Although cohabitation and childbearing within cohabitation have increased in Europe over recent decades, the variation across Europe remains remarkably wide. Most studies on union formation have not explicitly addressed the role of state policies in the development of cohabitation or discussed how countries have responded to changes in union formation by passing legislation. Here we discuss historical and theoretical issues relevant to the relationship between state policies and union formation and describe policies relating to cohabitation and marriage in nine Western European countries. Drawing on secondary sources and legal documents, we examine the quantity of regulations that mention cohabitation and the approach to cohabitation in 19 policy dimensions. We then place the countries along a continuum, from those that have equalized cohabitation and marriage to those that only regulate marriage. As a whole, this overview raises questions about the changing institution of marriage, as well as the increasing institutionalization of cohabitation.

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BACKGROUND Population projections using the cohort component method can be written as time-varying matrix population models. The matrices are parameterized by schedules of mortality, fertility, immigration, and emigration over the duration of the projection. A variety of dependent variables are routinely calculated (the population vector, various weighted population sizes, dependency ratios, etc.) from such projections. OBJECTIVEOur goal is to derive and apply theory to compute the sensitivity and the elasticity (proportional sensitivity) of any projection outcome to changes in any of the parameters, where those changes are applied at any time during the projection interval. METHODSWe use matrix calculus to derive a set of equations for the sensitivity and elasticity of any vector valued outcome ξ(t) at time t to any perturbation of a parameter vector θ(s) at any time s. RESULTSThe results appear in the form of a set of dynamic equations for the derivatives that are integrated in parallel with the dynamic equations for the projection itself. We show results for single-sex projections and for the more detailed case of projections including age distributions for both sexes. We apply the results to a projection of the population of Spain, from 2012 to 2052, prepared by the Instituto Nacional de Estadística, and determine the sensitivity and elasticity of (1) total population, (2) the school-age population, (3) the population subject to dementia, (4) the total dependency ratio, and (5) the economic support ratio. Caswell & Sánchez Gassen: The sensitivity analysis of population projections CONCLUSIONS Writing population projections in matrix form makes sensitivity analysis possible. Such analyses are a powerful tool for the exploration of how detailed aspects of the projection output are determined by the mortality, fertility, and migration schedules that underlie the projection.