For the sake of conversation, let's assume extremes to help understand what happens in normal ranges.
If the buyers premium was zero, I believe the buyer would spend the same bottom line. And the consignor would get more because the hammer would be higher.
If the buyers premium were 50%, I believe the buyer would still spend the same bottom line. And the consignor would get less because the hammer would be lower.
So, as the buyers premium increases, there is probably going to be no affect on what the buyer spends. However, the consignor would get less as BP increases.
In a roundabout way, the BP is actually a consignors fee.
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