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\noindent  22 November 2005.  Eric Rasmusen, Erasmuse@indiana.edu.  \\
 Http://www.rasmusen/org/GI/chap10\_mechanisms.pdf.

\begin{LARGE}
  \begin{center}
  { \bf  10   Mechanism Design  and Post-Contractual Hidden Knowledge  }
 \end{center}
 \end{LARGE}

\vspace{1in}

   \noindent
 {\bf  10.1  Mechanisms,  Unravelling,   the Maskin Matching Scheme,   and the
Revelation Principle }

 \bigskip

\noindent
This chapter  looks at mechanism design.   A mechanism is a set of rules that
one player constructs and another   freely accepts in order to convey
information from the second player to the first. The mechanism contains an
information    report  by the second player  and a mapping from each possible
report to some action by the first.

 Adverse selection models can be viewed as problems of mechanism design. {
Insurance Game III}  was about an   insurance company  which wanted to know
whether a customer was safe or not. In  equilibrium it offers two contracts, an
expensive  full-insurance contract  preferred  by the safe customers and a
cheap  partial-insurance contract preferred by the unsafe.   In the language of
mechanism design,  the insurance company sets up a game in which a customer
reports his type as Safe or Unsafe, whichever he prefers to report, and the
company then assigns him either partial or full insurance as a consequence.
The contract offers are   a mechanism for getting the agents to truthfully
report their types.

 Mechanism design goes beyond simple adverse selection. It can be useful even
when  players begin a game with symmetric information or when  both players have
hidden information that they would like to exchange.

Section 10.1  introduces  moral hazard with  hidden knowledge using Production
Game VIII. It shows how to construct an optimal mechanism, how a Maskin matching
scheme can attain the first-best when information is observable by the players
but unverifiable by courts, and how ``unravelling'' can reveal private
information if lying can be prevented but silence cannot. Section 10.2 designs a
mechanism  for  a product quality game invented by  Roger Myerson.  Section 10.3
uses diagrams to apply the model to sales quotas. Section 10.4 introduces a
mechanism that is multi-player and ``dominant-strategy'',  the  Groves
Mechanism,  for use when the problem is to  elicit truthful reports from not one
but $N$ agents  who need to decide whether to invest in a public good. Section
10.5  applies the principles of mechanism design  to   price discrimination.
Section 10.6  lays out a more complicated model,   of rate-of-return regulation
by a  government that constructs a mechanism to induce a regulated company to
reveal  how easily it could reduce its costs.




\bigskip
 \noindent
  {\bf Post-Contractual Hidden Knowledge}

   Information is complete in moral hazard games, but  in {\bf moral hazard with
hidden knowledge}, also called {\bf post-contractual adverse selection},     the
agent, but not the principal,  observes a move of Nature  after the game begins,
but before he takes his action.    Information is symmetric at the time of
contracting-- thus the   ``moral hazard''---  but becomes asymmetric later--
thus the ``hidden knowledge''.    From the principal's point of view, agents are
identical at the beginning of the game  but   develop  private types midway
through. His  chief concern is to  give them incentives to disclose   their
types later,  which gives the games  a flavor close to that of   adverse
selection.    The agent might  also have to   exert effort in the game, but
effort's  contractibility is  less important when the principal does not know
which effort is appropriate   because he is  ignorant of the state of the world
chosen by Nature.   The main difference   technically is  that if information is
symmetric at the start and only becomes asymmetric   after a contract is signed,
the participation constraint  is based on the agent's expected payoffs across
the different types of agent he might become. Thus, there is just one
participation constraint even if there are eventually $n$ possible types of
agents in the model, rather than the $n$ participation constraints that would be
required in a standard adverse selection model.

  What makes post-contractual hidden knowledge an ideal setting for the paradigm
of mechanism design is that the   problem is to  set up a contract that (a)
induces the agent to make  a truthful report to the principal, and (b)  is
acceptable  to both principal and agent.
 There is more hope for obtaining efficient outcomes    than in adverse
selection.  The advantage    is that information is symmetric at the time of
contracting, so  neither player can use    private information to extract
surplus from the other by choosing inefficient contract terms.

 For a comparison  between the two types of moral hazard, let us modify
Production Game VII   from Chapter 9  and  turn it into a slightly different
game of hidden knowledge.




\begin{center}
 {\bf  Production Game VIII:  Mechanism Design }
 \end{center}
{\bf Players}\\
The principal and the agent.

 \noindent
{\bf The Order of Play}\\
1 The principal offers the agent a wage contract of the form $w(q,m)$, where $q$
is output and $m$ is a message to be  sent by the agent.\\
2 The agent accepts or rejects the principal's offer.\\
 3 Nature chooses the state of the world $s$, according to probability
distribution $F(s)$, where the state $s$ is $good$ with probability 0.5 and
$bad$ with probability 0.5.  The agent observes $s$, but the principal does
not.\\
4 If the agent accepted, he exerts effort $e$ unobserved by the principal, and
sends  message $m \in \{good, bad\}$ to him. \\
 5 Output is $q(e, s)$, where $q(e, good) =3e$ and  $q(e,bad) =e$, and the wage
is paid.

\noindent
  {\bf Payoffs}\\
 If the agent rejects the contract,   $\pi_{agent} = \bar{U}=0$ and $
\pi_{principal} = 0$.\\
 If the agent accepts the contract, $\pi_{agent}= U(e,w,s)=w-e^2$  and
$\pi_{principal}= V(q - w)=q-w$.

 \bigskip

 This game is almost the same as Production Game VII. The big difference is that
now the agent does not know his type at the point in time at which he must
accept or reject the contract. A smaller difference is that we have added the
message $m$ which the agent sends to the principal. This message is cheap talk--
it does not affect payoffs directly and there is no penalty for lying. It is
useful as a modelling convenience, to indicate which output-wage combination the
agent chooses.

   The principal would like to know $s$ so he can tell which effort level is
appropriate. In an ideal world he would employ    an honest agent who always
chose $m=s$,   but in  noncooperative games we ordinarily assume that agents
have no moral sense. Since the agent's words are worthless, the principal must
try to design a contract that either provides incentive  for truth-telling or
takes lying into account. He  {\bf implements} a {\bf mechanism} to extract the
agent's information.

 In Production Game VII, the adverse selection version of the game,    the
optimal contracts  had to satisfy two participation constraints and two
incentive compatibility constraints.

 In Production Game VIII,  the moral hazard with hidden information version,
the optimal contract must satisfy just one participation constraint, with   the
same two incentive compatibility constraints.

 The first-best is unchanged from Production Game VII. The optimal effort and
output in the good state of the world are $e_g=1.5$ and  $q_g =4.5$, and in the
bad state they are $e_b=0.5$   and $q_b =0.5$. Also unchanged is that the
principal must solve the  problem:
 \begin{equation} \label{a1}
  \stackrel{Maximize}{q_g, q_b,w_g, w_b}  [ 0.5 (q_g - w_g) + 0.5 (q_b-w_b)],
 \end{equation}
 where the agent is paid under one of  two forcing contracts,  $(q_g, w_g)$ if
he reports $m=good$ and  $(q_b, w_b)$ if he reports $m=bad$,  where producing
the wrong output for a given contract results in boiling in oil.

 The contracts must induce  participation and self-selection. We can write the
constraints  in terms of the agent's payoff function from  in effect choosing
one of the two ($q,w$) contracts  by his choice of the report $good$ or $bad$.
The effort he will choose under those contracts will be
 $e=q/3$ for the good state and $e=q$ for the bad state,  the same as in
Production Game VII.

The self-selection constraints are the same as in Production Game VII.
  In the good state, the agent must choose the good-state contract,so
 \begin{equation} \label{a2}
{\displaystyle  \pi_{agent} (q_g, w_g|good) =  w_g -     \left( \frac{q_g}{3}
\right)^2  \geq  \pi_{agent} (q_b, w_b|good) = w_b -      \left( \frac{q_b}{3}
\right)^2  }
 \end{equation}
  and   in the bad state he must choose the bad-state contract, so
 \begin{equation} \label{a3}
\pi_{agent} (q_b, w_b|bad) =    w_b -    q_b^2  \geq \pi_{agent} (q_g, w_g|bad)
=    w_g -    q_g^2.
 \end{equation}

The participation constraints of Production Game VII now merge, though, because
at the time of contracting the agent does not know what the state will be. The
single participation constraint is therefore
\begin{equation} \label{a4}
{\displaystyle  0.5 \pi_{agent} (q_g, w_g|good) +0.5 \pi_{agent} (q_b, w_b|bad)
= 0.5 \left(  w_g -      \left( \frac{q_g}{3}   \right)^2  \right) + 0.5 \left(
w_b -      q_b^2    \right) \geq 0.}
 \end{equation}

  This single participation constraint is binding, since  the principal wants to
pay the agent as little as possible. The good state's self-selection constraint
will be binding. In the good state the agent  will   be tempted to take the
easier contract appropriate for the bad state (due to the ``single-crossing
property'' to be discussed in a later section ) and so the principal has to
increase the agent's payoff from the good-state contract to yield him at least
as much as in the bad state. He does not want to increase the surplus any more
than necessary, though, so the good state's self-selection constraint will be
exactly satisfied.      This gives us two equations,
\begin{equation} \label{a5}
\begin{array}{l}
0.5 \left(  w_g -    \left( \frac{q_g}{3}   \right)^2  \right) + 0.5 \left(  w_b
-   q_b^2  \right) = 0\\
  \\
       w_g -     \left( \frac{q_g}{3}   \right)^2 =    w_b -   \left( \frac{q_b}
{3}   \right)^2   \\
\end{array}
\end{equation}
 Solving them out yields $w_b = \frac{5}{9} q_b^2$ and $w_g = \frac{1}{9} q_g^2
+ \frac{4}{9} q_b^2$.

 Returning to the principal's maximization problem in (\ref{a1}) and subsituting
for  $w_b$ and $w_g$,  we can rewrite it as
 \begin{equation} \label{a6}
  \stackrel{Maximize}{q_g, q_b} \;\; \pi_{principal}=  \left[ 0.5 \left(q_g -
\frac{q_g^2}{9}   - \frac{4q_b^2}{9}  \right) + 0.5 \left(q_b  -\frac{5q_b^2}{9}
\right) \right]
 \end{equation}
 with no constraints.  The first-order conditions are
 \begin{equation} \label{a7}
  \frac{\partial  \pi_{principal}}{\partial q_g}=   0.5 \left(1 -   \left[
\frac{2}{9} \right] q_g
\right)        =0,
 \end{equation}
 so $q_g =4.5$, and
  \begin{equation} \label{a8}
   \frac{\partial  \pi_{principal}}{\partial q_b}=0.5 \left(   - \frac{8q_b}{9}
\right) + 0.5 \left( 1  -\frac{10q_b }{9}   \right)  =0,
 \end{equation}
so $q_b  = \frac{9}{18} =.5$. We can then find the wages that satisfy
the constraints, which are $w_g \approx 2.36$ and $w_b \approx 0.14$.

As in Production Game VII, in the good state the effort is at the first-best
level while in the bad state it is less. Unlike in Production Game VII   the
agent does not earn informational rents, because at the time of contracting he
has no private information. In Production Game VII  the wages were $w_g' \approx
2.32$ and $w_b'\approx 0.07$. Both   wages are higher in Production Game VIII,
but so is the effort and output required of the agent in the bad state. The
principal in Production Game VIII is less constrained,   and thus able to  (a)
come closer to the first-best when the state is bad,  and (b)    reduce the
rents to the agent.  Those are general features of moral hazard with hidden
knowledge.


 \bigskip
\noindent
{\bf Observable but  Nonverifiable Information and   the   Maskin Matching
Scheme}

  If the state or type is public information, then it is straightforward to
obtain the first-best using forcing contracts. What if the state is observable
by both principal and agent, but is not public information?

  The problem is that there are really three players involved in  the
contracting situation:  the principal who offers the contract, the agent who
accepts it--- and the court that enforces it.  If the courts cannot observe the
state, a contract conditioning the wage on the state   is unenforceable, no
better than having no contract at all. We say that the variable $s$ is {\bf
nonverifiable} if contracts based on it cannot be enforced. Most simply, the
variable would one whose value the court cannot measure accurately enough to be
useful-- a worker's intensity of effort, for example, as opposed to the number
of hours he was on the job. Or, the variable could be one which the court could
observe but which for some reason it will not allow to be used in a legal
contract--- the amount of cocaine in a package, for example.

It does seem, however, that even if the courts will not enforce a contract based
on a variable, if both the principal and the agent observe it they should be
able to come up with a more efficient contract than if just the agent observes
it. And indeed, mutual observability can help.
 Maskin (1977) suggested a matching scheme which would take the following two-
part  form for Production Game VIII:

\noindent
{\it  (1) Principal and agent simultaneously send messages $m_p$ and $m_a$ to
the court saying whether the state is good or bad.   If $m_p \neq m_a$, then no
contract is chosen and both players earn zero payoffs.     If  $m_p=m_a$, the
court enforced part (2) of the scheme. \\
  (2)  The agent is paid the wage $(w|q)$    with either the good-state forcing
contract $(2.25|4.5)$  or the bad-state forcing contract $(0.25|0.5)$, depending
on his report $m_a$, or is boiled in oil if he the output is inappropriate to
his report.   }

  There exists an equilibrium in which  both players are willing to  send
truthful messages, because a deviation would result in zero payoffs. The agent
earns a payoff of zero anyway  (the  numbers in the forcing contract  come from
making his participation constraint binding), but that is due to the open-set
problem and to our assumption that the principal has all of the bargaining
power.    The principal's payoff is positive and   efforts are at the first-best
level.

  Usually  this kind of scheme has multiple   equilibria, however,    perverse
ones in which both players send   false message which  match   and  inefficient
actions   result.  Here,  in a  perverse equilibrium  the principal  and agent
would always  send the message $m_p=m_a=bad$.    Even when the state was
actually good, the payoffs would be ($\pi_{principal} (good)=  0.5-0.25>0$ and
$\pi_{agent} (good)=  0.25- (0.5)^2=0$, so  neither player would have incentive
to deviate unilaterally and drive    payoffs to zero.\footnote{
 By the way, ``$m_p=m_a=good$ always''   could not happen in equilibrium because
the agent would prefer the  zero payoff from mismatching to being forced by the
threat of boiling oil to try  to attain $q_g$ when the state is bad.}

  Perhaps a  bigger problem than the multiplicity of equilibria is
  renegotiation due to players' inability to commit to the mechanism. Suppose
the  equilibrium says that  both players  will send truthful messages, but the
agent deviates and reports $m_a=bad$ even though the state is good. The court
will say that the contract is nullified, and I  told you  that the payoffs would
be zero.  But the court's decision is  not really the end of the story.  The
agent  could negotiate a new contract with the principal, something the
principal would be willing to do rather than give up the gains from trade. In
the  Maskin Matching Scheme the  punishment  for deviation  is costly to both
players, and so they will agree to bypass the scheme rather than inflict the
punishment.

 In this, the  Maskin scheme is similar to the Holmstrom Teams contract that was
discussed in Chapter 8, where if output was even a little too small, it was
destroyed rather than divided among the team members. There, a solution was to
have a third party who would receive the output if it was slightly too small,
and so would refuse to renegotiate it. The analogy here would be to write a
contract in which both principal and agent paid a third party if their
announcements disagreed. This sounds possible, yet we rarely observe this in
practice.

    Or do we? In his book {\it Wise Guys} (p. 57), Nicholas Pileggi quotes low-
level gangster Henry Hill  as saying that criminals need the protection of mafia
``wiseguys'' because they can't go to the police when they have  a dispute over
illegal activities:
\begin{quotation}
\begin{small}
``For instance, say I've got a  fifty-thousand-dollar hijack load, and when  I
make my delivery, instead of getting paid, I get stuck up.
What am I supposed to do? Go to the cops? Not likely. Shoot it out? I'm a
hijacker, not a cowboy. No.  The only way to guarantee that I'm not going ripped
off by anybody is to be established with a  member, like Paulie. Somebody who is
a made man. A member of a crime family. A soldier. Then ...    that's the end of
the ball game. Goodbye. They're  dead...  Of course,  problems can arise when
the guys sticking you up are associated with wiseguys too. Then there has to be
a  sit-down between your wiseguys and their wiseguys.  What usually happens then
is that the wiseguys divide whatever you stole for their own pocket, send you
and the guy who robbed you home with nothing. And if you complain, you're
dead.''
\end{small}
\end{quotation}
   The low-level gangsters have a strong incentive to report the same story, or
the higher-ups take away the property under dispute. This may sound familiar to
parents too--- ``If we can't resolve this, the toy is going in the closet for a
whole week.''  It is perhaps even the wisdom of Solomon-- see Glazer \& Ma
(1989) or  Baliga (2002) (which also has a nice exposition of mechanism design
using the author's own work to illustrate).



 \bigskip
\noindent
{\bf Unravelling: Information Disclosure when Lying Is Prohibited}

 \noindent
    There is another special case in which hidden information can be forced into
the open: when the agent is prohibited from lying  and only has a choice between
telling the truth or remaining silent.

  In Production Game VIII, this set-up would give the agent two possible message
sets. If the state were good, the agent's message would be taken from $m \in
\{good, silent\}$. If the state were $bad$, the agent's message would be taken
from  $m \in \{bad, silent\}$.

 The agent would have no reason to be silent if the true state were bad (which
means low output would be excusable), so his message then would be $bad$. But
then if the principal hears the message $silent$ he knows the state must be
good-- $good$ and $silent$ both  would occur   only when the state was good. So
the option to remain silent is worthless to the agent.

  This can be generalized.
Suppose   Nature uses the uniform distribution to assign the variable $s$ some
value in the interval $[0, 10]$ and the agent's payoff is increasing in the
principal's estimate of $s$.  Usually we assume that the agent can lie freely,
sending a message $m$ taking any value in $[0,10]$, but let us assume instead
that he cannot lie but he can conceal information. Thus, if $s = 2$, he can send
the uninformative message $m \geq 0$ (equivalent to no message), or the message
$m \geq 1$, or $m=2$, but not the lie that  $m \geq 4.36$.

 When $s=2$ the agent might as well send a message that  is the exact truth:
``$m=2$.''  If he were to choose the message ``$m \geq 1$''  instead, the
principal's first thought might be to estimate $s$ as the average value in the
interval $[1,10]$, which is 5.5. But the principal would realize that no agent
with a value of $s$ greater than 5.5 would want to send the message ``$m \geq
1$'' if  5.5 was  the resulting deduction. This realization restricts the
possible interval to [1, 5.5], which in turn has an average of 3.25. But then no
agent with $s > 3.25$ would send the message ``$m \geq 1$.''  The principal
would continue this process of logical {\bf unravelling} to conclude that $s=
1$.  The message ``$m \geq 0$'' would be even worse, making the principal
believe that $s = 0$. In this model, no news is bad news.   The agent would
therefore not send the message ``$m \geq 1$''  and  he would be indifferent
between ``$m=2$''  and ``$m \geq 2$'' because the  principal would make the same
deduction from either message.

   Perfect unravelling   is paradoxical, but that is because  the assumptions
behind the  last paragraph's  reasoning   are rarely satisfied in the real
world. In particular,  either unpunishable lying or  genuine ignorance allow
information to be concealed.  If the seller is free to lie without punishment
then  in the absence of other incentives  he always pretends that his
information is extremely favorable, so nothing he says conveys any information,
good or bad. If he really is ignorant in some states of the world, then his
silence could mean either that he has nothing to say or that he has nothing
favorable to report.   The unravelling argument fails because if he sends an
uninformative message the buyers will attach some probability to ``I don't
know'' instead of ``unfavorable news.''  Problem 10.1   explores unravelling
further.  For a careful discussion of the idea, see Milgrom (1981b).


\bigskip
  \noindent
 {\bf The Revelation Principle}

 \noindent
A principal might choose to offer a contract that induces his agent to lie in
equilibrium, since he can take lying into account when he designs the contract,
but this complicates the analysis. Each state of the world has a single truth,
but a continuum of lies. Generically speaking, almost everything is false.  The
following principle helps us simplify contract design.

 \noindent
 {\bf The Revelation Principle.} {\it For every contract $w(q,m)$ that leads to
lying (that is, to $m \neq s$), there is a contract $w^*(q,m)$ with the same
outcome for every $s$ but no incentive for the agent to lie.}



 Many  possible contracts  make  false messages   profitable for the   agent
because  when the state of the world is $a$ he receives a reward of $x_1$ for
the true report of $a$ and  $x_2>x_1$ for the false report of $b$.   A contract
which gives the agent the same reward of $x_2$ regardless of whether he reports
$a$ or $b$ would lead to exactly the same payoffs for each player while giving
the agent no incentive to lie. The revelation principle  notes that a   truth-
telling contract  like this  can always be found by imitating the relation
between states of the  world and payoffs in the equilibrium of a contract with
lying.   A {\bf direct mechanism}, in which the agents tell the truth in
equilibrium, can be found that is equivalent to any {\bf indirect mechanism} in
which they lie. The idea can also be applied to games in which two  players must
make reports to each other.

 In Production Game VIII, the Revelation Principle is not very useful since
there are only two possible states and the equilibrium is perfectly separating--
agents get different rewards in each state.  Where the Principle has bite is
when the equilibrium involves some pooling.
   Suppose we are trying to design a mechanism to make people with higher
incomes pay higher taxes,    but anyone who makes \$70,000 a year can claim he
makes \$50,000 and we do  not have the resources to  catch him, or  any other
variables to use for leverage to end up with   a fully  separating equilibrium.
 We could design a mechanism  in which higher reported incomes pay higher taxes,
but reports of \$50,000 would come from both people who truly have that income
and people whose income is \$70,000.  The revelation principle says that we can
rewrite the tax code to set the tax to be the same for taxpayers earning
\$70,000 and  for those earning \$50,000, and the same amount of taxes will be
collected without anyone having   incentive to lie. This would be better if we
are
 concerned with the effect on the moral climate of cheating on income taxes.
    Similarly, applied to children's education, the principle says that the
mother who agrees never to punish her daughter if she tells her  all her
escapades will never hear any untruths. That is fine for deterring lying, but
not very useful for deterring other misbehavior.

  Clearly, the principle's usefulness is not  so much to improve outcomes as
to simplify contracts. The principal (and the modeller) need only look at
contracts which induce truth-telling instead of more complex schemes in which
the principal knows lying is going on but adjusts rewards accordingly. Thus,
the relevant strategy space is shrunk.  and we can add a third constraint  to
the incentive compatibility and participation constraints to help calculate the
equilibrium:

\noindent
 (3) {\bf Truth-Telling.} The equilibrium contract makes the    agent willing to
choose $m =s$.

The  revelation principle says that a truth-telling equilibrium exists, but not
that it is unique.  It may well happen that the equilibrium is a weak Nash
equilibrium in which   the optimal contract gives the agent no incentive to lie
but also no incentive to tell the truth. This is similar to Section 4.3's open-
set problem;     the optimal contract   satisfies the agent's participation
constraint but makes him indifferent  between  accepting and rejecting the
contract. If agents derive the slightest utility from telling the truth,   then
truthtelling becomes a strong equilibrium, but if their utility from telling the
truth is really significant, it should be made an explicit part of the model. If
the utility of truth-telling is strong enough, in fact, agency problems and the
costs associated with them  disappear.   This is one reason  why morality is
useful to business.

    The Revelation Principle does depend heavily on an implicit assumption we
have made: the principal cannot breach his contract. In the example of tax
design, for example, if the government can freely modify the tax schedule after
the agents report their incomes, the  direct mechanism under which  agents pay
the same tax for income of either \$50,000 or \$70,000 will break down, because
the rich agents know that if they report their incomes of \$70,000 truthfully,
the government will change the tax schedule ex post and make them pay more.
Instead, the government will have to use an indirect mechanism in which taxes
are always higher for higher incomes and  those people with incomes of \$70,000
untruthfully report \$50,000.   Thus, throughout this chapter we will be
assuming that the principal can commit to his mechanism--- can commit to not
using all the information he receives from the agent.



\bigskip
\noindent
 {\bf  The Sender-Receiver Game  of Crawford and Sobel: Coarse Information
Transmission  }

 Even if the informed and uninformed players have different incentives and
can't commit to a mechanism,  if their incentives are close enough,   truthful
if imperfect messages can be sent in equilibrium.  Let us call the informed
player ``the sender'' and the uninformed player ''the receiver''. That is common
in this kind of model, instead of calling the two players the   agent and the
principal.   Suppose the information is the sender's type, $t$, uniformly
distributed on $[0,10]$.  The sender sends a message, $m$,  and the receiver
chooses an action,  $a$, in response,  where  $a$ and $m$ are also in  $[0,10]$.
At the extremes of payoff function similarity, it is clear what happens. Suppose
the sender  wants $a$ to be as close to $t$ as possible. If the sender also
wants $a$ to be close to $t$,  he will tell the truth: $m=t$. If he wants $a$ to
be as big as possible regardless of $t$, on the other hand, because his ideal
action is $a=10$, then he will lie  and  the receiver will ignore any message.
But what happens if, say, the sender's ideal action is $(t+1)$, so he doesn't
want $a$ to be too large, but he does want $a$ to be bigger than a fully-
informed receiver would choose?  Crawford \& Sobel (1982) discovered the answer
in a famous article titled ``Strategic Information Transmission.''   We will
see what happens in the game below, which I adapted  from Gibbons (1992, Section
4.3.A), a good place to go if you want to learn more.
 
  \begin{center} 
{\bf  The  Crawford-Sobel Sender-Receiver Game} 
 \end{center}
  {\bf Players}\\
   The sender and the receiver.   

\noindent 
{\bf The  Order of Play}\\
0 Nature chooses the sender's type to be $t \sim U[0, 10]$.\\
  1  The sender chooses message $m \in  [0, 10]$. \\ 
2   The receiver chooses action $a \in  [0, 10]$. \\
  
\noindent
 {\bf Payoffs}\\ 
 The payoffs are quadratic loss functions in which each player has an ideal
point and wants $a$ to be close to that ideal point.

 \begin{equation} \label{e200}
\begin{array}{ll}
\pi_{sender}& = \alpha - (a-[t+1])^2\\
& \\
\pi_{receiver} &= \alpha - (a-t)^2\\
\end{array}
\end{equation}

  First, let's see why perfect truthtelling cannot happen in equilibrium.
Suppose the receiver believed that the sender always sent $m=t$ and so chooses
$a=m$. Would the sender indeed be willing to tell the truth?

He would not. The sender would not always report $m=10$, because his ideal point
is $a=t+1$, rather than $a$ being as big as possible. If, however, the sender
thinks the  receiver will believe him, he will deviate to reporting $m=t+1$,
always exaggerating his type slightly.

 What if the receiver adapts to this, and chooses $a=m-1$? Then the sender would
change too, and send $m=t+2$, exaggerating more. This unravelling away from the
truth (as opposed to the unravelling {\it toward} the truth when outright lying
is forbidden) continues until the only message the sender reports is $m=10$,
regardless of his type and the receiver ignores it.  This explanation is
heuristic, and does not prove that there is no fully separating equilibrium, in
which each type of sender reports a different message, but  Crawford \& Sobel
(1982)  prove that this  is the case.




Thus, one
 equilibrium is the pooling equilibrium in which the sender's message is ignored
and the receiver chooses $a= Et = 5$.  This equilibrium could take either of two
forms:

\noindent
 {\bf Pooling Equilibrium 1} \\
{\bf   Sender:} Send $m=10$ regardless of $t$. \\
{\bf Receiver:} Choose $a=5$ regardless of $m$. \\
{\bf Out-of-equilibrium belief:} If the sender sends $m<10$, the receiver uses
passive conjectures and still believes that $t \sim U[0,10]$.

\noindent
 {\bf Pooling Equilibrium 2} \\
{\bf   Sender:} Send  $m$ using a mixed-strategy distribution  independent of
$t$ that has  the   support $[0,10]$ with positive density everywhere.  \\
{\bf Receiver:} Choose $a=5$ regardless of $m$. \\
{\bf Out-of-equilibrium belief:} Unnecessary, since any message might be
observed in equilibrium.

 In each of these two equilibria, the sender's action conveys no information and
is ignored by the receiver. The sender is happy about this if it happens that
$t=4$,  and the receiver is if $t=5$, but  averaging over all possible $t$, both
their  payoffs are lower than if the sender could commit to truthtelling.
There also, however, exists a partial pooling equilibrium   in which the sender
truthfully reports whether his type is in the  low interval  $[0,x]$  or  the
high interval $[x,10]$, with $x=3$.

\noindent
 {\bf Partial Pooling Equilibrium 3} \\
{\bf   Sender:} Send  $m=0$ if $t \in [0,3]$ or $m=10$ if $t \in   [3,10]$.   \\
{\bf Receiver:} Choose $a=1.5$ if $m<3$  and  $a=6.5$ if $m \geq 3$ \\
{\bf Out-of-equilibrium belief:}  If $m$ is something other than 0 or 10, then
$t \sim U [0, 3]$ if $m \in  [0,3)$   and  $t \sim U [3,10]$ if $a \in  [3,10]$.
 

  In effect, the Sender has reduced his message space to two messages,  LOW (=0)
and  HIGH (=10), in Equilibrium 3.  Rather than just testing that this is an
equilibrium, let us derive it, to show why the equilibrium interval-splitting
type is $x=3$.

First, note that the receiver's  optimal strategy in a a partially pooling
equilibrium is to choose his action to equal the expected value of the type in
the interval the sender has chosen. Thus,  if $m=0$, the receiver will choose
$a=x/2$ and if $m=10$ he will choose $a = (x+10)/2$.

 The receiver's equilibrium response determines the sender's payoffs from  his
two messages.    The payoffs between which he chooses are:
 \begin{equation} \label{e201}
\begin{array}{ll}
\pi_{sender, m=0} & {\displaystyle    = \alpha -  \left(\left[t+1 \right]-
\frac{x}{2}\right)^2}  \\
  & \\
\pi_{sender, m=10}& = {\displaystyle \alpha -  \left(\frac {10+x}{2} -\left[t+1
\right]\right)^2}
\end{array}
\end{equation}
    There exists a value $x$ such that if  $t=x$, the sender is indifferent
between $m=0$ and $m=10$, but if $t$ is lower he prefers $m=0$ and if $t$ is
higher he prefers $m=10$. To find $x$, equate the two payoffs in expression
(\ref{e201})  and simplify to obtain
 \begin{equation} \label{e202}
 {\displaystyle \left[t+1 \right]- \frac{x}{2}  
  =  \frac {10+x}{2} -\left[t+1 \right].  }
\end{equation}
 We   set $t=x$  at the point of indifference, and solving for $x$ then yields
$x=3$.

  Thus, the divergence in preferences of the sender and receiver coarsens the
message space, in effect. The sender will not send a truthful precise message,
but if expectations are right (so we have the partially pooling equilibrium) he
will send a truthful coarse message. If the true value of $t$ is small, the
sender will report the fairly precise information that $t$ lies in [0,3]. If $t$
is larger, it is harder to induce a truthful report, since the sender has a
tendency to exaggerate and report $t$ larger than it is, but the message can at
least rule out the interval [0,3).
 
If instead of wanting $(t+1)$ to be the action, the preferences of sender and
receiver diverged more--- say, to  $(t+8)$--- then there would only be the
uninformative pooling equilibrium. If they divereged less--- say, to $(t+0.1)
$--- then there would exist  other partially pooling equilibria that had more
than just two effective messages  and would  distinguish between three or more
intervals instead of between just two.

    In the Crawford-Sobel Sender-Receiver Game,   the receiver cannot commit to
the way he reacts to the message, so this is not a mechanism design problem. Nor
is the sender punished for lying, so the  unravelling argument for truthtelling
does not apply.  Nor do the players' payoffs depend directly on the message,
which might permit the signalling we will study in Chapter 11 to operate.
Instead, this is a {\bf cheap-talk game}, so called because of these very
absences: $m$ does not affect the payoff directly, the players cannot commit to
future actions, and  lying brings no directly penalty.  In Chapter 3 I alluded
to how cheap talk might help  select  among equilibria in the Battle of the
Sexes. In particular, knowing what the other player was thinking of doing would
help to avoid the mixed-strategy equilibrium, with its low payoff. Our sender
and receiver  are in a similar situation here: their interests are similar but
not identical,  and they could both benefit from some transfer of information.
If expectations are appropriate, they do so, in the partially pooling
equilibrium. If they do not expect the cheap talk to be informative, however, it
will not be, and coordination will fail.

 

\vspace{1in}


 \noindent
{\bf  10.2:  Myerson Mechanism Design  }

 Now let's look at another example, a classic one from Roger Myerson,
who uses a trading example to illustrate mechanism design in Sections 6.4  and
10.3 of his  1991 book.  A seller has 100 units of a good. If it is high
quality, he values it at 40 dollars per unit; if it is low quality, at 20
dollars. The buyer, who cannot observe quality before purchase, values high
quality at 50 dollars per unit and low quality at 30 dollars. For efficiency,
all of the good should be transferred from the seller to the buyer. The only way
to get the seller to truthfully reveal the quality of the good, however, is for
the buyer to say  that if the seller admits the quality is bad, he will buy more
units than if the seller claims it is good. Let us see how this works out.

  Depending on who offers the contract and when it is offered, various games
result.  We will look at one in which the seller makes the offer, and does so
before he knows whether his quality is high or low.

\begin{center}
{\bf The Myerson Trading Game:   Post-Contractual Hidden Knowledge}
\end{center}
 {\bf Players}\\
 A buyer and a seller.

 \noindent
{\bf The Order of Play}\\
1  The  seller offers the buyer a contract  $ \{ q_h, p_h,  q_l, p_l \}$  under
which   the seller  will    declare   his quality $m$ to be high or low,  and
the buyer   will then    buy  $q_l$ or $q_h$   units of the 100 the seller has
available, at price $p_l$ or $p_h$. The contract is $ \{ w(m) = q(m)p(m), q(m)
\}$. Zero is paid if the wrong output is delivered.  \\
 2 The buyer accepts or rejects the contract.\\
3  Nature chooses whether the  type of the seller's good,  $s$,    is High
quality (probability 0.2) or Low  (probability 0.8), unobserved by the buyer.\\
 4.  If  the contract was accepted by both sides,  the seller declares his type
to be L or H and sells at the appropriate quantity and price as stated in the
contract. \\

\noindent
{\bf Payoffs}\\
If the buyer rejects the contract,  $\pi_{buyer} =  0$, $\pi_{seller \; H} =
40*100$,  and $\pi_{seller \; L} =  20*100$. \\
    If the buyer accepts the contract and  the seller declares a type that has
price $p$ and   quantity $q$  then
 \begin{equation}   \label{e25}
  \pi_{buyer|  L} =   (30-p) q  \;\;\;  and\;\;\; \pi_{buyer|  H} =  (50-p) q
\end{equation}
 and
 \begin{equation}   \label{e26}
 \pi_{seller \; H} =  40(100-q) + pq  \;\;\;  and\;\;\;  \pi_{seller \; L} =
20(100-q) + pq.
 \end{equation}
 because the seller has an opportunity cost (a personal value or production
cost)  of 40 per high-quality unit and 20 per low-quality unit and

 \bigskip

 The seller wants to design a contract subject to two sets of constraints.
First, the buyer must accept the contract. Thus, the participation constraint
is\footnote{Another kind of participation constraint would apply if the buyer
had the option to reject purchasing anything, after accepting the contract and
hearing the seller's type announcement.  That would not make a difference here.
}
 \begin{equation}  \label{e27}
  \begin{array}{ll}
  (0.8) \pi_{buyer| seller\; L}   + (0.2) \pi_{buyer| seller\; H}   &\geq  0 \\
 &\\
 0.8   [ (30-p_l)q_l] + 0.2 [ (50-p_h)q_h]  &\geq  0   \\
\end{array}
   \end{equation}
 This constraint will be binding, since the seller has no reason to leave the
buyer any surplus. Notice, however, that if it is binding and both $q_l$ and
$q_h$ are positive, we can conclude that $p_l=30$ and $p_h=50$.

  There  might also  be a participation constraint for the seller himself,
because it might be that even when he designs the contract that maximizes his
payoff, his payoff is no higher than when he refuses to offer a contract. He can
always offer the  acceptable (if vacuous) null  contract,  $(q_l=0, p_l=0 ,
q_h=0, p_h=0)$, however, so we do not need to write out the seller's
participation constraint separately.

  Second, the seller must design a contract that will induce  himself to tell
the truth later once he discovers his type. This is, of course a bit unusual---
the seller is like a principal designing a contract for himself as agent. That
is why things are  different in this chapter than in the chapters on   moral
hazard.   What is happening is that the seller is trying to sell not just a
good, but a contract, and so he must make the contract attractive to the buyer.
Thus, he faces two incentive compatibility constraints: one for when he is low
quality,
 \begin{equation}  \label{e28}
 \begin{array}{ll}
 \pi_{seller \; L} (q_l, p_l )  &\geq  \pi_{seller \; L} (q_h, p_h )\\
 &\\
 20 (100-q_l) + p_l q_l   &\geq  20 (100-q_h) + p_h q_h    \\
 &\\
 20 (100-q_l) + 30 q_l   &\geq  20 (100-q_h) + 50q_h    \\
 &\\
  q_l   &  \geq  3q_h    \\
  \end{array}
\end{equation}
 and  one for when he has high quality,
  \begin{equation} \label{e29}
 \begin{array}{ll}
 \pi_{seller \; H} (q_h, p_h )& \geq \pi_{seller \; H} (q_l, p_l ) \\
 &\\
 40 (100-q_h) + p_h q_h &  \geq  40 (100-q_l) + p_l q_l   \\
&\\
 40 (100-q_h) + 50 q_h &  \geq  40 (100-q_l) + 30 q_l   \\
&\\
   q_h &  \geq  - q_l    \\
\end{array}
 \end{equation}

 The low-quality incentive compatibility constraint, inequality (\ref{e28}),
tells us that  $q_l>q_h$. The price of 50 that comes from claiming to have high
quality dominates the price of 30 that comes from claiming to have low quality.
If the seller cannot sell as great a quantity when he claims the quality is
high, though, he might admit to low quality (think of an extreme case such as
$q_l=100, q_h=5$).

  On the other hand,   the high-quality incentive compatibility constraint is
satisfied  for all possible $q_l$ and $q_h$, because the high-quality seller's
opportunity cost of 40 is greater than the price of 30 he could get by
pretending to have low quality.

   Thus, we know from the  low-quality incentive compatibility constraint that
we need  $q_l>q_h$, and in fact that $q_l = 3q_h$ at the optimum.  The seller's
payoff function is
  \begin{equation}  \label{e30}
 \begin{array}{ll}
 \pi_s  &=0.8 \pi_{seller \; L} (q_l, p_l ) +  0.2\pi_{seller \; H} (q_h, p_h
)   \\
 & \\
  &=  0.8 [(20)(100-q_l) +  p_lq_l  ]   +  0.2   [(40)(100-q_h) + p_hq_h  ]   \\
 &\\
  &=  0.8 [(20)(100-q_l) +  30q_l  ]   +  0.2   [(40)(100-q_h) + 50q_h  ]    \\
 \end{array}
 \end{equation}
  This is increasing in both $q_l$ and $q_h$, so the seller would like to choose
them to be as big as possible, subject to the constraints  that $q_l = 3q_h$,
$q_l \leq 100$, and $q_h \leq 100$. Thus, $q_l$ will be   the maximum possible,
the first-best level $q_l=100$, and $q_h$ will be $q_h =q_l/3= 33 \frac{1}{3}$.

   The equilibrium follows the general pattern for these games, though it has a
twist because the informed player (the seller) has all the bargaining power, so
it is the uninformed player (the buyer) whose participation constraint is
binding. The incentive compatibility constraint is binding for  the type with
the most temptation to lie, and not for the other type. Using the two binding
constraints, we can solve out for the values of  some  of the choice variables
in terms of other choice variables,   and then we can  maximize the payoff of
the player making the offer (the seller) to solve for  values of those remaining
variables.  That is a useful general method, even though  different  games will
have  their own special features.


The mechanism will not work if further offers can be made after the end of the
game.  The mechanism is not  first-best efficient; if the seller is high-
quality, then he only sells  $33 \frac{1}{3}$ units to the buyer instead of all
100, even though both realize that the buyer's value is 50 and the seller's is
only 40. If they could agree to sell the remaining $66 \frac{2}{3}$  units, then
the mechanism would not be incentive compatible in the first place, though,
because then the low-quality seller would pretend to be high-quality, first
selling  $33 \frac{1}{3}$  units and then selling the rest. The importance of
commitment is a general feature of mechanisms.

 A technical observation is that although we  specified the contract  in terms
of $(p, q)$, a price  per unit    $p$ and a quantity $q$ at that price, we could
have set it up  instead as  $(w, q)$, a total price amount $w$ for the quantity
$q$.  That would be more in the style of mechanism design, with its emphasis on
setting up  the outcome of every outcome as the  total transfers paid from one
player to another and how an allocation or other decision is made.





\vspace{1in}

\noindent
 {\bf 10.3: An Example of    Post-Contractual  Hidden Knowledge:{  The Salesman
Game}}

 \noindent
 Suppose the  manager of a company has told his salesman to investigate a
potential customer, who is either a $Pushover$ or a $Bonanza$.  If he is a
$Pushover$, the efficient sales effort is low and sales should be moderate.  If
he is a $Bonanza$, the effort and sales should be higher. This is similar to
Production Game VIII, not the same.

 In the Salesman Game, the  principal can perfectly deduce effort, even out of
equilibrium, we will look at pooling as well as separating equilibria,    and
for the analysis we will make use of    diagrams.

\begin{center}
 {\bf  The  Salesman Game}
 \end{center}
 {\bf Players}\\
A manager and a salesman.

\noindent
 {\bf The Order of Play}\\
1 The manager offers the salesman a contract of the form $[w(m), q(m)]$, where
$w$ is the wage,  $q$ is sales,  and $m$ is a message.\\
2 The salesman decides whether or not to accept the contract.\\
3 Nature chooses whether the customer type $t$ is a $  Bonanza$ or a $Pushover$
with probabilities 0.2 and 0.8.  The salesman observes the type, but the manager
does not.\\
 4 If the salesman has accepted the contract, he chooses his  effort $e$. His
sales level is  $q=e$, so his sales perfectly reveal  his effort. \\
 5 The salesman's wage is $w(m)$ if he chooses $e= q(m)$ and zero otherwise.

\noindent
 {\bf Payoffs}\\
 The manager is risk neutral and the salesman is risk averse. If the salesman
rejects the contract, his payoff is $\bar{U}= 8$ and the manager's is zero. If
he accepts the contract, then\\
 \begin{tabular}{ll}
$ \pi_{manager}$ & $= q - w$ \\
 $\pi_{salesman}$ &$ = U(e, w, \theta)$, where $\frac{\partial U}{\partial e} <
0, \frac{\partial^2 U}{\partial e^2} < 0, \frac{\partial U}{\partial w} > 0,
\frac{\partial^2 U}{\partial w^2} < 0 $
 \end{tabular}

 \bigskip

   Figure 1 shows the indifference curves of manager and salesman, labelled with
numerical values for exposition.  The manager's indifference curves are straight
lines with slope $1$  because he is acting on behalf of a risk-neutral company.
If the wage and the quantity both rise by a dollar, profits are unchanged, and
the profits do not depend directly on whether $s$ takes the value $Pushover$ or
$Bonanza$.

  The salesman's indifference curves also slope upwards, because he must receive
a higher wage to compensate for the extra effort that makes $q$ greater. They
are convex because the marginal utility of dollars is decreasing and the
marginal disutility of effort is increasing. As Figure 1 shows, the salesman has
two sets of indifference curves, solid for $Pushovers$ and dashed for
$Bonanzas$, since the effort that secures a given level of sales depends on the
state.

\includegraphics[width=150mm]{fig10-01.jpg}

\begin{center} {\bf Figure 1:  The Salesman Game  with  Curves for Pooling
Equilibria} \end{center}



  Because of the participation constraint, the manager must provide the salesman
with a contract giving him at least his reservation utility of 8, which is the
same in both states.  If the true state is that the customer is a $Bonanza$, the
manager would like to offer a contract that leaves the salesman on the dashed
indifference curve $\tilde{U_S}=8$, and the efficient outcome is ($q_2$,$w_2$),
the point at which the salesman's indifference curve is tangent to one of the
manager's indifference curves.  At that point, if the salesman sells an extra
dollar he requires an extra dollar of compensation.

   If it were common knowledge that the customer was a $Bonanza$, the principal
could choose $w_2$ so that $U(q_2, w_2, Bonanza)= 8$ and offer the forcing
contract
\begin{equation} \label{e1}
 \begin{array}{ll}
 w =& \left\{ \begin{array}{ll}
 0   & {\rm if} \; q < q_2 \\
 & \\
 w_2 & {\rm if} \;q \geq q_2 \\
\end{array} \right.
 \end{array}
 \end{equation}

 The salesman would accept the contract and choose $e = q_2$. But if the
customer were actually a $Pushover$, the salesman would still choose $e = q_2$,
an inefficient outcome that does not maximize profits.  High sales would be
inefficient because the salesman would be willing to give up more than a dollar
of wages to escape having to make his last dollar of sales.  Profits would  not
be  maximized, because the salesman achieves a utility of 17, and he would have
been willing to work for less.

    The revelation principle says that in searching for the optimal contract we
need only look at contracts that induce the agent to truthfully reveal what kind
of customer he faces. If it required more effort to sell any quantity to the
$Bonanza$, as shown in Figure 1, the salesman would always want the manager to
believe that he faced a $Bonanza$, so he could extract the extra pay necessary
to achieve a utility of 8 selling to $Bonanzas$.  The   optimal truth-telling
contract is the pooling contract that pays the intermediate wage of $w_3$ for
the intermediate quantity of $q_3$, and zero for any other quantity, regardless
of the message.  The pooling contract is a second-best contract, a compromise
between the optimum for $Pushovers$ and the optimum for $Bonanzas$. The point
$(q_3,w_3)$ is closer to $(q_1,w_1)$ than to $(q_2,w_2)$, because the
probability of a $Pushover$ is higher and the contract must satisfy the
participation constraint,
 \begin{equation}\label{e2}
 0.8 U(q_3, w_3, Pushover) + 0.2 U(q_3, w_3, Bonanza)   \geq 8.
 \end{equation}
   The nature of the equilibrium depends on the shapes of the indifference
curves. If they are shaped as in Figure 2, the equilibrium is separating, not
pooling, and there does exist a first-best, fully revealing  contract.


\includegraphics[width=150mm]{fig10-02.jpg}

\begin{center} {\bf Figure 2:  Indifference Curves for a Separating Equilibrium}
\end{center}



\vspace{.9in}

\begin{equation} \label{e3}
{\rm Separating\;\; Contract } \left\{ \begin{array}{ll} {\rm Agent\;\;
announces}\;\; {\it Pushover:} & \begin{array}{ll} w =& \left\{ \begin{array}{l}
0 \;  {\rm if}\; q < q_1 \\ w_1 \; {\rm if }\; q \geq q_1 \\
 \end{array} \right. \\ \end{array} \\
 & \\ {\rm Agent\;\; announces\;\;} { \it Bonanza:}  & \begin{array}{ll} w = &
\left\{ \begin{array}{l} 0 \; {\rm if}\; q < q_2 \\
 w_2\; {\rm if} \; q \geq q_2
\end{array} \right.
 \end{array}
\end{array} \right.
. \end{equation}

  Again, we know from the revelation principle that we can narrow attention to
contracts that induce the salesman to tell the truth. With Figure 2's
indifference curves, contract (\ref{e3}) induces the salesman to be truthful and
the incentive compatibility constraint is satisfied. If the customer is a
$Bonanza$,  but the salesman claims to observe a $Pushover$ and chooses $q_1$,
his utility is less than 8 because the point $(q_1,w_1)$ lies below the
$\tilde{U_S}=8$ indifference curve. If the customer is a $Pushover$ and the
salesman claims to observe a $Bonanza$, then although $(q_2,w_2)$ does yield the
salesman a higher wage than $(q_1,w_1)$, the extra income is not worth the extra
effort, because $(q_2,w_2)$ is far below the indifference curve $  {U_S}=8$.

Another way to look at a separating equilibrium  is to think of it as a choice
of contracts  rather than as one contract with  different wages for different
outputs. The salesman  agrees to work  for  the manager, and after he discovers
the customer's type   he chooses either the contract $(q_1,w_1)$ or the contract
$(q_2,w_2)$, where each is a forcing contract  that pays him 0 if after choosing
the contract $(q_i,w_i)$ he produces output of $q \neq q_i$. In this
interpretation,   the manager offers a {\bf menu of contracts} and the salesman
selects one of them after learning his type.

 Sales contracts in the real world  are often complicated because it is easy to
measure   sales  and hard to measure efforts when   workers  are out in the
field away from direct supervision. The Salesman Game is a real problem.   Gonik
(1978) describes hidden knowledge contracts used by IBM's subsidiary in Brazil.
Salesmen were first assigned quotas. They then announced their own sales
forecasts as a percentage of quota and chose from among a set of contracts, one
for each possible forecast. I'll invent  some numbers for illustration.  If
Smith were assigned a quota of 400 and he announced 100 percent, he would get
$w=70$ if he sold 400 and $w=80$ if he sold 450. If he had announced 120
percent, he would have gotten $w= 60$ for 400 and $w=90$ for 450. The contract
encourages extra effort when the extra effort is worth the extra sales.  The
idea here, as in The Salesman Game, is to reward salesmen not just for high
effort, but for appropriate effort.

   The Salesman Game illustrates a number of ideas.  It can have either a
pooling or a separating equilibrium, depending on the utility function of the
salesman. The revelation principle can be applied to avoid having to consider
contracts in which the manager must interpret the salesman's lies.  It also
shows how to use diagrams when the algebraic functions are intractable or
unspecified, a  problem that does not arise in most of the two-valued numerical
examples in this book.


 \vspace{1in}

 \noindent
{\bf  *10.4 The Groves Mechanism}

  \noindent
  Hidden knowledge is particularly important in public economics, the study of
government spending and taxation.
   In    Mirrlees (1971), a classic article in the optimal taxation literature,
citizens differ in their income-producing ability  and the government wishes to
demand higher taxes from the more able citizens. Since the government cannot
observe ability directly, this is a   problem of  hidden knowledge. An even
purer hidden knowledge problem  is choosing the level of  public goods  based on
private preferences. The government must decide whether it is worthwhile to buy
a public good based on the combined preferences of all the citizens, but it
needs to discover those preferences.   Unlike so far  in this chapter, a group
of agents  will now be involved, not just one agent. Moreover, here the
principal is an altruistic government that cares directly about the utility of
the agents  rather than a car buyer or an insurance seller who cares about the
agents' utility only in order to satisfy self-selection and participation
constraints.

 Our  example    is   adapted from   Varian (1992,  p. 426). The mayor of a town
is considering installing a streetlight costing \$100. Each of the five houses
near the light would be taxed exactly \$20, but the mayor will only install it
if he decides that the sum of the residents' valuations for it is greater than
the cost.


\begin{center}
 {\bf  The  Streetlight Game}
 \end{center}
 {\bf Players}\\
The mayor and 5 householders.

\noindent
 {\bf The Order of Play}\\
 0. Nature chooses the value $v_i$ that householder $i$ places on having a
streetlight installed, using distribution $f_i(v_i)$.   Only Householder $i$
observes $v_i$.   \\
1. The mayor announces a mechanism, $M$, which requires a householder who
reports $m$ to pay $w(m)$ if the streetlight is installed   and installs the
streetlight if $g(m_1, m_2,m_3, m_4, m_5) \geq 0$. \\
 2. Householder $i$ reports value $m_i$ simultaneously with all other
householders.\\
 3. If $g(m_1, m_2,m_3, m_4, m_5) \geq 0$,   the streetlight is built and
householder $i$ pays $w(m_i)$.

\noindent
 {\bf Payoffs}\\
 The mayor tries to maximize  social welfare, including the welfare of taxpayers
besides the 5 householders. His payoff is zero if the streetlight is not built.
Otherwise, it is
\begin{equation} \label{e3a}
 {\displaystyle \pi_{mayor}=   \left(  \sum_{i=1}^5   v_i   \right)   - 100, }
\end{equation}
 subject to the constraint that $ \sum_{i=1}^n  w  (m_i)\geq 100$ so he can
raise the taxes to pay for the light. \\
 The payoff of householder $i$ is zero if the streetlight is not built.
Otherwise it is
\begin{equation} \label{e3b}
 \pi_i (m_1, m_2, m_3, m_4, m_5) =   v_i - w(m_i).
\end{equation}



The mayor's  problem is to discover the valuations. If  he could  observe them
directly, he would    simply builds the streetlight if $\sum_{i=1}^5   v_i    >
100$).  Otherwise, he has a problem.  If he simply asks the householders  and
tells them that each must pay a tax of \$20 if the streetlight is built,  pro-
light householder  Smith might say that his valuation is \$5,000, and anti-light
householder Brown might say  that  he likes  darkness and would pay \$5,000 to
{\it not} have a streetlight, but all the mayor could conclude  would be that
Smith's valuation exceeded \$20 and Brown's did not.  Talk is cheap, and the
dominant strategy  would be to overreport or underreport.





\noindent
The flawed mechanism just described can be written as
 \begin{equation} \label{e4}% \label{e18}
M_1: \;\;\; \left(  w(m_i) = 20, Build \; iff \; \sum_{i=1}^5 m_i \geq 100
\right);
\end{equation}
that is,  each resident  pays \$20,  and the light is installed if the sum of
the valuations exceeds 100.

An alternative   is to make resident $i$ pay the amount of his message, or pay
zero if it is negative.  This mechanism is
\begin{equation}   \label{e5}% \label{e19}
 M_2: \;\;\; \left( w(m_i) = Max\{ m_i, 0\},  Build \; iff \;\sum_{j=1}^5 m_j
\geq 100 \right).
\end{equation}

 Mechanism M$_2$ has  no dominant strategy.   Householder $i$ would announce
$m_i=0$ if he thought the project would go through without his support, based on
his estimates of other  people's values, but he would announce up to his
valuation if necessary.


 If  all the householders knew each others' values perfectly, then there would
be a
 continuum of Nash equilibria that attained the efficient result, much as  in
The Holmstrom Teams Game of Chapter 8. If, for example, the values were known to
be $(10, 30, 30, 30, 80)$, one equilibrium would be to report $(0, 25, 25, 25,
25)$.  Since typically equilibria would be asymmetric, though, it is problematic
how  the equilibrium to be played out would become common knowledge, as well as
how  the householders know the $v$'s in the first place.    $M_2$ is a simple
mechanism, however, and it already teaches a lesson:   people are more likely to
report their true political preferences if they must bear part of the costs
themselves.

 It turns out, however, that  not only can a mechanism be found which makes
truth-telling a Nash equilibrium, one can be found which
  makes truth-telling  the best strategy for a player  regardless of what the
other players do---    a {\bf dominant-strategy mechanism}. Consider the
mechanism $M_3$.
 \begin{equation}  \label{e6}
 M_3: \;\;\; \left( w(m_i) =  100-\sum_{j\neq i} m_j ,  Build \; iff \; \sum_{j=
1}^5 m_j \geq 100 \right).
 \end{equation}
 Under mechanism M$_3$, player $i$'s message does not affect his tax bill except
by its effect on whether or not the streetlight is installed.  If player $i$'s
valuation is $v_i$, his full payoff is $v_i- 100 + \sum_{j\neq i} m_j $ if $m_i
+ \sum_{j\neq i} m_j \geq 100$, and zero otherwise. It is not hard to see that
he will be truthful in a Nash equilibrium in which the other players are
truthful, but we can go further: truthfulness is weakly dominant. Moreover, the
players are strictly better off telling the truth whenever lying would alter the
mayor's decision.

   Consider a numerical example.  Suppose   Smith's valuation is 40 and the sum
of the valuations is 110, so the project is indeed efficient. If the other
players report  their truthful sum of 70, Smith's payoff from truthful reporting
is his valuation of 40 minus his tax of 30. Reporting more would not change his
payoff, while reporting less than 30 would reduce it to 0.

  If we are wondering whether Smith's strategy is dominant, we must also
consider his best response when the other players lie.  If they underreported,
announcing 50 instead of the truthful 70, then Smith could make up the
difference by overreporting 60, but his payoff would be $-10$ ($=40 + 50 -100 $)
so he would do better to report the truthful 40, killing the project and leaving
himself with a payoff of 0. If the other players overreported, announcing 80
instead of the truthful 70, then Smith would benefit  if the project went
through, and he should report at least 20. Whether he reports 20, 21, 40, or
400, the streetlight is built and he pays a tax of 20 under mechanism $M_3$,
leaving him with  payoff of 20 (= 40-20).   In particular, he is willing to
report exactly 40,  so it is a weakly best response to the other players' lies.

 The problem with a dominant-strategy mechanisms like $M_3$ is that it is not
budget balancing.  This is not so bad if the budget had a surplus, as required
in our game rules above, but it turns out to have a deficit except in special
cases where it is perfectly balanced (e.g., $m_i=20$ for all 5 householders).
  The  first part of the tax $w(m)$  would collect  100 from each player, for
500, which leaves a surplus of 400 once we pay for the streetlight.  The second
part of $w(m)$, however,  would subtract each player's value  four times (1 for
each other player), subtracting $4  (\sum_{i=1}^5 m_i)$. If the project goes
through though,  then $\sum_{i=1}^5 m_i >100$, so the  budget would be left in
deficit.


In fact,  the total tax revenue could easily be negative too, because the
``taxes'' under $M_3$ are sometimes negative.  If $v=60$ for all five players,
for example, then $m=60$ and  $w(m)= -140 (= 100 -  4(60))$.




 Vickrey (1961) first suggested the non-budget-balancing mechanism for
revelation of preferences, but that was in the context of auctions, not public
economics.   We will see in Chapter 13 that the second-price auction has  the
same distinctive feature that a player's own report affects the  allocative
decision (whether the light is built, who wins the auction)  bu not the amount
he pays conditional on the decision being made (the tax, the price the auction
winner pays). The idea was rediscovered later and became known as the Groves
Mechanism  from Groves (1973).


\vspace*{1in}

 \noindent
{\bf  10.5:  Price Discrimination }

Now let's go to algebra to do a more conventional mechanism design problem,
where the agents not only select an action that reveals their information, but
must choose  to play the game in the first place.

 When a firm has market power --- most simply when it is a monopolist--- it
would like to   charge different    prices to different consumers. To the
consumer who would pay up to   \$45,000 for a  car, the firm would like to
charge    \$45,000; to the consumer who would pay up to \$36,000, the profit-
maximizing price is  \$36,000.  But how does the car dealer   know how much each
consumer is willing to pay?

 He does not, and that is what makes this a problem of mechanism design under
adverse selection.  The consumer who would be willing to pay  \$45,000  can hide
under the guise of being a less intense consumer, and despite facing a
monopolist he can end  up retaining consumer surplus -- an {\bf informational
rent}, a return to the consumer's private information about his own
type.\footnote{ A standard opening ploy  of  car salesman  is   to ask. ``So,
how much are you able to spend on a car today?''  My recommendation: don't tell
him. This may sound obvious, but remember it the next time your department
chairman asks you how high a salary it would take to keep you from leaving for
another university.   }

 Pigou   was   a contemporary of Keynes at Cambridge who   usefully divided
price discrimination into three types in 1920  but   named them  so obscurely
that I relegate his names to the endnotes and use better  ones  here:

\noindent
{\bf  1 Interbuyer Price Discrimination.  }  This is when the seller can charge
different prices to different  buyers. Smith's price  for a hamburger  is \$4
per burger, but Jones's is  \$6.

\noindent
{\bf  2 Interquantity Price Discrimination}  or  {\bf  Nonlinear Pricing}.
This is when the seller can charge different  unit prices for different
quantities.   A consumer can buy a  first   sausage for \$9, a  second sausage
for  \$4, and  a third sausage for \$3.  Rather than  paying  the ``linear''
total price of  \$9 for one sausage, \$18 for two, and \$27 for three,  he  thus
pays  the  nonlinear price of  \$9 for one sausage, \$13 for two, and \$16 for
three, the concave price path shown in Figure   3.

\noindent
 {\bf  3 Perfect Price Discrimination.}   This combines interbuyer and
interquantity price discrimination.  When the seller does have perfect
information and  can charge each buyer  that buyer's reservation price for each
unit bought,    Smith might end up paying  \$50 for his first hot dog and  \$20
for his second,   while next to him Jones pays \$4 for his first and \$3 for his
second.

\includegraphics[width=150mm]{fig10-03.jpg}

\begin{center} {\bf igure  3:  Linear and Nonlinear Pricing} \end{center}



  To illustrate price discrimination as mechanism design we will use a modified
version of  an   example  in   Chapter  14  of Hal  Varian's third edition
(Varian, 1992).

 \begin{center}
  {\bf  Varian's Nonlinear Pricing Game}
 \end{center}
 {\bf Players}\\
One seller and one buyer.

\noindent
{\bf The Order of Play }\\
0 Nature assigns   the   buyer a type, $s$. The buyer is  ``unenthusiastic''
with utility function  $u$    or   ``valuing''   with utility function  $v$,
with equal probability.  The seller does not observe Nature's move, but the
buyer does. \\
1 The  seller offers  mechanism  $\{ w_m, q_m \}$ under which the buyer can
announce his type as $m$ and buy amount  $q_m$ for lump sum $w_m$.\\
2 The buyer   chooses a message $m$ or rejects the mechanism entirely and does
not buy at all.  \\

\noindent
{\bf Payoffs}\\
 The seller  has a constant marginal cost of $c$, so his payoff  is
 \begin{equation}  \label{e6a}
  w_u + w_v - c \cdot (q_u+q_v).
 \end{equation}
 The buyers' payoffs are  $\pi_u=u (q_u) - w_u$ and $\pi_v=v (q_v) - w_v$ if $q$
is positive, and 0 if $q=0$, with $u', v'>0$ and  $u'', v'' <0$. The    marginal
willingness  to pay is greater  for  the valuing buyer: for  any   $q$,
 \begin{equation}  \label{e7}
  u' (q ) < v '(q )
 \end{equation}
Condition (\ref{e7}) is an example of    {\bf the single-crossing property},
which we will discuss at the end of this section. Combined with the assumption
that $v(0)=u(0) =0$, it also implies that
 \begin{equation}  \label{e7a}
 u  (q ) < v (q )
 \end{equation}
 for any value of $q$.      Figure 7a, some pages below, shows an example of
functions which satisfy it.



 To ease into the difficult problem of solving for the equilibrium mechanism,
let us start with   two simpler versions of the game that limit it to (a)
perfect price discrimination and (b) interbuyer discrimination.

\bigskip
 \noindent
{\bf Perfect Price Discrimination}

 The game would allow  perfect price discrimination if the seller did know
which buyer had which utility function.  He  can then  just maximize  profit
subject to the participation constraints for the two buyers:
 \begin{equation}  \label{e8}
 \stackrel{ Maximize}{w_u  ,w_v, q_u, q_v}  \;\;\;   w_u + w_v - c \cdot
(q_u+q_v).
 \end{equation}
 subject to
\begin{equation}  \label{e9}
 \begin{array}{ll}
 (a) & u (q_u) - w_u \geq 0 \;\; {\rm and} \\
 & \\
(b)&v (q_v) - w_v \geq 0.
 \end{array}
 \end{equation}
The constraints will be satisfied as equalities, since the seller will charge
all that the buyers will pay. Substituting  for $w_u$ and $w_v$ into the
maximand, the first order conditions become
\begin{equation}  \label{e10}
 \begin{array}{ll}
 (a) &u' (q_u^*) -c= 0 \;\;\; {\rm and} \\
 & \\
 (b)&v ' (q_v^*)-c=0.
\end{array}
\end{equation}

   Thus, the  seller will choose quantities so that each buyer's marginal
utility equals the marginal cost of production, and will choose prices so that
the entire consumer surpluses are eaten up: $w^*(q_u^*) =u  (q_u^*)$  and
$w^*(q_v^*) =v  (q_v^*)$.  Figure 4 shows this for the unenthusiastic buyer.


\includegraphics[width=150mm]{fig10-04.jpg}

\begin{center} {\bf Figure 4:  Perfect Price Discrimination} \end{center}



\bigskip

\noindent
{\bf Interbuyer Price Discrimination}

  The interbuyer price  discrimination problem arises when  the seller knows
which utility functions Smith and Jones have and can sell to them separately. If
he can choose $w_u$ and $w_v$ as before and use forcing contracts, this is the
same as the perfect price discrimination problem we just solved. If the seller
must charge each  buyer  a single price per unit and let the buyer choose the
quantity, however,  the problem is quite different:
\begin{equation}  \label{e11}
 \stackrel{ Maximize}{q_u,q_v, p_u  ,p_v}  \;\;\;   p_u q_u  + p_v q_v - c \cdot
(q_u+q_v),
 \end{equation}
 subject to the participation constraints
 \begin{equation}   \label{e12}
 \begin{array}{ll}
  & u (q_u) - p_u  q_u  \geq 0  \;\;\; {\rm and} \\
 & \\
 &v (q_v) - p_v q_v \geq 0
 \end{array}
\end{equation}
 and the incentive compatibility constraints
 \begin{equation}   \label{e13}
\begin{array}{ll}
  & q_u = argmax [u (q_u) - p_u  q_u]   \;\;\;  {\rm and} \\
 & \\
 & q_v = argmax [v (q_v) - p_vq_v] .
 \end{array}
 \end{equation}

This should remind you of   moral hazard. It is very like the problem of a
principal designing   two  incentive contracts  for two agents to induce
appropriate effort levels  given their different  disutilities of effort.


\noindent
 The  agents will solve their quantity choice problems in (\ref{e13}), yielding
\begin{equation}   \label{e14}
 \begin{array}{ll}
  &  u' (q_u) - p_u  =0 \;\;\;  {\rm and} \\
 & \\
  & v ' (q_v) - p_v=0.
 \end{array}
\end{equation}
Thus, we can simplify the original problem in (\ref{e11})  to
 \begin{equation}  \label{e15}
 \stackrel{ Maximize}{q_u,q_v}  \;\;\;  u'(q_u)  q_u  +  v '(q_v) q_v - c \cdot
(q_u+q_v),
 \end{equation}
 subject to the participation constraints
\begin{equation}  \label{e16}%   \label{e10.10}
 \begin{array}{ll}
   & u (q_u) - u' (q_u)  q_u  \geq 0\;\;\; {\rm and} \\
 & \\
 &v (q_v) - v ' (q_v) q_v \geq 0.
 \end{array}
\end{equation}

   The participation constraints will not be binding. If they were, then $ u(q)
/q  =  u'(q)$, but since $u''<0$ there is diminishing utility of consumption and
the average utility, $U(q)/q$, will be greater than the  marginal utility,
$u'(q)$. Thus we can solve problem (\ref{e15}) as if there were no constraints.
 The first-order conditions are
\begin{equation}   \label{e17}
 \begin{array}{ll}
   &  u''(q_u)  q_u   + u' =c \;\;\; {\rm and} \\
 & \\
  &v ''(q_v) q_v+v '  = c.
\end{array}
 \end{equation}
 This is  just  the `marginal revenue equals marginal cost' condition that any
monopolist uses,  but  one for each buyer instead of one for the entire market.

   The assumption of constant marginal cost helps make this problem easier,
because it makes it two independent problem, really. Choosing a contract for the
valuing customer is completely separate from choosing one for the unenthusiastic
customer.  If the cost function were a more general convex $c(q_u+q_v)$, on the
other hand, the two first-order conditions in (\ref{e17}) would have to be
solved together, because each condition would depend on both $q_u$ and $q_v$.



\bigskip
  \noindent
 {\bf Back to  Nonlinear Pricing}

Neither the perfect price discrimination nor the interbuyer problems are
mechanism design problems, since the seller  is perfectly informed about the
types of the buyers and has no need to worry about   designing incentives to
separate them. In the original game, however, separation  is the  seller's  main
concern. He must satisfy not just the participation constraints, but self-
selection constraints. The seller's problem is
 \begin{equation}   \label{e18}
 \stackrel{ Maximize}{q_u,q_v, w_u,  w_v}  \;\;\; w_u  + w_v  - c \cdot
(q_u+q_v),
 \end{equation}
 subject to  the participation constraints,
\begin{equation}   \label{e19}
 \begin{array}{l}
 (a)\;\;\;u (q_u) - w_u \geq 0\;\;\;  {\rm and} \\ (b)\;\;\;v (q_v) - w_v
\geq 0,
 \end{array}
 \end{equation}
and  the self-selection constraints,
 \begin{equation}  \label{e20}
 \begin{array}{l}
 (a) \;\;\;  u (q_u) - w_u \geq   u (q_v) - w_v   \\
 \\
(b) \;\;\;   v (q_v) - w_v  \geq   v (q_u) - w_u.  \\
 \end{array}
 \end{equation}

 Not all of these constraints will be binding.   If neither type had a binding
participation constraint, however,  the principal would be losing a chance to
increase his profits.   In a mechanism design problem like this, what always
happens is that the contracts are designed so that one type of agent is pushed
down to his reservation utility.

 Suppose  the optimal contract is in fact separating, and also that both types
accept a  contract.    At least one type  will  have a binding participation
constraint. Since the valuing consumer gets more consumer surplus from a given
$w$ and $q$ than an unenthusiastic consumer, it must be the unenthusiastic
consumer who is driven down to zero surplus for $(w_u, q_u)$. The valuing
consumer would get positive surplus from accepting that same contract, so his
participation constraint is not binding. To persuade the valuing consumer to
accept $(w_v, q_v)$ instead, the seller must give him that same positive surplus
from it. The seller will not be any more generous than he has to, though, so the
valuing consumer's self-selection constraint will be binding.


\noindent
 Rearranging our  two binding constraints and setting them out as equalities
yields:
 \begin{equation} \label{e21}
     w_u =u (q_u)
  \end{equation}
  and
\begin{equation} \label{e21a}
    w_v  =  w_u- v (q_u)   + v (q_v)
 \end{equation}

  This allows us to reformulate the seller's problem from (\ref{e18})   as
\begin{equation} \label{e22}
  \stackrel{Maximize}{q_u,q_v}  \;\;\; u(q_u)  + u(q_u) -v (q_u)   +v (q_v)   -
c \cdot (q_u+q_v),
  \end{equation}
 which has the first-order conditions
\begin{equation}  \label{e23}
\begin{array}{ll}
(a) &  u'(q_u)  - c  + [u'(q_u)  - v '(q_u)]  =0\\
 & \\
(b)  &   v' (q_v) -c =0\\
 \end{array}
\end{equation}

 The  first-order conditions  in (\ref{e23})  could be solved for  exact values
of $q_u$ and $q_v$ if we chose particular functional forms, but  they are
illuminating even if we do not. Equation (\ref{e23}b)  tells us that the
valuing  type of buyer buys a quantity such that his last unit's marginal
utility exactly equals the marginal cost of production; his consumption is at
the efficient level.   The unenthusiastic type, however,   buys less than his
first-best amount,  something we can deduce using the single-crossing  property,
assumption (\ref{e7}b),  that $u'(q) < v'(q)$, which implies from (\ref{e23}a)
that $u'(q_u)  - c >0$ and the unenthusiastic type has not bought enough to
drive his marginal utility down to marginal cost.   The intuition is that   the
seller must sell less than   first-best optimal to   the unenthusiastic type so
as not to make that contract  too attractive to the  valuing type. On the other
hand, making the valuing type's contract more valuable to him actually  helps
separation, so $q_v$ is chosen to maximize social surplus.

  The single-crossing property has another   important implication. Substituting
from first-order condition (\ref{e23}b)  into first-order condition (\ref{e23}a)
yields
\begin{equation}   \label{e24}%  \label{e10.18}
[u'(q_u)  - v' (q_v)]  + [u'(q_u)  - v '(q_u)]  =0
 \end{equation}
   The second term in square brackets  is  negative  by the single- crossing
property. Thus, the first term must be positive.  But since the single-crossing
property tells us that  $[u'(q_u)  - v' (q_u)] <0$,  it must be true, since
$v''<0$, that  if  $q_u  \geq q_v$  then   $[u'(q_u)  - v' (q_v)] <0$ -- that
is, that the first  term is negative. We cannot have that without contradiction,
so it must be that   $q_u  < q_v$. The   unenthusiastic buyer buys strictly less
than the valuing buyer.  This accords with our intuition, and also lets us know
that  the equilibrium is separating, not pooling (though we still have not
proven that the equilibrium involves both players buying  a positive amount,
something hard to prove elegantly since  one player buying zero  would be a
corner solution to our maximization problem).

\noindent
 {\bf A  Graphical Approach to the Same Problem}

  Under perfect price discrimination,  the seller would charge $w_u= A+B$ and
$w_v = A+B+J+K+L$ to the two buyers  for quantities $q_u^*$ and $q_v^*$, as
shown in Figure  5a.  An attempt to  charge $w_u^*  = A+B$  and $w_v^*   =
A+B+J+K+L$, however,   would simply lead to both buyers choosing to buy $q_u^*$,
which would yield the  valuing buyer a payoff of  $J+K$ rather than the zero  he
would get as a payoff from buying $q_v^*$. The seller's payoff from this pooling
equilibrium (which is the best pooling contract possible for him, since it
drives the unenthusiastic type to a payoff of zero) is $2(A+B)$.

\includegraphics[width=150mm]{fig10-05.jpg}

\begin{center} {\bf Figure 5:   The Varian Nonlinear Pricing Game} \end{center}



  The seller could separate the two buyers by   charging $w_u^* =A+B$  for
$q_u^*$   and $w_v^*   =A+B+ L$  for $q_v^*$, since the unenthusiastic buyer
would have no reason to switch to the greater quantity, and that  would
increase his profits   over pooling by amount $L$.

 Figure 5b shows, however,  that the seller  would do better to  slightly reduce
the quantity  sold to the unenthusiastic buyer, to below $q_u^*$,  and  reduce
the price to him by the amount of the dark shading. He could then sell $q_v^*$
to  the valuing buyer and raise the price to him by the light shaded area.  The
valuing buyer will   not be tempted to buy the smaller quantity  at the lower
price, and the seller will have gained profit by, loosely speaking, increasing
the size of the $L$ triangle.

 Continuing this process until profit is maximized is what we did earlier using
algebra. Our
  profit-maximizing mechanism   is shown  in Figure  5a as $w_u'  = A$  for
$q_u'$ and $w_v^* =   A+B+K+L$ for $q_v^*$.    The unenthusiastic buyer is left
with a binding participation constraint and  inefficiently low consumption,
because $w_u'  = A = u(q_u')$. The valuing buyer has a nonbinding participation
constraint, because   $w_v^*  = A+B+K+L < v(q_v^*) = A+B+ J+K+L$; he is left
with a surplus of $J$.  Moreover, he consumes the efficient amount for
him, which is $q_v^*$.  He also  has a binding self-selection constraint,
because he is
exactly indifferent between buying $q_u'$ and $q_v^*$. His choice is between a
payoff of $\pi_v(U)  =  (A+J)- A$ and $ \pi_v(V)    = (A+B+ J+K+L) - (A+B+K+L)$.
Thus, the diagram replicates the algebraic conclusions.


\bigskip

\noindent
 {\bf  The Single-Crossing Property}

Condition (\ref{e7}) is an example of    {\bf the single-crossing property},
since it implies that the indifference curves of the two agents cross  at most
one time.  Combined with the assumption that $v(0)=u(0) =0$, it   implies that
$u  (q ) < v (q )$ for any value of $q$, as stated in inequality  (\ref{e7a})
earlier. Thus, in Varian's Nonlinear Pricing Game it is unambiguous that the
valuing buyer has stronger demand than the unenthusiastic buyer.

When we say that   Buyer V's demand is stronger than Buyer U's, however,  there
are  two things we might mean:

1. Buyer V's {\it average demand} is stronger: $  \frac{v (q )}{q}>\frac{u  (q )
}{q} $. Buyer V would pay more for quantity $q$ than Buyer U would.

2. Buyer V's {\it marginal demand} is stronger: $  v' (q )> u'  (q )$. Buyer V
would pay more for an additional unit than Buyer U would.

  Definitions (1) and (2)  are not equivalent. In Figure 6a,   Buyer U is
willing to pay 5 per unit up to $q=4$, but only 1 per unit thereafter. Buyer V
is willing to pay only 2 per unit up to $q=10$, and 1 per unit thereafter. As a
result $u  (q ) > v (q )$, and U has the stronger demand by definition (1).  But
for $q \in [4, 10]$, Buyer V is willing to pay 2 per new unit while Buyer U is
only willing to pay 1, so in that interval Buyber V has the  stronger demand by
definition (2).

 It is the marginal demand that is more important to economic behavior and which
forms the basis for the single-crossing property.  The utility functions in
Figure 6b   satisfy the single-crossing property, even though it is not true
that $u  (q ) < v (q )$ ( because they don't satisfy $u(0)=v(0)=0$.) Buyer V is
always willing to pay 2  per unit, but Buyer U is only willing to pay 1 per
unit. To be sure,   $u  (q ) > v (q )$ for $u <12$, but that is only because
Buyer U starts with $u(0) =14$, a ``fixed benefit'' which is irrelevant to his
behavior.  Buyer U may be the happier person at a consumption level of zero, but
since that happiness is unaffected by his material circumstances, it is not
going to affect any economic predictions we might want to make.     Thus,
definition (2) is best:  when we say strong demand we should mean greater
marginal demand.  Indeed, that is what having a higher demand curve means in our
usual diagrams, since the Marshallian demand curve is  a marginal curve, showing
not the total amount a consumer spends on quantity Q but the amount extra he
willing to pay to buy a little more.

\includegraphics[width=150mm]{fig10-06.jpg}



\begin{center} {\bf Figure 6: Marginal versus Average Demand} \end{center}


 Figure 7a depicts functions which satisfy the assumptions of Varian's Nonlinear
Pricing Game: $u = \sqrt{q}$ and  $v = 2\sqrt{q}$. The two curves satisfy the
single-crossing property, condition (\ref{e7}), because  $v'(q) > u'(q)$ for all
$q$ and   $u(0)=0$ and $v(0)=0$.

\includegraphics[width=150mm]{fig10-07.jpg}

\begin{center} {\bf Figure 7: Two Depictions of the Single Crossing Property}
\end{center}



   Another way to think about the single-crossing property, a way where the name
``single crossing'' is clearly visible,  is using indifference curves.  The
point of this is that since   utility  is a more theoretical concept than the
idea of indifference between two consumption bundles, it is more reliable not to
use graphs in utility space.    The two goods in Varian's Nonlinear Pricing Game
are the commodity being sold and money, which enters  linearly in the form of $-
w$, the amount paid for the commodity. Another way to write the payoff functions
would have been as $\pi_u(q, money) = money + u(q)$, where $money = wealth-w(q)
$. Figure 7b shows how the buyers trade off money and the commodity. One
comparison is between the curves for which $\pi=10$, which both pass through the
point (0, 10) in $(q, money)$ space. The $\pi_u=10$  indfference curve then
descends more slowly than the $\pi_v=10$  curve because the commodity is not so
valued by Buyer U. Another comparison is between the two curves which contain
the point (4,8), which are  $\pi_u=10$ and $\pi_v=12$. These two curves also
cross only once, at that point. In fact, if you pick any one indifference curve
for Buyer U and any one for Buyer V, those curves will cross either not at all,
or once.

   It is often natural to assume that the single-crossing property holds, and it
is  a useful sufficient condition for separation to be possible, but it is not a
necessary condition.  If there are just two types of agents, as in many of our
models, what matters is that the incentive compatibility constraints hold at the
outputs that the principal specifies in the mechanism, not at all outputs. If
the V player actually has a smaller marginal than the  U player over a small
range of consumption near zero, that will not hurt separation if the two
contractual outputs are much larger.


\vspace{1in}

 \noindent
 {\bf  *10.6 Rate-of-Return Regulation and Government Procurement}

 \noindent
 The  central idea in both  government procurement and   regulation of natural
monopolies    is that the government   is trying to induce a private firm to
efficiently provide a good to the public while covering  the cost of production.
If information is symmetric, this is an easy problem; the government simply pays
the firm the cost of producing the good efficiently, whether the good be a
missile or electricity.  Usually, however, the firm has better information about
costs and demand than the government does.

The variety of ways   the firm might have better information and   the
government might extract  it   has  given rise to a large literature in which
moral hazard with hidden actions, moral hazard with hidden knowledge, adverse
selection, and signalling  all  put in appearances. Suppose the government wants
a firm to provide cable television service to a city. The firm knows more about
its costs before agreeing to accept the franchise (adverse selection),
discovers more after accepting it and beginning operations (moral hazard with
hidden knowledge),  and exerts greater or smaller effort to keep costs low
(moral hazard with hidden actions).     The government's  problem is to acquire
cable service at the lowest cost. It wants to  be generous enough  to induce the
firm to accept the franchise in the first place  but   no more generous than
necessary. It cannot simply agree to cover the firm's costs, because the firm
would always claim high costs and exert low effort.  Instead, the government
might auction off the right to provide the service, might allow the firm a
maximum price (a {\bf price cap}), or might agree to compensate the firm to
varying degrees for different levels of cost ({\bf rate-of- return regulation}).

 The problems  of regulatory franchises and   government procurement are the
same in many ways. If the government wants to purchase a   missile, it also has
the problem of how much to offer the firm. Roughly speaking, the equivalent of a
price cap is a flat price, and the equivalent of rate-of-return regulation is a
cost-plus contract, although the details differ in interesting ways. (A price
cap allows downwards flexibility in prices, and rate-of-return regulation allows
an expected but not guaranteed profit, for example.)

 Many of these situations are problems of moral hazard with hidden knowledge,
because  one player is trying to design a contract that   the other   will
accept   that will then  induce him to use his private information properly.


Although the literature on mechanism design  can be traced back to Mirrlees
(1971) and in 1979 Loeb and Magat suggested using a Groves Mechanism  to extract
information from regulated firms,  the extensive application  to regulation
began  with Baron \& Myerson's 1982 article, ``Regulating  a Monopolist with
Unknown Costs.''   McAfee \& McMillan (1988),    and   Laffont \& Tirole (1993)
provide 168-page,  and  702-page treatments  of the confusing array of possible
models and policies in their books on government regulation.   Here, we will
look at a version of the    model  Laffont and Tirole use to introduce their
book on pages 55 to 62.  This is  a two-type model in which  a special cost
characteristic and the effort of a firm is its private information  but its
realized cost is public and nonstochastic.  The model combines moral hazard and
adverse selection, but it will behave more like an adverse selection model.  The
government  will reimburse the firm's costs, but also fixes a  price (which if
negative becomes a tax)  that depends  on  the level of  the firm's costs.  The
questions the model hopes to answer are (a) whether   effort will be too high or
too low and  (b) whether the price is positive and rises with costs.




 The first version of the model will be one in which the government can observe
the firm's type and so the first-best can be attained. It will be  a benchmark
for our later versions.

\begin{center}
{\bf    Procurement I:  Full Information}\footnote{I have changed the notation
from the 3rd edition of this book. The expensiveness  variable $x$ replaces the
ability variable $a$;    $p$ replaces $s$; the type L firm becomes an expensive
firm. }
\end{center}
{\bf Players}\\
 The government and the firm.

\noindent
 {\bf The Order of Play}\\
 0 Nature assigns   the firm  expensive  problems with the project, which add
costs of $x$,  with probability $\theta$. A firm is thus  ``normal'', with type
$N$ and $s=0$, or   ``expensive'', with type $X$ and  $s=x$.    The government
and the firm both observe the type.  \\
 1 The government offers a contract $\{w(m) = c(m) + p(m), c(m)\}$ which pays
the firm its observed cost $c$ and a profit $p$ if it announces its type to be
$m$ and incurs cost $c(m)$,  and pays the firm zero otherwise. \\
2 The firm accepts or rejects the contract.\\
 3 If the firm accepts, it chooses    effort level $e$, unobserved by the
government. \\
 4 The firm finishes the   missile  at a cost   of  $c =\bar{c}+ s-e $,  which
is observed by the government, plus an additional unobserved cost\footnote{The
reader may ask why this disutility is specified as    $f(e- \bar{c})$ rather
than just $f(e)$. The reason is that we will later find an equilibrium cost
level of ($\bar{c}-e^* $), which would be negative if $c_0=0$. }
 of   $f(e- \bar{c})$.  The government reimburses $c(m)$ and pays $p(m)$.




\noindent
  {\bf Payoffs}\\
 Both firm and government are risk-neutral   and both receive payoffs of zero if
the firm rejects the contract. If the firm accepts, its   payoff is
\begin{eqnarray}    \label{e38}%  \label{e15.1}
\pi_{firm} & = &p - f(e-\bar{c})   \\
\end{eqnarray}
 where $f(e-\bar{c})$, the cost of effort, is increasing and convex, so $f'>0$
and $f''>0$. Assume   for technical convenience   that $f$ is increasingly
convex, so $f'''>0$.\footnote{The argument of $f$ is normalized to be $(\bar{c}-
e)$ rather than just $e$ to avoid clutter in the algebra later. The assumption
that $f'''> 0$  allows the use of first-order conditions   by making concave the
maximand in (\ref{e50}), which is  a difference of two concave functions.  It
will also make deterministic contracts superior to stochastic ones. See
Laffont \& Tirole (1993, p. 58).  } The government's payoff is
\begin{equation}  \label{e39}
\pi_{government}  = B -  (1+t) c  -t p  -  f,
 \end{equation}
 where $B$ is the  benefit of the   missile  and $t$ is the deadweight loss from
the taxation needed  for government spending.  This is substantial. Hausman \&
Poterba (1987) estimate the loss to be around \$0.30 for each \$1 of tax revenue
raised at the margin for the United States.



  The model differs from  most  other    principal-agent models in this book
(though not from  The Streetlight Game) because   the principal  cares about the
welfare of  the agent. If the government cared only about the value of the
missile and the cost to taxpayers, its payoff would be $[B-  (1+t) c - (1+t) p
] $. Instead, the payoff function    maximizes social welfare,  the sum of  the
welfares of  the taxpayers and the firm. The welfare of the   firm is $(p  - f )
$, and summing the two welfares yields    equation (\ref{e39}). Either kind of
government payoff function  may be realistic, depending on  the political
balance in the country being modelled,   and the model will have similar
properties whichever one is used. In the end, though, this model behaves in the
same way as one with a selfish principal, because though the government does
care about the welfare of the agent, the fact that taxation has deadweight loss
means that the government will want to pay the firm as little as possible.

   Assume for the moment that $B $ is large enough that the government
definitely wishes to build the   missile   (how large will   become apparent
later).  Cost, not output,  is the focus of this model. The optimal output is
one   missile  regardless of agency problems, but the government wants to
minimize the cost of producing the missile.

 In Procurement I,  whether the firm has expensive problems is observed by the
government, which can therefore specify a contract conditioned on the type of
the firm. The government pays   $p_N$ to a  normal firm with  the   cost $c_N$,
$p_X$ to an expensive firm    with   the  cost $c_X$, and   $p=0$ to a firm that
does not achieve  its  appropriate cost level.   The government thus maximizes
its payoff, equation (\ref{e39}), by choice of $p_X, p_N, c_X,$ and $c_N$,
subject to participation and incentive compatibility constraints.

 The expensive firm    exerts effort $e= \bar{c}+x-c_X$, achieves $c= c_X$,
generating unobserved effort  disutility $f(   e-\bar{c}) =  f(  x -  c_X  )$,
so its  participation constraint, that  type $X$'s   payoff from reporting that
it is type $X$,  is:
 \begin{equation}  \label{e40}%   \label{e15.6}
 \begin{array}{lll}
\pi_X(X)  &\geq & 0  \\
 &  &    \\
 p_X - f(  x -  c_X  ) &\geq & 0. \\
\end{array}
 \end{equation}
 Similarly,   in equilibrium  the normal firm  exerts effort $e= \bar{c} - c_N$,
so its participation constraint is
\begin{equation}  \label{e41}%   \label{e15.7}
 \begin{array}{lll}
 \pi_N(N)  &\geq & 0  \\
&  &    \\
p_N - f(  - c_N )& \geq&  0 \\
\end{array}
 \end{equation}
 The incentive compatibility constraints are trivial here: the government can
use a forcing contract that  pays a firm zero if it generates the wrong cost for
its type, since types are observable.

To make a firm's  payoff  zero and reduce the deadweight loss from taxation, the
government will  provide prices that do no more than equal the firm's disutility
of effort. Since there is no uncertainty,   we can invert the cost equation and
write it as $e=  \bar{c}+x -c  $ or $e= \bar{c} -c  $.   The prices will be
$p_X=f (e-\bar{c})=  f (x - c_X)$   and $ p_N=f (e- \bar{c})=  f (- c_N ) $.

Suppose the government knows  the firm has expensive problems. Substituting  the
price $p_X$ into the government's payoff function, equation (\ref{e39}),  yields
\begin{equation}  \label{e42}%   \label{e15.3}
\pi_{government}  = B -  (1+t) c_X - t f(x-c_X) - f(x - c_X).
 \end{equation}
 Since $f''>0$, the government's payoff function is concave, and standard
optimization techniques  can be used.  The first-order condition for  $c_X$   is
\begin{equation}   \label{e43}
 \frac{ \partial \pi_{government}}{\partial c_X} =   - (1+t)   +  (1+t)  f'(x  -
c_X)    = 0,
 \end{equation}
 so
 \begin{equation}  \label{e44}
 f'(x  -c_X)   =  1.
 \end{equation}
 Equation(\ref{e44}) is the crucial efficiency condition for effort.   Since the
argument of $f$ is $(e-\bar{c})$, whenever $f'=1$ the effort level is efficient.
At the   optimal effort level, the marginal disutility of effort equals the
marginal reduction in cost because of effort.  This is the first-best efficient
effort level, which we will denote by $e^*\equiv e:\{ f'(e- \bar{c})   =  1\}$.

  If we derived the first-order condition  for the normal firm  we would find
$f'(   -c_N)  = 1$  in the same way, so   $ c_N = c_X-x $. Also, if the
equilibrium disutility of effort is the same for both firms, then both must
choose the same effort, $e^*$, though the normal firm  can reach a lower cost
target with that effort.       The cost targets assigned to each firm are $c_X
= \bar{c}+ x - e^*$ and $c_N  =   \bar{c} - e^*$. Since both types must exert
the same effort, $e^*$, to achieve their different targets,   $p_X=    f(e^*-
\bar{c}) =  p_N$.   The two firms exert the same efficient effort level and are
paid the same   price to compensate for the disutility of effort. Let us call
this price level $p^*$.

 The assumption that $B$ is sufficiently large can now be made more specific: it
is that $B -  (1+t)c_X   -t f(e^*- \bar{c}) - f(e^*-\bar{c}) \geq 0$, which
requires that $B -  (1+t) (\bar{c}+ x - e^*)   -(1+t  ) p^* \geq 0$. If that
were not true, then the government would not want to build the missile at all if
the firm had an expensive cost function, as we will not treat of here.

\bigskip
  \noindent
 {\bf Procurement II: Incomplete Information (Adverse Selection)}

  In the second variant of the game,  the existence of expensive problems  is
not observed by the government, which must therefore provide incentives for the
firm  to volunteer  its  type if the normal firm  is to produce at lower cost
than  the firm with expensive problems.

 If the government offered the two contracts of Procurement I, both types of
firm would accept the expensive-cost contract, which has a
 price  of $p^* $ for a cost of $ c=\bar{c}+x - e^*$,   enough to compensate the
firm with expensive problems for its effort,  and  $p= 0$  for any other cost.
That is the cheapest pooling contract, since any contract that paid less would
violate the expensive-cost firm's participation constraint. It
is inefficient, though, because the normal firm can reduce costs to $ c=\bar{c}
+x - e^*$ by exerting effort lower than $e^*$.
The government would still be willing to build the missile, since the social
cost of having the normal firm build the missile inefficiently is still lower
than of having the expensive-cost firm build it efficiently.  But it will turn
out that   separating contracts will  yield higher welfare than the pooling
contract.

 First, let us establish that {\it some} pair of  separating  contracts is
better than the pooling contract, and then we will find the {\it optimal
separating contract}.  A separating contract menu superior to the pooling
contract would be a choice of (1)
      the  old pooling contract $(p^*, c=\bar{c}+x - e^*$), and (2)   a new
contract that offers a slightly  higher price  $p$   but requires reimbursable
costs  $c$ to be  slightly lower. By definition of $e^*$ in first-order
condition (\ref{e44}),  $f'(e^*-\bar{c}) =1$, so $f'( e'-\bar{c})  <1$ for the
effort the normal firm exerts in the old pooling contract. If the normal firm
increased its effort from $e'$ by some small amount $\Delta e$, costs would fall
by $(1) \Delta e$    but the firm would only have to be paid  $f'( e'-\bar{c})
\Delta e$ more to compensate for its extra disutility. Thus, there is  a new
contract that would draw the normal firm away from the old  pooling contract and
be preferred by the government.


We've shown that  there is a  pair of  separating  contracts that the government
likes better than the pooling contract, but not whether that pair is optimal. We
will therefore proceed to find
 the optimal  pair of contracts     ($c_N, p_N)$ ($c_X, p_X$) for firms that
announce $Normal$ or $Expensive$ (with      $p=0$ for other cost levels).
Adapting the government's payoff in  (\ref{e39}) to the  probability $\theta$ of
a expensive firm and probability $1-\theta$ of a normal firm, the government's
maximization problem under incomplete information is
 \begin{equation}   \label{e49}
 \stackrel{ Maximize}{c_N  , c_X  , p_N, p_X} \;\; \; \theta \left[  B -  (1+t)
c_X   - t p_X - f( x - c_X  )  \right] + \left[ 1- \theta \right] \left[  B -
(1+t) c_N   - t p_N- f(     - c_N  )  \right] .
 \end{equation}

  A separating contract  must satisfy  participation constraints and incentive
compatibility constraints for each type of   firm.  The participation
constraints are the same as in Procurement I: inequalities (\ref{e40}) and
(\ref{e41}):
\begin{equation}  \nonumber
\pi_X(X)  =
 p_X - f(  x -  c_X  )  \geq   0\;\; (\ref{e40})
\end{equation}
and
\begin{equation}  \nonumber
 \pi_N(N) =
p_N - f(  - c_N )  \geq   0 \;\; (\ref{e41})\\
\end{equation}



\noindent
 The incentive compatibility constraint for the expensive  firm   is
 \begin{equation}   \label{e45}
 \pi_X(X) =  p_X - f(  x  -  c_X  )  \geq  \pi_X(N)=   p_N - f(x  - c_N ),
 \end{equation}
and the incentive compatibility constraint  for the normal firm   is
\begin{equation}  \label{e46}%   \label{e15.9}
  \pi_N(N) = p_N - f( - c_N )  \geq\pi_N(X)=  p_X - f( - c_X  ).
\end{equation}

Since the  normal   firm  can achieve the same cost level as the expensive
firm   with less effort,  inequality (\ref{e46}) tells us that    if we  are to
have   $c_N < c_X$, as is necessary for us to have a separating equilibrium,  we
need $p_N>p_X$. The second  half of  inequality (\ref{e46}) must be positive.
If the expensive firm's participation constraint,  inequality (\ref{e40}), is
satisfied, then $p_X - f( - c_X  )>0$. This, in turn implies that    (\ref{e41})
is a strong inequality;   the normal firm's participation constraint is
nonbinding.

The expensive firm's participation constraint,  (\ref{e40}), will be binding
(and therefore satisfied as an equality), because the government wishes to keep
the price $p$ low   to reduce the deadweight loss  of  extra taxation, the $-
tp_{X}$  term in  problem (\ref{e49}).   The normal firm's incentive
compatibility constraint    must also  be binding, because if the pair ($c_N,
p_N$) were strictly more attractive for the normal firm, the government could
reduce the price $p_N$ and save on the $-tp_{N}$  term in  problem (\ref{e49}).
Constraint (\ref{e46}) is  therefore satisfied as an equality.  Knowing that
constraints (\ref{e40}) and (\ref{e46}) are binding, we can write,   from
constraint (\ref{e40}),
 \begin{equation}  \label{e47}
  p_X = f(   x   - c_X )
 \end{equation}
 and, making use of both (\ref{e40}) and (\ref{e46}),
 \begin{equation}  \label{e48}
 p_N = f (  - c_N )   + f (   x   - c_X ) - f (   -c_X  ).
 \end{equation}


  Substituting  for $p_X$ and $p_N$ from (\ref{e47}) and (\ref{e48})  into  the
maximization problem, (\ref{e49}),    reduces  the problem to
\begin{equation}  \label{e50}
 \begin{array}{ll}
 \stackrel{ Maximize}{c_N  , c_X }  & \theta [  B - (1+t) c_X   - t   f(   x  -
c_X ) - f(   x   - c_X   )   ] \\
 & \\
 &  +  [ 1- \theta ] [  B - (1+t) c_N  - t f(  - c_N ) - t f(   x   - c_X ) + t
f(    - c_X  ) - f(    - c_N  )  ] . \\
 \end{array}
\end{equation}

\noindent
 (1) The first-order condition with respect to $c_N  $ is
 \begin{equation}   \label{e51}%  \label{e15.14}
 (1- \theta) [   -  (1+t)  + t  f'(   - c_N  ) + f'(     - c_N  )  ] =0,
 \end{equation}
 which simplifies to
\begin{equation}  \label{e52}
 f'(   -c_N   )  =1.
 \end{equation}
Thus, as  in Procurement I,    $f'_N (e-\bar{c})  =1$. The  normal firm
chooses the efficient effort level $e^*$ in equilibrium, and   $c_N $ takes the
same value as it did in    Procurement I. Equation (\ref{e48}) can be rewritten
as
\begin{equation}  \label{e53}
 p_N= p^*   + f (   x   - c_X ) - f(    - c_X  ).
 \end{equation}

 Because $ f(  x   - c_X ) > f(    -c_X)$, equation (\ref{e53}) shows that $p_N
> p^*$. Incomplete information increases the   price for    the  normal firm,
which earns more than its reservation  utility in the game with incomplete
information. Since the expensive  firm   will earn exactly zero, this means that
the government is on average  providing its supplier with an above-market rate
of return, not because of corruption or political influence, but because that is
the   way to induce   normal suppliers to reveal that  they do not have
expensive problems. This should be kept in mind as an alternative  to the
product quality model of Chapter 5 and the efficiency wage model of Section 8.1
for why above-average rates of return persist.

\noindent
 (2) The first-order condition   with respect to $c_X  $ is
 \begin{equation}   \label{e54}%   \label{e15.16}
 \begin{array}{l}
\theta \left[    -  (1+t)  + t f'(  x  - c_X ) + f'(  x - c_X) \right] + \left[
1- \theta \right]   [   t f'(   x   - c_X ) -tf'(    - c_X  )  ] =0. \\
\end{array}
\end{equation}
 This can be rewritten as
 \begin{equation}   \label{e55}
 {\displaystyle  f'(  x    - c_X )=1 - \left(\frac{ 1-\theta}{\theta(1+ t)}
\right)   \left[ t f'(  x - c_X ) -tf'(  - c_X ) \right].}
 \end{equation}
Since the right-hand-side of equation (\ref{e55}) is less than one,  the
expensive firm   has a lower level of $f'$ than the normal firm, and if $f'$ is
lower and $f''>0$, effort must be less than the optimum,  $e^*$.    Perhaps this
explains the expression ``good enough for government work''-- though our model
can apply to any organization that is trying to buy goods instead of making them
internally.   Since, however,  the  expensive firm's participation constraint,
(\ref{e40}),  is satisfied as an equality, it must also be true that $p_X <
p^*$.  The expensive  firm's price is lower than under full information,
although since its effort is   lower  its payoff stays the same.

  I  have not yet said   whether the expensive firm's
  incentive compatibility constraint was binding. It is not:
  the  expensive firm    is not near being tempted to pick the  normal firm's
contract. This is a bit subtle. Setting  the left-hand-side of the incentive
compatibility constraint (\ref{e45})   equal to zero because the participation
constraint is binding for  the  expensive firm,   substituting in for $p_N$ from
equation (\ref{e48})  and rearranging  yields
\begin{equation}   \label{e56}
  f(     x  - c_N   )-  f( - c_N ) \geq   f(   x  - c_X ) -f(   - c_X ).
 \end{equation}
  As illustrated in Figure 8, inequality (\ref{e56}) is true as a strict
inequality, because $f$ is convex ($f''>0$) and so the increment in $f$'s value
starting frome the lower base $-c_X$ is smaller than starting from  $-c_N$.
Thus, the expensive firm's incentive compatibility constraint is nonbinding.

\includegraphics[width=150mm]{fig10-08.jpg}



\begin{center}
{\bf Figure 8: Why the Expensive Firm's Incentive Compatibility Constraint Is
Nonbinding}
\end{center}


 To summarize,  the government's optimal contract will induce the normal  firm
to  exert the first-best efficient effort level and achieve  the first-best cost
level, but  will yield that firm a positive profit.  The contract will  induce
the  expensive firm   to exert something less than the first-best effort level
and result in a cost level higher than the first-best, but its profit will be
zero.

 There is a tradeoff between  the government's two objectives of inducing the
correct amount of effort and minimizing the subsidy to the firm.  Even under
complete information, the government cannot provide a subsidy of zero, or the
firms will  refuse to build the   missile.  Under incomplete information, not
only must the subsidies be positive but the normal  firm earns {\bf
informational rents}; the government offers a contract that pays the normal firm
more  than  under complete information to prevent it from mimicking an expensive
firm   and  choosing an inefficiently low effort. The expensive   firm, however,
does choose an inefficiently low effort, because if it were assigned greater
effort it would have to be paid a greater subsidy, which would tempt the normal
firm to imitate it. In equilibrium, the government has compromised by having
some probability of  an inefficiently high subsidy  ex post, and some
probability of inefficiently low effort.

\bigskip

 In the last version of the game,  the firm's type is not known to either player
until after the contract is agreed upon. The firm, however, learns its type
before it must choose its effort level.

\begin{center}
 {\bf    Procurement III:  Moral Hazard with Hidden Information  }
 \end{center}
 \noindent
 {\bf The Order of Play}\\
 1 The government offers a contract $\{w(m) = c(m) + p(m), c(m)\}$ which pays
the firm its observed cost $c$ and a profit $p$ if it announces its type to be
$m$ and incurs cost $c(m)$,  and pays the firm zero otherwise. \\
2 The firm accepts or rejects the contract.\\
3 Nature assigns   the firm  expensive  problems with the project, which add
costs of $x$,  with probability $\theta$. A firm is thus  ``normal'', with type
$N$ and $s=0$, or   ``expensive'', with type $X$ and  $s=x$.
Only the  firm  observes its type.  \\
  4 If the firm accepted, it   announces its type to be $m$ and chooses
effort level $e$, unobserved by the government.  \\
 5 If the firm accepted, it  finishes the   missile  at a cost   of  $c =\bar{c}
+ x-e $ or $c=\bar{c}-e$ which is observed by the government, plus an additional
cost $f(e- \bar{c})$ that the government does not observe.  The government
reimburses $c$ and pays $p(c)$.

 The government's payoff function is exactly the same as in Procurement II,
equation (\ref{e49})
$$
 \stackrel{ Maximize}{c_N  , c_X  , p_N, p_X} \;\; \; \theta \left[  B -  (1+t)
c_X   - t p_X - f( x - c_X  )  \right] + \left[ 1- \theta \right] \left[  B -
(1+t) c_N   - t p_N- f(     - c_N  )  \right]  \;\;\; (\ref{e49}) .
$$





\noindent
 The incentive compatibility constraints are   exactly the same as in
Procurement II, equations (\ref{e45})  and  (\ref{e46}). For  the expensive  firm  the constraint is
\begin{equation}  \nonumber
 \begin{array}{lll}
  \pi_X(X) &\geq &  \pi_X(N)  \\
 &  &    \\
 p_X - f(  x  -  c_X  ) &\geq &  p_N - f(x  - c_N ),\;\;\;\;\;\;\;\;\; (\ref{e45})
 \end{array}
\end{equation}
and   for the normal firm  it  is
\begin{equation}   \nonumber
\begin{array}{lll}
 \pi_N(N)  &\geq & \pi_N(X)  \\
&  &    \\
 p_N - f( - c_N )& \geq&  p_X - f( - c_X  ) \;\;\;\;\;\;\;\;\; (\ref{e46})
 \end{array}
\end{equation}

  The difference from Procurement II is that now there is just one participation
constraint, not two, because the firm does not know its type at the time it
agrees to the contract:
 \begin{equation}  \label{e57}%   \label{e15.6z}
 \theta[ p_X - f(  x -  c_X  )  ] + [1-\theta][ p_N - f(  - c_N )] \geq  0.
 \end{equation}
 Thus, the maximization problem is less constrained in Procurement III. The two
participation  constraints of Procurement II jointly imply constraint
(\ref{e57}) is satisfied, but the reverse is not true. Now, the payoff of the
expensive type can be negative, so long as the payoff of the normal type is
positive enough to make the overall payoff non-negative.


 The participation constraint,  inequality (\ref{e57}),  will be binding,
because the government wants to  keep the deadweight loss from taxation low, and
it can now make the expensive firm's payoff   negative enough to   compensate
for   the normal firm's positive payoff. So long as the overall expected payoff
is zero, the firm will still agree to the contract.  There is no informational
rent ex ante, unlike in Procurement II.



 The  normal firm's incentive compatibility constraint   will still be binding.
That firm does not want to admit that it can reduce costs easily, so it has  a
strong incentive to imitate the expensive firm, and must be bribed not to. The
government will not want to  pay a bigger bribe than necessary.
 If   the   pair ($c_N, p_N$) were strictly more attractive for the normal firm
than ($c_X, p_X$), the government could reduce the  price $p_N$. Constraint
(\ref{e46}) is  therefore satisfied as an equality.


\noindent
 Knowing that   constraints    (\ref{e46}) and (\ref{e57})   are binding, we can
write   from constraint (\ref{e57}),
\begin{equation}  \label{e59}
 p_X =  f(  x -  c_X  )   - \frac{ [1-\theta][ p_N - f(  - c_N )] }{\theta}.
 \end{equation}
Substituting from (\ref{e59})  for $p_X$ into (\ref{e46}),  we get
\begin{equation}   \label{e60}
 p_N -f (  - c_N )  =   f(  x - c_X  )   - \frac{ [1-\theta][ p_N - f(  - c_N )]
}{\theta}  - f (   - c_X  ). \end{equation}
 This can be solved for $p_N$ to yield
  \begin{equation}  \label{e61}
p_N  =  \theta [f(x-c_X) - f( -c_X)] + f(- c_N ),
\end{equation}
 which when substituted into  (\ref{e59}) yields
   \begin{equation} \label{e62}
 p_X =    [1-\theta] [f(x-c_X) - f( -c_X)].
  \end{equation}


Substituting  for $p_N$ and $p_X$  from (\ref{e61}) and (\ref{e62})  into the
maximization problem reduces  the problem to
 \begin{equation}  \label{e64}%   \label{e15.13z}
 \begin{array}{ll}
\stackrel{ Maximize}{c_N  , c_X   } & \theta \left\{  B -  (1+t) c_X   - t
[1-\theta] [f(x-c_X) - f( -c_X)] - f( x - c_X  )  \right\} \\
 & + \left[ 1- \theta \right] \left[  B -  (1+t) c_N   - t \{ \theta [f(x-c_X) -
f( -c_X)] + f(- c_N ) \}- f(     - c_N  )  \right] .\\
 & \\
 \end{array}
 \end{equation}

\noindent
  The first-order condition with respect to $c_N  $ is
\begin{equation}  \label{e65}%   \label{e15.14z}
  (1- \theta) [   -  (1+t)  + t   f'(   - c_N  ) + f'(     - c_N  )  ] =0,
 \end{equation}
 just as under adverse selection,  which simplifies to
 \begin{equation}   \label{e66}%  {e15.15z}
 f'(   -c_N   )  =1.
 \end{equation}
 Thus,  the crucial efficiency  condition (\ref{e44}) is satisfied:     $f'(e -
c_0) =1$. The  normal firm   chooses the efficient effort level $e^*$ in
equilibrium, and   $c_N $ takes the same value as it did in   Procurement I and
II.

Now that we know that the effort level is $e^*$, we can say something about the
normal firm's payoff.

Since  $ p^* = f(- c_N )$, the  optimal price   equation (\ref{e61}) can be
rewritten as
 \begin{equation}   \label{e67}
  p_N  =   \theta [f(x-c_X) - f( -c_X)] + p^*   ,
  \end{equation}

 Because $ f(  x   - c_X ) > f(    -c_X)$, equation (\ref{e67}) shows that $p_N
> p^*$, even though $e_N=e^*$. The normal firm earns a positive payoff even
though information was symmetric at the start of the game.  It earns an
informational rent  because partway through the game it learns its type and the
government does not, and the government needs to pay   something to induce it to
admit to its low-cost type.
The expensive firm  must   earn less than its reservation utility  so that the
overall participation constraint will be satisfied as an equality.



The government could  have made the   expensive firm's payoff  negative in
either of two ways: (a) a lower $p_X$ than in Procurement II, or (b) a lower
$c_X$  than in Procurement II (and thus higher $e_X$). Option (b) is what  we
just derived.  It is beter than (a)  because it raises the expensive firm's
effort  to closer to the efficient level.  This increases the amount of surplus,
and   the government will get the entire surplus since there is now no
informational rent.   Hence, the expensive firm's effort is higher in
Procurement III than in Procurement II.


     Now that $c_X$ is higher   for the same $p_X$, the expensive cost-price
pair is less attractive than   in Procurement II. As a result, $\pi_N(X)$ is
lower, and  the government can make the normal firm's cost-price pair less
attractive too.  This is we found that though $c_N$ is unchanged from
Procurement II,  $p_N$ has  fallen.




To summarize,  the government's optimal contract in Procurement III  will induce
the normal  firm to  exert the first-best efficient effort level and achieve
the first-best cost level, but  will yield that firm a positive payoff, though
smaller than in Procurement II.  The contract will  induce the expensive   firm
to exert   less than the first-best effort level, though more than in
Procurement II,  and result in a cost level higher than the first-best and a
negative payoff. Overall, the firm's expected payoff will be zero.


 This is what one might expect of moral hazard with hidden information as
compared to adverse selection. Starting with symmetric information results in
less gaming at the time of contract formation, so  the principal's maximization
problem has three constraints to satisfy instead of four, and this results in
effort choices closer to the first-best. Effort choices still do not always
reach the first-best, however, because information  about player types does
become asymmetric midway  through  the game, and the contract has to be designed
to induce the informed player, the firm, to disclose its information.

 A practical implication is that the parties ought to agree on a contract as
early as possible in the procurement process, before one of them acquires an
informational advantage.  Of  course, this does not apply if one of them {\it
starts} with an informational advantage;  then, delay before agreement might
actually help, by giving the uninformed party time to  learn the facts it needs.


\bigskip

  A little reflection will provide a host of  additional ways to alter the
Procurement Game. What if the firm discovers its costs only after accepting the
contract?   What if two firms bid against each other for the contract? What if
the firm can bribe the government?  What if the firm and the government bargain
over the gains from the project  instead of the government being able to make a
take-it-or-leave-it contract offer? What if the game is repeated, so the
government can use the information it acquires in the second period? If it is
repeated, can the    government   commit to long-term contracts? Can it commit
not to  renegotiate? See   Laffont \& Tirole (1993) if these   questions
interest you.  If they merely confuse you, an aphorism by Doug Larson that
McAfee (2002, p. 202) quotes is an apt summary of the Procurement Game:
``Accomplishing the impossible means only that the boss will add it to your
regular duties.''



\newpage

\begin{small}

\bigskip
\noindent
 {\bf Notes}

 \noindent
{\bf N10.1   Production Game VIII, the Maskin Matching Scheme,  Unravelling, and
the Revelation Principle}
\begin{itemize}
\item
  The revelation principle was developed  in Myerson (1979), though the idea
can be traced back to  Gibbard (1973), and named in Myerson (1981).  Myerson's
game theory book is, as one might expect, a  good place to look for  further
details (Myerson [1991, pp. 258-63, 294-99]). See also the books by  Fudenberg
\& Tirole (1991a) and   Laffont \& Tirole (1993), Baron's chapter in the 1989
 {\it Handbook of Industrial Organization} edited by Schmalensee and Willig ,
the 2002 and 2005 books on the principal-agent problem by Laffont \& Martimort
and
Bolton  \&  Dewatripont,  and Stole's 2001 manuscript.



\item
  Levmore (1982) discusses hidden knowledge problems in tort damages, corporate
freezeouts, and property taxes in a law review article. A legal application of
the idea of the sender-receiver game, pointing out the usefulness of a mediator
as a mechanism is Brown \& Ayres (1994).



 \item
 Post-contractual adverse selection  is  common  in public policy. Should the
doctors who prescribe drugs also be allowed to sell them? The question trades
off the likelihood of overprescription against the potentially lower cost and
greater convenience of doctor-dispensed drugs. See ``Doctors as Druggists: Good
Rx for Consumers?''  {\it The Wall Street Journal}, June 25, 1987, p. 24.



  \item
  A hidden knowledge game requires that the state of the world matter to one of
the players' payoffs, but not necessarily in the same way as in Production Game
VII. The Salesman Game of Section 10.2 effectively uses the utility function
$U(e,w,\theta)$ for the agent and $V(q-w)$ for the principal. The state of the
world matters because the agent's disutility of effort varies across states. In
other problems, his utility of money might vary across states.


\item
  Eric Maskin came up the Maskin Matching Scheme in a 1977 MIT working paper,
``Nash Implementation and Welfare Optimality,''   The working paper became known
as a classic, but was not published until 22 years later,
 in  a 1999 issue of the  {\it  Review of Economic Studies}  with several other
articles on mechanism design.

\end{itemize}

\bigskip
\noindent
 {\bf N10.3 An Example of Post-Contractual Private Knowledge:   The Salesman
Game}
  \begin{itemize}

\item Sometimes students know more about their class rankings than the professor
does.  One professor of labor economics used a mechanism of the following kind
for grading class discussion. Each student $i$ reports a number evaluating other
students in the class.  Student $i$'s grade is an increasing function of the
evaluations given $i$ by other students and of the correlation between $i$'s
evaluations and the other students'. There are many Nash equilibria, but telling
the truth is a focal point.

\item In dynamic games of  moral hazard with hidden knowledge    the {\bf
ratchet effect} is important: the agent takes into account that his information-
revealing choice of contract this period will affect the principal's offerings
next period. A principal might allow high prices to a public utility in the
first period to discover that its costs are lower than expected, but in the next
period the prices would be reduced. The contract is ratcheted irreversibly to be
more severe. Hence, the company might not choose a contract which reveals its
costs in the first period.  This is modelled in Freixas, Guesnerie \& Tirole
(1985).

$\;\;\;$ Baron (1989) notes that the principal might purposely design the
equilibrium to be pooling in the first period so self selection does not occur.
Having learned nothing, he can offer a more effective separating contract in the
second period. \end{itemize}

\bigskip

\noindent
 {\bf N10.5} {\bf Price Discrimination}
 \begin{itemize}
\item
The  names for price discrimination in Part 2,  Chapter 17, Section 5 of Pigou
(1920) are:  (1) first-degree (perfect price discrimination), (2) second-degree
(interquantity price discrimination), and (3) third-degree (interbuyer price
discrimination).  These arbitrary names have plagued generations of students of
industrial organization, in parallel with the appalling   Type I and Type II
errors of  statistics (better named as False Negatives or Rejections,  and False
Positives or Acceptances). I  invented the terms  {\bf interbuyer price
discrimination} and {\bf interquantity price discrimination}  for this edition,
with the excuse that I think their meaning will be clear  to anyone who already
knows the concepts under their Pigouvian names.

   McAfee (2002, p. 261) has a  new taxonomy that may catch on: direct versus
indirect price discrimination. Direct price discrimination is based on observing
characteristics of the customer and charging him  a price for a given unit that
depends on those characteristics, an offer not open to everyone. Indirect price
discrimination makes offers open to everyone, but offers which will separate the
customers, whether by quantity desired, quality desired, attention paid to
newspaper coupons, or other unobservable  characteristics.

\item
A narrower category   of nonlinear pricing is the  {\bf quantity discount}, in
which  the  price per unit declines with the quantity bought.  Sellers are often
constrained to this, since if the price per unit rises with the quantity bought,
some means must be used to prevent a  canny consumer from buying two batches of
small quantities instead of one batch of a large quantity.

\item
Wilson's 1993 book, {\it Nonlinear Pricing}, is a good reference on  price
discrimination.

\item
In  Varian's Nonlinear Pricing Game  the probabilities of types for each player
are not independent, unlike in most games.  This does not make the game more
complicated, though. If the assumption were ``Nature assigns each buyer a
utility function $u$ or $v$ with independent probabilities of 0.5 for each
type,'' then there would be not just two possible states of the world in this
game-- $uv$ and $vu$ for Smith and Jones's types -- but four -- $uv, vu, uu$,
and $vv$. How would the equilibrium change?

\item The careful reader will  think, ``How can we say that Buyer V always gets
higher utility than Buyer U for given $x$?    Utility cannot be compared  across
individuals, and we could rescale Buyer V's utility function to make him always
have lower utility  without altering the essentials of the utility function.''
My reply is that   more generally we could set up the utility functions as
$v(x) +y$  and $u(x) +y$, with $y$ denoting  spending on all other goods (as
Varian does in his book).  Then  to say that   V always gets higher utility for
a given $x$ means that he always has a higher relative value than U does  for
good $x$  relative to   money.   Rescaling to give  V the utility function
$.001v(x) + .001 y$ would not alter that.

\item
 The notation I used in Varian's Nonlinear Pricing Game is  optimized for
reading.  If you wish to write this on the board or do the derivations for
practice,  use abbreviations like  $u_1$ for $u_1(x_1) $,  $a $ for $v(x_1)$,
and  $b $ for $v(x_2)$ to save writing.  The tradeoff between brevity and
transparency in notation is common, and must be made in light of whether you are
writing on a blackboard or on a computer, for just yourself or for the
generations.

\end{itemize}

\bigskip

\noindent
 {\bf N10.6} {\bf  Rate-of-Return Regulation and Government  Procurement}
  \begin{itemize}

 \item
 I  changed the notation from Laffont and Tirole and from my own previous
edition.   Rather than  assign each  type of firm a cost parameter $\beta$  for
a cost of $c= \beta -e$,  I now  assign each   type of firm an ability parameter
$a$, for a cost of $c=c_0-a -e$.   This will allow the desirable type of firm to
be the one with the  $High$ value of the type parameter, as in most  models.


\item
 In practice,   whether procurement is  by the government or by other large
organizations it must balance  a multitude of concerns, not just  incentives for
the supplier. Three of the most impoart concerns are giving performance
incentives to the government's own agents, the ones who arrange the procurement,
preventing corruption of those agents with  kickbacks (a bribe  awarded an agent
in consideration of getting a contract), and transaction costs of various sorts.
On this last, see    Bajari \&   Tadelis (2001). Much insight can be had from
auction theory as well, since auctions are frequently used for procurement.
  Paul Klemperer's  2004  {\it Auctions: Theory and Practice} (which is
relatively nontechnical),    Paul Milgrom's  1999   {\it   Auction Theory for
Privatization}  and 2004 {\it  Putting Auction Theory to Work}   are full of
practical advice based on sound theory.



\end{itemize}





\newpage

\noindent {\bf Problems}

\bigskip

 \noindent
 {\bf 10.1. Unravelling}  (hard) \\
  An elderly prospector owns a gold mine worth an amount $\theta$ drawn from the
uniform distribution $U[0,100]$ which nobody knows, including himself.  He will
certainly sell the mine, since he is too old to work it and it has no value to
him if he does not sell it.  The   several prospective buyers are all   risk
neutral. The prospector can, if he desires, dig deeper into the hill and collect
a sample of gold ore that will reveal the value of $\theta$. If he shows the ore
to the buyers, however, he must show genuine ore, since an unwritten Law of the
West says that fraud is punished by hanging offenders from joshua trees as food
for buzzards.

   \begin{enumerate}

 \item[(a)]
 For how much can he sell the mine   if he is clearly too feeble to have dug
into the hill and examined the ore? What is the price in this situation if, in
fact, the true value is $\theta=70$?

 \item[(b)]
For how much can he sell the mine  if he can dig the test tunnel at zero cost?
Will he show the ore? What is the price in this situation if, in fact, the true
value is $\theta=70$?

 \item[(c)]
 For how much can he sell the mine if, after digging the tunnel at zero cost and
discovering $\theta$, it costs him an additional 10 to verify the results for
the buyers?   What is his expected payoff?

\item[(d)]
 Suppose that  with probability 0.5 digging the test tunnel costs 5 for    the
prospector, but with probability 0.5 it costs  him 120. Keep in mind that the
0-100 value of the mine is  net of the buyer's digging cost. Denote the
equilibrium price that buyers will pay for the mine after the prospector
approaches them without showing ore by $P$. What is the buyer's posterior belief
about the probability it costs 120 to dig the tunnel, as a function of $P$?
Denote this belief by $B(P)$  (Assume, as usual, that all these parameters are
common knowledge, although only the prospector learns whether the cost is
actually 5 or 120.)

\item[(e)]
What is the prospector's expected payoff in the conditions of part (d) if (i)
the tunnel costs him  120,  or (ii) the tunnel costs him 5?

 \end{enumerate}

\bigskip
\noindent
 {\bf 10.2. Task Assignment } (medium) \\
 Table 1  shows the payoffs in the following game.   Sally  has been hired by
Rayco to do either Job 1, to do Job 2, or to be  a Manager.  Rayco believes that
Tasks 1 and 2 have equal probabilities of being the    efficient ones for Sally
to perform.    Sally knows which task is   efficient, but  what she would like
best is a job as Manager that gives her the freedom to choose rather than have
the job designed for the task.  The CEO of Rayco asks Sally which task is
efficient.  She can either reply  ``task 1,''   ``task 2,'' or be silent.   Her
statement, if she makes one, is an example of ``cheap talk,'' because it has no
direct effect on anybody's  payoff.  See Farrell \& Rabin (1996).

 \begin{center} {\bf   Table 1:  The Right To Silence Game payoffs}

 \begin{tabular}{lllccccc}
  &   &     &\multicolumn{5}{c}{\bf Sally's Job}\\
&   &   &  {\it Job 1}   &   &  {\it Job 2}   & & {\it Manager} \\
 &   && & &  & \\ &   &  Task 1 is efficient  (0.5) &  2, 5 &    &  $1, -2 $&  &
3, 3\\
 &  && & &  & \\
 & {Sally knows} &     &   &    &   & &   \\
&   && & &  & \\ &  &    Task 2 is efficient   (0.5)  & $1, -2 $  &    & 2, 5 &
& 3, 3 \\ &   && & &  & \\
\multicolumn{8}{l}{\it Payoffs to: (Sally, Rayco)}
\end{tabular}
 \end{center}

\begin{enumerate}
 \item[(a)]
If   Sally  did not have the option of speaking,  what would happen?

  \item[(b)]
There exist   perfect Bayesian  equilibria in which it does not matter how
Sally replies.    Find  one of these in which Sally speaks at least some of the
time, and    explain why  it is an equilibrium. You may assume that Sally is not
morally or otherwise bound to speak the truth.

 \item[(c)]
There exists a perverse variety of  equilibrium in which Sally  always tells the
truth  and never is silent.  Find an example of  this equilibrium, and  explain
why neither player would  have incentive to deviate to out-of-equilibrium
behavior.
 \end{enumerate}

%---------------------------------------------------------------

\bigskip
\noindent
{\bf 10.3. Agency Law } (easy)\\
 Mr. Smith is thinking of buying  a custom-designed machine from either Mr.
Jones  or Mr. Brown. This machine costs \$5,000 to build, and it is useless to
anyone but Smith.  It is common knowledge that with  90 percent probability the
machine  will be worth \$10,000 to Smith at the time of delivery, one year from
today, and with 10 percent probability it will only be worth \$2,000. Smith owns
assets of \$1,000.   At the time of contracting,  Jones and Brown  believe there
is  a 20 percent chance that Smith is actually acting as an ``undisclosed
agent'' for Anderson, who has assets of \$50,000.

  Find  the price be under the following two legal regimes: (a)  An undisclosed
principal is not responsible for the debts of his agent; and (b) even an
undisclosed principal is responsible for the debts of his agent. Also, explain
(as part [c]) which rule a moral hazard model like this would tend to support.

\bigskip

\noindent
{\bf 10.4.  Incentive Compatibility and Price Discrimination  } (medium) \\
 Two consumers have utility functions $u_1(x_1, y_1)=  a_1 log(x_1)  + y_1$ and
$u_2(x_2, y_2)=  a_2  log(x_2) + y_2$, where $7>a_2>a_1>2$. The price of  the y-
good is 1  and  each consumer has an initial wealth of 15.  A monopolist
supplies the x-good. He has  a constant  marginal cost of 1.2 up to  his
capacity constraint of 10. He will offer at most two price-quantity packages,
$(r_1,x_1)$ and $(r_2, x_2)$, where $r_i$ is the total cost of purchasing $x_i$
units.  He cannot identify which consumer is which, but he can prevent resale.

\begin{enumerate}

  \item[(a)]
 Write down the monopolist's profit maximization problem. You should have four
constraints plus the capacity constraint.

 \item[(b)]
 Which constraints will be binding at the optimal solution?

  \item[(c)]
 Substitute the binding constraints into the objective function. What is the
resulting expression? What are the first-order conditions for profit
maximization? What are the profit-maximizing values of $x_1$ and $x_2$?

 \end{enumerate}

\noindent
{\bf  10.5.    The Groves Mechanism}  (easy)\\
A new computer costing 10 million dollars  would benefit existing Divisions 1,
2, and 3 of a company with 100 divisions.  Each divisional manager knows the
benefit to his division (variables $v_i,  i = 1,...,3$), but nobody else does,
including the company CEO. Managers maximize the   welfare of their own
divisions. What    dominant strategy mechanism   might  the CEO   use to induce
the managers to tell the truth  when they report their valuations? Explain why
this mechanism will induce truthful reporting, and denote the reports by   $x_i,
i = 1,...,3$.   (You may assume that any  budget transfers to and from the
divisions in this mechanism are permanent-- that the divisions will not get
anything back later if the CEO collects more  payments than he gives, for
example.)

\bigskip
\noindent
 {\bf 10.6.  The Two-Part Tariff  }  (easy) (Varian 14.10, modified)\\
One way to price  discriminate is to charge a lump sum fee $L$ to have the right
to purchase a good, and then charge a per-unit charge $p$ for consumption of the
good after that. The standard example is an amusement park where the firm
charges an entry fee and a charge for the rides inside the park. Such a pricing
policy is known as a {\bf two-part tariff}. Suppose that all consumers have
identical utility functions given by $u(x)$ and that the cost of production is
$cx$. If the monopolist sets a two-part tariff, will it produce the socially
efficient level of output, too little, or too much?

%---------------------------------------------------------------

\bigskip
 \noindent
 {  \bf 10.7. Selling Cars } (medium) \\
 A car dealer must pay \$10,000 to the manufacturer for each car he adds to his
inventory. He faces three buyers. From the point of view of the dealer, Smith's
valuation is uniformly distributed between \$11, 000 and \$21,000, Jones's   is
between \$9,000 and \$11,000, and Brown's is between \$4,000 and \$12,000.  The
dealer's policy  is to make a separate  take-it-or-leave-it offer to each
customer, and he is smart enough to avoid making different offers to customers
who could resell to each other.  Use the notation that the maximum valuation is
$ \overline{V}$ and the range of valuations is $ R $.
\begin{enumerate}
  \item[(a)]
 What will the offers be?

 \item[(b)]
 Who is most likely to buy a car? How does this compare with    the outcome
with perfect price discrimination under full information? How does it compare
with the   outcome   when the dealer charges \$10,000 to each customer?

  \item[(c)]
 What happens to the equilibrium prices if with probability 0.25 each buyer has
a valuation of \$0, but the probability distribution remains otherwise the same?
What happens to the equilibrium expected profit?

  \item[(d)]
What happens to the equilibrium price the seller offers to seller Jones  if with
probability 0.25 Jones  has a valuation of \$30,000, but with probability 0.75
his valuation is uniformly distributed between \$9,000 and \$11,000 as before?
Show the relation between price and profit on a rough graph.

 \end{enumerate}

   %---------------------------------------------------------------

\newpage

\begin{center}
 {\bf  Regulatory Ratcheting:  A Classroom Game for  Chapter 10}
 \end{center}

 Electricity demand facing each of several firms  is perfectly inelastic  at 1
gigawatt per firm. The price   is chosen by the regulator. The regulator cares
about two things: (1) getting electrical service, and (2) getting it at the
lowest price possible. The utilities like  profit and dislike effort. Throughout
the game, utility $i$ has ``cost reduction'' parameter $x_i$, which it knows but
the regulator does not. This parameter is big if the utility can reduce its
costs with  just a little effort.

\noindent
 Each year, the following events happen.

\noindent
 1. The  regulator offers price $P_i$ to firm $i$.

\noindent
2. Firm $i$ accepts or rejects.

\noindent
 3. If Firm $i$ accepts, it secretly chooses its effort level $e_i$,

\noindent
  4. Nature secretly and randomly  chooses the economywide shock $u$ (uniform
from 1 to 6)  and Firm $i$'s  shock $u_i$ (uniform from 1 to 6) and announces
Firm $i$'s cost, $c_i$. That cost equals
 \begin{equation} c_i =  20 + u + u_i - x_i e_i.
  \end{equation}

\noindent
5. Firm $i$ earns a  period payoff of  0 if it rejects the contract. If it
accepts, its payoff is
 \begin{equation}
 \pi_i = p_i (1)   - c_i  - e_i^2
  \end{equation} The regulator earns a period payoff of 0  from firm $i$ if its
contract is rejected. Otherwise, its payoff from that firm is
 \begin{equation}
  \pi_{regulator}  (i) = 50 - p_i
  \end{equation}

All variables take integer values.

The game  repeats for as many years as the class has time for, with each firm
keeping the same value of $x$ throughout.


\end{small}

\end{document}


