A statistical thermodynamic model of transcriptional regulation is employed to investigate the likely effects of genetic variation on the stabilization of gene expression. The model is tailored to empirical data on the control of transcription of the hunchback gene by the morphogen Bicoid during Drosophila embryogenesis. Variable parameters include the number of binding sites for activator protein and the DNA-protein and protein-protein cooperative binding energies. Recursions are performed to derive transcriptional response curves over a range of concentrations of activator. Sigmoidal responses are indicative of threshold-dependent activation of gene expression, and the effects of variation of the parameters on the width and location of the threshold are considered. It is shown that there is a minimum threshold width (maximum switch sensitivity) that is a function of the number of binding sites and the level of response desired, but independent of the binding energies. This places a constraint on the evolution of sensitive genetic switches that generate discrete cell types. Inevitable trade-offs between threshold widths and locations for multiple target genes of most transcriptional activators are found to occur. These naturally lead to epistatic and pleiotropic effects, and may favor the generation of networks of compensatory mutations that together produce homeostatic developmental pathways.