We have a conceptual doubt as to the primal problem of BCC.

The objective is to maximise ratio of (weighted sum of outputs-u_o)/weighted sum of inputs.

Why are we using u_0 here?

We know that this is to account for variable return to scale but is BCC defining the efficiency in a different way? Rather than stating the efficiency to be weighted sum of outputs/weighted sum of inputs it is defining it to be (weighted sum of outputs-u_o)/weighted sum of inputs.

Suppose we have a stock which sells at 50 dollars for 1 unit and 400 dollars for 10 unit.

Now suppose we need to calculate its efficiency at 400 dollars for 10 unit

Would we define efficiency to be (weights*400 - u_0)/weights*10 or does BCC imply we are considering u_0 only when 50*10 is taken meaning define it to be weights*(500-10)/weights*input,so it becomes weights*(500)-u_0/weights*input where u_0=weights*10.