Appendix A: The input-output formulation

A.1 The material-balance identity

The basic idea of input-output analysis is to break the economy into a number of sectors and then ask: how much does each sector produce, and where do those goods go? We represent the output of each sector as a vector , where is the output of sector .

In principle, you could have one sector for every distinct good in the economy and track outputs in physical units — tons of steel, kilowatt-hours of electricity, number of trucks. In practice, tracking the millions of products produced each year is unrealistic, so we aggregate them into sectors. The aggregation is usually done in terms of dollars, so is measured in dollars per year.

Every unit of output goes to one of a few uses:

• Intermediate inputs. Goods used up in producing other goods. For example, steel is consumed to make a car, and electricity is consumed to smelt the steel.
• New capital formation. Goods installed as productive equipment. For example, a lathe bolted to a factory floor.
• Depreciation replacement. Replacing capital that has worn out. For example, a new truck replacing a retired one.
• Household consumption. Goods consumed by people rather than by industry. For example, food bought at a grocery store.
• Exports. Goods sold abroad rather than used domestically. For example, cars shipped to Europe. We assume a closed economy, so exports are zero throughout.

Mathematically, we write this as

where:

• is the dollars of good consumed per dollar of sector 's output. For example, the entry for (steel, auto manufacturing) gives how many dollars of steel go into each dollar of cars produced.
• is the dollar stock of capital good required per dollar of sector 's annual output. A sector growing at rate needs units of good per unit of new capacity, so gives the total flow of goods into new capital.
• is the per-year depreciation replacement flow. This is constructed by combining with per-good depreciation rates , which vary by asset class (vehicles wear out faster than buildings), so for any single scalar .
• is household consumption. In the balanced-growth calculation, is set to zero (all output reinvested) or to a small fixed share of output (Ramsey savings).
• is exports. We assume a closed economy, so throughout.

A.2 The balanced-growth path and the Perron eigenvalue

We want to find the fastest rate at which the economy can grow. If we set and — reinvesting all output and ignoring trade — then we can rearrange the material balance into a dynamical system:

We want the fastest-growing solution. It turns out this is a balanced-growth path, where every sector grows at the same constant rate :

Here is a fixed vector giving the relative size of each sector. We substitute this into the material balance to get a generalized eigenvalue problem:

By the Perron-Frobenius theorem, this has a unique largest eigenvalue with a strictly positive eigenvector . The eigenvalue is the maximum growth rate, and is the unique sectoral composition that can sustain it — the Von Neumann ray, following von Neumann (1945) and the Leontief (1951) input-output tradition.

A.3 Construction lags

In the material-balance equation, capital goods are treated as installed instantaneously: a dollar of commodity produced at time enters the capital stock at time . In reality, capital takes time to be transported, installed, and commissioned. A factory building delivered at time was started at time , where is the construction time. During those years the materials sit as work-in-progress, contributing nothing to current output.

We model this as a pure delay on the commodity being produced: the production lag is a property of commodity , not of the industry that uses it. A factory building takes the same time to construct whether it goes to a chemicals plant or an electronics plant. A unit of commodity produced at time enters service at time , so the material balance becomes

On a balanced-growth path we can expand:

and so defining , the eigenvalue problem becomes

which is a fixed-point problem in rather than a linear eigenvalue problem: the matrix on the right depends on the eigenvalue itself. We solve it by iteration: guess , build and , solve the resulting Perron problem for an updated , and repeat to convergence.

In practice we use two lag values: for construction commodities (NAICS 23, structures and buildings — about 49% of by total value) and for everything else. So is a binary diagonal with entries on construction rows and elsewhere.

The construction lag enters as the opportunity cost over and above the prevailing interest rate , which is already embedded in measured capital intensities . The inflation factor should therefore be , not , so that when the lag does not bind. In practice, however, the Von Neumann growth rate is so much higher than the real interest rate that we can ignore this effect.

A.4 Data sources

The intermediate-input matrix

We use the 2017 vintage of the BEA Supply-Use tables at the detail level. The Use table records, for each industry, how many dollars of each commodity it consumed as an intermediate input in a given year. It is a rectangular matrix with commodity rows and industry columns; the column totals include value added (compensation, gross operating surplus, taxes on production) and the row totals include final demand (household consumption, investment, government, exports).

To build a square technology matrix we restrict to the 398 commodity codes that also appear as industry codes (where the commodity and the industry that primarily produces it are both tracked at the detail level). Let be the dollar value of commodity used by industry , and let be industry 's total output. The technical coefficient is

the dollars of commodity consumed per dollar of output of industry . This is the standard Leontief construction: captures the recipe for producing each good in terms of every other good.

The capital-requirements matrix

The capital matrix records how many dollars of commodity are embodied as productive capital per dollar of output of industry . Building requires two ingredients: the capital intensity of each industry (how much PP&E per dollar of output) and the commodity composition of that capital (what the PP&E is made of).

Capital intensity comes from the BEA Fixed Assets tables, which report current-cost net stock of private fixed assets by industry. The Fixed Assets data are reported at a coarser summary level (~70 industries) than the detail IO tables, so each detail sector inherits the capital intensity of its parent summary sector. We compute , where is the sum of the equipment and structures net stock from Fixed Assets tables 3.1E and 3.1S, and is the summary-level gross output from the BEA summary Use tables.

For the commodity composition of capital, we use the BEA Detailed Nonresidential Fixed Assets tables, which break each industry's capital stock into roughly 60 asset types — specific categories of equipment (metalworking machinery, computers, trucks, electrical transmission equipment) and structures (manufacturing buildings, commercial warehouses, utility infrastructure). We map each asset type to the IO commodity that produces it using a concordance between BEA asset codes and the detail commodity classification. For example, asset code ET11 (autos) maps to commodity 336112 (light trucks and utility vehicles), and asset code EP1A (mainframe computers) maps to commodity 334111 (electronic computers).

For a given industry with capital intensity , the share of its capital stock in each asset type determines how much of each commodity is embodied as capital per dollar of output. Concretely, if industry has fraction of its capital in asset type , and asset maps to commodity , then . When detailed asset composition data is not available for a particular industry, we fall back to the economy-average investment composition vector , constructed from the final-demand columns of the Use table (nonresidential equipment and structures investment, BEA codes F02E00 and F02S00).

The depreciation matrix

The depreciation matrix records the annual depreciation replacement requirement: how many dollars of commodity must be produced each year just to replace the capital wearing out in industry , per dollar of 's output. It has the same structure as but applies asset-type-level depreciation rates rather than a single blended rate.

Depreciation rates come from the Fixed Assets tables 3.4E (equipment depreciation) and 3.4S (structures depreciation). Equipment typically depreciates at 10–20% per year; structures at 2–4% per year. We compute separate equipment and structures depreciation rates for each summary industry, then apply these to the corresponding asset categories when building . For industry with fraction of capital in asset type at depreciation rate :

This asset-level treatment matters because different industries have very different equipment-to-structures ratios. Utilities have most of their capital in long-lived structures (power plants, transmission lines) with depreciation rates around 3%, while electronics manufacturing has most of its capital in short-lived equipment with depreciation rates above 15%. A single blended depreciation rate per industry would miss this distinction.

Government infrastructure

Private fixed assets omit public infrastructure that private industry depends on — highways, water systems, and publicly owned electric utilities. We add these as additional capital requirements, allocated to the industries that use them:

• Highway and street capital ($4.9T) is allocated across industries in proportion to their use of transportation services (the sum of the transportation rows of ).
• Water supply and sewage capital ($1.5T) is allocated in proportion to utilities usage.
• Public electric utility capital ($0.7T) is allocated to the utilities sector itself.

These additions increase the total capital stock from about $27T (private PP&E alone) to about $34T.

Capacity utilization

The baseline model uses current capital productivity — how much output each industry actually produces per dollar of installed capital. But most plants do not run at full capacity. A factory might operate one shift per day, five days a week, producing well below what its installed equipment could deliver if run continuously.

We use data from the Census Bureau's Quarterly Survey of Plant Capacity (QPC), which reports two utilization benchmarks for manufacturing industries:

• Full-production utilization: the fraction of capacity at which the plant could operate sustainably under normal market conditions, with the maximum practical level of inputs. This closes the gap between current output and what the existing shift schedule and equipment could deliver without extending hours.
• Emergency utilization: the maximum output achievable during an emergency, with all constraints on hours and scheduling removed. This corresponds to wartime-style 24/7 operations, still including realistic downtime for maintenance.

The QPC also reports average weekly plant hours, which we use to compute a third benchmark: 168 hr/wk utilization, the theoretical maximum if every plant ran every hour of every week with no downtime at all.

For non-manufacturing sectors (mining, utilities, construction), the QPC does not apply. We supplement with Federal Reserve G.17 capacity utilization data (mining at 85%, utilities at 73%) and BLS Current Employment Statistics on average weekly hours (mining 46 hrs/wk, utilities 140 hrs/wk effective plant hours, construction 40 hrs/wk).

Utilization enters the model by scaling the capital matrices. A sector running at utilization multiplier produces times its baseline output from the same capital stock, so the effective capital requirement per unit of output divides by :

The intermediate-input matrix is unchanged: a ton of steel still requires the same physical inputs regardless of how many hours per week the mill runs.

Robot and compute sectors

To model the cost of replacing labor, we extend the 398-sector system by two additional sectors, robots and compute, producing a 400-sector system. These sectors are entered asymmetrically because the markets they represent are structurally different: a robot is a durable capital good purchased and operated by user firms, while compute is a flow service sold by data center operators to user firms.

The robot sector manufactures mechanical robot bodies without onboard compute. Each user sector that needs physical labor buys robot bodies, holds them as capital, and pays the electricity to run them. The compute sector is a data center operator. The AI brain that controls a robot — and the AI replacing every other worker — runs in a data center. User sectors do not own data center capital; they buy compute services as a flow input from the operator, who holds the capital and pays the electricity.

Each original sector acquires additional input requirements proportional to its labor share . The labor share is split into a physical fraction and an intellectual fraction . The physical-labor fractions come from BLS Current Employment Statistics, specifically the ratio of production and nonsupervisory employees to total employees by industry. All workers need data center compute for their AI brain; physical workers additionally need a robot body. Per dollar of output of sector :

where is robot body capital per dollar of physical labor replaced (central estimate 0.10, from a ~$30K body replacing ~4 FTEs at ~$70K/yr), is data center capital per dollar of labor replaced (central estimate 0.05, from $425K full DC capital per 8-GPU H100 server divided by 130 workers replaced at $70K/yr), and is the data center operator's capital-output ratio (from $425K capital per server divided by ~$140K/yr cloud-rental revenue at $2.50/H100/hr × 80% utilization). The form converts data center capital per dollar of labor replaced into the corresponding flow rate of compute services bought per year.

The robot and compute sectors need their own intermediate inputs and capital. The robot sector's column is a weighted average of proxy manufacturing sectors: machinery (40%), electronics (30%), motor vehicles (20%), and electrical equipment (10%). Its column uses the car factory proxy (BEA sector 3361MV — motor vehicle manufacturing), reflecting the similar physical plant required for assembling mechanical bodies. Robot operating energy enters as additional demand on the utilities row of for each original sector: a robot draws about 1 kW, which at 4 FTEs per robot and $70K/yr per FTE is about $0.0014 per dollar of physical labor replaced per year.

The compute sector's column is taken from BEA detail sector 518200 (Data processing, hosting, and related services), which captures the full operating recipe of a data center operator — advertising, real estate, computer services, and others — totaling about 45% of revenue. We override the electricity entry to 3.8% of revenue (reflecting AI hyperscale; BEA's 0.8% averages traditional data processing). The remaining service inputs survive post-AGI: data centers still consume them, and the sectors that produce them are themselves automated via the same labor-replacement extension. The compute sector's column reflects the AI data center capital mix: roughly 49% semiconductors, 10% electrical equipment, 7% construction, 7% machinery, with the remainder spread across other categories.

Appendix B: Von Neumann growth rates are similar across industrial economies

The main analysis uses US data — BEA input-output tables at 398-sector detail and BEA Fixed Assets capital stocks. A natural concern is whether the results are US-specific: perhaps the US IO structure is unusually favorable for self-reproduction, and other industrial economies would yield substantially different growth rates. This appendix checks the robustness of the – yr⁻¹ result across 18 OECD countries.

We compute free-labor Von Neumann growth rates using OECD Inter-Country Input-Output tables (2023 edition) at the 45-sector ISIC Rev 4 classification (2018 data year) combined with sector-level net capital stocks and depreciation from the OECD STAN database. The methodology follows the formulation in Appendix A but differs in three respects: (1) 45 sectors rather than 398, (2) STAN capital-stock data rather than BEA Fixed Assets, and (3) investment composition from GFCF final demand rather than detailed asset-by-industry tables.

Country	(yr⁻¹)	Doubling time (yr)	Avg
Germany	0.39	1.8	1.91
Austria	0.44	1.6	2.13
Italy	0.45	1.5	1.79
Norway	0.45	1.5	1.82
Sweden	0.49	1.4	1.64
Japan	0.50	1.4	1.73
Greece	0.51	1.4	1.76
Czechia	0.52	1.3	1.64
Denmark	0.53	1.3	1.71
United States	0.54	1.3	1.53
Slovakia	0.59	1.2	1.83
Belgium	0.64	1.1	1.33
Netherlands	0.64	1.1	1.40
New Zealand	0.64	1.1	1.35
Finland	0.66	1.0	1.69
France	0.68	1.0	1.80
United Kingdom*	0.92	0.8	1.13
Ireland*	1.14	0.6	1.11
Mean (excl. outliers)	0.54	1.3	1.69

Ireland's result is an artifact of multinational profit-shifting, which inflates measured output relative to physical capital actually located in Ireland. The United Kingdom's result also appears anomalously high due to a data-quality issue in the underlying capital-stock series. Both are excluded from the median.

The OECD 45-sector model systematically understates relative to the BEA 398-sector model. For the US, the OECD model gives 0.54 yr⁻¹ versus 0.61 yr⁻¹ from BEA — about 11% lower. The gap arises because aggregation into 45 sectors averages together high- and low- sub-industries, losing within-sector heterogeneity. Finer-grained IO tables would likely push the international results upward by a similar factor, placing the typical OECD economy closer to 0.6 yr⁻¹.

Appendix C: Resource extraction

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C.1 Minerals

This appendix estimates how much mining capital intensity would increase if the economy were 2x, 5x, or 10x its current size, assuming extraction scales proportionally.

The grade-cost scaling law

Energy and capital per tonne of metal scale roughly as an inverse power of ore grade, because lower grades mean processing more rock.

Mining has two stages. Mining and milling produces a concentrate at a standardized grade (e.g. 30% Cu for copper, 65% Fe for iron pellets). Smelting and refining processes the concentrate into metal. Smelting costs do not depend on ore grade, because the concentrate is the same regardless. Mining costs scale as roughly : halving the ore grade means processing twice the rock.

Two empirical results bracket this estimate. Fizaine and Court (2015) regressed energy cost of extraction against ore grade across 34 metals and found a full-supply-chain exponent of -0.60, shallower than because smelting dilutes the mining-specific effect. Koppelaar and Koppelaar (2016) found that for mining and milling alone, the relationship is steeper: roughly to , because lower grades also require finer particle sizes to liberate the mineral. We use as the basis for scaling below. It is intermediate between the two results and has the cleanest physical justification.

Skinner's mineralogical barrier

For geochemically scarce metals (Cu, Ni, Co, Zn, Sn, Pb, Au, Ag, PGMs), Skinner (1976) argued that crustal concentrations are bimodal. Metals exist either concentrated in discrete ore minerals (sulfides, oxides) at grades from roughly 0.1% up, or dispersed as atomic substitutions in common silicate minerals at average crustal abundance, with relatively little material in between.

Above the barrier, metals form distinct mineral grains that can be physically separated by flotation or gravity. Below it, metal atoms are locked in silicate crystal lattices, and liberation requires complete chemical destruction of the host rock. Energy costs jump by an estimated 100-1,000x.

For geochemically abundant metals (Fe, Al, Si, Mg, Ti, Mn), the distribution is unimodal and there is no barrier. For copper, the barrier is thought to lie around 0.1% Cu.

Iron

Iron ore grades are 30-65% Fe, reserves are 190 Gt of crude ore (88 Gt Fe content) against 2.5 Gt/yr extraction (USGS MCS (2025)), and resources exceed 800 Gt. The grade difference between best and worst iron ores is only about 2x (30% vs 65% Fe), so even at 10x extraction the capital intensity increase is at most 1.4x. Iron is geologically unconstrained at any plausible scaling factor.

Aluminum (bauxite)

Aluminum is the most abundant metal in the crust at roughly 8% by mass. Reserves are 29 Gt, resources 55-75 Gt, against roughly 450 Mt/yr extraction (USGS MCS (2025)). The USGS notes that most aluminum-producing countries have "essentially inexhaustible subeconomic resources" in clays, anorthosite, and coal fly ash. The binding constraint on aluminum is smelting energy (roughly 15 MWh per tonne of metal via Hall-Heroult), not geology. Grade decline in bauxite is modest even at scale.

Copper

Copper is the metal where grade decline matters most. Average mined grades have fallen from roughly 2% Cu in 1900 to roughly 0.5% today. Current production is roughly 23 Mt/yr of contained copper. Reserves are 980 Mt, identified resources roughly 2.1 Bt, and total resources (including undiscovered) estimated at 5-6 Bt (USGS MCS (2026); Singer (2017)).

The grade-tonnage relationship for copper is well characterized. Gerst (2008) built cumulative grade-tonnage curves for 1,778 Mt of mineable copper across four deposit types. Northey et al. (2014) found that mined grades are approaching the average resource grade of roughly 0.49% Cu: the easy high-grading is largely over. Calvo et al. (2016) documented a 25% ore grade decline across Chilean copper mines in ten years (2003-2013), accompanied by a 46% increase in energy consumption. The practical floor for conventional milling and flotation is roughly 0.2% Cu; below roughly 0.1%, Skinner's mineralogical barrier applies.

Nearly all copper ever mined is still in use in buildings, infrastructure, and equipment. Doubling the capital stock therefore requires roughly doubling cumulative extraction. At a 2x economy, cumulative copper extraction roughly doubles from the historical 800 Mt to 1,600 Mt. Drawing down the grade-tonnage curve, marginal grades decline to roughly 0.37% Cu, and by scaling, mining capital intensity rises by x. At 5x, marginal grades fall to roughly 0.25% Cu and intensity rises 2.0x. At 10x, cumulative extraction exceeds 5,000 Mt, marginal grades approach the mineralogical barrier at roughly 0.15% Cu, and intensity rises roughly 3.3x. At this point smooth extrapolation probably breaks down.

Aluminum is the primary substitute. It has 61% of copper's electrical conductivity per unit cross-section but one-third the density, so aluminum conductors carry the same current at roughly 1.6x the cross-sectional area and half the weight. Most overhead power transmission lines are already aluminum. Aluminum-wound motors and transformers are roughly 1.3x larger and heavier for the same output, acceptable for most industrial uses. Since aluminum is geologically unconstrained, copper scarcity has an effective ceiling: at sufficiently high copper prices, most electrical applications shift to aluminum at a modest size and weight penalty.

Nickel

Global nickel production is roughly 3.7 Mt/yr, with reserves of 130 Mt and resources exceeding 350 Mt. Two deposit types define supply: sulfide deposits (1-3% Ni, underground, conventional flotation) and laterite deposits (0.8-1.5% Ni, surface mining, acid leaching or pyrometallurgical processing). Indonesia now produces roughly 60% of global nickel from laterites.

Priester et al. (2019) found that nickel has the worst grade concentration of any of the nine metals they studied: only 1.7% of total nickel resources sit in the highest-grade decile, compared to 15-20% for iron, PGMs, and zinc. At a 2x economy, the ongoing shift to laterite processing continues with modest grade decline, and capital intensity rises roughly 1.5x. At 5x, marginal laterites at 0.7% Ni are required and intensity rises roughly 2.5x. At 10x, deep-sea nodules (which contain roughly 270 Mt of nickel in the Clarion-Clipperton Zone alone) become relevant and may hold intensity below 4x.

Lithium

Lithium reserves are 30 Mt, resources roughly 115 Mt, against extraction of only 240 kt/yr (USGS MCS (2025)). Reserve life is roughly 125 years. At 10x demand, resource life is still roughly 48 years. Capital intensity per tonne is roughly flat to 5x and rises modestly at 10x.

Cobalt

Cobalt is almost entirely a byproduct of copper and nickel mining and cannot scale independently of its host metals. Reserves are 11 Mt against extraction of roughly 290 kt/yr (USGS MCS (2025)). LFP cathodes are cobalt-free and already hold significant market share, and high-nickel NMC cathodes (NMC 811, NMC 9½½) progressively reduce cobalt content, so this geological constraint may become irrelevant as battery chemistry evolves.

Manganese

Reserves of 1,700 Mt at 35-54% Mn against production of roughly 20 Mt/yr, with massive additional resources on the deep-sea floor. Manganese is as geologically abundant as iron relative to its demand. Capital intensity scaling is negligible at any plausible economy size.

Rare earth elements

Global REE production reached roughly 390,000 tonnes REO in 2024, with reserves exceeding 90 Mt (USGS MCS (2025)). Total REE tonnage is not the constraint; the problem is the balance problem. Rare earth elements occur together in fixed geological ratios, and the valuable magnet elements (Nd, Pr, Dy, Tb) are a small fraction of total REO. Dysprosium is the most binding element: essential for high-temperature NdFeB permanent magnets (2-4% of magnet mass) and sourced primarily from ion-adsorption clays in southern China and Myanmar. At 5-10x magnet demand, getting enough Dy requires mining far more total REO than the market can absorb for other rare earths. In a post-AGI economy you mine all the REO you need to get enough Dy and stockpile or discard the rest; the cost is the mining and processing cost of 50-100x the mass of Dy you actually want. Substitution (ferrite magnets, RE-free motors) is available but carries significant performance penalties: ferrite magnets deliver roughly one-tenth the energy product of NdFeB.

Platinum group metals

Global PGM production is only 450-500 tonnes/year, concentrated in three deposits (Bushveld Complex in South Africa, Norilsk in Russia, Great Dyke in Zimbabwe). Iridium, essential for PEM electrolysers at roughly 400 kg/GW, is produced at only 7-8 tonnes per year globally. PGMs are severely constraining for specific applications at 5x+ scaling. Alkaline electrolysers (no PGMs needed) provide a substitution pathway.

Deep-sea mining

Polymetallic nodules on the abyssal ocean floor represent a large additional resource for several constrained metals. The Clarion-Clipperton Zone alone contains an estimated 21 Bt of nodules with roughly 5.95 Bt of manganese, 0.27 Bt of nickel, 0.23 Bt of copper, and 0.05 Bt of cobalt (ISA Technical Study 6 (2010); Hein and Mizell (2022)). For nickel and cobalt, the CCZ exceeds total terrestrial reserves.

Nodules sit on the seafloor surface at 4-6 km depth and are collected by vehicles that vacuum them up and pipe slurry to a surface vessel. No overburden removal, drilling, or blasting is required. As of 2025, no commercial deep-sea mining has occurred; the binding constraint is regulatory (the International Seabed Authority has not finalized exploitation rules) rather than technical. Deep-sea nodules do not appear in the capital intensity estimates in the summary table, which are based on terrestrial resources only.

Summary table

The table below estimates how much mining capital intensity per tonne of metal would increase at 2x, 5x, or 10x the current economy, based on scaling and grade-tonnage data where available. These do not account for technological improvement, substitution, or deep-sea mining.

Metal	Reserve life (yr)	Capital intensity multiplier @: 2×	Capital intensity multiplier @: 5×	Capital intensity multiplier @: 10×	Notes
Iron	76	1.0×	1.2×	1.4×	Abundant; infrastructure-limited
Aluminum	64	1.0×	1.2×	1.4×	Abundant; energy-limited (smelting)
Manganese	85	1.0×	1.0×	1.1×	Abundant; deep-sea floor backup
Lithium	125	1.0×	1.1×	1.3×	Abundant; permitting-limited
Copper	43	1.4×	2.0×	3.3×	Most binding bulk metal; Al substitution available
Nickel	35	1.5×	2.5×	4.0×	Worst grade distribution; deep-sea backup
Cobalt	38	1.7×	2.5×	4.0×	Being engineered out of batteries; deep-sea backup
Rare earths (Dy)	>230	1.7×	4.0×	7.0×	Cost is overproduction of unwanted REEs
PGMs (Ir)	100	1.7×	4.0×	7.0×	Severely concentrated; substitution available

The aggregate effect on the growth rate depends on how these metals weigh into the mining sector's capital requirements. Mining is roughly 4% of the investment supply chain. The growth economy is dominated by steel (iron), which faces negligible depletion. Copper is probably the single most important constraint: at a 2x economy, it adds perhaps 0.2x to the aggregate mining capital intensity multiplier; at a 5x economy, perhaps 0.7x. These translate to modest reductions in the growth rate (see the sensitivity table in the main text).

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C.2 Fossil fuels

Metals accumulate: nearly all the copper ever mined is still in use. Doubling the capital stock requires mining roughly as much copper again, and you can read the cost off the grade-tonnage curve. Oil and gas are mostly consumed. Only a fraction of historical extraction is embodied in the current capital stock, as the energy that went into constructing buildings, manufacturing equipment, smelting steel, and producing cement. The rest was burned in flow uses: transport, heating, consumer electricity.

The relevant question is therefore: how much oil must be extracted to build a 2x economy's capital stock? (Strictly, the Von Neumann growth path scales only the capital-goods sectors relevant to self-replication, not the full economy. Here we use doubling the economy as a rough proxy for whether growth can be sustained over a few doublings, but we are not (yet) considering the dynamics of this transition.)

Oil

Cumulative global oil extraction through 2024 is 1,572 Bb. Total recoverable resources are 3,100 Bb, of which 1,500 Bb remain. Global gross fixed capital formation has averaged 25% of GDP over the relevant period, and capital-forming sectors are more energy-intensive per dollar than the economy average by a factor of 1.5-2x. Roughly a third of cumulative oil consumption went to building the capital stock that currently exists: about 500 Bb.

The supply cost curve is well characterized. Middle East onshore conventional breaks even at $27/bbl, deepwater $43, North American shale $55, oil sands $65, Arctic $90. Today's 100 mb/d requires $45/bbl on average. The curve steepens above 110 mb/d as supply draws on oil sands, Arctic, and EOR-dependent resources.

A 2x economy requires roughly 500 Bb of oil embodied in new capital, plus flow consumption during the buildout (transport, chemical feedstocks, process heat not yet electrified). Call the total 700-1,000 Bb. Against 1,500 Bb remaining, this is feasible but moves cumulative extraction from 51% to 70-80% depleted, deep into the expensive part of the curve. Capital intensity per barrel roughly doubles.

A 5x economy requires roughly 2,500-3,500 Bb total. Against 1,500 Bb remaining under current technology, this is not possible. Even if kerogen oil shale becomes viable (adding perhaps 1,000-1,500 Bb of recoverable resources), a 5x economy is at the edge of what is geologically feasible, and only at capital intensities 3-5x current levels with EROIs approaching the minimum viable threshold.

A 10x economy is not possible under any assumptions about oil resources.

Natural gas

Cumulative global gas extraction through 2024 is roughly 150 Tcm (BGR (2017); Energy Institute (2024)). Total remaining technically recoverable resources are roughly 790 Tcm, of which 460 Tcm is conventional and 330 Tcm unconventional (IEA (2012)). Only 16% of the total endowment has been consumed, compared with 51% for oil. Gas skews more heavily toward heating and residential use, so the embodied fraction is lower: roughly one-quarter rather than one-third, giving about 38 Tcm embodied in the current capital stock.

A 2x economy requires roughly 75 Tcm total. Against 790 Tcm remaining, this barely moves cumulative extraction into the cost curve's shoulder. Capital intensity per unit of gas rises perhaps 1.3x. A 5x economy requires roughly 300 Tcm, pushing cumulative extraction to about 48% of the total endowment (comparable to where oil is today). Capital intensity roughly doubles. A 10x economy requires roughly 675 Tcm, bringing cumulative extraction to 88%. This is at the geological limit but not categorically impossible the way 5x is for oil; capital intensity would be 3-4x current levels.

Gas has far more headroom than oil. Geological constraints begin to bind only around 8-10x.

Coal

Cumulative coal extraction is roughly 400 Bt. Total recoverable resources exceed 10,000 Bt; proved reserves are 1,055 Bt (Energy Institute (2024)). One third embodied in capital gives 130 Bt. Coal could support many multiples of the current capital stock without geological constraints binding.

The cost-scaling mechanism is the transition from surface to underground mining. Powder River Basin surface mines produce at $11/short ton with productivity of 29 tonnes per miner-hour. Appalachian underground mines cost $60+/short ton at 2.4 tonnes per miner-hour: a 12x productivity gap. Depth compounds the penalty: costs roughly double from 200 m to 1,000 m. In China, average mining depth is 600 m and increasing at 15 m/year; over half of remaining resources sit below 1,000 m. Capital intensity rises 1.5x at 2x, 2.5x at 5x, and 4x at 10x as progressively deeper and thinner seams are exploited.

Summary

Fuel	Cumulative extraction	Remaining recoverable	Embodied in capital	Capital intensity multiplier @: 2×	Capital intensity multiplier @: 5×	Capital intensity multiplier @: 10×
Oil	1,572 Bb	1,500 Bb	500 Bb	2×	Not possible	Not possible
Natural gas	150 Tcm	790 Tcm	38 Tcm	1.3×	2×	3–4×
Coal	400 Bt	9,600+ Bt	130 Bt	1.5×	2.5×	4×

Oil cannot supply the energy needed to build more than about 2x the current capital stock. Gas has far more headroom: geological constraints begin to bind only around 8-10x, because only 16% of the total endowment has been consumed versus 51% for oil. Coal is not geologically constrained at any plausible economy size. These estimates exclude kerogen oil shale (roughly 6 trillion barrels in place globally, of which perhaps 1 trillion is technically recoverable at very high cost) and do not account for future discoveries or extraction technology improvements.

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C.3 Electrification

Oil and natural gas cannot support more than about 2x the current capital stock. The question for growth is whether replacing fossil fuel generation with solar, wind, and batteries significantly slows the economy's self-reproduction rate.

We construct explicit generation sectors from NREL ATB cost benchmark data, decompose each technology's capital cost into BEA IO commodities (e.g. solar modules to electronics, wind towers to primary metals, battery cells to primary metals + electronics), and replace fossil fuel generation in the IO model. Fossil fuel intermediate inputs to all sectors are zeroed and redirected to electricity at a 3:1 efficiency ratio, reflecting the thermodynamic advantage of electric motors over combustion. This increases total electricity demand by 1.42x.

Renewable generation has higher capital intensity than the current electricity mix ( for a 60/30/10 solar/wind/battery mix versus 7.6 for today's fossil-dominated grid), but the capital is composed of commodities the economy is already good at producing: steel, electronics, electrical equipment, fabricated metals, and construction. Using NREL ATB 2024 Moderate projections, at emergency utilization with central labor cost estimates:

Technology	Growth rate (yr⁻¹) w/projected tech: 2023	Growth rate (yr⁻¹) w/projected tech: 2030	Growth rate (yr⁻¹) w/projected tech: 2035
Solar	0.61	0.68	0.74
Solar + 4hr battery	0.37	0.45	0.49
Wind	0.57	0.65	0.67
Nuclear	0.48	0.50	0.51
Fossil baseline		0.79
Fossil + 5× oil & gas		0.58

Each row imagines the entire grid replaced by a single technology. A real grid mixes sources and needs storage, transmission, and backup not modeled here. Solar without storage is not independently dispatchable, and including batteries substantially reduces growth rates. A purely electrified grid using current technology grows materially slower than the fossil baseline, but the gap narrows with projected cost reductions.