ABSTRACT

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.

Keywords:

Innovation Clusters, General Purpose Technology, Technological Complementarity, Technology Adoption, Uncertainty, Critical Mass

1. Introduction

Since the Silicon Valley success story in the US, innovation clusters are seen as central to innovation policy and economic development worldwide. In France, 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 opportunities¹, 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 technologies². 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 importance of gathering information³. The second element, i.e. the strategic interaction, implies that incentive to adopt for a firm depends on the adoption decision of rival firms⁴.

The model we develop is inspired, on the one hand, by the work of [1] and, on the other, by the theoretical work on the adoption of new technologies, notably the important contribution of [16] and its variants in [17] and [19] . Then, in order to analyze the impact of innovation clusters, we consider a vertical relation between a GPT supplier (upstream sector) and GPT integrators (downstream sectors); we suppose the vertical relation is coordinated either within an innovation 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.

2. 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.

Let us suppose that the upstream sector is monopolized by a firm, called GPT-firm (indexed by g). In order to develop its technology with a quality z, $z>0$ (i.e. technological level), the GPT-firm invests $c^g\left(z\right)$ in R&D, with $c_z^g\left(z\right)>0$ and $c_{zz}^g\left(z\right)>0$ . To simplify matters, we assume that on the upstream market, the GPT’s firm may choose to produce low quality ( $\underset{\_}{z}$ ) or high quality ( $\bar{z}$ )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 ${\pi }^g\left(w,z\right)=r^g\left(w,z\right)-c^g\left(z\right)$, $r^g\left(w,z\right)$ 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, $k\ge 0$, 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 relationship7. 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, $\underset{\_}{z}$ or $\bar{z}$ . It has however an a priori belief $\theta$, $0\le \theta \le 1$, that the upstream GPT innovation is high quality $\bar{z}$ ; $\theta$ is assumed to follow a $f\left(\theta \right)$ distribution, $f\left(\theta \right)$ is a probability density function.

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, $0\le h\le 1$ ; 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 economic agents (downstream firm and customers) negotiate an efficient arrangement in order to maximize the gross surplus of the sector8. This assumes in particular that there is a monetary transfer from the customers to the downstream firm which ensures firm viability when gross profits do not cover fixed costs; our purpose here is to leave outside the scope of analysis all market imperfections. Thus, the gross surplus of the downstream sector is given by:

$CS\left(p^a\mathrm{,}k\mathrm{,}z\right)+\left(p^a-\gamma -w\right)X^a$ (1)

with $CS\left(p^a\mathrm{,}k\mathrm{,}z\right)$ the net consumer surplus, $\gamma \ge 0$ the unit production cost, $p^a>0$ the unit selling price of the downstream product; $p^a\ge \gamma +w$ . To simplify notations, it will be assumed in the rest of the paper that $\gamma =0$ without loss of generality. The demand of the GPT is given by $X^a=-C{S}_{p^a}$ with $C{S}_{p^a}<0$ . We suppose that one unit of GPT’s good leads to one unit of the downstream good.

The definition of z and k implies derivatives $C{S}_z>0$ and $C{S}_k>0$ . We also suppose that $C{S}_{zz}\ge 0$, $C{S}_{kk}\ge 0$, $C{S}_{p^a{p}^a}>0$, $C{S}_{p^{a}z}<0$, $C{S}_{p^{a}k}<0$, $C{S}_{kz}>0$ and $C{S}_{kk}-c_{kk}^a<0$ . The hypothesis $C{S}_{p^a{p}^a}>0$ implies that the demand $X^a$ is decreasing with respect to $p^a$ . Otherwise, we note $c^a\left(k\right)$ the R&D investment level, with $c_k^a>0$, $c_{kk}^a>0$ .

Timing of the game. We describe the vertical relation GPT and AS sectors as a four-stages sequential game:

- Stage 1 (R&D and upstream innovation). The upstream sector develops a product composed by GPT technology with quality $z>0$, $z\in \left\{\bar{z}\mathrm{,}\underset{\_}{z}\right\}$ and choose a wholesale price $w>0$ .

- Stage 2 (signal on the quality). The associated sector receive or not information related to quality z as a signal; the signal is perfect and comes with a probability h depending on the coordination mode. In the presence of the signal, the downstream firm is in perfect information; but in the absence of the signal, it is in imperfect information.

- Stage 3 (adoption decision and downstream R&D). The downstream sector observes the GPT’s wholesale price and decides or not to adopt the GPT. If it adopts, it choose the level $k\ge 0$ of its own associated technology and invests $c\left(k\right)$ in research and development9. Then, it fixes its price $p^a\ge 0$ .

- 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 $X^a$ 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.

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.

3. 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 $k\left(w\mathrm{,}z\right)$ and fixes its optimal price $p^{a\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}k\mathrm{,}z\right)$ which maximizes its profit ${\pi }^a\left(w\mathrm{,}{p}^a\mathrm{,}k\mathrm{,}z\right)$ . The choice of price is made by solving the maximizing problem of the downstream sector:

$r^a\left(w,p^a,k,z\right)=\underset{p^a}{\max }\left\{CS\left(p^a,k,z\right)+\left(p^a-w\right)X^a\right\}$ (2)

The first order condition leads to $\left(p^a-w\right)\left(-\frac{{\partial }^{2}CS}{\partial p^a{}^2}\right)=0$ ; we verify that the price is given by:

$p^{a*}=w$ (3)

The downstream sector’s gross surplus becomes $r^a\left(w,k,z\right)=CS\left(w,k,z\right)$ and its net profit ${\pi }^a\left(w,k,z\right)=CS\left(w,k,z\right)-c^a\left(k\right)$ ; from assumptions on CS, we deduce the following properties: $r_z^a>0$, $r_k^a>0$, $r_w^a<0$ ; the inter-sectoral technological complementarity is given by $r_{kz}^a\ge 0$ .

The optimal quality k is given by:

$k\left(w,z\right)=\underset{k}{\arg \max }\left\{{\pi }^a\left(w,k,z\right)\right\}$ (4)

respecting the first order condition [ $C{S}_k-c_k^a=0$ ] and the second order condition [ $C{S}_{kk}-c_{kk}^a<0$ ].

Note ${\pi }^{\max }\left(w,z\right)\equiv {\max }_k\left\{{\pi }^a\left(w,k,z\right)\right\}$ ; then the GPT of quality z is profitable (and will be adopted) iff ${\pi }^{\max }\left(w,z\right)>0$ ¹⁰.

Note that $\underset{\_}{k}\left(w\right)\equiv k\left(w,\underset{\_}{z}\right)$ and ${\underset{\_}{\pi }}^{\max }\left(w\right)\equiv {\pi }^{\max }\left(w,\underset{\_}{z}\right)$ respectively the optimal level of k and the associated maximum profit when the signal reveals $\underset{\_}{z}$ and $\bar{k}\left(w\right)\equiv k\left(w\mathrm{,}\bar{z}\right)$ and ${\bar{\pi }}^{\max }\left(w\right)\equiv {\pi }^{\max }\left(w\mathrm{,}\bar{z}\right)$ the optimal level of k and the associated maximum profit when the signal reveals $\bar{z}$ .

Remark 2. ${\underset{\_}{\pi }}^{\max }\left(w\right)$ and ${\bar{\pi }}^{\max }\left(w\right)$ are decreasing with respect to w.

Assumption 1. ${\underset{\_}{\pi }}^{\max }\left(0\right)>0$, ${\underset{\_}{\pi }}^{\max }\left(+\infty \right)<0$ and ${\bar{\pi }}^{\max }\left(0\right)>0$, ${\bar{\pi }}^{\max }\left(+\infty \right)<0$ .

By assuming ${\underset{\_}{\pi }}^{\max }\left(0\right)>0$ and ${\bar{\pi }}^{\max }\left(0\right)>0$, 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 know the quality. In this case, the GPT is always profitable regardless the quality1¹. However, by assuming ${\underset{\_}{\pi }}^{\max }\left(+\infty \right)<0$ and ${\bar{\pi }}^{\max }\left(+\infty \right)<0$, 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, $\underset{\_}{w}$ and $\bar{w}$, with $\underset{\_}{w}>0$ and $\bar{w}<+\infty$, such that: 1) the downstream firm adopts low-quality GPT technology $\underset{\_}{z}$ iff $w<\underset{\_}{w}$, 2) the downstream firm adopts high-quality GPT technology $\bar{z}$ iff $w<\bar{w}$ .

We will see that $\underset{\_}{w}<\bar{w}$ ; in other words, in perfect information, the willingness to pay high quality ( $\bar{z}$ ) is higher than the willingness to pay for low quality ( $\underset{\_}{z}$ ).

Remark 3. We verify here the first part of the technological complementarity hypothesis highlighted by [1] . Indeed, we show that $k_z\left(w,z\right)>0$ 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 $C{S}_k-c_k^a=0$ implies $C{S}_{kz}+k_{z}C{S}_{kk}-k_z{c}_{kk}^a=0$ ; which in turn implies that $k_z=\frac{C{S}_{kz}}{c_{kk}^a-C{S}_{kk}}$ . By hypothesis $C{S}_{kz}>0$ and $c_{kk}^a-C{S}_{kk}>0$, thus $k_z\left(w,z\right)>0$ . $\square$

We deduce the demand $X^a$ expressed by the downstream firm. It is a function of the both revealed quality z and wholesale price w of GPT good. Note that ${\underset{\_}{X}}^a\left(w\right)\equiv X^a\left(w\mathrm{,}\underset{\_}{z}\mathrm{,}\underset{\_}{k}\right)$ when the GPT good is of low quality and ${\bar{X}}^a\left(w\right)\equiv X^a\left(w\mathrm{,}\bar{z}\mathrm{,}\bar{k}\right)$ when it is of high quality; we distinguish the following three cases:

1) if $w<\underset{\_}{w}$ then ${\bar{X}}^a\left(w\right)=-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)$ and the profit is given by ${\bar{\pi }}^a\left(w\right)=CS\left(w,\bar{z},\bar{k}\right)-c\left(\bar{k}\right)$ while ${\underset{\_}{X}}^a\left(w\right)=-C{S}_{p^a}\left(w\mathrm{,}\underset{\_}{z}\mathrm{,}\underset{\_}{k}\right)$ with a profit given by ${\underset{\_}{\pi }}^a\left(w\right)=CS\left(w\mathrm{,}\underset{\_}{k}\mathrm{,}\underset{\_}{z}\right)-c\left(\underset{\_}{k}\right)$ ;

2) if $\underset{\_}{w}<w<\bar{w}$ then ${\bar{X}}^a\left(w\right)=-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)$ and the profit is given by ${\bar{\pi }}^a\left(w\right)=CS\left(w,\bar{z},\bar{k}\right)-c\left(\bar{k}\right)$ while ${\underset{\_}{X}}^a\left(w\right)=0$ and ${\underset{\_}{\pi }}^a\left(w\right)=0$ ;

3) if $w>\bar{w}$, the firm does not adopt whatever the quality of the GPT; ${\underset{\_}{X}}^a\left(w\right)={\bar{X}}^a\left(w\right)=0$ 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 $\theta$ that the technology is of high-quality. If it adopts, it choose its technology level $k^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)$ and its optimal price $p^a\left(w\mathrm{,}k\mathrm{,}\theta \right)$ so as to maximize its expected profit ${\Pi }^a\left(p^a\mathrm{,}k\mathrm{,}\theta \right)$ .

The choice of $p^a\left(w\mathrm{,}k\mathrm{,}\theta \right)$ is solution of the maximizing problem of the firm’s expected gross profit:

$\underset{p^a}{\max }{R}^a\left(w,p^a,k,\theta \right)$ (5)

with $\begin{array}{c}{R}^a\left(w,p^a,k,\theta \right)=\theta \left[CS\left(p^a,k,\bar{z}\right)+\left(p^a-w\right)X^a\right]\\ +\left(1-\theta \right)\left[CS\left(p^a,k,\underset{\_}{z}\right)+\left(p^a-w\right)X^a\right]\end{array}$ . The resolution

leads to a optimal price $p^a=w$ ; using this expression, the expected profit becomes ${\Pi }^a\left(w,k,\theta \right)=\theta CS\left(w,k,\bar{z}\right)+\left(1-\theta \right)CS\left(w,k,\underset{\_}{z}\right)-c^a\left(k\right)$ .

The optimal k is given by:

$k^{*}\left(w,\theta \right)=\underset{k}{\arg \max }\left\{{\Pi }^a\left(w,k,\theta \right)\right\}$ (6)

Let us note ${\Pi }^{\max }\left(w,\theta \right)={\max }_k{\Pi }^a\left(w,k,\theta \right)$ . The GPT technology is profitable iff ${\Pi }^{\max }\left(w,\theta \right)>0$ .

Definition 1. Let us consider $\tilde{w}\left(\theta \right)$ as the reserve price of the downstream firm in the absence of the signal; then the firm adopts iff $w<\tilde{w}\left(\theta \right)$ .

Lemma 2. $\tilde{w}\left(\theta \right)$ is increasing with $\theta$ .

Proof. For any given w, ${\Pi }_i^{\max }\left(w,\theta \right)=\theta CS\left(w,k^{*},\bar{z}\right)+\left(1-\theta \right)CS\left(w,k^{*},\underset{\_}{z}\right)-c^a\left(k^{*}\right)$ . Let us suppose that for any firm i with an a priori belief $\theta$, the GPT’s price is such as $w=\tilde{w}\left(\theta \right)$, then ${\Pi }_i^{\max }\left(\tilde{w}\left(\theta \right),\theta \right)=0$ because $\tilde{w}\left(\theta \right)$ is the price which cancels ${\Pi }_i^{\max }\left(w,\theta \right)$ . Moreover by assumption, the profit of associated firms decreases with w. Consider a firm j with belief ${\theta }^{\prime }$ such as ${\theta }^{\prime }<\theta$ ; let us assess the firm j’s profit ${\Pi }_j^{\max }$ in $\tilde{w}\left(\theta \right)$ ; we have

$\begin{array}{l}{\theta }^{\prime }CS\left(\tilde{w}\left(\theta \right),k^{*},\bar{z}\right)+\left(1-{\theta }^{\prime }\right)CS\left(\tilde{w}\left(\theta \right),k^{*},\underset{\_}{z}\right)-c^a\left(k^{*}\right)\\ =\left({\theta }^{\prime }-\theta \right)CS\left(\tilde{w}\left(\theta \right),k^{*},\bar{z}\right)+\left(\theta -{\theta }^{\prime }\right)CS\left(\tilde{w}\left(\theta \right),k^{*},\underset{\_}{z}\right)-c^a\left(k^{*}\right)\\ +\theta CS\left(\tilde{w}\left(\theta \right),k^{*},\bar{z}\right)+\left(1-\theta \right)CS\left(\tilde{w}\left(\theta \right),k^{*},\underset{\_}{z}\right)\\ =\left({\theta }^{\prime }-\theta \right)\left[CS\left(\tilde{w}\left(\theta \right),k^{*},\bar{z}\right)-CS\left(\tilde{w}\left(\theta \right),k^{*},\underset{\_}{z}\right)\right]<0\end{array}$ ; the profit of firm j

assessed in $\tilde{w}\left(\theta \right)$ is negative; knowing by hypothesis that the profit of j is decreasing with w and cancels in $\tilde{w}\left({\theta }^{\prime }\right)$, it is therefore necessary to reduce $\tilde{w}\left(\theta \right)$ to reach $\tilde{w}\left({\theta }^{\prime }\right)$, which implies $\tilde{w}\left({\theta }^{\prime }\right)<\tilde{w}\left(\theta \right)$ . Thus, we show that $\tilde{w}\left(\theta \right)$ increases with $\theta$ ; in particular, $\tilde{w}\left(0\right)=\underset{\_}{w}<\tilde{w}\left(1\right)=\bar{w}$ . $\square$

Remark 4. In imperfect information, the reserve price depends on the a priori belief of the downstream firm. A very pessimistic downstream firm ( $\theta \simeq 0$ ) will have a stricter adoption criterion because its reserve price will be lower; on the contrary a very optimistic downstream firm ( $\theta \simeq 1$ ) will have a higher reserve price. Uncertainty can reduce or increase incentives for adoption depending on whether the firm’s beliefs are pessimistic or optimistic. The lack of signal can also lead to suboptimal behavior for the downstream firm; in fact, a pessimistic firm might refuse a high-quality GPT $\bar{z}$ because the price w is such that $\tilde{w}\left(\theta \right)<w<\bar{w}$, whereas in reality the firm would gain ex-post to adopt the GPT because it is of high quality $\bar{z}$ . Similarly, an optimistic firm might adopt a low-quality GPT $\underset{\_}{z}$ because the price w is such that $\underset{\_}{w}<w<\tilde{w}\left(\theta \right)$ while it 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 $\theta$ .

1) if $w<\tilde{w}\left(\theta \right)$, the downstream firm adopts the GPT; the anticipated demand1³ on which it based its adoption decision at stage 3 is given by $\theta \left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\bar{z}\right)\right)+\left(1-\theta \right)\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}\underset{\_}{z}\right)\right)$ and its expected profit is given by ${\Pi }^a\left(w,\theta \right)=\theta CS\left(w,k^{*},\underset{\_}{z}\right)+\left(1-\theta \right)CS\left(w,k^{*},\underset{\_}{z}\right)-c^a\left(k^{*}\right)$ ;

2) if $w>\tilde{w}\left(\theta \right)$, the downstream firm does not adopt the GPT; its anticipated demand is zero as well as its profit.

4. Upstream Firm Equilibrium

The expected demand $X^g$ of the upstream firm depends on available informations; let us note $X^g\equiv {\underset{\_}{X}}^g$ when the firm produces low quality and $X^g\equiv {\bar{X}}^g$ 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 $X^g$ is identical to the downstream firm demand in stage 4. Then,

1) if $w<\underset{\_}{w}$, we have ${\bar{X}}^g\left(w\right)={\bar{X}}^a\left(w\right)=-C{S}_{p^a}\left(w,\bar{z},\bar{k}\right)$ and ${\underset{\_}{X}}^g\left(w\right)={\underset{\_}{X}}^a\left(w\right)=-C{S}_{p^a}\left(w,\underset{\_}{z},\underset{\_}{k}\right)$ ;

2) if $\underset{\_}{w}<w<\bar{w}$, ${\bar{X}}^g\left(w\right)={\bar{X}}^a\left(w\right)=-C{S}_{p^a}\left(w,\bar{z},\bar{k}\right)$ and ${\underset{\_}{X}}^g\left(w\right)={\underset{\_}{X}}^a\left(w\right)=0$ ;

3) if $w>\bar{w}$, ${\underset{\_}{X}}^g\left(w\right)={\underset{\_}{X}}^a\left(w\right)=0$ ;

the corresponding profits ${\pi }^g$ are given by $\left(w-c\right)X^g-c^g\left(z\right)$ .

Imperfect information. In the absence of a signal, the expected demand $X^g$ 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 $k^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)$ chosen by the θ-type downstream firm in the absence of signal. Then for an observed quality $\bar{z}$, we have ${\bar{X}}^a\left(w,\theta \right)=-C{S}_{p_a}\left(w,k^{*}\left(w,\theta \right),\bar{z}\right)$ and for an observed quality $\underset{\_}{z}$, we have ${\underset{\_}{X}}^a\left(w,\theta \right)=-C{S}_{p_a}\left(w,k^{*}\left(w,\theta \right),\underset{\_}{z}\right)$ ; therefore,

1) if $w<\underset{\_}{w}$, we have:

${\bar{X}}^g\left(w\right)={\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w,k^{*}\left(w,\theta \right),\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (7)

or

${\underset{\_}{X}}^g\left(w\right)={\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w,k^{*}\left(w,\theta \right),\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (8)

2) if $\underset{\_}{w}<w<\bar{w}$, we have:

${\bar{X}}^g\left(w\right)={\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (9)

or

${\underset{\_}{X}}^g\left(w\right)={\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (10)

3) if $w>\bar{w}$, there is no technology adoption and the upstream expected demand $X^g$ 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 $\left(1-h\right)$, its expected demand is the same as in imperfect information; so:

1) if $w<\underset{\_}{w}$, then

${\bar{X}}^g\left(w\mathrm{,}h\right)=h\left(-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)\right)+\left(1-h\right){\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (11)

or

${\underset{\_}{X}}^g\left(w\mathrm{,}h\right)=h\left(-C{S}_{p^a}\left(w\mathrm{,}\underset{\_}{k}\mathrm{,}\underset{\_}{z}\right)\right)+\left(1-h\right){\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (12)

2) if $\underset{\_}{w}<w<\bar{w}$

${\bar{X}}^g\left(w\mathrm{,}h\right)=h\left(-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)\right)+\left(1-h\right){\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (13)

or

${\underset{\_}{X}}^g\left(w\mathrm{,}h\right)=\left(1-h\right){\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$ (14)

3) If the price of the GPT is such that $w>\bar{w}$, 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. ${\bar{X}}^g-{\underset{\_}{X}}^g$ . To do this, let us consider two cases, either the upstream firm chooses a unique price $w^m$ regardless the GPT quality, either the upstream firm sets endogenous prices depending on the GPT quality:

1) the upstream firm sets a unique price $w^m=w^m\left(h,\underset{\_}{z}\right)=w^m\left(h,\bar{z}\right)$ for any given h; then expected demand gap¹⁴ according to values of $w^m$ are:

a) if $w^m<\underset{\_}{w}$, by using Equations (11) and (12), we have

${\bar{X}}^g-{\underset{\_}{X}}^g=h\left(-C{S}_{p^a}\left(w^m\mathrm{,}\bar{k},\bar{z}\right)-\left(-C{S}_{p^a}\left(w^m\mathrm{,}\underset{\_}{k}\mathrm{,}\underset{\_}{z}\right)\right)\right)$ (15)

b) if $\underset{\_}{w}<w^m<\bar{w}$, by using Equations (13) and (14), we have

${\bar{X}}^g-{\underset{\_}{X}}^g=h\left(-C{S}_{p^a}\left(w^m\mathrm{,}\bar{k},\bar{z}\right)\right)$ (16)

we know by assumptions that $\left(-C{S}_{p^a}\left(w^m,\bar{k},\bar{z}\right)\right)>0$, $\left(-C{S}_{p^a}\left(w^m,\underset{\_}{k},\underset{\_}{z}\right)\right)>0$ and $\left(-C{S}_{p^a}\left(w^m,\bar{k},\bar{z}\right)-\left(-C{S}_{p^a}\left(w^m,\underset{\_}{k},\underset{\_}{z}\right)\right)\right)>0$ ; so the effect of h on the demand gap is positive, i.e. $\frac{\partial \left({\bar{X}}^g-{\underset{\_}{X}}^g\right)}{\partial h}>0$ .

Let us calculate the profit gap, ${\bar{\pi }}^g-{\underset{\_}{\pi }}^g=\left(w^m-c\right)\left({\bar{X}}^g-{\underset{\_}{X}}^g\right)-\left(c^g\left(\bar{z}\right)-c^g\left(\underset{\_}{z}\right)\right)$ . The effect of h on the profit gap is given by $\frac{\partial \left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)}{\partial h}=\frac{\partial \left({\bar{X}}^g-{\underset{\_}{X}}^g\right)}{\partial h}\left(w^m-c\right)$ ; the previous result on the demand gap induces $\frac{\partial \left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)}{\partial h}>0$ .

These results show that when h increases, whatever the unique wholesale price $w^m$, including the one that maximizes the profit of low quality $\underset{\_}{z}$, 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 $\bar{z}$ ; 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; so we will have $w^m\left(h,z\right)=\arg {\max }_w{\pi }^g\left(w^m,k,z,h\right)$ .

Let us set respectively $w^m\left(h,\bar{z}\right)=w^{\bar{m}}\left(h\right)$ and $w^m\left(h,\underset{\_}{z}\right)=w^{\underset{\_}{m}}\left(h\right)$ the optimal price for high quality and low quality for a given h.

Assumption 2. Suppose, for any given h, $w^{\bar{m}}\left(h\right)\ge w^{\underset{\_}{m}}\left(h\right)>0$ .

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:

$\frac{\text{d}}{\text{d}h}\left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)=\left(w^{\bar{m}}-w^{\underset{\_}{m}}\right)\frac{\partial {\bar{X}}^g}{\partial h}+\left(w^{\bar{m}}-c\right)\frac{\partial }{\partial h}\left({\bar{X}}^g-{\underset{\_}{X}}^g\right)$ (17)

Proof.

$\frac{\text{d}{\bar{\pi }}^g}{\text{d}h}=\frac{\partial {\bar{\pi }}^g}{\partial h}+\frac{\partial {\bar{\pi }}^g}{\partial w^{\bar{m}}}\frac{\partial w^{\bar{m}}}{\partial h}$ (18)

By definition, $\frac{\partial {\bar{\pi }}^g}{\partial w^{\bar{m}}}=0$, so Equation (18) becomes:

$\frac{\text{d}{\bar{\pi }}^g}{\text{d}h}=\frac{\partial {\bar{\pi }}^g}{\partial h}=\left(w^{\bar{m}}-c\right)\frac{\partial {\bar{X}}^g}{\partial h}$ (19)

It’s the envelop theorem ; similarly, for the profit of low quality, we obtain:

$\frac{\text{d}{\underset{\_}{\pi }}^g}{\text{d}h}=\frac{\partial {\underset{\_}{\pi }}^g}{\partial h}=\left(w^{\underset{\_}{m}}-c\right)\frac{\partial {\underset{\_}{X}}^g}{\partial h}$ (20)

Using Equations (19) and (20), we write the profit differential:

$\frac{\text{d}}{\text{d}h}\left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)=\left(w^{\bar{m}}-w^{\underset{\_}{m}}\right)\frac{\partial {\bar{X}}^g}{\partial h}+\left(w^{\bar{m}}-c\right)\frac{\partial }{\partial h}\left({\bar{X}}^g-{\underset{\_}{X}}^g\right)$ (21)

The sign of profit differential $\frac{\text{d}}{\text{d}h}\left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)$ depends on signs of $\frac{\partial {\bar{X}}^g}{\partial h}$ and $\frac{\partial \left({\bar{X}}^g-{\underset{\_}{X}}^g\right)}{\partial h}$ . $\square$

Lemma 3. $\frac{\partial {\underset{\_}{X}}^g}{\partial h}<0$ and $\frac{\partial {\bar{X}}^g}{\partial h}>0$ .

Proof. It suffices to verify that with Equation (11) we have $\left(-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)\right)>{\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$, with Equation (13) we have $\left(-C{S}_{p^a}\left(w,\bar{k},\bar{z}\right)\right)>{\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\bar{z}\right)\right)f\left(\theta \right)\text{d}\theta$, with Equation (12) we have $\left(-C{S}_{p^a}\left(w\mathrm{,}\underset{\_}{k}\mathrm{,}\underset{\_}{z}\right)\right)<{\displaystyle {\int }_0^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$, and finally with Equation (14) we have $0<{\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta$ . Using the initial hypothesis $C{S}_{p^{a}k}<0$ (i.e. $-C{S}_{p^a}$ increases with k) and knowing that $\underset{\_}{k}<k^{*}\left(w,\theta \right)<\bar{k}$, we can easily verify inequalities resulting from equations (11), (13) and (12); for the Equation (14), we also verify that $\frac{\partial {\underset{\_}{X}}^g}{\partial h}<0$ because ${\displaystyle {\int }_{{\tilde{w}}^{-1}\left(w\right)}^1}\left(-C{S}_{p^a}\left(w\mathrm{,}{k}^{\mathrm{<mo>\; *\; </mo>}}\left(w\mathrm{,}\theta \right)\mathrm{,}\underset{\_}{z}\right)\right)f\left(\theta \right)\text{d}\theta >0$ . $\square$

Remark 5. Lemma 3 implies $\frac{\partial \left({\bar{X}}^g-{\underset{\_}{X}}^g\right)}{\partial h}>0$ but does not necessarily

implies ${\bar{X}}^g-{\underset{\_}{X}}^g>0$ when high quality and low quality are sold respectively at price $w^{\bar{m}}$ and $w^{\underset{\_}{m}}$ .

Lemma 4. $\frac{\text{d}\left({\bar{\pi }}^g-{\underset{\_}{\pi }}^g\right)}{\text{d}h}>0$ ; i.e. the incentive of the GPT-firm to sell high

quality rather than low quality increases with the probability of receiving the signal.

Proof. Lemma 4 is the consequence of lemma 3. $\square$

5. Effects of Innovation Clusters

We assume that the probability of receiving the signal h depends on the coordination mode of the vertical relation between the upstream firm (GPT supplier) and the downstream firm (integrator of GPT). We model innovation clusters by an increase of h compared to the market mechanism; in other words, if $h_1$ and $h_2$ correspond respectively to the signal probability on the market and within the cluster, then $\text{d}h=h_2-h_1>0$ . Indeed we assume that, in the presence of a innovation cluster, it is more likely for the downstream firm to receive information about the true quality of GPT technology than a firm on an arm’s length market; the economic literature on cluster policy argued that innovation clusters 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.

5.1. 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. The previous analysis of the profit gap with respect to h shows that this gap can become large enough for high values of h; in this case, the GPT firm would have to pay the extra R&D cost $c^g\left(\bar{z}\right)-c^g\left(\underset{\_}{z}\right)$ in order to produce high quality. On the other hand, the profit gap can sharply decrease for low values of h; the firm would then have an interest in favoring low quality. In particular, if h becomes zero, the GPT firm always chooses the low quality1⁵ because of the strict increase of R&D costs, $c^g\left(\bar{z}\right)-c^g\left(\underset{\_}{z}\right)$ .

Assumption 3. Assume that, in perfect information, i.e. $h=1$, the GPT firm always chooses the high quality $\bar{z}$ .

Note that assumption 3 ensures that in perfect information, the R&D cost for high quality always remains reasonable so that the profit ${\bar{\pi }}^g>0$ .

Proposition 1. There exists a limit value $h^{\mathrm{<mo>\; *\; </mo>}}$, such that if $h<h^{*}$ the GPT firm invests in low quality, if $h\ge h^{\mathrm{<mo>\; *\; </mo>}}$ the GPT-firm invests in high quality.

Proof. Proposition 1 is the consequence of lemma 4 and assumption 3. $\square$

Corollary 1. If the increase of h allows to move from a $h<h^{*}$ to a $h\ge h^{\mathrm{<mo>\; *\; </mo>}}$, the innovation cluster switches the GPT firm from low quality to high quality.

5.2. Downstream Adoption Behavior

From the point of view of the upstream firm, the expected level of R&D investment and the expected level of use of the GPT product in downstream depend on h, i.e. the probability of observing the signal in downstream.

Let us consider a downstream firm with a given type $\theta$ ; note $\tilde{k}\left(h\mathrm{,}\theta \right)$ its expected level of R&D investment and ${\tilde{X}}^a\left(h\mathrm{,}\theta \right)$ its expected level of GPT product utilization¹⁶. We analyze below the effect of increasing of h, $\text{d}h=h_2-h_1>0$ on $\tilde{k}\left(h\mathrm{,}\theta \right)$ and ${\tilde{X}}^a\left(h\mathrm{,}\theta \right)$ .

5.2.1. Investment Level in Downstream Quality

The expected quality of downstream technology is given by $\tilde{k}\left(h\mathrm{,}\theta \right)=hk\left(w^m\mathrm{,}z\right)+\left(1-h\right)k^{\mathrm{<mo>\; *\; </mo>}}\left(w^m\mathrm{,}\theta \right)$ ; the total differential1⁷ with respect to h is:

$\frac{\text{d}\tilde{k}}{\text{d}h}=\left(k-k^{\mathrm{<mo>\; *\; </mo>}}\right)+\left(\frac{\partial w^m}{\partial h}+\frac{\partial z}{\partial h}\frac{\partial w^m}{\partial z}\right)\left(h\left(\frac{\partial k}{\partial w^m}-\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^m}\right)+\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^m}\right)+h\left(\frac{\partial z}{\partial h}\frac{\partial k}{\partial z}\right)$ (22)

We distinguish here three effects of h on the expected quality of downstream technology:

1) $\left(k-k^{\mathrm{<mo>\; *\; </mo>}}\right)$ is the direct effect of h

2) $\left(\frac{\partial w^m}{\partial h}+\frac{\partial z}{\partial h}\frac{\partial w^m}{\partial z}\right)\left(h\left(\frac{\partial k}{\partial w^m}-\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^m}\right)+\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^m}\right)$ is the indirect effect of h through price.

3) $h\left(\frac{\partial z}{\partial h}\frac{\partial k}{\partial z}\right)$ 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. $h_1<h_2<h^{*}$ or $h^{*}\le h_1<h_2$ ; and the increase of h causes a switch from low quality to high quality upstream, i.e. $h_1<h^{*}\le h_2$ .

First case: the increase of h does not result in a change in upstream quality.

In this case, $z\left(h_1\right)=z\left(h_2\right)=\underset{\_}{z}$ or $z\left(h_1\right)=z\left(h_2\right)=\bar{z}$ . Note ${\tilde{k}}_{inf}\left(h\mathrm{,}\theta \right)$ the expected downstream quality when the upstream quality is $\underset{\_}{z}$ and ${\tilde{k}}_{sup}\left(h\mathrm{,}\theta \right)$ when the upstream quality is $\bar{z}$ ; we have:

$\frac{\text{d}{\tilde{k}}_{inf}}{\text{d}h}=\left(\underset{\_}{k}-k^{\mathrm{<mo>\; *\; </mo>}}\right)+h\left(\frac{\partial w^{\underset{\_}{m}}}{\partial h}\frac{\partial \underset{\_}{k}}{\partial w^{\underset{\_}{m}}}\right)+\left(1-h\right)\left(\frac{\partial w^{\underset{\_}{m}}}{\partial h}\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^{\underset{\_}{m}}}\right)$ (23)

$\frac{\text{d}{\tilde{k}}_{sup}}{\text{d}h}=\left(\bar{k}-k^{\mathrm{<mo>\; *\; </mo>}}\right)+h\left(\frac{\partial w^{\bar{m}}}{\partial h}\frac{\partial \bar{k}}{\partial w^{\bar{m}}}\right)+\left(1-h\right)\left(\frac{\partial w^{\bar{m}}}{\partial h}\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^{\bar{m}}}\right)$ (24)

Knowing that $\underset{\_}{k}-k^{*}<0$ and $\bar{k}-k^{*}>0$, the overall effect of h on ${\tilde{k}}_{inf}$ and on ${\tilde{k}}_{sup}$ depend on the both effect of h on the upstream wholesale price and effect of the upstream wholesale price on the choice of associated quality in

downstream. If these effects are negligible, we have $\frac{\text{d}{\tilde{k}}_{inf}}{\text{d}h}=\underset{\_}{k}-k^{\mathrm{<mo>\; *\; </mo>}}<0$ and $\frac{\text{d}{\tilde{k}}_{sup}}{\text{d}h}=\bar{k}-k^{\mathrm{<mo>\; *\; </mo>}}>0$ ; in other words, the increase of h reinforces the downstream

R&D investment when the GPT technology is of high quality and lowers it when the GPT technology is of low quality. On the contrary, if these two effects are not

negligible, then the overall effect of h on ${\tilde{k}}_{inf}$ depends on signs of $\frac{\partial w^{\underset{\_}{m}}\left(h\right)}{\partial h}$, $\frac{\partial \underset{\_}{k}}{\partial w^{\underset{\_}{m}}}$ and $\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^{\underset{\_}{m}}}$, while the overall effect of h on ${\tilde{k}}_{sup}$ depends on signs of $\frac{\partial w^{\bar{m}}\left(h\right)}{\partial h}$, $\frac{\partial \bar{k}}{\partial w^{\bar{m}}}$ and $\frac{\partial k^{\mathrm{<mo>\; *\; </mo>}}}{\partial w^{\bar{m}}}$ .

Assumption 4. $\forall h\in \left[\mathrm{0,1}\right]$, $\frac{\partial w^{\bar{m}}\left(h\right)}{\partial h}\ge 0$ and $\frac{\partial w^{\underset{\_}{m}}\left(h\right)}{\partial h}\le 0$ .

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 ${\tilde{k}}_{inf}$ and ${\tilde{k}}_{sup}$ :

1) Overall effect of h on ${\tilde{k}}_{inf}$

- if $\frac{\partial \underset{\_}{k}}{\partial w^{\underset{\_}{m}}}>0$ and $\frac{\partial k^{*}}{\partial w^{\underset{\_}{m}}}>0$, we verify that the indirect effect through price is negative; so there is a reinforcement of the direct negative effect of h on ${\tilde{k}}_{inf}$ .

- if $\frac{\partial \underset{\_}{k}}{\partial w^{\underset{\_}{m}}}<0$ and $\frac{\partial k^{*}}{\partial w^{\underset{\_}{m}}}<0$, we verify that the indirect effect through price is positive; which leads to a smaller decrease or an increase of ${\tilde{k}}_{inf}$ as h increases.

2) Overall effect of h on ${\tilde{k}}_{sup}$

- if $\frac{\partial \bar{k}}{\partial w^{\bar{m}}}>0$ and $\frac{\partial k^{*}}{\partial w^{\bar{m}}}>0$, we verify that the indirect effect through price is positive; this reinforces the positive direct effect of h on ${\tilde{k}}_{sup}$ .

- if $\frac{\partial \bar{k}}{\partial w^{\bar{m}}}<0$ and $\frac{\partial k^{*}}{\partial w^{\bar{m}}}<0$, we verify that the indirect effect through price is negative; this leads to a smaller increase or a decrease of ${\tilde{k}}_{sup}$ with respect to h.

Second case: The increase of h results in a switch from low quality to high quality in upstream

Let us evaluate the sign of ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)$ . We know that:

${\tilde{k}}_{sup}\left(h\mathrm{,}\theta \right)=h\bar{k}\left(w^{\bar{m}}\left(h\right)\right)+\left(1-h\right)k^{\mathrm{<mo>\; *\; </mo>}}\left(w^{\bar{m}}\left(h\right)\mathrm{,}\theta \right)$ (25)

${\tilde{k}}_{inf}\left(h\mathrm{,}\theta \right)=h\underset{\_}{k}\left(w^{\underset{\_}{m}}\left(h\right)\right)+\left(1-h\right)k^{\mathrm{<mo>\; *\; </mo>}}\left(w^{\underset{\_}{m}}\left(h\right)\mathrm{,}\theta \right)$ (26)

We can decompose and write:

${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)=\left({\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)\right)+\left({\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)\right)$ (27)

If the effect of the upstream price on the downstream quality and the effect of h on the upstream price are negligible, then $\bar{k}\left(w^{\bar{m}}\left(h_2\right)\right)=\bar{k}\left(w^{\bar{m}}\left(h_1\right)\right)=\bar{k}$, $\underset{\_}{k}\left(w^{\underset{\_}{m}}\left(h_2\right)\right)=\underset{\_}{k}\left(w^{\underset{\_}{m}}\left(h_1\right)\right)=\underset{\_}{k}$, $k^{*}\left(w^{\bar{m}}\left(h_2\right),\theta \right)=k^{*}\left(w^{\bar{m}}\left(h_1\right),\theta \right)=k^{*}\left(w^{\underset{\_}{m}}\left(h_2\right),\theta \right)=k^{*}\left(w^{\underset{\_}{m}}\left(h_1\right),\theta \right)=k^{*}$, with $\bar{k}>k^{*}>\underset{\_}{k}$ and $w^{\bar{m}}\left(h_2\right)=w^{\bar{m}}\left(h_1\right)=w^{\bar{m}}$, $w^{\underset{\_}{m}}\left(h_2\right)=w^{\underset{\_}{m}}\left(h_1\right)=w^{\underset{\_}{m}}$ with $w^{\bar{m}}>w^{\underset{\_}{m}}$ . In this case, using Equations (25) and (26), we show that the two right-side terms of the Equation (27) are positive; indeed:

${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)=\left(h_2-h_1\right)\left(\bar{k}-k^{*}\right)>0$ (28)

${\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)=h_1\left(\bar{k}-\underset{\_}{k}\right)>0$ (29)

Thus, we show that ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)>{\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)>0$ ; the switch from low quality to high quality reinforces the cluster’s positive impact on R&D investments in downstream.

On the contrary, if the effect of the upstream price on the downstream quality and the effect of h on the upstream price are not negligible, we can assess the overall effect of h on the expected investment in downstream; by using assumption (4), we have:

- if $\frac{\partial \bar{k}}{\partial w^{\bar{m}}}>0$ and $\frac{\partial k^{*}}{\partial w^{\bar{m}}}>0$, we show that the first right-side terms of the

Equation (27), i.e. ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)$, has an indefinite sign whereas the second term, ${\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)$, is positive; we conclude that the sign of ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)$ is indefinite. However, if the increase in upstream quality following the increase in the upstream price is very important, then we verify that ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)>0$ .

Proof. See Appendix A $\square$

- if $\frac{\partial \bar{k}}{\partial w^{\bar{m}}}<0$ and $\frac{\partial k^{*}}{\partial w^{\bar{m}}}<0$, we show that the first right-side terms of the

Equation (27), ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)$, has an indefinite sign whereas the second term, ${\tilde{k}}_{sup}\left(h_1\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)$, is negative. We conclude that the sign of ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)$ is indefinite. However, if the decrease in upstream quality following the increase in the upstream price is very important, then we verify that ${\tilde{k}}_{sup}\left(h_2\mathrm{,}\theta \right)-{\tilde{k}}_{inf}\left(h_1\mathrm{,}\theta \right)<0$ .

Proof. See Appendix A $\square$

Proposition 2. If $\frac{\partial w^m}{\partial h}=0$ or $\frac{\partial k}{\partial w^m}\ge 0$ or $\frac{\partial k}{\partial w^m}<0$ with $\left|\frac{\partial k}{\partial w^m}\right|\simeq 0$, the