Designs are explored that minimize the asymptotic variance of a single parameter in a dose-response study designed to estimate this parameter. An example is a design to find the dose producing 50% response. Uncertainty of parameter values of the dose-response curve is represented as a normal prior distribution. Because the integration of the criterion over the prior distribution is analytically untractable, numeric methods are used to find good designs. The extension to multi-stage experiments is straightforward. The normal prior distribution coupled with the asymptotically normal likelihood yields a normal posterior distribution that is used to optimize the succeeding stage. Simulation results suggest that the asymptotic methods are a good reflection of small sample properties of the designs, even with modest-sized experiments. If initial uncertainty of the parameters is large, two-stage designs can produce accuracy that would require a sample size fifty percent greater with a single-stage design.