Linear-quadratic approximation, external habit and targeting rules

We examine the linear-quadratic approximation of nonlinear dynamic stochastic optimization problems. A discrete-time version of Magill [1977a. A local analysis of N-sector capital accumulation under uncertainty. Journal of Economic Theory 15(2), 211-219] is generalized to models with forward-looking variables paying special attention to second-order conditions. This is the 'large distortions' case in the literature. We apply the approach to monetary policy in a DSGE model with external habit in consumption. We then develop a condition for 'target-implementability', a concept related to 'targeting rules'. Finally, we extend the approach to a comparison between cooperative and non-cooperative equilibria in a two-country model and show that the 'small distortions' approximation is inappropriate for this exercise.