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f;h. ^*kf^
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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Sloan School of Managcmont
Massachusetts histitntc of TrchnoloRy
Cambridge,
MA
02139
July 1984
Revised,
November 198G
SLOAN SCHOOL WP#1844-86
MASSACHUSETTS
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
INSTITUTE OF
7, ^^U-^--
A
Stochastic Theory of the Firm
Lode
Li*
Sloan School of ManaRoinrnt
Massachusetts histitute of Torhn()lnp:y
MA
Cambridge,
.Inly
02130
1984
Revised, Noveiiiher 1980
SLOAN SCHOOL WP#1844-86
Abstract
We
present a stochastic model of make-to-stork firms hasofl on a l^uffer flow system with
jumps. The cumulative production and the cumulative demand are poverned by two Poisson counting processes with random intensities parameterized by
price respectively.
Optimal operating and
])r()(hiction
capacity and
i)ricing policies (short-run dcci.sions)
and
opti-
mal capacity (long-run) decisions are explored by application nf a two-stage optimization
device.
Detailed computations regarding the Poisson
on the basic model with learning
Iiuffcr flow
system and a variation
effects are also presented.
Eriiaii f'liilar, Mirliaol Harrison, ami Morton Kamicn. I am
also gratefully acknowledge the referees' suggestions and
and comments.
the Associate Editor's detailed comments, which improved the article anil clarifieil the exposition sulistantially.
'This work has benefitpd ronsidcrahly from clisrn"!ion' with
greatly indebted for their
many
useful suggestions
I
DEC 2
1
1989
Introduction
1.
This article develops a stochastic model of a makc-to-stork firm
in a coiitiimons
time frame-
work. The model, on the one hand. Kcnerali/.rs the rlassiral rrnnnmir theory of monopoly to include
the dynamic aspects in the presence of both
[15], [19], [24]
and
demand and prodnrtion
All the
[25] for this line of litorattnr.
demand
single-period or a multiperiod setting with
mndds
nnn-rtainties. See
[2], [3], [8],
studied in the above articles have a
uncertainty only. De Vany introduces a quene-
and uses long-run
ing
model
On
the other hand, our basic model also parallels the type of models in inventory theory, but ex-
of make-to-order firms in
pands the framework
[G]
to explicitly consider capacity
and
averagi> profit criterion as the objective.
price as decision variables,
the btiffer system of continuous flows discussed in Harrison [11]
with jumps in which the explicit consideration of rapacity
The
are
basic assumption of our
model
is
is
hi particular,
(^xtended to the flow systems
anrl pric(> di^cisions
is
possible.
that the cumulative prodiiction antl cumulative
demand
two (Poisson) counting processes parameteri7,ed by iiroduction capacities and prices respectively.
Mimicking the
real life operation, the firm's decision
is
two
fold:
it
makes a
static design decision
(capacity decision) at time zero, and exercises the dynamic control of production rate and price over
time.
A
total discounted profit criterion
is
The optimal capacity
used.
decision and the optimal
operating and pricing policies are explicitly cliaracterized via a two-stage optimization procedure.
That
we
is,
find an ojitimal j^olicy for each given capacity level,
first
level to
maximize the
level
selected, the firm operates optimally thereafter.
is
The
article
obtained
in the first step
arranged as follows. Section
is
Though we make
profit function
2
is
and then
assuming that whatever the capacity
devoted to the formulation of the basic model.
a quite restrictive Poisson assumption on the prorluction and
the formulation readily extends to nuire general additive processes
explicitly
for
argument justifying the optimality
model with dynamic pricing and
unchanged over time.
of the l)arrier policy
its
economic
is
provided
imjilications.
Concluding remarks follow
in
Section
and an
Section 4 discusses the basic
hi Section 5,
<
we study an interesting
uiiiulative
production due
G.
Formulation
We
is
processes,
we consider a
of the inventory i)rocess
variation on the basic model where achievable capacity expands with
2.
it
3,
an optimal barrier policy and the value buu tiou under a barrier policy are
computed by the strong Markov and n-newal properties
to learning.
demand
Section
In
special case in which the firm sets the price at the begiiuiiug and kee])s
The condition
select a capacity
shall consider a firm that continually profluc(>s
placed in an inventory
while the inventory level
(fulfilled
demand)
liuffer
is
where
it
zero are not
is
and
sells a single
ccunmodity. The product
taken out to satisfy demand.
The cumulative
filled.
\u\int
Demands
that occur
(production) and output
are represented by the increasing non-negative integer-valued stochastic processes
1
.
A =
{A{t),t
>
0} and
B =
and the amount of demand
{D{t)j > 0}
fulfilled in the
A,
B
is
time interval
{0.t\.
-
Bit).
= x+
Z(t)
where x
where A{t) and D{t) drnnU- the ammuit
,
the anionnt of inventory at time
A{t)
We
7,ero.
the inventory level at time
(
is
(2.1)
assiune that
random
are independent Poisson processes with
Then
production
of
intensities {oii.t
>
D} and {0t,t
>
O}.
(2.2)
That
{/o a,ds.t
The
tively.
- J^a,d3,t >
{A(t)
is,
>
>
0} and {/^ ft,ds,t
only real
-
0} and {B{t)
J^f^.d.'^.t
>
()}
0} are compensators of the
requirement here
is
that i)rocesses
<i
and
are martingales,
munting processes
ft
hi other words,
A
and
B
respec-
he integrahle and he adapted
(
in
the sense to be explained shortly).
To have a model that embraces pricing and
demand and
analogs for the firm's
B
That
cost finictions.
is.
we need a family
of
tlemand processes
parameterized by prices, and a similarly parameteriz.efl family of pr<iduction processes A. For
the production process,
The value a
capacity.
We
is
a(>
let
0)
be the average outjiut rate when the firm working
<
that
Viewing «/
< a
«(
for all
input process which
demand
rate
function.
assumed
zero
to
(
>
as the actual jn'oduction rate the firm
0.
Note that
Poisson with rate a.
potential revenue rate p{P)P
be
whenever
it
willing or forced to
is
operating policy
is
(ccj)
and
(0t) are left
(2.4)
(rj,)
and
(/?()
<
Z(t)
a,
is
<
do
Also,
for all
t
>
assumed
to be
>
n and
ft,
all
is
bounded,
(,
we require
the potential
firm
is
a <leterministic decreasing
as usual.
may
set the
Management
demand
is
also
rate to be
'>
(selected at time zero), a feasible
(i*i.f1i) tliat jointly satisfy
the following:
limits.
Z
for all
t
is
ft,
t
>
0.
are functions of
based on only
Conditions (2.3) and (2.5) enstire that [n,] and
(ft,)
restriction (2.G) implies that backlogging
lost,
time
t.
control that the firm exercises at time
on hand are simply
/'(/^)
eonravc
and the capacity
are adapted with respect to
r»,/9,
at
A becomes
determined
so.
continuous and have right-hand
non-negative for
employs
tlien
0.
is
we assum(> the firm can control the average
That means the
/>()
Condition (2.4) implies that a, and
The
a
defined as a pair of stochastic [)rocesses
(2.3)
(2.6)
is
free to reject potential sales.
Given the inverse demand function
(2.5)
=
a,
if
through pricing and the inverse demand function
/?
The
is
at its full
a function of designed capacity factors such as capital and labor invested.
use a as a primitive parameter which measures the capacity of the firm which
at the beginning.
Z.
one must have stochastic
facility d(^sigii decisions,
and has no
effect
is
(Z{.t)..'<
th<- liistorical
<
t).
This says that the
information before time
(.
are intej^rabh" and predictable with respect to
not allowed: sales which cannot met from stock
on future demand.
For a fixed capacity and a fixed
but
in [U],
it
A
there exist processes Z,
B
and
X
the primitive netput process
our model parallels the
i)rice,
allows the system flow
How system described
l)ufferefi
One way
rejiresented by jumi) processes.
lie
to justify that
is
to generali/.e Harrison's proofs in ('Iiajiter 2 of [11]
is
in
DiO, oo),
by assuming
space of real-valued functions on D?+ that are
tiie
right continuous witli left limits, instead of ('^(O.oo), the space of r(^al-valued continuous ftinctions
on
We
3?.^..
show
shall
sample path construction of processes Z A and
a
to illustrate that there
.
is
no
Consider a flow system with
Z[i)
>
and
/9,
=
input up to time
amount
A(t)
is
Z(t)
if
=
and B{t)
t,
Wc
0.
up
>
t
-
B{t)
X{t)
=
X{())
is
=
L
is
increases only
increasing and
which
output
r»,
=
>
't
0,
/?,
= /?>
if
A{t) as the cumulative
lev(<l,
uj) to
time
Denote by L(t) the
t.
So the
input
actiiai
L(t) over [O.t]. Setting
+ Mt) -
Bit).
(2.7)
X{Q) + A{t) + B{t) = X{t) +
L
require the lost potential outjiut process
L
system
then given by
Z{t)
t
to time
for a simple flow
because of the buffer emptiness.
is
t
in
take X{0) as the initial inventory
as the cunuilative pnfentia/
of potential output lost
U
generali/atiou.
infinite buffer capacity
A(t) and the actual output B(t)
the inventory at time
We
make such
difficulty to
ROLL
with L{Q)
when Z =
=
Lit).
(2.8)
satisfy
and
D,
be consistent with the physical restriction Z{t)
0, to
>
for all
0.
We
will
show that the above conditions
uni(iuely determine
=
L{t)
L and
ftirther
imply
sup X-{m).
o<«<'
That
is,
L can be
concisely represented in terms of
flie
iniuutive
[)r()cess
X
The proof
involves
construction of sample paths only.
Denote by i
=
(xi,t
>
0), the generic
element of
i>i(r.)
=
D
Define inap])ings
iji.'p
:
D
^^
D
by
sup x~ and
(!<»</
<f>,(x)
for
t
>
0.
Fix X
E D and
let
/
=
ijiix)
imposition of a lower control barrier at
Proposition 2.1. Suppose x G
D
=
and
z
X,
=
+
V'/i-r)
=
x
il>{x)
/.s
'A(;r)
+
/
Then
z
is
obtained from x by
7,ero.
and x^
>
0.
Then
3
thr nnitpir funrtitm
I
such that
(2.9)
=
(2.10) zt
(2.11)
I
=
!/o
G
t
since
E D and
—
z'
z
[<,_i,t,)
=
first
_
z*
—
t*
yi^
tj'
G
is
7^
then
and
many
tj
ti,
=
left
=
\ I
(z*
=
h,-
^,,)(';,
<
yi
=
G
t
1/0
implies yi,
i
v,,
Wo want
implies
0.
This
I.
t
for all
/}.
=
is
for
+
x
l'.
wo enumerate them
[t,_i, (,)
and hence
=
t/j
j/(,
=
j*
contradicts the fact
''.-)
when
—
z'
and
0,
>
z
0,
0.
and identical reasoning shows that
is
>
0.
G
[',.
t
G
^
N
tln^t
so
tli;\t
t'
N =
where
=
/
—
=
z,_
So
0.
we
far,
<
t'
00.
tj
.
If t'
for all
=
inliuium
tlic
Let /
{1,2,...}.
=
or, o(iuivalently, (*
<
t'
/ since y,
is
z*
i|
i
Tims
^
1
=
j/,
Suppose
0.
+
-
<)f''.
iucroasos only
a contradiction.
> t\ and
(2.12)
<
show
to
-
(^''.
/*
that
=
=
Certainly ty + i
^
+
';,-)
side of (2.12)
show
t*).
for all
left
to
:
-
last ofjuation
that
=
-/^-)l(< --^^)
Wo know
0.
But the
does satisfy
I
sot z'
Then
=
-
this
and
tyjio.
first
continuous on the intnval
((/;,-/;,_)-(/,.
9 since
0.
(variation finite) function with
discontinuity points of
is
we can conclude
inf{<,
=
ran he vciifird that
It
I.
U(^LL and V F
^
=
is
In
otlier solution of (2.9)-(2.11)
is
0,
j^
=
t/
+
x
bo any
let /*
2.3 in (11).
—
_
/,*
=
z
Note that
0.
as well.
t*
N
!/f,._,
sup{iy
But fy+i <
is
=
<
if i/j.
O} and
note that
/,
=
i)(x)
with
inrirnsiii)^
0.
we note that y
I,
=
!/<,_,
Zi^_
The only thing
:
=
z
nwl
limits
and
term on the right side of the
the second term
{i
=
/
by proposition
have proved that
First,
when
= r —
Suppose
••
•
t/<__
so the
It
t>0
for all
Since y has at most countable
0.
as ti,t2,
for
=
1/
xi
>
To prove nniqneness,
(2.9)-(2.1I).
Setting
+
increases only
Fix X
Proof.
h^nd
right continuous with left
is
I
i
=
G
Now
/•
Thus,
let
=
implies yt ,^,
(*
oo.
some
for
(,.
=
= 00
0.
or /
empty.
Therefore y
=
0,
hence
l'
=
and the proof
I,
is
complete.
I
The other
is
results in
Chapter
2 of [ll|
ran
\>c siiiiilaily
noteworthy that the proof of the above i)iopositiou
'^eiier:ili/,ed
df>es not use the
trajectory of the difference of two Poisson [irorosses. and hence
n
to
is
space D.
Though we consider
tn the rase that x
tlie
E.
assumption that x
risult
remain valid
D.
is
It
the
as long as
a special case of Poisson. the risults in the paper are not hard
extend to the case of compound Poisson
procoss<>s. or
more
|.^eiieral
buffer system of additive
processes.
To complete onr formulation, we specify the
cost
C(a)
at time zero,
which
is
used to
also incurs a linear variable cost, say
1:
cost structure as follows.
l)uild cajiacity
o-.
and ({'t)
is
The
firm incurs a fixed
increasing in a.
dollars per uiut of actual pioduction.
A
The
firm
cost of h dollars
per unit time
is
incurred for each unit of prndurtioii
discoiinted revenue over the infinite horizon
=
TR{x)
where
r
>
is
a discount factor.
The
objective of the firm
(q:(,/?() to
is
[
to choose a
iiivnitmy.
Tlicrrforp the expected
is
+ hZ(t)dt]] + n{a).
production cajiacity a, and a pair of control processes
profit
= Tn{x) -T(nx)
n(x)
(2.1)-(2.2),
cost
'^~''\cdA{t)
maximize the expected discounted
such that assumptions
in
eAJ" r-"p{P,)dD{t)Y
The expected
= Ell
TC(x)
hcM
i?
and feasihihty constraints
(2.13)
(2.3)-(2.G) arc satisfied.
proposition transforms objective fiuiction (2.13) to an (vjuivalcnt form which
in finding the
is
The
following
easy to deal with
optimal operating policies.
Proposition 2.2. For any given policy
n{x)
(o:^/?,),
= V(x)-
h
x~ (^a).
(2.14)
where
V{x)
=
E,
y^
e-"{{p(ft,)
+ ^)dD{t)
- {r
+ ^)rM(0|}
(2.15)
.
Furthermore,
V{x)
Proof.
?,
{^"'--^'[(H/'r)
+
^)/^>
+
('•
+
^)'^>]dt^
Equation (2.14) can be obtained by substituting the iiivmtory
and applying the integration by parts formula
follows from the assumption that
{^,} and the
A and D
to
i-fjuation (2.1) into (2.13)
for the Riemanii-Sti(-ljos integrals.
are Poisson pn>ress(>s with
fact that the integrands in (2.15), r-''{p(fl,)
and right-limited processes adapted
(2.16)
Z hence
.
+
~)
and
random
-•-'''(,
+
'l),
Equation (2.1G)
intensity {oii}
and
are left-continuous
predictable.
I
The proposition says that with each
and an opportunity
loss
-
if
this luiit
unit
produced
to stock, the firm actually incurs a cost c
would be stockerl
the firm's actual gain would be selling price
/)
forever; with each tuiit sold
plus an opportunity gain - that
is
from stock,
equal to the
opportunity
loss
would otherwise
this unit
if
The second
stage
solve the problem,
we
optimal operating policy for each given capacity level
the profit value functions obtained in the
the capacity level
3.
A
Then condition
<
to
rate
«,
p
-]
<
a production capacity
/?),
profit 0.
—
,
and
and
unit.
/?(
it
=
/?l(o,/ij(^(t
ceases
if
Demands
policy since
n
w =
c
-\
or a
—
— )),
for
(
>
.
0.
are rejected only
E =
M/M/l/b
them without
Assume
{O,
1,
qtievie.
{0,/?},for
is
.
.
.
,
The
ft}.
ft
;»
A
t
>
shall follow
and the price remain
Q.
(3.1)
to select a selling price
in (2.15)
>
c,
(efiuivaleutly a potential
;>
is
of the
form
b
0,
(3.2)
and then q > m. Let's
barrier policy
is.
for
some
is
it
>
always
are unavailable in stock.
b if all
first
is
consider a class of
Z
is
=
0. ni
is
This
a
first
a:l[o,6)(^((
depleted by one
is
really a simple
In fact,
=
i/
is
inf{t
a birth and death process with finite state space
0(x,y)
reached and
>
=
under a
Markov process with
following computations have been derived in Li [IG] and
time at which state
— ))
capacity except
at full
other parameters are fixed.
the inventory constant process
Specifically,
h
and resum<'s when the inventory
when products
>
all
This means that production
T(y)
Then
we
the procedure that
proofs.
Denote the
.
unique
maximize
= E, U^e-"(qp, - vm,)dty
the inventory reaches level
barrier policy with parameter
state space
find a
at the beginning, aufl a pair of control processes (a/,^f)
depends on only one number
it
e
a,/?,
feasible policies, namely, barrier policies.
that
is
price flecision at time zero
its
The value function defined
V(x)
=
first
(2.4) reduces to
The firms problem
a design variable.
maximize
where q
.
Design Decision
unchanged over time.
demand
i.e
a capacity a to
sel(>ct
step assuming the firm op(>rates optimally whatever
first
assume that the firm resolves
First, let us
/? is
and then
this article.
Pricing As
where
sta^e involves rapacity selection
proc(>ed reversely,
rt.
This two-stage optimization approach
set.
is
first
to find the optimal operating policy, the production decisions
is
To
as well as pricing decisions.
throughout
zero.
problem involves a two-stage optimization. The
firm's
for the plant.
storked forever. Tlie vnlue V{x) ran be considered
by the operations after time
as the gross profit incurred
The
l^e
(),Z{t.)
=
its
Laplace transform by
y}.^n<\
E,[r-'''^''^]
we simply
list
LGmma
3,1.
<
For
<
x
b.
<'(^'0)
^ ^^^TIT^' ^ndO(x.h) = 4l!'
<:(b)
g(b)
(3.3)
where
e(x)
and
and
pi
<
with
p2 are the
pi
<
Y,
P2
>
^,
p
=
for r
>
-l)pl-{p;' - [)pI
(p-'
two roots of the
decreasing, hence, ^(6,0)
Following
=
(jiiaf/rafir
c<iu^tinn
(3.4)
Fiirtliermovr,
0.
strictly docrcnsing in b
is
P
'-
+
p'
g{)
and
is
0{oo.i))
strictly inrrrnsing,
=
value function under a harrier policy with jiarameter
(3.2), the
V{x)
= E,
=
U
w
7
-
e-^'[7/?l(o,A|(Z(«))
•
—
e{)
is
strictly
D.
b
ran he written as
"'r»l,,,»,(Z(0)Wt}
(3.5)
(-
V(x),
where
V{x)
=
wE, U^" e-"al^,^{Z(t))(lt\
-
7
•
^" r"'' flli^^{Z{t))dt\
^.
(3.6)
.
{
The term Ej [J^
output
Ej
lost
e~'''(^l[o)(Z{t))dt}
(/g
And V
e~'''al^i,^(Z{t))dt}
is
can be thought
as the
due to ceasing production when the inventory
is
(•xi)(-rt{>fl
discounted potential
total
level reaches the limit
the expected total discounted potential
sal(>s
lost
h,
while the term
due to stockout.
the profit gain, in expected present value terms, under the harrier [xilicy over that of the
potential production and
demand. By the strong Markov and the renewal properties
of process Z,
we have
Lemma
3.2.
E^
I
/
.
Thus the
explicit
e-''ft
l,o){Z(t))dt
\
=
—
'
,
'"""'"""'"} -/nr;
,-^»,„
form of
V
,
hence
V
,
can he obtained.
function under a barrier policy with a upper barrier
b
if it
is
We
shall
am/
3.7
'=•"
-,.„)
denote hy V' (V
necessary to do so.
)
the value
Proposition 3.1. There
is
an nptinial harnVr jmliry with mir
exist
riitir^il
invrntnry limit
th^t
b'
uniquely determined by the condition
+
0(b'
<-.
1,0)
w<lO{l>\Q) > -.
7
(3.9)
7
Proof. For a fixed i,
v\x) - K*-'(x) =
- v^-^{x) =
i/*(i)
r;
- -j
(o(h.o)
•
(3.10)
.
where
^_
G =
since
is
all
7M^^(-^(^))g(ft)
rn
TTT
.
...
>
.
.
'• for
>
j—
r^
b
1
the terms in the numerator and the denominator arc positive. So
same
exactly the
as that of O(b,0)
-
be the one such that r** + '(3;)
—
-. Since
<
V^' [r]
si^i of
strictly drcrcasinR. the
'?(•,()) is
and
tlic
-
V''{x)
K*-'(:r)
>
F
[x)
optimal
which
is
—V
6*
~'(z)
should
equivalent to
condition (3.9)
We now want
policies.
Under a
will
to
show that
barrier policy,
never get out again.
if
We
tunity loss to have
Markovian controls
extend the value
+
k)
=3
finictioii
+ hn.
V{b)
policy, the extra k units
them produced
is
i/;
=
c
+
Denote by V* the value function under
1).
Lemma
We
3.3.
record the following twf)
The
V
is
tlu-
with
for k
>
where /()
is
=
1)
+
feasible
all
x(>
b)
{O,
1,
.
.
.
,
6},
it
by
0.
would never have produced and the oppor-
kq.
>
for k
lemmas which can
af{x +
E —
nf states
{).
lie
and AVfx) the difference V(x)
proveij
i)y
ronrave with AV"(') f
direct verification.
1)
Define the operator F by
F/(x)
among
complete.
is
initial state
ojitimal barrier jiolicy.
strictly incrcasiufi: ;\nd
.<et
optimal
Similarly define
-.
V(~k) = V(0) -
V{x —
iudceil
is
our scttinp
in
the inventory level starts from the
V(b
meaning that under a barrier
optimal harrier policy
this
In other words, the class of
(lf{x
an function on integer values. Then
8
-
I)
-
{a
+
ft]f{x).
=
?/r
and
AV* (0) =
q.
—
Lemma
The function
3.4.
V
sntisfir^ the
(hffcrmrr r(junti<)ns
rV{x) - rV{x) =0, forO < X
<b -
I,
and
rV{x) - rV{x) <
One more
Lemma
3.5.
technical
Suppose
E,[e-''f(Z(t))\
=
lemma
/(•)
f(x) + E,
is
is
fnrx
n,
b.
needefl in the verificHtioii of o])hmality.
Thru
a function nn intrgors
U
>
e-"[a,/(Z(.) +
+
1)
/?,/(Z{.) -
1)
-
(n,
+
/?.
+
r)f(Z{s))\ds\
(3.11)
Proof.
By
integration by parts theorem,
we have
/ e-"df(Z{s)) = er"f{Z(t)) -
f(Zm
+
Jo
r
r'^' f(Z{s))ds.
/
(3.12)
Jn
Note that
E^l
= E,{[
e-"df(Z{,^))]
Jo
.-"[(/(Z(..-) + l)-/(Z(..-))),M(.)
Jn
+ (/(Z(..-)-l)-/(Z(..-))),/Z?(.01}
w
= E,{
e-''\a,f{Z(.) + I) f/?./(7(,,)-l)
(313)
Jo
-(a, +
The second equation
e~'^'\f(Z(s — —
)
Z. Taking
in (3.13)
Ej
follows from assumption (2.2)
—
1)
ft.)f{Z(s))]ds}.
f(Z(s — ))\
are
both
left
and that
— +
)
I)
—
f(Z(s — ))] and
continuous. riRlit-limitcrl and adapted with respect to
both sides of (3.12) noticing Z(0) —
of the
i'~'^'\f{Z{n
x. sulistitntins,' in
the expression obtained
and collecting the terms, we conclude the proof.
I
Note that as a special case of a barrier
])olicy,
F
is
the ^'ineiatnr of
Z and Lemma
3.5 implies
that
t-'^f{Z[t)]
I
c-"(r/ -
rf)(Z{.'<))ds
is
a inartm-alo.
Jo
Here
is
the
main
Proposition 3.2.
is
optimal
among
Proof. Denote by
Tiie hairier
all
F
result of the section.
pohcy with inventory
hniit [uiicjuely
ilefmninrd
l)y
condition (3.9)
the feasible policies.
the value fiuiction with arbitrary feasible policy
V,(x)
= E,[[
Jo
,
(re,,/?,).
r-"{q(^,-um.)ds + r-''V'{Z(t))l
Define
(3.14)
.
for
>
t
time
>
and i
This
0.
the valur funrtion for a hybrid pnhry that follows (a,,/?,)
is
up
to
yielding a inventory content of Z(t), and then cnforrcs the niitimal l)ariier jjolicy with value
t,
fimction
V*
By
thereafter.
(3.11),
E,[e-''V'{Zm = V'(x) + E,{ I
-{a, +
P,
e-''\a,V'(Z{.^)
+
+
1)
+
-
/^.V'iZ{.^)
1)
r)V'{Z{.^))]ds}
(3.15)
=
V'{x)
+ E,{ I e-"[(rV' Jo
-
- 0,)AV'{Z{s))) -
{0
-
rV'){Z{s))
{q/i
-
((r»
- r,.)AV'{Z{s) +
1)
wr,)\,h}.
Putting (3.15) into (3.14),
= V'(x) + E,{ I fr"\(YV' -
V,(x)
+
rV')(Z(.))
(/?
-
/?.)(
-
AV*(^(.'))
q)
Jo
+ {afor X
by
>
and
Lemma
a,){xv
(>
(3.16)
-
since
AK'(Z(.^) +
TV
- rF* <
and assumption
3.3, 3.4
we have F{x)
<
lim,_oo
VM <
0,
(2.5).
EA
lim
'-^
<
1))],/..}
V'(x).
w < AK* <
7,
<
and
<
a,
n.O
<
/?,
< ^
for a
>
Noticing that
-
I e-"{qft,
u,n,),h)
=
F{x),
Jo
V'i^-)-
I
The
6* as a
some monotonicity properties
following proposition gives
of the optimal inventory limit
fimction of other parameters.
Proposition 3.3. The optimal inventory
limit h' inrrmsrs
a.s
<y.r
nr h drrrmscs.
Proof. Let
=
/(6,a,r,;»,/i.r)
w
O(l>.0)
(3.17)
.
7
Note that
6—
1 is
then
b*
jumps only
optimal.
6—1
left (i.e.,
If
at the points
an increase
in
where / =
>
of the parameter.
an increasing
left
and f{b =
1)
Conversely,
According to rouditinn
some parameter causes
remains optimal to the right
f(b)
n.
<
if
0).
(i.e.,
—
f(l>
in this case b'
an increase
in
1)
is
a d(>cicasi' in
>
0,
and
/('»)
/
at
<
(3 9), f(b)
dc
df
0,
<
-f
dh
0,
df
—
<
dn
and
ft
10
>
1,
h is
=
0,
optimal to the
a decreasing right continuous step function
some parameter causes an
if
implies
the point where f(h)
0).
incicase in /, then
continuous step function of the parameter. However.
df
—
<
=
and
Of
—
=
dm
if
ft
=
1.
b' is
.
.
The
iutuition of proposition 3.3
An
(Hiite rlear.
is
inrioasc in piixlurtiou capacity a implies a
higher frequency of production and a longer time for a jjrodnrt staying in stock. Therefore the firm
and avoid higher
prefers a lower inventory limit to modify this effect
holding cost h or production cost
ambiguous. The only remark we can make
demand. However the
ftilfillerl
that
b*
is
to
have a lower inventory limit when facing a more
is
if
the firm incn-ases the price, then
elastic
market
anti a
higher one
far,
a unique optimal policy
is
again a function of a and
p,
and an
explicit
form of
optimal capacity can be determined by usual caloilus.
comes from the
it
on
tends
facing a
similar problem by approximating
We
Dut
shall
b, tlie
h' as
art"
p.
Assuming
made, the expected
it
can be obtained. Theoretically, the
it
is
a function of
The
not a trivial job.
fact that the first order derivative of the value function
not continuous because of the discontinuity of
fimction.
when
obtained for each specific capacity n and price
that the optimal operating policy always follows after the design decisions
is
effect of price
market.
less elastic
profit
decrease in
causes an increase in inventory limit simply because the firm
c
can afford a higher inventory and then has more
So
A
financial coat.
(t
is
j>.
De Vany (1976) avoids the
l)y
a continuous differentiable
or
l)alking value in his model,
difficulties
with respect to a or p
show that under a more rigorous mathematical treatment, the usual calculus
and the marginal revenue-marginal
cost interpretations can
still
be applied.
Let n* be the expected profit under the optimal barrier pojir y with capacity n and the price
p.
The
relations (2.14)
and
(3.5)
then imply
n
{x)
= V
h
- -X - (^a)
(x)
*"
(3.18)
=
+
K*(x)
-/?
-
To study the properties
of FI* as a fimction of
terms in (3.18) are assumed to be
Lemma
3.6.
As
n
-
-ri
rr(r»).
r
r
an<l
;».
it
is
sufficient to
examine
V
since the other
nice.
a function ofn, the
V
j'.s
mntiwiouri. strictly
inrrrr\^in>^. nitd '//fferenfiab/e
except
that
>
lim
•^I'tn
lim
"I""
dn
——
(3.19)
art
for epich discontinuity point an of b*
Proof. First note that as a function of a, the
is
V
(the value function with a fixed
upper barrier
6)
contLnvious and infinitely differentiable, and
V
- max{V',F^
V\
This fact immediately indicates the continuity of V* with respect to
because
if
the upper
boimd
(3.20)
..}.
<\.
of the production rate, a. increases to
11
n
The second
+
fi
(ft
>
assertion follows
0),
then the firm
rau do
by feasibly emplnyine;
at least as well
optimal poliry with rapacity a and
thf'
in fact,
it
is
strictly better off.
ft,
and that
V* and hence
and they
only when
ft*
are dV'''~^
^(ft*,0)
/da and
— w/q =
At each discontinuity
dV'''
/da
we have that
0,
—^ -
lim
a\no
So
a decreasing step function of n.
is
Fl*, is differentiable.
.
exist
pciint of
each discontinuity point
at
=
da
=
G >
and d0{h,O)/da <
G
where
is
the right and
ft*,
respectively. Using the fact (3.10)
^--
lim
n\aa
da
infinitely flifferentiable for
is
each continuity
at
—
da
[K* [x)
- V*
da
since
F*
last assertion, notice the relation (3 20). that
To prove the
any fixed
for
0.
the
derivatives
and that
jumps
ft*
ft*,
da
q
<
left
ft*,
-'(i)]
dO(b'.O)
G—
of
ixq
jjoint of
>
ft
1.
defined as in (3.10).
I
Lemma
3.6 guarantees the existence of the optimal capacity a'
nuity point of
B
and
A
dU/da. Let
respectively.
^ -
ft
'-" dB(t)]
E. I
P2
-{h' + D
Pi
e-"dA{t)\
= -
Jo
The conditions
1
r
for the
r
;>*
and the
+
r
and
(3.21)
A
is
(3.22),
we
+ B
when
see that
/? is
{'•+
D
and
h
dA
r
dp
-)—
strictly positive,
5
roiuUtions:
=
(3.24)
d(.
da
da
A
are functions of
0.
;)
only through
hence, a decreasing function of
;>,
as a fmiction of p.
Because otherwise, the
and then there would be room
for
12
improvement
liy
;>*
left
is
and
B
p.
can
is
an
However,
always located
side of (3.23)
changing
it
ft(p)-
large relative to a.lieuccv a decr(\asinK function of p.
the necessary condition for optimality (3.23) shows that the optimal
decreasing stretch of
tite
(3.23)
dA
ft,
.sati.-f/y
=",
r
h
+ -)
da
an increasing function of
increasing function of/? only
optiiitnl i-;ip;irify it'
dp
hdB
(P
From
(3.22)
optimal price and the optimal capacity an- as follows.
hdB
(p+-)—
shown that
(3.211
P7
Pi
Proposition 3.4. The optimal price
l)e
^
- (1-^2)^1^' '-(I-Pi)pV
a
f°°
E,[
- Pi )P2
-(*• + !)
P2
- (1
IPl
Jo
A=
occurrence at a conti-
its
Then
TOO
B =
and
us denote the expected total discounted actual sales and production by
at the
would be
Recall that f^{p)
is
.
the
mean
demand whereas D{p)
rate of potential
we
For this reason,
are actually fulfilled.
the
is
the decreasing portion and /?(p) as the patrntial f/e»ia;i(/ functions
effective
demand by
e
and that
e
Proposition 3.5.
potential elasticity
Proof.
Show
>
//" /?
e
demand hy
of potential
—
dp
p
,//?
=
— and
•
'
—
on
elasticity of
p
—
B
dp
ft
Denote the price
i.e.,
,
no
=
f
,
demands which
as the rffrrtivr (ie/najn/ function defined
D{p)
refer to
(-xpert(-(| total (iisroniitefl
a, then the price eluftticity of effective
demnnd
i
;s
.s;/ia//er
than the
in the absolute value.
only for zero
initial inventory.
f
Equation (3.21) then becomes
_
{I
B=
-
-
^{i
dB _\
dp
P2
r
Let
iPl
-
[l
Pi
-
Pi
I),
-
)P2
Pi
and
dft
ft
81 _dft
B
ft
d'L
dft
dp
r
dp
ft
r
Oft
dp
dp
which implies that
1-1
'''
since
dL/dft >
if /?
>
a and
=
dft /dp
<
1,1
'''
+
^
rB
'IL
'l^u
'''
Oft
dp
0.
I
To compare with the deterministic
+
r)
=
T^TT
(IB
I dp
e
This
price
is
is
analogous to the traditional monopoly result
set in the elastic
(3.23) as
r(dA/dp)+'j((0A/.3p)-(,')B/0p))
1
r(l
we can rewrite rouditinn
theory,
portion of the
demand
;»(l
+
=
-)
However the relevant marginal
cost in the stochastic
instead of that of output and
is
composed
of
in that
marginal revenue
function, and
two
is
a
jiarts.
marginal cost,
cost,
i.e.,
(3.25)
c.
model
i'([uals
markup over marginal
is
a cost of total discoinited actual sales
prodn<tion cost and inventory canying
cost.
Equation (3.24) can be viewed as a louf^-rmi condition since
capacity of the firm.
It
indicates that capacity
13
is
it
determines the production
exjjanded to the point where the expected
marginal revenue achieved through rethirtion of the
and can be rewritten
capacity,
p
=
limit
iiiV(Mil-oi7
cidAldn) + ^idAlda) -
tho marginal cost of
+ (.ICldn)
{<lBlf)n))
_,^-
^
^.
firm sets the price equal to the loug-nui marginal cost of total
The
('([iials
as
ntimerator of the right side in equation (3.2G)
is
(3.26)
(liscf)tuit(vl
the pr(-scut value of the
full
The
actual sales.
incremental cost of
increasing capacity which consists of the short-term marginal cost of output times the increase in
average total production stream induced by greater capacity, the increase
in
average total inventory
holding cost due to an increase in capacity, plus the marginal cost of capacity building. This
incremental cost of capacity
is
induce a unit increase in average total
Some
interesting remarks can be
period situation. First,
and
As a matter
sales.
drawn from our
moflol which copes with a stochastic multi-
necessary to formulate acttial versus potential production and demand,
to introduce a buffer stock
not always meet
/?.
it is
full
multiplied by l/{dB/f)ni). the incifuuent to capacity required to
demand even
if
Secondly, with stochastic variability, production does
possible.
expected value, that
in the sense of
is
does not necessarily equal
re*
of fact, the firm always has excess capacity in response to a stochastic situation in
the sense that a* always exceeds the
mean
rate of fulfilled dem;uid.
Finally, seeking a stochastic
theory of the firm, one inevitably arrives at a dynamic model of the firm where the decision process
is
split into
long-run design decisions and short-run operating derisions, because of the central role
of the inventories in responding to stochastic varial)ility.
4.
Dynamic Pricing
We
are
now
generality that
in a position to solve the basic
dynamic pricing
is
model foriiiuUted
be facilitated by considering a
The existence
is
class of
no longer easy
of
shown by
semi-Markov decision
the results parallel to those in the previous section.
of an
still
The
ilecisions,
the explicit
not impossible and our discussion
as
semi-Markov decision processes.
the contraction mai)ping fixed point
|)rocesses (see. eg.,
Also we shall leave the completeness of Markovian controls
For notation simplicity, we
if
problems known
of a finite stationary solution can be
theorem or the general theory
Section 2 with the further
and pricing
In the investigation of the optimal operating production
calculation as done in the previous section
will
in
allowed.
asid<'
[7].
[13], [14]
.md omit most
and
[21]).
of the proofs of
interest(-d readers are referred to [16].
denote the optimal value functions by
semi-Markov decision processes, the recuisive e([uations
V and
In the context
FI.
of the Hi'ms i)roblem can be specified
as follows
L
L
rV{x)=
max
f»'€[0,a|,<J'e(0,/9l,n,,,|(j')|
{/?'[;.(/?') -f
--
AF(x)l + -/(AV(.r
-f
I)
-
(c
+
-)]},
r
r
14
(4.1)
=
where AV{x)
Lemma
It is
V{x)
- V(x -
4.1. For fixed a, the va/iie function V{
easy to see that for each
a,
\
~
°'^^'
noticing that
a{x)
=
1).
a
j 0,
I)
>
c
AF(i +
1)
<
r
An(i) = AV{x) —
<
for
i
<
6
—
x, if rt(x) solves (4.1), fheii
d AV(x +
if
-.
Stippose
and a(h) =
1,
anJ mnrnve.
inrrr^ising
/s fttvirtly
)
a barrier policy with a inventory limit
+
+
J
b is
AU(x +
An(x +
the siuallost x .such that
That means
0.
Lemma
h.
rqnivalent
lently.
or,
^ or, equivaieut ly.
for fixed n, the
4.1 implies that h
is
1)
>
1)
<
c;
c,
An(i +
1)
<
then
c,
operating policy
tuiiqnely determined
is
still
by the
following conditions
>
An(ft)
The
andAn(ft +
c,
<
1)
r.
pricing policy can be solved by examining (4.1) in a similar fashion.
The
.'solution
0{x) should
satisfy
+
p
In the similar
way
that
- An{x) =
of proving Proposition 3.2,
Proposition 4.1. For
b,
P^
for
<
I
/..
we have
fixed a, the optimal o[)erating
pohcy
a h^rrirr
is
pnhry with inventory
limit
is
a,
where
n,
b is
=
alio.h)(Z{t-)).
(4.2)
uniquely determined so that
An(ft)
The optimal pricing policy
>
c,
and An(ft +
1)
<
(4.3)
,:.
is
b
and
ft(x) is
determined by the following short-run
r(uiilitions:
dp
p
+ ft— - An(x) =
0,
for
1
<
J
<
h.
(4.5)
dft
The implication
to a limit
of condition (4.3)
is
that the firm will produce to stock at
where the short-term marginal cost of prfiduction
increment of an additional unit of
total future profit
inventory limit
b
which
results hold in the
outjjiit at
case.
15
capacity
will
exceed the present value of the
the
moment. Regarding the optimal
solely determines the optimal operating policy, similar
dynamic pricing
its full
comparative
static
Proposition 4.2. The optimal inventory
On
the other hand, equation (4.5) ran
;,(1
where
t is
the price elasticity of potential
limit h »/rrrrasrs as
or h decrrnses.
written a?
I)e
+
Vr. r.
=
i)
demand
An(3:).
(4.6)
the inrviotis section. Here, U.{x)
rlefinerl as in
is
the total expected future profit given that there arc presently x units of prnrluct on hand. Suppose
one unit
Therefore
(4.6)
is
sold at the
is
moment, then
An(z) can be
the present value of net
i)rofit for this
unit
interpreted as the marginal cost of increase in sales
analogous to the traditional monopoly result (3.25). However
the actual production unit cost,
c,
it
is
is
l)y
;)+n(z —
1)
— 0(2).
one unit. Equation
noteworthy that instead
the correct marginal cost in a stochastic model should be the
imit cost of actual sales, An(3:), the increase in cost of selling one unit out of stock leaving the
capacity, the optimal operating and the optimal pricing i>olicies unchangefl for the ftiture.
Proposition 4.3. The price
that under certainty,
(fccrea.ses
a.s
the inventory /eve/ /ncjea.ses
aii'/
/.s
a/ways higher than
i.e.,
r(/9(l))>;.(/?(2))> ...>v{p{h))>p''.
where p
is
traditional
Lemma
Proof.
4.1
monopoly
(4.7)
price determined in (3.25).
shows that AV(x), hence AU(x).
condition for optimality says that
/)(!
+
1/f)
is
And
the second order
<^'<>ndition (4.5)
which determines
decreasing in
is
decreasing
in
/?.
x.
P(x) then implies that /?(x) increases as x increases, or p{(^{x)) flecrcascs as x increases.
Lemma
4.1
and condition
An(3;)
It
becomes obvious that
Finally,
imply that
(4.3)
p(ft(x))
> p
>
An(ft)
for
1
<
i
>
for
r.
<
1
<
<
x
/..
A.
I
Intuitively one can think that the
firm has to lower the price
line of
firm
is
is.
thinking
may be
more products
and encourage demand
misleading since following
pile u]) in inventory, the
for the sake
it,
one
may
<<{
reducing
its
gather that, with
more incentive the
holding cost. This
7,ero
inventory, the
would have a best situation and simulate the deterministic pricing decision. However, the
that, because of stochastic variability, the iimre proflnct the firm has
This can be seen from the fact that Tl(x)
lower stock on hand, the firm sets higher
— n(x —
1)
>
])rices to discoiu-age
lever anticipating higher profit in the future.
r
>
for
demand
<
x
<
h.
it
Therefore, with
in orrler to increase the stock
This procedure continues to the point
16
fact
on hand, the better off
at
which the
.
inventory level reaches
the deterministic case,
b
c.
of actual sales n(A)
and the marginal cost
It is
.
—
—
n{h
closest to that in
1) is
at this point that the firm r(>asos produrtiou ni)tinially and simulates
the deterministic pricing decision. In sum, the firm tiausfers the cost n( uncertainty to consumers
by raising
prices.
we derive the condition
Finally,
optimal capacity decision. The value function under
for the
the optimal operating and pricing policies
V
Lemma
cnntimiotis. strictly incrr^isin^.
4.2.
As
a function of a, the
V
is
possesses the
that
dV
for each discontinuity point
Still
as before,
we
let
anW
Lemma
3.6.
diffcrentiable except
—
lim
n\n„ d ft
da
aoofb.
we define
B=
In addition,
as in
OV
>
lim
"I'>n
same properties
TR
eA I
eA
A =
er''dD(t)\ ^nd
r-"dA(t)\.
I
be the present value of the expected total revenue.
-rl
TR = E,{j
c-"iiP,)dD(t
1
Proposition 4.4. The optimal capacity
dTR
—
—=
on
ct'
Inn^-run mudition
satisfies the
DC
dA
dA
dD
—
+ -(__- — + —
an
dn
da
h
c
(4.8)
)
iln
r
Condition (4.8) says that the firm expands
its
rapacity tn the point where the expected
marginal revenue achieved through reduction of the inventory limit
("(juals
the marginal cost of
capacity, which consists of production cost, holding cost and rapacity building cost, induced
by the
increase in the potential production rate.
5.
Learning Effects
Learning
effects
can be introduced into the model of the preceding section.
literature, learning effects are introduced
cumulated output or production
In
most of the
by the assumption that unit cost declines with the
and
(see, e.g., [l], [12]
learning effects into the ciirrent model in a
more
(23|).
way
direct
in
ac-
Hf)wever, we attempt to introduce
which productivity increases with
cumulative production.
Let ni{A(t—)) be the upper
bound
of the prod\iction rate that the firm
t,
where 7(a)
is
the capacity achievable at time zero, and n
is
increasing and concave in a, 7(0)
economic justification
is
simple.
The
is
firm builds
17
>
0,
and 7(a)
—
>
1
can achieve
as a
—
»
00.
the capacity achievable in the long
up an
ideal capacity level
n
at
time
So 0:7(0)
nm.
The
at the beginning,
and
its full
capacity production rate gets greater and greater apjiroaching the
management and the labor become more and more sophisticated
With learning
Section 2
is
effects
<
The
introduced this way, the modification
that the feasibility constraint (2.5) should
firm's goal
is
<
a,
rt7(A(<-))./?,
>
and
\)c
are
/?,
profit
U
model formulated
in the l)asic
in
altered to be
bounded,
to choose a pair of control processes (o:,,/?,)
a to maximize expected
rapacity as the
itleal
thrmigli actually producing.
providing the learning curve
for
all
t
>
0.
(5.1)
and a long-run achievable capacity
7(
).
For a fixed a, in order to investigate the optimal operating and pricing policies in the presence
of learning effect,
(x, a)
is
we need
to
expand the
state space to include cuiiiidative production.
a pair of non-negative integers with the
first
The system
the second representing the cumulative output.
said to be in state (x,a) at time
is
the firm has x rmits of product in stock and has produced n units
capacity production rate a'^(n) to start with from time
t
E,,A [
max
up
to time
t,
or say,
it
has
t
if
full
Denote the optimal value fimctions
on.
with learning by Tl(x,n) and V(x,n) satisfying relation (2.14).
V(x,a)=
Each state
element representing the inventory level and
i.e
,
r-^'[(r(f1,)
+ -)dD(t) -
(c
+ -)dA(t)]\
.
(5.2)
Let V^lx) be the value function with capacity
in
or
absence nf learning
tli<'
(-ffects as in
the previous
section. Simple observations indicate that
Lemma
and converges
5.1. V(x,n) increases
Proof. Starting with (x,n
+
1),
with starting state (x,a). This
to
V"(x)
as n i/jr/eases to infinity.
the firm would do at least as well by following the optimal policies
is
feasible since i{a
+
I
+
)
>
-i(ti
+
).
So V{x,a
+
1)
>
V(a;,a).
Similar argument implies that
V''''^"^x)
Letting a —* oo, hence a7(a)
<
<
—
V{x,
»
q,
lim[F''(x)
a)
< Vix)
V"{x)
is
< -^(a +
•)
<
n.
we have
-
V(x,a)]
< Yun[V"{x) -
nloo
since
since a7(a)
V'"''"'(.r)]
=
f»|oc
continuous in n as shown
in
Lemma
3.G
I
The recursive ecjuations that F(z,(t)"s satisfy can be specified as
rV(x, a)
=
max
'»'Gln,rri(n)|,/?'elO,;3l,o..,|(j')l
if^'lvift')
+ - - {V{x.
a)
- V(x -
I,
a))]
r
(5.3)
+ n'\(V{x+ l,a+
18
1)
-V(x.a)) -
(r
+
-)|}.
Lemma
For fixed a and
5.2.
concave for every a
Proof.
By double
>
Appendix C
induction. For details see
Examining recursive equation
is
strictly increasing
is
and
(5.3),
in [IC].
we have
and Irnrning curve i(), the optimal
fixed long-run achievable capacity n
Proposition 5.1. For
production policy
production, that
the optimal value fiuirtion V(-.ii)
-){).
0.
a barrier policy with inventory limit
l>(
).
a decreasing function of cumulative
is
=
«'
EE'^
•
l{,.„){Z{t-),A(t-)).
'^('^)
(5.4)
a=0 1=0
where b(a)
is
uniquely determined so that
n(b, a)
-
n(6
The optimal pricing policy
-
1,
a
oo
is
E
>
c,
and U(b +
I.
a)
-
n(b. a
l(..,.)(^(«-)-
[1, b(a)\,
is
fimction and
a
>
stock at
its full
(5.5)
(5.6)
+ ft^-{n{x,a)-n(x-l.a)\=0.
(5.7)
0.
oft, the optimal inventory limit b{A{t —
stochastic.
is
<c
determined by the following short-run condition
As a function
production
I)
Mt-)).
Regarding the operating pohcy, we note that the njiper harrier
before.
-
M")
- EE/'(/'(^"))
p
for each x
1)
is
/'(/?<)
where 0(x,a)
-
As a
fiuiction of the
))
is
is
uo longer a single number
cumulative actual production,
defined by the condition (5.5).
/'(•)
is
This again indicat(-s that the firm
a deterministic
will
capacity achievable up to a limit where the short-run marginal cost,
produce to
c, will
the present value of the total profit increment induced by an additional unit of output.
shown that
as a increases to infinity, th(^ optimal inventory limit h(ii) decreases to
inventory limit with capacity a and without learning effects. That
limit gradually as its
will
be
To
shift
upwards
as
stochastic since the accumulated
is,
/)'*,
It
exceed
can
l)e
the optimal
the firm lowers the inventory
production experience grows. Also, the downward-sloping upper barrier b()
as h,c, or
a decreases.
investigate the properties of the optimal jiricing policy with learning, rewrite condition
(5.7) as
r(l
+
-)
= n(x,a) - 0(1-
e
19
l.a),
-
(5.8)
where
is
demand
the price elasticity of potential
e is
The
as ])rfore.
— U{x —
diffeicnro. n(x, a)
the present value of the total future profit reduction rlue tn taking one miit out of stock.
be interpreted as the (opportunity) marginal cost of increase
n(x, a)
—
Y\(x
—
1,
a)
is
decreasing for any fixed a
p(/i(x, a))
Therefore the
effect,
<
-
p(ft(x
1,
>
<
1
unit.
By lemma
5.2,
x
<
and a
h(a)
>
0.
that in a stochastic environment the firm with lower stock on
hand has the
tendency to set a higher price to discourage demand and hence to achieve higher inventory
anticipating higher future profit,
tendency
is
U{x,a)
exists as that
still
moderated by the learning
- U(x -
Both terms on the
l,a
-
1)
=
effects.
[U(x,n)
To
prices, while the
in the
absence of learning.
level
However, this
see this, note that
- n{x -
right side of (5.9) are positive.
which determines the optimal
a),
can
This implies
0.
for
a)),
by one
in sales
1,
It
I, a)]
The
+
-
{n(x
term
first
second term
- n(x -
l.a)
is
-
l,a
(5.9)
1)].
the marginal cost of actual sales
reflects the profit gain
by learning. The
firm chooses the optimal inventory limit h(n) by jointly considering these two effects (see condition
(5.5) in Proposition 5.1).
Suppose
so chosen that
b is
-
n(b,a)
U(h
-
-
l.a
1)
=
r.
Then
n(6, a)
-
n(b
-
1,
a)
=
[n(ft, a)
-U{b- l.a-
1)]
-
\U(b
-
1.
-
-
-
<
c,
interesting point here
is
a)
n(ft
1,
a
1)]
and hence,
H/?(ft.a))
where p
is
the
monopoly
price
</,
under certainty satisfying
(3.25).
The
that
the firm with stochastic variability tends to transfer the cost of unrcrtaiuty to consumers, but the
presence of learning effects moderates this transfer. Tlnuefore the monopoly price with uncertainty
as well as learning effects
the firm
does not always dominate the iiiono])oly price under certainty.
become more and more mature, the
effect
of learning
is
When
weaker and weaker, and
this
moderation diminished gradually.
Proposition 5.2. The price
lative
dcrrerises
production but does not a/ways
a.s
the inventdry level iiirrenses far any
(/n/ninafe.s tlint
fixe(l level
of ciimu-
under certainty.
Similar to the case without learning effects, the long-run condition etpiating the long-term
marginal revenue and marginal cost of capacity determines the optimal capacity a*.
20
Proposition 5.3. The
optiinal ra/)ar;7y a' .safisfirs thr
dTR _
dm
,1A
h
8A
iiB
nc
da
r
da
da
da
TR = E{J^ e.-''p{f^,)dD{t)}. A = E{J^ rr"dA{t)}.
where
6.
cnii'litidii
mul
D = E{J^
e-"dB(t)}.
Concluding Remarks
The
article introduces a
model
of production firms
\>y
assuming that onunlative production
and cumulative demand are two counting processes with random intensity parameterized by production capacity and price respectively. The formulation
among production
of production firms, such as the distinction
mand
and actual
rate,
ca])turi-s
sales rate; the fhstinction
between
dynamic theory, with inventory tying together
rajjacity, actual
characteristics
production rate, de-
static design decisions (long-rxin decisions)
and dynamic operating decisions (short-run decisions), etc
tally
many important
one obtains a fundamen-
hi particular,
and
prrxluctirin decisions
jiricing decisions at
different points in time.
This
is
a natural generali7,ation of the classical (deterministic) moflel of the firm. In fact,
be shown that a sequence of the Poisson-Poisson problems formulated
in
Section 2
will
it
can
converge to a
deterministic model as certain parameters approach certain limits, kee])ing the variance approaching
On
zero.
the other hand, close to the special case described in Section 3
inventory and production control studied by Harrison in
potential input and cumulative potential
demand
is
simple.
However,
which are
in the diffusion
explicitly
accounted
model represents another
The
limit result helps
and
justifies a
a diffusion
single critical
number
model, the design decisions arc lumped into a single
for in
our model. Nevertheless.
limit of our basic
model
model of
modeled by a Brownian motion (with general
and variance parameters), and the optimal policy (involving a
drift
is
There the difference of cumulative
[ll|.
Li
shows
6*)
is
drift
very
term,
in [IG] that this diffusion
as certain jiarameteis apjiroach critical values.
one to better understand conditions
tui<ler
which the
difftision
model
applies,
very tractable approximation for general additive process formulation.
There are many other interesting
ciuestions,
explored in the continuation of this work.
cumulative production
A
Our
both niathematical and economic, that might be
nuxlel can be j,'euerali7,ed to the formulation where
and cumulative potential demand
D
are arliitrary additive processes.
Suppose, for example, the primitive processes are c(uui)ounfl Pnissou with absolutely continuous
jump
size distributions.
parameter
6*.
But,
it
The optimal operating
is
not clear
how
by basic price and capacity decisions.
many
different "kinds of capacity",
different
market
strategies.
policy would
still
be a barrier
families of jioteutial ])rocesses
This problem
and one can
is
not suri)rising
i)olicy
with one
critical
would be "parameterized"
-
also abstract different
in real life there
may be
'"kinds of business"
with
Secondly, the models in this stufly seem appropriate in the markets
21
characterized by non-durable goods since there
is
generalized to include the durable good markets
of price as well as cumulative
linear,
and there
more general
is
no
I)y
demand. Thirdly, the
trenri in the
demand
variable cost of prorluction
no cost associated with varying the production
cost structure.
As
a
first
production, and this does not create
step,
much
we can associate
22
rate.
is
They can be
is
a function
assumed
Certainly,
to be
we can have
a set-up cost with each restart of
difficulty at least in the
in Section 3.
pattern.
assuming that the demand rate
romputations as we carry out
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