Perturbation solution and welfare costs of business cycles in DSGE models

Lucas (1987) argues that the removal of cyclical fluctuations would barely improve economic welfare. He considers a risk-averse consumer valuing exogenously given streams of consumption Ct=Cegte−σ2/2+σϵt driven by an iid sequence of draws ϵt from a standard normal distribution. This allows him to change the size σ2 of the fluctuations without changing the mean of the process. In production economies, too, uncertainty is typically introduced in form of multiplicative, log-normally distributed shocks so that mean-preserving spreads can be analyzed in an analogous way. However, only few stochastic dynamic general equilibrium (DSGE) models admit an analytic solution. The most prevalent method to solve these models, perturbation methods, obtain an approximation to the stochastic model by perturbing the solution of the model’s deterministic counterpart with respect to the uncertainty parameter σ. Yet, widely available formulae to compute perturbation solutions are based only on the perturbation of σϵt and do not adequately capture the additional deviation −σ2/2 in the mean between the stochastic model and its deterministic version. We show within a model admitting an analytical solution that a second-order approximation of the welfare criterion also requires to perturb the mean. Thus, welfare measures based on the standard procedures for second-order solutions are (seriously) biased by a purely exogenous mean effect. We develop a general procedure of computing second-order accurate approximations of welfare gains or losses in the canonical DSGE model by extending the computation of second-order solutions pioneered by Schmitt-Grohé and Uribe (2004) to allow for mean preserving increases in the size of shocks. We apply our method to the model considered by Cho et al. (2015) and show that different from the results reported by these authors removing the cycle is always welfare improving. Welfare measures computed from weighted residuals methods confirm the logic behind our perturbation approach and verify the accuracy of our estimates.