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\begin{document}
 \parskip=12pt
 
\titlepage
         \vskip 50pt
         \begin{center}
\begin{large}
      {\bf PERFECTLY CONTESTABLE MONOPOLY AND ADVERSE SELECTION }
\end{large}
         \vskip 15pt

  
By Luis Fernandez and Eric Rasmusen
        \vskip 15pt
        {\it Abstract}\\
          \vskip 5pt
        \end{center}
        \par\noindent
 
 In a contestable market the possibility of ``hit-and-run'' entry
prevents the price from rising above average cost. A contestable
natural monopoly earns zero profits despite economies of scale.  We
show that informational imperfections can also result in a single
firm serving the entire market with zero profits.  This is possible
even under constant returns to scale, and when barriers to exit
preclude ``hit-and-run'' attacks and force potential entrants to
consider the post-entry response of the incumbent firm. Furthermore,
the equilibrium involves cross-subsidization, which is not possible
in conventional contestable markets.
 
\begin{small}
 
         
\noindent
  \hspace*{.2in} Fernandez: Dept. of Economics, Oberlin College,
Oberlin,
  Ohio 44074, (216)
775-8486. Fernandez@ocvaxa.cc.oberlin.edu. \\
 \hspace*{20pt}	Rasmusen:           \noindent 
\hspace*{20pt}	Professor of Business Econonomics and Publicy Policy and Sanjay Subheadar Faculty Fellow,   Indiana University,
Kelley School of Business, BU 456,   
  1309 E 10th Street,
  Bloomington, Indiana, 47405-1701.
  Office: (812) 855-9219.   Fax: 812-855-3354. Email: Erasmuse@indiana.edu; Erasmuse@Juno.com; Erasmusen@Yahoo.com (for attachments).   Web:  Php.indiana.edu/$\sim$erasmuse.  Copies of this paper can be found at 
       Www.bus.indiana.edu/$\sim$erasmuse/@Articles/Unpublished/cont.pdf. 
      
%Files: /wilson/contest.tex (Eric) and
%/research/wilson/monopoly/monover5.tex (Luis).
 
\noindent
 April 20, 1993\\
 
\end{small}
 
%---------------------------------------------------------------
 
\newpage
 \noindent
  {\bf 1. Introduction}%Section 1
 
 It has been widely accepted that the degree of market power
possessed by firms is inversely related to the number of firms in the
industry (Scherer (1980)).  This relationship has been thrown into
doubt by the concept of contestable markets (Baumol, Panzar, and
Willig (1982)). In a perfectly contestable market, there is no
relationship between the number of firms and the level of profits.
Profits are always zero in equilibrium, even where cost functions
lead the market to be served by a single firm.  The reason is the
existence of unobservable {\it potential} firms.
 
  The theory of contestable markets was developed under the
assumption of perfect information and has concentrated on the form of
the production function of a multiproduct firm.  We will use an
example to show that informational imperfections can also lead to
zero-profit monopoly equilibria.  Furthermore, this can happen even
though the technology exhibits constant returns to scale and exit
barriers preclude ``hit and run'' attacks by potential entrants,
forcing them, instead, to consider the incumbent's post-entry
response.  Our example is based on Spence's educational screening
model (Spence (1973)). Workers have heterogeneous marginal products
that cannot be observed before hiring takes place, and firms attempt
to {\it screen} workers by conditioning each wage offer on some
observable worker action called a {\it signal}. In our example,
despite the possibility of entry, exactly one firm will engage in
production, yet that firm will earn zero economic profits.
 
The organization of the paper is as follows: Section 2 presents
Spence's model along with our assumptions about the entry and exit of
firms from the market.  Section 3 presents conditions under which all
workers are employed by a single firm earning zero profits in
equilibrium. These conditions are presented in three steps: a
maximization problem that the equilibrium must satisfy, the
equilibrium outcome, and the equilibrium strategies. Section 4
contrasts our model of firm competition with a game of frictionless
entry and exit.  Section 5 summarizes the results.
 
%---------------------------------------------------------------
\bigskip
 \noindent
{\bf 2. A Model of Educational Screening}
 
 Adverse selection models have been used to analyze such markets as
business loans (Bester (1985)), insurance (Rothschild and Stiglitz
(1976)), corporate bonds (Leland and Pyle (1977)), and used cars
(Akerlof (1970)).  Our example is based on Spence's seminal model
(Spence (1973)) of education and the labor market.
 
 
 
 An infinite number of identical, risk-neutral firms may freely enter
as competitive buyers into the labor market.  Firms try to screen
workers by conditioning the wage on an observable activity of the
worker called a {\it signal}, whose level is denoted by the real
number $y$.  A firm may tender one or more {\it offers}, each
consisting of a wage-signal pair $(y,p)$, where $y \geq 0$ and $p
\geq 0$.  Such an offer means the firm will pay a wage $p$ to any
worker who signals at the level $y$.  The level of the signal has no
effect on the worker's productivity.
 
 There are two types of workers, who differ in their productivity and
their costs of signalling.  Proportion $1-\pi$ of the workers have a
``high'' productivity of 2, while proportion $\pi$ have a ``low''
productivity of 1. A worker's productivity cannot be observed before
he is hired. In order to simplify the exposition, we will assume the
preferences of the two types are represented by the following two
utility functions:

$$
\begin{array}{ll}
U_L(y,p) &= log (p) - y\\
U_H(y,p) &= log (p) - y/2,
\end{array}
$$

The choice of these particular functional forms is unimportant.  The
important properties of these utility functions are: (1) they are
increasing in p, (2) decreasing in y, (3) quasi-concave in both p and
y, and the indifference curves of low ability workers are steeper
than those of high ability workers.  The latter means the marginal
cost of signalling is higher for the low-ability workers than for
high-ability workers.
 
 
 Finally, we make two technical assumptions to deal with tie-breaking
when workers are indifferent between offers. The first kind of
tie-breaking arises when a worker faces two offers between which he
is indifferent.  In such a case, we assume that the worker chooses
the offer that requires less signalling.\footnote{This assumption,
which solves an open-set problem, would not be needed if there were a
continuum of worker types.  We will show later that it has no
substantive importance.} The second kind of tie-breaking arises when
the most attractive offer to some group of workers is tendered by two
different firms.  We will assume that each of these two firms has a
positive probability of attracting a worker.\footnote{This assumption
rules out the following bizarre story.  Suppose, contrary to the
assumption above, that workers who are otherwise indifferent between
firms follow a policy of going to the firm that offers the most
contracts. Two firms each start by offering a pair of contracts, a
profitable one that attracts high-ability workers and an unprofitable
one that attracts low-ability workers.  Neither firm drops the
unprofitable contract, because then the high-ability workers would
all depart for the other firm, which still offers two contracts. But
if even a single worker remains, dropping the unprofitable contracts
is a profitable deviation.}
 
 
 
\bigskip
 \noindent
{\bf 3. The Order of Play}
 
 
 The way that the market is organized is very important in situations
of asymmetric information. We will assume this labor market is a {\it
screening market}: the firms (the uninformed players) move first and
announce sets of offers, and the workers move last and choose among
the available offers.\footnote{In simpler adverse selection models
(as opposed to screening) only a price is announced (see, e.g.,
Akerlof [1970]).  A screening market is also different from a {\it
signalling} market where the workers move first and choose signals,
and the firms move second and tender offers after observing which
signals were chosen.  Stiglitz and Weiss (1989) discuss this crucial
distinction.} As a result, the workers are essentially passive
players.  The interesting game is played among the firms before the
workers make their selections.


  Firm $i$ begins the game endowed with a finite
set of {\it old offers}, denoted $O_i$.  These old offers are givens,
not moves of the game. They should be interpreted as offers that can
persist in a steady-state equilibrium, given the rules for entry and
exit.  The   game $W(O_1,O_2,\ldots)$ is then played as
follows:\footnote{This   process by which offers and counteroffers are
made,  follows  the specification of Wilson (1980) as elaborated
by Miyazaki (1977).} 
 
 \noindent
  \hspace*{.2in}(1) Each firm $i$ may simultaneously tender a set of new offers,
denoted $N_i$.\\
  \hspace*{.2in}(2) Each firm $i$ may simultaneously withdraw all or a subset $W_i$
of $O_i$.\\
 \hspace*{.2in} (3) Workers of each type simultaneously choose signal levels and
employers.\\
  \hspace*{.2in}(4) Wages are paid and profits are earned.\footnote{The sets $O_i$,
$N_i$ and/or $W_i$ may be empty, meaning that firm $i$ does not
tender any old offers, does not introduce any new offers, and/or does
not withdraw any old offers.}\\ 
 \hspace*{.2in}(5) At every decision node in the
game tree every player knows all previous moves made by all other
players. 


This  specification  is not the only way that  offers
and counteroffers could be made in a market. There are several distinct ways to
specify the order of play. None of them can be called ``correct,'' because
each is appropriate to the institutional structure of a particular
kind of market.   The order of play  used here implies that the market has the
following three features:
 
\noindent
 (A) When a firm introduces new offers, it cannot then withdraw this
offer before workers are hired; it cannot ``back out'' from its move.
 
\noindent
 (B) When a firm introduces new offers, its competitors have
sufficient advance notice to withdraw some or all of their old offers
before the workers make their choices.
 
\noindent
 (C) When a firm introduces new offers, its competitors do not have
sufficient advance notice to introduce any new offers of their own.
 
Feature (A) is  that legality, good industrial relations, or
administrative inertia require firms to give workers the opportunity
to accept new offers, rather than being able to withdraw them before
the workers have time to react. Under frictionless exit, no such
opportunity need be given.  Feature (B) is that a firm cannot
add new offers without its competitors becoming informed and being
able to react before workers choose employers.  Under frictionless
exit, such advance warning is not given. Feature (C) is  that
considerations of technology or timing do not allow offers to be made
instantly. This is the only one of the three features shared by
our model and the frictionless-exit model.
 
 Features (A) and (B) are not the only extra structure that can be
added to  a situation characterized by (C). Riley (1979) has proposed the ``reactive game'' in
which (A) still holds, but (B) and (C) do not: firms cannot withdraw
old offers, but they can make reactive new offers before workers
choose.  This kind of friction also adds enough structure to the game
for a pure-strategy equilibrium to exist under weak conditions
(Engers \& Fernandez (1987)).
 
What is remarkable about friction in this model is  that the result we will find, zero-profit monopoly,  would not be possible if new entrants
could then immediately exit at no cost.    In the usual contestable monopoly
market, by contrast, {\it frictionless} exit by potential entrants is 
needed   to obtain the same result.
  

%---------------------------------------------------------------
  \bigskip
 \noindent
{\bf 4. Equilibrium}


By an ``equilibrium outcome'' we will mean a 4-tuple,
$(y^*_L,p^*_L,y^*_H,p^*_H)$, for which there exists a sequence of old
offers, $\{O_1^*,O_2^*,\ldots\}$, and a pure-strategy subgame-perfect
equilibrium of the game $W(O_1^*,O_2^*,\ldots)$ in which in
equilibrium, (1) $N_i = W_i = \emptyset, \forall i$, and (2) the Lows
choose $(y^*_L,p^*_L)$ and the Highs choose $(y^*_H,p^*_H)$. The
first condition simply means that no firm $i$ wants to unilaterally
add or subtract from its set of old offers, $O^*_i$. If
$(y^*_L,p^*_L)$ = $(y^*_H,p^*_H)$, the equilibrium is said to be   ``pooling''; otherwise, it is said to be   ``separating.''
 
 
The equilibrium of our game is related to the solution of the
following ``Optimization Problem'' (OP).  The OP maximizes the
welfare of the high-ability worker among all pairs of offers (not
necessarily distinct) that are both incentive compatible and
profitable.  This optimal pair of offers constitute the offers chosen
by the high and low ability workers in any sub-game perfect
equilibrium of our game.  The OP is
 
 \pagebreak
 $$
 \begin{array}{cc}
Maximize & U_H(y_H,p_H)\\
 y_L,p_L,y_H,p_H
 \end{array}
 $$
 subject to:
 
\begin{tabular}{lll}
  (1)& $\pi (1 - p_L) + (1 - \pi)(2 - p_H) \geq 0$ &
(Non-negative
profits)\\
  (2)& $ U_L(y_L,p_L) \geq U_L(y_H,p_H)$ & (Lows do not prefer the
High offer)\\
 (3) & $U_H(y_H,p_H) \geq U_H(y_L,p_L)$ & (Highs do not prefer the
Low offer)\\
 (4) & $U_L(y_L,p_L) \geq U_L(0,1)$ & (Lows get their reservation
wage)\\
 (5) & $y_L \geq 0$ & (The Low signal is feasible).
 \end{tabular}
\bigskip
 
 For any particular value of $\pi$ this optimization problem has a
unique solution, but there are three qualitatively distinct solutions
over three different ranges of $\pi$.  These three solutions are
given in Lemma 1.
 
\bigskip
 
\noindent
 LEMMA 1: {\it The optimal arguments $y^*_L,p^*_L,y^*_H,p^*_H$ for
the Optimization Problem take the values shown in Table 1.}
 
 \begin{center}
  TABLE 1:\\
  SOLUTIONS TO THE  OPTIMIZATION PROBLEM
  $$
 \begin{array}{l|ll|ll}
 {\rm Fraction\; of\; Lows} & y_L^* & p_L^* &y_H^* & p_H^*\\
\hline
 \hline
\pi \leq 1/2  & 0  &2-\pi  &0 & 2-\pi\\
 \hline
 1/2 < \pi < 2/3 & 0 & \frac{2-\pi}{2\pi} &
log(\frac{\pi}{1-\pi}) &\frac{2-\pi}{2(1-\pi)} \\
 \hline
 \pi \geq 2/3 &0 &1 &log2 &2\\
 \hline
 \end{array}
 $$
 \end{center}
 
 \bigskip
 
 Lemma 1 tells us how workers react to offers.  Suppose that the
workers have the two (not necessarily distinct) offers $(0, p_L^*)$
and $(y_H^*, p_H^*)$ from which to choose.  When $\pi \leq 1/2$, both
types of workers choose the single offer $(0, p^*_L) = (y^*_H,p^*_H)$
and any firm that tenders this offer earns exactly zero profits.
When $1/2 < \pi < 2/3$, the Lows choose the offer $(0,p^*_L)$ and the
Highs choose the offer $(y^*_H,p^*_H)$.  Since in this case $1 <
p^*_L < p^*_H < 2$, the Highs are paid less than their marginal
product and the Lows are paid more than their marginal product; but
the two differentials exactly offset each other, so any firm that
tenders both offers exactly breaks even.
  When $\pi \geq 2/3$, the Lows and Highs again choose different
offers, but now every worker is exactly paid his marginal product.
Any firm that tenders either offer exactly breaks even.
 
   It can now be shown why our assumption that indifferent workers
choose the offer with the least signalling is non-substantive.
Suppose we did not have it, and $\pi \geq 2/3$.  The Lows would then
be indifferent between the two offers $(0,p^*_L)$ and
$(y^*_H,p^*_H)$, from which we have assumed they all pick the offer
with the lower signal, $(0, p^*_L)$. Suppose instead that some Lows
choose $(y^*_H,p^*_H)$.  Then no firm will want to tender
$(y^*_H,p^*_H)$ because it is unprofitable if even one Low chooses
it.  But a firm would be willing to tender $(y^*_H +\epsilon, p^*_H)$
for small $\epsilon$, because the Highs prefer it to $(0,p^*_L)$ but
the Lows do not. The only problem is that some other firm could now
tender the slightly more attractive offer of $(y^*_H + \epsilon/2,
p^*_H)$.  As a result, no equilibrium exists, in either pure or mixed
strategies.  But the problem is a modeling artifact.  If the set of
possible signal levels were discrete rather than continuous, so
signals could only rise by increments of $\epsilon$, the problem
would disappear.  Our tie-breaking assumption achieves the same
result more simply than a model with a large number of discrete
signal levels.
 
 
 
\bigskip
 
  The OP problem is static; entry and exit play no role.  We now come
to the most important part of this paper: the demonstration that for
a range of moderate parameters the equilibrium of our screening
market has the characteristics of a contestable-market equilibrium.
In what follows, an offer is ``active'' if some worker chooses it in
equilibrium; otherwise it
 is ``inactive.''  A firm is ``active'' if it tenders at least
one active offer in equilibrium, and therefore hires at least one
worker;
otherwise, the firm is ``inactive.''


\bigskip
 \noindent
 PROPOSITION 1.
 {\it Any  equilibrium outcome solves the  Optimization
Problem, and all firms earn zero profits.  In addition:
 (i) When $\pi \geq 2/3$, at least two firms are active;
 (ii) When $1/2 < \pi < 2/3$, exactly one firm is active;
  (iii) When $\pi \leq 1/2$, the number of active firms is
indeterminate.}
 
 \noindent
 PROOF: Suppose $(\ytil_L,\ptil_L,\ytil_H,\ptil_H)$ is a equilibrium
outcome of the game.  Then we claim this vector satisfies the five
constraints of the OP.  It satisfies constraint (5) trivially.  Since
the two offers $(\ytil_L,\ptil_L)$ and $(\ytil_H,\ptil_H)$ must be
best choices of the Lows and Highs, constraints (2) and (3) must be
satisfied as well.  The profitability constraint (1) must be
satisfied, for otherwise some firm could improve its profits by
unilaterally withdrawing all its offers.  Furthermore, (1) must be
binding as well.  Otherwise, the firms tendering offers would be
making strictly positive profits overall, and some entrant could
tender the two new offers $(\yhat_L,\phat_L)$ and $(\yhat_2,\phat_2)$
that attract both types of workers away from $(\ytil_L,\ptil_L)$ and
$(\ytil_H,\ptil_H)$ and yet earn positive profits.  Finally, suppose
constraint (4) is violated.  Then $0 = U_{L}(0,1) >
U_{L}(\ytil_L,\ptil_L) = \util$. Define $p' = e^{ \util/2}$.  Since
$p'<1$, any firm which in equilibrium was tendering nothing, but
which adds the offer $(0,p')$ will make positive profits, regardless
of which offers are subsequently withdrawn.
 
Suppose $(\ytil_L,\ptil_L,\ytil_H,\ptil_H)$ is not equal to the OP
solution $(y^*_L,p^*_L,y^*_H,p^*_H)$.  Then there exists an
$\epsilon$ such that $U_{H}(y^*_H,p^*_H- \epsilon) >
U_H(\ytil_H,\ptil_H)$ and $U_H(y^*_H,p^*_H-\epsilon) >
U_H(y^*_L,p^*_L)$.  Suppose a firm adds the two offers
$(y^*_L,p^*_L)$ and $(y^*_H,p^*_H-\epsilon)$.  Regardless of what old
offers the other firms withdraw, this firm will attract only the
Highs to $(y^*_H,p^*_H-\epsilon)$, which earns strictly positive
profits.  If the other firms all withdraw their old offers and the
Lows choose $(y^*_L,p^*_L)$, then our firm still earns positive
profits.  The contradiction shows $(\ytil_L,\ptil_L,\ytil_H,\ptil_H)$
must solve the OP, as claimed.
 
 The proposition's claims about the number of firms remain to be
proven.  If $1/2< \pi < 2/3$, then Table 1 tells us that a different
offer is chosen by each type and the offer accepted by the Lows
incurs losses for the offering firm (the wage of $\pi/(1-\pi)$
exceeds the marginal product of 1).  A firm that tenders only
$(y^*_H,p^*_H)$ cannot be part of the equilibrium because then any
firm offering $(y^*_L,p^*_L)$ would not attract enough Highs to break
even. Multiple firms offering both offers cannot be part of
equilibrium because each firm would want to unilaterally drop
$(y^*_L,p^*_L)$ in the withdrawal stage. The only other possibility
is for one firm to offer both offers, in which case no other firm is
active.
 
If $\pi \geq 2/3$, Proposition 1 claims there cannot be just one
active firm in equilibrium. If there were just one active firm
tendering both separating offers, that firm would drop the High offer
in the withdrawal stage, the Highs would accept the Low offer, and
the firm would earn positive profits.  Since there cannot be any
incentive to make new offers or withdraw old offers in a equilibrium,
there must be at least two active firms in equilibrium, with both
firms offering the High contract and at least one offering the Low
contract.
 
If $\pi \leq 1/2$, Proposition 1 claims that the number of active
firms is indeterminate. Clearly there could be two or more active
firms, each tendering the same pooling offer. Each would make zero
profits, and none could benefit by adding new offers, because any
offer that made profits by attracting away the Highs would make the
old pooling offer unprofitable. The old pooling offer would be
withdrawn, and the new offer would no longer be profitable. But there
could also be a single active firm, tendering the pooling offer $(0,
p_L^*$). In this case, some other firm would have to tender two
inactive offers: (0,1) and $(y^*_3,2)$, where $y^*_3$ is chosen so
that the Highs are just indifferent between $(0,p_L^*)$ and $(y^*_3,
2$). This would be an equilibrium because the active firm could not
benefit by adding to or withdrawing from its offer: the alternative
of (0,1) prevents it from profiting from the Lows by paying them less
than $p=1$ and the alternative of $(y^*_3, 2$) prevents them from
adding a more profitable pooling offer and withdrawing $(0, p_L^*$).
Thus, there can be either one or more firms active in equilibrium.
\\
 Q.E.D.
 
%---------------------------------------------------------------
 
\bigskip
 
When $\pi > 2/3$, there are at least two firms active in equilibrium,
an ordinary result. But whenever $\pi \leq 2/3$, there can be
monopoly in equilibrium, even though the production function shows
constant returns to scale. This range of $\pi$ can be further divided
into two smaller ranges with qualitatively different kinds of
monopoly.
 
  When $\pi \leq 1/2$, the situation is similar to a
perfect-information model with constant returns to scale. The number
of firms does not matter, but the possibility of entry does.  Profits
are zero whether one firm or many firms make the pooling offer.  That
is why we cannot determine the number of active firms for this
parameter range.
 
 When $1/2< \pi< 2/3$, the situation might be termed a ``natural
monopoly,'' since the unique equilibrium outcome is for only a single
firm to be active despite the absence of entry barriers of any kind.
The firm earns zero profits, however, even without government
regulation.  The results, if not the assumptions, recall the
paradigmatic contestable market: air service to a small town. Only
one airline will provide the service because of technological
economies of scale, but that airline cannot raise price above cost
without provoking entry.  In our model, a single firm hires all the
workers, but the reason is not economies of scale. It is, rather,
that by being the only active firm, the firm can internalize the
benefits of cross-subsidization. At the same time, the firm cannot
lower wages, or it will provoke entry.
 
 Cross-subsidization, in fact, is where contestable monopoly arising
from adverse selection differs most from contestable monopoly arising
from scale economies.  Baumol (1982) says that one of the three chief
welfare characteristics of a contestable market is the absence of
cross-subsidies: each product is sold at marginal
cost.\footnote{Baumol's other two welfare characteristics are
efficient production and zero profits.} Otherwise (in the markets he
is considering), an outsider would enter and undercut the price of
the product whose profits were cross-subsidizing the other product.
If the incumbent then lowered his price on that product, the entrant
would end up with no worse than zero profits.
 
  In the screening equilibrium described above, the High workers
subsidize the Low workers whenever $\pi < 2/3$, whether the market
contains one firm or several.  Should an entrant introduce an offer
that would attract just the Highs, the incumbent would withdraw all
active offers.  Both High and Low workers would choose the entrant's
offer, and the entrant's profits would be negative, not zero. The
firm, because it is the only active firm, can internalize the
benefits from cross-subsidization. Thus, the difference in
cross-subsidization from scale-economies contestable monopoly is not
just accidental, but is at the heart of adverse-selection contestable
monopoly.
  The type of cross-subsidization in the screening model differs in
the two parameter ranges. In the range $\pi \leq 1/2$, a single
pooling offer is made. That offer is profitable when it is accepted
by a High and unprofitable when accepted by a Low, but the firm does
not know whether a particular transaction is profitable or not. An
entrant might threaten to introduce a offer that would lure away the
Highs, but the incumbent's optimal response would be to withdraw
$(y^*_H,p^*_H)$, which would result in the entrant ultimately hiring
the Lows also, at a loss.  In the range $1/2<\pi < 2/3$, the contrast
with the perfect-information market is even more striking.  In that
range, the single, active firm makes two offers, one of which
attracts only Lows and is known to be unprofitable.  The firm tenders
that offer only to deter the Lows from accepting the profitable
offer, which would be unprofitable if it were accepted by Lows as
well as by Highs. Should an entrant enter with the offer
$(y^*_H,p^*_H)$, the incumbent's optimal response is to withdraw both
of its old offers, which leaves the entrant hiring both Highs and
Lows and earning negative profits.
 
%---------------------------------------------------------------
 
Proposition 1 describes the offers that are active in any
equilibrium, and thus characterizes the equilibrium outcome, but it
does not prove that a pure-strategy sub-game perfect equilibrium
exists not does it describe the equilibrium strategies. Given the
nonexistence result of Rothschild \& Stiglitz (1976) for insurance
markets similar to this labor market, the existence of a
pure-strategy equilibrium cannot be taken for granted. Proposition 2
in the Appendix provides the missing proof of the existence of a
pure-strategy equilibrium for our game. One interesting aspect of the
equilibrium strategies is that firms must tender inactive offers in
order to prevent deviations from equilibrium.
 
 
\bigskip
 
 Our model was designed to have the special feature of
cross-subsidization in a monopolized separating equilibrium, but some
of its other features can be found in simpler adverse selection
models with monopoly pooling equilibria. An example is the following
bid-ask spread model of the market for a security, which we will
merely sketch out here, since the results parallel those described
above.
 
Bagehot (1971) argued the bid-ask spread on a stock exchange exists
to guarantee zero profits to the marketmaker, who trades with whoever
appears at the market. Some of those who appear are informed traders,
and the marketmaker always loses in trades with them. The rest of
those who appear are uninformed traders, and the bid-ask spread
allows the marketmaker to profit in those trades.  The equilibrium is
pooling because there is no signal, and the marketmaker must offer
the same spread to both types. The uninformed effectively subsidize
the informed, but if the marketmaker tries to charge too high a
spread, he can be undercut by a competing marketmaker.
 
 The reason this market would be monopolized is the marketmaker can
make use of the volume of trades to learn the true value of the
security. If, for example, he finds many more traders are buying than
selling, he can conclude the uninformed traders are randomly
distributed on each side, but the informed traders realize the price
is too low. In response, he can raise the price. The marketmaker with
the greatest volume of trade can amass more information in this way,
set the price more accurately, and lower his bid-ask
spread.\footnote{While we have not seen the conclusion that such a
market is a natural monopoly published, it is unlikely the argument
is new. We discuss it here to contrast pooling monopoly with the more
complicated separating monopoly.}
 
 This securities market, like our labor market over most of its
parameter range, consists of one active firm earning zero profits and
cross-subsidizing across its transactions. The difference is that the
firm in the securities market does not know which transactions are
profitable and which are unprofitable. There is no possibility of an
entrant trying to skim off the trades with the uninformed traders,
and hence the details of entry and exit are not so important to the
model.
 
%---------------------------------------------------------------
\bigskip
 \noindent
{\bf 5. Concluding Remarks}
 
 
   The most important insight of the contestable-markets literature
has been that when we observe one firm monopolizing a market we
cannot immediately conclude there exists inefficiency or that the
firm is earning positive economic profits.  Instead, the conditions
of production may be such that it is most efficient for one firm to
serve the market, and other firms would enter if that one firm ever
tried to raise price above average cost.  The implication is that
policymakers ought to check the conditions of production---is entry
and exit costless, and are there economies of scale?
 
We have presented another reason why one firm might dominate a market
without earning positive profits or restricting its output.  In our
example, it is not the conditions of production so much as the
conditions of {\it distribution} that are important. The analyst need
not be concerned with production economies of scale---we have assumed
constant returns to scale--- but he must worry about whether
information problems make it important that only one firm operate.
 
 Our example has two features which are very different from a
standard contestable market.  First, we assume a certain friction in
the way the market operates: offers cannot be introduced and
withdrawn instantly. This ensures that equilibrium exists in an
adverse selection model like ours, but the idea of contestable
markets, like that of perfect competition, has usually been
associated with the absence of frictions. Second, cross-subsidization
occurs in equilibrium in our model. The single firm offers two
offers, one of which is profitable and the other, unprofitable.  This
cannot happen in a conventional contestable market; indeed, Baumol
(1982) says a chief conclusion of the theory is that no
cross-subsidization can occur in a perfectly contestable market.
Although economists normally associate cross-subsidization with
regulation, our model is one of {\it laissez faire} in which the
subsidy is paid out of pure self-interest.  Our model implies that if
one of a monopoly firm's products is observed to be profitable, that
does not mean the firm is making profits overall, for those profits
may be balanced by losses on another of its products.
 
%---------------------------------------------------------------
\newpage
\noindent
{\bf Appendix: Proofs of propositions}
\bigskip
 
\noindent
 PROOF OF PROPOSITION 1: Because both utility functions are
continuous, the constraint set is closed.  We claim the constraint
set is also bounded, which implies that an optimal solution
$(y^*_L,p^*_L,y^*_H,p^*_H)$ exists.  The incentive-compatibility
constraints (2) and (3) of the OP imply $y_L \leq y_H$ and $p_L \leq
p_H$, and constraints (1) and (4) imply $p_L \geq 1$ and $p_H \leq
2$.  Finally, constraints (2) - (4) together imply $U_H(y_H,p_H) \geq
U_H(0,1)$, which in turn implies $y_H \leq log 4$.  We have shown $1
\leq p_L \leq p_H \leq 2$ and $0 \leq y_L \leq y_H \leq log 4$, which
proves that the constraint set is bounded, as we claimed.  Note that
these bounds imply all four parameters, not just $y_L$, take
non-negative values.
 
 Second, we claim that constraints (1), (2), and (5) are binding at
any optimum. Suppose $(y_L,p_L,y_H,p_H)$ satisfies the constraints of
the OP, and $y_L > 0$.  Direct calculation reveals that the vector
$(0,p_L,y_H-y_L,p_H)$ also satisfies these constraints, but
$U_H(y_H-y_L,p_H) > U_H(y_H,p_H)$.  This shows that at any optimum,
constraint (5) is binding.  Next suppose $y_L = 0$, but $U_L(y_L,p_L)
> U_L(y_H,p_H)$.  It follows $y_H > log \frac{p_H}{p_L} = y_H'$.  The
vector $(y_L,p_L,y'_1,p_H)$ satisfies all the constraints of the OP,
but $U_H(y_H',p_H) > U_H(y_H,p_H)$.  This proves that at any optimum,
constraint (2) is also binding.  Finally suppose $y_L = 0$ and
$U_L(y_L,p_L) = U_L(y_H,p_H)$, but $\pi (1 - p_L) + (1-\pi)(2 - p_H)
> 0$.  Then $\beta = (2-\pi)/(\pi p_L + (1-\pi)p_H) > 1$.  The vector
$(y_L,\beta p_L,y_H, \beta p_H)$ satisfies all the constraints of the
OP, but $U_H(y_H,\beta p_H) > U_H(y_H,p_H)$.  This proves that at any
optimum, constraints (1), (2), and (5) are binding.
 
 It follows that $(y^*_L,p^*_L,y^*_H,p^*_H)$ is the solution of the
modified optimization problem
 $$
\begin{array}{cc}
Maximize & U_H(y_H,p_H)\\
 y_L,p_L,y_H,p_H
\end{array}
 $$
 subject to:
 
\begin{tabular}{lll}
  $(1')$& $\pi (1 - p_L) + (1 - \pi)(2 - p_H) = 0$ \\
  $(2')$& $ U_L(y_L,p_L) = U_L(y_H,p_H)$ \\
$ (3')$ & $U_H(y_H,p_H) \geq U_H(y_L,p_L)$ \\
$ (4')$ & $U_L(y_L,p_L) \geq U_L(0,1)$ \\
$(5')$ & $y_L = 0$ \\
 \end{tabular}
\bigskip
 
 Constraints ($1'$), ($2'$), and ($5'$) implicitly define $y_H$ and
$p_H$ as functions of $p_L$.  Specifically, $y_H(p_L) = log \frac{2 -
\pi - \pi p_L}{(1 - \pi)p_L}$ and $p_H(p_L) = \frac{2 - \pi - \pi
p_L}{(1 - \pi)p_L}$.  Substitution of these expressions into
constraint ($3'$) yields the inequality $p_L \leq 2 - \pi$; and their
substitution into constraint ($4'$) results in the inequality $p_L
\geq 1$.  Finally, $U_H(y_H(p_L),p_H(p_L)) = \frac{1}{2} log \frac{(2
- \pi -\pi p_L)p_L}{1-\pi}$, which is an increasing function of the
strictly concave function $V(p_L) = (2 - \pi)p_L -\pi {p_L}^2$.  It
follows that any solution of the OP is of the form: $(0, p^*_L,
y_H(p^*_L), p_H(p^*_L))$, where $p^*_L$ is a solution to the
one-variable constrained optimization problem
 $$
 \begin{array}{cc}
Maximize & (2 - \pi)p_L -\pi {p_L}^2\\
 p_L \in [1,2-\pi]&\\
\end{array}
 $$
 
\bigskip
 
It is simple to verify that this optimization problem has the
following solution: $p^*_L = 2-\pi$ when $\pi \leq 1/2$; $p^*_L =
\frac{2-\pi}{2 \pi}$ when $1/2 < \pi < 2/3$; and $p^*_L = 1$ when
$\pi \geq 2/3$.  The conclusions of Lemma 1 follow directly.\\
 {\it Q.E.D.}

\noindent
PROPOSITION 2:
 {\it Let $p_L^*, p_H^*, y_L^*$, and $y_H^*$ take the values in Table
1 and let $y^*_3 = 2 log(2/p^*_H) + y^*_H$.  Suppose Firm 1 tenders
the set of old offers $O^*_1$ $= \{(0,1), (0, p^*_L), (y^*_H,
p^*_H)\}$, Firm 2 tenders $O^*_2 = \{(0, 1), (y^*_3, 2) \}$, and all
remaining firms tender nothing. That is, $O_i^* = \emptyset$, for $i
> 2$. The Wilson-Miyazaki game $W(O_1^*, O_2^*, \ldots)$ has a
pure-strategy equilibrium in which no firm has an incentive to add
new offers or withdraw old ones.}
 
Before proving Proposition 2, it may be useful to describe the
equilibrium offers.  When $\pi \leq 1/2$, Table 1 tells us that $(0,
p^*_L) = (y^*_H,p^*_H)$, so the workers face three distinct offers---
$(0,1)$, $(0,p^*_L)$, and $(y^*_3,2)$--- from which both types of
workers choose $(0, p^*_L)$.  Only Firm 1 is active, and it makes
zero profits overall: losses on the Lows are offset by profits on the
Highs.  Firm 2's offers are inactive, but they are important in
constraining Firm 1's behavior. Figure 1 illustrates the offers. The
indifference curve of the Highs labeled $U_H$ passes through both
$(0, p^*_L)$ and $(y^*_3,2)$, while the indifference curve of the
Lows labeled $U_L$ passes through the offer $(0,p^*_L)$ but lies
above the offer $(y^*_3,2)$.
 
  \begin{center}
[SEE FIGURE 1]
   \end{center}
 
  When $1/2 < \pi < 2/3$, the workers face four distinct offers--
$(0,1)$, $(0,p^*_L,)$, $(y^*_H,p^*_H)$ and $(y^*_3,2)$-- from which
the Lows choose $(0,p^*_L)$ and the Highs choose $(y^*_H,p^*_H)$.
Firm 1 is again the only active firm, and it hires all of the
workers, both Highs and Lows.  It makes losses on the Lows which are
offset by the profits earned on the Highs. Figure 2 illustrates this.
The indifference curve of the Highs labeled $U_H$ passes through both
$(y^*_H,p^*_H)$ and $(y^*_3,2)$.  The indifference curve of the Lows
labeled $U_L$ passes through both $(0,p^*_L)$ and $(y^*_H,p^*_H)$.
The indifference curve of the Lows labeled $U_L'$ passes through
$(0,1)$ and lies below $(y^*_3,2)$.
 
  \begin{center}
[SEE FIGURE 2]
  \end{center}
 
  Finally, when $\pi >2/3$, Table 1 tells us that $(0,1) = (0,p^*_L)$
and $(y^*_H,p^*_H) = (y^*_3,2)$.  The workers face only two distinct
offers--- $(0,1)$ and $(y^*_H,p^*_H)$--- from which the Lows choose
$(0,1)$ and the Highs choose $(y^*_H,p^*_H)$.  Now both Firm 1 and
Firm 2 are active, and they each hire Lows with one of their offers
and Highs with the other, breaking even on each offer.  Figure 3
illustrates this.  The indifference curve of the Lows, labeled $U_L$,
goes through both $(0,1)$ and $(y^*_H,p^*_H)$, while the indifference
curve of the Highs, labeled $U_H$, goes through $(y^*_H,p^*_H)$ and
lies above the offer $(0,2-\pi)$ that is the pooling equilibrium
offer chosen when $\pi \leq 1/2$.
  \begin{center}
 [SEE FIGURE 3]
  \end{center}
 
%
\noindent
 PROOF OF PROPOSITION 2: We need to show that when Firm 1 tenders
$O^*_1$ as an old offer, Firm 2 tenders $O^*_2$ as an old offer, and
the other firms tender no old offers, no firm can make positive
profits by unilaterally deviating at any stage of the game--- whether
by tendering new offers and/or by withdrawing any of its old offers.
Moreover, since we are interested in a perfect equilibrium, any
deviant action is taken with the knowledge that the other firms will
react in later stages of the game.  Fortunately, we can sidestep the
complex maze of possible deviations by analyzing the possible
outcomes of any deviation.
 
 Suppose that some firm makes positive profits by deviating at some
stage of the game.  In this case, at the end of the game the Low
workers choose some offer $(\ytil_L,\ptil_L)$ and the Highs choose
$(\ytil_H,\ptil_H)$, where possibly $(\ytil_L,\ptil_L)=
(\ytil_H,\ptil_H)$. We will show the following: (i)
$(\ytil_L,\ptil_H,\ytil_H,\ptil_H)$ satisfies the constraints of the
OP and (ii) $U_H(\ytil_H,\ptil_H) < U_H(y^*_H,p^*_H)$, that is the
deviation hurts the Highs.  For this to happen, Firm 1 must withdraw
the offer $(y_H,p_H)$ and Firm 2 must withdraw the offer $(y_3,2)$,
since both offers generate identical levels of utility for the Highs.
But we will then show: (iii) it is impossible to induce both firms to
withdrawn both of these offers unilaterally.
 
(i) If constraint (1) of OP is violated, then total profits across
all firms are negative and some firm would do better by withdrawing
an active offer at stage 2.  Constraints (2) and (3) are the
self-selection constraints and are satisfied by the definition of
$(\ytil_L,\ptil_L)$ and $(\ytil_H,\ptil_H)$.  Constraint (5) is the
feasibility constraint and, therefore, must be satisfied. Finally,
constraint (4) must be satisfied if the offer $(0,1)$ is still being
tendered after stage 1.  But notice that after stage 1 at least one
firm must still be tendering $(0,1)$.  Since $(0,1)$ can never be
unprofitable, neither of these two firms benefits from withdrawing
this offer at stage 2.
 
(ii) It follows from Proposition 1 that $U_H(\ytil_H,\ptil_H) \leq
U_H(y^*_H,p^*_H)$.  Suppose $U_H(\ytil_H,\ptil_H) =
U_H(y^*_H,p^*_H)$.  Since the solution to the OP is unique, this
implies $(\ytil_L,\ptil_H,\ytil_H,\ptil_H)$ $ =
(y^*_L,p^*_L,y^*_H,p^*_H)$.  It now follows from the proof of
Proposition 1 that constraint (1) is binding--- that is, total
profits across firms equal zero.  Since the deviating firm is making
strictly positive profits by assumption, it follows some firm must be
making strictly negative profits.  But this in inconsistent with
optimal behavior at stage 2, since every firm can guarantee itself
zero profits by withdrawing all of its offers at that stage.  It
follows $U_H(\ytil_H,\ptil_H) < U_H(y^*_H,p^*_H)$.
 
(iii) Suppose no deviations occured at stage 1.  There is no
incentive for Firm 1 to unilaterally withdraw $(y^*_H,p^*_H)$ at
stage 2, since the Highs will just go to the offer $(y^*_3,2)$ and
the firm will be left hiring the Lows at a loss; and if Firm 1
withdraws both offers, then it makes no sales and therefore no
profits.  Likewise, there is no incentive for Firm 2 to unilaterally
withdraw the inactive offer $(y^*_3,2)$.  So any profitable deviation
from the equilibrium must begin at stage 1.  Suppose the new offers
were introduced by a firm other than Firm 1.  Since we have deduced
that the Highs must strictly prefer $(y^*_H,p^*_H)$ to any of these
new offers, Firm 1 is guaranteed to earn positive profits on
$(y^*_H,p^*_H)$ as long as it continues to tender $(0,p^*_L)$-- in
which case it is guaranteed to earn at least zero profits on both
offers.  So Firm 1 cannot be induced to withdraw $(y^*_H,p^*_H)$.  So
the new offers must have been added at stage 1 by Firm 1.
Furthermore, the Highs must consider any of these offers strictly
inferior to $(y^*_3,2)$.
 
We claim in this case, it is an equilibrium strategy for both firms
to continue to tender their old offers, in which case Firm 1 earns
zero profits, a contradiction.  For if Firm 1 continues to tender
$(y^*_H,p^*_H)$, then Firm 2 cannot do better by withdrawing
$(y^*_3,2)$.  And if Firm 2 continues to tender $(y^*_3,2)$ then Firm
1 cannot do better by withdrawing $(y^*_H,p^*_L)$ and
$(y^*_H,p^*_H)$.  For suppose Firm 1 withdraws $(y^*_H,p^*_H)$. It
then must lose the Highs to Firm 2. As for the Lows, either it also
loses the Lows to Firm 2; or it retains the Lows at $(y^*_L,p^*_L)$,
which results in a loss; or it hires the Lows at one of its new
offers.  But this new offer can attract the Lows away from Firm 2's
offer of $(0,1)$ only by paying the Lows more than 1, resulting in a
loss.  If Firm 1 continues to tender $(y^*_H,p^*_H)$, then it gains
nothing from withdrawing $(y^*_L,p^*_L)$.  For it can prevent the
Lows from choosing $(y^*_H,p^*_H)$--- which results in a loss--- only
by tendering a new offer that pays them more than $p^*_L$--- which
also results in a loss.
 
\noindent
{\it Q.E.D.}


\newpage 
\bigskip
\noindent
{\bf References }
 
{\bf Akerlof, George,} ``The Market for Lemons: Quality Uncertainty
and the Market Mechanism,'' {\it Quarterly Journal of Economics,}
August 1970, {\it 84}, 488-500.
 
{\bf Bagehot, Walter}, ``The Only Game in Town,'' {\it Financial
Analysts Journal,} March/April 1971, {\it 27}, 12-22.
 
{\bf Baumol, William}, ``Contestable Markets: An Uprising in the
Theory of Industry Structure,'' {\it American Economic Review}, March
1982, {\it 72}, 1-15.
 
{\bf Baumol, William, Panzar, John, and Willig, Robert}, {\it
Contestable Markets and the Theory of Market Structure}, New York:
Harcourt Brace Jovanovich, 1982.
 
{\bf Bester, H}, ``Screening versus Rationing in Credit Markets with
Imperfect Information,'' {\it American Economic Review}, September
1985, {\it 75}, 850-855.
 
{\bf Engers, Maxim and Fernandez, Luis}, ``Market Equilibrium
with
Hidden Knowledge and Self-Selection,'' {\it Econometrica,} March
1987.  {\it 55}, 425-39.
 
{\bf Fernandez, Luis and Rasmusen, Eric}, ``Equilibrium in Linear
Screening Markets when the Uninformed Traders can React to New
Offers,'' mimeo, UCLA AGSM, February 1989.
 
 {\bf Leland, Hayne and Pyle, David}, ``Informational
Asymmetries,
Financial Structure, and Financial Intermediation,'' {\it Journal
of
Finance,} May 1977, {\it 32,} 371-87.
 
{\bf Miyazaki, H.,} ``The Rat Race and Internal Labor Markets,''
{\it
Bell Journal of Economics}, Autumn 1977, {\it 8,} 394-418.
 
{\bf Rasmusen, Eric,} {\it Games and Information}, Oxford: Basil
Blackwell, 1989.
 
{\bf Riley, John,} ``Informational Equilibrium,'' {\it
Econometrica,}
March 1979, {\it 47}, 331-59.
 
{\bf Rosenthal R. and A. Weiss}, ``Mixed-Strategy Equilibrium in
a
Market with Asymmetric Information,'' {\it Review of Economic
Studies,}
1984, {\it 52}, 333-342.
 
{\bf Rothschild, Michael and Stiglitz, Joseph,} ``Equilibrium in
Competitive Insurance Markets: An Essay on the Economics of
Imperfect
Information,'' {\it Quarterly Journal of Economics,} November
1976,
{\it 90}, 629-49.
 
{\bf Scherer, Frederick}, {\it Industrial Market Structure and
Economic Performance,} second edition, Chicago: Rand McNally,
1980.
 
{\bf Spence, A.  Michael}, ``Job Market Signalling,'' {\it
Quarterly
Journal of Economics,} August 1973, {\it 87}, 355-74.
 
{\bf Stiglitz, Joseph and Weiss, Andrew}, ``Sorting Out the
Differences Between Screening and Signaling Models,'' 
in M. Dempster (ed.), {\it Papers in Commemoration of the
Economic 
Theory Seminars at Oxford University}, Oxford: Oxford University
Press, 1989.

{\bf Wilson, Charles}, ``The Nature of Equilibrium in Markets
with
Adverse Selection,'' {\it Bell Journal of Economics,} Spring
1980,
{\it 11}, 108-30.
 
 
\newpage
%---------------------------------------------------------------
\bigskip
 APPENDIX: {\bf Generalization}
 
   (In this section we suggest several generalizations that will
not
be made in the submitted version of this paper, but which might
be
the basis for future work. We include them in the working-paper
version in the hope of generating useful comment.)
 
The Spence screening model makes overly restrictive assumptions
about
the preferences of the sellers and the buyers.  It is possible to
greatly relax these assumptions and still be able to construct
examples of equilibria in which there is a single firm earning
zero profits.  The essential assumptions are these:
 
{\bf Assumption 1}: {\it There are two types of sellers, high
quality sellers (Highs) and low quality sellers (Lows).  The
proportion
of Highs is $\pi$ and the proportion of Lows is $1-\pi$.}
 
{\bf Assumption 2}: {\it The seller utility functions $U_L(y,p)$
and
$U_H(y,p)$ are differentiable,
strictly increasing in $p$, and
quasi-concave in both $p$ and $y$.}
 
{\bf Assumption 3}: {\it Suppose $y_L < y_H$, and $U_L(y_L , p_L
) <
U_L(y_H ,p_H )$.  Then $U_H(y_L , p_L ) < U_H(y_H , p_H )$.}
 
Assumption 3 is equivalent to assuming  at any offer the
indifference curves of the high quality sellers are flatter than
those of the low quality sellers.
 
{\bf Assumption 4}: {\it The buyer utility per unit of
consumption
purchased from a Low seller at the offer $(y,p)$ is $V_L(y) - p$,
and
the buyer utility per unit of consumption purchased from a High
seller at the offer $(y,p)$ is $V_H(y) - p$, where $V_L$ and
$V_H$
are differentiable and non-decreasing in $y$, and for every y,
$V_L(y) \leq V_H(y)$.}
 
We conjecture examples can also be constructed where there are
more than
two discrete types or where there is a continuous distribution of
seller types.

DIAGRAMS.
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