How John Nash Solved the Impossible Trinity

In the search for a sound basis of money one inevitably comes across the Impossible Trinity:

The Impossible trinity (also known as the Trilemma) is a trilemma in international economics which states that it is impossible to have all three of the following at the same time:

Although all three factors are favorable to society one cannot ensure 2 without excluding the possible of 1 of the trinity.

In Ideal Money John Nash makes an observation on axioms that seems relevant here:

I think there is a good analogy to mathematical theories like, for example, “class field theory”. In mathematics a set of axioms can be taken as a foundation and then an area for theoretical study is brought into being. For example, if one set of axioms is specified and accepted we have the theory of rings while if another set of axioms is the foundation we have the theory of Moufang loops.
So, from a critical point of view, the theory of macro-economics of the Keynesians is like the theory of plane geometry without the axiom of Euclid that was classically called the “parallel postulate”. (It is an interesting fact in the history of science that there was a time, before the nineteenth century, when mathematicians were speculating that this axiom or postulate was not necessary, that it should be derivable from the others.)
So I feel that the macroeconomics of the Keynesians is comparable to a scientific study of a mathematical area which is carried out with an insufficient set of axioms. And the result is analogous to the situation in plane geometry, the plane does not need to be really flat and the area within a circle can expand hyperbolically as a function of the radius rather than merely with the square of the radius. (This picture suggests the pattern of inflation that can result in a country, over extended time periods, when there is continually a certain amount of gradual inflation.)

This explanation/observation leads Nash to proclaim:

The missing axiom is simply an accepted axiom that the money being put into circulation by the central authorities should be so handled as to maintain, over long terms of time, a stable value.

The implications being that one can allow monetary policy to exist under proposed the scope of this problem, yet from a larger yet still relevant viewpoint Keynesian style banks can still be enticed (or forced) to enact monetary policy that tends towards a certain favorable type of stability. Exchange rate stability ensues as a product of a superior money in respect to each currency’s underlying basket of commodities it represents (without any persons actually knowing what those baskets consist of!).
This effectively addresses the Impossible Trinity.


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