# We Are All Socialist

In Marxist theory (from where the terms socialism and communism definitively originate), we essentially have the following idea and sets of definitions.  A society consists of people, and people consist of things called property (with the second consist in the sense of ownership).  Communism refers to a society in which society itself owns all of the property;  that is, in a communist society, each individual owns all of the property.  Note that this is equivalent to definition 2a in [1] for if there is no private property, then for every property, there is no person who does not own it (otherwise it would have been private).  And conversely, if everyone owns everything, then there is no property such that some person doesn’t own it (i.e. there is no private property).

It follows that we define a representative government to be communist if and only if it is a government of a communist society.  This way if we think of the government as a subset of society, then if members of the government own all of the property (i.e. communist government) and the government is a representative government, then all of the members own all of the property via the representation.  And conversely (and trivially) if the society is communist, then all members, and hence also those in the government, own all of the property.

In reality, no society or government is truly communist;  one can always find something not owned by all members of the government or society.  For example, one could argue that any individual $A$ owns their thoughts, and no other individual $B\neq A$ owns the thoughts of $A.$  So this would be one trivial counter example.  A capital society is defined as a society that is not a communist society.  That is, in a capital society, it is not the case that every member owns all property.  Ownership in a society may certainly change over time.  If it is heading in the direction of communism, we say the society is socialist.  If it is heading away from communism, then we say the society is capitalist.  If it is neither heading toward or away from communism, then we will call it static.

The claim that we are all socialist boils down to the fact that there is much consensus on the desire of public services including police, fire, medical, and education.  We pay taxes for these entities that serve us as needed.  In this sense, we all own them.  And we always want to see them improved.  For these things to exist, we need a government to oversee them.  Continually wanting to see them improved translates to us wanting a say in how governmental money is distributed to them.  Since this money comes from other members of society (as taxation for example), this amounts to us wanting more ownership over what was formerly some other person’s property.  It is this sense in which we are all socialist.

Just take a look at the chart previously posted.  It shows that relative to where we are now (the actual distribution of wealth), we want the wealth to be different (what we would like it to be).  We want more ownership over how resources are distributed in society.

Also consider provisions in the Affordable Care Act once stripped from its colloquial term “Obamacare”, which has lately had much negative connotation.  These polls suggest that most Americans support having more control/ownership of insurance companies in the sense of declaring how they can and cannot operate [2], [3].

# 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.