Managing the Human Fallback: Skill Investment Under Improving AI and Worker Mobility

# 1 Introduction
Source: [https://arxiv.org/html/2606.29111](https://arxiv.org/html/2606.29111)
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Singh, Ghosh, and Dai\\RUNTITLESkill Investment Under Improving AI and Worker Mobility\\TITLEManaging the Human Fallback: Skill Investment Under Improving AI and Worker Mobility\\ARTICLEAUTHORS\\AUTHORSimrita Singh∗Naireet Ghosh†Tinglong Dai‡

\\AFF

∗Leavey School of Business, Santa Clara University, Santa Clara, California, 95053[ssingh17@scu\.edu](https://arxiv.org/html/2606.29111v1/mailto:[email protected]) †Naveen Jindal School of Management, University of Texas at Dallas, Richardson, Texas, 75080[naireet\.ghosh@utdallas\.edu](https://arxiv.org/html/2606.29111v1/mailto:[email protected]) ‡Carey Business School, Johns Hopkins University, Baltimore, Maryland 21202,[dai@jhu\.edu](https://arxiv.org/html/2606.29111v1/mailto:[email protected])\\ABSTRACTWhen firms deploy autonomous AI, they must decide how much work to leave to the system and how much to keep workers engaged\. This decision affects current output and future human capital\. We develop a parsimonious two\-period model in which AI may outperform the worker when it functions, but may fail with positive probability\. A firm chooses worker engagement; engagement lowers current output for below\-benchmark workers, but changes future skill through learning and erosion\. We distinguish two dimensions of AI progress: capability, the system’s output when it works, and reliability, the probability that it works\. In a single\-firm benchmark, engagement is valuable only as fallback investment\. The firm engages the least\-skilled workers most, because they have the largest skill gaps and are least costly to bring toward a useful fallback level\. With worker mobility, engagement also affects labor\-market sorting: workers prefer jobs that build more valuable skill trajectories\. This sorting motive targets higher\-skill workers near the AI frontier, where skill gains are more valuable and engagement is less costly\. Mobility can therefore reverse the engagement pattern, shifting investment from the least\-skilled toward the most\-skilled workers below the AI benchmark\. Mobility also reshapes how AI progress affects engagement: greater capability raises engagement by increasing the value of the skill trajectory a firm offers, whereas greater reliability can raise or lower it because it reduces fallback need while also changing learning opportunities\. Under worker mobility, human–AI work design becomes a problem of human\-capital investment, in which allocating work today shapes future skill\.\\KEYWORDSHuman–AI interaction, service operations, human capital, workforce management

As of 2026, about three\-quarters of new code at Google is generated by AI and approved by engineers, up from about half in late 2025\(Pichai[2026](https://arxiv.org/html/2606.29111#bib.bib71)\)\. AI writes the code; a person reviews it and answers for it\(The Linux Kernel[2026](https://arxiv.org/html/2606.29111#bib.bib70)\)\. When the code is wrong or insecure, when a request falls outside what the model has seen, or when the assistant is unavailable, an engineer must step in and do the work unaided\. A firm that deploys an improving AI must decide how much human skill to keep in reserve for the moments AI falls short\.

The same division of labor extends to other settings in which a capable but imperfect AI shares a task with a person\. Waymo’s vehicles complete rides with no driver, at crash rates below human benchmarks, yet a vehicle that meets an ambiguous road calls a remote specialist for guidance\(Kusanoet al\.[2024](https://arxiv.org/html/2606.29111#bib.bib55), Waymo[2024](https://arxiv.org/html/2606.29111#bib.bib53)\)\. A radiologist adjudicates what an algorithm has flagged, and an underwriter signs the risk a model has scored\. In each case, AI absorbs more of the routine work, and human skill is most valuable exactly where AI reaches the limits of its competence, limits that arise less and less often as that absorption proceeds\.

It is tempting to view this division of labor as a transitional arrangement on the way to production with little human involvement\. Popular accounts take that view: Yuval Noah Harari warns of a coming “useless class”\(Harari[2017](https://arxiv.org/html/2606.29111#bib.bib68)\), and Sam Altman predicts that “whole classes of jobs” may disappear\(Altman[2025](https://arxiv.org/html/2606.29111#bib.bib69)\)\. For the firm itself, the decision is subtler\. Human skill remains valuable because AI is imperfect: the worker is the fallback when AI fails, faces a case outside its domain, or is unavailable\. But the skill is portable, so part of the return to preserving it accrues to the worker or another employer; and engaging the worker is costly, because AI often does this work better than the worker would\. Reducing engagement thus raises current output but erodes the skill the firm may later need\. The same progress that makes AI less costly to lean on also makes the skill it displaces harder to sustain\. We study how this friction shapes worker engagement: the extent to which a firm keeps the worker actively involved in AI\-assisted production\.111Institutions raise engagement through concrete instruments: AI\-free settings, a minimum number of cases handled without AI before access is granted, staged access, and protocols that have the worker commit to a judgment before consulting AI\(Kerenet al\.[2026](https://arxiv.org/html/2606.29111#bib.bib72)\)\. Shell, for example, requires early\-career employees to frame a problem before using AI\(Boston Consulting Group[2026](https://arxiv.org/html/2606.29111#bib.bib73)\)\.Engaging the worker lowers output today but preserves skill valued both inside the firm and in the labor market\.

A worker’s human capital is not fixed; sustained reliance on AI can weaken it\. In software, programmers who leaned on AI to learn a new library gained little speed on average yet ended up worse at reading and debugging code unaided\(Shen and Tamkin[2026](https://arxiv.org/html/2606.29111#bib.bib25)\)\. In education, high school students given an unguarded AI tutor solved more problems while they had it, but once it was withdrawn they scored below classmates who had never used it\(Bastaniet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib20)\)\. In medicine, after routine exposure to AI\-assisted colonoscopy, experienced endoscopists’ unassisted adenoma\-detection rate fell from 28\.4% to 22\.4%\(Budzyńet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib23)\)\. In a survey of business leaders, about half already observe deskilling, and more than 60% expect it to become a real threat within three to five years\(Boston Consulting Group[2026](https://arxiv.org/html/2606.29111#bib.bib73)\)\. Engagement is costly today, but disengagement erodes the human capital future production may need\.

Workers, in turn, value the skill a job builds\. Career advancement, learning, and skill development have become central to whether they take, keep, or leave a job\(Gallup[2024](https://arxiv.org/html/2606.29111#bib.bib49), Pew Research Center[2022](https://arxiv.org/html/2606.29111#bib.bib46)\)\. A job that routes most of its work to AI builds little human capital, and a mobile worker has reason to leave it for one that builds more\.

This matters most where firms cannot differentiate on pay\. When wages for comparable roles are standardized by salary bands, internal\-equity norms, and external market benchmarking\(Bakeret al\.[1988](https://arxiv.org/html/2606.29111#bib.bib12), Mas[2017](https://arxiv.org/html/2606.29111#bib.bib13)\), a firm cannot simply outbid a rival for a worker who could leave; what it can offer instead is the skill trajectory the job builds\. Engagement, the same lever that governs current output, then becomes the instrument through which a firm competes for mobile workers\.

We develop a parsimonious two\-period model of skill investment when an improving AI shares a task with a mobile worker\. A firm chooses how actively to involve a worker in an AI\-assisted task; engagement shapes current output and, through learning\-by\-doing and erosion, the worker’s future skill\. We model two attributes of AI along which it improves:*capability*, how good its output is when it functions, and*reliability*, how often it functions as intended\. Both are visible in the settings above: a more capable system handles more of the road, or more of the codebase, on its own, whereas a more reliable one fails or defers to a person less often\. We focus on below\-benchmark workers, whose unaided output falls short of what AI delivers; for them engagement lowers current output, because AI is faster or more accurate, but it builds skill the firm may later need\. Models of automation allocate tasks between people and machines but take the skill distribution as fixed\(Acemoglu and Restrepo[2018](https://arxiv.org/html/2606.29111#bib.bib16), Autoret al\.[2003](https://arxiv.org/html/2606.29111#bib.bib14)\); here engagement reshapes it\.

A firm engages such a worker, despite the current cost, because engagement builds the worker’s skill\. Engagement today sets the skill the worker carries forward, so the firm weighs current output against a future in which that skill may be its fallback, its draw in the labor market, or both\. Without mobility, that skill is worth preserving as a fallback for when AI fails\. Under mobility, two ex ante identical firms set engagement policies and workers sort across them by the skill trajectories those policies imply; engagement also attracts workers, whose skill the firm cannot fully keep\.

Our analysis yields three findings\. First, the two motives target opposite ends of the skill range\. Without mobility, the fallback motive drives engagement toward the lowest\-skilled workers: they sit furthest below AI, so they have the most room to grow, and a unit of engagement builds the most skill\. Mobility adds a sorting motive, strongest for higher\-skilled workers, for whom added skill is worth the most in the labor market and so does the most to draw them\. The same skill is a fallback where workers are weakest and a draw where they are strongest, so which workers a firm engages most can reverse, from the bottom of the skill range to the top, once workers are mobile\. This is the engagement reversal: the firm’s lever does not change, but the workers it engages most can shift from the least\- to the most\-skilled as the labor market opens\.

Second, the two dimensions of AI progress can move engagement in opposite directions\. Reliability sets how often human skill is called on\. Greater reliability makes the fallback less likely to be needed, weakening one reason to engage; but workers build skill best with a system that mostly works yet occasionally fails, so engagement can be highest at intermediate reliability\. A system that never fails leaves the worker nothing to stay ready for, while one that fails often is barely worth deploying; the room to build skill lies in between\. Capability does not by itself change how often workers are needed for fallback, but under mobility a more capable AI widens the gap from which workers learn, strengthening the trajectory a firm can offer\. Treating AI progress as a single index conflates channels that can point in opposite directions\.

Third, mobility reshapes engagement relative to the no\-mobility benchmark\. Because the skill a firm builds is general, it cannot capture the full return, so mobility should depress investment, asBecker \([1962](https://arxiv.org/html/2606.29111#bib.bib5)\)first argued\.222In Becker’s analysis of general \(fully portable\) training, a competitive firm bears none of its cost: because a trained worker can quit and command the now\-higher marginal product at any employer, wages rise to match it and the firm recoups nothing\. As Becker puts it, “the cost as well as the return from general training would be borne by trainees, not by firms” \(Becker[1962](https://arxiv.org/html/2606.29111#bib.bib5), p\. 13\)\. The worker therefore finances such training through lower wages while training and captures its return afterward; only firm\-specific training is shared between firm and worker\.We prove that this holds when workers respond weakly to differences in skill trajectories: engagement falls below the single\-firm benchmark\. When they respond strongly, the sorting motive can instead reverse the timing of engagement, so that mobility need not uniformly discourage investment in general skill\. Mobility can also break the symmetry between ex ante identical firms, with one investing in skill while the other draws on the shared labor pool\.

To our knowledge, this is among the first analytical models in which a firm’s engagement with an improving AI shapes the skill of mobile workers\. It identifies a sorting motive for skill investment, absent from existing work on AI\-assisted production, by which firms differentiate through the skill a job builds when they cannot compete on pay\(Bastani and Cachon[2025](https://arxiv.org/html/2606.29111#bib.bib32), Caosun and Aral[2026](https://arxiv.org/html/2606.29111#bib.bib26), Dai and Taylor[2025](https://arxiv.org/html/2606.29111#bib.bib34), Lu and Tomlin[2025](https://arxiv.org/html/2606.29111#bib.bib37), Sideriuset al\.[2026](https://arxiv.org/html/2606.29111#bib.bib31), Xuet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib38)\)\.

Together, these results recast the design of human–AI work as a human\-capital problem as much as an operational one\. Wherever an improving but imperfect AI shares a task with a mobile worker, whether in autonomous driving, software, radiology, underwriting, contact centers, or legal review, the firm’s engagement choice governs not only today’s output but the skill it will have on hand tomorrow and the workers it can keep\.

The remainder of the paper is organized as follows\.[Section2](https://arxiv.org/html/2606.29111#S2)reviews the related literature\.[Section3](https://arxiv.org/html/2606.29111#S3)presents the model\.[Section4](https://arxiv.org/html/2606.29111#S4)analyzes the single\-firm benchmark\.[Section5](https://arxiv.org/html/2606.29111#S5)studies how worker mobility reshapes engagement\.[Section6](https://arxiv.org/html/2606.29111#S6)discusses implications and concludes\.

## 2Related Literature

Our paper connects two questions that are often treated separately\. The first is how automation changes the use and preservation of human skill\. The second is how firms invest in human capital\. AI\-assisted work brings these questions together\. When the system outperforms the worker, engagement is costly today\. Yet engagement may also preserve skill that the firm needs as fallback and that the worker values in the labor market\. Our contribution is to model this tradeoff and show how worker mobility changes both the level of engagement and the workers to whom engagement is directed\.

The first question, how automation changes the use and preservation of human skill, has been studied from several angles\. The most direct is AI\-induced*deskilling*: more reliable systems erode the human capabilities needed when those systems fail\.Bainbridge \([1983](https://arxiv.org/html/2606.29111#bib.bib43)\)identifies the “ironies of automation,” in which more reliable systems leave operators less able to intervene when failures occur, andParasuraman and Riley \([1997](https://arxiv.org/html/2606.29111#bib.bib44)\)formalize the related taxonomy of automation misuse, disuse, and abuse\. Modern AI generates the same pattern:Bastaniet al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib20)\)find unrestricted generative\-AI access improves high\-school students’ performance during practice but lowers exam scores once access is removed;Poulidiset al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib21)\)show learners over\-request help even when they understand its long\-term cost; andDell’Acquaet al\.\([2023](https://arxiv.org/html/2606.29111#bib.bib22)\)find AI raises consultants’ productivity within its frontier but degrades it outside, because workers anchor to AI output\. Clinical evidence points the same way: endoscopists’ unassisted adenoma\-detection declined after routine AI\-assisted colonoscopy\(Budzyńet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib23)\), andAbdulnouret al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib27)\),Nataliet al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib24)\), andShen and Tamkin \([2026](https://arxiv.org/html/2606.29111#bib.bib25)\)document deskilling, never\-skilling, and mis\-skilling in medical training and software\. This literature establishes that reliance on automation can erode the skill needed in failure states, but it treats that erosion as a byproduct of automation rather than as something a firm controls\. We make erosion and learning*endogenous*to the firm’s engagement policy and ask which workers a profit\-maximizing firm keeps engaged\.

A second angle is human–AI co\-production, in which a central finding is that adding humans to AI need not improve output\.Vaccaroet al\.\([2024](https://arxiv.org/html/2606.29111#bib.bib30)\), in a meta\-analysis of experiments, find human–AI combinations often underperform the better of human or AI alone, especially when AI is stronger\.de Véricourt and Gürkan \([2026](https://arxiv.org/html/2606.29111#bib.bib35)\)provide an analytical explanation: when decision\-makers supervise AI output, verification bias can keep them from learning whether the machine outperforms their own judgment\.Gohet al\.\([2024](https://arxiv.org/html/2606.29111#bib.bib41)\)document a related pattern in a randomized clinical trial: physician access to an LLM did not improve diagnostic reasoning even though the LLM alone performed well, suggesting that integrating AI advice into expert judgment is itself difficult\.Agarwalet al\.\([2023](https://arxiv.org/html/2606.29111#bib.bib42)\)show a similar friction in radiology: AI predictions alone did not improve performance on average because radiologists underweighted the AI signal and did not combine it correctly with their own information, whereas contextual information improved performance\.Dai and Singh \([2025](https://arxiv.org/html/2606.29111#bib.bib40)\)show that when physicians exhibit anchoring bias, AI should serve as a gatekeeper for low\-risk patients and a second opinion for high\-risk patients\.

AI assistance also appears most helpful for less\-experienced workers:Brynjolfssonet al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib28)\)find generative AI raises customer\-support productivity on average, with gains concentrated among less\-experienced agents and small quality declines among the most skilled, andNiet al\.\([2024](https://arxiv.org/html/2606.29111#bib.bib29)\)find a similar skill\-dependent pattern in a field experiment at Alibaba\. This literature studies whether and how to pair people with AI; we study the dynamic skill consequences of keeping a worker engaged\. Our engagement variable is therefore distinct: not whether a worker has access to AI, but whether the worker stays actively involved in production when the AI baseline is already stronger\. Keeping a below\-benchmark worker engaged is operationally costly, pulling output away from the AI baseline, even as it builds skill through learning\-by\-doing\.

Closest to our model is a growing literature on the design of human–AI work\.Sideriuset al\.\([2026](https://arxiv.org/html/2606.29111#bib.bib31)\)develop a principal\-agent model in which workers exert costly effort to verify imperfect AI output and show profit\-maximizing compensation can be non\-monotonic in AI quality\.Caosun and Aral \([2026](https://arxiv.org/html/2606.29111#bib.bib26)\)develop a continuous\-time model in which a decision\-maker chooses AI usage intensity, showing forward\-looking adoption can rationally lead to long\-run skill loss, an “augmentation trap\.”Bastani and Cachon \([2025](https://arxiv.org/html/2606.29111#bib.bib32)\)identify a “human–AI contracting paradox”: as AI becomes more reliable, motivating human vigilance becomes prohibitively costly, so firms may prefer less reliable AI\.Dai and Taylor \([2025](https://arxiv.org/html/2606.29111#bib.bib34)\)study a principal\-agent model in which firms jointly choose AI temperature and effort\-contingent pay, showing endogenizing AI design can reverse classical delegation results by making temperature and worker effort complementary levers\.Lu and Tomlin \([2025](https://arxiv.org/html/2606.29111#bib.bib37)\)show fully disclosing an AI demand forecast can reduce managerial effort below what preserves the machine’s value, and characterize partial disclosure as the optimal design response\.Xuet al\.\([2025](https://arxiv.org/html/2606.29111#bib.bib38)\)show AI’s effect on entry\-level skill requirements and organizational span of control depends on whether firms deploy it as automation or augmentation\. These models study verification, contracting, disclosure, or AI\-use intensity, each within a single firm, with AI progress as a single dimension and the worker tied to one employer even as skill changes over the long run\. We ask how these forces interact, and our model brings them together: engagement trades current production against future skill; AI improves along two distinct dimensions, capability and reliability, which affect engagement through different channels; and workers sort across firms by the skill trajectories engagement creates\.

The second question, how firms invest in skill when workers are mobile, belongs to the human\-capital literature\. The economics of human capital, followingBecker \([1962](https://arxiv.org/html/2606.29111#bib.bib5)\)andMincer \([1974](https://arxiv.org/html/2606.29111#bib.bib6)\), establishes that skills accumulate through education and on\-the\-job training\.Acemoglu and Pischke \([1999](https://arxiv.org/html/2606.29111#bib.bib10)\)show labor\-market frictions, particularly compressed wage structures, enable firm\-sponsored general training that would unravel in a frictionless market\. The operations\-management literature has modeled closely related workforce\-learning and turnover tradeoffs with dynamic staffing and recruitment models:Gans and Zhou \([2002](https://arxiv.org/html/2606.29111#bib.bib50)\)on staffing when employees learn and turn over,Arlottoet al\.\([2014](https://arxiv.org/html/2606.29111#bib.bib51)\)on hiring and retention for heterogeneous workers who learn, andWhitt \([2006](https://arxiv.org/html/2606.29111#bib.bib52)\)on retention and contact\-center performance\. Workers may also sort across employers by the skill trajectories their jobs create\. We contribute to this literature by identifying a sorting channel for skill investment under wage standardization: when firms cannot differentiate through pay, they differentiate through the skill trajectories their engagement policies imply\. Unlike the classical prediction that mobility weakens incentives to invest in general skill, this channel can strengthen them, shifting engagement toward higher\-skill workers, the opposite end from the fallback motive, and it can lead ex ante identical firms to specialize\.

## 3Model

We study a firm that, over two periods, chooses worker engagement with an improving AI system\. The worker is below the AI benchmark \(the worker’s effective throughput is lower than that of AI when it functions\), so engagement lowers current output\. The value of engagement is instead dynamic: it slows erosion and can build skill that matters later as fallback capacity and, when workers are mobile, as a labor\-market draw\. The model therefore has three primitives: AI capability and failure risk, skill dynamics under engagement, and a wage schedule that values portable skill\.

### 3\.1Environment and Technology

The model spansTTperiods,t=1,2,…,Tt=1,2,\\ldots,T\. A firm employs a continuum of workers who do not interact in production, so we describe its problem for a single worker of skillst∈\[0,1\]s\_\{t\}\\in\[0,1\], measured as the worker’s unaided effective throughput on the task\. In software development, for example,sts\_\{t\}is the worker’s independent rate of correct routine code completion; lower skill means slower work, more mistakes, or more rework\. Production is assisted by an AI system indexed byAtA\_\{t\}\. The technology level determines two parameters: capabilityαt≜α​\(At\)∈\(0,1\)\\alpha\_\{t\}\\triangleq\\alpha\(A\_\{t\}\)\\in\(0,1\), measured on the same normalized effective\-throughput scale as worker skillsts\_\{t\}, and failure probabilityπt≜π​\(At\)∈\(0,1\)\\pi\_\{t\}\\triangleq\\pi\(A\_\{t\}\)\\in\(0,1\), the probability that AI cannot be used as the autonomous producer\. The path\{At\}t=1T\\\{A\_\{t\}\\\}\_\{t=1\}^\{T\}is exogenous and known to all agents\. In each period, the firm chooses an engagement levelht∈\[0,1\]h\_\{t\}\\in\[0,1\], which sets how much of the workflow is routed through the worker rather than left to the AI alone\. A higherhth\_\{t\}means deeper human review, more hands\-on work or verification, or more frequent escalation to the worker\.[Figure1](https://arxiv.org/html/2606.29111#S3.F1)shows the sequence of events within a period\.

s1s\_\{1\}s2s\_\{2\}s3s\_\{3\}g​\(s1,h1;A1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)g​\(s2,h2;A2\)g\(s\_\{2\},h\_\{2\};A\_\{2\}\)B​\(s3\)B\(s\_\{3\}\)post\-horizonPeriod 1Period 2AI technologyAtA\_\{t\}realizedFirm choosesengagementhth\_\{t\}Workers sort\(logit shareσtj\\sigma\_\{t\}^\{\\,j\}\)Output, wages,and profitsSkills updateviag​\(⋅\)g\(\\cdot\)Figure 1:Timing of the two\-period model\. Worker skill is carried forwards1→s2→s3s\_\{1\}\\\!\\to\\\!s\_\{2\}\\\!\\to\\\!s\_\{3\}; the terminal skills3s\_\{3\}is valued post\-horizon byB​\(⋅\)B\(\\cdot\), and within each period skill evolves according tost\+1=g​\(st,ht;At\)s\_\{t\+1\}=g\(s\_\{t\},h\_\{t\};A\_\{t\}\)\. The lower row lists the order of events within a period\. The sorting step is active under worker mobility and degenerate in the single\-firm benchmark\.Throughout the paper, we focus on the two\-period caseT=2T=2, which is sufficient to capture the key tradeoffs \(between current output and future skill, between the fallback and sorting motives, and between capability and reliability\) while keeping the analysis tractable and the results interpretable\.

###### Assumption 1

The capability functionα​\(⋅\)\\alpha\(\\cdot\)is strictly increasing and the failure probabilityπ​\(⋅\)\\pi\(\\cdot\)is strictly decreasing\. The technology path satisfiesA2≥A1A\_\{2\}\\geq A\_\{1\}, soα2≥α1\\alpha\_\{2\}\\geq\\alpha\_\{1\}andπ2≤π1\\pi\_\{2\}\\leq\\pi\_\{1\}\.

This assumption captures technological improvement along both dimensions: more advanced AI systems are both more capable when they function and less likely to fail\. We keep capability and reliability as separate parameters because they enter the model through distinct channels: capabilityαt\\alpha\_\{t\}governs the value of AI\-assisted output and the size of the performance gapαt−st\\alpha\_\{t\}\-s\_\{t\}, whereas reliability1−πt1\-\\pi\_\{t\}determines how often AI functions and how often workers must perform the task on their own\.

### 3\.2AI\-Assisted Production and the Engagement\-Output Tradeoff

Production depends on whether the AI system functions or fails\. Letq∈\[0,1\]q\\in\[0,1\]denote effective throughput: the fraction of a standardized task load completed correctly in a period, measured on the same scale as capabilityαt\\alpha\_\{t\}and worker skillsts\_\{t\}\. When AI functions \(is on\), which occurs with probability1−πt1\-\\pi\_\{t\}, the AI\-alone effective throughput isαt\\alpha\_\{t\}, and worker engagement modifies this throughput according to

qon​\(st,ht;At\)=αt\+δ​ht​\(st−αt\),\\displaystyle q^\{\\mathrm\{on\}\}\(s\_\{t\},h\_\{t\};A\_\{t\}\)=\\alpha\_\{t\}\+\\delta h\_\{t\}\(s\_\{t\}\-\\alpha\_\{t\}\),whereδ∈\(0,1\)\\delta\\in\(0,1\)measures the influence of worker intervention on AI\-on production\. Whenst\>αts\_\{t\}\>\\alpha\_\{t\}, engagement raises effective throughput by pulling it toward the worker’s skill; whenst<αts\_\{t\}<\\alpha\_\{t\}, engagement lowers effective throughput for the same reason\.

When AI fails \(is off\), which occurs with probabilityπt\\pi\_\{t\}, the worker performs the task independently, so effective throughput equals the worker’s own skill,qoff​\(st\)=stq^\{\\mathrm\{off\}\}\(s\_\{t\}\)=s\_\{t\}\.

Letλ\>0\\lambda\>0denote the standardized task load per worker per period, and letR\>0R\>0denote revenue per correctly completed task\. The firm’s expected operational revenue per period therefore equals

S​\(st,ht;At\)=λ​R​\[\(1−πt\)​\(αt\+δ​ht​\(st−αt\)\)\+πt​st\]\.\\displaystyle S\(s\_\{t\},h\_\{t\};A\_\{t\}\)=\\lambda R\\left\[\(1\-\\pi\_\{t\}\)\\big\(\\alpha\_\{t\}\+\\delta h\_\{t\}\(s\_\{t\}\-\\alpha\_\{t\}\)\\big\)\+\\pi\_\{t\}s\_\{t\}\\right\]\.
The production effect of engagement is∂S​\(st,ht;At\)/∂ht=λ​R​δ​\(1−πt\)​\(st−αt\)\\partial S\(s\_\{t\},h\_\{t\};A\_\{t\}\)/\\partial h\_\{t\}=\\lambda R\\delta\(1\-\\pi\_\{t\}\)\(s\_\{t\}\-\\alpha\_\{t\}\), so for every below\-benchmark worker \(st<αts\_\{t\}<\\alpha\_\{t\}\) engagement strictly lowers current operational revenue\. The analysis below focuses on this region, where engagement is costly today but can preserve or build skill\.

### 3\.3Skill Dynamics: Learning and Erosion

Engagement changes future skill through a tradeoff between learning and erosion\. When the firm leaves the task to AI and AI functions, the worker does less of the task and loses skill through passive reliance\. When the firm keeps the worker engaged, the worker learns by staying involved in AI\-assisted cases and by practicing independent execution in the states in which AI cannot be relied on\. For a below\-benchmark worker,st<αts\_\{t\}<\\alpha\_\{t\}, we capture these two forces with an erosion term,γ​\(1−ht\)​\(1−πt\)​\(αt−st\)\\gamma\(1\-h\_\{t\}\)\(1\-\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\), and a learning term,ϕ​ht​πt​\(1−πt\)​\(αt−st\)\\phi h\_\{t\}\\pi\_\{t\}\(1\-\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\)\. Both are proportional to the gapαt−st\\alpha\_\{t\}\-s\_\{t\}: workers with more room to grow learn more from engagement, but they also lose more when AI routinely does the work\. The productπt​\(1−πt\)\\pi\_\{t\}\(1\-\\pi\_\{t\}\)implies that learning is strongest at intermediate reliability\. If AI never works, the worker has little useful output to learn from; if it never fails, the worker gets little fallback practice\. The product form is a parsimonious reduced form for this exposure–practice complementarity; for the results most exposed to this specification, we report robustness checks that replaceπt​\(1−πt\)\\pi\_\{t\}\(1\-\\pi\_\{t\}\)with a generic learning profileℓ​\(πt\)\\ell\(\\pi\_\{t\}\)\.333We consider profiles such asℓ​\(π\)=πa​\(1−π\)c\\ell\(\\pi\)=\\pi^\{a\}\(1\-\\pi\)^\{c\},ℓ​\(π\)=κ\+π​\(1−π\)\\ell\(\\pi\)=\\kappa\+\\pi\(1\-\\pi\),ℓ​\(π\)=π\\ell\(\\pi\)=\\pi, andℓ​\(π\)=1−π\\ell\(\\pi\)=1\-\\pi, witha,c\>0a,c\>0andκ≥0\\kappa\\geq 0\. These alternatives capture asymmetric exposure–practice complementarity, baseline learning independent of failures, failure\-practice learning, and AI\-exposure learning, respectively\.The parametersγ\>0\\gamma\>0andϕ\>0\\phi\>0measure the rates of erosion and learning, respectively\.

Combining these forces yields the skill transition

st\+1=g​\(st,ht;At\)=\{max⁡\{st\+Γ​\(ht;At\)​\(αt−st\),0\}if​st<αt,stif​st≥αt,\\displaystyle s\_\{t\+1\}=g\(s\_\{t\},h\_\{t\};A\_\{t\}\)=\\begin\{cases\}\\max\\\{s\_\{t\}\+\\Gamma\(h\_\{t\};A\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\),\\,0\\\}&\\text\{if \}s\_\{t\}<\\alpha\_\{t\},\\\\\[4\.0pt\] s\_\{t\}&\\text\{if \}s\_\{t\}\\geq\\alpha\_\{t\},\\end\{cases\}where

Γ​\(ht;At\)=\(1−πt\)​\[ht​\(ϕ​πt\+γ\)−γ\]\.\\displaystyle\\Gamma\(h\_\{t\};A\_\{t\}\)=\(1\-\\pi\_\{t\}\)\\big\[h\_\{t\}\(\\phi\\pi\_\{t\}\+\\gamma\)\-\\gamma\\big\]\.For workers withst<αts\_\{t\}<\\alpha\_\{t\},Γ​\(ht;At\)\\Gamma\(h\_\{t\};A\_\{t\}\)represents the net fraction of the skill gapαt−st\\alpha\_\{t\}\-s\_\{t\}that the worker closes in one period \(whenΓ\\Gammais positive\) or the fraction by which the gap widens \(whenΓ\\Gammais negative\)\. For workers withst≥αts\_\{t\}\\geq\\alpha\_\{t\}, we setst\+1=sts\_\{t\+1\}=s\_\{t\}, that is, we assume workers already at or above AI’s capability level maintain their current skill\.

SolvingΓ​\(ht;At\)=0\\Gamma\(h\_\{t\};A\_\{t\}\)=0yields the engagement level that preserves worker skill,

hc​\(At\)≜γϕ​πt\+γ∈\(0,1\)\.\\displaystyle h^\{c\}\(A\_\{t\}\)\\triangleq\\frac\{\\gamma\}\{\\phi\\pi\_\{t\}\+\\gamma\}\\in\(0,1\)\.For workers withst<αts\_\{t\}<\\alpha\_\{t\}, skill declines whenht<hc​\(At\)h\_\{t\}<h^\{c\}\(A\_\{t\}\)and improves whenht\>hc​\(At\)h\_\{t\}\>h^\{c\}\(A\_\{t\}\)\. Engagement therefore functions as an investment in worker skill: for below\-benchmark workers, it lowers the current effective throughput of the AI\-assisted workflow, but raises future worker skill and hence future earning capacity\. The thresholdhc​\(At\)h^\{c\}\(A\_\{t\}\)is strictly decreasing inπt\\pi\_\{t\}, and therefore rises as AI becomes more reliable; that is, more engagement is required to maintain worker skill when failures become rarer\. This is the*automation paradox*\(Bainbridge[1983](https://arxiv.org/html/2606.29111#bib.bib43), Parasuraman and Riley[1997](https://arxiv.org/html/2606.29111#bib.bib44)\):πt\\pi\_\{t\}governs both how often human skill is needed and how often it is exercised, so gains in AI reliability simultaneously reduce the need for human skill and the opportunities to preserve it\. More generally, with learningϕ​ht​ℓ​\(πt\)​\(αt−st\)\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\), the threshold ishℓc​\(At\)=γ​\(1−πt\)/\[ϕ​ℓ​\(πt\)\+γ​\(1−πt\)\]h^\{c\}\_\{\\ell\}\(A\_\{t\}\)=\\gamma\(1\-\\pi\_\{t\}\)/\[\\phi\\ell\(\\pi\_\{t\}\)\+\\gamma\(1\-\\pi\_\{t\}\)\]; it rises with reliability \(i\.e\., falls inπt\\pi\_\{t\}\) wheneverℓ​\(π\)/\(1−π\)\\ell\(\\pi\)/\(1\-\\pi\)is increasing inπ\\pi, includingℓ​\(π\)=πa​\(1−π\)c\\ell\(\\pi\)=\\pi^\{a\}\(1\-\\pi\)^\{c\}on ranges wherea​\(1−π\)\>\(c−1\)​πa\(1\-\\pi\)\>\(c\-1\)\\pi\.

###### Assumption 2

For every AI technology levelAA,ϕ​π​\(A\)​\(1−π​\(A\)\)<1\\phi\\pi\(A\)\(1\-\\pi\(A\)\)<1\.

[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)rules out one\-period leapfrogging: a worker who starts below the AI benchmark remains below it after one period, even under full engagement\. The point of the assumption is economic as well as technical\. It keeps the analysis in the region in which engagement is costly in current production, so any positive engagement for a below\-benchmark worker must be justified by future skill rather than by an immediate operating gain\. The assumption also rules out an implausibly fast learning path in which one period of engagement turns a below\-benchmark worker into the new frontier\. The largest fraction of the skill gap closed in one period occurs under full engagement, where it equalsϕ​π​\(A\)​\(1−π​\(A\)\)\\phi\\pi\(A\)\(1\-\\pi\(A\)\); lower engagement closes a weakly smaller fraction because passive reliance erodes skill\. Becauseπ​\(1−π\)≤1/4\\pi\(1\-\\pi\)\\leq 1/4, the simpler restrictionϕ<4\\phi<4is a technology\-independent sufficient condition\. For the general learning profile, the analogous no\-leapfrogging condition isϕ​ℓ​\(π​\(A\)\)<1\\phi\\ell\(\\pi\(A\)\)<1for every AI technology levelAA\.

###### Assumption 3

The initial skill level satisfiess1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\,\\alpha\_\{1\}\), where

s¯≜γ​\(1−π2\)​α2\+γ​\(1−π1\)​α1​\[1\+γ​\(1−π2\)\]\[1\+γ​\(1−π2\)\]​\[1\+γ​\(1−π1\)\]\.\\displaystyle\\underline\{s\}\\triangleq\\frac\{\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\+\\gamma\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\\bigl\[1\+\\gamma\(1\-\\pi\_\{2\}\)\\bigr\]\}\{\\bigl\[1\+\\gamma\(1\-\\pi\_\{2\}\)\\bigr\]\\bigl\[1\+\\gamma\(1\-\\pi\_\{1\}\)\\bigr\]\}\.

[Assumption3](https://arxiv.org/html/2606.29111#Thmassumption3)keeps the zero\-skill boundary from binding on the relevant domain\. Even under complete disengagement in both periods, worker skill remains strictly positive\. Because engagement only builds skill for a below\-benchmark worker, disengagement is the binding case: the worst case ish1=h2=0h\_\{1\}=h\_\{2\}=0, ands¯\\underline\{s\}is constructed so that this path still stays in the interior\. In particular, for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\],

s2=g​\(s1,h1;A1\)\>s¯2≜γ​\(1−π2\)​α21\+γ​\(1−π2\),\\displaystyle s\_\{2\}=g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\>\\underline\{s\}\_\{2\}\\triangleq\\frac\{\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\}\{1\+\\gamma\(1\-\\pi\_\{2\}\)\},and henceg​\(s2,h2;A2\)\>0g\(s\_\{2\},h\_\{2\};A\_\{2\}\)\>0for everyh2∈\[0,1\]h\_\{2\}\\in\[0,1\]\. As a result, the main analysis can work with the smooth part of the skill transition rather than carrying themax⁡\{⋅,0\}\\max\\\{\\cdot,0\\\}operator through every comparative static\.

### 3\.4Worker Payoff

Workers earn a market\-clearing wage that depends on their skill level\. Because skill is general\-purpose and observable, an outside labor market pins each worker’s compensation to a common wage schedule: any firm employing a worker of skillsts\_\{t\}paysW​\(st\)W\(s\_\{t\}\)\. Standardized wages capture settings with salary bands, internal equity constraints, and market benchmarking\(Bakeret al\.[1988](https://arxiv.org/html/2606.29111#bib.bib12), Mas[2017](https://arxiv.org/html/2606.29111#bib.bib13)\), in which firms have limited ability to offer idiosyncratic premiums\. Because firms cannot bid for workers with wages, their main lever for attracting workers is job design, specifically how actively workers engage with AI\-assisted tasks: a policy that builds more skill leads to higher future wages, making the firm more attractive to workers\.

We maintain standardized wages throughout the main analysis\. Holding pay fixed isolates job design, and engagement in particular, as the lever firms use to attract workers, which is the mechanism we study\. The restriction is not essential to our conclusions: in[SectionOA\.8](https://arxiv.org/html/2606.29111#S8)we let firms attract workers with pay as well, offering a bonus above the market wage alongside their engagement choice, and a numerical study there shows our main results carry over\.

The firm’s per\-worker margin equals operational revenue minus wages,

M​\(st,ht;At\)=S​\(st,ht;At\)−W​\(st\)\.\\displaystyle M\(s\_\{t\},h\_\{t\};A\_\{t\}\)=S\(s\_\{t\},h\_\{t\};A\_\{t\}\)\-W\(s\_\{t\}\)\.Because the wage in periodttdepends only on entering skillsts\_\{t\}, not on the engagement chosen within the period, engagement affects the margin only through operational revenue\. SinceS​\(st,ht;At\)S\(s\_\{t\},h\_\{t\};A\_\{t\}\)is affine inhth\_\{t\}with slopeλ​R​δ​\(1−πt\)​\(st−αt\)\\lambda R\\delta\(1\-\\pi\_\{t\}\)\(s\_\{t\}\-\\alpha\_\{t\}\), the margin is strictly decreasing inhth\_\{t\}wheneverst<αts\_\{t\}<\\alpha\_\{t\}: for below\-benchmark workers, engagement reduces current output and lowers the firm’s current margin\.

Fix a realized engagement path and abstract from the idiosyncratic taste shocks introduced in the mobility model\. The deterministic portion of a worker’s payoff then has three components: wages earned in period 1, discounted wages earned in period 2, and a discounted terminal value capturing the labor\-market worth of the skill the worker carries beyond the model horizon\. A worker with initial skills1s\_\{1\}who experiences engagement levelsh1h\_\{1\}andh2h\_\{2\}has deterministic payoff

W​\(s1\)\+β​W​\(s2\)\+β2​B​\(s3\),\\displaystyle W\(s\_\{1\}\)\+\\beta\\,W\(s\_\{2\}\)\+\\beta^\{2\}\\,B\(s\_\{3\}\),wheres2=g​\(s1,h1;A1\)s\_\{2\}=g\(s\_\{1\},h\_\{1\};A\_\{1\}\)ands3=g​\(s2,h2;A2\)s\_\{3\}=g\(s\_\{2\},h\_\{2\};A\_\{2\}\)\. HereB:\[0,1\]→ℝ\+B:\[0,1\]\\to\\mathbb\{R\}\_\{\+\}is the post\-horizon value of skill, which we take to beC2C^\{2\}on\(0,1\]\(0,1\], strictly increasing, and strictly convex; in our comparative statics and numerical examples, we setB=WB=W, valuing post\-horizon skill by the same wage schedule used within the model\.

Because wages are determined by the market and do not depend on the firm’s engagement policy, this deterministic payoff differs across firms only through the skill trajectory each firm’s policy creates\. In the single\-firm benchmark of[Section4](https://arxiv.org/html/2606.29111#S4), a single employer operates, so the worker’s payoff does not enter the firm’s engagement decision\. When workers are mobile \([Section5](https://arxiv.org/html/2606.29111#S5)\), the terminal value drives sorting: a firm whose engagement policy yields higher future skill, and hence higher future wages and terminal value, attracts more workers\.

###### Assumption 4

The wage functionW:\[0,1\]→ℝ\+W:\[0,1\]\\to\\mathbb\{R\}\_\{\+\}isC2C^\{2\}on\(0,1\]\(0,1\], strictly increasing, strictly convex, withW​\(0\)=0W\(0\)=0\. The revenue scale satisfiesλ​R\>maxt=1,2⁡W​\(αt\)/αt\\lambda R\>\\max\_\{t=1,2\}W\(\\alpha\_\{t\}\)/\\alpha\_\{t\}\.

Convexity ofWWis consistent with the Mincerian earnings function\(Mincer[1974](https://arxiv.org/html/2606.29111#bib.bib6)\)and with evidence on convex returns to education\(Psacharopoulos and Patrinos[2018](https://arxiv.org/html/2606.29111#bib.bib11)\); the power wageW​\(s\)=sbW\(s\)=s^\{b\}withb\>1b\>1, used in our numerical examples, satisfies these conditions\. The revenue condition comes from settinght=0h\_\{t\}=0andst=αts\_\{t\}=\\alpha\_\{t\}in the revenue expression:S​\(αt,0;At\)=λ​R​αtS\(\\alpha\_\{t\},0;A\_\{t\}\)=\\lambda R\\alpha\_\{t\}, soM=λ​R​αt−W​\(αt\)M=\\lambda R\\alpha\_\{t\}\-W\(\\alpha\_\{t\}\), andM≥0M\\geq 0requiresλ​R\>W​\(αt\)/αt\\lambda R\>W\(\\alpha\_\{t\}\)/\\alpha\_\{t\}\. This is the tightest case; the margin is positive for all other\(st,ht\)\(s\_\{t\},h\_\{t\}\)in the relevant domain\. Employing a below\-benchmark worker is therefore always profitable, so the analysis concerns how much to engage the worker, not whether to employ one\.

The mobility variablesη\\etaandσtj\\sigma\_\{t\}^\{j\}are introduced formally in[Section5](https://arxiv.org/html/2606.29111#S5)\. The model primitives have direct empirical counterparts in task\-level workflow data, and their measurement and calibration are detailed in[SectionOA\.1](https://arxiv.org/html/2606.29111#S1a)\. A summary of the notation appears in[TableOA1](https://arxiv.org/html/2606.29111#S0.T1)of the online appendix\.

## 4The Single\-Firm Benchmark

We begin with the single\-firm benchmark, which isolates the fallback motive\. Without worker mobility, engagement matters only because it changes the skill the firm will have on hand later when AI fails\. This benchmark makes clear what worker mobility adds in[Section5](https://arxiv.org/html/2606.29111#S5): sorting and imperfect appropriability\. We first study a one\-period benchmark and then specialize to the two\-period setting\.

### 4\.1Single Period: No Skill Investment

Consider a single period with AI technology levelAA\. For a worker of skillss, the single firm chooses engagementh∈\[0,1\]h\\in\[0,1\]to maximize the per\-worker marginM​\(s,h;A\)=S​\(s,h;A\)−W​\(s\)M\(s,h;A\)=S\(s,h;A\)\-W\(s\)\. Because the wageW​\(s\)W\(s\)does not depend onhh, maximizing the margin is equivalent to maximizing operational revenue\.

That revenue is affine inhhwith slopeλ​R​δ​\(1−π​\(A\)\)​\(s−α​\(A\)\)\\lambda R\\delta\(1\-\\pi\(A\)\)\(s\-\\alpha\(A\)\), so engagement raises it only when the worker’s skill exceeds AI capability\. The single firm therefore engages a worker fully whens\>α​\(A\)s\>\\alpha\(A\)and not at all whens<α​\(A\)s<\\alpha\(A\), and is indifferent ats=α​\(A\)s=\\alpha\(A\)\. With no future skill to protect, it disengages every below\-benchmark worker, those withs<α​\(A\)s<\\alpha\(A\)\. Engagement here is purely static: the firm builds no skill, because no future state remains in which that skill could pay off\.

### 4\.2Two Periods: The Fallback Motive

We now set the horizon to two periods,t∈\{1,2\}t\\in\\\{1,2\\\}, with AI technology path\(A1,A2\)\(A\_\{1\},A\_\{2\}\)\. The single firm chooses\(h1sf,h2sf\)∈\[0,1\]2\(h\_\{1\}^\{\\mathrm\{sf\}\},h\_\{2\}^\{\\mathrm\{sf\}\}\)\\in\[0,1\]^\{2\}to maximize total discounted profit\. We focus on workers with initial skills1<α1s\_\{1\}<\\alpha\_\{1\}, because these are the workers for whom engagement involves a tradeoff: it lowers current production but preserves future skill\. By the no\-leapfrogging condition,g​\(s1,h1;A1\)<α1≤α2g\(s\_\{1\},h\_\{1\};A\_\{1\}\)<\\alpha\_\{1\}\\leq\\alpha\_\{2\}for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\], so engagement is operationally costly in both periods, and any positive first\-period engagement must reflect the firm’s goal of preserving fallback skill\. In the terminal period, the single firm has no future to protect, so the static logic above applies directly: it disengages every below\-benchmark worker, settingh2sf=0h\_\{2\}^\{\\mathrm\{sf\}\}=0whens2<α2s\_\{2\}<\\alpha\_\{2\}\(and is indifferent ats2=α2s\_\{2\}=\\alpha\_\{2\}\)\. The period\-1 problem is therefore

maxh1sf∈\[0,1\]⁡M​\(s1,h1sf;A1\)\+β​M​\(g​\(s1,h1sf;A1\),0;A2\)\.\\displaystyle\\max\_\{h\_\{1\}^\{\\mathrm\{sf\}\}\\in\[0,1\]\}\\;M\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\};A\_\{1\}\)\+\\beta\\,M\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\};A\_\{1\}\),\\,0;\\,A\_\{2\}\\bigr\)\.The objective is strictly concave inh1sfh\_\{1\}^\{\\mathrm\{sf\}\}, so the optimum is unique\. The first\-order condition balances the benefit of building worker skill against its cost\. The benefit is the extra operational revenue a more skilled worker generates as a fallback when AI fails; the cost is the current production forgone plus the higher wage the firm must later pay\. At an interior optimum, this givesW′​\(g​\(s1,h1;A1\)\)=λ​R​π2−λ​R​δ/\[β​\(ϕ​π1\+γ\)\]W^\{\\prime\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)=\\lambda R\\pi\_\{2\}\-\\lambda R\\delta/\[\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\]\. When the right\-hand side lies in the range of marginal wages,

λ​R​π2−λ​R​δβ​\(ϕ​π1\+γ\)∈\(W′​\(0\),W′​\(α1\)\),\\displaystyle\\lambda R\\pi\_\{2\}\-\\frac\{\\lambda R\\delta\}\{\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\}\\in\\bigl\(W^\{\\prime\}\(0\),\\,W^\{\\prime\}\(\\alpha\_\{1\}\)\\bigr\),\(1\)the right\-hand side has a unique preimage underW′W^\{\\prime\}, the*target skill*s∗s^\{\*\}:

s∗≜\(W′\)−1​\(λ​R​π2−λ​R​δβ​\(ϕ​π1\+γ\)\)∈\(0,α1\),\\displaystyle s^\{\*\}\\triangleq\(W^\{\\prime\}\)^\{\-1\}\\\!\\left\(\\lambda R\\pi\_\{2\}\-\\frac\{\\lambda R\\delta\}\{\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\}\\right\)\\in\(0,\\alpha\_\{1\}\),\(2\)the period\-2 skill the single firm would most like the worker to reach\. If the right\-hand side is weakly belowW′​\(0\)W^\{\\prime\}\(0\), then the objective is decreasing inh1sfh\_\{1\}^\{\\mathrm\{sf\}\}, so the firm setsh1sf​\(s1\)=0h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=0for every below\-benchmark worker\. If the right\-hand side is weakly aboveW′​\(α1\)W^\{\\prime\}\(\\alpha\_\{1\}\), then the objective is increasing inh1sfh\_\{1\}^\{\\mathrm\{sf\}\}, so the firm setsh1sf​\(s1\)=1h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=1for every below\-benchmark worker\. Condition[eq\.1](https://arxiv.org/html/2606.29111#S4.E1)rules out these two boundary cases and focuses attention on the case in which the target skills∗s^\{\*\}is well defined\. Even then, the target may lie outside a particular worker’s attainable interval\[g​\(s1,0;A1\),g​\(s1,1;A1\)\]\[g\(s\_\{1\},0;A\_\{1\}\),g\(s\_\{1\},1;A\_\{1\}\)\], giving the worker\-specific corner cases characterized below\.

###### Proposition 1

For eachs1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\alpha\_\{1\}\), the firm’s period\-1 engagementh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)is unique and drives period\-2 skill toward the targets∗s^\{\*\}, found by comparings∗s^\{\*\}with the worker’s attainable interval\[g​\(s1,0;A1\),g​\(s1,1;A1\)\]\[g\(s\_\{1\},0;A\_\{1\}\),\\,g\(s\_\{1\},1;A\_\{1\}\)\]:

1. \(i\)ifs∗≥g​\(s1,1;A1\)s^\{\*\}\\geq g\(s\_\{1\},1;A\_\{1\}\), so even full engagement falls short of the target, the firm engages fully,h1sf​\(s1\)=1h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=1;
2. \(ii\)ifg​\(s1,0;A1\)<s∗<g​\(s1,1;A1\)g\(s\_\{1\},0;A\_\{1\}\)<s^\{\*\}<g\(s\_\{1\},1;A\_\{1\}\), the target is attainable and engagement is interior, setting period\-2 skillg​\(s1,h1sf​\(s1\);A1\)=s∗g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);A\_\{1\}\)=s^\{\*\}, h1sf​\(s1\)=s∗−g​\(s1,0;A1\)\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\);\\displaystyle h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=\\frac\{s^\{\*\}\-g\(s\_\{1\},0;A\_\{1\}\)\}\{\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\};\(3\)
3. \(iii\)ifs∗≤g​\(s1,0;A1\)s^\{\*\}\\leq g\(s\_\{1\},0;A\_\{1\}\), so even full atrophy leaves the worker above the target, the firm does not engage,h1sf​\(s1\)=0h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=0\.

In the interior region,h1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)is strictly decreasing ins1s\_\{1\}\.

[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)shows that the firm follows a skill\-targeting rule: a single targets∗s^\{\*\}summarizes the period\-2 skill it wants each worker to reach, and it engages just enough to get there\. Whether a worker is engaged fully, partially, or not at all depends only on where the target sits relative to the skill the worker can attain\.

This characterization can equivalently be stated in terms of initial\-skill thresholds\. Define

sL≜s∗−ϕ​π1​\(1−π1\)​α11−ϕ​π1​\(1−π1\),sH≜s∗\+γ​\(1−π1\)​α11\+γ​\(1−π1\)\.\\displaystyle s\_\{L\}\\triangleq\\frac\{s^\{\*\}\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\}\{1\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\},\\qquad s\_\{H\}\\triangleq\\frac\{s^\{\*\}\+\\gamma\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\}\{1\+\\gamma\(1\-\\pi\_\{1\}\)\}\.\(4\)Full engagement arises fors1≤sLs\_\{1\}\\leq s\_\{L\}, no engagement fors1≥sHs\_\{1\}\\geq s\_\{H\}, and interior engagement fors1∈\(sL,sH\)s\_\{1\}\\in\(s\_\{L\},s\_\{H\}\), intersected with the feasible domains1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\alpha\_\{1\}\)\. Equivalently, the policy can be summarized by the thresholdssLs\_\{L\}andsHs\_\{H\}; a tabular summary is reported in[SectionOA\.3](https://arxiv.org/html/2606.29111#S3a)\.

###### Corollary 1

When the target skills∗s^\{\*\}is interior:

\(i\)The target skill satisfies

∂s∗∂π2=λ​RW′′​\(s∗\)\>0,∂s∗∂α2=0,\\displaystyle\\frac\{\\partial s^\{\*\}\}\{\\partial\\pi\_\{2\}\}=\\frac\{\\lambda R\}\{W^\{\\prime\\prime\}\(s^\{\*\}\)\}\>0,\\qquad\\frac\{\\partial s^\{\*\}\}\{\\partial\\alpha\_\{2\}\}=0,and the thresholdssL,sHs\_\{L\},\\,s\_\{H\}in[eq\.4](https://arxiv.org/html/2606.29111#S4.E4)are strictly increasing inπ2\\pi\_\{2\}and independent ofα2\\alpha\_\{2\}\.

\(ii\)For any fixeds1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\alpha\_\{1\}\), the period\-1 engagementh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)characterized in[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)is weakly increasing inπ2\\pi\_\{2\}on the feasible domain\(0,π1\]\(0,\\pi\_\{1\}\], with interior engagement strictly increasing inπ2\\pi\_\{2\}\. Asπ2\\pi\_\{2\}falls fromπ1\\pi\_\{1\}toward zero, the worker’s period\-1 engagement either remains constant or decreases\.

As AI becomes more reliable, the value of preserving fallback skill declines, and the firm withdraws engagement: if AI will rarely fail in the future, workers will rarely need to step in, so the firm has less reason to maintain worker skill today\. Future capability, by contrast, has no such effect\. Mathematically,α2\\alpha\_\{2\}enters the continuation payoff only as an additive constant, so it drops out of the first\-order condition for period\-1 engagement\. The reason is intuitive: because the firm disengages below\-benchmark workers in the terminal period, worker skill affects period\-2 production only when AI fails\. A more capable AI generates higher effective throughput when it functions, but that output does not depend on the worker’s skill\. The marginal value of worker skill in period 2 is therefore governed by how often AI fails \(π2\\pi\_\{2\}\), not by AI’s performance level when it functions \(α2\\alpha\_\{2\}\)\.[Figure2](https://arxiv.org/html/2606.29111#S4.F2)plots this policy: the next\-period skill it produces against the worker’s current skill\.

![Refer to caption](https://arxiv.org/html/2606.29111v1/x1.png)Figure 2:Single\-firm engagement targets the least\-skilled workers\. The figure traces the next\-period skills2s\_\{2\}produced by optimal period\-1 engagement against current skills1s\_\{1\}\. The firm engages the lowest\-skilled workers fully \(h1sf=1h\_\{1\}^\{\\mathrm\{sf\}\}=1\), building skill toward the targets∗s^\{\*\}; intermediate workers partially, holding skill ats∗s^\{\*\}; and the highest\-skilled not at all \(h1sf=0h\_\{1\}^\{\\mathrm\{sf\}\}=0\), letting skill erode toward the45∘45^\{\\circ\}line\. The shaded band is the skill built by engagement, widest for the least\-skilled\. Parameter values:W​\(s\)=s1\.2W\(s\)=s^\{1\.2\},α1=0\.5\\alpha\_\{1\}=0\.5,α2=0\.6\\alpha\_\{2\}=0\.6,π1=0\.45\\pi\_\{1\}=0\.45,π2=0\.35\\pi\_\{2\}=0\.35,ϕ=0\.5\\phi=0\.5,γ=0\.3\\gamma=0\.3,δ=0\.05\\delta=0\.05,λ​R=3\.65\\lambda R=3\.65,β=0\.95\\beta=0\.95\.This targeting of the least\-skilled is not special to the parameters of[Figure2](https://arxiv.org/html/2606.29111#S4.F2)\.

###### Proposition 2

In the absence of worker mobility, the firm’s engagement is nonincreasing in initial skill: it allocates the most engagement to the lowest\-skilled workers in the feasible domain, partial engagement to intermediate workers so as to bring next\-period skill exactly tos∗s^\{\*\}, and no engagement to workers whose future skill remains above the target even under atrophy\.

Intuitively, the firm engages only to secure a fallback for when AI is unavailable or unreliable, and that fallback is least expensive to build from below\. Two forces pull the same way\. The skill a unit of engagement adds is proportional to the gapα1−s1\\alpha\_\{1\}\-s\_\{1\}, so the firm gains more skill per unit of engagement from a worker who starts further down\. Because wages are convex, the marginal wage it pays to build that skill is also lower the lower the worker starts\. An already\-skilled worker offers neither saving, so the firm withholds engagement\.

The firm engages to secure adequate fallback capability at minimum cost, not to maximize skill\. Because AI handles most of the work, building a worker far above the target earns little return at a higher wage\. The firm aims only for the level at which a worker can take over when AI is unavailable or unreliable, and no higher\.

## 5Worker Mobility

Worker mobility adds a second motive to the firm’s engagement choice\. With two ex ante identical firms and a common market wage, workers choose between jobs on the basis of the skill trajectories those jobs build\. Engagement now affects profits in two ways\. It lowers the current margin on a below\-benchmark worker, as in the single\-firm benchmark, but it also makes the firm more attractive to workers who value the skill the job creates\. This sorting motive, absent without mobility because a worker has nowhere else to go, comes at a price: it weakens the firm’s hold on the skill it builds, because that skill is portable and moves with the worker\. The sorting motive operates even in the terminal period, in which a single firm disengages every below\-benchmark worker, and it targets the higher\-skilled workers, the opposite end of the spectrum from the least\-skilled workers a single firm favors\. Under mobility, AI capability and reliability move engagement in different directions\. In period 1, the sorting motive interacts with the fallback motive\. When workers respond weakly to the skill a job builds, mobility drains the firm’s return on portable skill and pulls engagement below the single firm’s level\. The analysis also illustrates three further possibilities: engagement can rise above that level, its timing can reverse so that a worker is engaged only after re\-sorting, and two ex ante identical firms can split apart\.

In each periodt∈\{1,2\}t\\in\\\{1,2\\\}, the two firms simultaneously choose engagement levelshtj,ht−j∈\[0,1\]h\_\{t\}^\{j\},h\_\{t\}^\{\-j\}\\in\[0,1\], and workers then sort between them\. A worker of skillsts\_\{t\}who joins firmjjis engaged at levelhtjh\_\{t\}^\{j\}and enters the next period with skillg​\(st,htj;At\)g\(s\_\{t\},h\_\{t\}^\{j\};A\_\{t\}\)\. LetVtj​\(st,htj\)V\_\{t\}^\{j\}\(s\_\{t\},h\_\{t\}^\{j\}\)denote the worker’s economic value from joining firmjj\. Because both firms pay the common market wageW​\(st\)W\(s\_\{t\}\), differences inVtjV\_\{t\}^\{j\}come only from the skill trajectory each engagement policy creates\. A worker’s realized valuation of firmjjis this value plus a private taste shock,Utj​\(st,htj\)=Vtj​\(st,htj\)\+η​εtjU\_\{t\}^\{j\}\(s\_\{t\},h\_\{t\}^\{j\}\)=V\_\{t\}^\{j\}\(s\_\{t\},h\_\{t\}^\{j\}\)\+\\eta\\varepsilon\_\{t\}^\{j\}, where the shocksεtj\\varepsilon\_\{t\}^\{j\}are drawn independently across workers and firms from a type\-I extreme\-value distribution\. The scaleη\>0\\eta\>0governs how strongly workers respond to the skill trajectories firms offer: asη\\etafalls, they track those trajectories ever more closely, and as it rises, private taste dominates\. This structure yields the familiar*logit share*\(Aksoy\-Piersonet al\.[2013](https://arxiv.org/html/2606.29111#bib.bib4), Basuroy and Nguyen[1998](https://arxiv.org/html/2606.29111#bib.bib3), McFadden[1974](https://arxiv.org/html/2606.29111#bib.bib2)\): a worker of skillsts\_\{t\}joins firmjjwith probability

σtj​\(htj,ht−j;st,At\)=11\+exp⁡\(\[Vt−j​\(st,ht−j\)−Vtj​\(st,htj\)\]/η\),\\displaystyle\\sigma\_\{t\}^\{j\}\(h\_\{t\}^\{j\},h\_\{t\}^\{\-j\};s\_\{t\},A\_\{t\}\)=\\frac\{1\}\{1\+\\exp\\\!\\left\(\\bigl\[V\_\{t\}^\{\-j\}\(s\_\{t\},h\_\{t\}^\{\-j\}\)\-V\_\{t\}^\{j\}\(s\_\{t\},h\_\{t\}^\{j\}\)\\bigr\]/\\eta\\right\)\},and, before the taste shock, the worker’s expected value𝔼​\[max⁡\{Ut1,Ut2\}\]\\mathbb\{E\}\\\!\\left\[\\max\\\{U\_\{t\}^\{1\},\\,U\_\{t\}^\{2\}\\\}\\right\]takes the log\-sum form

Vt​\(st,ht1,ht2\)=η​log⁡\[exp⁡\(Vt1​\(st,ht1\)/η\)\+exp⁡\(Vt2​\(st,ht2\)/η\)\]\+η​κ,\\displaystyle V\_\{t\}\(s\_\{t\},h\_\{t\}^\{1\},h\_\{t\}^\{2\}\)=\\eta\\log\\\!\\left\[\\exp\\\!\\left\(V\_\{t\}^\{1\}\(s\_\{t\},h\_\{t\}^\{1\}\)/\\eta\\right\)\+\\exp\\\!\\left\(V\_\{t\}^\{2\}\(s\_\{t\},h\_\{t\}^\{2\}\)/\\eta\\right\)\\right\]\+\\eta\\kappa,\(5\)the standard type\-I extreme\-value result\(McFadden[1980](https://arxiv.org/html/2606.29111#bib.bib1)\), withκ\\kappathe Euler–Mascheroni constant\. The economic valueVtjV\_\{t\}^\{j\}takes a different form in each period\. We begin with the terminal period, in which the firm’s own fallback motive is absent, so any engagement it offers can serve only to attract workers, isolating the sorting motive\. We take up the period\-1 problem, in which current output, future skill, and worker sorting all interact, in[Section5\.4](https://arxiv.org/html/2606.29111#S5.SS4)\.

### 5\.1Terminal Period: The Sorting Motive

In the terminal period, a firm without worker mobility has no reason to engage a below\-benchmark worker: with no future left to protect, engagement only sacrifices current output, so it disengages entirely\. Worker mobility reverses this\. Once a worker can choose between firms, a firm engages even though doing so earns it no fallback value of its own, because engagement raises the worker’s post\-horizon skill, and a worker who anticipates a stronger skill trajectory is more likely to join that firm\. Engagement thus becomes a tool to attract workers\. Two questions then organize the terminal analysis: whether a firm engages a below\-benchmark worker at all, settled here by a single threshold on the sorting friction, and which workers it engages most, taken up in[Section5\.2](https://arxiv.org/html/2606.29111#S5.SS2)\.

The worker’s economic valueV2jV\_\{2\}^\{j\}is the current wage plus the discounted post\-horizon value of the skill carried forward,

V2j​\(s2,h2j\)=W​\(s2\)\+β​B​\(g​\(s2,h2j;A2\)\),\\displaystyle V\_\{2\}^\{j\}\(s\_\{2\},h\_\{2\}^\{j\}\)=W\(s\_\{2\}\)\+\\beta B\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\),whereB​\(⋅\)B\(\\cdot\)is the post\-horizon skill\-value function introduced in[Section3\.4](https://arxiv.org/html/2606.29111#S3.SS4)\. Because both firms pay the common wage, it cancels out in the logit sorting probabilities, so a firm’s terminal\-period logit share depends only on the skill its engagement promises:

σ2j​\(h2j,h2−j;s2,A2\)=11\+exp⁡\(β​\[B​\(g​\(s2,h2−j;A2\)\)−B​\(g​\(s2,h2j;A2\)\)\]/η\)\.\\displaystyle\\sigma\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{1\+\\exp\\\!\\left\(\\beta\\bigl\[B\(g\(s\_\{2\},h\_\{2\}^\{\-j\};A\_\{2\}\)\)\-B\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\)\\bigr\]/\\eta\\right\)\}\.The firm that builds more skill draws the larger logit share, increasingly so asη\\etafalls\.

For a below\-benchmark worker \(s2<α2s\_\{2\}<\\alpha\_\{2\}\), firmj∈\{1,2\}j\\in\\\{1,2\\\}choosesh2j∈\[0,1\]h\_\{2\}^\{j\}\\in\[0,1\]to maximize expected profit, the product of its worker share and its per\-worker margin,

Π2j​\(h2j,h2−j;s2,A2\)=σ2j​\(h2j,h2−j;s2,A2\)​M​\(s2,h2j;A2\)\.\\displaystyle\\Pi\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)=\\sigma\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)\\,M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\.The firm is pulled in two directions\. A higherh2jh\_\{2\}^\{j\}lifts the worker’s future skill, and with it the firm’s shareσ2j\\sigma\_\{2\}^\{j\}; but because the worker sits below the benchmark, the sameh2jh\_\{2\}^\{j\}erodes the current marginM​\(s2,h2j;A2\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\. The best response weighs attracting workers against current output\.

Two ex ante identical firms could, in principle, support an asymmetric pure\-strategy equilibrium in the terminal\-period game, with one firm engaging more than the other\. The next lemma rules that out\.

###### Lemma 1

In the terminal period, any pure\-strategy Nash equilibrium of the game in which firms11and22simultaneously chooseh21,h22∈\[0,1\]h\_\{2\}^\{1\},h\_\{2\}^\{2\}\\in\[0,1\]to maximizeΠ21​\(h21,h22;s2,A2\)\\Pi\_\{2\}^\{1\}\(h\_\{2\}^\{1\},h\_\{2\}^\{2\};s\_\{2\},A\_\{2\}\)andΠ22​\(h22,h21;s2,A2\)\\Pi\_\{2\}^\{2\}\(h\_\{2\}^\{2\},h\_\{2\}^\{1\};s\_\{2\},A\_\{2\}\)is symmetric:h21⁣∗=h22⁣∗h\_\{2\}^\{1\*\}=h\_\{2\}^\{2\*\}\.

[Lemma1](https://arxiv.org/html/2606.29111#Thmlemma1)reflects a balance of two forces\. A firm that engages more attracts a larger share of workers but earns a thinner margin on each; a firm that engages less faces the reverse\. These two pulls offset, so neither firm can gain by engaging differently, and any gap between them unravels\. The balance holds only in the terminal period\. In period 1, by contrast, skill moves with the worker, so once workers re\-sort the other firm can draw on it without having paid to build it, and this spillover can break the symmetry between them \([Section5\.4](https://arxiv.org/html/2606.29111#S5.SS4)\)\.

Because[Lemma1](https://arxiv.org/html/2606.29111#Thmlemma1)rules out asymmetric pure\-strategy equilibria in the terminal period, it suffices to characterize symmetric candidates\. At a symmetric profileh2j=h2−j=hh\_\{2\}^\{j\}=h\_\{2\}^\{\-j\}=h, the own\-action marginal payoff∂Π2j​\(h2j,h2−j;s2,A2\)/∂h2j\\partial\\Pi\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)/\\partial h\_\{2\}^\{j\}of firmjjhas the same sign asF2​\(h;s2,A2\)−λ​R​δF\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\-\\lambda R\\delta, where

F2​\(h;s2,A2\)≜β​\(ϕ​π2\+γ\)2​η​B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)\.\\displaystyle F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\\triangleq\\frac\{\\beta\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}\\,B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)\\,M\(s\_\{2\},h;A\_\{2\}\)\.Thus an interior symmetric equilibrium solvesF2​\(h;s2,A2\)=λ​R​δF\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\. The termF2​\(h;s2,A2\)F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)is the marginal engagement gain: higher engagement builds more post\-horizon skill and makes the firm more attractive to workers\. The termλ​R​δ\\lambda R\\deltais the normalized marginal operating cost of engagement for a below\-benchmark worker\.

We impose a sufficient single\-crossing restriction that makes this comparison monotone\. There exists a cutoffs¯¯2∈\[s¯2,α2\)\\bar\{\\bar\{s\}\}\_\{2\}\\in\[\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)such that, withD2≡\(s¯¯2,α2\)D\_\{2\}\\equiv\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\), for everys2∈D2s\_\{2\}\\in D\_\{2\},

maxh∈\[0,1\]⁡\{B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)\}<λ​R​δϕ​π2\+γ\.\\displaystyle\\max\_\{h\\in\[0,1\]\}\\left\\\{\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\\,M\(s\_\{2\},h;A\_\{2\}\)\\right\\\}<\\frac\{\\lambda R\\delta\}\{\\phi\\pi\_\{2\}\+\\gamma\}\.\(6\)We call[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)the terminal single\-crossing condition\. Engagement raises future skill, and convexBBmakes that skill increasingly valuable; but engagement also lowers the firm’s current margin\. The condition ensures the margin decline dominates, so the marginal engagement gain falls with engagement and meets the costλ​R​δ\\lambda R\\deltaat most once\. Throughout the rest of[Section5](https://arxiv.org/html/2606.29111#S5), terminal\-period statements are understood fors2∈D2s\_\{2\}\\in D\_\{2\}\.

###### Proposition 3

Suppose the terminal single\-crossing condition in[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)holds\. Then the engagement gainF2​\(h;s2,A2\)F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)is strictly decreasing inhh, and the terminal\-period two\-firm game has a unique pure\-strategy Nash equilibrium, symmetric ath2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\), found by comparing the engagement gain at the boundaries with the operating costλ​R​δ\\lambda R\\delta:

1. \(i\)ifλ​R​δ≤F2​\(1;s2,A2\)\\lambda R\\delta\\leq F\_\{2\}\(1;s\_\{2\},A\_\{2\}\), both firms engage fully,h2∗=1h\_\{2\}^\{\*\}=1;
2. \(ii\)ifF2​\(1;s2,A2\)<λ​R​δ<F2​\(0;s2,A2\)F\_\{2\}\(1;s\_\{2\},A\_\{2\}\)<\\lambda R\\delta<F\_\{2\}\(0;s\_\{2\},A\_\{2\}\), both engage at the interior levelh2∗∈\(0,1\)h\_\{2\}^\{\*\}\\in\(0,1\)solvingF2​\(h2∗;s2,A2\)=λ​R​δF\_\{2\}\(h\_\{2\}^\{\*\};s\_\{2\},A\_\{2\}\)=\\lambda R\\delta;
3. \(iii\)ifF2​\(0;s2,A2\)≤λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\\leq\\lambda R\\delta, neither firm engages,h2∗=0h\_\{2\}^\{\*\}=0\.

[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)shows worker mobility creates a terminal\-period sorting motive even though engagement lowers the current margin\. At a symmetric profile, engaging more wins workers but loses margin, whereas engaging less saves margin but loses workers\.

For the power wageB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\},b\>1b\>1, which we use in several results below, the single\-crossing condition becomes explicit and fixes the domainD2D\_\{2\}\. The left\-hand side of[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)is then largest ath=0h=0and, under[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4), strictly decreasing ins2s\_\{2\}: it diverges ass2↓s¯2s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}and falls toward the benchmark, so the condition fails for the least\-skilled workers and holds for the most\-skilled\. The domainD2=\(s¯¯2,α2\)D\_\{2\}=\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)is therefore an upper\-tail band of below\-benchmark skills just under the AI frontier, where the cutoffs¯¯2\\bar\{\\bar\{s\}\}\_\{2\}is the unique skill at which the single\-crossing condition holds with equality;[OA2](https://arxiv.org/html/2606.29111#Thmclaim2)in the online appendix gives its closed\-form characterization\.444The restriction is an upper\-tail modest\-convexity condition, consistent with Mincerian wage profiles, which are convex but typically only modestly so\(Mincer[1974](https://arxiv.org/html/2606.29111#bib.bib6), Psacharopoulos and Patrinos[2018](https://arxiv.org/html/2606.29111#bib.bib11)\)\. In a power\-wage calibration,D2D\_\{2\}covers about73%73\\%of the below\-benchmark interval \([SectionOA\.4](https://arxiv.org/html/2606.29111#S4a)\)\. OutsideD2D\_\{2\},F2F\_\{2\}need not be monotone and the terminal equilibrium need not be unique;D2D\_\{2\}is the region on which single\-crossing yields a unique equilibrium\.

[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)states the no\-engagement boundary in general form asF2​\(0;s2,A2\)≤λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\\leq\\lambda R\\delta\. Under the power wage,F2​\(0;s2,A2\)F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)is proportional to1/η1/\\eta, so this boundary can be written as a threshold condition on the sorting friction\. Define

η¯​\(s2\)≜β​b​\(ϕ​π2\+γ\)2​λ​R​δ​\(g​\(s2,0;A2\)\)b−1​M​\(s2,0;A2\)\.\\displaystyle\\bar\{\\eta\}\(s\_\{2\}\)\\triangleq\\frac\{\\beta b\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\lambda R\\delta\}\\bigl\(g\(s\_\{2\},0;A\_\{2\}\)\\bigr\)^\{b\-1\}M\(s\_\{2\},0;A\_\{2\}\)\.\(7\)Equivalently,F2​\(0;s2,A2\)=λ​R​δ​η¯​\(s2\)/ηF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\\,\\bar\{\\eta\}\(s\_\{2\}\)/\\eta\.

###### Corollary 2

SupposeB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1\. Fors2∈D2s\_\{2\}\\in D\_\{2\},h2∗​\(s2\)\>0h\_\{2\}^\{\*\}\(s\_\{2\}\)\>0if and only ifη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)\.

Hereη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is the critical sorting\-friction threshold for engaging a worker of skills2s\_\{2\}: the firm engages forη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)and disengages forη≥η¯​\(s2\)\\eta\\geq\\bar\{\\eta\}\(s\_\{2\}\)\. Thusη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is the engagement gain ath=0h=0, expressed in units of the sorting friction\. As workers grow more responsive, that is, asη\\etafalls, engagement becomes profitable for a wider set of skills withinD2D\_\{2\}\.

A single firm’s workers cannot move to a rival, so it has no sorting motive; it engages a below\-benchmark worker in period 1 only when the fallback motive, the prospect of a more productive worker in period 2, justifies the cost\. With no future left, even that motive is absent, so a single firm setsh2sf=0h\_\{2\}^\{\\mathrm\{sf\}\}=0\. Under mobility, by contrast, a firm engages a worker withs2∈D2s\_\{2\}\\in D\_\{2\}whenever the engagement gain covers the cost,F2​\(0;s2,A2\)\>λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\>\\lambda R\\delta\. Engagement is then no longer a purely operational choice\. The question is not whether a firm engages, but whom\.

### 5\.2The Engagement Reversal

Mobility reverses which workers a firm engages\. Without it, the fallback motive engages the least\-skilled workers \([Section4](https://arxiv.org/html/2606.29111#S4)\): far below the AI benchmark, a unit of engagement closes the widest skill gap and buys fallback capability at the lowest wage\. In the terminal period the fallback motive is gone, and the sorting motive stands alone\. It points the other way, toward the high\-skill workers near the AI frontier, for two reasons\. First, engaging a below\-benchmark worker forgoes operational revenue in proportion to the skill gapα2−s2\\alpha\_\{2\}\-s\_\{2\}, so workers nearer the benchmark are less costly to engage, a force that points upward even when wages are linear\. Second, convex wages reinforce the pull: the marginal wageB′​\(s\)B^\{\\prime\}\(s\)rises with skill, so the portable skill that engagement builds is worth more to a higher\-skill worker\. A margin force works against both: higher\-skill workers also command higher current wages, compressing the firm’s per\-worker margin and dampening the gain from attracting more of them\. How strongly the net pull translates into engagement depends on the sorting frictionη\\eta: the lowerη\\eta, the more responsive workers are to the skill trajectories firms offer\. Mobility therefore shifts engagement from the bottom of the skill range toward an upper band that, for low enoughη\\eta, can reach the frontier \([Figure3](https://arxiv.org/html/2606.29111#S5.F3)\)\.

![Refer to caption](https://arxiv.org/html/2606.29111v1/x2.png)Figure 3:The engagement reversal\. Without mobility, the fallback motive engages the least\-skilled workers most; under mobility, the sorting motive engages the most skilled below\-benchmark workers\. The figure is schematic\.Under the power wageB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}, a firm engages a worker of skills2∈D2s\_\{2\}\\in D\_\{2\}when the sorting friction falls below a skill\-specific cutoff,η<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)\([Corollary2](https://arxiv.org/html/2606.29111#Thmcorollary2)\)\. For a givenη\\eta, the engaged workers are those whose cutoffη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)exceedsη\\eta, so which workers a firm engages is determined by how that cutoff varies with skill\. The cutoff and the engagement gainF2​\(0;s2,A2\)F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)are proportional, becauseF2​\(0;s2,A2\)=λ​R​δ​η¯​\(s2\)/ηF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\\,\\bar\{\\eta\}\(s\_\{2\}\)/\\etaand the factorλ​R​δ/η\\lambda R\\delta/\\etadoes not depend ons2s\_\{2\}\. The two therefore rise, fall, and peak together, so the targeting problem reduces to a single object: how the engagement gain ath=0h=0varies with skill\. That variation settles both which workers are engaged and how intensely\.

Whether the workers a firm engages most reach the AI frontier or sit just below it is governed by a single comparison, summarized by the scalarΔ2\\Delta\_\{2\}\. Engagement requiresη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\), so at a givenη\\etathe engaged workers are those with the highest cutoffs, an interval at the top of the admissible range\. Where this interval sits, at the very top or in a band just below the frontier, depends on the shape of the cutoffη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\), and that shape is determined by the sign ofΔ2\\Delta\_\{2\}:

Δ2≜\(b−1\)​\[1\+γ​\(1−π2\)\]​\(λ​R​α2−α2b\)\+α2​\(λ​R​π2−b​α2b−1\)\.\\displaystyle\\Delta\_\{2\}\\triangleq\(b\-1\)\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]\(\\lambda R\\alpha\_\{2\}\-\\alpha\_\{2\}^\{b\}\)\+\\alpha\_\{2\}\(\\lambda R\\pi\_\{2\}\-b\\alpha\_\{2\}^\{b\-1\}\)\.\(8\)
Intuitively,Δ2\\Delta\_\{2\}weighs the current\-period value of a worker’s skill near the AI frontier against the wage it commands there\. When that value keeps pace with the rising wage up to the frontier, the engaged interval reaches it; when the wage overtakes it first, the most\-engaged workers sit in a band just below\. The following proposition makes this precise\.

###### Proposition 4

SupposeB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1\. Fixη\\eta\. On the admissible domainD2D\_\{2\}, the engaged setE2​\(η\)≡\{s2∈D2:η<η¯​\(s2\)\}E\_\{2\}\(\\eta\)\\equiv\\\{s\_\{2\}\\in D\_\{2\}:\\eta<\\bar\{\\eta\}\(s\_\{2\}\)\\\}is an interval that widens asη\\etafalls, beginning from the workers with the highest cutoffη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)\. WhenΔ2≥0\\Delta\_\{2\}\\geq 0,η¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)rises monotonically over\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)and is largest at the benchmarkα2\\alpha\_\{2\}; whenΔ2<0\\Delta\_\{2\}<0,η¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is single\-peaked, reaching an interior maximum at somes2†∈\(s¯2,α2\)s\_\{2\}^\{\\dagger\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\.

The sign ofΔ2\\Delta\_\{2\}determines where engagement is most intense\. WhenΔ2≥0\\Delta\_\{2\}\\geq 0, the cutoff is highest at the AI frontier, so the most\-engaged workers sit at the top of the admissible range; whenΔ2<0\\Delta\_\{2\}<0, the cutoff peaks at an interior skills2†s\_\{2\}^\{\\dagger\}, and they sit in a band just below the frontier\.[OA3](https://arxiv.org/html/2606.29111#Thmclaim3)givess2†s\_\{2\}^\{\\dagger\}and the endpoints of the engaged interval in each case\.

Under one further condition,λ​R​π2≥b​α2b−1\\lambda R\\pi\_\{2\}\\geq b\\alpha\_\{2\}^\{b\-1\}, engagement rises monotonically with skill acrossD2D\_\{2\}: the firm engages the most\-skilled admissible workers most\. The condition rules out an offsetting channel\. A higher skills2s\_\{2\}raises the engagement gainF2F\_\{2\}by lifting the skill trajectory engagement builds, but it also raises the wage and so could erode the firm’s current margin\. The condition ensures the first effect dominates: even at the benchmark, where the wage slope is steepest, the extra current revenue a more skilled worker earns when AI fails covers the higher wage\. The gainF2​\(h;s2,A2\)F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)is then increasing ins2s\_\{2\}at every engagement levelhh\. Combined with the single\-crossing property, which makesF2F\_\{2\}decreasing inhh, a higher skill shifts the whole marginal\-gain curve upward without disturbing its single crossing with the costλ​R​δ\\lambda R\\delta\. Equilibrium engagement therefore rises weakly with skill: zero for the least\-skilled admissible workers, then interior, then full near the frontier\.

###### Corollary 3

SupposeB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1, and supposeλ​R​π2≥b​α2b−1\\lambda R\\pi\_\{2\}\\geq b\\alpha\_\{2\}^\{b\-1\}\. Then, onD2D\_\{2\}, the gain to attracting workers,F2​\(h;s2,A2\)F\_\{2\}\(h;s\_\{2\},A\_\{2\}\), is decreasing in engagement and increasing in skill, and equilibrium engagementh2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\)is continuous and nondecreasing in skill\. Two thresholdss¯¯2≤s^2D≤s¯2D≤α2\\bar\{\\bar\{s\}\}\_\{2\}\\leq\\hat\{s\}\_\{2\}^\{D\}\\leq\\overline\{s\}\_\{2\}^\{D\}\\leq\\alpha\_\{2\}partition the admissible workers into three consecutive bands: engagement is zero fors2≤s^2Ds\_\{2\}\\leq\\hat\{s\}\_\{2\}^\{D\}, interior fors^2D<s2<s¯2D\\hat\{s\}\_\{2\}^\{D\}<s\_\{2\}<\\overline\{s\}\_\{2\}^\{D\}, and full fors2≥s¯2Ds\_\{2\}\\geq\\overline\{s\}\_\{2\}^\{D\}, rising with skill across the interior band\. The lower thresholds^2D\\hat\{s\}\_\{2\}^\{D\}is the skill at which the engagement gain at zero engagement first covers the costλ​R​δ\\lambda R\\delta, so below it no worker is engaged; the upper thresholds¯2D\\overline\{s\}\_\{2\}^\{D\}is the skill at which the gain at full engagement covers the cost, so above it every worker is engaged fully\.

The precise threshold definitions, and the continuity and differentiability ofh2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\), are given in the proof of[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)\.

This is the engagement reversal\. In the single\-firm benchmark, engagement falls with skill because the firm is securing fallback capacity at the lowest cost \([Proposition2](https://arxiv.org/html/2606.29111#Thmproposition2)\)\. Under mobility, engagement rises weakly with skill onD2D\_\{2\}because the firm is competing through the skill trajectory the job offers\. The same choice variable serves a different economic purpose\.

### 5\.3Worker Mobility Separates the Effects of AI Capability and Reliability

Worker mobility also separates the effects of AI capability and AI reliability\. Without mobility, the two dimensions act differently: capability does not affect engagement at all, because a single firm values a worker’s skill only as a fallback for AI failures, although higher reliability reduces engagement monotonically \([Corollary1](https://arxiv.org/html/2606.29111#Thmcorollary1)\)\. Mobility changes both\. Because a firm now also engages to attract workers, capability matters: it raises the future skill a unit of engagement builds, which makes the job more attractive\. Reliability, by contrast, moves two margins at once\. It changes how much skill a unit of engagement builds, but it also changes the firm’s current operating margin on a below\-benchmark worker, and these two forces can point in opposite directions\. The net effect on engagement can therefore be non\-monotone\. We state the comparative statics for the power wageB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1and a worker of skills2∈D2s\_\{2\}\\in D\_\{2\}at which the terminal\-period equilibrium is interior,h2∗​\(s2\)∈\(0,1\)h\_\{2\}^\{\*\}\(s\_\{2\}\)\\in\(0,1\)\.

Capability and reliability diverge under mobility: a more capable AI moves engagement in one direction, although a more reliable AI can move it either way\. The next result makes the split precise for the power wage\. The proof of[Proposition5](https://arxiv.org/html/2606.29111#Thmproposition5)in the online appendix defines threshold valuesπ¯\+\\bar\{\\pi\}^\{\+\}andπ¯−\\bar\{\\pi\}^\{\-\}that separate the range in which a marginal increase in unreliability raises engagement from the range in which it lowers engagement\.

###### Proposition 5

Suppose the terminal single\-crossing condition in[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)holds ands2∈D2s\_\{2\}\\in D\_\{2\}is such that the terminal\-period equilibrium of[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)is interior,h2∗​\(s2\)∈\(0,1\)h\_\{2\}^\{\*\}\(s\_\{2\}\)\\in\(0,1\)\.

1. \(i\)A higher AI capability raises engagement: for local changes inα2\\alpha\_\{2\}that keeps2∈D2s\_\{2\}\\in D\_\{2\}and keep the equilibrium interior,∂h2∗​\(s2\)/∂α2\>0\\partial h\_\{2\}^\{\*\}\(s\_\{2\}\)/\\partial\\alpha\_\{2\}\>0\.555Part \(i\) does not rely on the power wage; it holds for anyBBsatisfying the terminal single\-crossing condition\.
2. \(ii\)Suppose additionally thatB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1\. For local changes inπ2\\pi\_\{2\}that keeps2∈D2s\_\{2\}\\in D\_\{2\}and keep the equilibrium interior, engagement rises with AI unreliability,∂h2∗​\(s2,π2\)/∂π2\>0\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)/\\partial\\pi\_\{2\}\>0, wheneverπ2<π¯\+\\pi\_\{2\}<\\bar\{\\pi\}^\{\+\}andπ2≤12\\pi\_\{2\}\\leq\\tfrac\{1\}\{2\}, and falls,∂h2∗​\(s2,π2\)/∂π2<0\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)/\\partial\\pi\_\{2\}<0, wheneverπ2\>π¯−\\pi\_\{2\}\>\\bar\{\\pi\}^\{\-\}\.

Capability moves engagement in one direction: under mobility, a more capable AI raises engagement, because it improves the skill trajectory the firm offers\. Reliability moves engagement in two directions at once, and its net effect splits into three regimes\. When AI is highly reliable, a marginal increase in unreliability raises engagement: it makes a unit of engagement build more skill, strengthening the sorting value of the skill the worker carries\. When AI is already unreliable, further unreliability lowers engagement: frequent failures erode the current margin by more than the added skill is worth, especially for workers far from the benchmark\. Between these regimes the two channels offset\. Reliability can therefore move engagement non\-monotonically, in contrast to the monotone decline without mobility, producing the hump\-shaped pattern in[Figure4](https://arxiv.org/html/2606.29111#S5.F4)on the interior of the curve, on whichs2∈D2s\_\{2\}\\in D\_\{2\}andh2∗​\(s2\)∈\(0,1\)h\_\{2\}^\{\*\}\(s\_\{2\}\)\\in\(0,1\)\.

![Refer to caption](https://arxiv.org/html/2606.29111v1/x3.png)Figure 4:Engagement can be highest at intermediate reliability\. Equilibrium engagementh2∗​\(s2;π2\)h\_\{2\}^\{\*\}\(s\_\{2\};\\pi\_\{2\}\)under mobility, plotted against AI unreliabilityπ2\\pi\_\{2\}\(so reliability is1−π21\-\\pi\_\{2\}\): as reliability falls from its highest level, a unit of engagement builds more skill and engagement rises, until frequent failures erode current output and engagement falls again, so it peaks nearπ2≈0\.5\\pi\_\{2\}\\approx 0\.5\. Parameter values:B​\(s\)=W​\(s\)=s1\.2B\(s\)=W\(s\)=s^\{1\.2\},s2=0\.30s\_\{2\}=0\.30,α2=0\.90\\alpha\_\{2\}=0\.90,ϕ=0\.5\\phi=0\.5,γ=0\.3\\gamma=0\.3,δ=0\.5\\delta=0\.5,λ​R=3\.0\\lambda R=3\.0,β=0\.95\\beta=0\.95,η=0\.21\\eta=0\.21\.The hump shape relies on the nature of learning, not on a specific functional form\. Building skill requires both working with a functioning AI and occasionally handling the fallback, so learning is strongest at intermediate reliability and weaker at either extreme\. The hump follows from this two\-sided pattern, not from the exact productπt​\(1−πt\)\\pi\_\{t\}\(1\-\\pi\_\{t\}\):[SectionOA\.6](https://arxiv.org/html/2606.29111#S6a)reproduces it for a range of two\-sided learning profiles and shows that it weakens or disappears when learning is one\-sided, drawing on AI exposure or fallback practice alone\.

### 5\.4Period 1: Mobility and the Fallback Motive

Period 1 combines the two forces the earlier subsections separated\. Engagement now lowers current output, builds fallback skill for the period\-2 state in which AI fails, and shapes how workers sort across firms\. Because that skill is portable, the other firm can draw on it after workers re\-sort in period 2\. Relative to the single firm, mobility therefore adds a sorting gain and an appropriability loss, and which force dominates depends on how readily workers respond to the skill trajectories firms offer\.

For the main text it suffices to study initial skillss1∈D1s\_\{1\}\\in D\_\{1\}, the region for which every continuation state reachable under a feasible period\-1 engagement lies in the terminal domainD2=\(s¯¯2,α2\)D\_\{2\}=\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\), so that[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)delivers a unique symmetric terminal equilibriumh2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\)at every reachable state\. The explicit cutoffs¯¯1\\bar\{\\bar\{s\}\}\_\{1\}definingD1D\_\{1\}is reported in[SectionOA\.5](https://arxiv.org/html/2606.29111#S5a), which also shows that this restriction is not narrow: across power\-wage examples satisfying the model assumptions and terminal\-domain conditions,D1D\_\{1\}is nonempty in95\.3%95\.3\\%of cases and covers a median81\.0%81\.0\\%of the feasible period\-1 skill interval\(s¯,α1\)\(\\underline\{s\},\\alpha\_\{1\}\)\.

A worker who joins firmjjenters period 2 with skillg​\(s1,h1j;A1\)g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)and is then engaged at the symmetric equilibrium levelh2∗​\(g​\(s1,h1j;A1\)\)h\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\\bigr\)of[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)\. The firm’s period\-2 profit, half the margin at that equilibrium, is

Π2∗​\(g​\(s1,h1j;A1\),A2\)≜12​M​\(g​\(s1,h1j;A1\),h2∗​\(g​\(s1,h1j;A1\)\);A2\),\\displaystyle\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)\\triangleq\\frac\{1\}\{2\}M\\\!\\Bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),\\,h\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\\bigr\);A\_\{2\}\\Bigr\),\(9\)and the worker’s period\-2 value, the log\-sum in[eq\.5](https://arxiv.org/html/2606.29111#S5.E5)at that equilibrium, is

V¯2​\(g​\(s1,h1j;A1\),A2\)=W​\(g​\(s1,h1j;A1\)\)\+β​B​\(ζ​\(s1,h1j;A1,A2\)\)\+η​log⁡2,\\displaystyle\\bar\{V\}\_\{2\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)=W\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\\bigr\)\+\\beta B\\\!\\bigl\(\\zeta\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\},A\_\{2\}\)\\bigr\)\+\\eta\\log 2,up to the type\-I extreme\-value normalization constant, where

ζ​\(s1,h1;A1,A2\)≜g​\(g​\(s1,h1;A1\),h2∗​\(g​\(s1,h1;A1\)\);A2\)\\displaystyle\\zeta\(s\_\{1\},h\_\{1\};A\_\{1\},A\_\{2\}\)\\triangleq g\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),\\,h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\);A\_\{2\}\\bigr\)is the post\-horizon skill the worker carries out of the model, as shaped by period\-1 engagement\.

A worker choosing a firm in period 1 weighs the current wage against the value expected from period 2 onward\. The wageW​\(s1\)W\(s\_\{1\}\)is common to both firms and cancels in the logit, exactly as in the terminal period\. The worker’s continuation value differs across firms only through the skill carried into period 2\. Because the worker re\-sorts at the start of period 2 and the terminal equilibrium at any reachable skill is symmetric \([Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)\), that continuation value is the log\-sum of[eq\.5](https://arxiv.org/html/2606.29111#S5.E5)evaluated at the terminal equilibrium, namelyV¯2\\bar\{V\}\_\{2\}as defined above\. The worker sorts in period 1 on these period\-2 values, choosing firmjjwith probability

σ1j​\(h1j,h1−j;s1,A1,A2\)=11\+exp⁡\(β​\[V¯2​\(g​\(s1,h1−j;A1\),A2\)−V¯2​\(g​\(s1,h1j;A1\),A2\)\]/η\)\.\\displaystyle\\sigma\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)=\\frac\{1\}\{1\+\\exp\\\!\\left\(\\beta\\Bigl\[\\bar\{V\}\_\{2\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\-j\};A\_\{1\}\),A\_\{2\}\\bigr\)\-\\bar\{V\}\_\{2\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)\\Bigr\]/\\eta\\right\)\}\.Regardless of which firm the worker chose in period 1, re\-sorting at the start of period 2 means both firms engage the worker identically; each firm therefore earns the period\-2 profitΠ2∗\\Pi\_\{2\}^\{\*\}of[eq\.9](https://arxiv.org/html/2606.29111#S5.E9), namely half the margin at the terminal equilibrium, irrespective of which firm built the skill the worker arrives with\. Period\-1 engagement thus determines who collects the current margin and what skill the worker carries forward, but not how the period\-2 profit on the worker is divided\. If the worker joins firmjj, the firm earns the current marginM​\(s1,h1j;A1\)M\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)and the discounted period\-2 profitβ​Π2∗​\(g​\(s1,h1j;A1\),A2\)\\beta\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\); if the worker joins the other firm, it earns no current margin but still collectsβ​Π2∗​\(g​\(s1,h1−j;A1\),A2\)\\beta\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\-j\};A\_\{1\}\),A\_\{2\}\\bigr\)on the skill the other firm built\. Firmjjtherefore choosesh1jh\_\{1\}^\{j\}to maximize

Φ1j​\(h1j,h1−j;s1,A1,A2\)≜\\displaystyle\\Phi\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)\\triangleq\{\}σ1j​\(h1j,h1−j;s1,A1,A2\)​\[M​\(s1,h1j;A1\)\+β​Π2∗​\(g​\(s1,h1j;A1\),A2\)\]\\displaystyle\\sigma\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)\\Bigl\[M\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\+\\beta\\,\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)\\Bigr\]\+\[1−σ1j​\(h1j,h1−j;s1,A1,A2\)\]​β​Π2∗​\(g​\(s1,h1−j;A1\),A2\)\.\\displaystyle\+\\bigl\[1\-\\sigma\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)\\bigr\]\\,\\beta\\,\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\-j\};A\_\{1\}\),A\_\{2\}\\bigr\)\.\(10\)
Whereas the terminal period has a unique symmetric equilibrium \([Lemma1](https://arxiv.org/html/2606.29111#Thmlemma1)\), period 1 admits both symmetric and asymmetric equilibria: because skill moves with the worker, a firm can let the other firm build it and draw on it after workers re\-sort, which can pull the two firms’ engagement apart\. We treat the two cases in turn\. This subsection characterizes the symmetric equilibrium and compares it to the single\-firm benchmark;[Section5\.5](https://arxiv.org/html/2606.29111#S5.SS5)shows by example when ex ante identical firms instead specialize into a skill\-builder and a free\-rider\.

LetF1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)denote the derivative ofΦ1j​\(h1j,h1−j;s1,A1,A2\)\\Phi\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)in its own actionh1jh\_\{1\}^\{j\}, evaluated at the symmetric profileh1j=h1−j=h1h\_\{1\}^\{j\}=h\_\{1\}^\{\-j\}=h\_\{1\}\. Unlike the terminal\-period gainF2F\_\{2\}, which was set against the separate operating costλ​R​δ\\lambda R\\delta,F1F\_\{1\}is the entire marginal payoff: it already embeds the operating cost of engaging a below\-benchmark worker, the sorting gain from a more attractive skill trajectory, and the continuation\-profit effect\. Because these enter with different weights, no common factor divides out to leave a constant cost on one side, as it did in period 2\. The interior condition is thereforeF1=0F\_\{1\}=0, the period\-1 counterpart ofF2=λ​R​δF\_\{2\}=\\lambda R\\delta, and a symmetric equilibrium must leave neither firm a profitable deviation in its own engagement, which characterizes the candidates by the sign ofF1F\_\{1\}at the boundaries and its roots in the interior\.

###### Lemma 2

Considers1∈D1s\_\{1\}\\in D\_\{1\}, and lets2∗≡g​\(s1,h1∗;A1\)s\_\{2\}^\{\*\}\\equiv g\(s\_\{1\},h\_\{1\}^\{\*\};A\_\{1\}\)\. At any symmetric profileh1j=h1−j=h1∗h\_\{1\}^\{j\}=h\_\{1\}^\{\-j\}=h\_\{1\}^\{\*\}at whichV¯2​\(s2,A2\)\\bar\{V\}\_\{2\}\(s\_\{2\},A\_\{2\}\)andΠ2∗​\(s2,A2\)\\Pi\_\{2\}^\{\*\}\(s\_\{2\},A\_\{2\}\)are differentiable ins2s\_\{2\}ats2=s2∗s\_\{2\}=s\_\{2\}^\{\*\}, a symmetric equilibrium must satisfyF1​\(h1∗;s1,A1,A2\)​\(h1−h1∗\)≤0F\_\{1\}\(h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)\(h\_\{1\}\-h\_\{1\}^\{\*\}\)\\leq 0for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\]; equivalently,h1∗=0h\_\{1\}^\{\*\}=0withF1​\(0;s1,A1,A2\)≤0F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)\\leq 0, orh1∗=1h\_\{1\}^\{\*\}=1withF1​\(1;s1,A1,A2\)≥0F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)\\geq 0, orh1∗∈\(0,1\)h\_\{1\}^\{\*\}\\in\(0,1\)withF1​\(h1∗;s1,A1,A2\)=0F\_\{1\}\(h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)=0\.

The lemma gives conditions that any symmetric equilibrium must satisfy\. These conditions are necessary but not sufficient: a profile meeting them need not be unique, and need not be an equilibrium\. The proposition below adds sufficient regularity conditions for both\. First,F1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)is strictly decreasing inh1h\_\{1\}, so the candidate conditions in[Lemma2](https://arxiv.org/html/2606.29111#Thmlemma2)select at most one symmetric candidate\. Second, for each firmjj, the payoffΦ1j​\(h1j,h1−j;s1,A1,A2\)\\Phi\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)is strictly concave inh1jh\_\{1\}^\{j\}for every fixedh1−jh\_\{1\}^\{\-j\}, so the first\-order condition is sufficient for a global best response\. Together, these two properties turn the necessary conditions in the lemma into a unique symmetric equilibrium\.

For tractability, we impose skill\-domain restrictions that keep terminal engagement fixed along every period\-1 skill trajectory starting froms1s\_\{1\}\. The issue is thatF1F\_\{1\},Π2∗\\Pi\_\{2\}^\{\*\}, andV¯2\\bar\{V\}\_\{2\}inherit the shape of the terminal policyh2∗​\(g​\(s1,h1;A1\)\)h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)\. By[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3), terminal engagement is zero below a lower skill cutoff, interior between the two cutoffs, and full above an upper cutoff\. If the reachable continuation states straddle one of these cutoffs, terminal engagement changes withh1h\_\{1\}, creating additional curvature and possible kinks\. We therefore focus on values ofs1s\_\{1\}for which every reachable continuation state lies entirely in a terminal corner region, soh2∗​\(g​\(s1,h1;A1\)\)h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)is either always zero or always one, independently ofh1h\_\{1\}\.

For the power\-wage setting, lets1Ds\_\{1\}^\{D\}be the period\-1 threshold below which even full engagement keeps the worker below the terminal no\-engagement cutoff, so terminal engagement is always zero, and lets1Fs\_\{1\}^\{F\}be the threshold above which even zero engagement keeps the worker above the terminal full\-engagement cutoff, so terminal engagement is always one\. These thresholds pull the domain\-restricted terminal cutoffss^2D\\hat\{s\}\_\{2\}^\{D\}ands¯2D\\overline\{s\}\_\{2\}^\{D\}of[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)back to period 1; their explicit formulas, and the requirement that each region be nonempty after intersection withD1D\_\{1\}, are given in[SectionOA\.5](https://arxiv.org/html/2606.29111#S5a), which also shows that the fixed\-terminal regionss1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ands1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}cover at least93\.8%93\.8\\%ofD1D\_\{1\}on average, with median coverage100%100\\%, across power\-wage examples satisfying the model assumptions and terminal\-domain conditions\.

###### Proposition 6

Suppose the conditions of[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)hold,s1∈D1s\_\{1\}\\in D\_\{1\}, ands1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ors1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}\. There is a thresholdη1​\(s1\)\>0\\eta\_\{1\}\(s\_\{1\}\)\>0such that, wheneverη≥η1​\(s1\)\\eta\\geq\\eta\_\{1\}\(s\_\{1\}\), the engagement gainF1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)is strictly decreasing inh1h\_\{1\}and each firm’s objectiveΦ1j​\(h1j,h1−j;s1,A1,A2\)\\Phi\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)is strictly concave inh1jh\_\{1\}^\{j\}, so the period 1 game has a unique symmetric equilibriumh1∗​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\), found by comparing the engagement gain at the boundaries with zero:

1. \(i\)ifF1​\(1;s1,A1,A2\)≥0F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)\\geq 0, both firms engage fully,h1∗​\(s1\)=1h\_\{1\}^\{\*\}\(s\_\{1\}\)=1;
2. \(ii\)ifF1​\(1;s1,A1,A2\)<0<F1​\(0;s1,A1,A2\)F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)<0<F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\), both engage at the interior levelh1∗​\(s1\)∈\(0,1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)\\in\(0,1\)solvingF1​\(h1;s1,A1,A2\)=0F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)=0;
3. \(iii\)ifF1​\(0;s1,A1,A2\)≤0F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)\\leq 0, neither firm engages,h1∗​\(s1\)=0h\_\{1\}^\{\*\}\(s\_\{1\}\)=0\.

Terminal engagement at the realized continuation state is then fixed: disengagement,h2∗​\(g​\(s1,h1∗​\(s1\);A1\)\)=0h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\}^\{\*\}\(s\_\{1\}\);A\_\{1\}\)\)=0, whens1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, and full engagement,h2∗​\(g​\(s1,h1∗​\(s1\);A1\)\)=1h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\}^\{\*\}\(s\_\{1\}\);A\_\{1\}\)\)=1, whens1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}\.

[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)shows that, on the fixed\-terminal regions, ex ante identical firms settle on a single common engagement level\. Intuitively, a fixed terminal engagement removes the curvature that portable skill would otherwise inject, so the period\-1 best responses are well behaved and the symmetric equilibrium is unique\. The thresholdη1​\(s1\)\\eta\_\{1\}\(s\_\{1\}\)is a regularity threshold for the period\-1 game, not a terminal\-engagement condition\. Terminal engagement is fixed by the skill restrictionss1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ands1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}; the role ofη≥η1​\(s1\)\\eta\\geq\\eta\_\{1\}\(s\_\{1\}\)is to makeF1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)strictly decreasing inh1h\_\{1\}andΦ1j​\(h1j,h1−j;s1,A1,A2\)\\Phi\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)strictly concave inh1jh\_\{1\}^\{j\}\. Ifη<η1​\(s1\)\\eta<\\eta\_\{1\}\(s\_\{1\}\),[Lemma2](https://arxiv.org/html/2606.29111#Thmlemma2)still gives necessary conditions for symmetric equilibria, but uniqueness and sufficiency are no longer guaranteed\.

We now compare period\-1 engagement under worker mobility with the single\-firm benchmark of[Section4](https://arxiv.org/html/2606.29111#S4)\. We focus on the regions1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, in which the worker is disengaged in period 2 both under mobility and without it\. The two period\-1 problems then share the same continuation, so the comparison isolates the forces mobility introduces in period 1: the sorting gain and the skill spillover, rather than differences in anticipated period\-2 engagement\.

###### Corollary 4

Suppose the conditions of[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)hold withs1∈D1s\_\{1\}\\in D\_\{1\}ands1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, and leth1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)denote the single firm’s period\-1 engagement \([Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)\)\. There exists a thresholdη2​\(s1\)≥η1​\(s1\)\\eta\_\{2\}\(s\_\{1\}\)\\geq\\eta\_\{1\}\(s\_\{1\}\)such that for allη≥η2​\(s1\)\\eta\\geq\\eta\_\{2\}\(s\_\{1\}\)the unique symmetric equilibrium satisfiesh1∗​\(s1\)≤h1sf​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)\\leq h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\), with strict inequality wheneverh1sf​\(s1\)∈\(0,1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\\in\(0,1\)\.666The restriction tos1<s1Ds\_\{1\}<s\_\{1\}^\{D\}does not tilt the comparison: the single firm’s policy bands do not depend onη\\eta, whereass1Ds\_\{1\}^\{D\}rises withη\\eta, so at the thresholdη2​\(s1\)\\eta\_\{2\}\(s\_\{1\}\)the region typically spans workers the single firm engages fully, partially, and not at all\. Its role is identification: it is the only region on which the firm\-under\-mobility’s terminal play matches the single firm’s invariable terminal disengagement, so the period\-1 gap is attributable to mobility alone\.

[Corollary4](https://arxiv.org/html/2606.29111#Thmcorollary4)isolates the region in which mobility works in the classical Becker direction\. When workers respond weakly to differences in skill trajectories, the sorting gain is small but the portability of skill remains\. A firm with mobile workers therefore invests no more, and typically less, than a single firm that captures the full return on the skill it builds\.

Mobility can also shift engagement across time\. In the single\-firm benchmark of[Section4](https://arxiv.org/html/2606.29111#S4), a below\-benchmark worker is never engaged in the terminal period; any engagement occurs only in period 1, as an investment in future fallback skill\. With mobility, by contrast, terminal engagement can arise even when firms do not engage the worker in period 1, because it attracts workers at once rather than serving as an investment whose return may spill over to the other firm\.

###### Example 1

For the parameterization reported in[SectionOA\.7](https://arxiv.org/html/2606.29111#S7), the unique symmetric equilibrium satisfiesh1∗​\(s1\)=0h\_\{1\}^\{\*\}\(s\_\{1\}\)=0while the induced continuation state satisfiesh2∗​\(g​\(s1,0;A1\)\)=1h\_\{2\}^\{\*\}\(g\(s\_\{1\},0;A\_\{1\}\)\)=1\. Mobility can therefore reverse the timing of engagement: a worker ignored before re\-sorting is fully engaged after re\-sorting\.

Because the example depends on the skill transition,[SectionOA\.7](https://arxiv.org/html/2606.29111#S7)tests robustness by replacing the baseline learning termϕ​ht​πt​\(1−πt\)​\(αt−st\)\\phi h\_\{t\}\\pi\_\{t\}\(1\-\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\)with alternative termsϕ​ht​ℓ​\(πt\)​\(αt−st\)\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\)\. The timing reversal persists for a wide range of exposure–practice profiles, such asℓ​\(π\)=πa​\(1−π\)c\\ell\(\\pi\)=\\pi^\{a\}\(1\-\\pi\)^\{c\}andℓ​\(π\)=\(κ\+π\)​\(1−π\)\\ell\(\\pi\)=\(\\kappa\+\\pi\)\(1\-\\pi\), that preserve sufficient terminal skill\-building atπ2=0\.35\\pi\_\{2\}=0\.35; it disappears when the alternative profile makes terminal skill\-building too weak\.

### 5\.5Asymmetric Specialization

Ex ante identical firms need not behave identically\. The reason is that skill, once built, is portable\. When a firm invests in a worker’s skill in period 1, the worker can switch firms in period 2 and carry that skill elsewhere\. The firm bears the cost of building skill but cannot fully capture its return\. No such force operates in the terminal period, because there is no period after it for workers to move into\.

Whether each firm’s engagement incentive rises or falls when the other firm engages more is measured by the cross\-partial of firmjj’s period\-1 objective,∂2Φ1j/∂h1j​∂h1−j\\partial^\{2\}\\Phi\_\{1\}^\{j\}/\\partial h\_\{1\}^\{j\}\\,\\partial h\_\{1\}^\{\-j\}\. When the other firm invests more heavily in skill, two opposing forces act on firmjj\. First, more continuation skill enters the shared labor pool once workers re\-sort, which weakens firmjj’s own incentive to build skill\. Second, the other firm’s engagement draws workers away in period 1, so firmjj’s engagement now applies to fewer workers, lowering its marginal cost of engaging and thus strengthening its incentive to engage\. Evaluating this cross\-partial at a symmetric profile \(the algebra is in[SectionOA\.2](https://arxiv.org/html/2606.29111#S2a)\), engagement choices are local strategic complements if and only if

2​β​Π2∗⁣′​\(g​\(s1,h1;A1\),A2\)​\(ϕ​π1\+γ\)<λ​R​δ\.\\displaystyle 2\\beta\\,\\Pi\_\{2\}^\{\*\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),A\_\{2\}\\bigr\)\(\\phi\\pi\_\{1\}\+\\gamma\)<\\lambda R\\delta\.\(11\)
The left side of[eq\.11](https://arxiv.org/html/2606.29111#S5.E11)is the pooled\-skill externality; the right side is the current\-period cost effect\. When the externality dominates and the condition fails, incentives are local strategic substitutes and asymmetric equilibria become possible, though the condition does not guarantee they exist\.

[Figure5](https://arxiv.org/html/2606.29111#S5.F5)illustrates such a case for an admissible power\-wage example\. Condition[eq\.11](https://arxiv.org/html/2606.29111#S5.E11)fails at the symmetric profile, so engagement choices are local strategic substitutes; each firm’s best response slopes downward, withB​R​\(0\)=1BR\(0\)=1andB​R​\(1\)=0BR\(1\)=0, and the two cross at one symmetric and two asymmetric equilibria\. At the asymmetric equilibria one firm engages fully and the other not at all, and the disengaged firm free\-rides on the skill the other builds, out\-earning it\.

![Refer to caption](https://arxiv.org/html/2606.29111v1/x4.png)Figure 5:Ex ante identical firms can specialize, and the free\-rider out\-earns the builder\. Each firm’s period\-1 best response declines in the other firm’s engagement, and the two cross at one symmetric equilibrium and a pair of asymmetric equilibria\(0,1\)\(0,1\)and\(1,0\)\(1,0\)in which one firm engages fully and the other not at all; at the asymmetric equilibria the disengaged firm earns7\.7327\.732per worker against the engaged firm’s7\.4657\.465, because it draws on the skill the other firm builds\. Parameter values:B​\(s\)=W​\(s\)=s1\.08B\(s\)=W\(s\)=s^\{1\.08\},α1=0\.76\\alpha\_\{1\}=0\.76,α2=0\.82\\alpha\_\{2\}=0\.82,π1=0\.72\\pi\_\{1\}=0\.72,π2=0\.48\\pi\_\{2\}=0\.48,ϕ=3\.4\\phi=3\.4,γ=0\.011\\gamma=0\.011,δ=0\.29\\delta=0\.29,λ​R=20\\lambda R=20,β=0\.5\\beta=0\.5,η=16\\eta=16,s1=0\.36s\_\{1\}=0\.36\.At the asymmetric profiles the firms specialize: one engages fully and builds skill, while the other disengages and free\-rides on the portable skill that enters the shared pool after re\-sorting\. Mobility reverses the payoff ranking, so the free\-rider out\-earns the builder \(7\.7327\.732versus7\.4657\.465\) despite offering the weaker skill trajectory\. Which firm plays which role depends on history or coordination, but in either case two ex ante identical firms split endogenously into a skill\-builder and a free\-rider\.

## 6Concluding Remarks

As AI handles more routine work, routing the rest to the machine is the obvious move\. Yet keeping the worker engaged, though it lowers current output, preserves the fallback skill the firm may need when AI fails, and, once workers are mobile, shapes the skill trajectory that determines which firm they join\.

The two motives point to different workers\. The fallback motive is strongest at the bottom of the skill distribution, where engagement builds the most capability for future failure states; the sorting motive is strongest near the AI frontier, where engagement is least costly and the portable skill it builds is most valuable in the labor market\. Worker mobility can therefore reverse who is engaged, depress investment by weakening the firm’s hold on the skill it builds, or sustain asymmetric specialization in which one of two ex ante identical firms builds skill and the other draws on it\. It also pulls the two dimensions of AI progress apart: under mobility a more capable AI raises engagement by strengthening the skill trajectory a job can offer, whereas a more reliable AI can raise or lower it, so engagement may be highest at intermediate reliability\.

The framework extends in several directions\. A longer horizon would let worker skill and AI capability co\-evolve, so that engagement today shapes the whole skill path rather than a single fallback\. Making AI progress itself respond to how intensively firms engage would close that loop\. A richer labor market, in which the sorting motive operates through wages as well as job design, would let firms compete on pay and skill trajectory at once\. Beyond these extensions, the central prediction is testable: because mobility can reverse engagement from the least\-skilled to the most\-skilled workers, it invites study wherever task\-level human–AI allocation and worker movement can be observed together\.

Human involvement in AI\-assisted production is an investment decision about fallback capacity and portable human capital\. Treating that decision as a cost to be minimized, routing each residual task to the machine, leaves a firm short of the fallback capacity it will need when the machine fails and short of the skill trajectory that keeps its strongest workers from leaving\. A radiology group that sends every routine scan to its AI may find, the first time a hard case arrives or the system is unavailable, that its own unaided reading has eroded below what the case demands, and that the strong readers whose skills it chose not to build have already left for a group that invested in them\.

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Online Appendix to “Managing the Human Fallback: Skill Investment Under Improving AI and Worker Mobility”

This online appendix supplements the main text\.[SectionOA\.1](https://arxiv.org/html/2606.29111#S1a)relates the model primitives to observable workflow data;[SectionOA\.2](https://arxiv.org/html/2606.29111#S2a)proves all results in the order they appear in the paper;[SectionOA\.3](https://arxiv.org/html/2606.29111#S3a)restates the single\-firm engagement policy as a skill\-transition table; and the remaining sections report the size of the terminal and period\-1 skill domains \([SectionsOA\.4](https://arxiv.org/html/2606.29111#S4a)and[OA\.5](https://arxiv.org/html/2606.29111#S5a)\), two robustness checks on the reliability pattern and the timing reversal \([SectionsOA\.6](https://arxiv.org/html/2606.29111#S6a)and[OA\.7](https://arxiv.org/html/2606.29111#S7)\), and an extension that lets firms attract workers with a pay bonus \([SectionOA\.8](https://arxiv.org/html/2606.29111#S8)\)\.

Table OA1:Summary of NotationSymbolDescriptionTime and AI technologyttPeriod index,T=2T=2AtA\_\{t\}AI technology level, periodttαt≜α​\(At\)\\alpha\_\{t\}\\triangleq\\alpha\(A\_\{t\}\)AI capabilityπt≜π​\(At\)\\pi\_\{t\}\\triangleq\\pi\(A\_\{t\}\)Failure probability \(1−πt1\-\\pi\_\{t\}reliability\)Worker skill and engagementsts\_\{t\}Worker skill,st∈\[0,1\]s\_\{t\}\\in\[0,1\]hth\_\{t\}Engagement,ht∈\[0,1\]h\_\{t\}\\in\[0,1\]hc​\(At\)h^\{c\}\(A\_\{t\}\)Skill\-preserving engagementSkill dynamicsδ\\deltaEffect of intervention on AI outputγ\\gammaSkill\-erosion rate, passive relianceϕ\\phiStrength of learning from engagementΓ​\(ht;At\)\\Gamma\(h\_\{t\};A\_\{t\}\)Net skill\-adjustment coefficientg​\(st,ht;At\)g\(s\_\{t\},h\_\{t\};A\_\{t\}\)Skill transition functionWorker mobilityη\\etaSorting\-friction parameterσtj\\sigma\_\{t\}^\{j\}Logit share of workers at firmjjProductionqon​\(st,ht;At\)q^\{\\mathrm\{on\}\}\(s\_\{t\},h\_\{t\};A\_\{t\}\)Throughput, AI functioningqoff​\(st\)q^\{\\mathrm\{off\}\}\(s\_\{t\}\)Throughput, AI failedS​\(st,ht;At\)S\(s\_\{t\},h\_\{t\};A\_\{t\}\)Operational revenue per workerFirm payoffsM​\(st,ht;At\)M\(s\_\{t\},h\_\{t\};A\_\{t\}\)Per\-worker margin,S−WS\-WW​\(s\)W\(s\)Market wage scheduleλ​R\\lambda RTask loadλ\\lambdatimes revenueRRβ\\betaDiscount factorB​\(s\)B\(s\)Post\-horizon value of skill## OA\.1Notation and Empirical Counterparts

[TableOA1](https://arxiv.org/html/2606.29111#S0.T1)lists the notation used throughout the paper\. Each primitive also has a direct empirical counterpart in task\-level workflow data, as we describe below\.

AI capabilityαt\\alpha\_\{t\}can be measured from tasks completed by AI with little or no human involvement, conditional on AI functioning, whereas failure probabilityπt\\pi\_\{t\}can be measured from cases in which AI cannot be used as the autonomous producer because it is unavailable, blocked, fails validation, or faces an edge case requiring human fallback\. Worker skillsts\_\{t\}can be measured from no\-AI holdout tasks or fallback tasks performed independently by the worker\. Variation in engagement across otherwise comparable AI\-on tasks can be used to estimateδ\\delta, by measuring how human involvement shifts realized throughput relative to the AI\-alone and worker\-alone benchmarks\. Operational records provideλ\\lambdaandRR, whereas salary bands and external labor\-market data can be used to calibrateW​\(⋅\)W\(\\cdot\)andη\\eta\.

The skill\-dynamics parameters require repeated measures of worker performance\. Declines in independent performance after periods of low engagement and high AI reliance can be used to estimate or calibrate the erosion rateγ\\gamma, whereas improvements after hands\-on engagement with AI\-assisted tasks can be used to estimate or calibrate the learning parameterϕ\\phi\. Existing empirical work provides guidance for these measurements: studies of AI\-assisted work use variation in AI access and workflow design to estimate changes in productivity or output, which speaks toαt\\alpha\_\{t\},sts\_\{t\}, andδ\\delta\(Brynjolfssonet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib28), Dell’Acquaet al\.[2023](https://arxiv.org/html/2606.29111#bib.bib22), Niet al\.[2024](https://arxiv.org/html/2606.29111#bib.bib29), Vaccaroet al\.[2024](https://arxiv.org/html/2606.29111#bib.bib30)\); studies that measure later unaided performance after AI exposure speak to skill erosion and learning, and therefore toγ\\gammaandϕ\\phi\(Bastaniet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib20), Budzyńet al\.[2025](https://arxiv.org/html/2606.29111#bib.bib23), Poulidiset al\.[2025](https://arxiv.org/html/2606.29111#bib.bib21), Shen and Tamkin[2026](https://arxiv.org/html/2606.29111#bib.bib25)\)\.

## OA\.2Proofs

We first record a fact used in several proofs below\.

###### Claim OA1

Under[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4), for eacht∈\{1,2\}t\\in\\\{1,2\\\}, allst∈\(0,αt\)s\_\{t\}\\in\(0,\\alpha\_\{t\}\), and allht∈\[0,1\]h\_\{t\}\\in\[0,1\],M​\(st,ht;At\)\>0M\(s\_\{t\},h\_\{t\};A\_\{t\}\)\>0\.

*Proof\.*Fixt∈\{1,2\}t\\in\\\{1,2\\\}andht∈\[0,1\]h\_\{t\}\\in\[0,1\]\. Define

f​\(s\)≜M​\(s,ht;At\)=S​\(s,ht;At\)−W​\(s\),s∈\[0,αt\]\.\\displaystyle f\(s\)\\triangleq M\(s,h\_\{t\};A\_\{t\}\)=S\(s,h\_\{t\};A\_\{t\}\)\-W\(s\),\\qquad s\\in\[0,\\alpha\_\{t\}\]\.For fixedhth\_\{t\},S​\(s,ht;At\)S\(s,h\_\{t\};A\_\{t\}\)is affine inss, whereasWWis strictly convex by[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4)\. Henceffis strictly concave on\[0,αt\]\[0,\\alpha\_\{t\}\]\. At the lower endpoint, usingW​\(0\)=0W\(0\)=0,f​\(0\)=λ​R​\(1−πt\)​αt​\(1−δ​ht\)\>0f\(0\)=\\lambda R\(1\-\\pi\_\{t\}\)\\alpha\_\{t\}\(1\-\\delta h\_\{t\}\)\>0, because1−πt\>01\-\\pi\_\{t\}\>0,αt\>0\\alpha\_\{t\}\>0, andδ​ht<1\\delta h\_\{t\}<1\. At the upper endpoint,f​\(αt\)=λ​R​αt−W​\(αt\)\>0f\(\\alpha\_\{t\}\)=\\lambda R\\,\\alpha\_\{t\}\-W\(\\alpha\_\{t\}\)\>0by revenue sufficiency in[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4)\. Strict concavity offfon\[0,αt\]\[0,\\alpha\_\{t\}\]with strictly positive endpoint values impliesf​\(s\)\>0f\(s\)\>0for everys∈\(0,αt\)s\\in\(0,\\alpha\_\{t\}\)\.*Q\.E\.D\.*

*Proof of[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)\.*Fixs1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\alpha\_\{1\}\)\. Under[Assumption3](https://arxiv.org/html/2606.29111#Thmassumption3), the zero floor in the skill transition never binds for anyh1∈\[0,1\]h\_\{1\}\\in\[0,1\], so

g​\(s1,h1;A1\)=s1\+\(1−π1\)​\(h1​\(ϕ​π1\+γ\)−γ\)​\(α1−s1\)\.\\displaystyle g\(s\_\{1\},h\_\{1\};A\_\{1\}\)=s\_\{1\}\+\(1\-\\pi\_\{1\}\)\\bigl\(h\_\{1\}\(\\phi\\pi\_\{1\}\+\\gamma\)\-\\gamma\\bigr\)\(\\alpha\_\{1\}\-s\_\{1\}\)\.The period\-1 single\-firm objective is

V​\(h1\)=M​\(s1,h1;A1\)\+β​M​\(g​\(s1,h1;A1\),0;A2\)\.\\displaystyle V\(h\_\{1\}\)=M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\+\\beta\\,M\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),\\,0;\\,A\_\{2\}\\bigr\)\.
The first term is affine inh1h\_\{1\}\. The continuation term is the sum of an affine function ofg​\(s1,h1;A1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)and−W​\(g​\(s1,h1;A1\)\)\-W\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)\. Becauseg​\(s1,h1;A1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)is affine inh1h\_\{1\}with strictly positive slope

∂g​\(s1,h1;A1\)∂h1=\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\>0,\\displaystyle\\frac\{\\partial g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}=\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\>0,andWWis strictly convex by[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4), the continuation term is strictly concave inh1h\_\{1\}\. HenceVVis strictly concave and has a unique maximizerh1sf​\(s1\)∈\[0,1\]h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\\in\[0,1\]\.

Differentiating gives

V′​\(h1\)\\displaystyle V^\{\\prime\}\(h\_\{1\}\)=λ​R​δ​\(1−π1\)​\(s1−α1\)\\displaystyle=\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(s\_\{1\}\-\\alpha\_\{1\}\)\+β​\[λ​R​π2−W′​\(g​\(s1,h1;A1\)\)\]​\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\\displaystyle\\quad\+\\beta\\bigl\[\\lambda R\\pi\_\{2\}\-W^\{\\prime\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)\\bigr\]\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)=\(1−π1\)​\(α1−s1\)​\{−λ​R​δ\+β​\(ϕ​π1\+γ\)​\[λ​R​π2−W′​\(g​\(s1,h1;A1\)\)\]\}\.\\displaystyle=\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)\\left\\\{\-\\lambda R\\delta\+\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\\bigl\[\\lambda R\\pi\_\{2\}\-W^\{\\prime\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)\\bigr\]\\right\\\}\.Strict concavity ofVVimpliesV′V^\{\\prime\}is strictly decreasing\. The optimum satisfies the standard Karush–Kuhn–Tucker conditions:h1sf​\(s1\)=0h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=0if and only ifV′​\(0\)≤0V^\{\\prime\}\(0\)\\leq 0;h1sf​\(s1\)=1h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=1if and only ifV′​\(1\)≥0V^\{\\prime\}\(1\)\\geq 0; otherwise the unique interior optimum solvesV′​\(h1\)=0V^\{\\prime\}\(h\_\{1\}\)=0, which is equivalent to

W′​\(g​\(s1,h1sf​\(s1\);A1\)\)=λ​R​π2−λ​R​δβ​\(ϕ​π1\+γ\)\.\\displaystyle W^\{\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);A\_\{1\}\)\\bigr\)=\\lambda R\\pi\_\{2\}\-\\frac\{\\lambda R\\delta\}\{\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\}\.
Next suppose[eq\.1](https://arxiv.org/html/2606.29111#S4.E1)holds, and lets∗s^\{\*\}be defined by[eq\.2](https://arxiv.org/html/2606.29111#S4.E2)\. BecauseWWis strictly convex,W′W^\{\\prime\}is strictly increasing, and the interior first\-order condition is equivalent tog​\(s1,h1sf​\(s1\);A1\)=s∗g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);A\_\{1\}\)=s^\{\*\}\. The boundary conditionsV′​\(0\)≤0V^\{\\prime\}\(0\)\\leq 0andV′​\(1\)≥0V^\{\\prime\}\(1\)\\geq 0translate intog​\(s1,0;A1\)≥s∗g\(s\_\{1\},0;A\_\{1\}\)\\geq s^\{\*\}andg​\(s1,1;A1\)≤s∗g\(s\_\{1\},1;A\_\{1\}\)\\leq s^\{\*\}, respectively\. This gives the three cases in the proposition\.

Solvingg​\(s1,h1sf​\(s1\);A1\)=s∗g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);A\_\{1\}\)=s^\{\*\}for the interior optimizer yields

h1sf​\(s1\)=s∗−g​\(s1,0;A1\)\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\),\\displaystyle h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=\\frac\{s^\{\*\}\-g\(s\_\{1\},0;A\_\{1\}\)\}\{\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\},which is[eq\.3](https://arxiv.org/html/2606.29111#S4.E3)\.

Finally, differentiating the interior closed form gives

d​h1sf​\(s1\)d​s1=s∗−α1\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)2<0,\\displaystyle\\frac\{dh\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\}\{ds\_\{1\}\}=\\frac\{s^\{\*\}\-\\alpha\_\{1\}\}\{\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)^\{2\}\}<0,becauses∗∈\(0,α1\)s^\{\*\}\\in\(0,\\alpha\_\{1\}\)\.*Q\.E\.D\.*

*Proof of[Corollary1](https://arxiv.org/html/2606.29111#Thmcorollary1)\.*Part \(i\)\.Under[eq\.1](https://arxiv.org/html/2606.29111#S4.E1),s∗s^\{\*\}is defined implicitly by

W′​\(s∗\)=λ​R​π2−λ​R​δβ​\(ϕ​π1\+γ\)\.\\displaystyle W^\{\\prime\}\(s^\{\*\}\)=\\lambda R\\pi\_\{2\}\-\\frac\{\\lambda R\\delta\}\{\\beta\(\\phi\\pi\_\{1\}\+\\gamma\)\}\.The right\-hand side does not depend onα2\\alpha\_\{2\}, so∂s∗/∂α2=0\\partial s^\{\*\}/\\partial\\alpha\_\{2\}=0\. Differentiating both sides inπ2\\pi\_\{2\}and usingW′′​\(s∗\)\>0W^\{\\prime\\prime\}\(s^\{\*\}\)\>0gives∂s∗∂π2=λ​RW′′​\(s∗\)\>0\\frac\{\\partial s^\{\*\}\}\{\\partial\\pi\_\{2\}\}=\\frac\{\\lambda R\}\{W^\{\\prime\\prime\}\(s^\{\*\}\)\}\>0\. The thresholdssLs\_\{L\}andsHs\_\{H\}defined in[eq\.4](https://arxiv.org/html/2606.29111#S4.E4)are affine functions ofs∗s^\{\*\}with strictly positive slopes

11−ϕ​π1​\(1−π1\)and11\+γ​\(1−π1\),\\displaystyle\\frac\{1\}\{1\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\}\\quad\\text\{and\}\\quad\\frac\{1\}\{1\+\\gamma\(1\-\\pi\_\{1\}\)\},respectively\. The first slope is positive by[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)\. Hence both thresholds are strictly increasing inπ2\\pi\_\{2\}and independent ofα2\\alpha\_\{2\}\.

Part \(ii\)\.Fixs1∈\(s¯,α1\)s\_\{1\}\\in\(\\underline\{s\},\\alpha\_\{1\}\)\. We show thath1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)is weakly increasing inπ2\\pi\_\{2\}on\(0,π1\]\(0,\\pi\_\{1\}\]\.

*Interior region\.*On the open set whereg​\(s1,0;A1\)<s∗​\(π2\)<g​\(s1,1;A1\)g\(s\_\{1\},0;A\_\{1\}\)<s^\{\*\}\(\\pi\_\{2\}\)<g\(s\_\{1\},1;A\_\{1\}\), the closed form[eq\.3](https://arxiv.org/html/2606.29111#S4.E3)gives

h1sf​\(s1\)=s∗​\(π2\)−g​\(s1,0;A1\)\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\.\\displaystyle h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=\\frac\{s^\{\*\}\(\\pi\_\{2\}\)\-g\(s\_\{1\},0;A\_\{1\}\)\}\{\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\}\.The denominator andg​\(s1,0;A1\)g\(s\_\{1\},0;A\_\{1\}\)do not depend onπ2\\pi\_\{2\}\. Differentiating inπ2\\pi\_\{2\},

∂h1sf​\(s1\)∂π2=1\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)⋅∂s∗∂π2\>0\\displaystyle\\frac\{\\partial h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\}\{\\partial\\pi\_\{2\}\}=\\frac\{1\}\{\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\}\\cdot\\frac\{\\partial s^\{\*\}\}\{\\partial\\pi\_\{2\}\}\>0by part \(i\)\.

*Boundary regions\.*The boundaryh1sf​\(s1\)=0h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=0holds wheng​\(s1,0;A1\)≥s∗​\(π2\)g\(s\_\{1\},0;A\_\{1\}\)\\geq s^\{\*\}\(\\pi\_\{2\}\), equivalentlys1≥sH​\(π2\)s\_\{1\}\\geq s\_\{H\}\(\\pi\_\{2\}\); the boundaryh1sf​\(s1\)=1h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=1holds wheng​\(s1,1;A1\)≤s∗​\(π2\)g\(s\_\{1\},1;A\_\{1\}\)\\leq s^\{\*\}\(\\pi\_\{2\}\), equivalentlys1≤sL​\(π2\)s\_\{1\}\\leq s\_\{L\}\(\\pi\_\{2\}\)\. Becauses∗​\(π2\)s^\{\*\}\(\\pi\_\{2\}\),sL​\(π2\)s\_\{L\}\(\\pi\_\{2\}\), andsH​\(π2\)s\_\{H\}\(\\pi\_\{2\}\)are all strictly increasing inπ2\\pi\_\{2\}by part \(i\), increasingπ2\\pi\_\{2\}can only move the worker from the no\-engagement region into the interior region \(oncesH​\(π2\)s\_\{H\}\(\\pi\_\{2\}\)rises pasts1s\_\{1\}\), and from the interior region into the full\-engagement region \(oncesL​\(π2\)s\_\{L\}\(\\pi\_\{2\}\)rises pasts1s\_\{1\}\)\. The reverse transitions are impossible\. Henceh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)is weakly increasing inπ2\\pi\_\{2\}on the feasible domain\(0,π1\]\(0,\\pi\_\{1\}\], with strict monotonicity in the interior region\. Equivalently, asπ2\\pi\_\{2\}falls fromπ1\\pi\_\{1\}toward zero,h1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)either remains constant or decreases\.*Q\.E\.D\.*

*Proof of[Proposition2](https://arxiv.org/html/2606.29111#Thmproposition2)\.*From the skill transitiong​\(s1,h1;A1\)=s1\+\(1−π1\)​\(h1​\(ϕ​π1\+γ\)−γ\)​\(α1−s1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)=s\_\{1\}\+\(1\-\\pi\_\{1\}\)\\bigl\(h\_\{1\}\(\\phi\\pi\_\{1\}\+\\gamma\)\-\\gamma\\bigr\)\(\\alpha\_\{1\}\-s\_\{1\}\),

∂g∂s1=1−\(1−π1\)​\(h1​\(ϕ​π1\+γ\)−γ\)≥1−\(1−π1\)​ϕ​π1\>0,\\displaystyle\\frac\{\\partial g\}\{\\partial s\_\{1\}\}=1\-\(1\-\\pi\_\{1\}\)\\bigl\(h\_\{1\}\(\\phi\\pi\_\{1\}\+\\gamma\)\-\\gamma\\bigr\)\\geq 1\-\(1\-\\pi\_\{1\}\)\\phi\\pi\_\{1\}\>0,where the bound usesh1≤1h\_\{1\}\\leq 1and the no\-leapfrogging[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)\. Thusggis strictly increasing ins1s\_\{1\}for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\], so by[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)the full\-engagement set\{s1:g​\(s1,1;A1\)≤s∗\}\\\{s\_\{1\}:g\(s\_\{1\},1;A\_\{1\}\)\\leq s^\{\*\}\\\}is a lower interval, and the no\-engagement set\{s1:g​\(s1,0;A1\)≥s∗\}\\\{s\_\{1\}:g\(s\_\{1\},0;A\_\{1\}\)\\geq s^\{\*\}\\\}is an upper interval, with the interior region between them\. On the interior,[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1)setsg​\(s1,h1sf​\(s1\);A1\)=s∗g\(s\_\{1\},h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);A\_\{1\}\)=s^\{\*\}; differentiating ins1s\_\{1\}and using∂g/∂h1=\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\>0\\partial g/\\partial h\_\{1\}=\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\>0,

d​h1sf​\(s1\)d​s1=−∂g/∂s1∂g/∂h1<0\.\\displaystyle\\frac\{dh\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\}\{ds\_\{1\}\}=\-\\frac\{\\partial g/\\partial s\_\{1\}\}\{\\partial g/\\partial h\_\{1\}\}<0\.Henceh1sfh\_\{1\}^\{\\mathrm\{sf\}\}equals11on the lower interval, strictly decreases across the interior region, and equals0on the upper interval, so it is nonincreasing ins1s\_\{1\}\.*Q\.E\.D\.*

*Proof of[Lemma1](https://arxiv.org/html/2606.29111#Thmlemma1)\.*Suppose, to the contrary, that\(h21⁣∗,h22⁣∗\)\(h\_\{2\}^\{1\*\},h\_\{2\}^\{2\*\}\)is a pure\-strategy Nash equilibrium withh21⁣∗\>h22⁣∗h\_\{2\}^\{1\*\}\>h\_\{2\}^\{2\*\}\. Fors2<α2s\_\{2\}<\\alpha\_\{2\}, the transitiong​\(s2,h2;A2\)g\(s\_\{2\},h\_\{2\};A\_\{2\}\)is strictly increasing inh2h\_\{2\}\. BecauseBBis strictly increasing, the deterministic worker valueV2j​\(s2,h2j\)=W​\(s2\)\+β​B​\(g​\(s2,h2j;A2\)\)V\_\{2\}^\{j\}\(s\_\{2\},h\_\{2\}^\{j\}\)=W\(s\_\{2\}\)\+\\beta B\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\)is also strictly increasing inh2jh\_\{2\}^\{j\}\. HenceV21​\(s2,h21⁣∗\)\>V22​\(s2,h22⁣∗\)V\_\{2\}^\{1\}\(s\_\{2\},h\_\{2\}^\{1\*\}\)\>V\_\{2\}^\{2\}\(s\_\{2\},h\_\{2\}^\{2\*\}\), and the logit sorting rule impliesσ21⁣∗\>1/2\>σ22⁣∗\\sigma\_\{2\}^\{1\*\}\>1/2\>\\sigma\_\{2\}^\{2\*\}\. Consider firm 1’s option to deviate unilaterally to the other firm’s engagement levelh22⁣∗h\_\{2\}^\{2\*\}, while firm 2 continues to playh22⁣∗h\_\{2\}^\{2\*\}\. The resulting action profile would be\(h22⁣∗,h22⁣∗\)\(h\_\{2\}^\{2\*\},h\_\{2\}^\{2\*\}\), at which the logit sorting probability isσ21=1/2\\sigma\_\{2\}^\{1\}=1/2by symmetry of the form, and firm 1’s own margin isM​\(s2,h22⁣∗;A2\)M\(s\_\{2\},h\_\{2\}^\{2\*\};A\_\{2\}\)\. The deviating payoff would therefore equal\(1/2\)​M​\(s2,h22⁣∗;A2\)\(1/2\)\\,M\(s\_\{2\},h\_\{2\}^\{2\*\};A\_\{2\}\), and Nash equilibrium requires that firm 1’s equilibrium payoff weakly exceeds this:

σ21⁣∗​M​\(s2,h21⁣∗;A2\)≥12​M​\(s2,h22⁣∗;A2\)\.\\displaystyle\\sigma\_\{2\}^\{1\*\}\\,M\(s\_\{2\},h\_\{2\}^\{1\*\};A\_\{2\}\)\\geq\\tfrac\{1\}\{2\}\\,M\(s\_\{2\},h\_\{2\}^\{2\*\};A\_\{2\}\)\.Analogously, considering firm 2’s option to deviate unilaterally toh21⁣∗h\_\{2\}^\{1\*\}while firm 1 continues to playh21⁣∗h\_\{2\}^\{1\*\}, Nash equilibrium requires

σ22⁣∗​M​\(s2,h22⁣∗;A2\)≥12​M​\(s2,h21⁣∗;A2\)\.\\displaystyle\\sigma\_\{2\}^\{2\*\}\\,M\(s\_\{2\},h\_\{2\}^\{2\*\};A\_\{2\}\)\\geq\\tfrac\{1\}\{2\}\\,M\(s\_\{2\},h\_\{2\}^\{1\*\};A\_\{2\}\)\.Both margins are strictly positive by revenue sufficiency in[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4)\. Multiplying the two no\-deviation inequalities and dividing through by the positive productM​\(s2,h21⁣∗;A2\)​M​\(s2,h22⁣∗;A2\)M\(s\_\{2\},h\_\{2\}^\{1\*\};A\_\{2\}\)\\,M\(s\_\{2\},h\_\{2\}^\{2\*\};A\_\{2\}\)givesσ21⁣∗​σ22⁣∗≥14\\sigma\_\{2\}^\{1\*\}\\,\\sigma\_\{2\}^\{2\*\}\\geq\\tfrac\{1\}\{4\}\. Butσ21⁣∗,σ22⁣∗\>0\\sigma\_\{2\}^\{1\*\},\\sigma\_\{2\}^\{2\*\}\>0andσ21⁣∗\+σ22⁣∗=1\\sigma\_\{2\}^\{1\*\}\+\\sigma\_\{2\}^\{2\*\}=1implyσ21⁣∗​σ22⁣∗≤1/4\\sigma\_\{2\}^\{1\*\}\\,\\sigma\_\{2\}^\{2\*\}\\leq 1/4, with equality only atσ21⁣∗=σ22⁣∗=1/2\\sigma\_\{2\}^\{1\*\}=\\sigma\_\{2\}^\{2\*\}=1/2\. This contradictsσ21⁣∗\>1/2\>σ22⁣∗\\sigma\_\{2\}^\{1\*\}\>1/2\>\\sigma\_\{2\}^\{2\*\}, so no asymmetric pure\-strategy Nash equilibrium exists\.*Q\.E\.D\.*

*Proof of[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)\.*Fixs2∈D2s\_\{2\}\\in D\_\{2\}\. SinceD2⊂\(s¯2,α2\)D\_\{2\}\\subset\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), the zero floor does not bind, so

g​\(s2,h;A2\)=s2\+\(1−π2\)​\[h​\(ϕ​π2\+γ\)−γ\]​\(α2−s2\),\\displaystyle g\(s\_\{2\},h;A\_\{2\}\)=s\_\{2\}\+\(1\-\\pi\_\{2\}\)\\bigl\[h\(\\phi\\pi\_\{2\}\+\\gamma\)\-\\gamma\\bigr\]\(\\alpha\_\{2\}\-s\_\{2\}\),and therefore

∂g​\(s2,h;A2\)∂h=\(1−π2\)​\(ϕ​π2\+γ\)​\(α2−s2\)\>0\.\\displaystyle\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial h\}=\(1\-\\pi\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(\\alpha\_\{2\}\-s\_\{2\}\)\>0\.Moreover,

∂M​\(s2,h;A2\)∂h=−λ​R​δ​\(1−π2\)​\(α2−s2\)<0,\\displaystyle\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial h\}=\-\\lambda R\\delta\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)<0,becauses2<α2s\_\{2\}<\\alpha\_\{2\}\.

First consider the engagement\-gain function

F2​\(h;s2,A2\)=β​\(ϕ​π2\+γ\)2​η​B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)\.\\displaystyle F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\frac\{\\beta\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)M\(s\_\{2\},h;A\_\{2\}\)\.Differentiating with respect tohhgives

∂F2​\(h;s2,A2\)∂h\\displaystyle\\frac\{\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\{\\partial h\}=β​\(ϕ​π2\+γ\)2​η​B′​\(g​\(s2,h;A2\)\)​\(1−π2\)​\(α2−s2\)\\displaystyle=\\frac\{\\beta\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)×\[\(ϕ​π2\+γ\)​B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)−λ​R​δ\]\.\\displaystyle\\quad\\times\\left\[\(\\phi\\pi\_\{2\}\+\\gamma\)\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}M\(s\_\{2\},h;A\_\{2\}\)\-\\lambda R\\delta\\right\]\.All terms outside the square brackets are strictly positive\. Therefore, condition[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)implies the expression in brackets is strictly negative for everyh∈\[0,1\]h\\in\[0,1\]\. HenceF2​\(h;s2,A2\)F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)is strictly decreasing on\[0,1\]\[0,1\]\.

Now consider firmjj’s terminal\-period payoff,

Π2j​\(h2j,h2−j;s2,A2\)=σ2j​\(h2j,h2−j;s2,A2\)​M​\(s2,h2j;A2\)\.\\displaystyle\\Pi\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)=\\sigma\_\{2\}^\{j\}\(h\_\{2\}^\{j\},h\_\{2\}^\{\-j\};s\_\{2\},A\_\{2\}\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\.Differentiating with respect toh2jh\_\{2\}^\{j\}and using the logit derivative, we obtain

∂Π2j∂h2j=σ2j​\(1−π2\)​\(α2−s2\)​\[\(1−σ2j\)​βη​B′​\(g​\(s2,h2j;A2\)\)​\(ϕ​π2\+γ\)​M​\(s2,h2j;A2\)−λ​R​δ\]\.\\displaystyle\\frac\{\\partial\\Pi\_\{2\}^\{j\}\}\{\\partial h\_\{2\}^\{j\}\}=\\sigma\_\{2\}^\{j\}\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)\\left\[\(1\-\\sigma\_\{2\}^\{j\}\)\\frac\{\\beta\}\{\\eta\}B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\)\(\\phi\\pi\_\{2\}\+\\gamma\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\-\\lambda R\\delta\\right\]\.Because the prefactor is strictly positive, the sign of∂Π2j/∂h2j\\partial\\Pi\_\{2\}^\{j\}/\\partial h\_\{2\}^\{j\}is the sign of the bracketed term\.

For any fixed action by the other firmh2−jh\_\{2\}^\{\-j\}, define

D​\(h2j;h2−j\)≜\(1−σ2j\)​βη​B′​\(g​\(s2,h2j;A2\)\)​\(ϕ​π2\+γ\)​M​\(s2,h2j;A2\)−λ​R​δ\.\\displaystyle D\(h\_\{2\}^\{j\};h\_\{2\}^\{\-j\}\)\\triangleq\(1\-\\sigma\_\{2\}^\{j\}\)\\frac\{\\beta\}\{\\eta\}B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\)\(\\phi\\pi\_\{2\}\+\\gamma\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\-\\lambda R\\delta\.We next showD​\(h2j;h2−j\)D\(h\_\{2\}^\{j\};h\_\{2\}^\{\-j\}\)is strictly decreasing inh2jh\_\{2\}^\{j\}\. First,σ2j\\sigma\_\{2\}^\{j\}is strictly increasing inh2jh\_\{2\}^\{j\}, becauseBBandggare strictly increasing in own engagement\. Hence1−σ2j1\-\\sigma\_\{2\}^\{j\}is strictly decreasing inh2jh\_\{2\}^\{j\}\. Second,

∂∂h2j​\[B′​\(g​\(s2,h2j;A2\)\)​M​\(s2,h2j;A2\)\]\\displaystyle\\frac\{\\partial\}\{\\partial h\_\{2\}^\{j\}\}\\left\[B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\right\]equals

B′​\(g​\(s2,h2j;A2\)\)​\(1−π2\)​\(α2−s2\)​\[\(ϕ​π2\+γ\)​B′′​\(g​\(s2,h2j;A2\)\)B′​\(g​\(s2,h2j;A2\)\)​M​\(s2,h2j;A2\)−λ​R​δ\]\.\\displaystyle B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\)\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)\\left\[\(\\phi\\pi\_\{2\}\+\\gamma\)\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\)\}M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\-\\lambda R\\delta\\right\]\.Sinces2∈D2s\_\{2\}\\in D\_\{2\}, condition[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)applies for every own actionh2j∈\[0,1\]h\_\{2\}^\{j\}\\in\[0,1\]\. Hence the expression in square brackets is strictly negative for every own actionh2j∈\[0,1\]h\_\{2\}^\{j\}\\in\[0,1\], regardless of the other firm’s action\. Therefore the derivative above is strictly negative\.

It follows that the product

\(1−σ2j\)​B′​\(g​\(s2,h2j;A2\)\)​M​\(s2,h2j;A2\)\\displaystyle\(1\-\\sigma\_\{2\}^\{j\}\)B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\bigr\)M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)is strictly decreasing inh2jh\_\{2\}^\{j\}, and soD​\(h2j;h2−j\)D\(h\_\{2\}^\{j\};h\_\{2\}^\{\-j\}\)is strictly decreasing in firmjj’s own engagement for every fixed action by the other firm\. Thus firmjj’s payoff is strictly quasiconcave in its own action\.

At a symmetric profileh2j=h2−j=hh\_\{2\}^\{j\}=h\_\{2\}^\{\-j\}=h, we haveσ2j=1/2\\sigma\_\{2\}^\{j\}=1/2\. Hence the first\-order condition for an interior symmetric equilibrium reduces toF2​\(h;s2,A2\)=λ​R​δF\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\. BecauseF2F\_\{2\}is strictly decreasing, there can be at most one interior symmetric equilibrium\. The boundary cases follow from the same strict\-quasiconcavity argument applied to each firm’s best response to the other firm’s boundary action\.

IfF2​\(0;s2,A2\)≤λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\\leq\\lambda R\\delta, then at the profile\(0,0\)\(0,0\)the sign\-determining termD​\(0;0\)D\(0;0\)is nonpositive\. SinceD​\(⋅;0\)D\(\\cdot;0\)is strictly decreasing, firmjj’s marginal payoff is negative for everyh2j\>0h\_\{2\}^\{j\}\>0\. Hence0is the unique best response to0, soh2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0is the unique symmetric equilibrium\.

IfF2​\(1;s2,A2\)≥λ​R​δF\_\{2\}\(1;s\_\{2\},A\_\{2\}\)\\geq\\lambda R\\delta, then at the profile\(1,1\)\(1,1\)the sign\-determining termD​\(1;1\)D\(1;1\)is nonnegative\. SinceD​\(⋅;1\)D\(\\cdot;1\)is strictly decreasing,D​\(h2j;1\)\>D​\(1;1\)≥0D\(h\_\{2\}^\{j\};1\)\>D\(1;1\)\\geq 0for everyh2j<1h\_\{2\}^\{j\}<1\. Hence firmjj’s payoff is increasing up toh2j=1h\_\{2\}^\{j\}=1, so11is the unique best response to11, andh2∗​\(s2\)=1h\_\{2\}^\{\*\}\(s\_\{2\}\)=1is the unique symmetric equilibrium\.

Finally,[Lemma1](https://arxiv.org/html/2606.29111#Thmlemma1)rules out asymmetric pure\-strategy equilibria\. Therefore the unique symmetric equilibrium identified above is the unique pure\-strategy Nash equilibrium of the terminal\-period two\-firm game\.*Q\.E\.D\.*

###### Claim OA2

SupposeB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1\. Fors2∈\(s¯2,α2\)s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), the left\-hand side of the terminal single\-crossing condition is maximized ath=0h=0and equals

Ψ​\(s2\)≡\(b−1\)​λ​R​\[\(1−π2\)​α2\+π2​s2\]−s2b\[1\+γ​\(1−π2\)\]​s2−γ​\(1−π2\)​α2\.\\displaystyle\\Psi\(s\_\{2\}\)\\equiv\(b\-1\)\\frac\{\\lambda R\\big\[\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\+\\pi\_\{2\}s\_\{2\}\\big\]\-s\_\{2\}^\{b\}\}\{\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]s\_\{2\}\-\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\}\.Moreover, under[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4),Ψ\\Psiis strictly decreasing,

lims2↓s¯2Ψ​\(s2\)=\+∞,lims2↑α2Ψ​\(s2\)=\(b−1\)​\(λ​R−α2b−1\)\.\\displaystyle\\lim\_\{s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}\}\\Psi\(s\_\{2\}\)=\+\\infty,\\qquad\\lim\_\{s\_\{2\}\\uparrow\\alpha\_\{2\}\}\\Psi\(s\_\{2\}\)=\(b\-1\)\(\\lambda R\-\\alpha\_\{2\}^\{b\-1\}\)\.Hence the terminal single\-crossing condition holds on a nonempty upper\-tail domain if and only if

\(b−1\)​\(λ​R−α2b−1\)<λ​R​δϕ​π2\+γ\.\\displaystyle\(b\-1\)\(\\lambda R\-\\alpha\_\{2\}^\{b\-1\}\)<\\frac\{\\lambda R\\delta\}\{\\phi\\pi\_\{2\}\+\\gamma\}\.\(OA1\)When this condition holds, there is a unique cutoffs¯¯2∈\(s¯2,α2\)\\bar\{\\bar\{s\}\}\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)solving

Ψ​\(s¯¯2\)=λ​R​δϕ​π2\+γ,\\displaystyle\\Psi\(\\bar\{\\bar\{s\}\}\_\{2\}\)=\\frac\{\\lambda R\\delta\}\{\\phi\\pi\_\{2\}\+\\gamma\},\(OA2\)and the terminal single\-crossing condition holds exactly fors2∈\(s¯¯2,α2\)s\_\{2\}\\in\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)\.

*Proof\.*For the power wage,

B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)=b−1g​\(s2,h;A2\)\.\\displaystyle\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}=\\frac\{b\-1\}\{g\(s\_\{2\},h;A\_\{2\}\)\}\.Thus the left\-hand side of the terminal single\-crossing condition is

\(b−1\)​maxh∈\[0,1\]⁡\{M​\(s2,h;A2\)g​\(s2,h;A2\)\}\.\\displaystyle\(b\-1\)\\max\_\{h\\in\[0,1\]\}\\left\\\{\\frac\{M\(s\_\{2\},h;A\_\{2\}\)\}\{g\(s\_\{2\},h;A\_\{2\}\)\}\\right\\\}\.Ons2∈\(s¯2,α2\)s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), we haveg​\(s2,h;A2\)\>0g\(s\_\{2\},h;A\_\{2\}\)\>0andM​\(s2,h;A2\)\>0M\(s\_\{2\},h;A\_\{2\}\)\>0\. Also,

∂M​\(s2,h;A2\)∂h=−λ​R​δ​\(1−π2\)​\(α2−s2\)<0\\displaystyle\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial h\}=\-\\lambda R\\delta\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)<0and

∂g​\(s2,h;A2\)∂h=\(1−π2\)​\(ϕ​π2\+γ\)​\(α2−s2\)\>0\.\\displaystyle\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial h\}=\(1\-\\pi\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(\\alpha\_\{2\}\-s\_\{2\}\)\>0\.Therefore

∂∂h​\[M​\(s2,h;A2\)g​\(s2,h;A2\)\]=Mh​g−M​ghg2<0\.\\displaystyle\\frac\{\\partial\}\{\\partial h\}\\left\[\\frac\{M\(s\_\{2\},h;A\_\{2\}\)\}\{g\(s\_\{2\},h;A\_\{2\}\)\}\\right\]=\\frac\{M\_\{h\}g\-Mg\_\{h\}\}\{g^\{2\}\}<0\.Hence the maximum is attained ath=0h=0\. Since

g​\(s2,0;A2\)=\[1\+γ​\(1−π2\)\]​s2−γ​\(1−π2\)​α2\\displaystyle g\(s\_\{2\},0;A\_\{2\}\)=\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]s\_\{2\}\-\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}and

M​\(s2,0;A2\)=λ​R​\[\(1−π2\)​α2\+π2​s2\]−s2b,\\displaystyle M\(s\_\{2\},0;A\_\{2\}\)=\\lambda R\\big\[\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\+\\pi\_\{2\}s\_\{2\}\\big\]\-s\_\{2\}^\{b\},the left\-hand side equalsΨ​\(s2\)\\Psi\(s\_\{2\}\)\.

It remains to showΨ\\Psiis strictly decreasing\. Letq≡1−π2q\\equiv 1\-\\pi\_\{2\},a≡1\+γ​qa\\equiv 1\+\\gamma q, and

N​\(s2\)≡λ​R​\(q​α2\+π2​s2\)−s2b,D​\(s2\)≡a​s2−γ​q​α2\.\\displaystyle N\(s\_\{2\}\)\\equiv\\lambda R\(q\\alpha\_\{2\}\+\\pi\_\{2\}s\_\{2\}\)\-s\_\{2\}^\{b\},\\qquad D\(s\_\{2\}\)\\equiv as\_\{2\}\-\\gamma q\\alpha\_\{2\}\.ThenΨ​\(s2\)=\(b−1\)​N​\(s2\)/D​\(s2\)\\Psi\(s\_\{2\}\)=\(b\-1\)N\(s\_\{2\}\)/D\(s\_\{2\}\), andD​\(s2\)\>0D\(s\_\{2\}\)\>0on\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. Thus the sign ofΨ′​\(s2\)\\Psi^\{\\prime\}\(s\_\{2\}\)is the sign ofN′​\(s2\)​D​\(s2\)−N​\(s2\)​D′​\(s2\)N^\{\\prime\}\(s\_\{2\}\)D\(s\_\{2\}\)\-N\(s\_\{2\}\)D^\{\\prime\}\(s\_\{2\}\)\. A direct calculation gives

N′​\(s2\)​D​\(s2\)−N​\(s2\)​D′​\(s2\)=−λ​R​q​α2​\(1\+γ\)\+s2b−1​\[b​γ​q​α2−a​\(b−1\)​s2\]\.\\displaystyle N^\{\\prime\}\(s\_\{2\}\)D\(s\_\{2\}\)\-N\(s\_\{2\}\)D^\{\\prime\}\(s\_\{2\}\)=\-\\lambda Rq\\alpha\_\{2\}\(1\+\\gamma\)\+s\_\{2\}^\{b\-1\}\\left\[b\\gamma q\\alpha\_\{2\}\-a\(b\-1\)s\_\{2\}\\right\]\.Writex=s2/α2∈\(0,1\)x=s\_\{2\}/\\alpha\_\{2\}\\in\(0,1\)\. Then

s2b−1​\[b​γ​q​α2−a​\(b−1\)​s2\]=α2b​xb−1​\[b​γ​q−a​\(b−1\)​x\]\.\\displaystyle s\_\{2\}^\{b\-1\}\\left\[b\\gamma q\\alpha\_\{2\}\-a\(b\-1\)s\_\{2\}\\right\]=\\alpha\_\{2\}^\{b\}x^\{b\-1\}\\left\[b\\gamma q\-a\(b\-1\)x\\right\]\.The functionxb−1​\[b​γ​q−a​\(b−1\)​x\]x^\{b\-1\}\\left\[b\\gamma q\-a\(b\-1\)x\\right\]is either nonpositive or attains its maximum atx=γ​q/ax=\\gamma q/a, where its value is

γ​q​\(γ​qa\)b−1<q​\(1\+γ\)\.\\displaystyle\\gamma q\\left\(\\frac\{\\gamma q\}\{a\}\\right\)^\{b\-1\}<q\(1\+\\gamma\)\.Usingλ​R\>α2b−1\\lambda R\>\\alpha\_\{2\}^\{b\-1\}from[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4)in the power\-wage case, we therefore haveN′​\(s2\)​D​\(s2\)−N​\(s2\)​D′​\(s2\)<0N^\{\\prime\}\(s\_\{2\}\)D\(s\_\{2\}\)\-N\(s\_\{2\}\)D^\{\\prime\}\(s\_\{2\}\)<0\. HenceΨ′​\(s2\)<0\\Psi^\{\\prime\}\(s\_\{2\}\)<0on\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\.

Finally, ass2↓s¯2s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}, the denominatorD​\(s2\)=g​\(s2,0;A2\)D\(s\_\{2\}\)=g\(s\_\{2\},0;A\_\{2\}\)converges to zero while the numerator remains positive by margin positivity\. HenceΨ​\(s2\)→\+∞\\Psi\(s\_\{2\}\)\\to\+\\infty\. At the upper endpoint,

D​\(α2\)=α2,N​\(α2\)=λ​R​α2−α2b,\\displaystyle D\(\\alpha\_\{2\}\)=\\alpha\_\{2\},\\qquad N\(\\alpha\_\{2\}\)=\\lambda R\\alpha\_\{2\}\-\\alpha\_\{2\}^\{b\},so

lims2↑α2Ψ​\(s2\)=\(b−1\)​\(λ​R−α2b−1\)\.\\displaystyle\\lim\_\{s\_\{2\}\\uparrow\\alpha\_\{2\}\}\\Psi\(s\_\{2\}\)=\(b\-1\)\(\\lambda R\-\\alpha\_\{2\}^\{b\-1\}\)\.Strict monotonicity then gives the claimed existence and uniqueness ofs¯¯2\\bar\{\\bar\{s\}\}\_\{2\}\.*Q\.E\.D\.*

*Proof of[Corollary2](https://arxiv.org/html/2606.29111#Thmcorollary2)\.*Fixs2∈D2s\_\{2\}\\in D\_\{2\}\. SinceD2⊂\(s¯2,α2\)D\_\{2\}\\subset\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), by the definition ofs¯2\\underline\{s\}\_\{2\},

g​\(s2,0;A2\)=\[1\+γ​\(1−π2\)\]​s2−γ​\(1−π2\)​α2\>0\.\\displaystyle g\(s\_\{2\},0;A\_\{2\}\)=\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]s\_\{2\}\-\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\>0\.Since

∂g​\(s2,h;A2\)∂h=\(1−π2\)​\(ϕ​π2\+γ\)​\(α2−s2\)\>0,\\displaystyle\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial h\}=\(1\-\\pi\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(\\alpha\_\{2\}\-s\_\{2\}\)\>0,we haveg​\(s2,h;A2\)\>0g\(s\_\{2\},h;A\_\{2\}\)\>0for everyh∈\[0,1\]h\\in\[0,1\]\. Thus the zero floor in the skill transition does not bind\.

At zero engagement, underB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\},

F2​\(0;s2,A2\)=β​b​\(ϕ​π2\+γ\)2​η​\(g​\(s2,0;A2\)\)b−1​M​\(s2,0;A2\)\.\\displaystyle F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)=\\frac\{\\beta b\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}\\bigl\(g\(s\_\{2\},0;A\_\{2\}\)\\bigr\)^\{b\-1\}M\(s\_\{2\},0;A\_\{2\}\)\.Therefore, by the definition ofη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)in[eq\.7](https://arxiv.org/html/2606.29111#S5.E7),

F2​\(0;s2,A2\)=λ​R​δ⋅η¯​\(s2\)η\.\\displaystyle F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\\cdot\\frac\{\\bar\{\\eta\}\(s\_\{2\}\)\}\{\\eta\}\.By[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3), the terminal\-period equilibrium satisfiesh2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0if and only ifF2​\(0;s2,A2\)≤λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\\leq\\lambda R\\delta\. Henceh2∗​\(s2\)\>0h\_\{2\}^\{\*\}\(s\_\{2\}\)\>0if and only ifF2​\(0;s2,A2\)\>λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\>\\lambda R\\delta\. Using the display above andη\>0\\eta\>0, this is equivalent toη¯​\(s2\)η\>1\\frac\{\\bar\{\\eta\}\(s\_\{2\}\)\}\{\\eta\}\>1, orη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)\.*Q\.E\.D\.*

*Shape of the terminal cutoff\.*Although the equilibrium characterization is restricted toD2D\_\{2\}, the cutoffη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is defined on the full smooth interval\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. The transitiong​\(s2,0;A2\)∝s2−s¯2g\(s\_\{2\},0;A\_\{2\}\)\\propto s\_\{2\}\-\\underline\{s\}\_\{2\}rises from0at the floor toα2\\alpha\_\{2\}at the benchmark, so withb\>1b\>1the cutoff vanishes ass2↓s¯2s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}\. The marginM​\(s2,0;A2\)M\(s\_\{2\},0;A\_\{2\}\)has slopeλ​R​π2−b​s2b−1\\lambda R\\pi\_\{2\}\-b\\,s\_\{2\}^\{b\-1\}, which decreases ins2s\_\{2\}: it rises far below the benchmark and turns down nearer it once the marginal wageb​s2b−1b\\,s\_\{2\}^\{b\-1\}overtakes the marginal current revenue a more skilled worker brings when AI fails,λ​R​π2\\lambda R\\pi\_\{2\}, which may or may not happen beforeα2\\alpha\_\{2\}\. How the cutoff varies over\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)is governed by the sign of the scalarΔ2\\Delta\_\{2\}defined in[eq\.8](https://arxiv.org/html/2606.29111#S5.E8)\. WhenΔ2≥0\\Delta\_\{2\}\\geq 0,η¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)rises monotonically across\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. WhenΔ2<0\\Delta\_\{2\}<0, it is single\-peaked, strictly increasing to a unique interior maximum ats2†s\_\{2\}^\{\\dagger\}and strictly decreasing beyond it\. WriteηU≜lims2→α2−η¯​\(s2\)\\eta^\{U\}\\triangleq\\lim\_\{s\_\{2\}\\to\\alpha\_\{2\}^\{\-\}\}\\bar\{\\eta\}\(s\_\{2\}\)for the cutoff at the benchmark and, in the single\-peaked case,η†≜η¯​\(s2†\)\\eta^\{\\dagger\}\\triangleq\\bar\{\\eta\}\(s\_\{2\}^\{\\dagger\}\)for its peak, with0<ηU<η†0<\\eta^\{U\}<\\eta^\{\\dagger\}\.

Because the equilibrium characterization applies onD2D\_\{2\}, the relevant object is the restriction ofη¯\\bar\{\\eta\}toD2D\_\{2\}\. DefineηD≡lims2↓s¯¯2η¯​\(s2\)\\eta^\{D\}\\equiv\\lim\_\{s\_\{2\}\\downarrow\\bar\{\\bar\{s\}\}\_\{2\}\}\\bar\{\\eta\}\(s\_\{2\}\)\. When a full\-domain threshold lies belows¯¯2\\bar\{\\bar\{s\}\}\_\{2\}, we truncate it at the admissible\-domain boundary: writes^2D≡max⁡\{s^2,s¯¯2\}\\hat\{s\}\_\{2\}^\{D\}\\equiv\\max\\\{\\hat\{s\}\_\{2\},\\bar\{\\bar\{s\}\}\_\{2\}\\\}ands^2L,D≡max⁡\{s^2L,s¯¯2\}\\hat\{s\}\_\{2\}^\{L,D\}\\equiv\\max\\\{\\hat\{s\}\_\{2\}^\{L\},\\bar\{\\bar\{s\}\}\_\{2\}\\\}\.

###### Claim OA3

SupposeB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1, and fixη\\eta\. On the admissible domainD2D\_\{2\}, the engaged setE2​\(η\)≡\{s2∈D2:η<η¯​\(s2\)\}E\_\{2\}\(\\eta\)\\equiv\\\{s\_\{2\}\\in D\_\{2\}:\\eta<\\bar\{\\eta\}\(s\_\{2\}\)\\\}is an interval\. IfΔ2≥0\\Delta\_\{2\}\\geq 0, thenE2​\(η\)E\_\{2\}\(\\eta\)is the high\-skill band\(s^2D,α2\)\(\\hat\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\)whenη<ηU\\eta<\\eta^\{U\}, and is empty otherwise\. IfΔ2<0\\Delta\_\{2\}<0ands2†\>s¯¯2s\_\{2\}^\{\\dagger\}\>\\bar\{\\bar\{s\}\}\_\{2\}, then the full\-domain peak lies insideD2D\_\{2\}, andE2​\(η\)E\_\{2\}\(\\eta\)is the high\-skill band\(s^2D,α2\)\(\\hat\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\)whenη≤ηU\\eta\\leq\\eta^\{U\}, a bounded band\(s^2L,D,s^2H\)\(\\hat\{s\}\_\{2\}^\{L,D\},\\hat\{s\}\_\{2\}^\{H\}\)containings2†s\_\{2\}^\{\\dagger\}whenηU<η<η†\\eta^\{U\}<\\eta<\\eta^\{\\dagger\}, and is empty whenη≥η†\\eta\\geq\\eta^\{\\dagger\}\. IfΔ2<0\\Delta\_\{2\}<0ands2†≤s¯¯2s\_\{2\}^\{\\dagger\}\\leq\\bar\{\\bar\{s\}\}\_\{2\}, then the full\-domain peak lies weakly belowD2D\_\{2\}, soη¯\\bar\{\\eta\}is strictly decreasing onD2D\_\{2\}, andE2​\(η\)=D2E\_\{2\}\(\\eta\)=D\_\{2\}whenη≤ηU\\eta\\leq\\eta^\{U\},E2​\(η\)=\(s¯¯2,s~2\)E\_\{2\}\(\\eta\)=\(\\bar\{\\bar\{s\}\}\_\{2\},\\tilde\{s\}\_\{2\}\)whenηU<η<ηD\\eta^\{U\}<\\eta<\\eta^\{D\}, wheres~2∈D2\\tilde\{s\}\_\{2\}\\in D\_\{2\}uniquely solvesη¯​\(s~2\)=η\\bar\{\\eta\}\(\\tilde\{s\}\_\{2\}\)=\\eta, andE2​\(η\)E\_\{2\}\(\\eta\)is empty whenη≥ηD\\eta\\geq\\eta^\{D\}\.

Asη\\etafalls, a firm first engages the admissible workers with the highest cutoff and then widens the band\. WhenΔ2≥0\\Delta\_\{2\}\\geq 0, these are the highest\-skill workers inD2D\_\{2\}\. WhenΔ2<0\\Delta\_\{2\}<0ands2†∈D2s\_\{2\}^\{\\dagger\}\\in D\_\{2\}, they are the workers around the peaks2†s\_\{2\}^\{\\dagger\}\. WhenΔ2<0\\Delta\_\{2\}<0ands2†≤s¯¯2s\_\{2\}^\{\\dagger\}\\leq\\bar\{\\bar\{s\}\}\_\{2\}, the peak is truncated away, so engagement starts just above the lower boundarys¯¯2\\bar\{\\bar\{s\}\}\_\{2\}\.[Proposition4](https://arxiv.org/html/2606.29111#Thmproposition4)is the qualitative summary of this characterization\.

*Proof of[Proposition4](https://arxiv.org/html/2606.29111#Thmproposition4)and[OA3](https://arxiv.org/html/2606.29111#Thmclaim3)\.*From[Corollary2](https://arxiv.org/html/2606.29111#Thmcorollary2), positive terminal\-period engagement is characterized by the cutoffη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)\. It remains to characterize the shape of this cutoff as a function ofs2s\_\{2\}\.

Because the multiplicative constant in[eq\.7](https://arxiv.org/html/2606.29111#S5.E7)is positive, the sign ofη¯′​\(s2\)\\bar\{\\eta\}^\{\\prime\}\(s\_\{2\}\)is the sign of the derivative of\(g​\(s2,0;A2\)\)b−1​M​\(s2,0;A2\)\\bigl\(g\(s\_\{2\},0;A\_\{2\}\)\\bigr\)^\{b\-1\}M\(s\_\{2\},0;A\_\{2\}\)\. Define

z0​\(s2\)≜g​\(s2,0;A2\)=\[1\+γ​\(1−π2\)\]​s2−γ​\(1−π2\)​α2\\displaystyle z\_\{0\}\(s\_\{2\}\)\\triangleq g\(s\_\{2\},0;A\_\{2\}\)=\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]s\_\{2\}\-\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}and

r0​\(s2\)≜M​\(s2,0;A2\)=λ​R​\[\(1−π2\)​α2\+π2​s2\]−s2b\.\\displaystyle r\_\{0\}\(s\_\{2\}\)\\triangleq M\(s\_\{2\},0;A\_\{2\}\)=\\lambda R\[\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\+\\pi\_\{2\}s\_\{2\}\]\-s\_\{2\}^\{b\}\.Thenη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is proportional toz0​\(s2\)b−1​r0​\(s2\)z\_\{0\}\(s\_\{2\}\)^\{b\-1\}r\_\{0\}\(s\_\{2\}\)\. Differentiating,

dd​s2​\[z0​\(s2\)b−1​r0​\(s2\)\]=z0​\(s2\)b−2​\[\(b−1\)​z0′​\(s2\)​r0​\(s2\)\+z0​\(s2\)​r0′​\(s2\)\]\.\\displaystyle\\frac\{d\}\{ds\_\{2\}\}\\left\[z\_\{0\}\(s\_\{2\}\)^\{b\-1\}r\_\{0\}\(s\_\{2\}\)\\right\]=z\_\{0\}\(s\_\{2\}\)^\{b\-2\}\\left\[\(b\-1\)z\_\{0\}^\{\\prime\}\(s\_\{2\}\)r\_\{0\}\(s\_\{2\}\)\+z\_\{0\}\(s\_\{2\}\)r\_\{0\}^\{\\prime\}\(s\_\{2\}\)\\right\]\.Becausez0​\(s2\)\>0z\_\{0\}\(s\_\{2\}\)\>0, the sign ofη¯′​\(s2\)\\bar\{\\eta\}^\{\\prime\}\(s\_\{2\}\)is the sign of

H​\(s2\)≜\(b−1\)​\[1\+γ​\(1−π2\)\]​r0​\(s2\)\+z0​\(s2\)​\(λ​R​π2−b​s2b−1\)\.\\displaystyle H\(s\_\{2\}\)\\triangleq\(b\-1\)\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]r\_\{0\}\(s\_\{2\}\)\+z\_\{0\}\(s\_\{2\}\)\(\\lambda R\\pi\_\{2\}\-bs\_\{2\}^\{b\-1\}\)\.Leta≜1\+γ​\(1−π2\)a\\triangleq 1\+\\gamma\(1\-\\pi\_\{2\}\)\. Because

r0′​\(s2\)=λ​R​π2−b​s2b−1,r0′′​\(s2\)=−b​\(b−1\)​s2b−2,\\displaystyle r\_\{0\}^\{\\prime\}\(s\_\{2\}\)=\\lambda R\\pi\_\{2\}\-bs\_\{2\}^\{b\-1\},\\qquad r\_\{0\}^\{\\prime\\prime\}\(s\_\{2\}\)=\-b\(b\-1\)s\_\{2\}^\{b\-2\},we haveH​\(s2\)=\(b−1\)​a​r0​\(s2\)\+z0​\(s2\)​r0′​\(s2\)H\(s\_\{2\}\)=\(b\-1\)ar\_\{0\}\(s\_\{2\}\)\+z\_\{0\}\(s\_\{2\}\)r\_\{0\}^\{\\prime\}\(s\_\{2\}\), and hence

H′​\(s2\)=b​a​r0′​\(s2\)\+z0​\(s2\)​r0′′​\(s2\)\.\\displaystyle H^\{\\prime\}\(s\_\{2\}\)=b\\,a\\,r\_\{0\}^\{\\prime\}\(s\_\{2\}\)\+z\_\{0\}\(s\_\{2\}\)r\_\{0\}^\{\\prime\\prime\}\(s\_\{2\}\)\.Differentiating once more gives

H′′​\(s2\)=−b​\(b−1\)​s2b−3​\[a​\(b\+1\)​s2\+\(b−2\)​z0​\(s2\)\]\.\\displaystyle H^\{\\prime\\prime\}\(s\_\{2\}\)=\-b\(b\-1\)s\_\{2\}^\{b\-3\}\\left\[a\(b\+1\)s\_\{2\}\+\(b\-2\)z\_\{0\}\(s\_\{2\}\)\\right\]\.The bracketed term is strictly positive for everys2∈\(s¯2,α2\)s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)and everyb\>1b\>1\. Ifb≥2b\\geq 2, this is immediate becausea\>0a\>0,s2\>0s\_\{2\}\>0, andz0​\(s2\)\>0z\_\{0\}\(s\_\{2\}\)\>0\. If1<b<21<b<2, thenz0​\(s2\)<a​s2z\_\{0\}\(s\_\{2\}\)<as\_\{2\}andb−2<0b\-2<0, so\(b−2\)​z0​\(s2\)\>\(b−2\)​a​s2\(b\-2\)z\_\{0\}\(s\_\{2\}\)\>\(b\-2\)as\_\{2\}\. Hence

a​\(b\+1\)​s2\+\(b−2\)​z0​\(s2\)\>a​\(b\+1\)​s2\+\(b−2\)​a​s2=a​\(2​b−1\)​s2\>0\.\\displaystyle a\(b\+1\)s\_\{2\}\+\(b\-2\)z\_\{0\}\(s\_\{2\}\)\>a\(b\+1\)s\_\{2\}\+\(b\-2\)as\_\{2\}=a\(2b\-1\)s\_\{2\}\>0\.ThereforeH′′​\(s2\)<0H^\{\\prime\\prime\}\(s\_\{2\}\)<0on\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), soHHis strictly concave\.

Moreover, ass2↓s¯2s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}, we havez0​\(s2\)↓0z\_\{0\}\(s\_\{2\}\)\\downarrow 0, and hence

lims2↓s¯2H​\(s2\)=\(b−1\)​\[1\+γ​\(1−π2\)\]​r0​\(s¯2\)\>0,\\displaystyle\\lim\_\{s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}\}H\(s\_\{2\}\)=\(b\-1\)\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]r\_\{0\}\(\\underline\{s\}\_\{2\}\)\>0,in which the inequality follows from margin positivity under[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4)\. At the upper endpoint,z0​\(α2\)=α2z\_\{0\}\(\\alpha\_\{2\}\)=\\alpha\_\{2\}andr0​\(α2\)=λ​R​α2−α2br\_\{0\}\(\\alpha\_\{2\}\)=\\lambda R\\alpha\_\{2\}\-\\alpha\_\{2\}^\{b\}, so

H​\(α2\)=\(b−1\)​\[1\+γ​\(1−π2\)\]​\(λ​R​α2−α2b\)\+α2​\(λ​R​π2−b​α2b−1\)=Δ2\.\\displaystyle H\(\\alpha\_\{2\}\)=\(b\-1\)\[1\+\\gamma\(1\-\\pi\_\{2\}\)\]\(\\lambda R\\alpha\_\{2\}\-\\alpha\_\{2\}^\{b\}\)\+\\alpha\_\{2\}\(\\lambda R\\pi\_\{2\}\-b\\alpha\_\{2\}^\{b\-1\}\)=\\Delta\_\{2\}\.
IfΔ2≥0\\Delta\_\{2\}\\geq 0, then strict concavity ofHH, together with the positive lower\-endpoint value, impliesH​\(s2\)\>0H\(s\_\{2\}\)\>0throughout\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. Thereforeη¯′​\(s2\)\>0\\bar\{\\eta\}^\{\\prime\}\(s\_\{2\}\)\>0throughout the domain, andη¯\\bar\{\\eta\}is strictly increasing\.

LetηU≜lims2→α2−η¯​\(s2\)\\eta^\{U\}\\triangleq\\lim\_\{s\_\{2\}\\to\\alpha\_\{2\}^\{\-\}\}\\bar\{\\eta\}\(s\_\{2\}\)\. Becausez0​\(s2\)↓0z\_\{0\}\(s\_\{2\}\)\\downarrow 0ass2↓s¯2s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}andb\>1b\>1, whereasr0​\(s2\)r\_\{0\}\(s\_\{2\}\)remains finite and positive, we havelims2↓s¯2η¯​\(s2\)=0\\lim\_\{s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}\}\\bar\{\\eta\}\(s\_\{2\}\)=0\. Thus, whenΔ2≥0\\Delta\_\{2\}\\geq 0, strict monotonicity implies ifη≥ηU\\eta\\geq\\eta^\{U\}, thenη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)never holds on\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), soh2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0for alls2∈\(s¯2,α2\)s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. Ifη<ηU\\eta<\\eta^\{U\}, strict monotonicity and the lower endpoint limit imply there exists a unique thresholds^2∈\(s¯2,α2\)\\hat\{s\}\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)such thatη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)if and only ifs2∈\(s^2,α2\)s\_\{2\}\\in\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\)\. By[Corollary2](https://arxiv.org/html/2606.29111#Thmcorollary2), this is equivalent to

h2∗​\(s2\)\>0⟺s2∈\(s^2,α2\)\.\\displaystyle h\_\{2\}^\{\*\}\(s\_\{2\}\)\>0\\quad\\Longleftrightarrow\\quad s\_\{2\}\\in\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\)\.
Now supposeΔ2<0\\Delta\_\{2\}<0\. Then continuity and the positive lower\-endpoint value ofHHimplyHHis positive nears¯2\\underline\{s\}\_\{2\}and negative atα2\\alpha\_\{2\}\. BecauseHHis strictly concave, its upper contour set\{s2∈\(s¯2,α2\):H​\(s2\)\>0\}\\\{s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\):H\(s\_\{2\}\)\>0\\\}is an interval\. Because this interval contains points arbitrarily close tos¯2\\underline\{s\}\_\{2\}but excludes points sufficiently close toα2\\alpha\_\{2\}, there exists a uniques2†∈\(s¯2,α2\)s\_\{2\}^\{\\dagger\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)such thatH​\(s2†\)=0H\(s\_\{2\}^\{\\dagger\}\)=0, withH​\(s2\)\>0H\(s\_\{2\}\)\>0fors2<s2†s\_\{2\}<s\_\{2\}^\{\\dagger\}andH​\(s2\)<0H\(s\_\{2\}\)<0fors2\>s2†s\_\{2\}\>s\_\{2\}^\{\\dagger\}\. Thereforeη¯\\bar\{\\eta\}is strictly increasing on\(s¯2,s2†\)\(\\underline\{s\}\_\{2\},s\_\{2\}^\{\\dagger\}\)and strictly decreasing on\(s2†,α2\)\(s\_\{2\}^\{\\dagger\},\\alpha\_\{2\}\)\.

Let

η†≜η¯​\(s2†\),ηU≜lims2→α2−η¯​\(s2\)\.\\displaystyle\\eta^\{\\dagger\}\\triangleq\\bar\{\\eta\}\(s\_\{2\}^\{\\dagger\}\),\\qquad\\eta^\{U\}\\triangleq\\lim\_\{s\_\{2\}\\to\\alpha\_\{2\}^\{\-\}\}\\bar\{\\eta\}\(s\_\{2\}\)\.Becauseη¯\\bar\{\\eta\}is strictly decreasing on\(s2†,α2\)\(s\_\{2\}^\{\\dagger\},\\alpha\_\{2\}\), we haveηU<η†\\eta^\{U\}<\\eta^\{\\dagger\}\. Moreover,ηU\>0\\eta^\{U\}\>0by margin positivity at the upper endpoint andz0​\(α2\)=α2\>0z\_\{0\}\(\\alpha\_\{2\}\)=\\alpha\_\{2\}\>0\. Hence0<ηU<η†0<\\eta^\{U\}<\\eta^\{\\dagger\}\.

Ifη≥η†\\eta\\geq\\eta^\{\\dagger\}, thenη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)never holds, soh2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0for alls2∈\(s¯2,α2\)s\_\{2\}\\in\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)\. If0<η≤ηU0<\\eta\\leq\\eta^\{U\}, then the lower endpoint limitlims2↓s¯2η¯​\(s2\)=0\\lim\_\{s\_\{2\}\\downarrow\\underline\{s\}\_\{2\}\}\\bar\{\\eta\}\(s\_\{2\}\)=0, strict increase on\(s¯2,s2†\)\(\\underline\{s\}\_\{2\},s\_\{2\}^\{\\dagger\}\), and strict decrease toηU\\eta^\{U\}on\(s2†,α2\)\(s\_\{2\}^\{\\dagger\},\\alpha\_\{2\}\)imply there exists a unique thresholds^2∈\(s¯2,s2†\)\\hat\{s\}\_\{2\}\\in\(\\underline\{s\}\_\{2\},s\_\{2\}^\{\\dagger\}\)such thatη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)if and only ifs2∈\(s^2,α2\)s\_\{2\}\\in\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\)\. IfηU<η<η†\\eta^\{U\}<\\eta<\\eta^\{\\dagger\}, then the cutoff crosses the levelη\\etaonce on each side ofs2†s\_\{2\}^\{\\dagger\}, so there exist unique thresholdss^2L∈\(s¯2,s2†\)\\hat\{s\}\_\{2\}^\{L\}\\in\(\\underline\{s\}\_\{2\},s\_\{2\}^\{\\dagger\}\)ands^2H∈\(s2†,α2\)\\hat\{s\}\_\{2\}^\{H\}\\in\(s\_\{2\}^\{\\dagger\},\\alpha\_\{2\}\)such thatη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)if and only ifs2∈\(s^2L,s^2H\)s\_\{2\}\\in\(\\hat\{s\}\_\{2\}^\{L\},\\hat\{s\}\_\{2\}^\{H\}\)\. Applying[Corollary2](https://arxiv.org/html/2606.29111#Thmcorollary2)gives the stated positive\-engagement regions\.

It remains to restrict these full\-domain regions to the admissible domainD2=\(s¯¯2,α2\)D\_\{2\}=\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)\. IfΔ2≥0\\Delta\_\{2\}\\geq 0, thenη¯\\bar\{\\eta\}is strictly increasing on\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)and has upper\-end limitηU\\eta^\{U\}\. Hence, ifη≥ηU\\eta\\geq\\eta^\{U\}, the inequalityη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)fails for everys2∈D2s\_\{2\}\\in D\_\{2\}, soE2​\(η\)=∅E\_\{2\}\(\\eta\)=\\emptyset\. Ifη<ηU\\eta<\\eta^\{U\}, the full\-domain positive\-engagement region is\(s^2,α2\)\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\)\. Intersecting withD2D\_\{2\}gives

E2​\(η\)=\(s^2,α2\)∩\(s¯¯2,α2\)=\(s^2D,α2\),\\displaystyle E\_\{2\}\(\\eta\)=\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\)\\cap\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)=\(\\hat\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\),wheres^2D=max⁡\{s^2,s¯¯2\}\\hat\{s\}\_\{2\}^\{D\}=\\max\\\{\\hat\{s\}\_\{2\},\\bar\{\\bar\{s\}\}\_\{2\}\\\}\.

Now supposeΔ2<0\\Delta\_\{2\}<0\. Ifs2†\>s¯¯2s\_\{2\}^\{\\dagger\}\>\\bar\{\\bar\{s\}\}\_\{2\}, the full\-domain peak lies insideD2D\_\{2\}\. Whenη≥η†\\eta\\geq\\eta^\{\\dagger\}, the inequalityη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)fails everywhere, soE2​\(η\)=∅E\_\{2\}\(\\eta\)=\\emptyset\. Whenη≤ηU\\eta\\leq\\eta^\{U\}, the full\-domain positive\-engagement region is\(s^2,α2\)\(\\hat\{s\}\_\{2\},\\alpha\_\{2\}\), and intersecting withD2D\_\{2\}givesE2​\(η\)=\(s^2D,α2\)E\_\{2\}\(\\eta\)=\(\\hat\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\)\. WhenηU<η<η†\\eta^\{U\}<\\eta<\\eta^\{\\dagger\}, the full\-domain positive\-engagement region is the bounded band\(s^2L,s^2H\)\(\\hat\{s\}\_\{2\}^\{L\},\\hat\{s\}\_\{2\}^\{H\}\), withs^2L<s2†<s^2H<α2\\hat\{s\}\_\{2\}^\{L\}<s\_\{2\}^\{\\dagger\}<\\hat\{s\}\_\{2\}^\{H\}<\\alpha\_\{2\}\. Sinces2†\>s¯¯2s\_\{2\}^\{\\dagger\}\>\\bar\{\\bar\{s\}\}\_\{2\}, the upper cutoffs^2H\\hat\{s\}\_\{2\}^\{H\}lies insideD2D\_\{2\}, whereas the lower cutoff may lie below the admissible boundary\. Therefore

E2​\(η\)=\(s^2L,s^2H\)∩\(s¯¯2,α2\)=\(s^2L,D,s^2H\),\\displaystyle E\_\{2\}\(\\eta\)=\(\\hat\{s\}\_\{2\}^\{L\},\\hat\{s\}\_\{2\}^\{H\}\)\\cap\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)=\(\\hat\{s\}\_\{2\}^\{L,D\},\\hat\{s\}\_\{2\}^\{H\}\),wheres^2L,D=max⁡\{s^2L,s¯¯2\}\\hat\{s\}\_\{2\}^\{L,D\}=\\max\\\{\\hat\{s\}\_\{2\}^\{L\},\\bar\{\\bar\{s\}\}\_\{2\}\\\}\.

Finally, supposeΔ2<0\\Delta\_\{2\}<0ands2†≤s¯¯2s\_\{2\}^\{\\dagger\}\\leq\\bar\{\\bar\{s\}\}\_\{2\}\. Then the peak is weakly below the admissible domain, soη¯\\bar\{\\eta\}is strictly decreasing onD2D\_\{2\}\. Its upper limit onD2D\_\{2\}isηD=lims2↓s¯¯2η¯​\(s2\)\\eta^\{D\}=\\lim\_\{s\_\{2\}\\downarrow\\bar\{\\bar\{s\}\}\_\{2\}\}\\bar\{\\eta\}\(s\_\{2\}\), and its lower limit at the AI benchmark isηU\\eta^\{U\}\. HenceηD\>ηU\\eta^\{D\}\>\\eta^\{U\}\. Ifη≤ηU\\eta\\leq\\eta^\{U\}, thenη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)for everys2∈D2s\_\{2\}\\in D\_\{2\}, soE2​\(η\)=D2E\_\{2\}\(\\eta\)=D\_\{2\}\. IfηU<η<ηD\\eta^\{U\}<\\eta<\\eta^\{D\}, strict monotonicity gives a uniques~2∈D2\\tilde\{s\}\_\{2\}\\in D\_\{2\}satisfyingη¯​\(s~2\)=η\\bar\{\\eta\}\(\\tilde\{s\}\_\{2\}\)=\\eta, andE2​\(η\)=\(s¯¯2,s~2\)E\_\{2\}\(\\eta\)=\(\\bar\{\\bar\{s\}\}\_\{2\},\\tilde\{s\}\_\{2\}\)\. Ifη≥ηD\\eta\\geq\\eta^\{D\}, the inequalityη<η¯​\(s2\)\\eta<\\bar\{\\eta\}\(s\_\{2\}\)fails throughoutD2D\_\{2\}, soE2​\(η\)=∅E\_\{2\}\(\\eta\)=\\emptyset\. These are exactly the three cases stated in the proposition\.*Q\.E\.D\.*

*Proof of[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)\.*OnD2⊂\(s¯2,α2\)D\_\{2\}\\subset\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\), the zero floor does not bind, andg​\(s2,h;A2\)\>0g\(s\_\{2\},h;A\_\{2\}\)\>0andM​\(s2,h;A2\)\>0M\(s\_\{2\},h;A\_\{2\}\)\>0for everyh∈\[0,1\]h\\in\[0,1\]\. Under the power wage,

F2​\(h;s2,A2\)=C​g​\(s2,h;A2\)b−1​M​\(s2,h;A2\),C=β​b​\(ϕ​π2\+γ\)2​η\>0\.\\displaystyle F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=C\\,g\(s\_\{2\},h;A\_\{2\}\)^\{b\-1\}M\(s\_\{2\},h;A\_\{2\}\),\\qquad C=\\frac\{\\beta b\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}\>0\.We first showF2F\_\{2\}is strictly increasing ins2s\_\{2\}for every fixedh∈\[0,1\]h\\in\[0,1\]\. Differentiating,

∂F2​\(h;s2,A2\)∂s2=C​g​\(s2,h;A2\)b−2​\[\(b−1\)​gs​M​\(s2,h;A2\)\+g​\(s2,h;A2\)​Ms\],\\displaystyle\\frac\{\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\{\\partial s\_\{2\}\}=C\\,g\(s\_\{2\},h;A\_\{2\}\)^\{b\-2\}\\Bigl\[\(b\-1\)g\_\{s\}M\(s\_\{2\},h;A\_\{2\}\)\+g\(s\_\{2\},h;A\_\{2\}\)M\_\{s\}\\Bigr\],where

gs\\displaystyle g\_\{s\}≡∂g​\(s2,h;A2\)∂s2=1−\(1−π2\)​\(h​\(ϕ​π2\+γ\)−γ\),\\displaystyle\\equiv\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial s\_\{2\}\}=1\-\(1\-\\pi\_\{2\}\)\\bigl\(h\(\\phi\\pi\_\{2\}\+\\gamma\)\-\\gamma\\bigr\),Ms\\displaystyle M\_\{s\}≡∂M​\(s2,h;A2\)∂s2=λ​R​\(π2\+δ​h​\(1−π2\)\)−b​s2b−1\.\\displaystyle\\equiv\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial s\_\{2\}\}=\\lambda R\\bigl\(\\pi\_\{2\}\+\\delta h\(1\-\\pi\_\{2\}\)\\bigr\)\-bs\_\{2\}^\{b\-1\}\.The functionh↦gsh\\mapsto g\_\{s\}is affine and decreasing, so its minimum on\[0,1\]\[0,1\]is attained ath=1h=1, wheregs=1−ϕ​π2​\(1−π2\)\>0g\_\{s\}=1\-\\phi\\pi\_\{2\}\(1\-\\pi\_\{2\}\)\>0by[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)\. Hencegs\>0g\_\{s\}\>0for allh∈\[0,1\]h\\in\[0,1\]\. The functionh↦Msh\\mapsto M\_\{s\}is nondecreasing, so its minimum is attained ath=0h=0, where

Ms=λ​R​π2−b​s2b−1\>λ​R​π2−b​α2b−1≥0,\\displaystyle M\_\{s\}=\\lambda R\\pi\_\{2\}\-bs\_\{2\}^\{b\-1\}\>\\lambda R\\pi\_\{2\}\-b\\alpha\_\{2\}^\{b\-1\}\\geq 0,usings2<α2s\_\{2\}<\\alpha\_\{2\},b\>1b\>1, andλ​R​π2≥b​α2b−1\\lambda R\\pi\_\{2\}\\geq b\\alpha\_\{2\}^\{b\-1\}\. ThusMs\>0M\_\{s\}\>0for allh∈\[0,1\]h\\in\[0,1\]\. Sinceg\>0g\>0andM\>0M\>0, every term in the bracket is positive, so

∂F2​\(h;s2,A2\)∂s2\>0for all​h∈\[0,1\]​and​s2∈D2\.\\displaystyle\\frac\{\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\{\\partial s\_\{2\}\}\>0\\qquad\\text\{for all \}h\\in\[0,1\]\\text\{ and \}s\_\{2\}\\in D\_\{2\}\.
By the terminal single\-crossing condition,∂F2​\(h;s2,A2\)/∂h<0\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)/\\partial h<0onD2D\_\{2\}\. Hence[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3)applies onD2D\_\{2\}: the equilibrium ish2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0whenF2​\(0;s2,A2\)≤λ​R​δF\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\\leq\\lambda R\\delta, ish2∗​\(s2\)=1h\_\{2\}^\{\*\}\(s\_\{2\}\)=1whenF2​\(1;s2,A2\)≥λ​R​δF\_\{2\}\(1;s\_\{2\},A\_\{2\}\)\\geq\\lambda R\\delta, and otherwise is the unique interior root ofF2​\(h;s2,A2\)=λ​R​δF\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\. On the interior region, implicit differentiation gives

h2∗⁣′​\(s2\)=−∂F2​\(h2∗​\(s2\);s2,A2\)/∂s2∂F2​\(h2∗​\(s2\);s2,A2\)/∂h\>0\.\\displaystyle h\_\{2\}^\{\*\\prime\}\(s\_\{2\}\)=\-\\frac\{\\partial F\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{2\}\);s\_\{2\},A\_\{2\}\)/\\partial s\_\{2\}\}\{\\partial F\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{2\}\);s\_\{2\},A\_\{2\}\)/\\partial h\}\>0\.On the two corner regions,h2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\)is constant\. Thereforeh2∗​\(s2\)h\_\{2\}^\{\*\}\(s\_\{2\}\)is nondecreasing onD2D\_\{2\}, and strictly increasing on the interior region\.

It remains to locate the regimes\. Sinces2↦F2​\(0;s2,A2\)s\_\{2\}\\mapsto F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)ands2↦F2​\(1;s2,A2\)s\_\{2\}\\mapsto F\_\{2\}\(1;s\_\{2\},A\_\{2\}\)are strictly increasing onD2D\_\{2\}, each crossesλ​R​δ\\lambda R\\deltaat most once\. Let

s^2D≡inf\(\{s2∈D2:F2​\(0;s2,A2\)\>λ​R​δ\}∪\{α2\}\),\\displaystyle\\hat\{s\}\_\{2\}^\{D\}\\equiv\\inf\\Bigl\(\\\{s\_\{2\}\\in D\_\{2\}:F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)\>\\lambda R\\delta\\\}\\cup\\\{\\alpha\_\{2\}\\\}\\Bigr\),and

s¯2D≡inf\(\{s2∈D2:F2​\(1;s2,A2\)≥λ​R​δ\}∪\{α2\}\)\.\\displaystyle\\overline\{s\}\_\{2\}^\{D\}\\equiv\\inf\\Bigl\(\\\{s\_\{2\}\\in D\_\{2\}:F\_\{2\}\(1;s\_\{2\},A\_\{2\}\)\\geq\\lambda R\\delta\\\}\\cup\\\{\\alpha\_\{2\}\\\}\\Bigr\)\.These definitions give the endpoint conventionss¯¯2≤s^2D≤s¯2D≤α2\\bar\{\\bar\{s\}\}\_\{2\}\\leq\\hat\{s\}\_\{2\}^\{D\}\\leq\\overline\{s\}\_\{2\}^\{D\}\\leq\\alpha\_\{2\}\. The ordering follows becauseF2​\(1;s2,A2\)<F2​\(0;s2,A2\)F\_\{2\}\(1;s\_\{2\},A\_\{2\}\)<F\_\{2\}\(0;s\_\{2\},A\_\{2\}\)for everys2∈D2s\_\{2\}\\in D\_\{2\}\. Thus the set where full engagement is optimal lies weakly above the set where engagement begins\. By[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3), the three regimes are

h2∗​\(s2\)=0on​\(s¯¯2,s^2D\],\\displaystyle h\_\{2\}^\{\*\}\(s\_\{2\}\)=0\\quad\\text\{on \}\(\\bar\{\\bar\{s\}\}\_\{2\},\\hat\{s\}\_\{2\}^\{D\}\],h2∗​\(s2\)∈\(0,1\)on​\(s^2D,s¯2D\),\\displaystyle h\_\{2\}^\{\*\}\(s\_\{2\}\)\\in\(0,1\)\\quad\\text\{on \}\(\\hat\{s\}\_\{2\}^\{D\},\\overline\{s\}\_\{2\}^\{D\}\),and

h2∗​\(s2\)=1on​\[s¯2D,α2\),\\displaystyle h\_\{2\}^\{\*\}\(s\_\{2\}\)=1\\quad\\text\{on \}\[\\overline\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\),with all intervals understood relative toD2D\_\{2\}\. Ifs^2D∈D2\\hat\{s\}\_\{2\}^\{D\}\\in D\_\{2\}, thenF2​\(0;s^2D,A2\)=λ​R​δF\_\{2\}\(0;\\hat\{s\}\_\{2\}^\{D\},A\_\{2\}\)=\\lambda R\\delta; ifs¯2D∈D2\\overline\{s\}\_\{2\}^\{D\}\\in D\_\{2\}, thenF2​\(1;s¯2D,A2\)=λ​R​δF\_\{2\}\(1;\\overline\{s\}\_\{2\}^\{D\},A\_\{2\}\)=\\lambda R\\delta\.

Finally, consider regularity\. On the two corner regions,h2∗h\_\{2\}^\{\*\}is constant and hence continuously differentiable with derivative zero\. On the interior region,\(h,s2\)↦F2​\(h;s2,A2\)\(h,s\_\{2\}\)\\mapsto F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)is continuously differentiable and∂F2/∂h<0\\partial F\_\{2\}/\\partial h<0, so the implicit function theorem impliesh2∗h\_\{2\}^\{\*\}is continuously differentiable there, withh2∗⁣′​\(s2\)\>0h\_\{2\}^\{\*\\prime\}\(s\_\{2\}\)\>0\.

Ifs^2D∈D2\\hat\{s\}\_\{2\}^\{D\}\\in D\_\{2\}, continuity ats^2D\\hat\{s\}\_\{2\}^\{D\}follows by the argument below\. Suppose, toward a contradiction, thath2∗​\(sn\)≥ε\>0h\_\{2\}^\{\*\}\(s\_\{n\}\)\\geq\\varepsilon\>0along a sequencesn↓s^2Ds\_\{n\}\\downarrow\\hat\{s\}\_\{2\}^\{D\}in the interior region\. Then

λ​R​δ=F2​\(h2∗​\(sn\);sn,A2\)≤F2​\(ε;sn,A2\)→F2​\(ε;s^2D,A2\)<F2​\(0;s^2D,A2\)=λ​R​δ,\\displaystyle\\lambda R\\delta=F\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{n\}\);s\_\{n\},A\_\{2\}\)\\leq F\_\{2\}\(\\varepsilon;s\_\{n\},A\_\{2\}\)\\to F\_\{2\}\(\\varepsilon;\\hat\{s\}\_\{2\}^\{D\},A\_\{2\}\)<F\_\{2\}\(0;\\hat\{s\}\_\{2\}^\{D\},A\_\{2\}\)=\\lambda R\\delta,a contradiction\. Henceh2∗​\(s2\)→0h\_\{2\}^\{\*\}\(s\_\{2\}\)\\to 0ass2↓s^2Ds\_\{2\}\\downarrow\\hat\{s\}\_\{2\}^\{D\}\. Similarly, ifs¯2D∈D2\\overline\{s\}\_\{2\}^\{D\}\\in D\_\{2\}, supposeh2∗​\(sn\)≤1−εh\_\{2\}^\{\*\}\(s\_\{n\}\)\\leq 1\-\\varepsilonalong a sequencesn↑s¯2Ds\_\{n\}\\uparrow\\overline\{s\}\_\{2\}^\{D\}in the interior region\. Then

λ​R​δ=F2​\(h2∗​\(sn\);sn,A2\)≥F2​\(1−ε;sn,A2\)→F2​\(1−ε;s¯2D,A2\)\>F2​\(1;s¯2D,A2\)=λ​R​δ,\\displaystyle\\lambda R\\delta=F\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{n\}\);s\_\{n\},A\_\{2\}\)\\geq F\_\{2\}\(1\-\\varepsilon;s\_\{n\},A\_\{2\}\)\\to F\_\{2\}\(1\-\\varepsilon;\\overline\{s\}\_\{2\}^\{D\},A\_\{2\}\)\>F\_\{2\}\(1;\\overline\{s\}\_\{2\}^\{D\},A\_\{2\}\)=\\lambda R\\delta,again a contradiction\. Henceh2∗​\(s2\)→1h\_\{2\}^\{\*\}\(s\_\{2\}\)\\to 1ass2↑s¯2Ds\_\{2\}\\uparrow\\overline\{s\}\_\{2\}^\{D\}\. Thereforeh2∗h\_\{2\}^\{\*\}is continuous onD2D\_\{2\}and continuously differentiable on each nonempty region\.*Q\.E\.D\.*

*Proof of[Proposition5](https://arxiv.org/html/2606.29111#Thmproposition5)\.*Part \(i\)\.Fixs2∈D2s\_\{2\}\\in D\_\{2\}and suppose the terminal\-period equilibrium is interior,h2∗​\(s2\)∈\(0,1\)h\_\{2\}^\{\*\}\(s\_\{2\}\)\\in\(0,1\)\. ThenF2​\(h2∗​\(s2\);s2,A2\)=λ​R​δF\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{2\}\);s\_\{2\},A\_\{2\}\)=\\lambda R\\delta\. For local changes inα2\\alpha\_\{2\}that keeps2s\_\{2\}in the admissible domainD2D\_\{2\}and keep the equilibrium interior, implicit differentiation gives

∂h2∗​\(s2\)∂α2=−∂F2/∂α2∂F2/∂h\.\\displaystyle\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}=\-\\frac\{\\partial F\_\{2\}/\\partial\\alpha\_\{2\}\}\{\\partial F\_\{2\}/\\partial h\}\.By[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3),∂F2/∂h<0\\partial F\_\{2\}/\\partial h<0\. It therefore suffices to show∂F2/∂α2\>0\\partial F\_\{2\}/\\partial\\alpha\_\{2\}\>0ath=h2∗​\(s2\)h=h\_\{2\}^\{\*\}\(s\_\{2\}\)\.

Recall that

F2​\(h;s2,A2\)=β​\(ϕ​π2\+γ\)2​η​B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)\.\\displaystyle F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\frac\{\\beta\(\\phi\\pi\_\{2\}\+\\gamma\)\}\{2\\eta\}B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)M\(s\_\{2\},h;A\_\{2\}\)\.The multiplicative constant is positive, so the sign of∂F2/∂α2\\partial F\_\{2\}/\\partial\\alpha\_\{2\}is the sign of

B′′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)​∂g​\(s2,h;A2\)∂α2\+B′​\(g​\(s2,h;A2\)\)​∂M​\(s2,h;A2\)∂α2\.\\displaystyle B^\{\\prime\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)M\(s\_\{2\},h;A\_\{2\}\)\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}\+B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}\.Now

∂g​\(s2,h;A2\)∂α2=\(1−π2\)​\(ϕ​π2\+γ\)​\(h−h2c\),h2c=γϕ​π2\+γ,\\displaystyle\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}=\(1\-\\pi\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(h\-h\_\{2\}^\{c\}\),\\qquad h\_\{2\}^\{c\}=\\frac\{\\gamma\}\{\\phi\\pi\_\{2\}\+\\gamma\},and

∂M​\(s2,h;A2\)∂α2=λ​R​\(1−π2\)​\(1−δ​h\)\.\\displaystyle\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}=\\lambda R\(1\-\\pi\_\{2\}\)\(1\-\\delta h\)\.Thus, after factoring out the positive term\(1−π2\)​B′​\(g​\(s2,h;A2\)\)\(1\-\\pi\_\{2\}\)B^\{\\prime\}\\\!\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\), the sign of∂F2/∂α2\\partial F\_\{2\}/\\partial\\alpha\_\{2\}is the sign of

B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)​\(ϕ​π2\+γ\)​\(h−h2c\)\+λ​R​\(1−δ​h\)\.\\displaystyle\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}M\(s\_\{2\},h;A\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(h\-h\_\{2\}^\{c\}\)\+\\lambda R\(1\-\\delta h\)\.
Ifh≥h2ch\\geq h\_\{2\}^\{c\}, this expression is strictly positive becauseB′\>0B^\{\\prime\}\>0,B′′\>0B^\{\\prime\\prime\}\>0,M\>0M\>0, and1−δ​h\>01\-\\delta h\>0\. Now considerh<h2ch<h\_\{2\}^\{c\}\. By condition[eq\.6](https://arxiv.org/html/2606.29111#S5.E6),

B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)<λ​R​δϕ​π2\+γ\.\\displaystyle\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}M\(s\_\{2\},h;A\_\{2\}\)<\\frac\{\\lambda R\\delta\}\{\\phi\\pi\_\{2\}\+\\gamma\}\.Becauseh−h2c<0h\-h\_\{2\}^\{c\}<0, multiplying both sides by\(ϕ​π2\+γ\)​\(h−h2c\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(h\-h\_\{2\}^\{c\}\)reverses the inequality and gives

B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)​\(ϕ​π2\+γ\)​\(h−h2c\)\>λ​R​δ​\(h−h2c\)\.\\displaystyle\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}M\(s\_\{2\},h;A\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(h\-h\_\{2\}^\{c\}\)\>\\lambda R\\delta\(h\-h\_\{2\}^\{c\}\)\.Therefore,

B′′​\(g​\(s2,h;A2\)\)B′​\(g​\(s2,h;A2\)\)​M​\(s2,h;A2\)​\(ϕ​π2\+γ\)​\(h−h2c\)\+λ​R​\(1−δ​h\)\>λ​R​δ​\(h−h2c\)\+λ​R​\(1−δ​h\)=λ​R​\(1−δ​h2c\)\>0,\\displaystyle\\begin\{aligned\} &\\frac\{B^\{\\prime\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}\{B^\{\\prime\}\(g\(s\_\{2\},h;A\_\{2\}\)\)\}M\(s\_\{2\},h;A\_\{2\}\)\(\\phi\\pi\_\{2\}\+\\gamma\)\(h\-h\_\{2\}^\{c\}\)\+\\lambda R\(1\-\\delta h\)\\\\ &\\qquad\>\\lambda R\\delta\(h\-h\_\{2\}^\{c\}\)\+\\lambda R\(1\-\\delta h\)\\\\ &\\qquad=\\lambda R\(1\-\\delta h\_\{2\}^\{c\}\)\>0,\\end\{aligned\}becauseδ∈\(0,1\)\\delta\\in\(0,1\)andh2c∈\(0,1\)h\_\{2\}^\{c\}\\in\(0,1\)\.

Hence∂F2/∂α2\>0\\partial F\_\{2\}/\\partial\\alpha\_\{2\}\>0ath=h2∗​\(s2\)h=h\_\{2\}^\{\*\}\(s\_\{2\}\)\. Because∂F2/∂h<0\\partial F\_\{2\}/\\partial h<0, implicit differentiation implies∂h2∗​\(s2\)∂α2\>0\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\}\)\}\{\\partial\\alpha\_\{2\}\}\>0\.

For Part \(ii\), which establishes the comparative static inπ2\\pi\_\{2\}under the power wageB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}withb\>1b\>1, we first define the threshold values used in the statement of[Proposition5](https://arxiv.org/html/2606.29111#Thmproposition5)\. Fixs2∈D2s\_\{2\}\\in D\_\{2\}, and define two functions ofπ2\\pi\_\{2\},

L\+​\(π2\)\\displaystyle L\_\{\+\}\(\\pi\_\{2\}\)≜\[λ​R​α2−s2b−λ​R​π2​\(α2−s2\)\]​ϕϕ​π2\+γ,\\displaystyle\\triangleq\\bigl\[\\lambda R\\alpha\_\{2\}\-s\_\{2\}^\{b\}\-\\lambda R\\pi\_\{2\}\(\\alpha\_\{2\}\-s\_\{2\}\)\\bigr\]\\,\\frac\{\\phi\}\{\\phi\\pi\_\{2\}\+\\gamma\},L−​\(π2\)\\displaystyle L\_\{\-\}\(\\pi\_\{2\}\)≜\[λ​R​α2−λ​R​δ​\(α2−s2\)−s2b−λ​R​π2​\(1−δ\)​\(α2−s2\)\]\\displaystyle\\triangleq\\bigl\[\\lambda R\\alpha\_\{2\}\-\\lambda R\\delta\(\\alpha\_\{2\}\-s\_\{2\}\)\-s\_\{2\}^\{b\}\-\\lambda R\\pi\_\{2\}\(1\-\\delta\)\(\\alpha\_\{2\}\-s\_\{2\}\)\\bigr\]×\[ϕϕ​π2\+γ\+\(b−1\)​max⁡\{γ,ϕ​\(1−2​π2\)\}​\(α2−s2\)s2−γ​\(1−π2\)​\(α2−s2\)\],\\displaystyle\\quad\\times\\left\[\\frac\{\\phi\}\{\\phi\\pi\_\{2\}\+\\gamma\}\+\(b\-1\)\\,\\frac\{\\max\\\{\\gamma,\\,\\phi\(1\-2\\pi\_\{2\}\)\\\}\\,\(\\alpha\_\{2\}\-s\_\{2\}\)\}\{s\_\{2\}\-\\gamma\(1\-\\pi\_\{2\}\)\(\\alpha\_\{2\}\-s\_\{2\}\)\}\\right\],both strictly decreasing inπ2\\pi\_\{2\}\. Each function tracks the net effect of reliability on the engagement gain through its two channels: a higherπ2\\pi\_\{2\}lets a unit of engagement build more skill, strengthening the skill\-value channel, but it also makes AI fail more often, eroding the current margin on a below\-benchmark worker \(the operating\-margin channel\)\.L\+L\_\{\+\}bounds this net effect from below andL−L\_\{\-\}from above, so a largeL\+L\_\{\+\}certifies that the skill\-value channel dominates, whereas a smallL−L\_\{\-\}certifies that the operating\-margin channel dominates\. Both fall inπ2\\pi\_\{2\}because more frequent failure steadily tilts the balance from skill building toward margin erosion\. Letπ¯\+\\bar\{\\pi\}^\{\+\}andπ¯−\\bar\{\\pi\}^\{\-\}, when they exist, denote the unique thresholds defined byL\+​\(π¯\+\)=λ​R​\(α2−s2\)L\_\{\+\}\(\\bar\{\\pi\}^\{\+\}\)=\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)andL−​\(π¯−\)=λ​R​\(1−δ\)​\(α2−s2\)L\_\{\-\}\(\\bar\{\\pi\}^\{\-\}\)=\\lambda R\(1\-\\delta\)\(\\alpha\_\{2\}\-s\_\{2\}\)\. If one equation has no solution in the relevant range, the corresponding sign region is empty\.

Part \(ii\)\.Fix a local range ofπ2\\pi\_\{2\}values over whichs2s\_\{2\}remains inD2D\_\{2\}, the terminal single\-crossing condition holds, and the equilibrium remains interior\. Holds2s\_\{2\}andα2\\alpha\_\{2\}fixed\. UnderB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\},

F2​\(h;s2,A2\)=β​b2​η​\(ϕ​π2\+γ\)​\(g​\(s2,h;A2\)\)b−1​M​\(s2,h;A2\)\.\\displaystyle F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)=\\frac\{\\beta b\}\{2\\eta\}\(\\phi\\pi\_\{2\}\+\\gamma\)\\bigl\(g\(s\_\{2\},h;A\_\{2\}\)\\bigr\)^\{b\-1\}M\(s\_\{2\},h;A\_\{2\}\)\.Because the equilibrium is interior,F2​\(h2∗​\(s2,π2\);s2,A2\)=λ​R​δF\_\{2\}\(h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\);s\_\{2\},A\_\{2\}\)=\\lambda R\\delta, and implicit differentiation gives

∂h2∗​\(s2,π2\)∂π2=−∂F2/∂π2∂F2/∂h\.\\displaystyle\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}=\-\\frac\{\\partial F\_\{2\}/\\partial\\pi\_\{2\}\}\{\\partial F\_\{2\}/\\partial h\}\.Becauses2∈D2s\_\{2\}\\in D\_\{2\}throughout this local range, the terminal single\-crossing condition in[eq\.6](https://arxiv.org/html/2606.29111#S5.E6)applies\. Hence, by[Proposition3](https://arxiv.org/html/2606.29111#Thmproposition3),∂F2​\(h;s2,A2\)/∂h<0\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)/\\partial h<0\. Therefore

sign⁡\(∂h2∗​\(s2,π2\)∂π2\)=sign⁡\(∂F2​\(h;s2,A2\)∂π2\|h=h2∗​\(s2,π2\)\)\.\\displaystyle\\operatorname\{sign\}\\\!\\left\(\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}\\right\)=\\operatorname\{sign\}\\\!\\left\(\\frac\{\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}\\Big\|\_\{h=h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\}\\right\)\.
For fixedhh, the derivatives of the skill transition and margin with respect toπ2\\pi\_\{2\}are

∂g​\(s2,h;A2\)∂π2=\(α2−s2\)​\{γ\+h​\(ϕ−γ−2​ϕ​π2\)\},∂M​\(s2,h;A2\)∂π2=−λ​R​\(α2−s2\)​\(1−δ​h\)\.\\displaystyle\\frac\{\\partial g\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}=\(\\alpha\_\{2\}\-s\_\{2\}\)\\\{\\gamma\+h\(\\phi\-\\gamma\-2\\phi\\pi\_\{2\}\)\\\},\\qquad\\frac\{\\partial M\(s\_\{2\},h;A\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}=\-\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\(1\-\\delta h\)\.Taking the logarithmic derivative ofF2F\_\{2\}with respect toπ2\\pi\_\{2\}therefore gives

𝒜​\(s2,π2,h\)\\displaystyle\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)≜1F2​\(h;s2,A2\)​∂F2​\(h;s2,A2\)∂π2\\displaystyle\\triangleq\\frac\{1\}\{F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\\frac\{\\partial F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}=ϕϕ​π2\+γ\+\(b−1\)​\(α2−s2\)​\{γ\+h​\(ϕ−γ−2​ϕ​π2\)\}g​\(s2,h;A2\)−λ​R​\(α2−s2\)​\(1−δ​h\)M​\(s2,h;A2\)\.\\displaystyle=\\frac\{\\phi\}\{\\phi\\pi\_\{2\}\+\\gamma\}\+\(b\-1\)\\frac\{\(\\alpha\_\{2\}\-s\_\{2\}\)\\\{\\gamma\+h\(\\phi\-\\gamma\-2\\phi\\pi\_\{2\}\)\\\}\}\{g\(s\_\{2\},h;A\_\{2\}\)\}\-\\frac\{\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\(1\-\\delta h\)\}\{M\(s\_\{2\},h;A\_\{2\}\)\}\.BecauseF2​\(h;s2,A2\)\>0F\_\{2\}\(h;s\_\{2\},A\_\{2\}\)\>0, the sign of𝒜​\(s2,π2,h\)\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)is the sign of∂F2/∂π2\\partial F\_\{2\}/\\partial\\pi\_\{2\}at that samehh\. Combined with the display above,∂h2∗​\(s2,π2\)/∂π2\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)/\\partial\\pi\_\{2\}has the sign of𝒜\\mathcal\{A\}at the equilibrium pointh=h2∗​\(s2,π2\)h=h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\. Becauseh2∗​\(s2,π2\)h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)has no closed form, we instead establish a sign for𝒜​\(s2,π2,h\)\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)that holds uniformly over allh∈\[0,1\]h\\in\[0,1\]; such a sign holds in particular ath=h2∗​\(s2,π2\)h=h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\), and the conclusion follows\.

We first show∂h2∗​\(s2,π2\)/∂π2\>0\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)/\\partial\\pi\_\{2\}\>0whenπ2<π¯\+\\pi\_\{2\}<\\bar\{\\pi\}^\{\+\}\. Supposeπ2≤1/2\\pi\_\{2\}\\leq 1/2\. Then

γ\+h​\(ϕ−γ−2​ϕ​π2\)≥0for all​h∈\[0,1\]\.\\displaystyle\\gamma\+h\(\\phi\-\\gamma\-2\\phi\\pi\_\{2\}\)\\geq 0\\qquad\\text\{for all \}h\\in\[0,1\]\.The middle term in𝒜\\mathcal\{A\}is therefore nonnegative, and the margin ratio strictly decreases inhh, because

∂∂h​\[λ​R​\(α2−s2\)​\(1−δ​h\)M​\(s2,h;A2\)\]=λ​R​\(α2−s2\)​δ​s2​\(s2b−1−λ​R\)M​\(s2,h;A2\)2<0,\\displaystyle\\frac\{\\partial\}\{\\partial h\}\\left\[\\frac\{\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\(1\-\\delta h\)\}\{M\(s\_\{2\},h;A\_\{2\}\)\}\\right\]=\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\\frac\{\\delta s\_\{2\}\(s\_\{2\}^\{b\-1\}\-\\lambda R\)\}\{M\(s\_\{2\},h;A\_\{2\}\)^\{2\}\}<0,in which the inequality follows from[Assumption4](https://arxiv.org/html/2606.29111#Thmassumption4), which impliesλ​R\>α2b−1\>s2b−1\\lambda R\>\\alpha\_\{2\}^\{b\-1\}\>s\_\{2\}^\{b\-1\}\. Therefore this ratio is maximized ath=0h=0, and for everyh∈\[0,1\]h\\in\[0,1\],

𝒜​\(s2,π2,h\)≥ϕϕ​π2\+γ−λ​R​\(α2−s2\)M​\(s2,0;A2\)=L\+​\(π2\)−λ​R​\(α2−s2\)M​\(s2,0;A2\)\.\\displaystyle\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)\\geq\\frac\{\\phi\}\{\\phi\\pi\_\{2\}\+\\gamma\}\-\\frac\{\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\}\{M\(s\_\{2\},0;A\_\{2\}\)\}=\\frac\{L\_\{\+\}\(\\pi\_\{2\}\)\-\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\}\{M\(s\_\{2\},0;A\_\{2\}\)\}\.BecauseM​\(s2,0;A2\)\>0M\(s\_\{2\},0;A\_\{2\}\)\>0andL\+L\_\{\+\}is strictly decreasing inπ2\\pi\_\{2\}, the right\-hand side is strictly positive precisely whenπ2<π¯\+\\pi\_\{2\}<\\bar\{\\pi\}^\{\+\}\. Thus𝒜​\(s2,π2,h\)\>0\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)\>0for allh∈\[0,1\]h\\in\[0,1\], and in particular ath=h2∗​\(s2,π2\)h=h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\), so∂h2∗​\(s2,π2\)∂π2\>0\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}\>0\. We next show∂h2∗​\(s2,π2\)/∂π2<0\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)/\\partial\\pi\_\{2\}<0whenπ2\>π¯−\\pi\_\{2\}\>\\bar\{\\pi\}^\{\-\}, by bounding𝒜\\mathcal\{A\}from above\. First,

γ\+h​\(ϕ−γ−2​ϕ​π2\)≤max⁡\{γ,ϕ​\(1−2​π2\)\}for all​h∈\[0,1\],\\displaystyle\\gamma\+h\(\\phi\-\\gamma\-2\\phi\\pi\_\{2\}\)\\leq\\max\\\{\\gamma,\\,\\phi\(1\-2\\pi\_\{2\}\)\\\}\\qquad\\text\{for all \}h\\in\[0,1\],andg​\(s2,h;A2\)≥g​\(s2,0;A2\)g\(s\_\{2\},h;A\_\{2\}\)\\geq g\(s\_\{2\},0;A\_\{2\}\), so the middle term in𝒜\\mathcal\{A\}is bounded above by\(b−1\)​\(α2−s2\)​max⁡\{γ,ϕ​\(1−2​π2\)\}/g​\(s2,0;A2\)\(b\-1\)\(\\alpha\_\{2\}\-s\_\{2\}\)\\max\\\{\\gamma,\\phi\(1\-2\\pi\_\{2\}\)\\\}/g\(s\_\{2\},0;A\_\{2\}\)\. Second, the margin ratio is decreasing inhh, so it is minimized ath=1h=1, and the negative margin term is bounded above by−λ​R​\(α2−s2\)​\(1−δ\)/M​\(s2,1;A2\)\-\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\(1\-\\delta\)/M\(s\_\{2\},1;A\_\{2\}\)\. Combining these bounds, for everyh∈\[0,1\]h\\in\[0,1\],

𝒜​\(s2,π2,h\)\\displaystyle\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)≤ϕϕ​π2\+γ\+\(b−1\)​\(α2−s2\)​max⁡\{γ,ϕ​\(1−2​π2\)\}g​\(s2,0;A2\)−λ​R​\(α2−s2\)​\(1−δ\)M​\(s2,1;A2\)\\displaystyle\\leq\\frac\{\\phi\}\{\\phi\\pi\_\{2\}\+\\gamma\}\+\(b\-1\)\\frac\{\(\\alpha\_\{2\}\-s\_\{2\}\)\\max\\\{\\gamma,\\,\\phi\(1\-2\\pi\_\{2\}\)\\\}\}\{g\(s\_\{2\},0;A\_\{2\}\)\}\-\\frac\{\\lambda R\(\\alpha\_\{2\}\-s\_\{2\}\)\(1\-\\delta\)\}\{M\(s\_\{2\},1;A\_\{2\}\)\}=L−​\(π2\)−λ​R​\(1−δ\)​\(α2−s2\)M​\(s2,1;A2\)\.\\displaystyle=\\frac\{L\_\{\-\}\(\\pi\_\{2\}\)\-\\lambda R\(1\-\\delta\)\(\\alpha\_\{2\}\-s\_\{2\}\)\}\{M\(s\_\{2\},1;A\_\{2\}\)\}\.BecauseM​\(s2,1;A2\)\>0M\(s\_\{2\},1;A\_\{2\}\)\>0andL−L\_\{\-\}is strictly decreasing inπ2\\pi\_\{2\}, the right\-hand side is strictly negative precisely whenπ2\>π¯−\\pi\_\{2\}\>\\bar\{\\pi\}^\{\-\}\. Thus𝒜​\(s2,π2,h\)<0\\mathcal\{A\}\(s\_\{2\},\\pi\_\{2\},h\)<0for allh∈\[0,1\]h\\in\[0,1\], and in particular ath=h2∗​\(s2,π2\)h=h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\), so∂h2∗​\(s2,π2\)∂π2<0\\frac\{\\partial h\_\{2\}^\{\*\}\(s\_\{2\},\\pi\_\{2\}\)\}\{\\partial\\pi\_\{2\}\}<0\.*Q\.E\.D\.*

*Proof of[Lemma2](https://arxiv.org/html/2606.29111#Thmlemma2)\.*Fixs1s\_\{1\}as stated and suppose\(h1∗,h1∗\)\(h\_\{1\}^\{\*\},h\_\{1\}^\{\*\}\)is a symmetric equilibrium\. For brevity, write

σ1j=σ1j​\(h1j,h1−j;s1,A1,A2\)\.\\sigma\_\{1\}^\{j\}=\\sigma\_\{1\}^\{j\}\(h\_\{1\}^\{j\},h\_\{1\}^\{\-j\};s\_\{1\},A\_\{1\},A\_\{2\}\)\.Thenx↦Φ1j​\(x,h1∗;s1,A1,A2\)x\\mapsto\\Phi\_\{1\}^\{j\}\(x,h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)attains its maximum over\[0,1\]\[0,1\]atx=h1∗x=h\_\{1\}^\{\*\}\. Differentiating[Section5\.4](https://arxiv.org/html/2606.29111#S5.Ex20)inh1jh\_\{1\}^\{j\},

∂Φ1j∂h1j=∂σ1j∂h1j​\[M​\(s1,h1j;A1\)\+β​Π2∗​\(g​\(s1,h1j;A1\),A2\)−β​Π2∗​\(g​\(s1,h1−j;A1\),A2\)\]\\displaystyle\\frac\{\\partial\\Phi\_\{1\}^\{j\}\}\{\\partial h\_\{1\}^\{j\}\}=\\frac\{\\partial\\sigma\_\{1\}^\{j\}\}\{\\partial h\_\{1\}^\{j\}\}\\Bigl\[M\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\+\\beta\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)\-\\beta\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\-j\};A\_\{1\}\),A\_\{2\}\\bigr\)\\Bigr\]\+σ1j​\[∂M​\(s1,h1j;A1\)∂h1j\+β​d​Π2∗​\(g​\(s1,h1j;A1\),A2\)d​h1\]\.\\displaystyle\+\\sigma\_\{1\}^\{j\}\\\!\\left\[\\frac\{\\partial M\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\}\{\\partial h\_\{1\}^\{j\}\}\+\\beta\\,\\frac\{d\\,\\Pi\_\{2\}^\{\*\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\bigr\)\}\{dh\_\{1\}\}\\right\]\.At the symmetric pointh1j=h1−j=h1∗h\_\{1\}^\{j\}=h\_\{1\}^\{\-j\}=h\_\{1\}^\{\*\}, the two continuation terms in the first bracket cancel,σ1j=1/2\\sigma\_\{1\}^\{j\}=1/2, and the logit derivative is∂σ1j/∂h1j\|sym=\(β/4​η\)​d​V¯2​\(g​\(s1,h1∗;A1\),A2\)/d​h1\\partial\\sigma\_\{1\}^\{j\}/\\partial h\_\{1\}^\{j\}\\big\|\_\{\\mathrm\{sym\}\}=\(\\beta/4\\eta\)\\,d\\bar\{V\}\_\{2\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\}^\{\*\};A\_\{1\}\),A\_\{2\}\\bigr\)/dh\_\{1\}, so the own\-action derivative at the symmetric profile equalsF1​\(h1∗;s1,A1,A2\)F\_\{1\}\(h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)\. \(By the differentiability assumption in the lemma,F1F\_\{1\}is well defined ath1∗h\_\{1\}^\{\*\}\.\) Optimality ofh1∗h\_\{1\}^\{\*\}on\[0,1\]\[0,1\]requires the directional derivative toward any feasibleh1∈\[0,1\]h\_\{1\}\\in\[0,1\]to be nonpositive:F1​\(h1∗;s1,A1,A2\)​\(h1−h1∗\)≤0F\_\{1\}\(h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)\(h\_\{1\}\-h\_\{1\}^\{\*\}\)\\leq 0for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\]\. Ifh1∗∈\(0,1\)h\_\{1\}^\{\*\}\\in\(0,1\), both directions are feasible, forcingF1​\(h1∗;s1,A1,A2\)=0F\_\{1\}\(h\_\{1\}^\{\*\};s\_\{1\},A\_\{1\},A\_\{2\}\)=0; ifh1∗=0h\_\{1\}^\{\*\}=0, onlyh1\>0h\_\{1\}\>0is feasible, givingF1​\(0;s1,A1,A2\)≤0F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)\\leq 0; ifh1∗=1h\_\{1\}^\{\*\}=1, onlyh1<1h\_\{1\}<1is feasible, givingF1​\(1;s1,A1,A2\)≥0F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)\\geq 0\.*Q\.E\.D\.*

*Proof of[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)\.*Fixs1∈D1s\_\{1\}\\in D\_\{1\}satisfying eithers1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ors1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}\. By[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2),

g​\(s1,1;A1\)=\[1−ϕ​π1​\(1−π1\)\]​s1\+ϕ​π1​\(1−π1\)​α1\\displaystyle g\(s\_\{1\},1;A\_\{1\}\)=\\bigl\[1\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\\bigr\]s\_\{1\}\+\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}and

g​\(s1,0;A1\)=\[1\+γ​\(1−π1\)\]​s1−γ​\(1−π1\)​α1\\displaystyle g\(s\_\{1\},0;A\_\{1\}\)=\\bigl\[1\+\\gamma\(1\-\\pi\_\{1\}\)\\bigr\]s\_\{1\}\-\\gamma\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}are strictly increasing ins1s\_\{1\}\. By the definitions in[eq\.OA3](https://arxiv.org/html/2606.29111#S5.E3),

g​\(s1D,1;A1\)=s^2D,g​\(s1F,0;A1\)=s¯2D\.\\displaystyle g\(s\_\{1\}^\{D\},1;A\_\{1\}\)=\\hat\{s\}\_\{2\}^\{D\},\\qquad g\(s\_\{1\}^\{F\},0;A\_\{1\}\)=\\overline\{s\}\_\{2\}^\{D\}\.Thuss1<s1Ds\_\{1\}<s\_\{1\}^\{D\}impliesg​\(s1,1;A1\)<s^2Dg\(s\_\{1\},1;A\_\{1\}\)<\\hat\{s\}\_\{2\}^\{D\}, ands1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}impliesg​\(s1,0;A1\)\>s¯2Dg\(s\_\{1\},0;A\_\{1\}\)\>\\overline\{s\}\_\{2\}^\{D\}\. Becauseg​\(s1,⋅;A1\)g\(s\_\{1\},\\cdot\\,;A\_\{1\}\)is increasing, the reachable continuation setI≜\[g​\(s1,0;A1\),g​\(s1,1;A1\)\]I\\triangleq\[g\(s\_\{1\},0;A\_\{1\}\),g\(s\_\{1\},1;A\_\{1\}\)\]is a compact interval\. Sinces1∈D1s\_\{1\}\\in D\_\{1\}, we haveI⊂D2I\\subset D\_\{2\}\. Ifs1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, thenI⊂\(s¯¯2,s^2D\)I\\subset\(\\bar\{\\bar\{s\}\}\_\{2\},\\hat\{s\}\_\{2\}^\{D\}\), so[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)givesh2∗​\(s2\)=0h\_\{2\}^\{\*\}\(s\_\{2\}\)=0for alls2∈Is\_\{2\}\\in I\. Ifs1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}, thenI⊂\(s¯2D,α2\)I\\subset\(\\overline\{s\}\_\{2\}^\{D\},\\alpha\_\{2\}\), so[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)givesh2∗​\(s2\)=1h\_\{2\}^\{\*\}\(s\_\{2\}\)=1for alls2∈Is\_\{2\}\\in I\. In either case, the inequalities are strict andIIis compact, soIIis bounded away from the relevant terminal cutoff\. Hence the terminal policy is constant on an open interval containingII\.

In the no\-terminal\-engagement cases1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, on this open interval,

Π2∗​\(s2,A2\)=12​M​\(s2,0;A2\),V¯2​\(s2,A2\)=s2b\+β​\(g​\(s2,0;A2\)\)b\+η​log⁡2\.\\displaystyle\\Pi\_\{2\}^\{\*\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}M\(s\_\{2\},0;A\_\{2\}\),\\qquad\\bar\{V\}\_\{2\}\(s\_\{2\},A\_\{2\}\)=s\_\{2\}^\{b\}\+\\beta\\bigl\(g\(s\_\{2\},0;A\_\{2\}\)\\bigr\)^\{b\}\+\\eta\\log 2\.In the full\-terminal\-engagement cases1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}, on this open interval,

Π2∗​\(s2,A2\)=12​M​\(s2,1;A2\),V¯2​\(s2,A2\)=s2b\+β​\(g​\(s2,1;A2\)\)b\+η​log⁡2\.\\displaystyle\\Pi\_\{2\}^\{\*\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}M\(s\_\{2\},1;A\_\{2\}\),\\qquad\\bar\{V\}\_\{2\}\(s\_\{2\},A\_\{2\}\)=s\_\{2\}^\{b\}\+\\beta\\bigl\(g\(s\_\{2\},1;A\_\{2\}\)\\bigr\)^\{b\}\+\\eta\\log 2\.In both cases,Π2∗​\(⋅,A2\)\\Pi\_\{2\}^\{\*\}\(\\cdot,A\_\{2\}\)andV¯2​\(⋅,A2\)\\bar\{V\}\_\{2\}\(\\cdot,A\_\{2\}\)areC2C^\{2\}on the open interval containingII, becauseg​\(s2,0;A2\)\>0g\(s\_\{2\},0;A\_\{2\}\)\>0fors2\>s¯2s\_\{2\}\>\\underline\{s\}\_\{2\}andg​\(s2,1;A2\)\>0g\(s\_\{2\},1;A\_\{2\}\)\>0throughout\.

In the no\-terminal\-engagement case,

Π2∗⁣′​\(s2,A2\)=12​\[λ​R​π2−b​s2b−1\],Π2∗⁣′′​\(s2,A2\)=−b​\(b−1\)2​s2b−2<0,\\displaystyle\\Pi\_\{2\}^\{\*\\prime\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}\\bigl\[\\lambda R\\pi\_\{2\}\-b\\,s\_\{2\}^\{b\-1\}\\bigr\],\\qquad\\Pi\_\{2\}^\{\*\\prime\\prime\}\(s\_\{2\},A\_\{2\}\)=\-\\frac\{b\(b\-1\)\}\{2\}s\_\{2\}^\{b\-2\}<0,and

V¯2′​\(s2,A2\)=b​s2b−1\+β​b​\[1\+γ​\(1−π2\)\]​\(g​\(s2,0;A2\)\)b−1\>0\.\\displaystyle\\bar\{V\}\_\{2\}^\{\\prime\}\(s\_\{2\},A\_\{2\}\)=b\\,s\_\{2\}^\{b\-1\}\+\\beta b\\bigl\[1\+\\gamma\(1\-\\pi\_\{2\}\)\\bigr\]\\bigl\(g\(s\_\{2\},0;A\_\{2\}\)\\bigr\)^\{b\-1\}\>0\.In the full\-terminal\-engagement case,

Π2∗⁣′​\(s2,A2\)=12​\[λ​R​π2\+λ​R​δ​\(1−π2\)−b​s2b−1\],Π2∗⁣′′​\(s2,A2\)=−b​\(b−1\)2​s2b−2<0,\\displaystyle\\Pi\_\{2\}^\{\*\\prime\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}\\bigl\[\\lambda R\\pi\_\{2\}\+\\lambda R\\delta\(1\-\\pi\_\{2\}\)\-b\\,s\_\{2\}^\{b\-1\}\\bigr\],\\qquad\\Pi\_\{2\}^\{\*\\prime\\prime\}\(s\_\{2\},A\_\{2\}\)=\-\\frac\{b\(b\-1\)\}\{2\}s\_\{2\}^\{b\-2\}<0,and

V¯2′​\(s2,A2\)=b​s2b−1\+β​b​\[1−ϕ​π2​\(1−π2\)\]​\(g​\(s2,1;A2\)\)b−1\>0\.\\displaystyle\\bar\{V\}\_\{2\}^\{\\prime\}\(s\_\{2\},A\_\{2\}\)=b\\,s\_\{2\}^\{b\-1\}\+\\beta b\\bigl\[1\-\\phi\\pi\_\{2\}\(1\-\\pi\_\{2\}\)\\bigr\]\\bigl\(g\(s\_\{2\},1;A\_\{2\}\)\\bigr\)^\{b\-1\}\>0\.
Now write

g1≜g​\(s1,h1;A1\),gh≜∂g​\(s1,h1;A1\)∂h1=\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\>0\.\\displaystyle g\_\{1\}\\triangleq g\(s\_\{1\},h\_\{1\};A\_\{1\}\),\\qquad g\_\{h\}\\triangleq\\frac\{\\partial g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}=\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\>0\.The derivativeghg\_\{h\}is constant inh1h\_\{1\}\. Therefore, on the terminal\-corner region under consideration, the marginal gain can be written as

F1​\(h1;s1,A1,A2\)=1η​A​\(h1\)\+C​\(h1\),\\displaystyle F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)=\\frac\{1\}\{\\eta\}A\(h\_\{1\}\)\+C\(h\_\{1\}\),where

A​\(h1\)≜β4​V¯2′​\(g1,A2\)​gh​M​\(s1,h1;A1\),\\displaystyle A\(h\_\{1\}\)\\triangleq\\frac\{\\beta\}\{4\}\\,\\bar\{V\}\_\{2\}^\{\\prime\}\(g\_\{1\},A\_\{2\}\)\\,g\_\{h\}\\,M\(s\_\{1\},h\_\{1\};A\_\{1\}\),and

C​\(h1\)≜−12​λ​R​δ​\(1−π1\)​\(α1−s1\)\+β2​Π2∗⁣′​\(g1,A2\)​gh\.\\displaystyle C\(h\_\{1\}\)\\triangleq\-\\frac\{1\}\{2\}\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)\+\\frac\{\\beta\}\{2\}\\Pi\_\{2\}^\{\*\\prime\}\(g\_\{1\},A\_\{2\}\)g\_\{h\}\.BothAAandCCareC1C^\{1\}on\[0,1\]\[0,1\]\. Moreover,

C′​\(h1\)=β2​Π2∗⁣′′​\(g1,A2\)​gh2≤−c<0,\\displaystyle C^\{\\prime\}\(h\_\{1\}\)=\\frac\{\\beta\}\{2\}\\Pi\_\{2\}^\{\*\\prime\\prime\}\(g\_\{1\},A\_\{2\}\)g\_\{h\}^\{2\}\\leq\-c<0,where

c≜β​b​\(b−1\)4​gh2​mins2∈I⁡s2b−2\>0\.\\displaystyle c\\triangleq\\frac\{\\beta b\(b\-1\)\}\{4\}g\_\{h\}^\{2\}\\min\_\{s\_\{2\}\\in I\}s\_\{2\}^\{b\-2\}\>0\.LetK≜maxh1∈\[0,1\]⁡\|A′​\(h1\)\|<∞K\\triangleq\\max\_\{h\_\{1\}\\in\[0,1\]\}\|A^\{\\prime\}\(h\_\{1\}\)\|<\\infty\. Then, for everyη≥2​K/c\\eta\\geq 2K/c,

F1′​\(h1;s1,A1,A2\)=1η​A′​\(h1\)\+C′​\(h1\)≤Kη−c≤−c2<0\\displaystyle F\_\{1\}^\{\\prime\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)=\\frac\{1\}\{\\eta\}A^\{\\prime\}\(h\_\{1\}\)\+C^\{\\prime\}\(h\_\{1\}\)\\leq\\frac\{K\}\{\\eta\}\-c\\leq\-\\frac\{c\}\{2\}<0for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\]\. HenceF1F\_\{1\}is strictly decreasing on\[0,1\]\[0,1\]\. Strict monotonicity ofF1F\_\{1\}gives uniqueness of the symmetric candidate selected by the variational inequality\. To show this candidate is an equilibrium, however, the first\-order condition must be sufficient for a global best response\. This is why we next establish strict concavity ofx↦Φ1j​\(x,y;s1,A1,A2\)x\\mapsto\\Phi\_\{1\}^\{j\}\(x,y;s\_\{1\},A\_\{1\},A\_\{2\}\)for each fixed other\-firm actionyy\. Under this concavity, for any candidatehhand any deviationz∈\[0,1\]z\\in\[0,1\],

Φ1j​\(z,h\)−Φ1j​\(h,h\)≤∂Φ1j∂x​\(h,h\)​\(z−h\)=F1​\(h;s1,A1,A2\)​\(z−h\)\.\\displaystyle\\Phi\_\{1\}^\{j\}\(z,h\)\-\\Phi\_\{1\}^\{j\}\(h,h\)\\leq\\frac\{\\partial\\Phi\_\{1\}^\{j\}\}\{\\partial x\}\(h,h\)\(z\-h\)=F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)\(z\-h\)\.Thus anyhhsatisfying the variational inequality is a best response to itself\. It remains to verify, for sufficiently largeη\\eta, each firm’s objective is strictly concave in its own action\. Fix any other\-firm actiony∈\[0,1\]y\\in\[0,1\], writexxfor the firm’s own action, and defineG​\(x\)≜g​\(s1,x;A1\)G\(x\)\\triangleq g\(s\_\{1\},x;A\_\{1\}\)\. Letσ=σ1j​\(x,y;s1,A1,A2\)\\sigma=\\sigma\_\{1\}^\{j\}\(x,y;s\_\{1\},A\_\{1\},A\_\{2\}\)and

Δ​\(x,y\)≜M​\(s1,x;A1\)\+β​Π2∗​\(G​\(x\),A2\)−β​Π2∗​\(G​\(y\),A2\)\.\\displaystyle\\Delta\(x,y\)\\triangleq M\(s\_\{1\},x;A\_\{1\}\)\+\\beta\\Pi\_\{2\}^\{\*\}\(G\(x\),A\_\{2\}\)\-\\beta\\Pi\_\{2\}^\{\*\}\(G\(y\),A\_\{2\}\)\.Then

Φ1j​\(x,y;s1,A1,A2\)=σ​Δ​\(x,y\)\+β​Π2∗​\(G​\(y\),A2\)\.\\displaystyle\\Phi\_\{1\}^\{j\}\(x,y;s\_\{1\},A\_\{1\},A\_\{2\}\)=\\sigma\\,\\Delta\(x,y\)\+\\beta\\Pi\_\{2\}^\{\*\}\(G\(y\),A\_\{2\}\)\.Differentiating the logit gives

σx=σ​\(1−σ\)​βη​V¯2′​\(G​\(x\),A2\)​gh\\displaystyle\\sigma\_\{x\}=\\sigma\(1\-\\sigma\)\\frac\{\\beta\}\{\\eta\}\\bar\{V\}\_\{2\}^\{\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}and

σx​x=σ​\(1−σ\)​\(1−2​σ\)​\(βη\)2​\(V¯2′​\(G​\(x\),A2\)​gh\)2\+σ​\(1−σ\)​βη​V¯2′′​\(G​\(x\),A2\)​gh2\.\\displaystyle\\sigma\_\{xx\}=\\sigma\(1\-\\sigma\)\(1\-2\\sigma\)\\left\(\\frac\{\\beta\}\{\\eta\}\\right\)^\{2\}\\bigl\(\\bar\{V\}\_\{2\}^\{\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}\\bigr\)^\{2\}\+\\sigma\(1\-\\sigma\)\\frac\{\\beta\}\{\\eta\}\\bar\{V\}\_\{2\}^\{\\prime\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}^\{2\}\.Since

Δx​\(x,y\)=∂M​\(s1,x;A1\)∂h1\+β​Π2∗⁣′​\(G​\(x\),A2\)​gh\\displaystyle\\Delta\_\{x\}\(x,y\)=\\frac\{\\partial M\(s\_\{1\},x;A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+\\beta\\Pi\_\{2\}^\{\*\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}and

Δx​x​\(x,y\)=β​Π2∗⁣′′​\(G​\(x\),A2\)​gh2,\\displaystyle\\Delta\_\{xx\}\(x,y\)=\\beta\\Pi\_\{2\}^\{\*\\prime\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}^\{2\},we obtain

∂2Φ1j∂x2=σ\{\\displaystyle\\frac\{\\partial^\{2\}\\Phi\_\{1\}^\{j\}\}\{\\partial x^\{2\}\}=\\sigma\\Biggl\\\{\(1−σ\)​\[\(1−2​σ\)​β2η2​\(V¯2′​\(G​\(x\),A2\)​gh\)2\+βη​V¯2′′​\(G​\(x\),A2\)​gh2\]​Δ​\(x,y\)\\displaystyle\(1\-\\sigma\)\\left\[\(1\-2\\sigma\)\\frac\{\\beta^\{2\}\}\{\\eta^\{2\}\}\\bigl\(\\bar\{V\}\_\{2\}^\{\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}\\bigr\)^\{2\}\+\\frac\{\\beta\}\{\\eta\}\\bar\{V\}\_\{2\}^\{\\prime\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}^\{2\}\\right\]\\Delta\(x,y\)\+2​\(1−σ\)​βη​V¯2′​\(G​\(x\),A2\)​gh​\[∂M​\(s1,x;A1\)∂h1\+β​Π2∗⁣′​\(G​\(x\),A2\)​gh\]\\displaystyle\+2\(1\-\\sigma\)\\frac\{\\beta\}\{\\eta\}\\bar\{V\}\_\{2\}^\{\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}\\left\[\\frac\{\\partial M\(s\_\{1\},x;A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+\\beta\\Pi\_\{2\}^\{\*\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}\\right\]\+βΠ2∗⁣′′\(G\(x\),A2\)gh2\}\.\\displaystyle\+\\beta\\Pi\_\{2\}^\{\*\\prime\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}^\{2\}\\Biggr\\\}\.Forx,y∈\[0,1\]x,y\\in\[0,1\], we haveG​\(x\),G​\(y\)∈IG\(x\),G\(y\)\\in I\. SinceV¯2\\bar\{V\}\_\{2\}andΠ2∗\\Pi\_\{2\}^\{\*\}areC2C^\{2\}on a neighborhood ofII, the quantities

V¯2′​\(G​\(x\),A2\),\|V¯2′′​\(G​\(x\),A2\)\|,\|Δ​\(x,y\)\|,and\|∂M​\(s1,x;A1\)∂h1\+β​Π2∗⁣′​\(G​\(x\),A2\)​gh\|\\displaystyle\\bar\{V\}\_\{2\}^\{\\prime\}\(G\(x\),A\_\{2\}\),\\quad\|\\bar\{V\}\_\{2\}^\{\\prime\\prime\}\(G\(x\),A\_\{2\}\)\|,\\quad\|\\Delta\(x,y\)\|,\\quad\\text\{and\}\\quad\\left\|\\frac\{\\partial M\(s\_\{1\},x;A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+\\beta\\Pi\_\{2\}^\{\*\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}\\right\|are bounded uniformly on\[0,1\]2\[0,1\]^\{2\}\. Also,\(1−σ\)≤1\(1\-\\sigma\)\\leq 1,\|1−2​σ\|≤1\|1\-2\\sigma\|\\leq 1, and the final term inside braces satisfiesβ​Π2∗⁣′′​\(G​\(x\),A2\)​gh2≤−2​c\\beta\\Pi\_\{2\}^\{\*\\prime\\prime\}\(G\(x\),A\_\{2\}\)g\_\{h\}^\{2\}\\leq\-2c\. Hence there exist finite constantsK1K\_\{1\}andK2K\_\{2\}, independent ofη\\eta, such that the expression inside braces is at mostK1η\+K2η2−2​c\\frac\{K\_\{1\}\}\{\\eta\}\+\\frac\{K\_\{2\}\}\{\\eta^\{2\}\}\-2c\. Forη≥max⁡\{1,K1\+K2c\}\\eta\\geq\\max\\left\\\{1,\\frac\{K\_\{1\}\+K\_\{2\}\}\{c\}\\right\\\}, this upper bound is at most−c<0\-c<0\. Sinceσ\>0\\sigma\>0, it follows that∂2Φ1j∂x2<0\\frac\{\\partial^\{2\}\\Phi\_\{1\}^\{j\}\}\{\\partial x^\{2\}\}<0for everyx,y∈\[0,1\]x,y\\in\[0,1\]\. Thus each firm’s objective is strictly concave in its own action, uniformly in the other firm’s action\.

The bounds above depend only on the primitives,s1s\_\{1\},A1A\_\{1\},A2A\_\{2\}, and the terminal corner under consideration, but not onη\\eta\. If both terminal\-corner cases are possible for different values of the cutoffs, takeη1​\(s1\)\\eta\_\{1\}\(s\_\{1\}\)to be the maximum of the corresponding bounds across the no\-terminal\-engagement and full\-terminal\-engagement cases\. Thus, for everyη≥η1​\(s1\)\\eta\\geq\\eta\_\{1\}\(s\_\{1\}\), whenevers1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ors1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}, the functionF1F\_\{1\}is strictly decreasing on\[0,1\]\[0,1\], and each firm’s objective is strictly concave in its own action\.

It remains to characterize the symmetric equilibrium\. Fixη≥η1​\(s1\)\\eta\\geq\\eta\_\{1\}\(s\_\{1\}\)\. By strict concavity, a firm has no profitable deviation from own actionhhagainst the other firm’s actionhhif and only if the directional derivative toward every feasible action is nonpositive\. By the computation in the proof of[Lemma2](https://arxiv.org/html/2606.29111#Thmlemma2), the own\-action derivative at the symmetric profile\(h,h\)\(h,h\)equalsF1​\(h;s1,A1,A2\)F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)\. Hence\(h,h\)\(h,h\)is a symmetric equilibrium if and only if

F1​\(h;s1,A1,A2\)​\(z−h\)≤0∀z∈\[0,1\]\.\\displaystyle F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)\(z\-h\)\\leq 0\\qquad\\forall z\\in\[0,1\]\.BecauseF1F\_\{1\}is strictly decreasing, exactly onehhsatisfies this condition\. IfF1​\(0;s1,A1,A2\)≤0F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)\\leq 0, thenh=0h=0satisfies the variational inequality, and everyh\>0h\>0hasF1​\(h;s1,A1,A2\)<0F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)<0, so no interior point or upper corner can satisfy it\. IfF1​\(1;s1,A1,A2\)≥0F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)\\geq 0, thenh=1h=1satisfies the variational inequality, and everyh<1h<1hasF1​\(h;s1,A1,A2\)\>0F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)\>0, so no other point can satisfy it\. Otherwise,F1​\(0;s1,A1,A2\)\>0\>F1​\(1;s1,A1,A2\)F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)\>0\>F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\), and continuity plus strict monotonicity gives a unique interior root ofF1​\(h;s1,A1,A2\)=0F\_\{1\}\(h;s\_\{1\},A\_\{1\},A\_\{2\}\)=0\. This root satisfies the variational inequality, while both corners fail\. This proves the stated characterization of the unique symmetric equilibriumh1∗​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)\.

Finally, the realized continuation stateg​\(s1,h1∗​\(s1\);A1\)g\(s\_\{1\},h\_\{1\}^\{\*\}\(s\_\{1\}\);A\_\{1\}\)lies inII\. Therefore, by the terminal\-corner argument above,h2∗​\(g​\(s1,h1∗​\(s1\);A1\)\)=0h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\}^\{\*\}\(s\_\{1\}\);A\_\{1\}\)\)=0whens1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, andh2∗​\(g​\(s1,h1∗​\(s1\);A1\)\)=1h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\}^\{\*\}\(s\_\{1\}\);A\_\{1\}\)\)=1whens1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}\.*Q\.E\.D\.*

*Proof of[Corollary4](https://arxiv.org/html/2606.29111#Thmcorollary4)\.*Fixs1∈D1s\_\{1\}\\in D\_\{1\}, and letI=\[g​\(s1,0;A1\),g​\(s1,1;A1\)\]I=\[g\(s\_\{1\},0;A\_\{1\}\),g\(s\_\{1\},1;A\_\{1\}\)\]be the set of period\-2 skills reachable froms1s\_\{1\}\. Sinces1∈D1s\_\{1\}\\in D\_\{1\}, we haveI⊂D2I\\subset D\_\{2\}\. On the regions1<s1Ds\_\{1\}<s\_\{1\}^\{D\},[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)impliesh2∗​\(g​\(s1,h1;A1\)\)=0h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)=0for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\]\. Equivalently, onII,Π2∗​\(s2,A2\)=12​M​\(s2,0;A2\)\\Pi\_\{2\}^\{\*\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}M\(s\_\{2\},0;A\_\{2\}\), and hence

Π2∗⁣′​\(s2,A2\)=12​\[λ​R​π2−b​s2b−1\]\.\\displaystyle\\Pi\_\{2\}^\{\*\\prime\}\(s\_\{2\},A\_\{2\}\)=\\frac\{1\}\{2\}\\bigl\[\\lambda R\\pi\_\{2\}\-b\\,s\_\{2\}^\{b\-1\}\\bigr\]\.Moreover, on this terminal corner,V¯2​\(s2,A2\)=s2b\+β​\(g​\(s2,0;A2\)\)b\+η​log⁡2\\bar\{V\}\_\{2\}\(s\_\{2\},A\_\{2\}\)=s\_\{2\}^\{b\}\+\\beta\(g\(s\_\{2\},0;A\_\{2\}\)\)^\{b\}\+\\eta\\log 2, soV¯2′​\(s2,A2\)\\bar\{V\}\_\{2\}^\{\\prime\}\(s\_\{2\},A\_\{2\}\)does not depend onη\\eta\.

Writeg1≡g​\(s1,h1;A1\)g\_\{1\}\\equiv g\(s\_\{1\},h\_\{1\};A\_\{1\}\)andgh≡\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)\>0g\_\{h\}\\equiv\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\)\>0\. The single firm’s period\-1 marginal payoff is

F1sf​\(h1\)=∂M​\(s1,h1;A1\)∂h1\+β​\[λ​R​π2−b​g1b−1\]​gh\.\\displaystyle F\_\{1\}^\{\\mathrm\{sf\}\}\(h\_\{1\}\)=\\frac\{\\partial M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+\\beta\\bigl\[\\lambda R\\pi\_\{2\}\-b\\,g\_\{1\}^\{b\-1\}\\bigr\]g\_\{h\}\.By[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1),h1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)is characterized by the Karush–Kuhn–Tucker conditions for this strictly concave single\-firm problem\.

Using the display forΠ2∗⁣′\\Pi\_\{2\}^\{\*\\prime\}above in the definition ofF1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\), we obtain

F1​\(h1;s1,A1,A2\)=1η​\[β4​V¯2′​\(g1,A2\)​gh​M​\(s1,h1;A1\)\]\+14​F1sf​\(h1\)\+14​∂M​\(s1,h1;A1\)∂h1\.\\displaystyle F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)=\\frac\{1\}\{\\eta\}\\left\[\\frac\{\\beta\}\{4\}\\bar\{V\}\_\{2\}^\{\\prime\}\(g\_\{1\},A\_\{2\}\)g\_\{h\}M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\right\]\+\\frac\{1\}\{4\}F\_\{1\}^\{\\mathrm\{sf\}\}\(h\_\{1\}\)\+\\frac\{1\}\{4\}\\frac\{\\partial M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}\.The term in square brackets is continuous on\[0,1\]\[0,1\], nonnegative, and independent ofη\\eta\. Let

KR≜maxh1∈\[0,1\]⁡β4​V¯2′​\(g​\(s1,h1;A1\),A2\)​gh​M​\(s1,h1;A1\)<∞\.\\displaystyle K\_\{R\}\\triangleq\\max\_\{h\_\{1\}\\in\[0,1\]\}\\frac\{\\beta\}\{4\}\\bar\{V\}\_\{2\}^\{\\prime\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),A\_\{2\}\)g\_\{h\}M\(s\_\{1\},h\_\{1\};A\_\{1\}\)<\\infty\.BecauseI⊂D2I\\subset D\_\{2\}is compact andη¯​\(s2\)\\bar\{\\eta\}\(s\_\{2\}\)is continuous onD2D\_\{2\},maxs2∈I⁡η¯​\(s2\)<∞\\max\_\{s\_\{2\}\\in I\}\\bar\{\\eta\}\(s\_\{2\}\)<\\infty\. Define

η2​\(s1\)≜max⁡\{η1​\(s1\),2​maxs2∈I⁡η¯​\(s2\),8​KRλ​R​δ​\(1−π1\)​\(α1−s1\)\}\.\\displaystyle\\eta\_\{2\}\(s\_\{1\}\)\\triangleq\\max\\left\\\{\\eta\_\{1\}\(s\_\{1\}\),\\;2\\max\_\{s\_\{2\}\\in I\}\\bar\{\\eta\}\(s\_\{2\}\),\\;\\frac\{8K\_\{R\}\}\{\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)\}\\right\\\}\.Thenη2​\(s1\)≥η1​\(s1\)\\eta\_\{2\}\(s\_\{1\}\)\\geq\\eta\_\{1\}\(s\_\{1\}\)\. For everyη≥η2​\(s1\)\\eta\\geq\\eta\_\{2\}\(s\_\{1\}\), the inequalityη\>η¯​\(s2\)\\eta\>\\bar\{\\eta\}\(s\_\{2\}\)holds for everys2∈Is\_\{2\}\\in I, so terminal engagement is zero throughout the reachable continuation set\. Thus the no\-terminal\-engagement case of[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)applies\.

For suchη\\eta, since

∂M​\(s1,h1;A1\)∂h1=−λ​R​δ​\(1−π1\)​\(α1−s1\),\\displaystyle\\frac\{\\partial M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}=\-\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\),we have, for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\],

KRη\+14​∂M​\(s1,h1;A1\)∂h1≤λ​R​δ​\(1−π1\)​\(α1−s1\)8−λ​R​δ​\(1−π1\)​\(α1−s1\)4<0\.\\displaystyle\\frac\{K\_\{R\}\}\{\\eta\}\+\\frac\{1\}\{4\}\\frac\{\\partial M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}\\leq\\frac\{\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)\}\{8\}\-\\frac\{\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)\}\{4\}<0\.Therefore

F1​\(h1;s1,A1,A2\)<14​F1sf​\(h1\)∀h1∈\[0,1\]\.\\displaystyle F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)<\\frac\{1\}\{4\}F\_\{1\}^\{\\mathrm\{sf\}\}\(h\_\{1\}\)\\qquad\\forall h\_\{1\}\\in\[0,1\]\.
We now compare the equilibrium conditions\. Ifh1sf​\(s1\)∈\(0,1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\\in\(0,1\), thenF1sf​\(h1sf​\(s1\)\)=0F\_\{1\}^\{\\mathrm\{sf\}\}\(h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\)=0, soF1​\(h1sf​\(s1\);s1,A1,A2\)<0F\_\{1\}\(h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\);s\_\{1\},A\_\{1\},A\_\{2\}\)<0\. Since[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)gives thatF1​\(h1;s1,A1,A2\)F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)is strictly decreasing inh1h\_\{1\}, the symmetric equilibrium under mobility must lie strictly belowh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\. Indeed,h1∗​\(s1\)=1h\_\{1\}^\{\*\}\(s\_\{1\}\)=1is impossible becauseF1​\(1;s1,A1,A2\)<0F\_\{1\}\(1;s\_\{1\},A\_\{1\},A\_\{2\}\)<0, and any interior root ofF1​\(h1;s1,A1,A2\)=0F\_\{1\}\(h\_\{1\};s\_\{1\},A\_\{1\},A\_\{2\}\)=0must occur to the left ofh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\.

Ifh1sf​\(s1\)=0h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=0, the single firm’s KKT condition givesF1sf​\(0\)≤0F\_\{1\}^\{\\mathrm\{sf\}\}\(0\)\\leq 0\. HenceF1​\(0;s1,A1,A2\)<0F\_\{1\}\(0;s\_\{1\},A\_\{1\},A\_\{2\}\)<0, and[Proposition6](https://arxiv.org/html/2606.29111#Thmproposition6)impliesh1∗​\(s1\)=0=h1sf​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)=0=h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\. Ifh1sf​\(s1\)=1h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)=1, thenh1∗​\(s1\)≤1=h1sf​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)\\leq 1=h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)trivially\. Thus, for allη≥η2​\(s1\)\\eta\\geq\\eta\_\{2\}\(s\_\{1\}\),h1∗​\(s1\)≤h1sf​\(s1\)h\_\{1\}^\{\*\}\(s\_\{1\}\)\\leq h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\), with strict inequality wheneverh1sf​\(s1\)∈\(0,1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)\\in\(0,1\)\.*Q\.E\.D\.*

*Derivation of the cross\-partial in[Section5\.5](https://arxiv.org/html/2606.29111#S5.SS5)\.*Writexxandyyfor the own and the other firm’s actions,P​\(x\)≜Π2∗​\(g​\(s1,x;A1\),A2\)P\(x\)\\triangleq\\Pi\_\{2\}^\{\*\}\(g\(s\_\{1\},x;A\_\{1\}\),A\_\{2\}\),Δ​\(x,y\)≜M​\(s1,x;A1\)\+β​P​\(x\)−β​P​\(y\)\\Delta\(x,y\)\\triangleq M\(s\_\{1\},x;A\_\{1\}\)\+\\beta P\(x\)\-\\beta P\(y\), andu​\(x,y\)≜β​\[V¯2​\(g​\(s1,y;A1\),A2\)−V¯2​\(g​\(s1,x;A1\),A2\)\]/ηu\(x,y\)\\triangleq\\beta\[\\bar\{V\}\_\{2\}\(g\(s\_\{1\},y;A\_\{1\}\),A\_\{2\}\)\-\\bar\{V\}\_\{2\}\(g\(s\_\{1\},x;A\_\{1\}\),A\_\{2\}\)\]/\\eta, so thatσ1j=\(1\+eu\)−1\\sigma\_\{1\}^\{j\}=\(1\+e^\{u\}\)^\{\-1\}andΦ1j=σ1j​Δ\+β​P​\(y\)\\Phi\_\{1\}^\{j\}=\\sigma\_\{1\}^\{j\}\\Delta\+\\beta P\(y\)\. Then∂Φ1j/∂x=σx​Δ\+σ1j​\[∂M/∂h1\+β​P′​\(x\)\]\\partial\\Phi\_\{1\}^\{j\}/\\partial x=\\sigma\_\{x\}\\Delta\+\\sigma\_\{1\}^\{j\}\[\\partial M/\\partial h\_\{1\}\+\\beta P^\{\\prime\}\(x\)\], and

∂2Φ1j∂x​∂y=σx​y​Δ−σx​β​P′​\(y\)\+σy​\[∂M​\(s1,x;A1\)∂h1\+β​P′​\(x\)\]\.\\displaystyle\\frac\{\\partial^\{2\}\\Phi\_\{1\}^\{j\}\}\{\\partial x\\,\\partial y\}=\\sigma\_\{xy\}\\,\\Delta\-\\sigma\_\{x\}\\,\\beta P^\{\\prime\}\(y\)\+\\sigma\_\{y\}\\,\\Bigl\[\\frac\{\\partial M\(s\_\{1\},x;A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+\\beta P^\{\\prime\}\(x\)\\Bigr\]\.Becauseuuis additively separable in\(x,y\)\(x,y\),ux​y=0u\_\{xy\}=0, and at a symmetric profileu=0u=0, where the logistic satisfiesσ1j=12\\sigma\_\{1\}^\{j\}=\\tfrac\{1\}\{2\}andd2​σ/d​u2=0d^\{2\}\\sigma/du^\{2\}=0; henceσx​y=0\\sigma\_\{xy\}=0there\. Moreover,σx=−σy=\(β/4​η\)​V¯2′​\(g​\(s1,h1;A1\)\)​gh\\sigma\_\{x\}=\-\\sigma\_\{y\}=\(\\beta/4\\eta\)\\bar\{V\}\_\{2\}^\{\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\bigr\)g\_\{h\}at the symmetric profile, withgh=\(1−π1\)​\(ϕ​π1\+γ\)​\(α1−s1\)g\_\{h\}=\(1\-\\pi\_\{1\}\)\(\\phi\\pi\_\{1\}\+\\gamma\)\(\\alpha\_\{1\}\-s\_\{1\}\), andP′=Π2∗⁣′​\(g​\(s1,h1;A1\),A2\)​ghP^\{\\prime\}=\\Pi\_\{2\}^\{\*\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),A\_\{2\}\\bigr\)g\_\{h\}\. Substituting,

∂2Φ1j∂x​∂y\|x=y=h1=−β4​η​V¯2′​\(g​\(s1,h1;A1\)\)​gh​\[∂M​\(s1,h1;A1\)∂h1\+2​β​Π2∗⁣′​\(g​\(s1,h1;A1\),A2\)​gh\],\\displaystyle\\frac\{\\partial^\{2\}\\Phi\_\{1\}^\{j\}\}\{\\partial x\\,\\partial y\}\\bigg\|\_\{x=y=h\_\{1\}\}=\-\\frac\{\\beta\}\{4\\eta\}\\bar\{V\}\_\{2\}^\{\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\bigr\)g\_\{h\}\\left\[\\frac\{\\partial M\(s\_\{1\},h\_\{1\};A\_\{1\}\)\}\{\\partial h\_\{1\}\}\+2\\beta\\,\\Pi\_\{2\}^\{\*\\prime\}\\\!\\bigl\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\),A\_\{2\}\\bigr\)g\_\{h\}\\right\],and inserting∂M/∂h1=−λ​R​δ​\(1−π1\)​\(α1−s1\)\\partial M/\\partial h\_\{1\}=\-\\lambda R\\delta\(1\-\\pi\_\{1\}\)\(\\alpha\_\{1\}\-s\_\{1\}\)and the value ofghg\_\{h\}yields the expression displayed in[Section5\.5](https://arxiv.org/html/2606.29111#S5.SS5)\.*Q\.E\.D\.*

## OA\.3Skill\-Transition Policy Without Worker Mobility

The single firm’s optimal period\-1 engagement policy, characterized in[Proposition1](https://arxiv.org/html/2606.29111#Thmproposition1), takes the firm’s target period\-2 skills∗s^\{\*\}and engages each worker just enough to reach it\.[TableOA2](https://arxiv.org/html/2606.29111#S3.T2)restates that policy as a partition of the initial\-skill range by the thresholdssLs\_\{L\}andsHs\_\{H\}of[eq\.4](https://arxiv.org/html/2606.29111#S4.E4): workers belowsLs\_\{L\}are engaged fully, workers abovesHs\_\{H\}are left alone, and workers in between are held exactly ats∗s^\{\*\}\.

Table OA2:Skill transition under optimal period\-1 engagement without worker mobilityTargeting regionInitial skills1s\_\{1\}Engagementh1sf​\(s1\)h\_\{1\}^\{\\mathrm\{sf\}\}\(s\_\{1\}\)Period\-2 skills2s\_\{2\}Full engagements1≤sLs\_\{1\}\\leq s\_\{L\}11g​\(s1,1;A1\)≤s∗g\(s\_\{1\},1;A\_\{1\}\)\\leq s^\{\*\}Partial engagementsL<s1<sHs\_\{L\}<s\_\{1\}<s\_\{H\}∈\(0,1\)\\in\(0,1\)s∗s^\{\*\}No engagements1≥sHs\_\{1\}\\geq s\_\{H\}0g​\(s1,0;A1\)≥s∗g\(s\_\{1\},0;A\_\{1\}\)\\geq s^\{\*\}
- •Notes\.s∗s^\{\*\}is the target period\-2 skill;sL≤sHs\_\{L\}\\leq s\_\{H\}are the initial\-skill thresholds defined in[eq\.4](https://arxiv.org/html/2606.29111#S4.E4); andg​\(s1,h1;A1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)is the skill\-transition map at period\-1 AI stateA1A\_\{1\}\. Equality in the last column holds at the thresholdss1=sLs\_\{1\}=s\_\{L\}ands1=sHs\_\{1\}=s\_\{H\}\.

## OA\.4Coverage of the Terminal Single\-Crossing Domain

The power\-wage analysis in the main text shows that the terminal single\-crossing condition holds on an upper\-tail domainD2=\(s¯¯2,α2\)D\_\{2\}=\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)\. To gauge the size of this restriction, we report two measures\. The first is the share of the full below\-benchmark interval covered byD2D\_\{2\},

α2−s¯¯2α2\.\\frac\{\\alpha\_\{2\}\-\\bar\{\\bar\{s\}\}\_\{2\}\}\{\\alpha\_\{2\}\}\.The second is the share of the smooth terminal domain covered byD2D\_\{2\},

α2−s¯¯2α2−s¯2,s¯2=γ​\(1−π2\)​α21\+γ​\(1−π2\)\.\\frac\{\\alpha\_\{2\}\-\\bar\{\\bar\{s\}\}\_\{2\}\}\{\\alpha\_\{2\}\-\\underline\{s\}\_\{2\}\},\\qquad\\underline\{s\}\_\{2\}=\\frac\{\\gamma\(1\-\\pi\_\{2\}\)\\alpha\_\{2\}\}\{1\+\\gamma\(1\-\\pi\_\{2\}\)\}\.The first measure is the intuitive coverage of below\-benchmark workers; the second isolates the additional restriction imposed by the single\-crossing condition, after the positivity requirement has already removed the interval belows¯2\\underline\{s\}\_\{2\}\.

Table OA3:Size of the terminal domainD2D\_\{2\}AI capabilityα2\\alpha\_\{2\}Cutoffs¯¯2\\bar\{\\bar\{s\}\}\_\{2\}Share of\(0,α2\)\(0,\\alpha\_\{2\}\)Share of\(s¯2,α2\)\(\\underline\{s\}\_\{2\},\\alpha\_\{2\}\)0\.500\.500\.1360\.13672\.8%72\.8\\%87\.0%87\.0\\%0\.600\.600\.1630\.16372\.8%72\.8\\%87\.0%87\.0\\%0\.700\.700\.1900\.19072\.9%72\.9\\%87\.1%87\.1\\%0\.800\.800\.2170\.21772\.9%72\.9\\%87\.1%87\.1\\%0\.900\.900\.2440\.24472\.9%72\.9\\%87\.1%87\.1\\%0\.950\.950\.2570\.25772\.9%72\.9\\%87\.1%87\.1\\%
Notes\.The table usesB​\(s\)=W​\(s\)=s1\.2B\(s\)=W\(s\)=s^\{1\.2\},π2=0\.35\\pi\_\{2\}=0\.35,ϕ=0\.5\\phi=0\.5,γ=0\.3\\gamma=0\.3,δ=0\.5\\delta=0\.5, andλ​R=3\.0\\lambda R=3\.0\. For each value ofα2\\alpha\_\{2\}, the cutoffs¯¯2\\bar\{\\bar\{s\}\}\_\{2\}solves[eq\.OA2](https://arxiv.org/html/2606.29111#S2.E2)at equality\.

Across these capability levels, the admissible terminal domain covers about three\-quarters of the full below\-benchmark interval and about seven\-eighths of the smooth terminal domain\. Thus the single\-crossing condition does not restrict attention to a narrow slice near the AI frontier; it removes the lowest\-skill region where the curvature term is largest, while leaving a broad upper band of below\-benchmark workers\.

## OA\.5Coverage of the Period\-1 Skill Domains

This appendix quantifies the period\-1 skill restrictions used in[Section5\.4](https://arxiv.org/html/2606.29111#S5.SS4)\. The main text restricts attention to initial skillss1∈D1s\_\{1\}\\in D\_\{1\}so that every continuation state reachable under feasible period\-1 engagement lies in the terminal domainD2D\_\{2\}\. It then further studies the regionss1<s1Ds\_\{1\}<s\_\{1\}^\{D\}ands1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}, where terminal engagement is fixed along the whole period\-1 skill trajectory\. The purpose of these restrictions is to avoid terminal\-policy kinks in the period\-1 objective, not to generate the mobility mechanism\.

We first record the explicit cutoffs\. The terminal\-period equilibrium characterization applies onD2=\(s¯¯2,α2\)D\_\{2\}=\(\\bar\{\\bar\{s\}\}\_\{2\},\\alpha\_\{2\}\)\. Becauseg​\(s1,h1;A1\)g\(s\_\{1\},h\_\{1\};A\_\{1\}\)is increasing inh1h\_\{1\}, and because[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)impliesg​\(s1,h1;A1\)<α1≤α2g\(s\_\{1\},h\_\{1\};A\_\{1\}\)<\\alpha\_\{1\}\\leq\\alpha\_\{2\}for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\], every reachable continuation state lies inD2D\_\{2\}as soon asg​\(s1,0;A1\)\>s¯¯2g\(s\_\{1\},0;A\_\{1\}\)\>\\bar\{\\bar\{s\}\}\_\{2\}\. Define

s¯¯1≜s¯¯2\+γ​\(1−π1\)​α11\+γ​\(1−π1\)andD1≜\(max⁡\{s¯,s¯¯1\},α1\)\.\\displaystyle\\bar\{\\bar\{s\}\}\_\{1\}\\triangleq\\frac\{\\bar\{\\bar\{s\}\}\_\{2\}\+\\gamma\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\}\{1\+\\gamma\(1\-\\pi\_\{1\}\)\}\\qquad\\text\{and\}\\qquad D\_\{1\}\\triangleq\\bigl\(\\max\\\{\\underline\{s\},\\bar\{\\bar\{s\}\}\_\{1\}\\\},\\alpha\_\{1\}\\bigr\)\.Fors1∈D1s\_\{1\}\\in D\_\{1\}we then haveg​\(s1,h1;A1\)∈D2g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\in D\_\{2\}for everyh1∈\[0,1\]h\_\{1\}\\in\[0,1\]; only the reachable subset ofD2D\_\{2\}is used, because[Assumption2](https://arxiv.org/html/2606.29111#Thmassumption2)impliesg​\(s1,h1;A1\)<α1≤α2g\(s\_\{1\},h\_\{1\};A\_\{1\}\)<\\alpha\_\{1\}\\leq\\alpha\_\{2\}, so the portion of the terminal domain aboveα1\\alpha\_\{1\}need not be reachable from period 1\.

The two fixed\-terminal thresholds pull the domain\-restricted terminal cutoffss^2D\\hat\{s\}\_\{2\}^\{D\}ands¯2D\\overline\{s\}\_\{2\}^\{D\}of[Corollary3](https://arxiv.org/html/2606.29111#Thmcorollary3)back to period 1:

s1D≜s^2D−ϕ​π1​\(1−π1\)​α11−ϕ​π1​\(1−π1\),s1F≜s¯2D\+γ​\(1−π1\)​α11\+γ​\(1−π1\)\.\\displaystyle s\_\{1\}^\{D\}\\triangleq\\frac\{\\hat\{s\}\_\{2\}^\{D\}\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\}\{1\-\\phi\\pi\_\{1\}\(1\-\\pi\_\{1\}\)\},\\qquad s\_\{1\}^\{F\}\\triangleq\\frac\{\\overline\{s\}\_\{2\}^\{D\}\+\\gamma\(1\-\\pi\_\{1\}\)\\alpha\_\{1\}\}\{1\+\\gamma\(1\-\\pi\_\{1\}\)\}\.\(OA3\)By construction,g​\(s1D,1;A1\)=s^2Dg\(s\_\{1\}^\{D\},1;A\_\{1\}\)=\\hat\{s\}\_\{2\}^\{D\}andg​\(s1F,0;A1\)=s¯2Dg\(s\_\{1\}^\{F\},0;A\_\{1\}\)=\\overline\{s\}\_\{2\}^\{D\}\. Hence, ifs1<s1Ds\_\{1\}<s\_\{1\}^\{D\}, then even full period\-1 engagement leaves the worker below the terminal no\-engagement cutoff:g​\(s1,h1;A1\)≤g​\(s1,1;A1\)<s^2Dg\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\leq g\(s\_\{1\},1;A\_\{1\}\)<\\hat\{s\}\_\{2\}^\{D\}for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\], soh2∗​\(g​\(s1,h1;A1\)\)=0h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)=0for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\]\. Similarly, ifs1\>s1Fs\_\{1\}\>s\_\{1\}^\{F\}, then even zero period\-1 engagement puts the worker above the terminal full\-engagement cutoff:g​\(s1,h1;A1\)≥g​\(s1,0;A1\)\>s¯2Dg\(s\_\{1\},h\_\{1\};A\_\{1\}\)\\geq g\(s\_\{1\},0;A\_\{1\}\)\>\\overline\{s\}\_\{2\}^\{D\}for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\], soh2∗​\(g​\(s1,h1;A1\)\)=1h\_\{2\}^\{\*\}\(g\(s\_\{1\},h\_\{1\};A\_\{1\}\)\)=1for allh1∈\[0,1\]h\_\{1\}\\in\[0,1\]\. After intersection withD1D\_\{1\}, the no\-terminal\-engagement region is nonempty only ifs1D\>max⁡\{s¯,s¯¯1\}s\_\{1\}^\{D\}\>\\max\\\{\\underline\{s\},\\bar\{\\bar\{s\}\}\_\{1\}\\\}and the full\-terminal\-engagement region only ifs1F<α1s\_\{1\}^\{F\}<\\alpha\_\{1\}; if either fails, the corresponding region is vacuous rather than inconsistent\.

We measure the size of the period\-1 terminal\-domain restriction by

ρ1≡\|D1\|α1−s¯=α1−max⁡\{s¯,s¯¯1\}α1−s¯,\\rho\_\{1\}\\equiv\\frac\{\|D\_\{1\}\|\}\{\\alpha\_\{1\}\-\\underline\{s\}\}=\\frac\{\\alpha\_\{1\}\-\\max\\\{\\underline\{s\},\\bar\{\\bar\{s\}\}\_\{1\}\\\}\}\{\\alpha\_\{1\}\-\\underline\{s\}\},wheneverD1D\_\{1\}is nonempty\. Thusρ1\\rho\_\{1\}is the share of the feasible period\-1 skill interval\(s¯,α1\)\(\\underline\{s\},\\alpha\_\{1\}\)covered byD1D\_\{1\}\.

We also measure the size of the two fixed\-terminal regions withinD1D\_\{1\}:

ρ1D≡\|D1∩\(−∞,s1D\)\|\|D1\|,ρ1F≡\|D1∩\(s1F,∞\)\|\|D1\|\.\\rho\_\{1\}^\{D\}\\equiv\\frac\{\|D\_\{1\}\\cap\(\-\\infty,s\_\{1\}^\{D\}\)\|\}\{\|D\_\{1\}\|\},\\qquad\\rho\_\{1\}^\{F\}\\equiv\\frac\{\|D\_\{1\}\\cap\(s\_\{1\}^\{F\},\\infty\)\|\}\{\|D\_\{1\}\|\}\.The first is the share ofD1D\_\{1\}for which every reachable terminal state falls in the no\-terminal\-engagement region; the second is the share for which every reachable terminal state falls in the full\-terminal\-engagement region\. The remaining share,1−ρ1D−ρ1F1\-\\rho\_\{1\}^\{D\}\-\\rho\_\{1\}^\{F\}, is the part ofD1D\_\{1\}where reachable continuation states may cross the terminal interior\-engagement band\.

We compute these shares for the power\-wage caseB​\(s\)=W​\(s\)=sbB\(s\)=W\(s\)=s^\{b\}\. The grid is

b∈\{1\.03,1\.05,1\.08,1\.12\},α1∈\{0\.55,0\.65,0\.75\},α2∈\{0\.70,0\.82,0\.90\},b\\in\\\{1\.03,1\.05,1\.08,1\.12\\\},\\quad\\alpha\_\{1\}\\in\\\{0\.55,0\.65,0\.75\\\},\\quad\\alpha\_\{2\}\\in\\\{0\.70,0\.82,0\.90\\\},π2∈\{0\.25,0\.35,0\.45,0\.55\},π1∈\{0\.45,0\.55,0\.65,0\.75\},\\pi\_\{2\}\\in\\\{0\.25,0\.35,0\.45,0\.55\\\},\\quad\\pi\_\{1\}\\in\\\{0\.45,0\.55,0\.65,0\.75\\\},ϕ∈\{0\.5,1\.2,2\.0\},γ∈\{0\.1,0\.2,0\.3\},δ∈\{0\.05,0\.10,0\.20,0\.50\},\\phi\\in\\\{0\.5,1\.2,2\.0\\\},\\quad\\gamma\\in\\\{0\.1,0\.2,0\.3\\\},\\quad\\delta\\in\\\{0\.05,0\.10,0\.20,0\.50\\\},λ​R∈\{3,5,10,20\},η∈\{0\.2,0\.5,1,2,5,10\},β=0\.95\.\\lambda R\\in\\\{3,5,10,20\\\},\\quad\\eta\\in\\\{0\.2,0\.5,1,2,5,10\\\},\\qquad\\beta=0\.95\.We retain parameter vectors satisfying the model assumptions, the AI path restrictionsα2≥α1\\alpha\_\{2\}\\geq\\alpha\_\{1\}andπ2≤π1\\pi\_\{2\}\\leq\\pi\_\{1\}, the terminal upper\-tail condition in[eq\.OA1](https://arxiv.org/html/2606.29111#S2.E1), and the monotone\-terminal\-policy conditionλ​R​π2≥b​α2b−1\\lambda R\\pi\_\{2\}\\geq b\\alpha\_\{2\}^\{b\-1\}\.

Table OA4:Size of the period\-1 domainD1D\_\{1\}GridNonemptyD1D\_\{1\}Meanρ1\\rho\_\{1\}10th pct\.Median90th pct\.Power\-wage grid95\.3%95\.3\\%72\.9%72\.9\\%36\.0%36\.0\\%81\.0%81\.0\\%95\.5%95\.5\\%
Notes\.ρ1=\|D1\|/\(α1−s¯\)\\rho\_\{1\}=\|D\_\{1\}\|/\(\\alpha\_\{1\}\-\\underline\{s\}\)is the share of the feasible period\-1 skill interval covered byD1D\_\{1\}\. The table reports statistics across admissible grid points\.

Table OA5:Coverage of the fixed\-terminal regions withinD1D\_\{1\}Sorting frictionη\\etaMeanρ1D\\rho\_\{1\}^\{D\}Meanρ1F\\rho\_\{1\}^\{F\}Meanρ1D\+ρ1F\\rho\_\{1\}^\{D\}\+\\rho\_\{1\}^\{F\}Medianρ1D\+ρ1F\\rho\_\{1\}^\{D\}\+\\rho\_\{1\}^\{F\}0\.20\.23\.6%3\.6\\%92\.1%92\.1\\%95\.7%95\.7\\%100\.0%100\.0\\%0\.50\.523\.9%23\.9\\%69\.9%69\.9\\%93\.8%93\.8\\%100\.0%100\.0\\%1\.01\.046\.9%46\.9\\%50\.2%50\.2\\%97\.1%97\.1\\%100\.0%100\.0\\%2\.02\.073\.8%73\.8\\%24\.6%24\.6\\%98\.4%98\.4\\%100\.0%100\.0\\%5\.05\.097\.0%97\.0\\%2\.8%2\.8\\%99\.8%99\.8\\%100\.0%100\.0\\%10\.010\.0100\.0%100\.0\\%0\.0%0\.0\\%100\.0%100\.0\\%100\.0%100\.0\\%
Notes\.The entries are average shares withinD1D\_\{1\}, averaged over admissible grid points withη\\etaheld at the row value\. The no\-terminal region isD1∩\(−∞,s1D\)D\_\{1\}\\cap\(\-\\infty,s\_\{1\}^\{D\}\); the full\-terminal region isD1∩\(s1F,∞\)D\_\{1\}\\cap\(s\_\{1\}^\{F\},\\infty\)\. The fixed\-terminal coverage is the share ofD1D\_\{1\}on which the terminal policy is constant along every feasible period\-1 skill trajectory\.

The period\-1 restrictions leave a sizable skill domain\. In the grid,D1D\_\{1\}is nonempty in nearly all admissible calibrations and covers a median of81\.0%81\.0\\%of the feasible period\-1 skill range\. WithinD1D\_\{1\}, the two fixed\-terminal regions together cover at least93\.8%93\.8\\%of the domain on average for every value ofη\\etareported in the table\. Thus the restrictions mainly exclude the small set of initial skills whose reachable continuation states cross a terminal engagement cutoff; they do not confine the analysis to a narrow slice of workers\.

## OA\.6Robustness of the Reliability Pattern to Alternative Learning Profiles

The reliability comparative static in[Section5\.3](https://arxiv.org/html/2606.29111#S5.SS3)is the result most directly tied to the learning profile in the skill transition\. The baseline model assumes that learning from engagement is proportional toπ​\(1−π\)\\pi\(1\-\\pi\)\. This appendix repeats the numerical exercise in[Figure4](https://arxiv.org/html/2606.29111#S5.F4)under alternative learning profiles\.

We replace the learning component in the skill transition withϕ​ht​ℓ​\(πt\)​\(αt−st\)\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\), so that forst<αts\_\{t\}<\\alpha\_\{t\},

gℓ​\(st,ht;At\)=st\+\[ϕ​ht​ℓ​\(πt\)−γ​\(1−ht\)​\(1−πt\)\]​\(αt−st\)\.g\_\{\\ell\}\(s\_\{t\},h\_\{t\};A\_\{t\}\)=s\_\{t\}\+\\left\[\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\-\\gamma\(1\-h\_\{t\}\)\(1\-\\pi\_\{t\}\)\\right\]\(\\alpha\_\{t\}\-s\_\{t\}\)\.The terminal\-period marginal engagement gain at a symmetric profile is then governed by

F2,ℓ​\(h;s2,π2\)=β2​η​\[γ\+ϕ​ℓ​\(π2\)1−π2\]​B′​\(gℓ​\(s2,h;A2\)\)​M​\(s2,h;A2\),F\_\{2,\\ell\}\(h;s\_\{2\},\\pi\_\{2\}\)=\\frac\{\\beta\}\{2\\eta\}\\left\[\\gamma\+\\frac\{\\phi\\ell\(\\pi\_\{2\}\)\}\{1\-\\pi\_\{2\}\}\\right\]B^\{\\prime\}\\\!\\left\(g\_\{\\ell\}\(s\_\{2\},h;A\_\{2\}\)\\right\)M\(s\_\{2\},h;A\_\{2\}\),which reduces to the baseline expression whenℓ​\(π\)=π​\(1−π\)\\ell\(\\pi\)=\\pi\(1\-\\pi\)\. For each learning profile and each value ofπ2\\pi\_\{2\}, we solve the terminal two\-firm game directly\. When the equilibrium is interior, it solves

F2,ℓ​\(h;s2,π2\)=λ​R​δ\.F\_\{2,\\ell\}\(h;s\_\{2\},\\pi\_\{2\}\)=\\lambda R\\delta\.
The numerical exercise uses the same parameters as[Figure4](https://arxiv.org/html/2606.29111#S5.F4):

B​\(s\)=W​\(s\)=s1\.2,s2=0\.30,α2=0\.90,ϕ=0\.5,γ=0\.3,δ=0\.5,B\(s\)=W\(s\)=s^\{1\.2\},\\quad s\_\{2\}=0\.30,\\quad\\alpha\_\{2\}=0\.90,\\quad\\phi=0\.5,\\quad\\gamma=0\.3,\\quad\\delta=0\.5,λ​R=3\.0,β=0\.95,η=0\.21\.\\lambda R=3\.0,\\quad\\beta=0\.95,\\quad\\eta=0\.21\.To isolate the shape of the learning profile from the scale of learning, each profile below is normalized so that

maxπ∈\[0,1\]⁡ℓ​\(π\)=14,\\max\_\{\\pi\\in\[0,1\]\}\\ell\(\\pi\)=\\frac\{1\}\{4\},the maximum of the baseline productπ​\(1−π\)\\pi\(1\-\\pi\)\. Hence all profiles satisfy the analogous no\-leapfrogging restrictionϕ​ℓ​\(π\)<1\\phi\\ell\(\\pi\)<1\.

Table OA6:Reliability pattern under alternative learning profilesLearning profileℓ​\(π\)\\ell\(\\pi\)Peakπ2\\pi\_\{2\}Peakh2∗h\_\{2\}^\{\*\}Positive\-engagement rangeπ​\(1−π\)\\pi\(1\-\\pi\)0\.5000\.5000\.6740\.674\(0\.132,0\.793\)\(0\.132,0\.793\)π1/2​\(1−π\)\\pi^\{1/2\}\(1\-\\pi\)0\.2530\.2530\.7660\.766\(0\.020,0\.634\)\(0\.020,0\.634\)π3/2​\(1−π\)\\pi^\{3/2\}\(1\-\\pi\)0\.7530\.7530\.8320\.832\(0\.289,0\.935\)\(0\.289,0\.935\)π​\(1−π\)2\\pi\(1\-\\pi\)^\{2\}0\.2960\.2960\.7530\.753\(0\.069,0\.551\)\(0\.069,0\.551\)π2​\(1−π\)2\\pi^\{2\}\(1\-\\pi\)^\{2\}0\.5000\.5000\.6740\.674\(0\.250,0\.685\)\(0\.250,0\.685\)\(0\.05\+π\)​\(1−π\)\(0\.05\+\\pi\)\(1\-\\pi\)0\.4500\.4500\.6780\.678\(0\.069,0\.770\)\(0\.069,0\.770\)π2​\(1−π\)\\pi^\{2\}\(1\-\\pi\)0\.7140\.7141\.0001\.000\(0\.419,0\.950\)\(0\.419,0\.950\)π\\pi0\.8020\.8021\.0001\.000\(0\.725,0\.950\)\(0\.725,0\.950\)1−π1\-\\pi0\.0010\.0010\.9720\.972\(0\.001,0\.355\)\(0\.001,0\.355\)0\.05\+π​\(1−π\)0\.05\+\\pi\(1\-\\pi\)0\.9170\.9171\.0001\.000\(0\.080,0\.950\)\(0\.080,0\.950\)
Notes\.Each profile is normalized to have maximum value1/41/4\. The table reports the peak of the equilibrium engagement curveh2∗​\(s2;π2\)h\_\{2\}^\{\*\}\(s\_\{2\};\\pi\_\{2\}\)and the range ofπ2\\pi\_\{2\}values over which engagement is positive, computed on a gridπ2∈\[0\.001,0\.950\]\\pi\_\{2\}\\in\[0\.001,0\.950\]\. The first six rows are exposure–practice profiles that keep learning limited when AI failures are either very rare or very frequent; all produce a hump\-shaped engagement curve\. The last four rows are extreme profiles\. When learning remains strong at high failure rates, as inπ2​\(1−π\)\\pi^\{2\}\(1\-\\pi\),π\\pi, or0\.05\+π​\(1−π\)0\.05\+\\pi\(1\-\\pi\), the high\-unreliability decline can disappear\. When learning is driven only by AI exposure, as in1−π1\-\\pi, engagement is highest at very high reliability rather than at intermediate reliability\.

The hump\-shaped reliability pattern does not depend on the literal productπ​\(1−π\)\\pi\(1\-\\pi\)\. It persists across a range of exposure–practice learning profiles, including asymmetric profiles that shift the peak toward higher or lower unreliability\. What matters is the economic structure: a small increase in unreliability must initially raise the skill built by engagement, while very frequent failures must eventually make the current operating margin too weak to justify engagement\.

The extreme profiles also clarify the boundary of the result\. If learning is purely failure\-practice based, or if learning remains available even when AI almost never functions, then the skill\-value channel can remain strong at highπ2\\pi\_\{2\}, and the high\-unreliability decline may disappear\. If learning is purely AI\-exposure based, engagement is strongest when AI is most reliable, and the intermediate\-reliability hump is lost\. Thus the nonmonotone reliability result is robust to the exact functional form of learning, but it does rely on the exposure–practice complementarity that motivates the skill transition\.

## OA\.7Robustness of the Timing Reversal to Alternative Learning Profiles

[Example1](https://arxiv.org/html/2606.29111#Thmexample1)shows that mobility can reverse the timing of engagement: the worker receives no engagement in period 1 but full engagement in period 2\. We show that this timing reversal is robust to the choice of learning profile, and does not hinge on the exact productπ​\(1−π\)\\pi\(1\-\\pi\)\.

We replace the learning component in the skill transition withϕ​ht​ℓ​\(πt\)​\(αt−st\)\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\(\\alpha\_\{t\}\-s\_\{t\}\), so that forst<αts\_\{t\}<\\alpha\_\{t\},

gℓ​\(st,ht;At\)=st\+\[ϕ​ht​ℓ​\(πt\)−γ​\(1−ht\)​\(1−πt\)\]​\(αt−st\)\.g\_\{\\ell\}\(s\_\{t\},h\_\{t\};A\_\{t\}\)=s\_\{t\}\+\\left\[\\phi h\_\{t\}\\ell\(\\pi\_\{t\}\)\-\\gamma\(1\-h\_\{t\}\)\(1\-\\pi\_\{t\}\)\\right\]\(\\alpha\_\{t\}\-s\_\{t\}\)\.The erosion term is unchanged\. To isolate the shape of the learning profile from the scale of learning, each alternative profile is normalized so thatmaxπ∈\[0,1\]⁡ℓ​\(π\)=1/4\\max\_\{\\pi\\in\[0,1\]\}\\ell\(\\pi\)=1/4, the maximum of the baseline profileπ​\(1−π\)\\pi\(1\-\\pi\)\. With the parameters in[Example1](https://arxiv.org/html/2606.29111#Thmexample1), this also preserves no\-leapfrogging, sinceϕ​maxπ⁡ℓ​\(π\)=0\.30<1\\phi\\max\_\{\\pi\}\\ell\(\\pi\)=0\.30<1\.

For each profile, we directly solve the two\-period mobility game using the same primitive parameters as in[Example1](https://arxiv.org/html/2606.29111#Thmexample1):

B​\(s\)=W​\(s\)=s1\.05,α1=0\.65,α2=0\.82,π1=0\.45,π2=0\.35,B\(s\)=W\(s\)=s^\{1\.05\},\\quad\\alpha\_\{1\}=0\.65,\\quad\\alpha\_\{2\}=0\.82,\\quad\\pi\_\{1\}=0\.45,\\quad\\pi\_\{2\}=0\.35,ϕ=1\.20,γ=0\.20,β=0\.95,δ=0\.05,λ​R=1\.5,η=2\.0,s1=0\.371\.\\phi=1\.20,\\quad\\gamma=0\.20,\\quad\\beta=0\.95,\\quad\\delta=0\.05,\\quad\\lambda R=1\.5,\\quad\\eta=2\.0,\\quad s\_\{1\}=0\.371\.The table reports the symmetric period\-1 equilibrium and the terminal equilibrium reached from the resulting continuation skill\.

Table OA7:Timing reversal under alternative learning profilesLearning profileℓ​\(π\)\\ell\(\\pi\)ℓ​\(π1\)\\ell\(\\pi\_\{1\}\)ℓ​\(π2\)\\ell\(\\pi\_\{2\}\)h1∗h\_\{1\}^\{\*\}h2∗h\_\{2\}^\{\*\}Timing reversalπ​\(1−π\)\\pi\(1\-\\pi\)0\.2480\.2480\.2270\.227011Yesπ1/2​\(1−π\)\\pi^\{1/2\}\(1\-\\pi\)0\.2400\.2400\.2500\.250011Yesπ3/2​\(1−π\)\\pi^\{3/2\}\(1\-\\pi\)0\.2230\.2230\.1810\.181011Yesπ​\(1−π\)2\\pi\(1\-\\pi\)^\{2\}0\.2300\.2300\.2500\.250011Yesπ2​\(1−π\)2\\pi^\{2\}\(1\-\\pi\)^\{2\}0\.2450\.2450\.2070\.207011Yes\(0\.05\+π\)​\(1−π\)\(0\.05\+\\pi\)\(1\-\\pi\)0\.2490\.2490\.2360\.236011Yes0\.05\+π​\(1−π\)0\.05\+\\pi\(1\-\\pi\)0\.2480\.2480\.2310\.231011Yesπ2​\(1−π\)\\pi^\{2\}\(1\-\\pi\)0\.1880\.1880\.1340\.13400Noπ\\pi0\.1130\.1130\.0880\.08800No1−π1\-\\pi0\.1380\.1380\.1630\.163011Yes
Notes\.Each nonbaseline profile is normalized so thatmaxπ∈\[0,1\]⁡ℓ​\(π\)=1/4\\max\_\{\\pi\\in\[0,1\]\}\\ell\(\\pi\)=1/4\. In every row, the symmetric period\-1 equilibrium ish1∗=0h\_\{1\}^\{\*\}=0, so the worker enters period 2 withs2=gℓ​\(s1,0;A1\)=0\.340s\_\{2\}=g\_\{\\ell\}\(s\_\{1\},0;A\_\{1\}\)=0\.340\. The terminal equilibriumh2∗h\_\{2\}^\{\*\}is then computed at this continuation skill\. A timing reversal occurs whenh1∗=0h\_\{1\}^\{\*\}=0andh2∗=1h\_\{2\}^\{\*\}=1\.

The timing reversal is not tied to the literal productπ​\(1−π\)\\pi\(1\-\\pi\)\. The period\-1 part of the reversal is especially robust: across all profiles in the table, the symmetric period\-1 equilibrium remainsh1∗=0h\_\{1\}^\{\*\}=0\. The terminal part depends on whether engagement builds enough skill at the terminal failure probabilityπ2=0\.35\\pi\_\{2\}=0\.35\. Balanced exposure–practice profiles preserve full terminal engagement and hence preserve the timing reversal\. The reversal disappears in extreme profiles, such as normalizedℓ​\(π\)=π\\ell\(\\pi\)=\\piorℓ​\(π\)=π2​\(1−π\)\\ell\(\\pi\)=\\pi^\{2\}\(1\-\\pi\), that assign too little learning to the terminal reliability level\. Thus the example relies on terminal skill\-building being strong enough to attract workers, but not on the exact baseline product form\.

## OA\.8Robustness to a Pay Bonus

In the baseline model, firms attract workers only through the skill trajectory their jobs build, because the wage scheduleW​\(⋅\)W\(\\cdot\)is set by the market and is the same at both firms\. Firms might also compete on pay\. We show that adding a simple cash instrument leaves our main conclusions intact\. In each periodt∈\{1,2\}t\\in\\\{1,2\\\}, each firmjjnow makes one extra choice alongside its engagementhtj∈\[0,1\]h\_\{t\}^\{j\}\\in\[0,1\]: a binary*bonus*decisionztj∈\{0,1\}z\_\{t\}^\{j\}\\in\\\{0,1\\\}: whether to pay a bonusω≥0\\omega\\geq 0on top of the market wage\. The bonus is pure cash and does not affect how skill evolves, so the skill dynamics of[Section3](https://arxiv.org/html/2606.29111#S3)are unchanged\. It matters in only two ways: it makes the firm more attractive to workers, and it lowers the firm’s per\-worker margin byω\\omega\. Settingω=0\\omega=0returns the model of[Section5](https://arxiv.org/html/2606.29111#S5)\.

In the terminal period, a worker’s value from joining firmjjbecomes

V2j​\(s2,h2j,z2j\)=W​\(s2\)\+ω​z2j\+β​B​\(g​\(s2,h2j;A2\)\),V\_\{2\}^\{j\}\\\!\\left\(s\_\{2\},h\_\{2\}^\{j\},z\_\{2\}^\{j\}\\right\)=W\(s\_\{2\}\)\+\\omega\\,z\_\{2\}^\{j\}\+\\beta\\,B\\\!\\left\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\\right\),and, because the common wage cancels when the worker compares the two firms, firmjj’s share is

σ2j=\[1\+exp⁡\(\{\[ω​z2−j\+β​B​\(g​\(s2,h2−j;A2\)\)\]−\[ω​z2j\+β​B​\(g​\(s2,h2j;A2\)\)\]\}/η\)\]−1\.\\sigma\_\{2\}^\{j\}=\\left\[\\,1\+\\exp\\\!\\Big\(\\big\\\{\[\\omega z\_\{2\}^\{\-j\}\+\\beta B\(g\(s\_\{2\},h\_\{2\}^\{\-j\};A\_\{2\}\)\)\]\-\[\\omega z\_\{2\}^\{j\}\+\\beta B\(g\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\)\]\\big\\\}/\\eta\\Big\)\\right\]^\{\-1\}\.Paying the bonus lowers the margin toM​\(s2,h2j;A2\)−ω​z2jM\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\-\\omega z\_\{2\}^\{j\}, so the firm’s terminal payoff isΠ2j=σ2j​\[M​\(s2,h2j;A2\)−ω​z2j\]\\Pi\_\{2\}^\{j\}=\\sigma\_\{2\}^\{j\}\\,\[\\,M\(s\_\{2\},h\_\{2\}^\{j\};A\_\{2\}\)\-\\omega z\_\{2\}^\{j\}\\,\]\. In the first period, letV¯2​\(⋅,A2\)\\bar\{V\}\_\{2\}\(\\cdot,A\_\{2\}\)andΠ¯2​\(⋅,A2\)\\bar\{\\Pi\}\_\{2\}\(\\cdot,A\_\{2\}\)denote the worker value and common per\-firm profit in the selected terminal\-period equilibrium, which already reflect the terminal bonus decision\. The worker’s first\-period value isV1j=W​\(s1\)\+ω​z1j\+β​V¯2​\(g​\(s1,h1j;A1\),A2\)V\_\{1\}^\{j\}=W\(s\_\{1\}\)\+\\omega z\_\{1\}^\{j\}\+\\beta\\,\\bar\{V\}\_\{2\}\\\!\\left\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\right\), the sharesσ1j\\sigma\_\{1\}^\{j\}follow the same logit, and firmjjchooses\(h1j,z1j\)\(h\_\{1\}^\{j\},z\_\{1\}^\{j\}\)to maximize

Φ1j=σ1j​\[M​\(s1,h1j;A1\)−ω​z1j\+β​Π¯2​\(g​\(s1,h1j;A1\),A2\)\]\+\(1−σ1j\)​β​Π¯2​\(g​\(s1,h1−j;A1\),A2\)\.\\Phi\_\{1\}^\{j\}=\\sigma\_\{1\}^\{j\}\\Big\[\\,M\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\)\-\\omega z\_\{1\}^\{j\}\+\\beta\\,\\bar\{\\Pi\}\_\{2\}\\\!\\left\(g\(s\_\{1\},h\_\{1\}^\{j\};A\_\{1\}\),A\_\{2\}\\right\)\\Big\]\+\\big\(1\-\\sigma\_\{1\}^\{j\}\\big\)\\,\\beta\\,\\bar\{\\Pi\}\_\{2\}\\\!\\left\(g\(s\_\{1\},h\_\{1\}^\{\-j\};A\_\{1\}\),A\_\{2\}\\right\)\.The last term is the free\-riding channel from[Sections5\.4](https://arxiv.org/html/2606.29111#S5.SS4)and[5\.5](https://arxiv.org/html/2606.29111#S5.SS5): if the worker joins the other firm, firmjjearns nothing today but must still attract the worker next period, against the skill the other firm built\.

The numerical extension preserves the main qualitative patterns\. The period\-1 comparison with the single\-firm benchmark is unchanged in the calibration in[FigureOA1](https://arxiv.org/html/2606.29111#S8.F1): first\-period engagement as a function ofη\\etais identical forω∈\{0,0\.05,0\.10\}\\omega\\in\\\{0,0\.05,0\.10\\\}, so the crossing pointη∗\\eta^\{\\ast\}is unaffected\. The terminal comparative statics in[Section5\.3](https://arxiv.org/html/2606.29111#S5.SS3)are also robust in the numerical exercises: greater AI capability still raises engagement, and the hump\-shaped response to reliability remains for moderate bonuses\. The free\-riding result in[Section5\.5](https://arxiv.org/html/2606.29111#S5.SS5)also persists: the asymmetric equilibrium in[Figure5](https://arxiv.org/html/2606.29111#S5.F5), in which one firm builds skill and the other free\-rides, remains an equilibrium even for a bonus worth about44%44\\%of the relevant per\-worker margin; in that equilibrium neither firm pays the bonus\.

![Refer to caption](https://arxiv.org/html/2606.29111v1/x5.png)Figure OA1:The bonus leaves the engagement margin unchanged\. First\-period mobility engagementh1∗h\_\{1\}^\{\\ast\}as a function of the worker\-response frictionη\\eta, for bonus levelsω∈\{0,0\.05,0\.10\}\\omega\\in\\\{0,0\.05,0\.10\\\}; the three curves coincide because the equilibrium bonus is not paid in this region\. Engagement crosses the single\-firm benchmarkh1sfh\_\{1\}^\{\\mathrm\{sf\}\}\(constant inη\\eta\) atη∗\\eta^\{\\ast\}: when workers respond weakly \(largeη\\eta\) mobility engages below the benchmark, as[Corollary4](https://arxiv.org/html/2606.29111#Thmcorollary4)establishes; when they respond strongly \(smallη\\eta\) it engages above the benchmark in this calibration, a numerical instance of the direction the analysis does not prove\. The crossing is unaffected by the bonus\. Parameter values:B​\(s\)=W​\(s\)=s1\.2B\(s\)=W\(s\)=s^\{1\.2\},α1=0\.5\\alpha\_\{1\}=0\.5,α2=0\.6\\alpha\_\{2\}=0\.6,π1=0\.45\\pi\_\{1\}=0\.45,π2=0\.35\\pi\_\{2\}=0\.35,ϕ=0\.5\\phi=0\.5,γ=0\.3\\gamma=0\.3,δ=0\.05\\delta=0\.05,λ​R=3\.65\\lambda R=3\.65,β=0\.95\\beta=0\.95,s1=0\.25s\_\{1\}=0\.25, givingh1sf≈0\.61h\_\{1\}^\{\\mathrm\{sf\}\}\\approx 0\.61andη∗≈6\.23\\eta^\{\\ast\}\\approx 6\.23\.Whether the bonus is worth paying depends on a single tension\. A firm that offers the bonus must pay it to every worker it employs, yet it only attracts the few extra workers it tips its way\. When workers respond weakly to what firms offer \(largeη\\eta\), too few are tipped to justify paying everyone, so firms leave the bonus off; when workers respond strongly \(smallη\\eta\), even a small bonus draws in many, so firms pay it\. Free\-riding and specialization arise only when workers respond weakly, exactly when the bonus goes unused\. The two levers therefore operate in separate regions and never interact\. Consistent with this separation, our numerical search finds asymmetric outcomes only through engagement\-based free\-riding, as in[Section5\.5](https://arxiv.org/html/2606.29111#S5.SS5); in those equilibria both firms make the same bonus choice\. We do not find equilibria in which one firm keeps workers with cash while the other builds skill\.

When workers respond strongly enough for the bonus to be used, both firms pay it\. Because both firms pay, neither gains a relative sorting advantage, so the bonus is largely passed through to workers as extra compensation\. In the numerical exercises, the bonus can substitute for some costly engagement of below\-benchmark workers, but it does not alter the core mechanisms: engagement remains concentrated where worker sorting is most valuable, AI capability and reliability have the same qualitative effects, and the free\-riding equilibrium persists\. Thus allowing firms to vary pay changes the division between cash and engagement in some regions, but it does not overturn the paper’s main engagement logic\.