Physical capital ($K_P$) represents traditional capital goods like machines, buildings, and equipment used in physical production. This is distinct from AI/intelligence capital. Higher $K_P$ means more physical infrastructure per worker - think factories with more machines, construction with better equipment, or restaurants with more kitchen tools. $K_P=1.38$ is calibrated to match the 2023 US economy's capital-to-labor ratio of about 4.6.
The substitution parameter ($\rho_P$) controls how easily physical capital can replace human labor in physical tasks. When $\rho_P=1$, robots can perfectly replace workers (perfect substitutes). When $\rho_P=0$, you get the standard Cobb-Douglas case (intermediate substitutability). When $\rho_P \to -\infty$, capital and labor are perfect complements - you need both in fixed proportions, like a driver and a truck. Our baseline $\rho_P=-0.67$ gives an elasticity of substitution $\sigma=0.6$, matching empirical estimates from US manufacturing where machines and workers are somewhat complementary but not perfectly so. Lower $\rho_P$ means it's harder for robots to replace physical workers.
The capital weight $\alpha_P$ represents the distribution parameter in the physical CES production function.
Choose how to aggregate capital and labor in physical production. CES: Standard production function. CES with scale (DS-weights): Adds θ and DS-style weights α^(1-ρ); no love-of-variety micro-foundation implied.
CES
CES with scale
$$P = \left[\alpha_P K_P^{\rho_P} + (1-\alpha_P) L_P^{\rho_P}\right]^{1/\rho_P}$$
AI capital ($K_I$) represents the effective computational capacity available for intelligence tasks - think of it as the total "brain power" from all AI systems combined. This includes hardware (GPUs, servers), software efficiency, and algorithmic improvements. $K_I=9$ represents an "abundant AI" scenario where there's 9 times more AI capital than human labor. Higher $K_I$ means more AI available to automate tasks.
Automation share ($\alpha_I$) is the fraction of intelligence tasks that have been automated by AI. This is the key parameter that increases as AI capabilities expand over time. $\alpha_I=0$ means no tasks are automated (all intelligence work done by humans), $\alpha_I=0.5$ means half of cognitive tasks are automated, and $\alpha_I=1$ means full automation of intelligence work. Our baseline $\alpha_I=0.2$ represents a near-term scenario where 20% of cognitive tasks can be done by AI. This parameter drives the main dynamics in our model - as it increases, workers reallocate from intelligence to physical tasks.
This gets changed during simulation on right.
Intelligence substitution parameter ($\rho_I$) controls how easily AI can replace humans in cognitive tasks and how different intelligence tasks substitute for each other. When $\rho_I=1$, AI can perfectly replace humans in intelligence tasks (perfect substitutes). When $\rho_I=0$, you get the standard Cobb-Douglas case (intermediate substitutability). Low $\rho_I$ means AI and humans are more complementary - they work better together. Our baseline $\rho_I=0.55$ gives an elasticity of substitution $\sigma=2.2$, reflecting high substitutability between intelligence tasks (consistent with how programming, writing, analysis, and other cognitive tasks can often substitute for each other).
Intelligence scaling parameter ($\theta_I$) captures diminishing returns to AI and labor in the production of intelligence. $\theta_I=1$ means constant returns - doubling both AI and human labor doubles intelligence. $\theta_I<1$ means decreasing returns, so that doubling both inputs increases intelligence less than proportionally. Our baseline $\theta_I=0.94$ reflects empirical estimates from professional services sectors. Lower $\theta_I$ implies stronger diminishing returns to AI investment in terms of producing intelligence.
$$I = \left[\alpha_I^{1-\rho_I} K_I^{\rho_I} + (1-\alpha_I)^{1-\rho_I} L_I^{\rho_I}\right]^{\theta/\rho_I}$$
Total labor supply ($L$) represents the size of the workforce, normalized to $L=1$ as baseline. Labor endogenously reallocates between physical and intelligence tasks to equalize marginal productivity (wages) across physical and intelligence sectors. As AI automates more intelligence tasks, workers optimally shift toward physical production. The model solves for the optimal allocation $\beta^*$ that balances wages between sectors. Increasing $L$ leads to more labor relative to capital, which makes workers less productive when they reallocate to the capital-limited physical sector.
The macro substitution parameter ($\rho$) is the most important parameter in the model - it determines whether we get intelligence saturation or explosive AI growth. When $\rho<0$, physical and intelligence inputs are complements, leading to intelligence saturation where additional AI yields diminishing returns. When $\rho>0$, they're substitutes, allowing unbounded growth as the growth effects of AI are not dragged down by physical production. Our baseline $\rho=-0.67$ ($\sigma=0.6$ elasticity) treats them as complements, reflecting that you can't build cars with pure intelligence - you need both smart control systems AND physical materials/assembly. This parameter determines whether we get the economists' 'bounded growth' view or the AI experts' 'singularity' scenario.
The physical weight $\tau$ represents the distribution parameter in the overall CES production function.
$$Y = \left[\tau P^{\rho} + (1-\tau) I^{\rho}\right]^{1/\rho}$$
Choose the AI development scenario. 'Fixed AI' holds AI capital and automation levels constant - useful for understanding static equilibrium effects of different automation levels. 'Exponential AI' simulates realistic AI development where both AI capital ($K_I$) grows exponentially (following observed cost reductions) and automation ($\alpha_I$) progresses over time as AI capabilities expand. Exponential mode captures the dynamic transition that we're likely experiencing now, while Fixed mode helps isolate the effects of different automation equilibria.
Fixed AI
Exponential AI
We increase the importance of intelligence in overall production.
Wage vs αI
Wage
Total Output vs αI
Total Output Y
P and I Components vs αI
P-nest Output
I-nest Output
Optimal Labor Allocation vs αI
β* (Optimal P-labor share)
Intelligence Saturation
Intelligence Saturation