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         \begin{document}

\begin{LARGE}


   
\parindent 24pt

 \parskip 10pt

 
  

\noindent  22 November 2005.  19 October 2006  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}

 For two 75 minute sessions. Price discrimination is in a separate section. 

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

   Information is complete in moral hazard games, but  in {\bf moral hazard with
hidden knowledge}, also called {\bf post-contractual adverse selection},     the
agent, but not the principal,  observes a move of Nature  after the game begins,
but before he takes his action.   

 Information is symmetric at the time of
contracting-- thus the   ``moral hazard''---  but becomes asymmetric later--
thus the ``hidden knowledge''.  

  From the principal's point of view, agents are
identical at the beginning of the game  but   develop  private types midway
through. 

 
 There is just ONE
participation constraint even if there are eventually $n$ possible types of
agents.

 There are still N incentive compatibility constraints. 
 
\newpage


\begin{center}
 {\bf  Production Game VIII:  Mechanism Design }
 \end{center}
 

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

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

 \bigskip

  
  The optimal efforts and
outputs  s  are $e_g=1.5$ and  $q_g =4.5$   or  $e_b=0.5$   and $q_b =0.5$. The
principal  solves 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}
 using   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.

\newpage

 The participation constraints of Production Game VII now merge.
\begin{equation} \label{a4}
{\displaystyle  0.5 \pi_{agent} (q_g, w_g|good) +0.5 \pi_{agent} (q_b, w_b|bad)
  \geq 0.}
 \end{equation}
\begin{equation} \label{a4}
{\displaystyle   0.5 \left(  w_g -      \left( \frac{q_g}{3}   \right)^2  \right) + 0.5 \left(
w_b -      q_b^2    \right) \geq 0.}
 \end{equation}


The self-selection constraints are the same as in Production Game VII.
  In the good state, the agent must choose the good-state contract,so
 \begin{equation} \label{a2}
{\displaystyle  \pi_{agent} (q_g, w_g|good)   \geq  \pi_{agent} (q_b, w_b|good)  }
 \end{equation}
 \begin{equation} \label{a2a}
{\displaystyle   w_g -     \left( \frac{q_g}{3}
\right)^2  \geq   w_b -      \left( \frac{q_b}{3}
\right)^2  }
 \end{equation}
  and   in the bad state he must choose the bad-state contract, so
 \begin{equation} \label{a3}
\pi_{agent} (q_b, w_b|bad) =    w_b -    q_b^2  \geq \pi_{agent} (q_g, w_g|bad)
=    w_g -    q_g^2.
 \end{equation}



  The single participation constraint is binding.

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

As in Production Game VII, in the good state the effort is at the first-best
level while in the bad state it is less. 

Unlike in Production Game VII   the
agent does not earn informational rents, because at the time of contracting he
has no private information.

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


 
\newpage


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

  

  If the courts cannot observe the
state, a contract conditioning the wage on the state   is unenforceable, no
better than having no contract at all. We say that the variable $s$ is {\bf
nonverifiable} if contracts based on it cannot be enforced.

 Even if the courts will not enforce a contract based
on a variable, if both the principal and the agent observe it they should be
able to come up with a more efficient contract than if just the agent observes
it. 


\noindent
  MASKIN: 

\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. \\

\noindent
  (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.   

 

   
 Renegotiation  is a problem as in the  Holmstrom teams model. 

\newpage
  

 {\it Wise Guys}   Nicholas Pileggi quotes low-
level gangster Henry Hill  saying :


\begin{quotation}
 
``For instance, say I've got a  fifty-thousand-dollar hijack load, and when  I
make my delivery, instead of getting paid, I get stuck up.
What am I supposed to do? Go to the cops? Not likely. Shoot it out? I'm a
hijacker, not a cowboy. No.  The only way to guarantee that I'm not going ripped
off by anybody is to be established with a  member, like Paulie. Somebody who is
a made man. A member of a crime family. A soldier. Then ...    that's the end of
the ball game. Goodbye. They're  dead...  Of course,  problems can arise when
the guys sticking you up are associated with wiseguys too. Then there has to be
a  sit-down between your wiseguys and their wiseguys.  What usually happens then
is that the wiseguys divide whatever you stole for their own pocket, send you
and the guy who robbed you home with nothing. And if you complain, you're
dead.''
 
\end{quotation}
   The low-level gangsters have a strong incentive to report the same story, or
the higher-ups take away the property under dispute.

 This may sound familiar to
parents too--- ``If we can't resolve this, the toy is going in the closet for a
whole week.''  

It is perhaps even the wisdom of Solomon.

 
 
\newpage

  \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}, in which the agents tell the truth in
equilibrium, can be found that is 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.

 
 

    The Revelation Principle does depend heavily on an implicit assumption we
have made: the principal cannot breach his contract. 

 



  
\newpage

  

\begin{center}
 {\bf   10.4 The Groves Mechanism:  The  Streetlight Game}
 \end{center}
  

\noindent
 {\bf The Order of Play}\\
 0. Nature chooses the value $v_i$ that each of 5 householders $i$ places on having a
streetlight installed, using distribution $f_i(v_i)$.   Only Householder $i$
observes $v_i$.   \\
1. The mayor announces a mechanism, $M$, which requires a householder who
reports $m$ to pay $w(m)$ if the streetlight is installed   and installs the
streetlight if $g(m_1, ..., m_5) \geq 0$. \\
 2. Householder $i$ reports value $m_i$ simultaneously with all other
householders.\\
 3. If $g(m_1, ..., m_5) \geq 0$,   the streetlight is built and
householder $i$ pays $w(m_i)$.

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




\newpage

  Each of 5 citizens has value $v_i$ from a streetlight, known only to himself. Each sends a message $m_i$ of his value to the mayor.  The streetlight costs 100.  What mechanism should the mayor use? He will  set tax $w(m)$ for message $m$ if he installs the streetlight. 
 
 
$$
M_1: \;\;\; \left(  w = 20, Build \; iff \; \sum_{i=1}^5 m_i \geq 100
\right);
$$
 


$$
 M_2: \;\;\; \left( w(m_i) = Max\{ m_i, 0\},  Build \; iff \;\sum_{j=1}^5 m_j
\geq 100 \right).
$$

$$
 M_3: \;\;\; \left( w(m_i) =  100- \sum_{j\neq i} m_j ,  Build \; iff \; \sum_{j=
1}^5 m_j \geq 100 \right).
$$

  Suppose the true values are  $(10, 30, 30, 30, 80)$. What message will be sent under each mechanism? 

M1: meaningless ones

M2: Nash equilibrium ones,e. g. (0, 25, 25,25,25)

M3: dominant strategy equilibrium ones, the truth

\newpage

$$
 M_3: \;\;\; \left( w(m_i) =  100- \sum_{j\neq i} m_j ,  Build \; iff \; \sum_{j=
1}^5 m_j \geq 100 \right).
$$

w1= 100- 170 = -70\\
w2 = 100-150 =-50\\
w3 = -50 \\
w4= -50\\
 w5 = 100- 100 =0 \\

 The sum of valuations is 180, so the streetlight is installed.  

 Each player is indifferent about his message in equilibrium, except that he does not want to reduce it so much that the streetlight would not be installed (it would have to be negative for that!)

What if player 2 thinks  the other players will overreport  a total of 200 instead of their true sum of 150? He's still happy for the project to go through (he gets an even bigger tax) so he won't underreport. 

What if player 2 thinks  the other players will UNDERrreport  a total of   60 instead of their true sum of 150?  He is sad. He could get the project to go through by reporting 40 instead of his true value of 30. But then his tax would be 100-60=40, and that's a little too high for him to want to lie. Instead, he'll tell the truth and kill the project.    

What if player 2 thinks  the other players will UNDERrreport  a total of   69 instead of their true sum of 150?  He is sad. He could get the project to go through by reporting 31 instead of his true value of 30. But then his tax would be 100-69=31, and that's a little too high for him to want to lie. Instead, he'll tell the truth and kill the project.
 


\newpage

A {\bf dominant-strategy mechanism},  $M_3$.
 \begin{equation}  \label{e6a}
 M_3: \;\;\; \left( w(m_i) =  100-\sum_{j\neq i} m_j ,  Build \; iff \; \sum_{j=
1}^5 m_j \geq 100 \right).
 \end{equation}
 
   Suppose   Smith's valuation is 40 and the sum
of the valuations is 110, so the project is indeed efficient. If the other
players report  their truthful sum of 70, Smith's payoff from truthful reporting
is his valuation of 40 minus his tax of 30. Reporting more would not change his
payoff, while reporting less than 30 would reduce it to 0.

What if  the other players lie?  If they underreported,
announcing 50 instead of the truthful 70, then Smith could make up the
difference by overreporting 60, but his payoff would be $-10$ ($=40 + 50 -100 $)
so he would do better to report the truthful 40, killing the project and leaving
himself with a payoff of 0.

 If the other players overreported, announcing 80
instead of the truthful 70, then Smith would benefit  if the project went
through, and he should report at least 20.

 Whether he reports 20, 21, 40, or
400, the streetlight is built and he pays a tax of 20 under mechanism $M_3$,
leaving him with  payoff of 20 (= 40-20).  

 In particular, he is willing to
report exactly 40,  so it is a weakly best response to the other players' lies.

 The problem with a dominant-strategy mechanisms like $M_3$ is that it is not
budget balancing. 

 This is not so bad if the budget had a surplus, as required
in our game rules above, but it turns out to have a deficit except in special
cases where it is perfectly balanced (e.g., $m_i=20$ for all 5 householders).

 

\newpage

\noindent
{\bf Unravelling:  Voluntary Statements,  Lying   Prohibited}

 \noindent
   Suppose 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  $m \in \{bad, silent\}$.
 

 
Suppose     $s$ is uniform on  $[0, 10]$ and the agent's payoff is increasing in the
principal's estimate of $s$. 

If $s = 2$, he can send
the uninformative message $m \geq 0$ (equivalent to no message), or the message
$m \geq 1$, or $m=2$, but not the lie that  $m \geq 4.36$.

 When $s=2$ the agent might as well send a message that  is the exact truth:
``$m=2$.''  

If he were to choose the message ``$m \geq 1$''  instead, the
principal's first thought might be to estimate $s$ as the average value in the
interval $[1,10]$, which is 5.5.

 But the principal would realize that no agent
with a value of $s$ greater than 5.5 would want to send the message ``$m \geq
1$'' if  5.5 was  the resulting deduction. 



  


 
 
\newpage

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

 Even if the informed and uninformed players have different incentives and
can't commit to a mechanism,  if their incentives are close enough,   truthful
if imperfect messages can be sent in equilibrium. 

 Let us call the informed
player ``the sender'' (AGENT)  and the uninformed player ''the receiver'' (PRINCIPAL).


   Crawford \& Sobel (1982)  ``Strategic Information Transmission.''   


  

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

  Perfect truthtelling cannot happen in equilibrium. 

 
One
 equilibrium is the pooling equilibrium in which the sender's message is ignored
and the receiver chooses $a= Et = 5$.   


\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 Sender has reduced his message space to two messages,  LOW (=0)
and  HIGH (=10). 

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

 The receiver's equilibrium response determines the sender's payoffs from  his
two messages.    The payoffs between which  the sender 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}



\newpage

  The payoffs between which  the sender chooses are:
 \begin{equation} \label{e201a}
\begin{array}{ll}
\pi_{sender, m=0} & {\displaystyle    = \alpha -  \left(\left[t+1 \right]-
\frac{x}{2}\right)^2}  \\
  & \\
\pi_{sender, m=10}& = {\displaystyle \alpha -  \left(\frac {10+x}{2} -\left[t+1
\right]\right)^2}
\end{array}
\end{equation}


    There exists a value $x$ such that if  $t=x$, the sender is indifferent
between $m=0$ and $m=10$, but if $t$ is lower he prefers $m=0$ and if $t$ is
higher he prefers $m=10$. 

To find $x$, equate the two payoffs in expression
(\ref{e201})  and simplify to obtain
 \begin{equation} \label{e202}
 {\displaystyle \left[t+1 \right]- \frac{x}{2}  
  =  \frac {10+x}{2} -\left[t+1 \right].  }
\end{equation}
 We   set $t=x$  at the point of indifference, and solving for $x$ then yields
$x=3$.

  Thus, the divergence in preferences of the sender and receiver coarsens the
message space, in effect. 


\newpage
 
\noindent
{\bf Comments on the Crawford and Sobel}
 
If instead of wanting $(t+1)$ to be the action, the preferences of sender and
receiver diverged more---

 say, to  $(t+8)$--- then there would only be the
uninformative pooling equilibrium. 

If they divereged less--- say, to $(t+0.1)
$--- then there would exist  other partially pooling equilibria that had more
than just two effective messages  and would  distinguish between three or more
intervals instead of between just two.

    In the Crawford-Sobel   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  signalling.

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 direct  penalty. 


 

 
\newpage

 \noindent
 {\bf   10.6 Rate-of-Return Regulation and Government Procurement}   (Baron \& Myerson's 1982   ``Regulating  a Monopolist with
Unknown Costs.'' )

 

  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  cannot just cover the firm's costs, because the firm
would always claim high costs and exert low effort. 

 Instead, the government
might auction off the right to provide the service, 

might allow the firm a
maximum price (a {\bf price cap}),

 or might agree to compensate the firm to
varying degrees for different levels of cost ({\bf rate-of- return regulation}).

Regulatory franchises are like  government procurement.  If the government wants to purchase a   missile, it also has
the problem of how much to offer the firm. Flat price, or cost-plus? 

  




 
\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. It will be  a benchmark
for our later versions.

\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 
 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

 

   Assume for the moment that $B $ is large enough that the government
definitely wishes to build the   missile.


 

FIRST BEST: The government pays   $p_N$ to a  normal firm with  the   cost $c_N$,
$p_X$ to an expensive firm    with   the  cost $c_X$, and   $p=0$ to a firm that
does not achieve  its  appropriate cost level. 

  The government thus maximizes
its payoff, equation (\ref{e39}), by choice of $p_X, p_N, c_X,$ and $c_N$,
subject to participation and incentive compatibility constraints.

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

\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 $p^*$.

  

\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.

 If the government offered the two contracts of Procurement I, both types of
firm would accept the expensive-cost contract, which has a
 price  of $p^* $ for a cost of $ c=\bar{c}+x - e^*$,   enough to compensate the
firm with expensive problems for its effort,  and  $p= 0$  for any other cost.

That is the cheapest pooling contract, since any contract that paid less would
violate the expensive-cost firm's participation constraint. 

It
is inefficient, though, because the normal firm can reduce costs to $ c=\bar{c}
+x - e^*$ by exerting effort lower than $e^*$.

The government would still be willing to build the missile, since the social
cost of having the normal firm build the missile inefficiently is still lower
than of having the expensive-cost firm build it efficiently. 

 But it will turn
out that   separating contracts will  yield higher welfare than the pooling
contract.

\newpage

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

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

 (2)   a new
contract that offers a slightly  higher price  $p$   but requires reimbursable
costs  $c$ to be  slightly lower. By definition of $e^*$ in first-order
condition (\ref{e44}), 


 $f'(e^*-\bar{c}) =1$, so $f'( e'-\bar{c})  <1$ for the
effort the normal firm exerts in the old pooling contract. If the normal firm
increased its effort from $e'$ by some small amount $\Delta e$, costs would fall
by $(1) \Delta e$    but the firm would only have to be paid  $f'( e'-\bar{c})
\Delta e$ more to compensate for its extra disutility. 

Thus, there is  a new
contract that would draw the normal firm away from the old  pooling contract and
be preferred by the government.

\newpage

  We
will   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 constraints are  
\begin {equation}
 \pi_X(X) =  p_X - f(  x  -  c_X  )  \geq  \pi_X(N)=   p_N - f(x  - c_N ),
 \end{equation}
\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.

\newpage

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}).


  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}
 \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}

\newpage

\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.  

\newpage

\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,  $e^*$.   

   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.
 
\newpage


 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.

    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.

 

 

\end{LARGE}

\end{document}




