## _wp0491

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---

### Research question and contribution
- Core question: Can tax reductions attract foreign capital and, at the same time, foster capital accumulation? Specifically, does a reduction in capital income taxes increase the level of capital, the rate of capital accumulation, or both?
- Contribution:
  - Two-country overlapping-generations Samuelson-Diamond model with endogenous growth, internationally mobile capital, and a public sector.
  - Incorporates technology and knowledge spillovers (learning-by-doing) to link capital mobility and growth.
  - Contrasts with short-period allocation models and infinite-lived-agent (Ramsey) models by allowing tax-induced changes in interest rates to affect saving behavior and long-run accumulation.

### Model setup — agents, markets, and taxes
- Agents and timing:
  - Two countries i = 1, 2; individuals live for two periods, work only in the first period and supply labor inelastically L_i_t.
  - Representative utility: u(c_i_1t, c_i_2t+1, L_i_t) + m(G_i_t+1), with u(·) and m(·) strictly increasing, strictly quasi-concave, twice continuously differentiable; lim_{c_i_1 → 0} u'_{c1}(·) = lim_{c_i_2 → 0} u'_{c2}(·) = +∞; lim_{G_i → 0} m'(G_i) = +∞.
- Capital and firms:
  - Capital perfectly mobile across countries; individuals immobile.
  - Firms produce a single homogeneous output combining mobile capital K_i_t+1 and local labor L_i_t+1; local productivity A_i may depend on domestic and cross-border capital (A_i(K_i_t) or A_i(K_i_t, K_j_t)).
  - Under linear homogeneity: F_i(A_i(K_i_t, K_j_t), K_i_t, L_i) = H_i(α_t) K_i_t where α_t = K_j_t / K_i_t.
  - Marginal products: r_i_t = r_i(α_t); w_i_t K_i_t = H_i(α_t) − r_i(α_t) = ω_i(α_t).
- Fiscal instruments:
  - Proportional source-based tax σ_i on income from capital invested in the country.
  - Linear residence-based tax ρ_i on residents’ income (tax on national savings), levied independently of source.
  - Repatriated return and after-tax returns:
    - β_ii_t = (1 − ρ_i_t)(1 − σ_i_t) r_i_t
    - β_ij_t = (1 − ρ_i_t)(1 − σ_j_t) r_j_t
  - With arbitrage (equalized after-residence tax returns): R_t+1 = (1 − σ_1_t+1) r_1_t+1 = (1 − σ_2_t+1) r_2_t+1.

### Individual behavior and saving
- Lifetime budget constraints:
  - w_i_t L_i = c_i_1t + S_i_t
  - S_i_t = S_ii_t + S_ij_t
  - c_i_2t+1 = [1 + (1 − ρ_i_t+1) r_i_t+1] S_ii_t + [1 + (1 − ρ_i_t+1) r_j_t+1] S_ij_t
- Consolidated present-value budget under equalized after-residence returns:
  - c_i_2t+1 = [1 + (1 − ρ_i_t+1) R_t+1] (w_i_t L_i − c_i_1t)
- First-order condition:
  - u'_{c1}(·) = u'_{c2}(·) [1 + (1 − ρ_i_t+1) R_t+1]
- Saving function (general):
  - S_i_t = ξ_i(w_i_t L_i, (1 − ρ_i_t+1) R_t+1) = ξ_i(w_i_t L_i, β_i_t+1)
  - 0 < ξ'_w_i_t < 1; sign of ξ'_{β_i_t+1} undetermined (substitution vs income effects).
- Homothetic utility special case:
  - S_i_t = s_i((1 − ρ_i_t+1) R_t+1) w_i_t L_i
  - Cobb-Douglas further implies s_i(·) constant (see Appendix I).

### Public sector and resource constraint
- Per-capita government budget (no population growth):
  - G_i_t+1 = σ_i_t+1 r_i(α_t+1) K_i_t+1 + ρ_i_t+1 r_i_t+1 S_ii_t + ρ_i_t+1 r_j_t+1 S_ij_t  (7)
  - First term: source-tax revenue; latter terms: residence-tax revenue on returns to national savings.
- Government debt ignored (equivalent to differential lump-sum taxes in overlapping-generations framework).

### Market equilibrium and accumulation mechanics
- National factor market conditions: wages and rental rates satisfy firms’ marginal productivity equalities (equation (6)).
- World capital market equilibrium (equalization of after-residence returns):
  - R_t+1 = (1 − σ_i_t+1) r_i(α_t+1) = (1 − σ_j_t+1) r_j(α_t+1)  (8)
- World investment = world saving:
  - K_t+1 = K_i_t+1 + K_j_t+1 = S_i_t + S_j_t = ξ_i(ω_i(α_t) K_i_t, (1 − ρ_i_t+1) R_t+1) + ξ_j(ω_j(α_t) K_j_t, (1 − ρ_j_t+1) R_t+1)  (9)
- Equilibrium capital ratio:
  - α_t+1 = K_j_t+1 / K_i_t+1 = α(σ_i_t+1, σ_j_t+1) (increasing in σ_i_t+1, declining in σ_j_t+1)
- Equilibrium worldwide return:
  - R_t+1 = R(σ_i_t+1, σ_j_t+1) (declining in source taxes; independent of the stock of capital)
- Law of capital accumulation (per country):
  - K_i_t+1 = (S_i_t + S_j_t) / (1 + α_t+1) = h_i(k_t, τ_i_t+1, τ_j_t+1)  (10)
  - K_j_t+1 = α_t+1 K_i_t+1 = h_j(k_t, τ_i_t+1, τ_j_t+1)
- Implications:
  - Stocks of capital in each country depend on both countries’ tax rates.
  - Changes in one country’s tax rates influence capital accumulation at home and abroad.
  - Existence/uniqueness of equilibrium accumulation path not guaranteed without further restrictions; Appendix I provides a Cobb-Douglas case with existence.

### Static (momentary) vs dynamic (intertemporal) effects
- Momentary equilibrium (single-period allocation) properties:
  - Given K_t and tax rates, the allocation K_i_t, K_j_t solves arbitrage condition (8): K_j_t / K_i_t = α(σ_i_t, σ_j_t).
  - Allocation depends on source taxes σ_i_t, σ_j_t but not on residence taxes ρ_i_t.
  - Static relocation effect: source-tax differentials reallocate existing world capital across countries in one period.
- Intertemporal equilibrium and growth:
  - Sequence of momentary equilibria with constant policies can generate a balanced growth path: K_i_{t+1} = (1 + g_i) K_i_t where g_i = g(τ_i, τ_j).
  - Growth rates depend on domestic and foreign tax rates — domestic taxes create a growth externality affecting both countries.
  - Static allocation and growth effects can move in different directions: a source tax cut can raise current domestic capital share while reducing saving and future growth depending on saving elasticity.

### Effects of residence-based taxes
- Residence taxes:
  - Do not affect the momentary allocation α (α = α(σ_i, σ_j)).
  - Affect growth via changes in β_i_{t+1} = (1 − ρ_i) R_{t+1} and thereby saving propensities s(·).
  - The sign of the effect of an increase in ρ_i on growth can be zero, positive, or negative depending on the elasticity of substitution (expenditure elasticity) — if elasticity < 1, higher residence taxes can boost growth.
  - Empirical evidence cited suggests elasticities of substitution are generally less than unity, making it empirically possible that increases in residence taxation may lead to faster growth while leaving α unchanged.

### Effects of source-based taxes (σ_i)
- Source taxes affect both:
  - The static allocation α(σ_i, σ_j) (direct relocation of existing capital).
  - Saving propensity s(·) indirectly via β and hence accumulation and growth.
- Cobb-Douglas special case (Appendix I) yields clear expressions:
  - Equilibrium α_t+1 = [ η_j (1−σ_j t+1) (Lj)^θ / ( η_i (1−σ_i t+1) (Li)^θ ) ]^{1/(2μ∗)}
  - World return R_t+1 = ² [ ηi (Li)^θ (1−σi t+1) ]^{1/2} [ ηj (Lj)^θ (1−σj t+1) ]^{1/2}
  - Under Cobb-Douglas homothetic utility s constant, growth rates depend on α and hence on source taxes.
- Non-monotonicity (Laffer-curve type):
  - There exists a national source-tax rate σ_i^∗ that maximizes the rate of economic growth given parameters.
  - Lower source taxes increase domestic capital share but do not necessarily raise growth; moving away from σ_i^∗ (including lowering taxes) can reduce growth at home and abroad.
- Distinction between domestic product and welfare:
  - A policy lowering source taxes can raise domestic product growth but reduce domestic saving and future national income (residents’ claims on product), lowering welfare.

### Comparison with infinite-lived Ramsey models
- Ramsey/infinite-life models predict capital taxes reduce net returns and the rate of growth with no long-run international effects because net-of-tax return equals time preference in steady state.
- Overlapping-generations framework here allows net-of-tax interest rates to differ from time preference; therefore, tax changes affect interest rates, saving, international capital allocation, and growth — generating persistent international interdependencies absent in Ramsey models.

### Policy-relevant conclusions and implications
- Distinguish policy objectives:
  - Maximizing domestic product growth is not equivalent to maximizing national income or welfare.
- International tax interdependencies:
  - Governments can use capital income taxes to compete for existing world capital (source-tax competition) and to influence accumulation and growth.
  - Pareto-optimal tax rates depend on trade-offs between static allocation benefits and dynamic accumulation effects.
- Key cautions:
  - A reduction in source-based taxes that raises current domestic share of world capital may lower domestic saving and future welfare.
  - Residence-based tax changes can influence growth without redistributing the existing stock of capital internationally.
  - Empirical magnitudes (e.g., elasticity of saving to after-tax returns, expenditure elasticities) critically shape policy outcomes.

### Appendix I — Cobb-Douglas example (existence and closed-form expressions)
- Consumer utility: ui = a log ci1t + (1−a) log ci2t+1 + b log(T − Li t) + γ log Gi t+1, with s = 1 − a.
  - Optimal saving: Si t = s wi t Li; ci1t = (1 − s) wi t Li.
- Production: Yi t = ηi (Li)^θ α μ∗ t Ki t with μ + μ∗ = 1 − ².
  - Marginal products: ri t = ² ηi (Li)^θ α μ∗ t; wi t = (1 − ²) ηi (Li)^θ α μ∗ t Ki t.
  - National savings: Si t = s(1 − ²) ηi (Li)^{1+θ} α μ∗ t Ki t.
- Equilibrium:
  - αt+1 closed form in σ_i, σ_j (A-4); R_t+1 closed form in σ_i, σ_j (declining in source taxes).
  - Accumulation equations yield Ki_{t+1} / Ki_t = 1 + g_i = s(1 − ²)(L)^{1+θ} [ α^{μ∗}_t + α^{1−μ∗}_t / (1 + α_t+1) ] in symmetric-normalized case.
- Existence of balanced growth path requires parameter restriction stated in Appendix I.

*Source: _wp0491 - References and Section V, Appendix I.*

### References .............................................................................................................

### _wp0491 - References .............................................................................................................

### Major elements listed in the unit
- References ............................................................................................................................. 26
- Figure
  - 1.  Momentary Equilibrium ............................................................................................15
- Page markers appearing in the extracted content: -3- ; -4-

### Introduction: scope and research question
- Many countries reduce capital income taxes to attract foreign capital and foster economic growth; examples cited include reductions in capital income taxes in the European Union and recent U.S. debate on corporate taxation.
- The core question: can tax reductions attract foreign capital and, at the same time, foster capital accumulation? Specifically, does a reduction in capital income taxes increase the level of capital, the rate of capital accumulation, or both?

### Literature gap and contribution
- Existing theoretical analyses typically:
  - Rely on short-period models focusing on the allocation of a given stock of capital and ignore long-term accumulation (examples: Gordon, 1983; Zodrow and Mieszkowski, 1986; Razin and Sadka, 1991; Wildasin 1991; Wilson 1986, 1991; Bucovetsky and Wilson, 1991).
  - When accumulation is considered, analysis often limits to small economies (Razin and Yuen, 1993, 1999; Turnovsky, 1996; Asea and Turnovsky, 1998) or to economies with no economic growth (Bovenberg 1986; Sibert, 1990; Ghosh 1991; Ha and Sibert, 1997).
- This paper examines international effects of domestic capital taxation on allocation and accumulation in an integrated world with large economies via:
  - A two-country model of endogenous growth with internationally mobile capital and tax decisions in one country affecting growth in the other.
  - Explicit incorporation of interdependence between capital mobility and growth through technology and knowledge spillovers, extending a learning-by-doing model to open economies (citing Bertola, 1993; Bernstein and Mohnen, 1998; Bayoumi, Coe, and Helpman, 1999).

### Model features and methodological differences
- The model is a two-country Samuelson-Diamond overlapping-generations framework (as in Buiter, 1981) that:
  - Incorporates a public sector and economic growth.
  - Differs from infinite-life-agent models (Razin and Yuen, 1996; Lejour and Verbon, 1997) because overlapping-generations allows first-period consumption to differ from second-period consumption and so net-of-tax interest rate need not equal the rate of time preference.
- Implication of model choice:
  - Changes in domestic tax rates affect the interest rate and individual saving behavior in each country, thereby influencing international allocation of capital and economic growth in different ways than in infinite-life-agent models.

### Main analytical insights and findings
- Taxes affect international allocation of capital, but the static effect (allocation of an existing stock) is not simply repeated generation after generation because:
  - Changes in capital income taxes also affect the rate of growth of capital.
  - The static allocation effect and the growth effect may not move in the same direction.
- Illustrative case:
  - A country that reduces source-based taxes can increase its share of current world capital, but depending on the elasticity of saving to net-of-tax returns, this may reduce its saving level and lower its economic growth rate; this outcome has some empirical support (e.g., Berheim, 2002).
- Distinction between output and welfare:
  - Output (domestic product) effects of tax policy and welfare (residents’ claims on product, national income) effects need not coincide.
  - A policy that raises domestic productivity and domestic product growth by lowering taxes can nevertheless reduce domestic saving and lower future claims of residents on product, thereby reducing welfare.

### Policy-relevant implications highlighted
- Source- versus residence-based capital taxation have different international effects; the paper examines both in separate sections (Sections III and IV).
- Policymakers focusing on boosting domestic product growth by cutting taxes should recognize potential trade-offs with national income and welfare due to effects on saving and cross-border capital allocation.

### Model organization (as described)
- Section II: Formal setup of the basic model of growth (two-country overlapping-generations Samuelson-Diamond type with public sector and growth).
- Sections III and IV: Examination of international effects of domestic source- and residence-based capital taxes.

### Representative citation and quoted finding preserved verbatim
- From Lejour and Verbon (1997): “...a higher domestic source-based capital-income tax will generate a lower growth rate of wealth” (p. 485)

*Document: _wp0491 - References .............................................................................................................*

### Section V summarizes the results and concludes.

### _wp0491 - Section V summarizes the results and concludes.

### The model (overview)
- Two-country overlapping generations model (Samuelson-Diamond type) with countries i = 1, 2.
- Individuals live for two periods; work only in the first period and supply labor inelastically L_i_t, earning wage rate w_i_t and wage income w_i_t L_i_t.
- Capital is perfectly mobile across countries; individuals are immobile.
- Firms combine internationally mobile capital K_i_t+1 with local labor L_i_t+1 to produce a single homogeneous output; output can be consumed privately or used by government to provide public good G_i_t+1 per head.
- Governments levy two capital-income taxes:
  - Proportional source-based tax σ_i on income from capital invested in the country.
  - Linear residence-based tax ρ_i on residents’ income independently of source (tax on national savings).
- Generation-t faces domestic tax vectors τ_i_t+1 = (ρ_i_t+1, σ_i_t+1).
- Repatriated return to capital invested in country i: r_i_t = (1 − σ_i_t) r_i_t, where r_i_t denotes firms’ marginal return to capital.
- After-tax returns accruing to national savings invested at home and abroad:
  - β_ii_t = (1 − ρ_i_t) r_i_t = (1 − ρ_i_t)(1 − σ_i_t) r_i_t
  - β_ij_t = (1 − ρ_i_t) r_j_t = (1 − ρ_i_t)(1 − σ_j_t) r_j_t
  - for i, j = 1, 2 and i ≠ j.

### A. Individuals (behavior and saving)
- Representative utility for generation born at t:
  - u(c_i_1t, c_i_2t+1, L_i_t) + m(G_i_t+1)
  - c_i_1t first-period consumption; L_i_t = L_i; c_i_2t+1 second-period consumption; G_i_t+1 public good.
  - u(·) and m(·) strictly increasing, strictly quasi-concave, twice continuously differentiable; lim_{c_i_1 → 0} u'_{c1}(·) = lim_{c_i_2 → 0} u'_{c2}(·) = +∞; lim_{G_i → 0} m'(G_i) = +∞. Public goods enter separably.
- Lifetime budget constraints:
  - w_i_t L_i = c_i_1t + S_i_t
  - S_i_t = S_ii_t + S_ij_t
  - c_i_2t+1 = [1 + (1 − ρ_i_t+1) r_i_t+1] S_ii_t + [1 + (1 − ρ_i_t+1) r_j_t+1] S_ij_t  (i ≠ j)
- With capital mobility and equilibrium equalization of after-residence tax returns (β_ii_t+1 = β_ij_t+1), r_1_t+1 = r_2_t+1 and:
  - R_t+1 = (1 − σ_1_t+1) r_1_t+1 = (1 − σ_2_t+1) r_2_t+1  (1)
  - R_t+1 is the economy-wide return to capital before residence tax and depends on both countries’ source taxes.
- Consolidated present-value budget constraint (individual indifferent to domestic vs. foreign allocation of saving):
  - c_i_2t+1 = [1 + (1 − ρ_i_t+1) R_t+1] (w_i_t L_i − c_i_1t)  (2)
- First-order condition for consumption choice:
  - u'_{c1}(·) = u'_{c2}(·) [1 + (1 − ρ_i_t+1) R_t+1]  (3)
- Saving function:
  - S_i_t = ξ_i(w_i_t L_i, (1 − ρ_i_t+1) R_t+1) = ξ_i(w_i_t L_i, β_i_t+1)  (4)
  - 0 < ξ'_w_i_t < 1; sign of ξ'_{β_i_t+1} is undetermined (substitution vs income effects).
- Homothetic utility special case:
  - S_i_t = s_i((1 − ρ_i_t+1) R_t+1) w_i_t L_i  (5)
  - Cobb-Douglas further simplifies s_i(·) to a constant (see Appendix I).

### B. Firms (production, returns, spillovers)
- Production function in country i: F_i(A_i, K_i_t, L_i); twice continuously differentiable; decreasing returns in capital and labor; local productivity factor A_i.
- Productivity incorporates national and cross-border spillovers:
  - A_i = A_i(K_i_t) (domestic spillovers) or A_i(K_i_t, K_j_t) (cross-border spillovers).
- Under linear homogeneity assumptions, output can be represented as:
  - F_i(A_i(K_i_t, K_j_t), K_i_t, L_i) = f_i(a_i(K_j_t / K_i_t)) K_i_t = H_i(α_t) K_i_t
  - α_t = K_j_t / K_i_t (i ≠ j)
- Competitive firms pay marginal products:
  - r_i_t = F'_{ik_t}|_{A_i} = r_i(α_t)
  - w_i_t K_i_t = H_i(α_t) − r_i(α_t) = ω_i(α_t)  (6)
- Comparative statics:
  - r'_i(α_t) > 0
  - ω'_i(α_t) = H'_iα(α_t) − r'_i(α_t) ≶ 0 (ambiguous)
  - Potential intergenerational conflicts: shifting α_t can raise private returns to capital while ambiguously affecting wages.
- Cobb-Douglas example (restrictive case):
  - Y_i_t = η_i^{1/(1−²_i)} (K_j_t / K_i_t)^{μ^*} K_i_t with parameters satisfying μ + μ^* = 1 − ².
  - r_i_t = ² η_i (L_i)^θ α_t^{μ^*}
  - w_i_t = (1 − ²) η_i (L_i)^θ α_t^{μ^*} K_i_t

### C. Public sector (revenues and financing)
- Government revenue constraint (per capita, no population growth, one unit output → one unit public good):
  - G_i_t+1 = σ_i_t+1 r_i(α_t+1) K_i_t+1 + ρ_i_t+1 r_i_t+1 S_ii_t + ρ_i_t+1 r_j_t+1 S_ij_t  (7)
  - First term: revenue from source-based tax σ_i_t+1; other terms: residence-tax revenue on returns to national savings invested at home and abroad.
- Government debt is ignored; in overlapping generations models debt is equivalent to differential lump-sum taxes (see Diamond (1973) and Atkinson and Sandmo (1980) remarks cited).

### D. Market equilibria and accumulation
- National factor market equilibrium conditions are given by (6) (wage and rental rate such that firms use available labor and capital services).
- World capital market equilibrium requires equality of after-residence tax returns across countries (condition (1) restated as):
  - R_t+1 = (1 − σ_i_t+1) r_i(α_t+1) = (1 − σ_j_t+1) r_j(α_t+1)  (8)
- World investment equals world saving:
  - K_t+1 = K_i_t+1 + K_j_t+1 = S_i_t + S_j_t  (9)
  - = ξ_i(ω_i(α_t) K_i_t, (1 − ρ_i_t+1) R_t+1) + ξ_j(ω_j(α_t) K_j_t, (1 − ρ_j_t+1) R_t+1)
- Equilibrium capital ratio α_t+1 determined by source taxes:
  - α_t+1 = K_j_t+1 / K_i_t+1 = α(σ_i_t+1, σ_j_t+1)  (increasing in σ_i_t+1, declining in σ_j_t+1)
- Equilibrium worldwide return:
  - R_t+1 = R(σ_i_t+1, σ_j_t+1)  (declining in source taxes; independent of the stock of capital)
- Law of capital accumulation (per country), using α_t+1 and R_t+1:
  - K_i_t+1 = K_t+1 / (1 + α_t+1) = (S_i_t + S_j_t) / (1 + α_t+1) = h_i(k_t, τ_i_t+1, τ_j_t+1)  (10)
  - K_j_t+1 = α_t+1 K_i_t+1 = α_t+1 (S_i_t + S_j_t) / (1 + α_t+1) = h_j(k_t, τ_i_t+1, τ_j_t+1)
  - τ_i_t+1 = (ρ_i_t+1, σ_i_t+1); k_t = (K_i_t, K_j_t) inherited stocks of capital.
- Key implications from accumulation equations:
  - Stocks of capital in each country depend on both countries’ tax rates.
  - Changes in one country’s tax rates influence capital accumulation at home and abroad.
  - Model does not guarantee existence or uniqueness of equilibrium accumulation path without further restrictions; subsequent analysis assumes a unique equilibrium with positive capital.

*Source: _wp0491 - Section V summarizes the results and concludes.*

### Appendix I presents a Cobb-Douglas version of this model where an equilibrium does exist.

### _wp0491 - Appendix I presents a Cobb-Douglas version of this model where an equilibrium does exist.

### III. Tax interdependencies in open economies — key mechanisms
- Domestic taxation of capital affects the international allocation of the existing stock of world capital.
- Domestic taxes affect international growth and capital accumulation over time by changing saving both at home and abroad.
- Distinction emphasized between:
  - Domestic product (political focus) and
  - National income (claims of residents on world product; welfare focus).
- Domestic taxation of capital has international distributional effects through changes in the international allocation of capital α_t and hence in factor prices. Example logic:
  - A higher domestic source-based tax in country i → relatively smaller stock of home capital (the ratio α_t decreases).
  - If ω_i0 α_t (α_t) < 0, higher domestic wages may result at home raising welfare of currently young at home while lowering welfare of young abroad via tax-generated capital inflows lowering their wages.
- Reference literature for existence and uniqueness of equilibria in overlapping generations models: Blanchard and Fisher (1989), Azariadis (1993); applications to open economies and endogenous growth: Buiter (1981), Buiter and Kletzer (1992).

*Findings enumerated in the text:*
- Static (one-period) and dynamic (intertemporal) effects of taxes are distinct and can lead to different assessments of tax policy.
- Residence taxes do not affect the momentary equilibrium allocation of the existing stock of capital; source-based taxes do.
- The static capital relocation effect (capital flows in response to source-tax differentials) is a pure single-period effect and does not depend on capital accumulation and economic growth.

### IV. The effect of taxation in a two-country world — momentary equilibria and allocation of capital
- Momentary (single-period) equilibrium defined as the international allocation of capital that, given tax rates, implies equilibrium in both countries’ national markets; characterized by prices {w_i_t, r_i_t} and consumption-savings allocations given fiscal policies {ρ_i_t, σ_i_t, G_i_t} and the inherited aggregate capital K_t.
- In momentary equilibrium:
  - Aggregate stock K_t is split between countries: K_t = K_i_t + K_j_t and K_j_t / K_i_t = α(σ_i_t, σ_j_t) (from arbitrage condition (8)), i, j = 1,2, i ≠ j.
  - Equilibrium capital in country i can be written K_i_t = d_i(σ_i_t, σ_j_t, K_t), i, j =1,2, i ≠ j.
  - The allocation {K_i_t, K_j_t} depends on source-based tax rates σ_i_t and σ_j_t, but not on residence taxation.
- Comparative static result for symmetric countries (differentiating condition (8) using (6)):
  - dK_j_t / dσ_i_t = − dK_i_t / dσ_i_t = − F'_i k_i_t | A_i^2 (1−σ_i_t) (F'_i k_i_t | A_i)́0 k > 0, i, j =1,2, i ≠ j.
  - As noted in text: by properties of production functions, (F'_i k_i_t | A_i)́0 k < 0.
  - Interpretation: In equilibrium, the amount of capital in each country is a declining function of the domestic source tax, but an increasing function of the other country’s tax rate.
- Cobb-Douglas example (see Appendix I):
  - Net returns expressed as r_1_t = ² η_1 (L_1)^θ ( (1−χ)/χ )^μ ∗ (1−σ_1_t) and r_2_t = ² η_2 (L_2)^θ ( χ/(1−χ) )^μ ∗ (1−σ_2_t) where χ = K_1 / K.
  - Figure 1 depiction: equilibrium when r_1_t = r_2_t at intersection I. An increase in domestic source tax (country 1) shifts its net-return curve down, reducing domestic capital share χ and lowering the international interest rate.
- Special case with Cobb-Douglas with no externalities and no economic growth (μ = μ^∗ = 0 and 0 < ² < 1):
  - An international equilibrium exists; changes in domestic tax rates reallocate capital toward the tax-reducing country (static relocation effect).

### IV.B Intertemporal equilibria, economic growth, and welfare — dynamics and growth externalities
- Intertemporal equilibrium: sequence of momentary equilibria in which national consumption profiles steadily grow given sequences of fiscal policies {ρ_i_t, σ_i_t, G_i_t}_t=0^∞.
- Using linear homogeneity of production and constant national policies: Y_1_t = F_1(A_1(K_1_t, K_2_t), K_1_t) = H_1(α) K_1_t and Y_2_t = F_2(A_2(K_1_t, K_2_t), K_2_t) = H_2(α) K_1_t. Domestic product in the two countries thus grows at a common rate, but national income may grow at different rates depending on domestic savings.
- Law of motion of capital (accumulation equations (10)):
  - K_i_{t+1} = h_i(K_i_t, K_j_t, τ_i, τ_j), i, j =1,2, i ≠ j where τ_i = (ρ_i, σ_i) and τ_j = (ρ_j, σ_j).
  - With K_j_t = α K_i_t and well-behaved utility functions, dynamics can be expressed as:
    - K_i_{t+1} = (1 + g_i) K_i_t, i =1,2 (equation (11)) where g_i = g(τ_i, τ_j) is constant (for an example see Appendix I).
- Key conceptual result: the growth rate in each country depends on domestic and foreign tax rates; domestic taxes create an intertemporal or “growth” externality.
- Caveat from text: model assumptions (perfect capital mobility, no transitional dynamics) imply strong international symmetry — changes in one country’s taxes have as strong an effect on domestic growth as on foreign growth. Introducing transitional dynamics and imperfect capital mobility would lead to differentiated country effects.
- Cobb-Douglas symmetric example with homothetic utility (derivation toward explicit growth expression):
  - Savings expressions (equation (5) specialization):
    - S_i_t = s(β_i_{t+1}) (1−²) η_i (L_i)^{1+θ} α μ^∗ K_i_t
    - S_j_t = s(β_j_{t+1}) (1−²) η_j (L_j)^{1+θ} μ (1/α) μ^∗ α K_i_t
    - where β_i_{t+1} = (1−ρ_i) R_{t+1}.
  - Substituting into (10) yields a simple expression for growth:
    - K_i_{t+1} / K_i_t = 1 + g_i = (1−²) s(β_i_{t+1}) (L)^{1+θ} μ α μ^∗ + α^{1−μ^∗} 1+α ¶ (equation (12))
    - For convenience η_i, η_j normalized to one and L_i = L_j = L.
- Notes from text:
  - For steady state existence, preferences must be homothetic (Buiter and Kletzer (1992)).
  - Appendix I provides a Cobb-Douglas example where a unique equilibrium exists.

*Source: _wp0491 - Appendix I presents a Cobb-Douglas version of this model where an equilibrium does exist.*

### appendix I. In this example, conditions on the parameters of the model can be found such that a

### _wp0491 - appendix I. In this example, conditions on the parameters of the model can be found such that a

### Effects of residence-based taxes on growth and capital allocation
- Residence taxes have no effect on the international allocation of capital α as α(σi,σj).
- Residence taxes affect the rate of growth through changes in the saving propensity factors {βi t+1}.
- The impact of an increase in residence taxes on the national (and foreign and international) rate of growth can be zero, positive or negative depending on whether the elasticity of substitution at home is equal to, smaller than, or greater than one (one being the value of the expenditure elasticity for homothetic utility functions).
- Higher residence taxes change intergenerational saving behavior and, if the elasticity of substitution is smaller than one, they can eventually boost economic growth.
- Empirical evidence cited in the paper suggests elasticities of substitution are generally less than unity, implying the empirical possibility that increases in residence taxation may lead to faster growth while leaving the international allocation of capital unchanged.

### Effects of source-based taxes (σi) — ambiguity and Cobb-Douglas special case
- Source taxes affect growth through two channels:
  - They influence the propensity to save s(·).
  - They affect the international allocation of capital α(·).
- In the special case of Cobb-Douglas utility functions:
  - s(·) becomes a constant s = 1 − a (propensity to save wage income).
  - National growth rates simplify to:
    - Ki t+1 / Ki t = 1 + gi = s(1−²)(L)1+θ [ α μ∗ + α 1−μ∗ / (1+α) ]
    - where α μ∗ = μ (1−σj t) (1−σi t) ¶1/2 (as defined in Appendix I) with i, j = 1,2 and i ≠ j.
  - Under Cobb-Douglas, gi is independent of residence-based tax rates ρi but depends on source taxation through α(σi,σj).
- There exists a national source-tax (or subsidy) rate σi∗ which maximizes the rate of economic growth given model parameters.
- Relation between source taxes and growth is non-monotonic (Laffer curve-type): a unique international allocation of capital maximizes growth externalities and requires specific national source tax rates.
- Lower source taxes increase the domestic stock of capital but do not necessarily raise the rate of economic growth; growth depends on the cross-country allocation of capital.
- Any change away from the growth-maximizing tax rate (including lower taxes or higher subsidies) may increase the domestic share of world capital but reduce the growth rate at home and abroad.
- A tax change can have different effects on domestic product and on national income (welfare). Example: with symmetric countries but country i levying a lower source tax than country j, an increase in country i’s source tax could raise domestic product growth but lower domestic saving and hence reduce national income under specific parameterizations.

### Comparison with infinite-lived Ramsey models
- Results differ from infinite-life Ramsey models (Barro and Sala-I-Martin; Lejour and Verbon) which predict positively correlated movements in r (net rate of return) and γ (rate of economic growth): capital taxes reduce net returns and the rate of growth.
- In Ramsey models:
  - The modified “golden rule” implies in the stationary state the net-of-tax interest rate in each country equals the rate of time preference θi:
    - βii t = (1−ρi t)(1−σi t) r i t = θi = 1,2
  - Domestic taxes have no long-run international effect because long-run equilibrium capital is determined by the modified golden rule (time preferences limit cross-country tax and growth interdependences).
  - Changes in σi t affect net-of-tax home returns and cause short-term capital movements, but the long-run equilibrium capital levels abroad remain unchanged.
- Overlapping-generations model in this paper differs:
  - First-period consumption differs from second-period consumption; net-of-tax interest rate need not equal the rate of time preference.
  - Changes in domestic tax rates affect interest rates and saving behavior, influencing international capital allocation and economic growth.
  - Effects on growth need not coincide with effects on capital allocation.

### Key conclusions and policy implications
- Distinguish between effects of domestic capital-income taxes on:
  - International allocation of capital.
  - Rate of economic growth.
- Increasing domestic source taxes reduces the stock of domestic capital, but taxation also influences capital accumulation via saving behavior; these effects can diverge.
- A tax policy that increases the domestic share of current world capital may not increase the future growth rate of that capital.
- Assessment of tax policy depends critically on the objective (e.g., maximizing domestic product vs. maximizing national income/welfare).
- International tax interdependencies complicate national policy design: governments can use capital income taxes both to compete for existing world capital and to affect capital accumulation over time; Pareto-optimal tax rates depend on these trade-offs.

### Appendix I — Two-country model with Cobb-Douglas utility and production
- Consumers:
  - Representative consumer utility:
    - ui = a log ci1t + (1−a) log ci2t+1 + b log(T − Li t) + γ log Gi t+1
  - Time endowment T, labor Li t = Li, and Gi t+1 are given; consumption choices are ci1t and ci2t+1.
  - Budget constraint yields optimal consumptions and savings:
    - ci1t = a wi t Li = (1−s) wi t Li
    - Si t = (1−a) wi t Li = s wi t Li
    - ci2t+1 = [1 + (1−ρi t+1) Rt+1] s wi t Li
    - where s = 1 − a.
  - Saving is linear in wage income; population normalized to one (no population growth).
- Firms:
  - Production function:
    - Yi t = [Ai]1/(1−²) (Ki t)² (Li)θ with ² > 0, θ < 1 and i = 1,2
  - Productivity factor:
    - Ai = ηi (Ki t)μ (Kj t)μ∗ with μ + μ∗ = 1 − ² and μ, μ∗, ηi > 0.
  - Aggregate expressions:
    - Yi t = ηi (Li)θ α μ∗ t Ki t
    - Yj t = ηj (Lj)θ α −μ∗ t Kj t
    - where αt = Kj t / Ki t, i ≠ j.
  - Factor prices (marginal products):
    - ri t = ² ηi (Li)θ α μ∗ t
    - rj t = ² ηj (Lj)θ α −μ∗ t
    - wi t = (1−²) ηi (Li)θ α μ∗ t Ki t
    - wj t = (1−²) ηj (Lj)θ α −μ∗ t Kj t
  - National savings:
    - Si t = s(1−²) ηi (Li)1+θ α μ∗ t Ki t
    - Sj t = s(1−²) ηj (Lj)1+θ α −μ∗ t Kj t
- World capital market equilibria and accumulation:
  - Equalization of net rates of return condition (A-2):
    - Rt+1 = ri t+1 = (1−σi t+1) ² ηi (Li)θ α μ∗ t+1 = (1−σj t+1) ² ηj (Lj)θ α −μ∗ t+1 = rj t+1
  - World savings equals next-period capital stocks (A-3):
    - Ki t+1 + Kj t+1 = Si t + Sj t
  - Equilibrium capital ratio (A-4):
    - αt+1 = Kj t+1 / Ki t+1 = [ ηj (1−σj t+1) (Lj)θ / ( ηi (1−σi t+1) (Li)θ ) ]1/(2μ∗) = α(σi t+1, σj t+1)
    - An increase in country i’s source tax (σi) with σj constant shifts capital to country j (raises αt+1).
  - Equilibrium world return to capital:
    - Rt+1 = ² [ ηi (Li)θ (1−σi t+1) ]1/2 [ ηj (Lj)θ (1−σj t+1) ]1/2
    - World-wide return depends only on national source taxes; an increase in national taxes lowers the equilibrium return.
  - Accumulation equations (symmetric normalization ηi = ηj = 1, Li = Lj = L):
    - Ki t+1 = s(1−²) / (1+αt+1) [ ηi (Li)1+θ α μ∗ t Ki t + ηj (Lj)1+θ α −μ∗ t Kj t ]
    - Kj t+1 = αt+1 s(1−²) / (1+αt+1) [ ηi (Li)1+θ α μ∗ t Ki t + ηj (Lj)1+θ α −μ∗ t Kj t ]
    - In symmetric constant-identical-tax case αt+1 = αt = 1 and gi = gj = g, where:
      - Ki t+1 / Ki t = 1 + gi = s(1−²)(L)1+θ [ α μ∗ t + α 1−μ∗ t / (1+αt+1) ]
  - Existence of balanced growth path requires parameter restriction:
    - For symmetric countries positive growth g > 0 if s(1−²)(Li)1+θ > (1+αt+1) / ( α μ∗ t + α 1−μ∗ t ). The paper assumes this holds.

*Source: _wp0491 - appendix I. In this example, conditions on the parameters of the model can be found such that a*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2004/_wp0491.pdf_