‘Hybrid’ competition, innovation outcomes and regulation: A duopoly model

Thomas LE TEXIER* Université de Rennes 1, CREM – UMR 6211 CNRS 7, place Hoche, 35065 Rennes Cedex, France [email protected]

Ludovic RAGNI Université de Nice – Sophia Antipolis, GREDEG – UMR 6227 CNRS 250, rue Albert Einstein, 06560 Valbonne Cedex, France [email protected]

*

Corresponding author

Tel. +33(0)223233006 Fax. +33(0)223233509 [email protected]

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Abstract

This paper presents a duopoly model in which a commercial organization and a community compete by providing digital products while being able to share their innovation outputs to develop their own activities. The commercial organization always benefits from either a ‘closed’ or an ‘open’ institutional regime shift. Our numerical analysis evidences that the ‘closed’ shift provides the best levels of innovation and welfare whereas it is not found to be profit-improving when product differentiation is small. This result partially qualifies the conventional idea according to which public policies may be designed to defend commercial interests rather than public ones.

JEL. classification: D43, L13, L86.

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1. Introduction

The introduction of new appropriation channels for goods has led to fierce opposition between traditional commercial-oriented players and less traditional ones. In the case of digital diffusion, the advent of the Internet as a new transactional space is not only directly based on new patterns of consumption but on productive ones too. Indeed, the development of new compression standards highlights the shift to the ‘dematerialization era’ which leads to the widespreading of digital files online and to new technological adoption issues (Shapiro and Varian, 1998; Varian, 2000; Hui and Png, 2003; Chellappa and Shivendu, 2005; Peitz and Waelbroeck, 2006; Belleflamme and Peitz, 2010). In this context, new competition patterns have also emerged inasmuch as non-commercial players have shown that their user-centric production activities may threaten those of traditional commercial entities (Toffler, 1980). The example of Napster represents an innovation case which has clearly shown that good ideas may be developed outside the boundaries of commercial organizations. The way commercial players first fought against that online music file-sharing before intending to integrate it into their business scheme reveals to what extent ‘outlaw’ innovations may be good enough to be considered for profit-enhancing purposes (Flowers, 2008; Schulz and Wagner, 2008). As an illustration, the success of iTunes as a commercial distributive digital platform mostly derives from that of its illegal ancestors, such as Napster, Gnutella and eDonkey. In a similar fashion, the developing success of the VoD – Video on Demand – commercial activity expresses the market-based attempts of some commercial players to react to the ‘pirate’ threat by exploiting the innovation outputs that online communities initially developed for their own – non-commercial – needs. As such, one may see a shift in the manner commercial organizations consider appropriation schemes. Therefore, an increasing number of commercial players are nowadays likely to co-operate with the external sources of innovation they previously competed with, evidencing other potential cases of either ‘democratized innovation’ (von Hippel, 1986; 1988; 2005) or ‘open innovation’ (Chesbrough, 2003; 2006).

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A first key research question aims at analyzing if such emerging hybrid – both private and public – production patterns are likely to provide the best outcomes to commercial players (Grand et al., 2004; Bonaccorsi et al., 2006; Economides and Katsamakas, 2006). Indeed, one may attempt to find out if developing an asset-sharing view of carrying out for-profit activities necessarily leads commercial organizations to achieve larger profits than they would within the old-fashioned ‘closed innovation’ framework. A second key research question is to study the impact of the closeness and openness of both commercial players and communities on their motives to innovate when they evolve on the same market. Indeed, such results have to be considered for policy applications to identify whether a ‘closed innovation’ or an ‘open innovation’ regime is likely to stimulate pro-innovation behaviors (Nelson, 1959; Merton, 1973; David, 1998; 2004; Nelson, 2004).

Although the ‘open innovation’ paradigm should obviously lead one to consider both market-based implications and organizational restructurings, our approach of such a paradigm rather deals with its consequences on competition outcomes in this article. Moreover, this work aims at measuring to what extent two competitors may – or may not – mutually benefit from sharing their innovation outputs when they evolve in an institutional context in which they are able to do so. In particular, it appears somehow relevant to study such potential cooperation patterns between commercial – for-profit – organizations and communities of users who are driven by non-pecuniary concerns inasmuch as one may observe the developing of institutional regimes within which the appropriating of external innovation outputs is likely to be stimulated. For illustration purposes, let us refer to the success of the ‘free/libre’ GNU/Linux software project over the Unix proprietary one which led to the creation of the Open Source Initiative in 1990’s. The goal of the Open Source Initiative was to clarify an ‘open source’ methodology to define a proxy to be exploited by both communities and software firms (Weber, 2004). From a practical point of view, the building of the Open Source Initiative and the ensuing labeling of specific ‘open source’ license terms can be seen as the developing of an open institutional regime which enabled players to benefit from mutualizing their innovation efforts.

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We analyze the impact of an institutional regime shift (i.e., ‘open shift’ or ‘closed shift’) on the profits of commercial organizations as well as on the level of innovation they are likely to provide. For this purpose, we present a duopoly model which depicts a ‘hybrid’ competition framework. By ‘hybrid’ competition framework, we here refer to a specific framework in which two types of producers (i.e., a commercial organization and a community) compete while being able to share – under some circumstances – their innovation outputs to develop their own activities. Competition outcomes (e.g., market shares) are identified according to the objective functions of each type of players, namely profit and surplus functions. We identify the optimal pricing strategy of the commercial organization and its related profit, as well as the optimal surplus of the community. The innovation levels delivered by the commercial organization and the community at the optimal state are also presented to provide regulatory insights. Our results reveal that the commercial organization always benefits from either a ‘closed’ or ‘open’ regime shift. They also evidence that the commercial organization and the community both have higher-leveled incentives to innovate when they act in a ‘closed regime’ framework. From a numerical illustration, we observe that the ‘closed shift’ provides the best levels of both global innovation and welfare whereas such a shift is not found to be profit-improving when product differentiation is small. This result clearly exhibits a potential conflict of interest between commercial players and policy makers and partially qualifies the conventional idea according to which public policies may be designed to defend private – commercial – interests rather than public ones.

The organization of the paper is as follows. Section 2 presents the settings of the model. In section 3 we analyze the optimal pricing and innovation strategies as well as the ensuing optimal profits and surpluses according to the three institutional regimes we specify. In section 4 we identify profitimproving and innovation-enhancing regimes and a numerical analysis is carried out to illustrate our comparative study. Welfare results are also discussed. Section 5 concludes and provides directions for further research.

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2. The model

We present a market in which two producers act as duopolists when providing digital products via their own dedicated distribution channels. Although each producer provides digital products users may adopt to meet their entertainment-related consumption expectations, they differ in their intrinsic nature. Indeed, we introduce two types of producers in the model, since one of the producers is set to be a commercial organization ( F ) and the other is set to be an online community ( C ). As such, both producers differ in the nature of their incentives to produce inasmuch as commercial organizations are driven by traditional profit-oriented motives whereas community-based production activities are rather driven by altruistic, ideological or signaling purposes (see Rossi 2006 and Flowers 2008 for general insights about motives and incentives to produce within online communities). From an output-related point of view, we suggest that both producers deliver different services according to their nature. We hence state that commercial organizations are more likely to produce high-quality digital goods and that communities are more likely to build up innovative distributive tools for users to easily get access to the content they provide. The products we consider here are not only batches of files but also the distribution channels that are developed by both organizations to diffuse digital goods. As a consequence, each organization sells a differentiated product in a ‘à la’ Hotelling framework, the firm (resp. the community) being located at 0 (resp. 1).

On the demand side, we consider adopters who are uniformly distributed on the Hotelling line and whose total mass is equal to 1. They adopt one product that is provided by either the commercial organization or the community. We define x as the location of each product on the line ( x   0;1 ).

Consumers whose x is close to 0 exhibit preferences for the digital product provided by the commercial organization whereas consumers whose x is close to 1 are more interested in adopting the digital product provided by the community. Utility functions are defined as follows:

r  qF   qC  tx  p r  qC   qF  t 1  x   s

Ux  

if adopts product from F if adopts product from C

1

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r is the gross utility adopters derive from digital consumption. tx (resp. t 1  x  ) is the transportation disutility adopters get from adopting the product provided by the commercial organization (resp. the community). x (resp. 1  x  ) represents the distance between any adopter’s ideal product and that provided by the commercial organization (resp. the community) and t is the traditional transportation cost parameter used when formalizing product differentiation. p ( p  0 ) is the price the firm charges to consumers when adopting their product and s ( s  0 ) represents the cost

adopters have to face when adopting the product provided by the community. For instance, s refers to the opportunity costs an adopter has to face when adopting a product provided by the community (e.g., the amount of time she needs to assimilate and to be able to efficiently use community-based services). qF ( qF  0 ) (resp. qC , qC  0 ) is the level of quality that is provided by the commercial organization (resp. the community) when releasing its product. Such levels are positively appreciated by adopters when consuming one of the two products. In our general framework ( 1 ), we also suppose that the commercial organization (resp. the community) can catch innovation outputs from the production activity of the the community (resp. the commercial organization) to enhance its own activity. In other words, we here define a situation in which both producers may freely appropriate innovation outputs that are developed outside their production boundaries.  (    0;1 ) represents the share of innovation outputs that the commercial organization appropriates from the production activity of the community and  (    0;1 ) represents the share of innovation outputs that the commercial organization appropriates from the production activity of the commercial organization. Such appropriation schemes are taken into account in the adopters’ decision-making and shape adoption patterns. Specific cases are next analyzed.

As we previously underlined, both producers (i.e., the commercial firm and the community) differ in their motives to provide digital products. The commercial firm is driven by pure for-profit purposes whereas the community is generally perceived to carry out production activities for rather altruistic or signaling reasons. Yet, although such a viewpoint may lead one to define distinct objective functions

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for these two types of producers, we consider that the objective function of the community can be defined as a traditional profit function. Indeed, although one may think that community-based activities are not carried out for pecuniary reasons, we observe that such activities require funds to be carried out, mostly for technical reasons. For instance, since servers are costly to acquire and maintenance activities are often required to avoid traffic overloads, financial aspects also have to be taken into account. As a consequence, both production activities are led so that these remain sustainable. In the case of commercial organizations, profit-maximizing decision-making is likely to shape the level of innovation effort. In the case of community-driven production activities, innovation decision-making is led for sustainability-related reasons and pecuniary resources are apprehended as a means of meeting technical constraints and dealing with external ‘disturbing’ effects (e.g., server crashes or legal penalties when activities are shown to be illegal). As such negative externalities are likely to occur at any time, we suggest that administrative players acting within online communities seek to raise funds from various sources to prevent uncertainty and meet technical imperatives. We define the producers’ objective functions as follows:

  nF p  1 2  qF2  2  S  nC a  1 2  qC

 2

 is the profit function of the commercial organization when supplying its product. We define nF ( nF  0;1 ) as the mass of consumers who adopt the product provided by the firm. S is the surplus function of the community when providing its product online. nC ( nC  0;1 ) is the mass of consumers who adopt the product released by the community. a is the marginal – pecuniary – reward the administrative staff of the community collect from the adopters. Contrary to p , a ( a  0 ) is not a retail price (i.e., a price that any adopter has to pay to get the product) but can be seen as a ‘participation’ price. By ‘participation’ price we mean the marginal reward the community earns from either adopters’ donations and adopters’ advertising banner ‘clicking’-behaviors. We suggest that such a ‘participation’ price is not set by the community from market-based mechanisms, as donations are not likely to be imposed within the community and advertising-issued revenues depend on the price

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set by external players. As generally assumed, we suppose that both producers face innovation production costs whose shapes are quadratic.

 and S differ in the nature of the gross benefit both organizations generate from production. Whereas p is a control variable endogeneously set by the commercial organization, a represents an exogenous variable that the community cannot set. Such a difference introduces in our model an asymmetry in the abilities a producer has to carry out innovation-based production activities while attempting to remain sustainable on the market. Indeed, the commercial producer can maximize her profit by setting both her retail price and quality whereas the community is only able to set the quality it provides to maximize its surplus.

We define the adoption decision process as a four-step game: -

at step t  0 , both the commercial organization and the community decide whether to produce or not to produce;

-

at step t  1 , the commercial organization and the community simultaneously set their qualities qF and qC ; producers fully have mutual knowledge of the qualities that are produced in the market and each producer can use them in the product they individually release if the institutional regime allows them to do so;

-

at step t  2 , the commercial organization sets its retail price p ;

-

at step t  3 , consumers adopt the product released by either the commercial organization or the community.

Our model stands in a framework in which all agents have both full and common knowledge of the production outcomes, whether they concern prices and qualities. We here suppose that their expectations about the way prices and qualities are defined do apply.

We consider three cases in our model. In the first case (case 1), the community integrates both its innovation effort (i.e., qC ) and that of the commercial organization (i.e., qF ) into its product whereas the commercial organization only uses its own innovation effort (i.e., qC ). Such a case refers to the

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actual situation one may easily observe in the market of digital goods in which community-based activities are presented to be illegal – ‘pirate’ – activities (e.g., peer-to-peer networks, illegal streaming websites). In the second case (case 2), communities are not able to appropriate innovation efforts that are made by commercial organizations. This particular case represents the situation one would observe if copyright enforcement is efficient enough to prevent community-based organizations from illegally appropriating commercial innovations, as officially wished by an increasing body of commercial players. In the third case (case 3), both producers are able to integrate both their own innovation effort and a – more or less high-leveled – share of that delivered by the competitor into their products. Such a third case depicts a situation in which outputs are shared and can be used by both producers. Our model aims at analyzing to what extent a paradigm shift may be profitable for the two types of producers. To put it simply, we here intend to know whether a ‘closed’ innovation paradigm (case 2) or an ‘open’ innovation paradigm (case 3) is likely to deliver (i) a better level of profit (resp. surplus) and (ii) a higher-leveled joint-innovation outcome than those achieved in the current – somehow ‘asymmetric’ – situation (case 1). Moreover, we analyze the impact of closeness and openness on the incentives of both producers to innovate when they are likely to evolve on the same market. Such results have to be considered for policy applications to measure whether a ‘closed’ or an ‘open’ regime stimulates pro-innovation production behaviors while being profit-improving.

3. Optimal quality and pricing strategies

We intend to identify the optimal quality strategies that the commercial organization and the community have to set to maximize their profit/surplus as well as the optimal retail price the commercial organization can charge its consumers. To do so, the four-step game we previously presented is solved by backward-induction. From the general case captured in 1 (3.1.), we consider three specific cases, namely the ‘current situation’ case (case 1), the ‘closed regime’ case (case 2) and the ‘open regime’ case (case 3). For these three cases, we identify the optimal outcomes (i.e., retail

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price, quality strategies, profit and surplus) both players achieve depending on the environmental parameters (i.e.,  and  ) they face (3.2.).

3.1 Analysis – General case

In the general case, the utility functions of the adopters are expressed as follows:

r  qF   qC  tx  p r  qC   qF  t 1  x   s

Ux  

if adopts product from F if adopts product from C

Here, values for  and  are not specified (i.e.,    0;1 and    0;1 ). The adopter’s choice is considered at step t  3 . The adopter who is indifferent between the two products is located at location x , given by:

 

r  qF   qC  t x  p  r  qC   qF  t 1  x  s We find that

x  1 2t   qF 1     qC 1     ( s  t )    p 2 

 3

All the adopters who are characterized by parameter x  x adopt the product that is provided by the commercial organization and all the other adopters adopt the product that is provided by the community. We make three assumptions about the structure of the market for digital products.

Assumption 1a. The market for digital products is fully-served, i.e., r is sufficiently large.

Assumption 1a stresses that all the adopters are likely to adopt one of the two products released on the market, i.e., r  qF   qC  tx  p  0 and/or

r  qC   qF  t 1  x   s  0 . Such fully-serving

adoption patterns occur when r is sufficiently large.

Assumption 1b. No competition crowding-out effect applies on the market for digital products.

Assumption 1b suggests that both producers share the market, i.e., x  0;1 . Assumption 1b implies that the difference between retail price p and opportunity costs s is lower and upper bounded so that

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qF 1     qC 1     t   p  s   qF 1     qC 1     t . Let us note that such an assumption is all the more likely to apply as transportation costs t are shown to be high-leveled.

Assumption 1c. The commercial organization and the community serve the market with differentiated

products whose level of differentiation t is sufficiently large, i.e., 1 3 1     s   t .



2



We here explicitly specify a level of differentiation t above which both commercial and communitybased players share the market for digital products.

Figure 1 presents the adoption patterns which apply under assumptions 1a, 1b and 1c.

– Insert Figure 1 here – Figure 1. Adoption patterns

Within this framework, the market shares nF and nC held by each type of producers are defined to be strictly positive and are expressed as follows:

1 2t   qF 1     qC 1     ( s  t )  p  1 2t    qF 1     qC 1     (  s  t )  p 

ni  

if i  F if i  C

nF  nC  1

 4

From  2  and  4  we obtain

  qF , qC , p   1 2t   qF 1     qC 1     ( s  t )  p  p  1 2  qF2  2  S  qF , qC , p   1 2t    qF 1     qC 1     (  s  t )  p  a  1 2  qC

 5

At step t  2 , the commercial producer sets her optimal pricing strategy. Let us bear in mind that the community is not able to design such a pricing strategy inasmuch as ‘participation’ price a is set as a parameter. The profit-maximizing program of the commercial producer is:

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max   qF , qC , p   1 2t   qF 1     qC 1     ( s  t )  p  p  1 2  qF2 p

It can be shown that the Nash equilibrium at t  2 is characterized by the following price: p *  qF , qC   1 2   qF 1     qC 1     ( s  t ) 

6

We verify that p*  qF ; qC  depicts a maximum state since  2 p 2    1 t   0 . Equation  6  provides preliminary insights about the impact of the quality of the products delivered by the two players on the retail price p*  p*  qF ; qC  charged by the commercial organization. On

the one hand, we observe that the level of such a price positively depends on both the quality the commercial organization sets when releasing its product (i.e., qF ) and the share of innovation outputs that it appropriates from the community’s activity (i.e.,  ). On the other hand, its level negatively depends on the quality of the innovation output provided by the community (i.e., qC ) as well as the share of commercial innovation outputs the community is able to integrate into its product (i.e.,  ). Overall effects are nevertheless more likely to be apprehended when optimal quality-based strategies are identified.

When the commercial organization charges retail price p  p*  qF , qC  , optimal levels of profit and surplus are defined so that

 *  qF ; qC   1 8t  1    qF  1    qC   s  t 2  1 2  qF2  * 2  S  qF ; qC   1 4t    1    qF  1    qC    s  3t   1 2  qC At step t  1 , both producers set their optimal quality strategies, namely, qF* and qC* . Optimal levels for quality are set according to the optimal retail price p*  p*  qF ; qC  the commercial organization is expected to charge at step t  2 .

The profit-maximizing program of the commercial producer is max  *  qF ; qC   1 8t  1    qF  1    qC   s  t   1 2  qF2 2

qF

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and the surplus-maximizing program of the community is defined by: max S *  qF ; qC   1 4t    1    qF  1    qC    s  3t   1 2  qC2 qC

The quality-related reaction functions of both producers are





 qF  qC   1 1   2  4t  1   1    qC  1    ( s  t )     qC  qF   1 4t 1    a

7

By solving system  7  we find that the Nash equilibria at step t  1 are as follows:





 q*  1  1    2  4t  4t    1    1    2 a  4t  s  t    F       *  qC  1 4t 1    a

8

Under assumption 1c, we find that  2 * qF2  1 4t  1     4t   0 and  2 S * qC2  1  0 . As   2

such, Nash equilibria qF* and qC* depict maximum states when the commercial producer (resp. the community) optimize her profit (resp. surplus) function.

Let us note that an asymmetry in the design of optimal quality strategies of the two producers occurs. Indeed, we see that the community does not take the quality strategy of the commercial organization into account whereas the commercial player integrates that of the community into her own quality strategy. Such an asymmetry here results from the differing abilities of both producers to set their prices at step t  2 (i.e., retail price p and ‘participation’ price a ). a being exogenous, the community is not able to align its level of quality to that of the commercial organization. When retail price p is set so that the profit of the commercial organization reaches its optimal state

 *   *  qF ; qC  , optimal quality strategies are defined so that the commercial organization benefits from a positioning advantage. This positioning advantage allows it to set its level of quality according to the level provided by the community and prevent the community from doing so. The optimal pricing strategy of the commercial organization thus allows it to consider the community as a productive entity which is not able to react to the quality strategy it sets.

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As already suggested by assumption 1c, we restrict our analysis to high levels of transportation costs. Moreover, we suppose that optimal quality levels are always found to be positive or null.

Assumption 2. Both producers serve the market with differentiated products so that optimal quality

strategies do not apply for negative levels, i.e., 1 2    s  s 2  1    a   t . 2





From assumption 2, we define an analytical framework within which the two types of producers maximize their profit/surplus by setting qualities whose levels are positive or null. Although qC*  1 4t 1    a is found to be positive or null for any values of t ,  and a , we characterize the values of

t

above which

qF*  1    1    a  4t  s  t   1     4t  2



1   2 a  4t  s  t   0 . We find that



2



1

 4t 1  0 ,

i.e.,

qF*  0 if and only 1 2    s  s 2  1    a   t . 2





From assumption 1c and assumption 2, we now restrict our analysis to transportation costs whose





levels are characterized so that t   max 1 3 1     s  , 1 2    s  s 2  1    a  ;   .





2





2





Some comments have to be made about the weight of the share of innovation outputs provided by the community (i.e., the commercial organization) the commercial organization (resp. the community) is able to integrate into its product (i.e.,  , resp.  ).





1 2  * qF    2 1   1     4at 1     4t   qC*    1 4t  a

 1   2 a  4t  s  t     2 1     1   2  4t  4t  8t  1    * qF          2    2 1     4t      * qC   0

15

From  8  , we unsurprisingly find that the optimal level of quality provided by the community negatively depends on the share of commercial appropriation  whereas the optimal level of quality set by the commercial organization positively depends on such a share. In a similar fashion, we observe that the optimal level of quality provided by the commercial organization negatively depends on the share  that the community appropriates from the commercial player. Yet, one striking result is that the way the commercial organization appropriates community-based innovation outputs does not directly shape the level of innovation outputs qC* the community sets. At optimal states, the pricing strategy p*  p*  qF ; qC  of the commercial organization is defined so that appropriating commercial innovation outputs for community-based purposes has a neutral effect on the community’s willingness to innovate. Finally, we also unsurprisingly observe that the ‘participation’ price a has a positive effect on the community’s willingness to innovate while it has a negative impact on the commercial organization’s.

Lemma

1.

At

the

optimal

state,

the

commercial

organization

 1    1    2 a  4t  s  t   1   2 a  4t  s  t     ;   q ; p    2 2 1     4t  4t 1     4t  8t        ** F

**

makes

strategies

profit

 2 1 2   1 a 4 t s t           2  whereas the community sets out a quality 32t 2  1     4t   qC**  1 4t 1    a

strategy

S 

and

out

1 

 **  

**

sets

1 32t 2

1   

a

2

makes

surplus

1    1    a 2  1     ( s  3t )    a. 1   2  4t   4t 2  1   2  4t  2

2

and

2

2

When both producers maximize their profit/surplus, they set out optimal pricing and innovation



strategies, namely qF** ; p**



and qC** . From  6  and  8  , we find that the optimal pricing strategy of

the commercial organization is such that

1    2 a  4t  s  t    . As a p  p q ; q    2 1     4t  8t   **

**

* F

* C

16

consequence,

the

optimal

 **   **  qF** , qC** , p**    surplus S S **

profit

p

**

** F

** C

,q ,q

by

the

commercial

organization

is

1 

 2 1 2   1    a  4t  s  t   and the optimal level of 2 32t  1     4t  2

achieved **

achieved

1

by

  32t 1   

the

2

a

2

is

1    1    a 2  1     ( s  3t )    a.  1   2  4t   4t 2  1   2  4t  2

2

community

2

2

Proposition 1. At the optimal state, the level of profit (resp. surplus) the commercial organization

(resp. the community) achieves is always positive if both products are differentiated enough (i.e.,





t   max 1 3 1     s  , 1 2    s  s 2  1    a  ;   ).



Proof

of



2

proposition



1.

2



From

assumption



1c



and

assumption

2,

we

find

that

 **   **  qF** , qC** , p**   0 and S **  S **  qF** , qC** , p**   0 .■

Proposition 1 highlights that both producers are able to make positive profit/surplus within the analytical framework we have previously defined. The commercial organization and the community are thus always shown to provide innovation outputs, their activities being sustainable if levels of appropriation innovation  ,     0;1 apply. As such, proposition 1 stresses that the provision of 2

highly differentiated products always leads both producers to serve the market at step t  0 . Our general analysis (i.e., for which  ,     0;1 ) reveals preliminary results about the weight of 2

innovation appropriation on the sustainability of both commercial and community-based activities. Indeed, the pricing strategy asymmetry we have introduced in our model also induces an asymmetry in the abilities of both producers to set out innovation strategies. Here, optimal pricing allows the commercial player to set out her optimal quality strategy by taking the level of quality that is provided by the community into account. We conversely find that such an optimal pricing strategy prevents the community from integrating the quality strategy of the commercial organization in its own innovation

17

decision-making. Commercial and community-based activities are nevertheless found to be both sustainable inasmuch as their related optimal level of profit/surplus are always shown to be positive.

3.2 Analysis – Specific cases

From the results of the general case we have obtained in the previous sub-section (3.1.), we consider three specific cases, namely the ‘current situation’ case (case 1), the ‘closed regime’ case (case 2) and the ‘open regime’ case (case 3). For these cases, we now specify the values of the share of commercial (resp. community-based) innovation outputs the community (resp. the commercial organization) is able to integrate into its product. Hence, table 1 presents the values’ specifications according to the institutional framework we consider.

– Insert Table 1 here – Table 1. Cases and environmental parameters

We distinguish the cases in which appropriation strategies cannot apply (i.e.,   0 and/or   0 ) from those in which such strategies are possible (i.e.,   0;1 and/or   0;1 ). Let us bear in mind that our analysis aims at identifying the type of regime that is likely to provide higher-leveled outcomes. As such, we do not consider  and  as control variables that can be set out by both players but as parameters whose values depend on the environmental conditions in which the commercial organization and the community carry out their activities. The outcomes reached by both players, as well as their quality/price strategies, are defined from the results stressed in lemma 1. For each case we have to consider values for t defined so that both assumption 1c and assumption 2 are respected. Such a range of values depends on the values for  and  . Therefore, the results that are found when the analysis is carried out in the ‘current situation’ case hold for values t defined so





that t  max 1 3  (1   ) 2  s  , 1 2    s  s 2  a  . The results that are found when the analysis  

is

carried

out

in

the

‘closed

regime’

case

hold

for

values

t

defined

so

that

18





t  max 1 31  s  , 1 2   s  s 2  a

 whereas those obtained in the ‘open regime’ framework









hold for values t defined so that t  max 1 3 1     s , 1 2   s  s 2  1    a 2

2

 .

Lemma 2a. In the ‘actual situation’ case (case 1), the commercial organization sets out strategies

q

** F

 1     a  4t  s  t 



; p**  

 **  

 1   2  4t  4t   

;

  1   2  4t  8t    

 a  4t  s  t 

and

makes

profit

1 

 1 2    a  4t  s  t  whereas the community sets out a quality strategy 2 2 32t  1     4t 

q  1  4t  a and makes surplus S  ** C

**

1 32t 2

 1     ( s  3t )  1    a 2   a . 2 2  1     4t   4t   1   2  4t  2

a

2

2

Such results are obtained by substituting  by value 0 and considering  so that   0;1 . Let us note that the quality strategy set out by the community is not here related to the level of innovation appropriation chosen by the commercial organization.

Lemma 2b. In the ‘closed regime’ case (case 2), the commercial organization sets out strategies

q

** F

  a  4t  s  t   a  4t  s  t   1  1  2 ** ;  a  4t  s  t   and makes profit    2   32t 1  4t  1  4t  8t   1  4t  4t



; p **  

whereas the community sets out a quality strategy S ** 

1 32t

2

a2 

a2

1  4t  4t 

2

qC**  1  4t  a

and makes surplus

1  ( s  3t )  a.  1  4t 



Such results are obtained by substituting both  and  by values 0. Let us note again that the quality strategy set out by the community is not here related to the level of innovation appropriation chosen by the commercial organization.

19

Lemma 2c. In the ‘open regime’ case (case 3), the commercial organization sets out strategies

 1    1   2 a  4t  s  t   1   2 a  4t  s  t    ;   qF ; p    1   2  4t  4t 1   2  4t  8t   **

**

makes

profit

1 

 2 1 2   1    a  4t  s  t   whereas the community sets out a quality 2 32t  1     4t 

 **  

2

qC**  1    a   4t 

strategy

S ** 

and

1 32t

1    2

a2 

makes

surplus

1    1    a 2  1     ( s  3t )   a. 2 2 2 1   4 t  1   4 t 4 t             2

2

and

2

2

Contrary to cases 1 and 2, we see that the quality strategy set out by the community is here negatively related to the level of innovation appropriation chosen by the commercial organization.

The outcomes reached by the commercial organization and the community in the three cases are likely to differ. Such varying results may highlight potential conflicts of interest between the two types of players. We next develop a comparative statics analysis to identify such conflicts.

4. Identifying profitable and innovation-improving regimes

Let us remind that we aim at analyzing to what extent a paradigm shift may be profitable for the two types of producers. Moreover, we here intend to know if a ‘closed’ innovation paradigm (case 2) or an ‘open’ innovation paradigm (case 3) is likely to deliver (i) a better level of profit (resp. surplus) and (ii) a higher-leveled joint-innovation outcome than those reached in the current –

somehow

‘asymmetric’ – situation (case 1). Figure 2 illustrates the ‘paradigm shift dilemma’ policy makers face when having to choose between a’ closed’ innovation regime and an ‘open’ one.

– Insert Figure 2 here – Figure 2. Paradigm shift dilemma

20

The way policy makers may decide to choose between the ‘closed’ regime and the ‘open’ regime depends on the outcomes these two are likely to provide. Switching from the current situation to a new regime directly affects the levels of price p** , profit  ** , surplus S ** and innovation outputs (i.e., qF** ,

qC** and Q**  qF**  qC** ). The surplus the commercial organization may derive from a regime switch as well as the global innovation effect such a switch leads to allow us to identify the regime which delivers the best outcomes. We next present a comparative statics analysis which presents the outcomes – gains or losses – the ‘closed regime’ and the ‘open regime’ provide when considering optimal surplus, prices and innovation outputs. We first present the general results and identify related conditions (4.1.). We then illustrate the equilibrium properties by considering a numerical example. Welfare issues are also discussed (4.2.).

4.1 Comparative statics – General results

We compare the optimal levels reached when shifting from the ‘actual situation’ framework to the ‘closed regime’ (resp. ‘open regime’) one. As such, we distinguish two cases, namely the ‘closed shift’ (denoted by  2,1 ) and the ‘open shift’ (denoted by  3,1 ). A comparative statics analysis enables us to measure the impact of the paradigm shift on the levels of optimal surpluses (  ** and S ** ), price ( p** ) and innovation outputs ( qF** , qC** and Q**  qF**  qC** ). For comparison purposes, we identify the range of values for t so that both assumption 1c and assumption 2 hold. In the ‘closed shift’ case, such values have to be defined so that





t  max 1 3  (1   ) 2  s  , 1 31  s  , 1 2    s  s 2  1    a  ,   2



i.e.,



t  max 1 31  s  , 1 2    s  s 2  1    a  . In the ‘open shift’ case, values for t have to be



defined so that



2







t  max 1 3 (1   ) 2  s  , 1 2    s  s 2  1    a  , 1 2    s  s 2  a  ,     2



i.e., t  max 1 3  (1   ) 2  s  , 1 2    s  s 2  a  .  

21

Proposition 2. The optimal price level of the commercial organization increases when a shift to either

the ‘closed regime’ or the ‘open regime’ applies. Proof of proposition 2. The impact of the ‘closed shift’ (resp. ‘open shift’) on the optimal price level

is  2,1 p** 

expressed

 a  4t  s  t    a  4t  s  t  1  4t  8t

1     4t  8t   2



1 8t

by

 1

 a  4t  s  t  

1  4t



1

1   

2

   4t 

(resp.

2 1   2 a  4t  s  t   a  4t  s  t   1     1   ** a ). From assumptions 1c  3,1 p    1   2  4t  8t 1   2  4t  8t 1   2  4t  8t      

and 2, we find that  2,1 p**  0 (resp.  3,1 p**  0 ).■ Proposition 2 highlights that switching to another regime (i.e., ‘closed shift’ or ‘open shift’) enables the commercial provider to charge a higher price than she does in the ‘actual situation’ context (i.e., case 1). Such a result – at least partially – explains why commercial players nowadays consider shifting to either a ‘closed regime’ or an ‘open regime’.

Proposition 3. The level of profit reached by the commercial provider increases when a shift to

whether the ‘closed regime’ or the ‘open regime’ applies. Proof of proposition 3. The impact of the ‘closed shift’ (resp. ‘open shift’) on the optimal level of

profit  2,1 **  

 3,1 p**  

of 1 32t

commercial

 a  4t  s  t 

2

2

1 32t

the

1 2

1   

2

organization

 1  1    2 1  4t 1     4t 

is

expressed

by (resp.

 1   2 a  4t  s  t   2   a  4t  s  t 2  ). From assumptions 1c and    4t  

2, we find that  2,1 **  0 (resp.  3,1 **  0 ).■ From proposition 3, we see that both regime shifts are likely to improve the level of profit of the commercial organization. As such, it qualifies the traditional view according to which profits only increase in a framework in which the ‘closed regime’ applies.

22

Proposition 4. The level of surplus the community generates from its activity decreases when a shift to

the ‘open regime’ applies. Such a decrease is also identified in the ‘closed shift’ case for low-leveled transportation costs.

Proof of proposition 4. The impact of the ‘open shift’ (resp. ‘closed shift’) on the optimal level of

1   2  1 1  1    2  1   2    2 a  a 2 (resp. surplus of the community is expressed by  3,1S  2 2 2 32t 1     4t   4t    **

 2,1S **   a





1  4t  4t    1  1 2



2

 2 2  1  1      a   4t   s  t   ). From assumptions 1c  4t 

and 2, we find that  3,1 S **  0 .  2,1 S **  0 if and only if a   4t 

2

 s  t   0 . We easily show from

assumption 2 that a   4t   s  t   0 for low values of parameter t (i.e., t  1 4  ).■ 2

Proposition 4 reveals that a regime shift is always detrimental to the level of surplus the community derives from its activity. Nevertheless, let us point out – from proposition 1 – that such a level is always found to be positive. Therefore, although a regime shift has a somehow detrimental impact on the community, it does not lead it to stop providing digital products.

Our surplus analysis stresses that the two types of producers both obtain positive surpluses when a regime shift applies and that they are likely to co-evolve whatever the regime may be. Such a shift may however shape their production patterns and influence their willingness to innovate.

Proposition 5. The level of innovation outputs the commercial provider delivers at the optimal state

increases when a shift to either the ‘closed regime’ or the ‘open regime’ applies. Proof of proposition 5. The impact of the ‘closed shift’ (resp. ‘open shift’) on the optimal level of

innovation  2,1qF** 

outputs

provided

by

the

commercial

organization

 a  4t  s  t   1     a  4t  s  t    a  4t  s  t   1  4t  4t

1     4t  4t   2

4t

is

expressed

1 1        2 1  4t 1     4t 

by (resp.

23

 q  ** 3,1 F

1    1   

2

a  4t  s  t  

1    2  4t  4t  

2     1     a  4t  s  t   1    1     1 a ). 1   2  4t  4t 1   2  4t  4t    

Again,

from assumptions 1c and 2, we find that  2,1qF**  0 (resp.  3,1qF**  0 ).■ Proposition 5 suggests – as this is the case for price levels – that both ‘closed’ and ‘open’ shifts lead the commercial provider to deliver a higher level of innovation outputs than that she delivers in case 1. Put it differently, switching to another regime motivates commercial players to provide higher-leveled qualities.

Proposition 6. The level of innovation outputs the community provides at the optimal state remains

the same when a shift to the ‘closed regime’ applies whereas it decreases when a shift to the ‘open regime’ applies. Proof of proposition 6. The impact of the ‘closed shift’ (resp. ‘open shift’) on the optimal level of

innovation outputs provided by the community is expressed by  2,1qC**   a  4t    a  4t   0 (resp.  3,1qC**   1    a   4t    a  4t     a   4t  ). As such, we find that  2,1qC**  0 and  3,1qC**  0 ).■ Proposition 6 stresses that a regime shift has no beneficial effect on the likelihood of the communities to level their innovation efforts up. Moreover, we point out that a shift to the ‘open regime’ negatively affects their willingness to innovate whereas a shift to the ‘closed regime’ is found to have a neutral effect on the level of innovation efforts they provide.

Propositions 5 and 6 evidence a conflict of interest between the two types of producers (i.e., the commercial organization and the community) about their willingness to innovate. Indeed, we find that commercial players are clearly motivated to innovate when a regime shift applies whereas communities reduce their innovation levels in such a framework. From a regulatory point of view, one may wonder if both regimes are likely to exhibit ‘pro-innovation’ (i.e., Q**  0 ) or ‘anti-innovation’ (i.e., Q**  0 ) effects.

24

Proposition 7. ‘Pro-innovation’ effects are always likely to apply in the case of the ‘closed shift’

whereas ‘anti-innovation’ effects may appear in the case of the ‘open shift’. Proof of proposition 7. See Appendix.

Proposition 7 suggests that the willingness to innovate of the two types of producers strongly depends on the nature of the regime shift (i.e., ‘closed shift’ or ‘open shift’). From a shifting-comparison viewpoint, we find that a shift to the ‘closed regime’ leads to the provision of higher-leveled outcomes whereas a shift to the ‘open regime’ leads to the provision of lower-leveled ones. Such a result may be taken into account by policy makers when designing suitable innovation-enhancing policies.

The following table (table 2) summarizes the main results we have obtained when analyzing optimal prices, surplus and innovation outcomes while considering the two types of regime shifts.

– Insert Table 2 here – Table 2. Regime shifts and gains

From table 2, we see that the ‘closed shift’ is more likely to improve the outcomes reached by the commercial provider at the optimal state than ‘the open shift’ does. Besides, proposition 7 evidences that the commercial organization and the community have higher-leveled incentives to innovate when they act in a ‘closed regime’ framework. Some results however remain unclear when we carry out a general comparative statics study. Indeed, the real impact of the ‘closed shift’ on the surplus of the community as well as that of the ‘open shift’ on the global innovation outcome are likely to depend on the values of the parameters of the model.

4.2 Numerical example

We illustrate the equilibrium properties by working on a numerical example. Some diagrams are also introduced. Let us suppose that r  10 , a  1 , s  0.5 ,   0.3 and   0.8 . Our analysis is here held in a framework in which assumptions 1a, 1b, 1c and 2 are satisfied when simultaneously dealing with

25

cases 1, 2 and 3. As such, we carry out the numerical analysis by considering values for t defined so that



t  max 1 3 (1  0.2) 2  0.5 , 1 3 1  0.5 , 1 2   0.5  

2 2 2  0.5   1  0.3  , 1 2   0.5   0.5   1 





, i.e., t  1 31  0.5   0.5 . For illustration purposes, we restrict our numerical analysis by considering values for t so that t  0.5;1.2 . Figures 3 and 4 exhibit optimal profits  ** (resp. surpluses S ** ) as functions of the transportation costs. We moreover consider the levels reached in the three regimes (i.e., cases 1 to 3).

– Insert Figure 3 here – Figure 3. Profits

From figure 3, as found in the general comparative statics study, we see that the level of profit the commercial organization reaches in case 1 is always lower than that reached in cases 2 and 3. Let us note that the profits are all increasing in the transportation costs. Besides, figure 3 shows that there is a value for t above (resp. below) which the commercial provider receives a higher-leveled profit when a ‘closed shift’ (resp. ‘open shift’) occurs. In our specific numerical example, such a level is equal to

t  0.62681 . We therefore point out that a ‘closed’ regime does not always allow commercial players to reap the highest levels of profits. Indeed, product differentiation has to be taken into account to measure to what extent a ‘closed shift’ has to be preferred to an ‘open’ one. This result presents the benefits the commercial organization may derive from using the innovation outputs delivered by community-based activities when transportation costs are high.

– Insert Figure 4 here – Figure 4. Community’s surpluses

Figure 4 illustrates the levels of surplus the community generates from its production activity according to the regime that applies. As already pointed out when analyzing profits, surpluses

26





functions are all increasing with transportation costs. We unsurprisingly find that the surplus of the community is higher in case 1 than it is when a regime shift occurs. Following a regime shift, surplus’ levels are moreover higher in an ‘open’ framework than in a ‘closed’ one. We nevertheless observe – as a result from assumption 2 – that the community gets positive surpluses whatever the nature of the regime may be.

Profits’ and surpluses’ analyses confirm the preliminary results we have obtained in the general framework. Indeed, we find that the community better benefits from the ‘actual situation’ framework whereas the commercial organization generates higher-leveled profits when a regime shift occurs. An ‘open shift’ notably enables the commercial provider to reach higher levels when product differentiation is small whereas a ‘closed shift’ allows her to receive bigger profits when transportation costs are large.

Optimal levels of global innovation outcomes Q** are represented in figure 5 as functions of parameter

t . Their studies allow us to better appreciate the relevancy of the public policies led to switching either to a ‘closed regime’ or an ‘open regime’.

– Insert Figure 5 here – Figure 5. Global innovation outcomes

Figure 5 shows that global innovation functions are all decreasing with transportation costs. It evidences that the ‘closed regime’ leads both producers to deliver higher-leveled innovation outputs than they would if the ‘actual situation’ and ‘open’ regimes apply. Such a result highlights that the ‘open shift’ generates a global ‘anti-innovation’ effect when taking into account the production activities of both players. The ‘closed shift’ delivers the best level of global innovation effort. These facts contribute to support the policies which have currently been carried out to prevent communities from developing activities that are based on – illegal – appropriation patterns. These findings conversely also qualify the idea according to which open-like cooperation schemes are likely to

27

provide higher innovation levels. Motives and incentives to innovate are thus shown to be weakened when the commercial organization and the community can share their innovation outputs. From a regulatory point of view, these results highlight that pro-innovation oriented policies should be based on a design upon which the mutual sharing of assets is not possible.

In contrast to the previous general framework we have specified, we branch out by discussing welfare outcomes to identify the nature of the regime shift that is likely to be socially-improving. Aggregate adopters surplus is equal to AS **  AS F**  ASC** . Producers surpluses are equal to the sum of the commercial organization’s profit  ** and the community’s surplus S ** . We define the optimal level of welfare as the sum of adopters and producers’ surpluses expressed by:

W **  AS **   **  S ** 

x

x 0

r  q

** F



  qC**  tx  p** dx  

1

x  x

r  q

** C



  qF**  t 1  x   s** dx   **  S ** ,

1 1 1 1 where x  qF** 1     qC** 1      s  t   p**  1  2t   . 2t 2t 2t 2

– Insert Figure 6 here – Figure 6. Welfare

Figure 6 presents the welfare levels public authorities have to consider depending on the nature of the institutional regime and the level of product differentiation. Welfare patterns are somehow similar to global innovation ones inasmuch as we find that the ‘closed regime’ is welfare-improving compared to the levels of welfare reached in the ‘actual situation’ and ‘open regime’ cases. Our numerical example notably reveals that the welfare levels obtained following an ‘open shift’ are always lower than those following a ‘closed shift’. Such results support the recent policies that are carried out to shape a copyright-enforcement – ‘closed’ – framework since the ‘closed shift’ provides the best level of welfare.

28

Our numerical illustration has evidenced that the ‘closed regime’ is always likely to provide the best levels of both global innovation outcomes and welfare. Public policies should therefore make efforts to define such a ‘closed’ framework and to prevent community-based players from keeping on illegally appropriating commercial outputs to develop their own activities. However, let us note that – contrary to conventional wisdom – implementing such policies does not always lead commercial organizations to achieve the best levels of profits. Indeed, we have found that commercial players better benefit from an ‘open shift’ than a ‘closed’ one when product differentiation is small. In this specific context, this result paradoxically exhibits a conflict of interest between commercial players and public regulators and qualifies the conventional idea according to which public policies may be shaped to defend private – commercial – interests rather than public ones.

5. Conclusion

In this article we have presented a model to identify the impact of a regime shift (i.e., ‘closed shift’ or ‘open shift’) on both commercial and innovation outcomes when an asymmetric appropriation scheme initially occurs. To do so, we have adopted an approach according to which online communities are not only driven by altruistic of self-oriented concerns but also by pecuniary motives inasmuch as such activities require funds to be undertaken, mostly for technical reasons. As such, we have intended to provide a rational view of community-based delivery of outputs. We have thus introduced a duopoly model in which the two types of players (i.e., the commercial organization and the community) compete by providing digital products. Moreover, our duopoly model considers three institutional cases – regimes – in which each player differs in her abilities to integrate both her own innovation effort and her competitor’s into her product. We have investigated to what extent a regime shift may improve the profit of the commercial organization as well as the level of joint-innovation effort which is provided by the two types of providers. Having first defined the levels of price, profit, community’s surplus and innovation efforts achieved at the optimal state, our comparative statics analysis has then evidenced the benefits a commercial

29

organization may derive from either a ‘closed’ or ‘open’ regime shift. Our study has also pointed out a potential conflict of interest between the two types of players concerning their likelihood to innovate. Indeed, we have clearly highlighted that a regime shift leads the commercial player to provide a higher level of innovation effort whereas it may negatively affect the willingness of the community to increase its own innovation effort. We have introduced a numerical illustration to analyze the impact of the regime shift on the global innovation outcome as well as discuss welfare results for regulatory purposes. Our results have shown that the ‘closed shift’ provides the best levels of both global innovation and welfare. Hence, our findings tend to support the recent public policies that have been pursued to build an institutional framework in which commercial innovations cannot be freely – illegally – appropriated by online – outlaw – communities. From a commercial point of view, we have found that an ‘open shift’ (resp. ‘closed shift’) leads the commercial organization to achieve larger profits when product differentiation is large (resp. small). Such a result exhibits a potential conflict of interest between commercial players and public authorities and partially qualifies the conventional idea according to which public policies may be designed to defend private – commercial – interests rather than public ones.

The model we have presented has led to better understand the impact of an institutional regime shift on commercial, innovation and social outcomes. Moreover, by considering innovation as both outputs and inputs, we have evidenced potential conflicts of interest which may emerge between commercial and community-based organizations concerning their incentives to innovate. Our results do not totally contrast with the previous findings that have stressed that an ‘open’ way of producing and innovating may lead a switching commercial organization to increase its profits. However, they evidence the difficulties to build efficient innovation-improving mechanisms when players are able to share their outputs to carry out their production activities. Some extensions of our model may be considered for further research. We may first extend our model to the case in which the adoption cost s that users have to face when adopting the digital product from the community is heterogeneous. By considering appropriation levels (i.e.,  and  ) no longer as parameters but as control variables, we could secondly identify additional results about the manner in

30

which the commercial organization and the community may strategically adapt to a framework in which the ‘open regime’ applies. One may finally find appropriate to analyze if the introduction of sharing ratios or pecuniary counterparts may improve both global innovation and welfare levels while being profit-enhancing. As such, introducing a ‘fair-use’ way of appropriating innovation outputs could deliver additional results concerning both surpluses and innovation outcomes.

The analysis of strategic interactions between organizations that differ in their intrinsic nature opens room for further research. It nevertheless contributes to make the innovation puzzle fuzzy when heterogeneous players are at stake and it points out the complex – yet critical – role of public authorities in designing an institutional regime in which profit-improving, innovation-enhancing and social-developing states have to be achieved.

31

TABLES AND FIGURES

Figure 1. Adoption patterns

Figure 2. Paradigm shift dilemma

32

Figure 3. Profits

Figure 4. Community’s surpluses

33

Figure 5. Global innovation outcomes

Figure 6. Welfare

34

 ,  

‘Actual situation’

‘Closed regime'

‘Open regime’

Case 1

Case 2

Case 3

  0,   0;1

  0, 

 0

  0;1 ,   0;1

Table 1. Cases and environmental parameters

‘Closed shift’

‘Open shift'

p**





 **





S **

 /?



qF**





qC**

0



Q**



 /

Table 2. Regime shifts and gains

35

APPENDIX

Proof of Proposition 7 As  2,1qF**  0 and  2,1qC**  0 , we easily find that  2,1Q**  0 . Results

are

more

complex

to

identify

when

dealing

with

the

 3,1Q**   3,1qF**   3,1qC**  a 1    1     1   1     4t   1     4t 





assumption

1c,

we

2

find





 3,1Q**  0

that

2

2

 

 3,1Q**  0 )

(resp.

1



if

‘open

shift’.

 4t 1 .

From

and

only

if

t  t   1 4 1     2     1    (resp. t  t   1 4 1     2     1    ). Let us keep in mind that we here restrict our analysis to transportation costs whose levels are





characterized so that t  max  1 3 1     s  , 1 2    s  s 2  a  ;   . Consequently, if





2













t   max 1 3 1     s  , 1 2    s  s 2  a  , an ‘anti-innovation’ effect is always shown to   2

apply. Let

us

firstly

1 3 1    ( X   0;1 )

2

compare

values

t

and

1 3 1   

2

 s .



We

find

that

 s   t   1 12  1     4s  3 1    2     . Introducing variables X  1  



and



Y  1

1 12   X 2  X  3 Y  1  4s  .

2

( Y   0;1 ),



we

have

to

analyze

the

sign

of

The discriminant of the quadratic polynomial can be written as

   3  Y  1  4 1 4 s   3  Y  1  4 s  3 Y  1  4 s  . For values of parameters  and 2

s set out so that we have 2   4 3 s   , 1 12   X 2  X  3  Y  1  4 s   0 . As a consequence, t   1 3 1     s  and an ‘anti-innovation’ effect here applies. For values of parameters  and



2



s set out so that we have 2   4 3  s   , we find the quadratic polynomial has two solutions for

36

X,

B1  1 2  3  Y  1  9  Y  1  16 s   0

namely

2



and



B2  1 2  3  Y  1  9  Y  1  16 s   1 . Therefore, t   1 3 1     s  . 2



Let

us

secondly

compare

values

2





t

and

1 2    s 

s2  a  .

s 2  a   t   1 4   2 s  2 s 2  a  1    3       .

1 2    s 

s2  a   t





if

We



value

for

a

is

We



1 2    s 





set

to

be

obtain

that

find

that

‘high’

(i.e.,

a  1 4 1    3      1    3       4 s  whereas 1 2    s  s 2  a   t  if value for





a is ‘low’ (i.e., a  1 4 1    3      1    3       4 s  ). From these results, we find that  3,1Q**  0 if

2   4 3 

s 

 2   4 3 s    1 4 1    3      1    3       4 s   a 2   4 3 s    1 4 1    3      1    3       4 s   a  t  1 4 1     2     1    ;   In a similar fashion, we find that  3,1Q**  0 if

 2   4 3 s    1 4 1    3      1    3       4 s   a  2 t   max 1 3 1     s  , 1 2    s  s 2  a  ; 1 4 1     2     1        





As such, we find that the ‘closed shift’ and the ‘open shift’ may differ in the nature of the innovation effect they are likely to provide. The ‘closed shift’ always generates a ‘pro-innovation’ effect whereas the ‘open shift’ may lead to an ‘anti-innovation’ one.

37

We can demonstrate that a shift to the ‘closed regime’ leads to the provision of higher-leveled outcomes than the ones reached out following an ‘open shift’ (i.e.,  3,1Q**   2,1Q** ). Indeed, to be able to compare the outcomes reached out in both regimes, we have to consider values for t defined so

that





t  max 1 3 1     s  , 1 31  s  , 1 2    s  s 2  1    a  , 1 2    s  s 2  a  ,



2





2





t  max 1 31  s  , 1 2    s  s 2  a  .





1 31  s   t   1 12   4  4s  3 1   

2

We







find

 3 1    2     .

Introducing



i.e.,

that again

variables

X  1   ( X   0;1 ) and Y  1   ( Y   0;1 ), we have to analyze the sign of 3 X 2  3 XY  4  4 s .

The discriminant of the quadratic polynomial is   9Y 2  12  4  4 s   0 and the quadratic polynomial has two solutions for

X , namely B3  1 6   3Y  9Y 2  12  4  4 s    0



B4  1 6   3Y  9Y 2  12  4  4 s    1 .



1 12   4  4s  3 1   

Therefore,



2



and

3 X 2  3 XY  4  4 s  0 ,

 3 1     2      0 and t   1 31  s  . We find that the values for t



that have to be considered to carry out a regime-comparison analysis are ‘too’ high for the ‘open shift’ to generate a ‘pro-innovation’ effect whereas the ‘closed shift’ is always found to do so for such values. Consequently,  3,1Q**   2,1Q** .■

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REFERENCES

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