## _wp06271

## Source details

**Canonical URL:** [_wp06271](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06271.pdf)

## Other formats

- [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06271.pdf.md)
- [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06271.pdf.json)

---

### I. Research question and approach
- Research question: What is the relation between the size of firms and the (optimal) taxes that they should pay?
- Modeling approach:
  - Firms modeled as asymmetric Cournot oligopolists with different, constant unit costs of production (αj).
  - Taxes studied in isolation: specific sales tax, ad valorem sales tax, and a hybrid profits tax (profits tax with incomplete deductibility of capital cost). Also consider combination of ad valorem sales tax and hybrid profits tax.
  - Key conceptual point: A hybrid profits tax (tax on pure profits with imperfect deductibility of capital costs) induces substitution away from capital, generating deadweight loss and not being optimally imposed at 100 percent.

### II. Model structure and welfare objective
- Market and firms:
  - Consumers purchase quantity M and pay price P; demand P(M) with P_M < 0 and P_MM + Σ_j m_j P_M + P_M < 0.
  - n firms, j = 1...n, with α1 = 1 (best-practice firm), αj > 1 for j>1.
  - Firms behave as Cournot oligopolists; jth firm produces mj and profits Πj = mj P - mj Cj.
- Production and factor costs:
  - Cost minimization: Cj = Cj*(w*, r*, αj). Simulations use Cobb-Douglas production with capital exponent ∈ = 0.25.
  - Market factor prices pre-tax: w = 1, r = 1.
- Social valuation:
  - Social value V = Producer surplus (Π) + Consumer surplus (S) + γ R, where γ is the social value of revenue and R is total tax receipts.
  - "Ideal Outcome": most efficient firm produces everything, regulated to price at marginal cost plus any sales tax; other firms exit or are induced to exit.
  - Practical constraint: tax systems typically limited to sales taxes and distortionary profits taxes with imperfect capital deductibility.

### III. Simulation framework and baseline parameterization
- Demand: P = 10 - 0.5 M.
- Factor prices: w = 1, r = 1.
- Profits tax treatment: net-of-tax factor prices w* = (1-τj) w and r* = (1-δ τj) r where τj is the profits tax rate and δ is the fraction of capital cost deductible.
- Production function: Cobb-Douglas with capital exponent ∈ = 0.25.
- Baseline cost and benchmarks:
  - If αj = 1 and no taxes, unit cost = 1.755 (computed with l = 1.316, k = 0.439).
  - Competitive case (many identical αj = 1): P = 1.755, M = 16.49, social welfare V = 67.98.
  - Monopoly case (single firm αj = 1): M = 8.245, P = 5.878, V = 50.99.
  - Benchmark oligopoly with three firms: α1 = 1, α2 = 1.20, α3 = 1.30: no taxes → M = 11.928, P = 4.036, V = 59.58.
- Social value of revenue γ examined over the range 1.15 through 2.5.
- Ideal Outcome calibration (specific tax that yields most efficient pricing and resulting output and welfare):
  - γ = 1.15 → Specific tax = 0.96, Total output = 14.57, Social Welfare = 69.16
  - γ = 1.30 → Specific tax = 1.56, Total output = 13.37, Social Welfare = 71.81
  - γ = 1.45 → Specific tax = 1.95, Total output = 12.59, Social Welfare = 75.23
  - γ = 1.60 → Specific tax = 2.25, Total output = 11.99, Social Welfare = 79.11
  - γ = 1.75 → Specific tax = 2.46, Total output = 11.57, Social Welfare = 83.28
  - γ = 1.90 → Specific tax = 2.64, Total output = 11.21, Social Welfare = 87.65
  - γ = 2.05 → Specific tax = 2.79, Total output = 10.91, Social Welfare = 92.16
  - γ = 2.20 → Specific tax = 2.91, Total output = 10.67, Social Welfare = 96.78
  - γ = 2.35 → Specific tax = 3.00, Total output = 10.49, Social Welfare = 101.47
  - γ = 2.50 → Specific tax = 3.09, Total output = 10.31, Social Welfare = 106.23
- Numerical solution details:
  - Computations in double-precision FORTRAN.
  - Grid search: ad valorem and profits taxes 0.0 to 0.99 in increments of 0.01; specific tax 0.0 to 3.5 in increments of 0.03.
  - Taxes bounded below by 0.0.

### IV. Specific sales tax — analytical characterization and numerical results
- Analytical highlights:
  - Firm profits with firm-specific specific tax tj: Πj = mj P - mj Cj - mj tj.
  - First-order condition: 0 = P + mj P_M - Cj - tj.
  - Aggregate quantity response to ti: dM/dti = 1/(1 + (1) n M P_M ) < 0 (expression preserved).
  - Optimality condition for firm-specific specific tax (for each i): 0 = −2 m_i + γ (m_i + t_i/P_M) + Ω (Ω collects i-independent terms).
  - Two analytic comparative-static cases depending on γ:
    - Case 1: (2 - γ) > 0 and γ > 1 → higher Ci ⇒ higher ti and lower mi.
    - Case 2: (2 - γ) < 0 → higher Ci ⇒ lower ti, but mi still lower.
- Numerical results (three firms α1 = 1, α2 = 1.20, α3 = 1.30; firm-specific specific taxes only). Selected rows from Table 2:
  - γ = 1.15:
    - t1 = 0.00, t2 = 0.00, t3 = 0.00
    - m1 = 4.56, m2 = 3.86, m3 = 3.51
    - Π1 = 10.40, Π2 = 7.45, Π3 = 6.16
    - V = 59.58, Φ = 0.000
    - Memo: V When t1 = t2 = t3 = 59.58
  - γ = 1.30:
    - t1 = 0.00, t2 = 0.36, t3 = 0.57
    - m1 = 5.03, m2 = 3.60, m3 = 2.83
    - Π1 = 12.63, Π2 = 6.50, Π3 = 4.01
    - V = 59.79, Φ = 0.017
    - Memo: V When t1 = t2 = t3 = 59.67
  - γ = 1.45:
    - t1 = 0.78, t2 = 0.99, t3 = 1.11
    - m1 = 4.44, m2 = 3.32, m3 = 2.73
    - Π1 = 9.86, Π2 = 5.51, Π3 = 3.72
    - V = 60.78, Φ = 0.077
    - Memo: V When t1 = t2 = t3 = 60.73
  - γ = 1.60:
    - t1 = 1.32, t2 = 1.44, t3 = 1.50
    - m1 = 4.05, m2 = 3.11, m3 = 2.64
    - Π1 = 8.21, Π2 = 4.83, Π3 = 3.48
    - V = 62.58, Φ = 0.154
    - Memo: V When t1 = t2 = t3 = 62.56
  - γ = 1.75:
    - t1 = 1.71, t2 = 1.77, t3 = 1.80
    - m1 = 3.78, m2 = 2.96, m3 = 2.55
    - Π1 = 7.15, Π2 = 4.38, Π3 = 3.25
    - V = 64.86, Φ = 0.223
    - Memo: V When t1 = t2 = t3 = 64.85
  - γ = 1.90:
    - t1 = 2.01, t2 = 2.04, t3 = 2.04
    - m1 = 3.59, m2 = 2.82, m3 = 2.47
    - Π1 = 6.43, Π2 = 3.99, Π3 = 3.06
    - V = 67.44, Φ = 0.280
    - Memo: V When t1 = t2 = t3 = 67.44
  - γ = 2.05:
    - t1 = 2.22, t2 = 2.22, t3 = 2.22
    - m1 = 3.45, m2 = 2.75, m3 = 2.40
    - Π1 = 5.96, Π2 = 3.78, Π3 = 2.88
    - V = 70.23, Φ = 0.327
    - Memo: V When t1 = t2 = t3 = 70.23
  - γ = 2.20:
    - t1 = 2.43, t2 = 2.40, t3 = 2.37
    - m1 = 3.30, m2 = 2.66, m3 = 2.37
    - Π1 = 5.45, Π2 = 3.54, Π3 = 2.81
    - V = 73.17, Φ = 0.365
    - Memo: V When t1 = t2 = t3 = 73.17
  - γ = 2.35:
    - t1 = 2.58, t2 = 2.52, t3 = 2.49
    - m1 = 3.20, m2 = 2.61, m3 = 2.32
    - Π1 = 5.11, Π2 = 3.42, Π3 = 2.70
    - V = 76.22, Φ = 0.397
    - Memo: V When t1 = t2 = t3 = 76.22
  - γ = 2.50:
    - t1 = 2.70, t2 = 2.64, t3 = 2.61
    - m1 = 3.14, m2 = 2.55, m3 = 2.26
    - Π1 = 4.92, Π2 = 3.26, Π3 = 2.56
    - V = 79.36, Φ = 0.424
    - Memo: V When t1 = t2 = t3 = 79.35
- Additional numerical observations:
  - No firm is driven out of the industry by the optimal specific taxes in these simulations; no t_i hits the algorithm maximum 3.50.
  - At relatively low γ some specific taxes are zero (e.g., t1 = 0.00 at γ = 1.30).
  - As γ increases, optimal specific taxes rise and social welfare V increases toward the Ideal Outcome benchmarks.
  - The fraction Φ (proportion of Ideal Outcome welfare improvement achieved) rises from 0.000 at γ = 1.15 to 0.424 at γ = 2.50.

### V. Ad valorem sales tax — analytical characterization and numerical results
- Analytical highlights:
  - After-tax profit: Πj ≡ (1 − tj) mj Pm − Cj.
  - First-order condition: (1 − tj) P − CMj = 0.
  - Change in market quantity in response to ti: expression (24) with sign < 0, depends on Ci and ti.
  - Optimal-tax first-order condition yields a locus between ti and mi; slope depends on si ≡ mi/M and η ≡ P/PM (η < 0).
  - Comparative statics:
    - At γ = 1 the two loci between ti and mi have equal slopes; slope sign can switch as γ increases toward 2.
    - Near γ = 1 less efficient firms face relatively high tax rates; near γ = 2 pattern reverses; for intermediate γ tax rate may be nonmonotonic in efficiency.
- Numerical results (selected Table 3 entries):
  - γ = 1.15:
    - t1 = 0.03, t2 = 0.08, t3 = 0.13
    - m1 = 4.74, m2 = 3.78, m3 = 3.12
    - Π1 = 10.91, Π2 = 6.58, Π3 = 4.22
    - V = 59.67, Φ = 0.009; V when equal = 59.64
  - γ = 1.30:
    - t1 = 0.28, t2 = 0.29, t3 = 0.30
    - m1 = 4.46, m2 = 3.40, m3 = 2.81
    - Π1 = 7.15, Π2 = 4.10, Π3 = 2.77
    - V = 61.15, Φ = 0.128; equal V = 61.14
  - γ = 1.45:
    - t1 = 0.38, t2 = 0.38, t3 = 0.38
    - m1 = 4.29, m2 = 3.16, m3 = 2.59
    - Π1 = 5.71, Π2 = 3.10, Π3 = 2.09
    - V = 63.68, Φ = 0.262; equal V = 63.68
  - γ = 1.60:
    - t1 = 0.44, t2 = 0.43, t3 = 0.42
    - m1 = 4.11, m2 = 2.99, m3 = 2.51
    - Π1 = 4.74, Π2 = 2.55, Π3 = 1.83
    - V = 66.74, Φ = 0.367; equal V = 66.73
  - γ = 2.50:
    - t1 = 0.56, t2 = 0.53, t3 = 0.52
    - m1 = 3.63, m2 = 2.65, m3 = 2.11
    - Π1 = 2.91, Π2 = 1.65, Π3 = 1.06
    - V = 89.03, Φ = 0.631; equal V = 88.87
- Comparative findings:
  - Social welfare is uniformly higher under the ad valorem tax than under the specific tax for the simulated parameterization.
  - Φ is much higher for ad valorem than for specific tax.
  - For these parameters constraining ad valorem taxes equal across firms has little effect on V.

### VI. Hybrid profits tax — analytical notes and numerical results
- Analytical highlights:
  - After-tax profit: Πj ≡ (1 − τj) mj Pm − Cj with capital deductibility parameter δ < 1.
  - First-order condition: (1 − τj) P − CMj = 0.
  - Change in total market quantity in response to τi: equation (34), sign < 0 and depends on αi, τi, and ki.
  - Total tax receipts: R ≡ Σj [τj mj Pm − (1 − δ)τj kj r αj].
  - Analytical tractability limited; simulations used.
- Numerical results (δ = 0.5 unless noted). Selected Table 4 entries:
  - γ = 1.15:
    - τ1 = 0.49, τ2 = 0.47, τ3 = 0.48
    - m1 = 4.51, m2 = 3.76, m3 = 3.36
    - Π1 = 5.18, Π2 = 3.75, Π3 = 2.93
    - V = 60.63, Φ = 0.110; equal-rate V = 60.63
  - γ = 1.30:
    - τ1 = 0.63, τ2 = 0.61, τ3 = 0.62
    - m1 = 4.47, m2 = 3.70, m3 = 3.27
    - Π1 = 3.70, Π2 = 2.67, Π3 = 2.03
    - V = 62.91, Φ = 0.272; equal-rate V = 62.91
  - γ = 1.60:
    - τ1 = 0.73, τ2 = 0.73, τ3 = 0.74
    - m1 = 4.47, m2 = 3.60, m3 = 3.12
    - Π1 = 2.69, Π2 = 1.75, Π3 = 1.27
    - V = 68.45, Φ = 0.454; equal-rate V = 68.45
  - γ ≥ 1.90:
    - Optimal τs produce τ1 = 0.83, τ2 = 0.86, τ3 = 0.98 and Π3 = 0.00 (exit); e.g., γ = 2.50 → V = 90.10; Φ = 0.654
- Sensitivity and memos:
  - Imperfect deductibility (δ = 0.5) substantially reduces welfare relative to δ = 1 (example: γ = 1.30, δ = 1 → V = 66.71 and Φ = 0.583 vs δ = 0.5 → V = 62.91 and Φ = 0.272).
  - Optimal τ is approximately constant across firms for a given γ, except where high τ triggers least efficient firm exit (γ ≥ 1.9).
  - Sensitivity to production parameter ∈:
    - Memo ∈ = 0.05, γ = 1.30: τ1 = 0.90, τ2 = 0.88, τ3 = 0.88; V = 75.44.
    - Memo ∈ = 0.5, γ = 1.30: τ1 = 0.48, τ2 = 0.48, τ3 = 0.51; V = 57.86.
- Comparative welfare:
  - Social welfare is always higher with the profits tax than with either sales tax alone for the simulated parameterization.

### VII. Two-tax package: hybrid profits tax plus ad valorem tax
- Simulation method: coarse grid over 0 to 0.99 in steps of 0.11, then refined in adjacent regions in 0.01 steps.
- Representative results (Table 5 highlights):
  - γ = 1.15:
    - τ: (0.49, 0.47, 0.47); t: (0.00, 0.00, 0.01)
    - T1/m1 = 1.26, T2/m2 = 1.06, T3/m3 = 1.01
    - m = (4.51, 3.77, 3.33); Π = (5.20, 3.76, 2.92)
    - V = 60.63; Φ = 0.110
  - γ = 1.30:
    - τ: (0.64, 0.57, 0.49); t: (0.00, 0.10, 0.20)
    - T/m = (1.78, 1.61, 1.61)
    - m = (4.82, 3.62, 2.64); Π = (4.17, 2.53, 1.42)
    - V = 63.04; Φ = 0.283
  - γ = 2.50:
    - τ: (0.75, 0.62, 0.50); t: (0.30, 0.42, 0.47)
    - T/m = (3.24, 3.07, 3.01)
    - m = (4.76, 2.64, 1.54); Π = (1.98, 0.77, 0.31)
    - V = 93.43; Φ = 0.726
- Comparative findings:
  - The two-tax package outperforms any single tax but only modestly beyond the best single tax (hybrid profits tax).
  - The package increases Φ by a little over 10 percent at the highest γ values relative to the best single tax.
  - The package falls materially short of the Ideal Outcomes, especially for low γ and by almost 30 percent even for highest γ.
  - Profits tax rates fall as firm efficiency falls; ad valorem tax rates rise as firm efficiency falls for these simulations.
  - Sum of profits and ad valorem taxes per unit output (Ti/mi) falls as firm efficiency falls.

### VIII. Interpretation and policy-relevant implications
- Cournot structure matters: own vs cross effects under Cournot competition produce firm-size dependent m_i terms in welfare FOCs, yielding differential taxation prescriptions.
- Comparative-static regimes depend on γ:
  - If (2 - γ) > 0 and γ > 1, higher-cost (smaller-output) firms face higher optimal specific taxes.
  - If (2 - γ) < 0, higher-cost firms face lower optimal specific taxes, though their output remains lower.
- Practical implications:
  - When model indicates smaller firms should be taxed less, it weakens arguments for lax enforcement on small firms due to administrative cost concerns; when the reverse holds, it motivates stronger enforcement targeting small firms.
  - A regime that applies a uniform tax with differing probabilities of contact is equivalent to differentiated always-paid taxes, linking results to enforcement intensity.
- Caveats:
  - Results depend on parameterization, notably the degree of imperfect competition and existence of pure profits; in perfect competition the hybrid profits tax has no role if sales taxes are available.
  - No theoretical or simulated evidence for a hump-shaped optimal tax pattern (intermediate firms taxed most) under the model assumptions.

*Content based solely on the supplied PDF section.*

### 1. Most Efficient Firm Pricing at Marginal Cost Plus a Specific Tax

### 1. Most Efficient Firm Pricing at Marginal Cost Plus a Specific Tax

### Section 1 — Most Efficient Firm Pricing at Marginal Cost Plus a Specific Tax to Yield the Ideal Outcome
- Section title and scope as listed in the source.
- Locator: page 8.

### Section 2 — Use of a Specific Tax Alone
- Section title and scope as listed in the source.
- Locator: page 14.

### Section 3 — Use of an Ad Valorem Tax Alone
- Section title and scope as listed in the source.
- Locator: page 18.

### Section 4 — Use of a Hybrid Profits Tax Alone
- Section title and scope as listed in the source.
- Locator: page 21.

### Section 5 — Simultaneous Use of a Hybrid Profits Tax and an Ad Valorem Tax
- Section title and scope as listed in the source.
- Locator: page 24.

### Figures
- Figure 1: Determination of the Firm’s Output and Specific Tax Rate in Case 1 — page 12.
- Figure 2: Determination of the Firm’s Output and Specific Tax Rate in Case 2 — page 13.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06271.pdf*

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

### _wp06271 - References..............................................................................................................

### I. Introduction — question and approach
- Research question: What is the relation between the size of firms and the (optimal) taxes that they should pay?
- Context: Tax laws and administrations often distinguish firms by size (Gauthier and Gersovitz (1997); Keen and Mintz (2004) noted as prior work).
- Modeling approach:
  - Firms modeled as asymmetric Cournot oligopolists with different, constant unit costs of production (αj).
  - Distribution of unit costs mediated by Cournot oligopoly determines firm size distribution.
  - Taxes studied in isolation: specific sales tax, ad valorem sales tax, and a hybrid profits tax (profits tax with incomplete deductibility of capital cost). Also consider combination of ad valorem sales tax and hybrid profits tax.
- Key conceptual point: A hybrid profits tax (tax on pure profits with imperfect deductibility of capital costs) induces substitution away from capital, generating deadweight loss and not being optimally imposed at 100 percent.

### II. Model structure and welfare objective
- Market and firms:
  - Consumers purchase quantity M and pay price P; demand curve P(M) with P_M < 0 and P_MM + Σ_j m_j P_M + P_M < 0 (stability condition in oligopoly).
  - n firms, j = 1...n, with α1 = 1 (best-practice firm), αj > 1 for j>1 (less efficient).
  - Firms behave as Cournot oligopolists; jth firm produces mj and profits Πj = mj P - mj Cj.
- Production and factor costs:
  - Cost minimization: Cj = Cj*(w*, r*, αj) with constant-returns-to-scale production function kj, lj and Cobb-Douglas form in simulations (see Section II.B).
  - Market factor prices pre-tax: r and w.
- Social valuation:
  - Social value V = Producer surplus (Π) + Consumer surplus (S) + γ R, where γ is the social value of revenue and R is total tax receipts.
  - "Ideal Outcome": most efficient firm produces everything, regulated to price at marginal cost plus any sales tax; other firms exit or are induced to exit.
  - Practical constraint: tax systems typically limited to sales taxes and distortionary profits taxes with imperfect capital deductibility.

### III. Simulation framework and baseline parameterization (Section II.B)
- Demand curve: P = 10 - 0.5 M.
- Factor prices: w = 1, r = 1.
- Profits tax treatment: under the profits tax the jth firm faces net-of-tax factor prices w* = (1-τj) w and r* = (1-δ τj) r where τj is the profits tax rate and δ is the fraction of capital cost deductible.
- Production function (simulations): Cobb-Douglas with capital exponent ∈ = 0.25:
  - m_j = α_j^{-1} (k_j^∈ l_j^{1-∈})? (production function specified as 1()/. jj j mkl α ∈−∈ = in source).
- Baseline cost and benchmark outcomes:
  - If αj = 1 and no taxes, unit cost = 1.755 (computed with l = 1.316, k = 0.439).
  - Competitive case (many identical αj = 1): P = 1.755, M = 16.49, social welfare V = 67.98.
  - Monopoly case (single firm αj = 1): M = 8.245, P = 5.878, V = 50.99.
- Benchmark oligopoly with three firms: α1 = 1, α2 = 1.20, α3 = 1.30:
  - No taxes: total M = 11.928, P = 4.036, social welfare V = 59.58 (about midway between monopoly and perfect competition).
- Social value of revenue γ examined over the range 1.15 through 2.5 (values bracket literature estimates).
- Ideal Outcome calibration (Table 1): for each γ the specific tax that makes the most efficient firm price at marginal cost plus that specific tax and resulting total output and social welfare:

  - γ = 1.15 → Specific tax = 0.96, Total output = 14.57, Social Welfare = 69.16
  - γ = 1.30 → Specific tax = 1.56, Total output = 13.37, Social Welfare = 71.81
  - γ = 1.45 → Specific tax = 1.95, Total output = 12.59, Social Welfare = 75.23
  - γ = 1.60 → Specific tax = 2.25, Total output = 11.99, Social Welfare = 79.11
  - γ = 1.75 → Specific tax = 2.46, Total output = 11.57, Social Welfare = 83.28
  - γ = 1.90 → Specific tax = 2.64, Total output = 11.21, Social Welfare = 87.65
  - γ = 2.05 → Specific tax = 2.79, Total output = 10.91, Social Welfare = 92.16
  - γ = 2.20 → Specific tax = 2.91, Total output = 10.67, Social Welfare = 96.78
  - γ = 2.35 → Specific tax = 3.00, Total output = 10.49, Social Welfare = 101.47
  - γ = 2.50 → Specific tax = 3.09, Total output = 10.31, Social Welfare = 106.23

- Numerical solution details:
  - Computations in double-precision FORTRAN.
  - Three-dimensional grid search over tax rates:
    - ad valorem and profits taxes: 0.0 to 0.99 in increments of 0.01.
    - specific tax: 0.0 to 3.5 in increments of 0.03.
  - Taxes bounded below by 0.0 (no subsidies in the general tax law).

### IV. Specific sales tax — analytical characterization (Section III)
- Profit and first-order conditions under a firm-specific specific tax tj:
  - Profits: Πj = mj P - mj Cj - mj tj.
  - First-order condition: 0 = P + mj P_M - Cj - tj.
- Key comparative-static derivatives (own vs cross effects) under Cournot competition:
  - Own effect includes an extra term 1/P_M relative to cross effects because tj directly affects the ith firm’s tax-inclusive marginal cost.
  - Aggregate quantity response to a change in tax ti is:
    - dM/dti = 1/(1 + (1) n M P_M ) < 0 (equation (12) in source; expression sign and dependence preserved as in source).
- Change in aggregate profits from a change in ti (equation (13)):
  - dΠ/dti = Σ_j (m_j P_M dm_j/dti + m_j P_M^2?) ⇒ reduces to expression written as 2 Σ_j m_j P_M dm_j/dti = ... with an explicit -2 m_i term highlighted (exact algebra in source).
  - Interpretation: effect decomposes into an i-independent summation and an i-dependent -2 m_i term (one half from direct effect on Πj when i=j, other half from direct effect on mj).
- Change in consumer surplus from a change in ti (equation (14)):
  - dS/dti = - P(M) dM/dti.
  - This change is independent of the particular firm i (impounded in Ω).
- Change in tax revenue from a change in ti (equation (16)):
  - dR/dti = Σ_j [m_j P_M dm_j/dti] + (m_i + t_i/P_M) (structure preserved as in source).
  - Decomposes into an i-independent summation plus an i-dependent (m_i + t_i/P_M) term.
- Optimality condition for firm-specific specific tax (equation (17)):
  - For each i = 1...n: 0 = −2 m_i + γ (m_i + t_i/P_M) + (terms independent of i, Ω)  (exact equation preserved as in source).
  - Ω collects all i-independent terms (including consumer surplus effect).
- Reduced system (equations (18) and (19)) for comparative statics between cost Ci, tax ti, and output mi:
  - (2) MM ii PP tm γ γγ −Ω = −  (equation (18) in source; preserves form)
  - iMii tPm PC = + −  (equation (19) in source; preserves form)
- Analytic cases for how optimal ti varies with Ci (firm inefficiency):
  - Case 1: (2 - γ) > 0 and γ > 1.
    - Both equations (18) and (19) have negative slopes; increasing Ci shifts only equation (19) down.
    - Result: higher Ci ⇒ higher ti and lower mi.
  - Case 2: (2 - γ) < 0.
    - Equation (18) has positive slope; (19) has negative slope.
    - Increasing Ci shifts only equation (19) down.
    - Result: higher Ci ⇒ lower ti, but mi still lower (tax reduction does not fully offset higher cost).
  - These analytic results are robust to demand functional form (do not rely on linear demand).

### V. Specific sales tax — numerical results (Table 2 summary and key findings)
- Simulation setup: three firms with α1 = 1, α2 = 1.20, α3 = 1.30; tax instruments are firm-specific specific taxes only.
- Main quantitative outcomes from Table 2 (rows correspond to different γ values). Columns preserved as in source: γ; specific taxes t1,t2,t3; quantities m1,m2,m3; after-tax profits Π1,Π2,Π3; social welfare V; Φ (fraction of Max ∆SW achieved); memo V when t1=t2=t3.

- Selected rows from Table 2 preserved exactly:

  - γ = 1.15:
    - t1 = 0.00, t2 = 0.00, t3 = 0.00
    - m1 = 4.56, m2 = 3.86, m3 = 3.51
    - Π1 = 10.40, Π2 = 7.45, Π3 = 6.16
    - V = 59.58, Φ = 0.000
    - Memo: V When t1 = t2 = t3 = 59.58

  - γ = 1.30:
    - t1 = 0.00, t2 = 0.36, t3 = 0.57
    - m1 = 5.03, m2 = 3.60, m3 = 2.83
    - Π1 = 12.63, Π2 = 6.50, Π3 = 4.01
    - V = 59.79, Φ = 0.017
    - Memo: V When t1 = t2 = t3 = 59.67

  - γ = 1.45:
    - t1 = 0.78, t2 = 0.99, t3 = 1.11
    - m1 = 4.44, m2 = 3.32, m3 = 2.73
    - Π1 = 9.86, Π2 = 5.51, Π3 = 3.72
    - V = 60.78, Φ = 0.077
    - Memo: V When t1 = t2 = t3 = 60.73

  - γ = 1.60:
    - t1 = 1.32, t2 = 1.44, t3 = 1.50
    - m1 = 4.05, m2 = 3.11, m3 = 2.64
    - Π1 = 8.21, Π2 = 4.83, Π3 = 3.48
    - V = 62.58, Φ = 0.154
    - Memo: V When t1 = t2 = t3 = 62.56

  - γ = 1.75:
    - t1 = 1.71, t2 = 1.77, t3 = 1.80
    - m1 = 3.78, m2 = 2.96, m3 = 2.55
    - Π1 = 7.15, Π2 = 4.38, Π3 = 3.25
    - V = 64.86, Φ = 0.223
    - Memo: V When t1 = t2 = t3 = 64.85

  - γ = 1.90:
    - t1 = 2.01, t2 = 2.04, t3 = 2.04
    - m1 = 3.59, m2 = 2.82, m3 = 2.47
    - Π1 = 6.43, Π2 = 3.99, Π3 = 3.06
    - V = 67.44, Φ = 0.280
    - Memo: V When t1 = t2 = t3 = 67.44

  - γ = 2.05:
    - t1 = 2.22, t2 = 2.22, t3 = 2.22
    - m1 = 3.45, m2 = 2.75, m3 = 2.40
    - Π1 = 5.96, Π2 = 3.78, Π3 = 2.88
    - V = 70.23, Φ = 0.327
    - Memo: V When t1 = t2 = t3 = 70.23

  - γ = 2.20:
    - t1 = 2.43, t2 = 2.40, t3 = 2.37
    - m1 = 3.30, m2 = 2.66, m3 = 2.37
    - Π1 = 5.45, Π2 = 3.54, Π3 = 2.81
    - V = 73.17, Φ = 0.365
    - Memo: V When t1 = t2 = t3 = 73.17

  - γ = 2.35:
    - t1 = 2.58, t2 = 2.52, t3 = 2.49
    - m1 = 3.20, m2 = 2.61, m3 = 2.32
    - Π1 = 5.11, Π2 = 3.42, Π3 = 2.70
    - V = 76.22, Φ = 0.397
    - Memo: V When t1 = t2 = t3 = 76.22

  - γ = 2.50:
    - t1 = 2.70, t2 = 2.64, t3 = 2.61
    - m1 = 3.14, m2 = 2.55, m3 = 2.26
    - Π1 = 4.92, Π2 = 3.26, Π3 = 2.56
    - V = 79.36, Φ = 0.424
    - Memo: V When t1 = t2 = t3 = 79.35

- Key numerical findings and patterns:
  - No firm is driven out of the industry by the optimal specific taxes in the simulations; no t_i reaches the algorithm maximum 3.50.
  - At relatively low γ (including γ = 1.30), some specific taxes are zero (e.g., t1 = 0.00 at γ = 1.30).
  - As γ increases, optimal specific taxes rise and social welfare V increases toward the Ideal Outcome benchmarks in Table 1.
  - The fraction Φ (proportion of the Ideal Outcome welfare improvement achieved) increases with γ from 0.000 at γ = 1.15 to 0.424 at γ = 2.50 in the specific-tax-only simulations.

### VI. Interpretation and policy-relevant implications (from analytic and simulated results)
- Cournot structure matters: the own vs cross effects of firm-specific taxes under Cournot competition produce firm-size dependent terms (involving m_i) in the social-welfare first-order conditions, generating nontrivial differential taxation prescriptions.
- Depending on γ (social valuation of revenue) two analytic cases yield opposite comparative statics:
  - If (2 - γ) > 0 and γ > 1, higher-cost (smaller-output) firms face higher optimal specific taxes.
  - If (2 - γ) < 0, higher-cost firms face lower optimal specific taxes, though their output is still lower.
- Practical interpretation:
  - When model suggests smaller (less efficient) firms should be taxed less than larger firms, it weakens arguments for lax enforcement on small firms due to administrative cost concerns.
  - Conversely, when small firms should be taxed more, it motivates stronger enforcement targeting small firms despite administrative cost.
  - A tax regime that is uniform but applied with different probabilities of contact (taxpayer pays only when contacted) is equivalent to a system of differentiated taxes that are always paid — implying interpretation of results in terms of differential administrative effort.

*Source: _wp06271 - References..............................................................................................................*

### Section II.B. that the grid constrains the tax rates to be nonnegative, and the binding nature of

### _wp06271 - Section II.B. that the grid constrains the tax rates to be nonnegative, and the binding nature of

### IV. ASYMMETRIC OLIGOPOLISTS AND THE AD VALOREM SALES TAX
- Setup and firm profit expression:
  - After-tax profit of firm j: Πj ≡ (1 − tj) mj Pm − Cj (equation (20) structure reflected).
  - First-order condition for firm j: (1 − tj) P − CMj = 0 (equation (21)).
  - Useful derivative identities: equations (22) and (23) relate dmj/dti to demand and cost derivatives.
  - Change in total market quantity in response to ti: expression (24) with sign < 0, depends on Ci and ti (hence on αi).
- Optimal-tax first-order condition:
  - Change in social welfare with respect to ti leads to equation (28), with Ω collecting terms independent of i.
  - Slope of the locus between ti and mi given by equation (29); si ≡ mi/M and η ≡ P/PM (η < 0).
  - Comparative statics:
    - At γ = 1 the two loci between ti and mi have equal slopes; slope sign can switch as γ increases toward 2.
    - Near γ = 1 less efficient firms face relatively high tax rates; near γ = 2 the pattern reverses (but less extreme than for specific tax).
    - For intermediate γ the tax rate may not be monotonic in firm efficiency.
- Simulation results (Table 3 highlights):
  - All tax rates are positive for the reported γ values.
  - Outputs and profits of all firms fall as γ rises (contrast with specific tax results for lowest γ).
  - Pattern of ad valorem rates by firm:
    - For γ = 1.15: t1 = 0.03, t2 = 0.08, t3 = 0.13; quantities m1 = 4.74, m2 = 3.78, m3 = 3.12; profits Π1 = 10.91, Π2 = 6.58, Π3 = 4.22; V = 59.67; Φ = 0.009; V when t1=t2=t3 = 59.64.
    - For γ = 1.30: t1 = 0.28, t2 = 0.29, t3 = 0.30; m1 = 4.46, m2 = 3.40, m3 = 2.81; Π1 = 7.15, Π2 = 4.10, Π3 = 2.77; V = 61.15; Φ = 0.128; V when equal = 61.14.
    - For γ = 1.45: t1 = 0.38, t2 = 0.38, t3 = 0.38; m1 = 4.29, m2 = 3.16, m3 = 2.59; Π1 = 5.71, Π2 = 3.10, Π3 = 2.09; V = 63.68; Φ = 0.262; equal V = 63.68.
    - For γ = 1.60: t1 = 0.44, t2 = 0.43, t3 = 0.42; m1 = 4.11, m2 = 2.99, m3 = 2.51; Π1 = 4.74, Π2 = 2.55, Π3 = 1.83; V = 66.74; Φ = 0.367; equal V = 66.73.
    - For γ = 2.50: t1 = 0.56, t2 = 0.53, t3 = 0.52; m1 = 3.63, m2 = 2.65, m3 = 2.11; Π1 = 2.91, Π2 = 1.65, Π3 = 1.06; V = 89.03; Φ = 0.631; equal V = 88.87.
  - Comparative performance:
    - Values of social welfare are uniformly higher under the ad valorem tax than under the specific tax.
    - Φ (fraction of the gain toward the Ideal Outcome) is much higher for ad valorem than for specific tax (compare Tables 2 and 3 in source).
  - Last-column observation: social welfare is not very different when all ad valorem taxes are constrained equal for these parameters.

### V. ASYMMETRIC OLIGOPOLISTS AND THE HYBRID PROFITS TAX
- Setup and firm profit expression:
  - After-tax profit (equation (30)): Πj ≡ (1 − τj) mj Pm − Cj = (1 − τj) mj Pm − (wj αjℓj + r αjkj(1 − δ)) with notation as in source (αj, ℓj, kj, w, r, δ < 1).
  - First-order condition: (1 − τj) P − CMj = 0 (equation (31)).
  - Derivative results: equations (32) and (33) for dmj/dti and related terms.
  - Change in total market quantity in response to τi: equation (34), depends positively on αi, τi, and ki; sign < 0.
- Tax receipts and welfare:
  - Total tax receipts: equation (35) R ≡ Σj [τj mj Pm − (1 − δ)τj kj r αj] (as given in source expression).
  - Analytical tractability is limited; simulation used for insights.
- Simulation results (Table 4 highlights; δ = 0.5 unless noted):
  - Optimal profits tax rates are high relative to typical practice:
    - For γ = 1.15: τ1 = 0.49, τ2 = 0.47, τ3 = 0.48; m1 = 4.51, m2 = 3.76, m3 = 3.36; Π1 = 5.18, Π2 = 3.75, Π3 = 2.93; V = 60.63; Φ = 0.110; equal-rate V = 60.63.
    - For γ = 1.30: τ1 = 0.63, τ2 = 0.61, τ3 = 0.62; m1 = 4.47, m2 = 3.70, m3 = 3.27; Π1 = 3.70, Π2 = 2.67, Π3 = 2.03; V = 62.91; Φ = 0.272; equal-rate V = 62.91.
    - For γ = 1.60: τ1 = 0.73, τ2 = 0.73, τ3 = 0.74; m1 = 4.47, m2 = 3.60, m3 = 3.12; Π1 = 2.69, Π2 = 1.75, Π3 = 1.27; V = 68.45; Φ = 0.454; equal-rate V = 68.45.
    - For γ ≥ 1.90: optimal τs produce τ1 = 0.83, τ2 = 0.86, τ3 = 0.98 and the least efficient firm’s after-tax profit Π3 = 0.00 (exit), V and Φ increase across γ (e.g., γ = 2.50: V = 90.10; Φ = 0.654).
  - Key conclusions drawn:
    - Profits tax rates are much higher than conventional observed rates (often ~30 percent).
    - Rates are significantly lower than 100 percent that would apply to pure profits if δ = 1.
    - Imperfect deductibility of capital (δ = 0.5) substantially reduces welfare relative to δ = 1 (example: for γ = 1.3, δ = 1 yields V = 66.71 and Φ = 0.583 versus V = 62.91 and Φ = 0.272 for δ = 0.5).
    - Optimal τ is approximately constant across firms for a given γ, except where high τ leads to least efficient firm exit (γ ≥ 1.9).
    - Sensitivity to production parameter ε: memo rows show alternative ε values:
      - Memo ε = 0.05, γ = 1.30: τ1 = 0.90, τ2 = 0.88, τ3 = 0.88; m1 = 4.65, m2 = 4.15, m3 = 3.89; Π1 = 1.08, Π2 = 1.03, Π3 = 0.91; V = 75.44.
      - Memo ε = 0.5, γ = 1.30: τ1 = 0.48, τ2 = 0.48, τ3 = 0.51; m1 = 4.43, m2 = 3.46, m3 = 2.85; Π1 = 5.10, Π2 = 3.11, Π3 = 1.99; V = 57.86.
  - Welfare comparisons:
    - Social welfare is always higher with the profits tax than with either sales tax alone for the simulated parameterization.
    - Only for highest γ values do Φ values for ad valorem and profits taxes lie near each other.
  - Outputs:
    - Outputs of all firms fall as γ rises until the least efficient firm exits; then outputs of remaining firms jump up.
    - Aggregate after-tax profits of all firms fall as γ rises.

### VI. ASYMMETRIC OLIGOPOLISTS AND THE HYBRID PROFITS AND AD VALOREM TAXES
- Combined-tax setup:
  - Total after-tax profits and total tax receipts used in simulations are given by equations (36) and (37) as in the source:
    - Profit expression combining τj and tj with production and factor cost terms (equation (36) structure).
    - Tax receipts expression R using τj and tj and factor cost terms (equation (37) structure).
  - Simulation method: grid search over taxes 0 to 0.99 in steps of 0.11, then refine in adjacent grid regions in 0.01 steps.
- Simulation results (Table 5 highlights):
  - The two-tax package outperforms any single tax but only modestly beyond the best single tax (the hybrid profits tax).
  - For γ = 1.15:
    - τ: (0.49, 0.47, 0.47); t: (0.00, 0.00, 0.01); total tax per unit T1/m1 = 1.26, T2/m2 = 1.06, T3/m3 = 1.01; m = (4.51, 3.77, 3.33); Π = (5.20, 3.76, 2.92); V = 60.63; Φ = 0.110.
  - For γ = 1.30:
    - τ: (0.64, 0.57, 0.49); t: (0.00, 0.10, 0.20); T/m = (1.78, 1.61, 1.61); m = (4.82, 3.62, 2.64); Π = (4.17, 2.53, 1.42); V = 63.04; Φ = 0.283.
  - For γ = 2.50:
    - τ: (0.75, 0.62, 0.50); t: (0.30, 0.42, 0.47); T/m = (3.24, 3.07, 3.01); m = (4.76, 2.64, 1.54); Π = (1.98, 0.77, 0.31); V = 93.43; Φ = 0.726.
  - Key comparative findings:
    - The package increases Φ by a little over 10 percent at the highest γ values relative to best single tax.
    - The package still falls materially short of the Ideal Outcomes (Table 1 in source), especially for low γ but even by almost 30 percent for highest γ.
    - Profits tax rates fall as firm efficiency falls; ad valorem tax rates rise as firm efficiency falls uniformly across γ for these simulations.
    - Sum of profits and ad valorem taxes per unit output (Ti/mi) falls as firm efficiency falls.
    - Outputs of the two less efficient firms fall as γ rises; output of most efficient firm rises up to γ = 1.6 (when its ad valorem tax becomes positive for first time); after-tax profits of all firms fall as γ rises.

### VII. CONCLUSIONS
- Overall findings:
  - The model investigates optimal firm-specific taxation under asymmetric firm efficiencies in an oligopoly.
  - There is no single universal pattern: even for one tax imposed alone (e.g., specific sales tax), optimal taxes may rise or fall with firm size depending on γ (social valuation of tax receipts).
  - For the simulated parameterization when the standard two-tax package (ad valorem sales tax + hybrid profits tax with imperfect capital deductibility) is used:
    - The profits tax tends to fall with firm size, while the ad valorem tax rises with firm size.
  - No theoretical or simulated evidence for a hump-shaped optimal tax pattern (intermediate firms taxed most) was found under these assumptions.
  - Practical tax policy implications would need to combine these modeling results with considerations of collection costs and exemptions (e.g., Keen and Mintz (2004) on optimal VAT thresholds).
- General caveat:
  - Results depend on the parameterization, particularly the degree of imperfect competition and existence of pure profits; in perfect competition the hybrid profits tax would have no role if sales taxes are available.

_Italic: Content based solely on the supplied PDF section._

---


_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06271.pdf_