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					            Co-Creation with Bandwagon Effects




                             Niladri Syam*
              Department of Marketing and Entrepreneurship
                         University of Houston
                          Houston, TX 77204
                        Email: nbsyam@uh.edu
                         Phone: 713-743-4568


                            Rambod Dargahi
              Department of Marketing and Entrepreneurship
                         University of Houston
                          Houston, TX 77204
                     Email: rdargahi@bauer.uh.edu
                         Phone: 713-743-4734


                             James D. Hess
              Department of Marketing and Entrepreneurship
                         University of Houston
                          Houston, TX 77204
                          Email: jhess@uh.edu
                         Phone: 713-743-4175


* Corresponding author

                     Please do not cite without permission



                            August 15, 2013
                          Co-Creation with Bandwagon Effects

                                              Abstract


We study the behavior of consumers who co-create products with a firm and ask how
bandwagon effects (social preferences) on the consumers affect their co-creation choices.
Consumers have to make three related decisions: (1) Should they just buy the standard, off-the-
shelf product or should they buy the customized product? (2) If they choose customization, what
is the optimal design of the customized product (customized design) and how should they split
the design task between themselves and the firm (extent-of-co-creation)? (3) In deciding on the
above, how should they take into account the preferences of other customers (bandwagon
effects)?

Our model allows us to ask how bandwagon effects determine how the market is endogenously
segmented among customers who choose the standard product, those who do the entire
customization task themselves (DIY-ers), and those who co-create the customized product with
the firm. We also analyze how bandwagon effects determine the optimal customized design and
the optimal extent-of-co-creation.

We find that for low levels of bandwagon intensity all co-creators will choose the customized
product at their ideal locations. Interestingly, our main result in this case finds that an increase in
the bandwagon intensity can either increase or decrease the mass of co-creators. When the fixed
setup cost of the customized product is large (small), then an increase in the bandwagon intensity
causes a decrease (increase) in the mass of co-creators. We have experimentally tested our main
result and found support for it.

For higher levels of bandwagon intensity co-creators may choose to customize at a location other
than their ideal location. Thus, bandwagon effects can explain why consumers may opt for less-
than-perfect customization even when they do choose customization over standardization.

We contribute by (1) developing an analytical model of how consumers co-create their ideal
products in the presence of social preferences, and (2) experimentally testing our model.



Keywords: Co-creation; Product Design; Social Preferences; Rational Expectations Equilibrium




1. Introduction



                                                   1
       Consider the following scenario: Bob, an avid dirt-biker and an amateur videographer

always wanted to combine his two hobbies on weekends. Some of the off-road trails he often

rides on are very scenic and Bob would like to share these sights with his friends. The major

problem is that he has to stop to capture the videos since the trails are very bumpy, making it

very difficult shoot the videos while riding. Calling on all his ingenuity, Bob has been able to

come up with an idea for a steadicam stabilizing mount that makes it easier to shoot videos while

on his dirt-bike. Now Bob is faced with the problem of finding a way to get his steadicam

produced. He recently heard of eMachineShop.com, an online vendor that allows people to

design and buy custom engineering products. eMachineShop.com’s user interface requires Bob

to translate his idea to an engineering design, and moreover allows buyers like Bob, who are not

completely familiar with manufacturing, to let the company build or partially build their designs.

At this point, Bob has to make some hard decisions.

       Though eMachineShop.com offers custom designs they also offer standard designs in

their ‘Store Buyers’ webpage. Bob could easily buy a product with a standard design similar to

his requirements even though it may not fully meet his exact requirements. Depending on their

skills, dirt-bikers ride on trails of wildly varying topographical features and the amount of

bumpiness they experience is an important consideration in the design of the steady-cam.

       Bob could instead opt for a custom design. Some aspects of design are more demanding

than others so Bob would have to figure out how much of the design task he would complete and

how much he would ask eMachineShop’s experts to do. If he is willing to devote considerable

time and effort, which in turn will depend on his cost of time and effort, he could do the entire

design task as a do-it-yourself (DIY) project.




                                                 2
       Finally, what about being able to sell the steadicam he creates? He noticed that in

eMachineShop’s ‘Store Sellers’ webpage the company features designs created by people like

Bob and these designs are available for sale to others who might have similar needs. If Bob ever

wanted to sell his co-created steadicam, clearly the demand would be higher if Bob is able to

reflect the preferences of others in his design. Since the sale to others has the potential for

considerable earnings, to the extent of even defraying his total cost of obtaining the steadicam,

this is an important consideration for Bob.

       In light of what has been said above, consumers who consider a customized product have

to make three cognate decisions. First, should they co-create a custom design according to their

specific preferences or should they just get the standard, off-the-shelf design? Second, if they

choose to co-create, how much of the co-creation should be done by them and how much should

be offloaded to the firm? Third, in developing their product design how should they account for

the preferences of other customers in order to be fashionable or to enhance resale value?

       Advances in information technology and manufacturing increasingly facilitate the

participation of the customer in the production of goods and services (Syam and Pazgal 2013). A

practical manifestation of co-creation is user design of products (Griffin and Hauser 1993; von

Hippel and Katz 2002; Randall, Terweisch and Ulrich 2007). Powered by the internet, instances

of user design of products continue to increase. Dahan and Hauser (2002) report how web-based

interfaces have been used for user-design of cameras, copier finishers, laptop bags, laundry

products, etc. Lulu.com allows everyone to become their own publisher and provides individuals

with printing and fulfillment infrastructure that, in the past, was the province of large publishing

houses. In this research, co-creation is conceptualized as a division of the production task




                                                  3
between the firm and the consumer. So the provision of customized products to the consumer is

the end which is achieved through various levels of co-creation between firm and customer.

       From the consumer’s point of view the co-creation of many customized products,

especially high ticket items, have an emotional or social aspect to them. For firms offering co-

created products and for academics seeking to understand them, it is important to understand the

personal and social motivation of the customer who co-creates a product that she will later

consume. Our focus is on the behavior of the consumer who co-creates a customized ideal

product when social factors influence the attractiveness of the product.

       Consider a potential homeowner jointly designing her home with a builder. On the one

hand the homeowner will want to personalize her home to her specific tastes, but on the other

hand she will also keep an eye on the likely tastes of other people. This could be for many

reasons. She may want to resell the home at some point in the future and this is likely to induce

her to reflect the preferences of others in her own design. More generally, she may change her

personalization decisions for the sake of fitting in with what she perceives are the preferences of

other people whose opinions she values. This is Leibenstein’s (1950) bandwagon effect that

       “represents the desire of people to purchase a commodity in order to get into ‘the swim of
       things’; in order to conform with the people they wish to be associated with; in order to
       be fashionable or stylish; or, in order to appear to be ‘one of the boys.’” (page 189).

       One can easily see this in the case of apparel where our desire to get uniquely customized

clothes may be tempered by our desire to fit in with our peer group. When social phenomena are

incorporated then each economic agent is required to account for the behavior of all other

economic agents in her decision making, and in our case the behavior in question is the product

choice of others. Since all consumers in the market are making a similar prediction about others’

behavior, consistency requires analysis in the spirit of rational expectations equilibrium models.


                                                 4
       Moreau and Herd (2010) and Moreau and Bonney and Herd (2011) have empirically

investigated some of the social forces in co-creation. However many aspects of how social

influences affect consumer’s co-creation decisions remain under-researched especially in terms

of building economic models of this phenomenon. Our model adds to the recent spate of work

that incorporates psychological and social forces in analytical models of marketing phenomena

(Amaldoss and Jain 2005). Syam et. al. (2008) have investigated how consumers’ feelings of

regret and miswanting can affect their design of the customized product, though the authors did

not explicitly model the division of the design task between firm and customer. Godes’ (2013)

investigation of how the extent of word-of-mouth communication among consumers affects the

optimal quality offered by the firm is similar in spirit, although the focal actor in our paper is the

consumer.

       In this research we adopt the consumer’s point of view and ask how she co-creates

products taking the firm’s decisions, like prices, as given. We address the following research

questions: (1) How do consumers decide whether they should just buy the standard, off-the-shelf,

product or the customized product? (2) If they choose customization, what is the optimal design

of the customized product (customized design) and how should they split the design task

between themselves and the firm (extent-of-co-creation)? (3) In deciding on the above, how

should they take into account the preferences of other customers (bandwagon effects)? The last

question is important. As already mentioned, at eMachineShop.com drill bits and other machine

tools co-created by the firm and a consumer can then be sold to other consumers via the firm’s

store. These now become the off-the-shelf or default standard products for others, and new

consumers visiting the site can co-create their own machine tools or merely buy one of the

standard tools on sale. At Threadless.com which co-creates customized T-shirts with consumers,



                                                  5
a shirt designed by a given consumer is then made available for sale to others. In both cases the

amateur designers whose designs of drill bits or T-shirts are sold get part of the proceeds from

the sale, and so their utility from co-creation can be greatly enhanced if they are able to reflect

the choices of other consumers in their co-created designs.

       We have two kinds of results pertaining to the segment of co-creators in the market and

their design of the co-created product. First, our most interesting result finds that, as the

bandwagon intensity increases the mass of co-creators can either decrease or increase. When the

setup cost for customization is large then an increase in the bandwagon intensity causes a

decrease in the mass of co-creators. When the setup cost for customization is large there is a

large segment of standardizers and the standard design serves as a focal point which becomes

more attractive as bandwagon intensity increases. On the contrary, when the setup cost for

customization is small then an increase in the bandwagon intensity causes an increase in the

mass of co-creators. This latter result is non-intuitive because, though a small setup cost implies

a large group of customizers, by definition there is large heterogeneity in the designs of the

various customized products and consumers cannot focus on any one design. We have

experimentally tested our main result and found support for it. Second, an increase in the

bandwagon intensity affects the design of the co-created (customized) product in a complex

manner. When bandwagon effects are absent and even for low levels of bandwagon intensity

consumers always choose the customized product at their ideal locations. Interestingly, for high

levels of bandwagon intensity consumers may choose to customize at a location other than their

ideal. Thus, we may see less-than-perfect customization because of bandwagon effects even in

situations where consumers would always choose perfect customization absent such bandwagon

effects. This result is similar to that in Syam et al. (2008) who find that consumers’ feelings of



                                                   6
regret could cause them to optimally choose less-than-perfect customization. Regret of losing out

on a better alternative is an individual-level emotion whereas the current paper incorporates

social aspects of attending to the choices of other people in the market.

       Our main contribution lies in (1) providing an analytical model of product co-creation in

the presence of social preferences on the part of consumers, and (2) experimentally testing the

main predictions of our analytical model. Our analytical model endogenizes the decision of the

customer to opt for the standard or the customized product. In the latter case, we endogenize the

optimal design of the customized product and the extent of co-creation of the customized

product. Incorporating social preferences in co-creation is important because of the increasing

role of consumers in designing their ideal products, and since consumers are social beings, the

role of social forces in their design choices is important to understand.




2. Co-Creation Model

       Some of the forces at work may best be seen by first ignoring the social aspect of co-

creation. This will serve as a benchmark case, to be contrasted with a situation where we

introduce bandwagon effects which are social preferences or other-regarding preferences. In

what follows, when we refer to the non-standard product we will use the terms customized

product and co-created product interchangeably. In both cases the product that is finally

consumed is customized to the customer’s preference but it may or may not be co-created with

the firm depending on whether or not it is optimal for the customer to do some of the design task

herself. Our focus is on investigating the behavior of customers who take the firm’s prices as

given and decide whether and how to co-create with the firm.




                                                  7
2.1 Co-creation without Bandwagon Effects

       In the current section we analyze the base case where consumers have only self-regarding

preferences, that is, there are no bandwagon effects. Consider a market where a monopolist

firm’s product has attribute level 0, and where a unit mass of consumers have preferences for

ideal products distributed uniformly on [0, 1] . The monopolist sells two types of products, a

standard product and a customized product that differs from the standard one. The consumption

utility of the self-focused consumer whose ideal is at x from buying the standard product is

                                      U Self  V  x  P .
                                        Std                                                    (1)

Here V is the consumer’s valuation for his ideal product, but the standard product at 0 is less

than ideal by an amount x, and P is the price of the standard product.

       Consider the customized product. Ignoring the social aspect for the time being, the

consumption utility of consumer x from buying a customized product z is

                   U Self                                                2
                     Custom  V  ( x  z)  (P  PS  PC k )  (z  k ) .                    (2)

The meanings of the variables will be explained subsequently. We allow the customer to

optimally choose the amount of customization she desires. She can get a customized product at

point z (≤ x), where z denotes the ‘customized design’ desired by her. Depending on parameters

the buyer at x may optimally choose to get her customized product exactly at her ideal location x,

in which case z=x. Said differently, the ‘customized design’ z is the consumer’s optimal design

of the product. Since a consumer’s social preferences are incorporated (Section 2.2) the optimal

design of the customized product may not be at her ideal location x.

       In the spirit of co-creation the customized product at z can be produced collaboratively by

the firm and customer. The ‘extent-of-co-creation’ will be denoted by k, 0≤ k ≤ z, where the

‘extent-of-co-creation’ is just the division of the manufacturing task between customer and firm

                                                  8
(see Anderson 2002). The firm will partially co-create the customized product’s design from 0 to

k and the customer x will complete the customized design from k to z. The extent-of-co-creation

k will be determined optimally depending on the price changed by the firm for its share of co-

creation and the customer’s own cost of co-creating. In many instances, for example at

eMachineShop.com, it is the customer who decides to what extent she would need the firm to

step in to help co-create the product. To summarize, there are two aspects of customization

choices for the consumer at x: she will optimally choose both the ‘customized design’ z and the

‘extent-of-co-creation’ k. The former determines her optimal design of the customized product

and the latter determines the division of the design task between the firm and the consumer.

       Since the manufacture of customized products involves a separate setup for different

customers, most sellers of customized products charge a setup price for them. This is true in the

case of eMachineShop.com where the vendor charges a setup price which is independent of the

amount of custom work that needs to be done. Indeed, the following quote from their website

makes this point abundantly clear:

        “In custom manufacturing - whether for machine parts from eMachineShop, printed
       brochures from a printer or stuffed teddy bears, setup costs are often the main factor in
       the price when ordering just one or a few parts…The reason is that a custom
       manufacturer has to go through several steps whether you order one or many parts”

       The setup price, PS, is charged to the customers regardless of the extent of customization

they require the firm to do and is added onto the standard price. Additionally, if the consumer

requires the firm’s help with customization then the firm changes a price of PC per unit of

customization, so that the total price to customize from 0 to k is P+PS+PC k. As an illustration,

eMachineShop’s pricing link specifies the following about its pricing:

       “Have eMachineShop staff make a CAD drawing (or convert from another format) for
       you for a small fee.” [our italics]
or,

                                                 9
       “Draw your part yourself in the free CAD Download (or use the import feature) to get an
       instant price.”[our italics]


       Of course, if the customer rationally chooses to do all the customization herself (k =0),

then she will not incur either the setup price or the unit price for customization. The customer

completes the product by bringing it from k, where the firm delivers it, to z at a cost that is

quadratic in the distance. The cost of product completion borne by the customer is (z  k ) 2 .

       The following result specifies the customization choices for the typical consumer whose

ideal product is located at x. Notice that not all consumers will want to customize and we will

return to this below.

       Proposition 1: Let 0<PC<1. If the consumer whose ideal is located at x chooses a
       customized product, the customized design and extent of co-creation are
                                   P                             P            P     
       a) z*=x and k*=0 when x  0, C  or b) z*=x and k*= x  C when x   C , 1 .
                                   2                            2            2 

Proof: See Appendix.

       Proposition 1 gives the customization choices of consumer x. This consumer has to make

two related decisions regarding her choice of customization. First, she has to determine the

customized design, in other words, the product design given by z*. Second, she has to determine

the extent-of-co-creation, in other words, the division of labor specifying how much she wants

the firm to build, k*, and how much she will do herself, z* k*.

       In the absence of social influences the typical consumer would optimally choose a

product customized to her ideal location, that is at z*=x. The consumers located further away

from the standard product at 0 will choose to co-create their ideal product such that the firm does

                                          P
an amount of production given by k *  x  C , and the consumer at x then completes the
                                          2

product by customizing the rest of the way from k* to x. It makes sense that the extent of co-

                                                 10
creation done by the firm decreases in the price PC and increases in the customer’s marginal cost

of self-production δ. Consumers who are located closer to the standard product at 0 will do all of

the production themselves. This is because, being located close to the standard product, their cost

of customization is not very large and they would like to avoid the setup cost of co-production.

We call these consumers the Do-It-Yourself (DIY) types.

       The above analysis assumed that the customer was buying a customized product.

However, the customer could choose to avoid customization altogether and merely buy the

standard product, although there is an opportunity cost of not having the ideal design. The

following proposition shows how the market is divided between the customized and the standard

products.

                                         PC
       Proposition 2: Let 0<PC<1 and            ˆ
                                              < x <1 where
                                          2
                                                           2
                                        x S
                                        ˆ     P   PC   /(1  P ) .
                                                                                                    (3)
                                                                   C
                                                     2  
                   P 
       a. For x  0, C  the consumer at x chooses a customized product over the standard
                   2 
          product. Her customization choices are k*=0 and z*=x. These consumers are Do-It-
          Yourself (DIY) consumers who get the perfectly ideal product at x, but do all the
          customization themselves by choosing the firm’s portion of customization k* to be
          zero.
                  P       
       b. For x   C , x  the consumer at x chooses the standard product 0.
                         ˆ
                   2 
       c. For x  x, 1the customer at x chooses a customized product over the standard
                   ˆ
                                                           P
          product. Her customization choices are k*= x  C and z*=x. These consumers are co-
                                                           2
          creators. They get their perfectly ideal product at x, but ask the firm to do part of the
          customization, k*>0, and then complete the product by bringing it to their ideal
          location x.

Proof: See Appendix.




                                                11
       Figure 1 shows how the market is segmented between the customers who prefer the

standard product and the customized product. Among those who prefer the customized product,

some choose to do all the customization themselves and are the DIYers. Other customers who

prefer the customized product over the standard product are the co-creators who will jointly

design the customized product with the firm.


                              Choose
                             Standard
           DIY                Product                           Co-Create

       0
                 PC                                    ^
                                                       x                              1
                 2
      Figure 1: Market segmentation between DIYers, standardizers and co-creators


       The market segmentation in Proposition 2 shows that there are two distinct, non-

                                                                               P 
connected segments of consumers who purchase the customized product. Those in 0, C 
                                                                               2 

customize the product without any help from the firm. On the other hand, those in x, 1
                                                                                   ˆ

                                                                           P 
customize the product with the help of the firm. The segment of DIYers in 0, C  has a very
                                                                           2 

small cost of customization since their design are very close to the standard and they will just

                                                                   P      
customize their ideal product themselves. The customers in segment  C , x  are farther away
                                                                         ˆ
                                                                    2 

from the standard product and so have a higher cost of customizing the product all by

themselves. If they chose to customize they would do it jointly in collaboration with the firm.

However, the misfit of their ideal product from the standard product is not very large and so they

cannot justify paying the setup price charged by the firm for the customized product. These

                                                12
consumers then opt for the standard product. The customers in segment x, 1 have ideals very far
                                                                       ˆ

away from the standard product and so have a very large marginal cost of customizing the

product all by themselves. They therefore prefer to have the firm co-create the customized

product. Moreover, since they have high misfit they are also willing to pay the setup price to get

an ideal product.

          Note the role of the setup price in creating the mass of standard product purchasers. If the

                              P         P     P 
setup price is very small, x  C or Ps  C 1  C  , then all consumers purchase a
                           ˆ
                               2        2    2 

customized product. Those closer to the firm will be DIYers and those further away are co-

                                                                                    P 
creators. On the other hand if the setup price is very large, x  1 or Ps  1  PC 1  C  , then
                                                              ˆ
                                                                                      4 

there will be no co-creators in the market. Those closer to the firm will be DIYers and those

further away will just purchase the standard product. Throughout, we assume setup costs are

neither very small nor very large.


2.2 Co-creation with Bandwagon Effects

          The valuation of many products has a social aspect. We assume that a person who

consumes design x gets additional utility if this product reflects the preferences of other

consumers in the market. In addition to wanting a fashionable product, the consumer may also

want to sell the product at some point in the future. This is the bandwagon effect, which we

model by computing the aggregate distinction of the chosen design from the products chosen by

others.

          Consider the standard product. The other-regarding or social utility from buying the

standard product for consumer x when there are bandwagon effects is



                                                   13
       Bandwagon
     U Std        V  P  x    - Expected distance from 0 to what others choose) .         (4)

In the above  is the bandwagon intensity parameter. It the degree to which consumers care

about reflecting the design choices of others in their own design.  is the value that consumer

derives when her choice of product is maximally fashionable in the sense that it is perfectly

consistent with that of others. Her happiness is diminished if there is some misfit between her

product at 0 and what others choose, and in that case gives additional utility “(- Expected

distance from 0 to what others choose).” Seen through the lens of the resale metaphor, “-

Expected distance from 0 to what others choose” can be thought of as the resale price adjusted

by the less fashionable product at 0. When the standard product is exactly the product desired by

others, then the consumer that buys design 0 is perfectly fashionable and gets the maximum

possible price of  from a resale. Of course, what others would choose is endogenously

determined and the typical consumer x calculates her utility based on how she expects the others

to behave. In a Rational Expectations Nash Equilibrium her expectation of others behavior has to

be consistent with the others’ true behavior.

       Similar to the case of the standard product, the social utility from buying the co-created

customized product for consumer x when there are bandwagon effects is

                  Bandwagon
                U Custom     V  ( x  z)  P  (PS  PC k )  (z  k ) 2
                                                                                                (5)
                       ( - Expected distance from z to what others choose).

       Note that here the distance is from z to what others chose. As in Section 2.1 we allow

consumers to choose their customized design at a location which is different than their ideal. We

assume that the customer at x wants to get the customized product at z≤ x, where the ‘customized

design’ will be optimally determined as in Section 2.1. The customized design z can be co-

created by the firm and customer, and the ‘extent-of-co-creation’ will be denoted by k, 0≤ k ≤ z.

                                                  14
       When there are bandwagon effects, the optimal ‘customized design’ z and the ‘extent-of-

co-creation’ k could depend on the bandwagon intensity . As with the case of choosing the

standard product, the term “ Expected distance from z to what others choose” can be thought

of as the resale price adjusted by the misfit of the owned product at z and others’ desired product.

When the customized product at z is exactly the product desired by others, there is no misfit and

consumer x gets the maximum possible price from a resale.

       The rational expectations Nash equilibrium calculation is complicated because consumers

have to form rational expectations of others’ behaviors along two dimensions: (1) The designs of

the customized product chosen by these other consumers who choose to customize and, (2) The

mass of the other consumers who choose to customize or to buy the standard product.

       Consider the following market segmentation when consumers have social preferences,

that is, they experience bandwagon effects. In the rational expectations Nash equilibrium

framework we proceed under the assumption that every consumer believes that all those

consumers who lie to the right of a product design  (ideal points higher than  ) will choose an

ideally customized product while those to the left of  (ideal points lower than  ) but above

PC/2 will choose the standard product. This belief will be shown to be consistent in equilibrium.


                                   Choose
                                   Standar
              DIY                     d                               Co-
                                   Product                       
                                                                     Create
          0         PC                                                                 1
                    2
                    Figure 2: Market segmentation with bandwagon effects




                                                15
                      P      
        Customers x   c ,   will purchase the standard product. For the focal standardizing
                       2 

consumer x who chooses the standard product at 0, we have

                          “Expected distance from 0 to what others choose” =
                    PC                                                               2
                                                      1               2 1 1  PC           (6)
                   
                   0
                    2
                         ( y  0)dy  PC (0  0)dy   ( y  0)dy       .
                                       2
                                                                     2 2 2  2 

The first integral on the right hand side is for all the other customers who have ideal points y

        PC
below      , and who choose to DIY customized products at y. Their distance from the focal
        2

standardizing consumer who gets the standard product at 0 is y-0 and the segment of such

              P 
consumers is 0, C  . The second integral is for all the other customers who have ideal points y
              2 

   P      
in  C ,   , and who are believed to choose the standard products at 0. Their distance from the
    2 

focal standardizing consumer who gets the standard product at 0 is (0-0). The third integral is for

all other customers who have ideal points y in [  , 1], and who are believed to get customized

products at y. Their distance from the focal standardizing consumer who gets the standard

product at 0 is equal to y-0.

        Customers x  [  , 1] will purchase the customized product at z. For the focal customizing

consumer who chooses the customized product at z we have

                         “Expected distance from z to what others choose” =
                 PC
                                                         z           1
               0
                 2 (z ( x )  y)dy   
                                      PC                               
                                         (z( x )  0)dy  (z  y)dy  ( y  z)dy.
                                                                     z
                                                                                              (7)
                                      2

                                                                                        P
The first integral on the right hand side is for customers y who have ideal points below C , and
                                                                                         2

who DIY to get customized products at y. Their distance from the focal customizing consumer
                                                     16
                                                                                  P 
who gets the customized product at z is z-y and the segment of such consumers is 0, C  . The
                                                                                  2 

                                                                        P      
second integral is for all the other customers who have ideal points in  C ,   and who choose
                                                                         2 

the standard product at 0. Their distance from the focal customizing consumer who gets the

customized product at z is z-0. The third and fourth integrals are for all other customers who

have ideal points y in [  , 1], and who choose to get customized products at y. Their distance

from the focal customizing consumer who gets the customized product at z depends on whether

y is less than z (third integral) or larger than it (fourth integral).

        For consumers with ideals in [ , 1] what is the best product design z? The consumer at x

will either choose her co-created customized product at her ideal location x or at a point z < x.

We will denote the former situation as ‘Ideal’ co-creation and the latter as ‘Non-Ideal’ co-

creation. It is not rational for her to choose customization at a point z > x because she will get a

non-ideal product and at a higher customization cost. The following lemma determines the

optimal design of the customized product for consumers with ideals in [ , 1] when there are

moderate levels of social bandwagon preferences.


        Proposition 3: Let the bandwagon intensity be small,  < 1  PC , then in a rational
        expectations Nash equilibrium the customized design chosen by consumers at x  [  , 1] is
        z*=x. Thus, there is Ideal co-creation.

Proof: See Appendix.

        Notice the force of Proposition 3 in ensuring that a consumer’s expectations of other

consumers’ behavior with respect to product design are rational. The consumer at x  [  , 1]

expects that all other consumers at y  [  , 1] will choose z(y)=y. Given these beliefs, she then

finds it optimal to design her customized product at z=x. Since x is just a generic consumer who

                                                    17
is designing her optimal customized product, the design given in proposition 3, z*=x, is a rational

expectations Nash equilibrium design.

       Proposition 3 also states that when the bandwagon intensity is low then the co-creation is

of the ‘Ideal’ co-creation type. The condition that bandwagon intensity is small has bite. We will

show in Section 4 that when bandwagon effects are higher, then it is possible to have equilibria

with Non-Ideal co-creation. Of course DIY consumers also get the customized products at their

ideal locations, but they are not co-creators since they do all the production themselves.

       Using z=x in (6), for the focal co-creating consumer we have

                  “Expected distance from z=x to what others choose” =
                PC
                                                   x              1
               02 (x  y)dy  PC
                                      ( x  0)dy   ( x  y)dy   ( y  x )dy 
                                                                  x
                                 2                                                                 (8)
                                             2
                           2 1 1  PC 
               x2  x               .
                           2  2 2  2 

       Substituting (6) and (8) into (4) and (5), and upon using z=x, we obtain the social utilities

from buying the standard and customized products:

                                                      2 1 1  PC  
                                                                    2
                 Bandwagon
               U Std       ( x )  V  P  x                 and                          (9)
                                                      2 2 2  2  
                                                                     

                                                                           2 1 1  PC   
                                                                                              2
     Bandwagon
   U Custom    ( x )  V  P  PS  PC k  ( x  k ) 2    - x 2  x                  .   (10)
                                                                           2 2 2  2        
                                                                                              

We must derive the optimal extent-of-co-creation k that the consumers in [,1] will choose by

maximizing utility given in (10). Taking the derivative of (10) with respect to k we can find the

optimal k, in other words, the ‘extent-of-co-creation’ that customer x desires:




                                                  18
                                  Bandwagon
                               U Custom
                                               PC  2( x  k )  0
                                      k                                                       (11)
                                                  P
                                      or k *  x  C
                                                   2
As a result, the optimal utility of consumers in [ , 1] is


                                            P 
                                                   2                 2 1 1  PC  
                                                                                   2
     Bandwagon
   U Custom    ( x )  V  P  PS  PC x   C     - x 2  x             .         (12)
                                             2                    2 2 2  2  
                                                                                    

        One can quickly check that in the limit without bandwagon effects (→0), upon

substituting k *  x  PC /(2) into (10) and equating with (9), the consumer who is indifferent

                                                                      P2 
between the standard and co-created products is given by x  x   Ps  c  /(1  PC ) . Thus, the
                                                             ˆ
                                                                      4 
                                                                         

identity of the indifferent customer in the limit without bandwagon effects is the same as that in

Proposition 2 in section 2.1, as is reasonable to expect. From the above discussion, when the

bandwagon intensity is low, that is when 0     , then the customer at x  [  , 1] will choose

  *             *            PC 
 z ( x )  x ; k ( x )  x     .
                             2 

        The utilities in (9) and (12) depend upon . We now consider how this threshold that

defines the number of co-creators is determined. It is clearly related to the bandwagon intensity

parameter . Solve for the consumer who is indifferent between the standard product and the

ideal co-created product. For consistency of our rational expectations Nash equilibrium the

consumer at x=  should be indifferent between buying the co-created product and the standard

               Bandwagon            Bandwagon
one, that is U Std         ( )  U Custom    ( ) . Consider the consumer at x=  . Substituting x=

 in (9) gives the optimal utility of choosing the standard product in terms of  :



                                                   19
                                                    2 1 1  P       
                                                                          2
                     Bandwagon
                   U Std       ( )  V  P         C          .                 (13)
                                                    2 2 2  2         
                                                                        
Substituting x=  in (12) gives the optimal utility of choosing the customized co-created product

in terms of  :


                                             P 2            3 2 1 1  PC  
                                                                            2
         Bandwagon
       U Custom    ( )  V  P  PS  PC   C                   .                (14)
                                              4             2   2 2  2  
                                                                             


The rationality condition gives us an implicit equation which defines the equilibrium threshold of

co-creating, denoted * . If we set (13) and (14) equal we obtain a quadratic equation in :

                                            P P       P 
                     2 2  (1  PC  )  C  C   C    PS  0 .
                                                2                                            (15)
                                            2        2  

Equation (15) can be solved to find the marginal consumer * who is indifferent between buying

the co-created (customized) product and the standard product, when consumers take into account

the product choices of other consumers in the market. Equation (15) has two roots and the

following result rules out the larger one,


       Proposition 4: In a rational expectations Nash equilibrium with small bandwagon
       intensity   (0,  ),   1  PC , the consumer who is indifferent between standardization
       and co-creation is given by
                                                            2           
                                                      PC     1  PC 
                                                                    ˆ
                                                                       x
                      * 1        1  PC             2             
                       1              1  1  8                      .
                           4                                    2                     (16)
                                                          1  PC       
                                                        1 
                                                                       
                                                                    
Proof: See Appendix.




                                                20
       Economic forces help determine * and therefore the proportion of consumers that chose

to co-create. It is easy to see, for example, if the setup cost of co-production, PS, increases, then

there is less co-creation: * increases in x which from (3) is positively related to setup cost, PS.
                                           ˆ

       More interestingly, social forces also affect which consumers choose to co-create. How

do the proportions of co-creators and standardizers change when the bandwagon intensity 

increases? Intuitively one might expect that as the bandwagon intensity grows more consumers

will want to imitate the cohort of customers that are buying the standard product. This intuition

can be mistaken, as seen in our principal theorem.

                                                                                 ˆ
       Theorem 1: Let bandwagon intensity be   (0,  ) . There is a setup cost PS such that
                             *
                   ˆ then   0 : when the setup cost is high then an increase in bandwagon
       a. If PS  PS
                           
          intensity decreases the number of co-creators;
                               *
                  ˆ       
       b. If PS  PS then      0 : when the setup cost is low then an increase in bandwagon
                           
           intensity increases the number of co-creators.

Proof: See Appendix.

       This result is intriguing because it shows that as the bandwagon intensity increases the

number of co-creators may either increase or decrease; this differential effect is determined by

the magnitude of the setup costs of co-production. Why? As the bandwagon intensity increases,

a typical consumer would increasingly like in her own product to reflect the preferences of

others. Since a large PS implies that * is also large, many consumers will naturally choose the

standard product to avoid the setup costs. The typical consumer then leans even more towards

the standard product when her bandwagon intensity increases because it is fashionable. Thus, the

segment of customers who choose the standard product increases as the bandwagon intensity

increases. On the other hand, when PS is small then * is also small, implying that many


                                                 21
consumers choose a co-created, customized product. A consumer who wishes to reflect others’

choices in her own will opt for a customized product over the standard one. Thus, the segment of

customers who choose the co-created customized products increases as bandwagon intensity

increases.

       Two aspects of Theorem 1 are important to note. First, the insight offered in Theorem 1b

would be difficult to guess without careful analysis. Theorem 1a says that as the bandwagon

intensity increases, more consumers will buy the standard product. Since there is one fixed

design for the standard product (at 0), one might guess that if enough people are buying the

standard product and the social preference (bandwagon intensity) increases then the attraction for

the standard product further increases. However, unlike the standard product, there is no fixed or

focal design of customized products, which by definition are different for different consumers.

Part 1b analyzes the case where a large proportion of the market is buying customized products

and the bandwagon intensity increases. The heterogeneity of custom designs means that there is

no fixed custom design for a customer to focus on as she decides whether to customize or

standardize. In this case consumers have to mentally do a rather complex calculation of how

close their customized design will be to the ‘average’ of all the custom designs of all the other

customizers in the market. As the social preference (bandwagon intensity) increases, more co-

created products are chosen. In Section 3, we find empirical support for this non-intuitive result.

       Second, we haven’t assumed an exogenous group of consumers of fixed size that

absolutely prefers the standard product. If the mass of standardizers is exogenously fixed, then it

is a forgone conclusion that this group, and its choice of product, becomes more and more

important as the intensity of social preference increases. In that case, as the bandwagon intensity

increases more consumers would like to purchase the standard product in order to be more



                                                22
fashionable. In our model the segment of customers choosing the standard product is

endogenously derived, so as the bandwagon intensity increases either more or fewer customers

will buy the standard product as detailed in Theorem 1. All consumers form rational expectations

of other consumers’ behaviors and make optimal choices based on maximizing their own

utilities, although these utilities have social components.

       Theorem 1 has a clear managerial implication for firms that wish to increase the number

of people that co-create products with them. Part (b) of the proposition states that the number of

                                    *
co-creators will increase (that is       0 ) when the setup cost of customization, PS, is small and
                                    

the firm is able to induce higher intensity of social preference among its customers. One obvious

manner in which the firm can influence a higher intensity of social preference is to encourage the

customers to focus on the resale value of their customized product. When the resale value is high

then consumers will want to attend to the preference of others in the market. Of course, the firm

usually gets a part of the resale price of custom designs created by consumers since the firm

usually displays these designs in their ‘buyer’s section’, as in the case of eMachineShop.com.

Thus, a combination of low setup cost of customization but a high resale price, part of whose

proceeds the firm will get, will lead to an increase in the mass of co-creators for the firm.

       Below, we will test the main implication of our model given in Theorem 1.

3. Experimental Study

The theory is tested in a 2×2 design experiment. One factor was setup cost (low versus high) and

the other was bandwagon intensity (absent versus present). Thirty-six students participants were

recruited from a major public university by providing extra credit in a course, as well as having

the opportunity to win $5 or $10 based on their performance in the experiment.


                                                 23
3.1 Procedure

       The experiment was done in a computer lab using z-Tree (Fischbacher 2007).

Participants designed a bicycle that they would buy for a niece, where the design attribute was its

color. The standard bicycle was a very pale green color (shade #0), but it could be customized to

darker shades of green, as shown in Figure 3. Participants were told the shade of green that their

niece found ideal, and were asked to make choices on behalf of the niece, regardless of their

personal tastes.




                       Figure 3: Shades of Green to Customize Bike Color


       The participants could either buy the standard bike for their niece, or incur the cost to add

layers of paint to bring the bike’s color closer to their niece’s ideal. Adding additional layers to

the bike could be done either by the local bike store or by themselves. The standard bike cost

$100, and the bike store charged a setup cost (low =$14 and high=$20) plus $7 for every layer

added by the store. They could also chose to paint layers by themselves at a cost; a cost table

was provided telling the total cost of DIY or co-creation. When setup costs are low, the theory

predicts a midrange threshold color for co-creation, so participants were told their niece preferred

a shade # 3 green (which is below the threshold). When setup costs were high, the niece wanted

a shade #7 green that is above the corresponding threshold. Participants needed to make three

decisions: (1) Will they choose the standard bike color or a customized color? (2) In the case

they choose the customized bike, how many layers of paint did they want to add to the standard

                                                 24
bike color? (3) Of the number of total layers that they want, how many will they paint

themselves (given that the store will paint the rest)? Each scenario was repeated three times in

order for the participants to have an opportunity to correct mistakes.

       After these decisions were recorded, participants were told that because their niece was

likely to outgrow her bike, they would also have to consider the resale value of the bike (the

niece could use the resale price for other things she wanted). The resale price depended upon

how different the bike is from the color of bikes chosen by others. They were told that every

shade of green from 0 to 10 was equally likely to be the ideal for someone else’s niece. Resale

value sheets were provided and the decisions were repeated for low and high setup cost.

Decisions were repeated for low and high setup costs, only now when resale value of the bike

was relevant. The theory predicts that for low setup costs, the threshold between standardization

and co-creation should diminish and the participant should switch to now wanting to co-create a

customized bike. On the other hand, for high setup costs, the resale value should motivate the

participant to buy a “popular” standard bike.

       The theory developed above implies the following hypotheses.
   H1: When setup costs are low, stronger bandwagon intensity increases the probability that a
         consumer co-creates.
   H2: When setup costs are high, stronger bandwagon intensity decreases the probability that a
        consumer co-creates.
As a consequence of these, we can also hypothesize:
   H3: Higher setup cost negatively moderates the effect of bandwagon intensity on probability
          of co-creators.



3.2 Logit Model of Co-Create or Standardize

       Experimental design predicts that the probability that co-creation is chosen depends upon

whether the setup cost is small or large and whether the bandwagon effect is absent or present,

                                                25
along with a moderated effect. The logit model of choice expresses the log odds of co-creation in

                                       dum             dum
terms of a factor  0  1 dum   2 PS   3 dum  PS , where dum and PSdum are the

dummy coded variables represent low and high values of bandwagon intensity and setup cost.

The coefficients i represent the effects of variables on log-odds, but our theory-driven

hypotheses are about probabilities. For example, the moderation effect on probabilities is not the

logit coefficient 3 of the interaction term ×PS on , but the difference in differences in

                              1                           1                     1                  1        
probabilities,                                                                                              . These
                       (  0  1   2   3 )         ( 0   2 )         (  0  1 )         ( 0 ) 
               1  e                              1 e                 1  e                  1 e          

are a nonlinear functions of all the parameters, but can be computed along with its standard

errors via bootstrapping (see Ai and Norton 2003, and Hess, Hu, Blair 2013).

                                                 Table 1
                         Logit Regression of Co-Creation versus Standard Product
                         as a function of Bandwagon Intensity  and Setup Cost PS
                                           Incremental effect
                                            on probability of       p-value        Logit Coefficients
   Hypothesis                                 co-creation                                 i

                                                  0.203*
                       PS=0                (0.095)
                                                                     0.0322
                                                 -0.501*
                       PS=1                (0.089)
                                                                     0.000
                                                 -0.705*                           -4.964*
                         ×PS                  (0.130)
                                                                     0.000
                                                                                   (1.245)
                                                                                   2.772*
                             PS                                                    (1.076)
                                                                                   1.299*
                                                                                  (0.643)
                                                                                   0.779*
                        Intercept                                                  (0.364)
                     Notes: Setup Cost PS and Bandwagon Intensity  are dummy coded {0,1}
                            Standard error in parentheses; Number of observations=144
                            Value of the log-likelihood= -64.505
                            Number of bootstrap resamples for Incremental Effects on Probability=5,000.



         The estimated logit coefficients and resulting incremental effects on the probability of co-

creating are found in Table 1. Figure 4 illustrates the findings.

                                                           26
                         Probability of Co-Creating Product




                        0.972                                 0.888

                                                                       Setup Cost PS
                                                                           High
                        0.685
                                                                           Low


                                                              0.472




                                Low                       High

                                      Bandwagon Intensity 

                Figure 4: The Effect of Bandwagon intensity on Co-Creating

       According to hypothesis H1, when setup costs are low then many customers choose to co-

create a customized product, although a wide variety of different designs are chosen (between *

and 1). If the bandwagon intensity increases, customers who previously chose the standard

product are drawn to co-creation because there are so many other co-created products being

used. In the experiment with setup costs low ($14), the person whose ideal is shade #3 green

tends to switch from the standard green to a co-created customized green when the bandwagon

effect is increased. The probability of co-creating rises by 0.203 (p=0.03) supporting H1.

       On the other hand, when setup costs are large, very few consumers want to co-create and

many choose the standard product to avoid the setup costs. Hypothesis H2 states that as the

bandwagon intensity increases, customers who previously co-created will be drawn toward the

large number of people choosing the standard product. In the experiment, when setup costs are



                                                  27
high ($20), the person whose ideal is shade #7 green tends to switch from a customized green

color to the standard color when the bandwagon effect is increased. The probability of co-

creating falls by 0.501 with p=0.00, thus supporting H2. The difference-in-difference in

incremental effects of bandwagon intensity for low and high setup costs, -0.705, is also highly

statistically significant, in line with the moderation hypothesis H3.


4. Higher Bandwagon Intensity

       The analysis in Section 2 was carried out when the bandwagon intensity of consumers is

low (<1-PC). In that case we saw that consumers always rationally choose Ideal co-creation

where the customized design is at their ideal location. In the current section we show that when

the bandwagon intensity is high then consumers may choose Non-Ideal co-creation. Since the co-

creation without bandwagon effects is always ideal co-creation, this shows that bandwagon

effects are the reason for Non-Ideal co-creation.

       We will construct an equilibrium where all co-creating consumers are doing Non-Ideal

co-creation, and demonstrate that, unlike the case with low bandwagon intensity, such an

equilibrium can be rational when bandwagon intensity is higher. Consider the market

segmentation in Figure 5 where all consumers located in [χ, 1] are Non-Ideal co-creators who

buy the imperfectly customized design at χ.

                            Choose
                           Standard                         Co-Create
            DIY             Product                        to Non-Ideal
                  Pc                             
        0                                                                           1
                  2
                  Figure 5: Market segmentation with non-ideal co-creators

                                                 28
       Consider the consumer located at x, ≤x≤1. She could have the following alternate belief

about how the market is segmented:

Belief NIC (Non-Ideal Co-creation): Consumer x expects that all other consumers y in [, 1] are
non-ideal co-creators who customize their product at 

       The following result shows that for higher bandwagon intensity this can be rational.


       Proposition 5: There exists a * such that 1-PC< * where there exists a rational
       expectations Nash equilibrium with the following characteristics:
                          P 
       1. Consumers in 0, C  are DIYers who customize the product at their ideal locations
                          2 
                                                            P        
          but do not co-create with the firm, consumers in  C , *  are standardizers and
                                                             2      
          consumers in [  *,1] are co-creators.

       2. All co-creators choose the customized product at *, that is, they are Non-Ideal co-
                                                                                                P 
          creators. The optimal choices for the co-creators are z* ( x )  * ; k * ( x )  *  C .
                                                                                                2 
       Proof: See Appendix.

       Proposition 5 states that Non-Ideal co-creation can be an equilibrium when the

bandwagon intensity exceeds 1-PC (although it may not be the unique one). This contrasts with

the case of low bandwagon intensity, β<1-PC, where Non-Ideal co-creation is not possible.

       It is important to note the difference between  * in section 2.2 and  * in the current

section. In section 2.2 all consumers x in [  * , 1] are Ideal co-creators who get the perfectly

designed customized product at their ideal locations x. In the current section all consumers in

[  * , 1] are Non-Ideal co-creators who get the imperfectly designed customized product at

 *  x . The critical difference between low and high bandwagon intensity lies in the optimal

design of the co-created product. When the bandwagon intensity is low (section 2.2) consumers

always choose the co-created customized product at their ideal location and therefore the design



                                                  29
of the co-created product does not vary with . In contrast when the bandwagon intensity is high,

the optimal design of the co-created product, given by * , depends on .

5. Conclusion

       Co-creation of goods and services jointly by the firm and consumers is growing in

popularity owing to increases in information technology and user interfaces. This research

focuses on the behavior of consumers who co-create custom designs with the firm when social

influences are important.

       We develop an analytical model of how social influences, specifically bandwagon

effects, determine the market segmentation among customers who choose the standard product,

customers who design their ideal customized product entirely by themselves (DIYers), and

customers who co-create their product with the firm. Our model also allows us to investigate

how social influences determine the optimal design of the customized product, that is, the

customized design, and the optimal extent-of-co-creation.

       We find that those who buy the standard product are sandwiched on either side by

customers who do all the customization by themselves (DIYers) and customers who will co-

create with the firm. Thus, there are two distinct, non-connected segments of consumers who

purchase the customized product. Our main result finds that, as the bandwagon intensity

increases the mass of co-creators can either decrease or increase. When the proportion of the

market that chooses the standard product is fairly large, then an increase in the bandwagon

intensity causes a decrease in the mass of co-creators. On the contrary, when the proportion of

the market that chooses the standard product is small, then an increase in the bandwagon

intensity causes an increase in the mass of co-creators. This result has been experimentally tested

and we have found support for it. Finally, we find that for high levels of bandwagon intensity

                                                30
consumers may rationally choose to customize the product at a location other than their ideal.

This contrasts with the case without bandwagon effects or when the bandwagon intensity is low,

in which case consumers will optimally design their customized products at their ideal locations.

Our contribution lies in developing an analytical model of product co-creation when consumer’s

social preferences are important, and secondly we experimentally test the main predictions of our

model.

         This paper focuses on co-production, leaving aside the process by which the product

design is created by an interaction of buyer and seller. For complex, multidimensional products,

consumers may not have the expertise to know what constitutes a feasible bundle of product

attributes and may require assistance from the seller on co-designing the customized product.

Since our focus is on the behavior of co-creating customers we also set aside the issue of how the

prices are determined by the seller. These and other issues are left for future research.




                                                 31
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                                               33
                                              Appendix
Proof of Proposition 1:

                 P 
Case a: Let x  0, C  . First, fix a z ≤x and determine the optimal k given z. From (2), the first
                 2 

                                                  U Self
                                                     Custom
order condition for an interior solution gives,                   =  PC  2(z  k ) =0, which implies
                                                         k

                P             P
that k (z)  z  C . But z ≤x< C , therefore k(z) is negative. Since k is the amount of
                2            2

customization to be done by the firm, it has to be non-negative and so an interior solution is

inadmissible. At a corner solution, we will either have k * (z) =0 or k * (z) =z. What is rational for

the consumer to do? From the consumer surplus (2), the consumer’s total cost of customization is

C(z, k)= PS  PC k  (z  k ) 2 . If we set k=0 the cost is C(z,0)   z 2 , recalling setup cost is

avoidable. On the other hand, if we set k  z the cost is C(z, z)= PS  PC z . Clearly, since z ≤

PC                                 P
   by assumption, therefore  z 2 ≤ C z ≤ PC z < PS  PC z .Thus, we have C(z, 0)≤ C(z, z),
2                                  2

and the consumer is better off DIYing, k*=0. Substituting this value into (2) gives

CSSelf                            2
  Custom  V  P  ( x  z )   z . Second, optimizing now with respect to design z, the first

                                                              1                    P    1
order condition for an interior solution of z gives z=           . Given that z ≤x< C <   , the interior
                                                              2                   2 2

solution is inadmissible, so the consumer will choose the largest possible z, z*=x.

                P     
Case b: Let x   C , 1 . First, fix a z ≤x and determine the optimal k given z:
                 2 




                                                    34
                                                          P
                                            0      if z  C ,
                                                          2
                                    k (z)  
                                            z  PC if z  PC .
                                            
                                                2        2

We analyze two subcases.

                P
Case b.i: If z ≤ C then k (z)  0 . Substituting this in the consumer’s surplus and following a
                2

logic similar to above, we can see that she will choose the maximum possible z consistent with

                   P                                                   P   P2
all constraints: z= C . Optimized consumer surplus is CSSelf
                                                        Custom  VPx C  C .
                   2                                                  2 4

                      P                            P
Case b.ii: Suppose z > C . Substituting k (z)  z  C the consumer surplus results in
                      2                           2

                                        P2
CSSelf
  Custom  V  P  x  (1  PC )z  PS  C . This is linear in z, and because 1-PC >0 the
                                        4

consumer will choose the largest possible z consistent with all constraints, z  x . Optimized

                                                  P2
consumer surplus is CSSelf
                      Custom  V  P  PC x  PS  C .
                                                  4

                                                 P                    P
       Will the consumer choose { k (z)  0 ; z = C } or { k (z)  x  C ; z  x }? Case b.i is
                                                 2                   2

                                P                                      P
preferred if PS  (1  PC )( x  C ) . Since PC<1 and by assumption x  C , the right hand side
                                2                                      2

is positive. Thus, as along as the setup cost is not too high, the optimal customization choice for

                               P
the consumer is { k * (z)  x  C ; z *  x }. 
                               2

                                 P 
Proof of Proposition 2: Let x  0, C  . If the consumer chooses customization, she will choose
                                 2 

k * (z)  0 and z *  x and the optimal consumer surplus is CSSelf                2
                                                              Custom  V  P   x . Notice that

                                                   35
since the consumer does not ask for any customization from the firm, she also does not pay the

                                        1                        P
setup costs. CSSelf > CSSelf if x<
               Custom   std               . This is true since x< C and PC<1.
                                                                2

                 P     
Now consider x   C , 1 . If the consumer chooses customization, she will choose
                  2 

                                                                                         2
 *          PC       *=x, so optimal consumer surplus is CSSelf
                                                                                        PC
k (z)  x 
            2
               and z                                       Custom  V  P  PC x  PS  4 .


                                      P2
CSSelf > CSSelf or V  P  PC x  PS  C > V  P  x if x> x . Thus, if x> x the consumer
  Custom   std
                                                           ˆ               ˆ
                                      4

                                            PC
will buy the customized product, and if            ˆ
                                               <x< x then she will buy the standard product. Finally
                                            2

                           PC                        P  P                     P 
we need to ensure that        < x <1. This reduces to C 1  C   PS  1  PC 1  C  . It can be
                                ˆ
                           2                         2    2                   4 

            P  PC               PC 
checked that C 1      1  PC 1      as long as PC  min{1, 2} .              
             2    2               4 



Proof of Proposition 3:

       We determine the equilibrium product design chosen by consumer x when she has social

preferences and takes other people’s designs into account. Because the cost of customization

increases with the distance of a consumer’s location from the location of the firm’s default

product at 0, it is conceivable that a given consumer x very distant from 0 will choose an

imperfectly customized product which lies to the left of her ideal location x. Of course, because

bandwagon effects matter, consumer x will take into account the co-creation choices of others in

designing her co-created product. Suppose consumer x has the following belief about other

consumers y in [  , 1].


                                                  36
        Belief IC (Ideal Co-creation): Consumer x expects that all other consumers y in [  , 1]
        perfectly customize their products at y


        We will show that this belief is rational by showing that, given this belief of others

behaviors, consumer x will also optimally choose to customize at her ideal location x. Since x is

a generic consumer in [,1], this will establish the rationality of this belief. Let customer x

customize her product at z, where z is in the interval [,1], that is  ≤ z≤ x. We will determine

the optimal z and k with the above segmentation. From equation (6) the expected distance from

her product at z(x) to what others choose is obtained by substituting z(y)=y because all other

customers y (≠x) are ideal co-creators who get the product at their ideal locations y.

“Expected distance from z to what others choose”=

                          Pc
                                                            z               1
                  IC   2  (z  y)dy  Pc (z  0)dy   (z  y)dy   ( y  z)dy .
                         0                   2
                                                                            z


Computing the integrals and simplifying gives

                                       Pc
                                                                  z                   1
                  IC  (zy    1 y 2 ) 2    zy Pc  (zy  1 y 2 )  ( 1 y 2  zy)
                                2       0                    2          2             z
                                                  2
                                                                                                  (17)
                                 2
                         P 
                     1  C   z2  z  1 2  1
                       2  2            2      2


The utility of the consumer is

                                                                  2                     
                                                       2 1  PC   z 2  z  1  2  1 
         U  V  x  z  P  PS  PC k  (z  k )                                  .
                                                          2  2              2       2
                                                      
                                                                                        
                                                                                         

Maximize this with respect to k and z. First, the optimal value of k, given z, is

        PC
k z        . Back substituting gives
        2




                                                       37
                                              2
                                             Pc    
                                                      1  PC 
                                                                2                   
                                                                                    
             U  V  x  z  P  PS  PC z               z2  z  1 2  1  .
                                             4       2  2             2      2
                                                                                    
                                                                                   

                                                             U
The marginal utility with respect to product location z is       1  PC  (2z  1) . Setting it
                                                             z

equal to zero and solving gives the interior optimal solution z  (1  (1  PC ) / ) / 2 . By

assumption   1  PC , and therefore this interior solution exceeds 1 and is inadmissible. Simple

                       U
re-arrangement gives       (1  PC  )  2(1  z)  0 . The value of z will be increased until the
                       z

constraint z ≤ x is binding. The equilibrium solution is

                                    z *  x ; k *  x  PC 
                                                        2 
                                                                                                 (18)
                                                           

This solution is consistent in equilibrium since consumer x’s Belief IC, that everyone in [,1] is

perfectly customizing, is correct. Given this supposition about others’ behaviors, she too finds it

optimal to customize perfectly at z*(x)=x.

    We will now show that it is not rational to expect that a sub-segment of [,1] will do non-

ideal co-creation when   1  PC . Nor is the belief that all consumers in [,1] will do non-ideal

co-creation. Consider the following market segmentation where consumers located at the farthest

right choose to buy the imperfectly customized product at χ. This is the sub-segment of non-ideal

co-creators. Consumers in [,] are ideal co-creators who buy the perfectly customized product.

That is, consumers in x  [,] buy the customized product at their ideal locations x.




                                                  38
                                                                           Ideal             Non-Ideal
                                                                        Co-Creation         Co-Creation
     DIY                          Standardize

            Pc
0                                                                                                        1
            2

             Figure 6: Market segmentation with ideal and non-ideal co-creators



       Consider the consumer located at x  ,1 in Figure 6. She could have the following

alternate belief about how the market is segmented.

       Belief M (Mixed Co-creation): Consumer x expects that all other consumers y in [, 1]
       are non-ideal co-creators who customize their product at while those in [, χ] are ideal
       co-creators who buy the perfectly customized product.

Suppose customer x in [] has Belief M, and customize her product at z, ≤ z≤ x. What is the

expected distance from her product at z to what others choose?

                    “Expected distance from z to what others choose”=
                       Pc
                                                                          1
                                                                                                  (19)
                M   2  (z  y)dy  Pc (z  0)dy   (z  y)dy   (z  )dy .
                       0                 2
                                                                           
                                                                                Pc
The first and third integrals follow because the DIY consumer y in [0,             ] and also consumer y
                                                                                2

in [  , χ] choose ideal co-creation at their ideal locations y. The fourth integral follows because

the all other customers in the interval [, 1] choose non-ideal co-creation at . Computing the

integrals and simplifying gives

                                       Pc
                                                                              1
                   M  (zy    1 y 2 ) 2    zy Pc  (zy  1 y 2 )  (z  ) y 
                                2       0                       2   
                                                  2
                                                                                                  (20)
                                  2
                          P 
                      1  C   1  2    1  2  z.
                        2  2   2           2



                                                       39
The utility of the consumer in this segment ≤x≤1 is

                                                                 2                    
                                                           1
                                                             P 
                                                      2        1  2    1  2  z .
        U  V  x  z  P  PS  PC k )  (z  k )                             
                                                           2 2     2         2
                                                       
                                                                                    
                                                                                       

Maximize this with respect to k and z. First, the optimal value of k given z is

        PC                                      Pc
k z        . That is, the customer produces      and asks the firm to do the rest. Back
        2                                      2

substituting gives utility as a function of only z:

                                      P2             P 
                                                             2                      
       U  V  x  z  P  PS  PC z  C      1  C   1  2    1  2  z  .       (21)
                                      4           2  2      2         2        
                                                                                   
To determine the optimal z, the marginal utility with respect to product location z is

                                    U
                                         1  PC   >0,                                   (22)
                                     z
which is positive, implying that consumer x will choose z*=x. Thus her choice is in contradiction

to her belief that all consumers in [, 1] will choose non-ideal co-creation at 

    What about non-ideal co-creation for all consumers in the segment farthest from the standard

product at 0? Consider the following market segmentation where = and all consumers located

in [χ, 1] are non-ideal co-creators who buy the imperfectly customized product at χ. Consider

Figure 5 and suppose the consumer located at x, ≤x≤1. She could have the following alternate

belief about how the market is segmented.

Belief NIC (Non-Ideal Co-creation): Consumer x expects that all other consumers y in [, 1] are
non-ideal co-creators who customize their product at 

Let customer x customize her product at z, ≤ z≤ x. What is the expected distance from her

product at z to what others choose?




                                                  40
                    “Expected distance from z to what others choose”=
                                                 Pc
                                          1 y 2 ) 2                     1
                          NIC  (zy                   zy Pc  (z  ) y 
                                          2       0                                               (23)
                                                            2
                                                                               .
                                         2
                                P    
                            1 C
                              2  2
                                        2    z
                                      
The utility of the consumer in this segment ≤x≤1 is

                                                        
                                                              P 
                                                                      2            
                                                                                   
        U  V  x  z  P  PS  PC k  (z  k ) 2     1  C    2    z  .
                                                             2  2 
                                                        
                                                                                  
                                                                                   

                                    PC
The optimal value of k is k  z         . Back substituting gives utility as a function of only z:
                                    2


                                           P2     
                                                        P 
                                                                2            
                                                                             
            U  V  x  z  P  PS  PC z  C     1  c    2    z  .                  (24)
                                           4     
                                                       2  2 
                                                                             
                                                                            

The marginal utility with respect to product location z is

                                         U
                                             1  PC   .                                        (25)
                                         z

For   1  PC this derivative is positive implying that consumer x will choose z*=x. Thus her

choice is in contradiction to her belief that all consumers in [, 1] will choose non-ideal co-

creation at 

        Finally, since the above derivation with Belief IC was carried out under the assumption

that all DIY consumers are perfectly customizing at their ideal locations, we need to confirm that

this is indeed rational for them to do. Consider the segmentation in Figure 6 above. Consider

               P                                                                      PC 
customer x in 0, C  who takes the behavior of the standardizing consumers in          2 ,   and the
               2                                                                            

ideally co-creating consumers in [  , 1] as given. Further x believes that all others DIY



                                                       41
                P 
consumers y in 0, C  are getting the perfectly customized product at their ideal locations y.
                2 

                                                                 PC
Suppose consumer x gets the customized product at z≤ x≤             . What is her optimal choice of z?
                                                                 2

                     “Expected distance from z to what others choose”=
                       z
                                         Pc
                                                                      1
                                                                                                  (26)
                    (z  y)dy   2  ( y  z)dy  Pc (z  0)dy   ( y  z)dy .
                      0              z                2
                                                                      
Computing the integrals and simplifying gives

                                 2
                        1  PC   2z PC  z 2  z ( 2  1)  1  1  2 .
                            
                        2  2 
                                                                                                  (27)
                                      2                       2 2

                                                P
The utility of the consumer in this segment ≤x≤ C is
                                                2

                                            2                                      
                               2   1  PC   z 2  1  2 PC   z 2  1  1  2 
      U  V  x  z  P  z         
                                    2  2 
                                                                                  .            (28)
                                                           2          2 2 
                                                                                   

Note that in (28) because the consumer x is DIYing, she can avoid setup cost. The marginal

                                                U                    P
utility with respect to product location z is       1  2z  (1  2 C  2  2z) . This gives the
                                                z                    2

                              1              P   1
interior solution as z 
                     ˆ                      C   . Notice that at z=0,
                           2(  )        2 2 

U                   P   1
          1  2(  C  ) >0 for  small. So the utility U increases at z=0. Clearly, if the
z z  0             2 2

                                       P
interior maximum z lies to the right of C then the optimal z is z*=x. Simple algebra shows
                 ˆ
                                        2


     ˆ
         P
that z ≥ c is equivalent to      1 1  1  PC  . Because
                                                               1  PC the right hand side exceeds 1,
        2                         2         
                                                

so the optimal z for consumer x in the DIY segment is z*=x, and we thus we have shown that it is

rational for the DIY consumers to perfectly customize at the ideal locations. 

                                                     42
Proof of Proposition 4:

                                                                      P P       P 
From equation (15) the equilibrium  * solves 2 2  (1    PC )  C  C   C    PS  0 .
                                                                          2           
                                                                      2        2  

There are two roots of this quadratic equation given by,

                                                             2
                   1  1  PC  1          1  PC   4P  P   P  8P
                  1                 1        C  C  C  S
                   4      4
                              
                                          
                                                     2  2  

               P 2
Using x   PS  C  /(1  PC ), the above becomes
      ˆ
                4 
                   

                                                      2           
                                                PC     1  PC 
                                                              ˆ
                                                                 x
                         1  1  PC            2             
                        1       1  1  8                      .
                         4                                2                             (29)
                                                    1  PC       
                                                  1 
                                                                 
                                                              

Since the analysis without bandwagon effects is the boundary case of the analysis with

bandwagon effects, as  approaches 0, the correct root for  should approach x . Notice the term
                                                                             ˆ

     1  PC
1          on the right hand side of (29) goes to infinity as →0. Also, using L’Hopital’s rule on
        

the term under the radical gives

                                   P  2 (1  P ) x 
                                  C          C ˆ
                                   2             
                                                                              4x
                                                                                  ˆ
                     lim   0                               = lim   0              =0.
                                      1  PC   
                                                    2                       1  PC 
                                     1                                  1 
                                                                                   
                                                                               
                                                                                    
                                               

If we consider the larger root in (29) (with the + sign) then the entire term inside the curly braces

approaches 2, so the larger root approaches  as →0, a contradiction. On the other hand, for the

smaller root, the term in curly braces on right hand side of (29) approaches 0 as →0. Combined


                                                        43
                                   1  PC   
with the fact that the multiplier 1 
                                             goes to infinity as →0, so the smaller root tends to
                                             
                                           

a finite limit. Thus, the smaller root is the correct one. Now for the root to be valid we need the

                                                                   2
                                                             PC  1  PC
                                                                        ˆ
                                                                           x
expression under the square root sign to be positive, 1  8  2     
                                                                             >0. It can be checked
                                                                         2
                                                               1  PC 
                                                              1 
                                                                      
                                                                    

that is will be the case for small β. Combining this with the requirement that β< 1-PC we can find

a critical value of β , say  , such that the smaller root in (29) identifies the consumer that is

indifferent between standardization and co-creation in equilibrium. 



Proof of Theorem 1:

From equation (15) the implicit equation defining the equilibrium  * is

                        P P       P                             P 2
2 2  (1    PC )  C  C   C    PS  0 . Using x   PS  C  /(1  PC ), the
                                                           ˆ
                        2  2
                           
                                         
                                   2                       
                                                                     4 
                                                                         

dividing line between standardizing and co-creation when =0, the above can be expressed as

                                                                       2
                            2                                 P 
                        2  (1  PC  )  x (1  PC )    C   0 .
                                              ˆ
                                                               2 

                               P
Notice that, by assumption, x  C , which implies that setup cost PS is sufficiently large that
                            ˆ
                                2

some customers want to avoid co-creation if  is near zero. For algebraic simplicity, let

        
          , where   [0, 1] . The implicit equation defining the equilibrium  can be expressed
     1  PC

as



                                                  44
                                       P 2              
                             x      C     2 2   F( ) .
                             ˆ           2                                                 (30)
                                                      
                                                        

Equation (30) implies that the equilibrium  is given by the intersection of a line x   and a
                                                                                    ˆ

                                                         2
                                  1 1      P 
parabola F() which has two roots,   1  2 C  , one positive and one negative. Note
                                  4 4        

                               2
that because x   C    C  , so the left hand side of (30), x   , exceeds F() at =0 for ≤1.
                   P      P
             ˆ                                              ˆ
                  2   2 

It is easy to see that F() is upward sloping at zero and is maximized at  = ¼ . See Figure 7.

       For the graph to look precisely like Figure 7 the positive root of parabola, where the

function F() meets the -axis, exceeds x or
                                        ˆ

                                                           P 2
                                                   2 PS  C
                                1 1      P                4 .
                                   1  2 C   x 
                                                 ˆ
                                4 4                (1  PC )

This is equivalent to
                                                           2        2
                             ˆ    1  PC              PC    PC 
                        PS  PS            1  1  8          .                      (31)
                                     4                2    2 
                                                            




                                                  45
                        F( )


                        ˆ
                        x

                                    x 
                                    ˆ

                                           b
                                                                       2
                                                                    PC    
                       
                            2                         2 2    
                                                                          
                                                                            
                  P                                                 2 
                2 
                 
                    C                                                      
                                               a



                                                                                    
                                1                                ˆ
                                                                 x
                                4

         Figure 7: Graphs of F(  ) and x   when x is smaller than root of F(  )
                                        ˆ          ˆ

       Let us investigate what happens when  increases from 0 given (31). The curvature of

F() sharpens and we move from the bold curve to the dashed curve in Figure 7. Note that the

parabola’s root where the function F()=0 is independent of the value of  whereas its intercept

increases in . The equilibrium  is given by the intersection of curve corresponding to F() and

the line x   . As  (equivalently ) increases the point of intersection moves from point ‘a’ to
         ˆ

                                                    
‘b.’ In other words, the equilibrium  decreases:       0.
                                                    

       Now consider the case where inequality (31) is not satisfied as in Figure 8 where the

                                                                      ˆ
positive root of parabola, where the function F()=0, is smaller than x .




                                                   46
              F


                 ˆ
                 x

                                   x 
                                   ˆ




                                         P  
                                                2

                            2 2     C  
                            
                                          2  



         
                      2
              PC 
              
              2                                               ˆ
                                                                 x
                                                                     c               
                               1
                               4



                                                                            d


             Figure 8: Graphs of F() and x   when x is larger than root of F()
                                          ˆ          ˆ


Again, when  increases from 0 (increasing ), the curvature of F() sharpens and we move from

the bold curve to the dashed curve. Now the intersection of F() and the line x   moves from
                                                                              ˆ

                                                       
point ‘c’ to ‘d.’ Here the equilibrium increases:          0.
                                                       

                                                                                    P 
       Notice that when =0, point a in Figure 7 and point c in Figure 8 are at x   C ,1 .
                                                                                ˆ
                                                                                     2 

When  increases, we need the location of * (either point a or c) to remain in the interval




                                                      47
 PC 
   ,1 and this will be true so long as  is not too large. That is, there is an open interval of
 2 

's,  ,     0 , so that * is neither too large nor too small. 
      0 ,

Proof of Proposition 5: Consider the market segmentation given in Figure 5. Consider the

consumer located at x, ≤x≤1 who has Belief NIC about all other consumers in [1]. Let

customer x customize her product at z, where z is in the interval [, 1], that is ≤ z≤ x.

Following the logic given in the proof of Proposition 3, the utility of this consumer is

                               P2     
                                            P 
                                                    2            
                                                                 
U  V  x  z  P  PS  PC z  C     1  c    2    z  and the marginal utility of
                               4     
                                           2  2 
                                                                 
                                                                

                                                                 U
consumer x with respect to her customized choice of z is             1  PC   . This is by assumption
                                                                 z

negative and so she will choose z*=χ, so her Belief NIC is rational when   1  PC .

                                                                                       P 
        The above analysis was carried out under the assumption that DIY consumers in 0, C 
                                                                                       2 

will choose their customized product at their ideal locations. Is this rational given the behavior of

                                                   P 
the co-creators in [, 1]? Consider consumer x in 0, C  who expects all other consumers y in
                                                   2 

this interval to choose the customized product at y. Suppose x chooses her customized product at

z ≤ x. What is her optimal choice of z?

                     “Expected distance from z to what others choose”=
                                                Pc
                                 z                                              1
                                0 (z  y)dy  z
                                                2
                                                     ( y  z)dy  Pc (z  0)dy   (  z)dy
                                                                  2
                                                                                                 (32)
                                     2
                           P      P      1
                        1  C   2 C    z  z 2  (1  ) .
                         2  2     2     2
The utility of this consumer is



                                                      48
                                             2                                     
                             2      1  PC   2z  PC    1   z 2  (1  ) 
      U  V  x  z  P  z          
                                     2  2 
                                                                                  .        (33)
                                 
                                                  2         2                   
                                                                                    

Note that in (33) since the consumer x is doing DIY she does not incur the setup cost nor does

she have to choose k. The marginal utility with respect to product location z is

U                  P
    1  2z  (1  C  2  2z) . This gives the interior solution as
z                   

~      1            P  1                                                   P
z                 
                          . Again, following the logic of Proposition 3, ~ > C is
                                                                           z
     2(  )       2 2                                                    2

                     1  PC                                            PC 
equivalent to   1 1       . If this is true then a consumer x in   0, 2  will optimally
                  2                                                        
                            

choose z*=x. We thus we have shown that it is rational for the DIY consumers to perfectly

customize at the ideal locations if bandwagon intensity is not too high. To summarize, if


  2  1 PC  1 then we can have a situation where consumers in 0, PC  are DIYers and
2 1 
                                                                        2

consumers in [χ, 1] are non-ideal co-creators.

       What determines the threshold * that defines the number of co-creators? After making

appropriate substitutions for z* and k* from Proposition 5 the utility of the consumer x in the co-

creation segment [, 1] is


                                                    P2    
                                                                P 
                                                                        2    
                                                                             
             Bandwagon
           U Custom     V  x    P  PS  PC   C    1  c    2  .                  (34)
                                                    4    
                                                               2  2 
                                                                             
                                                                            
                                                                      PC
Similarly, the utility of consumer x in the standardization segment       x   is
                                                                      2

                                                             2         
                      Bandwagon                     1  PC    2   
                                   V  x  P    
                    U Std                                   
                                                     2  2                                 (35)
                                                 
                                                                       
                                                                        


                                                 49
The consumer at x   must be indifferent between standardization and non-ideal co-creation

and therefore we can equate (34) and (35) after substituting x   . This gives the following

equation which implicitly defines *:

                                          P P       P 
                   2 2  (1  PC  )  C  C   C    PS  0 .                      (36)
                                           2  2
                                              
                                                           
                                                     2  


This equation is identical to that analyzed in the proof of Theorem 1, only here >1-PC. * is the

smaller of the two roots. 




                                                50

				
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