ABSTRACT: The Scarf algorithm was the first practical, almost surely convergent method for computing general equilibria of competitive models. The current focus of much computational research is computing equilibrium of dynamic stochastic models. While many of these models are examples of Arrow–Debreu equilibria, Scarf's algorithm and subsequent homotopy methods cannot be applied directly, because they have an infinite number of commodities. Many methods have been proposed for solving dynamic models and some work well on simple examples. However, all have convergence problems and are not likely to perform as well in models with heterogeneous agents, multiple goods, joint production, and other features often present in general equilibrium models. This paper discusses weaknesses of standard methods for solving dynamic stochastic models. We then present an alternative Negishi-style approach that combines convergent methods for solving finite systems of equations with convergent dynamic programming methods to produce more reliable algorithms for dynamic analyses. The dynamic programming step presents the key challenge, because most practical dynamic programming methods have convergence problems, but we argue that shape-preserving approximation methods offer a possible solution.

The Scarf (1967) algorithm was the first practical, surely convergent method for computing general equilibrium prices and, equivalently, systems of nonlinear equations in ℝn. This was followed by the development of almost surely convergent homotopy methods for solving nonlinear equations. This work gave us reliable and efficient methods for solving finite-dimensional systems of equations.