Solving RE models with discontinuous policy rules – an application to minimum wage setting in Germany

ABSTRACT

The legal regulations require the minimum wage in Germany to be adjusted biennially which gives rise to a policy discontinuity. From the perspective of rational expectations models, such policy features render standard local approximation techniques infeasible. The article presents a stylized model in which negotiated wages and corporate profits are the outcome of an optimization problem, while changes to the minimum wage are modelled by a discontinuous policy rule. Using the simple example of minimum wage setting in Germany, the article illustrates how such models can be solved using the method of undetermined coefficients and presents selected simulation results.

KEYWORDS:

• Rational expectations model
• discontinuous policy rule
• method of undetermined coefficients
• minimum wage setting

JEL CLASSIFICATIONS:

• E 10
• E 60

Acknowledgement

The author thanks Stephan Kohns, Hermann-Josef Hansen, Thomas Knetsch, Florian Kajuth, Daniel Radowski, Anke Wetzka, Thomas Haertel, Jochen Mankart, Stefan Goldbach, Markus Roth, Rachel Hand, Kilian Ruppert, and participants of Bundesbank’s Forschertreffen for very helpful comments. The opinions expressed in this article are those of the author and do not necessarily reflect the views of the Deutsche Bundesbank.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ The minimum wage commission comprises nine members: three representatives each from the representative organizations of the employers and employees, one chairperson, and two consulting (non-voting) members from academia.

² As a shortcut, continuous transition functions have frequently been used to model such non-linearities (see Rieth 2014 and Baldini and Ribeiro 2008; among others). Although easy to implement, important information may be lost when non-linearities are approximated in such a way. Consider, for instance, a model (as the one presented below) where some policy differs depending on the prevailing time period (even versus uneven). Applying a transition function to model this policy feature would then imply that the policy (in the steady state) refers to some ‘average period’, and neither to an even nor to an uneven period. While technically correct, this may be hard to justify from an economic point of view.

³ Subsequently referred to as ‘minimum wage rule’. Policymakers committing to policy rules are frequently encountered in economic modelling, in particular in the sphere of monetary and fiscal policy.

⁴ The negotiated wage setter can be thought of as an ‘aggregate’ of the employee and employer representatives.

⁵ For the sake of simplification, it is assumed that the number of negotiated wage and minimum wage recipients does not change over time such that and applies. Differences in wages could be motivated by heterogeneity of employees with respect to their labour productivity. The analyses abstract from the precise mechanisms that divide employees into recipients of the negotiated wage and minimum wage, respectively. Unemployment is also abstracted from.

⁶ The expression for results from solving and defining . It is assumed for simplicity that the ‘compound interest effect’, , is equal to zero.

⁷ The legal regulations on adjusting the minimum wage in Germany require the minimum wage commission to present biennial adjustment proposals to the Federal Government, which are subsequently implemented by the Federal Government. The negotiated pay rates index of the Federal Statistical Office (Tarifverdienstindex) is particularly important to the commission’s recommendations. However, the Federal Government is not bound to the commission’s biennial meeting schedule and implementations can in principle also occur on an annual basis. For adjustments based on , the stylized model assumes that the commission can condition its recommendations for the adjustment of the minimum wage in the year following its meeting on the development of the negotiated pay rates index.

⁸ As the minimum wage rule has been decided upon ex ante, the negotiated wage setter takes its functional form into account. It is assumed here that the minimum wage rule has already been substituted into the budget constraint (3).

⁹ Analyses based on quadratic loss functions are widespread in macroeconomics, in particular in the analysis of optimal monetary policy. See for applications Beck and Wieland (2007) and Gali and Monacelli (2005), among many others.

¹⁰ From an efficiency wage perspective, for instance, the wage-setting behaviour can also increase productivity and income. General equilibrium effects are, however, abstracted from which is standard in the literature that applies the method of undetermined coefficients.

¹¹ It is ensured that only meaningful parameter values are used for the simulations. A solution to the non-linear system is considered meaningful in an economic sense if positive income changes ceteris paribus also lead to positive changes in corporate profits (i.e. and ).

¹² The model was simulated for independent realizations of the income process each with dimension . The results presented are based on the average discounted total loss resulting from the simulations. The number of periods was selected such that all subsequent periods with are negligible for loss evaluation. In addition, the first 25% of the periods of each simulated process have been disregarded as ‘burn-in’ in order to ensure that the starting values required for the simulation of the paths do not have a significant impact on the qualitative results. The autocorrelation coefficient turned out not to be statistically significant for an AR(1) model with drift based on annual data. It is therefore not depicted in Table 1. Results are, however, similar when setting .

¹³ For details on the German Socio-Economic Panel Study (SOEP), see http://www.diw.de/en/soep.

¹⁴ The calibration has been chosen for expositional purposes: For , the negotiated wage setter endeavours to equally stabilize wages and corporate profits. Based on the assumption of risk-neutral employers and risk-averse employees, a negotiated wage setter representing both employees and employer representatives is likely to stabilize wages more strongly at the expense of corporate profits. This is the case for . Results are robust.