Joint Optimization of Human Headcount and Stochastic AI Resource Capacity

Joint Optimization of Human Headcount and Stochastic AI Resource Capacity Marco Montes de Oca EnFi, Inc. marco.montesdeoca@enfi.ai

arXiv:2608.00886v1 [cs.SE] 1 Aug 2026

Abstract Generative AI tools are rapidly becoming standard for coding and other business operations. However, while these tools can drive productivity, their unpredictable, and often significant, costs are putting severe financial strain on organizations. The leadership of such organizations must split a fixed budget between human headcount and generative-AI token capacity, yet standard deterministic planning ignores both the heavy-tailed volatility of token consumption and the cognitive cost of auditing AI-generated output. We model the joint allocation by intersecting a cognitive-friction-adjusted production function with a chance-constrained stochastic budget frontier. Per-engineer token usage is treated as i.i.d., non-negative, and right-skewed; no parametric family is assumed, as only its mean, variance, and skewness enter the analysis. The chance constraint is reduced to a deterministic equivalent via the Central Limit Theorem with a Cornish–Fisher skewness correction. Solving the resulting Lagrangian yields a closed-form optimal headcount, a transcendental condition for optimal per-capita token intensity, and a bordered-Hessian second-order condition that holds for any reasonable volatility-to-headcount ratio. Comparative statics show that rising individual token volatility raises optimal per-capita token intensity while contracting headcount: although human labor and tokens are complements in production, the stochastic budget makes them substitutes at the margin. Individual usage skewness acts as a pure deadweight tax that shrinks the budget without altering the substitution dynamics. Both conclusions presuppose that usage dispersion is invariant to the planned mean allocation. In the case where dispersion instead scales proportionally with the mean, the substitution reverses, so which regime applies is a sharp, empirically testable question.

Keywords: collective intelligence, human–AI collaboration, resource allocation, chanceconstrained optimization, production functions, stochastic programming, generative AI, software engineering teams

1

Frame of Reference and Core Assumptions

When scaling modern software engineering organizations1 , leadership must allocate a fixed monthly budget (B) across two primary variable inputs: human labor (headcount, N ) and generative AI computational resources (average monthly tokens per capita, T ). In collectiveintelligence terms, this is a question of optimal composition: how many human minds should 1 We focus on software creation, but the same dynamics apply to other business functions that may benefit from generative AI tool use.

1

1 · THE PROBLEM

2 · THE MODEL

A fixed monthly budget B is split between human headcount N and AI token capacity T.

An interior optimum sits strictly below token satiation β/λ. T

Two frictions break naïve planning: AI output needs human review (cognitive saturation) Token spend is heavy-tailed (a risk buffer is required)

3 · FINDINGS

(N ∗ , T ∗ )

N

Cognitive-friction production

∩ chance-constrained budget ⇒ optimal mix (N ∗ , T ∗ ).

Volatility σ↑ ⇒ N ∗ ↓ , T ∗ ↑ : heads and tokens substitute. Usage skewness γ1 is pure deadweight budget burned.

Figure 1: A fixed monthly budget is split between human headcount N and AI token capacity T under two frictions: AI output must be human-reviewed (cognitive saturation) and perengineer token spend is heavy-tailed (a risk buffer is required). Intersecting a cognitive-friction production function with a chance-constrained budget yields an interior optimum (N ∗ , T ∗ ). The two headline comparative statics follow: rising usage volatility substitutes headcount for percapita token intensity, and usage skewness is a pure budget deadweight. the collective contain, and how much machine throughput should flow through each [23, 7]. Traditional deterministic financial planning treats both inputs as predictable line items, but this assumption breaks down due to two structural characteristics: • Stochastic Token Volatility: Individual token consumption is highly volatile and prone to extreme right-hand tail events (e.g., recursive agent loops, automated mass-refactoring scripts, or deep repository re-indexing). • Cognitive Saturation: Generative AI is not a frictionless substitute for human labor. AI engines generate raw code drafts, context fragments, and configurations that demand human cognitive cycles to audit, debug, and validate. Large token volume shifts a human’s role from an active builder to a bottlenecked reviewer. This is the classic “irony of automation” [6], in which automating most of a task leaves the human holding its hardest residual. Our production function below captures this phenomenon quantitatively. We model the problem of deriving the optimal balance between human labor and token consumption as a stochastic program where the goal is to maximize a Cognitive-Friction Adjusted Production Function subject to a Chance-Constrained Stochastic Budget Constraint. Figure 1 summarizes this setup and the three findings the analysis delivers.

1.1

Core Modeling Assumptions

1. The i.i.d. Resource Pool: Individual engineer token consumption profiles (Ti ) are independent and identically distributed (i.i.d.) non-negative random variables with finite mean µ, variance σ 2 , and strictly positive standardized skewness γ1 , encoding high-token usage events through a long right-hand tail. No parametric family is assumed: the derivations below consume only these first three moments, so any standard non-negative right-skewed

2

distribution, such as the log-normal or the gamma, serves equally well as the motivating shape. 2. Asymptotic Convergence Qualification: The deterministic equivalent mapped via the Central Limit Theorem paired with the Cornish–Fisher expansion is an asymptotic approximation. For small teams (small N ), where individual skewness is most pronounced and the risk buffer is largest, the convergence rate is slower, meaning the boundary acts as an approximation that tightens as N scales. 3. Exogenous Volatility: The dispersion parameters σ and γ1 are exogenous: invariant to the chosen operational mean T . Note that this assumes nothing about volatility being controlled or limited, only that it does not vary with the planner’s allocation, as would be plausible if dispersion is driven by infrastructure-level events (runaway agent loops, mass refactors) rather than by planned intensity. This is a substantive scope condition rather than a mere analytical convenience: under a fixed-shape family (constant coefficient of variation), σ would instead scale with the mean and the marginal substitution results would reverse in sign, as Section 6.2 makes explicit. 4. Strict Binding Constraint: Because the marginal productivity of human headcount is strictly positive across the entire operational domain, the organization will always fully exhaust its budget space at equilibrium to maximize output.

2

The Production Field and Cognitive Friction

Let total organizational software output Q be mapped as a smooth scalar field over human labor and total organizational token usage T , where Q : R2++ → R+ . To capture true productionside coupling where individual cognitive overhead is governed by the per-capita density of the computational resource mix, we model the system using a Cobb–Douglas core [12] augmented with an exponential friction term: T

Q(N, T ) = AN α T β e−λ( N )

(1)

Where: • A > 0: The baseline structural efficiency of the engineering environment (codebase architecture cleanliness, documentation standards, automated testing maturity). • 0 < α, β < 1: The output elasticities of human labor and token leverage, respectively, establishing standard diminishing marginal returns for both inputs. • λ > 0: The Cognitive Overload Coefficient (dimensioned in tokens−1 ), representing the exact rate at which human oversight capacity is saturated by processing AI-generated context. To evaluate this from an operational management perspective, we transition to the per-capita token convention by substituting T = N · T , where T is the average monthly token consumption per engineer: NT

Q(N, T ) = AN α (N T )β e−λ( N ) = AN α+β T β e−λT 3

(2)

2.1

Step-by-Step Derivation of Marginal Productivities

We take the partial derivatives of Q(N, T ) with respect to our decision coordinates. Marginal Productivity of Tokens:   ∂ ∂Q ∂ −λT α+β β −λT β = AN [T ] · e +T · [e ] ∂T ∂T ∂T   ∂Q = AN α+β βT β−1 e−λT + T β (−λe−λT ) ∂T   β ∂Q α+β β −λT = AN T e −λ ∂T T

(3)

(4) (5)

Substituting the original production function (2) back into the equation isolates the final gradient:   β ∂Q = Q(N, T ) · −λ (6) ∂T T Marginal Productivity of Labor: Differentiating directly with respect to the headcount term N , treating the per-capita token allocation T as a fixed operational intensity: ∂Q = (α + β)AN α+β−1 T β e−λT (7) ∂N Multiplying the numerator and denominator by N reveals its proportional relationship to total output:   ∂Q (α + β)AN α+β T β e−λT α+β = = Q(N, T ) · (8) ∂N N N

2.2

Production Complementarity and Satiation

While the production function becomes multiplicatively separable in N and T after substitution, the inputs remain tightly coupled via their cross-partial derivative:      ∂2Q ∂ β Q(N, T ) · (α + β) β = Q(N, T ) · −λ = −λ (9) ∂N ∂T ∂N T N T Below the physical satiation threshold (T < β/λ), the cross-partial derivative is strictly positive ∂2Q ( ∂N ∂T > 0). This establishes N and T as structural Edgeworth complements in production [22, 17]. This means that hiring more human engineers directly expands the marginal productivity of tokens, as a larger human pool provides more cognitive capacity to safely ingest and ship AI output. Setting the token gradient ∂Q ∂T = 0 isolates the unconstrained token satiation point: β (10) λ This limit marks the structural peak where further token consumption yields a net decrease in organizational output due to cognitive overload. Because tokens carry a real financial cost Tsat =

4

160 120

optimum output Q ∗ at (N ∗ , T ∗ )

80

token-starved

60

budget pushes past satiation

40 20 0

lean team (N = 24) large team (N = 50)

0

10

20 30 40 50 60 Per-capita token intensity T (MTok/month)

friction-dominated

100

satiation T = β/λ

Organizational output Q

140

70

Figure 2: Output versus token intensity at fixed headcount, for a lean (N = 24) and a large (N = 50) team. Each curve is the production function Q(N, T ) = AN α+β T β e−λT ; the exponential cognitive-friction factor makes output rise, peak at the satiation intensity T = β/λ (dotted line), and decline in the friction-dominated region (shaded). Dots mark each team’s budget-feasible operating point at the same budget: the lean team can afford so many tokens per engineer that it is driven past satiation, while the large team is token-starved far to the left. Neither reaches the output Q∗ of the balanced optimum (N ∗ , T ∗ ) (dashed). Same illustrative parameters as Fig. 3. (Pt > 0), the budget-constrained economic optimum (T ∗ ) will sit strictly below this physical ceiling (10) (T ∗ < β/λ). Intuitive Example: Imagine an engineering organization where β = 0.4 and λ = 8 × 10−9 tokens−1 (equivalently 0.008 MTok−1 , the unit used in the figures). The unconstrained satiation point sits at Tsat = 0.4/(8 × 10−9 ) = 5 × 107 or roughly 50 million tokens/engineer/month, which is well within the range of present-day agentic usage. If tokens were entirely free (Pt = 0), leadership should cap individual allocation at that ceiling. Beyond it, engineers spend so much time reviewing automated AI output that total software shipping velocity slows down. Figure 2 makes this satiation concrete and previews the central substitution: at a fixed budget, a leaner team is driven past the satiation peak into the friction-dominated regime, while a larger team is left token-starved, so neither matches the balanced optimum of Section 5.

3

Derivation of the Stochastic Budget Frontier

The organization operates under a hard monthly budget B. Let S represent the average individual engineer salary and Pt represent the unit price per token. Because individual monthly token consumption values (Ti ) are random variables, total expenditure is also a random variable. To 5

protect the organization against cost overruns, leadership establishes an acceptable risk tolerance threshold ϵ (e.g., ϵ = 0.05, representing a strict 5% maximum allowable risk that token consumption spikes will cause a budget breach in any given month). This yields the following chance constraint [9]: ! N X P N · S + Pt Ti ≤ B ≥ 1 − ϵ (11) i=1

3.1

Step-by-Step Transformation to a Deterministic Equivalent

To make this probabilistic statement useful for algebraic optimization, we transform it into a deterministic equivalent using standard statistical properties. Step P 1: Isolate the Random Sum We isolate the summation of individual token consumpN tion, i=1 Ti , which contains all the stochastic volatility in our system. Subtract the fixed human payroll costs (N · S) and divide by the unit token price (Pt ) to isolate the aggregate random variable sum: ! N X B−N ·S P Ti ≤ ≥1−ϵ (12) Pt i=1

Step 2: Map Aggregate Moments via the Central Limit Theorem According to the Central Limit Theorem (CLT), the sum of independent, identically distributed random variables approaches a normal distribution as the sample size N scales. We calculate the aggregate expected mean and aggregate variance of our engineering pool: hP i P N • Aggregate Expected Mean: E T = N i i=1 E[Ti ] = N µ i=1 • Aggregate Variance: Var

P

N i=1 Ti

• Aggregate Standard Deviation:



√

=

PN

i=1 Var(Ti ) = N σ

Var =

√

2

Nσ

To maintain mathematical honesty regarding tail behavior in finite engineering populations, we apply a Cornish–Fisher expansion [13] to adjust for √ the non-zero population skewness (γ1 ). The aggregate skewness of the team scales as Γ1 = γ1 / N . This expansion adjusts the standard normal critical value into a skewness-adjusted quantile score (w1−ϵ ): w1−ϵ = z1−ϵ +

Γ1 2 γ1 2 (z1−ϵ − 1) = z1−ϵ + √ (z1−ϵ − 1) 6 6 N

(13)

Step 3: Z-Score Standardization We standardize our random sum into a standard normal distribution√by subtracting the aggregate mean (N µ) and dividing by the aggregate standard deviation ( N σ) on both sides of the inequality: ! ! P B−N ·S B−N ·S − Nµ − Nµ Ti − N µ Pt Pt √ √ √ P ≤ ≥ 1 − ϵ =⇒ Φ ≥1−ϵ (14) Nσ Nσ Nσ

6

Applying the inverse quantile function (Φ−1 ) to both sides isolates the scalar terms against our skewness-adjusted critical value: B−N ·S − Nµ Pt

√

Nσ

γ1 2 ≥ z1−ϵ + √ (z1−ϵ − 1) 6 N

(15)

Step 4: Clearing Denominators and Isolating Budget B Multiply both sides of the √ inequality by the aggregate standard deviation N σ:   √ γ1 B−N ·S 2 − N µ ≥ z1−ϵ + √ (z1−ϵ − 1) σ N (16) Pt 6 N √ √ Distributing σ N across the brackets causes the N terms in the skewness modifier to cancel out cleanly: 2 √ γ1 σ(z1−ϵ − 1) B−N ·S − N µ ≥ z1−ϵ σ N + Pt 6

(17)

Multiply the entire equation by the token price Pt , add N · S to both sides, and factor out headcount N from the baseline cost components. This yields the deterministic-equivalent budget frontier: 2 √ γ1 Pt σ(z1−ϵ − 1) B ≥ N (S + Pt µ) + z1−ϵ Pt σ N + 6

4

(18)

System Optimization and Equilibrium Derivation

To locate the coordinates that maximize total organizational software output within our safe financial boundary, we formulate the Lagrangian function (L) [11]. We define this over our production field and our binding deterministic budget constraint, treating our operational percapita token decision variable T as the runtime expected mean µ:     √ L(N, T, Λ) = AN α+β T β e−λT + Λ B − N (S + Pt T ) − z1−ϵ Pt σ N − Kγ (19) Where Λ is the Lagrange multiplier, and Kγ = deadweight term.

4.1

2 γ1 Pt σ(z1−ϵ −1) 6

collects the constant skewness

Step 1: Evaluating First-Order Conditions (FOC)

We take the partial derivatives of the Lagrangian with respect to our decision variables and set them identically to zero. ∂L Differentiating with respect to Token Intensity ( ∂T = 0):     β Q(N, T ) β − λ − ΛPt N = 0 =⇒ Λ = −λ Q(N, T ) · T Pt N T

7

(20)

∂L = 0): Differentiating with respect to Headcount ( ∂N     α+β z1−ϵ Pt σ Q(N, T ) · (α + β)  Q(N, T ) · − Λ S + Pt T + √ = 0 =⇒ Λ =  N 2 N √Pt σ N S + Pt T + z1−ϵ

(21)

2 N

4.2

Step 2: Equating the Marginal Conditions

Equating our two independent expressions for the marginal value of capital Λ and canceling out the total output scalar Q(N, T ) and the headcount parameter N from both sides yields the equilibrium condition: β T −λ

Pt

5

=

α+β √Pt σ S + Pt T + z1−ϵ 2 N

(22)

Closed-Form Solution Space

To resolve the precise operational headcount (N ∗ ) and per-capita token allocation (T ∗ ), the equilibrium relation must be solved simultaneously alongside our binding budget boundary as a non-linear 2 × 2 system. Headcount is treated as continuous throughout; at the organizational scales considered, rounding N ∗ to an integer perturbs the optimum negligibly. Figure 3 visualizes this system: the optimum is the point where the binding budget frontier is tangent to the highest production isoquant it can reach, and the stochastic risk buffer pulls that frontier strictly inside its deterministic counterpart.

5.1

Step 1: Solving for Optimal Headcount (N ∗ )

We isolate headcount by rearranging √ our binding budget boundary (18) into a standard quadratic form with respect to the variable N : √ (S + Pt T )N + (z1−ϵ Pt σ) N − (B − Kγ ) = 0 Applying the quadratic formula yields the exact structural mapping for headcount: p √ −z1−ϵ Pt σ + (z1−ϵ Pt σ)2 + 4(S + Pt T ∗ )(B − Kγ ) ∗ N = 2(S + Pt T ∗ )

5.2

(23)

(24)

Step 2: Solving for Optimal Token Intensity (T ∗ )

Cross-multiplying the equilibrium condition (22) from Section 4.2, canceling out matching βPt parameters, and grouping deterministic cost terms on the left side versus risk terms on the right side yields:   βS z1−ϵ Pt σ β − λS − λPt T − Pt α = √ λ− (25) T T 2 N Multiplying the entire equation by −T to clear all denominators leaves the transcendental condition: λPt T 2 + (λS + Pt α) T − βS = 8

z1−ϵ Pt σ √ (β − λT ) 2 N

(26)

Per-capita tokens T (MTok/month)

40

stochastic frontier (σ > 0) deterministic frontier (σ = 0) production isoquants

35 30

tokens/head surrendered to the risk buffer

25 20 15

(N ∗ , T ∗ )

10 5 0

35

40

45 Headcount N

50

55

Figure 3: The headcount–token tension. On the binding stochastic budget frontier (solid), every additional engineer lowers the affordable per-capita token allocation; the optimum (N ∗ , T ∗ ) is the tangency of this frontier with the highest attainable production isoquant. The risk buffer holds the stochastic frontier strictly inside the deterministic (σ = 0) frontier (dashed). Illustrative parameters (a premium agentic-coding regime, tokens in millions): α = 0.6, β = 0.4, λ = 0.008 MTok−1 (satiation β/λ = 50 MTok), S = $15k/month, Pt = $300/MTok, σ = 20 MTok, γ1 = 2, z0.95 = 1.645, B = $900k/month.

6

Optimization Rigor & Comparative Statics

6.1

Verification of the Bordered Hessian (Local Extremum)

To verify that our stationary coordinates (N ∗ , T ∗ ) represent a valid local maximum rather than a minimum or saddle point, we evaluate the determinant of the bordered Hessian matrix H. √ With the binding boundary (18) written as g(N, T ) = N (S + Pt T ) + z1−ϵ Pt σ N + Kγ − B = 0, the relevant second-order partials are:   ∂2Q (α + β) ∂Q β ∂2Q Q(α + β)(α + β − 1) ∂2Q 2 = Q δ − , = , = (27) 2 2 2 2 ∂T T ∂N N ∂N ∂T N ∂T

gN = S + Pt T +

z1−ϵ Pt σ √ , 2 N

gT = Pt N,

gN N = −

z1−ϵ Pt σ , 4N 3/2

gT T = 0,

gN T = Pt

(28)

Where we abbreviate the per-token marginal-output ratio δ ≡ Tβ − λ = Q1 ∂Q ∂T > 0 (strictly positive below satiation). The bordered Hessian is:

9



0 H = gN gT

gN QN N − ΛgN N QN T − ΛgN T

 gT QN T − ΛgN T  QT T

(29)

2 |H| = −gN QT T + 2gN gT (QN T − ΛPt ) − gT2 (QN N − ΛgN N )

(30)

The decisive simplification eliminates gN using the equilibrium condition (22) (Section 4.2), t gN = (α+β)P . Two intermediate identities follow directly from the first-order conditions. First, δ the cross term is governed by the firm’s returns to scale: (α + β) ∂Q 1 ∂Q (α + β − 1) ∂Q − = (31) N ∂T N ∂T N ∂T Second, the headcount-curvature term carries the budget buffer’s concavity. Substituting these and gT = Pt N , the output-proportional components of the determinant telescope cleanly (the m2 and m(m − 1) terms collapse, with m ≡ α + β). Utilizing T 2 δ 2 = (β − λT )2 , the determinant resolves to a single closed form: √   Pt3 Λz1−ϵ σ N (α + β)β 2 |H| = Pt Q(α + β) −1 − (32) (β − λT )2 4 | {z } | {z } QN T − ΛPt =

Production Curvature (>0)

Budget-Buffer Curvature (>0)

This decomposition is exact and assumes nothing about returns to scale. Two structural facts establish the sign: • The production-curvature term is unconditionally positive below satiation. The factor (α+β)β is minimized as T → 0, where it equals α+β β ; hence the bracket is bounded below (β−λT )2 2

by αβ > 0 and diverges as T → βλ . The strong log-concavity of output in tokens ( ∂ ∂Tln2Q = − Tβ2 < 0) is the engine driving this term.

√ • The budget-buffer term enters negatively. Because the risk cushion z1−ϵ Pt σ N is concave in headcount (gN N < 0), it renders the feasible region locally non-convex and acts against the interior maximum. The optimum is stable precisely when production curvature dominates this destabilizing force. Equating the two and using Λ/Q = δ/(Pt N ), the second-order condition |H| > 0 reduces to a clean, interpretable inequality:   β   − λ z1−ϵ σ T∗ (α + β)β √ (α + β) − 1 > (33) (β − λT ∗ )2 4 N∗ √ The left side is order-one and strictly positive; the right side scales as σ/ N ∗ . The candidate (N ∗ , T ∗ ) is therefore a verified local maximum for any reasonable volatility-to-headcount ratio, failing only in the pathological regime of extreme individual variance on a very small team (the same weakness noted earlier around our use of the CLT). For a global statement we reduce to one dimension along the binding boundary: substituting the closed-form N ∗ (T ) (24) collapses the system to a single continuous objective φ(T ) = Q(N ∗ (T ), T ) on the interval (0, β/λ). We do not prove φ globally unimodal; across the illustrative parameter ranges of Figs. 2–4 it numerically exhibits a single interior stationary point, which we accordingly treat as the global solution. 10

18.8 18.6 18.4 18.2 18.0 17.8 17.6

44 Optimal headcount N ∗

Optimal per-capita tokens T ∗ (MTok) Headcount lost to skewness deadweight

(a) Volatility substitutes heads for tokens

42 40 38 36 0

10 20 30 40 Token volatility σ (MTok)

50

(b) Kγ is pure deadweight

0.5 0.4 0.3 0.2 0.1 0.0

0

1 2 3 4 5 Token-usage skewness γ1

6

Figure 4: Volatility and skewness erode headcount. (a) As individual token volatility σ rises, optimal headcount N ∗ contracts (left axis) while optimal per-capita token intensity T ∗ expands (right axis). In this case, the stochastic budget turns two production complements into marginal substitutes, so squeezing σ is what funds a healthy headcount increase. (b) The skewness term Kγ is a pure deadweight tax: it grows linearly in usage skewness γ1 , shown here as the engineers’ worth of budget it consumes, independent of the allocation mix. Same illustrative parameters as Fig. 3.

6.2

∗

Comparative Statics: How Volatility Alters the Mix ( ∂T ) ∂σ

We analyze the behavior of our transcendental condition (26) under changes in stochastic variance (σ). Let T0 solve the deterministic baseline equation where f (T0 ) = 0 when σ = 0: • f (T ) describes an upward-opening parabola with a vertical intercept at f (0) = −βS < 0. Thus, at its positive root T0 , the function is strictly increasing: f ′ (T0 ) > 0. • g(T ) represents the stochastic modification layer. Because our economic optimum must sit below the physical satiation threshold (T ∗ < βλ ), the term (β − λT ) is strictly positive across the active domain. Hence, for any non-zero variance σ > 0, g(T ) > 0. At the original deterministic baseline, f (T0 ) = 0 < g(T0 ). Because f (T ) is monotonically increasing and g(T ) is monotonically decreasing on the interval T < βλ , the new intersection point T ∗ must lie strictly to the right of the baseline root: T ∗ > T0 =⇒

∂T ∗ >0 ∂σ

(34)

√ Two qualifications to keep in mind: First, g(T ) is evaluated holding N fixed, whereas N ∗ (T ) shifts along the binding frontier; the local claim, that any σ > 0 lifts g above f (T0 ) = 0 and pushes the root rightward, is unaffected, and the global monotonicity of T ∗ in σ is confirmed numerically across the range of Fig. 4(a). Second, the direction of the effect inherits the scope condition of Assumption 3, to which we return below.

11

The Substitution Mechanism This mathematical validation reveals the core resource allocation behavior under uncertainty: higher individual token volatility increases your optimal per-capita token intensity while contracting headcount. Figure 4(a) traces both optimal coordinates across the volatility axis, making the substitution explicit: the health√Pt σ iest headcount is bought not by spending more but by squashing σ. The active risk term z1−ϵ 2 N acts as a direct financial tax tied to human headcount. It dictates that hiring a human engineer requires holding back a defensive cash cushion to absorb potential token overruns within our risk tolerance ϵ. As variance (σ) increases, this safety buffer expands, making a human hire marginally more expensive. The optimization engine responds to this risk by substituting away from now-costlier human headcount (N ∗ ↓) and driving remaining resources into higher per-capita token intensity (T ∗ ↑). While these inputs are complements in production, they act as substitutes under a stochastic budget framework. Individual usage skewness (γ1 ), meanwhile, acts as a pure deadweight tax (Kγ ) that reduces absolute budget space uniformly without altering these marginal substitution dynamics. Figure 4(b) quantifies this loss directly: every unit of usage skewness burns a fixed amount of budget (budget that could otherwise fund engineers) regardless of how the remaining resources are split. Scope Condition: Where the Volatility Lives The direction of this substitution is conditional on Assumption 3: dispersion, whatever its magnitude, is invariant to the planned mean allocation. In the opposing fixed-shape regime (a constant coefficient of variation σ = cT , as implied by √a log-normal or gamma family with a frozen shape parameter) the risk buffer becomes z1−ϵ P cT N , so volatility taxes token intensity directly rather than taxing headcount through t √ the N buffer. Re-solving the system numerically in that regime reverses the substitution: rising c suppresses T ∗ while leaving N ∗ nearly unchanged. The model therefore delivers a sharp, testable dichotomy rather than one universal direction: whether volatility should push an organization toward higher or lower token intensity is decided by whether its spend dispersion scales with the planned allocation or is independent of it. This is the empirically testable question mentioned in the abstract. It could, for instance, be measured directly via usage telemetry.

7

Discussion and Related Work

Our contribution sits at the intersection of three literatures that, to our knowledge, have not previously been combined: the economics of production and input substitution, the empirical study of generative AI as a workplace input, and chance-constrained stochastic optimization. Treating AI token capacity as a stochastic, priced input jointly optimized with human headcount under a probabilistic budget is, we believe, novel; the paragraphs below position each component against its lineage and flag where our assumptions invite scrutiny. Production, substitution, and complementarity. Our output model is a Cobb–Douglas core [12] augmented with an exponential satiation term. Cobb–Douglas is the unit-elasticity special case of the constant-elasticity-of-substitution (CES) family introduced by Arrow, Chenery, Minhas, and Solow [4], which holds the elasticity of substitution between inputs constant; our friction term deliberately breaks that constancy near satiation. The Edgeworth complementarity we derive between headcount and tokens (Section 2.2) is the AI-era analogue of capital–skill complementarity, for which Ohanian, Orak, and Shen [19], extending the Krusell–Ohanian– Rı́os-Rull–Violante tradition, find continued strong empirical support: just as equipment capital 12

complements skilled labor, AI token capacity complements the human reviewers who ship its output. AI as a productive input, and the review bottleneck. The task-based view of automation [5, 1] holds that machines substitute for labor on codifiable tasks while humans retain comparative advantage on the rest; in our setting the residual human task is reviewing and validating AI output, and Acemoglu and Restrepo’s emphasis on labor’s shifting frontier contrasts instructively with our treatment of oversight saturation as the binding force. The cognitive-friction term e−λT is empirically motivated: randomized and field studies report large but bounded and heterogeneous gains from generative AI. For instance, roughly +15% support-agent throughput overall and +34% for novices [8], a 55.8% speed-up on a programming task [20], and a 40% time reduction with higher quality on writing tasks [18]. Dell’Acqua et al. [14] document a “jagged frontier” on which consultants using AI off its capability boundary were 19 percentage points less likely to be correct. That degradation-without-oversight is precisely the mechanism our friction term encodes. We note one genuine tension: Noy and Zhang [18] find AI substitutes for worker effort and reorients work toward editing rather than complementing skill. Our model is consistent with this once the two regimes are separated. Headcount and tokens are complements in production (more reviewers raise the volume of AI output that can be safely shipped), yet the stochastic budget makes them substitutes at the margin (Section 6.2), the very behavior those experiments observe at the task level. Stochastic budgeting under uncertainty. The probabilistic budget is a chance constraint in the sense introduced by Charnes and Cooper [9], and its reduction to a deterministic equivalent follows the program they later formalized [10] and that Prékopa [21] develops in depth. Our approach based on a Central Limit aggregation corrected for finite-team skewness via the Cornish–Fisher expansion [13, 16] imports a standard tool from value-at-risk practice, along with that tool’s known limitation: the truncated Cornish–Fisher inverse is accurate only over a bounded skewness range, so for very heavy-tailed token usage our deterministic frontier should be read as the tightening approximation of Assumption 2 rather than an exact boundary. Human oversight and collective intelligence. Bainbridge’s irony of automation [6], adopted in Section 1 as our central framing, is the qualitative foundation the e−λT term renders quantitative. Finally, the paper speaks to collective intelligence: where Woolley et al. [23] establish a measurable collective-intelligence factor in human groups and Bansal et al. [7] show that human– AI teams can beat either alone yet that uncalibrated reliance erodes the gain, we contribute the cost side of this via the budget-optimal composition of a human–AI collective, and how that composition should shift as usage volatility and skewness change. Limitations: homogeneous agents, team structure, and calibration. The collective modeled here is N exchangeable agents: cognitive friction depends only on per-capita intensity T , which rules out pooled review queues, dedicated reviewer roles, and specialization—precisely the division-of-labor structures that collective-intelligence research identifies as first-order for group performance [23]. The empirical record likewise resists homogeneity: novices gain roughly twice the average treatment effect [8], suggesting heterogeneous (βi , λi ). Extending the production field to a seniority mix with a review-pooling technology, and calibrating (α, β, λ) against

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measured usage telemetry and the experimental effect sizes cited above, are the two most valuable next steps. Appendix A assembles a plausibility calibration of (β, λ, σ/µ, γ1 ) from published telemetry and the cited effect sizes; full estimation requires panel usage data we do not possess, so all parameters in Figs. 2–4 remain illustrative.

Disclosure of Generative AI Use Claude (Claude Opus 4.8 and Claude Fable 5), large language models developed by Anthropic and accessed through the Claude Code command-line interface, were used in the preparation of this work. The author takes full responsibility for all content.

References [1] Daron Acemoglu and Pascual Restrepo. The race between man and machine: Implications of technology for growth, factor shares, and employment. American Economic Review, 108 (6):1488–1542, 2018. doi: 10.1257/aer.20160696. [2] Anthropic. Claude code: Manage costs effectively. https://docs.anthropic.com/en/ docs/claude-code/costs, 2025. Reports a mean of $6 per developer per active day with 90% of users below $12. Accessed June 2026. [3] Anthropic. Claude code documentation: Manage costs effectively. https://code.claude. com/docs/en/costs, 2026. Reports a mean of $13 per developer per active day, 90% of users below $30, and $150–$250 per developer per month at enterprise scale. Accessed June 2026. [4] Kenneth J. Arrow, Hollis B. Chenery, Bagicha S. Minhas, and Robert M. Solow. Capitallabor substitution and economic efficiency. The Review of Economics and Statistics, 43(3): 225–250, 1961. doi: 10.2307/1927286. [5] David H. Autor, Frank Levy, and Richard J. Murnane. The skill content of recent technological change: An empirical exploration. The Quarterly Journal of Economics, 118(4): 1279–1333, 2003. doi: 10.1162/003355303322552801. [6] Lisanne Bainbridge. Ironies of automation. Automatica, 19(6):775–779, 1983. doi: 10.1016/ 0005-1098(83)90046-8. [7] Gagan Bansal, Tongshuang Wu, Joyce Zhou, Raymond Fok, Besmira Nushi, Ece Kamar, Marco Tulio Ribeiro, and Daniel Weld. Does the whole exceed its parts? the effect of AI explanations on complementary team performance. In Proceedings of the 2021 CHI Conference on Human Factors in Computing Systems, CHI ’21. Association for Computing Machinery, 2021. doi: 10.1145/3411764.3445717. [8] Erik Brynjolfsson, Danielle Li, and Lindsey R. Raymond. Generative AI at work. The Quarterly Journal of Economics, 140(2):889–942, 2025. doi: 10.1093/qje/qjae044. [9] A. Charnes and W. W. Cooper. Chance-constrained programming. Management Science, 6(1):73–79, 1959.

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[10] A. Charnes and W. W. Cooper. Deterministic equivalents for optimizing and satisficing under chance constraints. Operations Research, 11(1):18–39, 1963. doi: 10.1287/opre.11.1. 18. [11] Alpha C. Chiang. Fundamental Methods of Mathematical Economics. McGraw-Hill, New York, 3 edition, 1984. [12] Charles W. Cobb and Paul H. Douglas. A theory of production. The American Economic Review, 18(1):139–165, 1928. [13] E. A. Cornish and R. A. Fisher. Moments and cumulants in the specification of distributions. Revue de l’Institut International de Statistique, 5(4):307–320, 1938. [14] Fabrizio Dell’Acqua, Edward McFowland III, Ethan Mollick, Hila Lifshitz-Assaf, Katherine Kellogg, Saran Rajendran, Lisa Krayer, François Candelon, and Karim R. Lakhani. Navigating the jagged technological frontier: Field experimental evidence of the effects of AI on knowledge worker productivity and quality. Technical Report Working Paper 24-013, Harvard Business School, 2023. [15] Finout. Claude code pricing 2026: Complete plans & cost guide. https://www.finout. io/blog/claude-code-pricing-2026, 2026. Grey literature; practitioner cost telemetry. Accessed June 2026. [16] Stefan R. Jaschke. The cornish-fisher-expansion in the context of delta-gamma-normal approximations. Journal of Risk, 4(4), 2002. [17] Paul Milgrom and John Roberts. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica, 58(6):1255–1277, 1990. [18] Shakked Noy and Whitney Zhang. Experimental evidence on the productivity effects of generative artificial intelligence. Science, 381(6654):187–192, 2023. doi: 10.1126/science. adh2586. [19] Lee E. Ohanian, Musa Orak, and Shihan Shen. Revisiting capital-skill complementarity, inequality, and labor share. Review of Economic Dynamics, 51:479–505, 2023. doi: 10. 1016/j.red.2023.05.002. [20] Sida Peng, Eirini Kalliamvakou, Peter Cihon, and Mert Demirer. The impact of AI on developer productivity: Evidence from GitHub Copilot. arXiv preprint arXiv:2302.06590, 2023. arXiv:2302.06590. [21] András Prékopa. Stochastic Programming, volume 324 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, 1995. [22] Paul A. Samuelson. Foundations of Economic Analysis. Harvard University Press, Cambridge, MA, 1947. [23] Anita Williams Woolley, Christopher F. Chabris, Alex Pentland, Nada Hashmi, and Thomas W. Malone. Evidence for a collective intelligence factor in the performance of human groups. Science, 330(6004):686–688, 2010. doi: 10.1126/science.1193147.

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A

A Plausibility Calibration from Published Telemetry

We lack the panel usage data required to estimate the model, but publicly reported statistics suffice to check that the illustrative parameters of Figs. 2–4 occupy a plausible region of parameter space. We record that exercise here; nothing in the body depends on it. Dispersion (σ/µ, γ1 ). Anthropic’s operator documentation for Claude Code reports two cross-sectional cost statistics for enterprise deployments: an earlier snapshot with a mean of $6 per developer per active day and 90% of users below $12 [2], and a later snapshot with a mean of $13 and 90% below $30 [3]. Taking cost as a proxy for token volume, each (mean, P90 ) pair pins down the shape of any two-parameter right-skewed family. A gamma fit yields shape k = 1.78 (σ/µ = 0.75, γ1 = 1.5) for the first snapshot and k = 0.99 (essentially exponential) (σ/µ = 1.0, γ1 = 2.0) for the second. A log-normal fit gives σ/µ = 0.91, γ1 = 3.5 for the first snapshot and is infeasible for the second: no log-normal attains P90 /mean > 2.27, whereas the reported ratio is 2.31. The published cross-sections are thus at least as dispersed as the most dispersed compatible log-normal. Two caveats apply: these are daily cross-sections, so monthly dispersion is lower to the extent individual usage is serially independent (and comparable to the extent heavy users are persistently heavy); and cost imperfectly proxies tokens under a mixed model portfolio. Within those limits, the illustrative σ/µ ≈ 1.1 and γ1 = 2 of Fig. 3 sit squarely inside the implied range (σ/µ ∈ [0.75, 1.0], γ1 ∈ [1.5, 3.5]). Output elasticity (β). Reading the cited experiments as comparisons of lower- versus higherintensity use, a doubling of effective AI input that raises output by a factor 1 + g implies β = log2 (1 + g): the +15% average and +34% novice gains of Brynjolfsson et al. [8] give β = 0.20 and 0.42, and the +40% writing gain of Noy and Zhang [18] gives β = 0.49. The illustrative β = 0.4 lies in the upper middle of this band. Friction (λ). Dell’Acqua et al. [14] establish that the friction-dominated regime is reachable in practice; for magnitude we anchor the satiation intensity Tsat = β/λ to the upper range of sustained productive consumption. Practitioner cost analyses place heavy sustained usage near 70 MTok per month, with multi-agent workflows adding 200–500% token overhead relative to the same task run single-agent [15], which is consistent with a productive ceiling of order 50–100 MTok per engineer-month. With β = 0.4 this brackets λ ∈ [0.004, 0.008] MTok−1 ; the figures use the upper end (Tsat = 50 MTok). Price regime (Pt ). The one deliberately stylized parameter is the token price. Observed median spend of $150–$250 per developer-month [3] is only 1–2% of a loaded salary, so at today’s median prices the budget trade-off is second-order and the optimum hugs satiation. The premium regime of Fig. 3 (Pt = $300/MTok-equivalent) instead describes the heavy tail in which the token line is a first-order budget item (frontier-model multi-agent workloads, for which fourfigure single sessions and heavy-user months exceeding $1,200 are already documented [15]) and, prospectively, organizations scaling per-capita compute faster than salaries. The comparative statics of Section 6 are price-regime invariant in direction; only the tightness of the interior optimum depends on Pt .

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