Discovery of complex-valued discounting and logarithmic utility via the Laplace equation: Proof of natural emergence

This paper challenges the long-standing paradigm that discount factors capturing time preference and shadow price are real numbers, and that utility functions are essentially arbitrary choices. Here, we prove discount factors must be complex numbers regarding the ubiquitous log-utility function, which we derive directly from the Laplace equation based on first principles. Our derivation adopts a minimal-assumption approach, requiring only the primitive of utility maximisation. This constitutes the first proof of the necessity of the imaginary unit $i=\sqrt{-1}$ in economics. Akin to the shift from classical probability to quantum amplitude, extending economic analysis to the complex plane may resolve persistent behavioural anomalies. These findings encourage reflection and discussion on discovery versus invention in social and behavioural sciences. While economics often relies on theoretical assumptions that resemble invention, our elegant proof of complex discounting compatible with logarithmic utility highlights a process of natural discovery, making room for new analytical techniques and a paradigm shift in economic modeling.