Emilie Dargaud
email: dargaud@gate.cnrs.fr.

Armel Jacques
email: armel.jacques@univ-reunion.fr.

Nicolas Fleury

Fabrice Gilles

José De Sousa

Julien Albertini

Etienne Lehmann

Laurent Simula

Alain Trannoy

Claudio Lucifora

Simone Moriconi

Bruno Van

Der Linden

Mathieu Bunel

Emilia Ene Jones

Yannick L'horty

Pascale Petit

Joseph Lanfranchi

Mathieu Narcy

Intermittent collusive agreements: antitrust policy and business cycles

Keywords: Collusion, antitrust policy, business cycles. JEL: K42, L22, L41

In this article we study collusive strategies when …rms face random demand ‡uctuations. This work analyzes the optimal level of …nes in times of crisis. We assume that detection probability may or may not depend on collusive price. When cartel is detectable irrespective of price they are less stable in booms than in recessions and prices can be counter-cyclical. This requires to set the highest achievable …nes. By contrast if detection probability depends on collusive prices intermittent collusive strategy can be implemented in which …rms collude on periods for which it is the most pro…table strategy. Firms can either charge procyclical or countercyclical collusive prices. Optimal level of …nes depends on demand state and it can be discounted if demand is low to support marginal deterrence e¤ect or during booms to decrease the harmfulness of anticompetitive conduct.

Introduction

This article discusses the relevance of state-dependent antitrust …nes. Facing ‡uctuating demand, cartel members can set the monopoly price for each state of demand or may decide to interrupt collusion in one of the demand states to avoid detection in that state since expected cartel detection and dissolution risk can dominate collusive payo¤s. Collusive behaviors rely on state-dependent …nes. Antitrust authority may …nd it preferable to adjust antitrust …nes to business cycles in order to modify the …rms collusive strategy. In such cases cartels may choose to hibernate in some periods leading to increased consumer surplus during those periods or, on the contrary, they can collude whatever the state of demand leading to increased detection probability. The implementation of state-dependent …nes may in ‡uence cartel members choice between intermittent collusive strategies (the cartel loses pro…t in one state but increases its survival rate) and constant collusive strategy with a shorter duration.

Our model is based on the approach adopted by [START_REF] Rotemberg | A supergame-theoretic model of price wars during booms[END_REF] [RS henceforth] in which we include an antitrust authority. We explore strategies implemented by collusive …rms when they face random demand ‡uctuations. Then we discuss the optimal level of cartel …nes in line with business cycle.

Our paper encompasses two strands of literature. The …rst is the literature tying collusion sustainability and random ‡uctuating demand. The second is the analysis on the adaptation of competition policy enforcement to cyclical changes of economy.

A recurring issue in industrial organization deals with sustainability of collusive agreements under business cycles. Some studies point out that recessions result in weakening of collusive sustainability whereas others underline the opposite result putting forward the argument that incentives to deviate from cartel are stronger during booms. We review major contributions on this topic in the next section. Our article contributes to this literature by including an active antitrust authority in RS. Antitrust authority is intended to …ght collusion and once detected, cartels are …ned and dissolved forever. Two di¤erent scenarii of cartel enforcement are studied. Under the …rst assumption (A1) implementation of cartel requires governance structure leading to hard evidence which can be detected regardless of price. Detection probability remains constant even if the cartel price equals marginal cost. Under this assumption primarily used to provide a benchmark, results are consistent with RS: collusive agreements are less likely to occur during booms. Under the more realistic assumption A2, detection probability is price dependent. 1 For the sake of simplicity we assume that detection probability is zero when …rms set price equal to marginal cost and is increased to for higher price. For some parameters values …rms can choose to implement collusion only during booms or recessions. Since they play an in…nitely game cartel detection is inescapable. Firms face a tradeo¤ between sustaining collusion regardless of demand to bene…t right away from higher price or implementing intermittent collusive strategy (…rms set collusive price only when it is the most pro…table strategy) leading higher but later expected pro…t. Consequently …rms may choose to set higher price only during booms and charge the marginal cost if demand is low, these results di¤er from RS. Firms may also choose the opposite strategy if antitrust …nes depend on business cycles and are discounted during low demand period. Finally if expected collusive pro…t is similar for both demand states …rms may choose to collude until cartel is detected and dissolved. Collusive …rms'strategies and price ‡uctuation according to business cycles rely on cartel enforcement including the imposition of …nes. Our paper contributes to the body of literature in introducing a new collusive strategy where …rms implement intermittent cartel. Some previous studies allow …rms to decrease the degree of collusion but without reverting to competitive equilibrium. This work contributes also to the determination of the optimal level of …nes during economic or …nancial crisis. Maintaining or releasing competition rules is a constant issue during each economic crisis. Arguments in favor of releasing competition rules are two-fold: (1) larger …nes in times of crisis may induce increased bankruptcy and (2) collusive behavior allows a faster and more e¢ cient reduction of over-capacities and provides industry to adapt more quickly to a new economic environment. Some speci…c clauses exist in the European Union such as a reduced antitrust …nes if companies can be made bankrupt because of punishment. 2

The role of competition policy is to promote competition not eliminate …rms from the market. European Union guidelines provide potential cooperation among competitors during crisis. [START_REF] Barjot | Why a special edition on cartels?[END_REF] and [START_REF] Barjot | Cartels et cartellisation : Des instruments contre les crises ?[END_REF] give as an example the capacity reduction agreement on arti…cial …bres during the 1980s, providing an answer to the structural crisis facing the European chemical industry, without using government funds. They point out that before 1945 cartels were legal in Europe and were considered to be in societies' interest (economics shock absorption, survival of businesses and jobs,...). The US antitrust authorities take a similar approach during some major crisis. As a consequence of the …nancial crisis of 1929 the National Industrial Recovery Administration 3 set up in 1933 by the US antitrust authorities putted as an aside the antitrust legislation. September 11 attacks committed in 2001 caused a serious decline in airline demand particularly in the United States. To respond to this drop the U.S. Department of Transportation allowed 4 an agreement between two Hawaiian airlines which received an antitrust immunity allowing them to coordinate air travel capacity and prices. Prices rose during the period of coordination and they still remained high after immunity expired [START_REF] Blair R | Collusive duopoly: The economic e¤ects of the Aloha and Hawaiian airlines'agreement to reduce capacity[END_REF][START_REF] Kamita | Analyzing the e¤ects of temporary antitrust immunity: The Aloha-Hawaiian immunity agreement[END_REF]. The temptation of releasing competition rules reoccurs regularly during crisis periods. 5 We put forward a third argument (quite di¤erent from the two previous reasons) to reduce …ne during recessions, which relies on marginal deterrence. Antitrust authority may induce the cartel to sustain collusive agreements even during recession since the consumer loss is lower than during high period demand and this increases the detection probability and cartel dissolution.

To the best of our knowledge [START_REF] Fabra | Antitrust …nes in times of crisis[END_REF] is the main paper which theoretically analyzes optimal lower …nes during recessions. They consider an incumbent …rm which, facing a potential entrant, can take a conduct violating antitrust laws. Antitrust authority can launch investigations and impose corporate and managerial …nes. They show that both …nes must be applied: a reduced corporate …ne to decrease 2 Some illustrative examples are provided in [START_REF] Fabra | Antitrust …nes in times of crisis[END_REF]. 3 The National Industrial Recovery Act was promoted to help the nation recover from the Great Depression and to abolish "unfair competition" on the basis of voluntary agreements established by …rms which set, inter alia, minimal wages. This law was declared unconstitutional in 1935. However later legislation incorporated many elements of the previous law. [START_REF] Cole | New deal policies and the persistence of the great depression: a general equilibrium analysis[END_REF], in a general equilibrium analysis, present data that show very little antitrust enforcement by the DoJ after 1935 and …nd that this more lenient cartelization policy is a key factor behind the weak recovery. 4 Against the consultative advice undertaken by the Department of Justice 5 However this toleration of crisis cartels is not systematically in place. [START_REF] Ghosal | The cyclical behavior of the Department of Justice's antitrust enforcement activity[END_REF] show that antitrust case activity of the US Department of Justice is countercyclical. the risk of social cost of bankruptcy and a managerial …ne to revert to the initial deterrence level. They argue that lower …nes during recessions are sometimes optimal. First it is harder to achieve deterrence in booms (anticompetitive practice is more bene…t) then higher …ne must be applied. Second the risk of bankruptcy facing the …rm if the …ne is high is lower during booms than recessions, then optimal …nes can be lower during recessions. Finally they show that varying policy according to the demand state can be welfare-maximizing. In contrast to this paper we consider a dynamic model. Both …nes act on collusive sustainability constraints: …ne imposed during recessions may in ‡uence collusive strategy in booms (and vice versa). Another relevant di¤erence relates to the marginal deterrence e¤ect which is not introduced in their model since only one anticompetitive strategy is achievable. Unlike their paper, …rms can implement di¤erent collusive strategies and besides the traditional deterrence e¤ect of …nes they can also in ‡uence collusive strategy and related social welfare. This marginal deterrence e¤ect produces the most interesting results under the assumption A2. Only one collusive strategy is implemented under the …rst assumption (A1) leading to set the highest achievable …nes. Under assumption A2 …nes produce a deterrent e¤ect on cartel formation (this induces highest level of punishment) and a marginal deterrent e¤ect relying on di¤erent anticompetitive e¤ects. This latter point may lead authority to vary …nes according to demand state. Contrary to earlier …ndings on traditional marginal deterrence6 the seriousness of infringements depends on parameters values then, relative to a constant collusive strategy, intermittent cartel can lead to higher or lower expected social welfare. Setting maximum …ne in one state and zero …ne in the other can be optimal to strengthen the marginal deterrence e¤ect but this is not in line with the traditional deterrence e¤ect on cartel formation.

Then antitrust authority must balance these di¤erent objectives (deter the cartel formation and decrease the harmfulness of anticompetitive conduct) and imposing one maximum …ne and an other lower …ne can be optimal. In our model, reducing …nes if demand is low can be optimal but authority may instead discount …nes during booms to induce the implementation of a less harmful intermittent collusive strategy. This article is organized as follows. In the next section we underline the related literature. Then we describe our model (section 3). Section 4 reports the results obtained under assumption A1. They serve as a benchmark to present in section 5 the results obtained with the alternative hypothesis. The conclusion follows in section 6.

Related literature

This paper contributes to two strands of literature. The …rst is the literature tying collusion sustainability and random ‡uctuating demand. The second analyzes the e¤ects of …nes on collusive strategies.

Collusion sustainability and business cycles

The relationship between collusion and business cycles is not a new issue. [START_REF] Kleinwächter | Die Kartelle: ein Beitrag zur Frage der Organisation der Volkswirthschaft[END_REF], the founder of cartel theory, describes cartels as children of distress ("Kinder der Not") since they might arise as the response to economic downturn in order to o¤set the drop in the price. Afterwards the literature on the relationship between cartel sustainability and business cycle is quite extensive.

In contrast to the argument that cartels arise during economic downturn, [START_REF] Green | Noncooperative collusion under imperfect price information[END_REF] suggest that low demand periods may trigger transitory price wars. Because of the inability of the …rms to determine if low demand is induced either by economic situation or by a deviation of the collusive path, period of weak demand requires punishment to dissuade the cheaters. 7Our article is in line with the works initiated by [START_REF] Rotemberg | A supergame-theoretic model of price wars during booms[END_REF] [RS]. They argue that cartels are likely to behave more competitively in periods of high demand since deviating from the collusive agreement results in maximum pro…t. This result is partly due to the fact that demands over various periods are uncorrelated, then future expected pro…t (whether …rms keep on collusion or deviate) is independent of current demand and cartel deviation is more attractive if such demand is high.

Several studies check the robustness of the results obtained by RS. [START_REF] Kandori | Correlated demand shocks and price wars during booms[END_REF] includes demand correlation over time and demonstrates that if the discount factor tends to unity or is close to n 1 n (n is the number of …rms) in which case the no deviation constraint is binding, prices disclose the same countercyclical movement as in RS. [START_REF] Haltiwanger | The impact of cyclical demand movements on collusive behavior[END_REF] allow for cyclical demand instead of random demand. They assume that movements in market demand are fully anticipated but each cycle is composed of …nite number of periods. In their model …rms can more easily collude during falling demand which does not necessarily contradict results of RS: the gain from deviation is highest when demand is strongest and begins to falls when demand is decreased. [START_REF] Bagwell | Collusion over the business cycle[END_REF] assume that the level of market demand swaps between slow-and fast-growth states, and the transition between states is determined by a Markov process. They show that if demand growth rates are strongly [weakly] correlated over time then collusion is easier to sustain if demand growth rate is strong [low]. [START_REF] Bernhardt | Collusion under risk aversion and …xed costs[END_REF] analyze collusion under demand uncertainty by risk-averse cartel and overturn the RS results when cartel members are risk adverse and face positive …xed operating costs.

Risk aversion implies that marginal gain from cheating on the cartel declines as demand is increasing. Fixed operating cost strengthens this e¤ect and allows to provide contradictory results of RS for some parameters values.

Staiger and Wolak (1992), [START_REF] Fabra | Collusion with capacity constraints over the business cycle[END_REF], [START_REF] Knittel | Tacit collusion in the presence of cyclical demand and endogenous capacity levels[END_REF] and Paha (2017) explore collusive pricing with capacity constrained …rms in the presence of demand uncertainty. Capacity constraint may constrain deviation opportunities of collusive …rms but also potential retaliation against deviating …rm.

The link between collusive opportunities and demand state depends on either …rm's capacities (if they are exogenous) or their expenditures (if they are endogenous). [START_REF] Montero | Output-expanding collusion in the presence of a competitive fringe[END_REF] analyze collusive agreement under demand uncertainty but the model di¤ers from RS in several ways. In their model collusive agreement is sustained by large …rms in the presence of a competitive fringe. Then cartels may lead to two opposite e¤ects. First members of the cartel may reduce their production so as to increase equilibrium price. Second cooperation among members may lead to outputexpanding strategies with the purpose of decreasing the production of the competitive fringe. The dominant e¤ect depends on the demand state.

These previous studies assume that the number of …rms is …xed, instead of [START_REF] Eswaran | Cartel unity over the business cycle[END_REF] and [START_REF] Bagliano | Liquidation risks in the Rotemberg-Saloner implicit collusion model[END_REF] who explicitly include in the context of RS the possibility of bankruptcy in recessions.

Then cartels may be more di¢ cult to sustain during recessions since the risk of bankruptcy induces …rms to place a greater weight on immediate pro…ts. To alleviate this problem, [START_REF] Eswaran | Cartel unity over the business cycle[END_REF] allows for statedependent market sharing strategies.8 [START_REF] Vasconcelos | Sustaining collusion in growing markets[END_REF] points out that market growth may trigger future entry.9 Potential entrant may alter sustainability of collusive agreement. Entry takes place earlier in growing market as the entrant pro…t is higher if collusive agreement is already sustained than if it is dissolved in response to deviation. Then deviation not only provides higher current pro…t, but also delays potential entry. Subsequently collusion may be more di¢ cult to sustain in growing markets when the number of …rms is endogenous.

Our article contributes to this literature by including an active antitrust authority. We show that, subject to certain conditions, …rms can implement intermittent collusive strategy: cartels can be bracketed for some periods (often, but not exclusively, when demand is low). This type of intermittent collusive strategy has never been studied before (in previous mentioned papers degree of collusion can be a¤ected but …rms do not switch to competitive equilibrium).

Marginal deterrence of collusion

Articles dealing with the …ght against cartels mainly focus on cartel deterrence. Few studies analyze marginal deterrence e¤ect: even if collusive agreements are not fully deterred the gravity of o¤ense can be reduced with appropriate competition policy and antitrust …nes. 10 Deterring the formation of collusive agreements requires the maximal level of …nes. But if perfect deterrence is not achievable authorities may set level of …nes depending on prices in order to decrease them and thus damages su¤ered by consumers. This issue is studied by [START_REF] Souam | Optimal antitrust policy under di¤erent regimes of …nes[END_REF] and [START_REF] Pénard | Collusion et politique de la concurrence en information asymétrique[END_REF] in static model (disregarding the no deviation constraint) and Houba, Motchenkova andWen (2010, 2018) in a dynamic setting.

There is an extensive literature on monetary penalty regimes which concentrates on welfare properties of such regimes 11 and some of these contributions show that revenue-based …nes can induce cartel to set a price above the monopoly price. 12 This anticompetitive e¤ect can be o¤set if method of computations of …nes is revised (Katsoulacos, Motchenkova and Ulph, 2020a) or if civil damages are included (Katsoulacos, Motchenkova and Ulph, 2020b). 13 [START_REF] Harrington | Post-cartel pricing during litigation[END_REF] indicates that methods for computing antitrust damages and especially for estimating the standard non-collusive price can in ‡uence post-cartel prices. [START_REF] Houba | The e¤ects of leniency on cartel pricing[END_REF] analyze how leniency programs a¤ect cartel pricing and Emons (2020) studies the impact of such programs on the degree of collusion. 14 Our contribution is quite di¤erent from the previous papers which mainly focus on cartel price when deterrence e¤ect is not prevailing. Since we assume that …ne does not depend on collusive price we disregard this feature. However we consider that …nes may depend on business cycles and then marginal deterrence e¤ect does not deal directly with cartel price (even if it is indirectly impacted) but with the number of demand states in which cartels operate. Another feature of our article is that the ranking of the impact of collusive strategies on welfare depends on parameters values unlike the previous papers for which lower collusive price has a positive e¤ect on welfare.

Our article features some similar points to those raised by [START_REF] Reuter | Endogenous cartel organization and antitrust …ne discrimination[END_REF]. He extends [START_REF] Green | Noncooperative collusion under imperfect price information[END_REF] model by involving the possibility of a trade association in collusion. This allows …rms to receive a signal about the aggregate demand state and this changes the length of price war. However cartel detection is increased then expected duration of cartel is reduced. As in our article several collusive strategies are achievable and expected …nes set by antitrust authority (…ne can be discounted or not for cartels which operate without trade association) impact the choice of the strategy adopted. Antitrust authorities face a trade-o¤ between inducing collusive …rms to avoid trade association in order to increase frequency of price wars and to decrease expected collusive prices or inducing …rms to fall back on trade association to boost detection probability and decrease the expected length of collusive agreement. Although our article similarly highlights a trade-o¤ for antitrust authority our model deviates however in a number of points from Reuter (2017). We assume perfect information, we do not include trade association and, more importantly, we focus on …nes based on business cycles.

The Model

We consider an n-…rm Bertrand competition setting over an in…nite time horizon. Firms produce homogeneous good with the same marginal cost c. At each period, …rms can decide to reach a collusive agreement.

Collusive outcomes are modelled on a grim trigger basis: as soon a …rm deviates from the agreement, all …rms will play competitively forever. 15 1 2 Bageri, [START_REF] Bageri | The distortive e¤ects of antitrust …nes based on revenue[END_REF], [START_REF] Katsoulacos | Penalizing cartels: the case for basing penalties on price overcharge[END_REF]. 1 3 [START_REF] Katsoulacos | Penalizing cartels -a spectrum of regimes[END_REF] provide an overview of the impact of di¤erent computations of …nes on collusive prices.

1 4 Multimarket colluding …rms pick the breadth of collusion and then the fraction of a continuum of markets on which they collude. Higher breadth leads to higher detection probability.

1 5 If we consider Nash equilibria of static game, at least two …rms set price equal to marginal cost and others can choose higher prices. In order to avoid that authority could think that those higher prices re ‡ect collusive strategies, we assume that without Fluctuating demand is characterized by the following function: Q (p; ) where p is the industry price and is a discrete random variable with the set of values: f l ; h g. Low [high] shock l [ h ] occurs with probability [1 ]. As in RS we assume that shocks are independently and identically distributed. The following assumptions apply: 8p: Q (p; h ) Q (p; l ) ; @Q @p (:) 0 and @ 2 Q @p 2 (p; ) p + 2 @Q @p (p; ) < 0. Thence monopoly price and pro…t (p m ( ) and m ( )) are well-de…ned. We keep demand function in the most general form throughout this article but we use sometimes a speci…c form: Q (p; ) = (A p) in order to illustrate the results.

We consider an antitrust authority intended to detect and …ght collusion. Each period, an active cartel is detected with probability which is not price dependent if …rms set higher price than static equilibrium price 16 (equal to marginal cost). Antitrust authority can launch investigations during deviation period if price exceeds marginal cost. 17 When colluding …rms set price equal to marginal cost, two cases are analyzed: Assumption A1: detection probability remains . This assumption is based on the following rationale.

Implementation of collusive agreements requires governance structure in which …rms delegates meet at the beginning of each period in order to …x the price according to . This generates hard evidence which can be found by the authority whatever the price is.

Assumption A2: detection probability is zero. We could assume that antitrust authority launches investigation only if consumers complain and this is not the case if price is competitive and thus when collusive agreement is bracketed. We favor the latter assumption which seems more realistic, assumption A1 is primarily used to provide a benchmark.

If a cartel is detected, authority launches investigation which leads to successful prosecution that results in a …ne f ( ) 2 [0; F ( )]. Prior to starting the game, antitrust authority discloses the amount of …nes incurred.

Fine can be state-contingent (f l f ( l ) may be di¤erent from f h f ( h )) but never exceeds the statutory maximum F ( ). Cartels are dissolved once they have been convicted and …rms can not collude again. Firms, consumers and antitrust authority face the same discount factor .

Denote the instantaneous pro…t and consumer surplus by and cs and the ‡ow of discounted pro…ts and consumer surplus by and CS. Superscripts m and nc designate respectively the monopoly and non cooperative situation.

Constant detection probability

Collusive outcomes are modelled on a grim trigger basis [START_REF] Friedman | A non-cooperative equilibrium for supergames[END_REF]): as soon a …rm deviates from agreement or if cartel is condemned, all other …rms will play competitively forever and obtain nc = 0.

collusion, all …rms play Bertrand Nash equilibrium and choose p = c. 1 6 Firm setting higher price than marginal cost faces the risk of being …ned even if its demand is 0. We do not consider the possibility that some …rms, alternatively, do not produce during given time to remove the potential …ne.

1 7 As in many other papers [START_REF] Spagnolo | Divide et Impera, Optimal Deterrence Mechanisms Against Cartels and Organized Crime[END_REF], [START_REF] Buccirossi | Optimal …nes in the era of whistle blowers -Should price …xers still go to prison?[END_REF], Chen and Rey (2013), [START_REF] Jensen | Fine schedule with heterogeneous cartels: are the wrong cartels deterred?[END_REF], Dargaud andJacques (2015, 2020)) we assume that the cartel can still be detected in a period of defection.

Denote the global cartel pro…t by ( ). Deviating …rm i obtains: d i ( ) = ( ) f ( ) since it captures global demand by setting p i = p ". Discounted deviation pro…t is the same: 18 d ( ) = ( ) f ( ).

Firms strategy on collusive path may depend on the value of . We …rst assume that is su¢ ciently high that collusion at the monopoly price is sustainable whatever the state of demand. Then we consider reduced values of for which collusive agreements could be sustainable at lower price.

High discount factor

According to assumption A1, and f ( ) are not price dependent then …rms choose the monopoly price p = p m if such price is sustainable. We de…ne the constant strategy of setting monopoly price regardless of demand as the "C strategy".

Associated discounted pro…ts are:

C = m ( l ) n f l + (1 ) m ( h ) n f h + (1 ) C , C = m ( l ) n f l + (1 ) m ( h ) n f h 1 (1 )
We have to determine C such as …rms behave like a monopoly for each state of demand.

C = max C l ; C h
where C l and C h are computed according to low and high demand respectively.

Low demand: Collusion on the monopoly price is sustainable i¤:

m ( l ) n f l + (1 ) C m ( l ) f l , C l (n 1) m ( l ) (1 ) f( + n 1) m ( l ) n f l + (1 ) [ m ( h ) n f h ]g
High demand: The collusion sustainability condition reads as:

m ( h ) n f h + (1 ) C m ( h ) f h , C h (n 1) m ( h ) (1 ) f[ ( m ( l ) n f l ) (1 ) n f h ] + (n ) m ( h )g
Increasing f l and/or f h makes collusion more di¢ cult to sustain regardless of demand state.

The following proposition establishes a comparison of collusion sustainability relative to business cycles.

Proposition 1 Under assumption A1, if collusion at the monopoly price is sustainable then collusive agreements are more di¢ cult to sustain in booms than in recessions.

Proof.

C h C l , (1 ) m ( h ) n f h + m ( l ) n f l 0
This condition is always checked if C > 0. Otherwise collusion is not sustainable and discount factors comparison is irrelevant.

However

m ( l ) n f l and m ( h ) n
f h can be negative. Firms cannot delete potential …nes by temporarily implementing competitive behavior, then expected …nes f l and f h seem …xed costs borne as long as cartel is active.

This result is consistent with RS. We assume that …rm can be …ned during deviation period then expected …ne is the same under collusion and deviation. Therefore if …rms rely on current pro…t to decide if they stick to collusive price they make the same choice as in RS's model without …ne. 19

Middle discount factor

If C l < < C h monopoly price is not sustainable if demand is high (and thus under the C strategy) but lower price can be sustainable (higher than the marginal cost). 20 If demand is low collusive price and individual pro…t are: p ( l ) = p m ( l ) and

m ( l ) n .
If demand is high p and stand for collusive price and global pro…t, they are computed maximising discounted collusive pro…t subject to the no-deviation constraint.

Expected discounted pro…t is equal to:

C = m ( l ) n f l + (1 ) n f h 1 (1 )
C is increasing in , then …rms choose the maximum sustainable price. The collusion sustainability condition reads as:

n f h + (1 ) C f h
The constraint is binding then:

= n (1 ) h m ( l ) n f l (1 ) f h i n 1 (1 ) (n )
is decreasing in f l and f h , so is collusive price p .

Then we obtain the following proposition which discloses that collusive price implemented during high demand period is a¤ected by amounts of …nes f h and f l .

Proposition 2 During booms, collusive price p is a decreasing function of the …nes f h and f l .

Even if f h and f l are too weak to deter collusive agreements, they can cushion their e¤ects by decreasing price during booms.

Expected discounted pro…t is:

C = [n 1 (1 ) (n 2 + 1)] m ( l ) [1 (1 )] (n 1) n [ f l + (1 ) f h ] [1 (1 )] [n 1 (1 ) (n )]

Level of …nes

On the basis of assumption A1, increasing f l and/or f h makes collusion more di¢ cult to sustain. Moreover p is decreasing in f l and f h . The next proposition establishes optimal level of …nes.

Proposition 3 Under assumption A1, optimal policy is to set the highest achievable …nes: f h = F h and

f l = F l .
As in many studies dealing with economic analysis of crime, optimal policy requires the maximum level of …ne. If maximum level of …nes does not depend on business cycle (F h = F l ), nor are …nes imposed on …rms.

Price-dependent detection probability

According to assumption A2, cartel can be detected and condemned only if price exceeds marginal cost. In this case and F are not price dependent then colluding …rms choose monopoly price. For periods in which price equals marginal cost …rms can not be convicted. We start with characterization of collusive agreements according to the level of …nes. Then we rank collusive strategies in accordance with consumers surplus and global welfare. Finally we show that setting only one maximum …ne can be optimal. These results are discussed with a general demand function and then illustrated with a speci…c one.

Collusive strategies

As in the previous section we …rst assume that is su¢ ciently high that cartel can choose the monopoly price.

Then we consider lower values of . Firms choose between three di¤erent collusive strategies: monopoly price can be set regardless of demand (C strategy), only if demand is high (H strategy) or low (L strategy).

Constant strategy

C is the same as in the previous section but expected …nes f l and f h are now quasi-…xed costs that must be paid only if …rms decide to set higher price than marginal cost. Firms prefer to switch to the H or L strategies if:

m ( l ) n f l < 0 or m ( h ) n f h < 0.
In the opposite case, the C strategy is similar to the one previously described.

Intermittent strategies

When collusion is intermittent …rms playing the H [L] strategy set monopoly price only if demand is high

[low], otherwise they charge marginal cost.

Discounted pro…ts are:

H = H + (1 ) h m ( h ) n f h + (1 ) H i , H = 1 1 +(1 ) h m ( h ) n f h i L = (1 ) L + h m ( l ) n f l + (1 ) L i , L = 1 + h m ( l ) n f l i
Firms playing the H [L] strategy can have incentives to deviate from the cartel if demand is high [low] but not otherwise, collusion sustainability conditions are: 1)n ( 1)] m ( l ) n(1 ) f l Threshold discount factor for each intermittent strategy increases in level of its …ne.

m ( h ) n f h + (1 ) H m ( h ) f h , H (n 1) m ( h ) [n(1 + ) ] m ( h ) n(1 )(1 ) f h m ( l ) n f l + (1 ) L m ( l ) f l , L (n 1) m ( l ) [(
For lower discount factor no price higher than the marginal cost is sustainable for each strategy (proof in Appendix 7.1). We reach the same result as in Bertrand competition with homogeneous product and constant demand function. If the monopoly price is not sustainable then nor is price higher than marginal cost.

Strategy choice

We derive equilibrium collusive choices, considering …xed level of …nes. We assume that …rms choose their collusive strategy before knowing the …rst value of .

If is su¢ ciently high then relevant comparisons are:

H C , (1 ) 1
+ ( 1)

m ( h ) n f h m ( l ) n f l C L , 1 + m ( h ) n f h m ( l ) n f l H L , 1 1 + 1 + (1 ) m ( h ) n f h m ( l ) n f l
The following conditions Proof. First observe that:

(1

) 1 + (1 ) < 1 < 1 +
Then, H C ) C L and …rms choose the H strategy.

Second the following inequality applies:

1 1 + 1 + (1 ) 1 + , 0 1
Then L > C ) L > H and …rms choose the L strategy.

Firms choose the H strategy if collusive pro…ts are signi…cantly higher when demand is high. This occurs when m ( h ) is well above m ( l ), f h is not too large compared to f l or if is not too high. Although collusion may be sustainable if demand is low …rms do not implement it and stay undetectable. In this case they can not be …ned and collusive agreement is not dissolved. In terms of expectation, …rms collude during 1= periods before cartel is detected and dissolved and they choose to collude only if demand is high. We get a di¤erent result to RS but we do not consider the same range of values for .

Corollary 5 When …rms choose the H strategy, cartel leads to an increase in price more important when demand is high.

This result can be reversed if f h is hugely higher than f l 23 in which case …rms favor collusion if demand is low and otherwise break o¤ the cartel (L strategy). Then we reach counter-cyclical price as in RS.

2 2 Proof in Appendix 7.2. 2 3 If f h = f l then the following inequality applies:

C L since 1 + > 1.
In such case the L strategy is not an equilibrium one.

In other cases, …rms choose the C strategy: collusion is sustainable regardless of demand but is more di¢ cult to sustain during booms.

In section 5.5 we illustrate these results with the following demand function: Q (p; ) = (A p). Before that, we rank collusive strategies according to consumers surplus and global welfare.

Consumer surplus and social welfare

Ranking of consumer surplus and social welfare is not obvious. Price e¤ect as well as length of cartel play an important role and impact on strategies di¤erently (for instance, if demand is low, …rms compete if they play the H strategy then consumer surplus should be higher compared to the C strategy but, overall, cartel is not the same length).

We still focus on su¢ ciently high values of to ensure that all collusive strategies can be applied.

Expected consumer surplus following cartel dissolution is: 24

CS = 1 X t=0 t [ cs nc ( l ) + (1 ) cs nc ( h )] = 1 1 [ cs nc ( l ) + (1 ) cs nc ( h )]
We then deduce expected consumer surplus relative to the various strategies: 1)cs m ( h )+( 1) CS 1 +( 1)

CS i = 8 > < > : cs m ( l )+(1 )cs m ( h )+ CS 1 (1 ) for i = C strategy cs nc ( l )+(
for i = H strategy cs m ( l )+(1 )cs nc ( h )+ CS 1 + for i = L strategy Proof. See Appendix 7.3
Social welfare is composed of consumers'surplus, producers'surplus and antitrust …nes (the …ne transfer is welfare neutral). We accordingly have: 1)

W i = 8 > < > : [cs m ( l )+ m ( l )]+(1 )[cs m ( h )+ m ( h )]+ CS 1 (1 ) for i = C strategy cs nc ( l )+(1 )[cs m ( h )+ m ( h )]+(1 ) CS 1 +(
for i = H strategy [cs m ( l )+ m ( l )]+(1 )cs nc ( h )+ CS 1 + for i = L strategy
Solving comparison of consumer surplus yields the following inequalities, where cs( i ) = cs nc ( i ) cs m ( i ):

CS H CS C , cs( l ) (1 ) 1 + (1 ) cs( h ) CS C CS L , cs( l ) 1 + cs( h ) CS H CS L , cs( l ) 1 1 + 1 + (1 ) cs( h )
We then deduce the following comparisons for welfare:

W H W C , cs( l ) m ( l ) (1 ) 1 + (1 ) ( cs( h ) m ( h )) W C W L , cs( l ) m ( l ) 1 + ( cs( h ) m ( h )) W H W L , cs( l ) m ( l ) 1 1 + 1 + (1 ) ( cs( h ) m ( h ))
The …rst point emerging from these computations is that ranking of surplus depends on parameters values. The H strategy seems to be less harmful than the C strategy since it is limited to only one state of demand. However, the H strategy is expected to be detected latter. Then this collusive strategy does not impact consumers when demand is low but consumers are expected to su¤er from collusion for a longer time when demand is high. The most damaging collusive strategy for consumers and global welfare depends on parameters values.

The second point is that these comparisons are di¤erent from borderlines de…ning …rms' choice. Equilibrium …rms'strategies diverge from socially optimal choices, which is to be expected since …rms can make collusive agreements. The choice of equilibrium collusive strategies depends on expected …nes, this is not the case for consumer surplus and social welfare. Then antitrust …nes can be used to impact the choice of the collusive strategy adopted and to reduce gravity of o¤ense. We deal with these points in the subsection 5.6.

Illustration with a speci…c demand function

In order to graphically illustrate the previous results and achieve further outcomes we consider the following speci…c demand function: Q (p; ) = (A p). We derive price, quantity and pro…ts compatible with monopoly equilibrium:

p m = A + c 2 , Q m ( ) = A c 2 and m ( ) = (A c) 2 4
Monopoly price does not depend on according to the speci…ed demand function.

Expressions of consumer surplus are:

cs nc ( ) = 1 2 (A c) 2 and cs m ( ) = 8 (A c) 2 then: cs m ( ) cs nc ( ) = 3 8 (A c)
Comparisons to collusive pro…ts lead to following results:

H C , h I 1 + (1 ) (1 ) l 4n f l (A c) 2 ! + 4n f h (A c) 2 C L , h II 1 + " l 4n f l (A c) 2 # + 4n f h (A c) 2 H L , h III 1 1 + (1 ) 1 + " l 4n f l (A c) 2 # + 4n f h (A c) 2
In the ( l ; h ) plane, each borderline de…ned by I , II and III is linear function and moves upward

[downward] by parallel translation with increased value of

f h [f l ].
Comparisons of consumer surplus and social welfare are given by:

CS H CS C , W H W C , h IV 1 + (1 ) (1 ) l CS C CS L , W C W L , h V 1 + l CS H CS L , W H W L , h V I 1 + (1 ) 1 + 1 l
The next graphical representations (…gures 1 to 4) illustrate equilibrium strategies and consumer surplus ranking for di¤erent parameters values. Overall we identify six speci…c zones (each …gure does not necessarily contains all areas):

Zone Equilibrium strategy Consumer and social surplus ranking

1 H L C H 2 H L H C 3 C L C H 4 C L H C 5 L Irrelevant 6 C* Irrelevant Since 1 + < 1 then W L > W C . Moreover W H < W L if < 1=2.
One of the main issues of this article is whether f l < f h must be implemented by antitrust authority. We start look at f l = f h and then relax this.

If

f l = f h then C L . Graphical representation in ( l ; h ) plane is straightforward. Firms choose
the H strategy is h is high and C strategy if h and l have similar values. H only depends on h , then the threshold is a minimal value H h de…ning an horizontal straight line. 26 In the same way C h and C l are horizontal straight line corresponding to the minimal values C h and C l . 27 L is a vertical straight line corresponding to the minimal value L l . 28 If f h = f l this latter threshold is irrelevant since …rms never choose the L strategy.

2 6 The expression of H h is:

(1 )(1 ) [n(1 + ) ] n+1 4n f h (A c) 2 . 2 7 With: C h = (1 ) n 1 (1 )(n ) h l 4n f l (A c) 2 (1 ) 4n f h (A c) 2 i ; C l = [n 1 (1 )( +n 1)] l + (1 ) 4n (A c) 2 [ f l +(1 )f h ] (1 ) (1 )
.

2 8 The expression of L l is:

(1 ) [( 1)n ( 1)] n+1

4n f l (A c) 2 .
Border lines set by the comparison of consumer surplus (or social surplus) are also straight lines. If the …nes are nil these border lines are exactly the same than the ones computed by pro…t comparisons: …rms' choices are diametrically opposed to welfare maximisation.

If

f h = f l > 0 the borderline I is below IV since 1 +(1 ) (1 )
> 1 and for the same level of …ne, the scale of downward move driven by f l is larger than the upward shift induced by f h . As a result, there is an area where …rms choose the socially optimal H strategy. In this …gure the L strategy is never sustainable 29 then we can easily de…ne the area where …rms play the C* strategy: at the left of the borderline

I if C l < < H and at the right if C l < < C h .
For lower values of f h the borderline I moves downward but the borderlines IV , V and V I are unchanged since they do not depend of f h . Then the scenario f l > f h 0 generates similar graphical representation to the previous one.

2 9 We assume that = 0:2 then low demand occurs on average every …ve periods and expected discount factor used to value the collusive L strategy is 5 ' 0:77. However in the basic model of price competition with homogeneous product, cartel between 5 …rms is sustainable if 0:8. Therefore even if = 0 the L strategy is not sustainable if = 0:95, = 0:2 and n = 5.

For lower values of f l the borderline I moves upward and may be over IV . The zone 2 (strategy choice is H and W H > W C ) disappears and the zone 3 emerges: strategy choice is C and W C > W H . Figure 2 illustrates the results obtained in such cases. We consider these parameters values: 30 = 0:8; = 0:01; Henceforward the L strategy is always sustainable ( is high and

f l = 0). If L > C then L > C . If
L < C and whenever the C strategy is no longer sustainable then the C* strategy can be sustained ( C l is below II ) and is selected over L. 31 The previous graphs based on a speci…c demand function support that intermittent collusive strategies H and L can be equilibrium choices. They will be used in the next section's discussion.

Level of …nes

We now analyze the optimal level of …nes. First we state results without specifying a demand function then we use the previous one which leads to more speci…c conclusions.

3 0 For L to be an equilibrium strategy we must assume f l < f h , but also modify some other parameters values. We strongly increase and we set f l = 0. is also increased to ensure that H is sustainable.

3 1 Comparison of pro…ts leads to the following result (see Appendix 7.4): The deterrence e¤ect (1) leads the antitrust authority to apply maximum …nes.

C L , l 4n (A c) 2 [1 (1 )] (1 + ) (n 1) [(1 + n) (1 ) (n 1) ] f h + f l
Proposition 6 Higher level of …ne

f h [f l ] increases values of C l , C h and H [ C l , C h and L ].
All collusive strategies are more di¢ cult to sustain for higher …nes.

The collusive price e¤ect (2) has a similar impact. Higher …nes reduce and constrain …rms to decrease collusive prices during booms under the C* strategy.

Implications of the marginal deterrence e¤ect (3) are obvious for collusive choice but are more ambiguous as regards the optimal …ne level.

Proposition 7 Higher level of f h or/and lower level of f l results in a switch of the borderlines I and II upward: …rms may switch from the H to the C strategy or from the C to the L strategy.

Implications for optimal level of …nes: Under assumption A1 optimal antitrust policy was to set the highest achievable …nes to maximize the deterrence and the collusive price e¤ects.

Under assumption A2 …rms may adopt di¤erent collusive strategies. Fines have a deterrent e¤ect and may change collusive strategy. The latter is somewhat parallel to marginal deterrence in some works dealing with crime. Lower value of …ne can modify collusive strategy and be welfare increasing. Here we consider a more complex than traditional model of marginal deterrence because the ranking of infringements according to the damages caused to the third party depends on the parameters values. The C strategy seems to be more harmful than the H strategy as collusion occurs at every state of demand but, depending on parameters values, the damaging e¤ects on welfare may be stronger with the H strategy (since detection occurs later).

Without specifying demand function and density function for l and h , only overall result can be obtained.

Proposition 8 Under assumption A2, the highest level of …nes might not always be optimal.

The ranking of infringements depends on parameters values (especially on l and h ) it is therefore not possible to decide which one of the two …nes should be below the highest achievable …ne. Nevertheless the following corollary can be stated:

Corollary 9 It may sometimes be optimal to set f l < f h .

So reducing …nes during recessions can sometimes increase both consumer surplus and social welfare.

This result is mainly based on the marginal deterrence e¤ect. Decreasing f l may induce …rms to switch from the H to the C strategy. Antitrust authorities wish that collusive …rms still maintain higher prices during recessions because on the one hand collusive impact on consumer surplus is lower than during high demand period and on the other hand antitrust authorities can more easily detect and break o¤ cartels. Lower level of …ne f l may also induce other cartels to switch from the C to the L strategy: in such cases collusion is only active during low demand period and is less harmful for consumers, cartels are expected to be dissolved latter but are less harmful when they are active.

Setting f l < F is optimal if there is su¢ ciently high density of …rms where marginal deterrence e¤ect exists and, on the contrary, su¢ ciently low density of …rms where lower level of …ne f l decreases deterrence e¤ect and where …rms choose the C strategy.

We illustrate these results with the previously used speci…c demand function (section 5.5).

Speci…c demand function Multi-e¤ect of …nes makes it di¢ cult to determine their optimal level, thereafter we concentrate on the marginal deterrence e¤ect (3). We assume that: min C h ; H h . We use the same values as in …gure 1. L is never sustainable and …nes only impact the borderline I . We set F l = F h = 2000: the …rst graphical representation shows the case in which f l = f h = F and this is clearly not the optimal policy. Lower level of f h moves the borderline I downward. In this case …rms may switch from the C strategy to the H one, this is socially optimal since W H > W C in this area. If min C h ; H h and f l > f h then the antitrust authority must set: f h = 0 and f l = F . Figure 3 illustrates the previous results 32 for = 0:2; = 0:01; = 0:95; n = 5; A c = 10; f h = 0; f l = 2000.

3 2 f h = 0 implies that H is always sustainable and C is sustainable at the right of the borderline I . Starting from f l = f h = F , lower value of f l while maintaining f h unchanged can also be considered.

Consumer surplus and social welfare decrease in a slight reduction of f l . The borderline I moves to the left then …rms may switch from the strategy H to the C strategy but it is not socially optimal. But the impact of f l on global welfare is a non-monotonic one. As long as I is on the right side of IV , lower value of f l decreases social welfare. In the opposite case, …rms can switch from H to the socially optimal C strategy.

This option is strengthened by a decreasing value of f l . If antitrust authority decides to decrease f l then it should set f l = 0 and maintain f h = F . Figure 4 illustrates the previous results 33 for = 0:2; = 0:01; If f h = f l = 0 then …rms' choices result to be the worst welfare situations. Then antitrust authority should move away as much as possible the borderline specifying the …rms'strategy and the one which ranks consumer or social surplus. The borderline I can be moved to the left by setting f h = F and f l = 0 or right by …xing the opposite. The optimal levels of …ne will either be f h = F and f l = 0 or f h = 0 and f l = F .

The choice between these two opposing options depends on relative weight of industries below or above the borderline IV .

With the …rst option f h = F and f l = 0 it is not optimal to …ne cartel during recessions (even if the antitrust authority still actively engages in detection in order to dissolve the cartel). This re ‡ects the marginal deterrence e¤ect (…rms may switch from the H strategy to the C strategy), not the consideration of potential risk of bankruptcy due to low demand. Collusion during low demand period decreases consumer surplus but this is o¤set by shorter collusion under the C strategy.

Lower values of : In this case e¤ects (1) and (2) should also be considered. In Figure 3 lower value of f h decreases H and moves H h downward. Then …rms may switch from the C* strategy to the strategy H generating an ambiguous e¤ect on social welfare. Moreover there is an area where …rms switch from no collusion to the strategy H leading to falling consumer surplus. In Figure 4 Once again setting one maximum …ne (f l or f h ) is optimal. Antitrust authority faces a tradeo¤ as regards the other …ne: e¤ects 1 and 2 encourage increased …ne but e¤ect 3 favors the opposite. For some parameters values a lower …ne f l remains optimal in which case the number of active cartels increases but the gravity and the duration of some infringements decrease.

If the L strategy can be an equilibrium then the situation is far more complicated. If antitrust authority previously focused on lower f l then this policy is now reinforced. By contrast if optimal policy was to set lower …ne f h then accounting for the L strategy provides an incentive to decrease f l and increase f h .

Conclusion

In this article we explore …rms behavior and cartel enforcement when colluding …rms face random demand ‡uctuations.

If a cartel is detectable irrespective of the collusive price (assumption A1) then we obtain similar results as RS without antitrust policy. Particularly, collusive agreements are less stable in booms than in recessions and prices can be counter-cyclical. In contrast, if for any period during which collusive …rms set competitive price the probability detection is zero (assumption A2), then …rms may implement intermittent collusive strategy. Firms set monopoly price only during periods where collusion is the most pro…table strategy and charge marginal cost otherwise. Antitrust authority is active then cartels do not last forever and …rms can not collude again if cartels are dissolved. Then …rms prefer to avoid the risk of cartel detection for any period in which collusion is less worthwhile. Collusive prices are procyclical if …rms play the strategy H. Firms set monopoly price only during booms and charge marginal cost if demand is low. By contrast they charge countercyclical collusive prices if they play the L strategy. Finally if collusive pro…ts are relatively similar for both demand states then …rms collude once the cartel is dissolved and cartel prices are weakly 34 procyclical if is high (C strategy) and countercyclical for lower values of (C* strategy).

Then we address cartel enforcement and level of …nes. Under Assumption A1, antitrust authorities aim to deter collusive agreements or at least minimize their negative impact. This requires to set the highest achievable …nes. Under assumption A2 …nes may also change collusive strategy and setting one maximum …ne is optimal. The other …ne level re ‡ects a tradeo¤ between deterrence e¤ect and incentive e¤ect driving colluding …rms to choose a less harmful strategy. Then optimal level of …nes may depend on the demand state. Reducing …nes if demand is low can support the marginal deterrence e¤ect, this is not related to the potential risk of bankruptcy (not included in this model). Perhaps less intuitively, antitrust authority may instead reduce …nes during booms in order …rms switch from the C strategy to the strategy H so that they do not collude during recessions. This theoretical article suggests that setting maximum …nes regardless of demand is not necessarily optimal. Depending on the demand state, lower …nes can change collusive behavior and lead to less harmful n f l + ( 1) L f l + 0 , L (n 1) [( 1) n (1 )] n (1 )

f l
The minimum value of L is reached for p = p m ( l ). Then the L strategy is easier to sustain when …rms set the monopoly price than a lower price.

If < L the L strategy is not sustainable.

Comparison of sustainability thresholds

If demand is high the collusive strategy i 2 fC; Hg is sustainable i¤:

m ( h ) n f h + (1 ) i m ( h ) f h
Deviation and current collusive pro…ts are the same with the C and H strategies. But the discounted expected collusive pro…t is di¤erent, then the following inequality applies:

Figure 1 Figure 1 :

 11 Figure 1 helps to illustrate some of the previous results if f h = f l = 2000; = 0:2; = 0:01; = 0:95; n = 5 and A c = 10.

= 0 :Figure 2 :

 02 Figure 2: Equilibrium strategies and consumer surplus ranking if f h > f l

Fines impacts :

 impacts Under assumption A2, …nes f h and f l may have three e¤ects. (1) First they shift the collusive sustainability thresholds (deterrence e¤ect). (2) Moreover they a¤ect collusive price p during booms under the C* strategy (collusive price e¤ect). (3) Lastly they can modify the choice of collusive strategy (marginal deterrence e¤ect).

Figure 3 :

 3 Figure 3: Equilibrium strategies and consumer surplus ranking if f l > f h = 0 Borderlines in Figure 1 are plotted with dots, the shifted borderline I is plotted with full line.

= 0 :Figure 4 :

 04 Figure 4: Equilibrium strategies and consumer surplus ranking if f h > f l = 0

  lower value of f l decreases C h and C l . Some …rms switch from the C* strategy to the C strategy and other switch from no collusion to the C or C* strategies: consumer surplus is reduced.

  Concerning the other comparison between C and L we can prove that 22 L If H C and > H …rms opt for the H strategy. If C < L and > L they choose the L strategy. If H < C , C L and > C h they adopt the C strategy.

	C C . For the case L L , H between L and C* but the following inequality applies: L L and H L C l C h if L C there are parameters values for which …rms must choose C are more easily checked if f l is high, f h is l i¤ C L and C C and they select the L strategy. Moreover for these values if L is not sustainable, neither is C*. Concerning the other case C L there may be parameters values for which C can not be sustain whereas L and C are sustainable. In such cases we obtain the following condition: C L , 1 + n f h m ( l ) n f l 5.3 Equilibria We focus on high values of . The following proposition establishes the equilibrium strategy of colluding …rms. low, is low, C Proposition 4

m ( h ) is high and m ( l ) is low.

We now turn to lower values of . In such cases the C strategy may not be sustainable and …rms can implement the C strategy. Note that C C . We can make comparisons between C with H and L , but the …rst comparison is useless since H C h , C H . 21 Then in the area where H is sustainable but not C, …rms favor H over C since H C C .

  Consumer surplus is cs m ( k ) throughout the collusive period and cs nc ( k ) afterwards, k 2 Consumer surplus is cs m ( h ) or cs nc ( l ) throughout the collusive period and cs nc ( k ) afterwards, k 2 fl; hg. Consumer surplus is sc nc ( h ) or sc m ( l ) throughout the collusive period and sc nc ( k ) afterwards, k 2 fl; hg.CS L = cs m ( l ) + (1 ) CS L + CS + (1 ) cs nc ( h ) + CS LReplacing for the derived values of allows us to re-write the previous condition as:With the speci…c demand function the condition reads as:

	L strategy: , CS L =	cs m ( l ) + (1 1		) cs nc ( h ) + +	CS
	7.4 Comparison of L and C
	L		C , 1	+		m ( l ) n		f l		m ( l ) n	f l + (1 1 (1	)	) n	f h
	,	1	+	n	f h			m ( l ) n	f l
		C	L , m ( l ) n	H	[1 [(1	C h , C (1 + n) (1 )] (1	+ ) (n 1) ] ) (n 1)	f h + f l
		C	L , l	n (A c) 4n 2 f l + (1 [1 [(1	) i )] (1 + n) (1 (1	m ( l ) + ) (n 1) ] f l ) (n 1)	f
	From which we deduce:
						L		C l , C	L
	7.3 Computations of expected consumer surplus
	C strategy: fl; hg.
		CS C =	cs m ( l ) + (1	) cs m ( h ) + (1	) CS C + CS
			, CS C =	cs m ( l ) + (1 1	) cs m ( h ) + CS (1 )
	H strategy: 1	) CS
							1		+ (1	)

H

If demand is low the collusive strategy i 2 fC; Lg is sustainable i¤:

m ( l ) CS H = cs nc ( l ) + CS H + (1 ) sc m ( h ) + (1 ) CS H + CS , CS H = cs nc ( l ) + (1 ) cs m ( h ) + (h + f l

Higher price may make customers suspicious that a cartel has formed.

[START_REF] Stigler | The optimum enforcement of laws[END_REF],[START_REF] Shavell | A note on marginal deterrence[END_REF],[START_REF] Wilde | Criminal choice, nonmonetary sanctions, and marginal deterrence: a normative analysis[END_REF],[START_REF] Mookherjee | Marginal deterrence in enforcement of law[END_REF] and[START_REF] Friehe | Marginal deterrence when o¤enders act sequentially[END_REF].

In the model presented in[START_REF] Green | Noncooperative collusion under imperfect price information[END_REF], …rms can not observe the demand shock or other …rm's quantity choices. Only the equilibrium price is public information. Since low prices can re ‡ect demand conditions or overproduction by competitors, then, to ensure that punishment mechanisms are credible, the regime switches to a reversionary episode in which …rms compete before to switch back to collusion.

This adjusting market share eliminates the risk of bankruptcy for …rms with …nancial weakness since the other producers can curtail their own products if demand is low to ensure the viability of these …rms. In return they obtain larger market shares when demand is high.

See also Correia-da-Silva,[START_REF] Pinho | Sustaining collusion in markets with entry driven by balanced growth[END_REF]. 1 0 See[START_REF] Stigler | The optimum enforcement of laws[END_REF],[START_REF] Shavell | A note on marginal deterrence[END_REF],[START_REF] Wilde | Criminal choice, nonmonetary sanctions, and marginal deterrence: a normative analysis[END_REF],[START_REF] Mookherjee | Marginal deterrence in enforcement of law[END_REF] and[START_REF] Friehe | Marginal deterrence when o¤enders act sequentially[END_REF] for studies on traditional marginal deterrence applied to crime.1 1 See e.g.[START_REF] Harrington | Optimal cartel pricing in the presence of an antitrust authority[END_REF][START_REF] Harrington | Comment on antitrust sanctions[END_REF],[START_REF] Buccirossi | Optimal …nes in the era of whistle blowers -Should price …xers still go to prison?[END_REF],[START_REF] Houba | Antitrust enforcement with price-dependent …nes and detection probabilities[END_REF],[START_REF] Bageri | The distortive e¤ects of antitrust …nes based on revenue[END_REF],[START_REF] Dargaud | Cartel punishment and the distortive e¤ects of …nes[END_REF],[START_REF] Katsoulacos | Penalizing cartels: the case for basing penalties on price overcharge[END_REF].

8 Notice that these pro…ts are the same for any deviating …rm given the model's symmetry, allowing us to drop index i to simplify notation.

9 It would be di¤erent if deviating …rm could not be …ned, considering for instance leniency program.

0 This case is similar to RS.

1 Proof in Appendix 7.2.

4 As with previous notations cs ( ) denotes the period surplus and CS ( ) the discounted surplus. Remind that superscript nc designates the non cooperative situation.

High value of ampli…es the negative impact of cartel then collusion is more harmful if demand is high.25 Moreover, relative to perfect competition, the consumer loss from monopoly situation is always higher than …rms'gain, then variations in social welfare and consumer surplus are perfectly lined up.2 5 We could obtain opposite result in looking at income e¤ect that could lead to higher marginal willingness to pay for low demand. In this article we consider partial equilibrium, we ignore income e¤ect then consumer surplus is strictly measured as the area below the downward-sloping demand curve.

h remains unchanged. The borderline I moves upward and is above IV .

4 In the example case used in this article collusive prices are constant since they do not depend on demand. For a more general demand function these prices will generally be procyclical.

strategy. Designing the optimal …nes in antitrust infringements is complex and requires a signi…cant knowledge of the distribution of industries according to the parameters values. Without such information, maximal …ne regardless of demand ‡uctuation is potentially a second-best optimum.

In this article we have exclusively focused on the level of …nes to …ght cartel facing demand ‡uctuations.

But antitrust authority may also adapt detection e¤orts to economic situation and we plan to include leniency programs for future research. We also intend to address the role of …nancial situation of …rms. Reducing …nes during recessions may help …rms to …ght bankruptcy and invest even during reduced availability of …nancing caused by economic crisis. However, in this article, we demonstrate that …nancial trouble is not the only possible justi…cation for lower …nes during recessions. Collusion is sustainable i¤:

Appendix

We need to determine :

Replacing for in the previous condition allows us to re-write the no-deviation condition during booms as:

The minimum value of H is reached for p = p m .

Then the H strategy is easier to sustain when …rms set the monopoly price than a lower price.

If < H the H strategy is not sustainable.

L strategy (collusion during recessions):

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