The aggregate production function (APF), typically written as (Y = F(K,L)), is one of macroeconomics’ most durable workhorses. It anchors growth accounting, provides the technological core of DSGE models, and underwrites a familiar distributional story: factor prices equal marginal products, and the profit rate is a scarcity signal for “capital.” Yet this convenience rests on a maintained hypothesis that becomes increasingly fragile in economies where capital is genuinely heterogeneous—and where a growing share of capital takes intangible and AI-specific forms.
In our recent work, “The Aggregate Production Function in the Age of AI: Heterogeneous Capital and Aggregation Artifacts,” we argue that, under modern capital heterogeneity (tangible, intangible, and AI capital), the APF is not merely an approximation with some error—it can become a structural artifact. In short: a statistical object that fits extremely well, even when its marginalist interpretation is structurally inconsistent with the underlying economy.
This post summarizes the paper’s core ideas, results, and methodological implications—especially for those working on structural change, productivity measurement, and complexity-based macro modeling.
1) Why the classical warning matters again: Cambridge, updated
The Cambridge Capital Controversy (CCC) established a deep point: when capital goods are heterogeneous, there is no guarantee that “capital” can be aggregated into a single quantity (K) that behaves monotonically with the profit rate (r). Classic phenomena such as:
- Reswitching: a technique can be cost-minimizing at low (r), dominated at intermediate (r), and cost-minimizing again at higher (r).
- Reverse capital deepening (RCD): higher (r) can be associated with more capital-intensive techniques (higher (K/L)), contradicting the neoclassical monotonicity required for a well-behaved demand for capital.
The mainstream response historically was pragmatic: these anomalies might exist in theory, but they are empirically rare—and the APF “works” well enough (Samuelson, 1966; Harcourt, 1972).
Our claim is that AI-era capital accumulation re-opens CCC not as a historical curiosity, but as a live identification problem for macroeconomic inference.
2) Why AI capital changes the aggregation game
The modern economy is increasingly shaped by intangible assets—software, data, R&D, organizational capital (Haskel & Westlake, 2018). AI capital is an extreme case: trained models, pipelines, and data infrastructures exhibit properties that strain standard convexity and rivalry assumptions:
- Obsolescence-driven depreciation (fast algorithmic and competitive replacement, not physical wear).
- Partial non-rivalry and spillovers (the same model can be deployed across many uses).
- Strong complementarity with skilled labor (and sometimes labor-saving in specific tasks).
- Non-convexities and threshold effects (platform dynamics, network effects, commoditization).
These features do not merely “complicate calibration.” They alter the ontology of capital: what counts as rent-bearing private capital versus socially available capacity becomes contingent on competition, regulation, and openness.
3) A new mechanism: capital precipitation
A central conceptual contribution of the paper is capital precipitation: an economic phase change in which assets transition abruptly from rent-generating private capital to latent common capacity.
We formalize two compartments (for each capital type (i)):
- (K^{priv}_i): valuable private capital, on balance sheets, earning rents.
- (K^{prec}_i): precipitated capital, whose market value collapses (≈0), while some technical capacity remains available.
Total potential capacity:
[
K^{pot}_i = K^{priv}_i + K^{prec}_i.
]
Precipitation differs from “depreciation.” Depreciation is gradual wear/obsolescence; precipitation is discontinuous devaluation induced by innovation shocks, competitive entry, open-source releases, or regulatory shifts (think: proprietary model value collapsing when a close open alternative appears; or stranded assets).
Crucially, precipitation implies the possibility of a systematic wedge between:
- Physical productivity (technological layer): (Q = F(K^{pot}, L)),
- Monetary productivity (valuation layer): (VA = P \cdot Q),
because increasing potential capacity can push prices downward, especially when precipitated AI capacity dilutes scarcity. This mechanism generates a structural version of the “productivity paradox”: physical output per worker may rise while measured value-added productivity stagnates or falls.
4) Method: combining a Sraffian multi-technique core with a dynamic ABM
The paper is deliberately methodological: it uses computational experiments where the data-generating process is known, allowing us to test whether APF estimation recovers meaningful structure.
4.1 Static model (Sraffian multi-technique)
We build a multi-sector, multi-technique Sraffian framework. Techniques differ in input-output coefficients, labor coefficients, and capital composition (tangible/intangible/AI shares). For any given profit rate (r), sectors select cost-minimizing techniques, generating a wage–profit frontier and implied “aggregate capital” patterns across (r).
4.2 Dynamic ABM (endogenous technique choice + innovation + precipitation)
We then embed this structure in a dynamic agent-based model with:
- heterogeneous firms within sectors,
- boundedly rational technique switching with inertia and switching costs,
- credit allocation, investment rules, and price formation,
- endogenous profit rate dynamics,
- innovation processes with an AI-acceleration regime,
- precipitation shocks affecting AI capital.
This ABM generates synthetic macro time series ((Y_t, K_t, L_t, r_t, \ldots)) on which one can estimate Cobb–Douglas or translog production functions—exactly as in applied macro—while knowing that the underlying economy violates the conditions needed for valid aggregation.
5) Core results: the APF can “fit perfectly” while meaning nothing structural
Result A: Reverse capital deepening is systematic (in the static experiments)
In Monte Carlo experiments with randomly generated economies, reverse capital deepening appears ubiquitously under baseline calibrations. Classical reswitching can be constructed in benchmark examples but is not typically observed in random draws—consistent with probabilistic arguments that multiple intersections of wage–profit curves require special structure (Schefold, 1976; Han & Schefold, 2006). The key point: you do not need frequent reswitching to break neoclassical monotonicity; RCD alone is enough.
Result B: The “APF illusion”—high (R^2) with economically nonsensical parameters
Estimating a Cobb–Douglas APF on ABM-generated data yields extremely high statistical fit (near-perfect (R^2)), while the estimated elasticities can be economically impossible (e.g., capital elasticity (>1), labor elasticity (<0)). Rolling-window estimates show extreme parameter instability, again hidden by consistently high (R^2).
This directly supports (and computationally deepens) the “accounting identity critique”: because value-added accounting embeds (VA \equiv wL + rK), regressions can mimic technology even when technology is not aggregable (Felipe & McCombie, 2013). Our contribution is to show how this illusion survives even under endogenous heterogeneity, innovation, and precipitation.
Result C: Precipitation generates a systematic wedge between physical and monetary productivity
Under AI shock scenarios, precipitation can raise physical productivity (more usable capacity), while monetary productivity falls (price compression and rent destruction). This is not measurement error in the usual sense; it is a structural outcome of capital transitioning from private scarcity to quasi-common capacity.
6) Implications: DSGE as “ideal gas,” ABM as “real gas”
A useful framing in the paper is the analogy to thermodynamics:
- DSGE models resemble an ideal gas: tractable, parsimonious, smooth aggregation, representative agents, small shocks around a steady state.
- AI-era economies resemble a real gas: interactions, heterogeneity, phase changes (precipitation), and non-convexities matter.
The conclusion is not “discard DSGE,” but recalibrate our epistemic stance: DSGE may remain a useful approximation regime, yet becomes fragile precisely where AI-driven structural change is most important. In those regimes, agent-based models are not optional ornamentation; they become a necessary complementary method for diagnosing aggregation breakdowns and policy-relevant nonlinearities.
7) What would an empirical program look like?
A key advantage of this framework is that it suggests testable signatures of aggregation artifacts in real data:
- Parameter instability in production-function estimation (rolling windows, structural breaks).
- Implausible elasticities coexisting with high fit (especially during AI adoption waves).
- Divergence between quantity-based and value-based productivity (sectoral output measures vs. value added).
- Episodes consistent with precipitation: abrupt devaluation of intangible/AI assets (competitive commoditization, open-source releases, regulatory revaluation), accompanied by price compression and shifting income shares.
The paper’s computational approach can be used as a calibration-and-detection laboratory: generate synthetic economies under competing structural hypotheses (smooth aggregable vs. precipitating heterogeneous), then compare the implied macro-statistical footprints with observed ones.
8) Where we go from here (SCCS angle)
From a complex-systems perspective, the paper points toward a macroeconomics where the primary state space is not ((K,L)), but a space of technique configurations—a distribution over heterogeneous productive methods and capital compositions, with endogenous transitions and revaluations. The interesting objects become:
- the topology of the technique space,
- switching and diffusion dynamics under frictions,
- emergent macro regularities (and their breakdown),
- institutional and market-structure parameters that govern precipitation and rent formation.
In other words, AI-era “capital” is increasingly about configuration, access, and appropriation, not just accumulation.
References (selected, as cited in the paper)
Baqaee, D. & Farhi, E. (2019). The macroeconomic impact of microeconomic shocks.
Brynjolfsson, E. et al. (2021). [on software/AI depreciation and intangibles].
Dosi, G. et al. (2010). [ABM and innovation dynamics].
Felipe, J. & McCombie, J.S.L. (2013). The aggregate production function and the measurement of technical change.
Granger, C.W.J. & Newbold, P. (1974). Spurious regressions in econometrics.
Han, Z. & Schefold, B. (2006). [probabilistic analysis of reswitching].
Harcourt, G.C. (1972). Some Cambridge controversies in the theory of capital.
Haskel, J. & Westlake, S. (2018). Capitalism without capital.
Hulten, C.R. (1978). Growth accounting with intermediate inputs.
Robinson, J. (1953). The production function and the theory of capital.
Samuelson, P.A. (1966). A summing up.
Schefold, B. (1976). [reswitching conditions].
Smets, F. & Wouters, R. (2007). Shocks and frictions in US business cycles: A Bayesian DSGE approach.
Sraffa, P. (1960). Production of commodities by means of commodities.
(Full bibliographic details are in the manuscript.)