Consider a bargaining set \(S\) with threat point \((0,0)\). Let \(a>0, b>0\), and let \(S^{\prime}=\{(a u, b v) \mid(u, v) \in S\}\). The set \(S^{\prime}\) is the set of utility pairs in \(S\) re-scaled by \(a\) for player I and by \(b\) for player II. Recall that one axiomatic property of the bargaining solution is independence from these scale factors.

Show that the Nash product fulfills this property, that is, the Nash bargaining solution from \(S\) obtained by maximising the Nash product re-scales to become the solution for \(S_{2}\).

Note: This is extremely easy once you state in the right way what it means that something maximizes something. You may find it useful to consider first the simpler case \(b=1\).