# Incomplete Regulation, Asymmetric Information, and Collusion-Proofness

Authors: Marco Meireles; Paula Sarmento
Journal: Economics Research International
 (2012)

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## Abstract

In an incomplete regulation framework, the regulator cannot replicate all the possible outcomes by himself since he has no influence over some firms in the market. Due to asymmetric information, it may be better for the regulator to allow the unregulated firms to extract a truthful report from the regulated firm through side-payments under collusion, and therefore the “collusion-proofness principle” may not hold. In fact, by introducing an exogenous number of unregulated firms, social welfare differences seem to favour a collusion-allowing equilibrium. However, such result will depend on the relative importance given by the regulator to the consumer surplus in the social welfare function.


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## Body

## 1. Introduction

This paper aims to evaluate the optimal incomplete regulation contract when it is possible to form a coalition between unregulated firms and the regulated firm, and the marginal costs of the regulated firm are private information, not available to the regulator and neither to the other competitors. We will also evaluate how such results may be influenced by the relative importance given by the regulator to the consumer surplus in the social welfare function.Most of the literature on economic regulation studied optimal contracts in the context of complete regulation, either between a regulator and a firm (monopoly) or between a regulator and all the firms in the market (usually a duopoly).Baron and Myerson [1], Laffont and Tirole [2], and Lewis and Sappington [3] focused on the design of an optimal regulation contract in a monopoly under asymmetric information. On the first two papers the regulator was not aware of the monopolist’s marginal costs, while in the last that private information was related to the market’s demand function. They all concluded that the regulator has to pay a price above the marginal cost to avoid an untruthful report by the monopolist. Lewis and Sappington [3], however, also stated that under some conditions (nondecreasing marginal costs), and in spite of the private information about the demand function, the same first-best optimum equilibrium may be achieved.Caillaud [4], Laffont and Martimort [5, 6], and Tangerås [7] extended optimal complete regulation under asymmetric information to a competition framework. Caillaud [4] focused on the informational effect that the existence of a competitive fringe had on the regulation of a dominant firm under asymmetric information. He concluded that the presence of a fringe is always welfare enhancing but its magnitude depends on the degree of information correlation and on the characteristics of the demand function. Both Laffont and Martimort [5, 6] and Tangerås [7] introduced the possibility of collusion between two firms in a context of complete regulation under asymmetric information. The first mainly emphasises the role of correlated information between the firms as a determinant of the strength of the coalition, concluding that in the presence of correlated private information the regulator can create a regulation contract which can replicate exactly the collusion outcome: the collusion-proofness principle. The second paper studied the incentives for collusion when a market is regulated through yardstick competition. Since firms decided to collude before knowing their own productivity, the collusion would be costly to society only if firms could commit to the side payments agreed.Still within the framework of complete regulation and asymmetric information, some literature focused on the effects of collusion on social welfare and analysed its potential benefits. While Che [8] studied a possible collusion between a regulator and a firm and how it might increase the regulator’s monitoring effort, Itoh [9], Lawarrëe and Shin [10], and Shin [11] analysed the possibility of collusion between agents, through side-payments. They all have concluded that by allowing side-payments between agents, the regulator is able to more easily induce truthful reports from the firms, reducing the informational rent.Biglaiser and Ma [12] and Aubert and Pouyet [13] are two very important contributions to the study of optimal regulation when the regulator is only able to make a contract with a dominant firm (incomplete regulation) under asymmetric information. Biglaiser and Ma [12] focus on optimal incomplete regulation when only the dominant firm has private information regarding the demand function and the unregulated competitor has some market power, acting as a Stackelberg follower. They proved that depending on the weight given by the regulator to consumer surplus on social welfare, the equilibrium outcome could be both separating and pooling or just separating. The Aubert and Pouyet’s [13] model is the closest to the one we propose. They have worked in a framework of incomplete regulation under asymmetric information and with the possibility of collusion between the two firms operating in the market. Due to product imperfect substitutability, the unregulated competitor has incentives to bribe the regulated firm such that it overstates her costs and produces less. They have concluded that it is not optimal to design a collusion-proof contract for regulation as it imposes both distortions at the bottom and at the top (inefficient and efficient regulated firms, resp.), while by allowing collusion the regulator may induce the nonregulated firm to indirectly tax its competitor. The Collusion-Proofness Principle will not hold in the incomplete regulation framework since the regulator is unable to contract with the unregulated firm, and therefore it has limited possibilities for contracts when compared to what can be achieved within the coalition of the two firms. The main difference between the Collusion-Proof contract and a contract that allows collusion is that in the first the regulator has to pay larger amounts to the regulated firm to ensure that it reveals its efficiency despite collusion while in the last, the regulator uses collusion to make the unregulated firm pay to ensure that the regulated firm reveals its efficiency.This paper’s main contribution is to study the role of collusion in the optimal design of incomplete regulation under asymmetric information in the presence of more than one unregulated firms. The market is composed of a dominant (regulated) firm and by a small number of unregulated firm, which produce the same homogeneous product (although different from the one produced by the regulated firm) and choose their quantities as Stackelberg followers. Similarly to Aubert and Pouyet [13], we assume that the only private information on the market is the marginal cost of the dominant firm, which can take one of two possible values. Additionally, we introduce differentiation among unregulated firms by admitting different marginal costs, which are publicly known. During entire the analysis we also use a similar social welfare function to the one used by Aubert and Pouyet [13], which allows us to compare the results directly. However, these assumptions may also be considered shortcomings and a reason for further research: generally the costs for the unregulated firms are also unknown and regulation is usually made in terms of price-cap, instead of quantities. As later showed in the paper, conclusions will also change dramatically by giving more weight to the consumer surplus in the social welfare function, emphasising the importance of the regulator’s priorities when ensuring competition in a market.This paper also has a wide range of empirical applicability. An example is the Portuguese fixed telecommunications market, which is constituted by a dominant regulated firm (Portugal Telecom) and by a small number of unregulated competitors. Once again, consumers view telecommunications services of Portugal Telecom as different from the other competitors. We may also apply the model to other sectors as Health and Education in Portugal. In each region, we could look at the health-care market as being composed of one (or two) dominant public (regulated) hospitals and by a small number of private hospitals that compete with most types of health services. The same can be said about the regional market for undergraduate degrees. Usually the public university is the dominant firm, regulated in terms of tuitions and there is also a small number of private universities that are free to impose the tuition they would like.The paper is organized as follows. In the next section we describe the model and its characteristics, also describing the timing to better understand the sequence of events. In later sections we analyse the optimal regulation outcome (quantities, prices, profits, transfer from the regulator, side-payment from the unregulated firms to the regulated firm and welfare level) in different frameworks within incomplete regulation: complete information, incomplete information in the absence of collusion and incomplete information with the possibility of collusion. Then we compare the outcome from contracts that are Collusion-Proof with the outcome obtained with contracts that allow collusion and assess which one entails a higher level of social welfare. Finally, we discuss the importance of the social welfare function to the robustness of such findings, by comparing the consumer surplus values in collusion-proofness and collusion-allowing equilibriums.


## 2. The Model

### 2.1. The Firms

The market is composed of a dominant regulated firmFA and by n unregulated firms, each denoted by FiB, where i=1,…,n. All the firms compete in quantities. Firm FA has a constant marginal cost θa which can take one of two possible values, θ̲ or θ¯, θ¯-θ̲≡Δθ>0. The value of this marginal cost is private information forFA; however, its distribution is public knowledge: with probability p̲  FA, it is efficient (θa=θ̲) and it is inefficient (θa=θ¯) with probability p¯=1-p̲. The unregulated firms have different marginal costs (θib), which are publicly known. For simplification, we have assumed that the unregulated firms are ordered from the most efficient to the least efficient, such that θ1b≤θ2b≤…≤θnb.


### 2.2. The Consumers

The consumers have two differentiated products available for consumption:qa is the quantity of the product produced by the regulated firm in market A and qib is the quantity of the product produced by the unregulated firm i, where Qb=∑i=1nqibrepresents the total quantity produced by all unregulated firms. The Gross Consumer Surplus is given by(1)GS(qa,Qb)=daqa+dbQb-12(qa)2-12(Qb)2-sqaQb,where the parameters∈(0,1) measures the degree of substitutability between the two products, whereas the parameter dj represents the size of marketj=A,B (Such Consumer Surplus function is a result from the standard quadratic utility function proposed by Dixit.) The inverse demand functions for both markets are given by(2)Pa(qa,Qb)=da-qa-sQb,Pb(qa,Qb)=db-Qb-sqa.For the time being, we assume thatdb-da=θ1b-θ̲=θnb-θ¯, which means that the difference between the marginal costs of the most efficient unregulated firm (F1b) and the efficient type of FA is equal to the difference between the cost of the most inefficient firm in market B (Fnb) and the inefficient type of the regulated firm, and they are both equal to the difference between the size of the two markets. (We will see later on that the conclusions remain the same if we disregard such assumption.)


### 2.3. The Regulator

Only firmFA is regulated by the regulator. The regulation contract is composed by a quantity-transfer pair {qa(θma),ta(θma)}θma∈{θ̲,θ¯}, which depends on the message sent by the regulated firm about its cost to the regulator (θma). For a given contract {qa,ta}, the total ex post profits are equal to(3)πa=[Pa(qa,Qb)-θa]qa-ta,πib=[Pb(qa,Qb)-θib]qib∀i=1,…,n.Similarly to Aubert and Pouyet [13], we have decided not to consider any relationship between the regulator and any other firm. However, that could be introduced into the model by extending the analysis to a lump-sum tax or taxes proportional either to profits or to output.We also assume that the firms’ profits do not enter the objective function of the regulator, so that this objective is reduced to the net consumer surplus and to the transfers paid by the regulated firm. Therefore, the rents left to the regulated firm are socially costly since they represent the informational rent paid by the regulator to induce a truthful report from the regulated firm. The regulator’s objective is to maximize social welfare, given by(4)W=GS(qa,Qb)-θaqa-Pb(qa,Qb)Qb-πa.


### 2.4. The Timing

In this model we have a sequential game, described as follows.(1)
Nature draws one of the two possible values forθa, only known to the regulated firm FA.(2)
The regulator proposes a contract{qa(θma),ta(θma)}θma∈{θ̲,θ¯}to the regulated firmFA, without knowing her marginal costs.(3)
The regulated firmFA decides whether to accept or reject this contract. If she refuses, she gets a reservation gain exogenously normalized to zero and the game stops. If FA accepts the contract, the game continues.(4)
After accepting the contract and before the report is sent to the regulator, the unregulated firms may try to payFA the amount b in order to induce her to always claim to be inefficient to the regulator: collusion. After the collusion agreement takes place, FAsends a message θma to the regulator, produces the corresponding quantity, and receives the transfer accordingly.(5)
All the unregulated firms act as Stackelberg followers, deciding their individual quantities simultaneously, given the quantity produced byFA.


## 2.1. The Firms

The market is composed of a dominant regulated firmFA and by n unregulated firms, each denoted by FiB, where i=1,…,n. All the firms compete in quantities. Firm FA has a constant marginal cost θa which can take one of two possible values, θ̲ or θ¯, θ¯-θ̲≡Δθ>0. The value of this marginal cost is private information forFA; however, its distribution is public knowledge: with probability p̲  FA, it is efficient (θa=θ̲) and it is inefficient (θa=θ¯) with probability p¯=1-p̲. The unregulated firms have different marginal costs (θib), which are publicly known. For simplification, we have assumed that the unregulated firms are ordered from the most efficient to the least efficient, such that θ1b≤θ2b≤…≤θnb.


## 2.2. The Consumers

The consumers have two differentiated products available for consumption:qa is the quantity of the product produced by the regulated firm in market A and qib is the quantity of the product produced by the unregulated firm i, where Qb=∑i=1nqibrepresents the total quantity produced by all unregulated firms. The Gross Consumer Surplus is given by(1)GS(qa,Qb)=daqa+dbQb-12(qa)2-12(Qb)2-sqaQb,where the parameters∈(0,1) measures the degree of substitutability between the two products, whereas the parameter dj represents the size of marketj=A,B (Such Consumer Surplus function is a result from the standard quadratic utility function proposed by Dixit.) The inverse demand functions for both markets are given by(2)Pa(qa,Qb)=da-qa-sQb,Pb(qa,Qb)=db-Qb-sqa.For the time being, we assume thatdb-da=θ1b-θ̲=θnb-θ¯, which means that the difference between the marginal costs of the most efficient unregulated firm (F1b) and the efficient type of FA is equal to the difference between the cost of the most inefficient firm in market B (Fnb) and the inefficient type of the regulated firm, and they are both equal to the difference between the size of the two markets. (We will see later on that the conclusions remain the same if we disregard such assumption.)


## 2.3. The Regulator

Only firmFA is regulated by the regulator. The regulation contract is composed by a quantity-transfer pair {qa(θma),ta(θma)}θma∈{θ̲,θ¯}, which depends on the message sent by the regulated firm about its cost to the regulator (θma). For a given contract {qa,ta}, the total ex post profits are equal to(3)πa=[Pa(qa,Qb)-θa]qa-ta,πib=[Pb(qa,Qb)-θib]qib∀i=1,…,n.Similarly to Aubert and Pouyet [13], we have decided not to consider any relationship between the regulator and any other firm. However, that could be introduced into the model by extending the analysis to a lump-sum tax or taxes proportional either to profits or to output.We also assume that the firms’ profits do not enter the objective function of the regulator, so that this objective is reduced to the net consumer surplus and to the transfers paid by the regulated firm. Therefore, the rents left to the regulated firm are socially costly since they represent the informational rent paid by the regulator to induce a truthful report from the regulated firm. The regulator’s objective is to maximize social welfare, given by(4)W=GS(qa,Qb)-θaqa-Pb(qa,Qb)Qb-πa.


## 2.4. The Timing

In this model we have a sequential game, described as follows.(1)
Nature draws one of the two possible values forθa, only known to the regulated firm FA.(2)
The regulator proposes a contract{qa(θma),ta(θma)}θma∈{θ̲,θ¯}to the regulated firmFA, without knowing her marginal costs.(3)
The regulated firmFA decides whether to accept or reject this contract. If she refuses, she gets a reservation gain exogenously normalized to zero and the game stops. If FA accepts the contract, the game continues.(4)
After accepting the contract and before the report is sent to the regulator, the unregulated firms may try to payFA the amount b in order to induce her to always claim to be inefficient to the regulator: collusion. After the collusion agreement takes place, FAsends a message θma to the regulator, produces the corresponding quantity, and receives the transfer accordingly.(5)
All the unregulated firms act as Stackelberg followers, deciding their individual quantities simultaneously, given the quantity produced byFA.


## 3. Optimal Regulation with Perfect Information

In this section we assume that all the firms’ marginal costs are known by all. The best response function for each of the unregulated firms is given by(5)qib=db-θib-sqa-∑j≠iqjb2,∀i=1,…,n.Since all unregulated firms decide their quantities simultaneously, it is possible to consider an aggregate best response function for market B, given by(6)Qb=1n+1[ndb-∑i=1nθib-nsqa].


### 3.1. Complete Regulation and Complete Information: The Social Optimum

The socially optimal quantities(qopta,qioptb,Qoptb)would be the ones that the regulator would ideally choose if he had complete information and complete regulation, becoming a reference benchmark to other possible situations. These optimal quantities are such that the prices equal the marginal costs (for the firms operating in market B, the optimal quantities are set by setting the price equal to the marginal cost of the least efficient firm (firm n). Hence, we are implicitly considering an exogenous process of entry and exit of firms in this market) and are given by(7)qopta=da-θa-s(db-θnb)1-s2,qioptb=θnb-θibQoptb=db-θnb-s(da-θa)1-s2.Notice that the only two relevant marginal costs for the optimal quantities are the ones related toFA and FnB since they are directly linked to the equilibrium prices for both markets. Such quantities also depend positively on the market dimension and negatively on the competitor’s market dimension. It is also worth mentioning that none of the first-best equilibrium prices depend on the number of firms in market B (n). Another interesting feature is that the quantity produced by each of the unregulated firms is directly given by the absolute costs advantage, leading to a quantity equal to zero for the least efficient firm.


### 3.2. Incomplete Regulation and Complete Information: The Second Best

In this section we assume that the regulator is only able to regulate one firm (FA), whose marginal costs’ are perfectly observable. As the rent of FA is socially costly, the regulator will decide to extract all profits from the regulated firm. By incorporating this constraint into the social welfare function stated previously, we reach the following solution (see Appendix A. for the objective function, constraints and first order conditions), for any θa={θ̲,θ¯}:(8)q*a(θa)=1(n+1)2-(ns)2×[(n+1)2(da-θa)-ns(ndb-∑i=1nθib)],qi*b(q*a(θa))=1(n+1)2-(ns)2×[(n+1)(db-s(da-θa))+(n+1-ns2)∑j≠iθjb+n(s2(n-1)-(n+1))θib],Q*b(q*a(θa))=n+1(n+1)2-(ns)2×[ndb-∑i=1nθib-ns(da-θa)].Notice that ifs=0 the two quantities q*a(θa), are the same as the equilibrium quantities in Section 3.1., meaning that when the two products are independent, the unregulated firms do not compete with the regulated firm, and therefore there is no way that the regulator can influence their behaviour in an incomplete regulation framework. However, when the two products are substitutes, and as the profits of the unregulated firms are not included in the social welfare function, the regulator wants to incentive firm FA to produce more such that the Consumer Surplus is greater, even if that means smaller profits for the unregulated firms. Also notice that all optimal quantities depend on the number of firms in market B (n).


## 3.1. Complete Regulation and Complete Information: The Social Optimum

The socially optimal quantities(qopta,qioptb,Qoptb)would be the ones that the regulator would ideally choose if he had complete information and complete regulation, becoming a reference benchmark to other possible situations. These optimal quantities are such that the prices equal the marginal costs (for the firms operating in market B, the optimal quantities are set by setting the price equal to the marginal cost of the least efficient firm (firm n). Hence, we are implicitly considering an exogenous process of entry and exit of firms in this market) and are given by(7)qopta=da-θa-s(db-θnb)1-s2,qioptb=θnb-θibQoptb=db-θnb-s(da-θa)1-s2.Notice that the only two relevant marginal costs for the optimal quantities are the ones related toFA and FnB since they are directly linked to the equilibrium prices for both markets. Such quantities also depend positively on the market dimension and negatively on the competitor’s market dimension. It is also worth mentioning that none of the first-best equilibrium prices depend on the number of firms in market B (n). Another interesting feature is that the quantity produced by each of the unregulated firms is directly given by the absolute costs advantage, leading to a quantity equal to zero for the least efficient firm.


## 3.2. Incomplete Regulation and Complete Information: The Second Best

In this section we assume that the regulator is only able to regulate one firm (FA), whose marginal costs’ are perfectly observable. As the rent of FA is socially costly, the regulator will decide to extract all profits from the regulated firm. By incorporating this constraint into the social welfare function stated previously, we reach the following solution (see Appendix A. for the objective function, constraints and first order conditions), for any θa={θ̲,θ¯}:(8)q*a(θa)=1(n+1)2-(ns)2×[(n+1)2(da-θa)-ns(ndb-∑i=1nθib)],qi*b(q*a(θa))=1(n+1)2-(ns)2×[(n+1)(db-s(da-θa))+(n+1-ns2)∑j≠iθjb+n(s2(n-1)-(n+1))θib],Q*b(q*a(θa))=n+1(n+1)2-(ns)2×[ndb-∑i=1nθib-ns(da-θa)].Notice that ifs=0 the two quantities q*a(θa), are the same as the equilibrium quantities in Section 3.1., meaning that when the two products are independent, the unregulated firms do not compete with the regulated firm, and therefore there is no way that the regulator can influence their behaviour in an incomplete regulation framework. However, when the two products are substitutes, and as the profits of the unregulated firms are not included in the social welfare function, the regulator wants to incentive firm FA to produce more such that the Consumer Surplus is greater, even if that means smaller profits for the unregulated firms. Also notice that all optimal quantities depend on the number of firms in market B (n).


## 4. Optimal Incomplete Regulation under Asymmetric Information and in the Absence of Collusion

We now assume that the regulator and all the unregulated firms do not know the firmFA’s actual marginal costs and only its distribution is publicly known. We also assume that the regulator can only regulate FA and the marginal costs of the unregulated firms are perfectly observable and that collusion cannot occur.According to the timing defined previously, afterFA decides which quantity to produce, the other firms will simultaneously choose how much they want to produce according to their respective best response functions.We denoteπa(θa,θma) as the profits for FA when she has a marginal cost of θa and reports to have a marginal cost of θma to the regulator, qa¯≡qa(θa¯) as the quantity that should be produced by the regulated firm when inefficient, and qa̲≡qa(θa̲) as the quantity produced when FA is efficient. Even in the presence of incomplete regulation, we can still assume that the revelation principle holds (see Green and Laffont [14] or Myerson [15], among others, on the revelation principle), and we can focus on direct and truthful contracts, especially on πa̲≡πa(θ̲a,θa̲)and πa¯≡πa(θa¯,θa¯) which represent the rents for the efficient and inefficient regulated firms at a truthful equilibrium, respectively. Let us also denote Qb¯and Qb̲as the total quantity produced in market B when FA is inefficient and efficient, respectively.Therefore, the regulator’s objective is to maximize social welfare, subject to incentive compatibility (ICC) and participation constraints (PC), as follows:(9)max⁡{qa̲,qa¯}E(W)=p̲{daqa̲+dbQb̲-12(qa̲)2-12(Qb̲)2-sqa̲Qb̲-θ̲qa̲-(db-Qb̲-sqa̲)Qb̲-πa̲12(Qb¯)2}+p¯{daqa¯+dbQb¯-12(qa¯)2-12(Qb¯)2-sqa¯Qb¯-θ¯qa¯-(db-Qb¯-sqa¯)Qb̲-πa¯12(Qb¯)2},s.t.πa̲≥0(PCforθ̲)πa¯≥0(PCforθ¯)πa̲≥πa¯+Δθqa¯(ICCforθ̲)πa¯≥πa̲-Δθqa̲(ICCforθ¯).As usual, only the PC for the inefficientFA and the ICC for the efficient regulated firm constraints are binding. Substituting these constraints into the objective function and using the first-order conditions, we get(10)qa̲=1(n+1)2-(ns)2×[(n+1)2(da-θ̲)-ns(ndb-∑i=1nθib)]=q*a(θ̲),qa¯=1(n+1)2-(ns)2×[(n+1)2(da-θ¯)-ns(ndb-∑i=1nθib)]-(n+1)2(n+1)2-(ns)2p̲p¯Δθ=q*a(θ¯)-(n+1)2(n+1)2-(ns)2p̲p¯Δθ.
Under asymmetric information and incomplete regulation and in the case where firms cannot collude we can verify that the standard no distortion at the top equilibrium in an adverse selection model holds. An efficient firm FA will produce the complete information output; however, the regulator needs to leave an information rent of Δθqa¯ in order to induce a truthful revelation of its efficiency. Since this rent increases with the quantity produced by the inefficient firm FA, this quantity will be distorted downwards relatively to the respective complete information quantity. This informational rent also increases with the probability of firm FA being efficient (p) and with the substitutability degree between the products A and B. The decrease in the quantity for an inefficient regulated firm does not have large consequences on the social welfare, since the risk of having an inefficient FA is lower and consumers can more easily compensate this decrease in qa¯ by consuming more from the other unregulated firms. The informational rent will also increase with the number of unregulated firms n. Once again, a lower qa¯ will lead to smaller distortions on the expected social welfare since the unregulated firms’ ability of capturing higher profits will be very low when they are many.At this point we should also notice that due to product substitutability there is a stake for collusion (Proof is in AppendixB). We have seen that the ICC for the efficient FA will lead to qa̲≥qa¯, also leading to πib¯≥πib̲,∀i=1,…,n, and therefore the unregulated firms have incentives to make the regulated firm always report to be inefficient. Since these unregulated firms’ profits increase will play an important role in the next section, we will denote Δπib≡πib¯-πib̲ as the increase in the profits of firm i in the unregulated sector when FA produces qa¯ instead of qa̲.


## 5. Optimal Regulation under Asymmetric Information and with the Possibility of Collusion

As it was previously mentioned, it is possible for the firms to collude such that the firmFA always reports to be inefficient to the regulator. Such decision takes place after the firm FA knows her true costs but before deciding which costs to report to the regulator. To model collusion under asymmetric information we used a methodology similar to that of Laffont and Martimort [5, 6] and Aubert and Pouyet [13]. Since θa is private information at the time when the collusion takes place, we can model the bargaining process between the firms by considering a hypothetical benevolent mediator whose objective is to maximize the collusion’s aggregate expected profits, subject to participation and incentive compatibility constraints. This mediator can be viewed as another principal to whom the revelation principle applies: the mediator will offer a collusive agreement such that all firms are willing to participate and firm FA truthfully reports its type to the coalition. We should also notice that since the marginal costs of all the unregulated firms are publicly known, they cannot deviate from their best response function to a certain report from the firm FA, leading to an immediate detection of the collusion by an antitrust authority. Conversely, this authority does not detect any type of collusion between the firms if they act according to the report made by FA to the regulator.The aim of this section is to analyse to what extent the collusion-proofness principle stated by Laffont and Martimort [5, 6] holds in the present framework. With complete regulation, there is no loss of generality if we focus only on regulation contracts that will replicate the collusion outcomes since the regulator is able to enforce any output to all firms in the market. However, as already pointed out by Aubert and Pouyet [13], in an incomplete regulation setting, the regulator is not able to replicate all possible outcomes using a regulation contract, and therefore we should not restrict our attention only to those contracts that avoid collusion. All situations that involve a side-payment from the unregulated firms to the regulated firm FA are out of the regulator’s reach since he cannot impose any tax on any firm FiBwhich could replicate those payments. By comparing the social welfare from Collusion-Proofness Contracts with the social welfare obtained through contracts that allow collusion, we are able to assess the robustness of the Collusion-Proof Principle.


### 5.1. Collusion-Proof Contracts

We start by focusing on Collusion-Proof Contracts, which are contracts that induce a passive response from the coalition. When designing such a contract, the mediator will ask firmFA to truthfully report its marginal costs to the regulator and no side-payment will be made from the unregulated firms to FA.Let us denoteθma(θa) as the report recommended by the mediator when the marginal cost of firm FA is θa, and let θnc(θa) be the report that firm FA would make to the regulator if no collusion occurs. In the case of collusion, the regulated firm will produce qa(θma), and each of the unregulated firms would have to produce qib(θma) and pay firm FA the amount bi(θa), where B(θa)=∑i=1nbi(θa) represents the aggregate side-payment made to firm FA.The mediator’s objective is to maximize the coalition’s aggregate profits subject to the participation constraints (PCs) for all firms and to the incentive compatibility constraints (ICCs) for both types of the regulated firm such that it will always reveal truthfully its marginal cost to the coalition. It can be written as(11)max⁡{(θma),bi(θa)}i=1nEθa[∑i=1nπib(θma)+πa(θma,θa)]s.t.πa(θma(θ̲),θ̲)+B(θ̲)≥πa(θnc(θ̲),θ̲),PCforFaπa(θma(θ¯),θ¯)+B(θ¯)≥πa(θnc(θ¯),θ¯),PCforFaπa(θma(θ̲),θ̲)+B(θ̲)≥πa(θma(θ¯),θ̲)+B(θ¯),ICCforFaπa(θma(θ¯),θ¯)+B(θ¯)≥πa(θma(θ̲),θ¯)+B(θ̲),ICCforFaπib(θma(θ̲))-bi(θ̲)≥πib(θnc(θ̲)),∀i=1,…,n,PCforFibπib(θma(θ¯))-bi(θ¯)≥πib(θnc(θ¯)),∀i=1,…,n,PCforFib.It is possible to prove that in the case of a Collusion-Proof contract, the regulator is able to use the information asymmetry within the coalition to create a regulation contract which leads to “individual” truthful revelation by firmFA. Hence, the ICC for the inefficient type will not be binding, and the collusive equilibrium would be the same as in complete information (See proof in Appendix C). This result is opposite to the one found by Laffont and Martimort [6] in a complete regulation context, where asymmetric information generated inefficiencies in the functioning of the coalition.By solving the mediator’s problem, we get the Collusion-Proofness Constraint which the regulator needs to include in his problem in order to induce a passive response from the coalition. This constraint can be written as (see proof in AppendixC):
(12)πa̲≥πa¯+Δθqa¯+∑i=1nΔπib.This result is consistent with the one reached by Aubert and Pouyet [13] for the case of two firms. Such constraint can be understood intuitively: an efficient firm FA is willing to report truthfully its marginal costs whenever its profits from telling the truth are greater than the ones she would get by pretending to be inefficient plus the highest amount of side-payment that the unregulated firms are willing to give to firm FA (which is equal to the increase on their profits when the regulated firms mislead the regulator).By solving the regulator’s problem and including the collusion-proofness constraint derived above, the best separating collusion-proof contract is characterized by the following rents and quantities (see proof in AppendixC):(13)qcpa̲=1(n+1)2-ns2(2+n)×[(n+1)2(da-θ̲)-(2+n)s(ndb-∑i=1nθib)],qcpa¯=1(n+1)2-ns2(n-2p̲p¯)×[∑i=1nθib(n+1)2(da-θ¯-p̲p¯Δθ)-s(n-2p̲p¯)×(ndb-∑i=1nθib)],πcpa¯=0,πcpa̲=Δθqcpa¯+∑i=1nΔπib.Notice that both quantities are distorted when compared with the complete information outcomes and that this best collusion-proof contract is separating only ifqcpa̲≥qcpa¯. Otherwise it entails pooling for a quantity given by (see proof in Appendix C):(14)qpoola=1(n+1)2-(ns)2×[(n+1)2(da-θ¯)-ns(ndb-∑i=1nθib)],which is the complete information outcome for an inefficientFA.With collusion-proof contracts, the regulator has to provideFA with an extra rent in order to compensate for any side-payment that she might get through collusion. Since such amount is affected by the quantity produced by both an efficient and an inefficient firm FA, they will be distorted downwards to minimize the cost of inducing truthful revelation, contrary to the standard result of no distortion at the top. In fact, to ensure collusion-proofness, the regulator has to reduce simultaneously the rent Δθqcpa¯ and the stake for collusion ∑i=1nΔπib. The first implies a decrease in qcpa¯ while the second requires a simultaneous decrease in qcpa̲ and an increase of qcpa¯ to bring these two quantities closer together. We have thus two opposite effects on qcpa¯. As in Aubert and Pouyet [13], we can verify that such quantity will be smaller than in the case with no possibility of collusion, meaning that the first effect dominates the second.The second objective of bringing both quantities closer together also conflicts with the condition of sustainability of a separating equilibrium for collusion-proof contracts:qcpa̲≥qcpa¯. Taken to an extreme, a pooling equilibrium may emerge. Such equilibrium will be especially desirable when the stake of collusion ∑i=1nΔπib is very large and very sensitive to the regulated quantities. This occurs when the profitability of the unregulated market is high (ndb-∑i=1nθib) and when FA  s scale of production (qa̲+qa¯) is low. Contrary to the findings of Aubert and Pouyet [13], the stake for collusion does not depend monotonically on the substitutability between the two products. It becomes larger for an increase in s when the products are almost independent (s close to zero), and it also becomes larger for a decrease in s for almost perfect substitutable products (s close to one). The stake for collusion is also larger for a small number of firms operating in the unregulated sector and decreases when n increases.


### 5.2. Collusion-Allowing Contracts

As already mentioned, with incomplete regulation we cannot rule out contracts that allow collusion since there are some outcomes only attainable by taxing the unregulated firms and therefore unattainable by the regulator. A potential benefit of contracts inducing active collusion is precisely the possibility of extracting some rents from the unregulated firms and direct them to the regulated firm through side-payments, as also stated by Tirole [16], Itoh [9], Lawarrëe and Shin [10], and Shin [11]. Allowing collusion, the regulated firm FA will have higher utility and therefore will be more willing to accept the regulation contract even if it yields higher taxes.As in the Aubert and Pouyet [13] model, it is possible to show that the revelation principle still applies to the present model framework, where the agent to be considered is the coalition as a whole and no longer the regulated (Proof is in Appendix D.) Hence, we can restrict our attention to direct mechanisms which induce active collusion and truthful reports from the coalition. The regulator must now include in his problem the incentive compatibility constraint and the participation constraint for the coalition, as well as the incentive compatibility constraints and participation constraints within the coalition, which will reflect the expected aggregate profits:(15)max⁡{qa̲,qa¯,bi(θ̲),bi(θ¯)}i=1nEθa(W)=p̲{daqa̲+dbQb̲-12(qa̲)2-12(Qb̲)2-sqa̲Qb̲-θ̲qa̲-(db-Qb̲-sqa̲)Qb̲-πa̲12}+p¯{daqa¯+dbQb¯-12(qa¯)2-12(Qb¯)2-sqa¯Qb¯-θ¯qa¯-(db-Qb¯-sqa¯)Qb̲-πa¯12}s.t.πa̲+∑i=1nπib̲≥πa¯+Δθqa¯+∑i=1nπib¯ICCforcoalition(θ̲)πa¯+∑i=1nπib¯≥πa̲-Δθqa̲+∑i=1nπib̲ICCforcoalition(θ¯)πa̲+∑i=1nπib̲≥0PCforcoalition(θ̲)πa¯+∑i=1nπib¯≥0PCforcoalition(θ¯)πa̲+B(θ̲)≥πa(θnc(θ̲),θ̲),PCforFaπa¯+B(θ¯)≥πa(θnc(θ¯),θ¯),PCforFaπa̲+B(θ̲)≥πa(θma(θ¯),θ̲)+B(θ¯),ICCforFaπa¯+B(θ¯)≥πa(θma(θ̲),θ¯)+B(θ̲),ICCforFaπib̲-bi(θ̲)≥πib(θnc(θ̲)),∀i=1,…,n,PCforFibπib¯-bi(θ¯)≥πib(θnc(θ¯)),∀i=1,…,n,.PCforFibGiven the monotonicity condition thatqaca̲≥qaca¯, the last six constraints will not be binding. Rearranging the other constraints it will yield the only two relevant conditions:(16)πa̲=Δθqa¯-∑i=1nπib̲,πa¯+∑i=1nπib¯=0.Hence, the best contract with active collusion is then characterized by the following rents and quantities:(17)qaca̲=1(1-s2)n(n+2)+1×[(n+1)2(da-θ̲)-s(n+2)(ndb-∑i=1nθib)]qaca¯=1(1-s2)n(n+2)+1×[(n+1)2(da-θ¯-p̲p¯Δθ)-s(n+2)(ndb-∑i=1nθib)]πaca̲=Δθqaca¯-∑i=1nπib̲πaca¯=-∑i=1nπib¯By comparing these quantities with the ones obtained for the collusion-proof contract, we can verify thatqaca̲=qcpa̲ and qaca¯<qcpa¯. This can be intuitively understood. By allowing collusion, the regulator is aware that FA will have higher profits when she reports to be inefficient due to the side-payments made by the other firms. Hence, instead of rewarding efficiency, the regulator penalizes even further FA if she reports to be inefficient. In another way, when allowing active collusion, the regulator is now internalizing not only the benefit that the consumers derive from qb but also the profits that this consumption generates, which will be forwarded to the regulated firm through side-payments. Hence, imposing lower quantities for FA will increase the profits of the unregulated firms, and therefore smaller will be the share that the regulator needs to transfer to FA.


### 5.3. Equilibrium Quantities Comparison

From this section onwards, due to algebraic complexity, we have decided to compare quantities, social welfare, and Consumer Surplus using graphical representations as a result of a computational simulation, using a set of parameters (See AppendixE. for parameter values used and detailed description of the simulation.) We will start by comparing FA’s and the unregulated firms’ equilibrium quantities for both collusion-proof and collusion-allowing contracts.First, we will analyse how the equilibrium quantities produced by firmFA change with the number of competitors.From the diagrams above. From Figure1, we can see that there is a negative (and convex) relationship between the quantities produced by FA and the amount of competitors she faces in the unregulated market. Although these reflect a relatively low substitutability between the products, we can see that when the number of competitors remains low but increases (between 1 and 10), it has a dramatic effect on the quantity produced by FA. The same happens for other values of the substitutability degree (as we will see later on), where the quantity of FA will tend to zero after a certain number of competitors.Figure 1The following graphs allow us to better understand how the two types of equilibrium quantities forFA and for all unregulated firms are related.We can see that for an efficientFA it is indifferent to allow collusion or not, as expected. However, if FA is inefficient then we have different levels of output for the two types of contracts. Her quantities tend to be lower when collusion is allowed, which is an evidence that the competitors are able to detect firm FA’s inefficiency and that the truth-telling constraint holds. We should also notice that the total output produced in market B increases with the number of firms belonging to that market and that it is slightly higher in collusion-allowing contracts (Figure 2).Figure 2The following diagrams compare firmFA’s equilibrium outcomes for different levels of the substitutability degree.We can immediately notice that when firmFA is efficient, her equilibrium quantities are approximately the same across the two possible situations, similarly to the result regarding the number of unregulated firms. However, when firm FA is inefficient the collusion-proof quantities are greater than the equilibrium quantities when collusion is allowed. Once more, this is evidence that when designing the collusive agreement, the unregulated firms are able to extract from FA truthful information about her costs (Figure 3).Figure 3The results obtained for increasing values of the substitutability degree are also very intuitive. IfFA is efficient, her equilibrium quantities will increase as the consumers start to perceive the products as close substitutes. The opposite happens when FA is inefficient, where her quantities decrease with the degree of substitutability and become zero after a certain threshold.It is also worth mentioning that the level of equilibrium quantities forFA also decrease when we increase the number of unregulated firms from two to five, and such pattern continues throughout the rest of the levels considered.


### 5.4. Social Welfare Differences

The goal of this section is to compare the levels of social welfare obtained by a collusion-proof contract and by a contract that allows collusion. Let us start by looking at the differences between the value of expected Welfare when we change the number of unregulated firms (Figure4).Figure 4It can be seen that the Welfare difference is always positive, meaning that the collusion-allowing equilibrium always entails a higher-expected Welfare than the one given by collusion-proof contracts. Furthermore, this value is increasing with the number of unregulated firms, and its level is slightly increasing with substitutability degrees. Such result can be justified in two ways. First, as we have argued before, the advantage of allowing collusion is that the unregulated firms in an effort to collude and decide side-payments will eventually lead firmFA to reveal its real costs, eliminating part of the information asymmetry. Such result would lead to a much higher cost for the regulator if collusion was not allowed. Secondly, as the number of unregulated firms increase the quantity and the relative weight of firm FA on the total output produced by both markets decrease, and therefore it will also decrease its importance in terms of social welfare.Next, we will analyse the influence of the substitutability degree on Welfare differences, similarly to Aubert and Pouyet [13]. Additionally, we will consider four different values for the number of firms in market B (Figure 5).Figure 5From the previous graph we can draw the same conclusions made previously for the number of unregulated firms. The Welfare differences are always positive, which means that the collusion-allowing equilibrium leads to greater social welfare. This Welfare difference increases with the degree of substitutability, which means that as the products are seen as close substitutes the two markets also seem closer and the weight thatFA has in the total output and in social welfare is much smaller. Additionally, by allowing collusion, the unregulated firms are able to decrease the information asymmetry in the economy, also decreasing the bias it will have on the Welfare. Once again, we can also notice that the level of Welfare difference increases with the number of unregulated firms.Finally, it is also worth mentioning that such analysis still holds when the range of values for the unregulated firms’ marginal costs increases, creating new unregulated firms that are more efficient than the efficientFA and other unregulated firms that are more inefficient than the inefficient regulated firm (Results and diagrams are in Appendix E.) The only difference lies on the level of values for the Welfare differences, which are still increasing and strictly positive but at even greater values, meaning that with increasing marginal cost differences across firms, not only the regulator does not need to impose costs on society by creating incentives to extract marginal cost information from FA, but also the risk that an effective collusion might appear will be much lower.


### 5.5. The Importance of Consumer Surplus Differences

From the previous section, a somewhat paradoxal result was raised: it is always Welfare improving to allow collusion between the regulated firm and the unregulated firms. Aubert and Pouyet [13] have already stated the failure of the collusion-proofness for some values of the substitutability degree, which was reinforced in this paper by admitting a new dimension to the model (an exogenous fixed number of unregulated firms). The reasons behind such conclusion were also already stated. In an incomplete regulation framework, the regulator has no control over most of the firms, and therefore he cannot replicate through contracts all the possible outcomes. Hence, it will be socially less costly to allow the unregulated firms, through side-payments, to extract FA’s information regarding her true marginal costs. In the light of such evidence, why do regulators still not allow collusion? The answer to this question may be found if we look only at the Consumer Surplus differences instead of social welfare differences (Figure 6).Figure 6The first and most important conclusion drawn from these diagrams is that active collusion will always lead to lower Consumer Surplus values. By allowing collusion, the aggregate quantity produced by both the regulated and unregulated firms will be lower (see the graphs in Section5.3) and therefore the prices will be higher, which together will harm consumers. However, the number of unregulated firms and the substitutability degree will have opposite effects on the consumer surplus. As the number of unregulated firms increases, the consumer surplus differences will decrease, while such differences will increase with the substitutability degree. Once again both results can be explained by the differences on the aggregate quantity produced. When the number of unregulated firms increases, there is a convergence between the levels of quantities produced and increases in the substitutability degree will lead to greater differences between the collusion-allowing and the collusion-proof outcomes.Hence, a new and important conclusion rises from this analysis: social welfare differences will depend heavily on the relative weight that the regulator allocates to Consumer Surplus on the social welfare function, also pointed out by Biglaiser and Ma [12].


## 5.1. Collusion-Proof Contracts

We start by focusing on Collusion-Proof Contracts, which are contracts that induce a passive response from the coalition. When designing such a contract, the mediator will ask firmFA to truthfully report its marginal costs to the regulator and no side-payment will be made from the unregulated firms to FA.Let us denoteθma(θa) as the report recommended by the mediator when the marginal cost of firm FA is θa, and let θnc(θa) be the report that firm FA would make to the regulator if no collusion occurs. In the case of collusion, the regulated firm will produce qa(θma), and each of the unregulated firms would have to produce qib(θma) and pay firm FA the amount bi(θa), where B(θa)=∑i=1nbi(θa) represents the aggregate side-payment made to firm FA.The mediator’s objective is to maximize the coalition’s aggregate profits subject to the participation constraints (PCs) for all firms and to the incentive compatibility constraints (ICCs) for both types of the regulated firm such that it will always reveal truthfully its marginal cost to the coalition. It can be written as(11)max⁡{(θma),bi(θa)}i=1nEθa[∑i=1nπib(θma)+πa(θma,θa)]s.t.πa(θma(θ̲),θ̲)+B(θ̲)≥πa(θnc(θ̲),θ̲),PCforFaπa(θma(θ¯),θ¯)+B(θ¯)≥πa(θnc(θ¯),θ¯),PCforFaπa(θma(θ̲),θ̲)+B(θ̲)≥πa(θma(θ¯),θ̲)+B(θ¯),ICCforFaπa(θma(θ¯),θ¯)+B(θ¯)≥πa(θma(θ̲),θ¯)+B(θ̲),ICCforFaπib(θma(θ̲))-bi(θ̲)≥πib(θnc(θ̲)),∀i=1,…,n,PCforFibπib(θma(θ¯))-bi(θ¯)≥πib(θnc(θ¯)),∀i=1,…,n,PCforFib.It is possible to prove that in the case of a Collusion-Proof contract, the regulator is able to use the information asymmetry within the coalition to create a regulation contract which leads to “individual” truthful revelation by firmFA. Hence, the ICC for the inefficient type will not be binding, and the collusive equilibrium would be the same as in complete information (See proof in Appendix C). This result is opposite to the one found by Laffont and Martimort [6] in a complete regulation context, where asymmetric information generated inefficiencies in the functioning of the coalition.By solving the mediator’s problem, we get the Collusion-Proofness Constraint which the regulator needs to include in his problem in order to induce a passive response from the coalition. This constraint can be written as (see proof in AppendixC):
(12)πa̲≥πa¯+Δθqa¯+∑i=1nΔπib.This result is consistent with the one reached by Aubert and Pouyet [13] for the case of two firms. Such constraint can be understood intuitively: an efficient firm FA is willing to report truthfully its marginal costs whenever its profits from telling the truth are greater than the ones she would get by pretending to be inefficient plus the highest amount of side-payment that the unregulated firms are willing to give to firm FA (which is equal to the increase on their profits when the regulated firms mislead the regulator).By solving the regulator’s problem and including the collusion-proofness constraint derived above, the best separating collusion-proof contract is characterized by the following rents and quantities (see proof in AppendixC):(13)qcpa̲=1(n+1)2-ns2(2+n)×[(n+1)2(da-θ̲)-(2+n)s(ndb-∑i=1nθib)],qcpa¯=1(n+1)2-ns2(n-2p̲p¯)×[∑i=1nθib(n+1)2(da-θ¯-p̲p¯Δθ)-s(n-2p̲p¯)×(ndb-∑i=1nθib)],πcpa¯=0,πcpa̲=Δθqcpa¯+∑i=1nΔπib.Notice that both quantities are distorted when compared with the complete information outcomes and that this best collusion-proof contract is separating only ifqcpa̲≥qcpa¯. Otherwise it entails pooling for a quantity given by (see proof in Appendix C):(14)qpoola=1(n+1)2-(ns)2×[(n+1)2(da-θ¯)-ns(ndb-∑i=1nθib)],which is the complete information outcome for an inefficientFA.With collusion-proof contracts, the regulator has to provideFA with an extra rent in order to compensate for any side-payment that she might get through collusion. Since such amount is affected by the quantity produced by both an efficient and an inefficient firm FA, they will be distorted downwards to minimize the cost of inducing truthful revelation, contrary to the standard result of no distortion at the top. In fact, to ensure collusion-proofness, the regulator has to reduce simultaneously the rent Δθqcpa¯ and the stake for collusion ∑i=1nΔπib. The first implies a decrease in qcpa¯ while the second requires a simultaneous decrease in qcpa̲ and an increase of qcpa¯ to bring these two quantities closer together. We have thus two opposite effects on qcpa¯. As in Aubert and Pouyet [13], we can verify that such quantity will be smaller than in the case with no possibility of collusion, meaning that the first effect dominates the second.The second objective of bringing both quantities closer together also conflicts with the condition of sustainability of a separating equilibrium for collusion-proof contracts:qcpa̲≥qcpa¯. Taken to an extreme, a pooling equilibrium may emerge. Such equilibrium will be especially desirable when the stake of collusion ∑i=1nΔπib is very large and very sensitive to the regulated quantities. This occurs when the profitability of the unregulated market is high (ndb-∑i=1nθib) and when FA  s scale of production (qa̲+qa¯) is low. Contrary to the findings of Aubert and Pouyet [13], the stake for collusion does not depend monotonically on the substitutability between the two products. It becomes larger for an increase in s when the products are almost independent (s close to zero), and it also becomes larger for a decrease in s for almost perfect substitutable products (s close to one). The stake for collusion is also larger for a small number of firms operating in the unregulated sector and decreases when n increases.


## 5.2. Collusion-Allowing Contracts

As already mentioned, with incomplete regulation we cannot rule out contracts that allow collusion since there are some outcomes only attainable by taxing the unregulated firms and therefore unattainable by the regulator. A potential benefit of contracts inducing active collusion is precisely the possibility of extracting some rents from the unregulated firms and direct them to the regulated firm through side-payments, as also stated by Tirole [16], Itoh [9], Lawarrëe and Shin [10], and Shin [11]. Allowing collusion, the regulated firm FA will have higher utility and therefore will be more willing to accept the regulation contract even if it yields higher taxes.As in the Aubert and Pouyet [13] model, it is possible to show that the revelation principle still applies to the present model framework, where the agent to be considered is the coalition as a whole and no longer the regulated (Proof is in Appendix D.) Hence, we can restrict our attention to direct mechanisms which induce active collusion and truthful reports from the coalition. The regulator must now include in his problem the incentive compatibility constraint and the participation constraint for the coalition, as well as the incentive compatibility constraints and participation constraints within the coalition, which will reflect the expected aggregate profits:(15)max⁡{qa̲,qa¯,bi(θ̲),bi(θ¯)}i=1nEθa(W)=p̲{daqa̲+dbQb̲-12(qa̲)2-12(Qb̲)2-sqa̲Qb̲-θ̲qa̲-(db-Qb̲-sqa̲)Qb̲-πa̲12}+p¯{daqa¯+dbQb¯-12(qa¯)2-12(Qb¯)2-sqa¯Qb¯-θ¯qa¯-(db-Qb¯-sqa¯)Qb̲-πa¯12}s.t.πa̲+∑i=1nπib̲≥πa¯+Δθqa¯+∑i=1nπib¯ICCforcoalition(θ̲)πa¯+∑i=1nπib¯≥πa̲-Δθqa̲+∑i=1nπib̲ICCforcoalition(θ¯)πa̲+∑i=1nπib̲≥0PCforcoalition(θ̲)πa¯+∑i=1nπib¯≥0PCforcoalition(θ¯)πa̲+B(θ̲)≥πa(θnc(θ̲),θ̲),PCforFaπa¯+B(θ¯)≥πa(θnc(θ¯),θ¯),PCforFaπa̲+B(θ̲)≥πa(θma(θ¯),θ̲)+B(θ¯),ICCforFaπa¯+B(θ¯)≥πa(θma(θ̲),θ¯)+B(θ̲),ICCforFaπib̲-bi(θ̲)≥πib(θnc(θ̲)),∀i=1,…,n,PCforFibπib¯-bi(θ¯)≥πib(θnc(θ¯)),∀i=1,…,n,.PCforFibGiven the monotonicity condition thatqaca̲≥qaca¯, the last six constraints will not be binding. Rearranging the other constraints it will yield the only two relevant conditions:(16)πa̲=Δθqa¯-∑i=1nπib̲,πa¯+∑i=1nπib¯=0.Hence, the best contract with active collusion is then characterized by the following rents and quantities:(17)qaca̲=1(1-s2)n(n+2)+1×[(n+1)2(da-θ̲)-s(n+2)(ndb-∑i=1nθib)]qaca¯=1(1-s2)n(n+2)+1×[(n+1)2(da-θ¯-p̲p¯Δθ)-s(n+2)(ndb-∑i=1nθib)]πaca̲=Δθqaca¯-∑i=1nπib̲πaca¯=-∑i=1nπib¯By comparing these quantities with the ones obtained for the collusion-proof contract, we can verify thatqaca̲=qcpa̲ and qaca¯<qcpa¯. This can be intuitively understood. By allowing collusion, the regulator is aware that FA will have higher profits when she reports to be inefficient due to the side-payments made by the other firms. Hence, instead of rewarding efficiency, the regulator penalizes even further FA if she reports to be inefficient. In another way, when allowing active collusion, the regulator is now internalizing not only the benefit that the consumers derive from qb but also the profits that this consumption generates, which will be forwarded to the regulated firm through side-payments. Hence, imposing lower quantities for FA will increase the profits of the unregulated firms, and therefore smaller will be the share that the regulator needs to transfer to FA.


## 5.3. Equilibrium Quantities Comparison

From this section onwards, due to algebraic complexity, we have decided to compare quantities, social welfare, and Consumer Surplus using graphical representations as a result of a computational simulation, using a set of parameters (See AppendixE. for parameter values used and detailed description of the simulation.) We will start by comparing FA’s and the unregulated firms’ equilibrium quantities for both collusion-proof and collusion-allowing contracts.First, we will analyse how the equilibrium quantities produced by firmFA change with the number of competitors.From the diagrams above. From Figure1, we can see that there is a negative (and convex) relationship between the quantities produced by FA and the amount of competitors she faces in the unregulated market. Although these reflect a relatively low substitutability between the products, we can see that when the number of competitors remains low but increases (between 1 and 10), it has a dramatic effect on the quantity produced by FA. The same happens for other values of the substitutability degree (as we will see later on), where the quantity of FA will tend to zero after a certain number of competitors.Figure 1The following graphs allow us to better understand how the two types of equilibrium quantities forFA and for all unregulated firms are related.We can see that for an efficientFA it is indifferent to allow collusion or not, as expected. However, if FA is inefficient then we have different levels of output for the two types of contracts. Her quantities tend to be lower when collusion is allowed, which is an evidence that the competitors are able to detect firm FA’s inefficiency and that the truth-telling constraint holds. We should also notice that the total output produced in market B increases with the number of firms belonging to that market and that it is slightly higher in collusion-allowing contracts (Figure 2).Figure 2The following diagrams compare firmFA’s equilibrium outcomes for different levels of the substitutability degree.We can immediately notice that when firmFA is efficient, her equilibrium quantities are approximately the same across the two possible situations, similarly to the result regarding the number of unregulated firms. However, when firm FA is inefficient the collusion-proof quantities are greater than the equilibrium quantities when collusion is allowed. Once more, this is evidence that when designing the collusive agreement, the unregulated firms are able to extract from FA truthful information about her costs (Figure 3).Figure 3The results obtained for increasing values of the substitutability degree are also very intuitive. IfFA is efficient, her equilibrium quantities will increase as the consumers start to perceive the products as close substitutes. The opposite happens when FA is inefficient, where her quantities decrease with the degree of substitutability and become zero after a certain threshold.It is also worth mentioning that the level of equilibrium quantities forFA also decrease when we increase the number of unregulated firms from two to five, and such pattern continues throughout the rest of the levels considered.


## 5.4. Social Welfare Differences

The goal of this section is to compare the levels of social welfare obtained by a collusion-proof contract and by a contract that allows collusion. Let us start by looking at the differences between the value of expected Welfare when we change the number of unregulated firms (Figure4).Figure 4It can be seen that the Welfare difference is always positive, meaning that the collusion-allowing equilibrium always entails a higher-expected Welfare than the one given by collusion-proof contracts. Furthermore, this value is increasing with the number of unregulated firms, and its level is slightly increasing with substitutability degrees. Such result can be justified in two ways. First, as we have argued before, the advantage of allowing collusion is that the unregulated firms in an effort to collude and decide side-payments will eventually lead firmFA to reveal its real costs, eliminating part of the information asymmetry. Such result would lead to a much higher cost for the regulator if collusion was not allowed. Secondly, as the number of unregulated firms increase the quantity and the relative weight of firm FA on the total output produced by both markets decrease, and therefore it will also decrease its importance in terms of social welfare.Next, we will analyse the influence of the substitutability degree on Welfare differences, similarly to Aubert and Pouyet [13]. Additionally, we will consider four different values for the number of firms in market B (Figure 5).Figure 5From the previous graph we can draw the same conclusions made previously for the number of unregulated firms. The Welfare differences are always positive, which means that the collusion-allowing equilibrium leads to greater social welfare. This Welfare difference increases with the degree of substitutability, which means that as the products are seen as close substitutes the two markets also seem closer and the weight thatFA has in the total output and in social welfare is much smaller. Additionally, by allowing collusion, the unregulated firms are able to decrease the information asymmetry in the economy, also decreasing the bias it will have on the Welfare. Once again, we can also notice that the level of Welfare difference increases with the number of unregulated firms.Finally, it is also worth mentioning that such analysis still holds when the range of values for the unregulated firms’ marginal costs increases, creating new unregulated firms that are more efficient than the efficientFA and other unregulated firms that are more inefficient than the inefficient regulated firm (Results and diagrams are in Appendix E.) The only difference lies on the level of values for the Welfare differences, which are still increasing and strictly positive but at even greater values, meaning that with increasing marginal cost differences across firms, not only the regulator does not need to impose costs on society by creating incentives to extract marginal cost information from FA, but also the risk that an effective collusion might appear will be much lower.


## 5.5. The Importance of Consumer Surplus Differences

From the previous section, a somewhat paradoxal result was raised: it is always Welfare improving to allow collusion between the regulated firm and the unregulated firms. Aubert and Pouyet [13] have already stated the failure of the collusion-proofness for some values of the substitutability degree, which was reinforced in this paper by admitting a new dimension to the model (an exogenous fixed number of unregulated firms). The reasons behind such conclusion were also already stated. In an incomplete regulation framework, the regulator has no control over most of the firms, and therefore he cannot replicate through contracts all the possible outcomes. Hence, it will be socially less costly to allow the unregulated firms, through side-payments, to extract FA’s information regarding her true marginal costs. In the light of such evidence, why do regulators still not allow collusion? The answer to this question may be found if we look only at the Consumer Surplus differences instead of social welfare differences (Figure 6).Figure 6The first and most important conclusion drawn from these diagrams is that active collusion will always lead to lower Consumer Surplus values. By allowing collusion, the aggregate quantity produced by both the regulated and unregulated firms will be lower (see the graphs in Section5.3) and therefore the prices will be higher, which together will harm consumers. However, the number of unregulated firms and the substitutability degree will have opposite effects on the consumer surplus. As the number of unregulated firms increases, the consumer surplus differences will decrease, while such differences will increase with the substitutability degree. Once again both results can be explained by the differences on the aggregate quantity produced. When the number of unregulated firms increases, there is a convergence between the levels of quantities produced and increases in the substitutability degree will lead to greater differences between the collusion-allowing and the collusion-proof outcomes.Hence, a new and important conclusion rises from this analysis: social welfare differences will depend heavily on the relative weight that the regulator allocates to Consumer Surplus on the social welfare function, also pointed out by Biglaiser and Ma [12].


## 6. Conclusion

The main purpose of this paper was to add a new dimension to the model put forth by Aubert and Pouyet [13], by introducing an exogenous number of unregulated firms in the market. This extra dimension was also able to produce a powerful argument against collusion even in situation when apparently it would be Welfare improving to allow it.Aubert and Pouyet [13] proposed a model of incomplete regulation in a duopoly with asymmetric information regarding the costs of the regulated firm, in a context of product differentiation. The main conclusion of that paper was the failure of the collusion-proofness principle, which stated that higher social welfare would be attained if the regulator designed a contract with the regulated firm that prevented any form of collusion between the two of them. This conclusion was also supported by Tirole [16], Itoh [9], Lawarrëe and Shin [10], and Shin [11].The model presented in this paper represents an extension, where the regulated firm faces competition fromn unregulated firms and analyses the robustness of the conclusions drawn by Aubert and Pouyet [13]. Although this new dimension leads to a much more mathematically demanding model, it has reinforced the failure of the collusion-proofness principle. As the number of unregulated firms increases, the Welfare difference between the collusion-allowing and the collusion-proof equilibriums is always positive and increasing. Such result may be justified by the decreasing weight of the regulated firm in the aggregated market and on the social welfare function and by the ability that the unregulated firms have to extract truthful information regarding the costs of the regulated firm. This result still holds when we consider the Welfare difference for different levels of the substitutability degree. Nevertheless, by analysing the Consumer Surplus differences, we were able to find the most powerful argument for the regulator to still not allow collusion. The results presented earlier depend heavily on the relative weight allocated to the Consumer Surplus in the social welfare function. This new fact brings a new perspective on the results reached by Aubert and Pouyet [13] and on the literature that stated the advantages of allowing collusion.However there is still some room for further research, by adding more dimensions to the present model. First and foremost, we could start by analysing the role of the weight of the Consumer Surplus in the social welfare function in the simpler framework of only two firms of Aubert and Pouyet [13]. Secondly, we can also assess what would happen if we had again only two firms (one regulated and the other unregulated) and both of them had marginal costs which were private information. Another way of improving the model is to introduce endogeneity in the number of unregulated firms operating in the market, by introducing entry and exit dynamics. Additionally, we could also introduce the possibility of collusion between the regulated firm and only some of the unregulated firms in the market.



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*Source: 818716-2012-02-09.xml*