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Product Safety, Liability Rules and Retailer Bankruptcy]
Marilyn J. Simon
Massachusetts Institute of Technology
Product Safety, Liability Rules and Retailer Bankruptcy
Marilyn J. Simon
Massachusetts Institute of Technology
Robert G. Wolf
Tufts University
Jeffrey M. Perloff
University of California at Berkeley
Number 304
July 1982
We would like to thank P. Diamond, J. Farrell,
Friedlaender A.M. Polinsky and R. Quandt for many
helpful comments.
A.
,
ALT. LIBRARIES
AUG 2 6
1982
RECEIVED
.
Product
Liability
Safety,
Rules
Marilyn
J.
Simon
Robert
G.
Wolf
Jeffrey
M.
Retailer
and
Bankrupcty
Perloff
Abstract
In this paper, we examine the levels of care which would be chosen by a
monopolistic manufacturer
turer and
retailer
can
competitive
and
affect
the
when
retailer,
both the manufac-
probability of an accident and when the
manufacturer cannot observe ex ante the care chosen by the retailer.
that
if retailer
bankruptcy
contract
between
the
retailer liability is superior
If retailer bankruptcy
to manufacturer liability.
rium
not possible,
is
manufacturer and
is
possible,
retailer will
profits for the retailer and inefficient levels of care.
on
the
equilibrium
contract
by
either
increasing
the
are
sufficiently
high,
the
positive
profits
and
the equilib-
include
positive
Society can improve
retailer's
liability or by increasing the retailer's "due care standard".
costs
We show
If
share
bankruptcy
inefficiencies
eliminated
074489
0744895
of
are
-1-
Introduction
I.
Whenever the manufacturer of a potentially hazardous product does not
directly
also
distribute
to
it
consumer,
the
there
exists
inducing proper precautionary behavior from the retailer.
problem
a
of
Ethical drugs are
dangerous unless the prescriptions are filled and the consumer informed about
their use,
in
appropriate manner.
an
equipment
require
they
delivered
are
deadly if the
and
inadequate
as
taken
installation,
falls
of dealer service
into
this
any
product
inspection
or
tools,
and
other
sufficient to insure
Electrical
defect.
Indeed,
use.
to
separate
by
serious
power
accompanied by inspections,
not
is
advice,
consumer
final
without
sale
education
and degree
a kind
Automobiles,
appliances
proper
which
can
be
installation,
when accompanied by
imposes
undue
category if manufacture and
risk
on
the
sale are under-
firms.
From either the perspective of social efficiency (i.e., the minimization
of the expected value of all accident
related costs) or the maximization of
manufacturer's profits, this problem can be solved by any of the negligencebased,
common-law product
For example,
precautions
liability rules or their contractual equivalents.
consider the case in which there were no cost effective safety
which
undertaken by the consumer and
can be
the
rule
that
the
manufacturer would be strictly liable for accident costs unless the retailer
used less than the efficient due care level in which case the retailer would
be liable.
in
Under these circumstances
equilibrium
inputs.
This
would
is
discussed in Brown
choose
true
of
[1973],
In
is
well known that the two parties
economically
the
any
it
of
the
efficient
negligence
addition without
based
levels
of
common
law
safety
rules
imposition by the courts of
any of these rules, the manufacturer and retailer would agree to a contingent
contract
which
would
be
one
of
these
rules
equivalent.
For
any
contract
-2-
which
miminize
not
did
producing
And second,
.
obligations.
tual
equilibria
care
all
there would
both
parties.
of
these
rules,
exist
one which did while
requires
however,
assumed observable by the parties,
is
key
two
least ex
at
agents are presumed to honor their legal or contrac-
In
optimal
and
costs
for
functioning
First,
assumptions.
post
profits
higher
proper
The
social
this
paper,
we
examine
legal
rules
(a)
whether or not
both
impact
the
market
on
information is ever
available about retailer care and, more centrally (b) when the retailer can
escape
liabilities
large
bankruptcy.
through
The market outcomes involve various contracts which create value in the
retail
not
in
franchise so that
go bankrupt
general,
adopt
rules
if
the
a defect
retailer exhibits some amount of care and does
causes
inefficient.
are
other
than
may,
It
those
loss
to
to
then,
which
consumer.
the
be
the
These equilibria,
advisable
parties
the
courts to
have
agreed.
for
would
This work is a special case of a more general principal-agent problem.
The same sort of incentive management situations occur whenever a retailer's
quality or quantity
mis-,
mal-,
or
inputs
are
nonfeasance
are
costly to observe and when the costs of his
borne
Creating
by others.
a
value to the
franchise which could be potentially forfeited can generate retailer behavior
desirable
from
the
perspective
Our model
efficiency.
of manufacturer
involves
a
profits
if
not
of economic
retailer whose care is not observable «c
ante and who has no exogenously given reason for staying in business.
other
hand,
safety
manufacturer's
our
because they are observable,
inputs
are
known
perhaps because of reputation.
by
all,
equity
could
be
financing.
handled
by
may
be
It
a
competitive
easiest,
firm
though,
with
for
perhaps
Further,
known that he will not declare bankruptcy to avoid liability.
condition
On the
it
is
This latter
sufficiently
large
purpose
both
the
of
-3-
as8umptions if we visualize the manufacturer as a monopolist.
As the single
firm in the market, his product reputation would be known, and given positive
profits,
would
he
The
bankruptcy.
received
results
literature
for
section
of
results
The
opt
not
in
our
II
when
bankrupcy.
derived
are
care
specific
without
observable
is
context.
ex
considering
retailer
replicate
post
The results when care
the
not
is
observable are a useful contrast with full retailer liability being both the
efficient legal rule and the equilibrium contract when retailer bankrupcy
not
III,
possible.
In
sections
III
and
positive retailer profits are
IV
bankruptcy is possible.
created
so
that
solvent
In
section
contract*
IV,
the
equilibrium contracts
In each case,
it
is
are
compared
section
retailer strat-
egies will be chosen, both when care is observed ex post and when it
In
is
with
the
is
not.
efficient
found that efficiency could be improved on by
either modifying the contractual due care standards or the extent of retailer
liability.
cussed.
In
The
Section V,
results
are
consumer
liability and bankruptcy costs are dis-
summarized
in
Section
VI.
.
-4-
The
II.
Structure of the Problem:
Liability Rules When Bankruptcy is not
Possible
Model:
The
In this section, we will present several
is
liability-sharing rules when
it
impossible or extremely costly for the retailer to declare bankruptcy to
liability.
escape
manufacturer
show
We
retailer
that
liability
Under retailer liability,
liability.
superior
Pareto
is
the manufacturer and
to
the
retailer will both exercise the optimal levels of care.
The monopoly quan-
tity of output, given expected unit costs, will be sold.
Under manufacturer
liability, expected unit costs are no longer minimized.
surplus are higher in the first case.
poly profits and consumers'
tive
retailers
Definitions
earn
profits
zero
Consequently, mono-
either
in
case.
Assumptions
and
Three possible assignments of liability are discussed:
ity,
manufacturer liability and
the manufacturer
assumed that
perfectly
Competi-
competitive
liability.
consumer
is
In
retailer liabileach
case,
retailer
The
retailers.
consumer
in his handling of
y,
retail
the
at
price,
is
monopolist, who sells his output to many
a
will
purchase
one
output from the manufacturer each period at the wholesale price, q.
use some care,
it
the product and sell
Let
p.
m
be
the
unit
of
He will
the output to the
markup,
p-q.
The probability of an accident, n;(x,y), depends on the care used by the
manufacturer, x, and the care used by the retailer,
y.
We assume that both x
x is observable by all agents and the observ-
and y are measured
in dollars,
ability of y
distinguishing characteristic of the models that
The
probability
convex.
for
is
the
Similarly,
a
function
is
decreasing
The cost of an accident
manufacturer,
the
given
retailers'
the
cost
is
A.
care
in
its
arguments
follow.
strictly
and
The cost-minimizing level of care
used
minimizing
by
care
the
is
retailer,
a
is
function
x*(y)
of
the
,
-5-
manufacturer's care:
y°
and
y*(x)
The efficient
.**
care
levels will be denoted x°
.
The
demand
consumers'
is
given
as
function
a
of
expected
the
full
price:
the retail price plus the expected uncompensated accident costs
1973].
For retailer liability and manufacturer liability, this is simply the
retail
price,
since
consumer
the
fully
is
compensated
accident.
The
demand
function is Q(p), where p is
and
the
total
quantity
is
Q
Case
1:
Retailer
Liability.
retailer
liability,
Under
profits
level
subject
two
to
care
to
retailers
earn
zero
entire
cost
given
of
the
care
the
y = y*(x)
becomes:
.
profits.
accidents,
chosen
First,
expected
expected
by
they
sold
the
profits,
and
Since
the
will
which
for
With
max
[1.1]
subject
an
maximize
retailers
second,
expected
are
selecting
the
competitive
a
retailers are paying the
expected
i.e.,
accident
constraint
first
the
costs
The second constraint gives the markup as a function of
the
retailer
of
period.
the level of care used by the manufacturer and the retail price.
markup
event
expected full price
will
minimize
manufacturer,
the
per
manufacturer
the
constraints:
maximize
of
output
of
the
the
in
[Oi,
optimizing
liability,
Q(p)(p - m -
profit
retailer's
the
manufacturer's
per
unit
problem
is
This is the
is
zero.
then:
x)
to:
m = y*(x) + n(x,y*(x))A.
[1.1a]
By substituting [1.1a]
problem
can
[1.2]
max
where
y*(x)
be
into the maximand, we see that the manufacturer's
simplified:
Q(p)(p - x - y*(x) - tc(x,v*(x))A)
is
chosen
to
minimize
expected
,
unit
costs,
x+y+ii(x,y)A.
-6-
solve
When we
problem,
this
we
see
that
the
manufacturer selects the
optimal level of care, given y, and that he selects the monopoly quantity of
given that
output,
are
expected unit
where
x°+y°+n;(x° ,y° )A,
It
the optimal
are
the
optimal
levels
of
care.
suprising that both the manufacturer and the retailer select
not
is
y°
and
x°
Expected unit costs
are minimized.
costs
The expected accident costs are being paid by the
levels.
care
retailer, providing him with the incentive to exercise care.
He will set the
marginal cost of care equal to the marginal reduction in his expected liability.
addition,
In
costs
as
increase
an
manufacturer
the
the
in
sees
a
reduction
expected
in
wholesale price he can charge at
any
accident
level of
output.
Therefore, he, too, has the incentive to minimize expected accident-
related
costs.
Case
Manufacturer
2:
Liability:
Given that we have assumed that
it
impossible for the manufacturer to
is
monitor the retailer's care, the retailer will use the minimum possible level
of care when the manufacturer is
liable
retail
price
i.e.,
price,
retailer
markup
problem
Q(p)( p - x - h(x,0)A)
max
His
the
the
manufacturer's
The
[2.1]
which
at
first
+
order
conditions
zero
profits
the
(x,0)A = 0, and
Q(p) = (p - x - n(x,0)A)Q'(p).
The manufacturer sees the full expected accident costs;
will
p,
are:
[2.2b]
he
is
.
1
x
earning
then:
[2.2a]
it
is
zero.
is
is
This level
Since the retailer's cost is zero, the
of care has been normalized to zero.
wholesale
for all accident costs.
minimize
expected
accident-related
costs
given
and,
that
therefore,
the
retailer
-7-
exercises
no
Expected
unit
retailer
liability.
Again,
costs
the
higher
are
surplus
polist's profits
retailers
In
lower
than
under
than
under
either
retailer
use
than
under
of output,
given
the
optimal
of
level
at
1,
a
liability
liability.
.
case
in
1,
higher price.
and
The
mono-
the
competitive
system.
problem under retailer
Under retailer liability,
given
care,
the
behavior
the
of
the
Also, the manufacturer uses the cost minimizing level of care.
manufacturer.
Total expected unit costs are minimized.
The manufacturer,
liability are OA.
costs.
unit
He
pays
Expected unit costs under retailer
in setting quantity, will consider total
himself,
x°
reduction in the wholesale price.
at a price P
level
we have graphed the manufacturer's
under manufacturer liability.
expected
liability
case
retailer
liability and
will
in
retailer
than under
profits
zero
1,
the monopoly
output
less
lower
is
are
earn
Figure
manufacturer
x=x*(0)
and
y=0,
Since expected unit costs are higher than
costs.
manufacturer will sell
Consumers'
under
manufacturer will sell
the
expected unit
equilibrium levels of care are:
The
care.
sees
and
y°+7t (x° ,y°
)A
as
a
With retailer liability, he will sell Q„
The manufacturer's profits plus consumer's surplus is ABCD.
.
Under manufacturer liability, the retailer will use the minimum level of
care,
0,
and
the manufacturer will
higher
than
manufacturer
total
sells
Q^
expected
units
unit
at
a
surplus
under
retailer
to
manufacturer's
costs
under
price
PM
liability.
manufacturer's profits have fallen.
superior
level of care,
given the
retailer
.
This
liability.
The
surplus
plus
Consumer's
.
This is contained in ABCD, profits plus
manufacturer's profits are A'B'C'D.
consumer's
the optimal
Expected unit costs for the manufacturer are OA'
retailer's behavior.
is
use
liability
Both
consumer's
surplus
and
Therefore, retailer liability is Pareto
whenever
it
is
not
feasible
for
the
Price
Expected Unit Cost
x*(0)+tt(x*(0) ,0)A
°+y°+TT (x°,y°)A
Q(P)
Quantity
Figure
1:
Retailer Liability and Manufacturer Liability,
when Retailer Bankrupcy is not feasible.
.
-8-
retailers
to
bankrupt
go
explanation
The
escape
to
liability.
straightforward.
is
With
retailer
liability,
the
retailer has the incentive to minimize his expected unit costs, y + n(x,y)A.
He will choose the optimal level of care, given the behavior of the manufac-
Accident related costs are also considered by the manufacturer, since
turer.
they
seen
are
increase
an
as
With manufacturer
care
of
manufacturer
the
accident
expected
costs
do
cannot monitor
not
enter
into
the
the
decision and total expected unit cost must exceed expected unit
retailer's
under
cost
when
liability,
retailer,
the
markup.
the
in
retailer
liability.
Since
costs
total
the
are
higher,
the
manufacturer will sell a lower quantity of output at a higher retail price.
Case
Care-based
3:
Contracts
Indemnification.
for
When the manufacturer can observe ex post the care used by the retailer,
a more
complex incentive system can be used.
If the retailer cannot go bankrupt to escape
can be based on his care level.
liability,
the
retailer
always be
can
The liability of the retailer
induced
use
to
the
optimal
of
level
care
When the retailer's share of liability can depend on his level of care,
the
manufacturer
can
negligence
system.
chooses
level
a
Otherwise,
the
The
of
up
set
retailer
care,
manufacturer
to
care. 6
retailer uses
The
the
show that
liable
is
retailer's costs are simply
,
y°
-t-rt(x°
,y° )A<y+rt(x° ,y)A,
the
all
accident
costs
he
optimal
the
for
resembles
all
.
select
If he chooses
for all y.
level
accident
the manufacturer pays
his expected unit costs are y+n(x° ,y)A.
y°<y
for
retailer will
the
y°
than
which
structure
liable
is
less
y,
straightforward
If
incentive
an
of
costs.
the optimal
all
if
care,
y°
It
.
is
level of
accident costs.
any care level below y°
Since y° minimizes x°+y+n(x ,y)A,
The retailer will always choose y°
.
,
-9-
the manufacturer minimizes expected unit costs and will
Similarly,
retailer,
a
if
liability
manufacturer select
the
retailer
The
y°
chooses
results
of
court
the
observe
can
system can be devised
and x°
at
cases
,
least
1
respectively.
y°
.
through
in
are
the
which the retailer and
the
the
care
used
The manufacturer is liable if
Otherwise
3
.
by
post
ex
therefore choose x°
the
retailer
summarized
in
is
liable.
Proposition
1:
Proposition 1:
When retailer bankruptcy is not possible retailer liability
Under retailer liability, the
is Pareto superior to manufacturer liability.
quality and quantity of output is higher, but the retail price is lower.
level of care, the
If the manufacturer can observe the retailer's
manufacturer can provide the retailer with the incentive to use the cost
minimizing level of care.
The quality and quantity of output will be the
same as under retailer liability.
-10-
Liability-Sharing
III.
Case
Care-based
4:
The
If
Arrangements
with
Bankruptcy
Possible.
Indemnification
when
Bankruptcy
is
Model
bankruptcy is possible and
care or
manufacturer
the
if
paying
accident
the
bankruptcy
in
liability
rule
reduced
The
and 3 are no longer
1
expecting zero profits
tive retailer,
care
of
never
monopolist
can
pay
damages
the
an
the
care-based
a
If the present discounted value of his expected
the
the
exercise
liability
guaranteeing the retailer positive expected profits in the
future.
fulfills
would
of
occurs.
and
he
declare
will
effects
retailer
contract
by
The competi-
he
that
accident
the
prefer bankruptcy
incentive
situation with
this
will
Given
model,
if
feasible.
future,
incurred.
our
In
remedy
the
in
accident,
an
diluted.
are
and
when
costs
event
the
consumer cannot observe retailer
the
if
liable when the retailer goes bankrupt,
is
equilibria described in cases
to
Possible
contractual
requirements
are
greater
future profits when
than or
equal
to
his
most
profitable bankruptcy strategy
the retailer will
exhibit contractual
"due
care" and not go bankrupt when an accident occurs.
To guarantee the
,
retailer positive expected profits, the manufacturer must control the retail
price of output
and
output each period.
retailer sells.
beginning
of
The
Each retailer will
quantity.
limit
time period
is
sell
one unit of
then defined by how much output the
The interest rate per period is r, decisions are made at the
each
period,
while
payments
are
made
at
the
the
The
end.
probability and cost of an accident are described in section II, above.
It
is
assumed that litigation is costless and that all accidents will result in the
manufacturer
and
retailers
paying
total
damages
of
A.
.
-11-
Manufacturer's
The
If
the
Problem
the manufacturer
manufacturer
can
can observe ex post
up
set
negligence system.
is
similar
the
to
The manufacturer chooses a "due care standard", y ,
used a level of care less than y
manufacturer must
conditional
system which
,
and
liable for some proportion of accident costs, 0A, only if he
the retailer is
The
incentive
an
care used by the retailer,
the
liability,
,
select
standard",
care
manufacturer's
the
and
6,
"due
the
level
of
retailer's
the
care,
x*.
The
retailer's liability must be sufficiently high that the retailer will choose
to use
the
care
the
expected
level, y
accident
rather than use some lower level of care and pay
,
,
costs.
addition,
In
the
retailer's
expected
future
profits must be sufficiently high that the retailer will not pursue a "fly-
by-night"
occurs
which he
in
would
choose
bankruptcy
after
an
accident
.
manufacturer's
The
[4.1]
max
subject
[4.2]
strategy
problem
then:
is
Q(p)[p-m-x*-n;(x*,y )A]
d
,
to:
y
d
y + 9n(x*,y)A,
<
for all y<y
d
,
and:
m-y,
°°
1
[4.3]
>
is
I
i=l
y
r
where k(x*,m,r)
max [m-y]
j.
\
^"ff^T
= k(x*,m,r),
In
occurring by the period (i.e.,
period
calculating k, profits
in each period, m-y,
interest rate and the probability of no accident
are discounted by both the
through
/
the present discounted value of the retailer's most prof-
itable bankruptcy strategy.
occurring
i
X
.)
(1—rc)
Solving
is
the
the
probability of no accident
expression
on
the
right
hand
.
)
-12-
side of [4.3], we see that:
^i**'
x* y^
k(x*,m,r) = max [m-y]
[4.3a]
3
r+it
(
,
y
Since
retailer's
the
share
liability,
of
does
9,
not
objective function or in constraint (4.3), constraint (4.2)
The retailer's conditional liability can be set equal to
standard"
high
as
Since
y*(x*).
as
function of the markup,
decreasing
is
1,
manufacturer's
the
constraint
(4.3)
will
appear
in
the
never binding.
with a "due care
profits
are
a
always hold as an
equality:
[4.3b]
y
m
=
,
- rk(x*,m,r).
m = M(x*,y ,r).
This can be rewritten as:
manufacturer
The
[4.4]
Q(p)[p-M(x*,y ,r)-x*-rt(x*,y )A]
d
with
Differentiating
[4.5]
maximize:
will
d
respect
to
price,
we
see
Q*(p)[p-M(x*,y ,r)-x*-rt(x*,y )A] + Q(p) =
d
d
that:
0.
The manufacturer will then choose x* and y, to minimize his expected
unit
costs:
min
[4.6]
C(x*,y
d
)
u(x*,y )A
= x* +
d
+
M(x*,y ,r).
d
He chooses the monopoly price and quantity of output, given his expected unit
costs, C(x*,y
d
).
Minimizing
[4.7]
1
+
n
these
x
costs
= (x*,y,)A
J
d
implies:
|^
9x
.
and:
[4.8]
1
+ n
(x*,y,)A
y^
"d'~
=
1
-
oM
5y^
.
-13-
side of condition
right hand
The
manufacturer's
the
bankruptcy strategy.
less than x*(y
From
will
level
care
The
[4.7]
is
negative 10
increase
the
i.e.,
,
value
of
increasing
optimal
the
of care chosen be the manufacturer is
level
then
,)
equation
we
[4.8],
that
see
manufacturer
the
choose
will
the
retailer's "due care standard" to be less than the cost minimizing level of
retailer care, given his care level, i.e.,
one
than
and
hand
right
the
side
of
a
of care,
he reduces the probability of an accident
will
increase
the
present
greater
When the manufacturer increases his
level
expected
is
?d
negative.
is
[4.8]
The explanation is straightforward.
—
since -
y <y*(x*),
d
discounted
value
therefore will
for every y.
retailer's
the
for
This
bankruptcy policy for every
y,
optimal bankruptcy policy.
When the manufacturer increases his care level,
he
increases
retailer
selecting
from
additional
profits
the
of
care,
since
if
cost
Similarly,
must
he
the
and
bankruptcy
he
will
the due
with
share
select
a
care standard
y
,
In
care
case,
this
standard"
given
to
When he
sub-optimal
prevent
considers
care
the
this
level.
increased the markup must
is
the cost to the manufacturer of
,
increase y, is greater than the resource cost.
"due
retailer
the
strategy.
increase by more than the increase in
suboptimal
increase the value of the
Therefore he will choose a
x.
retailer size will influence the degree of inefficiency.
As noted above, the retailer's volume is implicitly given by r, since this-is
the
interest
approaches
oM
per
0.
period
sales.
As
the
From [4.7] and [4.8] we note that if
were to approach
and ^
between
1,
...
retailer
-r—
X
gets
large,
r
were to approach zero
the equilibrium contract will approach the effi-
d
cient allocations of x and y
restrictions
on n(x.y). 11
.
These derivatives converge with certain
Inspection of
[4.3]
gives
the
intuition of these
-14-
results.
As
r
becomes small, the present value of any positive level of any
"non-negligent"
strategy as
profits
long
is
tortion
introduced
These
accident
the
as
n(x,y,)
dominates
zero and
not
by
results
are
tc
present value of the best bankruptcy
the
technology
(x,y)
preventing
summarized
is
is
sufficiently regular
bounded from below.)
retailer
in
bankruptcy
Proposition
If so,
(i.e.,
the dis-
vanishes.
2:
Proposition 2:
When it is possible for the retailer to go out of business
and when the manufacturer can select a liability sharing arrangement which
depends on the ex post observation of retailer care:
The manufacturer will choose a level of retailer liability and the "due
1.
care standard", y^
for the retailer, such that the retailer will always
,
choose to use y j
,
and avoid liability in the event of an accident.
2.
The manufacturer will choose a level of retailer care which is less than
optimal, given the care used by the manufacturer.
The manufacturer will choose a level of manufacturer care which is less
than that which is optimal, given that the retailer uses y j
3.
.
The manufacturer will sell the monopoly quantity of output, given his
expected unit costs.
These costs include the expenditures of both the
manufacturer and the retailer of accident avoidance, the total expected cost
of accidents and the profits per unit of output he must share with the
retailer.
4.
Generally, as the quantity of output handled by each retailer increases,
care used by the manufacturer and the retailer will approach the costminimizing levels of care.
5.
the
.
-15-
Indemnification when bankruptcy is possible and retailer care is
5:
unobservable.
Case
The manufacturer will maximize expected profits over p, x and 9, subject
The retailer will choose a level of care, y, to minimize
to two constraints:
expected
costs,
unit
and
retailer's
the
expected
future
profits
must
be
sufficiently high that he would not prefer bankruptcy to paying his share of
damages in the event of an accident.
9
cannot
The
care-based
be
manufacturer's
will
and
problem
Since care cannot be observed ex post,
enter
as
non-trivial
a
variable.
is:
max [p-m-[x+(l-9)rc(x,y)A]]Q(p)
[5.1]
subject
to:
[5.1a]
m - [y+9Tt(x,y)A]
[5.1b]
y = y(x,9),
where y(x,9)
r9A, and
given by the retailer's first order condition:
is
Since 9 is restricted to the unit interval, it is clear that
(x,y)A=0.
l+9it
>
y
the retailer will choose a level of care less than or equal to that which is
optimal given the care chosen by the manufacturer.
if
9=1.
substituting
By
function
[5.2]
The
He will choose y*(x) only
[5.1],
max
first
costs
[
problem
can
and
[5.1a]
be
rewritten
of the maximand
his
direct
and
is
the
objective
as:
indirect
His
costs.
direct
costs
are
His
his
While his indirect cost is the
the retailer's costs per unit plus the profits which are necesary to
prevent retailer bankrupcy.
[5.3]
into
the manufacturer's profits per unit.
avoidance costs plus his expected liability.
markup:
[5.1b]
p-[x+(l-9 )*(x,y(x,9) )A+m] ]Q(p)
terra
include
the
constraints
max
[
By substituting in constraint
p-[x+y(x,9)+it(x, y(x,9) )A+r9A] ]Q(p)
.
[5.1a], we get:
16-
differentiating
By
manufacturer's
order
first
with
[5.3]
respect
to
and
x,
p,
we
9,
find
the
conditions:
Q(p) + [p-[x+y+ x(x y)A+r9A]]Q (p) = 0,
,
[5.4]
1
5
From condition [5.4] and constraint [5.1b], we see that the manufacturer
will sell the monopoly quantity of output, given his expected unit costs.
note
costs
unit
this
his expected unit costs are the sum of his accident avoidance
case,
measures, x, his share of liability,
(
costs,
y+9u(x,y)A,
prevent
First
the per unit
and
bankruptcy,
order
+
[5.6]
1
the retailer's accident-related
[5.7]
[l^
with
(x,y)A =
-Pox
x
profits he must share with the retailer
r9A.
conditions
it
and the retailer's mark-
l-9)it(x,y)A,
The retailers mark-up is composed of:
up.
to
are:
C(x,9) = x+y(x,9)+it(x,y(x,9))A+r9A.
[5.5]
In
expected
total
his
that
We
(x,y)A]
y
||
respect
to
[l+n
(x,y)Al
J
L
J
y
x
,
and
imply:
9
and
= - rA.
The level of care chosen by the manufacturer is determined by condition
Using the retailer's first order condition, this can be rewritten:
[5.6].
[5.6']
1
+ u
The manufacturer
left
hand
side
x
H y (x,y)
= (1-9
(x,y)A
J
A p-.
°x
selecting the optimal
is
of
[5.6']
is
The
zero.
given y, when the
level of care,
left
hand
side
of
[5.6']
is
an
increasing function of x, so the manufacturer wili select a care level higher
than x*(y)
if and only
if the right hand side of [5.6']
is positive.
the manufacturer will
If the
select a level
right hand side of [5.6']
is
negative,
of care below x*(y)
is
positive (negative), and the manufacturer will
select
a
care
.
level
It
above
(below)
x*(y)
,
if
the
two
types
of
care
are
.
-17-
explanation
The
2
(substitutes).
complements
straightforward.
is
The manufacturer must
take
into
account the relationship between the care he chooses and the cost minimizing
care for the retailer.
in
manufacturer's
the
care
reduce
will
the
of
level
care
an increase
selected
by
the
The return to the manufacturer is then less than the reduction in
retailer.
expected
If the two types of care are substitutes,
accident
less than x*(y)
and
costs,
manufacturer will
the
If the two types
.
select
level
a
of care
of care are complements, the manufacturer
will see an increase in the retailers'
care when he increases his care.
The
return to care than exceeds the reduction in expected accident costs and the
manufacturer
select
will
profit-maximizing
The
care
a
period,
the
if
hand
x*(y)
than
arrangement
of
the
is
[5.7]
reduction
is
by
the increase
the
in
is
raised.
manufacturer's
interest rate
falls,
of
damages
by
paid
the left hand
retailer
the
and
the
levels
retailer approach x°
and
y°
approaches
one,
,
of
side of [5.7]
increases.
care
used
As
decreases and the share
As
by
The
expected
liability each period, resulting from the incentive effects of raising 9.
the
the
share with the retailers in each
proportion of damages paid by the retailers
side
given
is
The left hand side of [5.7]
profits which the manufacturer must
the
right
greater
liability-sharing
manufacturer's condition [5.7].
in
level
approaches
r
the
manufacturer
the cost minimizing levels of care.
zero,
and
9
the
Since the
interest rate is based on the time period between sales, a declining interest
interpreted as an
rate
can be
each
retailer.
must be earned
As
to
the
increase
retailer becomes
in
the
large,
prevent bankruptcy fall,
and
amount
the
it
per
the
optimal care.
unit
profits which
becomes less expensive to
the manufacturer to create the appropriate incentives.
firm approaches
of output handled by
The retailer's care
The care used by each
increases because he
.
-18-
is
paying
higher
a
fraction of
care approaches the optimal
approaches
These
level
the
as
accident
the
costs,
and
the
manufacturer's
right hand side of equation [5.6']
zero.
results
Proposition 3
and when the
:
are
summarized
in
Propositon
3:
When it is possible for the retailer to go out of business
manufacturer can select a liability-sharing arrangement:
1.
The manufacturer will select a liability-sharing arrangement with 9
less than one.
The retailer will not pay the entire cost of the accident.
Since the retailer is paying less than the full accident costs, he
2.
will choose a level of care less than that which is optimal given the care
selected by the manufacturer, i.e., y is less than y*(x)
3.
The relationship between the level of care chosen by the manufacturer and x*(y) depends on whether the two types of care are substitutes or
complements.
If the two types of care are substitutes (complements), the
manufacturer will select a level of care less than (greater than) x*(y).
4.
The
his expected
manufacturer
dents and the
manufacturer will sell the monopoly quantity of output, given
unit costs.
These costs include the expenditures of both the
and retailer on accident avoidance, the expected cost of acciprofits per unit of output he must share with the retailer.
As the quantity of output handled by each retailer increases, the
used by the manufacturer and the retailer will approach the cost
minimizing levels of care.
5.
care
.
-19-
Welfare
Social
IV.
and
Equilibrium
the
Contract
When the government can determine the liability-sharing arrangement and
when retailer care is observable ex post,
increasing
Let
At
y
I
,
retailer's
the
y'
"due
social welfare can be improved by
standard".
care
be the liability-sharing arrangement selected by the monopolist.
the monopolist's first order conditions hold.
defined by equations [4.1]
through [4.3]
implies that the manufacturer mini-
mizes his private expected unit costs over x and y
costs,
[4.6],
[6.1]
C(x,y
first
The
can
d
=
)
be
written
+
[x + y
d
Profit maximization as
,
.
The manufacturer's unit
as:
u(x,y >A]
+
d
[M(x,y ,r) - y
d
d
term on the right hand side of [6.1]
is
]
the
expected social cost
and the second term in the retailer's profits per period.
y,
A small change in
will result in no change in the manufacturer's expected unit costs, and
therefore there will be no change in the monopoly quantity of output.
Since
there is no change in the quantity of output sold, there is also no change in
the consumer's surplus.
in
the
"due
care
standard"
will
result
an
in
increase
Unit production costs plus unit
expected profits.
x+y+n(x,y)A,
5M
dy
Since as shown in footnote 9, ?
have
>
1,
an increase
d
the
in
retailer's
expected accident costs,
declined.
In Figure 2, we have graphed the manufacturer's problem at the "due care
standard" selected by the manufacturer, y \
manufacturer's expected unit costs, C(x,y
these costs,
with y"
must
,
C(x,y").
guarantee
bankruptcy.
changed,
C (x,y')=0.
so:
),
Therefore, OE
,
and at
y'j =
when he chose
is
yj + £
y'
•
OE is the
to minimize his
also his expected unit costs
When the retailer's standard increases, the manufacturer
the
retailer
higher
expected
future
profits
to
prevent
Since expected unit costs plus the retailer's profits has not
expected
unit
costs
must
fall
when
the
retailer's
profits
have
Price
Expected Unit Cost
x'+y
l
+7T(x'
,y')A
x"+y"+T7 (x" ,y")A
Quantity
Figure
2:
Increasing the Retailer's "due care Standard",
.
.
-20-
increased.
OA'
expected
is
unit
manufacturer's
equilibrium contract.
retailer faces
a
OP.
With 9',
respectively.
care
manufacturer will
the
manufacturer's
The
standard"
OA"
is
higher "due care standard".
With either system,
raised,
profits
the
the
x'+y'+i(x' ,y')A,
costs,
expected
OA
consumers'
and
costs
unit
retailers'
profits
retailers'
expected
the
when
the
must be greater than OA"
1
OQ units at
sell
with
surplus
retail price,
a
EFCP
are
A'B'CP.
are
profits
and
PCD,
With the
"due
increased
have
to
A"B"CP.
The manufacturer's profits and the consumer's surplus are unchanged,
but
increase
the
contract,
We
the
the
profits.
retailers'
selected
in
it
by
when
Thus,
1
standard has resulted
s
government
the
can
in
alter
an increase
the
increase the retailer's standard,
should always
in
equilibrium
over the level
monopolist.
the
can make
a
similar argument
liability-sharing
monopolist's
retailer
first
arrangement
the non-care-based
in
selected
by
the
case.
monopolist.
From condition
order conditions hold.
facturer minimizes his expected unit costs, C(x,9).
[5.4],
Let 9'
At
9',
be
the
the manu-
A small change in
9
will
result in no change in the manufacturer's expected unit costs, and therefore
there will be no change
the monopoly quantity of output.
in
Since there is
no change in the quantity of output sold, there is also no change in the con-
sumer's
surplus.
But,
an
increase
in
the proportion of damages paid by the
retailer will result in an increase in the retailer's expected profits.
Unit
expected
have
production
costs
plus
unit
declined.
Again,
when
the
arrangement,
it
should
accident
government
always
can
increase
9
costs,
determine
over
the
x+y+n(x,y)A,
the
level
liability-sharing
selected
by
the
monopolist
For any large change in y, or
problem is exacerbated.
At
9
,
we see that the monopoly quantity
the efficient contract, there will be a trade-off
-21-
between a decrease in the unit social costs and
quantity of output.
a
reduction in the monopoly
Proposition 4 follows.
Proposition 4:
When the manufacturer can use a care-based contract for
indemnification, society can always improve on the liability-sharing scheme
selected by the monopolist by increasing the retailer's "due care standard".
Similarly, when the manufacturer cannot use a care-based contract for
indemnification, society can always improve on the liability-sharing scheme
selected by the monopolist by increasing the proportion of damages that are
paid by the retailer.
,
:
-22-
Extensions
V.
Liability
Consumer
A.
allocation
The
resources
of
under
liability
consumer
depends
on
amount of information available to the consumer prior to purchase.
the
We will
consider the allocations which would result from two extreme assumptions.
increase
retailers,
many
are
assume
that
safety
the
in
level
individual
the
his
of
consumers
the
average
They know the
information.
there
can
one
First,
output,
of
care
retailer
retailer-specific
no
available.
is
therefore
and
Here,
rewarded
not
has
no
when
for
incentive
any
to
The manufacturer's optimization
He will set y equal to zero.
exercise care.
have
is
max Q(p
[7.1]
where p
CI
)[p-x]
full price":
is the "expected
C1
[7.1a]
CI
= p + Tt(x,0)A.
p
By solving this optimization, we see that, the manufacturer will use the
optimal
of
level
accident costs.
given
care,
since
y=0,
x*(0)-Hi(x*(0) ,0)A.
quality
manufacturer
output
Another
depends
on
expected
The manufacturer will sell the monopoly quantity of output,
given expected unit costs:
of
price
the
possible
quality of output
in
as
assumption
prior
to
is
This is the same quantity and
liability.
that
the
consumer
can
determine
the
Since the consumer can determine the
purchase.
expected full price of the output of any specific retailer, the consumer will
choose
the
retailer with the
will have the
care
used
by
manufacturer
expected
lowest
full
price,
and each retailer
incentive to use the cost-minimizing level of care, given the
the
will
manufacturer.
maximize
The
profits,
retailer
subject
to
will
the
choose
y*(x)
constraint
.
that
The
the
-23-
retailer
earns
max
[7.2]
profits.
zero
Q(p
C2
)(q - x)
subject
to:
[7.2a]
p = q + y*(x),
and where p
minimizing
[7.2b]
C2
is
optimization
is:
,
the expected
of
level
His
full price when the retailer uses the cost
care:
=p+it(x,y*(x))A.
p
By substituting the constraints into the maximand, we see that the cost-
minimizing quality of output
is
is
produced and the monopoly quality of output
sold.
Since the consumer can observe retailer care,
optimal
the
quality
price.
in
given the care used by the manufacturer.
of care,
level
of
the retailer chooses
output
sold
by
manufacturer will
the
affect
Similarly,
wholesale
the
The manufacturer will choose the cost-minimizing level of care.
the case of retailer liability,
expected unit
equilibrium
cost,
quality
and
As
the output will be produced at the lowest
the monopoly quantity of output
but
the
quantity
of
output
is
the
will be
same
as
sold.
in
The
retailer
liability.
These allocations differ from cases one and two,
the
distribution
manufacturer
compensated,
not
income
liability,
the
among
Under
income
as
the
the
consumers
whereas under consumer
compensated.
expected
of
consumer
respectively, only in
Under
consumers.
involved
liability,
liability,
other two systems, but
the
the
in
in
an
retailer
accident
individuals
consumers
will
or
be
involved are
have
the absence of
the
same
insurance,
.
-24-
Lheir
ex
J
differ.
incomes
post
Proposition
5
follows:
Proposition 5:
Consumer liability will result in the same quality and quantity of
output as retailer liability if the consumers can judge the quality of output
sold by each retailer.
When consumers know only the average quality of output available in the
market, consumer liability will result in the same quality and quantity of
output as manufacturer liability.
B
Bankruptcy
.
Costs
The retailer's decision can also be examined when there are set-up costs
financed
out
These set-up costs
goes bankrupt.
production
process
franchising
retailer
care
standard
equal
When
to
y°
a
or
,
arrangements
costs
these
the
in
such
as
a
suffi-
are
is
is
rule
with
observable
possible,
the
ex
constraint
of
o°
max (m-y-*(x,y)B)
care
the manufacturer
[4.3]:
.
1
I
[j^]
= k(x,m,B,r).
i=l
includes
least B, where k(x
due
post.
the
probability of
incurring
B
in
any period
which the firm continues to operate with a "fly-by-night" strategy.
at
retailer
"due care standard", to maximize [4.1] subject to the
y
[7.3]
care
arrangement
indemnity
m-y
Inequality
If
.
"negligence"
a
retailer
when
,
modification
- ±>
r
[7.3]
institutional
from
the
if
from non-convexities
arise
manufacturer.
a
unobservable
is
will choose to use
following
by
could
forfeited
be
manufacturer will either choose retailer liability, when
the
care-based
a
come
may
or
imposed
fee
ciently high,
which would
equity,
retailer
of
,ra
,B
,
r
)=0
,
in
If B is
then m is equal to y, and the manufacturer
will minimize social costs, setting y =y°
When retailer care in unobservable, retailer liability will be chosen if
the
cost
of bankruptcy to
the
retailer
is
at
least
A.
The manufacturer's
.
-25-
second
constraint
[7.4]
P -
becomes:
[g+y+9tt(x,y)A]
QA _ B
>
.
If B is greater than or equal to A,
on the unit
strategy for any
9=1,
minimizing
These
can
results
interval.
private
both
the retailer will not choose a bankruptcy
social
and
stated
be
The manufacturer will then choose
costs.
as:
Proposition
6:
When care-based
contracts
are
feasible
and
private
bankruptcy costs, B, are such that k(x,ra,B,r)<0, the equilibrium "due care
Likewise, when care is not observable ex post and B>A, the
standard" is y°
manufacturer will set 9=1 and the retailer will choose y°
With either
assumption, when the cost of bankruptcy is sufficiently high, the retailer
will be induced to choose the optimal level of care and the profit-maximizing
manufacturer will then also select the optimal level of care.
.
.
If
manufacturer
the
can
observe
post
ex
retailer's
the
care
and
the
retailer's cost of bankruptcy is less than B, k(x,m,B,r) is greater than zero
dM_
q
n
.
and consequently,
-=
manufacturer
choose
If
cost
the
will
choose
retailer
a
will
Proposition 4
From
[7.3]
and
[7.4]
care
a
can
then
we
observe
hand
right
the
can
arrangement
level
be
extended
calculate
with
the
less
8
same
analysis
as
for
Proposition
4,
than
the
[7.4]
see
and
one,
bankruptcy
low
manufacturer's
we
if
the
y*(x).
unit costs in these cases, just as for equation [4.6]
the
and
side of equations
include
to
y°
Therefore, the manufacturer
than
less
than
less
retailer care,
post
ex
on the unit interval.
9
liability-sharing
choose
standard"
care
than A,
less
is
must be positive for any
will
"due
a
manufacturer cannot
of bankruptcy
From [4.8], we see that the
greater than one.
is
and
that
expected
private
[5.5].
we
reach
costs.
Then using
a
similar
-26-
conclusion.
These
results
are
summarized
as:
Proposition 7
When the manufacturer can use a care-based contract for indemnification
and when the retailer's cost of bankruptcy is less than B, society can always
improve on the liability-sharing scheme selected by the monopolist by
increasing the retailer's "due care standard."
Similarly, when the manufacturer cannot use a care-based contract for
indemnification and when the retailer's cost of bankruptcy is less than A,
society can always improve on the liability-sharing scheme selected by the
monopolist by increasing the proportion of damages that are paid by the
retailer.
:
-27-
Conclusion
VI.
In
preceding
the
four
sections,
examined
we
equilibrium contracts
the
retailer when both the retailer and manufacturer can
for a manufacturer and
affect the probability of an accident and when the care used by the retailer
is
not
In
observable
Section
manufacturer
the
II,
and
ante.
ex
equilibrium
retailer
the
under
levels
care
liability
retailer
liability if retailer bankruptcy is not possible.
retailer liability,
optimal
the
derived
are
and
the
for
manufacturer
We have shown that under
quality of output will
be produced.
Under
both manufacturer liability and retailer liability, the monopoly quantity of
output, given expected unit costs,
to
show that
under
liability.
the
absence
of
The competitive
then straightforward
is
It
profits
are both higher
expected profits
retailers'
are
Retailer liability dominates manufacturer liability
zero under both systems.
in
produced.
surplus and manufacturer's
consumers'
retailer
is
retailer
bankruptcy.
In Sections III and IV, we examined the equilibrium contract if retailer
bankruptcy
is
possible.
If
the
manufacturer
observe
can
retailer
care
ex post, the equilibrium contract will involve suboptimal inputs to safety
and
positive
facturer's
retailer
care-based
"due care standard".
contract,
the
level
Society
profits.
can
profits
in
If
of
the manufacturer cannot
safety
inputs
equilibrium
liability-sharing
scheme
retailer's
of
share
on
the
manu-
indemnification contract by increasing the retailer's
relative
whether they are "substitutes" or "complements".
positive
improve
always
selected
damages.
and
by
society
the
use
the
to
care-based indemnity
a
optimum will
Again,
can
the
always
manufacturer
by
turn on
retailer earns
on
the
increasing
the
improve
-28-
Section
In
discussed.
can be
his
It
V,
is
consumer
shown
that
increased by either
"due
care
standard".
liability
if
and
positive
bankruptcy
bankruptcy costs are small,
increasing the
retailer's
costs
are
social welfare
share of liability or
-Fl-
FOOTNOTES
1.
See,
2.
Similar results have been reached in the principal-agent literature.
for example, Harris and Raviv (1978 and 1979) and Shavell (1979).
Here, "consumer liability" is used to mean that neither firm is liable.
Consumer liability will be discussed in Section V.
3.
It
is assumed that additional care by the
manufacturer or retailer will
reduce the probability of an accident at a decreasing rate, i.e., n
os
are both negative, and that
made about the sign
of u
°
4.
it
and n
xx
yy
and
ti
No assumption
is
r
are both positive.
r
xy
These optimal levels of care are given by the following first order
conditions.
x*(y)
is given by:
1+tx
(x*(y),y)A = 0.
1+ti
(x,y*(x))A =
5.
These optimal care levels are given by:
6.
See also Brown [1973].
7.
The results in cases
technologies.
y*(x)
differ.
given by:
0.
1
and
2
x°=x*(y
)
and y°=y*(x°).
may not be robust to alternative retailer
The retailer's ability to substitute away from the manufac-
turer's product will change the outcomes in these cases.
Shavell [1980,
is
1982]
In addition,
discusses some circumstances in which these outcomes may
.
-F2-
8.
See Shapiro [1980]
9.
This function can be inverted since
which is on the unit interval.
oy,
—
oM
= 1-rk
This implies that
m
»
(\
which equals
,
)
1
r+it
—
dy
,
greater than one.
is
d
When y, is increased the resulting increase in the markup is the increase in
the retailer's costs plus the increase in the profitability of the best
bankruptcy strategy, since m increases.
M
From equation (4.3a), we see that ?— = k /1-k
x
ox
m
7\
10.
From footnote 9, we see
.
that the denominator is positive and the numerator is -r(m-y)it
/(r+u) 2 >0.
'
x
is
11.
Neither of these derivatives, however, converge uniformly to the
that there exists oc>0,
there exists
p
= 1.
such that u(x,y,)>a, then lim
r*0
such that u (x,y)>p, for all y<y,
,
.
If we assume
Similarly, if
d
then
lim^—
= 0.
r*
From the retailers'
first order condition:
=£
ox
= - n
xy
/it
.
We have
yy
defined the inputs to be substitutes (complements) if an increase in x will
result in a decrease (increase) in y.
sub8titutes(complements) if and only if
13.
x
therefore positive.
respective values without some restrictions placed on n(x,y)
12.
M
The two types of care are
it
is
positive (negative)
Analagous results are obtained in Shavell [1980].
.
-F3-
14.
With
franchising fee, the retailer will place an amount F in escrow on
a
which he receives the competitive rate of interest,
r.
F
would be
transferred with any sale of the retail franchise and would therefore become
part of the equity of the firm under such a scheme.
The value of the
retailer's profit maximizing bankruptcy strategy would be the solution to:
k(m,x,F,r) = max
SagfaZJI
r+iux,y)
-
y
Retailer entry and profit maximization on the part of the manufacturer
together imply that F would be set such that k(m,x,F,r)=0, or that, from
[4.3], m=y
.
The manufacturer's problem is now:
max Q(p)[ p-y ,-x-rc(x,y )A]
Monopoly price is again chosen, but the equilibrium franchising scheme alters
the minimized private costs given by [4.6]
to the social costs.
manufacturer thus chooses y* = y° and x=x°
The appropriate franchising fee,
F*, can then be generated from:
.
The
k(y° ,x° ,F*,r) = 0.
To be regarded as a true bankruptcy costs, F must represent retailer net
wealth.
If there are limitations on the availability of such franchisees,
the above solutions cannot be reached.
Initial research indicates that
partial franchising may not improve social welfare.
,
BIBLIOGRAPHY
"Toward an Economic Theory of Liability," Journal of Legal
Brown, John P.:
Studies
,
June 1973, pp. 323-349.
Harris, M. and A. Raviv.
Economic Review
vol.
,
68,
no.
1,
American
March 1978, pp. 20-30.
"Optimal Incentive Contracts with Imperfect Information,"
:
Journal of Economic Theory
Oi, Walter:
"Some Results on Incentive Contracts,"
:
,
April 1979, pp. 231-259.
"The Economics of Product Safety," Bell Journal
and Management Science, Spring 1973, vol. 4, no.
1,
pp.
of Economics
3-28.
"Premiums for High Quality Products as Rents to Reputation,"
Shapiro, Carl:
Discussion Papers in Economics #6, Woodrow Wilson School, Princeton
University, 1981.
Shavell, Steven:
"Risk Sharing and Incentives in the Principal and Agent
Bell Journal of Economics
Relationship,"
:
,
Spring 1979, pp. 55-73.
"Strict Liability vs. Negligence," Journal of Legal Studies,
January 1980, pp. 1-25.
:
Spring 1982, pp.
"On Liability and Insurance,"
120-132.
36^2
003^
Bell Journal of Ecnoomics
/o-f i^Y-^-
HB31.M415 no.304
safety,
Simon, Marilyn/Product
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