Innovation Clusters Effects on Adoption of a General Purpose Technology under Uncertainty

This paper analyzes the effect of innovation clusters on the adoption of a general purpose technology (GPT) and on firms R&D investment levels in imperfect information situation. We developed a theoretical model of vertical relation, described as a four-step game between an upstream firm providing innovative GPT and an innovative downstream associated sector, integrator of this technology. The downstream sector ignores the quality of the GPT and we model the innovation cluster as a coordination mode of firms, improving the probability of the downstream firm to receive information about the quality of the GPT technology. Then, we determine firms equilibria (i.e. prices and technological qualities) and we showed that the effect of innovation clusters on the choice of qualities, the adoption behavior, levels of R&D investment as well as that social welfare depends on the quality of R&D activities carried out before the establishment of the clusters and a threshold effect (i.e. cluster critical mass); if the critical mass in terms of information sharing and interaction is not reached, the cluster may have negative effects. In other words, the consensual idea of expected positive effects of innovation clusters must be put into perspective.


Introduction
B. G. J. J. Iritié for example, the cluster policy concerns various technological domains and activity sectors. In this paper, we analyze theoretically the effect of the cluster policy on firms' behavior in technology adoption as well as on firms' R&D levels. To do this we will focus on nanotechnologies.
The nanotechnologies are currently qualified as general purpose technologies (GPT). The concept of GPT was introduced by [1]; it refers to all technologies characterized by their strong technological opportunities 1 , their potential use as a factor of production in a large number of activity sectors, their technological dynamism and their technological complementarity with existing or potential technologies 2 . Bresnahan and Trajtenberg modelled a vertical relation between an upstream sector producing a GPT (semiconductor) and downstream user sectors (computers, hearing aids, TV, scanners); this supplier-customers relation is coordinated by market mechanisms without contractual relations between firms. The authors described it as a simultaneous game in which each sector chooses its level of R&D investment, thus evaluating incentives to innovate of firms. Bresnahan and Trajtenberg showed that incentives to innovate in the two sectors remain socially weak due to the presence of technological complementarity and vertical and horizontal externalities; these externalities raise coordination problems between innovation actors and give strong motivations to create and increase the degree of cooperation and contracts, on the one hand between GPT sector and associated sectors, and on the other hand between associated sectors. This result constitutes the starting point of our analysis. Indeed, for us, an innovation cluster can be considered as a response to the coordination problem between innovators highlighted by the authors; it is supposed to be a localized platform that allows increased information sharing and firms' cooperation.
As a result, the innovation cluster should play a facilitating role, the importance of which should be assessed in knowledge creation and technologies adoption.
The literature on adoption of new technologies is abundant. In a synthesis, [15]emphasizes two relevant characteristic features of theoretical models: the uncertainty and the strategic interaction in the final product market. As regards the first, several works such as [16] [17] and [18] show that uncertainty on profitability of new technology can reduce or increase incentive for adoption according to whether beliefs are pessimistic or optimistics; this points out the impor- 1 In the French's territorial configuration of clusters, nanotechnologies are developed within the cluster MINALOGIC (Grenoble, Region Rhone-Alpes). Let us note that the technological opportunities of a sector represent the potential for technical progress in the corresponding activity; then the notion of technological opportunity refers, for example, to the fact that 1 Euro invested in research does not necessarily leads to the same gain of productivity according to the technological potential of the activity in which it is invested [2] [3]. [4] give an interesting discussion about the nature of nanotechnologies. For further reading and discussions, refer to [5]- [12]. 2 Following [1], other works have focused on the analysis of the characteristics of GPTs; [12] explains that GPTs involve both enormous technological and Hicksian complementarities, [11] and [13] emphasize the relevant changes brought about by the discovery of a GPT (e.g, structural changes, public policy changes, etc.), the transient decline in productivity at the macroeconomic level generally observed after the introduction of a GPT, called the "Solow Productivity Paradox" (see also [14]). tion cluster or by a classical market (i.e. outside the cluster). To capture the difference between the two coordination modes, it is assumed that their probabilities of receiving information about the upstream technology are different from each other. The vertical relation is described as a four-stage sequential game in which the downstream customer sector is confronted with the decision whether to adopt GPT innovation.
We solve the game and analyze the effect of innovation clusters on different equilibria. Our main results show that innovation clusters can positively or negatively influence the choice of qualities (i.e. technological levels), adoption behavior, upstream and downstream R&D investment levels as well as social welfare. However, this effect depends on the quality of the R&D activities carried out on the territory before the establishment of the cluster and on a threshold effect or cluster critical mass. If the critical mass is not reached, the innovation cluster can have negative effects. It can be deduced from this that the real issue for cluster policy is not only to increase the sharing of information and externalities of knowledge, but above all to allow this increase to be sufficient to reach a critical mass within the cluster; it is therefore necessary to put into perspective the expected positive effects of innovation clusters.
The rest of the paper is organized as follows. Section 2 presents the model. Sections 3 and 4 are devoted to the resolution of the game and to the determination of the equilibria of firms. We then analyze the effect of innovation clusters on the choice of upstream quality, on the downstream adoption behavior and on the social welfare in Section 5. In Section 6, we analyze an application with explicit functions. Finally Section 7 concludes the paper.

The Model
Setup. Let us consider the vertical relation between an upstream sector producing a good embodying a GPT and some downstream sectors, each developing an associated technology; there is technological complementarity between upstream technology and each downstream technology. 3 For example, the gathering of information by observing the experience of first adopters is named social learning in [20] and [21]; these authors showed that the social learning perspective delays adoption, except in the presence of explicit coordination of adopters. 4 [22] and [23] argued that rivalry can therefore accelerate adoption or delay it according to the advantage of the firm. To simplify matters, we assume that on the upstream market, the GPT's firm may choose to produce low quality ( z ) or high quality ( z ) 5 , its marginal cost of production is constant and given by c whatever quality z. GPT-firm sells its product to the user downstream sectors at a wholesale price w and realizes a net profit being its gross income. Now let us suppose a given downstream associated sector (AS) 6 with a representative firm (indexed by a) of all downstream firms in this sector. The downstream firm carries out its own R&D program enabling it to develop a technology with quality k, 0 k ≥ , incorporated into a (semi-finished) product. The adoption of the GPT, combined with associated technology, allows the downstream firm to produce and sell a final good on the downstream market. The example of nanotechnology illustrates the technological complementarity in this vertical relationship 7 . Note that the downstream firm does not necessarily know GPT true quality because of imperfect information (or uncertainty); it only knows that z can take two values, z or z . It has however an a priori belief θ , Before deciding whether to adopt, the downstream firm receives or not information in form of a signal on the quality of the GPT; the signal arrives randomly with a probability h, 0 1 h ≤ ≤ ; once the signal arrived, it is perfect and reveals the true quality of the GPT. In other words, in the presence of the signal, the downstream firm is, ex post, in perfect information. In the model, we assume that the occurrence probability h of the signal depends on the coordination mode of the vertical relation; by this, we model the difference between two modes of coordination of innovation actors, i.e. the arm's length market mechanism and the innovation clusters.
Let us suppose that on the market product of the downstream firm, all eco- 5 This assumption makes abusive all derivation notations with respect to z; but we maintain them to recall the general case. 6 As mentioned in the introduction, a general purpose technology (GPT) is used in several application sectors. However, if we consider that these application sectors (AS) are not necessarily interconnected and are independent of each other, they can be analyzed separately. Because of that, we do not model the horizontal externalities that could possibly exist. 7 Indeed, many research programs are currently being undertaken in the field of microprocessors composed of integrated circuits on the molecular or nano-metric scale by exploiting the properties of individual silicon atoms. If they are actually produced and marketed, the use of these next-generation processors by computer manufacturers could enable to manufacture computers with ultra-low power consumption. In this example, the GPT product is the microprocessor integrating nanotechnology whose quality z would be measured by its ability to lower energy consumption; the downstream semi-finished good would be all the technological environment of the computer (CPU and everything in it) without the microprocessor; the technological level of this environment is given by k. The performance of the final good (i.e. the complete computer) will therefore depends on the associated technological levels k and z.  -Stage 4 (revelation of z and downstream GPT demand). the quality z is revealed to all downstream sector's agents (seller and final consumers); Consumers express their demands a X of downstream good; the AS-firm buys the necessary quantity of the GPT input and satisfies the demand of its product. Remark 1. One can verify that even if the price p a is decided at stage 4, it would be the same as in stage 3; otherwise, we note that stage 4 suppose the GPT technology was adopted at stage 3. 8 Hence, in the rest of the paper, when we write gross profit (or respectively net profit) of the downstream firm, it is actually the gross profit (or respectively net profit) of surplus of the downstream sector. 9 There is an offset between stage 3 and stage 4 because R&D requires time. We use backward induction to determine equilibrium of the game. First, we determine price and quality of the downstream sector in stage 3 for any value of w chosen by the upstream firm in stage 1 (we will deduce the demand and the profit in stage 4). Then, we determine the fist stage's equilibrium of the upstream sector (expected demand, profit, quantity, price). Finally we proceed by static comparatives to analyse the effects of innovation clusters.

Downstream Firm Equilibrium
Perfect information. In the presence of the signal, the downstream firm observes the true quality of GPT. If it adopts technology, it chooses its technological level ( ) The first order condition leads to ( ) ; we verify that the price is given by: The downstream sector's gross surplus becomes  The optimal quality k is given by: π > , we suppose there is always a price low enough for the downstream firm to be ready to adopt the low quality; for this low price, the downstream firm is ready to adopt even if it does not 10 If ( ) max , 0 w z π < , the downstream sector does not adopt the GPT and chooses an opportunity action. We will normalize the opportunity cost to zero if technology z is an essential input; contrariwise if z is not an essential input, the downstream sector use an alternative technology. B. G. J. J. Iritié know the quality. In this case, the GPT is always profitable regardless the quality 11 . However, by assuming ( ) max 0 π +∞ < and ( ) max 0 π +∞ < , we suppose that for extremely high prices, adoption of GPT by downstream firm becomes unprofitable regardless of quality.
Lemma 1. There are two reserve prices, w and w , with 0 w > and w < +∞ , such that: 1) the downstream firm adopts low-quality GPT technology z iff w w < , 2) the downstream firm adopts high-quality GPT technology z iff w w < .
We will see that w w < ; in other words, in perfect information, the willingness to pay high quality ( z ) is higher than the willingness to pay for low quality ( z ).
Remark 3. We verify here the first part of the technological complementarity hypothesis highlighted by [1]. Indeed, we show that ( ) , 0 z k w z > for all given w; in other words, in perfect information, the incentive to R&D in the downstream sector increases with the level of quality of the GPT technology. 12 Proof. The first order condition We deduce the demand a X expressed by the downstream firm. It is a function of the both revealed quality z and wholesale price w of GPT good. Note that a a X w X w z k ≡ when it is of high quality; we distinguish the following three cases: and the profit is given by and the profit is given by 3) if w w > , the firm does not adopt whatever the quality of the GPT; ( ) ( ) 0 a a X w X w = = and profits are zero.
Imperfect information. In absence of a signal, the associated firm is in imperfect information; it does not observe the quality z of the GPT. So, the firm's adoption decision is based on its a priori belief θ that the technology is of high-quality. If it adopts, it choose its technology level ( ) * , k w θ and its optimal price ( ) , , a p w k θ so as to maximize its expected profit It is quite possible to assume that , which means that the adoption of GPT is profitable to the downstream firm if the quality is high ( z ) and unprofitable if the quality is low ( z ). [16] adopted this posture in its pioneering model of adoption and diffusion, a model generalized by [17]. 12 In our model and at this stage, we assume that the choice of z and w is exogenous. In their article, The choice of ( ) , , a p w k θ is solution of the maximizing problem of the firm's expected gross profit: The resolution leads to a optimal price a p w = ; using this expression, the expected profit be- The optimal k is given by: Moreover by assumption, the profit of associated firms decreases with w. Consider a firm j with belief θ ′ such as θ θ ′ < ; let us assess the firm j's profit would gain ex-post not to adopt the GPT because it is of low quality.
The purchasing behavior of the downstream firm in the absence of a signal depends on its reserve price, which is itself dependent on its a priori belief θ .
, the downstream firm adopts the GPT; the anticipated demand 13 on which it based its adoption decision at stage 3 is given by when the firm produces low quality and g g X X ≡ when it produces high quality.
Perfect information. In the presence of a signal, upstream and downstream firms have same informations; so the upstream expected demand g X is identical to the downstream firm demand in stage 4. Then, the corresponding profits g π are given by ( Imperfect information. In the absence of a signal, the expected demand g X of the upstream firm corresponds to the expectation of the downstream firm demand expressed in stage 4; indeed, at this stage, consumers observe the both GPT quality z and quality ( ) * , k w θ chosen by the θ-type downstream firm in the absence of signal. Then for an observed quality z , we have there is no technology adoption and the upstream expected demand g X is zero. Imperfect information with signal probability h. If the GPT firm takes into account the probability h for the downstream firm to receive a signal on quality z of its product, its expected demand is a function of h. Indeed, GPT-firm knows that with probability h its expected demand is the same as in perfect information, and with a probability ( ) 1 h − , its expected demand is the same as in imperfect information; so: 3) If the price of the GPT is such that w w > , then the downstream firm does not adopt and the expected demand of GPT is zero. Comparative static. Let us suppose that the downstream firm adopts GPT and let us analyze effects of the increase of h on the gap between the demand expected by the upstream firm when it produces high quality and the demand expected when it produces low quality, i.e. g g X X − . To do this, let us consider two cases, either the upstream firm chooses a unique price m w regardless the GPT quality, either the upstream firm sets endogenous prices depending on the GPT quality: 1) the upstream firm sets a unique price for any given h; then expected demand gap 14 according to values of m w are: a) if m w w < , by using Equations (11) and (12), we have < , by using Equations (13) and (14), we have Let us calculate the profit gap, These results show that when h increases, whatever the unique wholesale price m w , including the one that maximizes the profit of low quality z , the gap of demand expectations of the upstream firm increases as well as profits gap; as a result, the incentive to sell high quality GPT rather than low quality GPT increases with the probability h of receiving the signal. The upstream firm can therefore always have an incentive to switch to high quality z ; but this requires an increase in R&D costs, a wholesale price adjustment and a sufficient level of h so that the gross profit gap to be greater than the R&D cost gap.
2) the upstream firm sets endogenous prices according to the quality of GPT; Let us set respectively With assumption 2, we assume that for a given value of h, the upstream firm sells the high quality at a price at least equal to that of the low quality. Let us assess the effect of a variation of h on the profit gap between high and low quality; using the envelope theorem, we show that the total differential of the profit gap is given by: It's the envelop theorem ; similarly, for the profit of low quality, we obtain: Using Equations (19) and (20), we write the profit differential: The sign of profit differential Proof. It suffices to verify that with Equation (11) (11), (13) and (12); for the Equation (14), we also verify that 0 > when high quality and low quality are sold respectively at price m w and m w . Proof. Lemma 4 is the consequence of lemma 3.

Effects of Innovation Clusters
We assume that the probability of receiving the signal h depends on the coordi- give an advantage to firms, not only in terms of transaction costs, but also in terms of sharing valuable information because of geographical proximity and localized knowledge externalities (see e.g. [24] [25]). We will analyze innovation clusters effects on the choice of upstream quality and on the downstream adoption behavior of the GPT and on the social welfare.

Choice of Upstream Quality
We analyze here effects of an increase of h on the upstream firm behavior in its choice to produce high or low quality.  product utilization 16 . We analyze below the effect of increasing of h,

Investment Level in Downstream Quality
The expected quality of downstream technology is given by 15 In the absence of a signal, the GPT firm has no interest to choose the high quality because the downstream sector ignores the quality. If the GPT firm chooses high quality, it invests more fixed cost in R&D while it receives the same demand and income as low quality. However, in a dynamic model integrating reputational aspects, one could imagine that the GPT firm chooses the high quality in the absence of signal to build or preserve its reputation. 16 We note that ( ) ( We distinguish here three effects of h on the expected quality of downstream technology: is the indirect effect of h through change of upstream quality.
For this analysis of effects of an increase of h, one can distinguish two cases: the increase of h does not lead to a change in upstream quality, i.e.
Knowing that With assumption 4, we suppose that GPT firm raises its price with h when it chose to produce high quality and lowers its price with h when it chose to produce low quality. We suppose that the model fundamentals, i.e. functions of demand, cost, profit, surplus, ensure this assumption.
Given assumption 4, let us reassess the overall effect of h on inf k  and sup k  : If the effect of the upstream price on the downstream quality and the effect of h on the upstream price are negligible, then

Level of Use of GPT Good in Downstream
The total differential of ( ) , a X h θ  with respect to h is given by: the additional indirect effect of h through the variation of downstream product quality.
The first effect is interpreted as the observed upstream quality effect, the second as the price effect, the third as the upstream quality variation effect and the fourth as the downstream quality variation effect.
First case: the increase of h does not result in a change in upstream quality.
What is the overall effect of h on the adoption behavior of a downstream θ-type firm? 1) If the effect of the change in upstream price on the choice of downstream quality or the effect of the increase of h on the wholesale price is negligible, then only the direct effect and the indirect effect through price remain. Remark 6. With assumption 4 and lemma 3, we verify that in the two demand 19 We note that Secondly, considering Equations (12) and (14) . This result is also verified for Equation (14) and in this case, 2) If the effect of the change in the upstream price on the downstream quality is not negligible as well as the effect of h on the wholesale price, then given assumption (4) and knowing that 0 , , . We know that: We can decompose and write Thus, by shifting the upstream firm from low quality to high quality, the innovation cluster improves further its positive effect on the downstream level of use of GPT product.
2) If we do not neglect the effect of the upstream price on downstream quality and the effect of h on the upstream price, then given assumption (  In sum, we note that Propositions 2 and 3 show that the positive effect of innovation clusters on the level of use of new technology and on the level of R&D investment in downstream is subject to conditions, in particular that the increase of h has no effect on the upstream price and/or that the effect of the increase of the upstream price on the downstream technology quality is either positive or negative but negligible.

Welfare Implications
We suppose for the social planner, initiator of the innovation cluster policy, that 20 In Proposition 3, k represents the both k and * k and m w represents m w when the GPT-firm invests in high quality. Alternatively k represents the both k and * k and m w represents m w when the GPT-firm investsin low quality. the social welfare is the sum of the expected surpluses of upstream and down- By definition of k and k , we know that in Equation (50) Overall effect of h on a π and W W h is indefinite.
In sum, we showed that if the increasing of the upstream price significantly increases the downstream quality, the social surplus increases iff the upstream firm produces high quality.
Second case: The increase of h results in a switch from low quality to high quality in upstream.
Let us calculate We rewrite in the form:  Imperfect information. In the absence of the signal, the downstream firm bases its decisions on its a priori belief θ that GPT technology is high quality.
In order to choose its price, the firm maximizes its expected surplus 2) The levels of expected use of the GPT good in downstream when it is high quality and low quality are respectively given by