No such thing as a perfect hammer: comparing different objective function specifications for optimal control

The linear-quadratic (LQ) optimization is a close to standard technique in the optimal control framework. LQ is very well researched and there are many extensions for more sophisticated scenarios like nonlinear models. Usually, the quadratic objective function is taken as a prerequisite for calculating derivative-based solutions of optimal control problems. However, it is not clear whether this framework is so universal as it is considered. In particular, we address the question on whether the objective function specification and the corresponding penalties applied, are well suited in case of a large exogenous shock an economy can experience because of, e.g., the European debt crisis. While one can still efficiently minimize quadratic deviations in state and control variables around policy targets, the economy itself has to go through a period of turbulence with economic indicators, such as unemployment, inflation or public debt, changing considerably over time. In this study we test four alternative designs of the objective function: a least median of squares based approach, absolute deviations, cubic and quartic objective functions. The analysis is performed based on a small-scale model of the Austrian economy and finds that there is a certain trade-off between quickly finding optimal solution using the LQ technique (reaching defined policy targets) and accounting for alternative objectives, such as limiting volatility in the economic performance.