In the pursuit of the optimal level of debt that maximizes the firm's value, the predecessors to Modigliani and Miller (MM, 1958) acknowledged the existence of an optimum point, but they failed to calculate it. This failure primarily stems from their inability to find the mathematical expression...

This paper proposes an analytic representation of sequence-space Jacobians in heterogeneous agent models with aggregate shocks in continuous time. Our approach is based on a pen-and-paper perturbation of individual policy functions with respect to price changes, rather than numerical or...