**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**.