The long run equilibrium of a perfectly competitive market is well established. My question is - are the concepts of a long run equilibrium in a perfect competition extendable (analogous or otherwise) to an oligopoly, specifically considering the the Cournot model and the Bertrand model.
Suppose I want to find the equilibrium number of firms $ N $ in a Cournot model, for example - will the approach be similar to that of a perfect competition?
The 'long run' assumption is not about whether the firms already on the market are price takers (perfect competition) or oligopolists but whether entry to the market is free. If entry to the market is free then in the long run profits tends toward zero, as a profitable market makes it tempting for more firms to enter.
If you have special asymmetric conditions where some entrants have different cost functions than the new ones the zero profit condition may not hold even in the perfect competition case.
Taking the Cournot duopoly model as an example, the model has a solution in the sense that there exists a single price in the market, and at that price the two quantity-setting firms produce a given output that maximizes their profits given the existence of the other firm.
This solution stands as long as preferences and costs/technology do not change, which are also assumptions of the perfect competition long-run equilibrium.
But, elaborating on @denesp point in another answer, the additional requirement to consider the model solution as a long-run equilibrium, is that the firms remain two in number - namely that there exists a total barrier-to-entry in the market. With this additional assumption, we can speak of a duopoly long-run equilibrium.
I reiterate the point made above: In oligopolies, we use a concept of game theory called Nash Equilibrium. If a game (in this case and oligopoly) is in Nash Eq., this means no firm has unilateral incentives to deviate. That means in a sense if nothing else changes, the equilibrium is sustainable in the long run. No different parameters, no different answers.
I would like to elaborate further on your question regarding Cournot and Bertrand. There is something called a Bertrand Paradox in Industrial Organization, which assuming products are homogeneous (that is, if when choosing between purchasing products between different firms the consumer only cares about prices, and no other characteristic of this good) than modeling firm competition choosing quantity (Cournot) and modeling them competing choosing price (Bertrand) will yield completely different equilibria. This had a lot of attention a while back in Economics, but people don't take it too seriously anymore because it requires some very strong assumptions (like if one company charges a penny less than the other, it captures a 100% of the market, so demand is perfectly elastic and consumers have perfect knowledge of prices and no transaction costs) it isn't taken too seriously anymore. Instead, many of the present day applications of the Bertrand model are applied to heterogeneous (other aspects of the good beyond price are taken into account by the consumer, like quality, or simply distinct tastes and preferences) but somewhat substitutable goods (thus there is a cross price elasticity component). Those also have a Nash Eq., and brings forth some really interesting results.
