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### Key empirical findings on metals and oil shocks
- A one percent increase in copper price due to a supply shock:
  - Raises headline inflation by 0.03 percentage points after 12 months and 0.1 percentage points after 24 months (for countries with high exposure).
  - Raises core inflation by 0.03 percentage points after 12 months and 0.05 percentage points after 24 months (for countries with high exposure).
- For countries with low exposure to copper, the initial impact is about 0.01 percentage points on both headline and core inflation and becomes statistically insignificant after about 12 months.
- A one percent increase in oil prices:
  - Has a 0.05 percentage points impact on headline inflation after 12 months and also after 24 months, but becomes insignificant after 40 months.
  - Shows no significant impact on core inflation.
- Aggregate example from the network model:
  - A 10 percent supply-driven increase in metals prices would generate a 0.36 p.p. increase in China’s CPI, compared to a 0.1 p.p. increase for the US (average across countries implied by the network model).
- Median cross-country exposures:
  - The median country has a metal exposure of 0.03 (aggregate measure described below).
  - The distribution is wide: a country in the 90th percentile has an exposure five times larger than one in the 10th percentile.
- Median CPI exposure is three times larger for oil than for metals.

### Stylized facts about how metals and oil enter the economy
- Metals:
  - Mostly used for investment goods; embedded in production of machinery, electrical equipment, construction materials.
  - Direct input shares may be small in final consumption (example: metals 0.01 percent vs oil and coal products 2.6 percent in the US), but metals represent large direct intermediate input shares in investment-good sectors (e.g., more than 10 percent in electrical equipment and machinery).
  - Indirect exposure is important: fabricated metals and machinery show 28 and 46 percent shares once indirect effects are included; construction indirect exposure is 4 times direct exposure.
  - Because metals are embodied in investment goods, metals price shocks can produce persistent effects on core inflation.
- Oil:
  - Used primarily as a flow to produce energy, mostly in transportation and utilities.
  - Impacts headline inflation more immediately and does not significantly affect core inflation.
  - Indirect exposure is less prominent than for metals.

### Data and measurement
- US input-output data:
  - 3-digit US Bureau of Economic Analysis input-output tables for 2018.
  - Metals sector defined as the sum of the mining sector, except oil and gas, and the primary metals sector.
  - Oil proxy: sum of the oil and gas extraction sector and the petroleum and coal products sector (limited granularity prevents identifying crude oil precisely).
- OECD cross-country input-output data for other countries:
  - Metals: combined mining and quarrying of non-energy producing products sector and the basic metals sector.
  - Oil: mining and quarrying of energy-producing products sector plus coke and refined petroleum products manufacturing sector.
- Imports:
  - Imports of metals and oil by domestic sectors are accounted for when calculating intermediate inputs.
  - Methodology adapts a closed-economy Leontief inverse framework to include imports for stylized facts; main analysis includes an explicit tradable sector.

### Network metric and aggregation
- Sectoral exposure to metals (M) is captured by the Leontief inverse element (I−Ω)^{-1}_{iM} which reflects direct and indirect effects of an increase in metals prices on sector i’s intermediate input costs.
- CPI and GDP deflator exposures to a metals productivity shock Z_M:
  - dlogDGDP_t = ∑_k va_k · (I−Ω_total)^{-1}_{kM}
  - dlogCPI_t = ∑_k b_k · (I−Ω_total)^{-1}_{kM}
  - where va_k are value-added shares (for GDP deflator aggregation) and b_k are final consumption expenditure shares (for CPI aggregation).
- The Leontief inverse expansion: I + Ω + Ω^2 + Ω^3 + ... captures higher-order network effects.

### Theoretical framework highlights
- Model basis:
  - Small open economy production-network framework following Silva (2023).
  - Distinguishes domestically produced goods (N) and imported goods (M); separates imported metals used as intermediate inputs from domestically produced metals.
- Representative household:
  - Utility U(C_D, C_M) with price index approximated to first order as b_P = b_D^T b_P_D + b_M^T b_P_M, where b_D and b_M are expenditure shares of domestic and imported goods.
- Firms:
  - Representative firm i: Q_i = Z_i F_i( {L_{if}}_{f∈F}, {M_{ij}}_{j∈N}, {M_{im}}_{m∈M} ).
  - Marginal cost MC_i = MC_i(Z_i, P_D, P_M, W); constant returns to scale imply zero profits and marginal cost independence from scale.

### Equilibrium, nominal anchor, and price-index decomposition (sections 3.3–3.6)
- Market clearing and equilibrium:
  - Q_i = C_i + X_i + ∑_{j∈N} M_ji for each i∈N.
  - X_i is exogenous so there is always a price that clears the market for each domestically produced good.
  - Equilibrium obtained by taking factor prices (W) and expenditure (E) as given; households and firms optimize given these prices and market clearance.
- Nominal anchor (cash-in-advance):
  - Cash-in-advance constraint assumed: P C ≤ M^u = E.
  - In the SOE setup money supply (M^u) is exogenous and the central bank dictates nominal spending (E).
  - Equilibrium requires the binding cash-in-advance constraint P C = M^u = E.
- Log-linear approximation of changes in aggregate price index bP for perturbations (bZ, cW, bP_M):
  - bP = −[λ̄^T − e^{λ̄^T}] bZ + [Λ̄^T − e^{Λ̄^T}] cW + [bT_M + e^{bT_M}] bP_M,
  - with e^{λ̄^T} = x^T Ψ_D, e^{Λ̄^T} = x^T Ψ_D A, e^{bT_M} = bT_D Ψ_D Γ, and Ψ_D = (I − Ω)^{−1}.
- Mechanism under import-price shocks (metals), with cW = 0^F and bZ = 0^N:
  - bP = [b̄_T_M + b̄_T_D Ψ_D Γ] dP_M,
  - Two channels: direct import consumption (b̄_M) and indirect import consumption via domestic production network (b̄_T_D Ψ_D Γ).
  - Metals: imported metals price rises increase producer marginal costs via Γ_{hm}, raising P_h and propagating through Ψ_D to affect other domestic prices; metals also affect CPI through direct final consumption of metals (b_m).

### Empirical methods and figures
- Impulse response estimation:
  - Cumulative responses estimated using local projection, with 12 lags of shocks on the RHS.
  - Shaded areas in figures are 90% confidence bands based on Newey-West standard errors.
- Notable figures and specifications described:
  - Impulse responses of real copper and oil prices to positive copper and oil supply shocks.
  - Exposure measures constructed from OECD sectors: mining and quarrying non-energy products; basic metals; fabricated metal products (for metals including fabricated).
  - Robustness checks using consumption expenditure share of imported metals and oil (b_M) and net import share.
  - Instrument for metals: copper supply shock series from Baumeister, Ohnsorge, and Verduzco-Bustos (2024); instrument for oil: oil supply shocks from Baumeister and Hamilton (2019).

### Contributions to literature and policy implications
- Contributions:
  - Emphasizes metals as crucial inputs in intermediate and investment goods production and quantifies their role via production networks.
  - Shows that supply shocks to primary metals can significantly affect prices and core inflation via direct and indirect channels.
  - Offers a channel for inflation comovement across countries owing to metals’ central role in manufacturing and trade.
- Policy implications:
  - If the world economy becomes more metals intensive due to the energy transition, inflationary shocks could become more persistent.
  - Unlike oil supply shocks (more temporary and headline-focused), metals supply shocks can affect core inflation persistently; central banks may need to react to changes in core inflation driven by metals supply shocks.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024215-print-pdf.pdf*

### 0.03 percentage points after 24 months, respectively, and is quite persistent. Since copper

### wpiea2024215-print-pdf - 0.03 percentage points after 24 months, respectively, and is quite persistent. Since copper

### Key empirical findings on metals and oil shocks
- A one percent increase in copper price due to a supply shock:
  - Raises headline inflation by 0.03 percentage points after 12 months and 0.1 percentage points after 24 months (for countries with high exposure).
  - Raises core inflation by 0.03 percentage points after 12 months and 0.05 percentage points after 24 months (for countries with high exposure).
- For countries with low exposure to copper, the initial impact is about 0.01 percentage points on both headline and core inflation and becomes statistically insignificant after about 12 months.
- A one percent increase in oil prices:
  - Has a 0.05 percentage points impact on headline inflation after 12 months and also after 24 months, but becomes insignificant after 40 months.
  - Shows no significant impact on core inflation.
- Aggregate example from the network model:
  - A 10 percent supply-driven increase in metals prices would generate a 0.36 p.p. increase in China’s CPI, compared to a 0.1 p.p. increase for the US (average across countries implied by the network model).
- Median cross-country exposures:
  - The median country has a metal exposure of 0.03 (aggregate measure described below).
  - The distribution is wide: a country in the 90th percentile has an exposure five times larger than one in the 10th percentile.
- Median CPI exposure is three times larger for oil than for metals.

### Stylized facts about how metals and oil enter the economy
- Metals:
  - Mostly used for investment goods; embedded in production of machinery, electrical equipment, construction materials.
  - Direct input shares may be small in final consumption (example: metals 0.01 percent vs oil and coal products 2.6 percent in the US), but metals represent large direct intermediate input shares in investment-good sectors (e.g., more than 10 percent in electrical equipment and machinery).
  - Indirect exposure is important: fabricated metals and machinery show 28 and 46 percent shares once indirect effects are included; construction indirect exposure is 4 times direct exposure.
  - Because metals are embodied in investment goods, metals price shocks can produce persistent effects on core inflation.
- Oil:
  - Used primarily as a flow to produce energy, mostly in transportation and utilities.
  - Impacts headline inflation more immediately and does not significantly affect core inflation.
  - Indirect exposure is less prominent than for metals.

### Data and measurement
- US input-output data:
  - 3-digit US Bureau of Economic Analysis input-output tables for 2018.
  - Metals sector defined as the sum of the mining sector, except oil and gas, and the primary metals sector.
  - Oil proxy: sum of the oil and gas extraction sector and the petroleum and coal products sector (limited granularity prevents identifying crude oil precisely).
- OECD cross-country input-output data for other countries:
  - Metals: combined mining and quarrying of non-energy producing products sector and the basic metals sector.
  - Oil: mining and quarrying of energy-producing products sector plus coke and refined petroleum products manufacturing sector.
- Imports:
  - Imports of metals and oil by domestic sectors are accounted for when calculating intermediate inputs.
  - Methodology adapts a closed-economy Leontief inverse framework to include imports for stylized facts; main analysis includes an explicit tradable sector.

### Network metric and aggregation
- Sectoral exposure to metals (M) is captured by the Leontief inverse element (I−Ω)^{-1}_{iM} which reflects direct and indirect effects of an increase in metals prices on sector i’s intermediate input costs.
- Consumer price index (CPI) and GDP deflator exposures to a metals productivity shock Z_M:
  - dlogDGDP_t = ∑_k va_k · (I−Ω_total)^{-1}_{kM}
  - dlogCPI_t = ∑_k b_k · (I−Ω_total)^{-1}_{kM}
  - where va_k are value-added shares (for GDP deflator aggregation) and b_k are final consumption expenditure shares (for CPI aggregation).
- The Leontief inverse expansion: I + Ω + Ω^2 + Ω^3 + ... captures higher-order network effects.

### Theoretical framework highlights
- Model basis:
  - Small open economy production-network framework following Silva (2023).
  - Distinguishes domestically produced goods (N) and imported goods (M); separates imported metals used as intermediate inputs from domestically produced metals.
- Representative household:
  - Utility U(C_D, C_M) with price index approximated to first order as b_P = b_D^T b_P_D + b_M^T b_P_M, where b_D and b_M are expenditure shares of domestic and imported goods.
- Firms:
  - Representative firm i: Q_i = Z_i F_i( {L_{if}}_{f∈F}, {M_{ij}}_{j∈N}, {M_{im}}_{m∈M} ).
  - Marginal cost MC_i = MC_i(Z_i, P_D, P_M, W); constant returns to scale imply zero profits and marginal cost independence from scale.

### Contributions to literature and policy implications
- Contributions:
  - Emphasizes metals as crucial inputs in intermediate and investment goods production and quantifies their role via production networks.
  - Shows that supply shocks to primary metals can significantly affect prices and core inflation via direct and indirect channels.
  - Offers a channel for inflation comovement across countries owing to metals’ central role in manufacturing and trade.
- Policy implications:
  - If the world economy becomes more metals intensive due to the energy transition, inflationary shocks could become more persistent.
  - Unlike oil supply shocks (more temporary and headline-focused), metals supply shocks can affect core inflation persistently; central banks may need to react to changes in core inflation driven by metals supply shocks.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024215-print-pdf.pdf*

### 3.3    Equilibrium

### 3.3    Equilibrium

### Market clearing and equilibrium definition
- Market clearing conditions for sectoral output:
  - Q_i = C_i + X_i + ∑_{j∈N} M_ji for each i∈N.
  - X_i is exogenous so there is always a price that clears the market for each domestically produced good, even if the good is exported.
- Equilibrium is obtained by taking factor prices (W) and expenditure (E) as given to pinpoint feasible and equilibrium allocations:
  - Households choose (C^D, C^M) to maximize utility subject to budget constraints based on sequences (W, P^D, P^M, π) and exogenous parameters (T).
  - Firms choose (L_i, M_i) to minimize production costs given (W, P^D, P^M) and production technologies.
  - Market clearance is achieved given X.
  - The cash-in-advance constraint binds: P C = M^u = E.

### Nominal anchor (cash-in-advance)
- Cash-in-advance constraint assumed: P C ≤ M^u = E.
- In the small open economy (SOE) setup:
  - Money supply (M^u) is exogenous and the central bank dictates nominal spending (E) to maintain a benchmark.
  - By monitoring consumption (C) affected by real factors, the central bank implements the price level (P) accordingly.
- Conclusion: equilibrium requires the binding cash-in-advance constraint P C = M^u = E.

### Changes in the consumer price index (log-linear approximation)
- Log-linear approximation of changes in aggregate price index bP for perturbations (bZ, cW, bP_M) around an initial equilibrium:
  - bP = −[λ̄^T − e^{λ̄^T}] bZ + [Λ̄^T − e^{Λ̄^T}] cW + [bT_M + e^{bT_M}] bP_M,  (equation (3) in the source)
  - Definitions provided in the model:
    - e^{λ̄^T} = x^T Ψ_D
    - e^{Λ̄^T} = x^T Ψ_D A
    - e^{bT_M} = bT_D Ψ_D Γ
    - Ψ_D = (I − Ω)^{−1}
- Interpretation of terms:
  - First two terms: effects of shocks to sectoral productivity (bZ) and wages (cW).
  - Last term: effect of changes in import prices (bP_M), which influence inflation via intersectoral connections (Ψ_D) and the network-adjusted share of import consumption.
  - Note: input-output network Ω considers only domestic input-output linkages.

### Mechanism: focus on import price shocks (metals)
- Under fixed factor prices and no productivity shocks (cW = 0^F and bZ = 0^N), equation (3) reduces to:
  - bP = [b̄_T_M + b̄_T_D Ψ_D Γ] dP_M,  (equation (4) in the source)
  - where b̄_M is vector of imported consumption share, Ψ_D is domestic Leontief inverse matrix, bT_D is vector of domestic consumption shares, and Γ is vector of sectoral imported intermediate input shares.
- Two channels of import-price-driven inflation:
  - Direct import consumption: b̄_M — consumers purchasing imported goods directly.
  - Indirect import consumption: b̄_T_D Ψ_D Γ — consumers purchasing domestic goods that use imports as inputs (directly or indirectly), i.e., propagation through domestic production network.
- Application to metals:
  - A rise in imported metals prices increases producer h’s marginal costs via Γ_{hm}, raising product prices P_h and propagating through Ψ_D to affect prices of other domestically produced goods.
  - Metals can also affect CPI through direct consumer purchases of metals as final goods (b_m).
- Distinction from stylized-facts approach:
  - This SOE framework isolates the inflationary effect of an exogenous foreign supply shock (international prices) rather than productivity shocks captured in a closed-economy approach.
  - Particularly relevant because metal production is geographically concentrated; most economies are metals (oil) importers.

### Empirical implications highlighted for metals
- Metals are key intermediate inputs for investment goods in production networks; shocks to metals prices can elevate production costs and propagate persistently through the network to core and headline inflation.
- The framework suggests that if the world economy becomes more metals-intensive (e.g., due to the energy transition), metals price shocks could generate more persistent inflationary effects, posing challenges for central banks that historically ‘looked through’ oil price shocks.

*Source: wpiea2024215-print-pdf — sections 3.3–3.6 (Equilibrium, Nominal Anchor, Changes in the Price Index, The Mechanism).*

### References

### wpiea2024215-print-pdf - References

### References (selected thematic citations)
- Studies on sectoral and commodity shocks and markets:
  - Albrizio et al. 2023. “Sectoral Shocks and the Role of Market Integration: The Case of Natural Gas.” In AEA Papers and Proceedings, vol. 113. American Economic Association, 43–46.
  - Alvarez et al. 2023. “Geoeconomic Fragmentation and Commodity Markets.” IMF Working Paper 2023/201.
  - Baumeister, Ohnsorge, and Verduzco-Bustos. 2024. “Evaluating the Role of Demand and Supply Shocks in Copper and Aluminum Price Fluctuations.” Unpublished Manuscript.
  - Benguria, Saffie, and Urzúa. 2024. “The transmission of commodity price super-cycles.” Review of Economic Studies 91 (4):1923–1955.
  - Boer, Pescatori, and Stuermer. 2024. “Energy Transition Metals: Bottleneck for Net-Zero Emissions?” Journal of the European Economic Association 22.
- Methodological and empirical tools:
  - Jordà. 2005. “Estimation and inference of impulse responses by local projections.” American Economic Review 95 (1):161–182.
  - Kilian. 2009. “Not all oil price shocks are alike: Disentangling demand and supply shocks in the crude oil market.” American Economic Review 99 (3):1053–1069.
  - Baqaee and Farhi. 2019. “The Macroeconomic Impact of Microeconomic Shocks: Beyond Hulten’s Theorem.” Econometrica 87 (4):1155–1203.
  - Ha, Kose, and Ohnsorge. 2023. “One-stop source: A global database of inflation.” Journal of International Money and Finance 137:102896.
- Sector- and commodity-specific historical and theoretical studies:
  - Hamilton. 1983. “Oil and the macroeconomy since World War II.” Journal of Political Economy 91 (2):228–248.
  - Stuermer. 2018. “150 years of boom and bust: what drives mineral commodity prices?” Macroeconomic Dynamics 22 (3):702–717.
  - Jacks and Stuermer. 2020. “What drives commodity price booms and busts?” Energy Economics 85:104035.

### Data (empirical inputs and sources)
- Sample period and price indices:
  - Monthly data on nominal commodity prices and indexes for the period January 1996 to December 2019.
  - Oil prices: average of Brent and WTI.
  - Food prices: Food and Beverage Index from IMF Primary Commodity Price system.
- Inflation and control variables:
  - Headline CPI and core CPI from the global inflation database assembled by Ha, Kose, and Ohnsorge (2023).
  - Control variables: Global economic activity index from Baumeister, Korobilis, and Lee (2022); US 1-year treasury bill yield from FRED; bilateral exchange rates from BIS; Gilchrist and Zakrajšek (2012) excess bond premium (EBP) for the USA.
- Input-output and sectoral data:
  - Input-output data from the BEA for the US (71 sectors) and from the OECD (45 sectors) for cross-country comparisons for the year 2018.
  - Tables referenced for sectoral classification: Tables B.1 and B.3.
- Instruments for supply shocks:
  - Copper: estimated copper supply shock series from Baumeister, Ohnsorge, and Verduzco-Bustos (2024) for the same period.
  - Oil: oil supply shocks from Baumeister and Hamilton (2019).

### Definitions (sectoral classifications)
- OECD definitions (combination of Organisation for Economic Cooperation and Development (2023) and United Nations (2008)):
  - Mining and quarrying of non-energy producing products sector includes: mining of iron ores, non-ferrous metal ores, uranium and thorium ores, other non-ferrous metal ores, quarrying of stone, sand and clay, mining and quarrying n.e.c., mining of chemical and fertilizer minerals, extraction of peat, extraction of salt, and other mining and quarrying n.e.c.
  - Basic metals sector includes: manufacturing of basic iron and steel, manufacturing of basic precious and other non-ferrous metals, casting of metals, casting of iron and steel, and casting of non-ferrous metals.
  - Mining and quarrying of energy-producing products sector includes: mining of hard and lignite coal, extraction of crude petroleum and natural gas.
- US Bureau of Economic Analysis (2024) definitions:
  - Mining sector, except oil and gas: coal mining; iron, gold, silver, and other metal ore mining; copper, nickel, lead, and zinc mining; stone mining and quarrying; other nonmetallic mineral mining and quarrying.
  - Primary metals sector: iron and steel mills and manufacturing from purchased steel (including iron and steel mills and ferroalloy manufacturing and steel product manufacturing from purchased steel), nonferrous metal production and processing, and foundries (primary and secondary smelting and refining of non-ferrous metals; rolling, drawing, extruding, and alloying of non-ferrous metals).
  - Oil and gas extraction: defined straightforwardly as oil and gas extraction.
  - Non-durable goods manufacturing sector for petroleum and coal products (referred to as the “oil sector”): petroleum refineries; asphalt paving mixture and block manufacturing; asphalt shingle and coating materials manufacturing; other petroleum and coal products manufacturing.

### BEA and OECD sectoral classification (tables overview)
- BEA sectoral listing highlights:
  - Table B.1 and Table B.2 present BEA sectoral classification. The BEA sectoral classification table includes 66 sectors; sectors related to government are not included.
  - Examples of BEA sectors: Oil and gas extraction; Mining except oil and gas; Primary metals; Petroleum and coal products; Food and beverage and tobacco products; Motor vehicles, bodies, trailers, and parts; Utilities; Construction; Wholesale trade; Retail trade; Transportation sectors; Financial sectors; Health and education sectors; Accommodation and food services.
- OECD sectoral listing highlights:
  - Table B.3 lists OECD sectoral classification with sectors such as: Agriculture, hunting, forestry; Fishing and aquaculture; Mining and quarrying, energy producing products; Mining and quarrying, non-energy producing products; Food products, beverages and tobacco; Coke and refined petroleum products; Chemical and chemical products; Basic metals; Fabricated metal products; Computer, electronic and optical equipment; Electricity, gas, steam and air conditioning supply; Construction; Wholesale and retail trade; Transportation and storage; Information and communication; Financial and insurance activities; Real estate activities; Public administration and defence.

### Figures and empirical specifications (methods and key modelling notes)
- Impulse response estimation:
  - Cumulative responses estimated using local projection, with 12 lags of shocks on the RHS.
  - Shaded areas in figures are 90% confidence bands based on Newey-West standard errors.
- Figures and robustness checks described:
  - Figure B.1: Impulse Responses of Real Copper and Oil Prices to Positive Copper and Oil Supply Shocks, respectively.
  - Figure C.2: Exposure to metals (including fabricated metals) and heterogeneity in impulse responses; exposure based on the sum of three OECD sectors: mining and quarrying, non-energy producing products; basic metals; and fabricated metal products.
  - Figure C.3: Impulse response functions using consumption expenditure share of imported metals and oil (b_M), excluding indirect network linkages; red and blue lines denote exposure at the 90th/10th percentile of the sample in 2018.
  - Figure C.4: Impulse response functions using countries’ net import share of metals or oil; red and blue lines denote exposure at the 90th/10th percentile of the sample in 2018.
  - Figure C.5: Impulse responses of headline CPI and core CPI to a one percent increase in metal price index instrumented by copper supply shocks; top panel uses exposure to primary metals, bottom panel uses exposure to both primary and fabricated metals.
- Specific empirical notes:
  - Figure C.2 bottom panel: cumulative impulse responses of headline CPI inflation (left) and core (right) to a one percent increase in copper prices using the copper supply shock from Baumeister, Ohnsorge, and Verduzco-Bustos (2024) as an instrument.
  - Figure C.3 and C.4 x-axes denote months after the shock; shaded areas are 90% confidence bands.
  - Figure C.5: instrument uses the copper supply shock from Baumeister, Ohnsorge, and Verduzco-Bustos (2024).

*Beyond Energy: Inflationary Effects of Metals Price Shocks in Production Networks — Working Paper No. WP/2024/215*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024215-print-pdf.pdf_