Research/Appendix A
A Formal Model of Externality Correction under Producer Lobbying
The structural core of the Peace Dividend Mandate
Christopher Louis · Peace Economy InstituteORCID 0009-0008-4590-2792
Appendix A is bound into the working paper PDF and offered here on its own for readers who want only the model.
Abstract
One model of a defense contractor, a procuring government, and an accountable electorate produces four connected results. A linear Pigouvian charge fails against an amplified externality. A convex, conflict-indexed burden restores an interior optimum and can invert the producer’s lobbying incentive, including in the high-conflict region where a flat instrument points back toward war. Cost pass-through, which weakens the firm-level incentive, is the instrument that corrects the second, nested information asymmetry.
The active-conflict ratio is derived as a calibrated Pigouvian rate. The amplification assumption is examined against the empirical record. Scope conditions mark where each result depends on magnitudes not yet established. The propositions are offered for formal scrutiny and refutation. They are claims about structure, and structure is the only thing that can be proven in advance of implementation.
Try the model yourself.
Set the level of conflict, the pass-through share, and the ratio schedule, and watch Propositions A.1 to A.3 and Corollary A.5 recompute exactly.
Setup
H ≥ 0 is an index of the harm-generating activity the firm’s products sustain: scale and lethality of weapons output, and the conflict intensity that output underwrites. The firm’s operating margin m(H) is strictly increasing. The externality is amplified on an interval when m is strictly convex there, so the marginal profitability of harm itself rises with harm.
Social harm is D(H). The externality multiplier λ(H) is marginal external social cost per dollar of weapons spend. The Mandate imposes a burden B(H). A share φ of that burden is recovered in procurement prices. The firm retains m(H) − (1 − φ) B(H). The pass-through share φ is treated as a structural feature of procurement, not a free choice.
The propositions
Proposition A.1. If m is strictly convex and B(H) = τH, firm profit is strictly convex and its maximum on a compact interval is a corner. The firm chooses maximal harm whenever the average return to harm exceeds τ. Deterrence therefore requires a rate fixed by that average over the whole admissible range, not by local marginal damage. As the range grows, the required rate rises without bound whenever m′ does. The textbook instruction, set the rate equal to marginal damage, assumes a concave private objective and an interior optimum. Amplification removes both.
Proposition A.2. If B is twice differentiable and B″(H) > m″(H), profit is strictly concave, with a unique maximizer, interior under the stated boundary conditions, where m′ = B′. The dynamic ratio B(H) = ρ(H) s(H) generates that convexity when ρ′ > 0 and s′ ≥ 0, with ρ and s weakly convex and at least one of ρ″, s′, s″ strictly positive.
Proposition A.3. Lobbying direction is the sign of m′(H) − (1 − φ) B′(H). Inversion holds where the firm’s retained marginal burden exceeds its marginal margin. Under a constant charge, and with m′ rising, inversion is only an interval at the bottom of the range. It fails in the high-conflict region, which is where a static instrument points back toward war. A rising schedule can hold the inequality across the escalation range.
The schedule should not invert everywhere. Below a defensive baseline, retained marginal burden stays at or under marginal margin, so legitimate defense is not penalized at the margin. Above that baseline, it does not. The published ratio tiers are a discrete approximation to a schedule whose crossing sits at that baseline.
Proposition A.4 and Corollary A.5. With the government’s own payoff increasing in conflict capacity, a correction at only the lower layer leaves the government choosing H against an unobserved social cost. Transparency at only the upper layer leaves the contractor’s first-order condition untouched. Pass-through reaches both. As φ approaches zero, the firm bears the burden and inverts when B′ exceeds m′. As φ approaches one, the burden ceases to bear on the firm at all: its retained payoff no longer depends on B, its lobbying direction reverts to the uncorrected sign of m′, and the bite transfers to the government, whose payoff inverts when φ B′ exceeds g′. The same pass-through enters the public budget as a cost that rises with H.
Calibration, and what is not yet proved
Setting ρ(H) = λ(H) makes the dynamic burden a Pigouvian correction in ratio form. World military expenditure reached $2.89 trillion in 2025. The Institute for Economics and Peace put the economic impact of violence that year at $21.8 trillion, a gross ratio above seven to one. Two cautions keep the calibration conservative: the violence total is purchasing-power adjusted where the expenditure figure is not, and it counts military and security spending among its components. The external cost per dollar of spend therefore lies below the gross ratio. The working paper’s defensible range is five to seven to one. The active-conflict tier takes the lower bound, ρ ≈ 5.
Section A.6 examines the amplification assumption and weakens it against the record, from global convexity to amplification on the escalation range alone. Where even that is in doubt, the mechanism relocates rather than fails: high cost-plus insulation is high pass-through, and Corollary A.5 then carries the correction to the government and the public budget. Section A.7 marks each place a result depends on an empirical magnitude that has not been established. Those are the places a funder of the next paper, or a critic of this one, should look first. Section A.8 distinguishes a firm indifferent to the burden from a firm indifferent to conflict.
Suggested citation
Louis, Christopher. “Appendix A. A Formal Model of Externality Correction under Producer Lobbying.” In “The Peace Dividend Mandate: A Structural Framework for Incentivizing World Peace Through Market Mechanisms and Conflict Externality Pricing.” Working paper, Peace Economy Institute, September 2026.
© 2026 Christopher Louis / Peace Economy Institute. Released under a Creative Commons Attribution-NonCommercial license (CC BY-NC 4.0): shareable and adaptable for non-commercial purposes with attribution.