## wpiea2026188 ## Source details **Canonical URL:** [wpiea2026188](https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026188.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2026/english/wpiea2026188.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2026/english/wpiea2026188.pdf.json) --- ### 2.1 Environment — model primitives and markets - Time: two periods t ∈ {1,2}. Two shocks: public shock φ and private shock θ, each in [0,1]. Shocks realized and observable to all at t = 2; not contractible at t = 1. - Information structures at t = 1: - Uninformed I∅: θ and φ independent uniform on [0,1]; π(θ|I∅) = 1 for all θ ∈ [0,1]. - Informed Iθ: agents know θ; π(θ|Iθ) = 1 only at known θ (mass point). - Configurations I_L, I_I, I_S ∈ {I∅, Iθ} with I_L = I_I ⊆ I_S (speculators at least as informed as lenders/insurers). - Planner sets policy rate i1 at t = 1 and exchange rates {E1, E2} with E2 linear: E2 = αφ + βθ + γ. Ψ is set of possible E2 given information; ˜Ψ ⊆ Ψ is insurable set for which FX hedging contracts exist. - Households (measure one) maximize max {CHt, CFt, Nt} E[∑2 t=1 δt−1 U(CHt, CFt, Nt)] with - U(CHt, CFt, Nt) = σH log CHt + σF log CFt − Nt, σF = 1 − σH. - Budget constraint: Wt Nt + ΠHt + Et YFt + Tt + Dt ≥ PH CHt + Et CFt + (1 + it−1) Dt−1. Dollar prices normalized to 1; i0 = 0; D2 = 0. - Firms: monopolistic competition, price P_H set at t = 1 and rigid thereafter; production Y_Ht(j) = At N_t(j); aggregator with ε > 1; labor taxed at rate φ. - FX spot market (open at t = 1): - External lenders choose bond position B1 ≡ D1 E1 and may buy hedging contracts Q_L(E2) for E2 ∈ ˜Ψ at price p(E2) with constraint ∫˜Ψ p(E2) Q_L(E2) dE2 = 0. - Lenders maximize max {B1, Q_L(E2)} E[M2([η2 − (1 + i∗)] B1 + Q_L(E2)) | I_L] where η2 ≡ (1 − μ1) (1 + i1) E1 / E2. - Lenders’ FOC: E[M2 [η2 − (1 + i∗)] | I_L] = 0; result implies UIP premium ρ ≡ E[η2] − (1 + i∗). - FX hedging markets (continuum across E2 ∈ Ψ): insurers set prices p(E2) and quantities Q_I(E2) if they participate (d = 1); speculators and lenders demand Q_S(E2), ψ Q_L(E2) respectively; clearing Q_I(E2) = Q_S(E2) + ψ Q_L(E2). Density at each E2 is 1/ψ with ψ = (α + β) if I_I = I∅ and ψ = α if I_I = Iθ. - Lump-sum rebates: Tt = φ Wt Nt + μt−1 (1 + it−1) Dt−1. - Competitive equilibrium (Definition 1): set of quantities, prices, exchange-rate sets {Ψ, ˜Ψ}, and distributions {π(E2|I_I), π(E2|I_S)} satisfying agents’ optimization and market clearing, given foreigners’ SDF M2, information structures {I_L, I_I, I_S}, and policy instruments { i1, {Et}2 t=1, φ, μ1 } with E2 = αφ + βθ + γ. ### 2.2 Insurability and premia — characterization and geometry - Proposition 1 (Insurability condition): - E2 ∈ ˜Ψ iff π(E2|I_I) = π(E2|I_S) irrespective of θ. - Within ˜Ψ, (1 + i∗) p(E2) = π(E2|I_I). - Interpretation: insurers will offer contracts only when insurers and speculators agree on probabilities; otherwise insurers avoid large losses from informed speculators. - Speculators’ demand extremes imply insurers choose d = 1 only if they can set p(E2) such that π(E2|I_S) = (1 + i∗) p(E2) with probability one. - Proposition 2 (Insurability and premia) — assume α > β ≥ 0: - Lenders’ bond position satisfies an integral condition (exact integral expression preserved in source) involving m2 ≡ M2 / M and π(θ|I_I). - Cases: - (a) Full insurability: I_I = I_S ∈ {I∅, Iθ} and/or β = 0 → ˜Ψ = Ψ, ρ = 0. - (i) I_I = I_S = I∅: ˜Ψ = [γ, α + β + γ], m2 = 1, π(θ|I_I) = 1. - (ii) I_I = I_S = Iθ: ˜Ψ = [β θ + γ, α + β θ + γ], m2 = 1 (integrals over θ collapse). - (iii) I_I = I∅, I_S = Iθ, and β = 0: ˜Ψ = [γ, α + γ], π(θ|I_I) = 1 and lender integrals simplify. - (b) Partial insurability: I_I = I∅, I_S = Iθ, and β > 0 → Ψ = [γ, α + β + γ], ˜Ψ = [β + γ, α + γ], probability of E2 ∈ ˜Ψ is 1 − β α, upper/lower insurability contours φ = 1 − β α θ and φ = β α (1 − θ), m2 = m2(B1, η2, {η2} E2 ∈ ˜Ψ, β α), premium ρ = − Cov(m2, η2) / E[m2]. - Key implications: - Premia at t = 1 require asymmetric information in FX hedging market, β > 0 (exchange rate depends on private shock), and lenders’ dislike of future exchange-rate volatility. - With symmetric information or β = 0, full insurability and ρ = 0. - With asymmetric information and β > 0, central parallelogram region of (φ,θ) is insurable, tails uninsurable; premia stem from SDF deviations m2 ≠ 1 in uninsurable tails. - Geometry (E2 = αφ + βθ + γ, α > β ≥ 0): - Contours slope − β α. Panel A (β = 0): horizontal contours → full insurability. Panel B (β ∈ (0, α)): downward-sloping contours → insurable parallelogram ˜Ψ = [β + γ, α + γ] with uninsurable tails. ### 3.1 Planner problem — objective, wedges, constraints - Planner and insurers share information I_I. Constrained efficient allocation maximizes household welfare with implementability via policy instruments {i1, {Et}, φ, μ1}, E2 = αφ + βθ + γ. - AD wedge τ_Ht: τ_Ht ≡ 1 + (1/A_t) (U_Nt / U_Ht) = 1 − (1/A_t) C_Ht^{σ_H} for t ∈ {1,2}. - UIP wedge τ_Γ2: τ_Γ2 ≡ E[[η_2 − (1 + i^∗)]^{σ_F} C_F2]. - Planner’s period utility V(C_Ft, E_t P_H) ≡ U((σ_H/σ_F) E_t P_H C_Ft, C_Ft, (1/A_t)(σ_H/σ_F) E_t P_H C_Ft) with partial derivatives: - ∂V/∂C_Ft = σ_F C_Ft [1 + (σ_H/σ_F) τ_Ht] - ∂V/∂(E_t P_H) = σ_H P_H/E_t τ_Ht - Define χ ≡ (1 − μ_1) (1 + i_1) E_1 (= η_2 E_2), independent of shocks (φ,θ). - Planner problem (exact maximization and constraints preserved): - Maximize V_P = max_{C_F1, C_F2, E_1, α, β, γ, χ, P_H} V(C_F1, E_1 P_H) + δ ∫_{θ=0}^1 ∫_{φ=0}^1 V(C_F2, E_2 P_H) π(θ|I_I) dφ dθ - Subject to resource constraints B_0 (1 + i^∗) = (Y_F1 − C_F1) + (Y_F2 − C_F2)/E_2 χ for all (φ, θ), the premium constraint (aggregate across states), and E_2 = αφ + βθ + γ. - Three externalities identified: - AD externality: private agents ignore effect of consumption on AD wedges τ_Ht. - LC premium externality: private borrowing affects economywide premia and UIP wedge τ_Γ2 via B1 and m2 in uninsurable regions. - Information sensitivity (IS) externality: planner’s choices {α, β} affect insurability boundaries and premia; agents do not internalize this. ### 3.2 First-order conditions — four insights (assume α > β ≥ 0, P_H = 1) - Insight 1 — AD wedge at t = 1: - τ_H1 = 0. Planner uses exchange rate flexibility to eliminate AD externality at t = 1. - Insight 2 — χ internalizes AD and IS externalities: - Key FOC linking χ and C_F2 (exact integral condition preserved in source): B_1 ∫_{θ} ∫_{φ} (1/E_2) δ σ_F C_F2 [1 + (σ_H/σ_F) τ_H2] π(θ|I_I) dφ dθ = Λ [∫_{θ} ∫_{φ} (1/E_2) π(θ|I_I) dφ dθ + z_χ]. - Definitions preserved for B_1 and z_χ (integrals over uninsurable regions involving m2 − 1 and ∂m2/∂χ). - Λ ≠ 0 only if B_1 ≠ 0; z_χ captures IS externality through uninsurable regions. - Insight 3 — C_Ft internalizes AD and LC premium externalities: - FOC: σ_F C_F1 = ∫_{θ} ∫_{φ} (χ/E_2) δ σ_F C_F2 [1 + (σ_H/σ_F) τ_H2] π(θ|I_I) dφ dθ + Λ z_F1. - z_F1 defined as integrals over uninsurable regions involving ∂m2/∂B_1 [χ/E_2 − (1 + i^∗)]; captures LC premium externality. - Insight 4 — parameters {α, β, γ} internalize AD and IS externalities: - FOC for ω ∈ {α, β, γ} (exact integral equation preserved) equates effects of marginal E_2 changes on AD and resource returns to premium constraint terms including z_{ω1} and z_{ω2}. - z_{ω1} captures m2 − 1 and ∂m2/∂ω effects in uninsurable regions; z_{ω2} arises from changes in boundaries of ˜Ψ (non-zero only for ω ∈ {α, β}). - Additional notes: - First-order approach presumes interior solution; Euler and price-setting conditions remain operative and pin down implicit i_1, μ_1, and φ once planner variables chosen. ### 3.3 Policy impact of uninsurability — modifications, commitment, and lender models - Lemma 1 (Effective information symmetry): - Full insurability ˜Ψ = Ψ implies z_χ = z_F1 = z_{ω1} = z_{ω2} = 0 for all ω ∈ {α,β,γ} in case (a), except z_β2 may be non-zero in subcase (a)(iii). Thus LC and IS externalities vanish under full insurability (with stated exception). - Lemma 2 (Information asymmetry episode): changing from II = IS ∈ {I∅, Iθ} to II = I∅ and IS = Iθ may render prior premium ρ insufficient to satisfy premium constraint after the change iff condition (9) (exact integral inequality in source) holds. Three scenarios detailed depending on initial subcase and planner β choice. - Proposition 3 (Modified monetary policy): - If planner initially chooses β/α ∈ (0,1) and B1 > 0 under II = IS = I∅, then under II = I∅ and IS = Iθ there exist m(b)2 functions such that planner selects a different β/α. - If m(b)2 = 1, planner chooses same β/α; perturbations in m(b)2 alter FOCs via premium constraint and externality terms {zχ, zF1, zω1, zω2}, leading to adjustment in β/α generally. - Economic interpretation: planner changes relative sensitivity of E2 to θ vs φ (β/α) when insurability concerns and premia arise. - Corollary 1 (Aloof central banker): - There exist m(b)2 functions causing planner to prefer a lower β/α under asymmetric information; delegating policy to an aloof central banker who commits to lower β/α expands ˜Ψ and can reduce premia. Implementation relies on commitment to exchange-rate function at t = 2. - Proposition 4 (Log utility lenders — modeling option): - Foreign lenders with log utility and budget constraint imply m2 = 1 for E2 ∈ ˜Ψ and m2 given by the explicit expression in source for E2 ∉ ˜Ψ; premium constraint implies dχ/dB1 > 0. - Implications: m2 deviates from 1 in uninsurable regions only if κ > 0 (country large relative to lenders), premium increases with external debt B1, and switching from full to partial insurability raises premium, more so when B1 larger. - Simulations (parameter sets precisely listed) show insurability problems increase ρ, lower CF1, lower β/α, and lower welfare V_P; α and β fall but β falls relatively more. - Corollary 2 (Inframarginal lenders): - Presence of inframarginal lenders holding S1 modifies results by replacing B1 with B1 − S1 in Proposition 4. - Proposition 5 (One-sided constraint lenders — alternative modeling): - With lenders subject to a one-sided financial constraint, m2 = 1 on lower/central ranges and m2 > 1 in upper uninsurable tail; premium driven mainly by upper-tail large depreciations. - Planner’s policy may focus on mitigating largest depreciations; α and β chosen to address upper uninsurable region. - Overall policy implication: - Insurability concerns induce the planner to alter monetary policy design (especially β/α). Commitment and delegation can be welfare-relevant tools to expand insurability and reduce premia. ### 4.2 Fiscal policy — channels, endogenous policy, and constraints - Two fiscal-policy modeling options affecting FX insurability: - Option 1: exogenous government spending on imports G_Ft (dollar terms), subject to shocks. - Option 2: endogenous G_Ft where utility includes σ_Gt log G_Ft with σ_Gt ex-ante at t = 1 and shock-dependent at t = 2. - Government rebate updated: T_t = φ W_t N_t + μ_{t−1} (1 + i_{t−1}) D_{t−1} − E_t G_Ft. - No other shocks to tradable endowments or productivity in this section: Y_Ft = Y_F, A_t = A. - Proposition 6 (Exogenous fiscal policy): - Planner problem is isomorphic to subsection 3.1 with Y_Ft replaced by Y_F − G_Ft. - Fiscal policy raises premia through: 1. Pre-shock channel: higher G_F1 reduces Y_F1 or increases B_0 analogous to higher inherited external debt, raising B1 and premia exposure. 2. Shock channel: if private shock θ increases G_F2 (reducing private imports C_F2), planner may set β > 0 to make E2 sensitive to θ (to depreciate away debt when costly), which with asymmetric information generates uninsurability and a premium. - Implication: premia can be structural in countries whose fiscal policy depends on shocks known only to some FX hedging participants, and premia may surge episodically with regime shifts. - Proposition 7 (Endogenous fiscal policy): - Utility includes σ_Gt log G_Ft. Optimality conditions: - G_F1 = (σ_G1 / σ_F) C_F1 - G_F2 = (σ_G2 / σ_F) C_F2 [ 1 + (σ_H / σ_F) τ_H2 ] (equation (13); exact relations preserved) - If G_F2 varies with θ (case (b)), parameters {α,β} and insurability interact: G_F2 sensitivity affects and is affected by exchange-rate design, AD and IS externalities. - Planner can eliminate premium by constraining fiscal policy to be θ-insensitive. - Corollary 3 (Constrained fiscal policy): - Constraining G_F2 to be insensitive to θ (function only of φ) yields β = 0 in constrained-efficient allocation for subcase (a)(i), and premium ρ = 0 continues to satisfy premium constraint after information-sensitivity episode. - Modified condition for G_F2 given by equation (14) (exact expression preserved). - Trade-off: constraining fiscal responsiveness to private shocks can remove premia but may reduce welfare if σ_G2 ideally should respond to θ. - Simulation example: - Use σ_G2 = σ_G1 ( 1 + ξ (φ − 1/2) + ν (θ − 1/2) ) with parameter values listed. Moving from full to partial insurability raises ρ, lowers CF1, lowers β/α, and reduces V_P. Figure 4 (in source) illustrates that reducing sensitivity of G_F2 to θ lowers required β/α and mitigates insurability problem. - Policy takeaways: - Exogenous increases in G_F1 raise B1 and premia; monetary policy can partially offset for B1 > 0. - Endogenous fiscal policy internalizes insurability externalities; fiscal discipline pre-shock reduces vulnerability. - Fiscal rules constraining sensitivity of G_F2 to private shocks can achieve full insurability (β = 0, ρ = 0) at potential welfare cost if such sensitivity is optimal absent premia. Institutional design should weigh this trade-off. _italic line: Source — wpiea2026188 - 2.1 Environment; 2.2 Insurability and Premia; 3.1–3.3 Planner Problem & Policy Impact; 4.2 Fiscal Policy (PDF chapter)._ ### 2.1 Environment ### 2.1 Environment ### Shocks and information structures - Two periods t ∈ {1,2} and two shocks: a public shock φ and a private shock θ, each taking values within [0,1]. - Shocks are realized and observable to all agents at t = 2 and together characterize the economy’s state of nature (φ,θ) in that period. - Shocks are not contractible, so no hedging contracts exist to directly insure them at t = 1. - At t = 1, agents may have one of two information structures (sigma-algebras): the uninformed structure I∅ or the informed structure Iθ. - From the perspective of the uninformed agents, no shocks are known, and the two shocks follow independent uniform distributions over [0,1]. - From the perspective of the informed agents, the public shock follows a uniform distribution over [0,1] but these agents know the value of the private shock θ. - π(θ|I) is the probability density function of θ given information structure I: - π(θ|I∅) = 1 for all θ ∈ [0,1]. - π(θ|Iθ) = 1 only for the known value of θ and = 0 for other values (strictly speaking, there is a mass point at θ for the latter information structure). - Information configurations for lenders, insurers, and speculators at t = 1 are I_L, I_I, and I_S, each equal to one of {I∅, Iθ}, with I_L = I_I ⊆ I_S (speculators at least as informed as lenders and insurers). - No information revelation within t = 1; agents’ information structures do not get updated in this period. ### Monetary policy - The planner sets the policy rate i1 between the two periods and the exchange rates {E1, E2} in each period (local currency per dollar): - a single number at t = 1 and a function of (φ,θ) at t = 2. - Restriction: the planner uses a linear E2 = αφ + βθ + γ. - Ψ is the set of possible exchange rates E2 at t = 2 from the perspective of lenders and insurers at t = 1. - π(E2|I) is the probability density function of E2 given information structure I at t = 1. - π(E2|I∅) = ∫1 θ=0 π(E2|Iθ) dθ, where Iθ is evaluated at each θ. - Expectation operator for any variable x can be written equivalently as: - E[x|I] = ∫1 θ=0 ∫1 φ=0 x π(θ|I) dφ dθ = ∫E2 ∈ Ψ x π(E2|I) dE2. - The exchange rate is contractible, so FX hedging contracts may exist. - ˜Ψ denotes the insurable set, i.e., the set of exchange rates for which such contracts exist, where ˜Ψ ⊆ Ψ. ### Consumption and production - Household problem (continuum measure one) maximizes: - max {CHt, CFt, Nt} E[∑2 t=1 δt−1 U(CHt, CFt, Nt)] - U(CHt, CFt, Nt) = σH log CHt + σF log CFt − Nt and σF = 1 − σH. - Budget constraint: - Wt Nt + ΠHt + Et YFt + Tt + Dt ≥ PH CHt + Et CFt + (1 + it−1) Dt−1. - Income components: - Wt wage, Nt labor supply, ΠHt profits of home goods firms, YFt endowment of tradable goods, Tt lump-sum transfer, Dt local currency debt at end of period t. - Expenditure components: - PH (local price, fully rigid over time) and CHt consumption of home goods, CFt consumption of imports. - Dollar prices of tradable endowments and imports normalized to 1 and i0 to 0. - Debt can be non-zero initially but is extinguished at end of time, i.e., D2 = 0. - Households’ tradable endowments YFt are functions of (φ,θ) at t = 2; households have no access to insurance contracts. - First order conditions (FOCs): - C_Ht = σH σF E_t P_H C_Ft, - W_t = 1 σF E_t C_Ft, - 1 E_1 C_F1 = δ(1 + i1) E[1 E_2 C_F2]. - Home goods firms (continuum measure one) are monopolistically competitive, set prices at beginning of period t = 1, prices fully rigid thereafter. - Variety j ∈ [0,1] production: Y_Ht(j) = At N_t(j), where At is exogenous productivity and a function of (φ,θ) at t = 2. - Aggregator and demand: - Y_Ht = (∫1 0 Y_Ht(j) ε−1 ε dj) ε ε−1, - Y_Ht(j) = Y_Ht (P_H(j)/P_H) −ε, - P_H = (∫1 0 P_H(j) 1−ε dj) 1 1−ε, where ε > 1. - Labor taxed by planner at preset rate φ. - Variety price P_H(j) set to maximize expected profits: - max P_H(j) E[∑2 t=1 s_t [P_H(j) Y_Ht(j) − (1 + φ) W_t N_t(j)]], - where s_t ≡ δt−1 E1 C_F1 E_t C_Ft is households’ period-1 stochastic discount factor (SDF) for local currency income in period t. - FOC for price-setting: - P_H(j) = (1 + φ) ε ε−1 E[∑2 t=1 δt−1 1 E_t C_Ft W_t A_t Y_Ht] / E[∑2 t=1 δt−1 1 E_t C_Ft Y_Ht]. - All firms identical: P_H = P_H(j), Y_Ht = Y_Ht(j), N_t = N_t(j). - Labor market clears at the flexible wage W_t; home goods market clearing: Y_Ht = C_Ht. ### FX spot market - Open at t = 1. - Participants: households issuing local currency debt and a continuum measure one of external lenders absorbing that debt. - External lenders can: 1. Exchange currencies via position D1 in local currency bonds financed by −D1 E1 in dollar bonds. 2. Purchase hedging contracts over exchange rates for each E2 in insurable set ˜Ψ. - For each E2 ∈ ˜Ψ, lenders can purchase Q_L(E2) contracts at price p(E2) each yielding one dollar if that E2 materializes at t = 2. - Lenders maximize expected dollar profits for foreign owners: - max {B1, Q_L(E2)} E[M2([η2 − (1 + i∗)] B1 + Q_L(E2)) | I_L] - where η2 ≡ (1 − μ1) (1 + i1) E1 / E2 and B1 ≡ D1 E1. - Constraint: zero net purchases of available insurance contracts: - ∫E2 ∈ ˜Ψ p(E2) Q_L(E2) dE2 = 0. - M2 is foreigners’ period-1 SDF over dollar income at t = 2 (taken as exogenous function); μ1 tax on capital inflows set by planner at t = 1; i∗ foreign interest rate. - Lenders’ FOCs: - E[M2 [η2 − (1 + i∗)] | I_L] = 0, - Q_L(E2): - → ∞ if M2 > M (1 + i∗) p(E2) / π(E2|I_L), - ∈ R if M2 = M (1 + i∗) p(E2) / π(E2|I_L), - → −∞ if M2 < M (1 + i∗) p(E2) / π(E2|I_L), - where M is a positive constant. - Assumed SDF properties: M2 = M2(B1, η2, Q_L(E2)) continuously differentiable with: - dM2 / dQ_L(E2) < 0, - lim Q_L(E2) → ∞ M2 = 0, - lim Q_L(E2) → (Q_min(E2)) + M2 = ∞ for some Q_min(E2). - Combining conditions yields finite Q_L(E2) for each E2: - M2 = M (1 + i∗) p(E2) / π(E2|I_L) for all E2 ∈ ˜Ψ. - Solution to FX spot market yields UIP premium ρ ≡ E[η2] − (1 + i∗). ### FX hedging markets - Open to foreign agents at t = 1: continuum measure one of risk-neutral insurers and speculators plus external lenders. - Continuum of markets, one for each E2 ∈ Ψ; variables indexed by E2. - Density of insurers and speculators at each E2 is 1/ψ, where ψ is the width of support of E2 from insurers’ perspective: - ψ = (α + β) if I_I = I∅, - ψ = α if I_I = Iθ. - Market sequence within t = 1: - Step 1 (Insurers): decide participation d ∈ {0,1}. If d = 1, set price p(E2) and commit to sell quantity Q_I(E2) needed to fulfill all demands. Insurers maximize: - max p(E2) ∫1 θ=0 [(1 + i∗) p(E2) Q_I(E2) − π(E2|I_S) Q_I(E2)] π(θ|I_I) dθ. - If expected profits negative, insurers choose d = 0, offer no price, fulfill no contracts, earn zero. - Step 2 (Speculators and lenders): if d = 1, speculators and lenders observe p(E2) and set demands without observing each other. - Speculators maximize: - max Q_S(E2) [π(E2|I_S) Q_S(E2) − (1 + i∗) p(E2) Q_S(E2)]. - Lenders purchase quantity ψ Q_L(E2) of contracts derived from FX spot market analysis (ψ adjusts measure). - If d = 0, no contracts can be demanded. - Step 3 (Clearing): market clears, Q_I(E2) = Q_S(E2) + ψ Q_L(E2). - If d = 1, E2 ∈ ˜Ψ (insurable). If d = 0, E2 ∉ ˜Ψ and p(E2) = ∅, Q_I(E2) = Q_S(E2) = Q_L(E2) = 0. - Outcome depends on information structures of insurer and speculators; determines ˜Ψ. ### Lump sum rebates - Planner rebates to households in lump sum the revenues from taxes on labor and capital inflows: - Tt = φ Wt Nt + μt−1 (1 + it−1) Dt−1. ### Competitive equilibrium (Definition 1) - A competitive equilibrium comprises quantities: - { {CHt, CFt, Nt, YHt}2 t=1, D1, {Q_L(E2), Q_I(E2), Q_S(E2)} E2 ∈ Ψ }, - prices: - { PH, {W_t}2 t=1, {p(E2)} E2 ∈ Ψ }, - exchange rate sets { Ψ, ˜Ψ }, and probability distributions { π(E2|I_I), π(E2|I_S) } E2 ∈ Ψ - that satisfy optimization conditions and constraints of households, home goods firms, external lenders, insurers, and speculators, and market clearing for home goods, labor, local currency bonds, and FX hedging contracts, - given foreigners’ SDF M2, information structures {I_L, I_I, I_S}, and policy instruments { i1, {Et}2 t=1, φ, μ1 }, where E2 = αφ + βθ + γ for some {α, β, γ}. --- ### 2.2 Insurability and Premia ### Proposition 1 (Insurability condition) - E2 ∈ ˜Ψ if and only if π(E2|I_I) = π(E2|I_S) irrespective of θ. - Within that set, (1 + i∗) p(E2) = π(E2|I_I). - Interpretation: - For an exchange rate to be insurable, insurers and speculators need to agree on its probability—whether they have the same information structure or not. - Given price stickiness and market structure, insurers make large losses if speculators bet against them; insurers avoid such betting only if speculators agree with insurers on the probabilities. ### Sketch of proof and equilibrium implications - Speculators’ demands in step 2: - Q_S(E2) → ∞ if π(E2|I_S) > (1 + i∗) p(E2), - Q_S(E2) ∈ R if π(E2|I_S) = (1 + i∗) p(E2), - Q_S(E2) → −∞ if π(E2|I_S) < (1 + i∗) p(E2). - Insurers choose d = 1 only if they can set p(E2) such that π(E2|I_S) = (1 + i∗) p(E2) holds with probability one; otherwise they set d = 0. - Two insurer cases: 1. I_I = I_S ∈ {I∅, Iθ}: insurers and speculators either both uninformed or both informed. - Insurers can set p(E2) so (1 + i∗) p(E2) = π(E2|I_I) = π(E2|I_S). 2. I_I = I∅ and I_S = Iθ (insurers uninformed, speculators informed): - Insurers cannot generally set p(E2) to match π(E2|I_S) for all θ unless π(E2|I_S) does not vary with θ; otherwise insurers set d = 0. ### Proposition 2 (Insurability and premia) — assume α > β ≥ 0 - External lenders’ bond position satisfies: - 0 = ∫1 θ=0 ∫1− β α θ φ = β α (1 − θ) [η2 − (1 + i∗)] π(θ|I_I) dφ dθ + ∫1 θ=0 ∫β α (1 − θ) φ = 0 m2 [η2 − (1 + i∗)] π(θ|I_I) dφ dθ + ∫1 θ=0 ∫1 φ = 1− β α θ m2 [η2 − (1 + i∗)] π(θ|I_I) dφ dθ, - where m2 ≡ M2 / M. - Cases: (a) Full insurability: I_I = I_S ∈ {I∅, Iθ} and/or β = 0. - ˜Ψ = Ψ, so probability of E2 ∈ ˜Ψ is 1 and premium ρ = 0. - (i) If I_I = I_S = I∅: - ˜Ψ = [γ, α + β + γ], m2 = 1 and π(θ|I_I) = 1. - (ii) If I_I = I_S = Iθ: - ˜Ψ = [β θ + γ, α + β θ + γ], m2 = 1 and integrals over θ removed because π(θ|I_I) = 1 only at known θ. - (iii) If I_I = I∅, I_S = Iθ, and β = 0: - ˜Ψ = [γ, α + γ], π(θ|I_I) = 1 and second and third terms of the lender expression are zero. (b) Partial insurability: I_I = I∅, I_S = Iθ, and β > 0. - Exchange rate set Ψ = [γ, α + β + γ] and insurable set ˜Ψ = [β + γ, α + γ], while π(θ|I_I) = 1. - Probability of E2 ∈ ˜Ψ is 1 − β α. - Upper and lower insurability contours: φ = 1 − β α θ and φ = β α (1 − θ) respectively. - m2 = m2(B1, η2, {η2} E2 ∈ ˜Ψ, β α). - Premium: ρ = − Cov(m2, η2) / E[m2]. - Main implications: - For premia to arise at t = 1, necessary ingredients are: - asymmetric information in FX hedging market, - exchange rate must depend on the private shock at t = 2 (β > 0), - external lenders in FX spot market must dislike future exchange rate volatility. - With symmetric information or β = 0, full insurability obtains and ρ = 0. - With asymmetric information and β > 0 (panel B), there is partial insurability: a shaded central region of (φ,θ) space is insurable (˜Ψ), while tails are uninsurable. - In the insurable region m2 = 1 by the lenders’ FOC; any nonzero covariance generating premium ρ arises from uninsurable tails where m2 need not equal 1. - If external lenders generally have higher m2 during combinations of shocks with more depreciated exchange rates and lower η2, covariance is negative and premium positive. ### Geometry and intuition (linear E2 = αφ + βθ + γ, α > β ≥ 0) - Exchange rate contours in (φ,θ) space have slope − β α. - Panel A (β = 0): contours horizontal; exchange rate increases upward; shaded region covers whole panel → full insurability. - Panel B (β ∈ (0, α)): contours downward-sloping; shaded parallelogram region is insurable set ˜Ψ = [β + γ, α + γ]; tails (two triangular regions) are uninsurable. - Insurability depends on whether π(E2|I_I) = π(E2|I_S) irrespective of θ for each E2. --- *Italic line: Source — wpiea2026188 - 2.1 Environment (PDF chapter).* ### 3.1 Planner Problem ### 3.1 Planner Problem ### Definitions, information, and objective - Planner and insurers share the same information structure I_I. - Constrained efficient allocation comprises quantities: - {C_Ht, C_Ft, N_t, Y_Ht}_{t=1}^2, D_1, {Q_L(E_2), Q_I(E_2), Q_S(E_2)}_{E_2 ∈ Ψ} - prices {P_H, {W_t}_{t=1}^2, {p(E_2)}_{E_2 ∈ Ψ}} - exchange rate sets {Ψ, ̃Ψ} - probability distributions {π(E_2|I_I), π(E_2|I_S)}_{E_2 ∈ Ψ} - The allocation maximizes household welfare under full commitment given foreigners’ SDF M_2 and information structures {I_L, I_I, I_S}, subject to implementability via policy instruments {i_1, {E_t}_{t=1}^2, φ, μ_1}, where E_2 = αφ + βθ + γ for some {α, β, γ}. ### Wedges and planner period utility - AD (aggregate demand) wedges τ_Ht: - τ_Ht ≡ 1 + (1/A_t) (U_Nt / U_Ht) = 1 − (1/A_t) C_Ht^{σ_H} for t ∈ {1, 2} - Interpretation: compare consumption demand for home goods against labor disutility from producing those goods. - UIP wedge τ_Γ2: - τ_Γ2 ≡ E[[η_2 − (1 + i^∗)]^{σ_F} C_F2] - Interpretation: expected marginal utility of the consumption loss owing to premia paid by domestic agents to external lenders. - Planner’s period-t utility (incorporating households’ intratemporal FOCs): - V(C_Ft, E_t P_H) ≡ U((σ_H/σ_F) E_t P_H C_Ft, C_Ft, (1/A_t)(σ_H/σ_F) E_t P_H C_Ft) - Partial derivatives: - ∂V/∂C_Ft = σ_F C_Ft [1 + (σ_H/σ_F) τ_Ht] - ∂V/∂(E_t P_H) = σ_H P_H/E_t τ_Ht ### Auxiliary variable and planner problem formulation - Define χ ≡ (1 − μ_1) (1 + i_1) E_1 (= η_2 E_2), which aggregates t=1 policy settings and does not depend on shocks (φ, θ). - Planner problem: - Maximize V_P = max_{C_F1, C_F2, E_1, α, β, γ, χ, P_H} V(C_F1, E_1 P_H) + δ ∫_{θ=0}^1 ∫_{φ=0}^1 V(C_F2, E_2 P_H) π(θ|I_I) dφ dθ - Subject to: - B_0 (1 + i^∗) = (Y_F1 − C_F1) + (Y_F2 − C_F2)/E_2 χ for all (φ, θ) (resource constraints) - Premium constraint (aggregate across states) given by equation (3) (see source) ensuring external lenders receive the necessary premium to hold external debt at t=1 - E_2 = αφ + βθ + γ ### Role of choice variables and shocks - C_F2 is chosen as a function of (φ, θ); parameters {α, β, γ} determine sensitivity of E_2 to (φ, θ). - Productivity A_2 may vary with (φ, θ) at t = 2 and enters utility function. - External debt B_1 is in local currency; variation in E_2 across states alters dollar value of repayments and hence affects C_F2. ### Three externalities identified - AD externality: households do not internalize effect of individual consumption on AD wedges {τ_Ht}. - LC premium externality: households’ borrowing decisions may affect economywide premia and UIP wedge τ_Γ2 by affecting B_1 and thereby m_2 in uninsurable regions. - Information sensitivity (IS) externality: agents do not internalize that actions affecting monetary policy and exchange rates can generate premia; planner’s choices of {α, β} can affect insurability boundaries and the resulting premium. ### Background constraints remaining operational - Households’ Euler condition and firms’ price-setting equation do not appear explicitly in the planner problem but remain operative: once planner selects the variables, Euler condition backs out implicit policy rate i_1 and capital inflow tax μ_1, and price-setting determines implicit labor tax φ. --- ### 3.2 First Order Conditions — Four insights Assumption: solution satisfies α > β ≥ 0 (first-order approach); P_H normalized to 1. Insight 1 — AD wedge at t = 1: - τ_H1 = 0. - Comes directly from FOC for E_1: planner uses exchange rate flexibility to set C_H1 to eliminate the AD externality in period 1. Insight 2 — Setting of χ internalizes AD and IS externalities: - Key FOC condition combining χ and C_F2: - B_1 ∫_{θ=0}^1 ∫_{φ=0}^1 (1/E_2) δ σ_F C_F2 [1 + (σ_H/σ_F) τ_H2] π(θ|I_I) dφ dθ = Λ [∫_{θ=0}^1 ∫_{φ=0}^1 (1/E_2) π(θ|I_I) dφ dθ + z_χ] - Definitions: - B_1 = B_0 (1 + i^∗) + (C_F1 − Y_F1) = (Y_F2 − C_F2) / E_2 χ - z_χ = - ∫_{θ=0}^1 ∫_{φ=0}^{β/α (1−θ)} { [m_2 − 1] (1/E_2) + (∂m_2/∂χ) [χ/E_2 − (1 + i^∗)] } π(θ|I_I) dφ dθ - + ∫_{θ=0}^1 ∫_{φ=1−β/α θ}^1 { [m_2 − 1] (1/E_2) + (∂m_2/∂χ) [χ/E_2 − (1 + i^∗)] } π(θ|I_I) dφ dθ - Λ is multiplier on premium constraint (3). - Interpretations: - Λ ≠ 0 only if B_1 ≠ 0. - If B_1 = 0, χ set so premium constraint is non-binding (Λ = 0). If B_1 ≠ 0, changes in χ affect C_F2 and welfare. - With no externalities (τ_H2 = z_χ = 0), χ balances effect that increasing χ reduces C_F2 (if B_1 > 0) against providing sufficient returns for external lenders. - AD externalities at t = 2 enter via τ_H2 on LHS; IS externality enters via z_χ on RHS, reflecting uninsurable regions where m_2 may differ from 1 and where ∂m_2/∂χ matters. Insight 3 — Import consumption C_Ft internalizes AD and LC premium externalities: - FOC combining C_F1 and C_F2: - σ_F C_F1 = ∫_{θ=0}^1 ∫_{φ=0}^1 (χ/E_2) δ σ_F C_F2 [1 + (σ_H/σ_F) τ_H2] π(θ|I_I) dφ dθ + Λ z_F1 - z_F1 = - − [ ∫_{θ=0}^1 ∫_{φ=0}^{β/α (1−θ)} (∂m_2/∂B_1) [χ/E_2 − (1 + i^∗)] π(θ|I_I) dφ dθ - + ∫_{θ=0}^1 ∫_{φ=1−β/α θ}^1 (∂m_2/∂B_1) [χ/E_2 − (1 + i^∗)] π(θ|I_I) dφ dθ ] - Interpretations: - Without externalities (τ_H2 = z_F1 = 0), intertemporal profile of imports C_Ft determined solely by expected dollar value of repayments χ/E_2. - LC premium externality captured by z_F1: an increase in C_F1 raises B_1 and changes m_2 in uninsurable regions. Insight 4 — Parameters {α, β, γ} internalize AD and IS externalities: - For ω ∈ {α, β, γ}: - ∫_{θ=0}^1 ∫_{φ=0}^1 (∂E_2/∂ω) { (1/E_2) δ σ_H τ_H2 + B_1 χ (E_2)^{−2} δ σ_F C_F2 [1 + (σ_H/σ_F) τ_H2] } π(θ|I_I) dφ dθ = Λ [ ∫_{θ=0}^1 ∫_{φ=0}^1 (∂E_2/∂ω) χ (E_2)^{−2} π(θ|I_I) dφ dθ + z_{ω1} + z_{ω2} ] - z_{ω1} = - ∫_{θ=0}^1 ∫_{φ=0}^{β/α (1−θ)} { [m_2 − 1] (∂E_2/∂ω) χ (E_2)^{−2} − (∂m_2/∂ω) [χ/E_2 − (1 + i^∗)] } π(θ|I_I) dφ dθ - + ∫_{θ=0}^1 ∫_{φ=1−β/α θ}^1 { [m_2 − 1] (∂E_2/∂ω) χ (E_2)^{−2} − (∂m_2/∂ω) [χ/E_2 − (1 + i^∗)] } π(θ|I_I) dφ dθ - z_{ω2} = - (∂(−β/α)/∂ω) { [ lim_{φ→(β/α (1−θ))^-} − m_2 − 1 ] [χ/(β+γ) − (1 + i^∗)] ∫_{θ=0}^1 (1 − θ) π(θ|I_I) dθ + [ lim_{φ→(1 − β/α θ)^+} + m_2 − 1 ] [χ/(α+γ) − (1 + i^∗)] ∫_{θ=0}^1 θ π(θ|I_I) dθ } - Interpretations: - Without externalities (τ_H2 = z_{ω1} = z_{ω2} = 0), changes in {α, β, γ} move E_2 across states and affect C_F2 via resource constraints if B_1 ≠ 0; RHS ensures sufficient returns for external lenders. - AD externalities enter via τ_H2 on LHS. - IS externality first component z_{ω1}: if β > 0 initially, changes in parameters affect SDF-weighted returns in uninsurable regions because m_2 may differ from 1 and ∂m_2/∂ω may be non-zero. - IS externality second component z_{ω2}: non-zero only for ω ∈ {α, β} because these determine boundaries of insurable set ̃Ψ; changes affect slope −β/α and size of insurable set, expanding or shrinking uninsurable regions and thereby SDF-weighted returns and premium ρ. ### Additional notes - First-order approach assumes unique interior solution and that planner’s FOCs establish welfare maximum; Appendix B contains full FOCs. - For subcase (a)(ii), α > β ≥ 0 is trivially satisfiable; appendix discusses amendments when β > α ≥ 0 or when there are no premia and exchange rate function unrestricted. *Source: wpiea2026188 - 3.1 Planner Problem* ### 3.3 Policy Impact of Uninsurability ### 3.3 Policy Impact of Uninsurability ### Effective information symmetry and when externalities vanish - Lemma 1 (Effective information symmetry): For a country with full insurability ̃Ψ = Ψ, the constrained efficient allocation features no LC and IS externalities, with one exception. Specifically, - zχ = zF1 = zω1 = zω2 = 0 for all ω ∈ {α,β,γ} in all subcases of case (a), except that zβ2 may be non-zero for subcase (a)(iii). - Interpretation: - LC premium and IS externalities are generally zero throughout case (a) because of full insurability. - Subcases (a)(i) and (ii): full insurability arises from symmetric information, irrespective of the value of β. - Subcase (a)(iii): occurs when there is asymmetric information but the planner chooses β = 0; private information of speculators does not give superior information about the exchange rate. Moving β above zero may generate an IS externality. ### Feasibility under an episode of information asymmetry - Lemma 2 (Information asymmetry episode): Consider a change from II = IS ∈ {I∅, Iθ} to II = I∅ and IS = Iθ. The premium ρ from the constrained efficient allocation prior to the change is too low to satisfy constraint (3) after the change if and only if condition (9) holds: - 0 > ∫θ=0^1 ∫φ=β/α(1−θ)^{1−β/α(1−θ)} [1−π(θ|II)] [χE2 − (1 + i∗)] dφ dθ + ∫θ=0^1 ∫φ=0^{β/α(1−θ)} [m(b)2 − π(θ|II)] [χE2 − (1 + i∗)] dφ dθ + ∫θ=0^1 ∫φ=1−β/α θ^1 [m(b)2 − π(θ|II)] [χE2 − (1 + i∗)] dφ dθ - where the m(b)2 function corresponds to case (b) after the change, but all planner choice variables correspond to the allocation prior to the change. - Three scenarios identified: 1. Start in subcase (a)(i) with planner selecting β = 0. Change to asymmetric information (speculators more informed); country moves to subcase (a)(iii). Premium constraint (3) not violated because ̃Ψ = Ψ and m2 = 1 in all states (φ,θ). 2. Start in subcase (a)(i) with planner selecting β > 0. Change to asymmetric information creates uninsurable regions (̃Ψ ⊂ Ψ), moving country to case (b). Whether prior premium ρ suffices depends on m(b)2 values in uninsurable regions; condition (9) is the criterion. For this scenario π(θ|II) = 1 for all θ prior to change, so first term in (9) is zero; second and third terms correspond to uninsurable regions and capture lenders' gains/losses depending on η2 ≡ χE2 relative to (1 + i∗). 3. Start in subcase (a)(ii). Change makes speculators remain informed while external lenders and insurers become less informed. Condition (9) is again the criterion; here π(θ|II) = 1 only for the known value of θ and = 0 for other values, so all three terms in (9) may be non-zero. ### Modified monetary policy when insurability concerns arise - Proposition 3 (Modified monetary policy): - The constrained efficient allocation must satisfy FOCs (5)-(8). - Suppose for II = IS = I∅, the planner selects β/α ∈ (0,1) and B1 > 0. For II = I∅ and IS = Iθ, there exist m(b)2 functions such that the planner selects a different value of β/α. - Role of m(b)2: - If m(b)2 = 1, planner selects same β/α in asymmetric case (b) as in symmetric subcase (a)(i). - Perturbations in m(b)2 alter the system via the premium constraint (3) and LC and IS externalities {zχ, zF1, zω1, zω2} in FOCs (6)-(8), causing changes in all planner choice variables and the ratio β/α (except knife-edge cases). - Economic interpretation: - Change in β/α changes the relative sensitivity of exchange rate E2 to private shock θ vis-à-vis public shock φ. - Planner generally changes stabilization with respect to shocks (φ,θ) once concerns about insurability and the UIP premium arise. ### Aloof central banker and commitment - Corollary 1 (Aloof central banker): - Suppose for II = IS = I∅, planner selects β/α ∈ (0,1) and B1 > 0. For II = I∅ and IS = Iθ, there exist m(b)2 functions such that the planner selects a lower value of β/α. - Implementation: delegate monetary policy to a central banker who can commit to this lower β/α. - Mechanism and interpretation: - Decreasing β/α makes slopes −β/α gentler and expands the insurable set ̃Ψ. - Analysis of a small perturbation m(b)2 = 1 + ε ̃m2 yields: - At ε = 0, marginal impact on zβ2: dzβ2/dε = −1/(2α) { [χβ+γ − (1 + i∗)] limφ→(β/α(1−θ)) − ̃m2 + [χα+γ − (1 + i∗)] limφ→(1−β/α θ) + ̃m2 }. - If the term inside curly braces is negative, perturbation makes zβ2 larger, which (ceteris paribus) induces planner to decrease β and thus β/α. - Direction of change of β/α determined by weighted combination of multiple channels; simulation in subsection 4.1 shows an example where β/α decreases in case (b) relative to subcase (a)(i). - Commitment: implementation relies on planner’s ability to commit at t = 1 to its exchange rate function at t = 2; delegation to an “aloof central banker” who down-weights private shock θ is a case for monetary delegation distinct from Rogoff (1985) inflation bias rationale. ### How insurability concerns alter premia and allocations: applications and examples - Proposition 4 (Log utility modeling of external lenders): - Modeling option: lenders owned by foreigners with log utility maximizing max C∗F2 E[log C∗F2 | IL] subject to κΠ2 + Y∗F2 ≥ C∗F2 with κ > 0, Π2 = [η2 − (1 + i∗)]B1 + QL(E2). - For a country in case (b) with κ > 0, m2 equals: - m2 = 1 for E2 ∈ ̃Ψ - m2 = κ/(1−β/α B1 ∫θ=0^1 ∫φ=β/α(1−θ)^{1−β/α(1−θ)} [χE2 − (1 + i∗)] dφ dθ + Y∗F2 / (κB1 [χE2 − (1 + i∗)] + Y∗F2) ) for E2 ∉ ̃Ψ (presentation in source: m2 = 1 for E2 ∈ ̃Ψ κ 1−β/α B1 ∫... + Y∗F2 κB1 [χE2 − (1 + i∗)] + Y∗F2 for E2 ∉ ̃Ψ) - For any given parameters {α,β,γ}, premium constraint (3) yields dχ/dB1 > 0. - Implications: - Values of m2 deviate from 1 in uninsurable regions only if κ > 0 (country large relative to lenders’ foreign owners). - Premium increases with external debt B1; premium schedule is upward-sloping in external debt. - If country switches from full to partial insurability without changing {α,β,γ}, jump in premium is increasing in level of external debt. - Simulations (Figures 2 and 3) using example functions YF1 = Y, YF2 = Y − ξ(φ − 1/2) − ν(θ − 1/2), and A1 = A2 = A show: - Insurability problems cause higher premium ρ, lower imports CF1, lower value of β/α, and lower welfare VP. - Both α and β are lower in case (b) but β is reduced relatively more so β/α is lower. - Problems become more severe as κ increases and as inherited external debt B0 increases. - Parameter values used in figures: σH = 0.5, δ = 0.96, i∗ = 1/δ − 1, B0 = 0.5, Y∗F2 = 1, ξ = 0.6, ν = 0.3, A = 1 for Figure 2; and σH = 0.5, δ = 0.96, i∗ = 1/δ − 1, κ = 0.5, Y∗F2 = 1, ξ = 0.6, ν = 0.3, A = 1 for Figure 3. - Corollary 2 (Inframarginal lenders): - If marginal lenders follow first option and there also exist inframarginal lenders who hold dollar value S1 of local currency bonds, Proposition 4 continues to hold but with B1 replaced by B1 − S1. - Implication: premium may be higher if inframarginal investors hold less external debt; inframarginal investors affect premia only if some exchange rates are uninsurable. - Proposition 5 (One-sided constraint; second modeling option for external lenders): - Modeling option: lenders owned by foreigners maximizing max C∗F2 E[C∗F2^{1−ν}/(1−ν) | IL] subject to Π2 − ς Θ(Π2) (Π − Π2)^2/2 + Y∗2 ≥ C∗F2, with Θ(Π2) = 0 if Π2 ≥ Π and = 1 if Π2 < Π for some Π < 0, capturing a one-sided financial constraint. - Suppose for ς = 0, constrained efficient allocation for case (b) features β/α ∈ (0,1) and B1 > 0. There exist Π < 0 such that for ς > 0, constrained efficient allocation features β/α ∈ (0,1) and B1 > 0, while Θ(Π2) = 1 only for some E2 ∈ (α+γ, α+β+γ]. Also: - limν→0 m2 = - 1 for E2 ∈ [γ, α+γ] - 1 + ς Θ(Π2) ( Π − [χE2 − (1 + i∗)] B1 ) ≥ 1 for E2 ∈ (α+γ, α+β+γ]. - Implications: - Premium may be mostly determined by the upper tail of the exchange rate distribution (large depreciations). - Financial constraint binds only after large depreciations observed in upper uninsurable region, pushing m2 substantially above 1 there. - Monetary policy modifications may be driven mainly to mitigate largest depreciations; planner internalizing IS externality may select lower α and β mostly to address upper uninsurable region. *Source: IMF Working Paper — 3.3 Policy Impact of Uninsurability (source PDF content).* ### 4.2 Fiscal Policy ### 4.2 Fiscal Policy ### Overview - The section explores two modeling options for fiscal policy and their influence on FX market insurability problems. Both options relate to government spending on imports. - Option 1: fiscal policy is exogenous and subject to public and private shocks. - Option 2: fiscal policy is endogenous and the motivation for public spending is subject to public and private shocks. - Private shocks may reflect asymmetric information about political or rent-seeking pressures. - Government spending on imports is denoted G_Ft in dollar terms. Planner rebates to households are amended as: - T_t = φ W_t N_t + μ_{t−1} (1 + i_{t−1}) D_{t−1} − E_t G_Ft. - No other shocks to tradable endowments or productivity: Y_Ft = Y_F and A_t = A in households’ and firms’ maximization problems. ### Exogenous fiscal policy (Proposition 6) - Setup: - Fiscal policy follows the first option (exogenous). Government spending on imports G_Ft is given exogenously. - The planner problem is isomorphic to the planner problem in subsection 3.1 with Y_Ft replaced by Y_F − G_Ft and A_t = A. - Key findings: - Fiscal policy can be the underlying source of FX market premia. It increases premia through two channels: 1. Before shocks: an increase in G_F1 is identical to a reduction in Y_F1 and/or an increase in inherited external debt B_0. It tends to increase external debt B_1 issued at t = 1, which is subject to any premium on the debt. External debt affects the multiplier Λ via equation (6) and parameters {α,β,γ} in the exchange rate function via equation (8). For B_1 > 0, monetary policy may be useful to depreciate away the dollar value of external debt after adverse external shocks. 2. Shocks to G_F2 can make monetary policy sensitive to the private shock at t = 2 and generate a premium at t = 1. If private shock θ increases G_F2, reducing resources for private import consumption C_F2, the planner may set β > 0 in the exchange rate function (4) because depreciating away external debt is more valuable for larger θ. If the country is in case (b), sensitivity of the exchange rate to the private shock at t = 2 generates uninsurability in the FX hedging market and a premium on the FX spot market at t = 1. - Implication: - Premia may arise structurally in countries whose fiscal policy depends on shocks known beforehand only to a subset of FX hedging market participants, and may exhibit episodic surges when such shocks become more important or when there is risk of a regime shift increasing fiscal sensitivity to such shocks. ### Endogenous fiscal policy (Proposition 7) - Setup: - Fiscal policy follows the second option (endogenous). Government spending on imports and planner lump-sum rebates follow the same expressions as in option 1. - Households’ welfare amended: - E[∑_{t=1}^2 δ^{t−1} U(C_Ht, C_Ft, G_Ft, N_t)] - U(C_Ht, C_Ft, N_t) = σ_H log C_Ht + σ_F log C_Ft + σ_Gt log G_Ft − N_t, - where σ_Gt is a single number at t = 1 and a function of both shocks at t = 2. - Planner’s period-t utility function V is replaced as specified, with ∂V/∂G_Ft = σ_Gt / G_Ft. - Planner’s choice variables expanded to include {G_F1, G_F2}. - Optimality condition (combining FOCs for G_Ft and C_Ft): - G_F1 = (σ_G1 / σ_F) C_F1 - G_F2 = (σ_G2 / σ_F) C_F2 [ 1 + (σ_H / σ_F) τ_H2 ]. (Equation (13)) - Key findings: - If fiscal policy is endogenous, levels of government spending on imports are affected by FX market insurability problems. - Prior to shocks, G_F1 is tied to private import consumption C_F1. Since C_F1 is affected by AD and LC premium externalities (equation (7)), G_F1 is also affected by those externalities—fiscal discipline in normal times can help manage future shocks. - Sensitivity of G_F2 to shocks and parameters in the exchange rate function {α,β} depend on each other: if in case (b) and G_F2 varies with θ, {α,β} are affected by partial insurability because of AD and IS externalities. After shocks, E_2 affects desired G_F2 by affecting C_F2 and τ_H2. - The planner can eliminate the insurability problem and associated premium by constraining fiscal policy to be insensitive to the private shock. ### Constrained fiscal policy (Corollary 3) - Setup: - Fiscal policy follows the endogenous option but G_F2 is constrained to be insensitive to private shock θ while remaining a function of public shock φ. - Modified condition (13) replaced by: - G_F1 = (σ_G1 / σ_F) C_F1 - G_F2 = [ ∫_{θ=0}^1 σ_G2 π(θ|I_I) dθ ] / [ ∫_{θ=0}^1 σ_F C_F2 [ 1 + (σ_H / σ_F) τ_H2 ] π(θ|I_I) dθ ]. (Equation (14)) - Consequences: - The constrained efficient allocation for subcase (a) (i) features β = 0, and the premium ρ = 0 continues to satisfy the premium constraint (3) after the information sensitivity episode in Lemma 2. - Constraining G_F2 to be insensitive to θ achieves full insurability in the FX hedging market and eliminates the premium in the FX spot market. - Economic trade-off: - Welfare is hurt by constraining fiscal policy if σ_G2 ideally should vary with θ (e.g., π(θ|I_I) = 1 for all θ). - Welfare is improved by removing FX market premia if B_1 is positive. - Simulation example and comparative results: - Example functional form used: - σ_G2 = σ_G1 ( 1 + ξ (φ − 1/2) + ν (θ − 1/2) ). (Equation (15)) - For parameter values listed below, moving from full insurability subcase (a) (i) to partial insurability case (b) causes: - higher premium ρ, - lower imports C_F1, - lower value of β / α, - lower welfare V_P. - Figure 4 depiction: - Dashed contours: subcase (a) (i); solid contours: case (b). - Solid contours have gentler slope than dashed, indicating reducing sensitivity of G_F2 to θ lowers required β / α and reduces insurability problem. - Parameter values used in Figure 4: - σ_H = 0.5, σ_G = 0.5, δ = 0.96, i^∗ = 1/δ − 1, κ = 0.5, B_0 = 0.5, Y^∗_{F2} = 1, ξ = 0.6, ν = 0.3, Y_F = 1, and A = 1. ### Links to fiscal rules and institutions - The results connect literature on fiscal rules/institutions with international finance literature on FX premia. - Fiscal rules that reduce sensitivity of fiscal policy to private shocks can improve FX hedging market functioning and reduce FX spot premia. - Reducing premia may also reduce susceptibility to inframarginal return-insensitive investors (e.g., noise traders). ### Summary of policy-relevant implications - Exogenous fiscal policy that raises G_F1 increases external debt B_1 and can raise premia; monetary policy can partially offset by depreciating external debt when B_1 > 0. - Endogenous fiscal policy implies G_Ft choices internalize insurability externalities; fiscal discipline pre-shock reduces vulnerability. - Constraining G_F2 to be insensitive to private shocks can eliminate FX hedging market premia (β = 0, ρ = 0) but may reduce welfare if σ_G2 ideally should respond to θ. - Optimal institutional design and fiscal rules should weigh the trade-off between volatility reduction (insurability and lower premia) and the potential welfare loss from constraining countercyclical fiscal responses. *Source: IMF Working Paper — Section 4.2 Fiscal Policy (wpiea2026188).* ### References ### wpiea2026188 - References ### References list - Bibliographic entries [1] through [34] are provided, citing working papers, journal articles, chapters, and memos relevant to exchange rates, FX intervention, capital flows, market microstructure, macroprudential policy, and related theory. Key authors include Albagli; Basu; Gopinath; Bianchi; Cavallino; Sarno; Itskhoki; Farhi; Gabaix; Rey; and others. (Entries preserve numbering [1]–[34] and original bibliographic formatting.) ### Appendix A — Proofs of Results in Section 2 - Proof of Proposition 1: - Strategy: solve the FX hedging market backward; consider insurers choosing d = 1 so that E2 ∈ ̃Ψ, then check insurer incentives to prefer d = 0. - Lenders’ aggregate decisions satisfy dQ_L(E2)/dp(E2) < 0 and lim_{p(E2)→0} Q_L(E2) = ∞, with Q_L(E2) finite for p(E2) > 0; for π(E2|I_L) → 0, Q_L(E2) indeterminate if p(E2) → 0 and Q_L(E2) → Q_min(E2) if p(E2) > 0. - Insurers’ equilibrium pricing conditions (selected cases): - If π(E2|I_I) = π(E2|I_S) > 0 then insurers avoid p(E2) = 0 and obtain (1 +i^*)p(E2) = π(E2|I_I). - If π(E2|I_I) = π(E2|I_S) = 0 then insurers select p(E2) = 0 and (1 +i^*)p(E2) = π(E2|I_I). - When I_I = I_∅ and I_S = I_θ: - If π(E2|I_I) > 0 insurers avoid p(E2) = 0; insurers decide d = 1 only if π(E2|I_I) = π(E2|I_S), yielding (1 +i^*)p(E2) = π(E2|I_I). - If π(E2|I_I) = 0 insurers select p(E2) = 0, insurers decide d = 1 and (1 +i^*)p(E2) = π(E2|I_I). - Conclusion: these cases establish the proposition’s characterization of insurer pricing and insurability. ### Appendix B — Planner Problem and First-Order Conditions - Planner’s choice variables: {C_F1, C_F2, E1, α, β, γ, χ, P_H}. - Derived expressions (selected, preserving notation exactly): - i_1 = 1/(δ E1 C_F1 E[1/E2 C_F2] − 1) - μ_1 = 1 − δ χ C_F1 E[1/E2 C_F2] - φ = σ_F P_H ε−1/ε (1 + δ E1 C_F1 A_1 + δ E[E2 C_F2 A_2])−1 - E2 = αφ + β θ + γ - C_Ht = Y_Ht = σ_H/σ_F E_t C_Ft P_H, N_t = Y_Ht A_t, W_t = 1/σ_F E_t C_Ft for t ∈ {1,2} - D_2 = [B_0 (1 +i^*) + (C_F1 − Y_F1)] E1 - Premium constraint and associated objects: - The premium constraint integrals split into three region-terms based on the partition for α ≥ β ≥ 0. Probability density functions π(E2|I_θ) and π(E2|I_∅) are given piecewise with exact expressions involving α, β, γ. - First-order conditions (FOCs) excerpted (preserved form): - FOC w.r.t χ: B_1 χ ∫_θ ∫_φ Φ(φ,θ) π(θ|I_I) dφ dθ = Λ [ ∫_θ ∫_φ 1/E2 π(θ|I_I) dφ dθ + z_χ ] - FOCs w.r.t C_F1 and C_F2: σ_F C_F1 [ 1 + σ_H/σ_F τ_H1 ] = ∫_θ ∫_φ Φ(φ,θ) π(θ|I_I) dφ dθ + Λ z_F1 δ σ_F C_F2 [ 1 + σ_H/σ_F τ_H2 ] = Φ(φ,θ) / (E2 χ) - FOCs w.r.t ω ∈ {α, β, γ}: ∫_θ ∫_φ ∂E2/∂ω { 1/E2 δ σ_H τ_H2 + B_1 1/E2 Φ(φ,θ) } π(θ|I_I) dφ dθ = Λ [ ∫_θ ∫_φ ∂E2/∂ω χ(E2)^2 π(θ|I_I) dφ dθ + z_{ω1} + z_{ω2} ] - FOC w.r.t P_H: τ_H1 + δ ∫_θ ∫_φ τ_H2 π(θ|I_I) dφ dθ = 0, redundant if ∂m_2/∂α α + ∂m_2/∂β β + ∂m_2/∂γ γ + ∂m_2/∂χ χ = 0 (and this condition holds). - Boundary and corner adjustments: - If α = 0 or β = 0 binding: replace equalities with inequalities in corresponding FOC. - If β > α ≥ 0: premium constraint (B.1) becomes 0 = ∫_θ ∫_φ m_2[ χ/E2 − (1 +i^*) ] π(θ|I_I) dφ dθ and z_ω2 = 0, with explicit amended definitions for z_χ, z_F1, z_{ω1}, z_{ω2}. - Comparison to unrestricted exchange-rate / no-premia case: - Set m_2 = 1 and remove restricted E2 function; FOCs simplify (z_χ = z_F1 = 0) and FOCs w.r.t E2 become 1/E2 δ σ_H τ_H2 + B_1 1/E2 Φ(φ,θ) = Λ χ(E2)^2, and equation (B.2) applies: δ σ_H τ_H2 + B_1 χ/E2 δ σ_F C_F2 [ 1 + σ_H/σ_F τ_H2 ] = Λ χ E2. ### Appendix C — Proofs of Results in Sections 3–4 (Selected conclusions) - Lemma 1: - In subcases of Proposition 2 labeled (a)(i), (a)(ii), and (a)(iii), m_2 = 1 in every term of the premium constraint containing m_2, leading to z_χ = z_F1 = z_{ω1} = z_{ω2} = 0 in many subcases (with exceptions accounted for by β = 0). - Lemma 2: - Describes how a change in information structure from I_I = I_S ∈ {I_∅, I_θ} to I_I = I_∅ and I_S = I_θ can make the premium constraint violate equality (become > 0) depending on initial β; provides integral expressions (C.1)–(C.3) showing the change. - Proposition 3: - Perturbation approach: define m_2^(b) = 1 + ε ˜m_2(...) and show the planner’s reduced system in variables {C_F1, α, β, γ, χ, Λ} implies the planner may select different β/α in case (b) relative to subcase (a)(i) for nonzero perturbations ˜m_2. - Corollary 1: - There exist m_2^(b) functions such that the planner selects a lower β/α in case (b) than in subcase (a)(i); commitment matters because all LC and IS externality terms in FOCs multiply Λ, which is positive when B_1 > 0. - Proposition 4 (foreign lenders with linear preferences): - Foreigners’ FOC yields M_2 = κ Υ = κ / (κ Π_2 + Y^*_F2). - For case (b) with insurable set ̃Ψ, Π_2 = [η_2 − (1 + i^*)] B_1 + Q_L(E2) = Π for all E2 ∈ ̃Ψ; integrating yields expression (10) (as in text). - Substituting gives a premium constraint that, when differentiating, yields dχ/dB_1 = κ Y^*_F2 Z_1 / Z_2 and therefore dχ/dB_1 > 0 since {Z_1, Z_2} > 0. - Corollary 2: - If inframarginal external lenders hold S_1 of local bonds, replace B_1 by B_1 − S_1 in Proposition 4 results. - Proposition 5 (foreign lenders with non-linear preferences / demand externality parameter ς): - Foreigners’ M_2 given by (C.8) and m_2 expression (C.12); proof shows constrained-efficient allocation in case (b) must satisfy χ/(α+β+γ) < (1 + i^*). - For ς = 0 one can set Π = [χ/(α+β+γ) − (1 + i^*)] B_1 < 0 (equation (C.13)); for ς > 0 continuity arguments yield the expression (12) in the proposition. - Propositions 6 and 7 (domestic fiscal and government spending extensions): - Amend resource constraints and B_1 definitions to include government spending terms; FOCs w.r.t G_F1 and G_F2 produce expressions linking optimal government spending to private consumption terms, yielding condition (13). - Corollary 3: - Under restriction that G_F2 does not vary with θ and starting in subcase (a)(i), the constrained-efficient allocation features β = 0; relaxed planner problem (C.14) shows f(θ) must be constant across θ, hence β = 0 is optimal. The premium constraint remains satisfied after the information sensitivity shock when evaluated at that allocation. - Proposition 8: - Model amended with transfers X and transfer-relevant shock λ(θ,X); resource constraints and B_1 definitions updated. Planner’s FOC w.r.t X yields ∫_θ ∫_φ [λ_X(θ,X) − (1 + i^*)] Φ(φ,θ) / (E2 χ) π(θ|I_I) dφ dθ = 0 and together with FOC for C_F2 leads to covariance condition (16). - Proposition 9: - Planner selects institutional design ∆ ∈ {N, S, T} maximizing welfare at t = 0: - ∆ = N places country in case (b) at t = 1; t = 0 welfare equals t = 1 welfare. - ∆ = S places country in subcase (a)(i) at t = 1; planner subtracts secrecy cost to obtain V_S. - ∆ = T places country in subcase (a)(ii) at t = 1; planner integrates over θ and subtracts information-collection cost to obtain V_T. *Content and technical notation preserved exactly as in source PDF.* --- _Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026188.pdf_