## wpiea2026030-source-pdf ## Source details **Canonical URL:** [wpiea2026030-source-pdf](https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026030-source-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2026/english/wpiea2026030-source-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2026/english/wpiea2026030-source-pdf.pdf.json) --- ### Literature review on exchange rate policies and the role of oil - Paper positioning and core innovation - Sits at the intersection of: (i) macroeconomic management of commodity price shocks and (ii) theory of optimal exchange rate policy under financial frictions. - Core innovation: integrate oil as a productive input into a model with nominal rigidities and financial frictions that generate endogenous deviations from PPP and UIP. - Key claim: financial pressures that justify foreign-exchange intervention (FXI) can be endogenously generated by oil price shocks through the NFA channel, creating a UIP risk wedge even in the absence of exogenous financial shocks. - Calibration to a representative oil exporter (GCC) shows the FXI motive from oil shocks is quantitatively larger than that from standard noise-trader shocks. - Quantifies welfare costs of suboptimal policies: no FXI, currency pegs, and energy subsidies. - Strand 1 — Commodity price shocks and policy (selected literature and empirical points) - Cited works include: Catao and Chang (2013), Bergholt et al. (2019), Bergholt (2014), Omotosho (2022), Chan et al. (2024), Bjørnland et al. (2018), Fernáández et al. (2018). - Empirical point preserved: commodity price shocks explain roughly one-third of real economic fluctuations in emerging markets (Fernández et al. (2018) wording preserved). - Fiscal policy literature cited: Auclert et al. (2023), Hevia and Nicolini (2013), Mendes and Pennings (2025). - Prior exchange rate regime work for oil exporters cited: Al-Abri (2014), Jin and Xiong (2021), Faltermeier et al. (2022). - Distinctions of present approach: - Oil enters production within a nominal-rigidities, financial-frictions model producing endogenous UIP and PPP deviations. - Optimal exchange rate policy analyzed without exogenous financial shocks. - Framework analytically tractable, yielding clear optimal policy rules and welfare results. - Strand 2 — Financial-market frictions and FXI rationale - Key references: Gabaix (2014), Gabaix and Maggiori (2015), Chang et al. (2015), Cavallino (2019), Maggiori (2022), Camanho et al. (2022). - Emphasis: exchange rates driven by intermediaries’ risk-bearing capacity; FXI can correct UIP deviations created by exogenous currency-demand shocks. - Additional literature on interactions: Benes et al. (2015), Bianchi et al. (2021), Fanelli and Straub (2021), Mukhin (2022), Iovino and Sergeyev (2023), Egorov and Mukhin (2023), Ottonello et al. (2024). - Integrated Policy Framework (IPF) contributions: Basu et al. (2023, 2020) — commodity production present but not as an input; current paper extends by integrating commodity inputs into domestic production. - Note from Basu et al. (2020): permanent commodity-price shocks can justify FXI in shallow markets to reduce inefficient risk premia while preserving flexibility in deep FX markets. - Mechanism and policy-role summary from literature synthesis - Oil price shocks affect net foreign assets (NFA), generating endogenous UIP risk wedges even without exogenous financial shocks. - Interest rate (R_t) stabilizes domestic price dispersion under sticky domestic prices (1/R_t = β E_t [W_t / W_{t+1}] or 1/R_t = β E_t [C^H_t / C^H_{t+1}]). - FXI (F^*_t) neutralizes the UIP wedge and can restore first-best allocation. - Welfare comparisons evaluate first-best attainment vs suboptimal policies: no FXI, currency pegs, energy subsidies. ### The approximate Ramsey problem — formulation and optimal policy - Loss function and wedge definitions - Wedges: - v_t ≡ log(C_Ht / ̃C_Ht) - u_t ≡ log(C_Ft / ̃C_Ft) - w_t = log(L_t / O_t) − log(̃L_t / ̃O_t) - Second-order approximation of Ramsey loss: - 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + γ u_t^2 + (1−α) α (1−γ) (̄Y_H / ̄C_H) w_t^2 ] - Appendix A linkage: u_t = −w_t. - Reduced approximate Ramsey problem (Equation 16): - min_{u_t,v_t,f^*_t,b^*_t,e_t,o_t} 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + χ u_t^2 ] - subject to constraints: - e_t = ̃q_t + v_t − u_t - (̄C_H / ̄Y_H) v_t = −α u_t + o_t − ̃o_t - β b^*_t − b^*_{t−1} = −1 / ̄Y_H ( ̄C_F u_t + ̄P^*_O ̄O (o_t − ̃o_t) ) - E_t ∆u_{t+1} = ̄ω var_t(∆e_{t+1}) ( n^*_t + f^*_t − b^*_t ) - Definitions: - ̄Y_H = ̄C_H + ̄C^*_H - χ = γ + (1−α) α (1−γ) (̄Y_H / ̄C_H) - e_t = log ε_t, ̃q_t = log ̃Q_t, ̃o_t = log ̃O_t, o_t = log O_t - b^*_t = B^*_t − ̃B^*_t / ̄Y_H; n^*_t = N^*_t − ̃B^*_t / ̄Y_H - f^*_t = F^*_t / ̄Y_H; ̄ω = ω / β ̄Y_H - Proposition 1 — optimal policy and attainability of first-best - Optimal policy eliminates the consumption wedges v_t and u_t. - Monetary policy targets v_t. - FXI closes UIP gap by setting f^*_t = − n^*_t. - Nominal exchange rate e_t adjusts one-to-one with natural equilibrium exchange rate ̃q_t. - Closing u_t and v_t yields oil consumption following first-best path; productivity shocks affect oil required for production and first-best NFA path, potentially exacerbating UIP gap via market segmentation and arbitrageurs’ risk aversion; optimal FXI can eliminate these inefficiencies. - Policy assignment: - (i) Monetary policy controls domestic wedge (v_t) via domestic interest rate. - (ii) FXI manages external wedge (u_t) by adjusting government foreign asset holdings to offset NFA pressures. ### Applications — pegs and energy-price subsidies - Peg regimes (Trilemma and welfare) - Under a fixed exchange rate, the central bank cannot independently set interest rates; domestic interest rate converges to global rate. - Proposition 2: welfare loss from a peg is decreasing in openness γ and increasing in oil share in production (1−α), and in volatility of oil price and productivity. - Under peg: UIP risk wedge eliminated so foreign-goods-consumption wedge disappears, but monetary policy cannot close domestic consumption wedge; v_t is forced to move one-to-one with natural real exchange rate: v_t = − ̃q_t. - Natural real exchange rate (log-linear approximation, Equation 17): - ̃q_t = a_t − (1−α) p^*_Ot + α ∑_{i=0}^∞ E_t r^*_{t+i} - Price fixing as energy subsidy - Government fixes domestic oil price via state-contingent tax/subsidy τ_t: - ̄P_O = (1−τ_t) ε_t P^{*O}_t - Subsidy severs pass-through of global energy price and exchange rate fluctuations to domestic economy but distorts factor mix and creates misallocation (e.g., excessive oil intensity when P^{*O}_t is high). - Additional wedge under price fixing: labor-to-oil ratio deviation: - l_ot − ̃l_ot = − p^*_ot − ̃q_t − v_t - Loss function with energy subsidy: - 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + γ u_t^2 + (1−α) α (1−γ) (̄Y_H / ̄C_H) ( p^*_ot + ̃q_t + v_t )^2 ] - Key implication: even if consumption wedges are eliminated, administered energy prices create unavoidable distortion → first-best unattainable with subsidies. - Under some conditions, a peg combined with energy price subsidies may yield smaller welfare losses than a floating regime with same subsidies because v_t = − ̃q_t under a peg and third-term cost depends only on volatility of international oil price. ### Extended model — inflation, NKPC, and policy problem - Intermediate-good producer FOCs: - W_t = α P_It A_t (L_t / O_t)^{α−1} - P_Ot = (1−α) P_It A_t (L_t / O_t)^α - Calvo pricing and NKPC (Equation 18): - π_t = (1−θ)(1−βθ)/θ ( p_It − p_Ht ) + β E_t π_{t+1}; π_t = log(P_Ht / P_Ht−1) - Exporter pricing (destination currency): - P^*_Ht = ε_X / (ε_X − 1) P_It / ε_t - Approximate Ramsey problem extended (Equation 19): - min 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ)( v_t^2 + χ_1 π_t^2 ) + χ_2 u_t^2 ] - subject to constraints including exchange rate, UIP wedge, production linkage, NFA evolution, and NKPC: - ∆ e_t = ∆ ̃q_t + ∆ v_t − ∆ u_t + π_t − ∆ p^*_Ft - E_t ( ∆ u_{t+1} ) = ̄ω var_t( ∆ e_{t+1} ) ( n^*_t + f^*_t − b^*_t ) - (̄C_H / ̄Y_H) v_t = α ( ε_X ̄C^*_H / ̄Y_H − 1 ) u_t + o_t − ̃o_t - β b^*_t − b^*_{t−1} = − 1 / ̄Y_H ( ( ̄C^*_H (ε_X − 1) α + ̄C_F ) u_t + ̄P^*_O ̄O ( o_t − ̃o_t ) ) - π_t = (1−θ)(1−βθ)/θ ( v_t − (1−α) u_t ) + β E_t π_{t+1} - χ_1 and χ_2 definitions in source: - χ_1 = θ (1−βθ)(1−θ) / ε_H (ε_H + 1) - χ_2 = γ + (1−γ)(1−α) α (̄Y / ̄C_H) + (1−γ) α^2 / ε_X (̄C^*_H / ̄C_H) - Difference vs baseline: inflation appears in objective; weight on inflation depends on openness γ and χ_1. ### Ramsey and first-best conditions (model core equations preserved) - Household utility maximization: - max_{C^H_t, C^F_t, L_t, B_t} E_0 ∑_{t=0}^∞ β^t [(1−γ) log C^H_t + γ log C^F_t − L_t] - Household budget constraint (exact): - P^H_t C^H_t + P^F_t C^F_t + B_t / R_t = B_{t−1} + W_t L_t + Π_t + T_t + P^O_t O^w_t - Law of one price for foreign goods and oil: - P^F_t = ε_t P^{*F}_t and P^O_t = ε_t P^{*O}_t; P^H_t and P^{*F}_t are fully sticky and normalized to 1. - First-order conditions / expenditure switching: - C^F_t = γ W_t / ε_t - C^H_t = (1−γ) W_t - 1 / R_t = β E_t [W_t / W_{t+1}] (equivalently 1 / R_t = β E_t [C^H_t / C^H_{t+1}]) - γ / (1−γ) * (C^H_t / C^F_t) = ε_t - Firm production: - C^H_t + C^{*H}_t = A_t L_t^α O_t^{1−α} - Optimal labor–oil ratio: - L_t / O_t = α / (1−α) * ε_t P^{*O}_t / W_t - Arbitrageurs’ mean-variance condition (UIP deviation): - E_t [ Θ_{t+1} (R^*_t − R_t ε_t / ε_{t+1}) ] = ω D^*_t R^*_t var_t [ R_t ε_t / ε_{t+1} ] - Market clearing / NFA identity: - B^*_t = F^*_t + N^*_t + D^*_t - Home country NFA evolution (dollar terms): - B^*_t R^*_t − B^*_{t−1} = P^*_H_t C^{*H}_t − P^*_F_t C^F_t + P^*_O_t (O^w_t − O_t) - International risk-sharing with risk-sharing wedge: - E_t [ Θ_{t+1} R^*_t ] = 1 + (B^*_t − N^*_t − F^*_t) R^*_t / (ω σ^2_t) - where σ^2_t = var_t [ R_t ε_t / ε_{t+1} ] - Ramsey problem statement: - max_{R_t, F^*_t, ...} E_0 ∑_{t=0}^∞ β^t [(1−γ) log C^H_t + γ log C^F_t − L_t] subject to model constraints. ### Policy-relevant insights (from Ramsey analysis) - FXI can be welfare-improving when oil-induced NFA dynamics generate endogenous UIP wedges that monetary policy alone cannot neutralize. - Currency pegs block efficient expenditure switching; welfare cost of pegs declines with openness but rises with oil intensity. - Energy subsidies distort labor–oil ratio and fail to stabilize output gap—typically suboptimal relative to policies targeting UIP wedge and price dispersion. - Model tractability yields clear optimal policy rules: interest-rate policy to stabilize domestic price dispersion; FXI to neutralize UIP wedge and restore first-best. ### Exact algebraic definitions and planner optimality (selected preserved forms) - Planner optimality conditions (preserved algebraic forms): - β R^*_t E_t [ ˜C^F_t / ˜C^F_{t+1} ] = 1 - ˜C^H_t = (1−γ) α A_t ( ˜L_t / ˜O_t )^{α−1} - ˜C^F_t = γ α / (1−α) P^{*O}_t ˜O_t ˜L_t - Natural real exchange rate: - ˜Q_t = γ / (1−γ) * ˜C^H_t / ˜C^F_t ### Calibration — selected parameters (Table 1 excerpts preserved) - A. Preferences and Trade Structure - β 0.99 - γ 0.3 - ω 345 - ̄C^* 0.5 - ̄O_w 0.5 - B. Technology and Pricing - 1−α 0.17 - ε_H 6 - ε_X 1.5 - θ 0.75 - C. Policy Rules and Shocks - γ_π 2.0 - ρ_r 0.9 - ρ_a, ρ_po, ρ_n 0.9 - σ_a listed elsewhere as 0.0063 (see Shock calibrations) ### Shock calibrations (quarterly, GCC-focused) - Productivity shock: - Standard deviation: σ_a = 0.0063 (based on average standard deviation of TFP growth across GCC economies in Bannaga and Lezar (2024), assuming AR(1) TFP). - Oil price shock: - Standard deviation: σ_po = 0.158 (fitted AR(1) to quarterly real oil prices; standard deviation of residuals). - Noise-trader shock: - Standard deviation: σ_n = 0.100 (represents exogenous demand shock for foreign currency equivalent to 10 percent of quarterly GDP; interpreted as an upper bound). - Even with this large value, results indicate noise-trader shocks have negligible welfare implications in benchmark calibration. ### Interpretation of calibration - Discount factor β = 0.99 implies an annual real interest rate of approximately 4 percent (calculation noted in source). - Trade openness γ = 0.3 based on WITS. - Share of oil in production 1−α = 0.17 based on average ratio of energy expenditure to non-oil GDP. - ω = 345 chosen to satisfy ωVar_t(∆ln e_{t+1}) ̄Y = 0.06 consistent with market shallowness parameter Γ for emerging markets in Adrian et al. (2021). - Calvo θ = 0.75 implies average price duration of four quarters. ### Model implications on policy and price rigidity (summary) - Optimal policy can achieve first-best by closing u_t and v_t through monetary policy (interest rate) and FXI. - Mechanisms unaffected when relaxing full price rigidity; inflation dynamics incorporated via NKPC do not change core policy assignment: interest-rate policy stabilizes domestic wedge; FXI neutralizes UIP wedge. - Key wedges driven by wages (home goods consumption wedge) and energy prices (foreign goods consumption wedge). ### Quantitative results and welfare costs — regimes, wedges, and findings - Benchmark and wedges - Benchmark economy subject to productivity (A_t) and oil price (O_t) shocks; currency demand shocks (N_t) introduced to assess financial noise. - Welfare losses traced to: - Price stickiness → domestic inflation/output wedge v_t. - Input-mix distortions, foreign consumption distortions, export-pricing distortions → UIP/financial wedge u_t. - External wedges move proportionally with foreign-goods-consumption wedge when no energy subsidies and no DCP. - Policy regimes evaluated - No FXI: floating exchange rate; central bank follows Taylor rule. - No FXI + DCP: same but export prices invoiced in dominant currency. - Peg: exchange rate fixed. - Peg + Subsidy: peg plus government fixes domestic fuel price. - Welfare-cost comparisons (Figure 1 summary and preserved numeric descriptions) - Benchmark calibration: γ = 0.3, 1−α = 0.17. - Welfare losses with productivity and oil price shocks only "ranging from about 1 to 4 percent." - With only productivity shocks, welfare costs below 0.10 percent of permanent consumption. - Trade openness effects: - As γ increases, welfare weight on foreign-goods wedge increases; peg regimes tend to outperform floating regimes at higher γ because fixing exchange rate eliminates financial wedge u_t. - "No FXI + DCP" uniformly underperforms "No FXI". - At low openness, No FXI can deliver higher welfare because exchange rate flexibility allows expenditure switching and relative price adjustments that mitigate shock impacts. - Oil share in production effects: - All curves slope upward in 1−α because higher oil share amplifies transmission of oil price shocks to marginal costs. - Peg deteriorates sharply with higher oil intensity; Peg + Subsidy deteriorates more gradually. - Relative performance: - Peg + Subsidy lowest welfare loss among peg-type policies for baseline, but still inferior to integrated first-best policy using interest-rate policy and FXI. - Role and quantitative impact of noise–trader shocks - Introducing a noise–trader shock increases welfare losses only modestly under floating regimes. - Mechanism: noise-trader raises exchange-rate volatility → raises perceived risk for arbitrageurs → increases u_t. - Incremental welfare costs from adding a noise–trader shock are small (figure vertical axis range up to 0.1 percent CEV). - Under a peg the UIP wedge u_t is effectively closed so noise–trader shock generates near-zero incremental welfare loss. - Main quantitative contribution and interpretation - For commodity-exporting economies, endogenous financial pressures from real oil shocks are a quantitatively dominant rationale for active FXI, complementing literature focused on exogenous financial shocks. - Oil-price shocks act as first-order terms-of-trade and wealth shocks for oil exporters, inducing large trade-balance and exchange-rate volatility and sharp swings in NFA; without FXI, time-varying currency exposure leads to time-varying UIP wedge u_t and substantial welfare costs. ### Policy implications and recommendations (quantitative section) - Theoretical prescription: use interest-rate policy to close domestic output gap and FXI to neutralize UIP wedge to achieve first-best. - Practical caveats and guidance: - Fully integrated policy may be infeasible due to reserve, governance, or communication constraints. - Free float without FXI suffers welfare losses from volatile risk premia driven by oil price swings; hard peg eliminates premia but shifts volatility into inflation and output gap. - Combining peg with energy subsidy can sometimes outperform pure float by smoothing domestic marginal costs and reducing inflation and output volatility. - FXI should be used selectively and contingently, consistent with IMF 2023 IPF guidance: - Recommended only when specific financial frictions (e.g., shallow FX markets or elevated risk premia) materially impair market functioning. - Use of FXI should be contingent on clear evidence such risks have intensified and be supported by adequate reserves, strong institutional and governance frameworks, and transparent communication. - Regular or imprudent FXI can undermine market liquidity, constrain reserve capacity for future shocks, and weaken monetary policy signaling. - Scope and limits: - Framework isolates transmission mechanisms where oil price shocks via NFA dynamics and segmented FX markets generate policy motive for FXI in a stylized GE setting. - Analysis abstracts from issues like credibility/anchoring role of long-standing pegs, effects on capital costs and financial stability, transitional/reputational costs of regime changes. ### Extensions and robustness — expatriate workers, CES production, market depth, and multi-sector models - Model with expatriate workers - Expatriates accounted for 78% of labor force in GCC in 2024Q2 (contextual statistic preserved). - Expatriate labor treated as imported production input; expatriates remit earnings and do not consume domestically produced goods. - Under this specification, some wedges vanish (e.g., l_{FOt} − ˜l_{FOt} = 0) and second-order loss simplifies to: - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + ( γ + (1−σα)σα(1−γ) Ȳ_H Ȼ_H^{-1} ) u_t^2 ] - Natural real exchange rate expression: - ˜q_t = a_t − (1−α) p^*_Ot + σα ∑_{i=0}^∞ E_t r^*_{t+i} − α(1−σ) w^*_t - As σ → 0 (reduced household labor share relative to expatriates) effect of foreign rate volatility on peg welfare loss diminishes; oil-price volatility effect remains. - Extended model lemmas and second-order approximations - Lemma D.1–D.4 on price dispersion and relationships between price dispersion and inflation variance. - Resulting second-order loss function (intermediate form): - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) ( v_t^2 + χ_1 π_t^2 ) + χ_2 u_t^2 + ε_X (1−γ) Ȼ^*_H Ȼ_H^{-1} (p^*_Ht − ˜p^*_Ht)^2 ] - Using p^*_Ht − ˜p^*_Ht = α u_t reduces to equation (19). - Two-sector model with non-tradables (E) - Household and sectoral firm problems defined; equilibrium conditions and deviations identify wedge relationships (v_t, u_t) and constraints. - Quadratic loss function expressed as ˜L − L = (1/2) E_0 ∑ β^t x'_t H_{xx} x_t with explicit H_{xx} matrix; H_{xx} positive semidefinite ensuring minimum loss zero achievable under constraints. - CES production function (F) - CES production: Y_Ht = A_t ( α (a_L L_t)^σ + (1−α) (a_O O_t)^σ )^{1/σ}. - Ramsey objective reduces to (1/2) E_0 ∑ β^t [ (1−γ) v_t^2 + χ u_t^2 ] with χ defined: - χ = γ + (1−γ)/(1−σ) Ȳ_H Ȼ_H^{-1} α a_L^σ Ḷ^σ Ȳ_H^{−σ} [ 1 − α a_L^σ Ḷ^σ Ȳ_H^{−σ} ] - Constraint: Ȼ_H Ȳ_H^{-1} v_t = − α/(1−σ) u_t + o_t − ˜o_t - Main implication: first-best allocation achievable via optimal monetary policy and FXI even when oil and labor are not substitutable (σ → −∞); weight on foreign goods consumption wedge decreases as inputs become less substitutable. - Market depth and welfare loss (G) - UIP wedge scales with market depth via ω: - E_t(∆u_{t+1}) = ̄ω var_t(∆e_{t+1})(n^*_t + f^*_t − b^*_t) - Higher ω amplifies volatility of currency premia for given flows/NFA configuration. - Welfare losses under No FXI increase sharply and monotonically with ω. - Peg and Peg + Subsidy largely insensitive to ω because fixing the exchange rate eliminates UIP wedge; their welfare performance determined by domestic distortions v_t and π_t driven by ˜q_t. - Robustness message: case for FXI strengthens as financial markets become shallower and oil price shocks are prominent. - Figure G1 displays Welfare Costs (in % CEV) across regimes as ω varies; benchmark ω = 345. *Excerpted from provided sections of the source PDF (wpiea2026030-source-pdf).* ### Section 2 provides the literature review on exchange rate policies and the role of oil ### Section 2: Literature Review on Exchange Rate Policies and the Role of Oil ### Overview and paper positioning - Paper sits at the intersection of two strands of literature: (i) macroeconomic management of commodity price shocks and (ii) theory of optimal exchange rate policy under financial frictions. - Core innovation: integrate oil as a productive input into a model with nominal rigidities and financial frictions that generate endogenous deviations from purchasing power parity (PPP) and uncovered interest parity (UIP). - Key claim: financial pressures that justify foreign-exchange intervention (FXI) can be endogenously generated by oil price shocks through the NFA channel, creating a UIP risk wedge even in the absence of exogenous financial shocks. - Calibration to a representative oil exporter (GCC) shows the FXI motive from oil shocks is quantitatively larger than that from standard noise-trader shocks. - Quantifies welfare costs of suboptimal policies: no FXI, currency pegs, and energy subsidies. ### Strand 1 — Commodity price shocks and policy - Literature surveyed includes Catao and Chang (2013), Bergholt et al. (2019), Bergholt (2014), Omotosho (2022), Chan et al. (2024), Bjørnland et al. (2018), Fernáandez et al. (2018). - Empirical point: commodity price shocks explain roughly one-third of real economic fluctuations in emerging markets (Fernández et al. (2018) wording preserved). - Fiscal policy literature on commodity shocks cited: Auclert et al. (2023) (energy-importing economies), Hevia and Nicolini (2013), Mendes and Pennings (2025) (commodity-exporting economies). - Prior papers on exchange rate regime choice in oil exporters: Al-Abri (2014) — optimal regime depends on import pricing (producer vs local currency pricing); Jin and Xiong (2021) — regime-switching model; Faltermeier et al. (2022) — optimal response to commodity boom involves increasing foreign exchange reserves to stabilize real exchange rate and tradable production. - Distinctions of the present approach: - Introduces oil into production within a model with nominal rigidities and financial frictions that produce endogenous UIP and PPP deviations. - Analyzes optimal exchange rate policy without relying on exogenous financial shocks. - Framework is analytically tractable—yields clear optimal policy rules and welfare results. ### Strand 2 — Financial-market frictions and FXI rationale - Key references: Gabaix (2014), Gabaix and Maggiori (2015), Chang et al. (2015), Cavallino (2019), Maggiori (2022), Camanho et al. (2022). - These works emphasize that exchange rates are driven by intermediaries’ risk-bearing capacity as much as fundamentals; FXI can correct UIP deviations created by exogenous currency-demand shocks (noise-trader or portfolio flow shocks). - Further literature on FXI interactions with monetary policy, liquidity shocks, and capital flows: Benes et al. (2015), Bianchi et al. (2021), Fanelli and Straub (2021), Mukhin (2022), Iovino and Sergeyev (2023), Egorov and Mukhin (2023), Ottonello et al. (2024). - Integrated Policy Framework (IPF) contributions: Basu et al. (2023, 2020) develop a small open-economy model evaluating joint use of policy rate, FXI, capital controls, and macroprudential measures. In that literature, commodity production is present but not used as an input; the current paper extends this by integrating commodity inputs into domestic production. - Consistent note from Basu et al. (2020): for economies with large commodity sectors, permanent commodity-price shocks can justify FXI in shallow markets to reduce inefficient risk premia while preserving exchange-rate flexibility observed in deep FX markets. ### Model implications and mechanism - Mechanism emphasized: oil price shocks affect net foreign assets (NFA), generating endogenous UIP risk wedges even without exogenous financial shocks. - Policy lever roles: - Interest rate (R_t): used to stabilize domestic price dispersion under sticky domestic prices (equation (4): 1/R_t = β E_t [W_t / W_{t+1}] or 1/R_t = β E_t [C^H_t / C^H_{t+1}]). - FXI (F^*_t): used to neutralize the UIP wedge and restore the first-best allocation. - Welfare comparisons: - First-best attainable when interest rate stabilizes domestic price dispersion and FXI neutralizes endogenous UIP wedge. - Suboptimal policies assessed include: no FXI, currency pegs (which block efficient expenditure switching), and energy subsidies (which distort labor–oil ratio while failing to stabilize the output gap). ### Key model structure and exact conditions (selected equations and definitions preserved) - Household maximization (utility): - max_{C^H_t, C^F_t, L_t, B_t} E_0 ∑_{t=0}^∞ β^t [(1−γ) log C^H_t + γ log C^F_t − L_t] - Household budget constraint: - P^H_t C^H_t + P^F_t C^F_t + B_t / R_t = B_{t−1} + W_t L_t + Π_t + T_t + P^O_t O^w_t - Law of one price for foreign goods and oil: - P^F_t = ε_t P^{*F}_t and P^O_t = ε_t P^{*O}_t; P^H_t and P^{*F}_t are fully sticky and normalized to 1. - First-order conditions / expenditure switching: - C^F_t = γ W_t / ε_t - C^H_t = (1−γ) W_t - 1 / R_t = β E_t [W_t / W_{t+1}] (equivalently 1 / R_t = β E_t [C^H_t / C^H_{t+1}]) - γ / (1−γ) * (C^H_t / C^F_t) = ε_t - Firm production and input mix: - Production: C^H_t + C^{*H}_t = A_t L_t^α O_t^{1−α} - Optimal labor–oil ratio: - L_t / O_t = α / (1−α) * ε_t P^{*O}_t / W_t - Financial markets and UIP deviation: - Arbitrageurs’ mean-variance condition (right-hand side is UIP deviation): - E_t [ Θ_{t+1} (R^*_t − R_t ε_t / ε_{t+1}) ] = ω D^*_t R^*_t var_t [ R_t ε_t / ε_{t+1} ] - Noise traders zero-capital portfolio: N_t R_t = − ε_t N^*_t R^*_t - Market clearing / NFA identity: - B^*_t = F^*_t + N^*_t + D^*_t - Home country budget constraint (NFA evolution, dollar terms): - B^*_t R^*_t − B^*_{t−1} = P^*_H_t C^{*H}_t − P^*_F_t C^F_t + P^*_O_t (O^w_t − O_t) - International risk-sharing condition with risk-sharing wedge: - E_t [ Θ_{t+1} R^*_t ] = 1 + (B^*_t − N^*_t − F^*_t) R^*_t / (ω σ^2_t) - where σ^2_t = var_t [ R_t ε_t / ε_{t+1} ] - Ramsey policy problem (government chooses sequences of R_t and F^*_t to maximize household welfare subject to model constraints): - max_{R_t, F^*_t, ...} E_0 ∑_{t=0}^∞ β^t [(1−γ) log C^H_t + γ log C^F_t − L_t] subject to (3),(4),(5),(6),(7),(10),(11). - First-best allocation (planner with flexible prices and access to perfectly elastic dollar bonds R^*_t): - Planner problem: max E_0 ∑ β^t [(1−γ) log C^H_t + γ log C^F_t − L_t] subject to (6), (10) and given shocks {P^{*O}_t, O^w_t, C^{*H}_t, A_t, R^*_t} and initial B^*_{−1}, NPGC for B^*_∞. - Planner optimality conditions (preserved algebraic forms): - β R^*_t E_t [ ˜C^F_t / ˜C^F_{t+1} ] = 1 - ˜C^H_t = (1−γ) α A_t ( ˜L_t / ˜O_t )^{α−1} - ˜C^F_t = γ α / (1−α) P^{*O}_t ˜O_t ˜L_t - Natural real exchange rate definition: - ˜Q_t = γ / (1−γ) * ˜C^H_t / ˜C^F_t ### Policy-relevant insights (from literature synthesis and model setup) - FXI can be welfare-improving when oil-induced NFA dynamics generate endogenous UIP wedges that monetary policy alone (interest rate) cannot neutralize. - Currency pegs can be particularly costly in the presence of oil shocks because they block efficient expenditure switching; the welfare cost of pegs declines with openness but depends critically on oil intensity. - Energy subsidies distort the labor–oil input ratio and fail to stabilize the output gap—hence they are typically suboptimal relative to policies that target the UIP wedge and price dispersion. - Model tractability allows derivation of clear optimal policy rules: interest-rate policy to stabilize domestic price dispersion; FXI to neutralize UIP wedge and restore first-best. *Excerpted from Section 2 of the source PDF (literature review on exchange rate policies and the role of oil).* ### 4.2 The Approximate Ramsey Problem ### 4.2 The Approximate Ramsey Problem ### Loss function and wedges - Define wedges: - v_t ≡ log(C_Ht / ̃C_Ht) - u_t ≡ log(C_Ft / ̃C_Ft) - w_t = log(L_t / O_t) − log(̃L_t / ̃O_t) - Second-order approximation of the Ramsey loss (deviation from first-best): - 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + γ u_t^2 + (1−α) α (1−γ) (̄Y_H / ̄C_H) w_t^2 ] - Appendix A shows linkage between labor-to-oil wedge and foreign consumption wedge: - u_t = −w_t - Using u_t = −w_t, the approximate Ramsey problem can be rewritten as: - min_{u_t,v_t,f^*_t,b^*_t,e_t,o_t} 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + χ u_t^2 ] (Equation 16) - subject to: - e_t = ̃q_t + v_t − u_t - (̄C_H / ̄Y_H) v_t = −α u_t + o_t − ̃o_t - β b^*_t − b^*_{t−1} = −1 / ̄Y_H ( ̄C_F u_t + ̄P^*_O ̄O (o_t − ̃o_t) ) - E_t ∆u_{t+1} = ̄ω var_t(∆e_{t+1}) ( n^*_t + f^*_t − b^*_t ) - Definitions and parameters in constraints: - ̄Y_H = ̄C_H + ̄C^*_H - χ = γ + (1−α) α (1−γ) (̄Y_H / ̄C_H) - e_t = log ε_t - ̃q_t = log ̃Q_t - ̃o_t = log ̃O_t - o_t = log O_t - b^*_t = B^*_t − ̃B^*_t / ̄Y_H - n^*_t = N^*_t − ̃B^*_t / ̄Y_H - f^*_t = F^*_t / ̄Y_H - ̄ω = ω / β ̄Y_H ### Optimal policy and attainability of first-best - Proposition 1 (optimal policy): - Optimal policy eliminates the wedge in consumption of home goods v_t and foreign goods u_t. - Monetary policy targets v_t. - FXI closes the UIP gap by setting f^*_t = − n^*_t. - The nominal exchange rate e_t adjusts one to one with movement in natural equilibrium exchange rate ̃q_t. - Implications and mechanism: - Closing u_t and v_t leads oil consumption to follow first-best path. - Productivity shocks affect oil required for production and first-best NFA path, potentially exacerbating UIP gap via market segmentation and arbitrageurs’ risk aversion; optimal FXI can eliminate these inefficiencies. - Policy assignment logic: - (i) Monetary policy controls domestic wedge (v_t) via domestic interest rate and stabilizes domestic consumption (sticky prices). - (ii) FXI manages external wedge (u_t) by adjusting government foreign asset holdings to offset NFA pressures, neutralizing arbitrageurs’ required compensation and eliminating UIP risk premium. ### Applications — Peg regimes - Trilemma under a fixed exchange rate (peg): - Central bank loses ability to independently set interest rates and stabilize domestic consumption wedge. - Fixing the exchange rate eliminates currency risk for arbitrageurs, making supply of foreign exchange perfectly elastic and ensuring perfect capital mobility; domestic interest rate converges to global rate. - Proposition 2: - The welfare loss from the peg is decreasing in openness γ and increasing in the share of oil in production (1−α), and the volatility of oil price and productivity level. - Under a peg: - UIP risk wedge is eliminated → wedge in consumption of foreign goods disappears. - Monetary policy cannot close domestic consumption wedge; v_t is forced to move one-to-one with natural real exchange rate: - v_t = − ̃q_t - Natural real exchange rate log-linear approximation (Equation 17): - ̃q_t = a_t − (1−α) p^*_Ot + α ∑_{i=0}^∞ E_t r^*_{t+i} - Mechanism for comparative statics: - Higher γ reduces weight of domestic goods sector in aggregate welfare → peg cost falls. - Higher oil share (larger 1−α) amplifies effect of oil price shocks on ̃q_t → peg cost rises because variance of ̃q_t increases. ### Applications — Price fixing as energy subsidy - Government fixes domestic oil price via state-contingent tax/subsidy τ_t: - ̄P_O = (1−τ_t) ε_t P^*_Ot - Purpose: sever pass-through of global energy price and exchange rate fluctuations to domestic economy. - Distortion: - Subsidy severs price signal of true opportunity cost; firms base factor mix on administered price ̄P_O → misallocation (e.g., excessive oil intensity when P^*_Ot is high). - Additional wedge under price fixing: - Labor-to-oil ratio deviation: - l_ot − ̃l_ot = − p^*_ot − ̃q_t − v_t - Loss function with energy subsidy: - 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + γ u_t^2 + (1−α) α (1−γ) (̄Y_H / ̄C_H) ( p^*_ot + ̃q_t + v_t )^2 ] - Interpretation: third term is welfare cost from administered domestic energy prices in presence of oil price volatility (p^*_ot), real exchange rate movements (̃q_t), and domestic distortions (v_t). - Key implication: - Even if consumption wedges are eliminated, administered energy prices create an unavoidable distortion → first-best unattainable with subsidies. - Under some conditions, a peg combined with energy price subsidies may yield smaller welfare losses than a floating regime with same subsidies: - Under a peg: v_t = − ̃q_t ⇒ third term depends only on volatility of international oil price. - Under floating: exchange rate fluctuations can amplify welfare cost of oil price shocks; subsidies can reduce inflation volatility, potentially making peg + subsidy preferable to peg alone. ### Extended model — inflation, NKPC, and policy problem - Motivation: relax full price rigidity, allow inflation dynamics and export/import price fluctuations. - Intermediate good producer: - Production: Y_t = A_t L_t^α O_t^{1−α} - FOC results: - W_t = α P_It A_t (L_t / O_t)^{α−1} - P_Ot = (1−α) P_It A_t (L_t / O_t)^α - Home retailers (Calvo pricing): - Retailers set price with probability 1−θ each period; demand C_iHt = (P_iHt / P_Ht)^{−ε_H} C_Ht - Optimal price P^#_Ht and aggregate pricing condition lead to NKPC (Equation 18): - π_t = (1−θ)(1−βθ)/θ ( p_It − p_Ht ) + β E_t π_{t+1} - where π_t = log(P_Ht / P_Ht−1), p_It = log(P_It / ̄P_I), p_Ht = log(P_Ht / ̄P_H) - Exporters: - Demand C^*_Ht = P^*_Ht^{−ε_X} C^*_t - Exporter optimal price (flexible, destination currency): - P^*_Ht = ε_X / (ε_X − 1) P_It / ε_t - Note: alternative pricing assumptions (PCP vs DCP) affect attainability of first-best. - Home budget constraint and aggregate profits noted; exports, imports, and oil enter budget flows. - Approximate Ramsey problem in extended model (Equation 19): - min 1/2 E_0 ∑_{t=0}^∞ β^t [ (1−γ)( v_t^2 + χ_1 π_t^2 ) + χ_2 u_t^2 ] - subject to: - ∆ e_t = ∆ ̃q_t + ∆ v_t − ∆ u_t + π_t − ∆ p^*_Ft - E_t ( ∆ u_{t+1} ) = ̄ω var_t( ∆ e_{t+1} ) ( n^*_t + f^*_t − b^*_t ) - (̄C_H / ̄Y_H) v_t = α ( ε_X ̄C^*_H / ̄Y_H − 1 ) u_t + o_t − ̃o_t - β b^*_t − b^*_{t−1} = − 1 / ̄Y_H ( ( ̄C^*_H (ε_X − 1) α + ̄C_F ) u_t + ̄P^*_O ̄O ( o_t − ̃o_t ) ) - π_t = (1−θ)(1−βθ)/θ ( v_t − (1−α) u_t ) + β E_t π_{t+1} - χ_1 and χ_2 definitions: - χ_1 = θ (1−βθ)(1−θ) / ε_H (ε_H + 1) - χ_2 = γ + (1−γ)(1−α) α (̄Y / ̄C_H) + (1−γ) α^2 / ε_X (̄C^*_H / ̄C_H) - Differences vs baseline: - Inflation appears in objective function; weight on inflation depends on openness γ and χ_1 (which incorporates θ, price stickiness). - Inflation reflects discounted sum of current and future real marginal costs influenced by goods openness and price stickiness. ### Calibration (selected parameters from Table 1) - A. Preferences and Trade Structure - β 0.99 (Discount factor) - γ 0.3 (Trade openness (share of imports)) - ω 345 (Arbitrageur risk aversion (financial friction)) - ̄C^* 0.5 (Foreign demand) - ̄O_w 0.5 (Oil endowment) - B. Technology and Pricing - 1−α 0.17 (Share of oil in production) - ε_H 6 (Elasticity of substitution (domestic goods)) - ε_X 1.5 (Elasticity of export demand) - θ 0.75 (Calvo price stickiness parameter) - C. Policy Rules and Shocks - γ_π 2.0 (Taylor rule response to inflation) - ρ_r 0.9 (Taylor rule interest rate smoothing) - ρ_a, ρ_po, ρ_n 0.9 (Shock persistence (productivity, oil price, noise trader)) - σ_a (value listed but not completed in provided excerpt) *Source: IMF Working Paper — section 4.2 The Approximate Ramsey Problem (from provided PDF content).* ### 0.0063 Std. dev. of productivity shock ### wpiea2026030-source-pdf - 0.0063 Std. dev. of productivity shock ### Model implications on policy and price rigidity - The optimal policy can achieve the first-best outcome by closing the u_t and v_t gaps through a combination of monetary policy and FXI. - Therefore, the model’s implications remain unchanged even when the full price rigidity assumption is relaxed. - Key variables driving wedges: - wages, which are closely linked to the wedge in home goods consumption - energy prices, which are associated with the wedge in foreign goods consumption ### Calibration overview (quarterly frequency, GCC economies) - Objective: reflect key structural features of GCC economies. - Parameters summarized in Table 1 (in source). - Calibration notes: - Discount factor: β = 0.99 (implies an annual real interest rate of approximately 4 percent). - Trade openness parameter: γ = 0.3 (share of imported consumption goods in total consumption expenditure, from WITS). - Share of oil in production: 1−α = 0.17 (based on average ratio of energy expenditure to non-oil GDP; computed using GDP per unit of energy use from WDI). - Steady-state foreign aggregate demand and oil endowment: ̄C∗ = 0.5 and ̄Ow = 0.5 (calibrated jointly to match an oil export share in total exports of 0.6 and a non-oil GDP share in total GDP of 0.7). - Financial frictions parameter: ω = 345. - This value ensures that ωVar t (∆lne t+1 ) ̄Y = 0.06, consistent with the market shallowness parameter Γ for emerging market economies in Adrian et al. (2021). - Elasticity of substitution between domestic goods: εH = 6. - Elasticity of export demand: εX = 1.5. - Calvo parameter (nominal rigidities): θ = 0.75 (implying an average price duration of four quarters). - Monetary policy rule parameters: - Response to inflation: γπ = 2. - Interest rate smoothing coefficient: ρr = 0.9. - Persistence parameters for shocks: - ρa = 0.9 - ρpo = 0.9 - ρn = 0.9 ### Shock calibrations and interpretation - Productivity shock: - Standard deviation: σa = 0.0063. - Basis: average standard deviation of TFP growth across GCC economies reported in Bannaga and Lezar (2024), assuming TFP follows a trend-stationary AR(1) process. - Oil price shock: - Standard deviation: σpo = 0.158. - Basis: fitted AR(1) process to quarterly real oil prices; standard deviation of residuals. - Noise-trader shock: - Standard deviation: σn = 0.100. - Interpretation: represents an exogenous demand shock for foreign currency equivalent to 10 percent of quarterly GDP. - Calibration caveat: magnitude is difficult to identify empirically for GCC economies; this calibration can be interpreted as an upper bound. - Result: Even under this large value, the results indicate that noise-trader shocks have negligible welfare implications, consistent with the exchange-rate disconnect emphasized in the literature. *Source: wpiea2026030-source-pdf* ### 6.4 Quantitative Results and Welfare Costs ### 6.4 Quantitative Results and Welfare Costs ### Benchmark setup and identified wedges - The quantitative analysis uses a benchmark economy subject to both productivity (A_t) and oil price shocks (O_t), and then introduces currency demand shocks (N_t) to assess the quantitative importance of financial noise. - Welfare losses are traced to several key distortions ("wedges") that emerge from nominal and real rigidities: - Price stickiness in domestic goods markets → fluctuations in domestic inflation and output (sticky-price/output-gap wedge, v_t). - Input-mix distortions, foreign goods consumption distortions, and export-pricing distortions (UIP or financial wedge, u_t) → originate from financial frictions and absence of efficient international risk sharing. - When there are no energy subsidies or when export prices are not subject to dominant currency pricing (DCP), the external wedges move proportionally with the foreign-goods-consumption wedge; closing that wedge effectively eliminates the others. - Intuition on openness: A higher trade openness γ reduces the relative weight on domestic distortions and increases the weight on external distortions (u_t), because an open economy relies more heavily on foreign goods and international risk sharing. ### Policy regimes evaluated - Four policy regimes are considered: - No FXI: exchange rate freely floats; central bank follows a Taylor rule responding to inflation. - No FXI + DCP: same as No FXI but export prices are invoiced in a dominant currency (do not adjust to exchange rate). - Peg: exchange rate fixed. - Peg + Subsidy: peg plus government fixes domestic fuel price (energy-price subsidy). - Role of instruments: - Interest-rate rule primarily stabilizes v_t and π_t (domestic output-gap and inflation). - FXI directly addresses u_t (UIP/financial wedge). ### Welfare-cost comparisons and key quantitative findings (Figure 1 results) - Benchmark calibration: γ = 0.3 and 1−α = 0.17. - Welfare losses with productivity and oil price shocks only are economically significant, "ranging from about 1 to 4 percent." - With only productivity shocks, welfare costs remain small—below 0.10 percent of permanent consumption. - Trade openness (left subplot of Panel (A)): - As γ increases, the welfare weight on the foreign-goods wedge increases; peg regimes tend to outperform free-floating regimes at higher γ because fixing the exchange rate eliminates the financial wedge u_t. - Under a no-FXI regime, welfare losses rise with openness: although higher γ lowers the share of oil exports in the trade balance (reducing relevance of oil price shocks and stabilizing u_t), the increased welfare weight on u_t dominates, raising overall welfare costs with openness. - The "No FXI + DCP" regime uniformly underperforms "No FXI" because preset dollar prices erode the natural hedge from non-oil exports, amplifying UIP wedge fluctuations and introducing additional misalignment between marginal costs and revenues. - At low levels of openness, the No FXI regime can deliver higher welfare because exchange rate flexibility allows expenditure switching and relative price movements that mitigate shock impacts on domestic inflation; for an oil exporter, currency appreciation in response to higher global energy prices offsets part of the shock and allows substitution toward cheaper energy inputs, dampening domestic activity and stabilizing inflation. - Oil share in production (right subplot of Panel (A)): - All curves slope upward in 1−α because a higher oil share amplifies transmission of oil price shocks to marginal costs; oil input enters marginal costs with weight (1−α), so larger oil share implies stronger marginal-cost movements when oil prices fluctuate. - Under Calvo pricing, marginal-cost fluctuations increase output-gap and inflation volatility via the NKPC, raising welfare losses associated with price dispersion. - A peg deteriorates sharply with higher oil intensity because it rules out nominal exchange rate adjustment and transmits oil shocks directly into domestic marginal costs (u_t = 0 under peg), amplifying v_t and inflation. - Peg + Subsidy deteriorates more gradually as the subsidy cushions pass-through of oil prices into marginal costs. - Relative performance of regimes: - Peg + Subsidy delivers the lowest welfare loss among peg-type policies for the baseline calibration, but remains inferior to the integrated first-best policy that jointly targets both domestic and external wedges via interest-rate policy and FXI. ### Role and quantitative impact of noise–trader shocks (Panel (B)) - Introducing a noise–trader shock increases welfare losses only modestly under floating regimes. - Mechanism: - A noise–trader disturbance raises exchange-rate volatility → raises perceived risk for arbitrageurs → risk-averse intermediaries demand a higher premium → amplifies u_t. - The foreign-goods-consumption wedge depends on both exchange-rate volatility and the size of the underlying currency position (reflecting N_t and endogenous NFA position). - A surge in local currency demand appreciates the exchange rate and stimulates imports, partially offsetting arbitrageurs' position; this offset is smaller in relatively closed economies, so welfare losses from noise-trader shocks are larger when openness is low. - When labor has a higher share in production, appreciation reduces demand for domestic goods, lowers production costs, and boosts exports; the greater export response raises arbitrageurs' currency exposure, amplifying noise-trader effects and welfare loss. - Quantitative magnitudes in Panel (B): incremental welfare costs from adding a noise–trader shock are small (vertical axis range indicated up to 0.1 percent CEV in the figure), and under a peg the UIP wedge u_t is effectively closed so the noise–trader shock generates near-zero incremental welfare loss. ### Interpretation and main contribution - For commodity-exporting economies, endogenous financial pressures generated by real oil shocks are a quantitatively dominant rationale for active exchange rate intervention (FXI), complementing prior literature that highlighted exogenous financial shocks as motives for FXI. - Oil-price shocks act as first-order terms-of-trade and wealth shocks for oil exporters, inducing large trade-balance and exchange-rate volatility and sharp swings in net foreign asset positions. In the absence of FXI, risk-averse arbitrageurs must absorb time-varying currency exposure, producing a time-varying UIP wedge u_t and substantial welfare costs. ### Policy implications and recommendations - Theoretical result: monetary authorities can achieve first-best by using interest-rate policy to close the domestic output gap and FXI to neutralize the UIP wedge. - Practical caveats and guidance: - The fully integrated policy may not be feasible due to constraints on reserves, governance, or communication. - A free-floating regime without FXI suffers welfare losses due to volatile risk premia driven by oil price swings; a hard peg eliminates those premia but shifts volatility into inflation and output gap. - Combining a peg with an energy subsidy can, in some cases, outperform a pure float by smoothing domestic marginal costs in response to oil price shocks and reducing inflation and output volatility. - FXI should be used selectively and contingently, consistent with IMF 2023 IPF guidance: - Recommended only when specific financial frictions (e.g., shallow FX markets or elevated risk premia) materially impair market functioning. - Use of FXI should be contingent on clear evidence that such risks have intensified and be supported by adequate reserves, strong institutional and governance frameworks, and transparent communication to ensure policy credibility. - Regular or imprudent use of FXI can undermine market liquidity, constrain reserve capacity for future shocks, and weaken monetary policy signaling. - Scope and limits: - The analysis isolates transmission mechanisms through which oil price shocks, via net foreign asset dynamics and segmented FX markets, can generate a policy motive for FXI in a stylized general-equilibrium setting. - The framework abstracts from other important considerations (credibility and anchoring role of long-standing pegs, effects on capital costs and financial stability, transitional and reputational costs of regime changes) and is not a comprehensive prescriptive ranking of exchange rate regimes. *Source: 6.4 Quantitative Results and Welfare Costs (content unit from wpiea2026030-source-pdf).* ### 0. Additionally, using the relationshipe ### 0. Additionally, using the relationshipe ### Model with Expatriate Workers - Context and key assumption: - Expatriates accounted for 78% of the labor force in GCC countries in 2024Q2. - Expatriate labor is treated as an imported production input and expatriate workers do not consume domestically produced goods; all earnings are sent as remittances. - Firm’s problem (revised): - max_{C_Ht, L_Ht, L_Ft, O_t} C_Ht + ε_t P^*_Ht C^*_Ht − W_t L_Ht − ε_t W^*_t L_Ft − ε_t P^*_Ot O_t - Subject to: C_Ht + C^*_Ht = A_t [ L_Ht^σ L_Ft^{1−σ} ]^α O_t^{1−α}. (Equation (20)) - First-order conditions (FOCs) and implications: - FOCs with respect to L_HFt, L_FOt, O_t, and C_Ht yield: - W_t L^{1−α}_{FOt} = A_t σα L^{α(σ−1)}_{H Ft} - (1/σ − 1) W_t L_{H Ft} = ε_t W^*_t - (1/σα − 1) W_t L_{H Ft} = ε_t P^*_Ot L_{FOt} + ε_t W^*_t - Result: L_{FOt} = α(1−σ)/(1−α) P^*_Ot / W^*_t, which equals the first-best allocation of L_{FOt}; therefore l_{FOt} − ˜l_{FOt} = 0. - Using household optimality (C_Ft = γ W_t ε_t) and first-best relations, obtain ˜l_{H Ft} − l_{H Ft} = −(˜c_Ft − c_Ft). - Second-order approximation of the loss function: - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) ( (C_Ht − ˜C_Ht)/Ȼ_H )^2 + γ ( (C_Ft − ˜C_Ft)/Ȼ_F )^2 + (1−σα)σα(1−γ) (Ȳ_H / Ȼ_H) ( (L_{H Ft} − ˜L_{H Ft}) / Ḷ_{H F} )^2 ]. - Using wedges, simplified to equation (21): - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + ( γ + (1−σα)σα(1−γ) Ȳ_H Ȼ_H^{-1} ) u_t^2 ]. - Effect on natural real exchange rate and welfare under peg: - ˜q_t = a_t − (1−α) p^*_Ot + σα ∑_{i=0}^∞ E_t r^*_{t+i} − α(1−σ) w^*_t. - In this model ˜q_t is influenced by volatility of foreign wages; assuming foreign wages constant, as σ → 0 (reduced household labor share relative to expatriates) the effect of foreign rate volatility on welfare loss from pegging diminishes. - Influence of oil price volatility on welfare loss under a peg remains unchanged. ### The Extended Model (D) - Planner’s objective and variables: - L(x,λ,ε) = ∑_{t=0}^∞ β^t [ (1−γ) log C_Ht + γ log C_Ft − L_Ot/O_t + μ_t( P^*_Ht^{1−ε_X} C^*_t − P^*_Ft C_Ft + P^*_Ot (O_{wt} − O_t) − B^*_t R^*_t + B^*_{t−1} ) + ψ_t( A_t L_Ot^α O_t − ∫_0^1 p_{iHt}^{−ε_H} iHt di C_Ht − P^*_Ht^{−ε_X} C^*_t ) + Γ_t( 1 − ( ∫_0^1 p_{iHt}^{1−ε_H} iHt di )^{1/(1−ε_H)} ) ]. - x = {C_Ht, C_Ft, L_Ot, O_t, B^*_t, P^*_Ht, {p_{iHt}}_{i=0}^1 }_{t=0}^∞. - ε = { log(C^*_Ht / Ȼ^*_H), log(P^*_Ot / P̄^*_O), log(O_{wt} / Ȯ_{wt}), log(R^*_t / R̄^*) }_{t=0}^∞. - Key lemmas for price dispersion and approximations: - Lemma D.1: - ∫_0^1 (p_{iHt} − 1) di = ε_H/2 E_i ˆp_{iHt}^2 and ∫_0^1 (p_{iHt} − 1)^2 di = E_i ˆp_{iHt}^2, where ˆp_{iHt} = log p_{iHt}. - Lemma D.2: E_i ˆp_{iHt}^2 = var_i{ log p_{iHt} }. - Lemma D.3: ∑_{t=0}^∞ β^t var_i{ log p_{iHt} } = θ/(1−βθ)(1−θ) ∑_{t=0}^∞ β^t π_t^2. - Lemma D.4: (C_Ht − ˜C_Ht)/Ȼ_H × ∫_0^1 (p_{iHt} − 1) di = 0 up to second order. - Resulting second-order loss function (intermediate form): - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) ( v_t^2 + χ_1 π_t^2 ) + χ_2 u_t^2 + ε_X (1−γ) Ȼ^*_H Ȼ_H^{-1} (p^*_Ht − ˜p^*_Ht)^2 ]. - Where χ_1 = θ/(1−βθ)(1−θ) ε_H (ε_H + 1), χ_2 = γ + α(1−α)(1−γ) Ȳ Ȼ_H^{-1}. - Using p^*_Ht − ˜p^*_Ht = α u_t, the loss function reduces to equation (19). - Linear approximation of constraints and first-best: - First-best allocations satisfy a set of conditions (listed in the source) including: - ˜Q_t ≡ γ/(1−γ) ˜C_Ht / ˜C_Ft. - ˜C_Ht + C^*_Ht = A_t ˜L_Ot^α ˜O_t. - β R^*_t E_t [ P^*_Ft ˜C_Ft / (P^*_{Ft+1} ˜C_{Ft+1}) ] = 1. - ˜p_{iHt} = 1, ∀ i ∈ [0,1]. - Key equilibrium condition in log deviations: - ∆e_t = ∆˜q_t + ∆c_Ht − ∆˜c_Ht − (∆c_Ft − ∆˜c_Ft) + π_t − ∆p^*_Ft. - Price dispersion lemma: log Φ_t = 0 up to first order. - Home budget constraint in deviations: - β b^*_t − b^*_{t−1} = −1/Ȳ [ Ȼ^*_H (ε_X − 1)(p^*_Ht − ˜p^*_Ht) + Ȼ_F (c_Ft − ˜c_Ft) + P̄^*_O Ȯ (o_t − ˜o_t) ]. - Relationships used: l_ot − ˜l_ot = −(c_Ft − ˜c_Ft), p^*_Ht − ˜p^*_Ht = −α (l_ot − ˜l_ot). - With definitions v_t ≡ log(C_Ht/˜C_Ht) ≈ (C_Ht − ˜C_Ht)/Ȼ_H, u_t ≡ log(C_Ft/˜C_Ft) ≈ (C_Ft − ˜C_Ft)/Ȼ_F, the social planner’s problem simplifies to equation (19). ### A Two-Sector Model with Non-Tradables (E) - Household problem: - max_{C_Nt, C_Tt, L_t, B_t} E_0 ∑_{t=0}^∞ β^t [ (1−γ) log C_Nt + γ log C_Tt − ψ_l L_t^{1+σ_l} /(1+σ_l) ]. - Budget: P_Nt C_Nt + P_Tt C_Tt + B_t R_t = B_{t−1} + W_t L_t + Π_t + T_t + P_Ot O_{wt}. - Firm problem for j ∈ {N, T}: - max_{Y_jt, L_jt, O_jt} P_jt Y_jt − W_t L_jt − P_Ot O_jt subject to Y_jt = A_jt L_jt^{α_j} O_jt^{1−α_j}. - Assumptions and normalizations: - P_Nt = P^*_Tt = 1 for all t; P_Tt = ε_t P^*_Tt; P_Ot = ε_t P^*_Ot. - Define L_{jOt} ≡ L_jt / O_jt. - Equilibrium conditions (selected): - γ/(1−γ) C_Nt / C_Tt = ε_t. - C_Nt = A_Nt L_{NNOt}^{α_N} O_Nt; Y_Tt = A_Tt L_{TTOt}^{α_T} O_Tt. - B^*_t R^*_t − B^*_{t−1} = Y_Tt − C_Tt + P^*_Ot (O_{wt} − O_Nt − O_Tt). - Euler: E_t β [ C_Tt C_{Tt+1}^{−1} R^*_t ] = 1 + (B^*_t − N^*_t − F^*_t) R^*_t ω var_t(R_t ε_t ε_{t+1} ). - Factor price conditions: (1 − α_N) A_Nt L_{NNOt}^{α_N} = ε_t P^*_Ot; (1 − α_T) A_Tt L_{TTOt}^{α_T} = P^*_Ot. - Labor supply conditions linking marginal products to ψ_l and consumption shares. - Definitions of gaps and log deviations: - v_t = log C_Nt − log ˜C_Nt; u_t = log C_Tt − log ˜C_Tt; ∆l_{jOt} = log L_{jOt} − log ˜L_{jOt}; ∆o_{jt} = log O_{jt} − log ˜O_{Nt}. - Key identity: e_t − q_t = v_t − u_t. - Relations for sectoral deviations and constraints listed in source (see its system of equations). - Quadratic approximation of the loss function: - ˜L − L = (1/2) E_0 ∑_{t=0}^∞ β^t x'_t H_{xx} x_t, where x_t = (v_t, u_t, ∆l_{NOt}, ∆l_{TOt}, ∆o_{Nt}, ∆o_{Tt}). - H_{xx} matrix provided explicitly in the source (with entries involving 1−γ, γ, a = ψ_l σ_l Ḷ^{σ_l − 1}, Ḷ_N, Ḷ_T, Ȳ_T, Ȼ_T, α_N, α_T, etc.). - H_{xx} is positive semidefinite; loss function non-negative for any non-zero x_t; x_t = 0 satisfies constraints so minimum loss zero is achievable and solution unique. ### CES Production Function (F) - Motivation: - Oil often treated as an inelastic input; Cobb-Douglas may mischaracterize production. CES allows varying substitutability between labor and oil. - CES production: - Y_Ht = A_t ( α (a_L L_t)^σ + (1−α) (a_O O_t)^σ )^{1/σ}. - σ governs substitutability; σ = 0 reduces to Cobb-Douglas; σ → −∞ approaches Leontief with Y_Ht = A_t min{ a_L L_t, a_O O_t }. - Ramsey objective under CES: - Objective reduces to (1/2) E_0 ∑_{t=0}^∞ β^t [ (1−γ) v_t^2 + χ u_t^2 ]. - χ = γ + (1−γ)/(1−σ) Ȳ_H Ȼ_H^{-1} α a_L^σ Ḷ^σ Ȳ_H^{−σ} [ 1 − α a_L^σ Ḷ^σ Ȳ_H^{−σ} ]. - Constraint differencing: - Ȼ_H Ȳ_H^{-1} v_t = − α/(1−σ) u_t + o_t − ˜o_t. - Main implication: - First-best allocation achievable via optimal monetary policy and FX intervention even when oil and labor are not substitutable (σ → −∞). - Weight on the foreign goods consumption wedge decreases as inputs become less substitutable. ### Market Depth and the Welfare Loss (G) - Role of financial frictions parameter ω: - UIP wedge scales with market depth: - E_t(∆u_{t+1}) = ȯω var_t(∆e_{t+1})(n^*_t + f^*_t − b^*_t). - Higher ω amplifies volatility of currency premia for given flows/NFA configuration; oil price shocks generate first-order fluctuations in trade balance and NFA, so higher ω ⇒ larger, more volatile currency premium. - Welfare implications (qualitative and figure summary): - Welfare losses under No FXI regime increase sharply and monotonically with ω. - Peg and Peg + Subsidy regimes are largely insensitive to ω because fixing the exchange rate eliminates the UIP wedge; their welfare performance determined mainly by domestic distortions (v_t, π_t) that depend on the natural real exchange rate ˜q_t driven only by real fundamentals. - Robustness exercise message: the case for foreign exchange intervention strengthens as financial markets become shallower and oil price shocks are prominent; no FXI welfare cost originates primarily from unmanaged UIP risk premium, which is increasingly costly with imperfect currency markets. - Figure note: - Figure G1 shows Welfare Costs (in % CEV) across regimes (Peg, Peg + Subsidy, No FXI, No FXI + DCP, Optimal Policy) as ω varies; benchmark ω = 345. (Visual details and numeric plot points presented in the source figure.) *Source: wpiea2026030-source-pdf - 0. Additionally, using the relationshipe (Working Paper No. WP/2025/030)* --- _Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026030-source-pdf.pdf_