Lecture 5 on Auctions Monopoly We begin this lecture by comparing auctions with monopolies. We then discuss different pricing schemes for selling multiple units, the choice of how many units to sell, and the joint determination of price and quantity.

Prices and quantities An important issue for a monopolist is how to determine the quantity supplied. For example how does a multiunit auctioneer determine the number of units to be sold? Auctioneer should set a reservation prices that reflect the value of the auctioned item if it is not sold. This value represents the opportunity cost of auctioning the item. For example the item might be sold later at another auction, and perhaps used in the meantime. Should the auctioneer set a reservation above its opportunity cost? Can the auctioneer commit to setting a reservation price above its opportunity cost?

Auction Revenue What is the optimal reservation price in a private value, second price sealed bid auction, where bidders are risk neutral and their valuations are drawn from the same probability distribution function? Let r denote the reservation price, let v 0 denote the opportunity cost, let F(v) denote the distribution of private values and N the number of bidders. Then the revenue from the auction is:

Solving for the optimal reservation price Differentiating with respect to r, we obtain the first order condition for optimality below, where r 0 denotes the optimal reservation price. Note that the optimal reservation price does not depend on N. Intuitively the marginal cost of the top valuation falling below r, so that the auction only nets v 0 instead of r 0, equals the marginal benefit from extracting a little more from the top bidder when he is the only one to bidder to beat the reservation price.

The uniform distribution When the valuations are distributed uniformly with: then:

Intermediaries with market power We typically think of monopolies owning the property rights to a unique resource. Yet the institutional arrangements for trade may also be the source of monopoly power. If brokers could actively mediate all trades between buyers and sellers, then they could extract more of the gains from trade. How should a broker set the spread between the buy and sell price? A small spread encourages greater trading volume, but a larger spread nets him a higher profit per transaction.

Real estate agents Suppose real estate agencies jointly determined the fees paid by home owners selling their real estate to buyers. How should the cartel set a uniform price that maximizes the net revenue for intermediating between buyers and sellers? We denote the inverse supply curve for houses by f s (q) and the inverse demand curve for houses by f d (q). Writing price p = f s (q) means that if the price were p then suppliers would be willing to sell q houses. Similarly if p = f d (q), then at price p demanders would be willing to purchase q houses.

Optimization by a real estate cartel By convention the seller is nominally responsible for the real estate fees. Let t denote real estate fees and q the quantity of housing stock traded. The cartel maximizes tq subject to the constraint that t = f d (q) - f s (q), or chooses q to maximize: [f d (q) - f s (q)]q The interior first order condition is: [f d (q) + f’ d (q)q] = [f s (q) + f’ s (q)q] The marginal revenue from a real estate agency selling another unit (selling more houses at a lower price) is equated with the marginal cost of acquiring another house (and thus driving up the price of all houses being sold).

NYSE dealers In the NYSE dealers see the orders entering their own books, in contrast to the brokers and investors who place limit orders. The exchange forbids dealers from intervening in the market by not respecting the timing priorities of the orders from brokers and investors as they arrive. However dealers are expected to use their informational advantage make the market by placing a limit order in the limit order books if it is empty.

The gains from more information If dealers do not mediate trades, but merely place their own market orders, their ability to make rents is severely curtailed, but not eliminated. The trading game is characterized by differential information. 1.The order flow is uncertain, everyone sees past transaction prices and volume but only the dealer sees the existing limit orders, so the dealer is in a stronger position than brokers to forecast future transaction prices. 2.If valuations are affiliated then the broker is also more informed about the valuations of investors and brokers placing future orders.

Capital for startups While hard data are difficult to obtain, it seems that: 1. Less than 5% of of new firms incorporated annually are financed by professionally managed venture capital pools. 2. Venture capitalists are besieged with countless business plans from entrepreneurs seeking funding. 3. A tiny percentage of founders seeking financing attract venture capital.

Insiders Our work on bargaining and contracts explains why it is hard for entrepreneurs have difficulty funding their projects. Entrepreneurs typically sell their projects for less than its expected value or owns some of the project himself, thus accepting the risks inherent in it. Because raising outside funds is very costly, entrepreneurs might exchange shares in their projects for labor and capital inputs to known acquaintances, called insiders, rather than professionals. Marriage, kinship and friendship are examples of relationships that lead to inside contacts.

Risk sharing The entrepreneur offers shares to N insiders. We label the share to the n th insider by s n and the cost he incurs from becoming a partner by c n. Note that: The project that yields the net payoff of x, a random variable. Thus an insider accepting a share of s n in the partnership gives up a certain c n for a random payoff s n x. The payoff to the entrepreneur is then:

The cost of joining the partnership We investigate two schemes. 1. The entrepreneur makes each insider an ultimatum offer, demanding a fee of c n for a share of s n. This pricing scheme is potentially nonlinear in quantity and discriminatory between partners. 2. The entrepreneur sets a price p for a share in the firm, and the N insiders buy as many as they wish. (Note that it it not optimal for the entrepreneur to ration shares by under-pricing to create over-subscription.) In this case c n = p s n.

The merits of the two schemes The first scheme is more lucrative, since it encompasses the second, and offers many other options besides. However the first scheme might not be feasible: 1. For example if trading of shares amongst insiders can trade or contract their shares with each other, then the solution to the first scheme would unravel. 2. The first scheme may also be illegal (albeit difficult to enforce).

Two experiments In the experiments we will assume that the entrepreneur and the insiders have exponential utility functions. That is, for each n = 0,1,...,N, given assets a n the utility of the player n is: where the entrepreneur is designated player 0. We also assume that x is drawn from a normal distribution with mean and variance:

Solving the discriminatory pricing problem There are two steps: 1. Derive the optimal risk sharing arrangement between the insiders and the entrepreneur. This determines the number of shares each insider holds. 2. Extract the rent from each insider by a nonnegotiable offer for the shares determined in the first step.

Optimal diversification between the players For the case of exponential utility, the technical appendix shows that The more risk averse the person, the less they are allocated. If everyone is equally risk averse, then everyone receives an equal share (including the entrepreneur). Notice that in this case the formula does not depend on the wealth of the insider.

Optimal offers For the case of exponential utility, the certainty equivalent of the random payoff s n x is: The more risk averse the insider, and the higher the variance of the return, the greater the discounting from the mean return.

Solving the uniform pricing problem There are three steps: 1. Solve the demand for shares that each insider would as a function of the share price. 2. Find the aggregate demand for shares by summing up the individual demands. 3. Substitute the aggregate demand function for shares into the entrepreneur’s expected utility and optimize it with respect to price.

Demand for shares In the exponential case the demand for shares is Note that insider demand is 1. increasing in the net benefit of mean return minus price per share, 2. decreasing in risk aversion, 3. and decreasing in the return of the variance of the return too.

Summary We began this lecture by comparing auctions with monopoly, and establishing some close connections. We found the revenue equivalence theorem applies to multiunit auctions if each bidder only wants one item. Intermediaries exploit their monopolistic position, by creating a wedge between their buy and sell prices. Although fixed price monopolies create inefficiencies, by restricting supply, perfect price discriminators produce where the lowest value consumer only pays the marginal production cost, an efficient outcome.