1. Introduction

The rapid advance of artificial intelligence has prompted a large literature on the future of labor (Acemoglu and Restrepo, 2019; Korinek and Suh, 2024). A key insight from Kording and Marinescu (2025) is that AI's impact on aggregate output is bounded by the physical sector. When physical goods and intelligence services are complements (ρ < 0), increasing AI capital yields diminishing returns -- intelligence saturation.

This result rests on the assumption that K_P (physical capital accessible to production) is fixed or grows slowly, and that ρ_P (the elasticity of substitution between physical capital and labor) remains deeply negative. In other words, robots cannot meaningfully replace human labor in the physical sector.

We revisit this assumption. Industrial robot costs have fallen 82% since 1995 (ARK Invest, 2024). Humanoid robots entered serial production in 2024-2025 at price points declining ~40%/year (Goldman Sachs, 2024). Wright's Law -- the empirical regularity that costs fall by a constant percentage per doubling of cumulative production -- has held for robots with a ~50% learning rate. If this rate persists as production scales from thousands to millions of units per year, robot costs could follow the trajectory of solar photovoltaics, which fell 99.9% from 1976-2024.

We make three contributions. First, we extend the K&M model to allow time-varying (K_P(t), ρ_P(t)) trajectories. Second, we calibrate six scenarios to historical and projected data from ARK Invest, the IFR, Goldman Sachs, and Bank of America. Third, we characterize the critical boundary in parameter space that separates persistent saturation from saturation-breaking growth.

2. Model

We adopt the nested CES framework of Kording and Marinescu (2025). Final output combines physical goods P and intelligence services I:

The physical sector uses capital K_P and labor L_P:

The intelligence sector uses AI capital K_I and labor L_I with Dixit-Stiglitz returns to scale θ_I:

Labor is allocated between sectors to equalize wages, with total labor supply L fixed. The equilibrium labor allocation β* = L_P/L is found by Brent's method root-finding on the wage-equalization condition.

Our extension. We make K_P and ρ_P functions of time: K_P(t) follows a growth trajectory calibrated to technology cost data, and ρ_P(t) transitions from its initial value toward a scenario-specific endpoint reflecting changes in robot-labor substitutability.

2.1 Endogenous K_P via Wright's Law

Our central extension endogenizes K_P through a Wright's Law feedback loop. Unit robot cost follows a learning curve:

where Q(t) is cumulative robot production and b is the learning exponent. With b ≈ 1.0, the learning rate is ~50% per cumulative doubling, matching observed rates for industrial robots (ARK Invest, 2024). Effective physical capital K_P is inversely proportional to cost: K_P(t) = A · Q(t)^b, calibrated so K_P(0) = K_P,0.

Robot adoption is endogenous. The flow of new production responds to the wage-to-robot-cost ratio relative to initial conditions, with a logistic saturation cap:

where η is the demand elasticity. This creates the central feedback loop: lower costs → more adoption → higher cumulative production → lower costs. The system is solved by forward Euler integration.

2.2 Directed R&D

Following Acemoglu (2002), we model the endogenous allocation of R&D between physical and intelligence automation. A fixed R&D budget R is split between the two sectors:

where s is the current intelligence saturation level and γ controls the sensitivity of reallocation. As intelligence saturates (s → 1), the marginal value of intelligence R&D falls, and the model endogenously redirects innovation toward physical automation. Physical capital then grows at rate:

This generates the novel prediction that intelligence saturation accelerates physical automation -- a self-correcting dynamic absent from models with exogenous technology trajectories.

2.3 Empirical Calibration of ρ_P

The elasticity of substitution between robots and labor -- σ_P = 1/(1-ρ_P) -- is a critical parameter. Adachi (2025) provides the first clean causal estimates using a "Japan Robot Shock" instrument, finding σ_P up to 3.0 for production and material-moving occupations. This implies ρ_P ≈ 0.67 -- dramatically higher than the K&M default of -0.67 (strong complements). At this calibration, robots are strong substitutes for labor in physical tasks, fundamentally changing the saturation dynamics.

3. Scenarios

We define nine trajectories spanning exogenous, endogenous, and empirically calibrated physical automation futures:

The first six scenarios use exogenous K_P trajectories calibrated to historical analogs. The Wright's Law scenario endogenizes the cost-adoption feedback loop (Section 2.1). The Directed R&D scenario endogenizes the innovation allocation (Section 2.2). The Adachi Calibration applies empirical estimates of ρ_P from Adachi (2025) to test whether saturation holds under realistic substitution elasticities.

4. Results

4.1 Time Evolution

Figure 1 shows the evolution of key variables over 30 years under each scenario.

Under the Baseline and Persistent Scarcity scenarios, saturation reaches near 100% and wages remain flat. The Logistic Diffusion scenario delays but does not prevent saturation. In contrast, the Endogenous Learning scenario breaks saturation entirely: wages increase by an order of magnitude as both K_P growth and rising ρ_P loosen the physical bottleneck.

The three new scenarios reveal additional dynamics. The Wright's Law scenario generates accelerating K_P growth through the cost-adoption feedback loop: as robots become cheaper, adoption rises, cumulative production grows, and costs fall further. The Directed R&D scenario shows that intelligence saturation itself redirects innovation toward physical automation -- as the intelligence sector saturates, the marginal return to physical R&D rises, generating a self-correcting dynamic. Most strikingly, the Adachi Calibration (σ_P=3.0, ρ_P=0.67) breaks saturation even under moderate K_P growth of 7%/yr, because high substitutability means robots can effectively replace labor at current cost levels.

5. Empirical Calibration

The ~50% learning rate for industrial robots is high relative to other technologies, but only 2-3 cumulative production doublings have occurred. Bank of America projects that humanoid robot production could scale to 1M units/year by 2030, implying ~10 additional doublings within 15 years. At a 50% learning rate, this would reduce unit costs by ~99.9%, matching the solar PV trajectory. Our Endogenous Learning scenario is thus empirically grounded but represents the upper bound of plausible cost decline.

6. Discussion

Our results suggest that the intelligence saturation result of Kording and Marinescu (2025) is best understood as a conditional prediction: it holds if physical automation remains expensive and non-substitutable. Three lines of evidence challenge this condition.

First, the Wright's Law feedback loop (Section 2.1) generates accelerating cost decline as robot production scales. With a ~50% learning rate and cumulative production expected to grow from millions to tens of millions of units by 2035 (Goldman Sachs, 2024), the cost-adoption feedback creates explosive dynamics absent from exogenous-trajectory models.

Second, the Directed R&D mechanism (Section 2.2) provides a self-correcting dynamic: intelligence saturation itself redirects innovation toward physical automation. Following Acemoglu's (2002) price effect, when complementary inputs are scarce, R&D is directed toward improving those inputs. This means the very conditions that produce saturation also generate the economic incentive to break it.

Third, Adachi's (2025) empirical estimates of σ_P ≈ 3.0 for production occupations suggest that robots are already strong substitutes for labor in physical tasks. At these elasticities, even moderate K_P growth breaks saturation -- a finding that challenges the K&M assumption of strong complementarity (ρ_P = -0.67).

Several caveats apply. Wright's Law learning rates can break down at scale due to materials constraints or regulatory barriers. The Adachi (2025) estimates apply to production occupations and may not generalize to all physical tasks. The Directed R&D mechanism assumes costless reallocation of researchers, while in practice retooling takes time.

For policymakers, the key levers are robot subsidies, labor regulations, and technology adoption rates -- these shape both the rate of physical capital accumulation and the degree of robot-labor substitutability, which together determine whether a country or sector faces persistent saturation or saturation-breaking growth.

References