Scalable Allocation and Synchronous Growth of SMEs: A Priority Index with a Size Lens for Business Support Funds

This paper proposes an allocation rule for business support funds, especially for SMEs. The rule combines a value-per-euro index, a continuous weight (s/s̄)^(−λ) on firm size s, called a size lens, and a synchronous saturation of the budget. With diminishing returns within projects, the optimal allocation equalizes marginal returns across partially funded firms; ranking-and-filling is the limiting case and, with equal needs, spreads the budget over fewer firms. The size lens is the only continuous weight that is scale-coherent; its optimal exponent is minus the elasticity of additionality with respect to firm size, and the cost of misspecifying it grows with the dispersion of firm size. In a ranking mechanism, size-class intensities or weights create common notches in expected aid, whereas the lens creates a smooth slope or dispersed firm-specific thresholds. The few available per-euro estimates disagree on the sign of the size gradient, which must therefore be estimated programme by programme. A beneficiary-level description of Italian State aid shows that its high concentration arises mostly between measures. In a simulation calibrated on Italian manufacturing SMEs, the rule gains up to 22 percentage points of value relative to ranking-and-filling. Within a single call, a size lens and interpolated weights perform equally well and outperform class weights; as firms grow, they avoid bunching at the thresholds created by class weights, but when they favour larger firms they can reinforce eligibility thresholds set by law. Replication package: code (MIT licence), LaTeX sources, aggregate results and the public Eurostat data used. Italian version included.