The macroeconomics of automation

Layer 1: Overview

This paper asks a foundational question: can the economy-wide degree of automation be measured coherently from standard macroeconomic data, without relying on technology-specific proxies such as robot counts or AI investment surveys? Existing micro-level proxies are fragmented across technologies and difficult to aggregate, leaving it unclear how automation evolves at the macro level or how it relates to capital deepening, factor shares, and productivity growth. The authors, Hideki Nakamura, Masakatsu Nakamura, and Shota Moriwaki, address this by developing a task-based general equilibrium framework in which the aggregate degree of automation emerges endogenously and is fully identified from observable macroeconomic aggregates.

The theoretical architecture begins with a continuum of tasks, each exhibiting Leontief technology at the task level. Within each task, capital and labor are perfectly substitutable, but firms choose the least-cost input given factor prices. Tasks are ordered by the relative efficiency of capital to labor; as the wage-to-capital-service-price ratio rises with capital deepening, capital performs an expanding range of tasks. Aggregating task-level Leontief decisions over a firm generates a global (envelope) production function. The paper’s first main theorem shows that under a mild regularity condition on task efficiency orderings, this aggregation delivers a standard neoclassical production function. Its second set of results identifies the precise efficiency structure under which the aggregate function takes the CES form: that structure corresponds to a Pareto cumulative distribution of input efficiencies. This Pareto structure yields a clean closed-form relationship: the degree of automation is determined entirely by the capital-labor ratio (in efficiency units) and the elasticity of substitution. When the elasticity exceeds one, the degree of automation equals the capital income share; when the elasticity falls below one, it equals the labor income share. Neutral technical progress leaves the degree of automation unchanged at a given capital-labor ratio; capital-augmenting progress raises it; labor-augmenting progress lowers it.

The empirical application uses panel data from the 2023 Japan Industrial Productivity (JIP) database covering 52 manufacturing industries from 1994 to 2020 (N = 1,404 industry-year observations; two industries excluded for data quality). The CES production function is estimated via GMM using first-differenced factor-share equations derived from the normalized CES system (de La Grandville 1989 normalization), with five sets of instrumental variables drawn from lagged factor prices, information stock and its price, trade openness, workforce age composition, and part-time employment shares.

The main quantitative findings are as follows. Under the assumption of neutral technical progress, the elasticity of substitution sigma is significantly above one but close to one, ranging from 1.049 to 1.102 across the five IV sets (all significant at least at the 10 percent level). Under the assumption of capital-augmenting technical progress (gK > 0, gL = 0), sigma ranges from 1.035 to 1.068, again robustly greater than one. Capital-augmenting technical progress is statistically significant across all specifications; labor-augmenting technical progress cannot be confirmed in any specification. The average estimated degree of automation across the 52 industries over the full sample period is 0.417 (standard deviation 0.171, minimum 0.138, maximum 0.811). The average rises steadily from 0.407 in 1994 to 0.426 in 2020, temporarily declining around the 2008 financial crisis before recovering. Substantial heterogeneity persists across industries throughout the sample. The distribution shifts rightward over time but retains a fat left tail, with the mode just above 0.3 and several industries exceeding 0.7.

The two-level CES extension decomposes aggregate capital into industrial robots and other capital, exploiting a purpose-built robot capital stock constructed via the RAS and perpetual inventory methods (initial year 1985). Industrial robots account for only 0.44 percent of aggregate capital stock on average. The two-level estimation yields higher elasticities (sigma-a between 1.191 and 1.346 across IV sets for the composite-labor margin; sigma-b between 1.049 and 1.096 for the robots-other-capital margin). The degree of automation for the composite rises from 0.398 to 0.430 over the sample, a more pronounced increase than the standard CES estimate, reflecting robots’ amplifying role in automation.

The paper benchmarks three automation measures against an internal consistency criterion: the squared distance between the automation degree inferred from the capital-labor ratio and that inferred from output per worker, given the same CES structure. The Pareto-based measure (the paper’s preferred measure) achieves a distance of 0.0000319, far below the Cobb-Douglas alternative (0.002484) and the continuity-preserving alternative (0.00999), validating the Pareto efficiency-distribution assumption. The Cobb-Douglas alternative yields a mean automation of 0.500 rising from 0.454 to 0.529; the continuity alternative rises more sharply from 0.208 to 0.589 but is discontinuous and sometimes falls outside the unit interval.

For policy and theory, the paper’s framework implies that Japan’s sustained capital accumulation during its prolonged stagnation after 1990 translated into rising automation even without commensurate TFP growth, connecting automation dynamics to the “productivity paradox.” The model also shows that automation can rise alongside an increasing labor income share when sigma is below one, caution against interpreting a stable or rising labor share as evidence against ongoing automation. The degree of automation provides a unified lens connecting capital deepening, factor shares, and productivity in a single theory-consistent measure.

Layer 2: Deep Dive

What is the core identification strategy and what observables are used to infer the degree of automation?

The degree of automation is identified from the first-order conditions of the CES production function. Under the Pareto efficiency-distribution assumption, the CES structure implies a one-to-one mapping from the aggregate capital-labor ratio (in efficiency units), the share parameter s, and the elasticity of substitution rho to the degree of automation (Theorem 4, Eq. 25 and 31). In practice, the authors estimate the CES production function via GMM on first-differenced factor-share equations, recover rho and gK, and plug those into the formula for the degree of automation. No direct observation of tasks, robots (in the standard CES step), or technology-specific adoption decisions is required.

What are the main threats to identification and how do the authors address them?

The main threats are endogeneity of the output-to-labor and output-to-capital ratios (both simultaneously determined with factor prices) and measurement error in the capital-labor ratio (arising from industry classification changes and the RAS procedure used to construct robot data). The authors address endogeneity via GMM estimation using five distinct IV sets that include lagged factor prices, information stock and its price, trade openness, and workforce composition variables. They report that elasticity estimates are stable across all five IV sets and across alternative sample windows (including a longer 1973-2011 sample from pre-SNA-revision data), and conclude that measurement error is unlikely to drive the results. The overidentification test is not rejected for any IV set in the baseline CES specification (and for most in the two-level specification).

What theoretical result connects the degree of automation to factor income shares?

Corollary 1 establishes that under the Pareto efficiency structure (Eq. 22) with competitive factor markets, the degree of automation equals the capital income share when sigma > 1, and equals the labor income share when sigma < 1. This makes the degree of automation directly readable from income-share data in the theoretically preferred case (sigma > 1 for Japan). The empirical results are consistent with this: the average degree of automation across manufacturing industries is close to the average capital income share over the sample, providing a cross-check for Corollary 1.

Why does the paper use a Leontief production function at the task level while obtaining a CES function at the aggregate level?

The Leontief specification at the task level reflects the idea of a bottleneck in production: within a single narrowly-defined task, only capital or labor is used (once a task is automated, capital fully replaces labor in that task). Perfect substitutability between capital and labor operates at the extensive margin (which tasks are automated) rather than within a task. The aggregate (envelope) function, formed by varying the automation cutoff as the capital-labor ratio changes, generates any elasticity of substitution from zero to infinity. The Pareto efficiency-distribution assumption pins down the specific case of a CES aggregate.

How does the two-level CES extension work, and what does it add?

The two-level CES nests industrial robots and other capital into a capital composite at the inner level (robots vs. other capital, with elasticity sigma-b), then combines that composite with labor at the outer level (composite vs. labor, with elasticity sigma-a). Robot data for 52 industries are constructed via the RAS and perpetual inventory methods with an initial year of 1985. Because robots account for only 0.44 percent of aggregate capital on average, they have a small direct weight, but the two-level decomposition isolates their specific contribution to the automation margin. The two-level CES estimates sigma-a between 1.191 and 1.346 (higher than the standard CES estimates), and finds that the test of equality between sigma-a and sigma-b is rejected for three of five IV sets, suggesting the two elasticities genuinely differ. The average degree of automation rises more steeply under the two-level estimate (0.398 to 0.430) than under the standard CES estimate (0.407 to 0.426), indicating that explicitly accounting for robots reveals a more pronounced automation trend.

What is the paper’s internal consistency criterion, and how does it rank alternative automation measures?

Internal consistency is defined as the mean squared gap between the degree of automation inferred from the capital-labor ratio (Eq. 37, the paper’s preferred measure) and the degree of automation implied by observed output per worker given the same CES structure (Eq. 41). A smaller gap means the measure is more coherent with the CES framework from which it is derived. The Pareto-based measure achieves a distance of 0.0000319, more than seventy times smaller than the Cobb-Douglas alternative (0.002484) and over three hundred times smaller than the continuity-preserving alternative (0.00999). The authors therefore select the Pareto-based measure as most internally consistent with CES production.

What is documented about heterogeneity in automation across industries?

The degree of automation varies substantially across the 52 manufacturing industries, with a standard deviation of 0.171 and a range from 0.138 to 0.811 in the standard CES estimation. The kernel density in 1994 has a fat left tail with a mode just above 0.3, and several industries already exceed 0.7. The distribution shifts rightward by 2020 but remains dispersed. The authors split industries into those with an increasing capital income share (34 industries) and those with a decreasing share (18 industries) and test whether the elasticity of substitution differs between groups; they find no statistically significant difference for any IV set, implying the CES structure is uniform across industries even though automation levels differ.

How does the paper connect automation to TFP and the productivity paradox?

The theoretical framework shows that automation via task reallocation shifts the production function in a northeast direction in (k, y) space but does not shift it upward in a way that registers as TFP growth. Formally, increasing automation does not appear to impact TFP growth (citing Nakamura and Nakamura, 2008). The empirical finding that the degree of automation rose from 0.407 to 0.426 during Japan’s prolonged stagnation (1994-2020), a period of slow output-per-worker growth, is consistent with this: capital accumulation drove automation forward even though measured TFP growth was subdued. The paper thus links automation dynamics to Japan’s productivity paradox and implies that standard TFP accounting may understate the technological transformation underway.

What is the relationship between the elasticity of substitution and the direction of factor share changes under automation?

The CES framework implies that when sigma > 1 (capital and labor more substitutable), capital accumulation raises the capital income share and lowers the labor share; the degree of automation equals the capital income share. When sigma < 1, capital accumulation raises the wage-to-rental ratio by more, increasing the labor income share; the degree of automation equals the labor income share. In both cases automation rises with capital deepening. A key implication is that observing a stable or rising labor income share does not rule out rising automation when sigma is below one or close to one. The authors’ estimate of sigma slightly above one for Japanese manufacturing implies a slightly rising capital share, consistent with the panel-estimated trend (b-hat = 0.00102, t-value = 6.84).

What are the robustness checks and how stable are the estimates?

Robustness checks include: (1) five distinct IV sets spanning different combinations of lagged wages, capital rental prices, information stock, trade openness, and workforce composition; (2) estimation under both neutral and capital-augmenting technical progress assumptions; (3) estimation using a longer sample (1973-2011 using pre-SNA-revision data), which yields a sigma still significantly above one and close to one, with slightly larger capital-augmenting technical progress reflecting higher growth in that period; (4) estimation of the full CES production function equation simultaneously with the two FOC equations (Appendix E.2), yielding similar elasticity estimates; (5) a structural change test splitting industries by capital-share trend, finding no significant difference in elasticity between subgroups. Unit root tests (Harris-Tzavalis and augmented Dickey-Fuller) confirm stationarity of all key variables except the part-time ratio, which also passes the ADF test.

What are the caveats and acknowledged limitations?

The authors acknowledge several limitations. First, three conditions cannot be simultaneously satisfied: a CES aggregate, the degree of automation lying in the unit interval, and continuity of the automation measure at unit elasticity (sigma = 1). The preferred measure prioritizes the unit-interval restriction and sacrifices continuity at sigma = 1, making direct comparisons across the sigma < 1 and sigma > 1 cases problematic (an alternative continuous measure is derived in Appendix C but may fall outside the unit interval). Second, the framework abstracts from the creation of new tasks; changes in the total number of tasks over time would affect the automation measure. Third, the paper does not decompose automation by skill level; the observed differences between skilled and unskilled labor in automation suggest a need for nested CES structures in future work. Fourth, the two-level CES nesting (robots within capital composite) is dictated by data availability; alternative nestings, such as grouping robots and labor at the first level, are not separately identifiable.

How does this paper differ from and improve upon the prior literature?

The paper improves on micro-proxy approaches (robot counts, AI investment, task-exposure indices from Acemoglu-Restrepo 2020, Adachi 2025, etc.) by providing an aggregate, theory-consistent measure that does not require technology-specific data. It extends prior CES microfoundation work (Jones 2005 Pareto-Cobb-Douglas result, Growiec 2008 Weibull-CES results) by deriving the Pareto efficiency structure that yields CES specifically from task-level automation decisions. It improves on the authors’ own prior work (Nakamura and Nakamura 2008, Nakamura 2009, 2010) by providing a complete theoretical justification for input efficiencies, a full treatment of the elasticity of substitution, and an empirical implementation. Relative to Artuc et al. (2023) and Adachi (2025), which use Frechet distributions for task productivity, this paper uses a deterministic framework with Pareto-distributed input efficiencies and emphasizes aggregate-level identification rather than cross-occupational substitution.

What are the policy implications?

The paper does not make direct policy prescriptions, but its framework has several implications. First, policymakers tracking automation can use standard national accounts data (capital stock, labor input, output, factor shares) rather than waiting for technology-specific surveys, enabling faster and more comprehensive monitoring. Second, the result that automation can advance during periods of slow TFP growth suggests that technology policy focused solely on productivity metrics may underestimate the pace of labor displacement. Third, the finding that Japan’s capital accumulation drove automation even through prolonged stagnation implies that capital subsidies or policies encouraging investment could accelerate automation independent of TFP. Fourth, the model’s prediction that automation rises alongside increasing labor shares under low substitutability (sigma < 1) warns against complacency: labor-income gains and technology-driven labor displacement can coexist. Fifth, the need for future work on skill heterogeneity and task creation suggests that the framework can be extended to inform distributional policies.

Key Concepts

Degree of automation: In this paper, the share of the unit task continuum performed by capital rather than labor, denoted a_t, ranging from 0 to 1. It is determined endogenously in equilibrium by relative factor prices and increases with the capital-labor ratio. It is distinct from any technology-specific proxy and emerges as a function of aggregate macroeconomic observables.

Task-based production framework: A model in which output requires completing a continuum of tasks, each exhibiting Leontief technology at the task level (capital and labor are perfectly substitutable within a task, but the firm either fully automates a task or uses labor exclusively). Tasks are ordered by the relative efficiency of capital to labor, and firms choose the automation cutoff that minimizes cost given factor prices.

Pareto efficiency distribution: The specific parametric form of aggregate capital- and labor-input efficiency functions (Eq. 22) under which the task-level aggregation yields a CES production function at the macro level. The relationship between the degree of automation and aggregate input efficiencies follows a Pareto cumulative distribution, which also delivers the highest internal consistency among automation measures tested.

Internal consistency criterion: A criterion for selecting among automation measures, defined as the mean squared gap between the automation degree inferred from the capital-labor relationship and the automation degree implied by the output-per-worker relationship, within the same CES structure (Eq. 42). A smaller gap indicates that the measure is more coherent with the CES production framework from which it is derived.

Capital-augmenting technical progress: An exogenous shift in the efficiency of capital inputs (A_K,t) that raises the effective capital-labor ratio and therefore the degree of automation at any given physical capital-labor ratio. Distinguished from labor-augmenting and neutral technical progress. In the empirical estimation, capital-augmenting technical progress is statistically significant across all specifications, while labor-augmenting technical progress cannot be confirmed.

Two-level CES production function: An extension of the standard CES that nests industrial robots and other capital into a capital composite at the inner level (with substitution elasticity sigma-b), then combines the composite with labor at the outer level (with elasticity sigma-a). Allows separate identification of the automation role of robots versus other capital, yielding a more pronounced increase in the degree of automation than the standard CES when robots are explicitly accounted for.

Automation frontier: The marginal task at which the cost of capital use exactly equals the cost of labor use, i.e., the task a_t at which lambda(a_t)/theta(a_t) = w_t/R_t. Tasks with indices below this frontier are automated; tasks above are performed by labor. As the wage-to-rental ratio rises, the frontier expands (more tasks become automated), capturing the central mechanism by which capital deepening drives automation.