Concentration among exploiters

The prompt is based on a paragraph from Marxist and Austrian Class Analysis by Hans-Hermann Hoppe.

Exploitation, defined as nonproductive and noncontractual property acquisitions (e.g. theft, extortion), is only possible as long as there is anything that can be appropriated. Yet if there were free competition in the business of exploitation, there would be nothing left to expropriate. Thus, exploitation requires monopoly over some given territory and population; and the competition between exploiters is by its very nature eliminative and must bring about a tendency toward relative concentration of exploitative firms as well as a tendency toward centralization within each exploitative firm. Attempt to make a formal model of exploitation and its tendency toward concentration and centralization that clarifies these conceptions.

Here is one minimal formalization that captures the logic you want. It is not the only possible model, but it makes each claim explicit and shows where the tendency toward concentration and centralization comes from.

1. Exploitation as a contest over an appropriable surplus

Take a territory \(j\) with population \(n_j\). The territory yields an appropriable surplus \(V_j>0\) per period. This is not newly produced by the exploiter; it is a transfer extracted from the population.

Suppose \(m\) exploiters try to seize the right to exploit that territory. Let exploiter \(i\) choose coercive effort \(e_i\ge 0\). A simple contest payoff is

\[ u_i(e_i\;e_{-i}) = \frac{e_i}{\sum_{k=1}^m e_k}V_j - e_i \]

This says the prize is the right to expropriate \(V_j\), and coercive effort is pure dissipation.

In the symmetric Nash equilibrium of this Tullock contest,

\[
e_i^*=\frac{m-1}{m^2}V_j,\qquad
u_i^*=\frac{V_j}{m^2},\qquad
\sum_{i=1}^m e_i^*=\frac{m-1}{m}V_j.
\]

So as \(m\to\infty\),

\[
u_i^*\to 0,\qquad \sum_i e_i^*\to V_j.
\]

That is the clean formal version of your first point: under free entry into the business of exploitation, the exploitable surplus is dissipated in the struggle over who gets to seize it. There may still be a gross surplus \(V_j\), but there is no net exploitation rent left. Positive exploitation profit therefore requires exclusion of rivals, which is to say: monopoly jurisdiction over a territory and population.

So monopoly is not accidental here. It is a condition for exploitation to be a positive-return activity.

---

2. Why rivalry among exploiters is eliminative

Now suppose the total territory/population is normalized to 1, and exploiter \(i\) controls share \(s_i\), with \(\sum_i s_i=1\).

Let the net rent from exclusive control of share \(s\) be

\[
\pi(s)=\alpha s - F - c s^\eta,
\qquad 0<\eta<1,\ \alpha,F,c>0.
\]

Interpretation:

• \(\alpha s\): gross expropriable flow from the controlled population.
• \(F\): fixed cost of maintaining a coercive apparatus.
• \(c s^\eta\) with \(0<\eta<1\): variable coercion/administration cost with economies of scale.

Because \(\eta<1\), coercive control is cheaper per unit at larger scale.

Now compare two exploiters controlling \(a\) and \(b\) separately with one exploiter controlling \(a+b\). The gain from combination is

\[
\Delta(a,b)=\pi(a+b)-\pi(a)-\pi(b)
=F+c\big(a^\eta+b^\eta-(a+b)^\eta\big).
\]

Since \(0<\eta<1\), \(x^\eta\) is concave, so

\[
a^\eta+b^\eta>(a+b)^\eta,
\]

hence

\[
\Delta(a,b)>0.
\]

So one larger exploitative firm is more profitable than two smaller ones controlling the same total population. This is already a concentration result: conquest, annexation, cartelization, or merger is privately profitable because it removes duplicate coercive fixed costs and exploits scale economies in domination.

That is why competition among exploiters is eliminative rather than price-competitive. The rival is not someone to undercut; he is an obstacle to be absorbed or destroyed.

---

3. A dynamic concentration law

To make the tendency explicit, define per-unit fitness as

\[
w(s)=\frac{\pi(s)}{s}
=\alpha-\frac{F}{s}-c s^{\eta-1}.
\]

Then let shares evolve by a replicator-style law:

\[ \dot s_i = s_i\left[w(s_i)-\bar w\right],\qquad \bar w = \sum_k s_k w(s_k) \]

This says exploiters with above-average net rent expand at the expense of below-average ones.

Now

\[
w'(s)=\frac{F}{s^2}+c(1-\eta)s^{\eta-2}>0.
\]

So larger exploiters have higher fitness. Once an exploiter becomes larger, he has lower average coercion cost and therefore grows faster still. That is the formal engine of concentration.

A compact way to see concentration rise is through the Herfindahl index

\[
H=\sum_i s_i^2.
\]

Differentiating,

\[
\dot H
=2\sum_i s_i^2\big(w(s_i)-\bar w\big)
=2,\mathrm{Cov}_s\big(s_i,w(s_i)\big).
\]

Because \(w(s)\) is increasing in \(s\), that covariance is positive whenever sizes differ. Hence

\[
\dot H>0
\]

except at knife-edge equal-size configurations. So concentration rises endogenously.

Under these assumptions, fragmentation is unstable and monopoly or near-monopoly is the stable endpoint.

---

4. Why each exploitative firm tends toward internal centralization

Now treat each exploitative firm itself as a small political order. A firm of size \(s\) cannot directly coerce everyone; it must delegate power to local agents. But those agents are themselves potential exploiters. If left autonomous, they skim, withhold remittances, or break away.

Let \(z\ge 0\) measure centralization: surveillance, bureaucracy, monopoly of arms inside the organization, standardization of commands, suppression of local autonomy.

Let internal leakage be

\[
L(s,z)=\lambda \frac{s^2}{z},
\qquad \lambda>0,
\]

and let the cost of centralization be

\[
K(z)=\kappa z,
\qquad \kappa>0.
\]

Then firm net rent is

\[
\Pi(s,z)=\alpha s - F - c s^\eta - \lambda \frac{s^2}{z} - \kappa z.
\]

The firm chooses \(z\) to maximize \(\Pi\). The first-order condition is

\[
\frac{\partial \Pi}{\partial z}
=\lambda \frac{s^2}{z^2}-\kappa=0,
\]

so the optimal degree of centralization is

\[
z^*(s)=s\sqrt{\frac{\lambda}{\kappa}}.
\]

Therefore

\[
\frac{dz^*}{ds}>0.
\]

This is the centralization result: as the exploitative firm grows, it becomes optimal to concentrate command more tightly at the center, because the danger posed by semi-autonomous sub-exploiters rises with scale.

So concentration across firms and centralization within firms come from the same logic. External rivals must be excluded to preserve net rents; internal rivals must be subordinated to preserve remittances.

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5. What the model shows, in plain terms

Under these assumptions:

1. Exploitation requires exclusive jurisdiction.
   Free entry into coercive appropriation dissipates the exploitation rent.

2. Competition among exploiters is eliminative.
   Since larger coercive units are more profitable than smaller ones, rivals are to be annexed, not merely competed with.

3. There is a built-in tendency toward concentration.
   Fixed costs and scale economies in coercion make larger exploiters fitter, so concentration rises over time.

4. There is a built-in tendency toward centralization.
   Delegated coercers become internal threats, so growing exploitative firms centralize command.

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6. The shortest possible summary

A compact formal statement is this:

• External problem: exploiters contest monopoly rights over a transferable surplus; free entry dissipates the rent.
• Size effect: once monopoly exists, coercive profit is superadditive in size.
• Dynamic result: larger exploiters grow faster than smaller ones.
• Internal problem: delegation creates secondary exploiters inside the organization; growth raises the value of central command.

That yields exactly the structure you described: monopoly as a condition of exploitation, elimination as the form of competition among exploiters, concentration across firms, and centralization within each firm.

I can also turn this into a more rigorous theorem-proof version, or into a continuous-time war-state model with explicit conquest probabilities.

I took it up on the first offer here.