Benefits of a cartel among firms Suppose I have $n&gt;2$ firms selling differentiated products. These firms form a cartel for the price. The cartel has size $n_c$. Let $\pi_{i,m}$ be the payoff of a firm $i$ outside the cartel and $\pi_{j,c}$ be the payoff of a firm $j$ inside the cartel. 

I would like to know whether there exists a set of assumptions (a reference to a paper in the literature describing that set of assumptions and relative proofs is sufficient) under which 

For any firm $j$ outside the cartel:

(i) entering the cartel is weakly convenient in terms of        profits for any $n_c$, i.e. $\pi_{j,m}(n_c-1)\leq \pi_{j,c}(n_c)$ $\forall n_c$

(ii) the higher is $n_c$ the higher is the profit increase from entering the cartel for any $n_c$, i.e. $\pi_{j,c}(n_c)-\pi_{j,m}(n_c-1)\leq \pi_{j,c}(n_c+1)-\pi_{j,m}(n_c)$ $\forall n_c$

For any firm $i$ inside the cartel:

(i) the profit is increasing in $n_c$, i.e. $\pi_{i,c}(n_c-1)\leq \pi_{i,c}(n_c)$ $\forall n_c$

(ii) the higher is $n_c$ the higher is the profit increase from letting someone else entering the cartel, i.e. $\pi_{i,c}(n_c)-\pi_{i,c}(n_c-1)\leq \pi_{i,c}(n_c+1)-\pi_{i,c}(n_c)$ $\forall n_c$

All inequalities could hold also strictly.
Cartel size and collusive stability with non-capitalistic players lists the reasons why it's highly unlikely that your question finds an answer for profit-seeking firms (see Friedman, 1971 for a threshold on cartel stability)


  It is widely accepted from both the theoretical IO literature (e.g., Tirole, 1988) and policy reports (e.g., Ivaldi et al., 2003) that high market concentration is a facilitating factor for (tacit as well as explicit) collusion. In addition to coordination being likely more difficult in larger groups, the intuition that the incentive to collusion shrinks with too many competitors is fairly simple: as the number of firms grows larger, the individual share of cartel profits shrinks monotonically and therefore implicit collusion becomes harder to sustain (cf. Tirole, 1988, ch. 6).


So you may find "toy" answers for $n=4$ or the like but probably not for reasonably big $n$.

However, the same paper also proves that


  An increase in cartel size makes implicit collusion among labour-managed firms easier to sustain.


which is exactly what you're stating in your question.

As the authors sum up,


  In words, the critical threshold for $LM$ firms is decreasing and convex in n and tends to zero as n becomes arbitrarily large. On the other hand, the critical threshold for profit-seeking firms is increasing and concave in n and tends to one as n becomes arbitrarily large. To complete the description of the critical thresholds w.r.t. n, we may also observe that


So according to this reference, an answer to the question is yes, on the condition that the firms studied are not profit-seeking but $LM$-firms.



Note: A $LM$-firms is an enterprise that operates under the ultimate control 
of those who work in it, which as for consequences, among others, that the firm aims at maximising profit per worker rather than profit.