Concept of indirect optimization of the Cobb-Douglas function via reciprocal-variable transformation: Case of unknown constraint

We investigate an indirect optimization problem of a Cobb-Douglas function under an unknown budget constraint. The proposed approach relies on a reciprocal transformation applied to the first-order multivariate marginal conditions, enabling the analytical reconstruction of an implicit constraint from observable aggregated variables. To facilitate the identification of model parameters, goods are grouped into representative aggregates, allowing the estimation of the corresponding exponents while preserving the structural consistency of the Cobb-Douglas specification. The approach does not alter the classical Lagrangian framework but introduces a structural reparametrization that renders the constraint identifiable when initially unknown. A numerical example based on the thermal optimization of a freezer illustrates the analytical consistency, practical applicability, and reproducibility of the proposed method.