## _wp08209 ## Source details **Canonical URL:** [_wp08209](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08209.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08209.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08209.pdf.json) --- ### Introduction — stylized facts, motivation, and central argument - Stylized facts: - Government spending in developed countries is typically acyclical, whereas it displays a high degree of procyclicality in emerging markets (Table 1). - Key empirical references cited: Braun (2001); Gavin, Hausmann, Perotti and Talvi (1996); Gavin and Perotti (1997); Talvi and Vegh (2000); Kaminsky, Reinhart and Vegh (2004). - Two strands of theoretical explanations for fiscal procyclicality: - Political economy models: emphasize political pressures and common-pool problems (Lane and Tornell (1999); Talvi and Vegh (2000)) → suggest institutional reforms to fiscal arrangements. - Capital market imperfections: emphasize borrowing constraints, volatility of the tax base, and financial underdevelopment (Gavin, Hausman and Perotti (1996); Riascos and Vegh (2003)) → suggest improving financial markets and broadening financial instruments. - Central argument of the paper: - Fiscal procyclicality in emerging markets can stem from the inability of the monetary authority to commit to price stability (often due to fiscal dominance), because this affects the currency structure of public debt and the insurance available to the government. - Degree of market completeness is endogenous and influenced by monetary actions; local currency debt instruments provide insurance against real shocks associated with currency depreciations. - When monetary authority cannot commit, equilibrium can feature an "original sin" problem: inability of the government to borrow in local currency. - Nominal liabilities matter: negative real shocks are typically associated with currency depreciations, which lower the real value of nominal liabilities and thus provide partial insurance. - High-level policy implication: - Fiscal procyclicality may be a by-product of original sin and weak monetary institutions. Strengthening monetary institutions (improving commitment to price stability) can reduce original sin and improve the cyclical properties of fiscal policy, even without immediate financial deepening. ### Model: environment and key assumptions - Economy and agents: - Small open economy with flexible exchange rate and perfect foresight. - Continuum of identical households with separable utility U(c_t,g_t)=u(c_t)+v(g_t). - Households derive utility from private consumption c_t and public good g_t, supply one unit of labor inelastically, have perfect access to capital markets, hold foreign bonds k^h_t yielding interest r and nominal cash balances M^h_t, and face a cash-in-advance (CIA) constraint E_t c_t  M^h_{t-1} (Equation (9) and text). - Discount factor condition used for steady state: (1 + r) = 1. - Separability implies private consumption is constant over time as long as nominal interest rate is constant over time. - Firms: - Produce tradable good with labor only: y_t(l_t)=A_t (l_t)^, where 0 <  < 1 (Equation (14)). - First order condition for labor:  A_t (l_t)^{-1} = w_t (Equation (17)). - Government: fiscal and monetary authorities: - Fiscal authority levies nondistortionary taxes on firm profits and labor income; tax rate  is constant over time. Government expenditure is g_t. Can borrow externally in foreign currency (b_t) at interest rate r and in domestic currency (nominal D_{t-1}, real d_t). Tax revenues proportional to output: _t =  y_t (Equation (23)). Fiscal instrument of interest: g_t. - Monetary authority operates under flexible exchange rate; sets rate of money growth _{t+1} = (M_{t+1} - M_t) / M_t (Equation (24)). Timing: MA announces nominal money growth target at beginning of period; this announcement introduces a potential commitment problem. ### Borrowing, demand for local currency debt, and sovereign default - Demand for local currency debt (d): - Increasing in domestic bond nominal interest rate i^g_t and decreasing in expected depreciation rate "^e_t. - Formal partial derivatives: @d/@ i^g_t = - d'(...)(1 + "^e_t)  0; @d/@"^e_t = d'(...)(1 + "^e_t)^2  0 (Equation (19) and text). - Supply/demand for foreign currency instruments is perfectly elastic up to incentive-compatible debt levels. - Sovereign default and incentive compatibility (IC) constraint: - Sovereign debt contracts not enforceable; default generates a fiscal cost of  units of consumption in each period. - IC constraint: V^R_t  V^D_t (Equation (20)) yields an endogenous upper bound on total public debt. - In default, government expenditure becomes g^D_t = _t -  (Equation (21)) and the fiscal authority must balance budget every period. - Government budget constraint (when IC holds): - b_t = b_{t-1}(1 + r) + g_t - _t + _{MA,t} - (M_t - M_{t-1})/E_t + D_{t-1}/E_t (1 + i^g_{t-1}) - d_t (Equation (22) and text). - Assumption: nominal interest rate on domestic debt is constant over time: i^g_t = i^g for all t. ### Key mechanism: monetary commitment, currency of debt, and fiscal cyclicality - Monetary non-commitment implies an equilibrium inability to borrow in local currency (original sin emerges endogenously). - Currency composition matters because: - Negative shocks correlate with currency depreciations. - Local-currency nominal liabilities see their real domestic-currency burden fall with depreciation, providing insurance; foreign-currency liabilities do not. - Result: when local-currency borrowing is available (commitment), government can smooth spending; when unavailable (no-commitment), fiscal policy tends to be procyclical. ### Comparative cases analyzed - Commitment case: monetary authority can commit to pre-announced money growth path. - No-commitment case: monetary authority cannot commit; potential inflationary surprises to inflate away domestic-currency debt. - Key claim: across a range of productivity shock values, currency structure of public debt determines whether optimal fiscal policy is acyclical (Barro (1979)-type) or procyclical (emerging markets stylized fact). ### Distinguishing features relative to prior literature - Model attributes procyclicality to monetary policy credibility and endogenous original sin rather than to larger shocks or exogenously lower financial development. - Policy implication: improving monetary commitment/credibility can reduce original sin and enable more countercyclical fiscal policy even before financial deepening. ### Equilibrium path: competitive equilibrium and baseline relations - Definition: competitive equilibrium is a sequence for {c_t, k^h_t, b_t, d_t, m^h_t, g_t, k^f_t, l_t, ξ^f_t} and prices {E_t, i_t, i^g_t, w_t} satisfying household and firm maximization, government intertemporal budget constraint, and market clearing (l_t = 1; m^h_t = m_t). - With no uncertainty, agents perfectly anticipate depreciation rate "e_t" = "t. - Nominal interest rate relation: i_t = (1 + r)(1 + "t")^{-1} (equation (25)). - In equilibrium the depreciation rate "t must equal the rate of money growth ( "t = _t). - Real cash balances: m_t = m_{t-1} [ (1 + _t) / (1 + "t) ]. - Labor market equilibrium: l_t = 1 so output y_t = A_t. - Household intertemporal budget constraint (equation (27)): - Σ_{t=0}^1 c_t (1 + r)^{-t} = k^p_0 + Σ_{t=0}^1 [A_t (1 − ξ)] (1 + r)^{-t}. - Government fiscal constraint (equation (28)): - k^g_t = k^g_{t-1} (1 + r)^{-1} − g_t + ξ A_t − d_{t-1} [ (1 + i^g) / (1 + _t) − (1 + r) ]^{-1}. - k^g_t ≡ −(b_t + d_t) is government net external position. - Country net external position: k_t ≡ k^p_t + k^g_t. - Resource constraint (equation (29)): - k_t = k_{t-1} (1 + r)^{-1} + y_t − c_t − g_t + d_{t-1} I^g_{t-1}. ### Implication of monetary growth and nominal debt dynamics - By increasing money growth _t (thus depreciating currency), the government can reduce the real value of public liabilities, implying transfers from foreigners to domestic households — creating a potential commitment problem affecting currency composition of public debt. - Real cash balances dynamic around steady state is unstable unless "t = _t; hence in a nondivergent path real cash balances require equality of depreciation and money growth rates. ### Ramsey problem: commitment versus no-commitment (key results) - Commitment case: - Equilibrium: "e_t = ^A_t, i_t = (1 + r)(1 + ^A_t)^{-1}, eI^g_t = (1 + r)^{-1} (1 + i^g)/(1 + ^A_t). - Stationary first-order conditions: - u'(c_t) = u'(c_{t+1}) (36) → private consumption constant over time. - u'(g_t) = u'(g_{t+1}) [1 + (1 + r) ε_t] (37). - d_t, I_g elasticity: ξ_{d,I_g} = 1 (38). - ε_t [V^R_{t+1} − V^D_{t+1}] = 0 (39). - Result: there exists an optimal positive level of local currency–denominated debt; steady-state proportion ξ_ss = − d(I_g^*) / k^g_0 (equation (40)). - No-commitment case: - Announced ^A_0 irrelevant; demand for domestic currency debt depends on expected depreciation ^e. - With first-order conditions (49)–(52), any positive domestic debt d_t > 0 yields positive benefit of increasing depreciation → no interior solution; optimal to inflate debt away completely. - Equilibrium sets domestic bond demand to zero: - d_t(eI_g) = 0, ^e_t = (1 + r)^{-1} I_g (1 + i^g) for all t (equation (53)). - Thus proportion of local-currency denominated debt ξ = 0. ### Stationary economy characterization and welfare - For A_t = A constant and k^g_0 > − ζ / r (ensures no default in steady state): - Private consumption (both cases): c_t = r k^p_0 + A (1 − ξ) (equation (54)). - Government expenditure constant over time (Proposition 2). - Stock of domestic debt: - commitment case: d^* = d(I_g^*). - no-commitment case: d^* = 0. - Government expenditure level: - g_t = r k^g_0 + A ξ + d(I_g)[I_g] (equation (55)); with d(I_g) = d(I_g^*) under credible policy and d(I_g^*) = 0 otherwise. - Welfare: no-commitment yields lower welfare because benefits from borrowing in domestic currency vanish. ### Temporary shock (unexpected, lasting T periods) and procyclicality conditions - Shock specification: A_t = A_L for t ≤ T; A_t = A_H for t > T, with A_L < A_H and A_H steady-state value (equation (56)). - Private consumption constant after shock: c_t = c_0 with - c_0 = r k^p_0 + A_H (1 − ξ) − [A_H − A_L](1 − ξ)[1 − (1 + r)^{-T}] (equation (57)). - Government expenditure: - g_t constant over any interval where A_t constant (Proposition 1). Define g_1 for t ≤ T and g_2 for t > T. - For t > T: g_2 = r k^g_{T+1} + ξ A_H + d(I_g)[I_g] (equation (58)). - For t ≤ T government budget recursion: k^g_t = (1 + r) k^g_{t-1} − g_1 + ξ A_L + d I_g (equation (59)). - Recursive solution for k^g_{T+1} given by equation (60). - Procyclicality condition (Proposition 3): - There exists threshold k^g such that k^g_0 < k^g is necessary and sufficient for government expenditure to be procyclical (g_1 < g_2). - Threshold: - k^g = − [ ζ + d I_g − ξ (A_H − A_L)[1 − (1 + r)^{-T}] ] / r (equation (61)). - Comparative statics: k^g decreases in default cost ζ; increases in output fall magnitude (A_H − A_L) and persistence T. - When IC binds at t = T (procyclical case): - g_2 = ξ A_H − ζ (equation (62)). - g_1 solved as in equation (63). - Procyclicality in terms of currency composition (Proposition 4): - Necessary and sufficient condition for procyclical fiscal policy: - ξ_ss < [ (k^g_ss − k^g) / k^g_ss ] [ c_ss / (c_ss − c_0) ] (equation stated in prose). - Corollary 5 (classification by shock magnitude A_L): - There exist thresholds A_L and Ā_L such that: - i) if A_L > Ā_L: g_t constant regardless of monetary authority behavior. - ii) if A_L < A_L: fiscal policy procyclical regardless of monetary authority behavior. - iii) if A_L < A_L < Ā_L: fiscal policy acyclical if monetary authority can commit to price stability, procyclical otherwise. - Thresholds solve (equations (67) and (68) in the source). ### Policy implications and concluding remarks (compact) - Monetary policy credibility (ability to commit to money growth / price stability) influences the currency composition of public debt: - With commitment: positive optimal domestic-currency debt share and scope to smooth fiscal policy. - Without commitment: domestic-currency debt share collapses to zero, removing benefits from domestic-currency borrowing and potentially forcing procyclical fiscal policy. - Model links two stylized facts: - "Original sin" (currency-structure of public debt in emerging markets). - Procyclicality of fiscal policy in emerging economies. - Policy suggestion: strengthening monetary institutions toward committing to price stability can improve cyclical properties of fiscal policy in emerging markets; as original sin fades, cyclical properties should converge toward those of mature economies. ### Appendix A — highlights of proofs - Proposition 1 (IC cannot bind under constant output path): uses constancy of private consumption (Equations (36) or (49)), expressions (69)–(74) to show contradiction if IC binds under constant A_t. - Proposition 2 (IC as bound on public debt and stationary equilibrium): IC reduces to k^g_0  ( + dI_g)/r (Equation (76)); if k^g_0 > /r the constraint does not bind and government can borrow if MA can commit. - Proposition 3 (k^g_0 < k^g necessary and sufficient for procyclicality): derives k^g from Equations (77)–(83). - Proposition 4 and Corollary 5: express procyclicality in terms of ξ_ss and show RHS decreases in A_L (Equations (84)–(87)); sign conditions discussed with parameter assumptions 0 > r k^g_ss >  and dI_g  0. *Source: _wp08209 - 1. Introduction; 2.4 Equilibrium Path; Appendix A (excerpt).* ### 1. Introduction....................................................................................................... ### 1 Introduction ### Stylized facts and motivation - Empirical studies show government spending in developed countries is typically acyclical, whereas it displays a high degree of procyclicality in emerging markets (Table 1). - Key empirical references cited: Braun (2001); Gavin, Hausmann, Perotti and Talvi (1996); Gavin and Perotti (1997); Talvi and Vegh (2000); Kaminsky, Reinhart and Vegh (2004). - Two main strands of theoretical work seeking to explain fiscal procyclicality: - Political economy models: emphasize political pressures and common-pool problems (Lane and Tornell (1999); Talvi and Vegh (2000)). These suggest institutional reforms to fiscal arrangements. - Capital market imperfections: emphasize borrowing constraints, volatility of the tax base, and financial underdevelopment (Gavin, Hausman and Perotti (1996); Riascos and Vegh (2003)). These suggest improving financial markets and broadening financial instruments. ### Central argument of the paper - Novel mechanism: fiscal procyclicality in emerging markets can stem from the inability of the monetary authority to commit to price stability (often due to fiscal dominance), because this affects the currency structure of public debt and, hence, the insurance available to the government. - Degree of market completeness is endogenous and influenced by monetary actions; local currency debt instruments provide insurance against real shocks associated with currency depreciations. - When monetary authority cannot commit, equilibrium can feature an "original sin" problem: inability of the government to borrow in local currency. - Nominal (foreign- versus local-currency) liabilities matter: negative real shocks are typically associated with currency depreciations, which lower the real value of nominal liabilities and thus provide partial insurance and borrowing space in bad states. - These results do not rely on greater shock volatility or exogenously lower financial development in emerging markets; instead they hinge on monetary credibility/commitment. ### Policy implication (high level) - Fiscal procyclicality may be a by-product of original sin and weak monetary institutions. Strengthening monetary institutions (improving commitment to price stability) can reduce original sin and improve the cyclical properties of fiscal policy, even without immediate financial deepening. --- ### 2 Model: environment and key assumptions ### Economy and agents - Small open economy with flexible exchange rate and perfect foresight. - Continuum of identical households with separable utility: U(c_t,g_t)=u(c_t)+v(g_t). - Households: - Derive utility from private consumption c_t and public good g_t. - Supply one unit of labor inelastically. - Have perfect access to capital markets and face no borrowing constraints (can use entire income stream as collateral). - Hold foreign bonds k^h_t yielding interest r and hold nominal cash balances M^h_t. - Face a cash-in-advance (CIA) constraint for consumption: E_t c_t  M^h_{t-1} (Equation (9) and text). - Discount factor condition used for steady state: (1 + r) = 1. - Separability implies private consumption is constant over time as long as nominal interest rate is constant over time. - Firms: - Produce tradable good with labor only: y_t(l_t)=A_t (l_t)^, where 0 <  < 1 (Equation (14)). - Pay taxes on profits; dividends ^f_t distributed to households. - First order condition for labor:  A_t (l_t)^{-1} = w_t (Equation (17)). ### Government: fiscal and monetary authorities - Government comprises fiscal authority and monetary authority (MA). - Fiscal authority: - Levies nondistortionary taxes on firm profits and labor income; tax rate  is constant over time. - Government expenditure is the public good g_t that enters households' utility. - Can borrow externally in foreign currency (b_t) at interest rate r (perfectly elastic supply as long as IC constraint holds) and in domestic currency (nominal D_{t-1}, real d_t). - Tax revenues are proportional to output: _t =  y_t (Equation (23)). - Fiscal instrument of interest: government expenditure g_t. - Monetary authority: - Operates under flexible exchange rate; sets rate of money growth each period: _{t+1} = (M_{t+1} - M_t) / M_t (Equation (24)). - Timing: MA announces nominal money growth target at beginning of period; this announcement introduces a potential commitment problem. ### Borrowing, demand for local currency debt, and incentive compatibility - Demand for local currency debt (d) is: - Increasing in domestic bond nominal interest rate i^g_t and decreasing in expected depreciation rate "^e_t. - Formal partial derivatives: @d/@ i^g_t = - d'(...)(1 + "^e_t)  0; @d/@"^e_t = d'(...)(1 + "^e_t)^2  0 (Equation (19) and text). - Supply/demand for foreign currency instruments is perfectly elastic up to incentive-compatible debt levels. - Sovereign default and IC constraint: - Sovereign debt contracts are not enforceable; government can strategically default and be excluded from capital markets. - Default generates a fiscal cost of  units of consumption in each period (text). - Incentive Compatibility (IC) constraint: V^R_t  V^D_t (Equation (20)), which yields an endogenous upper bound on total public debt. - In default, government expenditure becomes g^D_t = _t -  (Equation (21)) and the fiscal authority must balance budget every period. ### Government budget constraint (when IC holds) - b_t = b_{t-1}(1 + r) + g_t - _t + _{MA,t} - (M_t - M_{t-1})/E_t + D_{t-1}/E_t (1 + i^g_{t-1}) - d_t (Equation (22) and text). - Here b_t: foreign currency debt; D_{t-1}: nominal domestic debt; d_t: real value of domestic debt; _{MA,t}: central bank transfers to household (money creation). - Assumption: nominal interest rate on domestic debt is constant over time: i^g_t = i^g for all t (final paragraph). --- ### 3 Mechanism, comparative cases, and main theoretical results ### Mechanism linking monetary commitment, currency structure of debt, and fiscal cyclicality - Monetary non-commitment (no-commitment) implies an equilibrium inability to borrow in local currency (original sin emerges endogenously). - Currency composition of public debt matters because: - Negative shocks are correlated with currency depreciations. - Nominal (foreign-currency) liabilities do not depreciate in domestic currency terms, while local-currency nominal liabilities see their real domestic-currency burden fall with depreciation, providing insurance. - Thus, when local-currency borrowing is possible, it helps smooth public spending across states; when it is not, fiscal policy tends to be more procyclical. ### Comparative cases studied (overview) - The paper compares two polar cases: - Commitment case: monetary authority can commit to pre-announced money growth path. - No-commitment case: monetary authority cannot commit; potential for inflationary surprise to inflate away domestic-currency debt. - Key claim: for a whole range of productivity shock values, the currency structure of public debt determines whether optimal fiscal policy is acyclical (as in Barro (1979)) or procyclical (as observed in emerging markets). ### Distinguishing features relative to prior literature - Unlike prior work that attributes procyclicality to larger shocks in EMEs or to exogenously lower financial development, this model roots procyclicality in monetary policy credibility and the endogenous emergence of original sin. - Policy implication emphasized: improving monetary commitment/credibility can reduce original sin and allow more countercyclical fiscal policies even before financial market deepening. --- *Source: _wp08209 - 1. Introduction* ### 2.4 Equilibrium Path ### _wp08209 - 2.4 Equilibrium Path ### Definition of competitive equilibrium and basic equilibrium conditions - A competitive equilibrium is a sequence for quantities {c_t, k^h_t, b_t, d_t, m^h_t, g_t, k^f_t, l_t, ξ^f_t} and prices {E_t, i_t, i^g_t, w_t} such that: - {c_t, m^h_t, k^h_t} solve the households' maximization problem given {E_t, i_t, w_t} and {g_t, ξ^f_t}. - {l_t} solves firms' maximization problem given {w_t}. - the government intertemporal budget constraint holds. - markets clear: l_t = 1; m^h_t = m_t. - With rational agents and no uncertainty, agents perfectly anticipate the depreciation rate "e_t" = "t". - Nominal interest rate relation: - i_t = (1 + r)(1 + "t")^{-1} (equation (25)). - In equilibrium the depreciation rate "t must equal the rate of money growth ( "t = _t ), standard in models with flexible exchange rates. - Real cash balances relationship (from m_t ≡ M_t/E_t): - m_t = m_{t-1} [ (1 + _t) / (1 + "t) ]. - Labor market equilibrium: l_t = 1 so output y_t = A_t. - Household intertemporal budget constraint (combining household and firm constraints and using ΔMA_t = M_t − M_{t-1}/E_t): - Σ_{t=0}^1 c_t (1 + r)^{-t} = k^p_0 + Σ_{t=0}^1 [A_t (1 − ξ)] (1 + r)^{-t} (equation (27)). - k^p_0 ≡ (k^h_0 + k^f_0) denotes private sector holdings of foreign assets. - Government fiscal constraint (using "t = _t and ΔMA_t substitution): - k^g_t = k^g_{t-1} (1 + r)^{-1} − g_t + ξ A_t − d_{t-1} [ (1 + i^g) / (1 + _t) − (1 + r) ]^{-1} (equation (28)); where k^g_t ≡ −(b_t + d_t) is government net external position. - Country net external position: k_t ≡ k^p_t + k^g_t. - Resource constraint (combining (27) and (28)): - k_t = k_{t-1} (1 + r)^{-1} + y_t − c_t − g_t + d_{t-1} I^g_{t-1} (equation (29)). - Last term represents the social benefit (cost) of borrowing from abroad in domestic currency. ### Implication of monetary growth and nominal debt dynamics - By increasing money growth _t (thus depreciating currency), the government can reduce the real value of public liabilities, implying transfers from foreigners to domestic households — creating a potential commitment problem affecting currency composition of public debt. - Real cash balances dynamic around steady state is unstable unless "t = _t, hence in a nondivergent path real cash balances require equality of depreciation and money growth rates. ### Ramsey problem: commitment vs no-commitment cases - Monetary authority can affect demand for domestic-currency debt through credibility of announced money growth _t = ^A_t. - Commitment case (monetary authority can commit): - In equilibrium "e_t = ^A_t, i_t = (1 + r)(1 + ^A_t)^{-1}, eI^g_t = (1 + r)^{-1} (1 + i^g)/(1 + ^A_t). - Government Ramsey problem written as dynamic program (equations (30)–(35)). - Key conditions: - Household budget constraint: k^p_0 + c + ... = k^p (equation (31)). - First-order condition for consumption: u'(c) = β [1 + β i] (equation (32)). - Fiscal constraint: k^g_0 + g + ... = k^g (equation (33)). - Opportunity cost of holding domestic debt: I_g = (1 + r)^{-1} (1 + i^g)/(1 + ^A) (equation (34)). - Incentive compatibility: V_D(k^p_0) ≤ V_R(k^p_0, k^g_0, ^A_0) (equation (35)). - First-order conditions in stationary economy: - u'(c_t) = u'(c_{t+1}) (36) — private consumption constant over time. - u'(g_t) = u'(g_{t+1}) [1 + (1 + r) ε_t] (37) — public-good consumption path depends on IC constraint. - d_t, I_g_t elasticity condition: ξ_{d,I_g} = 1 (38) defines optimal money growth level. - ε_t [V^R_{t+1} − V^D_{t+1}] = 0 (39). - Result: there exists an optimal positive level of local currency–denominated debt; steady-state proportion ξ_ss = − d(I_g^*) / k^g_0 (equation (40)). - If inflation path is constant, private sector path is independent of government problem; V_D(k^p_0) = V_R(k^p_0, k^g_0, ^A_0). - No-commitment case (monetary authority cannot commit): - Announced ^A_0 irrelevant; demand for domestic currency debt depends on expected depreciation ^e. - Ramsey problem modified (equations (42)–(48)), with I_g = (1 + r)^{-1} (1 + i^g)/(1 + ) (46) and eI_g = (1 + r)^{-1} (1 + i^g)/(1 + ^e) (47). - First-order conditions: - u'(c_t) = u'(c_{t+1}) (49). - u'(g_t) = u'(g_{t+1}) [1 + (1 + r) ε_t] (50). - d_t(eI_g) [1 + i^g)/(1 + _t)^2] = 0 (51). - ε_t [V^R_{t+1} − V^D_{t+1}] = 0 (52). - For any positive domestic debt d_t > 0, benefit of increasing depreciation rate is positive → no interior solution; optimal to inflate debt away completely. - Equilibrium depreciation rate sets demand for domestic bonds equal to zero: - d_t(eI_g) = 0, ^e_t = (1 + r)^{-1} I_g (1 + i^g) for all t (equation (53)). - Thus proportion of local-currency denominated debt ξ = 0. ### Stationary economy: characterization and welfare - For A_t = A constant and k^g_0 > − ζ / r (ensures no default in steady state): - Private consumption (same in both cases): - c_t = r k^p_0 + A (1 − ξ) (equation (54)). - Government expenditure constant over time (Proposition 2). - Stock of domestic debt: - commitment case: d^* = d(I_g^*). - no-commitment case: d^* = 0. - Government expenditure level: - g_t = r k^g_0 + A ξ + d(I_g) [I_g] (equation (55)); with d(I_g) = d(I_g^*) under credible policy and d(I_g^*) = 0 otherwise. - Welfare: inconsistent monetary policy (no-commitment) yields lower welfare because benefits from borrowing in domestic currency vanish. ### Temporary shock (unexpected, lasting T periods) and cyclical properties of fiscal policy - Shock: A_t = A_L for t ≤ T; A_t = A_H for t > T, with A_L < A_H and A_H steady-state value (equation (56)). - Private consumption constant after shock: c_t = c_0. Level c_0: - c_0 = r k^p_0 + A_H (1 − ξ) − [A_H − A_L](1 − ξ)[1 − (1 + r)^{-T}] (equation (57)). - Government expenditure behavior: - g_t constant over any interval where A_t constant (Proposition 1). - Define g_1 for t ≤ T and g_2 for t > T. - For t > T: - g_2 = r k^g_{T+1} + ξ A_H + d(I_g)[I_g] (equation (58)). - For t ≤ T government budget recursion: - k^g_t = (1 + r) k^g_{t-1} − g_1 + ξ A_L + d I_g (equation (59)). - Recursive solution for k^g_{T+1}: - k^g_{T+1} = k^g_0 (1 + r)^{T+1} + [ξ A_L + d I_g − g_1] [ (1 + r)^{T+1} − 1 ] / r (equation (60)). - Procyclicality condition (Proposition 3): - There exists threshold k^g such that k^g_0 < k^g is necessary and sufficient for government expenditure to be procyclical (g_1 < g_2). - Threshold: - k^g = − [ ζ + d I_g − ξ (A_H − A_L)[1 − (1 + r)^{-T}] ] / r (equation (61)). - Comparative statics: k^g decreases in default cost ζ; increases in output fall magnitude (A_H − A_L) and persistence T. - When IC binds at t = T (procyclical case): - g_2 = ξ A_H − ζ (equation (62)). - g_1 solved as: - g_1 = [ξ A_L + d I_g] + r k^g_0 + (ζ + d I_g) (1 + r)^{-T} [1 − (1 + r)^{-T}]^{-1} (equation (63)). - Government net external position dynamics (with definition ξ = − d_t / k^g_t): - k^g_t = k^g_{t-1} [ 1 − ξ_{t-1} ζ / ( 1 − (E_{t-1} (1 + )/E_t) ) ] (equation (64) as presented). - From cash-in-advance: E_{t-1} (1 + _t)/E_t = c_t / c_{t-1} (equation (65)). - Following a shock: - k^g_0 = k^g_ss − [ 1 − ξ_ss ] ζ [ 1 − c_0 / c_ss ] (equation (66) as combined). - Procyclicality in terms of currency composition (Proposition 4): - Necessary and sufficient condition for procyclical fiscal policy: - ξ_ss < [ (k^g_ss − k^g) / k^g_ss ] [ c_ss / (c_ss − c_0) ] (equation stated in prose). - Corollary 5 (classification by shock magnitude A_L): - There exist thresholds A_L and Ā_L such that: - i) if A_L > Ā_L: g_t constant regardless of monetary authority behavior. - ii) if A_L < A_L: fiscal policy procyclical regardless of monetary authority behavior. - iii) if A_L < A_L < Ā_L: fiscal policy acyclical if monetary authority can commit to price stability, procyclical otherwise. - Thresholds solve (equations (67) and (68) in the source). ### Policy implications and concluding remarks - Monetary policy credibility (ability to commit to money growth / price stability) influences the currency composition of public debt. - With commitment: positive optimal domestic-currency debt share and scope to smooth fiscal policy. - Without commitment: domestic-currency debt share collapses to zero, removing benefits from domestic-currency borrowing and forcing potentially procyclical fiscal policy. - Model links two stylized facts: - "Original sin" (currency-structure of public debt in emerging markets). - Procyclicality of fiscal policy in emerging economies. - Policy suggestion: strengthening monetary institutions toward committing to price stability can improve cyclical properties of fiscal policy in emerging markets; as original sin fades, cyclical properties should converge toward those of mature economies. *Source: _wp08209 - 2.4 Equilibrium Path (excerpt).* ### Appendix A ### _wp08209 - Appendix A ### Proof of Proposition 1 - Proves by contradiction that the IC constraint cannot bind if output follows a constant path (A_t = A). - From Equations (36) (or (49)), private consumption c_t follows a constant path. - Optimal consumption path under repayment solved from Equation (27): - c^R_t = r k^p_0 + r_1 Σ_{t=0} [A_t (1)]/(1 + r)^t (Equation (69)) - Under the assumption that  is constant and default does not affect access to capital markets, c^R_t equals consumption under default, so: - Σ_{j=0} u(c^R_{t+j})/(1 + r)^j = Σ_{j=0} u(c^D_{t+j})/(1 + r)^j (Equation (70)) - The IC constraint reduces to: - Σ_{j=0} v(g^R_{t+j})/(1 + r)^j  Σ_{j=0} v(g^D_{t+j})/(1 + r)^j (Equation (71)) - If the constraint binds at t and t+1, equality in (71) in both periods implies: - v(g^R_t) = Σ_{j=0} v(g^D_{t+j})/(1 + r)^j 1/(1 + r) Σ_{j=0} v(g^D_{t+1+j})/(1 + r)^j (Equation (72)) - Under default g_t =  A_H + dI_g  every period, so: - Σ_{j=0} v(g^D_{t+j})/(1 + r)^j = Σ_{j=0} v( A_t + dI_g )/(1 + r)^j (Equation (73)) - The RHS of (72) equals v( A_t + dI_g )/(1 + r), hence: - v(g^R_t) = v([ A_t + dI_g ])/(1 + r) (Equation (74)) - Since A_t constant implies RHS constant, this yields _t = 0 from Equation (37), contradicting the assumption that the IC constraint binds in the two periods. ### Proof of Proposition 2 - For any constant path of output, the IC constraint cannot bind; therefore government expenditure g_t is constant over time (from Equations (36) and (49)). - With g_t constant and private consumption equal under repayment and default, the IC constraint becomes: - v(g^R)/r  v(g^D)/r (Equation (75)) - Using g^D =  A  and g^R = r k^g_0 +  A + dI_g, the IC constraint is equivalent to: - k^g_0  ( + dI_g)/r (Equation (76)) - Thus the IC constraint is an upper bound on public debt (lower bound on net position). - If k^g_0 > /r the constraint does not bind; this ensures no default in stationary equilibrium. - If this condition holds, the government can borrow and d can be positive if the monetary authority can commit to a monetary policy. - From Equations (38) and (51): - d(I^g_t) = d(I^g__t) for the credible monetary authority case - d(I^g_t) = 0 otherwise ### Proof of Proposition 3 - Government expenditure is constant for t > T and for t < T. Denote g_1 for t < T and g_2 for t > T. - Government expenditure is procyclical if g_1 < g_2, which holds iff _T > 0. - g_1 < g_2 iff: - Σ_{j=0} v(g^R_{T+1+j})/(1 + r)^j = Σ_{j=0} v(g^D_{T+1+j})/(1 + r)^j (Equation (77)) - Rewriting using Equations (58) and (21): - v(r k^g_{T+1} +  A_H + dI_g)/r  v( A_H )/r (Equation (78)) - Which holds iff: - k^g_{T+1}  ( + dI_g)/r (Equation (79)) - If g_1 = g_2 = g, government budget constraint gives: - g = r k^g_0 + dI_g + Σ_{t=0}  A_t/(1 + r)^t (Equation (80)) - = r k^g_0 + dI_g +  A_L [1(1 + r)T] +  A_H [(1 + r)T] (Equation (81)) - Substituting into Equation (59) yields: - k^g_{T+1} = k^g_0 + [ A_L  A_H] [1(1 + r)T]/r = k^g (Equation (82)) - Combining (79) and (82) yields: - k^g_0  ( + dI_g) + [ A_L  A_H] [1(1 + r)T]/r = k^g (Equation (83)) - Therefore k^g_0 < k^g is a sufficient condition for the IC constraint to bind at time T+1 and induce procyclical fiscal policy (g_1 < g_2); it is also necessary. ### Proof of Proposition 4 - k^g_0 < k^g is necessary and sufficient for fiscal policy to be procyclical. - Using Equation (66) this condition can be expressed as: - k^g > k^g_ss  1_ss  1 c_0/c_ss  (Equation (84)) - Rewritten as: - _ss <  (k^g_ss k^g)/k^g_ss  c_ss/(c_ss c_0)  (Equation (85)) ### Proof of Corollary 5 - Need to show the RHS of Proposition 4's inequality is decreasing in A_L. - Differentiating RHS with respect to A_L yields: - dRHS/dA_L =  c_ss/(c_ss c_0) 1/k^g_ss dk^g/dA_L +  (k^g_ss k^g)/k^g_ss c_ss/(c_ss c_0)^2 dc_0/dA_L (Equation (86)) - Using Equations (54) and (57) simplifies this to: - dRHS/dA_L = (1) r k^g_ss c_ss (c_ss c_0)^2 [1(1 + r)T] [r k^g_ss +  + dI_g] (Equation (87)) - This is negative under the assumed parameter conditions: 0 > r k^g_ss >  and dI_g  0. *Source: _wp08209 - Appendix A* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08209.pdf_