Linear-Quadratic Approximation, Efficiency and Target-Implementability∗

We examine linear-quadratic (LQ) approximation of stochastic dynamic optimization problems in macroeconomics (and elsewhere), in particular in policy analysis using Dynamic Stochastic General Equilibrium (DSGE) models. We first examine the approximation to a general dynamic programming problem, which can be obtained via the Hamiltonian of the problem. We then define the problem that is solved by a social planner, given that the objective of the latter is to maximize average welfare; this yields the efficient solution. We move from this to the Ramsey problem, which has both the resource constraints and choices of the social planner, and in addition the constraints due to decentralized decision-making. We then comment on the LQ approximation when a tax or subsidy can be imposed such that the zero-inflation competitive steady state output level is equal to the efficient level. In addition we provide measures of smallness when there are ’small’ distortions between the efficient and the natural rate. We apply the Hamiltonian procedure to the ’large’ distortions Ramsey case of consumer habit. We then define the notion of Target-Implementability, which is also a sufficient condition for optimality, and test whether inflation is always zero at the optimum. Finally we apply all these ideas in the context of a two-bloc model. JEL Classification: E52, E37, E58