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\begin{LARGE}

   \parindent 24pt \parskip 10pt

 

\noindent   2 February 2006.  Eric Rasmusen, Erasmuse@indiana.edu.
Http://www.rasmusen.org. 
 Overheads for Chapter 10 of {\it Games and Information}.

 
 
  


\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}

\newpage
 


\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} 
  Agent rejects:   $\pi_{agent} = \bar{U}=0$ and $
\pi_{principal} = 0$. \\
 Agent accepts: $\pi_{agent}= U(e,w,s)=w-e^2$  and
$\pi_{principal}= V(q - w)=q-w$.

 

 

  Production Game VII, adverse selection version: two participation constraints and two
incentive compatibility constraints. PG VIII, moral hazard with hidden info:   one participation constraint. 

  

\newpage

 The first-best is unchanged from Production Game VII:  $e_g=1.5$ and  $q_g =4.5$,  $e_b=0.5$   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 self-selection constraints are the same as in Production Game VII.
 \begin{equation} \label{a2}
\begin{array}{ll}
  \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{array}
 \end{equation}
 \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
single participation constraint is  
\begin{equation} \label{a4}
\begin{array}{l}
   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{array}
 \end{equation}

\newpage

  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 $w_g = \frac{1}{9} q_g^2
+ \frac{4}{9} q_b^2$.

\newpage

 Returning to the principal's maximization problem in (\ref{a1}) and substituting
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.

 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  effort  in the bad state. The
principal in Production Game VIII can  (a)
come closer to the first-best when the state is bad,  and (b)    reduce the
rents to the agent.  

\newpage


\noindent
{\bf Observable but  Nonverifiable Information  }

  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?

  We say that the variable $s$ is {\bf
nonverifiable} if contracts based on it cannot be enforced.

 
 Maskin (1977) suggested a matching scheme  to achieve the first-best which would take the following two-part  form for Production Game VIII:

\noindent
{\  (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.   }

   

\newpage



  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
$\pi_{agent} (good)=  0.25- (0.5)^2=0$,

Neither player would have incentive
to deviate unilaterally and drive    payoffs to zero. 

  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.

  But  
agent  could negotiate a new contract with the principal.

 

The  Maskin scheme is  like the Holmstrom Teams contract, where if output was even a little too small, it was
destroyed rather than divided among the team members.

Solution:
  a third party who would receive the output if it was t oo small.

 


\newpage

\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.


\newpage


 
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 $s$. 

Assume the agent   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    $m \geq 4.36$.

 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 $s=
1$. 

\newpage


MODEL REPETITION:  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 $s$.  The agent   cannot lie but he can conceal information.

  

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.

\noindent
ANOTHER APPROACH  

The equilibrium is either fully separating or has some pooling. 

If it is fully separating, the agent's type is revealed, so it might as well be $m=s$. 

If it had some pooling, then two types  with $s_2>s_1$ would be pooled together and the principal's estimate of $s$ would be the average in the pool. Type $s_2$ would therefore deviate to $m=s_2$ to reveal his type. So the equilibrium must be perfectly separating. 

Where would this logic break down? --- 

   either unpunishable lying or  genuine ignorance.  


\newpage

  \noindent
 {\bf The Revelation Principle: We Can Restrict Attention to Direct Mechanisms }

 Let $w$ be the agent's wage, $q$ be output, $m$ be his message, and $s$ be his type. ALLOW COMMITMENT TO CONTRACTS. 

 \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.}


A  {\bf direct mechanism}--- agents tell the truth in
equilibrium--- can be found  equivalent to any {\bf indirect mechanism} in
which they lie.

  
   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.


 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.


    

\newpage

 
 
  \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}

  
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?

No.  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.

 


\newpage

 
\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.

\newpage
 
\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.  


 
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 $a = (x+10)/2$.

\newpage

\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 $a \in  [3,10]$.
 


 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 $x$   yields
$x=3$.

  

\newpage

    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.


  \newpage

  

 \noindent
{\bf  10.5:  Price Discrimination }

 

 Pigou   was   a contemporary of Keynes at Cambridge who   usefully divided
price discrimination into three types in 1920.

\noindent
{\bf  1 Interbuyer Price Discrimination.  }   

\noindent
{\bf  2 Interquantity Price Discrimination}  or  {\bf  Nonlinear Pricing}. 

\noindent
 {\bf  3 Perfect Price Discrimination.}   



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

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


 \newpage

 \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}

\newpage


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 $q$.      

\newpage

 
 \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}

\newpage

 

\noindent
{\bf Interbuyer Price Discrimination}

 
\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.

\newpage

\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}

\newpage


   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.

 
 \newpage

 
  \noindent
 {\bf   Nonlinear Pricing}

 
 \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.

\newpage

 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.

\newpage

\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}

\newpage

 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.

 Using the single-crossing  property,
assumption (\ref{e7}b),   $u'(q) < v'(q)$, 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.

\newpage

  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).

\newpage

\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}

 

\newpage


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

 


  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 $L$.

\newpage

 The  seller  would do even  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.



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

 


\newpage




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

 


  Our
  profit-maximizing mechanism   is shown  in Figure  5a as $w_u'  = A$  for
$q_u'$ and $w_v^* =   A+B+K+L$ for $q_v^*$. 

   Unenthusiastic buyer:  binding participation constraint,  inefficiently low consumption, because $w_u'  = A = u(q_u')$. 

Valuing buyer: 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 $J$. Efficient consumption: 
  $q_v^*$. 
B binding self-selection constraint,
because he is
  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.


\newpage

\noindent
 {\bf  The Single-Crossing Property}

 

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.

  

\includegraphics[width=150mm]{fig10-06.jpg}
\begin{center} {\bf Figure 6: Marginal versus Average Demand} \end{center}



\newpage

 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}

\newpage

 Another way to write the payoff functions
would have been as $\pi_u(q, money) = money + u(q)$, where $money = wealth-w(q)
$. 

 

 One
comparison is between the curves for which $\pi=10$, which both pass through the
point (0, 10) in $(q, money)$ space.


\includegraphics[width=140mm]{fig10-07.jpg}

 
 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.

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.

  


\newpage

 \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.
 .

  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  wants to  be generous enough  to induce the
firm to accept the franchise in the first place  but   no more generous than
necessary. 

 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}).

 
 

\newpage

 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.  


\begin{center}
{\bf    Procurement I:  Full Information} 

\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 $p(m)$.


\newpage

\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 $f'''>0$. 

 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.

\newpage

  

 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 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.


\newpage

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.
 
\end{LARGE}

\end{document}


