## 1. Growth Regressions: A Nonlinear Effect of Tax Revenues on the Growth Rate

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---

### Model framework and assumptions
- Dynamic general equilibrium model extending Barro (1990) for productive government spending with endogenous growth.
- Agents: many households and firms interacting with a government.
- Fiscal policy instruments:
  - Income tax policy (τ with 10<≤τ in DCE specification).
  - Three expenditure programs: productive spending (G), consumptive spending (H), and income transfers (σ(i)).
- Roles of expenditure types:
  - Consumptive government spending increases household utility directly (γ measures weight on public consumption).
  - Productive spending provides positive externalities to private firms and drives long-run growth (G enters production Y = A Gα K1−α with 0<α<1).
  - Income transfer σ(i) is linear in relative asset ownership σ(i) = σ(ai − a) with σ ≥ 0.
- Government budget decomposition:
  - G = θ ∫0I di ])([τ wLira+ ; H + transfers = (1 − θ) ∫0I di ])([τ wLira+ with 10<<θ.
- Key shorthand/notation preserved from source:
  - φ(τ, θ) ≡ A[τθ(1 − τ)]α−1 (as in text).
  - Aggregate asset K ≡ ∫I di ai; labor fixed at 1 = L.

### Main theoretical findings (exogenous vs endogenous fiscal policy)
- Under exogenous fiscal policy (τ, θ, σ treated as given):
  - An inverse U-shaped relationship exists between tax rates and growth rates.
  - Analytical conditions: ∂Γ/∂τ = )1(),()1(ττθτφταατ−−−=∂Γ∂ ; sign conditions:
    - ∂Γ/∂τ > 0 if ατ − << 10 (at low τ growth rises with τ).
    - ∂Γ/∂τ < 0 if 11<<−τα (at high τ growth falls with τ).
  - Benchmark τ* = α−1 marks the turning point of the inverse U.
  - Productive spending raises growth: ∂Γ/∂θ = (1 − α) θταφ > 0.
  - Redistribution (σ) reduces growth via distortionary taxation and moral hazard; adverse effect arises from parameter σ.
- Under endogenous (socially optimal, Ramsey) fiscal policy:
  - The relation between growth and tax rates is always negative; the nonlinear hump disappears.
  - Optimal tax and expenditure instruments satisfy first-order conditions that place the economy on the downward-sloping part of the growth–tax relation.
  - τ and θ are substitutes along the optimal path: 0<∂∂θτ (higher θ allows lower τ).
  - Optimal τ is within 110<<−<τα and optimal θ within 110<<−<θα (domain results from Appendix II).
  - When policy is optimally chosen, the increasing portion of the exogenous-policy hump is eliminated.

### Decentralized Competitive Equilibrium (DCE) — characterization and implications
- Household Euler equation (notation preserved): ])1)[(()(ρστ−−−= • ricic.
- Firm returns: r = α A (K G)α−1 ; wage w = (1 − α) A (K G)α.
- Aggregated DCE dynamics (given τ, θ, σ) include:
  - (1.1) ]),()[()(ρσθταφ−−= • icic
  - (1.2) )]([])1()()[,()()(iaaKiaiaic−+−+=+ • σααθτφ
- DCE is suboptimal because productive government spending generates positive externalities so r* > r when 10<<α, 0>τ, 0>θ.
- Growth rate expression (preserve original functional form): Γ is ρσθταφ−−=Γ),(.
  - Nonlinearity between Γ and τ arises even if γ = 0 or σ(i) = 0 provided θ ≠ 0 and α ≠ 1, 0.
- Interpretation:
  - At low τ, positive productivity effects of public production outweigh tax distortions → growth rises.
  - At high τ, distortionary taxation dominates → growth falls.
  - Nonlinearity can account for fragile empirical correlations between fiscal variables and growth.

### Ramsey optimal policy and equilibrium (setup and comparative statics)
- Government solves a Ramsey second-best problem maximizing aggregate lifetime utility ∫∫0∞ di dt [log(ci) + γ log(H)] subject to incentive constraints (equations (1.1)) and budget constraints (equation (1.2)).
- Key Ramsey properties (Appendix II and IV):
  - Optimal tax policy relation (preserved form): )1)(1( 1 1 1 θα θα τα τ −− −− = −− − .
  - No solution exists when ατ−<1; optimal τ lies in 110<<−<τα.
  - τ and θ move in opposite directions each period (0<∂∂θτ); τ and θ are substitutes along the optimal path.
  - Redistribution: total differentiation of (2.1) implies τ increases with )(iaa−; median voter less endowed than average implies higher τ and lower growth along the optimal path.
  - Special cases:
    - If γ = 0 and ατ−=1 in each period, then Ramsey implies 1=θ in all time periods (all tax revenues to productive spending) and constant τ, θ with no transitional growth dynamics.
    - If γ = 0 and public expenditures are not productive for firms (01=−α), then 0=τ in all periods (Ramsey tax rate zero) and no endogenous persistent growth.

### Long-run Ramsey equilibrium (Appendix IV) — existence and uniqueness
- Long-run conditions (tildes denote long-run values) include:
  - Equation (3.1) and Equation (3.2) as stated in the source (preserved).
  - From (3.1): z
~
 equals ρ plus σ: 0
~
)1(
~
>++−=ρσφαz (as derived in Appendix IV).
  - From (A.4): 0
~~
1
~
>
⎥
⎦
⎤
⎢
⎣
⎡
+−+=
ψ
γ
η
σ
σρ
z (as stated).
- If parameter values satisfy 0)1()1(
11
1
<−+−
−+
−
α
α
α
α
ααγγρ, then:
  - (i) A unique optimal long-run tax rate τ
~
 exists with 1
~
10<<−<τα.
  - (ii) A unique optimal long-run allocation θ
~
 exists with 1
~
10<<−<θα.
- (3.1) and (3.2) form a two-equation system in τ
~
 and θ
~
 with negative slopes; (3.1) is always steeper than (3.2) so at most one intersection (unique BGP).
- Steady-state (symmetric Ramsey) properties:
  - Consumption c, capital k, and asset a grow at the same constant positive rate; policy instruments are constant (0==•• θτ); social value of capital grows at a constant negative rate (•ψ negative in some cases).

### Empirical analysis and evidence (cross-country regressions and OECD exercises)
- Cross-country panel: 93 industrial and developing countries, 1990–2000 (variables averaged over the 1990s except LGDP initial level 1990). Controls: LGDP, openness, investment share, ICRG (rent-seeking index), regional dummies.
- Tax rate proxies: (i) tax revenue over GDP; (ii) tax revenue minus trade taxes over GDP.
- Growth regression specification: growth = a + b·tax + c·(tax)^2 + controls. Selected exact results from Table 1 (OLS; preserve reported coefficients, t-ratios, sample sizes, and Adj. R2):
  - Column (1) (All countries):
    - Tax = –0.0465 (–1.10)
    - LGDP = –1.3185** (–2.31)
    - ICRG = 0.1657** (2.47)
    - Sub-Saharan Africa = –1.9101** (–2.48)
    - Constant = 6.8347** (2.06)
    - Adj. R2 = 24.24 (%)
    - Number of observations = 93
  - Column (2) (Low and middle income):
    - Tax = –0.0793 (–1.20)
    - LGDP = –0.8285 (–1.23)
    - ICRG = 0.2257** (2.50)
    - Constant = 2.7899 (0.64)
    - Adj. R2 = 26.15 (%)
    - Number of observations = 66
  - Column (3):
    - Tax = 0.0008 (0.01)
    - (Tax)^2 = –0.001 (–0.39)
    - LGDP = –1.3456** (–2.33)
    - ICRG = 0.1649** (2.45)
    - Adj. R2 = 23.46 (%)
    - Number of observations = 93
  - Column (4) (Low and middle income, nonlinear):
    - Tax = 0.2851 (1.43)
    - (Tax)^2 = –0.0093* (–1.93)
    - LGDP = –0.8735 (–1.33)
    - Openness = –0.0203* (–1.77)
    - ICRG = 0.2220** (2.51)
    - Constant = 0.4646 (0.10)
    - Adj. R2 = 29.53 (%)
    - Number of observations = 66
  - Column (5) (Low and middle income; tax minus trade taxes):
    - Tax = 0.0981 (0.58)
    - (Tax)^2 = –0.0067 (–1.45)
    - ICRG = 0.2525** (2.93)
    - Openness = –0.0199* (–1.87)
    - Constant = –0.0750 (–0.02)
    - Adj. R2 = 31.29 (%)
    - Number of observations = 66
- Empirical takeaways from Table 1:
  - Linear tax term often insignificant; including quadratic term and restricting to low- and middle-income countries (column 4) produces a significant negative quadratic term, consistent with nonlinearity.
  - ICRG is robustly positive and significant across specifications.
  - Results sensitive to tax-rate definition; tax minus trade taxes (column 5) weakens quadratic significance.
- Tax rates vs productive government spending (OECD sample; Table 2 correlations for 26 OECD countries):
  - Correlation Theta(70–00) vs Distorting tax revenue (70–00): –0.4745 (n = 26)
  - Correlation Theta(80–00) vs Distorting tax revenue (80–00): –0.5241 (n = 26)
  - Correlation Theta(90–00) vs Distorting tax revenue (90–00): –0.4782 (n = 23)
  - Empirical implication: consistent negative correlations support model’s prediction that productive spending and distortionary tax rates are negatively related under optimal policy.
- Income redistribution, growth, and inequality (OECD correlations):
  - Table 3 (Sigma vs Growth; 25 OECD countries):
    - Sigma(70–00) vs Growth(70–00): –0.2510 (n = 25)
    - Sigma(80–00) vs Growth(80–00): –0.0759 (n = 24)
    - Sigma(90–00) vs Growth(90–00): –0.1105 (n = 22)
  - Table 4 (RGS vs Gini; 25 OECD countries):
    - RGS(70–00) vs Gini(70–00): –0.6254 (n = 25)
    - RGS(80–00) vs Gini(80–00): 0.1715 (n = 25)
    - RGS(90–00) vs Gini(90–00): 0.1718 (n = 22)
  - Interpretation: negative income-redistribution–growth association consistent with theory but sensitive to subperiod; strong negative RGS–Gini correlation for 1970–2000 but time-period dependence thereafter.
- Multiple equilibria evidence (Table 5; 25 OECD countries):
  - Sigma(70–00) vs GDP variance (70–00): 0.2043 (n = 25)
  - Sigma(80–00) vs GDP variance (80–00): 0.1930 (n = 25)
  - Sigma(90–00) vs GDP variance (90–00): 0.1183 (n = 22)
  - Result: positive but weak correlations; excluding outliers (Luxembourg and Denmark) renders correlation near zero, weakening empirical support for multiplicity.

### Conceptual and methodological implications
- Identification: treating fiscal instruments as exogenous vs endogenous is fundamental for identification in empirical endogenous growth models.
- Omitting interaction among instruments or endogeneity can bias estimates in general equilibrium contexts.
- Nonlinearity between fiscal variables and growth can induce statistical insignificance in cross-country analyses (explains fragile/mixed findings in the literature).
- τ and θ substitution along the optimal path implies potential colinearity and fragility in empirical specifications using aggregate fiscal measures.

### Policy implications and interpretation
- Policy evaluation must account for whether fiscal instruments are exogenously set or endogenously chosen through social optimization or political processes.
- Decomposing government spending into consumptive, productive, and redistributive components is crucial for assessing growth–equity trade-offs.
- Under optimal endogenous policy, the negative growth–tax relation suggests policymakers internalizing growth effects choose lower growth-reducing tax distortions than exogenous-policy regressions imply.
- Institutional quality (ICRG/rent-seeking ability) conditions the growth and redistributive effectiveness of fiscal policy.

*Source: IMF working paper section "1. Growth Regressions: A Nonlinear Effect of Tax Revenues on the Growth Rate" and related sections of _wp06165.*

### 1. Growth Regressions: A Nonlinear Effect of Tax Revenues on the Growth Rate................21

### 1. Growth Regressions: A Nonlinear Effect of Tax Revenues on the Growth Rate

### Model framework and assumptions
- Dynamic general equilibrium model extending Barro (1990) for productive government spending with endogenous growth.
- Agents: many households and firms interacting with a government.
- Fiscal policy instruments:
  - Income tax policy.
  - Three expenditure programs: consumptive spending, productive spending, and income transfer.
- Roles of expenditure types:
  - Consumptive government spending increases household utility directly.
  - Productive spending provides positive externalities to private firms and is an engine of long-run growth.
  - Income transfer is provided to individuals who own less than the average capital in the economy.
- The author explicitly decomposes allocative and redistributive government activities and considers both exogenous and endogenously chosen fiscal policy regimes.
- The analysis also examines implementation conditions for a decentralized competitive economy when fiscal policy is endogenously determined.

### Main theoretical findings
- Under exogenous fiscal policy:
  - An inverse U-shaped relationship exists between tax rates and growth rates.
- Under endogenous (socially optimal) fiscal policy:
  - The relation between the growth rate and tax rates is always negative; the nonlinear hump relation between growth and taxes disappears.
- The difference in empirical relations between growth and fiscal policy can be attributed theoretically to whether fiscal policy is treated as exogenous or endogenous.
- Productivities and net costs/benefits of government activities:
  - The model investigates the productivity associated with each policy instrument among the four categories of fiscal policy (income tax plus the three expenditure programs), and the net cost and benefit of government activities in terms of economic growth.
- The decomposition of government spending may affect the response of private sector investment to fiscal policy.

### Empirical analysis and evidence (cross-country regressions)
- Cross-country regressions mirror the theoretical distinction:
  - When fiscal policy is treated as exogenous in regressions, income tax rates show a nonlinear effect on growth (inverse U-shape).
  - When fiscal policy is endogenously chosen at a social optimum, the growth–tax relation is negative.
- Regression specifications include rent-seeking variables and regional dummies:
  - An index of rent-seeking activities is statistically significant, implying that a government’s implementation abilities conditionally affect economic growth and income distribution.
- Additional empirical associations identified:
  - A negative association between the growth rate and income redistribution, consistent with political economy literature.
  - A negative relation between consumptive spending and income inequality (consistent with Chu, Davoodi, and Gupta, 2004).
  - A negative relation between tax rates and productive government spending (reported for both endogenous and exogenous policy contexts; see Section III.2 and Table 2).

### Conceptual and methodological implications
- Identification issue: The treatment of policy instruments as exogenous or endogenous is fundamental for identification in empirical endogenous growth models.
- Studies focusing solely on expenditure or taxation may suffer from systematic biases in estimates in a general equilibrium framework.
- Differences between exogenous and endogenous fiscal policy formulations help explain fragile and mixed empirical results in the literature on fiscal policy and growth.

### Policy implications and interpretation
- Policy evaluation must account for whether fiscal instruments are treated as exogenous choices or are endogenously determined through political or social optimization processes.
- Decomposing government spending into consumptive, productive, and redistributive components is crucial for assessing growth and equity trade-offs.
- The negative growth–tax relation under optimal endogenous policy suggests that policymakers internalizing growth effects may optimally choose lower growth-reducing tax distortions than implied by exogenous-policy regressions.

*Source: IMF working paper section "1. Growth Regressions: A Nonlinear Effect of Tax Revenues on the Growth Rate" (Introduction and overview).*

### Section III solves and characterizes a socially optimal equilibrium and, thus, examines the

### _wp06165 - Section III solves and characterizes a socially optimal equilibrium and, thus, examines the

### Decentralized Competitive Equilibrium (DCE) — model setup
- Economy: closed, infinite horizon, perfect foresight, no population growth; infinite number of households and identical firms; government taxes household incomes to finance public production services G, public consumption services H, and transfers ∫I di σ(i).
- Households:
  - Supply labor inelastically (assumed 1 = L).
  - Index household i by asset ai relative to average asset a; aggregate asset K ≡ ∫I di ai.
  - Utility: maximize ∫0∞ e−ρt log(ci) + γ log(H) dt with 0>ρ and 0≥γ (γ measures weight on public consumption).
  - Transfer rule: σ(i) = σ(ai − a) (linear in relative asset ownership), with σ ≥ 0 (wealth/income transfer policy parameter).
  - Budget constraint (household i): ]([)(])()[1()()(iaaidwiraiaic−+++−=+ • στ, where ȧ denotes time derivative and initial ai at time 0 is given.
  - Euler equation: ])1)[(()(ρστ−−−= • ricic, capturing distortion from asset-contingent redistribution (σ term).
- Firms:
  - Production: Y = A Gα K1−α with 0<α<1 (Cobb–Douglas); G is aggregate productive public spending (pure public good).
  - Labor inelastic 1 = L. Profit maximization yields r = α A (K G)α−1 (marginal product) and wage w = (1 − α) A (K G)α.
- Government:
  - Tax revenue from each household i: ])([wLira+τ where 10<≤τ is the income tax rate; total revenues ∫0I di ])([τ wLira+.
  - Budget constraint (flow): ∫0I di ])([τ wLira+ = G + H + ∫0I di σ(i).
  - Decomposition: G = θ ∫0I di ])([τ wLira+ ; H + transfers = (1 − θ) ∫0I di ])([τ wLira+ with 10<<θ (fraction to productive spending).

### DCE dynamic system and characterization
- Aggregated DCE dynamic equations (given policy instruments τ, θ, σ):
  - (1.1) ]),()[()(ρσθταφ−−= • icic
  - (1.2) )]([])1()()[,()()(iaaKiaiaic−+−+=+ • σααθτφ
  - φ(τ, θ) ≡ A[τθ(1 − τ)]α−1 (notation preserved as in text: αατθτθτφ −−≡ 1 ][)1(),(AA).
- Key model implications under exogenous policy (τ, θ, σ):
  - Production function at equilibrium: Y = A[τθ]α K1−α.
  - Socially optimal marginal return to private capital A* = ∂(KA)/∂K = α α τθ − − = r* (notation preserved exactly as in source).
  - The privately perceived return in DCE: r = α α τθα − − = (expression as in text).
  - DCE is suboptimal because productive government spending creates positive externalities (spillovers) so r* > r when 10<<α, 0>τ, 0>θ.

### Growth rate relations and nonlinearities
- Economy growth rate Γ expressed as Γ = ρ + σ + θ + τ + φ (preserve original functional form as in text: Γ is ρσθταφ−−=Γ),(.)
- Redistribution (σ) effects:
  - Higher σ (stronger redistributive policy) reduces growth via:
    - Distortionary taxation (higher taxes to finance redistribution) that discourages investment.
    - Moral hazard at recipients reducing effort.
  - The adverse effect is from parameter σ, not directly from the size of income inequality (ai − a).
- Productive spending (θ) effects:
  - Positive relation: ∂Γ/∂θ = (1 − α) θταφ > 0 as given in text (0),()1(>−=∂Γ∂θθτφαθ).
  - Public production services (G) stimulate growth by raising private capital productivity (Barro-type result).
- Nonlinear relation between growth rate and tax rate τ:
  - Analytical expression preserved: ∂Γ/∂τ = )1(),()1(ττθτφταατ−−−=∂Γ∂ (equation as in source).
  - Sign conditions:
    - ∂Γ/∂τ > 0 if ατ − << 10 (i.e., at low tax rates an increase in τ raises growth).
    - ∂Γ/∂τ < 0 if 11<<−τα (i.e., at high tax rates an increase in τ lowers growth).
  - Inverse U-curve relationship between growth and τ with benchmark τ* = α−1 (benchmark related to productivity spillover from public capital).
  - Nonlinearity holds even if γ = 0 or σ(i) = 0 so long as θ ≠ 0 and α ≠ 1, 0 (i.e., productive public capital and spillovers present).
- Interpretation:
  - At low τ, public production services’ positive productivity effects outweigh tax distortions → growth rises.
  - At high τ, distortionary taxation dominates → growth falls.
  - Nonlinearity can explain weak/fragile empirical correlations between fiscal variables (including productive spending) and growth in cross-country studies.

### Empirical and methodological implications
- Nonlinearity between fiscal policy and growth may induce statistical insignificance in cross-country analyses (Levine and Renelt, 1992; Mendoza, Milesi-Ferretti, and Asea, 1997).
- Alternative explanations for empirical nonrobustness include:
  - Mix of taxes with negligible long-run effects on labor supply and savings (Mendoza et al., 1997).
  - Misallocation between productive and consumptive spending across countries (Alesina, 1999; Devarajan, Swaroop, and Zou, 1996).
  - Endogeneity of fiscal variables—necessitating optimal policy endogenization (motivation for Section III Ramsey analysis).

### Ramsey optimal policy and equilibrium (motivation and setup)
- Government endogenizes τ and θ by solving a Ramsey second-best problem under a benevolent government, subject to individual incentive compatibility (equations (1.1)) and budget/budget-like constraints (equation (1.2)).
- Government objective: maximize aggregate lifetime utility ∫∫0∞ di dt [log(ci) + γ log(H)] (notation preserved: [][][] dtdiHic I ∫∫ ∞ 00 log)(logγ).
- Ramsey equilibrium internalizes externalities from public capital spillover and public consumption while respecting decentralized agents’ decision rules.
- Purpose: characterize how endogenous optimal τ and θ alter growth implications relative to exogenous policy results and hence illustrate empirical endogeneity problems in growth regressions.

*Source: _wp06165 - Section III solves and characterizes a socially optimal equilibrium and, thus, examines the (IMF Working Paper PDF content provided)*

### Appendix II):

### Appendix II)

### Necessary conditions and transversality
- Euler-type first-order conditions and costate dynamics are given in equations (2.1)–(2.6) in the source.  
- The transversality condition ensuring bounded utility is:
  - ρρσθταφ<−−),(,  (equation (2.7))
- The model definitions and shorthand notation used in the conditions include expressions such as ),(θτφ≡ταττα)1(−−− and related mappings as presented in the source.

### A. Properties of economic policy along the optimal equilibrium path
- Ramsey tax policy in each time period is characterized by:
  - )1)(1( 1 1 1 θα θα τα τ −− −− = −− − .
- Domain and comparative-statics results:
  - Because the expression is negative, no solution exists when ατ−<1. This implies 01<−−τα. That is, the optimal tax rate, τ, is within the subset 110<<−<τα.
  - The optimal tax rate is higher than α−1 (the productivity of public production services) because the government provides public consumption services and transfer payments in addition to public production services.
  - For θ (share of tax revenues to public production), equation (2.2) implies 0)1(<−−θα, so θ is within the subset 110<<−<θα.
  - When policy is optimally chosen, the result must be on the downward-sloping part of the growth rate–tax rate relation (references: Devarajan, Swaroop, and Zou 1996; Hansson and Henrekson, 1994).
  - If government can choose policy optimally, the nonlinear inverse-U relation between growth and tax rate disappears (empirical support: Hansson and Henrekson (1994)).
- Interaction between policy instruments:
  - The atemporal condition implies the two instruments τ and θ move in opposite directions in each time period: 0<∂∂θτ.
  - Intuition: higher θ (larger share to public production) allows a lower τ because public production services stimulate private investment, expanding the tax base.
  - τ and θ are substitutes along the optimal path; this implies room for fiscal consolidation but risks colinearity in empirical work.
  - Empirical evidence may be fragile because τ and θ are hardly exogenous; cross-country regressions can be biased (references: Easterly and Rebelo, 1993; Devarajan, Swaroop, and Zou, 1996; Bleaney, Gemmell, Kneller (2001); Alesina and Perotti (1995); Gupta and others (2002)).
- Redistribution, inequality, and growth:
  - Total differentiation of (2.1) implies τ increases with )(iaa−. Thus individuals with capital below (above) the average prefer higher (lower) tax rates.
  - If the median individual is less endowed than the average, the voting majority leads to higher taxes; initial inequality harms growth because growth is negatively affected by τ along the optimal path (perspectives: Persson and Tabellini (1994); Alesina and Rodrik (1994); Benabou (1996)).
  - The model generalizes prior work by including explicit redistributive transfers and public production and consumption services, and by incorporating implementation conditions for competitive economy responses.

### B. Special case with no consumptive expenditure (γ = 0)
- Assumption: public consumption services offer no utility (0=γ).
- Logical possibilities from (2.1):
  - Either 0=+ac ac λλ, or 0),(=θτφ, or 0)1(=−−τα.
  - The first (0)(=+ac ac λλ) is ruled out by dynamics in (2.3)–(2.6).
  - The second (0),(=θτφ) cannot occur whenever the economy grows.
  - The remaining possibility is that the economy will not grow when 0=γ, consistent with Wager’s law.
- Constant optimal tax and expenditure shares:
  - A third possibility for 0=γ is 0)1(=−−τα (i.e., ατ−=1) in each period — the socially optimal tax rate of Barro (1990) and Barro and Sala-i-Martin (1992, 2004).
  - When 0=γ and ατ−=1, Ramsey policy implies 1=θ in all time periods: using all tax revenues for public production services is optimal.
  - Constant τ and θ imply a constant return to capital ),(θτφ; then (2.4) implies no transitional growth dynamics — immediate adjustment to steady state and balanced growth path.
  - Presence of nonproductive government expenditures opens the door for transitional dynamics; Fisher’s constant rule policy ceases to be optimal.
- Extreme case: if 0=γ and public expenditures are not productive for firms (01=−α):
  - Then 0=τ in all time periods: Ramsey tax rate is zero, implying zero tax revenues and zero transfer payments, and no endogenous persistent growth (via ),(θτφ).
- Policy implication: Government finds it optimal to redistribute only when it also provides public (production and consumption) services that generate endogenous growth — statement of Wager’s law.

### C. Symmetric long-run equilibrium (representative-agent case)
- Consider the special case where individuals are alike ex post at equilibrium (symmetric Ramsey equilibrium or representative agent economy).
- In symmetric equilibrium:
  - All individuals own ex post the same amount of capital; no actual transfers occur in equilibrium.
  - Symmetry conditions invoked: aia≡)(, cic≡)(, cc iλλ≡)(, and aa iλλ≡)(.
  - Transformations Kcz≡ and a a λψ≡ reduce dynamics of (2.1)–(2.6) to the four-equation system (A.1)–(A.4) in Appendix III in variables ψ, θ, τ, z.
- Steady-state symmetric Ramsey equilibrium (balanced growth path, BGP) characterization:
  - (i) Consumption c, capital k, and asset a grow at the same constant positive rate; kcz≡ is constant or 0=•z in (A.1).
  - (ii) Policy instruments do not change; 0==•• θτ in (A.2) and (A.3).
  - (iii) Social value of capital (k a λψ≡) grows at a constant negative rate; •ψ is negative (see (A.4)).
- Modeling note:
  - The symmetric steady state follows from identical rates of time preference across individuals. Heterogeneous time preferences would yield long-run equilibria where only patient agents hold capital (references: Bewley (1982); Epstein (1987)).

*Appendix II), _wp06165 - Appendix II):*

### Appendix IV solves for the

### _wp06165 - Appendix IV solves for the

### Long-run Ramsey equilibrium: analytical results
- Appendix IV solves for the long-run levels of z, τ, θ and the long-run rate of ψ and demonstrates the existence of a balanced growth path (BGP).
- Long-run conditions (tildes denote long-run values):
  - Equation (3.1): 0
~
)1(
)
~
1(
~
~
~
~
)1()(=
−
−
++−−−+−
φα
τργ
ησ
η
γσ
ηαγα
z (stated in the source).
  - Equation (3.2): )
~
1)(1(
)
~
1(
)
~
1(
)
~
1(
θα
θα
τα
τ
−−
−−
=
−−
−
. (stated in the source).
- From (3.1): z
~
 equals ρ plus σ: 0
~
)1(
~
>++−=ρσφαz (as derived in Appendix IV), so the long-run consumption-to-capital ratio z
~
 equals the discount factor 0>ρ and the effective redistributive parameter 0>σ.
- From (A.4): 0
~~
1
~
>
⎥
⎦
⎤
⎢
⎣
⎡
+−+=
ψ
γ
η
σ
σρ
z (stated in the source); this determines the long-run rate of ψ in terms of z
~
 and parameters.

### Existence and uniqueness of optimal long-run fiscal instruments (BGP characterization)
- If parameter values satisfy 0)1()1(
11
1
<−+−
−+
−
α
α
α
α
ααγγρ, then:
  - (i) A unique optimal long-run tax rate τ
~
 exists, where 1
~
10<<−<τα.
  - (ii) A unique optimal long-run allocation of tax revenues to productive government expenditures θ
~
 exists, where 1
~
10<<−<θα.
- Properties of the system (3.1) and (3.2):
  - (3.1) and (3.2) form a two-equation system in τ
~
 and θ
~
 only; solving them yields τ
~
, θ
~
, and ψ
~
.
  - Total differentiation results imply both equations have negative slopes:
    - Total differentiation of (3.1) yields 0
)
~
1)(1(
~
]
~
)
~
1)(1[(
~
~
~
<
−−−
−−−−−
=
∂
∂
ταατ
τατααθ
τ
θ.
    - Total differentiation of (3.2) yields 0
~
~
~
~
<−=∂∂
τθτθ.
  - (3.1) is always steeper than (3.2); hence any intersection can occur at most once.
  - Boundary behavior:
    - As )1(
~
ατ−→, (3.2) implies 1
~
→θ.
    - As 1
~
→τ, (3.2) implies αθ−→1
~
.
    - When αθ−→1
~
 and 0)1()1(
11
1
<−+−
−+
−
α
α
α
α
ααγγρ, 1
~
→τ cannot satisfy (3.1), implying a unique intersection exists (illustrated in Figure 2 of the source).

### Interpretation of ψ and steady-state considerations
- If 0=
•
z in the system implied by Equations (A.1) and (A.4), then 0=
•
ψ cannot be set; ψ should grow at a constant negative rate in that case (discussion in the source).
- Intuition: in standard long-term growth models the social price of capital aλ decreases when a increases so that 0=
•
ψ. With redistributive transfers included, aλ decreases at a higher rate than a increases, implying a negative long-run •ψ in some cases.

### Empirical observations and tests (summary of IV. EMPIRICAL OBSERVATIONS)
- Empirical aims reported:
  - Test nonlinearity of exogenous fiscal policy effects on economic growth.
  - Investigate whether productive government spending is negatively associated with distortionary tax rates (predicted in Section III with endogenous policy).
  - Test negative association between economic growth and redistributive policy (inequality and redistributive spending).
  - Assess relation between consumptive spending and income inequality.
  - Investigate possible existence of multiple equilibrium paths.
- Data and variables:
  - Cross-country panel of 93 industrial and developing countries between 1990 and 2000; variables averaged over the 1990s except LGDP (initial level, 1990).
  - Control variables: LGDP (log initial GDP per capita), openness (sum of imports and exports over GDP), investment share, ICRG (rule of law / rent-seeking index), and regional dummies (East Asia, sub-Saharan Africa, Latin America).
  - Two tax rate proxies: (i) tax revenue over GDP; (ii) tax revenue minus tax income from international trade over GDP.
  - Data sources: Penn World Tables version 6.1; IRIS-3 for rent-seeking indices; World Development Indicators; IMF Government Financial Statistics (GFS) for government spending classification following Bleaney, Gemmell, and Kneller (2001).
- Growth regression specifications and interpretation (Table 1 summary; ordinary least squares):
  - Model: growth = a + b·tax + c·(tax)^2 + controls.
  - Key reported coefficients and t-ratios (exact values from Table 1):
    - Column (1) (All countries): Tax = –0.0465 (–1.10); LGDP = –1.3185** (–2.31); ICRG = 0.1657** (2.47); Sub-Saharan Africa = –1.9101** (–2.48); Constant = 6.8347** (2.06); Adj. R2 = 24.24 (%); Number of observations = 93.
    - Column (2) (Low and middle income): Tax = –0.0793 (–1.20); LGDP = –0.8285 (–1.23); ICRG = 0.2257** (2.50); Constant = 2.7899 (0.64); Adj. R2 = 26.15 (%); Number of observations = 66.
    - Column (3): Tax = 0.0008 (0.01); (Tax)^2 = –0.001 (–0.39); LGDP = –1.3456** (–2.33); ICRG = 0.1649** (2.45); Adj. R2 = 23.46 (%); Number of observations = 93.
    - Column (4) (Low and middle income, nonlinear): Tax = 0.2851 (1.43); (Tax)^2 = –0.0093* (–1.93); LGDP = –0.8735 (–1.33); Openness = –0.0203* (–1.77); ICRG = 0.2220** (2.51); Constant = 0.4646 (0.10); Adj. R2 = 29.53 (%); Number of observations = 66.
    - Column (5) (Low and middle income; tax minus trade taxes): Tax = 0.0981 (0.58); (Tax)^2 = –0.0067 (–1.45); ICRG = 0.2525** (2.93); Openness = –0.0199* (–1.87); Constant = –0.0750 (–0.02); Adj. R2 = 31.29 (%); Number of observations = 66.
  - Empirical takeaways from Table 1:
    - Linear tax term often insignificant; inclusion of quadratic term and restriction to low- and middle-income countries (column 4) produces a significant negative quadratic term, suggesting nonlinearity is more likely in low- and middle-income countries.
    - ICRG is robustly positive and significant across specifications, indicating institutional quality matters for growth.
    - Results are sensitive to tax-rate definition; replacing tax revenue/GDP with tax minus trade taxes/GDP (column 5) yields similar but weaker quadratic significance.
- Relation between tax rates and productive government spending (OECD sample exercises):
  - The model with endogenous fiscal policy predicts tax rates are negatively related to productive government spending (see (3.2)).
  - Productive spending measure (Theta) = productive government spending over total government spending minus interest payments; classification follows Bleaney, Gemmell, and Kneller (2001); tax measure is effective tax rate = income tax revenue over GDP (GFS data).
  - Figure 3 in the source shows Tax Rate vs. Fraction of Productive Government Spending (visual).
  - Table 2 (Correlations for 26 OECD countries) reports:
    - Correlation Theta(70–00) vs Distorting tax revenue (70–00): –0.4745.
    - Correlation Theta(80–00) vs Distorting tax revenue (80–00): –0.5241.
    - Correlation Theta(90–00) vs Distorting tax revenue (90–00): –0.4782.
    - Number of observations: 26, 26, 23 respectively.
  - Empirical implication: strong negative correlations between tax rates and productive spending suggest the model’s prediction that higher productive spending is associated with lower tax rates in socially optimal allocations.
- Income redistribution, growth, and inequality (OECD sample exercises):
  - Income redistribution index Sigma defined as share of consumptive government spending in social security and welfare as a percentage of GDP over the Gini coefficient; variables averaged for periods 1970–2000, 1980–2000, 1990–2000.
  - Table 3 (Correlations between growth rate and sigma for 25 OECD countries):
    - Sigma(70–00) vs Growth(70–00): –0.2510.
    - Sigma(80–00) vs Growth(80–00): –0.0759.
    - Sigma(90–00) vs Growth(90–00): –0.1105.
    - Number of observations: 25, 24, 22 respectively.
  - Table 4 (Correlations between redistributive government spending (RGS) and Gini for 25 OECD countries):
    - RGS(70–00) vs Gini(70–00): –0.6254.
    - RGS(80–00) vs Gini(80–00): 0.1715.
    - RGS(90–00) vs Gini(90–00): 0.1718.
    - Number of observations: 25, 25, 22 respectively.
  - Interpretation:
    - Negative association between income redistribution index and growth is consistent with theoretical prediction; correlations are sensitive to time period.
    - Strong negative RGS–Gini correlation for 1970–2000 (–0.6254) but weak or positive correlations for later subperiods, indicating time-period dependence and potential institutional or implementation issues.
- Multiple equilibria evidence (GDP variance vs sigma):
  - Table 5 (Correlations between sigma and GDP variance for 25 OECD countries):
    - Sigma(70–00) vs GDP variance (70–00): 0.2043.
    - Sigma(80–00) vs GDP variance (80–00): 0.1930.
    - Sigma(90–00) vs GDP variance (90–00): 0.1183.
    - Number of observations: 25, 25, 22 respectively.
  - Finding: relation positive but weak; excluding outliers (Luxembourg and Denmark) yields correlation practically zero, weakening evidence for multiple equilibria empirically.
  - Theoretical note: multiple BGPs are well-documented in endogenous growth models with externalities; multiplicity can arise when income redistribution causes externalities, inefficiency, free riding, or moral hazard.

### Main conclusions (from V. CONCLUDING REMARKS)
- With exogenous fiscal policy, a nonlinear relation exists between the suboptimal long-run growth rate and distortionary tax rates because government policy can have mixed effects (productive spending positive; high taxation, rent-seeking, crowding out negative).
- With endogenously chosen fiscal policy at a social optimum, the relation between growth and tax rates is always negative; endogenous optimality removes the increasing portion of the hump in the growth-tax relation.
- Empirical implications:
  - Interaction among fiscal policies is complex and not well captured by simple linear models using aggregate fiscal measures.
  - Nonlinearity and endogeneity of policy variables can explain ambiguous empirical findings in the literature.
  - Cross-country regressions that assume common economic structure and fundamentals across countries can lead to incorrect statistical inferences.
  - Estimation techniques need to address endogeneity and nonlinearity when evaluating fiscal policy effects on growth.

*Source: _wp06165 - Appendix IV solves for the (PDF chapter/section).*

### References

### References

### Endogenous growth theory and public capital
- Aghion, Philippe, and Peter Howitt, 1998, Endogenous Growth Theory (Cambridge, Massachusetts: MIT Press).
- Barro, Robert J., 1990, “Government Spending in a Simple Model of Endogenous Growth,” Journal of Political Economy, Vol. 98, No. 5, pp. S103–25.
- Barro, Robert J., 1991, “Economic Growth in a Cross Section of Countries,” Quarterly Journal of Economics, Vol. 106 (May), pp. 407–43.
- Barro, Robert J., 1997, Determinants of Economic Growth: A Cross-Country Empirical Study (Cambridge: Massachusetts: MIT Press).
- Barro, Robert, and Xavier Sala-i-Martin, 1992, “Public Finance in Models of Economic Growth,” Review of Economic Studies, Vol. 99, No. 4, pp. 645–61.
- Barro, Robert, and Xavier Sala-i-Martin, 2004, Economic Growth, 2nd edition (Cambridge, Massachusetts: MIT Press).
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- Benhabib, Jess, and Robert Perli, 1994, “Uniqueness and Indeterminacy: On the Dynamics of Endogenous Growth,” Journal of Economic Theory, Vol. 63, No. 1, pp. 113–42.
- Benhabib, Jess, and Aldo Rustichini, 1996, “Social Conflict and Growth,” Journal of Economic Growth, Vol. 1, No. 1, pp. 125–42.
- Futagami, Koichi, Yuichi Morita, and Akihisa Shibata, 1993, “Dynamic Analysis of an Endogenous Growth Model with Public Capital,” Scandinavian Journal of Economics, Vol. 95, No. 4, pp. 607–25.
- Jones, Charles I., 1995, “R&D-Based Models of Economic Growth,” Journal of Political Economy, Vol. 103, No. 4, pp. 759–84.
- Lucas, Robert, 1989, “On the Mechanics of Economic Development,” Journal of Monetary Economics, Vol. 22, pp. 3–42.
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- Stokey, Nancy, and Sergio Rebelo, 1993, “Growth Effects of Flat-Rate Taxes,” Journal of Political Economy, Vol. 103, No. 3, pp. 519–50.

### Fiscal policy, expenditure composition, and growth
- Alesina, Alberto, 1999, “Too Large or Too Small Governments,” in Economic Policy and Equity, ed. by V. Tanzi, K. Chu, and S. Gupta (Washington: International Monetary Fund), pp. 216–49.
- Alesina, Alberto, Silvia Ardagna, Roberto Perotti, and Fabio Schiantarelli, 2002, “Fiscal Policy, Profits, and Investment,” American Economic Review, Vol. 92, No. 3, pp. 571–89.
- Alesina, Alberto, and Roberto Perotti, 1995, “Fiscal Expansion and Fiscal Adjustments in OECD Countries,” Economic Policy, Vol. 21, pp. 205–48.
- Alesina, Alberto, and Roberto Perotti, 1997, “Fiscal Adjustment of OECD Countries: Composition and Macroeconomic Effects,” Staff Papers, International Monetary Fund, Vol. 44 (June), pp. 210–48.
- Devarajan, Shantayanan, Vinaya Swaroop, and Heng-Fu Zou, 1996, “The Composition of Public Expenditure and Economic Growth,” Journal of Monetary Economics, Vol. 37, No. 2–3, pp. 313–44.
- Giavazzi, Francesco, Tullio Jappelli, and Marco Pagano, 2000, “Searching for Non-Linear Effects of Fiscal Policy: Evidence from Industrial and Developing Countries,” European Economic Review, Vol. 44, No. 7, pp. 1259–89.
- Kneller, Richard, Michael Bleaney, and Norman Gemmell, 1999, “Fiscal Policy and Growth: Evidence from OECD Countries,” Journal of Public Economics, Vol. 74, No. 2, pp. 171–90.
- Turnovsky, Stephen, and Walter Fisher, 1995, “The Composition of Government Expenditure and Its Consequences for Macroeconomic Performance,” Journal of Economic Dynamics and Control, Vol. 19, No. 4, pp. 747–86.
- Gupta, Sanjeev, Benedict Clements, Emanuele Baldacci, and Carlos Mulas-Granados, 2002, Expenditure Composition, Fiscal Adjustment, and Growth in Low-Income Countries, IMF Working Paper 02/77 (Washington: International Monetary Fund).
- Drazen, Allan, 2000, Political Economy in Macroeconomics (Princeton, New Jersey: Princeton University Press).
- Tanzi, Vito, and Howell H. Zee, 1997, “Fiscal Policy and Long-Run Growth,” Staff Papers, International Monetary Fund, Vol. 44 (June), pp. 179–209.
- Easterly, William, and Sergio Rebelo, 1993, “Fiscal Policy and Economic Growth: An Empirical Investment,” Journal of Monetary Economics, Vol. 32, No. 3, pp. 417–58.
- Peden, Edgar, 1991, “Productivity in the United States and Its Relationship to Government Activity: An Analysis of 57 Years, 1929–86,” Public Choice, Vol. 69, No. 2, pp. 153–73.
- Peltzman, Sam, 1980, “The Growth of Government,” Journal of Law and Economics, Vol. 23, No. 2, pp. 209–88.
- Alesina, Alberto, and Dani Rodrik, 1994, “Distributive Politics and Economic Growth,” Quarterly Journal of Economics, Vol. 109, pp. 465–90.
- Bewley, Truman, 1982, “An Integration of Equilibrium Theory and Turnpike Theory,” Journal of Mathematical Economics, Vol. 10, pp. 284–306.

### Inequality, redistribution, and human capital
- Benabou, Roland, 1996, “Inequality and Growth,” NBER Macroeconomics Annual, Vol. 11, pp. 11–74.
- Benabou, Roland, 2000, “Unequal Societies: Income Distribution and Social Contract,” American Economic Review, Vol. 90, No. 1, pp. 96–129.
- Bourguignon, Francois and Christian Morrisson, 1998, “Income Equality and Development,” Journal of Development Economics, Vol. 57, pp. 233–57.
- Chu, Ke-Young, Hamid Davoodi, and Sanjeev Gupta, 2004, “Income Distribution and Tax and Government Social-Spending Policies in Developing Countries,” in Inequality, Growth, and Poverty in an Era of Liberalization and Globalization, ed. by G. Cornia (New York: Oxford University Press).
- Deininger, Klaus, and Lyn Squire, 1996, “A New Data Set Measuring Income Inequality,” The World Bank Economic Review, Vol. 10, No. 3, pp. 565–91.
- Eicher, Theo S., and Cecilia Garcia-Penalosa, 2001, “Inequality and Growth: The Dual Role of Human Capital in Development,” Journal of Development Economics, Vol. 66, pp. 173–97.
- Glomm, Gerhard, and B. Ravikumar, 1992, “Public versus Private Investment in Human Capital: Endogenous Growth and Income Inequality,” Journal of Political Economy, Vol. 100, No. 4, pp. 813–34.
- Glomm, Gerhard, and B. Ravikumar, 1997, “Productive Government Expenditures and Long-Run Growth,” Journal of Economic Dynamics and Control, Vol. 21, No. 1, pp. 183–204.
- Fernandez, Raquel, and Richard Rogerson, 1995, “On the Political Economy of Education Subsidies,” Review of Economic Studies, Vol. 62, No. 2, pp. 249–62.
- Park, Hyun, and Apostolis Philippopoulos, 2003, “On the Dynamics of Growth and Fiscal Policy with Redistributive Transfers,” Journal of Public Economics, Vol. 87, No. 3, pp. 515–38.
- Pechman, Joseph, 1985, Who Paid the Taxes, 1966–85? (Washington: Brookings Institution).

### Institutions, governance, and social capital
- Knack, Stephen, and Philip Keefer, 1996, “Institutions and Economic Performance: Cross-Country Tests Using Alternative Institutional Measures,” Economics and Politics, Vol. 7, pp. 207–27.
- Knack, Stephen, and Philip Keefer, 1977, “Does Social Capital Have an Economic Payoff? A Cross-Country Investigation,” Quarterly Journal of Economics, Vol. 112, No. 4, pp. 1251–88.
- Mauro, Paolo, 1995, “Corruption and Growth,” Quarterly Journal of Economics, Vol. 110, No. 3, pp. 681–712.
- Krusell, Per, Vincenzo Quadrini, and Jose-Victor Rios-Rull, 1997, “Politico-Economic Equilibrium and Economic Growth,” Journal of Economic Dynamics and Control, Vol. 21, No. 1, pp. 243–72.
- Slemrod, Joel, William Gale, and William Easterly, 1995, “What Do Cross-Country Studies Teach about Government Involvement, Prosperity, and Economic Growth,” Brookings Papers on Economic Activity, Vol. 2, pp. 373–431.
- Buchanan, James M., and Richard E. Wagner, 1977, Democracy in Deficit: The Political Legacy of Lord Keynes (New York: Academic Press).
- Mueller, Dennis C., 2003, Public Choice III (Cambridge, Massachusetts: Cambridge University Press).
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- Persson, Torsten, and Guido Tabellini, 1994, “Is Inequality Harmful for Growth?” American Economic Review, Vol. 84, No. 3, pp. 600–21.
- Perotti, Roberto, 1993, “Political Equilibrium, Income Distribution, and Growth,” Review of Economic Studies, Vol. 60, No. 4, pp. 755–76.

### Taxation, optimal policy, and political economy of fiscal rules
- Chamley, Christophe, 1986, “Optimal Taxation of Capital Income in General Equilibrium with Infinite Lives,” Econometrica, Vol. 54, No. 3, pp. 607–22.
- Jones, Larry, Rodolfo Manuelli, and Peter Rossi, 1993, “Optimal Taxation in Models of Endogenous Growth,” Journal of Political Economy, Vol. 101, No. 3, pp. 485–517.
- Judd, Kenneth, 1985, "Redistributive Taxation in a Perfect Foresight Model," Journal of Public Economics, Vol. 28, pp. 59–84.
- Mendoza, Enrique, Gian Maria Milesi-Ferretti, and Patrick Asea, 1997, “On the Ineffectiveness of Tax Policy in Altering Long-Run Growth: Harberger’s Superneutrality Conjecture,” Journal of Public Economics, Vol. 66, No. 1, pp. 99–126.
- Schmitt-Grohe, Stephanie, and Martin Uribe, 1997, “Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability,” Journal of Political Economy, Vol. 105, No. 5, pp. 976–1000.
- Slemrod, Joel, William Gale, and William Easterly, 1995, “What Do Cross-Country Studies Teach about Government Involvement, Prosperity, and Economic Growth,” Brookings Papers on Economic Activity, Vol. 2, pp. 373–431.
- Peden, Edgar, 1991, “Productivity in the United States and Its Relationship to Government Activity: An Analysis of 57 Years, 1929–86,” Public Choice, Vol. 69, No. 2, pp. 153–73.
- Stokey, Nancy, and Sergio Rebelo, 1993, “Growth Effects of Flat-Rate Taxes,” Journal of Political Economy, Vol. 103, No. 3, pp. 519–50.

### Methodology, data sources, and sensitivity analysis
- Cooley, Thomas, and Stephen LeRoy, 1981, “Identification and Estimation of Money Demand,” American Economic Review, Vol. 71, No. 5, pp. 825–44.
- Heston, Alan, Robert Summers, and Bettina Aten, 2002, “Penn World Table Version 6.1,” Center for International Comparisons (Philadelphia: University of Pennsylvania).
- Levine, Ross, and David Renelt, 1992, “A Sensitivity Analysis of Cross-Country Growth Regressions,” American Economic Review, Vol. 82, No. 4, pp. 942–63.
- Hansson, Par, and Magnus Henrekson, 1994, “A New Framework for Testing the Effect of Government Spending on Growth and Productivity,” Public Choice, Vol. 81, No. 3-4, pp. 381–401.

### Financial development and markets
- Greenwood, Jeremy, and Boyan Jovanonic, 1990, “Financial Development, Growth, and the Distribution of Income,” Journal of Political Economy, Vol. 98, No. 5, pp. 1076–1107.

### Classic and historical references
- Wagner, Adolph, 1883, “Finanzwissenchaft,” 3rd edition, Winter (Leipzig).
- Peltzman, Sam, 1980, “The Growth of Government,” Journal of Law and Economics, Vol. 23, No. 2, pp. 209–88.
- Pechman, Joseph, 1985, Who Paid the Taxes, 1966–85? (Washington: Brookings Institution).

*References list from the source document.*

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