## _wp0988

## Source details

**Canonical URL:** [_wp0988](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0988.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0988.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0988.pdf.json)

---

### Chile's structural surplus rule
- Coverage and exclusions:
  - Covers the central government; main public sector elements left outside the rule are the central bank, public non-financial enterprises, the military sector, and municipalities.
- Target:
  - The central government’s overall structural balance target should equal a surplus of 1% (0.5% effective 2008) of actual GDP in every year.
- Definition and implementation:
  - Structural balance = structural revenues + interest on net government assets − actual expenditures on goods and services.
  - Structural revenue determined by two independent panels of experts; reflects:
    - what tax revenue would have been if the economy had operated at potential rather than actual output, and
    - what copper revenue would have been at a long-term reference world copper price rather than the actual price.
  - The rule specifies permissible annual expenditures on goods and services as a residual given the values of the target, structural revenues, the level of government assets, interest rates, and GDP.
  - No distinction is made between government consumption and investment expenditures in the rule.
- Asset accumulation and use:
  - A positive surplus target implies significant asset accumulation by the government; the 2006 Fiscal Responsibility Law introduced rules for investing surpluses (government pension fund, gradual central bank recapitalization, and a Fund for Economic and Social Stabilization).
  - In May 2007 a reduction in the surplus target from 1% to 0.5% of GDP was announced, effective in 2008; the additional resources are to be devoted primarily to education.

### Research questions addressed
- Question 1: Could the performance of the rule be improved by a more explicitly countercyclical stance (i.e., letting deficits respond more strongly to excess fiscal revenue than under the current rule)?
  - Trade-offs:
    - Current rule advantage: implies only small and gradual changes in fiscal instruments in response to shocks.
    - Potential advantage of more aggressive response: smaller volatility of GDP and inflation (example: responding to an increase in world copper prices by raising tax rates to dampen a boom).
- Question 2: Are there advantages to aligning the level of the surplus target more closely with the existing stock of net government debt?
  - Noted relationship:
    - There is a proportional long-run relationship between the surplus-to-GDP ratio and the government assets-to-GDP ratio.
    - If the targeted surplus-to-GDP ratio differs from the current assets-to-GDP ratio, actual short-run surpluses must vary over time until assets reach their long-run value, creating a fiscal policy–driven business cycle even without shocks.

### Analytical framework: two-country GIMF and model features
- Model:
  - A 2-country version (Chile and rest of the world) of the IMF’s Global Integrated Monetary and Fiscal Model (GIMF).
- Key structural features:
  - Dynamic general equilibrium with nominal and real rigidities.
  - Interest rate reaction function typical of Inflation-Targeting countries.
  - Non-Ricardian elements giving fiscal policy a prominent role:
    - overlapping generations of agents;
    - life-cycle income profiles;
    - liquidity-constrained consumers;
    - multiple distortionary taxes.
  - Explicit incorporation of a fiscal policy reaction function to represent Chile’s structural surplus rule.
- Copper market:
  - Global copper output modeled as an endowment, 38% of which accrues to Chile.
  - Copper price fluctuates with shocks to foreign industrial demand for copper; world copper market exhibits perfect price arbitrage.
  - Total copper revenue divided among domestic capital and labor, the domestic government, and foreigners in proportions observed in the data.

### Calibration specifics for Chile (preserved exactly)
- Model frequency and scale:
  - Quarterly model.
  - Chile represents one third of one percent of the world economy in GDP and population.
- Interest rates, risk premium, and debt:
  - Long-run world real interest rate: 3% per annum.
  - Risk premium calibrated to produce a 50 to 60 basis points premium over international interest rates at steady state net foreign liabilities to GDP ratio of 20 percent.
  - Government debt to GDP ratio: 0 in Chile and 50 percent in Foreign.
- Growth and inflation:
  - Real world growth rate: 2% per annum.
  - Population growth rate: 1% per annum.
  - Long-run inflation rate: Chile 3% per annum, rest of world 2% per annum.
- Non-Ricardian parameters and household preferences:
  - θ and χ set to 0.98125, corresponding to a 15 year planning horizon or average life expectancy 1/(1−θ) for θ, and a 15 year remaining working life for χ.
  - Intertemporal elasticity of substitution 0.2 (γ = 5).
  - Habit persistence v = 0.7.
  - Wage elasticity of labor supply targeted at 0.5 via η_OLG and η_LIQ.
  - Share ψ of liquidity constrained agents: 50 percent for Chile and 40 percent in Foreign.
  - Share of these agents in dividend income: half of their share in the population.
- Technology, elasticities, and markups:
  - Elasticities of substitution between capital, labor and copper: 1 in both tradables and nontradables.
  - Elasticities between domestic and foreign traded intermediates and final goods: 1.5.
  - Elasticity between tradables and nontradables: 0.8.
  - Copper demand shock persistence ρ_X = 0.95.
  - Markup of price over marginal cost: 10 percent in two manufacturing sectors and labor market.
  - Distribution and retail sectors markups: 5 percent.
  - Import agents markup: 2.5 percent.
- Shares and steady state targets:
  - Manufacturing labor income share: 55 percent in Chile, 64 percent in rest of world.
  - Nontradables labor shares: 64 percent in both Chile and Foreign.
  - Nontradables share in GDP: 50 percent.
  - Investment to GDP ratio: 20 percent in both countries.
  - Government spending shares in GDP: 12 percent in Chile and 18 percent in Foreign.
  - Government investment shares: 2 percent in Chile and 3 percent in Foreign.
  - Transfers to GDP: 10 percent in both countries.
  - Tax revenue composition (Chile): consumption tax revenue 50 percent, capital income tax revenue 14 percent, labor income tax revenue 15 percent, other (lump-sum) taxes 21 percent.
  - Tax revenue composition (Foreign): 30, 10, 30 and 30 (consumption, capital income, labor income, other).
- Capital and copper:
  - Annual depreciation rate of private capital calibrated at 10 percent.
  - Target steady state ratios: copper output to GDP 12.38 percent, copper exports to GDP 12.3 percent.
  - Steady state world copper price normalized to one.
  - Calibrated Chilean ratios to GDP: intermediate exports 8.7 percent, final goods exports 14 percent, final goods imports 5 percent.

### Fiscal rule parameterization, loss function, and numerical experiments
- Structural fiscal rule (rescaled form in model):
  - gs rat t = gs rat ∗ t + dtax (ˇτ t − ˇτpot t g ˇdp t) + dcop (ˇgX t − ˇgpot t g ˇdp t).
- Interpretation of coefficients:
  - d_tax = d_cop = 0 corresponds to a balanced-budget rule (procyclical).
  - d_tax = d_cop = 1 corresponds to Chile’s structural surplus rule (minimal short-run changes in fiscal instruments; countercyclical overall deficit).
  - d_tax and d_cop > 1 imply even more countercyclical fiscal policy.
- Loss function used to evaluate rules:
  - Loss = sd(π) + λ * sd(gdp), (125), where sd stands for standard deviation.
  - For each λ the weights d_tax and d_cop are chosen to minimize the loss function, tracing an inflation-output efficiency frontier.
- Numerical and qualitative findings:
  - Different fiscal policies affect mostly the volatility of output rather than inflation.
  - Chile’s rule (d_tax = d_cop = 1) performs far better than a balanced-budget rule (d_tax = d_cop = 0).
  - Chile’s structural surplus rule is very close to the efficiency frontier and implies a relative weight on output volatility in the policymaker’s objective function of approximately 0.13.
  - A rule increasing the weight on excess copper revenue to around two (example calibration: cob=2.1, tax=1.4 shown in Figure 2) reduces output volatility substantially but increases inflation volatility and leads to much higher volatility in tax rates, fiscal deficits, and government debt.
  - Volatility of fiscal instruments is minimized by the structural surplus rule with coefficients equal to one; deviations increase fiscal instrument volatility with opposite signs:
    - balanced budget rule → fiscal instruments very volatile and procyclical;
    - coefficients significantly > 1 → fiscal instruments very volatile but countercyclical.
- Reported table entries (preserved exactly as in source):
  - "Weight of Copper in Fiscal Rule=⇒Balanced0.4SFS12.1"
  - "Budget(Chile)"
  - Real GDP std. 1.05 0.72 0.58 0.58 0.38
  - Inflation std. 0.24 0.25 0.29 0.27 0.36
  - Deficit std. 0 0.89 1.07 1.41 1.96
  - Debt std. 0 9.9 10.5 16.2 19.2
  - ∆Tax std. 0.67 0.32 0.04 0.06 0.68
  - Output Weight in Loss Fct. -0.05 -0.13 >1

### Impulse-response scenarios (qualitative summaries)
- Structural surplus rule (SFS rule) — one standard deviation copper price shock (40 quarters):
  - GDP expands and inflation falls.
  - Monetary policy responds accommodatively, boosting investment.
  - Real exchange rate appreciates; net exports of non-copper goods and services decline.
  - Surge in government copper-related revenue reduces deficits and debt.
  - Tax rates change very little in the short run; as debt and interest charges fall, tax rates start to fall, stimulating consumption for an extended period.
  - Boom in GDP is more short-lived than the consumption boom; monetary stimulus dissipates quickly and real appreciation dominates over the medium term.
- Balanced-budget rule:
  - Surge in copper revenue is not allowed to affect deficits; it leads to an immediate reduction in taxes.
  - Result: amplification of the GDP boom relative to the SFS case.
- Strongly countercyclical (aggressive) fiscal rule:
  - Surge in copper prices is accompanied by an immediate increase in taxes and much higher fiscal surpluses.
  - Short-run effect on GDP is contractionary rather than expansionary.
  - Associated with much higher volatility in fiscal accounts and tax rates.

### Choice of surplus target and business cycle implications
- Chile’s pre-May 2007 target and asset context:
  - Chile’s government surplus to GDP target before May 2007 was 1%, and its government assets to GDP ratio was around 8%-9%.
  - Assumed long-run growth rates for Chile: technology 2%, population 1%, prices 3% per annum.
  - A 1% surplus target implies a long-run government assets to GDP ratio of 17.4%, representing substantial asset accumulation beyond current level.
  - The 0.5% surplus target adopted in May 2007 (effective 2008) is consistent with the current assets to GDP ratio and implies no significant changes in assets in the long run.
- Business cycle trade-offs of raising the surplus target:
  - Raising the surplus target above levels consistent with current asset stocks forces time-varying actual surpluses while assets converge, inducing a fiscal-policy-driven business cycle even without shocks.
  - Thought experiment (permanent increase of surplus target from 0.5% to 1% at time 0, lump-sum transfers used as fiscal instrument):
    - Transfers must be temporarily reduced to accumulate assets.
    - Temporary reduction in transfers crowds out consumption and crowds in investment and net exports due to non-Ricardian agent behavior.

### Policy conclusions (preserved conclusions from paper)
- Chile’s current structural surplus rule performs well if the policymaker places a small weight on output volatility relative to inflation volatility.
- A more aggressive countercyclical fiscal rule can lower output volatility, but with trade-offs: somewhat higher inflation volatility and much higher volatility of fiscal variables.
- Given Chile’s current stock of government assets, adopting a 0.5% surplus target starting in 2008 is desirable from a business cycle perspective because the earlier 1% target would have required significant further asset accumulation at the expense of greater volatility in fiscal instruments and GDP.

*Source: I. Introduction and Sections II–VI (excerpts) from the provided PDF content _wp0988.*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   44

### _wp0988 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   44

### Chile's structural surplus rule
- The rule covers the central government; main public sector elements left outside the rule are the central bank, public non-financial enterprises, the military sector, and municipalities.
- The central government’s overall structural balance target:
  - should equal a surplus of 1% (0.5% effective 2008) of actual GDP in every year.
- Structural balance defined as:
  - structural revenues plus interest on net government assets (which are positive in Chile) minus actual expenditures on goods and services.
- Structural revenue determined by two independent panels of experts; reflects:
  - what tax revenue would have been if the economy had operated at potential rather than actual output, and
  - what copper revenue would have been at a long-term reference world copper price rather than the actual price.
- The rule specifies permissible annual expenditures on goods and services as a residual given the values of:
  - the target, structural revenues, the level of government assets, interest rates, and GDP.
- No distinction is made between government consumption and investment expenditures in the rule.
- A positive surplus target implies significant asset accumulation by the government; the 2006 Fiscal Responsibility Law introduced rules for investing surpluses (government pension fund, gradual central bank recapitalization, and a Fund for Economic and Social Stabilization).
- In May 2007 a reduction in the surplus target from 1% to 0.5% of GDP was announced, effective in 2008; the additional resources are to be devoted primarily to education.

### Research questions addressed
- Question 1: Could the performance of the rule be improved by a more explicitly countercyclical stance (i.e., letting deficits respond more strongly to excess fiscal revenue than under the current rule)?
  - Trade-offs noted:
    - Current rule advantage: implies only small and gradual changes in fiscal instruments in response to shocks.
    - Potential advantage of more aggressive response: smaller volatility of GDP and inflation (example: responding to an increase in world copper prices by raising tax rates to dampen a boom).
- Question 2: Are there advantages to aligning the level of the surplus target more closely with the existing stock of net government debt?
  - Noted relationship:
    - There is a proportional long-run relationship between the surplus-to-GDP ratio and the government assets-to-GDP ratio.
    - If the targeted surplus-to-GDP ratio differs from the current assets-to-GDP ratio, actual short-run surpluses must vary over time until assets reach their long-run value, creating a fiscal policy–driven business cycle even without shocks.

### Analytical framework: two-country GIMF
- Model: a 2-country version (Chile and rest of the world) of the IMF’s Global Integrated Monetary and Fiscal Model (GIMF).
- Key model features:
  - Dynamic general equilibrium model with nominal and real rigidities useful for short-run business cycle analysis.
  - Interest rate reaction function common in Inflation-Targeting countries such as Chile.
  - Multiple non-Ricardian features that give an important role to fiscal policy:
    - overlapping generations of agents;
    - life-cycle income profiles;
    - liquidity-constrained consumers;
    - multiple distortionary taxes.
  - Incorporation of a fiscal policy reaction function to represent Chile’s structural surplus rule.

### Calibration specifics for Chile
- Calibrated to reproduce structural features of the Chilean economy, including:
  - breakdown of GDP into expenditure and income components;
  - breakdown of trade into raw materials, intermediates, and finished goods;
  - debt-to-GDP ratios of the public and private sectors;
  - trend real and nominal growth rates;
  - composition of tax revenue between labor, consumption, capital income, and other taxes;
  - composition of government outlays between expenditures on goods and services, transfers and interest expenses.
- World copper market added to standard GIMF:
  - Global copper output modeled as an endowment, 38% of which accrues to Chile, as in the data.
  - Copper price fluctuates with shocks to foreign industrial demand for copper; world copper market exhibits perfect price arbitrage.
  - Total copper revenue is divided among domestic capital and labor, the domestic government, and foreigners in proportions observed in the data.

### Paper organization
- Section II: presents the model.
- Section III: discusses calibration.
- Section IV: analyzes effects of different parameterizations of the structural surplus rule on business cycle volatility.
- Section V: analyzes consequences of choosing a government surplus target that is not aligned with the existing debt stock.

*Source: I. Introduction (excerpt) from the provided PDF content.*

### Section VI concludes.

### _wp0988 - Section VI concludes.

### Model setup and structure
- Two-country world: Chile and Foreign (rest of the world).
- Population growth rate: n (constant positive).
- Technology growth: world labor-augmenting technology Tt with trend growth gt = Tt/Tt−1.
- Two household types per country: overlapping generations (OLG) with finite planning horizons and external habit persistence, and liquidity constrained (LIQ) households.
  - OLG: constant probability of death (1−θ), average planning horizon 1/(1−θ).
  - Births per period: N(j) n t (1−ψ(j))(1−θ/n) for OLG and N(j) n t ψ(j)(1−θ/n) for LIQ.
- All households subject to uniform labor income tax τL,t and consumption tax τc,t.
- Goods and factor flows: manufacturers → distributors → retailers; import agents and foreign layers allow high trade-to-GDP ratios; cascading nominal rigidities across sectors affect aggregate inflation.

### Households — Overlapping Generations (OLG)
- Lifetime utility (representative age a at time t):
  - Equation (1): Et Σ∞ s=0 (βθ) s [1/(1−γ) ((cOLG a+s,t+s / hOLG a+s,t+s) ηOLG (1−ℓOLG a+s,t+s) 1−ηOLG) 1−γ + um/(1−γ) (Ma+s,t+s / PR t+s) 1−γ ].
- Habit formation:
  - hOLG a,t = (cOLG t−1 g t N n t−1 (1−ψ)) v (Equation (2)).
- Consumption aggregator and demand for variety i:
  - cOLG a,t = (∫0 1 (cOLG a,t(i)) σR−1/σR di) σR/(σR−1) (Equation (3)).
  - cOLG a,t(i) = (PR t(i)/PR t) −σR cOLG a,t (Equation (4)); PR t given by (5).
- Cashless limit for money demand: um → 0 (Woodford (2003) case) — money optimality conditions ignored.
- Assets: domestic government bonds Ba,t (domestic currency) and foreign bonds Fa,t (foreign currency); participation requires insurance premium (1−θ)/θ on financial wealth per period.
- Foreign exchange risk premium functional form:
  - ξf t = y1 + y2 (y4 − ca t/gdp t) y3 (Equation (6)), with constraint y1 = −y2 / y4 y3.
- Budget constraint (nominal) for OLG:
  - PR t cOLG a,t + Pt cOLG a,t τc,t + Ba,t + Et Fa,t = 1/θ [ it−1 Ba−1,t−1 + it−1(Foreign) Et Fa−1,t−1 (1+ξf t−1) ] + Wt Φa,t ℓOLG a,t (1−τL,t) + Σj ∫0 1 Dj a,t(i) di − Pt τOLG T a,t − Pt τOLG ls,t (Equation (8)).
- Consumption aggregation across generations:
  - cOLG t = N n t (1−ψ) (1−θ/n) Σ∞ a=0 (θ/n) a cOLG a,t (Equation (9)).
- Arbitrage / uncovered interest parity:
  - it = it(Foreign) ̃Et εt+1 (1+ξf t) (Equation (12)).
- Key aggregated OLG consumption rule (after rescaling by technology):
  - ˇcOLG t / Θt = ˇfw t + ˇhw L t + ˇhw K t (Equation (19)).
  - Financial wealth:
    - ˇfw t = 1/(πt gt n) [ it−1 ˇb t−1 + it−1(Foreign) εt (1+ξf t−1) ˇf t−1 / et−1 ] (Equation (20)).
  - Human wealth (labor):
    - ˇhw L t = (N(1−ψ)(ˇw t (1−τL,t))) + ̃Et θχ gt+1 / r t ˇhw L t+1 (Equation (21)).
  - Human wealth (capital/dividends):
    - ˇhw K t = (ˇdN t + ˇdT t + ˇdD t + ˇdR t + ˇdU t + ˇdM t + dX + dF t − ˇτT,t − ˇτOLG ls,t) + ̃Et θ gt+1 / r t ˇhw K t+1 (Equation (22)).
  - Marginal propensity to consume:
    - Θt = pR t + τc,t ηOLG + ̃Et θ jt / r t Θt+1 (Equation (23)).
- Intuition: OLG households discount future taxes at rate at least rt/θ (higher than market rt), so fiscal consolidations that tilt taxes forward lower human wealth and consumption for a given MPC.

### Households — Liquidity Constrained (LIQ)
- Utility similar to OLG but consumption constrained to current after-tax income plus transfers τLIQ T a,t.
  - LIQ utility (Equation (24)); habit hLIQ a,t in (25); CES aggregation (26).
- Budget constraint (nominal) summarized in (27).
- Aggregated first-order conditions:
  - ˇcLIQ t(i) = (PR t(i)/PR t) −σR ˇcLIQ t (Equation (28)).
  - ˇcLIQ t (pR t + τc,t) = ˇw t ℓLIQ t (1−τL,t) + ˇτT,t − ˇτLIQ ls,t (Equation (29)).
  - Labor-leisure condition for LIQ: ˇcLIQ t N ψ − ˇℓLIQ t = ηLIQ/(1−ηLIQ) ˇw t (1−τL,t)/(pR t + τc,t) (Equation (30)).
- Aggregate consumption and labor:
  - ˇC t = ˇcOLG t + ˇcLIQ t (Equation (31)).
  - ˇL t = ˇℓOLG t + ˇℓLIQ t (Equation (32)).

### Manufacturers and production structure
- Two manufacturing sectors J ∈ {N, T} (nontradables, tradables); continuum of varieties i ∈ [0,1].
- Demand aggregators for manufacturing output: Equations (33)–(36).
- Nested CES production: copper inputs XJ t(i) combined with capital-labor composite MJ t(i); MJ combines KJ and UJ with labor-augmenting productivity Tt.
  - ZJ t(i) = F(MJ t(i), XJ t(i)) specified in (37); MJ t(i) = F(KJ t(i), UJ t(i)) in (38).
- Copper demand shock process (Foreign copper share parameter):
  - αX t (Foreign) = (1−ρX) αX (Foreign) + ρX αX t−1 (Foreign) + eX (Equation (39)).
- Adjustment costs:
  - Inflation adjustment: GJ P,t(i) = φP J /2 ZJ t [ (P ̃J t(i)/P ̃J t−1(i) / (P ̃J t−1 / P ̃J t−2) − 1 ) ] 2 (Equation (40)).
  - Investment adjustment: GJ I,t(i) = φI /2 KJ t(i) (IJ t(i)/KJ t(i) − IJ t−1/KJ t−1) 2 (Equation (41)).
- Capital law of motion: KJ t+1(i) = (1−δ) KJ t(i) + IJ t(i) (Equation (42)).
- Dividend definition (after tax net cash flow) and firm objective:
  - DJ t(i) given in (43); firm maximizes Et Σ∞ s=0 ̃R t,s DJ t+s(i) subject to constraints (44).
- Key first-order conditions (rescaled):
  - Pricing condition (45).
  - Labor and copper demand first-order conditions (46)–(47).
  - Imputed real return to capital: rJ k,t = λJ t ̌F J K,t (Equation (48)).
  - Investment qJ t relation: qJ t = 1 + φI (ˇI J t / ˇK J t − ˇI J t−1 / ˇK J t−1) (Equation (49)).
  - Euler equation for capital (50).
  - Rescaled aggregate dividends ˇdJ t in (51).

### Copper sector
- Copper supply: exogenous endowment Xsup t (treated as constant endowment in this paper).
- Domestic copper price pX t; total price revenue distributed to foreigners fX t, domestic government gX t, and domestic factors dX t:
  - pX t Xsup t = gX t + dX + fX t (Equation (52)).
- Domestic net exports of copper: Xx t = pX t (Xsup t − Xdem t) (Equation (53)).
- Assumption: cyclical excess copper revenue goes to foreigners or government; dX t = dX (constant).

### Unions and wage setting
- Continuum of unions i ∈ [0,1]; monopolistic competition in labor supply, nominal wage rigidities with adjustment cost:
  - GU P,t(i) = φP U /2 U t / Tt [ (Vt(i)/Vt−1(i) / (Vt−1/Vt−2) − 1) ] 2 (Equation (58)).
- Demand for union labor varieties: UJ t(z,i) = (Vt(i)/Vt) −σU UJ t(z) (Equation (55)); aggregate wage index Vt in (56).
- First-order condition (rescaled) for wage setting:
  - [σU/(σU−1) ˇw t − ˇv t] = φP U σU−1 (πV t πV t−1)(πV t πV t−1 − 1) − Et θ gt+1/(n rt) φP U σU−1 ˇU t+1/ˇU t (πV t+1/πV t)(πV t+1/πV t−1) (Equation (60)).
- Union dividends: ˇdU t = (ˇv t − ˇw t) ˇU t − ˇG U P,t (Equation (61)).
- Wage inflation relation: ˇv t / ˇv t−1 = πV t / πt g t (Equation (62)).

### Import agents
- Two continua of import agents per country and destination market, J ∈ {T, D}; nominal rigidities via GJM P,t(i) in (67).
- CIF import price relation via PPP:
  - pJM,cif t (Chile) = pJH t (Foreign) e t (Equation (68)).
- Import agents’ FOC (rescaled):
  - [σJM/(σJM−1) pJM,cif t − pJM t] = φP JM σJM−1 (πJM t πJM t−1)(πJM t πJM t−1 − 1) − Et θ gt+1/(n r t) φP JM σJM−1 ˇY JM t+1/ˇY JM t (πJM t+1/πJM t)(πJM t+1/πJM t−1) (Equation (70)).
- Total dividends to OLG from import agents in Chile:
  - ˇdJM t (Chile) = [ (pJM t (Foreign) − pJM,cif t (Foreign)) ˇY JM t (Foreign) − ˇG JM P,t (Foreign) ] e t (Equation (72)).

### Distributors and final goods production
- Distributors produce final output combining domestic private output with public infrastructure and foreign final output; multiple CES nests and adjustment costs.
- Tradables composite YT t(i) combines domestic TH and foreign TF tradables with adjustment cost GTF,t(i) (Equations (77)–(82)).
- Tradables-nontradables composite YA t(i) (Equation (83)); marginal cost pA t in (84); input conditions (85)–(86).
- Private-public composite YDH t(i) = YA t(i) (KG t) αG S (Equation (87)) with pDH t condition (88).
- Final output Yt(i) combines domestic YDH and foreign YDF with adjustment cost GDF,t(i) (Equations (89)–(94)); input conditions (92)–(94).
- Distributor pricing FOC (rescaled):
  - [σD/(σD−1) pD t − 1] = φP D σD−1 (π t π t−1)(π t π t−1 − 1) − Et θ gt+1/(n r t) φP D σD−1 ˇY t+1/ˇY t (π t+1/π t)(π t+1/π t−1) (Equation (97)).
- Distributor dividends (rescaled): ˇdD t given in (98).

### Retailers
- Retailers buy final output from distributors and sell to households; face real quantity adjustment costs:
  - GR Y,t(i) = φC /2 C t ( (C t(i)/(g t n) − C t−1(i)/C t−1(i)) ) 2 (Equation (100)).
- Demand for retailer variety: C t(i) = (PR t(i)/PR t) −σR C t (Equation (99)).
- Retailer FOC (rescaled):
  - [ (σR−1)/σR pR t − 1 ] = φC (ˇC t − ˇC t−1 / ˇC t−1) ˇC t / ˇC t−1 − Et θ gt+1/(n r t) φC (ˇC t+1 − ˇC t / ˇC t) (ˇC t+1 / ˇC t) 2 (Equation (102)).
- Retail dividends and adjustment costs:
  - ˇdR t = (pR t − 1) ˇC t − ˇG R Y,t (Equation (103)).
  - ˇG R Y,t = φC /2 ˇC t (ˇC t − ˇC t−1 / ˇC t−1) 2 (Equation (104)).

### Government, fiscal rule, and monetary policy
- Government instruments: public investment Ginv t, public consumption Gcons t, transfers τT,t, taxes τL,t, τc,t, τk,t, τls,t; issues nominal one-period debt Bt at rate it.
- Public capital accumulation (rescaled):
  - ˇKG t+1 / gt+1 = (1−δ) ˇKG t + ˇGinv t (Equation (105)).
- Transfer rule to LIQ agents redistributing OLG dividends (rescaled):
  - ˇτT,t = ι (ˇdN t + ˇdT t + ˇdD t + ˇdM t + dX + dF t) + ˇcLIQ t / ˇC t ˇdR t + ˇℓLIQ t / ˇL t ˇdU t (Equation (106)).
- Rescaled aggregate real tax variable:
  - ˇτ t = τL,t ˇw t L t + τc,t ˇC t + τk,t Σj=N,T [ rJ k,t − δ qJ t ] ˇKJ t + ˇτls,t.
- Government budget constraint (rescaled):
  - ˇb t = it−1 / (πt g t n) ˇb t−1 + ˇGinv t + ˇGcons t − ˇτ t − ˇgX t (Equation (107)).
- Structural fiscal surplus rule (targeting rule):
  - gs rat t = gs rat ∗ t + dtax (ˇτ t − ˇτpot t g ˇdp t) + dcop (ˇgX t − ˇgpot t g ˇdp t) (Equation (108)).
  - gs rat t = −Bt − Bt−1 / GDP t = − ˇb t − ˇb t−1 / (πt g t n g ˇdp t) (Equation (109) definition).
  - Potential tax revenue ˇτpot t defined in (110).
  - Potential government copper revenue gpot X t defined in (111).
- Policy interpretations and calibrations:
  - Structural rule ensures dynamic stability (non-explosive government assets/GDP) and business cycle stabilization; default implementation uses labor tax τL,t as adjustment instrument, alternative instruments possible.
  - Chile example: structural surplus target gs rat reduced from 0.5 to 0 for 2009 fiscal stimulus (described qualitatively in text).
  - Relation between long-run surplus and government assets to GDP ratio:
    - gs rat ∗ = ̄π ̄g n − 1 / (̄π ̄g n) gassets rat ∗ (Equation (112)).
- Monetary policy rule (interest rate rule with smoothing):
  - it = (it−1) μi (̄r π4,t+4) 1−μi (π4,t+4 / ̄π) (1−μi) μπ (gdp t / gdp) (1−μi) μy (Equation (113)).
  - Four-quarter inflation π4,t = (πt πt−1 πt−2 πt−3) 1/4 (Equation (114)).

### Equilibrium and balance of payments (summary)
- An equilibrium requires optimal behavior by OLG and LIQ households subject to their constraints and budget sequences, optimal behavior by firms (manufacturers, unions, import agents, distributors, retailers), and a government policy sequence {Ginv s, Gcons s, τL,s, τc,s, τk,s, τls,s, is}∞ s=t such that:
  1. OLG households maximize (1) subject to (2), (3), (7) and (8).
  2. LIQ households maximize (24) subject to (25), (26) and (27).
- Asset market incompleteness: complete home bias in government debt, only nominal one-period foreign bonds traded internationally, equity not traded domestically (lump-sum dividends to households).
- Real variables rescaled by technology and population: notation ˇx t = x t /(Tt n t) with steady state ̄x; labor quantities rescaled only by n t.

*Source: _wp0988 - Section VI concludes.*

### 3. Manufacturers maximize the expected present discounted value of cash flows (44)

### 3. Manufacturers maximize the expected present discounted value of cash flows (44)

### Model agents and objectives
- Manufacturers maximize the expected present discounted value of cash flows (44) subject to:
  - definition of after-tax dividends (43),
  - capital accumulation process (42),
  - demands (36),
  - production functions (37), (38),
  - adjustment costs (40) and (41).
- The copper sector divides its revenues in accordance with (52).
- Unions maximize the expected present discounted value of cash flows (59) subject to demands (57) and adjustment costs (58).
- Import agents maximize the expected present discounted value of cash flows (69) subject to demands (66) and adjustment costs (67).
- Distributors maximize the expected present discounted value of cash flows (96) subject to:
  - demands (76),
  - production functions (77), (78), (79), (83), (87), (89), (90), (91),
  - adjustment costs (95).
- Retailers maximize the expected present discounted value of cash flows (101) subject to demands (99) and adjustment costs (100).

### Markets and clearing conditions
- Labor market clears:
  - U_t = U^N_t + U^H_t, (115)
  - L_t = \bar{OLG}_t + \bar{LIQ}_t,
  - U_t = L_t.
- Nontradables market clears:
  - \tilde{Z}^N_t = \tilde{Y}^N_t + ω_N + \tilde{G}^N_{P,t}. (116)
- Tradables market clears:
  - \tilde{Z}^T_t = \tilde{Y}^{TH}_t + ω_T + \tilde{G}^T_{P,t} + \tilde{Y}^{TX}_t. (117)
- Exports and imports clear:
  - \tilde{Y}^{TM}_t(Chile) = \tilde{Y}^{TX}_t(Foreign), (118)
  - \tilde{Y}^{DM}_t(Chile) = \tilde{Y}^{DX}_t(Foreign).
- Final output market clears:
  - \tilde{Y}_t = \tilde{C}_t + \tilde{I}^N_t + \tilde{I}^T_t + \tilde{G}^{inv}_t + \tilde{G}^{cons}_t + \tilde{Y}^{DX}_t + ω_D + \tilde{G}^{NI}_t + \tilde{G}^{TI}_t + \tilde{G}^{DP}_t + \tilde{G}^{UP}_t + \tilde{G}^{MP}_t + \tilde{G}^{RY}_t. (119)
- World copper market clears:
  - X^N_t(Chile) + X^T_t(Chile) + X^N_t(Foreign) + X^T_t(Foreign) = X^{sup}_t(Chile) + X^{sup}_t(Foreign). (120)

### Accounting: current account, international bonds, and GDP
- Combining market clearing, household and government budget constraints, and firm dividends yields the current account:
  - e_t \tilde{f}_t = i_{t-1}(Foreign) ε_t (1+ξ^f_{t-1}) π_t g_t n e_{t-1} \tilde{f}_{t-1} (121)
    + p^{TH}_t \tilde{Y}^{TX}_t + \tilde{d}^{TM}_t − p^{TF}_t \tilde{Y}^{TF}_t + \tilde{Y}^{DX}_t + \tilde{d}^{DM}_t − p^{DF}_t \tilde{Y}^{DF}_t + X^x_t − f^X_t.
- International bonds market clearing:
  - \tilde{f}_t(Chile) + \tilde{f}_t(Foreign) = 0. (122)
- Level of GDP:
  - g \tilde{dp}_t = \tilde{C}_t + \tilde{I}^N_t + \tilde{I}^T_t + \tilde{G}^{inv}_t + \tilde{G}^{cons}_t (123)
    + p^{TH}_t \tilde{Y}^{TX}_t + \tilde{d}^{TM}_t − p^{TF}_t \tilde{Y}^{TF}_t + \tilde{Y}^{DX}_t + \tilde{d}^{DM}_t − p^{DF}_t \tilde{Y}^{DF}_t + X^x_t.

### International relative prices and uncovered interest parity
- Real exchange rate based on final output numeraire:
  - e_t = E_t P_t(Foreign)/P_t.
- Together with uncovered interest parity (12):
  - e_{t+1}/e_t = ε_{t+1} π_{t+1}(Foreign)/π_{t+1} = r_t / r_t(Foreign). (124)

### Calibration: general design and key parameter values
- Model frequency and scale:
  - Quarterly model.
  - Chile represents one third of one percent of the world economy in GDP and population.
- Interest rates, risk premium, and debt:
  - Long-run world real interest rate: 3% per annum.
  - Risk premium calibrated to produce a 50 to 60 basis points premium over international interest rates at steady state net foreign liabilities to GDP ratio of 20 percent.
  - Government debt to GDP ratio: 0 in Chile and 50 percent in Foreign.
- Growth and inflation:
  - Real world growth rate: 2% per annum.
  - Population growth rate: 1% per annum.
  - Long-run inflation rate: Chile 3% per annum, rest of world 2% per annum.
- Non-Ricardian parameters:
  - θ and χ set to 0.98125, corresponding to a 15 year planning horizon or average life expectancy 1/(1−θ) for θ, and a 15 year remaining working life for χ.
  - This choice produces an elasticity of the real interest rate with respect to a 1 percentage point increase in government debt to GDP ratio of 4 basis points in U.S. calibrations.
- Household preferences:
  - Intertemporal elasticity of substitution 0.2 (γ = 5).
  - Habit persistence v = 0.7.
  - Wage elasticity of labor supply targeted at 0.5 via η_OLG and η_LIQ.
  - Share ψ of liquidity constrained agents: 50 percent for Chile and 40 percent in Foreign.
  - Share of these agents in dividend income: half of their share in the population.
- Technology and substitution elasticities:
  - Elasticities of substitution between capital, labor and copper: 1 in both tradables and nontradables.
  - Elasticities between domestic and foreign traded intermediates and final goods (long-run import demand elasticities): 1.5.
  - Elasticity between tradables and nontradables: 0.8.
  - Copper demand shock persistence ρ_X = 0.95.
- Price setting and markups:
  - Markup of price over marginal cost: 10 percent in two manufacturing sectors and labor market.
  - Distribution and retail sectors markups: 5 percent.
  - Import agents markup: 2.5 percent.
  - Inflation adjustment cost parameters chosen to yield plausible dynamics over the first two to three years following a shock.
- Shares and steady state targets:
  - Manufacturing labor income share: 55 percent in Chile, 64 percent in rest of world.
  - Nontradables labor shares: 64 percent in both Chile and Foreign.
  - Nontradables share in GDP: 50 percent.
  - Investment to GDP ratio: 20 percent in both countries.
  - Government spending shares in GDP: 12 percent in Chile and 18 percent in Foreign.
  - Government investment shares: 2 percent in Chile and 3 percent in Foreign.
  - Transfers to GDP: 10 percent in both countries.
  - Tax revenue composition (Chile): consumption tax revenue 50 percent, capital income tax revenue 14 percent, labor income tax revenue 15 percent, other (lump-sum) taxes 21 percent.
  - Tax revenue composition (Foreign): 30, 10, 30 and 30 (consumption, capital income, labor income, other).
- Capital income, profits, and depreciation:
  - Introduced fixed costs in manufacturing and distribution to partially or wholly eliminate steady state economic profits.
  - Annual depreciation rate of private capital calibrated at 10 percent.
- Copper and trade shares:
  - Target steady state ratios: copper output to GDP 12.38 percent, copper exports to GDP 12.3 percent.
  - Steady state world copper price normalized to one.
  - Calibrated Chilean ratios to GDP: intermediate exports 8.7 percent, final goods exports 14 percent, final goods imports 5 percent.
  - Copper demand and export/import calibration aimed to normalize initial steady state final output based real exchange rate.

*Source: IMF working paper excerpt (calibration and model equations as provided in the content unit).*

### 1. Taken together with net copper exports the current account equation then determines

### _wp0988 - 1. Taken together with net copper exports the current account equation then determines

### Assumptions and model setup
- Division of copper revenue in steady state:
  - Domestic factors of production receive 65 percent, with the remainder split evenly between the domestic government and the foreign private sector.
  - The net excess revenue following a shock is shared evenly between the domestic government and the foreign private sector.
- Monetary policy rule parameters:
  - μ_i = 0
  - μ_π = 1.0
  - "This produces reasonable monetary policy responses to the shocks we investigate."
- Fiscal instrument:
  - The labor income tax rate τ_L,t is chosen as the fiscal instrument that adjusts endogenously to satisfy the structural surplus rule.
- Shocks considered:
  - Focus on shocks to copper prices, calibrated to reproduce the unconditional variance and autocorrelation of international copper prices.

### Choice of countercyclical coefficients (d_tax and d_cop)
- Fiscal rule context:
  - d_tax = d_cop = 0 corresponds to a balanced-budget rule (procyclical: lower taxes or higher spending in response to a demand boom).
  - d_tax = d_cop = 1 corresponds to Chile’s structural surplus rule (minimal short-run changes in fiscal instruments; countercyclical overall deficit).
  - d_tax and d_cop > 1 imply even more countercyclical fiscal policy (countercyclical overall deficit and countercyclical fiscal instruments).
- Loss function used to evaluate rules:
  - Loss = sd(π) + λ * sd(gdp), (125)
  - sd stands for standard deviation.
  - For each λ the weights d_tax and d_cop are chosen to minimize the loss function; this traces an inflation-output efficiency frontier.
- Numerical and qualitative results reported:
  - Different fiscal policies affect mostly the volatility of output rather than inflation.
  - Chile’s rule (d_tax = d_cop = 1) performs far better than a balanced-budget rule (d_tax = d_cop = 0).
  - Chile’s structural surplus rule is very close to the efficiency frontier and implies a relative weight on output volatility in the policymaker’s objective function of approximately 0.13.
  - A rule increasing the weight on excess copper revenue to around two (example calibration: cob=2.1, tax=1.4 shown in Figure 2) reduces output volatility substantially but:
    - increases inflation volatility (albeit smaller),
    - leads to much higher volatility in tax rates, fiscal deficits, and government debt.
  - For the far left portion of the efficiency frontier the relative weight of output in the loss function is greater than one.
  - Volatility of fiscal instruments is minimized by the structural surplus rule with coefficients equal to one; deviations from this rule increase fiscal instrument volatility with opposite signs:
    - balanced budget rule → fiscal instruments very volatile and procyclical;
    - coefficients significantly > 1 → fiscal instruments very volatile but countercyclical.

- Table 1 (reported entries preserved exactly as in source):
  - "Weight of Copper in Fiscal Rule=⇒Balanced0.4SFS12.1"
  - "Budget(Chile)"
  - Real GDP std. 1.05 0.72 0.58 0.58 0.38
  - Inflation std. 0.24 0.25 0.29 0.27 0.36
  - Deficit std. 0 0.89 1.07 1.41 1.96
  - Debt std. 0 9.9 10.5 16.2 19.2
  - ∆Tax std. 0.67 0.32 0.04 0.06 0.68
  - Output Weight in Loss Fct. -0.05 -0.13 >1

### Impulse-response scenarios (qualitative summaries)
- Structural surplus rule (SFS rule) — impulse responses to a one standard deviation copper price shock (40 quarters):
  - GDP expands and inflation falls.
  - Monetary policy responds accommodatively, boosting investment.
  - Real exchange rate appreciates; net exports of non-copper goods and services decline.
  - Surge in government copper-related revenue reduces deficits and debt.
  - Tax rates change very little in the short run; as debt and interest charges fall, tax rates start to fall, stimulating consumption for an extended period.
  - Boom in GDP is more short-lived than the consumption boom; monetary stimulus dissipates quickly and real appreciation dominates over the medium term.

- Balanced-budget rule — impulse responses to a copper price shock:
  - Surge in copper revenue is not allowed to affect deficits; it leads to an immediate reduction in taxes.
  - Result is an amplification of the GDP boom relative to the SFS case.

- Strongly countercyclical (aggressive) fiscal rule — impulse responses:
  - Surge in copper prices is accompanied by an immediate increase in taxes and much higher fiscal surpluses.
  - Short-run effect on GDP is contractionary rather than expansionary.
  - Associated with much higher volatility in fiscal accounts and tax rates (see Figures 7 and 8).

### Choice of surplus target
- Chile’s targets and implied asset implications:
  - Chile’s government surplus to GDP target before May 2007 was 1%, and its government assets to GDP ratio was around 8%-9%.
  - Assumed long-run growth rates for Chile: technology 2%, population 1%, prices 3% per annum.
  - Calculation: a 1% surplus target implies a long-run government assets to GDP ratio of 17.4%, which represents substantial asset accumulation beyond current level.
  - The 0.5% surplus target adopted in May 2007 (effective 2008) is consistent with the current assets to GDP ratio and implies no significant changes in assets in the long run.
  - Advantages of the 0.5% target for business cycle stabilization:
    - Avoids requiring higher taxes and/or lower spending today to accumulate assets for the 1% target.
    - Prevents intertemporal effects on consumption and investment that would arise from demanding further asset accumulation.

- Thought experiment: permanent increase of surplus target from 0.5% to 1% at time 0 (lump-sum transfers used as fiscal instrument):
  - The rule induces a business cycle even absent shocks: transfers must be temporarily reduced to accumulate assets.
  - Due to the non-Ricardian nature of agents, temporary reduction in transfers crowds out consumption and crowds in investment and net exports.
  - This illustrates one cost of raising the surplus target above levels consistent with current asset stocks.

### Key quantitative values and model calibrations (preserved exactly)
- Revenue split in steady state: domestic factors 65 percent; remainder split evenly between domestic government and foreign private sector.
- Monetary rule parameters: μ_i = 0 and μ_π = 1.0.
- Loss function: Loss = sd(π) + λ * sd(gdp), (125).
- Chile’s pre-May 2007 surplus target: 1%.
- Chile’s government assets to GDP ratio (pre-May 2007): around 8%-9%.
- Assumed long-run growth rates for Chile: technology 2%, population 1%, prices 3% per annum.
- 1% surplus target ⇒ long-run government assets to GDP ratio of 17.4%.
- 0.5% surplus target adopted in May 2007 (effective 2008).

### Main conclusions (from paper)
- Chile’s current structural surplus rule performs well if the policymaker places a small weight on output volatility relative to inflation volatility.
- A more aggressive countercyclical fiscal rule can lower output volatility, but with trade-offs: somewhat higher inflation volatility and much higher volatility of fiscal variables.
- Given Chile’s current stock of government assets, adopting a 0.5% surplus target starting in 2008 is desirable from a business cycle perspective because the earlier 1% target would have required significant further asset accumulation at the expense of greater volatility in fiscal instruments and GDP.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0988.pdf*

### References

### _wp0988 - References

### References
- Blanchard, O.J., 1985, “Debt, Deficits, and Finite Horizons”, Journal of Political Economy, Vol. 93, pp. 223-247.
- Calvo, G.A., 1983, “Staggered Prices in a Utility-Maximizing Framework”, Journal of Monetary Economics, Vol. 12, pp. 383-398.
- Chang, Y. and Kim, S.-B., 2005, “On the Aggregate Labor Supply”, Federal Reserve Bank of Richmond Economic Quarterly, Vol. 91(1), pp. 21-37.
- Engen, E.M. and Hubbard, R.G., 2004, “Federal Government Debt and Interest Rates”, NBER Macroeconomics Annual, Vol. 19, pp. 83-138.
- Erceg, C., Guerrieri, L. and Gust, C., 2005, “Expansionary Fiscal Shocks And The Trade Deficit”, International Finance Discussion Paper 825, Board of Governors of the Federal Reserve System.
- Ireland, P., 2001, “Sticky-Price Models of the Business Cycle: Specification and Stability”, Journal of Monetary Economics, Vol. 47, pp. 3-18.
- Laubach, T., 2003, “New Evidence on the Interest Rate Effects of Budget Deficits and Debt”, Finance and Economics Discussion Series 2003-12, Board of Governors of the Federal Reserve System.
- Laxton, D. and Pesenti, P., 2003, “Monetary Rules for Small, Open, Emerging Economies”, Journal of Monetary Economics, Vol. 50(5), pp. 1109-1152.
- Marcel, M., M. Tokman, R. Valdés and P. Benavides (2001), “Balance Estructural: La Base de la Nueva Regla de Política Fiscal Chilena”, Economía Chilena, Vol. 4(3), pp. 5-27.
- Mendoza, E., 2005, “Real Exchange Rate Volatility and the Price of Nontradables in Sudden-Stop-Prone Economies”, NBER Working Paper No. 11691.
- Pencavel, J., 1986, “Labor Supply of Men”, in: O. Ashenfelter and R. Layards, eds., Handbook of Labor Economics, Vol. 1. Amsterdam: North Holland.
- Woodford, M., 2003, Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton: Princeton University Press.

### Appendix 1 — Population Growth (key definitions and identity)
- Population at time 0 equals N, composed of N(1−ψ) OLG households and Nψ LIQ households.
- Size of a new cohort born at time t: Nn_t (1−θ_n).
- Cohort size at time t+k: Nn_t (1−θ_n) θ^k.
- Summing cohorts at time t yields Nn_t, implying overall population growth rate n.
- Normalization: real quantities divided by technology T_t and by population n_t (real figures are absolute terms adjusted for population growth).

### Appendix 2 — Optimality Conditions for OLG Households (key equations and aggregations)
- Lagrangian for OLG households (marginal utility Λ_a,t and consumption price relation λ_a,t = Λ_a,t / P_t) given in equation (126) and (127).
- Utility and scaling definitions:
  - u_OLG_a,t = (c^OLG_{a,t}/h_t)^{η_OLG} (1−ψ^OLG_{a,t})^{1−η_OLG} (128)
  - h_t = (c^OLG_{t−1} / (g_t Nn_{t−1} (1−ψ)))^v (129)
- First-order conditions (consumption and labor supply):
  - η_OLG (u_OLG_{a,t})^{1−γ} / c^OLG_{a,t} = λ_{a,t} (p^R_t + τ_{c,t}) (130)
  - (1−η_OLG) (u_OLG_{a,t})^{1−γ} / (1−ψ^OLG_{a,t}) = λ_{a,t} w_t Φ_{a,t} (1−τ_{L,t}) (131)
- Combined consumption–labor relation:
  - c^OLG_{a,t} / (1−ψ^OLG_{a,t}) = (η_OLG / (1−η_OLG)) * (w_t Φ_{a,t} (1−τ_{L,t})) / (p^R_t + τ_{c,t}) (132)
- Aggregate and normalized versions:
  - c^OLG_t Nn_t (1−ψ) − ψ^OLG_t = η_OLG/(1−η_OLG) w_t (1−τ_{L,t}) (p^R_t + τ_{c,t}) (133)
  - ˇc^OLG_t N(1−ψ) − ˇψ^OLG_t = η_OLG/(1−η_OLG) ˇw_t (1−τ_{L,t}) (p^R_t + τ_{c,t}) (134)
- Assumptions on labor productivity:
  - Φ_{a,t} = κ χ^a (135)
  - Nn_t (1−ψ) (1−θ_n) Σ_{a=0}^∞ (θ_n)^a Φ_{a,t} = Nn_t (136)
  - κ = (n − θ χ) / (n − θ) (137)
  - Definition of effective aggregate labor supply: Σ_{a=0}^∞ (θ_n)^a (ψ^OLG_{a,t} Φ_{a,t}) ≡ ψ^OLG_t (138)
- First-order conditions for domestic and foreign bonds:
  - λ_{a,t} = β E_t λ_{a+1,t+1} i_t / π_{t+1} (139)
  - λ_{a,t} = β E_t λ_{a,t+1} i^*_t ε_{t+1} / [(1+ξ^f_t) π_{t+1}] (140)
  - Uncovered interest parity: i_t = i^*_t ̃E_t ε_{t+1} (1+ξ^f_t) (141)
- Expression for marginal utility λ_{a,t} in terms of consumption and parameters (142).
- Generation-specific consumption Euler equation:
  - ̃E_t c^OLG_{a+1,t+1} = ̃E_t j_t c^OLG_{a,t}, where j_t given in (143).

### Appendix 3 — Consumption and Wealth (aggregation of lifetime wealth and consumption)
- Budget constraint (nominal) for OLG household (144).
- Decomposition of after-tax wage income (145) and link to consumption via (146).
- Dividend redistribution to LIQ agents and lump-sum taxes definitions (147)-(148).
- Total nominal wage and dividend income of age-a households (149).
- Alternative budget constraint form (150).
- Forward substitution and no-Ponzi condition:
  - No-Ponzi condition: lim_{s→∞} ̃E_t ̃R_{t,s} [B_{a+s,t+s} + E_{t+s} F_{a+s,t+s}] = 0 (152)
  - Inherited nominal financial wealth: FW_{a−1,t−1} = (1/θ) [i_{t−1} B_{a−1,t−1} + i^*_{t−1} E_t F_{a−1,t−1} (1+ξ^f_{t−1})] (153)
  - Lifetime human wealth: HW_{a,t} = ̃E_t Σ_{s=0}^∞ ̃R_{t,s} Inc_{a+s,t+s} (154)
- Core consumption–wealth identity (nominal discounted form):
  - ̃E_t Σ_{s=0}^∞ ̃R_{t,s} [P_{t+s} c^OLG_{a+s,t+s} (p^R_{t+s} + τ_{c,t+s})^{η_OLG}] = HW_{a,t} + FW_{a−1,t−1} (155)
- Recursive representation for the discount factor Θ_t (158) and compact relation:
  - P_t c^OLG_{a,t} Θ_t = HW_{a,t} + FW_{a−1,t−1} (159)
- Real aggregate human wealth hw_t and its decomposition:
  - hw_t = Nn_t (1−ψ) (1−θ_n) Σ_{a=0}^∞ (θ_n)^a HW_{a,t} / P_t (160)
  - Labor component (recursive, scaled): ˇhw^L_t = (N(1−ψ) ˇw_t (1−τ_{L,t})) + ̃E_t θ χ g_{t+1} r_t ˇhw^L_{t+1} (161)
  - Capital/dividend component (recursive, scaled): ˇhw^K_t = (Σ_{j=N,T,D,M,X,F} ˇd_{j,t} (1−ι) + c^OLG_t / C_t (d_{R,t} − τ_{ls,t}) + ψ^OLG_t L_t d_{U,t}) + ̃E_t θ g_{t+1} r_t ˇhw^K_{t+1} (162)
  - ˇhw_t = ˇhw^L_t + ˇhw^K_t (163)
- Aggregation of financial wealth and real scaling:
  - Total nominal financial wealth aggregation described with B_t−1 and FW_{t−1} definitions (FW_{t−1} = [i_{t−1} B_{t−1} + i^*_{t−1} E_t F_{t−1} (1+ξ^f_{t−1})]).
  - Real financial wealth per scaled units:
    - ˇfw_t = FW_{t−1} / (P_t T_t n_t) = (1 / (π_t g_t n)) [i_{t−1} ˇb_{t−1} + i^*_{t−1} ε_t ˇf_{t−1} / e_{t−1} (1+ξ^f_{t−1})] (164)
- Final real consumption identity:
  - ˇc^OLG_t Θ_t = ˇhw_t + ˇfw_t (165)
- Note: linearized form derivable by linearizing individual age-group constraints and aggregating; use of ̃E_t in nonlinear equations valid up to first-order approximations.

### Appendix 4 — Optimality for Manufacturing Firms (firm problem, constraints, and FOCs)
- Firm objective: Max_{P ̃J_s(i), U^J_s(i), I^J_s(i), K^J_{s+1}(i)} E_t Σ_{s=t}^∞ ̃R_{t,s} D^J_{t+s}(i).
- Dividend specification and tax/price components provided; τ_{k,t} enters dividend term.
- Output equals demand constraint:
  - F(K^J_t(i), U^J_t(i), X^J_t(i)) = Z^J_t(i), with nested CES structures and sectoral indexes J ∈ [N, T].
- Capital accumulation and adjustment costs:
  - K^J_{t+1}(i) = (1−δ) K^J_t(i) + I^J_t(i)
  - 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
  - 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
- Lagrangian formulation introduces multipliers Λ^J_t (nominal marginal cost) and q^J_t (Tobin’s q).
- First-order condition w.r.t. P ̃^J_t(i) (after symmetry and rescaling) yields:
  - [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) − E_t θ g_{t+1} n 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) } (166)
- First-order conditions for inputs, investment, and q:
  - ˇv_t = ˇλ^J_t ˇF^J_{U,t} (167)
  - p_X_t = ˇλ^J_t ˇF^J_{X,t} (168)
  - q^J_t = 1 + φ_I (ˇI^J_t / ˇK^J_t − ˇI^J_{t−1} / ˇK^J_{t−1}) (169)
  - q^J_t = θ r_t E_t [ q^J_{t+1} (1−δ) + r^J_{k,t+1} − τ_{k,t+1} (r^J_{k,t+1} − δ q^J_{t+1}) ] + θ r_t E_t (ˇI^J_{t+1}/ˇK^J_{t+1}) φ_I (ˇI^J_{t+1}/ˇK^J_{t+1} − ˇI^J_t / ˇK^J_t) − θ r_t E_t φ_I/2 (ˇI^J_{t+1}/ˇK^J_{t+1} − ˇI^J_t / ˇK^J_t)^2 (170)
- Definitions and auxiliary relations:
  - r^J_{k,t} = ˇλ^J_t ˇF^J_{K,t} (171)
  - ˇF^J_{U,t}, ˇF^J_{K,t}, ˇF^J_{X,t} specified in (172)-(174) as functions of sectoral parameters α, ξ and aggregate variables.

*Source: _wp0988 - References (PDF chapter/section)._

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp0988.pdf_