We initiate the study of congestion games with variable demands in which the players strategically choose both a nonnegative demand and a subset of resources. The players’ incentives to use higher demands are stimulated by nondecreasing and concave utility functions. The payoff for a player is defined as the difference between the utility of the demand and the associated cost on the used resources. Although this class of noncooperative games captures many elements of real-world applications, it has not been studied in this generality in the past. Specifically, we study the fundamental problem of the existence of pure Nash equilibria, PNE for short. We call a set of cost functions consistent if every congestion game with variable demands and cost functions from the set possesses a PNE. We show that only affine and homogeneous exponential functions are consistent. En route, we obtain novel characterizations of consistency for congestion games with fixed but resource-dependent demands.