## wpiea2023161-print-pdf ## Source details **Canonical URL:** [wpiea2023161-print-pdf](https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023161-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2023/english/wpiea2023161-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2023/english/wpiea2023161-print-pdf.pdf.json) --- ### Model overview (Sections 2.1, 2.3) - Structure and agents: - Three-period model t ∈ {0,1,2} of a small open economy with dominant currency pricing and domestic and external financial frictions. - Agents: households, tradable sector firms, housing sector firms, domestic banks, global financiers. - Planner instruments: monetary policy rate i_t, capital inflow taxes φ_t, domestic debt taxes θ (household and housing-sector), and FX intervention FXI_t / O_{t+1}. - Key market and pricing features: - Dominant currency pricing: prices sticky in dollars for imports and for exports; domestically-sold component of exports priced in domestic currency. - Two noncontingent assets: local currency bond Q_{t+1} and dollar bond (−Q_{t+1}/E_t). - Asset market segmentation: domestic agents trade only local currency bond; global financiers trade both but face binding portfolio friction Γ ≥ 0. - Fraction λ ∈ (0,1) of global financiers domestically owned → effective FX mismatch for representative household. - Timeline and shocks: - Timeline: t = 0 (ex ante policies, price-setting), t = 1 (shock realization H or L), t = 2 (final consumption, exchange rate). - Shocks considered individually: productivity, world interest rate i^*_t, foreign appetite for domestic assets (surge/taper tantrum S_1), external pledgability of collateral κ_q (sudden stop). - Households and budget: - Welfare E0[∑_{t=0}^2 β^t U(C_Ht, C_Ft, C_Rt, N_t)] with U = α_H log C_Ht + α_F log C_Ft + (1 − α_H − α_F) log C_Rt − N_t. - Budget constraint: W_t N_t + Π_Tt + Π_Bt + λ Π_FIt + Π_Rt + T_t + D_HHt+1 ≥ P_Ht C_Ht + E_t C_Ft + P_Rt C_Rt + (1 + θ_HHt−1) (1 + ρ_t−1) D_HHt. - Tradable and housing sectors: - Tradable: monopolistic competition, pre-set prices at t = 0; production Y_Tt(j) = A_t N_t(j); exports priced in dollars P_X, domestic tradable P_H. - Housing: two subsectors h ∈ {Linear, Concave} with outputs Y^Linear_{R,t+1} = k^Linear_t and Y^Concave_{R,t+1} = G(k^Concave_t) (G′ > 0, G′′ < 0, G′(0)=1). Linear subsector can be taxed θ^{Linear}_{Rt} ∈ R; concave subsector θ^{Concave}_{Rt} ≡ 0. - Linear-subsector borrowing constraint: D^{Linear}_{R2} ≤ κ_q q_1 k^{Linear}_1. - Banks and global financiers: - Banks intermediate between global financiers and domestic borrowers; aggregate domestic debt D_{t+1} = D_HHt+1 + D^{Linear}_{Rt+1} + D^{Concave}_{Rt+1}. - Banks’ profit Π_{Bt+1} = (ρ_t − i_t) D_{t+1}; external borrowing constraint D_2 ≤ κ_H P_H may bind (sudden stop when κ_H falls). - Global financiers face balance-sheet friction Γ: demand for local currency bonds determined by Γ and yields endogenous premium η_{t+1} = (1−φ_t) (1 + i_t) (E_t / E_{t+1}). - Market clearing and FX mismatch: - Q_{t+1} + F_{t+1} + O_{t+1} = D_{t+1}. - Effective interest I_t ≡ λ (1 + i^*_t) + (1 − λ) η_{t+1}; FX mismatch λ determines macro responses. *Source: wpiea2023161-print-pdf - 2.1 Model Overview* ### Planner problem and wedges (Section 2.3) - Competitive equilibrium objects: quantities {C_Ht, C_Ft, C_Rt, N_t, k^Linear_t, k^Concave_t, Y_Ht, Y_Xt, Y^Linear_Rt, Y^Concave_Rt, Q_{t+1}, D_{t+1}}_{t=0}^2 and prices {P_H, P_X, {ρ_t}_{t=0}^1, {W_t, E_t, P_Rt, q_t}_{t=0}^2} satisfying agents’ optimization, constraints, and market clearing given initial dollar debts and policy instruments {i_t, φ_t, θ^{HH}_t, θ^{Linear}_{Rt}, FXI_t}_{t=0}^1. - Four wedges summarizing deviations from first-best (focus on first three): - AD wedge τ_{Ht} = 1 + (1/A_t) (U_{Nt}/U_{Ht}) = 1 − (1/A_t) C_{Ht}^{α_H}. Externalities: AD externality and pecuniary AD externality via exchange rate effects on banks’ constraint. - UIP wedge τ_{Γ,t+1} = (1−λ) E_t[ [η_{t+1} − (1 + i^*_t)]^{α_F} C_{Ft+1} ]. Positive when transfers to foreign-owned intermediaries reduce welfare; salient when Γ > 0. - Housing wedge τ_{Rt} = [1 − G′(1 − k^Linear_{t−1})]^{α_R} C_{Rt}. Captures pecuniary production externality through land pricing and borrowing constraints. - Terms-of-trade wedge τ_{Xt} (planner ignores this when export demand unit-elastic). - Constrained-efficient allocation: - Ramsey planner maximizes household welfare subject to implementability via {i_t, φ_t, FXI_t, θ^{HH}_t, θ^{Linear}_{Rt}}_{t=0}^1 and ignoring effect of policies on pre-set export price P_X. - Key constraints (in dollar terms) include resource constraint, occasionally-binding external borrowing constraint, Gamma equations (binding), domestic housing borrowing constraint, and equalization condition E_1 η_1 across states. *Source: wpiea2023161-print-pdf - 2.3 Planner Problem* ### General conditions and traditional prescription (Section 3.1) - First-order approach yields pivotal results: - Exchange rate flexibility eliminates AD wedges in periods with no shocks: τ_H0 = τ_H2 = 0. - Period-1 tradeoff (combining FOCs for E_1 and η_1) links τ_H1 to pecuniary AD externality, depreciation effects on dollar wealth and external repayments, and shallow-market friction terms. - First-order conditions for import consumption and relationships with capital inflow taxes φ_t derive constrained-efficient φ_0 and φ_1 (Equations (13)-(14)). - FX intervention optimality conditions (Equations (15)-(16)) show FX intervention smooths external premia and allows planner to adjust external debt positions without changing consumption. - Proposition 1 (Traditional prescription): constrained-efficient allocation achievable with only policy rate and exchange rate flexibility if a set of equalities/conditions (Equations (19)-(23)) hold that eliminate need for φ_t, FXI_t, and housing debt taxes. - Examples where traditional prescription suffices: - Productivity shock (Lemma 1): for B_0 = 0 and {κ_H, κ_q} large (constraints slack), policy rate and exchange rate flexibility suffice; AD, UIP, and housing wedges stabilized at zero; terms-of-trade wedge not stabilized. - World interest rate shock (Lemma 2): for λ = 1 and Γ = 0, policy rate, exchange rate flexibility, and ex ante capital controls suffice; AD, UIP, and housing wedges stabilized. *Source: wpiea2023161-print-pdf - 3.1 General Conditions* ### Foreign appetite shocks and optimal tool mixes (Sections 4.1, 4.1 Proposition 2) - Setup: non-fundamental local-currency surges S_H1 > 0 and taper tantrums S_L1 < 0; assume B0 = 0 and {κ_H, κ_q} large so pecuniary AD/production externalities slack; relevant externalities: AD and local-currency premium. - Role of (1−λ) Γ: - If (1−λ) Γ = 0: supply of external funds perfectly elastic up to limit; foreign appetite shocks do not shift supply curve. - If (1−λ) Γ > 0: supply curve upward-sloping and foreign appetite shocks shift it horizontally; UIP/premium externality becomes state-dependent. - Perfect stabilization results: - Lemma 3: if (1−λ) Γ = 0, AD and UIP wedges stabilized to zero, imports set to C^*, FX intervention indeterminate and can be zero, policy rate set to (1 + i_t) = 1/β, E_t = α_F A C_F. - Proposition 2 (Perfect stabilization) for (1−λ) Γ > 0 and symmetric shocks S_H1 = − S_L1 > 0: - AD wedges τ_Ht = 0 for t ∈ {0,1,2}. - Planner uses FX intervention FXI_t and capital inflow taxes φ_t jointly ex post; ex ante φ_0 = 0. - Policy rate and exchange rate do not vary across states: (1 + i_t) = 1/β for t ∈ {0,1}, E_t = α_F A C_F. - UIP wedge between periods 1 and 2 set to τ_Γ2 = − (1−λ) Γ S1 / (2 α_F C_F). - Optimal FXI_0 = −B1 and FXI_1 = S1 / 2 − B2 with B1 = (1−λ) Γ (S1)^2 / 4 (1 + 1/β + 1/β^2) and B2 = (1 + 1/β) B1 and C_Ft = C^* + B1 for t ∈ {0,1,2}. - Missing-tools corollary: - If either FX intervention or capital inflow taxes unavailable ex post, perfect stabilization impossible; absence of either tool leads to imperfect stabilization of AD wedges, imports, policy rates, and exchange rates. *Source: wpiea2023161-print-pdf - 4.1 Externalities and Policy Tools* ### Sudden stops, housing, and tool substitutability (Sections 4.1, 6.1–6.3) - Sudden stop: reduction in κ_H in period-1 L state that can make external borrowing constraint bind (Ψ_B^L > 0); then domestic borrowing rate ρ_t > policy rate i_t in L state. - Housing-sector interactions: - If housing pledgability κ_q sufficiently large, capital inflow taxes φ_t and household debt taxes θ_HHt are perfect substitutes (Equation (33)), allowing FX intervention + either tax to achieve welfare-equivalent outcomes (Proposition 7, κ_q ∈ [κq, ∞) case). - If κ_q below threshold (κ_q ∈ [0, κq)), substitutability breaks: household debt taxes may fail to prevent housing constraint (6) from binding in L state, producing non-zero housing wedges τ_Rt and potentially making ex ante household/housing debt taxes useful. - Mechanisms in sudden stops: - Binding external constraint reduces C_F1 in L state; marginal value of a dollar z_1 satisfies z^L1 > E_0[z_1 η_1] / E_0 η_1 > z^H1 (Lemma 4). - Exchange rate usually more depreciated in L state (Assumption 1: E^L1 > E^H1), generating expenditure switching and ambiguous net effect on housing constraint binding. - Policy implications: - When both FX intervention and capital inflow taxes available, joint ex post use is effective for foreign-appetite shocks with shallow FX markets. - When housing constraint can bind, capital inflow taxes and household debt taxes are not perfect substitutes; ex ante macroprudential measures (household/housing debt taxes or ex ante φ0) may be warranted to avoid domestic credit crunch and binding housing constraints. - Central bank cannot typically absorb private-sector external debt in sudden stops → FX intervention often unavailable in sudden-stop analysis; φ^L1 becomes redundant if external constraint binds ex post. *Source: wpiea2023161-print-pdf - 4.1, 6.1–6.3* ### Case for an ex ante capital inflow tax (Section 5.3) - Special case (Proposition 3): λ = 1 and Γ = 0 (full effective FX mismatch, deep FX markets): - Optimal φ0 > 0, φ1 = 0; AD wedge τ_LH1 > 0, τ_HH1 = 0; UIP wedges zero. - Intuition: with Γ = 0 no premium externality, positive AD wedge in L state calls for φ0 to shift consumption and borrowing from period 0 to period 1. - Welfare effects of changing λ (Lemma 5): - Planner-valued welfare change from marginal change in λ equals β B1 Cov(z1, η1) + Γ { β(B1)2 E0 z1 + β2 E0[ z2 (B2)2 ] }. - If Γ = 0 and Cov(z1, η1) < 0 (Assumption 1 implies this), planner prefers lower λ. - If Γ > 0, second term positive and expression ambiguous. - Perturbation results: - Deep FX markets Γ = 0 (Proposition 4): under Perturbation 1 (marginal reduction in λ with E^s_1 fixed and no FXI), planner reduces ex ante φ0 when sufficient condition (26) holds. - Shallow FX markets Γ > 0 (Proposition 5): set of sufficient conditions (27)–(31) determine whether planner reduces φ0 after lowering λ; conditions more stringent and gains from reducing λ can be offset by amplified premium externalities. - Numerical insights: - With Γ = 0: as λ decreases, φ0 decreases and welfare increases. - With Γ > 0: starting at λ = 1, decreasing λ first reduces then increases φ0 as premium externalities emerge; welfare can decrease with lower λ. - Policy takeaway: - Role and sign of optimal ex ante capital inflow tax φ0 depend jointly on FX mismatch λ and FX market depth Γ. - Countries with deep FX markets more likely to benefit from φ0 reductions when reducing FX mismatch; shallow FX markets can make reducing λ harmful and increase optimal φ0. *Source: wpiea2023161-print-pdf - 5.3 Case for an Ex Ante Capital Inflow Tax* ### Key policy conclusions and practical implications (Sections 4–7) - Traditional prescription (policy rate + exchange rate flexibility) effective for some shocks (productivity, some world rate shocks) and parameter configurations (e.g., Γ = 0, λ = 1). - For foreign-appetite shocks when (1−λ) Γ > 0 (shallow FX markets with foreign participation): - Joint ex post use of FX intervention and capital inflow taxes is optimal to stabilize AD and manage UIP/premium externality (Proposition 2). - Absence of either tool leads to imperfect stabilization (Corollary 1). - Housing-sector frictions and domestic borrowing constraints: - When housing pledgability κ_q is low, domestic asset-price amplification can make household/housing debt taxes non-substitutable with capital inflow taxes; ex ante macroprudential measures may be required (Proposition 7). - Sudden stops amplify interplay of AD, pecuniary AD, premium, and pecuniary production externalities; optimal policy depends on λ, Γ, κ_H, κ_q. - Practical considerations for policymakers: - Measurement of wedges (τ_Ht, τ_{Γ,t}, τ_{Rt}) essential for tool calibration. - Credible communication and coordination across agencies needed when using joint instruments (FX intervention, inflow taxes, debt taxes, policy rate). - Market development and tool usage are endogenous; reforms that deepen FX markets (reduce Γ) change optimal policy mixes. *Source: wpiea2023161-print-pdf - Sections 4–7 conclusion summary* _Italic: Source: wpiea2023161-print-pdf_ ### 2.1 Model Overview ### 2.1 Model Overview ### Model structure and key features - A three-period model of a small open economy with dominant currency pricing and a combination of domestic and external financial market frictions. - Agents: households, tradable sector firms, housing sector firms, domestic banks, and global financiers. - Planner instruments: monetary policy rate, capital inflow taxes, domestic debt taxes on the borrowing of households and housing sector firms, and FX intervention. - Dominant currency pricing: prices are sticky in dollars for imports and for exports of home-produced tradable goods; the domestically-sold component of exports has sticky prices set in domestic currency. - Two noncontingent assets: a local currency bond and a dollar bond. - Asset market segmentation: domestic agents can trade only the local currency bond; global financiers can trade both bonds but face a portfolio friction that always binds. - Global financiers borrow in dollars on the world market and lend at a premium in local currency to domestic banks; the premium reflects FX market depth and depends on the portfolio friction severity. - Domestic ownership of a fraction of global financiers implies that the representative household may have effective FX mismatch on its external borrowing. ### Financial structure (figure 4 description) - Domestic agents: Households, Housing sector, Domestic banks. - Global side: Financial intermediaries, World capital markets. - Policy and regulatory levers shown: Capital controls, Macroprudential measures, FX intervention, Monetary policy. ### Timeline and shock structure (figure 5 description) - Timeline: t = 0 (ex ante policies and price-setting), t = 1 (shock realization, borrowing constrained decisions), t = 2 (final consumption and exchange rate determination). - Shocks take one of two values: high, “H”, or low, “L”, and strike in period 1, after which all uncertainty is resolved. - Some policies are set in period 0 (ex ante); others are implemented in periods 1 and 2 (ex post). - Considered shocks (typically one at a time): - Productivity shocks. - World interest rate shocks. - Foreign appetite for domestic assets (leading to a small-to-moderate “taper tantrum”). - External pledgability of domestic collateral (leading to a severe “sudden stop”). ### Households: preferences and budget - Welfare function: - E0[∑_{t=0}^2 β^t U(C_Ht, C_Ft, C_Rt, N_t)] - U(C_Ht, C_Ft, C_Rt, N_t) = α_H log C_Ht + α_F log C_Ft + (1 − α_H − α_F) log C_Rt − N_t. - Budget constraint (as presented in text; symbols preserved): - W_t N_t + Π_Tt + Π_Bt + λ Π_FIt + Π_Rt + T_t + D_HHt+1 ≥ P_Ht C_Ht + E_t C_Ft + P_Rt C_Rt + (1 + θ_HHt−1) (1 + ρ_t−1) D_HHt. - Income components: W_t wage, N_t labor supply, Π_Tt profits of tradable firms, Π_Bt profits of domestic banks, λ fraction of global financiers owned by domestic households times Π_FIt, Π_Rt transfer from housing firms (period 2 only), T_t lump-sum transfer from planner, D_HHt+1 domestic-currency debt. - Prices and variables: P_H domestic price of home-produced tradable good, E_t exchange rate in units of local currency per dollar, dollar price of imports normalized to 1, C_F imports consumption, P_R and C_R price and consumption of nontradable housing services, θ_HHt household debt tax between periods t and t+1, ρ_t interest rate offered by domestic banks between those periods. ### Tradable sector firms - Market structure: monopolistically competitive; set prices at the beginning of period t = 0 and prices are fully rigid thereafter. - Production: for varieties j ∈ [0,1], Y_Tt(j) = Y_Ht(j) + Y_Xt(j) = A_t N_t(j), where A_t is productivity. - Domestic and export pricing: domestically-sold goods priced P_H(j) in domestic currency; exported goods priced P_X(j) in dollars. - Aggregators: - Y_Ht = (∫_0^1 Y_Ht(j)^{(ε−1)/ε} dj)^{ε/(ε−1)} - Y_Xt = (∫_0^1 Y_Xt(j)^{(ε−1)/ε} dj)^{ε/(ε−1)} - Market clearing: labor taxed at rate φ; flexible wage W_t; Y_Ht = C_Ht; Y_Xt = C^* / P_X where C^* is index of foreign demand and export demand is unit-elastic in P_X. ### Housing sector firms - Perfect competition and flexible rental prices. - Two subsectors (Kiyotaki and Moore (1997) framework): Linear and Concave. - Production across subsectors h ∈ {Linear, Concave}: - Y^h_{R,t+1} = k^h_t for h = Linear - Y^h_{R,t+1} = G(k^h_t) for h = Concave, where G is continuously differentiable with G(0) = 0, G′ > 0, G′′ < 0, and G′(0) = 1. - Financing: housing firms borrow from domestic banks; maximize expected profits: - E_t[P_R,t+1 Y^h_{R,t+1} + q_{t+1} k^h_t] − (1 + θ^h_{Rt}) (1 + ρ_t) q_t k^h_t. - Planner can impose debt taxes on linear subsector θ^{Linear}_{Rt} ∈ R; concave subsector unregulated θ^{Concave}_{Rt} ≡ 0. - Lump-sum transfers: planner rebates revenues from housing subsector debt taxes to the same subsector; in period 2 firms remit final assets to households. - Initial conditions: D^{Linear}_{R0} = − D^{Concave}_{R0} > 0 (linear subsector has inherited debt; concave subsector has inherited assets). - Borrowing constraint for linear subsector between periods 1 and 2: - D^{Linear}_{R2} ≤ κ_q q_1 k^{Linear}_1, where κ_q represents pledgability of land; RHS tightens when land price declines. - Market clearing in land: k^{Linear}_t + k^{Concave}_t = 1. - Housing services clearing: Y^{Linear}_{Rt} + Y^{Concave}_{Rt} = C_Rt. ### Domestic banks - Banks borrow from global financiers and lend to domestic agents; transfer funds in local currency. - Aggregate domestic debt at end of each period: - D_{t+1} = D_HHt+1 + D^{Linear}_{Rt+1} + D^{Concave}_{Rt+1}. - Banks’ profit: - Π_{Bt+1} = (ρ_t − i_t) D_{t+1}, where i_t is the domestic policy rate. - Banks’ occasionally-binding external borrowing constraint between periods 1 and 2: - D_2 ≤ κ_H P_H, where κ_H captures external pledgability of domestic tradable goods. - Under some conditions the constraint becomes tighter in dollar terms when the exchange rate depreciates, depending on extent of FX mismatch. - Interest rates: if banks’ constraints do not bind, competition sets ρ_t = i_t; if constraints bind, ρ_t > i_t to clear domestic debt market. - A reduction of κ_H in the period-1 L state represents a “sudden stop shock”. ### Global financiers and portfolio/frictional structure - Two categories of global financiers: - Optimizing financiers choose Q_{t+1} in local currency bonds and −Q_{t+1} / E_t in dollar bonds to maximize dollar profits, subject to a balance sheet friction (Gabaix and Maggiori (2015)-style). - Objective (as in text): max_{Q_{t+1}} 1/(1 + i^*_t) Q_{t+1} / E_t E_t[ (1 − φ_t) (1 + i_t) E_t / E_{t+1} − (1 + i^*_t) ] subject to same LHS ≥ 1/(1 + i^*_t) Γ(Q_{t+1} / E_t)^2. - i^*_t is the dollar interest rate; φ_t is the capital inflow tax announced in period t and applies to repayments made to financial intermediaries in period t+1. - In absence of shocks, i^*_t = 1/β − 1 for all t ∈ {0,1}; world interest rate shocks shift i^*_1 away from this value. - Γ ≥ 0 parameter captures balance sheet friction severity; constraint always binds and yields intermediaries’ demand for local currency bonds. - Γ = 0 ⇒ “deep FX markets”; Γ > 0 ⇒ “shallow FX markets”. - Non-optimizing financiers with exogenous stochastic demands for local currency debt in period 1: hold F_2 in local currency bonds, equal to S_1 = F_2 / E_1 in dollar value. Variations in S_1 across period-1 states are “foreign appetite shocks”: positive in H state = “surge”; negative in L state = “taper tantrum”. - FX intervention by planner: planner takes position O_{t+1} in local currency bonds and FXI_t = − O_{t+1} / E_t in dollar bonds, with all carry profits and losses rebated to households. Availability of unrestrained FX intervention depends on the shock (paper-specific assumption). ### Market clearing and FX mismatch - Local currency debt market clearing: - Q_{t+1} + F_{t+1} + O_{t+1} = D_{t+1}. - FX mismatch: - Fraction λ ∈ (0,1) of global financiers are owned by domestic households; remaining (1 − λ) owned by foreigners. - Representative household effectively faces that a fraction λ of its external debt is in FX (repayments follow the dollar interest rate), while (1 − λ) of external debt is effectively in local currency (repaid at domestic policy rate). - The FX mismatch parameter λ determines macroeconomic responses to shocks. ### Lump-sum transfers and rebates - Planner rebates in lump sum revenues from taxes on labor, capital inflows, household debt, and carry profits from FX intervention: - T_t = φ W_t N_t + φ_{t−1} (1 + i_{t−1}) (Q_t + F_t) + θ_{HH,t−1} (1 + ρ_{t−1}) D_HHt + O_t [ (1 + i_{t−1}) − (1 + i^*_{t−1}) E_t / E_{t−1} ]. *Source: wpiea2023161-print-pdf - 2.1 Model Overview* ### 2.3 Planner Problem ### 2.3 Planner Problem ### Competitive equilibrium (definition and objects) - A competitive equilibrium is a set of quantities - {C_Ht, C_Ft, C_Rt, N_t, k^Linear_t, k^Concave_t, Y_Ht, Y_Xt, Y^Linear_Rt, Y^Concave_Rt, Q_{t+1}, D_{t+1}}_{t=0}^2 - and prices - {P_H, P_X, {ρ_t}_{t=0}^1, {W_t, E_t, P_Rt, q_t}_{t=0}^2} - that satisfy optimization conditions and constraints of households, tradable-sector firms, housing-sector firms, domestic banks, and global financiers, and market clearing for tradable goods, housing services, labor, land, and local currency bonds, taking as given dollar values of all initial and terminal debt stocks and the terminal land price, and policy instruments {i_t, φ_t, θ^{HH}_t, θ^{Linear}_{Rt}, FXI_t}_{t=0}^1. - The set of equations characterizing the competitive equilibrium is provided in subsection 2.2 and in appendix A.1. ### Four wedges summarizing deviations from first-best - The paper defines four wedges that summarize distance from the frictionless first-best frontier and identifies key externalities associated with each wedge. The analysis mainly focuses on the first three wedges. - AD wedge (home consumption) - τ_{Ht} = 1 + (1/A_t) (U_{Nt}/U_{Ht}) = 1 − (1/A_t) C_{Ht}^{α_H}. - Interpretation: Positive if pre-set domestic price P_H is inappropriately high (domestic demand for home-produced tradable goods is excessively low relative to cost of production). - Externalities: - AD externality: households do not internalize impact of consumption on time path of aggregate demand, which determines appropriateness of pre-set price P_H. - Pecuniary AD externality: households do not internalize impact of their decisions on exchange rate E_t which affects domestic banks’ external borrowing constraint. - UIP wedge (external premium on local currency bonds) - τ_{Γ,t+1} = (1−λ) E_t[ [η_{t+1} − (1 + i^*_t)]^{α_F} C_{Ft+1} ], where η_{t+1} = (1−φ_t) (1 + i_t) (E_t / E_{t+1}). - Applies only if some global financiers are foreign-owned, i.e., λ < 1. - Interpretation: Positive wedge implies net reduction in welfare due to transfer of resources to foreign-owned intermediaries. - Externalities: - Local currency premium externality when FX market is shallow (Γ>0): premium η_{t+1} is endogenous to external debt, and households do not internalize that their borrowing affects the external premium. - Planner behavior: Given FX market shallowness and endogenous premium, the planner may optimally set the wedge at non-zero level. - Housing wedge - τ_{Rt} = [1 − G′(1 − k^Linear_{t−1})]^{α_R} C_{Rt}. - Interpretation: Positive if land usage shifts from the linear to the concave subsector; production of housing services is maximized when linear subsector uses all land, but may be reduced by macroprudential taxes and/or binding borrowing constraints. - Externality: - Pecuniary production externality: housing firms do not internalize impact of land usage in periods 0 and 1 on land price q_1, which enters their borrowing constraint. - Terms of trade wedge (export production) - τ_{Xt} = (1 − 1/γ) + (1/P_X) (1/A_t) (U_{Nt}/U_{Ft}) = − (1/P_X) (1/A_t) C_{Ft}^{α_F}, where γ ≡ (1/P_X) Y_{Xt} dY_{Xt}/d(1/P_X). - Interpretation: Arises from stickiness of price of exported tradable goods; standard New Keynesian terms-of-trade externality. - Note: Under unit elastic export demand, 1 − 1/γ is zero and the wedge is always negative. The planner ignores this externality in defining the constrained efficient allocation. ### Constrained efficient allocation (definition and planner problem) - Definition: A constrained efficient allocation is a set of quantities and prices (same lists as for competitive equilibrium) which maximizes household welfare under full commitment, subject to the restriction that the allocation can be implemented via policy instruments {i_t, φ_t, FXI_t, θ^{HH}_t, θ^{Linear}_{Rt}}_{t=0}^1 in a competitive equilibrium, and the additional constraint that the planner ignores impact of policies on pre-set export price P_X. - Ramsey planner problem (integrated frictions): - V_P = max_{C_{Ft}, P_H, E_t, η_{t+1}, FXI_t, k^Linear_{t−1}} E_0[ ∑_{t=0}^2 β^t V(C_{Ft}, E_t P_H, k^Linear_{t−1}, A_t) ] - Subject to constraints (all written in dollar terms), including resource constraints, external borrowing constraints, Gamma equations, domestic housing sector borrowing constraints, and equilibrium conditions across states s ∈ {L, H}. ### Key constraints and equilibrium conditions (as in text) - Resource constraint (equation (2)) — one for each state s ∈ {L, H}: - (1 + i^*_{−1}) B_0 = [C^* − C_{F0}] + [C^* − C_{F1}] − (1−λ) FXI_0 [η_1 − (1 + i^*_0)] I_0 - + [C^* − C_{F2}] − (1−λ) FXI_1 [η_2 − (1 + i^*_1)] I_0 I_1 - Occasionally-binding external borrowing constraint (equation (3)) — one per state s ∈ {L, H}: - B_1 / I_0 + [C_{F1} − C^*] + (1−λ) FXI_0 [η_1 − (1 + i^*_0)] ≤ κ_H P_H / E_1 - Gamma equations (equations (4)-(5)) — always-binding: - Γ (B_1 + FXI_0) = E_0[η_1 − (1 + i^*_0)] - Γ (B_2 + FXI_1 − S_1) = η_2 − (1 + i^*_1) for s ∈ {L, H} - Domestic housing sector borrowing constraint (equation (6)) — occasionally-binding, one per state: - χ_1[ (1 + i^*_{−1}) B^Linear_{R0} − \hat P_{R0} ] + χ_1 \hat q_0 (k^Linear_0 − 1) − \hat P_{R1} k^Linear_0 + \hat q_1 (k^Linear_1 − k^Linear_0) ≤ κ_q \hat q_1 k^Linear_1 for s ∈ {L, H} - Equalization condition across period-1 states (equation (7)): - E_1 η_1 is equalized across period-1 states s ∈ {L, H}. - Notation and fixes: - B_{t+1} ≡ D_{t+1} / E_t is economy-wide dollar debt at end of period t. - I_t ≡ λ (1 + i^*_t) + (1 − λ) η_{t+1} is representative household’s effective interest rate on external borrowing between t and t+1. - χ_{t+1} ≡ (1 + ρ_t) (E_t / E_{t+1}); \hat P_{Rt} ≡ P_{Rt} / E_t; \hat q_t ≡ q_t / E_t. - Fixes: (1 + i^*_{−1}) B^Linear_{R0} = − (1 + i^*_{−1}) B^Concave_{R0}; B^Linear_{R3} = B^Concave_{R3} = 0; k^Linear_{−1} = 1; \hat q_2 = 0. - The dollar value of initial economy-wide debt repayments is fixed at (1 + i^*_{−1}) B_0, and final debt B_3 = 0. - If λ < 1, there may be carry profits or losses from FX intervention. ### Shock transmission, supply curve, and policy implications - In absence of shocks: - Consumption is flat over time and across states; external debt decreases smoothly from B_0 to 0. - Shocks of interest (productivity shocks, world interest rate shocks, taper tantrums, sudden stops) may alter tightness of multiple constraints, affecting allocations and welfare via changes in multiple externalities. - Transmission highlights: - Productivity shocks affect indirect utility V_P via the AD wedge. - An increase in the world interest rate shifts up the supply-of-funds curve irrespective of FX market depth and changes the fundamental cost of external borrowing in the resource constraint (2) and Gamma equations (4)-(5). - The supply curve has slope (1−λ) Γ, which is positive only if λ < 1 and Γ > 0. - A non-fundamental taper tantrum shifts up the supply curve only if (1−λ) Γ > 0. Unlike the world interest rate shock, the taper tantrum’s impact on carry profits/losses and external borrowing costs is manipulable by the planner (see equation (5)). - A sudden stop shifts leftward the external borrowing limit of the supply curve; magnitude depends on effective FX mismatch and exchange rate depreciation (see equation (3)). - All shocks that alter financing conditions cause volatility in domestic asset prices, and the housing constraint (6) may bind in some circumstances. - Policy use: - The integrated model can be used to analyze trade-offs and interactions among externalities and policy tools. - The following sections of the paper explore general trade-offs at the constrained efficient allocation and establish conditions under which traditional policy prescriptions (monetary policy and exchange rate flexibility) can achieve that allocation, and when additional instruments (capital inflow taxes, FX intervention, domestic debt taxes) may be warranted. *Italic: Source: wpiea2023161-print-pdf - 2.3 Planner Problem* ### 3.1 General Conditions ### wpiea2023161-print-pdf - 3.1 General Conditions ### First-order conditions, wedges, and exchange rate flexibility - Planner problem solved via the first order approach (Farhi and Werning (2016) method); all instruments assumed available and borrowing constraints, if binding, bind only in the L state. - Exchange rate flexibility eliminates AD wedges in periods with no shocks: - τ_H0 = τ_H2 = 0. (Equation (8)) - With given import consumption C_Ft, policy rate and exchange rate flexibility can set home-produced tradable consumption C_Ht = (α_H/α_F) E_t C_Ft to eliminate the AD externality in initial and terminal periods. - In period 1 (when shocks strike), exchange rate flexibility can generate or reduce other macro distortions. Combining FOCs for E_1 and η_1 yields the period-1 tradeoff (Equation (9)): - τ_H1 = Ψ_B κ_H β I_0 α_H E_1 − (1−λ) (B_1 + FXI_0) η_1 α_H [ z_1 − E_0[z_1 η_1] / E_0 η_1 ] + y_E1 - Interpretation of terms in (9): - First term: pecuniary AD externality; Ψ_B is multiplier on period-1 external borrowing constraint. If constraint binds, depreciation tightens external constraint and reduces C_F1. - Second term: impact of depreciation on external debt repayments and on dollar wealth; z_t is marginal value of a dollar and may vary across states. - Square-bracketed term [ z_1 − E_0[z_1 η_1] / E_0 η_1 ] implies depreciation in one state must be offset by appreciation in the other to satisfy repayment commitments and global financiers’ period-0 expected returns. - Third and fourth terms: shallow-market friction (if (1−λ) Γ > 0) and FX market shallowness affecting land prices and the housing constraint through y_ηt; y_E1 captures easing of effective dollar interest rate for housing via depreciation when domestic housing borrowing constraint binds. ### Marginal value of a dollar and related FOCs - Marginal value definitions (Equation (10)): - z_1 = 1/(β I_0) [ Φ + Ψ_B + Φ_I1 (1−λ) Γ (B_2 + FXI_1) + Γ I_0 y_η2 ] - z_2 = Φ / (β^2 I_0 I_1) - z_t components: - Φ: multiplier on the resource constraint (marginal value of relaxing the resource constraint). - z_1 additionally includes marginal values of relaxing the external borrowing constraint, the Gamma equation, and the housing constraint. - FOCs for import consumption {C_Ft}_t=0^2 (Equations (11)-(12)): - α_F C_F0 = β E_0[ I_0 z_1 ] + β (1−λ) Γ (B_1 + FXI_0) E_0[z_1 η_1] / E_0 η_1 + y_F01 + (α_H/α_F) τ_H0 (Equation (11)) - α_F C_Ft = z_t + y_Ft1 + (α_H/α_F) τ_Ht for t ∈ {1,2}. (Equation (12)) - Interpretation: - In periods 1 and 2, marginal utility from import consumption balanced against z_t, AD wedge τ_Ht, and y_Ft (non-zero only if housing sector constraint binds). - In period 0, marginal utility from imports balanced against expected period-1 dollar marginal value, period-1 temptation to depreciate (to reduce repayments on period-0 external debt), and impact on period-1 housing constraint. ### Capital inflow taxes and household Euler comparisons - Constrained-efficient ex ante and ex post capital inflow taxes derived by comparing FOCs for C_Ft against household Euler conditions (Equations (13)-(14)): - (1−φ_0) β E_0[ I_0 z_1 ] + β (1−λ) Γ (B_1 + FXI_0) E_0[z_1 η_1] / E_0 η_1 + y_F01 + (α_H/α_F) τ_H0 = β E_0[ η_1 z_1 + y_F11 + (α_H/α_F) τ_H1 ] (Equation (13)) - (1−φ_1) z_1 + y_F11 + (α_H/α_F) τ_H1 = β η_2 z_2 + y_F21 + (α_H/α_F) τ_H2. (Equation (14)) - Rationale for capital inflow taxes: - Households do not internalize externalities generating wedges; macroprudential taxes may be needed to stabilize AD wedges over time. - Inflows taxes may also address externalities related to local currency debt’s role in marginal dollar value, expected temptation to depreciate period-0 local-currency external debt, and pecuniary production externality in housing. ### FX intervention FOCs and role of FXI - Constrained-efficient ex ante and ex post FX intervention conditions combine FOCs for {FXI_t, η_{t+1}} (Equations (15)-(16)): - (1−λ)[ E_0[ η_1 − (1 + i^*_0) ] E_0[z_1 η_1] / E_0 η_1 + E_0[z_1 { η_1 − (1 + i^*_0) } ] ] + Γ E_0{ y_η1 η_1 } / E_0 η_1 = 0. (Equation (15)) - (1−λ)[ [ η_2 − (1 + i^*_1) ] + Γ S_1 / 2 ] + Γ y_η2 I_0 I_1 2 Φ = 0. (Equation (16)) - Interpretation: - FX intervention allows the planner to change external debt position without altering consumption; smoothing external premia is a primary justification for FX intervention. - Equation (15): set FXI_0 so external debt at end of period-0 allows appropriate dollar wealth transfers across period-1 states. - Equation (16): set FXI_1 so premium between periods 1 and 2 equals world interest rate unless non-fundamental foreign appetite shock generates carry profit/loss or housing constraint binds. ### Housing debt taxes and wedges - Constrained-efficient ex ante and ex post housing debt taxes from FOCs for {k_Linear_t}_t=0^1 (Equations (17)-(18)): - τ_R1 = Ψ_L R π_L1 / β { [ χ_L1 ȷq_0 − ȷP_LR1 − ȷq_L1 ] + ∂(χ_L1 ȷq_0)/∂k_Linear0 ( k_Linear0 − 1 ) − ∂ȷP_LR1/∂k_Linear0 k_Linear0 } (Equation (17)) - τ_R2 = Ψ_R / β^2 { [ ȷq_1 + ∂ȷq_1/∂k_Linear1 ( k_Linear1 − k_Linear0 ) ] − κ_q [ ȷq_1 + ∂ȷq_1/∂k_Linear1 k_Linear1 ] + ∂(χ_1 ȷq_0)/∂k_Linear1 ( k_Linear0 − 1 ) } (Equation (18)) - Sign properties established: - { ∂(χ_1 ȷq_0)/∂k_Linear0, ∂ȷq_1/∂k_Linear1, ∂(χ_1 ȷq_0)/∂k_Linear1 } > 0 - ∂ȷP_LR1/∂k_Linear0 = 0 at k_Linear0 = k_Linear1 = 1 - ∂ȷP_LR1/∂k_Linear0 < 0 for k_Linear0 < 1 - Interpretation: - τ_R2 > 0 in period-1 L state if k_Linear1 < 1 (linear housing subsector borrowing-constrained); the pecuniary production externality ∂ȷq_1/∂k_Linear1 > 0 tightens borrowing constraint. - Ex ante housing debt taxes may be effective where ex post taxes are ineffective when constraint binds; hedging motive and effects on period-0 land price and period-1 rents discussed. - Note: if housing constraint not binding ex post, constrained-efficient allocation of remainder unaffected by presence of housing sector. ### Constrained-efficient policy mix and the traditional prescription - The FOCs lead to conclusions on constrained-efficient policy mix across shocks and country characteristics. - Traditional prescription: only policy rate and exchange rate flexibility. - Proposition 1 (Traditional prescription) — constrained-efficient allocation achievable with only policy rate and exchange rate flexibility if all of the following hold (conditions (19)-(23)): - (i) Capital inflow taxes are zero when: - β E_0[ I_0 z_1 ] + β (1−λ) Γ (B_1 + FXI_0) E_0[z_1 η_1] / E_0 η_1 + y_F01 + (α_H/α_F) τ_H0 = β E_0[ η_1 z_1 + y_F11 + (α_H/α_F) τ_H1 ]. (Equation (19)) - z_1 + y_F11 + (α_H/α_F) τ_H1 = β η_2 z_2 + y_F21 + (α_H/α_F) τ_H2 for each state {L,H}. (Equation (20)) - (ii) FX intervention can be set to zero when: - (1−λ) [ Γ B_1 E_0[z_1 η_1] / E_0 η_1 + E_0[ z_1 { η_1 − (1 + i^*_0) } ] ] = 0 at FXI_0 = 0. (Equation (21)) - τ_Γ2 = α_F C_F2 / ( (1−λ) Γ (B_2 − S_1) ) for each state {L,H}. (Equation (22)) - (iii) Housing debt taxes are zero when τ_R1 = 0 and τ_R2 = 0 for each state {L,H}. (Equation (23)) - Interpretation of conditions: - (19)-(20): ensure ex ante and ex post capital inflow taxes unnecessary by balancing AD wedges, UIP temptation, and housing wedge incentives over time. - (21)-(22): ensure ex ante and ex post FX intervention unnecessary if other tools can achieve constrained-efficient UIP wedge and premia outcomes. - (23): ensures no need for housing debt taxes when housing constraint not binding ex post. ### Examples of shocks where traditional prescription suffices - Assumptions for examples: B_0 = 0 and { κ_H, κ_q } sufficiently large so constraints (3) and (6) are slack. - Lemma 1 (Productivity shock: A_1). Suppose λ ∈ [0,1] and Γ ∈ [0,∞). Policy rate and exchange rate flexibility suffice and are set as: - (1 + i_0) = 1 / ( β A_0 A_L1 A_H1 π_H1 A_L1 + π_L1 A_H1 ), (1 + i_1) = 1 / β, and E_t = α_F A_t C^* for t ∈ {0,1,2}. - Import consumption stabilized at C^*. - Period-1 dollar price of land does not vary across states. - AD, UIP, and housing wedges stabilized: { { τ_Ht }_{t=0}^2, { τ_Γt }_{t=1}^2, { τ_Rt }_{t=1}^2 } = 0. - Terms-of-trade wedges not stabilized: τ_Xt = − 1/α_F P_X1 A_t C^*. - Key insight: irrespective of effective FX mismatch and FX market depth, traditional prescription manages productivity shocks; exchange rate depreciates proportionally to the shock and planner stabilizes AD wedge but cannot stabilize terms-of-trade wedge. - Lemma 2 (World interest rate shock: i^*_1). Suppose λ = 1 and Γ = 0. Policy rate, exchange rate flexibility, and ex ante capital controls suffice and are set as: - (1 + i_0) = E_0[ η_1 C_F0 (C_F1)^2 ] / ( β E_0[ η_1 C_F1 ] ), (1 + i_1) = 1 / β, and E_t = α_F A C_Ft for t ∈ {0,1,2}. - φ_0 = 1 − β E_0[ η_1 C_F0 / C_F1 ]. - Consumption of imports follows 1 / C_F0 = E_0[ 1 / C_F1 ] and C_F2 / C_F1 = β (1 + i^*_1). - Period-1 dollar price of land follows ȷq_1 = β α_R / α_F C_F1. - AD, UIP, and housing wedges stabilized: { { τ_Ht }_{t=0}^2, { τ_Γt }_{t=1}^2, { τ_Rt }_{t=1}^2 } = 0. - Terms-of-trade wedges not stabilized: τ_Xt = − 1/α_F P_X1 A C_Ft. - Key insight: with full FX mismatch (λ = 1) and deep FX markets (Γ = 0), world interest rate shocks can be handled using ex ante capital inflow taxes plus the traditional prescription ex post; ex post capital inflow taxes do not vary with world interest rate in this framework. ### When traditional prescription is insufficient - The paper previews that for destabilizing foreign appetite shocks (surges and taper tantrums) affecting local currency debt, capital inflow taxes and FX intervention should be used jointly ex post, and instead of the policy rate and exchange rate flexibility, especially for EMDEs with foreign participation and when FX market depth and borrowing constraints interact with shocks. - The analysis in following sections focuses on foreign appetite shocks, their interaction with FX market depth, abstracting from external and domestic borrowing constraints in the initial analysis. *Source: wpiea2023161-print-pdf - 3.1 General Conditions* ### 4.1 Externalities and Policy Tools ### 4.1 Externalities and Policy Tools ### Setup and assumptions - Consider non-fundamental local currency “surges”, i.e., S_H1 > 0, and “taper tantrums”, i.e., S_L1 < 0. - Assume B0 = 0 and that {κ_H, κ_q} are sufficiently large so constraints (3) and (6) are slack (pecuniary AD and pecuniary production externalities not salient). - Domestic debt taxes are not needed: Ψ_B = {τ_Rt}2 t=1 = {θ_Rt}1 t=0 = {θ_HHt}1 t=0 = 0. - Relevant externalities for the planner: AD and local currency premium externalities. - Policy tool set considered: FX intervention, capital inflow taxes, policy rate, and exchange rate flexibility. ### Role of (1−λ) Γ and foreign appetite shocks - Impact of foreign appetite shocks depends on whether (1−λ) Γ is zero or positive: - If (1−λ) Γ = 0: - Supply of external funds is perfectly elastic up to the external debt limit; foreign appetite shocks do not shift the supply curve. - If (1−λ) Γ > 0: - Supply curve is upward-sloping up to the external debt limit; foreign appetite shocks horizontally shift the upward-sloping portion of the supply curve. - Practical interpretation: foreign appetite shocks ≈ hot money flows into/out of local currency government debt; central banks can use FX reserves (at opportunity cost of the dollar interest rate) to intermediate without necessarily exhausting reserves. Capital inflow controls can be imposed at the border as taxes or subsidies. ### Perfect stabilization result (special cases and proposition) - Lemma 3 (Elastic external supply). Suppose (1−λ) Γ = 0. Irrespective of foreign appetite shocks: - AD and UIP wedges are stabilized: {τ_Ht}2 t=0 , {τ_Γt}2 t=1 = 0. - Imports stabilized: {C_Ft}2 t=0 = C*. - FX intervention indeterminate and can be set to zero: {FXI_t}1 t=0 = 0. - Capital inflow taxes are zero except for {λ = 1, Γ > 0}, in which case set {φ_0 = 0, φ_1 = β Γ S1 }. - Policy rate and exchange rate do not vary across states: {(1 + i_t) = 1/β}1 t=0 and {E_t}2 t=0 = α_F A C_F. - If Γ = 0 (deep FX markets), foreign appetite shocks and FX intervention have no macroeconomic impact; FX intervention indeterminate and can be set to zero. - If λ = 1 and Γ > 0 (full effective FX mismatch with shallow FX markets), foreign appetite shock becomes domestically internalized; FX intervention indeterminate; capital inflow taxes become domestic financial instruments addressing the non-fundamental disruption. - Proposition 2 (Perfect stabilization). Suppose (1−λ) Γ > 0 and S_H1 = −S_L1 > 0. Then: - AD wedges and imports are stabilized, but UIP wedges between periods 1 and 2 are not. - FX intervention and capital inflow taxes are jointly used ex post while policy rate and exchange rate do not vary across states. - Ex ante capital inflow taxes are set to zero. - Policy/control values: - τ_Ht = 0 for t ∈ {0,1,2} - τ_Γ1 = 0 and τ_Γ2 = − (1−λ) Γ S1 / (2 α_F C_F) - φ_0 = 0 and φ_1 = β Γ S1 / 2 - FXI_0 = −B1 and FXI_1 = S1 / 2 − B2 - (1 + i_t) = 1/β for t ∈ {0,1}, and E_t = α_F A C_F for t ∈ {0,1,2} - Definitions used: - B1 = (1−λ) Γ (S1)^2 / 4 (1 + 1/β + 1/β^2) - B2 = (1 + 1/β) B1 - C_Ft = C_F = C* + B1 for t ∈ {0,1,2} - Mechanism and intuition: - Planner stabilizes AD wedge and manipulates UIP wedge to expand resource constraint at expense of non-optimizing global financiers. - During inflow shocks: planner accumulates FX reserves and imposes an inflow tax. - During outflow shocks: planner borrows FX reserves and imposes an inflow subsidy. - Joint revenues from inflow taxes and carry profits from FXI augment the resource constraint in each state by 1/4 (1−λ) Γ (S1)^2. - Ex ante FX intervention fully absorbs period-0 debt position: (B1 + FXI_0) = 0, yielding ex ante capital inflow tax = 0. - FX intervention and capital inflow taxes are distinct costly tools that together manage premium externality ex post; once UIP wedges are set, policy rate need not move across states. ### Missing tools and imperfect stabilization - Corollary 1 (Missing tools). Suppose (1−λ) Γ > 0. If either FX intervention or capital inflow taxes are not available ex post: - There is imperfect stabilization of AD wedges, import consumption, policy rates, and exchange rates across period-1 states and over time. - Key implications when tools absent: - Policy rate cannot substitute for missing tool because it addresses a different margin. - Without capital inflow taxes: external premium becomes connected to policy rate; FX intervention cannot stabilize AD wedges without eliminating carry profits. - Without FX intervention: economy-wide external debt position connects to import consumption; capital inflow taxes cannot earn revenues from non-optimizing financiers without distorting imports and AD wedges. - Comparative notes: - If only FX intervention available: relates to Cavallino (2019) and Fanelli and Straub (2021). - If only capital inflow taxes available: relates to Bianchi and Lorenzoni (2022) (they shock Γ instead and do not permit local currency external borrowing). - Policy takeaway: when shallow-market friction leaves a country vulnerable to foreign appetite component of global cycle, both FX intervention and capital inflow taxes should be used jointly ex post. ### Sign of the ex ante capital inflow tax under asymmetric shocks - Moving from symmetric to asymmetric shocks (S_H1 / ||S_L1|| ≠ 1) affects ex ante capital inflow tax and ex post policy mix. - When ex post use of FX intervention and capital inflow taxes permitted: - Tools continue to follow equations (14) and (16) and remain jointly used. - z_1 varies across period-1 states, so policy rate and exchange rates vary across states (but are not main tools). - Ex ante capital inflow tax deviates from zero as soon as ex post stabilization is imperfect (but remains small relative to ex post tax). - When FX intervention is not permitted: - Ex ante capital inflow tax sign depends on expected relative sizes of taper tantrum vs inflow surge: - Ex ante capital inflow tax is positive if expected taper tantrum is large relative to expected inflow surge. - Ex ante capital inflow tax is negative if expected inflow surge is large relative to expected taper tantrum. - Intuition: constrained efficient level of period-0 debt depends on expected period-1 cost of external financing (and thus local currency premium externalities); expectations vary with country conditions and stage in global financial cycle. - Takeaway: joint ex post use of capital inflow taxes and FX intervention is robust when shocks are non-fundamental foreign appetite into local currency external debt; sign of ex ante capital inflow tax depends on expectations about future capital flows and availability of FX intervention. ### Sudden stop risks: externalities, tools, and ex post destabilization (link to section 5) - Sudden stop shock defined as reduction of κ_H in period-1 L state that makes external borrowing constraint (3) bind, with Ψ_B^L > 0. - Assume B0 > 0 and, absent shock, B0 > B1 > B2 > B3 = 0. Shock alters {B1, B^L2, B^H2}, maintains inequalities, and causes C^L_F1 < C^H_F1. - Assume κ_q sufficiently large so housing constraint (6) slack; pecuniary production externalities not salient and housing debt taxes not needed: {τ_Rt}2 t=1 = {θ_Rt}1 t=0 = {θ_HHt}1 t=0 = 0. - Relevant externalities: AD, pecuniary AD, and local currency premium externalities. - Tool set for sudden stop analysis: capital inflow taxes, policy rate, exchange rate flexibility; exclude FX intervention ({FXI_t}1 t=0 = 0) since central bank typically cannot absorb private-sector external debt in sudden stops. - Ex post capital inflow taxes redundant in period-1 L state because binding constraint implies domestic borrowing rate must exceed policy rate to prevent households' debt exceeding external debt limit. Therefore anchor policy rate and set φ^L1 = 0: - (1 + i^L1) = η^L2 E^L2 / E^L1 and φ^L1 = 0. - Effect of sudden stop on supply curve depends on (1−λ) Γ as before: - If (1−λ) Γ = 0: sudden stop causes no change in external premium. - If (1−λ) Γ > 0: reduction in external debt reduces the premium. - Lemma 4 (Marginal value of a dollar). The marginal value of a dollar satisfies: - z^L1 > E_0[z_1 η_1] / E_0 η_1 > z^H1. - Interpretation: import consumption declines in period-1 L state, requiring exchange rate depreciation to eliminate AD externality, balanced against other externalities. - Signs for AD wedges in period-1 states (inserting lemma into equations (9) and (12)): - τ_H1 = Ψ_B κ_H α_H β I_0 / E_1 − (1−λ) B1 / η_1 α_H [ z_1 − E_0[z_1 η_1] / E_0 η_1 ] = { τ_LH1 > 0, τ_HH1 = 0 if λ = 1 τ_LH1 S0, τ_HH1 > 0 if λ ∈ [0,1). } - If λ = 1 (full effective FX mismatch): exchange rate movements do not change FX value of external debt repayments; AD wedge positive in period-1 L state, zero in H state. - If λ < 1: second term non-zero; policy rate loosened to depreciate exchange rate more in L state, reducing AD wedge and making sign ambiguous; if λ low, depreciation motive to reduce FX value of local currency external debt can dominate pecuniary AD externality and AD wedge can become negative in L state; planner appreciates exchange rate in H state so AD wedge unambiguously above zero there. - Assumption 1 (Ex post depreciation). At t = 1, exchange rate is more depreciated in L state than in H state: E^L1 > E^H1 ⇔ η^L1 < η^H1. - Combining Lemma 4 and Assumption 1 implies local currency premium lower precisely when marginal value of a dollar is higher: - Cov(z_1, η_1) < 0. *Source: wpiea2023161-print-pdf - 4.1 Externalities and Policy Tools* ### 5.3 Case for an Ex Ante Capital Inƒow Tax ### 5.3 Case for an Ex Ante Capital Inflow Tax ### Special case: full effective FX mismatch and deep FX markets (Proposition 3) - Parameterization: {λ = 1, Γ = 0}. - Optimal policy under these assumptions: - Ex ante capital inflow tax is positive: φ0 > 0. - Ex post capital inflow taxes are zero: φ1 = 0. - AD wedges: {τLH1 > 0, τHH1 = 0, {τHt}t=0,2 = 0}. - UIP wedges: {τΓt}2t=1 = 0. - Intuition: - No local currency premium externalities when Γ = 0. - Positive AD wedge in period-1 L state → planner imposes φ0 to induce households to consume and borrow less in period 0 and shift demand into period 1. - Relation to literature: accords with subsection 5.3 of Farhi and Werning (2016) who assume full FX mismatch and deep FX markets. ### Welfare effects of changing FX mismatch (Lemma 5) - Starting at a constrained efficient allocation, marginal change in λ gives planner-valued welfare change: - βB1 Cov(z1, η1) + Γ { β(B1)2 E0 z1 + β2 E0 [ z2 (B2)2 ] } (equation (25)) - Interpretation: - If FX markets are deep (Γ = 0), only the first term applies; it is negative (given Assumption 1: Cov(z1, η1) < 0), so planner prefers lower λ. - Reason: more debt effectively in local currency reduces FX value of repayments in period-1 L state → relaxes constraints in that state. - If FX markets are shallow (Γ > 0), the second term applies and is positive; average local currency premium can exceed world rate, so shifting debt into local currency can increase average FX repayments and hurt welfare → expression (25) ambiguous. ### Perturbation approach to isolate effects (Perturbation 1) - Define a marginal reduction in λ while the planner reoptimizes policy tools subject to: - Exchange rates {EL1, EH1} held fixed and {FXIt}1t=0 = 0. - Rationale: - Fixing exchange rates ensures increases in import consumption in period-1 L state feed directly into reduction in AD wedge, removing equation (9) from consideration. - Keeps right-hand side of borrowing constraint (3) unchanged (so applicable for a range of borrowing-constraint formulations). ### Deep FX markets result (Proposition 4) - Assumption: Γ = 0. - After Perturbation 1, planner achieves a preferred allocation and sets a lower ex ante capital inflow tax φ0 provided sufficient condition (26) holds: - IL0 (CLF1)2 [ β (CHF2)2 (CHF1)2 (1 + αH/αF) + 1 ] − IH0 (CHF1)2 > 0. (equation (26)) - Interpretation and intuition: - Setting Γ = 0 implies ηL1 < (1 + i*0) < ηH1. - Shifting more external debt into local currency reduces FX repayments in L state and increases them in H state. - Relaxation in L state allocated to higher import consumption → lower AD wedge in L state (by Perturbation 1). - Planner lowers φ0 if marginal-utility-weighted reduction in AD wedge in L offsets weighted increase in H (smoothing mitigates H-state impact). - Condition (26) holds trivially at λ = 1 because IL0 = IH0 and CLF1 is pushed down by sudden-stop shock relative to CHF1. - Condition may hold for all λ ∈ [0,1] or be violated for low λ. ### Shallow FX markets result (Proposition 5) - Assumption: Γ > 0. - After Perturbation 1, planner achieves a preferred allocation provided condition (27) holds: - B1 [ zL1 πL1 [ (1 + i*0) − ηL1 ] − zH1 [ πL1 [ (1 + i*0) − ηL1 ] + Γ B1 ] ] > [ β πL1 αF CLF2 Γ (κH EL1)2 + β πH1 αF CHF2 Γ (BH2)2 ]. (equation (27)) - Planner sets a lower ex ante capital inflow tax φ0 provided sufficient conditions (28)–(31) all hold: - Condition (28): - πL1 [ (1 + i*0) − ηL1 ] 1 (CLF1)2 [ IL0 + (1 − λ) Γ B1 ηL1 E0 η1 ] ω1 − [ πL1 [ (1 + i*0) − ηL1 ] + Γ B1 ] 1 (CHF1)2 [ IH0 + (1 − λ) Γ B1 ηH1 E0 η1 ] > 0. (equation (28)) - Condition (29): - zL1 πL1 [ (1 + i*0) − ηL1 ] − zH1 [ πL1 [ (1 + i*0) − ηL1 ] + Γ B1 ] − Γ B1 E0 [ z1 η1 ] / E0 η1 > 0. (equation (29)) - Condition (30): - Γ [ πL1 IL0 zL1 + πH1 IH0 zH1 + (1 − λ) Γ B1 E0 [ z1 η1 ] / E0 η1 − 2 (1 − λ) E0 [ z1 η1 ] ] ≥ 0. (equation (30)) - Condition (31): - Γ β BH2 [ RH1 BH2 + 2 CHF2 ] / ω2 ≥ 0, where ω1 = (CHF2)2 (CHF1)2 (1 + αH/αF) + β [ (RH1)2 + 2 (1 − λ) Γ CHF2 ] / β [ (RH1)2 + 2 (1 − λ) Γ CHF2 ], RH1 = 1/β + 2 (1 − λ) Γ BH2, and ω2 is defined in appendix B. (equation (31)) - Interpretation and comparative statics: - The set of sufficient conditions is larger and more easily violated than the deep-market condition (26). - As λ decreases (more debt effectively local-currency), UIP wedge and local-currency premium externality gain salience → planner may need φ0 to address this externality. - Condition (27) ensures redistribution of FX value of repayments from L to H is desirable because marginal value of dollar is higher in L; but Γ > 0 increases external repayments in both states, complicating the trade-off. - Conditions (28)–(31) capture: shallow-market amplification of repayments (28); amplified H-state repayments and depreciation temptation (29); households’ period-0 consumption vs. premium externality (30); and period-1 H-state borrowing/premia trade-offs via Gamma equation (5) (31). - Conditions are trivially satisfied when Γ = 0; must be checked when Γ > 0. - If λ is high, households face higher expected local-currency borrowing rate and may consume too little (favoring smaller role for φ0). If λ is low, premium externality may dominate and reverse conclusions. ### Simulations and numerical insights (figures referenced) - Deep FX markets (Γ = 0), Figure 10: - As λ decreases: - AD wedge in period-1 L state decreases. - AD wedge in period-1 H state increases. - Ex ante capital inflow tax decreases. - Welfare increases. - Shallow FX markets (Γ > 0), Figure 11: - Starting at λ = 1, decreasing λ first decreases then increases φ0 as premium externalities become salient. - Welfare decreases as λ decreases. - Overall: simulations illustrate theoretical results that the usefulness of φ0 depends jointly on FX mismatch λ and FX market depth Γ. ### Comparison to related normative literature and policy implications - Contrast with other models: - Bianchi and Lorenzoni (2022) rationalize a capital inflow tax in a model with nominal rigidities and all external debt in FX. - This model differs by: occasionally-binding external borrowing constraint, inclusion of a domestic borrowing constraint, and imperfect FX mismatch where local-currency debt incurs a higher premium than FX debt (consistent with empirical literature). - Policy implications: - When FX markets are deep (Γ = 0), ex ante capital inflow taxes are effective and may be substituted by ex ante FX mismatch regulation (next subsection). - When FX markets are shallow (Γ > 0), reducing FX mismatch can exacerbate premium externalities and increase the planner’s need for φ0. - Thus, the role and optimal level of φ0 depend on joint state of λ and Γ; countries with shallow FX markets are more likely to retain FX mismatch and rely on ex ante capital inflow taxes. *Source: IMF working paper section 5.3 (wpiea2023161-print-pdf).* ### 6.1 Externalities and Policy Tools ### 6.1 Externalities and Policy Tools ### Key assumptions and setup - Assumption 2 (Positive housing debt): It is not possible for the linear housing subsector to repay all its inherited debt by the end of period 1: - 1 β [ ( 1 +i ∗ −1 ) B Linear R0 − ̂ P R0 ] − ̂ P R1 >0. - Relevant externalities the planner must handle: - AD externality - Pecuniary AD externality - Premium externality - Pecuniary production externality ### Equivalence and limits of policy tools (capital inflow taxes vs household debt taxes) - In the absence of a binding domestic borrowing constraint, capital inflow taxes φt and household debt taxes θHHt are perfect substitutes with the equivalence: - (1−φ0) αF CF0 = (1 +θHH0) βE0 [ η1 αF CF1 ] - (1−φ1) αF CF1 = (1 +θHH1) βη2 αF CF2 .(33) - Consequence: prior sections focused on capital inflow taxes and set household debt taxes to zero. - When the housing constraint binds in the period-1L state: - Capital inflow tax disconnects global financial conditions from all domestic borrowing interest rates (households and housing sector). - Without capital inflow tax, the domestic policy rate is equated to external returns. - Household debt tax can stabilize the borrowing rate for households, but land prices and returns in the housing sector remain tied to the policy rate because land is priced by the marginal productivity of the unregulated concave subsector. - Variation in land prices is welfare-irrelevant if housing constraint is slack; if it binds (ΨL R >0), welfare is affected and perfect substitutability breaks. - Separate housing debt tax θLinear R1 can attempt to alter borrowing rate for linear housing subsector, but is ineffective if the housing constraint binds. - If the external borrowing constraint also binds, the capital inflow tax φ1 becomes redundant; when both external and domestic borrowing constraints bind, neither the capital inflow tax nor the housing debt tax are effective. --- ### 6.2 Foreign Appetite Shocks and Housing Markets ### Setup and Proposition summary - Consider symmetric foreign appetite shocks (surges or taper tantrums). - Proposition 2 (from earlier subsection 4.2): for a country with shallow FX markets but no binding borrowing constraints, use capital inflow taxes and FX intervention jointly ex post, without need for ex ante capital inflow tax. - Proposition 7 (Substitutability). Suppose that (1−λ) Γ > 0 and S H1 = −S L1 > 0. There exists κq > 0 such that: - (i) For κq ∈ [κq, ∞), capital inflow taxes and household debt taxes are perfect substitutes in welfare terms, achieving zero housing wedges and housing debt taxes, i.e., - {{τRt}^2_{t=1}, {θLinear Rt}^1_{t=0}} = 0. - (ii) For κq ∈ [0, κq), capital inflow taxes and household debt taxes are not perfect substitutes. Use of capital inflow taxes achieves zero housing wedges and housing debt taxes. Use of household debt taxes is associated with a binding housing constraint, non-zero AD wedges, and violation of conditions (19) and (23) (except knife-edge case where yE1 is zero and {yFt}^1_{t=0} terms balance in condition (19)). ### Policy mixes and implications for housing prices and constraints - If land pledgability parameter κq is above threshold κq, FX intervention can be combined with household debt taxes instead of capital inflow taxes without welfare impact. - If planner uses capital inflow taxes alongside FX intervention as in Proposition 2: - All terms in housing constraint (6) identical across period-1 states: - dollar value of rents ̂PR1 = αR αF CF - housing sector returns {χ1 = χ2 = 1/β} - dollar price of land ̂q1 = β αR αF CF - Therefore, if housing constraint does not bind in period-1H state, it does not bind in L state either. - If planner replaces capital inflow taxes with household debt taxes and calibrates ex post policy rate to offer necessary external premia and varies ex post household debt tax to stabilize borrowing rate for households: - χ1 = (1 + i0) = 1/β and χ2 = (1 + i1) = 1/β − ΓS1/2 - θHH0 = 0 and θHH1 = βΓS1 / (2 − βΓS1) - Rents stabilized: ̂PR1 = αR αF CF - Housing sector repayments between periods 0 and 1, χ1, stabilized - Expected housing sector returns destabilized due to destabilized policy rate across period-1 states - Dollar price of land becomes: - ̂q1 = 2β αR αF CF / (2 − βΓS1) - Dollar price of land increases with inflow surge and decreases with taper tantrum. - If κq ∈ [κq, ∞): - Housing constraint does not bind in period-1L state; movement in domestic asset prices not associated with allocation changes; no additional policy tools beyond ex post FX intervention and household debt taxes required. Conditions (19) and (23) hold; no role for ex ante household or housing debt tax. - If κq ∈ [0, κq): - Capital inflow taxes and household debt taxes are not perfect substitutes. - Using capital inflow taxes with FX intervention stabilizes dollar price of land across states, ensuring housing constraint never binds. - Replacing capital inflow taxes with household debt taxes can reduce dollar price of land during taper tantrum sufficiently to make housing constraint (6) bind in period-1L state. - Binding housing constraint implies violation of condition (23) from Proposition 1. - When housing constraint binds, terms {yEt, yηt, yFt} enter FOCs for constrained efficient allocation; AD wedge destabilized across states as planner attempts to support land prices in period-1L by depreciating exchange rate and easing effective dollar interest rate for housing market in that state. - Condition (19) from Proposition 1 violated; an ex ante household debt tax becomes useful. - Characterization near threshold κq (illustration): - When housing constraint just binding (κq = κq, ΨL R = 0, {kLinear t = 1}^1_{t=0}), condition: - 1 − κq / κq ̂q1 > ∂̂q1 / ∂kLinear1 - required for term multiplying ΨR β^2 on RHS of equation (18) to be positive. - Term multiplying ΨL R πL1 β on RHS of equation (17) has same sign as hedging motive equal to deviation of dollar land price in period-1L state from its average: - [ χL1 ̂q0 − ̂PL R1 − ̂qL1 ] = E0 ̂q1 − ̂qL1 > 0. - For κq marginally below κq: - τR2 > 0 and τR1 > 0 - first inequality indicates kLinear,L1 < 1 - second indicates a positive ex ante housing debt tax ### Broader insight on multiple frictions and policy calibration - Constrained efficient level of ex ante macroprudential measures depends on: - Domestic considerations (e.g., Kiyotaki and Moore, 1997) - Global foreign appetite shocks and FX market depth (e.g., Gabaix and Maggiori, 2015; Cavallino, 2019; Fanelli and Straub, 2021) - Two tools—capital inflow taxes and household debt taxes—appear equally adept when only shallow-market friction considered, but diverge when domestic borrowing constraint risk exists because they differentially affect transmission of global financial conditions into domestic asset prices. - If domestic leverage is sufficiently high relative to collateral value, tools are no longer perfect substitutes in welfare terms; case for additional ex ante debt taxes arises. - Welfare impact of non-fundamental foreign appetite shocks should include cost of domestic credit crunch in addition to inflow tax revenues and carry profits/losses from shocks. --- ### 6.3 Sudden Stops and Housing Markets ### Mechanisms and ambiguous outcomes - A sudden stop shock that generates a binding external borrowing constraint (3) in period-1L state causes domestic borrowing rate to exceed policy rate in that state: - (1 + ρL1) > (1 + iL1) - Expected returns on housing jump in L state relative to H state, causing dollar price of land to be lower in L state: - χL2 > χH2 and ̂qL1 < ̂qH1 - Lower import consumption in L state reduces dollar value of rents: - ̂PL R1 < ̂PH R1 - Combined mechanisms make linear housing subsector’s borrowing constraint more likely to bind in period-1L state than in H state (similar to Caballero and Krishnamurthy, 2001). - Countervailing force: Assumption 1 states exchange rate depreciates when external borrowing constraint binds (consistent with EMDE experience). Depreciation generates expenditure-switching from import consumption to consumption of non-tradable housing services, increasing housing rents and land prices relative to past debt repayments, making linear housing subsector’s borrowing constraint less likely to bind in period-1L state than in H state. - Net effect ambiguous due to nominal rigidities. ### Policy implications and interaction with FX market characteristics - Severity of ex post sudden stop and constrained efficient ex post depreciation depends on effective FX mismatch λ and shallow-market friction Γ. - Policy reforms to improve FX market depth: - Enable direct reduction of premium externalities - By inducing planner to impose stricter ex ante FX mismatch regulations, indirectly reduce pecuniary AD externalities - If reducing FX mismatch results in more ex post depreciation in a sudden stop and/or a smaller ex post jump in domestic borrowing rate above policy rate, it could reduce pecuniary production externalities in housing sector - Actions to reduce external FX market frictions affect incidence and severity of frictions in domestic credit markets. --- ### Conclusion (section 7 summary) - Policy advice to small open economies must account for heterogeneous financial market characteristics and evolving transmission of external financial shocks due to market development and crises. - The welfare-optimizing framework with dominant currency pricing and multiple frictions shows constrained efficient policy mix depends on configuration of frictions. - Joint use of multiple policy tools may ease some frictions while exacerbating others; model provides integrated optimal policy guidance beyond existing literature. - Traditional prescription—monetary policy and exchange rate flexibility—remains appropriate for some shocks; for other shocks or friction configurations, recommended tools may include FX intervention, capital inflow taxes, and taxes on housing sector debt in addition to or instead of policy rate adjustments. - Existence of each friction can alter calibration of tools used to handle other frictions. - Practical translation requires careful judgments on: - Measurement of wedges - Credible communication about joint use of policy tools - Coordination between government agencies - Endogeneity of market development to tool usage - Spillovers to other countries *Source: wpiea2023161-print-pdf - 6.1 Externalities and Policy Tools* ### References ### wpiea2023161-print-pdf - References ### Online appendix A — Model details and planner FOCs - Model periods and key state indexing: periods t∈{0,1,2}; period-1 states s∈{L,H}. - Households: intratemporal conditions and Euler conditions presented for C_Ht, C_Ft, C_Rt, W_t; stochastic discount factor σ_t ≡ β^t α_F E_t C_Ft. - Tradable-sector firms: price-setting FOCs yield - P_H(j) = (1 + φ) ε/(ε−1) E_0[Σ_{t=0}^2 β^t 1/E_t C_Ft W_t A_t Y_Ht] / E_0[Σ_{t=0}^2 β^t 1/E_t C_Ft Y_Ht] - analogous expression for P_X(j); with identical firms implying P_H(j)=P_H, P_X(j)=P_X. - Relation determining P_X/P_H in terms of A_t, C_Ft and expectations across t=0,1,2 is given exactly in the appendix. - Housing sector: - Local currency debt dynamics for each subsector h∈{Linear,Concave} specified with exact expressions for D_{hRt+1} including terms (1+θ^h_{Rt-1})(1+ρ_{t-1})D^h_{Rt}, q_t L^h_t, P_{Rt} Y^h_{Rt}, T^h_{MPMt}, and Π^h_{Rt}. - Lump-sum transfer T^h_{MPMt} = θ^h_{Rt-1} (1+ρ_{t-1}) D^h_{Rt}. - Linear subsector FOCs include conditions linking expected prices and q_t; concave subsector FOCs involve G′(k^Concave_t). - Borrowing constraint for linear subsector binds when D^{Linear}_{R2} = κ_q q_1 k^{Linear}_1 (otherwise an inequality). - Global financiers: demand for local currency bonds from binding constraints yields Γ Q_{t+1} E_t = E_t[(1−φ_t)(1+i_t) E_t/E_{t+1} − (1+i^*_t)]; realized profit Π_{FIt+1} given accordingly. - Constrained efficient allocation: - Planner ignores impact of policy on P_X (P_X fixed) and indirect utility V(...) specified. - Partial derivatives relevant to planner: ∂V/∂C_Ft = α_F/C_Ft[1 + (α_H/α_F) τ_{Ht}], ∂V/∂(E_t P_H)= α_H/P_H E_t τ_{Ht}, ∂V/∂k^{Linear}_{t-1}= τ_{Rt}. - System of reduced equilibrium equations (2)-(7) defined in variables {C_Ft, P_H, E_t, η_{t+1}, FXI_t, k^{Linear}_t}. - Expressions for wages, outputs, price P_X, interest relationships, χ_{t+1}, and q_t provided in closed form across t∈{0,1,2}. - Policy instruments and occasionally-binding constraints: - Planner uses either capital inflow taxes φ_t (with θ_{HHt} ≡ 0) or household debt taxes θ_{HHt} (with φ_t ≡ 0). - Occasionally-binding constraints assumed slack in all periods/states except the period-1 L state, where they may bind. - If banks’ external borrowing constraint binds in the period-1 L state (Ψ^L_B > 0) then capital inflow taxes and household debt taxes are set to zero (ineffective/redundant). - If linear housing subsector’s domestic borrowing constraint binds in period-1 L state (Ψ^L_R > 0), housing debt taxes are set to zero; if it does not bind (Ψ^L_R = 0) housing debt taxes optimally zero as well. In period-1 H state, k^{Linear,H}_1 cannot exceed 1, so housing debt tax is zero. - Planner FOCs: - Full set of FOCs for {E_0, E_1^s, E_2}, {η_{t+1}}, {C_Ft}, {k^{Linear}_t}, {FXI_t}, and P_H are derived; P_H normalized to 1 after noting FOC redundancy. - FOC for E^s_1 (s∈{L,H}): τ^s_{H1} = Ψ^s_B κ^s_H β I^s_0 α_H E^s_1 + π^L_1 β π^s_1 α_H Ψ^L_R [E^s_1 ∂χ^L_1/∂E^s_1 ((1+i^*_{−1}) B^{Linear}_{R0} − ˆP_{R0}) + E^s_1 ∂(χ^L_1 ˆq_0)/∂E^s_1 (k^{Linear}_0 −1)] + Θ^s Λ^{E^s_1} η^s_1, with Θ^H = −1, Θ^L = 1. - FOCs for η^s_1 and η^s_2 produce linear conditions Ω_0 = β(1−λ)(B_1+FXI_0) z^s_1 + β y^s_{η1} + β y^s_{η1} η_1 + Θ^s Λ^{E^s_1} π^s_1 (and analogous for η^s_2) used to derive main-text equations. - FOCs for FXI_t explicitly: FXI_0: Γ Ω_0 = −β(1−λ) E_0[z_1 {η_1 − (1+i^*_0)}]; FXI^s_1: Γ Ω^s_1 = −Φ^s (1−λ) [η^s_2 − (1+i^*_1)] I_0/I^s_1. - Long-form housing sector derivatives and their dependence on whether capital inflow taxes (φ_t) or household debt taxes (θ_{HHt}) are used: - Explicit partial derivatives for ∂(χ^L_1 ˆq_0)/∂C^s_{Ft}, ∂(χ^L_1 ˆq_0)/∂E^s_1, ∂(χ^L_1 ˆq_0)/∂η^s_2, ∂(χ^L_1 ˆq_0)/∂k^{Linear}_0, ∂ˆq^s_1/∂C^s_{Ft}, ∂ˆq^s_1/∂η^s_2, ∂ˆq^s_1/∂k^{Linear,s}_1, ∂ˆP_{R0}/∂C_{F0}=α_R/α_F, and ∂ˆP^s_{R1}/∂C^s_{F1}=α_R/α_F 1/Y_{R1} are presented. - If {φ_t ∈ R, θ_{HHt} ≡ 0} and Ψ^L_B = 0: χ^s_1 = 1/(β E_0[E^s_1 C_{F0}/E_1 C_{F1}]), χ^s_2 = C^s_{F2}/(β C^s_{F1}) and several cross-partial derivatives with respect to η vanish. - If {φ_t ≡ 0, θ_{HHt} ∈ R} and Ψ^L_B = 0: χ^s_{t+1} = η^s_{t+1} for t∈[0,1], implying many derivatives with respect to C_F and E^s_1 vanish. - If Ψ^L_B > 0 and φ^L_t = θ^L_{HHt} = 0, then χ^L_2 = C^L_{F2}/(β C^L_{F1}). ### Online appendix B — Proofs of results in the main text - Proof of Proposition 1 (summary of logic and conclusions): - Capital inflow taxes: conditions (19)-(20) in main text follow by substituting zero capital controls into equations (13)-(14). - FX intervention: - Step 1: condition (21) obtained by substituting equation (4) into equation (15). - Step 2: For ex post FX intervention, three cases show condition (22) suffices for FXI_1 = 0: - Case (a): (1−λ) Γ > 0. Substitution leads to condition (22). - Case (b): λ = 1. Condition (22) trivially holds; FXI_1 indeterminate and can be set to zero. - Case (c): {λ < 1, Γ = 0}. FXI_1 indeterminate and can be set to zero. - Housing debt taxes: - Subsection A.2 implies θ^{Linear}_{R1} = 0 for both period-1 states. - θ^{Linear}_{R0} = 0 requires G′(k^{Concave}_0) = 1 ⇒ k^{Linear}_0 = 1 and τ_{R1} = 0. - From equation (17), τ_{R1} = 0 arises if either Ψ^L_R = 0 (housing constraint does not bind) or χ^L_1 ˆq_0 − ˆP^L_{R1} − ˆq^L_1 = 0, which implies Ψ^L_R = 0 under maintained assumptions. Therefore Ψ^L_R = 0 and inserting into equation (18) yields τ_{R2} = 0. - Proof of Lemma 1 (key steps and conclusions): - Step 1: Allowing capital inflow taxes, setting S_1 = 0, and rewriting FOC (16) yields (1−λ)[η_2 − (1+i^*_1)] = 0, implying I_1 = (1+i^*_1) irrespective of λ. - Step 2: Relaxed planner problem without constraint (7). Modified FOCs summarized: - τ_{H1} = 0. - α_F C_{F0} = β E_0[I_0 z_1] + β(1−λ) E_0[η_1 − (1+i^*_0)] z_1 / (1 + α_H/α_F τ_{H0}). - (1−φ_0) related expression equals β E_0[η_1 z_1/(1+α_H/α_F τ_{H1})]. - (1−λ) [E_0[η_1 − (1+i^*_0)] z_1 + E_0[z_1{η_1 − (1+i^*_0)}]] = 0. - With constraints (3) and (6) slack, Ω_0 = β(1−λ)(B_1+FXI_0) z_1. - From (5), (10), and (B.1): α_F C_{Ft} = z_t = Φ/(β I_0) for t∈{1,2}. - C_{F1} = C_{F2}. - Resource constraint implies C_{F1} = C^* − 1/(1+β) [ (1−λ)(B_1+FXI_0) η_1 + (1+i^*_0)[λ(B_1+FXI_0) − FXI_0] ] with B_1 = C_{F0} − C^* and B_0 = 0. - Step 3: For relevant cases of the relaxed planner problem, obtain (B_1+FXI_0)=0, E_0[η_1−(1+i^*_0)]=0, z_1 equalized across period-1 states, and {η^L_1, η^H_1} not determined: - Case (a) λ = 1: leads to {C_{Ft}}_{t=0}^2 = C^*, B_1 = B_2 = 0; E_0[η_1 − (1+i^*_0)] = 0. - Case (b) {λ < 1, Γ = 0}: contradictions force E_0[η_1 − (1+i^*_0)] = 0 and (B_1+FXI_0)=0; then {C_{Ft}} = C^*, B_1 = B_2 = 0; FXI_0 and FXI_1 indeterminate and can be set to zero. - Case (c) {λ < 1, Γ > 0}: similar conclusions proceed (proof continues beyond provided excerpt). - Logical implications: - Several results hinge on whether Γ > 0 or Γ = 0, and on λ = 1 versus λ < 1. - When Γ = 0, FXI tools often become indeterminate in FOCs and can be set to zero. - Equality of certain consumption variables across states and periods (e.g., C_{F1} = C_{F2}, and ultimately C_{Ft} = C^* in many relaxed-planner cases) follows from the FOCs and resource constraint under the assumptions described. *Italic: Source — wpiea2023161-print-pdf (online appendix and references).* ### 0. Equation (B.10) then establishes thatC ### 0. Equation (B.10) then establishes thatC ### Major analytical results and solution structure - Equation (B.10) implies C_F1, and hence z1, are equalized across period-1 states while {η_L1, η_H1} are not determined. - Equation (B.8) implies C_F0 = C_F1. Substitution yields {C_Ft}_{t=0}^2 = C^∗, which implies B1 = FXI0 = 0 and B2 = 0; with η2 = (1 + i^∗_1) substituted into equation (5) giving FXI1 = 0. - Constrained efficient allocation solves the relaxed planner problem and thus the original Ramsey planner problem. Where {η_L1, η_H1} remain undetermined there is a continuum of solutions; the unique solution to the original problem sets {η_L1, η_H1} so that constraint (7) holds, derived from {τ_Ht}_{t=0}^2 = 0. ### Solutions for the productivity shock (determination of η) - η_H1 = (1 + i^∗_0) A_L1 (π_H1 A_L1 + π_L1 A_H1) - η_L1 = (1 + i^∗_0) A_H1 (π_H1 A_L1 + π_L1 A_H1) - With these and equation (B.1) it follows {τ_Γt}_{t=1}^2 = 0. - Substituting into (B.4) and (14) yields φ_0 = φ_1 = 0. - Policy rate and land-price expressions in subsection A.2 produce {i_t}_{t=0}^1 provided in the lemma and \hat{q}_1 = β α_R α_F C^∗ equalized across period-1 states. - With constraints (3) and (6) slack, {τ_Rt}_{t=1}^2 = 0 and {θ^Linear_{R0}, θ^Linear_{R1}} = 0. ### Proof of Lemma 2 (λ = 1 and capital inflow taxes allowed) - FX intervention is indeterminate and can be set to zero if λ = 1 and capital inflow taxes available. - With λ = 1 and constraints (3) and (6) slack: - {τ_Ht}_{t=0}^2 = 0, {τ_Γt}_{t=1}^2 = 0, {τ_Rt}_{t=1}^2 = 0, {θ^Linear_{R0}, θ^Linear_{R1}} = 0. - {E_t}_{t=0}^2 follows expression in lemma. - Equation (10) yields z1 = Φ and z2 = Φ β (1 + i^∗_1)^{-1}? (text: z2 = Φ β(1+i^∗_1) — preserve exact form as provided). - {C_Ft}_{t=0}^2 path matches lemma. - With Γ = 0, equations (5) and (14) give η2 = (1 + i^∗_1) and φ_1 = 0; equation (13) determines φ_0 per lemma. - Policy rate and land price expressions produce {i_0, i_1, \hat{q}_1} in lemma. ### Proof of Lemma 3 (setting (1−λ) Γ = 0 in (16)) - Equation (B.1) obtained by setting (1−λ) Γ = 0 in (16) rather than S1 = 0. - Solutions again leave {η_L1, η_H1} undetermined; unique original solution sets them so constraint (7) holds, resulting in η_L1 = η_H1 = (1 + i^∗_0). - Substituting into (B.4) and (14) gives φ_0 = 0 and φ_1 = 1 − β η2; for λ < 1 this equals zero, and for λ = 1 equals β Γ S1. - UIP wedges are zero from η_L1 = η_H1 = (1 + i^∗_0) and equation (B.5). ### Proof of Proposition 2 (modified conditions with Γ and I terms) - Combining (5) and (16) yields (B.11): - FXI1 = S1^2/2 − B2, - η2 = 1/β − Γ S1/2, - I1 = 1/β − (1−λ) Γ S1/2. - Replacements for (B.7) and (B.10) are (B.12) and (B.13) respectively: - α_F C_Ft = z_t = Φ β^2 I_0 / I_1 for t ∈ {1,2} (B.12). - C_F1 expression in (B.13) (preserved verbatim). - Under the assumption S_H1 = − S_L1 so (S_H1)^2 = (S_L1)^2, (B.13) equalizes C_F1 and z1 across period-1 states while leaving {η_L1, η_H1} undetermined. - Equation (B.8) gives C_F0 = C_F1; substitution into (B.13) yields expressions for {C_Ft}_{t=0}^2 and {B_t}_{t=1}^2 in lemma. - Unique original solution sets η_L1 = η_H1 = (1 + i^∗_0); substitution yields expressions for {φ_t, i_t}_{t=0}^1 and UIP wedges per lemma. ### Corollary 1 — two cases ruling out perfect stabilization without tools - Case (a): If FX intervention unavailable ex post (FXI1 = 0 and its FOC removed), equations (2) and (5) imply (B.14): - B2 I1 = [C^∗ − C_F2], with definitions for B2, I0, η2, I1 given in text. - If perfect stabilization of imports and exchange rates (constraint (7)) imposed, B2 and C_F2 should be equalized across period-1 states but (1−λ) Γ > 0 and varying S1 prevent I1 from being equalized; contradiction — hence perfect stabilization not constrained efficient without ex post FX intervention. - Case (b): If capital inflow taxes and household debt taxes unavailable ex post (φ_1 = 0) and additional constraint (B.15): C_F2 = β η2 C_F1 ∀ s: - FOCs modified to (B.16)-(B.18); (16) replaced by (B.19). - With constraints (3) and (6) slack and τ_H2 = 0, combining (B.19) and (B.17) yields (B.20). Resource constraint becomes (B.21). - Imposing perfect stabilization requires equalization that forces (B2 + FXI1 − S1) to vary across states via (B.20), producing contradiction with (B.15). Therefore perfect stabilization not constrained efficient if ex post capital inflow taxes unavailable. ### Lemma 4 and Proposition 3 (housing constraint slack, κ_q large) - With κ_q sufficiently large so housing constraint (6) is slack, from (9): - τ_LH1 / π_L1 = Ψ_L B / κ_LH β I_L0 α_H E_L1 π_L1 − (1−λ) B1 α_H Y, - τ_HH1 / π_H1 = (1−λ) B1 α_H Y (B.22), where Y defined in (B.23). - Feasible configurations from (B.23): (i) z_L1 > E_0[z1 η1]/E_0 η1 > z_H1; (ii) equalities; (iii) opposite inequalities. - A proof by contradiction establishes (i) must hold (Y > 0) when external constraint binds. - Proposition 3 (λ = 1, Γ = 0, Assumption 1): - [η_H1 − (1 + i^∗_0)] = (π_L1/π_H1) [(1 + i^∗_0) − η_L1] > 0. - τ_LH1 = Ψ_L B / κ_LH β I_L0 α_H E_L1 > 0 and τ_HH1 = 0 (B.25). - φ_0 expression given; define \tilde{φ}_0 = 1 − 1/E_0 z1 E_0[ z1 / (1 + α_H/α_F τ_H1) ] with η values set to (1 + i^∗_0); \tilde{φ}_0 > 0 and φ_0 > \tilde{φ}_0 > 0. - φ_L1 = 0 and φ_H1 = 0; λ = 1 implies UIP wedges are zero. ### Welfare derivatives and planner first-order conditions (Lemma 5 and Propositions 4–5) - Lemma 5: derivative of planner value V_Planner with respect to λ given explicitly (preserving exact expression). Substitution of resource constraint (2), external borrowing constraint (3), Gamma equations (4)-(5), equation (10), and covariance definition yields lemma. - Proposition 4 (Γ = 0, Assumption 1): perturbation analysis holds E_L1, E_H1 and Γ = 0 fixed; η_L1, η_H1 fixed; system of equations listed (consistently preserved). Differential expressions for dC_F0/dλ, dC_LF1/dλ, dC_HF1/dλ, dφ_0/dλ given in exact forms; definitions X, ∆, ̃z, ̂z, and sign conditions noted. Planner welfare change: - dV_Planner = − π_L1 B1 [(1 + i^∗_0) − η_L1] β ∆ dλ, so for dλ < 0 planner attains preferred allocation. - Proposition 5 (Γ > 0): η_L1, η_H1 may vary and are determined via expressions with ΓB1; full system listed; differential expressions preserved; Z_1, Z_2, Z_3 decomposition with conditions (28)-(31) provide sufficient conditions for dφ_0 < 0 when dλ < 0. Planner welfare change expression given; condition (27) sufficient for planner to prefer dλ < 0. ### Intermediary shutdown and effective parameters (Lemma 6 and Proposition 6) - Shutting down fraction ξ ∈ [0,1] of domestic-owned global financiers changes effective parameters: - ̂λ ≡ (1−ξ) λ / [(1−ξ) λ + (1−λ)] ∈ [0, λ] and d ̂λ / dξ = − (1−λ) λ / [(1−ξ) λ + (1−λ)]^2 < 0. - ̂Γ ≡ Γ / [(1−ξ) λ + (1−λ)] ∈ [Γ, Γ/(1−λ)] and d ̂Γ / dξ = Γ λ / [(1−ξ) λ + (1−λ)]^2 > 0. - d ̂Γ / d ̂λ = − Γ / (1−λ) < 0. - Planner FOC w.r.t. ̂λ provided exactly; substitution of constraints and Gamma equations yields condition (iii) in lemma. - Proposition 6: for Γ > 0 planner selects ̂λ in (̂λ_{Γ=0}, λ]; marginal reductions in ̂λ produce preferred allocations only up to a point; explicit inequality with Cov(z1, η1) and terms B1, B2 given. ### Housing sector tool availability and constraint binding (Proposition 7) - Step 1: configurations of tools and whether housing constraint may bind in L state. - Case (a): capital inflow taxes and FX intervention available, household debt taxes not available: - From Proposition 2 and A.2: χ_1 = χ_2 = 1/β; ̂P_R1 = α_R / α_F C_F; ̂q_1 = β α_R / α_F C_F; k^Linear_0 = k^Linear_1 = 1. - Housing constraint terms equalized across period-1 states; if κ_q such that constraint slack in H state it is slack in L state. Use of capital inflow taxes and FX intervention achieves zero housing wedges and household debt taxes: { {τ_Rt}_{t=1}^2, {θ^Linear_{Rt}}_{t=0}^1 } = 0. - Case (b): household debt taxes and FX intervention available, capital inflow taxes not available: - With φ_0 = φ_1 = 0, household debt taxes required: θ_HH0 = 0 and θ_HH1 = β Γ S1^2 − β Γ S1 (expression preserved). - From A.2: χ_1 = 1/β, χ_2 = 1/β − Γ S1^2, ̂P_R1 = α_R / α_F C_F, ̂q_1 = 2 β α_R / α_F C_F / (2 − β Γ S1), k^Linear_0 = k^Linear_1 = 1. - ̂q_1 not equalized across period-1 states: lower in L state when S1 < 0. Under Assumption 2, left-hand side of (6) identical across states but right-hand side lower in L state; constraint may bind in L even if slack in H. There exists κ_q threshold separating κ_q ∈ [κ_q, ∞) (constraint slack both states) and κ_q ∈ [0, κ_q) (constraint binds in L only). When constraint binds in L, Ψ_L^R > 0, y_E1 ≠ 0, τ_H1 ≠ 0, y_Ft non-zero, θ_HH0 ≠ 0, and {τ_LR2 > 0, τ_HR2 = 0} so ex ante household debt tax may be optimal. - Step 2: combining findings — for κ_q ∈ [κ_q, ∞) constrained efficient allocation and welfare identical whether capital inflow taxes or household debt taxes available; for κ_q ∈ [0, κ_q) allocations and welfare differ by available tools. ### Notes on parameterization (beginning of Appendix C) - The document proceeds to C Parameterization of Simulations and Table 1: Parameter Values (table not reproduced here). *Italic: Content derived verbatim from the provided IMF PDF section.* ### Section 3Section 4 ### Integrated Monetary and Financial Policies for Small Open Economies — Section 3Section 4 ### Model parameters (expenditure shares, factors, and initial conditions) - α_H Expenditure share of tradable goods1/31/3 - α_F Expenditure share of imports1/31/3 - α_R Expenditure share of housing services1/31/3 - β Discount factor0.80.8 - C* World demand level11 - i*0 Initial world interest rate1/β-11/β-1 - A0 Initial level of productivity11 - B0 Initial debt level00.6 - B_R0 Initial housing sector debt levelNANA - λ Domestic share of intermediaries0.8[0, 1] - Γ Balance sheet friction0.9{0, 0.0025} ### Shock and scenario parameters - π Probability of good/bad shock0.50.5 - κ_H Bank Debt limitNA{0.025, 1} - S_1 Foreign risk appetiteS_L1 =−1,S_H1 ∈[0.1,3] ### Miscellaneous - NA73 *Integrated Monetary and Financial Policies for Small Open Economies — Working Paper No. WP/2023/161* --- _Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023161-print-pdf.pdf_