“Efficiency droop,” i.e., a decline in brightness of light-emitting diodes (LEDs) at high electrical currents, limits the performance of all commercial LEDs and has limited the output power of submicrometer LEDs and lasers to nanowatts. We present a fin p-n junction LED pixel that eliminates efficiency droop, allowing LED brightness to increase linearly with current. With record current densities of 1000 kA/cm2, the LEDs transition to lasing, with brightness over 20 uW. Despite a light extraction efficiency of only 15%, these devices exceed the output power of any previous electrically driven submicrometer LED or laser pixel by 100 to 1000 times while showing comparable external quantum efficiencies. Modeling suggests that spreading of the electron-hole recombination region in fin LEDs at high injection levels suppresses the nonradiative Auger recombination processes. Further refinement of this design is expected to enable a new generation of high-brightness LED and laser pixels for macro- and microscale applications.