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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 0 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}