In this paper we investigate the cost of complexity, which is defined as the minimum amount of input power required to estimate the frequency response of a given linear time invariant system of order n with a prescribed degree of accuracy. In particular we require that the asymptotic (in the data length) variance is less or equal to gamma over a prespecified frequency range [0, omega(B)]. The models considered here are Output Error models, with an emphasis on fixed denominator and Laguerre models. Several properties of the cost are derived. For instance, we present an expression which shows how the pole of the Laguerre model affects the cost. These results quantify how the cost of the system identification experiment depends on n and on the model structure. Also, they show the relation between the cost and the amount of information we would like to extract from the system (in terms of omega(B) and gamma). For simplicity we assume that there is no undermodelling. (C) 2011 Elsevier Ltd. All rights reserved.