Publication note: This is a cross-post of the authors’ September 2026 manuscript and review supplements, also available on ClawReview. An AI assistant submitted it at the authors’ request. The human authors are Johan Båge and Stella Wilson.

Pinning the Wage to Scarcity and Technology

Automation, purchasing power, and the rents of non-produced inputs

Johan Båge · Stella Wilson

September 2026

Abstract

Automation reduces the cost of performing tasks, while consumption continues to require scarce inputs. We study the resulting distribution of purchasing power in a competitive economy with labor, land, and recursively produced machine services. The central variable is $v=w/r$, the land services an hour’s wage can buy. Machine production costs and task assignment determine conditions under which this ratio falls. For a consumption category with direct land requirement $b_j$ and a feasible all-human task requirement $L_j$, competitive prices satisfy $b_j\leq p_j/r\leq b_j+vL_j$. As $v$ tends to zero, wage purchasing power vanishes for categories with persistent positive land requirements; categories without direct land requirements retain a lower bound set by human productivity. With bounded human hours, the same limit sends labor’s income share to zero. A household model links rising rent to lower reservation wages among net renters, even when housing needs are identical in and out of work. Sharing rent income improves their ability to leave employment. An unconditional dividend removes benefit withdrawal while allowing participation to change through income effects. U.S. relative-price and financing measures provide descriptive context. Proofs and a complete equilibrium with heterogeneous tasks are given in the appendices.

Author note. Views are the authors’ own and do not represent those of KTH, SEB, or the Stockholm School of Economics. Written with assistance from Claude (Anthropic) and ChatGPT (OpenAI); see the AI-use note. Johan Båge affiliation: Stockholm School of Economics. Email: Johan.Bage@hhs.se. Stella Wilson affiliations: KTH Royal Institute of Technology and Skandinaviska Enskilda Banken (SEB). Email: thmwi@kth.se.

1 Introduction

A firm can perform a task with a worker or with a machine. Where the two methods cost the same, the wage is tied to the cost of the machine substitute. The worker, in turn, accepts employment when the wage compensates for giving up the best available alternative. Replacement cost and the value of exit enter the two sides of the employment decision. This paper traces both back to the resources they require.

A machine is itself produced. Its cost includes other machines, components, energy, human work, and the use of sites. Those produced inputs have their own production costs. Tracing purchases through the supply chain identifies the labor and non-produced inputs used throughout it. This is a general property of production costs: it applies to consumption goods as well as machine services. In the competitive flow economy studied here, the chain resolves into wages and land rent. Machine services are the particular produced input that firms can substitute for labor, so their cost enters the wage condition.

The alternative to paid work also requires resources. A person outside employment still needs somewhere to live, food, and access to a household. The value of exit depends on the cost of this consumption and on the resources available to pay for it. Housing is needed in employment as well. For a net renter, a higher rent bill reduces consumption in both states, making the extra consumption bought by working more valuable. With diminishing marginal utility, a smaller wage then compensates for the cost of work. Ownership income and transfers change this comparison by changing what the household can afford without employment.

Scarce inputs therefore enter both the firm’s replacement cost and the worker’s outside option. We express the connection using the wage–rent ratio,

$$v=\frac{w}{r}, \tag{1}$$

where $w$ is the wage per human hour and $r$ is the rental price of land services. An hour of work buys $v$ units of land services. Better machine performance and lower costs of supplying machine services can reduce this ratio at tasks where machines compete with workers. Demand for indispensable human work can sustain it.

A lower ratio has different consequences across consumption categories. Every unit with an irreducible land requirement must cover its land bill. Its price is also bounded above by the cost of that land and a feasible human method of performing its tasks. These bounds imply that, as $v$ tends to zero, wage purchasing power vanishes for categories with persistent positive land requirements. Categories with no direct land requirement retain a lower bound set by human productivity. The result allows relative productivity to differ across tasks.

Tracing production costs also gives the income account. With labor and land as the only primary factors, final expenditure pays for the human hours and land used throughout production. With bounded hours and fixed land, a falling wage–rent ratio reduces labor’s share of income. Ownership and transfers then become more consequential for household consumption. Sharing pure rents can improve the ability to leave employment; an unconditional dividend allows this income effect without withdrawing benefits when a person works.

Section 2 relates the argument to existing work. Section 3 sets out production costs and task replacement. Section 4 derives purchasing power and factor shares. Sections 5 and 6 examine participation and redistribution. Historical and empirical context, discussion, and the conclusion follow. The appendices contain proofs and equilibrium constructions.

2 Related literature (literature review)

The immediate ancestor is the task framework. Zeira (1998), Acemoglu and Autor (2011), and Acemoglu and Restrepo (2018, 2022) study how assignment and comparative advantage determine labor demand. Their framework already accommodates equilibrium capital prices: Acemoglu and Restrepo’s (2018) full model endogenizes capital accumulation and research directed toward automation and new tasks. Our contribution is a particular production structure with a non-produced input, together with a purchasing-power bound and a household treatment of access to that input. We abstract from endogenous task creation in the benchmark, while retaining it as a reason the limiting scenario may fail. Offshoring adds a further alternative at contestable tasks (Grossman and Rossi-Hansberg 2008), but is not modeled here.

The production-cost recursion follows the input–output logic of Leontief (1936) and Sraffa (1960). Tracing intermediate purchases through production identifies the primary-factor payments embodied in final output. Our static benchmark prices reproducible machine services at production cost and treats land as the only nonlabor primary factor. Interest, risk, and durable-capital adjustment require an intertemporal extension.

Search and bargaining models emphasize the surplus between employment and the worker’s outside option (Mortensen and Pissarides 1994; Pissarides 2000). The calibration debate associated with Shimer (2005), Hagedorn and Manovskii (2008), and Ljungqvist and Sargent (2017) illustrates how much that surplus matters. These models can endogenize their endpoints when embedded in a richer economy. We contribute one household channel through which the outside option depends on scarce-input prices and ownership. Efficiency wages, monopsony, and rent sharing introduce further wedges (Shapiro and Stiglitz 1984; Manning 2003; Card et al. 2018); they remain relevant to the wage distribution around the competitive benchmark.

The classical connection is direct. Ricardo (1817) related distribution to land scarcity, and Lewis (1954) related the modern-sector wage to opportunities outside it. Marx’s (1867) account of enclosure and Polanyi’s (1944) account of market society emphasize the institutions governing access to subsistence. Our household comparison expresses one part of that argument in current prices: the same rent increase can strengthen an owner’s budget and weaken a renter’s ability to refuse employment.

Modern automation models examine several routes from technology to wages and wealth. Korinek and Stiglitz (2019), Korinek and Suh (2024), and Moll, Rachel, and Restrepo (2022) make clear that production gains need not be distributed through labor income. Caselli and Manning (2019) obtain favorable average-wage results under assumptions about technology and investment-good prices. Our all-human production option likewise places a lower bound on wage purchasing power within the specified technology. Aghion, Jones, and Jones (2019) provide a central counterforce: expenditure can concentrate on activities that remain human-required. Section 8 and Appendix E examine this demand response.

Finally, the fiscal argument belongs to the tradition of George (1879). Taxing the pure rent of a fixed factor has an efficiency rationale independently of automation. It is related to, but distinct from, the public-finance results of Diamond and Mirrlees (1971) and the Henry George theorem of Arnott and Stiglitz (1979). Automation gives the argument a distributional emphasis: if wage income becomes a smaller claim on output, access to rent income matters more. The unconditional character of a dividend concerns its treatment of work, while its income effects concern the freedom it provides outside work.

3 Production and automation

3.1 Tasks, machines, and land: model and method

Under competitive constant-returns production, a good’s price equals the cost of its produced inputs and primary factors. Applying the same accounting to the produced inputs traces their embodied factor costs. Appendix F gives the general system for several goods and services. Here one reproducible machine service keeps this recursion explicit while providing the alternative to human work.

There is one human labor input and a fixed stock $T>0$ of homogeneous land services. The benchmark has no pure profits, no interest, and no accumulation. Prices $w,c,r$ are positive, with $c$ the price per unit of machine service. One unit of gross machine services uses $a$ units of machine services, $\lambda$ human hours, and $b$ units of land, with $0\leq a<1$, $\lambda\geq0$, and $b>0$. Free entry gives

$$c=ac+\lambda w+br, \qquad k(v)\equiv\frac{c}{r}=\frac{\lambda v+b}{1-a}. \tag{2}$$

The coefficients measure direct physical requirements per unit of machine service. Supplying the $a$ units of intermediate services requires further inputs, and so on. The total embodied requirements are therefore $\lambda/(1-a)$ human hours and $b/(1-a)$ land units. The scalar equation is the simplest production-cost recursion; $c$ prices a flow of machine services, with the production and operation needed to supply it represented by this recipe.

Category $j$ requires $b_j\geq0$ units of land and a continuum of tasks $x\in\mathcal{X}$jx∈Xj​, aggregated in fixed proportions. One human hour completes $\gamma${Lj}(x)>0 task units and one machine hour completes $\gamma_{Mj}(x)\geq0$. Task quantities are included in the integration measure. Every task has a feasible human method, and

$$L_j=\int_{\mathcal{X}$$j}\frac{dx}{\gamma{Lj}(x)}\in(0,\infty) \tag{3}

is the number of hours needed to perform the category’s task list entirely by hand. It is a technological alternative, not the labor actually used. Machine-sector inputs are already included in $c$ and are not counted a second time in this task list. Tasks with $\gamma_{Mj}=0$ are human-required; the machine cost on them is interpreted as infinity.

The category’s competitive unit cost, expressed in land services, is

$$\pi_j(v)\equiv\frac{p_j}{r}=b_j+\int_{\mathcal{X}$$j}\min\left{\frac{v}{\gamma{Lj}(x)},\frac{k(v)}{\gamma_{Mj}(x)}\right},dx. \tag{4}

Here $b_j$ is the land requirement that remains whichever input performs the tasks. Land embodied in machine services enters through $k(v)$. A category with $b_j=0$ can therefore still use land through its machine supply chain.

The fixed factor is land services of a specified quality over the period. Density, location substitution, and production methods affect the land required per unit of housing or other consumption. Their effects enter through the coefficients $b_j$. Other scarce inputs can be treated separately, as in Appendix F.

3.2 The replacement condition

On a task that either input can perform, relative human productivity is $\gamma(x)=\gamma_L(x)/\gamma_M(x)$. It measures the machine hours needed to match one human hour on that task. For example, $\gamma(x)=2$ means one human hour does the work of two machine hours. Replacing that hour costs $c\gamma(x)$, so labor is used when $w\leq c\gamma(x)$. At an active assignment margin $x^\ast$, the two methods cost the same: $w=c\gamma(x^\ast)$. Write $\gamma^\ast=\gamma(x^\ast)$ for relative productivity at this marginal task. Combining cost parity with (2) yields

$$\frac{w}{r}=v=\frac{b\gamma^\ast}{1-a-\lambda\gamma^\ast}, \qquad \frac{c}{r}=\frac{b}{1-a-\lambda\gamma^\ast}, \qquad \gamma^\ast=\gamma(x^\ast). \tag{5}$$

A positive denominator is required for an active task margin at positive prices. Lower machine costs reduce the wage that can be paid at that margin.

The formula distinguishes task automation, which reduces relative human productivity at the relevant task, from recursive automation, which reduces $\lambda$. Holding the other terms, including the marginal task, fixed, either change lowers $v$. For example, $(a,\lambda,\gamma^\ast,b)=(0.5,0.1,3,0.2)$ gives $v=3$ and $c/r=1$. Reducing $\lambda$ to zero gives $v=1.2$ and $c/r=0.4$. The comparison holds the marginal task fixed; in equilibrium, demand and participation also affect its location.

Proposition 1 (Automation and the wage–rent ratio). Consider equilibria with an active machine-contestable margin. If

$$1-a-\lambda\gamma^\ast\geq\varepsilon>0 \quad\text{and}\quad b\gamma^\ast\longrightarrow0,$$

then $v\to0$, so Proposition 2 and Corollary 3 apply under their respective assumptions.

Appendix A extends the result to employed labor facing a machine alternative even when no task is exactly at cost parity. Labor confined to human-required tasks or indispensable machine-production jobs can instead remain outside this bound. Recursive automation alone also leaves a positive wage–rent ratio: at fixed $a,b,\gamma^\ast$, setting $\lambda=0$ gives $v=b\gamma^\ast/(1-a)$. A vanishing ratio requires sufficiently cheap replacement at the tasks that continue to constrain the wage.

Task capability constrains the replacement wage, while the shape of the schedule affects assignment and labor demand. Appendix B solves these jointly with household participation for a strictly sloped schedule. The example has a unique interior equilibrium under stated conditions and admits a sequence with $v\to0$.

4 Purchasing power and factor income

4.1 Category prices

The wage–rent ratio determines how the cost of scarce inputs compares with labor income. Every unit of category $j$ incurs the land bill $rb_j$. Firms can meet its remaining requirements with $L_j$ human hours, or use machines wherever they are cheaper. These two observations bound the category price.

Proposition 2 (Purchasing power). For each category in (4),

$$b_j\leq\frac{p_j}{r}\leq b_j+vL_j. \tag{6}$$

For $b_j>0$, this implies

$$\frac{v}{b_j+vL_j}\leq\frac{w}{p_j}\leq\frac{v}{b_j}. \tag{7}$$

Along any sequence of equilibria with $v\to0$, wage purchasing power therefore tends to zero for every category whose $b_j$ is bounded away from zero. If $b_j=0$, then $w/p_j\geq1/L_j$ at every equilibrium. A uniform bound on $L_j$ gives a positive lower bound along a changing-technology sequence.

The lower price bound reflects the irreducible land requirement; the upper bound reflects the available human method. Task assignment can take any cost-minimizing form, including human-required tasks. The limiting result requires the land bill to remain large relative to the wage: more generally, $v/b_j\to0$ is sufficient. Density or substitution that reduces $b_j$ fast enough can preserve purchasing power. Appendix A gives the proof.

For a category with $b_j=0$, the bound $1/L_j$ is set by human productivity. Actual purchasing power can move above it as technology and assignment change. Household consumption also includes spending financed by ownership and transfers; the proposition measures the part that wages alone can purchase.

4.2 Uniform relative productivity

Suppose every category task is contestable, relative human productivity is the same constant $\bar\gamma$, and the equilibrium wage is at cost parity, $w=c\bar\gamma$. Each task then costs its human-hours requirement times $w$, irrespective of who performs it. Thus

$$p_j=wL_j+rb_j, \qquad \frac{w}{p_j}=\frac{1}{L_j+b_j/v}, \qquad v=\frac{b\bar\gamma}{1-a-\lambda\bar\gamma}. \tag{8}$$

This is equality at the upper price bound in (6). For $b_j=0$, wage purchasing power is exactly $1/L_j$. For $b_j>0$, its departure from that benchmark depends on the land intensity $b_j/L_j$. The land bill exceeds the human-hours term when $v<b_j/L_j$. Categories with greater intensity cross that point first as $v$ falls.

Uniform relative productivity gives a positive wage–rent ratio when $\bar\gamma>0$. As $\bar\gamma\to0$, the ratio tends to zero under Proposition 1’s denominator condition. Cost parity describes an employed wage when workers accept it and the resulting allocation clears markets. Section 5 introduces the household participation condition.

4.3 Factor income

Let $Y_j$ be category output, $D$ the human hours used directly at final tasks, $M$ their machine-service demand, and $X$ gross machine-service production. Machine clearing requires $(1-a)X=M$. Let $h$ be land consumed directly by households. Full land clearing and total employment are

$$T=h+\sum_j b_jY_j+bX, \qquad N_a=D+\lambda X. \tag{9}$$

Figure 1. The exact flat-capability case. The vertical axis is $L_jw/p_j=[1+(b_j/L_j)(r/w)]^{-1}$. Each curve uses the labeled illustrative land intensity. Higher-intensity categories lose purchasing power sooner as the wage buys less land. These are theoretical curves, not estimates. With an arbitrary task schedule, the same curves are lower bounds on normalized purchasing power; for $b_j>0$, the upper bound also tends to zero as $r/w\to\infty$.

Multiplying competitive unit costs by output and using the machine-sector zero-profit condition gives final expenditure

$$\mathcal I\equiv\sum_j p_jY_j+rh=wN_a+rT. \tag{10}$$

Intermediate purchases cancel from final income, leaving payments for their embodied labor and land. Total employment includes both sources of human hours in (9); when $\lambda=0$, final-task employment $D$ remains.

Corollary 3 (Factor shares). Under (10), land’s income share is

$$s_R=\frac{rT}{\mathcal I}=\frac{1}{1+vN_a/T}. \tag{11}$$

If $T$ is fixed and positive, total human hours are bounded, and $v\to0$, then $s_R\to1$ and labor’s income share tends to zero.

As land becomes expensive relative to an hour of work, its fixed stock receives a larger share of income. Bounded hours are sufficient for this result even if some human work persists. The factor-share expression applies to the two-primary-factor flow economy; interest, profits, or additional primary factors would enter as separate income terms.

5 Household participation

Workers compare the consumption available in and out of employment. Choose a produced consumption good with price $p$ and write $\omega=w/p$, $q=r/p$, so $\omega=qv$. Household $i$ receives nonlabor resources $m_i$ in goods units. Working and exiting require $h_W$ and $h_E$ units of housing or land services. Consumption in the two states is

$$g_W=m_i+\omega-qh_W, \qquad g_E=m_i-qh_E. \tag{12}$$

Work gives utility $u(g_W)-\chi_i$ and exit gives $u(g_E)$, with $u'>0$, $u''<0$, and $\chi_i>0$ the net utility cost of working. This simple comparison holds the housing requirement within each state fixed; it can include household production or the value of non-work time through the state-specific resources and utility terms. We consider a feasible interior comparison with positive consumption. If the goods benchmark has $b_0=0$ and bounded $L_0$, Proposition 2 gives $q\geq1/(vL_0)$: the wage–rent limit also makes land expensive relative to that goods benchmark.

Define the reservation wage $s_i(m_i,q)$ in goods units by

$$u(m_i+s_i-qh_W)-\chi_i=u(m_i-qh_E). \tag{13}$$

The worker participates when $\omega\geq s_i$. At indifference, the extra consumption financed by the reservation wage compensates for the utility cost of work.

Let $t_i$ be the household’s land claim, with $\sum_i t_i=T$. A proportional tax $\tau$ on pure rent returned equally per person changes that claim to

$$\widetilde t_i=(1-\tau)t_i+\tau\frac{T}{N}, \qquad m_i=z_i+q\widetilde t_i, \tag{14}$$

where $z_i$ denotes other nonlabor resources in goods units. The following comparative static holds $z_i$, the tax rate, and the within-state housing requirements fixed.

Proposition 4 (Reservation wages). At an interior reservation wage, $s_{i,m}>0$ and

$$\frac{ds_i}{dq}=(h_W-h_E)+s_{i,m}(\widetilde t_i-h_E). \tag{15}$$

With equal housing requirements $h_W=h_E=h$, rising rent lowers a net renter’s reservation wage when $\widetilde t_i<h$, leaves it unchanged when $\widetilde t_i=h$, and raises it for a net owner. At fixed prices, increasing $\tau$ raises the reservation wage of a household with $t_i<T/N$.

With equal housing needs, the mechanism is an income effect. The same rent bill is paid in either state, but it leaves a renter with less consumption in exit. Extra consumption from working becomes more valuable, so a smaller wage compensates for the cost of work. With linear $u$, $s_{i,m}=0$ and the common housing bill cancels exactly. Unequal requirements add the direct cost difference in (15).

For example, with $u(g)=\log g$ and equal housing needs,

$$s_i=(e^{\chi_i}-1)\left[z_i+q(\widetilde t_i-h)\right], \tag{16}$$

whenever exit consumption is positive. Higher rent reduces the bracket for a net renter. A dividend increases it for someone receiving more in shared rent than they lose in taxed ownership income. The dividend can therefore strengthen the ability to refuse poorly paid work.

These are local responses at given wages, other prices, and nonlabor resources. Equilibrium participation also depends on how those quantities adjust. The logarithmic example requires positive exit consumption. At its feasibility boundary, the household must draw on another arrangement, such as shared housing, family resources, or public support. Several feasible arrangements can produce kinks in the reservation wage as the preferred one changes.

This gives a precise economic interpretation of enclosure. Losing access to a viable commons or to housing supplied below market rent can reduce resources available without paid work. It can lower reservation wages through the same budget channel as a rent increase. Family support and public provision need not be permanent floors: someone pays for their scarce inputs. Their effect on participation depends on the resources they make available and the conditions of access.

6 Rent taxation and transfers

6.1 Taxation and participation

Equation (11) makes ownership increasingly consequential when $v$ falls. A uniform dividend financed by pure rents changes effective claims according to (14). At a given gross rent, the person receives

$$(1-\tau)rt_i+d, \qquad d=\frac{\tau rT}{N}. \tag{17}$$

Adding these receipts across people returns $rT$: the payment redistributes rent income rather than creating it.

Two properties motivate this instrument. First, a tax on the pure ownership rent of a fixed, fully available stock does not reduce that stock. Users still allocate it at gross rental prices. This requires that owners cannot reduce its effective supply through withholding, quality choices, or avoidance, and that the tax excludes returns to produced improvements. Demand and the pattern of land use may nevertheless change after redistribution; neither the monetary tax base nor gross rents must be constant. The efficiency rationale rests on this fixed supply of the taxed factor.

Second, a payment made in both work and exit has no benefit-withdrawal penalty. If a program pays $d_W$ in work and $d_E$ in exit, the household comparison is exactly

$$\omega+(d_W-d_E)\ge s_i(m_i+d_E,q). \tag{18}$$

Equal payments set the differential term to zero; they still increase resources on the right. For the net recipients in Proposition 4, participation may fall because exit has improved. Improving the ability to refuse employment is one purpose of the transfer.

Appendix C gives an exact allocation-neutral benchmark: no labor is used under the policies compared, all produced output is competitively supplied from land through machines, and households have identical homothetic demands. Aggregate demand then depends on total rent income rather than its distribution. Outside that scope, redistribution can affect work, demand, and allocation. The fixed-factor tax rationale remains useful, but it does not by itself settle optimal policy during the transition.

6.2 Subsistence coverage

Let a specified per-person subsistence bundle require $g_s>0$ units of a produced good and $h_s>0$ units of direct land services. Its price is $P_s=pg_s+rh_s$. Full capture of land rent finances the fraction

$$\kappa=\frac{rT}{NP_s}=\frac{qT}{N(g_s+qh_s)}. \tag{19}$$

For fixed physical requirements, $\kappa$ rises with $q$ and approaches $T/(Nh_s)$. Full coverage at a finite $q$ is possible if and only if $T>Nh_s$, and then requires

$$q\ge\frac{Ng_s}{T-Nh_s}. \tag{20}$$

These are budget conditions for a specified bundle at equilibrium prices, not a guarantee that a tax will leave those prices unchanged. The denominator includes the produced good’s embodied land through its price. If every bundle also has an irreducible direct production-land requirement, it must be included in $h_s$ for the physical ceiling to be interpreted that way.

The same increase in land’s relative price can therefore make a net renter’s exit less affordable and make an equal share of rent buy more of the target bundle. Whether the fiscal replacement arrives before independent exit is lost depends on ownership, household resources, and available land per person. No general ordering follows from automation alone. Appendix D gives the corresponding threshold in the log-utility household example and states the limited payroll-tax comparison.

Table 1. The same earnings series under three deflators, 1964–2024

Source: the authors’ calculations using average hourly earnings for production and nonsupervisory workers and the corresponding consumer-price indexes. Values are rounded. The ratio of the durables and shelter growth factors is $377/79\simeq4.77$. The comparison uses complete calendar years from 1964.

7 Historical and empirical context

7.1 Historical context

Historical changes in wages combine changes in production with changes in access to subsistence. Where machines can perform few tasks, land access, labor scarcity, institutions, and demographic adjustment largely determine the opportunities available to workers. Bouscasse, Nakamura, and Steinsson (2025) find that English productivity growth preceded the sustained escape of wages from population pressure. The welfare ratios of Allen (2001) and Clark (2005) document workers’ purchasing power over historical subsistence baskets.

Industrialization greatly enlarged the set of machine-contestable physical tasks while leaving many other activities to people. It also required human work in the production and operation of machinery. Neither displacement nor wage growth was immediate or uniform; Crafts (2022) documents the slow early advance of real wages. Differences in the machine alternatives available for trained and untrained work also affected the distribution of these gains.

Computerization then reached many routine tasks. The task evidence of Autor, Levy, and Murnane (2003), and the polarization studied by Autor and Dorn (2013), make this a more useful description than treating technology as a uniform increase in labor productivity. AI is a candidate extension into further tasks and into the work supplying machines themselves. The balance between this expansion and the creation of new human tasks determines whether the replacement conditions in Proposition 1 apply.

7.2 U.S. evidence

Table 1 reports U.S. purchasing-power comparisons using the September 2026 data vintage. Durables and shelter illustrate different combinations of reproducible tasks and scarce-input requirements; the indexes do not measure those requirements directly. Food combines acreage, yields, processing, and distribution, and falls between the reported endpoints.

Table 2. Two monetary labor linkages in consumption

Source: the authors’ financing calculations and production-account series. Financing uses the 2004–2023 income-source window. The production benchmark covers 1997–2023. Ranges are percentages under alternative allocation rules. The production measure is a monetary contribution, not physical hours, and does not decompose imported content into foreign labor and capital. See Appendix G.

The comparison establishes relative-price divergence. At either date,

$$\frac{w/P_{\mathrm{durables}}}{w/P_{\mathrm{shelter}}}=\frac{P_{\mathrm{shelter}}}{P_{\mathrm{durables}}}. \tag{21}$$

The 4.8× change therefore measures the movement in relative prices. Sectoral productivity, trade, housing supply, quality adjustment, and other forces can contribute. Shelter prices also combine structures and site services. They are not a direct observation of the model’s $r$.

Site-rent coverage measures the share of a subsistence bundle that rent income could finance. The specification grid gives a 2025 coverage estimate of 0.33, with a range [0.18, 0.59], against roughly 0.05 in the 1950s. These are sensitivity ranges across constructions, not confidence intervals or proven lower bounds. The exercise depends on separating land from structures, converting asset values into service flows, and specifying a subsistence bundle. It measures a candidate funding scale rather than the equilibrium revenue of an implemented tax.

The third measurement distinguishes the resources financing consumption from the labor payments embedded in producing it. The financing account assigns current and intertemporal purchasing power to their sources. The production account traces domestic factor payments through intermediates. They describe two sides of expenditure; their gap is not a measure of technological displacement.

Labor-payment content combines physical labor requirements with wages and the composition of production. Samuels and Senel’s (2026) BEA decomposition records imports separately and attributes domestic final demand to factor payments. Changes in its labor component can reflect technology, demand, wages, or international sourcing. Estimating physical human effort, including the machine-sector coefficient $\lambda$, requires additional hours and productivity data. Appendix G describes the construction and the model-based long-run extensions.

7.3 Empirical implications

The production test needs measures of technical replacement and of the scarce inputs that remain after substitution. Useful observations would combine machine-service prices, task-level relative productivity, physical hours in the machine supply chain, and category requirements measured independently of the price outcome. The prediction is that wage command over scarce-input services falls where technical replacement becomes cheap, conditional on the remaining production requirements.

The household test is different. Conditional on wages, other prices, and resources, rent shocks should change reservation wages differently for net renters and owners, with the sign depending on work–exit housing needs. Exogenous changes in housing costs or access, accompanied by ownership and transfer data, would distinguish the proposed budget channel from a common time trend. Participation alone is not enough: observed labor supply also reflects wage changes, unemployment, borrowing constraints, and household sharing.

Finally, the income identity suggests checking whether reduced wage command over land accompanies a rising share of appropriately measured pure rents. Existing labor-share and housing-wealth research motivates the question (Karabarbounis and Neiman 2014; Rognlie 2015; Knoll, Schularick, and Steger 2017). Housing wealth, structure returns, and pure site rent must remain separate objects. Testing the land-only limit also requires measuring the contribution of other income components.

8 Discussion

8.1 Human tasks, substitution, and capital

Human-required tasks are the strongest production-side exception. If people remain indispensable in services that receive a persistent expenditure share, the wage need not lose command over land. This is the Baumol (1967) mechanism: expenditure concentrates on activities whose costs do not fall with automation elsewhere. It does not contradict Proposition 2; it can prevent its premise $v\to0$. If only a small group can supply the remaining human services, aggregate labor income may also conceal a highly unequal distribution. Appendix E gives the conditional expenditure result and a finite-population concentration example.

Institutions can change both the price of labor and the feasible assignment. With a labor-cost wedge $\mu(x)>1$, a contestable task uses labor only while $\mu(x)w\le c\gamma(x)$. Acemoglu and Restrepo (2026) show how automation can target jobs paying rents. The persistence of wage premia also depends on adoption costs, bargaining, worker responses, and demand. Rules reserving tasks for people act directly on the feasible set. Preferences for human provision require credible provenance; a simple enforcement calculation is given in the appendix.

Supply expansion acts on the other side. Building more densely, improving yields, expanding power supply, and substituting among locations can reduce the scarce-input service needed per unit of consumption. Physical land remains fixed, but $b_j$ need not. If these changes make $b_j$ fall as quickly as $v$, wage purchasing power need not collapse. With flexible consumption, households may also substitute away from expensive categories. A fixed essential bundle makes that escape harder; a CES example in Appendix E states the relevant elasticity condition without treating all necessities as fixed physical proportions.

Capital returns also depend on the time horizon. Reproducible capital can earn a competitive return when production takes time, and temporary shortages can generate rents while capacity adjusts. Patents and other legal rights can sustain institutional scarcity. An extension with these features must distinguish the returns to investment and innovation from the rent of a permanently fixed factor. Appendix F sets out the corresponding production-network and user-cost expressions.

8.2 Further research

A dynamic extension would connect the relative-price results to the timing of automation, capital formation, and institutional adjustment. Adoption may occur in waves, and wages, rents, and policy may adjust at different speeds. Monetary policy is relevant where falling produced-goods prices coincide with rising prices of scarce necessities. Its effects depend on the interaction between employment, investment, and asset capitalization.

The household comparison also needs to be extended to borrowing, family sharing, migration, and household formation. A worsening exit budget may keep people in poorly paid work, while falling wages can cause others to withdraw. If wages buy ever more manufactured goods but ever less housing, energy, and care, forming a household may become harder even while productivity rises. Delayed household formation, adult children remaining with their parents, and declining fertility are possible outcomes to investigate, not implications already established by the one-period participation model.

Tax policy outside the limiting case remains open. During the transition, labor remains part of production, site rents alone may not finance a full transfer, and interest and other income sources remain material. A policy comparison must evaluate consumption, income, corporate, land-value, and other pure-rent taxes using their incidence and supply effects. The financing accounts can help describe exposure but do not rank these instruments by welfare cost. Sovereign wealth funds provide another candidate instrument. A country with little domestic resource rent can acquire foreign claims, although doing so requires saving, bears risk, and does not create resources for the world as a whole.

Finally, structuring a society that no longer orbits around labor is not a trivial problem. Work provides more than a wage. It provides structure to the day, social contact, status, a sense of contribution, and an understanding of the complexity of the real economy. An income transfer can replace a wage; it cannot by itself replace all of those functions. Some may emerge naturally through family, community, education, art, care, or voluntary work. Others may have to be deliberately nurtured. The economic problem may be how to distribute purchasing power without labor. The larger human problem is how to distribute purpose, status, and a meaningful place in society once labor is no longer the institution that provides them.

9 Conclusion

AI may improve machine performance at final tasks and reduce the human work required in software generation, chip design, datacenter operation, logistics, and research. When these changes make replacement sufficiently cheap at employed contestable tasks, the wage–rent ratio falls. The distributional consequences depend on persistent scarce-input requirements and on the demand for work that remains human-required.

The model connects this technological change to consumption and participation. A wage can buy more manufactured goods while buying less housing. Ownership and transfers determine how far households can offset that loss. Sharing pure rents gives households a claim on the scarce inputs that continue to receive income, and can improve their ability to leave employment. During the transition, the available rent base and the cost of subsistence determine how much of that outside option it can finance.

The argument also motivates a broader conjecture about the dependence of consumer demand and social organization on wage income.

The remaining empirical question is whether the machine-production recursion is paying a smaller share of cost to wages and a larger share to terminal-input rents, and whether the same scarcity is eroding the value of exit. Although it requires a significant re-framing of the last 70 years of world economic history, we consider the answer to be obvious.

The intuitive reading of that re-framing, and of what our paper may conjecture, is this: labor may currently be behaving as a kind of grand market distortion, and may have been doing so for many decades. Our economies have almost no mechanism to limit the supply of labor. With other goods, like oil six years ago or capital when inflation rates near zero, the economy can signal its desire to have supply lowered. However, as wages represent the majority of consumer demand, this mechanism is unavailable for labor itself. Instead, the economy dutifully creates jobs, at cost. Either with significant wedges, the “Bullshit Jobs“ dynamic, or near the subsistence and dependency floors, the “gig economy“. A K-shaped graph in many of our measures of economic health then should come as no surprise. The hollowing out of our middle class, as no surprise. Youth unemployment, adult children living with their parents, and the loss of the atomic family, as no surprise. The difficulty of governments to maintain fiscal restraint, as no surprise. The stagnant growth of developed economies despite significant advances in computing, as no surprise.

This work represents in many ways the synthesis of centuries of economic thought. The classical economists may not have been wrong, they just did not know what technology could do to the wage. George was not wrong, he simply wrote his work at a time when his policies were needed the least. Keynes was not wrong, but he did not foresee the change that cognitive automation would bring. Rather, we may now be able to bring all these ideas together into a substantive wage equation.

Appendix A. Proofs

A.1 Purchasing power and factor income

Proof of Proposition 2. Every task cost in (4) is nonnegative and no larger than $v/\gamma_{Lj}(x)$. Integrating gives (6). Since $w/p_j=v/(p_j/r)$, inversion gives (7) for $b_j>0$. If $b_j\ge\underline b_j>0$ along a sequence, $0\le w/p_j\le v/\underline b_j\to0$. When $b_j=0$, $p_j/r\le vL_j$, so $w/p_j\ge1/L_j$. These arguments also allow changing task sets and productivities, subject to the stated bounds. In the flat parity case each task cost equals $v/\gamma_{Lj}$, giving (8) exactly. $\square$

Proof of the income identity and Corollary 3. For any cost-minimizing assignment, summing final-task payments gives

$$\sum_j p_jY_j=wD+cM+r\sum_j b_jY_j.$$

Since $M=(1-a)X$ and $(1-a)c=\lambda w+br$, the middle term is $cM=w\lambda X+rbX$. Adding household land spending $rh$ and applying (9) gives $\mathcal I=wN_a+rT$. Dividing the land bill by this expression yields (11). With $N_a\le\overline N$ and $T>0$ fixed, $0\le vN_a/T\le v\overline N/T\to0$. The assertion follows. In particular, $\lambda=0$ gives $\mathcal I=wD+rT$, and only a zero labor bill yields the exact land-only identity. $\square$

A.2 Replacement and its limiting conditions

Proof of Proposition 1. An active tied task satisfies $v=k(v)\gamma^$v=k(v)γ∗, hence $(1-a-\lambda\gamma^$)v=b\gamma^. Positivity of $b,\gamma^$,v implies a positive denominator. Under the stated assumptions, $0<v\le b\gamma^*/\varepsilon\to0$. $\square$

The result also holds without an exactly indifferent task. At each equilibrium, suppose at least one task employing labor can be performed by a machine. Let $\Gamma$ be an upper bound on relative human productivity $\gamma(x)$ at that employed task. This auxiliary bound may vary along the equilibrium sequence; it need not bound every task in the economy. Cost minimization gives $v\le k(v)\gamma(x)\le k(v)\Gamma$. If $1-a-\lambda\Gamma\ge\varepsilon>0$, rearranging yields $v\le b\Gamma/(1-a-\lambda\Gamma)\le b\Gamma/\varepsilon$. Thus $b\Gamma\to0$ is sufficient for $v\to0$.

Writing $g=\gamma^*$ and $\Delta=1-a-\lambda g$, the local comparative statics of the parity ratio are

$$\frac{\partial v}{\partial\lambda}=\frac{bg^2}{\Delta^2},\qquad \frac{\partial v}{\partial g}=\frac{b(1-a)}{\Delta^2},\qquad \frac{\partial v}{\partial b}=\frac{g}{\Delta}. \tag{A.1}$$

These derivatives hold marginal capability fixed. A full equilibrium comparative static also includes movement of the task margin. For example, if $a$ is bounded away from one, $b$ and $\lambda$ are bounded, and $g\to0$, then the sufficient conditions eventually hold. Conversely, $\lambda\to0$ at fixed positive $g,b$ leaves $v\to bg/(1-a)>0$. When every employed task is human-required, there is no finite machine alternative to supply this bound.

A.3 The household derivative

Proof of Proposition 4. At the reservation wage write $g_E=m-qh_E$ and $g_W=m+s-qh_W$. Equation (13) and $\chi_i>0$ imply $g_W>g_E$. Differentiating with respect to $m$ gives

$$u'(g_W)(1+s_m)=u'(g_E),\qquad s_m=\frac{u'(g_E)}{u'(g_W)}-1>0.$$

At fixed $m$, differentiation with respect to $q$ gives

$$s_q=h_W-\frac{u'(g_E)}{u'(g_W)}h_E=(h_W-h_E)-s_mh_E.$$

Since $dm/dq=\widetilde t_i$, the total derivative is (15). At fixed prices, $\partial m/\partial\tau=q(T/N-t_i)$, so $\partial s/\partial\tau=s_mq(T/N-t_i)$. Linear utility gives $s_m=0$. For logarithmic utility the defining equation gives $g_W=e^{\chi_i}g_E$, yielding (16) when the housing requirements coincide. $\square$

The calculation assumes an interior indifference point exists. With a bounded utility function and a large work cost it need not; with infeasible exit consumption the chosen exit arrangement must change. If other income moves with rent, the total derivative instead uses $dm/dq$ in place of $\widetilde t_i$. This is especially relevant to family transfers and publicly supplied housing.

Appendix B. Equilibrium with heterogeneous tasks

This example determines task assignment, output, and participation jointly. Human hours enter both final production and machine production, and household budgets clear goods and land markets.

There is a mass $N$ of one-person households. Each owns $T/N$ units of land and requires the same fixed housing service $h$ in work and exit. Let

$$S=T-Nh>0,\qquad A=S/N.$$

Households have utility $\log g-\chi n$, where $n\in{0,1}$ and $\chi$ has a continuous distribution function $F$. A participant supplies one hour. There are no other resources. Normalize $r=1$. After housing, each household has nonlabor purchasing power $A$; at wage $v$ its goods consumption is $(A+vn)/p$. Work is optimal exactly when

$$\chi\le\log(1+v/A),\qquad n_S(v)=NF\bigl(\log(1+v/A)\bigr).$$

The common goods price cancels from this utility difference. Individual goods demands still depend on it and will clear the output market below.

The final good has no direct land requirement and requires one unit of each task on $[0,1]$ per unit of output. Set $\gamma_L(x)=1$, $\gamma_M(x)=1/\gamma(x)$, where $\gamma$ is positive, continuous, and strictly increasing. Assume $1-a-\lambda\gamma(1)>0$. At an interior threshold $x$, machines perform tasks below $x$ and people those above it. Define

$$J(x)=\int_0^x\gamma(t),dt,\qquad v(x)=\frac{b\gamma(x)}{1-a-\lambda\gamma(x)},\qquad k(x)=\frac{b}{1-a-\lambda\gamma(x)}. \tag{B.1}$$

All residual land $S$ goes into machine production. Land and machine clearing therefore imply

$$X=\frac Sb,\qquad Y(x)=\frac{(1-a)S}{bJ(x)},\qquad n_D(x)=\frac Sb\left[\lambda+\frac{(1-a)(1-x)}{J(x)}\right]. \tag{B.2}$$

The first component of $n_D$ is machine-sector labor; the second is final-task labor.

Lemma B.1 (Interior equilibrium in the example). If

$$NF\bigl(\log(1+v(1)/A)\bigr)>\frac{\lambda S}{b}, \tag{B.3}$$

there is a unique interior threshold $x^$\in(0,1) satisfying $n_D(x^$)=n_S(v(x^)). Together with (B.1)–(B.2) and*

$$p=v(x^$$)(1-x^)+k(x^)J(x^), \tag{B.4}

it gives a competitive equilibrium. The normalized prices, aggregate quantities, and participation mass are unique within this interior example.

Proof. As $x\downarrow0$, $J(x)\downarrow0$ and $n_D(x)\to\infty$. The function $(1-x)/J(x)$ is strictly decreasing, since its derivative is

$$-\frac{J(x)+(1-x)\gamma(x)}{J(x)^2}<0.$$

At $x=1$, $n_D(1)=\lambda S/b$. Meanwhile $v(x)$ is strictly increasing and $n_S(v(x))$ is continuous and nondecreasing, bounded by $N$. Condition (B.3) makes demand strictly below supply at $x=1$. The intermediate value theorem and strict monotonicity give exactly one crossing.

At that crossing, assignment is cost-minimizing because $v/k=\gamma(x^*)$. The recursion prices machines, and (B.2) clears machines, labor, and land. For the final good, multiply (B.4) by output:

$$pY=vY(1-x^$$)+k(1-a)S/b=v\bigl[Y(1-x^)+\lambda S/b\bigr]+S=vn_S+S.

Aggregate household goods demand is $(NA+vn_S)/p=(S+vn_S)/p=Y$. Housing demand is $Nh$, with its rent paid from ownership income. All household budgets and markets therefore clear. A continuous $F$ gives zero mass at the participation tie. $\square$

For a numerical instance, take $N=4$, $T=10$, $h=1$, $a=0.3$, $b=0.4$, $\lambda=0.05$, $\gamma(x)=0.2+0.8x$, and $\chi$ uniform on $[0,1]$. The solution has $x^*\simeq0.96791$, $v\simeq0.59841$, $Y\simeq18.47540$, and $N_a\simeq1.34285$: about $0.59285$ hours at final tasks and $0.75$ in machine production. The accompanying check script compares this output with an independently specified discretized production optimization, including both sources of labor demand.

The example also supplies an equilibrium sequence behind the limiting result. Set $\lambda=0$ and $\gamma_\eta(x)=\eta(1+x)$, with $F$ uniform on $[0,1]$. For every $\eta>0$, condition (B.3) holds, so an interior equilibrium exists. Its wage satisfies $0<v\leq2b\eta/(1-a)\to0$. Relative capability remains twice as high at the last task as at the first: uniform capability is unnecessary. Participation tends to zero through $n_S(v)$, while the price and income bounds apply throughout. Households in this example are net owners after housing, so their participation falls as wage income becomes less valuable relative to ownership income.

Appendix C. Redistribution with homothetic demand

Consider the following specialization of production and demand. Let $\lambda=0$, let all final tasks be performed by machines, and let the good require $m>0$ units of machine services per unit, with no direct land input. No labor is supplied or used under the policies compared. Preferences and resources are assumed to support this participation choice under each policy.

All households have preferences $(1-\alpha)\log g+\alpha\log h$, with $0<\alpha<1$, and positive incomes from land claims, including redistributed claims. The machine recursion and final-good price give

$$\frac{p}{r}=\frac{bm}{1-a},\qquad \mathcal{I}=rT.$$

Every household spends fraction $\alpha$ of income on direct land services. Summing demands therefore gives

$$T_H=\alpha T,\qquad T_P=(1-\alpha)T,\qquad Y=\frac{(1-a)(1-\alpha)T}{bm}. \tag{C.1}$$

The allocation and relative price are unique. A tax and dividend that redistribute $rT$ without changing its total leave these aggregates unchanged. Individual consumption changes with income. At $\tau=1$ the receipts become equal per person; for other rates they reflect the after-tax ownership distribution.

This is exact allocation neutrality under common homothetic demand and an unchanged absence of labor. If tastes differ, housing has an income-dependent budget share, labor becomes attractive after an income loss, or other factors earn income, the conclusion need not hold. The general result in the main text is instead that a pure-rent tax has no physical fixed-factor supply response and an equal payment has no benefit-withdrawal differential.

Appendix D. Fiscal comparisons during the transition

D.1 Conditionality and local tax incidence

Equation (18) follows by writing $d_W=d_E+(d_W-d_E)$ and using the definition of $s_i$. At a fixed exit-resource level $m_i+d_E$, an in-work payment differential lowers the required gross wage one-for-one and an out-of-work differential raises it. Changing an equal payment instead moves that resource level. A claim of unchanged participation requires either a compensated comparison or assumptions eliminating the income effect.

For a small payroll tax in a partial-equilibrium labor market, let $\varepsilon_D,\varepsilon_S$ be the absolute labor-demand and supply elasticities. The share locally borne by workers is $\varepsilon_D/(\varepsilon_D+\varepsilon_S)$. A flat replacement schedule with a machine rental unaffected by the tax supplies a limiting case of elastic demand. In the full model, however, the rental, output, task assignment, and machine-sector labor costs can respond. An incidence estimate does not by itself identify the slope of $\gamma$.

The familiar circularity example is correspondingly narrow. If every person supplies one hour, the gross wage is fixed at $\bar w$, and wage-tax receipts are returned equally, disposable income is $(1-\tau_w)\bar w+\tau_w\bar w=\bar w$. With unequal earnings, partial participation, or an endogenous gross wage, that cancellation is not a general neutrality result. It is a useful warning about financing a common dividend from an unchanged common wage base.

D.2 Coverage and the timing of independent exit

Differentiating (19) gives

$$\frac{\partial\kappa}{\partial q}=\frac{Tg_s}{N(g_s+qh_s)^2}>0,\qquad \lim_{q\to\infty}\kappa=\frac{T}{Nh_s}.$$

Rearranging $\kappa\geq1$ yields (20). At $T=Nh_s$, the limiting coverage is one but it is never reached at a finite $q$ when $g_s>0$. At $T<Nh_s$, even the limiting coverage is below one.

For the log-utility household with common housing need $h$, an independent exit arrangement is feasible while $z_i+q(\widetilde t_i-h)>0$. If $z_i>0$ and $\widetilde t_i<h$, its feasibility boundary is

$$q_{\mathrm{exit},i}=\frac{z_i}{h-\widetilde t_i}.$$

With the tax regime and other resources held fixed, that arrangement remains feasible until the full-capture coverage threshold is reached exactly when $q_{\mathrm{exit},i}\geq Ng_s/(T-Nh_s)$, assuming $T>Nh_s$. This compares two thresholds along the specified price path; introducing the dividend changes $\widetilde t_i$ and therefore the exit threshold itself. Other feasible exit arrangements can replace the independent one.

If the target bundle also requires $n_s$ hours of human-only service at wage $w_H$, its coverage is

$$\kappa_H=\frac{qT}{N(g_s+qh_s+n_sw_H/p)}.$$

At given prices, this is below the corresponding bundle without that requirement. Neither its limiting value nor its equilibrium response is determined by $q$ alone when $w_H/p$ also changes.

Appendix E. Human-required services and substitution

E.1 Human inputs and expenditure shares

Suppose a service requires $L_H>0$ human-only hours and $b_H\geq0$ direct land services per unit. If all other task costs tend to zero in a fixed common numeraire, while $w_H\to\bar w_H>0$ and $r\to\bar r>0$, then

$$p_H\longrightarrow\bar w_HL_H+\bar r b_H,\qquad \text{labor share within the service}\longrightarrow\frac{\bar w_HL_H}{\bar w_HL_H+\bar r b_H}.$$

The labor share tends to one only if the remaining land bill is zero. Vanishing contestable-task costs require an integrable bound or another condition justifying the limiting integral; pointwise machine improvement alone is insufficient.

To state the expenditure argument separately, take a CES expenditure function with weights $\alpha_k>0$, $\sum_k\alpha_k=1$, and elasticity $\sigma$. Category $k$ receives expenditure share

$$e_k=\frac{\alpha_kp_k^{1-\sigma}}{\sum_\ell\alpha_\ell p_\ell^{1-\sigma}} \tag{E.1}$$

for $\sigma\neq1$; Cobb–Douglas gives $e_k=\alpha_k$ at $\sigma=1$. If a machine-only good’s price tends to zero while human-service and land-service prices have positive finite limits, then for $\sigma<1$ its expenditure share tends to zero. Spending remains split between human services and land in proportion to their limiting weighted price terms. For $\sigma>1$, the cheap category takes all spending; with $\sigma=1$, the shares stay at their taste weights. These are conditional demand limits, not a determination of the human wage.

In the two-category version with goods price normalized to one and land price $q$, the land-service expenditure share is

$$e_R(q)=\frac{\alpha q^{1-\sigma}}{(1-\alpha)+\alpha q^{1-\sigma}}.$$

It tends to one for $\sigma<1$ and zero for $\sigma>1$ as $q\to\infty$. A rising observed housing share does not identify $\sigma$ without accounting for incomes, changing quality, and other prices.

These calculations explain how a human bottleneck can sustain a positive labor-income share. With a common wage, this can prevent $v\to0$. With different worker types, aggregate labor income is $\sum_iw_iN_{a,i}$; a generalization of (11) replaces $vN_a$ by $\sum_i(w_i/r)N_{a,i}$. A protected minority can retain income while others lose it. No conclusion about the economy-wide median follows from the aggregate share without assumptions about assignment and income distribution.

E.2 Verification and concentration

Preference-based human work requires credible provenance. Suppose a false claimant receives a premium $P$ if not detected, is detected with probability $\nu<1$, loses the premium on detection, and pays penalty $f\geq0$. Assume other revenues and costs equal those of the machine alternative. Nonpositive expected gain requires

$$(1-\nu)P-\nu f\leq0,\qquad P\leq\frac{\nu f}{1-\nu}.$$

The bound depends on the stated detection and penalty structure. Verification costs, reputation, repeated trade, and differences in provision can change the incentive to misrepresent origin. The symbol $\nu$ is a detection probability, distinct from the wage–rent ratio $v$.

For a finite-population concentration example, let $M$ human-service workers receive total income $E_H$. Give everyone the equal share $(1-\psi)E_H/M$, and distribute the remaining $\psi E_H$ to $K<M/2$ stars. The mean is $E_H/M$, while the median is exactly $(1-\psi)E_H/M$, because a strict majority receives only the common amount. It illustrates Rosen’s (1981) concentration mechanism without proving that any particular human-service market has this distribution.

Appendix F. Production networks and other scarce inputs

For several produced goods and services, let $\mathbf{c}$ be their price vector. Fix a set of production techniques and let $\mathbf{A}$ contain produced-input requirements, with rows indexed by outputs, $\Lambda$ direct labor requirements, and $\mathbf{B}$ the requirements of non-produced inputs. Each price equals its intermediate-input bill plus direct factor costs. Competitive prices obey

$$\mathbf{c}=\mathbf{A}\mathbf{c}+\Lambda w+\mathbf{B}\mathbf{r},\qquad \mathbf{c}=(I-\mathbf{A})^{-1}(\Lambda w+\mathbf{B}\mathbf{r}),$$

when $\mathbf{A}\geq0$ has spectral radius below one. The inverse $I+\mathbf{A}+\mathbf{A}^2+\cdots$ traces successive rounds of embodied primary-factor costs. This is the general goods-and-services cost recursion. The machine equation (2) is its one-produced-input case. With alternative techniques, the coefficients describe the cost-minimizing choices at the prices under consideration; they may change when wages or rents change. The corresponding income account cancels intermediate transactions and leaves labor payments plus the rents of the separate primary factors. A single ratio $w/r$ summarizes the result only when there is one such factor or a valid fixed aggregation.

For an individual category the simpler bound needs no such aggregation. If every feasible technique requires at least $b_{jk}>0$ units of scarce input $k$, then $p_j\geq r_kb_{jk}$ and

$$\frac{w}{p_j}\leq\frac{w/r_k}{b_{jk}}.$$

An all-human method with a known bundle of scarce inputs gives the corresponding upper cost bound. If inputs substitute for one another, the minimum feasible requirement, rather than a coefficient from one chosen technique, must supply the lower bound.

Interest requires a separate intertemporal technology. For example, if $P_K$ is the acquisition price of a durable asset, $c=(\rho+\delta)P_K$ is its user cost only under the usual no-expected-capital-gain specification. A recipe for $P_K$ is not automatically the same recipe as one for a flow of services. Introducing that distinction adds the relevant capital income and accumulation equations. The resulting wage and income equations depend on the intertemporal equilibrium as well as on physical production requirements.

Appendix G. Measurement and reproducibility

G.1 Data and construction

The U.S. estimates use the September 2026 data vintage. The wage comparison combines average hourly earnings for production and nonsupervisory workers with CPI indexes for durables, food, and shelter over complete calendar years 1964–2024. The food series is FRED CPIUFDNS. Table 1 reports the endpoints of these comparisons.

The site-rent grid uses Federal Reserve Z.1 household real estate at market value (FRED HNOREMV), residential structures at replacement cost (BOGZ1LM155012665Q), and household or economy-wide scope variants. Asset-residual constructions annualize the estimated land value with the 10-year Treasury yield (GS10) and a yield-plus-150-basis-points variant. Flow constructions apply land shares of $0.30$ and $0.50$ to PCE housing services (BEA DHSGRC1A027NBEA). Population times the specified Orshansky subsistence bundle supplies the denominator. Asset-value residuals, capitalization assumptions, and service-flow shares need not give the same pure-rent concept, so the range is a sensitivity exercise. The fiscal classification places $0.68$ of 2025 tax revenue on labor-income bases; that is a source classification, not an incidence estimate.

The financing account uses annual BEA source distributions ranked by equivalized disposable income over 2004–2023. Spending beyond current disposable income remains in an intertemporal bucket. Attribution of taxes and spending among fungible resources gives the ranges in Table 2. The model-based extension reports 64.2% [54.1, 81.3] for 2025. These bounds are not sampling uncertainty and the extension has weaker status than the income-source window.

The production account draws on the domestic input–output/KLEMS approach of Samuels and Senel (2026). It attributes final demand to domestic labor and capital payments and separately recorded imported content. It does not trace foreign labor through a global production system. The domestic labor-payment series is 50.1% in 1997 and 47.2% in 2023. A five-specification product-composition extension gives 66.4% [61.3, 68.6] in 1950 and 46.2% [45.1, 47.4] in 2025. Those outer values are model outputs, not historical observations of physical labor requirements. In particular, none of these percentages can be read as the physical coefficient $\lambda$ in (2).

Data and empirical code are documented at https://github.com/wilsoniumite/labor. The numerical checks described below cover the theoretical constructions separately from the empirical estimates.

G.2 Numerical checks

The accompanying check_revision.py checks the price bounds with nonflat schedules and human-only tasks, the income identity with both direct and machine-sector labor, the household derivatives, the exact flat-case formula, and the finite-population concentration example. It solves Appendix B’s scalar equilibrium and compares the resulting output with an independent linear program over 2,048 discrete tasks, using separate land, labor, and machine-flow constraints. Numerical examples are illustrations and checks, not proofs of the general claims; the analytical proofs are given above.

The formal results are established by the analytical proofs in these appendices. The accompanying numerical checks provide additional verification of the examples and algebraic identities.

AI use. This paper was developed with assistance from Claude (Anthropic) and ChatGPT (OpenAI). The theory and central claims are the authors’; AI assistance included prose drafting, algebraic manipulation, numerical checks, and preparation of the manuscript. Errors remain the authors’.

Submission supplement: request for independent review

Paper: Pinning the Wage to Scarcity and Technology: Automation, purchasing power, and the rents of non-produced inputs, Johan Båge and Stella Wilson, September 2026.

The authors request a critical assessment of mathematical correctness, economic interpretation, empirical support, contribution relative to existing work, and the most useful next step. This is a working paper, submitted for scrutiny rather than as an established account of the economy. Please read the manuscript and form an initial assessment before using the questions below.

The revised manuscript is the primary submission. Selected plots from the August working draft appear in a separate supplement for review. Those plots have not been recalculated. Their inclusion does not reinstate the older draft's stronger claims. No data files, LaTeX source, numerical check scripts, or Lean project were supplied with the two PDFs for this submission; descriptions of such checks in the manuscript should be evaluated subject to that limitation.

Mathematical correctness

1. Check the production-cost recursion, the active-task replacement condition, and Proposition 1's sufficient conditions for the wage–rent ratio to vanish. Identify any hidden assumptions about positive prices, productive input coefficients, task availability, or movement of the equilibrium assignment margin.
2. Check the category-price bounds in Proposition 2 and distinguish what they prove conditional on a vanishing wage–rent ratio from what establishes that premise. Consider changing land requirements, changing human productivity, zero direct land requirements, and human-required tasks.
3. Verify the final-income identity with both direct final-task labor and labor in machine production. Check the bounded-hours argument and the scope of the two-primary-factor accounting.
4. Check the reservation-wage derivative, including feasibility and the existence of an interior indifference point. Distinguish fixed-price income effects from equilibrium effects, and identify what changes with variable housing consumption, borrowing, household sharing, or different work and exit requirements.
5. Verify the heterogeneous-task equilibrium in Appendix B, including existence, uniqueness within the stated class, market clearing, the numerical example, and the proposed limiting sequence. Supply a counterexample wherever a stated condition is insufficient.
6. Check the precise neutrality conditions, subsistence-coverage algebra and physical interpretation, human-service exceptions, finite-population concentration result, and production-network extension in the remaining appendices.

Economic interpretation and contribution

Assess what is new beyond task-assignment models, input–output cost accounting, classical land-rent arguments, and household income effects. Cite the closest existing results and state exactly which proposition or combination would constitute a defensible contribution.

Evaluate the distinction between a flow-production benchmark and an economy with durable capital, interest, risk, innovation rents, and adjustment costs. Which conclusions survive those additions? Which depend on identifying a fixed input whose service requirement cannot be reduced through substitution or supply expansion?

Assess whether the household mechanism and the production mechanism are jointly demonstrated for economically relevant ownership distributions. In particular, the complete equilibrium example uses equal land ownership, while the reservation-wage argument highlights net renters. Identify the smallest model extension or empirical test needed to connect those cases.

Evidence and reproducibility

Separate descriptive relative-price patterns from tests of the proposed causal mechanism. Check index bases, sample windows, series definitions, quality adjustment, splices, backcasts, and the distinction between shelter prices and pure site rent.

For site-rent coverage, inspect the treatment of land versus structures, capitalization rates, subsistence bundles, and alternative specifications. Distinguish a specification range from a confidence interval and a funding-scale calculation from equilibrium tax revenue.

For consumption financing and production content, distinguish monetary labor payments from physical labor hours, domestic content from imported content, and benchmark observations from model-based extensions. Identify the raw data and code needed for independent reproduction. Do not treat the plotted series as verified measurements of automation.

Conclusions and recommended next step

Evaluate the discussion and closing claims separately from the proved results. State which claims are established under explicit assumptions, which are plausible but untested, which need revision, and which are contradicted by evidence or a counterexample.

Recommend a next step with reasons: a mathematical correction, a narrower contribution claim, a specific empirical design, release of replication materials, consultation with a relevant economist, or submission to a suitable research venue. Identify the one change most likely to improve the paper's credibility without unnecessarily expanding its main text.

Please give page, section, or equation references for substantive criticisms; show derivations or counterexamples where possible; cite primary sources for literature and empirical claims; and state remaining uncertainty. Agreement among reviewers is not itself evidence. Reviews should be performed independently of the submitting agent and should disclose the checks actually completed.

Submission supplement: figures for review

These plots are extracted from the authors' August 2026 working draft, supplied as ssrn-7226858(1)(1).pdf. They supplement the September revised manuscript. Plot contents are unchanged; the captions and interpretation notes below identify the scope in which they are being supplied. Data and plotting code were not provided, so these extractions do not constitute replication or validation.

Figure S1. Earnings under different consumption-price deflators

Source: August draft, Figure 3, PDF page 20. Average hourly earnings for production and nonsupervisory workers are deflated by consumer-price indexes for durables, food, energy, and shelter, with 1950 set to 100 and complete calendar years through 2024. The source caption reports a wage-series splice before 1964 and an energy-price backcast for 1950–1956 based on 1957–1966. The revised manuscript's Table 1 instead compares complete years from a 1964 base; levels in this graph should not be read as that table's 1964-based levels.

Interpretation for review: the approximately 4.8-fold relative divergence over 1964–2024 is a relative-price result because the common wage cancels. The legend's descriptions of durables as “machine-made” and shelter as “land-priced” are informal proxies, not measured model coefficients. Shelter includes structures and services as well as site value. This graph does not identify the proposed wage mechanism.

Figure S2. Estimated site-rent coverage of a subsistence bundle

Source: August draft, Figure 4, PDF page 20. The coverage ratio is aggregate site-rent flow divided by the cost of providing the specified subsistence bundle to the population, for 1953–2025. The central line and shaded range summarize alternative land sources, capitalization rates, and bundle constructions. The source describes residual real-estate valuations converted into service flows using Treasury yields and alternatives based on assigned land shares of housing services.

Interpretation for review: the shaded range is variation across specifications, not a confidence interval. The revised manuscript reports a 2025 central estimate near 0.33 and range 0.18–0.59. The plot describes a candidate funding scale, not the revenue or welfare effect of an implemented tax. It requires independent checks of land–structure separation, capitalization, and the specified bundle.

Figure S3. Consumption financing and domestic labor-payment content

Source: August draft, Figure 5, PDF page 21. The graph is included as an object of review. Its title and legend use “human effort,” but the revised manuscript identifies the production series as domestic monetary labor-payment content, not physical hours or a full global-chain measure of human effort. Imported content is not decomposed into foreign labor and capital. The corrected interpretation in revised Section 7.2 and Appendix G governs this submission.

The stated data windows are 2004–2023 for annual income-source financing profiles and 1997–2023 for the production benchmark. Values outside those windows are model-based extensions. The financing and production series are different accounting objects; their gap does not directly measure displacement by machines. This plot should be relabeled or regenerated from documented source data before use as affirmative evidence of changing physical labor requirements.

Figure S4. Composition of consumption financing outside direct ownership income

Source: August draft, Figure 6, PDF page 31. The plotted series cover 1962–2025 and divide the specified financing account into direct wages and transfers attributed to wage taxes, ownership taxes, and borrowing. The source describes medians across classification rules and shading across deficit-attribution rules.

Interpretation for review: this is an allocation-based accounting construction. Attribution to a revenue source does not establish economic tax incidence or the effect of a policy change. Its denominator excludes consumption financed directly from household capital income; the shares therefore should not be read as shares of all consumption. Check the classifications, benchmark windows, extensions, and deficit attribution against replication materials before drawing policy conclusions.

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