Transcript lecture 6 - ComLabGames

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 v0 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:

Frv0 NFr 
1 Fr
r NN 1 vFvN2
1 Fv
F vdv
r
N
N1
Solving for the optimal reservation price
Differentiating with respect to r, we obtain the first
order condition for optimality below, where r0 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 v0 instead
of r0, 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.

ro v 0 
F 
ro 1 F
ro 
The uniform distribution
When the valuations are distributed uniformly with:
F
v 
v v 
/v v 
then:
r v v 0 
/2
o
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 fs(q)
and the inverse demand curve for houses by fd(q).
Writing price p = fs(q) means that if the price were p
then suppliers would be willing to sell q houses. Similarly
if p = fd(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 = fd(q) - fs(q), or chooses q
to maximize:
[fd(q) - fs(q)]q
The interior first order condition is:
[fd(q) + f’d(q)q] = [fs(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 nth insider by sn and the
cost he incurs from becoming a partner by cn. Note
that:
N
n1 sn  1, with sn  0
The project that yields the net payoff of x, a random
variable.
Thus an insider accepting a share of sn in the
partnership gives up a certain cn for a random payoff
sn x.
The payoff to the entrepreneur is then:
x  n1 cn  sn x 
N
The cost of joining the partnership
We investigate two schemes.
1. The entrepreneur makes each insider an
ultimatum offer, demanding a fee of cn for a
share of sn. 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
cn = p sn.
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 an
the utility of the player n is:
un
a n exp
n a n 
where the entrepreneur is designated player 0.
We also assume that x is drawn from a normal
distribution with mean and variance:

, 2 
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
s on
s 
n
N
k 0 k
1
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 snx is:
sn  
n s n 2
2
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
sn 
p
p
n 2
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.