Machine endowment cost model: task assignment between humans and machines

The paper addresses when tasks should be assigned to humans versus machines and why some tasks can be automated while others cannot. It situates the study within the observed phenomenon of job/task polarization, where technology tends to automate middle-skill tasks, leaving humans concentrated in low- and high-skill roles. Building on prior classifications of tasks (routine vs nonroutine; cognitive vs manual), the study argues that automation decisions hinge on two task dimensions: cognitive requirements and manual flexibility. The purpose is to develop a theoretical cost-based framework that determines an economically efficient human–machine task allocation, to analyze how this allocation shifts with technological change and wage evolution, and to identify conditions under which task polarization emerges. The work is important for understanding technological impacts on labor markets and for guiding industrial human–robot collaboration and education/training policy.

Prior economics literature distinguishes routine from nonroutine tasks (Autor et al., 2003) and classifies tasks along cognitive and manual dimensions (Acemoglu and Autor, 2011; Autor, 2015), finding that routine middle-skill work is most susceptible to automation and documenting job polarization across countries (Goos and Manning, 2007; Goos et al., 2009, 2014; Autor et al., 2006). Research links rising wages and education to cognitive skill demand (Yamaguchi, 2012; Michaels et al., 2014; Frey and Osborne, 2017; Alabdulkareem et al., 2018). More recent work in engineering focuses on human–robot task allocation, proposing complexity- or capability-based assignment rules (Ranz et al., 2017; Malik and Bilberg, 2019; Yuan et al., 2020). Existing macro models of automation and polarization (Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2018b, 2018c) generally do not explicitly model the joint two-dimensional flexibility–cognition requirements at the task level. This paper synthesizes these strands by formalizing a two-dimensional endowment model that captures both flexibility and cognitive requirements and relates them to costs and wages.

The study develops a machine endowment cost model based on two task dimensions: cognitive endowment g and manual flexibility endowment f. A machine capable of performing a task requiring (g, f) incurs cost C(g, f) that increases in both g and f, with a CES-like structure governed by an elasticity/exponent parameter p > 1. Technology and cost parameters include: α representing overall technology level (reduces costs as it rises), a the coefficient on cognitive cost, and b the coefficient on flexibility cost. Comparative statics yield ∂C/∂g > 0, ∂C/∂f > 0, ∂C/∂a > 0, ∂C/∂b > 0, and ∂C/∂α < 0. Human wages for tasks requiring (g, f) depend only on cognitive ability via W(g, f) = W(g) with minimum wage/base c > 0 and a cognition premium β > 0, using the same exponent p for analytical comparability; flexibility does not directly raise human wages in the model. The machine production possibility curve (MPPC) is defined by the locus of task endowments where machine cost equals human wage, C(g, f) = W(g). The MPPC in (g, f) space divides tasks: points below are economically assigned to machines; points above to humans (Proposition 1). The paper derives expressions for the MPPC and analyzes how it shifts with parameters (Proposition 2): as technology improves (higher α) or as wages rise (higher c or β), the MPPC moves upward/outward, expanding the set of tasks assigned to machines; increasing machine cost coefficients a or b shifts the MPPC inward. The model then considers aggregate endowment θ = g + f for isolines in (g, f) space. By comparing C(g, θ − g) with W(g) along an isoline, the paper establishes an upper bound θ_u above which tasks must be assigned to humans (Proposition 3) and a lower bound θ_l below which tasks must be assigned to machines (Proposition 5). Comparative statics for θ_u and θ_l (Propositions 4 and 6) show they rise with α, c, β and fall with a, b. Within the intermediate range θ_l < θ < θ_u, intersections of the isoline with the MPPC determine assignment regions; depending on parameters, there can be one or two intersections, leading to cases with or without polarization. When two intersections occur, low- and high-g tasks on the isoline are assigned to humans while midrange tasks are assigned to machines, yielding task polarization (Proposition 7). The paper characterizes how the intersection points g1, g2, g3, g4 shift with parameters (Proposition 8), providing a reallocation rule as technology and wages evolve. A numerical illustration uses parameters a = 2, b = 3, c = 50, α = 2, β = 0.8 to compute lower and upper bounds and to map the model’s prescriptions to engineering task-allocation studies.

- The machine production possibility curve (MPPC), defined by equality of machine cost and human wage, provides a clear partition of the task space: tasks below the curve are economically assigned to machines; tasks above to humans (Proposition 1). - Technological progress (higher α) and wage growth (higher c and β) expand the domain of machine-performed tasks by shifting the MPPC outward; increases in machine cost coefficients (a for cognition, b for flexibility) contract it (Proposition 2). - Direction of technological change matters: improving flexibility (lower b) primarily expands machine feasibility for high-flexibility tasks; improving cognition (lower a or higher α) expands feasibility for higher-cognition tasks, potentially into high-skill domains (e.g., AI-enabled tasks). - Aggregate endowment analysis identifies thresholds: a lower bound θ_l below which all tasks are assigned to machines (Proposition 5) and an upper bound θ_u above which all tasks are assigned to humans (Proposition 3). Both bounds increase with α, c, and β and decrease with a and b (Propositions 4 and 6). - Task polarization arises for intermediate aggregate endowments when the isoline intersects the MPPC twice: low- and high-end tasks (on the same θ) go to humans, while midrange tasks go to machines (Proposition 7). Comparative statics of intersection points (Proposition 8) show that advances in α or increases in c and β widen the machine-assignment middle segment, while increases in a or b shrink it. - Empirical alignment: A numerical example (a=2, b=3, c=50, α=2, β=0.8) yields a lower bound of 28 and an upper bound of 92, consistent with engineering studies. Malik and Bilberg (2019) allocate low-flexibility tasks to machines (g=0, f≤28). Yuan et al. (2020) consider both flexibility and cognition, aligning with the model’s lower/upper bounds and indicating potential polarization within the intermediate range (e.g., θ=87 partitioned across low-skill humans, machines, and high-skill humans depending on g).

The findings provide a theoretical basis for assigning tasks between humans and machines by explicitly modeling two-dimensional task requirements and comparing machine costs with human wages. This addresses the research question of when tasks should be automated by supplying a transparent decision rule via the MPPC and by identifying how economic and technological parameters shift that rule. The results clarify why middle-skill tasks are frequently automated: machines optimized for moderate levels of cognition and flexibility can be cheaper than humans, while tasks demanding very high flexibility or very high cognition tend to remain human-performed due to steep machine cost escalations. The framework connects labor-market forces—minimum wage levels and the rate at which wages rise with cognitive ability—to the direction and magnitude of technological change and automation, offering explanations for cross-country variation in observed skill-biased technological change. Practically, the model can guide industrial task design and human–robot collaboration by mapping specific task endowments to cost-effective assignment, and it informs education policy by highlighting the value of combined cognitive and manual flexibility training to maintain employability as technology advances.

The paper develops a machine endowment cost model that formalizes task allocation between humans and machines using two task dimensions: cognitive ability and manual flexibility. By deriving the machine production possibility curve (MPPC), it offers a direct criterion for assignment and demonstrates how technological progress and wage dynamics shift the boundary between human and machine tasks. The aggregate endowment analysis identifies lower and upper thresholds that guarantee assignment to machines or humans, respectively, and delineates the intermediate region where task polarization can occur. The framework explains observed patterns of middle-skill automation and highlights how the direction of technological advances (cognitive vs flexibility) and wage structure shape future allocations. It provides actionable insights for firms and for education systems to prepare middle-skilled workers with both cognitive and manual capabilities. Future research could empirically estimate model parameters across sectors, incorporate additional task dimensions (e.g., social/interpersonal), and test the model’s predictions with longitudinal task- and wage-level data.

The study is theoretical and does not analyze new empirical datasets; parameter values in the numerical example are illustrative. The model abstracts task requirements into two dimensions (cognition and flexibility), omitting other factors such as social/interactive skills, safety, regulation, or organizational constraints. Human wages are modeled as a function of cognition only, which simplifies heterogeneous labor markets. The mapping from engineering task complexity scores to endowments is approximate and not empirically calibrated, which may affect generalizability across industries and contexts.