In this paper a family of functional iterations is introduced. One member of this family is the Newton-Raphson method and another member, obtained from a generalization of Steffensen's method to a system of equations, has been considered in [7]. The general member of the family is derived from a regulafalsi construction, due to Gauss, for a particular choice of points in the iteration. From the computational point of view, all the members of the family of iterations, except the Newton-Raphson method, have the property that the partial derivatives of the system of equations are used almost never if a computing device with unlimited precision is utilized. Further, the asymptotic speed of convergence for any member is at least of order two. In view of the difficulties of obtaining the functional form of the second order partials of the likelihood function for general linear and nonlinear simultaneous systems, the method proposed here may be recommended in the computation of full information maximum likelih Dod estimates. Even if the partials of the system of equations are easily calculated, then some member of the family may still lead to convergence if the Newton-Raphson method does not. Practically speaking, the proposed method can be used to determine an approximate solution and this approximate solution will be closer to the solution if the precision of the computations is higher.

In this paper the question of induced factor augmenting technical change in the context of the profit maximizing firm is addressed. The Kennedy innovation possibility frontier is employed to describe the opportunities available to the firm for factor augmentation and from it the direction of factor augmentation can be chosen. In addition, this opportunity curve can be shifted inwards or outwards according to the expenditure on research and development. The selection of the direction and extent of technical change is first determined via a myopic decision rule and then by maximization of the present value of the stream of net revenue from production and sale of output less the cost of technical advance. In the latter problem the maximum principle of Pontryagin is employed. Questions regarding the existence of stationary states and stability are resolved, and the optimal solutions are compared with the myopic decision rules. IN THIS PAPER further results relating Hicks' theory of induced technical change to the traditional theory of the profit maximizing firm are presented. The mechanism for technical change is disembodied factor augmentation instead of, as presumed in our earlier work, alteration of the parameters of the production function. It turns out, however, that many of the conclusions regarding the effects of technical progress are invariant to the way in which that progress is represented. The assumption of factor augmentation pursued here also yields results not obtained under the alternative model of parametric change studied earlier. Moreover, by using factor augmentation as the vehicle for representing technical change, our results are closely related to and can be compared with those of others working in the area of induced innovation, although those models are macroeconomic in focus [2, 3, 4, 5, 6, 7]. In our analysis we envisage a single firm to whom the state of technology is an endogenous variable, alterable at a positive cost; the firm strives to maximize a discounted profits stream over the indefinite but certain future. This problem differs from the one faced by the firm in the more traditional analysis insofar as the firm selects not only the optimal levels of the factors of production, and thereby the level of production, but the optimal technology as well. The desired optimal policies are derived from the appropriate maximization problem. These policies are then compared with those the firm would pursue if it were following a myopic policy of maximizing the instantaneous rate of profit growth under the same opportunities. In addition to discussion of the optimal rate and direction of technical change, the effects of technical change on the firm are also investigated. In particular, the impact upon the rate of production, cost of production, and factor shares are described.