Social change manifests itself in changing behavior. With respect to fertility, that means changes in the level and timing of age-specific birth rates. Demographic models with exponential change in the age pattern of fertility have frequently been used by demographers, but have not been closely examined. Here, in both continuous and discrete models, we emphasize the identities between the exponential pattern of age-change and the stable intrinsic rate of growth over time. Analytically, exponential age-change can be used to shift the mean age of fertility without changing the level of fertility, and to relate population momentum to an initial stable population’s growth rate. Exponential age-change in fertility is also shown to be closely linked to proportional age-change. Comparisons are made between those two principal analytical patterns under three types of constraints: a shared growth rate, a shared fertility level, and a shared birth sequence. Despite the conceptual difference between those approaches, new and significant relationships are found that link them together.