Many seemingly different questions that interest demographers can be phrased as the same technical question: how, within a given demographic model, would variable "y" change if the age- or time-specific function "f" were to change arbitrarily in shape and intensity?  At present demography lacks the machinery to answer this question in analytical and general form. 

This paper suggests a method, based on modern functional calculus, for deriving closed-form expressions for the sensitivity of demographic variables to changes in input functions or schedules.  It uses this "causal linkage method" on three bodies of theory: stable population analysis, nonstable or transient population analysis, techniques for the estimation of incomplete demographic data. 

In stable theory, closed-form expression are obtained for the response of the intrinsic growth rate, birth rate, and age composition to  arbitrary marginal changes in the age patterns of fertility and mortality. 

In nonstable theory, expressions are obtained for the transient response of the age composition to time-varying changes in the birth sequence, and to age-specific fertility and mortality patterns.  The  problem of "bias" in period vital rates is also looked at.