We study the sustainability of collusion with optimal penal codes in markets where demand growth triggers the entry of a new firm. In contrast to grim trigger strategies, optimal penal codes make collusion easier to sustain before entry than after. This conclusion is robust to changes in the number of entrants and to the consideration of price-setting instead of quantity-setting. A comparison is given between different reactions of the incumbents to entry in terms of sustainability of collusion, incumbents’ profits, entrant’s profits, consumer surplus and social welfare. One of our findings is that the incumbent firms may prefer competition to collusion.
Demand growth is usually understood as facilitating collusion, by reducing the short-term gains of deviating relative to the costs of future retaliation (Ivaldi et al. 2007). However, demand growth may attract new firms to the market, which hinders collusion.  Hence, the ultimate impact of demand growth on the likeliness of collusion is not straightforward.
The empirical evidence is mixed. For example, Dick (1996) found that Webb-Pomerene cartels are more frequent in growing industries. In contrast, Asch and Seneca (1975) concluded that firms whose growth is slow have more incentives to collude than those whose growth is fast. Symeonidis (2003) provided empirical evidence suggesting that the relationship between the sustainability of collusion and demand growth is not monotonic. More precisely, collusion is more sustainable in markets with moderate growth than in stagnant or declining markets, or in markets with rapid growth. 
Capuano (2002) and Vasconcelos (2008) studied this issue in a theoretical model where demand growth induces the entry of a new firm. Comparing the sustainability of collusion before and after entry, Vasconcelos (2008) concluded that collusion is more difficult to sustain before entry, i.e. when there are less firms in the market. The rationale for this result lies in the possibility that incumbents have of profitably delaying entry by disrupting the collusive agreement. As shown in this paper, however, this result crucially depends on the fact that Vasconcelos (2008) did not consider optimal penal codes (Abreu 1986, 1988).
In this paper, we investigate the case in which firms adopt a severe (Abreu-type) punishment strategy, which induces the lowest possible continuation payoff. The fact that a well-designed punishment is a threat that is not carried out (because it induces compliance with the collusive agreement) is an obstacle to its empirical observation. Still, there is some empirical support for the fact that firms engage in finite length price wars. In the words of Levenstein and Suslow (2006, 54): “one of the most clearly established stylized facts is that cartels form, endure for a period, appear to breakdown, and then re-form again.” 
The structure of our model is the same as in Vasconcelos (2008). The industry is composed of two incumbent firms and one potential entrant who set quantities in an infinite number of periods. Market demand is linear in price and grows at a constant rate. Production costs are null. To become active, the potential entrant has to incur a fixed setup cost. There is a trade-off in the choice of the entry period: on the one hand, postponing entry implies the loss of the current period’s profits; on the other hand, it decreases the discounted value of the entry cost.
At the beginning of the game, the incumbent firms agree to produce quantities that maximize their joint profit in each period. Unilateral deviations originate a harsh punishment, which induces a null (or security level) continuation value after a deviation. A null continuation value can be imposed through reversion to an equilibrium in which firms have null profits forever. Alternatively, it can be imposed through stick-and-carrot strategies according to which firms have high losses in one or more punishment periods and have positive profits when collusion is restored. 
Assuming that the incumbents accommodate the entrant in their collusive agreement, we conclude, in contrast to Vasconcelos (2008), that collusion is more difficult to sustain after entry than before entry. The result of Vasconcelos (2008) does not hold with optimal penal codes because a deviation before entry ceases to be an effective way of delaying entry and enjoying additional pre-entry profits. In our setting, a deviation can still delay entry, as the entrant would avoid entering the market in the punishment phase. However, the fact that the penal code implies a null (or security level) continuation value means that the punishment absorbs all the gains from delaying entry.
The idea that firms benefit from establishing collusive agreements is widely accepted. Interestingly, however, we conclude that the incumbent firms may prefer competition to collusion. Two conditions are necessary for this to be the case: (i) entry occurs later under competition; and (ii) individual profits are greater under competition with two firms than under collusion with three firms. These conditions imply that, under competition, the incumbents enjoy a longer pre-entry period, in which, although competing, they obtain higher profits than if they were colluding after entry.
As explained by Harrington (1989), Stenbacka (1990), Friedman and Thisse (1994) and Vasconcelos (2004), immediate accommodation of the entrant in the collusive agreement is just one of the possible cartel reactions to the entry of a new firm. Therefore, we compare our base scenario of immediate accommodation, which is also the scenario that is considered by Vasconcelos (2008), with three alternative reactions to entry: discontinuance of collusion, entry deterrence and gradual accommodation.
Firstly, we consider that the collusive agreement is discontinued when entry occurs. In this scenario, relative to immediate accommodation, post-entry profits are lower and, as a result, entry occurs later. If the entry delay is sufficiently high, the incumbents prefer to discontinue collusion than to immediately accommodate the entrant. We also conclude that this reaction to entry makes the collusive agreement easier to sustain because then there is only collusion before entry. Notice, however, that collusion before entry becomes slightly more difficult to sustain as a result of the decrease of post-entry profits.
Alternatively, the cartel may respond to entry by implementing the following predatory strategy: if entry occurs, the incumbents start a punishment process, as if one of them had defected. As a result, if the new firm effectively entered the market, the discounted value of the flow of its profits would be null (or equal to the security level, which is assumed to be lower than the entry cost). Therefore, entry would not be profitable. This is the best scenario for the incumbents because the severity of this threat allows them to effectively deter entry. For the same reason, this is the worst scenario for the potential entrant.  An interesting aspect of this scenario is that incumbents effectively deter entry at no cost and charge the monopoly price in all periods. Of course, it is necessary that the incumbents coordinate on this extreme form of punishment.
Finally, we analyze the sustainability of a collusive agreement that gradually accommodates the entrant (Friedman and Thisse 1994). In this scenario, firms maximize their joint profits in all periods, but the entrant is not treated as a full partner immediately after entering the market. In the first period, the profit of the entrant can be as low as the Cournot profit. Then, as time passes, the entrant is gradually accommodated in the collusive agreement, in the sense that its share of cartel profits increases until reaching a fraction not higher than one third of the monopoly profit. When compared to immediate accommodation, gradual accommodation is preferred by the incumbents because: (i) each incumbent receives a greater share of the monopoly profit during the accommodation phase; and (ii) entry is delayed. For the same reasons, the entrant prefers immediate accommodation. As a result, the entrant has more incentives to disrupt the collusive agreement than the incumbents. This imbalance implies that collusion is less likely to be sustained when accommodation is gradual than when it is immediate.
In terms of social welfare, collusion is detrimental relative to competition, independent of the reaction of the cartel to entry. Naturally, it is less so if the agreement is discontinued following entry. The remaining collusive scenarios are equivalent in terms of output and prices, as firms produce the monopoly output in every period. Still, entry deterrence is slightly preferred to accommodation because the entry cost is not incurred. Gradual accommodation is socially preferred to immediate accommodation because the entry cost is incurred later in time.
The finding that collusion is more difficult to sustain after entry than before entry extends to a scenario in which firms set prices instead of quantities. In such a scenario, collusion is even more difficult to sustain both before and after entry because the one-shot gain from deviating is higher, while the continuation value after a deviation is the same. We remark that, in the case of price-setting firms, permanent reversion to competition is an optimal penal code, in the sense that it also implies a null continuation value.
We also demonstrate that the same conclusion (that collusion is more difficult to sustain after entry than before entry) is robust to the consideration of any number of incumbents and any number of entrants. However, in our environment, the greater the total number of firms (independent of whether they are incumbents or entrants), the harder it is to sustain collusion.
The contribution of Capuano (2002) is closely related to our study, since he also examined the sustainability of collusion when firms use stick-and-carrot strategies to punish deviations. There are, however, important differences worth mentioning. Firstly, Capuano (2002) considered that the incumbents immediately accommodate the entrant in their collusive agreement, while we also study alternative cartel reactions to entry and compare them in terms of surplus for firms and consumers. Secondly, we analyze the willingness of the incumbents to establish the collusive agreement by comparing their discounted flow of profits under collusion and competition. Thirdly, Capuano (2002) assumed that entry occurs when the discounted value of the profits of the entrant becomes positive, while we consider that entry occurs when the discounted value of the entrant’s profits is maximal.  Fourthly, Capuano (2002) did not consider that, above a certain threshold level of output, the price is zero.  This apparently insignificant technical point is a key issue in the construction of the penal code. Finally, we study the sustainability of collusion with several entrants, while Capuano (2002) restricted his analysis to the case of one single entrant.
The remainder of the paper is organized as follows. Section 2 presents the assumptions of the model and derives the optimal entry period. Section 3 describes the collusive agreement in the scenario of immediate accommodation and analyzes its sustainability before and after entry. Section 4 studies alternative reactions to entry that may delay or even deter the entry of a new firm. Section 5 studies two extensions of the baseline model: the case in which firms set prices instead of quantities; and the case in which there is an arbitrary number of incumbents and entrants. Finally, Section 6 summarizes the main conclusions of the paper. Appendix A extends the description of the punishment strategy to the case in which the security level is strictly positive. Appendix B contains most of the proofs.
Consider an industry with two incumbents (firms 1 and 2) and a potential entrant (firm 3) that sell a homogeneous product in an infinitely repeated setting. In each period, the active firms simultaneously choose the quantities to produce. Production costs are null. Market demand in period is given by :
1st: firm 3 decides whether or not to enter the market in the current period;
2nd: the active firms simultaneously choose their output levels.
Assumption 1. Demand does not grow too fast: . 
Lemma 1. If firm 3 expects its profit in each period t to be given by , it enters the market at:
The following assumption guarantees that , which means that the incumbents are alone in the market at (even if the entrant expects to collude after entry). 
Assumption 2. The entry cost is not too small: .
3 Collusion with entry accommodation
In this section, we investigate the sustainability of a collusive agreement with an optimal penal code, assuming that the entrant is included in the cartel immediately after entering the market. Firms are assumed to maximize their joint profits and inflict the harshest possible punishment on deviators.
3.1 Collusive agreement
At the beginning of period , the incumbents establish the following collusive agreement, which extends to firm 3 after it enters the market. In each period:
Produce the quantities that maximize their joint profit, if all firms adhered to the agreement in the previous period.
Start a severe punishment, which reduces the continuation value to zero, if the agreement was violated in the previous period.
3.2 Punishment strategies
In this section, we describe some punishment processes that induce a zero continuation value after a deviation.  In Section 3.3, we show that, despite being very harsh, these punishments are credible as firms do not gain by unilaterally deviating.
3.2.1 Stick-and-carrot punishment with observable dissipative costs
If firms have access to some kind of unproductive activities that are observable (e.g. dissipative advertising, donations to charity or destruction of equipment),  it is possible to construct an optimal penal code with a single punishment period, or with any desired number of punishment periods.
In the case of a single punishment period, the penal code could be as follows:
If a deviation occurs in period t, then, in period , the active firms produce quantities that completely satiate the market (implying that the price is zero), even if one of the firms deviates and produces nothing:
and spend an amount in dissipative advertising, (calculated in Appendix A), which implies a zero continuation value.
If no firm deviates from the punishment at , firms return to collusion in period . Otherwise, the punishment is restarted.
3.2.2 Stick-and-carrot punishment with side-payments
In a recent empirical study on the duration of international cartels, Levenstein and Suslow (2011) found that “one-third of the cartels compensate members when realised sales differ from proposed allocations; […] cartels that retaliate in response to deviations are significantly more likely to break up.” 
In the context of our model, instead of requiring that the deviator makes observable and unproductive expenses, the other cartel members could demand a side-payment.  The penal code would be exactly as stated above, except for the fact that the deviator would transfer the amount to the remaining firms that were active in period t (divided in equal parts), instead of spending it in unproductive activities.
In our environment, the incentives to comply with the collusive agreement would remain the same because firms are indifferent between “burning money” and transferring it to rivals. However, since side-payments are usually perceived as a “smoking gun”, the expected cost of cartel indictment (fines, or a bad image) should be taken into account.
3.2.3 Stick-and-carrot punishment with production costs
Suppose that the marginal production cost is strictly positive and constant in output, . In this case, the following penal code would induce a zero continuation value after a deviation: 
In period , the firms that complied with the collusive agreement in period t produce quantities that completely satiate the market (driving the price to zero), even if the deviator produces nothing: 
In period , the firm that deviated in period t produces the quantity that originates a loss of the same magnitude as the discounted value of future profits (so that its value of being in the market becomes null):
where is the collusive profit if firms have unit costs equal to c, and denotes the number of active firms in period s.
If no firm deviates from the punishment at , the collusive agreement is restored in period . Otherwise, the punishment is restarted.
3.3 Credibility of the punishment
We must guarantee that firms do not deviate in the punishment phase. Firms adhere to the punishment if and only if the following incentive compatibility constraint (henceforth, ICC) is satisfied:
By construction, the optimal penal code gives a zero continuation value after a deviation ( ). This allows us to write the ICC  as:
Suppose that firms could exit the market and receive a strictly positive residual value, (which may result, for example, from the sale of machinery, plants, copyrights or other kinds of licenses). In this case, their security level would coincide with this residual value, i.e. , and the ICC would become , which is even more easily satisfied. In the opposite case, in which exit is costly (e.g. due to legal requirements such as compensation to employees, clients or suppliers, or from broken contracts), for the security level to become negative, it would also be necessary to introduce fixed production costs (otherwise, instead of incurring the cost of exiting, firms would produce zero forever at no cost).
We will now study whether firms, knowing that the penal code is credible, have incentives to deviate from the collusive agreement.
3.4 Sustainability of collusion
The collusive agreement is sustainable if firms have no incentives to deviate either before or after entry (which occurs at ).
Collusion is sustainable after entry if and only if the following incentive compatibility constraint holds for all :
Proposition 1. Collusion is sustainable after entry if and only if: .
Lemma 2. Let . If the ICC  is satisfied in the period that immediately precedes entry, , it is satisfied in all previous periods, .
Considering the ICC  at , we obtain the critical discount factor above which collusion is sustainable before entry (under the hypothesis that it is sustainable after entry).
Proposition 2. The ICC  is satisfied for all if and only if: .
Our conclusion is the opposite of the one obtained by Vasconcelos (2008).  In his model, after a deviation from the collusive agreement, firms permanently revert to Cournot equilibrium (grim trigger strategies). In that setting, a deviation effectively delays entry, and this delay is profitable because Cournot profits with two firms are greater than collusion profits with three firms. As a result, the incumbents have an additional incentive to deviate before entry. In our model, however, the incumbents cannot benefit from delaying entry through a deviation from the collusive agreement because the ensuing punishment absorbs all the continuation value. Firms deviate if and only if the single-period deviation profit exceeds the value of colluding forever.
Corollary 1. Collusion is sustainable (before and after entry) if and only if: .
4 Can the incumbents profitably delay entry?
In this section, we consider four alternative cartel reactions to entry and compare them in terms of incumbents’ profits, entrant’s profits, consumer surplus and total surplus: 
collusion before entry and competition after entry;
collusion before entry and punishment if and when entry occurs;
gradual accommodation of the entrant in the collusive agreement.
The remaining alternative scenarios correspond to plausible cartel reactions to entry, previously considered and compared by Harrington (1989), Friedman and Thisse (1994) and Vasconcelos (2004). See also the discussion by Harrington (1991b).
We could have also considered a scenario of competition before entry and collusion after entry (possibly in response to fiercer competition). The condition for sustainability of collusion would be the same as in the baseline scenario of immediate accommodation. However, the incumbents would be worse off than in the baseline scenario because they would have lower profits before entry and the same level of profits after entry.
4.1 No collusion
If there is no collusion, the present value of the profits of each incumbent is
Under collusion with immediate accommodation, this value is given by:
Proposition 3. The incumbents are better off under competition than under collusion with entry accommodation if and only if: .
It may seem counterintuitive that the incumbent firms prefer not to collude. Notice, however, that: (i) a collusive agreement with immediate accommodation of the entrant induces an earlier entry (at rather than at ); and (ii) each incumbent profits more under competition between the two incumbents than under a collusive agreement that includes the three firms. Of course, the entrant surely prefers the collusive scenario.
The parameter values for which the incumbents prefer competition to collusion with accommodation of the entrant are represented in Figure 1. In the painted area, the collusive agreement with entry accommodation would be sustainable, since , but the incumbents prefer to compete rather than to collude and accommodate the entrant.
It should be clear that this result does not depend on the penal code. It also holds under grim trigger punishment strategies because the present values of profits under collusion and competition do not depend on the punishment strategy.
4.2 Discontinuance of collusion
Suppose now that the incumbents combine to discontinue the collusive agreement after firm 3 enters the market.  More precisely, the collusive agreement established by the incumbents is the following:
Before entry, produce the quantities that maximize joint profits if the agreement was honored in the previous period.
Before entry, engage in a punishment that absorbs all the continuation value if there was a defection in the previous period. 
After entry, switch to stage Nash-Cournot equilibrium.
Before entry, the incumbents abide by the collusive agreement if and only if:
Lemma 3. Let . If the ICC  is satisfied in the period that immediately precedes entry, , it is satisfied in all previous periods, .
Proof. Follow the same steps of the proof of Lemma 2.
Substituting and the expressions for profits in the ICC , we obtain the following result.
Proposition 4. The collusive agreement with discontinuance of collusion after entry is sustainable if and only if: .
To investigate whether the incumbents prefer discontinuance of collusion to immediate accommodation, we compare the present value of profits in the two scenarios.
Proposition 5. The incumbents are better off discontinuing the collusive agreement when entry occurs than accommodating the entrant if and only if: .
The parameter values for which the incumbents prefer discontinuance of collusion to accommodation of the entrant are represented in Figure 2. In the painted area, they prefer discontinuance of collusion (collusion is sustainable in both scenarios), i.e. .
The complex pattern in Figure 2 results from the discrete nature of time, which implies that a slight change in parameter values may lead to a discrete change of the entry period. We can distinguish two types of lines: the oblique lines, corresponding to jumps in or ; and the almost vertical lines, corresponding to the condition , for different values of the entry delay, .
The relative desirability of accommodating entry or discontinuing the collusive agreement after entry depends on the type of agent (incumbent, entrant or consumer), as will be discussed in Section 4.5.
4.3 Entry deterrence
Produce quantities that maximize the industry profit, if the agreement was honored in the previous period and firm 3 did not enter the market.
Start a punishment that reduces the continuation value to zero, if one incumbent deviated in the previous period or if firm 3 entered the market.
Proposition 6. The collusive agreement with entry deterrence is sustainable if and only if: .
This is the best collusive scenario for the incumbents, as each incumbent receives half of the monopoly profit in all periods. On the other hand, this is the worst scenario for the entrant. Of course, if the incumbents are not able to commit to this punishment, the entrant will anticipate that the incumbents will not punish entry (as they receive a higher profit by sharing the market than by punishing entry), and, therefore, will enter.
Predatory behavior toward new entrants is not rare. In his study of the cartel formed in the late nineteenth century by British shipping firms, Morton (1997) reported that when a new firm entered the market, the cartel either started a price war or admitted the entrant to the cartel without conflicts. Morton (1997) concluded that the reaction to entry depended on the characteristics of the entrant. Firms with low financial resources, little experience and without an established customer base were more likely to be preyed upon. Strong competitors were more likely to be accepted in the cartel. 
Another case in point is the pre-insulated pipe cartel in the 1990s, in which the firm ABB took the bulk of the costs of running the cartel. In particular, when Powerpipe, a firm outside the cartel, tried to expand its activities, ABB used large resources to try to eliminate the maverick from the market. This predatory activity was multidimensional and costly, involving, for instance, a systematic campaign of luring away key employees of Powerpipe, including its then managing director (for details, see Ganslandt, Persson, and Vasconcelos (2012)).
4.4 Gradual accommodation
Based on the proposal of Friedman and Thisse (1994), we now consider the case in which the entrant is gradually accommodated in the collusive agreement. Firms maximize the industry profit in all periods (before and after entry), but there is an adjustment phase during which the entrant receives a smaller share of the industry profit than the incumbents.
Let and , respectively, denote the output and profit of firm in period t. We assume that the entrant never profits less than under competition nor more than the incumbents, i.e. . Moreover, we assume that the entrant’s share of the cartel profit is non-decreasing along the accommodation phase. Formally, letting denote the entrant’s share of the cartel profit s periods after entering the market, we assume that is non-decreasing in s and that .
Let denote the optimal entry period.  The assumption that , , implies that .
After entry, each incumbent receives half of the difference between the monopoly profit and the profit of the entrant:
Since the entrant’s share of the cartel profit is increasing in time, it is in the entry period that the ratio between the one-shot gain from deviating (difference between the deviation profit and the collusive profit) and the continuation value under collusion is the highest.
Proposition 7. The entrant is the most tempted to deviate in the entry period, .
In contrast, since the incumbents’ share of the cartel profit is decreasing, their incentives to disrupt the collusive agreement (after entry) increase as time passes.
Proposition 8. After entry, the incentives for the incumbents to deviate from the collusive agreement are non-decreasing over time.
The assumption that the entrant’s share of the cartel profit is not greater than the share of each incumbent implies that the entrant has more incentives to deviatethan the incumbents.
Lemma 4. In any period after entry, , the incumbents have less incentives to deviate than the entrant.
We conclude that the critical ICC for collusion to be sustainable after entry is the one for the entrant in the entry period, .
Corollary 2. Collusion is sustainable after entry if the entrant does not deviate in the entry period, . This is the case if and only if: .
Finally, we need to analyze the incentives for the incumbents to collude before entry. They comply with the collusive agreement in period , with , if and only if:
Lemma 5. If the ICC  is satisfied in the period that immediately precedes entry, , it is satisfied in all previous periods, , with .
Thus, the necessary and sufficient condition for collusion to be sustainable before entry is the ICC  evaluated at , which can be rewritten as follows:
Proposition 9. The ICC for the entrant to abide by the collusive agreement in the entry period, , is the most difficult to satisfy.
These results suggest that the increase over time of the profit share of the entrant hurts the sustainability of collusion relative to a situation in which the entrant’s profit share is constant over time.
Consider an equivalent sharing of the discounted value of cartel profits using constant profit shares. More precisely, let be the share of cartel profit that accrues to the entrant. By construction, with this constant profit-sharing rule, the discounted value of the entrant’s and incumbent’s profits is the same as with the sharing rule defined by the sequence .
It follows that the entry period is not affected, since the mapping from the entry period to the entrant’s discounted sum of profits is preserved. However, profit sharing in constant proportions has an advantage: collusion is easier to sustain. Instead of a situation in which only the entrant’s ICC at is binding, we have the entrant’s ICCs equally binding at every period, but with a lower critical value of the adjusted discount factor.
In our environment, therefore, there seems to be no rationale for gradual accommodation of the entrant in the collusive agreement.
4.5 Welfare analysis
Let us now compare the different cartel reactions to entry in terms of: incumbents’ profits, entrant’s profits, consumer surplus and total surplus.
4.5.1 Incumbents’ surplus
Entry deterrence is undoubtedly the scenario that gives the highest payoff to incumbents since they receive half of the monopoly profit in every period.
The incumbents prefer gradual to immediate accommodation ( ) because: (i) gradual accommodation delays entry; and (ii) the share of incumbents in the industry profit is greater during the accommodation process than after entry.
The comparison between the scenarios of no collusion and discontinuance of collusion is also immediate ( ) since: (i) entry occurs in the same period; (ii) profits after entry are the same; but (iii) profits before entry are lower under competition.
Remark 1. The preferences of the incumbents regarding the different scenarios satisfy the following partial ordering:
4.5.2 Entrant’s surplus
Obviously, the worst scenario for the entrant is the one in which the incumbents adopt a predatory behavior and deter entry ( ).
For the entrant, it is irrelevant whether the incumbents compete from the beginning of the game or discontinue collusion when entry occurs since entry takes place in the same period and the flow of the entrant’s profits is the same ( ).
The reason for the entrant to prefer collusion with gradual accommodation to no collusion is easy to understand. Suppose that the entrant was forced to enter at . Even in this case, collusion with gradual accommodation would be preferable because, in all periods , profits would be greater than in the absence of collusion. The possibility of entering at only increases the comparative value of collusion with gradual accommodation ( ).
It is straightforward that the entrant prefers to be accommodated in the collusive agreement immediately after entry rather than gradually ( ).
Remark 2. The preferences of the entrant regarding the different scenarios satisfy the following complete ordering:
4.5.3 Consumers’ surplus
We measure consumer welfare as the discounted sum of each period’s consumer surplus:
The competitive scenario is the best for consumers because output is the highest in all periods. For the same reason, consumers prefer discontinuance of collusion to entry accommodation, as there is competition after entry ( ). They are indifferent between collusion with entry deterrence or entry accommodation (immediate or gradual) because total output is at the monopoly level in all periods ( .
Remark 3. The preferences of consumers regarding the different scenarios satisfy the following complete ordering:
4.5.4 Social welfare
We define social welfare as the sum of consumers’ surplus with the discounted sum of the industry profits. As production costs are null, social welfare is increasing in total output. In addition, ceteris paribus, the later the entry occurs (i.e. the lower the discounted value of the entry cost is), the higher the social welfare is.
It is clear that social welfare is higher if there is no collusion than if there is collusion with immediate accommodation because the output is higher in every period and the entry occurs later. Likewise, discontinuance of collusion is socially better than entry accommodation, but worse than competition ( ).
The output is the same under immediate accommodation, gradual accommodation and entry deterrence. However, social welfare is higher when accommodation is gradual than when it is immediate because the entry cost is incurred later; and it is even higher when entry is deterred because the entry cost is not even incurred ( ).
The comparison between discontinuance of collusion and entry deterrence is not straightforward. On the one hand, total output is greater when collusion is discontinued. On the other hand, the entry cost is not incurred under entry deterrence. We find that the output effect more than compensates the entry cost effect.
Remark 4. The different scenarios satisfy the following complete ordering in terms of social welfare:
In this section, we consider one variation and one extension of the baseline model. More precisely, we analyze the case in which firms set prices instead of quantities and allow for an arbitrary number of firms (incumbents and entrants).
5.1 Price-setting firms
Until now, we have considered that the decision variable of firms is the quantity to produce and sell in the market. Now, we will consider that firms set prices.
In this case, since firms produce homogeneous goods, Bertrand competition implies that all firms receive zero profits. Thus, one way of implementing a null continuation value after a deviation is through the permanent reversion to Bertrand competition after a deviation. Grim trigger strategies constitute, therefore, an optimal punishment.
With firms setting prices instead of quantities, collusion becomes harder to sustain. The reason is simple. The collusive profits and the continuation value after a deviation do not depend on whether firms choose prices or quantities. However, the one-shot deviation profit is higher when firms set prices since the deviator is able to receive the monopoly profit by slightly undercutting the collusive price.
For the sake of completeness, we derive the critical discount factor for collusion to be sustainable (before and after entry) when firms set prices.
Proposition 10. If firms set prices, collusion is sustainable after entry if and only if: .
Comparing the critical discount factor for collusion to be sustainable after entry when firms set quantities (Proposition 1) and when firms set prices (Proposition 10), it immediately follows that collusion is more difficult to sustain in the latter case.
Proposition 11. If firms set prices, collusion is sustainable before entry if and only if: .
We conclude that collusion is more difficult to sustain when firms set prices than when they set quantities, both before and after entry. Regardless of the firms’ decision variable, the ICC for collusion to be sustainable after entry is more demanding than the ICC for collusion to be sustainable before entry.
5.2 Multiple entrants
The assumption that there is a single entrant may not be the most suitable to study a market that expands forever. We now study the sustainability of collusion when there are multiple entrants.  More precisely, we consider an industry with incumbent firms and gentrants, focusing our attention on the scenario in which entrants are immediately accommodated in the collusive agreement.
As in the single-entrant case, the active firms produce quantities that maximize the industry profit as long as no firm defects; if some firm breaks the collusive agreement, a harsh punishment is started, which drives the continuation value of all active firms to zero. Such an optimal penal code can be constructed as in the case of a single entrant (see Section 3.2 and Appendix A).
For simplicity, we assume that all entrants incur the same entry cost, . This generates some indeterminacy in the order of entry, which is not relevant for the sustainability of collusion.  Whenever it is optimal for one firm to enter, but not for a second firm to enter simultaneously, all inactive firms will want to be that single entrant. We assume that there exists a simple coordinating device that randomly and equiprobably selects one of the inactive firms to become active.
We start by obtaining the (optimal) entry period of the last entrant. Since there are no more potential entrants in the market, the last entrant will start its activity when the discounted value of its profits, , is maximal. Using expression , we obtain the optimal entry period of the last entrant under collusion:
The entry period of firm is only slightly more difficult to obtain. After the entry of firm , the remaining firms have an expected value given by the simple average between and , which we denote by . The entry of firm will occur, therefore, in the first period, , that verifies the following condition: ; or (if there is no such period) at . The same logic applies to earlier entrants.
To study the sustainability of collusion with entrants, it is necessary to consider incentive compatibility conditions. We start by obtaining the critical discount factor for collusion to be sustainable after entry, i.e. in periods :
Consider now a period , with and . The n active firms abide by the collusive agreement in this period if and only if the following ICC is satisfied:
Lemma 6. If and collusion is sustainable in period , it is sustainable .
Since the periods that immediately precede each entry are the critical moments for collusion to be sustainable before entry, we can replace in the ICC  that applies to periods in which there are active firms. We obtain a single ICC for each value of . Comparing these ICCs, we conclude that they become stricter as the number of active firms increases. Finally, we compare the ICC before the last entry with the ICC after entry to arrive at the following result.
Proposition 12. If collusion is sustainable after entry, it is also sustainable before entry. Collusion is globally sustainable if and only if: .
We conclude that, with optimal penal codes, collusion is more difficult to sustain after entry than before entry regardless of the number of entrants. We also conclude that the higher the total number of firms, the harder it is to sustain collusion.
Observe that the critical discount factor depends on the total number of firms (N) but not on the split between incumbents and entrants ( ). The reason for this is the following. As stated in Proposition 12, the binding ICC for collusion to be sustainable is the one corresponding to periods after entry. After the last entry, all firms are symmetric (entrants and incumbents). Thus, the incentives for the incumbents to comply with the collusive agreement are exactly the same as those of the entrants.
In this paper, we have studied the sustainability of collusion when the market growth triggers entry. In a similar model, Vasconcelos (2008) assumed that, after a deviation from the collusive agreement, firms permanently reverted to the Cournot equilibrium (grim trigger strategies). However, firms can increase the sustainability of the collusive agreement by adopting Abreu-type punishment strategies. Motivated by this idea, we have modified the model of Vasconcelos (2008) by considering penal codes that drive the continuation value after a deviation to zero. 
Following Vasconcelos (2008), we started by considering that the incumbents immediately accommodate the entrant in a more inclusive agreement. In contrast to Vasconcelos (2008), we concluded that collusion is more difficult to sustain after entry than before entry. This finding conforms to the idea that the higher the number of firms in the market, the less sustainable is collusion. The origin of the discrepancy between the results of Vasconcelos (2008) and ours is the ability of the incumbents to profitably delay entry by deviating from the collusive agreement. With grim trigger strategies, a deviation may profitably delay entry (since individual profits are higher when two firms compete than when three firms collude). In contrast, with optimal penal codes, breaking the agreement leads to a punishment that absorbs all future profits. As a result, the entry delay is irrelevant.  The finding that, with optimal penal codes, collusion is more difficult to sustain after entry than before entry is robust to the consideration of price-setting instead of quantity-setting and to the number of incumbents and entrants in the market.
Surprisingly, incumbents may prefer to compete (since the beginning of the game) rather than establish a collusive agreement that accommodates entry. This result is, again, explained by the fact that entry occurs later under competition than under collusion, together with the fact that competition before entry is more profitable than collusion after entry.
We have studied alternative reactions to entry and compared them in terms of the surplus of firms and consumers. The incumbents’ surplus is the highest when they are able to deter entry by regarding entry as a deviation from the collusive agreement. Not surprisingly, this is worst scenario for the entrant. The entrant is best off when the incumbents immediately accommodate entry. Depending on the parameters of the model, the incumbents may prefer discontinuance of collusion or immediate accommodation, while the entrant always prefers immediate accommodation. Finally, incumbents prefer gradual to immediate accommodation, while the entrant prefers the opposite. Consumers are best off when firms compete in all periods, since the incumbents and the entrant sell homogeneous products and competition is the scenario in which total output is the highest. Consumers are indifferent between collusion with entry deterrence or (immediate or gradual) entry accommodation, since total output is always at the monopoly level. Among the collusive scenarios that were considered, discontinuance of collusion after entry is the one that harms consumers the less.
Our results embody some important competition policy implications. First, by proposing a framework wherein the number of market participants depends on the evolution of demand, our analysis warns that market growth is potentially detrimental to collusion because entry becomes easier in growing markets, and future entry may hinder firms’ ability to engage in a collusive agreement. Second, we offer a more comprehensive analysis than previous works, which often focus attention on a specific strategy regarding incumbents’ reaction to entry (e.g. entry accommodation). More specifically, by comparing alternative cartel reactions to entry, we show, among other things, that incumbents can coordinate on carefully designed punishment processes that threaten the potential entrant with the prospect of a null post-entry discounted value of the flow of profits, thereby deterring entry for any level of the entry cost (as long as this cost is positive). As a consequence, a significant degree of collusion can be sustained in equilibrium by incumbents in a growing market even if the entry costs are low. Lastly, and perhaps most importantly, the obtained results are important for the evaluation of the coordinated effects of mergers in markets with growing demand.  In particular, it is shown that, in markets where demand growth may trigger future entry, whether the most severe coordinated effects arise after or before entry takes place crucially depends on whether deviations trigger optimal (i.e. security level) punishments or not.
Optimal penal code with a positive security level
We assume that the security level is relatively low, so that the entrant does not profit by entering the market, deviating and exiting immediately afterward.
Temporary reversion to zero profit equilibrium
The penal code is as follows. If a deviations occurs in period t, and there are active firms in the market:
In periods (with T calculated below), firms produce quantities that completely satiate the market (implying that the price is zero), even if one of the firms deviates and produces nothing:
At , firms produce quantities such that profits in this period are equal to
If no firm deviates along the punishment path, the collusive agreement is reinstated in period T. Otherwise, the punishment is restarted.
If collusion is reinstated in a period T after entry (i.e. ), the continuation value after the deviation (discounted to ) is given by:
Stick-and-carrect punishment with dissipative costs
Suppose now that firms have access to some kind of dissipative cost. Consider further that the punishment lasts for one single period (in the case of being respected by all firms). 
Suppose that one of the cartel members deviates in period t. Denote by the value of the dissipative cost incurred in the punishment period, . This value must be such that the continuation value after a deviation is equal to the security level, v. In addition, the output of each firm in the punishment period must be high enough for the market price to be null, even if there is a deviation. Therefore, as production costs are null, the profit of each firm in the punishment period is .
Keep in mind that the decision of firm 3 regarding the entry period only depends on the flow of profits that it expects to obtain after entry. Recall that if the firm expects to get one third of the monopoly profit in all periods, entry occurs in period . The firm does not care about the existence of a deviation in any period because the collusive agreement would be restored at . The entry decision is only affected if one incumbent deviates at . In that case, the punishment takes place at and firm 3 delays its entry to period .
Suppose that the deviation occurs at . The punishment will take place at ; there will be collusion with two firms from until ; entry will occur at ; and there will be collusion with three firms from onward. In this case, for the continuation value (after the deviation at t) to be equal to the security level, v, the dissipative cost must satisfy the following condition:
If a firm deviates after entry, at , there will also be collusion with three firms from onward. Therefore, the same expression for applies.
In short, the amount spent on dissipative costs in the punishment period, , is equal to
Proof of Lemma 1
Firm 3 prefers to enter at T rather than at if and only if:
Stick-and-carrot punishment with production costs
Suppose that firms have constant unit costs equal to c>0. Suppose further that, in period t, one of the nt (active) firms deviates from the collusive agreement. Let us describe a punishment scheme that drives the continuation value of the deviator to zero.
In period t + 1, the firms that complied with the collusive agreement in period t must produce quantites that satiate the market (even if the deviator produces nothing), i.e.
Proof of Proposition 1
In period t, the profit function of firm i is given by:
Proof of Lemma 2
Substituting in eq. , we obtain the deviation quantity, , and the corresponding profit, . Substituting the expressions for profits in ICC , we obtain
Proof of Proposition 2
Collusion is sustainable at (and, therefore, at all ) if and only if:
Proof of Proposition 3
The incumbents prefer no collusion to collusion with immediate accommodation if and only if:
Proof of Proposition 4
Replacing the expressions for profits, we can write the ICC  as follows:
Proof of Proposition 5
The present value of profits of an incumbent firm in the collusive agreement with immediate accommodation of the entrant is
Proof of Proposition 6
Since the continuation value after a deviation is null, the incumbents are willing to collude in period if and only if:
Proof of Proposition 7
The collusive and deviating profit of the entrant in period t can be written as follows:
Proof of Proposition 8
In period , the collusive and the deviation profits of an incumbent are, respectively, given by:
Proof of Lemma 4
Consider a period , with . The incumbents abide by the collusive agreement if and only if inequality  is satisfied:
Proof of Lemma 5
Proof of Proposition 9
According to Corollary 2, collusion is sustainable after entry if the ICC for the entrant to abide by the collusive agreement is satisfied in period , i.e. if:
Replacing , , above, we obtain for sustainability after entry and for sustainability before entry. Since , the proof is finished.□
Proof of Remark 4
We only need to show that .
In the scenario of discontinuance of collusion, the sum of the incumbents’ profits with the consumer surplus is
Proof of Proposition 10
If firms are price-setters, collusion is sustainable in period if the following ICC is satisfied:
Proof of Proposition 11
If firms are price-setters, the collusive agreement is sustainable in period if and only if:
Proof of Lemma 6
Substituting the expressions  and  for profits in ICC , we obtain
Proof of Proposition 12
Using Lemma 6, we know that collusion is sustainable for , if inequality  is satisfied. Expanding the sums, we obtain
This work was financed by FEDER, through the Operational Program for Competitiveness Factors (COMPETE), and by National Funds, through Fundação para a Ciência e a Tecnologia (FCT), through projects PTDC/IIM-ECO/5294/2012 and PEst-OE/EGE/UI4105/2014. Joana Pinho is also grateful to FCT for her post-doctoral scholarship (SFRH/BPD/79535/2011). We are grateful to Emilie Dargaud and two anonymous referees for their helpful comments and suggestions. We also thank participants in seminars at U. Porto, the 13th SAET Conference on Current Trends in Economics and the 40th EARIE Annual Conference in Rome.
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