Decision making often involves the accumulation of information over time, but acquiring information typically comes at a cost. Little is known about the cost incurred by animals and humans for acquiring additional information from sensory variables due, for instance, to attentional efforts. Through a novel integration of diffusion models and dynamic programming, we were able to estimate the cost of making additional observations per unit of time from two monkeys and six humans in a reaction time (RT) random-dot motion discrimination task. Surprisingly, we find that the cost is neither zero nor constant over time, but for the animals and humans features a brief period in which it is constant but increases thereafter. In addition, we show that our theory accurately matches the observed reaction time distributions for each stimulus condition, the time-dependent choice accuracy both conditional on stimulus strength and independent of it, and choice accuracy and mean reaction times as a function of stimulus strength. The theory also correctly predicts that urgency signals in the brain should be independent of the difficulty, or stimulus strength, at each trial.