We study linear bilevel and pricing problems in which the upper- and lower-level constraints’ right-hand sides are perturbed. In this setting, it is an important question, also for the validity of numerical solution schemes, if the solution-set mapping of the parametric bilevel problem is calm at the zero-perturbation. We provide the complete picture both for linear bilevel as well as for pricing problems. If the result is positive or not depends on whether the problems have coupling constraints or not and on whether the perturbation is allowed to take place in both levels and if they lead to relaxations or tightenings of the respective constraints. In particular, the solution-set mapping is calm for linear bilevel problems without coupling constraints and for pricing problems if the upper-level problem is not tightened by the perturbation. For the negative results, we provide illustrative counterexamples.