Author’s Accepted Manuscript Pricing Competition with inventory considerations in a hazard rateprone market of durables Konstantin Kogan
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S01651889(16)301579 http://dx.doi.org/10.1016/j.jedc.2016.09.015 DYNCON3355
To appear in: Journal of Economic Dynamics and Control Received date: 10 March 2016 Revised date: 28 July 2016 Accepted date: 30 September 2016 Cite this article as: Konstantin Kogan, Pricing Competition with inventory considerations in a hazard rateprone market of durables, Journal of Economic Dynamics and Control, http://dx.doi.org/10.1016/j.jedc.2016.09.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
PRICING COMPETITION WITH INVENTORY CONSIDERATIONS IN A HAZARD RATEPRONE MARKET OF DURABLES Konstantin Kogan Department of Management, BarIlan University
Abstract This paper addresses Bertrandtype pricing competition between two firms producing partially differentiated durables over a finite planning horizon. The demand for durables, characterized by increasing returns of scale to a price reduction, is led by the hazard rate. While the effect of inventories on pricing of nondurables is widely recognized, the management and marketing literature typically overlooks this effect in regard to horizontally competing firms for durables. In this paper we show that the pricing trajectory of durables may significantly alter when inventory dynamics are accounted for. In particular, the price may hike upwards before dropping; gradually grow; or even stay at the same level over the entire product life while it would only decline if inventories and related costs are disregarded. Furthermore, the wellknown, optimal pricing strategy of following the pattern of sales does not necessarily confirm even for symmetric equilibria when the competing firms have either an inventory surplus or shortage. Keywords: competition, pricing dynamics, inventory, durables
1. Introduction This research is related to the extensive body of literature on differential games in economics, marketing and management sciences in general, and Bertrandtype pricing competition in particular. Since a significant part of that literature has addressed horizontal competition between firms selling both durable and nondurable goods, we only mention here some of the works on this subject; for a more detailed picture please refer to the excellent reviews by Dockner et al. (2000), Jørgensen and Zaccour (2004), and Long (2010). Equilibrium pricing policies for nondurables have been studied according to various dynamics: learningbydoing (Stokey, 1986; Besanko et al. 2010; Kogan and El Ouardighi, 2016); sticky prices (Fershtman and Kamien, 1987; Piga, 2000); and costly output adjustments 1
(Dockner, 1992; Jun and Vives, 2004). Much attention also has been given to pricing competition for durables in a marketing channel along with advertising policies and supply chain interactions. Sales dynamics of durables have been the central point of modeling efforts that resulted in diverse diffusion models inspired by the seminal work by Bass (1969) and its generalization (Bass et al, 2004). Clarke and Dolan (1984), Eliashberg and Jeuland, 1986, Krishnamoorthy et al. 2010; Chutani and Sethi, 2012; Shalchi Tousi et al, 2015 present just a few examples of innovation diffusion and new product growth models with and without competition; for a recent extensive review of such models we refer to Peres et al. (2010). Overall, as summarized by Krishnan (1999), unless the time horizon is infinite and the prices therefore steady, pricing policies are shown to follow the pattern of sales that have either a single peak or are dropping due to demand saturation over a finite planning horizon. Notably, while the effect of inventories on pricing of nondurables is widely recognized (see, for example, literature reviews by Vives, 1999; Elmaghrabi and Keskinosak, 2003; and Talluri and Van Ryzin, 2004), the management and marketing literature typically overlooks such an important factor when considering horizontal competition for durables. The goal of this paper is to study pricing patterns under horizontal competition for durables. In particular, we seek to answer the following two questions: (i)
Are these patterns different from those that sales exhibit when pricing decisions disregard inventories?
(ii)
How and when are pricing patterns affected by inventory surpluses and shortages?
In this research, we show that inventory considerations may have a critical impact on the resulting pricing strategies of horizontally competing firms. Specifically, we find that the price of partially differentiated durables may climb before dropping, only grow and even remain unchanged while if inventories are not taken into account, prices would only drop over time. The demand for durables is viewed in our model as a hazard rate which is a modification of the Bass model. This type of modeling has been used to investigate monopolist pricing (Sethi and Bass, 2003); duopolistic competition (Eliashberg and Jeuland,1986); and the multifactor effect on the pricing of durables (Raymond et al. 1993). Modern durables become obsolete very quickly. Therefore, we consider a finite planning horizon and assume a zero discount rate over 2
that horizon. Referring to an equilibrium in a game theoretical sense and not to a steady state, we study pricing equilibria between two competing firms that have been affected by an initial surplus/shortage of inventories. We assume that demand uncertainty arising due to a newly released durable can be resolved at the beginning of the selling season and the firms can deploy sufficient capacity to produce the goods with respect to the ongoing demand, but not necessarily with respect to demand that may emerge in the first days of sales. Consequently, to offset possible differences between their production capacity and initial interest in the product, the firms estimate and prepare in advance initial stocks (e.g., with a newsvendortype model). Clearly, such estimates are never perfect and the firms may initially find themselves with excessive inventories as well as with stock outs especially when last minute orders can be placed online. An extreme example of the latter scenario occurred during the fifth Harry Potter book release. The book sold over a million copies in one day through Amazon.com alone, breaking many sale records and resulting in massive stock outs. Ten million copies were sold in the first three days (English version alone). Clearly, the greater the difference between the initial interest and its estimate, the larger the shortage/surplus the firm will have to start off the season and the longer it will take to balance production and demand for the new product and thereby eliminate the surplus/shortage. For the new Apple and Samsung smartphone presentations, enormous initial inventories were provided. In particular, in September 2013, Apple sold a combined 9 million iPhone 5S and 5C smartphones in the course of the first full weekend of the handsets' sale. That was enough for Apple to have already sold out its initial stock of the iPhone 5S. Similarly, the first day sales of Galaxy S5 doubled those of Samsung's Galaxy S4 and several million units were sold within a few days. Samsung Experience stores saw long lines of customers anxious to get their hands on the 2014 flagship. As a result, while some stores still had excessive inventories, others such as the Samsung Experience store in Paris, sold its entire 800unit inventory. Notably, the price history of the two aggressively competing brands with smartphones iPhone 5S and Samsung S5 frequently shows similar trends with price hikes and drops as shown in Figure 1, (based on the UK data: pricespy.co.uk). Thus, production and inventories may not always match demand thereby presenting one of the possible causes affecting the pricing competition on the smartphone market. This is to say, given an initial inventory surplus/shortage, the competing 3
firms face a challenging problem of coordinated production, pricing and inventory control to maximize their profit. Although the mathematical structure of the model that we develop does not allow for deriving closedloop and/or feedback strategies, the states of the competing firms are rarely known to implement such strategies. In particular, accurate information about real life hazard rates, demands and inventory levels is not quite easy to acquire. Moreover, firms frequently distort these factors to mislead their competitors. Therefore, decisions are typically made using some estimates that are insufficient for a feedbackbased type of decision making. In such an environment, an openloop control and/or its closedloop representation involving estimates of the current state is a more realistic approach. Therefore, in the next section we present our model followed with openloop equilibrium conditions and their properties in Section 3. Section 4 focuses on symmetric production and pricing equilibria, whose patterns are then studied in Section 5. A closedloop representation of the equilibria is developed in Section 6. Section 7 summarizes our results. 400 300 Samsung Galaxy S5 SMG900F 16GB
200
Apple iPhone 5s 16GB
100 0 May
Jun
Jul
Aug
Sep
Oct
Nov
Figure 1: Apple iPhone 5s and Samsung Galaxy S5 price history (in pounds) for MayNovember 2015.
2. The model Consider a duopoly with two firms operating in a Bertrand competition setting with partially substitutable goods. At each period of time t over a finite planning horizon, T, firm i=1,2 produces durables of type i and chooses simultaneously a price, sold.
Each
customer
can
buy
only
one 4
durable.
The
( )
for the durables to be
Bertrand’s
demand
rate,
( ( )
( ))
, for durables of type
at time t depends negatively on the price
( ) set by firm i producing it and positively on its competitor’s price
( ), j=3i. The sales are
modelled with a hazard rate approach. The approach assumes that the length of time t until an event (sale of product i) occurs is a random variable with density fi(t) and cumulative distribution ( )
( )
Fi(t). The hazard rate
( )
( )
(the conditional probability that a customer will buy a
new product of type i given that she has not bought either i or j until now) is determined by the demand rate, ( ) ( )
( ( )
( )
( )).
(1)
Equation (1) implies the higher the product i price, the lower the conditional probability that a customer will buy this product; vice versa,
the competitor’s price increase amplifies the
probability that a customer will buy product i. Letting the market size be equal to one, the probability that a customer has bought a product of type i by time t , Fi(t) is viewed as the ( ) is the fraction of those
fraction of the market that has bought this product and 1 Fi(t)
who have not bought either type of the durable. In an empirical study of consumer decisions to purchase durable goods, Raymond et al. (1993) obtain statistically significant results based on an exponential hazard rate in exogenous variables. In terms of our exogenous variables, pi(t), ( )
i=1,2, their model implies that the hazard rate is
( )
. Adopting this ( )
result with respect to equation (1), entails an exponential demand,
( )
. Notably,
the exponential model of demand has been extensively used to reflect the response to a reduction in price that follows increasing returns to scale (see, for example, Hanssens and Parsons (1993), Song et al. (2008), Chen et al. (2006)). Introducing the cumulative hazard function, ̇( ) and, since
( ) (
( )
( )
, Yi(0)=0,
( ))
( )
( )
∫
( )
, we then have
and j=3i ( ),
( )
(2) ( )
( ( )
( ))
and, with
respect to (1), the instantaneous sales are, ( )
( )
( )
( )
( )
and j=3i.
(3)
The firms’ inventory level, Xi(t), i=1,2 is due to the classical balance equation between the production rate, ui(t)≥0, and the sales rate, ( ), 5
̇( )
( )
( )
( )
( )
( )
,
( )
, i=1,2.
(4)
As discussed in the previous section, each firm may have either an initial surplus of inventories, , or shortage,
, which is backlogged (e.g., online placed orders) and supplied when
the firm catches up. We denote by ci the product i marginal cost without inventory consideration; indicates the unit surplus (holding) cost per time unit; and
the shortage (backlog) cost per
time unit. Consequently, the inventory cost is ( ( ))
( )
( ) i=1,2,
(5)
where ( ) and
( ).
(6)
We assume the durable becomes obsolete by time T so that any leftovers are salvaged with zero margin. Each firm maximizes its expected cumulative profit ()
()
) ( )
∫ ( ( )
subject to (2)(5),
( ( ))
(7)
.
3. Nash equilibrium properties In this section we analyze openloop control strategies. The Hamiltonians for the differential game (2)(7) are (the time index is henceforth omitted for convenience): (
)
( )
(
)
)
and j=3i,
where the costate variables ̇
{ ̇
,
,
̇ ̇ ,
( )
(8)
satisfy the following equations: 0,
(9a)
;
(9b) (
)
)(
)
),
( )
.
From (9b) and (9d), we observe that, ̇
and ( )
(( ̇
,
( ) ( ) ( ) ( )
(
(
; (9c) (9d)
and ). 6
, and therefore (10)
The equilibrium price is found from (
)
, or, equivalently, (
)
(
(
)(
)
)=0.
(11)
It is straightforward to verify if the costate (the shadow price of saturating the demand by one more unit) is negative,
<0, the equilibrium price defined by (11) satisfies the sufficient
optimality conditions. Specifically, (
( (
)
(
(
(
)
(
)
(
(
)( )(
) )
)),
)).
When accounting for (11), we find that (
(
)(
)
)
,
(12)
In addition, (
)
,
(13)
which implies that the differential game (2)(7) is controllable by both firms. Accordingly, we next consider the operating conditions characterized by
<0 for 0≤t
shadow price of increasing the rate of inventory accumulation by one more product unit,
, is
equal to the cost incurred by the firm during a period of time when one product unit is held/backlogged. Therefore, it is sufficient, for example, to require such pricing that, as long as the firms have at least one product unit in surplus/shortage, condition with respect to (10) and the transversality condition
holds. Indeed, this ( )
immediately results in
for 0≤t
)
,
(14)
where the markup effect of negative costate are myopic in terms of inventories, pricing the durables, we have
is explicitly observed. Furthermore, if the firms , i.e., they do not take stocks into account when
under an obvious condition of 7
. We can also
readily find from (14) that myopic pricing in terms of demand saturation dynamics, 
leads to lower pricing,
,
.
Turning now to the production equilibrium conditions, we observe that the production rate
related terms of the Hamiltonian are linear. Therefore the firm’s equilibrium output is ̅
() ( ) ( )
{
(15)
where ̅ is the maximal production rate and the intermediate production rate, ̅, is found by differentiating condition (15iii) over an interval of time, i.e., ̇ respect to (9a), can hold only if
, that, with
. This, when accounting for (4), readily leads to
. Our observations from (15), when taking into account condition (9a), lead to the following two conclusions. ( )=0 and
Lemma 1. Let there exist a time point, t1, such that ( )=0 and
inventory state and costate are
( )=0. Then equilibrium
( )=0 for
production rate of durable i is given by
and the equilibrium .■
for
The second observation relates the nonzero inventory state and costate. Lemma 2. If
( )
( )
, then
; otherwise, if
( )
( )
, then
.
Proof: See Appendix. ■ From Lemmas 1 and 2, it is straightforward to conclude that
is a continuous variable,
monotonically increasing and negative as long as the available stock is not depleted and monotonically decreasing and positive as long as the firms have stock outs. We next turn to pricing dynamics. Strategic instantaneous complementarity of pricing policies is readily observed from
(see (13)). Furthermore, applying implicit
differentiation to equation (11), we obtain, (
)(
)
(
)(
)
=0.
Or equivalently, (
(
)
(
)
(
)(
)
8
)
(
)(
)
(
)(
)
.
for 0≤t
Given
. Consequently, the firms’ pricing strategies
follow a similar pattern – if one firm raises its price then the other does so as well. We subsequently identify symmetric pricing patterns. Lemma 3. Symmetric equilibrium price pi=pj is given by ̇
(
̇
(
)(
)),
and j=3i,
(16)
with the terminal condition, ( )
.
(17)
)( ̇
( ̇ ̇ ̇
̇)
̇ ̇
Proof: Differentiating equation (14) over an interval of time, we have
(
), which with respect to pi=pj, (9c) and (2) results in
(
)
)
(( ̇
)
)
(
(
).
)(
Using (14) again to substitute it into the last expression, we obtain ̇ ̇
((
̇
)
)(
(
(
)(
(
(
)
))=
)),
as stated in this lemma. The terminal price is obtained from (14) by setting t=T and recalling the transversality conditions, (
) ( )
( )
( )
and ( )
=0, which leads to
, i.e., .
( ( )
)
( )
■
An immediate corollary is in order. Corollary 1. Given conditions of Lemmas 1, the equilibrium price for a durable, i, determined by Lemma 3 transforms into ( ̇
(
with the same terminal condition (17).
)(
)) for
.
(18)
■
Unlike complementing pricing interactions, production rates of the firms can become strategic substitutes. In particular, considering production dynamics that are not necessarily symmetric,
we
( rule,
find )
(
by
differentiating
the (
and
instantaneous )
sales
. Using the chain
) , we derive that the instantaneous sales are substitutes,
9
that,
, when
, and complements
, i=1,2, j=3i for 0≤t
, when
for a period of time and thereby, according to (15iii), the production rate is equal to the sales rate, we observe strategic production interactions analogous to those of the instantaneous sales, , when
, and
, when
.
Given the implications of Lemma 2, we now can show that equilibria even without requiring,
is true for symmetric
and therefore the sufficient optimality
conditions (12) always hold. Indeed, from (10) it is readily seen that since everywhere,
is continuous as well. Substituting (14) into (10) we have
̇
(
(
)
),
(19)
which, with respect to the transversality condition, leads to ̇ ( )
the costate to change, we need ̇ )
. Therefore integrating
for tdt. On the other hand, for this trend, ̇
(19) backward from t=T, we have
(
is continuous
, of
to hold at some point of time, i.e., according to (19), (
, which cannot hold for
)
̇ (
. Thus,
)
and
proceeding with the integration to the beginning of the planning horizon we conclude as follows. for 0≤t
Lemma 4. Given the equilibrium price is symmetric, pi=pj, then ■
sufficient optimality conditions (12) always hold.
4. Symmetric equilibria Based on Lemmas 2, 3, and 4, we now assume identical initial inventories and unit costs,
( )
( )
, i=1,2 to determine symmetric equilibria with pricing
affected by inventories over the entire planning horizon. Theorem 1. Let functions (
̇
, )
(
̇ (
x*=∫
)
( )
∫
and positive constants x* and x** be given by ( ( (
( (
)
))( (
and x**= ̅
))( ∫
)), (
If x ≥x*, then the symmetric equilibrium production and prices are: and
,
for
. 10
( )
)),
)
. (20)
( ) ∫
. (21) (
)
.(22)
If x ≤ x**, then the symmetric equilibrium production and prices are: ̅ and
,
for
.
■
Proof: See Appendix.
Theorem 1 formalizes equilibrium production and pricing for those cases when the initial inventory level is so over/underestimated that the firms are not able to fully eliminate their surplus/shortage until the end of the planning horizon. We henceforth refer to such initial inventories as excessive surpluses/shortages, x≥x* and x≤x**, respectively. Based on Lemmas 14, the next theorem considers equilibria under moderate surpluses/shortages as well as no initial inventories at all. Theorem 2. Let x*, x** be defined by Theorem 1 and (
̇ If 0
)
(
and ) (
̇
)(
( )
)),
.
(23)
satisfy the system of two equations
(
∫
(
be given by
(
∫ )
(
)
,
(
(
))(
( )
)),
( ); (24)
and the symmetric equilibrium production and prices are: for 0
(
,
for 0 If x**
)
(
∫
for
)
for
;
.
and
satisfy the system of two equations
∫
( (
̇
)
)
(
(
∫
)
,
(
and
(25)
(
))(
)),
( )
( ); (26)
and the symmetric equilibrium production and prices are: ̅ for 0
(
,
for 0 Otherwise, if x=0, then
)
(
∫
for (
)
)
; ∫
(
Proof: See Appendix. ■
5. Pricing and Sales Patterns
11
)
for
and
. and
for
;
.
Theorems 1 and 2 unveil the effect of inventories on equilibrium pricing determined by equations (24)(26). Although these nonlinear differential equations are not analytically, solvable, it is possible to investigate their implications in regard to underlying pricing patterns. Pricing without inventory consideration. We start off from an ideal condition of no initial surplus/shortage of inventories, in which case ( )
the equilibrium pricing curve is concave, always monotonically decreasing toward
,
as formalized in the proposition below. ( )
Proposition 1. Let
(
)(
). If there is no initial inventory, x=0, or the
inventories are disregarded when pricing, then ̇
and ̈
for 0≤t≤T,
and
.
(27)
Proof: Differentiating (18) with respect to time we have, ̈
̇
(
)
(
)(
(
)(
))
(
),
(28)
First, we observe from (18) that the sign of ̇ is fully determined by ( )
(
)(
),
(29)
which, with respect to (28) implies that if ̇ a time point t such that
, then ̈
, i.e,
. That is, if there exists
, then the price can only decrease from that point on.
Substituting the terminal price (17) into (29), we find ( )
( ( )
and therefore, ̇ ( )
)(
)
,
. The question then is if there could be a point of time t, such that ( )
. To study that, we next integrate equation (18) backwards from (17), which, with respect to ̇ ( )
, implies the price initially grows for t≤T. For this trend to
change we need that at some point, say t*, ̇ ( ) ( )
as defined by
, but then with respect to (28) ̈
. This entails
( )
and thereby
and it is easy to verify that all the higher
order derivatives are equal to zero as well. To resolve this uncertainty more formally in a close proximity to
, we analyze the stability of differential equation (18).
Let us introduce a new variable, ( ) which ̇
( )
( )
(
). Then ̇
. Assume T tends to infinity. Then
. Moreover, 12
̇,
( )
( ) and
determines a steady state at
̇

Therefore
(
)
(
(
)
)
(
)
.
( )
is asymptotically stable and
. Thus,
when integrating (18) backward, the product price increases starting from toward
(30) and
( )
, but never reaches this level over a finite planning horizon.
■
According to Proposition 1, when there is either no inventory or the firms are myopic in terms of accounting for inventories, the longer the planning horizon, the closer the price at t=0 to ( )
. Furthermore, with respect to (30), parameter a visibly affects this convergence,
as Figure 2 demonstrates.
Figure 2. Price convergence to
( ) (b=0.5, c=0.1,T=30)
for a=0.5 and a=0.1.
Parameter b, on the other hand, which reflects the level of partial substitution of the products produced by the two firms, impacts the diapason of the possible price markup. Namely, the greater the competition between the firms by means of fuller substitutability of their products, the closer b is to one and the narrower the range [c+1,
) for the price to adjust.
Pricing under inventory surpluses We next analyze the effect of excessive inventories on pricing dynamics along with sales patterns. 13
Proposition 2. Let the firms have excessive inventory, x≥x*, pi as determined by Theorem 1 and ( )
(
(
))(
). Then the evolution of the equilibrium price is
described as follows:
(
If
)( (
when ( )
)
(
), then
) ( )
and ̇
( ) at an interior time point
, there will be a price peak, (
( )
0
and
) ( )
, so that, ̇
for 0≤t
for t*
Otherwise, if ( )
) ( )
, the maximal markup is at t=0, ̇
and ̈
for 0≤t≤T.
If
(
)(
)
(
), then
, the maximal markup is at t=T and ̇
(
)(
)
(
), then
for 0≤t≤T.
for 0≤t≤T.
If
Proof: We first differentiate equation (20), ( ̈
(
) ̇
)
(
(
(
(
))
))(
)
( ̇
).
)(
(31)
and define function: ( )
(
(
))(
).
(32) ( ( )
Substituting the terminal price from (20) into (31) and (32) we find ( ) )
and with respect to (20), ̇ ( ) (
) ( )
(
( )
if )(
)
(
)
.
Assume (33) holds. We next integrate equation (20) backwards from to ̇ ( ) ̈( )
)(
(33) ( )
, which, due
, implies the price initially grows for t≤T. Moreover, from (31) it follows that . For this trend to change we need that at some point, say t*, ̇ ( )
, i.e., from (20)
we have, ( )
(
) ( )
,
(34)
which can happen only when ( ) (
) ( )
(
)
. Furthermore, from (31) we observe that
̈( )
. Consequently, we find the price evolves concavely and if it
grows from the very beginning, ( )
(
) ( )
14
, there exists a maximal markup price at
t*: 0
(
, we determine the upper bound,
)
. The next question is whether the price, after growing to peak by
and then decreasing
(moving in the backward direction), can increase again? For this to happen, we need that at another point of time t
, but now along with ̈
. According to (31), ̈
, when
the following holds (
) ̇
( ̇
By taking into account ̇
).
)(
(35)
, we readily observe that (35) never holds. Thus, there can be at
most one price peak which has a concave shape. (
Consider now the opposite case to (33), ̇( )
(
. An immediate observation is if
equal to
( ) over the entire planning horizon, ̇
)(
. Let
)
)(
(
)
(
) and therefore
), then the price remains (
)(
)
(
) and
therefore, when moving in the backward direction, the price decreases at least around t=T, as ̇( )
. Again, for this trend to change and the price to start to increase, we need that at some
point of time ̇
along with ̈
, which according to (35) never holds.
■
Proposition 2 shows that when the available stock is excessive, in addition to a single price peak anywhere along the planning horizon, every monotone trend whether displaying an increasing, decreasing or timeinvariant price, is possible. In particular, high inventory holding cost can induce the firms to initially markdown goods to stimulate higher sales and quicker surplus depletion. Then the price would gradually grow. On the other hand, low holding costs may induce pricing similar to what occurs when there are no inventory concerns starting with a high markup price that then monotonically decreases. An interesting outcome is due to an intermediate cost relationship, ( )
(
) ( )
(
)(
)
(
),
inducing initially stimulating, low price. Then, the price, after a gradual increase and significant surplus reduction would decrease to compensate for saturating sales, as presented in Figure 3. Figure 3 illustrates concave templets of pricing with (subscript 1) and without (subscript 2) inventory consideration when inventory costs induce a hike in price at an interior part of the planning horizon. Another nontrivial outcome is due to the last condition of Proposition 2, 15
(
which shows that a delicate cost balance,
)(
)
(
), leads to a timeinvariant
pricing.
a)
b)
c)
Figure 3. Equilibria over time under inventory surpluses – subscript 1 and without inventory consideration – subscript 2: a) Prices; b) Sales; c) Market shares. (b=0.5, c=0.1, a=0.5, h+=0.05, T=10) When inventory surplus is moderate, 0
̇
is determined by ̃ ( )
). Consequently, the highest possible price peak is still ) and the lowest peak is at t*=0:
(
(
(which occurs at
. This is to say, the equilibrium price
either gradually decreases over the two trajectories or there is an interior markup so that the price first increases and then drops at the first trajectory, which is followed from t=t1 by a further decline along the second trajectory. 16
The described pricing strategy is outlined by Krishnan (1999) as following the pattern of sales induced by the generalized Bass Model without competition and inventory concerns. Notably, our result is due to the effect of inventories without which the price would only decline as shown by Proposition 1 and which Figure 2 illustrates. To investigate if the equilibrium pricing strategy follows the pattern of sales, we next differentiate symmetric sales from (3): ̇
̇)
) ̇
((
(
)
(36)
which naturally implies that whenever the price grows, ̇
, the sales must drop. Notably, the
opposite relationship is not always true. Specifically, even when (
̃ )(
)
(
)
(
(
)
we obtain ̇
)
)
(
, but
, or equivalently ̃(
(
)),
(37)
and therefore, ̃̇
. Furthermore, from ̇
attained when t=t1, i.e., when ̃ ( )
̇
, the maximal value for ̃ is
. Accordingly, for (37) and thereby ̇
to hold, it
is necessary that (
(
)
which never holds even if
)
(
)
(
)
(
(
))
.
. That is, we derive the following result.
Proposition 3. As long as the firms have no inventory shortage the equilibrium sales always decline over time, i.e., ̇
■
.
With respect to Propositions 1 and 2, Proposition 3 indicates that, under inventory surpluses, the equilibrium price follows the sales pattern when either inventories are not taken into consideration or when the maximal markup is at t=0. In all the other cases, the pricing pattern is very different from the sales pattern. For example, when
(
)(
)
(
),
the price peak is at t=T and the price grows over the planning horizon; on the other hand, the sales exhibit the opposite trend and decline all the time. We next investigate the relationship between the pricing trajectories with and without inventory consideration,
and

, respectively. Let the price decreases , ̇
, at least at a
final part of the planning horizon where there exists a point t, such that the price at this point, with and without inventories accounted for, is identical
17
= 
. Compare the rate of change
and ̇ 
from this point on in the backward direction. Since, ̇ have
< 
at tdt if ̇ ( )> ̇ ( ) (
̇( ) where ( )
) ( )
at this point t, we
, which by accounting for (24) and (23) implies
̃( )
(
̇ ( )
) ( )
( )
. (38)
, as shown in Proposition 1. Since
̃
(
(
))(
)<
(
)(
),
as long as the firms have inventories, i.e., t
( )
( )
,leads to the following conclusion.
Proposition 4. Let x>0. As long as the firms have a surplus of durables and account for it when pricing, they price their durables lower than if they would by disregarding the surplus, which results in a higher market share and hazard rate. But the instantaneous sales do not necessarily increase: < 
,
> 
, and
> 
.
■
In accordance with Proposition 4, Figure 3 shows p1F2. While the sales gained by pricing with inventory consideration (subscript 1) first dominate f1>f2, they then become lower than those when inventories are not accounted for (subscript 2), f1
to the fact that a lower price implies a higher hazard rate led demand, in Proposition 4. As a result, market share, (
)
, since the sales component,
)
, as stated
, grows, but not necessarily sales, , decreases, reflecting saturation of the
market of durables and thereby offsetting the increase in demand,
(
)
.
Pricing under inventory shortages We next analyze the effect of excessive shortages on pricing dynamics along with sales patterns. We first find that in such a case the price drops all the time. Proposition 5. Let the firms have excessive shortage of inventory, x≤x**, and pi be determined by Theorem 1. Then the equilibrium price is decreasing over the entire planning horizon, ̇ for 0≤t≤T. Proof: We first differentiate (21),
18
( ̈
(
) ̇
)
(
(
(
))
))(
(
).
)(
)
( ̇ (39)
Substituting the terminal price (17) into (21), we have (
̇( )
)(
(
)
. Note ( )
and according to (39), ̈ ( ) ( )
)
(
(
, where
))(
).
(40)
Therefore, for the price to increase, we need at some point, say t*, in addition to ̈ ( ) have ̇ ( )
. However, from (39) we observe that if ̇ ( ) (
̈( )
) ( )
(
)
(i.e.,
( )
to ), then
. Consequently, the price declines over the entire planning
horizon toward the terminal value of c+1.
■
Figure 4 illustrates prices declining over time when the firms have an excessive shortage of inventories. Similar to the effect of a moderate surplus, when the shortage is moderate, x**
, i.e., (
(
)
)(
(
)
)
,
or equivalently (
(
)
)
(
(
)).
(41)
The next proposition shows that if (41) holds at t=T, i.e., if (
then ̇
)(
) (
)
,
(42)
over the entire horizon.
Proposition 6. Let x≤x**. If
(
)(
) (
19
)
, then
̇
for 0≤t≤T.
Proof: Recalling that ̇
(see Proposition 5), to prove this proposition, we show that the
maximal value of the righthand side of (41) is attained at t=T. Indeed, differentiating the righthand side of (41) we find that it is increasing over time ( when
(
) ̇
)
(
̇(
(
(
)
(
)
)
(
(
̇
)(
)
)
)). Since the latter always holds if
,
, assuming
, then we
need to show that (
)
.
holds. According to (21),
(43) ̇
, implies
Consequently, for (43) to hold and thus ̇ true at t=T, as stated in this proposition.
. Therefore,
̇ ̇
.
for 0≤t≤T, it is sufficient to require that (43) is ■
Figure 4. Equilibria over time under inventory shortages – subscript 1, without inventory consideration – subscript 2: a) Prices; b) Sales; c) Market shares. (b=0.5, c=0.1,a=0.3, h=0.4) Proposition 6 derives sufficient condition (42), which implies that if the firms experience excessive shortages and this condition holds, the equilibrium price follows the pattern which is , while ̇ ( )
exactly opposite to that of the sales. That is, ̇
as illustrated with Figure 4.
This outcome is very different from what we found under excessive stocks where the periods of decreasing prices, ̇
, were always associated with sales decline, ̇ ( )
.
We next compare pricing trajectories with and without accounting for shortages, 
, respectively. Let us assume that at a point t, we have 20
= 
and ̃
(
and
(
))(
). Similar to the analysis under a surplus, we compare in the backward
direction the rate of change from this point on. According to Propositions 1 and 5, ̇ ̇
> 
over the entire planning horizon. Therefore,
, at tdt if ̇ < ̇ 
and at point
t, which by accounting for (23) and (26) leads to the requirement ) ( )̃
(
̇( )
( )
(
̇ ( )
) ( )
( )
. (44)
Since ̃
(
(
))(
)>
(
)(
)
we readily observe that (44) always holds. This, along with the fact that
, ( )
( )
(see Theorem 2), leads to the following result. Proposition 7. Let x<0. As long as the firms have a shortage of durables and take it into consideration, they price their durables higher than what they would do if they were to disregard the shortage. As a result the market share and hazard rate are lower, but the instantaneous sales do not necessarily decrease, > 
,
< 
, and
< 
.
■
Comparing the outcomes of Propositions 4 and 7, we find that the impact of inventory surplus and shortage on the firms’ pricing strategies are completely different. When the firms price their durables myopically without taking inventories into consideration, the price is always lower if the firms experience shortages and higher in case of inventory surpluses than the equilibrium price with inventories taken into account. Moreover, the myopic pricing improves the market share under shortages and deteriorates it under surpluses.
6. Closedloop representation Section 5 focuses on the evolution of pricing over time under the assumption that only initial inventories of the competing firms are known. In this section we assume that the cumulative demands of competitors over time can be also estimated as, for example, their expected values. Notably, while the effect of the state of inventories on pricing is explicated by Theorems 1 and 2, the impact of the state of cumulative demands is not readily observed from there. To understand such an effect, we next transform the equilibria determined by Theorems 1 and 2 into a closedloop form. 21
Similar to the maximum principle for a closedloop control, we attempt to express the costate via the state of cumulative demands. Assume the costate variable cumulative hazard rate,
( )
( ). Then ̇
, only:
depends on the
̇ and when accounting for (2)
and (9c), we have (
)
(
Next, substituting (
)
( )
)
(
)
.
(45)
with ( ), we obtain
with (14) and (
(
( )
)
) ( )
(
)
,
(46)
i.e., with respect to the transversality condition from (9c), (
( )
) ( )
( ( ))
,
.
(47)
( ) is determined by Theorems 1 and 2 depending on the initial inventory x, which
Note that
also implies t1=t1(x) when the inventories are moderate. The solution to the differential equation (47) is (
( )
)
,
(48)
where the integration constant C=C(x) is derived from ( ( )) ( )(
:
)
.
(49)
Consequently, by substituting (48) into (14) and accounting for Theorems 1 and 2, we derive a closedloop pricing strategy for different inventory states as summarized in the following proposition. Proposition 8. Let constant C be determined by (49) and t1(x) by Theorem 2. Then a closedloop representation of the symmetric pricing equilibria is: If x≥x*, then (
)
(
)
(
)
(
( ))
)(
(
)
);
(50)
If 0
(
)( ( )
(
)
), as long as (51)
If x≤x**, then 22
;
(
)
(
)
(
(
( ))
(
)(
)
);
(52)
If 0>x>x**, then (
)
(
(
)( ( )
)
), as long as
;
(53) If either x=0 or (
, then
)
(
(
)(
)
), as long as
. (54)
Proof: To prove the derived closedloop pricing, we need to show that it satisfies the optimality condition (14) with the evolution of the costates and states as determined by Theorems 1 and 2. Consider, for example, equation (50). It is identical to (14) for x≥x* if ( )
(
) (
)(
(
)
, or after simple manipulations, if )
.
(55)
First, by accounting for (49), we find from (55) at t=T that the transversality condition, ( )
, holds. Furthermore, differentiating (55) we obtain (
)
((
Next, using (50) we substitute (
(
)
)
̇
(56) ) with
(
(
( (
(10c), while
(
)(
and by accounting for (2) we find )
̇ .
) ̇
)
)
( (
(
)), which along with
( )
) in (56)
) is identical to
), as shown in Theorem 1. Similarly, (50)  (54) are verified. ■
Closedloop pricing equilibria presented in Proposition 8 show the explicit influence of two dynamic factors: the holding/backlog inventory cost over the time remaining until the end of the planning horizon and the state of the cumulative demand. Specifically, as long as the firms have a surplus, (i.e., until increased by (
)
(
) the price of their goods is decreased by
(
), and
) when the firms have shortages. On the other hand, according to (49),
, and therefore,
(
)(
(
)
)
, which implies that the current
state of the accumulated demand (hazard rate) always increments the price. This effect of inventories, however, diminishes with time as well as the demand accumulation ̇ >0 saturates, 23
due to ̈
. We thus conclude from Proposition 8 that the pricing pattern is due to the
interaction between the declining effect of the surplus/backlog costs (proportional for shortages and inversely proportional for surpluses) and increasingly saturating cumulative demand. The latter, however, does not necessary imply saturation of sales. For example, even under shortages (see (52) and (53)), when the two trends are not conflicting with each other, the sales can grow over time. This occurs when the initial markup price is high, entailing low sales. The price then gradually drops which, as shown in Proposition 6, can reignite the sales resulting in two diverging patterns of pricing and selling. Finally, turning to the production equilibria, we recall that the firms either do not produce until the surplus is depleted, ui=0, or produce at the maximal rate as long as there is a shortage. (
Then they trace the sales,
)
, until the end of the planning horizon. That is, the
closedloop equilibrium production is: if Xi>0, then ui=0; if Xi<0, then Xi=0, ui( )=
(
(
)(
(
)
))(
)
̅; and as long as,
.
7. Conclusions In this paper we address the effect of inventories on pricing competition for partially substitutable durables. When the competing firms are either myopic in terms of inventory consideration or have no surplus/shortage of inventories, the equilibrium pricing curve is always monotonically decreasing following the pattern of saturating sales. Furthermore, the greater the returns of scale in customer demand, the later and the more abruptly the price drops by the end of the product life. In addition, the greater the competition between the firms by means of fuller substitutability of their products, the narrower the range for the price to adjust. We show that when the inventories are excessive, there can be a price peak anywhere along the planning horizon. In addition, every monotone trend is possible including growing, declining and timeinvariant prices. In particular, a high inventory holding cost can induce the firms to initially markdown to stimulate higher sales and quicker surplus depletion. Then the price will gradually grow. On the other hand, low holding costs may induce a pricing pattern similar to that without inventory concerns, starting from a highest price which then monotonically decreases. The most convoluted pricing strategy arises in an intermediate cost 24
relationship. In such a case, initially stimulating low price, after a gradual increase and surplus reduction can decrease to compensate for saturating sales. Notably, such pricing policies are typically attributed to the corresponding sales pattern, while we find this is not generally the case when inventories are taken into consideration. In particular, under inventory surpluses, the equilibrium price follows the sales pattern when either the inventories are myopically not taken into consideration or the maximal markup is at the very beginning of the planning horizon. In all the other cases, the pricing pattern is very different from the sales pattern. For example, when the inventory holding costs are high, the price grows over the planning horizon while the sales exhibit the opposite trend of declining all the time. In contrast, when the firms have excessive shortages of inventories, the price declines over time while the sales, on the contrary, grow. We develop a closedloop representation of the pricing equilibria and show the explicit effect of the two dynamic factors: the holding/backlog inventory cost over the time left until the end of the planning horizon and the state of the cumulative demand. We show that the pricing pattern is due to the interaction between two trends, conflicting in case of surpluses and complementing in case of shortages. Specifically, the interaction is due to the declining effect over time of the surplus/backlog costs  proportional for shortages and inversely proportional for surpluses and of the increasingly saturating cumulative demand. The latter, however, does not necessarily imply saturation of sales. For example, even under shortages, when the two trends are not conflicting with each other, the sales can grow over time. Overall, we find a very contrasting impact of inventory surplus/shortage on the firms’ pricing strategies. When the firms price their durables myopically without taking inventories into consideration, their prices are always lower under shortages and higher under surpluses than the equilibrium price derived when inventories are taken into account. Moreover, the myopic pricing improves the market share under shortages and deteriorates it under surpluses.
Appendix Proof of Lemma 2: The proof is by contradiction. Assume that
( )
, but
the optimality conditions (15) as well as the transversality condition from (9a), Then from (15), we have,
( )
and, due to (4),
̇( )
( ) ( )
and 0, hold.
. That is, the
positive initial inventory level cannot decrease, which, with respect to (9a), induces the costate 25
̇ ( )
to monotonically grow, ( )
. The process will continue for any t>0 resulting in ( )
0. This contradicts the transversality condition of
that when there is a shortage,
0. Similarly, one can verify
, the costate must be positive to ensure production at the
maximum rate and that the shortage is being taken care of. ■ ( )
Proof of Theorem 1: Given
( )
, consider symmetric pricing, p1=p2, for the
case when the initial inventories cannot be fully depleted by the demand. This implies, ̇ (
,
)
(see Lemmas 2), and condition (9a)) for ( )
(see (15ii)), and from (4), (
∫ where
)
∫
(
)
∫
(
)
(
∫
,
)
for
, i.e.,
, . Consequently, the threshold surplus inventory level, x* is
given by (
∫
(
In addition, given (
̇
)
x*
), according to Lemma 3,
)
(
(
(
))(
)) and
( )
,
as stated in this theorem and all equilibrium conditions are met. Similarly, we identify symmetric equilibrium when initial shortage, , cannot be eliminated. This implies and condition (9a)) for ( )
̅ (̅
where
(
) (
∫
∫
(
)
∫
(
)
(
,
̅ for
,
∫
̇
)
( )
(see Lemma 2
(see (15i)), and according to (4),
, i.e., )
), . Consequently, the threshold shortage inventory level, x** is
given by ̅
(
∫
)
(
Finally, with respect to ̇
( )
(
as stated in this theorem.
)
(
(
x**>0. ) and Lemma 3 we have (
■ 26
))(
)),
( )
,
Proof of Theorem 2: Given initial stocks x>0 of both firms, according to Lemma 2, the ,
( )=0, where it remains unchanged according to Lemma 1, i.e.,
until reaching the zero level, (
, which grows linearly because ̇ ( )
( )
equilibrium production is due to ) for 0
( )=0 for
and
.
By accounting for (15), this costate behavior corresponds to producing nothing up to depleting the stock by the switching point,
, thereby
. Then, according to Lemma 1, the firms trace the
demand to ensure zero stocks over the remaining time: for 0
(
,
)
.
Note, it follows from Theorem 1, that if 0
̇ (
̇
)
)
(
(
(
(
and into (18) for (
)(
))(
)) for 0
)) for
, ,
,
respectively, where the terminal price is determined by (17). Finally, due to (14), the two pricing functions before and after t1 are equated at t=t1 while full inventory depletion by t1 implies: ( )
(
∫
)
(
∫
)
( )
,
which ensures all equilibrium conditions are met. Similarly, based on Lemmas 14, the equilibrium pricing and production (
̇ ̅ for 0
)
,
(
(
( (
))(
)) for 0
,
)
are verified for x**
■
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