## 1. OverlappingGenerationsHouseholds

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### Model overview and key ingredients
- World: 2 countries — the United States (U.S.) and the rest of the world (RW).
- Agents: Two household types in each country: overlapping generations households (OLG) with finite planning horizons, and liquidity constrained households (LIQ) without access to financial markets.
- Growth and rescaling:
  - World grows at constant rate g = T_t / T_{t−1}; real variables rescaled by T_t (notation ˇx_t = x_t / T_t); steady state denoted ̄x.
- Asset markets and trade:
  - Incomplete asset markets; complete home bias in government debt (one-period nominal domestic-currency bonds).
  - Only internationally traded asset: one-period nominal bonds denominated in U.S. currency.
  - Complete home bias in ownership of domestic firms; equity not traded domestically—households receive lump-sum dividends.
- Production and rigidities:
  - Manufacturers produce tradable and nontradable goods; distributors assemble goods combining private output with publicly provided infrastructure capital.
  - Government investment finances public capital stock.
  - Multiple nominal and real rigidities: habit persistence, investment adjustment costs, import adjustment costs, sticky goods prices, sticky nominal wages (unions).
- Policy interactions:
  - Monetary policy: reaction function stabilizing inflation and output.
  - Fiscal policy: stabilizes government balance and debt; medium- and long-run dynamics depend on fiscal policy; short-run dynamics determined by monetary–fiscal interaction.
  - Government investment treated as productive input, enabling analysis of spending-cut consolidations.

### Non-Ricardian features (four key model ingredients)
- Finite planning horizons (OLG): agents discount future tax liabilities at a higher rate than the market real interest rate due to death probability.
- Lifecycle income decline: productivity declines with age, increasing effective discounting of future labor income tax liabilities.
- Liquidity-constrained agents (LIQ): optimizing but without access to financial markets; consumption adjusts one-for-one with after-tax income.
- Distortionary taxes: uniform labor income tax τ_{L,t} and uniform consumption tax τ_{c,t} that distort labor effort and consumption choices.

### OLG household structure and consumption technology
- Utility and money:
  - Lifetime expected utility E_t Σ_{s=0}^∞ (βθ)^s [ 1/(1−γ) ( (c^OLG_{a+s,t+s}/h^OLG_{a+s,t+s})^{η_OLG} (1−c^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: β; θ<1; γ>0; 0<η_OLG<1; money demand considered in cashless limit u_m → 0.
- Habit:
  - h^OLG_{a,t} = ( c^OLG_{t−1} n(1−ψ) )^v with habit persistence parameter v.
- CES retail aggregator:
  - c^OLG_{a,t} = μ ( ∫_0^1 (c^OLG_{a,t}(i))^{(σ_R−1)/σ_R} di )^{σ_R/(σ_R−1)}.
  - Demand for variety: c^OLG_{a,t}(i) = μ ( P^R_t(i)/P^R_t )^{−σ_R} c^OLG_{a,t}.
  - Aggregate retail price P^R_t = ( ∫_0^1 (P^R_t(i))^{1−σ_R} di )^{1/(1−σ_R)}.

### Financial instruments, income and budget constraints
- Assets: domestic government bonds B_{a,t} and foreign bonds F_{a,t} (U.S. currency); nominal exchange rate E_t; nominal net foreign assets E_t F_{a,t}.
- Gross nominal interest rates: i^*_t (U.S.), i_t (RW).
- Insurance contract: participation costs premium (1−θ)/θ on household financial wealth per period alive; insurer encashes household wealth on death.
- Productivity: Φ_{a,t} = Φ_a = κ χ^a with χ<1; population average normalized to one.
- Taxes/transfers: lump-sum transfers τ^{OLG}_T_{a,t} (redistributed to LIQ), labor tax τ_{L,t}, consumption tax τ_{c,t} payable on pre-incentive price P_t; final goods price P_t chosen as numeraire.
- Nominal budget constraint (abbreviated):
  - P^R_t c^OLG_{a,t} + P_t c^OLG_{a,t} τ_{c,t} + B_{a,t} + E_t F_{a,t} = (1/θ)[ i_{t−1} B_{a−1,t−1} + i^*_{t−1} E_t F_{a−1,t−1} ] + W_t Φ_{a,t} c^OLG_{a,t} (1−τ_{L,t}) + Σ_{j=N,T,D,R,U,M} ∫_0^1 D^j_{a,t}(i) di − τ^{OLG}_T_{a,t}.

### Aggregation and first-order conditions
- Aggregation across cohorts:
  - c^OLG_t = n(1−ψ)(1−θ) Σ_{a=0}^∞ θ^a c^OLG_{a,t}.
- Key first-order conditions (rescaled):
  - Variety demand: ˇc^OLG_t(i) = ( P^R_t(i)/P^R_t )^{−σ_R} ˇc^OLG_t.
  - Consumption/leisure intratemporal condition:
    - ˇc^OLG_t n(1−ψ) − c^OLG_t = [ η_OLG/(1−η_OLG) ] ˇw_t (1−τ_{L,t}) / (p^R_t + τ_{c,t}).
  - Covered/uncovered interest parity arbitrage for foreign currency bonds:
    - i_t = i^*_t ε_{t+1}.

### Consumption, wealth decomposition, and marginal propensity to consume
- Aggregate consumption related to wealth:
  - ˇc^OLG_t / Θ_t = ˇfw_t + ˇhw_t, where Θ_t is marginal propensity denominator.
- Definitions:
  - Θ_t = (p^R_t + τ_{c,t}) / η_OLG + θ j_t r_t / Θ_{t+1}.
  - ˇfw_t = (1/π_t) g [ i_{t−1} ˇb_{t−1} + i^*_{t−1} ε_t ˇf_{t−1} e_{t−1} ].
  - ˇhw_t = ˇhw^L_t + ˇhw^K_t with recursive components specified in the source.
  - j_t expression depends on μ, β, i_t/π_{t+1}, real wages, consumption growth, habit persistence v, and χ.
- Economic implications:
  - Households discount future tax liabilities at rate at least r_t/θ (higher than market rate r_t) due to mortality and productivity decline χ<1.
  - Government debt may be perceived as net wealth when households do not expect to bear future tax burden; greater myopia (lower θ) increases perceived net wealth from outstanding government debt.
  - Marginal propensity to consume simplifies to (1−βθ) in the simplest case (log utility, exogenous labor supply, no consumption taxes).
  - Intertemporal elasticity of substitution 1/γ and habit persistence v affect responsiveness to real interest rate and past consumption.

### Representative agent alternative (for policy comparison)
- Infinite horizon representative agent model identical except:
  - θ and χ set to 1.
  - Calibrated with higher share of LIQ households to mimic short-run non-Ricardian behavior.
  - Third specification: positive deviations from target net foreign liabilities/GDP raise external interest rate.
  - Consumption system replaced by ˇc^OLG_{t+1} = j_t g ˇc^OLG_t with χ=1 in j_t.

---

### Liquidity Constrained Households (LIQ) and Aggregate Household Sector
- LIQ households:
  - Objective similar to OLG except no money holdings.
  - Budget constraint restricts consumption to at most current income (after-tax wage income + government transfers τ^{LIQ}_{T,a,t}).
  - LIQ agents solve intratemporal consumption–leisure optimization (not simple rule-of-thumb).
- Aggregated LIQ conditions (rescaled):
  - Variety demand: ˇc^{LIQ}_t(i) = ( P^R_t(i)/P^R_t )^{−σ_R} ˇc^{LIQ}_t.
  - Budget/income (first-order) form:
    - ˇc^{LIQ}_t (p^R_t + τ_{c,t}) = ˇw_t c^{LIQ}_t (1−τ_{L,t}) + ˇτ_{T,t}.
  - Consumption/leisure intratemporal condition:
    - ˇc^{LIQ}_t n ψ − c^{LIQ}_t = [ η_{LIQ} / (1−η_{LIQ}) ] ˇw_t (1−τ_{L,t}) / (p^R_t + τ_{c,t}).
- Aggregate household sector:
  - ˇC_t = ˇc^{OLG}_t + ˇc^{LIQ}_t.
  - L_t = c^{OLG}_t + c^{LIQ}_t.

---

### Firms, unions, retailers and distributors — structure and price dynamics
- Firms: continuum of firms per sector, monopolistic competition in output, competitive input markets, sectoral price and investment adjustment costs.
- Manufacturers:
  - Two sectors J∈[N,T]; CES production combining capital and union labor with elasticity ξ_J.
  - Price adjustment costs and investment adjustment costs specified; capital law K^J_{t+1}(i) = (1−δ) K^J_t(i) + I^J_t(i).
  - First-order conditions include sectoral Phillips curve for inflation π^{˜J}_t with inflation adjustment cost parameter φ_{P^{˜J}}.
- Unions:
  - Demand for labor varieties with elasticity σ_U; nominal wage rigidities yield a Phillips curve for wage inflation π^U_t.
- Import agents:
  - CES demand for imported varieties with elasticity σ_JM; border import price P_{M,cif,t}; purchasing power parity p^{JM,cif}_t = p^{JH*}_t e_t.
  - Nominal price rigidities and a Phillips curve for import price inflation π^{JM}_t.
- Distributors:
  - Multi-stage nested CES production combining domestic tradables, foreign tradables, nontradables, and public capital K^G_t.
  - Import adjustment costs G^T_{F,t}(i) and distributor price adjustment costs; distributor profit maximization yields final goods Phillips curve π_t.

### Retailers (quantity adjustment and optimization)
- Household demand for retailer varieties:
  - C_t(i) = μ (P^R_t(i) / P^R_t)^−σ_R C_t. (43)
- Quantity adjustment costs:
  - G^RY,t(i) = φ_C / 2 · C_t · [ μ ( (C_t(i)/g) − C_{t−1}(i) ) / C_{t−1}(i) ]^2. (44)
- Retailer first-order condition (quantity adjustment Phillips-like):
  - ( σ_R − 1 ) / σ_R · p^R_t−1 = φ_C · μ · [ ( Č_t − Č_{t−1} ) / Č_{t−1} ] · ( Č_t / Č_{t−1} ) − θ g_r_t φ_C μ [ ( Č_{t+1} − Č_t ) / Č_t ] · ( Č_{t+1} / Č_t )^2. (45)

---

### Government fiscal framework and public capital
- Fiscal instruments:
  - Taxes τ_{L,t}, τ_{c,t}; transfers τ_{T,t}; government consumption G^{cons}_t (exogenous, unproductive); government investment G^{inv}_t (augments public infrastructure K^G_t).
- Transfers rule (redistribution of dividends):
  - τ̌_{T,t} = ι [ ď_Nt + ď_Tt + ď_Dt + ď_Mt ] + č_{LIQ,t} Č_t ď_{R,t} + c_{LIQ,t} L_t ď_{U,t}. (46)
  - Redistributed share parameter ι ≤ ψ.
- Public infrastructure law of motion:
  - Ǩ^G_{t+1} / g_{t+1} = (1 − δ_G) Ǩ^G_t + Ǧ^{inv}_t. (47)
- Government budget and surplus targeting:
  - Real government budget constraint: b̌_t = (i_{t−1} / π_t^g) b̌_{t−1} − š_t with b̌_t = B_t / P_t T_t. (48)
  - Primary surplus: š_t = τ_{L,t} w̌_t L_t + τ_{c,t} Č_t − Ǧ^{cons}_t − Ǧ^{inv}_t. (49)
  - Fiscal rule ensuring non-explosive debt: š_t − (i_{t−1} − 1) b̌_{t−1} / π_t^g ǧ_dp_t = − b̌_t + b̌_{t−1} / π_t^g ǧ_dp_t = r_t. (50)

---

### Monetary policy, equilibrium identities, and calibration
- Monetary rule:
  - i_t = (i_{t−1})^{μ_i} (r^{smooth}_t π_{t+1})^{1−μ_i} (π_{t+1}/̄π)^{(1−μ_i)μ_π μ_g [gˇdp_t gˇdp_{t-1}]^{(1−μ_i)μ_ygr}}. (51)
  - r^{smooth}_t = (r_{t-1} r_t r_{t+1})^{1/3}. (52)
- Key equilibrium and accounting relations include market clearing, net foreign asset evolution, and international bond market clearing (equations (53)–(59)).
- Calibration highlights (annual model; preserve exact numeric values):
  - U.S. represents 25 percent of world GDP.
  - U.S. long-run government debt to GDP = 60 percent.
  - U.S. net foreign liabilities (NFL) to GDP = 55 percent.
  - RW net foreign assets to GDP = 18.3 percent.
  - RW government debt to GDP = 30 percent.
  - Share of liquidity-constrained agents ψ: U.S.: 33 percent; RW: 50 percent.
  - U.S. ratios to GDP of intermediate and final goods imports and of intermediate goods exports = 6 percent.
  - Initial steady state final output based real exchange rate e = 1.
  - Steady state world real growth rate fixed at 1.5% per annum or ̄g = 1.015.
  - Steady state inflation ̄π = 1.02 (2% per annum).
  - Long-run real interest rate ̄r = 1.02 (2% per annum).
  - Steady-state nominal growth rate gπ = 1.0353.
  - Average remaining time at work 20 years → χ = 0.95.
  - Planning horizon 1/(1−θ) = 10 years → θ = 0.9.
  - Calibration target: a one percentage point increase in government debt to GDP in the U.S. → approximately four basis points increase in the U.S. (and world) real interest rate.
  - Intertemporal elasticity of substitution = 0.25 → γ = 4.
  - Habit persistence v = 0.4.
  - Frisch elasticity targeted at 0.5.
  - Elasticities ξ_{ZN} = ξ_{ZT} = 1; ξ_T = ξ_D = 1.5; ξ_A = 0.5.
  - Adjustment costs φ_I = 10; φ_{FT}, φ_{FD}, φ_C = 5.
  - Markups: 20 percent in manufacturing and labor market; 5 percent in distribution and retail; 2.5 percent for import agents.
  - Inflation adjustment cost parameter = 10 for all sectors except import agents in RW (zero there).
  - Manufacturing labor income share set to reach aggregate labor income share 64 percent.
  - Nontradables share in GDP = 50 percent.
  - Steady state investment to GDP = 16 percent.
  - Steady state government spending to GDP = 18 percent.
  - Implied consumption to GDP ratios: U.S. = 65.7 percent; RW = 66.1 percent.
  - Private capital depreciation δ = 10 percent (annual).
  - Productive public spending to GDP = 3 percent; other government spending = 15 percent.
  - Public capital depreciation δ_G = 0.04.
  - Public capital stock to GDP ratio implied = 54.5 percent.
  - Long-run elasticity of GDP w.r.t. K_G targeted at 0.14 → α_G set accordingly.
  - Monetary policy parameters: μ_i = 0.25; μ_π = 0.6; μ_{ygr} = 0.25.
- Alternative representative agent model (REP) adjustments:
  - ψ^{*} increased from 33 to 50 percent for U.S.
  - Parameter governing speed with which NFL returns to long run set as small as possible to generate current account persistence.

---

### Fiscal experiment, comparative predictions (FH vs REP), and quantitative implications
- Long-run identity linking NFL and current account:
  - CDEF_t = NFL_t − NFL_{t−1}. (60)
  - Along balanced growth path: CDEF/GDP = (gπ − 1)/gπ NFL/GDP = 0.0341 NFL/GDP. (61)
  - Numeric implication: A 10 percentage point increase in the NFL ratio → a 0.341 percentage point increase in the current account deficit ratio in the very long run.
- Government deficit to debt identity:
  - GDEF/GDP = (gπ − 1)/gπ GDEBT/GDP = 0.0341 GDEBT/GDP. (62)
- Fiscal experiment (permanent reduction in government savings ratio by 0.5 percentage points with front-loaded short-run expansion):
  - Short-run: both FH (finite-horizon) and REP models predict similar gradual increase in government debt to GDP of 14.7 percentage points (∆(GDEF/GDP)/0.0341).
  - Labor tax rate declines by about 5 percentage points in the first 2 years, then rises gradually to service higher debt.
- Divergent medium- and long-run outcomes:
  - FH model:
    - Long-run labor tax rate ≈ 1.5 percentage points higher.
    - Real interest rates rise by about 40 basis points, causing strong crowding-out of investment, consumption and GDP.
    - Permanently higher current account deficit and NFL; spillovers to RW via international equalization of real rates.
  - REP model:
    - Long-run labor tax rate ≈ 0.5 percentage points higher.
    - Non-liquidity-constrained agents increase savings after ~5 years; NFL changes little; minimal long-term crowding out.
- Short-run dynamics and monetary policy:
  - Short-run consumption boom in FH due to non-internalization of future tax burden; REP shows similar short-run behavior given higher ψ but diverges after ~5 years.
  - Short-run GDP constrained by monetary policy rule; holding nominal short-term interest rate fixed for the first 2 years roughly doubles the short-run GDP effects.
- Real exchange rate and trade dynamics:
  - FH: short-run appreciation with trade deficits; over time real exchange rate depreciates as trade balance adjusts to finance larger interest obligations on higher NFL.
  - REP: much smaller real exchange rate movements.
- Sensitivities:
  - Exchange rate response sensitive to elasticity ξ_T (base ξ_T = 1.5).
  - Short-run GDP response sensitive to monetary policy accommodation.
  - Crowding-out magnitude sensitive to planning horizon θ (longer horizons → results closer to REP).
  - REP long-run predictions for NFL and real rates less sensitive to parameter changes.

### Policy implication from fiscal simulations
- Fiscal deficits may have contributed to deterioration of U.S. current account deficits since 2000; given long lags, main current account effects of fiscal policies may still be forthcoming.
- Improving U.S. fiscal deficits should be part of a policy package to raise national savings and help resolve global current account imbalances.
- Fiscal improvement options include higher taxes or lower government spending; cutting productive government investment has adverse supply-side long-run effects (see Section V).

---

### V. Implications of Government Spending Cuts (select quantitative findings)
- Experiment: permanent 10 percent reduction in government investment (solid lines) versus permanent 10 percent reduction in government consumption (dashed lines); government balance held fixed via labor tax profile.
- 10 percent reduction in government investment:
  - Long-run decline in real GDP in the U.S. of about 1.4 percent.
  - Short run: demand falls; lower public investment reduces lifetime wealth, limiting crowding-in of private consumption.
  - Initial decline in real interest rates stimulates consumption and investment; over time supply-side effects dominate, reducing private investment and consumption.
  - External: RW experiences higher imports, lower exports and an appreciation in the real exchange rate; short- and medium-term spillovers to RW GDP are positive.
- 10 percent reduction in government consumption:
  - Medium- and long-run GDP effects opposite to cutting government investment.
  - Frees resources for private consumption and investment; short-run investment may decline as consumption boom raises interest rates.
  - Part of GDP increase represents higher exports to RW and leads to real exchange rate depreciation.

---

### VI. Implications of Private Sector Savings Behavior
- Permanent one percentage point reduction in U.S. rate of time preference (lower U.S. private savings):
  - World savings falls; permanently higher real interest rates; declines in output and consumption in both U.S. and RW.
  - Medium- and long-run effects similar to lower public sector savings.
- Permanent one percentage point increase in RW rate of time preference (higher RW private savings):
  - Lower RW consumption crowds in investment and expands production; over time RW consumption rises but lags production increases.
  - Persistent rise in savings produces a 100 basis points reduction in real interest rates in both U.S. and RW.
  - U.S. effects: consumption boom and eventually higher GDP as lower real rates stimulate investment; short run: U.S. GDP may decline as monetary policy resists inflation.
  - U.S. NFL position path: initially declines then rises, stabilizing at a new steady-state value almost 10 percentage points higher.
  - Current account deteriorates by around 0.35 percentage points in the long run, and by significantly more in the short run.
  - Sharp U.S. real exchange rate appreciation accompanies generated trade deficits.
- Interpretation:
  - Higher RW savings (private and public) consistent with elevated U.S. NFL position and low world real interest rates.
  - Major drivers of large U.S. current account deficits include low U.S. private and public savings and increased RW savings/investment gap.

---

### VII. Conclusion and extensions
- Model summary:
  - Open economy business cycle model combining short-run monetary policy analysis with a finite-horizon OLG framework for medium- and long-term fiscal analysis.
- Key conclusions:
  - Short-run dynamics similar to representative agent models with liquidity-constrained consumers.
  - Medium- to long-run predictions differ: in finite-horizon model, higher government deficits strongly reduce current account balances and private capital accumulation; fiscal consolidation can materially improve current account imbalances.
  - Cutting productive government investment poses trade-offs between short-term demand management and longer-term supply losses (e.g., permanent −1.4 percent U.S. GDP for a 10 percent cut in government investment).
  - Higher RW savings have important roles in explaining low world real rates and rising U.S. NFL.
- Ongoing extensions:
  - Multi-country setting for more detailed accounting of savings behavior across countries.
  - Linearized version for Bayesian estimation and richer shock identification.

*Source: _wp07202 - 1. OverlappingGenerationsHouseholds (excerpt) — IMF working paper content unit*

### 1. OverlappingGenerationsHouseholds .................   8

### 1. OverlappingGenerationsHouseholds

### Model overview
- World: 2 countries — the United States (U.S.) and the rest of the world (RW).
- Agents: Two household types in each country: overlapping generations households (OLG) with finite planning horizons, and liquidity constrained households (LIQ) without access to financial markets.
- Growth: World grows at constant rate g = T_t / T_{t−1}; real variables are rescaled by T_t (notation ˇx_t = x_t / T_t), steady state denoted ̄x.
- Asset markets: incomplete. Complete home bias in government debt (one-period nominal domestic-currency bonds). Only internationally traded asset: one-period nominal bonds denominated in U.S. currency. Complete home bias in ownership of domestic firms; equity not traded domestically—households receive lump-sum dividends.
- Production structure: manufacturers produce tradable and nontradable goods; distributors assemble goods combining private output with publicly provided infrastructure capital; government investment finances public capital stock; multiple layers of nominal price rigidities and real rigidities (habit persistence, investment adjustment costs, import adjustment costs).

### Household types, demographics, and taxes
- Birth and cohort sizes:
  - U.S. OLG births per period: n^* (1−ψ^*)(1−θ)
  - RW OLG births per period: n (1−ψ)(1−θ)
  - U.S. LIQ births per period: n^* ψ^* (1−θ)
  - RW LIQ births per period: n ψ (1−θ)
- Mortality and planning horizon:
  - Constant probability of death each period: (1−θ)
  - Average planning horizon: 1/(1−θ)
- Lifecycle productivity: labor productivity declines at a constant rate over agents’ lifetimes (stylized lifecycle income).
- Taxes: uniform labor income tax and uniform consumption tax apply to both household types.

### Non-Ricardian features (four key model ingredients)
- Finite planning horizons (OLG): agents discount future tax liabilities at a higher rate than the market real interest rate because they attach positive probability to not surviving to pay future taxes.
- Lifecycle income decline: average agent expects declining labor productivity, further raising the effective discounting of future labor income tax liabilities.
- Liquidity-constrained agents (LIQ): optimizing but without access to financial markets; consumption adjusts one-for-one with after-tax income.
- Distortionary taxes: labor and consumption taxes are distortionary because labor effort and consumption respond to tax-induced relative price movements.

### Rigidities, preferences, and technology
- Preferences: CRRA utility used to highlight the role of the intertemporal elasticity of substitution in fiscal shock propagation.
- Endogenous labor supply and endogenous capital formation.
- Technology: both traded and nontraded goods; public infrastructure is an essential input into private output.
- Nominal rigidities: multiple levels of sticky goods prices and sticky nominal wages (unions), enabling meaningful business cycle dynamics beyond one-period price stickiness.
- Real rigidities: habit persistence in consumption, investment adjustment costs, and import adjustment costs.
- Retailers: flexible prices but real adjustment costs in sales volume, contributing to inertial consumption dynamics.

### Fiscal and monetary policy interactions
- Monetary policy: reaction function from state-of-the-art monetary theory stabilizes inflation and output.
- Fiscal policy: stabilizes government balance and therefore government debt; medium- and long-run dynamics depend only on fiscal policy, while short-run dynamics are determined by the interaction of monetary and fiscal policies.
- Government expenditure: model allows for government investment in productive infrastructure (unlike models that treat all government spending as waste), enabling meaningful analysis of spending-cut consolidations.

### Comparative model implications and motivation
- Motivation: assess whether post-2001 U.S. fiscal deterioration contributed substantially to the increase in U.S. current account deficits, or whether fiscal consolidation in the U.S. would significantly reduce global current account imbalances.
- Contrast with infinite-horizon (Ricardian) models: models with infinite planning horizons and liquidity-constrained subsets (e.g., SIGMA, GEM) produce powerful short-run fiscal effects but negligible medium- and long-run effects of higher government debt on current account balances.
- Role of OLG: overlapping generations structure uniquely produces medium- and long-run crowding-out effects of government debt and endogenously determines net foreign liability positions as a function of government debt.
- Quantitative expectations preview:
  - The OLG model and an infinite-horizon model with liquidity-constrained agents behave similarly in the very short run.
  - In the infinite-horizon model, medium- and long-run effects of higher government debt are negligible.
  - In the finite-horizon (OLG) model, plausible planning horizons imply an economically significant link between higher government deficits and larger current account deficits, with crowding-out effects on physical capital stocks.

### Structural aggregation and modeling choices
- Firms: manufacturers hold finite planning horizons (aligned with myopic OLG owners); unions face nominal wage rigidities; manufacturers face nominal price rigidities and real capital accumulation rigidities.
- Trade and distribution: import agents and distributors operate with pricing-to-market and nominal rigidities; final goods trade intermediated by final goods import agents.
- Aggregation: per capita variables are only considered at the disaggregated household level; all aggregate variables are absolute.
- Solution approach: results presented for the perfect foresight case; log-linear approximations are straightforward (Technical Appendix).

*Source: _wp07202 - 1. OverlappingGenerationsHouseholds (excerpt) — IMF working paper content unit*

### 1.  Overlapping Generations Households

### 1.  Overlapping Generations Households

### OLG household preferences and consumption technology
- Representative agent of age a at time t derives utility from consumption c^OLG_{a,t} relative to habit h^OLG_{a,t}, leisure (1−c^OLG_{a,t}), and real balances (M_{a,t}/P^R_t).
- Lifetime expected utility:
  - E_t Σ_{s=0}^∞ (βθ)^s [ 1/(1−γ) ( (c^OLG_{a+s,t+s}/h^OLG_{a+s,t+s})^{η_OLG} (1−c^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: β (discount factor); θ<1 (degree of myopia); γ>0 (coefficient of relative risk aversion); 0<η_OLG<1.
  - Money demand considered in the cashless limit u_m → 0 (Woodford (2003) case; Technical Appendix discussed).

- Consumption habit (external “catching up with the Joneses”):
  - h^OLG_{a,t} = ( c^OLG_{t−1} n(1−ψ) )^v, where v parameterizes habit persistence.

- Consumption aggregator and retail price:
  - c^OLG_{a,t} is a CES aggregate over retail varieties c^OLG_{a,t}(i) with elasticity σ_R:
    - c^OLG_{a,t} = μ ( ∫_0^1 (c^OLG_{a,t}(i))^{(σ_R−1)/σ_R} di )^{σ_R/(σ_R−1)}.
  - Demand for individual variety:
    - c^OLG_{a,t}(i) = μ ( P^R_t(i)/P^R_t )^{−σ_R} c^OLG_{a,t}.
  - Aggregate retail price level:
    - P^R_t = ( ∫_0^1 (P^R_t(i))^{1−σ_R} di )^{1/(1−σ_R)}.

### Financial instruments, income and budget constraint
- Assets: domestic government bonds B_{a,t} (domestic currency) and foreign bonds F_{a,t} denominated in U.S. currency; nominal exchange rate vis-à-vis U.S. is E_t; nominal net foreign assets in domestic currency E_t F_{a,t}.
- Gross nominal interest rates: i^*_t (U.S.), i_t (rest of world).
- Participation in financial markets requires an insurance contract costing premium (1−θ)/θ on household financial wealth per period alive; insurer encashes household wealth on death.
- Income sources: labor income from unions, dividend income from firms; productivity of age a: Φ_{a,t} = Φ_a = κ χ^a, with χ<1; overall population average productivity normalized to one.
- Taxes and transfers:
  - Lump-sum transfers τ^{OLG}_T_{a,t} to government (redistributed to LIQ agents).
  - Labor income taxed at τ_{L,t}; consumption taxed at τ_{c,t}.
  - Consumption tax τ_{c,t} payable on pre-incentive price P_t (price at which retailers purchase from distributors); aggregate final goods price P_t (determined by distributors) chosen as numeraire.
- Notation conventions:
  - 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}; final goods inflation π_t = P_t / P_{t−1}.
  - Gross nominal exchange rate depreciation ε_t = E_t / E_{t−1}.
  - Real exchangerate vis-à-vis U.S.: e_t = (E_t P^*_t)/P_t.
  - Real domestic bonds b_t = B_t / P_t; real internationally traded bonds f_t = F_t / P^*_t.
  - Real interest rate r_t = i_t / π_{t+1}.
- Nominal budget constraint:
  - P^R_t c^OLG_{a,t} + P_t c^OLG_{a,t} τ_{c,t} + B_{a,t} + E_t F_{a,t} = (1/θ)[ i_{t−1} B_{a−1,t−1} + i^*_{t−1} E_t F_{a−1,t−1} ] + W_t Φ_{a,t} c^OLG_{a,t} (1−τ_{L,t}) + Σ_{j=N,T,D,R,U,M} ∫_0^1 D^j_{a,t}(i) di − τ^{OLG}_T_{a,t}.

### First-order conditions and aggregation
- Aggregation accounts for cohort sizes at birth and remaining size of each generation:
  - c^OLG_t = n(1−ψ)(1−θ) Σ_{a=0}^∞ θ^a c^OLG_{a,t}.
- First-order conditions (rescaled by technology):
  - Variety demand: ˇc^OLG_t(i) = ( P^R_t(i)/P^R_t )^{−σ_R} ˇc^OLG_t.
  - Consumption/leisure intratemporal condition:
    - ˇc^OLG_t n(1−ψ) − c^OLG_t = [ η_OLG/(1−η_OLG) ] ˇw_t (1−τ_{L,t}) / (p^R_t + τ_{c,t}).

- Covered/uncovered interest parity (arbitrage for foreign currency bonds):
  - i_t = i^*_t ε_{t+1}.

### Consumption as function of wealth and human wealth decomposition
- Aggregate consumption of OLG households expressed as function of financial wealth ˇfw_t and human wealth ˇhw_t, marginal propensity to consume 1/Θ_t:
  - ˇc^OLG_t / Θ_t = ˇfw_t + ˇhw_t.
- Definitions and recursive components:
  - Θ_t = (p^R_t + τ_{c,t}) / η_OLG + θ j_t r_t / Θ_{t+1}.
  - ˇfw_t = (1/π_t) g [ i_{t−1} ˇb_{t−1} + i^*_{t−1} ε_t ˇf_{t−1} e_{t−1} ].
  - ˇhw_t = ˇhw^L_t + ˇhw^K_t.
  - ˇhw^L_t = ( n(1−ψ) (ˇw_t (1−τ_{L,t})) ) + θ χ g / r_t ˇhw^L_{t+1}.
  - ˇhw^K_t = ( ˇd^N_t + ˇd^T_t + ˇd^D_t + ˇd^R_t + ˇd^U_t + ˇd^M_t − ˇτ_{T,t} ) + θ g / r_t ˇhw^K_{t+1}.
  - j_t = [ μ β ( i_t / π_{t+1} ) ]^{1/γ} ( (p^R_t + τ_{c,t}) / (p^R_{t+1} + τ_{c,t+1}) )^{1/γ} ( (ˇw_{t+1} g (1−τ_{L,t+1}) (p^R_t + τ_{c,t})) / (ˇw_t (1−τ_{L,t}) (p^R_{t+1} + τ_{c,t+1})) )^{(1−η_OLG)(1−1/γ)} ( (ˇc^OLG_t / g ˇc^OLG_{t−1}) )^{ v η_OLG (1−1/γ) χ^{(1−η_OLG)(1−1/γ)} }.

- Economic intuition:
  - Financial wealth equals current liabilities of government and foreign households; government servicing of liabilities via taxes affects human wealth components and Θ_t.
  - Households discount future tax liabilities at rate at least r_t/θ (higher than market rate r_t) because of mortality and productivity decline (χ<1).
  - Government debt can be net wealth if households do not expect to bear future tax burden; greater myopia (lower θ) increases perceived net wealth from outstanding government debt.
  - Fiscal expansion via lower taxes tilts tax profile to distant future, increasing debt and (for households that discount future taxes at higher rates) increasing human wealth and thus consumption for a given marginal propensity to consume.
  - Marginal propensity to consume 1/Θ_t:
    - In simplest case (log utility, exogenous labor supply, no consumption taxes) equals (1−βθ).
    - Endogenous labor supply: η_OLG (consumption share) is key determinant of consumption vs leisure split and appears in Θ_t.
    - Time profile of consumption taxes affects Θ_t; balanced-budget shift of consumption taxes from present to future increases Θ_t.
    - Intertemporal elasticity of substitution 1/γ influences responsiveness of consumption to real interest rate r via j_t; for γ>1 income effect of r increase dominates substitution effect, increasing marginal propensity to consume and offsetting contractionary effects on ˇhw_t.
    - Habit persistence enters through the j_t term.

### Representative agent alternative for policy simulations
- Infinite horizon representative agent model identical except:
  - θ and χ set to 1.
  - Calibrated with higher share of LIQ households to mimic short-run non-Ricardian behavior.
  - Requires third specification: positive deviations from target net foreign liabilities/GDP raise external interest rate.
  - Consumption system replaced by:
    - ˇc^OLG_{t+1} = j_t g ˇc^OLG_t, where in j_t expression χ=1.

---

### 2.  Liquidity Constrained Households (LIQ)
- LIQ households: objective function identical to OLG households except no money; budget constraint restricts consumption to at most current income (after-tax wage income + government transfers τ^{LIQ}_{T,a,t}).
- Unlike “rule of thumb” agents, LIQ agents solve intratemporal consumption-leisure optimization.
- Aggregated first-order conditions (rescaled by technology) with aggregate lump-sum transfer ˇτ_{T,t} from OLG to LIQ:
  - Variety demand: ˇc^{LIQ}_t(i) = ( P^R_t(i)/P^R_t )^{−σ_R} ˇc^{LIQ}_t.
  - Budget/income constraint in first-order form:
    - ˇc^{LIQ}_t (p^R_t + τ_{c,t}) = ˇw_t c^{LIQ}_t (1−τ_{L,t}) + ˇτ_{T,t}.
  - Consumption/leisure intratemporal condition:
    - ˇc^{LIQ}_t n ψ − c^{LIQ}_t = [ η_{LIQ} / (1−η_{LIQ}) ] ˇw_t (1−τ_{L,t}) / (p^R_t + τ_{c,t}).

---

### 3.  Aggregate Household Sector
- Aggregate consumption and labor supply are additive across OLG and LIQ households:
  - ˇC_t = ˇc^{OLG}_t + ˇc^{LIQ}_t.
  - L_t = c^{OLG}_t + c^{LIQ}_t.

---

### B.  Firms and Unions

### General structure
- Each sector: continuum i∈[0,1] of firms/agents, perfectly competitive in input markets, monopolistically competitive in output markets.
- Nominal rigidities: manufacturers, unions, import agents, distributors. Retailers face real rigidities (due to adjustment costs and incentives).
- Fixed cost of production calibrated to match steady state labor and capital shares in GDP.
- Sectors pay out net cashflow as dividends each period to OLG households and maximize present discounted value of dividends. Discount rate includes θ so firms’ discount θ/r_t equates to pricing kernel for nonfinancial income of myopic households (β θ (λ_{a+1,t+1}/λ_{a,t})), consistent with household Euler equation λ_{a,t} = β (λ_{a+1,t+1} r_t).

### 1. Manufacturers
- Two manufacturing sectors J∈[N,T]; prices indexed by ˜J∈[N,TH].
- Demand for variety i:
  - Z^J_t(i) = ( P^{˜J}_t(i) / P^{˜J}_t )^{−σ_J} Z^J_t.
- Technology: CES production in capital K^J_t(i) and union labor U^J_t(i) with elasticity ξ_J and labor-augmenting technology T_t:
  - Z^J_t(i) = F(K^J_t(i), U^J_t(i)) = [ (1−α_{U^J})^{1/ξ_J} (K^J_t(i))^{(ξ_J−1)/ξ_J} + (α_{U^J})^{1/ξ_J} (T_t U^J_t(i))^{(ξ_J−1)/ξ_J} ]^{ξ_J/(ξ_J−1)}.
- Price adjustment costs (inflation adjustment quadratic in changes in inflation rate):
  - 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 ).
- Investment adjustment costs quadratic:
  - G^J_{I,t}(i) = φ_I / 2 K^J_t(i) ( I^J_t(i)/K^J_t(i) − I^J_{t−1}/K^J_{t−1} )^2.
- Capital law of motion:
  - K^J_{t+1}(i) = (1−δ) K^J_t(i) + I^J_t(i), with δ depreciation rate.
- Dividends D^J_t(i) = P^{˜J}_t(i) Z^J_t(i) − V_t U^J_t(i) − P_t I^J_t(i) − P_t G^J_{I,t}(i) − P^{˜J}_t G^J_{P,t}(i) − P^{˜J}_t T_t ω_J.
- Net output in sector J = max(0, Z^J_t(i) − T_t ω_J).
- Firm maximizes Σ_{s=0}^∞ ˜R_{t,s} D^J_{t+s}(i) subject to (25)-(29), with ˜R_{t,s} = Π_{l=1}^s θ i_{t+l−1} for s>0 ( =1 for s=0).
- First-order conditions include optimal labor, investment, capital choices and a Phillips curve for sectoral inflation π^{˜J}_t:
  - [ σ_J/(σ_J−1) λ^J_t p^{˜J}_{t−1} ] = φ_{P^{˜J}} (σ_J−1) [ π^{˜J}_t / π^{˜J}_{t−1} ] [ π^{˜J}_t / π^{˜J}_{t−1} − 1 ] − θ g / r_t φ_{P^{˜J}} (σ_J−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 ).

### 2. Unions
- Manufacturers demand CES aggregate of labor varieties from unions with elasticity σ_U:
  - U_t(i) = ( V_t(i)/V_t )^{−σ_U} U_t.
- Nominal wage rigidities: G^U_{P,t}(i) same functional form as (27).
- Union optimization: maximize present discounted value of nominal wages charged minus wages paid and inflation adjustment costs.
- First-order condition yields a Phillips curve for wage inflation π^U_t similar to (33).

### 3. Import Agents
- U.S. and RW own continua of import agents indexed by J∈[T,D]; distributors demand CES aggregate of imported varieties with elasticity σ_JM:
  - Y^{JM}_t(i) = ( P^{JM}_t(i) / P^{JM}_t )^{−σ_JM} Y^{JM}_t.
- Nominal price rigidities: G^{JM}_{P,t}(i) as in (27).
- Border (cif) import price denoted P_{M,cif,t}; purchasing power parity:
  - p^{JM,cif}_t = p^{JH*}_t e_t, p^{JM,cif*}_t = p^{JH}_t e_t.
- Import agent maximizes present discounted value of nominal revenue minus input costs and inflation adjustment costs; first-order condition gives Phillips curve for import price inflation π^{JM}_t similar to (33).

### 4. Distributors (final goods producers)
- Customers demand CES aggregate of distributed varieties with elasticity σ_D:
  - D_t(i) = ( P_t(i) / P_t )^{−σ_D} D_t.
- Production technology nested across stages:
  - Stage 1: tradables composite Y^T_t(i) combines foreign tradables Y^{TF}_t(i) with domestic tradables Y^{TH}_t(i) with adjustment cost G^T_{F,t}(i).
    - Y^T_t(i) = [ (α_{TH})^{1/ξ_T} (Y^{TH}_t(i))^{(ξ_T−1)/ξ_T} + (1−α_{TH})^{1/ξ_T} ( Y^{TF}_t(i) (1 − G^T_{F,t}(i)) )^{(ξ_T−1)/ξ_T} ]^{ξ_T/(ξ_T−1)}.
  - Stage 2: tradables-nontradables composite Y^A_t(i):
    - Y^A_t(i) = [ (1−α_N)^{1/ξ_A} (Y^T_t(i))^{(ξ_A−1)/ξ_A} + (α_N)^{1/ξ_A} (Y^N_t(i))^{(ξ_A−1)/ξ_A} ]^{ξ_A/(ξ_A−1)}.
  - Stage 3: combine Y^A_t(i) with public capital K^G_t:
    - Y^{DH}_t(i) = Y^A_t(i) ( K^G_t )^{α_{GS}}.
  - Stage 4: combine private-public composite with foreign final output Y^{DF}_t(i) (with adjustment cost G^D_{F,t}(i)) to produce domestic final output Y_t(i):
    - Y_t(i) = [ (α_{DH})^{1/ξ_D} (Y^{DH}_t(i))^{(ξ_D−1)/ξ_D} + (1−α_{DH})^{1/ξ_D} ( Y^{DF}_t(i) (1 − G^D_{F,t}(i)) )^{(ξ_D−1)/ξ_D} ]^{ξ_D/(ξ_D−1)}.
- Import adjustment cost for intermediates:
  - G^T_{F,t}(i) = φ_{FT}/2 (R^T_{t−1})^2 / (1 + (R^T_{t−1})^2), where R^T_t = ( Y^{TF}_t(i)/Y^T_t(i) ) ( Y^{TF}_{t−1}/Y^T_{t−1} ).
- Public infrastructure K^G_t is identical for all firms, provided free to end user, enters like technology with decreasing returns (keeps constant returns to scale at firm level). S is technology scale factor to normalize relative economy size n/(n+n^*).
- Nominal price rigidities for distributors G^D_{P,t}(i) same as (27).
- Distributor profit maximization: maximize present discounted value of nominal revenue P_t(i) Y_t(i) minus nominal input costs P^{TH}_t Y^{TH}_t(i) + P^{TF}_t Y^{TF}_t(i) + P^N_t Y^N_t(i) + P^{DF}_t Y^{DF}_t(i), fixed cost P_t T_t ω_D, and inflation adjustment costs P_t G^D_{P,t}(i).
- First-order conditions yield a Phillips curve for final goods inflation π_t similar to (33) and input demand conditions (listed in Technical Appendix).

*Source: _wp07202 - 1.  Overlapping Generations Households (PDF chapter/section).*

### 5.  Retailers

### 5. Retailers

### Retailer demand for output varieties
- Household demand for the output varieties C_t(i) supplied by retailers follows directly from (9) and (20) as
  - C_t(i) = μ (P^R_t(i) / P^R_t)^−σ_R C_t. (43)

### Retailer quantity adjustment costs
- Retailers face quantity adjustment costs that take the form
  - G^RY,t(i) = φ_C / 2 · C_t · [ μ ( (C_t(i)/g) − C_{t−1}(i) ) / C_{t−1}(i) ]^2. (44)
- Note: The presence of the growth term in (44) ensures that adjustment costs are zero along the balanced growth path.

### Retailer optimization: first-order condition
- The optimization problem of retailers consists of maximizing the present discounted value of nominal revenue P^R_t(i) C_t(i) minus nominal costs of inputs P_t C_t(i), minus nominal quantity adjustment costs P_t G^RY,t(i).
- The first-order condition for the retailer’s problem has the form
  - ( σ_R − 1 ) / σ_R · p^R_t−1 = φ_C · μ · [ ( Č_t − Č_{t−1} ) / Č_{t−1} ] · ( Č_t / Č_{t−1} )
    − θ g_r_t φ_C μ [ ( Č_{t+1} − Č_t ) / Č_t ] · ( Č_{t+1} / Č_t )^2. (45)

### Government — Fiscal policy summary
- Fiscal policy instruments specified:
  - Taxes: τ_{L,t} and τ_{c,t}
  - Transfers: τ_{T,t}
  - Government consumption spending: G^{cons}_t (exogenous and unproductive)
  - Government investment spending: G^{inv}_t (augments public infrastructure capital K^G_t)
- Transfers rule (redistribution of dividends):
  - τ̌_{T,t} = ι [ ď_Nt + ď_Tt + ď_Dt + ď_Mt ] + č_{LIQ,t} Č_t ď_{R,t} + c_{LIQ,t} L_t ď_{U,t}. (46)
  - The rule specifies that dividends of the retail and union sector are redistributed in proportion to LIQ agents’ share in consumption and labor supply, while the redistributed share of dividends in the four remaining sectors is ι ≤ ψ.

### Public infrastructure accumulation
- Government investment augments publicly provided infrastructure capital K^G_t with law of motion
  - Ǩ^G_{t+1} / g_{t+1} = (1 − δ_G) Ǩ^G_t + Ǧ^{inv}_t. (47)

### Government budget constraint and surplus targeting
- The government issues nominally non-contingent one-period debt B_t at gross nominal interest rate i_t. Letting b̌_t = B_t / P_t T_t, the real government budget constraint is
  - b̌_t = (i_{t−1} / π_t^g) b̌_{t−1} − š_t. (48)
- The primary surplus is
  - š_t = τ_{L,t} w̌_t L_t + τ_{c,t} Č_t − Ǧ^{cons}_t − Ǧ^{inv}_t. (49)
- Fiscal policy targets a possibly time-varying government surplus to GDP ratio r_t, which automatically ensures a non-explosive government debt to GDP ratio:
  - š_t − (i_{t−1} − 1) b̌_{t−1} / π_t^g ǧ_dp_t = − b̌_t + b̌_{t−1} / π_t^g ǧ_dp_t = r_t. (50)
- Implementation: The rule can be implemented by adjusting one of the tax rates to generate sufficient revenue, or by reducing one of the expenditure items.

*Source: _wp07202 - 5.  Retailers*

### 2.  Monetary Policy

### 2.  Monetary Policy

### Monetary policy rule and nominal/real rate dynamics
- Monetary policy uses an interest rate rule similar to Orphanides (2003) but with no unchanging steady state real interest rate in the non-Ricardian model.
- The term proxying the steady state nominal interest rate includes a moving average of past and future real interest rates to track the evolution of the equilibrium real interest rate:
  - i_t = (i_{t−1})^{μ_i} (r^{smooth}_t π_{t+1})^{1−μ_i} (π_{t+1}/̄π)^{(1−μ_i)μ_π μ_g [gˇdp_t gˇdp_{t-1}]^{(1−μ_i)μ_ygr}, (equation label (51) in source)
  - r^{smooth}_t = (r_{t-1} r_t r_{t+1})^{1/3}. (52)
- A government policy is defined as a sequence of instruments {G^{inv}_s, G^{cons}_s, τ^{L}_s, τ^{c}_s, i_s}_{s=t} such that (46), (48), (49), (50), (51) and (52) hold at all times.

### Equilibrium and balance of payments (market clearing and accounting relations)
- Market clearing and aggregation identities (as stated):
  - U_t = U^N_t + U^H_t = L_t = c^{OLG}_t + c^{LIQ}_t. (53)
  - ˇZ^N_t = ˇY^N_t + ω_N + ˇG^N_{P,t}. (54)
  - ˇZ^T_t = ˇY^{TH}_t + ˇY^{TF*}_t + ω_T + ˇG^{TH}_{P,t}. (55)
  - ˇY_t = ˇC_t + ˇI^N_t + ˇI^H_t + ˇG^{inv}_t + ˇG^{cons}_t + ˇY^{DF*}_t + ω_D + ˇG^{NI}_t + ˇG^{TI}_t + ˇG^{D}_{P,t} + ˇG^{U}_{P,t} + ˇG^{M}_{P,t} + ˇG^{RY}_t. (56)
- Net foreign asset evolution:
  - e_t ˇf_t = i^{*}_{t−1} ε_t π_t g e_{t−1} ˇf_{t−1} + p^{TH}_t ˇY^{TF*}_t + ˇd^{TM}_t − p^{TF}_t ˇY^{TF}_t + ˇY^{DF*}_t + ˇd^{DM}_t − p^{DF}_t ˇY^{DF}_t. (57)
- International bond market clearing:
  - ˇf_t + ˇf^{*}_t = 0. (58)
- Level of GDP:
  - gˇdp_t = ˇC_t + ˇI^N_t + ˇI^H_t + ˇG^{inv}_t + ˇG^{cons}_t + p^{TH}_t ˇY^{TF*}_t + ˇd^{TM}_t + ˇY^{DF*}_t + ˇd^{DM}_t − p^{TF}_t ˇY^{TF}_t − p^{DF}_t ˇY^{DF}_t. (59)

### Calibration: country shares, preferences, technologies, and policy rule parameters
- Two-country world: U.S. and Rest of World (RW); annual model.
- Key population, debt and shares:
  - U.S. represents 25 percent of world GDP.
  - U.S. long-run government debt to GDP = 60 percent.
  - U.S. net foreign liabilities (NFL) to GDP = 55 percent.
  - RW net foreign assets to GDP = 18.3 percent.
  - RW government debt to GDP = 30 percent.
- Share of liquidity-constrained agents ψ:
  - U.S.: 33 percent.
  - RW: 50 percent.
- Trade share calibration:
  - U.S. ratios to GDP of intermediate and final goods imports and of intermediate goods exports = 6 percent.
  - Initial steady state final output based real exchange rate e = 1.
- Steady state growth, inflation, and interest:
  - Steady state world real growth rate fixed at 1.5% per annum or ̄g = 1.015.
  - Steady state inflation in each country at 2% per annum or ̄π = 1.02.
  - Long-run real interest rate equalized across countries, assumed value 2% per annum or ̄r = 1.02.
  - Steady-state nominal growth rate gπ = 1.0353 (2 percent inflation plus 1.5 percent productivity growth).
- Preference and non-Ricardian parameters:
  - Average remaining time at work of 20 years → χ = 0.95.
  - Planning horizon 1/(1−θ) = 10 years → θ = 0.9.
  - Calibration target: a one percentage point increase in government debt to GDP in the U.S. → approximately four basis points increase in the U.S. (and world) real interest rate.
  - Intertemporal elasticity of substitution = 0.25 → γ = 4.
  - Habit persistence v = 0.4.
  - Frisch elasticity of labor supply targeted at 0.5.
- Technology and substitution elasticities:
  - ξ_{ZN} = ξ_{ZT} = 1.
  - ξ_T = ξ_D = 1.5 (long-run price elasticities of demand for imports).
  - ξ_A = 0.5.
- Adjustment cost parameters:
  - Investment adjustment costs φ_I = 10.
  - Trade and consumption adjustment costs φ_{FT}, φ_{FD}, φ_C = 5.
- Price setting and markups:
  - Markup = 20 percent in two manufacturing sectors and in the labor market.
  - Markup = 5 percent in distribution and retail sectors.
  - Markup = 2.5 percent for import agents.
  - Inflation adjustment cost parameter = 10 for all sectors except import agents in RW (zero there).
- Income shares and steady-state shares:
  - Manufacturing labor income share parameters set to ensure labor income share of 64 percent in aggregate.
  - Nontradables share in GDP = 50 percent.
  - Steady state investment to GDP = 16 percent.
  - Steady state government spending to GDP = 18 percent.
  - Implied consumption to GDP ratios: U.S. = 65.7 percent; RW = 66.1 percent.
- Capital and profits calibration:
  - Annual depreciation rate of private capital δ = 10 percent.
- Public capital and productivity:
  - Productive public spending to GDP = 3 percent; other government spending = 15 percent.
  - Depreciation rate of public capital δ_G = 0.04.
  - Given these and 1.5 percent per annum real growth, public capital stock to GDP ratio implied = 54.5 percent.
  - Long-run elasticity of GDP with respect to K_G targeted at 0.14 → α_G set accordingly.
- Monetary policy rule parameters (annual model):
  - Interest rate smoothing μ_i = 0.25.
  - Coefficient on inflation μ_π = 0.6.
  - Coefficient on output growth μ_{ygr} = 0.25.
- Alternative representative agent model (REP) adjustment:
  - Share of U.S. liquidity constrained agents ψ^{*} increased from 33 to 50 percent.
  - Parameter governing speed with which NFL returns to long run set as small as possible to generate current account persistence.

### Implications of U.S. government deficits: steady-state identities and model comparison
- Long-run link between NFL stock and current account deficit:
  - CDEF_t = NFL_t − NFL_{t−1}. (60)
  - Along the balanced growth path (NFL_{t−1} = NFL_t / gπ), dividing by nominal GDP yields:
    - CDEF/GDP = (gπ − 1)/gπ NFL/GDP = 0.0341 NFL/GDP. (61)
  - Numeric interpretation: A 10 percentage point increase in the NFL ratio → a 0.341 percentage point increase in the current account deficit ratio in the very long run.
- Government deficit to debt identity:
  - GDEF/GDP = (gπ − 1)/gπ GDEBT/GDP = 0.0341 GDEBT/GDP. (62)
- Fundamental model distinction (FH finite-horizon vs REP representative-agent):
  - REP models: consumers not liquidity-constrained save to offset future tax burdens; long-run NFL and real interest rates are specified independently of government debt.
  - FH models: consumers are disconnected from future generations; higher government debt crowds out investment in physical capital and foreign assets; long-term positive causal relationship between government debt and NFL; long-term causal relationship between government savings and the current account deficit.
  - FH models imply equilibrium real interest rates are endogenous and linked to savings-investment fundamentals; REP models tie long-run real rate to time preference and productivity growth.

### Quantitative fiscal experiment and comparative predictions (FH vs REP)
- Fiscal experiment setup:
  - Permanent reduction in government savings ratio by 0.5 percentage points.
  - Short-run fiscal expansion assumed larger to accelerate debt increase: −2.5 for the first 2 years, then gradually falling to −0.5 by the 5th year.
  - Reduction in government savings results from lower labor income tax rates.
- Common short-run responses:
  - Both models predict a gradual increase in government debt to GDP ratio of 14.7 percentage points (∆(GDEF/GDP)/0.0341).
  - Labor tax rate declines by about 5 percentage points in the first 2 years, then rises gradually to pay additional interest on rising debt.
- Long-run and medium-term divergences:
  - Long-run labor tax rate:
    - FH model: about 1.5 percentage points higher.
    - REP model: about 0.5 percentage points higher.
  - FH model: real interest rates rise by about 40 basis points, leading to strong crowding-out of investment, consumption and GDP; permanently higher current account deficit and NFL position; spillovers to RW due to international equalization of real rates.
  - REP model: non-liquidity-constrained agents increase savings to offset lower government savings after about 5 years; NFL changes little; minimal long-term crowding out.
- Short-run consumption and output:
  - FH model: short-run consumption boom because consumers do not fully internalize future tax burden; private savings rise by less than decline in government savings.
  - REP model: similar short-run behavior given 50 percent liquidity-constrained share, but diverges beyond ~5 years.
  - Short-run GDP responses are constrained by the monetary policy rule (designed to keep inflation near target); holding nominal short-term interest rate fixed for the first 2 years approximately doubles the short-run GDP effects.
- Real exchange rate dynamics:
  - FH model: short-run appreciation consistent with trade deficits following lower national savings; over time real exchange rate depreciates as trade balance improves to finance larger interest obligations on higher NFL.
  - REP model: much smaller real exchange rate movement.
- Parameter sensitivity and robustness notes:
  - Exchange rate response magnitude sensitive to elasticity of substitution between imported goods and domestic tradables (base case ξ_T = 1.5); lower ξ requires larger real exchange rate changes for same trade volume profile.
  - Short-run GDP response sensitive to monetary policy accommodation.
  - Crowding-out magnitude sensitive to planning horizon; longer planning horizons yield results closer to REP.
  - Sensitivity to consumption interest-rate responsiveness: low sensitivity requires larger real rate changes to equilibrate world savings and investment.
  - REP model long-term predictions for NFL and real rates are far less sensitive to parameter changes due to strong long-run assumptions.
- Policy implication from simulations:
  - Fiscal deficits may have contributed to deterioration of U.S. current account deficits since 2000; given long lags, main current account effects of fiscal policies may still be forthcoming.
  - Improving U.S. fiscal deficits should be part of a policy package to raise national savings and help resolve global current account imbalances.
  - Fiscal improvement options include higher taxes or lower government spending; consequences of lower government spending are analyzed in the following section (in source).

*Source: _wp07202 - 2.  Monetary Policy*

### Section 6 therefore explores the role of private sector savings behavior in bringing about

### Section 6 therefore explores the role of private sector savings behavior in bringing about

### V.  Implications of Government Spending Cuts
- Setup:
  - Extended the basic FH model to allow for productive government investment.
  - Simulation: permanent 10 percent reduction in government investment (solid lines) versus permanent 10 percent reduction in government consumption (dashed lines).
  - Government balance held fixed by computing implied profile of labor tax rates to isolate pure expenditure-cut effects.

- Findings for a 10 percent reduction in government investment:
  - Long-run decline in real GDP in the U.S. of about 1.4 percent.
  - Short run: reduction in aggregate demand; crowding in of private consumption limited by negative effects of lower public investment on lifetime household wealth.
  - Lower demand → decline in real interest rates → stimulates consumption and investment initially.
  - Over time: reduced public capital stock → supply-side effects dominate → significant declines in private sector investment and consumption.
  - External effects: reduction in supply to RW eventually results in higher imports, lower exports and an appreciation in the real exchange rate.
  - Short-term and medium-term spillover effects on GDP in RW are positive.

- Findings for a 10 percent reduction in government consumption:
  - Medium- and long-run effects on GDP are opposite to those from cutting government investment.
  - Less government consumption frees resources for private consumption and investment.
  - Investment declines in the short run as consumption boom raises interest rates.
  - Part of the increase in GDP represents an increase in supply to RW (higher exports) and causes a depreciation in the real exchange rate.

### VI.  Implications of Private Sector Savings Behavior
- Model context:
  - FH models produce well-defined steady states where countries can be net debtors or creditors depending on public and private savings rates.
  - Figure 8 simulations: U.S. NFL position increases due to private sector savings behavior.
    - Solid lines: permanent one percentage point reduction in the U.S. rate of time preference (reduces U.S. private sector savings).
    - Dashed lines: permanent one percentage point increase in the RW rate of time preference (raises RW private sector savings).

- Findings for a permanent one percentage point reduction in U.S. rate of time preference (lower U.S. private savings):
  - World savings falls.
  - Permanently higher real interest rates.
  - Decline in output and consumption in both the U.S. and RW.
  - Medium- and long-run effects broadly similar to effects of lower public sector savings.

- Findings for a permanent one percentage point increase in RW rate of time preference (higher RW private savings):
  - Lower level of consumption in RW that crowds in investment and expands production.
  - Over time consumption in RW rises but lags increases in production.
  - Persistent rise in savings results in a 100 basis points reduction in real interest rates in both the U.S. and RW.
  - Effects in the U.S.:
    - Consumption boom in the U.S. and eventually higher levels of GDP as lower real interest rates stimulate investment.
    - Short run: U.S. GDP declines as monetary policy is assumed to resist inflationary pressures.
    - If policy delayed hikes in nominal interest rates, lower real interest rates in short run would expand GDP (parallel to fiscal multiplier discussion in Section 4.2).
    - U.S. NFL position path: declines initially but then shows a steady rise that stabilizes at a new higher steady-state value that is almost 10 percentage points higher.
    - Current account deteriorates by around 0.35 percentage points in the long run, and by significantly more in the short run.
    - Sharp U.S. real exchange rate appreciation, consistent with generated trade deficits from savings–investment dynamics.

- Interpretation and broader implications:
  - Higher RW savings is more consistent with both an elevated U.S. NFL position and low world real interest rates observed in recent years.
  - Major driving forces behind large U.S. current account deficits include:
    - Low U.S. private and public sector savings rates.
    - Factors that increased the gap between savings rates and investment in RW (modeled here as higher private sector savings).
  - Analysis abstracts from changes in public sector savings and perceived investment opportunities, but transmission to the U.S. operates through world savings and the world real interest rate.
  - Footnotes note that a significant part of increased RW savings has been due to public rather than private sector savings, and that shifts in asset preferences or perceived investment attractiveness may matter but are unlikely to be the whole story given persistence in the U.S. NFL position.

- Comparative model capabilities:
  - FH model can account for the combination of low real world interest rates and a rising U.S. NFL position.
  - REP models that fix the NFL steady-state value and pin the long-run real interest rate to the rate of time preference and growth struggle to account for these simultaneous features.

### VII.  Conclusion
- Model description:
  - Developed an open economy business cycle model suitable for joint analysis of fiscal and monetary policies.
  - Combines short-run monetary policy analytical frameworks with a finite horizon model for medium- and long-term fiscal analysis.

- Key conclusions:
  - Short-run dynamics from the framework are virtually identical to a representative agent model with liquidity constrained consumers.
  - Medium- to long-run predictions differ substantially from representative agent models:
    - In finite horizon model, higher government deficits have a strong negative effect on the current account and private capital accumulation.
    - Fiscal consolidation can substantially improve current account imbalances.
  - Model can analyze consequences of cuts in productive government investment, useful for evaluating trade-offs between short-term demand management and longer-term supply repercussions.
  - Role of private sector savings:
    - Higher RW savings rates (private and public) must have played an important role in recent years.
    - This is consistent with both current account imbalances and low world real interest rates.
    - The effect has likely been strong enough that negative effects of U.S. fiscal policies analyzed in Section 4 have not yet materialized, except for larger current account imbalances.

- Extensions in progress:
  - Extend model to a multi-country setting for more detailed accounting of causes and consequences of private and public savings behavior in the U.S. and RW.
  - Develop a linearized version to estimate with Bayesian methods and account for a larger set of shocks relevant to monetary and fiscal policy analysis.

*IMF Working Paper (content unit)._wp07202*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2007/_wp07202.pdf_