Who Prices Cognitive Labor in the Age of Agents? A Position on Compute-Anchored Wages

# Who Prices Cognitive Labor in the Age of Agents? A Position on Compute-Anchored Wages
Source: [https://arxiv.org/html/2605.05558](https://arxiv.org/html/2605.05558)
###### Abstract

A natural intuition about the economics of AI agents is that, because agents can be replicated at near\-zero marginal cost, they constitute a labor input in infinitely elastic supply, and therefore drive cognitive\-labor wages to zero\. We argue this framing is wrong in mechanism but partially correct in conclusion, and that the correction matters for both theory and policy\.Agents are not labor; they are a production technology that converts compute capitalKcK\_\{c\}into effective units of cognitive laborLAL\_\{A\}\.Once this is recognized, the elastic\-supply margin that anchors the equilibrium wage migrates from the labor market to the compute capital market\. Building on the textbook factor\-pricing framework\[mankiw2020\], we derive a*Compute\-Anchored Wage*\(CAW\) bound stating that, on tasks where human and agent cognitive labor are substitutes, the competitive human wage is bounded above byλ⋅k⋅rc\\lambda\\cdot k\\cdot r\_\{c\}, wherercr\_\{c\}is the rental rate of compute capital,kkis the compute intensity of one effective agent\-labor unit, andλ\\lambdais the relative human\-to\-agent productivity\. We generalize the result through CES aggregation, separate substitutable from complementary tasks \(yielding a directional inversion of skill\-biased technical change\), and discuss factor\-share consequences\. The position is concise:*the price\-setter for cognitive labor is no longer the labor market\.*

## 1Introduction

The standard textbook account of wage determination, as presented bymankiw2020, has two ingredients: a downward\-sloping labor demand curve given by the marginal product of labor, and a labor supply curve determined by household time allocation and demographics\. Equilibrium wages clear the labor market\.

The arrival of capable AI agents disturbs this picture, and the analytical question is where the disturbance enters\. It is tempting to model agents as a new labor input — a substitute for human cognitive labor, reproducible at near\-zero marginal cost, with a supply curve horizontal at zero — and to read off a collapsing wage from that horizontal supply\. We argue that this accounting misplaces the elastic margin\. Agents are not a labor input; they are a technology that converts compute capital into effective cognitive labor\. Their supply elasticity is inherited from the supply elasticity of compute capital, which is finite and governed by physical and political\-economic constraints \(semiconductor fab capacity, electricity, water, land, geopolitics\)\. The correct model therefore reroutes the price\-determination question through the compute capital market, not the labor market\.

### A concrete instance\.

Consider a junior contract\-review paralegal whose work consists largely of clause extraction, redlining against templates, and summary memos\. A frontier large language model performs each of these tasks at quality close to or above the paralegal’s, at a marginal compute cost on the order of single\-digit dollars per labor\-hour\-equivalent at 2024–2025 prices \(Section[7](https://arxiv.org/html/2605.05558#S7)\)\. The competitive wage on this paralegal’s substitutable hours is therefore not set by the supply of paralegals; it is set by the rental rate of compute capital, scaled by an algorithmic constant\. The same logic applies to first\-pass equity\-research drafting, customer\-support triage, and a long list of cognitive tasks that the popular discussion classifies as “automatable” without identifying the specific market in which the new price is determined\.

This position paper makes that rerouting explicit, derives a closed\-form ceiling on competitive wages \(the Compute\-Anchored Wage, henceforth CAW\), generalizes the substitution structure via a CES aggregator, separates wage effects across heterogeneous tasks, calibrates the bound to current compute prices, and discusses limitations and policy\. The argument is consciously a*position*: it does not claim empirical settlement, but rather that the analytical primitive — where to put the elastic supply — has been miscoded in much current discussion, and that fixing this miscoding yields a sharp, testable prediction about cognitive\-labor pricing\. The mathematical content of the bound is standard cost minimization; the substantive content is the identification of the elastic margin\.

### Roadmap\.

Section 2 reviews the related literature and locates our contribution relative to the task\-based automation framework ofacemoglu2018,acemoglu2022and the capital\-skill complementarity tradition ofkorv2000\. Section 3 sets up the textbook factor\-pricing model\. Section 4 reformulates AI agents as a capital\-to\-labor conversion technology\. Section 5 derives the CAW bound\. Section 6 generalizes to imperfect substitution via CES\. Section 7 calibrates the bound\. Section 8 visualizes the migration of the price\-setter\. Section 9 develops the cross\-task heterogeneity prediction\. Section 10 discusses macro factor\-share consequences and policy levers\. Section[11](https://arxiv.org/html/2605.05558#S11)states limitations\.

## 2Related Work

The position developed here intersects five strands of literature\. We summarize each, and then state explicitly what the Compute\-Anchored Wage \(CAW\) framework adds\.

### Textbook factor pricing\.

The marginal\-productivity theory of factor pricing in competitive markets, codified inmankiw2020, supplies the entire formal apparatus we use\. We make no modifications to the underlying theory; the only re\-coding is in how an AI agent enters the production function\.

### Skill\-biased and routine\-biased technical change\.

katz1992document the rising college premium and frame technology–skill complementarity as the principal driver\.autor2003sharpen this into a task\-based account, in which information technology substitutes for routine cognitive and manual tasks while complementing non\-routine analytic and interpersonal tasks\. Subsequent empirical work \(autor2013;autor2015;goldinkatz2008\) traces job polarization and the long\-run race between education and technology\. Our framework predicts a directional inversion*within*cognitive labor itself: the salient axis becomes the substitutable–complementary mix on which an occupation is exposed to AI agents, rather than its position on a one\-dimensional skill ladder\. The relevant primitive is the elasticity of substitution between human and agent labor on each task,σj\\sigma\_\{j\}, indexed by the taskjj\.

### Capital\-skill complementarity\.

korv2000show that a CES production function in which capital equipment complements skilled labor and substitutes for unskilled labor matches the joint behavior of the skill premium and capital–output ratios over decades\. Our setup is structurally close: we treat compute capitalKcK\_\{c\}as a factor that, when paired with the inference stackϕ\\phi, becomes a quasi\-substitute for human cognitive labor on a measurable subset of tasks\. The crucial difference is that in KORV the substitution margin is between physical equipment and*unskilled*labor; in CAW the margin is between compute capital and*cognitive*labor previously regarded as the complement of capital\. CAW thus inherits KORV’s machinery but inverts the sign of the implied skill premium on the substitutable margin\.

### Tasks and automation \(Acemoglu–Restrepo\)\.

The closest theoretical antecedent is the task\-based automation framework ofacemoglu2018,acemoglu2019,acemoglu2020,acemoglu2022, in which capital automates a contiguous subset of tasks, displacing labor on the automated margin and reinstating labor through the creation of new tasks\. Our framework can be read as a specialization of A–R to the case where the automating capital is compute and the displaced labor is cognitive; the price effect we isolate \(WH≤λ​k​rcW\_\{H\}\\leq\\lambda kr\_\{c\}\) is the explicit equilibrium consequence of A–R’s displacement effect when the automating capital is in elastic but finite supply\. We add three things to A–R: \(i\) we identify a specific elastic margin, namely the compute capital market, that anchors the equilibrium wage on automated tasks and traces the price\-determination problem to a measurable rental ratercr\_\{c\}; \(ii\) we make the task partition explicit through an elasticity\-of\-substitution parameterσ\\sigmathat is in principle estimable from observed factor demands; and \(iii\) we discuss the political economy of compute\-market concentration as a determinant ofrcr\_\{c\}that has no analogue in standard A–R\. The reinstatement effect of A–R is consistent with our complementary\-task subsetTCT\_\{C\}and is preserved\.

### AI in macroeconomics and growth\.

aghion2017model AI as a sequence of new general\-purpose technologies that automate task production and study balanced\-growth implications\.korinek2019examine distributional consequences of AI under capital\-labor substitution, andtrammellkorinek2023survey the macroeconomics of transformative AI\.korinek2023discusses the use of large language models specifically\. CAW operates at a different level of abstraction: rather than modeling growth dynamics, it identifies an equilibrium pricing relation that any of these dynamic models must satisfy on the substitutable margin in any period\.

### General\-purpose technologies\.

bresnahan1995formalize GPTs as innovations whose value derives from co\-invention in downstream sectors\.goldfarb2023provide empirical support for treating machine learning as a GPT\. CAW is consistent with the GPT view but is silent on diffusion dynamics; it concerns the equilibrium price relation once an AI agent technology is deployed at scale\.

### Empirical exposure of occupations to LLMs\.

A nascent empirical literature measures occupation\-level exposure to large language models\.eloundou2023estimate that 80% of US workers could see at least 10% of their tasks affected, and that 19% could see 50% affected\.felten2023provide an alternative occupational exposure score\.brynjolfsson2023document a 14% productivity gain among customer\-support agents using a generative\-AI assistant, with the largest gains for less\-experienced workers\.noyzhang2023find a roughly 40% time reduction and 18% quality improvement on professional writing tasks\. These studies provide the natural empirical input to theTST\_\{S\}/TCT\_\{C\}partition and to estimatingσ\\sigmaon each task\.

### Compute supply\.

sevilla2022document the doubling time of frontier\-model training compute;cottier2024estimate the rising cost of frontier\-model training\. These pin down the empirical content of thercr\_\{c\}time path that drives CAW\. We do not model compute supply explicitly but rely on the stylized fact that it is finite and only moderately elastic in the medium run because of fab capacity, energy, water, land, and policy\.

### Declining labor share\.

karabarbounis2014,piketty2014, andautorvanreenen2020document the long\-run decline in the labor share and the rise of superstar firms\.susskind2020provides a non\-technical synthesis\. CAW refines this discussion by identifying a specific channel—the compute share of capital income—through which capital\-income concentration is now operating, and by predicting that the same mechanism will compress wages within cognitive labor\.

### Summary of contribution\.

The CAW position contributes a single analytical re\-coding—agents are a capital\-to\-labor conversion technology, not a labor input—and traces its equilibrium consequences\. Relative to the closest antecedent \(A–R\), we \(i\) identify the compute capital market as the elastic margin that prices substitutable cognitive labor; \(ii\) proposeσ\\sigmaon each task as the empirical primitive that replaces the binary “automated/not” coding; and \(iii\) connect the wage bound to a quantitatively tractable factor pricercr\_\{c\}for which there is now an active spot and contract market\. The remainder of the paper develops the consequences of that re\-coding\.

## 3Setup: Factor Markets in the Mankiw Framework

Following the textbook factor\-market model, consider a representative competitive firm with a constant\-returns\-to\-scale production technology

whereKKis physical capital andLLis labor\. Profit maximization in a competitive output market with pricePPyields

W\\displaystyle W=P⋅∂F∂L,\\displaystyle=P\\cdot\\frac\{\\partial F\}\{\\partial L\},r\\displaystyle r=P⋅∂F∂K\.\\displaystyle=P\\cdot\\frac\{\\partial F\}\{\\partial K\}\.\(2\)Factor prices equal the value of the marginal product\. Equilibrium in the labor market is

Ld​\(W\)=Ls​\(W\),L^\{d\}\(W\)=L^\{s\}\(W\),\(3\)withLsL^\{s\}determined by household time allocation and demographic factors\. The wageWWis set by where these curves intersect\.

The question is what happens when an AI agent becomes available as a partial substitute forLL\. The temptation is to add a new labor typeLAL\_\{A\}with infinitely elastic supply at zero price, mechanically forcingW→0W\\to 0on the substitutable margin\. We now argue this addition is the wrong primitive\.

## 4Reformulation: Agents as Capital\-to\-Labor Conversion

###### Definition 1\(Agent labor\)\.

An*agent labor unit*LAL\_\{A\}is the cognitive\-labor\-equivalent output produced when a fixed bundle of compute capitalkk\(GPU\-hours, energy, memory, bandwidth, model weights amortized as IP rents\) is operated for one unit of time\. Formally,LA=ϕ​\(Kc\)L\_\{A\}=\\phi\(K\_\{c\}\), whereϕ\\phiis increasing and at the relevant scale approximately linear,ϕ​\(Kc\)≈Kc/k\\phi\(K\_\{c\}\)\\approx K\_\{c\}/k\.

The augmented production function is

Y=F​\(Ko,LH,LA\)=F​\(Ko,LH,ϕ​\(Kc\)\),Y=F\(K\_\{o\},L\_\{H\},L\_\{A\}\)=F\\bigl\(K\_\{o\},L\_\{H\},\\phi\(K\_\{c\}\)\\bigr\),\(4\)whereKoK\_\{o\}is non\-compute capital,LHL\_\{H\}is human cognitive labor, andKcK\_\{c\}is compute capital with rental ratercr\_\{c\}determined in the compute capital market\. The firm’s first\-order conditions become

WH\\displaystyle W\_\{H\}=P⋅∂F∂LH,\\displaystyle=P\\cdot\\frac\{\\partial F\}\{\\partial L\_\{H\}\},\(5\)rc\\displaystyle r\_\{c\}=P⋅∂F∂LA⋅ϕ′​\(Kc\)\.\\displaystyle=P\\cdot\\frac\{\\partial F\}\{\\partial L\_\{A\}\}\\cdot\\phi^\{\\prime\}\(K\_\{c\}\)\.\(6\)The crucial observation:LAL\_\{A\}has no household supply curve\. Its supply derives entirely from the supply ofKcK\_\{c\}, which is governed by fab capacity, energy infrastructure, data\-center construction lead times, and policy\. These are finite and relatively inelastic in the short run, only moderately elastic in the long run\.

## 5The Compute\-Anchored Wage Bound

We state the central proposition first under the strong, illustrative case of perfect substitution, then generalize\.

###### Assumption 1\(Perfect substitutability on the margin\)\.

There exists a set of tasks on which one unit of human cognitive labor andλ\\lambdaunits of agent labor are perfect substitutes inFF, so that effective cognitive labor on these tasks aggregates asLeff=LH\+λ−1​LAL\_\{\\text\{eff\}\}=L\_\{H\}\+\\lambda^\{\-1\}L\_\{A\}\. Equivalently, one human unit produces the same effective cognitive output asλ\\lambdaagent units, so thatλ\>1\\lambda\>1corresponds to humans being more productive per unit, andλ<1\\lambda<1to agents dominating\.

###### Proposition 1\(Compute\-Anchored Wage\)\.

Under Assumption[1](https://arxiv.org/html/2605.05558#Thmassumption1)and competitive factor markets, the equilibrium human cognitive wage on the substitutable task set satisfies

WH≤λ⋅k⋅rc\\boxed\{\\;W\_\{H\}\\;\\leq\\;\\lambda\\cdot k\\cdot r\_\{c\}\\;\}\(7\)in any equilibrium withLH\>0L\_\{H\}\>0\. Moreover, in any equilibrium with bothLH\>0L\_\{H\}\>0andLA\>0L\_\{A\}\>0, the bound holds with equality,WH=λ​k​rcW\_\{H\}=\\lambda kr\_\{c\}\.

###### Proof sketch\.

The unit cost of effective cognitive labor on the substitutable task ismin⁡\{WH,λ​k​rc\}\\min\\\{W\_\{H\},\\,\\lambda kr\_\{c\}\\\}, sinceλ\\lambdaunits ofLAL\_\{A\}substitute for one unit ofLHL\_\{H\}, and each unit ofLAL\_\{A\}requireskkunits of compute at rental ratercr\_\{c\}\. Three cases:

- •IfWH<λ​k​rcW\_\{H\}<\\lambda kr\_\{c\}: firms substitute fully towardLHL\_\{H\}on this task, soLAd=0L\_\{A\}^\{d\}=0onTST\_\{S\}\. The bound is slack but nonbinding because no agent labor is used\.
- •IfWH\>λ​k​rcW\_\{H\}\>\\lambda kr\_\{c\}: firms substitute fully towardLAL\_\{A\}, soLHd=0L\_\{H\}^\{d\}=0onTST\_\{S\}\. Any positive supply ofLHL\_\{H\}at this wage on this task is unemployed; the wage cannot be sustained in any equilibrium with positive employment ofLHL\_\{H\}onTST\_\{S\}\. HenceWH≤λ​k​rcW\_\{H\}\\leq\\lambda kr\_\{c\}in any such equilibrium\.
- •Interior coexistence \(LH,LA\>0L\_\{H\},L\_\{A\}\>0onTST\_\{S\}\) requires the marginal indifferenceWH=λ​k​rcW\_\{H\}=\\lambda kr\_\{c\}\.

The detailed cost\-minimization derivation is in Appendix B\. ∎

###### Corollary 1\(Migration of the price\-setter\)\.

The equilibrium wage on substitutable cognitive tasks is determined by the parameters of the compute capital market\(k,rc\)\(k,r\_\{c\}\)and the technology parameterλ\\lambda\. The labor supply curveLHsL\_\{H\}^\{s\}does*not*appear in the binding condition\. The price\-setting margin has migrated from the labor market to the compute capital market\.

This is the formal content of the position\. Note carefully what it does not say\. It does not say wages go to zero; they go toλ​k​rc\\lambda kr\_\{c\}, which can be high or low depending on compute\-market conditions\. It does not say humans become unemployed; they may relocate to complementary tasks \(Section[9](https://arxiv.org/html/2605.05558#S9)\)\. It does not say this happens in all sectors—only on tasks where Assumption[1](https://arxiv.org/html/2605.05558#Thmassumption1)approximately holds\.

## 6CES Generalization: Imperfect Substitution

Perfect substitution overstates the case\. Generalize via constant elasticity of substitution\. Let

Leff=A​\[α​LHρ\+β​LAρ\]1/ρ,σ=11−ρ,L\_\{\\text\{eff\}\}=A\\bigl\[\\alpha L\_\{H\}^\{\\rho\}\+\\beta L\_\{A\}^\{\\rho\}\\bigr\]^\{1/\\rho\},\\qquad\\sigma=\\frac\{1\}\{1\-\\rho\},\(8\)withσ∈\(0,∞\)\\sigma\\in\(0,\\infty\)the elasticity of substitution between human and agent cognitive labor, and CES weightsα,β\>0\\alpha,\\beta\>0\. The effective unit cost of an agent labor unit is its compute cost,WAeff≡k​rcW\_\{A\}^\{\\text\{eff\}\}\\equiv kr\_\{c\}, sinceLA=Kc/kL\_\{A\}=K\_\{c\}/kand the rental rate ofKcK\_\{c\}isrcr\_\{c\}\. Cost minimization of \([8](https://arxiv.org/html/2605.05558#S6.E8)\) for one unit ofLeffL\_\{\\text\{eff\}\}delivers the conditional factor demands and the relative\-wage condition

WHWAeff=αβ​\(LHLA\)ρ−1=αβ​\(LHLA\)−1/σ\.\\frac\{W\_\{H\}\}\{W\_\{A\}^\{\\text\{eff\}\}\}=\\frac\{\\alpha\}\{\\beta\}\\left\(\\frac\{L\_\{H\}\}\{L\_\{A\}\}\\right\)^\{\\rho\-1\}=\\frac\{\\alpha\}\{\\beta\}\\left\(\\frac\{L\_\{H\}\}\{L\_\{A\}\}\\right\)^\{\-1/\\sigma\}\.\(9\)With normalizationα/β=λ\\alpha/\\beta=\\lambdaon the perfect\-substitute limit, \([9](https://arxiv.org/html/2605.05558#S6.E9)\) reduces toWH=λ​WAeffW\_\{H\}=\\lambda W\_\{A\}^\{\\text\{eff\}\}asσ→∞\\sigma\\to\\infty, recovering the CAW bound \([7](https://arxiv.org/html/2605.05558#S5.E7)\)\.

###### Proposition 2\(Compute\-driven wage compression\)\.

Hold the human\-labor supplyLHsL\_\{H\}^\{s\}and the level of demand forLeffL\_\{\\text\{eff\}\}fixed\. Consider an exogenous increase in compute capital supplyK¯c\\bar\{K\}\_\{c\}\(equivalently, a fall inkkviaϕ\\phiimprovement\) at a fixed rental ratercr\_\{c\}that lowersWAeffW\_\{A\}^\{\\text\{eff\}\}\. Then in the new equilibrium the human cognitive wage falls with semi\-elasticity

∂log⁡WH∂log⁡WAeff=1−1σ⋅∂log⁡\(LH/LA\)∂log⁡WAeff,\\frac\{\\partial\\log W\_\{H\}\}\{\\partial\\log W\_\{A\}^\{\\text\{eff\}\}\}\\;=\\;1\-\\frac\{1\}\{\\sigma\}\\cdot\\frac\{\\partial\\log\(L\_\{H\}/L\_\{A\}\)\}\{\\partial\\log W\_\{A\}^\{\\text\{eff\}\}\},\(10\)and in the polar limit of perfectly elasticLHsL\_\{H\}^\{s\}this collapses to∂log⁡WH/∂log⁡WAeff=1\\partial\\log W\_\{H\}/\\partial\\log W\_\{A\}^\{\\text\{eff\}\}=1\. Asσ→∞\\sigma\\to\\infty, the CES bound collapses to the perfect\-substitute CAW bound \([7](https://arxiv.org/html/2605.05558#S5.E7)\); asσ→0\\sigma\\to 0\(Leontief\),LHL\_\{H\}andLAL\_\{A\}are used in fixed proportion, the binding factor is whichever is shorter in supply, and the comparative\-static channel fromWAeffW\_\{A\}^\{\\text\{eff\}\}toWHW\_\{H\}vanishes\.

The CES form makes precise an empirically tractable claim:*the magnitude of CAW pressure on a given task is governed by the elasticity of substitution between human and agent cognitive labor on that task*\. This is the right object for empirical estimation, replacing the binary “AI replaces / does not replace” framing common in policy discourse\.eloundou2023,felten2023,brynjolfsson2023, andnoyzhang2023provide the natural empirical input\.

## 7A Numerical Calibration of CAW

To give the bound \([7](https://arxiv.org/html/2605.05558#S5.E7)\) empirical traction, we now plug in plausible 2024–2025 numbers\. The exercise is illustrative, not estimation; the goal is to show thatλ​k​rc\\lambda kr\_\{c\}takes economically meaningful values that vary by orders of magnitude across tasks\.

### Compute rental ratercr\_\{c\}\.

On\-demand H100 GPU rental from major cloud providers traded in the$​2\\mathdollar 2–$​5\\mathdollar 5/GPU\-hour range in 2024, with multi\-year contract pricing closer to$​1\.50\\mathdollar 1\.50/GPU\-hour\. We takerc=$​2r\_\{c\}=\\mathdollar 2/GPU\-hour as a midpoint\. Frontier\-model inference is typically priced per million tokens; converting to GPU\-hour\-equivalents using public throughput benchmarks for a 70B\-class model on an H100 yields roughly the same order of magnitude\.

### Compute intensitykk\.

For a frontier reasoning agent producing sustained, high\-quality output, current inference stacks consume on the order of0\.50\.5–22H100\-hours of compute to deliver one “hour” of effective senior\-knowledge\-worker output, depending on whether the workload is interactive \(heavy KV\-cache reuse\) or batched\. We takek=1k=1H100\-hour per agent\-labor\-hour for a frontier model andk=0\.05k=0\.05H100\-hour per agent\-labor\-hour for a small distilled model on a substitutable subtask\.

### Productivity ratioλ\\lambda\.

The empirical literature on LLM productivity gains\[brynjolfsson2023,noyzhang2023\]reports time savings of 14–40% on substitutable tasks, with quality at or above human baseline\. We takeλ∈\{0\.5,1\.0,2\.0\}\\lambda\\in\\\{0\.5,1\.0,2\.0\\\}to span the cases where agents are absolutely more productive \(λ<1\\lambda<1\), at parity \(λ=1\\lambda=1\), and where humans retain a productivity edge \(λ\>1\\lambda\>1\)\.

### Implied CAW\.

Combining these, the boundWH≤λ​k​rcW\_\{H\}\\leq\\lambda kr\_\{c\}implies the per\-hour ceilings shown in Table[1](https://arxiv.org/html/2605.05558#S7.T1)\.

Table 1:Illustrative CAW ceilingλ​k​rc\\lambda kr\_\{c\}in US$/hour on substitutable cognitive tasks, under 2024–2025 compute prices\. Read each cell as:*the binding human wage on tasks where this\(λ,k,rc\)\(\\lambda,k,r\_\{c\}\)configuration applies*\.Two implications are immediate\. First, on tasks where small distilled models suffice \(high\-volume classification, summarization, first\-pass document review\), CAW is already binding well below any plausible human reservation wage; the wage on such tasks is effectively pinned at the marginal\-product floor\. Second, on tasks requiring frontier\-model reasoning, CAW currently sits between roughly $1 and $10/hour\. Any human cognitive labor on tasks where Assumption[1](https://arxiv.org/html/2605.05558#Thmassumption1)approximately holds and the frontier model is competent must be priced under that ceiling to retain employment\. Asϕ\\phiimproves andkkfalls, every cell of Table[1](https://arxiv.org/html/2605.05558#S7.T1)moves down monotonically\.

### Sensitivity\.

The ceiling is linear in each ofλ\\lambda,kk, andrcr\_\{c\}\. A doubling of compute prices \(e\.g\., from a supply shock or geopolitical disruption\) doubles CAW; a halving ofkkfrom algorithmic improvement halves it\. Thus the empirical content of the position is the joint trajectory\(kt,rc,t\)\(k\_\{t\},r\_\{c,t\}\)on a task\-by\-task basis, withσ\\sigmagoverning how rapidly the relevant occupational wage tracks that trajectory\.

## 8Visualizing the Migration of the Price\-Setter

Figure[1](https://arxiv.org/html/2605.05558#S8.F1)contrasts the textbook labor market \(left\) with the CAW\-anchored cognitive labor market onTST\_\{S\}\(right\)\. On the right, the human\-labor supply curve is shown but does*not*determine the equilibrium wage; the wage is pinned by the horizontal CAW line atλ​k​rc\\lambda kr\_\{c\}, which is itself the projection of the equilibrium rental rate from the compute capital market\.

LLWWLdL^\{d\}LsL^\{s\}W∗W^\{\*\}L∗L^\{\*\}\(a\) Textbook labor marketWage set byLd∩LsL^\{d\}\\cap L^\{s\}LHL\_\{H\}WHW\_\{H\}LHdL\_\{H\}^\{d\}LHsL\_\{H\}^\{s\}CAW:λ​k​rc\\lambda kr\_\{c\}W¯H\\bar\{W\}\_\{H\}LHd​\(W¯H\)L\_\{H\}^\{d\}\(\\bar\{W\}\_\{H\}\)\(b\) Cognitive labor onTST\_\{S\}Wage capped bycompute capital marketmigrationof theprice\-setterFigure 1:Migration of the price\-setting margin\.\(a\)In the textbook framework, the wage is determined by the intersection of labor demand and labor supply\.\(b\)On substitutable cognitive tasksTST\_\{S\}, the human\-labor supply curveLHsL\_\{H\}^\{s\}is no longer the binding constraint \(drawn dashed\); the equilibrium wage ceiling is the horizontal CAW lineλ​k​rc\\lambda kr\_\{c\}, wherercr\_\{c\}is the rental rate of compute capital determined in a separate market\. Employment onTST\_\{S\}is given byLHd​\(W¯H\)L\_\{H\}^\{d\}\(\\bar\{W\}\_\{H\}\)\.
## 9Task Heterogeneity: A Directional Inversion of SBTC

katz1992and the subsequent skill\-biased technical change literature document that information technology has historically complemented high\-skill cognitive labor and substituted for routine labor\[autor2003,autor2013,autor2015\]\. The CAW framework predicts a directional inversion*within*cognitive labor\.

Partition tasks into two sets:

- •Substitutable cognitive tasks \(TST\_\{S\}\):drafting, code generation against specifications, summarization, first\-pass analysis, scheduling, retrieval, classification\. OnTST\_\{S\},σ\\sigmais large, CAW binds, andWHW\_\{H\}is anchored byλ​k​rc\\lambda kr\_\{c\}\. Empirical exposure scores fromeloundou2023andfelten2023are useful proxies for membership inTST\_\{S\}\.
- •Complementary cognitive tasks \(TCT\_\{C\}\):judgment under deep uncertainty, accountability and legal liability, relational and political work, cross\-domain integration, taste, principal\-agent monitoring, and any task where∂F/∂LH\\partial F/\\partial L\_\{H\}is increasing inLAL\_\{A\}\. OnTCT\_\{C\},σ\\sigmais small or the cross\-partial∂2F/∂LH​∂LA\>0\\partial^\{2\}F/\\partial L\_\{H\}\\partial L\_\{A\}\>0, andWHW\_\{H\}*rises*withLAL\_\{A\}\.

The wage distribution across cognitive workers therefore widens, but the relevant axis is no longer traditional skill—it is theTST\_\{S\}/TCT\_\{C\}exposure mix of each occupation\. To make this concrete, consider two occupations with similar formal credentials:

- •A junior contract\-review paralegal whose work is roughly 80% onTST\_\{S\}\(clause extraction, redlining against templates, summary memos\) and 20% onTCT\_\{C\}\(escalation judgment\)\.
- •A senior litigation associate whose work is roughly 30% onTST\_\{S\}\(document review, brief drafting\) and 70% onTCT\_\{C\}\(case strategy, client management, courtroom work\)\.

Under CAW, the paralegal’s wage is dominated byλ​k​rc\\lambda kr\_\{c\}for the substitutable component and is squeezed downward askkfalls; the associate’s wage is dominated by complementary tasks and may rise\. This is consistent with the early empirical findings ofbrynjolfsson2023that productivity gains are concentrated in less\-experienced workers but does not by itself imply that less\-experienced workers gain in compensation: the same tasks that they used to monopolize have been priced down\. Two occupations with identical traditional skill requirements but differentTST\_\{S\}/TCT\_\{C\}shares will diverge sharply in compensation\. This is a testable cross\-sectional prediction distinct from canonical SBTC\.

## 10Macro Implications: Factor Shares

Aggregating across tasks, define the labor sharesL=WH​LH/Ys\_\{L\}=W\_\{H\}L\_\{H\}/Y\. As compute substitutes for human cognitive labor onTST\_\{S\}, the wage billWH​LHW\_\{H\}L\_\{H\}shrinks on those tasks, while the compute rental billrc​Kcr\_\{c\}K\_\{c\}grows\. The capital share rises\. Recipients of the rising capital share are owners of compute infrastructure, energy producers, and holders of model intellectual property; these need not coincide with the historical owners of physical capital\.

This connects the CAW framework to the literature on declining labor shares\[karabarbounis2014,autorvanreenen2020\]and the long\-run dynamics emphasized bypiketty2014, with an important refinement: under CAW, the capital share rises specifically through the*compute*channel\. Policy interventions targeting that channel \(compute taxation, public compute provision, antitrust on accelerator markets, energy policy\) have first\-order effects on the cognitive\-labor wage distribution that interventions targeting the labor market do not\. We discuss four such levers in turn\.

### Compute taxation\.

A tax onrcr\_\{c\}raises the CAW ceiling proportionally\. Incidence depends on the elasticity of compute supply and on the elasticity of demand for substitutable cognitive output\. With moderately inelastic compute supply in the short run, much of the tax falls on compute owners; in the long run, with more elastic capacity expansion, incidence shifts toward output prices and ultimately toward human wages onTST\_\{S\}via the bound\. Pigouvian arguments based on energy externalities and Ramsey arguments based on the relative inelasticity ofKcK\_\{c\}supply both push toward positive optimalτc\>0\\tau\_\{c\}\>0\.

### Public compute provision\.

A public option that supplies compute at marginal cost compresses the markup component ofrcr\_\{c\}and tightens the CAW ceiling\. The effect is symmetric to compute taxation in sign onrcr\_\{c\}but distributionally different: public provision lowersWHW\_\{H\}onTST\_\{S\}, raising the consumer surplus of cognitive output buyers without redistributing to workers onTST\_\{S\}\.

### Antitrust on accelerator markets\.

If the accelerator market is concentrated,rcr\_\{c\}contains a markup over marginal cost\. Antitrust enforcement that erodes that markup again lowers the CAW ceiling\. The substantive question is whether the resulting consumer\-surplus gains in cognitive output exceed the wage compression onTST\_\{S\}; under standard welfare assumptions they do, but the distributional consequences are large\.

### Energy policy\.

A binding fraction ofrcr\_\{c\}is electricity cost\. Policy that lowers the levelized cost of electricity to data centers \(transmission build\-out, nuclear permitting, renewable subsidies\) operates throughrcr\_\{c\}in the same direction as capacity expansion\. A binding fraction also operates as a regional siting question: the cognitive\-labor wage compression is geographically uneven if data\-center electricity is cheap in one region and expensive in another, but CAW is a price relationship that propagates through tradable cognitive output and is therefore largely national to global, not local\.

## 11Limitations and Boundary Conditions

We owe an honest accounting of what would invalidate or complicate the position\.

1. 1\.Jevons effects\.A fall in the unit cost of cognitive labor may expand demand for cognitive output enough to raise totalLHL\_\{H\}employment even onTST\_\{S\}tasks\. CAW bounds the*wage*, not the wage*bill*; whether total compensation rises or falls depends on the demand elasticity for cognitive output\.
2. 2\.Ricardian comparative advantage\.Even if agents are absolutely more productive on every task, humans retain employment via comparative advantage\. But comparative advantage determines*allocation*, not*price*; wages on the substitutable margin remain anchored\.
3. 3\.Non\-productivity wage components\.Liability, accountability, signaling, trust, and physical co\-presence add a non\-marginal\-product premium to human labor that the production\-function setup does not capture\. CAW bounds the marginal\-product component of wages, not the total\.
4. 4\.Compute\-market structure\.The derivation assumes competitive compute markets\. If compute is monopolized, vertically integrated with model providers, or politically rationed,rcr\_\{c\}is no longer a competitive rental rate and the bound is replaced by a markup\-adjusted version\. The position then becomes a claim about*political economy*as much as about prices\.
5. 5\.Endogenousϕ\\phi\.Algorithmic improvement reduceskkover time\. CAW is therefore a moving target, declining secularly even at constantrcr\_\{c\}\. The relevant long\-run object is the joint trajectory of\(kt,rc,t\)\(k\_\{t\},r\_\{c,t\}\)\. A first\-pass dynamic statement: holdingrcr\_\{c\}constant, exponential improvement inϕ\\phiat rateggimplies the CAW trajectoryW¯H​\(t\)=λ​k0​e−g​t​rc\\bar\{W\}\_\{H\}\(t\)=\\lambda k\_\{0\}e^\{\-gt\}r\_\{c\}, which converges to zero ast→∞t\\to\\inftyunless arrested by hardware bottlenecks or model\-quality saturation\.
6. 6\.Task boundaries are themselves endogenous\.The partitionTS∪TCT\_\{S\}\\cup T\_\{C\}shifts with the capability frontier; tasks migrate fromTCT\_\{C\}toTST\_\{S\}as agents improve\. The empirical content of the framework depends on a measurable, time\-indexed task taxonomy\.
7. 7\.Heterogeneity ofKcK\_\{c\}\.As noted in the remark onKcK\_\{c\}heterogeneity in the reformulation,KcK\_\{c\}is a composite of physical compute \(rival, competitive\), model\-weight IP \(non\-rival, often monopolistic\), and sunk training capital\. A more refined version of CAW would carry these as separate factors with their own pricing equations\.

## 12Conclusion

The position can be stated in one sentence:*on tasks where AI agents substitute for human cognitive labor, the equilibrium wage ceiling is set in the compute capital market, not the labor market\.*The standard textbook framework already contains all the machinery needed to see this; the only required correction is to recognize agents as a capital\-to\-labor conversion technology rather than a labor input\. Once that correction is made, the Compute\-Anchored Wage boundWH≤λ​k​rcW\_\{H\}\\leq\\lambda kr\_\{c\}follows directly from competitive cost minimization, and a number of empirical and policy questions reorient accordingly: the relevant elasticity to estimate isσ\\sigmaacross task categories; the relevant macro outcome is the compute share of capital income; the relevant policy levers are compute\-market levers, not labor\-market levers\. The numerical calibration in Section[7](https://arxiv.org/html/2605.05558#S7)suggests CAW is already binding on the high\-volume substitutable margin and approaching binding on frontier\-substitutable tasks at present compute prices\. The mathematical content of the bound is standard cost minimization; the substantive content is the identification of the elastic margin\.

## References

\\beginappendix

## 13Notation Summary

For convenience we collect the symbols used throughout the paper\.

- •YY— aggregate output of the representative competitive firm\.
- •KK,LL— generic physical capital and labor in the textbook setup\.
- •KoK\_\{o\}— non\-compute physical capital\.
- •KcK\_\{c\}— compute capital \(GPUs, accelerators, data\-center capacity, energy, model\-weight IP rents\)\.
- •rcr\_\{c\}— competitive rental rate of compute capital, in $/compute\-unit\-hour\.
- •LHL\_\{H\}— human cognitive labor, in labor\-hours\.
- •LAL\_\{A\}— agent \(cognitive\-labor\-equivalent\) units, satisfyingLA=ϕ​\(Kc\)≈Kc/kL\_\{A\}=\\phi\(K\_\{c\}\)\\approx K\_\{c\}/k\.
- •kk— compute intensity: units of compute capital required per effective agent\-labor unit, in compute\-units per agent\-labor\-hour\.
- •ϕ\\phi— capital\-to\-labor conversion technology embedding model architecture and inference stack\.
- •λ\\lambda— relative human\-to\-agent productivity on the substitutable task set; one human\-labor unit produces the same effective output asλ\\lambdaagent\-labor units\.λ\>1\\lambda\>1corresponds to humans being more productive per unit;λ<1\\lambda<1to agents dominating\.
- •WHW\_\{H\}— human cognitive wage, in $/labor\-hour\.
- •WAeff≡k​rcW\_\{A\}^\{\\text\{eff\}\}\\equiv kr\_\{c\}— effective agent unit wage, in $/agent\-labor\-hour\.
- •σ\\sigma— elasticity of substitution between human and agent cognitive labor in the CES aggregator\.
- •TST\_\{S\},TCT\_\{C\}— substitutable and complementary cognitive task sets\.

## 14Detailed Derivation of the CAW Bound

We expand the proof sketch of Proposition[1](https://arxiv.org/html/2605.05558#Thmproposition1)\. Under Assumption[1](https://arxiv.org/html/2605.05558#Thmassumption1), the effective cognitive\-labor input on the substitutable task set isLeff=LH\+λ−1​LAL\_\{\\text\{eff\}\}=L\_\{H\}\+\\lambda^\{\-1\}L\_\{A\}\. A profit\-maximizing firm chooses\(LH,LA\)\(L\_\{H\},L\_\{A\}\)to minimize the cost of producing one unit ofLeffL\_\{\\text\{eff\}\}:

minLH,LA≥0⁡WH​LH\+rc​k​LAs\.t\.LH\+λ−1​LA≥1\.\\min\_\{L\_\{H\},\\,L\_\{A\}\\geq 0\}\\;W\_\{H\}L\_\{H\}\+r\_\{c\}\\,k\\,L\_\{A\}\\quad\\text\{s\.t\.\}\\quad L\_\{H\}\+\\lambda^\{\-1\}L\_\{A\}\\geq 1\.Since the constraint is linear and the objective is linear, the solution is a corner whenever the per\-unit costsWHW\_\{H\}\(per unit ofLHL\_\{H\}\) andλ​k​rc\\lambda kr\_\{c\}\(per unit of effective labor delivered throughLAL\_\{A\}\) are unequal:

- •IfWH<λ​k​rcW\_\{H\}<\\lambda kr\_\{c\}: full substitution towardLHL\_\{H\}, soLAd=0L\_\{A\}^\{d\}=0onTST\_\{S\}\.
- •IfWH\>λ​k​rcW\_\{H\}\>\\lambda kr\_\{c\}: full substitution towardLAL\_\{A\}, soLHd=0L\_\{H\}^\{d\}=0onTST\_\{S\}\. Any positive supply ofLHL\_\{H\}at this wage on this task is therefore unemployed; the wage cannot be sustained in any equilibrium withLH\>0L\_\{H\}\>0onTST\_\{S\}\.
- •Interior coexistence requiresWH=λ​k​rcW\_\{H\}=\\lambda kr\_\{c\}\.

Across all cases consistent with positive employment,WH≤λ​k​rcW\_\{H\}\\leq\\lambda kr\_\{c\}on the substitutable margin, with equality whenever both factors are simultaneously employed\. WhenLA=0L\_\{A\}=0everywhere \(e\.g\., compute is unavailable or prohibitively expensive\), the standard labor\-market wage prevails and the bound is slack\.

## 15CES Algebra

Cost minimization of the CES aggregator \([8](https://arxiv.org/html/2605.05558#S6.E8)\) subject to producing one unit ofLeffL\_\{\\text\{eff\}\}yields the conditional factor demands

LH=1A​\(αWH\)σ​Λ,LA=1A​\(βWAeff\)σ​Λ,L\_\{H\}=\\frac\{1\}\{A\}\\Bigl\(\\frac\{\\alpha\}\{W\_\{H\}\}\\Bigr\)^\{\\sigma\}\\Lambda,\\qquad L\_\{A\}=\\frac\{1\}\{A\}\\Bigl\(\\frac\{\\beta\}\{W\_\{A\}^\{\\text\{eff\}\}\}\\Bigr\)^\{\\sigma\}\\Lambda,whereΛ\\Lambdais the dual CES price index \(a function ofWHW\_\{H\}andWAeffW\_\{A\}^\{\\text\{eff\}\}alone\)\. Taking the ratio gives \([9](https://arxiv.org/html/2605.05558#S6.E9)\)\.

The two limits are immediate\. Asσ→∞\\sigma\\to\\infty\(perfect substitutes\), the relative wage \([9](https://arxiv.org/html/2605.05558#S6.E9)\) forcesWH→\(α/β\)​WAeffW\_\{H\}\\to\(\\alpha/\\beta\)\\,W\_\{A\}^\{\\text\{eff\}\}, which under the normalizationα/β=λ\\alpha/\\beta=\\lambdarecovers the perfect\-substitute CAW bound\. Asσ→0\\sigma\\to 0\(Leontief\), the conditional demand forLHL\_\{H\}is fixed by the technology in proportion toLAL\_\{A\}, soLHL\_\{H\}andLAL\_\{A\}are used together in fixed ratio; the binding factor is then whichever is shorter in supply\. If human\-labor supply is the binding constraint at the prevailing demand forLeffL\_\{\\text\{eff\}\}, thenWHW\_\{H\}is determined byLHsL\_\{H\}^\{s\}and the comparative\-static channel fromWAeffW\_\{A\}^\{\\text\{eff\}\}toWHW\_\{H\}vanishes; if compute is the binding constraint, thenWHW\_\{H\}tracksWAeffW\_\{A\}^\{\\text\{eff\}\}scaled by the technology proportion\.

For Proposition[2](https://arxiv.org/html/2605.05558#Thmproposition2), holdingLHsL\_\{H\}^\{s\}and the demand forLeffL\_\{\\text\{eff\}\}fixed, totally differentiating \([9](https://arxiv.org/html/2605.05558#S6.E9)\) with respect toWAeffW\_\{A\}^\{\\text\{eff\}\}yields

d​log⁡WHd​log⁡WAeff=1−1σ⋅d​log⁡\(LH/LA\)d​log⁡WAeff\.\\frac\{d\\log W\_\{H\}\}\{d\\log W\_\{A\}^\{\\text\{eff\}\}\}=1\-\\frac\{1\}\{\\sigma\}\\cdot\\frac\{d\\log\(L\_\{H\}/L\_\{A\}\)\}\{d\\log W\_\{A\}^\{\\text\{eff\}\}\}\.Under perfectly elasticLHsL\_\{H\}^\{s\},WHW\_\{H\}tracksWAeffW\_\{A\}^\{\\text\{eff\}\}one\-for\-one and the second term vanishes\. Under inelasticLHsL\_\{H\}^\{s\}, the cross\-derivative term partially offsets the direct effect, with the magnitude controlled by1/σ1/\\sigma\.