tailieunhanh - Recursive macroeconomic theory, Thomas Sargent 2nd Ed - Chapter 3

Chapter 3 Dynamic Programming This chapter introduces basic ideas and methods of dynamic programming. 1 It sets out the basic elements of a recursive optimization problem, describes the functional equation (the Bellman equation), presents three methods for solving the Bellman equation | Chapter 3 Dynamic Programming This chapter introduces basic ideas and methods of dynamic It sets out the basic elements of a recursive optimization problem describes the functional equation the Bellman equation presents three methods for solving the Bellman equation and gives the Benveniste-Scheinkman formula for the derivative of the optimal value function. Let s dive in. . Sequential problems Let p G 0 1 be a discount factor. We want to choose an infinite sequence of controls ut 0 o to maximize 00 2 r xt ut t 0 subject to xt 1 g xt ut with x0 given. We assume that r xt ut is a concave function and that the set xt 1 xt xt 1 g xt ut ut G Rk is convex and compact. Dynamic programming seeks a time-invariant policy function h mapping the state xt into the control ut such that the sequence us 0 o generated by iterating the two functions ut h xt xt i g xt ut starting from initial condition x0 at t 0 solves the original problem. A solution in the form of equations is said to be recursive. To find the policy function h we need to know another function V x that expresses the 1 This chapter is written in the hope of getting the reader to start using the methods quickly. We hope to promote demand for further and more rigorous study of the subject. In particular see Bertsekas 1976 Bertsekas and Shreve 1978 Stokey and Lucas with Prescott 1989 Bellman 1957 and Chow 1981 . This chapter covers much of the same material as Sargent 1987b chapter 1 . 82 Sequential problems 83 optimal value of the original problem starting from an arbitrary initial condition x G X. This is called the value function. In particular define 00 V x0 max 5 3r xt ut us x n 1 n 0 t o where again the maximization is subject to xt 1 g xt ut with x0 given. Of course we cannot possibly expect to know V x0 until after we have solved the problem but let s proceed on faith. If we knew V x0 then the policy function h could be computed by solving for each x G X the problem max

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