Abstract
Linear-quadratic (LQ) optimization is a fairly standard technique in the optimal control framework. LQ is very well researched, and there are many extensions for more sophisticated scenarios like nonlinear models. Conventionally, 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 as universal as it is considered to be. In particular, we address the question 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 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. 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 illustrates a certain trade-off between quickly finding an optimal solution using the LQ technique (reaching defined policy targets) and accounting for alternative objectives, such as limiting volatility in economic performance. As an implication, we argue in favor of the considerably more flexible optimization technique based on heuristic methods (such as Differential Evolution), which allows one to minimize various loss function specifications, but also takes additional constraints into account.
Highlights
Today, several countries in the European Union face difficulties in mitigating their public budget deficit and debt issues, which were triggered by the last economic crisis
We argue in favor of the considerably more flexible optimization technique based on heuristic methods, which allows one to minimize various loss function specifications, and takes additional constraints into account
The question that we address here is whether least median of squares (LMS)-style-shaped objective function serves the goal of mitigating instability due to a one-time shock or this approach may even increase the volatility of the resulting states and controls obtained by the optimal control exercise
Summary
Several countries in the European Union face difficulties in mitigating their public budget deficit and debt issues, which were triggered by the last economic crisis. For the Austrian economy (and other countries of the Euro zone) such an event has a one-time (nonrecurring) negative impact on the budget balance. The optimal control framework is a well-known tool to address such a fiscal policy question (see, e.g., Feichtinger and Hartl 1986; Neck et al 2008; Neck 2009). A ‘traditional’ way to consider optimal control problems is the linear quadratic (LQ) optimization technique. This technique is mainly based on works by Pontryagin et al (1962) and Bellman (1957). A squared outlier influences the objective function considerably, forcing an active use of control variables, which might be undesirable in certain situations. In case of a large exogenous shock, a policy maker faces an additional task of mitigating the effects of this shock without putting the stability of the whole system at risk
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