Uniform Price Auction question. Why price is #Seller + 1st highest of #Buyers

[econStudent1]

Hello everyone,

I'm trying to read an economics paper on competitive auction pricing and I'm still coming up to speed with some of the concepts.

As the title says, why is the clearing price of a uniform price auction the S + 1st highest of B buyers values?

In this case each Seller has only 1 unit and each buyer demands only 1 unit.

From a basic Supply/Demand curve I can see that the eq. would be where the S intersects Buyer demand but why S + 1st?

Maybe its something obvious but I appreciate the help. Not too much information online about uniform price auctions.

 

[256bits]

I guess you are relating this to flips of artistic works or comic book collections or building flips where the price at sale rises at each transaction and the final selling price after 3 or 4 tranactions is many multiples of what the original seller received. Why did not the original seller hold out for the final price and get the whole return and not just a small part? Stupid original seller.

Point is the value of an article is fixed by say an evaluator or what you paid for it. Say that price is P. You want to sell at S, some value above P. All buyers below S drop out and look elsewhere. A buyer B1, offers you S+$, since he wants it and you sell at S+$ since it is above your S and you are happy with the extra money. Buyer B1, then tries to sell the article at S+$ +$ to make himself a profit. All buyers below S+$+$ drop out and look elsewhere, but buyer B2 offers B1 S+$+$+$ and the article is sold. B1 is happy with his profit. B2 then attepmts to sell again at and the process continues until there are no more buyers for a certain price. Many buyers at a low price and fewer at a higher price.

How does that relate to a uniform price auction? Especially where Buyers demand only 1 unit each. Buyers are willing to pay a certain price for a product. As the bid for the product goes up, the number of buyers submitting that bid goes down. If a buyer B1 submits a bid at a price of S + x for the whole lot then the whole lot may be sold at that price. Buyers submitting a bid at a lower price loose out. Buyers submitting a bid at a higher price for partial lots receive their lots or partial lots and buyer B1 may receive nothing at all. As long as the whole lot is sold. Note that if buyer B2 submits a bid above that of buyer B1 of S+y, the auctioneer may decide to sell at that price if there are enough buyers above B2 where the whole lot can be sold, in which case all buyers above B2 receive their lots at the price of B2, and B2 may receive only part of his lot.

For buyers wanting only 1 item lot, the whole lot can be sold at the bid of S+ 1st highest bid, which means the lowest bid above S. All buyers get their 1 lot items at S+1st, the whole lot is sold, and no buyer looses out.

I believe this would work only with many sellers of 1 item and many buyers of 1 item. If the number of items being sold is less than the number of buyers, the auctioneer could still sell the whole lot at a higher price than S + 1st.
If there are more items than buyers at S, than the price S drops to bring in more buyers.

 

[econStudent1]

For buyers wanting only 1 item lot, the whole lot can be sold at the bid of S+ 1st highest bid, which means the lowest bid above S. All buyers get their 1 lot items at S+1st, the whole lot is sold, and no buyer looses out.

I believe this would work only with many sellers of 1 item and many buyers of 1 item. If the number of items being sold is less than the number of buyers, the auctioneer could still sell the whole lot at a higher price than S + 1st.
If there are more items than buyers at S, than the price S drops to bring in more buyers.

Thanks a lot for your response!

Does your explanation assume that there are people willing to pay at S + 1? Also, why does this work specifically with buyers only demanding 1 unit? What difference does demanding 1 unit make vs. demanding more units?

 

[256bits]

Thanks a lot for your response!

Does your explanation assume that there are people willing to pay at S + 1? Also, why does this work specifically with buyers only demanding 1 unit? What difference does demanding 1 unit make vs. demanding more units?

Yeah, you are right. I did not actually answer your question with an explanation of why it is S+1 instead of just S. I have to think about that now or go back to some economic material. ( I think it has somehing to do with the number of buyers being one less than number of sellers but don't quote me on that yet, as that doesn't make sense to me either )

Does your explanation assume that there are people willing to pay at S + 1.
There are also buyers willing to pay S+2 but less than number of buyers willing to pay S+1. Similarily less buyers bidding S+3, even less at S+4, etc. It is assumed that as the bid increases, the number of buyers submitting that bid diminishes in a kind of pyramid fashion, and also the number of units they want to acquire diminishes respectively.


As for buyers bidding on more than one unit.
The same pyramid relationship for buyers and the amount of bid is assumed here also. The auctioneer determines the highest bid at which all of the lot can be sold.
Lets say 100 items need to be sold. The lowest bid is at X1 for a partial lot of units. The auctioneer works his way down from the top bid. If it is X5 for 10 units then of course all units will not be sold. Perhaps the next lower bid from the top is from 2 buyers at X4 for 10 units and 20 units. Here, the auctioneer could sell 10 ( from highest bid ) and 30 ( 10+20) = 40 units at price X4 - still not enough to sell the whole lot. And he continues down the bids until he finds a value where the lot can be sold - call it X. any buyer who bid below X gets none of the units, Those buyers bidding above X should get all the units of their bid, but the buyer bidding at X might get all or only a partial lot of his units. Thus the whole 100 lot is sold at a price X. The auctioneer is actually moving along the demand curve until it intersects the supply curve,

Hope that makes aense this time.

 

[256bits]

Here are some sites that offer much more analysis than that which I have given you, especially in real market scenarios, that I found interesting and hope you do too.

http://www.cramton.umd.edu/papers20...ng-behavior-in-electricity-markets-hawaii.pdf

http://www.udesa.edu.ar/files/UAEconomia/Seminarios%20y%20Actividades/Seminarios%20Permanentes/2003/VARGAS.PDF

http://www.market-design.com/files/ausubel-schwartz-ascending-auction-paradox.pdf

http://www.epsa.org/industry/primer/?fa=prices

http://www.math.cornell.edu/~mec/Winter2009/Spulido/Auctions/dutch.html

As you can see, even game teory is used to analyze auctions.

 

[Physics Monkey]

It would help to provide the reference you are referring to. According to wikipedia http://en.wikipedia.org/wiki/Uniform_price_auction, a uniform price auction functions as follows.

There are S units to be sold and B buyers. Each buyer $b_i \,(i=1,...,B)$ submits a secret bid offering to pay $p_i$ per unit for $s_i \leq S$ units. The auctioneer then looks at the sealed bids starting with the highest and proceeding down until a price is found such that all S units can be sold. That price is then clearing price.

If, as you seem to imply, each buyer only offers to purchase one unit, then the price should be $p_{S}$ (assuming I've ordered the prices from largest to smallest). That is the first bid where are S units can be sold. This is how I interpreted the wikipedia article. Now another possibility is that the auctioneer charges the lowest price such that only S units will be sold but does not require this price to be among the bids offered. In that case the price can be anything p with $p_S \geq p > p_{S+1}$ since p must be less than or equal to $p_S$ to secure S buyers but greater than $p_{S+1}$ to avoid S+1 buyers. Hence you could take $p = p_{S+1} + \epsilon$ where $\epsilon$ is a small amount, like a cent or something, so that the price is effectively $p_{S+1}$. Without further information I can't determine which scenario best describes the situation.

Hope this helps.

 

[Pyrrhus]

Get from the library -> Vijay Krishna Auction Theory