Predicting economies of scale in Brazilian municipal expenditure: Panel econometrics and interpretable machine learning

Municipal size is a persistent question in urban planning and public finance because small jurisdictions may face high per capita costs, whereas larger jurisdictions may face congestion and administrative complexity. This study compares panel econometric and interpretable machine-learning models for real municipal operational expenditure per capita in Brazil. The analysis uses 49,228 municipality-year observations for 5,568 municipalities from 2013 to 2021 and evaluates prediction through municipality-holdout, temporal-holdout, and state-block designs. Histogram Gradient Boosting performs best for unseen municipalities and held-out states, while a municipality fixed-effects-style predictor performs best for later-year prediction of municipalities already observed in the panel. The results indicate strong economies of scale among small municipalities followed by a broad low-cost range rather than one universal optimum. Controlled econometric and spatial-error models imply minima near 143,000 to 176,000 residents, while a reduced-form pooled OLS model excluding grants implies about 232,000 residents. The strongest boosting models show no clear interior upturn within the evaluated grid, with predicted expenditure remaining within 10% of the minimum above approximately 48,000 to 53,000 residents. Grouped SHAP results rank population first, followed by grants and local economic capacity. Residual Moran tests show that unexplained expenditure differences remain spatially clustered. The findings support spatially informed expenditure benchmarking and caution against consolidation policies based on one fixed population threshold.