# Continuously Increasing Price in an Inventory Cycle: An Optimal

Authors: Prafulla  Joglekar; Patrick Lee; Alireza M. Farahani
Journal: Journal of Applied Mathematics and Decision Sciences
 (2008)

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## Abstract

Operations researchers have always assumed that when a product's unit cost
is constant and its demand curve is known and stationary, a retailer of the product would find it optimal to replenish the inventory with a fixed quantity and to sell the product always at a fixed price. We present, with proof, a model that shows that, in such a case, an e-tailer is better off using a continuously increasing price strategy than using a fixed price strategy within each inventory cycle. Sensitivity analysis shows that this strategy is particularly profitable when demand is highly price sensitive and the inventory ordering and carrying costs are high.


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## Body

## 1. Introduction

When  a retailer of a product
buys the product at a constant unit cost, incurs a fixed cost per order, stores
the product at a constant carrying cost per unit of inventory per year, and
faces a deterministic and constant demand rate over an infinite horizon, the
economic order quantity (EOQ) model tells us that the retailer’s optimal
strategy is to buy a fixed quantity every time he or she buys. Ignoring
inventory related costs, classical price theory tells us that when a product’s
demand is price sensitive but the demand curve is known and stationary, the retailer’s
optimal strategy is to charge a single price throughout the year. Whitin [1]
was the first one to integrate the concepts of inventory theory with the
concepts of price theory to investigate the simultaneous determination of price
and order quantity decisions of a retailer. Although he never stated so
explicitly, Whitin [1] assumed that when all assumptions of the EOQ model are
valid except that demand is price sensitive, with a known and stationary demand
curve, a retailer’s optimal strategy would be, once again, to buy a fixed
quantity every time he buys and to sell it at a single price throughout the
inventory cycle.Kunreuther and Richard  [2] sought to show that when demand is price
elastic, centralized decision-making (using simultaneous determination of
optimal price and order quantity) was superior to the common practice of
decentralized decision-making whereby the price decisions of a retailer were
made by the marketing department and given that price, the order quantity
decisions were made by the operations department. Although Kunreuther and
Richard [2] were perhaps unaware of Whitin’s [1] paper, their model was very
similar to Whitin’s [1] model. Assuming a known and stationary demand curve
along with the remaining conditions of the EOQ model, they asserted: “given constant
marginal costs of holding and purchasing the goods, the firm will want to
maintain the same price throughout the year” [3, page 173]. Thus, Kunreuther
and Richard [2] explicitly stated their assumption that the retailer’s selling
price would be constant throughout the year. What they did not realize is that,
even though marginal holding costs are constant per unit, a firm’s holding
costs at any particular time within an inventory cycle are a function of
inventory on hand, which itself is a function of the time from the beginning of
the inventory cycle. As our research shows, in this situation, a constant price
throughout the year is not optimal.Over the five decades since
Whitin’s [1] work, numerous authors
[4–10]
have used Whitin’s [1] and Kunreuther and Richard’s [2] models as foundations
to their own models.  However, none of
these authors have questioned Whitin’s [1] and Kunreuther and Richard’s [2]
assumption that the retailer’s optimal strategy would be to sell the product at
a single price throughout the inventory cycle. Considering a situation of price
sensitive demand, Abad [10, 11] found that, in the case of a temporary sale with
a forward buying opportunity, a retailer’s optimal strategy is to charge two
different prices during the last inventory cycle of the quantity bought on sale—a low price at
the beginning of the inventory cycle and a higher price starting somewhere in
the middle of the cycle. Yet, Abad [10, 11] did not consider a similar strategy
in every regular inventory cycle of a product with price sensitive demand.In this paper, we show that,
when demand is price-sensitive, Whitin’s [1] and Kunreuther and Richard’s [2]
assumption of a single price throughout an inventory cycle leads to suboptimal
profits for the retailer. During an inventory cycle, a retailer’s inventory
level and carrying costs are a declining function of time. When a retailer
faces a price-insensitive demand, his selling price is set arbitrarily, since
any optimizing model would push the price to infinity. In other words, in that
situation, price is not seen as a decision variable for any mathematical model.
Given an arbitrary price (and corresponding demand), the retailer’s only
strategy is to minimize his inventory ordering and holding costs by using the
EOQ model. However, in today’s e-commerce environment, when demand is price
sensitive, an e-tailer (e.g., Amazon.com) can adopt a continuously increasing
price strategy to minimize the impact of the time-dependent inventory carrying
costs. An e-tailer can easily quote different prices for a product to an
on-line shopper depending on the time at which the shopper is ready to order
the product.The idea underlying our
continuously increasing price strategy is to charge a relatively low selling
price at the beginning of an inventory cycle when the on-hand inventory is
large. The low price would generate high demand. Consequently, the inventory
level as well as the marginal inventory carrying costs would decline rapidly.
However, as the inventory cycle progresses, and as the on-hand inventory and
the marginal carrying costs decline, the e-tailer can charge increasingly
higher prices to maximize profit.With the widespread use of
revenue management or yield management techniques [12–16] in the airline, car rental, and hotel industries today, a
time-dependent (or dynamic) pricing strategy has become commonly adopted. Revenue
management techniques are typically applied in situations of fixed, perishable
capacity and a possibility of market segmentation [15].
However, in recent years, retail and other industries have begun to use dynamic
pricing policies in view of their inventory considerations. Elmaghraby and Keskinocak
[17] present a comprehensive review of the academic literature on dynamic
pricing in the presence of inventory considerations. They find that the bulk of
academic literature on dynamic pricing is focused on two market environments.
The first is an environment where there is no opportunity for inventory
replenishment over the selling horizon and demand is independent over time. In
the second environment, inventory replenishment is possible, but customers
behave myopically and are unconcerned about future price decreases. Our
research is not related to either of these two environments.Elmaghraby and Keskinocak’s [17]
review suggests that the paper of Rajan et al. [18] may be the only study in
the available literature that is relevant to the scope and purpose of our
study. Rajan et al. [18] also focus on the price changes that can occur during
a seller’s inventory cycle. However, they focus on a seller who sells a single
perishable product that deteriorates over time and may or may not experience a
reduction in its value due to that deterioration. We show that even for a
nonperishable product with replenishment possibility, a dynamic pricing
strategy makes sense. We present a model that shows that, when the demand curve
for a nonperishable product is known and stationary, a retailer is better off
using a continuously increasing price strategy within each inventory cycle.
Because the continuously increasing price policy we are envisioning is likely
to be practical primarily in online retail environments, we say that an
e-tailer is better off using a continuously increasing price strategy within
each inventory cycle.The applicability of our model
and many other models we have referenced here may be limited by the fact that
they all assume deterministic customer behavior and any lack of competitive
reactions to one’s actions. To the extent that customer behavior is stochastic
and competitors do react to a firm’s actions, these models may be
oversimplifications of reality. On the other hand, without some simplification no
theoretical or practical model can be built.In what follows, first we
recapitulate Whitin’s [1] and Kunreuther and Richard’s [2] fixed price
strategy model. In the next section, we present our own model. Then, we provide
a mathematical proof that the retailer’s profit under the continuously
increasing price strategy is higher than the profit under the fixed price
strategy. Next, we provide several numerical examples considering a linear
demand curve and varied values of relevant parameters. The final section
provides the conclusions of our analysis along with some directions for future
research.


## 2. The Fixed Price Model

Both papers of Whitin [1] and
Kunreuther and Richard [2] consider a situation where all the other
assumptions of the EOQ model are valid but demand is price sensitive, with a
known and stationary demand curve. Whitin’s [1] notation is different from
Kunreuther and Richard’s [2] notation. There are also some minor differences
in the details of the two models. However, the following captures the basics of
both the models. Although the model is applicable to any form of the demand
function,  for simplicity, we use a linear demand function.Let the following notations hold,C = retailer’s known
and constant unit cost of buying the product, S = retailer’s known
and constant ordering cost per order, I = retailer’s
carrying costs per dollar of inventory per year, P1 = retailer’s
selling price per unit in this model.It is assumed thatP1>C, D1 = retailer’s
annual demand as a function of the selling price, P1. D1=a−bP1, where a and b are nonnegative constants, a representing the theoretical maximum annual
demand (at the hypothetical price of $0 per unit) and b representing the
demand elasticity (i.e., the reduction in annual demand per dollar increase in
price).Note that sinceD1 must be positive for the conceivable range of values of P1,
a>bP1 for that range of values of P1, and
since P1>C, it follows that a>bC. Let T = the duration
of retailer’s inventory cycle, and Q = retailer’s order
quantity per order in this model. Then, Q=D1T1. Then Z1 = retailer’s
profit per year under  this model (2.1)Z1=(P1−C)D1−IC(D1T2)−ST=(P1−C−ICT2)(a−bP1)−ST. Differentiating Z1 with respect to P1 and T, the first-order conditions for the maximization of this function are (2.2)P1=(12)(ab+C)+ICT4,(2.3)T={2SIC(a−bP1)}1/2. From (2.2) and (2.3), we can
see that T satisfies the 3rd
degree polynomial (2.4)T3−2[a−bCbIC]T2+8SI2C2b=0. Assuming T1  is a
positive root of (2.4), substituting the optimal price from (2.2) and T1  from (2.4) in
(2.1), we obtain Z1,
the retailer’s optimal profit from this model, as (2.5)Z1=(b4)[(ab)−C−(ICT12)]2−ST1.We have verified that the second-order
conditions (not presented here) fulfil the requirements for a local maximum of
the annual profit. Thus, by solving for (2.2) and (2.3) simultaneously, one can
obtain the optimal values of the price and the inventory cycle time.
Unfortunately, to obtain a closed form solution to these equations, one must
solve a cubic equation that has three possible roots for the value of T.
Of course, we would be interested in only the real root(s). Depending on the
coefficients of the cubic equation, the closed form solutions may be of real
numbers and/or complex numbers. Hence, in our numerical examples, we rely on
Excel Solver to obtain the real and optimal solution to this problem. Of
course, when there are multiple real solutions to a cubic equation, Excel gives
only one of those solutions. In theory, one should use a software such as
MathCAD to obtain all of the possible solutions and then determine which one is
the global optimal. We did carry out this strategy for several numerical
examples. In all the cases, what we found was that only one (if any) of the
three solutions results in a real and feasible, hence optimal, solution.
Therefore, in Excel Solver, we input feasibility conditions such as price must
be greater than unit cost, order quantity must be positive, and annual profit
must be positive. Excel also needs reasonable starting values of the decision
variables. Once care is taken to input these conditions and reasonable starting
values, in our experimentation, Excel Solver has never failed to return the
best real solution (if any). Hence, we believe that practicing managers would
be adequately served by the use of Excel Solver. They need not worry about obtaining
all real roots of the cubic equation.


## 3. The Continuously Increasing Price Model

We
retain all of the assumptions of the foregoing model, except that now we assume
that the retailer uses a continuously increasing price strategy within each
inventory cycle.Let
us add the following notation:t = time elapsed from
the beginning of an inventory cycle, P2(t) = the retailer’s selling
price at time t=f+gt, where f and g are nonnegative decision variables, and f>C. f represents the
retail price at the beginning of each inventory cycle and g represents
the annual rate of increase in the retail price throughout an inventory cycle.Given that price is a function of time,
now the retailer’s annual demand rate will also be a function of time. Hence,
we should redefine demand as(3.1)D2(t)=a−bP2(t)=a−bf−bgt. Also let X(t) = retailer’s inventory at time t,  T = the
duration of retailer’s inventory cycle.Since
at the beginning of the inventory cycle, the retailer orders a quantityQ to
meet the cycle time demand, then (3.2)Q=X(0)=∫0TD2(t)dt=∫0T(a-bf-bgt)dt=(a−bf)T−(bg2)T2, and
at time t, the retailer’s on-hand
inventory is given by (3.3)X(t)=∫0TD2(t)dt−∫0tD2(t)dt=(a−bf)T−(bg2)T2−(a−bf)t+(bg2)t2. Let Y = retailer’s
profit per inventory cycle. Y is given by (3.4)Y=∫0T[P2(t)−C]D2(t)dt−IC∫0TX(t)dt−S. When simplified, this yields (3.5)Y=(a−bf)(f−C−ICT2)T+(a−2bf+bC)gT22+bg(IC−g)T33−S. Thus, the retailer’s annual profit under this model, Z2, is given by (3.6)Z2=YT=(a−bf)(f−C−ICT2)+(a−2bf+bC)gT2+bg(IC−g)T23−ST. To derive the
first-order conditions for the maximization of Z2, differentiating Z2 with respect to f and equating it to
zero, we obtain (3.7)f*=(12)(ab+C)+(IC4−g*2)T. Differentiating Z2 with respect to g and equating
it to zero, we obtain (3.8)f*=(12)(ab+C)+(IC3−2g*3)T. From (3.7) and (3.8), we can see that the optimal values of g and f are given by (3.9)g*=IC2,f*=(12)(ab+C).An interesting thing to note here is that the
optimal value of the price increase per year,g*, is one half of the cost of carrying one unit for
one year. In other words, the retailer’s optimal selling price at the beginning
of the cycle will be f* while
his optimal selling price for a unit at the end of the cycle will be (f*+ICT/2).
This makes intuitive sense. A unit in stock at the beginning of an inventory
cycle, if unsold, will be carried (on an average) over one half of the duration
of the cycle. Hence, it would cost the retailer ICT/2 to carry it, whereas a unit in stock at the end
of the cycle will not incur any additional carrying cost since it will be simply sold.
Hence, the retailer can afford to charge the amount ICT/2 less for a unit at the beginning of the cycle (and
save that carrying cost if the unit sells immediately) compared to the price
for a unit at the end of the cycle. In the process, the retailer benefits from
the greater demand generated by that lower price in the early part of an
inventory cycle.DifferentiatingZ2 with respect
to T, equating it to zero, and substituting the optimal values of f and g from (3.9), we see that, in this model, T satisfies the 3rd-degree polynomial (3.10)T3+[3(bC−a)2bIC]T2+6SbI2C2=0.Given that (3.10) is a cubic
equation, we have three possible roots for the value of T. Of course, we
would be interested in only the real root(s). As we said in the context of the
fixed price model, there is no simple closed form solution to obtain the real
root(s) of a cubic equation. Hence, in our numerical examples, we rely on Excel
Solver to obtain the real and optimal solutions to this problem. As in the case
of the fixed price model, we input several feasibility conditions and provide
reasonable starting values for the decision variables. We are happy to report
that, in our experimentation with this model also, Excel Solver has never
failed to return the best real solution (if it exists). Hence, we believe that
practicing managers would be adequately served by the use of Excel Solver.We tried to verify whether the second-order
conditions of this model (not presented here) fulfil the requirements for a
local maximum of the annual profit. Unfortunately, the results are not conclusive.
They indicate either a local maximum or an inflection point. Thus, once again, in
theory, we will have to first obtain a numerical solution toT and then
verify if that solution gives a local maximum for the retailer’s profit by
perturbing the solution value of T and checking its impact on the profit
value. In our numerical experiments with this model, the solution given by
Excel Solver was always a local optimum (if it existed) rather than an inflection
point.Note that ifT2 is a positive solution of (3.10), substituting
optimal f*,g*, and T2 in (3.6), we
get Z2, the optimal profit from this model, as (3.11)Z2=(b4)[(ab)−C]2+(b4)[(ab)−C]ICT2+(b4)(ICT223)−(ST2).Now let us
prove thatZ2 of
(3.11), the retailer’s optimal profit from the continuously increasing price
model, is greater than or equal to Z1 of (2.5), the
retailer’s optimal profit from the fixed price model.


## 4. Proof ofZ2≥Z1

To simplify
(2.4) and (3.10), let us define (4.1)u=2(a−bC)bIC,v=8SI2C2b.Note thatv is clearly greater than zero. Also, we
have noted that for both D1
and D2
to be positive, we
must have a>bC. It follows that u>0.Now we can
rewrite (2.4) and (3.10) as
2.4′T3−uT2+v=0,3.6′T3−34uT2+34v=0. We are interested in the real positive roots of (2.4′) and (3.6′) since the
maximizing cycle time T for the
profit functions (2.1) and (3.6) are among these roots, respectively.The following lemmas present some interesting results on
these positive real roots. For convenience, let us designate the left-hand side
of (2.4′) as ϕ1 and
the left-hand side of (3.6′) as ϕ2.Lemma 4.1.
Equation (3.6′) has  two distinct positive real roots
if and only if u3>12v.Proof.
From the first- and second-order conditions for the maximization ofϕ2, we see that ϕ2 has
a local maximum at T=0 and a local
minimum at T=(u/2). Note that at T=0, the polynomial ϕ2(0)=(3/4)v>0.
Also, limT→∞ϕ2(T)=∞.
Since the only local minimum ofϕ2(T) is at T=u/2>0 with value ϕ2(u/2)=(−u3/16)+(3/4)v,
we see that (3.6′) can have two distinct positive real roots if and only if (−u3/16)+(3/4)v<0,
that is, if and only if u3>12v.Lemma 4.2.
Ifu3>12v,
then (2.4′) has two distinct positive real roots.Proof.
The polynomialϕ1(T)=T3−uT2+v behaves similar to ϕ2(T) at points T = 0 and infinity. Also, ϕ1(T)  has a local minimum only at T=(2/3)u.
The local minimum of ϕ1(T) is ϕ1((2/3)u)=(−4u3/27)+v,
and with u3>12v,
this minimum is negative. Thus, in this case, (2.4′) has two distinct positive real roots.
Observe that the functionsϕ1 and ϕ2 provide the same type of information about Z1 and Z2 as the first derivatives of Z1 and Z2 do. Figure 1 shows the general forms of ϕ1 and ϕ2 as functions of T.
As can be seen from Figure1, at the smaller roots, T1 and T2,
of (2.4′) and (3.6′), respectively, ϕ1 and ϕ2 are declining with an
increase in T. Thus, at T1 and T2 we have local maxima of the profit functions Z1 and Z2, respectively. At the larger roots of (2.4′) and (3.6′), ϕ1 and ϕ2 are increasing which
indicates local minima of Z1 and Z2 at those values of T. Since our research is focused on the
maximization of Z1 and Z2, in the remainder of this
paper, we only consider the smaller positive roots of (2.4′) and (3.6′).Figure 1
ϕ1 and ϕ2
as functions of T.We next state a lemma that gives a lower bound for these
roots.Lemma 4.3.
Ifu3>12v and T1 and T2 are the smaller of the two positive roots of (2.4′)
and (3.6′), respectively, then v/u<T1,T2.Proof.
Sinceu3>12v,
then v/u<(1/2)u<(2/3)u.
Consider the interval (v/u,(2/3)u);
note that ϕ1(v/u)>0 and ϕ1((2/3)u)<0.
Since ϕ1(T) is continuous on the interval (v/u,(2/3)u),
we see that the root T1∈(v/u,(2/3)u).
Therefore, v/u<T1.
We can similarly prove thatv/u<T2.Lemma 4.4.
Ifu3>12v and T1 and T2 are the smaller of the two positive roots of (2.4′)
and (3.6′), respectively, then T1<T2.Proof.
We prove this lemma by contradiction.
Conditionu3>12v insures the existence of two positive roots of
(2.4′) and (3.6′).
SupposeT2≤T1, then   (4.2)ϕ1(T2)=T23−uT22+v=T23−34uT22−14uT22+34v+14v=(T23−34uT22+34v)−14uT22+14v=φ2(T2)−14uT22+14v=−14uT22+14v, where ϕ2(T2)=0.
BecauseT1 is the smallest positive root of ϕ1(T),
and T2≤T1,
we must have ϕ1(T2)≥0,
which implies that −(1/4)uT22+(1/4)v≥0,
but this requires that T2≤v/u which contradicts Lemma 4.3. Therefore, T1<T2.Theorem 4.5.
Supposea>bC and 8(a−bC/bIC)3>96S/I2C2b.
LetT1 be the
smallest positive real root of (2.4′), and let T2
be the smallest positive real root of (3.6′), then the
maximized profit of (3.11) is larger than the maximized profit of (2.5).Proof.
To show that differenceZ2−Z1 is positive, we need to show that (4.3)Z2−Z1=b4(a−bCb)IC[T2−T1]+b4I2C2[T223−T124]+S[1T1−1T2]>0. For that to be true, it suffices to show that T1<T2.
Using the substitutionu=2(a−bC/bIC),
and v=8S/I2C2b in (2.4′) and (3.6′), we can apply Lemma 4.4 to see
that T1<T2.
Thus, we have established that the maximized profit under the
continuously increasing price strategy is larger than the maximized profit
under the fixed price strategy.Corollary 4.6.
LetP1* be the optimal price given by (2.2). For f* and g* given by (3.9),  respectively, and the optimal times T1
and T2 in Theorem 4.5, one has the inequality f*<P1*<f*+g*T2.Proof.
From (2.2) and (3.9), we see that P1*−f*=ICT1/4>0. Thus P1*>f*.
On the other
hand, sinceT2>T1, (4.4)f*+g*T2−P1*=12(ab+C)+IC2T2−12(ab+C)−ICT14=IC4(2T2−T1)>0. Corollary 4.6 implies that the range of optimal prices over the
inventory cycle in the continuously increasing price model includes the optimal
price for the fixed price model. The numerical example in Table 1 below also
confirms the above result.Table 1
A numerical example.
Assumptions common to both modelsC = $7/unit;a=50,000 units/year;b=5000 units/year;S = $400/order;I = $40/dollar/yearOptimal decisions and consequences under the two modelsFixed price modelContinuously increasing price modelPercent difference between the two modelsOptimal decisionsCycle time (T)0.2053 years0.2093 years1.95%Price at the beginning of the cycle (f)$8.64/unit$8.50/unit−1.62%Price increase rate per year (g)None$1.40/unitNAConsequencesOrder quantity (Q)1392 units/order1416 units/order1.72%Price at the end of the cycle (=f+gt)$8.64/unit$8.79/unit1.74%Annual demand6790 units/year6775 units/year−0.22%Profit per cycle [Y(T)]$1487.96/cycle$1524.47/cycle2.45%Profit per year (Z)$7249.24/year$7284.32/year0.48%


## 5. Numerical Example: Base Case

Consider
a situation where the retailer’s cost of a product is $7 per unit. The
theoretical maximum annual demand is 50,000 units and annual demand declines at the
rate of 5000 units for each dollar’s increase in the price. The ordering costs
are $400 per order and the carrying costs are $0.40 per dollar of inventory per
year. We will refer to this set of assumptions as the base case.Table1 summarizes the base case assumptions, the optimal decisions and the consequences
under the two models. As can be seen there, in the fixed price model, the
optimal retail price is $8.64 per unit and the optimal inventory cycle time is
0.2053 years. This means that the retailer would order 1392 units per order and
would obtain a per cycle profit of $1487.96. The retailer’s annual demand is 6790 units and his profit under this strategy is $7249.24 per year.In
the continuously increasing price strategy model, the optimal retail price is
$8.50 per unit at the beginning of the cycle and that price increases at the
rate of $1.40 per year. The optimal cycle time is 0.2093 years. This means that
the retail price at the end of the cycle is $8.79 per unit, the retailer would
order 1416 units per order, and his per cycle profit under this strategy would
be $1524.47. The
retailer’s annual demand would be 6775 units and his annual profit would be $7284.32.In
addition to reporting these numbers, Table1 also presents the percent
differences between the two models for each relevant decision and consequence.
Observe that, in percent terms, the differences are small. In comparison with
the fixed price strategy, the continuously increasing price strategy results in
a slightly longer cycle time. At the beginning of an inventory cycle, under the
continuously increasing price model, the retail price is smaller than what it
is under the fixed price model. However, by the end of the cycle, the retail
price under the continuously increasing price model is larger than what it is
under the fixed price model. As a result, the annual demand is smaller under
the continuously increasing price model. The per cycle profit shows a 2.45%
improvement in the continuously increasing price model compared to the per
cycle profit under the fixed price model. However, the annual profit is only
0.48% greater under the continuously increasing price model. Although this
increase in the annual profit is modest, it is clear that, for an e-tailer, the
continuously increasing price strategy is more profitable than the fixed price
strategy. Furthermore, in a competitive market, every modest increase in profit
is desirable.Arguably, one may wonder if the magnitude
of the profit difference between the two models is worth all the fuss. Similarly,
one may also argue that the continuously increasing price strategy model may be
costlier to implement because it is more complex and because it would call for
additional efforts in communicating the pricing strategy to the consumer.
However, in a computerized environment, this model is no more difficult to
program than the fixed price model, and for an e-tailer to quote a
time-dependent price is no more costly than quoting a fixed price when a
customer is shopping at the e-tailer’s website.The
specific numerical results we have obtained are a function of the numerical
assumptions we have made. Hence, in order to identify the circumstances under
which the continuously increasing price strategy would be particularly
desirable, we carried out a sensitivity analysis, as described in the following
section.


## 6. Sensitivity Analysis

Table2 summarizes the results of an analysis where we increased the value of each
one of our parameters, one at a time, by 10% while maintaining the values of
the other parameters constant. In each case, Table 2 shows the consequences of
these changes on the retailer’s annual profits under the two models and the
percentage increase in the annual profit that the retailer obtains by using the
continuously increasing price strategy as against using the fixed price
strategy. For comparison purposes, the first row of Table 2 repeats the profit
results of the two models in the base case.Table 2
Sensitivity analysis.
Changed assumption (s)A comparison of the annual profit under the two modelsFixed price modelContinuously increasing price modelPercent difference between the two modelsNone (base case)$7249.24/year$7284.32/year0.48%C = $7.70/unit$2993.58/year$3048.31/year1.83%a=55,000 units/year$15,339.34/year$15,364.48/year0.16%b=5500 units/year$2568.27/year$2623.70/year2.16%S = $440/order$7059.27/year$7098.11/year0.55%I = $0.44/dollar/year$7059.27/year$7098.11/year0.55%While other things
remaining unchanged from the base case, when the retailer’s unit cost for the
product goes up by 10% (from $7 per unit to $7.70 per unit), a retailer who
uses the fixed price strategy would obtain an annual profit of $2993.58 while a retailer
who uses the continuously increasing price strategy would realize a profit of $3048.31. Thus, in this
situation, the continuously increasing price strategy affords an increase of
1.83% over the profit provided by the fixed price strategy. When these results
are compared with the base case results, we can conclude that, while other things remaining
the same, when a retailer’s product cost increases, the continuously increasing
price strategy becomes substantially more desirable to use.Table2 also indicates that, while
other things remaining unchanged, if the theoretical maximum demand increases
by 10%, the advantage of the continuously increasing price strategy reduces
substantially to only 0.16% compared to the profit a retailer can make using
the fixed price strategy. On the other hand, when demand elasticity increases
by 10%, the continuously increasing price strategy seems to be the most
desirable compared to all the cases of sensitivity analysis we have presented.
In this case, it provides an annual profit that is 2.16% higher than the profit
provided by the fixed price strategy. Other things remaining unchanged, when the ordering cost increases by 10%, the advantage of the continuously increasing price strategy goes up slightly to 0.55% compared to its base case advantage of 0.48%. In terms of the retailer’s profits from the two strategies, the impact of a 10% increase in inventory carrying costs is identical to that of a 10% increase in the ordering costs.Thus,
while other things
remaining the same, high price elasticity favors the continuously increasing
price strategy the most, followed by high unit cost and then by high ordering
and carrying costs. It seems appropriate to conclude that the continuously
increasing price strategy is particularly desirable for e-tailers of undifferentiated commodities
(high price elasticity), and e-tailers whose suppliers have considerable
pricing power (high product cost). It should also be attractive to e-tailers of
imported products (high ordering costs), e-tailers who also manufacture the
product (high set up costs), e-tailers of perishable products (high carrying
costs), and e-tailers of products subject to sudden obsolescence (high carrying
costs). Note that Rajan et al. [18] have studied the case of perishable
products in greater depth. Our findings are consistent with Rajan et al.’s [18]
finding that perishable products that only deteriorate but do not lose any value call for an increasing price within an inventory cycle. However, Rajan et al.
[18] also found that if a perishable product loses value (i.e., the product
simply could not be sold at its original selling price), then it may be optimal
to reduce its price as time increases within an inventory cycle. We have not
considered the case of products losing any value. On the other hand, we have
shown that even for a nonperishable product, the increasing price strategy is
optimal within an inventory cycle.Of course, our
sensitivity analysis focused on changes in one parameter at a time. When
several parameters are favorable to the continuously increasing price strategy,
the gains offered by this strategy need not remain modest.


## 7. Conclusion

Traditionally, operations researchers (Whitin
[1], Kunreuther and
Richard [2], and others) have assumed
that when a product’s demand curve is known and stationary, a retailer of the
product would find it optimal to buy a fixed quantity every time he buys and to
sell the product at a fixed price throughout the year. We find that this
assumption leads to suboptimal pricing policy. Instead, we have shown that,
within each inventory cycle, a continuously increasing price strategy is more
desirable than the fixed price strategy. Our model resulted in an optimal
increase per year of one half of unit holding cost, which made intuitive sense.A continuously increasing price strategy
might have been deemed impractical in the past. However, today’s e-tailers can
easily adopt this strategy. Elmaghraby and Keskinocak’s [17] review of
dynamic price models indicates that a number of industries are already using
continuously changing pricing strategies.Our numerical example suggests that often the
advantage of a continuously increasing price strategy is only modest. However,
sensitivity analysis shows that this strategy is particularly desirable when
demand is highly price sensitive or when an e-tailer’s supplier commands great
pricing power. E-tailers facing high inventory ordering and carrying costs will
also find this strategy fairly attractive.While the continuously increasing price
strategy may not be practical for a brick-and-mortar retailer, such a retailer
could use the dual price strategy model developed by Joglekar [3]. Contending that, during any
inventory cycle, the retailer’s inventory carrying costs are a declining
function of time, Joglekar [3] showed that a retailer who sets two
different prices at two different points in an inventory cycle obtains a
greater profit than a retailer using a single fixed price throughout the cycle.
Our numerical analysis of Joglekar’s [3] model (not presented here) showed that a continuously increasing price
strategy is superior to his model. However, Joglekar’s [3] model’s gains are only slightly inferior to
the model we have presented here. Hence, a brick-and-mortar retailer may be
well served by that model.There are several directions in which this
research can be extended. First, we have assumed that the e-tailer obtains the
product from a vendor. The model can be easily extended to a situation where
the e-tailer is also a manufacturer of the product. We have already cited
numerous works that have continued to use the single price assumption. Clearly,
each one of those models needs to be revisited in light of our findings here.
Finally, a number of recent models considering the coordination pricing and order quantity decisions across a supply chain [19–22] have assumed a fixed price for each participant of the supply chain. These models also need to be updated by considering a continuously
increasing price strategy.



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*Source: 483267-2008-05-25.xml*