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---

### Box 4.3 — Summary of conceptual findings
- Strategic complementarity in corporate tax setting is the dominant expectation: empirical and theoretical evidence suggests that a one percentage point cut in the average foreign statutory rate typically induces a positive response at home (examples reported: between 0.34 and 0.67 percentage points in advanced-country samples; between 0.25 and 0.3 points when developing countries are included).
- In a Stackelberg (sequential-move) context, the low tax country never gains from being the follower: the leader can use a minimum to induce the other to reduce its rate, benefiting the leader and harming the low tax country. If the low tax country is the leader, introducing a formal minimum cannot make it better off.
- Repeated-game dynamics can complicate outcomes: imposing a minimum may reduce the scope for sustaining cooperative outcomes by lowering punishment for defection.
- Overall welfare effects of a minimum corporate tax rate are ambiguous in general: depending on modeling assumptions, a modest minimum could benefit both high- and low-tax countries, only one, or neither. Yet the balance of evidence on strategic complementarity and profit shifting suggests a real possibility that both may benefit from a modest minimum.

### Model setup and key equations (tax competition with profit shifting)
- Representative multinational profits:
  - Low-tax country profit: π (strictly positive).
  - High-tax country profit: Π (strictly positive).
- Cost of shifting a proportion s of profit abroad: (δ/2) s^2.
- Optimized shifting out of the high tax country: amount (T−t)Π/δ.
- Welfare in the low tax country (recipient of shifted profit):
  - w(t,T) = (1−t) π + λ t ( π + (T−t)/δ Π ). (equation (2))
  - Best response of the low-tax country:
    - t(T) = 1/2 [ ( (λ−1)/λ ) δ θ + T ], where θ ≡ π/Π. (equation (3))
- Welfare in the high tax country (donor of shifted profit):
  - Baseline combined expression:
    - W(T,t) = (1−T) Π + (1/2) (T−t)^2 / δ Π + Λ T ( 1 − (T−t)/δ ) Π. (equation (5))
  - Generalized with social valuation parameter α ∈ [0, 1/2]:
    - W(T,t) = (1−T) Π + α (T−t)^2 / δ Π + Λ T ( 1 − (T−t)/δ ) Π. (equation (6))
    - Interpretation: α = 1/2 corresponds to full social valuation of the net private gain from shifting; α = 0 corresponds to zero social valuation.
- High-tax country’s best response:
  - T(t) = 1/2 [ ( (Λ−1)/(Λ−α) ) δ + ( (Λ−2α)/(Λ−α) ) t ]. (equation (7))
  - Slope T′(t) = (Λ−2α)/2(Λ−α) > 0; hence strategic complementarity for the high-tax country given Λ > 2α.

### Nash equilibrium characterization and comparative statics
- Existence and uniqueness:
  - If (Λ−1)/Λ > (λ−1)/λ θ, then there is a unique Nash equilibrium with t_N < T_N. (Proposition 1; condition (8))
  - Interpretation: the country setting the lower tax rate is relatively small in global profits (small θ) and/or has relatively low marginal valuation of public spending (λ).
- Nash rates (explicit expressions):
  - t_N = δ ( Λ−α )/( 3Λ−2α ) [ ( (λ−1)/λ ) 2 θ + ( (Λ−1)/(Λ−α) ) ]. (equation (9))
  - T_N = T(t_N).
- Comparative statics:
  - Both t_N and T_N increase with Λ and λ.
  - Both t_N and T_N decrease with θ.
  - Both t_N and T_N increase with α.
- Role of α:
  - A higher α shifts the high-tax country’s best response upwards but flattens it (reduces responsiveness), damping the low-tax country’s gain from a minimum-induced increase in t.

### Policy implications and interpretation
- A discretely binding minimum tax rate μ has no effect until it reaches the lower Nash tax rate t_N; beyond that point the low-tax country’s rate will be μ.
- The high-tax country is unambiguously better off when minima force an increase in the low-tax rate (from ∂W/∂t . (δ/Π) = (Λ−2α) T + 2α t > 0): minima that increase t raise the high-tax country’s welfare.
- The scope for Pareto-improving minima depends on Λ, λ, θ, α, and the discrete level μ relative to t_N and T_N.
- Policymaking must account for strategic complementarity, profit shifting behavior, altered best-response dynamics, and repeated-game incentives.
- Modest minima could plausibly benefit both high- and low-tax countries, but this is model- and parameter-dependent.

---

### Section 3.2 — Implications of a Minimum Rate: mechanism, welfare effects, and critical minimum rates
- Setup:
  - A constraint that no tax rate may be set below μ is imposed, with μ taken to exceed t_N so it binds on the low tax country in Nash equilibrium.
  - The high tax country sets its rate according to its best response T(·).
- Low-tax-country welfare when constrained to μ:
  - w(μ) ≡ w(μ, T(μ)) δ/Π = δθ + δ ( λ(Λ−1) / 2(Λ−α) + (λ−1)θ ) μ − ( λΛ / 2(Λ−α) ) μ^2.  (equation (10))
- Qualitative pattern (Figure 1 summary):
  - W(μ) (high-tax-country welfare) is strictly increasing and convex for μ > t_N.
  - w(μ) (low-tax-country welfare) increases for μ just above t_N, continues rising for some range, reaches a peak at μ* (Pareto efficient minimum), and thereafter declines.
  - At μ = μ** welfare in the low tax country equals its Nash welfare w_N; for μ > μ** the low tax country is worse off than in Nash.
  - μ** is termed the maximal Pareto dominant minimum rate.
- Interaction with high-tax-country constraint:
  - If μ exceeds μ̄, the minimum also binds the high tax country; for μ > μ̄ both countries set their rate at the minimum and welfare in each country rises indefinitely with μ (within model assumptions).
  - μ̄ = ( (Λ−1) / Λ ) δ.  (equation (11))
  - Analysis focuses on μ ≤ μ̄ because μ > μ̄ corresponds to effective harmonization.

### Characterization of critical rates (Proposition 2)
- Pareto efficient minimum rate:
  - max(μ*, μ̄), where
    - μ* = δ [ ( (λ−1)(Λ−α)^2 θ + λ(Λ−1) ) / (2λΛ) ].  (equation (12))
- Maximal Pareto dominant minimum rate:
  - max(μ**, μ̄), where
    - μ** = δ [ ( (Λ−α)^2 / Λ(3λ−2α) ) ( (λ−1)/λ 4θ + (2(Λ−1)/(Λ−α)) ) ] > μ*.  (equation (13))
  - μ** = t_N + 2(μ* − t_N) (since w(μ) is quadratic).
- Three possible alignment cases (using μ* < μ**):
  - Case U (Unanimity): μ̄ ≤ μ*. Occurs iff
    - ( (Λ−1)/Λ ) / ( (λ−1)/λ ) < ( (Λ−α)/Λ )^2 4θ.  (equation (14))
  - Case PC (Potential conflict): μ* ≤ μ̄ < μ**. Low-tax country always gains relative to Nash, but marginal increases in μ are not always unanimously approved.
  - Case PL (Potential loss): μ** < μ̄. There exist minima that bind only the low tax country but reduce its welfare below Nash. Occurs iff
    - ( (Λ−1)/Λ ) / ( (λ−1)/λ ) > ( (Λ−α)/Λ )^2 4θ.  (equation (15))
- Comparative statics:
  - μ* and μ** increase with the low-tax country’s marginal valuation of public spending and with the low-tax country’s size.
  - The slope T′(t) (increasing in Λ, decreasing in α) critically shapes gains from a minimum.

### Calibration: closed-form relationships and illustrative numeric results
- Relationships relative to t_N:
  - μ* = ( (3Λ − 2α) / 2Λ ) t_N.  (equation (16))
  - μ** = ( 2(Λ − α) / Λ ) t_N.  (equation (17))
  - Gaps between μ*, μ**, and t_N depend only on α/Λ and are strictly decreasing in α/Λ.
- Illustrative benchmark: t_N = 12.5%.
  - Extreme parameter values:
    - If α/Λ = 0.5 (α = 0.5, Λ = 1), no Pareto gain from a minimum (best response flat).
    - If α = 0 or Λ → ∞ (T′(t) = 1/2 maximum), upper bounds for μ* and μ** are reached.
    - Under such extremes, Pareto efficient μ* = 18.75% and both countries gain from any minimum up to 25%.
  - Examples with Λ = 1.5:
    - If α/Λ = 1/3 (α = 0.5, Λ = 1.5): μ* = 14.6% and μ** = 16.7%.
    - If α/Λ = 1/6 (α = 0.25, Λ = 1.5): μ* = 16.7% and μ** = 20.8%.
  - Interpretation: with t_N = 12.5%, a low-tax country might benefit from a minimum close to 17% and could gain (Pareto) up to about 20% in plausible calibrations.
- Table 1 summary (selected implications):
  - For a wide range of parameterizations with t_N = 12.5%, the high Nash rate T_N, μ̄, μ*, μ**, and the applicable case (U, PC, PL) are reported.
  - Illustrative finding: it is only at minimum levels roughly in the order of 17−24% that the low tax country may be made worse off than in the Nash equilibrium for plausible parameter values; losses relative to Nash are associated with higher values of the high Nash rate T_N (e.g., T_N ≥ 20%).
  - The scope for dissonant interests increases with α and with greater differences in country scale or revenue valuation.

### Conclusions: policy implications and caveats
- A minimum corporate tax rate changes the strategic game of international tax competition; strategic responses by both affected and unaffected countries are central to assessing outcomes.
- With tax rates as strategic complements and profit-shifting considerations, the most plausible outcome is a gain for the low-tax country from an infinitesimally binding minimum; an infinitesimally binding minimum is then Pareto-improving.
- Key policy questions for discrete minima are the levels of μ* and μ**.
- The slope of the high-tax country’s best response is a critical determinant of μ* and μ**; from the low-tax country’s perspective only the magnitude of that slope matters.
- Quantitative takeaways:
  - Simulations suggest plausible parameter values with t_N = 12.5% yield μ* around 15% and μ** around 17% in some calibrations, implying both countries may gain from minima materially above 12.5%.
  - Given the policy debate ranges (12.5–20%), these differences are salient: national interests may align around minima substantially above the lowest proposed levels.
- Limitations:
  - Model limitations include details of Pillar Two not captured, two-country assumption, and other simplifications.
  - Extending to N-country settings could amplify spillback effects and raise the maximal Pareto dominant rate.

*Source: wpiea2021250-print-pdf - Box 4.3 and section 3.2.*

### Box 4.3.

### Box 4.3.

### Summary of conceptual findings
- Strategic complementarity in corporate tax setting is the dominant expectation: empirical and theoretical evidence suggests that a one percentage point cut in the average foreign statutory rate typically induces a positive response at home (examples reported: between 0.34 and 0.67 percentage points in advanced-country samples; between 0.25 and 0.3 points when developing countries are included).
- In a Stackelberg (sequential-move) context, the low tax country never gains from being the follower: the leader can use a minimum to induce the other to reduce its rate, benefiting the leader and harming the low tax country. If the low tax country is the leader, introducing a formal minimum cannot make it better off.
- Repeated-game dynamics can complicate outcomes: imposing a minimum may reduce the scope for sustaining cooperative outcomes by lowering punishment for defection.
- Overall welfare effects of a minimum corporate tax rate are ambiguous in general: depending on modeling assumptions, a modest minimum could benefit both high- and low-tax countries, only one, or neither. Yet the balance of evidence on strategic complementarity and profit shifting suggests a real possibility that both may benefit from a modest minimum.

### Model setup and key equations (tax competition with profit shifting)
- Representative multinational profits in the low-tax country: π (strictly positive); in the high-tax country: Π (strictly positive).
- Cost of shifting a proportion s of profit abroad: (δ/2) s^2. Profits shifted out of the high tax country occur in amount (T−t) s Π/δ, which, when optimized, yields shifting in amount (T−t)Π/δ.
- Welfare in the low tax country (recipient of shifted profit):
  - w(t,T) = (1−t) π + λ t ( π + (T−t)/δ Π ). (equation (2))
  - Best response of the low-tax country (maximizing w with respect to t):
    - t(T) = 1/2 [ ( (λ−1)/λ ) δ θ + T ], where θ ≡ π/Π. (equation (3))
- Welfare in the high tax country (donor of shifted profit), allowing for potential social discounting of private gains from profit shifting:
  - Baseline expression combined and simplified:
    - W(T,t) = (1−T) Π + (1/2) (T−t)^2 / δ Π + Λ T ( 1 − (T−t)/δ ) Π. (equation (5))
  - Generalized with social valuation parameter α ∈ [0, 1/2]:
    - W(T,t) = (1−T) Π + α (T−t)^2 / δ Π + Λ T ( 1 − (T−t)/δ ) Π. (equation (6))
    - Interpretation: α = 1/2 corresponds to full social valuation of the net private gain from shifting; α = 0 corresponds to zero social valuation.
- High-tax country’s best response (differentiating (6)):
  - T(t) = 1/2 [ ( (Λ−1)/(Λ−α) ) δ + ( (Λ−2α)/(Λ−α) ) t ]. (equation (7))
  - Slope T′(t) = (Λ−2α)/2(Λ−α) > 0; hence strategic complementarity for the high-tax country given Λ > 2α.

### Nash equilibrium characterization and comparative statics
- Existence and uniqueness condition:
  - If (Λ−1)/Λ > (λ−1)/λ θ, then there is a unique Nash equilibrium with t_N < T_N. (Proposition 1; condition (8))
  - Interpretation: the country setting the lower tax rate is relatively small in global profits (small θ) and/or has relatively low marginal valuation of public spending (λ).
- Nash rates (explicit expressions):
  - t_N = δ ( Λ−α )/( 3Λ−2α ) [ ( (λ−1)/λ ) 2 θ + ( (Λ−1)/(Λ−α) ) ]. (equation (9))
  - T_N = T(t_N) (i.e., T evaluated at t_N).
- Comparative statics of Nash rates:
  - Both t_N and T_N increase with Λ and λ.
  - Both t_N and T_N decrease with θ.
  - Both t_N and T_N increase with α: a higher social valuation attached to outward profit shifting (higher α) induces the high-tax country to raise its rate, which in turn induces the low-tax country to raise its rate.
- Role of α:
  - A higher α shifts the high-tax country’s best response upwards but flattens it (reduces responsiveness), because convexity of profit shifting in T−t implies reduced incremental shifting when t is higher; thus higher α can damp the gain of the low-tax country from a minimum-induced increase in t.

### Policy implications and interpretation
- A discretely binding minimum tax rate μ has no effect until it reaches the lower Nash tax rate t_N; beyond that point the low-tax country’s rate will be μ.
- The high-tax country is unambiguously better off when minima force an increase in the low-tax rate (from ∂W/∂t . (δ/Π) = (Λ−2α) T + 2α t > 0): minima that increase t raise the high-tax country’s welfare.
- The scope for Pareto-improving minima depends critically on how the minimum affects the low-tax country’s welfare; outcomes depend on Λ, λ, θ, α, and the discrete level at which μ is set relative to t_N and T_N.
- Practical policymaking must account for:
  - Strategic complementarity and profit shifting behavior.
  - The potential for minima to alter best-response dynamics and repeated-game incentives.
  - The possibility that modest minima could plausibly benefit both high- and low-tax countries, but that this is model- and parameter-dependent.

*Source: wpiea2021250-print-pdf - Box 4.3.*

### 3.2    Implications of a Minimum Rate

### 3.2    Implications of a Minimum Rate

### Analysis: mechanism, welfare effects, and critical minimum rates
- Setup:
  - A constraint that no tax rate may be set below μ is imposed, with μ taken to exceed t_N so it binds on the low tax country in Nash equilibrium.
  - The high tax country sets its rate according to its best response T(·).
- Low-tax-country welfare when the minimum constrains its rate to μ:
  - w(μ) ≡ w(μ, T(μ)) δ/Π = δθ + δ ( λ(Λ−1) / 2(Λ−α) + (λ−1)θ ) μ − ( λΛ / 2(Λ−α) ) μ^2.  (equation (10))
- Qualitative pattern (Figure 1 summary):
  - W(μ) (high-tax-country welfare) is strictly increasing and convex for μ > t_N.
  - w(μ) (low-tax-country welfare) increases for μ just above t_N, continues rising for some range as higher μ induces a higher high-tax rate that cushions revenue loss, then reaches a peak at μ* (Pareto efficient minimum), and thereafter declines.
  - At μ = μ** welfare in the low tax country equals its Nash welfare w_N; for μ > μ** the low tax country is worse off than in Nash.
  - μ** is termed the maximal Pareto dominant minimum rate.
- Interaction with high-tax-country constraint:
  - If μ exceeds μ̄, the minimum also binds the high tax country; for μ > μ̄ both countries set their rate at the minimum and welfare in each country rises indefinitely with μ (within model assumptions).
  - μ̄ = ( (Λ−1) / Λ ) δ.  (equation (11))
  - Analysis focuses on μ ≤ μ̄ because μ > μ̄ corresponds to effective harmonization.
- Characterization of critical rates (Proposition 2):
  - (a) Pareto efficient minimum rate: max(μ*, μ̄), where
    - μ* = δ [ ( (λ−1)(Λ−α)^2 θ + λ(Λ−1) ) / (2λΛ) ].  (equation (12))
  - (b) Maximal Pareto dominant minimum rate: max(μ**, μ̄), where
    - μ** = δ [ ( (Λ−α)^2 / Λ(3λ−2α) ) ( (λ−1)/λ 4θ + (2(Λ−1)/(Λ−α)) ) ] > μ*.  (equation (13))
  - μ** = t_N + 2(μ* − t_N) (since w(μ) is quadratic).
- Three possible cases for alignment of interests (using μ* < μ**):
  - Case U (Unanimity): μ̄ ≤ μ*.  A higher minimum is always in both countries' interests. Occurs iff
    - ( (Λ−1)/Λ ) / ( (λ−1)/λ ) < ( (Λ−α)/Λ )^2 4θ.  (equation (14))
  - Case PC (Potential conflict): μ* ≤ μ̄ < μ**.  Low-tax country always gains relative to Nash, but marginal increases in μ are not always unanimously approved.
  - Case PL (Potential loss): μ** < μ̄.  There exist minima that bind only the low tax country but reduce its welfare below Nash. Occurs iff
    - ( (Λ−1)/Λ ) / ( (λ−1)/λ ) > ( (Λ−α)/Λ )^2 4θ.  (equation (15))
- Comparative statics:
  - μ* and μ** are greater when the marginal valuation of public spending in the low-tax country is higher and when the low tax country is larger (other things equal).
  - The magnitude of the slope T′(t) of the high-tax best response (increasing in Λ, decreasing in α) critically shapes the gains from a minimum.

### Calibration: relationships, numeric expressions, and illustrative quantitative results
- Closed-form relationships relative to the unconstrained low-tax rate t_N:
  - μ* = ( (3Λ − 2α) / 2Λ ) t_N.  (equation (16))
  - μ** = ( 2(Λ − α) / Λ ) t_N.  (equation (17))
  - Gaps between μ*, μ**, and t_N depend only on the ratio α/Λ and are strictly decreasing in α/Λ.
- Illustrative benchmark: t_N = 12.5% (the paper’s chosen illustrative low tax rate).
  - Extreme parameter values:
    - If α/Λ = 0.5 (α = 0.5, Λ = 1), no Pareto gain from a minimum (best response flat).
    - If α = 0 or Λ → ∞ (T′(t) = 1/2 maximum), upper bounds for μ* and μ** are reached.
    - Under such extremes, Pareto efficient μ* = 18.75% and both countries gain from any minimum up to 25%.
  - Examples with Λ = 1.5:
    - If α/Λ = 1/3 (α = 0.5, Λ = 1.5): μ* = 14.6% and μ** = 16.7%.
    - If α/Λ = 1/6 (α = 0.25, Λ = 1.5): μ* = 16.7% and μ** = 20.8%.
  - Interpretation: with t_N = 12.5%, a low-tax country might benefit from a minimum close to 17% and could gain (Pareto) up to about 20% in plausible calibrations.
- Table 1 summary (selected implications):
  - For a wide range of parameterizations with t_N = 12.5%, the high Nash rate T_N, μ̄, μ*, μ**, and the applicable case (U, PC, PL) are reported (see source table).
  - Illustrative finding: it is only at minimum levels roughly in the order of 17−24% that the low tax country may be made worse off than in the Nash equilibrium for plausible parameter values; losses relative to Nash are associated with higher values of the high Nash rate T_N (e.g., T_N ≥ 20%).
  - The scope for dissonant interests increases with α and with greater differences in country scale or revenue valuation.

### Conclusions: policy implications and caveats
- Purpose and strategic responses:
  - A minimum corporate tax rate changes the strategic game of international tax competition; strategic responses by both affected and unaffected countries are central to assessing outcomes.
  - With tax rates as strategic complements and profit-shifting considerations, the most plausible outcome is a gain for the low-tax country from an infinitesimally binding minimum; an infinitesimally binding minimum is then Pareto-improving.
- Key policy questions for discrete (non-infinitesimal) minima:
  - Critical questions are the levels of the Pareto efficient minimum μ* and the maximal Pareto dominant minimum μ**.
  - The slope of the high-tax country’s best response (driven by the high-tax country’s valuation of private profit-shifting gains relative to public loss) is a critical determinant of μ* and μ**; from the low-tax country’s perspective only the magnitude of that slope matters.
- Quantitative takeaways and salience for policy debate:
  - Simulations suggest plausible parameter values with t_N = 12.5% yield μ* around 15% and μ** around 17% in some calibrations, implying both countries may gain from minima materially above 12.5%.
  - Given the policy debate ranges (12.5–20%), these differences are salient: national interests may align around minima substantially above the lowest proposed levels.
- Limitations and extensions:
  - Model limitations include details of Pillar Two not captured, two-country assumption, and other simplifications.
  - Intuition suggests moving to N-country settings (e.g., introducing a “middle tax” country) could amplify spillback effects and raise the maximal Pareto dominant rate, potentially increasing the scope for congruent interest in more ambitious minimum rates.

*Source: IMF Working Paper (section 3.2, “Implications of a Minimum Rate”) — wpiea2021250-print-pdf.*

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*Source: wpiea2021250-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021250-print-pdf.pdf_