The mathematical notion of the production function is developed to relate input and output rates for unconstrained technological possibilities. It is shown that the structure of production may be expressed interchangeably, either by postulating the existence of a production functionΦ (X) satisfying certain properties or by a family of production possibility setsL (U), whenU is output rate,X is a vector of input rates,L (U) is the set of input rate vectors yielding at leastU, and\(\Phi (X) = \mathop {Max U}\limits_{L(U) \supset X} \). A class of production functions called homothetic is defined. This class is particularly useful in economic studies.