Tax reform as a constrained optimization problem: a piecewise-linear framework and software implementation

Technical Summary: Tax Reform as a Constrained Optimization Problem

Problem Statement
Income tax codes in developed nations have evolved into complex, interacting systems of brackets, benefits, and deductions. This complexity creates an "implementation gap" where theoretical optimal-taxation principles (e.g., Mirrlees, 1971; Saez, 2001) diverge sharply from statutory reality. A primary symptom is the emergence of "marginal-rate spikes"—effective tax rates exceeding 80% for low- and middle-income households due to the simultaneous phase-out of multiple benefits. These spikes blunt work incentives and are often unintended by-products of ad-hoc historical reforms.

Existing tools for tax policy face three limitations:

1. Microsimulation models (e.g., EUROMOD, TAXSIM) evaluate analyst-supplied proposals but cannot generate them.
2. Numerical optimal-taxation models optimize abstract, smooth schedules under stylized assumptions, ignoring the specific statutory rules of a real code.
3. Reinforcement learning approaches return black-box policies without certifying optimality or identifying infeasibility.

Consequently, reform efforts often rely on incremental adjustments rather than systematic redesign, failing to meet stated objectives regarding efficiency, equity, and simplicity.

Methodology: A Piecewise-Linear Framework
The authors propose casting tax reform as a constrained optimization problem. The core modeling insight is that under four mild assumptions, any statutory income tax code can be reduced to a finite-dimensional, continuous, piecewise-linear function for each taxpayer group.

• Assumptions:

  1. Additivity: Total tax pressure is the sum of individual rule pressures.
  2. Scalar Input: Each rule levies taxes on a single input (e.g., income).
  3. Piecewise Linearity: Marginal pressure is constant on intervals; absolute pressure is a continuous, piecewise-linear function.
  4. Group Heterogeneity: Eligibility and parameters depend on taxpayer characteristics, partitioning the population into finite groups.

• Mathematical Formulation:
  The total tax liability $f_k(x_i)$ for a taxpayer $i$ in group $k$ is expressed as:
  $$f_k(x_i) = \sum_{j} \sum_{b \in \mathcal{B}_{j,k}} \alpha_{j,b,k} g_b(x_{ij}; \Phi_{j,k}) + Z_k$$
  where $\alpha$ are marginal rates, $g_b$ are bracketing functions defined by a support set $\Phi$ (cutoffs), and $Z_k$ is a lump-sum transfer.

• Optimization Structure:

  • Decision Variables: The legislatable parameters of the code: rates ($\alpha$), bracket cutoffs ($\Phi$), and lump-sum transfers ($Z$).
  • Constraints: Political and distributional requirements, including:
    • Income Guarantees: Bounds on net income changes (e.g., no household loses more than $x\%$).
    • Marginal Constraints: Caps on effective marginal tax rates.
    • Budget Constraints: Limits on total revenue loss/gain.
    • Rule Protection: Hard constraints to keep specific legacy rules intact.
  • Objectives: Minimizing revenue loss, minimizing system complexity (weighted count of active rules), or minimizing poverty gaps.

• Extensions:

  • Complexity: Modeled via a weighted-cardinality objective (minimizing active rules) using binary variables, transforming the problem into a Mixed-Integer Linear Program (MILP).
  • Dynamic Bracketing: Treating cutoff points as decision variables to select the optimal support grid.
  • Behavioral Responses: Incorporating elasticity of taxable income. This introduces a product of decision variables (rate $\times$ income response), resulting in a nonconvex Mixed-Integer Quadratically Constrained Program (MIQCP). The authors also provide a fixed-point iterative algorithm (Algorithm 1) as an LP-only alternative.
  • Infeasibility Certification: If constraints conflict, the solver returns an Irreducible Infeasible Subsystem (IIS), providing a certificate that no reform satisfies the specific set of commitments.

Key Contributions

1. Modeling Result: A proof (Proposition 1) that statutory codes satisfying the four assumptions reduce to a single piecewise-linear function per group, enabling reform to be framed as a Linear Program (LP) or MILP.
2. Inverse Recovery: A method (Proposition 2) to recover current system parameters from observed tax liabilities, with conditions for unique identifiability.
3. Reform-Optimization Framework: A library of constraints and objectives that allows policymakers to search the space of implementable codes for optimal candidates or receive a certificate of infeasibility.
4. Case Study: A near-complete reconstruction of the Dutch income tax code (15 rules, 13,500 weighted taxpayers) demonstrating the generation of certified reforms.
5. Software Implementation: The release of TaxSolver, an open-source, solver-agnostic Python package.

Results: The Dutch Case Study
The framework was applied to the Dutch tax system, characterized by high marginal-rate spikes (>80%) for many households. The authors generated reform candidates that:

• Smoothed Marginal Rates: Capped marginal pressure at 55–80%, eliminating the spikes caused by simultaneous benefit phase-outs.
• Protected Households: Guaranteed that no household's net income fell below 95% of its current level.
• Reduced Complexity: Roughly halved the number of active rules (from a complexity weight of 89 to ~32–39) via a lexicographic two-stage procedure (first minimizing revenue loss, then minimizing rule count within a bounded revenue slack).
• Budgetary Feasibility: Operated within a $\pm 1.5\%$ budget band, with a secondary objective allowing an additional 1% slack to purchase simplicity.

The study demonstrated that the framework could produce "certified" reforms—complete statutory proposals that are provably optimal within the modeled space—or identify when specific policy goals (e.g., extreme rate caps combined with strict income protection) are mutually incompatible.

Significance
The paper positions itself as a bridge between optimal-tax theory and the political reality of statutory reform. By treating the statutory code itself as the decision variable rather than an abstract schedule, the methodology offers a tractable way to navigate the "implementation gap." It provides a rigorous mechanism to:

• Generate reform proposals that satisfy hard political constraints (e.g., "no household loses income") rather than just optimizing a welfare function.
• Certify optimality or infeasibility, moving beyond heuristic searches that cannot prove a solution is best or that a problem is unsolvable.
• Quantify the trade-off between system complexity and fiscal outcomes, allowing policymakers to explicitly "buy" simplicity with controlled revenue losses.

The authors note that the methodology is currently in active use by the Dutch Ministry of Finance, developed in close collaboration with the authors, validating its practical utility in a real-world policy environment.