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\begin{LARGE}

\noindent
 February 12, 2003 \\
  Eric Rasmusen, Erasmuse@indiana.edu 

notes on procurement

Varian.

 Reading for Monday. 

 Data assignment this weeekend. 
 
 
\newpage

  

\noindent
{\bf    Procurement I: Perfect Information } 
 
  

\noindent
 {\bf The Order of Play}\\
 0 Nature determines whether the firm has special  problems that add
costs   of $x$, which
has probability $\theta$, or no special problems, which has
probability
$(1-\theta)$.  We will call these ``special'' and ``normal'' firms,
with the understanding that ``special''  problems may be the norm in
engineering projects.   The government and the firm both observe this
move.    \\
 1 The government offers a contract    agreeing to cover the
firm's cost  $c$  of producing a cruise missile  and specifying an
additional
price  $p(c) $  for each cost level that the firm might report.\\
 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 cruise missile  at a cost   of  $c =c_0+ x-e
$ or $c=c_0-e$ which is observed by the government, plus an additional
cost $f(e- c_0)$ that the government does not observe.  The government
reimburses $c$ and pays $p(c)$.

\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{equation} \label{e15.1}
 \pi_{firm} =p - f(e-c_0), 
\end{equation}
 where $f(e-c_0)$, the cost of effort, is increasing and convex, so
$f'>0$ and $f''>0$. Assume, too, for technical convenience,  that $f$
is increasingly convex, so $f'''>0$.\footnote{The argument of $f$ is
normalized to be $(c_0-e)$ rather than just $e$ to avoid clutter in
the algebra later. The assumption that $f'''> 0$  allows the use of
first-order conditions   by making concave the maximand in
(\ref{e15.13}), which is  a difference of two concave functions. See
p. 58 of  Laffont \& Tirole (1993).  } The government's payoff is
   \begin{equation} \label{e15.2}
   \pi_{government}  = B -  (1+\lambda) c  -\lambda p  -  f,
  \end{equation}
  where $B$ is the  benefit of the cruise missile  and $\lambda$ is
the deadweight loss from the taxation needed  for government
spending.\footnote{Hausman \& Poterba (1987) estimate this loss to be
around \$0.30 for each \$1 of tax revenue raised at the margin for the
United States.}

\newpage

  The model differs from  other    principal-agent models in this book
because   the principal  cares about the welfare of  the agent. If the
government cared only about the value of the cruise missile and the
cost to taxpayers, its payoff would be $[B-  (1+\lambda) c -
(1+\lambda) p  ] $. Instead, the payoff function    maximizes social
welfare,  the sum of  the welfares of  the taxpayers and     the firm.
The welfare of the   firm is $(p  - f )$, and summing the two welfares
yields    equation (\ref{e15.2}). Either kind of government payoff
function  may be realistic, depending on  the political balance in the
country being modelled,   and the model will have similar properties
whichever one is used.

\newpage
 

 

   In this first variant of the game,  whether the firm has special
problems is observed by the  government, which can therefore specify a
contract conditioned on the type of the firm. The government pays
prices of $p_N$ to a  normal firm with  the   cost $\underline{c}$,
$p_S$ to a  special firm    with   the  cost $\overline{c}$, and a
 price  of $p=0$ to a firm that does not achieve  its  appropriate
cost level.

  The participation constraints will   be binding for both types of
firms, and to make a firm's  payoff  zero    the government will
provide prices that exactly  cover the firm's disutility of effort.
Since there is no uncertainty  we can invert the cost equation and
write it as $e=   c_0+x -c  $ or $e= c_0 -c  $.   The prices will be
$p_S= f (    x  - \overline{c})$   and $ p_N= f (   - \underline{c} )
$.

Suppose the government knows  the firm has special problems.
Substituting  the subsidy into the government's payoff function,
equation (\ref{e15.2}),  yields
\begin{equation} \label{e15.3}
   \pi_{government}  = B -  (1+\lambda) \overline{c} - \lambda  f(
c_0+
 x  -\overline{c}) - f( (c_0+  x  -  \overline{c})-c_0   ).
  \end{equation}
   Since $f''>0$, the government's payoff function is concave, and
standard optimization techniques  can be used.  The first-order
condition for  $\overline{c}$   is
 \begin{equation} \label{e15.4}
  \frac{ \partial \pi_{government}}{\partial \overline{c}} =   -
(1+\lambda)   +  (1+\lambda)  f'(    x  -\overline{c})    = 0,
  \end{equation}
so
 \begin{equation} \label{e15.5}
         f'(   x  -\overline{c})   =  1.
  \end{equation}


\newpage

$$
         f'(   x  -\overline{c})   =  1.
$$ 
  Since  $f'(  x  -\overline{c})  =  f'([c_0+  x  -\overline{c}] -c_0)
$ and $  c_0 + x  -\overline{c}=e$,  equation (\ref{e15.5}) says that
$\overline{c} $ should be chosen so that $ f'(e-c_0)   =  1$; 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 $e^*\equiv e:\{ f'(e-
c_0)   =  1\}$.

 Exactly the same is true for the normal firm, so $f'(   x  -
\overline{c})   = f'(      -\underline{c})  = 1$ and $ \underline{c} =
\overline{c}-x $.    The cost targets assigned to each firm are
$\overline{c}  = c_0+ x - e^*$ and $\underline{c}  =   c_0 - e^*$.
Since both types must exert the same effort, $e^*$, to achieve their
different targets,   $p_S=    f(e^*- c_0) =  p_N$.   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 $B -  (1+\lambda)\overline{c}   -\lambda f(e^*-
c_0) - f(e^*-c_0) \geq 0$, which   requires that $B -  (1+\lambda)
(c_0+ x - e^*)   -(1+\lambda  ) p^* \geq 0$.




\newpage
\noindent
{\bf     Procurement II: Incomplete Information} 

  In the second variant of the game,  the existence of special
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 special
problems.

 The government could use a pooling contract, simply providing   a
price  of $p^* $ for a cost of $ c=c_0+x - e^*$,   enough to
compensate the    firm with special problems for its effort, with $p=
0$  for any other cost.    Both types would accept this, but   the
normal firm could exert   effort less than $e^*$ and still  get costs
down enough to receive the  price.  (Notice that this is the cheapest
possible pooling contract; any cheaper contract would be rejected by
the firm with special  problems.) Thus, if the government would build
the cruise missile  under full information knowing that the firm has
special problems, it would also build it under incomplete information,
when the firm might have special problems.

  The pooling  contract, however, is not optimal.    Instead, the
government could offer a choice between   the contract $(p^*, c=c_0+x
- e^*$) and a new contract that offers a higher price but requires
reimbursable costs to be lower. By definition of $e^*$,  $f'(c_0+x-
e^*-c_0) =1$, so $f'(c_0 -e^*-c_0)  <1$, which is to say that the
normal firm's marginal disutility of effort when it exerts just enough
effort to get costs down to $c=c_0+x - e^*$ is less than 1. This means
that if the government  can offer  a new contract with slightly lower
$c$ but slightly higher $p$ that will be acceptable to the normal firm
but will have a lower combined  expense of $(p+c)$.  This tell us that
a separating contract exists that is superior to the pooling contract.

\newpage

 Let us  therefore find the optimal contract   with values
($\underline{c}, p_N)$ and ($\overline{c}, p_S$)   and  $p=0$ for
other cost levels.  It will turn out that  the  ($\overline{c}, p_S$)
part of the  optimal separating contract will not be the same as the
pooling contract in the previous paragraph, because  to find the
optimal separating contract it is not enough to find the    optimal
``new contract;''  we  need to find the  optimal {\it pair} of
contracts, and by finding a new contract for the special-problems firm
too, we will be able to reduce the government's expense from the
normal firm's contract.

  A separating contract  must satisfy  participation constraints and
incentive compatibility constraints for each type of   firm.  The
firm with special  problems  exerts effort $e= c_0+x-\overline{c}$,
achieves $c= \overline{c}$,  generating unobserved effort  disutility
$f(   e-c_0) =  f(  x -  \overline{c}  )$     and  participation
constraint
  \begin{equation} \label{e15.6}
  p_S - f(  x -  \overline{c}  ) \geq  0.  
\end{equation}
Similarly,   in equilibrium  the normal firm  exerts effort $e= c_0 -
\underline{c}$, so its participation constraint is
  \begin{equation} \label{e15.7}
    p_N - f(      - \underline{c} ) \geq  0.
 \end{equation}

\noindent
  The incentive compatibility constraint for the firm with special
problems is
 \begin{equation} \label{e15.8}
  p_S - f(  x  -  \overline{c}  ) \geq   p_N - f(x  - \underline{c} ),
  \end{equation}
 and  for the normal firm  it is
  \begin{equation} \label{e15.9}
    p_N - f(       - \underline{c} ) \geq  p_S - f( -
\overline{c}  ).
 \end{equation}

\newpage
   

Since the  normal   firm  can achieve the same cost level as the
special    firm   with less effort,  inequality (\ref{e15.9}) tells
us that    if we  are to have   $\underline{c} < \overline{c}$, as is
necessary for us to have a separating equilibrium,  we need $P_N>P_S$.
The second  half of  inequality (\ref{e15.9}) must be positive, If the
special-firm participation constraint,  inequality (\ref{e15.8}),  is
satisfied, then $p_S - f( - \overline{c}  )>0$. This, in turn implies
that    (\ref{e15.7}) is a strong inequality;   the normal firm's
participation constraint is nonbinding.

The special firm's participation constraint,  (\ref{e15.6}), will be
binding  (and therefore satisfied as an equality), because the
government will  reduce the subsidy  as much as possible in order to
avoid the deadweight loss  of taxation.   The incentive compatibility
constraint for the normal firm  must also  be binding, because if the
pair ($\overline{c}, p_N$) were strictly more attractive for the
normal firm, the government could reduce the subsidy $p_N$. Constraint
(\ref{e15.9}) is  therefore satisfied as an equality.\footnote{ The
same argument does not hold for the  special firm, 
because if $p_S$ were reduced, the participation constraint would be
violated.}     Knowing that   constraints (\ref{e15.6}) and
(\ref{e15.9}) are binding, we can write   from constraint
(\ref{e15.6}),
      \begin{equation} \label{e15.11}
    p_S = f(   x   - \overline{c} )  
 \end{equation}
 and, making use of both (\ref{e15.6}) and (\ref{e15.9}),   
\begin{equation} \label{e15.12}
    p_N = f (      - \underline{c} )   + f (   x   -
\overline{c} ) - f (   -   -\overline{c}  ).
 \end{equation}

\newpage
  
\noindent
 From (\ref{e15.2}), 
the government's maximization problem under incomplete information is 
   \begin{equation} \label{e15.10}
 \stackrel{ Maximize}{\underline{c}  , \overline{c}  , p_N, p_S} \;\;
\; \theta \left[  B -  (1+\lambda) \overline{c}   - \lambda p_S - f(
 x     - \overline{c}  )  \right] + \left[ 1-
\theta \right] \left[  B -  (1+\lambda) \underline{c}   - \lambda p_N-
f(         - \underline{c}  )  \right] .
       \end{equation}
 Substituting  for $p_S$ and $p_N$  from (\ref{e15.11}) and
(\ref{e15.12}) reduces  the problem to
\begin{equation} \label{e15.13}
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c} }  & \theta [  B -
(1+\lambda) \overline{c}   - \lambda (  f(   x  - \overline{c} )
- f(   x   - \overline{c}   )   ] +  [ 1- \theta ] [  B -
(1+\lambda) \underline{c}\\
 & \\
   &    - \lambda f(      - \underline{c} )
- \lambda f(   x   - \overline{c} ) + \lambda f(        -
\overline{c}  )
  - f(        - \underline{c}  )  ] . \\
 \end{array}
       \end{equation}


\newpage


$$
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c} }  & \theta [  B -
(1+\lambda) \overline{c}   - \lambda (  f(   x  - \overline{c} )
- f(   x   - \overline{c}   )   ] +  [ 1- \theta ] [  B -
(1+\lambda) \underline{c}\\
 & \\
   &    - \lambda f(      - \underline{c} )
- \lambda f(   x   - \overline{c} ) + \lambda f(        -
\overline{c}  )
  - f(        - \underline{c}  )  ] . \\
 \end{array}
  $$

\noindent
 (1) The first-order condition with respect to $\underline{c}  $ is 
\begin{equation} \label{e15.14}
  (1- \theta) [   -  (1+\lambda)  + \lambda      f'(       -
\underline{c}  ) + f'(         - \underline{c}  )  ] =0,
       \end{equation}
 which simplifies to 
\begin{equation} \label{e15.15}
     f'(       -\underline{c}   )  =1.      
                \end{equation}
 Thus, as earlier,    $f'_N  =1$. The  normal firm   chooses
the efficient effort level $e^*$ in equilibrium, and   $\underline{c}
$ takes the same value as it did in    Procurement I.   
 Equation
(\ref{e15.12}) can be rewritten as
\begin{equation} \label{e15.15a}
    p_N= p^*   + f (   x   - \overline{c} ) - f(    -
\overline{c}  ).
 \end{equation}

 Because $ f(  x   - \overline{c} ) > f(    -\overline{c})$, equation
(\ref{e15.15a}) shows that $p_N > p^*$. Incomplete information
increases the subsidy to the  normal firm, which earns more than its
reservation  utility in the game with incomplete information. Since
the   firm  with special problems will earn exactly its reservation,
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 special costs. 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.



\newpage

$$
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c} }  & \theta [  B -
(1+\lambda) \overline{c}   - \lambda (  f(   x  - \overline{c} )
- f(   x   - \overline{c}   )   ] +  [ 1- \theta ] [  B -
(1+\lambda) \underline{c}\\
 & \\
   &    - \lambda f(      - \underline{c} )
- \lambda f(   x   - \overline{c} ) + \lambda f(        -
\overline{c}  )
  - f(        - \underline{c}  )  ] . \\
 \end{array}
    $$

\noindent
(2) The first-order condition   with respect to $\overline{c}  $ is
  \begin{equation} \label{e15.16}
\begin{array}{l}
 \theta \left[    -  (1+\lambda)  + \lambda f'(  x  -
\overline{c} )     + f'(  x     - \overline{c})     \right] + 
  \left[ 1- 
\theta \right]   [   \lambda f'(   x   - \overline{c} ) 
  +f'(        - \overline{c}  )  ] =0. \\
 \end{array} 
       \end{equation}
 This can be rewritten as
   \begin{equation} \label{e15.17}
f'(  x    - \overline{c} )=1 - \left(\frac{ 1-\theta}{\theta(1+
\lambda)}
\right)       \left[ \lambda
f'(      x - \overline{c} ) + f'(      - \overline{c} )
\right].
        \end{equation}
  Since the right-hand-side of equation (\ref{e15.17}) is less than
one,  the special firm   has a lower level of $f'$ than
the normal firm, and must be exerting effort less than $e^*$  since
$f''>0$. Perhaps this explains the expression ``good enough for
government work''.  Also since the  special firm's participation
constraint,
(\ref{e15.6}),  is satisfied as an equality, it must also be true that
$p_S < p^*$.  The special  firm's price is lower than
under full information, although since its effort is also lower, its
payoff stays the same.

\newpage

  We must also see that the incentive compatibility constraint for the
firm with special problems is satisfied as a weak inequality; the firm
with special problems 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{e15.8})   equal to zero
because the participation constraint is binding for  the firm with
special problems, substituting in for $p_N$ from equation
(\ref{e15.12})  and rearranging  yields
  \begin{equation} \label{e15.18}
    f(    x  - \underline{c} )-  f(     - \underline{c} ) 
\geq       f(   x  - \overline{c}
) -f(       - \overline{c} ).
 \end{equation}
 This is true, and true as a strict inequality, because $f''>0$ and
the arguments of $f$ on the left-hand-side of equation (\ref{e15.18})
take larger values than on the right-hand side, as shown in Figure
10.7.


 
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\newpage

 To summarize,  the government's optimal contract will 

(1)induce the
normal  firm to  exert the first-best efficient effort level 
and achieve  the first-best cost level,

(2)  will yield that firm a
positive profit. 

 (3)   will  induce the   firm with special
costs  to exert something less than the first-best effort level

(4)  
result in a special-firm 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
cruise 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
with  more  then  under complete information to prevent it from
mimicking a   firm with special problems  by choosing an inefficiently
low effort. The   firm with special problems, 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
  
\begin{center}
{\bf    Procurement III:  Moral Hazard with Hidden Information  } 
\end{center}
\noindent
 {\bf The Order of Play}\\
1 The government offers a contract    agreeing to cover the
firm's cost  $c$  of producing a cruise missile  and specifying an
additional
price  $p(c) $  for each cost level that the firm might report.\\
 2 The firm accepts or rejects the contract.\\
 3 Nature determines whether the firm has special  problems that add
costs   of $x$, which
has probability $\theta$, or no special problems, which has
probability
$(1-\theta)$.  We will call these ``special'' and ``normal'' firms,
with the understanding that ``special''  problems may be the norm in
engineering projects.   The government and the firm both observe this
move.    \\
 4 If the firm accepts, it chooses    effort level $e$, unobserved
by the government. \\
 5 The firm finishes the cruise missile  at a cost   of  $c =c_0+ x-e
$ or $c=c_0-e$ which is observed by the government, plus an additional
cost $f(e- c_0)$ that the government does not observe.  The government
reimburses $c$ and pays $p(c)$.


 \newpage

  The  contract  must satisfy one overall  participation constraint
and
and two incentive compatibility constraints, one for each type of
firm.  The
participation constraint is
  \begin{equation} \label{e15.6z}
 \theta[ p_S - f(  x -  \overline{c}  )  ] + [1-\theta][
    p_N - f(      - \underline{c} )] \geq  0.
 \end{equation}

\noindent
  The incentive compatibility constraints are the same as before:  for
the  special firm,
 \begin{equation} \label{e15.8z}
  p_S - f(  x  -  \overline{c}  ) \geq   p_N - f(   - x    -
\underline{c} ),
  \end{equation}
 and  for the normal firm, 
  \begin{equation} \label{e15.9z}
    p_N - f(        - \underline{c} ) \geq  p_S - f(     -
\overline{c}  ).
 \end{equation}

\newpage
   
  
  As before, constraint (\ref{e15.6z})
will be binding  (and therefore satisfied as an equality), because
the government will  reduce the  price  as much as possible in order
to avoid the deadweight loss  of taxation.   The  normal firm's
incentive
compatibility constraint    must also  be binding, because
if the   pair ($\overline{c}, p_N$) were strictly more attractive for
the normal firm, the government could reduce the  price $p_N$.
Constraint
(\ref{e15.9z}) is  therefore satisfied as an equality.\footnote{ The
same argument does not hold for the firm with special costs, because
if $p_S$ were
reduced, the participation constraint would be violated.xxx check
this}     Knowing
that   constraints (\ref{e15.6z}) and   (\ref{e15.9z}) are
binding, we can write   from constraint (\ref{e15.6z}),
      \begin{equation} \label{e15.11z}
    p_S =  f(  x -  \overline{c}  )   - \frac{ [1-\theta][
    p_N - f(      - \underline{c} )] }{\theta}. 
 \end{equation}
 Substituting from (\ref{e15.11z})  for $p_S$ into (\ref{e15.9z}),  we
get
\begin{equation} \label{e15.12z}
    p_N -f (      - \underline{c} )  =   f(  x -
\overline{c}  )   - \frac{ [1-\theta][
    p_N - f(      - \underline{c} )] }{\theta}  - f (   -    
\overline{c}  ).
 \end{equation}
 This can be solved for $p_N$ to yield
\begin{equation} \label{e10.80}
    p_N  =  \theta [f(x-\overline{c}) - f( -\overline{c})] + f(-
\underline{c} ),
 \end{equation}
 which when substituted into  (\ref{e15.11z}) yields
  \begin{equation} \label{e10.81}
    p_S =    [1-\theta] [f(x-\overline{c}) - f( -\overline{c})]. 
 \end{equation}

\newpage

\noindent
 From (\ref{e15.2}), 
the government's maximization problem under incomplete information is 
   \begin{equation} \label{e15.10z}
 \stackrel{ Maximize}{\underline{c}  , \overline{c}  , p_N, p_S} \;\;
\; \theta \left[  B -  (1+\lambda) \overline{c}   - \lambda p_S - f(
 x     - \overline{c}  )  \right] + \left[ 1-
\theta \right] \left[  B -  (1+\lambda) \underline{c}   - \lambda p_N-
f(         - \underline{c}  )  \right] .
       \end{equation}
 Substituting  for $p_N$ and $p_S$  from (\ref{e10.80}) and
(\ref{e10.81}) reduces  the problem to
\begin{equation} \label{e15.13z}
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c}  , p_N, p_S} &
\theta \left\{  B -  (1+\lambda) \overline{c}   - \lambda   [1-\theta]
[f(x-\overline{c}) - f( -\overline{c})] - f(
 x     - \overline{c}  )  \right\} \\
 & + \left[ 1-
\theta \right] \left[  B -  (1+\lambda) \underline{c}   - \lambda \{
\theta [f(x-\overline{c}) - f( -\overline{c})] + f(- \underline{c} )
\}-
f(         - \underline{c}  )  \right] .\\
 & \\
 \end{array}
       \end{equation}

\newpage


$$
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c}  , p_N, p_S} &
\theta \left\{  B -  (1+\lambda) \overline{c}   - \lambda   [1-\theta]
[f(x-\overline{c}) - f( -\overline{c})] - f(
 x     - \overline{c}  )  \right\} \\
 & + \left[ 1-
\theta \right] \left[  B -  (1+\lambda) \underline{c}   - \lambda \{
\theta [f(x-\overline{c}) - f( -\overline{c})] + f(- \underline{c} )
\}-
f(         - \underline{c}  )  \right] .\\
 & \\
 \end{array}
$$

\noindent
 (1)  The first-order condition with respect to $\underline{c}  $ is 
\begin{equation} \label{e15.14z}
  (1- \theta) [   -  (1+\lambda)  + \lambda   f'(       -
\underline{c}  ) + f'(         - \underline{c}  )  ] =0,
       \end{equation}
just as under adverse selection,  which simplifies to 
\begin{equation} \label{e15.15z}
     f'(       -\underline{c}   )  =1.      
                \end{equation}
 Thus, as earlier,    $f'_N =1$. The  normal firm   chooses
the efficient effort level $e^*$ in equilibrium, and   $\underline{c}
$ takes the same value as it did in   Procurement I.  Equation
(\ref{e15.12z}) can be rewritten as
\begin{equation} \label{e15.15az}
    p_N= p^*   + f (   x   - \overline{c} ) - f(    - \overline{c}  ).
 \end{equation}

 Because $ f(  x   - \overline{c} ) > f(    -\overline{c})$,
equation (\ref{e15.15az}) shows that $p_N > p^*$.  
The normal firm 
earns more than its reservation  utility, even under complete
information. The special firm  must therefore earn less than its
reservation utility, so that the overall participation constraint will
be satisfied as an equality.

 \newpage

$$
 \begin{array}{ll}
 \stackrel{ Maximize}{\underline{c}  , \overline{c}  , p_N, p_S} &
\theta \left\{  B -  (1+\lambda) \overline{c}   - \lambda   [1-\theta]
[f(x-\overline{c}) - f( -\overline{c})] - f(
 x     - \overline{c}  )  \right\} \\
 & + \left[ 1-
\theta \right] \left[  B -  (1+\lambda) \underline{c}   - \lambda \{
\theta [f(x-\overline{c}) - f( -\overline{c})] + f(- \underline{c} )
\}-
f(         - \underline{c}  )  \right] .\\
 & \\
 \end{array}
$$

\noindent
 (2) 
 The first-order condition with respect to  $\overline{c}  $ is
  \begin{equation} \label{e15.16z}
\begin{array}{l}
 \theta \left\{    -  (1+\lambda)  - \lambda (1-\theta)[-f'(  x  -
\overline{c} )     + f'(      - \overline{c})]   + f'(  x  -
\overline{c} )  \right\} +  \lambda \left[ 1- \theta \right]   [
f'(
x   - \overline{c} ) -f'(        - \overline{c}  )  ] =0. \\
 \end{array} 
       \end{equation}
 This can be rewritten as
   \begin{equation} \label{e15.17z}
xcxcvxcvcx f'(    x  - \overline{c} )=1 - sdfsfsdfsdfdsf
        \end{equation}
   

The ultimate effect: the participation constraint is binding, and
total cost of p and c is less for the government than under incomplete
information, so the deadweight loss of taxation is less too. The
general features of the contract are similar, but the special firm now
earns a loss, rather than breaking even.

 

\newpage
 
Additional ways to
alter the Procurement Game.

What if the firm discovers its costs only after accepting the
contract? (we did this) 

  What if two firms bid against each other for the contract?

What if the firm can bribe the government?  

What if the firm and the
government bargain over the gains from the project  instead of the
government being able to make a take-it-or-leave-it contract offer?


What if the game is repeated, so the government can use the
information it acquires in the second period? 

If it is repeated, can
the    government   commit to long-term contracts? 

Can it commit not
to  renegotiate? 

See  Spulber (1989)  and Laffont \& Tirole (1993) if
these   questions   interest you.



\end{document}