ABSTRACT
Consider a group of commodities that have in common ingredients which satisfy certain requirements. Thus the minimum cost diet assumes that people require minimal amounts of certain nutrients. Different foods contain different amounts of these nutrients so a particular food can be described by its nutrients. This poses the problem of finding the cheapest collection of foods that can satisfy these minimal nutritional requirements. Indicator functions are useful for solving this problem. Let there be n basic requirements. The indicator function Φ(S) is n-vector with coordinates equal to 0 or 1. If the commodity S has characteristic j, then the jth coordinate of Φ(S) is 1 and if it does not have characteristic j, then the jth coordinate of Φ(S) is 0. Therefore, the number of different commodities defined by the presence or absence of a characteristic is 2 n −1, excluding the commodity without any characteristics. We may say two commodities S and T are complements if their indicator functions are orthogonal. This means Φ(S)'Φ(T) = 0 if and only if S ⋂ T = Ø and S ⋃ T = N, where N is the commodity with all n characteristics that all coordinates of Φ(N) are 1. This notation can also measure the degree to which two commodities are substitutes using the scalar product Φ(S)'Φ(T). This measure is between 0 and the number of their common features. Hence if S and T have n − 1 features in common so that either S or T must posses all n features and the other n − 1 features, then their scalar product is n − 1. It is the maximal degree of substitutability that can occur in this model. Substitutability between two commodities is least when their indicator functions are orthogonal so that the two commodities have nothing in common.
