Endogenous Grid Method

Solving dynamic economic models with realistic heterogeneity, nonlinearities, occasionally binding constraints, and discrete-continuous choices often involves solving high-dimensional systems of nonlinear equations. The endogenous gridpoint method offers an efficient alternative to traditional numerical approaches for problems in which at least one policy function satisfies a first-order condition (Euler equation). It speeds up the solution of the Euler equation for the associated policy function by reversing the standard solution logic. Instead of solving for the choice that corresponds to a beginning-of-period state, it chooses a grid for the end-of-period endogenous state variables and uses the Euler equation to solve for the associated values of the beginning-of-period endogenous state variables. The intuition is possibly best conveyed in the case of an optimal saving problem. Rather than solving for next-period wealth a′(a) on a grid for current wealth a, it solves for its inverse a = a′ on an exogenous grid for a′. The advantage is that the Euler equation is often linear in, or a closed-form function of, a but not a′. This significantly reduces interpolation errors and computational costs by eschewing numerical root-finding and the associated repeated evaluation of the Euler equation and the associated expectation. An additional advantage in models with exogenous borrowing constraints is that the location of the constraint on a′ is known, and the corresponding value of initial wealth can be computed exactly. While the method was originally introduced in the basic, concave, and differentiable framework with a single endogenous state and control variable, the computational and accuracy gains relative to other methods increase with the model complexity. The endogenous grid method extends naturally to multidimensional models that satisfy concavity and differentiability. Typically, its gains more than offset the higher interpolation costs stemming from the fact that the method generates nonrectangular grids in the multidimensional case. The endogenous grid method can also be extended to a large class of nonconcave and nondifferentiable problems, such as those with both discrete and continuous choices, for which the Euler equation is only necessary for an optimum. The speed and accuracy gains of the endogenous grid method are extremely high for such problems compared to alternative solution methods, which typically involve some form of grid search.