Testing by Competitors in Enforcement of Product Standards

Erica L. Plambeck and Terry A. Taylor

Graduate School of Business, Stanford University

Haas School of Business, University of California, Berkeley

Firms have an incentive to test competitors’ products to reveal violations of safety and environmental

standards, in order to have competitors’ products blocked from sale. This paper shows that testing by

a regulator crowds out testing by competitors, and can reduce firms’ efforts to comply with the product

standard. Relying on competitor testing (i.e., having the regulator test only to verify evidence of violations

provided by competitors) is most effective in large or concentrated markets in which firms have strong

brands and high quality, and for standards that are highly valued by consumers. Under those conditions,

firms tend to test competitors’ products and exert high compliance effort. Conversely, unless compliance

is highly valued by consumers, a firm with low quality does not draw testing from competitors, and so

does not comply. Enforcing a product standard through competitor testing encourages entry by such low-

quality, noncompliant firms and can reduce quality investment by incumbents. Stripping offending products

of labels (such as “Energy Star”), instead of blocking them from the market, eliminates the problem of entry

by low-quality, noncompliant firms, but may reduce incumbents’ compliance efforts.

1. Introduction

This paper derives insights from a game-theoretic model of firms’ efforts to comply with a product standard

and to test competitors’ products for violations. Each firm chooses how much cost to incur to increase the

likelihood that its product complies with the standard. Compliance is a “credence” attribute (Darby and

Karni 1973): in the normal course of use, consumers do not directly observe whether a product complies

with the standard. However, testing by a competitor or a regulator may detect a violation, causing a product

to be blocked from the market (or alternatively, in §4.4, stripped of a label signifying compliance with a

voluntary standard).

The paper is motivated by the potentially important role of firms’ testing of competitors’ products in

the enforcement of product standards, such as those regulating energy efficiency, hazardous substances and

safety. Historically, the U.S. and E.U. have relied to large extent on testing by competitors in enforcement

of energy efficiency performance standards (Wiel and McMahon 2005, Gaffigan 2007, Department of En-

ergy 2010). Since 2002, the E.U. has restricted the use of increasingly many hazardous substances in an

increasingly wide variety of products. In addition, the E.U. has required products to be labeled with the

manufacturer’s identity, which promotes testing and reporting by competitors, and helps regulators prevent

the sale of products that violate the restrictions on hazardous substances or other safety or environmental

standards. These regulations have substantially increased the frequency of products being blocked from

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E.U. markets due to safety violations, to more than 20,000 such measures during 2004-2015 (Croft and

Strongman 2004, Kapoor 2012, European Commission 2012 and 2015). In 2002, authorities blocked the

sale of Sony PlayStation consoles due to a violation of the E.U.’s new Restrictions on Hazardous Substances

in Electronics (RoHS), reputedly in response to a tip from one of Sony’s competitors about the cadmium in a

peripheral cable (Hess 2006), causing Sony to miss $110 million in revenue (Shah and Sullivan 2002). Since

then, testing by competitors has become increasingly common in the E.U. consumer electronics industry,

according to (Green Supply Line 2006, Smith 2008). In the U.S. and E.U., some consumer product manu-

facturers test competitors’ products and report violations of safety standards to have competitors’ products

blocked from those markets (Overfelt 2006, Ross 2007), though data on firms’ reporting of competitors’

violations is not publicly available.1

For government regulatory authorities, testing to detect a violation is costly and difficult because there

are many potential failure modes (specific ways in which a product might fail to meet a standard) to be

tested. For example, to determine whether or not one personal computer is RoHS-compliant would require

disassembly and testing of approximately 3,000 constituent materials for each of six restricted substances,

which would cost regulatory authorities as much as $200,000 (Bruschia 2008).2 As a second example, U.S.

energy efficiency standards require a product to have power consumption below specified thresholds in a

variety of different operating modes and ambient conditions. Additional failure modes–not addressed in

those detailed specifications–can cause a violation due to insufficient efficiency in actual use.3

In contrast, when a firm identifies how a competitor’s product violates a standard and reports that in-

formation to the regulator, the regulator can cheaply and reliably verify that the product is noncompliant

(Bruschia 2008, Smith 2008). Regulators follow up on credible reports that specify precisely how a prod-

uct violates the standard and are well-supported by testing data (Bruschia 2008, Smith 2008). Why aren’t

regulators flooded with false or nonspecific reports of violations? Only a correct, specific report enables

the regulator to verify noncompliance. Providing fraudulent information to a regulatory authority in order

1Regulatory authorities in the U.S. and E.U. encourage competitor testing by offering anonymity for

firms that report competitors’ violations of restrictions on hazardous substances (Bruschia 2008, Smith

2008). While “it’s not unusual for competitors to turn each other in” to the U.S. Consumer Product Safety

Commission (CPSC) for violating product safety standards (Overfelt 2006), anonymous reports from com-

petitors are explicitly excluded from the public database of reports about safety violations in consumer

products (see www.saferproducts.gov/faq-business.aspx).2The E.U.’s REACH regulation has expanded the number of restricted substances–from six under RoHS

to 62 under REACH as of October 2016. In addition, REACH requires that all chemicals in a product be

clearly identified and registered. Thus, REACH is multiplying the number of possible failure modes for a

product. Whereas RoHS applies to electronics, REACH applies to all products.3Whirlpool identified such a failure mode in refrigerator-freezers of its competitor LG, motivating the

U.S. Department of Energy to prevent LG from selling those refrigerator-freezers under the Energy Star

label (Vestel 2009, GAO 2010, Brown 2012).

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to damage a competitor is illegal in the E.U., U.S. and many other countries; in the U.S., for example, the

penalties for doing so include fines and imprisonment for up to five years (see Subsection 18 of U.S. Code

§ 1001(a)). Therefore, in the model in this paper, each firm decides how much to spend on testing each

competitor’s product. A firm submits evidence to the regulator that a competitor’s product is noncompliant

if and only if its testing produces that evidence. The evidence characterizes the mode by which the prod-

uct fails to meet the standard and, upon obtaining that evidence, the regulator verifies that the product is

noncompliant and prevents its sale.

In testing a product to detect a violation of a product standard, competitors tend to be more efficient

than the regulator, for the following reasons. Firms typically have equipment and trained staff for testing

the compliance of their own products, so can test competitors’ products at little additional cost (Wiel and

McMahon 2005, Gaffigan 2007, Smith 2008). They often purchase competitors’ products to evaluate other

quality characteristics, which reduces their procurement cost to test compliance (Day 2007). Through their

own compliance efforts, firms develop a better understanding than the regulator of when and how a com-

petitor’s product is likely to be noncompliant, so can better target their testing efforts (Hess 2006). For

example, a firm’s understanding of a competitor’s suppliers’ reputation and capabilities can provide insight

into which components or constituent materials are likely to cause noncompliance. Unlike firms, govern-

ment regulatory authorities lack a profit motive for efficiency in testing, and their budget for testing must be

raised through taxes that distort the economy and reduce social welfare (Polinsky 1980).

This paper addresses standards for credence attributes that consumers are willing to pay for, and ones

they are not. Credence attributes with private benefits for consumers include, for example, restrictions on

toxic and endocrine-disrupting substances in children’s products, and energy efficiency. (Energy efficiency

is a credence attribute because, though a consumer may look at a monthly household electricity bill, she

cannot easily evaluate the energy efficiency of an individual product (Ko and Simons 2016).) Other product

standards cover credence attributes with a social or environmental benefit, for which consumers might be

unwilling to pay a premium. For example, RoHS aims to mitigate the environmental impact of electronics

waste, and many survey respondents indicate they will not pay more for such “green” electronics (Saphores

et al. 2007).

Literature

Risk that a product is defective (violates a standard) arises from operational challenges, notably the diffi-

culties of supply chain management to ensure that all components of a product meet design specification.

Conformance-quality (i.e., product conformance to design specification) effort and inspection effort are

key operational decisions that reduce the probability a product is defective. An extensive operations man-

agement literature addresses variants of conformance-quality and inspection effort, e.g., statistical process

3

control (Porteus and Angelus 1997) and contractual incentives for suppliers in conjunction with: inspec-

tion of suppliers’ output (Baiman et al. 2000, Balachandran and Radhakrishnan 2005, Babich and Tang

2012), auditing of suppliers’ conformance-quality capability (Hwang et al. 2006), or investment in suppli-

ers’ conformance-quality capability (Zhu et al. 2007). “Compliance” effort in this paper represents all the

effort that a firm exerts to reduce the probability its product violates a standard, including conformance-

quality and inspection effort. In addition, this paper extends the operations management literature–which

focuses on inspection of one’s own or one’s suppliers’ products–by incorporating inspection of competitors’

products. The model in this paper is similar to those in (Baiman et al. 2000, Balachandran and Radhakr-

ishnan 2005, Hwang et al. 2006, Babich and Tang 2012) in that a firm’s entire output is either defective

or conforming and, in the former event, inspection effort (testing expenditure) increases the probability of

detecting the defect.

Papers surveyed in (Cohen 1999) model a regulator choosing effort to detect a violation, in game the-

oretic equilibrium with a potential violator. Those papers adopt a variety of assumptions regarding the

regulator’s objective function and whether or not the regulator moves first by committing to a detection-

effort level. For example, in (Mookherjee and Png 1992), the regulator moves first and maximizes social

welfare, whereas in (Boyer et al. 2000) the regulator moves simultaneously with the potential violator and

maximizes the regulator’s own utility (the expected fine less cost of detection-effort). Rather than restrict

attention to one of those various formulations, this paper treats testing by the regulator as a parameter for

sensitivity analysis. One may interpret that parameter as an initial commitment by the regulator or as the

level of regulator testing anticipated by the firms. The paper identifies conditions under which relying on

competitor testing (having the regulator do zero testing to detect violations, only verify reports of violations)

is effective in enforcing a product standard. Similarly, Mookherjee and Png (1992) show that a regulator

should only verify reports of violations, in a setting in which a violator chooses the severity of his violation,

a victim reports a violation with high probability, and the cost of verification is low.

Whereas much of the literature on regulation or voluntary standards for products’ credence attributes

assumes perfect monitoring and compliance (see Roe and Sheldon 2007, Heyes and Martin 2016, and papers

surveyed therein), as notable exceptions, Mason (2011) emphasizes that tests for credence quality are noisy,

(McCluskey and Loureiro 2005) treat testing by a regulator as a parameter for sensitivity analysis, and

(Feddersen and Gilligan 2001) consider an exogenous probability of monitoring by an activist. To the best

of our knowledge, this paper is the first to address firms’ testing of credence attributes in competitors’

products.

The literature on whistleblowing models whistleblowing by firms in cartels (see Spagnolo 2008, Bigoni

et al. 2012, and papers surveyed therein) or by employees (Austen-Smith and Feddersen 2008, Ting 2008).

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Whereas that literature focuses on a potential whistleblower’s decision whether or not to report a violation

to a regulator, this paper focuses on firms’ efforts to detect competitors’ violations; in the setting of this

paper, reporting a detected violation is optimal. The whistleblowing literature and this paper are consistent

in assuming that any report of a violation is correct, and the regulator acts in response to a report.

In (Li and Peeters 2014), a firm decides whether to test the quality of a competitor’s product and re-

port that information to consumers. Their results are discussed in §4.4, which also addresses reporting to

consumers.

Overview of Main Results

The first part of §3 shows that testing by a regulator crowds out testing by competitors and can reduce

compliance. Specifically, a low level of testing by the regulator fails to increase firms’ compliance efforts

or the detection probability for a noncompliant product (i.e., the probability that, in the event that a product

violates the standard, the regulator or a competitor detects that violation). It simply causes the firms to do

less testing. A high level of testing by the regulator causes firms not to test, and can strictly reduce all

firms’ compliance efforts. These results, combined with the observation that firms can detect violations in

competitors’ products at lower cost and more effectively than a regulator, suggest that social welfare might

be improved by relying on competitor testing (having the regulator do zero testing to detect violations).

Therefore, the second part of §3 identifies conditions under which relying on competitor testing is ef-

fective in enforcing a product standard. When the regulator does not test, competitor testing occurs if and

only if at least two competing firms have sufficiently high product quality. That quality threshold is low–

competitor testing tends to occur–in a large or concentrated market. With symmetric firms, the detection

probability for a noncompliant product initially increases and then decreases with the number of competi-

tors, and also is nonmonotonic in the market size and a firm’s quality. Nevertheless, each firm’s compliance

effort decreases with the number of competitors, increases with the size of the market, and increases with

its product quality.

§4 shows that those results largely hold in extensions of the model with consumers forming rational

expectations about the likelihood that a product is compliant, endogenous qualities and quantities, fines,

fixed costs of testing, and duplication in firms’ testing activities. Further, §4 builds on §3 by providing

guidance for a regulator that relies on competitor testing.

In particular, §4 provides insight into whether a regulator should strip offending products of labels

(such as "Energy Star"), instead of blocking them from the market. The blocking penalty enforced through

competitor testing increases the expected profit of firms with low quality because they do not draw testing

from competitors and benefit when their competitors are blocked from the market. Consequently, blocking

encourages entry by low-quality, noncompliant firms. Switching to the labeling penalty eliminates such

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entry, but reduces incumbents’ compliance efforts.

§5 explains how the results apply to settings with third-party testing.

2. Model

N firms compete in a market of size m governed by a product standard. Each firm i ∈ N ≡ {1, ..,N}

makes an effort ei ∈ [0,1] to comply with the standard and incurs a cost ci(ei) that is positive, strictly

increasing and convex. Firm i produces mqi units of its product, which is compliant with probability ei.

Then each firm i tests the product of competitor-firm j at level ti j ≥ 0, for j ∈ N \i, and incurs a cost

Σ j∈N \ jti j. The regulator tests the product of firm i at level tRi ≥ 0 for i ∈N , and incurs a cost Σi∈N tRi. Let

ti =<t1i, t2i, .., ti−1,i, ti+1,i, .., tNi, tRi> denote the vector of testing levels. If firm i’s product is noncompliant,

then with probability di(ti) testing provides evidence of this noncompliance to at least one party that tested

firm i at a strictly positive level. A firm submits evidence to the regulator that a competitor’s product is

noncompliant if and only if its testing produces that evidence. The evidence characterizes the mode by

which the product fails to meet the standard and, upon obtaining that evidence, the regulator confirms that

the product is noncompliant and prevents its sale. Thus, with probability

si (ei, ti)≡ 1−di (ti)(1− ei)

firm i successfully brings its product to market; with probability di (ti)(1− ei) firm i sells nothing. That is,

the sales quantity for firm i ∈N is the random variable mqiwith

qi≡{

qi with probability si (ei, ti)0 with probability 1− si (ei, ti) .

We assume that for a given vector of the firms’ compliance and testing efforts, the qi

and qj

for j 6= i

are independent. The detection probability for a noncompliant product di(ti) is componentwise strictly

increasing, continuously differentiable and satisfies di(0, ..,0) = 0 and limt ji→∞ di(ti) = d for j ∈ {R,N \i}

where d ∈ (0,1). Furthermore, to the extent that the detection probability for firm i is already high, additional

testing is less effective: for j ∈N ∪R and i ∈N \ j,

(∂/∂ t ji)di(ti)< (∂/∂ t ji)di(t′i) for ti, t

′i ∈ RN+1

+ such that di(ti)> di(t′i). (1)

As in the workhorse model of vertically-differentiated quality used in (Motta 1993) and references

therein, the product of firm i has “quality” ui, and the mass m of consumers in the market are differenti-

ated by a willingness-to-pay-for-quality parameter α , uniformly distributed on [0,1], such that a consumer

with parameter α that purchases product i at price pi has expected utility

αui− pi, (2)

and will buy the product on the market with the maximum (2), if that is nonnegative, and otherwise will not

buy a product. Under the standard condition Σi∈N qi < 1, the unique market equilibrium price per unit for

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each firm i’s product is

pi = ui− Σj∈N

min(ui,u j)q j; (3)

(3) is derived in the appendix. Note that qi is the share of the consumer market captured by firm i if it

successfully brings its product to market; we will refer to parameter qi as firm i’s market share.

Hence firm i’s expected profit (gross of fixed production costs) is

πi =

[ui(1−qi)− Σ

j∈N \imin(ui,u j)s j (e j, t j)q j

]si (ei, ti)mqi− ci(ei)−Σ j∈N \iti j. (4)

Each firm i ∈N chooses its compliance ei ∈ [0,1] and testing of competitors ti j > 0 for j ∈N \i to maxi-

mize (4), given its beliefs about competitors’ compliance and testing decisions. (After exerting compliance

effort ei, firm i does not know with certainty whether its product is actually compliant (unless ei ∈ {0,1});

instead the firm knows this likelihood ei. The other firms j 6= i cannot observe ei. Whether or not a product

is compliant (a credence attribute) can be observed only through testing. Hence although compliance ef-

fort occurs prior to testing, for purposes of analysis, this is a simultaneous-move game.) We focus on pure

strategy Nash equilibria in compliance and testing by the firms.

As motivated by the review of related literature, we treat the regulator testing tRi of each firm i ∈N as

a parameter for sensitivity analysis, and assume that the firms correctly anticipate {tRi}i∈N .

Quantity, Quality and Consumers’ Beliefs about Compliance: Exogenous in §3, Endogenous in §4

Production quantity is not a decision variable in the base model analyzed in §3. That is equivalent to

assuming that each firm produces at capacity, as in (Chod and Rudi (2005), Anand and Girotra (2007), and

Swinney et al. (2011)), in which case mqi may be interpreted as the capacity of firm i. That assumption is

reasonable under a new product standard because firms’ compliance efforts and the potential for blocking

increase the expected selling price, which tends to strengthen a firm’s incentive to produce at capacity.

Regarding consumers’ utility from compliance with the product standard, the base model of consumer

utility in (2) with fixed {ui}i∈N has two interpretations. First, consumers are indifferent to whether a product

is compliant with the standard (which might be the case with a standard that benefits the environment and

does not directly benefit a consumer). Second, consumers trust that products on the market comply with the

standard, in which case ui reflects the utility a consumer perceives in having a compliant product from firm

i.

§4 extends the model and analysis, to allow consumers to value compliance and have rational expecta-

tions about the likelihood that a product in the market is compliant, to allow a firm to choose its product

quality apart from compliance, and to allow a firm to choose its production quantity, among other extensions.

3. Results

3.1. Testing by the Regulator

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What is the impact of regulator testing on the firms’ testing and compliance efforts? Proposition 1 shows

that a small amount of regulator testing reduces firms’ testing and has no impact on compliance; a large

amount of regulator testing prevents firms’ testing and can reduce compliance.

In order to state Proposition 1, consider an initial equilibrium in compliance and testing by the firms

{ei, t ji}i∈N , j∈N \i when the regulator does not test (tRi = 0 for i ∈N ) and define

τi ≡max{tRi : di(0, ...,0, tRi)≤ di(t1i, ..., tNi,0)} for i ∈N . (5)

Lemma 1 implies existence of that initial equilibrium {ei, t ji}i∈N , j∈N \i.

Lemma 1 For any given amount of regulator testing {tRi}i∈N , an equilibrium exists in the firms’ testing

and compliance efforts.

The proofs of the results in this section are in the appendix.

Proposition 1a considers an increase in regulator testing from zero (tRi = 0 for i ∈ N ) to a “small”

amount (tRi ≤ τi for i ∈ N ). “Small” is a relative term, meaning that the probability that the regulator

would detect a noncompliant product if competitors did not test, di(0, ...,0, tRi) in (5), is smaller than the

initial probability that the competitors detect a noncompliant product when the regulator does not test,

di(t1i, ..., tNi,0) in (5). Insofar as the regulator is less effective than firms in testing, and insofar as firms are

motivated to test their competitors, each τi is actually large. Similarly, Proposition 1b considers an increase

in regulator testing from zero (tRi = 0 for i ∈N ) to a “large” amount (tRi > τ i for i ∈N ), where

τ i ≡min{tRi : (∂/∂ t ji)di(0, ..0, tRi)≤ 1/(mu j) for j ∈N \i}. (6)

Proposition 1 (a.) A “small” amount of testing by the regulator (tRi≤ τi for i∈N ) reduces the equilibrium

testing by each firm while preserving each firm’s equilibrium compliance and detection probability. (b.) A

“large” amount of testing by the regulator (tRi > τ i for i ∈N ) ensures that the firms do not test, t ji = 0 for

i ∈N , j ∈N \i and can strictly reduce the unique equilibrium compliance ei of every firm i ∈N .

The rationale for Proposition 1a is that testing by the regulator discourages firms from testing competi-

tors’ products, by reducing the marginal probability that a firm can “knock out” a competitor through its

own testing effort. As the regulator increases testing of firm i from tRi = 0 to a “small” level tRi ≤ τi, each

competitor firm j optimally reduces its testing of firm i to the extent that the detection probability for firm

i remains the same.4 Every firm’s optimal compliance effort and testing of competitors other than firm i

depend on the detection probability for firm i, not how that detection probability is achieved through reg-

ulator versus competitors’ testing of firm i. Hence for every firm, the initial (with zero regulator testing)

4The preservation of the detection probability relies on the assumptions that the detection function is

differentiable, componentwise strictly increasing, and satisfies (1), which are used only in the proof of

Proposition 1a. §4.7 relaxes those assumptions and extends Proposition 1a.

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equilibrium compliance effort and testing of competitors other than firm i remains optimal, so the detection

probability for a noncompliant product for each firm also remains the same as in the initial equilibrium.

The rationale for Proposition 1b is similar in that at sufficiently large levels of regulator testing, the firms

cease to test. To prove that regulator testing can strictly decrease compliance effort by all firms in the market,

the proof of Proposition 1b provides a simple example with N = 2 firms, in which the regulator applies the

same level of testing to both firms (tR1 = tR2), and the firms are symmetric except that a violation by firm

2 is more difficult to detect, especially for the regulator. Without testing by the regulator (tR1 = tR2 = 0),

each firm tests its competitor. When the regulator applies a large level of testing (tR1 = tR2 > max(τ1,τ2)),

the firms cease to test (t12 = t21 = 0) in the unique equilibrium. Because it is easier to detect a violation

by firm 1, the regulator’s common testing level increases the detection probability of firm 1 and decreases

the detection probability for firm 2. The latter causes firm 2 to decrease its compliance effort. The decrease

in firm 2’s detection probability makes firm 2 more likely to bring its product to market, which reduces

the expected market price, and thereby causes firm 1 to decrease its compliance effort. Firm 1 is no longer

motivated to test firm 2’s product because the regulator is doing so and because firm 1’s detection probability

is higher and its compliance lower, so firm 1 is less likely to bring its product to market, and hence less likely

to benefit from knocking firm 2 out of the market.5

In short, Proposition 1 shows that a small amount of regulator testing will be ineffectual, and a large

amount can be counterproductive. (Why would a regulator test at an ineffectual or counterproductive level?

A small budget earmarked for testing might cause a regulator to choose an ineffectually small level of

testing. Inability to commit to a testing level before firms choose their compliance efforts and utility from

detecting violations might cause a regulator to choose a high testing level that reduces firms’ compliance

efforts.)

The policy implication of Proposition 1 is that relying on competitor testing (setting regulator testing

tRi = 0 for i ∈N )6 might be socially optimal. If firms test competitors when the regulator does not test,

introducing a relatively “small” level of testing by the regulator reduces social welfare by increasing the

total social cost of testing (crowding out the presumably more efficient testing by firms) without improving

detection or compliance, according to Proposition 1a.7 To potentially increase detection or compliance,

5The regulator’s being less effective in testing than the firms is not essential. The online supplement

provides an example wherein testing by a more effective regulator reduces every firm’s compliance effort.

However, that one firm has a lower detection probability under regulator testing, as in the example, is

necessary for regulator testing to strictly reduce every firm’s compliance effort.6To implement this in practice, government could give a regulator responsibility and budget only to

verify reported violations.7To be precise, a sufficient condition for nonzero regulator testing (setting tRi > 0 for at least one i ∈N

instead of tRi = 0 for i ∈N ) to strictly reduce social welfare is: (7) in Proposition 2a below, which ensures

τi > 0 for at least one i ∈N ; tRi ≤ τi for i ∈N ; and (∂/∂ tRi)di(ti) < (∂/∂ t ji)di(ti) for all i, j ∈N with

9

the regulator must incur an even higher testing cost, which is not worthwhile insofar as firms’ detection

probabilities and compliance efforts are large without regulator testing.

Therefore, in §3.2, we assume that the regulator does not test, and characterize conditions under which

all firms’ products are tested by competitors and the firms’ compliance efforts are large. Under those condi-

tions, regulator testing would crowd out testing by competitors and potentially reduce social welfare. When

those conditions do not hold, regulator testing, targeted so as to complement competitor testing, can be

useful, as explained at the end of §3.2.

3.2. Testing Only by Competitors

Proposition 2 highlights the critical importance of firms’ quality levels {ui}i∈N in stimulating competitor

testing and hence compliance. The quality parameter ui can be interpreted as the utility that a consumer

expects from firm i’s product, which reflects the strength of firm i’s brand, the consumer’s utility from

compliance with the standard (assuming the consumer trusts that a product on the market is compliant), and

the consumer’s utility from other quality attributes of the product.

Proposition 2 (a.) In any equilibrium, at least one firm tests a competitor (tnk > 0 for some n ∈N and

k ∈N \n) if and only if at least two firms have sufficiently high quality

min(ui,u j)> 1/{q jmqi(∂/∂ ti j)d j(t j)|t j=0} for some j ∈N and i ∈N \ j. (7)

(b.) In any equilibrium, if firm j’s quality is sufficiently low

u j ≤ 1/{q j maxi∈N \ j[mqi(∂/∂ ti j)d j(t j)|t j=0]}, (8)

then firm j draws no testing and does not comply (ti j = 0 and e j = 0 for all i ∈N \ j).

The proof shows that firm i tests firm j’s product (or vice versa) if both have sufficiently high quality

(7) and no other party tests their products. The robust underlying phenomenon is that in a market with

products of vertically-differentiated quality, the boost in the price of product i from knocking out product j

is proportional to the quality of the lower-quality product min(ui,u j).8 Consequently, firm i is motivated to

test firm j ’s product insofar as both firms’ products are of high quality. As a further consequence, if firm j

has sufficiently low quality (8), no competitor tests its product, so firm j has no incentive for compliance.

i 6= j and ti ∈R+N , meaning that the regulator is less efficient than the firms in testing.

8As shown in the derivation of equilibrium prices in the Appendix, a market allocates higher quality

products to consumers with higher willingness-to-pay-for-quality α . If firm i knocks out a higher-quality

competitor j (one with u j > ui), then consumers with higher α buy firm i’s product, so firm i’s price pi

increases by an amount proportional to firm i’s own quality ui. The market price pi for firm i’s product

depends on the quality levels of firms with lower quality (not the quality levels of firms with higher quality)

because the price of each product is determined by its marginal consumer’s indifference between purchasing

that product or the next lower-quality product (or no product). If firm i knocks out a lower-quality competitor

j (one with u j < ui), the marginal consumer for firm i’s product remains the same but her alternative is worse

by an amount proportional to u j, the quality of the product knocked out of the market, so the price she is

willing to pay for product i increases by an amount proportional to u j.

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Under the additional assumption c′i(0) = 0 for i ∈N , meaning that a firm can at low cost achieve some

positive probability that its product will be compliant, (7) is the necessary and sufficient condition for at

least one firm to exert some compliance effort. Under the additional assumption, any firm that draws testing

exerts strictly positive compliance effort.

Proposition 2 shows that testing of competitors occurs in a large or concentrated market. Specifically,

(7) tends to hold when the market size m is large or market share is concentrated in the hands of two firms

(so max j∈N , i∈N \ j{q jqi} is large). The rationale is that a firm j tends to draw testing when its market share

q j is large, so that knocking firm j out of the market substantially increases the market price for every other

firm’s product (3). Furthermore, a firm i is motivated to test competitors insofar as the market m or firm i’s

own market share qi is large, because the increase in firm i’s per-unit price pi from knocking out a competitor

is multiplied by firm i’s production quantity mqi.

Proposition 2 also shows that testing of competitors occurs when one firm has insider knowledge that

makes a small amount of testing likely to be effective. Specifically, condition (7) holds and (8) is violated

when (∂/∂ ti j)d j(t j)|t j=0 is sufficiently large. In practice, that could occur because firm i has insider knowl-

edge (perhaps a tip-off from a supplier or other channel partner) regarding how competitor j’s product is

likely to be noncompliant, so that with minimal testing expenditure, firm i has a substantial probability of

detecting a violation by competitor j.

As a corollary to Proposition 2b, enforcement of a product standard through competitor testing strictly

improves the profitability of firms with low quality (8). The counterfactual scenario has no product standard

or, equivalently, a standard that is not enforced, so that firm i’s expected profit is (ui−Σ j∈N min(ui,u j)q j)mqi.

Corollary A product standard, enforced through competitor testing, increases the expected profit of firms

with low quality (8), which do not comply with the standard.

The rationale is that a low-quality firm spends nothing on compliance, may choose not to test, and sells at a

higher price when its higher-quality competitors knock each other out of the market.

Focusing on the case where the firms are symmetric, Proposition 3 characterizes the impact of the

number of firms and their quality on the firms’ equilibrium compliance and testing efforts, and the resulting

detection probability for a noncompliant product. To ensure existence of a unique symmetric equilibrium,

in which each firm exerts compliance effort e and tests each competitor at level t, Proposition 3 assumes

that the cost of compliance is strictly convex, high compliance effort is costly and the detection probability

function is component-wise sufficiently concave, for e ∈ [0,1)

c′′(e)> 0 and lime↑1c(e)> mu (9)

(∂ 2/∂ t2i j)d(t j)<−max{1/d(t j),mu/[(1−d)3c′′(e)]}N(∂/∂ ti j)d(t j)

2. (10)

Convexity in compliance cost and concavity in detection probability are natural in that one would expect a

11

firm to prioritize the most cost effective activities. For example, to increase the likelihood of detecting non-

compliance in a competitor product, a firm could: gather additional information about what failure modes

are most likely (e.g., by identifying a competitor’s riskiest suppliers); or increase the scope or sophistication

of its tests. Similarly, to increase the likelihood that its product complies with the standard, a firm could:

increase its care in product design, supplier selection or manufacturing; or subject its own product to more

rigorous inspection. To the extent that activities differ in their cost effectiveness and that a firm prioritizes

its activities accordingly, that convexity and concavity become more pronounced. In (10), greater convexity

in the compliance cost lessens the degree of concavity required in the detection probability function.

Proposition 3 Suppose the firms are symmetric and (9)-(10) hold. In the unique symmetric equilibrium,

firms test (t > 0) if and only if u> 1/[mq2(∂/∂ ti j)d j(t j)|t j=0].Compliance e increases with the firms’ quality

levels u and with the market size m, and decreases with the number of firms N. The detection probability for

a noncompliant product increases with the number of firms N for N ≤ N and decreases with N for N ≥ N.

The basic reason that equilibrium compliance effort e increases with the firms’ quality levels u and with

market concentration (a smaller number N of firms supplying the market) is that those exogenous factors

drive up the market price (3). Hence each firm has greater value from bringing products to market, and

correspondingly stronger incentive for compliance effort to increase the probability of doing so. Similarly,

an increase in the market size m increases each firm’s value from bringing a product to market, which

stimulates compliance effort.

However, the proof of that monotonicity in equilibrium compliance effort e is subtle because, in contrast,

equilibrium testing t and the resulting detection probability for a noncompliant product are not monotonic

in the market size m, quality u, and number of firms N. Equilibrium testing t increases with the number

of firms N for N ≤ N and decreases with N for N ≥ N because reduced compliance strengthens a firm’s

incentive for testing when compliance is high (as is the case with few firms N ≤ N) and weakens this

incentive when compliance is low (as is the case with many firms N ≥ N). Reduced compliance has two

countervailing effects on testing. First, a competitor’s reduced compliance strengthens a firm’s incentive for

testing because testing is more likely to reveal noncompliance. This testing-incentive-strengthening effect

is strong when the testing firm’s compliance is high because then the firm is likely to bring products to

market, so the firm’s incentive for testing is sensitive to its competitor’s reduced compliance. Second, a

firm’s reduced compliance weakens its incentive to test because the firm is more prone to being blocked

from the market. This testing-incentive-weakening effect is weak when the competitor firm’s compliance

is high because then it is unlikely that testing will be effective in blocking the competitor firm’s products

from the market, so the incentive for testing is insensitive to the reduction to the testing firm’s reduced

compliance.

12

A synthesis of the propositions in §3.1 and §3.2 reveals the conditions under which regulator testing

can be useful, as opposed to socially inefficient. The latter occurs when firms’ quality levels are high

and the market size is large, because then–without regulator testing–all firms’ products would be tested by

competitors and their compliance efforts would be large (Proposition 3). Hence regulator testing would

crowd out competitor testing (Proposition 1) and likely reduce social welfare as explained in the last two

paragraphs of §3.1. When, to the contrary, the firms’ quality levels are low or the market size is small,

regulator testing can be useful. First, because firms do not test competitors’ products (Proposition 2a),

regulator testing is needed to detect and block noncompliant products from the market. Second, a regulator’s

sufficiently intense testing of a firm’s product pushes the firm, that otherwise would not exert compliance

effort, to do so. More generally, these benefits are captured by targeted regulator testing of the product of

any firm with low quality or low market share (Proposition 2b). Such targeted regulator testing complements

firm testing, which is directed to products of competitors with high quality and market share. If the social

cost of noncompliance is sufficiently high, it may be useful for the regulator to test the products of a broader

set of firms, including products that would otherwise be tested by competitors. For this to be socially

beneficial, the regulator must test those products at a high level, as otherwise its testing will simply crowd

out the more efficient testing by competitors (Proposition 1a).

4. Extensions

4.1. Endogenous Quality: Entrants

The Corollary to Proposition 2b suggests that a product standard, enforced through competitor testing

(alone), will cause entry by low-quality firms with little incentive for compliance. Proposition 4 confirms

that such entry occurs, in the following, extended version of our model. Suppose that a firm can enter the

market with production quantity mq. An entrant chooses whether to develop a product with low quality ul

or high quality uh, where ul < uh,

ul ≤ 1/[mq2 max j∈N \i(∂/∂ t ji)di(ti)|ti=0], (11)

and the entry cost is kl or kh depending on whether the entrant chooses low or high quality, respectively,

where kl < kh. A firm enters if doing so yields nonnegative expected profit. Initially (prior to adoption of

the product standard), the market is in an equilibrium with more than two high-quality firms; a sufficient

condition for existence of such an equilibrium is that their entry cost kh ≤ min(uh(1− 3q)mq,kluh/ul).

Similarly, assume that kl < ul(1− 2q)mq so a low-quality firm might enter. Finally, assume that testing

costs are sufficiently low that, after adoption of the product standard, at least one incumbent firm does some

testing.

Proposition 4 A product standard, enforced through competitor testing, causes entry by at least one firm

13

with low quality ul (the type of firm that does not comply) and no firms with high quality uh if detection is

sufficiently easy

(∂/∂ ti j)d j(t j)≥ γ if d j(t j)≤ d, (12)

and compliance is sufficiently costly

ci(ei)≥ c for ei ≥ e,

for i ∈N , where γ ∈ (0,∞), d < 1, c ∈ (0,∞) and e ∈ (0,1).

The proofs of all the results in this section are in the online supplement.

4.2. Endogenous Quality: Incumbents

We now turn to the impact of a product standard, enforced through competitor testing, on the quality chosen

by incumbent firms. Suppose that each firm i ∈ N privately chooses a quality level ui ∈ {u1,u2, ..,uM}

simultaneously with compliance ei. The cost to firm i of choosing quality un is kni . If a firm is indifferent

between two quality levels, the firm chooses the higher quality level.

Our main insight is that enforcement of a product standard through competitor testing reduces quality.

Proposition 5 establishes that the quality reduction occurs for all firms, when firms are symmetric. As in the

above Corollary and Proposition 4, the counterfactual scenario has no product standard or, equivalently, a

standard that is not enforced.

Proposition 5 If the firms are symmetric, a product standard, enforced through competitor testing, reduces

quality in any symmetric equilibrium.

As is clear from the proof, a parallel results holds with asymmetric firms: a product standard, enforced

through competitor testing, reduces the quality of firms with highest equilibrium quality in the counterfactual

scenario.

In reality, firms may observe competitors’ investments in quality (or advertising expenditures, or other

costly efforts to strengthen a brand) before deciding how much to test competitors’ products. Consider,

therefore, a sequential-move version of the game, with observable investment in quality followed by testing.

A new product standard, enforced through competitor testing, can strictly reduce the quality investment of

every incumbent firm, and do so to greater extent than in the initial version of the game with unobserved

quality; a numerical example is given in the online supplement. Quality investment tends to be lower in the

sequential-move game with observable quality than in the initial version of the game because lowering one’s

quality reduces the incentive for testing by competitors. The caveat is that lower quality may signal lower

compliance effort, and thereby, indirectly, increase the incentive for testing by competitors.

4.3. Consumer Utility from Compliance

14

Suppose that a consumer who purchases product i at price pi has expected utility (2) in the event that the

product is noncompliant, and has expected utility

αui(1+∆)− pi (13)

in the event that the product is compliant, where ∆≥ 0. We refer to ∆ as a consumer’s utility from compli-

ance. Recall that consumers have heterogeneous levels of the parameter α, which may be attributed to their

heterogeneous income levels; a consumer with high income has low price-sensitivity, corresponding to high

α.

Further suppose that consumers form expectations regarding firm i’s compliance effort ei, the testing

applied to firm i by its competitors and the regulator ti, and the resulting probability ei/si

(ei, ti

)that firm

i’s product is compliant given that it is available for sale in the market. Hence, the expected utility of a

consumer with valuation parameter α from purchasing firm i’s product is αui

[1+∆ei/si

(ei, ti

)]− pi. We

focus on rational expectations equilibria that are consistent, i.e., satisfy ei = ei and ti = ti for all i ∈ N.

Therefore the market equilibrium price for product i and firm i’s expected profit are given by (3) and (4)

respectively, with un replaced by

un ≡ un

[1+∆en/sn

(en, tn

)]for n ∈ {i, j}. (14)

All of the preceding Propositions hold, with explicit incorporation of the utility from compliance. Propo-

sition 1 holds with u j(1+∆) replacing u j in (6). Proposition 2a holds under the conservative (for purposes

of evaluating the effectiveness of competitor testing) assumption that if multiple equilibria exist, including

one with zero testing, then zero testing occurs. Even without that assumption, (7) is sufficient for at least

one firm to test a competitor, and (7) with min(ui,u j)(1+∆) replacing min(ui,u j) is sufficient for no firm

to test a competitor. Proposition 2b holds with u j(1+∆) replacing u j in (8). Proposition 3 holds with the

following generalized conditions to ensure a unique equilibrium: In (9), mu∆/(1− d)3 replaces 0; in (10),

u(1+∆)2 replaces u, and (1−d)3c′′(e)−mu∆ replaces (1−d)3c′′(e). Furthermore, Proposition 3 extends in

that compliance e increases with a consumer’s utility from compliance ∆. Proposition 4 holds. Proposition

5 holds with an upper bound on ∆.

In sum, relying on competitor testing tends to be effective when consumers value compliance (∆ is large)

in that competitor testing stimulates high compliance effort (Proposition 3). Conversely, when consumers

value neither compliance nor the product itself (∆ and ui for i ∈N are small), relying on competitor testing

is ineffective because firms choose not to test competitors’ products (Proposition 2).

We adopt this generalized formulation in all subsequent sections.

4.4. Other Penalties for Noncompliance

In the event that a firm’s product is shown to be noncompliant, instead of blocking the sale of the prod-

uct, a regulator could simply notify consumers that the product is noncompliant. Notification could be

15

accomplished by publishing a list of noncompliant products or preventing a firm from labeling its product as

compliant. For example, the U.S. Department of Energy employs this “labeling” approach in enforcement

of the voluntary Energy Star standard. In this labeling scenario, firm i always sells mqi units, but at a reduced

price in the event that its product is shown to be noncompliant. The unique market equilibrium price per unit

for each firm i’s product is

pi = ui− Σj∈N

min(ui,u j)q j,

where u j is the random variable

u j ≡{

u j with probability s j (e j, t j)u j with probability 1− s j (e j, t j) ,

with u j defined in (14). For a given vector of the firms’ compliance and testing efforts, we assume that ui

and u j for j 6= i are independent. Therefore, firm i’s expected profit (gross of fixed production costs) is

πi =

[ui(1−qi)− Σ

j∈N \i

{s j (e j, t j)min(ui, u j)+ [1− s j (e j, t j)]min(ui,u j)

}q j

]si (ei, ti)

×mqi+

[ui(1−qi)− Σ

j∈N \i

{s j (e j, t j)min(ui, u j)+ [1− s j (e j, t j)]min(ui,u j)

}q j

]× [1− si (ei, ti)]mqi− ci(ei)−Σ j∈N \iti j. (15)

Another motivation for this labeling scenario is that, in the absence of a government-imposed standard, firms

may test competitors’ products in order to detect and inform consumers of a defect or false claim. For ex-

ample, competitor testing alone serves to enforce voluntary product labeling standards, devised by industry

and other non-governmental organizations, for “natural” and “organic” personal-care products (Story 2008,

Struck 2008).

Suppose that when a firm’s product is shown to be noncompliant, the firm incurs a fine f ≥ 0. That

introduces the term− f [1−si (ei, ti)] into the objective function for firm i in (4) for the scenario with blocking

and in (15) for the scenario with labeling. That “fine” f may also incorporate costs associated with civil

lawsuits, criminal penalties, long-term reputational damage, or (with blocking) the cost to safely destroy

and dispose of a product shown to be noncompliant.

The propositions in §3 hold with the fine f ≥ 0 and blocking. Propositions 1, 2, and 3 hold. Proposition

3 extends in that the symmetric equilibrium compliance increases with the fine f , and there exists f such

that the symmetric equilibrium detection probability increases with f for f ≤ f and decreases with f for

f ≥ f .

With the penalty of the fine f ≥ 0 and labeling, the propositions in §3 largely hold. Proposition 1a

and the first part of Proposition 1b hold. Proposition 2b holds when minn∈N {un}/(1+∆) is added to the

right hand side of (8). The comparative statics in Proposition 3, including those for the fine, hold for the

symmetric equilibrium with the largest compliance level; nonuniqueness arises because of existence of an

equilibrium with zero compliance and testing. Existence of that equilibrium nullifies Proposition 2a.

16

Should a regulator strip offending products of labels (such as “Energy Star”), instead of blocking them

from the market? Doing so reduces compliance effort by incumbent firms, in the context of Proposition 3.

Specifically, for any fine f ≥ 0, any symmetric equilibrium compliance under labeling is lower than the

unique symmetric equilibrium compliance under blocking. However, with asymmetric firms, the extended

Proposition 2b implies that the switch to labeling might cause firms with low–but not too low–quality to draw

testing by a competitor and exert some compliance effort. Labeling reverses the results in the Corollary and

Proposition 4: A product standard, enforced through competitor testing and labeling, decreases the expected

profit of firms that do not comply and, in the setting of Proposition 4, does not cause entry by low-quality,

noncompliant firms. The rationale for that reversal is that, in the labeling scenario, a firm that draws testing

from competitors and therefore chooses to exert compliance effort earns higher profit; consumers become

willing to pay a higher price for its product than if the firm did not draw testing. To summarize, the switch

to labeling eliminates the problem of entry by low-quality, noncompliant firms, but may reduce incumbents’

compliance efforts.

Another argument for blocking is that in a labeling scenario, with a different sequence of decisions than

assumed in this paper, a firm may choose not to report a competitor’s violation. In (Li and Peeters 2014), an

incumbent firm decides whether to incur cost to learn the quality level of an entrant’s product and whether to

report that information to consumers. Afterwards, the incumbent and entrant set their prices, then produce to

meet consumer demand. The main result in (Li and Peeters 2014) is that, in a specified parameter region, the

incumbent chooses not to report a defect in the entrant’s product, because doing so would cause the entrant

to set a lower price and thereby reduce the incumbent’s sales. In contrast, in this paper, firms’ production

quantities are set in advance, so reporting a competitor’s violation to consumers is always optimal.

4.5. Endogenous Production Quantities

Suppose that each firm i ∈N chooses its quantity mqi, unobserved by its competitors, at cost Ci(qi), where

Ci(·) is a strictly increasing function. That is equivalent to having each firm i ∈N choose its target market

share qi, because the market size m is an exogenous parameter. The lead time for production is suffi-

ciently long that a firm chooses its quantity prior to observing which products are blocked. For purposes

of analysis, this is captured by assuming that firms make quantity, in addition to compliance and testing,

decisions in a simultaneous-move game. The propositions in §3 are robust in this extension. Proposi-

tion 1 holds. Proposition 2 holds where: in the sufficient condition for a firm to draw testing (7), qn for

n ∈ { j, i} is firm n’s equilibrium production quantity when no firm draws testing; the sufficient condition

for no firm to draw testing is (7) being violated where (1+∆)m replaces q jmqi; and (8) is replaced by

u j ≤ 1/{(1+∆)mmaxi∈N \ j[(∂/∂ ti j)d j(t j)|t j=0]}. The online supplement provides numerical results con-

sistent with Proposition 3.

17

4.6. Fixed Costs for Testing

Suppose that each firm i ∈N must incur a fixed cost φi ≥ 0 in order to test the product of any competitor-

firm j ∈N \i at a strictly positive level ti j > 0. This captures the setting in which firm i needs to acquire

equipment and/or technical expertise to test its competitors’ products; φi represents the fixed cost associated

with obtaining this competency. The base model formulation in §2, wherein φi= 0, represents the case where

the firm already has this competency, for example, due to prior investments in its own product development

process (Wiel and McMahon 2005). Alternatively, φi = 0 may arise because firm i relies on an outside

laboratory to conduct tests. Similarly, suppose the regulator incurs a fixed cost φR ≥ 0 if tR j > 0 for any

j ∈N .

The propositions in §3 largely hold. Proposition 1a holds, with a more complex expression for τi reflect-

ing that a marginal increase in regulator testing can motivate a discontinuous drop in a firm’s testing from

a strictly positive level to zero. Proposition 1b holds. In fact, a stronger version of Proposition 1b holds

in that, for some parameters, introducing testing by the regulator strictly reduces the detection probability

for a noncompliant product for every firm, in addition to strictly reducing compliance effort for every firm.

Regarding Proposition 2a, (7) is sufficient for at least one firm to test a competitor provided that the fixed

cost of testing is not too large, and (7) with min(ui,u j)(1+∆) replacing min(ui,u j) is sufficient for no firm

to test a competitor. Proposition 2b holds with u j(1+∆) replacing u j in (8). Proposition 3 holds provided

that the fixed cost of testing is not too large.

4.7. Alternative Detection Probability Function

In the base model formulated in §2, the detection probability function is component-wise strictly increasing.

That requires that when multiple firms test the product of firm i, their testing activities are not perfectly

duplicative. That is natural if the firms have different sources of information regarding what failure modes

are likely or different testing capabilities. The base model also captures a scenario wherein firms conduct

the same noisy test that randomly fails to detect a violation, so independent repetition of the test increases

the probability that at least one detects a violation. In another scenario, firms pay a testing service provider

to test the product of a competitor, the testing service provider prioritizes the most cost-effective testing

activities, and therefore the detection probability is a strictly increasing and concave function of firms’ total

expenditure Σi∈N \ jti j; that no-duplication scenario also is captured in the base model.

Let us now consider the case of perfect duplication in testing activities that is not captured in the base

model. Suppose that the detection probability for a noncompliant product di(ti) = Di(maxn∈R∪N \i{tni}),

where Di(·) is strictly increasing, strictly concave, differentiable and satisfies Di(0) = 0 and limt→∞ Di(t) =

d, where d ∈ (0,1). Further, in (10), D(t) replaces d(t j), D′(t) replaces (∂/∂ ti j)d(t j), and D′′(t) replaces

(∂ 2/∂ t2i j)d(t j). That alternative detection function is not componentwise strictly increasing, is not continu-

18

ously differentiable, and does not satisfy (1).

Nevertheless, the preceding Propositions hold. The caveat is that Proposition 1a is modified in that

the expression for τi is more complex (reflecting that a marginal increase in regulator testing of firm i can

motivate a discontinuous drop in competitor testing from a high level to zero, and thus strictly reduce the

detection probability for firm i) and that “maintains the equilibrium testing” replaces “reduces the equilib-

rium testing.” The central message of Proposition 1a, that a small of amount of testing by the regulator may

be socially inefficient, continues to hold, but because such regulator testing duplicates rather than crowds

out testing by the firms. Propositions 1b, 2, 3, 4 and 5 hold. The modified Propositions described in the

preceding extensions subsections also hold.

5. Concluding Remarks

Insights from the stylized model in this paper suggest that testing by government regulatory authorities to

detect violations of a product standard may be detrimental to social welfare–when that testing is directed

toward products that would otherwise be tested by competitors. Testing by a regulator crowds out testing

by competitors and thereby increases the overall social cost of testing, because regulator testing is less

efficient, as documented in §1. With a small budget–as is common in practice (Bruschia 2008, Smith 2008)–

regulator testing fails to improve firms’ compliance efforts and fails to increase the detection probability for

a noncompliant product. With a large budget, regulator testing can strictly reduce firms’ compliance efforts

and strictly reduce the detection probability for a noncompliant product.

Competitor testing alone is effective in enforcing a product standard if consumers highly value the

product or compliance with the product standard, or if the market is concentrated or large. Market integration

and harmonization of product standards (increasing the size of the market governed by a product standard)

spurs firms to test their competitors and exert greater compliance effort, as does each of the other listed

factors.

When relying on competitor testing to detect violations, stronger penalties may be detrimental. Increas-

ing the fine increases compliance effort, but can reduce competitor testing and the detection probability for

a noncompliant product. Blocking the sale of a product shown to be noncompliant–rather than simply pub-

licizing that the product is noncompliant–increases compliance effort by incumbent firms, but may cause

entry by low-quality, noncompliant firms that do not draw testing from competitors.

A caveat is that the stylized model in this paper does not account for collusion or reputational concerns

that, in reality, deter some firms from reporting competitors’ violations. For example, German automakers

secretly agreed to commonly adopt technology inadequate to clean diesel exhaust to emissions standards.

None of the automakers reported a competitor’s violation, because doing so would reveal its own violation.

The collusion came to light after an NGO’s testing detected violations (Dohmen and Hawranek 2017, Ewing

19

2017). Similarly, firms might fail to report competitors’ violations because they have a common reputation

(meaning that a firm would be harmed by publicity of a competitor’s violation, as in some food industries

(Winfree and McClusky 2005)) or a common supplier (so a firm might be harmed if a competitor’s violation

was traced to the common supplier (Lawrence et al. 2017)).

Another caveat is that with a voluntary standard like Energy Star, relying on competitor testing is effec-

tive only if consumers have favorable beliefs. In a pernicious equilibrium, consumers think that products

are noncompliant, firms have no incentive to test competitors’ products and–without testing–have no incen-

tive to comply with the standard. Indeed, the U.S. relied on competitor testing for enforcement of Energy

Star and consumers trusted the Energy Star label–until a General Accountability Office report and related

publicity of violations damaged that trust. In response, the U.S. Department of Energy instituted a new

requirement for firms to pay for third-party testing of their products to use the Energy Star label (Gaffigan

2007, GAO 2010, Rosner 2010).

Third-party testing has qualitatively the same impacts on firms’ compliance and testing efforts as does

regulator testing. Specifically, in the model in this paper, one may interpret the parameter tRi as the level

of third-party (e.g., independent testing laboratory or NGO) testing for firm i’s product. Proposition 1

holds with “third-party” substituted for “regulator” and for any specified levels of {tRi}i∈N . Of course, the

specified testing levels {tRi}i∈N and detection probability functions {di()}i∈N should reflect the objectives,

capabilities, and budget constraints of the third party, which differ from those of a regulator. For example, a

fiscally-constrained government may impose higher tRi with third-party testing than with regulator testing,

by requiring firm i to pay for the third-party testing–if firm i can afford to do so. To avoid putting small

producers out of business, the U.S. exempts them from some third-party product safety testing requirements

(Nord 2010). The results in this paper suggest that such small producers do not draw testing by competitors,

so regulator or third-party testing is needed to incentivize them to comply with credence standards.

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Appendix

Derivation of Market Equilibrium Prices (3): Index the firms such that quality ui increases with i. In any

equilibrium, prices pi increase with i, consumers with low willingness-to-pay-for-quality α ∈ [0,1−ΣNj=1q

j)

do not purchase, and the marginal consumer with α = 1−ΣNj=1q

jpurchases the product with lowest price

and quality, indexed l= argmin{u j : qj> 0},which implies pl = ul(1−ΣN

j=1qj), establishing (3) for product

23

i = l. For every other product in the market, i > l with qi> 0, the marginal buyer has α = 1−ΣN

j=iq jand

is indifferent between product i and the one with next-highest price and quality, indexed k = argmax{u j :

qj> 0, j < i}:

ui(1−ΣNj=iq j

)− pi = uk(1−ΣNj=iq j

)− pk. (16)

Having established (3) for product l, proceed by induction, assuming (3) holds for product k:

pk = uk− Σj∈N

min(uk,u j)q j. (17)

Together, (16) and (17) imply (3) for product i.

Proof of Lemma 1: A pure strategy Nash equilibrium exists in a game in which every player has a compact,

convex strategy set and a utility function that is continuous in all players’ strategies and quasiconcave in his

own strategy (Theorem 8.D.3 in Mas-Colell et al. 1995). The assumption that ci(·) is convex for i ∈N

ensures that it is continuous and, with the assumption that di(ti) is continuously differentiable for i ∈N ,

ensures that firm i’s objective function (4) is continuous in all the firms’ compliance and testing strategies

{ek, tk j}k∈N , j∈N \k. To establish that firm i’s objective function (4) is quasiconcave in firm i’s compliance

and testing strategy {ei, ti j} j∈N \i, observe that the assumption that di(ti) is componentwise strictly increas-

ing, continuously differentiable and satisfies (1) implies

(∂ 2/∂ tni∂ tki)di(ti)< 0 for n,k ∈N \i, (18)

and, in particular, d j(t j) is a strictly concave function of ti j for i ∈N and j ∈N \i. A sum of concave

functions is concave. A concave function remains concave when multiplied by a nonnegative constant. A

convex function (ci(·)) becomes concave when multiplied by a negative constant. Therefore, it remains to

show that eid j(t j) is a concave function of {ei, ti j} for each j ∈N \i. The Hessian matrix of eid j(t j),[ei(∂

2/∂ t2i j)d j(t j) (∂/∂ ti j)d j(t j)

(∂/∂ ti j)d j(t j) 0

],

is negative semidefinite due to (18), d j(t j) being componentwise strictly increasing, and the nonnegativity

of ei, so eid j(t j) is indeed a concave function of {ei, ti j}. To see that each firm i ∈N has a compact, convex

strategy set, recall that firm i∈N is constrained to choose ei ∈ [0,1] and ti j ≥ 0 for j ∈N \i. Firm i∈N is

also, effectively, constrained to choose ti j ≤ ui for j ∈N \i; by inspection of (4), firm i would have a strictly

negative objective value if firm i set ti j > ui for any j ∈N \i, whereas firm i is guaranteed a nonnegative

objective value by setting ei = 0 and ti j = 0 for all j ∈N \i. Therefore, by the aforementioned theorem, a

pure strategy Nash equilibrium exists in the firms’ testing and compliance efforts {ek, tk j}k∈N , j∈N \k.�Proof of Proposition 1: (a.) Each firm j’s objective π j is quasiconcave in its compliance and testing

{e j, t ji}i∈N \ j as shown in the proof of Lemma 1. Hence {e j, t ji} j∈N , i∈N \ j is an equilibrium if and only if

24

the first order conditions for compliance and testing hold for each firm j ∈N :

∂π j

∂e j

=

[u j(1−q j)− Σ

i∈N \ jmin(u j,ui)si (ei, ti)qi

]d j(t j)mq j− c′j(e j)

{ ≤ 0 if e j = 0= 0 if e j ∈ (0,1)≥ 0 if e j = 1,

(19)

∂π j

∂ t ji

=min(u j,ui)s(e j, t j)qimq j(1− ei)∂di(ti)

∂ t ji

−1{ ≤ 0 if t ji = 0= 0 if t ji > 0 for i ∈N \ j. (20)

Recall that {e j, t ji} j∈N , i∈N \ j denotes the initial equilibrium in compliance and testing by the firms when the

regulator does not test, tRi= 0 for i∈N . Consider the introduction of regulator testing tR=<tR1, tR2, .., tRN>

that satisfies tRi ≤ τi for i ∈N . We will construct best response testing levels for the firms {t ′ji}i∈N , j∈N \i

that are reduced from the initial testing levels so as to preserve the detection probability for a noncompliant

product:

t ′ji ≤ t ji for j ∈N , i ∈N \ j (21)

t ′ji < t ji for i ∈N with tRi > 0, for some j ∈N \i (22)

di(t′i, tRi) = di(ti,0) for i ∈N , (23)

with vector notation ti =<t1i, t2i, .., ti−1,i, ti+1,i, .., tNi> for the initial equilibrium testing of firm i by other

firms and t′i =<t ′1i, t′2i, .., t

′i−1,i, t

′i+1,i, .., t

′Ni> for their best response testing of firm i after the introduction

of regulator testing. We will then prove that {e j, t′ji} j∈N , i∈N \ j is an equilibrium, i.e., each firm’s initial

compliance level is preserved in equilibrium. To do so, it will be sufficient to prove that

(∂/∂ t ji)di(t′i, tRi)≤ (∂/∂ t ji)di(ti,0) and that inequality

holds with equality if t ′ji > 0, for j ∈N , i ∈N \ j. (24)

Together, (23), (24) and the fact that (19)-(20) hold at the initial equilibrium {e j, t ji} j∈N , i∈N \ j with zero

regulator testing imply that (19)-(20) hold for {e j, t′ji} j∈N , i∈N \ j with regulator testing tR.

The construction of τi in (5) implies that di(0, tRi) ≤ di(ti,0) for i ∈ N . If tRi = 0, then set t′i = ti.

If tRi > 0, then 0 < di(0, tRi) ≤ di(ti,0), so at least one competitor tested firm i in the initial equilibrium.

Let Ni denote the number of firms that tested firm i in the initial equilibrium. Suppose that the firms are

indexed so that firms 1, ..,Ni are doing that testing, i.e., t ji > 0 for j = 1, ..,Ni and t ji = 0 for j > Ni. Let

T1(t, ti) denote the vector of competitor testing levels for firm i with first component t and second through

Nth components identical to those of ti. In other words, T1(t, ti) transforms a vector ti by substituting t for

its first component. Set

t ′1i =min{t ≥ 0 : di(T1(t, ti), tRi)≥ di(ti,0)}.

As di is continuous and componentwise strictly increasing, one of the following two cases must occur. In

the first case, di(T1(t′1i, ti), tRi) = di(ti,0). In the second case, di(T1(t

′1i, ti), tRi)> di(ti,0) and t ′1i = 0. In the

first case, set t′i =T1(t′1i, ti). In the second case, sequentially reduce the testing of firm i by firms j = 2,3, ...

in the same manner, until the detection probability for a noncompliant product by firm i falls to the initial

25

level di(ti,0). Specifically, for j ∈N \i, let T j(t, ti) denote the vector of competitor testing levels for firm

i with components 1 through j− 1 set to zero, component j set to t, and all other components identical to

those of ti. In other words, T j(t, ti) transforms a vector ti by substituting t for its jth component, and 0 for

components 1 through j−1. Iteratively, starting with j = 2, set

t ′ji =min{t ≥ 0 : di(T j(t, ti), tRi)≥ di(ti,0)}.

If di(T j (t′ji, ti), tRi) = di(ti,0), then stop with t′i = T j (t

′ji, ti). Otherwise, di(T j (t

′ji, ti), tRi) > di(ti,0) and

t ′ji = 0. Increment j = j+ 1, until di(T j (t′ji, ti), tRi) = di(ti,0) and then stop with t′i = T j (t

′ji, ti). Observe

that di(T j (t′ji, ti), tRi) = di(ti,0) is achieved for j ≤ Ni because di is continuous and component-wise strictly

increasing, and di(0, tRi) ≤ di(ti,0). Follow this process to construct t′i for each i ∈N with the properties

(21)-(23).

It remains to prove (24). The proof is by contradiction, and uses the assumptions that di is contin-

uously differentiable, component-wise strictly increasing and satisfies (1). Suppose (∂/∂ t ji)di(t′i, tRi) >

(∂/∂ t ji)di(ti,0). Then, because (∂/∂ t ji)di is continuous with respect to t ji and di is component-wise

strictly increasing, there exists some ε > 0 such that, with t′′i defined to have t ′′ji = t ′ji + ε and t ′′ki = t ′ki

for k ∈N \{i, j}, (∂/∂ t ji)di(t′′i , tRi) > (∂/∂ t ji)di(ti,0) and di(t

′′i , tRi) > di(t

′i, tRi). The latter, with (1) and

(23), implies that (∂/∂ t ji)di(t′′i , tRi) < (∂/∂ t ji)di(ti,0), a contradiction. Similarly, for the case that t ′ji > 0,

suppose (∂/∂ t ji)di(t′i, tRi)< (∂/∂ t ji)di(ti,0). Then, because (∂/∂ t ji)di is continuous with respect to t ji and

di is component-wise strictly increasing, there exists some ε > 0 such that, with t′′i defined to have t ′′ji =

t ′ji− ε and t ′′ki = t ′ki for k ∈N \{i, j}, (∂/∂ t ji)di(t′′i , tRi) < (∂/∂ t ji)di(ti,0) and di(t

′′i , tRi) < di(t

′i, tRi). The

latter, with (1) and (23), implies that (∂/∂ t ji)di(t′′i , tRi)> (∂/∂ t ji)di(ti,0), a contradiction.

(b.) First, we show that if tRi > τ i for i ∈ N , then any equilibrium has t ji = 0 for i ∈ N and j ∈

N \i. Each τ i is a finite, nonnegative constant, due to our assumptions that di(ti) is componentwise strictly

increasing, continuously differentiable and satisfies (1) and limt ji→∞ di(ti) = d for j ∈ {R,N \i}, which

imply that limtRi→∞(∂/∂ t ji)di(0, ..,0, tRi) = 0. Those assumptions also imply that for any ti with regulator

testing tRi > τ i,

mu j(∂/∂ t ji)di(ti)−1≤ mu j(∂/∂ t ji)di(0, ..,0, tRi)−1< 0 for every j ∈N \i. (25)

Also observe that

(∂/∂ t ji)π j =min(ui,u j)s j (e j, t j)qimq j(1− ei)(∂/∂ t ji)di(ti)−1≤ mu j(∂/∂ t ji)di(ti)−1. (26)

Therefore, tRi> τ i implies that (∂/∂ t ji)π j < 0, so in any equilibrium t ji= 0. Second, we provide an example

with N = 2 firms and regulator testing tRi > τ i for i ∈ {1,2} in which every firm’s unique equilibrium com-

pliance ei is strictly lower than with zero regulator testing. Let m= u1 = u2 = 1, q1 = q2 = 0.48, d = 0.99,

ci(e) = 0.15e2 for i ∈ {1,2}, d1(t21, tR1) = min(√

8(t21+ tR1) ,

2d

(√[300(t21+ tR1)]2+300(t21+ tR1)−300(t21+ tR1)

))and d2(t12, tR2) = min

(√2t12+ tR2/150 ,

26

2d

(√[300(t21+ tR1)]2+300(t21+ tR1)−300(t21+ tR1)

)). Then, under regulator testing (tR1, tR2) =

(0.12,0.12), the unique equilibrium has zero testing by the firms and compliance (e1,e2)= (0.0835,0.0213).

Under (tR1, tR2) = (0,0), the unique equilibrium has testing by the firms (t21, t12) = (0.07850,0.00188) and

compliance (e1,e2) = (0.0866,0.0380).�Proof of Proposition 2: The proof has five steps. First, we establish properties of an equilibrium. Second,

we establish properties of the detection probability function d j(t j). Third, we establish that (7) implies that

in any equilibrium either tn j > 0 for some n ∈N \ j or tki > 0 for some k ∈N \i. Fourth, we establish that

if in equilibrium tnk > 0 for some k ∈N and n ∈N \k, then (7) holds. Fifth, we establish that (8) implies

that firm j draws no testing. Thus, Steps 1 to 4 address part (a.) and Step 5 addresses part (b.). First,

any equilibrium {e j, t ji} j∈N , i∈N \ j must satisfy the first order necessary conditions for compliance by firm

j ∈N and testing by firm i ∈N \ j:

∂π j

∂e j

=

[u j(1−q j)− Σ

l∈N \ jmin(ui,ul)sl (el, tl)ql

]d j(t j)mq j− c′j(e j)

{ ≤ 0 if e j = 0= 0 if e j ∈ (0,1)≥ 0 if e j = 1,

(27)

∂πi

∂ ti j

=min(ui,u j)s(ei, ti)q jmqi(1− e j)∂d j(t j)

∂ ti j

−1{ ≤ 0 if ti j = 0= 0 if ti j > 0. (28)

If firm j draws no testing (ti j = 0 for i ∈N \ j), then d j(t j) = 0 and consequently, in equilibrium e j = 0.

Second, observe that our assumption that d j(t j) is componentwise strictly increasing, continuously dif-

ferentiable, and satisfies (1) implies that (∂ 2/∂ tn j∂ tk j)d j(t j)< 0 for n ∈N \ j and k ∈N \ j. This implies

(∂/∂ ti j)d j(t j)|t j=0> (∂/∂ ti j)d j(t j)|t j=t jwhere tn j > 0 for some n∈N \ j. Third, consider the case in which

(7) holds. Consider a j ∈N and i ∈N \ j such that the inequality in (7) holds. Suppose in equilibrium

tn j = tki = 0 for all n ∈N \ j and k ∈N \i. This implies that in equilibrium e j = ei = 0 (from Step 1). Let

{ei, ti}i∈N denote this equilibrium. Then

(∂/∂ ti j)πi|(en,tn)=(en ,tn) for n∈N =min(ui,u j)q jmqi(∂/∂ ti j)d j(t j)|t j=0−1> 0, (29)

where the equality follows from the expression for (∂/∂ ti j)πi in (28), and the inequality in (29) follows

from the inequality in (7). Because (29) contradicts (28), we conclude that in any equilibrium tn j > 0 for

some n ∈N \ j and/or tki > 0 for some k ∈N \i. Thus, if tn j = 0 and for all n ∈N \{i, j} and tki = 0 and

for all k ∈N \{i, j}, then ti j > 0 and/or t ji > 0; that is, firm i tests firm j’s product (or vice versa) if no

other party tests firm i or j. Fourth, consider the case in which in equilibrium tnk > 0 for some k ∈N and

n ∈N \k. Let {ei, ti}i∈N denote this equilibrium. Suppose (7) does not hold. For j ∈N and i ∈N \ j

such that ti j > 0,

(∂/∂ ti j)πi|(en,tn)=(en ,tn) for n∈N =min(ui,u j)si(ei, ti)q jmqi(1− e j)(∂/∂ ti j)d j (t j)−1 (30)

<min(ui,u j)q jmqi(∂/∂ ti j)d j(t j)|t j=0−1≤ 0, (31)

where the equality follows from the expression for (∂/∂ ti j)πi in (28), the first inequality follows from Step

2, and the second inequality follows from (7) being violated. Because (31) contradicts that (28) holds with

27

equality, we conclude that (7) holds. Fifth, consider the case in which (8) holds. Suppose in equilibrium that

firm j draws testing ti j > 0 for some i ∈N \ j. Let {ei, ti}i∈N denote the equilibrium. Then (31) holds for

the same reasons in Step 4, with the exception that the last inequality holds because (8) holds. We conclude

that firm j does not draw testing (ti j = 0 for all i ∈N \ j), and so does not comply e j = 0 (from Step 1).�Lemma 2 is useful in the proof of Proposition 3. Let s0(e, t) denote s(e, t), d0(t) = d(t), d1(t) =

(∂/∂ ti j)d(t), and d2(t) = (∂ 2/∂ t2i j)d(t) for i ∈N and j ∈N \i, where t=<t, t, ..t,0> denotes the vec-

tor wherein each firm exerts testing effort t and the regulator does not test. Let

f1(e, t) = mus0(e, t)q2(1− e)d1(t)−1

f2(e, t) = mu(1− [1+(N−1)s0(e, t)]q)qd0(t)− c′(e).

f1(e, t) is the first derivative of firm i’s profit function with respect to ti j and f2(e, t) is the first derivative

with respect to ei, when each firm j ∈N chooses compliance e j = e and testing t jk = t for k ∈N \ j. Define

for t > 0,

e0(t) =

(2d0(t)−1−

√1−4d0(t)/[mud1(t)q2]

)/[2d0(t)]

e0(t) =

(2d0(t)−1+

√1−4d0(t)/[mud1(t)q2]

)/[2d0(t)],

and let e0(0) = limt↓0 e0(t) and e0(0) = limt↓0 e0(t). Let t denote the unique solution to d0(t)/d1(t) =

muq2/4. If t > t, then no value of e satisfies f1(e, t) = 0; otherwise, f1(e, t) = 0 has two roots in e : e0(t)

and e0(t). Note that f2(e, t) is strictly increasing in t and strictly decreasing in e with lime↑e f2(e, t) < 0.

Let t = inft≥0{t : f2(0, t) > 0}. If t > t, then let e0(t) denote the unique solution to f2(e, t) = 0,and note

that e0(t)> 0;otherwise, let e0(t) = 0. Let e(d) = e0(t)|t such that d0(t)=d , e(d) = e0(t)|t such that d0(t)=d , e(d) =

e0(t)|t such that d0(t)=d , d = d0(t) and d = d0(t). Our assumption that d(·) is continuously differentiable and

component-wise strictly increasing implies existence of e(d) and e(d) for d ∈ [0,d]. Note that in the sym-

metric case, inequality (7) simplifies to u> 1/[mq2(∂/∂ ti j)d(t j)|t j=0].

Lemma 2 Suppose the firms are symmetric and (9)-(10) hold. If (7) is violated, then the unique symmet-

ric equilibrium in compliance and detection probability (e, d) = (0,0). Otherwise, the unique symmetric

equilibrium (e, d) has d ∈ (0,d] and one of the following:

e= e(d) = e(d) (32)

e= e(d) = e(d). (33)

Further, e(·) is continuous and strictly increasing and e(·) is continuous and strictly decreasing; e(d) is

continuous on d ∈ [0,d) and increasing in d, strictly so on d ∈ (d,d). Finally, e(0) = 0 and e(0)< 0< e(0).

Proof of Lemma 2: The proof proceeds in seven steps. First, we establish necessary conditions for a sym-

metric equilibrium. Second, we show that these conditions are sufficient. Third, we establish properties of

28

the functions e(d), e(d) and e(d). Fourth, we show that f1(e0(t), t) is strictly decreasing in t. Fifth, we show

that if (7) is violated, then the unique symmetric equilibrium has zero compliance and testing. Sixth, we

show that if (7) is satisfied, then a symmetric equilibrium must satisfy either (32) or (33). Seventh, we show

that if (7) is satisfied, then a unique symmetric equilibrium exists. First, we establish necessary conditions

for a symmetric equilibrium. In a symmetric equilibrium (e, t), each firm j ∈N chooses compliance e j = e

and testing t jk = t for j ∈N and k∈N \ j. If firm i anticipates that the remaining firms j ∈N \i will choose

compliance e j = e and testing t jk = t for j ∈N and k ∈N \ j, then for compliance ei = e and testing ti j = t

for j ∈N to be a best response for firm i, the following first order conditions must be satisfied

(∂/∂ ti j)πi|en=en=e, tnl=tnl=t for n∈N and l∈N \n = f1(e, t)≤ 0 (34)

(∂/∂ei)πi|en=en=e, tnl=tnl=t for n∈N and l∈N \n = f2(e, t)≤ 0, (35)

where (34) must hold with equality if t > 0 and (35) must hold with equality if e > 0.Second, we establish

that any solution to (34)-(35) is a symmetric equilibrium. If firm i anticipates that the remaining firms

j ∈N \i will choose compliance e j = e and testing t jk = t for k ∈N \ j, then any solution to the first order

conditions for firm i must for j ∈N \i have ti j = τ for some τ ≥ 0. We can write firm i’s expected profit

under compliance ei and testing level τ as

πi(ei,τ) = u{1− [1+(N−1)so(e, t,τ)]q}mqs0(ei, t)− c(ei)− (N−1)τ, (36)

where so(e, ta, tb) = 1− do(ta, tb)(1− e), where do(ta, tb) denotes the detection probability when all firms

but one chooses testing level ta, one firm chooses testing level tb and the regulator does not test, do(ta, tb) =

d(ta, ..ta, tb, ta, ..ta,0). The assumptions that c(·) is strictly convex and (∂ 2/∂ t2i j)d(t j) <

−(N − 1)mu(∂/∂ ti j)d(t j)2/[(1− d)2c′′(e)] imply that πi is jointly strictly concave in (ei,τ). Therefore,

if (34)-(35) are satisfied, then firm i’s best response is (ei,τ) = (e, t). Third, we establish properties of the

functions e(d), e(d) and e(d). Because (∂ 2/∂ t2i j)d(t j) < −N(∂/∂ ti j)d(t j)

2/d0(t), e0(t) is strictly increas-

ing and e0(t) is strictly decreasing in t. Therefore, because d(·) is component-wise strictly increasing and

continuously differentiable, e(d) is continuous and strictly increasing in d and e(d) is continuous and strictly

decreasing in d. Because f2(·, ·) is continuous, e0(t) is continuous on t ∈ [0,∞). By the implicit function

theorem, e0(t) is strictly increasing in t for t ∈ (t,∞)

(∂/∂ t)e0(t) = {mu[(N−1)q2s0(e, t)2d0(t)2

]+ s0(e, t)2c′′(e)}−1

× [mu(N−1)qs0(e, t)2[1−Nq+2(N−1)(1− e)d0(t)q]d1(t) (37)

> 0.

Therefore, because d(·) is component-wise strictly increasing and continuously differentiable, e(d) is con-

tinuous and strictly increasing in d ∈ (d,d). Let fn(e,0) denote limt↓0 fn(e, t) for n= 1,2. Fourth, we estab-

29

lish that f1(e0(t), t) is strictly decreasing in t. Note that

(∂/∂ t) f1(e0(t), t) = mu{−[1− e0(t)]2d1(t)2−{(1−2d0(t)[1− e0(t)])−∆[1−2e0(t)]}d1(t)e′0(t)

+ [s0(e0(t), t)+∆e0(t)][1− e0(t)]d2(t)}q2. (38)

Using (37) and (38), with some effort it is possible to show that (∂ 2/∂ t2i j)d(t j) < −2mu×

(∂/∂ ti j)d(t j)2/[(1− d)3c′′(e)] for e ∈ [0,1) implies that (∂/∂ t) f1(e0(t), t) < 0. Fifth, suppose (7) is vi-

olated. Then f1(0,0) ≤ 0 and f2(0,0) ≤ 0, so (e, t) = (0,0) is an equilibrium. By Proposition 2, no equi-

librium exists with t > 0, so (e, t) = (0,0) is the unique equilibrium. Sixth, suppose (7) holds. Because

f1(0,0)> 0, (e, t) = (0,0) is not an equilibrium. Because f2(e,0)< 0 for e ∈ (0,1], an equilibrium cannot

have t = 0. Thus, in any equilibrium (34) must hold with equality. Thus, a symmetric equilibrium must

satisfy t ∈ (0, t], and e= e0(t) = e0(t) or e= e0(t) = e0(t). Therefore, a symmetric equilibrium must satisfy

d ∈ (0,d], and (32) or (33). Seventh, suppose (7) holds. From Step 6, this implies that a symmetric equilib-

rium (e, t)must have t > 0. From the analysis in Step 1, a symmetric equilibrium must have (e, t) = (e0(t), t)

where t satisfies

f1(e0(t), t) = 0, (39)

and, from Step 2, any such solution is an equilibrium. To establish that there exists an unique symmetric

equilibrium it is sufficient to show that there exists a unique solution to (39). Assumption (7) implies

f1(e0(0),0)> 0; further, limt↑∞ f1(e0(t), t)< 0. Therefore, the existence of a unique solution to (39) follows

from the fact that f1(e0(t), t) is strictly decreasing in t (as shown in Step 4). Finally, it is straightforward to

verify that (7) implies e0(0)< 0< e0(0), which in turn implies e(0)< 0< e(0).�Proof of Proposition 3: From Lemma 2, the unique symmetric equilibrium detection probability d > 0

(equivalently, t > 0) if and only if (7) holds. Further, if (7) is violated, then the unique symmetric equilibrium

compliance e= 0. Therefore, in the remainder, we consider the case where (7) holds. First, we demonstrate

the comparative statics for the number of firms N. By the implicit function theorem, e(d) is decreasing in

N. Let N = maxN∈{2,3,..}

{N : e(d)≥ 1−1/(2d)

}. With some abuse of notation, let (e(N), d(N)) denote the

unique symmetric equilibrium. If N ≤ N, then d is the unique solution to e(d)− e(d) = 0. Further,

e(d)− e(d)≥ 0 if and only if d ∈ [0, d]. (40)

For any N0 < N1 ≤ N, 0=[e(d(N0))− e(d(N0))

]|N=N0

≤[e(d(N0))− e(d(N0))

]|N=N1

, where the inequal-

ity follows because e(d) is decreasing in N. This implies that

d(N0)≤ d(N1) (41)

(from (40)). Thus,

e(N0) = e(d(N0))≥ e(d(N1)) = e(N1), (42)

where the inequality follows from (41) and the fact that e(·) is decreasing. By similar argument, for any

30

N < N2 < N3,

d(N2)≥ d(N3) (43)

e(N2)≥ e(N3). (44)

Further, for any N1 ≤ N < N2,

e(N1)≥ 1−1/(2d)≥ e(N2). (45)

Together (42), (44) and (45) imply that e is decreasing in N. Together (41) and (43) imply that d is increasing

in N for N ≤ N and decreasing in N for N ≥ N, where N = argmaxN∈{2,3,..}{d}. Second, we demonstrate

the comparative statics for the quality level u. By the implicit function theorem, e(d) is increasing in u.With

some effort, one can show that e(d) is increasing in u and that e(d) is decreasing in u for d ∈ [0,d]; recall

that d ≤ d (by Lemma 2). Because e(·) and e(·) are continuous, e(0) < e(0), e(d) < e(d), and d satisfies

either e(d) = e(d) or e(d) = e(d),

e(d)< e(d) for d ∈ [0, d). (46)

With some abuse of notation, let (e(u), d(u)) denote the unique symmetric equilibrium under quality level

u. From (46), for any uh > ul and any d < d(ul)

e(d)|u=uh≤ e(d)|u=ul

< e(d)|u=ul≤ e(d)|u=uh

. (47)

If e(uh) = e(d(uh))|u=uh, then e(d(uh))|u=uh

= e(d(uh))|u=uhand (47) together imply that d(uh) ≥ d(ul).

Therefore, e(uh) = e(d(uh))|u=uh≥ e(d(ul))|u=uh

≥ e(d(ul))|u=ul= e(ul). It remains to show that if e(uh) =

e(d(uh))|u=uh, then e(uh) ≥ e(ul). Suppose e(uh) = e(d(uh))|u=uh

and e(ul) = e(d(ul))|u=ul. Let D(ua,ub)

denote the unique solution to

e(D)|u=ua− e(D)|u=ub

= 0. (48)

Note that when ua = ub = un for n ∈ {h, l}, there is only one solution to (48) and d(un) =D(un,un).We will

show that

d(ul)≥ D(ul,uh). (49)

The proof is by contradiction. Suppose that d(ul)< D(ul,uh). Then

e(D(ul,uh))|u=ul< e(D(ul,uh))|u=ul

≤ e(D(ul,uh))|u=uh, (50)

where the first inequality holds because the continuity of e(·) and e(·), e(0) > e(0), and the uniqueness of

(e, d) imply e(d) < e(d) for d > d; the second inequality holds because e(d) is decreasing in u. Because

(50) contradicts the definition of D(ul,uh), we have established (49). By similar argument,

d(uh)≥ D(ul,uh). (51)

We conclude that e(ul) = e(d(ul))|u=ul≤ e(D(ul,uh))|u=ul

= e(D(ul,uh))|u=uh≤ e(d(uh))|u=uh

= e(uh),

where the first inequality follows from (49) and e(·) being decreasing; the second inequality follows from

(51) and e(·) being increasing. By similar argument, if e(uh) = e(d(uh))|u=uhand e(ul) = e(d(ul))|u=ul

, then

e(ul) ≤ e(uh). Third, by argument parallel to that for the comparative statics for u, e is increasing in the

market size m.�

31

Online Supplement

Base Model: Alternative Example in Which Regulator Testing Reduces Compliance

In this example, testing by a regulator that is more effective in testing than the firms reduces every firm’s

compliance effort. Parameters are as in the proof of Proposition 1b, except that d1(t21, tR1) =

min(√

8t21+2400tR1 ,2d

(√[300(t21+ tR1)]2+300(t21+ tR1)−300(t21+ tR1)

)), d2(t12, tR2) =

min(√

2(t12+ tR2) ,2d

(√[300(t21+ tR1)]2+300(t21+ tR1)−300(t21+ tR1)

)). Under regulator testing

(tR1, tR2) = (0.0004,0.0004), the unique equilibrium has zero testing by the firms and compliance (e1,e2) =

(0.0835,0.0213). Under (tR1, tR2) = (0,0), the unique equilibrium has testing by the firms (t21, t12) =

(0.07850,0.00188) and compliance (e1,e2) = (0.0866,0.0380).

Endogenous Quality: Entrants

Proof of Proposition 4: First consider the adoption of a product standard, enforced through competitor

testing. We will refer to this as “adoption of a product standard.” The proof proceeds in four parts. In the

first part, we establish two properties of the equilibrium prior to the adoption of the product standard. Let

(Nh, Nl) denote the number of high- and low-quality firms prior to the adoption of the product standard.

Prior to the adoption of the product standard, the profit of an entering high-quality firm is

[uh(1− Nhq)−ulNlq]mq− kh (52)

and the profit of a low-quality firm is

ul[1− (Nh+ Nl)q]mq− kl. (53)

Because, by assumption, a high-quality firm enters, (52) must be nonnegative:

(uhmq− kh)/(uhmq2)≥ Nh+(ul/uh)Nl. (54)

Similarly, if Nl > 0, then (53) must be nonnegative:

(ulmq− kl)/(ulmq2)≥ Nh+ Nl. (55)

Furthermore, as (53) strictly decreases with the number of firms, and we have assumed that the potential to

earn a nonnegative profit attracts entry, an additional low-quality firm cannot enter and achieve nonnegative

profit:

ul[1− (Nh+(Nl+1))q]mq− kl < 0. (56)

In the second part, we show that there exist γ ∈ (0,∞), d < 1, c ∈ (0,∞) and e ∈ (0,1) such that after

the adoption of the product standard, the success probability of any high-quality firm is strictly less than

σ ≡min(σ1,σ2,σ3),

where σ1 ≡ kh/[uh(1−q)mq], σ2 ≡ (Nh−2)/(Nh−1), and σ3 ≡ uh(ul(1−2q)mq−kl)/[ul(uh(1−q)mq−

kh)], with the exception that if only one high-quality firm does testing, it will have a success probability of 1.

Our assumptions that Nh > 2 and kl < ul(1−2q)mq ensure that σ is strictly positive. Let c= uh(1−q)mq,

1

γ = 1/[uh(1− e)(1− d)mq2], and select d < 1, and e ∈ (0,1) such that 1− d(1− e) = σ . Let i denote the

index of an incumbent firm that tests competitors’ products. Firm i must have high quality because (1) and

(31), as in the proof of Proposition 2, imply that in equilibrium, for any low-quality firm L and any other

firm j ∈N \L,

(∂/∂ tL j)πL < ulmq2(∂/∂ tL j)d j(t j)|t j=0−1, (57)

wherein assumption (11) implies the right hand side is negative, so tL j = 0 (i.e., no low-quality firm will

test a competitor). Let j 6= i denote the index of any other high-quality firm. The proof proceeds in three

steps. First, we show, in equilibrium, that high-quality firm j chooses compliance e j < e. If the high-quality

firm j chooses compliance e j ≥ e, then because c j(e j)≥ c, the firm’s cost of compliance c j(e j) exceeds its

expected revenue. If instead the high-quality firm chooses compliance e j = 0, its expected revenue exceeds

its cost of compliance (because c j(0) = 0). We conclude that, in equilibrium, e j < e. Second, we show that,

in equilibrium, the probability that a high-quality firm’s noncompliance is detected d j(t j)> d. Recall that j

and i 6= j denote the indices of high-quality firms. Because di(ti)< d and e j < e, condition (12) implies that

firm i’s expected profit is strictly increasing in ti j if d j(t j)≤ d.We conclude that, in equilibrium, d j(t j)> d.

Third, because s j (e j, t j) < 1− d(1− e) = σ and because the choice of the index j was arbitrary among

high-quality firms, we conclude that the success probability of any high-quality firm is strictly less than σ

(with the exception that if only one incumbent firm does testing, it will have a success probability of 1).

In the third part, we show that no additional high-firms will enter following the adoption of the product

standard. From the previous step, after the adoption of the product standard, the success probability of a

high-quality entrant would be strictly less than σ , so its expected profit would be strictly less than

uh(1−q)σmq− kh; (58)

σ ≤ σ1 implies that (58) is negative, so no additional high-firms will enter.

In the fourth part, we show that at least one additional low-quality firm will enter following the adoption

of the product standard. From the previous step, σ ≤ σ1 implies that no additional high-quality firms will

enter. Proposition 2 and our assumption (11) imply that low-quality firms will not comply, test, or draw

testing from competitors. If Λ low-quality firms enter, the expected profit of an entering low-quality firm

is strictly greater than ul[1− (σ(Nh− 1)+ 1+ Nl +Λ)q]mq− kl. Therefore, to establish that at least one

additional low-quality firm will enter following the adoption of the product standard, it is sufficient to show

that

ul[1− (σ(Nh−1)+1+ Nl+1)q]mq− kl ≥ 0. (59)

First, suppose that Nl > 0. Together Nh > 2 and σ ≤ σ2 imply that Nh+ Nl ≥ σ(Nh−1)+1+ Nl+1. This,

together with (55) implies (59). Second, suppose that Nl = 0. Then σ ≤ σ3 and (54) imply (59).�

Endogenous Quality: Incumbents

2

Proof of Proposition 5: Without loss of generality, order the quality levels such that u1 < u2 < .. < uM.

For generality, we begin the proof by allowing the firms to be asymmetric. Let {ui}i∈N denote equilibrium

quality levels of the firms under no product standard. Let uh = maxi∈N ui, i.e., the highest equilibrium

quality level is uh. Let H denote the index of a firm with this quality level, uH = uh. If firm H anticipates that

the remaining firms j ∈N \H will choose quality u j = u j, where u j ≤ uh, then for quality uH = uh to be a

best response for firm H it must be that firm H’s profit under quality uH = uh is greater than under quality

uH = un for n> h :

[uh(1−qH)−Σ j∈N \H u jq j]mqH − khH > [u

n(1−qH)−Σ j∈N \H u jq j]mqH − knH , (60)

where the inequality is strict because a firm that is indifferent between two quality levels chooses the higher

quality level. Inequality (60) implies

knH − kh

H > (un−uh)(1−qH)mqH for all n> h. (61)

We will show that under a product standard enforced by competitor testing, in any equilibrium firm H’s

quality is reduced from its level under no standard. Let {ui, ei, ti j}i∈N , j∈N \i denote equilibrium quality,

compliance and testing levels of the firms under a product standard enforced by competitor testing. To

establish that firm H’s quality is reduced from its level under no standard uH ≤ uH , suppose to the contrary

that uH = un where n> h. Under competitor testing, if firm H anticipates that the remaining firms j ∈N \H

will choose quality u j = u j, compliance e j = e j and testing t ji = t ji for i ∈N \ j, then for quality uH = un,

compliance eH = eH and testing tHi = tHi for i ∈N \H to be a best response for firm H it must be that[un(1−qH)− Σ

j∈N \Hmin(un, u j)s j(e j, t j)q j

]sH(eH , tH)mqH − cH(eH)− Σ

j∈N \HtH j− kn

H

> max(eH ,tH j) j∈N \H∈[0,1]×RN+1

+

{[uh(1−qH)− Σ

j∈N \Hmin(uh, u j)s j(e j, t j−, tH j)q j

]sH(eH , tH)mqH

− cH(eH)− Σj∈N \H

tH j

}− kh

H

≥[

uh(1−qH)− Σj∈N \H

min(uh, u j)s j(e j, t j)q j

]sH(eH , tH)mqH − cH(eH)− Σ

j∈N \HtH j− kh

H , (62)

where t j−, tH j denotes the vector t j wherein tH j is replaced by tH j. Inequality (62) implies[(un−uh)(1−qH)− Σ

j∈N \H[min(un, u j)−min(uh,u j)]s j(e j, t j)q j

]sH(eH , tH)mqH ≥ kn

H − khH . (63)

It is straightforward to verify that the right hand side of (61) is larger than the left hand side of (63). Conse-

quently, inequality (61) implies that (63) is violated. We conclude that firm H’s quality is reduced from its

level under no standard uH ≤ uh = uH .

Suppose the firms are symmetric. Let u denote a symmetric equilibrium quality level under no product

standard. By parallel argument to that above, if u= uh is an equilibrium under no product standard, then

kn− kh > (un−uh)(1−q)mq for all n> h. (64)

3

Let (u, e, t) denote symmetric equilibrium quality, compliance and testing levels under a product standard

enforced by competitor testing. By parallel argument to that above, if a symmetric equilibrium under com-

petitor testing has quality u= un for n> h, then

(un−uh)[1−q− (N−1)s(e, t)q]s(e, t)mq≥ kn− kh. (65)

It is straightforward to verify that the right hand side of (64) is larger than the left hand side of (65). Con-

sequently, inequality (64) implies that (65) is violated. We conclude that the equilibrium quality under

competitor testing is reduced from its level under no standard u≤ uh = u.�We provide an example in which a product standard, enforced through competitor testing, strictly re-

duces the equilibrium quality investments by all firms in the sequential move game with observable quality,

and does so to greater extent than in the initial game with unobservable quality. There are M = 3 three

possible quality levels u1 = 6, u2 = 8, u3 = 10 with costs k1 = 0.3, k2 = 0.5, k3 = 0.8. There are N = 2

symmetric firms with m= 1, q1 = q2 = 0.4, dn(tln) =min(√

tln,0.99) for n= 1,2 and l 6= n, c′n(0) = 10 for

n = 1,2. As the firms are symmetric, if an asymmetric equilibrium exists, then a second equilibrium exists

with exchange in the firms’ strategies. Focusing on equilibria in which firm 1 invests weakly more in quality

than than firm 2, the unique equilibrium has quality investments (u1,u2) = (10,8) with no product standard,

(u1,u2) = (8,8) with the product standard and unobservable quality, and (u1,u2) = (8,6) in the sequential

move game with the product standard and observable quality.

Consumer Utility from Compliance

Proposition 1. (a.) With the substitution of constants {ui}i∈N for {ui}i∈N , the proof in Lemma 1 that

each firm j’s objective function π j is quasiconcave in its compliance and testing {e j, t ji}i∈N \ j holds. In

the second sentence of the proof “a consistent rational expectations equilibrium” replaces “an equilibrium”

and in (19) and (20), un replaces un for n ∈ { j, i}, where un ≡ un[1+∆en/sn(en, tn)] for n ∈ N . After

(20), the following is inserted “The expressions follow from differentiating the expected profit function ((4)

with {ui}i∈N substituted for {ui}i∈N ) with respect to the corresponding decision variables (with {ui}i∈N

constant, because consumers do not observe {e j, t ji}i∈N \ j) and then substituting u j for u j to reflect the

consistency of consumers’ beliefs, i.e., that e j = e j and t j = t j for j ∈N .” All the remaining arguments

in the proof remain valid and, as the equilibrium compliance and detection probabilities are preserved by

construction, so are consumers’ rational expectations in equilibrium. (b.) For the first part, in the definition

τ i and (25), u j(1+∆) replaces u j. In (26), min(ui, u j) replaces min(ui,u j), and mu j(1+∆) replaces mu j. For

the second part, in the numerical example, ∆= 0.001. Then, under regulator testing (tR1, tR2) = (0.12,0.12),

the unique equilibrium has zero testing by the firms and compliance (e1,e2) = (0.0842,0.0213). Under

(tR1, tR2) = (0,0), the unique equilibrium has testing by the firms (t21, t12) = (0.07841,0.00188) and com-

pliance (e1,e2) = (0.0868,0.0380).

4

Proposition 2. In the proof’s first step’s first sentence “any consistent rational expectations” replaces “any.”

In (27), (28) and (30), un replaces un for n ∈ { j, i}, where un ≡ un[1+∆en/sn(en, tn)] for n ∈N . At the

end of Step 1, the following is inserted “The expressions follows from differentiating the expected profit

function (4) for with respect to the corresponding decision variable (with u j constant, because consumers

do not observe ti j) and then substituting u j for u j to reflect the consistency of consumers’ beliefs, i.e., that

e j = e j and t j = t j for j ∈N .” In (31), (1+∆)min(ui,u j) replaces min(ui,u j).

Proposition 3. See the proof of the more expansive result Proposition 3A in the next subsection.

Proposition 4. In (58) and the definitions of σ1 and c, (1+∆)uh replaces uh. In (57), (1+∆)ul replaces ul.

Proposition 5. In inequality (65), (un−uh)[1+∆e/s(e, t)] replaces (un−uh). Let

∆=

{(N−1)q if q≤ 1/(2N−1)(1−q)[(4N−3)q−1]/[4(N−1)q] otherwise,

and note ∆> 0. It remains to show that for ∆≤ ∆, the right hand side of (64) is larger than the left hand side

of (65), or equivalently

1−q≥ [s(e, t)+∆e][1−q− (N−1)s(e, t)q]. (66)

Inequality (66) holds for all ∆≤ ∆ if

1−q≥ ∆+ maxs∈[0,1]

f (s), (67)

where f (s) = s[1− q− (N − 1)sq]. It is straightforward to verify that the inequality in (67) holds with

equality.

Other Penalties for Noncompliance

Proposition 1. (a.) With the substitution of constants {ui}i∈N for {ui}i∈N , the proof in Lemma 1 that

each firm j’s objective function π j is quasiconcave in its compliance and testing {e j, t ji}i∈N \ j holds. In

the second sentence of the proof “a consistent rational expectations equilibrium” replaces “an equilibrium”

and in (19) and (20), un replaces un for n ∈ { j, i}, where un ≡ un[1+∆en/sn(en, tn)] for n ∈ N . After

(20), the following is inserted “The expressions follow from differentiating the expected profit function ((4)

with {ui}i∈N substituted for {ui}i∈N ) with respect to the corresponding decision variables (with {ui}i∈N

constant, because consumers do not observe {e j, t ji}i∈N \ j) and then substituting u j for u j to reflect the

consistency of consumers’ beliefs, i.e., that e j = e j and t j = t j for j ∈N .” For the blocking scenario, the

extension to the fine is captured by adding the term + f/(mq j) to term in square brackets in (19). The proof

in the labeling scenario follows that in the blocking scenario, where in (19) the term is square brackets is

replaced by

[(u j−u j)(1−q j)−Σi∈N \ j{[min(u j,ui)−min(u j,ui)]si (ei, ti)+[min(u j,ui)−min(u j,ui)][1− si (ei, ti)]qi+ f/(mq j)]

and in (20), min(ui,u j)s(ei, ti) is replaced by

([min(ui,u j)−min(ui,u j)]si (ei, ti)+ [min(ui,u j)−min(ui,u j)] [1− si (ei, ti)]).

5

(b.) For the blocking scenario, the proof is unchanged by the fine f ≥ 0. For the first part of the proof, for

the labeling scenario, it is straightforward to show that (∂/∂ ti j)πi ≤maxn∈N mun(1+∆)(∂/∂ ti j)d j(t j)−1.

The conclusion that in any equilibrium the firms do not test follows by the same argument as in the blocking

scenario.

Proposition 2. The extension to ∆ ≥ 0 is addressed in the previous subsection. For the blocking scenario,

the extension to the fine is captured by adding the term + f d j(t j) to the middle equation in (27). For the

labeling scenario, Proposition 2b holds when minn∈N {un}/(1+∆) is added to the right hand side of (8).

Only Steps 1 and 5 are relevant. The proof in the labeling scenario follows that in the blocking scenario,

where in (27) the term is square brackets is replaced by

[(u j−u j)(1−q j)−Σi∈N \ j{[min(u j,ui)−min(u j,ui)]si (ei, ti)+[min(u j,ui)−min(u j,ui)][1− si (ei, ti)]qi+ f/(mq j)]

and in (28), min(ui,u j)s(ei, ti) is replaced by

([min(ui,u j)−min(ui,u j)]si (ei, ti)+ [min(ui,u j)−min(ui,u j)] [1− si (ei, ti)]),

where un ≡ un[1+∆en/sn(en, tn)] for n ∈N . This change is also made on the right hand side (30), with the

addition that en and tn are replaced by en and tn for n ∈ {i, j}. In (31) min(ui,u j) is replaced by [u j(1+∆)−

minn∈N {un}].

Proposition 3A generalizes Proposition 3 to include the fine f and the labeling penalty for noncompliance,

which are described in §4. We generalize the right hand side of (10) by replacing the numerator mu with

(mu+ f ).

Proposition 3A Suppose the firms are symmetric and (9)-(10) hold. The unique symmetric equilibrium

compliance effort in the blocking scenario is higher than the compliance effort in any symmetric equilibrium

in the labeling scenario. In the labeling scenario, an equilibrium in compliance and testing (e, t) = (0,0). In

the blocking scenario, in the unique symmetric equilibrium, firms test (t > 0) if and only if (7). The unique

symmetric equilibrium in the blocking scenario has the following properties: The detection probability for a

noncompliant product increases with the number of firms N for N ≤N and decreases with N for N ≥N. The

detection probability for a noncompliant product increases with the fine f for f ≤ f and decreases with f for

f ≥ f . Compliance e increases with the firms’ quality levels u, a consumer’s utility from compliance ∆ and

the fine f , and decreases with the number of firms N. For the labeling scenario the symmetric equilibrium

with the largest compliance level also has these properties.

Lemma 2A is useful in the proof of Proposition 3A. Let s0(e, t) denote s(e, t), d0(t) = d(t), d1(t) =

(∂/∂ ti j)d(t), and d2(t) = (∂ 2/∂ t2i j)d(t) for i ∈N and j ∈N \i, where t=<t, t, ..t,0> denotes the vector

wherein each firm exerts testing effort t and the regulator does not test. Let

f1(e, t) = mu[IBs0(e, t)+∆e]q2(1− e)d1(t)−1

f2(e, t) = {mu[IB+∆e/s0(e, t)](1− [1+(N−1)s0(e, t)]q)q+ f}d0(t)− c′(e),

6

where IB = 1 in the blocking scenario and IB = 0 in the labeling scenario. f1(e, t) is the first derivative of

firm i’s profit function with respect to ti j and f2(e, t) is the first derivative with respect to ei, when each firm

j ∈N chooses compliance e j = e and testing t jk = t for k ∈N \ j. Define for t > 0,

e0(t) =

(IB[2d0(t)−1]+∆−

√(IB+∆)2− 4[IBd0(t)+∆]

mud1(t)q2

)/(2[IBd0(t)+∆])

e0(t) =

(IB[2d0(t)−1]+∆+

√(IB+∆)2− 4[IBd0(t)+∆]

mud1(t)q2

)/(2[IBd0(t)+∆]),

and let e0(0) = limt↓0 e0(t) and e0(0) = limt↓0 e0(t). If limt↓0 d1(t)< 4∆/[muq2(IB+∆)2], then f1(e, t)< 0

for (e, t)∈ [0,1)× [0,∞). Suppose instead that limt↓0 d1(t)≥ 4∆/[muq2(IB+∆)2] and let t denote the unique

solution to

[IBd0(t)+∆]/d1(t) = muq2(IB+∆)2/4.

If t > t, then no value of e satisfies f1(e, t) = 0; otherwise, f1(e, t) = 0 has two roots in e : e0(t) and e0(t).

Note that f2(e, t) is strictly increasing in t and strictly decreasing in e with lime↑e f2(e, t)< 0. Let

t = inft≥0{t : f2(0, t)> 0}.

If t > t, then let e0(t) denote the unique solution to

f2(e, t) = 0,

and note that

e0(t)> 0;

otherwise, let e0(t)= 0. Let e(d)= e0(t)|t such that d0(t)=d , e(d)= e0(t)|t such that d0(t)=d , e(d)= e0(t)|t such that d0(t)=d ,

d= d0(t) and d= d0(t).Our assumption that d(·) is continuously differentiable and component-wise strictly

increasing implies existence of e(d) and e(d) for d ∈ [0,d]. Note that in the symmetric case, inequality (7)

simplifies to u> 1/[mq2(∂/∂ ti j)d(t j)|t j=0].

Lemma 2A Suppose the firms are symmetric and (9)-(10) hold. (a.) In the blocking scenario, if (7) is

violated, then the unique symmetric equilibrium in compliance and detection probability (e, d) = (0,0).

Otherwise, the unique symmetric equilibrium (e, d) has d ∈ (0,d] and one of the following:

e= e(d) = e(d) (68)

e= e(d) = e(d). (69)

Further, e(·) is continuous and strictly increasing and e(·) is continuous and strictly decreasing; e(d) is

continuous on d ∈ [0,d) and increasing in d, strictly so on d ∈ (d,d). Finally, e(0) = 0 and e(0)< 0< e(0).

(b.) In the labeling scenario, a symmetric equilibrium in compliance and detection probability (e, d) =

(0,0). The following are necessary and sufficient conditions for a symmetric equilibrium (e, d) with d ∈

(0,d]: any (e, d) which satisfies (68) or (69) is a symmetric equilibrium; any symmetric equilibrium (e, d)

with d ∈ (0,d] satisfies (68) or (69).

7

Proof of Lemma 2A: The proof proceeds in eight steps. The first three steps consider both the blocking and

labeling scenarios. First, we establish necessary conditions for a symmetric equilibrium. Second, we show

that these conditions are sufficient. Third, we establish properties of the functions e(d), e(d) and e(d).

Steps 4 to 7 address the blocking scenario. Fourth, we show that f1(e0(t), t) is strictly decreasing in

t. Fifth, we show that if (7) is violated, then the unique symmetric equilibrium has zero compliance and

testing. Sixth, we show that if (7) is satisfied, then a symmetric equilibrium must satisfy either (68) or (69).

Seventh, we show that if (7) is satisfied, then a unique symmetric equilibrium exists.

Step 8, addresses the labeling scenario, establishes that (e, d) = (0,0) is an equilibrium, and establishes

necessary and sufficient conditions for an equilibrium with d > 0.

First, we establish necessary conditions for a symmetric equilibrium. In a symmetric equilibrium (e, t),

each firm j ∈N chooses compliance e j = e and testing t jk = t for j ∈N and k ∈N \ j. If firm i anticipates

that the remaining firms j ∈N \i will choose compliance e j = e and testing t jk = t for j ∈N and k ∈N \ j,

then for compliance ei = e and testing ti j = t for j ∈N to be a best response for firm i, the following first

order conditions must be satisfied

(∂/∂ ti j)πi|en=en=e, tnl=tnl=t for n∈N and l∈N \n = f1(e, t)≤ 0 (70)

(∂/∂ei)πi|en=en=e, tnl=tnl=t for n∈N and l∈N \n = f2(e, t)≤ 0, (71)

where (70) must hold with equality if t > 0 and (71) must hold with equality if e> 0.

Second, we establish that any solution to (70)-(71) is a symmetric equilibrium. If firm i anticipates that

the remaining firms j ∈ N \i will choose compliance e j = e and testing t jk = t for k ∈ N \ j, then any

solution to the first order conditions for firm i must for j ∈N \i have ti j = τ for some τ ≥ 0. We can write

firm i’s expected profit under compliance ei and testing level τ as

πi(ei,τ) = (1− IB)u(1−Nq)q+[u− (1− IB)u]{1− [1+(N−1)so(e, t,τ)]q}

×mqs0(ei, t)− [1− s0(ei, t)] f − c(ei)− (N−1)τ, (72)

where so(e, ta, tb) = 1−do(ta, tb)(1−e), where do(ta, tb) denotes the detection probability when all firms but

one chooses testing level ta, one firm chooses testing level tb and the regulator does not test, do(ta, tb) =

d(ta, ..ta, tb, ta, ..ta,0). The assumptions that c(·) is strictly convex and (∂ 2/∂ t2i j)d(t j) < −(N− 1)mu(IB+

∆)(∂/∂ ti j)d(t j)2/[(1− d)2c′′(e)] imply that πi is jointly strictly concave in (ei,τ). Therefore, if (70)-(71)

are satisfied, then firm i’s best response is (ei,τ) = (e, t).

Third, we establish properties of the functions e(d), e(d) and e(d). Because (∂ 2/∂ t2i j)d(t j) <

−IBN(∂/∂ ti j)d(t j)2/[IBd0(t)+∆], e0(t) is strictly increasing and e0(t) is strictly decreasing in t. There-

fore, because d(·) is component-wise strictly increasing and continuously differentiable, e(d) is continuous

and strictly increasing in d and e(d) is continuous and strictly decreasing in d. Because f2(·, ·) is continu-

ous, e0(t) is continuous on t ∈ [0,∞). By the implicit function theorem, e0(t) is strictly increasing in t for

8

t ∈ (t,∞)∂ e0(t)

∂ t=

(∂/∂ t) f2(e, t)

−(∂/∂e) f2(e, t)

= {mu[∆{(N−1)s0(e, t)2q− (1−q)[1−d0(t)]}d0(t)q+ IB(N−1)q2s0(e, t)2d0(t)2

]+ s0(e, t)2c′′(e)}−1× [muq{[IBs0(e, t)2+∆e][1−Nq+(N−1)(1− e)d0(t)q]

+ [IBs0(e, t)2+∆es0(e, t)](N−1)(1− e)d0(t)q}+ s0(e, t)2 f ](N−1)d1(t) (73)

> 0,

where the inequality follows from the assumption c′′(e)>mu∆/(1−d)3. Therefore, because d(·) is component-

wise strictly increasing and continuously differentiable, e(d) is continuous and strictly increasing in d ∈

(d,d). Let fn(e,0) denote limt↓0 fn(e, t) for n= 1,2.

Steps 4 to 7 address the blocking scenario. Fourth, we establish that f1(e0(t), t) is strictly decreasing in

t. Note that

(∂/∂ t) f1(e0(t), t) = mu{−[1− e0(t)]2d1(t)2− (1−2d0(t)[1− e0(t)])d

1(t)e′0(t)

+ s0(e0(t), t)[1− e0(t)]d2(t)}q2. (74)

Using (73) and (74), with some effort it is possible to show that (∂ 2/∂ t2i j)d(t j) < −2mu(∂/∂ ti j)d(t j)

2

/[(1−d)3c′′(e)] for e ∈ [0,1) implies that (∂/∂ t) f1(e0(t), t)< 0.

Fifth, suppose (7) is violated. Then f1(0,0)≤ 0 and f2(0,0)≤ 0, so (e, t) = (0,0) is an equilibrium. By

Proposition 2, no equilibrium exists with t > 0, so (e, t) = (0,0) is the unique equilibrium.

Sixth, suppose (7) holds. Because f1(0,0)> 0, (e, t) = (0,0) is not an equilibrium. Because f2(e,0)< 0

for e ∈ (0,1], an equilibrium cannot have t = 0. Thus, in any equilibrium (70) must hold with equality.

Thus, a symmetric equilibrium must satisfy t ∈ (0, t], and e= e0(t) = e0(t) or e= e0(t) = e0(t). Therefore,

a symmetric equilibrium must satisfy d ∈ (0,d], and (68) or (69).

Seventh, suppose (7) holds. From Step 6, this implies that a symmetric equilibrium (e, t) must have

t > 0. From the analysis in Step 1, a symmetric equilibrium must have (e, t) = (e0(t), t) where t satisfies

f1(e0(t), t) = 0, (75)

and, from Step 2, any such solution is an equilibrium. To establish that there exists an unique symmetric

equilibrium it is sufficient to show that there exists a unique solution to (75). Assumption (7) implies

f1(e0(0),0)> 0; further, limt↑∞ f1(e0(t), t)< 0. Therefore, the existence of a unique solution to (75) follows

from the fact that f1(e0(t), t) is strictly decreasing in t (as shown in Step 4). Finally, it is straightforward to

verify that (7) implies e0(0)< 0< e0(0), which in turn implies e(0)< 0< e(0).

Step 8, addresses the labeling scenario, establishes that (e, d) = (0,0) is an equilibrium, and establishes

necessary and sufficient conditions for an equilibrium with d > 0. Assumption (∂/∂ ti j)d(t j) < ∞ implies

that f1(0,0) ≤ 0 and f2(0,0) ≤ 0, so (e, t) = (0,0) is an equilibrium; equivalently, (e, d) = (0,0) is an

9

equilibrium. Because f2(·,0) is strictly decreasing, f2(e,0)< 0 for e> 0; this implies that any equilibrium

with t = 0 must have e = 0. From the analysis in Step 1, a symmetric equilibrium with t > 0 must have

(e, t) = (e0(t), t) where t satisfies (75) and, from Step 2, any such solution is an equilibrium. Therefore, a

symmetric equilibrium with d ∈ (0,d] must satisfy (68) or (69).�Proof of Proposition 3A: The proof proceeds in three parts. Part A establishes the properties of the sym-

metric equilibrium in the blocking scenario when ∆≥ 0. Part B establishes the properties of the symmetric

equilibrium in the labeling scenario. Part C establishes that the symmetric equilibrium compliance effort in

the blocking scenario is higher than any symmetric equilibrium compliance effort in the labeling scenario.

Part A: First, we generalize the Proposition 3 proof of comparative statics to allow for ∆≥ 0. Regarding

the comparative statics for the number of firms N, the expression 1−1/(2d), which appears in the definition

of N and in (45) is replaced by 1− [1+∆]/[2(d+∆)]. The proof of the comparative statics for the quality

level u and market size m is unchanged. Second, we establish the comparative statics for the fine f . By the

implicit function theorem, e(d) is increasing in f . By argument parallel to that for the comparative statics

for N, e is increasing in f and d is increasing in f for f ≤ f and decreasing in f for f ≥ f , where f =

argmax f∈[0,∞){d}. Third, we establish the comparative statics for the consumer’s utility from compliance

∆. By the implicit function theorem, e(d) is increasing in ∆. With some effort, one can show that e(d) is

increasing in ∆ and that e(d) is decreasing in ∆ for d ∈ [0,d]. By argument parallel to that for the comparative

statics for u, e is increasing in ∆.

Part B: From Lemma 2, in the labeling scenario, a symmetric equilibrium in compliance and detection

probability (e, d) = (0,0). The proof that the comparative statics in Part A apply to the maximal symmetric

equilibrium in compliance and detection probability (eM, dM) parallels the proof in Part A. The primary

change is that (e, d), which denotes the unique solution to (68) or (69) in the blocking scenario, is replaced

by (eM, dM), which denotes the maximal solution to (68) or (69) in the labeling scenario. In addition, the

expression 1−1/(2d), which appears in the definition of N and in (45) is replaced by 1/2.

Part C: Because f2(e, t) is larger in the blocking scenario, e0(t) and hence e(d) is larger in the scenario

with blocking. Because f1(e, t) is larger in the blocking scenario, e0(t) is larger and e0(t) is smaller in the

blocking scenario for t ∈ [0, t]; thus, e(d) is larger and e(d) is smaller in the blocking scenario for d ∈ [0,d].

By argument parallel to that for the comparative statics for u in the proof of Proposition 3, it follows that

the symmetric equilibrium compliance effort in the blocking scenario is higher than the maximal symmetric

equilibrium compliance effort in the labeling scenario.�Corollary and Proposition 4. Labeling reverses these results: A product standard, enforced through com-

petitor testing and labeling, decreases the expected profit of firms that do not comply and, in the setting of

Proposition 4, does not cause entry by low-quality, noncompliant firms. The proof is as follows: Our as-

10

sumption (11) and Proposition 2 imply that a low-quality firm L will not comply. Therefore, in any rational

expectations equilibrium, uL = ul so firm L cannot obtain a higher selling price by testing and reporting on

any competitor, its expected profit (15) would strictly decrease with any testing, and it does not test. There-

fore firm L has the same profit with or without the product standard, expressed in (53). From (56), prior to

the adoption of the product standard, entry of an additional low quality firm would result in negative profit

for the low quality firm. That profit could only decrease due to entry by additional firms of high quality.

Therefore, no firm of low quality will enter.

Endogenous Production Quantities

Proposition 1. The statement of the Proposition is unchanged, except that “any initial equilibrium in com-

pliance and testing by the firms (e, t)” is replaced by “any initial equilibrium in quantity, compliance and

testing by the firms (q, e, t).” (a.) Let {q j, e j, t ji} j∈N , i∈N \ j denote the initial rational expectations equilib-

rium when the regulator does not test. Let {t ′ji}i∈N , j∈N \i be constructed as in the proof of Proposition 1a

in the Appendix, with the substitution of constants

ui ≡ ui

[1+∆ei/si

(ei, ti

)]for i ∈N (76)

for {ui}i∈N , as in the above "Consumer Utility from Compliance" extension. Recall that si

(ei, ti

)=

si

(ei, t′i

)because the construction of {t ′ji}i∈N , j∈N \i preserves the detection probabilities (23). For brevity

of notation in subsequent analysis, let di ≡ di(ti,0) = di(t′i, tRi) denote the detection probability for each firm

i∈N that is preserved after the introduction of regulator testing tRi, by construction of the other firms’ test-

ing levels {t ′ji} j∈N \i.We need to show that {q j, e j, t ji} j∈N , i∈N \ j is a rational expectations equilibrium, and

will do so by proving that (q j, e j,{t ′ji}i∈N \ j) is a best response for firm j ∈N when the other firms adopt

the strategies {qk, ek, t′ki}k∈N \ j, i∈N \k and consumers’ rational expectations regarding the initial equilibrium

compliance and detection probabilities are preserved. For i ∈N \ j define the function δi : R+ → R+ by

δi(t) =

di(ti)|t ji=t, tRi=0, tki=tki for k∈N \{ j,i} . Define the function δ ′i : R+ → R+ by δ ′i (t) =

di(ti)|t ji=t, tRi, tki=t ′ki

for k∈N \{ j,i} wherein tRi is the newly-introduced level of regulator testing for firm i ∈

N \ j. Observe that those functions satisfy

δi(t ji) = δ′i (t′ji) = di for i ∈N \ j. (77)

Suppose that

δi(0)≤ δ′i (0) for i ∈N \ j; (78)

(we will later prove (78)). The assumptions that di() is componentwise strictly increasing, continuously

differentiable, and satisfies limt ji→∞ di(ti) = d for j ∈ {R,N \i} imply that δ ′i (0) < d and δi() is strictly

increasing, continuous and satisfies limt−→∞ δi(t) = d for j ∈ {R,N \i}. Hence there exists a threshold

11

βi ≥ 0 such that δi(βi) = δ ′i (0). Furthermore, the assumptions that di() is componentwise strictly increasing,

continuously differentiable and satisfies (1) imply that (d/dt)δi(βi+ t) = (d/dt)δ ′i (t) and

δi(βi+ t) = δ′i (t) for t ≥ 0, i ∈N \ j. (79)

Together, (79) and (77) imply that

βi = t ji− t ′ji for i ∈N \ j. (80)

(By construction, t ji− t ′ji ≥ 0 for i ∈N \ j.) Because {q j, e j, t ji} j∈N , i∈N \ j is a rational expectations equi-

librium when the regulator does not test, (q j, e j,{t ji}i∈N \ j) maximizes firm j’s expected profit

{u j(1−q j)− Σi∈N \ j

min(u j, ui)[1−(1− ei)δi(t ji)]qi}[1−(1−e j)d j]mq j−c j(e j)−Σi∈N \ jt ji−C j(q j), (81)

subject to q j ∈ [0,1], e j ∈ [0,1], t ji ≥ 0 for i ∈N \ j. We now show that (q j, e j,{t ′ji}i∈N \ j) maximizes

{u j(1−q j)− Σi∈N \ j

min(u j, ui)[1−(1− ei)δ′i (νi)]qi}[1−(1−e j)d j]mq j−c j(e j)−Σi∈N \ jνi−C j(q j)+Σi∈N \ j (t ji− t ′ji).

(82)

subject to q j ∈ [0,1], e j ∈ [0,1] and νi ≥ 0 for i ∈N \ j. Equalities (79) and (80) imply that for any νi =

t ji− (t ji− t ′ji)≥ 0, δ ′i (νi) = δi(t ji). Hence for any νi = t ji− (t ji− t ′ji)≥ 0, q j ∈ [0,1] and e j ∈ [0,1], (81) and

(82) are equal. Furthermore, observe that (q j, e j,{t ji}i∈N \ j)maximizes (81) subject to t ji ≥ t ′ji for i∈N \ j,

q j ∈ [0,1] and e j ∈ [0,1]; recall that t ji ≥ t ′ji by construction for i ∈N \ j. Therefore νi = t ji− (t ji− t ′ji) = t ′ji

for i ∈N \ j, q j and e j maximize (82) subject to νi ≥ 0 for i ∈N \ j, q j ∈ [0,1] and e j ∈ [0,1]. When firms

i ∈N \ j adopt the strategies {qk, ek, t′ki}k∈N \ j, i∈N \k and consumers have rational expectations that firms

will adopt the strategies {qk, ek, t′ki}k∈N , i∈N \k in response to the introduction of regulator testing {tRi}i∈N ,

equations (76), (77) and the definition of δ ′i () imply that the expected profit function for firm j is (82)

minus the constant Σi∈N \ j (t ji− t ′ji). Subtracting a constant does not change the set of optimal solutions, so

(q j, e j,{t ′ji}i∈N \ j) is a best response for firm j ∈N

Finally, suppose (78) does not hold. Together, δi(0) > δ ′i (0) and the assumptions that di() is compo-

nentwise strictly increasing, continuously differentiable, and satisfies limt ji→∞ di(ti) = d for j ∈ {R,N \i}

imply existence of a threshold γi > 0 such that δi(0) = δ ′i (γi). Furthermore, the assumptions that di()

is componentwise strictly increasing, continuously differentiable and satisfies (1) imply that for t ≥ 0,

(d/dt)δi(t) = (d/dt)δ ′i (γi+ t) and δi(t) = δ ′i (γi+ t). The latter, at t = t ji , is δi(t ji) = δ ′i (γi+ t ji), which

is inconsistent with (77), γi > 0, δ ′i () strictly increasing (because di() is componentwise strictly increasing)

and the fact that t ′ji ≤ t ji by construction. Hence (78) must hold. (b.) For the first part, in the definition τ i

and (25), u j(1+∆) replaces u j. In (26), min(ui, u j) replaces min(ui,u j), and mu j(1+∆) replaces mu j. For

the second part, with the addition to the numerical example of Ci(0) = 0, Ci(0.48) = 0.001 and Ci(qi) = 1

for qi /∈ {0,0.48} for i= 1,2 and ∆= 0, the proof holds.

Proposition 2. In the proof’s first step’s first sentence “any consistent rational expectations” replaces “any.”

In (28), the previously displayed equation, and the first line of (31), un replaces un for n ∈ { j, i}, where

12

un ≡ un[1+∆en/sn(en, tn)] for n ∈ N . At the end of Step 1, the following is inserted “The expressions

follows from differentiating the expected profit function (4) for with respect to the corresponding decision

variable (with u j constant, because consumers do not observe ti j) and then substituting u j for u j to reflect

the consistency of consumers’ beliefs, i.e., that e j = e j and t j = t j for j ∈N .” Throughout {ei, ti, qi}i∈N

replaces {ei, ti}i∈N . In (29) and (31), (en, tn,qn) = (en, tn, qn) replaces (en, tn) = (en, tn), and q jmqi replaces

q jmqi. Step 3 is replaced with the following: “Third, consider the case in which (7) holds wherein qn for

n ∈ { j, i} is firm n’s equilibrium production quantity when no firm draws testing. Suppose in equilibrium

tn j = 0 for all j ∈N and i ∈N \ j. This implies that in equilibrium e j = 0 for all j ∈N (from Step 1).

Let {ei, ti, qi}i∈N denote this equilibrium. Then (29) holds by the same argument in the base case. Because

(29) contradicts (28), we conclude that in any equilibrium tn j > 0 for some n ∈N \ j and/or tki > 0 for some

k ∈N \i.” Further, (1+∆)min(ui,u j) replaces min(ui,u j) in the second line of (31). The second inequality

in (31) holds for the same reasons as in the base case and because q jqi ≤ 1. This last inequality holds

because, in equilibrium, every firm n ∈N chooses a quantity qn ∈ [0,1) because Cn(qn) strictly increases

with qn and if a firm were to supply qn ≥ 1 to the market, its price (3) would be zero.

Fixed Costs for Testing

Proposition 1. (a.) Let {t ′ji}i∈N , j∈N \i be constructed as in the proof of Proposition 1a in the Appendix,

with the substitution of constants

ui ≡ ui

[1+∆ei/si

(ei, ti

)]for i ∈N

for {ui}i∈N , as in the above "Consumer Utility from Compliance" extension. Recall that si

(ei, ti

)=

si

(ei, t′i

)because the construction of {t ′ji}i∈N , j∈N \i preserves the detection probabilities: (23). Define

t =<t1, ..., tN> and t′(tR) =<t′1, ..., t′N>, where, as in the proof of Proposition 1a in the appendix, ti rep-

resents the initial equilibrium testing of firm i by the other firms when the regulator does not test, and t′i is

the vector (t ′1i, ..., t′Ni), the candidate equilibrium testing of firm i by the other firms with regulator testing tR,

where tR =<tR1, tR2, .., tRN>. Let π0j (tR) denote the expected profit for firm j, assuming: regulator testing

tR; firms i ∈ N \ j test competitors according to t′(tR); firms i ∈ N \ j maintain their initial equilibrium

compliance levels; firm j chooses not to test competitors (avoiding the fixed cost φ j); and firm j chooses

compliance e j to maximize its expected profit. Define ε j = π j−π0j (0) where π j denotes the expected profit

of firm j in the initial equilibrium and, by construction, π0j (0) is the maximum expected profit of firm j if

firm j chooses not to incur the fixed cost φ j to test competitors, the regulator does not test, and the other

firms i ∈N \ j have the initial equilibrium testing and compliance levels. Extend definition (5) to define τi

as the minimum of argmax{tRi : di(0, tRi)≤ di(ti,0)} and

argmax{‖tR‖∞: π

0j (tR)−π

0j (0)≤ ε j for j ∈N such that t ji > 0 for any i ∈N }, (83)

13

where ‖tR‖∞= max{|tR1| , .., |tRN |}. The proof of Proposition 1a in the appendix establishes that, with reg-

ulator testing tRi ≤ τi for i ∈N , the (lowered) testing levels t′(tR) maintain the initial detection probability

for a noncompliant product and, with the initial equilibrium compliance levels, constitute an equilibrium

for the firms–if no firm j ∈ N can achieve strictly greater expected profit with a unilateral deviation

from testing competitors to not doing so (avoiding the fixed cost φ j). Constraining tRi to be smaller than

(83) for i ∈ N ensures that π j ≥ π0j (tR) for every firm j that was testing in the initial equilibrium, i.e.,

every firm j with t ji > 0 for any i ∈N . That means that with a unilateral deviation to cease testing com-

petitors, firm j has weakly lower expected profit π0j (tR) than in the initial equilibrium without regulator

testing, π j. Clearly, firm j has greater expected profit in the candidate equilibrium with regulator testing

than in the initial equilibrium without regulator testing. Therefore, no firm deviates to avoid the fixed cost

of testing; the candidate equilibrium is indeed an equilibrium. A sufficient condition for τi to be strictly

positive is that firm i draws testing from competitors in the initial equilibrium, and a firm j ∈N chooses

to incur the fixed cost φ j only if, by doing so, the firm strictly increases its expected profit. (b.) For the

first part, in the definition τ i and (25), u j(1+∆) replaces u j. In (26), min(ui, u j) replaces min(ui,u j), and

mu j(1+ ∆) replaces mu j. For the second part, we provide an example with N = 2 in which increasing

regulator testing from tRi = 0 to tRi > 0 for i ∈ N strictly reduces the compliance effort for both firms,

strictly reduces the detection probability for a noncompliant product for both firms, and causes the firms

not to test, in equilibrium. Let ∆ = φR = 0, φ1 = φ1 = 0.01, m = u1 = u2 = 1, q1 = q2 = 0.48, d = 0.99,

cn(e) = 0.1e2 and di(t ji, tRi) =min(√

20(t ji+ tRi),2d

(√[50(t ji+ tRi)]2+50(t ji+ tRi)−50(t ji+ tRi)

))for

i ∈ {1,2} and j 6= i. Under regulator testing tR1 = tR2 = 0, the unique equilibrium has testing by the firms

t12 = t21 = 0.0235, compliance e1 = e2 = 0.583 and detection probability d1 = d2 = 0.686. Then, under

regulator testing tR1 = tR2 = 0.0151, the unique equilibrium has zero testing by the firms, compliance

e1 = e2 = 0.473 and detection probability d1 = d2 = 0.550.

Proposition 2. Replace the second half of the outline of the proof with: “Third, we establish that if (7) holds

and the fixed cost of testing is not too large, then, in any equilibrium, at least one firm tests a competitor:

tnm > 0 for some n ∈N and m ∈N \n. Fourth, we establish that if (7′), defined as (7) with min(u j,ui)(1+

∆) replacing min(u j,ui), is violated, then in equilibrium no firm tests a competitor: tnk = 0 for all k ∈N

and n ∈ N \k. Fifth, we establish that if (8′), defined as (8) with u j(1+∆) replacing u j, holds, then in

equilibrium firm j does not draw testing: ti j = 0 for all i ∈N \ j.”

Modify Step 1 as follows: In the first sentence “any consistent rational expectations” replaces “any.” In

(27), (28) and (30), un replaces un for n ∈ { j, i}, where un ≡ un[1+∆en/sn(en, tn)]. In (28), “≤ 0 if ti j = 0”

is eliminated.

Replace Step 3 with the following: “Third, we establish that if (7) holds and the fixed cost of testing

14

is not too large, then at least one firm tests a competitor. More precisely, consider a j ∈N and i ∈N \ j

such that the inequality in (7) holds. We will show that there exists φ i > 0 such that if φi < φ i, then, in

any equilibrium, at least one firm tests a competitor: tnm > 0 for some n ∈ N and m ∈ N \n. Suppose

in equilibrium ti j = ti j = 0 for all i ∈ N and j ∈ N \i. This implies that in equilibrium e j = 0 for all

j ∈N (from Step 1) and therefore e j = 0. Let {ei, ti}i∈N denote this equilibrium. Then (29) holds, where

the equality in (29) follows from (28), and the inequality in (29) follows from the inequality in (7). Let

ψi(ei, ti) = πi(ei, ti)+ I{ti j>0 for any j∈N \i}φi, where ti =<ti1, ti2, .., ti,i−1, ti,i+1, .., tiN> denotes firm i’s testing

levels. Thus, ψi(ei, ti) denotes firm i’s expected profit under φi = 0, and πi(ei, ti) denotes firm i’s expected

profit under φi > 0. Observe that under e j = e j = ei = 0 and t jk = t jk = ti j = 0 for all j ∈N \i and k ∈N \ j,

ψi(ei, ti) and πi(ei, ti) are strictly decreasing in ei, so firm i’s best response compliance ei = 0. Let

φ i = maxti∈[0,∞)N−1

ψi(0, ti)−ψi(0,0).

Inequality (29) implies that φ i > 0. Further, φi < φ i implies

maxti∈[0,∞)N−1

πi(0, ti)≥ maxti∈[0,∞)N−1

ψi(0, ti)−φi > maxti∈[0,∞)N−1

ψi(0, ti)−φ i = ψi(0,0) = πi(0,0).

Therefore, a best response by firm i to e j = e j = ei = 0 and t jk = t jk = ti j = 0 for all j ∈N \i and k ∈N \ j

cannot have ti = 0. We conclude, in any equilibrium, at least one firm tests a competitor: tnm > 0 for some

n ∈N and m ∈N \n.”

Modify Step 4 by replacing (7) with (7′) and by replacing min(ui,u j)(1+∆) with min(ui,u j) in (31).

Step 4 shows that it cannot be that both in equilibrium tnk > 0 for some k ∈N and n ∈N \k and (7′) is

violated. We conclude that if (7′) is violated, then in equilibrium tnk = 0 for all k ∈N and n∈N \k.Modify

Step 5 by replacing (8) with (8′).

Proposition 3A. The result holds provided that the firms’ fixed cost of testing φ < φ , for some φ > 0. The

proof of Proposition 3A is unchanged. Modify the proof of Lemma 2A by inserting the following imme-

diately after Step 2: “Steps 1 and 2 have established necessary and sufficient conditions for a symmetric

equilibrium under φ = 0. In this next step, we show that there exists φ > 0 such that the necessary and suf-

ficient conditions for a symmetric equilibrium under φ ∈ (0,φ) are identical to the necessary and sufficient

conditions under φ = 0.Again suppose firm i anticipates that the remaining firms j ∈N \i will choose com-

pliance e j = e and testing t jk = t for j ∈N and k ∈N \ j. By the same argument in Step 2, under φ > 0, any

solution to the first order conditions for firm i must for j ∈N \i have ti j = τ for some τ ≥ 0.We generalize

the definition in (72) by adding the term −I{τ>0}φ to the right hand side. Let ψi(ei,τ) = πi(ei,τ)+ I{τ>0}φ .

Thus, ψi(ei,τ) denotes firm i’s expected profit under φ = 0, and πi(ei,τ) denotes firm i’s expected profit

under φ > 0. Step 2 establishes that under φ = 0, there exists a unique best response for firm i, which we de-

note (ei,τ) = (e, τ). Under φ > 0, there exists a best response for firm i; to see this observe that there exists

ς ∈ (0,∞) such that if τ ≥ ς , then πi(ei,τ)< 0; therefore, in terms of maximizing πi(ei,τ) one can, without

15

loss of generality, restrict attention to (ei,τ) ∈ [0,e)× [0,ς ]. Let (ei,τ) = (e, τ) denote a best response for

firm i under φ ∈ (0,φ).We will show that there exists φ > 0 such that if φ ∈ (0,φ), then

(e, τ) = (e, τ), (84)

that is, there exists a unique best response for firm i under φ ∈ (0,φ) and it is identical to the best response

under φ = 0. If τ > 0, then let

φ = ψi(e, τ)− maxei∈[0,1]

ψi(ei,0),

and observe that φ > 0. First, suppose τ > 0 and that there exists a best response under φ ∈ (0,φ) with

τ = 0. Then

πi(e, τ) = ψi(e, τ)−φ > ψi(e, τ)−φ = maxei∈[0,1]

ψi(ei,0) = maxei∈[0,1]

πi(ei,0) = πi(e, τ),

which contradicts that (ei,τ) = (e, τ) is a best response under φ ∈ (0,φ). Therefore, if τ > 0, then any best

response under φ ∈ (0,φ)must have τ > 0, which implies that the best response is unique and that it satisfies

(84). Second, suppose τ = 0 and that there exists a best response under φ ∈ (0,φ) with τ > 0. Then

πi(e, τ) = πi(e,0) = ψi(e,0)> ψi(e, τ)> ψi(e, τ)−φ = πi(e, τ),

which contradicts that (ei,τ) = (e, τ) is a best response under φ ∈ (0,φ). Therefore, if τ = 0, then any best

response under φ ∈ (0,φ)must have τ = 0, which implies that the best response is unique and that it satisfies

(84). We conclude that there exists φ > 0 such that the necessary and sufficient conditions for a symmetric

equilibrium under φ ∈ (0,φ) are identical to the necessary and sufficient conditions under φ = 0.”

Alternative Detection Probability Function

Proposition 1. (a.) Recall that {ei, t ji}i∈N , j∈N \i denotes the initial equilibrium in compliance and testing

by the firms when the regulator does not test. Note for each i ∈N that t ji > 0 for at most one j ∈N \i.We

will show that {ei, t ji}i∈N , j∈N \i remains as an equilibrium when the regulator tests at a sufficiently small

level. Let tR =<tR1, tR2, .., tRN>. For tR such that tRi ≤ max j∈N \i{t ji} for i ∈ N , let π0j (tR) denote the

expected profit for firm j, assuming: regulator testing tR; firms i ∈N \ j maintain their initial equilibrium

testing and compliance levels; firm j chooses compliance e j and testing {t ji}i∈N \ j to maximize its expected

profit, subject to the constraint that t ji = 0 for at least one i ∈N such that t ji > 0 (if t ji = 0 for all i ∈N ,

then the constraint is vacuous). Define ε j = π j−π0j (0) where π j denotes the expected profit of firm j in

the initial equilibrium and, by construction, π0j (0) is the maximum expected profit of firm j when firm j

does not test at least one competitor it tested in the initial equilibrium, the regulator does not test, and the

other firms i ∈N \ j have the initial equilibrium testing and compliance levels. The right hand side of (5)

simplifies to max j∈N \i{t ji}. Extend definition (5) to define τi as the minimum of max j∈N \i{t ji} and

argmax{‖tR‖∞: π

0j (tR)−π

0j (0)≤ ε j for j ∈N such that t ji > 0 for any i ∈N }, (85)

where ‖tR‖∞= max{|tR1| , .., |tRN |}. Constraining tRi to be smaller than (85) for i ∈N ensures that π j ≥

π0j (tR) for every firm j that was testing in the initial equilibrium, i.e., every firm j with t ji > 0 for any i∈N .

16

That is, firm j has weakly lower expected profit when it does not test one or more firms it tested in the initial

equilibrium. Therefore, for firm j, choosing the compliance and testing levels in the initial equilibrium is

a best response to regulator testing tR and testing and compliance by other firms {ei, tin}i∈N \ j, n∈N \i. We

conclude that under regulator testing tRi ≤ τi for i ∈N , {ei, t ji}i∈N , j∈N \i is an equilibrium in compliance

and testing for the firms. (b.) For the first part, our assumptions regarding Di(·) imply that τ i is a finite,

nonnegative constant and that inequalities (25) and (26) hold. The conclusion that tRi > τ i implies that in

any equilibrium t ji = 0 follows. For the second part, the numerical example has the same firm equilibrium

testing and compliance levels when Di(t) = min(√

βit,2d(√(300t)2+300t− 300t) for i ∈ {1,2}, where

β1 = 8 and β2 = 2, and the non-zero regulator testing levels are (tR1, tR2) = (0.1200,0.0004).

Proposition 2. The conclusion of Step 2 holds because D j(·) is strictly concave.

Lemma 2. See the proof of the more expansive result Lemma 2A.

Proposition 3. See the proof of the more expansive result Proposition 3A.

Proposition 4. Inequality (57) follows from (31) and strict concavity of D j(·). In the second step, let i denote

the index of a firm that applies the greatest testing to firm j: ti j ≥ tk j for k ∈N \ j.

Proposition 5. The proof is unchanged.

Consumer Utility from Compliance: The proofs of Propositions 1, 2, 3 and 5 hold with the modifications

described in the Consumer Utility from Compliance subsection. In the numerical example in the proof of

Proposition 1, the detection probability function and non-zero regulator testing levels are as described in the

Alternative Detection Probability Function proof.

Other Penalties for Noncompliance: The proofs of Propositions 1, 2 3A, 4 and 5 and the Corollary hold

with the modifications described in the Other Penalties for Noncompliance subsection. Regarding Lemma

2A and Proposition 3A, in a symmetric equilibrium (e, t), each firm chooses compliance e j = e and testing

t j, j+1 = t and t jl = 0 for j ∈N and l ∈N \{ j, j+1}, where by convention tN,N+1 = tN1.

Lemma 2A. The proof holds with the following modifications: Let s0(e, t) = 1−D(t)(1−e), d0(t) =D(t),

d1(t) =D′(t), and d2(t) =D′(t). Note f1(e, t) is the first derivative of firm i’s profit function with respect to

ti,i+1 and f2(e, t) is the first derivative with respect to ei, when each firm j ∈N chooses compliance e j = e

and testing t j, j+1 = t and t jl = 0 for j ∈N and l ∈N \{ j, j+1}, where by convention tN,N+1 = tN1.

In Step 1, recall that in a symmetric equilibrium (e, t), each firm chooses compliance e j = e and testing

t j, j+1 = t and t jl = 0 for j ∈N and l ∈N \{ j, j+1}. If firm i anticipates that the remaining firms j ∈N \i

will choose compliance e j = e and testing t j, j+1 = t and t jl = 0 for j ∈N and l ∈N \{ j, j+1}, then any

solution to the first order conditions for firm i must for l ∈N \{i, i+1} have til = τ for some τ, where τ = 0

or τ > t. Further, for compliance ei = e and testing ti,i+1 = t and til = 0 for l ∈N \{i, i+ 1} to be a best

17

response for firm i, the following first order conditions must be satisfied

(∂/∂ til)πi|en=en=e, tn,n+1=tn,n+1=t, tnk=tnk=0 for n∈N and k∈N \{n,n+1} ≤ 0 (86)

(∂/∂ til)πi| en=en=e, tn,n+1=tn,n+1=t, tnk=0 for n∈N and k∈N \{n,n+1},thk =0 for h∈N \i and k∈N \{h,h+1}, tik =τ>t for k∈N \{i,i+1}

< 0 (87)

(∂/∂ ti,i+1)πi|en=en=e, tn,n+1=tn,n+1=t, tnk=tnk=0 for n∈N and k∈N \{n,n+1} = f1(e, t)≤ 0

(∂/∂ei)πi|en=en=e, tn,n+1=tn,n+1=t, tnk=tnk=0 for n∈N and k∈N \{n,n+1} = f2(e, t)≤ 0,

where the last two inequalities correspond to (34)-(35), and the inequality corresponding to (34) must hold

with equality if t > 0 and the inequality corresponding to (71) must hold with equality if e> 0.

In Step 2, because d(ti) = D(maxn∈R∪N \i{tni}), where D(·) is strictly concave,

(∂/∂ til)πi|til=0,tl−1,l≥0 ≤ (∂/∂ tl−1,l)πi|til=0,tl−1,l≥0;

consequently, (34) implies (86). Because D(·) is strictly concave and τ > t, (34) implies (87). If firm

i anticipates that the remaining firms j ∈ N \i will choose compliance e j = e and testing t j, j+1 = t and

t jl = 0 for j ∈N and l ∈N \{ j, j+ 1}, then any solution to the first order conditions for firm i must for

l ∈N \{i, i+1} have til = τ for some τ, where τ = 0 or τ > t. We can write firm i’s expected profit under

compliance ei and testing levels ti,i+1 and τ as

πi(ei, ti,i+1,τ) = (1− IB)u(1−Nq)mq+[u− (1− IB)u]{1− [1+(N−2)so(e,max(t,τ))+ so(e, ti,i+1)]q}

×mqs0(ei, t)− [1− s0(ei, t)] f − c(ei)− (N−2)τ− ti,i+1,

The assumptions that c(·) is strictly convex and D′′(t)<−2(N−1)mu(IB+∆)D′(t)2/[(1−d)2c′′(e)] imply

that πi is jointly strictly concave in (ei, ti,i+1,τ). This, together with that fact that (34) implies (86) and (87),

implies that if (70)-(71) are satisfied, then firm i’s best response is (ei, ti,i+1,τ) = (e, t,0).

In Steps 3, 4 and 7, D′(t) replaces (∂/∂ ti j)d(t j), and D′′(t) replaces (∂ 2/∂ t2i j)d(t j). In Step 3, “D(·) is

strictly increasing” replaces “d(·) is component-wise strictly increasing.”

Endogenous Production Quantities: The proofs of Propositions 1 and 2 hold with the modifications de-

scribed in the Endogenous Production Quantities subsection. The numerical results regarding Proposition 3

hold with D(t) = 2d(√(200t)2+200t−200t).

Fixed Costs for Testing: The proofs of the modified Propositions 1, 2 and 3 hold with the modifications

described in the Fixed Costs for Testing subsection.

18