Chapter 21 Monopoly. Auctions and Monopoly Prices and Quantities Segmenting the Market. 1. Auctions and Monopoly.
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We begin this chapter by putting auctions in a more general context to highlight the similarities and differences between auctions and monopolies. In this spirit we investigate the sale of multiple units by auction, to see when the selling mechanism affects the outcome, and how.
Within the context of a multiple unit auction we derive our first result in finance, the efficient markets hypothesis, that in its simplest form, states prices of stocks follow a random walk.
The two main differences distinguishing models of monopoly from a auction models are related to the quantity of the good sold:
Monopolists price discriminate through market segmentation, but auction rules do not make the winner’s payment depend on his type. However holding auctions with multiple rounds (for example restricting entry to qualified bidders in certain auctions) segments the market and thus enables price discrimination.
Suppose there are exactly Q identical units of a good up for auction, all of which must be sold.
As before we shall suppose there are N bidders or potential demanders of the product and that N > Q.
Also following previous notation, denote their valuations by v1 through vN.
We begin by considering situations where each buyer wishes to purchase at most one unit of the good.
Suppose each bidder:
 knows her own valuation
 only want one of the identical items up for auction
 is risk neutral
Consider two auctions which both award the auctioned items to the highest valuation bidders in equilibrium.
Then the revenue equivalence theorem applies, implying that the mechanism chosen for trading is immaterial (unless the auctioneer is concerned about entry deterrence or collusive behavior).
By a multiunit demander we mean that each bidder might desire (and bid on) all Q units for himself. We now drop the assumption that N > Q.
Relaxing the assumption that each bidder demands one unit at most seriously compromises the applicability of the Revenue Equivalence theorem.
Typically auctions will not yield the same resource allocation even if the usual conditions are met (private valuations, risk neutrality, lowest feasible expects no rent from participation).
Consider a third price sealed bid auction for two units where there are two bidders, each of whom wants two units. Thus N = Q = 2. Each bidder submits two prices.
We suppose the first bidder has a valuation of v11 for his first unit and v12 for for his second, where v11 > v12 say. Similarly the valuations of the second bidder are v21 and v22 respectively, where v21 > v22.
The arguments given for single unit second price sealed bid auctions apply to the highest price of each bidder. One of his prices is highest valuation.
There is some probability that each bidder will win one unit, and in this case the price paid by one of the bidders will be determined by his second highest bid. Recognizing this in advance, he shades his valuation on his second highest bid.
A Vickery auction is a sealed bid auction, and units are assigned according to the highest bids (as usual).
Each bidder pays for the (sum of the) price(s) for the losing bid(s) his own bids displaced. By definition the losing bids he displaced would have been included within the winning set of bids if the bidder had not participated in the auction, and everybody else had submitted the same bids. In a single unit auction this corresponds to the second highest bidder.
The total price a bidder pays in a Vickery auction for all the units he has won is the sum of the bids on the units he displaced.
A Vickery auction is the multiunit analogue to a second price auction, in that the unique solution (derived from weak dominance) is for each bidder to truthfully report his valuations.
This implies that a Vickery auction allocates units efficiently, in contrast to many multiunit auction mechanisms.
This section of the chapter analyzes how the determination of quantity impacts on the monopolist’s optimization problem. We begin with a discussion of the reservation price in an auction, before moving on to monopoly supply. Although traditional arguments suggest that monopolists are inefficient, we argue the monopolist has an incentive to be as efficient as a competitive industry.
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Perfect price discrimination is often hard to impose directly. However quantity discounting, product bundling and dynamic pricing strategies sometimes provide the means for achieving its objective of value maximization.
To profitably engage in explicit price discrimination, the monopolist must be able to
1. Identify the individual reservation prices by his clientele for his goods
2. Prevent resale from customers with low reservation prices to potential customers with high reservation prices.
3. Be free of incrimination from laws of price discrimination.
When the monopolist knows the distribution of demand but not the characteristics of individual demanders, or alternatively is subject to laws against price discrimination, it can sometimes segment the market to increase its profits.
We first consider a geographically isolated retail market monopolized by a firm selling kitchen and laundry detergents or bathroom toiletries to two types of consumers, large volume commercial buyers and small volume households.
The commercial demanders are willing to search over a wider area for suppliers, and consider a greater range of close substitutes (paper towels versus blow dry).
Households have less incentive to search for these low cost items, rarely consider substitute products, and limited space to store these items; household rental rates for inventory storage are typically greater commercial property rates (per cubic foot).
Suppose the reservation value of a commercial demander is vc and the reservation price of a household is vh where vc < vh.
We also assume a commercial demander would buy k units if the price is less than its reservation value, whereas a household would only buy one unit.
Commercial and household demanders are distributed in proportion p and (1 – p) respectively throughout the local market catchment area.
Unit (wholesale) costs for the monopolist are c, where c < vc.
If the firm adopts a uniform pricing policy, then the maximum monopoly profits are found by charging a high price and only serving households, or charging a low price to capture all the local demand:
max{p(vc – c) + (1  p)k(vc – p), p(vh – c) }
If the firm charges a high price for single units and a discount price for bulk orders of k units then the maximum monopoly profits are
p(vh – c) + (1  p)k(vc – p)
Comparing the net profits of the two, we see that discounting bulk orders is profitable.
Here perfect price discrimination is achieved without resort to charging households and commercial demanders different prices!
Note that if vc > vh then segmenting the market in this way cannot be achieved unless the monopolist can restrict the number of individual units purchased separately (which is typically infeasible).
This result on segmentation can be extended to monopoly markets with several consumer types. We only assume that the consumer types demanding more units have lower reservation values. The same logic applies.
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inverse demand curve
Uniform price solution
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marginal revenue curve
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Uniform quantity solution
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residual inverse demand
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Uniform price solution
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New marginal revenue curve
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