in both regimes, which is convenient analytically. As, however, the optimal levels of q will not be the same in the two regimes, the quantities, profits and abatement levels will not be identical.

II. WELFARE ANALYSIS:CONSTRAINED SOCIAL PLANNER
Given the market equilibrium in Lemma 1 for some N, we study the regulator's welfaremaximizing choice of fines and audit probability. That is, in this section we assume that the regulator is a 'constrained social planner' that takes the market size N as given. Further, the regulator acts as a 'Stackelberg leader' that chooses its policy anticipating firms' reaction to its policy, identified in Lemma 1. In other words, given the fines, the audit probability and the regime, firms choose the symmetric Cournot oligopoly quantities and level of abatement derived in the previous section.
To identify the regulator's objective, we follow most of the literature in economics and assume that the regulator is a utilitarian (e.g. Mookherjee and Png 1995) that maximizes the difference between the benefits and the costs to society. Then the expected cost of enforcement for the regulator is given by C(.): Cðq,δ,zÞ¼gq δ qNa 1 z¼SR , δ ∈ f0,1g and g> 0, where 1 A takes the value 1 when condition A is true, and 0 otherwise. Here C(.) is the product of the cost per audit gq δ , where g>0 is a scalar, and the expected number of audits qNa 1 z¼SR . The parameter δ determines the structure of costs-that is, whether they are fixed or variable in firm size (which is measured by q). When δ = 0, costs are fixed in the sense that firm size does not affect enforcement costs; in this case g is exactly the marginal cost of audit. If δ = 1, then costs are linear in firm size. 9 Importantly, equation (3) also helps to identify the benefit of self-reporting, first recognized in Kaplow and Shavell (1994). Under self-reporting (z = SR), costs become a function of a, for in expectation, the regulator need only audit the proportion a of firms who have not selfreported causing harm.
The benefit to society from this industry is given by where Q=qN is the equilibrium market size in the symmetric equilibrium characterized in Lemma 1. In this benefit function we assume that fines are transfers from firms to society; the net cost to society of a fine is therefore zero. Given these costs and benefits, in the SR regime, the regulator chooses q and F to maximize while in the NR regime, the regulator maximizes Note that the cost differential between the two welfare functions is critical to the wellknown result that self-reporting is optimal. Under self-reporting, the regulator needs to audit only those firms that do not cause harm (with probability a). Under the noreporting regime, the regulator must always audit. Before proceeding to analyse the socially optimal choices, we make the following assumptions for any regime z 2 {NR, SR}. Assumption 1. The parameters in our model possess the following properties.
1. Demand is sufficiently strong; that is, β−c>k, so full abatement is feasible (for firms). 2. h>k. 3. hF−kg<kF. 4. The fine F is less than the level of monopoly profit.
While we leave the algebra to the Appendix, the intuitive justification for these assumptions is straightforward. Assumption 1(a) ensures that firms produce a positive quantity even under full abatement. Assumption 1(b) ensures that the marginal benefit from abatement (a reduction in h) is greater than the marginal cost of abatement, ka, for all a. Hence society wants to provide incentives for abatement (through regulation), instead of the alternative, complete deregulation. Assumption 1(c) ensures that full abatement is not optimal for the regulator. Note that if full abatement is optimal under self-reporting, then there is no longer any gain from self-reporting because the enforcement costs are identical in both regimes. This can be observed easily from equation (3). As we are interested in evaluating self-reporting, we do not explore the case where full abatement is optimal. Assumption 1(d) ensures that when the regulator imposes a fine, it is always feasible for the firm to pay it. This follows because profits are highest under a monopoly.
Under these assumptions the regulator's welfare-maximizing problem involves choosing q and F to maximize W z , z ∈ fNR,SRg, subject to the constraint q≤k/F (for a=1 if and only if q=k/F, and it is never optimal to raise q once a=1).
Our first step in identifying the welfare-maximizing policy involves the following result concerning the optimal fine in the NR regime.
Lemma 2. Regardless of the cost structure, the fine F NR , which applies to unreported harm in both the self-reporting and no-reporting regimes, is maximal.
Given this result, herein the fine F NR is the maximal fine F. A direct consequence of Lemmas 1 and 2 is that we can write the equilibrium level of abatement in each regime as These expressions yield various observations that aid our subsequent characterization of the welfare-maximizing policies. First, it follows that when q SR > q NR , abatement will be higher in the SR regime. In this case we say that enforcement is higher in the SR regime compared to the NR regime. The reverse will be true when q SR < q NR . Second, when q z ¼ 0, abatement is zero. Therefore a policy that implements q z ¼ 0 is effectively the laissez-faire policy; if a policy implements q z > 0, then some regulation or market [JULY intervention is welfare-maximizing. Accordingly, an increase in q z can be described as an increase in enforcement.

Fixed enforcement costs
Let q Ã z represent the welfare-maximizing audit probability. When enforcement costs are fixed with respect to firm size (δ=0), the socially optimal audit probability possesses the following characteristics with respect to the level of competition and the level of harm. Proposition 1. The welfare-maximizing audit probability in the no-reporting regime, q NR , may be higher or lower than the welfare-maximizing audit probability in the selfreporting regime, q SR . Whether q NR is greater or smaller than q SR depends on the level of harm h, the cost of enforcement g, and the level of market competition N. Specifically, there exist thresholds h 1 ,h 2 ,h 3 on the level of harm that are functions of g, with h 1 ðgÞ > h 2 ðgÞ > h 3 ðgÞ > 0, such that the following hold.
1. If the harm is sufficiently high so that h ≥ h 1 ðgÞ, then there exist N 1 , N 2 ,N 3 , with N 3 > N 2 > N 1 ≥ 1, such that we have the following.
1. If the market is sufficiently concentrated (N ≤ N 1 ), then the audit probability is the same in both regimes: , then the laissez-faire policy of q Ã NR ¼ 0 is preferred under the NR regime, but regulation is still welfaremaximizing under the SR regime; that is, q Ã SR > 0. 2. If the harm is moderately high so that h 2 ðgÞ ≤ h < h 1 ðgÞ, then there exist N 2 , N 3 , with N 3 > N 2 ≥ 1, such that we have the following.
1. If the market is sufficiently concentrated, in the sense that N ≤ N 2 , then enforcement is higher in the NR regime (q Ã NR ≥ q Ã SR > 0). 2. If the market is moderately concentrated in the sense that N 2 < N < N 3 , then enforcement is higher in the SR regime (q Ã SR > q Ã NR > 0). 3. If the market is sufficiently competitive in the sense that N ≥ N 3 , then the laissezfaire policy (q Ã NR ¼ 0) is welfare-maximizing under the NR regime, but regulation is still welfare-maximizing under the SR regime (q Ã SR > 0). 3. If the harm is moderately low so that h 3 ðgÞ ≤ h< h 2 ðgÞ, then for any level of competition (for all N), enforcement is higher in the SR regime and there exists an N 3 ≥ 1 such that if N ≥ N 3 , then the laissez-faire policy is welfare-maximizing under the NR regime (q Ã NR ¼ 0). 4. If the harm is sufficiently low so that h ≤ h 3 ðgÞ, then for all levels of market concentration, the laissez-faire policy is welfare-maximizing under the NR regime (q Ã NR ¼ 0), but regulation is always welfare-maximizing under the SR regime (q Ã SR > 0).
Proposition 1 is illustrated in Figure 1. Panel (a) depicts the optimal enforcement in (h,N)-space, as described in the proposition, and panel (b)  probabilities fq Ã NR , q Ã SR g generally differ, because while the marginal social benefit from q is the same in either regime, the costs of enforcement differ at the margin. As first explained by Kaplow and Shavell (1994), on the one hand, the marginal enforcement cost tends to be lower with self-reporting because an increase in the probability of audit applies only to deterred firms. On the other hand, the marginal enforcement cost tends to be higher with self-reporting because an increase in the probability enlarges the pool of firms subject to audit by making harm less likely. The magnitude of the former effect is decreasing in q (for, as enforcement is tightened, an increasing proportion of firms generate no harm), while the magnitude of the latter effect is increasing in q. It follows that under the conditions of the proposition, there exists a (unique) point at which marginal costs in the two regimes coincide.
Proposition 1 offers three key insights into optimal audit probabilities under the SR and NR regimes.
First, under the SR regime, regardless of the level of harm or the market's concentration, it is always optimal to provide incentives for abatement by auditing firms. In contrast, auditing is not always optimal in the NR regime (for some market structures). Thus implementing a self-reporting regime permits welfare-enhancing regulation in circumstances where the laissez-faire outcome is preferred to a no-reporting regime (i.e. no regulation is optimal in a second-best sense because no-reporting is too costly).
Second, whether or not auditing is optimal depends on both the level of competition N and the level of harm h, because the total harm to society is proportional to Qh. However, this does not mean (as is the case in, for example, Polinsky and Shavell (2000) and much of the remaining deterrence literature) that a higher total harm Qh implies a greater willingness to audit on the part of the regulator. Given some level of harm h 0 > h 1 , the total harm under a monopoly, Q M h 0 , is less than the total harm under a more competitive market, Q c h 0 , without enforcement. Nevertheless, in the NR regime, the regulator may choose to audit the monopolistic market (where total harm is lower) but not the more competitive market (where total harm is higher) if the latter case falls in the region where N > N 3 whereas the former case occurs in the region where N < N 3 in Figure 1. Indeed, it is only when the harm is sufficiently large and the market sufficiently concentrated that the audit probability is positive under both regimes. The audit probability may even attain its maximum, q=k/F, if the market is sufficiently concentrated, a case that would essentially amount to continuous monitoring (Dechenaux and Samuel 2019). Thus to determine whether or not auditing is optimal, the regulator must account for both the level of competition and the per unit harm h; the total harm Qh is not sufficient.
Third, although implementing the SR regime always allows for harm-reducing regulation (regardless of the level of harm or market concentration), this does not imply that the abatement level under the NR regime is always lower than that produced under the SR regime. As seen in Figure 1(b), if N < N 2 , then implementing an SR regime can lower abatement (relative to the 'status quo' NR regime), whereas the opposite is true if N > N 2 . Consequently, when the level of competition is sufficiently high, the level of abatement under self-reporting will be closer to full abatement, whereas when the level of competition is low, the level of abatement under no-reporting more closely approximates full abatement. 10 Thus competition is 'good' for self-reporting in the sense that if markets are sufficiently competitive, then the efficiency gains from self-reporting can be realized fully without raising harm. Indeed, if a regulator were constrained (perhaps politically) by the notion that any new policy implemented must lower the harm-a concern raised in Economica  For the welfare-maximizing audit probabilities identified in Proposition 1, the following comparative static result holds at an interior maximum.
1. Under both the NR and SR regimes: 1. q Ã z is strictly decreasing in market competition N and the slope of the demand curve γ; 2. q Ã z is increasing in the level of harm h and the strength of the demand β−c; 3. q Ã z may be increasing with respect to the cost of abatement k-that is, ∂q Ã z =∂k is ambiguous in sign. 2. The fine rate F can affect q Ã z differently in the NR and SR regimes: 1. q Ã NR is increasing in F if the marginal benefit of an increase in q, Φ q , is inelastic with respect to q, and is decreasing in F otherwise; 2. q Ã SR is decreasing in F.
The comparative statics with respect to h and β−c are intuitive. As the harm increases, the regulator needs to increase the audit intensity. Similarly, when demand is strong (i.e. β−c large), quantity produced increases, and consequently the harm also increases. Thus audit intensity also rises. The effect of competition on the audit probability, however, is particularly interesting. Increases in competition, as measured by N, increase the marginal cost of raising the audit probability. Consequently, the optimal audit probability declines with N (Figure 1(b)).
An increase in the fine rate has competing effects, which implies that whether the fine and audit rate are complements or substitutes depends on the reporting regime. On the one hand, an increase in F incentivizes firms to increase abatement. On the other hand, this increase in abatement induces firms to lower their output. The proof of Proposition 2establishes that in the NR regime, the balance of these competing effects depends on whether the marginal social benefit, Φ q , is elastic or inelastic with respect to the probability of audit. In the inelastic case, an increase in the fine rate increases the optimal audit probability, so that the fine and the optimal audit probability are complements. In the SR regime, an increase in F has a third effect: it increases the marginal cost of raising the audit probability (C qF > 0). This third effect is sufficient to ensure that in the SR regime, the fine rate and the audit rate are substitutes in optimal enforcement. To summarize, under the SR regime, F and the optimal audit probability are always substitutes; hence an increase in F allows the planner to lower q, thereby reducing enforcement costs. Such cost savings may not be enjoyed under no-reporting since q Ã NR and F may be complements. To our knowledge, this relationship between fine rates and optimal enforcement under self-reporting has not been explored previously in the literature.
Proposition 2 may also be used to understand the comparative static properties of N 2 , the critical value of N at which q Ã NR ¼ q Ã SR , and N 3 , at which q Ã NR ¼ 0. In particular, the comparative statics of N 3 are identical in sign to those of q Ã , while for N 2 , only the comparative statics effects for k and F can possibly differ in sign from those of q Ã . This implies that as the harm increases, the range of competition (N ≥ N 3 ) in which the laissez-faire policy is optimal is smaller. In other words, even for relatively high levels of competition, q Ã NR > 0. Intuitively, because the harm is higher, the regulator chooses to audit under the NR regime even when the level of competition is relatively high.
Further, Proposition 2 also claims that N 2 (if it exists) is also increasing in h. Recall that q Ã NR > q Ã SR for N < N 2 . Thus as the harm increases, the interval of N for which q NR is higher than q SR (and hence abatement is higher in the NR regime) is larger. Accordingly, when h is large, a switch from the NR to the SR regime will lower the level of abatement for even moderately competitive industries (N ∈ ðN 1 ,N 2 Þ). Whereas when the harm is low (h< h 2 ), a switch to the SR regime increases abatement for all levels of market concentration.
Finally, both N 1 and N 2 are decreasing in γ. Recall that γ is the slope of the demand curve. Thus when demand is more inelastic, the range of competition over which selfreporting yields a higher level of abatement grows.

Variable enforcement costs
We now consider the case where costs are variable (i.e. δ=1 in equation (3)). Analogous to the previous subsection, we characterize the optimal audit probability as a function of h and N in the following proposition.
Proposition 3. The welfare-maximizing audit probability in the no-reporting regime, q NR , may be higher or lower than the welfare-maximizing audit probability in the selfreporting regime, q SR . Whether q NR is greater or smaller than q SR depends on the level of harm h, the cost of enforcement g, and the level of market concentration N. Specifically, there exist thresholdsh 1 ðgÞ,h 2 ðgÞ,h 3 ðgÞ on the level of harm that are functions of g, with h 3 ðgÞ >h 2 ðgÞ >h 1 ðgÞ > 0, such that the following hold.
1. If the harm is sufficiently high so that h >h 3 ðgÞ, then for all levels of market concentration, the level of enforcement is higher in the NR regime (q NR ¼ k=F >q SR ). 2. If the harm is moderately high so thath 2 ðgÞ < h ≤h 3 ðgÞ, then there exists an N 2 > 1 such that we have the following.
1. If the market is sufficiently concentrated in the sense that N ≤ N 2 , then enforcement is higher in the SR regime (q SR ≥q NR > 0). 2. If the market is sufficiently competitive, in the sense that N > N 2 , then the level of enforcement is higher in the NR regime (q NR >q SR > 0).
3. If the harm is moderately low so thath 1 ðgÞ< h ≤h 2 ðgÞ, then there exist N 1 ,N 2 , with N 2 > N 1 ≥ 1, such that we have the following.
1. If the market is sufficiently concentrated, in the sense that N ≤ N 1 , then the laissezfaire policyq NR ¼ 0 is welfare-maximizing under the NR regime, but some regulation is still welfare-maximizing under the SR regime (q SR >q NR ¼ 0). 2. If the market is moderately competitive, in the sense that N ∈ ðN 1 ,N 2 , then the level of enforcement is higher in the SR regime (q SR ≥q NR > 0). 3. If the market is sufficiently competitive, in the sense that N > N 2 , then the level of enforcement is higher in the NR regime (q NR >q SR > 0).
4. If the harm is sufficiently low so that h ≤h 1 ðgÞ, then for all market structures, the laissez-faire policyq NR ¼ 0 is welfare-maximizing under the NR regime but some regulation is still welfare-maximizing under the SR regime (q SR >q NR ¼ 0).
We illustrate the salient features of this proposition graphically in Figure 2. Panel (a) depicts optimal enforcement in (h,N)-space, and panel (b) showsq NR andq SR as functions of N for the case in whichh 2 ðgÞ < h <h 3 ðgÞ.
The main lesson from Proposition 3 is that the results with respect to N are qualitatively the 'inverse' of the case where costs are fixed. Specifically, as seen in Figure 2(b), given some level of harm h, at higher levels of competition (i.e. N > N 2 ) the optimal level of enforcement is lower under the SR regime than under the NR regime. In contrast, when costs were assumed fixed, enforcement was higher under the SR regime than under the NR regime for higher levels of competition. Accordingly, when costs are variable, a regime switch from NR to SR in a highly competitive industry will lower abatement when costs are variable, whereas when costs are fixed, a switch from NR to SR will likely raise abatement in a highly competitive industry. Further, as can be seen in Figure 2(b), when h ∈ðh 2 ðgÞ,h 3 ðgÞÞ, for lower levels of competition enforcement is higher in the SR regime, whereas for higher levels of competition enforcement is lower in the SR regime. The main message from our analysis is that the impact of self-reporting on compliance depends critically on both competition and the structure of the marginal cost of enforcement (i.e. whether it is fixed or variable). This is especially the case for moderate levels of harm betweenh 1 ðgÞ andh 3 . When the harm is sufficiently high or low (cases (a) and (d) in Proposition 3), however, the audit probability does not depend on market structure.
The following proposition further highlights the distinction between the cases δ=0 and δ=1. Proposition 4. At an interior solutionq z ∈ ð0, k=FÞ, z 2 {NR,SR}, the welfaremaximizing audit probability under variable costs possesses the following comparative static properties.

Under both NR and SR:
1.q z is strictly increasing in market competition N and the the level of harm h; 2.q z is independent of the slope of the demand curve γ. The fine rate F, the strength of the demand β−c, and the cost of abatement k,c a n each affectq z differently in the NR and SR regimes, but the relative magnitudes of these effects in the two regimes, and their signs, are ambiguous. Figure 2(b): optimal enforcement is increasing in the level of competition N. The intuition underlying this finding is that in the variable cost case, an increase in N has two effects on the marginal cost of raising the audit probability C q . First, a higher N increases C q , as increasing proportionally the fraction of firms that are audited implies a larger absolute number of extra audits, the larger is N. Second, however, higher competition endogenously reduces output per firm q, thereby reducing the per-firm audit cost. In contrast, in the fixed cost case, only the first of these effects applies.

Proposition 4 proves a clear visual feature in
A key difference between the fixed and variable cost cases is that the demand parameters β−c and γ do not enter the cost function in the fixed cost case, but do in the variable cost case. As a result, whereas a steepening of the demand curve increases the optimal audit probability in the fixed case, it has no impact on the optimal audit probability in the variable case. Also, whereas an increase in the strength of demand increases the optimal audit probability in the fixed case, its impact in the variable case becomes ambiguous in sign. For parameters such as β−c that interact with optimal enforcement in a complex way, it is possible, for prescribed parameter values, that the sign of the comparative statics effect differs between the NR and SR regimes. However, as the relative magnitudes of the effect between regimes is also complex, the divergence in signs (when present) can itself go in either direction, depending on parameter values.
Finally, note that when costs are variable, the optimal audit probability and the (maximal) fine may be complements or substitutes in either regime. Whereas when enforcement costs are fixed, the optimal audit probability and fine are necessarily substitutes in the SR regime (Proposition 2).

III. WELFARE ANALYSIS:UNCONSTRAINED SOCIAL PLANNER
We now assume that the social planner can choose N as well as q in both the NR and SR regimes. When moving from the NR regime to an SR regime, the regulator faces a compromise. Simultaneously increasing N as well as q would potentially stimulate competition and reduce harm, but both acts would also raise the marginal cost of enforcement. Therefore if this latter effect were too large, then social welfare might instead be maximized by increasing one of N and q, and decreasing the other choice variable. Accordingly, the route to maximizing social welfare is not immediately obvious. Here we show that if the social planner can choose N, then there will more competition but higher levels of harm in the (socially optimal) SR regime. This result is summarized in the next proposition.
Proposition 5. Letq,N denote the socially optimal level of auditing and market size. If enforcement costs are fixed (δ=0), then this socially optimal policy for an unconstrained social planner possesses the following characteristics: Proposition 5 reveals an important finding: the socially optimal market size is higher when self-reporting policies can be implemented. Specifically, in the fixed cost case, a welfare-maximizing regulator will, if switching from the NR regime to the SR regime, choose to lower the audit probability, as a consequence of which market competition increases, as does the level of harm. Since welfare is always raised under self-reporting, it follows that the socially optimal policy consists of implementing self-reporting. Given Proposition 5 this, in turn, implies an increase in the market size N and therefore a reduction in prices and larger consumer surplus.
Some intuition for this finding comes from Figure 3, which depicts the social optimum in Proposition 5. The two lines q Ã NR ðNÞ and q Ã SR ðNÞ depict the optimal choice of audit probability for a given N in the NR and SR regimes. The two lines N Ã NR ðqÞ and N Ã SR ðqÞ depict the regulator's optimal choice of N for a given q (these functions have been inverted to be drawn in (N,ρ)-space). The optimal pair ðN z ,q z Þ, z 2 {NR,SR}, are found at the intersection of q Ã z ðNÞ with N Ã z ðqÞ. The optimal N Ã in the NR regime is seen to be non-monotonic in q. At low values of q, the industry generates large amounts of harm, inducing a regulator to restrict its size. At large values of q, the marginal enforcement cost of raising N becomes increasingly high, again leading a regulator to wish to restrict N. The highest optimal choices of N therefore arise for intermediate values of q at which both harm and marginal enforcement costs are not too high.
Switching to the SR regime alters the trade-off between harm and marginal enforcement costs. Note that whereas self-reporting can be associated with either higher or lower marginal enforcement costs with respect to increases in q, self-reporting is always associated with lower marginal enforcement costs with respect to increases in N. Intuitively, following an increase of DN in N, an extra q DN firms must be audited under no-reporting, but only an extra q DN(1−a) firms must be audited under self-reporting. Consistent with this point, in Figure 3 we see that N Ã SR ðqÞ > N Ã NR ðqÞ for every value of q such that a>0. The higher optimal N under self-reporting acts to reduce q, for-as we proved in Proposition 2-the optimal audit probability is decreasing in N. As well as the optimal N being always higher under self-reporting, it is also seen in Figure 3 to vary to a greater degree in the choice of q. The interaction between q and N in the cost function is given by C qN ¼ N À1 C q > 0. Hence C N is more sensitive to variation in q the higher is C q .  The greater variability in the optimal N Ã under self-reporting therefore implies that C q must be higher under self-reporting than under no-reporting. This, in turn, implies that q Ã SR ðN Ã Þ< q Ã NR ðN Ã Þ, which places N Ã SR ðqÞ and N Ã NR ðqÞ at values below N 2 in Figure 1. Accordingly, with reference to Figure 3, when switching from the NR regime to the SR regime, there are two effects on q, both of which are negative. The first is a discrete downwards jump when switching from the line q Ã NR ðNÞ to the line q Ã SR ðNÞ at N ¼N NR , and the second is a move to the right along the line q Ã SR ðNÞ fromN NR toN SR . Similar intuitions apply to the variable cost case (δ=1), as depicted in Figure 4. It can be shown that the social optimum again lies in the region whereq NR ðNÞ>q SR ðNÞ, albeit this occurs forÑ NR > N 2 rather thanÑ NR < N 2 . An important difference, however, is that the optimal audit probability is increasing in N. This implies that in a switch from the NR regime to the SR regime, although the optimal q falls on account of the move downwards fromq NR ðNÞ toq SR ðNÞ, this effect is offset by an upward movement in q along the linẽ q SR ðNÞ arising from an increase in N. Accordingly, whether the optimal q increases or decreases from a switch from no-reporting to reporting remains unclear. Intractability precludes a more definite answer. 12 Finally, note that the policy in Proposition 5 can be implemented by introducing a fixed entry cost y>0. Under this policy, firms enter the industry until their profits are zero, given the fixed cost y. As profits tend to zero with N, the regulator can choose q ¼q SR and an entry fee y to achieveN SR . 13

IV. CONCLUSION
Although economic analyses of self-reporting show that implementing such a policy always raises welfare, there is still considerable dispute regarding its overall effectiveness (Toffel and Short 2011). Many empirical studies find little evidence that implementing self-reporting improves compliance rates (e.g. Esbenshade 2004;Vidovic and Khanna 2007). And other studies find that compliance falls under self-reporting (King and Lenox 2000). Consequently, some regulators have considered eliminating their self-reporting policies altogether (Toffel and Short 2011). In light of this debate, our paper makes a contribution towards understanding these empirical findings and their implications for evaluating the impact of self-reporting. We show that the impact of self-reporting on compliance is affected by strategic market forces so that whether the optimal level of compliance is higher or lower under noreporting than under self-reporting depends on the level of competition. Since many regulatory agencies regulate firms in oligopolistic contexts, our findings suggest that selfreporting, though welfare-increasing, need not raise compliance and lower the harm. Accordingly, it may not be appropriate to evaluate the effectiveness of self-reporting by examining whether compliance rises or falls post-implementation.
Our results also imply that regulators introducing self-reporting need to consider the level of competitiveness in order to determine whether harm will rise or fall. This is especially important in the unconstrained social planner's problem where we show that a regulator chooses more competition while also 'permitting more harm' (Proposition 5). This suggests an important policy implication: that there may need to be more coordination between competition (antitrust) authorities and other regulatory bodies that correct for externalities.
The result that under self-reporting the optimal 'permissible' level of harm may be higher than under no-reporting is related to findings in Innes (1999). Albeit for very different reasons, he finds that the level of 'care' in preventing accidents is always lower under self-reporting than under no-reporting. Notably, we find instead that the level of care (abatement) may be higher or lower under self-reporting than under no-reporting depending on the level of competition. This suggests that market characteristics should not be ignored when evaluating the benefits of enforcement policies such as self-reporting.
Our paper also identifies new benefits that are achieved from implementing selfreporting. First, under self-reporting, the optimal audit rate and F are always substitutes (when costs are fixed). Thus an increase in F lowers the audit rate, thereby reducing enforcement costs. However, such cost savings will not always be realized in the NR regime, as there the optimal audit rate and the fine need not be substitutes. Second, when both the audit probability and the market size N can be chosen by the regulator (the unconstrained case), the optimal market size-and therefore also consumer surplus-will be higher in a self-reporting regime than in a no-reporting regime. Although previous literature has examined some aspects of optimal enforcement in oligopolies (e.g. Baumann and Friehe 2016), no study considers characteristics of the optimal market size in relation to enforcement. As we see, studying this problem reveals an important finding concerning the benefit of self-reporting.
We end by noting some extensions and ideas for future work. First, we did not consider the possibility of free entry and exit in this market. This could be undertaken by assuming that there is a fixed exogenous cost that is incurred by firms on entry. In this case, the number of firms that enter the industry depends on this cost, similar to the analysis in note 13 (except that here entry costs are exogenous, whereas there they are chosen by the regulator). Once N is determined, our results are broadly similar to the constrained regulator's choices in that if the harm is sufficiently large (small) then the optimal enforcement under the self-reporting regime is higher (lower) than the optimal enforcement in the no-reporting regime. Consequently, when the harm is high (low), fewer (more) firms enter the industry under the self-reporting regime. Second, while selfreporting generates a welfare surplus in a model with homogeneous firms, it may not do so if firms are sufficiently differentiated. Intuitively, in a vertically differentiated Bertrand duopoly, a firm's decisions to self-report will be a best response to the other firm's decision to report. Hence the impact on welfare is unclear. Finally, whereas we consider a 1. Quantity (and hence profits) are positive, that is, q>0 when a=1. Substituting a=1 into the function for quantity yields q ¼ 2ðβ À c À k=2Þ γð1 þ NÞ >0o r2 ðβ À cÞÀk >0: 2. As discussed in the main text.
3. Full abatement is not socially optimal for the regulator. To ensure this, 2γk : At a=1, the above expression must be negative, or which implies that hF−kg<kF. 4. Monopoly profit is given by max q π z ¼ðβ À γq À m z Þq. We assume that monopoly profit exceeds the fine rate F, so a firm can always afford to pay the fine F when levied.

PREAMBLE TO PROOFS
The following expressions and their derivatives are utilized in the proofs of Propositions 1-5.

W=Φ−C, where
Φðq,NÞ¼Φ ¼ Nqðq,NÞ wðq,NÞ, The case of no-reporting corresponds to φ=0, and the self-reporting case to φ=1. Next, we establish the expressions for the following derivatives: Proof of Lemma 1. Profit is given by π i,z ¼ðβ À γQ À m z Þq i , z 2 {NR,SR}. Differentiating with respect to q i gives the first-order condition β À γQ À m z À γq i ¼ 0. Imposing q i ¼ q z for all i (such that Q ¼ Nq z ) and solving for q z , the results in the lemma follow.
Proof of Lemma 2. Note that q is always post-multiplied by F in Φ, but not in C, where q appears both independently and post-multiplied by F. Accordingly, social welfare can be written as Consider lowering q and increasing F, holding qF constant. Then Φ(qF) is unchanged, but C(q,qF) falls (thereby increasing W), as ∂Cðq, qFÞ ∂q This observation implies that W must be maximized with respect to F NR at the maximal choice F NR ¼ F.
Proof of Proposition 1. We first characterize the optimal q Ã . Using the expressions in the preamble, the first-order condition for q is ðÞ : Setting q=k/F in the first-order condition (A5), we solve for N to obtain N 1 ðφÞ¼ Fðh À kÞ 2ðβ À cÞÀk ðÞ 2γgkð1 þ φÞ À 1: (A6) Next we prove the following claim.
Proof . Using the derivatives in (A1)-(A3), we obtain Rewriting the first-order condition in (A5) as NðΦ q =N À C q =NÞ¼0, an increase in N causes Φ q =N to decrease (Claim 1), thereby forcing C q =N to decrease also in order to restore the firstorder condition. As C q =N ¼ g 1 þð2a À 1Þφ ðÞ is independent of N, for it to fall, it must be that q (and hence a) falls. It follows that q=k/F for all N ≤ N 1 ðφÞ.A sN 1 ðτÞ is decreasing in φ, it follows that q=k/F for all φ (and therefore in both the NR and SR regimes) when Fðh À kÞ 2ðβ À cÞÀk ðÞ 2γgk À 1: Finally, we need to ensure that N 1 > 1, which holds if ≡ h 1 ðgÞ: We now turn to N 2 . Setting q=k/(2F)(a=1/2) and N=1 in equation (A5), we obtain F 8ðβ À cÞð3k À 4hÞÀ4hk þ 5k 2 ÀÁ 64γk À g ¼ 0: From Claim 1, for N 2 ≥ 1 we must therefore have ≡ h 2 ðgÞ: (A7) Turning to N 3 , setting q=φ=0 in equation (A5) implies that at an optimum, Proof of Proposition 2. (Comparative statics of q)Let ɛ a,b ≡ ðb=aÞð∂a=∂bÞ be the elasticity of a with respect to b. We first prove a claim. Claim 2. ɛ Φq,N < 1 for N≥1.
Proof. Using equation (A5), ɛ Φq,N <1 when it holds that ! : Note that for N≥1, the left-hand side does not exceed 1/2, and the right-hand side necessarily exceeds 1/2, hence the inequality holds as claimed.
Using the implicit function theorem in equation (A5), we have that for an arbitrary exogenous variable x, As W qq < 0 is the second-order condition for a maximum, the sign of ∂q Ã z =∂x is the sign of W qx . Noting that W qx ¼ Φ qx À C qx , we have where the sign of W qN follows from Claim 2. It follows that ∂q Ã z =∂N< 0, ∂q Ã z =∂γ < 0, ∂q Ã z =∂h > 0, ∂q Ã z =∂ðβ À cÞ > 0, ∂q Ã NR =∂F≷0 , ɛ Φq,q ≷ À 1, ∂q Ã SR =∂F< 0, and ∂q Ã z =∂k≷0. Cðq,N;φÞ¼gNqðq,NÞ q 1 À½1 À aðqÞφ ðÞ : The first-order condition with respect to q is On calculation it can be observed that both the left-and right-hand side terms in equation (A9) are proportional to N/(γ (1+N)). Cancelling this term, we write (A9) as ðβ À mÞw q Àð1 À aÞwF |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl ffl} where MC ¼ MCðq,N;φÞ¼g ½1 þð2a À 1Þφðβ À mÞÀð1 À aÞ½1 Àð1 À aÞφqF ðÞ : Hence, as MC NR ¼ MCðq, N;0Þ and MC SR ¼ MCðq, N;1Þ, we may write Hence it is clear that marginal benefit is increasing in N, while marginal cost is constant in N. Thus, given the assumption of concavity of welfare with respect to q, it follows that q NR and q SR at an interior solution that satisfies equation (A10) are increasing in N.
Next, we establish the following points.
First, we show that q SR >0 for all N. Since MC SR ¼ 0a tq=0, as long as MBj q¼0,N¼1 > 0, we have q SR > 0 for all N: which is strictly positive because the right-hand side is increasing in h and positive at the smallest value of h, namely h=β−c. Thus because MB is increasing in N, MB>MC at q=0, therefore q SR > 0.
Next, at q=0 and N→∞, we have If this expression is less than g(β−c), then q SR > q NR ¼ 0 for all N. Simplifying this condition yields h < gðβ À cÞk Fðβ À cÞþFk ≡h 1 ðgÞ: Therefore if h <h 1 ðgÞ, then for all N, q SR > q NR ¼ 0.
Next, at q=0 and N=1, MB is given by equation (A11), which is less than g(β−c) if and only if h < ðβ À cÞk ðβ À c þ kÞF g þ F 2 ¼h 2 ðgÞ >h 1 ðgÞ: If h ∈½h 1 ðgÞ,h 2 ðgÞ, then as MB is increasing in N and given the properties of MC z concerning q, there exist N 1 , N 2 , with N 1 <N 2 , such that q SR > q NR ¼ 0 for all N < N 1 , q SR > q NR > 0ifN ∈ ½N 1 , N 2 and q SR < q NR if N > N 2 .Ifh >h 2 ðgÞ, then q NR >0, and there exists an N 2 such that q NR <q SR if and only if N< N 2 . Finally, if MB at q=k/F and N=1 is greater than MC NR at q=k/F, then for all N we have q NR ¼ k=F> q SR .A tq=k/F, MB ¼ Fðh À kÞ=k ðÞ β À k À 1 2 k ÀÁ and MC ¼ g β À k À 1 2 k ÀÁ . Therefore MB>MC if h ≥ gk F þ k≡ h 3 ðgÞ > h 2 ðgÞ, then q NR ¼ k=F> q SR .
Proof of Proposition 4. From the first-order condition (A10) we have that for an arbitrary exogenous variable x, ∂q z ∂x ¼À MB x À MC x MB q À MC q , z ∈ fNR, SRg: As MB q À MC q < 0 at a maximum, ∂q z =∂x takes the sign of MB x À MC x . We then have MB N À MC N ¼ðβ À mÞw qN Àð1 À aÞw N F> 0, MB βÀc À MC βÀc ¼ F ðh À kÞþNðh À qFÞ ðÞ kð1 þ NÞ À g 1 Àð1 À 2aÞφ ðÞ ≷0, MB k À MC k ≷0: It follows that ∂q z =∂N > 0, ∂q z =∂h >0, ∂q z =∂γ ¼ 0, ∂q z =∂ðβ À cÞ≷0, ∂q=∂F≷0 and ∂q=∂k≷0. Proof of Proposition 5. Using the characterization of the regulator's objective function provided in the subsection 'Preamble to proofs' above, the first-order conditions for {q,N} can be written as where, when δ=0,  Proof. The proof of this claim follows directly from the first-order conditions. At any solution, That is, at the optimal solution, the marginal rate of substitution between q and N with respect to Φ must equal their rate of substitution with respect to costs. A straightforward calculation shows that keeping the total costs fixed at C 0 gives q ¼ C 0 gN 1 Àð1 À aÞφ ðÞ : Therefore N and q are substitutes, and at the optimum, ∂N/∂q<0. Claim 4. At the social optimum (in the SR regime),q SR > k=ð2FÞ.
Proof. As C is homogeneous of degree 1 in N, we have C N ¼ C=N,s oC ¼ NC N . Hence By similar reasoning, at q ¼ q Ã , W ¼ Φ À qΦ q 1 Àð1 À aÞφ 1 þð2a À 1Þφ : It follows that at a social optimum, NΦ N ¼ qΦ q 1 Àð1 À aÞφ 1 þð2a À 1Þφ : Noting that 1 Àð1 À aÞφ 1 þð2a À 1Þφ ≤ 1, it must hold that qΦ q À NΦ N ≥ 0. Using the derivatives in (A1)-(A3), we obtain qΦ q À NΦ N ¼ so it must hold, at a social optimum, that which is equivalent to w À β À m 2 > 0: Define ζ ≡ w À β À m 2 : Then Thus if inequality (A14) is not satisfied at the highest value of β (which is h), then it is not satisfied for all β. Similarly, if inequality (A14) is not satisfied at the lowest value of N (which is 1), then it is not satisfied for all N. Thus we have that if 2h À að2k À qFÞ 8 Àð1 À aÞðh À qFÞ ≤ 0, (A15) then q cannot be part of a social optimum. Moreover, if inequality (A15) holds at some q 0 , then it holds for all q ≤ q 0 (so a social optimum must satisfy q>q 0 ). Set q=k/(2F). Then inequality (A15) becomes À 8h À 5k 32 <0: Hence q=k/(2F) cannot be part of a social optimum. Rather, it must hold at a social optimum that q> k=ð2FÞ. cases is maximal. Thus although we have not yet derived this result, we make this assumption to avoid excessive notation. 7. Strictly, firms are indifferent between self-reporting or not when q SR F NR ¼ F SR . In keeping with the mechanism design literature, however, we assume that when they are indifferent, firms report truthfully. 8. This result also follows from the revelation principle. 9. If δ 2 (0,1), then costs are concave in firm size. We do not analyse this interior case here. 10. The second -best level under costly enforcement is higher or lower under the SR or the NR regime because of the differential structure of enforcement costs between the two regimes. 11. Kaplow and Shavell (1994) show that, in general, the audit probability under self-reporting may be higher or lower than the probability under no-reporting. As the fine is always maximal, it implies that the level of harm may be higher or lower under self-reporting than in the no-reporting regime. Proposition 1 'tightens' their result and shows that whether the audit probability in one regime is higher or lower than in the other depends on the level of competition (see Figure 1). 12. We note as an aside, however, that in Figure 4-and in other numerical examples that we have tried-we observe the outcomeq NR >q SR , consistent with Proposition 5. 13. A straightforward calculation shows thatN in Proposition 5 is implemented by an entry cost yðNÞ¼γ À1 ðβ À m z Þ 2 ð1 þNÞ À2 , z 2 {NR, SR}.