Demographics in Demographic—Economic Models: A Reply to Madden

In his response to our call for increased attention to the demographic aspects of demographic-economic models, Madden (1988) has formalised several aspects of the problem by presenting a mathematical comparison of the early Madden-Batey model (Madden and Batey, 1980) and the Hynes Jackson alternative (Hynes and Jackson, 1988). We welcome the increased formality of presentation. Madden's comments, however, imply that conceptual problems exist only in a special case. Because Madden's mathematical formulation is more precise than was our less formal note, we shall employ his notation throughout. Instead of focusing on semantic distinctions among phrases such as "drawn from", "derives from", and "are taken up by", which serve little purpose, we begin by repeating our initial concern with the Madden-Batey model, "... unemployed others are treated inconsistently ... . If the negative values in the household columns [a and /3 in Madden's notation] are equal, the total number of unemployed nonheads will never change. If these values are not equal, then an implausible statement is made about the demographic system" (Hynes and Jackson, 1988, page 1534). First, it is clear that we were aware that a and /? could have unequal values, and that the resulting system would be mathematically solvable. We were simply not satisfied with the conceptual interpretations and implications of unequal "unemployed other" coefficients. When the values of a and ft are not equal, the employment status of a nonhead will necessarily (and can only) change when the employment status of a household head changes. From Madden's equations (13)-(16), we can write