What is the major difference between the notion of rational expectations equilibrium and Nash equilibium? And why do we, only, have a rational expectation about the first moment (the price) in the former one? Do they coincide or they constitute a completely different framework? They seem so similar to me in some articles, that I can not distinguish their features if they are any!
Hi: They are pretty close to completely different. I don't know the least thing about game theory but a nash equilibrium describes what happens in a game when two or more people have a certain type of expectation. RE is one type of expectation but if you google for RE you'll see that it's a field in itself. It happens to have an application in game theory but it's a seperate field developed by macro-economists in the early 70's. RE founder could be viewed as Lucas or John Muth depending on who you ask. Muth came up with the concept in 1961. Lucas applied it to macro-economics in the early 70's and it took off from there.
I think you just need to compare model-specific definitions carefully, rather than vague notions.
Nash equilibirum (NE) typically is a strategy profile such that every player's strategy is actually a best response to other players' strategies, i.e. that no player in this equilibrium has any incentive to deviate.
Rational expectations equilibirum (REE) in macroeconomics is typically defined as a set of prices (actually price functions of some state variable) and agents' decisions (i.e. strategies in the language of game theory rather than decision theory) and perceived laws of motion for some state variables, such that given prices and these perceived laws of motions (i.e. expectations), agents maximize their respective objective functions, markets clear and, last but not least, the perceived law of motion coincides with the actual law of motion in this equilibrium (this is the actual law of motion).
Note that while NE concretely specifies the number of agents (i.e. players) involved (at least typically it's a finite number of players) in a game studied by game theory, REE can be more abstract as it typically involves representative households and representative firms, which means that the all the households and firms can be represented by one household and one firm. Additionally, in the latter there's this impersonal aspect of market equilibirum: magic happens and markets clear. Finally, there's this explicit intertemporal dimension in the latter involving the coincidence between perceived laws of motion and the actual ones. In contrast, NE doesn't involve any of these, but in case of imperfect information, expectations (beliefs) are explicitly taken into consideration in the notions of, say, weak perfect bayesian equilibirum, in which additionally implausible Nash equilibria are ruled out.
"And why do we, only, have a rational expectation about the first moment (the price) in the former one?"
I don't know what you exactly mean by the moment in this context, but in the model in which prices are flexible, price isn't even a state variable (if the states are minimally, parsimoniously defined), but, say, capital or technology can be the only state variables, so rational expectations need only primarily involve capital and technology rather than prices. But in the models with sticky prices, prices can actually be the state variables, so the expectations can primarily involve them.