tailieunhanh - Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 5

Chapter 5 Linear Quadratic Dynamic Programming . Introduction This chapter describes the class of dynamic programming problems in which the return function is quadratic and the transition function is linear. This specification leads to the widely used optimal linear regulator problem | Chapter 5 Linear Quadratic Dynamic Programming . Introduction This chapter describes the class of dynamic programming problems in which the return function is quadratic and the transition function is linear. This specification leads to the widely used optimal linear regulator problem for which the Bellman equation can be solved quickly using linear algebra. We consider the special case in which the return function and transition function are both time invariant though the mathematics is almost identical when they are permitted to be deterministic functions of time. Linear quadratic dynamic programming has two uses for us. A first is to study optimum and equilibrium problems arising for linear rational expectations models. Here the dynamic decision problems naturally take the form of an optimal linear regulator. A second is to use a linear quadratic dynamic program to approximate one that is not linear quadratic. Later in the chapter we also describe a filtering problem of great interest to macroeconomists. Its mathematical structure is identical to that of the optimal linear regulator and its solution is the Kalman filter a recursive way of solving linear filtering and estimation problems. Suitably reinterpreted formulas that solve the optimal linear regulator also describe the Kalman filter. - 107 - 108 Linear Quadratic Dynamic Programming . The optimal linear regulator problem The undiscounted optimal linear regulator problem is to maximize over choice of ut 0 the criterion - x tRxt utQut t o subject to xt 1 Axt But x0 given. Here xt is an n x 1 vector of state variables ut is a kx 1 vector of controls R is a positive semidefinite symmetric matrix Q is a positive definite symmetric matrix A is an n x n matrix and B is an n x k matrix. We guess that the value function is quadratic V x -x Px where P is a positive semidefinite symmetric matrix. Using the transition law to eliminate next period s state the Bellman equation becomes -x Px max -x Rx u Qu Ax