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The household composition matrix and its application to migration analysis and population projection
Abstract
A matrix showing the age composition of households is presented, and the impact of household migration on the age distribution of the population is simulated.
An indexing method, adopted from mathematical logic, is propounded to express any type of household composition as a single, natural number. Application of the indexing method yields a matrix whose elements denote average numbers of persons in given ages, in households with housing needs for a specified number of rooms. The matrix has similar properties to the household composition matrix introduced recently in connection with the projection of population and households. It is demonstrated that if the matrix is modified so as to fulfill the Stolper‐Samuelson condition, known from economic theory, then a simultaneous projection of population and housing needs can be performed.
Summary Composition of households by age of head and by age of other household members has recently been presented in a convenient algebraic expression, the household composition matrix. It has been shown that this matrix operates as a linear transformation from the vector of household distribution by age of head to the vector of population age distribution. A further analysis will show that the first row of the matrix may be interpreted as representing a vector of average household fertility rates. If the linear relationship between household and population distributions is fully implemented, then a relationship between household fertility and the size of the youngest age group can be derived. If w is the population age distribution and w (1) is the number of persons in the youngest age group, then: where α is the first row of the household composition matrix with its first element eliminated, C is the household composition matrix with its first row and first column eliminated, and Ψ is the vector w with its first element, w (1) eliminated. Extension of this result will enable simultaneous projection of population and households, suitable for computer application to conventional five-year age groups.
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