## wpiea2025208-source-pdf

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---

### Main research question and contribution
- Examines how domestic interconnectedness of the commodity sector conditions the transmission of commodity price shocks, arguing that interconnectedness matters more than sector size.
- Emphasizes production linkages measured by the Network-Adjusted Value-Added Share (NAVAS) rather than conventional measures like Domar weights.
- Combines panel local projections for OECD countries with a dynamic small open economy model with production networks to identify transmission channels.

### Empirical setup, data, and NAVAS definition
- Sample and classification:
  - Unbalanced annual panel of 66 countries (37 AE, 29 EMDE) over 1990 to 2018 (extended through 2023 when available).
- Primary data sources and measurements:
  - IMF’s Commodity Terms of Trade database; country-level export commodity prices proxied by the Commodity Net Export Price Index (weighted by net exports as a share of GDP, with rolling windows for time-varying weights).
  - Global Macro Database (GMD, Müller et al.2025); variables deflated using the U.S. consumer price index (CPI) and expressed in U.S. dollars.
  - OECD Input-Output Tables (IOT) 2018 edition to construct production-network linkages and NAVAS.
- NAVAS (notation and formulas preserved):
  - Commodity sector value-added share: a_{N+1} = 1→!_{i} P^{M}_{N+1,i} / P_{N+1}Q_{N+1}
  - NAVAS:  ̃a_{N+1} = sum_{i=1}^{N+1} ”_{N+1,i} a_{i}, where ” = (I→#)→1 and # denotes the IO structure with elements #_{i,j}.
- NAVAS interpretation:
  - Captures share of total factor income generated by the commodity sector accounting for direct and indirect upstream and downstream linkages.
  - Can be high because of a large commodity-sector value-added share, reliance on value-added–intensive intermediate suppliers, greater reliance on domestic intermediates, or supplying key inputs that feed back into commodity production.

### Stylized empirical findings (panel local projections)
- Identification and methods:
  - Instrumental variable local projection (LP-IV) panel framework (Jordà 2005) with interaction terms; shocks identified via Baumeister and Guérin 2021 (demand) and Baumeister and Hamilton 2019 (supply).
  - Structural shocks scaled to produce a 5 percent contemporaneous increase in real commodity prices for impulse response interpretation.
- Key empirical results:
  - Demand-driven shocks:
    - Generate positive consumption responses.
    - NAVAS comoves positively with the consumption response; interaction coefficient θ(1) (NAVAS) amplifies the effect.
    - Commodity sector size (Domar weight) has a modest and statistically insignificant effect at all horizons in many specifications.
  - Supply-driven shocks:
    - Generate negative consumption responses on average.
    - NAVAS has a positive, large, and significant effect on consumption at all horizons after controlling for commodity sector size; countries at the 10th percentile of NAVAS show persistently negative consumption responses while those at the 90th percentile show statistically insignificant or upward-trending responses.
  - Interaction results:
    - NAVAS interaction effects are markedly larger than corresponding size (Domar weight) interactions and often of opposite sign.
  - Robustness:
    - Results robust to alternative identification (Schmitt-Grohé and Uribe 2018) and to inclusion of labor market slack interactions (slack interaction statistically insignificant in demand-driven estimates; size interaction loses significance when slack included).

### Cross-country NAVAS and size patterns (descriptive statistics)
- Aggregate comparisons:
  - On average, the commodity sector’s NAVAS is 31% higher in EMDEs than in AEs.
  - Commodity sectors are generally larger and somewhat more interconnected within domestic production networks in EMDEs compared to AEs.
  - In AEs, commodity sector size tends to be small, but NAVAS (which averages around 0.6) indicates greater importance for macroeconomic fluctuations than size alone.
  - Right tail of NAVAS distribution in AEs overlaps with left tail in EMDEs, implying some AEs have commodity sectors more interconnected than many EMDEs.
- Representative NAVAS values (selected entries from NAVAS tables, exact values preserved):
  - Advanced economies (selected): Australia Aggregate 0.89, Energy 0.91, Metals 0.88, Agriculture 0.88; United States Aggregate 0.81, Energy 0.76, Metals 0.82, Agriculture 0.90; France Aggregate 0.65, Energy 0.33, Metals 0.65, Agriculture 0.81.
  - Emerging and developing economies (selected): Argentina Aggregate 0.93, Energy 0.92, Metals 0.91, Agriculture 0.94; Russia Aggregate 0.93, Energy 0.95, Metals 0.89, Agriculture 0.89; Saudi Arabia Aggregate 0.97, Energy 0.98, Metals 0.84, Agriculture 0.91.
- Summary descriptive statistics (Table VI and VII exact means and ranges preserved):
  - AEs: Size mean (Energy, Metals, Agriculture, Aggregate) = 0.05, 0.04, 0.04, 0.13; NAVAS mean (Energy, Metals, Agriculture, Aggregate) = 0.50, 0.61, 0.78, 0.61; NAVAS Min = 0.21, 0.21, 0.55, 0.22; NAVAS Max = 0.94, 0.88, 0.90, 0.89.
  - EMDEs: Size mean (Energy, Metals, Agriculture, Aggregate) = 0.14, 0.08, 0.15, 0.39; NAVAS mean (Energy, Metals, Agriculture, Aggregate) = 0.66, 0.73, 0.86, 0.80; NAVAS Min = 0.23, 0.37, 0.65, 0.50; NAVAS Max = 0.98, 0.95, 0.96, 0.97.

### Theoretical model, calibration, and mechanisms
- Model features:
  - Dynamic small open economy (SOE) model with N+1 domestic production sectors, production networks (Leontief structure), labor and intermediate inputs (domestic and imported), representative household with foreign bond holdings Bt.
  - Foreign assets Bt denominated either in commodity units (P_{N+1}) or importable units (P_M); adjustment cost g(Bt) = ε2(Bt → ̄B)2.
  - Exogenous processes: logP_{N+1,t} = ↼N+1 logP_{N+1,t→1} + πN+1,t; logZi,t = ↼Z logZi,t→1 + πi,t.
- Calibration highlights (Table I exact parameters preserved):
  - ω = 0.961: Discount rate; matches interest rate r = 4%.
  - ε = 2: Intertemporal elasticity of substitution.
  - θ = 0.000742: Bond holdings adjustment cost.
  - εN+1 = 0.53: Commodity price persistence.
  - π = 0.15: Commodity price elasticity to demand shifter.
  - ςi = 3: Labor and intermediate inputs elasticity.
  - φi = 0.6: Domestic and imported inputs elasticity.
  - φD i = 0.2: Elasticity across domestic intermediate inputs.
  - Datasource: OECD data covering 66 countries and 44 sectors; six commodity sectors aggregated into one (benchmark: one commodity sector + 38 non-commodity sectors); calibration matches sectoral final consumption shares, IO linkages, and commodity sector net exports from 2018.
- Two primary transmission mechanisms:
  - Income effect:
    - Higher global commodity prices raise real wages via increased labor demand in the commodity sector.
    - Greater NAVAS reduces the economy-wide real wage response because commodity-sector suppliers raise its marginal cost; nominal wage pass-through from commodity prices falls with larger ̃a_{N+1}.
  - Wealth/valuation effect via Net Foreign Assets (NFA):
    - Valuation of NFA changes with commodity price movements (PN+1/Pt), generating wealth effects that influence consumption.
    - If downstream propagation is strong, PN+1/Pt can fall on impact (negative valuation) or rise (positive valuation), producing differing immediate wealth effects and consumption dynamics.
    - Wealth effects dominate consumption responses in the calibrated model.

### Model results and channel decomposition
- Main experiment:
  - First-period percentage change in real consumption following a 1 percent increase in terms of trade (1 percent ToT shock).
  - Model reproduces empirical pattern: higher NAVAS associated with stronger consumption–price comovement; EMDEs generally exhibit higher NAVAS and higher correlations, though some AEs also display elevated NAVAS.
- NAVAS versus sector size:
  - No clear relationship between consumption responses and commodity sector size (Domar weight) in simulations; NAVAS is the stronger predictor.
- Valuation vs income decomposition:
  - Valuation effect:
    - Positive relationship between valuation effect and NAVAS when foreign assets are denominated in commodity units; commodity-exporting economies that hold positive NFA in commodity terms see PN+1/Pt increases raise NFA real value and boost consumption.
    - Most AEs (often net debtors) experience negative consumption responses reflecting increased value of their debt when commodities price rise.
  - Income effect:
    - Higher NAVAS dampens real wage response; less interconnected commodity sectors require larger wage increases to clear profits.
- Sensitivity to asset denomination:
  - If NFAs are denominated in units of the importable good (P_M), the positive relationship between NAVAS and consumption responses also holds for valuation effect, but overall consumption responses on impact can be negative for all countries because PM/P decreases on impact.
  - If assets are denominated in domestic price index Pt (no valuation effect), the consumption–NAVAS relationship is negative in the model, contradicting empirical evidence and highlighting the role of NFA valuation.
- Productivity shocks comparison:
  - Productivity shocks in the commodity sector can produce consumption responses qualitatively similar to ToT shocks when assets are denominated in units of the importable good, but the underlying channels differ (productivity shocks affect marginal costs and downstream linkages more directly).

### Case studies: Kazakhstan (KAZ) versus South Africa (ZAF)
- Commonalities and differences:
  - Both net commodity exporters with commodity sectors accounting for 39 percent of GDP.
  - NAVAS: Kazakhstan = 0.90; South Africa = 0.73.
- Impulse-response findings to a 1 percent ToT shock (assets denominated in commodity units):
  - Income effects:
    - Both countries experience positive real wage responses; larger income effect in South Africa because its commodity sector is less interconnected.
  - Valuation and consumption:
    - Kazakhstan: PN+1,t / Pt increases more, producing a positive wealth shock and a large positive consumption response on impact; NFA valuation supports consumption.
    - South Africa: Aggregate price index P increases more, PN+1,t / Pt declines on impact producing a negative wealth shock and a negative consumption response on impact; South Africa increases current account and smooths consumption over time at smaller magnitudes.
- Implication:
  - Production network structures (NAVAS) and the currency/units in which foreign assets are denominated fundamentally alter macroeconomic effects of commodity ToT shocks.

### Policy implications and recommendations
- Macroeconomic and monetary policy frameworks should be adapted to account for the structure of domestic production networks.
- Central banks should account for production network structures (NAVAS) when calibrating responses to commodity price movements to reduce risk of policy miscalibration.
- Incorporating NAVAS into policy analysis can enhance macroeconomic stability across both advanced and emerging market economies, regardless of net commodity trade position.

### Key methodological notes and robustness
- Empirical identification follows Kilian (2009a) separation of demand-driven and supply-driven shocks, using Baumeister and Guérin 2021 and Baumeister and Hamilton 2019 instruments.
- NAVAS constructed via OECD IOT 2018 and Leontief inverse to capture full upstream linkages.
- Panel local projections include country fixed effects, lagged dependent variables, lagged shocks, and interactions; time fixed effects excluded because first principal component explains approximately 81% of variation in country-level commodity export prices.
- Shock residuals’ white-noise properties validated by Ljung-Box diagnostics for all but one country (Latvia).

*Source: IMF Working Paper WP/25/208, "Commodity-driven Macroeconomic Fluctuations: Does Size Matter?", Patricia Gomez-Gonzalez, Maximiliano Jerez-Osses, Vida Maver, Jorge Miranda-Pinto, Jean-Marc Natal (October 2025).*

### Section 1

### Commodity-driven Macroeconomic Fluctuations: Does Size Matter?

### Main research question and contribution
- Examines how domestic interconnectedness of the commodity sector conditions the transmission of commodity price shocks, arguing that interconnectedness matters more than sector size.
- Emphasizes production linkages measured by the Network-Adjusted Value-Added Share (NAVAS) of the commodity sector rather than conventional measures like Domar weights.
- Combines panel local projections for OECD countries with a dynamic small open economy model with production networks to identify transmission channels.

### Key empirical setup and data
- Unbalanced annual panel of 66 countries, classified into AE and EMDEs according to the IMF taxonomy.
- Sample period: 1990 to 2018 (extended through 2023 when data availability permits).
- Country split: 37 countries in the AE group and 29 in the EMDE group.
- Primary data sources:
  - IMF’s Commodity Terms of Trade database; country-level export commodity prices proxied by the Commodity Net Export Price Index (weighted by net exports as a share of GDP, with rolling windows for time-varying weights).
  - Global Macro Database (GMD, Müller et al.2025) for macro indicators and controls; all variables deflated using the U.S. consumer price index (CPI) and expressed in U.S. dollars.
  - OECD Input-Output Tables (IOT) 2018 edition to construct production-network linkages and NAVAS.

### NAVAS: definition and interpretation
- NAVAS captures the share of total factor income generated by the commodity sector, accounting for all direct and indirect linkages (upstream and downstream).
- Notation and formulas preserved from the source:
  - Commodity sector value-added share: a_{N+1} = 1→!_{i} P^{M}_{N+1,i} / P_{N+1}Q_{N+1}
  - NAVAS:  ̃a_{N+1} = sum_{i=1}^{N+1} ”_{N+1,i} a_{i}, where ” = (I→#)→1 and # denotes the IO structure with elements #_{i,j}.
- NAVAS can be high because of:
  - The commodity sector itself generating a large share of value-added.
  - Reliance on intermediate inputs from sectors that create a lot of value-added.
  - Greater reliance on domestic rather than imported intermediate inputs.
  - Supplying key inputs to sectors that feed back into commodity production.
- Mechanisms when commodity prices increase:
  - Upstream channel: higher production costs for sectors using commodities as inputs (amplified when commodity sector is an important supplier — high NAVAS).
  - Wage/channel: higher commodity-sector revenues raise labor demand and nominal wages; however, greater NAVAS reduces the economy-wide real wage response because suppliers raise marginal costs of the commodity sector.
  - Net effect (as argued): the second effect tends to dominate, so higher NAVAS leads to smaller overall increases in wages and prices, effectively making the commodity sector a cost absorber and increasing real wages and the real price of commodities — important for NFA valuation.

### Stylized empirical findings
- Commodity sectors are generally larger and somewhat more interconnected within domestic production networks in EMDEs compared to AEs (consistent with prior literature).
- Using panel local projections (Jordà 2005) and exogenously identified commodity shock series (Baumeister and Guérin 2021; Baumeister and Hamilton 2019):
  - Demand-driven shocks:
    - Generate positive consumption responses.
    - The commodity sector’s interconnectedness (NAVAS) comoves positively with the consumption response.
    - Commodity sector size (Domar weight) has a modest and statistically insignificant effect at all horizons.
  - Supply-driven shocks:
    - Generate negative consumption responses.
    - NAVAS has a positive, large, and significant effect on consumption at all horizons, after controlling for commodity sector size.
- Overall implication: NAVAS is a key variable explaining cross-country heterogeneity in consumption responses to commodity terms of trade fluctuations; reliance on Domar weights alone understates the true transmission mechanism.

### Theoretical model and transmission channels
- Builds a dynamic small open economy (SOE) model with production networks, extending Silva et al. (2024) to analyze income and wealth channels.
- Two primary transmission mechanisms identified:
  - Income effect:
    - Higher global commodity prices raise real wages via increased labor demand in the commodity sector.
    - Greater NAVAS reduces the economy-wide real wage response because commodity-sector suppliers raise its marginal cost.
  - Wealth/valuation effect via Net Foreign Assets (NFA):
    - Valuation of NFA changes with commodity price movements, generating wealth effects that influence consumption.
    - If downstream propagation is strong, NFA valuation drops on impact, producing a negative wealth effect that induces consumption to decrease on impact, increase NFA, and smooth consumption increases into the future.
    - The comovement between NAVAS and these wealth effects is positive.
- Model calibration and simulations:
  - Wealth effects dominate consumption responses in the calibrated model.
  - Positive comovement between NAVAS and consumption responses on impact is observed, aligning with empirical results.
  - Relationship between commodity sector size and consumption responses on impact is weak in simulations, reinforcing NAVAS as the stronger transmission mechanism.
- Comparison with productivity shocks:
  - When NFA are valued in units of the importable good and their valuation follows an increasing path, impact responses of consumption to positive terms-of-trade and productivity shocks are qualitatively similar.
  - The positive relationship between NAVAS and the consumption response on impact requires the presence of wealth effects through NFA.
  - Whether the valuation path of NFA is increasing or decreasing is irrelevant for the NAVAS–consumption relationship.

### Methodological notes and robustness
- Empirical identification distinguishes shock origin following Kilian (2009a): separate analyses for demand-driven fluctuations (Baumeister and Guérin 2021) and supply-side disturbances (Baumeister and Hamilton 2019).
- NAVAS is constructed using OECD IOT 2018 and Leontief inverse to capture full upstream linkages.
- The paper documents NAVAS for both AE and EMDE economies by country and sector and estimates panel local projections following Jordà (2005) with external instruments for robustness.

*Source: IMF Working Paper WP/25/208, "Commodity-driven Macroeconomic Fluctuations: Does Size Matter?", Patricia Gomez-Gonzalez, Maximiliano Jerez-Osses, Vida Maver, Jorge Miranda-Pinto, Jean-Marc Natal (October 2025).*

### Section 2

### wpiea2025208-source-pdf - Section 2

### Commodity sector size and network-adjusted VA share (NAVAS)
- In the sample, the average Domar weight of commodity sectors in EMDEs is:
  - twice as large for metals,
  - three times as large for energy,
  - almost four times as large for agriculture,
  compared to AEs (see Tables VI and VII in the Appendix).
- Sectoral size alone is an incomplete measure of systemic relevance and total exposure to commodity price shocks.
- NAVAS comparisons across country groups:
  - On average, the commodity sector’s NAVAS is 31% higher in EMDEs than in AEs.
  - Energy exhibits the biggest difference in average NAVAS across country groups.
  - Metals and agricultural products exhibit the smallest NAVAS differences across country groups.
- Distributional heterogeneity of NAVAS across sectors:
  - Agricultural sectors show consistently high NAVAS and smaller variation across countries, indicating substantial use of domestic capital and labor in a network sense and structural importance across EMDEs and AEs.
  - Energy sectors show greater dispersion in domestic factor use as indicated by higher standard deviations in both country groups.
  - Metals display moderate NAVAS and comparatively lower variability in both country groups.
- Empirical notes:
  - Detailed country-specific NAVAS values—aggregate and disaggregated by energy, metals, and agricultural products—are presented in Tables IV and V in the Appendix.
  - Figure XIX of the Appendix plots the relationship between commodity sector size and NAVAS:
    - Across country groups, size and NAVAS are correlated.
    - Within AEs there is practically no correlation between size and NAVAS.
    - Within EMDEs there is only a weak positive relationship between size and NAVAS, indicating larger commodity sectors in EMDEs tend to be somewhat more central buyers of inputs in the domestic production network.
- Aggregate observation:
  - Commodity sectors are much larger and slightly more interconnected within the production network in EMDEs relative to AEs.
  - In AEs, commodity sector size tends to be small, but NAVAS (which averages around 0.6) indicates greater importance for macroeconomic fluctuations than size alone would suggest.
  - The right tail of NAVAS distribution in AEs markedly overlaps with the left tail in EMDEs, implying some AEs have commodity sectors more interconnected than many EMDEs—so size-based indicators could underestimate shock impacts.

### NAVAS and the correlation between commodity prices and aggregate consumption
- Data and measurement:
  - Figure II shows correlation between NAVAS (horizontal axis) and the correlation between countries’ cyclical consumption and cyclical commodity prices, measured annually over 1990–2023 for 66 countries.
  - Commodity prices are measured by the Commodity Net Export Price Index, weighted by net exports as a share of GDP and deflated using the U.S. consumer price index (CPI).
  - Countries in the AE group are blue; EMDEs are red. Triangles: commodity net importers. Circles: commodity net exporters.
- Empirical pattern:
  - Countries with more interconnected commodity sectors (higher NAVAS) tend to exhibit stronger co-movement between aggregate consumption and commodity prices.
  - Examples:
    - Advanced economies such as Australia, New Zealand, and Canada display higher NAVAS and stronger comovements.
    - EMDEs like Bulgaria, Hungary, Poland, and South Africa display lower NAVAS and weaker comovements.
  - The sign of the correlation is not simply determined by net importer/exporter status:
    - Many net importers exhibit a positive correlation between commodity price shocks and consumption.
    - Many net exporters show a negative correlation.
  - This counterintuitive pattern motivates deeper empirical panel projections and a theoretical small-open economy model.

### Panel local projections: methodology, identification, and empirical results
- Empirical strategy:
  - Use instrumental variable local projection (LP-IV) panel framework (Jordà 2005) with interaction terms to isolate NAVAS’s additional effect alongside baseline commodity price shocks.
  - Dynamic regression specification (cumulative change in log real consumption as dependent variable) includes:
    - Country fixed effects (μi).
    - Lagged dependent variables to control for dynamic persistence.
    - Lagged values of the commodity price shock (πi,t) to capture delayed transmission effects.
    - Interaction terms: πi,t ↑ NAVASi and πi,t ↑sizei, measured as deviations from cross-country annual means to isolate within-year cross-sectional heterogeneity.
  - Dependent variable defined as cumulative change in log real consumption for country i, denominated in U.S. dollars.
  - Time fixed effects are not included because a principal component analysis indicates the first factor accounts for approximately 81% of total variation in country-level commodity export prices.
- Shock identification (instrumental variables):
  - Two instruments used to capture different shock sources:
    - Demand instrument: real commodity price factor in Baumeister and Guérin 2021 (common demand factor from 23 industrial and agricultural commodities).
    - Supply instrument: oil supply shocks from Baumeister and Hamilton 2019 (proxy for broader supply-side disturbances).
  - Structural shocks are scaled to produce a 5 percent contemporaneous increase in real commodity prices for impulse response interpretation, implemented within the LP-IV framework following Stock and Watson 2018.
  - Robustness checks:
    - Identification strategy of Schmitt-Grohé and Uribe 2018 (assumes limited country market power) used to derive country-specific commodity price residuals; aggregated results provided in the Appendix.
    - Using oil supply shocks as instruments yields closely mirroring results.
- Key empirical findings:
  - General:
    - NAVAS plays a significant role in shaping transmission of commodity price shocks to consumption, independent of Domar weight (size).
    - Figures III and IV present cumulative impulse responses of aggregate consumption following a structural shock that increases commodity prices by 5% on impact, comparing countries at the 10th and 90th percentiles of NAVAS.
  - Demand-driven shocks (Figures III):
    - Real consumption responds positively and significantly on impact to a demand-driven 5% commodity price increase, with effects persisting over time.
    - Countries at the 90th percentile of NAVAS exhibit a stronger consumption response than those at the 10th percentile; the difference is statistically significant across horizons of interest.
    - Interaction coefficient θ(1) (NAVAS) amplifies the effect of commodity price shocks on domestic consumption.
    - Size interaction coefficients (ς) generally suggest a dampening effect and are modest relative to NAVAS amplification; some size estimates are not statistically significant.
    - Example: Thailand’s commodity sector is six times larger than Switzerland’s, but NAVAS values are nearly identical (0.68 and 0.65), yielding very similar impacts of commodity price shocks on consumption.
  - Supply-driven shocks (Figures IV):
    - Oil supply shock that raises real commodity prices produces more ambiguous effects dependent on NAVAS.
    - Countries at the 10th percentile of NAVAS experience a persistently negative and statistically significant consumption response, indicating economies with weakly integrated commodity sectors contract.
    - Countries at the 90th percentile show statistically insignificant responses, with an upward trajectory across most horizons—higher interconnectedness may help buffer adverse effects of supply-driven price shocks.
    - Panel (b) shows a negative and statistically significant direct effect of commodity price shocks on consumption across most horizons, while the interaction with NAVAS yields positive and consistently significant coefficients.
    - NAVAS interaction effects are markedly larger than corresponding size (Domar weight) interactions and often sign opposite to size coefficients.
- Interpretation:
  - Differences in commodity sector linkages drive heterogeneity in macroeconomic responses via interplay between real income and real wealth effects:
    - Demand-side price increases: consumption responses are positive across countries but vary with NAVAS; NAVAS acts as amplifier.
    - Supply-side price increases: resemble negative wealth shocks where adverse wealth effects can outweigh positive income effects; higher NAVAS can mitigate contractionary effects.
  - Relying solely on Domar weights may understate true transmission mechanisms; incorporating NAVAS is crucial to analyze propagation of commodity price shocks.

### Transition to the dynamic model
- Purpose:
  - Develop a quantitative model to understand mechanisms driving empirical relationships—specifically, why countries with higher NAVAS display stronger consumption responses to commodity terms-of-trade shocks.
- Model setup highlights:
  - Builds on Silva et al. 2024, adapted for multi-sector open economy with N+1 domestic production sectors.
  - Firms combine labor, domestic intermediate inputs, and imported intermediates to produce output.
  - Representative household:
    - Supplies labor, consumes a composite bundle of goods, and accesses international financial markets at fixed interest rate r.
    - Solves intertemporal optimization: max {Ct, Bt} →t=0 E0 Σt=0 θt C1→ω t →1 1→↼ subject to budget constraint P t C t + P N+1,t (B t + g(B t)) ↓W t  ̄ L + (1+r)P N+1,t B t→1, given B →1.
    - Parameters and variables: θ (discount factor), ↼ (inverse of intertemporal elasticity of substitution), Ct aggregate consumption, Pt associated price index.
- Model’s aim:
  - Quantitatively capture propagation of commodity price shocks through production networks, labor markets, and asset positions to explain aggregate consumption responses documented empirically.

*Source: wpiea2025208-source-pdf - Section 2*

### Section 3

### wpiea2025208-source-pdf - Section 3

### Model setup
- Foreign asset position Bt is denominated in units of the commodity good, priced at PN+1,t.  
- Adjustment costs on asset holdings: g(Bt) = ε2(Bt → ̄B)2, where ↽>0 determines cost intensity and ̄B is the steady-state level of debt.  
- Nominal wage: Wt. Labor supply ̄L is fixed.  
- Household cost-minimization for aggregate consumption PC subject to the consumption bundle C and imported bundle CM (Equations (4)–(5)). Consumption shares θi and θM satisfy ΣN+1 i=1 θi + θM = 1. Sector N+1 is the commodity sector; sector M is the imported goods bundle.  
- Sectoral production: Qi ̄Qi = Zi( ai (Li ̄Li)ωi↑1ωi + (1→ai)(Mi ̄Mi)ωi↑1ωi )ωiωi↑1 (Equation (6)). ai is labor share; (1→ai) is intermediate inputs share. Elasticity between labor and intermediates: ⇀i.  
- Intermediate input bundle Mi composed of domestic MD i and imported MM i inputs (Equation (7)); domestic-intermediate shares ⇁D i and elasticity φi between domestic and imported inputs.  
- Domestic intermediate composite MD i: MD i ̄MD i = ( ΣN+1 j=1 ⇁ij (Mij ̄Mij)εD i↑1 εD i )−εD i εD i↑1 (Equation (8)); ΣN+1 j=1 ⇁ij = 1. Elasticity among domestic intermediates: φD i.  
- Exogenous processes: logP↓N+1,t = ↼N+1 logP↓N+1,t→1 + πN+1,t; logZi,t = ↼Z logZi,t→1 + πi,t. Parameters ↼N+1 and ↼Z govern persistence; πN+1,t and πi,t are i.i.d. shocks.  
- Market-clearing conditions (Equations (9)–(12)):  
  - Qi,t = Ci,t + ΣN+1 j=1 Mji,t, for i = 1,...,N (9).  
  - QN+1,t = CN+1,t + Xt + ΣN+1 j=1 Mj,N+1,t (10).  
  - ̄L = ΣN+1 i=1 Li,t (11).  
  - Bt = (1+r)Bt→1 → g(Bt) + Xt → PM,t PN+1,t ( ΣN+1 i=1 MiM,t + CM,t ) (12). Trade balance enters evolution of foreign asset position.

### Static model intuition
- Aggregate consumption identity (Equation (13)):  
  Ct = Wt ̄L Pt + PN+1,t Pt ((1 + r)Bt→1 → (Bt + g(Bt))). Global commodity price in domestic currency: PN+1 = eP↓N+1; with numeraire e = 1, PN+1 = P↓N+1. Commodity price shocks affect aggregate consumption via the real wage W/P and valuation of net foreign assets via PN+1/P.  
- Aggregate price index (Equation (14) and differential form (15)):  
  logPt = ΣN i=1 (θi logPi → θi logθi) + θN+1 logPN+1 + θM logPM → (θN+1 logθN+1 + θM logPM) (14). With normalization PM = 1 and e = 1, dlogPt = ΣN+1 i=1 θi dlogPi and dlogPt = [1 ̃a N+1 ΣN+1 i=1 θi ̃ai ] dlogPN+1 = θ↔ ̃a ̃a ̃a N+1 dlogPN+1 (15). Coefficient b↓ ̃a ̃a ̃aN+1 captures amplification via sectoral linkages.  
- Nominal wage response (Equation (16)): dlogWt = 1 ̃a N+1 dlogPN+1. Larger ̃a N+1 implies smaller pass-through from commodity prices to wages.  
- Real wage response (Equation (17)): dlog(Wt Pt) = (1→θ↔ ̃a) ̃a N+1 dlogPN+1. Real wage decreases in θ↔ ̃a.  
- Valuation effect (Equation (18)): dlog(PN+1 Pt ) = ̃a N+1 θ↔ ̃a. Larger ̃a N+1 or smaller θ↔ ̃a lead to stronger valuation effect.  
- Distinction: commodity price shock vs commodity-sector productivity shock (Equations (19)–(20)). Productivity shocks change marginal costs directly and involve downstream linkage terms ”N+1,N+1 and ”i,N+1, giving greater role to downstream linkages compared to global commodity price shocks.

### Calibration (model inputs)
- Datasource: OECD data covering 66 countries and 44 sectors; calibration matches sectoral final consumption shares, IO linkages, and commodity sector net exports from 2018. Six commodity sectors aggregated into one for tractability (benchmark: one commodity sector + 38 non-commodity sectors).  
- Key calibrated parameters (Table I):  
  - ω = 0.961: Discount rate. Match interest rate r = 4%.  
  - ε = 2: Intertemporal elasticity of substitution. Uribe and Schmitt-Grohé 2017.  
  - θ = 0.000742: Bond holdings adjustment cost. Schmitt-Grohé and Uribe 2003.  
  - εN+1 = 0.53: Commodity price persistence. Uribe and Schmitt-Grohé 2017.  
  - π = 0.15: Commodity price elasticity to demand shifter. Baumeister and Hamilton 2019.  
  - ρ1 = Export demand elasticity. Cobb-Douglas Foreign Demand.  
  - ̄B = Country-specific steady-state asset level. Trade balance/GDP (M ̈uller et al. 2025).  
  - ςi = 3: Labor and intermediate inputs elasticity. Silva et al. 2024.  
  - φi = 0.6: Domestic and imported inputs elasticity. Silva et al. 2024.  
  - φD i = 0.2: Elasticity across domestic intermediate inputs. Silva et al. 2024.

### Results — aggregate consumption responses
- Main experiment: first-period percentage change in real consumption following a 1 percent increase in terms of trade (1 percent terms-of-trade shock).  
- Empirical match: Model simulations closely mirror data (Figure V). Higher NAVAS associated with stronger consumption-price co-movement; EMDEs generally exhibit higher NAVAS and higher correlations, though some AEs also show elevated NAVAS and strong co-movement.  
- NAVAS vs commodity sector size: No clear relationship between consumption responses and commodity sector size (Domar weight) — Figure VI shows heterogeneity.  
- Interpretation: Two transmission channels (valuation of NFA and real wage/income effect) interact with production-network structures and downstream passthrough to determine consumption responses.

### Disentangling channels (valuation vs income)
- Static decomposition identifies: valuation effect (NFA real value via PN+1/P) and income effect (real wage). Figure VII panels (a) and (b) report first-period percentage changes in real consumption to a 1 percent commodity price shock.  
- Valuation effect (panel a): Positive relationship between valuation effect and commodity-sector NAVAS. When foreign assets are denominated in commodity units, a rise in commodity prices that increases PN+1/P raises the real value of NFA for commodity-exporting economies, producing a positive wealth effect that tends to boost consumption in EMDEs (which often hold positive NFA in commodity terms). Most AEs experience negative consumption responses reflecting net debtor positions (negative wealth shock on impact) as the value of their debt increases. In dynamic settings, a stronger downstream propagation can reverse the path of PN+1/P, making it increasing and incentivizing countries to increase NFA on impact to finance increased future consumption (PIH and consumption smoothing).  
- Income effect (panel b): Higher commodity-sector NAVAS dampens real wage response. In EMDEs, higher commodity prices typically raise commodity-sector revenues and labor demand, pushing up wages and supporting consumption. In AEs, higher input costs in downstream sectors compress margins and dampen labor income, often producing negative impact on consumption.  
- Sensitivity to foreign-asset denomination: If NFAs are denominated in units of the importable good, the positive relationship between consumption responses and NAVAS in Figure VII panel (a) also holds (see Appendix Figure XIV), but the overall consumption response on impact is negative for all countries. All countries decrease consumption on impact when NFA are denominated in units of the importable good because PM/P decreases on impact (PM fixed while P increases due to downstream propagation); thereafter PM/P increases, incentivizing countries to increase NFA on impact by consuming less now to smooth consumption over time (PIH).

### Case studies: Kazakhstan vs South Africa
- Both are net commodity exporters with commodity sectors accounting for 39 percent of GDP, but differ in interconnectedness: NAVAS = 0.90 for Kazakhstan and NAVAS = 0.73 for South Africa. Model uses foreign assets valued in units of the commodity good.  
- Impulse responses to a 1 percent terms-of-trade shock (Figure VIII):  
  - Both countries experience a positive income effect (real wages rise in both). The income effect is larger in South Africa because its commodity sector is less interconnected (consistent with Equation (17)). Less interconnectedness implies larger wage increase needed to ensure zero profits in the commodity sector.  
  - Divergent consumption responses: Kazakhstan exhibits a large positive consumption response on impact, while South Africa displays a negative impact response.  
  - Mechanism: PN+1,t Pt increases more in Kazakhstan (supporting NFA valuation) while South Africa’s aggregate price index P increases substantially more, temporarily declining PN+1,t Pt and producing a negative wealth shock. Consequently, Kazakhstan experiences a positive wealth shock and consumption increases; South Africa endures a negative wealth shock and consumption falls.  
- Implication: Production network structures (NAVAS) and the currency/units in which foreign assets are denominated fundamentally alter macroeconomic effects of commodity terms-of-trade shocks.

_Italic source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025208-source-pdf.pdf_

### Section 4

### wpiea2025208-source-pdf - Section 4

### Impulse response evidence: Kazakhstan and South Africa
- Figure VIII compares responses to a terms of trade shock for Kazakhstan and South Africa.
- Key dynamics illustrated:
  - Consumption and savings (panel (a)):
    - Kazakhstan: increased real wages lead to higher consumption while NFA remains unchanged.
    - South Africa: increasing path for Pn+1/Pt makes saving and smoothing consumption more attractive; South Africa increases its current account and is able to increase consumption for a longer period of time, albeit by a smaller amount.
  - Real wage responses (panel (b)) are shown separately for KAZ and ZAF.
  - Price ratio Pn+1/Pt (panel (c)) differs across the two economies and drives valuation effects.
- Note on calibration: both economies are calibrated to start with an equal initial trade balance and use each economy’s input-output structure.

### Terms of trade shocks versus productivity shocks
- Kehoe and Ruhl (2008) point out that terms of trade shocks are not productivity shocks: terms of trade shocks have no effect on real GDP while productivity shocks do change real GDP.
- Silva et al. (2024) extend this distinction to multisector small open economies with production networks.
- The study contrasts consumption responses to:
  - A 1 percent terms of trade (ToT) shock (benchmark).
  - A 1 percent ToT shock with assets denominated in imported prices Pm rather than Pn+1.
  - A 1 percent ToT shock with assets denominated in Pt (no valuation effect).
  - A 1 percent productivity shock in the commodity sector (benchmark model).
- Figure IX findings (first-period response of real consumption, in percent, across NAVAS):
  - Panel (a) ToT shock (benchmark model): shows a positive relationship between commodity sector NAVAS and consumption responses when assets are denominated in commodity units.
  - Panel (b) ToT shock (assets denominated in Pm): also shows a positive relationship between NAVAS and consumption responses.
  - Panel (c) ToT shock (no valuation effect; assets denominated in Pt): shows that without NFA valuation effects, the relationship between NAVAS and consumption is negative.
  - Panel (d) Productivity shock: shows a positive relationship between NAVAS and consumption responses when assets are denominated in units of the commodity good.
- A key comparative result:
  - Panels (b) and (d) show that a productivity shock in the commodity sector can produce consumption responses qualitatively similar to a ToT shock when foreign assets are denominated in units of the importable good: both shocks cause consumption to decrease with impact for all countries and by similar magnitudes.
  - Despite similar consumption responses in magnitude, the underlying channels differ.

### Mechanisms emphasized by the model
- Commodity sector linkages amplify the wealth effect of ToT shocks by increasing the value of net foreign assets (NFA) of the economy.
- A highly interconnected commodity sector mitigates the response of the real wage (income effect).
- Overall, the wealth effect dominates, explaining the positive relationship observed in the data between commodity sector linkages (NAVAS) and aggregate consumption responses.
- When valuation effects are absent (assets denominated in Pt), only income effects operate and the consumption–NAVAS relationship is negative, contradicting empirical evidence from Figures II, III, and IV.

### Policy implications
- Macroeconomic and monetary policy frameworks should be adapted to account for the structure of domestic production networks.
- Central banks should account for production network structures when calibrating their response to commodity price movements.
- Adjusting policy frameworks to incorporate network structure can reduce the risk of policy miscalibration and enhance macroeconomic stability across both advanced and emerging market economies, regardless of their net commodity trade position.

### Appendix A.1 — Intuition and examples for Commodity Sector Network-Adjusted Value-Added Share (NAVAS)
- NAVAS captures factor demand from the commodity sector including the commodity sector’s suppliers’ factor usage.
- Simplified three-sector example: sectors C (commodity), A, and B with value-added shares vector a = (0.3, 0.3, 0.6)↔ (ordering C, A, B).
- If N+1,i = 0 for all i (commodity sector does not buy intermediates), then ̃aN+1 = aN+1 = 0.3.
- Example 1 — Supplier centrality heterogeneity:
  - IO matrix #1 and #2 both have sector C buying from all sectors equally, but differ in supplier centrality of C.
  - Greater domestic supplier importance (in #2) raises NAVAS due to expanded indirect labor use via input–output linkages.
  - NAVAS outcomes in the example:
    - Commodity sector does not buy domestic intermediates: ̃aN+1 = aN+1 = 0.3.
    - Lower supplier centrality case: NAVAS = 0.78.
    - Higher supplier centrality case: NAVAS = 0.85.
- Example 2 — Customer centrality to labor-intensive suppliers with fixed supplier centrality:
  - Two IO matrices #3 and #4 keep commodity sector supplier centrality constant but change where C buys intermediates (toward labor-intensive sector B or less labor-intensive sector A).
  - NAVAS varies with the labor intensity of upstream suppliers even if supplier centrality is fixed.
  - NAVAS outcomes in the example:
    - When C buys more from labor-intensive sector B: NAVAS = 0.79.
    - When C buys more from less labor-intensive sector A: NAVAS = 0.77.

*Source: IMF staff calculations and figures from the provided section.*

### Section 5

### Section 5

### NAVAS by Country and Commodity (2018)
- Table IV (Advanced Economies NAVAS, 2018) reports NAVAS by country for Aggregate, Energy, Metals, Agriculture. Selected entries (exact values preserved):
  - Australia: Aggregate 0.89, Energy 0.91, Metals 0.88, Agriculture 0.88
  - Canada: Aggregate 0.74, Energy 0.78, Metals 0.63, Agriculture 0.79
  - France: Aggregate 0.65, Energy 0.33, Metals 0.65, Agriculture 0.81
  - Japan: Aggregate 0.70, Energy 0.72, Metals 0.62, Agriculture 0.88
  - United States: Aggregate 0.81, Energy 0.76, Metals 0.82, Agriculture 0.90
  - (Full table lists NAVAS for 66 AE countries across Aggregate, Energy, Metals, Agriculture.)

- Table V (Emerging and Developing Economies NAVAS, 2018) reports NAVAS by country for Aggregate, Energy, Metals, Agriculture. Selected entries (exact values preserved):
  - Argentina: Aggregate 0.93, Energy 0.92, Metals 0.91, Agriculture 0.94
  - Brazil: Aggregate 0.85, Energy 0.79, Metals 0.83, Agriculture 0.89
  - China: Aggregate 0.84, Energy 0.71, Metals 0.82, Agriculture 0.91
  - India: Aggregate 0.82, Energy 0.48, Metals 0.71, Agriculture 0.96
  - Russia: Aggregate 0.93, Energy 0.95, Metals 0.89, Agriculture 0.89
  - Saudi Arabia: Aggregate 0.97, Energy 0.98, Metals 0.84, Agriculture 0.91
  - (Full table lists NAVAS for EMDE countries across Aggregate, Energy, Metals, Agriculture.)

### Heterogeneity in Commodity Linkages — Descriptive Statistics
- Advanced Economies (Table VI):
  - Size (Energy, Metals, Agriculture, Aggregate) mean: 0.05, 0.04, 0.04, 0.13
  - NAVAS (Energy, Metals, Agriculture, Aggregate) mean: 0.50, 0.61, 0.78, 0.61
  - Median Size: 0.04, 0.04, 0.04, 0.13
  - Median NAVAS: 0.48, 0.60, 0.78, 0.62
  - SD Size: 0.05, 0.03, 0.03, 0.06
  - SD NAVAS: 0.22, 0.13, 0.07, 0.15
  - Min Size: 0.00, 0.00, 0.00, 0.04
  - Min NAVAS: 0.21, 0.21, 0.55, 0.22
  - Max Size: 0.23, 0.11, 0.12, 0.31
  - Max NAVAS: 0.94, 0.88, 0.90, 0.89

- Emerging and Developing Economies (Table VII):
  - Size (Energy, Metals, Agriculture, Aggregate) mean: 0.14, 0.08, 0.15, 0.39
  - NAVAS (Energy, Metals, Agriculture, Aggregate) mean: 0.66, 0.73, 0.86, 0.80
  - Median Size: 0.10, 0.05, 0.11, 0.31
  - Median NAVAS: 0.67, 0.77, 0.88, 0.82
  - SD Size: 0.18, 0.06, 0.14, 0.22
  - SD NAVAS: 0.21, 0.16, 0.08, 0.12
  - Min Size: 0.00, 0.01, 0.01, 0.12
  - Min NAVAS: 0.23, 0.37, 0.65, 0.50
  - Max Size: 0.95, 0.23, 0.74, 1.05
  - Max NAVAS: 0.98, 0.95, 0.96, 0.97

### Panel Local Projections — Shock Identification and Results
- Identification approach:
  - International commodity prices are treated as exogenous for small open economies.
  - Commodity price dynamics are modeled with an AR(1) specification; residuals from this AR(1) capture unanticipated changes and display white-noise properties per Ljung-Box diagnostics.
  - Exception: Latvia is the only country for which the null hypothesis of no serial autocorrelation is rejected (one out of 66 countries).

- Shock scaling and impulse responses:
  - Commodity price shock scaled to increase the real commodity price by 5 percent on impact.
  - Impulse responses presented with 68 and 90 percent confidence intervals.
  - In economies with lower NAVAS, the consumption response is significantly negative and persistent.
  - In economies with higher NAVAS, the consumption response is positive but statistically insignificant.
  - Panel local projections present consumption coefficient estimates at annual horizons for:
    - Direct commodity price shock (red in figures)
    - Interaction with NAVAS (blue)
    - Interaction with Domar weight (orange)
    - Interaction with labor market slack (green, where included)

- Country-specific and aggregate residuals:
  - Country-specific surprise components are extracted from an AR(1) process and aggregated by country group.
  - Pre-COVID trough at onset of the Global Financial Crisis (GFC) followed by rebound in 2009, especially among AEs.
  - Renewed spike in residuals during 2015–2016 (sharp decline in oil prices and accommodative policies).
  - 2020s: sharp increase in residuals in 2020 (COVID-19 supply chain disruptions), steep decline in 2021–2022 (post-pandemic inflationary pressures), partial recovery in 2023 with residuals remaining in negative territory.

### Robustness: Interaction with Labor Market Slack
- Labor market slack proxy:
  - Slack measured as the deviation of each country’s unemployment rate from its trend.
- Results:
  - Including interaction between commodity price shock and labor market slack leaves main results robust.
  - Slack interaction is statistically insignificant in the demand-driven estimates (highlighted in green in Figure XII, panel (b)).
  - Interaction with sectoral size loses significance across most horizons (increase in standard errors).
- LP-IV (supply-driven) results:
  - Consumption responses to supply-driven commodity price increases remain robust and closely resemble baseline results.
  - For supply-driven shocks, coefficients remain statistically significant across most horizons even when labor market slack is included.
  - Standard errors surrounding interaction with sectoral size increase in both demand- and supply-driven cases.

### Theoretical Model — Key Equations and Mechanisms
- Price and cost relationships:
  - Total differentiating sectoral marginal costs yields:
    - dlogP_i = a_i dlogW + sum_{j=1}^{N+1} ψ_{ij} dlogP_j + ϖ_i dlogP_M → dlogZ_i for all i = 1,2,...,N+1 (Equation numbering and symbols preserved as in source).
  - Definitions:
    - a_i = WL_i / P_i Q_i = WL_i / TC_i
    - ϖ_i = P_M M_iM / TC_i
    - ψ_{ij} = P_j M_{ij} / TC_i
  - Stacked system and inversion lead to expressions for dlogP in terms of dlogW and dlogZ, with network-adjusted labor share vector defined as ā where typical element ā_i = sum_{h=1}^{N+1} ψ_{ih} a_h.

- Terms of trade (ToT) shock (A.4.1):
  - Under numeraire choice P_M = 1, and with dlogP_{N+1} = ā_{N+1} dlogW, solving yields:
    - dlogW = 1 / ā_{N+1} dlogP_{N+1}
    - dlogP_i = (sum_{h=1}^{N+1} ψ_{ih} a_h) * (1 / ā_{N+1}) dlogP_{N+1}
    - Aggregate dlogP_t = sum_{i=1}^{N+1} θ_i dlogP_i = sum_{i=1}^{N+1} θ_i ā_i (1 / ā_{N+1}) dlogP_{N+1}

- Productivity shock to the commodity sector (A.4.2):
  - Positive productivity shock impacts real wages more strongly when the commodity sector is an important supplier of intermediates (dependence on ψ_{i,N+1} and ψ_{N+1,N+1}).
  - Solution under dlogZ_{N+1} nonzero and dlogP_{N+1}=ā_{N+1} dlogW gives:
    - dlogW = 1 / (ā_{N+1} ψ_{N+1,N+1}) dlogZ_{N+1}
    - dlogP_i = (sum_{h=1}^{N+1} ψ_{ih} a_h) * (1 / (ā_{N+1} ψ_{N+1,N+1})) ψ_{i,N+1} dlogZ_{N+1}
    - Aggregate dlogP_t = sum_{i=1}^{N+1} θ_i dlogP_i = sum_{i=1}^{N+1} θ_i [ā_i (1 / (ā_{N+1} ψ_{N+1,N+1})) ψ_{i,N+1}] dlogZ_{N+1}

### Denomination of Assets and Interpretation of Terms of Trade Changes (A.5)
- Two cases for asset denomination and implications for the trade balance and asset valuation:
  - Assets denominated in commodity units (P_{N+1} units):
    - An increase in the terms of trade lowers the relative price of imports and raises the domestic valuation of foreign assets when both are measured against the rising price of the commodity good.
    - Expressions preserved from source:
      - P_{N+1,t} B_t = (1+r) P_{N+1,t} B_{t→1} → P_{N+1,t} g(B_t) + P_{N+1,t} X_t → P_{M,t} (sum_{i=1}^{N+1} M_{iM,t} + C_{M,t}) ./(...)
      - Trade Balance B_t = (1+r) B_{t→1} → g(B_t) + X_t → P_{M,t} / P_{N+1,t} (sum_{i=1}^{N+1} M_{iM,t} + C_{M,t}) ./(...)

  - Assets denominated in importable units (P_M units):
    - An increase in the terms of trade enhances the value of exports and increases the real value of foreign assets relative to the importable good.
    - Expressions preserved from source:
      - P_{M,t} B_t = (1+r) P_{M,t} B_{t→1} → P_{M,t} g(B_t) + P_{N+1,t} X_t → P_{M,t} (sum_{i=1}^{N+1} M_{iM,t} + C_{M,t}) ./(...)
      - Trade Balance B_t = (1+r) B_{t→1} → g(B_t) + P_{N+1,t} / P_{M,t} X_t → (sum_{i=1}^{N+1} M_{iM,t} + C_{M,t}) ./(...)

*Source: Section 5 of the provided PDF content.*

### Section 6

### Section 6

### A.5.1 Euler Equation for Different Assets denomination
- Denominated in units of exportable good (P_{N+1})
  - Budget constraint:
    - P_t C_t + P_{N+1,t} (B_t + g(B_t)) ↓ W_t ̄ L + (1+r) P_{N+1,t} B_{t→1}. (29)
  - Lagrangian:
    - L = E_0 ↑ " t=0 θ_t  C_{1→ω t →1 1→↼ + ▷_t  W_t ̄ L + (1+r) P_{N+1,t} B_{t→1} → P_t C_t → P_{N+1,t} (B_t + g(B_t))   (30)
  - FOCs:
    - ◁L ◁C_t: θ_t C_{→ω t →▷ t} P_t = 0 ↗▷_t = θ_t C_{→ω t} P_t (31)
    - ◁L ◁B_t: →▷_t P_{N+1,t} (1 + g↔(B_t)) + θ_{t+1} E_t [▷_{t+1} (1 + r) P_{N+1,t+1}] = 0 (32)
  - Euler Equation (general prices):
    - Substitute (36) into (37) and simplify:
      - C_{→ω t} P_t = θ(1 + r) / (1 + g↔(B_t)) E_t  C_{→ω t+1} P_{t+1} · P_{N+1,t+1} / P_{N+1,t} . (33)

- Denominated in units of importable good (P_M)
  - Budget constraint:
    - P_t C_t + P_{M,t} (B_t + g(B_t)) ↓ W_t ̄ L + (1+r) P_{M,t} B_{t→1}. (34)
  - Lagrangian:
    - L = E_0 ↑ " t=0 θ_t  C_{1→ω t →1 1→↼ + ▷_t  W_t ̄ L + (1+r) P_{M,t} B_{t→1} → P_t C_t → P_{M,t} (B_t + g(B_t))   . (35)
  - FOCs:
    - ◁L ◁C_t: θ_t C_{→ω t →▷ t} P_t = 0 ↗▷_t = θ_t C_{→ω t} P_t, (36)
    - ◁L ◁B_t: →▷_t P_{M,t} (1 + g↔(B_t)) + θ_{t+1} E_t [▷_{t+1} (1 + r) P_{M,t+1}] = 0. (37)
  - Euler Equation:
    - Substitute (36) into (37) and simplify:
      - C_{→ω t} P_t = θ(1 + r) / (1 + g↔(B_t)) E_t  C_{→ω t+1} P_{t+1} · P_{M,t+1} / P_{M,t} . (38)

- Special case (denominated in P_t)
  - If the export good is the numeraire so that P_{N+1,t} ↘ 1 for all t, (38) collapses to:
    - C_{→ω t} P_t = θ(1 + r) / (1 + g↔(B_t)) E_t  C_{→ω t+1}  . (39)
  - Purpose: "Here to get a deeper understanding of the mechanism behind a ToT and a Productivity shock."

### Figures and empirical patterns (as presented)
- Figure XIV: Consumption Response and NAVAS: Assets Denominated in Units of the Imported Good
  - NAVAS axis values shown: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
  - Consumption Response in % scale: -4.00, -3.50, -3.00, -2.50, -2.00, -1.50, -1.00
  - Groups plotted: AEs, EMDEs

- Figure XV: Mechanism reactions to a ToT shock and a productivity shock, no valuation effect
  - Panels show responses over Periods 1 to 20.
  - Response scales include:
    - -0.05 to 0.30 (Response, %)
    - -0.06 to 0.14 (Response, %)
    - -0.10 to 0.60 (Response, %)
    - -1.20 to 0.00 (Response, %)
  - Series labeled: Agg. consumption KAZ, Agg. consumption ZAF, Current account KAZ, Current account ZAF, Real Wage KAZ, Real Wage ZAF, P_{n+1}/P_t KAZ, P_{n+1}/P_t ZAF
  - Two shock types shown:
    1) Commodity price shock no valuation effect
    2) Productivity shock no valuation effect

- Figure XVI: Mechanism reactions to a ToT shock and a productivity shock
  - Panels show responses over Periods 1 to 20.
  - Response scales include:
    - -0.05 to 0.40 (Response, %)
    - -0.06 to 0.14 (Response, %)
    - -2.00 to 2.50 (Response, %)
    - -1.20 to 0.00 (Response, %)
  - Series labeled: Agg Consumption KAZ, Agg Consumption South Af, Current account KAZ, Current account South Afr, Real Wage KAZ, Real Wage ZAF, P_{n+1}/P_t KAZ, P_{n+1}/P_t ZAF
  - Two shock types shown:
    1) Commodity price shock
    2) Productivity shock

- Figure XVII: Mechanism reactions to a ToT shock with assets denominated on P_m (units of the foreign good)
  - Panels show responses over Periods 1 to 20.
  - Response scales include:
    - -2.00 to 2.00 (Response, %)
    - 0.00 to 0.35 (Response, %)
    - -1.20 to 0.00 (Response, %)
  - Series labeled: Agg. consumption KAZ, Agg. consumption ZAF, Current account KAZ, Current account ZAF, Real Wage KAZ, Real Wage ZAF, P_m/P_t KAZ, P_m/P_t ZAF

- Figure XVIII: Consumption Responses against Wealth and Income NAVAS Productivity shock
  - Panel (a) Valuation Effect:
    - Valuation NAVAS axis: 0.4 0.6 0.8 1 1.2 1.4 1.6
    - Consumption response in % scale: -5.00, -4.00, -3.00, -2.00, -1.00, 0.00
    - Groups: AEs, EMDEs
  - Panel (b) Income Effect:
    - Income NAVAS axis: 0 0.5 1 1.5 2 2.5
    - Consumption response in % scale: -5.00, -4.00, -3.00, -2.00, -1.00, 0.00
    - Groups: AEs, EMDEs
  - Note: "NAVAS operates through two tranmission channels, the valuation of NFA, shown in panel (a), and real wages, shown in panel (b). Both panels display the first-period percentage response of real consumption to a 1 percent commodity sector productivity shock."

- Figure XIX: Size and NAVAS of the Commodity Sector
  - Size axis values shown: 0 0.2 0.4 0.6 0.8 1 1.2
  - NAVAS axis values shown: 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00
  - Groups: AEs, EMDEs
  - Note: "Size is the ratio between commodity sectors total sales to GDP in 2018. NAVAS is the network-adjusted value-added share of the commodity sector in 2018."

- Figure XX: Productivity shock no valuation
  - NAVAS axis values shown: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
  - Consumption Response in % scale: 0.00, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30
  - Groups: AEs, EMDEs
  - Note: "NAVAS is the network-adjusted value-added share of the commodity sector in 2018."

- Data and sources referenced in figures: "Source: OECD and IMF staff calculations."

*Commodity-driven Macroeconomic Fluctuations: Does Size Matter? Working Paper No. WP/2025/208*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025208-source-pdf.pdf_