Human-AI Substitution Model (HAT)

Sources#

• A New Look at AI's Impact on Jobs: Firm-Level AI Spending and Workforce Adjustment
• AI-Augmented Human Resource Management? Insights from German companies
• Return of the solo author: The changing division of labor in science in the age of generative AI
• State of AI 2026: The Builder's Economy
• The Human-AI Substitution Principle: When will you be replaced by AI in your organization?
• The Shift to Agentic AI: Evidence from Codex
• Voice AI in Firms: A Natural Field Experiment on Automated Job Interviews
• When Better Opportunities Backfire: Service Worker Responses to Parallel AI Deployment in Customer Acquisition

Summary#

Banerjee & Singh's Human-AI Task Allocation (HAT) model (arXiv 2607.20781) opens with the organizational-complements argument stated as its motivating premise: "Organizations do not replace employees simply because AI becomes technically capable of performing a task. Replacement is an organizational decision that depends simultaneously on expected costs, risks, organizational structure, managerial coordination, and the relative economic characteristics of human and AI labor." It then builds a hierarchical optimization around that premise. What it contributes is vocabulary and a parameter list — not evidence.

Its named result, the Human-AI Substitution Principle, says to replace the human iff the AI's risk-adjusted cost is lower, C'_ik + λR'_ik < C_ij + λR_ij. Everything else (abrupt transitions, organizational flattening, middle-management vulnerability, hybrid structures, irreversibility, an upskilling contest) is derived from that comparison plus one cost-asymmetry assumption. The paper's Table 3 turns those results into seven falsifiable predictions, and this page checks each against what the vault measures. Only flattening (P3) has confirming evidence. Middle-management vulnerability (P2), the most-quoted result, rests on a hand-set feasibility curve. The model also folds the entire organizational-complements list into one coordination parameter and assumes it small, and its linear objective cannot represent three things the vault's causal evidence found: a task split by stage, a variance advantage, and feedback from reassignment onto the human's remaining work.

Evidence note. practitioner-opinion — a formal model with no empirical data, the same tier and the same shape as The Tragedy of the Cognitive Commons. Every result is a theorem conditional on stated assumptions; the paper is disciplined about saying so ("the results should be interpreted as conditional statements… The purpose of the HAT framework is not to claim universal inevitability of AI substitution"). Where the vault holds a measurement bearing on one of its predictions, the measurement wins and is named below.

The machinery, and the assumption that carries it#

An organization has depth D managerial layers over line workers at level D+1, with level sizes derived from spans of control (e_i = ∏ s_ℓ, after Garicano 2000). A human line worker costs Min$ + ½·Δ$·D·μ (compensation rising in skill μ); a manager costs C_{D+1}·(1+r_0 D)^{D-i+1}, so human cost escalates super-exponentially toward the top. An AI agent costs T_k/n_k + M'_k + ½·Δ$'·D·μ' — a fixed training cost amortized over n_k deployments, plus a marginal operating cost, plus a capability term. Both sides are risk-adjusted at the level of the individual agent: C̃ = C + λR, where λ is the firm's risk sensitivity and AI risk decomposes as R' = ω₁R^(rel) + ω₂R^(comp) + ω₃R^(rep) — reliability, compliance, reputation.

Assumption 1 ("Human-AI Cost Asymmetry") carries the whole paper: (i) Δ$' ≤ Δ$, AI capability costs grow no faster than human skill costs; (ii) T_k/n_k → 0 as deployments scale, so the AI baseline approaches a capability-independent floor M'_k; (iii) r'_0 ≤ r_0, AI coordination cost escalates no faster with depth than human coordination cost. The named result, the Human-AI Substitution Principle (Theorem 5), is then one line: replace the human iff C'_ik + λR'_ik < C_ij + λR_ij. Everything else — abrupt transitions (Thms 7, 14), flattening (Thm 4, Cor. 1), middle-management vulnerability (Cor. 2), hybrid organizations (Thms 10–11), irreversibility (Cor. 5), an upskilling-vs-AI-investment Nash equilibrium (Thm 15) — is derived from that comparison plus Assumption 1.

Where the organizational complements went#

The complements are inside r'_0, and the model assumes them away. Table 1's own observable gloss for r'_0 is: "Integration effort, workflow orchestration, monitoring, human oversight requirements." That is the complements list, verbatim, compressed into one scalar — and Assumption 1(iii) asserts it is no larger than the human coordination multiplier. The paper concedes the assumption is unfalsifiable as stated: r'_0 "is not directly observable and should be calibrated as a scenario parameter," bracketed between 0 (fully scalable AI coordination) and r_0 (human-equivalent). The vault has the measurement it declines to bracket above r_0. In Jabarian & Henkel's experiment, automating the information-collection stage made the remaining human stage 2.8× slower (interview→offer 2.62 → 7.24 median days) and end-to-end time-to-hire longer (20 → 24 days, p=0.033); oversight capacity does not expand when output does. A single measured case is not a refutation, but it is one instance of r'_0 > r_0 on the margin the model needs it small.

The seven predictions, against what the vault measures#

Table 3 is the paper's most useful part — falsifiable predictions with named empirical signatures. Four of the seven can be checked against evidence already here.

One caveat that travels with P2, the paper's most-quoted result. Theorem 8 proves only that cost-based substitution pressure rises monotonically toward the top — on cost alone the CEO is the most attractive replacement, and the calibrated example shows exactly that ($60.6k of gap at the line-worker level rising to $173.7k at the CEO). The intermediate peak requires a second curve, substitution feasibility F(i), that falls near the top. F(i) is neither derived nor measured: in the calibration it is five numbers set by hand (F = 0.50 / 0.80 / 1.00 / 0.55 / 0.25 from line worker to CEO), chosen so that the product peaks at level 3. Corollary 2 is honest about this — it says middle managers are most exposed if a single-crossing condition holds — but the headline travels without the conditional.

Two structural results the vault's evidence contradicts#

Extreme-point optimality. Theorem 6 puts a linear objective on a simplex, so the optimum sits at a vertex: the whole task goes to the single lowest-risk-adjusted-cost agent, and mixtures arise only from exact cost ties or externally imposed constraints (Thm 11). The vault's only randomized substitution result is the opposite shape — the hiring experiment split one task by stage, automating information collection and leaving evaluation entirely human, and that split is what produced +12% offers and +18% retention. HAT can represent it only as a constraint imposed from outside the optimization, never as the optimum. The paper's limitation 4 concedes the point ("best interpreted as benchmark structural tendencies rather than literal descriptions of operational firms") and limitation 5 concedes the single-task abstraction that rules out stage decomposition.

And the model cannot express a variance advantage at all. This is worth stating plainly, because the vault's best causal evidence says the winning margin was variance. Three separate reasons the cost conditions cannot encode it:

1. Output quality is assumed identical. §5.1 derives the objective from V = B − C − λR and drops B because "B is fixed for a given task." Both agents produce the same benefit by construction. Controlled variance is a claim about the dispersion of output quality across repetitions — the model has neither repetitions nor variable quality.
2. R is a level, not a moment. λ is called risk sensitivity and C + λR has the shape of a mean-variance functional, but Table 1 defines R as error/turnover/absenteeism for humans and reliability/compliance/reputational failure for AI — an expected-loss level. No second moment appears anywhere in the paper.
3. Even a generous encoding collapses the distinction. One could set R' < R to say the AI is more reliable — but that is a mean shift in effective cost, indistinguishable from the AI simply being cheaper. The model cannot separate "AI is better on average" from "AI is more consistent," which is precisely the distinction the experiment identifies: the AI beat only 61%/64% of human recruiters on topic coverage and vocabulary richness while beating 100% on question-guideline similarity and 83% on order adherence. Not more capable — less dispersed.

The direction is not incidental. Every worked example in the paper runs the other way (R'_5k = $20k against R_5j = $5k), and all seven predictions are built on λ(R' − R) > 0 — AI as the riskier party whose risk penalty is what delays substitution. A substitution model in which AI's reliability is the barrier has no way to describe the case where AI's reliability is the product.

A third gap, measured from the human side (Fu et al., October 2026). HAT prices every agent's per-task cost and benefit as fixed while tasks are reassigned between them. A regression discontinuity at a Chinese insurance brokerage measures what happens when that assumption is broken. Routing cold leads to an AI voice agent improved the leads left for humans, and human conversion on the most-improved calls fell (−4.5pp at the first three calls of the shift, relative to the same-day average), which fits workers' reservation value rising with the quality of the next expected draw. Allocating the cheap tasks to AI changed the human's productivity on the tasks that remained, a feedback a model with exogenous C and fixed B cannot contain. One firm and one cutoff, with a firm-affiliated co-author; details and caveats are on that page.

The calibration's own headline, and what the vault says happened. §5.7 calibrates a five-level, 341-person professional-services firm (Min$ = $70k, Δ$ = $12k, r_0 = 0.10; Min$' ≈ $9.95k, Δ$' = $0.5k, λ = 1.5, R' = $20k vs R = $5k) and computes the skill threshold at μ* ≈ −1.52. Since skill is non-negative, every line worker sits in the vulnerability zone — the model's verdict is that a mid-sized advisory firm should already have replaced all 256 analysts, driven almost entirely by the $70k-vs-$10k baseline gap. Raising compliance risk to a healthcare-like setting moves it only to μ* ≈ −0.46, still negative. Against that: at intensively AI-adopting US firms, entry-level headcount grew 12.0% over 24 months and entry-level share rose 1.15pp. The model is normative and the measurement is descriptive, so this is not a formal contradiction — but it is the most checkable claim the paper makes, and the vault's best firm-level evidence points the other way.

What the model is good for#

What the model is genuinely good for, stripped of its predictions: it names the levers cleanly. Substitution decisions decompose into nominal cost, risk sensitivity λ, and risk-profile difference R' − R; the three AI risk components map one-to-one onto technical, social, and legal-regulatory accountability (§5.2, and see Human-AI Accountability Redesign); and governance investments — auditability, regulatory engagement, transparency — become parameters of the substitution boundary rather than obstacles to it. That decomposition is portable whether or not any theorem built on it survives contact with data.

Open Questions#

• HAT's P2 (middle-management vulnerability) is the vault's most-quoted-but-least-tested substitution claim, and nothing here can settle it: Ramp × Revelio resolves seniority only to entry-level / non-entry / manager-plus, which cannot distinguish "middle layers thinned first" from "manager-plus grew more slowly than the bottom." Falsifiable with a level-resolved employment panel (Revelio or matched employer-employee data cut by reporting depth, not seniority band) tracking layer counts before and after intensive AI adoption. The same instrument would settle P4, since it could compare regulated against unregulated industries on the same measure — though P4 now has its first field evidence from German HR (see the ledger row), which confirms the direction on self-report and leaves the panel test outstanding.
• Corollary 5 claims irreversibility: once an AI agent is strictly cheaper on risk-adjusted grounds, the optimal allocation never reverts, given fixed human costs and negligible switching costs. The vault has no case either way, and this is falsifiable only by a future event — a documented instance of a firm re-staffing with humans a role it had already automated, for reasons other than a regulatory shock or a rise in AI risk (both of which the corollary's own conditions exempt). Watch for it in the same firm-level adoption panels; a reversal with human costs and regulation unchanged would falsify the corollary, and a long clean run of non-reversal would be weak support for it.

Connections#

• Organizational Complements to AI — the thesis this model formalizes: its motivating premise is that page's argument, and its coordination parameter r'_0 is that page's complements list compressed into one scalar and assumed no larger than the human equivalent
• The Tragedy of the Cognitive Commons — the vault's other no-new-data framework paper, its sibling in method, and its collision partner on P7: upskilling as a private effort choice against a commons the individual does not control
• The Solo-Authorship Rebound — P1's first labor-composition outcome that breaks sharply at a capability event rather than ramping, in the ledger row above; also the substitution decision observed one level down from the firm, inside a single knowledge-production team, where the replaced party is a coauthor and the "organization" doing the deciding is one researcher
• Role Averaging, Not Role Elimination — the practitioner counterweight to the extreme-point result: the model's optimum assigns a whole task to one agent, while the observed pattern is roles recomposing around a retained judgment half
• Firm AI-Spend Intensity and Headcount Growth — the vault's only firm-level measurement of the outcome the predictions describe; counter-evidence on P1, P2 and P6, and the entry-level growth that the calibration's μ* ≈ −1.52 says should not have happened
• AI-Native Organization — the 1–4-management-layers and widening-spans figures that are the only confirming evidence for P3
• Controlled Variance: AI's Edge as Reduced Dispersion — the randomized case the model cannot express: a stage-split task as the optimum, and a variance (not mean) advantage, plus one measured instance of r'_0 > r_0
• Human-AI Accountability Redesign — where the three AI-risk components (reliability, compliance, reputation) map one-to-one onto technical, social and legal-regulatory accountability
• Exposure Taxonomy: Observed, Theoretical, Reported, Anticipated — the theory-side counterpart to that page's instrument disagreement: the skill threshold μ* is as sign-indeterminate as the exposure instruments are, though P5 states it as determinate
• Returns to Expertise in Agentic Coding — expertise amplifying the agent, a complementarity the linear objective cannot represent (the paper's limitation 3)
• Experimental Learning Impact of Generative AI — the mechanism-level check on P7: whether upskilling under substitution pressure works depends on the augmentation/automation split, a distinction the private-effort formulation has no term for
• Parallel AI Routing and the Reservation-Value Backfire — the measured break of the fixed-per-task-cost assumption: routing cheap tasks to AI changed the human's productivity on the tasks that remained

Sources#

• The Human-AI Substitution Principle: When will you be replaced by AI in your organization? — Bonny Banerjee (Independent Scholar) & Shreya Singh (Chennai Mathematical Institute), The Human-AI Substitution Principle: When will you be replaced by AI in your organization?, arXiv 2607.20781 (2026-07-22, 68pp). practitioner-opinion — a formal model with no empirical data, tiered alongside The Tragedy of the Cognitive Commons: How AI Could Disrupt the Regeneration of Professional Expertise as the vault's other no-new-data framework paper; every result is a theorem conditional on Assumption 1, and the paper says so itself. §3 the HAT model (span-of-control level sizes eq. 1; human cost eqs. 2–4; AI cost eqs. 5–9; the three-component AI risk decomposition eq. 8); §3.4 Assumption 1 (i) Δ$' ≤ Δ$, (ii) T_k/n_k → 0, (iii) r'_0 ≤ r_0, with its own "maintained modelling premise" hedge; §3.7 Table 1 (parameter → observable-construct map, incl. the r'_0 gloss); §4.2.2 Theorem 5 (the Substitution Principle); §4.2.3 Theorem 6 (extreme-point optimality); §4.2.4 Theorem 7 + Corollary 1; §4.3 Theorem 8 + Corollary 2 (middle-management vulnerability, conditional on single-crossing); §4.3.2 Corollary 3 (the μ* protection/vulnerability threshold); §4.4 Theorems 10–12 (risk-adjusted substitution, hybrid optima, full-automation condition); §4.5 Theorems 13–15 + Corollaries 5–6; §5.1 the V = B − C − λR value identity that fixes B; §5.2 accountability mapped onto the three risk components; §5.4 Table 2 (organizational mechanisms); §5.6 Table 3 (predictions P1–P7); §5.7 the calibrated five-level firm and μ* ≈ −1.52; §5.8 the ten stated limitations. Parse notes: Table 1's λ row was truncated mid-cell at ingest (full value: "Firm's tolerance for operational, regulatory, or strategic risk.") and a stray <formula><loc_…> wrapper duplicating an equation block was cleaned during ingest with the math preserved; Tables 2 and 3 were verified verbatim and are the only tables quoted here. The F(i) feasibility values behind the middle-management illustration are from §5.7 prose, and are the paper's own hand-set assumption rather than a derived or measured quantity
• The Shift to Agentic AI: Evidence from Codex — the OpenAI internal function-level adoption series (legal and recruiting ~0 → ~75% Codex output-token share, the steepest stretch 20% → 75% within a month), cited for the P1 adoption-margin row only; full treatment at Organizational Complements to AI
• A New Look at AI's Impact on Jobs: Firm-Level AI Spending and Workforce Adjustment — Kharazian, Simon & Stevens (Ramp × Revelio, June 2026): the high-intensity event study (0.003 → 0.452 over 24 months), seniority-band headcount changes (manager-plus +6.5%, entry-level +12.0%) and the $2.78 vs $33.67/month intensity gate, cited for P1, P2, P6 and the calibration check; full treatment at Firm AI-Spend Intensity and Headcount Growth
• State of AI 2026: The Builder's Economy — ICONIQ Growth, State of AI 2026: The Builder's Economy (2026-07-08, empirical survey): management-layer and span-of-control figures cited for P3; full treatment at AI-Native Organization
• Voice AI in Firms: A Natural Field Experiment on Automated Job Interviews — Jabarian & Henkel, Voice AI in Firms (arXiv 2607.28222, empirical, pre-registered RCT): the 2.62 → 7.24-day evaluation stage and 20 → 24-day time-to-hire (the r'_0 > r_0 instance), the stage split behind +12% offers and +18% retention, and the 61%/64% vs 100%/83% dispersion figures; full treatment at Controlled Variance: AI's Edge as Reduced Dispersion
• AI-Augmented Human Resource Management? Insights from German companies — Kalff & Simbeck, arXiv 2607.13839 (empirical, self-reported, N=410 German HR managers plus interviews and group discussions): cited for the P2, P3, P4, P6 and P7 ledger rows (the clerical layoff COM1, the micro-management reading COD1/COM4, the compliance-avoiding adoption pattern and tool-use shares, the SME returns gap and the 183-of-410 informal use, the reskilling intent); full treatment at Organizational Complements to AI
• Return of the solo author: The changing division of labor in science in the age of generative AI — Akira Matsui, arXiv 2607.10780 (2026-07-12; empirical, 300M+ OpenAlex works, 26 fields): §Results (the 23-of-26 positive breaks and the field ordering), SI §L (the balanced-venue attenuation +1.72 → +0.75 pp/yr), SI §M (the event study, −2.9 pp in 2022 then +0.1 in 2023). Cited here only for the P1 ledger row; identification caveats and full treatment at The Solo-Authorship Rebound
• When Better Opportunities Backfire: Service Worker Responses to Parallel AI Deployment in Customer Acquisition — Fu, Luo, Ramasubbu, Shen, Wang & Wang, arXiv 2610.25603 (2026-10-19, empirical, RDiT on one Chinese insurance brokerage; firm-affiliated co-author). Cited for the fixed-per-task-cost gap only (−4.5pp human conversion at the first three calls of the shift); full treatment on Parallel AI Routing and the Reservation-Value Backfire