Continued from the previous post.
Nice, yes, please make a more rigorous theorem-proof version.
Absolutely. Here is a more theorem-proof version.
I will state a minimal model, then prove four claims:
- exploitation rents are dissipated under free entry into coercive appropriation;
- exclusive jurisdiction is therefore necessary for positive exploitation profit;
- exploitative organizations have a structural tendency toward concentration;
- conditional on growth, each exploitative organization has a structural tendency toward internal centralization.
0. Environment
There is a unit mass of subjects spread over a unit territory. A firm can obtain revenue only by noncontractual transfer, not by production. So all revenue is a coercively appropriated transfer from subjects.
Let \(s\in[0,1]\) denote the measure of territory-population under a firm’s control.
Assume that under exclusive jurisdiction over size \(s\), a firm obtains gross transfer revenue
\[
R(s)=as,\qquad a>0.
\]
To maintain coercive control over size \(s\), it incurs operating cost
\[
C(s)=F+cs^\eta,\qquad F>0,\ c>0,\ 0<\eta<1.
\]
Hence operating profit under exclusive control is
\[
\pi(s)=R(s)-C(s)=as-F-cs^\eta.
\]
The assumption \(0<\eta<1\) is the scale-economy assumption: coercive cost rises with size, but less than proportionally.
1. Competition for a given territory dissipates rents
Fix one territory of size \(s\), with prize
\[
V=R(s)=as.
\]
There are \(m\ge 2\) potential exploiters. If they contest the territory, each chooses coercive effort \(e_i\ge 0\), and the probability that firm \(i\) wins exclusive jurisdiction is
\[
p_i(e)=\frac{e_i}{\sum_{k=1}^m e_k},
\]
with payoff
\[
u_i(e)=p_i(e)V-e_i.
\]
This is a pure transfer contest: effort does not create output, it only affects who gets to seize it.
Theorem 1 (contest rent dissipation)
For fixed \(m\), the contest has a unique symmetric Nash equilibrium with
\[
e_i^*=\frac{m-1}{m^2}V
\]
for every \(i\). Equilibrium expected profit is
\[
u_i^*=\frac{V}{m^2},
\]
and aggregate dissipation is
\[
\sum_{i=1}^m e_i^*=\frac{m-1}{m}V.
\]
Hence as \(m\to\infty\),
\[
u_i^*\to 0
\quad\text{and}\quad
\sum_i e_i^*\to V.
\]
Proof
In a symmetric profile \(e_i=e\) for all \(i\), player \(i\)'s payoff from deviation \(x\) is
\[
u_i(x;e)=\frac{x}{x+(m-1)e}V-x.
\]
FOC:
\[
\frac{(m-1)e}{(x+(m-1)e)^2}V-1=0.
\]
At a symmetric equilibrium, set \(x=e\). Then
\[
\frac{(m-1)e}{(me)^2}V=1
\quad\Longrightarrow\quad
e=\frac{m-1}{m^2}V.
\]
Second derivative is negative, so this is optimal. Then
\[
u_i^*=\frac{1}{m}V-\frac{m-1}{m^2}V=\frac{V}{m^2}.
\]
And
\[
\sum_i e_i^*=m\frac{m-1}{m^2}V=\frac{m-1}{m}V.
\]
Taking \(m\to\infty\) gives the limit. ∎
Corollary 1 (free entry eliminates net exploitation rents)
Suppose entry into the contest requires cost \(k>0\), and entrants choose whether to enter before the contest. Then free entry implies the equilibrium number of entrants \(m^*(k)\) satisfies
\[
\frac{V}{(m^*(k))^2}\approx k.
\]
So as \(k\downarrow 0\), \(m^*(k)\to\infty\), and expected net profit converges to \(0\).
Proof
By Theorem 1, post-contest gross expected profit per entrant is \(V/m^2\). Under free entry, entry continues until \(V/m^2\le k\), with equality up to integer rounding. As \(k\to 0\), the implied \(m\) diverges and net profit vanishes. ∎
Interpretation
This formalizes the claim that exploitation as a profitable activity requires restricted entry into coercive appropriation. If coercive appropriation itself is competitively open, the prize is dissipated in the struggle over who gets it.
So the relevant “market structure” of exploitation is not competitive supply of transfers, but exclusionary monopoly over a territory and population.
2. Concentration across exploiters
Now consider an industry of exploiters dividing total controlled size \(S\le 1\). Suppose this size is partitioned among \(n\) firms as \(s_1,\dots,s_n\), with
\[
\sum_{i=1}^n s_i=S.
\]
Theorem 2 (superadditivity of exploitative profit)
For any nontrivial partition \(s_1,\dots,s_n\) of \(S\),
\[
\pi(S)>\sum_{i=1}^n \pi(s_i)
\qquad\text{whenever }n\ge 2.
\]
Hence a single exploiter controlling all \(S\) is strictly more profitable than any fragmentation of that same total territory-population.
Proof
Compute
\[
\pi(S)-\sum_{i=1}^n \pi(s_i) = \big(aS-F-cS^\eta\big)-\sum_{i=1}^n \big(as_i-F-cs_i^\eta\big).
\]
Using \(\sum_i s_i=S\),
\[
\pi(S)-\sum_i \pi(s_i) = (n-1)F + c\left(\sum_{i=1}^n s_i^\eta - S^\eta\right).
\]
Since \(0<\eta<1\), the map \(x\mapsto x^\eta\) is strictly concave, so for any nontrivial partition,
\[
\sum_{i=1}^n s_i^\eta > \left(\sum_{i=1}^n s_i\right)^\eta = S^\eta.
\]
Also \((n-1)F>0\). Hence the difference is strictly positive. ∎
Corollary 2
Merger, conquest, annexation, cartelization under one command, or any other elimination of separate exploitative apparatuses is privately profitable for exploiters.
This gives a formal concentration logic: the total exploiter surplus is maximized by unification, not fragmentation.
3. A dynamic law of concentration
To move from comparative statics to dynamics, define per-unit fitness
\[
w(s)=\frac{\pi(s)}{s}=a-\frac{F}{s}-cs^{\eta-1}.
\]
Lemma 1
\(w'(s)>0\) for all \(s>0\).
Proof
Differentiate:
\[
w'(s)=\frac{F}{s^2}+c(1-\eta)s^{\eta-2}>0.
\]
Since \(F>0\), \(c>0\), and \(0<\eta<1\), both terms are strictly positive. ∎
Thus larger exploiters have strictly higher per-unit net return.
Now let exploiters’ shares evolve by a replicator law:
\[
\dot s_i=s_i\big(w(s_i)-\bar w\big),
\qquad
\bar w=\sum_{k} s_k w(s_k).
\]
This says firms with above-average per-unit exploitation return grow at the expense of those with below-average return.
Define concentration by the Herfindahl index
\[
H(s)=\sum_i s_i^2.
\]
Theorem 3 (concentration rises along the dynamic)
Along the replicator dynamic,
\[
\dot H = 2\sum_i s_i^2\big(w(s_i)-\bar w\big)
= 2,\mathrm{Cov}_s(s_i,w(s_i)).
\]
Because \(w\) is strictly increasing, \(\dot H\ge 0\), with strict inequality whenever at least two active firms have different sizes.
Proof
Differentiate:
\[
\dot H = 2\sum_i s_i \dot s_i
= 2\sum_i s_i^2\big(w(s_i)-\bar w\big).
\]
Since \(\sum_i s_i=1\), this is exactly the covariance of \(s_i\) and \(w(s_i)\) under the share weights \(s_i\). Because \(w\) is strictly increasing in \(s\), larger shares are paired with larger fitness. Therefore the covariance is nonnegative, and is strictly positive unless all positive-share firms have equal size. ∎
Theorem 4 (fragmented equal-share states are unstable; monopoly is stable)
Any rest point with \(k\ge 2\) active firms of equal size \(1/k\) is unstable. Each monopoly state \(s=(1,0,\dots,0)\) is locally asymptotically stable.
Proof
A \(k\)-firm equal-share state is a rest point because all active firms have the same fitness. To show instability, perturb two firms:
\[
s_1=\frac1k+\varepsilon,\qquad s_2=\frac1k-\varepsilon,
\]
with \(\varepsilon>0\) small, leaving others unchanged. Since \(w\) is strictly increasing,
\[
w(s_1)>w(s_2).
\]
Hence the larger firm has a higher growth rate than the smaller, so the perturbation expands rather than contracts. Therefore the equal-share state is unstable.
For a monopoly state, say \(s_1=1\), \(s_j=0\) for \(j\ge 2\), consider a nearby state with \(s_j>0\) small. Then for each small rival \(j\),
\[
\dot s_j=s_j\big(w(s_j)-\bar w\big).
\]
Since \(w(s)\) is increasing and \(s_j\ll 1\), one has \(w(s_j)<w(1)\), and \(\bar w\) is close to \(w(1)\). Hence \(\dot s_j<0\). Small rivals shrink, so the monopoly state is locally asymptotically stable. ∎
Interpretation
The theorems show not merely that concentration is profitable, but that under a simple evolutionary law it is dynamically selected. Competition among exploiters is eliminative because larger coercive units are fitter than smaller ones.
4. Centralization within each exploitative firm
Now take one exploiter of size \(s\). It must rule through subordinate coercive agents. Those agents can skim transfers, retain local control, or secede. Let \(z\ge 0\) denote the intensity of centralization: surveillance, standardization, command concentration, monopoly of force inside the organization, audit capacity, and suppression of local autonomy.
Write the firm’s problem as
\[
\max_{z\ge 0}\ \Pi(s,z)=\pi(s)-L(s,z)-K(z),
\]
where:
- \(L(s,z)\) is internal leakage/loss;
- \(K(z)\) is the resource cost of centralization.
Assume:
- \(K'(z)>0,\ K''(z)\ge 0\);
- \(L_s(s,z)>0\): larger domain creates more internal leakage risk;
- \(L_z(s,z)<0\): more centralization reduces leakage;
- \(L_{zz}(s,z)>0\): diminishing returns to centralization;
- \(L_{sz}(s,z)<0\): the marginal benefit of centralization is larger in larger organizations.
Condition 5 is the key “centralization tendency” assumption.
Theorem 5 (monotone comparative statics of centralization)
Under assumptions 1–5, the optimal centralization choice \(z^*(s)\) is nondecreasing in \(s\). If the optimum is interior and \(L_{sz}<0\) strictly, then \(z^*(s)\) is strictly increasing in \(s\).
Proof
The objective is
\[
\Pi(s,z)=\pi(s)-L(s,z)-K(z).
\]
Its cross-partial is
\[
\Pi_{sz}(s,z)=-L_{sz}(s,z)>0.
\]
So \(\Pi\) has increasing differences in \((s,z)\). By monotone comparative statics, the argmax correspondence \(z^*(s)\) is nondecreasing in \(s\). If the optimum is interior and the inequality is strict, then the first-order condition
\[
-K'(z)-L_z(s,z)=0
\]
implies that a rise in \(s\) raises the marginal benefit of \(z\), so the optimal \(z\) rises strictly. ∎
Parametric corollary
Take
\[
L(s,z)=\lambda \frac{s^\beta}{z},\qquad \beta>1,\ \lambda>0,
\qquad
K(z)=\kappa z,\qquad \kappa>0.
\]
Then
\[
\Pi(s,z)=as-F-cs^\eta-\lambda \frac{s^\beta}{z}-\kappa z.
\]
Corollary 3
The optimal centralization level is
\[
z^*(s)=\sqrt{\frac{\lambda}{\kappa}}, s^{\beta/2},
\]
so \(z^*(s)\) is strictly increasing in firm size.
Proof
FOC:
\[
\frac{\partial \Pi}{\partial z} = \lambda \frac{s^\beta}{z^2}-\kappa=0.
\]
Hence
\[
z^*(s)=\sqrt{\frac{\lambda}{\kappa}}, s^{\beta/2}.
\]
Since \(\beta>1\), this is strictly increasing in \(s\). ∎
Interpretation
Internal centralization is not an arbitrary managerial taste. It is the endogenous response to the fact that a large exploitative firm must itself govern subordinate exploiters.
So the same organization that eliminates external rivals also suppresses internal rivals.
5. Main synthesis theorem
We can now combine the pieces.
Theorem 6 (tendency toward concentration and centralization)
Suppose:
- coercive appropriation over any given territory is contestable as in Theorem 1;
- net operating profit under exclusive jurisdiction is \(\pi(s)=as-F-cs^\eta\) with \(0<\eta<1\);
- firm shares evolve by the replicator dynamic;
- internal centralization is chosen optimally as in Theorem 5.
Then:
- profitable exploitation requires exclusion of rival exploiters from each territory-population;
- the exploiter industry has a structural tendency toward concentration;
- along any path of concentration, each surviving exploiter has a structural tendency toward higher internal centralization.
Proof
(1) follows from Corollary 1: under free entry into coercive appropriation, exploitation rents are dissipated.
(2) follows from Theorems 2–4: unified coercive organizations are more profitable than fragmented ones, and the dynamic raises concentration over time.
(3) follows from Theorem 5: as each surviving exploiter grows in size, its optimal centralization rises. ∎
6. What has been formally shown
In theorem-proof terms, the argument is:
- Exploitation is a pure transfer prize.
- Free entry into contesting that prize dissipates the prize.
- Therefore positive exploitation profit requires monopoly jurisdiction.
- Given monopoly jurisdiction, coercive profit is superadditive in scale, so fragmentation is privately dominated by concentration.
- Under a simple growth law, larger exploiters outgrow smaller ones, so concentration is dynamically selected.
- As firms grow, the marginal value of central control rises, so internal centralization also increases.
That gives a clean formal statement of your original conception:
competition among exploiters is eliminative externally and centralizing internally.
The next natural step would be one of these two:
- a state-formation version, where firms are interpreted as proto-states and annexation is literal war;
- a general equilibrium version, where subjects reduce output when predation rises, so excessive exploitation can shrink the future tax base.
I can write either one in the same theorem-proof style.
