David's First Math Problem

Last night, I had a math problem.

Sort of. Not really. I'm not saying it's a hard math problem, or an interesting math problem. It may be a profoundly weird way to look at a math problem, and the way I solved it may be a really silly and convoluted solution to the math problem. But it stuck with me until I just had to tackle it -- and I haven't done any real math in about twenty years.

Okay, let's get to it.

At an auction, the amount an item sells for is known as the "hammer price". But it is not necessarily the case that either the buyer pays, or the seller receives, the hammer price. Rather, there is both a buyer's premium (BP) and a seller's premium (SP) that is calculated as a percentage of the hammer. The buyer pays the hammer plus the BP; the seller receives the hammer minus the SP. In other words:

• Final buyer's price = Hammer + (Hammer x BP)
• Final seller's final return = Hammer - (Hammer x SP)

Imagine two different auction premium models. In the first, the buyer pays no premium (BP = 0), and the seller pays a premium of 50% (SP = .5). In the second, the buyer pays a premium of 30% (BP = .3) and the seller plays a premium of 20% (SP = .2). Assuming the final buyer's price remains constant, which model is better for the seller?*

The answer is obviously the second -- but not by as much as might appear at first glance. Since the buyer pays no premium in the first model, they'll pay a higher hammer price than in the second, which partially offsets the first model's higher seller's premium. It's just not enough. To give an example, for a buyer willing to pay $130 for an item:

• In model one, the item would hammer for $130 (130 + (130 x 0)). The seller would get half of that, or $65 (130 - (130 x .5).
• In model two, the item would hammer for $100 to yield the same final buyer's price of $130 (100 + (100 x .3)). The seller would get 80% of the $100 hammer, or $80 (100 - (.2 x 100)).

So model two is better than model one for the seller by a ratio of 16/13 (80/65 = 160/130 = 16/13).

That was not the math problem.

I looked at that number: 16/13. And I thought, that's an ugly number. It expresses out to 1.23076923077 -- yuck!** Moreover, it's an ugly number that, at first glance, has relatively little to do with the relatively nice numbers we saw in the inputs -- .3, .2, .5. Where on earth did 16/13 come from?

16/13, I knew, was not just the ratio that applied to the particular example buyer's price I chose ($130). It was the ratio for any buyer's final price fed into these two premium models. The seller's return would always be 16/13 times higher in model two compared to model one.

What I wanted was (and this was the math problem) was to write some sort of equation or proof that would spit out that ugly 16/13 number, not just for one particular buyer's price but generally.

Now all of this thinking occurred last night, in bed, right around when the daylight's savings time switch happened. It was pretty unproductive.

But this afternoon, I decided to actually take out a pen and paper and really see if I could do this. Here were my steps.

Fb = Final Buyer's Price

Fs = Final Seller's return

H = Hammer.

(1) Fb1 = H + (H x 0) = H

(2) Fs1 = H - (H x .5) = Fb1 - (Fb1 x .5) = .5Fb1

Since The Final Buyer's Price in the first model is just the hammer, we can substitute it in for the hammer in calculating the first model's Final Seller's Return. The Seller pays a 50% premium, so they receive half the Final Buyer's Return aka half the hammer price.

(3) Fb2 = H + (H x .3) = 1.3H

(4) 10Fb2 = 13H, divide both by 13 --> (10/13)Fb2 = H

Not as neat as the first model, but that also lets us express H as a function of Fb2.

(5) Fs2 = H - (H x .2) = .8H

(6) Fs2 = .8((10/13)Fb2 = (8/13)Fb2

(7) Fs2 = (8/13)Fb2 and Fs1 = (1/2)Fb1.

But by stipulation, Fb1 and Fb2 are the same (the buyer is paying the same final price in each model). Meaning:

(8) The ratio of Fs2/Fs1 = (8/13)/(1/2), multiply those by 2 and you get (16/13)/1, or just 16/13.

Tada!

And listen: maybe you're good at math. Maybe you do math all the time. I'm not and I don't. I did not enjoy math in school, and I have generally avoided math as an adult. Indeed, I've written about my complex relationship with my own math educational trajectory, which while not exactly a tragedy was not exactly a triumphant tale either. So I felt very good about being able to work this out, and to stretch some muscles that had lain dormant for quite some time -- even though to be honest I'm still not entirely sure why this particular hypothetical sunk its claws into me so.

Yay me!

* If you're wondering where these two "models" came from, model two represents the rough premiums charged by a standard commercial auction house, and model one represents a charity auction where returns are split 50/50 between the consigner and the charity beneficiary. My initial inquiry was to ask just how much the consigner is losing by doing the charity auction (with the answer being "they are losing, but not as much as they might think upon seeing that steep 50% seller's premium").

** In hindsight, while writing this post, I realized that just by flipping the ratio -- 13/16 -- it would be at least a little less ugly (13/16 = .8125).