O-Ring Automation

Source: Joshua S. Gans, Avi Goldfarb; NBER Working Paper 34639 · 2026

We study automation when tasks are quality complements rather than separable. Production requires numerous tasks whose qualities multiply as in an O-ring technology. A worker allocates a fixed endowment of time across the tasks performed; machines can replace tasks with given quality, and time is allocated across the remaining manual tasks. This “focus” mechanism generates three results. First, task-by-task substitution logic is incomplete because automating one task changes the return to automating others. Second, automation decisions are discrete and can require bundled adoption even when automation quality improves smoothly. Third, labour income can rise under partial automation because automation scales the value of remaining bottleneck tasks. These results imply that widely-used exposure indices, which aggregate task-level automation risk using linear formulas, will overstate displacement when tasks are complements. The relevant object is not average task exposure but the structure of bottlenecks and how automation reshapes worker time around them.

Introduction. Automation is often evaluated through a task lens: identify what tasks are performed in a job, determine which tasks can be performed by machines, and infer how labour demand and wages change as machines take over (Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2019b). A common formalisation, going back to Zeira (1998), abstracts from cross-task complementarities: tasks (or activities) are effectively separable, so automating one task does not change the productivity of labour on the others. Many production processes are not like that. In a wide class of processes, tasks are quality complements. A single weak link can spoil the product, delay the workflow, or undermine the value of effort elsewhere. Kremer (1993) formalised this idea with the O-ring production function, where output is multiplicative in task qualities. Under such a technology, a change in the quality of one task scales the marginal value of quality in every other task. This paper studies automation in a stripped-down environment designed to isolate that interaction. A production process consists of n essential tasks. In the benchmark, a single worker performs all tasks and allocates a fixed time endowment h across them. More time on a task increases its quality. Automation becomes available as a capital alternative for any task: for a rental cost r per task, a machine completes that task at fixed quality. The firm chooses which tasks to automate. If at least one task remains manual, the firm and worker then bargain bilaterally over wages (we use Nash bargaining). When a task is automated, the worker reallocates their fixed time to the remaining manual tasks, increasing their quality.1 Because output is the product of task qualities, this reallocation changes the return to automating any given task. The model delivers three messages. First, task-by-task substitution logic is incomplete. Automating a task not only replaces the quality of that task; it also changes the worker’s time allocation and thus the quality of all remaining manual tasks. This generates a “focus” mechanism: automating some tasks can raise the worker’s contribution on the remaining tasks. Second, automation decisions are discrete and bundle-like. Even when automation quality improves smoothly, the firm’s optimal number of automated tasks changes via thresholds. Moreover, because the base level of output rises with automation, it can be unprofitable to automate the first task at the margin while still being optimal to automate many tasks in total. This undermines simple marginal adoption rules. Third, labour income need not fall under partial automation. When not all tasks are automated, increases in automation quality can raise labour income because automation scales the value of the remaining labour bottlenecks. This is the opposite of what a purely substitutive, separable task accounting would suggest. The paper is organised as follows. Section 2 sets up the model and defines the equilibrium concept for labour income. Section 3 solves the worker’s time allocation for any set of automated tasks. Section 4 characterises the firm’s automation choice, first for homogeneous automation quality and then for heterogeneous task-specific quality. Section 5 derives implications for labour income. Section 6 discusses what this implies for prospective and retrospective task-based assessments of automation. Section 7 concludes.

Method. A firm produces a single good (price normalised to one) using a process composed of n ≥1 essential tasks, indexed by s ∈{1, . . . , n}. Each task produces a quality qs > 0. Output is the product of task qualities, as in an O-ring technology:

A single worker supplies a fixed time endowment h > 0 (hours) to the firm. If task s is performed manually, the worker assigns hs ≥0 hours to it and produces quality where a > 0 is labour productivity (common across tasks). Time is scarce: s∈M hs = h, (3) For each task s, there exists a capital technology that can perform the task at fixed quality θs > 0 at a rental cost r > 0 per task. If task s is automated, its quality is qK s = θs, (4) and it requires no worker time. Let A denote the set of automated tasks and let k = |A| be the number of automated tasks. Then |M| = n −k. The firm chooses which tasks to automate (i.e. chooses A), hires the worker for h hours when at least one task remains manual, and time is allocated efficiently across manual tasks.2 A competitive “wage equals marginal product” benchmark is not very informative in this environment. When more than one task is manual, output is multiplicative in the task-level hour allocations, so changes in one task’s hours scale the productivity of all other hours. Moreover, when k < n the firm cannot produce without the worker, so a naive “pay the worker their marginal contribution” rule would assign the worker essentially all net output Y (A)−r|A|, making labour the residual claimant on capital, which is not an appealing outcome. Instead, we assume the firm and worker engage in bilateral Nash bargaining.3 If at least one task remains manual (k < n), the worker is required for production and the firm and worker engage in bilateral wage bargaining. We assume efficient bargaining: conditional on reaching agreement, the time allocation across manual tasks maximises output given A. This allows us to solve for the efficient time allocation and output first, and then pin down the division of surplus. Let β ∈(0, 1) denote the worker’s bargaining weight and let u0 ≥0 denote the worker’s disagreement payoff (the value of spending the h hours elsewhere). If bargaining fails, production does not occur when k < n and the firm earns zero (and incurs no rental costs). We model wage determination using the Nash bargaining solution (Nash, 1950). Let W(A) denote the total wage bill paid to the worker when the automation set is A. When k < n, Nash bargaining chooses W(A) to maximise W(A) −u0 β Y (A) −r|A| −W(A) 1−β. (5) Whenever the match surplus is positive (i.e. Y (A)−r|A| > u0), the solution is interior and yields W(A) = u0 + β Y (A) −r|A| −u0 , (6) and firm profit Π(A) = Y (A) −r|A| −W(A) = (1 −β) Y (A) −r|A| −u0 . (7) If all tasks are automated (k = n), the worker is not required. In that case, there is no wage bargaining, the firm produces Y (A) = Qn s=1 θs, and earns Π(A) = Y (A) −rn. The worker is not employed by the firm and receives u0 elsewhere. When k < n, an associated per-hour wage is w(A) = W(A)/h. To focus on the interesting case, we make the following assumption throughout:

Assumption 1 (Gains from trade). For any automation choice A with k < n, the match surplus is strictly positive: Y (A) −r|A| > u0.

Hence, whenever the worker is required for production, the Nash bargaining solution is interior and agreement is reached.

Discussion. The model is deliberately spare, so the implications are correspondingly sharp. In an O-ring job with fixed worker time, task-based inferences that treat tasks independently can be wrong because tasks may be quality complements and because worker time can be reallocated. This contrasts with much empirical work on the labour market impact of automation, which often assumes task separability. Specifically, the standard task-based approach typically assumes that a job can be decomposed into a vector of discrete, independent tasks, and that the automation of one task represents a pure substitution of capital for labour, leaving the production function of the remaining human tasks largely unchanged (e.g., Autor et al., 2003; Acemoglu and Restrepo, 2018, 2019a).

This “additive” view of the labour process simplifies aggregation but ignores the interdependence of work activities. Here we consider the empirical implications of the model and, in particular, the interpretation of recent studies. We consider, first, studies that anticipate the impact of artificial intelligence (or AI) on labour markets, followed by those that study past automation and find evidence consistent with some of the issues raised by the model here.

6.1 Studies that anticipate AI’s impact By assuming task separability, researchers have developed tools for anticipating the impact of task automation on labour markets. However, these tools will be misleading under an O-ring production function. Starting with the probabilistic classifications of occupation risk developed by Frey and Osborne (2017), and including the patenttext analysis of Webb (2020) and the rubrics of Felten et al. (2021), researchers have sought to quantify the “exposure” of human labour to machine substitution. These papers do not account for two core features of the mechanism in this paper: (i) tasks are quality complements (O-ring production), and (ii) the worker has a fixed endowment of time that must be allocated across remaining manual tasks. Under this framework, the automation of a task is not merely a subtraction of labour demand; it is a reallocation event. When a machine takes over a subset of tasks, the worker’s fixed time endowment is concentrated onto the remaining manual tasks, endogenously raising their quality and, potentially, the marginal product of the worker. This “focus effect” implies that standard empirical measures of exposure and wage impacts are biased: prospective indices that treat tasks as additive bundles will systematically overstate displacement risks by failing to account for the rising value of human comparative advantage in remaining “bottleneck” tasks.6 Endogenous human quality. A common benchmark compares machine quality to “human quality” on the task. With fixed worker time, human quality on a task is a(h/m), where m is the number of manual tasks after automation. That is an endogenous equilibrium object. Automating some tasks increases human quality on the remaining tasks, potentially making those remaining tasks less susceptible to automation than they were initially. An exposure calculation that holds human quality fixed will tend to overstate the degree of job automation. Put differently, prospective studies typically compare machine capability (θ) to a fixed human capability (ahuman) and classify a task as “exposed” if θ > ahuman. In the O-ring fixed-time model, human capability is dynamic: as tasks are automated, m decreases and effective human quality a(h/m) rises. A task that looks automatable today (when the worker is split across many tasks) may become non-automatable tomorrow (when the worker focuses on only a few tasks).

Conclusion. This paper shows that the economics of automation differ when tasks are quality complements and worker time is fixed. Automating a task does not merely substitute capital for labour on that task. It also changes the allocation of labour time across the remaining tasks and, therefore, the quality of those tasks, which feeds back into the return to automation itself. Three results are central. First, task-by-task substitution logic is incomplete: automating a task changes the worker’s time allocation and, in turn, the quality of all remaining manual tasks, creating a ”focus” mechanism that standard exposure measures miss. Second, optimal automation is discrete and can require bundled adoption rather than marginal one-by-one adoption, even when automation quality improves smoothly. Third, conditional on partial automation, improvements in automation quality can raise bargained labour income because automation scales the value of remaining labour bottlenecks and the worker captures a share of the resulting match surplus. A practical implication is that prospective measures of automation exposure may be misleading when applied to jobs with quality-complementary tasks. The linear aggregation formulas used in widely-cited exposure indices are mathematically inconsistent with O-ring production: they treat a job as the sum of its task-level risks rather than as a system governed by bottlenecks. When tasks are complements and worker time is fixed, automating some tasks raises the quality of the remaining tasks, potentially making them harder to automate than they initially appeared. Exposure indices that hold human task quality fixed will therefore tend to overstate displacement risk. The relevant object is not the average automatability of a job’s tasks but the structure of its bottlenecks and how automation reshapes the allocation of worker time around them.