Kennard R. Laviers, PhD (Computer Science), and Maria Siakola
The complete equation set of the model, block by block, as implemented in the model code (Appendix B of the paper lists the source files). One annual loop from 2025 to 2050; every index equals one at the 2025 anchor unless stated; the years 2019 to 2024 are the record (observed series where one exists, reconstruction otherwise), not simulation. Parameter values are the model defaults; the calibration section of the paper says where each comes from.
| Symbol | Meaning | 2025 |
|---|---|---|
| $C$, $C_{\mathrm{eff}}$ | installed compute stock; the part that can be powered | $1$ |
| $R$ | cumulative algorithmic efficiency | $1$ |
| $C_{\mathrm{train}}$ | frontier training compute | $1$ |
| $A$ | capability index (effective frontier compute) | $1$ |
| $h_{50}$, $h_{80}$ | METR task horizon at 50% and 80% success, hours | $3.5$, $0.7$ |
| $\tau$ | technically substitutable share of the 2025 cognitive task stock | $0.083$ |
| $D$ | share of the 2025 task stock performed by AI ($D \le \tau$) | $0.02$ |
| $T$ | cognitive task volume (induced demand included) | $1$ |
| $H$ | human cognitive task demand | $1$ |
| $w_h$ | cognitive wage | $1$ |
| $P_{\mathrm{cost}}$, $P_{\mathrm{AI}}$ | unit cost of AI output; price charged | $0.25$ |
| $L_c$, $L_p$, $U$, $N$ | cognitive and physical employment, displaced pool, left the workforce (shares of the workforce) | $0.45$, $0.55$, $0$, $0$ |
| $\varphi$ | political friction on diffusion | $0$ |
| $f$ | accountability floor (share of cognitive work that stays human) | $0.15$ |
Compute is a vintage stock. Each year's additions form a vintage $v$ with quantity $q_v$, book value $B_v$ and straight-line depreciation over the book life $L_{\mathrm{book}}$; quantity wears at $\delta_{\mathrm{phys}}$:
Firm power capacity $K$ is built through a commitment pipeline with channels $\ell$ (grid, five-year lag, 60%; behind-the-meter gas, two years, 30%; retrofit, one year, 10%):
Electricity carries a scarcity markup when headroom falls short of target, and grid capital is socialised into household rates:
Power is drawn only by running compute: training, research and experiments always, inference as used.
Algorithmic efficiency grows at a rate produced by research effort with diminishing returns to the idea stock (fishing-out exponent $\phi$) and a soft limit:
Effective compute is allocated by shares that sum to one: the frontier run $s_{\mathrm{front}} = \min(s_{\max},\, s_{0}\, C_{\mathrm{train}} / (C_{\mathrm{eff}}/C_{\mathrm{eff},0}))$, which grows with the race from $0.04$ to a cap of $0.3$ and takes its share from inference; research agents $s_{\mathrm{res}} = 0.1$; experiments $s_{\mathrm{exp}} = 0.2$; inference $s_{\mathrm{inf}} = 1 - s_{\mathrm{front}} - s_{\mathrm{res}} - s_{\mathrm{exp}}$. Research agents scale with the compute allocated to them (none without research compute). One efficiency index $R$ enters three channels with separate exponents, $R^{a_{\mathrm{train}}}$ in capability, $R^{a_{\mathrm{inf}}}$ in the inference cost and capacity, $R^{a_{\mathrm{agent}}}$ in agent productivity (all $1$ by default). The AI-attributable part of progress is the counterfactual difference, $\text{recursion} = d\ln R - d\ln R\,|_{L_{\mathrm{AI}}=0}$, not the labor share. Only the frontier run is built regardless of returns, and only to the extent lab budgets hold: race additions carry a factor $\min(1, \text{AI revenue} / \text{peak revenue to date})$, floored at $0.1$.
The reference shares $\bar s_{\mathrm{res}} = 0.10$ and $\bar s_{\mathrm{exp}} = 0.20$ are fixed across scenarios: agents and experiments are measured in units of the 2025 compute at the reference allocation and $\kappa$ is calibrated there, so moving a budget between researchers and experiments changes research effort (divided by each scenario's own allocation, the share would cancel). $E$ is a CES aggregate of research labor $L$ (the human research workforce $H_r$, $1$ in 2025 and falling to the frontier handful per §4, plus AI agents, whose count scales with effective compute and efficiency and whose quality rises with capability) and experiment compute, with elasticity $1/(1-\rho) = 0.5$, so labor cannot substitute for experiments without limit. $\kappa$ is calibrated so that progress is $0.9$ log units a year in 2025.
Capability is effective frontier compute; frontier training compute is its own state whose growth decays toward the stock's:
Recursive-improvement regimes, selected by research_regime:
with each discovery deployed from $t + \ell$ over $m$ years as $\ln J / m$ per year onto $\ln R$ (probability $1 - p_{hw} - p_{en}$), onto $-\ln c_{hw}$ ($p_{hw}$) or onto $\ln \eta$ ($p_{en}$); $h_0 = 0.08$, $\xi = 0.25$, $h_{\max} = 0.35$, $\mu_J = 1.1$, $\sigma_J = 0.5$, $\ell = 2$, $m = 3$, $p_{hw} = 0.25$, $p_{en} = 0.15$. The diminishing regime is the equation above with $P_t = 1$ and no discoveries.
Three capabilities: task competence $c_t$ on the 50% horizon, autonomous reliability $a_t$ on the 80% horizon times the autonomy discount $\delta$ (what substitutability follows), research autonomy $r_t$ on the unattended horizon against the research midpoint $\kappa_r h_{\mathrm{mid}}$ (what the research workforce follows, §4):
The task horizon is anchored to METR and substitutability to the 80% horizon:
with $h_0 = 3.5$ h, $k_h = 1.9$, $r_{80} = 0.2$, $h_{\mathrm{mid}} = 60$ h, $w = 2$.
Vendor unit cost is the carrying cost of running compute over effective compute. The human review, rework and accountability time per AI task is not vendor cost: it is the buyer's own labor, $\rho_t$ task-hours per AI task paid at the human wage, so the buyer's all-in cost per delivered AI task is the vendor price plus $\rho_t w_h$. The vendor price adds a slice of the gap to the human alternative net of review labor, taken by concentration (rent) or by scarcity when last year's inference demand exceeded powered supply:
Two tiers: the frontier tier is the share $\phi = (\tau - \tau_{\mathrm{open}})/\tau$ of AI volume only the frontier can do yet, at $k_f$ times the compute per task; each tier is priced cost-plus with its own scarcity $s_c$, $s_f$ on its own capacity, the frontier also with rent:
with the commodity cost $c$ such that the tier-weighted cost is $P_{\mathrm{cost}}$ (and $P_{\mathrm{cost}}$ scaled in 2025 so that the tier-weighted posted price meets the anchor), the multiple splitting the calibrated inference need between the tiers rather than scaling it, and the frontier's capacity share adjusting to its need share with inertia 0.5. The formula above is the volume-weighted average of the two tiers. The cost-plus floor $F_t$ is the vendors' pricing power: gross margin $m_t$ over compute cost, 0.40 in 2025 (the price of $0.10\, w_h$ is cost $0.06$ plus the margin) eroding to 0.25. AI services outside the 2025 task stock are a share of household consumption rising with the capability gap already closed, priced at a premium over the task price fixed in 2025 by the services' share of inference compute, and are vendor income in GDP:
with $\sigma_0 = 0.0005$ (household-paid AI on the accounting bridge; 0.0035, sized to total vendor revenue, in the services_total_revenue_sizing sensitivity), $\sigma_{\max} = 0.03$ a prior, $\kappa_s$ set in 2025 by a services share of inference compute of 0.6, and $s_{\mathrm{pool}} = 0.6$ the most of powered inference consumer usage can run on, so that the budget sets the price per unit of usage rather than a budget over a collapsing price setting an unbounded volume. Tasks and services clear against one powered pool: each demands a volume at its base price, $V^{\mathrm{AI}}_\star$ and $V^{\mathrm{s}}_\star$, and when their sum exceeds the capacity $K_{\mathrm{inf}}$ one clearing premium $\mu \ge 1$ rations both (budget-fixed demand falls as $1/\mu$), each price bounded by the human alternative:
and when both prices sit at their bounds the remaining shortfall is rationed pro rata. The task side's premium $\pi$ (§3) is the same object seen from the task budget. Services revenue is booked on delivered volume only; planned spending that buys nothing is saved. The tiers (above) share the pool for delivery and are priced by their own scarcity states, so borrowing between tiers is allowed and not separately constrained. Services volume counts in inference need and energy; the capital block, the frontier race budget and the antitrust rule judge on $R^{\mathrm{AI}}_{\mathrm{tot}}$.
Deployment concentration $\mathrm{conc} = \mathrm{HHI} \cdot (\tau - \tau_{\mathrm{open}})/\tau$ is the share of substitutable tasks only the frontier can do, with $\tau_{\mathrm{open}}$ evaluated at the open-weights capability lag. The number of frontier actors follows run cost against budgets with a Pareto tail:
Diffusion is logistic toward the ceiling, gated by cost advantage and slowed by political friction:
with lag $= 6$ years (5% to 95% of the ceiling at full pressure).
Human task demand, the human share of the cognitive bundle, the AI share by volume, the wage and cognitive employment:
Human cognitive demand is direct tasks plus review labor on delivered AI output, and employment and the wage follow the total:
$\psi$ is the human volume pass-through on AI-induced volume and $\gamma$ the capability gap already closed, the share of the automatable stock AI can do (the capability sigmoid itself). Induced tasks are drawn from the same capability distribution as the stock, so once AI can do a task better than people the induced volume of it goes to AI too: the pass-through decays with $\gamma$ and $H \to (1-D)\,T_{\mathrm{trend}} \to f\,T_{\mathrm{trend}}$, the accountability floor of trend volume. $\kappa_c = 0$ (preset fixed_complementarity) is the sustained structure in which humans keep $\psi$ of every induced task however capable AI becomes. $\tau$ and $D$ are shares of the 2025 task stock; trend volume is served by humans one for one; the AI share by volume counts the induced volume. Cognitive jobs gone are $L_0 - L_c$.
The research and engineering workforce (share $r_0 = 0.057$ of the 2025 workforce) follows the same capability sigmoid with the accountability floor replaced by the frontier handful $\eta_r$, the share of the 2025 field that stays whatever AI can do:
$H_r$ (2025 = 1) is the human research labor in the recursion of §2, $L = H_r + s_{\mathrm{AI}} L_{\mathrm{AI}}$, and the AI share of research labor is $s_{\mathrm{AI}} L_{\mathrm{AI}} / L$. Delivered AI volume is $V_{\mathrm{AI}} = (1-h)\,T$ capped by powered inference capacity in the same year:
and the same $V_{\mathrm{AI}}^{\mathrm{del}}$ sets inference compute need, AI revenue and running energy. The premium $\pi$ clears the planned cognitive budget $N_c^{\mathrm{plan}}$ against the delivered quantity: humans are paid $w_h H u$, AI receives $P_{\mathrm{AI}}\,\pi\, V_{\mathrm{AI}}^{\mathrm{del}} u$, sales $N_c = P_c\, T_{\mathrm{del}}\, u$ equal their sum, and the remainder of the plan the wage cap leaves unbought, $N_c^{\mathrm{plan}} - N_c \ge 0$, is unspent and saved. Demand at the posted price ($\pi = 1$) is the shortage signal the lagged scarcity markup of §3 reacts to.
Displacement is gross: the fall in cognitive employment plus the task volume newly automated and shed as headcount, split between entrants (foregone hiring, capped by the entry cohort) and incumbents:
Transition matrix over $C$ (cognitive employed), $P$ (physical employed), $S$, $L$ (short- and long-term pools) and $N$ (out of the workforce), beginning stocks to end stocks:
with $h_S$ the hires taken from the short-term pool first and $h_L$ the rest; $C^d$ and $P^d$ are the demands from the fixed point; $j_0 = 0.4$, $\ell = 0.3$, $f_l = 0.5$, $x = 0.1$. The end stocks sum to one on every row; unfilled vacancies $v - h$ are reported; the pool is tracked by origin sector; re-employed workers carry a wage discount of $0.3$ that closes at $0.12$ a year. With physical automation off, $P^d = \infty$ and physical work takes anyone the pools release at rate $j$ (the residual treatment of physical employment).
Joint closure. The matrix is evaluated inside the yearly fixed point, so production and income are computed from matched stocks rather than from labor demand. With $\phi_h$ the hours flexibility (0.05):
The wage $w_h$ clears on demand ($H_{\mathrm{tot}}$); wages are paid for hours done; the planned cognitive budget clears against $T_{\mathrm{del}}$ as in §3, so the fixed point is on the plan and realised sales can sit below it at convergence. Entrants are a decomposition under a stationary workforce: the cohort $c_{\mathrm{entry}} C_t$ replaces retirements one for one, $\text{entry}_t$ is the retirements not replaced, and
Nominal flows in 2025-GDP units, solved as a damped fixed point each year:
Real income is reported three ways: nominal in the 2025 wage numeraire, deflated by a chained Fisher basket over three goods, and deflated by physical prices only:
with $X^{hh}_c = X_c - u_t X_c / N^{\mathrm{plan}}_c$ the household's cognitive spending net of its pro rata share of the unspent task budget $u_t$, $X^{hh}_s = R^{\mathrm{s}}$ delivered services revenue at $P^{\mathrm{s}}_{\mathrm{eff}}$, and $X^{hh}_p = X_p$. The chained real-output index is the Fisher volume index over the same three goods (cognitive final demand $X_c / P_c$, services $R^{\mathrm{s}} / P^{\mathrm{s}}_{\mathrm{eff}}$, physical $N_p / P_p$).
Review binds delivered AI output: with $\bar H$ the matched hours, review is staffed first and AI output beyond what can be reviewed is lost, $V^{\mathrm{AI}} \leftarrow \min\!\big(V^{\mathrm{AI}},\ \min(\rho V^{\mathrm{AI}}, \bar H) / \rho\big)$, with the shortfall recorded as ai_unreviewed_lost.
Climate (off in the baseline): $g_t$ is the gross drift on physical unit cost ($r_d$ 0.003 in the central preset), $a$ the adaptation share of physical value added spent each year (physical demand funded from saving, $e_a = 6$ avoided per unit spent), $d_t$ the net damage; the electricity price carries its own drift $(1 + r_E)^t$ ($r_E$ 0.005 central).
Composite stress and the reactions it drives, each with a lag $\lambda$:
A rising floor can roll adoption back ($D \le \tau$); above $U = 0.15$ the run is flagged as outside the historical record.
Return on replacement value against a hurdle that includes obsolescence; plans are judged on $n_h = 4$ years of extrapolated revenue growth, adjusted with inertia, and land with a two-year commitment lag, gated by power committed two years out:
Impairment follows IAS 36: each vintage is written down to the higher of value in use (margin per unit times quantity times an annuity over remaining life) and fair value (replacement cost less a haircut) when book exceeds it; write-downs accumulate.
For cognitive group $g$ with exposure $x_g$, floor $f_g$, lead $\ell_g$ years and demand elasticity $\varepsilon_g$, with $s(t)$ the economy-wide AI share by volume:
The three research and engineering groups (computer and mathematical, architecture and engineering, life/physical/social science) are then rescaled to sum to $L_r^{\mathrm{tot}}$ and the other cognitive groups to the layer's cognitive total less $L_r^{\mathrm{tot}}$, so the occupation layer and the aggregate series carry one definition.
A trade is devalued when $s_g > 0.5$; the stranded skill premium is the present value of the wage premium over the remaining careers of the 2025 holders, weighted by training years. Physical groups use a robot exposure $x^r_g$ and floor $f^r_g$ against the robot task share $D_p$ in the same form, normalised to 2025, and are renormalised to the model's physical employment $L_p$.
Robots perform a share $D_p$ of 2025 US physical task-hours (70 million physical jobs in BLS major groups 31-53 at 1,800 hours a year, 126 billion task-hours). $D_p$ is the sum over four classes $k \in \{\text{industrial}, \text{mobile}, \text{vehicles}, \text{general}\}$, each with its own fleet, cost, reliability, ceiling and lag (robotics dossier). The 2025 share of each class is its US fleet converted to delivered task-hours:
with $N_k$ units in operation, $H_k$ operating hours a year, $u_k$ uptime, $e_k$ human task-hours replaced per robot-hour and $\rho_k$ reliability (the share of a task-hour delivered unattended). Industrial arms (0.42M units) and mobile logistics robots (0.9M) account for almost all of it; autonomous vehicles are a few thousand units and general manipulation delivers nothing yet.
The substitutable physical share $\tau_p$ has a pre-AI automation base and rises with the same 80% horizon as cognitive work, with a much later midpoint and a higher floor:
Each mature class has a fixed addressable ceiling $A_k$ (industrial 0.10, mobile 0.12, vehicles 0.10 of physical task-hours); the general class takes whatever the capability ceiling leaves, $A_{\mathrm{gen}} = \tau_p - \sum_{k\ \mathrm{mature}} D_k$, so the AI-driven class is the one that moves with $h_{80}$. Reliability is a state, logistic toward a ceiling, and the effective cost per delivered task-hour divides the all-in cost per robot-hour by it:
with, for general manipulation, $c_{2025} = 1.0$ (RaaS at about the loaded wage per robot-hour), $\rho_{2025} = 0.30 \to 0.85$ at $r = 0.35$, $\delta = 0.15$, so $\hat c_{\mathrm{gen},2025} = 3.3$; industrial arms start at $\hat c = 0.29$ and mobile robots at $0.50$, already below the wage. Diffusion is logistic within each class, gated by its cost advantage over the physical wage, slowed by friction, scaled by a common factor $s_r$ (1 in the baseline, 0 in robots_frozen) and capped by fleet buildout in relative and absolute terms (the absolute cap $g^{abs}_k$ lets a class start from a few thousand units):
with lags of 15, 10, 12 and 8 years, relative caps 0.15, 0.30, 0.60 and 0.35, absolute caps 0.3, 0.6, 0.4 and 1.0 percentage points of physical task-hours a year. $c_r$ is the fleet-average effective cost that enters the physical unit cost below.
Physical employment is demand-determined, not a residual. The physical unit cost mixes humans and robots, labor demand follows physical volume $q_p$ and the wage with hiring inertia, and the people available for physical work beyond that demand join the pool, while demand above the people available is filled from the pool at the job-finding rate:
Robot capital: the fleet is a stock $K_r = D_p\, q_p / N_{p,0}$ with unit capital value $P_K = \kappa_K\, c_r\, W_{p,0} / (r + \delta_r)$ ($\kappa_K = 0.7$, $\delta_r = 1/8$, $W_{p,0}$ the 2025 physical wage bill); purchases $I_{r,t+1} = P_K \big(K_{r,t} - (1-\delta_r) K_{r,t-1}\big)^+$ are capped at $\phi_r$ (0.5) of the private saving left after AI and power capex, with the year's class additions scaled back pro rata when the cap binds; book $B_{r,t} = \min\big((1-\delta_r) B_{r,t-1} + I_{r,t},\ K_{r,t} P_{K,t}\big)$ with the difference an impairment; the fleet's electricity $E_r = e_r E_{0}\, K_r / K_{r,0}$ ($e_r = 0.03$) enters the power pipeline's demand and the compute build's power gate; and $I_r$ is physical-goods investment demand in $Y = C_h + G + I_{\mathrm{other}} + I_{\mathrm{AI}} + I_r$.
Physical labor income is the physical labor share of value added times $\eta_p / \eta_{p,0}$; the remainder accrues to robot owners. Physical productivity growth raises capacity rather than cutting labor demand at given volume. With $D_p = 0$ (the no_robotics preset) $L_p = a_p$ and the residual treatment of physical employment is recovered.
Five household quintiles $q$ receive the model's three income flows with 2025 shares from the record: cognitive wage shares $c_q = (0.02, 0.05, 0.12, 0.26, 0.55)$, physical wage shares $p_q = (0.08, 0.16, 0.22, 0.26, 0.28)$, capital ownership $k_q = (0.01, 0.02, 0.04, 0.10, 0.83)$. With $W_c$ the human cognitive wage bill, $W_p$ the physical wage bill, $Tr$ transfers and $K = Y_{hh} - W - Tr$ capital payouts:
with $u$ the universal share of transfers (transfer_universal_share, 0) and $\beta$ the broadened share of payouts (ownership_broadening, 0; 0.5 in citizen_dividend). Real income per quintile uses the paper's lenses, $Y_q / P_p$ and $Y_q / P_p \times M_{sat}/M_p$, each indexed to its own 2025 value. The necessities basket costs $n\,Y_{3,2025}$ in 2025 ($n = 0.55$) and is repriced at $P_{nec} = (1-e)\,P_p + e\,P_{elec}(1+\text{uplift})$, $e = 0.08$; with lognormal dispersion $\sigma = 0.35$ inside a quintile the share able to afford it is
and concentration is reported as the top-quintile shares of disposable income and of payouts and the between-quintile Gini.