Definition 1. We define a static capital system as a triple with counting measure where and is called the monetary constant, is a collection of subsets of such that elements of which are called owners, and is called the worth of for an owner
Note we are not requiring to be closed under any operations (i.e. it is not an algebra of sets). Suppose we have two structures on and Let (i.e. a multi-valued map into ). Such a function is called a trade (and may correspondingly be thought of as a change of ownership). We define the trade utility of a trade as a map by
Again, need not be in but we can of course still define the counting measure on it.
Definition 2. A composite trade is a map where and are trades.
Note that since it is defined on the image of simply evaluates on all sets in
Definition 3. Let be a continuum of static capital systems. We say is a capital system if
- for every and there is a unique trade
- (i.e. );
- if and are trades such that then for all
Example 4. A capital system is in a socialist state at time if for all We may further say is socialist during if is in a socialist state for all A capital system is in a communist state at time if Similarly we have the definition for communist during a set
Note that by this definition a communist state implies a socialist state. In the above regards, a communist state can be thought of as having a single owner (say, “the people”), and socialist state has owners with equal worth.
Definition 5. A dynamic capital system is a capital system where is a static capital system for all where and are comparable (in the inclusion sense) for all In particular the function defined by is called the monetary policy. If is strictly increasing during an interval, we say is expansionary during that interval. Similarly it is contractionary if it is strictly decreasing on some interval.
Definition 6. A dynamic capital system is rational if for all
Of course if is we have and thus the condition is satisfied for this case:
So in a rational dynamic capital system we have the inequality
with If exists and is finite, then the rational dynamic capital system is said to have an end game.