Estimation of Production Possibility Surface from Empirical Data

A production possibility surface is the frontier surface determined by the available resources in an economy and the technology applied for production. The surface is in the non-negative orthant, meaning thereby that neither the resources nor the quantity produces may take on a negative value. Mathematically, the shape of the surface is ellipsoidal. It is interesting to fit a production possibility surface to empirical data on two counts: first, it is an example of a nonlinear surface fitting that cannot appreciably be approximated by a simple linearization method, and second, that it is an example of the surface that cannot ordinarily be fitted by most of the conventional non-linear regression algorithms. These algorithms fail mostly because they approach the best fitting surface from both sides, interior and exterior of the surface, and thereby land themselves into the problem of finding square root of a negative quantity. Statistical fitting of ellipse (or ellipsoid) to empirical data remained unattractive for a long time. However, in the last two decades or so, it attracted many researchers especially to solve the pattern recognition problems. Bookstein (1979), Sampson (1982), Taubin (1991), Rosin (1993), Gander et al. (1994), Kanatani (1994), Pilu (1996) and Matei and Meer (2000) are some of the important works on this problem. In this paper we propose a new, simple iterative method for our limited purpose at hand. For an illustration, we estimate the parameters of the ellipse by fitting the parametric equations to the simulated data. A computer program (FORTRAN) implementing the proposed method is given