## _wp1034 — Dynamic Stability and Business Cycle Stabilization ## Source details **Canonical URL:** [_wp1034 — Dynamic Stability and Business Cycle Stabilization](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1034.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1034.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1034.pdf.json) --- ### Dynamic stability: fiscal objective and long-run relationships - Fiscal policy objective: - Ensure a non-explosive government-debt-to-GDP ratio by adjusting tax rates or reducing expenditure to stabilize the interest-inclusive government surplus-to-GDP ratio gssrat_t at a long-run level chosen by policy. - Government surplus (level) (preserved notation): - gs_t = − [ ˇb_t − ˇb_{t−1} / π_t^{gn} ] = ˇτ_t + ˇg_X_t − p_G_t ˇG_t − ˇΥ_t − i_{t−1} / π_t^{gn} ˇb_{t−1}. (Equation (246)) - Government surplus-to-GDP ratio: - gsrat_t = −100 (B_t − B_{t−1}) / (P_t gdp_t) = 100 ˇgs_t / gˇdp_t. (Equation (247)) - The model allows gssrat_t to follow an exogenous stochastic process. - Debt–surplus long-run relationship: - gssrat_t = −4 [ ˇπ_t^{gn−1} / ˇπ_t^{gn} ] ˇbssrat_t. (Equation (248)) - Interpretation preserved: for a given nominal growth rate, choosing surplus target gssrat_t implies a debt target ˇbssrat_t and keeps debt from exploding. ### Business-cycle stabilization: structural fiscal surplus rule and trade-offs - Structural fiscal surplus targeting rule (preserved functional form): - gsrat_t = gssrat_t + d_debt (β b_rat_t − β bssrat_t) + d_gdp ln(gˇdpfisher_t / gˇdppot_t) + d_tax (βτ_t − βτpott) gˇdp_t + d_rawmat (βgX_t − βgpotX,t) gˇdp_t. (Equation (249) structure preserved) - Trade-offs in parameter choices: - Setting d_debt = 0: - Ensures non-explosive government-debt-to-GDP ratio of βbssrat_t. - Yields long-run autoregressive coefficient on debt at 1/(βπ_tgn) that is very close to one. - Setting d_debt > 0: - Ensures faster convergence of debt. - Increases volatility of government surpluses. - Balanced-budget rule (d_debt = d_gdp = d_tax = d_rawmat = 0): - Described as highly procyclical and undesirable. - Practical calibration and policy variants: - OECD-style calibrations available for d_gdp for many countries. - Some countries use d_gdp = 0 and d_tax = 1 (excess tax revenue in booms used to pay down debt), reducing variability of fiscal instruments and output relative to balanced-budget rule. - A counter-cyclical rule would set d_tax > 1. ### Instrumenting the targeting rule (examples and implications) - Targeting versus instrument distinction: - Rule (249) is a targeting rule; instruments must be specified (default instrument: labor tax rate τL,t). - Examples of instrument comovement (preserved forms): - τc,t = τ̄c + d_ctax (τL,t − τ̄L). (Equation (258)) - τk,t = τ̄k + d_ktax (τL,t − τ̄L). (Equation (259)) - d_ctax = d_ktax = 1 → perfect comovement of τL,t, τc,t, τk,t. - d_ctax = d_ktax = 0 → only labor tax rates change. ### Monetary policy rule and equilibrium-rate proxy - Monetary rule (preserved full form): - i_t = E_t(i_{t−1})^{δ_i} (r_eq_t ̃π_t)^{1−δ_i} [ (̃π_t / βπ_t)^{(1−δ_i)δ_π} (gˇdpfisher_t / gˇdppot_t)^{(1−δ_i)δ_y} (gˇdpfisher_t / gˇdpfisher_{t−4})^{(1−δ_i)δ_ygr} (ε_t / βε_t)^{δ_e} ] (1 + S_int_t). (Equations (260)-(261)) - ̃π_t = π_t^{δ ̃π} π_{t+1}^{1−δ ̃π}. - Important model-specific exception: - Non-Ricardian features imply no unchanging steady-state GDP or real interest rate. - Proxy r_eq_t uses geometric moving averages of world pre-risk-premium real interest rate r_world_t and of the risk premium ξ_ma_t: - r_eq_t = r_world_t ξ_ma_t. (Equation (262)) - r_world_t = Π_{ ̃N j=1} [ r_ma(j)_t ]^{gdp_ss(j) / Σ_{ ̃N i=1} gdp_ss(i)}. (Equation (263)) - r_ma(j)_t = [ r_preξ(j)_t + r_ma(j)_{t−1}^{k_r} ]^{1/(1+k_r)}. (Equation (264)) - ξ_ma_t = [ ξ_t + ξ_ma_{t−1}^{k_r} ]^{1/(1+k_r)}. (Equation (265)) - Policy special cases: - Exchange-rate targeting: δ_i = 1 and δ_e −→ ∞. - Quarterly versions replace one-year-ahead inflation with a one-year-ahead four-quarter geometric moving average. ### Shocks: transitory and unit-root components (specification) - Generic two-component shock specification (preserved): - x_t = (1 − ρ_x) ̃x_t + ρ_x x_{t−1} + u_x_t; ln(̃x_t) = ln(̃x_{t−1}) + u_{̃x_t}. (Equations (266)-(267)) - Policy and relative-price unit roots: - ln(βπ_t) = ln(βπ_{t−1}) + u_{π_t}. (268) - gssrat_t = gssrat_{t−1} + u_{gss_t}. (269) - ln(̃p_y_t) = ln(̃p_y_{t−1}) + u_{py_t}, y ∈ {I,G,exp}. (270) - Representative transitory shock forms preserved (examples): - S_int_t = (1 − ρ_int) + ρ_int S_int_{t−1} + u_int_t. (271) - S_inv_t = (1 − ρ_inv) + ρ_inv S_inv_{t−1} + u_inv_t. (272) - ξ_f_t = ρ_fxp ξ_f_{t−1} + u_fxp_t. (274) - ξ_b_t = ρ_gbp ξ_b_{t−1} + u_gbp_t. (275) - Productivity catch-up feature: - US: A_J(US)_t = (1 − ρ_AJ(US) + e_AJ(US)_t) ̃A_J(US)_t + ρ_AJ(US) A_J(US)_{t−1}. (277) - Country j: A_J(j)_t = (1 − ρ_AJ(j)) ̃A_J(j)_t + catchup(j) * (A_J(US)_t − ̃A_J(US)_t) + ρ_AJ(j) A_J(j)_{t−1} + e_AJ(j)_t ̃A_J(j)_t. (278) - Catching-up can be turned off (modular). ### Balance of payments and current account (expressions) - Current-account and bond-market clearing (preserved nominal expression): - e_t ˇf_t = i_{t−1}( ̃N) ε_t (1 + ξ_f_{t−1}) / π_t^{gn} e_{t−1} ˇf_{t−1} + p_TH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_TX_t(1,j) + ˇd_TM_t − p_TF_t ˇY_TF_t + p_DH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_DX_t(1,j) + ˇd_DM_t − p_DF_t ˇY_DF_t + ˇX_x_t + ˇd_F_t − ˇf_X_t. (281) - Σ_{ ̃N j=1} ˇf_t(j) = 0. (282) - ca_t = e_t ˇf_t − e_{t−1} ˇf_{t−1} / π_t^{gn}. (283) - GDP definition used in surplus and structural rule: - gˇdp_t = p_C_t ˇC_t + p_I_t ˇI_t + p_G_t ˇG_t + ˇX_x_t + p_TH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_TX_t(1,j) + ˇd_TM_t − p_TF_t ˇY_TF_t + p_DH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_DX_t(1,j) + ˇd_DM_t − p_DF_t ˇY_DF_t. (284) ### Calibration: steady-state and key parameter values (annual version; raw-materials sector excluded) - Regions modeled: five-region model — United States (US), emerging Asia (AS), euro area (EU), Japan (JA), remaining countries (RC). - World steady-state growth and rates: - World technology growth g = 1.015 (1.5% p.a.). - World population growth n = 1.01 (1% p.a.). - Steady-state inflation rates: 2.0% in US, AS, EU and RC; 1% in JA. - Long-run real interest rate βr = 1.03 (3% p.a.), equalized across countries. - Household calibration: - χ = 0.95 (average remaining time at work of 20 years). - θ = 0.9 (planning horizon 1/(1 − θ) = 10 years). - γ = 4 (intertemporal elasticity of substitution 0.25). - Labor-supply elasticity targeted at 0.5 via η. - Shares of liquidity-constrained agents ψ: 25% in US, EU, JA; 50% in AS and RC. - Dividend share of liquidity-constrained agents ι: half population shares in all regions (specific table entries: 0.125, 0.25, 0.125, 0.125, 0.25). - Fiscal calibration target: - A one percentage point increase in U.S. government-debt-to-GDP raises U.S. (and world) real interest rate by approximately three to four basis points (calibration target aligning with cited literature). - Technology and markups (selected elasticities and markups): - ξ_ZN = ξ_ZT = 1; ξ_NM = ξ_TM = ξ_T = ξ_I = ξ_C = 0.75; ξ_A = ξ_G = 0.5. - Steady-state markups: EN = ET = EU = 1.1; EI = EC = ER = 1.05; ENM = ETM = 1.025. - Expenditure and factor shares (selected ratios): - Share in World GDP: 27.4, 12.3, 22.0, 9.1, 29.3 (five-region ordering preserved). - Consumption / GDP: 65.1, 59.2, 58.1, 59.8, 59.1. - Private Investment / GDP: 17.2, 25.0, 18.3, 21.0, 19.0. - Government Spending / GDP: 17.5, 16.0, 23.5, 19.5, 22.0. - Government Investment / GDP: 2.5, 4.0, 3.0, 2.5, 2.0. - Public capital and depreciation: - δ_G = 0.04 (4% p.a.). - α_G = 0.1 (elasticity of aggregate output w.r.t. public capital; implied by cited estimate 0.14 and model simulations). - Financial accelerator calibration: - Leverage (corporate debt / corporate equity) = 100 in all sectors and regions. - Annual bankruptcy rate = 8 percent. - Steady-state external finance premium = 1.5 percent. - Monetary and fiscal rule parameters (selected table entries): - Monetary: δi = 0.715, 1, 0.343, 0.392, 0.715 (regional entries); δπ = 1.034, 0, 1.483, 0.913, 1.034; δ̃π = 0.216, 1, 0.237, 0.216, 0.216; δygr = 0.25, 0, 0, 0; δe = 0106 000 (table formatting preserved). - Fiscal: dgdp = 0.34, 0.25, 0.49, 0.33, 0.30; ddebt = 00000; dtax = 00000; drawmat = 00000; dctax = 00000; dktax = 00000 (table formatting preserved). ### Model structure, modularity, and applications - Core blocks included in fiscal analyses: - Fiscal-rule block, shocks block, and balance-of-payments block are core and included in the application “Fiscal Stimulus to the Rescue?”. - Modularity highlights: - OLG households, tradables manufacturing, capital-goods producers and financial accelerator are core and present in the application “Fiscal Stimulus to the Rescue?”. - Raw-materials sector, catching-up technology feature, and certain wage rigidities are modular and can be turned off; raw-materials sector typically omitted unless focus on raw materials. - Retail sector is not core but never dropped; needed to generate inertial consumption dynamics. - Reported practical users: - Central banks and IMF Area Departments have used/adapted GIMF (examples in source): HKMA, CBR, BdP (PESSOA), Banco Central de Reserva de Perú, Banque de France; and multiple IMF country and policy studies. *Source: _wp1034 — 1. Dynamic Stability; 2. Business Cycle Stabilization; calibration and model notes (IMF Working Paper chapter/section).* ### 1. Dynamic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45 ### 1. Dynamic Stability ### Major Sections and Subsections - 2. Business Cycle Stabilization .................................................. 45 - D. Monetary Policy .................................................................. 47 - XVII. Shocks ......................................................................... 48 - XVIII. Balance of Payments ..................................................... 49 - XIX. Calibration .................................................................. 50 - XX. Applications of GIMF ....................................................... 52 - A. Central Banks Using GIMF .............................................. 52 - B. IMF Area Departments Using GIMF .................................. 53 ### Appendices (Content Areas) - 1. Population Growth .......................................................... 55 - 2. Optimality Conditions for OLG Households .......................... 56 - 3. Consumption and Wealth .................................................. 58 - 4. Manufacturers .............................................................. 62 - 5. Entrepreneur’s Problem - Lognormal Distribution .................. 64 - 1. Basic Properties ofΓandG ............................................. 64 - 2. Basic Properties of the Lognormal Distribution ................ 64 - 3. Derivations .............................................................. 64 - 4. Final Equation System ................................................ 66 ### References and Supporting Material - References ....................................................................... 67 ### Tables (Listed) - 1. Long Run Growth Rates and Interest Rates .......................... 68 - 2. Utility Functions ........................................................ 68 - 3. Elasticities of Substitution ........................................... 68 - 4. Steady State Markups .................................................... 68 - 5. Steady State Expenditure to GDP Ratios .......................... 69 - 6. Steady State Factor Shares and Depreciation Rates .......... 69 - 7. Miscellaneous Steady State Ratios and Parameters .......... 69 - 8. Financial Accelerator Sector ......................................... 70 - 9. Monetary Rule Parameters ............................................. 70 - 10. Fiscal Rule Parameters ................................................ 70 ### Figures (Listed) - 1. Goods and Factor Flows in GIMF .................................... 71 - 2. Trade Matrix: Intermediate Goods ................................ 72 - 3. Trade Matrix: Investment Goods .................................. 72 *Source: _wp1034 - 1. Dynamic Stability (PDF chapter/section).* ### 4. Trade Matrix: Consumption Goods . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 73 ### 4. Trade Matrix: Consumption Goods ### I. Introduction - Presents the theoretical structure of the International Monetary Fund’s Global Integrated Monetary and Fiscal Model (GIMF), a multicountry dynamic general equilibrium model used for policy and risk analysis. - Key historical and planned uses: - Used extensively inside the IMF and at a small number of central banks for policy and risk analysis. - Produced a significant number of GIMF-based IMF Working Papers and Special Issues Papers. - GIMF simulations used for World Economic Outlook scenario analyses and internal risk assessment analyses since 2008. - Future uses include forecasting exercises based on filtering historical data. - Traditional strength: - Ability to analyze fiscal policy questions due to incorporation of non-Ricardian features that make spending-based and revenue-based fiscal measures non-neutral. - Intensive deployment for analyzing short-run effectiveness of fiscal stimulus packages. - Also useful for long-run sustainability questions: link between fiscal deficits and real interest rates, crowding out, and current account deficits. - Recent and ongoing extensions: - Maco-financial linkages: financial accelerator mechanism for the non-financial corporate sector—roles for corporate net worth, corporate leverage, external finance premia, and bankruptcies (described at length in the document). - Banking sector under development: intermediates funds between households and non-financial corporate sector; has its own net worth and leverage. - Raw-materials sector: generic raw-materials sector with inelastic supply and demand price elasticity calibratable by raw material; motivated by 2008 oil-price shocks. - Modularity: - GIMF is highly modular; sectors can be turned on or off depending on application complexity. - Modular structure operational in TROLL versions; DYNARE users: similar modular structure in development. - At the end of each agent section the paper comments on modularity and which features can be turned off depending on question; also comments on features present in the version used for the paper “Fiscal Stimulus to the Rescue?”. ### II. Model Overview - World structure and notation: - World consists of Ñ countries. - Domestic economy indexed by j = 1; foreign economies by j = 2,...,Ñ. - All parameters except gross population growth n and gross technology growth g can differ across countries. - Figure 1 (in source) illustrates flows for the two-country case. - Agents and sectors: - Two types of households: overlapping generations (OLG) with finite planning horizons and liquidity-constrained (LIQ) households who consume after-tax income each period. - OLG household death probability per period: (1 − θ(j)), implying average planning horizon 1/(1 − θ(j)). - Birth counts each period: - OLG births: N(j) n_t (1 − ψ(j)) (1 − θ(j)^n) - LIQ births: N(j) n_t ψ(j) (1 − θ(j)^n) - Aggregation implies total agents in country j: N(j) n_t. - Manufacturers produce tradable and nontradable goods; buy capital services, labor, and raw materials; subject to nominal price-setting rigidities and real rigidities in hiring and raw-material use. - Capital goods producers face investment adjustment costs. - Entrepreneurs finance capital with internal and external financing; capital income tax levied on capital goods producers (without Financial Accelerator) or on entrepreneurs (with Financial Accelerator). - Unions have nominal wage rigidities and sell labor. - Import agents: domestically owned, located in each export destination; when pricing-to-market is assumed they face nominal rigidities in foreign currency. - Distributors assemble nontradable and tradable goods; changes in imported input volumes are subject to adjustment costs. - Publicly provided capital stock (infrastructure) is combined with private sector output; maintained by government investment financed by tax revenue and government debt. - Two-layer import structure: upstream import agents for producers and a second set of import agents for final output—critical to generate high trade-to-GDP ratios in small open economies. - Consumption and investment goods producers combine domestic and foreign output; face nominal rigidities in price setting. - Retailers are monopolistically competitive facing real rigidities: flexible prices but costly adjustments in sales volume—contributes to inertial consumption dynamics. - Trends and normalization: - World technology grows at constant rate g = T_t / T_{t−1}. - Population grows at constant rate n. - Real variables x_t rescaled by technology T_t and population n_t: x̄_t = x_t / (T_t n_t); steady state denoted by x̄. - Quantities of labor rescaled only by n_t. - Asset markets and financial assumptions: - Asset markets incomplete. - Complete home bias in government debt: nominally non-contingent one-period domestic-currency bonds. - Only internationally traded assets: nominally non-contingent one-period bonds denominated in currency of Ñ. - Complete home bias in ownership of domestic firms; equity not traded in domestic financial markets—households receive lump-sum dividend payments. - Financial accelerator and banking sector details discussed in model extensions. - Pricing and numeraire conventions: - Retail price index: P_Rt; retail variety prices P_Rt(i); stochastic elasticity of substitution σ_Rt. - Consumption tax τ_c,t payable on pre-incentive price P_Ct; retailers offer incentives that affect effective retail purchase price P_Rt. - Chosen numeraire: P_t (final output price level). - Notation: real wage w_t = W_t / P_t; relative price p_x,t = P_x,t / P_t; gross inflation π_x,t = P_x,t / P_x,t−1; gross nominal exchange rate depreciation ε_t = E_t / E_t−1. - Trade and adjustment costs: - Changes in volume of imported inputs (both upstream and final) are subject to adjustment costs—important for replicating trade-to-GDP ratios. - Additional modeling choices: - Habit persistence not used to generate consumption inertia due to preference constraints (balanced growth and ability to aggregate across generations); retail-sector sales adjustment costs used instead. ### III. Overlapping Generations Households - Preferences and utility: - Representative OLG household of age a derives utility at time t from consumption c_OLG_{a,t}, leisure (S_Lt − ℓ_OLG_{a,t}), and real balances (M_{a,t} / P_Rt). - Lifetime expected utility: - E_t Σ_{s=0}^∞ (β_t θ)^s [ (1 / (1 − γ)) ( c_OLG_{a+s,t+s}^{η_OLG} (S_L{t+s} − ℓ_OLG_{a+s,t+s})^{1−η_OLG} )^{1−γ} + u_m/(1−γ) (M_{a+s,t+s} / P_R{t+s})^{1−γ} ] - Parameters: θ < 1 (degree of myopia), γ > 0 (coefficient of relative risk aversion), 0 < η_OLG < 1, u_m > 0, β_t stochastic discount factor. - Cashless limit and money: - Analysis focuses on the cashless limit u_m → 0 (Woodford): money demand equation becomes redundant and inflation not directly distortionary for consumption-leisure decision. - Cashless limit adopted to avoid implausible indirect distortionary effects of inflation via government budget constraint. - GIMF is not designed to quantify costs of inflation. - Consumption aggregation and retail varieties: - Consumption c_OLG_{a,t} is CES aggregate over retailed consumption varieties with stochastic elasticity σ_Rt: - c_OLG_{a,t} = [ ∫_0^1 c_OLG_{a,t}(i)^{(σ_Rt − 1)/σ_Rt} di ]^{σ_Rt/(σ_Rt − 1)} - Individual demand: c_OLG_{a,t}(i) = (P_Rt(i) / P_Rt)^{-σ_Rt} c_OLG_{a,t} - Aggregate retail price index: P_Rt = [ ∫_0^1 P_Rt(i)^{1−σ_Rt} di ]^{1/(1−σ_Rt)} - Assets and interest rates: - Two bond types households can hold: - Domestic bonds: B_{a,t} (and, in Financial Accelerator version, B_N_{a,t} + B_T_{a,t} issued by banks lending to nontradables and tradables sectors). - Foreign bonds: F_{a,t} denominated in currency of Ñ. - Nominal exchange rate E_t; nominal net foreign assets in domestic currency E_t F_{a,t}. - Gross nominal interest rates on assets held from t to t+1: - Domestic: i_t / (1 + ξ_b,t) - Foreign: i_t(Ñ) (1 + ξ_f,t) - ξ_b,t is domestic risk premium (ξ_b,t < 0 when private sector faces larger marginal funding rate than public sector). - ξ_f,t is foreign exchange risk premium. - For country Ñ: i_t = i_t(Ñ) and ξ_f,t = 0 (in the version noted). - Foreign exchange risk premium functional forms: - Nonlinear asymmetric form: - ξ_f,t = y_1 + y_2 ( (ca_t / gdp_t − y_4)^{y_3} ) + S_fx,t - S_fx,t mean-zero risk premium shock; y_1 − y_4 parameters; y_1 constrained by y_1 = −y_2 / (−y_4)^{y_3} to generate zero premium at zero current account. - ca_t / gdp_t is current account-to-GDP ratio. - Suitable for emerging markets because asymmetry allows steeply increasing premium at large current account deficits. - Linear option available: - ξ_f,t = −y_1 (ca_t / gdp_t) + S_fx,t - Domestic risk premium: - Can be made to depend on government-debt-to-GDP ratio or treated as an exogenous stochastic process. - Example: recent financial events may be characterized by a persistent negative shock to ξ_b,t. - Financial market participation and insurance: - Households entering financial markets pay an insurance premium (1 − θ)/θ on financial wealth each period while alive; insurance encashes entire financial wealth at death. - Income sources and taxation: - Households receive labor income W_t Φ_{a,t} ℓ_OLG_{a,t} (pre-tax) and lump-sum dividends D_{j,a,t}(i) from a broad set of sectors: N, T, D, C, I, R, U, M, X, F, K, EP (entrepreneurs only in Financial Accelerator version). - In Financial Accelerator version, OLG households receive remuneration for bankruptcy monitoring rbr_{a,t} = p_Nt rbr_{N,a,t} + p_THt rbr_{T,a,t}. - Households liable to pay lump-sum transfers τ_OLG^T_{a,t} (a small share of dividend income) which government redistributes to LIQ agents. - Taxes: labor income taxed at τ_L,t; consumption taxed at τ_c,t; lump-sum taxes τ_ls,OLG_{a,t}; transfers Υ_OLG_{a,t}. - Age-specific productivity profile: - Productivity of age group a: Φ_a = κ χ^a, with χ < 1. - Under average productivity normalized to one, κ = (n − θ χ) / (n − θ). - Retailer incentives and pricing: - Retailers face costs of rapidly adjusting sales volume and offer incentives/disincentives incorporated into effective retail purchase price P_Rt. - Consumption tax τ_c,t payable on pre-incentive price P_Ct. - P_Ct is marginal cost of retailers who combine consumption goods producer output (price P_t) and raw materials used directly by consumers (price P_Xt). - Household budget constraint (nominal): - P_Rt c_OLG_{a,t} + P_Ct c_OLG_{a,t} τ_c,t + P_t τ_ls,OLG_{a,t} + P_t τ_OLG^T_{a,t} + B_{a,t} + B_N_{a,t} + B_T_{a,t} + E_t F_{a,t} = (1/θ)[ i_{t−1} B_{a−1,t−1} + i_{t−1} (1 + ξ_b,t−1)^{-1} (B_N_{a−1,t−1} + B_T_{a−1,t−1}) ] + i_{t−1}(Ñ) E_t F_{a−1,t−1} (1 + ξ_f,t−1)^{-1} + W_t Φ_{a,t} ℓ_OLG_{a,t} (1 − τ_L,t) + Σ_{j} ∫_0^1 D_{j,a,t}(i) di + P_t rbr_{a,t} + P_t Υ_OLG_{a,t} - (Sum over j = N,T,D,C,I,R,U,M,X,F,K,EP) - First-order conditions and aggregation: - First-order conditions for goods varieties and consumption/leisure: - ˇc_OLG_t(i) = (P_Rt(i) / P_Rt)^{−σ_Rt} ˇc_OLG_t - ˇc_OLG_t N (1 − ψ) S_Lt − ˇℓ_OLG_t = η_OLG / (1 − η_OLG) ˇw_t (1 − τ_L,t) / (p_Rt + p_Ct τ_c,t) - (Notation: variables with ˇ are real variables rescaled by technology and population as defined.) - Uncovered interest parity (arbitrage condition for foreign currency bonds): - i_t = i_t(Ñ) (1 + ξ_f,t) (1 + ξ_b,t) E_t E_t[ε_{t+1}] - Consumption Euler equation for each generation: - E_t c_{a+1,t+1} = E_t j_t c_{a,t}, where - j_t = [ β ˇr_t ]^{1/γ} [ (p_Rt + p_Ct τ_c,t) / (p_R{t+1} + p_C{t+1} τ_c,{t+1}) ]^{1/γ} [ χ g ˇw_{t+1} (1 − τ_L,{t+1}) (p_Rt + p_Ct τ_c,t) / (ˇw_t (1 − τ_L,t) (p_R{t+1} + p_C{t+1} τ_c,{t+1}) ) ]^{(1−η_OLG)(1−1/γ)} - Real interest rate notation: - ˇr_t = E_t i_t / π_{t+1} (1 + ξ_b,t) = r_t (1 + ξ_b,t), where r_t is real interest rate in terms of final output payable by the government; ˇr_t is real rate payable by private sector. - Production-based real exchange rate vis-à-vis Ñ: - e_t = (E_t P_t(Ñ)) / P_t - Real assets: - Real domestic bonds: b_t = B_t / P_t - Real foreign bonds: f_t = F_t / P_t(Ñ) - Discount factors: - Subjective nominal discount factor: - R̃_{t,s} = Π_{l=1}^s [ θ (1 + ξ_b,t+l−1) i_{t+l−1} ] for s > 0 (equals 1 for s = 0) - Market nominal discount factor: - R_{t,s} = Π_{l=1}^s [ (1 + ξ_b,t+l−1) i_{t+l−1} ] for s > 0 (equals 1 for s = 0) - Subjective real discount factor: - r̃_{t,s} = Π_{l=1}^s [ θ ˇr_{t+l−1} ] for s > 0 (equals 1 for s = 0) - Market real discount factor: - r_{t,s} = Π_{l=1}^s [ 1 / ˇr_{t+l−1} ] for s > 0 (equals 1 for s = 0) - Subjective discount factors incorporate probability of death θ, making near-term receipts relatively more valuable. - Aggregation across age cohorts: - OLG aggregate consumption: - c_OLG_t = N n_t (1 − ψ) (1 − θ^n) Σ_{a=0}^∞ θ^n^a c_OLG_{a,t} - Implications for κ calibration given χ and θ under average productivity normalization. - Comments on aggregation and linearization: - Several optimality conditions are nonlinear Euler equations; aggregation requires nonlinear transformations valid under certainty equivalence. - Tractable aggregate consumption conditions exist for perfect foresight and first-order approximations; stochastic applications use linear approximations. - For exposition, optimality conditions presented in nonlinear form with expectations operator E_t understood in that fashion. *Source: _wp1034 - 4. Trade Matrix: Consumption Goods, IMF Working Paper (PDF chapter/section).* ### Appendix 3. The final result expresses current aggregate consumption ofOLGhouseholds as a ### Appendix 3. The final result expresses current aggregate consumption of OLG households as a ### OLG households: consumption, financial and human wealth - Key identity (after rescaling by technology): - ˇc_OLG_t Θ_t = ˇfw_t + ˇhw_t (equation (20)) - Financial wealth (rescaled): - ˇfw_t = 1/π_t gn [ i_{t−1} ˇb_{t−1} + i_{t−1} (1 + ξ^b_{t−1}) ( ˇb^N_{t−1} + ˇb^T_{t−1} ) + i_{t−1} ( ̃N ) (1 + ξ^f_{t−1}) ε_t ˇf_{t−1} e_{t−1} ] (equation (21)) - Human wealth decomposition: - ˇhw_t = ˇhw^L_t + ˇhw^K_t (equation (22)) - Labor component: - ˇhw^L_t = ( N(1−ψ)( ˇw_t (1−τ_{L,t}) S^L_t ) ) + E_t θ χ_g ˇr_t ˇhw^L_{t+1} (equation (23)) - Capital/dividend component: - ˇhw^K_t = Σ_{j=N,T,D,C,I,R,U,M,X,F,K,EP} ˇd^j_t + r ˇbr_t − ˇτ^OLG_{T,t} − ˇτ^{ls,OLG}_t + ˇΥ^OLG_t + E_t θ g ˇr_t ˇhw^K_{t+1} (equation (24)) - Marginal propensity to consume: - Θ_t = p^R_t + p^C_t τ_{c,t} η_OLG + E_t θ_j_t ˇr_t Θ_{t+1} (equation (25)) - In simplest case (log utility, exogenous labor supply): 1/Θ_t = (1−βθ) - Consumption tax timing affects Θ_t; balanced-budget shift of consumption taxes from future to present reduces Θ_t - Intertemporal elasticity of substitution 1/γ matters: for γ > 1 the income effect of r tends to increase the marginal propensity to consume - Behavioral and discounting intuition: - Households discount future tax liabilities at least at rate ˇr_t/θ, higher than market rate ˇr_t, and labor-income component discounted at ˇr_t/(θ χ) due to lifetime labor income decline - Fiscal consolidation that tilts taxes to near future reduces human wealth as households discount future taxes more heavily than government discounting of surpluses; for ξ^b_t < 0 effect partly due to borrowing spread - Modularity: - OLG households are core to GIMF; cannot be removed (present in “Fiscal Stimulus to the Rescue?”) ### Liquidity-constrained (LIQ) households - Preferences nearly identical to OLG households: - Lifetime utility: E_t Σ_{s=0}^∞ (βθ)^s [ 1/(1−γ) ( c^{LIQ}_{a+s,t+s} )^{η_{LIQ}} ( S^L_{t} − ℓ^{LIQ}_{a+s,t+s} )^{1−η_{LIQ}} ] (equation (26)) - Consumption limited to current income; budget constraint (rescaled): - p^R_t c^{LIQ}_{a,t} + p^C_t c^{LIQ}_{a,t} τ_{c,t} ≦ W_t Φ_{a,t} ℓ^{LIQ}_{a,t} (1−τ_{L,t}) + τ^{LIQ}_{T,a,t} + Υ^{LIQ}_{a,t} − τ^{ls,LIQ}_{a,t} (equation (28)) - Aggregated first-order conditions (rescaled): - ˇc^{LIQ}_t (i) = ( P^R_t (i) / P^R_t )^{−σ_R_t} ˇc^{LIQ}_t (equation (29)) - ˇc^{LIQ}_t (p^R_t + p^C_t τ_{c,t}) = ˇw_t ℓ^{LIQ}_t (1−τ_{L,t}) + ˇτ^{LIQ}_{T,t} + ˇΥ^{LIQ}_t − ˇτ^{ls,LIQ}_t (equation (30)) - Labor FOC: ˇc^{LIQ}_t N ψ S^L_t − ˇℓ^{LIQ}_t = [ η_{LIQ} / (1−η_{LIQ}) ] ˇw_t (1−τ_{L,t}) / (p^R_t + p^C_t τ_{c,t}) (equation (31)) - Alternative "rule-of-thumb" variant: drop equation (31) and set exogenous labor supply - Modularity: - Share of LIQ agents not core but critical for short-run fiscal effects; marginal propensity to consume out of current income = 1; included in short-run fiscal analyses including “Fiscal Stimulus to the Rescue?” ### Aggregate household sector - Aggregation: - ˇC_t = ˇc^{OLG}_t + ˇc^{LIQ}_t (equation (32)) - ˇL_t = ˇℓ^{OLG}_t + ˇℓ^{LIQ}_t (equation (33)) - Note: distinction of generations can be dropped if agents identical (footnote 12) ### Manufacturers: demand, technology, adjustment costs, pricing - Two manufacturing sectors J ∈ {N, T} (nontradables, tradables); continuum of firms i ∈ [0,1] - Demand for varieties and aggregation: - Y^J_t(z) = ( ∫_0^1 Y^J_t(z,i)^{σ_J_t−1 / σ_J_t} di )^{σ_J_t / (σ_J_t−1)} (equation (34)) - Variety demand: Y^J_t(z,i) = ( P^̃J_t(i) / P^̃J_t )^{−σ_J_t} Y^J_t(z) (equation (35)) - Price index: P^̃J_t = ( ∫_0^1 P^̃J_t(i)^{1−σ_J_t} di )^{1/(1−σ_J_t)} (equation (36)) - Production technology (with raw materials): - Z^J_t(i) = F(K^J_t(i), U^J_t(i), X^J_t(i)) = T [ (1−α_X^J_t)^{1/ξ_XJ} M^J_t(i)^{(ξ_XJ−1)/ξ_XJ} + (α_X^J_t)^{1/ξ_XJ} X^J_t(i)^{(1−G^J_X,t)/ξ_XJ} ]^{ξ_XJ/(ξ_XJ−1)} (equation (38), with M^J_t(i) defined) - Adjustment costs: - Inflation adjustment (Rotemberg form option): G^J_{P,t}(i) = φ^P_J /2 Z^J_t [ ( P^̃J_t(i)/P^̃J_{t−1}(i) / (P^̃J_{t−1}/P^̃J_{t−2}) − 1 )^2 ] (equation (40)); alternative target-based form (41) - Raw-materials adjustment: G^J_{X,t}(i) = φ^J_X /2 [ ( X^J_t(i)/(gn) − X^J_{t−1} ) / X^J_{t−1} ]^2 (equation (42)) - Labor hiring adjustment: G^J_{U,t}(i) = φ_U /2 U^J_t [ ( U^J_t(i)/n − U^J_{t−1}(i) ) / U^J_{t−1}(i) ]^2 (equation (43)); φ_U can be set to 0 - Firm dividends and optimization: - D^J_t(i) = P^̃J_t(i) Z^J_t(i) − V_t U^J_t(i) − P_X_t X^J_t(i) − R^J_{k,t} K^J_t(i) − P^̃J_t T_t ω_J − P^̃J_t G^J_{P,t}(i) − V_t G^J_{U,t}(i) (equation (45)) - Firms maximize E_t Σ ̃R_{t,s} D^J_{t+s}(i) subject to production and demand (equation (46)) - Sticky inflation pricing condition (Rotemberg form): - E^J_t λ^J_t p^̃J_{t−1} = φ^P_J (E^J_{t−1}) [ ( π^̃J_t / π^̃J_{t−1} − 1 ) (π^̃J_t / π^̃J_{t−1}) ] − E_t θ gn ˇr_t φ^P_J (E^J_{t−1}) (p^̃J_{t+1}/p^̃J_t) (ˇZ^J_{t+1}/ˇZ^J_t) (π^̃J_{t+1}/π^̃J_t) (π^̃J_{t+1}/π^̃J_{t−1} − 1) (equation (47)) - Sticky prices variant given by equation (48) - Factor demand FOCs: - Labor demand condition involves λ^J_t, ˇv_t and ˇF^J_{U,t} with adjustment cost terms (equation (49) and marginal product ˇF^J_{U,t} (50)) - Raw materials: p_X_t = λ^J_t ˇF^J_{X,t} (equation (51)-(52)) - Capital: r^J_{k,t} = λ^J_t ˇF^J_{K,t} (equation (53)-(54)) - Aggregate dividends rescaled: - ˇd^J_t = p^̃J_t ˇZ^J_t − ˇv_t ˇU^J_t − p_X_t ˇX^J_t − r^J_{k,t} ˇK^J_t − ˇv_t ˇG^J_{U,t} − p^̃J_t ˇG^J_{P,t} − p^̃J_t ω_J (equation (57)) - Market-clearing: - ˇZ^N_t = ˇY^N_t + ω_N + ˇG^N_{P,t} + r ˇcu^N_t + ˇS_{N,nwyshk,t} (equation (60)) - ˇZ^T_t(1) = ˇY^{TH}_t(1) + ω_T(1) + ˇG^T_{P,t}(1) + r ˇcu^T_t + ˇS_{T,nwyshk,t} + ̃p^{exp}_t Σ_{j=2}^{ ̃N} ˇY^{TX}_t(1,j) (equation (61)) - Modularity: - Tradables manufacturing is core and cannot be removed; nontradables can be removed but are important for real exchange rate effects ### Capital goods producers (with and without Financial Accelerator) - With Financial Accelerator: - Installation equation: ̃K^J_{t+1} = ̃K^J_t + S^{inv}_t I^J_t (equation (62)) - Investment adjustment cost: G^J_{I,t} = φ_I /2 I^J_t [ ( I^J_t/(gn) − I^J_{t−1} ) / I^J_{t−1} ]^2 (equation (63)) - q-equation (solution): q^J_t S^{inv}_t = p_I_t + φ_I p_I_t [ ... ] − E_t θ gn ˇr_t φ_I p_I_{t+1} [ ... ] (equation (66)) - Physical capital evolution: K^J_{t+1} = (1−δ^J_K) K^J_t + S^{inv}_t I^J_t (equation (67)) - Depreciation shock: δ^J_{K,t} = δ^J_K + S^{nwkshk}_t (equation (68)) - Utilization relation: ˇK^J_t = u^J_t K^J_t (equation (69)) - Dividend: ˇd^K_J_t = q^J_t S^{inv}_t ˇI^J_t − p_I_t ( ˇI^J_t + ˇG^J_{I,t} ) (equation (70)) - Without Financial Accelerator: - Physical capital accumulation (equation (71)), depreciation shocks (72), investment adjustment cost (73) - Capital utilization costs: a(u^J_t) = 1/2 φ^J_a σ^J_a u^{J,2}_t + φ^J_a (1−σ^J_a) u^J_t + φ^J_a (σ^{J,2}_a/2 − 1/2) (equation (74)) - Investment q-equation (77) and q-dynamic relation (78) - Utilization FOC: r^J_{k,t} = φ^J_a σ^J_a u^J_t + φ^J_a (1−σ^J_a) (equation (79)) - Resource cost of utilization: rˇcu^J_t = a(u^J_t) ˇK^J_t / p^̃J_t (equation (80)) - Dividend: ˇd^K_J_t = (1−τ_{k,t}) ( r^J_{k,t} u^J_t − a(u^J_t) ) + τ_{k,t} δ^J_{K,t} q^J_t ˇK^J_t − p_I_t ˇI^J_t + ˇG^J_{I,t} (equation (81)) - Modularity: - This sector is core and present in “Fiscal Stimulus to the Rescue?” ### Entrepreneurs and banks (Financial Accelerator) - Entrepreneur balance sheet: - Q^J_t K^J_{t+1}(j) = N^J_t(j) + B^J_t(j) (equation (82)) - Normalized: q^J_t ˇK^J_{t+1}(j) gn = ˇn^J_t(j) + ˇb^J_t(j) (equation (83)) - Idiosyncratic shock ω^J_{t+1} (unit-mean lognormal) with s.d. ln(ω) = σ^J_{t+1}; distribution F^J_{t+1}(·) - Entrepreneurs choose utilization u^J_t and rents; capital utilization cost a(u^J_t) scaled by ω^J_t (equation (84)-(86)) - Debt contract & bankruptcy cutoff: - Bankruptcy cutoff ω̄^J_{t+1} given by Ret^J_{k,t} ω̄^J_{t+1} Q^J_t K^J_{t+1}(j) = i^J_{B,t+1} B^J_t(j) (equation (90)) - Bank zero-profit (participation) condition (state-by-state): - (1−F(ω̄)) i^J_{B,t+1} B^J_t(j) + (1−E^J_{t+1}) ∫_0^{ω̄} ω f(ω) dω Ret^J_{k,t} Q^J_t K^J_{t+1}(j) = ˇi_t B^J_t(j) (equation (91)-(92)) - Definitions for lender shares: - Γ(ω̄) ≡ ∫_0^{ω̄} ω f(ω) dω + ∫_{ω̄}^∞ f(ω) dω (equation (93)) - G(ω̄) = ∫_0^{ω̄} ω f(ω) dω (equation (94)) - Net lender share = Γ(ω̄) − E G(ω̄) - Entrepreneur share = 1 − Γ(ω̄) = ∫_{ω̄}^∞ (ω − ω̄) f(ω) dω (equation (95)) - Contract optimality and FOCs yield λ_t = Γ′ / (Γ′ − E G′) (equation (99)) and the loan-contract condition (100) - Aggregated lender zero-profit normalized condition: - q^J_{t−1} ˇK^J_t gn / ˇn^J_{t−1} rˇet^J_{k,m1,t} / ˇr_{m1,t} (Γ^J_t − E^J_t G^J_t) − q^J_{t−1} ˇK^J_t gn / ˇn^J_{t−1} + 1 = 0 (equation (101)) - Closed-form expressions (PNORM/Φ) for Γ, G and derivatives: - z̄^J_t = ln(ω̄^J_t) + 1/2 (σ^J_t)^2 / σ^J_t (equation (102)) - f(ω̄) = (1 / (√(2π) ω̄ σ^J_t)) exp( −1/2 ( z̄^J_t )^2 ) (equation (103)) - Γ^J_t = Φ( z̄^J_t − σ^J_t ) + ω̄^J_t ( 1 − Φ( z̄^J_t ) ) (equation (104)) - G^J_t = Φ( z̄^J_t − σ^J_t ) (equation (105)) - Γ′^J,t = 1 − Φ( z̄^J_t ) (equation (106)) - G′^J,t = ω̄^J_t f(ω̄^J_t) (equation (107)) - Aggregate entrepreneur net worth evolution (nominal): - N^J_t = ret^J_{km1,t} Q^J_{t−1} ˇK^J_t (1 − E^J_t G^J_t) − ˇi_{t−1} B^J_{t−1} − P^̃J_t div^J_t + S^J_{nwyshk,t} (equation (108)) - Normalized evolution: ˇn^J_t = ˇr_{m1,t} gn ˇn^J_{t−1} + q^J_{t−1} ˇK^J_t [ rˇet^J_{km1,t} (1 − E^J_t G^J_t) − ˇr_{m1,t} ] − p^̃J_t [ dˇiv^J_t ] + ˇS^J_{nwyshk,t} (equation (109)) - Dividend rules and smoothing: - Aggregated dividends and dividend policy via smoothed income and net worth (equations (110)-(116)) - Typical parameter ranges: S^J_{nwd} in [0, 0.05], θ^J_{nw} in [0, 0.05]; smoothing parameters k_{incJ}, k_{nw} typically around 3 (backward-looking) or around 10 (very slow-moving) - Modularity: - Financial Accelerator is part of the core; present in “Fiscal Stimulus to the Rescue?” ### Raw-materials producers (oil-like sector) - Supply and storage: - Exogenous endowment X^{exog}_t; firms choose stored stock O_t - Storage cost G^O_t = φ_O /2 (T_t n_t) O_t^2 − κ_o O_t (equation (118)) - Profit maximization: Max Σ ̃R_{t,s} P_X_{t+s} [ X^{exog}_{t+s} − (O_{t+s} − O_{t+s−1}) − G^O_{t+s} ] (equation (119)) - FOC: 1 − κ_o + φ_O ˇO_t = E_t θ ˇr_t p_X_{t+1} / p_X_t (equation (120)) - Sales: ˇX^{sup}_t = ˇX^{exog}_t − [ ˇO_t − ˇO_{t−1} ]/gn − ˇG^O_t (equation (121)) - Sales, revenue shares and world arbitrage: - Value of exports: ˇX^x_t = p_X_t ( ˇX^{sup}_t − ˇX^{dem}_t ) (equation (122)) - Steady-state shares: ˇd_X = s^x_d p_X X^{sup} (equation (123)); public/foreign shares via s^x_f (equations (124)-(126)) - International arbitrage: p_X_t = p_X_t( ̃N ) e_t (equation (127)) - Market-clearing world: Σ_{ ̃N j=1} ( ˇX^{sup}(j)_t − ˇX^{dem}(j)_t ) = 0 (equation (131)) - Modularity: - Raw-materials sector not core; typically omitted unless focus on raw materials; not present in “Fiscal Stimulus to the Rescue?” ### Unions: wage setting and rigidities - Labor demand by firms and aggregation: - U^J_t(z) = ( ∫_0^1 U^J_t(z,i)^{σ_{U,t}−1 / σ_{U,t}} di )^{σ_{U,t} / (σ_{U,t}−1)} (equation (132)) - Demand: U^J_t(z,i) = ( V_t(i) / V_t )^{−σ_{U,t}} U^J_t(z) (equation (133)) - Wage index: V_t = ( ∫_0^1 V_t(i)^{1−σ_{U,t}} di )^{1/(1−σ_{U,t})} (equation (134)) - Wage rigidities (three types): - Sticky wage inflation (Rotemberg): G^U_{P,t}(i) = φ^P_U /2 U_t T_t [ ( V_t(i)/V_{t−1}(i) / (V_{t−1}/V_{t−2}) − 1 )^2 ] (equation (136)) - Real wage rigidities (costly change in real wage V_t/P^c_t): G^U_{P,t}(i) = φ^P_U /2 U_t T_t ( π^{rw}_t(i) − 1 )^2 (equation (137)) - Union optimization FOC (sticky wage inflation): - E^U_t ˇw_t ˇv_{t−1} = φ^P_U (E^U_{t−1}) [ π^V_t / π^V_{t−1} ( π^V_t / π^V_{t−1} − 1 ) ] − E_t θ gn ˇr_t φ^P_U (E^U_{t−1}) (ˇv_{t+1}/ˇv_t)(ˇU_{t+1}/ˇU_t) (π^V_{t+1}/π^V_t)(π^V_{t+1}/π^V_{t−1} − 1) (equation (139)) - Real union dividends to households: - ˇd^U_t = ( ˇv_t − ˇw_t ) ˇU_t − ˇv_t ˇG^U_{P,t} (equation (141)) - Labor-market clearing: - ˇL_t = ˇU^N_t + ˇU^T_t + ˇG^N_{U,t} + ˇG^T_{U,t} + ˇG^U_{P,t} (equation (143)) - Modularity: - Not core but required for sticky wages; typically present (including “Fiscal Stimulus to the Rescue?”) ### Import agents, distributors, and trade integration - Import agents: - CES aggregation for imports Y^{JM}_t(1,j) from varieties i (equation (144)-(147)) - Nominal rigidities and pricing FOC (Rotemberg form) (equation (148) and (151)) - CIF price relation: p^{JM,cif}_t(1,j) = p^{JH}_t(j) ̃p^{exp}_t(j) e_t(1)/e_t(j) (equation (149)) - Real dividends to OLG households in country j: ˇd^{JM}_t(1,j) = ( p^{JM}_t(1,j) − p^{JM,cif}_t(1,j) ) ˇY^{JM}_t(1,j) − p^{JM}_t(1,j) ˇG^{JM}_{P,t}(1,j) (equation (152)) - Market-clearing: ˇY^{JX}_t(j,1) = ˇY^{JM}_t(1,j) + ˇG^{JM}_{P,t}(1,j) (equation (155)) - Distributors: multi-stage technology - Foreign input composite Y^{JF}_t(1) (CES with elasticity ξ_{JM}) combining imports from j with share ̃ζ^J(1,j) (equations (156)-(160)) - Tradables composite Y^T_t produced from domestic Y^{TH}_t and foreign Y^{TF}_t with share ̃α^T_H_t, spillover and adjustment costs G^T_{F,t} (equations (162)-(173)) - Tradables-nontradables composite Y^A_t (CES with ξ_A) with input share dynamics ̃α^T_t and ̃α^N_t (equations (174)-(180)) - Private-public composite / domestic final output Z^D_t = Y^A_t ( K_{G1,t} )^{α_{G1}} ( K_{G2,t} )^{α_{G2}} S (equation (181)) - Distributor first-order condition linking p_{DH,t} and p_{A,t} (equation (182)) - Distributor aggregated dividends (zero in equilibrium): ˇd^D_t = p_{DH,t} ˇZ^D_t − p_N_t ˇY^N_t − p^{TH}_t ˇY^{TH}_t − p^{TF}_t ˇY^{TF}_t (equation (183)) - Market-clearing: ˇZ^D_t = ˇY^{IH}_t + ˇY^{CH}_t + ̃p^{exp}_t Σ_{j=2}^{ ̃N } ˇY^{DX}_t(1,j) (equation (184)) - Modularity: - Distributors are core; public capital inclusion optional but important for fiscal multipliers ### Investment and consumption goods producers - Investment goods producers: - CES in domestic and foreign final outputs with share α^I_H_t and elasticity ξ_I; foreign share dynamics and adjustment costs G^I_{F,t} (equations (189)-(196)) - Pricing FOC (Rotemberg form) (equation (200)) - Rescaled aggregated dividends ˇd^I_t and net output ˇX^I_t defined (equations (201)-(203)) - Market-clearing: ˇZ^I_t − ˇG^I_{P,t} − ω_I = ̃p_I_t ̆p_I_t ( ˇI_t + ˇG^N_{I,t} + ˇG^T_{I,t} + ˇY_{GI,t} ) (equation (205)) - Modularity: part of core; used to distinguish investment vs consumption trade shares - Consumption goods producers: - Analogous CES structure with α^C_H_t, ξ_C, spillovers and adjustment costs G^C_{F,t} (equations (210)-(217)) - Pricing FOC (equation (221)) - Aggregated dividends ˇd^C_t (equation (222)) - Market-clearing: ˇZ^C_t = ˇC^{ret}_t + ˇY_{GC,t} + ω_C + ˇG^C_{P,t} + ˇG^C_t (equation (223)) - Modularity: core; distinguishes consumption vs investment in trade flows ### Retailers - Technology: CES in consumption goods and direct raw materials with adjustment cost G^C_{X,t}(i) (equations (224)-(225)) - Optimal input choice: ˇX^C_t / ˇC^{ret}_t = α_{X^C,t} ( (1−α_{X^C,t}) (1−G^C_{X,t}) ) ( p_X_t ̃O^C_t )^{−ξ_{XC}} (equation (226)) - Marginal cost p_C_t expression (equation (227)) - Demand for retail varieties from households: C_t(i) = ( P^R_t(i) / P^R_t )^{−σ_R_t} C_t (equation (228)) - Quantity adjustment costs (real rigidities) for retailers: - G^C_t(i) = φ_C /2 C_t [ ( C_t(i)/(gn) − C_{t−1}(i) ) / C_{t−1}(i) ]^2 (equation (229)) - Retailer pricing FOC (equation (231)) - Real dividends and rescaled adjustment costs: - ˇd^R_t = ( p^R_t − p_C_t ) ˇC_t − ˇG^C_t (equation (232)) - ˇG^C_t = φ_C /2 ˇC_t ( ( ˇC_t − ˇC_{t−1} ) / ˇC_{t−1} )^2 (equation (233)) - Modularity: - Retail sector not core but never dropped; needed to generate inertial consumption dynamics ### Government: production, budget, fiscal policy - Government production of Z^G_t from Y^G_C_t and Y^G_I_t (CES, equations (234)-(238)) - Final price p_G_t = ̃p_G_t p_{ZG,t} with ̃p_G_t = 1/T^G_t unit-root relative price shock (equation (238)) - Government demand ˇG_t = ˇG_{cons,t} + ˇG_{inv,t}; market-clearing ˇZ^G_t = ̃p_G_t ˇG_t (equation (240)) - Public capital accumulation (rescaled): - ˇK_{G1,t+1} gn = (1−δ_{G1}) ˇK_{G1,t} + ˇG_{inv,t} (equation (241)) - Optional second productive public capital if α_{G2} > 0: ˇK_{G2,t+1} gn = (1−δ_{G2}) ˇK_{G2,t} + ˇG_{cons,t} (equation (242)) - Transfers and redistribution rule: - ˇτ_{T,t} = ι ( ˇd_N_t + ˇd_T_t + ˇd_D_t + ˇd_C_t + ˇd_I_t + ˇd_M_t + d_X + ˇd_F_t + ˇd_K_t + ˇd_{EP,t} ) + ˇc^{LIQ}_t ˇC_t ( ˇd^R_t + ˇΥ_t − ˇτ^{ls}_t ) + ˇℓ^{LIQ}_t ˇL_t ˇd^U_t (equation (243)) - Parameter ι typically calibrated relative to ψ (share of LIQ agents) - Tax bases and rescaled aggregate real tax variable: - ˇτ_t = τ_{L,t} ˇw_t ˇL_t + τ_{c,t} p_C_t ˇC_t + ˇτ^{ls}_t + τ_{k,t} Σ_{j=N,T} [ u^J_t r^J_{k,t} − δ^J_{K,t} q^J_t − a(u^J_t) ] ˇK^J_t (equation (244)) - Government real budget constraint (rescaled): - ˇb_t + ˇτ_t + ˇg_X_t = i_{t−1}/π_t gn ˇb_{t−1} + p_G_t ˇG_t + ˇΥ_t (equation (245)) - Modularity: - Government budget and fiscal policy equations are core and included in fiscal analyses including “Fiscal Stimulus to the Rescue?” - Model assumptions on fiscal policy: - Two key assumptions mentioned: dynamic stability and business-cycle stabilization (text ends before detailed rule specification) *Source: _wp1034 - Appendix 3. The final result expresses current aggregate consumption of OLG households as a; canonical PDF content as provided.* ### 1. Dynamic Stability ### 1. Dynamic Stability ### Fiscal policy objective and mechanism - Fiscal policy ensures a non-explosive government-debt-to-GDP ratio by adjusting tax rates to generate sufficient revenue, or by reducing expenditure, in order to stabilize the overall, interest inclusive government surplus-to-GDP ratiogs rat t at a long-run level ofgss rat t chosen by policy. ### Government surplus (level) - The government surplus is given by - gs t =−  ˇ b t − ˇ b t−1 π t gn  = ˇτ t + ˇg X t −p G t ˇ G t − ˇ Υ t − i t−1 −1 π t gn ˇ b t−1 ,(246) ### Government surplus-to-GDP ratio - Its ratio to GDP (gdp t will be defined below) is - gs rat t =−100 B t −B t−1 P t gdp t = 100 ˇgs t g ˇ dp t ,(247) - The model allows for the possibility that gss rat t follows an exogenous stochastic process. ### Debt-to-GDP notation and long-run relationship - Current value and long-run target for the government-debt-to-GDP ratio: - ˇ b rat t and ˇ bss rat t , expressed as a share of annual GDP. - Relationship between long-run government balance and government-debt-to-GDP ratios: - gss rat t =−4 π t gn−1 π t gn ˇ bss rat t .(248) - Interpretation: Here π t is the inflation target of the central bank. For a given nominal growth rate, choosing a surplus target gss rat t implies a debt target ˇ bss rat t and therefore keeps debt from exploding. *Source: _wp1034 - 1. Dynamic Stability* ### 2. Business Cycle Stabilization ### 2. Business Cycle Stabilization ### Structural fiscal surplus rule and debt dynamics - Structural fiscal surplus targeting rule (notation preserved from source): - gsrat_t = gssrat_t + d_debt (β b_rat_t − β bssrat_t) + d_gdp ln(gˇdpfisher_t / gˇdppot_t) + d_tax (βτ_t − βτpott) gˇdp_t + d_rawmat (βgXt − βgpotX,t) gˇdp_t. (Equation (249) structure and terms preserved as in source.) - Setting d_debt = 0 ensures a non-explosive government-debt-to-GDP ratio of βbssrat_t, but yields a long-run autoregressive coefficient on debt at 1/(βπ_tgn) that is very close to one. - Setting d_debt > 0: - Ensures faster convergence of debt. - Increases volatility of government surpluses. - Balanced budget rule corresponds to d_debt = d_gdp = d_tax = d_rawmat = 0 and is described as highly procyclical and undesirable. ### Fiscal responses to the business cycle (components and moving averages) - Output-gap response (d_gdp term): - Uses current and potential Fisher-weighted GDP gˇdpfisher_t and gˇdppot_t. - Potential GDP is proxied by a moving average: gˇdppot_t = [gˇdpfisher_t + gˇdppot_{t−1}^{k_gdp}]^{1/(1+k_gdp)} (preserved structure per (250)). - Model allows for unit-root shocks to technology and to savings; potential GDP is subject to nonstationary shocks, justifying moving-average proxy. - Tax-revenue-gap response (d_tax term): - Potential tax revenue βτpott = τL,t taxbase_maL,t + τC,t taxbase_maC,t + τK,t taxbase_maK,t + βτls. (Equation (251)) - Moving-average tax bases: - taxbase_maL,t = [βw_t βL_t taxbase_maL,t−1]^{1/(1+kLτ)} with parameter kLτ embedded (preserved form from (252)). - taxbase_maC,t = [pCt βCt taxbase_maC,t−1]^{1/(1+kCτ)} (preserved form from (253)). - taxbase_maK,t = Σ_{i∈N,T} [ (u_it r_{i,k,t} − δ J K_t q_it − a(u_it)) βK_it taxbase_maK,t−1 ]^{1/(1+kKτ)} (preserved form from (254)). - Raw-materials revenue-gap response (d_rawmat term): - Potential raw-materials revenue gpotXt = [e_t pX,ma_t( ̃N) βXsup,ma_t − β dX ]^{1−sxf} (preserved structure per (255)). - Moving averages: - pX,m a_t( ̃N) = [pX_t( ̃N) + pX,ma_{t−1}( ̃N)^{k_px}]^{1/(1+k_px)} (preserved form from (256)). - βXsup,ma_t = [βXsup_t + βXsup,ma_{t−1}^{k_yx}]^{1/(1+k_yx)} (preserved form from (257)). - Practical calibrations: - OECD estimates can calibrate d_gdp for many countries; some countries implement structural fiscal surplus rules with d_gdp = 0 and d_tax = 1. - With d_tax = 1 and d_gdp = 0, during a boom excess tax revenue is used to pay down debt, reducing variability of fiscal instruments and output relative to a balanced budget rule. - A counter-cyclical rule would set d_tax > 1. ### Instrumenting the targeting rule - The fiscal rule (249) is a targeting rule, not an instrument rule. - Default instrument: labor tax rate τL,t, but other instruments or combinations may be used. - Examples of instrument comovement: - τc,t = τ̄c + d_ctax (τL,t − τ̄L). (Equation (258)) - τk,t = τ̄k + d ktax (τL,t − τ̄L). (Equation (259)) - d_ctax = d_ktax = 1 → perfect comovement of τL,t, τc,t, τk,t. - d_ctax = d_ktax = 0 → only labor tax rates change. ### Monetary policy rule - Interest-rate rule features interest-rate smoothing and responds to: - (i) deviations of one-year-ahead year-on-year inflation π_{t+1} from inflation target βπ_t, - (ii) the output gap, - (iii) year-on-year growth rate of Fisher-weighted GDP, - (iv) deviations of current exchange-rate depreciation ε_t from long-run value βε_t = βπ_t/βπ_t( ̃N). - Monetary shocks S_int_t are autocorrelated. - Important model-specific exception: non-Ricardian features imply no unchanging steady-state GDP or real interest rate; proxy r_eq_t ̃π_t includes geometric moving averages of the world pre-risk-premium real interest rate r_world_t and of the risk premium ξ_ma_t: - r_eq_t = r_world_t ξ_ma_t. (Equation (262)) - r_world_t = Π_{ ̃N j=1} [ r_ma(j)_t ]^{gdp_ss(j) / Σ_{ ̃N i=1} gdp_ss(i)} (Equation (263)). - r_ma(j)_t = [ r_preξ(j)_t + r_ma(j)_{t−1}^{k_r} ]^{1/(1+k_r)} (Equation (264)). - ξ_ma_t = [ ξ_t + ξ_ma_{t−1}^{k_r} ]^{1/(1+k_r)} (Equation (265)). - Complete monetary rule (preserved from source): - i_t = E_t(i_{t−1})^{δ_i} (r_eq_t ̃π_t)^{1−δ_i} [ (̃π_t / βπ_t)^{(1−δ_i)δ_π} (gˇdpfisher_t / gˇdppot_t)^{(1−δ_i)δ_y} (gˇdpfisher_t / gˇdpfisher_{t−4})^{(1−δ_i)δ_ygr} (ε_t / βε_t)^{δ_e} ] (1 + S_int_t). (Equations (260)-(261)) - ̃π_t = π_t^{δ ̃π} π_{t+1}^{1−δ ̃π}. - Exchange-rate targeting special case: modeled with δ_i = 1 and δ_e −→ ∞. - Quarterly versions replace one-year-ahead inflation with a one-year-ahead four-quarter geometric moving average of inflation (footnote preserved). ### Shocks: transitory and unit-root components - Many shocks (β_t, α_CH_t, α_IH_t, α_TH_t, α_XC_t, α_XN_t, α_XT_t, X_sup_t, σ_N_t, σ_T_t, E_N_t, E_T_t, ˇS_{N,nwd}_t, ˇS_{T,nwd}_t, ˇS_{N,nwy}_t, ˇS_{T,nwy}_t, ˇS_{N,nwk}_t, ˇS_{T,nwk}_t, ˇG_cons_t, ˇG_inv_t, and foreign counterparts) have both transitory and unit-root components: - x_t = (1 − ρ_x) ̃x_t + ρ_x x_{t−1} + u_x_t; ln(̃x_t) = ln(̃x_{t−1}) + u_{̃x_t}. (Equations (266)-(267)) - Policy variables gssrat_t and βπ_t: - Transitory components endogenously given by fiscal/monetary rules. - Permanent components are unit roots: - ln(βπ_t) = ln(βπ_{t−1}) + u_{π_t}. (268) - gssrat_t = gssrat_{t−1} + u_{gss_t}. (269) - Relative price processes ̃p_y_t, y ∈ {I,G,exp} assumed unit roots: ln(̃p_y_t) = ln(̃p_y_{t−1}) + u_{py_t}. (270) - Interest-rate, investment, labor supply, foreign exchange risk premium, government risk premium and markup shocks assumed transitory; markup shocks serially uncorrelated. Representative forms: - S_int_t = (1 − ρ_int) + ρ_int S_int_{t−1} + u_int_t. (271) - S_inv_t = (1 − ρ_inv) + ρ_inv S_inv_{t−1} + u_inv_t. (272) - S_L_t = (1 − ρ_L) + ρ_L S_L_{t−1} + u_L_t. (273) - ξ_f_t = ρ_fxp ξ_f_{t−1} + u_fxp_t. (274) - ξ_b_t = ρ_gbp ξ_b_{t−1} + u_gbp_t. (275) - E_i_t = βE_i (1 + u_{=i}_t), i = U,C,I (276) (notation preserved). - Productivity shocks: country-specific technology can follow U.S. with a catchup parameter catchup(j) ∈ [0,1]: - US: A_J(US)_t = (1 − ρ_AJ(US) + e_AJ(US)_t) ̃A_J(US)_t + ρ_AJ(US) A_J(US)_{t−1}. (277) - Country j: A_J(j)_t = (1 − ρ_AJ(j)) ̃A_J(j)_t + catchup(j) * (A_J(US)_t − ̃A_J(US)_t) + ρ_AJ(j) A_J(j)_{t−1} + e_AJ(j)_t ̃A_J(j)_t. (278) - Stationary shock to investment goods price also allows catchup to U.S. (279)-(280). - Modularity: catching up feature can be turned off in applications. ### Balance of payments and current account - Current-account expression (nominal/notation preserved): - e_t ˇf_t = i_{t−1}( ̃N) ε_t (1 + ξ_f_{t−1}) / π_t^{gn} e_{t−1} ˇf_{t−1} + p_TH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_TX_t(1,j) + ˇd_TM_t − p_TF_t ˇY_TF_t + p_DH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_DX_t(1,j) + ˇd_DM_t − p_DF_t ˇY_DF_t + ˇX_x_t + ˇd_F_t − ˇf_X_t. (281) (preserved) - Market-clearing for international bonds: Σ_{ ̃N j=1} ˇf_t(j) = 0. (282) - Current account balance: ca_t = e_t ˇf_t − e_{t−1} ˇf_{t−1} / π_t^{gn}. (283) - GDP expression: - gˇdp_t = p_C_t ˇC_t + p_I_t ˇI_t + p_G_t ˇG_t + ˇX_x_t + p_TH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_TX_t(1,j) + ˇd_TM_t − p_TF_t ˇY_TF_t + p_DH_t ̃p_exp_t Σ_{ ̃N j=2} ˇY_DX_t(1,j) + ˇd_DM_t − p_DF_t ˇY_DF_t. (284) - Modularity: this block is core to GIMF. ### Calibration: steady-state and key parameter values (annual version, raw-materials sector excluded) - Regions: five-region model: United States (US), emerging Asia (AS), euro area (EU), Japan (JA), remaining countries (RC). - World steady-state parameter assumptions: - World technology growth rate g = 1.015 (1.5% p.a.). - World population growth rate n = 1.01 (1% p.a.). - Steady-state inflation rates: 2.0% in US, AS, EU and RC; 1% in JA. - Long-run real interest rate βr = 1.03 (3% p.a.), equalized across countries. - Household calibration: - Average remaining time at work of 20 years → χ = 0.95. - Planning horizon 1/(1 − θ) = 10 years → θ = 0.9. - Intertemporal elasticity of substitution 0.25 → γ = 4. - Labor-supply elasticity targeted at 0.5 via leisure share parameter η. - Shares of liquidity-constrained agents: 25% in US, EU, JA; 50% in AS and RC. - Shares in dividend income equal to half their population shares in all regions. - Fiscal calibration target: a one percentage point increase in U.S. government-debt-to-GDP raises U.S. (and world) real interest rate by approximately three to four basis points (calibration target aligning with Laubach (2003), Engen and Hubbard (2004), Gale and Orszag (2004)). - Technology and markups: - Elasticities: ξ_ZN = ξ_ZT = 1; ξ_NM = ξ_TM = ξ_T = ξ_I = ξ_C = 0.75; ξ_A = ξ_G = 0.5. (Table 3 and related text preserved.) - Steady-state markups: in tradables and nontradables manufacturing βE_N and βE_T, and union wage setting βE_U = 1.1; investment and consumption goods production βE_I and βE_C, and retailing βE_R = 1.05; import agents E_NM and E_TM = 1.025. (Table 4 descriptions preserved.) - Expenditure and factor shares: - Investment-to-GDP ratio very high in AS; reflected by higher capital share and depreciation rate in AS. - Depreciation rates: conventional 10% p.a. for US, EU, JA, RC; 12% for AS (supports high investment-to-GDP in AS). - Nontradables labor share 6 percentage points higher than average; tradables labor share 6 percent lower than average. - Government public capital depreciation δ_G = 0.04 (4% p.a., Kamps (2004)). - Elasticity of aggregate output w.r.t. public capital α_G = 0.1 (implied by Ligthart and Suárez (2005) estimate of 0.14 and model simulations). - Financial accelerator calibration: - Leverage (corporate debt / corporate equity) = 100 in all sectors and regions. - Share of firms that go bankrupt in any given year = 8 percent. - Steady-state external finance premium = 1.5 percent. - These fix steady-state firm riskiness σ_N and σ_T, bankruptcy monitoring costs E_N and E_T, and steady-state shares of net worth distributed as dividends ˇS_{N,nwd} and ˇS_{T,nwd}. - Monetary rule parameters: listed in Table 9 (regional calibrations; AS assumed fixed exchange rate). - Fiscal rule parameters: listed in Table 10; target surplus-to-GDP ratios consistent with calibrated government-debt-to-GDP ratios; OECD estimates used for countercyclical coefficients. ### Model structure, modularity, and applications - Fiscal rule block, shocks block, and balance-of-payments block are part of the core of GIMF and included in the application “Fiscal Stimulus to the Rescue?”. - Catching-up feature of technology shocks is modular and can be turned off (it is turned off in “Fiscal Stimulus to the Rescue?”). - Central banks using GIMF (summary of engagement): - Hong Kong Monetary Authority (HKMA): started 2008Q3/4, produced HKMA working paper, uses model internally. - Central Bank of Russia (CBR): started 2008Q4, using production version for policy simulations; Russia is a test case for the oil sector. - Banco de Portugal (BdP): started 2007, adapted GIMF (PESSOA), production model for Portugal. - Banco Central de Reserva de Perú: started late 2008, converted to DYNARE-based version calibrated to Peru. - Banque de France (BdF): started 2008Q4, collaborating on translation to DYNARE. - IMF Area Departments using GIMF: multiple country and policy studies (excerpts listed in source). *Content unit: _wp1034 - 2. Business Cycle Stabilization (extracted from provided IMF chapter).* ### References ### _wp1034 - References ### References - Bernanke, B.S., Gertler, M. and Gilchrist, S. (1999), “The Financial Accelerator in a Quantitative Business Cycle Framework”, in: John B. Taylor and Michael Woodford, eds., Handbook of Macroeconomics, Volume 1C.Amsterdam: Elsevier. - Blanchard, O.J. (1985), “Debt, Deficits, and Finite Horizons”,Journal of Political Economy, Vol. 93, pp. 223-247. - Christiano, L., Motto, R. and Rostagno, M. (2007), “Financial Factors in Business Cycles”, Working Paper. - Engen, E.M. and Hubbard, R.G. (2004), “Federal Government Debt and Interest Rates”,NBER Macroeconomics Annual, Vol. 19, pp. 83-138. - Faruqee, H. and Laxton, D. (2000), “Life-Cycles, Dynasties, Saving: Implications for Closed and Small, Open Economies”, IMF Working Paper Series, WP/00/126. - Gale, W. and Orszag, P. (2004), “Budget Deficits, National Saving, and Interest Rates”, Brookings Papers on Economic Activity, Vol. 2, pp. 101-187. - Kamps, C. (2004), “New Estimates of Government Net Capital Stocks for 22 OECD Countries 1960-2001”, IMF Working Paper Series, WP/04/67. - Laubach, T. (2003), “New Evidence on the Interest Rate Effectsof Budget Deficits and Debt”, Finance and Economics Discussion Series 2003-12, Board of Governors of the Federal Reserve System. - Ligthart, J.E. and Suárez, R.M.M. (2005), “The Productivity of Public Capital: A Meta Analysis”, Working Paper, Tilburg University. ### Key Tables and Numerical Parameters - Table 1: Long Run Growth Rates and Interest Rates - World Technology Growth g1.015 1.015 1.015 1.015 1.015 - World Population Growth n1.01 1.01 1.01 1.01 1.01 - Steady State Inflation Rate π1.02 1.02 1.02 1.01 1.02 - Long Run Real Interest Rate r1.03 1.03 1.03 1.03 1.03 - Forex Risk Premium ξf 00000 - Government Risk Premium ξb 00000 - Table 2: Utility Functions - Average Planning Horizon in Years (θ= 0.9)1010 101010 - Average Remaining Working Life (χ= 0.95)2020 202020 - Intertemporal Elasticity of Substitution (γ= 4)0.25 0.25 0.25 0.25 0.25 - Labor Supply Elasticity (endogenizes ηOLG, ηLIQ) 0.5 0.5 0.5 0.5 0.5 - Share of Liquidity Constrained Agents ψ0.25 0.50 0.25 0.25 0.50 - Dividend Share of Liq. Constrained Agents ι0.125 0.25 0.125 0.125 0.25 - Table 3: Elasticities of Substitution - Nontradables: Capital-Labor ξZN 11111 - Tradables: Capital-Labor ξZT 11111 - Nontradables Import Agents: Different Countries ξNM 0.75 0.75 0.75 0.75 0.75 - Tradables Import Agents: Different Countries ξTM 0.75 0.75 0.75 0.75 0.75 - Distributors: Home-Foreign Tradables ξT 0.75 0.75 0.75 0.75 0.75 - Inv. Goods Producers: Home-Foreign Tradables ξI 0.75 0.75 0.75 0.75 0.75 - Cons. Goods Producers: Home-Foreign Tradables ξC 0.75 0.75 0.75 0.75 0.75 - Distributors: Tradables-Nontradables ξA 0.5 0.5 0.5 0.5 0.5 - Government: Consumption-Investment Goods ξG 0.5 0.5 0.5 0.5 0.5 - Table 4: Steady State Markups - Nontradables Manufacturing EN 1.1 1.1 1.1 1.1 1.1 - Tradables Manufacturing ET 1.1 1.1 1.1 1.1 1.1 - Union Wage Setting EU 1.1 1.1 1.1 1.1 1.1 - Investment Goods Production EI 1.05 1.05 1.05 1.05 1.05 - Consumption Goods Production EC 1.05 1.05 1.05 1.05 1.05 - Retail Sector ER 1.05 1.05 1.05 1.05 1.05 - Nontradables Import Agents ENM 1.025 1.025 1.025 1.025 1.025 - Tradables Import Agents ETM 1.025 1.025 1.025 1.025 1.025 - Table 5: Steady State Expenditure to GDP Ratios - Share in World GDP 27.4 12.3 22.0 9.1 29.3 - Consumption / GDP 65.1 59.2 58.1 59.8 59.1 - OLG Consumption / GDP 51.3 34.3 45.8 46.9 34.0 - LIQ Consumption / GDP 13.8 24.9 12.3 12.9 25.1 - Private Investment / GDP 17.2 25.0 18.3 21.0 19.0 - Government Spending / GDP 17.5 16.0 23.5 19.5 22.0 - Government Investment / GDP 2.5 4.0 3.0 2.5 2.0 - Government Consumption / GDP 15.0 12.0 20.5 17.0 20.0 - Government Transfers / GDP 20.0 10.0 20.0 20.0 20.0 - Trade Balance / GDP 0.2 -0.2 0.1 -0.3 -0.1 - Exports / GDP 11.7 26.8 17.5 10.8 21.9 - Final Goods Exports / GDP 8.3 20.5 13.7 8.0 9.6 - Intermediate Goods Exports / GDP 3.4 6.3 3.8 2.8 12.3 - Imports / GDP 11.5 27.0 17.4 11.0 21.9 - Consumption Goods Imports / GDP 5.2 5.7 6.8 3.8 9.1 - Investment Goods Imports / GDP 2.6 6.4 4.0 1.6 7.5 - Intermediate Goods Imports / GDP 3.7 14.9 6.6 5.6 5.3 - Tradables Demand Effects of Technology κ1 1 1 1 1 - Nontradables Demand Effects of Technology ̃κ1 1 1 1 1 - Table 6: Steady State Factor Shares and Depreciation Rates - Labor Income / GDP 60 54 60 60 60 - Nontradables Labor Income / GDP 66 60 66 66 66 - Tradables Labor Income / GDP 54 48 54 54 54 - Depreciation Rate of Private Capital δK 0.1 0.12 0.1 0.1 0.1 - Nontradables Output / Manufacturing Output 50 50 50 50 50 - Consumption Goods Input / Government Output 50 50 50 50 50 - Table 7: Miscellaneous Steady State Ratios and Parameters - Government Debt / GDP 50 55 60 75 60 - Net Foreign Assets / GDP -28.0 27.5 -13.0 42.5 11.1 - Labor Income Taxes / Total Taxes 40 40 40 40 40 - Capital Income Taxes / Total Taxes 10 10 10 10 10 - Consumption Taxes / Total Taxes 25 25 25 25 25 - Lump-Sum Taxes / Total Taxes 25 25 25 25 25 - Depreciation Rate of Public Capital δG 0.04 0.04 0.04 0.04 0.04 - Output Elasticity w.r.t. Public Capital (αG = 0.1) 0.14 0.14 0.14 0.14 0.14 - Table 8: Financial Accelerator Sector - Leverage in Nontradables in % 100 100 100 100 100 - Leverage in Tradables in % 100 100 100 100 100 - Annual Bankruptcy Rate in Nontradables in % 8 8 88 8 - Annual Bankruptcy Rate in Tradables in % 8 8 88 8 - External Finance Premium in Nontradables in % 1.5 1.5 1.5 1.5 1.5 - External Finance Premium in Tradables in % 1.5 1.5 1.5 1.5 1.5 - Table 9: Monetary Rule Parameters - δi 0.715 1 0.343 0.392 0.715 - δπ 1.034 0 1.483 0.913 1.034 - δ̃π 0.216 1 0.237 0.216 0.216 - δy 00 000 - δygr 0.25 0 000 - δe 0106 000 - Table 10: Fiscal Rule Parameters - dgdp 0.34 0.25 0.49 0.33 0.30 - ddebt 00000 - dtax 00000 - drawmat 00000 - dctax 00000 - dktax 00000 ### Figures and Trade Matrices - Figure 1: Goods and Factor Flows in GIMF - Diagrammatic representation includes: OLG HH (HO), OLG HH (RW), LIQ HH (HO), LIQ HH (RW), Unions (HO), Unions (RW), Inv. Producers (HO/RW), Manufacturers (HO/RW), Nontradables (HO/RW), Tradables (HO/RW), Retailers (HO/RW), Import Agents (HO/RW), Gov’t (HO/RW), World Oil Market, Sticky Prices, Sticky Wages, Import Adj. Costs, Inv.Adj.Costs, EPs, K, G, Y flows. - Figure 2: Trade Matrix: Intermediate Goods - Global Bilateral Trade Flows - Intermediate Goods (in % of World GDP) - Displayed bilateral flows include: 0.11% , 0.13% , 0.41% , 0.10% , 0.10% , 0.02% , 0.60% , 0.09% , 0.15% , 0.02% , 0.03% , 0.03% , 1.33% , 1.20% , 0.27% , 0.78% , 0.24% , 0.11% , 0.07% , 0.50% - Figure 3: Trade Matrix: Investment Goods - Global Bilateral Trade Flows - Investment Goods (in % of World GDP) - Displayed bilateral flows include: 0.21% , 0.08% , 0.4% , 0.23% , 0.18% , 0.02% , 1.02% , 0.12% , 0.13% , 0.05% , 0.07% , 0.07% , 0.29% , 0.44% , 0.003% , 0.27% , 0.17% , 0.15% , 0.04% , 0.70% - Figure 4: Trade Matrix: Consumption Goods - Global Bilateral Trade Flows - Consumption Goods (in % of World GDP) - Displayed bilateral flows include: 0.35% , 0.16% , 0.61% , 0.46% , 0.35% , 0.14% , 0.06% , 1.25% , 0.2% , 0.1% , 0.07% , 0.07% , 0.13% , 0.29% , 0.86% , 0.01% , 0.61% , 0.14% , 0.2% , 0.1% , 0.73% *Source: _wp1034 - References (PDF).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1034.pdf_