optimal modular production strategies under market uncertainty: a real options perspective

9
Optimal modular production strategies under market uncertainty: A real options perspective Su Xiu Xu a,n , Qiang Lu a , Zhuoxin Li b a Department of Economics and Management, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, PR China b McCombs School of Business, The University of Texas at Austin, Austin, TX, USA article info Article history: Received 29 April 2010 Accepted 27 April 2012 Available online 22 May 2012 Keywords: Modular production Modularization Real options Market uncertainty Managerial flexibility abstract In this paper, we develop a product modularization model based on real options theory to help a firm determine the optimal modular production strategies under market uncertainty: whether/when to exercise the option of modularization. We find that market volatility has great impacts on the optimal degree of product modularity and the timing for modular production. The results show that when market is more volatile, it is optimal for a firm to postpone modularization, and when a firm’s investment efficiency at the preparation stage is higher, the firm can start modular production earlier with relatively low product modularity. Also, an increase in market uncertainty will stimulate the firm to improve its product modularity. Finally, by comparing the predictions from the widely used net present value method (NPV) to the results from our real options model, we argue that traditional NPV method underestimates a firm’s value for modular production and might mislead a firm to modularize earlier. & 2012 Elsevier B.V. All rights reserved. 1. Introduction Due to its significant benefits such as cost advantage, economic mass customization, product development flexibility, and modular innovation (Ulrich and Eppinger, 2000; Sanchez and Mahoney, 1996; Baldwin and Clark, 1997; Schilling, 2000; Mikkola, 2003), modularity has been widely used as a dominant strategy for product design and production. In the extant litera- ture, there are two broad research themes on modularity. The first stream of research investigate the conditions under which mod- ular architectures are superior to integrated ones (Ulrich and Eppinger, 2000; Baldwin and Clark, 2000; Langlois, 2002). The second theme in the literature focuses on the ‘‘power of modularity’’ from the real options perspective (Baldwin and Clark, 2000; Gamba and Fusari, 2009). Despite increasing interests in modularity, little attention has been paid to modularization timing in either a qualitative or quantitative way. Furthermore, most quantitative literatures derive the optimal product modularity based on the assumption of complete information. However, in reality, it is impossible to obtain perfect information about the complexity of design tasks (Schaefer, 1999). Incorrect estimation of the optimal number of modules and interfaces might lead firms to the mistake of idealized modularity (Ethiraj and Levinthal, 2004). A con of modularity is that modularity strategies might be costly. Increasing product modularity is not trivial because firms need to invest in acquiring the appropriate technologies and related designing skills to enable higher modularity (Sanchez and Mahoney, 1996; Baldwin and Clark, 2000; Mukhopadhyay and Setoputro, 2005). When compared to these visible costs, another component of efforts in modularization preparation stage as hidden costs or coordination costs is a quite large portion of the total investment costs and plays an important role in better modular production outputs. Brusoni and Prencipe (2001) point out that modular product architecture does not define specific information structures of coordinating participants. Instead, they found that product modularization calls for highly interactive coordination activities. In this paper, we adopt a real options approach to analyze a firm’s optimal modular production strategies within a framework where the firm can decide whether/when to exercise the option of modularization. We assume that the firm can benefit from the marginal cost advantage associated with modularization only by investing in the redesign of the existing product architecture. We also assume that redesigning the existing product architecture causes a loss of the expected profit of existing products. Thus, the firm’s decision of whether/when to exercise the option of mod- ularization involves a trade-off, which is affected by the oppor- tunity costs including investment cost and expected profit of the existing products, the marginal cost advantage associated with Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/ijpe Int. J. Production Economics 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.05.009 n Correspondence to: Rm. 8-24, Haking Wong Building, Pokfulam Road, Hong Kong. Tel: þ852 5315 6986; fax: þ852 2858 6535. E-mail address: [email protected] (S.X. Xu). Int. J. Production Economics 139 (2012) 266–274

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Page 1: Optimal modular production strategies under market uncertainty: A real options perspective

Int. J. Production Economics 139 (2012) 266–274

Contents lists available at SciVerse ScienceDirect

Int. J. Production Economics

0925-52

http://d

n Corr

Kong. T

E-m

journal homepage: www.elsevier.com/locate/ijpe

Optimal modular production strategies under market uncertainty: A realoptions perspective

Su Xiu Xu a,n, Qiang Lu a, Zhuoxin Li b

a Department of Economics and Management, Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen, PR Chinab McCombs School of Business, The University of Texas at Austin, Austin, TX, USA

a r t i c l e i n f o

Article history:

Received 29 April 2010

Accepted 27 April 2012Available online 22 May 2012

Keywords:

Modular production

Modularization

Real options

Market uncertainty

Managerial flexibility

73/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.ijpe.2012.05.009

espondence to: Rm. 8-24, Haking Wong Bu

el: þ852 5315 6986; fax: þ852 2858 6535.

ail address: [email protected] (S.X. Xu).

a b s t r a c t

In this paper, we develop a product modularization model based on real options theory to help a firm

determine the optimal modular production strategies under market uncertainty: whether/when to

exercise the option of modularization. We find that market volatility has great impacts on the optimal

degree of product modularity and the timing for modular production. The results show that when

market is more volatile, it is optimal for a firm to postpone modularization, and when a firm’s

investment efficiency at the preparation stage is higher, the firm can start modular production earlier

with relatively low product modularity. Also, an increase in market uncertainty will stimulate the firm

to improve its product modularity. Finally, by comparing the predictions from the widely used net

present value method (NPV) to the results from our real options model, we argue that traditional NPV

method underestimates a firm’s value for modular production and might mislead a firm to modularize

earlier.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Due to its significant benefits such as cost advantage,economic mass customization, product development flexibility,and modular innovation (Ulrich and Eppinger, 2000; Sanchezand Mahoney, 1996; Baldwin and Clark, 1997; Schilling, 2000;Mikkola, 2003), modularity has been widely used as a dominantstrategy for product design and production. In the extant litera-ture, there are two broad research themes on modularity. The firststream of research investigate the conditions under which mod-ular architectures are superior to integrated ones (Ulrich andEppinger, 2000; Baldwin and Clark, 2000; Langlois, 2002). Thesecond theme in the literature focuses on the ‘‘power ofmodularity’’ from the real options perspective (Baldwin andClark, 2000; Gamba and Fusari, 2009).

Despite increasing interests in modularity, little attention hasbeen paid to modularization timing in either a qualitative orquantitative way. Furthermore, most quantitative literaturesderive the optimal product modularity based on the assumptionof complete information. However, in reality, it is impossible toobtain perfect information about the complexity of design tasks(Schaefer, 1999). Incorrect estimation of the optimal number of

ll rights reserved.

ilding, Pokfulam Road, Hong

modules and interfaces might lead firms to the mistake ofidealized modularity (Ethiraj and Levinthal, 2004).

A con of modularity is that modularity strategies might becostly. Increasing product modularity is not trivial because firmsneed to invest in acquiring the appropriate technologies andrelated designing skills to enable higher modularity (Sanchezand Mahoney, 1996; Baldwin and Clark, 2000; Mukhopadhyayand Setoputro, 2005). When compared to these visible costs,another component of efforts in modularization preparation stageas hidden costs or coordination costs is a quite large portion of thetotal investment costs and plays an important role in bettermodular production outputs. Brusoni and Prencipe (2001) pointout that modular product architecture does not define specificinformation structures of coordinating participants. Instead, theyfound that product modularization calls for highly interactivecoordination activities.

In this paper, we adopt a real options approach to analyze afirm’s optimal modular production strategies within a frameworkwhere the firm can decide whether/when to exercise the option ofmodularization. We assume that the firm can benefit from themarginal cost advantage associated with modularization only byinvesting in the redesign of the existing product architecture. Wealso assume that redesigning the existing product architecturecauses a loss of the expected profit of existing products. Thus, thefirm’s decision of whether/when to exercise the option of mod-ularization involves a trade-off, which is affected by the oppor-tunity costs including investment cost and expected profit of theexisting products, the marginal cost advantage associated with

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S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274 267

modular production as well as the parameters characterizing thestochastic properties of the profit flow.

The rest of this paper is organized as follows: In Section 2, wereview related research on modularity and applications of realoptions theory to product management. Section 3 formulates thereal options model for modular production. In Section 4, weprovide some significant results. An example is illustrated inSection 5. In Section 6, we further discuss the implications of theresults derived from the proposed model and point out futureresearch directions.

2. Modularity and real option theory

Schilling (2000) defines modularity as ‘‘a continuum describ-ing the degree to which a system’s components can be sepa-rated and recombined, and it refers both to the tightness ofcoupling between components and the degree to which therules of the system architecture enable (or prohibit) the mixingand matching of components’’ (p. 312). For detailed discussionon the potential benefits and costs of modularity strategies,readers can refer to Mikkola and Gassmann (2003). Benefits ofmodularity include shorter product life cycles, increased num-ber of product variants, increased product development flex-ibility, technological upgrading of products, reducing thenumber of suppliers, and reducing costs of development andproduction (Sanchez and Mahoney, 1996; Baldwin and Clark,1997; Schilling, 2000; Sanchez, 2002; Mikkola and Gassmann,2003), to name a few.

Modularity has been measured by a number of methods andcriteria on different objects in the literature. Gershenson et al.(1999) gives a four-step process to measure the product mod-ularity considering the dependency and similarity of componentsand processes. Mikkola and Gassmann (2003) propose a mod-ularization function which is a function of the number of standardcomponents in the product, the number of new-to-firm compo-nents (which depends on whether the firms have had priorknowledge and application of these components in existingproduct architectures), the degree of coupling (which dependson the number of interfaces between the components), and thesubstitutability factor (the number of product families madepossible by the number of interfaces). Frenken (2006) developsan inverse function of system pleiotropy to measure systemmodularity in terms of fitness landscape. In this paper, modularitylevel can be adopted using any of the measures given above.Basically, the term ‘‘modularity’’ in our paper refers to a designstructure that improves a firm’s profit flow due to the basicbenefits of modularity such as cost advantages and productdevelopment flexibility. At the same time, modularity can helpthe firm make rapid and effective responses to unpredictableevolution of market demand and technology by modifying part ofthe components without affecting other components throughloose-coupling merit.

A real option is a right without an obligation to take specificfuture actions in managing a real asset depending on howuncertain conditions evolve (Amram and Kulatilaka, 1999).Real options have been used to recognize and capture the latentvalue in a number of domains, including real estate, researchand development, product development, and natural resources(Trigeorgis, 1993; Dixit and Pindyck, 1994; Amram andKulatilaka, 1999; Brennan and Trigeorgis, 2000; Huchzermeierand Loch, 2001). Methods for valuing options for real assets havebeen developed (Trigeorgis, 1993; Dixit and Pindyck, 1994;Trigeorgis, 1996), and applied to engineering projects (Park andHerath 2000, Baldwin and Clark 2000) and supply chain riskmanagement (Cucchiella and Gastaldi, 2006; Wong et al., 2011;

Tang and Musa, 2011). The central idea of real option approach isto proactively use strategic flexibility (e.g., abandonment, delay,and increasing investment, see Huchzermeier and Loch, 2001) torecognize and capture the latent project value hidden in dynamicuncertainties. In this paper, we mainly follow the basic real optiontheory by Dixit and Pindyck (1994) to set up our optimal modularproduction model.

To our best knowledge, Baldwin and Clark (2000) are amongthe first to model the value of modularity in design as realoptions. In their study, two basic types of modules are identified:hidden modules, meaning that the design decisions in thosemodules do not affect decisions in other modules; visible modules,meaning that those modules represent a set of design rules(visible information) that hidden modules should obey. They alsoexplained the dynamic evolution of a modular design through aset of operators, which are also generic design structures denomi-nated splitting, substitution, augmenting, excluding, inversion, andporting. Based on real options theory, Baldwin and Clark (2000)use splitting and substitution as examples of the six modularoperators to describe the operations by which modular designsevolve and the value enhances accordingly. Further, Gamba andFusari (2009) provide a general real option valuation approach forthe modularization in the design of a system and sequentiallyexamined the value added through all the six operators thatdeveloped by Baldwin and Clark (2000) for modular designevolution.

However, especially for modular production, the trade-offbetween marginal cost advantage associated with modulariza-tion and opportunity cost has not been well studied in theliterature. For the discussed reasons, this paper develops a realoption model of modular production with consideration ofopportunity cost. For the valuation method, as discussed in mostfinancial economics and managerial literatures (e.g., see Dixit andPindyck, 1994; Trigeorgis, 1996; Luehrman, 1998; Huchzermeierand Loch, 2001), the standard NPV method fails to capture theoption value of timing flexibility and can result in the exercise ofinvestment earlier than the optimal time, thereby causing anopportunity cost. Therefore, the proposed model in this paperformulated based on the real option theory can help a firm dobetter decisions about whether/when to exercise the option ofmodularization.

3. The base model

In this section, a basic modular production problem has threeimportant aspects to be formulated: (a) a firm is currentlyoperating a series of products (product family) of low modularity;(b) improves the modularity of existing products with a certainproportion leading to different profit flow for low-modular andhigh-modular productions (i.e., the advantages of modularity);and (c) transfers to high-modular production causing an oppor-tunity cost that includes an investment sunk cost and a loss ofexpected profit of the existing low-modular products. For easierreadability of the text, we still adopt the term ‘‘modular produc-tion’’ or ‘‘modularization’’ as the representative of high-modularproduction.

In this paper, we follow the basic real option theory of Dixitand Pindyck (1994). To develop a model, we first need to note thatthe future value for profits is difficult to observe, and so it hasuncertainty. To introduce the influence of market uncertainty, weassume that firm’s future profit flow Pt follows a GeometricBrownian Motion (GBM) on Rþ over time, which is the contin-uous-time formulation of the random walk:

dPt ¼ mPtdtþsPtdzt , m40, s40, ð1Þ

Page 3: Optimal modular production strategies under market uncertainty: A real options perspective

S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274268

where m is the mean-drift of expected future change and s is thevolatility of a random walk, which are both exogenously deter-mined parameters1 ; and zt is a standard Brownian motion.

The current GBM setting is the standard setting in real optionstheory as a good approximation for uncertainty (Abel and Eberly,1994; Dixit, 1989; Li and Kouvelis, 1999). The parameters m ands reflect the nature of the profit flow. Higher level of a mean-drift(m) characterizes drastic increase of profit flow. The uncertaintyrate (s) represents how much the profit flow changes over thetime. We also assume mor (r is riskless rate) for the conver-gence like Dixit and Pindyck (1994). The modularity of currentproducts is determined by the firm’s previous environmental orstrategic situation. As market demand changes and technologyupgrades, a firm may improve the modularity of existing productsto make more profit. Therefore, a firm will make profits, m0Pt andmPt , with current production and modular production. Here, m0

represents current modular degree and m represents improvedmodular degree. As discussed earlier, the advantage from mod-ularity allows us to assume 0om0omr1: Simply, if the condi-tion is not satisfied, a firm will not consider modularizationbecause the modular production does not give any incentive tothe firm. From these profit flows, a firm has profits for currentproduction and modular production as follows:

CðPtÞ ¼ E

Z L

tm0Pte�rðt�tÞdt

� �¼

m0Ptð1�e�ðr�mÞLÞ

r�m ,

VmðPtÞ ¼ E

Z L

tmPte�rðt�tÞdt

� �¼

mPtð1�e�ðr�mÞLÞ

r�m,

where L is the life-span of this product family (calculated fromnow on), CðPtÞ represents the expected profit of the presentproduction mode in time interval (t, L), and VmðPtÞ representsthe expected profit of modular production mode in time interval(t, L).

Assume the life-span (L) of this product family follows aPoisson process with associated parameter (g), then the twoequations above can be rearranged as follows:

CðPtÞ ¼

Z 1t

ge�gtm0Ptð1�e�ðr�mÞtÞ

r�mdt¼ m0Pt

r�m, ð2Þ

VmðPtÞ ¼mPt

r�m , ð3Þ

where r represents discount rate, satisfying that r¼ gþr andr4m (convergence condition).

In addition, the firm is currently operating with a low mod-ularity as generating a profit flow before the optimal modulariza-tion timing. The expected profit of current production in timeinterval (0, t) can be expressed as follows:

Vm0ðPtÞ ¼ E

Z t

0e�rtm0Ptdt�

� �: ð4Þ

However, we here need to discuss an investment cost for themodular production. When a firm improves its product modular-ity, this procedure causes a certain amount of product architec-ture redesign costs and related other costs, i.e., coordination cost,learning cost, material cost and an organizational cost. Suchinvestments are largely irreversible due to the complex inter-dependencies among organizational and technological elements(Kogut and Kulatilaka, 2001). Therefore, modular production is

1 The estimation of drift m and volatility s requires firm-specific historical

series data. In our real options setting, we can estimate the parameters in the

following way: First, using the logarithm of Pt, we can estimate m�ðs2=2Þ as the

average of ln Pt�ln Pt�1. Then we can get an estimation of the volatility s by

taking the standard deviation of ln Pt�ln Pt�1. Similarly, we can get the estimate of

drift m using the same historical series data.

not trivial. As we mentioned in the previous section, firms need toinvest in acquiring the appropriate technologies and relateddesigning skills to enable higher modularity (Sanchez andMahoney, 1996; Baldwin and Clark, 2000; Mukhopadhyay andSetoputro, 2005). This cost is thus assumed to be increasing andconvex in m. Here, convexity of the cost function is used toexpress the diminishing marginal returns from modularity invest-ment expenditures in practice. Thus, investment cost is given by

Im ¼b

lml, l41, b40, 0om0omr1, ð5Þ

where l represents the efficiency parameter, higher investmentefficiency has higher l because of ð@I=@lÞo0.

According to (2) and (5), the opportunity cost of modularproduction I is expressed as

I¼ CðPtÞþ Im ð6Þ

Finally, based on (2)–(5), a real option model of optimalmodularity is derived as follows:

FðPtÞ ¼maxt

Vm0ðPtÞþE e�rt

� �VmðPtÞ�CðPtÞ�Im½ �

� �¼max

tE

Z t

0e�rtm0PtdtþE e�rt

� � mPt

r�m�m0Pt

r�m �b

lml

� �� �� ,

ð7Þ

where FðPtÞ is the value function to maximize the total profit withrespect to any possible modularization time t, and recall that Pt isthe firm profit flow at time t.

The stochastic dynamic programming problem given in (7)leads to the following optimal modular production strategies.

4. Optimal modular production strategies

In this section, we aim at solving two key decision problems:(a) what’s the optimal product modularity (i.e., modular degree);and (b) what’s the optimal modularization timing.

Let Pn

T be the optimal profit flow threshold for initiatingmodularization. Hence, T is the optimal modularization timingat this threshold and is different to the time t which representsany possible modularization time t.

Lemma 1. Vm0ðPtÞ is expressed as

Vm0ðPtÞ ¼ E

Z t

0e�rtm0Ptdt

� �¼

m0Pt

r�m 1�Ptb1�1ðPn

T Þ1�b1

h iPt oPn

T

0 Pt ZPn

T

8<:

where

b1 ¼1

2�

ms2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffims2�

1

2

� �2

þ2r

s2

s41:

Proof. For easier readability of the text, all proofs are shown inthe Appendix A. The proof of Lemma 1 is given in Appendix A.1.

Lemma 2. E½e�rt� is given by

E e�rt� �

¼

Pt

Pn

T

�b1

Pt oPn

T

1 Pt ZPn

T

8<:

Proof. See Appendix A.2.

In Lemmas 1 and 2, we provide the basis for calculating thesolution of (7). Next we will establish the following set of resultscharacterizing the dynamics of the optimal modular production.First, we introduce the result of optimal product modularity.

Page 4: Optimal modular production strategies under market uncertainty: A real options perspective

S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274 269

Theorem 1. The optimal product modularity reads as

m¼Pt

bðr�mÞ

h i1=l�1Pt obðr�mÞ

1 Pt Zbðr�mÞ

8<:

Proof. See Appendix A.3.

The theorem above gives us a way to calculate the optimalproduct modularity given the values of related parameters.Specifically, the threshold of Pt for maximum modularity (m¼1)is bðr�mÞ: If the value of Pt exceeds this threshold, then maximummodularity policy is optimal. In the situation when the value ofPt is less than bðr�mÞ, the optimal modular degree depends on thevalue of the threshold ðPn

T Þ for initiating modularization,and exogenous variables such as market uncertainty (s) andinvestment efficiency (l).

Theorem 2. The optimal modularization timing and the option

value at this timing are given as follows:

(i)

Under the condition of Pt Zbðr�mÞ or m¼1:the optimal threshold (Pn

T ) for initiating modularization reads as

Pn

T ¼b1bðr�mÞ

lðb1�1Þð1�m0Þwhere Pn

T 4bðr�mÞ,

and the option value of modular production at this timing,

GðPtÞ , reads as

GðPtÞ ¼

blðb1�1Þ bðr�mÞrPt oPn

T

ð1�m0ÞPt

r�m � bl Pt ZPn

T

8><>:

(ii)

Under the condition of Pt obðr�mÞ or mo1:the optimal threshold ðPn

T Þ for initiating modularization reads as

Pn

T ¼ bðr�mÞ~b1m0l

ð ~b1�1Þðl�1Þ

!l�1

where Pn

T obðr�mÞ,

and the option value of modular production at this timing,

GðPtÞ , reads as

GðPtÞ ¼

m0Pn

T

ð ~b1�1Þðr�mÞPt oPn

T

Ptr�m

�l=l�1b1=1=l 1� 1

l

� ��

m0Pt

r�m Pn

T rPt obðr�mÞ

8><>:

where

~b1 ¼1

2�

~mV�mð ~sV�sÞ2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~mV�mð ~sV�sÞ2

�1

2

" #2

þ2ðr�mÞð ~sV�sÞ2

vuut 41,

~mV ¼l

l�1mþ 1

2

ll�1

ll�1�1

� �s2or

ðConvergence conditionÞ,

~sV ¼l

l�1s:

Proof. See Appendix A.4.

Fig. 1 demonstrates the structure of the optimal modulariza-tion strategies. As shown in Fig. 1(a), modularization is worth-while in the regions M1 and M2. However, in the regions W1 andW2, waiting is the optimal decision currently due to the value ofpostponement option. Moreover, the regions below the dashedline exist if mo1 and the remaining regions occur if m¼1.Fig. 1(b) illustrates the optimal deepness of modularization whenit occurs. Specifically, optimal modularity (mn) increases with the

value of profit flow if Pt oP0 and maintains the maximum level(mn¼1) if Pt 4P0.

Theorem 2 explains when a firm starts to implement mod-ularization optimally. The optimal threshold (Pn

T ) for initiatingmodularization is discussed in two separate regions: maximummodularity (m¼1) and partial modularity (mo1). Moreover, ineach region the upper part in the GðPtÞ represents the option valuethat a firm may have when a firm’s profit flow at time t is lowerthan the modularization timing threshold ðPn

T Þ, and the bottom ofthe value function represents the option value when a firm’sprofit flow at time t is higher than the threshold ðPn

T Þ.According to Lemmas 1 and 2 and Theorem 2, we derive the

total profit as follows.

Corollary 1. The total profit, FðPtÞ , is expressed as follows:

(i)

Under the condition of Pt Zbðr�mÞ or m¼ 1 :

FðPtÞ ¼

m0Pt

r�m 1�Ptb1�1ðPn

T Þ1�b1

h iþ Pt

Pn

T

�b1 blðb1�1Þ bðr�mÞrPt oPn

T

ð1�m0ÞPt

r�m �bl Pt ZPn

T

8><>:

where Pn

T ¼b1bðr�mÞ

lðb1�1Þð1�m0Þ.

(ii)

Under the condition of Pt obðr�mÞ or mo1 :

FðPtÞ ¼

m0Pt

r�m 1�Ptb1�1ðPn

T Þ1�b1

h iþ Pt

Pn

T

�b1 m0Pn

ð ~b1�1Þðr�mÞPt oPn

T

Ptr�m

�l=l�1b1=1�l 1� 1

l

� ��

m0Pt

r�m Pn

T rPt obðr�mÞ

8>><>>:

where Pn

T ¼ bðr�mÞ~b1m0l

ð ~b1�1Þðl�1Þ

�l�1.

Theorem 3. The optimal threshold ðPn

T Þ for modular production is

strictly increasing as the market volatility (s) increases; and the

optimal partial modular degree (mo1) is positively related to

market volatility (s).

Proof. See Appendix A.5.

From the first part of Theorem 3, we can conclude thatincreased market uncertainty, measured by the volatility co-efficient s, will postpone the timing of when to switch to modularproduction. In another word, our model supports the view thatincreased uncertainty stimulates the current relatively low mod-ularity (m0) production as it increases the value of postponingirreversible investments for the establishment of a product familyof high-modularity. In this respect our model is consistent withthe common theory of real options: a high uncertainty increasesthe value of postponement option (Dixit and Pindyck, 1994).

The second part of Theorem 3 confirms that @mn=@s40 underthe condition of Pt obðr�mÞ where it is optimal for a firm toredesign a product family into the one of higher modularity asmarket uncertainty s increases. Thus, increased market uncer-tainty will stimulate modularization in the sense of increasingproduct modularity. Intuitively, this can be explained as follows.The introduction of a production mode with relatively high partialmodularity (mo1) means a switch from a profit flow associatedwith current product family of relatively low modularity (m0) to aprofit flow with high-modular production. This switch in thepresent mode to modular production is undertaken when thediscounted value of the cost savings associated with modulariza-tion dominates relative to the proportion of opportunity cost thatincludes adjusted irreversible investment expenditure and theexpected profit of the present product family of low modularity.An increase in market uncertainty will amplify the dominance ofthe discounted value of these cost savings, which will induce thefirm to increase its product modularity. Therefore, modularizationin the sense of increasing its product modularity can be used as ageneral strategy in response to the increased market uncertainty.However, it should be emphasized that our approach permits the

Page 5: Optimal modular production strategies under market uncertainty: A real options perspective

Pt P0

M1

M2

W1

W2

* 1

1( 1)(1 )

P0PT m0

1* 1 0

1( 1)( 1)

mPT

P00

Pt

P0Pt

0

m*

1P0

Fig. 1. Structure of optimal strategies (M—modularization, W—waiting, P0¼ bðr�mÞ).

S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274270

distinction of the effects of increased volatility on the optimalproduct modularity and on modularization timing.

Theorem 4. The optimal threshold ðPn

T Þ for starting modularization

is strictly decreasing as investment efficiency (l) increases; the

optimal partial product modularity (mo1) is negatively related to

the investment efficiency (l).

Proof. See Appendix A.6.

In Appendix A.6, we show that the feasible region is l41=ð1�yÞ (where y¼ ~b1m0=ð ~b1�1Þo1) under the condition of partialmodularity (mo1), and the optimal threshold Pn

T is decreasedwhen investment efficiency increases. Also, under the conditionof maximum modularity (m¼1), the nature of the relationshipbetween the optimal threshold Pn

T and investment efficiency isinvariable. This finding is intuitive in many real projects. Higherinvestment efficiency increases an opportunity cost of delaydecision that stimulate the firm to initiate modularization earlierfor capturing the substantial investment value. The second part ofTheorem 4 confirms that @mn=@lo0 in the feasible regionl41=ð1�yÞ. In our model, the optimal partial modularity levelmn is an endogenous variable which depends on the optimalthreshold Pn

T for initiating modularization and other relatedparameters. In Appendix A.6, we shows that in the feasible regionl41=ð1�yÞ, an inverse relationship between mn and l exists,that is mn ¼ yþy=ðl�1Þ. Therefore, Theorem 4 demonstratesthat when investment efficiency is high at preparation stage, afirm should modularize earlier with a relatively low productmodularity.

Theorem 5. The optimal threshold ðPn

NPV Þ for initiating modulariza-

tion derived by the net present value method (NPV) must be less than

the threshold ðPn

T Þ derived by real option theory. That is, Pn

NPV oPn

T .

Proof. See Appendix A.7.

Theorem 5 implies that the traditional NPV method is alwaysless rigorous than the real option theory. In this respect our modelis consistent with the common theory of real options. As men-tioned in the preceding section, the standard NPV method fails tocapture the value of managerial flexibilities and can induce thethreshold of investment before the optimal time (i.e., Pn

NPV oPn

T ),thereby causing a loss of an investment opportunity value (seeDixit and Pindyck, 1994; Luehrman, 1998; Huchzermeier andLoch, 2001).

So far we have commented on the impact of the marketuncertainty on the optimal threshold for initiating modularizationand the optimal product modularity. According to Corollary 1 we

can also directly draw conclusions on the effects of the marginalcost savings associated with modular production. Clearly, themarginal cost savings reduce the threshold for introducingmodular production and thereby speed up the product architec-ture redesign as well as the organizational redesign. Once thisoptimal threshold has been reached the optimal product mod-ularity is invariant with respect to these profit flow advantages.Besides, we also analyze the effects of investment efficiency onthe optimal modularization strategies (timing and level) anddraw a comparison between our model and the traditional NPVmethod.

5. An illustrative example

An example is used next to illustrate the potential of realoptions to increase the flexibility value of a modular productionsystem. In most cases, the process of adjusting real options intothe practice of strategic decision-making is far from smooth(Kemna, 1993). Based on the application of a real options methodto strategy planning at Shell International Petroleum Company,Kemna (1993) recommended adopting a few simple options.Therefore, this example is simplified in order to keep theillustration legible, without changing the basic option-like char-acteristics of the decision problem.

Consider a firm who is investing in a modularization project toimprove the existing product architecture to gain highercredibility and bigger foothold in the competitive and changingmarketplace. One key reason to choose modular architecture asthe new product architecture is that ‘‘modularity is tolerant ofuncertainty’’ (Baldwin and Clark, 2000), which is justified by apreliminary investigation and feasibility study. This projectrequires the firm to redesign the product architecture and processarrangement, procure materials and services, and produce newproducts. The life-cycle of this modular architecture is expectedto be 8 years. At the end of the planning horizon, the new productarchitecture will become obsolete with zero salvage value. Thefirm will operate the modular production system for 8 years andtherefore absorbs the dynamic uncertainties associated withmarket environment and firm operation. The tasks of strategicproject planners are to choose proper modularity measurementand analyze the optimal timing of modular production and theresulting modularity deepness. Hence, the first question theproject planners struggle with is: How to measure the modularityappropriately? That is, which modularity measurement is appro-priate for the company? The planners should consider the specificcharacteristics of the products, the firm and the market.

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S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274 271

To facilitate the understanding of modularity measurement,we here briefly describe the modularity function presented byMikkola and Gassmann (2003) and widely used in practices.Specifically, modularization function (the below equation) is usedto examine the degree of modularization embedded in productarchitectures and trade-offs caused by components, interfaces,the substitutability factor of NTF (new-to-firm) components, anddegree of coupling. See Mikkola and Gassmann (2003) for itsformulation and detailed illustration:

MðnNTF Þ ¼ e�n2NTF

=2Nsd,

where M(nNTF) is the modularity function, nNTF is the number ofNTF components, N is the total number of components, s is thesubstitutability factor, and d is the degree of coupling.

Having addressed the modularity measurement, the plannersturn their attention to project parameter estimation. According tothe selected modularity measurement, the modularity level ofexisting products is estimated to be 0.1 ðm0 ¼ 0:1Þ. Also, ananalysis reveals that the maximum of modularity level of pro-ducts is less than one ðmo1Þ. Using the data of similar projectsand the life-cycle of this modular architecture, the risk adjusteddiscount rate for net modular production profits can be estimated(r is estimated to be 8%). An analysis reveals that effectivelyforecasting market uncertainty could make the differencebetween profit and loss for the firm. To evaluate the influenceof market uncertainty, the planners can view that firm’s futureprofit flow follows a Geometric Brownian Motion (GBM). There-fore another issue faced by the project planners is how toproperly parameterize the variables of drift (m) and volatility(s). The estimation of m and s requires firm-specific historicalseries data. Here, m and s are estimated to be 0.01 and 0.5,respectively (See Section 3 for the estimation approach).Similarly, the parameters related to the investment costs of modularproduction can also be appropriately evaluated (i.e., l¼4 andb¼100). In summary, all the relevant parameters are given asfollows: m0 ¼ 0:1, r¼ 8%, m¼ 0:01, s¼ 0:5, l¼ 4, and b¼ 100.

Using the information estimated or provided, the plannersneed to select an appropriate method for project evaluation. Thetraditional NPV method assumes that the project will have toinitiate immediately, irrespective of its future NPV. That is, giventhe received information, a positive NPV results in project initia-tion. However, the planners have readily recognized that post-poning the investment of modular production could lead to ahigher NPV in the future due to a potential increase in firm profitflow. Therefore, in the real options setting, there is no obligationto start the modular production, but only a right which will onlybe exercised if the future NPV is positive.

Based on the received information, the project planners can workout the timing of exercising the option and the resulting modularitydeepness. In this example, the threshold of starting modular produc-tion obtained by real options method is Pn

T ¼ 0:0174, which is largerthan the threshold ðPn

NPV ¼ 0:0022Þ derived by NPV method. More-over, the option value at initiation time is 0.023, which is added byflexible planning strategy. But the flexibility value of traditional NPVevaluation is zero because any positive NPV will lead to projectinitiation. In addition, the optimal modularity deepness is 0.136(4m0) when modularization initiates at the timing of Pn

T ¼ 0:0174:The flexibility provided by the real options strategy method

adds both monetary value and strategic value to the modulariza-tion project. By recognizing and designing different possiblemeasurement and evaluation approaches through the project,the firm shifts from managing and planning in a narrow and‘‘myopic’’ way to a broader spectrum of alternatives and scenariosfrom which the firm may select. For example, this firm can furtherconsider the impact of multiple types of uncertainties on themodularization strategies. In particular, one vital thing to keep in

mind is that the value of flexibility is captured only by identifyingand modeling this latent project value.

6. Conclusions and further research

This paper develops a product modularization model based onreal options theory to help a firm determine the optimal modularproduction strategies under market uncertainty: whether/whento exercise the option of modularization. Managerial flexibilityincurred by the uncertainty is involved in our model using a realoption theory. Even though many modularity models are estab-lished partially in the real world, there exist only few models.Furthermore, the efforts that a firm is willing to invest in order toimprove the profit flow of the existing products has been over-looked in the literatures. The proposed valuation model fills thisgap among researches for modularization under market uncer-tainty. Based on the proposed model, we have studied the effectsof volatility factor on optimal product modularity and modular-ization timing strategies.

The results show that when market is more volatile, it isoptimal for a firm to postpone modularization, and when a firm’sinvestment efficiency at the preparation stage is higher, the firmcan start modular production earlier with relatively low productmodularity. An increase in market uncertainty will stimulate thefirm to improve its product modularity. By comparing the predic-tions from the widely used net present value method (NPV) to theresults from our real options model, we argue that traditional NPVmethod underestimates a firm’s value for modular production andmight mislead a firm to modularize earlier. Finally, an example isprovided to illustrate the flexibility value of a modular productionsystem added by real options.

The model proposed in this paper makes a theoretical step furthertoward better understanding the effects of market variability on thevalue of managerial flexibility in modularization projects. Severalissues still remain to be explored. First, one can study the impact ofmultiple types of uncertainties (e.g., market payoff variability, budgetvariability, product performance variability, market requirementvariability, and schedule variability, see Huchzermeier and Loch(2001)) on the optimal modularization strategies. Second, our modelcan be extended to strategic competition environment (e.g., oligopolymarket) where the firm is a strategic option holder with the optimalthresholds for modularization and optimal product modularity asstrategic instruments. Additionally, the relationship between modu-larity and outsourcing can be another interesting extension. Mikkola(2003) points out that outsourcing decisions are usually madesimultaneously with the design of modular product architectureswhere specialization of knowledge is reached through specializationof labor. Consequently, the mix strategies considering modularity andoutsourcing under dynamic uncertainties can be another interestingresearch question.

Acknowledgments

The work was supported by a grant from the National NaturalScience Foundation of China (No. 70872027). The authors thankthe reviewers for their critical but constructive comments.

Appendix A

A.1. Proof of Lemma 1.

As shown in Dixit and Pindyck (1994), in the continuousregion (0, t), the Bellman equation of Vm0

ðPtÞ can be rearranged

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S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274272

as follows:

Vm0ðPtÞ ¼m0Ptdtþe�rdtE½Vm0

ðPtþdPtÞ�:

Using Ito’s lemma, the Bellman equation above becomes

Vm0ðPtÞ ¼m0Ptdtþð1�rdtÞ Vm0

ðPtÞþmPtV0m0ðPtÞdtþ

1

2s2P2

t V 00m0ðPtÞdt

� �:

Then we conclude that Vm0ðPtÞ satisfies the following differential

equation:

1

2s2Pt

2V 00m0ðPtÞþmPtV

0m0ðPtÞ�rVm0

ðPtÞþm0Pt ¼ 0: ðA:1Þ

Solving (A.1), the complete solution is given by

Vm0ðPtÞ ¼ A1Pb1

t þA2Pb2t þ

m0Pt

r�m , ðA:2Þ

where

b1 ¼1

2�

ms2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffims2�

1

2

� �2

þ2r

s2

s41,

b2 ¼1

2�

ms2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffims2�

1

2

� �2

þ2r

s2

so0:,

In addition, Vm0ðPtÞ satisfies the following boundary conditions:

Vm0ðPn

T Þ ¼ 0, and Vm0ð0Þ ¼ 0:

Then we have A2¼0 and

A1 ¼�m0ðP

n

T Þ1�b1

r�m:

Plugging the expressions above for A1 and A2 into (A.2), we have

Vm0ðPtÞ ¼ E

Z t

0e�rtm0Ptdt

� �¼

m0Pt

r�m 1�Ptb1�1ðPn

T Þ1�b1

h iPt oPn

T

0 Pt ZPn

T

8<:

A.2. Proof of Lemma 2.

Let DðtÞ ¼ E½e�rt�, the Bellman equation of D(t) can be read as

DðtÞ ¼ e�rdtE½DðPtþdPtÞ�:

By the similar way to the proof of Lemma 1, we derive that D(t)satisfies the following differential equation:

12s

2Pt2D00ðtÞþmPtD

00ðtÞ�rDðtÞ ¼ 0: ðA:3Þ

In addition, D(t) satisfies the following boundary conditions:

Dð0Þ ¼ 0, and DðTÞ ¼ 1: ðA:4Þ

Solving (A.3) and(A.4), we obtain that

E e�rt� �

¼

Pt

Pn

T

�b1

Pt oPn

T

1 Pt ZPn

T

8<:

A.3. Proof of Theorem 1.

Consider the mapping

HðmÞ ¼mPt

r�m�b

lml:

By the first and second order conditions, we have that

H0ðmÞ ¼Pt

r�m�bml�1 and H00ðmÞ ¼�bðl�1Þml�2o0:

Thus, the optimal modularity level is

m¼Pt

bðr�mÞ

h i1=l�1Pt obðr�mÞ

1 Pt Zbðr�mÞ

8<:

A.4. Proof of Theorem 2.

(i) Under the condition of Pt Zbðr�mÞ or m¼1:Let GðVÞ ¼ V�Im,where

V ¼1�m0

r�mPt , Im ¼

b

l:

Here G(V) represents the option value of modular production atthe investment launch time t. Hence, in the continuous region(t, N), we have the following Bellman equation:

rGdt¼ E½dG�: ðA:5Þ

Using Ito’s lemma, we have

E dG� �

¼ mG0dtþ12s

2G00dt: ðNote that, E dzt

� �¼ 0:Þ ðA:6Þ

The above Eqs. (A.5) and (A.6) result in the following differentialequation:

12s

2G00dtþmG0dt�rG¼ 0,

with boundary conditions as follows:

Gð0Þ ¼ 0,

GðVnÞ ¼ Vn

�Im ðvalue-matching conditionÞ,

G0ðVnÞ ¼ 1 ðsmooth-pasting conditionÞ:

Solving the differential equation above, we derive that

GðVÞ ¼ BVb1 ,

where

Vn¼

b1

b1�1Im,

B¼ðb1�1Þb1�1

ðb1Þb1 Im

b1�1: ðA:7Þ

According to (A.7), we have

1�m0

r�mPn

T ¼b1b

ðb1�1Þl:

Thus, if m¼1, then the optimal threshold ðPn

T Þ is

Pn

T ¼b1bðr�mÞ

lðb1�1Þð1�m0Þwhere Pn

T 4bðr�mÞ,

and GðPtÞ reads as

GðPtÞ ¼

blðb1�1Þ bðr�mÞrPt oPn

T

ð1�m0ÞPt

r�m � bl Pt ZPn

T

8><>:

(ii) Under the condition of Pt obðr�mÞ or mo1:Taking Lemma 1, we derive that m¼ ½Pt=bðr�mÞ�1=l�1,and

VmðPtÞ�CðPtÞ�ImðPtÞ ¼Pt

r�m

� �l=l�1

bl=l�1 1�1

l

� ��

m0Pt

r�m :

Let:

Gð ~V ,~IÞ ¼ ~V�~I , where ~V ¼Pt

r�m

� �l=l�1

b1=1�l 1�1

l

� �and ~I ¼

m0Pt

r�m :

Here Gð ~V Þ represents the option value of modular production atthe investment launch time t.

Using Ito’s lemma, we have that

d ~V ¼ ~mV~V dtþ ~sV

~V dz,

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S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274 273

where

~mV ¼l

l�1mþ 1

2

ll�1

ll�1�1

� �s2or, ~sV ¼

ll�1

s: ðA:8Þ

Also, we obtain that

d~I ¼ m~Idtþs~Idz: ðA:9Þ

Thus, ~V and ~I follow the Geometric Brownian Motion thatexpressed in (A.8) and (A.9) respectively. Note that, we assume~mV or for the convergence.

As shown in Dixit and Pindyck (1994), in the case of variableinvestment cost, the optimal decision depends on the value ofv¼ ~V =~I :

Let

Gð ~V ,~IÞ ¼ g~V~I

!~I ¼ gðvÞ~I ,

where g is a function of v. Then, g(v) follows the followingdifferential equation (see Dixit and Pindyck, 1994):

12ð~s2

V�2 ~sVsþs2Þv2g00ðvÞþð ~mV�mÞvg0ðvÞ�ðr�mÞgðvÞ ¼ 0

with boundary conditions as follows:

gðvÞ ¼ v�1 ðvalue-matching conditionÞ,

g0ðvÞ ¼ 1 ðsmooth-pasting conditionÞ:

Solving the differential equation above, the optimal vn is given by

vn ¼~Vn

~In¼

~b1

~b1�1, ðA:10Þ

where ~b1 is the larger root of the following characteristicequation:

1

2ð ~sV�sÞ2 ~bð ~b�1Þþð ~mV�mÞ ~b�ðr�mÞ ¼ 0: ðA:11Þ

Solving the characteristic equation above, we have

~b1 ¼1

2�

~mV�mð ~sV�sÞ2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~mV�mð ~sV�sÞ2

�1

2

" #2

þ2ðr�mÞð ~sV�sÞ2

vuut 41:

Based on (A.8)–(A.10), the optimal threshold is obtained asfollows:

Pn

T ¼ bðr�mÞ~b1m0l

ð ~b1�1Þðl�1Þ

!l�1

where Pn

T obðr�mÞ,

and further we derive

GðPtÞ ¼

m0Pn

T

ð ~b1�1Þðr�mÞPt oPn

T

Pt

r�m

�l=l�1b1=1�l 1� 1

l

� ��

m0Pt

r�m Pn

T rPt obðr�mÞ

8><>:

A.5. Proof of Theorem 3.

(i) Under the condition of Pt Zbðr�mÞor m¼1:According to Theorem 2, we derive that

@Pn

T

@b1

¼�ðb1�1Þ�2 bðr�mÞlð1�m0Þ

o0:

Let Q ðb,sÞ is the characteristic equation of b, then we have

Q ¼1

2s2bðb�1Þþmb�r¼ 0,

@Q

@b1

@b1

@sþ@Q

@s¼ 0:

Since that ð@Q=@sÞ ¼ sbðb�1Þ40, and ð@Q=@b1Þ40, then wederive ð@b1=@sÞo0.Thus, we obtain

@Pn

T

@s ¼@Pn

T

@b1

@b1

@s 40:

(ii) Under the condition of Pt obðr�mÞ or mo1:According to Theorem 2, we have

@Pn

T

@ ~b1

¼�ð ~b1�1Þ�2ðl�1Þ

~b1

~b1�1

!l�2

bðr�mÞðm0ll�1Þl�1o0:

Let Q ð ~b,sÞ is the characteristic equation of ~b. Based on(A.8)–(A.11), we have

Q ¼ 12 ðl�1Þ�2s2 ~bð ~b�1Þþ ðl�1Þ�1mþ1

2lðl�1Þ�2s2h i

~b�ðr�mÞ ¼ 0,

@Q

@ ~b1

@ ~b1

@s þ@Q

@s ¼ 0:

Because of@Q@s ¼ ðl�1Þ�2 ~bsð ~bþl�1Þ40, and @Q

@ ~b1

40,then we deriveð@ ~b1=@sÞo0

Thus, we get

@Pn

T

@s¼@Pn

T

@b1

@b1

@s40

and further

@m

@s ¼@m

@Pn

T

@Pn

T

@s 40:

A.6. Proof of Theorem 4.

(i) Under the condition of Pt Zbðr�mÞor m¼1:Taking the derivative of Pn

T given in Theorem 2, with respect tol, we obtain

@Pn

T

@l¼�l�2 b1bðr�mÞ

ðb1�1Þð1�m0Þo0:

(ii) Under the condition of Pt obðr�mÞ or mo1:Let

y¼~b1m0

~b1�1, x¼ l�1:

Rearranging the optimal threshold given in Theorem 2, we obtain

Pn

T ¼ bðr�mÞ yþyx

� �x

, 0oyo1, x40:

Taking the derivative of Pn

T given above, with respect to x, we have

@Pn

T

@x¼ Pn

T ln yþyx

� ��

1

xþ1

� �:

Let

HðxÞ ¼ ln yþyx

� ��

1

xþ1,

then we obtain

@HðxÞ

@x¼

1

xþ1

1

xþ1�

1

x

� �o0,,

limx-0

HðxÞ ¼ þ1 and limx-þ1

HðxÞ ¼ lnyo0:

Thus, there exists unique xn such that HðxnÞ ¼ 0.

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S.X. Xu et al. / Int. J. Production Economics 139 (2012) 266–274274

Further, we derive that

@Pn

T

@x40 if xoxn;

@Pn

T

@xo0 and x4xn:

Since Pn

T 4 limx-0Pn

T ðxÞ ¼ bðr�mÞ if xoxn (infeasible region), weconclude that

@Pn

T

@xo0 and Pn

T oPn

T ð ~xÞ ¼ bðr�mÞ if x4 ~x ðfeasible regionÞ

where

~x ¼y

1�y4xn:

That is, in the feasible region (l41=1�y), Pn

T satisfies @Pn

[email protected] to Theorem 1, the optimal partial modularity level is

expressed as mn ¼ yþðy=l�1Þ.Thus, mn satisfies @mn=@lo0 where l4 ð1=1�yÞ.

A.7. Proof of Theorem 5

(i) Under the condition of Pt Zbðr�mÞ or m¼1:According to NPV method and Theorem 2, we have GðVÞ ¼

V�Im ¼ 0 and Pn

NPV ¼bðr�mÞlð1�m0Þ

.Thus, we derive

Pn

NPV ¼bðr�mÞlð1�m0Þ

oPn

T ¼b1bðr�mÞ

lðb1�1Þð1�m0Þ:

(ii) Under the condition of Pt obðr�mÞor mo1:Similarly, let Gð ~V ,~IÞ ¼ ~V�~I ¼ 0, and then we have Pn

NPV ¼

bðr�mÞðm0l=l�1Þl�1.Thus, we derive

Pn

NPV ¼ bðr�mÞ m0ll�1

� �l�1

oPn

T ¼ bðr�mÞ~b1m0l

ð ~b1�1Þðl�1Þ

!l�1

:

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