Hundreds of papers have been written on the numerical solution of dynamic optimization problems based on the method of dynamic programming. These papers have appeared in journals in economics, engineering and operations research, applied mathematics, and other fields of applied science. Averaging 25 papers per year since the publication of Bellman ( 1957) would make the total number close to 1,000. A survey of ten papers on the subject can be found in Taylor and Uhlig ( 1990). My approach is to bypass the value function and the associated Bellman equation and to concentrate on the first-order conditions for optimum based on the Lagrange method. In previous chapters, I have demonstrated that knowledge of the value function is unnecessary in solving dynamic optimization problems. In this chapter, I show that the Lagrange method is computationally more efficient and more accurate in many applications. It is more efficient because resources are wasted to obtain and store information about the value function, except for a vector of some of its derivatives, which is the Lagrange function. By using the same computational effort without seeking the value function, one can make the numerical result more accurate. Because I rely on first-order conditions, I must check whether the solution obtained is a maximum by using the results presented in section 2.5 in case there is any doubt.
I discuss numerical methods for solving the first-order conditions of equations (2.15) and (2.16) in discrete time, or (7.21) and (7.22) in continuous time, and ignore the value function in dynamic programming. As a numerical solution of the Bellman equation for the value function V has been an active area of research for over three decades, a numerical solution of the two first-order conditions derived by the method of Lagrange multipliers is an important area of research if one wishes to solve dynamic optimization problems by that method. In sections 2.4 and 7.4, I have provided numerical methods for solving the first-order conditions of stochastic dynamic optimization problems in dis-