To demonstrate the difference between the transient responses of systems of varying complexity, a hierarchy of energy balance climate models is analyzed for abrupt and linearly increasing radiative forcings. Only the simplest case (a zero-dimensional mixed-layer ocean) exhibits the e-folding behavior that is a property of linear first-order systems. In each case considered, the natural time scales that occur in the systems are illustrated and physically interpreted. For the case of linearly increasing radiative forcing, the asymptotic solutions are derived by employing the Laplace transform approach. The time scales are related to the asymptotic solutions as illustrated in the text. In a companion paper (Morantine and Watts, this issue), analytical solutions for the transient responses valid for all values of time are presented for the models considered in this paper.