## wpiea2026067-online-appendix ## Source details **Canonical URL:** [wpiea2026067-online-appendix](https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026067-online-appendix.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2026/english/wpiea2026067-online-appendix.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2026/english/wpiea2026067-online-appendix.pdf.json) --- ### 2.1 Households and consumption - Consumption aggregation and price indices - Aggregate consumption: - C_t = [ γ^{1/χ}(C_{T,t})^{(χ−1)/χ} + (1−γ)^{1/χ}(C_{N,t})^{(χ−1)/χ} ]^{χ/(χ−1)} with γ ∈ [0,1], χ>0. - Sectoral demand: - C_{T,t} = γ (P_{T,t}/P_t)^{−χ} C_t. - C_{N,t} = (1−γ) (P_{N,t}/P_t)^{−χ} C_t. - CPI: - P_t = [ γ P_{T,t}^{1−χ} + (1−γ) P_{N,t}^{1−χ} ]^{1/(1−χ)}. - Tradable aggregate (home vs foreign): - C_{T,t} = [ (1−α)^{1/η} C_{Ht}^{(η−1)/η} + α^{1/η} C_{Ft}^{(η−1)/η} ]^{η/(η−1)} with α ∈ [0,1], η>0. - CES sub-aggregates: - C_{Ht} = [ ∫_0^1 C_{Ht}(j)^{(ε−1)/ε} dj ]^{ε/(ε−1)}, etc., with ε>1. - Domestic/imported tradable demand: - C_{Ht} = (1−α) (P_{Ht}/P_t)^{−η} C_t, C_{Ft} = α (P_{Ft}/P_t)^{−η} C_t. - Tradable CPI: - P_{T,t} = [ (1−α) P_{H,t}^{1−η} + α P_{F,t}^{1−η} ]^{1/(1−η)}. - Foreign households - Foreign consumption aggregator mirrors home: - C^*_t = [ γ^{1/χ}(C^*_{T,t})^{(χ−1)/χ} + (1−γ)^{1/χ}(C^*_{N,t})^{(χ−1)/χ} ]^{χ/(χ−1)}. - Foreign tradable aggregator with home bias: - C^*_{T,t} = [ (1−α)^{1/η} C^*_{Ft}^{(η−1)/η} + α^{1/η} C^*_{Ht}^{(η−1)/η} ]^{η/(η−1)}. - Household types: HTM and perpetual youth - Hand-to-mouth (HTM): - Consume all income each period: - (1 + τ_{C,t}) C^H_t = (1−τ_{l,t}) W_t N_t + T_t. - Perpetual youth (P): - Maximize: - E_0 ∑_{t=0}^∞ (β θ)^t [ 1/(1−σ) C_{P,t}^{1−σ} − 1/(1+ν) N_{P,t}^{1+ν} ]. - Face taxes τ_C and τ_l; hold domestic government debt B_{H} and international bond B_{I} (real terms). - Aggregate consumption share: - C_t = ω C^H_t + (1−ω) C^P_t with ω the HTM share. - Perpetual youth budget and adjustment costs - Budget constraint (real terms, scaled by (1−ω)): - (1 + τ_{C,t}) C^P_t + B_{H,t}/(1−ω) + B_{I,t}/(1−ω) + φ/(1−ω)^2 (B_{I,t} − B_{I,ss})^2 = R_{H,t−1}/(θ(1−ω)π_t) B_{H,t−1} + R_{I,t−1}/(θ(1−ω)π_t) B_{I,t−1} + (1−τ_{l,t}) W_t N_t + Π_t/(1−ω) + T_t. - Adjustment cost φ on international bond holdings captures capital control frictions; φ set at very low levels in baseline calibration. - Extra return factor 1/θ reflects inheritance/new households in Blanchard (1985) framework. - Labor supply and unions - Labor supplied to a union that maximizes weighted household utility; union faces wage adjustment cost φ_w. - With flexible wages (φ_w = 0), post-tax real wage: - (ε_w/(ε_w − 1)) C^σ N^ν = (1−τ_{l,t}) w^{real}_t. - Intertemporal optimality and international linkages - Euler equation (perpetual youth): - 1 = β E_t[ (C^P_{t+1}/C^P_t)^{−σ} R_{H,t}/π_{t+1} ]. - FOC for international bond with φ: - β λ_{t+1}/λ_t R_{I,t}/π_{t+1} = 1 + φ (B_{I,t} − B_{I,ss}). - Wedge between international and domestic government bond: - R_{I,t}/R_{H,t} = 1 + φ (B_{I,t} − B_{I,ss}). - Foreign perpetual youth budget and UIP intuition - Foreign budget (accounts for exchange rate ε_t): - C^*_{P,t} + B_{F,t}/(1−ω) + B_{I,t} ε_t/(1−ω) = R_{F,t−1}/(θ(1−ω)π^*_t) B_{F,t−1} − R_{I,t−1} ε_t/(θ(1−ω)π^*_t) B_{I,t−1} + W^*_t N^*_t + Π^*_t/(1−ω) + T^*_t. - Foreign FOC for international bond: - ε_t/ε_{t+1} R_{I,t}/π^*_{t+1} = R_{F,t}/π^*_{t+1} = λ^*_t/(β λ^*_{t+1}). - Real exchange rate and link to interest rates - Real exchange rate: - Q_t = ε_t P^*_t / P_t. - With φ: - Q_{t+1}/Q_t R_{F,t}/π^*_{t+1} = R_{H,t}/π_{t+1} (1 + φ (B_{I,t} − B_{I,ss})). - Collapses to standard UIP when φ → 0. - Present-value consumption for perpetual youth - Consumption relation: - C^P_t ∑_{τ=0}^∞ (θ β^{1/σ})^τ (Q_{t+τ})^{1−1/σ} = R_{t−1}/((1−ω) π_t) B_{H,t−1} + R_{I,t−1}/((1−ω) π_t) B_{I,t−1} + ∑_{τ=0}^∞ θ^τ Q_{t+τ} (Y^P_{t+τ}). - Q_{t+τ} defined as product of π and R terms across horizons. ### 2.2 Tariffs, prices, and exchange rates - Tradable price indices with ad-valorem tariff τ_t: - Home tradable price index: - P_{T,t} = [ (1−α) P_{H,t}^{1−η} + α ((1 + τ_t) P_{F,t})^{1−η} ]^{1/(1−η)}. - Foreign tradable price index with τ^*_t: - P^*_{T,t} = [ (1−α) P^{* 1−η}_{F,t} + α ((1 + τ^*_t) P^*_{H,t})^{1−η} ]^{1/(1−η)}. - Tradable inflation and terms of trade - Terms of trade S_t = P_{F,t} / P_{H,t}. - Relative tradable price: - P_{T,t}/P_{H,t} = [ (1−α) + α ((1 + τ_t) S_t)^{1−η} ]^{1/(1−η)}. - Tradable inflation (Π_{T,t} = P_{T,t}/P_{T,t−1}, Π_{H,t} = P_{H,t}/P_{H,t−1}): - Π_{T,t} = Π_{H,t} [ ((1−α) + α ((1 + τ_t) S_t)^{1−η}) / ((1−α) + α ((1 + τ_{t−1}) S_{t−1})^{1−η}) ]^{1/(1−η)}. - Foreign tradable inflation analogous using S_t^{−1}. - Tariff shock effects - Tariff shock creates an immediate wedge between domestic price inflation Π_H and tradable consumer price inflation Π_T; this feeds into aggregate CPI inflation via tradables share γ. - Tariff does not directly enter household budget constraint except through relative price effects (see Lemma A.1). - Non-tradable and aggregate inflation - Non-tradable inflation Π_{N,t} driven by domestic real marginal costs MC_{N,t}. - Aggregate inflation: - Π_t = [ γ (Π_{T,t} P_{T,t−1}/P_{t−1})^{1−χ} + (1−γ) (Π_{N,t} P_{N,t−1}/P_{t−1})^{1−χ} ]^{1/(1−χ)}. - Real exchange rate decomposition - Q_t = ε_t P^*_t / P_t decomposed as: - Q_t = (ε_t P^*_{T,t} / P_{T,t}) × [ γ + (1−γ) (P^*_{N,t}/P^*_{T,t})^{1−χ} ]^{1/(1−χ)} / [ γ + (1−γ) (P_{N,t}/P_{T,t})^{1−χ} ]^{1/(1−χ)}. - Internal tradable/non-tradable price wedges affect aggregate RER even if tradable prices obey law of one price. ### 2.3 Firms and production - Production technology and taxes/subsidies - Firm production: - Y_{k,t}(i) = A_{k,t}(i) K_{k,t}(i)^{μ_k} N_{k,t}(i)^{1−μ_k}. - Firms face investment subsidies φ_t, corporate tax τ_K, and consumption ad-valorem tax τ_C. - Depreciation P_k δ K_{k,t} deducted from revenues. - Firm optimization and marginal cost - Labor FOC: - φ_t (1−μ) A_{k,t} K_{k,t}^{μ} N_{k,t}^{−μ} = (1−τ_{K,t}) w_t. - Post-tax marginal cost: - φ_t = (1−τ_{k,t}) MC_{k,t} = (1−τ_{k,t}) w_t (1−μ)^{-1} A_{k,t}^{−1} K_{k,t}^{−μ} N_{k,t}^{μ}. - Investment FOC and Tobin's q include P_{k,t}/P_t and adjustment κ. - Transmission and markup shocks - Investment cost may use P_T or P_N depending on sector; tradable investment can use imported goods (P_H). - Lower markups (higher MC) raise q and desired investment and draw labor into the sector. - Simulated markup shock M^ε_t follows AR(1) (section 4). - Market clearing - Aggregate production: - Y_{H,t} = I_{H,t} + C_{H,t} + C^*_{H,t} + I^*_{H,t}. - Home tradable market clearing: - Y_{H,t} = (1−α) (P_{H,t}/P_{T,t})^{−η} (C_{T,t} + I_{T,t}) + α (P_{H,t}/(ε_t P^*_{T,t}))^{−η} (C^*_{T,t} + I^*_{T,t}). - Non-tradable clearing: - Y_{N,t} = C_{N,t} + I_{N,t}. - Pricing and optimal price setting (Rotemberg) - Retailer price adjustment cost φ_π/2 (P_{H,t+j}(i)/P_{H,t+j−1}(i) − 1)^2 Y_{H,t+j}; FOC yields pricing relation with adjustment terms. - Without adjustment costs: - P_{H,t}/P_t = ε/(ε−1) MC_t/P_t (standard markup rule). - Analogous for non-tradable and foreign sectors. - Profits to perpetual youth households - Sectoral profits: - Π_{k,t} = P_{k,t}/P_t [ (1−τ_{K,t}) [ (1−(ε−1)/ε)(1−μ) Y_{k,t} + μ Y_{k,t} ] + δ τ_{k,t} K_t − I_{k,t} ]. - Aggregate profits Π_t = Π_{H,t} + Π_{N,t}. ### 2.4 Accounting identities and policy rules - Aggregates - GDP: - Y_t = GDP_t = (P_{H,t}/P_t) * Y_T + (P_{N,t}/P_t) Y_N. - Aggregate investment: - I_t = (P_{H,t}/P_t) * I_H,t + (P_{N,t}/P_t) I_N,t. - Trade and current account - Current account balance: - CAB_t = (Y_t − C_t) + (R_{I,t−1} − 1) B_{t−1}/π_t. - Net international debt evolution: - B_t − B_{t−1} = CAB_t. - Fiscal accounts and government bond holdings - Government revenues: - Rev_t = τ_t P_{F,t}/P_t C_{F,t} + τ_{C,t} C_t + τ_{L,t} W_t N_t + τ_{K,t} ( Π_t − (δ) K_t + (1−φ_t) I_t ). - Government expenditure: - E_t = T_t + φ_t I_t. - Government budget constraint: - B_{H,t} + E_t = Rev_t + R_{t−1}/π_t B_{H,t−1} + B^ε_I. - Government holdings of international bond B^ε_I evolve: - B_{G,I} − R_{I,t−1}/π_t B_{G,I} = ε_{G,t}. - Fiscal rule (debt-stabilizing): - E_t / Y_t = ρ_F (Rev_t / Y_t) − ρ_B B_{H,t} / Y_t. - Monetary policy - Taylor-type rule: - R_{H,t} = R_{ss} ( π_t / π_{ss} )^{ψ_π}. ### 3 Parameters (calibrated values shown) - Preferences & Households - Discount Factor β = 0.995 - Inter-temporal substitution σ = 2.0 - Inverse Frisch Elasticity ν = 0.33 - Habit Persistence ρ = 0.7 - Hand-to-Mouth Share ω = 0.3 - Technology & Capital - Capital Share μ_k = 0.33 - Depreciation Rate δ = 0.025 (Quarterly) - Inv. Adjustment Cost κ = 3.0 - Openness & Sectors - Tradable Share γ = 0.5 - T/NT Elasticity χ = 0.5 - Trade Elasticity η = 2.0 - Openness α = 0.3 - Nominal Frictions & Policy - Price Elasticity ε = 6.0 - Rotemberg Parameter (prices) φ_π = 70 - Rotemberg Parameter (wages) φ_w = 70 - Taylor Rule (Inf) ψ_π = 1.5 ### 4 Shock processes - AR(1) processes (logs or levels) - Sector-specific productivity (logs): - ln(A_{T,t}) = ρ_A ln(A_{T,t−1}) + (1−ρ_A) ln(Ȧ_T) + ε_{A,T,t}. - ln(A_{N,t}) = ρ_A ln(A_{N,t−1}) + (1−ρ_A) ln(Ȧ_N) − ε_{A,N,t}. - Investment subsidy shocks: - τ_{sub,T,t} = ρ_{τ,sub} τ_{sub,T,t−1} + (1−ρ_{τ,sub}) ȳ_{τ,sub,T} + ε_{τ,sub,T,t}. - Markup shock on tradable: - M^ε_t = ρ_M M^ε_{t−1} + ε_{M,t}. - Tariffs and factor taxes AR(1); labor and capital tax shocks enter with negative sign (tax cut): - τ_t = ρ_τ τ_{t−1} + ε_{τ,t}. - τ_{L,t} = ρ_{τ,L} τ_{L,t−1} − ε_{τ,L,t}. - τ_{K,t} = ρ_{τ,K} τ_{K,t−1} − ε_{τ,K,t}. - Government reserve accumulation: - B_{g,I,t} = ρ_{gov} B_{g,I,t−1} − ρ_{g,stab} ( B_{g,I,t−1} / GDP_{t−1} ) + ε_{gov,t}. *Source: wpiea2026067-online-appendix - 2.1 Households and consumption (PDF chapter).* ### 5.1 Micro-industrial policy: productivity related developments - Temporary vs permanent 1% tradable productivity boost - Shock: 1% boost to tradable sector productivity; permanent case sets ρA = 1, temporary case sets ρA = 0. - Dynamics (perpetual youth households): - Temporary shock: households save part of the windfall, generating a current account surplus that exceeds the initial increase in investment (partly due to investment adjustment frictions). - Permanent shock: saving initially rises by less than in the temporary case, then rises alongside higher real interest rates and a higher marginal product of capital. - Reported magnitudes presented in figures for CAB(%GDP), Saving(%GDP), Inv.(%GDP) across quarters. - Non-tradable productivity fall vs tradable productivity rise - Paradox: A 1% fall in A_N can mirror the current account impact of a 1% rise in A_T. - Mechanism: - Low χ (< 1) → tradables and nontradables complements; rise in nontradable prices when A_N falls leads households not to substitute away. - RER appreciates, increasing relative value of domestic output and accompanied by decline in current account. - Variables highlighted: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Relative nontradable price, RER(-is appreciation). ### 5.2 Micro-industrial policy: Subsidies - Investment subsidy (φ) and financing modalities - Policy: subsidy to investment encourages capital deepening; effects similar to productivity-induced capital deepening. - Financing cases: 1. Tax-financed (via τc): - Immediate cost lowers consumption for perpetual youth and HTM households; aggregate saving declines. 2. Debt/deficit-financed (low ρg): - Future tax liabilities partially discounted by perpetual youth (death probability θ); households save less than subsidy cost, lowering aggregate saving. - Reported variables: CAB(%GDP), Saving(%GDP), Inv.(%GDP) under "De-cit-nanced" and "Tax-nanced" scenarios. ### 5.3 Micro-industrial policy: Production targets - Mandated output increase / permanent fall in tradable markup - Permanent fall in tradable markup → lower tradable price, higher tradable production and factor use (firms accept lower profits). - Indirect effects: - Relative shortage of nontradables → sharp rise in relative prices and RER appreciation due to inelastic nontradable demand. - Increased investment and labor demand; higher expected future income induces dissaving, lowering current account. - Variables shown: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Relative nontradable price(%), RER(-is appreciation). ### 5.4 Macro-industrial policy: Forced saving and capital controls - Preference shock to discount rate β and capital flow regimes - Policy: persistent increase in saving via higher β (forced saving). - Two φ cases: 1. Free financial flows (φ = 0): - Saving ↑ → CAB ↑; domestic and global interest rates ↓; exchange rate initially depreciates before investment rises. 2. Strict capital flow restrictions (high φ, "CFM"): - Households cannot accumulate foreign assets; import demand reduction → large RER appreciation. - Domestic interest rates fall below international bond rate; domestic investment rises and CAB little changed. - Variables: CAB(%GDP), Saving(%GDP), InterestRates (Home / International bond), RER(-is appreciation). ### 5.5 Macro-industrial policy: reserve accumulation - Government accumulation of international bonds with capital controls - Setup: φ = 1 for high private transaction costs; shock εg causes persistent government accumulation of B_{G,I}. - Effects: - Funded mainly through domestic bond issuance; higher domestic interest rates needed to induce private purchase of new net supply. - Domestic interest rates rise above global bond rate; persistent depreciation; fall in domestic absorption from lower consumption and investment. - Variables: CAB(%GDP), Saving(%GDP), Inv.(%GDP), InterestRates (Home / Intlbond), RER(-is appreciation). ### 5.6 Combined Micro and Macro industrial policy - Interaction: tradable production targets + forced saving - Micro: lower tradable markups → capital deepening expectations lower saving relative to investment; nontradable shortage raises relative prices and causes RER appreciation. - Macro: forced saving via β shock lowers consumption persistently (no capital controls in this combination). - Interaction effects: - Forced saving increases saving relative to investment, counteracting dissaving from production-target expectations. - Forced saving counteracts the RER appreciation seen in standalone micro-IP case. - Variables illustrated: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Inflation(ppt), Relative nontradable price, RER(-is appreciation), ToT(-is appreciation), Real interest rate. ### 6.1 Consumption smoothing (model extension) - Habit formation and lifetime budget with perpetual youth - Household utility with habits: - E0 ∑∞_{t=0} β^t [1/(1−σ) (C_t(i) − ρ C_{t−1})^{1−σ} − 1/(1+φ) N_t^{1+φ}] - Habits reference aggregate consumption ρ C_{t−1}. - Define C̃_t = C_t − ρ C_{t−1}. - Budget constraint in C̃_t includes consumption tax (1 + τC,t), aggregate bond holdings B_T,t, investment I_t (1 − φ_t), wages, profits, transfers, and adjustment costs on international bond holdings. - FOCs: - C̃_t^{−σ} = (1 + τC,t) λ_t - λ_t = β R_t / Π_{t+1} λ_{t+1} - Forward iteration and lifetime budget constraint involve θ, Q_{t+τ}, and R_{t−1} (1−ω) θ π_t B_T,t−1; surviving households and new entrants treated explicitly to eliminate θ term in aggregate consumption function. ### A.1 Lemma (Tariffs and the budget constraint) - Lemma A.1: - Imposition of tariffs does not alter the standard form of the household budget constraint; tariff effects operate only through relative and aggregate price adjustments. - Real cost of tradable consumption: - P_{H,t}/P_t C_{H,t} + (1 + τ_t) P_{F,t}/P_t C_{F,t} - Rewriting with C_H and C_F as functions of C_T and S_t and algebra using α, η, and tariff term (1 + τ_t) demonstrates the household budget constraint reduces to C_T; hence tariffs enter only via price effects. *Source: wpiea2026067-online-appendix - 2.1 Households and consumption (PDF chapter).* ### 2.1 Households and consumption ### 2.1 Households and consumption ### Consumption bundles and price indices - Aggregate consumption C_t is a CES aggregate of tradable and non-tradable goods: - C_t = [ γ^{1/χ}(C_{T,t})^{(χ−1)/χ} + (1−γ)^{1/χ}(C_{N,t})^{(χ−1)/χ} ]^{χ/(χ−1)} with γ ∈ [0,1], χ>0. - Sectoral demand: - C_{T,t} = γ (P_{T,t}/P_t)^{−χ} C_t. - C_{N,t} = (1−γ) (P_{N,t}/P_t)^{−χ} C_t. - CPI: - P_t = [ γ P_{T,t}^{1−χ} + (1−γ) P_{N,t}^{1−χ} ]^{1/(1−χ)}. - Tradable bundle across home (H) and foreign (F) varieties: - C_{T,t} = [ (1−α)^{1/η} C_{Ht}^{(η−1)/η} + α^{1/η} C_{Ft}^{(η−1)/η} ]^{η/(η−1)} with α ∈ [0,1], η>0. - Sub-aggregates (CES across varieties): - C_{Ht} = [ ∫_0^1 C_{Ht}(j)^{(ε−1)/ε} dj ]^{ε/(ε−1)}, etc., with ε>1. - Demand for domestic and imported tradables: - C_{Ht} = (1−α) (P_{Ht}/P_t)^{−η} C_t, C_{Ft} = α (P_{Ft}/P_t)^{−η} C_t. - Tradable CPI: - P_{T,t} = [ (1−α) P_{H,t}^{1−η} + α P_{F,t}^{1−η} ]^{1/(1−η)}. ### Foreign households - Foreign consumption mirrors home preferences: - C^*_t = [ γ^{1/χ}(C^*_{T,t})^{(χ−1)/χ} + (1−γ)^{1/χ}(C^*_{N,t})^{(χ−1)/χ} ]^{χ/(χ−1)}. - Foreign tradable aggregator with analogous home bias: - C^*_{T,t} = [ (1−α)^{1/η} C^*_{Ft}^{(η−1)/η} + α^{1/η} C^*_{Ht}^{(η−1)/η} ]^{η/(η−1)}. ### Household types: hand-to-mouth and perpetual youth - Two household types in each economy: - Hand-to-mouth (HTM): credit constrained, consumes all income each period. - (1 + τ_{C,t}) C^H_t = (1−τ_{l,t}) W_t N_t + T_t. - Perpetual youth (P): can hold positive or negative net assets, maximizes: - E_0 ∑_{t=0}^∞ (β θ)^t [ 1/(1−σ) C_{P,t}^{1−σ} − 1/(1+ν) N_{P,t}^{1+ν} ]. - Faces sales tax τ_C on consumption, labor tax τ_l, and can hold domestic government debt B_{H} and international bond B_{I} (real terms). - Aggregate consumption share: - C_t = ω C^H_t + (1−ω) C^P_t with ω the HTM share. ### Perpetual youth household budget and adjustment costs - Budget constraint (real terms, scaled by (1−ω) factors): - (1 + τ_{C,t}) C^P_t + B_{H,t}/(1−ω) + B_{I,t}/(1−ω) + φ/(1−ω)^2 (B_{I,t} − B_{I,ss})^2 = R_{H,t−1}/(θ(1−ω)π_t) B_{H,t−1} + R_{I,t−1}/(θ(1−ω)π_t) B_{I,t−1} + (1−τ_{l,t}) W_t N_t + Π_t/(1−ω) + T_t. - Adjustment cost φ on changing international bond holdings captures capital control frictions and aids stationarity; φ is set at very low levels in baseline calibration. - Perpetual youth extra return factor 1/θ reflects inheritance/new households in Blanchard (1985) framework. ### Labor supply and unions - Labor from both household types supplied to a union that maximizes weighted household utility, faces wage adjustment costs. - Union maximization (representative form) includes adjustment cost term φ_w on wage changes. - With flexible wages (φ_w = 0), post-tax real wage satisfies: - (ε_w/(ε_w − 1)) C^σ N^ν = (1−τ_{l,t}) w^{real}_t. ### Intertemporal optimality and international financial linkages - Euler equation for perpetual youth household: - 1 = β E_t[ (C^P_{t+1}/C^P_t)^{−σ} R_{H,t}/π_{t+1} ]. - FOC for international bond with capital control friction φ: - β λ_{t+1}/λ_t R_{I,t}/π_{t+1} = 1 + φ (B_{I,t} − B_{I,ss}). - Wedge between international bond and domestic government bond: - R_{I,t}/R_{H,t} = 1 + φ (B_{I,t} − B_{I,ss}). - Foreign perpetual youth household budget (accounts for currency rate ε_t): - C^*_{P,t} + B_{F,t}/(1−ω) + B_{I,t} ε_t/(1−ω) = R_{F,t−1}/(θ(1−ω)π^*_t) B_{F,t−1} − R_{I,t−1} ε_t/(θ(1−ω)π^*_t) B_{I,t−1} + W^*_t N^*_t + Π^*_t/(1−ω) + T^*_t. - Foreign FOC for international bond (UIP intuition): - ε_t/ε_{t+1} R_{I,t}/π^*_{t+1} = R_{F,t}/π^*_{t+1} = λ^*_t/(β λ^*_{t+1}). - Real exchange rate: - Q_t = ε_t P^*_t / P_t. - Link between real interest rates and real exchange rate (with φ): - Q_{t+1}/Q_t R_{F,t}/π^*_{t+1} = R_{H,t}/π_{t+1} (1 + φ (B_{I,t} − B_{I,ss})). - Collapses to standard UIP when φ → 0. ### Consumption of perpetual youth household (present-value expression) - Consumption relation (accounting for asset endowment and discounted future income Y_P): - C^P_t ∑_{τ=0}^∞ (θ β^{1/σ})^τ (Q_{t+τ})^{1−1/σ} = R_{t−1}/((1−ω) π_t) B_{H,t−1} + R_{I,t−1}/((1−ω) π_t) B_{I,t−1} + ∑_{τ=0}^∞ θ^τ Q_{t+τ} (Y^P_{t+τ}). - Q_{t+τ} defined as product of π and R terms across horizons. --- ### 2.2 Tariffs, prices, and exchange rates ### Tradable price indices with tariffs - Home tradable price index with ad-valorem tariff τ_t on foreign good: - P_{T,t} = [ (1−α) P_{H,t}^{1−η} + α ((1 + τ_t) P_{F,t})^{1−η} ]^{1/(1−η)}. - Foreign tradable price index with tariff τ^*_t on imports from Home: - P^*_{T,t} = [ (1−α) P^{* 1−η}_{F,t} + α ((1 + τ^*_t) P^*_{H,t})^{1−η} ]^{1/(1−η)}. ### Tradable inflation and terms of trade - Define terms of trade S_t = P_{F,t} / P_{H,t}. - Relative tradable price: - P_{T,t}/P_{H,t} = [ (1−α) + α ((1 + τ_t) S_t)^{1−η} ]^{1/(1−η)}. - Tradable inflation (Π_{T,t} = P_{T,t}/P_{T,t−1}, Π_{H,t} = P_{H,t}/P_{H,t−1}): - Π_{T,t} = Π_{H,t} [ ((1−α) + α ((1 + τ_t) S_t)^{1−η}) / ((1−α) + α ((1 + τ_{t−1}) S_{t−1})^{1−η}) ]^{1/(1−η)}. - Foreign tradable inflation analogous, using inverse terms of trade S_t^{−1}. ### Tariff shock effects - A tariff shock generates an immediate wedge between domestic price inflation Π_H and tradable consumer price inflation Π_T, which feeds into aggregate CPI inflation according to tradables share γ. - Tariff does not directly enter household budget constraint except via relative price effects (see Lemma A.1). ### Non-tradable and aggregate inflation - Non-tradable inflation Π_{N,t} driven by domestic real marginal costs MC_{N,t}. - Aggregate inflation Π_t (with P_t = [ γ P_{T,t}^{1−χ} + (1−γ) P_{N,t}^{1−χ} ]^{1/(1−χ)}): - Π_t = [ γ (Π_{T,t} P_{T,t−1}/P_{t−1})^{1−χ} + (1−γ) (Π_{N,t} P_{N,t−1}/P_{t−1})^{1−χ} ]^{1/(1−χ)}. ### Real exchange rate decomposition - Real exchange rate Q_t = ε_t P^*_t / P_t decomposed as: - Q_t = (ε_t P^*_{T,t} / P_{T,t}) × [ γ + (1−γ) (P^*_{N,t}/P^*_{T,t})^{1−χ} ]^{1/(1−χ)} / [ γ + (1−γ) (P_{N,t}/P_{T,t})^{1−χ} ]^{1/(1−χ)}. - Internal tradable/non-tradable price wedges affect aggregate RER even if tradable prices obey law of one price. --- ### 2.3 Firms and production ### Production technology and taxes/subsidies - Firms producing tradable (H) and non-tradable (N) goods: - Y_{k,t}(i) = A_{k,t}(i) K_{k,t}(i)^{μ_k} N_{k,t}(i)^{1−μ_k}. - Firms face investment subsidies φ_t, corporate tax τ_K, and consumption ad-valorem tax τ_C. - Investment adjustment costs and standard tax treatment: depreciation P_k δ K_{k,t} deducted from revenues. ### Firm optimization and marginal cost - First-order condition for labor: - φ_t (1−μ) A_{k,t} K_{k,t}^{μ} N_{k,t}^{−μ} = (1−τ_{K,t}) w_t. - Post-tax marginal cost: - φ_t = (1−τ_{k,t}) MC_{k,t} = (1−τ_{k,t}) w_t (1−μ)^{-1} A_{k,t}^{−1} K_{k,t}^{−μ} N_{k,t}^{μ} (interpretable as post-tax marginal cost). - Investment FOC and Tobin's q include price-of-output terms P_{k,t}/P_t and investment adjustment parameter κ. ### Transmission and markup shocks - Cost of investment may be P_T or P_N depending on sector; tradable investment can use imported goods (P_H). - Directive to produce at lower markups (higher MC) increases q and desired investment and draws labor into sector. - Simulated markup shock M^ε_t follows AR(1) in shock processes (section 4). ### Market clearing - Aggregate production must satisfy consumption and investment demands; tradable investment uses tradable output aggregator: - Y_{H,t} = I_{H,t} + C_{H,t} + C^*_{H,t} + I^*_{H,t}. - Home tradable market clearing (domestic use + exports): - Y_{H,t} = (1−α) (P_{H,t}/P_{T,t})^{−η} (C_{T,t} + I_{T,t}) + α (P_{H,t}/(ε_t P^*_{T,t}))^{−η} (C^*_{T,t} + I^*_{T,t}). - Non-tradable clearing: - Y_{N,t} = C_{N,t} + I_{N,t}. ### Pricing and optimal price setting (PCP with Rotemberg costs) - Retailer faces price adjustment costs φ_π/2 (P_{H,t+j}(i)/P_{H,t+j−1}(i) − 1)^2 Y_{H,t+j} and chooses sequence {P_{H,t+j}(i)}. - FOC (aggregate form) yields pricing relation with price adjustment terms; without adjustment costs reduces to: - P_{H,t}/P_t = ε/(ε−1) MC_t/P_t (standard markup rule). - Analogous equations for non-tradable and foreign sectors. ### Aggregate profits distributed to perpetual youth households - Sectoral profits Π_{k,t}: - Π_{k,t} = P_{k,t}/P_t [ (1−τ_{K,t}) [ (1−(ε−1)/ε)(1−μ) Y_{k,t} + μ Y_{k,t} ] + δ τ_{k,t} K_t − I_{k,t} ]. - Aggregate profits Π_t = Π_{H,t} + Π_{N,t}. --- ### 2.4 Accounting identities and policy rules ### Aggregates - Aggregate output: - Y_t = GDP_t = (P_{H,t}/P_t) * Y_T + (P_{N,t}/P_t) Y_N. - Aggregate investment: - I_t = (P_{H,t}/P_t) * I_H,t + (P_{N,t}/P_t) I_N,t. ### Trade and current account - Current account balance: - CAB_t = (Y_t − C_t) + (R_{I,t−1} − 1) B_{t−1}/π_t. - Net international debt evolution: - B_t − B_{t−1} = CAB_t. ### Fiscal accounts and government bond holdings - Government revenues: - Rev_t = τ_t P_{F,t}/P_t C_{F,t} + τ_{C,t} C_t + τ_{L,t} W_t N_t + τ_{K,t} ( Π_t − (δ) K_t + (1−φ_t) I_t ). - Government expenditure: - E_t = T_t + φ_t I_t. - Government budget constraint: - B_{H,t} + E_t = Rev_t + R_{t−1}/π_t B_{H,t−1} + B^ε_I. - Government holdings of international bond B^ε_I evolve: - B_{G,I} − R_{I,t−1}/π_t B_{G,I} = ε_{G,t}. - Government fiscal rule (debt-stabilizing): - E_t / Y_t = ρ_F (Rev_t / Y_t) − ρ_B B_{H,t} / Y_t. - Note: lower ρ makes stock of debt adjust more to shocks to expenditure or revenue. ### Monetary policy - Monetary policy (Taylor-type rule): - R_{H,t} = R_{ss} ( π_t / π_{ss} )^{ψ_π}. --- ### 3 Parameters (calibrated values shown) - Preferences & Households - Discount Factor β = 0.995 - Inter-temporal substitution σ = 2.0 - Inverse Frisch Elasticity ν = 0.33 - Habit Persistence ρ = 0.7 - Hand-to-Mouth Share ω = 0.3 - Technology & Capital - Capital Share μ_k = 0.33 - Depreciation Rate δ = 0.025 (Quarterly) - Inv. Adjustment Cost κ = 3.0 - Openness & Sectors - Tradable Share γ = 0.5 - T/NT Elasticity χ = 0.5 - Trade Elasticity η = 2.0 - Openness α = 0.3 - Nominal Frictions & Policy - Price Elasticity ε = 6.0 - Rotemberg Parameter (prices) φ_π = 70 - Rotemberg Parameter (wages) φ_w = 70 - Taylor Rule (Inf) ψ_π = 1.5 --- ### 4 Shock processes - Exogenous processes are AR(1) in logs or levels: - Sector-specific productivity (logs) AR(1): - ln(A_{T,t}) = ρ_A ln(A_{T,t−1}) + (1−ρ_A) ln(Ȧ_T) + ε_{A,T,t}. - ln(A_{N,t}) = ρ_A ln(A_{N,t−1}) + (1−ρ_A) ln(Ȧ_N) − ε_{A,N,t}. - Foreign counterparts analogous with * superscripts. - Investment subsidy shocks (persistent): - τ_{sub,T,t} = ρ_{τ,sub} τ_{sub,T,t−1} + (1−ρ_{τ,sub}) ȳ_{τ,sub,T} + ε_{τ,sub,T,t}. - τ^*_{sub,T,t} analogous. - Markup shock on tradable sector: - M^ε_t = ρ_M M^ε_{t−1} + ε_{M,t}. - Tariffs and factor taxes follow AR(1); labor and capital tax shocks enter with negative sign (tax cut): - τ_t = ρ_τ τ_{t−1} + ε_{τ,t}. - τ_{L,t} = ρ_{τ,L} τ_{L,t−1} − ε_{τ,L,t}. - τ_{K,t} = ρ_{τ,K} τ_{K,t−1} − ε_{τ,K,t}. - Government reserve accumulation (home government international bond holdings B_{g,I}): - B_{g,I,t} = ρ_{gov} B_{g,I,t−1} − ρ_{g,stab} ( B_{g,I,t−1} / GDP_{t−1} ) + ε_{gov,t}. --- *Source: wpiea2026067-online-appendix - 2.1 Households and consumption (PDF chapter).* ### 5.1 Micro-industrial policy: productivity related developments ### 5.1 Micro-industrial policy: productivity related developments ### Temporary vs permanent 1% tradable productivity boost - Shock: 1% boost to tradable sector productivity; permanent case sets ρA = 1, temporary case sets ρA = 0. - Dynamics (perpetual youth households): - Temporary shock: households save part of the windfall, generating a current account surplus that exceeds the initial increase in investment (partly due to investment adjustment frictions). - Permanent shock: saving initially rises by less than in the temporary case (anticipation of higher future income), then rises alongside higher real interest rates and a higher marginal product of capital. - Key reported magnitudes: 1% productivity shock (temporary vs permanent) — effects presented in figures on CAB(%GDP), Saving(%GDP), Inv.(%GDP) across quarters. ### Non-tradable productivity fall vs tradable productivity rise - Paradox: A negative shock to non-tradable productivity (A_N fall by 1%) can mirror the current account impact of a positive tradable shock (A_T rise by 1%). - Mechanism: - Low elasticity of substitution between sectors (χ < 1). - When A_N falls, the relative price of nontradables surges. - With tradables and nontradables as complements, households do not substitute away from now-expensive nontradables. - Result: RER appreciates, increasing the relative value of domestic output and accompanied by a decline in the current account balance. - Variables highlighted: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Relative nontradable price, RER(-is appreciation) over quarters. --- ### 5.2 Micro-industrial policy: Subsidies ### Investment subsidy (φ) and financing modalities - Policy: Simple subsidy to investment (φ) encourages capital deepening; similar impacts to productivity simulations due to productivity via capital deepening. - Two financing cases: 1. Tax-financed: subsidy fully funded via a tax on consumption (τc). - Cost of subsidy immediately lowers consumption: - Perpetual youth households bear cost immediately and reduce consumption. - HTM households lower consumption given income constraints. - Aggregate saving declines consistent with immediate tax burden. 2. Debt/deficit-financed: subsidy initially heavily deficit financed via low ρg (reducing the increase in taxes as government debt rises). - Future tax liabilities are partially discounted by perpetual youth households due to probability of death θ. - Households therefore save less than the cost of the subsidy, lowering aggregate saving. - Reported variables: CAB(%GDP), Saving(%GDP), Inv.(%GDP) under "De-cit-nanced" and "Tax-nanced" scenarios across quarters. --- ### 5.3 Micro-industrial policy: Production targets ### Mandated output increase / permanent fall in tradable markup - Simulation: Permanent fall in the markup in the tradable sector to approximate mandated production targets that sacrifice short-run profit. - Direct effects: - Lowers price of tradable output. - Increases production and factor inputs (firms accept lower profits). - Indirect effects: - Improved demand and output for tradables creates a relative shortage of nontradables. - Inelastic demand for nontradables → sharp rise in relative prices and RER appreciation. - Increased investment and labor demand to alleviate nontradable shortages. - Overall price level rises; higher expected future income from capital deepening induces dissaving, lowering the current account balance. - Variables shown: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Relative nontradable price(%), RER(-is appreciation) over quarters. --- ### 5.4 Macro-industrial policy: Forced saving and capital controls ### Preference shock to discount rate (β) and capital flow regimes - Policy: Persistent preference shock to household discount rate β to increase saving and lower interest rates (forced saving). - Two cases approximating capital flow regimes via φ (adjustment cost on international bond holdings): 1. Free financial flows (φ = 0): - Persistent increase in saving → current account balance rises. - Domestic and global (international bond yield) interest rates lower. - Exchange rate initially depreciates before investment rises due to lower interest rates. 2. Strict capital flow restrictions (high φ, "CFM" case): - Households cannot accumulate foreign assets to match desired saving. - Reduction in import demand from consumption suppression → large RER appreciation. - Domestic interest rates fall below international bond rate; domestic investment rises sufficiently so current account balance is little changed. - Variables highlighted: CAB(%GDP), Saving(%GDP), InterestRates (Home / International bond), RER(-is appreciation). --- ### 5.5 Macro-industrial policy: reserve accumulation ### Government accumulation of international bonds with capital controls - Setup: φ set to 1 to introduce high costs for private sector international bond transactions; shock εg causes persistent government accumulation of international bonds (higher BG,I). - Financing and equilibrium effects: - Accumulation funded primarily through domestic bond issuance. - Higher domestic interest rates required to induce private sector to purchase new net supply of bonds (private sector cannot sell down international bond holdings due to high transaction costs). - Result: domestic interest rates rise above global bond rate; persistent depreciation; fall in domestic absorption from lower consumption and investment. - Variables presented: CAB(%GDP), Saving(%GDP), Inv.(%GDP), InterestRates (Home / Intlbond), RER(-is appreciation) over quarters. --- ### 5.6 Combined Micro and Macro industrial policy ### Interaction: tradable production targets + forced saving - Micro policy: Lower desired markups in tradable sector (tradable output targets increase) → capital deepening expectations lower saving relative to investment; relative shortage of nontradables raises relative prices and causes RER appreciation. - Macro policy: Additional forced saving via intertemporal preference shock (to β) lowers consumption persistently (no capital controls in this combination). - Interaction effects: - Forced saving increases saving relative to investment, counteracting dissaving from production-target expectations. - Forced saving counteracts the RER appreciation that occurs in the standalone micro-IP case. - Variables illustrated: CAB(%GDP), Inv.(%GDP), Saving(%GDP), Inflation(ppt), Relative nontradable price, RER(-is appreciation), ToT(-is appreciation), Real interest rate across quarters. - Note: Simulated as a highly persistent (but declining) fall in desired tradable sector markup; "with additional forced saving" includes an additional shock to intertemporal preferences lowering consumption. --- ### 6.1 Consumption smoothing (model extension) ### Habit formation and lifetime budget constraint with perpetual youth - Household utility: - E0 ∑∞_{t=0} β^t [1/(1−σ) (C_t(i) − ρ C_{t−1})^{1−σ} − 1/(1+φ) N_t^{1+φ}] - Habits reference aggregate (not individual) consumption in previous period. - Definitions and substitutions: - Replace C̃_t = C_t − ρ C_{t−1}. - Budget constraint written in terms of C̃_t and includes consumption tax (1 + τC,t), aggregate bond holdings B_T,t, investment I_t (1 − φ_t), wage income, profits Π_t, transfers T_t, and adjustment costs on international bond holdings. - First-order conditions: - C̃_t^{−σ} = (1 + τC,t) λ_t - λ_t = β R_t / Π_{t+1} λ_{t+1} - Forward iteration and lifetime budget constraint: - Present-value expressions involve survival probability θ, pricing kernels Q_{t+τ}, and the term R_{t−1} (1−ω) θ π_t B_T,t−1. - For surviving households the consumption equation includes an infinite sum ∑_{τ=0}^∞ (θ β^{1/σ})^τ (...) and adjustment for new entrants (weighting by θ and (1 − θ)), leading to elimination of θ term on return to B_T,t−1 in aggregate consumption function. - Applicable to households who have survived the previous period and inherited assets; new entrants have no assets but receive wages and profits. --- ### A.1 Lemma (Tariffs and the budget constraint) - Lemma A.1: The imposition of tariffs does not alter the standard form of the budget constraint; effects operate through relative and aggregate price adjustment. - Real cost of consumption of tradables expressed as: - P_{H,t}/P_t C_{H,t} + (1 + τ_t) P_{F,t}/P_t C_{F,t} - Rewriting with relative price and substitution for C_H and C_F as functions of total consumption C_T and terms of trade S_t: - Expressions involve parameters α, η and the tariff term (1 + τ_t) raised to powers (1 − η) and further algebra shows the household budget constraint reduces trivially to C_T. - Conclusion: inclusion of tariffs changes household outcomes only via aggregate and relative price effects, not the structural form of the budget constraint. *Source: wpiea2026067-online-appendix - 5.1 Micro-industrial policy: productivity related developments* --- _Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026067-online-appendix.pdf_