Optimal Lending Rate under Endogenous Credit Risk, Reserve Requirements, and Demand Elasticity: Closed-Form Solutions and Analytical Regime Classification

This paper develops an integrated theory of the optimal lending rate from the perspective of a lender who maximizes expected real profit. The model incorporates expected inflation, endogenous delinquency, operating costs, reserve-requirement-adjusted funding cost, competition, collateral recovery, and taxes. Three analytical regimes are distinguished: an elementary exact regime, a closed-form regime using the Lambert W function, and a finite local closure regime using nilpotent algebra. The work organizes these regimes into a single architecture, derives closed-form formulas and comparative-statics results, and proposes a path for empirical validation in the Dominican Republic. The classical literature on Fisher, Ramsey, Stiglitz-Weiss, credit risk, and corporate finance provides the conceptual building blocks integrated here into a single analytical structure, using elementary calculus (Regime I), the Lambert W function (Regime II), and nilpotent operational algebra SON (Regime III). For calibrated Dominican parameters, the model predicts an optimal commercial lending rate of approximately 24–25%, closely matching observed averages of 22–26% during 2022–2024.