Automation, AI, and the Intergenerational Transmission of Knowledge

I acknowledge the financial support of IESE through the High Impact Initiative-course 2024/2025. I declare we have no relevant or material financial interests that relate to the research described in this paper.

Introduction

Recent advances in artificial intelligence (AI) have fueled optimistic forecasts of dramatic increases in productivity and economic growth. Unlike traditional automation, which can only follow predefined instructions, AI technologies learn and adapt from examples, enabling them to perform non-codifiable work once considered inherently human [13, 25, ?]. As a result, many technologists foresee economic transformations that rival or surpass those of the Industrial Revolution [12, 53, 10].

However, the widespread adoption of AI may carry unintended long-term costs. By enabling senior workers to perform more tasks on their own, these technologies may reduce entry-level opportunities and threaten the implicit contract under which younger workers accept lower wages in exchange for training and prospects for future promotion [69]. Such arrangements have historically been crucial for transmitting tacit knowledge—insights that resist codification and develop through repeated observations of practical successes and failures—raising concerns about how future generations will acquire essential expertise [17, 18].

To analyze these concerns, this paper develops a model to explore how advanced automation can disrupt the intergenerational transmission of knowledge and impact long-run economic growth. While existing studies on AI’s economic impact emphasize immediate productivity gains or direct labor-market effects, they overlook the critical role interpersonal interactions play in transferring valuable workplace skills. My framework addresses this gap by introducing automation and AI within a growth model of knowledge diffusion (as in, e.g., [54, 30]). This enables me to capture both automation’s immediate productivity impact and its long-run consequences for the skills of future generations.

The baseline model comprises two overlapping cohorts—novices and experts. Novices are born identical, whereas experts vary in skill. Production requires expert knowledge and the execution of routine tasks, the latter of which can be performed either by novices or machines. By undertaking these routine tasks, novices interact with randomly matched experts, whose skills they observe and assimilate. This gives rise to an apprenticeship-like arrangement wherein novices accept lower wages in exchange for learning opportunities. Novices also independently attempt to innovate by generating random ideas. Thus, when novices transition to expert status, their skill is determined either by the highest skill among the experts they encountered or by the quality of their innovative idea—whichever is greater. Long-run growth arises from both the diffusion of best practices as novices learn from experts and novices’ independent innovations.

The model remains agnostic regarding whether routine tasks involve codifiable or non-codifiable knowledge. Instead, its central feature is that these tasks are entry-level activities typically performed by novices, requiring minimal prior experience. Consequently, the model’s insights broadly apply to various forms of automation but hold particular relevance for AI, given AI’s distinctive ability to automate a wide and expanding range of tasks. In contrast, experts’ skills embody deeply experiential, non-codifiable knowledge that can only be developed through extensive practice and direct interactions with more experienced individuals.

The analysis of the baseline model reveals that the economy converges to one of three distinct long-run regimes, depending on the initial stock of knowledge and the costs of machines. In the Learning Breakdown (LB), extensive automation severely restricts novice-expert interactions, gradually eroding tacit knowledge and ultimately resulting in economic stagnation. Under the Constrained Learning (CL) regime, moderate levels of automation allow some interactions between novices and experts to persist, sustaining limited yet positive long-term growth. Finally, in the Full Learning (FL) regime, high automation costs lead to frequent novice-expert interactions, ensuring robust knowledge transmission and maximal long-run economic growth.

Building on this characterization, I then examine how one-time improvements in automation shape the economy’s long-run trajectory. The analysis reveals a critical trade-off: while cheaper automation delivers immediate productivity benefits, these short-term gains may impose considerable long-term costs. By limiting novice-expert interactions, increased automation may disrupt the transmission of tacit knowledge, permanently lowering the economy’s growth potential. This finding aligns with anecdotal evidence from sectors such as finance and medicine, where the practical skills of junior professionals have deteriorated following the widespread adoption of automated financial tools and robotic surgical systems [17, 18].

To quantify the potential costs highlighted by the analysis above, I perform simple back-of-the-envelope calculations informed by recent debates on the potential economic impact of generative AI. This exercise is neither a formal calibration nor an attempt to closely match empirical data. Moreover, it deliberately abstracts from possible compensating factors, such as the emergence of new junior-level roles or AI’s potential to stimulate innovation. Instead, its purpose is to illustrate how recent advances in AI might hinder long-run economic growth by disrupting the intergenerational transmission of knowledge.

The model naturally yields a simple expression capturing how automating entry-level tasks reduces the economy’s steady-state growth rate. Specifically, the reduction in the annual long-run output growth rate, denoted by $\Delta g^{Y}$ and expressed in percentage points per year, is given by:

where $g^{Y}$ is the economy’s annual steady-state growth rate before the automation shock, $a$ represents the fraction of entry-level tasks automated due to the shock, $x$ denotes the share of steady-state growth driven by the diffusion of tacit knowledge, $\theta$ captures the dispersion of tacit knowledge across experts, and $T$ corresponds to the duration (in years) individuals spend at each career stage—first as novices, then as experts.

I set the baseline growth rate $g^{Y}$ to 2%, reflecting the average per-capita growth rate in the U.S. over the past 150 years [49]. I choose $T = 20$, implying that individuals spend 20 years as novices followed by another 20 years as experts. Following [54], [55], and [30], I adopt $\theta= 0.5$ as my baseline specification. Additionally, guided by empirical estimates from the international trade and knowledge diffusion literature ([33]; [66]), I assume that about two-thirds ($x = 65\%$) of steady-state growth is driven by the transmission of tacit knowledge.

Finally, motivated by recent forecasts assessing the potential one-time productivity gains from AI adoption, I consider two illustrative scenarios for the fraction $a$ of entry-level tasks that are automated. The first is a conservative scenario ($a = 5\%$), aligned with [1] cautious projection of modest productivity improvements of approximately 0.71% over ten years. The second is an aggressive scenario ($a = 30\%$), reflecting [8] more optimistic forecasts of more extensive automation and substantial productivity gains of around 7% over the same period.

The results show that automating 5% of entry-level jobs reduces annual output growth by roughly 0.05 percentage points, while automating 30% leads to considerably larger losses—around 0.35 percentage points per year. When combined with the initial productivity gains forecasted by [1] and [8], these steady-state losses underscore the tradeoff between short-term benefits and long-term costs mentioned above. Under the conservative scenario ($a = 5\%$), output initially rises by 0.71% to 1.18% after ten years relative to no automation, but ultimately declines by 3.31% to 3.76% after a century. In the aggressive scenario ($a = 30\%$), initial gains are significantly higher—between 7% and 11.67% within a decade—yet long-run losses mount sharply, with output falling between 16.28% and 19.78% below the no-automation baseline after 100 years.

Thus far, the analysis has concentrated primarily on the negative implications of automating entry-level tasks. However, automation represents only one dimension of AI’s broader capabilities. Another important dimension involves recent advancements in decision-support systems, commonly known as AI co-pilots, which offer scalable access to specialized expertise previously attainable only through extensive hands-on experience ([13]; [46]). Consequently, these systems have the potential to mitigate some adverse consequences of entry-level automation by democratizing access to expert-level knowledge.

As a final step, I examine the impact of co-pilots in the intergenerational transmission of tacit knowledge, drawing on the approach introduced by [46]. I demonstrate that AI co-pilots can indeed partially mitigate the negative effects of automating entry-level tasks by expanding access to specialized expertise. However, their introduction is not always beneficial: by providing access to expert insights, co-pilots may inadvertently diminish younger workers’ incentives to invest in hands-on learning. Thus, whereas entry-level automation reduces the supply of apprenticeship-like contracts offered by experts, AI co-pilots diminish novices’ demand for these arrangements. Both mechanisms undermine interpersonal knowledge transfer, potentially constraining long-term economic growth.

In sum, this paper challenges the optimistic view that automation and AI will necessarily lead to sustained long-term productivity growth. My results indicate that for AI to increase long-run growth, it must either foster the emergence of new junior-level roles that preserve novice-expert interactions or substantially increase the economy’s underlying rate of innovation. Policymakers, therefore, face a delicate balancing act: promoting AI adoption while safeguarding entry-level opportunities. Effective policy responses might include targeted subsidies for mentorship and training programs, taxes on entry-level automation, or explicit incentives for the development of AI systems designed to complement rather than replace human labor. Without thoughtful intervention, the long-term economic benefits of AI risk being substantially diminished by a silent erosion of tacit expertise.

Related Literature

This paper bridges two strands of literature. First, it extends the literature on automation and AI by analyzing how these technologies influence the intergenerational transmission of tacit knowledge, with significant implications for long-term economic growth. Second, it contributes to the literature on idea flows and economic growth by explicitly modeling how automation and AI affect knowledge diffusion across generations.

The literature on automation has largely adopted a task-based framework, pioneered by [72], [15], and [2]. In this approach, automation technologies do not replace entire occupations but rather specific tasks within jobs, particularly those that are routine and easily codifiable. Building on this framework, [5], [6], [7], [3], and [14] examine how technological innovations—such as automation and labor-augmenting technologies—displace or complement human labor in existing tasks, generate new ones, and reshape employment patterns, wages, inequality, and productivity. Complementary research by [1] and [8] applies this task-based approach to forecast potential one-time productivity increases from AI adoption.¹

More recently, [45], [46], [47] build on the literature on knowledge hierarchies developed by [38] and [40], [41] to study AI’s distinctive features. As highlighted earlier, AI differs from earlier automation technologies in its ability to acquire and deploy non-codifiable knowledge. The knowledge-hierarchies framework is a specialized version of the task-based approach, placing non-codifiable knowledge at its core and deriving task structures explicitly from organizational considerations ([42, 46]). Leveraging this perspective, [45], [46], [47] analyze AI’s implications for occupational choices, organizational structures, labor income, and international trade patterns. Their findings reveal mechanisms absent from conventional task-based analyses, emphasizing the critical roles of AI autonomy, communication, and endogenous organizational responses.

I contribute to this literature by examining how automation and AI influence the intergenerational transmission of knowledge, thereby shaping the distribution of human skills across generations. I do so by embedding automation into an overlapping-generations framework, where the skill levels of successive cohorts critically depend on the extent of interactions between novices and experts. My baseline model builds upon the traditional task-based approach, remaining agnostic as to whether routine tasks involve codifiable or tacit knowledge. Consequently, the model’s insights apply broadly to various forms of automation but hold particular relevance for AI, given its unique capability to automate an expanding set of tasks. In contrast, my extension analyzing AI co-pilots draws inspiration from [46], particularly their treatment of these systems.

[52] also investigate how automation—and specifically AI—can affect human skills. In a recent pre-print, they present neuroscientific evidence that individuals consistently relying on AI for tasks such as essay writing experience a marked decline in neural connectivity and cognitive engagement compared to those performing these tasks without AI support. While complementary, their findings differ fundamentally from the mechanism highlighted in this paper. My analysis emphasizes how automation indirectly reshapes skill distribution by disrupting intergenerational knowledge transmission through diminished novice-expert interactions. In contrast, [52] highlight a direct channel, showing that frequent personal AI use impairs cognitive skill development at the individual level. Taken together, these findings illustrate two distinct yet interconnected pathways through which AI-driven automation could affect human capital.

The literature on idea flows and economic growth emphasizes how knowledge diffuses through interactions among individuals and firms, shaping productivity and growth. [51] and [33] introduce extreme value theory for modeling the diffusion of ideas across firms and economies. Building on this foundation, [54] develops a model where growth arises from continuous exchanges of ideas among heterogeneous agents. [55] extend this framework by explicitly modeling individuals’ optimal allocation of time between producing output and learning from others. [28] explore how knowledge transmission within hierarchical organizations shapes career trajectories as workers advance by learning from coworkers.² My paper contributes to this literature by examining how automation and AI affect these interpersonal interactions—a crucial channel overlooked by existing models.

In the context of idea flows and economic growth, the paper closest to mine is [30] (2018). They also develop an overlapping-generations model in which novices learn from experts, emphasizing the critical role European apprenticeship institutions played in fostering innovation and economic growth leading up to the Industrial Revolution. However, my analysis differs from theirs along three key dimensions.

First, whereas [30] (2018) highlight how medieval institutions—such as clans and guilds—helped alleviate incomplete-contract problems in apprenticeship contracts, I abstract from these incentive conflicts and instead study how modern automation and AI technologies reshape these arrangements. Second, their model captures medieval economic conditions, explicitly featuring population growth and Malthusian constraints. By contrast, my framework aligns more closely with [54], capturing contemporary economies by abstracting from these features. Third, while [30] allow novices to pay upfront for apprenticeships and incorporate parental altruism to guarantee intergenerational knowledge transmission, my model assumes wealth-constrained novices and no altruism. Collectively, these differences yield fundamentally different long-run outcomes and comparative statics from those obtained by [30].

Finally, my paper also contributes to the literature on the economics of apprenticeships. As argued by [58], apprenticeship-like arrangements suffer from contractual incompleteness due to the tacit nature of the knowledge involved. This incompleteness generates various incentive conflicts, which have been extensively studied in the literature (e.g., [39]; [37]), including the paper by [30] mentioned above.

I contribute to this body of work by explicitly examining the impact of automation and AI—an aspect that, to my knowledge, has not yet been explored. To isolate this novel contribution, I abstract from issues related to contract incompleteness, implicitly assuming they do not affect my main qualitative conclusions. Although modeling these incentive conflicts explicitly would enhance realism, it would also significantly complicate the analysis and obscure the core mechanisms and insights I want to emphasize here. Exploring how the incomplete nature of apprenticeship contracts interacts with automation to shape the transmission of knowledge across generations remains an interesting avenue for future research.

Roadmap

The rest of the paper is organized as follows. Section 2 outlines four motivating ideas underpinning the analysis. Section 3 introduces the baseline model, with a focus on the automation of entry-level tasks. Section 4 characterizes the long-run equilibrium of the baseline model and examines the implications of a one-time improvement in automation. Section 5 presents back-of-the-envelope calculations that illustrate how AI-driven entry-level automation could hinder long-term growth by disrupting the intergenerational transmission of tacit knowledge. Section 6 explores the implications of AI co-pilots, and Section 7 concludes.

Four Motivating Ideas

This section describes four motivating ideas. First, advanced automation technologies, especially AI, have fueled optimistic forecasts of dramatic increases in productivity. Second, the most valuable workplace skills remain largely tacit, typically transmitted from experts to novices through personal interactions. Third, automation threatens the expert-novice relationship, potentially weakening the transmission of tacit knowledge. Fourth, despite such disruption, AI also opens new pathways to democratize tacit-like expertise through scalable decision-support systems (commonly referred to as “AI co-pilots”).

Automation, AI, and the Promise of Growth.—As highlighted in the Introduction, recent breakthroughs in AI have dramatically expanded the scope of automation beyond repetitive, clearly defined tasks. Modern AI technologies can now effectively perform sophisticated work—such as complex problem-solving, original research, and creative tasks—that have traditionally been considered the exclusive domain of humans ([13, 25, ?]). Such expanded capabilities drive optimistic forecasts of substantial increases in productivity and economic growth ([12, 53, 10]).

AI has the potential to enhance productivity through two distinct channels ([8]). First, it can directly do so by automating tasks and reducing production costs. Some estimates suggest that this channel could lead to one-off productivity gains realized over ten years, equivalent to between 0.68 and 3.4 percentage points per year throughout this period ([43, 57, 8]).³ Second, and perhaps more importantly, AI’s ability to automate the production of ideas could stimulate innovation, potentially generating a permanent increase in productivity growth ([49]). However, despite widespread speculation, reliable estimates quantifying this last channel remain unavailable.

Tacit Knowledge and the Role of Apprenticeships.—Codifiable knowledge consists of explicit rules and procedures that can be readily communicated through manuals, books, and databases. By contrast, non-codifiable (or tacit) knowledge refers to intuitive skills and insights that individuals possess but find difficult to clearly articulate ([65, 36]).

Although recent technological advances, such as the Information and Communication Technology (ICT) revolution, have enhanced the codifiability of knowledge ([29, 36]), the most valuable workplace skills remain predominantly tacit ([56, 18]). Formal education, which focuses on codifiable knowledge, provides basic qualifications—what [18] (2024b, p. 4) calls “table stakes”—but true competence emerges only when workers integrate foundational knowledge with deeper insights gained through practice and hands-on collaboration with more experienced individuals.

This explains why apprenticeships have historically been crucial for skill transmission—and remain essential today. Though formal apprenticeships have declined, informal apprenticeships remain prevalent across modern economies. Medical training, particularly in surgery, exemplifies this through the principle “see one, do one, teach one” ([18]). Similar arrangements characterize professions such as law, consulting, and finance, where junior associates perform routine tasks under the supervision of senior colleagues, exchanging their labor for training, experience, and eventual promotion ([39]). These exchanges naturally arise because novices typically lack the financial resources to directly compensate experts for training, and the intangible, tacit nature of the knowledge they seek prevents its use as collateral ([20]).

Due to their importance in transmitting tacit knowledge, apprenticeships have long played a pivotal role in fostering innovation and driving economic growth. [44], for example, identifies England’s apprenticeship system as a “neglected factor” underpinning the British Industrial Revolution. Similarly, [30] argue that European apprenticeship institutions significantly contributed to the continent’s relative rise in technological creativity, population growth, and per-capita income in the centuries preceding the Industrial Revolution.

Advanced Automation May be Disrupting the Intergenerational Transmission of Knowledge.— Because non-codifiable knowledge spreads primarily through direct experience, its preservation depends crucially on the continual renewal of knowledgeable individuals across generations ([36]).⁴ Advanced automation technologies may disrupt this renewal by enabling experts to operate independently, reducing juniors’ opportunities for hands-on learning ([18]).

Anecdotal evidence from investment banking and urological surgery illustrates this disruption ([19, 17, 18]). Junior analysts in investment banking are increasingly distanced from senior bankers, who now rely on AI for tasks that were previously delegated to juniors. Similarly, surgical residents are being sidelined from hands-on participation in complex procedures, as attending surgeons are turning instead to robotic systems. This lack of practical experience can significantly compromise juniors’ productivity and performance once they advance to more senior roles. [18] (p. 9) vividly captures this concern through a conversation with a chief of surgery, who highlights that most new surgeons lack critical skills:

I mean these guys can’t do it. They haven’t had any experience doing it. They watched it happen. Watching a movie doesn’t make you an actor.

Recent advances in Generative AI may further exacerbate these disruptions by directly reducing entry-level job opportunities. [23], for instance, leverage the unexpected release and rapid adoption of ChatGPT as a natural experiment, analyzing weekly job-posting data from LinkUp. Using the measure of occupational exposure to Generative AI developed by [35], they find that a one-standard-deviation increase in exposure led firms to reduce weekly job postings for entry-level white-collar positions by approximately 18%. Even more dramatically, Dario Amodei, CEO of Anthropic, recently suggested that AI could soon eliminate half of all entry-level white-collar jobs ([70]).

Decision-Support AI and Democratized Expertise.—While advanced automation threatens knowledge transmission by reducing direct interactions between experts and novices, AI simultaneously creates new opportunities to democratize expertise through decision-support systems [13]. These systems—commonly known as AI co-pilots—could partially offset the erosion of knowledge by enhancing the capabilities of less experienced individuals.

AI co-pilots have the potential to democratize expertise precisely because they can acquire tacit-like knowledge and efficiently scale its application through computational resources. This allows a broad range of individuals to perform sophisticated tasks without personally acquiring the underlying tacit expertise. However, this knowledge remains implicit, embedded within the AI’s complex internal structures, which users can apply but cannot explicitly interpret or fully comprehend. Consequently, AI-embedded knowledge exhibits a hybrid nature, combining the scalability typical of codifiable knowledge with the opacity and experiential features of non-codifiable knowledge [46].

Recent experimental evidence supports the democratizing potential of AI co-pilots [31, 60, 61, 71, 25]. For instance, [25] find that introducing an AI-based conversational assistant increased productivity by 30% among lower-skilled, less-experienced customer support agents, doubling the 15% average gain. Similarly, [60] show that AI assistance reduced the correlation in individuals’ performance across successive writing tasks from 0.41 in the control group (without AI) to 0.14 in the treatment group (with AI), highlighting AI’s potential to narrow performance gaps.

The Baseline Model: Automation of Entry-Level Work

Motivated by the central role of apprenticeship-like arrangements in transmitting tacit knowledge and the threat posed by automation, this section develops a framework to formally analyze how automating entry-level tasks affects intergenerational knowledge transmission and impacts long-run economic growth. While the framework applies broadly to various forms of automation, its insights are especially relevant for AI due to its unique capacity to automate a wider range of tasks.

Later in the paper, I explicitly extend this baseline model to incorporate AI’s unique potential to democratize tacit-like expertise through scalable decision-support systems. I then explore how the hybrid nature of AI-embedded knowledge affects intergenerational knowledge transmission.

Formal Description

The Environment.— Time is discrete and infinite, indexed by $t = 0, 1, 2, \ldots$. The economy is populated by overlapping generations. At each point in time, two cohorts coexist: young agents (“novices”) and old agents (“experts”). Each individual lives for two periods: first as a novice, then as an expert, before retiring. Both cohorts have unit mass, so the total population remains constant at two. All individuals seek to maximize income.

Novices at time $t$ are born identical and without wealth. By contrast, experts in the same period are heterogeneous in skill. Specifically, expert $i$ at time $t$ has a skill level $q_{i,t}$, drawn from a Fréchet distribution with scale parameter $k_t$, shape parameter $1/\theta> 1$, and location parameter $q^{\min} > 0$. The cumulative distribution function (CDF) of $q_{i,t}$ is, therefore:

so $\mathbb{E}_t[q_{i,t}] = q^{\min} + k_t\Gamma(1-\theta)$, where $\Gamma(s) \equiv\int_0^\infty x^{s-1}e^{-x}\mathrm{d}x$ is the Euler’s gamma function.

I refer to $k_t$ as society’s stock of tacit knowledge at time $t$. A higher $k_t$ raises the average expert skill level, shifting the entire distribution of skills upward. This knowledge stock evolves endogenously as novices gain experience interacting with experts in the preceding period and attempt to innovate on their own (as explained below). The initial stock of knowledge $k_0 > 0$ is exogenously given.

Production, Technology, and Labor Demand.— There is a single final good, which serves as the numeraire. Its production is linear in expert skill but requires the completion of a continuum of routine tasks with total measure $N > 1$. Formally, the output produced by an expert with skill $q_{i,t}$ at time $t$ is given by:

Routine tasks can be performed either by novices or machines. For simplicity, experts do not supply routine labor themselves. Experts incur labor costs when hiring novices and rental costs when using machines. Additionally, delegating tasks to novices generates supervision costs, reflecting the difficulties experts face in identifying and correcting mistakes made by novices. These supervision costs increase as novices are assigned more tasks, reflecting the greater complexity involved in effectively overseeing novices who take on greater responsibilities.

Formally, the total cost faced by expert $i$ at time $t$, when assigning a measure $L_{i,t} \in[0,N]$ of tasks to novices (and thus, the remaining measure $N-L_{i,t}$ to machines), is:

where $w_t$ denotes the novice wage, $r_t$ is the rental rate of machines, and $c > 0$ governs the magnitude of supervision costs. I assume that novices cannot identify or select experts based on skill ex ante, explaining why all experts pay the same wages.

Moreover, following [3], I assume that each machine is produced instantaneously using $\rho> 0$ units of the final good and fully depreciates after one period. As a result, the rental rate of machines remains constant at $r_t = \rho$ for all $t \ge0$. This simplifying assumption allows me to focus explicitly on the dynamics of knowledge accumulation rather than physical capital accumulation.

Experts seek to maximize their income during the final period of their economic lives. Therefore, expert $i$’s optimization problem at time $t$ is:

I assume $q^{\min}\geq\rho N$, ensuring that production is profitable even for experts possessing the lowest possible skill level $q^{\min}$.

The solution to this problem depends solely on current prices and parameters—not on individual skill or time. Thus, I omit subscripts and denote the common solution as $L^d(w_t,\rho)$, given explicitly by:

Intuitively, when novice wages exceed the rental cost of machines, experts assign all routine tasks to machines. Conversely, when novice wages are sufficiently low, all experts delegate all tasks to novices. Between these extremes, experts split tasks between novices and machines to optimally balance supervision, labor, and machine costs.

Finally, since all experts find it profitable to produce and all of them—regardless of individual skill—use the same amount of novices and machines, $L^d(w_t,\rho)$ also constitutes the aggregate demand for novice labor.

Novices’ Labor Supply and Skill Acquisition.— A novice chooses her labor supply $l_t$ to maximize expected lifetime income:

where $\mu>0$ captures the disutility of labor, $\beta\in(0,1)$ is the discount factor, and $\mathbb{E}_t[I^*_{i,t+1}\mid l_t]$ denotes the novice’s expected income as an expert in the next period. This expectation depends on the novice’s current labor effort, as each task allows her to interact with a randomly matched expert, whose skill she observes and assimilates. This random matching process captures the idea, introduced earlier, that novices are unable to distinguish experts based on their skill levels ex ante.

Moreover, after this learning period—and before becoming an expert—the novice independently attempts innovation by generating a new idea. The quality of this idea, denoted by $\iota_{i,t}$, is random and depends on society’s current stock of tacit knowledge $k_t$. Specifically, $\iota_{i,t}\sim\operatorname{Fréchet}(\nu^\theta k_t,1/\theta,q^{\min})$, where $\nu\geq0$ captures society’s capacity for innovation.⁵ The novice’s skill level upon becoming an expert in the next period is thus determined by the highest skill among experts she meets and her own generated idea:

Since each expert requires the same amount of novice labor, novice-expert interactions are unbiased, with each encounter representing an independent draw from the current distribution of expert skills. Hence, $q_{1,t}, \ldots, q_{l_t,t} \sim_{\mathrm{i.i.d.}} G_t$. Additionally, since the quality of the novice’s own idea is also independently distributed, her future skill level as an expert—conditional on her labor supply $l_t$—is distributed according to:

where $F_t$ denote the CDF of idea quality at time $t$.

To accommodate a continuous labor supply $l_t \in\mathbb{R}_+$, I interpret $q_{i,t+1} = \max\{q_{1,t}, \ldots, q_{l_t,t}, \iota_{i,t}\}$ using fractional-order statistics, which generalizes traditional order statistics to non-integer sample sizes [67]. This formulation is well defined for all $l_t > 0$ (including $l_t < 1$) and coincides with the standard maximum distribution when $l_t \in\mathbb{N}$.

Since future income is linear in skill, i.e., $I^{*}_{i,t+1} = q_{i,t+1} - C(\cdot)$, and $\mathbb{E}_t[q_{i,t+1} \mid l_t] = q^{\min} + (l_t+\nu)^\theta k_t\Gamma(1-\theta)$, the problem faced by a novice at time $t$ is:

where $C(\cdot)$ does not depend on $l_t$, as experts use the same amount of novices and machines, regardless of their skill.

As evident from (3), novices may accept lower wages if compensated through valuable learning opportunities. In a sense, novices “purchase” knowledge through their labor, giving rise to an apprenticeship-like arrangement. However, as novices are born without wealth, wages cannot fall below zero; in other words, novices cannot pay upfront for training.

Given that the objective in (3) is strictly concave in $l_t$, its solution is unique and, conditional on $(w_t,k_t)$, time-invariant. Moreover, as novices are identical and each cohort has unit mass, the aggregate labor supply at time $t$, denoted by $L^s(w_t,k_t)$, coincides with the individual solution to problem (3).

Competitive Equilibrium.—Since novices lack initial wealth, they cannot pay for training directly, and the labor market may fail to clear at a non-negative wage. In such cases, I assume proportional rationing. That is, each novice supplies a fixed fraction of their desired labor, determined by the economy-wide imbalance between supply and demand. This ensures that all novices are still exposed to a representative cross-section of experts, and the resulting distribution of expert skill retains its Fréchet form over time.

I can now define a competitive equilibrium. For this definition, let $\omega(k_t)$ be the (possibly negative) wage that clears the labor market if the stock of knowledge is $k_t$; that is, $L^d(\omega(k_t),\rho) = L^s(\omega(k_t),k_t)$. In Appendix A, I show that for any $k_t > 0$, a unique such $\omega(k_t)$ exists.

Definition 1 (Competitive Equilibrium). A competitive equilibrium is a sequence $\{w_t^*, k_t^*\}_{t \ge0}$ of wages and knowledge levels, along with a sequence of novice labor $\{L_t^*\}_{t \ge0}$, such that for all $t \ge0$:

• Given $\{w_t^*, k_t^*\}_{t\geq0}$, experts and novices behave optimally, i.e.,

  $L^d(w_t^*, \rho)$ is given by (2) and $L^s(w_t^*, k_t^*)$ by the solution to (3).

• Markets clear (possibly via proportional rationing):

  • If $\omega(k_t^*) \geq0$, then $w_t^* = \omega(k_t^*)$ and $L_t^* = L^d(w_t^*, \rho) = L^s(w_t^*, k_t^*)$.

  • If $\omega(k_t^*) < 0$, then $w_t^* = 0$ and $L_t^* = L^d(0, \rho) = \lambda_t^* L^s(0, k_t^*)$, where $\lambda_t^* = L^d(0, \rho)/L^s(0, k_t^*)$.

• Knowledge evolves according to $k_{t+1}^* = (L_t^* + \nu)^\theta k_t^*$, with $k_0^* = k_0 > 0$.

As shown in Appendix A, for any initial stock of knowledge $k_0$, a competitive equilibrium always exists and is unique. Note also that the law of motion of knowledge implies that:

Some Additional Definitions.— For future reference, society’s per-period welfare at time $t$ is defined as the aggregate income earned by experts and novices, net of machine costs, labor disutility, and supervision costs. Formally, $W_t^* = W(L_t^*, k_t^*; \rho)$, where:

The equilibrium growth rate of the stock of knowledge at time $t$ is given by $g_t^k = (k_{t+1}^* - k_t^*)/k_t^* = (L_t^*+\nu)^\theta- 1$. Equilibrium aggregate output at time $t$, in turn, equals experts’ expected skill $Y_t^* = \mathbb{E}_t[Y_{i,t}] = \mathbb{E}_t[q_{i,t}] = q^{\min} + k_t\Gamma(1-\theta)$. Hence, the growth rate of output is:

When knowledge growth stalls ($g_t^k=0$), output growth also ceases ($g_t^Y=0$). Conversely, as knowledge accumulates ($k_t^* \to\infty$), the influence of the lower bound $q^{\min}$ becomes negligible, and output growth converges precisely to knowledge growth:

Discussion and Interpretation

Before proceeding, I discuss the motivation behind several simplifying assumptions in my modeling approach.

On the Nature of Tasks and Roles.— The model remains agnostic regarding whether routine tasks involve codifiable or non-codifiable knowledge, making it broadly applicable to various forms of automation. The critical feature of these tasks is that they represent entry-level work typically assigned to novices, requiring minimal prior hands-on experience.

For example, robotic systems like the da Vinci Surgical System commonly perform repetitive, well-defined tasks—such as controlled incisions, suturing, and tissue manipulation—that are predominantly codifiable and traditionally performed by resident surgeons. Similarly, in law, AI increasingly handles tasks such as document review and initial contract drafting. While these legal tasks involve nuanced judgments—and thus resist full codification—they remain sufficiently bounded in scope for junior associates, precisely because they require limited experiential knowledge.

In contrast, experts’ skills represent deeply experiential, non-codifiable knowledge. For instance, attending surgeons must manage unforeseen complications and anatomical variations, while senior lawyers handle complex client relationships, sophisticated legal analyses, and settlement negotiations—all of which require extensive practical experience.

On Supervision Costs.—The assumption that assigning tasks to novices—but not machines—incurs supervision costs reflects novices’ higher error rates and lower reliability compared to automated systems. For instance, a senior investment banker highlighted the advantages of technologies like Factset and CapIQ—platforms that automate the collection, interpretation, and calculation of financial metrics—over junior analysts ([19], p. 413):

This [Factset and CapIQ] is great because right now when I ask like five different analysts to run the same analysis I get five different answers. I can’t tell which one is right and it’s always slightly off, especially if it’s a major analysis.

Similarly, surgical residents typically operate more slowly and make more mistakes compared to robotic systems, potentially prolonging anesthesia times and increasing blood loss ([18]).

However, none of the key results qualitatively change if experts also incur positive and weakly increasing supervision costs when using machines, provided these costs are not prohibitively high. This scenario might be relevant with generative AI, which remains prone to errors and hallucinations compared to technologies that follow explicit instructions ([13]; [21]).

On Fréchet, Fractional-Order Statistics, and Novice-Expert Matching.—The assumptions that expert skills follow a Fréchet distribution and that fractional-order statistics can be employed are primarily adopted for analytical tractability, as they ensure the distribution of expert skills remains Fréchet over time. Both assumptions are standard in the literature on knowledge diffusion and economic growth (e.g., [54]; [55]; [30]; [26]).

A side-effect of adopting the Fréchet distribution is that it implies unbounded support, suggesting—somewhat unrealistically—that all productive knowledge already exists at the outset. [30] show, however, that unbounded support can be naturally interpreted as an approximation of their more realistic “Da Vinci model.” In this model, highly advanced ideas—such as Da Vinci’s visionary inventions—initially exist only theoretically and become feasible only as society’s knowledge accumulates over time.

Finally, the assumption that novices cannot observe experts’ skills before matching is also adopted for analytical convenience, as it is also essential to preserve the Fréchet distribution over time. In reality, novices likely have partial information about expert quality, allowing more skilled experts to attract novices at lower wages—much as prestigious law firms or investment banks leverage reputational advantages to hire juniors at reduced pay. Explicitly solving for equilibrium in such a setting would require numerical methods—significantly reducing transparency—though the key insights of the baseline model would likely remain broadly relevant.⁶

On the Production Function.— The production function used in this paper has two simplifying properties, chosen explicitly to maintain analytical tractability. First, it is linear in expert skill, facilitating the computation of expected outcomes. Second, and more importantly, it implies that all experts hire an identical number of novices, irrespective of their own skill levels. This assumption ensures the expert skill distribution retains its Fréchet structure over time, greatly simplifying equilibrium analysis.

Allowing for a more general production function would introduce skill-dependent hiring decisions, requiring numerical methods to solve the equilibrium and thereby reducing transparency.⁷ Nevertheless, we can briefly speculate on the likely implications of this extension.

The qualitative impact of working with a more general production function would depend crucially on two factors: (i) whether more skilled experts hire more novices—accelerating knowledge transmission—or fewer novices—slowing it down; and (ii) whether high- or low-skilled experts respond more strongly to reductions in automation costs. If high-skilled experts replace novices more rapidly, disruptions to intergenerational knowledge transmission would intensify, exacerbating long-term growth losses. Conversely, if low-skilled experts are more responsive, the negative impacts on knowledge diffusion would be partially mitigated.

Finally, for simplicity, the model assumes that all entry-level tasks have identical learning value. Although real-world tasks vary in complexity and educational content, introducing such heterogeneity would significantly complicate the analysis without yielding additional insights.⁸ Moreover,

where $\gamma_l(s)$ and $\gamma_m(s)$ denote the productivity of novices and machines for task $s$, respectively, and $a_{i,t}(s)\in\{0,1\}$ indicates whether expert $i$ allocates task $s$ to a novice ($a_{i,t}(s)=1$) or to a machine ($a_{i,t}(s)=0$). The baseline model’s production function emerges as a special case, setting $\gamma_l(s)=\gamma_m(s)=1$ for all $s\in[0,1]$. Under this generalized specification, the shape of $\gamma_l(s)$ and $\gamma_m(s)$ determines whether higher-skilled or lower-skilled experts hire more novices, and similarly, whether the labor demands of higher-skilled or lower-skilled experts respond more strongly to reductions in $\rho$. given the lack of systematic evidence on precisely which entry-level tasks—high- or low-learning-value—are most likely to be automated, I abstract from these differences, implicitly assuming that automation affects roughly equal proportions of both task types.

On the Zero-Lower Bound on Wages.— The assumption that wages cannot become negative is primarily adopted for realism; it is not essential to the model’s core insights (as discussed in Section 4.1) and can be dispensed with, along with proportional rationing, if unpalatable.

Nevertheless, the assumption is strongly justified on practical grounds. Novices typically lack the financial resources to compensate experts directly and upfront for training (as discussed in Section 2). Moreover, they face severe liquidity and borrowing constraints, as the intangible and tacit nature of the knowledge they seek prevents its use as collateral. Finally, novices must earn at least subsistence wages during training, further reinforcing the practical infeasibility of negative wages.⁹

In most circumstances, reaching the zero-lower bound should not be interpreted literally, but rather as a scenario in which novices work extremely long hours or undertake demanding tasks relative to their earnings. Such conditions are prevalent in fields such as law, investment banking, and consulting, where intense labor effort often serves as implicit payment for training, experience, and potential promotion.

The Impact of Automating Entry-Level Work

In this section, I characterize the economy’s long-run equilibrium outcomes. I then analyze how improvements in the automation of entry-level work—modeled as a reduction in machine costs—can disrupt intergenerational knowledge transmission, ultimately reducing long-term growth. These results set the stage for Section 5, which provides numerical illustrations on the potential magnitude of these disruptions in the context of recent advances in AI.

Long-Run Equilibrium Outcomes

In the long run, the economy converges to one of three stable regimes—Learning Breakdown (LB), Constrained Learning (CL), and Full Learning (FL)—depending on the cost of machines and the initial stock of knowledge. Each regime differs in how routine tasks are allocated between novices and machines, which influences the effectiveness with which tacit knowledge is transmitted across generations.

Proposition 1. Fix an initial knowledge stock $k_0 > 0$ and let:

ends_mid=1

Except for the knife-edge parameter values described in Remarks 1 and 2 below, the competitive equilibrium converges to one of the following stable long-run regimes:

1. Learning Breakdown (LB). If $\rho\le\underline{\rho}$ or $k_0 < k^{\dagger}$, the economy converges asymptotically to the LB regime, characterized by:

During the transition, $k_t^*$, $L_t^*$, and $w_t^*$ are strictly decreasing, weakly decreasing, and weakly increasing in time, respectively.

2. Constrained Learning (CL). If $\underline{\rho} < \rho< \bar{\rho}$ and $k_0 > k^{\dagger}$, the economy reaches in finite time the CL regime, characterized by:

During the transition, $k_t^*$ and $L_t^*$ strictly increase in time, while $w_t^*$ strictly decreases in time.

3. Full Learning (FL). If $\rho\ge\bar{\rho}$ and $k_0 > k^{\dagger}$, the economy reaches in finite time the FL regime, characterized by:

During the transition, $k_t^*$ and $L_t^*$ strictly increase in time, while $w_t^*$ strictly decreases in time.

Remark 1 (Knife-Edge Stable Equilibrium). When $\rho= \underline{\rho}$ and $k_0 \ge k^{\dagger}$, the economy immediately (at $t = 0$) reaches a stable steady state characterized by $L_t^* = 1-\nu$, $w_t^* = 0$, and $k_t^* = k_0$ for all $t \ge0$.

Remark 2 (Unstable Equilibrium). When $\rho< \underline{\rho}$ and $k_0 = k^{\dagger}$, the economy immediately reaches an unstable steady state characterized by $L_t^* = 1-\nu$, $w_t^* = \rho-c(1-\nu)$, and $k_t^* = k^{\dagger}$ for all $t \ge0$.

Proof. See Appendix A. □

Proposition 1 is illustrated in Figure 1. The figure depicts the parameter combinations of initial knowledge stock $k_0$ and machine costs $\rho$ that lead to each of the three long-run regimes.

As the figure shows, when machines are sufficiently cheap ($\rho< \underline{\rho}$), the economy converges to the Learning Breakdown (LB) regime. Even at zero wages, experts do not allocate enough tasks to novices to sustain knowledge accumulation ($L^d(0,\rho)=\rho/c<1-\nu$).¹⁰ Consequently, tacit knowledge erodes over time, novice labor declines, and wages rise, shrinking output. This occurs because each generation is less knowledgeable than the previous one, so novices see fewer learning opportunities and therefore demand higher wages over time. These rising wages then make substituting novice labor with machines increasingly attractive, further accelerating the erosion of knowledge. In the long run, knowledge converges to the minimal skill level $q^{\min}$, and long-run output growth ceases.

The outcome changes when machine prices rise to intermediate levels ($\underline{\rho} < \rho< \bar{\rho}$). Experts are then willing to allocate a larger share of tasks to novices—enough to sustain knowledge accumulation over time—as long as wages remain sufficiently low. In this scenario, the long-run outcome depends on the initial knowledge stock. If $k_0 > k^{\dagger}$, the economy converges to the Constrained Learning (CL) regime, where knowledge and output grow, though at a relatively moderate rate. Otherwise, the economy again descends into the LB regime, characterized by the erosion of knowledge and the eventual stagnation of growth.

The intuition for why the initial stock of tacit knowledge matters is as follows. When $k_0 > k^{\dagger}$, the first generation of novices is willing to accept very low wages in exchange for learning opportunities. Experts respond by hiring more novices, who in turn become highly skilled. This dynamic reinforces itself over time, with each cohort willing to work for less to learn from increasingly knowledgeable predecessors, leading to steady knowledge accumulation. Eventually, however, wages hit the zero lower bound. At that point, novices would pay to displace machines and learn more—but lack the means to do so. Consequently, machines remain in use, limiting the growth of knowledge and output. The solid lines with circle markers in Figure 2, panels (a) and (b), illustrate the resulting wage and knowledge dynamics in this case.

By contrast, when $k_0 < k^{\dagger}$, the initial generation of novices is less eager to learn and demands Notes. Parameter Values: $N = 3$, $\theta= 0.5$, $c = 1$, $\mu= 1$, $\beta= 0.96^{20}$, $\rho= 1.2$, $\nu= 0$, and $q^{\min} = 3.6$. These parameter values imply that $k^{\dagger} = 2.04$. The solid line with the marker corresponds to $k_0 = 2.09 > k^{\dagger}$, so the economy converges to the CL regime. The solid line without a marker corresponds to $k_0 = 1.99 < k^{\dagger}$, so the economy converges to the LB regime.

higher wages. Experts respond by using more machines, thereby reducing available learning opportunities. Each new generation of experts thus becomes less knowledgeable than the previous one, prompting subsequent cohorts of novices to demand even higher wages. This negative feedback loop results in declining knowledge and ultimately a collapse in growth, as illustrated by the solid lines without markers in Figure 2, panels (a) and (b).

Finally, as Figure 1 shows, when machine prices are sufficiently high ($\rho\ge\bar{\rho}$), the economy converges to the Full Learning (FL) regime. In the particular case shown in the figure, convergence occurs regardless of the initial knowledge stock, as the condition $\rho\ge\bar{\rho}$ alone ensures $k_0 > k^{\dagger}$. More generally, however, both high machine costs and a sufficiently large initial knowledge stock are required—mirroring the logic of the CL regime. In the FL regime, experts assign all routine tasks to novices, maximizing novice learning. As a result, each generation reaches its full potential, and output grows at the highest feasible rate.

As noted earlier, the assumption of non-negative wages is primarily made for realism. Beyond its practical justification (see Section 3.2), this assumption ensures the existence of the Constrained Learning (CL) regime—a weak balanced growth path (BGP) where both machines and novices perform a fraction of routine tasks in steady state. Without it, the economy would inevitably converge to either the Learning Breakdown (LB) or Full Learning (FL) regime, ruling out this empirically plausible intermediate scenario. However, relaxing the zero lower bound on wages would not alter the equilibrium’s core mechanics or qualitative insights.

In sum, Proposition 1 characterizes how machine costs and the initial stock of knowledge (along with society’s capacity for innovation) jointly determine the dynamics of intergenerational knowledge transmission, and consequently shape long-run equilibrium outcomes. Building on this characterization terization, the next section shows that improvements in automation can boost short-term productivity but ultimately disrupt intergenerational knowledge transfer, significantly undermining long-term growth and welfare.¹¹

The Impact of Improvements in Automation

Consider an economy that at time $\tau- 1$ is either in the FL regime, the CL regime, or approximately at the LB regime.¹² At the start of period $\tau$, the costs of machines permanently fall from $\rho$ to a lower level $\rho^{\prime} < \rho$. I restrict attention to cases where $\rho^{\prime} < \bar{\rho}$; otherwise, improvements in automation would be irrelevant, as machines would remain unused.

Proposition 2. Let $\mathcal{W}^{*}_{t}(\rho) \equiv W(L^{*}_{t}(\rho), k^{*}_{t}(\rho); \rho)$ be society’s equilibrium per period welfare at time $t$ given machine cost $\rho$.

• Suppose the economy is initially in the CL or FL regime at time $\tau- 1$. Then, a permanent reduction in machine costs at the beginning of time $\tau$ immediately increases welfare but eventually reduces it in the long run:

• Suppose instead the economy is approximately in the LB regime at time $\tau- 1$. Then, a permanent reduction in machine costs at the beginning of time $\tau$ permanently improves welfare:

Proof. See Appendix A.

Proposition 2 shows that the welfare implications of advances in entry-level automation crucially depend on the economic regime prevailing at the time of the shock. When the stock of tacit knowledge has not been depleted, cheaper automation technologies introduce a fundamental trade-off: they immediately boost efficiency but simultaneously disrupt intergenerational knowledge transmission, harming future generations. In contrast, once tacit knowledge has been depleted and intergenerational learning has ceased, this trade-off disappears. At this point, further advancements in automation become unequivocally beneficial, as little human capital remains to be lost.

These insights are illustrated in Figure 3. Panel (a) highlights the first result from Proposition 2, showing the consequences of an automation shock when the economy begins in the CL regime. The Notes. Parameter Values: Both panels use $N = 3$, $\theta= 0.5$, $c = 1$, $\mu= 1$, $\beta= 0.9620$, $\rho= 1.2$, $\nu= 0$, and $q^{\min} = 3.6$. These values imply $k^{\dagger} = 2.04$. In panel (a), the initial capital stock is $k_{0} = 2.09 > k^{\dagger}$, so the economy converges to a CL regime before the automation improvement. In panel (b), $k_{0} = 1.99 < k^{\dagger}$, leading to convergence to an LB regime instead (convergence is only approximate in this case, as it occurs asymptotically). Each panel compares the baseline case with no change in automation costs ($\rho= \rho' = 1.2$) to cases in which the price of machines falls to $\rho' = 1.05$ and $\rho' = 0.75$ at time $\tau$.

figure compares a baseline scenario—in which machine costs remain constant at $\rho= \rho' = 1.2$—with two scenarios featuring an improvement in automation at time $\tau$, lowering machine costs to either $\rho' = 1.05$ or $\rho' = 0.75$.

As the figure shows, the drop in machine costs immediately reduces production costs, increasing per-period welfare—particularly raising the income of current experts. However, these short-term benefits come at the expense of subsequent cohorts. By replacing novice labor with machines, automation reduces critical interactions between novices and experts, causing all subsequent cohorts—including today’s novices—to become progressively less skilled. Consequently, the initial welfare gains enjoyed by today’s experts are gradually offset—and eventually reversed—by the erosion of future human capital. In the figure, when machine prices drop to $\rho' = 1.05$, the economy still converges to a CL regime, but with slower growth. In contrast, when they drop to $\rho' = 0.75$, skill erosion is more severe, and the economy now converges to the LB regime in the long run.

Panel (b) illustrates the second result from Proposition 2, depicting an economy that is already approximately in the LB regime at time $\tau$. Here, further automation causes little additional harm to learning, as the stock of tacit knowledge has already been almost depleted. In this case, automation simply lowers production costs, producing a lasting gain in welfare. Hence, in this scenario, there is no such trade-off between current and future generations.

As discussed in Section 2, recent analyses forecast significant productivity gains from AI adoption over the next decade ([43]; [57]; [8]). A central insight from these studies is that these anticipated improvements are fundamentally oneoff, primarily reflecting efficiency gains realized through initial AI adoption. In the context of my model, these productivity gains map directly onto the immediate welfare improvements described in Proposition 2—benefits that predominantly accrue to the current generation of skilled experts.

However, these immediate benefits must be carefully balanced against significant potential long-term costs. Automating entry-level tasks inherently reduces novice-expert interactions, disrupting the transmission of knowledge across generations that sustains productivity growth over the longer run. Indeed, as documented in Section 2, a decline in these interactions is already impairing skill acquisition among junior professionals in sectors such as finance and medicine, highlighting a tangible risk of skill erosion.

More broadly, Proposition 2 challenges the optimistic notion that automation and AI will necessarily lead to sustained long-term productivity growth. Although conventional wisdom often views AI as a lasting driver of innovation and prosperity, this paper highlights an important countervailing mechanism: automating entry-level tasks gradually reduces novice-expert interactions, weakening society’s foundational stock of tacit knowledge. Overlooking this mechanism could lead policymakers to substantially overestimate AI’s long-term economic benefits, potentially fostering policies or investments that unintentionally weaken, rather than strengthen, future productivity growth.

Quantitative Illustration

Building on the theoretical model developed in Sections 3 and 4, this section performs simple back-of-the-envelope calculations informed by recent debates on the potential economic impact of generative AI. This exercise is neither a formal calibration nor an attempt to closely match empirical data. Moreover, it deliberately abstracts from possible compensating factors, such as the emergence of new junior-level roles or AI’s potential to stimulate innovation. Instead, its purpose is to illustrate how recent advances in AI might hinder long-run economic growth by disrupting the intergenerational transmission of knowledge.

Specifically, I first calculate the potential reduction in steady-state growth arising from AI-driven entry-level automation. Then, drawing on recent estimates by [1] and [8], I explore the trade-off between short-run productivity gains and long-run growth losses, illustrating how output evolves over time under different plausible automation scenarios.

Steady State Losses

To quantify the potential steady-state growth losses arising from automation, I rely on a minimal set of assumptions fully consistent with—but not requiring—the complete structure of the baseline model developed earlier. Specifically, the analysis builds on three core premises:

• (a) OLG Structure: Each generation participates in the economy for exactly two periods—initially as novices and subsequently as experts.

(b) Steady-State Growth: In steady-state, the growth rate of aggregate output equals the growth rate of aggregate productivity, which is also equal to the growth rate of the stock of tacit knowledge.

(c) Evolution of Tacit Knowledge: Society’s tacit knowledge, denoted by $k_t$, corresponds to the scale parameter of a Fréchet distribution and evolves according to:

where $L_t$ denotes the measure of entry-level tasks performed by novices, $1/\theta$ is the shape parameter of the Fréchet distribution, and $\nu> 0$ is a parameter capturing society’s capacity for innovation.

Under these assumptions, the annual steady-state growth rate of output is initially given by:

while after the automation improvement, it becomes:

In these expressions, $T$ represents the number of calendar years corresponding to one OLG period, $L^*$ denotes the steady-state measure of tasks performed by novices prior to automation, and $a \in(0,1)$ denotes the fraction of these tasks automated following AI’s adoption.

The reduction in the steady-state growth rate resulting from increased automation (expressed in percentage points per year) is then given by:

Here, $x$ denotes the fraction of steady-state growth attributable to the diffusion of tacit knowledge, while $1-x$ represents the fraction attributable to the creation of new ideas.

Thus, to explore the long-term growth consequences of the automation shock, I only need five pieces of information:

• (i) The conversion of model periods into calendar years, $T$.

• (ii) The baseline annual growth rate of output prior to the automation shock, $g^Y$.

• (iii) A plausible value for the parameter $\theta$.

• (iv) The fraction of steady-state growth driven by diffusion, $x$.

• (v) The proportion $a$ of entry-level tasks that become automated due to AI.

I set $T = 20$, implying individuals spend roughly ages 25 to 45 as novices and ages 45 to 65 as experts. Regarding the baseline annual growth rate, I follow [49] and set it to $g^Y = 2\%$, reflecting the average per-capita growth rate observed in the U.S. over the past 150 years.

I also adopt $\theta= 0.5$ as my preferred specification, in line with [54], [55], and [30]. [54] initially calibrated this parameter to match observed earnings dispersion in contemporary U.S. data. However, for robustness, I also consider an alternative scenario with $\theta= 0.28$, as used by [28].

I set $x = 65\%$ as my baseline, with robustness checks at $x = 50\%$ and $x = 80\%$. This baseline reflects evidence from [33], who estimate that international technology diffusion alone accounts for roughly 40% of productivity growth in the U.S., and [66], who provides a lower estimate of about 25% for developed economies. Taking the midpoint (32.5%) and doubling it—assuming domestic diffusion contributes equally—results in the baseline choice of $x = 65\%$. The robustness checks capture a more conservative scenario ($x = 50\%$), assuming a smaller domestic contribution, and a more generous scenario ($x = 80\%$), reflecting potentially larger domestic diffusion.

Finally, regarding the proportion $a$ of entry-level tasks automated by AI, I consider three distinct scenarios:

Scenario 1 - $a = 5\%$: This scenario is loosely inspired by the [1]. He uses estimates from [35] to conclude that approximately 19.9% of tasks are currently exposed to AI, and then multiplies this figure by the fraction of AI-exposed tasks currently profitable to automate (23%), as estimated by [68].¹³

Scenario 2 - $a = 30\%$: This scenario is based on [8]. Their calculation employs a broader measure of exposure by [64], who find that around 60% of employment-weighted tasks are exposed to AI. [8] also assume it will be profitable to automate approximately half of these tasks, driven by rapid anticipated reductions in computing costs (about 22% annually, as projected by [24]). These estimates are also roughly in line with recent industry reports, such as [43], which suggests that around a quarter of employment-weighted tasks could potentially be automated using AI.

Scenario 3 - $a = 50\%$: This scenario reflects Dario Amodei’s recent suggestion that generative AI could automate approximately half of all entry-level white-collar jobs ([70]). While this estimate may represent a particularly aggressive automation scenario, incorporating it highlights the potential upper bound of economic disruption.

In all three scenarios, I assume that the tasks being automated are exclusively entry-level tasks (by definition in the Amodei case). This assumption is reasonable, given that entry-level tasks are among the easiest and most cost-effective to automate initially. Nevertheless, this assumption abstracts from potential task re-bundling or the emergence of new junior-level roles, both of which could partially offset the reduction in novice learning opportunities.

Notes. Remaining parameter values: $g^{Y} = 2\%$ and $T = 20$. The baseline scenario is indicated in bold.

The results of this numerical exercise are summarized in Table 1. In the baseline scenario with $\theta= 0.5$ and $x = 65\%$, annual growth losses range from 0.05 percentage points when $a = 5\%$ to 0.35 percentage points when $a = 30\%$. Under the more extreme scenario proposed by [12] ($a = 50\%$), annual growth losses would reach approximately 0.6 percentage points. These losses become larger when $x = 80\%$, as a higher proportion of baseline growth then stems from diffusion—the very channel disrupted by automation. Conversely, losses diminish when $\theta= 0.28$, since a lower $\theta$ shifts a greater share of growth toward innovation, thus mitigating automation’s negative impact.

	$\theta= 0.5$			$\theta= 0.28$
	$x = 50\%$	$x = 65\%$	$x = 80\%$	$x = 50\%$	$x = 65\%$	$x = 80\%$
$a = 5\%$	0.0321	0.0544	0.0829	0.0061	0.0154	0.0325
$a = 30\%$	0.1986	0.3450	0.5421	0.0365	0.0950	0.2072
$a = 50\%$	0.3399	0.6033	0.9787	0.0613	0.1620	0.3640

Short-Run Gains versus Long-Run Losses

The numerical results in the previous subsection highlight a critical tension: scenarios involving extensive automation lead to higher immediate productivity improvement, but simultaneously cause greater disruptions to the intergenerational transmission of knowledge, undermining long-run growth.

For example, [1] considers a relatively modest automation scenario ($a = 5\%$), forecasting cumulative TFP growth of approximately 0.71% over ten years—or about 0.07 percentage points annually. According to my estimates in Table 1, this leads to relatively small long-run annual growth losses of around 0.05 percentage points. Conversely, the more aggressive scenario by [8] ($a = 30\%$) forecasts a cumulative TFP increase of approximately 7% over the same period (about 0.68 percentage points annually), translating into substantially larger long-run annual growth losses—about 0.35 percentage points.

To explicitly illustrate the tradeoff between short-run gains versus long-run losses, I combine my baseline estimates of long-run losses with the short-run productivity improvements estimated by [1] and [8], analyzing the evolution of output at different horizons. This exercise is intended for illustrative purposes only, as the one-time productivity estimates borrowed from these authors originate from frameworks conceptually distinct from mine. For one, in the model developed in Sections 3 and 4, one-time automation improvements primarily manifest as increases in per-period welfare rather than direct output gains. Nevertheless, for simplicity, here I treat these automation improvements as directly contributing to aggregate output.

I evaluate output with and without automation at three distinct horizons—10, 50, and 100 years— under the assumption that disruptions to knowledge transmission materialize gradually over time. During the first ten years, the economy experiences only the initial productivity boosts of [1] and [8]. Between years 10 and 20, half of the estimated long-run productivity loss is phased in. After year 20, the full productivity loss applies permanently. Formally:

where $\psi\in\{0.0071, 0.07\}$ and $\Delta g^{Y} \in\{0.0005, 0.0035\}$ depending on the scenario. Output in the no-automation baseline after $\tau$ years is simply $Y_{\tau}^{\mathrm{no-AI}} = (1+g^{Y})^{\tau}$.

An additional consideration relates to the issue of capital deepening. Since my model assumes complete capital depreciation each period, increases in TFP directly translate into equivalent output gains. By contrast, [1] and [8] incorporate capital deepening into their frameworks, implying that their TFP increases result in more-than-proportional output gains. For consistency, I initially use their TFP estimates directly, but later conduct a robustness check in which I convert their TFP estimates into GDP-equivalent terms.

The results are summarized in Table 2, which compares output with automation relative to the no-automation baseline. The table also reports the “break-even year,” defined as the point at which cumulative losses due to reduced long-term growth fully offset initial short-term productivity gains.

Scenario	10 years	50 years	100 years	Break-Even Year
Acemoglu (2024) ($a = 5\%$)	+0.710 %	-1.154 %	-3.757 %	$\sim29$
Aghion and Bunel (2024) ($a = 30\%$)	+7.000 %	-4.964 %	-19.775 %	$\sim35$
Scenario	10 years	50 years	100 years	Break-Even Year
Acemoglu (2024) ($a = 5\%$)	+1.180 %	-0.693 %	-3.308 %	$\sim37$
Aghion and Bunel (2024) ($a = 30\%$)	+11.667 %	-0.820 %	-16.276 %	$\sim48$

In the scenario inspired by [1], output initially exceeds the no-automation baseline by 0.71% at year 10, reflecting purely short-term productivity gains. However, after 50 years, output falls 1.15% below the baseline, and by 100 years, the gap widens to 3.76%, with a break-even occurring around year 29. Under the more aggressive scenario based on [8], output initially surpasses the baseline by 7% at year 10 but subsequently declines, falling 4.96% below baseline after 50 years and 19.76% after 100 years. In this scenario, the break-even point occurs later, around year 35.

I now conduct the robustness check, where I convert Acemoglu’s (2024) and Aghion and Bunel’s (2024) one-time TFP increases into GDP-equivalent figures. Specifically, I divide their TFP measures

by the labor share, set at 60%, consistent with [1]. This adjustment implicitly assumes that the capital stock grows in proportion to TFP.

Consequently, the 0.71% TFP increase over ten years from [1] translates into a 1.18% GDP increase, while the 7% TFP increase from [8] translates into an 11.67% GDP increase. Thus, this robustness check provides a more generous estimate of the short-run benefits of automation, with revised productivity boosts of $\psi\in\{0.0118, 0.1167\}$, while leaving the estimated long-run growth losses unchanged at $\Delta g^Y \in\{0.0005, 0.0035\}$.

The results of this robustness exercise are presented in tbl:3. The table presents results similar to those in Table 2, albeit with modestly smaller long-run output losses and delayed break-even points. These findings reinforce the conclusion that for AI to sustainably enhance long-term growth, it must either (i) foster the emergence of new junior-level roles to preserve novice-expert interactions, or (ii) substantially increase the economy’s underlying rate of innovation (captured by the parameter $\nu$).

Moving Beyond Automation: Decision-Support AI and Democratized Expertise

The results presented thus far suggest potentially significant disruptions arising from an extensive automation of entry-level tasks, highlighting a critical trade-off between immediate productivity gains and the long-term erosion of skills. Nevertheless, focusing exclusively on entry-level automation does not fully capture the transformative potential of contemporary AI. The next section broadens the analysis by explicitly examining how the hybrid nature of AI-embedded knowledge not only democratizes tacit-like expertise but also reshapes intergenerational knowledge transmission.

AI Decision-Support Systems (AI Co-Pilots)

As discussed in Section 2, AI co-pilots provide scalable access to sophisticated expertise previously attainable only through extensive hands-on experience. Hence, AI co-pilots could partially offset the loss of tacit knowledge caused by entry-level automation by independently enhancing the skills of future cohorts. However, as this section demonstrates, AI co-pilots are not unambiguously beneficial. By offering experts alternative access to tacit expertise, these systems inadvertently diminish novices’ incentives for hands-on learning, as novices anticipate relying on AI assistance in the future. This effect undermines human-driven knowledge transmission and may constrain long-term growth through a mechanism distinct from the one discussed earlier: whereas the automation of entry-level tasks reduces the supply of apprenticeships, AI co-pilots reduce novices’ demand for them.

The Setting

The model is identical to that of Section 3, except that experts now have the option to augment their skills by paying to access an AI co-pilot. However, because the reasoning behind AI recommendations remains opaque, novices cannot indirectly acquire this knowledge by observing experts who use these systems. Consequently, subsequent generations must independently incur the same costs to access and benefit from AI-generated expertise.

Formally, an expert with skill $q_{i,t}$ can access an AI co-pilot with skill level $z_{\mathrm{AI}}$ at a cost $\zeta\rho> 0$, thereby increasing her effective skill to $\max\{q_{i,t}, z_{\mathrm{AI}}\}$. Nevertheless, novices learn only from the expert’s original skill $q_{i,t}$, not from the AI-enhanced skill $\max\{q_{i,t}, z_{\mathrm{AI}}\}$.

Given that expert income is linear in skill, experts decide whether to employ the AI co-pilot following a threshold rule: experts with skills $q_{i,t} \le\phi_{\mathrm{AI}} \equiv z_{\mathrm{AI}} - \zeta\rho$ use the co-pilot, while those with higher skills do not. I assume $\phi_{\mathrm{AI}} > q^{\min}$ to ensure the co-pilot meaningfully enhances the skills of at least some experts. Thus, an expert’s optimal income is given by $I^{*}_{i,t} = \max\{q_{i,t}, \phi_{\mathrm{AI}}\} - C(\cdot)$, where $C(\cdot)$—which is independent of $q_{i,t}$—denotes the costs of hiring novices or renting machines to perform the routine tasks required for production.

This modification alters novices’ expected returns from apprenticeship-based learning. Specifically, a novice’s expected income at time $t+1$ conditional on supplying $l_t$ units of labor at time $t$ is now given by:

Accordingly, the novice optimization problem at time $t$ becomes:

The competitive equilibrium is defined exactly as in Section 3, except that novices’ labor supply now follows equation (6) rather than (3). Since the tacit-like expertise provided by AI cannot be transmitted among humans, the dynamics of tacit knowledge accumulation remain unchanged, with $k^{*}_{t+1} = (L^{*}_{t}+\nu)^{\theta}k^{*}_{t}$. Nonetheless, AI co-pilots alter the effective distribution of expert skills, modifying aggregate output (gross of AI costs) at time $t$ to:

This approach of modeling AI co-pilots is inspired by [?]. The access cost, $\zeta\rho$, represents the computational resources—or “inference compute”—required to use the system. Moreover, the assumption that experts who employ AI have their skills enhanced to $\max\{q_{i,t},z_{\mathrm{AI}}\}$ captures the intuitive idea that experts whose tacit knowledge already surpasses the AI’s capability gain no additional insights from it, having internalized comparable expertise. This formulation is also consistent with the experimental evidence presented in Section 2, which highlights that AI co-pilots predominantly benefit less-skilled individuals.

Finally, the assumption that experts can readily leverage AI-generated insights, yet the reasoning behind these insights remains opaque, closely resonates with the discussion of scalability and interpretability of AI systems in Section 2. As noted there, AI-embedded expertise can be easily duplicated, transferred, and scaled using computational resources, facilitating a broad democratization of sophisticated skills. However, the inherent complexity of these internal representations makes them largely inaccessible to human interpretation ([?]). Thus, modeling AI-enhanced expertise as simultaneously scalable and opaque effectively captures both the hybrid nature of AI knowledge and its democratizing potential.

The Impact of AI Co-Pilots

Having established the formal model, I now turn to the implications of AI co-pilots for long-run economic outcomes. As the next proposition show, AI co-pilots enhance the baseline skill level, thereby partially offsetting the adverse effects of the Learning Breakdown (LB) regime. Nevertheless, these systems may also crowd out critical hands-on learning, potentially trapping the economy in an equilibrium marked by persistent knowledge stagnation and zero long-run growth.

Proposition 3. Fix an initial knowledge stock $k_{0}>0$. Recall that $\underline{\rho}\equiv c$ and $\bar{\rho}\equiv cN$, and define $k^{\dagger}_{\mathrm{AI}}$ as the unique solution to:

so it satisfies $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ whenever $k^{\dagger}>0$. Except for the knife-edge parameter values described in Remarks 1 and 2 below, the competitive equilibrium converges to one of the following stable long-run regimes:

1. Mitigated Learning Breakdown (MLB). If $\rho\leq\underline{\rho}$ or $k_{0}<k^{\dagger}_{\mathrm{AI}}$, the economy converges asymptotically to the MLB regime. This regime is identical to the LB regime of Proposition 1 except that long-run output converges to $Y^{*}_{t}\to z_{\mathrm{AI}}$, rather than $Y^{*}_{t}\to q^{\min}$.

2. Constrained Learning (CL). If $\underline{\rho}<\rho<\bar{\rho}$ and $k_{0}>k^{\dagger}_{\mathrm{AI}}$, the economy reaches in finite time the CL regime described in Proposition 1.

3. Full Learning (FL). If $\rho\ge\bar{\rho}$ and $k_0 > k_{\mathrm{AI}}^{\dagger}$, the economy reaches in finite time the FL regime described in Proposition 1.

Remark 1 (Knife-Edge Stable Equilibrium). When $\rho= \bar{\rho}$ and $k_0 \ge k_{\mathrm{AI}}^{\dagger}$, the economy immediately (at $t = 0$) reaches a stable steady state characterized by $L_t^* = 1 - \nu$, $w_t^* = 0$, and $k_t^* = k_0$ for all $t \ge0$.

Remark 2 (Unstable Equilibrium). When $\rho< \rho$ and $k_0 = k_{\mathrm{AI}}^{\dagger}$, the economy immediately reaches an unstable steady state characterized by $L_t^* = 1 - \nu$, $w_t^* = \rho- c(1 - \nu)$, and $k_t^* = k_{\mathrm{AI}}^{\dagger}$ for all $t \ge0$.

Proof. See Appendix B.

Proposition 3 is illustrated in Figure 4, which shows the economy’s long-run equilibrium outcomes as a function of machine costs ($\rho$) and the initial knowledge stock ($k_0$). Comparing Figure 4 with Figure 1—which depicts the analogous long-run outcomes without AI co-pilots—reveals two key differences.

First, the Mitigated Learning Breakdown (MLB) regime replaces the Learning Breakdown (LB) regime. These two regimes differ only in their long-run equilibrium output: whereas output converges to the minimal skill level $q^{\min}$ in the LB regime, it converges instead to the higher, AI-provided skill level $z_{\mathrm{AI}}$ in the MLB regime. Thus, AI co-pilots partially mitigate disruptions from automation

by democratizing access to expertise.

Second, Figure 4 identifies a distinct region—indicated by a combined dotted and cross-hatched pattern—where adopting AI co-pilots shifts the economy from the Constrained Learning (CL) regime, characterized by sustained knowledge growth, into the MLB regime, characterized by persistent stagnation. This finding highlights that introducing AI co-pilots need not be unambiguously beneficial. Crucially, this stagnation emerges through a mechanism fundamentally different from that previously discussed in Section 4.

Specifically, entry-level automation reduces novice-expert interactions by diminishing experts’ supply of apprenticeship opportunities, as automation enables experts to perform more without novice assistance. In contrast, AI co-pilots suppress novices’ demand for apprenticeships, as novices anticipate future reliance on AI to complement their own skills. Though distinct, these two mechanisms both weaken interpersonal knowledge transfer, progressively eroding the economy’s stock of tacit human knowledge.

It is important to emphasize that the mechanism by which AI co-pilots undermine intergenerational knowledge transmission relies fundamentally on their opacity. Since novices cannot indirectly acquire AI-generated expertise simply by observing experts using these systems, each generation must independently bear the cost of accessing that expertise.

By contrast, if AI co-pilots were interpretable—meaning novices could understand and internalize AI-generated insights merely by interacting with human experts—the resulting dynamics would differ substantially. Interpretability would effectively raise the minimum skill level $q^{\min}$ of the experts’ skill distribution to the AI-provided skill level $z_{\mathrm{AI}}$, permanently elevating society’s baseline stock of tacit knowledge. Such an improvement would unambiguously mitigate the Learning Breakdown (LB) regime and, by pushing society’s knowledge stock beyond critical thresholds, might even enable a transition from stagnation to sustained long-term growth.

Consequently, a direct policy implication emerging from this analysis is for policymakers to actively promote the interpretability of AI systems—defined as their ability to articulate reasoning processes in terms understandable to humans ([32]). Should interpretability prove unattainable,¹⁴ an alternative policy approach would involve incentivizing advancements that expand the scope of embedded tacit-like knowledge in AI systems—in other words, raising the AI-provided skill level, $z_{\mathrm{AI}}$.

Final Remarks

In this paper, I develop a model to examine how advanced automation influences the intergenerational transmission of tacit knowledge. The analysis reveals that automating entry-level tasks yields immediate productivity gains, yet simultaneously undermines long-term growth by eroding the skill base of future generations. Back-of-the-envelope calculations indicate that AI-driven automation of entry-level work could lower the long-run U.S. annual growth rate by approximately 0.05 to 0.35 percentage points, depending on the scale of automation. Although AI co-pilots, by providing scalable access to sophisticated expertise, may partly offset these adverse effects, they can also inadvertently discourage younger workers from investing in hands-on learning.

Taken together, these findings challenge overly optimistic views that automation and AI will automatically sustain productivity growth. The analysis identifies two critical pathways for AI to genuinely enhance long-term economic prosperity: either the emergence of new junior-level roles that preserve robust novice-expert interactions or significant improvements in the economy’s underlying innovation rate. Policymakers thus face a delicate balancing act: encouraging AI adoption while preserving crucial entry-level opportunities. Effective policy interventions might include targeted subsidies for mentorship and training programs, taxes designed to discourage excessive entry-level automation, explicit incentives for developing interpretable AI systems, and support for AI tools that complement—rather than replace—human labor. Without deliberate interventions, the subtle yet significant erosion of tacit knowledge could substantially diminish AI’s long-term economic potential.

Appendix

The Baseline Model: The Competitive Equilibrium

Existence and Uniqueness of a Competitive Equilibrium

Recall that $\omega(k_t)$ denotes the (potentially negative) wage that solves $L^{d}(\omega(k_t),\rho)=L^{s}(\omega(k_t),k_t)$. To establish the existence and uniqueness of a competitive equilibrium, it suffices to show that, for any given $k_t>0$, such a wage $\omega(k_t)$ exists and is unique.

Existence of the competitive equilibrium then follows directly from the existence of $\omega(k_t)$, as this ensures that the three defining conditions of a competitive equilibrium are well-defined and satisfied. Uniqueness also follows directly: for any given knowledge stock $k_t^*$, there is precisely one wage-labor pair $(w_t^*,L_t^*)$ consistent with equilibrium. Since knowledge evolves deterministically, it follows that, from any initial knowledge stock $k_0>0$, the economy traces out a unique deterministic trajectory $\{w_t^*,k_t^*,L_t^*\}_{t\geq0}$.

To formally establish the existence and uniqueness of $\omega(k_t)$, fix any $k_t>0$. Observe that labor demand $L^{d}(w_t,\rho)$ is continuous, weakly decreasing in $w_t$, bounded between 0 and $N$, and satisfies $\lim_{w_t\to-\infty}L^{d}(w_t,\rho)=N$. Meanwhile, labor supply $L^{s}(w_t,k_t)$ is strictly increasing in $w_t$, and satisfies $\lim_{w_t\to-\infty}L^{s}(w_t,k_t)=0$ and $\lim_{w_t\to\infty}L^{s}(w_t,k_t)=\infty$. Thus, for each $k_t>0$, there exists a unique real number $\omega(k_t)$ that satisfies $L^{d}(\omega(k_t),\rho)=L^{s}(\omega(k_t),k_t)$.

I conclude this section by showing two additional properties of $\omega(k_t)$: first, that $\omega(k_t)$ is strictly decreasing in $k_t$, and second, that $\lim_{k_t\to\infty}\omega(k_t)=-\infty$. Because $L^{d}(w_t,\rho)$ does not depend on $k_t$ and $L^{s}(w_t,k_t)$ is strictly increasing in $k_t$, it follows directly that $\omega(k_t)$ is strictly decreasing in $k_t$. Moreover, because $L^{d}(w_t,\rho)$ is bounded and $\lim_{k_t\to\infty}L^{s}(w_t,k_t)=\infty$, if $\omega(k_t)$ were bounded below, say $\omega(k_t)\geq w$, then for $k_t$ large enough we would have $L^{s}(w,k_t)>N\geq L^{d}(\omega(k_t),\rho)$, contradicting market clearing. Hence, since $\omega(k_t)$ is strictly decreasing in $k_t$ and no lower bound exists, it must be that $\lim_{k_t\to\infty}\omega(k_t)=-\infty$. □

Proof of Proposition 1

Recall that the law of motion of knowledge in equilibrium is $k_{t+1}^*=(L_t^*+\nu)^\theta k_t^*$, with $k_0^*=k_0>0$. Given that $\theta\in(0,1)$, it follows immediately that:

Recall also that $\omega(k_t)$ is strictly decreasing in $k_t$, and that:

With these definitions in place, I now proceed to prove Proposition 1. First, I characterize the economy’s long-run equilibrium outcomes through the following five lemmas. Afterward, I describe the transition dynamics toward the steady state.

Lemma A.2.1. If $\rho< \underline{\rho}$, then the competitive equilibrium converges asymptotically to the LB regime. In this regime, $L_t^* \to\rho/(c+\mu) < 1-\nu$, $w_t^* \to\mu\rho/(c+\mu)$, and $k_t^* \to0$, so $g_t^k \to0$ and $g_t^Y \to0$.

Proof. Note first that since $\rho> 0$, then for $\rho< \underline{\rho} = c(1-\nu)$ it must be that $\nu< 1$. With this in mind, notice that $L^d(w_t,\rho) \leq L^d(0,\rho) = \rho/c < 1-\nu$, as $\rho< \underline{\rho} = c(1-\nu)$. Because $L_t^* = L^d(w_t^*,\rho)$, it immediately follows that $L_t^* < 1-\nu$ for all $t \geq0$. Given that $k_{t+1}^* = (L_t^*-\nu)^\theta k_t^*$, with $k_0^* = k_0 > 0$, it must be that $k_t^* \leq(1-\epsilon)^{\theta t} k_0$ for some $\epsilon> 0$. Therefore, $\lim_{t\to\infty} k_t^* = 0$, implying that both $g_t^k$ and $g_t^Y$ converge to zero. Moreover, since $\lim_{t\to\infty} k_t^* = 0$, it also follows that $\lim_{t\to\infty} L^s(w_t^*,k_t^*) = L^s(w_t^*,0) = w_\infty^s/\mu$. Thus, asymptotically, the labor market clears at a strictly positive wage, with $L^s(w_\infty^s,0) = L^d(w_\infty^s,\rho) = L_\infty^* = \rho/(c+\mu) < 1-\nu$ and $w_\infty^s = \mu L_\infty^*$.

Lemma A.2.2. If $\underline{\rho} < \rho< \overline{\rho}$, then the competitive equilibrium converges in finite time to a CL regime if $k_0 > k^\dagger$, and converges asymptotically to the LB regime if $k_0 < k^\dagger$. In the CL regime, $L_t^* = \rho/c \in(1,N)$, $w_t^* = 0$, $g_t^k = \left(\frac{\rho}{c}+\nu\right)^\theta-1$, and $\lim_{t\to\infty} g_t^Y = \left(\frac{\rho}{c}+\nu\right)^\theta-1$. The outcome in the LB regime is the same as in Lemma A.2.1.

Proof. Consider first the case $\nu> 1$. Since $L_t^* \geq0$ and $k_{t+1}^* = (L_t^*+\nu)^\theta$, this implies that $k_t^* \geq(1+\epsilon)^{\theta t}k_0$ for some $\epsilon> 0$, so $\lim_{t\to\infty} k_t^* = \infty$, and hence $\lim_{t\to\infty}\omega(k_t^*) = -\infty$. Thus, there must exist a finite time $T \geq0$ at which $\omega(k_T^*) \leq0$. From time $T$ onward, the economy permanently settles into the CL regime, characterized by $w_t^* = 0$, $L_t^* = \rho/c \in(1-\nu,N)$ (due to proportional rationing), and $g_t^k = \left(\frac{\rho}{c}+\nu\right)^\theta-1 > 0$. The fact that $\lim_{t\to\infty}g_t^Y = \left(\frac{\rho}{c}+\nu\right)^\theta-1$, then follows because $\lim_{t\to\infty} k_t^* = \infty$.

Now consider $\nu\leq1$. Note then that $k_{t+1}^* = k_t^*$ if and only if $L_t^* = 1-\nu$. Additionally, recall from the definition of a competitive equilibrium that $L_t^* = L^d(w_t^*,\rho)$. Hence, if $L_t^* = 1-\nu$, it must be the case that $w_t^* = \rho-c(1-\nu)>0$. Because the labor market clears at a strictly positive wage when $L_t^* = 1-\nu$, it follows that $L^d(\rho-c,\rho) = L^s(\rho-c,k_t^*) = 1-\nu$. Using the first-order condition from problem (3), I find that:

Thus, when $\underline{\rho}<\rho$, the only knowledge level at time $t$ consistent with $L_t^*=1-\nu$ is exactly $k_t^*=k^\dagger$.

Next, recall from Appendix A.1 that $\omega(k_t)$ is strictly decreasing in $k_t$, with $\lim_{k_t\to\infty}\omega(k_t)=-\infty$. Therefore, when $k_t^*>k^\dagger$, I have $w_t^*=\max\{0,\omega(k_t^*)\}<\rho-c(1-\nu)$ and $L_t^*>1-\nu$, and thus $k_{t+1}^*>k_t^*$. Consequently, $w_{t+1}^*<w_t^*$ and $L_{t+1}^*>1-\nu$. Therefore, whenever $k_0>k^\dagger$, it follows that $L_t^*>1-\nu$ for all $t\geq 0. This implies that $k_t^*\geq(1+\epsilon)^{\theta t}k_0$ for some $\epsilon>0$, so $\lim_{t\to\infty}k_t^*=\infty$. Thus, by the same argument given above, there exists a finite time $T$ such that the economy permanently settles into the CL regime.

Conversely, if $k_t^*<k^\dagger$, then $w_t^*=\omega(k_t^*)>\rho-c(1-\nu)$, implying $L_t^*<1-\nu$, which in turn ensures $k_{t+1}^*<k_t^*<k^\dagger$. However, if so, then $w_{t+1}^*>w_t^*$ and $L_{t+1}^*<1-\nu$. Thus, if $k_0<k^\dagger$, it follows that $L_t^* < 1-\nu$ for all $t \ge0$. Following an argument identical to the one used in Lemma A.2.1, I conclude that $\lim_{t\to\infty} k_t^* = 0$, implying that the economy asymptotically converges to the LB regime described in Lemma A.2.1.

Lemma A.2.3. If $\bar{\rho} \le\rho$, then the competitive equilibrium converges in finite time to a FL regime if $k_0 > k^\dagger$, and converges asymptotically to the LB regime if $k_0 < k^\dagger$. In the FL regime, $L_t^* = N$, $w_t^* = 0$, $g_t^k = (N+\nu)^\theta- 1$, and $\lim_{t\to\infty} g_t^Y = (N+\nu)^\theta- 1$. The outcome in the LB regime is the same as in Lemma A.2.1.

Proof. The proof closely follows that of Lemma A.2.2. The only difference arises when iterating forward from an initial knowledge stock $k_0 > k^\dagger$. In this case, once the equilibrium wage reaches zero at some finite time $T \ge0$, equilibrium novice labor remains constant at $L_t^* = N \le\rho/c$ for all periods $t \ge T$.

Lemma A.2.4. If $\rho< \bar{\rho}$ and $k_0 = k^\dagger$, the economy immediately (at $t = 0$) reaches an unstable equilibrium with $L_t^* = 1-\nu$, $w_t^* = \rho-c(1-\nu)$, and $k_t^* = k^\dagger$ for all $t \ge0$.

Proof. In the proof of Lemma A.2.2, I have already shown that if $\rho< \bar{\rho}$ and $k_t^* = k^\dagger$, the unique competitive equilibrium must feature $L_t^* = 1-\nu$ and $w_t^* = \rho-c(1-\nu) > 0$, which implies $k_{t+1}^* = k_t^*$. Thus, if the initial knowledge stock is exactly $k_0 = k^\dagger$, the economy immediately settles into the stationary equilibrium described in the statement of this lemma. However, this equilibrium is unstable, since any small deviation from $k_0 = k^\dagger$ induces the economy to converge toward either the LB, CL, or FL regimes, as established in Lemmas A.2.2-A.2.3.

Lemma A.2.5. If $\rho= \bar{\rho}$ and $k_0 \ge k^\dagger$, the economy converges immediately (at $t = 0$) to a stable but nongeneric equilibrium with $L_t^* = 1-\nu$, $w_t^* = 0$, and $k_t^* = k_0$ for all $t \ge0$. In contrast, if $\rho= \bar{\rho}$ and $k_0 < k^\dagger$, the economy converges to the LB regime described in Lemma A.2.1.

Proof. By an argument analogous to the one I provided in the proof of Lemma A.2.2, when $\rho= \bar{\rho}$ and $k_t^* = k^\dagger$, the equilibrium novice labor satisfies $L_t^* = L^d(0,\rho) = L^s(0,k_t^*) = 1-\nu$, with equilibrium wage $w_t^* = 0$. As a result, when $k_t^* > k^\dagger$, the equilibrium wage remains at $w_t^* = 0$, and the market experiences rationing, i.e., $L_t^* = L^d(0,\rho) = 1-\nu< L^s(0,k_t^*)$. Thus, the knowledge stock remains constant at $k_{t+1}^* = k_t^*$. Consequently, if $\rho= \bar{\rho}$ and $k_0 \ge k^\dagger$, the economy immediately converges to the steady-state equilibrium with $L_t^* = 1-\nu$, $w_t^* = 0$, and $k_t^* = k_0$ for all $t \ge0$.

Conversely, if $\rho= \bar{\rho}$ and $k_t^* < k^\dagger$, I have $w_t^* = \omega(k_t^*) > 0$ and thus $L_t^* < 1-\nu$. Hence, $k_{t+1}^* < k_t^*$, so $L_{t+1}^* < 1-\nu$. Thus, whenever $\rho= \bar{\rho}$ and $k_0 < k^\dagger$, then $L_t^* < 1-\nu$ for all $t \ge0$. Following an identical reasoning as in the proof of Lemma A.2.1, I have that $\lim_{t\to\infty} k_t^* = 0$, so the economy asymptotically converges to the LB regime described in Lemma A.2.1.

Finally, I characterize the transition dynamics to the steady state:

Lemma A.2.6.

• In the transition to the LB regime, $k_t^*$ strictly decreases in time, $L_t^*$ weakly decreases in time, and $w_t^*$ weakly increases in time.

• In the transition to the CL or FL regime, $k_t^*$ and $L_t^*$ strictly increase in time, while $w_t^*$ strictly decreases in time.

Proof. First, consider the transition to the LB regime. In this transition, $k_t^*>k_{t+1}^*$ for all $t\geq0$ (by Lemmas A.2.1–A.2.3 and A.2.5). Since I previously established that the function $\omega(k_t)$ is strictly decreasing in $k_t$, it immediately follows that $\omega(k_t^*)<\omega(k_{t+1}^*)$. Given that $w_t^*=\max\{0,\omega(k_t^*)\}$, this implies that equilibrium wages $w_t^*$ are weakly increasing over time. Furthermore, since $L_t^*=L^d(w_t^*,\rho)$ for all $t\geq0$, and $L^d(w_t,\rho)$ is strictly decreasing in $w_t$, it follows that $L_t^*$ is weakly decreasing in time.

Next, consider the transition to the CL or FL regime. By definition, during this transition, equilibrium wages are strictly positive ($w_t^*=\omega(k_t^*)>0$) for all $t\geq0$, as the economy has not yet reached a steady state. Furthermore, knowledge strictly increases over time ($k_t^*<k_{t+1}^*$ for all $t\geq0$, by Lemmas A.2.1–A.2.3). Because $\omega(k_t)$ is strictly decreasing in $k_t$, it follows directly that equilibrium wages $w_t^*$ are strictly decreasing over time. And since $L_t^*=L^d(w_t^*,\rho)$ for all $t\geq0$, and $L^d(w_t,\rho)$ is strictly decreasing in $w_t$, it follows that $L_t^*$ is strictly increasing in time. $\square$

Proof of Proposition 2

To start, note that:

where $L_t^*(\rho)$ and $k_t^*(\rho)$ denote the equilibrium labor and knowledge at time $t$ given machine cost $\rho$. Also, for future reference, I denote by $w_t^*(\rho)$ the equilibrium wage at time $t$ given machine cost $\rho$. Finally, recall that I restrict attention to cases where $\rho'<\bar{\rho}$, since otherwise, automation improvements have no effect—machines would remain unused.

With this established, I now proceed to prove Proposition 2 using the following two lemmas:

Lemma A.3.1. Suppose the economy is initially in the CL or FL regime at time $\tau-1$. Then, a permanent reduction in machine costs at the beginning of time $\tau$ immediately increases welfare but eventually reduces it in the long run:

Proof. I first establish that $\Delta W_\tau^*>0$. Because the reduction in the cost of machines occurs at the beginning of period $\tau$, I have that $k_\tau^*(\rho)=k_\tau^*(\rho')$. Hence,

Now, recall that $\omega(k_t,\rho)$ denotes the (potentially negative) wage that solves $L^d(\omega(k_t,\rho),\rho)=L^s(\omega(k_t,\rho),k_t)$, where I explicitly note its dependence on $\rho$. Because $L^d(w_t,\rho)$ is strictly increasing in $\rho$ for any given $w_t$, it follows that $\omega(k_t,\rho)$ is also strictly increasing in $\rho$.

Note then that if the economy is initially in the CL or FL regime at time $\tau-1$, wages have already reached zero, i.e., $w^*_{\tau-1}(\rho)=0$, which implies $w^*_{\tau}(\rho)=0$ as well. Since $\omega(k_t,\rho)$ is strictly increasing in $\rho$, it follows that after the reduction in $\rho$, wages remains at the zero lower bound at time $\tau$, i.e., $w^*_{\tau}(\rho')=\max\{0,\omega(k^*_{\tau},\rho')\}=0$. This is because $\omega(k^*_{\tau},\rho')<\omega(k^*_{\tau},\rho)\leq0$, where the final inequality holds due to the earlier observation that $w^*_{\tau-1}(\rho)=w^*_{\tau}(\rho)=0$. The fact that $w^*_{\tau}(\rho')=0$ then implies that $L^*_{\tau}(\rho')=\rho'/c$ for all $\rho'<\bar{\rho}$.

Next, consider each initial regime separately. If the economy is initially in the FL regime at $\tau-1$, then $L^*_{\tau}(\rho)=N$. Evaluating $\Delta\mathcal{W}^*_{\tau}$ when $L^*_{\tau}(\rho)=N$ and $L^*_{\tau}(\rho')=\rho'/c$ yields:

Similarly, if the economy is initially in the CL regime, then $L^*_{\tau}(\rho)=\rho/c$. Evaluating $\Delta\mathcal{W}^*_{\tau}$ when $L^*_{\tau}(\rho)=\rho/c$ and $L^*_{\tau}(\rho')=\rho'/c$ gives:

where the last inequality follows because $\rho'<\rho<\bar{\rho}$ in this case. Thus, in both the CL and FL regimes at time $\tau-1$, it holds that $\Delta\mathcal{W}^*_{\tau}>0$.

I now show that the permanent reduction in machine costs at the beginning of time $\tau$ eventually decreases welfare in the long run. To begin, note that the one-off welfare gain from cheaper machines is bounded above:

It follows immediately that $\Delta\mathcal{W}^*_t\leq\Gamma(1-\theta)[k^*_t(\rho')-k^*_t(\rho)]+B$.

Next, observe that equilibrium novice labor remains permanently lower from period $\tau$ onward. Specifically, the original path involves a constant amount of novice labor $L^*_t(\rho)=L^{\mathrm{old}}\in\{\rho/c,N\}>1$ for all $t\geq\tau$, while under the reduced machine cost scenario, the path involves a sequence $\{L^*_t(\rho')\}_{t=\tau}^{\infty}$, where $L^*_t(\rho')\leq\rho'/c<L^{\mathrm{old}}$ for all $t\geq\tau$ (note that the economy need not immediately settle into a steady state following the reduction in $\rho$).

Since knowledge evolves according to $k^*_{t+1}=(L^*_t+\nu)^\theta k^*_t$, the knowledge stocks under the two paths evolve as follows:

Because $L^{\mathrm{old}}>1-\nu$ and $L^{\mathrm{old}}>\rho'/c$, the difference $k^*_t(\rho')-k^*_t(\rho)$ decreases strictly over time and becomes arbitrarily negative as $t\to\infty$. Given that $\Gamma(1-\theta)>0$ and $B<\infty$, this implies there must exist a finite date $T\in(\tau,\infty)$ such that $\Gamma(1-\theta)[k^*_t(\rho')-k^*_t(\rho)]+B<0$ for all $t\geq T$. Since this expression provides an upper bound on $\Delta\mathcal{W}^*_t$, the desired result immediately follows. ∴

Lemma A.3.2. Suppose the economy is approximately in the LB regime at time $\tau- 1$. Then, a permanent reduction in machine costs at the beginning of time $\tau permanently improves welfare:

Proof. Given that the economy is approximately in the LB regime at time $\tau- 1$, this implies that $\rho\le(c+\mu)(1-\nu) \le c+\mu$ (otherwise, the economy would have converged either to the CL or FL regime). Moreover, $k^{*}_{t}(\rho)<\varepsilon$ and $k^{*}_{t}(\rho')<\varepsilon$ for all $t\ge\tau$, where $\varepsilon>0$ but arbitrarily small. This also implies that $L^{*}_{t}(\rho)<\rho/(c+\mu)+\varepsilon$ and $L^{*}_{t}(\rho')<\rho'/(c+\mu)+\varepsilon$. Using this expressions on (7) and taking $\varepsilon\to0$, yields:

where the second-to-last equality follows because $\rho'<\rho\le c+\mu$, and the last inequality follows because $N>1$ and $\rho>\rho'$.

AI Decision-Support Systems

Existence and Properties of $k^{\dagger}_{\mathrm{AI}}$

Recall that Proposition 3 defines $k^{\dagger}_{\mathrm{AI}}$ as the unique solution to:

and states that $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ whenever $k^{\dagger}>0$. In this appendix, I demonstrate both the uniqueness of the solution to (8) and verify that $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ whenever $k^{\dagger}>0.

To establish uniqueness, first note that the left-hand side of (8) approaches $-\infty$ as $k\to-\infty$ and $+\infty$ as $k\to+\infty$. Moreover, it is strictly increasing in $k$, as its derivative is positive: $\beta\theta\gamma(1-\theta,u(k))+\beta u(k)^{-\theta}e^{-u(k)}>0$. Thus, the left-hand side of (8) crosses zero exactly once, from below, ensuring uniqueness.

Next, to verify that $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ whenever $k^{\dagger}>0$, note that $k^{\dagger}$ is the unique solution to:

This implies that $k^{\dagger}=0$ if and only if $\rho=(c+\mu)(1-\nu)$, and $k^{\dagger}>0$ whenever $\rho<(c+\mu)(1-\nu)$. Similarly, from (8), I have that $k^{\dagger}_{\mathrm{AI}}=0$ if and only if $\rho=(c+\mu)(1-\nu)$, and $k^{\dagger}_{\mathrm{AI}}>0$ whenever $\rho<(c+\mu)(1-\nu)$. Therefore, it remains only to show that $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ for all $\rho<(c+\mu)(1-\nu)$.

To show this, note that for all $k>0$, the left-hand side of (9) is strictly pointwise higher than the left-hand side of (8). This inequality holds since $\phi_{\mathrm{AI}}>\zeta_{\mathrm{AI}}$ implies $u(k)<\infty$, so:

Given that the left-hand side of (8) crosses zero exactly once and from below, this immediately implies $k^{\dagger}_{\mathrm{AI}}>k^{\dagger}$ whenever $\rho<(c+\mu)(1-\nu)$, as desired.

Proof of Proposition 3

The proof closely parallels that of Proposition 1; therefore, I outline only the key steps.

First, if $\rho< \underline{\rho}$, then $L_t^* < 1-\nu$ for all $t \ge0$, and thus $\lim_{t \to\infty} k_t^* = 0$ (given that $k_{t+1}^* = (L_t^* + \nu)^\theta k_t^*$). Consequently, the economy converges asymptotically to the MLB regime. This regime is identical to the LB regime from Proposition 1, except that long-run output converges to $Y_t^* \to z_{\mathrm{AI}}$, rather than $Y_t^* \to q^{\min}$.

Next, suppose that $\rho\ge\underline{\rho}$. Observe that $k_{t+1}^* = k_t^*$ if and only if $L_t^* = 1-\nu$. Moreover, whenever $L_t^* = 1-\nu$, it must be that $w_t^* = \rho- c(1-\nu) \ge0$. Thus, the first-order condition from problem (6) implies that the unique knowledge level at time $t$ consistent with $L_t^* = 1-\nu$ is precisely $k_t^* = k_{\mathrm{AI}}^\dagger$, where $k_{\mathrm{AI}}^\dagger$ solves:

It then follows—by exactly the same logic as in the proof of Proposition 1—that whenever $k_0 > k_{\mathrm{AI}}^\dagger$, then $\lim_{t \to\infty} k_t^* = \infty$, so the economy reaches either the CL or the FL regime in finite time. Conversely, whenever $k_0 < k_{\mathrm{AI}}^\dagger$, then $\lim_{t \to\infty} k_t^* = 0$, and the economy converges asymptotically to the MLB regime. ∎