Reserves, stocks and depletion: Depletion: fixed stock or rising cost

3.1 Two ways to frame depletion: a fixed stock or an opportunity cost

Tilton describes two mental models. The fixed-stock paradigm starts from a true premise: the Earth is finite, so its nickel or copper is finite too. Demand is a flow that recurs every year, so it must eventually exhaust the stock, and exponential growth brings the day closer. Tilton and Guzmán compare humanity to mice that have found a warehouse full of cheese: fat and multiplying until the last crumb is gone. On this view recycling and conservation only postpone the end.

The opportunity-cost paradigm asks what society must give up to obtain one more tonne, and whether that sacrifice is rising. It has three measures: the real price, the real cost of the marginal producer, and the value of marginal reserves in the ground. Real prices are used most because the data exist.

Tilton and Guzmán find four flaws in the fixed-stock view. Metals are not destroyed by use, so they can be recovered; the lead once added to petrol still exists, it is just too costly to collect. Society needs services such as light and power, not particular materials, so substitutes count. The crust holds enormous quantities: its copper would last 84 million years at current output. And long before the last tonne, rising cost would choke off demand. The threat is economic depletion, not physical exhaustion.

With a fixed stock, scarcity is certain and only its date is open. With opportunity cost, a mineral can grow more or less available, depending on whether depletion, which raises costs, outruns technology, which lowers them. Barnett and Morse found that resources became less scarce between 1870 and 1957 despite soaring use.

• Tilton (2003), On Borrowed Time? · pdf p. 30
• Tilton & Guzmán (2016), Mineral Economics and Policy · pdf pp. 223-230
• Radetzki & Wårell (2021), A Handbook of Primary Commodities in the Global Economy · pdf pp. 173, 177-178

3.2 The Hotelling rule prices a finite stock, but its assumptions fail

Harold Hotelling's 1931 paper asked how the owner of a fixed, known stock of ore should time extraction. An ordinary competitive firm produces until the cost of one more unit equals the price. A mine faces an extra cost: a tonne sold today cannot be sold later. Tilton and Guzmán call this user cost; others say Hotelling rent or scarcity rent. It equals the present value of future profit given up by producing now.

The rule is an arbitrage condition. Ore in the ground is an asset whose value per tonne is price minus extraction cost. If that value rose more slowly than the interest rate, the owner would mine now and invest the proceeds; if faster, he would wait. In equilibrium the value in the ground rises at the rate of interest. As an illustration, with a net value of $1,000 a tonne and a real interest rate of 5 per cent, the owner is indifferent between mining this year and next only if next year's net value is expected to be $1,050. Applied to a whole industry, prices should climb as the stock dwindles.

Radetzki and Wårell call the early 1980s the rule's triumph and perversion: analysts read it as saying oil prices themselves must rise at the rate of interest, and Exxon and the World Bank forecast endlessly rising real oil prices.

The rule needs a known stock of uniform quality, no uncertainty and fixed technology. Exploration and innovation break all three. A series of prices paid for US oil reserves from 1982 to 2003 shows no upward trend. Pindyck and Rubinfeld's estimates, cited by Tilton and Guzmán, put user cost at 5 to 30 per cent of price for most minerals and 40 to 50 per cent for oil and gas; Cairns found under 5 per cent for Canadian nickel. Tilton and Guzmán suspect even these ranges are too high.

• Tilton & Guzmán (2016), Mineral Economics and Policy · pdf pp. 136-137, 141, 223, 240-241
• Radetzki & Wårell (2021), A Handbook of Primary Commodities in the Global Economy · pdf pp. 176-179

3.3 The cumulative availability curve turns depletion into a race

Tilton and Skinner proposed a better tool in 1987, the cumulative availability curve. It ranks all deposits of a mineral by cost and shows how much could be recovered profitably over all time at each price. Unlike a textbook supply curve, which is a flow per year, it is a stock. It rises because cheap deposits are used first.

The curve sorts the forces into three groups. Geology sets its shape. Demand, including population, income and recycling, sets how fast society climbs it. Input costs and technology shift it, and over the long run innovation pushes it down. Future scarcity is a race between the climb and the shift. Conservation and recycling slow the climb; only new technology moves the curve.

Shape is the knowable part, because it reflects geology laid down long ago. Yaksic and Tilton estimated it for lithium. Known deposits on land form a rising curve, but behind them lies seawater, with an estimated 44.8 billion tonnes recoverable even if only a fifth of its lithium is captured, at 7 to 10 dollars per pound of lithium carbonate in 2007 money. Their most aggressive demand case needed 17.5 million tonnes of primary lithium by 2100. Seawater thus sets a ceiling on lithium's long-run price.

Two features make a curve less benign. By-products face a step if demand outgrows what host-metal mining releases. And the Yale geologist Brian Skinner argues that for scarce metals such as copper, lead, tin and zinc, metal may be distributed by grade in two humps: rich ore deposits, a gap, then vast amounts spread thinly through ordinary rock. Getting copper from common silicate rock rather than sulphide ore could take one or two orders of magnitude more energy. If so the curve has a cliff, though no one knows where.

• Tilton & Guzmán (2016), Mineral Economics and Policy · pdf pp. 231-237, 239-240
• Radetzki & Wårell (2021), A Handbook of Primary Commodities in the Global Economy · pdf pp. 182-183
• Conway (2023), Material World · pdf p. 215
• Humphreys (2015), The Remaking of the Mining Industry · pdf pp. 231-232

3.4 From The Limits to Growth to the Simon-Ehrlich bet

The public argument is old. Tilton and Guzmán trace it from Malthus through Jevons's 1865 warning about British coal, the American conservation movement and the Paley Commission of 1952. Its best-known modern statement is The Limits to Growth (1972), a computer model commissioned by the Club of Rome. Copper reserves then covered about 30 years of output, and Conway reports that even the model's optimistic case gave copper no more than 48 years, until 2022. Tilton notes that the authors tried to allow for discovery by multiplying known reserves by five. The 1973-74 boom and OPEC's price rises seemed to confirm the book.

Radetzki and Wårell's reply is that its authors ignored history: from 1925 to 2000, reserves had kept pace with ninefold growth in copper output and 24-fold growth in oil. Tilton and Guzmán note a pattern in who takes which side. Pessimists are often geologists and engineers, optimists often economists. Radetzki and Wårell add that the alarm grows loudest during booms. In 2024, two years after copper was supposed to run out, the USGS put world reserves at 980 million tonnes.

Neither side owns the lessons. Tilton and Guzmán quote Neumayer: the pessimists have been wrong so far, but to conclude there is nothing to worry about repeats their error of extrapolating the past. Humphreys expects depletion to add to costs for some metals. Conway argues that copper's real limit may be how much more blasting and digging societies will tolerate, as Chile and Peru weigh the environmental cost of new mines against the copper that electrification needs.

• Tilton & Guzmán (2016), Mineral Economics and Policy · pdf pp. 221-224, 230-231
• Conway (2023), Material World · pdf pp. 210-214, 220
• Radetzki & Wårell (2021), A Handbook of Primary Commodities in the Global Economy · pdf pp. 172, 174
• Tilton (2003), On Borrowed Time? · pdf pp. 32-33
• Humphreys (2015), The Remaking of the Mining Industry · pdf p. 232

See it in the table

Check your understanding

Under the opportunity-cost paradigm, which observation would be the strongest evidence that copper is becoming scarcer?

A mine owner expects the net value of ore in the ground to rise from $1,000 to $1,020 a tonne next year, while the real interest rate is 5 per cent. What does the Hotelling logic imply?

Which policy shifts the cumulative availability curve downward rather than slowing society's movement up it?