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|DEC 21
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PROVISION OF QUALITY AND POWER OF
INCENTIVE SCHEMES IN REGULATED INDUSTRIES
Jean-Jacques Laffont
Jean Tirole
No.
528
August 1989
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
PROVISION OF QUALITY AND POWER OF
INCENTIVE SCHEMES IN REGULATED INDUSTRIES
Jean-Jacques Laffont
Jean Tirole
No.
528
August 1989
MIT *«R*«irs
DEC 2 1 1989
flEOSVED
PROVISION OF QUALITY
AND POWER OF INCENTIVE SCHEMES IN REGULATED INDUSTRIES
by
Jean-Jacques Laffont
**
and
Jean Tirole
April 1989
*
Research support from the Taussig visiting professorship at Harvard
University, the Guggenheim Foundation,
the Pew Charitable Trust/Ford
Foundation ENS Program, the Center for Energy Policy Research at M.I.T.,
the National Science Foundation, and the French Ministry of Education is
gratefully acknowledged.
**
GREMAQ, Universite des Sciences Sociales, Toulouse.
***
Harvard University and Massachusetts Institute of Technology.
Abstract
We study the incentives of a regulated monopoly to supply quality.
an experience
good,
the
current
level of sales yields
For
information about
no
quality and the cost reimbursement rule is the only instrument to achieve the
conflicting goals of provision of quality and cost reduction.
for
quality
moves
optimal
contracts
increase in the discount factor raises
toward
the
cost-plus
A high concern
contracts;
and
power of incentive schemes.
an
In
contrast, for a search good, direct sales incentives can be provided to supply
quality;
cost-plus
whether
or
a
high
fixed-price
quality
concern
contracts
then
drives
depends
quality are net substitutes or net complements.
optimal
on
contracts
whether
toward
quantity
and
1
.
Introduction
.
An unregulated monopolist may have two incentives
the
and
incentive"
"sales
the
quality
when
(experience good)
then
quality
is
a reduction in quality
incentive
the monopolist has no
,
consumers
observed by
is
consumers may repeat their purchase in the future.
is
,
quality:
and thus revenue as the monopoly price exceeds marginal cost.
reduces sales,
contrast,
When
incentive."
"reputation
observed by consumers before purchasing (search good)
In
provide
to
with
linked
the
monopolist's
desire
to
only
purchasing
after
supply quality unless
to
The provision of quality
keep
its
reputation
and
preserve future profits.
In
this
paper,
we
investigate
whether
Before doing so,
quality exist in a regulated environment.
and verifiable
distinguish between observable
observable
by
consumers
either
before
incentives
similar
or
quality.
after
it
to
consumption.
useful to
is
Quality
provide
is
usually
Quality
is
furthermore verifiable if its level can be (costlessly) described ex-ante in a
When quality is verifiable, the
contract and ascertained ex-post by a court.
regulator can impose a quality target to the regulated firm or more generally
reward or punish the firm directly as a function of the level of quality.
For
instance a regulatory commission may dictate the heating value of gas or may
punish
outages.
an
electric
Formally,
utility
on
the
basis
of
the
number
and
intensity
of
the regulation of verifiable quality is analogous to the
regulation of a multiproduct firm, as the level of quality on a given product
may be treated as the quantity of another, fictitious product.
This paper will be concerned with observable but unverifiable quality.
The effectiveness of a new weapons system,
regulated television station,
See
Sappington
theories of the
regulation.
the
level
of
the quality of broadcasting by a
services
enjoyed by a
for
and Laffont-Tirole
[1983]
[1988]
regulation of a multiproduct firm with
railroad
information-based
and without cost
passenger or the probability of a core melt-down at a nuclear plant are hard
to quantify and include in a formal contract.
As Kahn [1988, p.
22]
argues:
But it is far more true of quality of service than of price
that the primary responsibility remains with the supplying
company instead of with the regulatory agency, and that the
agencies, in turn, have devoted much more attention to the
The reasons for this are fairly
latter than to the former.
Service standards are often much more difficult to
clear.
specify by the promulgation of rules.
When quality is unverif iable
unregulated firm
First,
regulation.
Second,
to
provide
,
the regulator must recreate the incentives of an
quality without
throwing
away
the
benefits
of
it must reward the regulated firm on the basis of sales.
the threat of nonrenewal of the regulatory license,
of second sourcing
or of deregulation makes the regulated firm concerned about its reputation as
supplier of quality.
The focus of our analysis is the relationship between quality concern and
power
of
optimal
incentive
An
schemes.
incentive
scheme
high-
is
powered if the firm bears a high (low) fraction of its realized costs.
fixed-price contract is very high-powered,
and a cost-plus
(low-)
Thus a
contract is very
low-powered.
The link between quality and the power of incentive schemes has been much
For instance, there has been a concern that "incentive regulation"
discussed.
(understand:
high-powered
incentive
schemes)
conflicts
with
safe
the
operation of nuclear power plants by forcing management to hurry work,
There have been accounts
shortcuts and delay safety investments.
switch
to
after
its
a
high-powered incentive scheme
privatization
(Vickers -Yarrow [1988,
2
p.
produced
228]).
2
a
for
poor
Similarly,
British Telecom
record
on
Kahn [1988,
the
I,
that
(price
quality
p.
24]
take
the
caps)
front
contends
It is not surprising that the dissatisfaction with the quality performance
subsequently led to the costly development and monitoring of quality indices
to be included in the incentive schemes.
that,
under cost-of -service regulation (a very low-powered Incentive scheme),
In the matter of quality "far more than in the matter of price,
of
the
monopolist on the one hand and the
nearly coincident than in conflict."
consumer on the
intuition
monetary
costs
these
is
paid
the
Second,
by
consumers
First,
incomplete.
costs.
regulation,
are
and more
other
interest
are
more
Kahn's intuition is that the regulated
monopolist does not suffer from incurring monetary costs
because
the
some
through
components
importantly,
to
direct
of
under
enhance quality
charges.
quality
pure
This
Involve
non
cost-of -service
regulated firm does not gain from providing costly services
either so that a low perceived cost of supplying quality does not imply a high
incentive to supply quality.
3
Last,
in the context of military procurement,
Scherer [1964, pp. 165-166] has suggested that
There is reason to believe that the use of fixed-price
contracts would not greatly reduce the emphasis placed on
quality in weapons development projects, although it might
affect certain marginal tradeoff decisions with only a minor
expected impact on future sales.
To give formal arguments to assess the relevance of these perceptions, we
introduce two related natural monopoly models of an experience and of a search
good.
Whether
the
power
of
regulatory
contracts
when
decreases
quality
becomes more desirable depends crucially on whether contractual incentives can
be based on sales (on top of cost) or not, i.e., on whether the regulated firm
3
In practice, one does not observe pure cost-of -service regulation.
Due to
the regulatory lag, the regulated firm is, like an unregulated monopolist, the
residual claimant for the revenue it generates and costs It incurs between
rate reviews (the differences being that the prices are fixed and that the
regulated monopolist is concerned about the ratchet effect)
thus actual
cost-of -service incentive schemes are not as low-powered as one might believe.
We will not try to study (variants of) cost-of-service regulation, but will
rather focus on optimal regulation.
See Joskow and Rose [1987] for empirical evidence on the level of
services under cost-of-service regulation.
;
supplies
a
experience
search
good,
the
experience
an
good.
is
then
reputation
the
fixed amount
a
The supplier's
incentive,
of
a
as
although other
has
no
incentive to provide
losing
of
procurement model,
interpretations
and
latter
the
possible
are
former model
The
consumers who observe quality before purchasing.
thought
regulator
possibility
the
i.e.,
an
regulated
from the
the
,
of
our static search good model has the firm sell to
In contrast,
future sales.
two-period model
our
unverif iable
ex-ante
is
alternative than to accept the product.
quality
In
regulator purchases
quality
Because
firm.
or
as
best
regulation model,
a
particular,
(in
is
regulated
some
products are experience goods)
To separate issues in this paper we choose cost functions for which the
incentive-pricing dichotomy
(Laf font-Tirole
holds.
[1988])
Pricing
not
is
used to extract the rent due to asymmetric information.
In the case of an experience
pood
,
we argue
incentives
that
quality and those to reduce cost are inherently in conflict.
has
a
types
but
single
instrument
the
the
cost
reimbursement rule
High-powered incentives schemes
of incentives.
increase
--
firm's
perceived
cost
of
--
The regulator
more
concerned
decreases,
schemes.
about
which
the
induces
future,
the
provide both
to
induce cost reduction
providing
quality.
crowding-out effect implies that the more important quality is,
power of an optimal incentive scheme.
supply
to
This
the lower the
We also show that when the firm becomes
its
perceived
regulator
to
cost
offer
of
more
quality
supplying
powerful
incentive
We thus find Scherer's suggestion quite perceptive.
In the case of a search good
,
the crowding-out effect is latent but has
no influence on the power of incentive schemes.
In our model,
the regulator
can separate the two incentive problems because it has two instruments:
reimbursement rule and sales incentives.
The incentive to provide quality is
provided through a reward based on a quality index,
sales
corrected by the price
charged by
cost
the
firm.
which is
As
in
the
the
level
case
of
of
an
experience
cost-reducing activity
the
good,
encouraged
is
through
the
cost
reimbursement rule, which is now freed from the concern of providing the right
This dichotomy does not, however,
quality incentives.
in
the
of
desirability
schemes;
it has
decrease
the
has
quality
effect
no
on
imply that an increase
power
the
of
an indirect effect because higher services may
and
cost
reducing marginal
of
level
optimal
which
output,
thus
affects
turn
in
changes
regulator's
the
incentive
increase
or
value
of
the
arbitrage
between
incentives and rent extraction.
While there exists a vast literature on the provision of quality by an
monopoly,
unregulated
devoted
this
to
that
assume
discriminate
Mussa-Rosen
4
surprisingly
issue
the
in
a
monopolist
among
consumers
and
[1978])
little
regulated
offers
with
investigate
theoretical
environment.
range
a
different
of
Besanko
verifiable
tastes
effect of
the
work of Lewis
and
unverifiable quality.
Sappington
[1988]
,
who
for
et
been
has
al
.
[1987]
qualities
quality
(a
to
la
imposing minimum quality
standards or price ceilings (see also Laffont [1987]).
the
research
Closer to our paper is
examine
both verifiable
and
Sales depend on a demand parameter, price and quality.
When quality is unverifiable, Lewis and Sappington assume that the regulator
monitors prices (but not cost, quantity or quality) and operates transfers to
the firm.
To give incentives to provide quality the regulator allows prices
in excess of the (known) marginal cost.
A higher mark-up above marginal cost
raises the dead-weight loss due to pricing but also raises quality.
Lewis and
Sappington show that the rent derived by the firm from its private information
about the demand parameter is higher when quality is verifiable than when it
is not.
Our work differs from theirs in that,
among other things, we allow
cost observation and focus on the effect of quality concerns on the power of
incentive schemes.
4
See,
e.g., Tirole [1988, chapters
2
and
3]
The paper is organized as follows.
His imperfect knowledge of the production
and output of a natural monopoly.
and
of
the
activities
cost-reducing
demand
and
function
makes
provision
3
problems
quality/services
of
cannot be disentangled from other costs,
Section
the
of
inducing
non
trivial.
monetary services enter total cost but
Services are monetary or nonmonetary;
aggregate accounting data.
develops a single product,
2
The regulator observes the total cost, price
static model of sales incentive.
technology
Section
cannot be recovered from the
i.e.,
solves for the optimal incentive scheme.
Section 4 shows that the optimum can be implemented by a scheme that is linear
realized cost and a quality
in both
Section
sales data.
5
index
that
is
computed from price
and
links variations in the quality concern and the slope
Section
(power) of incentive schemes.
6
discusses reputation incentives.
It
develops a model of an experience good in which a quality choice has permanent
This model is one of moral hazard (unverif iable intertemporal choice
quality.
of
information about future
The observation of quality today reveals
effects.
quality).
In
many
contrast,
models
reputation
of
in
the
industrial
organization literature have assumed that the firm can be "born" a high-
or
low-quality producer, and can, at a cost masquerade as a high-quality producer
if
it
is
a
low-quality
producer.
Appendix
B
stages
a
variant
of
the
reputation models of Kreps-Wilson [1982], Milgrom-Roberts [1982] and Holmstrom
[1982]
in a regulatory context and obtains
moral-hazard model.
Section
7
results
similar to
those
in the
concludes the paper and suggests some desirable
extensions
Part A:
2.
Search goods.
The model
.
We consider a natural monopoly producing a single commodity in quantity q
with observable but unverif iable quality/services
model,
we briefly discuss
the methodology.
s.
We assume
Before describing the
that
the
firm's
cost
increases with output and with the level of services, decreases with the cost
reducing activity e and depend on some privately known technological parameter
j3;
C - C(q,s,e,/?).
On the demand side, we assume that the regulator does not
perfectly know the demand curve; otherwise he would be able to infer the exact
level of services provided by the firm from the price and output data; we thus
posit an inverse demand function p - P(q,s,0). where P decreases with q and
increases with
and
s,
demand parameter which
a
is
8
known by the
is
Our model is thus one of two-dimensional moral hazard
only.
especially two-dimensional adverse selection
(/?
(e
firm
and s)
and
In order to obtain a
and 6).
closed form solution, we specialize it to linear cost and demand functions,
which enable us to reduce the problem to a one-dimensional adverse selection
one
Quantity and quality are said to be gross complements if an increase in
marginal willingness to pay:
quality raises the consumers'
„
.
- ^- >
.
Quantity and quality are net complements if an increase in quality raises the
net
marginal
willingness
cost:
marginal
b
2,
c)
a ^ s g_
=
—dsdq
—
pay,
to
3(p-C
-
—
^
ds
i.e.,
the
difference
between
price
and
)
^— >
0.
As we will see,
the effect of quality
i
j
concerns on the power of incentive schemes depends on whether quantity and
quality are net complements or net substitutes.
Let us now describe the model in more detail.
The (variable) cost function is
C - (/3+s-e)q,
(2.1)
where
fi
in
[/?,/?]
is
an intrinsic cost parameter known only to the firm and e
is the firm's cost-reducing activity or effort and is also unobservable by the
regulator.
Note
See
Sappington
theories of the
regulation.
that
(2.1)
assumes
that
the
cost of providing quality
and Laf font-Tirole
for
[1983]
[1988]
regulation of a multiproduct firm with
is
information-based
and without cost
The remark below shows that a relabelling of variables allows the
monetary.
cost of providing quality to be nonmonetary.
The
cost
regulator.
verifiable
is
C
by
and,
accounting
convention,
the
Similarly, we assume without loss of generality that the regulator
receives directly the payment pq made by the consumers
good,
born by
where p
is
the
After paying C
price.
good's
regulator pays a net transfer
t
to
the
exchange
in
for
and receiving pq
,
the
the
firm (that will depend on observable
and verifiable variables C,p,q).
Letting V( e ) (with
of effort,
> 0, V" > 0,
yj>'
\j>"'
denote the firm's disutility
^ 0)
the firm's utility or rent is
U - t-^(e).
(2.2)
Consumers
observe
the
quality before
purchasing
the
(search)
good
and
derive from the consumption of the commodity a gross surplus:
S g (q,s,0)
(2.3)
where
A,B,h,k
- (A+ks-hOq
known
are
2
-
positive
fq
-
I^Mi
constants
2
,
and
8
in
[8,6]
is
a
demand
parameter.
The inverse demand curve is then
(2.4)
p -
p
A+ks-htf-Bq.
Quantity and quality are always gross complements in this model, and are net
complements if k >
1
and net substitutes if k <
The technical assumption
optimal.
i]>"'
>
1.
We will of course focus on
ensures that stochastic mechanisms are not
Note that S g differs from
at q - 0.
One can think of S g as a local
approximation in the relevant range.
Or one might allow services to affect
consumers even in the absence of consumption. The reader should be aware that
the Spencian comparison between the marginal willingnesses to pay for quality
of the marginal and the average consumers under unregulated monopoly requires
that S° be equal to zero at q — 0.
parameters that put the problem in the relevant range
^
—>
—>
0,
^
Q
Let
>
1+A
1
be
cost
social
the
public
of
funds.
The
consumers '/taxpayers' net surplus is:
(2.5)
S
n (A+ks-h0)q
2
2
-
-
§q
-pq- (1+A) (C-pq+t)
(2.6)
S
Remark
:
n (A+ks-h0)q
(ks ' hg)
2
-
Suppose further that the firm exerts a second
services
Then the disutility of effort is V( e+S )
to
consumers
vividly that cost-reducing activities
yj>'
an
:.
increase
per unit of output.
s
Letting e « e+s
•
becomes C - (/?+s-e)q and the disutility of effort is
substitute
.
Suppose that the accounting cost is C — (/3-e)q, where e
that provides
s
+A(A+ks-h0-Bq)q- (1+A) (C+t)
our model covers the case of a nonmonetary cost
the effort to reduce cost.
of effort
2
-
fq
As mentioned above,
of providing quality.
type
From
(2.5) can be rewritten:
(2.4),
are
.
includes the taxes needed by the regulator to finance the firm.
(2.5)
is
-Oi^Ml
(e)
V"(
the accounting cost
e)
,
This remark shows
•
and the provision of services
(s)
marginal
disutility
a utilitarian regulator maximizes
the sum of
in
services
raises
the
(e+s) of exerting effort to reduce cost.
Under complete information
,
consumer and producer surpluses under the constraint that the firm be willing
to participate:
jw - (l+A)(A+ks-htf)q
Max
(2.7)
{
q s e
.
,
)
-
B(| + A)q
(ks " hg)
2
2
-
L
-(l+A)((^+s-e)q+t)+t-^(e)j,
s.t.
g
A
is
funds
strictly positive because distortive
taxes
are
used
to
raise
public
t-iKe) > 0,
(2.8)
where (2.8) normalizes the firm's reservation utility to be zero.
program
the
enough,
(2
(
.
7)
,
(2
.
8)
concave
is
)
and
its
For B large
(interior)
maximum
is
characterized by the first-order conditions:
(2.9)
(l+A)p-ABq - (1+A) (/3+s-e)
(2.10)
(l+A)kq-k(ks-hfl) - (1+A)q
(2.11)
V'(e) - q
(2.12)
t
- 4>(e).
equates the marginal social utility of the commodity (composed of
(2.9)
the
utility
marginal
marginal
gain
of
to
-r-(Apq)
the
its
commodity
marginal
to
consumers
social
cost
S
and
((1+A)C
financial
the
Similarly,
).
(2.10) equates the marginal social utility of service quality to its marginal
social cost.
(2.11)
marginal utility
equates the marginal disutility of effort
\f>'
(e)
to
its
And (2.12) says that no rent is left to the firm.
q.
Equations
(2.9)
P" c
i
and (2.10) are most easily interpreted in the following
forms
(2.10'
v
f£f
'3s
)
(a number
shadow
cost
r;
'
h 2_
Bq
--or price-marginal
between
optimal
where
+ A |H q" - (1+A)
j£.
v
'
3s
5s
The Lerner index
the
i
^ - TXT ^
1+A T)
p
(2.9')'
v
level
of
and
1
--
is equal to a Ramsey index
times) the inverse of the elasticity of demand.
of services
public
cost ratio
funds
equates
times
the
marginal cost of quality.
10
the
marginal
increase
in
gross
surplus
revenue
to
plus
the
And
the
social
For the record, it is worth comparing the regulated level of quality with
that chosen by an unregulated monopoly.
linear
are
services,
in
either zero
monopoly solution is
the
if quantity
Spence
in
an
[1975],
Quality
maximal
or
(if
is
an
and quality are net complements.
may
monopolist
unregulated
"bang-bang."
substitutes
quantity and quality are net
if
upper bound on quality exists)
As
Because the cost and demand functions
over-
undersupply
or
quality for a given quantity.
Optimal regulation under asymmetric information
3.
We now assume that the regulator faces
He knows neither
information.
nor
8
e
and
/J
on p €
F.
and
8
[{),&]
asymmetry of
However,
s.
the regulator is
Bayesian statistician who maximizes
a
The firm knows
E [8,8].
and cannot observe
8
welfare and has a prior distribution
F„ on
a multidimensional
Faced with this informational gap,
he observes C, p and q.
assumed to behave as
fi
.
expected social
and a prior distribution
before contracting.
The regulator knows that consumers equate their marginal utility of the
commodity to the price, hence:
p - A+ks-h0-Bq.
(3.1)
Using
the
observability
of
p
and
unobservable service quality level
q,
it
is
possible
in the consumers'
s
to
eliminate
the
gross valuation of the
commodity which becomes:
- |q 2 +pq
S g (p,q)
(3.2)
-
|(p-A+Bq)
2
.
Similarly, the cost function becomes:
C -
(3.3)
Note
(/J
that
combination 7 *
+
(2
/?
M
'
-e)q + q
and
+
-r-8
8
k
'
p-A+Bq
8
enter
.
-
This
the
cost
function
only
feature which also holds
11
through
the
for
firm's
the
linear
and
regulator's
objective
functions
below)
(see
single -dimensional adverse-selection problem,
closed-form solution.
Consider
now
reduces
model
the
and will enable us
to
to
obtain
a
a
9
the
firm's
objective
function
(recalling
our
accounting
convention that the regulator pays the cost and receives the revenue)
U -
(3.4)
t-V-(e)
-
p-A+Bq
t-V>
_
k
C
qj
From the revelation principle we can restrict the problem of control of
the firm to the analysis of direct and truthful revelation mechanisms.
For
simplicity
F(-)of 7 on [7,7] -
we
assume
P + % B.p
+£
the monotone hazard rate property:
that
»
the
cumulative
distribution
(the convolution of F,
d(F(-v)/f (7))
——
—f-
and F„) satisfies
Appendix A.l derives
> 0.
sufficient conditions on the primitive distributions
function
F,
and F„
for
this
to
hold.
The
first-
and second-order
conditions
of
incentive
compatibility are
(see Appendix A. 2):
E+Bq_
(3.5)
V
(3 .6)
E±|3
where t
dt/d7, etc...
.
gj
<
A more general formulation of the consumers' tastes would, lead to a truly
two-dimensional adverse-selection problem.
The qualitative results would be
similar, but the technical analysis would be greatly complicated (See Laffpnt,
Maskin and Rochet [1987], for example).
This assumption avoids bunching without significant loss for the economics of
the problem.
12
The social welfare function can be written (see (2.7)):
2
W - |q + (1+A)pq
(3.7)
The
i(p-A+Bq) 2
-
regulator maximizes
1+A)
(
7 -e) q+q
(
p-A+Bq
(
(3
.
5)
,
(3
.
6)
)
+0(e))-AU.
k
expected social welfare
conditions
compatibility
incentive
the
-
function under the
and
individual
the
rationality constraint of the firm:
U( 7 ) >
(3.8)
for any 7.
The maximization program is, using U as a state variable,
Max
(3.9)
s
.
2
J j|q
2
+(l+A)pq- i(p-A+Bq) - (1+A)
(
7 -e)q+q
p-A+Bq
+y.(e)
k
•AUjdF(
Ac 7 )
first-order
incentive
t.
(3.10)
U( 7 ) - -V-'(e)
(3.11)
e-1 <
(3.12)
U( 7 ) > 0.
Equation
(3.10)
another
is
compatibility constraint
(3.5).
version
Moreover,
of
the
because U( 7 )
is
decreasing,
IR-constraint (3.8) reduces to the boundary condition (3.12).
A. 2,
(3.11)
is a rewriting of the second-order condition (3.6).
the
From Appendix
We ignore it
in a first step and later check that it is indeed satisfied by the solution of
the subconstrained program.
For A and B large enough,
the program is concave
and the optimum is characterized by its first-order conditions
A. 3).
(see Appendix
Let ^( 7 ) denote the Pontryagin multiplier associated with (3.10) and H
the Hamiltonian associated with the program {(3.9),
(3.10),
From the
(3.12)).
Pontryagin principle, we have:
11
We implicitly assume here that it is worth producing for any 7 in
13
[
7 7]
,
£<7) -
(3.13)
|5 -
-
A
-
From the transversality condition and (3.13) we derive:
(3.14)
- AF( 7 >.
/,( 7 )
Maximizing the Hamiltonian with respect to
q,
p
D
e,
we
after some
get
algebraic manipulations:
£^±B3
ABq - (1+A) 7 . e+
(3.15)
(1+A)p
(3.16)
(l+A)kq-k(p-A+Bq) - (1+A)q
(3.17)
V'(e) - q
Equations
-
-
^ ££$
and
(3.15)
w
l4
(e).
(3.16)
which correspond to the maximizations with
respect to q and p coincide with
effort
the
e,
about
reminiscent
of
and
quantity and quality
price,
information
(2.9)
technology
the
(although
are
demand
and
implied by)
not
(2.10).
the
That
same
as
parameter.
the
for
is,
under
given
complete
result
This
incentives -pricing
result for multiproduct firms obtained in Laffont-Tirole [1988],
a
is
dichotomy
Appendix A.
derives a more general class of cost functions for which the dichotomy holds
in this quality problem.
To extract part of the firm's rent,
a given output level
Appendix A.
p(7)
>
0,
q(f)
(compare (3.17) and (2.11)), except when
shows
3
<
the effort is distorted downward for
that
and e(7)
for
<
the
solution to
((3.15),
7-7.
(3.16),
(3.17)),
In particular the neglected second-order
0.
condition for the firm (l-e(7) > 0) is satisfied.
Next we
compare
information about
ft
and
the
levels
of
quality
6:
14
under
complete
and
incomplete
Proposition
of quality
level
The
1:
is
lower under
incomplete
information if and only
than under complete
information
quantity and
if
quality are net complements.
Proof
See Appendix A. 5.
:
information makes
Incomplete
rent extraction difficult
and
power of incentive schemes, i.e., leads to a decrease in effort.
and
cost
marginal
reduces
output.
If
quantity
reduces
This raises
quality
and
the
net
are
complements, lower services are desirable; and conversely for net substitutes.
Note that asymmetric information lowers quality exactly when the unregulated
monopolist oversupplies quality.
To conclude
this
section,
for a search good sales
quality in the same way cost is an indicator of effort
an indicator
are
(and quality)
of
The
.
regulation of quality and effort under asymmetric information is therefore in
the
spirit
of
the
regulation
of
a
multi-product
firm
(see
Laffont-Tirole
[1988]).
4.
Implementation of the optimal regulatory mechanism
For each announcement of the firm's
consumers'
achieve,
technological parameter and of the
the regulator imposes
taste parameter,
.
a quantity to produce and a market price
a
level of average cost to
to charge.
An appropriate
net transfer t(7) is offered to induce truthful behavior.
This
Then,
(4.1)
transfer can be reinterpreted
as
v
follows.
Let z
"—
q
-
•"-
—
-,
-**.
k
the first-order incentive compatibility condition is (see Appendix A. 2):
g+^
(7 . z)
g_
o
or
15
dt dz
(4.2)
(7-z) <
-i/>'
Differentiating (4.2) we obtain:
2
d t
(4.3)
-V>"(7-z)
d^
1
kd
7
dz
>
From the second-order condition -j—
d7
^ >
—
dz^
r\
A. 3.
Therefore
and
1
dz
ds
— -j—
<
d7
-j—
-
d7
from Appendix
rr
t"
0,
the transfer as a function of z is a convex and
i.e.,
decreasing function (see Figure
1)
FIGURE
As in Laffont and Tirole
HERE.
1
the convex non- linear transfer function
[1986],
t(z) can be replaced by a menu of linear contracts
t(z, 7 ) - a(7)+b(7)(z( 7 )-z),
where
z(7)
is
announced value
the
ex-post value.
of —
-
——
-^-
;
and
-*-^-
z
is
the
observed
The transfer is therefore a function of a performance index
which subtracts from the realized average cost an approximation of the service
quality inferred from market data.
In other words,
the
firm is
offered a
choice in a menu of linear contracts and is rewarded or penalized according to
deviations
from
an
index
aggregating
data
cost
and
service
quality
data
inferred by the observation of the market price and quantity and the a priori
knowledge on the demand function.
The
coefficient b(7)
sharing
the
overrun
v
i>'
(e
(7)) where e (7)
is the
in
the
performance
index
is
solution of the regulator's optimization program.
Indeed,
*
.-.
Max
-ja( 7 )+V>'(e (7) ) (z( 7 ) -7+e)
3X <c
-j>(e)
}
Ie,7)
implies
\p'
(e
(7))
-
V'
(
e)
an d
therefore
16
e
-
e
(7)
and a(7)+^"e (7)(z(7)-z)
<^\s
t-ljJ
(Y~z)
=
constant
/N
^/l—
t-^(y-z)
-
constant
">
FIGURE
16A
1
+}/>'
(e
(j))z(-y) - 0.
If a(«)
(7))z(7) -
any
for
7-7.
then
7 or a(7) - t(7),
is chosen so that a(7)+V>' (e
Alternatively, the menu of linear contracts
can
be
decomposed
into
a
*
linear sharing of total costs overruns with a coefficient
sharing
and a linear
coefficient b„(7> -
t
V'
overruns
of
e
(
in
service
the
quality
™ Ce
-
——
(y)
v
'
q
(7)
i/>'
—
b, (7)
index
'
with
a
(7)) or
- a( 7 ) + b (7)(C(7)-C)+b ( 7 )
2
1
p(7)+Bq( 7 )
k
p+Bq
k
Since the firm is risk neutral we see that our analysis extends immediately to
the case in which random additive disturbances affect total
inverse demand function
Proposition
and
C
the
P.
implemented
be
The optimal regulatory scheme can
2.
cost
menu of contracts that are linear in realized cost
through
a
and
a
in
quality index equal to sales corrected by the price level:
t
Clearly,
- S- bl C +b -^9Li
2
parameters
the
schemes, b, (7) -
7
,
.
characterizing
the
power
of
the
and b„(7) - q(7)b, (7), are related.
comparative statics of these coefficients with respect to
a
incentive
If we study the
parameter
x
we
have
d_
dx
From
f(e> -
ip
r
(e)
xM r Mx
q
q
-
^
f
^(e),
W ^ $T
'
%
+
-
tf x
and
d_ V»'(e)
dx
The comparative
-^xTTxl W'
statics
of
2
-*'*'")
b„(7)
17
-
Tp'
(e
(7))
is
the
same
as
the
comparative statics of
as long as
i/>"'
*-
e
(7)
not to °
s
and the comparative statics of b. (7) is identical
l ar E e
which we will assume in Section
>
5
(of course,
no such assumption is needed if one studies the slope of the incentive schemes
respect
with
to
marginal
cost)
that
Note
.
and b„
b.
then positively
are
correlated over the sample of types.
Concern for quality and the power of incentive schemes
5.
This
section
studies
effects
the
of
increase
an
.
in
the
consumers'
marginal surplus from quality (3S°/3s) and of an increase in the marginal cost
of supplying quality (dC/ds) on the power of incentive schemes.
analysis
of
information,
the
level
these
effects,
in particular the
of quality or by
must
change
the
not
where
does not affect
£,
,
<
and
The
(their proofs are provided
gross surplus S^ is replaced by S^ -
>
£, „
the
is
(1/
12
.
a
parameter
indexing
(3.15) and (3.16) respectively.
the
The first-order conditions (3.15) to
s.
fgff
are unchanged except that <^— B
the
Note that a change in v
inverse demand curve and therefore does not change
information revealed by q and p about
1+A~)
-
—j—
~\
V
(dt
and <^—
~C\
r->
must be added to
Differentiating (3.15) to (3.17) totally with
respect to v yields:
12
of
:
marginal gross surplus with respect to quality)
(3.17)
structure
cost-reducing effort.
data about
cost
the
Suppose first that the consumers'
S^+£(ks-h# ,v)
the
Information revealed by the demand data about
three changes we consider yield identical results
in Appendix A6)
affect
For a clean
Subscripts here denote partial derivatives,
18
Proposition
3:
An increase in the concern for quality (in the sense of an
increase
in
raises
u)
power
the
incentive
of
schemes
if
quantity and quality are net complements and lowers the power
of incentive schemes if they are net substitutes.
in the marginal
Next let us consider an increase
keep
the
structure of information constant,
into C m C+m(ks-h0 p) where m,
,
to
an
increase
in
because
unchanged
verifiable)
,
>
and m.
marginal
cost.
term
ks-h#
the
The
„
we
To
transform the cost function
(an increase in p corresponds
>
structure
equal
is
cost of quality.
to
of
information
p-A+Bq
and
is
is
kept
therefore
.
Propos i tion 4:
An increase in the marginal cost of supplying quality (in the
sense
an
of
increase
in
p)
lowers
the
power
of
incentive
schemes if quantity and quality are net complements and raises
the power of incentive schemes if they are net substitutes.
let us consider an increase in k keeping
Last,
keeping
r-
r-
constant.
The point of
constant is to leave the asymmetry of information unaffected:
facing demand parameter
by choosing
services
s
8
,
the firm can claim that the demand parameter is f
satisfying ks-h0
through the demand data p and q.
Proposition
5,:
When
An increase
— ks(#)-h# without being
detected
Thus the "concealment set" is invariant.
in k keeping h/k
constant raises
the
power of
incentive schemes if quantity and quality are net complements
and lowers
the
power of
substitutes
19
incentive
schemes
if
they
are
not
Proposition
admits several interpretations.
5
First, an
increase
corresponds to an increase in the marginal willingness to pay for the
if and only if k > 1.
from
(3.16).
To see this, note that 3p/5k -
Proposition
5
thus
willingness to pay for the good tilt the
price contract for all k.
that
says
optimal
increases
5S
as
5
sS
3k3s h
f3
toward
incentive schemes.
fixed
a
—
sign 1-2(1+A)
J
k-1
.(using
(3.16)).
constant''
shows that an increase in the
the
p
marginal
the
marginal
valuation
marginal
For k >
1,
valuation
quality
for
has an ambiguous effect on the power of incentive schemes:
an increase in
good
13
i
Proposition
k
(1+A) (k-l)q/k
in
contracts
2
sign
Second,
-
ks-h0
in
For k
<
1,
quality
lowers
the
power
of
but "not too large," it
raises
the
power
of
for
incentive schemes.
The intuitions for all these propositions are similar.
An increase in
the concern for quality (or a decrease in the marginal cost of supplying
quality) makes higher quality socially desirable.
If quality and quantity are
net complements, higher quantity is also socially more desirable.
Effort
becomes then more effective since it affects more units of the product.
To
encourage effort more powerful incentive schemes are used.
Part B.
6.
Experience goods
.
Reputation incentives
.
In some industries the sales incentive
is
limited
quantity purchased is fixed or inelastic, e.g., because it
because
is
an
either
the
experience
good or the buyer is the regulator himself, as in the case in procurement,
both.
Suppose that the buyer (regulator) buys a fixed amount from
13
the
This result is similar to the one obtained when unverifiable quality
an issue (see our [1986] paper).
20
or
firm.
is
not
The
firm's
incentive
main
provide
to
quality
is
then
threat
the
of
jeopardizing future trading opportunities with the buyer rather than current
In general
ones.
quality and
future
has
supplied
First,
sales.
punishing the agent,
that link current
develop
This
past.
mechanism
reputation
a
for
when the latter
by not trading with him,
the
in
buyer may
the
instance,
for
quality
low
can think of two mechanisms
one
terms,
likely
is
to
be
particularly powerful when the buyer oversees many agents and thus has a good
opportunity
develop
to
such
reputation.
a
industrial organization tradition,
profitability of future
the buyer may infer
from
trade
the
and
Second,
closer
to
the
information about the
observation of current quality.
We
will focus on this second mechanism.
industrial organization literature has
The
reasons why a buyer may find future
quality.
On the one hand,
may have
permanent
identified two
trade undesirable
after observing poor
the quality of the product supplied by the seller
characteristics;
that
is,
the
seller
commits
investments that affect the quality level over several periods.
hand,
long-term
On the other
intertemporal link may be human capital rather than technological
the
investment.
today's
informational
The
choice
of
seller
then
quality
signals
and
conveys
his
competence
information
or
about
diligence
tomorrow's
through
future
quality even though he will manufacture a possibly brand-new product using new
In this section, we focus on a permanent and unverifiable choice of
machines.
Appendix B
technology.
quality producer.
The
develops
a
model
of
reputation
two models yield similar results.
for
being
a
high
We will emphasize
the new insights added by the new model in the Appendix.
For the seller to care about future trade,
trade creates a rent.
of rents
(due,
necessity
for
for
the
it must be the case that this
This rent can be an informational rent, but other types
instance,
buyer
to
to
bargaining power of the
seller,
or
to
"efficiency
scheme
to
create
offer
an
incentives) are consistent with the model.
21
wage"
the
The model has two periods,
- 1,2.
r
In period
seller (regulated
the
1,
firm) produces one unit of the good for the buyer (regulator),
(6.1)
- ^-e-j+s,
Cj_
where
is
C,
the first-period, verifiable cost,
an efficiency parameter,
/9,
the firm's effort to reduce the first-period cost,
As in the sales incentive model of Sections
monetary cost,
The variables
/?,
,
and
e,
s
the
monotone
and
through
2
^(e^
it(s)
formalized as a
is
s
5,
(with
> 0, V" >
\i'
r
d
property
rate"
0.
the regulator
with density f (/3.
on [p.,fi.
F(/3..)
dfi
involves disutility
the level of "care."
s
are private information to the firm;
"hazard
c
.
>
.,
t(P^)
1
that
)
F(^)1
V
Effort
0.
e,
1
With probability
^ 0).
the product "works" and yields a gross social surplus
e [0,1],
e..
interpreted as a non-monetary cost.
but can alternatively be
has a prior cumulative distribution
satisfies
at cost:
S.
;
with
probability (1-tt(s)), the product is defective and yields gross social surplus
We
0.
will
respectively.
to
say
that
We assume that
avoid a corner
produces
firm
the
>
it'
solution at
0,
-
s
it"
<
0;
and
0)
a
high
as well as
it"'
<
not
verifiable
by
a
court
that
so
the
(0)
item
- +» (in order
a
is
sufficient
Whether the product
condition for the regulator's program to be concave).
is
it'
(which
works or is defective is observed at the end of period
quality
low
or
1
regulatory
by the regulator, but
contract
cannot
be
contingent on the quality outcome.
Parties have a discount factor
the
simplest
model,
we
assume
between the two periods.
>
S
that
the
second-period
firm's
To obtain
product
defective if and only if the first-period product is defective (that is,
first-
and
second-period
quality
outcomes
both
are
first-period level of care and are perfectly correlated.
what
matters
for
the
results
that
is
expected quality level in period
2)
won't be asked to produce in period
2
a
higher
s
in
determined
the
the
As is easily seen,
period
Our assumption implies
.
by
is
1
raises
that
the
the
firm
if its product is defective in period 1;
22
in this case,
the second-period social welfare and firm's rent are normalized
denote the second-period expected rent for
and W„ >
Let £L >
to be zero.
the firm and expected social welfare.
W„ are independent of
Let
e,
{s (/?.),
and U,
functions,
with type
/9,
/?,
(/?, )
14
For simplicity, we assume that U and
?
.
denote the firm's first-period care and effort levels
)
denote the firm's first-period rent.
(/?, )
Note that a firm
can always duplicate the cost and probability of high quality of
a firm with type
(/9,-d/L)
by choosing levels {s(/L
-d/9..)
,
e..
(/9.-d;9..
)
}
so that
the incentive compatibility constraint for effort is:
U (^ ) - -V'(e (^ )).
1
1
1
1
(6.2)
Next consider
level of care by
by
At
1.
the
firm's
the
choice
To reach the same cost,
1.
such
margin,
Suppose
of care.
changes
it
raises
the
the firm must increase its effort
not
do
that
affect
the
firm's
The
rent.
incentive compatibility constraint with respect to effort is thus:
Sn' (s)U
(6.3)
2 -V>'
(e
L
)
- 0.
suppose that the firm cannot produce in period
Next,
(and thus invested)
2
if it hasn't produced
The individual rationality constraint says
in period 1.
that the firm must obtain at least its reservation utility, which we normalize
14
To give an example:
the firm produces one unit of the same
suppose that (a)
good or a related good in period
2
at cost C~ -
/3
?
-e„, where ^„e[^„,^„]
second-period efficiency parameter, is uncorrelated with
the firm only) between the two periods
(/3„
and is learned (by
might reflect the new input costs);
the second-period gross surplus is equal to S„
(b)
/?,
is the
(possibly equal to
S.. )
if
the firm has produced a high quality item in period 1, and zero otherwise; and
(c)
the regulator offers the second-period contract at the beginning of
period 2 (no commitment).
In this simple example, U„ and W_ are computed as
in Laffont-Tirole
to
~{}j
,
[1986].
Furthermore,
lL is independent of S„.
23
if S«
is
sufficiently large relative
at zero:
for all @*
(6.4)
U (^ )+5U tt(s(^ )) > 0.
2
1
1
1
As
usual,
is
i
only:
individual
rationality constraint
binding
is
at
/L
-
j}.
15
U (^ )+5U 7r(s(^ )) 1
2
1
1
(6.5)
Note
the
:
type
that
rent
first-period
"buys
/L
for
in";
an
0.
that
expected
is,
he
is
willing
second-period
trade
to
a
negative
Expected
rent.
social
welfare can then be written:
±
(6.6)
[
|
7
r(s)(S +5W )-(l+A)(^ -e +s+V (e ))-AU (^ )]f(^ )d^
1
2
1
1
1
1
1
1
1
)
The regulator maximizes
and
u(/L )f (/9,
respectively.
H-
(6.7)
)
denote
(6.6)
the
subject to (6.2),
multipliers
(6.3),
and (6.5).
constraints
of
.
(6.2)
Let
and
/*(/?-,)
(6.3)
The Hamiltonian is:
[7r(s)(S +5w )-(l+A)(^ -e +s+V-(e ))-AU
1
2
1
1
1
1
]f-^'
(e
1
)
+i/f[$7r'(s)U -^'(e )].
2
1
1
We have:
Be3cause
/9,
is a free
boundary and
F(/?,
)
Indeed it can be checked ex post that
- 0, we thus obtain
-7-5—
<
0.
Such "buy in" phenomena are typical of non-commitment models, in which firms
trade off current losses and future rents.
See, e.g., Riordan-Sappington
[1988].
24
/i(^) - AF(^
(6.9)
L
).
Taking the derivatives of H with respect to the control variables
e,
and
s
yields
(6.10)
r
(6.11)
tt'
we
will
As
-
(V
i
-
f(^y ^(^)
ITA
(s)(S +6W )-(l+A)+^57r"(s)U
see
in
proof
the
of
TTA
next
the
conditions for maximization are satisfied.
is
thus
given by
(6.10),
(6.3),
and
^"< e
l>
- 0.
2
2
1
"
proposition,
The solution
(6.11).
second-order
the
(/L
(e..
We now derive
)
,
s(/L
)
,^(^-,
)
comparative
the
statics results using the interpretation of the optimal incentive scheme as a
menu of linear contracts (see Appendix A. 7).
Proposition
6.:
contract tends toward a cost-plus contract
incentive scheme decreases for all
a)
the discount factor
b)
(if
5w
2
-5-pr-
5
:
/3. )
(the
slope of the
when
decreases;
au
>
2
and -^— -
See footnote 14 for an example)
0.
the social value of quality S.
Proof
The first-period
Optimal cost reimbursement rules are linear.
increases.
See Appendix A. 8.
Thus,
low-powered
illustrates
when quality becomes very
incentive
scheme
to
important,
supply
the crowding- out effect,
more
the
care
firm must be
(result
b)).
given
a
This
according to which care and effort are
substitutes, so that the production of more quality is obtained at the expense
of
cost-reducing
activities.
result
Last,
25
a)
formalizes
the
reputation
Far-sighted firms can be given high-powered incentive schemes.
argument.
7
Concluding remarks
.
.
We have analyzed the circumstances under which quality concerns call for
low-powered
schemes.
incentive
For
experience
an
good,
lack
the
of
informational value of the current sale indicator makes the cost reimbursement
rule
the
only
instrument
achieve
to
conflicting goals
the
We have also argued that steeper
incentive
schemes
optimal if the supplier is sufficiently eager to preserve his reputation.
a
there
search good,
can be provided.
on
the
power
low-powered
of
is
no
"crowding-out effect"
There is, however,
incentive
incentive
of
A high concern for quality leads to low-powered
quality and cost reduction.
incentive schemes.
of provision
schemes;
schemes
if
a new,
a
direct sales
For
incentives
"scale effect" of quality concern
high
and only
as
are
if
concern
for
quality
quantity and quality
leads
are
to
net
substitutes
The paper has considered the important polar cases of a search good with
scale effects
(in which a higher output makes
cost reduction more valuable)
and of an experience good without scale effect (for which ouput was taken as
given)
.
Understanding the crowding-out and the scale effects makes it easy to
extend the theory to cover the other two polar cases. Consider first a search
good and assume that the cost-reducing activity affects the fixed cost rather
than the marginal cost:
output changes,
18
C — cq+/?-e+s.
17
When quality becomes more desirable,
but the optimal effort is unaffected.
Because direct sales
incentives can be given and because there is no scale effect,
the
power of
This technology is chosen so as to keep the adverse selection one -dimensional
and as to yield a closed- form solution.
The analysis, which follows the lines
of Section 3 is left to the reader.
18
For the above cost function, output increases,
complements as long as they are gross complements.
26
as
the
goods
are
net
incentive schemes is independent of the demand for quality.
an experience good with variable scale such that a)
Second,
consider
a higher quality raises
effort reduces
the marginal
the demand for
the good and b)
such a good,
higher valuation for quality raises demand for the good and
a
the
creates a scale effect that may offset the crowding-out effect.
Effort
reduces
marginal
cost
Search Good
Experience Good
+ is q and s
net complements
Ambiguous
Thus we have:
— if q and s
net substitutes
Effort
reduces
fixed
cost
Table
For
cost.
-
Effect of an increase in the demand for quality on the power of
incentive schemes
1:
We should also note that scale effects can take other forms than the one
obtained in this paper.
If the two moral hazard variables
2
the cost function (3 C/3s3e * 0)
,
s
and
e
interact in
an increase in the demand for quality has a
direct effect on effort, not only an indirect one through the complementarity
or substitutability of quality and output.
a
is
good working hypothesis,
one
While ruling out this interaction
can think of situations
in which effort
produces a new technology that reduces the marginal cost of providing services
2
(3
C/dsde < 0)
effect,
that
.
We conjecture that in such situations,
leads
to
high-powered
incentive
schemes
there is a new scale
when
the
demand
for
quality is high.
Our
theory may
regulatory
also
hierarchies,
shed
for
light
some
instance
objectives than maximizing welfare.
ones
on
in
the
which
For instance,
27
behavior of more
regulators
complex
have
other
suppose that the regulator
derives perks from the supplier's delivering high-quality products.
the
regulator values quality more
than the public
at
large.
will then lobby in favor of low-powered incentive schemes
experience
Defense
good.
officials
Our
theory
who
value
thus
offers
quality
a
highly
fixed-price contracts into cost-plus contracts.
Last,
the
distinction between
clue
as
sometimes
to
if
That is,
The
the
why
manage
regulator
good is an
Department
to
of
transform
19
verifiable
and
unverifiable
quality
is
extreme. More generally one would want to allow quality to be verifiable at a
cost.
monitor
It would be worthwhile to analyze the relationship between expenses
quality,
quality
and
reputation
concerns
and
power
of
to
incentive
schemes
See also Scherer [1964, pp. 33-34 and 236-239]. DoD officials are well-known
favor performance over cost.
They often feel that fixed-price contracts
encourage contractors to make "uneconomic" reliability tradeoffs and make them
reluctant to make design improvements.
to
28
REFERENCES
S.
and L. White
"Monopoly and Quality
Donnenfeld,
D.,
[1987]
Besanko,
Effects and Remedies," Quarterly Journal of Economics, 102,
Distortion:
743-768.
Guesnerie, R. and J. -J. Laffont [1986] "A Complete Solution to a Class of
Principal-Agent Problems with an Application to the Control of a
Self-Managed Firm," Journal of Public Economics 25, 329-369.
,
A Dynamic Perspective,"
Holmstrom, B. [1982] "Managerial Incentive Problems:
in Essays in Economics and Management in Honor of Lars Wahlbeck
Swedish School of Economics.
Helsinki:
Rose
"The
Effects
of Economic
and N.
P.
[1987]
Schmalensee and R.
Willig (eds.),
forthcoming in R.
Industrial Organization
Joskow,
Kahn,
A.
[1988]
Cambridge:
Kreps
The Economics of Regulation:
MIT Press.
Regulation,"
Handbook of
Principles and Institutions
D. and R. Wilson [1982] "Reputation and Imperfect Information," Journal
of Economic Theory, 27, 253-279.
,
Laffont, J. -J. [1987] "Optimal Taxation of a Non Linear Pricing Monopolist,"
Journal of Public Economics 33, 137-155.
,
Laffont, J. -J., E. Maskin and J.-C. Rochet [1987] "Optimal Non Linear Pricing
with Two -Dimensional Characteristics," Chapter 8 of T. Groves, R. Radner
Incentives and Economic Mechanisms,
and S. Reiter (eds.), Information
University of Minnesota Press.
,
Laffont, J. -J. and J. Tirole [1986] "Using Cost Observation
Firms," Journal of Political Economy, 94, 614-641.
to
Regulate
Laffont, J. -J. and J. Tirole [1988] "The Regulation of Multiproduct Firms,
Theory," forthcoming, Journal of Public Economics.
I:
Lewis, T. and D. Sappington [1988a] "Monitoring Quality Provision in Regulated
Markets," Discussion Paper, U.C. Davis and Bellcore.
Lewis, T. and D. Sappington [1988b] "Regulating a
Demand," American Economic Review, 78, 986-998.
Monopolist with Unknown
Milgrom,
and J.
P.
Roberts
"Predation,
Reputation
[1982]
Deterrence," Journal of Economic Theory, 27, 280-312.
Riordan, M. and D. Sappington
Journal of Economics
[1988]
"Second
M.
and S. Rosen [1978] "Monopoly
Economic Theory, 23, 301-317.
Mussa,
and
Sourcing,"
Product
and
forthcoming
Quality,"
Entry
Rand
Journal
of
Sappington, D. [1983] "Optimal Regulation of a Multiproduct Monopoly with
Unknown Technological Capabilities," Bell Journal of Economics, 14,
453-463.
29
Scherer, F. [1964] The Weapons Acquisition Process:
Economic Incentives
Graduate School of Business Administration, Harvard University.
Spence,
A.M.
Tirole, J.
Press
Quality
"Monopoly,
417-429.
[1975]
Economics
,
6,
[1988]
The
Theory
Vickers, J. and G. Yarrow
Cambridge:
MIT Press.
of
[1988]
and
Industrial
Regulation,"
Organization,
Privatization:
30
An
Bell
Journal
Cambridge:
Economic
of
MIT
Analysis
APPENDIX A.l.
Let 7 -
ft
+
v-6
and assume for notational simplicity that h -
reasonable in this problem to assume that the distributions of p and
and F^
(
•
)
It
k.
is
F,
8,
(
•
Then
are independent.
.+»
F(7) ^ Prob(7 < 7) -
-1
-J
f( 7 )
-
f (6)f
2
J
1
Prob(0 < 1 < 0+d0)Prob(/9 < 7-0)
I
-00
f (B)? (-r-e)dB,
- CO
2
l
(T6)d6,
and
£ {6)F
J
d_
F( 7 )
d7
f(7)
-
2
1
.
l
/3
is
uniformly distributed)
,
F(7)
f(7)
£ (6)£' (T6)d8
2
1
.2
f (B)f
J2
uniformly
-r-
J
+00
J
In particular, if
1 -B)dO
(
1 -B)dB)
l
(
distributed
1
>
(or,
symmetry
by
,
is
0.
More generally, if one of the densities is non increasing
0)
if
(fC
<
or
f'„
<
F(') satisfies the monotone hazard-rate condition.
APPENDIX A. 2.
v
The firm is faced with the revelation mechanism t(7)
'C"
,
p(7)
For simplicity we assume that the mechanism is differentiable
and Laffont [1984] for a proof of this).
q(7)
(see
,
X
(7)
Guesnerie
The firm chooses the announcement
which maximizes its objective function, i.e., solves:
31
,
7
(A2.1)
Max{t(7)-^(7 +
*&+&*&
jjtf))}
-
7
<->
Max|t(7)-V'(7-z( r))},
:
7
where
C,~.
,~v
z(7) m -(7)
-
p(7)-A+Bq(7)
r
£ a
•
The first-order condition of incentive compatibility is:
(A2.2)
t( 7 )+V-' (7-z(7))z(7)
- 0,
and the second-order condition is
(A2.3)
z( 7 ) > 0.
(A2.2) and (A2.3) constitute
necessary
and
sufficient
conditions
for
incentive compatibility (see Guesnerie-Laffont [1984]).
Note that z(y) — 7-e(7)
(A2.4)
So the second-order condition can be rewritten:
.
4-1 < 0.
APPENDIX A.
3.
The regulator maximizes
H
with
respect
to
{q,p,e}.
The
matrix
second-derivatives of H with respect to these variables is (divided by f )
B-B
(1+A)
2( 1+A)B
2
k
-
B
-
(1+A)
(1 + A)-B
-
Ugl
1+A
-1
d+A)r
1+A
32
AF\A"'
of
[This matrix differs from the Jacobian of equations (3.15) through
that (3.16) was used to obtain equation (3.15).]
(3.17)
For the sub-matrix in
in
(q,p)
to be semi-definite negative it must be the case that
(A3.1)
B > 1
-
2(1 A)
^
and
(A3. 2)
B(1+2A) > (1+A)
:
The determinant is:
A - -(l+A)qA +(l+A)'
1
wh ere A,
is the determinant of the submatrix in (q,p).
A <
Or:
<-> qA, > 1+A.
Solving (3.15) and (3.16), we get:
A q - (l+A)(A-( 7 -e))„
x
To sum up, sufficient conditions for the second-order
conditions
to
be
satisfied are:
A > 7+1
B > Max^l
Max/l
-
2(1+A)
k
<
,
1+A >
1+2A
:
1
-
)
Moreover, the solution obtained in Section
incentive constraint e
<
satisfied
is
1
by
3
is
the
valid if the second-order
solution
first-order equations.
Differentiating (3.15),
(A3. 3;)
-fA +
(3.16) and (3.17), we obtain:
H±A>l Bq+(1+A)
1
-
jMp + (1+A)e - 1+A
33
of
the
above
(A3. A)
((l+A)(k-l)-Bk)q-kp -
(A3. 5)
q
-
V-"(e)
+
A
F( 7 )
1+A f(7)
\i'"(e)
TT\*
(e)
fF(7)1
d^ f(7)
Hence,
P -
(1+A)(k-1)-Bk (1+A) ^"< e > +
+
TTxIW)*'" (e)
F(7) ]
f(7)J
T^" (e) ^
where A is the determinant of the system which is negative since we are
maximum.
Since V" >
if B > 1,
which,
implies that
(A3. 5).
8S g
^
0,
V>'"
>
and
>0,p>0
d 7 |p^-j
is automatically satisfied in
kq(l-(l+A)(l
In that case,
-
?-)
)
> 0.
the
for
relevant
From (A3. A), q <
the second-order condition of
range:
and
incentive
a
^ 1+A)(k-1)
>
B
at
e
(3.16)
<
from
compatibility
is satisfied.
Appendix
A. 4.
The dichotomy result between optimal
incentive schemes obtained in Section
3
pricing
and
quality
and
optimal
can be generalized as follows:
Let C(/3,e,q,s) denote a general cost function and let
obtained by inverting the demand function
q -
s
—
£(0,q,p)
be
D(p,s,0).
The derived cost function is:
C(0,0,e,q,p) - C(^e,q,aJ,q,p)).
Let
e
-
E(/?,
,C,q,p) be the solution of
C(£,0,e,q,p) - C.
The problem is here genuinely two-dimensional.
constraints can be written:
34
The first-order incentive
U^ - -V'(e)E^,
>
V
-
where subscripts denote partial derivatives.
A sufficient condition for the dichotomy result is
8
«g
2&
3q
dp
h
B
" dq
20
d
l±
n
dp
which requires (from Leontieff's theorem) that
exists
there
A(',«).
r(',«)
such that
C -
C(A(£,e),r(0,e),p,q),
a special case of which is
C - C($(j3,0,e),p,q).
In our example, we have $(fi,8,e) —
Appendix A.
fi
+ t—
-
e.
5.
Since the equations defining quality and price
are
complete information (dichotomy result) the comparison
incomplete information can be easily obtained
information gives a lower level
of
effort
20
by
same
the
between
observing
conditionally
under
complete
that
on
as
the
and
incomplete
level
of
the derivatives of the Hamiltonian with respect to
associated with the incentive constraints and
therefore with asymmetric information.
If these conditions hold,
terms
p and q involve no
35
This decrease
production (equation (3.17)).
(3.15) and (3.16) to a decrease of q
of
effort
(reinforcing
initial
the
decrease
effort which is therefore an unconditional
itself
of
leads,
from
decrease
effort)
and
decrease
in
to
of
an
increase in p.
For any
7
let
us
call
(de)
infinitesimal
an
effort.
Differentiating (3.15) and (3.16) yields:
kS§ -
&
B^
de
+
- (1+A)
1
-
11
de*
The conclusion follows from the fact that e and q are lower
under
incomplete
information from the first- and second-order conditions.
Appendix A.
6
Proof of Propositions
3.,
4,
and
5.
We offer a single proof to the three
for the shadow price of (3.10)
(A6.1)
2
H - |q +(l+A)pq
(1+A)
A
Letting A <
|V
(
-
7 -e)q+q
propositions.
(see (3.14)),
After
substituting
the Hamiltonian becomes:
i(p-A+Bq) 2 +t (p-A+Bq
,
1/)
p-A+Bq
+m(p-A+Bq,p)+V'(e)
k
(e)
denote the determinant of
control variables (q,p,e), we have
36
the
Jacobian
with
respect
to
the
2
31
a
5q
(A6.2)
|-
2
2
2
h
3q3p
1
a
A
3q3p
h
aqax
2
ai
2
31
1
a
3p3x
2
a
3p
2
2
a
2
h
a
2
31
h
3e3p
a
3eaq
dedx
for x - v, p, k.
2
2
—
2
=—g- But = ^— - x(1+A),
'
3e3q
aea\
(A6.3)
3
and
de
dx
sign
sign
H
dedx
'
—
for x - v
2
31
2
31
3q3p
3q3x
2
31
2
31
,
p,
k.
Hence,
3p3x
3p
The propositions follow from:
o
|_H_-
(1+A) |\
.
lj-B(l +mn (l + A)-£
11 )
2
^-| - -(l+m 11 (l+A)-£
3p
Q
)
2
2
_ R»
3q3i/
31
3q3k
n
12
1+A
-
"2
„ 3JH_
_ a
3p3»/
(p- A+2B q)
k
[To obtain Proposition 5,
2
2
Z
.
3JH_ _
'
3q3/>
a
;
cm
"
h
spin; -
12
_ D
3
H
,
3p3p*
i+A
-£*-
use (3.16) to substitute (p-A+Bq)]
37
Appendix A.
The linearity of contracts.
7:
From Laf font-Tirole [1986] we know that we can decentralize a
contract with a menu of linear contracts if
transfer
the
t(C,)
non- linear
is
convex.
From the first-order conditions of incentive compatibility
dc i
1
/
N
+ *,, (e
}
i dX-
hidt
a*
dC.
>
with
from second-order incentive compatibility conditions.
2
dt
- -V
(ep
d t
and
1
,„•
dC,
1
dC
Hence
-d/L
t(C,
)
is convex iff e,
<
0.
Differentiating
first-order
the
conditions
we
get:
2
-
:
1-
FC/^)
^l>
"^"V ITA^d^f(^)
-
<
APPENDIX
F(
d
A
(
(S-j+fii^)
(n"+vSn"' U
2
)
2
(1+A)V>"+A
j^j
V"'+^"'
(V>")'
0.
A. 8.
Proof of Proposition
6:
Differentiating (6.10),
yields
(A8.1)
)"
(^7r"U
V>"+
A
F
_v_
1+A
f"
1+A
V"
A
de
l
-
"
V
TTA
38
A
du
-
(6.11),
and
(6.3)
totally
3W
(A8.2)
7r"(S
1
+
(A8.3)
+5W )+i/5U
2
[i/7r"U
V»"de
2
ds + tt'+
7r"'
2
-,
2
6
as7
dS.
]d$+67r"iLdi/ - 0.
+7r'W
- (7r'U )dS+(SU 7r")ds.
1
2
2
Substituting:
C«
~\
3e
(A8.4)
sign
(A8.5)
sign
aw,
sign
as.
n'
+8
< o
asT
3e,
- sign(-7r"7r' U-S.-fi^w'"
35
using (6.11), V"' —
assumptions V"' —
— 0-
an<3 f"'
aR d it"
tt'
+6U
2
(7r")
2
") > 0,
it is easily
Furthermore,
seen
that
the
guarantee that the second-order conditions are
^
satisfied.
§
Appendix
§
§
Quality, asymmetric information and reputation.
B:
We consider a two-period model in which the firm is, with probability
a high-quality producer,
i.e.,
generates a social surplus
With probability 1-x it is a low-quality producer, i.e.,
when
S
x,
producing.
generates
a
social
the
social
To avoid signaling issues which would complicate the analysis, we
assume
T
surplus
S
surplus is
JJ
<
S
S
.
,
unless it puts some effort
s
>
0,
in which
that the firm does not know in period 1 if it
is
producer.
Holmstrom
(Thus
the
model
is
closer
to
a
high
case
or
a
[1982] 's
low-quality
than
to
Kreps-Wilson [1982]'s and Milgrom-Roberts [1982]'s)„
Moreover, quality cannot
be contracted upon.
production
At the
end
of
period
regulator discovers the quality level.
39
1,
if
occurs,
the
the firm has a cost function
In period 1,
where
/?,
known
is
and
to
C.
observed
the firm is putting the extra effort
Let
regulator,
s
The regulator cannot commit for period
e
fi~
[P,fi]
and
e,
-
In period
a
is
e..
when
e+s
to make sure that first-period
be the transfer given to the firm in period
t,
cost parameter
the
Total effort for the firm is e. - e or
cost-reducing effort.
is high.
by
quality
1.
the
firm
has
a
with a (common knowledge) distribution
F(«)
and
a
2.
2
cost function:
C
/?„
2
- P -e
2
2
.
will be learned by the firm at the beginning of date
2.
Therefore
at
that time we will be in a one-period adverse-selection problem (Laffont-Tirole
[1986]).
For any expected social surplus
S
in period 2, second-period welfare is:
a
W*(S) -
{s-(l+A)(V-(e*(^ ))+ 9 -e*(^ ))-AU*(^ )|dF(^ )
2
2
2
2
2
/
J
where (according to Laffont-Tirole [1986])
*-C*0» 2 »
-l
-
ITX^ *"<*>>>>
h
U*(/3
Let t
,
C
2
)
-
,
U
-
J
V'(e*(? 2 ))d£ 2
.
be the expected transfer, cost and rent
in
this
second-period mechanism.
To limit the analysis to the most interesting cases, we postulate
40
optimal
Assumption
1:
If
S
- xS H +(l-x)S L
If
S
-
Assumption
is
1
S
,
W*(S) >
for any F(-) on
[§,J3]
,
W*(S) <
for any F(«) on
[/3,/9]
L
means that for an expected quality defined by the prior,
worth realizing the project.
However,
if
the
firm
is
known
it
be
to
a
low-quality firm, it is not worth doing it, even if attention is restricted to
the best types (close to
/?)
.
Assumption
1
requires that
not
(.fi-fi)
be
"too
large.
r,
Assumption
2:
Assumption
,„H „L.
<. (l-g)(l-x)
- (S -S ).
y-7-r
s
2
means that the extra effort needed to upgrade quality is not
H I
too high compared to the social gain (S -S
).
Without this assumption, it
is
never optimal to induce upgrading of quality in the first period.
Two policies must be considered.
policy
B,
s
is
In policy A, s is not induced,
and
in
induced (inducing randomization by the firm is never optimal).
Policy A:
Social welfare is
H
L
H
W - xS +(l-x)S -(l+A)(^ -e +t )+5x(S -(l+A)(t*+C*))+U,
1
1
1
where U is the firm's intertemporal rent, and we use the fact that in period 2
the regulator knows whether he is facing a high-quality firm.
Social welfare must be maximized under the
individual
rationality
(IR)
and incentive compatibility (IC) constraints.
If the firm does not produce in period
and the firm can expect a rent U
1,
in period 2.
takes the form
41
the
regulator
learns
nothing
Accordingly, the IR-constraint
(IR
t^fie^+SxV
A)
> 6U
The IC constraint Is
(IC
t
A)
1
-V'(e
1
In regime I,
tj>'
t
(e,
-
V'(e
x
U - 5U
(IC
1
1
where $(e
1
1
or
e,
- e
)+5(l-x)U
^
)
or5<
> 5(l-x)U
)
(l-x)U
S
de
>
W
> W
,
Policy
B:
1
11
-V'( e
)
<
6,,
is binding,
(IC.)
and e(S) > e
W
V>(e+s)
*»
.
5(e(«)) - 5(l-x)U
-??
+s)+5U
.
In regime II,
W
-V'(e
can be rewritten with $(e)
)
$(e
with
1
is not binding and therefore
(IC.)
-
)
> t
)+5xU
*
.
and effort is defined by
,
Welfare is then
- xS H +(l-x)S L -(l+A)(/3 -e*+V(e*))+5xW*(S H )-A(l-x)5u"
1
H
L
H
- xS +(l-x)S -(l+A)(/9 -e*(5)+V'(e*(5)))+5xW*(S )-A(l-x)5U*.
1
but regime
I
is not feasible for 6 > 5,
.
Social welfare is now:
W -
,H
i
S
'-(l+A)(^ -(e -s)+t )+5
1
1
1
S
42
H
-
H L
(1-x) (S -S )
-
(1+A) (t*+C*) +U
with the constraints
(IR
)
t -V>(e )+5U*
1
1
)
t
fi
(IC
fi
-V'(e
1
1
)
> £U*
>
+ 5U
In regime III,
(IC
t
)
1
-V'(e
1
-s)+5xU
.
is not binding,
and e^ - e
*
,
- V>(e,).
t,
fi
Regime III prevails as long as
5( e;L -s) - V(e 1 )-V'(e 1 -s) < £(l-x)U*
or
> 5„ with
S
If
< 5„,
6
> 5„.
5,
(IC
R)
is
binding and
is defined by
e..
5(e -s) - 6(l-x)U*
1
or
- e(S)+s with s+e(S) <
e
e
(regime IV).
1
For
<
6
6
defined by
— e(5), it is not possible to induce quality.
s
Welfares are
W
111 «
W
TV
xv
-
S
S
H
H
1
H
n
*
1
FIGURE
1:
W
> W
H
L
))
H
n
-(l+A)(^ -e(5)+V'(e(5)+s))+5(W (S )-(l-x)(S -S
Effort levels are summarized in Figure
Lemma
H
-(l+A)(^ -e*+s+V»(e*))+5(W*(S )-(l-x)(S -S
2
,
43
2;
HERE
T
L
))
II
III
/f-K^e(6)+s
' Iv
i
->
6
6.
FIGURE
43A
2
Proof
:
W
111
^
L
- (l-fi)(l-x)(S H -S )-(l+A)s
1
+ (l-x)5(S
H
-(l+A)E(V>(e*(/3
2
)+/?
-e*(/?
2
2
))).
The sum of the first two terms is positive in view of A.
2
and
the
third
term is positive in view of A.l.
Q.E.D.
< W
As W
regime III is optimal for
,
6
> 5„.
For
6
<
S,
regime
I
is
is
to
clearly optimal.
Lemma
2:
a)
There exists
e [5,5„) such that for
b
5
>
6
optimum
the
induce quality and
:,
Proof
:
- e(5)+s < e
if
ft
<
S
- e*
if
5
>
S
H
L
and
e,
— e
a)
At
5
b)
H
jg- - xW*(S )-A(l-x)U*
IV
^-
for
Q
.
A
S
,
S
< $ the optimum is not to
If
-
Q
"*
< AU
b)
-S
<
S
induce
,
5
n
,
W
- W
.
Apply Lemma
1.
- U*(S H )-(l-x)(S H -S L )-(l+A)(^(e(5)+s)-l)^f
and
de
(l-x)U*
dS " V(e(<$)+s)-y>' (e(6))
dW
IV
d$
/ KH
(l+A)(l-x)U (l-^(e(g)+s))
44
..
vW „H
„L
quality
If
S
H
-S
L
IV
< XV*
,
I
^r~
d6
> ?£-, hence the result.
q6
Q.E.D.
Thus, we get:
Proposition
7:
a)
Incentives are higher when one does
want
not
to
induce
incentive
scheme
quality.
b)
Conditional on
quality,
inducing
becomes more high-powered when
Proof
a)
:
follows from the fact that
b) follows from
de
-tt
>
e,
5
the
increases.
< e
when quality is induced.
.
Q.E.D.
The results are thus very similar to those of the moral hazard
Section
factor.
is
6.
A difference is that incentives are no monotonic in
model
the
For a low discount factor, an extremely low-powered incentive
needed for the firm to exert
s.
The loss of incentive to
too costly, which yields a corner solution:
to provide quality and /L
fixed-price contract.
reduce
Because the firm is
is common knowledge,
not
of
discount
scheme
cost
is
induced
the first-period contract is
a
But as long as the discount factor is sufficiently high
to make it worthwhile to induce quality,
with the discount factor.
45
incentives to
reduce
cost
increase
2385 001
Date Due
&-j
g6
1990
582
,
ROW-
l|RLt8
W
3
2003
Lib-26-67
MIT LIBRARIES DUPL
3
I
TOAD DDS7TOD5
7