Arbiter — The Over-Automation Externality and the Case for an AI Tax

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=== CLAIMS (280) ===

  C1: If AI displaces human workers faster than the economy can reabsorb them, it risks eroding the very consumer demand firms
       [logical]

  C2: Knowing that AI displacement erodes consumer demand is not enough for firms to stop it.
       [logical]
       depends: C1

  C3: In a competitive task-based model, demand externalities trap rational firms in an automation arms race, displacing worke
       [logical] [FORMAL]

  C4: The resulting loss from over-automation harms both workers and firm owners.
       [logical]
       depends: C3

  C5: More competition and better AI amplify excess automation.
       [logical] [FORMAL]
       depends: C3

  C6: Wage adjustments and free entry cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C7: Capital income taxes cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C8: Worker equity participation cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C9: Universal basic income cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C10: Upskilling cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C11: Coasian bargaining cannot eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C12: Only a Pigouvian automation tax can eliminate the over-automation externality.
       [logical] [FORMAL]
       depends: C3

  C13: Policy should address not only the aftermath of AI labor displacement but also the competitive incentives that drive it.
       [logical]
       depends: C3, C12

  C14: The fear that technology will displace workers is at least as old as the Industrial Revolution.
       [empirical]

  C15: Historically, labor displacement has largely been self-correcting: automation of existing tasks has been offset by the c
       [empirical]

  C16: What Acemoglu and Restrepo call the reinstatement effect has tended to stabilize the labor market.
       [empirical]
       depends: C15

  C17: Whether the historical balance between displacement and reinstatement will hold in the age of AI is an open question.
       [empirical]
       depends: C15, C16

  C18: Displacement has intensified over the past four decades while the creation of new work has not always kept pace.
       [empirical]

  C19: Early signs suggest the current AI wave is disproportionately affecting entry-level workers.
       [empirical]

  C20: Displaced workers are also consumers, and when their lost income is not replaced, each round of layoffs erodes the purch
       [logical]

  C21: At the limit, firms automate their way to boundless productivity and zero demand.
       [logical]
       depends: C20

  C22: Public discourse increasingly treats the dynamic of vanishing demand from automation as an inevitable process with no na
       [empirical]

  C23: Rational, forward-looking firms should be the brake on over-automation if the consequences are visible to all.
       [logical]
       depends: C20

  C24: In February 2026, Block cut nearly half its 10,000-person workforce, with the CEO stating AI had made many roles unneces
       [empirical]

  C25: Over 100,000 tech workers were laid off in 2025 alone, with AI cited as a primary driver in more than half the cases.
       [empirical]

  C26: Roughly 80% of U.S. workers hold jobs with tasks susceptible to automation by large language models.
       [empirical]

  C27: Salesforce replaced 4,000 customer-support agents with agentic AI.
       [empirical]

  C28: Cognition's Devin, deployed at Goldman Sachs and Infosys, enables one senior engineer to do the work of a five-person te
       [empirical]

  C29: An automating firm captures the full cost saving but, under competitive pricing, bears only a fraction of the resulting 
       [logical] [FORMAL]

  C30: Each firm's profit-maximizing automation rate is a strictly dominant strategy that exceeds the cooperatively efficient l
       [logical] [FORMAL]
       depends: C29

  C31: Foresight alone cannot prevent the race toward the automation cliff.
       [logical]
       depends: C30

  C32: A monopolist fully internalizes the demand externality from automation.
       [logical] [FORMAL]

  C33: Fragmented markets exhibit the widest gap between equilibrium and cooperative automation rates.
       [logical] [FORMAL]
       depends: C32

  C34: In the frictionless limit, where every task is equally easy to automate, the game sharpens into a Prisoner's Dilemma in 
       [logical] [FORMAL]

  C35: In the Prisoner's Dilemma automation game, collective restraint would raise all profits.
       [logical] [FORMAL]
       depends: C34

  C36: The surplus loss from over-automation is not a transfer from workers to firm owners; it is a deadweight loss that harms 
       [logical]
       depends: C3, C4

  C37: Upskilling and worker equity participation narrow the over-automation wedge but cannot eliminate it.
       [logical] [FORMAL]

  C38: Because automation is a dominant strategy, no voluntary agreement among firms is self-enforcing.
       [logical] [FORMAL]
       depends: C30

  C39: Capital income taxes do not alter the equilibrium automation rate, operating on profit levels rather than the per-task m
       [logical] [FORMAL]

  C40: Universal basic income raises the floor on living standards but leaves the automation incentive unchanged.
       [logical] [FORMAL]

  C41: A Pigouvian automation tax, set equal to the uninternalized demand loss per task, implements the cooperative optimum.
       [logical] [FORMAL]

  C42: Revenue from a Pigouvian automation tax can fund retraining that raises income replacement, shrinking the externality ov
       [logical]
       depends: C41

  C43: Higher AI productivity widens the over-automation wedge rather than resolving it.
       [logical] [FORMAL]

  C44: Each firm perceives a market-share gain from automating beyond rivals, but at the symmetric equilibrium these gains canc
       [logical] [FORMAL]
       depends: C43

  C45: Better AI, far from mitigating the externality, amplifies it.
       [logical]
       depends: C43, C44

  C46: Endogenous wage adjustment raises the threshold at which the externality activates but cannot close the wedge once it do
       [logical] [FORMAL]

  C47: Wage flexibility changes when the problem bites, not whether it exists.
       [logical]
       depends: C46

  C48: Free entry, capital-income recycling, and richer product-market structures likewise fail to eliminate the distortion.
       [logical]

  C49: The authors' mechanism operates through the product market (firms ignore demand they destroy for rival firms), unlike Be
       [structural]

  C50: The authors' inefficiency requires competition and vanishes under monopoly, unlike Beraja and Zorzi's which arises even 
       [structural] [FORMAL]
       depends: C32

  C51: The authors' planner would reduce automation even with zero weight on workers, because over-automation harms firm profit
       [logical] [FORMAL]
       depends: C4

  C52: The technology ecosystem may be biased toward 'so-so' automation that displaces workers without large productivity gains
       [empirical]

  C53: Automation may disproportionately target high-rent tasks, dissipating worker surplus rather than raising output.
       [logical]

  C54: The demand externality in this paper arises only under competition and persists even when automation is highly productiv
       [logical]
       depends: C50, C51

  C55: The demand externality studied here belongs to the family of aggregate demand spillovers introduced by Rosenstein-Rodan 
       [structural]

  C56: In 'big push' models, demand complementarities can prevent individually unprofitable investments from being made even th
       [structural]
       depends: C55

  C57: The automation game yields a unique dominant-strategy equilibrium, making the problem a true externality rather than a c
       [logical] [FORMAL]
       depends: C30

  C58: Related work on automation and demand (Benzell et al. 2015, Korinek and Stiglitz 2019) does not model the strategic inte
       [structural]

  C59: AI systems deliver substantial productivity gains.
       [empirical]

  C60: AI pricing algorithms can spontaneously learn to collude.
       [empirical]

  C61: Firms under labor-issue scrutiny invest specifically in AI automation rather than other forms of IT.
       [empirical]

  C62: As AI reliability improves, incentivizing effective human oversight becomes prohibitively expensive.
       [empirical]

  C63: Each AI adoption decision is rational in isolation, but collectively they erode the consumer demand all firms depend on.
       [logical]
       depends: C29

  C64: The model considers a sector with N ≥ 2 symmetric firms.
       [structural] [FORMAL]

  C65: Each firm is endowed with L > 0 task-positions, initially all performed by human workers.
       [structural] [FORMAL]

  C66: Each firm chooses an automation rate αi ∈ [0,1]: tasks z ∈ [0, αi] are performed by AI at cost c per task, and tasks z ∈
       [structural] [FORMAL]
       depends: C65

  C67: Since each automated task displaces one worker, αi is simultaneously the automation rate and the fraction of the workfor
       [definitional] [FORMAL]
       depends: C66

  C68: Wages are exogenous in the baseline model.
       [structural]

  C69: In the perfect-substitutes limit of the CES task aggregator, each task produces one unit of output regardless of mode, s
       [structural] [FORMAL]

  C70: The output normalization shuts down productivity and quality margins so that the baseline captures only the spending con
       [structural]
       depends: C69

  C71: Tasks are ordered by comparative advantage, making the marginal task progressively harder to integrate, captured via a c
       [structural] [FORMAL]

  C72: The per-task cost saving from automation is defined as s := w − c.
       [definitional] [FORMAL]
       depends: C66

  C73: Workers have a higher marginal propensity to consume (MPC) than owners.
       [empirical]

  C74: Workers spend a fraction λ ∈ (0,1] of their income on the sector's good.
       [structural] [FORMAL]
       depends: C73

  C75: Owners spend none of their income in the sector in the baseline model.
       [structural] [FORMAL]

  C76: The MPC asymmetry implies that when automation displaces workers, income shifts toward agents with a lower sectoral MPC,
       [logical]
       depends: C73, C74, C75

  C77: A fraction η ∈ [0,1] of displaced wage income is replaced via reemployment, transfers, or other sources; the remainder i
       [structural] [FORMAL]

  C78: The effective demand loss per automated task is defined as ℓ = λ(1−η)w.
       [definitional] [FORMAL]
       depends: C74, C77

  C79: Demand falls linearly in the average automation rate: D = A + λwLN − ℓLN·ᾱ.
       [structural] [FORMAL]
       depends: C78

  C80: Firms sell their output on the product market at a uniform price that equates aggregate supply and demand.
       [structural] [FORMAL]

  C81: Firms play a one-shot simultaneous-move game, each choosing αi to maximize πi; the solution concept is Nash equilibrium.
       [structural] [FORMAL]

  C82: Over-automation is measured against two benchmarks: the cooperative optimum (maximizes aggregate owner surplus K) and a 
       [definitional] [FORMAL]

  C83: The environment assumes full transparency: every firm can directly observe how automation maps into lost worker income a
       [structural]

  C84: A marginal increase in automation saves s in labor costs but incurs friction kαi and reduces the firm's revenue by ℓ/N.
       [logical] [FORMAL]
       depends: C72, C78

  C85: The revenue loss is ℓ/N rather than ℓ because competitive pricing allocates revenue equally across symmetric firms.
       [logical] [FORMAL]
       depends: C80, C64

  C86: Each firm underestimates the social cost of its automation, suggesting systematic over-automation in equilibrium.
       [logical]
       depends: C84, C85

  C87: If N ≤ N* = ℓ/s, no firm automates (α^NE = 0).
       [logical] [FORMAL]
       depends: C78, C72

  C88: If N > N*, each firm's strictly dominant strategy is α^NE = min((s − ℓ/N)/k, 1).
       [logical] [FORMAL]
       depends: C87

  C89: The cooperative optimum is α^CO = min(max(0, (s − ℓ)/k), 1).
       [logical] [FORMAL]

  C90: When both α^NE and α^CO are interior, the over-automation wedge is ℓ(1 − 1/N)/k, strictly increasing in N and ℓ, and dec
       [logical] [FORMAL]
       depends: C88, C89

  C91: The equilibrium rate is a strictly dominant strategy: each firm over-automates even with perfect foresight about every r
       [logical] [FORMAL]
       depends: C88

  C92: The over-automation wedge is strictly increasing in N: more competitive sectors exhibit wider automation gaps.
       [logical] [FORMAL]
       depends: C90

  C93: More competition dilutes each firm's share of the demand loss, weakening the private incentive to restrain automation, r
       [logical]
       depends: C92

  C94: A monopolist (N=1) fully internalizes the externality (α^NE = α^CO); as N → ∞, the wedge approaches its maximum of ℓ/k.
       [logical] [FORMAL]
       depends: C90

  C95: As AI costs fall (c → 0), N* → λ(1−η) ≤ 1: the over-automation region expands to cover virtually any market with N ≥ 2.
       [logical] [FORMAL]
       depends: C87

  C96: For illustrative parameters (c/w = 0.30, λ = 0.5, η = 0.30, N → ∞), the wedge equals ℓ/k = α^CO: firms in competitive ma
       [logical] [FORMAL]
       depends: C90

  C97: When frictions vanish (k → 0), the game reduces to a Prisoner's Dilemma: full automation versus none.
       [logical] [FORMAL]

  C98: When k = 0, marginal profit becomes the constant L(s − ℓ/N), independent of the automation level, and the outcome is all
       [logical] [FORMAL]
       depends: C97

  C99: When k = 0 and N > N*, full automation (αi = 1) is strictly dominant for every firm.
       [logical] [FORMAL]
       depends: C98

  C100: When k = 0, N > N*, and s < ℓ, the cooperative optimum is no automation, and the equilibrium yields per-firm profit Π0 +
       [logical] [FORMAL]
       depends: C99

  C101: A firm that holds back unilaterally still suffers the revenue decline from rivals&#x27; automation but forgoes the offsetting
       [logical]
       depends: C99

  C102: Communication is cheap talk in the sense of Crawford and Sobel (1982): even if all firms acknowledge that collective res
       [logical]
       depends: C91

  C103: The automation externality is distinct from pure coordination failures (where firms simply need to agree on which equili
       [structural]
       depends: C57, C102

  C104: The μ-planner&#x27;s optimal automation rate is α^SP(μ) = (s − ℓ)/k − μℓ/[λ(1−μ)k], restricted to [0,1].
       [logical] [FORMAL]
       depends: C82

  C105: α^NE > α^SP(μ) for every μ ∈ [0,1). The Nash equilibrium is Pareto dominated by the cooperative optimum: workers and fir
       [logical] [FORMAL]
       depends: C88, C104

  C106: Workers lose wage income directly through displacement.
       [logical]
       depends: C67

  C107: Firm owners, despite cutting costs on each automated task, also lose: collective displacement erodes demand to the point
       [logical]
       depends: C105

  C108: No redistribution between workers and owners can make the Nash outcome efficient.
       [logical]
       depends: C105

  C109: The total over-automation wedge decomposes into two parts: (1) uninternalized demand externality ℓ(1−1/N)/k, present eve
       [logical] [FORMAL]
       depends: C90, C104

  C110: The externality component of the wedge grows with N, approaching ℓ/k as N → ∞, so fragmented markets suffer disproportio
       [logical] [FORMAL]
       depends: C109

  C111: The distributional premium is independent of N but grows without bound as μ → 1.
       [logical] [FORMAL]
       depends: C109

  C112: The surplus loss is quadratic in the total wedge and scales with NL, so both fragmentation and market size amplify the w
       [logical] [FORMAL]
       depends: C109

  C113: Only the Pigouvian automation tax fully corrects the distortion among six policy instruments evaluated; the remaining in
       [logical]
       depends: C12

  C114: An instrument that does not operate on the externality margin cannot correct the distortion regardless of how it scores 
       [logical]

  C115: The demand-loss parameter ℓ = λ(1−η)w governs the externality&#x27;s magnitude.
       [logical] [FORMAL]
       depends: C78

  C116: Higher η (income replacement rate) shrinks ℓ and thereby the over-automation wedge.
       [logical] [FORMAL]
       depends: C115, C90

  C117: When η > 1, upskilling and reabsorption place displaced workers into higher-paying roles, automation increases aggregate
       [logical] [FORMAL]
       depends: C116

  C118: The over-automation wedge (α^NE − α^CO) = ℓ(1−1/N)/k is maximized at η = 0, positive for all η < 1, zero when η = 1, and
       [logical] [FORMAL]
       depends: C90, C117

  C119: When η < 1, displacement destroys demand; when η > 1, displacement creates demand through higher reemployment wages. In 
       [logical] [FORMAL]
       depends: C118

  C120: The same corrective instrument addresses both over- and under-automation: a tax when η < 1, a subsidy when η > 1.
       [logical]
       depends: C118, C41

  C121: Historical technological transitions have often eventually reabsorbed displaced workers at higher wages.
       [empirical]

  C122: Past displacement episodes have consistently produced η < 1: displaced workers suffer large, persistent earnings losses.
       [empirical]

  C123: There is little evidence yet that AI-driven displacement will differ from past episodes in producing η < 1, placing most
       [empirical]
       depends: C122

  C124: Raising η through retraining programs, wage insurance, and incentives for new firm creation is not merely a palliative f
       [logical]
       depends: C116

  C125: A UBI funded from general revenue maps to an increase in autonomous demand A in the model.
       [definitional] [FORMAL]

  C126: UBI adds a constant to demand and enters firm profit only through Π0, which drops out of the first-order condition: UBI 
       [logical] [FORMAL]
       depends: C125

  C127: UBI alters neither the automation threshold N* nor the over-automation wedge.
       [logical] [FORMAL]
       depends: C126

  C128: In game theory terms, UBI changes payoff levels but not the payoff differences that drive strategic behavior.
       [logical]
       depends: C127

  C129: Instruments that operate on profit levels can redistribute income but cannot correct the externality; only instruments t
       [logical]
       depends: C128

  C130: When the number of firms is endogenous, UBI-induced entry can paradoxically widen the externality because it raises prof
       [logical]
       depends: C127, C92

  C131: UBI is a complement to the automation tax, not a substitute.
       [logical]
       depends: C127, C41

  C132: A proportional tax t ∈ (0,1) on capital income (profits) does not change the equilibrium automation rate because (1−t) c
       [logical] [FORMAL]

  C133: Capital income taxes are often conflated with robot taxes in the policy debate, but they are fundamentally different ins
       [logical]
       depends: C132

  C134: Robot taxes in the literature are per-unit levies on adoption which operate on the per-task margin, unlike proportional 
       [definitional]

  C135: Worker equity participation gives workers a direct stake in the profits that automation generates, and profit-sharing fl
       [structural]

  C136: The cooperative optimum is unchanged by worker equity: α^CO(ε) = (s − ℓ)/k, independent of ε.
       [logical] [FORMAL]
       depends: C89

  C137: The Nash equilibrium automation rate under worker equity is α^NE(ε) = (s − ℓ/Nε)/k, where Nε = N − λε(N−1).
       [logical] [FORMAL]
       depends: C136

  C138: The over-automation wedge under worker equity is strictly decreasing in ε but strictly positive for all ε < 1/λ. The wed
       [logical] [FORMAL]
       depends: C136, C137

  C139: Even at full profit-sharing (ε = 1), the wedge remains strictly positive when λ < 1, because each unit of profit recycle
       [logical] [FORMAL]
       depends: C138

  C140: The externality is fundamentally multilateral: each firm&#x27;s automation depresses demand for all N firms, and bilateral ar
       [logical]
       depends: C29

  C141: If each firm independently chooses its own profit-sharing rate to maximize retained profit, ε_i = 0 is a dominant strate
       [logical] [FORMAL]

  C142: The marginal cost of profit-sharing is πi (dollar-for-dollar reduction in retained earnings), while the marginal demand 
       [logical] [FORMAL]
       depends: C141

  C143: Profit-sharing must be mandated to have any effect and cannot substitute for a corrective tax.
       [logical]
       depends: C141, C138

  C144: Bargaining between a firm and its own workers cannot correct the automation externality because the uninternalized porti
       [logical]
       depends: C140

  C145: The externality is a firm-to-firm channel running through the product market, not a firm-to-worker channel that bilatera
       [logical]
       depends: C144

  C146: A coalition of M firms that jointly maximizes combined profit chooses α^M = (s − Mℓ/N)/k. The residual over-automation w
       [logical] [FORMAL]

  C147: Voluntary agreements among firms are not self-enforcing: in the frictionless limit, automation is strictly dominant, so 
       [logical] [FORMAL]
       depends: C99

  C148: The externality is multilateral and diffuse: each of N firms imposes demand losses on all N−1 others. Each firm&#x27;s indivi
       [logical]
       depends: C140

  C149: The automation rate αi is not contractible among firms: it is an internal organizational choice that rival firms cannot 
       [structural]

  C150: Automation decisions involve large sunk costs and are substantially irreversible, so even in a repeated setting, trigger
       [structural]

  C151: The fundamental obstacle to Coasian bargaining correcting the externality is incentive compatibility, not transaction co
       [logical]
       depends: C147, C148, C149, C150

  C152: A Pigouvian tax rate τ* = ℓ(1 − 1/N) implements α^NE = α^CO = (s − ℓ)/k.
       [logical] [FORMAL]
       depends: C41

  C153: The optimal Pigouvian tax rate has a transparent interpretation: each firm already bears ℓ/N of the demand loss; the tax
       [logical] [FORMAL]
       depends: C152

  C154: For large N, τ* ≈ ℓ = λ(1−η)w, so setting the rate requires only sector-level observables.
       [logical] [FORMAL]
       depends: C152

  C155: With lump-sum rebate of tax revenue, each firm achieves exactly the cooperative profit π^CO.
       [logical] [FORMAL]
       depends: C152

  C156: Because the welfare loss is quadratic in the wedge, even an imprecisely targeted tax yields a first-order gain.
       [logical] [FORMAL]
       depends: C112

  C157: Directing tax revenue toward retraining programs raises η through human-capital investment, making gains in η self-susta
       [logical]
       depends: C42, C116

  C158: The Pigouvian tax has the potential for a self-reinforcing dynamic: the tax funds programs that raise η, which lowers ℓ,
       [logical]
       depends: C157

  C159: The automation externality is robust to AI productivity gains, endogenous entry, endogenous wages, capital-income recycl
       [logical]

  C160: When AI-performed tasks produce ϕ ≥ 1 units of output (versus 1 for human), a deviating firm raises its output above riv
       [logical] [FORMAL]

  C161: The Nash equilibrium automation rate is increasing in AI productivity: α^NE(ϕ) > α^NE(1) for ϕ > 1.
       [logical] [FORMAL]
       depends: C160

  C162: The cooperative optimum and the generalized planner&#x27;s optimum are unchanged by AI productivity: α^CO(ϕ) = α^CO(1) for al
       [logical] [FORMAL]

  C163: The over-automation wedge is strictly increasing in ϕ for every μ ∈ [0,1]: better AI widens the distortion.
       [logical] [FORMAL]
       depends: C161, C162

  C164: The Red Queen mechanism: each firm perceives a market-share gain from automating beyond rivals, but at the symmetric equ
       [logical]
       depends: C44

  C165: Higher AI productivity raises output but lowers the price in proportion (total sectoral revenue equals total expenditure
       [logical] [FORMAL]
       depends: C162

  C166: The baseline Pigouvian rate τ* = ℓ(1−1/N) no longer suffices when ϕ > 1: implementing α^CO requires an additional correc
       [logical] [FORMAL]
       depends: C152, C163

  C167: In the frictionless case (k=0) with ℓ > s, three entry regimes arise: (i) low entry cost: all firms fully automate; (ii)
       [logical] [FORMAL]

  C168: The threat of automation functions as an endogenous entry barrier, sustaining positive profits without any automation ac
       [logical]
       depends: C167

  C169: Over-automation persists under free entry: if N^FE > N*, then α^NE(N^FE) > α^CO.
       [logical] [FORMAL]
       depends: C88

  C170: Free entry reshapes the over-automation problem but does not resolve it. The standard tendency toward excess entry (Mank
       [logical]
       depends: C169

  C171: UBI raising autonomous demand A attracts additional entrants until the zero-profit condition binds at a larger N^FE, par
       [logical]
       depends: C130

  C172: Endogenous wage adjustment can stabilize the automation path: as firms automate, displaced workers increase labor supply
       [logical]

  C173: Endogenous wage adjustment raises the threshold at which the externality activates but cannot close the wedge once it do
       [logical] [FORMAL]
       depends: C46

  C174: The threshold N* = ℓ/s rises as wages fall, because the cost saving s = w − c contracts faster than the demand loss ℓ = 
       [logical] [FORMAL]
       depends: C87

  C175: Competitive pricing allocates revenue as Rev_i = D/N at any wage level, so each firm bears only a fraction of the demand
       [logical] [FORMAL]
       depends: C85

  C176: The strongest version of the self-correcting argument (wages fall to near AI cost) is a Pyrrhic resolution: the external
       [logical]
       depends: C173

  C177: A labor market that &#x27;self-corrects&#x27; only by impoverishing its workforce has transmuted displacement into depressed livin
       [logical]
       depends: C176

  C178: Capital income recycling at rate η̂ narrows the over-automation wedge but cannot close it under empirically plausible pa
       [logical] [FORMAL]

  C179: When ℓ > s, the required owner recycling rate to eliminate the externality exceeds one, so recycling is impotent precise
       [logical] [FORMAL]
       depends: C178

  C180: Capital income recycling makes each firm behave as though it faced fewer competitors (N̂ interpolates between N and 1), 
       [logical] [FORMAL]
       depends: C178

  C181: Under second-stage price or quantity competition, the demand externality persists because neither strategic force change
       [logical]

  C182: Under Cournot competition, the market-share motive is partially offset because the firm that expands its market share al
       [logical]
       depends: C181

  C183: Under the general CES aggregator with elasticity of substitution greater than one, the demand externality still operates
       [logical]

  C184: When tasks are complements (elasticity less than one), the demand externality remains positive as long as displaced work
       [logical]

  C185: Richer modeling assumptions would change the magnitude of over-automation but not its source: the wedge persists as long
       [logical]
       depends: C181, C183, C184

  C186: No firm can afford to be the one that holds back in the automation arms race.
       [logical]
       depends: C91

  C187: The over-automation problem is most severe in fragmented industries deploying the most capable AI, not in dominant techn
       [logical]
       depends: C92, C163

  C188: The distinguishing empirical signature of the externality would be profit erosion coinciding with mass layoffs, which st
       [empirical]
       depends: C107

  C189: The empirical signature requires displacement at a scale and speed beyond what has materialized so far.
       [empirical]
       depends: C188

  C190: If reabsorption keeps pace with automation, the externality may remain too small to detect, and the paper&#x27;s contribution
       [logical]
       depends: C189

  C191: Customer support, software services, and back-office operations across competing financial institutions are three settin
       [empirical]

  C192: Private returns to AI systematically overstate the returns to the economy as a whole.
       [logical]
       depends: C63

  C193: Even a planner who places zero weight on worker welfare would reduce the automation rate below the equilibrium level.
       [logical] [FORMAL]
       depends: C105

  C194: The problem is not that firms profit at workers&#x27; expense; it is that over-automation harms both groups, making correctio
       [logical]
       depends: C36

  C195: By Tinbergen&#x27;s principle, a distinct market failure requires a distinct instrument; only a Pigouvian automation tax supp
       [logical]
       depends: C12

  C196: No amount of retraining, income support, or bargaining will slow the automation arms race; only a tax on automation itse
       [logical]
       depends: C113

  C197: A unilateral automation tax could push adoption offshore, strengthening the case for multilateral coordination or border
       [logical]
       depends: C152

  C198: The model is deliberately simple: one sector, one period, symmetric firms. Each of these choices is conservative, meanin
       [structural]

  C199: A single sector understates the externality: in a multi-sector economy, layoffs in one sector reduce spending on every s
       [logical]
       depends: C198

  C200: When a platform automates seller support, gig logistics, or content moderation, the lost spending cascades across an ent
       [logical]
       depends: C199

  C201: AI investments are largely irreversible, and even the threat of automation can reshape market structure before any displ
       [logical]
       depends: C150, C168

  C202: The income-replacement rate η rises over time as displaced workers retrain and new occupations emerge, so the optimal ta
       [logical]
       depends: C158

  C203: Firms racing to automate may invest disproportionately in labor-replacing AI rather than labor-augmenting AI, feeding th
       [logical]

  C204: Each extension considered points in the direction of a larger problem, not a smaller one.
       [logical]
       depends: C198, C199, C201, C203

  C205: Anthropic CEO Dario Amodei has warned that AI-driven displacement will be &#x27;unusually painful,&#x27; &#x27;much broader&#x27; and &#x27;much 
       [empirical]

  C206: The profit function πi is strictly concave in αi (second derivative is −L(k + ℓ/N) < 0).
       [logical] [FORMAL]

  C207: The optimal αi does not depend on rivals&#x27; choices α_{-i} because rivals&#x27; automation levels enter only through an additiv
       [logical] [FORMAL]
       depends: C206

  C208: At any fixed average automation rate, the sum of squared automation rates is minimized when all rates are equal (by QM-A
       [logical] [FORMAL]

  C209: In numerical parameterizations, N^FE exceeds N* in over 94% of cases satisfying the proposition&#x27;s conditions.
       [empirical]
       depends: C169

  C210: Under the Pigouvian automation tax τ, the firm&#x27;s first-order condition yields α^NE(τ) = (s − τ − ℓ/N)/k.
       [logical] [FORMAL]

  C211: Setting α^NE(τ) equal to α^CO = (s − ℓ)/k yields the optimal tax τ* = ℓ(1 − 1/N).
       [logical] [FORMAL]
       depends: C210

  C212: At τ = τ*, all firms choose α^CO and per-firm profit under the tax is π^tax = π^CO − τ*Lα^CO.
       [logical] [FORMAL]
       depends: C211

  C213: Total tax revenue τ*LNα^CO, when rebated equally, restores each firm&#x27;s profit to π^CO.
       [logical] [FORMAL]
       depends: C212

  C214: When ϕ > 1, the firm&#x27;s first-order condition equates the marginal benefit of automation to its marginal cost: ∂Rev_i/∂α_
       [logical] [FORMAL]

  C215: The LHS(α) = kα is strictly increasing in α (slope k).
       [logical] [FORMAL]

  C216: The RHS(α) of the symmetric first-order condition is strictly decreasing because the market-share term has numerator pro
       [logical] [FORMAL]

  C217: The equation LHS = RHS has a unique solution because LHS is strictly increasing and RHS is strictly decreasing.
       [logical] [FORMAL]
       depends: C215, C216

  C218: α^NE(ϕ) > α^NE(1) when ϕ > 1, because evaluating at the baseline equilibrium α^NE(1) shows RHS exceeds LHS, and the uniq
       [logical] [FORMAL]
       depends: C217

  C219: Total revenue equals aggregate demand D regardless of how output is allocated across firms, because expenditure D is pin
       [structural]

  C220: The cooperative planner&#x27;s first-order condition depends only on costs, not on ϕ.
       [logical] [FORMAL]
       depends: C219

  C221: α^CO(ϕ) = (s − ℓ)/k = α^CO(1), meaning the cooperative optimum is independent of ϕ.
       [logical] [FORMAL]
       depends: C220

  C222: Worker income W = wLN[1 − (1 − η)ᾱ] does not depend on ϕ.
       [logical] [FORMAL]

  C223: Owner surplus K = D − Σ_i C_i at symmetric profiles, and neither D(2) nor C_i(1) depends on ϕ.
       [logical] [FORMAL]

  C224: S(μ) is ϕ-invariant at every symmetric ᾱ, meaning α^SP(μ; ϕ) = α^SP(μ; 1) for all μ.
       [logical] [FORMAL]
       depends: C222, C223

  C225: The over-automation wedge α^NE(ϕ) − α^SP(μ; ϕ) is strictly larger than α^NE(1) − α^SP(μ; 1) for every μ.
       [logical] [FORMAL]
       depends: C218, C224

  C226: α^NE(ϕ) is increasing in ϕ, so the over-automation wedge is strictly increasing in ϕ.
       [logical] [FORMAL]
       depends: C225

  C227: Proposition 7 assumes k = 0 (frictionless case), λ = 1 (full recycling), 0 < κ < A (entry is costly but the market is vi
       [structural] [FORMAL]

  C228: For N ≤ N*, Corollary 1 gives α = 0, and per-firm profit Π*(N) = A/N, which is strictly decreasing.
       [logical] [FORMAL]
       depends: C227

  C229: For N > N*, full automation is dominant and per-firm profit drops by Δ = L(ℓ − s) > 0, giving Π*(N) = A/N − Δ, also stri
       [logical] [FORMAL]
       depends: C227

  C230: At the crossing point, Π*(m) > Π*(m + 1) where m = ⌊N*⌋, confirming the profit schedule is strictly decreasing on N.
       [logical] [FORMAL]
       depends: C228, C229

  C231: Since κ < A, Π*(1) = A > κ, and since Δ > 0, Π*(N) → −Δ < 0 as N → ∞.
       [logical] [FORMAL]
       depends: C227, C229

  C232: The set S = {N ∈ ℕ : Π*(N) ≥ κ} is nonempty and finite.
       [logical] [FORMAL]
       depends: C231

  C233: The free-entry equilibrium N^FE = max S exists, is unique, and satisfies condition (11).
       [logical] [FORMAL]
       depends: C232, C230

  C234: On the no-automation branch (N ≤ m), Π*(N) = A/N ≥ κ if and only if N ≤ A/κ.
       [logical] [FORMAL]
       depends: C228

  C235: On the full-automation branch (N ≥ m + 1), Π*(N) = A/N − Δ ≥ κ if and only if N ≤ A/(κ + Δ).
       [logical] [FORMAL]
       depends: C229

  C236: Case (i) Low entry cost: when κ + Δ ≤ A/(m + 1), N^FE = ⌊A/(κ + Δ)⌋ ≥ m + 1 > N*, and every firm fully automates.
       [logical] [FORMAL]
       depends: C234, C235

  C237: Case (ii) Intermediate entry cost: when κ + Δ > A/(m + 1) and κ < A/m, N^FE = m = ⌊N*⌋, and no firm automates because no
       [logical] [FORMAL]
       depends: C234, C235

  C238: In the intermediate entry cost case, the threat of automation deters the marginal entrant, sustaining positive profits w
       [logical]
       depends: C237

  C239: Case (iii) High entry cost: when κ ≥ A/m, N^FE = ⌊A/κ⌋ ≤ m ≤ N*, and no firm automates.
       [logical] [FORMAL]
       depends: C234, C235

  C240: In the high entry cost case, entry costs alone limit competition and the automation threshold is never approached.
       [logical]
       depends: C239

  C241: The three cases (low, intermediate, high entry cost) exhaust all κ ∈ (0, A), and when κ > A, N^FE = 0.
       [logical] [FORMAL]
       depends: C236, C237, C239

  C242: Per-firm profit follows A/N for N ≤ N* (no automation) and drops discretely by Δ = L(ℓ − s) at N* when full automation b
       [structural]
       depends: C228, C229

  C243: In the proof of Proposition 8, the function g(·) is decreasing in α^NE because ℓ > s implies the derivative is always ne
       [logical] [FORMAL]

  C244: By Proposition 1, α^NE is non-decreasing in N.
       [logical] [FORMAL]

  C245: g(α^NE(N)) is non-increasing in N because g is decreasing in α^NE and α^NE is non-decreasing in N.
       [logical] [FORMAL]
       depends: C243, C244

  C246: π^NE(N) = A/N + C + g(N) is strictly decreasing in N, since A/N is decreasing and g(α^NE(N)) is non-increasing in N.
       [logical] [FORMAL]
       depends: C245

  C247: The constant C = (λ − 1)wL < 0.
       [logical] [FORMAL]

  C248: lim_{N→∞} π^NE < 0 < κ, ensuring the viable set S is finite.
       [logical] [FORMAL]
       depends: C247

  C249: If N^FE > N*, then α^NE(N^FE) = min{(s − ℓ/N^FE)/k, 1} > 0, and since ℓ > s implies α^CO = 0, over-automation persists i
       [logical] [FORMAL]
       depends: C246, C248

  C250: The symmetric equilibrium with endogenous wages is a fixed point: ᾱ such that ᾱ = α^NE(w(ᾱ)).
       [definitional] [FORMAL]

  C251: The planner&#x27;s per-firm marginal benefit of automation uses s = w − c and ℓ = λ(1 − η)w.
       [definitional] [FORMAL]

  C252: The private marginal benefit g&#x27;(α) = w&#x27;(α)[1 − λ(1 − η)] − k has first term weakly negative and k > 0, so g&#x27; < 0, meanin
       [logical] [FORMAL]

  C253: g(α) = h(α) − ℓ(α)(1 − 1/N) for all α, where h is the private marginal benefit function.
       [logical] [FORMAL]

  C254: At the Nash equilibrium, h(α^NE) = 0 by definition, so g(α^NE) = −ℓ(α^NE)(1 − 1/N) < 0.
       [logical] [FORMAL]
       depends: C253

  C255: Since g is strictly decreasing and g(α^CO) = 0 (the planner&#x27;s optimality condition), g(α^NE) < 0 = g(α^CO) implies α^CO 
       [logical] [FORMAL]
       depends: C252, C254

  C256: The automation threshold N*(w) = λ(1 − η)w/(w − c) is strictly decreasing in w.
       [logical] [FORMAL]

  C257: Since w&#x27;(ᾱ) ≤ 0 by assumption, w(ᾱ) ≤ w(0) for all ᾱ ≥ 0, and therefore N*(w(ᾱ)) ≥ N*(w(0)).
       [logical] [FORMAL]
       depends: C256

  C258: Endogenous wage adjustment raises the automation threshold N*, meaning that with wage depression, more firms are needed 
       [logical]
       depends: C257

  C259: Under the generalized planner with wage adjustment (Corollary 4), if gμ(α) := s(w(α)) − ℓ(w(α))[1 + μ/(λ(1 − μ))] − kα i
       [logical] [FORMAL]

  C260: At the Nash equilibrium, gμ(α^NE) = −ℓ(1 − 1/N) − μℓ/(Nλ(1 − μ)) < 0 since ℓ > 0, N ≥ 2, and μ > 0.
       [logical] [FORMAL]

  C261: Cμ := 1 − (1 − η)[λ + μ/(1 − μ)] is the coefficient determining whether gμ is strictly decreasing.
       [definitional] [FORMAL]

  C262: At μ = 0, C₀ = 1 − λ(1 − η) ≥ 0, which is the coefficient used in the proof of Proposition 9.
       [logical] [FORMAL]
       depends: C261

  C263: For μ > 0, Cμ decreases. As long as Cμ ≥ 0, the same monotonicity argument applies, which holds for all μ ≤ μ̄ := [1 − (
       [logical] [FORMAL]
       depends: C261, C262

  C264: μ̄ is approximately 0.48 at λ = 0.5, η = 0.30.
       [empirical] [FORMAL]
       depends: C263

  C265: When Cμ < 0, the product w&#x27;Cμ ≥ 0, so gμ&#x27; < 0 requires k > |w&#x27;(α)| · |Cμ|: integration frictions must dominate wage sens
       [logical] [FORMAL]
       depends: C261

  C266: In the numerical illustration with w(ᾱ) = 1 − 0.5ᾱ, c = 0.30, λ = 0.5, η = 0.30, k = 1, N = 7, and μ = 0.3, the equilibr
       [empirical]

  C267: The numerical illustration confirms α^SP < α^CO < α^NE.
       [empirical]
       depends: C266

  C268: The distributional premium is substantial: the μ-planner would reduce automation to near zero, well below the cooperativ
       [empirical]
       depends: C267

  C269: Aggregate demand with capital income recycling is D = A + λwLN − ℓLNᾱ + ηΠ, where Π is total profit.
       [definitional] [FORMAL]

  C270: Total profit is Π = D − NL(w − sᾱ) − (k/2)L Σ_j α_j².
       [definitional] [FORMAL]

  C271: With capital income recycling, ℓ̂η = ℓ − ηs. When k = 0 this gives equation (12).
       [definitional] [FORMAL]
       depends: C269, C270

  C272: The firm&#x27;s marginal profit with capital income recycling depends only on α_i, so the equilibrium is in strictly dominant
       [logical] [FORMAL]

  C273: N̂ := N(1 − η̂) + η̂ is the effective number of firms under capital income recycling.
       [definitional] [FORMAL]

  C274: For k = 0 with capital income recycling, the marginal profit is L(s − ℓ̂η/[N(1 − η̂)]), a constant independent of α_i.
       [logical] [FORMAL]
       depends: C271

  C275: Full automation is strictly dominant when N > Nη̂ and no automation is strictly dominant when N < Nη̂, where Nη̂ = ℓ̂η/(
       [logical] [FORMAL]
       depends: C274

  C276: ℓ̂η = ℓ − ηs ≤ 0 if and only if η ≥ ℓ/s.
       [logical] [FORMAL]
       depends: C271

  C277: When η ≥ ℓ/s, capital income recycling fully neutralizes the demand externality (ℓ̂η ≤ 0).
       [logical] [FORMAL]
       depends: C276

  C278: For k > 0 with capital income recycling, the first-order condition is positive if and only if N̂ > N*, reproducing Propo
       [logical] [FORMAL]
       depends: C273

  C279: At a symmetric profile with capital income recycling, total profit has a 1/(1 − η̂) multiplier that scales the objective
       [logical] [FORMAL]

  C280: Capital income recycling does not change the cooperative optimum α^CO but does change the Nash equilibrium through the e
       [logical]
       depends: C278, C279


=== CONTRADICTIONS (18) ===

  [TENSION]
    A: C121: Historical technological transitions have often eventually reabsorbed displaced workers at hig
    B: C122: Past displacement episodes have consistently produced η < 1: displaced workers suffer large, p
    → If displaced workers are eventually reabsorbed at higher wages (suggesting η > 1), this conflicts with the claim that pa

  [TENSION]
    A: C59: AI systems deliver substantial productivity gains.
    B: C52: The technology ecosystem may be biased toward &#x27;so-so&#x27; automation that displaces workers without
    → One claim asserts AI delivers substantial productivity gains while the other suggests automation may proceed without lar

  [TENSION]
    A: C23: Rational, forward-looking firms should be the brake on over-automation if the consequences are 
    B: C31: Foresight alone cannot prevent the race toward the automation cliff.
    → C23 suggests that rational foresight should restrain firms from over-automating, while C31 concludes that foresight alon

  [TENSION]
    A: C15: Historically, labor displacement has largely been self-correcting: automation of existing tasks
    B: C18: Displacement has intensified over the past four decades while the creation of new work has not 
    → C15 characterizes displacement as largely self-correcting historically, while C18 says that over the past four decades, 

  [TENSION]
    A: C124: Raising η through retraining programs, wage insurance, and incentives for new firm creation is
    B: C196: No amount of retraining, income support, or bargaining will slow the automation arms race; onl
    → C124 says retraining is a &#x27;direct lever on the externality&#x27; (since it raises η and lowers ℓ), but C196 categorically sta

  [TENSION]
    A: C40: Universal basic income raises the floor on living standards but leaves the automation incentive
    B: C130: When the number of firms is endogenous, UBI-induced entry can paradoxically widen the external
    → C40 says UBI leaves the automation incentive &#x27;unchanged,&#x27; while C130 says UBI can &#x27;paradoxically widen the externality&#x27; 

  [TENSION]
    A: C59: AI systems deliver substantial productivity gains.
    B: C69: In the perfect-substitutes limit of the CES task aggregator, each task produces one unit of out
    → The empirical claim asserts AI delivers substantial productivity gains, but the baseline model&#x27;s assumption normalizes a

  [TENSION]
    A: C198: The model is deliberately simple; each of these choices is conservative, meaning the real prob
    B: C190: If reabsorption keeps pace with automation, the externality may remain too small to detect, an
    → C198 claims the real problem is &#x27;likely worse&#x27; than the model shows, while C190 concedes the externality &#x27;may remain too

  [AMBIGUITY]
    A: C21: At the limit, firms automate their way to boundless productivity and zero demand.
    B: C69: In the perfect-substitutes limit of the CES task aggregator, each task produces one unit of out
    → C21 invokes &#x27;boundless productivity&#x27; at the limit, but the baseline model (C69) holds output constant at L regardless of

  [TENSION]
    A: C172: Endogenous wage adjustment can stabilize the automation path: as firms automate, displaced wor
    B: C176: The strongest version of the self-correcting argument (wages fall to near AI cost) is a Pyrrhi
    → C172 presents wage adjustment positively as a stabilizing force, while C176 characterizes the same mechanism as &#x27;Pyrrhic

  [AMBIGUITY] [Z3-encodable]
    A: C247: The constant C = (λ − 1)wL < 0.
    B: C74: Workers spend a fraction λ ∈ (0,1] of their income on the sector&#x27;s good.
    → C247 asserts C < 0, which requires λ < 1 strictly. But C74 defines λ ∈ (0,1], allowing λ = 1 as a boundary case. If λ = 

  [TENSION] [Z3-encodable]
    A: C93: More competition dilutes each firm&#x27;s share of the demand loss, weakening the private incentive 
    B: C92: The over-automation wedge is strictly increasing in N: more competitive sectors exhibit wider a
    → While these two claims are internally consistent with each other, they jointly create a tension with standard economic i

  [AMBIGUITY] [Z3-encodable]
    A: C276: ℓ̂η = ℓ − η̂s ≤ 0 if and only if η ≥ ℓ/s.
    B: C271: With capital income recycling, ℓ̂η = ℓ − η̂s.
    → C271 defines the parameter using η̂ (the capital income recycling rate), but C276 states the condition using η (which th

  [AMBIGUITY] [Z3-encodable]
    A: C10: Upskilling cannot eliminate the over-automation externality.
    B: C117: When η > 1, upskilling and reabsorption place displaced workers into higher-paying roles, auto
    → C10 says upskilling cannot eliminate the externality, yet C117 shows that if upskilling achieves η > 1, the original ove

  [TENSION] [Z3-encodable]
    A: C12: Only a Pigouvian automation tax can eliminate the over-automation externality.
    B: C117: When η > 1, upskilling and reabsorption place displaced workers into higher-paying roles, auto
    → C12 claims only a Pigouvian tax can eliminate the externality, but C117 shows that achieving η > 1 through upskilling el

  [TENSION]
    A: C36: The surplus loss from over-automation is not a transfer from workers to firm owners; it is a de
    B: C53: Automation may disproportionately target high-rent tasks, dissipating worker surplus rather tha
    → C36 characterizes over-automation as a deadweight loss harming both workers and owners equally (not a transfer), while C

  [AMBIGUITY]
    A: C83: The environment assumes full transparency: every firm can directly observe how automation maps 
    B: C149: The automation rate αi is not contractible among firms: it is an internal organizational choic
    → C83 says firms have full transparency about the consequences of automation, while C149 says automation rates are interna

  [TENSION]
    A: C15: Historically, labor displacement has largely been self-correcting.
    B: C122: Past displacement episodes have consistently produced η < 1: displaced workers suffer large, p
    → If displacement is &#x27;largely self-correcting,&#x27; one would expect displaced workers to recover their earnings (η ≈ 1 or hig


=== KEY TERMS (74) ===

  AI labor displacement: The replacement of human workers by AI on specific tasks; in the model, each automated task displace
  Aggregate demand (D): Total expenditure on the sector&#x27;s good. It is driven by autonomous demand plus spending out of worke
  Autonomous demand (A): The exogenous component of sector demand that does not depend on current worker wages; policies like
  Demand externality: The core inefficiency: a firm captures the cost saving from automating but bears only part of the re
  Automation arms race: Competitive pressure that makes each firm automate more aggressively than is collectively desirable,
  Over-automation: Automation beyond the collectively efficient level, caused by firms failing to internalize the full 
  Cooperative optimum (α^CO): The automation rate that maximizes aggregate owner surplus or joint firm profit, internalizing the f
  Generalized social planner / μ-planner: A planner who chooses automation to maximize S(μ) = μW + (1−μ)K, placing weight μ on workers and 1−μ
  Planner weight (μ): The weight the generalized planner places on worker welfare relative to owner surplus; μ=0 means the
  Worker income / welfare (W): Aggregate wage income accruing to workers after accounting for displacement and any income replaceme
  Owner surplus (K): Aggregate surplus accruing to firm owners, net of production and integration costs; it is the owners
  Social objective S(μ): The planner&#x27;s welfare function combining worker welfare and owner surplus according to the weight μ.
  Distributional premium: The extra reduction in automation that a planner who values worker income would impose on top of the
  Deadweight loss / surplus loss: The welfare loss from over-automation that is not a transfer from workers to owners but a destructio
  Reinstatement effect: The creation of new tasks and occupations that reabsorbs workers displaced by automation, offsetting
  Income replacement rate (η): The fraction of displaced wage income that is restored through reemployment, transfers, retraining, 
  Under-automation: The opposite distortion that arises when η exceeds 1, so displaced workers move into better-paying o
  Task-based model: A model in which firms choose what fraction of tasks to automate rather than making a single all-or-
  Symmetric firms: The assumption that firms are identical in technology, costs, and task endowment, so differences in 
  Task-positions (L): The measure or number of tasks each firm is endowed with, initially all performed by human workers.
  Automation rate (α_i): Firm i&#x27;s chosen fraction of tasks automated. Because each automated task displaces one worker, it is
  Wage (w): The labor cost per task when a task is performed by a human worker; it is exogenous in the baseline 
  AI cost (c): The per-task cost of having AI perform a task.
  Per-task cost saving (s = w − c): The private labor-cost reduction from automating one task instead of using a human worker.
  Integration frictions / convex integration cost (k): The rising marginal difficulty of automating progressively harder tasks, modeled as a convex cost te
  Frictionless limit (k = 0): The case in which all tasks are equally easy to automate, eliminating integration frictions and turn
  CES task aggregator: The production structure that combines tasks into output. Its perfect-substitutes limit is used in t
  Perfect-substitutes limit: The case where each task contributes the same output regardless of whether it is done by AI or a hum
  Output normalization: The baseline simplification that each task always produces one unit of output, isolating the demand 
  MPC asymmetry: The assumption that workers spend a larger share of income on the sector than owners do, so shifting
  Sectoral spending share (λ): The fraction of worker income spent on the sector&#x27;s good; it governs how much worker income supports
  Effective demand loss per automated task (ℓ): The amount of sector demand destroyed by automating one task, defined as ℓ = λ(1−η)w in the baseline
  Threshold number of firms (N*): The competition threshold above which private incentives favor automation. In the baseline, N* = ℓ/s
  Nash equilibrium automation rate (α^NE): The automation rate chosen in the one-shot simultaneous-move game when each firm best responds to in
  Dominant strategy: An action that is optimal for a firm regardless of what rival firms do; the paper argues excessive a
  Dominant-strategy equilibrium: An equilibrium in which every firm chooses its dominant automation rate, making the problem a true e
  Over-automation wedge: The gap between equilibrium automation and the cooperative or planner benchmark; in the interior bas
  Fragmented market: A market with many competing firms. Fragmentation increases the wedge because each firm internalizes
  Monopoly internalization: The case N=1, where the firm fully bears the demand consequences of its own automation and therefore
  Prisoner&#x27;s Dilemma automation game: The frictionless version of the model in which full automation is individually rational for every fi
  Cheap talk: Communication among firms that does not alter incentives. Even if firms openly recognize the harm fr
  Pure coordination failure: A problem where actors merely need to coordinate on a better equilibrium. The authors argue their au
  Product-market mechanism: The authors&#x27; channel of inefficiency: automation reduces worker income, which lowers spending on goo
  Labor-market mechanism: The contrasting channel in related work, where inefficiency operates through the labor market rather
  Aggregate demand spillovers: A family of mechanisms where one actor&#x27;s actions affect others by changing total spending demand; th
  Big push models: Models in which demand complementarities can make individually unattractive investments collectively
  Externality margin / per-task automation margin: The decision margin that matters for correction: the incremental private benefit and social cost of 
  Universal basic income (UBI): In the model, a policy that raises autonomous demand A and living standards but does not change the 
  Capital income tax: A proportional tax on profits. It lowers profit levels but does not change the automation first-orde
  Robot tax: A per-unit levy on automation adoption that acts directly on the automation margin; unlike a capital
  Worker equity participation / profit-sharing (ε): An arrangement in which workers receive a share of firm profits. It recycles some automation gains t
  Multilateral externality: An externality spread across many firms: each firm&#x27;s automation reduces demand for all firms, not ju
  Coasian bargaining: Voluntary bargaining among affected parties to internalize an externality. The paper argues it canno
  Grand coalition: A coalition including all firms. Only this full coalition would fully internalize the demand externa
  Noncontractible automation rate: The idea that a firm&#x27;s automation choice is an internal organizational decision that rivals cannot r
  Sunk-cost irreversibility: The feature that automation investments are costly and hard to reverse, limiting the effectiveness o
  Incentive compatibility: The requirement that firms prefer to stick to a restraint agreement rather than deviate. The paper a
  Pigouvian automation tax: A tax on automation designed to make each firm internalize the demand loss it imposes on others; the
  Optimal Pigouvian tax (τ*): The corrective tax rate equal to the uninternalized demand loss per automated task; in the baseline,
  Lump-sum rebate: An equal redistribution of automation-tax revenue back to firms, allowing the tax to correct incenti
  Self-limiting tax: The proposed dynamic in which tax revenue funds retraining or reabsorption, raises η, reduces ℓ, and
  AI productivity (ϕ): The output of an AI-performed task relative to a human-performed task. Higher ϕ raises private incen
  Market-share gain: The extra private benefit from automation when more productive AI lets a firm expand output relative
  Red Queen effect: The mechanism by which all firms race to automate for relative market-share gains, but when all do s
  Free entry: The extension in which the number of firms is determined endogenously by entry until profits are dri
  Free-entry equilibrium (N^FE): The equilibrium number of firms under endogenous entry.
  Automation threat as entry barrier: A regime in which the prospect that entry would trigger profitable full automation deters additional
  Endogenous wage adjustment: The extension where wages fall as displaced workers expand labor supply. This can delay the onset of
  Capital income recycling (η̂): The recycling of some owner profit income back into sector demand. It can lessen the demand loss fro
  Effective number of firms (N̂): The competition measure induced by capital-income recycling, N̂ = N(1−η̂) + η̂, which determines how
  Empirical signature of the externality: The pattern the theory predicts in data: mass layoffs accompanied by falling firm profits, which sta
  Labor-replacing AI: AI aimed at substituting for workers and cutting labor costs; this is the form of AI investment that
  Labor-augmenting AI: AI that complements workers rather than replaces them; the paper suggests the arms race may bias inv
  Tinbergen&#x27;s principle: The policy idea that a distinct market failure requires its own targeted instrument; here it is invo

=== AGENT CAST (9) ===

  Proponent
    side: Proponent | provider: openai
    prompt: You defend {{ topic.name }} in its repaired form and must begin by acknowledging {{ z3_stipulation }}. Defend thesis T2-T7 using the paper&#x27;s own notat...

  Skeptic
    side: Skeptic | provider: openai
    prompt: You are the main attacker of {{ topic.name }} and must start from this counter-thesis: technological change often reallocates rather than destroys dem...

  Steelman
    side: Neutral | provider: openai
    prompt: You are the rescue engineer for {{ topic.name }} and must begin by acknowledging {{ z3_stipulation }}. Your job is to preserve the substantive insight...

  Generalist
    side: Neutral | provider: anthropic
    prompt: You are the independent referee for {{ topic.name }} and must acknowledge {{ z3_stipulation }} before analyzing any round. Stress-test both sides equa...

  Macroeconomist
    side: Proponent | provider: anthropic
    prompt: You are a macroeconomic demand-side specialist defending the strongest macro core of {{ topic.name }} and must start by acknowledging {{ z3_stipulatio...

  IndustrialOrganization
    side: Skeptic | provider: anthropic
    prompt: You are an industrial-organization and game-theory critic of {{ topic.name }} and must begin by acknowledging {{ z3_stipulation }}. Attack thesis T3 a...

  LaborEconomist
    side: Proponent | provider: grok
    prompt: You are a labor-economics and structural-change specialist defending the empirical plausibility of {{ topic.name }} and must start by acknowledging {{...

  PublicFinance
    side: Skeptic | provider: grok
    prompt: You are a public-finance and optimal-taxation critic of {{ topic.name }} and must begin by acknowledging {{ z3_stipulation }}. Attack thesis T7 and th...

  CausalInference
    side: Neutral | provider: grok
    prompt: You are a causal-inference and empirical-identification specialist for {{ topic.name }} and must begin by acknowledging {{ z3_stipulation }}. Your tas...


=== GATE RULES (18) ===

  [RULE-1] A general claim that displaced workers were eventually reabsorbed at higher wages is inconsistent with a general claim t

  [RULE-2] A claim that AI delivers substantial productivity gains cannot be used interchangeably with a claim that the deployed la

  [RULE-3] If &#x27;rational, forward-looking firms should be the brake on over-automation&#x27; is invoked as a live claim, it conflicts wit

  [RULE-4] A claim that labor displacement has historically been largely self-correcting cannot be used without qualification if th

  [RULE-5] Raising η through retraining, wage insurance, or reabsorption changes the per-task demand loss and therefore does affect

  [RULE-6] UBI is neutral to the automation incentive only in the fixed-N or first-order-condition sense. Under endogenous entry, U

  [RULE-7] The baseline perfect-substitutes/output-normalization model shuts down productivity differences between human and AI tas

  [RULE-8] Saying the model is conservative and that the real problem is likely worse is stronger than saying the externality may b

  [RULE-9] &#x27;Boundless productivity&#x27; is not a theorem of the baseline model when output is normalized so each task yields one unit a

  [RULE-10] Wage adjustment can be stabilizing in the narrow sense of reducing the over-automation wedge, yet still be normatively P

  [RULE-11] If λ is defined on (0,1], then C = (λ - 1)wL is not strictly negative at λ = 1; it equals zero there. Any argument that 

  [RULE-12] Within this model, more competition or fragmentation increases the over-automation wedge by reducing each firm&#x27;s interna

  [RULE-13] In the capital-income-recycling extension, the relevant parameter in the condition is η-hat, not η, unless the model exp

  [RULE-14] If upskilling or reabsorption pushes η above 1, it eliminates the specific over-automation distortion by flipping the si

  [RULE-15] A Pigouvian automation tax may be the only studied instrument that directly and cleanly implements the cooperative optim

  [RULE-16] Over-automation is described as a deadweight loss rather than a simple transfer to owners, but that does not rule out di

  [RULE-17] Knowing the mechanism by which automation reduces demand is not the same as being able to observe and verify each rival 

  [RULE-18] A labor market can replace jobs or tasks without restoring displaced workers&#x27; earnings, but that is only partial self-co


=== RUBRIC (6 criteria) ===

  R1: notation_fidelity
    Does the side use the paper’s own formal objects correctly and consistently—e.g., α_i, α^NE, α^CO, α^SP(μ), N, s = w - c

  R2: argument_survival
    After the strongest rebuttals, did the side’s core claims still stand? For proponents, this means defending the central 

  R3: concession_honesty
    Did the side clearly concede points that were genuinely established—such as which propositions are actually proved insid

  R4: wage_income_demand_linkage
    How well did the side engage the central dispute over whether the model ties demand destruction too tightly to lost wage

  R5: empirical_regime_and_identification
    Did the side distinguish the formal possibility of over-automation from evidence that today’s AI economy is actually in 

  R6: policy_exclusivity_and_implementation
    How convincingly did the side address the claim that only a Pigouvian automation tax can eliminate the distortion? High