 # A polynomial optimization approach to principal-agent problems

Renner, Philipp Johannes; Schmedders, Karl (2013). A polynomial optimization approach to principal-agent problems. Swiss Finance Institute Research Paper 12-35, University of Zurich.

## Abstract

This paper presents a new method for the analysis of moral hazard principal-agent problems. The new approach avoids the stringent assumptions on the distribution of outcomes made by the classical first-order approach and instead only requires the agent's expected utility to be a rational function of the action. This assumption allows for a reformulation of the agent's utility maximization problem as an equivalent system of equations and inequalities. This reformulation in turn transforms the principal's utility maximization problem into a nonlinear program. Under the additional assumptions that the principal's expected utility is a polynomial and the agent's expected utility is rational in the wage, the final nonlinear program can be solved to global optimality. The paper also shows how to first approximate expected utility functions that are not rational by polynomials, so that the polynomial optimization approach can be applied to compute an approximate solution to non-polynomial problems. Finally, the paper demonstrates that the polynomial optimization approach, unlike the classical approach, extends to principal-agent models with multi-dimensional action sets.

## Abstract

This paper presents a new method for the analysis of moral hazard principal-agent problems. The new approach avoids the stringent assumptions on the distribution of outcomes made by the classical first-order approach and instead only requires the agent's expected utility to be a rational function of the action. This assumption allows for a reformulation of the agent's utility maximization problem as an equivalent system of equations and inequalities. This reformulation in turn transforms the principal's utility maximization problem into a nonlinear program. Under the additional assumptions that the principal's expected utility is a polynomial and the agent's expected utility is rational in the wage, the final nonlinear program can be solved to global optimality. The paper also shows how to first approximate expected utility functions that are not rational by polynomials, so that the polynomial optimization approach can be applied to compute an approximate solution to non-polynomial problems. Finally, the paper demonstrates that the polynomial optimization approach, unlike the classical approach, extends to principal-agent models with multi-dimensional action sets.