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\begin{LARGE}

   \parindent 24pt \parskip 10pt

 

\noindent    30 January 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.

 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,  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).

\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}

\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.

\newpage

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

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


  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.


\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}


 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.

 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.

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.

\newpage

  \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 $e^*$.

   But    separating contracts will  yield higher welfare than the pooling
contract.

\newpage

   A separating contract menu superior to the pooling
contract would be a choice of

 (1)
      the  old pooling contract $(p^*, c=\bar{c}+x - e^*$), OR

 (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.

\newpage


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.

\newpage

 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.

 
 
 
\end{LARGE}

\end{document}


