The auroral current density-voltage and energy flux density-voltage relationships are derived under the assumption that magnetospheric electrons above the auroral acceleration region are described by the &kgr; distribution function. To illustrate the effects of this boundary condition on auroral precipitation, a two-dimensional model of auroral electrodynamics similar to that of Lyons [1980> has been developed by imposing current continuity in a region of converging electric field in the auroral zone. The current carried by precipitating magnetospheric electrons inside auroral arcs connects to return current regions at the arc edges via ionospheric Pedersen currents. A key feature is the ability to parameterize the magnetospheric boundary electron population as either a &kgr; or Maxwellian distribution. For equal density and temperature a &kgr;=5 boundary condition yields a parallel potential drop ~5% larger and a precipitating energy flux structure ~5% wider than those of its Maxwellian counterpart. More significant differences emerge between these two distributions when the density is held constant but the spectral shape of the distributions are allowed to differ. The &kgr; distribution results predict up to double the peak auroral energy flux and as much as a 20--30% increase in the latitudinal width of the auroral energy flux as compared with the Maxwellian results. The width and intensity of the inverted V structures in the model results are found to be closely related to the level of thermal energy flux outside the inverted V. ¿ 1999 American Geophysical Union