A note on computing partial derivatives of the value function by simulation

The problems involving incentive compatibility constrains in the form of participation constrains have received wide attention in the literature due to the recent advances in the dynamic optimization techniques. Often the optimality conditions for this class of problems involve partial derivatives of the optimal value function with respect to some of the endogenous state variables. In this note we suggest an algorithm for computing these partial derivatives by simulation. The attractive features of the algorithm include its rather wide scope of applicability and simplicity of implementation. Furthermore, the suggested method does not su¤er from the curse of dimensionality and therefore it is particularly convenient for the models involving many state variables. (JEL C63, E10) Keywords: dynamic participation constrains, numerical algorithm, simulation. 1 Introduction The purpose of this note is to propose a simple algorithm for computing partial derivatives of the optimal value function. The problems involving incentive compatibility constrains in the form of participation constrains have received wide attention in the literature due to the recent advances in the dynamic optimization techniques. Often the optimality conditions for this class of problems involve partial derivatives with respect to some of the endogenous state variables of the optimal value function corresponding to the dynamic programming formulation of an outside option. Although many numerical methods can provide an approximation for the value function, there is no reason to believe that a derivative of this approximation will be close in any sense to the actual value of the derivative. In this note we suggest an algorithm for computing these partial derivatives by simulation. This issue has been previously considered by Marcet and Marimon (1992) who solved numerically a growth model with capital accumulation under one-sided lack of commitment. To circumvent the problem of …nding the values of the derivatives Marcet and Marimon (1992) School of Economics, Australian School of Business, The University of New South Wales, Sydney NSW 2052, Australia. Tel.: (+61) 9385 3351; Email: [email protected]