Abstract
This paper develops a two-frontier task model in which robots and AI displace labor along distinct task continua — a physical sector and a cognitive sector. Within a nested CES production structure with two skill types and worker mobility across sectors, the partial effects of robot and AI shocks on the skill premium have opposite signs under pure displacement, and remain opposite under symmetric reinstatement regimes. Aggregate automation is therefore not a sufficient statistic for inequality: any test that lumps robots and AI into a single index masks economically meaningful heterogeneity.
Setup
Two sectors, Physical ($P$) and Cognitive ($C$), and two skill types, High ($H$) and Low ($S$). Each sector uses a nested CES production function with capital, surviving labor, and reinstated labor modules. The within-sector labor aggregator nests skill types with elasticity $\varepsilon_w > 1$.
A robot shock is an increase in $I_R$, the physical frontier (holding $I_A$ fixed). An AI shock is an increase in $I_A$, the cognitive frontier (holding $I_R$ fixed). Each frontier marks the fraction of its task continuum that has been automated.
Key Reduction
Wage equalization across sectors and the skill comparative-advantage assumption imply a constant cross-sector skill ratio,
$$\frac{r_C}{r_P} = \kappa^{\varepsilon_w - 1},$$
which depends only on the skill comparative-advantage parameter $\kappa$ and the labor elasticity $\varepsilon_w$ — not on automation levels, prices, or output. This collapses the two-dimensional labor-allocation problem $(\lambda_S, \lambda_H)$ to a single dimension, enabling a clean chain-rule decomposition of the skill-premium effects of each frontier.
Main Result
Proposition 1 (Opposite Skill Premium Effects). Under pure displacement ($\delta_R = \delta_A = 0$), inner elasticity $\sigma > 1$, outer elasticity $\rho > 1$, and Cobb-Douglas final aggregation,
$$\frac{\partial \mathcal{P}}{\partial I_R} > 0 \qquad \text{and} \qquad \frac{\partial \mathcal{P}}{\partial I_A} < 0.$$
A robot shock reduces surviving physical-labor productivity $\mathcal{A}{SP}$, pushing workers out of the physical sector — the resulting reallocation raises the skill premium. An AI shock reduces surviving cognitive-labor productivity $\mathcal{A}{SC}$, pushing workers into the physical sector — the resulting reallocation lowers the skill premium. The opposite-sign result is robust to the nested CES structure (the outer elasticity $\rho$ scales the magnitude but not the sign) and to symmetric reinstatement regimes.
Calibration
The calibrated configuration is strongly asymmetric across frontiers:
- $\hat{\delta}_R \approx 1.13$ — robots reinstate substantial new physical tasks per task displaced (strong reinstatement)
- $\hat{\delta}_A \approx 0$ — AI displaces cognitive tasks with negligible reinstatement
This asymmetry is illustrated below: in Panel A the physical-task identity extends past $1$ (robots create new tasks beyond the surviving range), while in Panel B the cognitive labor band is simply compressed with no offsetting task creation. Drag the sliders to see how each frontier reshapes its task continuum.
Implication
Aggregate automation indices conflate two technologically distinct frontiers. Empirical work and policy that treat automation as a single phenomenon will systematically miss the opposite skill-premium responses to robots and AI. The model rationalizes apparent “no effect” findings on composite automation indices: each frontier moves the skill premium, but in opposite directions — the canceled sum looks like nothing happened.