A new FDH‑RBF hybrid lets AI approximate non‑convex production frontiers more faithfully than convexity-constrained hybrids, improving frontier estimation where marginal returns rise; applying it to Chinese cities suggests medium-sized cities lag in efficiency.

Main Finding

The paper introduces FDH‑RBF, a hybrid method that combines Free Disposal Hull (FDH) preprocessing with a Radial Basis Function (RBF) network to estimate production frontiers without imposing convexity. FDH‑RBF preserves monotonicity while allowing non‑convex production possibility sets (PPS) and produces a smooth, differentiable frontier. In simulations (Cobb–Douglas setups) and an application to Chinese cities, FDH‑RBF outperformed convexity‑constrained DEA+ANN hybrids in approximating frontiers with increasing marginal returns and revealed a “medium‑sized efficiency trap” among Chinese cities.

Key Points

• Motivation

  • Standard DEA (CCR/BCC) enforces convexity, which implies nonincreasing marginal products and can be misspecified when technologies exhibit increasing marginal returns (non‑convex PPS).
  • Pure AI methods (ANN, RBF) can model non‑convex surfaces but are sensitive to training data that violate monotonicity (identical inputs with different outputs).
  • Prior DEA+ANN hybrids used DEA to enforce monotonicity, but that imports convexity into the AI stage and prevents capturing non‑convexities.

• Methodological innovation

  • Replace convexity‑imposing DEA preprocessing with FDH, which enforces free disposability and inclusion of observed DMUs but not convexity.
  • Train an RBF network on the FDH‑filtered (monotonicity‑preserving, non‑convex) reference set to obtain a smooth frontier that is differentiable (enabling marginal product analysis).
  • RBF chosen for its localized kernels and established approximation properties (Girosi & Poggio 1990), reducing spurious oscillations in sparse regions.

• Advantages claimed

  • Better approximation accuracy in settings with local convexity violations (e.g., increasing marginal returns, agglomeration economies).
  • Smooth, interpretable frontier (unlike stepwise FDH) permitting marginal product estimation.
  • Computationally efficient: FDH dominance checks are simple; RBF training complexity comparable to standard ANNs and avoids the heavy LP enumeration or facet elimination required by some non‑convex estimators.

• Empirical finding

  • Application to Chinese cities suggests agglomeration economies yield increasing marginal returns; FDH‑RBF uncovers a pattern where medium‑sized cities systematically underperform (a “medium‑sized efficiency trap”).

Data & Methods

• Preprocessing: Free Disposal Hull (FDH)

  • Uses pairwise dominance to retain monotonicity (free disposability) without imposing convexity.
  • Produces a non‑convex reference set of observed or FDH‑efficient points for training.

• Smoothing / Frontier estimation: Radial Basis Function (RBF) network

  • Trained on FDH reference-set points to produce a continuous, differentiable approximation of the frontier.
  • RBF’s localized kernels help avoid spurious global oscillations, especially in sparsely sampled regions.

• Benchmarking & validation

  • Simulation experiments using Cobb–Douglas production functions (including cases with increasing marginal returns) to compare FDH‑RBF against DEA‑based preprocessing + ANN and other baselines.
  • Application to real data: city‑level inputs/outputs in China to study agglomeration economies and efficiency patterns.

• Theoretical/algorithmic notes

  • FDH preserves only free disposability and data inclusion; avoids convexity bias.
  • RBF has theoretical best‑approximation properties for continuous functions in classes considered.
  • Complexity: FDH dominance checks are low‑cost; RBF training similar to standard ANNs; overall more tractable than combinatorial non‑convex frontier methods relying on many LPs or facet computations.

Implications for AI Economics

• Methodological

  • When the economic technology may be non‑convex (increasing marginal returns, strong agglomeration/network effects), researchers should avoid automatic convexity assumptions in preprocessing. FDH‑RBF provides a practical alternative that combines economic shape axioms (free disposability, monotonicity) with smooth ML approximators.
  • Smooth, differentiable frontiers from FDH‑RBF enable analysis of marginal products and other derivative‑based economic quantities (elasticities, marginal returns) that stepwise estimators (pure FDH or tree‑based methods) cannot deliver directly.
  • The framework is modular: other smoothers or shape‑constrained ML methods could be swapped for RBF if desired (the paper uses RBF for computational/theoretical convenience).

• Empirical & policy

  • For urban and regional economics (agglomeration, scaling laws) and sectors with network effects, FDH‑RBF can yield more accurate efficiency and frontier estimates, affecting policy inferences (e.g., where to invest, which city sizes to target).
  • The “medium‑sized efficiency trap” result (medium cities underperforming relative to small/large counterparts) suggests targeted policies may be needed to help medium cities capture agglomeration benefits.

• Practical guidance and caveats

  • Use FDH‑RBF when convexity is suspect; validate with simulations and holdout tests since model performance depends on sample coverage and noise structure.
  • Data quality matters: monotonicity violations in data (measurement error, heterogeneity) can still affect RBF training; FDH preprocessing reduces but does not eliminate all issues.
  • Consider robustness checks: alternative smoothers, cross‑validation, shape‑constrained ML approaches (e.g., monotone or concave/convex neural nets) and sensitivity to kernel choices.
  • Future research areas: asymptotic/statistical properties of FDH‑RBF, robustness to measurement error, comparisons with recently developed shape‑constrained ML frontier estimators, and scalability to very large n or high‑dimensional input spaces.

Overall, FDH‑RBF provides a principled hybrid for estimating non‑convex production frontiers while preserving monotonicity and yielding smooth, economically interpretable surfaces—particularly relevant whenever increasing marginal returns or other non‑convexities are plausible.