- Email: [email protected]
Multiperiod monopoly under uncertainty
JOURNAL
OF ECONOMIC
THEORY
Multiperiod
5,
524-536 (1972)
Monopoly
under
EDWARD Department
of Economics,
University
Uncertainty*
ZABEL
of Rochester,
Rochester,
New
York
14627
Received February 23, 1972
1. INTRODUCTION
In a previous paper [6] the author considered the role of uncertainty of demand in the theory of monopoly when the horizon consistsof a single period. The aim here is to extend the analysisto allow any finite number of periods in the horizon. Two alternative models are proposed. The first assumesmultiplicative demand, following lines suggestedby Nevins [4, 51 and the author [6]. The second assumes additive demand, the approach introduced by Mills 12,31. The progress we achieve in the two models reflects the fact that the first is intrinsically more difficult of analysis. In the model with multiplicative demand, while it is possibleto show the existence of optimal solutions having qualitative properties analogous to some of those obtained in the single-period model, we have not been able to obtain corresponding uniqueness properties. The assumptions which guarantee uniquenessin the single-period case do not appear to be sufficient in the multiperiod setting, nor does any reasonableextension of these assumptionssuffice. Some additional research on the uniquenessquestion seemswarranted. Questions of existence and uniqueness are more fully resolved in the caseof additive demand. With an appropriate extension of assumptions, we derive properties of optimal solutions in any finite horizon and show that these solutions are uniquely determined. The plan of the paper is the following. In the next section we introduce the models. In Section III, we assumedemand is multiplicative, but, since results are lessconclusive in this model, we only briefly outline properties of behavior and list some qualifications about uniqueness of solutions. Then, for the additive model, Section IV gives a more detailed derivation of optimal behavior. Section V is a concluding section. * The author gratefully acknowledges the support of the National Science Foundation under grant GS-2666 to the University of Rochester.
524 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction id any form reserved.
MONOPOLY UNDER UNCERTAINTY
525
II. MULTIPERIOD MODELS OF MONOPOLY Since features common to both models are explained fully in the author’s earlier paper, we give only a brief specification here. Assume that the monopolist wishes to maximize discounted expected profit with respect to periodic output q and price p. Periodic demand 5 is a random variable, depending on price, and decisions are taken before the demand becomes known. Each period the firm incurs a variable production cost c(q) with properties c(0) = 0, c’(q) > 0, and c”(q) 3 0. The monopolist also incurs a linear storage c,:t h * ( y - f), defined over y 3 t where y, the starting inventory, is the sum of output q and x, the inventory of commodity on hand at the beginning of a period. In the multiplicative case, let .$ = mu where 71is a random variable with density function #(v) which is defined over 0 < 77 < jl < co and has mean value equal to one. Also, let Y(a) be the corresponding probability distribution function. The function u(p) gives the expected demand and also represents the demand function in the absence of uncertainty, or the riskless demand. The demand assumptions require that the density function of demand #$;p) = #(&(p))/u(p) for u(p) > 0. Assume that the positive constants a and b are quantity and price intercepts so that u(0) = a and u(b) = 0. Riskless demand is taken to be downward sloping, u’(p) > 0. Other characteristics of demand will be discussed later. For additive demand let 4 = u(p) + 9 where again q is a random variable with density function #(v). In this case, it is customary to choose expectation ;i = 0, which requires q to extend over negative values and, consequently, introduces the possibility of negative demands. To avoid this problem we restrict 11to be nonnegative, requiring ;i > 0.l We assume that u(p) has the properties listed above, except that expected demand equals u(p) + +j. Now, also, the density of demand c#;P) = #([ - u), where 0 < 77 < ,8 and u < [ < /3 + u. Let ft(x) represent the maximum discounted expected profit over an t-period horizon, depending on initial inventory x. Then
where the maximization occurs over b 3 p > 0 and y 2 x and Gt(p; y, x) specifies maximum expected profit over the horizon for given p, y and x. 1However, whether we choose the mean to be positive or zero, other problems arise which we discuss later in appropriate places.
526
ZABEL
Using the relationships plicative demand, WP; Y, 4 = -4~ + JI’”
between d([;~)
and #(‘(rl), we obtain for multi-
- 4 [Prl u - h . (Y - 74 +
+ [PY + dd9l[l
- y/(V/~H,
and, for additive demand, G~(P; Y, x) = -4~
- 4 + 1:”
[P(U + ‘I) - h * (Y
+ &l(U - @ + a1 flh) + [PV + ~h-dW - WY - 41, where&(x) = 0. Note that the particular formulations given in (2) and (3) imply that demand is not back-logged, i.e., when 5 > y, sales are lost forever. As occasion arises, we shall attach distinguishing subscripts to decision variables and parameters. In the next section, using (1) and (2), we discuss properties of optimal decisions in any period in any horizon, when demand is multiplicative.
III.
MULTIPLICATIVE
DEMAND
The general procedure here is to assume, initially, those conditions which ensure regular behavior in the single-period horizon and to examine the usefulness of these assumptions in the multiperiod case. As the first step, then, we take optimal behavior in the single-period horizon, established previously [6], as given and explore properties of behavior as the horizon is extended. More exactly, in reference to single-period behavior, we begin by introducing assumptions which give uniqueness of solutions, profitability of output, positive prices for all inventories, and prices varying inversely with inventory levels. That is, we assume @[(p + h)u]/dp2 < 0, b > c’(O), (a + hu’) > 0, and densities of demand, such as the uniform or the exponential, which insure &G(p,*(y); y, X) < 0 [6, pp. 208-2121. Under these circumstances fi(x) is a concave, differentiable function. Turning to the two-period horizon, it is not difficult to show that the given assumptions assure the existence of a finite optimal price pZ*(y) for all y. Taking DIGS as the derivative of Gsr(p;y, x) with respect to price,
MONOPOLY
UNDER
527
UNCERTAINTY
the optimal price either equals zero or satisfies DIG.&,*(y); y, X) = 0 depending on whether DIG,(O; y, x) is negative or nonnegative. A multiperiod analog of the condition (a + hu’) > 0, which is [a + s&u’] > 0, where st = CL-2 CP now suffices to guarantee D,G,(O; y, x) > 0 for all y. Uniqueness of pZ*(y) depends on the sign pattern of the second partial derivative D,,G,(p; y, x). An examination of this derivative shows that the assumption d2[(p + h)uJ/dp2 < 0 is not sufficient to determine its sign in general so that the possibility arises that p2*(y) is not uniquely determined. However, if we evaluate the derivative at p2*(y), assume that the demand function is concave, u” < 0, and that the density #(q) is either uniform or exponential, then D,;G,(p,*; y, X) < 0 which guarantees the uniqueness of optimal price in these cases. Moreover, as in the singleperiod horizon, the sign of dp,*/dy is indeterminate for an arbitrary density. Again, if we take the density to be uniform or exponential, dp,*ldy
< 0.
The essential difference so far between price behavior in the single- and two-period horizons is that stronger conditions are needed to ensure uniqueness of price in the first period of the two-period horizon. Not only do we need stronger conditions on expected demand, such as concavity, but also conditions on the density of demand, such as requiring the density to be uniform. More serious difficulties arise in the determination of optimal starting stock or output. While we may show the existence of finite optimal starting stock, the question of uniqueness is unresolved. Define G,(p,*(y); y, x) = H,(y, X) which specifies expected discounted profit given y, x, and optimal price p2*( y). Now an examination of the partial derivative D,H,( y, X) shows that positive output is profitable at least for some x and that, in any event, optimal output is finite. When output is positive, optimal starting stock y2*(x) satisfies &H,(y,*, x) = 0, and its uniqueness would be assured if D,,H,( y2*, x) -=c0. However, we have not been able to derive this inequality, even when expected demand is concave and the density is uniform or exponential. Nor have we been able to specify any other reasonable assumptions which would provide the desired inequality. The source of difficulty appears to be the assumption of multiplicative demand, as we attempt to show now, and confirm later when we introduce additive demand. Obtaining D,,H2(y
x)=
~~G,(P,*;Y,
4
D22G2(~2*;~9
9 &G2(~2*;
xl
-
Y,
4
W,&~(P~*;Y,
41" (4
we see that D,,H, has the opposite sign of the numerator D,,Gz < 0 at an optimal solution.
in (4) since
528
ZABEL
The difficulty in obtaining a sign of the numerator lies in the fact that the second derivative f;(y - 7~) is weighted differently in the corresponding integrals in DllGz , D,,G, and D,,Gz , having the weights, respectively, of y2, 7, and 1, a consequence of the assumption of multiplicative demand. However, if we consistently choose the largest optimal starting stock y2*(x), along with the largest optimal pricep,*(y), it is possible to continue the induction for then the appropriate function ft(x) is concave and differentiable and the arguments developed in the two-period horizon will apply. Moreover, additional properties of optimal starting stock are also obtainable. In particular, using arguments developed previously [6, pp. 212-2161, we may show that 1 >, dy,*/dx 3 0, dp,*/dx < 0, and that analogous conclusions concerning equilibrium inventory xte apply. In general, we have not been able to extend the single-period horizon results on parameter changes to the multiperiod case since they depend on additional properties of derivatives which are not now available. Another gap in the analysis is that, except in the special case of constant marginal production cost, we have not been able to provide inter-period comparisons of optimal decisions. As we show in the next section, the peculiar difficulties, arising in the multiplicative demand case, do not appear when demand is additive.
IV.
ADDITIVE
DEMAND
In this section, the major objective is to show that greater progress in analyzing behavior may be achieved if demand is additive rather than multiplicative. We use essentially the same assumptions as in the preceding section, but as observed there, assumptions which give regularity of behavior in the single-period horizon do not always suffice when the horizon is lengthened. The procedure here, then, is to introduce directly multiperiod analogs of these assumptions. Also, as in the preceding section, two kinds of assumptions are used. The first kind, the assumptions of concave expected demand and particular densities, function mainly to assure uniqueness of behavior. In the additive demand case, these assumptions guarantee uniqueness by providing definite signs to derivatives involving both positive and negative terms. As noted earlier, these uniqueness assumptions are not sufficient in the case of multiplicative demand. The second kind involve conditions on parameters which provide convenience in exposition. These may be relaxed at a cost only of a more tedious argument. We leave these details to the reader. In this category
MONOPOLY
UNDER
529
UNCERTAINTY
are the assumptions, b > c’(O), which guarantees the profitability of positive output at some initial inventories and [a + s,hu’] > 0, which implies that optimal price is always positive. Since we take the expectation ?j to be positive, we also need an assumption here to assure that 7 is not too large, otherwise the firm will always choose price at its maximum value. For expositional convenience only, we introduce an extreme assumption which guarantees that the maximum price is never chosen when inventory is positive. Since the parametric form of this assumption depends on the density function, we introduce it as a condition on a partial derivative, i.e., D,G,(b; y, x) < 0 for all positive y. We discuss some implications of this assumption later. We give the major outcomes of this section in three theorems. The first states properties of optimal output and price in any period of any finite horizon. In other words, since outcomes are independent of chronological dates, it states optimal behavior if t periods remain in the horizon. The second gives interperiod comparisons of behavior, i.e., it gives variations in behavior when the number of remaining periods in the horizon changes. The third theorem examines variations in behavior with a change in holding cost. In stating Theorem 1 we use the notation pto and ~2 = u(p,O) where [alto + (pto + s&z) $1 + +j = 0. Since the full proof of the theorem is rather lengthy, involving an induction, we omit many of the details in the interest of brevity. THEOREM 1. Assuming u(p) is concave, #(q) is either the uniform or exponential density, b > c’(O), [a + s,hu’] > 0 and D,Gt(b; y, x) < 0, optimal behavior is uniquely determined, with the properties
(4 (b) (cl (4 (4 (0 64 00 Proof.
b > pt*(y) > 0 and y > u(pt*) ify > 0, as y -+ 0, pt*(y) -+ b and u(p,*) + 0, as Y - ~0, Pi* dpt*(yYdy -=c 0,
-+ pt” and u(P,*)
-
uto,
qt*(x) > 0 undy,*(x) > x, $0 < x < x6*, qt*(x) = 0 andy,*(x) = x, ifx > xt*, 0 > dq,*/dx > -1 and 1 > dy,*/dx > 0, $0 < x < xt*, dpt*(yt*(xNldx
-=c0.
Using (3) for t = 1 and differentiating W,(P;
Y, 4 = b + (P + h)
with respect to p,
~‘1WY - 4
+ j:-” rl dyh) +
YD
-
WY
- 43.
(5)
530
ZABEL
Since [a+s,hu’] > 0, clearly, D,G,(O; y,x) > 0, and given D,Gt(b; y, x) < 0, an optimal price b > pi*(y) > 0 exists for all y.2 Now differentiating (5) with respect top, &WP;
y, 4 =
[2u’
+
(p
+
WI
qy
-
24) -
(p
+
h)(d)2
$h
-
24) <
0.
(6)
Thus pi*(y) is unique and satisfies DIG,(p,*(y); y, x) = 0. From (5), an immediate property of optimal price is that y > u(p,*), for, otherwise, D,Gr > 0 when y > 0. From this result we also obtain that as y--f 0, u-+0 and pl* + b. Moreover, the equation DIG, = 0 gives that as -+ ulO. In fact, if /3 < co, it easily follows Y a, pl* - plo and 4pl*) that the set (plo, ulo) is obtained whenever y 2 /I + ur”. Hereafter, to avoid excessive detail, when j3 < co, we only explicitly consider y < p + ulO. Finally, since dp,*/dy = -D,,G,/D,,G, and &GI(P,*;
Y, 4
=
(PI*
+
4
u’$(Y
-
4
+
[] -
which is indeterminate in sign for an arbitrary density, indeterminate in sign. However, it is not difficult D2,Gl(p,*; y, x) < 0 whenever #(T) is either the uniform density and thus dp,*/dy < 0 in these instances. Turning to output behavior, define Gl(p,*( y); y, x) differentiating with respect to y, D,H,(y,
x) = -c’(y - x) + pl*[l -
‘U(Y
- 41 -
VY
41, (7)
-
is also to show that or exponential
dp,*/dy
= H1( y, x) and ~Y((Y
- 4. (8)
From results on price behavior, D,H,( y, x) < 0 as y + co, D,H,(x, x) < 0 as x -+ co, and D,H,(y, 0) = D,H,(x, x) > 0 as y -+ 0 and x -+ 0. Finally, by arguments analogous to ones applying when demand is multiplicative, D,,H,( y, x) < 0 and dH,(x, x)/dx < 0. These character* Without the condition D,Gt(b; y, x) < 0, there is a possibility that pi*(y) = b over some interval of y. As noted earlier, this problem arises since 3 > 0. That is, notice from (5) that l),G1(b; y, x) < (b + h)u’Y + 3. Consequently, if +j = 0, then D,Gl(b; y, x) < 0 for all y. However, if 3 = 0, the sign of D,G,(O; y, x) is now in doubt since then the sum of the last two terms in (5) is negative, opening a possibility that pr*(y) = 0 over some or all y. Consequently, whether i is zero or positive, we need additional conditions on the variation of demand to prevent boundary solutions. Also, as noted, when 15> 0, the parametric from of the condition DIGt(b; y, n) < 0 depends on the particular density NT). To give an idea of what the condition entails in the case of an explicit density, suppose that #(T) is the uniform density defined over 0 < 7 < /3. Then, from (S), DrGr(b; y, x) = y/B . [(b + R)u’ + b - y/2]. Thus, in particular, if (b + h) u’(b) < -8, we have &G,(b; y, x) < 0 for all y > 0. In general, as the reader may observe later, since f:(x) < b for all t, &Cl@; y, x) < 0 if [(1 -
a)b + h] u’(b)
<
-8.
531
MONOPOLY UNDER UNCERTAINTY
istics imply uniqueness of starting stock and output, with starting stock and output having the properties listed in the theorem,andpricep,*(y,*(x)) varying inversely with X. Thus, the theorem is true for I = 1. In considering the two-period horizon we need properties of fi(x) = G(PI*(YI*(x)); YI*, x). Omitting details it is not difficult to show that fl(x) is a concave, differentiable function, --h
Y, 4 = [u + (P + h) ~‘1 Y + iv- 7 dY(rl) + ~$1 - Yl 0 Y--u - cd s 0 fi’(Y - (u + 9)) dwl).
(9)
In conjunction withf,‘(x) > --h, the condition [a + s,hu’] > 0 and the assumption DIG,(b; y, x) < 0 give the existence of optimal price b > pZ*(y) > 0 satisfying D,G,(p,*; y, x) = 0 for all y. Now differentiating (9), &G(P;
Y, 4 = W
+
(P
+
4
~“1
‘y
-
(P
+
~)W2
$
+ 402 Iv- f;(Y - (u + 4) dwl) 0
which in general is indeterminate in sign but is negative if we evaluate the derivative at pZ*( y), given a concave expected demand and a density I,$($ which is either the uniform or the exponential. We omit the details. By the negative sign of DllG2 , pZ*(y) is uniquely determined and, for reasons noted in the single-period case, has the properties y > u(p,*), andp,* + b and II + 0 as y -+ 0. Now since D,,Gz(pz*; y, 4 =
u’# + [1 ‘ul Y-U - au’ I o fXY - tu + 7)) fltT4 - &flYO) *
(p2*
+
4
(11)
is negative if the density is either the uniform or the exponential, dp,*/dy < 0. Moreover, from (9) note that as y + co, p2* -+p: and dP2
*) +
u2O.
Next define G2(p2*; y, x) = H2( y, x) and differentiating, D,Kty,
4 = -c’(y
- x) + p2*[1 - !t’j - hY
+ rx p’tY
- (u + 4) dw7),
(12)
532
LABEL
which has the appropriate limits: D,H,( y, x) < 0 as y + co, D,H,(x, X) < 0 as x - co, D,H,(y, 0) > 0 as y -+ 0 and D,H,(x, x) > 0 as x -+ 0. With these limits there exists a finite, optimal starting stock yZ*(x) and output q2*(x) for each x. The critical step is to show that D&L&*, x) < 0 for y,* > x and dDIHz(xz*, x,*)/dx < 0 where D,H,( yZ*, x) = 0 and WUx, *> x2*) = 0 for then y,* and q2* are uniquely determined with properties listed in the theorem and pz* varies inversely with x. Since this step represents the major departure from the case of multiplicative demand, we show the derivation of dD1Hz(x2*, x,*)/dx < 0 in some detail, given expected demand is concave and Z+(T) is either the uniform or exponential density. A similar argument will give x) < 0. As in the case of multiplicative demand, D~&(Y~*, dD1H2(x2*, x,*)/dx
< 0 if det[DijGz(pz*;
x2*, x,*)1 > 0 for i, j = 1,2.
In particular, the determinant is positive if both D,,G,/u’ and u’D,,G, exceed -D,,G, , since each of these terms are positive, where &,Gd~z*;
xz*, x2*> = -c(Pz*
+ h) $
+ a I;*-“f;(x,*
- (u + 4) dY(d
+ d’(O)
#.
(13) On comparing (13) with (1 i), DzzGz is seen to be negative whenever DzlGz is negative. In fact, in this comparison it is also easily seen that u’D,,G, 3 -D,,G, in any event. Now note that y
+ D,,G,
= PU + “Pi; + h)
4 y + [I _ lp]
-_adsx2*-u h’(x,* u’
(u + 11))dwi9.
0
(14)
Next, using (12), solve the equation D,H2(x2*, x2*) = 0 for the integral overf,’ and substitute in (14) to obtain, after cancellations and collections of terms, (D,,G,/u’)
Turning D,G,(p,*;
+ D,,G,
= 1 + y-t
[PZ* -
c’(O)lW/u’).
to (9) and using the same substitution x2*, x2*) = 0, it follows that pz* 2 c’(O)
and
D,,G,lu’
(1%
in the equation
+ D,,G, I=- 0.
x 2*)/dx < 0 and the theorem is proved for t = 1,2. Thus, dD,H,(x,*, It is now straightforward to continue the induction to any arbitrary finite horizon. Consequently, Theorem 1 is correct for t representing any finite number of periods remaining in the horizon.
533
MONOPOLY UNDER UNCERTAINTY
We now prove the theorem providing interperiod comparisons of optimal behavior. In this theorem we use the notation u&) = u(p,*(y,*(x))) and let x; represent equilibrium inventory. As in the author’s previous paper [6, p. 2141, equilibrium inventory is that level of initial inventory such that output equals expected demand if that equality exists or which equals zero otherwise. THEOREM 2.
Given the conditions of Theorem 1,
(a) P~*(Y> > P,*_~(Y),$0 d Y < VT-~where AL (b)
yt* (4 > Y,*_~(x>md Yt*W - W) if&%4
> d-l(x)
> x,*_, , - ~t-k4
> 4
(cl
xt* > 4-l 7
(d)
xte > x& , ifx;-,
> 0.
Proof. From (5) and (9) it is apparent that DIG&*; y, x) > 0 when y is sufficiently small, for then jfl’(y - (u + 7)) dY(q) > 0. Consequently, pZ*(y) > pi*(y) in an initial interval of starting stock [O, ylo], which obviously contains the interval [0, x1*] sincef,‘(x) > 0 in the latter interval. To prove x2* > x1* it suffices to show &&(x1*, x1*) > 0. We explicitly derive this inequality only for the uniform density since an analogous argument applies to the exponential. Similarly, since arguments are again analogous, we omit the derivation of yz*(x) > yl*(x). The argument we now develop applies to any arbitrary uniform density so for notational convenience we choose p = 2 and ;i = 1. Using properties of the uniform density, D,H,(x,*, x1*) = 0 and (S), we obtain c’(O) + h = (PI* + Wl* - uJ2. For the uniform density we may also rewrite (9) as
M%(P,*; xl*, xl*) = CP~*+ h> ~2’ (xl* 2- u2) + [xl* - (xl* 4- u2Y] 21*-7+ -- oluz‘ 2 Io fi’(xl* - (~2 + 4) drl, (16) where [x1* - (x1* - 2.#/4] > 0 when x1* - u1 < 2. Solving the equation - (u2 + ~7))dq and substituting D,Gz(ps*; xl*, x1*) = 0 for 01/2 . jfl’(xl* this expression, along with the one for c’(0) + h, into (I2),
DdW,*,
x1*1 = (Pz* - pl*) + (pl* + h) (x1* 2- 4) + -$
647./5/3-15
[xl* _ (x1* 4 .2)2]
(17)
534
ZABEL
after collection of terms. Now sincep,*(x,*) > pl*(xl*) and u1 > u2 and since ul’/uZ’ < 1 from the concavity of u(p), it easily follows that ul’DIHz(xl*, xi*) < 0 on observing
W%P,*; xl*, xl*) = (PI* + h) u,‘(x,* - U,)/2 + [Xl * - (x1 * - U,)2/4] = 0. Consequently, D,H2(xl*, xi*) > 0. Next, assume y2*(x) - u2 < vl*(x) - ul. Then since y,* > yi*, we must havep,* < pl*, u2 > u1 andf,‘(y,* - u2) > 0. Now note from (8) and (12) that, in these circumstances, it is not possible for both
ww,*, 4 = 0
and
4~2(Y,*,
-4 = 0
to be satisfied. Thus yZ*(x) - u2 > yi*(x) - u1 . To provide a comparison of equilibrium inventories, notice that J+*(X) - uZ(x) > y,*(x) - ur(x) implies q2*(xle) - [u2(xle) + +j] > ql*(xle) - [u,(x,~) + ;i] = 0 which implies x2e > xle. Again, it is straightforward to continue the induction so that Theorem 2 is proved for any t. Some amplification of Theorem 2 may be helpful. First, while p,*(v) exceeds pcl(y) in an initial interval of y, note from Theorem 1 that pto < pfwl . Thus, pt* eventually falls below pEl , reflecting the potential extra storage charges incurred in the longer horizon. Second, property (c) states that the difference between the starting stock and expected demand increases with an addition to the remaining number of periods in the horizon. In general, however, p,*(y,*(x)) > pfPl(y$Ll(x)) is not available in this interval because of the inequality v,*(x) > J&(X) and the inverse relationship between price and starting stock. Finally, the interperiod comparisons provided in Theorem 2, or which are implied in the analysis, giving convergent or uniformly convergent sequences of decisions, optimal expected profits and optimal marginal expected profits, suggest that the analysis may be readily extended to the infinite horizon. In the infinite horizon behavior would then be identical in each period, having properties listed in Theorem 1. We omit the details. The final theorem computes effects of a change in the holding cost. Other parameter changes may be handled similarly with analogous results. For example, we could as well consider changes in the discount factor or production cost or examine the impact of taxes. We leave the details to the reader. In this theorem qte = qt*(xte) and ptC = pt*(xte), i.e., equilibrium quantity and price. THEOREM
3. Given the conditions of Theorem 1,
535
MONOPOLY UNDER UNCERTAINTY
(a) $,*/ah -=c0, aqt*lall < ‘0, and ap,*/ah < 0, if0 < x < (b) dx,“/dh < 0, ifxte > 0, (c)
xt*,
dq,“/dh and dp,“ldh are indeterminate in sign if xtc > 0.
Proof. Since previous equations are only explicitly given for t = 1,2, we prove the theorem for t = 2, though the same argument applies for t being any finite integer. In the interval of initial inventory [0, x,*), optimal starting stock and price simultaneously satisfy D~Gz(pz*; yz*, x, h) = 0
and
D,Gz(pz*; yz*, x, h) = 0.
Differentiating,
+,*/ah = (--D,,G&G, %,*/ah = (--D,,GhG,
+ 4,G&&)IA + D,,Wd-UA
< 0, < 0,
(18) (19)
where A = D,,G,D,,G2 - (D,,G,)2 > 0. The inequalities in (18) and (19) follow since DlhG, = -u’D,~G, < 0, which is easily confirmed, and since D,,G, + u’DlzGz < 0 and u’D,,G, + D,,G, > 0, which were shown earlier. Reference to Fig. 2 [6, p. 2151 will now confirm that dxze/dh < 0 since in that figure q2*(x) shifts downward by (18) and z(~(x) shifts upward by (19). An algebraic confirmation of this conclusion is readily obtained. In particular, referring to Eq. (29) in the author’s paper [6, p. 2171, dxze/dh has the opposite sign of (u’ap,*/ah - aq2*/ah) which is positive. Moreover, Eq. (30) of that paper [6, p. 2181 will convince the reader that the direction of change in equilibrium output and price are indeterminate. The interesting development in this theorem is that ap,*/ah has a definite sign which is not true in the case of multiplicative demand even in the single-period horizon [6, pp. 216-2171. Theorem 3 also affords some explanation of observed “stickiness” in administered prices. That is, while changes in parameters do lead to changes in price and output, part of the adjustment occurs in the desired holding of inventory. In fact, in Nevins’ numerical examples, for changes in the holding cost and discount factor, most of the adjustment occurs in equilibrium inventory, and only little in equilibrium price and output [5, pp. 81-821. As a final exercise we could compare behavior under risky and riskless monopoly. The major element in this comparison is the inequality [u + ;T + pt*u’]/u’ < c’(y,* - x). We omit the derivation of this inequality and its full implications, which would follow a similar treatment in the author’s earlier paper [6, pp. 214-2161. Essentially, the outcome is that prices tend to be lower and output higher than in riskless monopoly.
536
ZABEL
V. CONCLUSIONS With the assumptions of a concave expected demand function and particular densities of demand, we have demonstrated the existence and characteristics of uniquely determined decisions in any horizon in the case of additive demand. While it is possible to derive some characteristics of optimal behavior in any horizon when demand is multiplicative, uniqueness properties are not readily available. Apart from analytic ease, other important differences distinguish these two cases. In particular, results by various authors [ 1, 4-61 indicate clearly that optimal behavior is directly affected by the form of demand uncertainty. Multiplicative demand tends to induce high prices and low outputs and conversely for additive demand. The reason for this difference seems to hinge on the variance of demand. lf we let a2 be the variance of the density #(T) and u2 be the variance of demand, then it is easily shown, that in the additive demand, a2 = S2, and in multiplicative demand, u2 = 62[~(p)]2. Thus, with additive demand variance is constant at all prices, while with multiplicative demand it varies inversely with price, specifically, tending to zero as price tends to its maximum amount. Similarly, relative variance, a2/u2, is constant with multiplicative demand but varies directly with price in additive demand, going to infinity as price goes to its maximum. Consequently, on either an absolute or relative basis, high prices tend to be less risky in multiplicative than in additive demand. REFERENCES 1. S. KARLIN AND C. R. CARR, Prices and optimal inventory policy, in “Studies in Applied Probability and Management Science” (K. J. Arrow et al., Eds.), pp. 159-172, Stanford University Press, 1962. 2. E. S. MILLS, Uncertainty and price theory, Quart. J. Econ. 73 (1959), 116-130. 3. E. S. MILLS, “Price, Output and Inventory,” Wiley, New York, 1962. 4. A. NEVINS, “A Simulation Study of Price, Output and Inventory Under Uncertainty,” Doctoral dissertation, University of Rochester, 1964. 5. A. NEVMS, Some effects of uncertainty: Simulation of a model of price, Quart. J. Econ.
80 (1966),
73-87.
6. E. ZABEL, Monopoly
and uncertainty,
Rev. &on.
Studies
37 (1970),
205-219.
OF ECONOMIC
THEORY
Multiperiod
5,
524-536 (1972)
Monopoly
under
EDWARD Department
of Economics,
University
Uncertainty*
ZABEL
of Rochester,
Rochester,
New
York
14627
Received February 23, 1972
1. INTRODUCTION
In a previous paper [6] the author considered the role of uncertainty of demand in the theory of monopoly when the horizon consistsof a single period. The aim here is to extend the analysisto allow any finite number of periods in the horizon. Two alternative models are proposed. The first assumesmultiplicative demand, following lines suggestedby Nevins [4, 51 and the author [6]. The second assumes additive demand, the approach introduced by Mills 12,31. The progress we achieve in the two models reflects the fact that the first is intrinsically more difficult of analysis. In the model with multiplicative demand, while it is possibleto show the existence of optimal solutions having qualitative properties analogous to some of those obtained in the single-period model, we have not been able to obtain corresponding uniqueness properties. The assumptions which guarantee uniquenessin the single-period case do not appear to be sufficient in the multiperiod setting, nor does any reasonableextension of these assumptionssuffice. Some additional research on the uniquenessquestion seemswarranted. Questions of existence and uniqueness are more fully resolved in the caseof additive demand. With an appropriate extension of assumptions, we derive properties of optimal solutions in any finite horizon and show that these solutions are uniquely determined. The plan of the paper is the following. In the next section we introduce the models. In Section III, we assumedemand is multiplicative, but, since results are lessconclusive in this model, we only briefly outline properties of behavior and list some qualifications about uniqueness of solutions. Then, for the additive model, Section IV gives a more detailed derivation of optimal behavior. Section V is a concluding section. * The author gratefully acknowledges the support of the National Science Foundation under grant GS-2666 to the University of Rochester.
524 Copyright All rights
0 1972 by Academic Press, Inc. of reproduction id any form reserved.
MONOPOLY UNDER UNCERTAINTY
525
II. MULTIPERIOD MODELS OF MONOPOLY Since features common to both models are explained fully in the author’s earlier paper, we give only a brief specification here. Assume that the monopolist wishes to maximize discounted expected profit with respect to periodic output q and price p. Periodic demand 5 is a random variable, depending on price, and decisions are taken before the demand becomes known. Each period the firm incurs a variable production cost c(q) with properties c(0) = 0, c’(q) > 0, and c”(q) 3 0. The monopolist also incurs a linear storage c,:t h * ( y - f), defined over y 3 t where y, the starting inventory, is the sum of output q and x, the inventory of commodity on hand at the beginning of a period. In the multiplicative case, let .$ = mu where 71is a random variable with density function #(v) which is defined over 0 < 77 < jl < co and has mean value equal to one. Also, let Y(a) be the corresponding probability distribution function. The function u(p) gives the expected demand and also represents the demand function in the absence of uncertainty, or the riskless demand. The demand assumptions require that the density function of demand #$;p) = #(&(p))/u(p) for u(p) > 0. Assume that the positive constants a and b are quantity and price intercepts so that u(0) = a and u(b) = 0. Riskless demand is taken to be downward sloping, u’(p) > 0. Other characteristics of demand will be discussed later. For additive demand let 4 = u(p) + 9 where again q is a random variable with density function #(v). In this case, it is customary to choose expectation ;i = 0, which requires q to extend over negative values and, consequently, introduces the possibility of negative demands. To avoid this problem we restrict 11to be nonnegative, requiring ;i > 0.l We assume that u(p) has the properties listed above, except that expected demand equals u(p) + +j. Now, also, the density of demand c#;P) = #([ - u), where 0 < 77 < ,8 and u < [ < /3 + u. Let ft(x) represent the maximum discounted expected profit over an t-period horizon, depending on initial inventory x. Then
where the maximization occurs over b 3 p > 0 and y 2 x and Gt(p; y, x) specifies maximum expected profit over the horizon for given p, y and x. 1However, whether we choose the mean to be positive or zero, other problems arise which we discuss later in appropriate places.
526
ZABEL
Using the relationships plicative demand, WP; Y, 4 = -4~ + JI’”
between d([;~)
and #(‘(rl), we obtain for multi-
- 4 [Prl u - h . (Y - 74 +
+ [PY + dd9l[l
- y/(V/~H,
and, for additive demand, G~(P; Y, x) = -4~
- 4 + 1:”
[P(U + ‘I) - h * (Y
+ &l(U - @ + a1 flh) + [PV + ~h-dW - WY - 41, where&(x) = 0. Note that the particular formulations given in (2) and (3) imply that demand is not back-logged, i.e., when 5 > y, sales are lost forever. As occasion arises, we shall attach distinguishing subscripts to decision variables and parameters. In the next section, using (1) and (2), we discuss properties of optimal decisions in any period in any horizon, when demand is multiplicative.
III.
MULTIPLICATIVE
DEMAND
The general procedure here is to assume, initially, those conditions which ensure regular behavior in the single-period horizon and to examine the usefulness of these assumptions in the multiperiod case. As the first step, then, we take optimal behavior in the single-period horizon, established previously [6], as given and explore properties of behavior as the horizon is extended. More exactly, in reference to single-period behavior, we begin by introducing assumptions which give uniqueness of solutions, profitability of output, positive prices for all inventories, and prices varying inversely with inventory levels. That is, we assume @[(p + h)u]/dp2 < 0, b > c’(O), (a + hu’) > 0, and densities of demand, such as the uniform or the exponential, which insure &G(p,*(y); y, X) < 0 [6, pp. 208-2121. Under these circumstances fi(x) is a concave, differentiable function. Turning to the two-period horizon, it is not difficult to show that the given assumptions assure the existence of a finite optimal price pZ*(y) for all y. Taking DIGS as the derivative of Gsr(p;y, x) with respect to price,
MONOPOLY
UNDER
527
UNCERTAINTY
the optimal price either equals zero or satisfies DIG.&,*(y); y, X) = 0 depending on whether DIG,(O; y, x) is negative or nonnegative. A multiperiod analog of the condition (a + hu’) > 0, which is [a + s&u’] > 0, where st = CL-2 CP now suffices to guarantee D,G,(O; y, x) > 0 for all y. Uniqueness of pZ*(y) depends on the sign pattern of the second partial derivative D,,G,(p; y, x). An examination of this derivative shows that the assumption d2[(p + h)uJ/dp2 < 0 is not sufficient to determine its sign in general so that the possibility arises that p2*(y) is not uniquely determined. However, if we evaluate the derivative at p2*(y), assume that the demand function is concave, u” < 0, and that the density #(q) is either uniform or exponential, then D,;G,(p,*; y, X) < 0 which guarantees the uniqueness of optimal price in these cases. Moreover, as in the singleperiod horizon, the sign of dp,*/dy is indeterminate for an arbitrary density. Again, if we take the density to be uniform or exponential, dp,*ldy
< 0.
The essential difference so far between price behavior in the single- and two-period horizons is that stronger conditions are needed to ensure uniqueness of price in the first period of the two-period horizon. Not only do we need stronger conditions on expected demand, such as concavity, but also conditions on the density of demand, such as requiring the density to be uniform. More serious difficulties arise in the determination of optimal starting stock or output. While we may show the existence of finite optimal starting stock, the question of uniqueness is unresolved. Define G,(p,*(y); y, x) = H,(y, X) which specifies expected discounted profit given y, x, and optimal price p2*( y). Now an examination of the partial derivative D,H,( y, X) shows that positive output is profitable at least for some x and that, in any event, optimal output is finite. When output is positive, optimal starting stock y2*(x) satisfies &H,(y,*, x) = 0, and its uniqueness would be assured if D,,H,( y2*, x) -=c0. However, we have not been able to derive this inequality, even when expected demand is concave and the density is uniform or exponential. Nor have we been able to specify any other reasonable assumptions which would provide the desired inequality. The source of difficulty appears to be the assumption of multiplicative demand, as we attempt to show now, and confirm later when we introduce additive demand. Obtaining D,,H2(y
x)=
~~G,(P,*;Y,
4
D22G2(~2*;~9
9 &G2(~2*;
xl
-
Y,
4
W,&~(P~*;Y,
41" (4
we see that D,,H, has the opposite sign of the numerator D,,Gz < 0 at an optimal solution.
in (4) since
528
ZABEL
The difficulty in obtaining a sign of the numerator lies in the fact that the second derivative f;(y - 7~) is weighted differently in the corresponding integrals in DllGz , D,,G, and D,,Gz , having the weights, respectively, of y2, 7, and 1, a consequence of the assumption of multiplicative demand. However, if we consistently choose the largest optimal starting stock y2*(x), along with the largest optimal pricep,*(y), it is possible to continue the induction for then the appropriate function ft(x) is concave and differentiable and the arguments developed in the two-period horizon will apply. Moreover, additional properties of optimal starting stock are also obtainable. In particular, using arguments developed previously [6, pp. 212-2161, we may show that 1 >, dy,*/dx 3 0, dp,*/dx < 0, and that analogous conclusions concerning equilibrium inventory xte apply. In general, we have not been able to extend the single-period horizon results on parameter changes to the multiperiod case since they depend on additional properties of derivatives which are not now available. Another gap in the analysis is that, except in the special case of constant marginal production cost, we have not been able to provide inter-period comparisons of optimal decisions. As we show in the next section, the peculiar difficulties, arising in the multiplicative demand case, do not appear when demand is additive.
IV.
ADDITIVE
DEMAND
In this section, the major objective is to show that greater progress in analyzing behavior may be achieved if demand is additive rather than multiplicative. We use essentially the same assumptions as in the preceding section, but as observed there, assumptions which give regularity of behavior in the single-period horizon do not always suffice when the horizon is lengthened. The procedure here, then, is to introduce directly multiperiod analogs of these assumptions. Also, as in the preceding section, two kinds of assumptions are used. The first kind, the assumptions of concave expected demand and particular densities, function mainly to assure uniqueness of behavior. In the additive demand case, these assumptions guarantee uniqueness by providing definite signs to derivatives involving both positive and negative terms. As noted earlier, these uniqueness assumptions are not sufficient in the case of multiplicative demand. The second kind involve conditions on parameters which provide convenience in exposition. These may be relaxed at a cost only of a more tedious argument. We leave these details to the reader. In this category
MONOPOLY
UNDER
529
UNCERTAINTY
are the assumptions, b > c’(O), which guarantees the profitability of positive output at some initial inventories and [a + s,hu’] > 0, which implies that optimal price is always positive. Since we take the expectation ?j to be positive, we also need an assumption here to assure that 7 is not too large, otherwise the firm will always choose price at its maximum value. For expositional convenience only, we introduce an extreme assumption which guarantees that the maximum price is never chosen when inventory is positive. Since the parametric form of this assumption depends on the density function, we introduce it as a condition on a partial derivative, i.e., D,G,(b; y, x) < 0 for all positive y. We discuss some implications of this assumption later. We give the major outcomes of this section in three theorems. The first states properties of optimal output and price in any period of any finite horizon. In other words, since outcomes are independent of chronological dates, it states optimal behavior if t periods remain in the horizon. The second gives interperiod comparisons of behavior, i.e., it gives variations in behavior when the number of remaining periods in the horizon changes. The third theorem examines variations in behavior with a change in holding cost. In stating Theorem 1 we use the notation pto and ~2 = u(p,O) where [alto + (pto + s&z) $1 + +j = 0. Since the full proof of the theorem is rather lengthy, involving an induction, we omit many of the details in the interest of brevity. THEOREM 1. Assuming u(p) is concave, #(q) is either the uniform or exponential density, b > c’(O), [a + s,hu’] > 0 and D,Gt(b; y, x) < 0, optimal behavior is uniquely determined, with the properties
(4 (b) (cl (4 (4 (0 64 00 Proof.
b > pt*(y) > 0 and y > u(pt*) ify > 0, as y -+ 0, pt*(y) -+ b and u(p,*) + 0, as Y - ~0, Pi* dpt*(yYdy -=c 0,
-+ pt” and u(P,*)
-
uto,
qt*(x) > 0 undy,*(x) > x, $0 < x < x6*, qt*(x) = 0 andy,*(x) = x, ifx > xt*, 0 > dq,*/dx > -1 and 1 > dy,*/dx > 0, $0 < x < xt*, dpt*(yt*(xNldx
-=c0.
Using (3) for t = 1 and differentiating W,(P;
Y, 4 = b + (P + h)
with respect to p,
~‘1WY - 4
+ j:-” rl dyh) +
YD
-
WY
- 43.
(5)
530
ZABEL
Since [a+s,hu’] > 0, clearly, D,G,(O; y,x) > 0, and given D,Gt(b; y, x) < 0, an optimal price b > pi*(y) > 0 exists for all y.2 Now differentiating (5) with respect top, &WP;
y, 4 =
[2u’
+
(p
+
WI
qy
-
24) -
(p
+
h)(d)2
$h
-
24) <
0.
(6)
Thus pi*(y) is unique and satisfies DIG,(p,*(y); y, x) = 0. From (5), an immediate property of optimal price is that y > u(p,*), for, otherwise, D,Gr > 0 when y > 0. From this result we also obtain that as y--f 0, u-+0 and pl* + b. Moreover, the equation DIG, = 0 gives that as -+ ulO. In fact, if /3 < co, it easily follows Y a, pl* - plo and 4pl*) that the set (plo, ulo) is obtained whenever y 2 /I + ur”. Hereafter, to avoid excessive detail, when j3 < co, we only explicitly consider y < p + ulO. Finally, since dp,*/dy = -D,,G,/D,,G, and &GI(P,*;
Y, 4
=
(PI*
+
4
u’$(Y
-
4
+
[] -
which is indeterminate in sign for an arbitrary density, indeterminate in sign. However, it is not difficult D2,Gl(p,*; y, x) < 0 whenever #(T) is either the uniform density and thus dp,*/dy < 0 in these instances. Turning to output behavior, define Gl(p,*( y); y, x) differentiating with respect to y, D,H,(y,
x) = -c’(y - x) + pl*[l -
‘U(Y
- 41 -
VY
41, (7)
-
is also to show that or exponential
dp,*/dy
= H1( y, x) and ~Y((Y
- 4. (8)
From results on price behavior, D,H,( y, x) < 0 as y + co, D,H,(x, x) < 0 as x -+ co, and D,H,(y, 0) = D,H,(x, x) > 0 as y -+ 0 and x -+ 0. Finally, by arguments analogous to ones applying when demand is multiplicative, D,,H,( y, x) < 0 and dH,(x, x)/dx < 0. These character* Without the condition D,Gt(b; y, x) < 0, there is a possibility that pi*(y) = b over some interval of y. As noted earlier, this problem arises since 3 > 0. That is, notice from (5) that l),G1(b; y, x) < (b + h)u’Y + 3. Consequently, if +j = 0, then D,Gl(b; y, x) < 0 for all y. However, if 3 = 0, the sign of D,G,(O; y, x) is now in doubt since then the sum of the last two terms in (5) is negative, opening a possibility that pr*(y) = 0 over some or all y. Consequently, whether i is zero or positive, we need additional conditions on the variation of demand to prevent boundary solutions. Also, as noted, when 15> 0, the parametric from of the condition DIGt(b; y, n) < 0 depends on the particular density NT). To give an idea of what the condition entails in the case of an explicit density, suppose that #(T) is the uniform density defined over 0 < 7 < /3. Then, from (S), DrGr(b; y, x) = y/B . [(b + R)u’ + b - y/2]. Thus, in particular, if (b + h) u’(b) < -8, we have &G,(b; y, x) < 0 for all y > 0. In general, as the reader may observe later, since f:(x) < b for all t, &Cl@; y, x) < 0 if [(1 -
a)b + h] u’(b)
<
-8.
531
MONOPOLY UNDER UNCERTAINTY
istics imply uniqueness of starting stock and output, with starting stock and output having the properties listed in the theorem,andpricep,*(y,*(x)) varying inversely with X. Thus, the theorem is true for I = 1. In considering the two-period horizon we need properties of fi(x) = G(PI*(YI*(x)); YI*, x). Omitting details it is not difficult to show that fl(x) is a concave, differentiable function, --h
Y, 4 = [u + (P + h) ~‘1 Y + iv- 7 dY(rl) + ~$1 - Yl 0 Y--u - cd s 0 fi’(Y - (u + 9)) dwl).
(9)
In conjunction withf,‘(x) > --h, the condition [a + s,hu’] > 0 and the assumption DIG,(b; y, x) < 0 give the existence of optimal price b > pZ*(y) > 0 satisfying D,G,(p,*; y, x) = 0 for all y. Now differentiating (9), &G(P;
Y, 4 = W
+
(P
+
4
~“1
‘y
-
(P
+
~)W2
$
+ 402 Iv- f;(Y - (u + 4) dwl) 0
which in general is indeterminate in sign but is negative if we evaluate the derivative at pZ*( y), given a concave expected demand and a density I,$($ which is either the uniform or the exponential. We omit the details. By the negative sign of DllG2 , pZ*(y) is uniquely determined and, for reasons noted in the single-period case, has the properties y > u(p,*), andp,* + b and II + 0 as y -+ 0. Now since D,,Gz(pz*; y, 4 =
u’# + [1 ‘ul Y-U - au’ I o fXY - tu + 7)) fltT4 - &flYO) *
(p2*
+
4
(11)
is negative if the density is either the uniform or the exponential, dp,*/dy < 0. Moreover, from (9) note that as y + co, p2* -+p: and dP2
*) +
u2O.
Next define G2(p2*; y, x) = H2( y, x) and differentiating, D,Kty,
4 = -c’(y
- x) + p2*[1 - !t’j - hY
+ rx p’tY
- (u + 4) dw7),
(12)
532
LABEL
which has the appropriate limits: D,H,( y, x) < 0 as y + co, D,H,(x, X) < 0 as x - co, D,H,(y, 0) > 0 as y -+ 0 and D,H,(x, x) > 0 as x -+ 0. With these limits there exists a finite, optimal starting stock yZ*(x) and output q2*(x) for each x. The critical step is to show that D&L&*, x) < 0 for y,* > x and dDIHz(xz*, x,*)/dx < 0 where D,H,( yZ*, x) = 0 and WUx, *> x2*) = 0 for then y,* and q2* are uniquely determined with properties listed in the theorem and pz* varies inversely with x. Since this step represents the major departure from the case of multiplicative demand, we show the derivation of dD1Hz(x2*, x,*)/dx < 0 in some detail, given expected demand is concave and Z+(T) is either the uniform or exponential density. A similar argument will give x) < 0. As in the case of multiplicative demand, D~&(Y~*, dD1H2(x2*, x,*)/dx
< 0 if det[DijGz(pz*;
x2*, x,*)1 > 0 for i, j = 1,2.
In particular, the determinant is positive if both D,,G,/u’ and u’D,,G, exceed -D,,G, , since each of these terms are positive, where &,Gd~z*;
xz*, x2*> = -c(Pz*
+ h) $
+ a I;*-“f;(x,*
- (u + 4) dY(d
+ d’(O)
#.
(13) On comparing (13) with (1 i), DzzGz is seen to be negative whenever DzlGz is negative. In fact, in this comparison it is also easily seen that u’D,,G, 3 -D,,G, in any event. Now note that y
+ D,,G,
= PU + “Pi; + h)
4 y + [I _ lp]
-_adsx2*-u h’(x,* u’
(u + 11))dwi9.
0
(14)
Next, using (12), solve the equation D,H2(x2*, x2*) = 0 for the integral overf,’ and substitute in (14) to obtain, after cancellations and collections of terms, (D,,G,/u’)
Turning D,G,(p,*;
+ D,,G,
= 1 + y-t
[PZ* -
c’(O)lW/u’).
to (9) and using the same substitution x2*, x2*) = 0, it follows that pz* 2 c’(O)
and
D,,G,lu’
(1%
in the equation
+ D,,G, I=- 0.
x 2*)/dx < 0 and the theorem is proved for t = 1,2. Thus, dD,H,(x,*, It is now straightforward to continue the induction to any arbitrary finite horizon. Consequently, Theorem 1 is correct for t representing any finite number of periods remaining in the horizon.
533
MONOPOLY UNDER UNCERTAINTY
We now prove the theorem providing interperiod comparisons of optimal behavior. In this theorem we use the notation u&) = u(p,*(y,*(x))) and let x; represent equilibrium inventory. As in the author’s previous paper [6, p. 2141, equilibrium inventory is that level of initial inventory such that output equals expected demand if that equality exists or which equals zero otherwise. THEOREM 2.
Given the conditions of Theorem 1,
(a) P~*(Y> > P,*_~(Y),$0 d Y < VT-~where AL (b)
yt* (4 > Y,*_~(x>md Yt*W - W) if&%4
> d-l(x)
> x,*_, , - ~t-k4
> 4
(cl
xt* > 4-l 7
(d)
xte > x& , ifx;-,
> 0.
Proof. From (5) and (9) it is apparent that DIG&*; y, x) > 0 when y is sufficiently small, for then jfl’(y - (u + 7)) dY(q) > 0. Consequently, pZ*(y) > pi*(y) in an initial interval of starting stock [O, ylo], which obviously contains the interval [0, x1*] sincef,‘(x) > 0 in the latter interval. To prove x2* > x1* it suffices to show &&(x1*, x1*) > 0. We explicitly derive this inequality only for the uniform density since an analogous argument applies to the exponential. Similarly, since arguments are again analogous, we omit the derivation of yz*(x) > yl*(x). The argument we now develop applies to any arbitrary uniform density so for notational convenience we choose p = 2 and ;i = 1. Using properties of the uniform density, D,H,(x,*, x1*) = 0 and (S), we obtain c’(O) + h = (PI* + Wl* - uJ2. For the uniform density we may also rewrite (9) as
M%(P,*; xl*, xl*) = CP~*+ h> ~2’ (xl* 2- u2) + [xl* - (xl* 4- u2Y] 21*-7+ -- oluz‘ 2 Io fi’(xl* - (~2 + 4) drl, (16) where [x1* - (x1* - 2.#/4] > 0 when x1* - u1 < 2. Solving the equation - (u2 + ~7))dq and substituting D,Gz(ps*; xl*, x1*) = 0 for 01/2 . jfl’(xl* this expression, along with the one for c’(0) + h, into (I2),
DdW,*,
x1*1 = (Pz* - pl*) + (pl* + h) (x1* 2- 4) + -$
647./5/3-15
[xl* _ (x1* 4 .2)2]
(17)
534
ZABEL
after collection of terms. Now sincep,*(x,*) > pl*(xl*) and u1 > u2 and since ul’/uZ’ < 1 from the concavity of u(p), it easily follows that ul’DIHz(xl*, xi*) < 0 on observing
W%P,*; xl*, xl*) = (PI* + h) u,‘(x,* - U,)/2 + [Xl * - (x1 * - U,)2/4] = 0. Consequently, D,H2(xl*, xi*) > 0. Next, assume y2*(x) - u2 < vl*(x) - ul. Then since y,* > yi*, we must havep,* < pl*, u2 > u1 andf,‘(y,* - u2) > 0. Now note from (8) and (12) that, in these circumstances, it is not possible for both
ww,*, 4 = 0
and
4~2(Y,*,
-4 = 0
to be satisfied. Thus yZ*(x) - u2 > yi*(x) - u1 . To provide a comparison of equilibrium inventories, notice that J+*(X) - uZ(x) > y,*(x) - ur(x) implies q2*(xle) - [u2(xle) + +j] > ql*(xle) - [u,(x,~) + ;i] = 0 which implies x2e > xle. Again, it is straightforward to continue the induction so that Theorem 2 is proved for any t. Some amplification of Theorem 2 may be helpful. First, while p,*(v) exceeds pcl(y) in an initial interval of y, note from Theorem 1 that pto < pfwl . Thus, pt* eventually falls below pEl , reflecting the potential extra storage charges incurred in the longer horizon. Second, property (c) states that the difference between the starting stock and expected demand increases with an addition to the remaining number of periods in the horizon. In general, however, p,*(y,*(x)) > pfPl(y$Ll(x)) is not available in this interval because of the inequality v,*(x) > J&(X) and the inverse relationship between price and starting stock. Finally, the interperiod comparisons provided in Theorem 2, or which are implied in the analysis, giving convergent or uniformly convergent sequences of decisions, optimal expected profits and optimal marginal expected profits, suggest that the analysis may be readily extended to the infinite horizon. In the infinite horizon behavior would then be identical in each period, having properties listed in Theorem 1. We omit the details. The final theorem computes effects of a change in the holding cost. Other parameter changes may be handled similarly with analogous results. For example, we could as well consider changes in the discount factor or production cost or examine the impact of taxes. We leave the details to the reader. In this theorem qte = qt*(xte) and ptC = pt*(xte), i.e., equilibrium quantity and price. THEOREM
3. Given the conditions of Theorem 1,
535
MONOPOLY UNDER UNCERTAINTY
(a) $,*/ah -=c0, aqt*lall < ‘0, and ap,*/ah < 0, if0 < x < (b) dx,“/dh < 0, ifxte > 0, (c)
xt*,
dq,“/dh and dp,“ldh are indeterminate in sign if xtc > 0.
Proof. Since previous equations are only explicitly given for t = 1,2, we prove the theorem for t = 2, though the same argument applies for t being any finite integer. In the interval of initial inventory [0, x,*), optimal starting stock and price simultaneously satisfy D~Gz(pz*; yz*, x, h) = 0
and
D,Gz(pz*; yz*, x, h) = 0.
Differentiating,
+,*/ah = (--D,,G&G, %,*/ah = (--D,,GhG,
+ 4,G&&)IA + D,,Wd-UA
< 0, < 0,
(18) (19)
where A = D,,G,D,,G2 - (D,,G,)2 > 0. The inequalities in (18) and (19) follow since DlhG, = -u’D,~G, < 0, which is easily confirmed, and since D,,G, + u’DlzGz < 0 and u’D,,G, + D,,G, > 0, which were shown earlier. Reference to Fig. 2 [6, p. 2151 will now confirm that dxze/dh < 0 since in that figure q2*(x) shifts downward by (18) and z(~(x) shifts upward by (19). An algebraic confirmation of this conclusion is readily obtained. In particular, referring to Eq. (29) in the author’s paper [6, p. 2171, dxze/dh has the opposite sign of (u’ap,*/ah - aq2*/ah) which is positive. Moreover, Eq. (30) of that paper [6, p. 2181 will convince the reader that the direction of change in equilibrium output and price are indeterminate. The interesting development in this theorem is that ap,*/ah has a definite sign which is not true in the case of multiplicative demand even in the single-period horizon [6, pp. 216-2171. Theorem 3 also affords some explanation of observed “stickiness” in administered prices. That is, while changes in parameters do lead to changes in price and output, part of the adjustment occurs in the desired holding of inventory. In fact, in Nevins’ numerical examples, for changes in the holding cost and discount factor, most of the adjustment occurs in equilibrium inventory, and only little in equilibrium price and output [5, pp. 81-821. As a final exercise we could compare behavior under risky and riskless monopoly. The major element in this comparison is the inequality [u + ;T + pt*u’]/u’ < c’(y,* - x). We omit the derivation of this inequality and its full implications, which would follow a similar treatment in the author’s earlier paper [6, pp. 214-2161. Essentially, the outcome is that prices tend to be lower and output higher than in riskless monopoly.
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V. CONCLUSIONS With the assumptions of a concave expected demand function and particular densities of demand, we have demonstrated the existence and characteristics of uniquely determined decisions in any horizon in the case of additive demand. While it is possible to derive some characteristics of optimal behavior in any horizon when demand is multiplicative, uniqueness properties are not readily available. Apart from analytic ease, other important differences distinguish these two cases. In particular, results by various authors [ 1, 4-61 indicate clearly that optimal behavior is directly affected by the form of demand uncertainty. Multiplicative demand tends to induce high prices and low outputs and conversely for additive demand. The reason for this difference seems to hinge on the variance of demand. lf we let a2 be the variance of the density #(T) and u2 be the variance of demand, then it is easily shown, that in the additive demand, a2 = S2, and in multiplicative demand, u2 = 62[~(p)]2. Thus, with additive demand variance is constant at all prices, while with multiplicative demand it varies inversely with price, specifically, tending to zero as price tends to its maximum amount. Similarly, relative variance, a2/u2, is constant with multiplicative demand but varies directly with price in additive demand, going to infinity as price goes to its maximum. Consequently, on either an absolute or relative basis, high prices tend to be less risky in multiplicative than in additive demand. REFERENCES 1. S. KARLIN AND C. R. CARR, Prices and optimal inventory policy, in “Studies in Applied Probability and Management Science” (K. J. Arrow et al., Eds.), pp. 159-172, Stanford University Press, 1962. 2. E. S. MILLS, Uncertainty and price theory, Quart. J. Econ. 73 (1959), 116-130. 3. E. S. MILLS, “Price, Output and Inventory,” Wiley, New York, 1962. 4. A. NEVINS, “A Simulation Study of Price, Output and Inventory Under Uncertainty,” Doctoral dissertation, University of Rochester, 1964. 5. A. NEVMS, Some effects of uncertainty: Simulation of a model of price, Quart. J. Econ.
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