Optimal Taxation in an RBC Model: A Linear-Quadratic Approach

We reconsider the optimal taxation of income from labor and capital in the stochastic growth model analyzed by Chari et al. (1994, 1995), but using a linear-quadratic (LQ) approximation to derive a log-linear approximation to the optimal policy rules. The example illustrates how inaccurate “naive” LQ approximation — in which the quadratic objective is obtained from a simple Taylor expansion of the utility function of the representative household — can be, but also shows how a correct LQ approximation can be obtained, which will provide a correct local approximation to the optimal policy rules in the case of small enough shocks. We also consider the numerical accuracy of the LQ approximation in the case of shocks of the size assumed in the calibration of Chari et al. We find that the correct LQ approximation yields results that are quite accurate, and similar in most respects to the results obtained by Chari et al. using a more computationally intensive numerical method. ∗Presented by the second author as a Plenary Lecture at the 10th Annual Conference on Computing in Economics and Finance, Amsterdam, Netherlands, July 8-10, 2004. We thank Vasco Curdia and Mauro Roca for excellent research assistance, Ken Judd, Jinill Kim, Andy Levin and Willi Semmler for helpful comments, and the National Science Foundation for research suppport. Linear-quadratic (LQ) optimal-control problems have been the subject of an extensive literature. It is not clear, however, how likely it is that optimal policy problems with explicit microfoundations — that is, policy problems in which both the assumed objective of policy and the constraints on possible outcomes are derived from an explicit account of the decision problems of private agents — should take this form. Elsewhere (Benigno and Woodford, 2005b), we show that it is possible in a broad class of models to derive an LQ problem that locally approximates an exact policy problem, in the sense that the solution to the LQ problem represents a local linear approximation to the solution to the exact problem, that will describe it with arbitrary accuracy in the case of small enough random disturbances. It does not generally suffice for this purpose to define an LQ problem in which the objective is a local quadratic approximation to the exact objective and the constraints are local linear approximations to the exact constraints. Nonetheless, we show that it is quite generally possible to derive a correct LQ approximation, if sufficient care is taken in the choice of the quadratic objective. Here we illustrate both the potential problems with naive LQ approximation and the application of our own method in the context of a well-known example, the analysis of dynamic optimal taxation of income from labor and capital in an RBC model, treated by Chari et al. (1994). The example is of interest not only because it is a simple case in which naive LQ approximation would lead to extremely incorrect conclusions, but also because the paper of Chari et al. is often cited as evidence that log-linearization is dangerous in the context of optimal tax policy problems, even if it can be used with fair accuracy in other contexts (such as the approximate characterization of the aggregate fluctuations implied by an RBC model). In fact, Chari et al. use a minimum-weighted-residual method that is computationally more difficult than ours to numerically characterize the optimal dynamics of capital and labor taxes, and state that they do so because a log-linear approximation Important references include Bertsekas (1976), Chow (1975), Hansen and Sargent (2004), Kwakernaak and Sivan (1972), and Sargent (1987). Judd (1989) is an example of an earlier treatment of optimal taxation of capital and labor income using LQ methods. The problem with “naive” LQ approximation of this sort is discussed, for example, by Judd (1996, sec. 4; 1999, pp. 505-508), who argues for perturbation techniques as an alternative that avoids this problem. As noted above, Judd (1989) analyzes dynamic optimal taxation of income from labor and capital using LQ methods, but using an ad hoc quadratic objective that is not explicitly derived from microfoundations. See, e.g., Kim and Kim (2003) and Albanesi (2003).