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Technical Appendix for “Impeding the Juggernaut of Innovation

Diffusion: A Production Constrained Model”

Section A. Proofs for Theorems, Lemmas and Corollaries

Constant annual production policy

Proof of Theorem 1:

We consider a new product market where the diffusion of innovation can be characterized by

the Bass (1969) model. Here, if the installed production capacity is sufficiently high then we

should not have any customers waiting to purchase once they have made the mental commitment

to adopt. Any production capacity C(t) greater than Cm satisfies the no waiting constraint and

results in a Bass like diffusion. However, this comes at the cost of carrying excess inventory. As

shown in figure 4, by setting A1=A3+A4, we derive an expression for the minimum capacity

Cm*. Using equation 1, Area A1 as shown in figure 4 can be represented as

∫ ) )

)) (E1)

To calculate A3+A4 as shown in figure 4, only A3 needs to be computed. The unimpeded

diffusion curve described by Bass is symmetric and bell shaped, as shown in figure 4.

Specifically, the adoption process is symmetric around time-to-peak sales T* up to time 2T

*

(Mahajan, Muller, and Bass 1990). We exploit this analytical property to derive an expression

for Cm*. The symmetry of the curve allows us to double the area of A3 to get the total area. To

calculate A3, we first evaluate A2+A3+A5 and then subtract A2 and A5.

) ) ) (E2)

) ) ) (E3)

We equate (E1) and (E3) to calculate Cm*

) ) )

)

) ))

) (E4)

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Figure 4: Constant annual production (reproduced from main paper)

Proof of Corollary 1:

Equation E4 can be rewritten for easier managerial interpretation. Specifically, we express

Cm* as a combination of two terms as expressed by equation E5.

)

(E5)

Where,

[

] [{

( ) ))

)}

)

] (E6)

Equation E6 can be rewritten to show that is always positive.

[

)] [ ) )] (E7)

Since ,

) is always positive. The term inside the second square bracket in

E7 can be shown to be > 0 by rewriting as an inequality:

) ) )

)

(E8)

The inequality in equation E8 is always true as average sales up to peak is always greater

than average sales up to any time less than T*. Hence proved that is always a positive term.

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Linearly increasing production policy

Proof of Theorem 2:

For a new product market with a linearly increasing production policy and initial capacity a,

an expression for minimum production rate b* can be derived as follows. As shown in figure T-1,

A1=A3 is the necessary condition to have unconstrained diffusion. Area A1 can be calculated by

adding area A1 and A2 followed by subtraction of area A2. Area A1+A2 corresponds to the area

of a quadrilateral whereas area A2 can be calculated by employing equation 1. Consequently,

Area A1 is given by equation E9 as follows:

)

) ) (E9)

Area A3 can be calculated as (A2+A3+A4)-(A2+A4). Similar to the calculation of A2,

(A2+A3+A4) can be represented as m×F( ) by employing equation 1. Area A4 corresponds to

the area of quadrilateral. Thus, Area A3 is represented by equation E10 as follows:

[ ) )] [ ) ] [

) )] (E10)

We equate E9 and E10 and substitute for CT1 and CT2:

)

(

) )

[ ) )]

) [ ) ] (E11)

Simplifying the expression above, the minimum production rate b*

needed for unconstrained

diffusion is represented by equation E12.

[ ) ] (E12)

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Figure T-1: Linearly increasing production

Proof of Corollary 2:

On the other hand, if the rate of capacity change b is known, then we can provide an

expression to determine the initial installed capacity. Equation E12 can be rewritten to provide

an expression for minimum initial industry capacity needed for unimpaired and impeded

diffusion in terms of the diffusion parameters, time and the constant production rate b as:

[{ )

}

] (E13)

Linearly decreasing production policy

Proof of Lemma 1:

For a new product market with a linearly decreasing production rate b, an expression for

minimum initial capacity a*

can be derived as follows. As shown in figure T-2, the necessary

condition to have unconstrained diffusion is when area A1equals A3. Area A1 can be calculated

by adding area A1 and A2 followed by subtraction of area A2. Area A1+A2 corresponds to the

area of a quadrilateral whereas area A2 can be calculated by employing equation 1 is given by:

) ) (E14)

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Area A3 can be calculated by evaluating A2+A3+A4 and subtracting A2 and A4. Area

A2+A3+A4 represents the entire area under the diffusion curve and is equal to market size m. A2

is given by equation 1, whereas A4 corresponds to a triangle with area

) ).

Thus,

) [

) )] (E15)

Thus, equating areas A1 and A3,

) ) ) [

) )] (E16)

Since at T2, capacity is zero, equation E16 can be simplified by cancelling common terms and

substituting T2 =a/b. Thus, the minimum initial installed capacity is represented as:

√ (E17)

Proof of Theorem 3:

We employ Lemma 1 to evaluate the condition under which minimum installed capacity a*

is less than unconstrained peak sales. Specifically, we compare the expression for peak sales

from Bass (1969) with equation E17. Under the condition of unconstrained diffusion, the impact

of social influence from adopters, q, proposed by Bass (1969) is equivalent to the term q2 in our

C-WBM model. The necessary condition therefore is expressed as √

)

which can be further simplified and expressed in terms of a minimum production rate

b* as given by equation E18.

)

(E18)

Proof of Theorem 4:

We invoke Lemma 1 to derive an expression for the minimum initial capacity needed for

unconstrained diffusion while satisfying the condition that production stops before peak sales

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occurs. By rearranging equation E17, we derive an expression for T2 and compare it with time

taken to reach unconstrained peak sales, T*. Specifically, we take the square on both sides of

E17, substitute T2 =a/b (at T2, capacity is zero), and rewrite T2 as shown in E19.

(E19)

From Bass (1969), we know that time to peak sales can be expressed in terms of p and q

as

) (

). The impact of social influence from adopters, q, in the standard Bass model is

captured by the term q2 in our C-WBM model. Thus, equating E19 with time to reach peak sales

yields an expression for minimum initial capacity given by E20 such that T2 < T*.

)

)

(E20)

Figure T-2: Linearly decreasing production

Exponentially increasing annual production policy

Proof of Theorem 5:

For an exponentially increasing policy with a production rate b, an expression for minimum

initial capacity a* can be derived as follows. As shown in figure T-3, the necessary condition to

have unimpeded diffusion is when area A1 equals A3. Area A1 can be calculated by computing

A1+A2 followed by subtraction of A2. Area A1+A2 corresponds to the area under the curve

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between interval [0, T1] whereas area A2 can be calculated by employing equation 1. Therefore,

Area A1 is given by equation E21 as follows:

∫

) (E21)

Solving the integral and imposing the limits, A1, can be expressed as:

[ ] ) (E22)

Area A3 can be calculated as (A2+A3+A4)-(A2+A4). Similar to the calculation of A2,

(A2+A3+A4) can be represented as m×F( ). Area A4 corresponds to the area under the curve

between interval [T1, T2]. Thus, Area A3 is represented by equation E22 as follows:

) ) ∫

(E23)

Solving the integral and applying the limits, A3 can be expressed as:

) )

[ ] (E24)

Given a rate of production b, minimum value of initial installed capacity, a*, needed for

unconstrained diffusion can be calculated by equating E23 with E24 and represented as:

)

[ ] (E25)

Figure T-3: Exponentially growing production

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S-shaped annual production policy

Proof of Theorem 6:

For an S-shaped production policy with a production rate b, an expression for minimum

initial capacity a* can also be derived as follows. In such a production environment, as shown in

figure T-4, no customer waiting will occur, if area A1 is equal to area A3. Area A1 can be

calculated as (A1+A2)–(A2). Area A3 can be calculated as (A2+A3+A4)-(A2)-(A4).

) (E26)

Where ∫

)

Integral can be solved by rewriting I as:

∫

)

[∫ (

)

∫

)

(E27)

The first part of equation E27 equals T1; the second part of equation E27 can solved by a

simple substitution of , and . Thus, substituting and solving for the

limits of the integral leads to A1, where:

[

)] ) (E28)

Similarly, A3 can be calculated and shown to be equal to:

) ) )

[ (

)] (E29)

Equating equation E28 and E29 and simplifying for a, the minimum initial capacity for a given

production growth rate of b is represented by E30:

)

[

(

)] (E30)

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Figure T-4: S-shaped production

Two-regime production policy

Proof of Lemma 2:

In hi-tech markets, production may be carried out in discrete chunks (see figure T-5). In such

a two-regime production environment, there are two questions of interest. First, how long should

one wait (T1), before installed capacity has to be expanded to avoid customer waiting? A second

related question is related to the minimum capacity that needs to be added at T1 in order to avoid

customers waiting over the remaining diffusion life cycle. To address the first question, we

derive an expression for the longest time T1 that one can wait before capacity needs to be

expanded in order to avoid waiting customers. T1 can be evaluated in close form, given CT1 is

known. At T1, ) = C T1, and as per Bass (1969) ) can be expressed in terms of p, q, m,

and T1 as shown in E31.

)

(

)

(

) )

) (E31)

By substituting

)

) equation E31 can be rearranged in a

quadratic form shown by E32.

) (E32)

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If , the equation will have imaginary roots and T1 cannot be calculated. Our

computational experience with real world product data shows that is always lower than 0.25

and that both roots of the quadratic equation are always positive. As noted earlier, the term q in

the standard Bass model maps on to its counterpart, q2 in our C-WBM model. Thus, taking the

larger of the two roots can be expressed in terms of the original parameters p, q2, and m.

Therefore, T1max

is expressed as shown in equation E33:

[

)]

(

[ ) ]

{

[(

[ ) ]) √(

[ ) ]

)]

}

)

(E33)

Proof of Theorem 7:

To address the second question, an expression for minimum additional capacity can be

derived by invoking Lemma 2. Specifically, at T1<=T1max

, capacity has to be expanded to CT2 to

avoid waiting. As shown in figure T-5, to derive CT2, we once again use the symmetric property

of a Bass diffusion curve. Specifically, in order to ensure unimpeded diffusion, areas (A1+A3)

should be equal to areas (A6 +A7). Area A1 can be evaluated as (A1+A4) – (A4) whereas A3

can be evaluated as (A3+ A5) - (A5-A4). Area A1+A3 is thus represented by equation E34 as

follows:

{ } { ) )} (E34)

Area A6 can be calculated as (A5+A8+A6)-(A5-A8). Thus, A6+A7 can be calculated by

exploiting the curve symmetry and doubling A6 as represented by equation E35. T* represents

time to peak sales as discussed previously and m×F(T*) is the cumulative sales up to T

*.

[ ) ) ) ] (E35)

Equating equation E34 and E35, results in an expression with two variables as shown by

equation E36.

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) [ ) )] (E36)

Given CT1 is a known quantity, CT2 can be expressed as:

[ ) )]

) (E37)

Minimum additional capacity is then a difference between CT2 and CT1 and can be

represented as:

) ) )

) (E38)

Figure T-5: Two-regime production

Section B: Simulation Settings for Parameter Estimation with Early Sales Data

To assess the ability of our model to identify installed production capacity from early sales

observations, we employ the C-WBM model with constant annual production to generate

synthetic data. Using different values of p, q2, and m to seed the diffusion environment, we set up

diffusion scenarios under two different production settings. Specifically, under setting A, we

employ values of p=0.03, q2=0.38, m=22,389, and a value of C=1,250 such that diffusion occurs

with a time to peak sales in the unconstrained process at period seven. The first occurrence of

backorder appears at period six (TB), right before peak sales have occurred. Under setting B, we

employ values of p=0.013, q2=0.38, m=40,001, and a value of C =2,300 to set up diffusion that

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has a time to peak sales in the unconstrained process at period 11. The first occurrence of

backorder appears at period 12 (TB), right after peak sales has occurred.

There is a material difference between employing data points after the first occurrence of the

backorder and those prior to the onset of this. From the point of first occurrence of the backorder,

the “true” diffusion process is not only impacted but is also impeded. Prior to the onset of

backorders, both the “true” and the capacity constrained diffusion curves are identical.

Conceptually, this means that prior to the first occurrence of the backorder we need to be able to

estimate the minimum supply needed for “true” unimpeded diffusion. Consequently, this

suggests that we may be able to deploy our production capacity constrained diffusion model to

help managers and analysts get an early handle on the industry supply required. Subsequent to

the onset of backorders, the observed sales will diverge from the “true” unimpeded and

unimpaired sales. From this point forward, we now have information that can help us understand

the specific industry level supply that is in play (which led to the divergence). Therefore, it

would be possible with our C-WBM model to unearth the magnitude of the industry capacity.

Under each of the two constrained diffusion settings, we estimate p, q2, m, and C with limited

data, going back up to five periods, starting from TB and going forward two periods from TB.

Thus, our experimental design has two different constrained diffusion settings and set of eight

limited data points resulting in 16 scenarios.

Reference

Mahajan, V., Muller, E., Bass, F.M. 1990. New product diffusion models in marketing:A review

and directions for research. Journal of Marketing. 54(1), 1-26.

Balakrishnan, P.V. (Sundar), Pathak, S.D. 2013. Impeding the Juggernaut of Innovation

Diffusion: A Production Constrained Model. Production and Operations Management,

Forthcoming.