Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/auctioningincentOOIaff
HB31
.M415
\-W6>H&
working
department
of economics
AUCTIONING INCENTIVE CONTRACTS
By
Jean-Jacgues LAFFONT
and
Jean TIROLE
W.P.
#403
November 1985
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
AUCTIONING INCENTIVE CONTRACTS
By
Jean-Jacques LAFFONT
and
Jean TIROLE
W.P.
#403
November 1985
AUCTIONING INCENTIVE CONTRACTS
by
Jean-Jacques LAFFONT*
and
Jean TIROLE**
October 1985
Support from Commissariat du Plan, Direction de
la Prevision, the National Science Foundation,
is gratefully acknowledged.
GREMAQ, Universite des Sciences Sociales de Toulouse,
France
M.I.T.,
Cambridge, MA, U.S.
Abstract
This paper draws
a
remarkably simple bridge between auction
theory and incentive theory.
It considers the
auctioning of an
indivisible project between several firms. The firms have private information about their future cost at the bidding stage,
and
the selected firm ex-post invests in cost reduction. We show
that
:
The optimal auction can be implemented by a dominant strategy
auction which uses both information about the first bid and the
1
second bid.
2)
The winner faces
3)
The fixed transfer to the winner decreases with his announ-
a
(linear]
incentive contract.
ced expected cost and increases with the second lowest announced
expected cost.
4
The share of cost overruns born by the winner decreases
with the winner's announced expected cost.
-1-
1
Introduction
-
In Laffont and Tirole
between
a
Government and
a
(1984)
we studied the optimal contract
firm realizing
a
project for the Govern-
both the firm and the Government are risk neu-
ment.
In our model
tral,
but the Government faces a double asymmetry of information,
not knowing the exact productivity parameter of the firm and not
observing its effort level. With socially costly transfers the optimal strategy for an utilitarian Government is to offer
so-called incentive contracts each composed of
a
a
menu of
fixed payment,
function of the announced cost, and of a linear sharing of overruns,
i.e.
of the difference between ex-post observed costs and announ-
ced costs. The optimal contract trades off the truthfull eli-
citation of information about productivity (which would lead
to
a
cost plus contract) and the ex-post inducement of an appro-
priate level of effort (which would lead to
tract)
'
a
fixed price con-
.'
When several firms are possible candidates to realize the
project,
it has been argued by Demsetz
(1968)
that competitive
bids should be elicited. In this paper we extend our analysis to
the case of multiple firms. However, the model is meant to for-
malize
a
one shot project and we leave aside the questions rai-
sed by the auctioning of an activity repeated over time
(see
Williamson (1976)).
In section
2
project which has
we set up the model
a
;
n
firms can realize
a
fixed large value for the Government. Under
perfect information, the Government would select the most efficient firm and impose an optimal level of effort independently
of the efficiency level of the firm. Section 3 characterizes the
optimal Bayesian auction for an utilitarian Government. Under
some assumptions which exclude bunching and random auctions, the
best auction awards the project to the firm which announces the
smallest expected cost. 'The type of contract which associates
the Government with the selected firm is then similar to the one
•2-
derived
in
the case of
a
single firm.
It can be written as the
sum of a fixed payment function of the announced cost and of
linear sharing of overruns
sharing rule is
a
;
a
the coefficient characterizing this
function of the announced cost. This dichotomy
property is the main result of this paper which can the rely on
Laffont and Tirole (1984) for a detailed study of the optimal
r
.
contract. Section
4
constructs
a
dominant strategy-auction which
implements the optimum and section
5
concludes.
-3-
2
The model
-
We assume that
firm
i,
1,...,n,
=
i
for a cost
C
e
able to realize the
is
Each
indivisible project
:
(1)
where
firms can participate in the auction.
n
1
1
=
-
B
e
1
manager i's ex-post effort level and
is
firm i's
is
B
efficiency parameter.
The efficiency parameters (B
the same distribution with
F(
f
.
)
on the interval
and
which is bounded below by
Moreover, F(.)
are drawn independently from
cumulative distribution function
a
[B_,B]
)
a
a
dif f erentiable density function
strictly positive number on
is common knowledge and satisfies the
[_B,B].
monotone
hazard rate property.
Assumption
A1
F
1
facilitates the analysis by preventing bunching in the
2)
one firm problem
Manager
i,
(2)
where
t
t
decreasing
is non
j
:
i
=
1
-
is the net
1
,
.
^(e
.
.
,
n
,
has the utility function
1
)
(i.e.
in addition to cost)
that he receives from the Government and
of effort,
ty
Assumption
2
>
'
0,
i|>"
'
i
ty'
'
>
:
0.
\|/(ei)
monetary transfer
is his disutility
Moreover, we postulate
:
>
A2 facilitates the analysis by ensuring that in the one
firm problem there is no need for using stochastic mechanisms
(and similarly in the n-firm case,
see Appendix 1).
-4-
Let
be the social utility of the project.
S
Under perfect
information the Government selects the firm with the lowest parameter
say firm
6,
i,
and makes
transfer only to that firm.
a
The gross payment made by the Covernment to firm
The social cost of one unit of money is 1+A,
>
A
is
i
,
+ C
t
.
so that the
social net utility of the project for an utilitarian Government
is
:
S-(
(3)
+ A) (t
1
S-AU
X
1
+C 1 )+t 1 -*(e 1
-( 1+A) (C
1
+*(e
=
)
1
)
denotes firm i's utility level.
where U
Procurement in which the principal is a private firm would lead
to
different objective function of the type, profits minus trans-
a
None of our results would be affected
fers.
;
the important feature
is that the principal dislikes transfers.
Each firm has outside opportunities (its individual ratio-
nality level) normalized to zero. Under perfect information, the
optimal level of effort of the firm selected should be determined by
i>
'
(
e
* i
)
=
1
(marginal disutility of effort equal marginal
cost savings), and the net transfer to that firm should be
However, the Government observes neither (8
)
nor
(e
* i
\p(e
).
).
Ex-
post he observes the realized cost of the firm which has been
chosen to carry out the project. To select a firm and regulate
it,
-
3
he organizes an auction as explained below.
The optimal Bavesian auction
3
parameters
Firms bid simultaneously
etticiency parametei
usiy by
Dy announcing efficiency
n
1
= 8. Let x (B) be the probabil
,...,!")
probability that firm i is selec...,B
t=
1
B
)
ted to carry out the project. We must have
V
(4)
x^B)
<
1
for any
8
i=1
(5)
x
1
(8)
>
i
=
1
,
.
.
.
,n
for any
6
-5-
Since both the Government and managers are risk neutral there
is no need to consider random transfers. Let t (6) be manager i's
expected transfer as a function of the announced bids. Ex-ante
manager i's expected utility is
E
(6)
^
where
-
(8)
x
1
1
<B)^[e
)}
-i
,6
(B,..-,6
=
8
1
{t
:
,...,6)
The ex-post observability of cost enables us to rewrite
6
)
as
:
E
(7)
where C
is
(S)
{t
x
-
(B)
x
x
1
(B)^(6 -C (B)
)}
the cost level that the Government requires the
firm to reach given announcements
We look for mechanisms
(
x
B
1
(
B
)
,
.
C
1
(
B
,
)
t
1
B
(
)
)
which induce
truthtelling Bayesian Nash equilibrium. From appendix
sary condition for truthtelling is
2,
a
a
neces-
:
(8)
_i_ e .t^B)
_1
1
3B
=
——
3B
B
1
E
[x
1
(B)4'(B
1
1
-C (B)
)
]
.
,
-i
B
dF a.e
for
i
=
1
,
.
.
n
(Differentiability of these expectations almost everywhere
results from their monotonicity which in turn stems from incentive
compatibility)
If moreover the following condition
sufficient
(9)
is
(9)
satisfied
(8)
is
:
^-r
>
1
~
3B
-^—
•
SB
1
E
x
X
(B)
<
g-i
first ignore the second order conditions
(9) and check later that they are indeed satisfied at the optimum.
In the sequel we
-6-
Let U
B
(
be manager i's expected utility level when tel-
)
ling the truth
^(B 1
CO)
)
=
E
[t
B"
From
1
1
IT(B
(11)
)
1
1
(B
)
)
]
1
and (8) we have
(7)
1
(B)-x (BI^(B -C
= - E
From (10) we see that
:
[x
x
X
(B)*' (B -C (B)
]
non increasing in
is
U
)
so that
B
manager i's individual rationality level is satisfied if it is
satisfied at
=6.
B
decreasing in
U
it will be tight at B,
,
U
(12)
X
As the Government's objective function is
(B)
=0
Suppose for simplicity that
with type
B
=
i
n
1
so large that even a firm
is
S
i.e.
is worth selecting.
The maximand of an utilitarian Government is
n
(£
(13)
x
)
S-M+A) J
t
X
xi
)
£"x
7
i=1
(t^xV)
£
=
i=l.
n
X
U -( l+X)
1=1
i= l
X
-(l+X) J x C +
n
Y,
n
X
i=l
i=1
n
(
n
n
1
:
x
Y
1
1
(C +
1
)
iJ;
i=l
The Government's optimization problem under incomplete infor-
mation is then
:
n
n
Max
(14)
(x
1
(.),C
E
1
(.),U
1
(.
6
j)
V
(
x
x
(B))S -
X
U
£
1
1
(B
)
J
i
=
i
l
=
l
n
(
1+X)
x
£
'
i=
S.T.
1
(B)[c
i
1
(B)+^(B -C
1
(B))]
-7-
U^B 1
(15)
= - E
)
,
(16) U
1
{x
1
^)^'
1
(8
1
^C (B)
)
dF a.e
}
=
(B)
i
=
l,...,n
i
=
1
i
=
1,...,n
-l
,
.
.
.
,
n
V
-
(17)
x
1
+
(B)
1
for any
>
B
i=1
(18)
x
1
for any
>
(B)
6
Without loss of generality we can assume that
and
tp
'
(
)
(
=
(otherwise in the next proofs write
=
)
\\>
=
\\>(x)
vp {
)
+
= ip'(O)
ip'(x)
with
\jj(x)
vp
)
(
=
lp'U) with \p'(0) = 0).
+
We now show that program (14) can be simplified by conside-
which are functions of
ring functions C
(B)
Lemma
1
,
only,
i
=
Proof
:
Suppose the contrary. We show that the optimal value of
Under A2
.
1
,
.
.
.
,
at the optimum C
X
1
(B
)
e
E
B"
(x
x
(B)
only.
is a function of
and choose C
(B))
contradiction. Denote
a
1
1
(B
such that
)
:
1
C^B 1
(19)
)
=
E
C
i
{B" /x
i
X
(B)
(B)=l}
Observe that from the linearity of the program in
optimum can be chosen so that the
Then,
since
i
B
n.
program (14) can be increased,
1
*i
B
\p
(
)
=
0,
(x
(8))
(x
(8)),
the
are zeros and ones.
{x
E
(20)
1
(B)ip(B
1
-C
exceeds
4-(B
1
by Jensen's inequality,
>
1
X
1
(B
1
E
)*(
(B
l
1
(B)
(B)=l}
the expression in (20)
quantity
1
-C
(B)
)
=
)
1
X
1
(S
1
1
1
-C
)i>lB
(8
)
(B)=1}
The maximand of
can therefore be replaced by the larger
(14)
:
n
n
L
)S-
x
i=l
L,
e
i = l
B
{Xu
.
(B
+
)
(i +
X)x
(B
(c
)
(B
)
+
Moreover, the constraints can be relaxed. Since
and by A2,
(23)
-C
:
{B" /x
(
1
E
)
{B~ /x
l
E
1
1
Since Y"
(22)
X^B
=
(BJ )}
i
E
(21)
1
ip
{x
E
B"
'
'
'
>
0,
1
1
(B)\|»'
(B
we have similarly
-C
1
(B) )}
-c
\|>'(0)
(B
=
1
(B
-C
1
1
(B
)
)
1
Therefore, since the objective function is decreasing in U
the constraints
(24)
)
:
X^B 1 )^*
>
i|>(B
1
U (B
1
)
are relaxed if we replace
1
1
1
= - X (B )*'(B -C
For given optimal (x
* i
(C
by
:
1
(B
(.))
optimization with respect to
programs of the type
1
(15)
)
and therefore given X
(B
* i
the
(.),
can be decomposed into
))
:
£
(25) Max
S.T.
1
{-XU (B
1
)-(1+X)X
1
(B
i
)
(C
1
1
(B
1
)
+ '('(B -C
1
(B
1
))}f
(B^dB
1
n
,
)
)1
•9-
1
1
1
= - X (6
(26)
U (8
(27)
u'fB) =
)
1
-)\p'
1
-C
(8
1
(B
1
)
as the state variable and C
Considering U
variable, the Hamiltonian of this program is
(28)
H
1
1
[-\U (&
=
1
)-{
1
]+\)X (B
1
-
1
(B
y
1
)
(C
1
)(-X (B
1
1
(B
as the control
:
1
)+iKB -C
1
1
-C
)*'(B
1
1
1
(B
)
)
IftB
)
1
)
1
(B
)
)
)
The Pontryagin principle gives
1
(29
(J^B
oo:
(1 + X)
(31)
p
1
=
)
(
ftB
X
W
1
(B
1
)
1
-C (B
1
)
)
)f (6
1
1
=
)
1
(B
vj
1
)*"(B -C
1
1
(B
))
=
(B)
Integrating (29) and using the transversality condition (31)
we have
:
U^B 1
(32)
)
=
FtB
X
1
*i
The optimal cost function C
by
(B
l
is therefore determined
)
:
F(6l)
i
(33)
(
1+X){1-\p' (B -C
*iX
i
(B
=
)))
—
X
f (B
1
We then substitute the (C
optimal x
(
B
x
(26)
(6) )S - X
+
(1+X)
E
v
(B
i
1
)
into (14) to solve for the
,
the Lagrangian can be written
x
I
1
1
(b
)
rd 1
C^MJJ
-
1
)
JdB
1
i=1
n
-
*i1
)
n
l.
L
))
i
*"(B -C
1
Integratxng
[
1
(B
1
.
x
i| 1
1
(B)
1
r
i
(B) lc*
(B
-]
1
)
+
"'
1
(B
-
C*i
(S
1
))
+
E
6
x
1
!]
y (B) (1 - Z
i=
x
1
x
(B)
)
-10-
where y(6)
f(6i)...f(6
constraint
1
associated with
i
x
-.£.,
x
and
>
(6)
v
1
f
(S)
(
6i
.
)
.
.
(Bj the multiplier
f
0.
>
(B)
is the multiplier associated with the
)
_1
Integrating with respect tc
the coefficient of X
B
(B
is
)
then
1
(34)
f (6
i
)-f(6
f (B
1
)[E
1
FIB
S -I
1
B
x
For
i
(35)
i
(B)
-C* i
!
v
l
(B
v
1
(dividing
1
)+\p(B
-C*
1
F(B
1
(B
)
At an interior optimum in C
i.e.
(1+A)i|>"f
+
)
+
1
H-
(B
-C*i
1
(B
)
=0 by complementary slackness
(B)
]
)
f(B
cave,
1
(S)J
to be selected we must have
i
C* i (6
_i
can equal one only if
S-(1+X)[c*
m:
1
(B
1
- E
(B)
1
¥' (B
)
XFvp
'
'
'
>
check that Al is sufficient for C
— ' (B
X
X
-c
1
))
1
(B
))
:
>E
B
)
Y(B)
>0
_i
the Hamiltonian must be con-
,
0.
by f(B
(34)
Then, differentiating
*i
we
(33),
to be non decreasing in
i
B
.
Consequently, using A1 again, we see the left hand side of (35)
is a non decreasing function of
B
.
Therefore, we must choose x *i (B) as
x*
(36)
x
=
(B)
1
if
1
<
B
min
:
B
kr'i
=
1
if
>
B
min
B
Mi
1
Hence, the (x
everywhere and
* i
(
c
(.))
(B
i
))
and X
(B
)
are non increasing almost
are non decreasing in
where. The neglected second order conditions
satisfied. We have obtained
i
B
(9)
almost everyare therefore
:
Under AT, A2, an (interior) optimal auction awards
the contrac- to the firm announcing the lowest cost parameter .The
cost level (and consequently the effort level) required from the ma-
Theorem
1
:
nager selected is
must be
a
a
solution of (33). The transfer to the firm
solution of
,(8).
This result deserves some comments. The optimal auction is
deterministic. The level of effort determined by equation (33)
-1 1-
Actually, the effort level is identical to the one we obtained
Intuitively the
with only one firm (Laffont and Tirole (1984)).
reason is as follows. Competition amounts for the best firm to
_
a
*
truncation of the interval [S,B] to [6,8
lowest bid.
*
where
]
the second
is
6
But the distribution plays a role in equation
F(8
only through the value of
which is independent of
(33)
(as
8
1
f
is well
known,
the term
(3
comes from the trade-off between lowe-
?
ring the effort distorsion at
B
-weighted by the density f(6
)-
and lowering the rent of the firms which are more efficient than
6
-weighted by the c.d.f. F(8
ner's productivity parameter
F(6
)
)).
as n grows the win-
Of course,
becomes close in probability to
8
£
becomes close to zero and the effort level is close to the
optimal one. Competition solves asymptotically the moral hazard
problem by solving the adverse selection problem (this extreme
result clearly relies on risk neutrality).
4
-
a)
Implementation by
From (10) the system of transfers of the optimal Bayesian
auction is such that
37)
dominant strategy auction
a
t*
1
(6
1
)
=
:
t*
E
6
x
(B)
=
U*
1
^1
)
+ X*
1
(B
1
1
)ty(B
-C
1
(B
1
))
-l
Using (11) and (12) we have
r
(38)
t
B
i,„i
x
X
(6
x
*i,
x
i
,?ri ,, *i
i
i
*ii ,7ri,
i
x
(B
(6)iJj(8-.Cn
(6)^'(B-C
(B ))d8+X
>
,
N
.,.,
dominant strategy auction of the Vickrey
type which implements the same cost function (and effort function
We now construct
a
and also selects the most efficient firm.
i
12-
Let
B-
r
r4
(39)
= i'is
1
-^
i,„i
)
+
)
*' (B
1
-C*
1
(B
1
)
)dB
X
1
B
i f
40)
t
1
i
=
min
k
B
Mi
otherwise.
=
When firm
with B^
= mi n
k
B
wins the auction its transfer is equal to the
individually rational transfer plus the rent the firm gets when the
distribution is truncated at
B
.
Therefore we are back to the
monopoly case, except that the truncation point B^ is random.
But for any truncation point,
truthtelling is optimal in the mo-
nopoly case. Therefore, truthtelling is
Appendix
3
/
)
a
dominant strategy (see
.
.
If
1
=
B
B,
X
from (39), U (B) =
for any
D
)
(
and therefore
also in expectation. The dominant strategy auction is individually rational.
Using the distribution of the second order statis-
tics it is easy to check that E
t
1
(
B
)
x
= t
1
(
)
Therefore, the dominant strategy auction costs the same
expected transfers than
the
optimal Bayesian auction; this is
somewhat analogous to the revenue equivalence theorem.
Implicit in the approach with mechanisms that we have followed is the fact that if the selected firm ex-post cost differs
b)
from C
* i
i
(B
)
it incurs an infinite penalty.
nism is not robust to the introduction of
e
(uncorrelated with
8
a
However, this mecha-
random disturbance
into the cost function (C
=
e^e l
)
-13-
We will now rewrite the transfer function in
is
a
form which
closer to actual pratice and is robust to cost disturbances.
Let
1
(41)
s
where K(6
X
=
)
t(B)
=
(B,C)
i^MB^C
(B
+
KIB^IC^IB
1
))
-C)
)
noting that the optimal
[0,1],
€
1
*
cost function
C
independent of
is
It is easy to check that
i.
still induces the appropriate
(46)
ex-post effort level and truthtelling
sed into a transfer
into
computed at the time of the auction and
(B)
t
The transfer is now decompo-
.
sharing of overruns (meaningfull when costs are random)
a
determined by the coefficient K(S
).
The auction selects the best firm on the market and awards
the winner an incentive contract to induce
a
second best level
Thus we generalize the result in Laffont and Tirole
of effort.
that the optimal allocation can be implemented by offering
(1984)
the firm(s) a menu of linear contracts.
In particular,
winner's bid
(B
)
decreases with the
the fixed transfer t
and increases with the second bid
(B
).
The
slope of the incentive scheme K decreases with the winner's bid.
Thus, the contract moves toward a fixed-price contract when the
winning bid decreases.
Lastly we can note that firm i's expected cost, if it is
chosen,
is an increasing function of
B
.
This implies that the
optimal auction can have the firms announce expected cost C
(41)
can then be rewritten
(42)
where C ai
s
<
3
i
(C
C
,C)
aj
<
C
= t(c
ak
ai
,C
3J
for all k
+
)
ji
K( C
i,j.
ai
)(C
ai
-C)
14-
5
-
Conclusion
This paper draws
thecy
a
remarkably simple bridge between auct.on
and incentive theory. The optimai. auction tru:
ates
the
cost
uncertainty range for the successful bidder's intrinsic
is the second bidder's intiinsic
from [_B,B] to [_8,B ], where 6
cost.
The principal offers the successful bidder the optimal in-
centive contract for
a
monopolist with unknown cost in
[_6,B
Previous work then implies that the successful bidder faces
].
a
linear cost reimbursement contract, the slope of which depends
only on his bid, and not on the other bidders'
bids.
Technically, the optimal auction can be viewed by each
bidder as the optimal monopoly contract with
a
random upward
truncation point. This trivially implies that the auction is
dominant strategy auction. Our dichotomy
-
a
the winner's fixed
transfer depend on the first and the second bids, and the slope
of his incentive scheme depends only on the first bid - comes
from the fact that the ex-post incentive constraint is downward
binding
while bidding leads to an upward truncation of the con-
ditional distribution. Lastly we should note that the optimal
allocation can be implemented by a dominant strategy auction which
.—
uses information about the first and second bids
-15-
ApDendix
1
:
Non optimal itv of random schemes
the one firm problem the optimization program of the
In
Government is
{S-XU(6)-( 1+A)[C(B)+vp(B-C(8)
Max
C(
.
)
,U(
.
)
]}f (B)dS
)
S.T.
U(B)
=
-
*'
(B-C(B)
U(B) - 0.
*
Suppose that the optimal control
C
(B)
is random.
We show
*
that replacing it by EC (B)
improves the program a contradiction
From Jensen's inequality and
-EvMB-C (B)
<
)
-
H-CB-EC
>
i>"
(B)
The maximand can be replaced by the larger quantity
AU(B)
{S -
If C
-
(B)
U(B) = -
M+A) [EC*(B)
+
^(6-EC*(B)
E\|>'
]f (B)
is
random the constraint AT.
E\|>'
(B-C (B)
)
From Jensen's inequality and
-
)
(B-C (B)
)
<
-
i|/'
(6-EC
4>
'
'
'
>
2
dB
must be written'
16-
As the maximand is decreasing
in U,
the value of the
*
*
program is increasing
if
we replace -Ei[)'(B-C (8))
by -\Jj'(B-EC
The same property holds in the competitive case.
(8)).
17-
ADoendix
2
Characterization of truthtellinq
:
To mini'aize technicalities we assume that
di
a.e.
erentiable
f
It can be
.
For truthtelling to be
be the best answer
1
^ ^ ^
1
1
1
i.e.
=.E
)
1
must
B
when it assumes that the other
i
must be the solution of
{t
B'
6
shown that it is indeed the case.
best Bayesian Nash strategy,
for firm
firms are truthful,
Max UIB
a
the allocation is
1
1
Cfif
,6"
1
1
)^x (B
1
,B~
1
1
)rp(B
-C i
1
(B
,B"
1
))}
1
The objective function satisfies the conditions CS_, CS
of Guesnerie and Laffont
,^/sx 1
8
3B
1
3U/
3U/
3
36
,
1
3U/
gt
= -
V
E
i.e.
<
i
3C\
8t
(1984),
=
E
i
B
x
i>"
>
-i
Transfers must satisfy the first order condition of incentive compatibility
(A2.1)
:
—
E.ti(S)
8S 1
B
-
If
C
-E
=
_1
3B
1
a
[x
B
a.e.,
1
1
(B)vKB -C (B»]
-i
B
is non decreasing in
increasing in
1
B
a.e.
and if E
x
(B)
then following the proof of theorem
Guesnerie and Laffont (1984) we show that these functions
C
(
B
)
is non
can be implemented by transfers solutions of A2
.
1
2
x
in
(B),
-18-
Appendix
3
Implementation in dominant strategy
:
Let us check that indeed truthtell ing is
dominant stra-
a
tegy.
Suppose first that by announcing the truth, manager
i
wins
Since the auction is individually rational and lo-
the auction.
sers get nothing he cannot gain by lying and losing the auction
What about an answer
r
6
but such that
B
Manager i's program is
1
<
B
)
i|>'
(B^C* 1
6
?
iMB^C* 1
-
^
1
)
))
Min B k
k?i
The first order condition
(A3. 2)
min
k^i
:
{t^B 1 ^" 1
Max
(A3.1)
<
6
^
for an interior solution is
*i
1
)
)
(
1-^U
1
1
(B
)
)-\p' (B
...
-C
1
^(B
:
1
)
)
dB
/
i
+
tp'
(B
-C
* X
i
* i
-
i
(B
dC
))^1
-i1
(B
)
=
dB
Hence
B
=
is the only
B
solution of the first order
condition. Moreover the second order condition at
B
=
B
is sa-
* i
tisfied since
——
>
dB
~
i
1
0.
Finally, observe that
a
loser
to bid less than his true parameter.
he gains nothing.
of the auction does not want
If he still loses the auction
If he gains the auction,
his transfer does not
compensate him for his increased effort level.
Let
B
1
be his true parameter and let his announcement
smaller than the smallest announcement
B
.
B
1
be
-19-
|3
x
,1
U
,ol
(6
=
)
i
\|>(B
-C«i,;i
(B
'
))
+
1
-C*
1
1
1
(B ))dB
1
->Ke -B
1
+B
1
n 1
<
1
1
-(B -? )^' (B
1
-^ ^
1
1
)
)
*' (B
+
1
-C„
1
,
n
(B
1
,
)
,o
)d3
,
nl
<
as
1
1
-(B -! )*' (B
(from (33)
^—
de
,
dB
Since
/
1
B
>
B
D
,
-
C^tB 1 J+te^-B 1
)
=
i±
)
1
1
-
1
U^B 1
—^
dC
dB
)
<
0.
i
—
•
)*'
(B^-C^tB
1
)
1
-C*
1
(6
1
20-
Footnotes
In Laffont and Tirole
1)
but with
is
If Al
2)
(1985) we study dynamic contracting
single firm.
a
not satisfied the methods described in Guesnerie
and Laffont (1984) must be used.
From the revelation principle, we know that there is no
3)
loss of generality in restricting the analysis to these
direct mechanisms (see e.g. Riley and Samuelson (1981)).
Note that as
4)
n
is
large 6^ is close to
6
in probability
and the transfer converges to the disutility of the "second
lowest bidder"
A precursor to our paper is Mc Afee
5)
also Fishe
/
-
-
Mc Millan
(1984)
(see
Mc Afee (1983)). Their paper looks at the trade-
off between giving the agent incentives for cost reduction,
stimulating bidding competition and sharing risk. They restrict the contract to be linear in observed cost and in the
successful bidder's bid ("first bid auction")". Our paper
shows that, under risk neutrality, the optimal contract can
indeed be taken to be linear in observed costs. However the
coefficient of risk sharing should decrease with the successful bidder
'
s
bid.
framework complementary to ours. Bidders submit bids and then learn their
true marginal cost. Each firm's private information at the
bidding stage is a signal correlated with final cost. Thus
they are interested in repeated adverse selection, as opposed
Riordan and Sappington (1985) consider
a
to ex-ante adverse selection and ex-post moral hazard.
In
the independent signals case they consider two classes of
U35
3
Lis
21
auctions. One is
a
first price auction, in
which the successful bidder's allocation depends only on
his bid,
not on the other bidder's bid. The ether is a
second price Ruction, in which the winner ge:-
corresponding to the second bid,
of the first bid.
the terms
independently of the value
Riordan and Sappington show that the
revenue eauivalence theorem does not hold.
*
*
*
-22-
Ref erences
Demsetz,
and Economics
Guesnerie,
"Why Regulate Utilities", Journal of Law
(1968),
H.
11,
,
55-66.
and J.J. Laffont
R.
"A Complete Solution to a
(1984),
Class of Principal Agent Problems with an Application to
the Control of
nomics
Fishe,
25,
,
a
Self-Managed Firm", Journal of Public Eco -
329-369.
and R.P. Mc Afee (1983),
R.P.
"Contract Design under Un-
certainty", Mimeo.
Laffont, J.J., and J. Tirole (1984),
"Using Cost Observation to
Regulate Firms", GREMAQ, P.P. 84
Political Economy
.
Laffont, J.J., and J. Tirole (1985),
Contracts", GREMAQ P.P.
Mc Afee,
R.P.,
forthcoming Journal of
,
85
and J. Mc Millan
"The Dynamics of Incentives
.
"Bidding for Contracts
(1984),
A Principal Agent Analysis", Mimeo
Riley, J., and W. Samuelson (1981),
Economic Review
Riordan, M.
Mimeo
,
and D.
,
71,
:
.
"Optimal Auctions", American
381-392.
Sappington (1985),
"Franchise Monopoly",'
.
"Franchise Bidding for Natural Monopolies In General and with Respect to CATV", Bell Journal of Econo -
Williamson, 0.
mics,
7,
(1976),
73-104.
MIT LIBRARIES
3
TDfiD DDM
477
TBfl
Date Due
^07
'88
APR&
Lib-26-67
6°
<K
l/v
fl