Multi-step perturbation solution of nonlinear differentiable equations applied to an econometric analysis of productivity

Fourth-order multi-step perturbation (MSP) is described and applied as a general method for numerically solving nonlinear, differentiable, algebraic equations which are first-order conditions of economic optimization problems. MSP is first described at a general level and is, then, applied to estimating production-function models, using annual US total manufacturing KLEMS data from 1949 to 2001. The application continues by comparing total factor productivity based on the best estimated model with standard Solow-residual productivity. The optimization problem is the classic firm problem of maximizing output for a given production function, given input prices, and a given cost of inputs. If started sufficiently closely to the correct solution, usual iterative methods, such as quasi-Newton methods, can quickly compute accurate solutions of such problems. However, finding good starting points can be difficult, especially in high-dimensional problems. By contrast, MSP automatically provides a good starting point and iterates a finite number of times over preset steps so that, unlike in usual iterative methods, convergence or divergence is not an issue. Although, as in any numerical method, MSP accuracy is limited by the problem’s condition and floating-point accuracy, in practice, at least as implemented here, MSP can quickly compute solutions of nearly single-precision or higher accuracy.