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Sloan School of Managcmont

Massachusetts histitntc of TrchnoloRy

Cambridge, MA 02139

July 1984
Revised, November 198G







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A Stochastic Theory of the Firm

Lode Li*

Sloan School of ManaRoinrnt

Massachusetts histitute of Torhn()lnp:y

Cambridge, MA 02130

.Inly 1984
Revised, Noveiiiher 1980



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 Pois-
son counting processes with random intensities parameterized by ])r()(hiction capacity and
price respectively. Optimal operating 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 Iiuffcr flow system and a variation
on the basic model with learning effects are also presented.

'This work has benefitpd ronsidcrahly from clisrn"!ion' with Eriiaii f'liilar, Mirliaol Harrison, ami Morton Kamicn. I am
greatly indebted for their many useful suggestions and comments. I also gratefully acknowledge the referees' suggestions and
the Associate Editor's detailed comments, which improved the article anil clarifieil the exposition sulistantially.

DEC 2 1 1989

1. Introduction

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 demand and prodnrtion nnn-rtainties. See [2], [3], [8],
[15], [19], [24] and [25] for this line of litorattnr. All the mndds studied in the above articles have a
single-period or a multiperiod setting with demand uncertainty only. De Vany introduces a quene-
ing model of make-to-order firms in [G] and uses long-run averagi> profit criterion as the objective.
On the other hand, our basic model also parallels the type of models in inventory theory, but ex-
pands the framework to explicitly consider capacity and price as decision variables, hi particular,
the btiffer system of continuous flows discussed in Harrison [11] is (^xtended to the flow systems
with jumps in which the explicit consideration of rapacity anrl pric(> di^cisions is possible.

The basic assumption of our model is that the cumulative prodiiction antl cumulative demand
are 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 used. The optimal capacity decision and the optimal
operating and pricing policies are explicitly cliaracterized via a two-stage optimization procedure.
That is, we first find an ojitimal j^olicy for each given capacity level, and then select a capacity
level to maximize the profit function obtained in the first step assuming that whatever the capacity
level is selected, the firm operates optimally thereafter.

The article is arranged as follows. Section 2 is devoted to the formulation of the basic model.
Though we make a quite restrictive Poisson assumption on the prorluction and demand processes,
the formulation readily extends to nuire general additive processes In Section 3, we consider a
special case in which the firm sets the price at the begiiuiiug and kee])s it unchanged over time.
The condition for an optimal barrier policy and the value buu tiou under a barrier policy are
explicitly computed by the strong Markov and n-newal properties of the inventory i)rocess and an
argument justifying the optimality of the l)arrier policy is provided Section 4 discusses the basic
model with dynamic pricing and its economic imjilications. hi Section 5, we study an interesting
variation on the basic model where achievable capacity expands with < uiiiulative production due
to learning. Concluding remarks follow in Section G.

2. Formulation

We shall consider a firm that continually profluc(>s and sells a single ccunmodity. The product
is placed in an inventory liuffer where it is taken out to satisfy demand. Demands that occur
while the inventory level is zero are not filled. The cumulative \u\int (production) and output
(fulfilled demand) are represented by the increasing non-negative integer-valued stochastic processes


A = {A{t),t > 0} and B = {D{t)j > 0} , where A{t) and D{t) drnnU- the ammuit of production
and the amount of demand fulfilled in the time interval {0.t\. Then the inventory level at time ( is

Z(t) = x+ A{t) - Bit). (2.1)

where x is the anionnt of inventory at time 7,ero. We assiune that

A, B are independent Poisson processes with random intensities {oii.t > D} and {0t,t > O}.

That is, {A(t) - J^a,d3,t > 0} and {B{t) - J^f^.d.'^.t > ()} are martingales, hi other words,
{/o a,ds.t > 0} and {/^ ft,ds,t > 0} are compensators of the munting processes A and B respec-
tively. The only real requirement here is that i)rocesses 0. Note that if a, = a for all t > 0. tlien A becomes the potential
input process which is Poisson with rate a. Also, we assum(> the firm can control the average
demand rate /? through pricing and the inverse demand function /'(/^) is a (■) and the capacity '>■ (selected at time zero), a feasible
operating policy is defined as a pair of stochastic [)rocesses (i*i.f1i) tliat jointly satisfy the following:

(2.3) (ccj) and (0t) are left continuous and have right-hand limits.

(2.4) (rj,) and (/?() are adapted with respect to Z .

(2.5) < a, < r»,/9, > n and ft, is bounded, for all t > 0.

(2.6) Z(t) is non-negative for all t.

Condition (2.4) implies that a, and ft, are functions of (Z{.t)..'< < t). This says that the
control that the firm exercises at time t is based on only th c, and then q > m. Let's first consider a class of
feasible policies, namely, barrier policies. A barrier policy is. for some h > 0. ni = a:l[o,6)(^(( — ))
and /?( = /?l(o,/ij(^(t — )), for ( > 0. This means that production is always at full capacity except
that it ceases if the inventory reaches level b and resums lost due to stockout.
And V is 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 / e-''ft ■ l,o){Z(t))dt \ = — ' , am/ 3.7

. '"""'"""'"} -/nr; ,-^»,„ -,.„) ■ '=•"

Thus the explicit form of V , hence V , can he obtained. We shall denote hy V' (V ) the value
function under a barrier policy with a upper barrier b if it is necessary to do so.

Proposition 3.1. There exist an nptinial harnVr jmliry with mir riitir^il invrntnry limit b' th^t
is uniquely determined by the condition

0(b' + 1,0) -. (3.9)

7 7

Proof. For a fixed i,

v\x) - K*-'(x) = i/*(i) - v^-^{x) = r; • (o(h.o) - -j . (3.10)


^_ 7M^^(-^(^))g(ft) . . . ...

G = TTT rn r^ j— > '•■ for b > 1

since all the terms in the numerator and the denominator arc positive. So tlic si^i of F [x) — V ~'(z)
is exactly the same as that of O(b,0) — -. Since '?(•,()) is strictly drcrcasinR. the optimal 6* should
be the one such that r** + '(3;) - V^' [r] < and V''{x) - K*-'(:r) > which is equivalent to
condition (3.9)

We now want to show that this optimal harrier policy is iudceil optimal among all feasible
policies. In other words, the class of Markovian controls in our scttinp is complete.

Under a barrier policy, if the inventory level starts from the . b) by

V(b + k) =3 V{b) + hn. for k > 0.

meaning that under a barrier policy, the extra k units would never have produced and the oppor-
tunity loss to have them produced is i/; = c + -. Similarly define

V(~k) = V(0) - kq. for k > {).

Denote by V* the value function under tlu- ojitimal barrier jiolicy. and AVfx) the difference V(x) —
V{x — 1). We record the following twf) lemmas which can lie proveij i)y direct verification.

Lemma 3.3. The V is strictly incrcasiufi: ;\nd ronrave with AV"(') f 1) = ?/r and AV* (0) = q.

Define the operator F by

F/(x) = af{x + 1) + (lf{x - I) - {a + ft]f{x).

where /(■) is an function on integer values. Then


Lemma 3.4. The function V sntisfir^ the (hffcrmrr r(junti and i > 0. This is the valur funrtion for a hybrid pnhry that follows (a,,/?,) up to
time t, yielding a inventory content of Z(t), and then cnforrcs the niitimal l)ariier jjolicy with value
fimction V* thereafter. By (3.11),

E,[e-''V'{Zm = V'(x) + E,{ I e-''\a,V'(Z{.^) + 1) + /^.V'iZ{.^) - 1)

-{a, + P, + r)V'{Z{.^))]ds}

= V'{x) + E,{ I e-"[(rV' - rV'){Z{s)) - ((r» - r,.)AV'{Z{s) + 1)

- {0 - 0,)AV'{Z{s))) - {q/i - wr,)\,h}.
Putting (3.15) into (3.14),

V,(x) = V'(x) + E,{ I fr"\(YV' - rV')(Z(.)) + (/? - /?.)( AV*(^(.')) - q)

+ {a- a,){xv - AK'(Z(.^) + 1))],/..} < V'(x).


for X > and (> since TV - rF* < 0, w < AK* < 7, and < a, < n.O < /?, < ^ for a >
by Lemma 3.3, 3.4 and assumption (2.5). Noticing that

lim EA I e-"{qft, - u,n,),h) = F{x),
'-^ Jo

we have F{x) < lim,_oo VM < V'i^-)-


The following proposition gives some monotonicity properties of the optimal inventory limit
6* as a fimction of other parameters.

Proposition 3.3. The optimal inventory limit h' inrrmsrs a.s .0) . (3.17)


Note that b* jumps only at the points where / = n. According to rouditinn (3 9), f(b) = implies

6— 1 is optimal. If an increase in some parameter causes a d(>cicasi' in / at the point where f(h) = 0,

then 6—1 remains optimal to the right (i.e., f(l> — 1) > 0, and /('») < 0). and h is optimal to the

left (i.e., f(b) > and f{b = 1) < 0). in this case b' is a decreasing right continuous step function

of the parameter. Conversely, if an increase in some parameter causes an incicase in /, then b' is

an increasing left continuous step function of the parameter. However.

df df df Of

— < 0, -f < 0, — < if ft > 1, and — = if ft = 1.

dc dh dn dm


The iutuition of proposition 3.3 is (Hiite rlear. An inrioasc in piixlurtiou capacity a implies a
higher frequency of production and a longer time for a jjrodnrt staying in stock. Therefore the firm
prefers a lower inventory limit to modify this effect and avoid higher financial coat. A decrease in
holding cost h or production cost c causes an increase in inventory limit simply because the firm
can afford a higher inventory and then has more ftilfillerl demand. However the effect of price on
b* is ambiguous. The only remark we can make is that if the firm incn-ases the price, then it tends
to have a lower inventory limit when facing a more elastic market anti a higher one when facing a
less elastic market.

So far, a unique optimal policy is obtained for each specific capacity n and price p. Assuming
that the optimal operating policy always follows after the design decisions art" made, the expected
profit is again a function of a and p, and an explicit form of it can be obtained. Theoretically, the
optimal capacity can be determined by usual caloilus. Dut it is not a trivial job. The difficulties
comes from the fact that the first order derivative of the value function with respect to a or p is
not continuous because of the discontinuity of h' as a function of (t or j>. De Vany (1976) avoids the
similar problem by approximating b, tlie l)alking value in his model, l)y a continuous differentiable
fimction. We shall 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

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