J Syst Sci Syst Eng DOI: 10.1007/s11518-008-5062-1
ISSN: 1004-3756 (Paper) 1861-9576 (Online) CN11-2983/N
OPTIMAL PRICING OF A PERSONALIZED PRODUCT Suresh P. SETHI 1 1
School of Management, The University of Texas at Dallas 800 W. Campbell Rd., Richardson, TX 75080
[email protected] ( )
Abstract This paper deals with optimal pricing of a personalized product such as a personal portrait or photo. A new model of the pricing structure inspired by two real-life cases is introduced to the literature and solved to obtain optimal photo sitting fees and the final product price. A sensitivity analysis with respect to the problem parameters is performed. Keywords: Personalized product, pricing, salvage loss
1. Introduction
product is sold to the customer for whom it is
This paper is concerned with optimal pricing
intended.
of very personalized products such as personal
To the best of our knowledge, this type of
portraits or photos that a customer does not buy
problem has not been modeled or studied in the
before seeing the results and their salvage value
literature. A 2003 survey on personalized
is zero or even negative, if the customer decides
products by Murthi and Sarkar (2003) does not
not to buy after viewing the results. The product
mention the situation described here.
is typically produced in two stages: i) photo
A model will be motivated and developed in
sitting during which the photos of the customer
the next section by two instances that the author
are taken and ii) developing the photos and
has personally encountered. The first instance is
preparing a package for a possible purchase by
that of a photo studio, which is in the business of
the customer. The unit production cost of the
providing professional quality portraits. Its
second stage is substantially higher than that of
pricing structure is as follows. A customer sits
the first stage.
for a fixed number of poses for no charge. At the company
time of the sitting, the customer knows that a
producing the personalized product, the bulk of
specified package of developed photos, large
the revenue comes from selling the product and
and small, will be made available to him in a
not from the sitting fees, since only a fraction of
few days at a given price, and that he has an
the
From
the
viewpoint
of
the
eventually
option to purchase or not after viewing the
purchase the product. Thus, the price of the
package. In some studios, the customer also
product must factor in the probability that the
knows that he will get one free photo of the
customers
who
sit
would
© Systems Engineering Society of China & Springer-Verlag 2007
Electronic copy available at: http://ssrn.com/abstract=1094175
Optimal Pricing of a Personalized Product
studio’s choice, if he comes to view the photos
agreed, and the plaque was purchased for a total
when developed. If the customer decides not to
of $50 and no shipping fee as long as payment
purchase the package after viewing the photos,
was made promptly.
he goes home possibly with a free photo, and the
One
could
think
of
other
cases
of
personalized products that are offered this way
package is destroyed. Some studios may allow the customer to
along with a variety of related pricing structures,
bargain, but this is not normally the case. In
but they will not be covered in this paper. Nor
others, a smaller package or CD containing the
will this paper study the economics of such
photos may be offered at a much reduced price
markets. Rather, the purpose of this brief paper
after the studio’s first attempt to sell the original
is to introduce a type of problem that to our
package to the customer at the regular price
knowledge has not yet been studied in the hope
fails.
of inspiring more detailed exploration of the
The second instance is that of a company
problem area. To this end, the paper will develop
that makes quality plaques of published material
a simple mathematical model and solve it
including photographs that can be displayed. In
explicitly to obtain the optimal sitting fee and
my case, a business magazine published a
package price in the next section. We will
one-page article about one of my professional
conclude the paper in Section 3 with a
areas of specialty along with my photo. Shortly
discussion on a variety of extensions worthy of
after the publication, I got a phone call from a
further research.
company offering a display plaque mounted on wood highlighting the name of the publication,
2. The Model and Its Solution
the article including my photo, and a plate
Let S denote the sitting fee. The paper
containing my name, affiliation, and date. Upon
assumes that the customers are sensitive to this
receipt, I had a few days in which to decide to
fee and the customer demand, i.e., the number of
buy it at the original price quoted at $159 plus
customers who would walk into the studio for a
$20 for shipping. If I decided not to purchase the
sitting each day is a downward sloping function D( S ) with D ′( S ) < 0. It also assumes that
plaque, there was no charge. I was instructed to destroy the plaque by removing the article with box cutters and return it to the company so there was no return cost to the company. When the plaque arrived, I was disappointed with the reproduction quality of the photo. After discussing this with the company representative, they offered to redo it. However, I was not convinced that the reproduction would match the quality of the published photo in the magazine article, so I offered $50 for the existing plaque instead. To my surprise, the company quickly
every one who comes for a sitting would come to view the photos when developed, at least out of sheer curiosity. Note that the case when every viewer gets a free photo can be included in our model by having a negative value of S. Of course, if this free photo incentive is designed to ensure that more of the sitters would come to view the developed photos, then our model needs to introduce a viewing demand function dependent on the incentive offered. But this case will not be addressed in this brief paper, where
JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING
Electronic copy available at: http://ssrn.com/abstract=1094175
SETHI
the incentive would be another decision variable separate from the sitting fee. Let the package price be denoted by P ≥ 0. We will assume that the probability a randomly chosen viewer buys the package is a decreasing function of the price P. Let this function be denoted by θ ( P) with θ ′( P) < 0. We should mention that from the studio’s perspective, this is a reasonable assumption even when the customers are heterogeneous. On the other hand, from a customer’s viewpoint, the probability of purchase may depend on several other factors such as the perceived quality of the photos at the time of viewing, which in general is a random variable, and his own sensitivity to the price as it varies from customer to customer. Let the cost of producing a package from photo sitting to development be denoted by
In order to check the second-order conditions for maxima, we partially differentiate (1) once more with respect to P and S and make use of (2) and (3) to obtain
∂ 2π ∂P
2
∂ 2π ∂S
2
=
D(S ) ⎡ 2(θ ′( P ))2 − θ ( P )θ ′′( P ) ⎤⎦ , θ ′( P ) ⎣
=
2( D ′( S ))2 − D ( S ) D ′′( S ) . D ′( S )
Since θ ′ < 0 and D ′ < 0, both these quantities are negative if 2(θ ′)2 − θθ ′′ > 0 and 2( D ′) 2 − DD ′′ > 0. These conditions roughly mean that
these functions are not too concave. Of course, that clearly holds in the special case when θ and D are convex functions. We can now state the following proposition. Proposition 1 Let P∗ and S ∗ be unique
P∗ > 0,
C > 0, and let the salvage loss or cost of
solutions of (2) and (3) and let
destroying the package be σ ≥ 0. With this
2(θ ′( P∗ ))2 > θ ( P∗ )θ ′′( P∗ )
notation, we can write the expression for the
D ( S ∗ ) D′′( S ∗ ). Then, P∗ and S ∗ maximize
studio’s daily profit from this photo operation
the studio’s daily profit given in (1).
scheme as
Henceforth, we assume that θ ( P)
π ( S , P) = (θ P + S ) D( S ) − (C + (1 − θ )σ ) D( S ). (1) The first-order condition to maximize the profit with respect to S and P ≥ 0 give rise to the relations ( P + σ )θ ′( P) + θ ( P) = 0,
(2)
and
D( S ) satisfy the conditions 2(θ ′)2 > θθ ′′, 2( D′)2 > DD ′′.
(4)
Next, we derive a result concerning how P∗ and S ∗ change when the cost parameters C and
σ change. Proposition 2 The optimal price P∗ decreases
in σ and is independent of C The optimal
and [ S + Pθ ( P) − C − (1 − θ ( P ))σ ]D ′( S ) + D ( S ) = 0, (3) provided
and 2( D′( S ∗ ))2 >
P > 0,
meaning that there is an
interior optimal solution. Recall that we allow negative values of the sitting fee, so the variable S is not constrained.
JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING
sitting fee S ∗ increases in C and σ . Proof. That P∗ is independent of C and S ∗
increases in C are obvious from the first-order conditions (2) and (3). For the sensitivity of P∗ with respect to σ , we implicitly differentiate (2) with respect to σ and obtain
Optimal Pricing of a Personalized Product
θ ′( P∗ ) ∂P∗ =− . ∂σ 2θ ′( P∗ ) + ( P∗ + σ )θ ′( P∗ )
(2) and (3) conclude that
Since P∗ + σ = −θ ( P∗ ) /θ ′( P∗ ) from (2), we can use it to obtain (θ ′( P∗ )) 2 ∂P∗ =− < 0. ∗ ∂σ 2(θ ′( P ))2 − θ ( P∗ )θ ′′( P∗ )
(5)
We can now derive the following results regarding the sensitivity of the optimal price and the sitting fee with respect to the parameters α ,
Similarly from (3) and the use of (5), we can derive [1 − θ ( P∗ )]( D ′( S ∗ )) 2 ∂S ∗ = > 0. ∂σ 2( D′( S ∗ ))2 − D ( S ∗ ) D′′( S ∗ )
σα −1 ⎫ ]⎪ ⎪⎧η[C + σ − (1/α )e P∗ = 1/α − σ , S ∗ = max ⎨ ⎬. η −1 ⎪⎩ ⎭⎪ (11)
(6)
β , and η . Proposition 3 The optimal price P∗ decreases in α and is independent of β in Case 1 and of η in Case 2. Proposition 4 The optimal sitting fee S ∗ decreases in α . It also decreases in the price
This completes the proof.
■
Next we consider two special cases of θ ( P)
and D ( S ), which allow us to obtain explicit solutions for P∗ and S ∗ . In both cases, we set
sensitivity parameters β in Case 1 and η in Cases 2.
The results derived in Propositions 2.1–2.4 are intuitively appealing. Note that Pθ ( P ) is
(7)
the revenue from the sale and it is maximized at
As for the demand for sitting, we consider the exponential demand
fact that each sale saves the product from being
θ ( P) = e
−α P
, α > 0.
D( S ) = Me− β S , β > 0, M > 0, defined for −∞ < S < ∞ demand
(8)
and the isoelastic
D( S ) = MS −η , η > 1, M > 0
1/α . The optimal price (1/α − σ ) reflects the destroyed resulting in a savings of σ . The optimal sitting fee is one way of recouping the costs C and σ , so it increases in those parameters. On the other hand, when
α increases, the probability that a customer
(9)
will buy the product intended for him decreases.
defined for S ≥ 0. In the second case, η > 1 is being assumed so that the demand is elastic. We note that θ ( P) in (7) and D( S ) defined in (8) and in (9) satisfy the assumption (4). Case 1. Exponential Demand: In this case, (2) and (3) can be easily solved to give
This means that the sitting fee should be lowered in order to increase the number of sittings. That the sitting fee decreases in the price sensitivity parameters should be intuitively obvious. Finally, in Case 2, we see that if 1 C ≤ θ ( P∗ ) P∗ − (1 − θ ( P∗ ))α = e(σα −1) − σ ,
α
P∗ = 1/α − σ , S ∗ = 1/β + C + σ − (1/α )eσα −1 . (10)
then the optimal sitting fee should be zero. This
Case 2. Isoelastic Demand: Since the domain
changing a positive sitting fee decreases the
of the demand function is S ≥ 0, we can from
demand for sitting and, in turn, decreases profit.
is the case where the unit cost is quite low and
JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING
SETHI
This may very well be the case in the first instance of the studio that was described in the Introduction.
an ex-post inelastic demand. Also in the second stage, there may also be the possibility of bargaining. More generally, one could look into formulating the problem as
3. Conclusions and Extensions
games between the buyer and the seller.
We have studied a new pricing structure used
From the point of view of the viability of
by the sellers of very personalized products. We
business, one may look into the amount of
have formulated a problem that is simple enough
investment needed and return on this investment
to enable us to obtain explicit optimal price and
over a number of years.
sitting fees in two special cases. While we have
Finally, it would be of interest to characterize
assumed the sitting demand to depend only on
features that lead to the kind of pricing structure
the sitting fee, it is more realistic to have it also
studied here. This may also involve empirical
depend on the product price. One could consider
studies of companies that are already using these
introducing different qualities offered at the time
pricing schemes.
of sitting and different number of poses taken for the customer to choose from. In this case,
Acknowledgement
there may be a fixed fee for each sitting and
The author wishes to thank Ernan Haruvy,
variable fee depending on the number of poses
B.P.S. Murthi, Jun Zhang, Xiuli He, and Sridhar
taken and the quality chosen. The company
Seshadri for their helpful suggestions.
could offer a variety of packages to choose from and their prices. In the case of the photo studio,
References
if the offered package is framed, then the frame
[1] Murthi, B.P.S. and Sarkar, S. (2003). The
does not need to be destroyed. This situation
role of the management sciences in research
could be treated by having a negative salvage
on personalization. Management Science,
loss.
49 (10): 1344-1362
In the second stage, i.e., at the time of viewing, the customer would be able to assess
Suresh P. Sethi is Charles & Nancy Davidson
the quality of the product. This is a random
Distinguished
variable, especially in the case of the photo
Management and Director of the Center for
studio example, where the photo may offer a
Intelligent Supply Networks in the School of
unique prospective that may be hard to
Management at The University of Texas at
reproduce and thus difficult to pass up. The
Dallas, Richardson, TX. He earned his Ph.D. in
probability that the customer will buy in this
Operations Research from Carnegie Mellon
extension would definitely depend on the
University in 1972. He has written 5 books and
realized quality. This means that the final
published more than 300 research papers in the
demand depends on the sitting fee, the product
fields
price, and the purchase probability. This could
management, finance and economics, marketing,
allow us to have an ex-ante elastic demand and
and optimization theory. He serves on the
JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING
of
Professor
manufacturing
of
and
Operations
operations
Optimal Pricing of a Personalized Product
editorial board of such journals as Journal on
Fellow (2001). Two conferences were organized
Decision and Risk Analysis and Automatica. He
and two books edited in his honor in 2005-6.
is a Departmental Editor of Production and
He is a member of AAAS, CORS, DSI,
Operations
INFORMS, IIE, ORSI, POMS, SIAM, and
Management.
Recent
honors
include: POMS Fellow (2005), INFORMS
IEEE.
Fellow (2003), AAAS Fellow (2003), IEEE
JOURNAL OF SYSTEMS SCIENCE AND SYSTEMS ENGINEERING