Rejoinder to Professor Mckenzie

Abstract

PROFESSOR MCKENZIE objects to my contention that every cost function can be extended on the grounds that in some cases the set of input vectors producing unit output may not be bounded. In particular he quotes the where we may wish to consider disposal processes. Later the Cobb-Douglas production function is instanced as a simple case which does not fall within the scope of my theorem. Consider first the Cobb-Douglas function, since this will serve to illustrate very well the way I would deal with disposal processes. Notice first that, however realistic an approximation the Cobb-Douglas function may be to production functions of the real world over part of its range, it cannot possibly continue to be a good approximation to anything likely to be observed in a region where any factor price approaches zero. This is because in this region the number of units of input of at least one factor will begin to exceed the number of elementary particles in the universe. This absurdity is, of course, precisely why I thought it reasonable to assume the boundedness of the set of input vectors. But let us suppose (impossibly) that the laws of physics are truly reflected by the Cobb-Douglas function over the whole range and that no marginal product can ever be zero for a finite input of the factor. What we should then observe is not an infinite input of a factor but that the factor price could never attain the value zero. There would be a natural bound, different from zero, on the economically significant domain of factor prices and a corresponding natural bound on the quantity of input, both determined by the maximum quantity of the factor in existence in any one country. Suppose now we consider the reduced domain of definition of factor prices which will in this be contained in the positive orthant. Let S1 be the corresponding reduced set of minimum cost techniques. Now extend the cost function as required by defining it as the support function of the reduced set Si. If an inverse exists to the extended cost function, there must also be an inverse to the cost function thought of as defined only over the economically significant range. Disposal processes of any kind can be treated the same way. This leads to the following proposition.