Why not just have one less feature before softmax? [D]

Softmax has N inputs and N outputs but it's output only has N-1 degrees of freedom because of the condition that the sum of outputs must be equal to one. Based on this we can figure out that actually we can make due with only N-1 inputs by making an assumption that logits before softmax must sum up to zero (though it can be any other constant value) and have the last logit be calculated as minus sum of all the other logits. In theory it should remove "unnecessary" parameters from the last layer before softmax (however few of them may there be) and maybe speed up model convergence a little (my intuition might be wrong about that). Is there any good reason not to do it besides any benefit being negligable in almost all situations?