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1. Elasticity and sensitivity analyses are used widely in evolutionary biology, ecology and population management. However, almost all applications ignore density dependence, despite the widespread assumption that density dependence is ubiquitous. We assess whether this matters by comparing density-dependent and density-independent elasticity analyses for the LPA model of Tribolium. 2. Density-independent elasticities of λ are a poor indicator of the effects of changes in demographic parameters on population size, even for populations at stable equilibrium. With non-equilibrium dynamics, the divergence can be particularly large. In the extreme, a change in a demographic parameter with a positive effect on individual fitness can reduce mean population size, so even the sign of a density-independent elasticity may be wrong. Elasticities of larval, pupal and adult numbers are not proportional to each other, neither are they proportional to elasticities of total population size. 3. A full density-dependent analysis is therefore vital when concerned with effects on population numbers, as in population management, pest control and prediction of population effects of toxins. 4. When examining the consequences for individual fitness of changes in demographic parameters, density-independent elasticities provide a more useful approximation to the density-dependent values. However, they fail to detect cases where non-equilibrium dynamics means that particular life histories gain an advantage by exploiting predictable periods when density dependence is relaxed. 5. This phenomenon can produce a marked change in the pattern of elasticities as a bifurcation is crossed. The corresponding changes in selection pressures may act to stabilize dynamics in some circumstances and destabilize them in others. There is no single answer to the question of whether selection should favour equilibrium or non-equilibrium dynamics.