OPTIMAL PRICING OF A PERSONALIZED

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.


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


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


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


of

Professor

manufacturing

of

and

Operations

operations


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
