# Training for Obsolescence? The AI-Driven Education Trap

Andrew Peterson

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Andrew Peterson. Training for Obsolescence? The AI-Driven Education Trap. 2025. hal-05233991

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# Training for Obsolescence?
The AI-Driven Education Trap

● Andrew J. Peterson *

August 27, 2025

## ABSTRACT

Artificial intelligence simultaneously transforms human capital production in schools and its demand in labor markets. Analyzing these effects in isolation can lead to a significant misallocation of educational resources. We model an educational planner whose decision to adopt AI is driven by its teaching productivity, failing to internalize AI's future wage-suppressing effect on those same skills. Our core assumption, motivated by a pilot survey, is that there is a positive correlation between these two effects. This drives our central proposition: this information failure creates a skill mismatch that monotonically increases with AI prevalence. Extensions show the mismatch is exacerbated by the neglect of unpriced non-cognitive skills and by a school's endogenous over-investment in AI. Our findings caution that policies promoting AI in education, if not paired with forward-looking labor market signals, may paradoxically undermine students' long-term human capital, especially if reliance on AI crowds out the development of unpriced non-cognitive skills, such as persistence, that are forged through intellectual struggle.

JEL Codes: J24, J31, O33, D82

## 1 Introduction

As a general-purpose technology, artificial intelligence is poised to simultaneously transform the production of human capital in schools and alter the demand for it in the labor market (Agrawal et al., 2018; Acemoglu and Restrepo, 2018). While the ultimate magnitude of its macroeconomic impact remains a subject of debate (Acemoglu, 2025), these two effects are typically analyzed in isolation. This separation obscures a critical policy dilemma: AI makes certain skills easier to teach, yet these may be the very skills it simultaneously devalues through automation. Consider programming. While AI tutoring systems can make learning to code more accessible and efficient, AI-based coding assistants are also automating routine programming tasks, suppressing the value of those same entry-level skills. This creates a powerful, technology-driven incentive for educational systems to specialize in precisely the wrong areas, generating a systematic skill mismatch.

This paper formalizes how AI's dual impact can generate an economically inefficient skill mismatch stemming from a core information failure. A wedge between the optimal and actual allocation of human capital arises

<footnote>*Assistant Professor (Maître de conférences), University of Poitiers.</footnote>
because educational planners observe AI's immediate, positive effect on teaching productivity, while its future, wage-suppressing effect in the labor market remains unpriced. Because no forward market exists to transmit this future price signal, a decision that is statically optimal for the educational institution leads to a dynamically inefficient outcome. Our central research question is therefore: *How does a naive educational planner, who observes AI's benefits as a teaching tool but not its future substitutive pressure in the labor market, misallocate student learning time, and how does this mismatch evolve with technological progress?*

To formalize this information failure, we develop a two-stage model of skill acquisition and labor market competition, motivated by initial evidence from a pre-registered survey. The framework isolates two opposing channels of AI's influence: its productivity-enhancing effect as an educational input, which we parameterize by $K_e$, and its substitutive effect as labor-replacing capital in the firm's production function, parameterized by $K$. Our analysis hinges on contrasting the choices of two types of planners. We demonstrate that when the educational planner is “naive”, reacting only to current productivity signals without forecasting future wage effects, a perverse outcome emerges when compared to an “informed” planner who anticipates AI's full impact. This behavior, consistent with models of multitasking under misaligned incentives (Holmstrom and Milgrom, 1991), creates a wedge between the institution's private optimum and the social optimum. Under standard concavity assumptions, our central proposition shows that the proliferation of AI capital ($K$) causes a monotonically increasing mismatch between the skills supplied by the education system and those optimally valued by the labor market. The education system systematically over-invests in the very skills AI is making obsolete.

Building on this core framework, we explore several extensions that yield further insights into this skill mismatch. First, we show that misallocation is amplified when we account for unpriced, non-cognitive skills. Because these skills (e.g., persistence) are often jointly produced through the same intellectual effort that educational AI aims to reduce, a naive planner's optimization inadvertently crowds out their development. Second, we endogenize the costly adoption of educational AI and find a bias toward over-investing in technologies that enhance easily measured skills at the expense of these unpriced dimensions. Finally, motivated by debates over skills like coding, where AI may simultaneously increase the utility of basic literacy while devaluing intermediate proficiency, we explore non-monotonic wage effects. We show how a naive planner can fall into “substitution traps,” directing students into a middle-skill bracket that is maximally exposed to automation, whereas a socially optimal plan would target either basic or advanced training.

This paper connects and contributes to four distinct strands of literature. First, we build on the literature of general-purpose technologies to frame AI's dual-use nature (Bresnahan and Trajtenberg, 1995). Second, we ground our assumptions about AI's impact on wages in the task-based framework of labor economics (Acemoglu and Autor, 2011). Third, our modeling of skill acquisition is informed by the economics of education, particularly research on education production functions (Hanushek, 2020). Finally, our model of mismatch, driven by a failure to anticipate future price changes, is conceptually related to classic “cobweb” models (Ezekiel, 1938). Our contribution is distinct, however, as the source of error is not a failure to anticipate the aggregate supply response of other agents, but a failure to connect the dual, cross-domain impacts of a single technological shock.

The remainder of the paper is structured as follows. Section 2 reviews the relevant literature. Section 3 provides the empirical motivation for our core assumptions. Section 4 develops our baseline model to isolate the core education trap, then presents several extensions that analyze the role of non-cognitive skills, endogenous technology adoption, and non-monotonic wage effects. Section 5 discusses policy implications and concludes.
# 2 Related Literature

## 2.1 AI as a General-Purpose Technology and the Mismatch Problem

Our central contribution is to formally connect two bodies of literature that have largely developed in parallel: the impact of artificial intelligence (AI) on labor market demand and its impact on the educational production function. We structure this review to directly motivate our model's core assumptions and extensions. First, we review the literature on AI's substitutive effects on labor to ground our assumptions about wage pressure. Second, we survey the evidence on AI's productivity-enhancing role in education. Finally, we connect our approach to classic models of educational choice under information frictions and motivate the model's key extensions concerning unpriced skills and endogenous technology adoption.

## 2.2 The Labor Market Channel: Skill Substitution and Wage Effects

Research on technological change has progressed from a canonical skill-biased view to a task-based framework that treats jobs as bundles of activities with varying susceptibility to automation (Acemoglu and Autor, 2011). Within this framework, AI acts as a general-purpose technology whose frontier algorithms substitute for routine, codifiable cognition while complementing non-routine analytical, creative, and socio-emotional tasks (Autor, 2015). Formal models decompose AI's impact into a displacement effect, which substitutes for human labor in existing tasks, and a productivity effect, which lowers costs, expands output, and creates new human-centered tasks (Acemoglu and Restrepo, 2019; Bessen, 2019).

Empirical evidence confirms this heterogeneity. Patent-based exposure indices show that tasks vulnerable to current AI are concentrated in clerical, administrative, and certain analytical occupations, whereas tasks intensive in problem-solving or interaction remain comparatively insulated (Webb, 2019; Felten et al., 2021). Studies of industrial robots estimate significant wage losses borne by routine task specialists (Acemoglu and Restrepo, 2020). Collectively, these findings indicate that AI capital exerts downward wage pressure on automatable skills while potentially raising the marginal product of complementary skills. This literature often treats the supply of skills as given or slow-moving, leaving the mechanisms of educational response under-theorized, which directly motivates our model's Assumption 6 (Asymmetric Workplace Substitution).

## 2.3 The Education Channel: Asymmetric Productivity Gains

Within the framework of education production functions (Hanushek, 2020), a rapidly growing literature documents AI's capacity to lift teaching productivity, particularly where learning objectives are structured, rule-based, and easily assessed (Fazlollahi et al., 2022). While the effects of education technology are often mixed (Escueta et al., 2017), compelling evidence for gains in structured domains predates the current AI wave. A meta-analysis found intelligent tutoring systems were already achieving effectiveness nearly on par with one-on-one human tutoring (VanLehn, 2011), a benchmark of high pedagogical effectiveness (Bloom, 1984). Recent advances in generative AI appear to amplify these efficiencies. Randomized trials demonstrate dramatic learning gains from AI tutors in subjects like mathematics and English, sometimes equivalent to one to two years of conventional schooling, in both high-resource and developing-country settings (Kestin et al., 2024; De Simone et al., 2025; Henkel et al., 2024).

However, a crucial strand of this research provides a cautionary note, showing that such pedagogical efficiency does not guarantee deeper or more durable learning (Gerlich, 2025; Carter et al., 2017), and that EdTech has been shown to focus disproportionately on STEM Alam and Mohanty (2022). These heterogeneous returns provide a strong incentive for educators, especially those evaluated on narrow,
quantitative metrics, to expand AI-mediated instruction where its effects are most visible. This empirical regularity supplies the microfoundation for our model's Assumption 2 (*Asymmetric AI Complementarity in Education*), which posits that AI enhances the marginal product of teaching time for certain kinds of skills more than others.

## 2.4 Modeling Educational Choice under Uncertainty and Information Frictions

The core friction in our model is rooted in the challenge of making human-capital decisions amidst rapid technological change, a modern incarnation of the classic “race between education and technology” (Goldin and Katz, 2008; Autor et al., 2020). Skill mismatch, that is the divergence between competencies supplied by education and those demanded by firms, carries sizable welfare losses (Quintini, 2011; Brunello and Wruuck, 2021). Classic “cobweb” models show how enrollment decisions that naïvely extrapolate current wage premia produce cyclical skill imbalances (Freeman, 1976; Ryoo and Rosen, 2004). This problem is one of decision-making under severe uncertainty, where point forecasts of future returns are unreliable guides for policy (Manski, 2004; Frank et al., 2019). Our model builds on this cobweb tradition but identifies a critically different source of friction. In the classic framework, the forecasting error stems from a failure to anticipate the *endogenous supply response* of other agents to a public price signal. By contrast, the mechanism we propose is driven by a failure to anticipate the dual, cross-domain consequences of a single technological shock. The planner’s error is not a failure to predict the actions of their peers, but a failure to connect AI’s immediate, positive productivity effect in education with its future, negative substitution effect in the labor market. This behavioral assumption is consistent with institutional incentives: educational planners are often evaluated on immediate, measurable learning outcomes (e.g., test scores), making them rationally responsive to local productivity signals while underweighting distant and unpriced labor-market externalities. This distinction motivates our central modeling choice: contrasting a planner who reacts only to the immediate productivity signal with a fully-informed planner who anticipates AI’s complete general equilibrium impact on wages. The divergence in their choices provides a novel channel for how a general-purpose technology can systematically generate human capital misallocation, a friction distinct from those in the canonical literature.

## 2.5 Displacement of Non-Cognitive Skills as a Negative Externality

Our first extension introduces a critical, often unpriced, dimension of human capital, reflecting a foundational tension in educational philosophy: the risk of prioritizing narrow vocational training over the holistic development of citizens (Dewey, 2024). More recent literature confirms that the social value of education extends far beyond wages. Schooling generates substantial non-pecuniary benefits, including better health, more stable families, and higher levels of social trust, that represent a significant, if unpriced, return on educational investment (Oreopoulos and Salvanes, 2011).

The economic literature has documented the immense and growing labor market value of these non-cognitive skills (Deming, 2017; Lundberg, 2017), which are often poorly measured and inadequately incentivized within education systems that prioritize cognitive assessments (Heckman and Kautz, 2012, 2013). The intense focus on AI-teachable skills (our skill A) thus risks displacing the development of these competencies, creating a negative externality by degrading the production of a valuable social good with proven long-run benefits (Jackson, 2018; Peterson, 2025).

Recent research provides a specific mechanism for this, showing that reliance on AI can promote “cognitive offloading,” which undermines the deep processing required for durable learning (Jose et al., 2025). For instance, evidence suggests that using AI as a “crutch” for problem-solving can inhibit the development of
underlying conceptual understanding, with students performing worse when the tool is removed (Kosmyna et al., 2025). This aligns with findings that frequent AI tool usage can be negatively correlated with critical thinking abilities, mediated by this increase in cognitive offloading (Gerlich, 2025). Consequently, the net effect of AI may depend on whether it is used to develop metacognitive skills like self-regulation or merely to find efficient shortcuts (Zhou et al., 2024), that might even harm non-cognitive skills. This dynamic provides a clear illustration of a classic multitasking problem (Holmstrom and Milgrom, 1991): when a planner's focus is drawn to an easily measured task (teaching skill A), the introduction of a technology that boosts its productivity will predictably divert resources from other valuable but unmeasured tasks. The extension in Section 4.4 of our model formalizes how this distortion amplifies the social cost of the initial mismatch.

## 2.6 Endogenous Technology Adoption and Directed Change

This threat to unpriced virtues becomes particularly acute when we consider the active, costly choice of AI adoption by educational institutions. We frame this within the logic of principal-agent problems, where the decision-maker (e.g., a school administrator) optimizes over a narrow set of objectives. A school leader, assessing which tasks have high suitability for machine learning (Brynjolfsson et al., 2018), may choose to adopt an AI teaching tool based on its documented ability to improve test scores in skill A. However, this ignores negative externalities on the development of other, unpriced skills (for contextual evidence on unintended consequences of rapid ed-tech adoption, see West et al. 2023). This logic mirrors the framework of directed technical change (Acemoglu, 2002), where innovation is endogenously directed toward factors with higher perceived (and privately captured) profitability, even if this leads to socially suboptimal outcomes. This underpins an extension of our model (Section 4.5) where the school's chosen AI intensity ($a^*$) is inefficiently high because it ignores this collateral social cost, a specific case of the general finding that decentralized adoption of transformative technologies with unpriced risks tends to be too fast (Acemoglu and Lensman, 2024).

## 2.7 Non-Monotonic Returns and Within-Skill Polarization

Finally, our extension of non-monotonic skill returns refines the canonical task-based model of labor market polarization (Acemoglu and Autor, 2011). While existing models effectively explain the 'hollowing out' of the occupational structure by skill level, our extension examines how automation can create a "barbell effect" within a skill category. Recent theory provides a direct foundation for this mechanism; for instance, Acemoglu and Loebbing (2022) show that when automation targets tasks of "interior" or medium complexity, it naturally generates polarization by pushing human labor to the simplest and most advanced tasks. This aligns with firm-level evidence demonstrating that generative AI disproportionately boosts the productivity of lower-skilled agents while having little effect on top performers, effectively creating a non-linear return to skill within the same role (Brynjolfsson et al., 2023). Our model formalizes this dynamic, showing how a naive planner might inefficiently over-produce for the middle-skill tier.

# 3 Empirical Motivation

To motivate the core mechanism of our model, we present evidence from a small, pre-registered pilot survey. The goal is to provide prima facie evidence for our model's central tension: that the skills for which AI most enhances teaching (our 'education channel,' Assumption 2) are also the most substitutable by AI in the workplace (our 'workplace substitution channel,' Assumption 6). A convenience sample of 20 educators rated the extent to which AI will improve teaching efficiency for a set of 194 skills drawn
primarily from O*NET (National Center for O*NET Development, 2025). Participants each rated eight randomly-assigned skills on a 5-point scale.<fnref n="1" /> To create a proxy for future AI-driven skill substitution in the workplace, we develop an ‘AI Skill Disruption Index.’ Our approach joins a recent literature that uses Large Language Models (LLMs) for labor market analysis (Eloundou et al., 2024), but is distinct in both its aim and methodology. Conceptually, our index is designed to be *forward-looking*, ranking skills by their substitutability over the next five-to-ten years, rather than by current task exposure. Methodologically, we construct the index using an Elo-style rating system designed to produce a more robust rank ordering than direct scoring. While LLM-based measures reflect public information and likely fail to capture some labor market dynamics, they provide a scalable and consistent proxy for widespread expectations. Full details on our pre-registration and instruments are in Appendix C and our OSF repository.

Our first pre-registered hypothesis ($H_1$) posits a positive correlation between a skill's amenability to AI-assisted teaching and its vulnerability to AI-driven substitution. We test this using the pre-specified Kendall's $\tau_b$ rank correlation on the 90 skills for which we have educator ratings. We find a moderate, positive, and statistically significant association ($\tau_b = 0.377, p < 0.001$). This result indicates that the skills receiving the highest ratings for AI-assisted teaching are also the ones our index identifies as most exposed to workplace automation.

<figure><img src="image_2.png" /><figcaption>Figure 1: Correlation between Perceived AI Teaching Impact and AI Skill Disruption</figcaption></figure>

*Notes*: Each point represents one of 90 unique skills. The y-axis shows the mean rating from our survey ('AI Teaching Impact'); the x-axis shows our LLM-derived workplace 'AI Skill Disruption Index.' The line shows the OLS fit (however inference is based on Kendall's $\tau_b$, not the OLS slope.) The pre-specified Kendall's $\tau_b$ rank correlation for these data is 0.377, *p* < 0.001.

Figure 1 illustrates this positive relationship. The association is robust across a comprehensive set of pre-registered and supplementary checks detailed in D. The correlation holds when using alternative specifications (Pearson's *r*, Spearman's $\rho$), is not driven by influential outliers, and remains significant after

<footnote n="1">¹The options ranged from (a) significant negative effect (30% or greater reduction in efficiency) to (e) significant positive effect (30% or greater improvement), along with 'no significant change' / 'no answer'.</footnote>
including skill-category fixed effects. Finally, to assess if this relationship holds across the broader skill space, we extend the analysis to the full set of 194 skills. We address the missing survey data for 104 skills using multiple imputation (m = 25), based on an LLM-derived education impact index, with details provided in Appendix D.1. Pooling the results confirms a statistically significant, positive association (pooled $\tau_b$ = 0.295, *p* < 0.001), suggesting the finding is not an artifact of the specific skills rated by our sample.

To account for the non-independence of individual ratings within our survey, we supplement our main skill-level analysis with a rating-level OLS regression. As detailed in Appendix Table 4, the positive association between our indices remains strong and significant ($\beta$ = 0.231, *p* < 0.001) after including skill-type fixed effects and using standard errors clustered by respondent and separately, two-way clustered by respondent and skill.<fnref n="2" />

Our second pre-registered hypothesis ($H_2$) posits that educators perceive AI as less effective for, or potentially even harmful to, teaching non-cognitive skills (e.g., perseverance, ethical reasoning). To test this formally while accounting for the fact that each educator rated multiple skills (so the ratings are not independent), we use a within-educator analysis.<fnref n="3" /> Wilcoxon signed-rank test, using the *n* = 20 respondents who rated at least one skill in both categories. The result is statistically significant and confirms our hypothesis: educators rate AI's teaching effectiveness significantly lower for non-cognitive skills (*W* = 22, exact *p* = 0.001). This finding provides the empirical motivation for our model's extension on the under-provision of non-cognitive skills (Assumption 8).

Finally, we investigate a possible mechanism driving the positive correlation in $H_1$. We posit this correlation is explained by a pre-registered distinction between whether a skill is 'closed' (procedural, verifiable) or 'open' (subjective, context-dependent). As detailed in D.6, our analysis supports this: closed skills are perceived as both significantly more amenable to AI-assisted teaching and significantly more vulnerable to workplace automation. This provides a direct empirical basis for our model's central claim that the 'education' (Assumption 2) and 'substitution' (Assumption 6) channels may operate strongly on the same set of skills. It also clarifies our $H_2$ result, insofar as educators may find AI less useful in teaching non-cognitive skills that are useful for solving open tasks.<fnref n="4" />

While exploratory, these findings suggest the tension we model is not a theoretical curiosity but is rooted in plausible perceptions about AI's dual role in education and the workplace. The very skills that AI is perceived to be the most useful in helping to teach (procedural, 'closed' skills) appear to be the same ones most susceptible to workplace automation. This evidence thus motivates our theoretical exploration of a 'training for obsolescence' trap, where a naive planner's focus on AI's educational benefits creates a mismatch with labor market needs. is small and convenience-based, and our LLM-derived index proxies expectations, not realized outcomes. The evidence should therefore be viewed as suggestive and is intended solely to motivate our modeling choices.

<footnote n="2">²With only 20 respondent clusters, two-way cluster-robust SEs can be anti-conservative. We therefore (i) report two-way clustered SEs (respondent and skill) and (ii) report wild cluster bootstrap *p*-values using Rademacher weights and $B=9,999$ replications (Cameron et al., 2008). Conclusions are unchanged.</footnote>

<footnote n="3">³Our pre-analysis plan specified a pooled Mann-Whitney *U* test on the skill-level data. However, that approach does not account for the non-independence of ratings from the same respondent. We therefore present the more appropriate within-subjects Wilcoxon signed-rank test, which directly addresses this data structure. For transparency, the pre-registered test yielded a descriptively similar (but statistically invalid) conclusion (*U* = 646).</footnote>

<footnote n="4">⁴Conversely, LLMs excel at closed tasks where reasoning steps can be automatically verified. This verifiability allows for powerful training techniques like “process supervision,” where the model is rewarded for each correct step, not just the final outcome (Lightman et al., 2023). This approach has proven highly effective in domains like mathematics, where models can generate and validate their own training data at scale (Trinh et al., 2024).</footnote>
# 4 Model

Our model analyzes the tension between two countervailing effects of AI on skill formation. The first is an *education channel*, where AI enhances the productivity of teaching certain skills. The second is a *workplace substitution channel*, where AI may substitute for those same skills in the labor market. This creates a potential conflict for an educational planner: the skills that become easiest to teach with AI may simultaneously become the least valuable in an AI-rich future.

To formalize this, we construct a two-stage model. In the first stage, an educational planner allocates student time between two skills, A and B. For simplicity, we will speak as though there is a school planner that decides the allocation of student time. To fix ideas, one might think of skill A as routine cognitive tasks (e.g., coding, data entry) and skill B as non-routine interpersonal or creative tasks (e.g., negotiation, critical thinking), but the model is general. The crucial asymmetry is that AI is more complementary to teaching skill A, but also a stronger substitute for it in the workplace. In the second stage, wages are determined by marginal productivity in a competitive labor market. The core of our analysis compares the choices of a “naive” planner, who focuses only on the immediate teaching benefits of AI, with an “informed” planner, who correctly anticipates AI’s wage-suppressing effects. The divergence between their allocations defines the skill mismatch that we analyze.

## 4.1 The Education Stage

We consider a unit-mass continuum of identical students. Each student has a total time endowment T to be allocated between learning skill A ($t_A$) and skill B ($t_B$):

$$t_A + t_B = T, \quad t_A, t_B \ge 0.$$

Skill acquisition is also influenced by the level of educational AI, $K_e$, which we assume is positively linked to the level of AI in the general economy, K, via a strictly increasing and twice continuously differentiable ($C^2$) function $K_e = g(K)$, with $g(0) = 0$ and $g'(K) > 0$.

The resulting skill levels are given by the production functions:

$$A = f_A(t_A, K_e), \quad B = f_B(t_B, K_e),$$

where $f_A$ and $f_B$ are strictly increasing in their respective time inputs and weakly increasing in $K_e$. Given the unit mass of identical students, these individual skill levels also represent the aggregate supplies of skills, $L_A$ and $L_B$.

We first analyze the behavior of a “naive” (or myopic) planner. This planner’s objective is to maximize the value of student skills based on a set of fixed price signals, $p_A$ and $p_B$, which do not change with the general level of AI, K. This behavior can be understood through the lens of a classic multitasking problem (Holmstrom and Milgrom, 1991). This reflects the difficulty of basing incentives on complex future outcomes, which leads to rewards being tied to simpler, measurable proxies like skill outputs valued at current prices. To anchor our analysis, we assume these fixed prices correspond to the true market wages at an initial level of AI capital, $K_0 \ge 0$, such that $p_A = w_A(K_0)$ and $p_B = w_B(K_0)$.<fnref n="5" />

The naive planner therefore solves:

<footnote n="5">⁵The wage functions $w_j(K)$ are formally defined in the following subsection, where we detail the labor market structure.</footnote>
$$ \max_{t_A \in [0, T]} U_{\text{naive}}(t_A; K) := p_A f_A(t_A, g(K)) + p_B f_B(T - t_A, g(K)). $$

Let the solution to this problem be $t_A^*(K)$. The behavior of this planner is governed by the following assumptions:

**Assumption 1 (Uniform Concavity of Skill Production).** For any given level of educational capital $K_e$, the skill production functions $f_A(t_A, K_e)$ and $f_B(t_B, K_e)$ are twice continuously differentiable ($C^2$) and uniformly strictly concave in teaching time. That is, there exists a constant $\lambda > 0$ such that for all $t_A, t_B \in [0, T]$ and all $K \ge 0$:

$$ \frac{\partial^2 f_A}{\partial t_A^2}(t_A, K_e) \le -\lambda \quad \text{and} \quad \frac{\partial^2 f_B}{\partial t_B^2}(t_B, K_e) \le -\lambda. $$

This assumption ensures the naive planner's objective function is strictly concave in $t_A$. The second derivative of the objective, $\partial^2 U_{\text{naive}}/\partial t_A^2 = p_A \frac{\partial^2 f_A}{\partial t_A^2} + p_B \frac{\partial^2 f_B}{\partial t_B^2}$, is therefore bounded above by $-(p_A + p_B)\lambda < 0$. This guarantees a unique solution $t_A^*(K)$ and ensures the non-vanishing second derivative required to apply the Implicit Function Theorem.

**Assumption 2 (Asymmetric AI Complementarity in Education).** For the naive planner, the marginal return to allocating time towards skill A increases with the level of AI capital K. This is formalized by requiring the planner's objective function, $U_{\text{naive}}(t_A; K)$, to have strictly increasing differences in $(t_A, K)$. Given $K_e = g(K)$ with $g'(K) > 0$, this requires that for all $t_A \in [0, T]$ and $K \ge 0$:

$$ \frac{\partial^2 U_{\text{naive}}}{\partial K \partial t_A} = g'(K) \left( p_A \frac{\partial^2 f_A}{\partial t_A \partial K_e} - p_B \frac{\partial^2 f_B}{\partial t_B \partial K_e} \right) > 0. $$

With $p_A, p_B > 0$, this captures the intuition that AI is more complementary to the time spent teaching skill A than skill B, weighted by their fixed prices. The minus sign on the second term arises from the chain rule, as differentiating the term $f_B(T - t_A, K_e)$ with respect to $t_A$ yields $-\partial f_B/\partial t_B$.

**Assumption 3 (Interior Allocation).** We assume the relative prices and marginal productivities at the boundaries are such that the planner always finds it optimal to devote some time to both skills, guaranteeing that the naive planner's optimal choice, $t_A^*(K)$, is in the interior $(0, T)$.<fnref n="6" />

The preceding assumptions ensure that the planner's choice $t_A^*(K)$ is a continuously differentiable function of K. We can therefore analyze its response to changes in AI capital by applying the Implicit Function Theorem to the first-order condition. This yields the standard comparative static result, where the sign of the response is determined by the sign of the objective function's cross-partial derivative:

$$ \frac{dt_A^*}{dK} = - \frac{\partial^2 U_{\text{naive}}/\partial K \partial t_A}{\partial^2 U_{\text{naive}}/\partial t_A^2}. \qquad (1) $$

Under these assumptions, the naive planner perceives a growing advantage in teaching skill A as AI advances. Assumption 2 ensures that the planner's objective function, $U_{\text{naive}}$, has strictly increasing differences in $(t_A, K)$. It follows from Equation (1) that this positive cross-partial derivative, combined with the strict

<footnote n="6">⁶Formally, this requires $\frac{\partial f_B/\partial t_B|_{t_B=T}}{\partial f_A/\partial t_A|_{t_A=0}} < \frac{p_A}{p_B} < \frac{\partial f_B/\partial t_B|_{t_B=0}}{\partial f_A/\partial t_A|_{t_A=T}}$ for all $K \ge 0$. This strong assumption is made for clarity. The model's qualitative results on the direction of mismatch hold on any interval of K where the solutions are interior; if an allocation hits a boundary, the comparative statics simply become weak inequalities thereafter.</footnote>
concavity from Assumption 1, directly implies that the unique optimal allocation $t_A^*(K)$ is strictly increasing in $K$.<fnref n="7" />

## 4.2 The Labor Market and Wage Determination

The skills produced in the education stage, $(L_A, L_B)$, are supplied inelastically to a competitive labor market. A representative firm uses these skills alongside general AI capital $K$ to produce a single output good $Q$:

$$Q = F(L_A, L_B, K),$$

where $F$ is strictly concave in $(L_A, L_B)$ and continuously differentiable. The wage for each skill $j \in \{A, B\}$ is equal to its marginal product:

$$w_j(K) = \frac{\partial F}{\partial L_j}(L_A, L_B, K).$$

In our partial equilibrium analysis, planners take these wage functions $w_j(K)$ as given.<fnref n="8" />

**Assumption 4 (Positive Wages).** For all $K \ge 0$, the equilibrium wage functions are bounded below by a strictly positive constant, i.e., there exists a $\underline{w} > 0$ such that $w_A(K) \ge \underline{w}$ and $w_B(K) \ge \underline{w}$.

This is a mild regularity condition, satisfied if skills A and B are essential inputs in a standard neoclassical production function. It ensures that skills always have positive economic value and that the planner's objective function is well-behaved.

**Assumption 5 (Unique Interior Solution for the Informed Planner).** Let

$$U_{\inf}(t_A; K) := w_A(K) f_A(t_A, g(K)) + w_B(K) f_B(T - t_A, g(K)).$$

For each $K \ge 0$, $U_{\inf}(\cdot; K)$ is strictly concave in $t_A$ and admits a unique optimizer $t_A^*(K) \in (0, T)$. These properties obtain under Assumptions 1 and 4; interiority holds whenever the boundary inequalities in Assumption 3 are satisfied.

The core tension of the model comes from how AI capital $K$ affects these wages.

**Assumption 6 (Asymmetric Workplace Substitution).** In the labor market, AI capital is a stronger substitute for skill A than for skill B. We assume the wage functions $w_j(K)$ are twice continuously differentiable ($C^2$), and that their total derivatives with respect to $K$ satisfy:

$$w'_A(K) < 0 \quad \text{and} \quad w'_B(K) \ge 0 \quad \text{for all } K \ge 0.$$

For consistency with Assumption 4, the condition $w'_A(K) < 0$ requires that $\lim_{K \to \infty} w_A(K) \ge \underline{w}$, which implies that $w'_A(K)$ must approach zero for large $K$.

This captures the central friction of the model: the very technology that enhances educational productivity for skill A simultaneously erodes its value in the labor market. This assumption can be understood as the net result from a richer general equilibrium model. In such a setting, the total derivative $w'_A(K)$ would capture

<footnote n="7">⁷Our assumptions of uniform concavity and a globally signed cross-partial derivative are sufficient for this result. While general theorems on monotone comparative statics (e.g., Topkis's Theorem) would ensure the monotonicity of the solution set, our stricter assumptions guarantee a *unique* and continuously differentiable solution path. This allows us to use the Implicit Function Theorem (Eq. (1)) to prove strict monotonicity. Note that the IFT only requires the weaker, "local" condition that the cross-partial has the appropriate sign when evaluated along the optimal path; our global assumption is a convenient simplification.</footnote>

<footnote n="8">⁸A general equilibrium model would endogenize wages based on the aggregate skill supply. Our partial equilibrium approach deliberately abstracts from this channel to isolate the core tension between AI's enhancement of educational productivity and its substitution for skills in the labor market.</footnote>
both a direct, negative substitution effect ($\partial w_A / \partial K$) and indirect effects via changes in the aggregate skill supply. Our assumption posits that the former effect dominates, leading to an overall decline in skill A's wage.

## 4.3 Skill Mismatch

In contrast to the naive planner, an *informed (forward-looking) planner* anticipates how AI will alter the returns to skills in the labor market (Assumption 6). Using $U_{\text{inf}}(t_A; K)$ as defined in Assumption 5, the planner solves

$$ \max_{t_A \in [0, T]} U_{\text{inf}}(t_A; K), \quad \text{where } U_{\text{inf}}(t_A; K) = w_A(K)f_A(t_A, g(K)) + w_B(K)f_B(T - t_A, g(K)). $$

Let the solution be $t_A^\dagger(K)$. As formalized in Assumption 7, the wage-suppressing channel dominates for $K \ge K_0$.

**Assumption 7 (Dominant Workplace Substitution).** We assume that as AI advances beyond the baseline $K_0$, the negative effect of workplace substitution on the marginal return to teaching skill A outweighs the positive effect from educational complementarity. This ensures the informed planner's incentive to allocate time to skill A decreases as K increases. Formally, the objective function $U_{\text{inf}}(t_A; K)$ has decreasing differences in $(t_A, K)$ for $K \ge K_0$:

$$ \frac{\partial^2 U_{\text{inf}}}{\partial K \partial t_A}(t_A, K) \begin{cases} \le 0 & \text{if } K = K_0 \\ < 0 & \text{if } K > K_0 \end{cases} \quad \text{for all } t_A \in [0, T]. $$

This condition reflects the central trade-off. The cross-partial derivative decomposes into two competing forces:

$$ \begin{align*} \frac{\partial^2 U_{\text{inf}}}{\partial K \partial t_A} &= \underbrace{w_A'(K) \frac{\partial f_A}{\partial t_A} - w_B'(K) \frac{\partial f_B}{\partial t_B}}_{(a) \text{ Negative Workplace Channel}} \Big|_{t_B=T-t_A} \\ &\quad + \underbrace{g'(K) \left( w_A(K) \frac{\partial^2 f_A}{\partial t_A \partial K_e} - w_B(K) \frac{\partial^2 f_B}{\partial t_B \partial K_e} \right)}_{(b) \text{ Positive Education Channel}} \Big|_{t_B=T-t_A}. \end{align*} $$

Assumption 7 posits that the negative workplace channel (a) is strong enough to dominate the positive education channel (b) for all $K > K_0$.

The divergence between the choices of the naive and informed planners defines our measure of skill mismatch. We define the mismatch as the normalized difference in time allocation:

$$ \text{Mismatch}(K) := \frac{t_A^*(K) - t_A^\dagger(K)}{T}. $$

As we will show in Proposition 1, the difference $t_A^*(K) - t_A^\dagger(K)$ is zero at $K = K_0$ and strictly positive for $K > K_0$. The mismatch is therefore a direct measure of the naive planner's over-allocation to skill A relative to the informed benchmark.

To analyze the evolution of mismatch, we anchor the planners' decisions by assuming their incentives align at an initial level of AI capital, $K_0 \ge 0$. We set the naive planner's fixed prices equal to the true wages at this point: $p_A = w_A(K_0)$ and $p_B = w_B(K_0)$. This construction ensures that at $K = K_0$, the planners' problems are
identical, so $t_A^*(K_0) = t_A^+(K_0)$ and the mismatch is zero. For any $K > K_0$, their incentives diverge, leading to our first result.

**Proposition 1 (Mismatch Increases with AI Capital).** Under Assumptions 1–7, for $K \ge K_0$:

i) The naive allocation $t_A^*(K)$ is strictly increasing in $K$.

ii) The informed allocation $t_A^+(K)$ is weakly decreasing in $K$, and strictly decreasing for $K > K_0$.

Therefore, the skill mismatch, Mismatch(K), is strictly increasing for all $K \ge K_0$.

*Proof.* The proof is located in A.1. □

## 4.4 Extension: Unpriced Dimensions of Learning

In our baseline model, a mismatch arises because the naive planner uses fixed price proxies instead of the true, K-dependent wages. We now introduce a second source of mismatch: *unpriced non-cognitive skills* (e.g., perseverance, creativity), denoted $C$. Even a planner who perfectly foresaw wages would still under-provide these skills if their value is not part of the planner's objective. We show how this non-cognitive skills mismatch is amplified by AI.

The naive planner's problem remains unchanged, maximizing utility based on fixed prices ($p_A, p_B$). The informed planner, however, now optimizes a richer objective function that includes the value of non-cognitive skills, weighted by a parameter $\Gamma > 0$. Let the production of these skills be denoted by the function $C(t_A, K)$, which depends on the allocation of time to skills A and B, and the level of educational AI. The underlying production technology is $f_C(t_A, t_B, K_e)$. Imposing the time constraint $t_B = T - t_A$, we define the composite function as:

$$C(t_A, K) := f_C(t_A, T - t_A, g(K)).$$

The planners' problems are:

$$\text{Naive Planner:} \quad \max_{t_A \in [0, T]} U_{\text{naive}}(t_A; K) := p_A f_A(t_A, K_e) + p_B f_B(T - t_A, K_e),$$

$$\text{Informed Planner:} \quad \max_{t_A \in [0, T]} S(t_A; K) := U_{\text{inf}}(t_A, K) + \Gamma C(t_A, K).$$

The naive planner's allocation, $t_A^*(K)$, is unchanged. The informed planner's solution is now denoted $t_A^+(K)$. To analyze the resulting behavior, we introduce the following assumptions about the underlying production technology $f_C$.

**Assumption 8 (Properties of Non-Cognitive Skill Production).** The non-cognitive skills production function $f_C(t_A, t_B, K_e)$ has the following properties:

(a) Time allocation effect: Reallocating time from skill B to skill A is detrimental to non-cognitive skill formation. We model this by assuming the marginal product of time spent on skill B is greater than that spent on skill A. Formally, for any given $K_e$:

$$\frac{\partial f_C}{\partial t_B} > \frac{\partial f_C}{\partial t_A}.$$
This implies that for the composite function $C(t_A, K)$, the partial derivative with respect to its first argument, holding $K$ constant, is strictly negative:

$$ \frac{\partial C}{\partial t_A} = \frac{\partial f_C}{\partial t_A} - \frac{\partial f_C}{\partial t_B} < 0. $$

(b) Direct AI effect: For a fixed time allocation and for all $K \ge 0$, an increase in educational AI is detrimental to non-cognitive skills production. Formally, $C(t_A, K)$ is strictly decreasing in $K$ for any given $t_A$:

$$ \frac{\partial C}{\partial K} = g'(K) \frac{\partial f_C}{\partial K_e} < 0. $$

This requires $\frac{\partial f_C}{\partial K_e} < 0$ as $g'(K) > 0$.

(c) Synergistic harm: The negative marginal effect of AI on non-cognitive skills is stronger when more time is allocated to skill A. This synergy is rooted in the underlying technology: the complementarity between educational AI ($K_e$) and time is weaker for skill A than for skill B, i.e., $\partial^2 f_C / (\partial K_e \partial t_A) < \partial^2 f_C / (\partial K_e \partial t_B)$. This implies that the composite function $C(t_A, K)$ has strictly decreasing differences:

$$ \frac{\partial^2 C}{\partial K \partial t_A} = g'(K) \left( \frac{\partial^2 f_C}{\partial K_e \partial t_A} - \frac{\partial^2 f_C}{\partial K_e \partial t_B} \right) < 0. $$

(d) Smoothness and concavity: The underlying production function $f_C(t_A, t_B, K_e)$ is twice continuously differentiable ($C^2$) and weakly concave in its time inputs $(t_A, t_B)$. This ensures that the composite function $C(t_A, K)$ is $C^2$ and weakly concave in $t_A$.

**Assumption 9 (Interior Allocation with Non-Cognitive Skills).** For all $K \ge 0$, the parameters and functions are such that the informed planner's choice, $t_A^\dagger(K)$, is always in the interior $(0, T)$.

**Remark 1 (Concavity with Non-cognitive Skills).** The informed planner's new objective function is $S(t_A, K) = U_{inf}(t_A, K) + \Gamma C(t_A, K)$. Since $U_{inf}$ is strictly concave in $t_A$ (from prior assumptions) and $\Gamma C(t_A, K)$ is weakly concave in $t_A$ (by Assumption 8(d)), their sum $S(t_A, K)$ remains strictly concave in $t_A$ for any given $K$. This, along with Assumption 9, ensures a unique interior optimum $t_A^\dagger(K)$ that satisfies the first-order condition $\partial S / \partial t_A = 0$.

**Remark 2 (Uniform curvature of S).** By Assumptions 1 and 4, for all $(t_A, K)$,

$$ \frac{\partial^2 U_{inf}}{\partial t_A^2}(t_A, K) = w_A(K) \frac{\partial^2 f_A}{\partial t_A^2} + w_B(K) \frac{\partial^2 f_B}{\partial t_B^2} \le -(w_A(K) + w_B(K)) \underline{\lambda} \le -2\underline{w} \underline{\lambda} < 0. $$

Since $C$ is weakly concave in $t_A$ (Assumption 8(d)), $\partial^2 C / \partial t_A^2 \le 0$, hence

$$ \frac{\partial^2 S}{\partial t_A^2}(t_A, K) = \frac{\partial^2 U_{inf}}{\partial t_A^2} + \Gamma \frac{\partial^2 C}{\partial t_A^2} \le -2\underline{w} \underline{\lambda} < 0. $$

Therefore the IFT applies along $(t_A^\dagger(K), K)$ and $t_A^\dagger$ is $C^1$ in $K$.

The inclusion of non-cognitive skills reinforces the informed planner's tendency to shift away from skill A as AI advances. The new objective function $S(t_A, K)$ now has strictly decreasing differences in $(t_A, K)$, which we formalize in the following lemma. The necessary conditions for applying the Implicit Function Theorem (e.g., strict concavity) are therefore met for the informed planner's problem, ensuring that the optimal allocation $t_A^\dagger(K)$ is continuously differentiable.
**Lemma 1** (AI Reinforces Shift from Skill A). Under Assumptions 7 and 8(c), for all $K \ge K_0$, the informed planner's objective function $S(t_A, K)$ has strictly decreasing differences in $(t_A, K)$:

$$ \frac{\partial^2 S}{\partial K \partial t_A} < 0. $$

*Proof.* The cross-partial derivative of $S(t_A, K)$ is the sum of the cross-partials of its components:

$$ \frac{\partial^2 S}{\partial K \partial t_A} = \frac{\partial^2 U_{\inf}}{\partial K \partial t_A} + \Gamma \frac{\partial^2 C}{\partial K \partial t_A}. $$

By Assumption 7, the first term is non-positive ($\le 0$) at $K = K_0$ and strictly negative for $K > K_0$. By Assumption 8(c), the second term is strictly negative ($< 0$) for all $K \ge 0$, since $\Gamma > 0$. At $K = K_0$, the sum is non-positive plus strictly negative, which is strictly negative. For $K > K_0$, the sum of two strictly negative terms is strictly negative. Thus, the inequality holds for all $K \ge K_0$. $\square$

By Lemma 1, the objective function $S(t_A, K)$ has strictly decreasing differences. Given the strict concavity of $S$ in $t_A$ (Remark 1) and an interior solution (Assumption 9), we can apply the Implicit Function Theorem to find the sign of the comparative static:

$$ \frac{dt_A^\ddagger}{dK} = - \frac{\partial^2 S / \partial K \partial t_A}{\partial^2 S / \partial t_A^2} < 0 \quad \text{for all } K \ge K_0. $$

The numerator is strictly negative by Lemma 1, and the denominator is strictly negative by strict concavity. Thus, the informed planner's allocation to skill A, $t_A^\ddagger(K)$, is strictly decreasing in $K$ on $[K_0, \infty)$.

**Proposition 2** (Non-Cognitive Skills Under-provision and Its Amplification by AI). Under Assumptions 1–9, anchor the planners at $K_0$ by setting $p_j = w_j(K_0)$. Let $C^*(K) = C(t_A^*(K), K)$ and $C^\dagger(K) = C(t_A^\dagger(K), K)$ be the levels of non-cognitive skills produced under the naive and informed planners, respectively.

i) **(Initial Under-provision)** For any $\Gamma > 0$, the naive planner under-provides non-cognitive skills at the initial state:

$$ C^*(K_0) < C^\dagger(K_0). $$

ii) **(Growing Under-provision)** For $K \ge K_0$, the gap in non-cognitive skills, defined as $G(K) := C^\dagger(K) - C^*(K)$, is strictly increasing in $K$.

*Proof.* The proof is located in A.2. $\square$

*Interpretation of Proposition 2(ii).* The result that the non-cognitive skills gap widens as AI capital $K$ increases is driven by two reinforcing effects.

First, the time allocation paths diverge. The naive planner, responding to AI's pedagogical efficiency for skill A, increases time allocated to it ($dt_A^\dagger/dK > 0$). In sharp contrast, the informed planner internalizes all of AI's consequences: the falling wage for skill A, the direct harm to non-cognitive skills, and the opportunity cost of time shifted away from skill B. As a result, the informed planner's allocation to skill A, $t_A^\dagger(K)$, is strictly decreasing in $K$ (by Lemma 1). These opposing allocation paths ensure their choices diverge as AI advances.

Second, the direct marginal harm of educational AI is greater under the naive plan. The naive planner operates at a higher allocation to skill A. Per Assumption 8c, this is the domain where educational AI is most
damaging to non-cognitive skills. Consequently, as $K_e$ increases with $K$, the negative impact on non-cognitive skills from $K_e$ itself (Assumption 8b) is disproportionately severe under the naive planner's strategy.

This highlights a second wedge in educational choices, beyond the skill-wage mismatch. Even if the school had correct wage forecasts ($w_A, w_B$), it would still under-provide unpriced non-cognitive skills unless they appear explicitly in its objective.

## 4.5 Extension: Endogenous AI Adoption and Its Cost

In the main model, educational AI capital $K_e$ rises exogenously with general AI capital $K$. However, schools often make active, costly decisions about adopting specific AI teaching tools. This extension analyzes this endogenous choice, leading to a new dimension of mismatch focused on AI intensity.<fnref n="9" />

To isolate this mechanism, we analyze the choice of AI intensity $\alpha \in [0, 1]$ at a marginal cost $c \ge 0$. We hold the broader economic context fixed: workplace AI is at a level $K > K_0$, setting a suppressed market wage for skill A, $w_A(K)$. The naive school, however, continues to base its decisions on the historical, pre-disruption price $p_A = w_A(K_0)$, ensuring its price signal is misaligned and overly optimistic ($p_A > w_A(K)$).

The school and planner optimize over both time allocation $t_A$ and AI intensity $\alpha$. While these choices are jointly determined, we can isolate the core mechanism of AI adoption mismatch by analyzing the choice of $\alpha$ conditional on a time allocation $t_A > 0$. This simplification is reasonable because the fundamental wedge driving over-adoption (the school's neglect of AI's negative impact on non-cognitive skills) persists for any given time allocation. A full joint optimization would not overturn the qualitative result that the school over-invests in $\alpha$ relative to the planner, provided the partial derivatives that define the core trade-offs ($\partial f_A / \partial \alpha > 0$ and $\partial f_C / \partial \alpha < 0$) retain their signs at the new optimum.

For this extension only, we modify the production functions to isolate the role of AI intensity. We model AI as a tool adopted specifically for teaching skill A, while skill B is taught traditionally, and allow the production of non-cognitive skills to depend on both time allocation and AI intensity.

**Assumption 10 (Technology with Endogenous AI Intensity).** The skill and non-cognitive skill production functions for this extension are:

$$ 
\begin{align*}
A &= f_A(t_A, \alpha) \\
B &= f_B(t_B) \\
C &= f_C(t_A, T - t_A, \alpha)
\end{align*}
$$

These functions are twice continuously differentiable.

(a) AI enhances Skill A: AI intensity is productive for skill A, but with diminishing returns ($\partial f_A / \partial \alpha > 0$ and $\partial^2 f_A / \partial \alpha^2 < 0$).

(b) AI harms Non-Cognitive Skills: AI intensity is detrimental to the production of non-cognitive skills ($\partial f_C / \partial \alpha < 0$).

(c) Non-decreasing marginal harm: The marginal harm of AI on non-cognitive skills is non-decreasing in intensity. Formally, we assume $f_C$ is weakly concave in $\alpha$ ($\partial^2 f_C / \partial \alpha^2 \le 0$). Since $\partial f_C / \partial \alpha$ is negative by

<footnote n="9">⁹In this section, for brevity, we refer to the “naive planner” as the *school* and the “informed planner” simply as the *planner*. This reflects the context of an individual school making a direct technology adoption choice.</footnote>
part (b), its magnitude (the marginal harm) is $|\frac{\partial f_C}{\partial \alpha}| = -\frac{\partial f_C}{\partial \alpha}$. The derivative of this magnitude with respect to $\alpha$ is therefore $-\frac{\partial^2 f_C}{\partial \alpha^2} \ge 0$.

The core friction is that the naive school's optimization ignores the negative impact of its AI choice ($\alpha$) on non-cognitive skills (C), while the informed planner internalizes it. The school and planner choose their optimal AI intensity by equating marginal benefit to marginal cost.

The fundamental asymmetry between the school and the planner has two sources: (i) the school uses an inflated price for skill A ($p_A > w_A(K)$), and (ii) it ignores AI's harm to non-cognitive skills, which the planner internalizes via the term $\Gamma \frac{\partial f_C}{\partial \alpha} < 0$. Since AI enhances skill A ($\frac{\partial f_A}{\partial \alpha} > 0$), the school's perceived marginal benefit from AI adoption strictly dominates the planner's for all $\alpha \in [0, 1]$:

$$ \mathcal{M}\mathcal{B}_{\text{school}}(\alpha) = p_A \frac{\partial f_A}{\partial \alpha} > w_A(K) \frac{\partial f_A}{\partial \alpha} > w_A(K) \frac{\partial f_A}{\partial \alpha} + \Gamma \frac{\partial f_C}{\partial \alpha} = \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha). $$

From Assumption 10, both marginal benefit curves are strictly decreasing in $\alpha$.<fnref n="10" /> Hence, for any given marginal cost $c$, each agent's optimal $\alpha$ is unique.

**Assumption 11 (Common Interior Cost Set).** AI adoption has linear cost $C(\alpha; c) = c\alpha$ with $c > 0$. To ensure a non-empty range of costs for which both agents choose interior intensities $\alpha \in (0, 1)$, we assume the single overlap condition

$$ \mathcal{M}\mathcal{B}_{\text{planner}}(0) > \mathcal{M}\mathcal{B}_{\text{school}}(1). $$

Then the common interior set

$$ C := (\mathcal{M}\mathcal{B}_{\text{school}}(1), \mathcal{M}\mathcal{B}_{\text{planner}}(0)) $$

is non-empty, and any $c \in C$ yields a unique interior solution for both agents.<fnref n="11" />

**Proposition 3 (AI Adoption Mismatch).** Under Assumptions 10 and 11, with $p_A > w_A(K)$ and $\Gamma > 0$, fix any $t_A > 0$. For any $c \in C$: Let $C^*(c) = f_C(t_A, T - t_A, \alpha^*(c))$ and $C^\dagger(c) = f_C(t_A, T - t_A, \alpha^\dagger(c))$ be the non-cognitive skills produced under the school's and planner's choices, respectively.

i) Systematic Over-adoption: The school's chosen AI intensity is strictly greater than the planner's: $\alpha^*(c) > \alpha^\dagger(c)$.

ii) Impact of Falling AI Costs: As the marginal cost of AI, c, falls:

(a) Both chosen intensities increase: $\frac{\partial \alpha^*}{\partial c} < 0$ and $\frac{\partial \alpha^\dagger}{\partial c} < 0$.

(b) Widening adoption mismatch: The adoption mismatch, $\alpha^*(c) - \alpha^\dagger(c)$, strictly increases if and only if the planner's marginal benefit curve is steeper than the school's at their respective optima:

$$ |\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^\dagger)| > |\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^*)|. $$

(c) Widening non-cognitive skill gap: If condition (b) holds, then by Assumption 10(b), the gap in non-cognitive skills, $C^\dagger(c) - C^*(c)$, strictly increases as c falls.

<footnote n="10">¹⁰Strictly, $\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha) = p_A \frac{\partial^2 f_A}{\partial \alpha^2} < 0$ by Assumption 10(a). Then $\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha) = w_A(K) \frac{\partial^2 f_A}{\partial \alpha^2} + \Gamma \frac{\partial^2 f_C}{\partial \alpha^2} < 0$ since $w_A(K) > 0$, $\frac{\partial^2 f_A}{\partial \alpha^2} < 0$ by Assumption 10(a), and $\frac{\partial^2 f_C}{\partial \alpha^2} \le 0$ by Assumption 10(c).</footnote>

<footnote n="11">¹¹An interior solution for a given agent requires $c \in (\mathcal{M}\mathcal{B}(1), \mathcal{M}\mathcal{B}(0))$. Since we established that $\mathcal{M}\mathcal{B}_{\text{school}}(\alpha) > \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha)$ for all $\alpha$, the school's interval is to the right of the planner's. The intervals have a non-empty intersection if and only if the end of the planner's interval, $\mathcal{M}\mathcal{B}_{\text{planner}}(0)$, is greater than the start of the school's, $\mathcal{M}\mathcal{B}_{\text{school}}(1)$.</footnote>
*Proof.* The proof is located in A.3. □

*Discussion of the widening condition (ii-b).* Because both marginal benefit slopes are negative, the condition $|\mathcal{M}B'_{\text{planner}}(\alpha^\dagger)| > |\mathcal{M}B'_{\text{school}}(\alpha^*)|$ is equivalent to $\mathcal{M}B'_{\text{planner}}(\alpha^\dagger) < \mathcal{M}B'_{\text{school}}(\alpha^*)$. Substituting $\mathcal{M}B'_{\text{planner}} = w_A(K) f_A'' + \Gamma f_C''$ and $\mathcal{M}B'_{\text{school}} = p_A f_A''$, the adoption gap widens when

$$w_A(K) f_A''(\alpha^\dagger) + \Gamma f_C''(\alpha^\dagger) < p_A f_A''(\alpha^*).$$

Intuitively, the planner's curve must be locally steeper (more negative): either diminishing returns in teaching A are strong at $\alpha^\dagger$ or the curvature penalty from C (a more negative $f_C''$ weighted by $\Gamma$) is large. A sufficient (global) condition is $|\mathcal{M}B'_{\text{planner}}(\alpha)| > |\mathcal{M}B'_{\text{school}}(\alpha)|$ for all $\alpha$, which obtains if $\Gamma|f_C''(\alpha)|$ uniformly dominates $(p_A - w_A(K))|f_A''(\alpha)|$.

*Intuition for the widening non-cognitive skill gap (ii-c).* Two reinforcing channels generate the widening in $C^\dagger - C^*$ as $c$ falls. First, by (ii-b), the school's adoption rises faster than the planner's, so $\alpha^* - \alpha^\dagger$ increases. Second, by Assumption 10(c), the marginal harm $|C'(\alpha)|$ is non-decreasing in $\alpha$, so additional adoption at the school's already higher $\alpha^*$ inflicts at least as much (and typically more) damage. Together, faster movement and higher marginal harm yield a strictly larger gap in C.

## 4.6 Extension: Non-Monotonic Returns to Skill A

In our baseline model, we assumed AI is a straightforward substitute for skill A. We now relax this assumption to explore a scenario where the impact of AI on the returns to skill is non-monotonic. The intuition is that AI may displace certain skills or skill-levels while complementing others. Specifically, we conceptualize a three-tiered structure for the value of skill A. First, basic 'literacy' in skill A remains valuable, as it enables workers to collaborate with others, and to effectively prompt, oversee, or use AI tools.<fnref n="12" /> Second, there is an "AI-substitution trap," that is, there exists an intermediate skill range where human abilities are directly replaced by AI, yielding low returns. Finally, advanced expertise (for designing, developing, and supervising complex systems, for example) continues to command a significant wage premium.

**Remark 3 (Setup for the Piecewise Extension).** This extension modifies the baseline in two ways. First, we model the return to skill A as a piecewise lump-sum payment, $w_A(A; K)$, rather than a continuous marginal price. Consequently, we relax the anchoring condition that sets the initial mismatch to zero. Second, the notation $t_A^\dagger(K)$ will henceforth refer to the informed planner's choice under this new wage schedule.

### 4.6.1 Model Specification: A Piecewise Wage Structure

We formalize this with a piecewise wage schedule that determines the lump-sum payment, $w_A(A; K)$, a worker receives for achieving a skill level $A = f_A(t_A, K_e)$. Let there be two skill thresholds, $0 < \underline{A} < \bar{A}$. The wage structure is:

$$w_A(A; K) = \begin{cases} w_A(K), & 0 \le A < \underline{A} \quad (\text{basic skill 'literacy'}) \\ \tilde{w}_A(K), & \underline{A} \le A < \bar{A} \quad (\text{AI-substitution trap}) \\ \bar{w}_A(K), & A \ge \bar{A} \quad (\text{advanced expertise}) \end{cases} \quad (2)$$

<footnote n="12">¹²For instance, as AI comes to write more code, it may be useful for many people to have basic knowledge of coding to oversee the process, do minimal debugging, evaluate outputs or potential pitfalls, or have certain conceptual frameworks or logical reasoning processes.</footnote>
*Wage determination at thresholds.* We assume the wage schedule is right-continuous in skill A. This means a student achieving exactly a skill threshold earns the wage corresponding to the start of the next tier (For instance, a student at exactly $\bar{A}$ earns the expert wage $\bar{w}_A(K)$.) This provides a clear rule for payments at boundary points and is a standard technical convention.<fnref n="13" />

This structure reflects the potential for an “AI-substitution trap” and a high-skill premium, where we assume $\bar{w}_A(K) > \bar{w}_A(K) \ge \tilde{w}_A(K)$ for all relevant K. Furthermore, we assume that while AI capital K substitutes for basic and intermediate skills, it may complement advanced expertise. Formally, the wage levels evolve with K such that $\frac{dw_A}{dK} < 0$ and $\frac{d\tilde{w}_A}{dK} < 0$, but the change in the expert wage, $\frac{d\bar{w}_A}{dK}$, is ambiguous. This allows for both the case where AI suppresses all wages and the case where it creates a “super-skill premium” by complementing top-tier human capital.

This piecewise scheme departs from the baseline’s continuous marginal-product pricing. Such a structure can arise from several market features, such as institutional wage-setting practices, the discrete nature of tasks that AI can automate once a skill threshold is met, certification requirements at levels $\underline{A}$ and $\bar{A}$, or labor market frictions that lead to segmented returns. It models a reality where remuneration is based on achieving competency brackets rather than a per-unit price for skill.<fnref n="14" />

To isolate the effect of the new wage structure, we retain the core assumptions governing skill production and the naive planner’s behavior. Specifically, the production functions $f_A$ and $f_B$ remain strictly concave (Assumption 1), and the naive planner’s objective still exhibits increasing differences (Assumption 2). The wage for skill B, $w_B(K)$, remains positive and non-decreasing in K.

**Assumption 12 (Reachable Skill Thresholds).** For any given K in the domain of analysis, the skill production function $f_A(t_A, g(K))$ is such that the basic and advanced skill thresholds are attainable but not trivial. Formally, $f_A(0, g(K)) < \underline{A}$ and $f_A(T, g(K)) > \bar{A}$. Because $f_A$ is continuous and strictly increasing in its first argument, there exist unique time allocations $\tau_A(K) \in (0, T)$ and $\tau_{\bar{A}}(K) \in (0, T)$ satisfying:

$$f_A(\tau_A(K), g(K)) = \underline{A} \quad \text{and} \quad f_A(\tau_{\bar{A}}(K), g(K)) = \bar{A},$$

with $\tau_A(K) < \tau_{\bar{A}}(K)$. Furthermore, these threshold times are weakly decreasing in K:

$$\frac{d\tau_{\bullet}}{dK} = - \frac{\left. \frac{\partial f_A}{\partial K_e} \right|_{(\tau_{\bullet}(K), g(K))}}{\left. \frac{\partial f_A}{\partial t_A} \right|_{(\tau_{\bullet}(K), g(K))}} g'(K) \le 0, \quad \bullet \in \{\underline{A}, \bar{A}\},$$

with strict inequality whenever $\frac{\partial f_A}{\partial K_e} > 0$ at the evaluation point.

The central departure lies in the wage structure for skill A. A significant implication of this piecewise formulation is that the Dominant Wage-Substitution Effect (Assumption 7), which previously ensured that the informed allocation $t_A^\dagger(K)$ was weakly decreasing in K, no longer guarantees such monotonic behavior. The informed planner’s optimal time allocation may now respond non-monotonically to changes in K.

<footnote n="13">¹³An alternative rule, such as left-continuity, would alter the specific set of candidate allocations (e.g., to the tiers’ upper boundaries $\{\tau_A(K), \tau_{\bar{A}}(K), T\}$), but it would not change the fundamental nature of the planner’s problem as a discrete choice among a few salient options.</footnote>

<footnote n="14">¹⁴This structure simplifies the labor market to focus on the planner’s problem. We assume firms take this wage schedule for skill A as given, potentially due to institutional factors or industry-wide certification standards. Firms continue to hire skill B until its wage equals its marginal product. We abstract from the firm’s profit maximization problem with respect to skill A under this non-linear wage scheme.</footnote>
### 4.6.2 Planner objectives and optimal strategies

The naive planner's objective remains to maximize the perceived value of skills based on fixed, linear price signals:

$$ \max_{t_A \in [0, T]} U_{\text{naive}}(t_A; K) := p_A f_A(t_A, g(K)) + p_B f_B(T - t_A, g(K)). $$

The naive planner's objective is misspecified in two ways, distinct from simply failing to forecast wage changes. First, the planner misunderstands the structure of remuneration, optimizing against a linear, per-unit price $p_A$ when the market actually offers a lump-sum payment for achieving a competency bracket. Second, the level of the price signal is incorrect, as $p_A$ is a fixed historical value that does not account for the non-monotonic returns created by AI. Because the perceived marginal benefit $\partial U_{\text{naive}}/\partial t_A$ is increasing in K (increasing differences; Assumption 2), the optimal allocation $t_A^*(K)$ is strictly increasing in K.

The informed planner, by contrast, seeks to maximize the actual, non-linear income of students:

$$ \max_{t_A \in [0, T]} U_{\text{inf}}(t_A; K) := w_A(f_A(t_A, g(K)); K) + w_B(K)f_B(T - t_A, g(K)). $$

This objective function is discontinuous and not globally concave, rendering standard first-order methods inapplicable. The discontinuities in $w_A(\cdot; K)$ induce jumps in $U_{\text{inf}}$ at the time thresholds $\tau_A(K)$ and $\tau_{\bar{A}}(K)$. Specifically, there is a non-positive jump at $\tau_A(K)$ and a strictly positive jump at $\tau_{\bar{A}}(K)$:

$$ U_{\text{inf}}(\tau_A(K); K) - \lim_{t_A \uparrow \tau_A(K)} U_{\text{inf}}(t_A; K) = \tilde{w}_A(K) - \underline{w}_A(K) \le 0, $$

$$ U_{\text{inf}}(\tau_{\bar{A}}(K); K) - \lim_{t_A \uparrow \tau_{\bar{A}}(K)} U_{\text{inf}}(t_A; K) = \bar{w}_A(K) - \tilde{w}_A(K) > 0. $$

However, within each of the three skill tiers (for a fixed K), the wage component $w_A(A; K)$ is constant in A. The derivative of the informed planner's objective is therefore determined entirely by the returns to skill B:

$$ \frac{d}{dt_A} U_{\text{inf}}(t_A; K) = -w_B(K) \frac{\partial f_B}{\partial t_B}(T - t_A, g(K)) < 0. $$

(The strict inequality holds by Assumption 4 and the properties of $f_B$ from the baseline model). This implies that any optimal allocation must occur at the lower boundary of a tier, reducing the planner's problem to a discrete comparison of candidate allocations.

**Lemma 2 (Candidate Allocations for the Informed Planner).** Fix K satisfying Assumption 12. The informed planner's objective function $U_{\text{inf}}(t_A; K)$ is strictly decreasing in $t_A$ on the interior of each of the three skill tiers. Consequently, a maximizer for $U_{\text{inf}}(t_A; K)$ exists and must belong to the set of the tiers' lower boundaries:

$$ \arg \max_{t_A \in [0, T]} U_{\text{inf}}(t_A; K) \subseteq \{0, \tau_A(K), \tau_{\bar{A}}(K)\}. $$

*Proof.* On the interior of each of the three skill tiers defined by Eq. (2), $w_A(A; K)$ is constant, so the objective's derivative is:

$$ \frac{d}{dt_A} U_{\text{inf}}(t_A; K) = -w_B(K) \frac{\partial f_B}{\partial t_B} < 0 $$

As the function is strictly decreasing within each tier, any potential maximizer must lie at the lower boundary of a tier. This reduces the candidate set to $\{0, \tau_A(K), \tau_{\bar{A}}(K)\}$, as $t_A = T$ is strictly dominated by $t_A = \tau_{\bar{A}}(K)$. The optimization problem over the interval $[0, T]$ is therefore equivalent to maximizing the objective over this finite set, which guarantees that a solution exists. $\square$
**Remark 4 (Existence without semicontinuity).** By Lemma 2, any maximizer lies in the finite set {$0, \tau_A(K), \tau_{\bar{A}}(K)$}. A maximizer therefore exists by direct comparison of the objective function's value at these points.

The planner's problem therefore reduces to comparing the utility at a discrete set of candidate points. Since the utility function is strictly decreasing in $t_A$ within each tier, the utility over any continuous skill range is maximized at its starting point. This initially suggests a candidate set of {$0, \tau_A(K), \tau_{\bar{A}}(K)$}. However, the strategy of targeting the substitution trap at $\tau_A(K)$ is strictly dominated by the basic literacy strategy at $t_A = 0$.

**Lemma 3 (Barbell Optimality).** If $\underline{w}_A(K) \ge \tilde{w}_A(K)$, the informed planner's optimal allocation lies in {$0, \tau_{\bar{A}}(K)$}. Targeting the substitution trap is strictly dominated.

*Proof.* By Lemma 2, any maximizer must lie in {$0, \tau_A(K), \tau_{\bar{A}}(K)$}. We show that the candidate allocation $t_A = \tau_A(K)$ is strictly dominated by $t_A = 0$. Comparing the utility at these two points yields:

$$U_{\inf}(0; K) - U_{\inf}(\tau_A(K); K) = (\underline{w}_A(K) - \tilde{w}_A(K)) + w_B(K)[f_B(T, g(K)) - f_B(T - \tau_A(K), g(K))].$$

The first term, $\underline{w}_A(K) - \tilde{w}_A(K)$, is non-negative by assumption. Because $\tau_A(K) > 0$ and $f_B$ is strictly increasing in time, the second term, which represents the gain in skill B from reallocating time away from skill A, is strictly positive. The entire expression is therefore strictly positive, proving that $t_A = \tau_A(K)$ is suboptimal. The candidate set thus reduces to {$0, \tau_{\bar{A}}(K)$}. □

**Decision rule.** The informed planner chooses the advanced skill tier if the wage premium for expertise outweighs the value of the foregone production of skill B. This choice is determined by the sign of $\Delta(K)$:

$$\Delta(K) := \underbrace{\underline{w}_A(K) - \underline{w}_A(K)}_{\text{Wage Gain from Skill A}} - \underbrace{w_B(K) [f_B(T, g(K)) - f_B(T - \tau_A(K), g(K))]}_{\text{Opportunity Cost (Value of Lost Skill B)}}. \quad (3)$$

The planner's optimal choice is to set $t_A^\dagger(K) = \tau_A(K)$ if $\Delta(K) > 0$, and $t_A^\dagger(K) = 0$ if $\Delta(K) \le 0$.

**Mismatch in the Piecewise Setting.** In this extension, we define the mismatch as the simple difference in time allocations:

$$\text{Mismatch}(K) := t_A^*(K) - t_A^\dagger(K).$$

Crucially, we do not impose the anchoring condition from the baseline model, so it is possible that $\text{Mismatch}(K_0) \ne 0$.

*Note on normalization.* For comparability with the baseline model, one may use the normalized mismatch $(t_A^*(K) - t_A^\dagger(K))/T$. All qualitative results in this section remain unchanged.

**Proposition 4 (Barbell choices and non-monotonic mismatch).** Under the assumptions of this section and Eq. (2):

i) The naive allocation $t_A^*(K)$ is strictly increasing in K, a consequence of the increasing differences of $U_{naive}$ (Assumption 2) combined with strict concavity and interiority (Assumptions 1 and 3).

ii) $t_A^\dagger(K) \in \{0, \tau_{\bar{A}}(K)\}$ (Lemma 3) and $\tau_{\bar{A}}(K)$ is weakly decreasing in K (Assumption 12).

iii) Let $\Delta(K)$ be defined in (3). The mismatch, $t_A^*(K) - t_A^\dagger(K)$, evolves as follows:
- Monotonically increasing mismatch: If the sign of $\Delta(K)$ is constant for all $K \ge K_0$, the informed planner's strategy does not switch between its "barbell" options. In this case, the mismatch is strictly increasing in $K$.

- Non-monotonic mismatch: If $\Delta(K)$ changes sign at some $K^*$, the informed allocation $t_A^\dagger(K)$ jumps discontinuously. This introduces a discontinuity into the mismatch function, which is therefore non-monotonic.

*Proof.* i) This follows directly from the strict concavity of $U_{\text{naive}}$ in $t_A$ (Assumption 1) and its strictly increasing differences in $(t_A, K)$ (Assumption 2), which guarantee a unique, interior, and strictly increasing solution via the Implicit Function Theorem.

ii) By Lemma 3, the informed planner's optimal allocation $t_A^\dagger(K)$ must belong to the set $\{0, \tau_A^\dagger(K)\}$. By Assumption 12, the threshold time $\tau_A^\dagger(K)$ is weakly decreasing in $K$.

iii) The mismatch is $\text{Mismatch}(K) = t_A^*(K) - t_A^\dagger(K)$. If $\Delta(K)$ does not change sign for $K \ge K_0$, the planner's optimal strategy $t_A^\dagger(K)$ does not switch between 0 and $\tau_A^\dagger(K)$.

*Case 1:* If $t_A^\dagger(K) = 0$, then $\text{Mismatch}(K) = t_A^*(K)$, which is strictly increasing by (i).

*Case 2:* If $t_A^\dagger(K) = \tau_A^\dagger(K)$, its derivative is $\frac{d\text{Mismatch}}{dK} = \frac{dt_A^\dagger}{dK} - \frac{d\tau_A^\dagger}{dK}$. By part (i), $\frac{dt_A^\dagger}{dK} > 0$. By Assumption 12, $\frac{d\tau_A^\dagger}{dK} \le 0$. Thus, $\frac{d\text{Mismatch}}{dK} > 0$, so the mismatch is strictly increasing.

Finally, if $\Delta(K)$ changes sign at some $K^*$, $t_A^\dagger(K)$ switches between 0 and $\tau_A^\dagger(K)$, introducing a discontinuity into $\text{Mismatch}(K)$, which is therefore non-monotonic.

□

**Remark 5 (Drivers of $\Delta'(K)$).** The sign of $\Delta'(K)$ determines whether the planner is becoming more or less likely to target the advanced skill tier as AI progresses. Let $\tau = \tau_A^\dagger(K)$ and $K_e = g(K)$. Differentiating (3) and writing $\tau = \tau_A^\dagger(K)$ and $K_e = g(K)$ yields:

$$ \begin{align*} \Delta'(K) &= \underbrace{\overline{w}'_A(K) - w'_A(K)}_{(a) \text{ Direct wage effect}} \\ &\quad + \underbrace{w'_B(K)[f_B(T - \tau, K_e) - f_B(T, K_e)]}_{(b) \text{ Wage effect on skill B value}} \\ &\quad + \underbrace{w_B(K) \frac{\partial f_B}{\partial t_B}(T - \tau, K_e)(-\tau'(K))}_{(c) \text{ Time-saving effect on skill B}} \\ &\quad + \underbrace{w_B(K) g'(K) \left[ \frac{\partial f_B}{\partial K_e}(T - \tau, K_e) - \frac{\partial f_B}{\partial K_e}(T, K_e) \right]}_{(d) \text{ AI-education effect on skill B}}. \end{align*} $$

The sign of $\Delta'(K)$ is generally ambiguous. Term (a) is ambiguous because $\overline{w}'_A(K)$ is ambiguous. Term (b) is non-positive since $w'_B(K) \ge 0$ and the term in brackets is negative. Term (c) is non-negative, as it is a product of non-negative terms: $w_B(K) > 0$, $\partial f_B / \partial t_B > 0$, and $-\tau'(K) \ge 0$. The sign of term (d) depends on the cross-partial derivative of $f_B$. Using the fundamental theorem of calculus, we can write the term in brackets as $-\int_{T-\tau}^T \frac{\partial^2 f_B}{\partial t_B \partial K_e}(s, K_e) ds$. Thus, the sign of (d) is determined by the sign of this cross-partial over the relevant time interval. Given these conflicting and ambiguous forces, $\Delta(K)$ can evolve non-monotonically.
### 4.6.3 The Dynamics of Mismatch and Policy Implications

An increase in $K$ alters the planner's trade-off. While the associated rise in educational AI makes reaching the advanced skill threshold easier (i.e., $\tau_A^K$) is weakly decreasing), the corresponding wage changes can shift the incentive entirely. For instance, a sharp drop in the expert wage $\bar{w}_A(K)$ could render the pursuit of expertise suboptimal, causing the planner to abruptly shift the allocation $t_A^K$ from $\tau_A^K$ to 0. Consequently, the mismatch, $t_A^K - \bar{t}_A^K$, no longer widens monotonically; it can shrink or grow discontinuously as the informed planner shifts its target between the two ends of the barbell.

The optimal strategy may therefore shift discontinuously. For instance, a sharp drop in $\bar{w}_A(K)$ could make aiming for the advanced tier newly suboptimal, causing the planner to shift focus toward basic literacy, thereby decreasing $t_A^K$ even as $K$ rises. Consequently, the mismatch, $t_A^K - \bar{t}_A^K$, no longer widens monotonically as in our baseline model. It can shrink or grow depending on whether a change in $K$ causes the informed planner to shift its target from one end of the barbell to the other.

Although the mismatch is no longer monotonic, it persists and the welfare loss it generates is potentially more severe. The harm is no longer simply over-investment, but *mis-targeted* investment. The primary danger is the naive planner pushing an entire cohort of students into the “AI-substitution trap”, that is towards a low-return region that the informed planner would deliberately avoid. This cautions against one-size-fits-all educational mandates that may generate “intermediate” levels of skill. Instead, policy may be better directed toward subsidizing skills in which humans retain a comparative advantage ($B$) or creating specialized programs that support the difficult jump to true expertise, preventing students from being stranded in the unproductive middle.

This core insight is not an artifact of the discrete thresholds in our piecewise model. A similar mismatch dynamic would arise with a smooth, “valley-shaped” wage schedule, where wages are low for intermediate skills and higher for basic and advanced skills. In such a scenario, a naive planner optimizing against a perceived linear price could still inefficiently target the low-return region of intermediate skills, while an informed planner would pursue a barbell strategy. The central lesson holds: a failure to recognize the non-monotonic returns created by AI can lead to costly misallocations of human capital.<fnref n="15" />

## 5 Discussion and Conclusion

This paper formalizes an 'AI-Driven Education Trap,' a systematic coordination failure where the technology that boosts teaching productivity for a given skill simultaneously erodes that skill's market value. Our central result (Proposition 1) is that when educational planners are naive to future wage effects, the resulting skill mismatch grows monotonically with the prevalence of AI-enhanced capital. The mechanism is rooted in a dual friction: an information wedge, where planners rely on misaligned prices, and an incentive wedge, where they ignore unpriced externalities. This structure distinguishes our model from classic cobweb dynamics and is motivated by our empirical finding of a positive correlation between skills amenable to AI-assisted teaching and those vulnerable to workplace substitution. Importantly, the naive planner's allocation is rational under accountability regimes that reward measured outputs, so the failure is institutional rather than behavioral.

<footnote n="15">¹⁵This “low-skill and super-skill advantage” resonates with labor-market narratives of “routine-biased technical change.”</footnote>
The net societal impact of AI in education can be decomposed into three competing channels. A change in social surplus, ΔW, from an increase in AI capital can be conceptualized as:

$$ \Delta W(K) = \underbrace{\Delta \Pi_E(K)}_{\text{Teaching Gains}} + \underbrace{\Delta \Pi_L(K)}_{\text{Substitution Losses}} + \underbrace{\Gamma \Delta C(K)}_{\text{Non-Cognitive Externality}} $$

A naive planner, focusing only on the first term (ΔΠ<sub>E</sub>), makes choices that are locally rational but socially inefficient because they fail to internalize the negative labor-market effects captured in ΔΠ<sub>L</sub> and the unpriced harm to non-cognitive skills, ΓΔC. This framework clarifies that effective policy must target one or more of these specific wedges to be effective.

Our extensions demonstrate how this core friction may be exacerbated by additional considerations. First, the presence of unpriced, non-cognitive skills increases the harm of a naive shift towards skill A, creating another source of mismatch rooted in the under-provision of these crucial abilities. Furthermore, when technology adoption is endogenous, schools systematically over-invest in AI intensity (α* > α<sup>†</sup>), guided by an inflated price signal for skill A (Proposition 3). Finally, a non-monotonic, tiered wage structure creates a slightly different problem in the form of a 'substitution trap.' Here, the risk is not so much over-investment but mis-targeted investment into a low-return skill bracket that an informed planner would strategically avoid, stranding students in an unproductive middle ground (Proposition 4).

Our findings could also be framed through the classic lens of comparative advantage. As AI automates routine cognitive tasks (skill type A), the comparative advantage of human capital shifts toward areas resistant to automation, such as inter-personal skills (B) and non-cognitive abilities (C). The policy challenge, therefore, is to realign educational incentives with this evolving frontier of human comparative advantage, rather than simply embracing technology for its immediate productivity benefits in teaching. The non-monotonic extension suggests this frontier may itself be 'barbell-shaped,' with human comparative advantage persisting in specific levels of expertise.

For example, to mitigate the information wedge, policy can focus on reducing the friction caused by backward-looking price signals. Creating and disseminating forward-looking wage forecasts (ŵ<sub>j</sub>(K)), perhaps through partnerships between labor ministries and industry associations, could provide planners with better information. The objective is not clairvoyance but rather to provide credible signals about the evolving structure of returns, including the likely sign of w'<sub>A</sub>(K), but also highlighting potential non-monotonicities like the 'substitution trap'. To be effective and avoid replacing static expectations with a false sense of certainty, these signals should be presented as scenario-based dashboards that incorporate measures of uncertainty, rather than as single point estimates embedded in high-stakes accountability metrics, which would be susceptible to Goodhart's Law.

Correcting the incentive wedge associated with unpriced non-cognitive skills requires moving beyond impractical textbook solutions like a Pigouvian tax. A more realistic approach is to leverage existing institutional structures, namely accreditation standards and targeted funding incentives. Accrediting bodies, for instance, could be directed to update their review criteria to require institutions to provide clear evidence of programs that foster durable skills like resilience and complex communication. Under this model, institutional legitimacy would depend not just on adopting technology, but on demonstrating how that technology is integrated into a holistic curriculum. Symmetrically, policymakers could tie a portion of discretionary or innovation-focused grants to the successful implementation of such programs. Rather than creating a new, complex tax, this strategy repurposes existing financial streams to directly reward institutions that cultivate the human-centric skills least susceptible to automation.
Given the challenges of precise wage forecasting and pricing externalities, direct institutional guardrails offer a reasonable, if imperfect, policy response. Such policies could mandate balanced curricula, shifting instructional focus from skills susceptible to AI substitution toward more durable, human-centric abilities. They should also govern the adoption of educational AI, evaluating its success based on its holistic impact on student development, including non-cognitive skills like resilience, and not merely on narrow gains in testable outputs. For technical fields with non-monotonic returns, this approach supports a 'barbell' strategy: pairing universal digital literacy with selective pathways to deep expertise, thereby helping students avoid the middle-skill 'substitution trap.' To prevent such regulations from becoming rigid, they must be designed as adaptive rules, for instance, by incorporating review triggers keyed to shifts in skill-specific wage premiums.

The trap we identify is unlikely to be uniform in its effects. Low-resource schools may face the strongest incentives to adopt off-the-shelf AI that boosts easily measured skills, potentially exacerbating inequality. More subtly, the type of AI deployed may differ systematically: affluent students may be exposed to AI in a human-capital intensive environment that foster metacognition (a complement to skill C), while disadvantaged students receive minimal AI tools that provide simple answers, promoting cognitive offloading (a substitute for skill C). This could create a new digital divide in developmental impact, widening disparities in durable, non-cognitive abilities.

This paper's theoretical framework generates several testable predictions. (1) Using proxies for AI prevalence (K), one should find that instructional time in naive institutions (e.g., those with weak industry ties) increasingly diverges from forward-looking indicators of skill demand. (2) The adoption of specific AI teaching tools should be highest for skills whose wage premia are forecast to decline. (3) Provided that non-cognitive outcomes can be reliably measured, the gap in non-cognitive outcomes between students with high versus low exposure to skill-A-focused AI should widen as the real cost (c) of the technology falls. (4) In settings with tiered returns to A, cohort skill distributions should bunch at the thresholds $\underline{A}$ and $\bar{A}$, with a valley in between. (5) The adoption gap, $a^*(c) - a^\dagger(c)$, widens as the marginal cost of AI (c) falls if the planner's marginal benefit curve is steeper than the school's at their respective optimal choices (i.e., $|MB'_{\text{planner}}(a^\dagger)| > |MB'_{\text{school}}(a^*)|$, per Proposition 3). This structural condition is testable via estimation of the underlying production functions $f_A$ and $f_C$.

Our analysis abstracts from general equilibrium effects on capital prices, the rich dynamics of student heterogeneity, and the political economy of school governance, all important areas for future work. While general equilibrium forces can alter magnitudes, the signs of our core comparative statics follow from decreasing-differences conditions and are robust to modest relaxations. Our model provides a clear warning and a corresponding policy principle: to prevent training for obsolescence, the adoption of educational AI must be steered by forward-looking signals and safeguarded by incentives that protect the holistic development of human capital, effectively guiding students toward their uniquely human comparative advantages.

# A Proofs

## A.1 Proof of Proposition 1

*Proof.* The proof demonstrates the strictly divergent paths of the naive and informed planners' allocations for $K > K_0$. We establish the monotonicity of each allocation, $t_A^*(K)$ and $t_A^\dagger(K)$, by analyzing the sign of the cross-partial derivative of each planner's objective function, $U(t_A, K)$, with respect to time allocation $t_A$ and AI capital $K$. As established by the comparative static result in Equation (1), the sign of this cross-partial
derivative determines the sign of $dt_A^*/dK$. Assumptions 1, 3, and 5 ensure that unique interior solutions exist and that the required second-order conditions hold.

Part (i): Naive Planner's Allocation $t_A^*(K)$ is Strictly Increasing.

The naive planner's objective function $U_{naive}(t_A; K)$ has the cross-partial derivative:

$$ \frac{\partial^2 U_{\text{naive}}}{\partial K \partial t_A} = g'(K) \left( p_A \frac{\partial^2 f_A}{\partial t_A \partial K_e} - p_B \frac{\partial^2 f_B}{\partial t_B \partial K_e} \right). $$

By Assumption 2 (Asymmetric AI Complementarity), this expression is strictly positive for all $K \ge K_0$. By Assumption 1, $U_{naive}$ is strictly concave, ensuring the second-order condition $\partial^2 U_{naive}/\partial t_A^2 < 0$ holds. From Equation (1), it follows that:

$$ \frac{dt_A^*}{dK} = - \frac{\partial^2 U_{\text{naive}}/\partial K \partial t_A}{\partial^2 U_{\text{naive}}/\partial t_A^2} = - \frac{(> 0)}{(< 0)} > 0. $$

Thus, the naive allocation $t_A^*(K)$ is strictly increasing in $K$ for all $K \ge K_0$.

Part (ii): Informed Planner's Allocation $t_A^+(K)$ is non-increasing

The informed planner's objective $U_{inf}(t_A; K)$ has the cross-partial derivative:

$$ \begin{align*} \frac{\partial^2 U_{\text{inf}}}{\partial K \partial t_A} &= \underbrace{w'_A(K) \frac{\partial f_A}{\partial t_A} - w'_B(K) \frac{\partial f_B}{\partial t_B}}_{\text{(a) Wage Channel}} \\ &\quad + \underbrace{g'(K) \left( w_A(K) \frac{\partial^2 f_A}{\partial t_A \partial K_e} - w_B(K) \frac{\partial^2 f_B}{\partial t_B \partial K_e} \right)}_{\text{(b) Education Productivity Channel}}. \end{align*} $$

Assumption 7 (Dominant Wage-Substitution Effect) specifies the sign of this expression. It posits that the negative wage channel (a) dominates the education channel (b) as $K$ increases. Specifically:

* For $K = K_0$, $\partial^2 U_{inf}/\partial K \partial t_A \le 0$.

* For $K > K_0$, $\partial^2 U_{inf}/\partial K \partial t_A < 0$.

The second-order condition for a maximum must be satisfied. We differentiate the first-order condition, $\frac{\partial U_{inf}}{\partial t_A} = w_A(K) \frac{\partial f_A}{\partial t_A} - w_B(K) \frac{\partial f_B}{\partial t_B}$, with respect to $t_A$. Using the chain rule on the second term, where $t_B = T - t_A$ implies $\frac{dt_B}{dt_A} = -1$, we get:

$$ \frac{\partial^2 U_{\text{inf}}}{\partial t_A^2} = w_A(K) \frac{\partial^2 f_A}{\partial t_A^2} - w_B(K) \left( \frac{\partial^2 f_B}{\partial t_B^2} \frac{dt_B}{dt_A} \right) = w_A(K) \frac{\partial^2 f_A}{\partial t_A^2} + w_B(K) \frac{\partial^2 f_B}{\partial t_B^2}. $$

By Assumption 4, $w_A(K) > 0$ and $w_B(K) > 0$. By Assumption 1, $\partial^2 f_A/\partial t_A^2 < 0$ and $\partial^2 f_B/\partial t_B^2 < 0$. Therefore, $\partial^2 U_{inf}/\partial t_A^2 < 0$ as required. Applying Equation (1) again:

$$ \frac{dt_A^\dagger}{dK} = - \frac{\partial^2 U_{\text{inf}}/\partial K \partial t_A}{\partial^2 U_{\text{inf}}/\partial t_A^2}. $$
This implies $\frac{dt_A^\dagger}{dK}(K_0) \le 0$ and $\frac{dt_A^\dagger}{dK} < 0$ for $K > K_0$. Therefore, the informed allocation $t_A^\dagger(K)$ is weakly decreasing at $K = K_0$ and strictly decreasing for $K > K_0$.

## Conclusion on Mismatch

First, we establish the initial condition. By construction, at $K = K_0$, the naive planner's prices match the market wages ($p_j = w_j(K_0)$), making the two planners' objective functions identical: $U_{naive}(t_A; K_0) = U_{inf}(t_A; K_0)$. Since both functions are strictly concave and yield unique interior solutions (by Assumptions 3 and 5), their allocations must coincide: $t_A^\dagger(K_0) = t_A^\dagger(K_0)$. This means the mismatch is zero at $K_0$.

To show that the mismatch, defined as $\text{Mismatch}(K) = (t_A^\dagger(K) - t_A^\dagger(K))/T$, is strictly increasing for $K \ge K_0$, we examine its derivative:

$$ \frac{d\text{Mismatch}}{dK} = \frac{1}{T} \left( \frac{dt_A^\dagger}{dK} - \frac{dt_A^\dagger}{dK} \right). $$

We analyze the sign of this derivative for the domain $K \ge K_0$. From part (i), we established that $\frac{dt_A^\dagger}{dK} > 0$ for all $K \ge K_0$. From part (ii), we established that $\frac{dt_A^\dagger}{dK} \le 0$ for all $K \ge K_0$ (specifically, $\frac{dt_A^\dagger}{dK}(K_0) \le 0$ and $\frac{dt_A^\dagger}{dK} < 0$ for $K > K_0$).

The derivative of the mismatch is therefore the difference between a strictly positive term and a non-positive term:

$$ \frac{d\text{Mismatch}}{dK} = \frac{1}{T} \left( (\text{positive}) - (\text{non-positive}) \right) > 0 \quad \text{for all } K \ge K_0. $$

Since the derivative is strictly positive for all $K \ge K_0$, we conclude that $\text{Mismatch}(K)$ is strictly increasing for $K \ge K_0$. Furthermore, since $\text{Mismatch}(K_0) = 0$, it follows that $\text{Mismatch}(K) > 0$ for all $K > K_0$. □

## A.2 Proof of Proposition 2

We first prove a supporting lemma regarding the ordering and divergence of the time allocations.

**Lemma 4 (Ordering and Divergence of Optimal Allocations).** Under Assumptions 1–9 and for any $\Gamma > 0$, we have $t_A^\dagger(K) > t_A^\dagger(K)$ for all $K \ge K_0$. Furthermore, the gap $\Delta(K) := t_A^\dagger(K) - t_A^\dagger(K)$ is strictly increasing for all $K \ge K_0$.

*Proof.* The proof proceeds in two steps.

**Step 1: Ordering at $K = K_0$.** The naive optimum $t_A^\dagger(K_0)$ satisfies the first-order condition $\partial U_{naive}/\partial t_A = 0$ at $(t_A, K) = (t_A^\dagger(K_0), K_0)$. By the anchor assumption $p_j = w_j(K_0)$, this implies $\partial U_{inf}/\partial t_A = 0$ at the same point. Evaluating the gradient of the informed planner's objective, $S$, at the naive optimum yields:

$$ \left. \frac{\partial S}{\partial t_A} \right|_{t_A=t_A^\dagger(K_0), K=K_0} = \left. \frac{\partial U_{inf}}{\partial t_A} \right|_{t_A=t_A^\dagger(K_0), K=K_0} + \Gamma \left. \frac{\partial C}{\partial t_A} \right|_{t_A=t_A^\dagger(K_0), K=K_0}. $$

By Assumption 8(a), $\partial C/\partial t_A < 0$. Since $\Gamma > 0$, we have $\partial S/\partial t_A < 0$ at $t_A = t_A^\dagger(K_0)$. As $S(\cdot, K_0)$ is strictly concave in $t_A$ (Remark 1), its unique maximizer $t_A^\dagger(K_0)$ must lie to the left of $t_A^\dagger(K_0)$, thus $t_A^\dagger(K_0) < t_A^\dagger(K_0)$.

**Step 2: Strict divergence for $K \ge K_0$.** Let $\Delta(K) := t_A^\dagger(K) - t_A^\dagger(K)$. From (1) and Assumption 2, we have $\frac{dt_A^\dagger}{dK} > 0$ for all $K \ge K_0$. Moreover, by Lemma 1 (numerator < 0) and Remark 2 (denominator < 0), the
Implicit Function Theorem yields

$$ \frac{dt_A^{\ddagger}}{dK} = - \frac{\frac{\partial^2 S}{\partial K \partial t_A}}{\frac{\partial^2 S}{\partial t_A^2}} < 0 \quad \text{for all } K \ge K_0. $$

Hence

$$ \Delta'(K) = \frac{dt_A^*}{dK} - \frac{dt_A^{\ddagger}}{dK} > 0 \quad \text{for all } K \ge K_0, $$

so $\Delta(K)$ is strictly increasing on $[K_0, \infty)$. Together with $\Delta(K_0) > 0$ from Step 1, this implies $\Delta(K) > 0$ for all $K \ge K_0$. □

*Proof of Proposition 2.* Let $C(t_A, K) := f_C(t_A, T - t_A, g(K))$.

### Part (i): Initial Under-provision at $K = K_0$

By Lemma 4, we have $t_A^{\ddagger}(K_0) < t_A^*(K_0)$. By Assumption 8(a), $C(t_A, K)$ is strictly decreasing in its first argument, $t_A$. It follows directly that $C(t_A^{\ddagger}(K_0), K_0) > C(t_A^*(K_0), K_0)$, which is $C^{\ddagger}(K_0) > C^*(K_0)$.

### Part (ii): Growing Under-provision for $K \ge K_0$

We analyze the derivative of the gap $G(K) := C^{\ddagger}(K) - C^*(K) = C(t_A^{\ddagger}(K), K) - C(t_A^*(K), K)$. By the chain rule:

$$ G'(K) = \underbrace{\left( \frac{\partial C}{\partial t_A} \Big|_{t_A^{\ddagger}} \frac{dt_A^{\ddagger}}{dK} - \frac{\partial C}{\partial t_A} \Big|_{t_A^*} \frac{dt_A^*}{dK} \right)}_{(1) Time Allocation Divergence} + \underbrace{\left( \frac{\partial C}{\partial K} \Big|_{t_A^{\ddagger}} - \frac{\partial C}{\partial K} \Big|_{t_A^*} \right)}_{(2) Direct AI Harm Amplification}. $$

We show that both terms are strictly positive for all $K \ge K_0$.

*Term (1) is strictly positive.*

By Assumption 2 and Equation (1), $\frac{dt_A^*}{dK} > 0$; by Lemma 1 and Remark 2, $\frac{dt_A^{\ddagger}}{dK} < 0$. From Assumption 8(a), we have $\frac{\partial C}{\partial t_A} < 0$. Therefore, the first part of the term, $(\frac{\partial C}{\partial t_A})_{t_A^{\ddagger}}(\frac{dt_A^{\ddagger}}{dK})$, is a product of two negatives and is positive. The second part, $-(\frac{\partial C}{\partial t_A})_{t_A^*}(\frac{dt_A^*}{dK})$, is $-(\text{negative}) \times (\text{positive})$, which is also positive. Their sum is strictly positive.

*Term (2) is strictly positive.* Assumption 8(c) states that $\partial^2 C/(\partial K \partial t_A) < 0$. This implies that the marginal harm of AI on non-cognitive skills, $\partial C/\partial K$, is a strictly decreasing function of $t_A$. Since Lemma 4 established that $t_A^*(K) > t_A^{\ddagger}(K)$, it follows that the marginal harm is less severe at the lower allocation $t_A^{\ddagger}(K)$:

$$ \frac{\partial C}{\partial K} \Big|_{t_A=t_A^{\ddagger}(K)} > \frac{\partial C}{\partial K} \Big|_{t_A=t_A^*(K)}. $$

Rearranging this inequality shows that Term (2) is strictly positive.

Since both terms (1) and (2) are strictly positive for all $K \ge K_0$, their sum $G'(K)$ is strictly positive. This proves that the gap in non-cognitive skills strictly widens as AI capital increases. □
### A.3 Proof of Proposition 3

For any $c \in C$, a fixed time allocation $t_A > 0$, and a general AI capital level $K > K_0$, the naive school (superscript *) and the informed planner (superscript †) choose AI intensity $\alpha \in [0, 1]$ to solve their respective problems. By construction of the set $C$, both solutions are interior.

For the fixed time allocation $t_A > 0$, write

$$C(\alpha) := f_C(t_A, T - t_A, \alpha), \quad C'(\alpha) = \frac{\partial f_C}{\partial \alpha}(t_A, T - t_A, \alpha).$$

The corresponding marginal benefit functions for adopting AI are:

$$\begin{aligned} \mathcal{M}\mathcal{B}_{\text{school}}(\alpha) &:= p_A \frac{\partial f_A}{\partial \alpha}(t_A, \alpha) \\ \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha) &:= w_A(K) \frac{\partial f_A}{\partial \alpha}(t_A, \alpha) + \Gamma \frac{\partial f_C}{\partial \alpha}(t_A, T - t_A, \alpha) \end{aligned}$$

The application of the Implicit Function Theorem requires the marginal benefit curves to be strictly decreasing, which is the standard second-order condition for a unique maximum. Assumption 10(a,c) guarantees this for both agents. The school's marginal benefit slope, $\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha) = p_A \frac{\partial^2 f_A}{\partial \alpha^2}$, is strictly negative by part (a) of the assumption. The planner's slope, $\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha) = w_A(K) \frac{\partial^2 f_A}{\partial \alpha^2} + \Gamma \frac{\partial^2 f_C}{\partial \alpha^2}$, is also strictly negative, as it is the sum of a strictly negative term (from part a) and a non-positive term (from part c). Since both objective functions are strictly concave in $\alpha$, the uniqueness of the interior optima is ensured, and the IFT can be validly applied.

*Proof of part i) – Systematic Over-adoption.* We first establish that the school's perceived marginal benefit from AI is strictly greater than the planner's for any level of adoption $\alpha$. The difference is:

$$\mathcal{M}\mathcal{B}_{\text{school}}(\alpha) - \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha) = \underbrace{(p_A - w_A(K))}_{>0 \text{ by premise}} \underbrace{\frac{\partial f_A}{\partial \alpha}}_{>0 \text{ by Asm. 10(a)}} - \underbrace{\Gamma \frac{\partial f_C}{\partial \alpha}}_{<0 \text{ by Asm. 10(b)}}.$$

The entire expression is the sum of two strictly positive terms, so $\mathcal{M}\mathcal{B}_{\text{school}}(\alpha) > \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha)$ for all $\alpha \in [0, 1]$.

Since the solutions are interior by assumption, we use the FOCs. We evaluate the planner's marginal benefit at the school's optimum $\alpha^*$:

$$\mathcal{M}\mathcal{B}_{\text{planner}}(\alpha^*) < \mathcal{M}\mathcal{B}_{\text{school}}(\alpha^*) = c.$$

Since the planner's FOC is $c = \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha^\dagger)$, we have $\mathcal{M}\mathcal{B}_{\text{planner}}(\alpha^*) < \mathcal{M}\mathcal{B}_{\text{planner}}(\alpha^\dagger)$. As $\mathcal{M}\mathcal{B}_{\text{planner}}(\cdot)$ is a strictly decreasing function, this inequality implies $\alpha^*(c) > \alpha^\dagger(c)$. $\square$

*Proof of part ii)(a) – Impact of Falling AI Costs.* We apply the Implicit Function Theorem to the school's FOC, $G(\alpha^*, c) = \mathcal{M}\mathcal{B}_{\text{school}}(\alpha^*) - c = 0$.

$$\frac{d\alpha^*}{dc} = -\frac{\partial G/\partial c}{\partial G/\partial \alpha^*} = -\frac{-1}{\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^*)} = \frac{1}{\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^*)}.$$

Since $\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^*) < 0$ by Assumption 10(a), we have $\frac{d\alpha^*}{dc} < 0$. The same logic yields $\frac{d\alpha^\dagger}{dc} = \frac{1}{\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^\dagger)} < 0$. Thus, as the cost $c$ falls, AI adoption increases for both agents. $\square$
*Proof of part ii)(b) – Widening Adoption Mismatch.* The mismatch in AI intensity is $\Delta_{\alpha}(c) = \alpha^{*}(c) - \alpha^{\dagger}(c)$. From part (i), $\Delta_{\alpha}(c) > 0$. The mismatch widens as cost falls if its derivative with respect to cost is strictly negative, i.e., $\frac{d\Delta_{\alpha}}{dc} < 0$.

$$ \frac{d\Delta_{\alpha}}{dc} = \frac{d\alpha^{*}}{dc} - \frac{d\alpha^{\dagger}}{dc} = \frac{1}{\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})} - \frac{1}{\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})}. $$

Since both marginal benefit derivatives are negative, the condition $\frac{d\Delta_{\alpha}}{dc} < 0$ is equivalent to:

$$ \frac{1}{\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})} < \frac{1}{\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})} \iff \mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger}) < \mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*}). $$

As both derivatives are negative, this is equivalent to requiring the planner's marginal benefit curve to be locally steeper at their optimum than the school's curve at their optimum:

$$ |\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})| > |\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})|. $$

□

*Proof of part ii)(c) – Widening Gap in Non-Cognitive Skills.* The gap in non-cognitive skills is $G_C(c) = C(\alpha^{\dagger}(c)) - C(\alpha^{*}(c))$. From part (i), $\alpha^{*} > \alpha^{\dagger}$, and from Assumption 10(b), $C'(\alpha) = \frac{\partial f_C}{\partial \alpha} < 0$, so the gap $G_C(c)$ is strictly positive.

The gap widens as cost $c$ falls if its derivative with respect to cost is strictly negative, i.e., $\frac{dG_C}{dc} < 0$. Differentiating with respect to $c$:

$$ \frac{dG_C}{dc} = \left. \frac{\partial C}{\partial \alpha} \right|_{\alpha^{\dagger}} \frac{d\alpha^{\dagger}}{dc} - \left. \frac{\partial C}{\partial \alpha} \right|_{\alpha^{*}} \frac{d\alpha^{*}}{dc}. $$

Using the result from part (ii)(a) that $\frac{d\alpha}{dc} = \frac{1}{\mathcal{M}\mathcal{B}'(\alpha)}$, and noting that both $\frac{\partial C}{\partial \alpha}$ and $\mathcal{M}\mathcal{B}'(\alpha)$ are negative, we can write the expression in terms of absolute values:

$$ \frac{dG_C}{dc} = \frac{\left| \frac{\partial C}{\partial \alpha} \right|_{\alpha^{\dagger}}}{|\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})|} - \frac{\left| \frac{\partial C}{\partial \alpha} \right|_{\alpha^{*}}}{|\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})|}. $$

The gap widens if this expression is negative. We need to show that if the adoption mismatch widens (the condition from part ii-b), then:

$$ \frac{\left| \frac{\partial C}{\partial \alpha} \right|_{\alpha^{\dagger}}}{|\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})|} < \frac{\left| \frac{\partial C}{\partial \alpha} \right|_{\alpha^{*}}}{|\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})|}. $$

This condition holds under the widening condition from part (ii)(b), namely $|\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})| > |\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})|$, which implies

$$ \frac{1}{|\mathcal{M}\mathcal{B}'_{\text{school}}(\alpha^{*})|} > \frac{1}{|\mathcal{M}\mathcal{B}'_{\text{planner}}(\alpha^{\dagger})|} $$
Furthermore, by Assumption 10(c), the marginal harm $|\frac{\partial C}{\partial \alpha}|$ is non-decreasing in $\alpha$. Since part (i) established $\alpha^* > \alpha^\dagger$, we have

$$|\frac{\partial C}{\partial \alpha}|_{\alpha^*}| \geq |\frac{\partial C}{\partial \alpha}|_{\alpha^\dagger}| > 0$$

Combining these facts yields

$$\frac{|\frac{\partial C}{\partial \alpha}|_{\alpha^*}|}{|\mathcal{M}B'_{\text{school}}(\alpha^*)|} > \frac{|\frac{\partial C}{\partial \alpha}|_{\alpha^*}|}{|\mathcal{M}B'_{\text{planner}}(\alpha^\dagger)|} \geq \frac{|\frac{\partial C}{\partial \alpha}|_{\alpha^\dagger}|}{|\mathcal{M}B'_{\text{planner}}(\alpha^\dagger)|}.$$

Thus the second term in $\frac{dC_C}{dc}$ is strictly larger than the first, so $\frac{dC_C}{dc} < 0$ and the non-cognitive skill gap widens as $c$ falls. □

# B Corner Solutions and KKT Conditions

The main text relies on interior-solution assumptions (Assumptions 3, 5, 9, and 11) for analytical clarity, which permits the use of the Implicit Function Theorem for strict comparative-static results. This appendix demonstrates that the paper’s qualitative conclusions are robust to the inclusion of corner solutions. We formalize the Karush-Kuhn-Tucker (KKT) conditions for each optimization problem. Given that all objective functions are strictly concave, these conditions are necessary and sufficient for a global maximum, providing a complete characterization of optimal behavior.<fnref n="16" />

## B.1 KKT Conditions for the Time-Allocation Problem

For any of the planners, the problem is to maximize a strictly concave objective function, which we denote generically as $U(t_A, K)$ (representing $U_{naive}$, $U_{inf}$, or $S$ from the main text), subject to $t_A \in [0, T]$. The Lagrangian for this problem is $\mathcal{L} = U(t_A, K) + \mu_0 t_A + \mu_T (T - t_A)$, where $\mu_0, \mu_T \ge 0$ are the Lagrange multipliers for the constraints $t_A \ge 0$ and $t_A \le T$, respectively. The KKT conditions for an optimal allocation $t_A^*$ are:

$$\frac{\partial U}{\partial t_A}(t_A^*, K) + \mu_0 - \mu_T = 0 \qquad (4)$$

$$\mu_0 t_A^* = 0, \quad \mu_T (T - t_A^*) = 0 \qquad (5)$$

$$t_A^* \in [0, T], \quad \mu_0, \mu_T \ge 0 \qquad (6)$$

These conditions characterize the solution as follows:

* **Interior Solution ($t_A^* \in (0, T)$):** Requires $\mu_0 = \mu_T = 0$, leading to the standard first-order condition $\frac{\partial U}{\partial t_A}(t_A^*, K) = 0$.

* **Corner Solution at 0 ($t_A^* = 0$):** Requires $\mu_T = 0$, implying $\frac{\partial U}{\partial t_A}(0, K) = -\mu_0 \le 0$. The marginal utility of allocating the first unit of time to skill A is non-positive.

<footnote n="16">¹⁶Feasible sets are nonempty compact intervals with interior points, so Slater’s condition holds. With concave objectives and affine constraints, the KKT conditions are necessary and sufficient for global optimality. The strict concavity of the objective functions further ensures this optimum is unique.</footnote>
* Corner Solution at T ($t_A^* = T$): Requires $\mu_0 = 0$, implying $\frac{\partial U}{\partial t_A}(T, K) = \mu_T \ge 0$. The marginal utility of allocating time to skill A remains non-negative at the upper bound.

## B.2 Robustness of Propositions 1 and 2

The logic of monotone comparative statics ensures the qualitative direction of the effects remains robust, but the strict monotonicity claimed in the propositions may fail. For the naive planner, the assumption of increasing differences ensures that by Topkis's Theorem, the optimal allocation $t_A^*(K)$ is a non-decreasing function of K. Conversely, for the informed planners, the assumption of decreasing differences ensures their optimal allocations, $t_A^\dagger(K)$ and $t_A^\ddagger(K)$, are non-increasing functions of K. Because the objectives are strictly concave in $t_A$, the argmax correspondences are single-valued for each K, so the weak monotonicity statements above apply to these (single-valued) selections even when the solutions lie at the bounds.

Therefore, the mismatch, $t_A^*(K) - t_A^\dagger(K)$, is a non-decreasing function of K. Proposition 1 claims the mismatch is strictly increasing. This strict result holds only as long as the solutions remain interior. If the allocations reach boundaries (e.g., $t_A^*(K) = T$ and $t_A^\dagger(K) = 0$ for sufficiently large K), the allocations become locally constant, and the mismatch ceases to strictly increase. Thus, the results generalize only to weak monotonicity when interiority assumptions are relaxed.

## B.3 KKT Conditions for the AI-Intensity Problem

In Section 4.5, an agent chooses AI intensity $\alpha \in [0, 1]$ to maximize a strictly concave objective $\Pi(\alpha) = B(\alpha) - c\alpha$. The Lagrangian is $L = B(\alpha) - c\alpha + \lambda_0\alpha + \lambda_1(1 - \alpha)$, with multipliers $\lambda_0, \lambda_1 \ge 0$. Let $MB(\alpha) := \partial B/\partial \alpha$. The KKT conditions for an optimal intensity $\alpha^*$ are:

$$ MB(\alpha^*) - c + \lambda_0 - \lambda_1 = 0 \qquad (7) $$

$$ \lambda_0 \alpha^* = 0, \quad \lambda_1 (1 - \alpha^*) = 0 \qquad (8) $$

$$ \alpha^* \in [0, 1], \quad \lambda_0, \lambda_1 \ge 0 \qquad (9) $$

These conditions define cost thresholds for adoption:

* No Adoption ($\alpha^* = 0$): Occurs if $c \ge MB(0)$. The marginal cost meets or exceeds the marginal benefit of the first unit of AI.

* Interior Solution ($\alpha^* \in (0, 1)$): Occurs if $MB(1) < c < MB(0)$, in which case the FOC is $MB(\alpha^*) = c$.

* Full Adoption ($\alpha^* = 1$): Occurs if $c \le MB(1)$. The marginal cost is at or below the marginal benefit of the last available unit of AI.

## B.4 Robustness of Proposition 3

The result of systematic over-adoption is robust only in the sense of weak inequality. The wedge in perceived marginal benefits, $MB_{school}(\alpha) > MB_{planner}(\alpha)$, guarantees that $\alpha^*(c) \ge \alpha^\dagger(c)$ for all c.

However, the strict inequality claimed in Proposition 3(i), $\alpha^*(c) > \alpha^\dagger(c)$, does not hold globally if corner solutions are permitted. If the cost c is sufficiently low such that $c \le MB_{planner}(1)$, then it must be that $c < MB_{school}(1)$, and both agents choose full adoption: $\alpha^*(c) = \alpha^\dagger(c) = 1$. Symmetrically, if $c \ge MB_{school}(0)$ (and hence $c \ge MB_{planner}(0)$), both agents choose no adoption: $\alpha^*(c) = \alpha^\dagger(c) = 0$. Thus, strict inequality can also fail at sufficiently high costs.
In this case, the cost $c$ lies outside the interior range specified by Assumption 11. Although the school's marginal incentive to adopt remains strictly higher, a boundary condition ($\alpha = 1$ for sufficiently low costs or $\alpha = 0$ for sufficiently high costs) can become binding for both agents, forcing their choices to coincide. The strict conclusions rely critically on Assumption 11 ensuring interior solutions.

In summary, relaxing the interior-solution assumptions does not alter the paper's qualitative conclusions. The strict comparative statics derived via the Implicit Function Theorem become the weak inequalities predicted by the more general theory of monotone comparative statics, which naturally accommodates boundary solutions. When an optimal choice is at a boundary, the relevant margin is locally inelastic, but the underlying economic forces pushing the allocation in a specific direction persist. The KKT conditions presented here provide the complete and formally correct characterization of the agents' behavior across the entire parameter space.

# C Empirical Methodology and Data

## C.1 Data and Summary Statistics

Our analysis uses data on 194 skills. The 'AI Teaching Impact' variable is the mean educator rating of AI's potential to improve teaching efficiency for a given skill, based on our survey that yielded ratings for 90 unique skills. The AI Workplace 'Skill Disruption Index' is the first dimension of a principal component analysis of LLM-Elo scores from six different models, described in greater detail below. 'Open Skill' is an indicator variable we coded in the Pre-analysis plan, where 1 indicates an 'open' skill (e.g., Critical Thinking) and 0 indicates a 'closed' skill. Table 1 provides summary statistics.

<table><caption>Table 1: Summary Statistics</caption><thead><tr><th>Variable</th><th>N</th><th>Mean</th><th>Std. Dev.</th><th>Min</th><th>P25</th><th>P50</th><th>P75</th><th>Max</th></tr></thead><tbody><tr><td colspan="9"><em>Panel A: Survey skills sample</em></td></tr><tr><td>AI Teaching Impact</td><td>90</td><td>0.05</td><td>1.13</td><td>-2.00</td><td>-1.00</td><td>0.00</td><td>1.00</td><td>2.00</td></tr><tr><td>AI Workplace Skill Disruption Index</td><td>90</td><td>-0.42</td><td>2.34</td><td>-4.54</td><td>-2.55</td><td>-0.06</td><td>1.51</td><td>3.36</td></tr><tr><td>Open Skill</td><td>88</td><td>0.53</td><td>0.50</td><td>0.00</td><td>0.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr><tr><td colspan="9"><em>Panel B: Full sample with imputed data</em></td></tr><tr><td>AI Teaching Impact (imputed)</td><td>194</td><td>0.12</td><td>0.83</td><td>-2.00</td><td>-0.44</td><td>0.21</td><td>0.60</td><td>2.00</td></tr><tr><td>AI Workplace Skill Disruption Index</td><td>194</td><td>0.00</td><td>2.26</td><td>-4.54</td><td>-1.77</td><td>0.43</td><td>1.71</td><td>4.45</td></tr><tr><td>Open Skill</td><td>147</td><td>0.60</td><td>0.49</td><td>0.00</td><td>0.00</td><td>1.00</td><td>1.00</td><td>1.00</td></tr></tbody></table>

*Notes:* Panel A includes the 90 skills rated by survey participants. Panel B includes the full set of 194 skills, with missing 'AI Teaching Impact' scores imputed using an LLM-based measure of teaching impact as described in Appendix D.1. The 'Open Skill' indicator variable has fewer observations because skills were excluded if they were labeled with an 'intermediate' classification according to the pre-analysis plan.

## C.2 LLM Prompts

To ensure methodological transparency, we present the core prompts used to generate the pairwise comparisons for our two indices. Full details and scripts are available in the OSF repository.

### Workplace AI Skill Disruption Index Prompt

This prompt directs the LLM to evaluate which of two skills would be more exposed to AI-driven substitution in the workplace.
### System Prompt:

Assume you are a worker with an average level of expertise in your role, completing a task that requires the skill or knowledge in question. You will evaluate the extent to which access to AI, such as a large language model (LLM), along with other existing software or hardware tools, will impact your ability to perform tasks associated with this skill or knowledge over the next 5-10 years.

While they will take more time to be developed, you can also consider the impact of LLM-Powered Applications or AI-enabled systems that combine LLMs with systems to interact with knowledge-bases, data, documents or APIs in the workplace, properly discounted.

Evaluate the following aspects for each skill or knowledge, while considering also the challenges and realistic limitations in AI and LLMs.

1. Time savings and efficiency: To what extent will AI reduce the time or effort required to perform tasks associated with this skill/knowledge over the next 5-10 years?

2. Quality improvements: How significantly will AI improve the accuracy, reliability, or other measures of quality of outputs related to this skill/knowledge over the next 5-10 years?

Reflect carefully on each skill independently and consider the extent to which human workers using them will be exposed to impacts by AI in the next 5-10 years. 'Exposure' includes economic effects from both labor-augmenting and labor-displacing impacts (Brynjolfsson et al., 2018; Felten et al., 2018; Webb, 2020). Compare the two skills and provide a structured evaluation.

### Return your answer in the following format:

- Reflect on each skill, describing the extent of expected AI impact across the three aspects above.

- Provide a final answer using one of the following:

1. if skill 1 is more exposed,

2. if skill 2 is more exposed, or

3. if both are equally exposed.

### User Prompt Example:

Skill 1: [Basic Skill 1] -- [Subskill 1]

Description: [Description of Skill 1]

Skill 2: [Basic Skill 2] -- [Subskill 2]

Description: [Description of Skill 2]

Jobs depending on which skill/ knowledge are more exposed to impact by AI such as LLMs, or by AI-powered applications, either through labor-augmenting or labor-displacing effects

## Education AI Impact Prompt

This prompt, used for the imputation robustness check, asked the LLM to act as an analytical assistant focused on educational technology. The goal was to determine which of two skills would be more positively
impacted by AI-driven improvements in teaching methods, with the LLM returning a structured JSON object.

**System Prompt:**

You are a thoughtful, analytical assistant.

Your task is to evaluate the potential impact of AI on improving or reducing
the efficiency of teaching specific skills over the next 5-10 years.

You will be given two skills, each with a description. Your goal is to
determine which skill would be more positively impacted by AI-driven teaching
improvements over the next 5-10 years.

Return your answer as structured JSON with the fields:

- reflection: reasoning about both skills.

- answer: 1 if Skill 1 is more positively impacted,
2 if Skill 2 is more positively impacted, or
0 if both are equally impacted.

**User Prompt Example:**

Skill 1: [Basic Skill 1] -- [Subskill 1]

Description: [Description of Skill 1]

Skill 2: [Basic Skill 2] -- [Subskill 2]

Description: [Description of Skill 2]

## C.3 LLM-Based Index Construction

To create a fine-grained ranking, we generated our indices using a comparative judgment approach based on an Elo rating system, which forces trade-offs and avoids the limited variance often seen in direct LLM scoring.

**Elo Rating Procedure** The procedure is as follows:

1. Initialization: Each of the 194 skills starts with an Elo score of $R_0 = 1000$.

2. *Pairwise Comparison:* In each round, two skills, $S_A$ and $S_B$, are randomly selected and presented to an LLM, which judges the pair based on a specific criterion (see prompts).

3. *Symmetric Comparison:* To mitigate order bias and reduce stochastic drift, each pair is presented to the LLM twice in reverse order, ($S_A, S_B$) and ($S_B, S_A$). Inconsistent judgments are treated as a draw.

4. *Score Update:* Scores are updated using the standard Elo formula: $R'_{A} = R_{A} + K(W_{A} - E_{A})$, where $W_{A}$ is the outcome (1 for a win, 0.5 for a draw, 0 for a loss), $E_{A} = (1 + 10^{(R_{B}-R_{A})/400})^{-1}$ is the expected outcome, and $K = 32$ is the learning rate factor.

5. *Final Score:* The process was repeated for 5,000 comparisons and the final Elo scores serve as the index value.

### Index Aggregation
We used this Elo procedure to create two indices. For the workplace AI Skill Disruption Index, we generated Elo ratings from six LLMs (GPT-4o-mini (OpenAI), Claude-Haiku (Anthropic), Gemini-2-Flash (Google), Llama-3.3-70b (Meta AI), Mistral-small-2409 (Mistral AI), and Qwen-2.5-72B (Alibaba)). We then standardized the scores from each model and used the first principal component (PC1) as our final
index, capturing the shared variance across models. For the Education AI Impact LLM ratings, which are
used only for the imputation described in D.1, we followed a similar procedure.

# D Robustness and Supplementary Analysis

To ensure the validity of our main finding (a positive correlation between AI's teaching value and its workplace skill disruption) we conduct a series of robustness checks. These tests, summarized in Table 2, confirm that the result is stable across different correlation measures, is not driven by influential outliers, and holds after accounting for skill characteristics and the non-independence of survey ratings.

## D.1 Multiple Imputation of Missing Survey Data

To test whether our primary finding generalizes beyond the 90 human-rated skills, we imputed the missing 'AI Teaching Impact' scores for the remaining 104 skills in our sample. We employed Multiple Imputation by Chained Equations (MICE), which generates multiple complete datasets to ensure that final estimates properly account for the uncertainty inherent in the imputation process. Our procedure used scikit-learn's 'IterativeImputer' with a 'BayesianRidge' estimator to create $m = 25$ distinct datasets.

The imputation models used a separate, LLM-derived 'AI Teaching Impact' index to predict missing survey ratings; the procedure does not include our 'AI Workplace Skill Disruption Index' (the main independent variable) to avoid mechanical correlation.

After imputation, we performed our pre-specified Kendall's $\tau_b$ analysis on each of the 25 datasets and combined the results using Rubin's rules (Rubin, 1987). Because standard large-sample approximations for the variance of Kendall's $\tau_b$ may be unreliable here, we adopted a more robust non-parametric approach. Within each of the $m = 25$ imputed datasets, we estimated the sampling variance of $\tau_b$ using 1,000 bootstrap replications. The point estimates and their corresponding bootstrapped variances were then pooled across the datasets to produce a single, valid coefficient and confidence interval.

This analysis confirms that the positive correlation found in our survey sample holds in the expanded dataset (pooled $\tau_b = 0.295$, $p < 0.001$). This suggests that the relationship is not an artifact of the particular subset of skills rated by educators and may extend across the broader skill space.

## D.2 Stability of Correlation Measures

Our pre-registered analysis plan specified Kendall's $\tau_b$ to test $H_1$. We supplement this with two common alternatives, Pearson's $r$ and Spearman's $\rho$, to ensure the finding is not an artifact of the chosen statistic. As shown in Panel B of Table 2, the results for the survey-only sample ($N = 90$) demonstrate a stable and statistically significant positive relationship across all three measures. The primary pre-registered estimate is $\tau_b = 0.377$, with a 95% bias-corrected and accelerated (BCa) bootstrap confidence interval of [0.251, 0.486]. The parametric Pearson's $r = 0.524$ and rank-based Spearman's $\rho = 0.519$ are of similar magnitude and also highly significant. To confirm the parametric significance test, a permutation test on Pearson's $r$ (10,000 permutations) also yields $p < 0.0001$.

## D.3 Influence Diagnostics

To check whether our results are driven by a few influential observations, we calculated Cook's distance for each skill in a simple OLS regression of the workplace skill disruption index on the teaching value score. Using a standard threshold of $D_i > 4/n$, we identified two influential points in the survey-only sample
<table><caption>Table 2: Robustness of Correlation Between AI Teaching Impact and Workplace Skill Disruption</caption><thead><tr><th rowspan="2">Method</th><th colspan="2">Survey-only sample (N = 90)</th><th colspan="2">Full Imputed Sample (N ≈ 194)</th></tr><tr><th>Estimate</th><th>95% C.I.</th><th>Estimate</th><th>95% C.I.</th></tr></thead><tbody><tr><td>Panel A: Pre-registered Analysis (Kendall's τ<sub>b</sub>)</td><td></td><td></td><td></td><td></td></tr><tr><td>BCa Bootstrap / Pooled Bootstrap</td><td>0.377***</td><td>[0.251, 0.486]</td><td>0.295***</td><td>[0.190, 0.400]</td></tr><tr><td>Panel B: Alternative Correlation Metrics</td><td></td><td></td><td></td><td></td></tr><tr><td>Pearson's r</td><td>0.524***</td><td>[0.360, 0.654]</td><td>0.439***</td><td>[0.294, 0.563]</td></tr><tr><td>Spearman's ρ</td><td>0.519***</td><td>[0.356, 0.656]</td><td>0.430***</td><td>[0.277, 0.562]</td></tr><tr><td>Panel C: Sensitivity to Influential Observations</td><td></td><td></td><td></td><td></td></tr><tr><td>Kendall's τ<sub>b</sub> (Outliers Removed)</td><td>0.416***</td><td>[0.300, 0.515]</td><td>0.329***</td><td>[0.220, 0.438]</td></tr><tr><td>Pearson's r (Outliers Removed)</td><td>0.582***</td><td>[0.446, 0.690]</td><td>0.494***</td><td>[0.337, 0.624]</td></tr></tbody></table>

Notes: The table reports correlation coefficients. For the survey-only sample, estimates and Bias-Corrected and accelerated (BCa) confidence intervals are based on 5,000 bootstrap replications. For the full imputed sample, estimates are pooled from 25 imputed datasets using Rubin's rules; for Kendall's τ<sub>b</sub>, within-imputation variance is estimated via 1,000 bootstrap replications. Panel C repeats analyses with influential points (identified by Cook's distance) removed. *** p < 0.001, ** p < 0.01, * p < 0.05.

(N = 90): 'Asking questions to ensure clarity' and 'The ability to maintain composure, regulate emotions...'. As shown in Table 2, removing these two points strengthens both the Kendall's τ (from 0.377 to 0.416) and Pearson's r (from 0.524 to 0.582). For the full imputed dataset, a similar analysis identifies and removes influential points from each of the 25 imputed datasets. In this case, removing the outliers strengthens the pooled correlation, with Kendall's τ<sub>b</sub> increasing from 0.295 to 0.329 and Pearson's r increasing from 0.439 to 0.495. This confirms that the underlying positive association is robust and not an artifact of a few influential points.

## D.4 Subgroup and Controlled Analysis

The survey was administered in both English and French. The correlation remains positive and statistically significant within each language subgroup. For the English subset (N = 63), Kendall's τ is 0.482 (p < 0.001). For the French subset (N = 27), Kendall's τ is 0.385 (p = 0.009).

Next, we use OLS to test whether the positive association persists after controlling for key covariates. As shown in Table 3, the AI Teaching Impact score remains a significant, positive correlate of the AI Skill Disruption Index even after including fixed effects for O*NET skill types (β = 0.758, p < 0.001). Model (2), which reverses the specification, likewise shows a significant partial correlation (β = 0.242, p < 0.001), confirming the robustness of the association within this descriptive framework.

## D.5 Rating-level analysis

Our primary analysis averages the ratings for each skill, which is appropriate for testing our hypothesis at the skill level. However, this approach treats each skill's average score as an independent observation, potentially overlooking that multiple ratings come from the same individual. To ensure our findings are not biased by this non-independence, we conduct a supplementary analysis at the rating level (N = 152).

As shown in Table 4, we regress the raw AI Teaching Impact score on our AI Workplace Skill Disruption Index, including fixed effects for skill type and survey language. Crucially, we cluster the standard errors to account for intra-rater correlation. Model (1) clusters standard errors by respondent, while Model (2) employs a two-way clustering by both respondent and skill (although respondent clusters number only 20; two-way clustered p-values may be anti-conservative in this small-G setting.) The positive relationship remains highly significant in both specifications (β = 0.231, p < 0.001), confirming that the association
<table><caption>Table 3: AI Teaching Impact and Workplace Skill Disruption, OLS</caption><thead><tr><th></th><th>(1)<br>Dep. Var.:</th><th>(2)<br>Dep. Var.:</th></tr><tr><th></th><th>AI Skill Disruption Index</th><th>AI Teaching Impact</th></tr></thead><tbody><tr><td>AI Teaching Impact</td><td>0.758***<br>(0.179)</td><td></td></tr><tr><td>AI Workplace Skill Disruption</td><td></td><td>0.242***<br>(0.055)</td></tr><tr><td>Skill Type FE</td><td>Yes</td><td>Yes</td></tr><tr><td>Intercept</td><td>-0.828<br>(0.528)</td><td>0.155<br>(0.189)</td></tr><tr><td>Observations</td><td>90</td><td>90</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.470</td><td>0.273</td></tr></tbody></table>

Notes: OLS results on the human-rated skills sample (N=90). Model (1) regresses the AI Workplace Skill Disruption Index on the AI Teaching Impact score. Model (2) reverses the specification. Both models include fixed effects for the eight O*NET skill types. Heteroskedasticity-robust standard errors (HC1) in parentheses. *** p < 0.001, ** p < 0.01, * p < 0.05.

holds even when accounting for the survey's rating structure. This provides strong evidence that our main skill-level result is robust.

<table><caption>Table 4: Rating-Level OLS with Clustered Standard Errors</caption><thead><tr><th></th><th>(1)<br>Dependent Variable: AI Teaching Impact</th><th>(2)</th></tr></thead><tbody><tr><td>AI Workplace Skill Disruption</td><td>0.231***<br>(0.045)</td><td>0.231***<br>(0.055)</td></tr><tr><td>Skill Type FE</td><td>Yes</td><td>Yes</td></tr><tr><td>Language FE</td><td>Yes</td><td>Yes</td></tr><tr><td>Intercept</td><td>-0.042<br>(0.252)</td><td>-0.042<br>(0.188)</td></tr><tr><td>Observations</td><td>152</td><td>152</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.302</td><td>0.302</td></tr><tr><td>Clustering</td><td>Respondent</td><td>Respondent &amp; Skill</td></tr></tbody></table>

Notes: OLS results on the un-aggregated, rating-level data (N=152). The dependent variable is the raw 'AI Teaching Impact' score. Both models include fixed effects for O*NET skill types and survey language. Standard errors, reported in parentheses, are clustered by respondent in Model (1) and two-way clustered by respondent and skill in Model (2), although with only 20 respondent clusters, two-way clustered inference may be anti-conservative; results should be interpreted with caution. *** p < 0.001, ** p < 0.01, * p < 0.05.

## D.6 Mechanism: The Role of 'Open' vs. 'Closed' Skills

To understand the mechanism driving our main finding, we explore a pre-registered distinction between skills: whether they are 'closed' (procedural, verifiable) or 'open' (subjective, context-dependent). We posit this dichotomy explains the correlation found in $H_1$. Specifically, we test two propositions that underpin our model's core tension: (A) that 'closed' skills are more amenable to AI-assisted teaching (motivating Assumption 2), and (B) that 'closed' skills are also more vulnerable to workplace automation (motivating
Assumption 6). A skill being 'open' also relates to our H₂ finding, as non-cognitive skills are often crucial for solving open-ended problems.

We test these propositions using the rating-level data. As shown in Table 5, we regress our two main indices on an indicator variable for 'Open Skill'. In both models, the coefficient for 'Open Skill' is negative and highly significant. This indicates that, relative to 'closed' skills, 'open' skills are perceived as both less effectively taught with AI's help and less susceptible to workplace automation. This provides direct evidence for the model's central friction: the skills where planners might see the most educational benefit from AI are precisely those most susceptible to devaluation in the labor market. The inference is robust to using wild cluster bootstrap p-values to account for the small number of respondent clusters (Cameron et al., 2008).

<table><caption>Table 5: Mechanism Tests: OLS Regressions on 'Open' Skill Classification</caption><thead><tr><th></th><th>(1)<br>Dep. Var.:<br>AI Teaching Impact</th><th>(2)<br>Dep. Var.:<br>AI Skill Disruption</th></tr></thead><tbody><tr><td>Open Skill</td><td>-0.837***<br>(0.232)</td><td>-3.377***<br>(0.436)</td></tr><tr><td>Intercept</td><td>0.527***<br>(0.165)</td><td>1.623***<br>(0.260)</td></tr><tr><td>Observations</td><td>152</td><td>152</td></tr><tr><td>Adj. R<sup>2</sup></td><td>0.113</td><td>0.441</td></tr></tbody></table>

Notes: OLS results on the rating-level data (N=152). Standard errors, in parentheses, are two-way clustered by respondent and skill. To address the small number of respondent clusters (G=20), we also compute wild cluster bootstrap p-values (clustering by respondent, using Rademacher weights and B = 9,999 replications). These confirm the significance of the 'Open Skill' coefficient in both models (Model 1: *p* = 0.001; Model 2: *p* < 0.001). *** *p* < 0.001, ** *p* < 0.01, * *p* < 0.05.

<table><caption>Table 6: OLS Regressions of AI Skill Disruption controlling for 'Open' skill</caption><thead><tr><th></th><th colspan="2">Dependent variable: AI Skill Disruption Index</th></tr><tr><th></th><th>(1)</th><th>(2)</th></tr></thead><tbody><tr><td>AI Teaching Impact</td><td>1.054***<br>(0.171)</td><td>0.653***<br>(0.152)</td></tr><tr><td>Open Skill</td><td></td><td>-2.830***<br>(0.448)</td></tr><tr><td>Intercept</td><td>-0.525<br>(0.347)</td><td>1.279***<br>(0.286)</td></tr><tr><td>Observations</td><td>152</td><td>152</td></tr><tr><td>Adjusted R<sup>2</sup></td><td>0.251</td><td>0.525</td></tr></tbody></table>

Notes: The unit of observation is a respondent-skill rating (N=152). Standard errors, in parentheses, are clustered by respondent (G=20). *** *p* < 0.001, ** *p* < 0.01, * *p* < 0.05. Wild cluster bootstrap p-values (respondent-level, Rademacher, B = 9,999): AI Teaching Impact (*p* < 0.001), Open Skill (*p* < 0.001).

Finally, we examine the extent to which this 'open/closed' distinction explains the positive correlation from H₁. Table 6 presents a rating-level regression of the AI Skill Disruption Index on the AI Teaching Impact score, with standard errors clustered by respondent. Column (1) shows the baseline positive association. Column (2) adds the 'Open Skill' indicator as a control. While the coefficient on 'AI Teaching Impact' attenuates by
nearly 40% (from 1.054 to 0.653), it remains large and highly significant, a finding confirmed by the wild cluster bootstrap (*p* < 0.001). This suggests that while the ‘open/closed’ dichotomy is a key explanatory factor, it does not fully mediate the relationship. A skill’s fundamental nature is clearly a powerful predictor of both its perceived teachability by AI and its vulnerability in the labor market.

# E Declarations

## E.1 Data & Code Availability

All data and code required to reproduce the findings, along with the pre-registration plan, are available via an OSF repository. Upon publication, the complete replication package will be released publicly under an open license.

## E.2 Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## E.3 Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the authors used ChatGPT (OpenAI) to provide suggestions for improving the human-authored text and for copy-editing. After using this service, the authors reviewed and edited the content as needed and take full responsibility for the content of the published article.

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