# 3.8. Auctions as a Way to Find Equilibrium Price

#cfa #cfa-level-1 #economics #microeconomics #reading-13-demand-and-supply-analysis-introduction #study-session-4

Sometimes markets really do use auctions to arrive at equilibrium price. Auctions can be categorized into two types depending on whether the value of the item being sold is the same for each bidder or is unique to each bidder. The first case is called a common value auction in which there is some actual common value that will ultimately be revealed after the auction is settled. Prior to the auction’s settlement, however, bidders must estimate that true value. An example of a common value auction would be bidding on a jar containing many coins. Each bidder could estimate the value; but until someone buys the jar and actually counts the coins, no one knows with certainty the true value. In the second case, called a private value auction, each buyer places a subjective value on the item, and in general their values differ. An example might be an auction for a unique piece of art that buyers are hoping to purchase for their own personal enjoyment, not primarily as an investment to be sold later.

Auctions also differ according to the mechanism used to arrive at a price and to determine the ultimate buyer. These mechanisms include the ascending price (or English) auction, the first price sealed bid auction, the second price sealed bid (or Vickery) auction, and the descending price (or Dutch) auction.

Perhaps the most familiar auction mechanism is the ascending price auction in which an auctioneer is selling a single item in a face-to-face arena where potential buyers openly reveal their willingness to buy the good at prices that are called out by an auctioneer. The auctioneer begins at a low price and easily elicits nods from buyers. He then raises the price incrementally. In a common value auction, buyers can sometimes learn something about the true value of the item being auctioned from observing other bidders. Ultimately bidders with different maximum amounts they are willing to pay for the item, called reservation prices, begin to drop out of the bidding as price rises above their respective reservation prices.4 Finally, only one bidder is left (who has outbid the bidder with the second highest valuation) and the item is sold to that bidder for his bid price.

Sometimes sellers offer a common value item, such as an oil or timber lease, in a sealed bid auction. In this case, bids are elicited from potential buyers, but there is no ability to observe bids by other buyers until the auction has ended. In the first price sealed bid auction, the envelopes containing bids are opened simultaneously and the item is sold to the highest bidder for the actual bid price. Consider an oil lease being auctioned by the government. The highest bidder will pay his bid price but does not know with certainty the profitability of the asset on which he is bidding. The profits that are ultimately realized will be learned only after a successful bidder buys and exploits the asset. Bidders each have some expected value they place on the oil lease, and those values can vary among bidders. Typically, some overly optimistic bidders will value the asset higher than its ultimate realizable value, and they might submit bids above that true value. Because the highest bidder wins the auction and must pay his full bid price, he may find that he has fallen prey to the winner’s curse of having bid more than the ultimate value of the asset. The “winner” in this case will lose money because he has paid more than the value of the asset being auctioned. In recognition of the possibility of being overly optimistic, bidders might bid very conservatively below their expectation of the true value. If all bidders react in this way, the seller might end up with a low sale price.

If the item being auctioned is a private value item, then there is no danger of the winner’s curse (no one would bid more than their own true valuation). But bidders try to guess the reservation prices of other bidders, so the most successful winning bidder would bid a price just above the reservation price of the second-highest bidder. This bid will be below the true reservation price of the highest bidder, resulting in a “bargain” for the highest bidder. To induce each bidder to reveal their true reservation price, sellers can use the second price sealed bidmechanism (also known as a Vickery auction). In this mechanism, the bids are submitted in sealed envelopes and opened simultaneously. The winning buyer is the one who submitted the highest bid, but the price she pays is not equal to her own bid. She pays a price equal to the second-highest bid. The optimal strategy for any bidder in such an auction is to bid her actual reservation price, so the second price sealed bid auction induces buyers to reveal their true valuation of the item. It is also true that if the bidding increments are small, the second price sealed bid auction will yield the same ultimate price as the ascending price auction.

Yet another type of auction is called a descending price auction or Dutch auction in which the auctioneer begins at a very high price—a price so high that no bidder is believed to be willing to pay it.5 The auctioneer then lowers the called price in increments until there is a willing buyer of the item being sold. If there are many bidders, each with a different reservation price and a unit demand, then each has a perfectly vertical demand curve at one unit and a height equal to his reservation price. For example, suppose the highest reservation price is equal to $100. That person would be willing to buy one unit of the good at a price no higher than$100. Suppose each subsequent bidder also has a unit demand and a reservation price that falls, respectively, in increments of $1. The market demand curve would be a negatively sloped step function; that is, it would look like a stair step, with the width of each step being one unit and the height of each step being$1 lower than the preceding step. For example, at a price equal to $90, 11 people would be willing to buy one unit of the good. If the price were to fall to$89, then the quantity demanded would be 12, and so on.

In the Dutch auction, the auctioneer would begin with a price above $100 and then lower it by increments until the highest reservation price bidder would purchase the unit. Again, the supply curve for this single unit auction would be vertical at one unit, although there might be a seller reserve price that would form the lower bound on the supply curve at that reserve price. A traditional Dutch auction as just described could be conducted in a single unit or multiple unit format. With a multiple unit format, the price quoted by the auctioneer would be the per-unit price and a winning bidder could take fewer units than all the units for sale. If the winning bidder took fewer than all units for sale, the auctioneer would then lower the price until all units for sale were sold; thus transactions could occur at multiple prices. Modified Dutch auctions (frequently also called simply “Dutch Auctions” in practice) are commonly used in securities markets; the modifications often involve establishing a single price for all purchasers. As implemented in share repurchases, the company stipulates a range of acceptable prices at which the company would be willing to repurchase shares from existing shareholders. The auction process is structured to uncover the minimum price at which the company can buy back the desired number of shares, with the company paying that price to all qualifying bids. For example, if the share price is €25 per share, the company might offer to repurchase 3 million shares in a range of €26 to €28 per share. Each shareholder would then indicate the number of shares and the lowest price at which he or she would be willing to sell. The company would then begin to qualify bids beginning with those shareholders who submitted bids at €26 and continue to qualify bids at higher prices until 3 million shares had been qualified. In our example, that price might be €27. Shareholders who bid between €26 and €27, inclusive, would then be paid €27 per share for their shares. Another Dutch auction variation, also involving a single price and called a single price auction, is used in selling US Treasury securities.6The single price Treasury bill auction operates as follows: The Treasury announces that it will auction 26-week T-bills with an offering amount of, say,$90 billion with both competitive and non-competitive bidding. Non-competitive bidders state the total face value they are willing to purchase at the ultimate price (yield) that clears the market (i.e., sells all of the securities offered), whatever that turns out to be. Competitive bidders each submit a total face value amount and the price at which they are willing to purchase those bills. The Treasury then ranks those bids in ascending order of yield (i.e., descending order of price) and finds the yield at which the total $90 billion offering amount would be sold. If the offering amount is just equal to the total face value bidders are willing to purchase at that yield, then all the T-bills are sold for that single yield. If there is excess demand at that yield, then bidders would each receive a proportionately smaller total than they offered. As an example, suppose the following table shows the prices and the offers from competitive bidders for a variety of prices, as well as the total offers from non-competitive bidders, assumed to be$15 billion:

Discount Rate Bid (%)Bid Price per $100Competitive Bids ($ billions)
Cumulative Competitive Bids
($billions) Non-competitive Bids ($ billions)
Total Cumulative Bids
($billions) 0.173199.9125010101525 0.174199.9120015251540 0.175199.9115020451560 0.176099.9110012571572 0.177099.9105010671582 0.178099.910005721587 0.179099.9095010821597 At yields below 0.1790 percent (prices above 99.90950), there is still excess supply. But at that yield, more bills are demanded than the$90 billion face value of the total offer amount. The clearing yield would be 0.1790 percent (a price of 99.9095 per $100 of face value), and all sales would be made at that single yield. All the non-competitive bidders would have their orders filled at the clearing price, as well as all bidders who bid above that price. The competitive bidders who offered a price of 99.9095 would have 30 percent of their order filled at that price because it would take only 30 percent of the$10 billion ($90 billion –$87 billion offered = $3 billion, or 30 percent of$10 billion) demanded at that price to complete the $90 billion offer amount. That is, by filling 30 percent of the competitive bids at a price of 99.9095, the cumulative competitive bids would sum to$75 billion. This amount plus the $15 billion non-competitive bids adds up to$90 billion.

EXAMPLE 8

# Auctioning Treasury Bills with a Single Price Auction

The US Treasury offers to sell $115 billion of 52-week T-bills and requests competitive and non-competitive bids. Non-competitive bids total$10 billion, and competitive bidders in descending order of offer price are as given in the table below:

Discount
Rate
Bid (%)
Bid Price
per $100 Competitive Bids ($ billions)
Cumulative
Competitive
Bids
($billions) Non-competitive Bids ($ billions)
Total
Cumulative
Bids
($billions) 0.157599.842512 0.158099.842020 0.158599.841536 0.159099.841029 0.159599.84055 0.160099.840015 0.160599.839510 1. Determine the winning price if a single price Dutch auction is used to sell these T-bills. 2. For those bidders at the winning price, what percentage of their order would be filled? ## Solution to 1: Enter the non-competitive quantity of$10 billion into the table. Then find the cumulative competitive bids and the total cumulative bids in the respective columns:

Bid Price per $100Competitive Bids ($ billions)
Cumulative Competitive Bids
($billions) Non-competitive Bids ($ billions)
Total Cumulative Bids
($billlions) 99.842512121022 99.842020321042 99.841536681078 99.8410299710107 99.8405510210112 99.84001511710127 99.83951012710137 Note that at a bid price of 99.8400 there would be excess demand of$12 billion (i.e., the difference between $127 billion bid and$115 billion offered), but at the higher price of 99.8405 there would be excess supply. So the winning bid would be at a price of 99.8400.

## Solution to 2:

At a price of 99.8400, there would be $15 billion more demanded than at 99.8405 ($127 billion minus $112 billion), and at 99.8405 there would be excess supply equal to$3 billion. So the bidders at the winning bid would have only 315 , or 20 percent, of their orders filled.