In this paper, we investigate the efficiency of response-adaptive locally optimum designs. We focus on two-stage adaptive designs, where after the first stage the accrued data are used to determine a locally optimum design for the second stage. On the basis of an explicit expansion of the information matrix, we compare the variance of the maximum likelihood estimates obtained from a two-stage adaptive design and a fixed design without adaptation. For several one-parameter models, we provide explicit expressions for the relative efficiency of these two designs, which is seen to depend sensitively on the statistical problem under investigation. In particular, we show that in non-linear regression models with moderate or large variances the first-stage sample size of an adaptive design should be chosen sufficiently large in order to address variability in the interim parameter estimates. These findings support the results of recent simulation studies conducted to compare adaptive designs in more complex situations. We finally present an application to a real clinical dose-finding trial aiming at the estimation of the smallest dose achieving a certain percentage of the maximum treatment effect by using a three-parameter Emax model. Copyright © 2012 John Wiley & Sons, Ltd.