Are you familiar with general equilibrium theory in microeconomics? And the Arrow–Debreu model although fundamental and crucial, has no room for money in value theory. Gerard Debreu himself said "An important and difficult question...is not answered by the approach taken here: the integration of money in the theory of value". Mathematically, the model can reach equilibrium without variables implying the use of money. Since price itself is really just a ratio. Currency doesn't have to be an element. The paradox of the use of metallic currency in trade has puzzled the classic economists since the beginning. Monetary trade involves one party to a transaction giving up something desirable for something useless (a metal token which one has no immediate use) in the hope of retrading it in an unforeseen future date. The problem became more puzzling when fiat money becomes the norm, money is essentially just information bits, hence why currencies evolved and take place everywhere even before the modern concept of central banks or any other financial institutes existed, or the link between debt and banknotes.Nazka231 wrote:This part: "Recent researches have told us lots of theories are very limited in when and where they can be applied. It's even possible to design a theory without currency at all (actually a lot of classic theories don't), and still have emerging behavior like economy cycles."
If you have information about that, I will be glad to read them.
With CGE (Computable General Equilibrium) models become the mainstream, and the forerunner of computational economics, researchers start to look into the very core of monetary theory, what is money, and why is there money? There have been many attempts through out the centuries, but more recently (since 1990s) we can now use fast simulation and experiments to explore this question with testable data instead of just mathematical equation. Ross M. Starr published a book called "Why is There Money?" just a few years back, exploring some of the issue and advancement from the perspective of General Equilibrium theory. (You probably have to check university libraries, since it's an academic publishing for the most part, and require some knowledge of monetary theory to understand it). Starr himself has a brief for students for his economic courses in his website, with less scary math you can check here. It contain a very interesting "bargain type" trading post economy system experiment based on A-D model where students can participate in which doesn't involved money. It also has plenty of references you can check if you are interested in the academic side. For more recent development (pass 2000s), you can check "Handbook of Computational Economics", with recent updated vol 3 in 2014. (probably still need access from university libraries I image).
In a general sense, what I meant is that since late 19th century to early 20th century where static mathematically models had dominate the economic theories where most economics textbook today still quote them regularly, but now since late 1990s, the mainstream economic researches have been shifted toward dynamic models where theories can be tested and experiment upon. Thus although many theories where finding equilibrium is still the keys for solving equations have given ways to computerized models where algorithm is more important, where they can be combined with field studies data, even big data to be verified. In the process, we've known the limitation of static models, and find the discrepancy in them toward newer dynamic models, where many old models are now considered to be special cases rather than general solutions.