MANUFACTURING & SERVICE
OPERATIONS MANAGEMENT
informs
Vol. 10, No. 1, Winter 2008, pp. 1–18
issn 1523-4614 eissn 1526-5498 08 1001 0001
®
doi 10.1287/msom.1060.0140
© 2008 INFORMS
Structural Properties of Buyback Contracts for
Price-Setting Newsvendors
Yuyue Song
Faculty of Business Administration, Memorial University of Newfoundland, St. John’s, Newfoundland A1B 3X5, Canada,
[email protected]
Saibal Ray, Shanling Li
Desautels Faculty of Management, McGill University, Montreal, Quebec H3A 1G5, Canada
{[email protected], [email protected]}
T
his paper studies a buyback contract in the Stackelberg framework of a manufacturer (leader) selling to
a price-setting newsvendor retailer (follower). Using an analytical model that focuses on a multiplicative
demand form, we generalize previous results and produce new structural insights. A novel transformation
technique first enables us to establish the unimodality of the profit functions for both channel partners, under
relatively mild assumptions. Further analysis identifies the necessary and sufficient condition under which the
optimal contract for the manufacturer (wholesale and buyback prices) is distribution free, i.e., independent of
the uncertainty in customer demand. A specific instance of the above condition is also necessary and sufficient
for a no-buyback contract to be optimal from the manufacturer’s perspective. We then prove that the optimal
performance of the decentralized channel for distribution-free buyback contracts depends only on the curvature
of the deterministic demand part. In addition, some of the optimal decisions and relevant profit ratios for
buyback contracts in our setting are shown to be identical to those for their deterministic price-only counterparts.
Key words: price-setting newsvendor; buyback contract; supply chain performance; demand curvature
History: Received: November 12, 2004; accepted: October 12, 2006. Published online in Articles in Advance
December 11, 2007.
1.
Introduction
price-independent, end customer demand with order
quantity as the only retail decision (e.g., Pasternack
1985, Lariviere 1998). The contrast between the two
paradigms, then, is obvious.
The most common contract in practice is a priceonly type, in which the manufacturer charges a perunit wholesale price for the quantity ordered by the
retailer. Such contracts have been analyzed before
in Stackelberg frameworks, with the manufacturer as
the leader and the fixed-price newsvendor retailer as
the follower. These models clearly show that coordination and arbitrary allocation of profits among the
channel partners cannot be achieved concurrently in
such settings. The optimal decisions and the performance of the decentralized channel then depend crucially on the coefficient of variation of the random
demand (Lariviere and Porteus 2001). A number of
contracts have been proposed by researchers that can
attain coordination and arbitrary division of profits
among the channel partners in “selling to a fixed-price
newsvendor” paradigm (Cachon 2003, §2).
Recent operations management (OM) literature has
focused a great deal on developing effective contracting schemes between channel partners in decentralized settings. In particular, there is a growing
interest in modelling realistic contracts like price-only,
buyback, and consignment/revenue-sharing (refer to
Cachon 2003 for more details). The widespread use of
these contracts can be attributed to their simplicity in
terms of administration, compared to complex coordinating contracts.1 One particularly interesting feature
of some of these contracting models is that they are
based on the framework of price-sensitive and stochastic end customer demand (Yano and Gilbert 2003).
Their particular demand characteristics require retailers to simultaneously decide on the optimal price and
order-quantity levels. Note that the traditional fixedprice newsvendor framework assumes stochastic, but
1
Coordinating contracts are those that allow a decentralized chain
to attain the profit performance of a centralized one (Lariviere 1998).
1
2
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
One of the most popular of these coordinating contracts is the buyback type. In buyback contracts, the
retailer still pays the wholesale price but is assured
of some financial restitution from the manufacturer
in the form of a per-unit buyback price (less than
wholesale price) for any unsold items at the end
of the season. Buybacks have been used extensively
in sectors like publishing, fashion apparel, and cosmetics (Kandel 1996, Emmons and Gilbert 1998). In
this case, contrary to price-only contracts, the manufacturer shares the risk of a poor demand outcome.
Pasternack (1985) and Lariviere (1998) analyzed this
contract in model settings similar to the one for priceonly contracts above. The profit function for the manufacturer is then complicated and not well behaved
(Lariviere 1998, Theorem 6). However, they show that
a properly designed buyback contract induces the
retailer to order the same as under the centralized
system, and coordination ensues. Depending on the
values of the contract parameters, the optimal total
channel profit can also be allocated arbitrarily among
the partners. Moreover, the channel-coordinating contract parameters are independent of the randomness
in end customer demand.
In this paper, we also analyze a buyback contract in
a two-echelon Stackelberg framework, but for a pricesetting newsvendor retailer who faces price-sensitive
stochastic demand. The retailer bases his optimal
price and order-quantity decisions on the buyback
contract (wholesale and buyback prices) offered by
the manufacturer before any uncertainty is resolved.
It has already been shown that such a contract cannot attain coordination in our setting (Bernstein and
Federgruen 2005, Cachon 2003). Hence, we do not
address coordination issues in this paper. Instead, we
concentrate on identifying some structural properties
of the optimal decentralized system when the retail
demand can be expressed in a multiplicative form.2
We are specifically interested in: (i) Characterizing the
behavior of the manufacturer’s profit function and its
optimal contract parameters, (ii) comparing the optimal decisions and profits of the centralized system
to those of a decentralized one, and (iii) comparing
the optimal profits of the two channel partners in a
decentralized setting.
The above issues have been previously tackled in
a similar framework by Granot and Yin (2005, henceforth referred to as G&Y). They assume certain specific
deterministic demand functions and a uniformly distributed random part for their multiplicative demand.
G&Y then prove that the manufacturer’s profit function is well behaved for each demand form, and
present the associated optimal decision values for
both channel partners. It is clear from their analysis
that, for isoelastic demand function, buybacks should
not be introduced by the manufacturer at optimality. G&Y also ascertain the optimal efficiency3 and the
optimal profit allocation of the decentralized system
for each of their demand settings. The multiplicative
demand in our analysis, on the other hand, covers
a much wider range of deterministic functions and
randomness distributions compared to G&Y. Consequently, we are able to generate analytical results
with greater scope and applicability. Moreover, we
integrate various demand-specific insights from G&Y
(and other related models, refer below) and extend
them to develop fairly broad structural properties
of optimal buyback contracts. In that sense, our primary contributions are in unifying previous results
and in pushing the envelope of generality within this
area. We also contribute methodologically to the literature by proposing a change of variables, based on
a one-to-one relationship, so that the manufacturer’s
profit function can be rewritten in terms of retail decisions, i.e., what decisions to induce from the retailer.
This transformation procedure helps us to demonstrate that the profit function for the manufacturer is
well behaved even in our extended demand setting.
Specifically, there is always a unique contract that will
induce the optimal retail response in terms of end
customer price and order quantity from the manufacturer’s perspective.
More importantly, we derive a necessary and sufficient condition on the shape of the deterministic demand form under which the manufacturer’s optimal
contract decisions are distribution free. A particular
instance of the above condition is also necessary and
2
Multiplicative (additive) demand = product (sum) of a pricesensitive deterministic function and a nonprice-sensitive positive
random variable.
3
The ratio of the optimal total profit of a decentralized chain
(retailer + manufacturer) to that of a centralized one.
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
sufficient for buybacks not to be offered by the manufacturer at optimality. Many of the demand forms
in the literature, including those used by G&Y, do
satisfy our condition. However, to the best of our
knowledge, this is the first paper to recognize the
shared trait among the various demand functions and
to relate it to the nature of the optimal contract. One
of the advantages of the distribution-free property is
that a manufacturer can design optimal contracts for
multiple independent retailers without knowing their
market uncertainties.4 The optimality of no-buyback
contracts, on the other hand, may be the underlying
reason why buyback offers are not as prevalent as
expected. We also show that the optimal efficiency
and the optimal profit allocation of the decentralized chain for distribution-free buyback contracts are
defined only by the curvature of the deterministic
demand part. In fact, some of the optimal decisions
and performance ratios in our framework turn out to
be similar to those for the corresponding deterministic price-only setting. However, most of our structural
insights are valid only for buyback contracts in multiplicative demand paradigms and do not hold for
additive demand or for price-only schemes (G&Y also
conclude likewise).
In addition to G&Y, there are two other streams
of research that are related to our work. The first
one consists of models that also study buyback contracts in stochastic, price-sensitive settings. For example, Kandel (1996) notes that the resulting problem is
quite complex, and that buybacks may be unable to
achieve coordination. Padmanabhan and Png (1997),
and subsequently Wang (2004), study the joint effect
of demand uncertainty and retail competition on
manufacturer profitability for such contracts. Marvel
and Peck (1995) deal with valuation uncertainty and
demand uncertainty simultaneously in their model.
They illustrate that only valuation uncertainty results
in price-only contracts from the manufacturer, while
only demand uncertainty encourages the manufacturer to also offer buybacks. Note that there is a
fundamental distinction between the demand frameworks of Marvel and Peck (1995) and most papers in
4
Note that if a manufacturer offers such a contract, it may imply
its willingness to enter into a contract without precise knowledge
of optimal profit levels, which still depend on the degree of uncertainty (Lariviere 1998).
3
the OM literature, including our own. The OM literature normally assumes that the number of arrivals,
i.e., demand, is random and/or price dependent, but
that all arrivals buy the product. However, the number of arrivals and the number of purchases are independent for Marvel and Peck (1995). Specifically, the
random number of arrivals is independent of the
retail price, while the purchase decision of a customer, once she or he arrives, depends on whether
the retail price is less or more than the customer’s
valuation of the product.5 Recently, Bernstein and
Federgruen (2005) have shown that buybacks cannot coordinate a setting similar to that in this paper
unless the manufacturer’s decisions are made contingent on the retail price chosen. Nevertheless, for
a specific multiplicative demand form, Emmons and
Gilbert (1998) demonstrate that if the wholesale price
is set sufficiently high, but not necessarily optimally,
both channel partners may be better off with buyback
contracts than with price-only ones. Note that—unlike
our paper—none of the above papers addresses the
structural characterization of the contract and performance of the optimal decentralized system.
The other related research stream addresses issues
similar to our research but focuses on different contracting schemes. The primary examples of this stream
are Wang et al. (2004) and Petruzzi (2004). Both of
these papers are also based on decentralized pricesetting newsvendor frameworks with multiplicative
retail demand forms. Wang et al. (2004) analyze consignment6 contracts in this setting for an isoelastic deterministic demand function. They are able to
determine the optimal decision variable values for
both channel partners and evaluate the optimal performance of the decentralized system. The primary
contribution of Petruzzi (2004) lies in his comparison of the optimal decisions and profit performance
of consignment contracts with those of price-only
schemes. In contrast to the above two papers, we
5
Refer to G&Y (p. 767) for more details. We compare our results to
those of Marvel and Peck (1995) in §2.
6
In such contracts, the retailer offers to deduct a percentage from
the selling price and to remit the balance to the manufacturer. On
this basis, the manufacturer decides on the optimal price and stocking level for the end consumers.
4
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
focus on buyback contracts. We demonstrate the similarities and differences among optimal buyback, consignment, and price-only contracts in §2.4, based on
our numerical study.
1.1. Model Framework
Our basic model framework involves a manufacturer
selling to a price-setting newsvendor retailer in a
risk-neutral Stackelberg setting. The retail demand
function Xp comprises two elements: dp, a deterministic function of the retail price p; and , a positive random variable having support on L U (L ≥ 0
and U ≤ ), with mean , density f u, and distribution F u. For analytical convenience, we assume
f u to be positive for u ∈ L U , and zero otherwise.
Moreover, suppose that limu→0+ f u > 0 if L = 0. Any
unmet demand for the retailer is lost. The buyback
agreement between the two parties consists of the
manufacturer offering to charge a per-unit wholesale
price w for the quantity ordered, and paying a buyback price of b per unit of leftover inventory at the
end of the selling season to the retailer. Based on these
contract parameters, the retailer then chooses its optimal retail price p and the optimal order quantity y
before the onset of the selling season. The production
cost for the manufacturer is c per unit. Notice that, in
our setting, w and b do not explicitly depend on p;
furthermore, p > w > c > 0 and w > b ≥ 0 for realistic solutions. For expositional simplicity, suppose
that there is no salvage value for the leftovers, and
that any unmet demand has no associated goodwill
penalty other than the foregone revenue p. We briefly
discuss the effects of relaxing the last two assumptions in §2.5.
Before proceeding to the detailed analysis, we present the following mild assumption about . Note that
throughout this paper we use increasing/decreasing
in the weak sense, unless otherwise stated. As in
Lariviere and Porteus (2001), we define gz = zf z/
1−F z as the generalized failure rate of the demand
distribution for , and assume that:
Assumption 1. gz is increasing for z ∈ L U .
The above increasing generalized failure rate (IGFR)
property of is indeed satisfied by most of the theoretical distributions used in the previous literature
(refer to Lariviere and Porteus 2001 and Lariviere 2006
for details).
The organization of the remainder of this paper is
as follows. Building on the above framework, in §2
(except §2.6) we analyze the multiplicative retail demand scenario Xp = dp, which is widely used
for studying price-setting newsvendor paradigms
(Petruzzi and Dada 1999, Wang et al. 2004, G&Y,
and references therein). Specifically, §§2.1, 2.2, and
2.3 examine the characterizations of profit functions,
optimal decisions, and optimal performance measures,
respectively, for both centralized and decentralized
systems. In §2.4, we compare the optimal decisions and
profits for buyback contracts with those for optimal
price-only and consignment schemes, while in §2.5 we
study the effects of positive salvage value/goodwill
penalty on our model. Section 2.6 uses a numerical
example to illustrate that the results for multiplicative demand are, in general, not valid for the additive
form, i.e., Xp = dp + (Petruzzi and Dada 1999,
G&Y). Our concluding remarks and future research
directions are presented in §3. Note that the detailed
proofs for all theorems and propositions are provided
in the appendix.
2.
Model Analysis
Throughout this section, with the exception of §2.6,
we assume the multiplicative demand form Xp =
dp for our analysis. Also, without loss of generality,
suppose that the mean of for multiplicative demand
is one, i.e., = 1. Let P u be the lowest positive retail
price, if any, such that dP u = 0. If dp > 0 for all
p > 0, we define P u = +. Thus, the feasible range of p
is 0 P u . In addition, we assume that the demand
function dp exhibits the following properties:
Assumption 2. The demand function dp is positive
and strictly decreasing for p ∈ 0 P u and limp→P u pdp =
0. The elasticity p = −pd p/dp of the demand
function is increasing for p ∈ 0 P u . Moreover, p/p
is monotone (can be either increasing or decreasing) and
convex, while p1 − 1/p is strictly increasing for p ∈
0 P u .
The above assumption is directly related to the curvature of the demand function, which is defined as
Ep = dpd p/d p2 (Bresnahan and Reiss 1985,
Cowan 2004).7 For example, increasing p and
7
Ep can be defined as the elasticity of the slope of the inverse
demand pd, i.e., Ep = −dp
d/p
d =dpd p/d p2 ).
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
strictly increasing p1 − 1/p are equivalent to
Ep ≤ 1 + 1/p and Ep < 2, respectively. Similarly,
Ep ≤ 1 Ep ≥ 1 and E p ≥ 0 are synonymous
with p/p being increasing (decreasing) and convex, respectively. Assumption 2 basically implies that
the curvature of dp should not be highly positive
and it should increase in p. Such an assumption is
common in the literature (Cowan 2004 and references
therein), and a large family of dps satisfies it. Examples include:
• dp = ap−k a > 0 k > 1, dp = a − kp a > 0
k > 0 > 0, dp = ae−kp a > 0 k > 0
• dp = a − pk a > 0 k > 1, dp = a − ekp a > 0
k > 0.
However, demand functions like dp = a − k lnp
a > 0 k > 0 do not satisfy Assumption 2. As will
become evident later on, the curvature of the demand
function plays an important role in our analysis.
2.1. Characterization of the Profit Functions
For any given buyback contract w b offered by the
manufacturer, the retailer needs to determine its optimal order quantity y and the optimal retail price p.
The retailer’s expected profit is given by
y
p y = pdp 1 − dp
y
+b
dp − wy
(1)
dp
+
z
where z = z u − zf u du and z = 0 z − u ·
f u du are defined based on Petruzzi and Dada
(1999).8 In (1), the first term represents the expected
sales revenue from consumers dpy/dp is the
expected shortage quantity), the second term denotes
the expected buyback revenue dp y/dp is the
expected leftover amount), and the third term is the
purchase cost. The expressions z and z simplify to
z
z = zF z −
uf u du and
0
(2)
z
z = 1 −
uf u du − z!1 − F z"#
0
8
Note that f z = 0 for any z ∈ !0 L" ∪ !U +. Hence, for expositional convenience, we use 0 in place of L and for U as the limits
of the integration without affecting the analysis.
5
If z represents the stocking factor =y/dp, then
z!1 − F z"/1 − z is the elasticity of expected
sales with respect to the stocking factor, i.e., percent increase (decrease) in expected sales from percent increase (decrease) in the stocking factor (refer
to Petruzzi 2004). For the purpose of exposition,
we introduce the expression V z = 1 − z!1 − F z"/
1 − z−1 ∀ z ∈ L U , which is a one-to-one function of the above elasticity. Taking partial derivatives
of the retailer’s expected profit function with respect
to p and y, we then obtain the following:
y/dp
% py
y
d p
uf udu V
= dp
+p −b
%p
dp
dp
0
% p y
y
y
=p 1−F
+ bF
− w#
%y
dp
dp
Before proceeding to the detailed characterization of
p y, we present a result concerning the location of
the maximizer of p y that helps us to establish the
feasible ranges for p and y.
Lemma 1. Any maximizer p∗ y ∗ of p y must satisfy L < y ∗ /dp∗ < U .
In the sequel, we assume y/dp ∈ LU . The feasible range of y is then 0d0+ U , and for any given
y ∈ 0d0+ U , the feasible range of p is d −1 y/L
d −1 y/U . If L = 0, we then define d −1 y/L = 0, and if
U = +, let d −1 y/U = +.
Theorem 1. Under Assumptions 1 and 2, for any given
order quantity y ∈ 0 d0+ U , the retailer’s expected
profit function p y is unimodal in p. There exists a
unique retail price py that maximizes p y, and py
is strictly decreasing. Moreover, py y has a unique
maximizer.
Therefore, for any given w b, the corresponding
optimal p y should satisfy the first-order conditions
% p y/%p = 0 and % p y/%y = 0. These two equations can be simplified to express w b in terms of
optimal p y as follows:


1
y
y
1
−
V
F

dp
dp 
p
w


=p
(3)
#


b
y
1
V
1−
p
dp
The manufacturer’s goal is to maximize its expected
profit, w − cy − bdp y/dp, by optimally selecting the wholesale price w and the buyback price b,
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
6
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
while keeping in mind the retailer’s optimal actions.
Researchers normally try to solve the manufacturer’s
maximization problem directly in terms of w and b
(e.g., G&Y). This involves using % p y/%p = 0 and
% p y/%y = 0 to express the optimal p and y in
terms of w and b, substituting them in the manufacturer’s profit function, and then optimizing over w
and b. The process is quite cumbersome, and oftentimes results in a messy analysis. To circumvent this
problem, we use (3) to express the manufacturer’s
maximization problem in terms of p and y. Substitution of (3) in w −cy −bdp y/dp and subsequent
simplification yield the following tractable form for
the manufacturer’s profit function:
1
y
1−
− cy# (4)
&p y = pdp 1 − dp
p
Taking partial derivatives with respect to p and y, we
obtain the following first-order conditions for &p y:
%&p y
= dp
uf u du
%p
0 y
p V
· 1−
+ 1 − p p
dp
%&p y
1
y
=p 1−F
1−
− c#
%y
dp
p
y/dp
Theorem 2. Under Assumptions 1 and 2, for any given
y ∈ 0 d0+ U , there exists a unique retail price P y
that maximizes &p y, and P y is strictly decreasing.
Furthermore, &P y y has a unique maximizer y D .
Note that the determinant of the Hessian of the
manufacturer’s profit function is always negative
for a buyback contract associated with a fixed-price
newsvendor retailer (Lariviere 1998). Any structural
characterization of the profit function then becomes
difficult. However, when the retailer is a price-setting
newsvendor and the end customer demand is of the
multiplicative form, the manufacturer’s profit function exhibits a “nice” unimodal structure for an extensive family of demand forms and distributions. This
property allows us to identify the optimal contract
through standard optimization approaches, as shown
in the next section.
In a centralized system, there is a single decision
maker for both the retailer and the manufacturer, and
no financial transaction transpires between the two
parties. The channel as a whole has to decide on the
optimal retail price pC , and the optimal order quantity y C . The expected profit of the centralized system
can be expressed as )p y = pdp!1 − y/dp" −
cy, which is similar to the retailer’s profit function
in (1) with b = 0 and w = c. Hence, following Theorem 1, we have (refer also to Petruzzi and Dada 1999):
Theorem 3. Under Assumptions 1 and 2, for any given
order quantity y ∈ 0 d0+ U , the centralized system’s
expected profit )p y is unimodal in p. There exists a
unique price p̃y that maximizes )p y, and p̃y is
strictly decreasing. Furthermore, )p̃y y has a unique
maximizer y C .
2.2. Optimal Decision Variable Values
Based on Theorems 2 and 3, we derive the following
optimal decision variable values:
Corollary 1. The optimal decisions for the centralized
system are to charge a price pC to the consumers, and
to order a quantity y C , where pC y C can be obtained
from the simultaneous solutions of %)p y/%p = 0 and
%)p y/%y = 0.
The manufacturer’s optimal contract and the optimal retailer response in the decentralized system are given by:
• Retailer: Charge a retail price pD to the end consumers
and order y D from the manufacturer, where y D is the solution to d&P y y/dy = 0 and pD = P y D .
• Manufacturer: Offer the contract w D b D to the retailer, where the optimal wholesale price w D and the optimal buyback price b D can be determined by substituting
pD y D in (3). Specifically,
p + dp/d p − c and
wD = p −
1 + dp/d p
*p=pD +
(5)
dp/d p + p D
#
b = p−
1 + dp/d p
D
*p=p +
In what follows, we study the behavior of the optimal contract w D b D . According to Pasternack (1985)
and Lariviere (1998), the channel-coordinating buyback contract is independent of the demand distribution in a fixed-price newsvendor setting. Moreover,
Kandel (1996), Marvel and Peck (1995), and G&Y provide specific examples of the optimality of a no-buyback contract, i.e., b D = 0, from the manufacturer’s
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
7
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
perspective. We are interested in identifying a general condition under which the distribution-free property of the optimal contract and/or the optimality
of a no-buyback contract are valid for our modelling
framework.
Theorem 4. The manufacturer’s optimal contract parameters are independent of the demand distribution iff the
demand elasticity p = −p/A + Bp, where A and B are
constants, i.e., dp/d p = A + Bp is linear in terms of p.
Moreover, a no-buyback contract is optimal from the
manufacturer’s viewpoint iff p = constant, i.e., A = 0.
In other words, b D = 0 iff dp is isoelastic.
Based on our definition of the curvature Ep = dp ·
d p/d p2 of a demand function, if dp/d p =
A + Bp, then Ep = 1 − B ∀ p. Therefore, a distributionfree buyback contract is optimal for the manufacturer
iff the demand function has a constant curvature. Most of
the demand functions used in the literature for studying joint pricing-inventory decisions do exhibit constant curvatures (refer to Emmons and Gilbert 1998,
Petruzzi and Dada 1999, Wang et al. 2004, Petruzzi
2004, G&Y). Hence, in those cases the manufacturer
can set the optimal contract without any knowledge
about the distribution of . Some of the examples are
given in Table 1.
Note that the optimal price-only contracts for the
above demand functions may still be dependent on
the distribution of (refer to §2.4 for more details).
The optimal buyback contract for any dp that does
not satisfy the condition in Theorem 4, e.g., dp = a −
k lnp a > 0 k > 0), will also depend on the demand
distribution.
Moreover, only when dp = a!p1/B " a > 0 B < 0,
i.e., isoelastic demand function, does it make sense
for the manufacturer not to offer buyback opportunities to the retailer. The optimal buyback contract
then equates to a price-only one. For multiplicative
demand, the usual assumption about dp in the literature is actually isoelastic (Petruzzi and Dada 1999,
Table 1
Wang et al. 2004). Note that the curvature of an isoelastic demand function is a positive constant. The
optimality of a no-buyback contract seems to contradict Marvel and Peck (1995), according to whom the
manufacturer should always offer buybacks when the
uncertainty stems only from the number of arrivals.
As has been detailed by G&Y (pp. 771–772), Marvel
and Peck’s (1995) result is due to the fact that, with
only arrival uncertainty, their model is equivalent to a
fixed-price buyback model. Their result of full-credit
buyback is then the same as the complete consignment contract already proposed in the OM literature
for such cases. The numerical example in Marvel and
Peck (1995, pp. 704–712) shows that the optimal buyback price may be zero in the presence of both arrival
and valuation uncertainties when the first type of
uncertainty is low.
Most of our subsequent analysis of the multiplicative demand function (remainder of §§2.2, 2.3, and 2.4)
is based on the assumption that p = −p/A+Bp.
To satisfy Assumption 2, we require A ≤ 0 and B +
1 > 0. Depending on the values of A and B, we then
determine the following closed-form expressions for
the optimal decision variables (refer to the appendix
for detailed derivations). In Table 2, Z1 ∈ L U is the
unique solution of !1 − F z" + c/A1/V z + B = 0,
and Z2 ∈ L U is the unique solution of 1+BV z = 0.
The optimal values for commonly used demand
functions in the literature can be deduced by substituting for A and B in Table 2 (e.g., from Table 1).
We present only the detailed expressions for dp =
a − kp in Table 3, where = 1 denotes linear dp,
and Z1 ∈ L U is the unique solution of !1 − F z" −
ck/a1/V z + 1/ = 0.
We generate additional insights by analyzing the
buyback contract in terms of retail order quantity y.
Note that the manufacturer’s overall optimal contract
is the one that results in an optimal retail order of
y D and maximizes its own profit. For any arbitrary
Examples of Demand Functions Satisfying Theorem 4
Isoelastic
p
dp/d p
Ep
A
B
ap−k a > 0 k > 1
k
−p/k
k + 1/k
0
−1/k
kp
p
−
−a/k + p/
−1/k
1
−1/k
0
−a/k + p/
1 − 1/
−a/k
1/
−kp
Exponential
Linear-power
dp
ae
a > 0 k > 0
a − kp a > 0 k > 0 ∈ − −1 ∪ 0 Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
8
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
Table 2
Optimal Decision Variable Values for p = −p/A + Bp
p = −p/A + Bp
Retail order quantity
wD
bD
−AV Z1 1 + BV Z1 y C = dpC Z1
N/A
N/A
−A1 + 1 + BV Z1 1 + B1 + BV Z1 y D = dpD Z1
c − A
>0
1 + B
−A
>0
1 + B
y C = dpC Z2
N/A
N/A
y D = dpD Z2
c
>0
1 + B
0
Retail price
A<0
pC =
Centralized
pD =
Decentralized
A=0
c
1 − F Z2 c
pD =
1 + B1 − F Z2 pC =
Centralized
Decentralized
retail order quantity (say y0 ), let us denote the manufacturer’s optimal contract by wy0 by0 . This is
the contract that enables the manufacturer to induce
the optimal retail order of y0 , while also maximizing
its own profit &p y0 . Under the condition of Theorem 4, i.e., dp/d p = A + Bp A ≤ 0 B + 1 > 0, we
have (refer to the appendix for details)
A
F Zy0 1+B
A
by0 = −
1+B
wy0 = P y0 !1 − F Zy0 " −
and
(6)
where Zy0 = y0 /dP y0 is the optimal stocking factor corresponding to y0 , and P y0 is the solution to
1 + BV y0 /dp + !1 + p/A + Bp" = 0 (note that
py0 wy0 by0 = P y0 ). It is relatively straightforward to show that P y0 > wy0 > by0 . Interestingly, the optimal buyback price to induce any desired
level of retail order quantity is a constant (as long as
the corresponding retail price is set optimally). Moreover, this constant value depends only on the form of
dp, and does not require any knowledge about y0 or
the distribution of . On the other hand, the optimal
wholesale price depends on y0 , the form of dp, and
Table 3
the distribution of . Only at the optimal y0 = y D does the wholesale price become a distribution-free
constant =w D . Rather intuitively, wy0 is decreasing
in y0 , i.e., the higher the desired retail order, the lower
the optimal wholesale price charged by the manufacturer should be. Especially for isoelastic demand (i.e.,
A = 0, B = −1/k < 0), the manufacturer’s optimal contract should not involve any buyback offer, regardless of the desired retail order size. Therefore, the
optimality of distribution-free or zero-buyback prices
are valid under more general conditions than is evident from Table 2. Note that in the case of isoelastic
demand, even the elasticities of the retailer’s order
with respect to w and that of expected sales with
respect to the stocking factor are distribution-free constants (refer to the appendix).
Although the coordinating buyback contract for a
fixed-price newsvendor and the optimal contract in
our framework are both distribution free, their behaviors manifest an important difference. In the former
scenario, it has been shown that the manufacturer
only derives a profit on the units sold to the end customers, while the retailer’s true procurement cost per
unit is less than the unit production cost c (Lariviere
Optimal Decision Variable Values for dp = a − kp
Retail price
dp = a − kp
Centralized
Decentralized
V Z1 a
k + V Z1 + + 1V Z1 a
k + 1
+ V Z1 Wholesale price, w D
Buyback price, b D
Z1
N/A
N/A
Z1
a
1
+ c
+ 1 k
a
1
k + 1
Retail order quantity
a
+ V Z1 a 2
+ V Z1 Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
9
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
Table 4
Comparisons of Optimal Decision Variable Values and Profits for p = −p/A + Bp
D C
p = −p/A + Bp
pC /pD
y C /y D
/
1 + BV Z1 2+B
2
A < 0, B = 0
∈
<1
1 + B1/B > 1
1
1+1/B
1 + BV Z1 + 1
1 + B
e
A=0
A < 0, B = 0
D / D
1+B
1+B
V Z1 <1
V Z1 + 1
1998). In our setting, w D − b D = c/1 + B (refer to
Table 2). Therefore, if 0 < B + 1 < 1, then w D − b D > c. The manufacturer then not only makes a profit on
each unit sold, but also does so on each leftover item.
Clearly, the retailer’s true unit cost for procurement
is also greater than c. For example, if dp = a − kp
a > 0 k < 0 < −1, then w D − b D = c/1 + 1/. For
any < −1, w D − b D is greater than c.
2.3. Performance of the Optimal Contract
In this section, we compare: (i) The optimal decisions
and profits of the centralized system to those of the
decentralized one, and (ii) the optimal profits of the
two channel partners in the decentralized system.
Let & D = &pD y D and D = pD y D represent the
optimal expected profits for the manufacturer and
the retailer, respectively, in the decentralized scenario;
and )C = )pC y C and )D = & D + D denote the
optimal expected profits in the centralized and the
decentralized system, respectively.
Proposition 1. For p = −p/A + Bp A ≤ 0 B +
1 > 0, the optimal decision/profit comparisons are as follows in Table 4.
From Table 4 it is clear that the decentralized system is not coordinated by an optimal buyback contract. Recall that for p = −p/A + Bp, the curvature
of dp is given by Ep = 1 − B ∀ p. This implies that
the optimal efficiency )D /)C and the optimal allocation of profits & D / D in the above table are dependent
only on the curvature of dp. As the next section shows,
both of these profit ratios are exactly the same as
those for a corresponding optimal deterministic priceonly contract. In a deterministic setting, the curvature
of dp is a measure of the relative market power of
the two channel partners (Bresnahan and Reiss 1985).
Because it is the curvature that determines the price
e
2
e
1
change necessary to stimulate additional sales, it is
intuitive that this feature plays an important role in
profit measures (refer to Malueg 1994 and Cachon and
Lariviere 2005 for details). Due to the distribution-free
nature of the optimal contract, it seems that this property carries over to the stochastic buyback scenario.
A more detailed look at Table 4 reveals the following:
Corollary 2. Because B ∈ −1 , the ratio )D /)C
lies in the range !2/e 1 and is unimodal in terms of B.
The ratio is decreasing for B ∈ −1 0 and increasing for
B ∈ !0 , and attains its minimum value of 2/e at B = 0.9
On the other hand, & D / D lies in the range 0 and is
linearly increasing in B.
Because there is a one-to-one relation between B
and Ep in our setting, we can explain the above
corollary in terms of the demand curvature Ep ∈
− 2.10 When the demand curvature is high 1 <
E < 2, e.g., fairly convex isoelastic dp = ap−k k > 1,
the retailer’s optimal profit outperforms that of the
manufacturer. Specifically, for quite positive demand
curvatures (k ≈ 1 or E ≈ 2), the channel does not
lose much from decentralization, but almost all of
it is captured by the retailer. As the demand curvature decreases, the decentralized channel becomes
less efficient, but the manufacturer’s relative profit
performance improves. When the demand curvature is not too high—specifically, exponential demand
E = 1—the two partners derive exactly equal profits
and the decentralized system proves the most inefficient. For demand functions less convex than exponential (i.e., E < 1), the manufacturer becomes more
9
Note that limB→−1 )D /)C and limB→ )D /)C are both 1.
10
Concave, linear, and convex demand functions exhibit negative,
zero, and positive curvatures, respectively. Moreover, dp is log
concave iff Ep ≤ 1, i.e., all dps no more convex than dp = ae−kp
are log concave (Cowan 2004).
10
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
profitable compared to the retailer. For slightly convex/linear demand functions 0 ≤ E < 1, like dp =
a − kp a k > 0 ≥ 1, the decentralized channel
remains relatively inefficient. As the demand function becomes concave, like dp = a − kp a k > 0
0 < < 1, the profit allocation begins to skew
more towards the manufacturer, and the decentralized system as a whole improves its performance.
For demand functions with highly negative curvatures ( ≈ 0 or E ≈ −), the total decentralized channel is, again, almost as profitable as the centralized
one; however, almost the entire profit is now captured
by the manufacturer. Obviously, we can substitute the
different values of B from Table 1, e.g., B = 1 for linear,
B = 0 for exponential, to evaluate the optimal performance ratios for the common deterministic demand
forms.
2.4. Comparison with Two Related Contracts
In this section, we compare our results for the buyback contract with two other contracts widely used
in practice: price-only (contract parameter: wholesale
price) and consignment (contract parameter: revenue
share). First note that iff dp is isoelastic, price-only
and buyback contracts are the same at optimality
(Theorem 4). Petruzzi (2004) demonstrates that isoelastic dp is also the only scenario when optimal
price-only and consignment contracts result in: (i) the
same expected profits for the two channel partners,
and (ii) the same level of retail price and availability
(i.e., retail stock) for the consumers. Hence, it is only
under isoelastic dp that managers of both channel
partners and the consumers are indifferent to the type
of contract—price-only, buyback, or consignment—in
use.11 However, in what follows, we show that the
distribution-free properties of the optimal buyback
contract may not hold for the other two contracts.
Here, we focus on an illustrative counterexample.
Suppose that Xp = dp, where dp = 500 − 20p
and ∼ UniformL U . Moreover, assume c = 1 and
let s CC be the optimal revenue share for the consignment contract CC. For all other optimal values, we
use the same notation as in the buyback case, but use
the superscripts W and CC to denote price-only and
11
Note from Table 4 that because B < 0 for isoelastic dp, therefore
D > &D.
Figure 1
Optimal Contract Parameters and Retail Prices for Buyback,
Consignment, and Price-Only Contracts X p ! = 500 −
20p!, ! ∼ UniformL U
w D, b D, wW
(a) 13
s CC
wD
0.88
bD
0.84
11
s CC
9
0.80
wW
0.76
0.72
7
0, 10
1, 9
2, 8
3, 7
4, 6
4.99, 5.01
L, U
(b) 20
pD
18
pW
16
pCC
14
pC
12
10
0, 10
1, 9
2, 8
3, 7
4, 6
4.99, 5.01
L, U
consignment contracts, respectively. The three optimal contracts are presented in Figure 1(a). The figure clearly shows that w D and b D are independent of
the distribution, while w W and s CC are not. That is,
Theorem 4 is not valid for price-only and consignment contracts. For high demand variability (variance
decreases over the horizontal axis, while the mean
remains constant), we expect the retailer’s risk of having leftovers, and hence its use of the buyback offer,
to be high. In that case, the optimal price-only contract must charge a relatively low wholesale price to
compensate for not providing any buyback opportunity. However, as the variability decreases, so does
the retailer’s leftover risk. Consequently, the retailer’s
need to use buybacks is comparatively reduced, even
though it is offered. This knowledge allows the manufacturer to raise the optimal price for the priceonly contract. Hence, although w W > w D , when the
demand is almost deterministic, the manufacturer is
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
unable to extract much premium for offering buybacks, i.e., w W ⇓ w D as variance tends to zero.
We have carried out extensive numerical experiments to compare the performance of the three contracts at optimality. The basis of comparison is two
widely used demand functions that were indicated in
the last section: Linear dp = a − kp and exponential dp = ae−kp ; and two common forms of distribution used in the OM literature for : Uniform L U and exponential 1 (refer also to G&Y and Petruzzi
2004 for similar analysis). We studied all four possible
demand function-distribution combinations by varying the values of k and distribution parameters (i.e., L
and U or 1) in each case, while keeping c and a fixed
at 1 and 500, respectively. Specifically, we changed k
in the range 10–50 for linear, and 0.5–5 for exponential; in the case of Uniform , we varied L and U as
shown in Figure 1, while for exponential , we altered
1 from 0.5 to 10. Our results suggest that:
• Consignment contracts result in the lowest retail
prices for end consumers, and buyback contracts in
the highest ones; the optimal retail prices associated
with price-only schemes, on the other hand, lie somewhere in between the two. Obviously, all optimal
decentralized retail prices are greater than the optimal centralized one (pD > pW > pCC > pC ; refer to Figure 1(b)).
• Both buyback and consignment contracts boost
the optimal retail order quantities compared to priceonly contracts; the outcome is increased availability
for the consumers. However, the comparison between
y CC and y D is inconclusive. The optimal centralized
order size is greater than all optimal decentralized
ones y CC > y W y D > y W y C > max*y D y CC +.
• From an expected profit perspective, the retailer’s
most preferred contract is a consignment one, and
the least preferred is a buyback one CC > W >
D . The order reverses when seen from the manufacturer’s perspective & D > & W > & CC . It seems that the
expected total channel profit is greater for a consignment contract than for a buyback scheme, although
both profits outperform that for a price-only scenario
)CC > )D > )W . Note that the degree of improvement (deterioration) in the manufacturer’s (retailer’s)
profit due to introduction of buybacks in a price-only
contract, i.e., & D /& W ( D / W ), is dependent on the
distribution of .
11
• In the case of consignment contracts, the retailer’s
expected profit is higher than that of the manufacturer for both linear and exponential dps. In the
other two contracts, the demand form plays an important role in the optimal profit allocation. As discussed
before, for buyback contracts, the expected profits for
the two parties are the same for exponential demand,
and the manufacturer makes more for linear demand.
For price-only ones, the manufacturer profits more
than the retailer for linear dp, but less for exponential dp.
• The optimal efficiency and the optimal profit allocation between the channel partners for price-only and
consignment contracts ()W /)C , )CC /)C , & CC / CC ,
and & W / W ) are all dependent on the distribution of .
Therefore, the distribution-free profit ratio property
of the optimal buyback contract (Proposition 1) also
does not hold for the other two contracts.
Last, an interesting result of our model setting is
that some of the optimal decisions and performance
ratios for buyback contracts are exactly similar to their
deterministic price-only counterparts. The former can
then be determined merely by solving a deterministic problem. Note that G&Y prove this property for
certain specific forms of dp and ∼ Uniform0 U .
However, as they conjecture, the result is valid for
more general settings. Specifically, (for all satisfying
Assumption 1):
Theorem 5. If p = −p/A+Bp A ≤ 0 B +1 > 0,
then: (i) The optimal wholesale price, (ii) the optimal efficiency )D /)C , and (iii) the optimal profit allocation
between the channel partners & D / D for a buyback contract in our setting, are the same as those for the corresponding optimal price-only contract in a deterministic
model. In particular, the value of the optimal wholesale
price w D in our buyback contract is the same as the optimal
retail price in a deterministic centralized system.
2.5.
Effects of Positive Salvage Value and
Goodwill Penalty
This section demonstrates the effects of including a
positive salvage value and/or goodwill penalty for
lost sales in the model framework of §1.1, all other
conditions remaining the same. First, let us assume
that the leftover inventory at the end of the selling
season (if any) can be salvaged at a positive value of v
per unit, but the goodwill penalty is still zero. For
12
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
expositional clarity, also suppose that both channel
partners have the same opportunity to recoup the
value of the unsold items, but it is the manufacturer
that does the actual salvaging. In that case, it is natural to presume that p > w > c > v > 0 and b > v (refer
to G&Y for a more detailed discussion). In what follows, we briefly show that under the above assumptions most of our previous results remain valid.12
= w − v, b̂ = b − v, ĉ = c − v, and
Let p̂ = p − v, w
ˆ
dp̂ = dp̂ + v. It is straightforward to establish that
ˆ p̂ satisfies all the conditions of Assumption 2 withd
out any additional requirement. We can also show
that the structure of the retailer’s profit function in (1),
as well as that of the manufacturer’s transformed
profit function in (4), remain exactly the same with
ˆ p̂,
b̂, ĉ, and d
p, w, b, c, and dp replaced by p̂, w,
respectively (note that only the manufacturer receives
the salvage revenue vdp y/dp). Consequently,
both Theorems 1 and 2 are still valid for a positive
salvage value, as is Theorem 3 (assuming a positive v
for the centralized system). The optimal decisions for
the channel partners can then be determined following Corollary 1. Most importantly, as in the first part
of Theorem 4, we can prove that the manufacturer’s
optimal contract parameters are distribution free iff
ˆ p̂/dˆ
p̂ is a linear function of p̂ =Â + Bp̂, where
d
 = A + Bv < 0 and B = B.13 This means that, as
before, only contracts for demand functions with constant curvatures, e.g., those in Table 1, do not depend
on the distribution of . The distribution-free optimal
contract for the manufacturer with a positive v turns
out to be w D = c − A/1 + B and b D = v − A/1 + B.
Because the results in §2.3 are only related to B = B,
all of them continue to hold. The main change in the
results due to a positive salvage value is that the optimal buyback price is now always positive, even for an
isoelastic demand function, i.e., A = 0. Because the manufacturer in this case is able to generate revenues from
returns, it is intuitive that he will be more willing to
offer buybacks.
However, unlike the positive salvage value scenario, the analysis of the model becomes quite complicated whenever there is a positive goodwill penalty
12
13
The detailed analysis is available from the authors upon request.
ˆ p̂/dˆ
p̂ = dp/d p = A + Bp = A + Bv + B p̂. As
Note that d
before, A ≤ 0 and B + 1 > 0.
cost associated with lost sales. In fact, many of the
insights from previous sections are, in general, not
true anymore. For example, we can easily show that
the optimal buyback price can then be positive for
isoelastic dp even for v = 0. Moreover, both elements of the manufacturer’s optimal contract might
be dependent on the distribution of for demand
functions with constant curvatures. Both of the above
are true for a scenario with v = 0, an isoelastic dp,
and ∼ UniformL U (refer also to G&Y, Footnote 1).
2.6. Illustration of the Additive Demand Scenario
In this section, our intent is to illustrate that the structural results of an optimal buyback contract established for multiplicative demand are not valid for the
additive form. The model framework remains as in
§1.1, except that Xp = dp + , where dp = a − kp
a > 0 k > 0 (Petruzzi and Dada 1999). Note that in
this case the mean of is positive, but not necessarily
equal to one. Due to lack of space, we do not present
any detailed analysis for this model. When we use
the results of Chen et al. (2005), Petruzzi and Dada
(1999), and our transformation technique, as in (3),
the manufacturer’s profit function can be written as
d zF z − z
&p z = !dp + z − z" p − −
k
kF z
− c!dp + z"
(7)
where z = y − dp represents the riskless leftovers at
the end of the period. The profit-maximizing p and z
then enable us to decide on the optimal decisions for
both channel partners. The analysis of the manufacturer’s problem in (7) is considerably more intricate
than that for the multiplicative demand model. Most
of the properties for multiplicative demand cease to
hold, even when ∼ UniformL U (G&Y also indicate likewise).
2.6.1. Illustrative Numerical Example. Figure 2
uses Xp = 500 − 20p + , ∼ UniformL U and
c = 1. The notation for both buyback and price-only
contracts remains the same as before. Clearly, the optimal buyback price is positive, and w D , b D , and w W
are now all dependent on L and U . Although we do
not show it, )D /)C and & D / D are also functions
of the stochasticity of the demand function. That is,
neither Theorem 4 nor Proposition 1 is valid for an
optimal buyback contract associated with an additive
demand.
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
Figure 2
Optimal Contract Parameters and Retail Prices for Buyback and Price-Only Contracts X p ! = 500 − 20p + !,
! ∼ UniformL U
(a) 15
wD
14
13
wW
12
bD
11
10
0, 100
20, 80
40, 60
49.99, 50.01
L, U
(b) 22
pD
20
pW
18
16
pC
14
12
0, 100
20, 80
40, 60
49.99, 50.01
L, U
3.
Concluding Discussion
In this paper, we have analyzed a buyback contract
for a decentralized supply chain. The retailer in our
framework faces multiplicative (price-sensitive and
stochastic) demand, while the manufacturer is the
Stackelberg leader offering the contract. We show that
the profit functions for both channel partners are unimodal for a large family of demand functions and randomness distributions. This allows us to determine
the unique optimal retail decisions and closed-form
expressions for the optimal contract parameters. From
a methodological perspective, our contribution is the
transformation procedure (Equation (3)), which simplifies the characterizations of profit functions and
optimal decisions. Note that this technique may be
helpful in analyzing multiechelon supply chains, especially where both echelons have an equal number
of decision variables (also see Lariviere and Porteus
2001).
13
Our further analysis sheds light on a number of
structural properties pertaining to the optimal decentralized channel. First, the optimal contract is independent of the uncertainty in customer demand, i.e.,
distribution free, iff the deterministic part of the
demand, dp, has a constant curvature. We show that
some of the most-used demand functions in the literature do satisfy the curvature property. Second, in
particular, iff dp is isoelastic and the salvage value is
zero, the optimal contract should not involve buyback
opportunities, and the manufacturer’s optimal strategy is just to offer a price-only contract. The expected
profits of the channel partners, as well as the price
and availability to the end consumers, would then
be the same as for an optimal consignment contract.
Third, the performance of the decentralized channel,
specifically the optimal efficiency and the optimal
profit allocation, for a distribution-free buyback contract depends only on the curvature of dp, and is
unaffected by the randomness in demand. The curvature (somewhat loosely) represents the relative market power of the channel partners in our setting. The
optimal efficiency approximates one for demand functions with large curvatures, either positive or negative, while the decentralized system proves quite
inefficient when dp is slightly convex/linear. On
the other hand, the relative profit performance of
the manufacturer, compared to the retailer, improves
with a decrease in the curvature of dp. Fourth,
the distribution-free properties are unique to buyback
contracts in multiplicative demand settings (with zero
goodwill penalty), and do not hold for price-only
and consignment schemes, or for additive demand.
We numerically compare the optimal decisions and
profits for buyback, price-only, and consignment contracts in our setting. This shows which of the three
contracts is preferable from the viewpoint of the consumers, the channel partners, and the entire chain.
Lastly, we demonstrate that certain distribution-free
decisions and profit ratios of an optimal buyback
contract in our framework exactly replicate the corresponding ones for a price-only contract in a deterministic environment.
Note that we use the IGFR property of only for
proving the unimodality of the profit functions. Most
of our results will hold for non-IGFR , as long as
the profit functions are well behaved. Our research
14
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
also clearly shows that for certain stochastic buyback
models, it is the deterministic demand part—in particular, its curvature—that drives many of the results.
This finding has important managerial implications in
terms of what type of contract to offer and the role of
uncertainty. Firms need to understand, maybe empirically, the form of dp in these cases.
Finally, we would like to reiterate our contributions
compared to G&Y. G&Y’s model setting and research
objectives resemble ours, and they prove/conjecture
a number of our results. However, their analysis is
based on three specific forms of dp—isoelastic, linear, and exponential—and ∼ Uniform0 U . Evidently, our framework is considerably broader in
scope. Furthermore, we derive the necessary and sufficient conditions for the structural properties of the
optimal contract that are not available in G&Y. This
means that we go beyond pointing out when the contract exhibits particular properties. Our results also
suggest that the investigation of scenarios not covered by the conditions (e.g., nonconstant curvatures)
will depend on the specifics of the setting. Overall, we
believe that this paper significantly generalizes previous insights regarding optimal buyback schemes and
provides a theoretically stronger basis for analysis of
decentralized price-setting newsvendor paradigms.
3.1. Future Research Opportunities
The most important extension to our modelling framework would be a detailed investigation of buyback
contracts in additive demand settings. We have shown
that none of the major results from multiplicative
demand remain valid under that scenario. As Petruzzi
and Dada (1999) have pointed out, any change in
price affects the two most common measures of uncertainty—variance and coefficient of variation—differently for additive and multiplicative demand forms.
This naturally entails some structural differences in
the behavior of the two models that need to be studied. It may also be important to understand whether
our results are valid in more comprehensive frameworks like risk-averse newsvendors, or competitive
retailers, or multiperiod models.
icantly improved the relevance, positioning, and presentation of the paper. Help from Xiaoqing Jing is also gratefully acknowledged. This research is supported by grants
from the Natural Sciences and Engineering Research Council (NSERC) of Canada and Fonds de Recherche sur la
Société et al Culture (FQRSC) of Québec.
Appendix
Proof of Lemma 1. As ∈ L U , the retailer’s order
quantity y ≥0 should satisfy dpL ≤ y ≤ dpU , i.e., L ≤
y/dp ≤ U . First note that because p∗ y ∗ is the maximizer
of p y, we have y ∗ > 0 and p∗ < P u . If either y ∗ = 0 (zero
order quantity) or p∗ = P u (zero demand), then the retailer’s
optimal profit p∗ y ∗ is less than or equal to zero. On the
other hand, based on the expression of p y, the optimal profit should be positive because p y*y=0+ = 0 and
% p y/%y*y=0+ = p − w > 0. This is a contradiction. Thus,
we always have y ∗ > 0 and p∗ < P u .
Suppose that there exists a pair p∗ y ∗ such that it is a
maximizer of py and y ∗ /dp∗ = L. We consider two cases:
If L > 0, then % py/%p*p=p∗ y=y∗ + = Ldp∗ > 0. This is a contradiction with the optimality of p∗ y ∗ . If L = 0, then y ∗ = 0,
which is not possible. Thus, y ∗ /dp∗ = L is impossible. Similarly, if y ∗ /dp∗ = U , then % py/%y*p=p∗ y=y∗ + = b −w < 0,
which contradicts the optimality of p∗ y ∗ . Note that the following lemma about the behavior of is
subsequently used in other proofs.14
+
Lemma 2. Let z = z u − zf u du and V z = 1 −
z!1 − F z"/1 − z−1 for any z ∈ L U . Under Assumption 1, V z is strictly decreasing for z ∈ L U .
−1
Proof. Note
z that V z=1−z!1 − F z"/1−z =
1−z/ 0 uf u du. To prove
the lemma, we then need to
z
show that V
z
=
1
−
F
uf
u
du − 1 − zf z < 0. As
0
z
1 − z = 0 uf u du + z!1 − F z" (refer to (2)),
z the above
inequality is equivalent to Hz = !1 − gz" 0 uf u du −
z2 f < 0. Because Hzz=L = −L2 f L ≤ 0, to prove the
lemma, it is sufficient to study the behavior of H z for z ∈
L U . Differentiating Hz we have
z
H z = −zf z − zf zgz − z2 f z − g z
uf u du
0
z
uf u du
= −z!1 − F z"g z − g z
0
= −!1 − z"g z#
From Assumption 1 we know that g z ≥ 0 and g 0+ > 0
if L = 0 (recall that limu→0 f u > 0 if L = 0).
We consider two cases based on the value of L. If L > 0,
then HL < 0 and H z ≤ 0 for any z ∈ L U ; hence,
Hz < 0 for any z ∈ L U . If L = 0, then H0 = 0 and
g 0+ > 0, which implies that H z < 0 for any z ∈ 0 U ,
i.e., Hz < 0 for any z ∈ L U . Therefore, V z < 0 for any
z ∈ L U . Acknowledgments
The authors thank the two anonymous referees, the Editor,
and Senior Editor, for their valuable support, which signif-
14
Without loss of generality, we present Lemma 2 for = 1. Recall
that f u > 0 for u ∈ L U and f u = 0 for u ∈ !0 L" ∪ !U +.
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
15
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
Proofs of Theorems 1 and 3. Notice that
y/dp
y
d p
% p y
uf u du V
= dp
+ p − b
%p
dp
dp
0
and V z is strictly decreasing on L U (Lemma 2). As
dp/d p is monotone by Assumption 2, d p/dp is also
monotone (can be either increasing or decreasing). In the
following, we consider two cases.
Case 1. d p/dp is increasing. In this case
p − b
d p
d p
d p
=p
−b
dp
dp
dp
is decreasing. For any given y ∈ 0 d0+ U , let py be the
maximizer of p y. If there exists a y0 ∈ 0 d0+ U such
that
d p
d p
1+p
−b
>0
dp
dp
for any p ∈ d −1 y0 /L d −1 y0 /U , then
1+p
d p
d p
−b
>0
dp
dp
for any y ≥ y0 and any p ∈ d −1 y/L d −1 y/U . Let ymin be
the minimum value among all these y0 (if there is no such
y0 , let ymin = d0+ U ). Then, for any y ∈ ymin d0+ U , the
maximizer of p y is the py such that y/dpy = U , i.e.,
py = d −1 y/U . For any y ∈ 0 ymin , the maximizer py of
p y should then satisfy
y
d p
d p
V
+p
−b
= 0#
(A1)
dp
dp
dp
It is obvious that py is unique and py > b for any y ∈
0 ymin . Taking derivatives with respect to y on both sides
of (A1), we get
−yd p
d p
d p
y
V
+
p
−
b
p
y
dp
dp2
dp
dp
1
y
= 0#
+V dp dp
Hence, p
y < 0, i.e., py is strictly decreasing. Rewriting
the above equation about p
y, we also get that y/dpy
is strictly increasing, i.e.,
1
d pyy −
p y ≥ 0#
dpy
dpy2
In summary, we can conclude that py is continuous and
strictly decreasing on 0 d0+ U .
Next we study the behavior of py y on 0 d0+ U .
We first consider the case of y ∈ 0 ymin . If we take derivatives of py y with respect to y, we have
d py y
y
y
= py 1 − F
+ bF
−w
dy
dpy
dpy
and
d 2 py y
y
y
−
f
=
p
y
1
−
F
dy 2
dpy
dpy
d pyy 1
−
p
y
#
· !py − b"
dpy
dpy2
Thus, d 2 py y/dy 2 < 0, because
1
d pyy −
p y ≥ 0
dpy
dpy2
py > b
and
p
y < 0#
This implies that py y is strictly concave on 0 ymin .
Next we consider the case of y ∈ ymin d0+ U . As
y/dpy = U and % p y/%p*p=py+ > 0, we get
d pyy
% py % py = p
y
+
≤ b −w < 0
dy
%p *p=py+
%y *p=py+
(note that % p y/%y*p=py+ = b − w).
Therefore, there is a unique maximizer of py y on
0 d0+ U . We complete the proof for Case 1.
Case 2. d p/dp is decreasing. Note that both V z and
d p/dp are now decreasing. The remainder of the proof
for this case is almost identical to the first case. Due to space
constraints, we choose to omit it.15
The proof of the centralized system is exactly the same
as above, with b = 0 and w = c. Proof of Theorem 2. Note that
y/dp
%&p y
= dp
uf u du
%p
0
y
d p
dp V
+
1
+
p
#
· 1+
d p
dp
dp
Based on Assumption 2, we can see that
dp d p
y
1+
+
1
+
p
V
d p
dp
dp
is decreasing with respect to p (!1 + dp/d p
" is the first
derivative of !p + dp/d p"). Hence, for any given y ∈
0 d0+ U , there exists a P y that maximizes &p y. The
remaining argument for the proof is almost identical to the
one for Theorem 1, and, hence, is not presented. Proof of Theorem 4. From the first-order conditions of
&p y, i.e., %&p y/%p = 0 and %&p y/%y = 0, we have
1 + p d p/dp and
V = −
1 + dp/d p
*p=pD +
c
#
F = 1−
p + dp/d p *p=pD +
Note that, at optimality, both V and F are functions of pD .
Substitution of V and F in (3) enables us to generate the
15
Details of any proof not presented in the appendix are available
from the authors on request.
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
16
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
expressions for w D and b D that are shown in (5) (for any dp
satisfying Assumption 2). Clearly, b D is a constant iff the first
derivative of b D in the above expression is zero, i.e.,
dp
dp = 0#
(A2)
p
+
d p
d p *p=pD +
D
Similarly, w is a constant iff the following condition is satisfied.
dp dp
−
c
(A3)
p
+
D = 0#
d p
d p
*p=p +
From the above discussion, the sufficient condition is obvious. Now we show the necessary part. As pD is a function
of the distribution F (this is true based on a uniformly distributed ) and not a constant, b D is distribution free iff (A2)
is satisfied, and w D is distribution free iff (A3) is satisfied.
If (A2) is satisfied, we claim that dp/d p
= 0. Suppose
that dp/d p
is nonzero on some open interval, then we
obtain p + dp/d p = 0 on this interval. This is a contradiction. Hence, we have dp/d p is linear in terms of p. Similarly, if (A3) is satisfied, we can also show that dp/d p
is linear in terms of p.
The optimality of no buyback, i.e., b D = 0, is a special case
of the above. Note that
dp/d p + p D
#
b = p−
1 + dp/d p
*p=pD +
Because dp/d p = A + Bp, b D = −A/1 + B. Hence, b D = 0
iff A = 0. Optimal Buyback Contract for an Arbitrary Retail Order
Quantity of y0
We consider two cases based on the value of A, where
dp/d p = A + Bp A ≤ 0 B + 1 > 0.
Case A < 0. From our analysis in §2.1, it is clear that, for
any given y0 , P y0 is the solution of
y0
p
%&p y0 = 1 + BV
+ 1+
= 0# (A4)
%p
dp
A + Bp
The above equation implies that
A + 1 + BP y0 V Zy0 = −
1 + B!A + BP y0 "
where Zy0 = y0 /dP y0 is the optimal stocking factor for
a given y0 . Substitution of V Zy0 in (3) would result in
a contract such that: (i) the optimal retail order is y0 , and
(ii) the manufacturer’s profit &p y0 is maximized. Note
that the contract also implies that the optimal retail price
is py0 wy0 by0 = P y0 . We denote the corresponding
contract parameters by wy0 and by0 , which are provided
in (6). Rewriting (A4), we obtain
V z
A
P y0 = −
1+
#
1+B
1 + BV z *z=Zy0 +
Substitution of P y0 in wy0 and subsequent differentiation
with respect to y0 yields
AZ y0 f z!1+BV z"−!1−F z"V z w y0 =
#
1+B
!1+BV z"2
*z=Zy0 +
Because Z y0 = y0 /dpy0 > 0 based on the proof of Theorem 1, V # is strictly decreasing by Lemma 2, A < 0 and
1+B > 0, we have w y0 < 0.
Case A = 0, B < 0. In this case, the expressions for the
optimal contract simplify to wy0 = P y0 1−F Zy0 and
by0 = 0. Moreover, because V Zy0 is now −1/B ∀y0
(note that for A = 0, 1+BV Zy0 = 0), so the elasticity of
expected sales with respect to the stocking factor, given by
1−1/V Zy0 , is 1+B > 0, a constant. Also, the optimal
stocking factor for a given y0 , Zy0 = V −1 −1/B ∀y0 , i.e.,
Z y0 = 0.
On the other hand, the elasticity of retail order y0 with
respect to w (say, 4y0 ) can be determined from:
4y0 = −wy0 y0
w
1
= −wy0 #
y0
y0 w y0 Based on the expression of wy0 and keeping in mind
that Z y0 = 0, we can show that 4y0 = −P y0 /y0 P y0 .
Because for A = 0, dp = ap1/B a > 0 B < 0, and Zy0 =
V −1 −1/B, so P y0 = y0B /!aV −1 −1/B"B . Differentiating
P y0 and simplifying, we then have y0 = −1/B > 0, a
constant.
Optimal Decision Variable Values for Table 2
Case A < 0. The optimal pD y D should satisfy the firstorder conditions (FOCs) %&py/%p = 0 and %&py/%y = 0.
When dp/d p = A + Bp < 0 A ≤ 0 B + 1 > 0, the FOCs
of &p y shown just before Theorem 2 result in
y
p
!1 + B"V
+1+
= 0 and
dp
A + Bp
(A5)
y
1−F
!B + 1p + A" − c = 0#
dp
From the first equation in (A5), we derive
p=−
A 1 + BV y/dp + 1
#
1 + B 1 + BV y/dp
Substituting this into the second equation in (A5), we obtain
y
c
1
1−F
+
B+
= 0#
dp
A
V y/dp
As V z is decreasing, it is then easy to show that the
above equation has an unique solution Z1 , which lies within
L U . Hence, y D /dpD = Z1 and
pD = −
A 1 + BV Z1 + 1
#
1+B
1 + BV Z1 Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
Obviously, y D = dpD Z1 . The transformation in (3) then
simplifies to


p + A + BpF y/dpV y/dp
w
#
=
(A6)
b
p + A + BpV y/dp
When we substitute pD into the second equation in (A5),
it gives us F Z1 = 1 + c/A1 + BV Z1 /V Z1 . Substituting both pD and F Z1 into the first equation in (A6), we
obtain w D in Table 2. On the other hand, based on the first
equation in (A5) and the second equation in (A6), we have
the expression for b D . The expressions for pC and y C can
likewise be deduced from the FOCs % p y/%p = 0 and
% p y/%y = 0, i.e.,
y
p
y
V
+
= 0 and
1−F
p −c = 0 (A7)
dp
A+Bp
dp
although the optimal Z1 still remains the solution of !1 −
F z" + c/A!B + 1/V z" = 0.
Case A = 0. The proof is somewhat different in this case.
The FOCs (%&p y/%p = 0 and %&p y/%y = 0) are now
y
1 + B 1 + BV
= 0 and
dp
(A8)
y
1−F
p1 + B − c = 0#
dp
Because 1 + B > 0, the first equation implies that Z2 is
the solution to 1 + BV z = 0. The expression for pD is
generated by substituting Z2 in the second equation. The
expressions for y D , w D , and b D then follow (e.g., because
b D = pD + BpD V Z2 and BV Z2 = −1, b D = 0). In the centralized case also, the optimal Z2 is the solution to 1 +
BV z = 0, and pC can be obtained by substituting Z2 in
%)p y/%y = 0, i.e., p!1 − F Z2 " − c = 0, while y C = dpC Z2 .
Proof of Proposition 1. All the ratios for pC /pD follow
from Table 2. For other ratio values, we consider three cases.
Case A < 0, B = 0. The general demand function in this
case can be written as dp = a!p + A/B1/B ", a > 0. Therefore, we focus on p + A/B. Substituting the expressions for
pC and pD from Table 2, we can show that
pD +
A
A
=
B
B1 + B!1 + BV Z1 "
pC +
A
A
=
#
B
B!1 + BV Z1 "
and
(A9)
This implies that y C /y D = 1 + B1/B (y C = dpC Z1 , y D =
dpD Z1 ). For comparing & D and D , let us substitute w D
and b D from Table 2 in (1) and simplify using (2). Then we
obtain (recall that dp/d p = A + Bp)
A
c −A D
dpD Z1 −
y
1+B
1+B
c
= pD dpD !
− Z1 " −
yD
1+B
pD y D = pD dpD !
−Z1 "−
17
A
!y D − dpD Z1 "
1+B
A
c
!
− Z1 " −
yD
= dpD pD +
1+B
1+B
1
d
=
dpD pD + !
− Z1 " − cy D
1+B
d
+
=
1
&pD y D 1+B
where p y and &p y are given by (1) and (4), respectively. Hence, & D / D = 1 + B.
Note that y D /dpD = y C /dpC = Z1 ; hence,
)C
pC dpC !1−Z1 "−cy C y C pC /Z1 !1−Z1 "−c
#
= D
=
D
)
p dpD !1−Z1 "−cy D y D pD /Z1 !1−Z1 "−c
By substituting for pC , pD , and y C /y D in the above ratio (from
FOC we know 1 + BV Z1 /AV Z1 = −1 − F Z1 /c), we
have
1 − Z1 −1
)C
!1 − F Z1 "Z1
1/B
#
=
1
+
B
!1 − Z1 "!1 + 1 + BV Z1 "
)D
−1
!1 − F Z1 "Z1 1 + BV Z1 Further simplifications using the relations in (2) result in
Z1
uf u du
)C
1/B
0
#
= 1 + B
Z1
1 − Z1 )D
uf u du +
0
1 + BV Z1 Now from the definition of
ZV z in Lemma 2, we know
that 1 − Z1 /V Z1 = 0 1 uf u du. Hence, )C /)D =
1 + B1+1/B /2 + B.
Case A = 0. The analysis in this case is exactly the same
as above except that the FOC is 1 + BV Z2 = 0, and we
should take due note of the optimal prices in Table 2.
Case A < 0, B = 0. The demand function in this case
would be dp = a!ep/A ", a > 0. Note from Table 2 that
pC = −AV Z1 and pD = −A!1 + V Z1 ". Thus, y C /y D =
C
D
dpC /dpD = ep −p /A = e. We can then apply the same
technique as above to prove the expression for & D / D in
Table 4 and also evaluate )C /)D .
The expressions for )D /)C in Table 4 then follow for all
three cases. Proof of Theorem 5. In a deterministic price-only setting, the profit function for the retailer, for a given wholesale price w, can be written as p = p − wdp. For
dp/d p = A + Bp A ≤ 0 B + 1 > 0, we can show that
d p/dp = 0 has a unique solution and the optimal retail
price (for a given w) is given by pw = w − A/B + 1 or
wp = A + B + 1p. Replacing it in the manufacturer’s profit
function, we have &p = wp − cdp. When we differentiate &p with respect to p and equate it to zero, we can
show that the overall optimal retail price is given by p∗ =
c − AB + 2/B + 12 . Replacing it in wp, we obtain the
18
Song, Ray, and Li: Structural Properties of Buyback Contracts for Price-Setting Newsvendors
Manufacturing & Service Operations Management 10(1), pp. 1–18, © 2008 INFORMS
optimal wholesale price in a deterministic setting as w ∗ =
c − A/B + 1 = w D . The ratio between the manufacturer’s
and the retailer’s optimal profit in a deterministic price-only
setting is equivalent to w ∗ − c/p∗ − w ∗ . Substituting the
values of p∗ and w ∗ , and simplifying we find the ratio to
be 1 + B = & D / D (Table 4). We can also show that the
centralized system in the deterministic case results in the
unique optimal retail price of p̃∗ = c −A/B +1 = w ∗ = w D .
The comparison between the optimal profits of the centralized system and the decentralized system in the deterministic price-only case is given by p∗ − cdp∗ /p̃∗ − cdp̃∗ .
When B = 0, dp = a!p + A/B1/B " a > 0. Substituting the
values of p̃∗ and p∗ and simplifying, we obtain the ratio
as 2 + B/1 + B1+1/B = )D /)C (Table 4). When B = 0, we
have dp = aep/A a > 0 and the ratio value is 2/e = )D /)C
(Table 4). References
Bernstein, F., A. Federgruen. 2005. Decentralized supply chains
with competing retailers under demand uncertainty. Management Sci. 51(1) 18–29.
Bresnahan, T. F., P. C. Reiss. 1985. Dealer and manufacturer margins. RAND J. Econom. 16(2) 253–268.
Cachon, G. P. 2003. Supply chain coordination with contracts. A. G.
de Kok, S. Graves, eds. Supply Chain Management: Design,
Coordination and Operation, Chapter 6. Elsevier, Amsterdam,
The Netherlands.
Cachon, G. P., M. A. Lariviere. 2005. Supply chain coordination with
revenue-sharing contracts: Strengths and limitations. Management Sci. 51(1) 31–44.
Chen, Y., S. Ray, Y. Song. 2005. Optimal pricing and inventory control policy in periodic-review systems with fixed ordering cost
and lost sales. Naval Res. Logist. 53(2) 117–136.
Cowan, S. 2004. Demand shifts and imperfect competition. Working paper, Department of Economics, University of Oxford,
Oxford, UK.
Emmons, H., S. M. Gilbert. 1998. The role of return policies in pricing and inventory decisions for catalogue goods. Management
Sci. 44(2) 276–283.
Granot, D., S. Yin. 2005. On the effectiveness of returns policies
in the price-dependent newsvendor models. Naval Res. Logist.
52(8) 765–779.
Kandel, E. 1996. The right to return. J. Law Econom. 39 329–356.
Lariviere, M. A. 1998. Supply chain contracting and coordination
with stochastic demand. S. Tayur, M. Magazine, R. Ganeshan,
eds. Quantitative Models for Supply Chain Management, Chapter 8. Kluwer, Boston, MA.
Lariviere, M. A. 2006. A note on probability distributions with
increasing generalized failure rates. Oper. Res. 54(3) 602–604.
Lariviere, M. A., E. L. Porteus. 2001. Selling to the newsvendor:
An analysis of price-only contracts. Manufacturing Service Oper.
Management 3(4) 293–305.
Malueg, D. A. 1994. Monopoly output and welfare: The role of
the curvature of the demand function. J. Econom. Ed. 25(3)
235–250.
Marvel, H. P., J. Peck. 1995. Demand uncertainty and returns policies. Internat. Econom. Rev. 36(3) 691–714.
Padmanabhan, V., I. P. L. Png. 1997. Manufacturer’s returns policies
and retail competition. Marketing Sci. 16(1) 81–94.
Pasternack, B. A. 1985. Optimal pricing and return policies for perishable commodities. Marketing Sci. 4(2) 166–176.
Petruzzi, N. C. 2004. Newsvendor pricing, purchasing and consignment: Supply chain modeling implications and insights.
Working paper, College of Business, University of Illinois,
Urbana-Champaign, IL.
Petruzzi, N. C., M. Dada. 1999. Pricing and the newsvendor problem: A review with extensions. Oper. Res. 47(2) 183–194.
Wang, H. 2004. Do returns policies intensify retail competition?
Marketing Sci. 23(4) 611–613.
Wang, Y., J. Li, Z. Shen. 2004. Channel performance under consignment contract with revenue sharing. Management Sci. 50(1)
34–47.
Yano, C. A., S. M. Gilbert. 2003. Coordinated pricing and production/procurement decisions: A review. A. Chakravarty,
J. Eliashberg, eds. Managing Business Interfaces: Marketing, Engineering and Manufacturing Perspectives. Kluwer Academic Publishers, Boston, MA.