The effective elasticity of highly charged stiff polyelectrolytes is studie
d in the presence of counterions, with and without added salt. The rigid po
lymer conformations may become unstable due to an effective attraction indu
ced by counterion density fluctuations. Instabilities at the longest (inter
mediate) length scales may signal a collapse to globule (necklace) states,
respectively. In the presence of added salt, a generalized electrostatic pe
rsistence length is obtained, which has a nontrivial dependence on the Deby
e screening length. [S0031-9007(99)09257-1].