  We investigate the set $I$ of parameters $\kappa$ for which the singular
value of $z\mapsto e^z+\kappa$ converges to $\infty$. The set $I$ consists of
uncountably many parameter rays, plus landing points of some of these rays. We
show that the parameter rays have Hausdorff dimension 1, while the ray
endpoints in $I$ alone have dimension 2. Analogous results were known for
dynamical planes of exponential maps; our result shows that this also holds in
parameter space.
