# Capitalism

Definition 1.  We define a static capital system as a triple $(X,\Sigma,\mu)$ with counting measure $\mu$ where $\mu(X)=m$ and  is called the monetary constant, $\Sigma$ is a collection of subsets of $X$ such that $\mu(\cup_{A\in\Sigma}A)=m,$ elements of which are called owners, and $\mu(A)$ is called the worth of $A$ for an owner $A.$

Note we are not requiring $\Sigma$ to be closed under any operations (i.e. it is not an algebra of sets).  Suppose we have two structures on $X,$ $(X,\Sigma_1,\mu)$ and $(X,\Sigma_2,\mu).$   Let $f:\Sigma_1\to P(\Sigma_2)$ (i.e. a multi-valued map into $\Sigma_2$).  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 $f$ as a map $u_f:\Sigma_1\to\mathbb{Z}$ by

$\displaystyle u_f(A)=\mu\left(\bigcup f(A)\right)-\mu(A).$

Again, $\cup f(A)$ need not be in $\Sigma_2,$ but we can of course still define the counting measure on it.

Definition 2.  A composite trade is a map $g\circ f:\Sigma_1\to P(\Sigma_3)$ where $f:\Sigma_1\to P(\Sigma_2)$ and $g:\Sigma_2\to P(\Sigma_3)$ are trades.

Note that $g\,\circ:P(\Sigma_2)\to P(\Sigma_3)$ since it is defined on the image of $f.$  $g\,\circ$ simply evaluates $g$ on all sets in $f(A).$

Definition 3.  Let $(X,\Sigma_t,\mu)_{t\geq 0}$ be a continuum of static capital systems.  We say $(X,\Sigma_t,\mu)_{t\geq 0}$ is a capital system if

1. for every $t\geq 0$ and $\varepsilon\geq t$ there is a unique trade $f_{t,\varepsilon}:\Sigma_t\to P(\Sigma_{t+\varepsilon});$
2. $f_{t,0}=1$ (i.e. $f_{t,0}(A)=\{A\}$);
3. if $f_{t,\varepsilon_1}$ and $f_{\varepsilon_1,\varepsilon_2}$ are trades such that $\varepsilon_1+\varepsilon_2=\varepsilon,$ then $f_{t,\varepsilon}=f_{\varepsilon_1,\varepsilon_2}\circ f_{t,\varepsilon_1}$ for all $t,\varepsilon_1,\varepsilon_2\geq 0.$

Example 4.  A capital system is in a socialist state at time $t$ if $\mu(A)=\mu(B)$ for all $A,B\in\Sigma_t.$  We may further say $(X,\Sigma_t,\mu)_{t\geq 0}$ is socialist during $T\subseteq[0,\infty)$ if $(X,\Sigma_t,\mu)$ is in a socialist state for all $t\in T.$  A capital system is in a communist state at time $t$ if $\Sigma_t=\{X\}.$ Similarly we have the definition for communist during a set $T\subseteq[0,\infty).$

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 $(X_t,\Sigma_t,\mu)_{t\geq 0}$ where $(X_t,\Sigma_t,\mu)$ is a static capital system for all $t$ where $\mu(X_t)=m_t$ and $X_t,X_s$ are comparable (in the inclusion sense) for all $s,t\geq 0.$  In particular the function $m:[0,\infty)\to\mathbb{N}$ defined by $m(t)=m_t$ is called the monetary policy.  If $m_t$ is strictly increasing during an interval, we say $(X_t,\Sigma_t,\mu)$ is expansionary during that interval.  Similarly it is  contractionary if it is strictly decreasing on some interval.

Definition 6.  A dynamic capital system $(X_t,\Sigma_t,\mu)_{t\geq 0}$ is rational if $u_{f_{t,\varepsilon}}\geq 0$ for all $t,\varepsilon\geq 0.$

Of course if $\varepsilon$ is $0$ we have $f_{t,0}=1$ and thus the condition is satisfied for this case:

$\displaystyle u_{f_{t,0}}(A)=\mu\left(\bigcup \{A\}\right)-\mu(A)=0.$

So in a rational dynamic capital system we have the inequality

$\displaystyle\mu(A)\leq\mu\left(\bigcup f_{t,\varepsilon}(A)\right)\leq m_{t+\varepsilon}$

with $A\in\Sigma_t.$  If $\lim_{t\to\infty}m_t$ exists and is finite, then the rational dynamic capital system $(X_t,\Sigma_t,\mu)$ is said to have an end game.

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