## Section 8 concludes (wpiea2024081-print-pdf)

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### Empirical evidence on oil shocks and wage adjustment (US and euro area)
- Identification and data:
  - Surprise changes in oil futures prices around OPEC meeting announcements (Känzig (2021)); one-day window using WTI crude futures starting in 1983 (NYMEX).
  - Oil price surprise series 1983–2019 aggregated quarterly used as instrument in a proxy SVAR.
  - Proxy SVAR variables: real oil prices (deflated by core or headline prices), oil production, global industrial production, real GDP, core or headline CPI inflation, short-term policy rate, nominal wage inflation.
  - Logs for quantities and relative prices; policy rates in percentage points; price and wage inflation as log differences.
  - Confidence bands: moving block bootstrap (block size 24); 68 and 90 percent bands from 10,000 bootstrap replications.

- Baseline SVAR results (US):
  - An oil supply-driven 10 percent increase in oil prices with a 2-year half life:
    - Increases core inflation by about 0.1 percentage point for a few quarters.
    - Reduces the nominal wage by about 0.2 percent on impact.
    - Implied producer real wage falls by roughly 0.3 percent over the first 2 years and by nearly 0.6 percent after 5 years.
    - Real GDP falls by more that 0.4 percent after 2 years.
  - Interpretation: limited ability of US households and labor unions to achieve compensation for higher prices through higher wages.

- Baseline SVAR results (euro area):
  - Nominal wage growth persistently increases (consistent with more centralized wage bargaining and higher unionization).
  - Rising wages put upward pressure on core inflation; ECB raises interest rates significantly.
  - Output contraction about the same as in the US despite lower oil dependence.
  - Implied producer real wage: point estimate modestly positive but not statistically significant.
  - Interpretation: mild evidence of indexation mechanisms in European wage setting compensating households for energy price shocks.

- Headline vs core inflation and real consumer wage:
  - Impact on headline inflation stronger on impact in both US and euro area — up to 1 (0.5) percent in the same quarter — but much less persistent.
  - Implied consumer real wage (nominal wage divided by headline CPI) falls notably more initially than implied producer real wage in the US.
  - Smaller pass-through to headline inflation relative to core inflation in the euro area may reflect higher excise taxes on liquid fuels and more rigid downstream retail prices.
  - Regardless of core or headline inflation used, implied real wage in the euro area falls less than in the US.

- Robustness checks:
  - Imposing mean reversion of real wage impulse responses — US real producer wages still decline significantly; Europe rises somewhat but not significant at the 90 percent level.
  - Excluding shocks smaller than one standard deviation — no evidence that shock size matters for transmission to real wages.
  - Alternative identification with sign restrictions (Baumeister and Hamilton (2019)) — impact on real wages remains robust.
  - Note: Känzig shocks may contain a demand component (oil and global industrial production do not necessarily fall immediately), reinforcing interpretation that evidence lends little support for strong indexation of wage formation to headline inflation.

### Simple closed economy model with energy — structure and key primitives
- Households:
  - Preferences: E_t sum_{s=0}^∞ β^s ( (C_{t+s}(h) − κ C_{t+s−1}(h))^{1−1/σ} / (1−1/σ) − χ_0 N_{t+s}(h)^{1+χ} /(1+χ) ), with 0≤β,κ<1 and σ,χ,χ_0 >0.
  - Budget constraint: P_{C,t} C_t(h) + B_t(h) + T_t = (1+τ_w) W_t(h) N_t(h) + R_{K,t} K + I_{t−1} B_{t−1}(h) + D_t + Ξ_t(h).
  - Labor differentiated with elasticity parameter θ_w >0; aggregate N_t as specified.
  - Wages set in Calvo fashion (probability 1−ξ_w to reoptimize); non-reoptimizers index wages by ˜Π_{W,t} = Π_y^{ι_y} Π_{Y,t}^{ι_y} Π_{C,t}^{ι_c} Π_{Y,t−1}^{1−ι_y−ι_c}.
  - Perfect insurance across idiosyncratic wage risk (Ξ_t transfers); static subsidy to labor income τ_w ≡ (1+θ_w)/θ_w.

- Firms and production:
  - Final goods combine non-energy Y_t and energy OC,t with CES (parameters ω_c, η_c); objective includes (1−τ_{C,t}) P_{O,t} O_{C,t}.
  - Intermediate firms combine energy O_{Y,t} and capital-labor bundle V_t = K_t(f)^α N_t(f)^{1−α}; CES parameters ω_y, η_y.
  - Profits include markup and energy subsidy τ_{Y,t}; Calvo pricing with probability 1−ξ; non-optimizers index prices by ˜Π_{Y,t}.
  - Energy supply exogenous endowment Y_{O,t}; market clearing Y_{O,t} = O_{C,t} + O_{Y,t}.

- Monetary and fiscal authorities:
  - Monetary policy: I_t = I + ψ_π (Π_{Y,t} − Π) + ψ_y (Y_t / Y_{pot,t} − 1), with ψ_π >1, ψ_y ≥0; central bank responds to producer (core) inflation Π_{Y,t}.
  - Fiscal authority: lump-sum taxes T_t or debt B_{G,t} finance subsidies τ_{C,t}, τ_{Y,t} with B_{G,t} = I_{t−1} B_{G,t−1} + τ_{Y,t} P_{O,t} O_{Y,t} + τ_{C,t} P_{O,t} O_{C,t} − T_t.
  - Ricardian equivalence holds.

### Transmission of energy shocks and subsidies — equivalence results
- Proposition 1 (exact statement):
  - Any sequences {τ_{C,t}, τ_{Y,t}} and {̊τ_{C,t}, ̊τ_{Y,t}} yield identical equilibrium allocations iff (1−τ_{C,t})/(1−τ_{Y,t}) = (1−̊τ_{C,t})/(1−̊τ_{Y,t}) for all t.
  - Immediate implication: subsidies work only if differentiated between households and firms; identical rates have no effect on allocations.

- Corollary 1 (linearized statement):
  - Subsidizing energy use by households is equivalent to taxing energy use by firms at the same rate (in linearized model).

- Intuition:
  - Only effective energy costs (1−τ_{C,t}) P_{O,t} and (1−τ_{Y,t}) P_{O,t} matter; P_{O,t} never appears independently.
  - A household subsidy lowers household effective energy price but raises pre-subsidy P_{O,t}, increasing unit energy cost to firms — a household subsidy can act like a tax on firms.
  - To first order, τ_{C,t} = − τ_{Y,t}.
  - Equivalence holds for allocations and all prices except pre-subsidy energy price.
  - Caveats: derived in closed-economy with exogenous energy and Ricardian equivalence; equivalence may break if energy production responds or in small open economy.

### Log-linearized model with full price flexibility (key results)
- Flexible-price energy market clearing:
  - −y_{O,t} = Ψ_{oc} (p^f_{O,t} − τ_{C,t}) + Ψ_{oy} (p^f_{O,t} − τ_{Y,t}), Ψ_{oc}, Ψ_{oy} > 0.
  - Potential pre-subsidy energy price p^f_{O,t} is the only endogenous variable; falls in energy endowment or increases in subsidies raise p^f_{O,t}.

- Labor market and real wages (full flexibility):
  - Producer demand real wage: w^f_t = −(ω_y/(1−ω_y)) (p^f_{O,t} − τ_{Y,t}) − α n^f_t.
  - Household supply real wage: w^f_t = χ n^f_t + ω_c (p^f_{O,t} − τ_{C,t}) + (1/σ) c^f_t.
  - Consumer subsidy τ_{C,t} reduces net energy price to households (outward labor supply), raises energy price paid by firms (downward labor demand) → product real wage depressed; hours ambiguous.

- Condition ensuring product real wage decline with consumer subsidy:
  - Product real wage declines if ω_c [1−(1/σ) η_c] Ψ_{oy} + [ω_y/(1−ω_y)][(1/α) − 1][χ + 1/σ] + (1/σ) η_y ] Ψ_{oc} > 0.
  - Stronger sufficient condition: η_c < σ (energy and non-energy goods are complements in household preferences).

- Headline and core inflation; wage dynamics:
  - Headline inflation: π_{C,t} = (1 − ω_c) π_{Y,t} + ω_c (π_{O,t} − Δτ_{C,t}).
  - Price Phillips curve: π_{Y,t} = β E_t π_{Y,t+1} + κ_p [ω_y (p_{O,t} − τ_{Y,t}) + (1 − ω_y) (w_t + α n_t)], κ_p ≡ (1−ξ)(1−βξ)/ξ.
  - Wage Phillips curve: π_{W,t} − ̃π_{W,t} = β E_t{π_{W,t+1} − ̃π_{W,t+1}} + κ_w [χ n_t + (1/σ) c_t − w_{C,t}], with κ_w defined; stronger indexation to headline inflation amplifies transmission from energy shocks to nominal wages.

- Gap representation:
  - Pre-subsidy energy price deviation: ˆp_{O,t} = Ψ_{on} ˆn_t + Ψ_{ow} ˆw_t, Ψ_{on}, Ψ_{ow} > 0.
  - Price and wage Phillips curves in gap form: π_{Y,t} = β E_t π_{Y,t+1} + κ_p [Ψ^{mpl}_n ˆn_t + Ψ^{mpl}_w ˆw_t]; π_{W,t} − ̃π_{W,t} = β E_t{π_{W,t+1} − ̃π_{W,t+1}} + κ_w [Ψ^{mrs}_n ˆn_t + Ψ^{mrs}_w ˆw_t].

- Central analytical result (transmission of consumer energy subsidies to core inflation):
  - If central bank focuses on closing the output/employment gap, then consumer energy subsidies lower potential real producer wage w^f_t (under complementarity or condition (21)).
  - Wage law-of-motion yields E_t Σ_{j=0}^∞ β^j ˆw_{t+j} = D w^f_t with D < 0.
  - Combining with price Phillips curve implies subsidies increase core inflation.
  - Proposition 2 (exact statement): Under (i) energy and non-energy goods are complements, (ii) dynamic component of wage indexation relies on producer price inflation, (iii) monetary authority focuses on closing output gap — an increase in household energy subsidies leads to an increase in core inflation.

- Persistence and pre-subsidy price effects:
  - More persistent subsidy (higher ρ) → more lasting increase in core inflation; ρ → 1 implies persistent wage gap and core inflation deviation.
  - Subsidies raise potential pre-subsidy energy price; with sticky prices/wages, equilibrium pre-subsidy price rises further.

### Key takeaways from analytical section
- In global or highly segmented energy markets, general equilibrium makes consumer energy subsidies likely counterproductive.
- Endogenous adjustment of pre-subsidy energy prices makes a consumer subsidy akin to a firm energy tax.
- Under empirical wage indexation (linked to producer-price inflation), consumer subsidies exacerbate policy tradeoffs by increasing core inflation for a given output gap.
- For consumer subsidies to be potentially productive, wages need strong indexation to headline inflation.
- Analytical results align with prior literature: negative energy supply shocks can raise core inflation when policy targets employment; consumer subsidy acts similarly to a negative shock to producers.

### Can energy subsidies reduce inflation? (Section 5.3 — simulations and findings)
- Simulation setup:
  - Two subsidy types: to households and to firms.
  - Initial subsidy value set at 17%.
  - Subsidies have same persistence as energy shocks.
  - Baseline energy supply shock calibrated to produce a 50% increase in real energy prices absent fiscal measures.
  - Subsidies ≈ 1 percent worth of annual GDP in the first year of deployment.

- Macroeconomic effects of household energy subsidies:
  - Direct: mitigate cost of living increases; headline price level responds downward; lower consumer prices reduce wage pressure so real producer wage falls.
  - Indirect/counterproductive: households cut energy consumption less → pre-subsidy energy price faced by firms rises → firms’ marginal cost and core inflation increase.
  - Reduction in headline inflation is substantially reduced; after about two years the cost of living index can be higher than without the subsidy.
  - Activity impact: subsidies ≈ 1 percent of annual GDP in year one have very small impact on economic activity.
  - Quantitative note: a 17% subsidy halves the impact of a 50% increase in the pre-subsidy energy price as faced by an atomistic subsidized agent.

- Macroeconomic effects of firm energy subsidies:
  - Mirror image due to equivalence result: subsidizing household energy prices ↔ taxing energy prices for firms.
  - Firm subsidies drive up pre-subsidy energy price, shifting burden from subsidized to non-subsidized agents.
  - Firm subsidies can bring core inflation down, aiding monetary policy’s medium-term stabilization.
  - Short-term cost to households: higher per-unit energy prices.

- Robustness: alternative calibrations (Figure 7 experiments)
  - Wage indexation flip:
    - Baseline: wages indexed to core inflation (ιy = 0.75, ιc = 0).
    - Alternative: wages indexed to headline inflation (ιy = 0, ιc = 0.75).
    - Result: subsidy achieves sizable wage moderation but still raises producers’ prices because household subsidy increases energy use and energy price paid by firms.
  - Energy substitutability in production:
    - Assume ηy → 1 (easy substitution) together with wages indexed to headline inflation.
    - Only under these simultaneous, highly unrealistic assumptions does a household subsidy lead to a fall in core inflation and keep headline price level below the no-subsidy case.

- Optimal energy subsidy (representative-agent model):
  - Planner chooses consumer energy subsidy τC,t (negative values → subsidy to firms) and interest rate Rt under full commitment.
  - If subsidies available, optimal policy uses subsidies to firms rather than households:
    - Contains increase in marginal cost and lowers core inflation.
    - Requires less contraction in economic activity.
  - Conclusion: consumer energy subsidies are counterproductive for stabilization in representative-agent environment.

### TANK extension (hand-to-mouth households) — key results
- TANK modifications:
  - Keynesian households (subscript K): consume disposable income each period, hold no assets, budget PC,t CK,t = Wt NK,t − PC,t TK,t, no bargaining power, work same hours as average Ricardian household.
  - Ricardian households (superscript R) retain baseline behavior.
  - Taxes: Tj,t − TjY = tj,t + φj (BG,t − BG/PY) with φK = 0, φR > 0; baseline tj = 0.
  - Equivalence results continue to hold under this transfer formulation.

- TANK calibration:
  - Share of Keynesian households λ = 0.4.
  - Relative consumption CK/CR = 0.5.
  - Steady state debt-to-output BG/Y = 2.4 (consistent with debt-to-annual-GDP ratio of 60%).
  - φK, φR = 0, 0.01.

- Energy subsidies vs transfers (TANK results):
  - Presence of Keynesian agents amplifies aggregate demand effects: deeper fall in output; core inflation increase more short-lived; consumption inequality rises.
  - Energy subsidies more counterproductive for core inflation in TANK than representative-agent model, though they curb consumption inequality somewhat.
  - Targeted lump-sum transfers to Keynesian households (adjusted to prevent any increase in consumption inequality) outperform energy subsidies:
    - Better at limiting contraction in output and consumption.
    - Much less inflationary.
  - Rationale: transfers do not distort relative energy prices, preserving incentives to save energy and limiting rise in pre-subsidy energy price and firms’ marginal cost.

- Optimal subsidies in TANK:
  - Optimal monetary policy alone: more focused on core inflation; simple rule is a worse approximation when hand-to-mouth agents exist; optimal policy cannot prevent significant rise in consumption inequality.
  - Adding subsidies:
    - Short-term: steep, short-lived consumer subsidy shields Keynesian households at peak energy prices.
    - Medium-term: small persistent firm subsidy helps prevent significant rise in core inflation via forward-looking price setting.
  - If targeted transfers are available to prevent any rise in consumption inequality, optimal policy subsidizes only firms (similar to representative-agent optimal policy).
  - Welfare gains from including transfers are large even if transfers not chosen fully optimally.
  - Policy implication: consumer subsidies can shield vulnerable households, but transfers are generally more effective and less inflationary when feasible.

### Open economy extension — setup and key mechanisms
- Two-country model (Home and Foreign); world population normalized to unity; Home size 0< ζ <1.
- Consumption aggregator for firm j and non-energy composition defined (Equations (36)–(37)).
- Prices in destination currency (local currency pricing) with Calvo probability ξ_m for exports.
- Energy endowments tradable with law of one price: P_O,t = ε_t P^*_O,t, ε_t is nominal exchange rate (Home currency per unit of Foreign).
- Financial intermediation friction (Gabaix and Maggiori (2015)):
  - Financiers take positions F_t in domestic currency and −F_t/ε_t in foreign currency; can divert up to min{1, Γ|F_t P_Y,t|}.
  - Incentive compatibility: V_t(k) ≥ Γ F_t(k)^2 / P_Y,t.
  - Representative financier value: V_t(k) = 1/I_t E_t[ I_t F_t(k) − I^*_t ε_{t+1} F_t(k) / ε_t ].
  - Γ parameter captures FX market shallowness.

- World energy market clearing:
  - ζ Y_{O,t} + (1−ζ) Y^*_{O,t} = ζ(O_{C,t} + O_{Y,t}) + (1−ζ)(O^*_{C,t} + O^*_{Y,t}).

- Net foreign assets evolution (Home):
  - B_t = [ (1−ζ) I_{t−1} + ζ (ε_t/ε_{t−1}) I^*_{t−1} ] B_{t−1} + ε_t P^*_M,t M^*_t − P_M,t M_t + P_O,t (Y_{O,t} − O_{Y,t} − O_{C,t}).

- Risk-augmented UIP:
  - I_t = E_t[ I^*_t ε_{t+1} / ε_t ] − Γ I_t B_t / P_Y,t.
  - World energy prices affect net foreign assets and risk premia; energy price increases raise (lower) the premium for net energy importers (exporters), causing depreciation (appreciation).

- Breakdown of equivalence:
  - In open economy, energy supply is not fixed due to imports; household and producer subsidies are distinct instruments and can be complementary.
  - Small open economy limit (ζ→0): pre-subsidy energy price (in foreign currency) does not depend on that country's subsidies.
  - Additional channels:
    - Exchange rate channel: subsidies that increase domestic energy use worsen energy trade balance → exchange rate depreciation → raises domestic-currency energy cost to firms and price of non-energy imports → weakens subsidy efficacy.
    - Spillovers: foreign subsidies can amplify domestic inflation for importers; exporters may benefit via appreciation offsetting world price rise.

### Open economy calibration and differentiated responses
- Calibration highlights:
  - Relative country size ζ = 0.001.
  - Share of energy production Y_O/Y = 0.05, 0.15.
  - Share of imports in non-energy consumption ω = 0.1.
  - Elasticity of substitution η = 0.8.
  - Calvo probability for exports ξ_m = 0.9.
  - Indexation for export prices ι_m, ι^*_m = 0.5, 0.5.
  - FX market shallowness Γ = 0.02.

- Energy importers vs exporters (small open economy):
  - Importers:
    - Activity contracts; core inflation persistently increases; reactions stronger than closed economy due to terms-of-trade deterioration.
    - Households borrow from abroad; net foreign assets fall; exchange rate weakens; exchange rate pass-through to energy prices is complete → larger inflation rise.
  - Exporters:
    - Positive wealth shock from higher world energy prices; non-energy output declines less; consumption stronger.
    - Terms-of-trade improvement → accumulate foreign assets → exchange rate appreciation.
    - With large enough energy trade surplus and openness, core inflation may decline.

- Role of openness to non-energy trade:
  - Greater openness allows trade balance to absorb energy shocks, bringing importer/exporter responses closer; can induce Dutch disease for exporters in medium run; stronger pass-through increases divergence of core inflation responses.

- Role of FX market depth (Γ):
  - Small Γ (deep FX markets) → easier cross-border flows; larger divergence in net foreign assets but smaller UIP risk-premium effect → smaller exchange rate adjustment → more similar core inflation dynamics across countries.

### Domestic and multilateral effects of subsidies (open economy)
- Consumer subsidies in small open economy importer:
  - When only Home uses subsidy:
    - Effective at limiting fall in consumption and activity.
    - World energy prices unaffected by small Home subsidies, but core inflation still rises due to exchange rate depreciation from worsening net foreign assets.
    - Increase in core inflation smaller than in closed economy; overall macro effect more favorable for SOE.
  - When rest of world (RoW) subsidizes households:
    - RoW subsidies raise global energy prices, worsening Home outcomes.
    - Home matching subsidies limit activity contraction but deepen net foreign asset deterioration, depreciation, and core inflation increases.
    - If all countries introduce consumer subsidies, outcome: little improvement in activity and significant persistent increase in core inflation — worse for net energy importers.

- Producer subsidies in small open economy importer:
  - Both household and producer subsidies are expansionary when implemented by small country in isolation.
  - Producer subsidies yield lower core inflation compared to consumer subsidies by mitigating firms’ marginal cost rise.
  - If RoW subsidizes firms:
    - World energy prices rise, harming Home activity and pushing inflation up.
    - Producer subsidies by many countries depress consumption in importers but still keep core inflation lower than no-subsidy case.
    - Effectiveness reduced with shallow FX markets or greater non-energy trade openness.

- Spillovers to exporters:
  - Exporters benefit from higher world energy prices: improved terms of trade, exchange rate appreciation, and lower core inflation.
  - Little difference whether RoW subsidies target producers or consumers; main effect is higher world energy prices.

### Conclusions and policy guidance (Section 8)
- Main findings:
  - Consumer energy subsidies are often counterproductive as inflation-fighting tools when applied globally or in segmented markets, and less effective than targeted transfers at preventing increases in consumption inequality.
  - Key mechanism: impact on pre-subsidy energy price shifts burden from households to firms.
  - Consumer subsidies may be more effective in a small open economy well connected to world energy markets, but conditions for reducing inflation are restrictive due to endogenous exchange rate adjustments.

- Policy guidance and implications:
  - Cushioning vulnerable consumers is a valid objective; policies should aim for the lowest possible efficiency cost and be feasible to implement.
  - Targeting vulnerable households while preserving incentives to reduce energy consumption is challenging and may require unconventional measures.
  - Policy design and evaluation must account for general equilibrium consequences, especially when many countries implement policies.
  - Further research needed on relief policy design when transfers cannot be precisely targeted due to imperfect government information.

*Source: wpiea2024081-print-pdf — Section 8 concludes (and supporting sections 4.2, 5.3, 7.1–7.5, Appendices).*

### Section 8 concludes.

### Section 8 concludes.

### Empirical evidence on oil shocks and wage adjustment (US and euro area)
- Identification strategy and data:
  - Use surprise changes in oil futures prices around OPEC meeting announcements as identified by Känzig (2021); the window is one day, using WTI crude futures starting in 1983 traded at the New York Mercantile Exchange (NYMEX).
  - Series of oil price surprises spanning from 1983 to 2019 aggregated at quarterly frequency and used as an instrument in a proxy SVAR.
  - Proxy SVAR variables include: real oil prices (deflated by either core or headline prices), oil production, global industrial production, real GDP, core or headline CPI inflation, the short-term policy rate, and nominal wage inflation.
  - All quantities and relative prices are in logs, policy rates in percentage points, price and wage inflation measured as log differences.
  - Confidence bands computed using a moving block bootstrap (block size 24); bands reported at the 68 and 90 percent levels based on 10,000 bootstrap replications.

- Baseline SVAR results (US):
  - An oil supply-driven 10 percent increase in oil prices with a 2-year half life:
    - Increases core inflation by about 0.1 percentage point for a few quarters.
    - Reduces the nominal wage by about 0.2 percent on impact.
    - Implied producer real wage (accumulated nominal wage growth minus accumulated core inflation) falls by roughly 0.3 percent over the first 2 years and by nearly 0.6 percent after 5 years.
    - Real GDP falls by more that 0.4 percent after 2 years.
  - Interpretation: limited ability of US households and labor unions to achieve compensation for higher prices through higher wages.

- Baseline SVAR results (euro area):
  - Oil price path similar to US, but:
    - Nominal wage growth persistently increases (consistent with more centralized wage bargaining and higher unionization, implying more indexation).
    - Rising wages put upward pressure on core inflation; ECB raises interest rates significantly.
    - Output contraction is about the same as in the US despite euro area being less dependent on oil.
    - Implied producer real wage: point estimate modestly positive but not statistically significant.
  - Interpretation: mild evidence in favor of some indexation mechanisms in European wage setting that compensate households for loss of purchasing power due to energy price shocks.

- Headline vs core inflation and real consumer wage:
  - Appendix A.3: Impact on headline inflation is stronger on impact in both US and euro area — up to 1 (0.5) percent in the same quarter — but much less persistent.
  - Implied consumer real wage (nominal wage divided by headline CPI) falls notably more initially than implied producer real wage in the US.
  - Smaller pass-through to headline inflation relative to core inflation in the euro area than in the US may be partly explained by higher excise taxes on liquid fuels and more rigid downstream retail prices.
  - Regardless of using core or headline inflation in the SVAR, the implied real wage in the euro area falls less than in the US.

- Robustness checks:
  - Alternative SVARs reported in Figure 3 and Appendix A.3:
    - Imposing mean reversion of the real wage impulse responses (including real wage directly and imposing stability) — US real producer wages still decline significantly; Europe rises somewhat but not significant at the 90 percent level.
    - Non-linearities: excluding shocks smaller than one standard deviation (impulses scaled to imply the same initial oil price reaction) — no evidence that shock size matters for transmission to real wages.
    - Alternative identification with sign restrictions following Baumeister and Hamilton (2019) where oil production declines strongly on impact — impact on real wages remains robust.
  - Note: Känzig shocks do not necessarily lead to immediate reduction in oil and global industrial production (they may rise initially), suggesting identified shocks may contain a demand component; this strengthens interpretation that evidence lends little support for strong indexation of wage formation to headline inflation.

### Simple closed economy model with energy — structure and key primitives
- Households:
  - Representative continuum of households maximizing:
    - E_t sum_{s=0}^∞ β^s ( (C_{t+s}(h) − κ C_{t+s−1}(h))^{1−1/σ} / (1−1/σ) − χ_0 N_{t+s}(h)^{1+χ} /(1+χ) )
    - Parameters: 0≤β,κ<1 and σ,χ,χ_0 >0.
  - Budget constraint (period t):
    - P_{C,t} C_t(h) + B_t(h) + T_t = (1+τ_w) W_t(h) N_t(h) + R_{K,t} K + I_{t−1} B_{t−1}(h) + D_t + Ξ_t(h).
  - Labor differentiated with constant elasticity parameter θ_w >0; aggregate labor:
    - N_t = [ ∫_0^1 N_t(h)^{1/(1+θ_w)} dh ]^{1+θ_w}.
  - Wages set in Calvo fashion: each period household h can reoptimize with probability 1−ξ_w; non-reoptimizers index wages by:
    - ˜Π_{W,t} = Π_y^{ι_y} Π_{Y,t}^{ι_y} Π_{C,t}^{ι_c} Π_{Y,t−1}^{1−ι_y−ι_c} (expressed in source as Π̃_{W,t} = Π_{ι_y} Y,t Π_{ι_c} C,t Π_{1−ι_y−ι_c}).
  - Assumptions: perfect insurance across idiosyncratic wage risk (Ξ_t transfers), static subsidy to labor income τ_w ≡ (1+θ_w)/θ_w.

- Firms and production:
  - Final goods producers combine non-energy inputs Y_t and energy OC,t with CES aggregator (parameters ω_c, η_c), maximize P_{C,t} C_t(j) − P_{Y,t} Y_t(j) − (1−τ_{C,t}) P_{O,t} O_{C,t}(j). τ_{C,t} is a proportional net subsidy to household energy consumption.
  - Non-energy goods producers aggregate intermediate varieties with elasticity θ and maximize revenues minus intermediate costs.
  - Intermediate goods producers combine energy O_{Y,t} and capital-labor bundle V_t; CES aggregator parameters ω_y, η_y; V_t(f)=K_t(f)^α N_t(f)^{1−α}. Profits include (1+τ) markup term and energy subsidy τ_{Y,t} to energy use in production. Prices set in Calvo fashion with probability 1−ξ of reoptimization; non-optimizers index prices by ˜Π_{Y,t}.
  - Energy supply modeled as exogenous endowment Y_{O,t}; firms and households face energy market clearing Y_{O,t} = O_{C,t} + O_{Y,t}.

- Monetary and fiscal authorities:
  - Monetary policy: Taylor-type rule:
    - I_t = I + ψ_π (Π_{Y,t} − Π) + ψ_y (Y_t / Y_{pot,t} − 1),
    - with ψ_π >1, ψ_y ≥0. Central bank responds to producer (core) inflation Π_{Y,t} rather than headline inflation Π_{C,t}.
  - Fiscal authority: lump sum taxes T_t or debt B_{G,t} finance subsidies τ_{C,t}, τ_{Y,t}. Government budget constraint:
    - B_{G,t} = I_{t−1} B_{G,t−1} + τ_{Y,t} P_{O,t} O_{Y,t} + τ_{C,t} P_{O,t} O_{C,t} − T_t.
  - Ricardian equivalence holds; sequence of distortionary taxes irrelevant for equilibrium as long as transversality satisfied.

### Transmission of energy shocks and subsidies — analytical insights, equivalence results
- Key analytical result (Proposition 1, exact statement):
  - Proposition 1. In a New Keynesian model with energy, any sequence of subsidy pairs {τ_{C,t}, τ_{Y,t}}_{t=0}^∞ yields identical equilibrium allocations as an alternative sequence {̊τ_{C,t}, ̊τ_{Y,t}}_{t=0}^∞ if and only if
    - (1−τ_{C,t})/(1−τ_{Y,t}) = (1−̊τ_{C,t})/(1−̊τ_{Y,t}) for all t = 0,1,....
  - Immediate implication: energy subsidies work only if they are differentiated between households and firms; if subsidized at exactly the same rate, the subsidy has no effect on allocations.

- Linearized corollary (exact statement):
  - Corollary 1. In a linearized New Keynesian model with energy, subsidizing energy use by households is equivalent to taxing energy use by firms at the same rate.

- Intuition and mechanisms:
  - What matters for allocations is the effective energy costs faced by households and firms, (1−τ_{C,t}) P_{O,t} and (1−τ_{Y,t}) P_{O,t}, because P_{O,t} never appears independently in equilibrium conditions.
  - Introducing a subsidy to households lowers their effective energy price but endogenously raises the pre-subsidy energy price P_{O,t}, increasing unit energy cost to firms — thus a household subsidy can act like a tax on firms.
  - To first order in the linearized model, τ_{C,t} = − τ_{Y,t}.
  - Equivalence applies to allocations and to all prices except the pre-subsidy energy price; endogenous adjustment of the pre-subsidy price is central to the equivalence.
  - Caveats: derived in a closed-economy model with exogenous energy endowment and Ricardian equivalence. If energy production responds or in an open economy with a small country, equivalence may break down. The intuition remains useful for large or coordinated subsidies, especially short-run when global energy supply is very inelastic.

*Source: wpiea2024081-print-pdf - Section 8 concludes.*

### 4.2    Log-linearized Model with Full Price Flexibility

### 4.2    Log-linearized Model with Full Price Flexibility

### Flexible-price equilibrium and energy market clearing
- Analysis focuses on energy shocks and energy subsidies in the flexible price and wage equilibrium; with monopolistic competition effects eliminated by subsidies, this equilibrium coincides with the first-best allocation provided that energy subsidies are not utilized.
- For analytical exposition, habit persistence and the dynamic component of price indexation are abstracted from by setting κ=ι= 0.
- Log-deviations from steady state use lowercase letters; all real prices are normalized by the producer price index P_{Y,t}.
- Superscript f denotes flexible price equilibrium allocations and prices ("potential" levels).
- Energy market clearing condition (flexible prices and wages) can be written as
  - −y_{O,t} = Ψ_{oc} (p^f_{O,t} − τ_{C,t}) + Ψ_{oy} (p^f_{O,t} − τ_{Y,t}), where Ψ_{oc}, Ψ_{oy} > 0.
  - Implication: the potential pre-subsidy energy price p^f_{O,t} is the only endogenous variable in (18); a fall in the energy endowment or an increase in either type of energy subsidy raises p^f_{O,t}.

### Labor market and real wages under full flexibility
- Producer "demand" real wage (what firms would pay) under full price and wage flexibility:
  - w^f_t = −(ω_y/(1−ω_y)) (p^f_{O,t} − τ_{Y,t}) − α n^f_t.  (Equation (19))
- Household "supply" real wage (marginal rate of substitution expressed in product real wage terms):
  - w^f_t = χ n^f_t + ω_c (p^f_{O,t} − τ_{C,t}) + (1/σ) c^f_t.  (Equation (20))
- Intuition for a consumer energy subsidy τ_{C,t}:
  - Reduces net energy price to households → shifts labor supply outward.
  - Raises energy price paid by firms → shifts labor demand down.
  - Net effect: product real wage is depressed; effects on hours worked are ambiguous.

### Conditions ensuring product real wage decline with consumer subsidy
- With wealth effects, product real wage necessarily declines if α = 0 (linear in labor).
- With diminishing returns to labor (α > 0), consumption c^f_t may rise and shift labor supply inward, but under reasonable calibrations the fall in labor demand dominates and product real wage still falls.
- Formal sufficient condition (Appendix A.7): product real wage declines if
  - ω_c [1−(1/σ) η_c] Ψ_{oy} + [ω_y/(1−ω_y)][(1/α) − 1][χ + 1/σ] + (1/σ) η_y ] Ψ_{oc} > 0.  (Equation (21))
- A sufficient (stronger) condition: η_c < σ, i.e., energy and non-energy goods are complements in household preferences (Pareto-Edgeworth complements).

### Log-linearized model: headline and core inflation, wage dynamics
- Headline inflation:
  - π_{C,t} = (1 − ω_c) π_{Y,t} + ω_c (π_{O,t} − Δτ_{C,t}), where π_{O,t} ≡ log(P_{O,t}/P_{O,t−1}) − log(Π).  (Equation (22))
  - Consumer energy subsidies can reduce or offset the first-round effect if Δτ_{C,t} = Δp_{O,t}.
- Core (sticky) inflation via price Phillips curve:
  - π_{Y,t} = β E_t π_{Y,t+1} + κ_p [ω_y (p_{O,t} − τ_{Y,t}) + (1 − ω_y) (w_t + α n_t)], where κ_p ≡ (1−ξ)(1−βξ)/ξ.  (Equation (23))
  - Real marginal cost has energy and non-energy components; a rise in energy price increases the energy component and can raise core inflation depending on wage and employment adjustments.
- Nominal wage dynamics via wage Phillips curve:
  - π_{W,t} − ̃π_{W,t} = β E_t {π_{W,t+1} − ̃π_{W,t+1}} + κ_w [χ n_t + (1/σ) c_t − w_{C,t}], where κ_w ≡ (1−ξ_w)(1−βξ_w)(1 + (1+θ_w)/θ_w χ)/ξ_w, π_{W,t} ≡ log(W_t/W_{t−1}) − log(Π), ̃π_{W,t} = ι_y π_{Y,t} + ι_c π_{C,t}, and w_{C,t} ≡ w_t − ω_c (p^f_{O,t} − τ_{C,t}).  (Equation (24))
  - Energy shocks raise nominal wages via: (i) gap between consumer real wage (falls) and marginal rate of substitution for those who can reoptimize; (ii) indexation mechanism—stronger indexation to headline inflation amplifies transmission.
  - Hence subsidies more effective at moderating wage pressures when there is high indexation to headline inflation.

### Gap representation (real wage gap and employment gap)
- Define deviations from potential with hats: ˆx_t ≡ x_t − x^f_t.
- Energy market clearing in levels:
  - y_{O,t} = v_t − η_y [ (O_Y/Y_O + ω_y O_C/Y_O) (p_{O,t} − τ_{Y,t} − p_{V,t}) ] − η_c O_C/Y_O (p_{O,t} − τ_{C,t}), where p_{V,t} ≡ α r_{k,t} + (1 − α) w_t.  (Equation (25))
  - A negative energy supply shock y_{O,t} or increases in subsidies require higher pre-subsidy real energy prices, a decline in employment, or a decline in non-energy factor prices.
- In gap form, pre-subsidy energy price deviation:
  - ˆp_{O,t} = Ψ_{on} ˆn_t + Ψ_{ow} ˆw_t, where Ψ_{on}, Ψ_{ow} > 0.  (Equation (26))
- Price and wage Phillips curves in gap variables:
  - π_{Y,t} = β E_t π_{Y,t+1} + κ_p [Ψ^{mpl}_n ˆn_t + Ψ^{mpl}_w ˆw_t].  (Equation (27))
  - π_{W,t} − ̃π_{W,t} = β E_t {π_{W,t+1} − ̃π_{W,t+1}} + κ_w [Ψ^{mrs}_n ˆn_t + Ψ^{mrs}_w ˆw_t].  (Equation (28))
- With ι_c = 0 (non-optimized wages indexed to core and steady-state inflation), these equations share structure with standard New Keynesian formulations where policymakers trade off stabilizing price inflation, employment, and wage inflation.

### Transmission of consumer energy subsidies to inflation (central analytical result)
- Consider central bank focusing on keeping non-energy output v_t (and employment n_t) at potential.
- Under this policy, price Phillips curve implies
  - π_{Y,t} = κ_p Ψ^{mpl}_w E_t Σ_{j=0}^∞ β^j ˆw_{t+j}.  (Equation (29))
- Law of motion for real producer wage (combining (27) and (28)):
  - E_t{w_{t+1}} − [1 + 1/β + κ/β] w_t + (1/β) w_{t−1} = −(κ/β) w^f_t, where κ ≡ κ_w [1 − Ψ^{mrs}_w] + κ_p (1 − ι_y) Ψ^{mpl}_w.  (Equation (30))
  - Solution:
    - w_t = R_2 w_{t−1} + (1 − R_2) [(1 − β R_2)/(1 − ρ β R_2)] w^f_t, where ρ is the AR(1) coefficient driving the subsidy and 0 < R_2 < 1 is the lower root of the characteristic equation.  (Equation (31))
  - Implication:
    - E_t Σ_{j=0}^∞ β^j ˆw_{t+j} = D w^f_t, with D < 0.  (Equation (32))
- Since consumer energy subsidies push down the potential real producer wage w^f_t (for realistic parametrizations and complementarity condition), combining (32) with (29) implies subsidies increase core inflation.
- Formal statement (Proposition 2):
  - Assume (i) energy and non-energy goods are complements in household preferences, (ii) the dynamic component of wage indexation, if present, relies on producer price inflation, (iii) the monetary authority focuses on closing the output gap. Then, in a linearized New Keynesian model with energy, an increase in energy subsidies to households leads to an increase in core inflation.
  - Complementarity assumption (i) can be replaced by the weaker restriction given by formula (21).

### Additional implications of the wage law-of-motion
- Persistence effect: the more persistent the subsidy (higher ρ), the more lasting the increase in core inflation; in the limit ρ → 1 (permanent subsidy) the wage gap and core inflation deviation remain positive before converging to zero.
- Pre-subsidy energy price effect: if the central bank stabilizes employment, energy subsidies raise the energy price gap ˆp_{O,t}; since subsidies also raise potential pre-subsidy energy prices, they increase even more in the equilibrium with sticky prices and wages.

### Key takeaways (summary of Section)
- In global or highly segmented energy markets, general equilibrium mechanisms make consumer energy subsidies likely counterproductive.
- Endogenous adjustment in pre-subsidy energy prices makes a consumer subsidy akin to a tax on energy use by firms.
- Under an empirically relevant wage indexation formulation (indexation linked to producer-price inflation), consumer energy subsidies exacerbate policy tradeoffs from energy supply shocks by increasing core inflation for a given output gap.
- For such subsidies to potentially be productive, wages need to be strongly linked to headline inflation (indexation to headline inflation can materially change transmission).
- The analytical results are consistent with prior literature showing that negative energy supply shocks can raise core inflation under a policy that targets employment at potential; in this model a consumer subsidy acts similarly to a negative shock to energy supply to producers and thus boosts inflation.

*Source: 4.2 Log-linearized Model with Full Price Flexibility (excerpt).*

### 5.3    Can Energy Subsidies Reduce Inflation?

### 5.3 Can Energy Subsidies Reduce Inflation?

### Overview
- The model considers two subsidy types: to households and to firms. When used in simulations:
  - Initial subsidy value is set at 17%.
  - Subsidies are assumed to have the same persistence as energy shocks.
- Baseline energy supply shock calibrated to produce a 50% increase in real energy prices absent fiscal measures.
- Subsidies amount to about 1 percent worth of annual GDP in the first year of deployment.

### Macroeconomic effects of household energy subsidies
- Direct effects:
  - Mitigate increase in the cost of living for households; headline price level responds downward.
  - Lower consumer prices reduce wage pressure, so real producer wage falls.
- Indirect (counterproductive) effects:
  - Households’ incentive to cut energy consumption is blunted, raising the pre-subsidy energy price faced by firms and moving allocation away from efficiency.
  - Firms’ marginal cost and core inflation increase.
  - The achieved reduction in headline inflation is substantially reduced; after about two years the cost of living index can be higher than without the subsidy.
- Activity impact:
  - Despite subsidies equaling about 1 percent of annual GDP in year one, they have very small impact on economic activity.
- Quantitative calibration note:
  - A 17% subsidy halves the impact of a 50% increase in the pre-subsidy energy price as faced by an atomistic subsidized agent.

### Macroeconomic effects of firm energy subsidies
- Mirror image of household subsidies due to equivalence result in Section 4.1 (subsidizing household energy prices ↔ taxing energy prices for firms).
- Firm subsidies:
  - Drive up the pre-subsidy energy price, shifting adjustment burden from subsidized to non-subsidized agents.
  - Can successfully bring core inflation down, aiding monetary policy’s medium-term stabilization objectives.
  - Come at a short-term cost to households who face higher per-unit energy prices.

### Robustness: alternative calibrations (Figure 7 experiments)
- Wage indexation flip:
  - Baseline: nominal wages dynamically indexed to core inflation (ιy = 0.75, ιc = 0).
  - Alternative: wages indexed to headline inflation (ιy = 0, ιc = 0.75).
  - Result: subsidy achieves sizable wage moderation but still raises producers’ prices because household subsidy increases energy use and the energy price paid by firms.
- Energy substitutability in production:
  - Further assume energy is easy to substitute with capital or labor (ηy → 1) together with wages indexed to headline inflation.
  - Result: only under these two simultaneous, highly unrealistic assumptions does a household subsidy lead to a fall in core inflation and keep headline price level below the no-subsidy case.

### Optimal energy subsidy (representative-agent model)
- Social welfare maximization setup:
  - Planner can choose consumer energy subsidy τC,t (negative values interpreted as subsidy to firms) and interest rate Rt under full commitment.
- Optimal policy findings:
  - If only monetary policy is chosen optimally (no subsidies), outcomes similar to earlier benchmark: rule 13 is a reasonable approximation.
  - If subsidies are available, the optimal policy uses subsidies to firms rather than households:
    - Helps contain increase in marginal cost and lower core inflation.
    - Requires less contraction in economic activity.
  - Conclusion: consumer energy subsidies are counterproductive for stabilization in the representative-agent environment.

### TANK extension: key model modifications
- Introduce Keynesian (rule-of-thumb) households (subscript K) who:
  - Do not perform intertemporal optimization; consume real disposable income each period.
  - Hold no assets; nominal budget constraint: PC,t CK,t = Wt NK,t − PC,t TK,t.
  - Have no bargaining power and work same hours as average Ricardian household, earning economy-wide wage Wt determined by Ricardian agents.
- Ricardian households (superscript R) retain baseline behavior.
- Taxes evolve as:
  - Tj,t − TjY = tj,t + φj (BG,t − BG/PY) for j ∈ {K,R}
  - With φK = 0, φR > 0, and tj = 0 in the baseline formulation.
- Equivalence results (Proposition 1 and Corollary 1) continue to hold under this transfer formulation.

### TANK calibration
- Share of Keynesian households; λ = 0.4
- Relative consumption of Keynesian households; CK/CR = 0.5
- Steady state debt-to-output ratio; BG/Y = 2.4 (consistent with a debt-to-annual-GDP ratio of 60%)
- Response of taxes to public debt; φK, φR = 0, 0.01

### Energy subsidies versus transfers (TANK results)
- Presence of Keynesian agents amplifies aggregate demand effects of an energy price shock:
  - Fall in output is deeper; increase in core inflation is more short-lived.
  - Consumption inequality rises because Keynesian households cannot smooth consumption.
- Policy comparisons:
  - Energy subsidies are more counterproductive for core inflation in the TANK environment than in the representative-agent model, though they somewhat curb increases in consumption inequality.
  - Targeted lump-sum transfers to Keynesian households (adjusted to prevent any increase in consumption inequality, i.e., CK,t/CR,t = CK/CR) are more effective than energy subsidies at:
    - Limiting contraction in output and consumption.
    - Being much less inflationary.
  - Rationale: transfers do not distort relative energy prices faced by households and thus preserve incentives to save energy; pre-subsidy energy price and firms’ marginal cost rise less.

### Optimal subsidies in the TANK model
- Optimal monetary policy alone:
  - More focused on keeping core inflation down relative to the simple policy rule; the simple rule is a worse approximation when hand-to-mouth agents exist.
  - Even optimal monetary policy cannot prevent a significant increase in consumption inequality.
- Adding subsidies as an instrument:
  - Short-term use: a steep, short-lived increase in consumer energy subsidy shields Keynesian households when energy prices peak.
  - Medium-term use: subsequently a small persistent subsidy to firms helps prevent a significant rise in core inflation via forward-looking price setting.
- Role of transfers:
  - If the planner also has access to targeted transfers that prevent any increase in consumption inequality, then optimal policy subsidizes only firms (similar to representative-agent optimal policy).
  - Welfare gains from including transfers are large even if transfers are not chosen fully optimally.
- Policy implication:
  - Consumer energy subsidies can help shield vulnerable households, but only when targeted transfer instruments are unavailable; transfers are generally more effective and less inflationary.

*Source: IMF Working Paper chapter section “5.3 Can Energy Subsidies Reduce Inflation?” and subsequent sections in the provided content.*

### 7.1    Modifications to the Baseline Model

### 7.1    Modifications to the Baseline Model

### Model setup and international structure
- Two-country open economy extension: Home and Foreign (asterisk for Foreign variables).
- World population normalized to unity; Home economy size 0< ζ <1.
- Both countries are isomorphic; presentation focuses on Home agents.

### Consumption and production linkages
- Final consumption aggregator for each firm j:
  - C_t(j) = [ (1−ω_c)^(1/η_c) C_N,t(j)^(η_c−1)/η_c + ω^(1/η_c) C_O,t(j)^(η_c−1)/η_c ]^(η_c/(η_c−1)). (Equation (36) in source)
- Non-energy consumption component combines domestic and imported non-energy goods:
  - C_N,t(j) = [ (1−ω)^(1/η) Y_t(j)^(η−1)/η + ω^(1/η) M_t(j)^(η−1)/η ]^(η/(η−1)). (Equation (37) in source)
  - Parameters: 0< ω <1 and η >0.

### Pricing, energy trade, and exchange rate
- Intermediate goods producers sell domestically and abroad; prices in destination currency (local currency pricing) with Calvo probability of not reoptimizing equal to ξ_m.
- Energy endowments internationally tradable; law of one price: P_O,t = ε_t P^*_O,t.
  - ε_t defined as Home currency units per unit of Foreign currency (nominal exchange rate).

### International financial intermediation friction
- Financiers intermediate international flow of funds with limited risk-bearing capacity (Gabaix and Maggiori (2015) framework).
- Each period financiers take positions F_t in domestic currency bonds and −F_t/ε_t in foreign currency bonds.
- They can divert up to min{1, Γ|F_t P_Y,t|} of assets; incentive compatibility constraint:
  - V_t(k) ≥ Γ F_t(k)^2 / P_Y,t. (Equation (38) in source)
- Representative financier maximizes:
  - V_t(k) = 1/I_t E_t[ I_t F_t(k) − I^*_t ε_{t+1} F_t(k) / ε_t ]. (Equation (39) in source)
- After profits/losses financiers transfer them uniformly to Home and Foreign households.

---

### 7.2    Key Equilibrium Conditions and Analytical Insights

### World energy market clearing
- World market clearing condition (energy, since energy is freely tradable):
  - ζ Y_{O,t} + (1−ζ) Y^*_{O,t} = ζ(O_{C,t} + O_{Y,t}) + (1−ζ)(O^*_{C,t} + O^*_{Y,t}). (Equation (40) in source)
- Implication: energy prices affected by both countries’ energy endowments and energy demand; uncoordinated subsidies can affect world energy prices.

### Net foreign assets evolution (Home perspective)
- B_t evolves as:
  - B_t = [ (1−ζ) I_{t−1} + ζ (ε_t/ε_{t−1}) I^*_{t−1} ] B_{t−1}
    + ε_t P^*_M,t M^*_t − P_M,t M_t
    + P_O,t (Y_{O,t} − O_{Y,t} − O_{C,t}). (Equation (41) in source)
  - Terms labeled respectively: gross interest payment on foreign assets; non-energy trade balance; energy trade balance.

### Risk-augmented UIP and implications
- Imperfect intermediation yields risk-augmented UIP:
  - I_t = E_t[ I^*_t ε_{t+1} / ε_t ] − Γ I_t B_t / P_Y,t. (Equation (42) in source)
- Implication: world energy prices affect net foreign assets which directly affect countries’ risk premia.
  - An increase in energy prices tends to increase (decrease) the premium for net energy importers (exporters), leading to depreciation (appreciation) of their exchange rates.

### Breakdown of equivalence between consumer and producer energy subsidies
- In an open economy, energy supply for a country is not fixed due to imports; thus subsidies to households and producers are distinct and potentially complementary instruments.
- In the small open economy limit (ζ→0), pre-subsidy energy price (in foreign currency) does not depend on subsidies by that country.
- Additional transmission channels introduced:
  - Exchange rate channel: subsidies that increase domestic energy use worsen energy trade balance → exchange rate depreciation (especially if FX markets are shallow, Γ high) → raises domestic currency energy cost to firms and increases price of non-energy imports, weakening subsidy efficacy against energy-driven inflation.
  - Spillovers from foreign subsidies: if rest of world is large enough to influence world energy price, foreign subsidies can amplify domestic inflation pressures for energy importers; for net energy exporters, exchange rate appreciation may offset higher world energy prices and lower domestic inflation.

---

### 7.3    Calibration

### Key calibrated parameters for the open economy extension (as presented in Table 3)
- Relative country size; ζ = 0.001
- Share of energy production (energy importers vs exporters); Y_O/Y = 0.05, 0.15
- Share of imports in non-energy consumption; ω = 0.1
- Elasticity of substitution btw. non-energy inputs to consumption; η = 0.8
- Calvo probability in non-energy price setting for exports; ξ_m = 0.9, 0.9
- Indexation for non-energy prices for exports; ι_m, ι^*_m = 0.5, 0.5
- FX market shallowness; Γ = 0.02

### Calibration choices and rationale
- Focus in most simulations on a small open economy (ζ set very small).
- Two alternative steady state values for energy endowment to represent energy trade surplus or deficit worth about 5% of output.
- FX market depth parameter Γ corresponds to median estimate for advanced economies reported by Chen et al. (2023).
- Import content of non-energy consumption reflects typical values for small open economies.
- Low elasticity η chosen to reflect low sensitivity of trade volumes to exchange rate.
- High Calvo probabilities and substantial dynamic indexation for prices sold to foreign markets to match slow, delayed exchange rate pass-through to consumer prices.

---

### 7.4    Energy Importers vs Exporters

### Differential responses to an identical foreign energy supply cut (small open economy)
- Energy importers:
  - Economic activity contracts; core inflation persistently increases.
  - Reactions stronger than in closed economy due to sharp deterioration in terms of trade.
  - Households borrow from abroad; net foreign assets fall; exchange rate weakens.
  - Exchange rate pass-through to energy prices is complete → relative price and inflation rise more than in closed economy.

- Energy exporters:
  - Face positive wealth shock from higher global energy prices.
  - Non-energy output declines much less; consumption stronger.
  - Terms of trade improvement → accumulation of foreign assets → exchange rate appreciation.
  - With large enough energy trade surplus and sufficient openness (baseline calibration), core inflation may decline.

### Role of openness to non-energy trade
- Greater openness (higher share of non-energy trade) allows non-energy trade balance to absorb energy trade imbalances, bringing importer and exporter responses closer.
- High openness can induce Dutch disease for exporters: output may fall below importer in medium run.
- Greater openness implies stronger pass-through of exchange rate movements to core inflation, increasing divergence of core inflation responses.

### Role of FX market depth (Γ)
- If Γ is very small (deep FX markets):
  - Easier cross-border flows; net foreign assets positions of exporters and importers diverge more.
  - Net foreign assets have smaller effect on UIP risk premium; exchange rate adjustment is much smaller.
  - Core inflation dynamics between exporters and importers become more similar.

---

### 7.5    Domestic and Multilateral Effects of Energy Subsidies

### Consumer (household) energy subsidies in a small open economy energy importer
- When only Home economy uses the subsidy:
  - Subsidies effective at limiting fall in consumption and economic activity (small country).
  - World energy prices unaffected by small Home subsidies, but core inflation still rises due to exchange rate depreciation driven by worsening net foreign assets.
  - Stronger activity raises labor demand; smaller fall in real producer wage than in closed economy case.
  - Increase in core inflation is smaller than in closed economy; overall macro effect of consumer subsidies more favorable in SOE than in closed economy.

- When rest of world (RoW) subsidies households:
  - RoW subsidies raise global energy prices, exacerbating negative consequences for energy importer Home country.
  - Home can limit activity contraction by matching subsidies, but at cost of deeper deterioration in net foreign assets, exchange rate depreciation, and further increase in core inflation.
  - If all countries introduce consumer subsidies globally, outcome: very little improvement in activity and significant persistent increase in core inflation — worse for net energy importers due to negative wealth transfer and additional inflationary exchange rate channel (especially if FX markets are shallow).

### Producer (firm) energy subsidies in a small open economy energy importer
- Both household and producer subsidies are expansionary when implemented by a small country in isolation.
- Key difference: producer subsidies help achieve lower core inflation compared to consumer subsidies.
  - Rationale: producer subsidies mitigate energy-price-driven increase in firms’ marginal costs, offsetting higher import prices from exchange rate depreciation.
- If RoW subsidizes firms:
  - World energy prices rise, harming Home economic activity and pushing inflation up.
  - Producer subsidies implemented globally depress consumption in energy importers.
  - Nonetheless, producer subsidies still help keep core inflation lower than the no-subsidy case, despite higher world energy prices and weaker exchange rate.
  - Effectiveness may be reduced in energy importers with shallow FX markets or greater non-energy trade openness.

### Spillovers to energy exporting countries from foreign subsidies
- Energy exporters benefit from subsidy-driven higher world energy prices:
  - Improved terms of trade, exchange rate appreciation, and lower core inflation.
  - Higher energy prices and exchange rate appreciation hurt non-energy output, but positive wealth transfer raises consumption — net benefit for exporter.
- There is little difference whether RoW subsidies target producers or consumers; key is that world energy prices increase.

---

### 8    Conclusions

### Main findings on energy subsidies and inflation
- Consumer energy subsidies, widely used in recent global energy shocks, are often counterproductive as inflation-fighting tools when applied globally or in segmented markets, and less effective at preventing increases in consumption inequality than targeted transfers.
- Key mechanism of ineffectiveness: impact on pre-subsidy energy price shifts burden from households to firms.
- Consumer subsidies may work better in a small open economy well connected to world energy markets, but conditions for reducing inflation are restrictive due to endogenous exchange rate adjustments.

### Policy guidance and implications
- Cushioning vulnerable consumers is a valid objective; policies should aim to achieve it at lowest possible efficiency cost and be feasible to implement.
- Targeting vulnerable households while preserving incentives to reduce energy consumption is challenging and may require unconventional measures.
- Policy design and evaluation need to account for general equilibrium consequences, especially when policies are implemented by a large group of countries.
- Further research is needed on relief policy design when transfers cannot be precisely targeted due to imperfect government information.

*Source: wpiea2024081-print-pdf — Section 7.1–8 (open economy extension, calibration, simulation results, and conclusions).*

### References

### wpiea2024081-print-pdf - References

### Proxy-SVAR methodology (Appendix A.1)
- Structural SVAR representation used:
  - A0 Yt = sum_{j=1}^p Aj Y_{t−j} + B εt (equation A.1).
  - Reduced form: Yt = sum_{j=1}^p Dj Y_{t−j} + μt with μt = A0^{-1} B εt and Dj = A0^{-1} Aj (equation A.2).
- Identification target: an oil supply shock identified using an external instrument — oil price changes around OPEC’s announcements.
- Instrument validity conditions stated exactly:
  - E[ε_oil_t, z'_t] ≠ 0
  - E[ε_others_t, z'_t] = 0
- Baseline VAR specification includes eight macroeconomic variables:
  - US (EA) core CPI
  - the fed funds rate (EONIA) as proxy for the monetary policy rate
  - US (EA) gross domestic product
  - US (EA) nominal wage growth
  - global oil production
  - global industrial production
  - the WTI (Brent) oil price
  - (real wage calculated post-estimation, deflated with core CPI index)
- Two-stage proxy-SVAR procedure described:
  - First stage: regress reduced-form VAR innovation for WTI (Brent) oil prices on the oil price surprise (instrument) to identify impact of OPEC announcements on oil prices.
  - Second stage: regress predicted values from first stage on remaining VAR innovations to recover impact coefficients (columns of B) up to scaling.
- Impulse response computation:
  - IR_0 = B_i for t = 0 (i-th column of B)
  - IR_t = D · IR_{t−1} for t = 1,2,...,H

### Data used in VAR (Appendix A.2)
- Sample period: 1983Q1 - 2019Q4.
- Instrument:
  - Oil supply news: OPEC oil supply news shock from K̈anzig (2021).
- VAR variables and transformations (selected exact entries preserved):
  - PROD: World crude oil production (WDPCOBD.P) — Datastream — 100*log
  - GIP: Industrial production of OECD + 6 (Brazil, China, India, Indonesia, Russia and South Africa) from Baumeister and Hamilton (2019) — Baumeister’s webpage — 100*log
  - POIL US: Spot Crude Oil Price: West Texas Intermediate (WTISPLC) deflated by US core or headline CPI according to VAR specifications — FRED — 100*log
  - POIL EA: Brent Forties Oseberg month 1 Europe, deflated by EA core or headline CPI according to VAR specifications — PCPS, IMF — 100*log
  - WAGE US: Employment Cost Index: Wages and Salaries: Private Industry Workers (ECIWAG), spliced by Average Hourly Earnings of Production and Nonsupervisory Employees, Total Private (AHETPI) before 2001 — FRED — 400*∆log
  - WAGE EA: EU27: Wages & Salaries: Business Economy spliced by Average compensation per head before 2001 — Haver, AWM and IMF staff calculation — 400*∆log
  - GDP US: Real Gross Domestic Product — FRED — 100*log
  - GDP EA: Real GDP — Haver, AWM and IMF staff calculation — 100*log
  - FFR: Federal Funds Effective Rate — FRED — NA
  - US1Y: Market Yield on US Treasury Securities at 1-Year Constant Maturity — FRED — NA
  - EONIA: Euro Interbank Offered Rate, spliced by short-term (3-month) interest rate (STN) before 1999 — FRED, AWM and IMF staff calculation — NA
  - CPI US: Consumer Price Index for All Urban Consumers: All Items in US City Average — FRED — 400*∆log
  - CPI EA: HICP: European Index of Consumer Prices spliced by AWM data before 2000 — Haver, AWM and IMF staff calculation — 400*∆log
  - Core US: Consumer Price Index for All Urban Consumers: All Items Less Food and Energy in US City Average — FRED — 400*∆log
  - Core EA: HICP: Total excl Energy spliced by AWM data before 2001 — Haver, AWM and IMF staff calculation — 400*∆log

### Empirical robustness (Appendix A.3)
- Headline vs. core inflation:
  - Appendix reports SVAR results replacing core inflation with headline inflation for US and EA.
  - Figures referenced with exact confidence-band construction: Confidence bands are the 68 and 90 percent level bands based on 10,000 bootstrap replications.
- Imposing stationary real wage:
  - Robustness step replaces log-difference in nominal wages with level of real wages (nominal wage deflated by core CPI).
  - US: relative to baseline, real wages drop almost immediately driven by increase in headline inflation.
  - EA: initial fall in point estimate not significant; real wages do not decline.
- Testing for non-linearities:
  - Focus on announcements with oil price response bigger than one standard deviation (calculated using whole sample of oil price surprises).
  - Results: qualitatively the same as baseline; quantitatively notable visible differences only for EA real wage response but not statistically significant.
  - Conclusion: baseline results driven mostly by announcements with large price impacts (i.e., big shocks).
  - Additional test using local projections (Jordà (2005) approach): interacted oil price with 4-quarter core inflation moving average, lagged by one quarter; instrumented interaction with K̈anzig oil shock series interacted with the 4-quarter core inflation moving average, lagged by one quarter. Result: interaction term has no statistical significance for either US or EA.
- Alternative identification scheme (Appendix A.3.4):
  - Robustness using Baumeister and Hamilton (2019) supply-shock series as instrument (Bayesian approach imposing priors on price elasticities and sign restrictions).
  - Main empirical differences under this instrument:
    - Oil production declines strongly.
    - Real GDP eventually declines in both US and EA, with decline stronger and more precisely estimated for Europe.
    - Impact on core inflation: relatively muted in the US; stronger in Europe, suggesting relatively more indexation to headline in Europe.
    - Real (producer) wages rise in Europe, contrary to the US where they decline — consistent with baseline text results.

### Model: Equilibrium conditions and equivalence result (Appendices A.4 and A.5)
- Aggregate equilibrium conditions:
  - Full set of equilibrium conditions jointly determine evolution of real per capita allocations and real prices for given initial conditions and exogenous paths Y_{O,t}, τ_{C,t}, and τ_{Y,t}.
  - Representative equations preserved exactly (selected highlights):
    - Aggregate production: Y_t / P^#_{Y,t} = [ (1−ω_y)^{1/η_y} V_t^{(η_y−1)/η_y} + ω^{1/η_y} y_{O}^{(η_y−1)/η_y} Y_t ]^{η_y/(η_y−1)} (equation A.3)
    - V_t = K_t^{α} N_t^{1−α} (equation A.4)
    - Price dynamics and price dispersion index provided (equations A.7–A.10).
    - Headline price level and inflation: P_{C,t} defined via energy share ω_c and tax τ_{C,t} (equation A.11) and Π_{C,t} = P_{C,t} / P_{C,t−1} Π_{Y,t} (equation A.12).
    - Marginal cost MC_t and labor/ capital relations (equations A.13–A.14).
    - Wage dynamics with Calvo-style notation (equations A.15–A.19).
    - Consumption aggregator and optimal composition of consumption (equations A.20–A.21).
    - Marginal utility Λ_t and Euler equation Λ_t = E_t Λ_{t+1} I_t / Π_{C,t+1} (equations A.22–A.23).
    - Energy market clearing Y_{O,t} = O_{C,t} + O_{Y,t} (equation A.24).
    - Monetary policy rule: I_t = I + ψ_π (Π_{Y,t} − Π_Y) + ψ_y [ Y_t / Y^f_{t−1} ] (equation A.25).
- Model dimensionality:
  - Complementing flexible-price (potential) block yields in total forty-six equilibrium conditions determining twenty-three endogenous variables Y_t, C_t, Λ_t, V_t, O_{Y,t}, O_{C,t}, N_t, W_t, R_{K,t}, MC_t, P_{C,t}, P_{V,t}, P_{O,t}, Π_{Y,t}, ˜Π_{Y,t}, ˜P_{Y,t}, P^#_{Y,t}, Π_{C,t}, ˜W_{C,t}, W_{C,t}, Π_{W,t}, ˜Π_{W,t}, I_t, and their twenty-three potential equivalents (superscript f).
- Equivalence result (proof sketch, Appendix A.5):
  - Key step: pre-subsidy energy price P_{O,t} and potential level P^f_{O,t} never appear independently in equilibrium conditions but always net of subsidies.
  - Define net-of-subsidy energy prices:
    - P_{OC,t} ≡ (1−τ_{C,t}) P_{O,t}
    - P_{OY,t} ≡ (1−τ_{Y,t}) P_{O,t}
    - Similarly for potential block: P^f_{OC,t} ≡ (1−τ_{C,t}) P^f_{O,t}, P^f_{OY,t} ≡ (1−τ_{Y,t}) P^f_{O,t}
  - Using these variables allows elimination of pre-subsidy energy prices P_{O,t} and P^f_{O,t} from equilibrium conditions, underpinning the proposition.

*Italic: Content derived from the "References" and Appendix sections of wpiea2024081-print-pdf.*

### Appendix A.4.  This modified set of equations will be isomorphic to the original one if we add

### Appendix A.4. This modified set of equations will be isomorphic to the original one if we add the following two restrictions on net energy prices

### Isomorphism and role of subsidies
- Two restrictions on net energy prices (equation (A.26)) bind energy subsidies into the modified system:
  - P_OC,t / P_OY,t = P^f_OC,t / P^f_OY,t = (1−τ_C,t) / (1−τ_Y,t). (A.26)
- Implication: only the ratio of energy price wedges created by subsidies, (1−τ_C,t)/(1−τ_Y,t), matters for allocations.
  - A given path of this ratio is consistent with infinitely many paths of τ_C,t and τ_Y,t individually; the specific split is irrelevant for allocations and for real prices other than the pre-subsidy real energy price P_O,t (and its potential value P^f_O,t).
- This establishes Proposition 1.

### Linearized model — structure and variables
- Linearizing around the non-stochastic steady state yields a system of exactly seventeen linearized equilibrium equations (A.27)–(A.43).
- The system describes the dynamics of these seventeen endogenous variables, all in log-deviations from steady state:
  - y_t, c_t, v_t, o_Y,t, o_C,t, n_t, w_t, r_K,t, mc_t, mrs_t, π_Y,t, π_O,t, p_O,t, π_C,t, π_W,t, ̃π_W,t, i_t.
- Under no price/wage rigidities (ξ, ξ_w → 0), the same system describes the potential equilibrium including potential output y^f_t entering the monetary policy rule (A.43).

### Equilibrium with full price and wage flexibility
- With ξ, ξ_w → 0, mc^f_t = 0 and w^f_t = mrs^f_t; resulting equilibrium conditions include:
  - Labor demand / supply pair: (A.44) and (A.45).
- By substituting linearized relations (A.27), (A.29), (A.37), (A.38), (A.28), (A.30), the labor-supply schedule in terms of (n^f_t, w^f_t, p^f_O,t − τ_·,t) is obtained (A.46).
- Energy market clearing in flexible-price equilibrium leads to (A.47) and, after eliminating capital rental and non-energy inputs, to (A.48).
- The three static equations (A.44), (A.46), (A.48) determine (n^f_t, w^f_t, p^f_O,t) for given realizations of y_O,t, τ_C,t, τ_Y,t and can be solved analytically.

### Closed-form implications for pre-subsidy energy price and wages
- Closed-form solution for pre-subsidy real energy price p^f_O,t (equation (A.49)) expressed via two positive coefficients Ψ_oc and Ψ_oy:
  - Ψ_oc defined in (A.50).
  - Ψ_oy defined in (A.51).
- Signs:
  - Ψ_oy > 0.
  - Ψ_oc > 0 (proof given by inequality chain in (A.52)).
- Corollary: a fall in energy endowment y_O,t or an increase in any energy subsidy raises the potential pre-subsidy real energy price p^f_O,t.
- Real producer wage solution: w^f_t equation (A.53) depends on Ψ_τc (A.54) and Ψ_τy (A.55).
  - Consumer energy subsidies reduce potential real producer wages iff condition (A.56) (equivalently the parameter restriction (21) in main text) holds.

### Aggregate demand implications of energy shocks
- Aggregate IS curve (abstracting away from habits) (A.57):
  - y_t = E_t y_{t+1} − σ(i_t − E_t π_C,t+1) − ω_c η_c E_t{Δp_O,t+1 − Δτ_C,t+1}.
- Interpretation:
  - If central bank stabilizes the real interest rate, the formula implies a positive relationship between consumer-relevant energy prices and activity.
  - However, efficiency requires output to contract after a negative energy supply shock for realistic parametrization; hence stabilizing real interest rate is generally not approximating optimal policy.
- Rewriting IS in terms of core inflation (A.58):
  - y_t = E_t y_{t+1} − σ(i_t − E_t π_Y,t+1) + ω_c(σ − η_c) E_t{Δp_O,t+1 − Δτ_C,t+1}.
  - If policy rate perfectly tracks expected core inflation and σ > η_c (energy and non-energy goods are complements), energy price increases reduce output and consumer energy subsidies stimulate activity.

### Gap representation of sticky price and wage equilibrium
- Energy market clearing in gap form yields (A.59) and, comparing with flexible-price equilibrium, the relation (A.60) or compact form (A.61):
  - ˆp_O,t = Ψ_on ˆn_t + Ψ_ow ˆw_t, with Ψ_on (A.62) and Ψ_ow (A.63).
- Real marginal cost expressed as (A.64) and converted to gap form (A.65) with coefficients Ψ_mpl_n (A.66) and Ψ_mpl_w (A.67).
- Price Phillips curve in gap form (A.68):
  - π_Y,t − β E_t π_Y,t+1 = κ_p (Ψ_mpl_n ˆn_t + Ψ_mpl_w ˆw_t), κ_p ≡ (1−ξ)(1−βξ)/ξ.
- Wage Phillips curve derived via labor supply and mrs definitions → gap form (A.71) with Ψ_mrs_n (A.72) and Ψ_mrs_w (A.73), then
  - π_W,t − ̃π_W,t = β E_t{π_W,t+1 − ̃π_W,t+1} + κ_w (Ψ_mrs_n ˆn_t + (Ψ_mrs_w − 1) ˆw_t), κ_w defined below (A.74).
- Real wage dynamics and combination with Phillips curves produce (A.76) and, assuming wage indexation to core inflation ̃π_W,t = ι_y π_Y,t, yield (A.77).

### Proof of Proposition 2 — central bank closing employment gap
- If central bank closes the employment gap, real wage dynamics reduce to a second-order difference equation (A.78) with κ defined in (A.79).
- Characteristic roots r_1, r_2 given by (A.81); both real and 0 < r_2 < 1.
- Solving yields closed-form expressions (A.84)–(A.89) tying actual real wage w_t to potential wage w^f_t via autoregressive structure and coefficients r_2, A.
- Core inflation response (A.88): π_Y,t = κ_p Ψ_mpl_w ∑_{j=0}^∞ β^j ˆw_{t+j}.
  - Since κ_p, Ψ_mpl_w > 0, core inflation sign matches discounted sum of wage gaps.
- Under Proposition 2 assumptions, consumer energy subsidies lower potential real producer wages and hence increase core inflation (detailed algebra in A.10).

### TANK extension (hand-to-mouth households)
- Introduces Keynesian (hand-to-mouth) agents consuming disposable income and working same hours as Ricardian agents:
  - C_K,t = W_C,t N_K,t − T_K,t (A.90); N_K,t = N_R,t (A.91); taxes evolve T_K,t − T^Y_K = t_K,t (A.92).
- Aggregates: C_t = λ C_K,t + (1−λ) C_R,t and N_t = λ N_K,t + (1−λ) N_R,t (A.93)-(A.94).
- Optimal reset wage condition (A.95) and marginal utility Λ_t redefined for Ricardians (A.96); all assets are priced by Ricardian households.
- The TANK extension yields a model with fifty-six equations and the same four exogenous drivers Y_O,t, τ_C,t, τ_Y,t, t_K,t.
- Social welfare is population-weighted average of Keynesian and Ricardian utilities; Keynesian utility aggregates as in (A.97) (no wage dispersion term for Keynesians).

### Open economy extension
- Two-country setup: Home and Foreign. Real exchange rate Q_t ≡ ε_t P^*_Y,t / P_Y,t.
- Non-energy consumption aggregator and composite consumption modified (A.98)–(A.99); consumer price and consumption composition adjusted (A.100)–(A.103); core inflation Π_N,t defined in (A.104).
- Wage indexation to core/headline modified (A.105).
- Firms produce for domestic and foreign markets; aggregate production (A.106).
- Export price setting under local currency pricing and export price dispersion described by (A.107)–(A.111).
- Monetary policy targets core inflation and responds to total output (domestic sales + exports) gap; rule (A.112).
- World energy market clearing and law of one price for energy: (A.113) and (A.114).
- External sector: UIP and net foreign asset accumulation (A.115)–(A.116).
- Accounting of equations and variables:
  - The open economy extension introduces sixty-eight new equilibrium conditions (fifteen Home equations (A.98)–(A.112), their fifteen Foreign counterparts, four cross-border equations (A.113)–(A.116), and thirty-four potential-equilibrium counterparts).
  - It drops twenty-eight Home conditions and their Foreign/potential counterparts.
  - Net increase of forty equations corresponds to twenty new endogenous variables and their twenty potential counterparts (listed collectively in the text).

*Source: Appendix A.4–A.12, wpiea2024081-print-pdf.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024081-print-pdf.pdf_