A Theory of Price Caps on Non-Renewable Resources

Layer 1: Overview

This paper asks what the optimal response of an exhaustible-resource producer is to sanctions in the form of a price cap, and how a sanctioning coalition should set the cap. The motivation is the $60-per-barrel cap on seaborne Russian crude imposed by the G7, EU and Australia in December 2022 (with $100/barrel for high-value and $45/barrel for low-value refined products), whose stated aim was to cut Russian revenue without triggering a global supply shock. The authors (two of whom were involved in designing the policy) argue that static models, frictionless Hotelling models, and truncated-supply-curve intuitions are all inadequate, and build a dynamic structural model.

Model setup: A petrostate extracts an exhaustible resource (reserves normalized to 1) whose price follows a Cox-Ingersoll-Ross (Feller square-root) process, estimated on monthly real oil prices 1973-2024 (deflated WTI), yielding long-run mean p̃=$76 (2024 prices), volatility ς=2.43, and mean reversion D=0.21 annually, implying a price half-life of ln2/D = 3.6 years and a right-skewed Gamma limiting distribution. Preferences are CRRA with γ=2 (baseline); marginal extraction cost M=$19/barrel (Osintseva 2021); real discount rate 3%; and non-oil income τ=2, implying commodity sales fund between 1/3 and 1/2 of state income. A two-period model first shows that sufficiently severe financial frictions (low saving returns, high borrowing rates, fixed participation costs Φ) make the producer endogenously live hand-to-mouth (Propositions 1-2), consuming oil proceeds directly; the infinite-horizon model takes this as given.

Main findings: (1) Even without physical adjustment costs, optimal supply is highly inelastic — supply falls sharply below $40/barrel and reaches zero just below $30 — matching Russia’s observed price-insensitivity. A novel decomposition attributes the shape to four forces: time-the-market, revenue-smoothing, precautionary, and non-homotheticity effects, with their balance governed by γ. (2) A perfect (universal, credible, permanent) price cap shifts the supply curve OUTWARD — the producer extracts MORE — because the cap removes price upside, making reserves less valuable (non-homotheticity) and, under market power, eliminating the point of restricting supply (a binding cap means cutting volume no longer raises price). (3) Consequently a binding perfect cap can LOWER and stabilize world prices, and the stabilizing benefit is LARGER the greater the producer’s market power (demand elasticity calibrated to 1/ϵ=0.25; short-run literature range [0.07,0.14]). (4) An imperfect (leaky and/or temporary) cap produces highly state-dependent behavior: when the market is already tight (reference price high, above ~$150/barrel in the calibration), the producer optimally ‘shuts in,’ cutting output toward the shadow-fleet capacity κ and selling only outside the cap — DESTABILIZING the market exactly when prices are high. With κ=0.01 (about one-third of normal extraction), a leaky cap reduces the welfare damage to the producer by about two-thirds relative to a perfect cap, even though contemporaneous profits fall up to 50% when shutting in. (5) The authors introduce a ‘sanctions possibility frontier’ trading producer harm v(p̄) against the excess probability of a price shock ϕ(p̄) (P(price>$120), ~12% historically). The optimal cap is HIGHER (less aggressive) the greater the leakage; preferences (weight λ) matter mainly at intermediate leakage. Policy corollary: effective enforcement is a precondition for setting a low cap.

Layer 2: Deep Dive

What is the core conceptual contribution about how a price cap operates?

The paper argues a price cap is not a truncation of the existing supply curve but a fundamental change to the stochastic environment the producer faces. By capping prices at min{p,p̄}, it eliminates the upside of high prices, lowers the value of reserves, and reduces uncertainty. Because the environment changes, the policy rules must be recomputed rather than read off the pre-policy supply curve adjusted with a vertical segment above p̄.

Why does a perfect price cap make the producer extract MORE, counter to policymaker intuition?

Two mechanisms. First, the non-homotheticity effect: with outside income τ>0, less valuable reserves are depleted faster, so capping the price (which lowers reserve value) raises the extraction rate. Second, for a producer with market power, a binding cap removes the incentive to restrict supply — curbing volume no longer raises the (capped) price, rendering market power ineffective. The supply curve under a binding cap closely follows the no-volatility supply curve.

What are the four forces in the supply-curve decomposition and what governs them?

(1) Time-the-market: sell more when prices are high. (2) Revenue-smoothing: with γ>1 the income effect dominates, so the producer extracts more when prices are low/expected to rise to smooth revenue. (3) Precautionary: price volatility induces conservation (extract less today); found quantitatively small. (4) Non-homotheticity: a permanently less valuable resource (low or capped price) is extracted faster, like greater impatience. Their balance is governed by preferences, specifically γ (inverse IES). Higher γ strengthens revenue-smoothing and weakens time-the-market; as γ→0 the model collapses to the frictionless Hotelling benchmark with infinitely elastic supply.

What is the empirical evidence presented, and what is the identification?

Section 2.5 tests whether financially constrained producers have more inelastic supply. Using 53 OPEC supply-news announcements 1984-2017 (from Känzig 2021) as price shocks, the authors examine production changes in 70 non-OPEC countries in the month after versus before each announcement. The dependent variable is the change in log production, sign-flipped so that producing more when prices fall (or less when prices rise) counts negatively. Regressing on the share of years a country had above-median debt-to-GDP yields a negative coefficient of -0.026 (std err 0.010), consistent with financially constrained countries having more inelastic supply. A country-risk-premium measure (Damodaran 2022) gives a similar but noisier result. Identification rests on OPEC announcements being exogenous price-news shocks to non-OPEC producers; threats include the announcements not being clean exogenous shocks and the debt-to-GDP dummy proxying other country characteristics — the paper treats this as motivating, not causal-structural, evidence.

How does the model incorporate market power and how is it endogenous?

World demand is isoelastic: pw=δ(r+y)^(-ϵ), where r is stochastic rest-of-world residual supply, y is producer output, and 1/ϵ is demand elasticity. The effective elasticity εD=ϵ·y/(r+y) depends on the producer’s market share, so market power evolves endogenously with past extraction (Cournot intuition). Market power makes the producer more conservationist in normal times, exerting upward price pressure. 1/ϵ is set to 0.25; the process for r is estimated by simulated method of moments so the laissez-faire equilibrium price matches the estimated oil-price process.

How is the ’leaky’ cap modeled and what is the shut-in strategy?

A shadow-fleet parameter κ∈[0,1] is the fraction of reserves exportable outside the cap per unit time (κ=0 is a perfect cap). With market power plus leakage, when the market is tight and prices are high, the producer optimally cuts output toward κ, selling only outside the regime at elevated prices (‘shut-in’). In the calibration with κ=0.01 (about a third of normal extraction), shut-in to κ is optimal when prices exceed ~$150/barrel; between $60 and $120 the cap still expands supply. So the cap stabilizes near the $76 long-run average but destabilizes when prices are already high.

What is the welfare and profit impact of a leaky cap?

Shutting in is not driven by higher contemporaneous profits — those fall by up to 50% relative to a perfect cap unless prices already exceed ~$150 — but by a more spread-out production profile that raises intertemporal welfare. Producer welfare rises with κ. Quantitatively, a leaky cap with κ=0.01 reduces the welfare damage inflicted on the producer by about two-thirds relative to a perfect cap, showing leakage sharply blunts the sanction.

How is cap non-credibility (temporariness) modeled?

Cap removal is a Poisson event with intensity λ, so duration is exponentially distributed. With a perceived 50% probability of removal within the first year, λ=0.69. Expecting the cap to be temporary makes the producer more inclined to shut in and keep barrels underground for extraction after removal, reinforcing the shadow-fleet mechanism and further weakening the cap’s stabilization effect; intertemporal welfare effects are significantly diminished.

What is the sanctions possibility frontier and how is the optimal cap chosen?

The policymaker minimizes v(p̄)+λ·ϕ(p̄), where v is proportional producer welfare loss from the value function and ϕ is the excess probability of an oil shock (P(pw>$120), baseline ~12% matching history). For each leakage level κ, the sanctions possibility frontier maps achievable (v,ϕ) combinations across cap levels. With a perfect cap the frontier is upward-sloping (no trade-off) and the optimum is the lowest cap above marginal cost. With leakage it becomes downward-sloping, creating a trade-off, and the frontier steepens as κ rises. Example: at κ=1/6, a cautious policymaker (λ=2) picks $55/barrel while an aggressive one (λ=1) picks $20; as leakage grows both converge to about $100. The optimal cap rises with leakage; preferences matter mainly at intermediate leakage.

How does this paper relate to and differ from prior work?

It contrasts with the frictionless Hotelling (1931) model (perfectly elastic supply) and with Anderson, Kellogg & Salant (2018), who derive inelasticity from geological well-pressure constraints — here inelasticity comes instead from financial frictions and market power. It differs from Stiglitz (1976), who found market power irrelevant to extraction quantity, because of positive marginal costs, financial frictions, and non-oil income. It complements empirical work (Babina et al. 2023 on market fragmentation and discounts), Salant (2023) on pre-announcement, Sappington & Turner (2023, static Cournot), Wachtmeister et al. (2023, quantitative), and Cardoso et al. (2024, endogenous shadow fleet). No separate drilling decision is modeled, for parsimony.

What robustness checks are reported?

Results are robust to: (a) excluding US/UK from the cross-country regression or using a 6-month horizon; (b) using the Damodaran country-risk-premium measure; (c) an alternative increasing, L-shaped marginal-cost curve with a 3% capacity constraint (Rystad/Wachtmeister data, M(y)=1.5+sqrt(0.25/(0.03-y))) — all conclusions hold, except predicted extraction is capped at the 3% capacity limit; and (d) HARA utility (nesting CRRA and CARA), available on request. The constant-marginal-cost main specification is chosen because it more clearly exposes the incentive to increase extraction (medium-term view).

What are the scope conditions and caveats on the policy conclusions?

The stabilizing-cap result requires the cap to be ’not too leaky’ and credible. The destabilizing shut-in only kicks in at high reference prices (above ~$150 in calibration). The financial-frictions/hand-to-mouth assumption is motivated by sanctioned petrostates specifically (frozen reserves — $300bn of Russian central-bank reserves frozen — sanctioned banks, war financing); it may apply less to unconstrained producers. The model is partial equilibrium (no general-equilibrium world economy, no strategic multi-state interaction, no endogenous shadow-fleet investment in the main analysis), and abstracts from storage and from a separate drilling margin. The policymaker objective is assumed linear in (v,ϕ).

Key Concepts

Price cap (as a tool of statecraft): In this paper, a sanction that lets the producer sell only at or below a ceiling p̄ when using coalition-controlled services, so the price received is pr=min{p,p̄}. Crucially it is interpreted not as a truncation of the supply curve but as a fundamental change to the stochastic environment, eliminating price upside and reducing reserve value and uncertainty.

Endogenous hand-to-mouth behavior: The result (Propositions 1-2) that sufficiently severe financial frictions — low saving returns, high borrowing costs, and/or fixed participation costs Φ — make the producer optimally consume oil proceeds period-by-period without using financial markets, regardless of its preferences. This is taken as the operating assumption for the dynamic model.

Non-homotheticity effect: With outside (non-oil) income τ>0, a permanently less valuable resource — whether from a low permanent price or a binding cap — is extracted faster, because reserve depletion is a less threatening prospect. It makes the producer behave as if more impatient and is a key driver of the outward supply shift under a cap.

Shut-in strategy: Under a leaky cap with market power, the producer sharply cuts extraction toward the shadow-fleet capacity κ when prices are already high, selling only outside the cap at elevated prices. It lowers contemporaneous profits (up to 50%) but raises intertemporal welfare via a more spread-out production profile; it destabilizes the market precisely when it is tight.

Shadow fleet / leakage (κ): The fraction of reserves the producer can export outside the cap regime per unit time (κ∈[0,1]); κ=0 is a perfect cap. For Russia it represents non-coalition tanker/insurance capacity; the paper notes the share of Russian oil outside the cap rose from about 20% (April 2022) to 67% (August 2024).

Sanctions possibility frontier: A novel menu, for each leakage level κ, of the achievable combinations of damage inflicted on the producer (v) and the probability of an oil-market shock (ϕ) across cap levels. Upward-sloping under a perfect cap (no trade-off; pick lowest cap), it becomes downward-sloping and steeper under leakage, making the optimal cap preference-dependent and increasing in leakage.

Reference price: The hypothetical equilibrium price that would prevail if the producer did not exercise market power — a monotone transformation of the state variable rt. It measures market tightness cleaned of the sanctioned producer’s endogenous decisions, and the cap’s price-lowering effect is larger when the reference price is high.