Minimum prices and optimal production under multiple sources of risk: a note


Eeckhoudt, L.; Hansen, P.

European Review of Agricultural Economics 16(3): 411-418

1989


In a competitive market where only output price is uncertain, risk averse producers always respond positively to a better protection against downward price fluctuations. This note analyses the robustness of this result when there is more than one source of risk and concludes that: (1) if the financial and technological risks enter additively into the objective function their joint consideration does not affect the result obtained with only financial uncertainty; and (2) with a multiplicative specification, matters are less clear cut. Sufficient conditions to retain the classical result are presented but it is shown through a numerical example that when the conditions are not met, counterintuitive results may occur.

Minimum
prices
and
optimal
production
under
multiple
sources
of
risk:
a
note
L.
EECKHOUDT
and
P.
HANSEN
Catholic
Faculties
of
Mons
and
Lille;
Rutgers
University
(received
November
1987;
final
version
accepted
March
1989)
Summary
In
a
competitive
market
where
only
output
price
is
uncertain,
risk
averse
producers
always
respond
positively
to
a
better
protection
against
downward
price
fluctuations.
In
this
note
we
analyse
the
robustness
of
this
result
when
there
is
more
than
one
source
of
risk.
1.
Introduction
The
exisiting
economic
literature
strongly
indicates
that,
at
least
in
the
short-
run,
guaranteed
minimum
prices
stimulate
the
output
level
of
competitive
producers.
For
instance,
Belongia
(1983)
made
the
point
that
by
raising
the
expected
end
of
season
price,
a
price
support
program
positively
affects
the
value
of
resources
used
in
production
through
a
higher
derived
demand
(provided
of
course
that
the
supply
of
these
resources
is
not
perfectly
elastic).
From
a
very
different
approach,
Eeckhoudt
and
Hansen
(1980)
have
shown
with
the
help
of
a
static
model
that
risk
averse
competitive
producers
facing
price
uncertainty
would
unambiguously
increase
their
output
if
they
bene-
fitted
from
a
higher
minimum
guaranteed
price.
Much
of
the
literature
devoted
to
the
short-run
impact
of
minimum
prices
assumes
that
the
firm
faces
only
one
source
of
uncertainty
which
is
financial
in
its
nature:
the
unit
price
it
will
receive
for
its
output
(see
Eeckhoudt
and
Hansen,
1980).
Although
such
an
assumption
can
be
regarded
as
reasonable
in
the
industrial
field
where
the
relationship
between
output
and
inputs
can
be
controlled
to
a
wide
extent,
it
is
much
less
natural
in
the
agricultural
sector.
Indeed
when
they
make
their
planting
decisions,
farmers
are
uncer-
tain
both
about
the
amount
of
output
they
will
obtain
and
about
the
unitary
price
of
this
output.
Hence
the
purpose
of
this
note
is
to
examine
whether
the
The
authors
thank
the
Editor
and
an
anonymous
referee
for
comments
and
suggestions
which
have
led
to
substantial
improvements
in
the
text.
Euro.
R.
agr.
Eco.
16
(1989)
411-418
0165-1587/89/0016-0411
$2.00
©
Mouton
de
Gruyter
6
412
L.
Eeckhoudt
and
P.
Hansen
positive
relationship
between
the
level
of
output
and
a
better
protection
against
downward
price
fluctuations
that
holds
in
the
presence
of
one
risk
is
also
valid
under
the
more
general
setting
of
simultaneous
sources
of
uncertainty.
In
this
way,
our
paper
is
in
line
with
recent
developments
in
risk
theory
where
the
importance
of
multiple
risks
has
been
stressed
(see
for
example
Doherty
and
Schlesinger,
1983;
and
Ross,
1981).
The
major
results
of
the
note
are
as
follows:
—if
the
financial
and
technological
risks
enter
additively
into
the
objective
function
their
joint
consideration
does
not
affect
the
result
obtained
with
only
financial
uncertainty;
—with
a
multiplicative
specification,
matters
are
less
clear-cut.
We
present
sufficient
conditions
to
retain
the
classical
result
but
we
show
through
a
numerical
example
that
when
the
conditions
are
not
met,
counterintuitive
results
may
occur.
2.
The
basic
model
At
the
beginning
of
the
season,
the
farmer
plans
an
output
level
q
which
induces
perfectly
known
variable
costs
c(q)
since
they
are
assumed
to
be
paid
immediately.
Besides,
he
has
to
incur
non-random
fixed
costs
denoted
B.
The
marginal
costs
(c'(q))
are
non-negative
and
non-decreasing
(c"(q)?:
0).
At
the
end
of
the
season,
the
initially
planned
output
q
will
yield
a
random
amount
q
r
depending
upon
q
and
a
random
term,
e":
4,
=
qr(q,
(I)
aq
with
and
ae
>
0.
aq
It
is
assumed
that
the
observed
discrepancy
between
realized
and
actual
output
does
not
induce
adjustment
costs.
The
next
period
realised
harvest
will
be
sold
at
a
random
price
which
can
be
expressed
in
current
monetary
units
by
appropriate
discounting.
The
present
value
of
the
future
random
price
is
denoted
13.
Hence
the
present
value
of
profits
ir
is
given
by:
ir
=04,(q,
c(q)—
B
(2)
The
subjective
joint
probability
about
e
and
p
is
denoted
f(p,
E)
and
its
marginal
densities
by
g(p)
and
h(e).
Given
these
specifications,
the
expected
utility
of
the
decision
maker
is
written:
/4-.)
E[1.1]
=
e)
—c(q)
e)dp
de
(
3
)
J
o
and
the
first
order
condition
for
a
maximum
with
respect
to
the
decision
Minimal
prices
and
optimal
production
413
variable
q
is:
+
J
f
+
CO
OD
0
U'
(p
c'(q))
f(p,
c)
dp
de
=0
Risk
aversion
combined
with
non-decreasing
marginal
costs
guarantees
that
the
second
order
condition
for
a
maximum
is
verified.
The
optimal
value
of
q
will
be
denoted
q*.
Notice
that
for
the
equality
sign
to
prevail
in
(4),
it
is
necessary
that
p—
—c'(q)
aq
be
positive
for
some
realisations
of
p
and
e
and
negative
for
others.
This
observation
will
be
important
later
on
in
the
discussion.
When
a
price
support
program
is
implemented
at
the
level
p
m
(smaller
than
the
expected
price),
the
expected
utility
is
transformed.
Indeed
for
all
realisations
of
p
below
p
m
,
the
farmer
received
p
m
while
his
receipts
are
not
affected
every
time
p
exceeds
p
m
.
Hence
the
new
E(U)
is
given
by:
E[U]
=
f
+
co
.
ip
m
co
0
U(Prn
Cir(Ch
c(q)
BAP,
dP
dE
.
i+.
+
U(p
qr(q,
E)
c(q)
B)f(p,
E)
dp
de
(
5
)
OD
Pm
and
the
corresponding
first-order
condition
is:
r
+.
i
`
Pm
tY
pm
co
0
(p
m
-
1374
—c'(q))f(p,
e)
dp
de
+
f
U
P
'
(p
al
31
c'(q))f(p,
e)
dp
dc
=
0
co
Pm
a
q
(
5
')
The
subscript
behind
U
indicates
at
which
prices
the
profits
entering
into
U'
are
evaluated.
We
notice
that
the
left
hand
side
of
(5')
is
a
function
of
p
m
and
q
so
that
we
denote
it
L(q,p
m
).
Then
it
is
easily
shown
that
sign
(
d
d
p
q
_
sign
(
aL
because
by
the
second-order
condition
L
L
<0.
aq
The
comparative
statics
analysis
would
be
rather
cumbersome
with
a
too
general
model.
This
is
the
reason
why
we
now
examine
two
different
specifications
of
the
function
q
f
,
namely
an
additive
one:
qh.=q+i
with
E(Z)=0
(6)
(4)
.
1
"
co
Pm
U;
(p
c'(q))
f(p,
E)
dp
dE
=
0
(
7
)
+
op
+a,
414
L.
Eeckhoudt
and
P.
Hansen
and
a
multiplicative
one:
q
r
=
co
with
E(Z)=
1
and
e
0
(6')
For
each
specification,
one
has
of
course
q
r
3
0.
An
additive
disturbance
Although
this
assumption
is
not
very
realistic
since
it
implies
a
disturb-
ance
irresponsive
to
the
level
of
the
planned
output
we
start
by
discussing
this
case
in
order
to
have
a
better
understanding
of
the
problem.
From
(6),
aq
is
equal
to
unity,
so
that
(5')
becomes
1
-(q,
P.)
=
f
+
-:
f
up_
(pEn_clo
ffp,
dp
de
For
the
first
order
condition
to
hold,
it
is
necessary
that
p
n
,
<
cr(q*).
Then
aL
f
aPm
=
{U;'.,(p
n
,—e(q*))(q
+E)+U'
m
if(p,
e)
dp
dE
(8)
-
0
is
always
positive
because
of
risk
aversion
and
the
fact
that
p
rn
c'(q*)
is
*
negative.
We
thus
conclude
that
even
if
p
and
e
are
not
independent
dq
dp
n
,
is
always
positive
and
the
result
obtained
in
the
case
of
only
price
uncertainty
remains
true.
A
multiplicative
disturbance
Because
of
(6')
and
using
the
same
procedure
as
above
aL
op.
is
now
equal
to:
(U'
I
L(p
n
,E—
c'(q))(qE)+
U'
pr
.E)
f(p,
e)
dp
de
(
9
)
The
sign
of
(9)
is
no
longer
well
determined
under
all
circumstances
since
depending
upon
the
realized
value
of
6,
p
cn
E
can
be
greater
or
smaller
than
c'(q).
Obviously,
p
rn
e
must
be
smaller
than
c'(q)
for
the
lowest
values
of
e
(otherwise
the
first-order
condition
could
not
hold)
but
for
sufficiently
large
e,
p
cn
E—c'(q)
may
become
positive
and
the
whole
integral
may
be
negative.
If
dq*
this
is
the
rase
one
obtains
dpn,
<0
and
the
price
support
program
may
have
a
perverse
effect
on
planned
output.
Minimal
prices
and
optimal
production
415
In
order
to
clarify
matters,
we
first
present
a
sufficient
condition
under
which
the
classical
result
de/dp
m
>
0
holds.
In
order
to
do
so,
we
rewrite
(9)
as:
f
+
j•
p,
33
U"
o
E
{1
(p
—c'(q))
q)
f(p,
e)
dp
de
(10)
U'
r
o
-
Since
U'
r
„„.E
is
always
positive,
the
sign
of
(10)
is
uniquely
determined
by
that
of
the
expression
in
brackets
in
which
one
recognises
partial
relative
risk
aversion,'
that
is
minus the
absolute
risk
aversion
(U"/U')
multiplied
by
a
monetary
value,
here
(p
r
o
c'(q)).q.
We
thus
easily
obtain:
Proposition
1:
If
partial
relative
risk
aversion
is
always
smaller
than
unity
in
absolute
value,
dq/dp„,
is
positive.
Of
course,
not
all
utility
functions
exhibit
a
coefficient
of
partial
relative
risk
aversion
that
satisfies
the
above
condition.
2
However,
as
Cheng
et
al.
(1987)
have
convincingly
argued
such
a
property
is
highly
desirable
to
yield
meaningful
behavioural
results.
When
the
sufficient
condition
presented
above
is
not
satisfied,
it
is
possible
to
construct
examples
where
de/dp„,
becomes
negative
despite
the
presence
of
risk
aversion.
As
can
be
seen
from
(9),
a
negative
correlation
between
p
and
e
may
very
well
give
rise
to
the
paradoxical
result.
Indeed
if
low
values
of
p
are
most
of
the
time
associated
with
high
E'S,
when
the
price
support
is
effective,
a
is
likely
to
be
high
so
that
the
positive
terms
[p
r
o
c'(q*)]
are
more
heavily
weighted
than
the
negative
ones
and
the
perverse
effect
on
planned
output
may
take
place.
To
illustrate
this
point,
let
us
consider
two
decision-makers
(D.M.)
who
face
similar
conditions
except
for
one
of
them.
These
D.M.'s
(denoted
A
and
B)
share
the
same
views
about
the
possible
value
of
p
(80
or
220)
and
of
e
(.5
or
1.5)
and
these
equally
likely
events
have
the
same
marginal
densities
for
A
and
B.
However
A
and
B
disagree
about
the
correlation
between
p
and
e.
For
instance
A
is
located
in
a
closed
economy
so
that
when
E
is
high
(for
example
favourable
climatic
conditions)
p
is
low
and
conversely.
On
the
contrary
B
operates
in
an
open
economy
so
that
he
faces
world
market
prices
for
his
output
and
these
prices
are
uncorrelated
with
the
specific
production
conditions
prevailing
in
his
region.
The
views
of
A
and
B
are
expressed
in
the
following
joint
densities.
A
P
N
.5
1.5
g(P)
130
0
.50 .50
220
.50
0
.50
h(e)
..50
.50
B
.5
P
1.5
g(p)
80
.25
.25
.50
220
.25
.25
.50
h(c)
.50
.50
416
L.
Eeckhoudt
and
P.
Hansen
Except
for
their
beliefs
about
13
and
"i
the
two
decision
makers
are
otherwise
similar
in
that
they
share
the'same
quadratic
utility
function:'
U=
n—
fie
with
U'
=
1
—2/3n>0
for
all
n.
Of
course,
/3
is
the
parameter
of
risk
aversion.
The
two
D.M.'s
also
have
a
constant
marginal
cost
c'=
113,
and
B
=
0
for
each
of
them
(this
assumption
is
made
to
simplify
the
computations).
These
values
of
q*
are
compatible
with
positive
marginal
utilities'
and
as
expected
q*
is
decreasing
in
/3
the
degree
of
risk
aversion.
Notice
also
that
q
i
t
is
much
lower
than
q:
despite
the
fact
that
E(pE)
is
higher
for
B
than
for
A.
5
The
rationale
is
simply
that
Var(131)
is
much
higher
also
for
B
so
that
risk
averse
producers
reduce
the
output
level.
Let
us
now
consider
a
support
price
program
at
the
level
of
82.
6
This
affects
the
joint
densities
in
tables
A
and
B
where
p
=80
is
now
replaced
by
p
=
82.
Consequently,
one
obtains:
We
see
that
the
two
producers
have
reacted
differently
to
the
agency's
price
program:
A
has
lowered
its
planned
output
while
B
has
increased
it.
The
surprising
result
for
A
stems
again
from
the
impact
of
the
program
on
the
variance
of
the
marginal
receipt.
Before
the
intervention
the
density
of
the
marginal
receipt
for
A
was
Pe
prob
120
1/2
110
1/2
while
after
the
program
implementation,
it
has
become
pe
prob
123
1/2
110
1/2
exhibiting
a
higher
expected
value
but
also
a
greater
dispersion
which
itself
results
from
the
negative
correlation
between
p
and
e.
Conclusion
It
is
generally
admitted
that
at
least
in
the
short-run
minimum
prices
stimulate
output
on
competitive
markets.
This
result
is
indeed
robust
when
Minimal
prices
and
optimal
production
417
there
is
only
one
source
of
randomness
for
example
in
the
price
itself.
However,
in
many
agricultural
environments,
the
price
uncertainty
is
linked
to
technological
or
climatic
factors
which
create
a
discrepancy
between
planned
and
realized
outputs.
When
these
two
sources
of
uncertainty
are
combined,
the
relationship
between
the
level
of
p
m
and
planned
output
is
less
clearcut.
In
some
cases
(additive
risks
or
partial
relative
risk
aversion
smaller
than
unity
for
multiplicative
risks)
a
more
generous
protection
against
downward
price
fluctuations
stimulates
output.
However,
when
these
condi-
tions
are
not
satisfied,
a
negative
correlation
between
unitary
output
price,
p,
and
the
random
'technological'
element,
a
may
give
rise
to
a
perverse
response
in
planned
output.
NOTES
1.
This
concept
which
is
an
extension
of
relative
risk
aversion
was
introduced
by
Menezes
and
Hanson
(1970)
as
well
as
Zeckhauser
and
Keeler
(1970).
It
is
becoming
more
familiar
in
the
economic
literature.
For
its
recent
use
in
related
contexts
see
Briys
and
Eeckhoudt
(1985)
or
Cheng
et
al.
(1987).
2.
It
is
worth
noticing
however
that
the
very
much
used
logarithmic
utility
function
does
have
a
coefficient
of
partial
relative
risk
aversion
that
is
always
smaller
than
unity.
3.
For
the
quadratic
utility,
partial
relative
risk
aversion
is
not
everywhere
smaller
than
unity.
4.
When
one
is
working
with
a
quadratic
utility
function,
one
must
always
check
that
at
the
highest
possible
realization
of
profits
U'
remains
positive.
This
is
the
case
here.
5.
i
is
the
random
marginal
receipt
attached
to
planning
one
more
unit
of
output.
6.
Let
us
stress
that
we
discuss
here
only
the
short
run
effect
on
output.
We
do
not
consider
more
general
problems
such
as
the
impact
on
welfare.
These
aspects
are
analyzed
for
example
in
Kirschke
(1985)
or
Newbery
and
Stiglitz
(1981).
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