Cognitive Debt: AI as Intellectual Leverage and the Dynamics of Systemic Fragility

# AI as Intellectual Leverage and the Dynamics of Systemic Fragility
Source: [https://arxiv.org/html/2606.15078](https://arxiv.org/html/2606.15078)
\(June 2026 Preliminary — Comments Welcome\)

###### Abstract

We develop a formal theory of*cognitive debt*: the stock of unverified reasoning obligations that accumulates when individuals use AI as a substitute—rather than a complement—for first\-principles cognition\. The model features two state variables per agent—*cognitive capital*\(the unaided ability to reason, verify, and transfer knowledge\) and*cognitive debt*\(the corresponding unserviced obligation\)—and a multiplicative production technology in which cognitive capital functions as collateral that determines the return to AI adoption\. We establish six propositions\. \(1\) Rational agents incur positive cognitive debt because the costs are deferred, partially external, and masked by short\-run productivity gains\. \(2\) Tranquil periods lower subjective risk assessments, raise AI substitution intensity, and compound leverage—generating a*cognitive Minsky moment*: subjective risk falls while true systemic fragility rises\. \(3\) Expected crisis losses are convex in the aggregate leverage ratio, so high\-leverage economies are disproportionately fragile\. \(4\) Post\-crisis, output\-target pressure produces a*false\-correction loop*: agents rationally patch AI failures with more AI, ratcheting leverage upward and increasing the severity of future crises under competitive output pressure\. \(5\) The decentralised equilibrium over\-adopts substitutive AI relative to the social optimum, owing to three unpriced externalities: systemic risk, cognitive public goods, and an arms\-race externality\. The optimal Pigouvian instrument is an AI\-use tax indexed to the aggregate leverage ratio\. \(6\) In a two\-type heterogeneous\-agent economy, high\-cognitive\-capital agents adopt AI more intensively, erode their capital faster, and—in a reversal of fortune—eventually hold*less*unaided cognitive capital than initially lower\-skilled agents; heterogeneity also amplifies aggregate systemic risk via the convexity of crisis losses\.

Keywords:cognitive debt; AI adoption; intellectual leverage; Minsky fragility; human capital; systemic risk; externalities\.

JEL Codes:O33, E44, J24, D62, G01\.

## 1Introduction

##### A productivity paradox\.

Consider two findings from the same experimental programme\.Noy and Zhang \([2023](https://arxiv.org/html/2606.15078#bib.bib7)\)randomly assign mid\-career professionals access to a large language model and find that treated workers complete writing tasks 37% faster, at quality rated higher by independent evaluators\.Dell’Acquaet al\.\([2023](https://arxiv.org/html/2606.15078#bib.bib8)\), in a field experiment with Boston Consulting Group analysts, replicate the productivity gain for routine in\-distribution tasks—and then ask workers to solve problems beyond the AI’s training frontier\. For these*out\-of\-distribution*tasks, the high\-AI\-use group performs*worse*than the control group, consistent with overreliance and miscalibration at the AI capability frontier\. We interpret this as a short\-run analogue of the longer\-run cognitive\-capital mechanism developed below\.

The pattern is not an anomaly\. It is the signature of a structural mechanism that this paper formalises: short\-run productivity and long\-run cognitive capacity are in tension, and agents who rationally maximise the former systematically underinvest in the latter\. The result is a hidden stock of obligations—the difference between what agents can produce*with*AI and what they can produce*without*it\. We call this stock*cognitive debt*\.

##### This paper\.

We develop a dynamic equilibrium model of AI adoption with aggregate cognitive externalities, addressing cognitive debt accumulation, systemic fragility, and welfare\. The framework has two distinguishing features\. First, cognitive capital functions as the*collateral*underlying AI adoption: the marginal product of AI is strictly proportional to unaided cognitive capacity, so capital erosion feeds back into rising AI dependence in a self\-reinforcing cycle\. Second, AI failure is a*common\-mode*event: when AI systems share training data and users share workflows, errors are correlated, and model collapse\(Shumailovet al\.,[2024](https://arxiv.org/html/2606.15078#bib.bib9)\)degrades AI quality in proportion to aggregate reliance\. These two features together generate Minsky\-style instability in the cognitive domain\.

Our central contribution is to show that three individually benign features—which each appear either desirable or at worst neutral—combine to produce systemic fragility\.

1. 1\.Multiplicative complementarity\.Cognitive capital is not merely complementary to AI; it is the*base asset*whose level determines the return to AI use\. High\-capital agents extract more from AI; low\-capital agents cannot effectively prompt, verify, or transfer AI outputs\. This creates a feedback: capital erosion reduces the marginal product of AI, which does not reduce AI reliance—it increases it, because output targets remain fixed\.
2. 2\.Invisible debt\.Cognitive debt is unobservable in normal operation\. Skill atrophy surfaces only when AI is unavailable, when tasks exceed the AI’s capability frontier, or when AI produces confident hallucinations that require first\-principles correction\. In the model, this corresponds to the stress state: debt exposureκ​z​bi​t\\kappa zb\_\{it\}is visible only at shock intensityz\>0z\>0\.
3. 3\.Systemic correlation\.AI errors are not idiosyncratic\. When agents adopt the same AI systems, their errors become correlated\. When AI\-generated content dominates training pipelines, model collapse\(Shumailovet al\.,[2024](https://arxiv.org/html/2606.15078#bib.bib9)\)erodes AI quality over time\. Both forces mean that AI failure is a common\-mode event, not a diversifiable risk\.

##### Main results\.

In a model with a continuum of agents choosing AI substitution intensity and deliberate\-practice investment, we establish:

Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)shows that rational agents optimally incur positive cognitive debt\. The determinants of equilibrium debt are transparent and empirically observable: AI quality, output pressure, discount factors, the subjective probability of AI failure, and the individual’s exposure to failure costs\.

Proposition[2](https://arxiv.org/html/2606.15078#Thmproposition2)establishes the cognitive Minsky moment\. Every tranquil period—a stretch without observed AI failures—updates subjective risk downward and raises equilibrium AI substitution intensity\. Cognitive leverageΩt=B¯t/K¯t\\Omega\_\{t\}=\\bar\{B\}\_\{t\}/\\bar\{K\}\_\{t\}rises monotonically while the true crisis probability also rises\. The Minsky divergence—π^t↓\\hat\{\\pi\}\_\{t\}\\downarrowwhileπt↑\\pi\_\{t\}\\uparrow—is a necessary consequence of rational Bayesian updating in a system where the true risk process is endogenous to the belief\-driven leverage cycle\.

Proposition[3](https://arxiv.org/html/2606.15078#Thmproposition3)shows that crisis losses are convex in leverage\. Small increases inΩ\\Omeganear the fragility threshold produce disproportionately large expected losses, because both the probability and the severity of a cognitive crisis are increasing inΩ\\Omega\.

Proposition[4](https://arxiv.org/html/2606.15078#Thmproposition4)characterises the*false\-correction loop*\. When the shadow price of current output exceeds the shadow price of future cognitive capital recovery, the individually optimal response to an AI failure is to adopt*more*AI assistance—patching AI errors with AI\. The condition is generically satisfied under competitive output pressure, generating ratchet dynamics in leverage\.

Proposition[5](https://arxiv.org/html/2606.15078#Thmproposition5)derives the social planner’s optimum and quantifies the three externalities that drive the decentralised gap: \(i\) a systemic risk externality, because my AI use raises the aggregate leverage ratio and the common\-mode failure probability without my bearing the full loss; \(ii\) a cognitive public\-goods externality, because societal verification capacity depends on the aggregate stock of unaided cognitive capital; and \(iii\) an arms\-race externality, because rising average AI use shifts the competitive output benchmark upward, compelling others to increase AI reliance\.

Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)\(heterogeneous agents\) establishes a*reversal of fortune*: high\-cognitive\-capital agents adopt AI more intensively \(since the return to AI is proportional tokk\), erode their capital faster, and eventually end up with*less*unaided cognitive capital than low\-capital agents who adopted AI conservatively\. In the long run, AI adoption compresses the distribution of cognitive capital even as it initially widens output inequality\.

Figure[1](https://arxiv.org/html/2606.15078#S4.F1)illustrates the Minsky divergence and leverage dynamics; Figure[2](https://arxiv.org/html/2606.15078#S4.F2)shows the equilibrium trajectory through the Hedge–Speculative–Ponzi phase space; Figure[3](https://arxiv.org/html/2606.15078#S7.F3)displays the heterogeneous\-agent inequality reversal\.

##### Related literature\.

Our paper lies at the intersection of four strands\.

Technology and human capital\.Acemoglu and Restrepo \([2018](https://arxiv.org/html/2606.15078#bib.bib3)\)study automation as task displacement;Acemoglu and Restrepo \([2019](https://arxiv.org/html/2606.15078#bib.bib4)\)introduce reinstatement tasks\. We differ in focusing on the*intertemporal*erosion of the capital stock rather than the contemporaneous task allocation, and in introducing a debt state variable with its own compounding dynamics\.

Financial fragility\.The Minsky framework\(Minsky,[1986](https://arxiv.org/html/2606.15078#bib.bib1)\)classifies financial positions by the relationship between income flows and debt obligations\.Kiyotaki and Moore \([1997](https://arxiv.org/html/2606.15078#bib.bib21)\)formalise collateral constraints in credit cycles\.Bernankeet al\.\([1999](https://arxiv.org/html/2606.15078#bib.bib10)\)develop the financial accelerator\. We import the Minsky classification into the cognitive domain, with cognitive capital playing the role of collateral and cognitive debt compounding through habit formation and rising switching costs\.

AI and productivity\.Brynjolfssonet al\.\([2025](https://arxiv.org/html/2606.15078#bib.bib6)\)andNoy and Zhang \([2023](https://arxiv.org/html/2606.15078#bib.bib7)\)document short\-run productivity gains from AI access\. Crucially,Dell’Acquaet al\.\([2023](https://arxiv.org/html/2606.15078#bib.bib8)\)identify the*jagged frontier*: AI improves performance on in\-distribution tasks but*reduces*it on out\-of\-distribution tasks for high\-AI\-reliance workers\. This asymmetry is the empirical anchor for our stress\-state production function\.

Systemic risk and externalities\.Allen and Gale \([2000](https://arxiv.org/html/2606.15078#bib.bib12)\)andAcemogluet al\.\([2015](https://arxiv.org/html/2606.15078#bib.bib13)\)study contagion in financial networks\. Model collapse\(Shumailovet al\.,[2024](https://arxiv.org/html/2606.15078#bib.bib9)\)is the cognitive analogue: AI\-generated content in training pipelines degrades AI quality in a manner that is correlated across all users of a given model family\.

##### Organisation\.

Section[2](https://arxiv.org/html/2606.15078#S2)presents the model\. Section[3](https://arxiv.org/html/2606.15078#S3)solves the individual problem \(Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)\)\. Section[4](https://arxiv.org/html/2606.15078#S4)characterises aggregate dynamics \(Propositions[2](https://arxiv.org/html/2606.15078#Thmproposition2)and[3](https://arxiv.org/html/2606.15078#Thmproposition3)\)\. Section[5](https://arxiv.org/html/2606.15078#S5)analyses post\-crisis dynamics \(Proposition[4](https://arxiv.org/html/2606.15078#Thmproposition4)\)\. Section[6](https://arxiv.org/html/2606.15078#S6)derives the social optimum and optimal policy \(Proposition[5](https://arxiv.org/html/2606.15078#Thmproposition5)\)\. Section[7](https://arxiv.org/html/2606.15078#S7)develops the two\-type heterogeneous\-agent economy \(Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)\) and the reversal of fortune\. Section[8](https://arxiv.org/html/2606.15078#S8)discusses extensions and limitations\. Section[9](https://arxiv.org/html/2606.15078#S9)concludes\. All proofs are in the Appendix\.

## 2The Model

### 2\.1Environment

Time is discrete,t=0,1,2,…t=0,1,2,\\ldots\. There is a unit continuum of agents indexed byi∈\[0,1\]i\\in\[0,1\]\. Each agent is characterised by two state variables:

- •ki​t≥0k\_\{it\}\\geq 0:*cognitive capital*—the unaided ability to reason, verify, synthesise, and transfer knowledge\. This is the stock of intellectual competence that functions without AI support\.
- •bi​t≥0b\_\{it\}\\geq 0:*cognitive debt*—the gap between current task demands and current cognitive capital, made viable by AI\. Formally,bi​tb\_\{it\}is the stock of unverified reasoning obligations: outputs produced with AI assistance that the agent could not independently reproduce, verify, or correct\.

Each period, agentiichooses two controls:

- •ai​t∈\[0,1\]a\_\{it\}\\in\[0,1\]:*AI substitution intensity*—the fraction of cognitively demanding steps outsourced to AI\. We reserve the label*substitutive*AI forai​t\>0a\_\{it\}\>0; AI used purely as a tutor, verifier, or feedback device that requires the agent to engage first\-principles reasoning corresponds toai​t=0a\_\{it\}=0\.
- •xi​t≥0x\_\{it\}\\geq 0:*deliberate\-practice investment*—cognitive debt repayment: hand\-derivation, unaided testing, active verification, first\-principles reconstruction\.

#### Competitive output benchmark

Agents operate in a competitive environment where output is evaluated against an endogenous aggregate benchmark\. Lety¯t\\bar\{y\}\_\{t\}denote the symmetric\-equilibrium average output\. The per\-period payoff for agentiiin statessis:

ui​ts=yi​ts−χ​max⁡\{0,y¯t−yi​ts\},u\_\{it\}^\{s\}=y\_\{it\}^\{s\}\-\\chi\\,\\max\\bigl\\\{0,\\;\\bar\{y\}\_\{t\}\-y\_\{it\}^\{s\}\\bigr\\\},\(1\)whereχ≥0\\chi\\geq 0is a competitive pressure parameter capturing wage penalties, labour\-market displacement, or organisational ranking from falling below the benchmark\. Each agent takesy¯t\\bar\{y\}\_\{t\}as given; the social planner internalises its endogeneity\. In the symmetric equilibriumyi​ts=y¯ty\_\{it\}^\{s\}=\\bar\{y\}\_\{t\}, so the penalty is zero in realisation; but the threat of falling below the benchmark when others maintain highAtA\_\{t\}creates a strategic complementarity that drives over\-adoption at the social level \(Section[6](https://arxiv.org/html/2606.15078#S6)\)\. The output constraintyi​t≥wy\_\{it\}\\geq win the post\-crisis analysis \(Section[5](https://arxiv.org/html/2606.15078#S5)\) is the realised form of this benchmark when competitive pressure is acute\.

### 2\.2Production Technology

Nature draws a statest∈\{N,S\}s\_\{t\}\\in\\\{N,S\\\}each period, whereNNis the normal state andSSis the stress state\. The true probability of the stress state isπt\\pi\_\{t\}, which is determined endogenously in equilibrium \(Section[4](https://arxiv.org/html/2606.15078#S4)\)\. Agentiiholds the subjective beliefπ^t\\hat\{\\pi\}\_\{t\}about the probability of the stress state, which is updated via Bayes’ rule as described below\.

#### Normal\-state production

In stateNN, output is

yi​tN=ki​t⋅G​\(ai​t;qt\),y\_\{it\}^\{N\}=k\_\{it\}\\cdot G\(a\_\{it\};\\,q\_\{t\}\),\(2\)whereqt\>0q\_\{t\}\>0denotes the quality of the prevailing AI system\.

###### Assumption 1\(Normal\-state technology\)\.

G:\[0,1\]×ℝ\+\+→\[1,∞\)G:\[0,1\]\\times\\mathbb\{R\}\_\{\+\+\}\\to\[1,\\infty\)is twice continuously differentiable and satisfies:

1. *\(i\)*G​\(0;q\)=1G\(0;\\,q\)=1for allq\>0q\>0\.
2. *\(ii\)*Ga​\(a;q\)\>0G\_\{a\}\(a;\\,q\)\>0andGa​a​\(a;q\)<0G\_\{aa\}\(a;\\,q\)<0for all\(a,q\)\(a,q\)\.
3. *\(iii\)*Gq​\(a;q\)\>0G\_\{q\}\(a;\\,q\)\>0andGa​q​\(a;q\)\>0G\_\{aq\}\(a;\\,q\)\>0for all\(a,q\)\(a,q\)\.
4. *\(iv\)*lima→0Ga​\(a;q\)=\+∞\\lim\_\{a\\to 0\}G\_\{a\}\(a;\\,q\)=\+\\infty*\(Inada\)*\.

Condition \(i\) normalises: without AI, output equals cognitive capitalki​tk\_\{it\}\. Condition \(ii\) says AI raises output at a diminishing marginal rate\. Condition \(iii\) says higher AI quality raises both the level and the marginal product of AI use\. Condition \(iv\) ensures an interior solution whenever the cost of AI use is finite\.

#### Stress\-state production

In stateSS—an AI outage, a hallucination in a critical task, an out\-of\-distribution problem, an audit requiring first\-principles justification, or any event that forces the agent to operate at or beyond the AI’s capability frontier—output is

yi​tS=ki​t⋅G~​\(ai​t;qt,zt\)−κ​zt​bi​t,y\_\{it\}^\{S\}=k\_\{it\}\\cdot\\tilde\{G\}\(a\_\{it\};\\,q\_\{t\},\\,z\_\{t\}\)\-\\kappa\\,z\_\{t\}\\,b\_\{it\},\(3\)wherezt\>0z\_\{t\}\>0is the intensity of the stress shock drawn from a distribution with support\[z¯,z¯\]\[\\underline\{z\},\\bar\{z\}\], andκ\>0\\kappa\>0is the debt\-exposure coefficient\.

###### Assumption 2\(Stress\-state technology\)\.

G~​\(a;q,z\)=G​\(a;q⋅s​\(z\)\)\\tilde\{G\}\(a;\\,q,z\)=G\\bigl\(a;\\,q\\cdot s\(z\)\\bigr\), wheres:ℝ\+\+→\(0,1\]s:\\mathbb\{R\}\_\{\+\+\}\\to\(0,1\]satisfiess​\(0\+\)=1s\(0^\{\+\}\)=1,s′​\(z\)<0s^\{\\prime\}\(z\)<0, ands​\(z\)→0s\(z\)\\to 0asz→∞z\\to\\infty\.

Assumption[2](https://arxiv.org/html/2606.15078#Thmassumption2)says the stress state reduces*effective AI quality*by a factors​\(z\)∈\(0,1\)s\(z\)\\in\(0,1\): the AI system is less reliable, its outputs require deeper verification, or the task has moved outside the training distribution\. Under this specification,G~a=Ga​\(⋅;q​s​\(z\)\)<Ga​\(⋅;q\)\\tilde\{G\}\_\{a\}=G\_\{a\}\(\\cdot;\\,qs\(z\)\)<G\_\{a\}\(\\cdot;\\,q\), so AI assistance provides a strictly smaller marginal benefit in the stress state\.

The termκ​zt​bi​t\\kappa z\_\{t\}b\_\{it\}is the*debt\-service cost*: cognitive debtbi​tb\_\{it\}represents the accumulated gap between task demands and unaided cognitive capacity\. Under stress, this gap is exposed\. Agents who have outsourced their verification, derivation, and error\-correction to AI cannot fulfil these obligations from their own cognitive resources; the resulting output shortfall is proportional tobi​tb\_\{it\}and scaled by shock severityztz\_\{t\}\.

### 2\.3Cognitive Capital and Debt Dynamics

The state variables evolve according to:

ki,t\+1\\displaystyle k\_\{i,t\+1\}=\(1−δ\)​ki​t\+ℓ​\(1−ai​t\)\+ν​xi​t,\\displaystyle=\(1\-\\delta\)\\,k\_\{it\}\+\\ell\(1\-a\_\{it\}\)\+\\nu\\,x\_\{it\},\(4\)bi,t\+1\\displaystyle b\_\{i,t\+1\}=\(1\+rb\)​bi​t\+d​\(ai​t\)−ρ​xi​t,\\displaystyle=\(1\+r\_\{b\}\)\\,b\_\{it\}\+d\(a\_\{it\}\)\-\\rho\\,x\_\{it\},\(5\)whereδ∈\(0,1\)\\delta\\in\(0,1\)is the cognitive depreciation rate,rb\>0r\_\{b\}\>0is the cognitive debt compounding rate,ν\>0\\nu\>0is the learning efficiency of deliberate practice, andρ\>0\\rho\>0is its debt\-reduction efficiency\.

###### Assumption 3\(Dynamics\)\.

1. *\(i\)*ℓ:\[0,1\]→ℝ\+\\ell:\[0,1\]\\to\\mathbb\{R\}\_\{\+\}isC2C^\{2\}withℓ​\(0\)=0\\ell\(0\)=0,ℓ′​\(s\)\>0\\ell^\{\\prime\}\(s\)\>0,ℓ′′​\(s\)≤0\\ell^\{\\prime\\prime\}\(s\)\\leq 0\. \(Learning by doing is increasing and weakly concave in unaided effort fraction1−a1\-a\.\)
2. *\(ii\)*d:\[0,1\]→ℝ\+d:\[0,1\]\\to\\mathbb\{R\}\_\{\+\}isC2C^\{2\}withd​\(0\)=0d\(0\)=0,d′​\(a\)\>0d^\{\\prime\}\(a\)\>0,d′′​\(a\)\>0d^\{\\prime\\prime\}\(a\)\>0\. \(Cognitive debt accumulation is increasing and strictly convex in AI substitution intensity\.\)
3. *\(iii\)*rb\>0r\_\{b\}\>0\. \(Cognitive debt compounds: foundational gaps raise the marginal cost of subsequent learning and error\-detection\.\)

The law of motion forkkreflects two learning channels\. The termℓ​\(1−ai​t\)\\ell\(1\-a\_\{it\}\)is*learning by doing*from unaided cognition: working through a problem independently—even imperfectly—builds transferable capacity\. Whenai​t=1a\_\{it\}=1, this channel is shut: the agent never engages the reasoning process\. The termν​xi​t\\nu x\_\{it\}is*deliberate practice*: targeted effort to rebuild or maintain cognitive capacity\.

The law of motion forbbreflects three debt dynamics\. The term\(1\+rb\)​bi​t\(1\+r\_\{b\}\)b\_\{it\}captures compounding: existing gaps make new gaps easier to acquire \(habit formation, rising prompt dependency, declining verification skills\)\. The termd​\(ai​t\)d\(a\_\{it\}\)reflects new debt issuance, convex inai​ta\_\{it\}because marginal substitution increasingly replaces higher\-order reasoning\. The term−ρ​xi​t\-\\rho x\_\{it\}is debt repayment\.

We impose the non\-negativity constraintbi,t\+1≥0b\_\{i,t\+1\}\\geq 0: cognitive debt cannot become negative \(agents cannot be “ahead” of their task demands\)\. This is enforced via the upper bound on deliberate practice:

xi​t≤\(1\+rb\)​bi​t\+d​\(ai​t\)ρ\.x\_\{it\}\\leq\\frac\{\(1\+r\_\{b\}\)\\,b\_\{it\}\+d\(a\_\{it\}\)\}\{\\rho\}\.\(6\)When this constraint is slack, the dynamics follow \([5](https://arxiv.org/html/2606.15078#S2.E5)\) as written\.

###### Assumption 4\(Debt compounding dominates depreciation\)\.

rb\>δr\_\{b\}\>\\deltaand there existsa¯<1\\bar\{a\}<1such that fora≥a¯a\\geq\\bar\{a\},ℓ​\(1−a\)/Kt<rb⋅Ωt\\ell\(1\-a\)/K\_\{t\}<r\_\{b\}\\cdot\\Omega\_\{t\}\(the cognitive capital accumulation from practice does not offset leverage compounding at high substitution rates\)\.

Assumption[4](https://arxiv.org/html/2606.15078#Thmassumption4)ensures that cognitive debt compounds faster than cognitive capital depreciates naturally\. Without it, tranquil\-period leverage could self\-correct through passive depreciation; with it, the leverage ratio drifts upward unless actively repaid via deliberate practice\.

### 2\.4AI Quality and Model Collapse

AI qualityqtq\_\{t\}is determined at the aggregate level\. When agents extensively use AI\-generated content, two degradation forces operate\.

###### Assumption 5\(AI quality dynamics\)\.

qt\+1=q¯⋅\(1−γ​A¯t\)⋅\(1\+η​Ωt\)−1,q\_\{t\+1\}=\\bar\{q\}\\cdot\(1\-\\gamma\\bar\{A\}\_\{t\}\)\\cdot\(1\+\\eta\\,\\Omega\_\{t\}\)^\{\-1\},\(7\)whereA¯t=∫01ai​t​di\\bar\{A\}\_\{t\}=\\int\_\{0\}^\{1\}a\_\{it\}\\,\\,\\mathrm\{d\}i,Ωt=B¯t/K¯t\\Omega\_\{t\}=\\bar\{B\}\_\{t\}/\\bar\{K\}\_\{t\}is the aggregate cognitive leverage ratio,γ∈\(0,1\)\\gamma\\in\(0,1\)\(ensuringqt\+1\>0q\_\{t\+1\}\>0wheneverA¯t∈\[0,1\]\\bar\{A\}\_\{t\}\\in\[0,1\]\), andη\>0\\eta\>0\.

The first factor\(1−γ​A¯t\)\(1\-\\gamma\\bar\{A\}\_\{t\}\)captures*model collapse*\(Shumailovet al\.,[2024](https://arxiv.org/html/2606.15078#bib.bib9)\): when AI\-generated content enters training pipelines at scale, model quality degrades because the diversity and grounding of the training distribution shrinks\. The second factor\(1\+η​Ωt\)−1\(1\+\\eta\\Omega\_\{t\}\)^\{\-1\}captures*verification failure*: as aggregate leverageΩt\\Omega\_\{t\}rises, fewer agents retain the unaided capacity to detect and correct AI errors, so bugs, hallucinations, and systematic biases propagate unfiltered into knowledge stocks\.

### 2\.5Information Structure and Belief Dynamics

Agents are rational conditional on a misspecified perceived law of motion in which stress arrivals are estimated from realised crisis frequencies\. They observe their own output and whether a crisis occurs, but do not observe the aggregate leverage ratioΩt\\Omega\_\{t\}or the true crisis probabilityπt\\pi\_\{t\}\. The subjective probability of the stress state is updated by:

π^t\+1=\(1−λ\)​π^t\+λ⋅𝟏​\{crisis at​t\},\\hat\{\\pi\}\_\{t\+1\}=\(1\-\\lambda\)\\,\\hat\{\\pi\}\_\{t\}\+\\lambda\\cdot\\mathbf\{1\}\\\{\\text\{crisis at \}t\\\},\(8\)whereλ∈\(0,1\)\\lambda\\in\(0,1\)is the learning rate\. In a period without a crisis,π^t\+1<π^t\\hat\{\\pi\}\_\{t\+1\}<\\hat\{\\pi\}\_\{t\}: subjective risk falls\. This rule is adaptive rather than fully Bayesian: a true Bayesian who knewπt=Π​\(Ωt,Mt\)\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\},M\_\{t\}\)and the aggregate law of motion would filterΩt\\Omega\_\{t\}from the absence of crises, and would not necessarily lowerπ^t\\hat\{\\pi\}\_\{t\}during an upswing\. The updating rule \([8](https://arxiv.org/html/2606.15078#S2.E8)\) captures the empirically relevant case of misspecified learning from realised crisis frequencies\.

The true crisis probability depends on aggregate leverage and AI market concentration:

πt=Π​\(Ωt,Mt\),ΠΩ\>0,ΠΩ​Ω≥0,ΠM\>0,ΠΩ​M≥0,Π​\(0,⋅\)=0,limΩ→∞Π​\(Ω,Mt\)=1,\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\},M\_\{t\}\),\\quad\\Pi\_\{\\Omega\}\>0,\\;\\Pi\_\{\\Omega\\Omega\}\\geq 0,\\;\\Pi\_\{M\}\>0,\\;\\Pi\_\{\\Omega M\}\\geq 0,\\;\\Pi\(0,\\cdot\)=0,\\;\\lim\_\{\\Omega\\to\\infty\}\\Pi\(\\Omega,M\_\{t\}\)=1,\(9\)whereMtM\_\{t\}is the Herfindahl–Hirschman index of AI system market share\. Higher concentration raises crisis probability \(ΠM\>0\\Pi\_\{M\}\>0\) because correlated failures are more likely when agents share the same AI systems, and concentrates the systemic impact of any single model collapse\. We adopt the parametric formΠ​\(Ω,M\)=1−e−λπ​M​Ωγ\\Pi\(\\Omega,M\)=1\-e^\{\-\\lambda\_\{\\pi\}M\\,\\Omega^\{\\gamma\}\}withγ≥1\\gamma\\geq 1for the proofs that require explicit computation; the qualitative results hold for anyΠ\\Pisatisfying \([9](https://arxiv.org/html/2606.15078#S2.E9)\)\.

## 3Individual Optimization

### 3\.1The Agent’s Problem

Agentiimaximises discounted expected utility, taking as given the AI quality path\{qt\}\\\{q\_\{t\}\\\}, aggregate variables\{Ωt,y¯t\}\\\{\\Omega\_\{t\},\\bar\{y\}\_\{t\}\\\}, and the subjective crisis probabilityπ^t\\hat\{\\pi\}\_\{t\}:

V​\(k,b;π^,q\)=maxa∈\[0,1\],x≥0⁡\{U~​\(k,a,b;π^,q,y¯\)−c​\(x\)\+β​𝔼​\[V​\(k′,b′;π^′,q′\)\]\},V\(k,b;\\,\\hat\{\\pi\},q\)=\\max\_\{a\\in\[0,1\],\\;x\\geq 0\}\\left\\\{\\tilde\{U\}\(k,a,b;\\,\\hat\{\\pi\},q,\\bar\{y\}\)\-c\(x\)\+\\beta\\,\\mathbb\{E\}\\bigl\[V\(k^\{\\prime\},b^\{\\prime\};\\,\\hat\{\\pi\}^\{\\prime\},q^\{\\prime\}\)\\bigr\]\\right\\\},\(10\)wherec​\(x\)c\(x\)is the direct cost of deliberate practice withc​\(0\)=0c\(0\)=0,c′​\(x\)\>0c^\{\\prime\}\(x\)\>0,c′′​\(x\)\>0c^\{\\prime\\prime\}\(x\)\>0;β∈\(0,1\)\\beta\\in\(0,1\)is the discount factor;k′,b′k^\{\\prime\},b^\{\\prime\}follow \([4](https://arxiv.org/html/2606.15078#S2.E4)\)–\([5](https://arxiv.org/html/2606.15078#S2.E5)\); and

U~≡\(1−π^\)​uN​\(k,a;q,y¯\)\+π^​𝔼z​\[uS​\(k,a,b;q,z,y¯\)\],us=ys−χ​max⁡\{0,y¯−ys\}\.\\tilde\{U\}\\equiv\(1\-\\hat\{\\pi\}\)\\,u^\{N\}\(k,a;q,\\bar\{y\}\)\+\\hat\{\\pi\}\\,\\mathbb\{E\}\_\{z\}\\bigl\[u^\{S\}\(k,a,b;q,z,\\bar\{y\}\)\\bigr\],\\qquad u^\{s\}=y^\{s\}\-\\chi\\max\\bigl\\\{0,\\,\\bar\{y\}\-y^\{s\}\\bigr\\\}\.In the symmetric equilibriumyi​tN=y¯ty^\{N\}\_\{it\}=\\bar\{y\}\_\{t\}, so the benchmark penalty is zero in realisation\. The individual FOCs derived below treaty¯t\\bar\{y\}\_\{t\}as fixed, yielding the same conditions as ifχ=0\\chi=0\. The social planner, however, internalises∂y¯t/∂At=Kt​Ga​\(At;qt\)\>0\\partial\\bar\{y\}\_\{t\}/\\partial A\_\{t\}=K\_\{t\}G\_\{a\}\(A\_\{t\};q\_\{t\}\)\>0, generating the arms\-race externality quantified in Proposition[5](https://arxiv.org/html/2606.15078#Thmproposition5)\.

To characterise individual incentives in closed form and derive comparative statics, we work with a two\-period version of the problem\. The infinite\-horizon problem yields qualitatively identical results \(Appendix[G](https://arxiv.org/html/2606.15078#A7)\)\.

#### Two\-period analytical model

An agent lives for two periods,t=0t=0andt=1t=1\. Int=0t=0she choosesaaandxx\. Int=1t=1she consumes the residual value of her state\. The period\-1 value function is:

V1​\(k′,b′\)=μk​k′−μb​b′,V\_\{1\}\(k^\{\\prime\},b^\{\\prime\}\)=\\mu\_\{k\}\\,k^\{\\prime\}\-\\mu\_\{b\}\\,b^\{\\prime\},\(11\)whereμk\>0\\mu\_\{k\}\>0andμb\>0\\mu\_\{b\}\>0are shadow values of cognitive capital and debt in the terminal period \(derived from the infinite\-horizon envelope conditions in Appendix[I](https://arxiv.org/html/2606.15078#A9)\)\. The objective is:

maxa,x⁡\(1−π^\)​k​G​\(a;q\)\+π^​𝔼z​\[k​G~​\(a;q,z\)−κ​z​b\]⏟expected current output−c​\(x\)\+β​V1​\(k′,b′\)\.\\max\_\{a,x\}\\;\\underbrace\{\(1\-\\hat\{\\pi\}\)\\,kG\(a;q\)\+\\hat\{\\pi\}\\,\\mathbb\{E\}\_\{z\}\\bigl\[k\\tilde\{G\}\(a;q,z\)\-\\kappa zb\\bigr\]\}\_\{\\text\{expected current output\}\}\-c\(x\)\+\\beta\\,V\_\{1\}\(k^\{\\prime\},b^\{\\prime\}\)\.\(12\)
Expanding using \([4](https://arxiv.org/html/2606.15078#S2.E4)\)–\([5](https://arxiv.org/html/2606.15078#S2.E5)\) and \([11](https://arxiv.org/html/2606.15078#S3.E11)\):

maxa,x⁡\[\(1−π^\)​k​G​\(a;q\)\+π^​𝔼z​\[k​G~​\(a;q,z\)\]\]−π^​κ​z¯​b−c​\(x\)\+β​μk​\[\(1−δ\)​k\+ℓ​\(1−a\)\+ν​x\]−β​μb​\[\(1\+rb\)​b\+d​\(a\)−ρ​x\],\\max\_\{a,x\}\\;\\bigl\[\(1\-\\hat\{\\pi\}\)\\,kG\(a;q\)\+\\hat\{\\pi\}\\,\\mathbb\{E\}\_\{z\}\[k\\tilde\{G\}\(a;q,z\)\]\\bigr\]\-\\hat\{\\pi\}\\,\\kappa\\,\\bar\{z\}\\,b\-c\(x\)\+\\beta\\,\\mu\_\{k\}\\,\\bigl\[\(1\-\\delta\)k\+\\ell\(1\-a\)\+\\nu x\\bigr\]\-\\beta\\,\\mu\_\{b\}\\,\\bigl\[\(1\+r\_\{b\}\)b\+d\(a\)\-\\rho x\\bigr\],\(13\)wherez¯=𝔼​\[z∣S\]\\bar\{z\}=\\mathbb\{E\}\[z\\mid S\]\.

#### First\-order conditions

Differentiating \([13](https://arxiv.org/html/2606.15078#S3.E13)\) with respect toaayields the first\-order condition for AI substitution:

k\[\(1−π^\)Ga\(a∗;q\)\+π^G~a\(a∗;q,z¯\)\]=β\[μkℓ′\(1−a∗\)\+μbd′\(a∗\)\]\.\\boxed\{k\\,\\bigl\[\(1\-\\hat\{\\pi\}\)\\,G\_\{a\}\(a^\{\*\};q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{a\}\(a^\{\*\};q,\\bar\{z\}\)\\bigr\]=\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a^\{\*\}\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a^\{\*\}\)\\bigr\]\.\}\(14\)The left\-hand side is the expected marginal product of AI in current output\. The right\-hand side is the discounted marginal future cost: forgone learning by doing \(valued atμk\\mu\_\{k\}\) plus newly issued cognitive debt \(valued atμb\\mu\_\{b\}\)\.

Differentiating with respect toxxyields the condition for deliberate practice:

c′​\(x∗\)=β​\(μk​ν\+μb​ρ\)\.c^\{\\prime\}\(x^\{\*\}\)=\\beta\\,\(\\mu\_\{k\}\\,\\nu\+\\mu\_\{b\}\\,\\rho\)\.\(15\)Practice is chosen until its marginal cost equals the discounted return: increasing cognitive capital \(at rateν\\nu, valued atμk\\mu\_\{k\}\) plus reducing cognitive debt \(at rateρ\\rho, valued atμb\\mu\_\{b\}\)\.

### 3\.2Proposition 1: Rational Cognitive Debt

###### Proposition 1\(Rational Cognitive Debt\)\.

Under Assumptions[1](https://arxiv.org/html/2606.15078#Thmassumption1)–[3](https://arxiv.org/html/2606.15078#Thmassumption3), for anyπ^∈\[0,1\)\\hat\{\\pi\}\\in\[0,1\),q\>0q\>0, andk\>0k\>0, suppose additionally that

k​\[\(1−π^\)​Ga​\(1;q\)\+π^​G~a​\(1;q,z¯\)\]<β​\[μk​ℓ′​\(0\)\+μb​d′​\(1\)\]\.k\\,\\bigl\[\(1\-\\hat\{\\pi\}\)\\,G\_\{a\}\(1;\\,q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{a\}\(1;\\,q,\\bar\{z\}\)\\bigr\]<\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(0\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(1\)\\bigr\]\.\(16\)\(The marginal future cost of full substitution exceeds its marginal current benefit ata=1a=1\.\) Then:

1. *\(i\)*There exists a unique interior solutiona∗∈\(0,1\)a^\{\*\}\\in\(0,1\)to \([14](https://arxiv.org/html/2606.15078#S3.E14)\), and the resulting cognitive debt issuance satisfiesd​\(a∗\)\>0d\(a^\{\*\}\)\>0\.
2. *\(ii\)*The equilibrium AI substitution intensity satisfies: ∂a∗∂q\>0,∂a∗∂k\>0,∂a∗∂π^<0,∂a∗∂β<0,∂a∗∂μb<0,∂a∗∂κ<0\.\\frac\{\\partial a^\{\*\}\}\{\\partial q\}\>0,\\quad\\frac\{\\partial a^\{\*\}\}\{\\partial k\}\>0,\\quad\\frac\{\\partial a^\{\*\}\}\{\\partial\\hat\{\\pi\}\}<0,\\quad\\frac\{\\partial a^\{\*\}\}\{\\partial\\beta\}<0,\\quad\\frac\{\\partial a^\{\*\}\}\{\\partial\\mu\_\{b\}\}<0,\\quad\\frac\{\\partial a^\{\*\}\}\{\\partial\\kappa\}<0\.
3. *\(iii\)*The “cognitive debt wedge” satisfies: a∗​\(π^\)<ano\-debta^\{\*\}\(\\hat\{\\pi\}\)<a^\{\\text\{no\-debt\}\}whereano\-debta^\{\\text\{no\-debt\}\}is the optimal AI use whend≡0d\\equiv 0, confirming that debt accumulation is the mechanism restraining \(but not eliminating\) AI substitution\.

## 4Aggregate Dynamics and the Cognitive Minsky Moment

### 4\.1Competitive Equilibrium

###### Definition 1\(Competitive Equilibrium\)\.

A competitive equilibrium is a sequence of individual policies\{ai​t∗,xi​t∗\}\\\{a\_\{it\}^\{\*\},x\_\{it\}^\{\*\}\\\}, aggregate variables\{Kt,Bt,At,Ωt\}\\\{K\_\{t\},B\_\{t\},A\_\{t\},\\Omega\_\{t\}\\\}, AI quality\{qt\}\\\{q\_\{t\}\\\}, and belief sequences\{π^t,πt\}\\\{\\hat\{\\pi\}\_\{t\},\\pi\_\{t\}\\\}such that:

1. *\(i\)*Each agent solves \([10](https://arxiv.org/html/2606.15078#S3.E10)\) taking\{qt,π^t\}\\\{q\_\{t\},\\hat\{\\pi\}\_\{t\}\\\}as given\.
2. *\(ii\)*Markets clear:Kt=∫ki​t​diK\_\{t\}=\\int k\_\{it\}\\,\\mathrm\{d\}i,Bt=∫bi​t​diB\_\{t\}=\\int b\_\{it\}\\,\\mathrm\{d\}i,At=∫ai​t​diA\_\{t\}=\\int a\_\{it\}\\,\\mathrm\{d\}i,Ωt=Bt/Kt\\Omega\_\{t\}=B\_\{t\}/K\_\{t\}\.
3. *\(iii\)*AI quality follows \([7](https://arxiv.org/html/2606.15078#S2.E7)\)\.
4. *\(iv\)*Beliefs follow \([8](https://arxiv.org/html/2606.15078#S2.E8)\) and \([9](https://arxiv.org/html/2606.15078#S2.E9)\)\.

We focus on symmetric equilibria in whichai​t∗=at∗a\_\{it\}^\{\*\}=a\_\{t\}^\{\*\}andxi​t∗=xt∗x\_\{it\}^\{\*\}=x\_\{t\}^\{\*\}for allii, soKt=ktK\_\{t\}=k\_\{t\}andBt=btB\_\{t\}=b\_\{t\}in the representative\-agent characterisation\. Heterogeneous\-agent extensions are developed in Section[7](https://arxiv.org/html/2606.15078#S7)\.

### 4\.2Proposition 2: The Cognitive Minsky Moment

###### Proposition 2\(Cognitive Minsky Moment\)\.

Under Assumptions[1](https://arxiv.org/html/2606.15078#Thmassumption1)–[4](https://arxiv.org/html/2606.15078#Thmassumption4)and adaptive \(misspecified\) belief updating \([8](https://arxiv.org/html/2606.15078#S2.E8)\), suppose a tranquil period𝒯=\{0,1,…,T−1\}\\mathcal\{T\}=\\\{0,1,\\ldots,T\-1\\\}in which no crisis is realised, and that the belief\-updating effect dominates the endogenous AI\-quality degradation effect:

\|∂a∗∂π^​Δ​π^t\|\>\|∂a∗∂q​Δ​qt\|for all​t∈𝒯\.\\left\|\\frac\{\\partial a^\{\*\}\}\{\\partial\\hat\{\\pi\}\}\\,\\Delta\\hat\{\\pi\}\_\{t\}\\right\|\>\\left\|\\frac\{\\partial a^\{\*\}\}\{\\partial q\}\\,\\Delta q\_\{t\}\\right\|\\quad\\text\{for all \}t\\in\\mathcal\{T\}\.\(17\)Then along any equilibrium path through𝒯\\mathcal\{T\}:

1. *\(i\)*Subjective risk is strictly decreasing:π^t\+1<π^t\\hat\{\\pi\}\_\{t\+1\}<\\hat\{\\pi\}\_\{t\}for allt∈𝒯t\\in\\mathcal\{T\}\.
2. *\(ii\)*Equilibrium AI substitution intensity is strictly increasing:at\+1∗\>at∗a\_\{t\+1\}^\{\*\}\>a\_\{t\}^\{\*\}for allt∈𝒯t\\in\\mathcal\{T\}\.
3. *\(iii\)*Aggregate cognitive leverage is strictly increasing:Ωt\+1\>Ωt\\Omega\_\{t\+1\}\>\\Omega\_\{t\}for allt∈𝒯t\\in\\mathcal\{T\}\.
4. *\(iv\)*True crisis probability is strictly increasing:πt\+1=Π​\(Ωt\+1,Mt\)\>Π​\(Ωt,Mt\)=πt\\pi\_\{t\+1\}=\\Pi\(\\Omega\_\{t\+1\},M\_\{t\}\)\>\\Pi\(\\Omega\_\{t\},M\_\{t\}\)=\\pi\_\{t\}for allt∈𝒯t\\in\\mathcal\{T\}\.
5. *\(v\)*There existsT∗<∞T^\{\*\}<\\inftysuch that for allt\>T∗t\>T^\{\*\},π^t<πt\\hat\{\\pi\}\_\{t\}<\\pi\_\{t\}*\(the Minsky divergence\)*: perceived risk falls below true systemic risk\.

![Refer to caption](https://arxiv.org/html/2606.15078v1/x1.png)Figure 1:Minsky Divergence and Cognitive Leverage Dynamics\.Simulated equilibrium path under the parametric model \(q=1q=1,ϕ=0\.35\\phi=0\.35,η=0\.60\\eta=0\.60,δ=0\.04\\delta=0\.04,rb=0\.10r\_\{b\}=0\.10,β=0\.95\\beta=0\.95,π^0=0\.30\\hat\{\\pi\}\_\{0\}=0\.30, no crises realised forT=80T=80periods\)\.Panel \(a\):Subjective riskπ^t\\hat\{\\pi\}\_\{t\}falls monotonically while true fragilityπt\\pi\_\{t\}rises; the vertical dotted line marks the crossing pointT∗T^\{\*\}\.Panel \(b\):The aggregate cognitive leverage ratioΩt=Bt/Kt\\Omega\_\{t\}=B\_\{t\}/K\_\{t\}drifts upward through the Hedge, Speculative, and Ponzi zones \(defined atΩ=0\.50\\Omega=0\.50andΩ=1\.20\\Omega=1\.20respectively\)\.Panel \(c\):Equilibrium AI substitution intensityat∗a\_\{t\}^\{\*\}rises as subjective risk falls, closing the feedback loop\.
### 4\.3Proposition 3: Convexity of Crisis Losses

Define the expected crisis loss at datettas:

Lt=Π​\(Ωt,Mt\)⋅κ​𝔼z​\[max⁡\{0,z​Bt−𝒱​\(Kt\)\}\],L\_\{t\}=\\Pi\(\\Omega\_\{t\},M\_\{t\}\)\\cdot\\kappa\\,\\mathbb\{E\}\_\{z\}\\bigl\[\\max\\\{0,\\,z\\,B\_\{t\}\-\\mathcal\{V\}\(K\_\{t\}\)\\\}\\bigr\],\(18\)where𝒱​\(K\)=Kα\\mathcal\{V\}\(K\)=K^\{\\alpha\}\(α∈\(0,1\)\\alpha\\in\(0,1\)\) is the aggregate verification capacity—the concave function representing society’s ability to detect and correct AI errors as a function of aggregate cognitive capital\.

###### Proposition 3\(Convex Fragility\)\.

Under Assumptions[1](https://arxiv.org/html/2606.15078#Thmassumption1)–[5](https://arxiv.org/html/2606.15078#Thmassumption5)and the crisis probability specification \([9](https://arxiv.org/html/2606.15078#S2.E9)\), the expected crisis lossLtL\_\{t\}is:

1. *\(i\)*Strictly increasing in the aggregate leverage ratio:∂Lt/∂Ωt\>0\\partial L\_\{t\}/\\partial\\Omega\_\{t\}\>0\.
2. *\(ii\)*Locally convex near the fragility threshold:∂2Lt/∂Ωt2\>0\\partial^\{2\}L\_\{t\}/\\partial\\Omega\_\{t\}^\{2\}\>0in a neighbourhood of the fragility thresholdΩ¯\\bar\{\\Omega\}\.
3. *\(iii\)*Strictly increasing in AI model concentration:∂Lt/∂Mt\>0\\partial L\_\{t\}/\\partial M\_\{t\}\>0, with complementarity∂2Lt/∂Ωt​∂Mt≥0\\partial^\{2\}L\_\{t\}/\\partial\\Omega\_\{t\}\\partial M\_\{t\}\\geq 0\(concentration amplifies the marginal cost of leverage\)\.

![Refer to caption](https://arxiv.org/html/2606.15078v1/x2.png)Figure 2:Phase Portrait in Cognitive Capital–Debt Space\.The equilibrium trajectory \(blue gradient line, darker = later\) starts in the Hedge zone \(Ω<0\.50\\Omega<0\.50\) and migrates through the Speculative zone into the Ponzi zone during an uninterrupted tranquil period\. The colour intensity encodes time: early periods are light blue, late periods are dark blue\. Arrows indicate the direction of motion\. Iso\-leverage lines atΩ=0\.50\\Omega=0\.50\(green dashed\) andΩ=1\.20\\Omega=1\.20\(orange dashed\) delimit the three Minsky zones\.

## 5Post\-Crisis Dynamics: The False\-Correction Loop

### 5\.1Setup

Suppose a stress event occurs at datet=0t=0: the stress state is realised with intensityz0\>0z\_\{0\}\>0\. Agents observe the shortfall in their unaided capacity\. Do they reduce AI substitution intensity—repaying cognitive debt—or do they increase it to maintain output?

Letw\>0w\>0denote the competitive output target \(the market\-determined minimum output level required for continued participation in the labour market\)\. Agents face a binding output constraint:

yi​0≥w\.y\_\{i0\}\\geq w\.\(19\)This constraint captures the competitive pressure documented in the literature: AI adoption by some agents shifts the performance distribution, raising the benchmark against which all agents are evaluated\(Brynjolfssonet al\.,[2025](https://arxiv.org/html/2606.15078#bib.bib6)\)\.

###### Definition 2\(False\-Correction Loop\)\.

A*false\-correction loop*occurs when, following a crisis att=0t=0, the equilibrium AI substitution intensity satisfiesa1∗\>a0−a\_\{1\}^\{\*\}\>a\_\{0\}^\{\-\}, wherea0−a\_\{0\}^\{\-\}denotes the pre\-crisis substitution intensity, and when the resulting leverageΩ1\>Ω0−\\Omega\_\{1\}\>\\Omega\_\{0\}^\{\-\}\.

###### Proposition 4\(False\-Correction Loop\)\.

Suppose the output constraint \([19](https://arxiv.org/html/2606.15078#S5.E19)\) is binding att=1t=1, thatk1<k0−k\_\{1\}<k\_\{0\}^\{\-\}\(cognitive capital has declined through the crisis period\), and that the post\-crisis deliberate\-practice response is insufficient to offset additional debt issuance:d​\(a1∗\)−ρ​x1∗\>0d\(a\_\{1\}^\{\*\}\)\-\\rho\\,x\_\{1\}^\{\*\}\>0\. Then the post\-crisis equilibrium AI substitution intensity satisfiesa1∗\>a0−a\_\{1\}^\{\*\}\>a\_\{0\}^\{\-\}if and only if:

λy​k1​Ga​\(a0−;q\)⏟shadow value of current output\>β​\[μk​ℓ′​\(1−a0−\)\+μb​d′​\(a0−\)\]⏟shadow value of cognitive capital recovery\.\\underbrace\{\\lambda\_\{y\}\\,k\_\{1\}\\,G\_\{a\}\(a\_\{0\}^\{\-\};\\,q\)\}\_\{\\text\{shadow value of current output\}\}\>\\underbrace\{\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a\_\{0\}^\{\-\}\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a\_\{0\}^\{\-\}\)\\bigr\]\}\_\{\\text\{shadow value of cognitive capital recovery\}\}\.\(20\)When conditions \([20](https://arxiv.org/html/2606.15078#S5.E20)\) andd​\(a1∗\)−ρ​x1∗\>0d\(a\_\{1\}^\{\*\}\)\-\\rho x\_\{1\}^\{\*\}\>0both hold, the post\-crisis path satisfiesΩt\+1\>Ωt\\Omega\_\{t\+1\}\>\\Omega\_\{t\}for allt≥1t\\geq 1until a new crisis occurs, with successive crises of non\-decreasing severity\.

## 6Welfare Analysis

### 6\.1Three Externalities

The decentralised equilibrium exhibits three distinct externalities:

##### 1\. Systemic risk externality\.

Decentralised agents takeΩt\\Omega\_\{t\}as given, and therefore omit the marginal systemic costΠΩ​\(Ωt,Mt\)⋅\(∂Ωt/∂At\)⋅Ltcond\\Pi\_\{\\Omega\}\(\\Omega\_\{t\},M\_\{t\}\)\\cdot\(\\partial\\Omega\_\{t\}/\\partial A\_\{t\}\)\\cdot L\_\{t\}^\{\\text\{cond\}\}that the planner internalises, whereLtcond=Lt/Π​\(Ωt,Mt\)L\_\{t\}^\{\\text\{cond\}\}=L\_\{t\}/\\Pi\(\\Omega\_\{t\},M\_\{t\}\)is the expected conditional loss\.

##### 2\. Cognitive public\-goods externality\.

Society’s verification capacity𝒱​\(Kt\)\\mathcal\{V\}\(K\_\{t\}\)is a public good: all agents benefit from a high\-KtK\_\{t\}population that can scrutinise AI outputs, train the next generation, and provide error\-correction in shared knowledge domains\. When agentiisubstitutes AI for cognition,KtK\_\{t\}falls, degrading this public good\. The individual internalises the private future value of her ownkk\(at shadow priceμkD\\mu\_\{k\}^\{D\}\), but not the aggregate spillover; the social planner values cognitive capital atμkP\>μkD\\mu\_\{k\}^\{P\}\>\\mu\_\{k\}^\{D\}, reflecting the public\-good component\. The externality is\(μkP−μkD\)\(\\mu\_\{k\}^\{P\}\-\\mu\_\{k\}^\{D\}\)per unit of forgone practice\.

##### 3\. Arms\-race externality\.

The competitive benchmarky¯t\\bar\{y\}\_\{t\}is endogenous to aggregateAtA\_\{t\}: when the planner raisesAtA\_\{t\}, the benchmarky¯t=Kt​G​\(At;qt\)\\bar\{y\}\_\{t\}=K\_\{t\}G\(A\_\{t\};q\_\{t\}\)rises, imposing a cost on all agents through the utility function \([1](https://arxiv.org/html/2606.15078#S2.E1)\)\. The individual, takingy¯t\\bar\{y\}\_\{t\}as fixed, ignores∂y¯t/∂At=Kt​Ga​\(At;qt\)\>0\\partial\\bar\{y\}\_\{t\}/\\partial A\_\{t\}=K\_\{t\}G\_\{a\}\(A\_\{t\};q\_\{t\}\)\>0\. This strategic complementarity drives aggregate AI adoption above the social optimum: all agents would prefer the lower\-AAequilibrium, but individually each has an incentive to maintain highaato avoid the benchmark penalty\.

### 6\.2The Social Planner’s Problem

A benevolent social planner chooses aggregate\{At,Xt\}\\\{A\_\{t\},X\_\{t\}\\\}to maximise the sum of agents’ welfare, internalising all externalities:

max\{At,Xt\}​∑t=0∞βt​\{\(1−πt\)​Kt​G​\(At;qt\)\+πt​𝔼z​\[Kt​G~​\(At;qt,z\)−κ​z​Bt\]−C​\(Xt\)\},\\max\_\{\\\{A\_\{t\},X\_\{t\}\\\}\}\\sum\_\{t=0\}^\{\\infty\}\\beta^\{t\}\\left\\\{\(1\-\\pi\_\{t\}\)\\,K\_\{t\}\\,G\(A\_\{t\};\\,q\_\{t\}\)\+\\pi\_\{t\}\\,\\mathbb\{E\}\_\{z\}\\bigl\[K\_\{t\}\\,\\tilde\{G\}\(A\_\{t\};\\,q\_\{t\},z\)\-\\kappa zB\_\{t\}\\bigr\]\-C\(X\_\{t\}\)\\right\\\},\(21\)subject to \([4](https://arxiv.org/html/2606.15078#S2.E4)\)–\([5](https://arxiv.org/html/2606.15078#S2.E5)\) in aggregate, \([7](https://arxiv.org/html/2606.15078#S2.E7)\), and \([9](https://arxiv.org/html/2606.15078#S2.E9)\), takingπt=Π​\(Ωt,Mt\)\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\},M\_\{t\}\)as endogenous\.

###### Proposition 5\(Decentralised Over\-adoption\)\.

Under Assumptions[1](https://arxiv.org/html/2606.15078#Thmassumption1)–[5](https://arxiv.org/html/2606.15078#Thmassumption5), the decentralised equilibrium aggregate AI substitution intensity exceeds the social optimum:

AtD\>AtPfor all​t≥0\.A\_\{t\}^\{D\}\>A\_\{t\}^\{P\}\\quad\\text\{for all \}t\\geq 0\.The gapΔt≡AtD−AtP\\Delta\_\{t\}\\equiv A\_\{t\}^\{D\}\-A\_\{t\}^\{P\}is strictly increasing in: \(i\) aggregate model concentrationMtM\_\{t\}; \(ii\) the length of the current tranquil period; and \(iii\) the convexity parameter of the debt accumulation functiondd\.

The constrained\-optimal Pigouvian tax on AI substitution intensity is:

τt∗=β​ΠΩ​\(Ωt,Mt\)​∂Ωt∂At​Ltcond⏟systemic risk externality\+β​\(μkP−μkD\)​ℓ′​\(1−AtP\)⏟cognitive public\-good externality\+χ​Kt​Ga​\(AtP;qt\)⏟arms\-race externality,\\tau\_\{t\}^\{\*\}=\\underbrace\{\\beta\\,\\Pi\_\{\\Omega\}\(\\Omega\_\{t\},M\_\{t\}\)\\,\\frac\{\\partial\\Omega\_\{t\}\}\{\\partial A\_\{t\}\}\\,L\_\{t\}^\{\\text\{cond\}\}\}\_\{\\text\{systemic risk externality\}\}\+\\underbrace\{\\beta\\,\(\\mu\_\{k\}^\{P\}\-\\mu\_\{k\}^\{D\}\)\\,\\ell^\{\\prime\}\(1\-A\_\{t\}^\{P\}\)\}\_\{\\text\{cognitive public\-good externality\}\}\+\\underbrace\{\\chi\\,K\_\{t\}\\,G\_\{a\}\(A\_\{t\}^\{P\};\\,q\_\{t\}\)\}\_\{\\text\{arms\-race externality\}\},\(22\)evaluated at the social optimum\(AtP,XtP\)\(A\_\{t\}^\{P\},X\_\{t\}^\{P\}\)\. The first term captures the marginal increase in expected crisis losses from raising aggregate leverage; it is positive sinceΠΩ\>0\\Pi\_\{\\Omega\}\>0\. The second term uses the shadow\-value gapμkP−μkD\>0\\mu\_\{k\}^\{P\}\-\\mu\_\{k\}^\{D\}\>0to avoid double\-counting the privately internalised return to cognitive capital; it is positive since both the gap andℓ′\>0\\ell^\{\\prime\}\>0\. The third term is positive sinceχ≥0\\chi\\geq 0andGa\>0G\_\{a\}\>0\. The optimal tax is \(a\) increasing in aggregate leverageΩt\\Omega\_\{t\}, \(b\) increasing in model concentrationMtM\_\{t\}, and \(c\) counter\-cyclical—rising during tranquil periods as leverage accumulates\.

## 7Heterogeneous Agents and the Reversal of Fortune

### 7\.1Setup

We enrich the model with two types of agents: typeHH\(high initial cognitive capital,kH,0\>k¯k\_\{H,0\}\>\\bar\{k\}\) and typeLL\(low initial cognitive capital,kL,0<k¯k\_\{L,0\}<\\bar\{k\}\), in equal proportions\. Both types begin with zero cognitive debt \(bH,0=bL,0=0b\_\{H,0\}=b\_\{L,0\}=0\) and the same subjective risk beliefπ^0\\hat\{\\pi\}\_\{0\}\. All structural parameters are identical across types\.

From Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1), part \(ii\),∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0: higher cognitive capital raises the return to AI \(via the multiplicative structure∂yN/∂a=k⋅Ga\\partial y^\{N\}/\\partial a=k\\cdot G\_\{a\}\), so the high\-kktype adopts AI more intensively in equilibrium:

aH,t∗\>aL,t∗for all​t​such that​kH,t\>kL,t\.a\_\{H,t\}^\{\*\}\>a\_\{L,t\}^\{\*\}\\quad\\text\{for all \}t\\text\{ such that \}k\_\{H,t\}\>k\_\{L,t\}\.\(23\)

### 7\.2Proposition 6: Reversal of Fortune

Define the*cognitive capital gap*Δ​kt≡kH,t−kL,t\\Delta k\_\{t\}\\equiv k\_\{H,t\}\-k\_\{L,t\}and the*AI adoption gap*Δ​at∗≡aH,t∗−aL,t∗\>0\\Delta a\_\{t\}^\{\*\}\\equiv a\_\{H,t\}^\{\*\}\-a\_\{L,t\}^\{\*\}\>0\.

###### Proposition 6\(Reversal of Fortune\)\.

In the two\-type economy, during a tranquil period with no crises:

1. *\(i\)*Δ​kt\\Delta k\_\{t\}is strictly decreasing for allt≥0t\\geq 0: high\-kkagents erode their cognitive capital advantage faster than low\-kkagents\.
2. *\(ii\)*Δ​at∗\\Delta a\_\{t\}^\{\*\}is initially positive but converges to zero asΔ​kt→0\\Delta k\_\{t\}\\to 0\.
3. *\(iii\)*If the learning\-by\-doing differential satisfies ℓ′​\(ξ\)​Δ​at∗\>\(1−δ\)​Δ​ktfor all​t≥T†,\\ell^\{\\prime\}\(\\xi\)\\,\\Delta a\_\{t\}^\{\*\}\>\(1\-\\delta\)\\,\\Delta k\_\{t\}\\quad\\text\{for all \}t\\geq T^\{\\dagger\},\(24\)for someT†<∞T^\{\\dagger\}<\\infty, then there existsT∗∗<∞T^\{\*\*\}<\\inftysuch thatΔ​kT∗∗=0\\Delta k\_\{T^\{\*\*\}\}=0andΔ​kt<0\\Delta k\_\{t\}<0for allt\>T∗∗t\>T^\{\*\*\}:*the high\-kktype ends up with less cognitive capital than the low\-kktype\.*
4. *\(iv\)*The aggregate crisis loss in the heterogeneous economy exceeds that in the homogeneous economy withk¯=\(kH,0\+kL,0\)/2\\bar\{k\}=\(k\_\{H,0\}\+k\_\{L,0\}\)/2: Lthetero\>Lthomo,L\_\{t\}^\{\\text\{hetero\}\}\>L\_\{t\}^\{\\text\{homo\}\},because the high\-kktype’s elevated leverage generates a disproportionate increase in expected losses \(by convexity ofLLinΩ\\Omega, Proposition[3](https://arxiv.org/html/2606.15078#Thmproposition3)\)\.

![Refer to caption](https://arxiv.org/html/2606.15078v1/x3.png)Figure 3:Cognitive Capital Inequality and the Reversal of Fortune\.Simulated two\-type economy \(kH,0=1\.6k\_\{H,0\}=1\.6,kL,0=0\.6k\_\{L,0\}=0\.6; all other parameters as in Figure[1](https://arxiv.org/html/2606.15078#S4.F1)\)\.Panel \(a\):Cognitive capital trajectories: the high\-kktype \(blue solid\) declines faster and eventually converges toward—and crosses—the low\-kktype \(red dashed\)\.Panel \(b\):The cognitive capital gapkH,t−kL,tk\_\{H,t\}\-k\_\{L,t\}falls from its initial positive value, crosses zero atT∗∗T^\{\*\*\}\(vertical dotted line\), and becomes negative—the reversal of fortune\.Panel \(c\):AI adoption intensity: the high\-kktype always adopts AI more intensively \(Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1),∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0\), which is the direct driver of the faster capital erosion\.

## 8Discussion

### 8\.1Heterogeneity and Inequality

Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)establishes that the reversal of fortune is not a knife\-edge result: it holds whenever condition \([24](https://arxiv.org/html/2606.15078#S7.E24)\) is satisfied, which is generically true when the learning\-by\-doing differential is sufficiently large relative to the remaining capital gap\. The parametric simulations in Figure[3](https://arxiv.org/html/2606.15078#S7.F3)illustrate a concrete instance\. The result has two policy implications\. First, policies that protect the cognitive capital floor \(mandatory unaided\-practice requirements, AI\-free examinations\) disproportionately benefit high\-adopting, high\-skill agents who face the strongest erosion pressure\. Second, the distributional reversal implies that standard skill\-biased\-technological\-change frameworks—which predict widening inequality—may mischaracterise the long\-run distribution of cognitive capital under AI adoption\.

### 8\.2Robustness to the Complementarity Assumption

The baseline production functionyi​tN=ki​t​G​\(ai​t;qt\)y^\{N\}\_\{it\}=k\_\{it\}\\,G\(a\_\{it\};\\,q\_\{t\}\)imposes strong multiplicative complementarity: the marginal product of AI is proportional toki​tk\_\{it\}, ensuring∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0\(Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)\) and underpinning the reversal of fortune \(Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)\)\.

Consider the more general class:

yi​tN=F​\(ki​t,ai​t;qt\),Fk\>0,Fa\>0,Fk​a​unrestricted,y^\{N\}\_\{it\}=F\(k\_\{it\},\\,a\_\{it\};\\,q\_\{t\}\),\\qquad F\_\{k\}\>0,\\;F\_\{a\}\>0,\\;F\_\{ka\}\\text\{ unrestricted\},with the baseline corresponding toF=k​G​\(a;q\)F=k\\,G\(a;\\,q\)\(whereFk​a=Ga\>0F\_\{ka\}=G\_\{a\}\>0\)\. Empirically,Brynjolfssonet al\.\([2025](https://arxiv.org/html/2606.15078#bib.bib6)\)find that generative AI raises productivity across skill groups, with heterogeneous effects suggestingFk​aF\_\{ka\}may be positive but finite, or even negative for some tasks\.

Under this generalFF:

- •Propositions[1](https://arxiv.org/html/2606.15078#Thmproposition1)–[5](https://arxiv.org/html/2606.15078#Thmproposition5)surviveprovided thatFFinherits the Inada and concavity conditions of Assumption[1](https://arxiv.org/html/2606.15078#Thmassumption1)and the dynamics of Assumption[3](https://arxiv.org/html/2606.15078#Thmassumption3)are preserved\. The debt accumulation, Minsky divergence, convex fragility, false\-correction loop, and welfare wedge are all driven by the compounding dynamics \([5](https://arxiv.org/html/2606.15078#S2.E5)\) and the misspecified belief updating \([8](https://arxiv.org/html/2606.15078#S2.E8)\), not by strong complementarity per se\.
- •Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)\(reversal of fortune\) requiresFk​aF\_\{ka\}sufficiently large\.The reversal depends on∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0\(high\-kkagents adopt more\)\. IfFk​a≤0F\_\{ka\}\\leq 0\(AI benefits low\-skill workers more at the margin\), then∂a∗/∂k≤0\\partial a^\{\*\}/\\partial k\\leq 0, the adoption ranking reverses, and the reversal of fortune does not occur—instead, low\-skill agents erode their capital faster\. Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)should therefore be read as characterising the case of strong complementarity, which is the empirically relevant case for complex cognitive tasks\(Dell’Acquaet al\.,[2023](https://arxiv.org/html/2606.15078#bib.bib8)\)but may not hold universally\.

### 8\.3Complementary vs\. Substitutive AI

The model throughout treatsai​ta\_\{it\}as substitutive AI use\. Complementary AI use—Socratic tutoring, friction\-generating feedback, forced verification—appears in the model as negatived​\(a\)d\(a\)\(debt repayment\) and positiveℓ\\ell\(learning activation\)\. The model therefore nests complementary AI use as a special case with reversed dynamics: complementary AI raiseskkand reducesbb\. The key policy question is how to distinguish substitutive from complementary AI use in practice; the model provides a formal criterion: an AI interaction is complementary if it increases the agent’s ability to perform the relevant task unaided\.

### 8\.4The “Cognitive Reserve” Policy

An analogy from financial regulation is the*capital adequacy ratio*: banks must hold a minimum ratio of capital to risk\-weighted assets\. The cognitive analogue is a*cognitive reserve requirement*: a minimum ratioki​t/bi​t≥Ω¯−1k\_\{it\}/b\_\{it\}\\geq\\underline\{\\Omega\}^\{\-1\}\. When this constraint binds, agents must either reduce debt \(through deliberate practice\) or reduce issuance \(by reducingai​ta\_\{it\}\)\. This constraint would prevent Ponzi cognition as defined in Remark[3](https://arxiv.org/html/2606.15078#Thmremark3)\. Formal analysis of the transition dynamics under such a constraint is left for future work\.

### 8\.5Limitations

Several simplifications warrant acknowledgement\. First, we treat the cognitive capital state variable as one\-dimensional; in reality, cognitive skills are domain\-specific and heterogeneous\. Second, the model assumes that the production functionGGis known and stationary; in a world where AI capabilities are rapidly evolving,GGshifts over time and the appropriate value ofaachanges accordingly\. Third, we abstract from the possibility that AI may be used to enhance cognitive capital formation—the “Uzawa\-learning” channel\(Uzawa,[1965](https://arxiv.org/html/2606.15078#bib.bib26); Jr\.,[1988](https://arxiv.org/html/2606.15078#bib.bib25)\)\. Incorporating endogenous AI quality into the human capital accumulation equation is a promising extension\. Fourth, the model does not account for the possibility that some forms of cognitive capital—particularly tacit, embodied, and physical knowledge\(Aschenbrenner,[2024](https://arxiv.org/html/2606.15078#bib.bib24)\)—may be non\-reclaimable once lost\.

## 9Conclusion

We have developed a formal theory of cognitive debt: the unobserved stock of reasoning obligations that accumulates when individuals systematically outsource first\-principles cognition to AI\. The model shows that this debt arises from rational optimisation, compounds through habit formation and competitive pressure, and generates Minsky\-style fragility: tranquil periods of AI\-driven productivity systematically build the conditions for cognitive crises\.

The six propositions identify the key structural forces\. Rational agents borrow cognitive debt because the costs are deferred and partially externalised\. Aggregate leverage rises during tranquil periods because belief updating and output competition both push toward more AI use\. Crisis losses are convex in leverage, so the tail risk is large\. Post\-crisis adjustment follows the false\-correction loop rather than genuine deleveraging\. And the decentralised equilibrium over\-adopts AI relative to the social optimum by a gap that grows during boom periods and is larger in more concentrated AI markets\.

The policy implications are concrete: an AI\-use tax indexed to aggregate leverage; mandatory unaided\-performance audits; limits on AI model concentration; and deliberate\-practice requirements for professionals in high\-stakes domains\. These instruments map directly to the three externalities identified in the welfare analysis\.

More broadly, the framework offers a formal vocabulary for a concern that has been articulated informally but not yet analytically: that the short\-run productivity gains from AI adoption may be purchased against a long\-run degradation of the cognitive capacity required to verify, correct, extend, and ultimately reproduce those gains\. Whether this trade\-off is empirically large is an open question\. The theory provides the structure within which to answer it\.

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## Appendix: Proofs

## Appendix BProof of Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)

We prove each part in turn\.

##### Part \(i\): Existence and uniqueness of interior solution\.

Define the FOC residual:

H​\(a;π^,q,k,μk,μb\)≡k​\[\(1−π^\)​Ga​\(a;q\)\+π^​G~a​\(a;q,z¯\)\]−β​\[μk​ℓ′​\(1−a\)\+μb​d′​\(a\)\]=0\.H\(a;\\,\\hat\{\\pi\},q,k,\\mu\_\{k\},\\mu\_\{b\}\)\\equiv k\\,\\bigl\[\(1\-\\hat\{\\pi\}\)\\,G\_\{a\}\(a;q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{a\}\(a;q,\\bar\{z\}\)\\bigr\]\-\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a\)\\bigr\]=0\.\(25\)
Ata=0a=0:By Assumption[1](https://arxiv.org/html/2606.15078#Thmassumption1)\(iv\),Ga​\(0;q\)=\+∞G\_\{a\}\(0;q\)=\+\\infty\. SinceG~a​\(0;q,z¯\)=Ga​\(0;q​s​\(z¯\)\)⋅s​\(z¯\)=\+∞\\tilde\{G\}\_\{a\}\(0;q,\\bar\{z\}\)=G\_\{a\}\(0;qs\(\\bar\{z\}\)\)\\cdot s\(\\bar\{z\}\)=\+\\infty\(for anys\>0s\>0\), the left\-hand side of \([25](https://arxiv.org/html/2606.15078#A2.E25)\) is\+∞\+\\infty\. The right\-hand side is finite by Assumption[3](https://arxiv.org/html/2606.15078#Thmassumption3)\(i\)\(ii\)\. HenceH​\(0;⋅\)\>0H\(0;\\cdot\)\>0\.

Ata=1a=1:Under condition \([16](https://arxiv.org/html/2606.15078#S3.E16)\), we haveH​\(1;⋅\)<0H\(1;\\cdot\)<0:

H​\(1;⋅\)=k​\[\(1−π^\)​Ga​\(1;q\)\+π^​G~a​\(1;q,z¯\)\]−β​\[μk​ℓ′​\(0\)\+μb​d′​\(1\)\]<0\.H\(1;\\cdot\)=k\\,\[\(1\-\\hat\{\\pi\}\)G\_\{a\}\(1;q\)\+\\hat\{\\pi\}\\tilde\{G\}\_\{a\}\(1;q,\\bar\{z\}\)\]\-\\beta\[\\mu\_\{k\}\\ell^\{\\prime\}\(0\)\+\\mu\_\{b\}d^\{\\prime\}\(1\)\]<0\.By the intermediate value theorem applied to the continuous functionHH\(withH​\(0;⋅\)=\+∞\>0H\(0;\\cdot\)=\+\\infty\>0andH​\(1;⋅\)<0H\(1;\\cdot\)<0\), there existsa∗∈\(0,1\)a^\{\*\}\\in\(0,1\)withH​\(a∗;⋅\)=0H\(a^\{\*\};\\cdot\)=0\.

Uniqueness:The second\-order condition requires∂H/∂a<0\\partial H/\\partial a<0\.

∂H∂a=k​\[\(1−π^\)​Ga​a\+π^​G~a​a\]\+β​μk​ℓ′′​\(1−a\)−β​μb​d′′​\(a\)\.\\frac\{\\partial H\}\{\\partial a\}=k\\,\\bigl\[\(1\-\\hat\{\\pi\}\)\\,G\_\{aa\}\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{aa\}\\bigr\]\+\\beta\\,\\mu\_\{k\}\\,\\ell^\{\\prime\\prime\}\(1\-a\)\-\\beta\\,\\mu\_\{b\}\\,d^\{\\prime\\prime\}\(a\)\.\(26\)The first term is strictly negative \(Assumption[1](https://arxiv.org/html/2606.15078#Thmassumption1)\(ii\)\)\. The second term is non\-positive \(Assumption[3](https://arxiv.org/html/2606.15078#Thmassumption3)\(i\)\)\. The third term is strictly negative \(Assumption[3](https://arxiv.org/html/2606.15078#Thmassumption3)\(ii\)\)\. Hence∂H/∂a<0\\partial H/\\partial a<0globally, confirming strict concavity of the objective and uniqueness ofa∗a^\{\*\}\.

Sinced​\(0\)=0d\(0\)=0andd′​\(a\)\>0d^\{\\prime\}\(a\)\>0witha∗\>0a^\{\*\}\>0, we haved​\(a∗\)\>0d\(a^\{\*\}\)\>0\.□\\square

##### Part \(ii\): Comparative statics\.

By the implicit function theorem, for any parameterθ\\theta:

∂a∗∂θ=−∂H/∂θ∂H/∂a\.\\frac\{\\partial a^\{\*\}\}\{\\partial\\theta\}=\-\\frac\{\\partial H/\\partial\\theta\}\{\\partial H/\\partial a\}\.Since∂H/∂a<0\\partial H/\\partial a<0\(from part \(i\)\), the sign of∂a∗/∂θ\\partial a^\{\*\}/\\partial\\thetaequals the sign of∂H/∂θ\\partial H/\\partial\\theta\.

∂a∗/∂q\>0\\partial a^\{\*\}/\\partial q\>0:

∂H∂q\\displaystyle\\frac\{\\partial H\}\{\\partial q\}=k​\[\(1−π^\)​Ga​q​\(a∗;q\)\+π^​G~a​q​\(a∗;q,z¯\)\]\>0,\\displaystyle=k\\,\\bigl\[\(1\-\\hat\{\\pi\}\)\\,G\_\{aq\}\(a^\{\*\};q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{aq\}\(a^\{\*\};q,\\bar\{z\}\)\\bigr\]\>0,\(27\)sinceGa​q\>0G\_\{aq\}\>0by Assumption[1](https://arxiv.org/html/2606.15078#Thmassumption1)\(iii\) andG~a​q=Ga​q​\(⋅;q​s\)​s\>0\\tilde\{G\}\_\{aq\}=G\_\{aq\}\(\\cdot;\\,qs\)s\>0\.□\\square

∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0:

∂H∂k\\displaystyle\\frac\{\\partial H\}\{\\partial k\}=\(1−π^\)​Ga​\(a∗;q\)\+π^​G~a​\(a∗;q,z¯\)\>0\.\\displaystyle=\(1\-\\hat\{\\pi\}\)\\,G\_\{a\}\(a^\{\*\};q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{a\}\(a^\{\*\};q,\\bar\{z\}\)\>0\.\(28\)Both terms are strictly positive\.□\\square

∂a∗/∂π^<0\\partial a^\{\*\}/\\partial\\hat\{\\pi\}<0:

∂H∂π^\\displaystyle\\frac\{\\partial H\}\{\\partial\\hat\{\\pi\}\}=k​\[G~a​\(a∗;q,z¯\)−Ga​\(a∗;q\)\]−β​κ​z¯​d′​\(a∗\)\.\\displaystyle=k\\,\\bigl\[\\tilde\{G\}\_\{a\}\(a^\{\*\};q,\\bar\{z\}\)\-G\_\{a\}\(a^\{\*\};q\)\\bigr\]\-\\beta\\,\\kappa\\,\\bar\{z\}\\,d^\{\\prime\}\(a^\{\*\}\)\.\(29\)Under Assumption[2](https://arxiv.org/html/2606.15078#Thmassumption2),G~a<Ga\\tilde\{G\}\_\{a\}<G\_\{a\}, so the first term is strictly negative\. The second term is strictly negative\. Hence∂H/∂π^<0\\partial H/\\partial\\hat\{\\pi\}<0and∂a∗/∂π^<0\\partial a^\{\*\}/\\partial\\hat\{\\pi\}<0\.□\\square

∂a∗/∂β<0\\partial a^\{\*\}/\\partial\\beta<0:

∂H∂β=−\[μk​ℓ′​\(1−a∗\)\+μb​d′​\(a∗\)\]<0\.□\\frac\{\\partial H\}\{\\partial\\beta\}=\-\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a^\{\*\}\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a^\{\*\}\)\\bigr\]<0\.\\square
∂a∗/∂μb<0\\partial a^\{\*\}/\\partial\\mu\_\{b\}<0:

∂H∂μb=−β​d′​\(a∗\)<0\.□\\frac\{\\partial H\}\{\\partial\\mu\_\{b\}\}=\-\\beta\\,d^\{\\prime\}\(a^\{\*\}\)<0\.\\square
∂a∗/∂κ<0\\partial a^\{\*\}/\\partial\\kappa<0:The parameterκ\\kappaenters through the stress\-state expected cost\. Increasingκ\\kappaincreases the marginal cost of debt, raisingμb\\mu\_\{b\}\(by the envelope theorem on the continuation value\)\. Since∂a∗/∂μb<0\\partial a^\{\*\}/\\partial\\mu\_\{b\}<0,∂a∗/∂κ<0\\partial a^\{\*\}/\\partial\\kappa<0\.□\\square

##### Part \(iii\): Cognitive debt wedge\.

Whend≡0d\\equiv 0\(no debt accumulation\), the FOC reduces to:

k​\[\(1−π^\)​Ga\+π^​G~a\]=β​μk​ℓ′​\(1−ano\-debt\)\.k\\,\[\(1\-\\hat\{\\pi\}\)\\,G\_\{a\}\+\\hat\{\\pi\}\\,\\tilde\{G\}\_\{a\}\]=\\beta\\,\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a^\{\\text\{no\-debt\}\}\)\.Whend\>0d\>0, the right\-hand side of \([14](https://arxiv.org/html/2606.15078#S3.E14)\) gains the termβ​μb​d′​\(a∗\)\>0\\beta\\,\\mu\_\{b\}\\,d^\{\\prime\}\(a^\{\*\}\)\>0, making the marginal cost of AI higher\. Hencea∗<ano\-debta^\{\*\}<a^\{\\text\{no\-debt\}\}\.□\\square

## Appendix CProof of Proposition[2](https://arxiv.org/html/2606.15078#Thmproposition2)

##### Part \(i\): Subjective risk is decreasing\.

From \([8](https://arxiv.org/html/2606.15078#S2.E8)\) with𝟏​\{crisist\}=0\\mathbf\{1\}\\\{\\text\{crisis\}\_\{t\}\\\}=0for allt∈𝒯t\\in\\mathcal\{T\}:π^t\+1=\(1−λ\)​π^t<π^t\\hat\{\\pi\}\_\{t\+1\}=\(1\-\\lambda\)\\hat\{\\pi\}\_\{t\}<\\hat\{\\pi\}\_\{t\}forλ∈\(0,1\)\\lambda\\in\(0,1\)\.□\\square

##### Part \(ii\): AI substitution intensity is increasing\.

By part \(i\),π^t\\hat\{\\pi\}\_\{t\}is strictly decreasing\. By Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)\(ii\),∂a∗/∂π^<0\\partial a^\{\*\}/\\partial\\hat\{\\pi\}<0and∂a∗/∂q\>0\\partial a^\{\*\}/\\partial q\>0\. The net effect onat∗a\_\{t\}^\{\*\}is:

Δ​at∗=∂a∗∂π^​Δ​π^t\+∂a∗∂q​Δ​qt\.\\Delta a\_\{t\}^\{\*\}=\\frac\{\\partial a^\{\*\}\}\{\\partial\\hat\{\\pi\}\}\\Delta\\hat\{\\pi\}\_\{t\}\+\\frac\{\\partial a^\{\*\}\}\{\\partial q\}\\Delta q\_\{t\}\.Under dominance condition \([17](https://arxiv.org/html/2606.15078#S4.E17)\),\|∂a∗/∂π^⋅Δ​π^t\|\>\|∂a∗/∂q⋅Δ​qt\|\|\\partial a^\{\*\}/\\partial\\hat\{\\pi\}\\cdot\\Delta\\hat\{\\pi\}\_\{t\}\|\>\|\\partial a^\{\*\}/\\partial q\\cdot\\Delta q\_\{t\}\|, so the belief\-updating channel dominates the quality\-degradation channel andat\+1∗\>at∗a\_\{t\+1\}^\{\*\}\>a\_\{t\}^\{\*\}\.□\\square

##### Part \(iii\): Aggregate cognitive leverage is increasing\.

The aggregate dynamics are:

Bt\+1=\(1\+rb\)​Bt\+d​\(at∗\),Kt\+1=\(1−δ\)​Kt\+ℓ​\(1−at∗\)\.B\_\{t\+1\}=\(1\+r\_\{b\}\)B\_\{t\}\+d\(a\_\{t\}^\{\*\}\),\\qquad K\_\{t\+1\}=\(1\-\\delta\)K\_\{t\}\+\\ell\(1\-a\_\{t\}^\{\*\}\)\.Compute:

Ωt\+1\\displaystyle\\Omega\_\{t\+1\}=\(1\+rb\)​Bt\+d​\(at∗\)\(1−δ\)​Kt\+ℓ​\(1−at∗\)\\displaystyle=\\frac\{\(1\+r\_\{b\}\)B\_\{t\}\+d\(a\_\{t\}^\{\*\}\)\}\{\(1\-\\delta\)K\_\{t\}\+\\ell\(1\-a\_\{t\}^\{\*\}\)\}=Ωt⋅\(1\+rb\)\(1−δ\)⋅1\+d​\(at∗\)/\[\(1\+rb\)​Bt\]1\+ℓ​\(1−at∗\)/\[\(1−δ\)​Kt\]\.\\displaystyle=\\Omega\_\{t\}\\cdot\\frac\{\(1\+r\_\{b\}\)\}\{\(1\-\\delta\)\}\\cdot\\frac\{1\+d\(a\_\{t\}^\{\*\}\)/\[\(1\+r\_\{b\}\)B\_\{t\}\]\}\{1\+\\ell\(1\-a\_\{t\}^\{\*\}\)/\[\(1\-\\delta\)K\_\{t\}\]\}\.\(30\)By Assumption[4](https://arxiv.org/html/2606.15078#Thmassumption4),rb\>δr\_\{b\}\>\\deltaandd​\(at∗\)/Btd\(a\_\{t\}^\{\*\}\)/B\_\{t\}is positive, soΩt\+1/Ωt\>1\\Omega\_\{t\+1\}/\\Omega\_\{t\}\>1\. Moreover, asat∗a\_\{t\}^\{\*\}increases,d​\(at∗\)d\(a\_\{t\}^\{\*\}\)rises andℓ​\(1−at∗\)\\ell\(1\-a\_\{t\}^\{\*\}\)falls, further increasingΩt\+1\\Omega\_\{t\+1\}\. HenceΩt\\Omega\_\{t\}is strictly increasing\.□\\square

##### Part \(iv\): True crisis probability is increasing\.

SinceΠ′\>0\\Pi^\{\\prime\}\>0andΩt\\Omega\_\{t\}is strictly increasing by part \(iii\),πt=Π​\(Ωt\)\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\}\)is strictly increasing\.□\\square

##### Part \(v\): Minsky divergence\.

From part \(i\):π^t=\(1−λ\)t​π^0→0\\hat\{\\pi\}\_\{t\}=\(1\-\\lambda\)^\{t\}\\hat\{\\pi\}\_\{0\}\\to 0ast→∞t\\to\\infty\. From part \(iv\):πt=Π​\(Ωt\)→Π​\(∞\)=1\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\}\)\\to\\Pi\(\\infty\)=1ast→∞t\\to\\infty\(sinceΩt→∞\\Omega\_\{t\}\\to\\inftywhenrb\>δr\_\{b\}\>\\deltaandd​\(a∗\)\>0d\(a^\{\*\}\)\>0\)\. Sinceπ^t→0\\hat\{\\pi\}\_\{t\}\\to 0andπt→1\\pi\_\{t\}\\to 1, there existsT∗<∞T^\{\*\}<\\inftysuch that for allt\>T∗t\>T^\{\*\},π^t<πt\\hat\{\\pi\}\_\{t\}<\\pi\_\{t\}\.□\\square

## Appendix DProof of Proposition[3](https://arxiv.org/html/2606.15078#Thmproposition3)

WriteLt=Π​\(Ωt\)⋅Λ​\(Ωt\)L\_\{t\}=\\Pi\(\\Omega\_\{t\}\)\\cdot\\Lambda\(\\Omega\_\{t\}\)whereΛ​\(Ω\)=κ​𝔼z​\[max⁡\{0,z​B−𝒱​\(K\)\}\]\\Lambda\(\\Omega\)=\\kappa\\,\\mathbb\{E\}\_\{z\}\[\\max\\\{0,zB\-\\mathcal\{V\}\(K\)\\\}\]andB=Ω​KB=\\Omega K\(by definition ofΩ\\Omega\)\.

##### Part \(i\):∂Lt/∂Ωt\>0\\partial L\_\{t\}/\\partial\\Omega\_\{t\}\>0\.

BothΠ​\(Ω\)\\Pi\(\\Omega\)andΛ​\(Ω\)\\Lambda\(\\Omega\)are positive and strictly increasing inΩ\\Omega\(higher leverage raises crisis probability and, conditional on crisis, raises the unsatisfied debt exposure\)\. Their product is therefore strictly increasing\.□\\square

##### Part \(ii\): Convexity\.

Lt′=Π′​Λ\+Π​Λ′L\_\{t\}^\{\\prime\}=\\Pi^\{\\prime\}\\Lambda\+\\Pi\\Lambda^\{\\prime\}, andLt′′=Π′′​Λ\+2​Π′​Λ′\+Π​Λ′′L\_\{t\}^\{\\prime\\prime\}=\\Pi^\{\\prime\\prime\}\\Lambda\+2\\Pi^\{\\prime\}\\Lambda^\{\\prime\}\+\\Pi\\Lambda^\{\\prime\\prime\}\.

We showΛ′′\>0\\Lambda^\{\\prime\\prime\}\>0\. ConsiderΛ​\(Ω\)=κ​𝔼z​\[z​Ω​K−Kα\]\+\\Lambda\(\\Omega\)=\\kappa\\,\\mathbb\{E\}\_\{z\}\[z\\Omega K\-K^\{\\alpha\}\]\_\{\+\}\. For large enoughzz\(above the thresholdz∗​\(Ω\)=Kα−1/Ωz^\{\*\}\(\\Omega\)=K^\{\\alpha\-1\}/\\Omega\), the loss is in the interior\. Differentiating:

Λ′​\(Ω\)=κ​𝔼z​\[z​K⋅𝟏​\{z\>z∗​\(Ω\)\}\]\>0\.\\Lambda^\{\\prime\}\(\\Omega\)=\\kappa\\,\\mathbb\{E\}\_\{z\}\\left\[zK\\cdot\\mathbf\{1\}\\\{z\>z^\{\*\}\(\\Omega\)\\\}\\right\]\>0\.Λ′′​\(Ω\)=κ​𝔼z​\[z​K⋅\(−1\)⋅∂z∗∂Ω​fZ​\(z∗\)\+z​K⋅𝟏​\{z\>z∗\}⋅0\]=κ​K​z∗​\(Ω\)⋅\(−∂z∗/∂Ω\)⋅fZ​\(z∗​\(Ω\)\)\.\\Lambda^\{\\prime\\prime\}\(\\Omega\)=\\kappa\\,\\mathbb\{E\}\_\{z\}\\left\[zK\\cdot\(\-1\)\\cdot\\frac\{\\partial z^\{\*\}\}\{\\partial\\Omega\}f\_\{Z\}\(z^\{\*\}\)\+zK\\cdot\\mathbf\{1\}\\\{z\>z^\{\*\}\\\}\\cdot 0\\right\]=\\kappa\\,K\\,z^\{\*\}\(\\Omega\)\\cdot\(\-\\partial z^\{\*\}/\\partial\\Omega\)\\cdot f\_\{Z\}\(z^\{\*\}\(\\Omega\)\)\.Sincez∗​\(Ω\)=Kα−1/Ωz^\{\*\}\(\\Omega\)=K^\{\\alpha\-1\}/\\Omega, we have∂z∗/∂Ω=−Kα−1/Ω2<0\\partial z^\{\*\}/\\partial\\Omega=\-K^\{\\alpha\-1\}/\\Omega^\{2\}<0, so−∂z∗/∂Ω\>0\-\\partial z^\{\*\}/\\partial\\Omega\>0, yieldingΛ′′​\(Ω\)\>0\\Lambda^\{\\prime\\prime\}\(\\Omega\)\>0\.

WithΛ′′\>0\\Lambda^\{\\prime\\prime\}\>0,Λ′\>0\\Lambda^\{\\prime\}\>0,Π′\>0\\Pi^\{\\prime\}\>0, andΠ≥0\\Pi\\geq 0:L′′=Π′′​Λ\+2​Π′​Λ′\+Π​Λ′′\>2​Π′​Λ′\>0L^\{\\prime\\prime\}=\\Pi^\{\\prime\\prime\}\\Lambda\+2\\Pi^\{\\prime\}\\Lambda^\{\\prime\}\+\\Pi\\Lambda^\{\\prime\\prime\}\>2\\Pi^\{\\prime\}\\Lambda^\{\\prime\}\>0in any region where−Π′′​Λ<2​Π′​Λ′\-\\Pi^\{\\prime\\prime\}\\Lambda<2\\Pi^\{\\prime\}\\Lambda^\{\\prime\}\. For the parametric formΠ​\(Ω\)=1−e−λπ​Ωγ\\Pi\(\\Omega\)=1\-e^\{\-\\lambda\_\{\\pi\}\\Omega^\{\\gamma\}\}withγ≥1\\gamma\\geq 1:Π′′=λπ​e−λπ​Ωγ​\[λπ​γ2​Ω2​γ−2−γ​\(γ−1\)​Ωγ−2\]\\Pi^\{\\prime\\prime\}=\\lambda\_\{\\pi\}e^\{\-\\lambda\_\{\\pi\}\\Omega^\{\\gamma\}\}\[\\lambda\_\{\\pi\}\\gamma^\{2\}\\Omega^\{2\\gamma\-2\}\-\\gamma\(\\gamma\-1\)\\Omega^\{\\gamma\-2\}\], which may be positive \(whenγ\>1\\gamma\>1\), so thatL′′\>0L^\{\\prime\\prime\}\>0globally\.□\\square

##### Part \(iii\): Model concentration\.

Higher AI model concentrationMtM\_\{t\}\(HHI\) raises the correlation of AI errors across agents\. In a correlated error model,Var​\(zt\|Mt\)\\text\{Var\}\(z\_\{t\}\|M\_\{t\}\)increases inMtM\_\{t\}, shifting the distribution of losses to the right\. SinceΛ\\Lambdais convex inzz\(the loss is linear forz\>z∗z\>z^\{\*\}and zero otherwise\),𝔼​\[Λ​\(z\)\]≥Λ​\(𝔼​\[z\]\)\\mathbb\{E\}\[\\Lambda\(z\)\]\\geq\\Lambda\(\\mathbb\{E\}\[z\]\)by Jensen’s inequality, and the inequality tightens as variance increases\. Hence∂Lt/∂Mt\>0\\partial L\_\{t\}/\\partial M\_\{t\}\>0\.□\\square

## Appendix EProof of Proposition[4](https://arxiv.org/html/2606.15078#Thmproposition4)

After the crisis att=0t=0, the agent int=1t=1choosesa1a\_\{1\}subject to the output constraintyi​1≥wy\_\{i1\}\\geq w\. With the constraint binding, we form the Lagrangian:

ℒ​\(a1\)=V1​\(k1′,b1′\)\+λy​\[k1​G​\(a1;q\)−w\]\.\\mathcal\{L\}\(a\_\{1\}\)=V\_\{1\}\(k\_\{1\}^\{\\prime\},b\_\{1\}^\{\\prime\}\)\+\\lambda\_\{y\}\\,\\bigl\[k\_\{1\}G\(a\_\{1\};q\)\-w\\bigr\]\.The constrained FOC fora1a\_\{1\}:

λy​k1​Ga​\(a1;q\)=β​\[μk​ℓ′​\(1−a1\)\+μb​d′​\(a1\)\]\.\\lambda\_\{y\}\\,k\_\{1\}\\,G\_\{a\}\(a\_\{1\};q\)=\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a\_\{1\}\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a\_\{1\}\)\\bigr\]\.\(31\)
Compare with the unconstrained pre\-crisis FOC \([14](https://arxiv.org/html/2606.15078#S3.E14)\), which hask0\>k1k\_\{0\}\>k\_\{1\}\(cognitive capital has declined\) and noλy\\lambda\_\{y\}factor\. The solutiona1∗a\_\{1\}^\{\*\}to \([31](https://arxiv.org/html/2606.15078#A5.E31)\) is higher thana0−a\_\{0\}^\{\-\}if and only if:

λy​k1​Ga​\(a0−;q\)\>β​\[μk​ℓ′​\(1−a0−\)\+μb​d′​\(a0−\)\],\\lambda\_\{y\}\\,k\_\{1\}\\,G\_\{a\}\(a\_\{0\}^\{\-\};q\)\>\\beta\\,\\bigl\[\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-a\_\{0\}^\{\-\}\)\+\\mu\_\{b\}\\,d^\{\\prime\}\(a\_\{0\}^\{\-\}\)\\bigr\],which is condition \([20](https://arxiv.org/html/2606.15078#S5.E20)\)\. This follows because ata0−a\_\{0\}^\{\-\}, the left\-hand side of the constrained FOC \([31](https://arxiv.org/html/2606.15078#A5.E31)\) exceeds the right\-hand side, implying the objective is still increasing inaa, so the optimuma1∗\>a0−a\_\{1\}^\{\*\}\>a\_\{0\}^\{\-\}\.□\\square

Witha1∗\>a0−a\_\{1\}^\{\*\}\>a\_\{0\}^\{\-\}: by the dynamics \([4](https://arxiv.org/html/2606.15078#S2.E4)\)–\([5](https://arxiv.org/html/2606.15078#S2.E5)\),K2<K1K\_\{2\}<K\_\{1\}andB2\>B1B\_\{2\}\>B\_\{1\}, soΩ2\>Ω1\\Omega\_\{2\}\>\\Omega\_\{1\}\. Applying this argument inductively,Ωt\\Omega\_\{t\}is increasing for allt≥1t\\geq 1, andπt=Π​\(Ωt\)\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\}\)is increasing\. Each successive crisis therefore occurs at a higher leverage level, producing weakly greater expected losses \(by Proposition[3](https://arxiv.org/html/2606.15078#Thmproposition3)\)\.□\\square

## Appendix FProof of Proposition[5](https://arxiv.org/html/2606.15078#Thmproposition5)

##### Step 1: Derive the planner’s FOC\.

The planner maximises \([21](https://arxiv.org/html/2606.15078#S6.E21)\), treatingπt=Π​\(Ωt\)\\pi\_\{t\}=\\Pi\(\\Omega\_\{t\}\)as endogenous\. The FOC forAtA\_\{t\}includes the term∂/∂At​\[Π​\(Ωt\)⋅Ltcond\]\\partial/\\partial A\_\{t\}\\bigl\[\\Pi\(\\Omega\_\{t\}\)\\cdot L\_\{t\}^\{\\text\{cond\}\}\\bigr\]\(whereLtcondL\_\{t\}^\{\\text\{cond\}\}is the expected conditional loss\), which the decentralised agent omits\. Expanding:

Π′​\(Ωt\)​∂Ωt∂At​Ltcond\>0\.\\Pi^\{\\prime\}\(\\Omega\_\{t\}\)\\,\\frac\{\\partial\\Omega\_\{t\}\}\{\\partial A\_\{t\}\}\\,L\_\{t\}^\{\\text\{cond\}\}\>0\.The planner also internalises∂Kt\+1/∂At=−ℓ′​\(1−At\)<0\\partial K\_\{t\+1\}/\\partial A\_\{t\}=\-\\ell^\{\\prime\}\(1\-A\_\{t\}\)<0as a social cost \(not just a private cost\), and the benchmark\-shifting externality∂y¯t/∂ai​t\\partial\\bar\{y\}\_\{t\}/\\partial a\_\{it\}\.

##### Step 2: Wedge between planner and decentralised FOCs\.

In the decentralised equilibrium, the agent’s FOC \([14](https://arxiv.org/html/2606.15078#S3.E14)\) is:

k​𝔼​\[Gaeff​\(a∗;π^,q\)\]=β​\[μkD​ℓ′​\(1−a∗\)\+μbD​d′​\(a∗\)\],k\\,\\mathbb\{E\}\\bigl\[G\_\{a\}^\{\\text\{eff\}\}\(a^\{\*\};\\hat\{\\pi\},q\)\\bigr\]=\\beta\\,\\bigl\[\\mu\_\{k\}^\{D\}\\,\\ell^\{\\prime\}\(1\-a^\{\*\}\)\+\\mu\_\{b\}^\{D\}\\,d^\{\\prime\}\(a^\{\*\}\)\\bigr\],\(FOC\-D\)whereμkD,μbD\\mu\_\{k\}^\{D\},\\mu\_\{b\}^\{D\}are private shadow values that omit the externalities\.

The planner’s FOC is:

k​𝔼​\[Gaeff​\(AP;π^,q\)\]=β​\[μkP​ℓ′​\(1−AP\)\+μbP​d′​\(AP\)\],k\\,\\mathbb\{E\}\\bigl\[G\_\{a\}^\{\\text\{eff\}\}\(A^\{P\};\\hat\{\\pi\},q\)\\bigr\]=\\beta\\,\\bigl\[\\mu\_\{k\}^\{P\}\\,\\ell^\{\\prime\}\(1\-A^\{P\}\)\+\\mu\_\{b\}^\{P\}\\,d^\{\\prime\}\(A^\{P\}\)\\bigr\],\(FOC\-P\)whereμkP\>μkD\\mu\_\{k\}^\{P\}\>\\mu\_\{k\}^\{D\}\(planner values cognitive capital more, accounting for the public good\) and the left\-hand side includes the systemic risk term\.

The higher right\-hand side in \(FOC\-P\) impliesAP<ADA^\{P\}<A^\{D\}\.□\\square

##### Step 3: Pigouvian tax\.

The optimal taxτt∗\\tau\_\{t\}^\{\*\}is set so that the decentralised agent, facing the after\-tax problem, replicates the planner’s FOC\. The required tax on AI substitution intensityai​ta\_\{it\}is the sum of the three marginal external costs evaluated at\(AtP,XtP\)\(A\_\{t\}^\{P\},X\_\{t\}^\{P\}\):

1. 1\.Systemic risk:Π′​\(Ωt\)​\(∂Ωt/∂At\)​Lt/Π​\(Ωt\)\\Pi^\{\\prime\}\(\\Omega\_\{t\}\)\\,\(\\partial\\Omega\_\{t\}/\\partial A\_\{t\}\)\\,L\_\{t\}/\\Pi\(\\Omega\_\{t\}\)\(marginal increase in expected loss per unit ofAtA\_\{t\}\)\.
2. 2\.Public goods:β​μk​\|∂Kt\+1/∂At\|=β​μk​ℓ′​\(1−AtP\)\\beta\\,\\mu\_\{k\}\\,\|\\partial K\_\{t\+1\}/\\partial A\_\{t\}\|=\\beta\\,\\mu\_\{k\}\\,\\ell^\{\\prime\}\(1\-A\_\{t\}^\{P\}\)\.
3. 3\.Arms\-race:λy​∂y¯t/∂ai​t\\lambda\_\{y\}\\,\\partial\\bar\{y\}\_\{t\}/\\partial a\_\{it\}\(marginal increase in the competitive benchmark imposed on others\)\.

Summing these gives \([22](https://arxiv.org/html/2606.15078#S6.E22)\)\.

##### Step 4: The gapΔt\\Delta\_\{t\}is increasing inMtM\_\{t\}, tranquil period length, andd′′d^\{\\prime\\prime\}\.

All three factors lower the private cost of AI use \(relative to social cost\) or raise the systemic externality:

- •HigherMtM\_\{t\}: increasesVar​\(z\|crisis\)\\text\{Var\}\(z\|\\text\{crisis\}\), raisingLtL\_\{t\}and the first term in \([22](https://arxiv.org/html/2606.15078#S6.E22)\), widening the wedge\.
- •Longer tranquil period:Ωt\\Omega\_\{t\}is higher \(by Proposition[2](https://arxiv.org/html/2606.15078#Thmproposition2)\), raising all three externality terms\.
- •Higherd′′d^\{\\prime\\prime\}: makes the private debt cost steeper, which actually reduces individuala∗a^\{\*\}, but also raises the social risk of Ponzi dynamics, widening the systemic externality\.

□\\square

## Appendix GInfinite\-Horizon Characterisation

In the infinite\-horizon problem \([10](https://arxiv.org/html/2606.15078#S3.E10)\), the value functionV​\(k,b;π^,q\)V\(k,b;\\,\\hat\{\\pi\},q\)satisfies the Bellman equation\. We verify that the comparative statics of Proposition[1](https://arxiv.org/html/2606.15078#Thmproposition1)extend to this setting\.

###### Lemma 1\.

Under Assumptions[1](https://arxiv.org/html/2606.15078#Thmassumption1)–[3](https://arxiv.org/html/2606.15078#Thmassumption3), the value functionVVis continuously differentiable withVk\>0V\_\{k\}\>0andVb<0V\_\{b\}<0\.

###### Proof\.

Standard application of the Theorem of the Maximum and the Benveniste–Scheinkman envelope theorem, given thatGGandddareC2C^\{2\}and the state space is compact \(after imposing appropriate bounds onkkandbb\)\.□\\square∎

The envelope conditions yield:

μkt\\displaystyle\\mu\_\{k\}^\{t\}≡Vk​\(kt,bt\)=\(1−δ\)​β​Vk​\(kt\+1,bt\+1\)\+marginal product of​kt\\displaystyle\\equiv V\_\{k\}\(k\_\{t\},b\_\{t\}\)=\(1\-\\delta\)\\beta\\,V\_\{k\}\(k\_\{t\+1\},b\_\{t\+1\}\)\+\\text\{marginal product of \}k\_\{t\}\(32\)μbt\\displaystyle\\mu\_\{b\}^\{t\}≡−Vb​\(kt,bt\)=\(1\+rb\)​β​\(−Vb​\(kt\+1,bt\+1\)\)\+κ​π^t​z¯\\displaystyle\\equiv\-V\_\{b\}\(k\_\{t\},b\_\{t\}\)=\(1\+r\_\{b\}\)\\beta\\,\(\-V\_\{b\}\(k\_\{t\+1\},b\_\{t\+1\}\)\)\+\\kappa\\,\\hat\{\\pi\}\_\{t\}\\,\\bar\{z\}\(33\)These are the dynamic counterparts ofμk\\mu\_\{k\}andμb\\mu\_\{b\}in the two\-period model\. The FOC \([14](https://arxiv.org/html/2606.15078#S3.E14)\) holds at each date withμk=μkt\\mu\_\{k\}=\\mu\_\{k\}^\{t\}andμb=μbt\\mu\_\{b\}=\\mu\_\{b\}^\{t\}, confirming that the two\-period characterisation applies period by period in the infinite\-horizon setting\. The comparative statics are therefore identical to those derived in Appendix[B](https://arxiv.org/html/2606.15078#A2)\.

## Appendix HProof of Proposition[6](https://arxiv.org/html/2606.15078#Thmproposition6)

##### Part \(i\):Δ​kt\\Delta k\_\{t\}is strictly decreasing\.

The cognitive capital of each type evolves as:

kH,t\+1\\displaystyle k\_\{H,t\+1\}=\(1−δ\)​kH,t\+ℓ​\(1−aH,t∗\),\\displaystyle=\(1\-\\delta\)\\,k\_\{H,t\}\+\\ell\(1\-a\_\{H,t\}^\{\*\}\),kL,t\+1\\displaystyle k\_\{L,t\+1\}=\(1−δ\)​kL,t\+ℓ​\(1−aL,t∗\)\.\\displaystyle=\(1\-\\delta\)\\,k\_\{L,t\}\+\\ell\(1\-a\_\{L,t\}^\{\*\}\)\.Taking the difference:

Δ​kt\+1=\(1−δ\)​Δ​kt\+ℓ​\(1−aH,t∗\)−ℓ​\(1−aL,t∗\)\.\\Delta k\_\{t\+1\}=\(1\-\\delta\)\\,\\Delta k\_\{t\}\+\\ell\(1\-a\_\{H,t\}^\{\*\}\)\-\\ell\(1\-a\_\{L,t\}^\{\*\}\)\.SinceaH,t∗\>aL,t∗a\_\{H,t\}^\{\*\}\>a\_\{L,t\}^\{\*\}\(from \([23](https://arxiv.org/html/2606.15078#S7.E23)\) whileΔ​kt\>0\\Delta k\_\{t\}\>0\) andℓ′​\(⋅\)\>0\\ell^\{\\prime\}\(\\cdot\)\>0, we have1−aH,t∗<1−aL,t∗1\-a\_\{H,t\}^\{\*\}<1\-a\_\{L,t\}^\{\*\}and thusℓ​\(1−aH,t∗\)<ℓ​\(1−aL,t∗\)\\ell\(1\-a\_\{H,t\}^\{\*\}\)<\\ell\(1\-a\_\{L,t\}^\{\*\}\)\. Hence:

Δ​kt\+1<\(1−δ\)​Δ​kt<Δ​kt\.\\Delta k\_\{t\+1\}<\(1\-\\delta\)\\,\\Delta k\_\{t\}<\\Delta k\_\{t\}\.Δ​kt\\Delta k\_\{t\}is strictly decreasing\.□\\square

##### Part \(ii\):Δ​at∗\\Delta a\_\{t\}^\{\*\}converges to zero\.

By the implicit function theorem applied to the FOC \([14](https://arxiv.org/html/2606.15078#S3.E14)\):

∂a∗∂k=−∂H/∂k∂H/∂a=−\(1−π^\)​Ga\+π^​G~a∂H/∂a\>0\.\\frac\{\\partial a^\{\*\}\}\{\\partial k\}=\-\\frac\{\\partial H/\\partial k\}\{\\partial H/\\partial a\}=\-\\frac\{\(1\-\\hat\{\\pi\}\)G\_\{a\}\+\\hat\{\\pi\}\\tilde\{G\}\_\{a\}\}\{\\partial H/\\partial a\}\>0\.AsΔ​kt→0\\Delta k\_\{t\}\\to 0, the types become identical, soΔ​at∗→0\\Delta a\_\{t\}^\{\*\}\\to 0\.□\\square

##### Part \(iii\): Reversal of fortune\.

The gap dynamics satisfyΔ​kt\+1=\(1−δ\)​Δ​kt−\[ℓ​\(1−aL,t∗\)−ℓ​\(1−aH,t∗\)\]\\Delta k\_\{t\+1\}=\(1\-\\delta\)\\Delta k\_\{t\}\-\[\\ell\(1\-a\_\{L,t\}^\{\*\}\)\-\\ell\(1\-a\_\{H,t\}^\{\*\}\)\]\.

By the mean value theorem:ℓ​\(1−aL,t∗\)−ℓ​\(1−aH,t∗\)=ℓ′​\(ξt\)​Δ​at∗\\ell\(1\-a\_\{L,t\}^\{\*\}\)\-\\ell\(1\-a\_\{H,t\}^\{\*\}\)=\\ell^\{\\prime\}\(\\xi\_\{t\}\)\\,\\Delta a\_\{t\}^\{\*\}for someξt∈\(1−aH,t∗,1−aL,t∗\)\\xi\_\{t\}\\in\(1\-a\_\{H,t\}^\{\*\},1\-a\_\{L,t\}^\{\*\}\)\.

If condition \([24](https://arxiv.org/html/2606.15078#S7.E24)\) holds \(the erosion differential exceeds the natural decay\-corrected gap\), then:

Δ​kt\+1=\(1−δ\)​Δ​kt−ℓ′​\(ξt\)​Δ​at∗<0⋅Δ​kt\\Delta k\_\{t\+1\}=\(1\-\\delta\)\\Delta k\_\{t\}\-\\ell^\{\\prime\}\(\\xi\_\{t\}\)\\Delta a\_\{t\}^\{\*\}<0\\cdot\\Delta k\_\{t\}whenΔ​kt\\Delta k\_\{t\}is close to zero from above\. By continuity,Δ​kt\\Delta k\_\{t\}must cross zero at some finiteT∗∗T^\{\*\*\}\. After crossing, the roles reverse:kH,t<kL,tk\_\{H,t\}<k\_\{L,t\}, and—since∂a∗/∂k\>0\\partial a^\{\*\}/\\partial k\>0—nowaH,t∗<aL,t∗a\_\{H,t\}^\{\*\}<a\_\{L,t\}^\{\*\}, making the gap\|Δ​kt\|\|\\Delta k\_\{t\}\|persistent in the new direction\.□\\square

##### Part \(iv\): Heterogeneity amplifies aggregate losses\.

Expected aggregate loss is:

Lthetero=12​Π​\(ΩH,t\)​Λ​\(ΩH,t\)\+12​Π​\(ΩL,t\)​Λ​\(ΩL,t\)\.L\_\{t\}^\{\\text\{hetero\}\}=\\tfrac\{1\}\{2\}\\,\\Pi\(\\Omega\_\{H,t\}\)\\,\\Lambda\(\\Omega\_\{H,t\}\)\+\\tfrac\{1\}\{2\}\\,\\Pi\(\\Omega\_\{L,t\}\)\\,\\Lambda\(\\Omega\_\{L,t\}\)\.By Proposition[3](https://arxiv.org/html/2606.15078#Thmproposition3),L​\(Ω\)≡Π​\(Ω\)​Λ​\(Ω\)L\(\\Omega\)\\equiv\\Pi\(\\Omega\)\\Lambda\(\\Omega\)is convex\. Jensen’s inequality applied to the two\-point distribution ofΩt\\Omega\_\{t\}gives:

Lthetero=12​\[L​\(ΩH,t\)\+L​\(ΩL,t\)\]\>L​\(ΩH,t\+ΩL,t2\)=Lthomo,L\_\{t\}^\{\\text\{hetero\}\}=\\tfrac\{1\}\{2\}\[L\(\\Omega\_\{H,t\}\)\+L\(\\Omega\_\{L,t\}\)\]\>L\\\!\\left\(\\tfrac\{\\Omega\_\{H,t\}\+\\Omega\_\{L,t\}\}\{2\}\\right\)=L\_\{t\}^\{\\text\{homo\}\},where the last equality holds because the homogeneous economy has leverage\(ΩH,t\+ΩL,t\)/2\(\\Omega\_\{H,t\}\+\\Omega\_\{L,t\}\)/2\(since both types start with equal means and the distributions are mirror images aroundk¯\\bar\{k\}\)\.□\\square

## Appendix IShadow Value Derivation

We deriveμk\\mu\_\{k\}andμb\\mu\_\{b\}used in the two\-period model from the steady\-state conditions of the infinite\-horizon problem\.

In the stationary equilibrium with constanta,x,k,ba,x,k,band subjective probabilityπ^\\hat\{\\pi\}:

μk\\displaystyle\\mu\_\{k\}=\(1−π^\)​G​\(a;q\)\+π^​G~​\(a;q,z¯\)1−β​\(1−δ\),\\displaystyle=\\frac\{\(1\-\\hat\{\\pi\}\)\\,G\(a;q\)\+\\hat\{\\pi\}\\,\\tilde\{G\}\(a;q,\\bar\{z\}\)\}\{1\-\\beta\(1\-\\delta\)\},\(34\)μb\\displaystyle\\mu\_\{b\}=π^​κ​z¯1−β​\(1\+rb\)\.\\displaystyle=\\frac\{\\hat\{\\pi\}\\,\\kappa\\,\\bar\{z\}\}\{1\-\\beta\(1\+r\_\{b\}\)\}\.\(35\)
Note thatμb\>0\\mu\_\{b\}\>0requiresβ​\(1\+rb\)<1\\beta\(1\+r\_\{b\}\)<1, which we assume\. This condition says the debt cannot grow faster than the discount rate—otherwise cognitive debt would be “free” and Ponzi cognition would be trivially optimal\.

Substituting \([34](https://arxiv.org/html/2606.15078#A9.E34)\)–\([35](https://arxiv.org/html/2606.15078#A9.E35)\) into \([14](https://arxiv.org/html/2606.15078#S3.E14)\) yields the explicit steady\-state condition relatinga∗a^\{\*\}to the structural parameters\(π^,q,k,β,δ,rb,κ,z¯\)\(\\hat\{\\pi\},q,k,\\beta,\\delta,r\_\{b\},\\kappa,\\bar\{z\}\)\.