A Mathematical Model for Human Capital Reproduction in Enterprises under Artificial Intelligence Adoption

Introduction. At the current stage of scientific and technological progress, the rapid development of AI technologies is gradually becoming a key driver of socio-economic development. For instance, the spread and adoption of AI technologies are projected to boost the global economy by 15% by 2035. At the same time, some researchers [1] point to a negligible impact of AI on productivity, suggesting the presence of the Solow paradox [2] in this new technological landscape. This is due to the fact that, despite the significant qualitative shift in new technologies from basic algorithms to large language models, existing production and business processes [3] — as well as management approaches — rely heavily on human involvement and may constrain the pace and scale of implementation. At the same time, transforming these processes is impossible without assessing the prospects and key factors involved in their implementation. AI technologies are adopted directly at the enterprise level, where they drive the transformation of production processes. In this context, the development of human capital — a crucial factor of production — through the expansion of workers' knowledge and competencies serves as a key condition for realizing the potential of AI [4] and overcoming the Solow paradox. The foregoing underscores the importance of studying reproduction at the enterprise level.

In classical theory, human capital is viewed through the lens of investments in education, knowledge, and skills that yield future economic returns. The theory pioneers, T. Schultz and G. Becker, define human capital as an accumulated stock of knowledge, skills, competencies, ideas, and health capable of generating a future income stream [5]. They treat training expenditures as investments rather than mere costs, an approach that enables the use of quantitative methods to evaluate the efficiency of management decisions at both the individual and enterprise levels.

Economic-mathematical modeling is one of the main tools for researching human capital. It formalizes the relationship between various factors of its development and socio-economic results. In some papers [6], human capital is considered as a factor of endogenous growth, including the accumulation of professional knowledge and skills. It is interpreted as a source of long-term productivity.

At the same time, modern research highlights the increasing non-linearity of processes related to the development and utilization of human resources, driven by the rapid obsolescence of competencies, the emergence of new technologies, and workforce mobility [7]. This underscores the need for models capable of describing how the factors affecting enterprise personnel change over time, while also considering the competitive environment.

To construct a dynamic economic-mathematical model of human capital reproduction, it is required to identify the key factors driving its change under modern conditions and to determine how they are described in the literature.

One of the most common models for assessing it through the return to education is the Mincer equation [8]. It relates salary levels to length of education and work experience. Empirical evidence confirms the correlation of education indicators with income and productivity growth, making investment in training an important source of human capital accumulation [9]. However, some studies have shown that returns to education are heterogeneous across cohorts, occupations, and economic sectors [10], which may reflect their nonlinearity and dependence on technological change [11]. This implies that assessing human capital through static regression models fails to fully capture the processes of its reproduction at the enterprise level, where factors such as the rate of learning, the obsolescence of competencies, and competitive conditions must be taken into account.

The current stage of scientific and technological progress intensifies the dynamics of workforce development, as employee competencies can both accumulate and become obsolete due to new technologies and changing professional requirements. Consequently, it is appropriate to view them as an asset subject to depreciation [12]. Some research papers indicate that the rate of competence obsolescence varies across education levels and task types, potentially accelerating in occupations involving non-routine analytical and creative functions [13]. Amidst the acceleration of scientific and technological progress, the “half-life” of high-tech knowledge is shortening, thereby increasing the importance of continuous learning and retraining.

For micro-level modeling, this circumstance indicates that an enterprise human capital must be described not only through the mechanism of competence accumulation but also through the mechanism of competence obsolescence and value loss — factors that drive depreciation. In this context, depreciation is not a one-off or random occurrence but a systematic process reflecting the mechanism for subsequently recouping losses [14] — losses that can arise even when investments in employee training are increasing.

A distinct area of contemporary research focuses on assessing the impact of AI technologies on the workforce. Empirical studies on the adoption of AI assistants based on large language models in certain occupations demonstrate their ability to accelerate work processes and boost productivity, particularly among less experienced employees [15][16]. This effect stems not only from workers' immediate access to information but also from knowledge accumulation: employees who actively utilize recommendations from AI assistants show sustained performance improvements over time, thereby contributing to the overall accumulation of human capital.

For economic-mathematical modeling, this implies that AI can act as a distinct factor in human capital development, affecting its dynamics over time. It is worth noting that the greatest gains are observed among groups with low initial skill levels, whereas the increase may be more modest for highly skilled workers. Consequently, AI contribution to the human resource dynamics model should be accounted for as a systematic source of acceleration — one potentially dependent on the state of the system (specifically, the level of human capital and competencies) and the context of production tasks.

Another factor affecting the development of an enterprise human capital is intercompany mobility of workers. Traditionally, it is seen as a risk factor for the return on investment in training, since the enterprise incurs costs associated with updating competencies. However, some of them may not pay off due to the dismissal of employees [17]. Modern research shows that mobility is simultaneously a channel for the dissemination of knowledge and practices between organizations. A study using the example of manufacturing industries showed that hiring workers from more productive enterprises leads to an increase in the productivity of the host enterprise, and the effect is asymmetric in nature, since the influx from productive enterprises is more pronounced and significant [18]. These studies also emphasize the importance of industry specificity, since transitions within an industry ensure the transfer of the most appropriate competencies.

Thus, the dynamics of an enterprise human capital is determined not only by internal investments but also by the external inflow or outflow of employees possessing the requisite competencies. Consequently, an enterprise can accumulate human resources by attracting workers with the required skills, yet it can also lose them through staff turnover.

The examined development factors and characteristics of human capital make it possible to formulate key requirements for its reproduction model, which encompasses the processes of formation, accumulation, and utilization. First, the model must be dynamic, that is, it should describe workforce potential as a system that evolves over time, wherein mechanisms of accumulation and the technologically driven loss of employee competencies operate simultaneously [12][19].

Second, given the diminishing marginal returns as one approach leading industry practices, the dynamics of human capital must be nonlinear and account for the limitations of competence accumulation [20].

Third, amid the spread of AI technologies, their contribution must be singled out as a factor capable of influencing knowledge dissemination within an enterprise with effects that may vary across occupational groups [21].

Finally, the model must account for the external competitive environment [18], as inter-firm mobility drives the inflow or outflow of workers with the requisite competencies. This necessitates considering interactions between enterprises and allows for shifting from an equation for an individual firm to a systemic formulation involving a set of competitors. A convenient basis for this approach is the use of relative human resource metrics, interpreted as a firm's position within the industry, which provides comparability among enterprises and imposes a natural constraint on dynamics.

Despite the substantial body of research on human capital and modeling the factors driving its development, most studies fail to fully account for the interconnected processes occurring at the enterprise level, namely corporate training, competence depreciation, adoption of new technologies, and workforce mobility. Consequently, enterprises lack formalized tools for assessing how training investments, AI adoption, and workforce mobility jointly shape the trajectory of human capital development over time. Development of an economic-mathematical model of human capital reproduction will enable a comprehensive assessment of its current and future state, thereby providing the scientific validity of management decisions at enterprises.

The objective of this article is to develop and test a dynamic economic-mathematical model of human capital reproduction at the enterprise level, taking into account the key factors of its transformation. To achieve this aim, the following tasks must be addressed:

• to validate the factors of reproduction of human resources of enterprises in the context of the spread of AI technologies;
• to construct a dynamic model of human capital reproduction based on an ODE system;
• to pilot-test a dynamic model of human capital reproduction based on the implementation of three forecast scenarios;
• to develop proposals and practical recommendations for improving human resource management at enterprises based on the results of the pilot implementation.

Materials and Methods. Human capital indicators and dynamics factors. To build a dynamic economic-mathematical model of human capital reproduction, comparable indicators of the level of employee competencies were introduced, and the factors affecting their change over time were formalized. In the framework of this study, human potential is considered at the enterprise level, taking into account the occupational structure of personnel and the competitive environment.

If a set of enterprises in an industry is represented as j = 1, 2, …, N, and the occupational groups of workers as
i =1, 2, …, M, then the human capital indicator for occupational group i at the enterprise j at time t takes the form:

(1)

Value H_ji(t) (1) is interpreted as an aggregate quantitative assessment of the level of knowledge, skills, experience, and applied competencies of employees within a specific occupational group. In practice, H_ji(t) can be derived from internal HR metrics [19] (such as qualification status and competency assessment results), labor productivity indicators, training records, and other HR analytics tools. For convenience, H_ji can be measured in arbitrary units (normalized indices).

For the enterprise under study, the indicator is expressed as equation (2).

(2)

To account for the enterprise competitive position, an industry-wide maximum for human capital within an occupational group i is introduced:

(3)

Hereafter, value H_{max j} will be treated as a fixed industry frontier for occupational group i over the modeling horizon. In practice, H_{max i} (3) may be defined as the maximum value within the observed sample of enterprises at the base time t₀ (representing an industry best-practice benchmark) or as a normative (target) competency level. This assumption allows h_ji(t) to be interpreted as the enterprise position relative to the established benchmark and enables the use of autonomous dynamics for h_ji(t). Accounting for the endogenous, time-varying frontier H_{max i}(t), calculated as the maximum current state among the enterprises, requires a separate modification of the model and lies beyond the scope of this article.

A relative indicator of human capital is determined based on this measure [20]:

(4)

For the enterprise in question,

(5)

Consequently, value h_ji = 1 (4) corresponds to the leader level (the maximum observed in the industry), while a decrease in h_ji reflects a lag in the quality and relevance of the competencies of the respective occupational group. The use of a relative scale provides comparability between enterprises and occupational groups despite differences in absolute metrics. It also accounts for growth constraints, as approaching the frontier is accompanied by diminishing opportunities to further accelerate the accumulation of human resources.

Key controllable inputs shaping human capital dynamics are defined by the following indicators:

• investments in the education and development of the occupational group i [22][5]:

(6)

• level of adoption and use of AI technologies in the group operations [21]:

(7)

Variables EDU_i(t) and AI_i(t) characterize, respectively, the intensity of in-house training (6) and the degree of using AI tools in the activities of occupational group i (7). In applied calculations, they can be set on the basis of corporate statistics. They are considered as manageable parameters of personnel policy and digital transformation.

To describe the reaction of human capital to management actions and the external environment, the following parameters are introduced:

• α_i — efficiency coefficient of educational investments for group , defined as the rate or efficiency of competence accumulation at a given intensity of EDU_i;
• β_i — coefficient representing the impact of AI adoption on the dynamics of the workforce in group i, defined in terms of accelerated learning, increased efficiency in knowledge application, and the diffusion of best practices within the enterprise;
• δ_i> 0 — human capital depreciation coefficient for group , defined as the rate at which competences become obsolete due to technological changes and evolving task requirements;
• γ_ji — inter-firm interaction/mobility coefficient, representing the contribution of human resources from enterprise j to the dynamics of the enterprise under consideration regarding group i. It reflects the intensity and direction of competence reallocation via the labor market.

In an applied context, coefficient γ is linked to wage differentiation across enterprises, as wage disparities are a key factor in inter-firm worker mobility. Specifically, the transition attractiveness coefficient can be defined as:

(8)

where δ_ij — mobility intensity parameter (propensity to switch jobs) for occupational group i; P_ij — average wage of group i employees at enterprise j; P_i — average wage of group employees across enterprises in the industry.

This characteristic reflects the increasing attractiveness of a job switch as the wage gap widens in favor of the receiving enterprise.

The proposed system of indicators is designed for practical use by enterprises in personnel management and strategic planning for human capital development.

Taken together, this formalization of human potential factors establishes a foundation for constructing a dynamic model that enables the analysis of development trajectories h_i(t) under various strategies for employee training and AI adoption, as well as different competitive conditions.

A dynamic model of human capital reproduction within an enterprise. Nonlinear logistic-type models are widely used in applied problems involving the management of mass processes. In these models, the rate of change of a given indicator is proportional to its current level and/or the intensity of external action and the system remaining capacity. As an example, there is a model of an advertising campaign involving two competing firms, formulated as a Cauchy problem for a system of nonlinear ODE [23]. Here, the growth rate of the customer base for each product is determined by advertising intensity and inter-product interaction, while growth is constrained by a factor of the form (N₀ – N₁ – N₂), reflecting the finite market capacity. A closed-form solution for the case of constant coefficients was derived in the cited work.

Taking into account the aforementioned factors and indicators, including (6), (7), and (8), the dynamics of the relative level of the enterprise human capital for occupational group are described by the following differential equation:

(9)

Equation (9) reflects nonlinear dynamics in which the rate of change in human capital is determined simultaneously by internal enterprise decisions (training and digital technologies), structural losses (depreciation), and the external competitive environment (inter-firm mobility of workers).

Thus, several key levers can be identified within the framework of human resource management. First, an enterprise can affect trajectory h_i(t) through training intensity (EDU_i(t)) and competency updating, thereby increasing the contribution of training to human capital accumulation. Second, accelerating this accumulation is possible through the implementation of AI technologies (AI_i(t)), which can enhance task execution efficiency. Third, it is particularly important to take into account human capital depreciation. Thus, at high values of δ_i, maintaining and increasing h_i(t) requires systematic investments; otherwise, the trajectory may shift toward stagnation or decline. Finally, inter-firm mobility introduces an external component to the dynamics, as competitive labor market conditions can amplify or dampen the impact of internal investments, necessitating the alignment of AI training and implementation policies with strategies for talent retention and acquisition.

Empirical data. Given the difficulty of obtaining corporate data, the results of a survey of employees from three machine-building enterprises were used to test the model. All respondents (n = 617) belong to the same occupational group: engineers. The enterprises analyzed are comparable in size and operate in the same industry. The survey questions addressed the intensity of formal and informal employee training as well as the adoption of AI technologies. Coefficients α and β were derived by calculating and normalizing indices for training and artificial intelligence adoption. The employee survey was conducted in May–June 2026.

Research Results. An ODE system for a set of competing enterprises. Equation (9) includes an inter-firm component that depends on the human capital levels of other enterprises in the industry via variables of the form h_ji(t). Consequently, when the competitive environment is treated as endogenous, trajectory h_i(t) of the enterprise in question cannot be determined without describing the dynamics of h_ji(t) for other firms. In this sense, the model of a single enterprise is open, as its right-hand side is determined by variables external to the enterprise — variables that, in reality, also change over time.

Two practical applications of the model are possible.

1. Trajectories h_ji(t) are specified exogenously; for example, based on expert scenarios, industry forecasts, or historical data (with short-term trend extrapolation). In this case, the equation for the enterprise in question is solved as a problem with a prescribed external impact from the competitive environment.
2. If the aim is to model competitive dynamics (including feedback loops, catch-up effects, competence spillovers, and shifts in the industry balance of power), then competitor trajectories cannot be specified arbitrarily and must be derived from equations of the same type. This necessitates formulating an equation for each firm and constructing a system of interconnected ODE.

In the systemic formulation, the dynamics of relative workforce potential is specified for all enterprises in the industry. For each enterprise, human capital is described by an equation analogous to (9), where the inter-firm flow reflects the influence of other labor market participants. Thus, inter-firm mobility acts as a mechanism of system connectivity, since changes in the workforce potential of one organization become a factor affecting the dynamics of other organizations.

For specific occupational group , the system for enterprises can be represented as follows:

(10)

In this system, h_ki(t) — relative human capital of the k-th firm within group i; EDU_ki(t) and AI_ki(t) — firm-specific control inputs; coefficients γ_jk,i reflect inter-firm linkages (intensity or direction of competence spillovers), causing the dynamics of each firm to depend on the trajectories of the others.

Data, parameter calibration, and confidentiality. The proposed ODE system (10) allows for a systematic problem formulation across a set of enterprises. However, its practical implementation faces a fundamental limitation: the completeness of data for the enterprise under analysis differs significantly from that of its competitors. Internal HR data (competency assessments, training, AI adoption, staff turnover, and reasons for departure) may be available only for the enterprise being analyzed, whereas competing organizations typically do not publish such information. It is either fragmentary or protected as trade secrets. Consequently, even when the ODE system for enterprises is correctly formulated, issues arise regarding the identifiability of competitors' parameters and, in a number of cases, the impossibility of directly empirically calibrating the entire system.

Calibrating the dynamic model at the enterprise level requires four types of data: human capital metric H_ji(t) / h_ji(t), training intensity EDU, AI usage, mobility and attractiveness parameters γ, including components related to remuneration and personnel flows.

For the enterprise under analysis, some of this data can be obtained from internal sources, such as assessments of competency proficiency levels, calculations of staff training indices based on volume and coverage, information on the use of AI technologies, HR data regarding hiring and turnover, etc.

For assessing competing enterprises, the most critical issue is that the following data are either unpublished or published only in part:

• assessment of competency proficiency levels and competency matrices by occupational group, which form the basis for H_ji(t);
• information on staff training, including coverage and efficiency, which forms the basis for EDU_ji(t) and α_i;
• actual level of AI technology adoption and usage in the workplace, which serves as the basis for AI_ji(t) and β_i;
• dynamics of labor flows between specific pairs of firms and across groups, which forms the basis for γ;
• wage structure by occupational group, which can be partially derived from vacancy surveys but is rarely comparable or comprehensive.

Individual public indicators, even when available, cannot replace the key metrics required to directly estimate parameters in competitor equations.

The limited observability of competitors stems not only from the difficulty of obtaining their data but also from the economic nature of that data. For enterprises, indicators H(t), EDU(t), AI(t) and parameters α, β, δ, γ effectively describe the internal performance of human capital management and the adoption of new technologies, as well as the company position in the labor market. Disclosing such data creates competitive risks, as it allows other market participants to draw conclusions regarding weaknesses in staff training and AI adoption, as well as vulnerability to talent attrition. This can facilitate the development of strategies to poach personnel.

Therefore, for competing enterprises, direct estimation of α, β, δ from open sources is usually impossible. As a result, the parameters of competitors in the system model have to be set in the following ways:

• scenario-based — through several realistic sets of parameters for competitors;
• intervally — through upper/lower boundaries based on industry research and indirect indicators;
• indirectly — through the selection of parameters that harmonize the model with available aggregates: productivity growth rates, dynamics of revenue per employee, public data on hiring/vacancies, etc.

Validation of a dynamic model of human capital reproduction based on three scenarios, along with recommendations for enterprises. To validate system model (10), the human capital of three enterprises (N = 3) within the same industry and occupational group (M = 1) is analyzed. The dynamics is described by the relative human capital indicator h_N(t) ∈ [0; 1], interpreted as the position of the N-th enterprise relative to a fixed industry frontier. The employee mobility rate is set as a constant equal to 0.05. The modeling horizon is five years (T 5), with the year serving as the unit of time.

The initial conditions are selected such that the first enterprise (Firm 1) is located near the frontier:

The depreciation of competencies is defined by parameter δ = 0.012 (1.2% per year), which aligns with empirical estimates of competency obsolescence rates for university-educated workers [12].

Control inputs EDU_N and AI_N are specified as constants (scenario-based intensity levels) throughout the modeling horizon. The following values are adopted for the baseline configuration:

Training efficiency coefficients α_N and AI adoption coefficients β_N are obtained through a survey of employees at three enterprises and are presented in Table 1.

Presented below there are several scenarios regarding the dynamics of enterprise human capital over a five-year period.

Scenario 1 (Fig. 1) illustrates the dynamics in the absence of inter-firm interaction (γ = 0). Growth in relative human capital potential is observed across all three enterprises, as the combined contribution of the controlled factors (EDU, AI) exceeds the losses due to depreciation δ. The enterprise (Firm 1) located close to the frontier exhibits the expected slowdown in the rate of human capital growth due to the multiplier (1 – h₁), with gradually approaching 1 over the five-year span h₁(t). The human capital of another enterprise (Firm 2) grows faster than that of the third enterprise (Firm 3), reaching a higher trajectory, h₂(t), by the end of the forecast horizon. This corresponds to higher learning efficiency values (α₂) given comparable control inputs. The human capital of the enterprise (Firm 3) grows from a lower initial level and narrows the gap with the enterprise (Firm 2) over the five-year horizon, yet remains behind it.

Scenario 2 (Fig. 2) shows the impact of wage differentiation on dynamics via component γ_j→k. A wage increase at Firm 2 creates an additional advantage for human capital accumulation driven by inter-firm flows: trajectory h₂(t) rises compared to the baseline case (other things being equal). Simultaneously, Firms 1 and 3 face an adverse external effect

(partial outflow of human capital to Firm 2), which slows their human capital growth rates relative to Scenario 1. At the selected mobility scale δ^mob = 0.05 и and a five-year horizon, the effect is moderate. However, it alters the interpretation, since even with identical investments in training and AI adoption, a competitive advantage can be generated through the labor market and the firm's compensation policy.

Scenario 3 (Fig. 3) shows an active corporate compensation policy: while Firm 2 maintains a wage advantage, Firm 3 increases the intensity of training and AI adoption. As a result, trajectory h₃(t) accelerates, and over a five-year horizon, Firm 3 significantly narrows the gap with Firms 1 and 2. Furthermore, Figure 3 shows that by the end of the period, the human capital of Firm 3 approaches the level of Firm 2 and has the potential to surpass it in the future. This indicates that targeted investments in competency development, and the implementation of new technologies can partially or fully offset unfavorable conditions in the competition for personnel stemming from wage disparities.

The research findings allow for several conclusions with practical implications for managing the human resource potential of enterprises. First, an enterprise position relative to the industry frontier determines the maximum rate of human capital development: enterprises closer to the frontier (such as Firm 1 in this instance) exhibit a slower increase in indicator h(t) due to the saturation effect dictated by factor (1 – h). This implies that for industry leaders, the priority shifts from rapidly building competencies to maintaining the level already achieved, whereas catch-up strategies are relevant for lagging enterprises.

Second, the results indicate that labor market competition can significantly alter human capital dynamics, even given identical EDU and AI interventions. Specifically, raising wage levels (Scenario 2) boosts the enterprise human potential development trajectory through the external component γ, which reflects the inter-firm flow of workers possessing the required, up-to-date competencies. Consequently, training and AI adoption programs should be considered in conjunction with staff retention measures, as these elements collectively shape the overall outcome regarding human capital dynamics.

Finally, Scenario 3 demonstrates the compensatory potential of managed investments in training. Specifically, increasing the intensity of training and AI technology adoption makes it possible to partially offset the adverse effects of the competitive environment and accelerate convergence with the frontier, even given a less favorable wage position. This indicates that an effective human capital management strategy in the context of emerging technologies must rely on the coordinated use of development levers (training and AI adoption) and retention levers (working conditions and competitive pay).

Scenarios involving a reduction in investment in an enterprise human capital and the consequent potential for stagnation or degradation are examined. The analysis is based on a baseline setup corresponding to Scenario 1, in which wage levels across the enterprises are equal (P₁ = P₂ = P₃). The study focuses on Firm 2, for which management errors are modeled sequentially while the parameters of Firms 1 and 3 remain constant.

Under the scenario of reduced investment in staff training at Firm 2 (shown in Fig. 4 a), growth slows. Instead of the 0.2537-point increase seen in the baseline scenario, the enterprise records a gain of only 0.1323 points. When investment in AI adoption ceases, growth is curtailed to just 0.1704 points (Fig. 4 b). Figure 4 c illustrates the scenario of wage cuts, which also slows growth over the five-year period, resulting in a gain of 0.1962 points. However, if all these management measures are applied simultaneously, the enterprise starts to lose human resources, as shown in Figure 4 d — with a loss of 0.1919 points.

The presented results show that when only internal investments at Firm 2 are reduced, indicator h₂(t) continues to rise, albeit significantly more slowly than in the baseline scenario. At the same time, the trajectories of Firms 1 and 3 coincide with the baseline ones, since, given equal wages, the inter-firm component γ is zero, and the dynamics of the firms become autonomous. In the scenario involving a wage reduction at Firm 2, a redistribution of human capital occurs through inter-firm mobility. The most critical situation arises under the combined scenario. With simultaneous cutbacks in training and AI adoption, compounded by a wage disadvantage, Firm 2 enters the state of human capital degradation.

Table 2 presents the numerical results of these management decisions for all the enterprises analyzed.

The results make it possible to assess the relative importance of the factors under consideration for the dynamics of human capital. When these factors are varied in isolation, the greatest slowdown in human capital accumulation is observed upon the complete cessation of investment in training. However, this result is scenario-dependent. It is determined by the selected values α₂, β₂, mobility intensity, and other parameters. Therefore, it should not be interpreted as a universal relationship among factors applicable to all enterprises. Thus, the proposed ODE system can be used to identify conditions leading to the degradation of labor resources and to assess the sensitivity of individual enterprises to various management decisions. Consequently, the algorithm proposed by the authors can serve as a tool for preliminary risk assessment in human capital management decision-making.

Discussion. In this study, an economic-mathematical model of human resource reproduction has been developed based on a first-order differential equation, enabling the description of the dynamics of their changes over time. Classical models of human capital accumulation are also described by differential equations. Notable examples include the Lucas [24] model, which accounts for the allocation of time between work and training, and the Ben-Porath [25] model, which incorporates investments in human capital and accounts for its depreciation. Thus, the proposed model, on the one hand, is built on the basis of established postulates of the mainstream; on the other hand, it takes into account the impact of AI technologies on human resources and the state of the competitive environment, which seems relevant under modern conditions of scientific, technical and socio-economic development. The applied ODE system provides a comprehensive, dynamically consistent description of the dynamics of reproduction of an enterprise human capital under the effect of investments in employee training, the adoption of AI technologies, and changes in the competitive environment.

The results of testing the developed model across three scenarios, using three enterprises as case studies, align with findings from previous research. In particular, the enterprise-level saturation effect in the medium term, as described in [26] was confirmed. These research results regarding the positive impact of raising wage levels, acting as an exogenous factor, on human capital development (facilitated by employee training, AI adoption, and staff support tools) align with findings reported by other authors [27]. Furthermore, testing the proposed model confirmed the importance of boosting employee training and adopting AI for human resource development, specifically for enterprises starting with low initial levels of human capital [21].

Overall, the proposed dynamic economic-mathematical model of human capital reproduction comprehensively accounts for the most significant factors and their dynamic changes. It is systemic in nature and can have practical value for improving the quality of management decisions at enterprises.

Conclusion. The article proposes an approach to modeling the reproduction of an enterprise human capital within the digital economy, based on the premise that employee competencies are formed and lost over time through the combined influence of in-house training, the adoption of AI technologies, technology-driven skill obsolescence, and inter-firm mobility. Unlike static assessments, the proposed economic-mathematical formulation describes the dynamics of human capital as a nonlinear process of constrained growth relative to a fixed industry frontier. This provides comparability among firms and reflects the saturation effect as they approach best practices.

It is shown that accounting for inter-firm mobility necessitates a shift from an equation describing a single organization to a system of interconnected ODE representing a set of firms. In this systemic framework, the competitive environment becomes endogenous: changes in one firm human capital affect the trajectories of others through parameters of inter-firm interaction. To specify these links in practice, it is proposed to define mobility coefficients based on wage differentials, thereby allowing labor flows to be interpreted as an outcome of labor market competition.

It is particularly emphasized that calibrating model parameters for competing firms is constrained by data availability: internal metrics regarding competencies, training, actual AI usage, and detailed statistics on personnel turnover are generally not published and are considered commercially sensitive information. Consequently, in industry-level modeling, parameters for competitors can often only be defined through scenarios, ranges, or indirect estimates based on publicly available indicators — a factor that must be taken into account when interpreting the results.

A numerical experiment involving three enterprises demonstrated practical interpretability of the model and the importance of jointly accounting for the factors involved. In the baseline scenario, excluding inter-firm flows, dynamics is driven by differences in the efficiency of learning and AI adoption amidst depreciation, with growth rates slowing as the frontier is approached. In the scenario involving wage increases at a specific enterprise, the firm trajectory is bolstered by an inter-firm component while competitors experience slower growth, confirming the role of compensation policy as a driver of competence dynamics. Finally, the compensating strategy scenario show that increased investment in training and AI can accelerate convergence with the frontier and partially offset the adverse effects of competition for talent.

An analysis of adverse management decisions reveals that cutting investment in training or AI adoption in isolation primarily slows the accumulation of human resources. However, when combined with reduced labor market competitiveness, stemming from a weakened wage position and inter-firm mobility issues, such decisions can push a firm into a state of decline. Thus, the human capital reproduction model proposed by the authors can be used not only to analyze growth trajectories but also to identify the conditions leading to a loss of human capital potential and to assess management risks.

Prospects for further research into human capital reproduction involve the empirical validation of the model using corporate panel data, the expansion of the range of occupational groups, the refinement of the specification of inter-firm flows (including sectoral proximity and task heterogeneity), and the development of robust parameter calibration procedures for situations where data on competitors is incomplete.