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. $K_P=1.5$ is calibrated to match the 2023 US economy's capital-to-labor ratio of about 5.8. Increasing $K_P$ helps maintain wages as workers shift to physical tasks during AI automation.
Capital share ($\alpha_P$) determines how much physical production relies on capital versus labor in the CES production function. It's the relative importance of machines vs workers in creating physical output. $\alpha_P=0$ means production uses only labor (pure handicraft economy), $\alpha_P=1$ means only capital matters (fully automated physical production). $\alpha_P=0.7$ reflects that in the US economy, about 42% of income goes to capital owners while 58% goes to workers. This parameter affects how easily capital can replace workers in physical tasks.
The substitution parameter ($\rho_P$) controls how easily physical capital can replace human labor in physical tasks. When $\rho_P\to 1$, robots can perfectly replace workers (perfect substitutes). When $\rho_P=0$, you get the standard Cobb-Douglas case where both are needed in fixed proportions. When $\rho_P\to -\infty$, capital and labor are perfect complements - you need both, 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.
Choose how to aggregate capital and labor in physical production. CES (Constant Elasticity of Substitution) is the standard economic production function where substitutability between inputs stays constant. DS (Dixit-Stiglitz) is similar but adds a scaling parameter $\theta_P$ that allows for increasing or decreasing returns to scale - useful for modeling situations where doubling all inputs might more than double output (economies of scale) or less than double it (congestion effects). Most economists use CES; DS gives more flexibility.
CES
DS
$$P = \left[\alpha_P K_P^{\rho_P} + (1-\alpha_P) L_P^{\rho_P}\right]^{1/\rho_P}$$
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 about 6 times more AI capital than physical capital. Higher $K_I$ means more tasks can be automated and AI can substitute for human cognitive work more easily. Unlike physical capital, AI capital has been growing exponentially - costs halving every 3 months in some domains.
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 $\alpha_I=0.9$ represents a near-future scenario where 90% 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. High $\rho_I$ means AI and human intelligence are nearly perfect substitutes - AI can easily replace human cognitive work. Low $\rho_I$ means they're more complementary - humans and AI 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). This parameter critically affects wage dynamics during automation.
Choose how to aggregate AI capital and human intelligence. CES (Constant Elasticity of Substitution) treats AI and human intelligence with fixed substitutability. DS (Dixit-Stiglitz) adds a scaling parameter $\theta_I$ that captures the power-law relationships observed in AI research - where doubling compute might give sublinear performance improvements. This reflects empirical AI scaling laws showing diminishing returns to additional compute, consistent with findings that model performance scales as a power of training compute, data, and parameters.
CES
DS
Intelligence scaling parameter ($\theta_I$) captures the diminishing returns to AI observed in machine learning research. $\theta_I=1$ means constant returns - doubling AI compute doubles intelligence output. $\theta_I<1$ means decreasing returns, consistent with AI scaling laws where performance follows power laws in compute. Our baseline $\theta_I=0.94$ reflects empirical estimates from professional services sectors. Lower $\theta_I$ means stronger diminishing returns to AI investment, leading to earlier intelligence saturation. This parameter bridges economic modeling with computer science scaling laws, explaining why exponential AI progress might not translate to exponential economic growth.
$$I = \left[\alpha_I^{1-\rho_I} K_I^{\rho_I} + (1-\alpha_I)^{1-\rho_I} L_I^{\rho_I}\right]^{\theta/\rho_I}$$
Choose how to combine physical and intelligence outputs into final production. CES (Constant Elasticity of Substitution) provides smooth substitution with constant elasticity throughout. PSI (S-curve) has fundamentally different behavior: it acts like perfect complements when intelligence is low (you need both physical and intelligence inputs), but approaches perfect substitutes when intelligence is very high (pure intelligence can completely replace physical production). This captures scenarios where AI might initially complement human work but eventually completely automate entire sectors, achieving full saturation that CES cannot model.
CES
PSI
Total labor supply ($L$) represents the size of the workforce, normalized to $L=1$ as baseline. Unlike traditional models where labor allocation is fixed, here labor endogenously reallocates between physical and intelligence tasks to equalize marginal productivity (wages) across sectors. As AI automates more intelligence tasks, workers optimally shift toward physical production. Increasing $L$ simulates population growth or immigration; decreasing $L$ simulates demographic decline. The model solves for the optimal allocation $\beta^*$ that balances wages between sectors.
Physical weight ($\tau$) determines how much the economy depends on physical versus intelligence inputs in final production. $\tau=0$ would be a pure virtual economy (like a metaverse) where only intelligence matters. $\tau=1$ would be a pure physical economy where intelligence adds no value. Our baseline $\tau=0.18$ reflects that about 26% of US workers are in manual occupations. This parameter is crucial for wage effects: higher $\tau$ protects workers during automation by giving them more physical tasks to move into. Lower $\tau$ makes automation more beneficial for output but potentially harmful for wages.
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 AI replaces 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.
$$Y = \left[\tau P^{\rho} + (1-\tau) I^{\rho}\right]^{1/\rho}$$
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.