Should we reallocate patent fees to the universities?

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Should we reallocate patent fees to theuniversities?Elsa Martin a & Hubert Stahn ba Centre d'Economie et de Sociologie appliquées à l'Agricultureet aux Espaces Ruraux (CESAER), UMR 1041 of the INRA, AgroSupDijon, 26, Bd du Docteur Petitjean, BP 87 999, 21 079 DijonCedex, Franceb Groupement de Recherche en Economie Quantitative d'Aix-Marseille (GREQAM), UMR 6579 of the CNRS, Université de laMéditerranée, Château Lafarge, Route des Milles, 13 290, LesMilles, France

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To cite this article: Elsa Martin & Hubert Stahn (2011): Should we reallocate patent fees to theuniversities?, Economics of Innovation and New Technology, 20:7, 681-700

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Economics of Innovation and New TechnologyVol. 20, No. 7, October 2011, 681–700

Should we reallocate patent fees to the universities?

Elsa Martina* and Hubert Stahnb

aCentre d’Economie et de Sociologie appliquées à l’Agriculture et aux Espaces Ruraux (CESAER),UMR 1041 of the INRA, AgroSup Dijon, 26, Bd du Docteur Petitjean, BP 87 999, 21 079 Dijon

Cedex, France; bGroupement de Recherche en Economie Quantitative d’Aix-Marseille (GREQAM),UMR 6579 of the CNRS, Université de la Méditerranée, Château Lafarge, Route des Milles,

13 290 Les Milles, France

(Received 25 September 2009; final version received 16 September 2010 )

In knowledge economies, patent agencies are often viewed as a relevant instrument ofan efficient innovation policy. This paper brings a new support to that idea. We claim thatthese agencies should play an increasing role in the regulation of the relation betweenprivate R&D labs and public fundamental research units especially concerning the ques-tion of the appropriation of free usable research results. Since these two institutions workwith opposite institutional arrangements (see P.S. Dasgupta and P.A. David. 1987. Infor-mation disclosure and the economics of science and technology. In Arrow and the accentof modern economic theory, ed. G.R. Feiwel, 519–42. New York: State University ofNew York Press), we essentially argue that there is, on the one hand, an over-appropriation of these results while, on the other hand, there is also an under-provisionof free usable results issued from more fundamental research. We show how a publicpatent office can restore efficiency.

Keywords: science and technology; patent agency; innovation policy

JEL Classification: O31; H42

1. IntroductionShould patents fees only cover the patents offices examination costs or should these fees bea major lever of an innovation policy in a knowledge based economy? In order to answerthis question, several economists suggest, in the line of Scotchmer (1999), that these fees,especially their modulation, render the innovation process more efficient.1 However, mostof these arguments are worked out in a technological world where knowledge satisfies thepatentability criteria of utility, novelty and non-obviousness. In this paper, we suggest a newargument in favor of the use of patents fees as a policy instrument by looking upstream, atthe interaction between science and technology.

In fact, Dasgupta and David (1987), based on the Arrovian tradition of analysis of theeconomics of knowledge, proposed to define science and technology neither according to

*Corresponding author. Email: [email protected]

ISSN 1043-8599 print/ISSN 1476-8364 online© 2011 Taylor & Francishttp://dx.doi.org/10.1080/10438599.2010.526310http://www.tandfonline.com

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682 E. Martin and H. Stahn

the types of knowledge (general principles versus applied knowledge) they produce, noron the methods of inquiry (focus versus broader perspective) they adopt, but as distinctinstitutional arrangements, broadly corresponding to non-market and market allocationmechanisms.2 The former activity is usually associated to the universities while the secondis related to private R&D labs. Of course, this distinction is nowadays less clear-cut since theuniversities compete with the R&D lab to patent ideas or even manage their own portfolio ofpatents. However, a recent study of Thursby, Fuller, and Thursby (2009) suggests that thisactivity, when the patent characteristic is rather applied, follows from a consulting activity,i.e. which responds to a market mechanism.3 So, even if the borders are dim, in this paperwe look at the universities in perhaps a more narrow sense, i.e. as a community producingnon-directly marketable knowledge and being organized by its own rules.

It is however largely recognized, especially by several empirical studies, that academicresearch has a real effect on corporate patents. According to Jaffe’s (1989) seminal con-tribution, the elasticity of economically useful knowledge – measured by the amount ofinduced corporate patents – with respect to academic research is quite important,4 even ifthis contribution is relatively scattered. For instance, Klevorick et al. (1995) observed thatthe direct impact of recent university research in most industries is small when assessedrelatively to other sources of scientific knowledge, while Cohen, Nelson, and Walsh (2002)argued, by making use of the Carnegie Mellon database, that the basic research stronglyaffects industrial R&D. This means that the channels are not those that are expected in thesense that there is not a direct causal link from fundamental research to new applied projectsand patents. Their survey however indicates (see Cohen, Nelson, and Walsh 2002, Table5) that the key channels of the impact of the university research essentially goes throughpublished papers, reports, public conferences and informal exchanges, that is to say throughtotally free and open scientific results.

So, even if there is a debate that tries to identify which of these indirect channelsis the more efficient5 or to which firm it is mostly helpful,6 we must recognize that (i)the production of knowledge which encompasses both science and technology involvesvarious actors with different motivations and (ii) the transformation of basic knowledgeinto product or process innovations is not as mechanical as predicted in the said linearmodel of innovation. All the basic ingredients inducing inefficiency are therefore present.Our claim is that:

• patent fees can be viewed as tools that help to restore efficiency in the process ofprivatization of free scientific knowledge,

• and that these fees can be reallocated to the universities in the form of an incentivescheme which increases the set of usable, patentable, knowledge.

To be more precise, we consider, in the sense of Dasgupta and David (1987), two basicinstitutional arrangements: the university and private labs.

We first look at the university (in, perhaps, some narrow sense) which produces anamount of generic basic knowledge with its own system of incentives and we assume thatthis knowledge is freely available and only more or less usable, i.e. not directly marketable.But this community by developing its own Common with its own rules, in the sense ofthe Common-Pool Resource approach (Hess and Ostrom 2003, 2007), also produced onlyindirectly – because it is less valuated by its own system of incentives – potential R&Dsolutions which are freely available.

This by-product, which has all the features of a new Common, largely attracts what wecall private labs, i.e. institutions which have the ability to access to this knowledge (or to a

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Economics of Innovation and New Technology 683

part of it), to use it in order to solve market based R&D puzzles and, by the way, to partiallyprivatize this free knowledge.7 In other words, basic research implicitly contributes to acommon resource that helps private labs to solve market based R&D puzzles and for whichthese last institutions are ready to compete because of the expected market gains.

This reference to a common resource may conduct the reader to think that conducting aR&D process can be assimilated to exploiting a natural resource for sale at the market priceat perhaps some constant marginal cost. This is typically not the picture that one has fromsuch an innovation process since these freely available embryonic inventions often requirefurther developments for commercial success:8 they must be accommodated at some costs.Let us take the example of chopping down trees. To be more in phase with an innovationprocess, the idea is that the forest is located on a mountain: as the uphill slope becomessteeper, the extraction becomes harder. In other words, we assume that the easiest potentialR&D solutions will be treated first, while the others will be postponed until later due toincreasing accommodation costs. If one now has in mind that all R&D labs access thisknowledge at the same time, one can expect that the profitability of these private labs is notonly decreasing with the amount of free knowledge they accommodate but also with thestock of free ideas which is appropriated by all these units. This is the only assumption thatwe make on the process of nurturing an idea through to a viable patent and we try to keepthe functional forms introduced in our model as general as possible in order to encompassthe largest set of situations.

If we continue our metaphor of tree chopping, this also means that the extraction inducesan external cost since each exploiter, when he chops trees downhill on the mountains, doesnot take into account the cost that is supported by other exploiters that have to chop treesuphill. This external effect leads to the so-called overexploitation of the forest. If we nowcome back to our innovation process, the appropriation of this new Common induces anegative externality since the marginal accommodation cost of a given R&D lab is relatedto a remaining amount of freely available basic results. This external effect therefore leads toan excessive patenting behavior which typically corresponds to what Hardin (1968) denotesas a Tragedy of the Commons. But following Heller and Eisenberg (1998) which takes somelessons from the biomedical research, we must also concede that more intellectual propertyrights may lead paradoxically to fewer usable products for improving human health (i.e.induces an Anticommon Tragedy). This is why we suggest that a cautious choice of thepatents fees contributes to an adjustment of the private appropriation costs to the social one.

If we go a step further, we even observe that this Common which is appropriated bythe private labs is not as limited as expected. In fact this set of knowledge is indirectly (thestock of knowledge is not directly equal to the number of patentable ideas as in the naturalresource case) produced by the academic research sector; it can therefore be increased byresearch programs that are proposed to the universities. In fact, if these programs are nottoo circumscribed to operational knowledge, we can expect that these results meet, at leastpartially, the incentives schemes of the academic research and therefore increase the by-product of this research, i.e. the set of freely available usable knowledge which is sharedby the R&D labs. But, we must also accept the idea that the researchers who contribute tothese programs are nevertheless forced to deviate from the optimal strategy dictated by pureresearch criteria and therefore support a cost which must, at least, be covered by a financialtransfer. This therefore suggests that the patents fees, which are levied in order to reduce theappropriation costs, must be allocated to these programs: it not only increases the positiveexternality of academic research on technological innovation but it also reduces the socialappropriation costs of scientific knowledge, by extending the Common which is shared bythe private R&D labs.

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Our paper mainly illustrates this prospect. We first introduce, in Section 2, our basicassumptions on the behavior of the university and of the private labs in order to emphasize,on the one hand, the possibility for the university to deviate from its optimal strategy and,on the other hand, the external appropriation costs which are at hand in the private sector.By keeping the stock of freely available usable results as given, we characterize in Section 3the strategic knowledge appropriation behaviors which lead to a Tragedy of the Commons.We then propose in Section 4 a patent fiscal scheme which has the property to implementan optimal allocation and to be accepted by all private labs. In Section 5, we then show thata patent agency even has the ability to couple this fiscal scheme with the sponsorship ofa targeted research program, in order to implement an optimal provision and allocation ofthe more or less applied and freely available ideas induced by academic research. Finally,Section 6 concludes the paper and an appendix is dedicated to the proofs of the differentpropositions.

2. The modelIn order to illustrate this feature, we need two kinds of actors: the university and the privateR&D labs. We mainly look at their strategic interactions. This is why we do not directlyaddress the question of the production of basic and applied knowledge, which tackles thequestion of the scientists’ decision9 in both the university and the private R&D laboratories.We rather take some macroscopic view which takes these decisions as granted and whichgives us the opportunity to focus on the problems induced by their interactions. So let usnow present the assumptions we made on the university and the private R&D labs.

2.1. The universityWe assume that there is a public institution – a university, for instance – which conductsfundamental research on the basis of public funds. As noted in the introduction, this isperhaps a narrow view of the activity of such an institution since it also conducts moreapplied and even patentable research. But since these activities are mostly driven by financialincentives and consulting behavior, we implicitly identify these activities to those done byprivate labs.10 Under this assumption, we can say that this institution operates by its ownrules, typically a peer validation one, which provide incentives to produce basic research,and, by the way, to produce a set of freely available usable ideas for the industry. Since weconcentrate our attention on the interactions between the university and the private labs,we do not however explicitly model the behavior of the researchers within the university.We simply assume that a given stock of public funds generates an amount r of researchand, as a by-product, an amount y = K(r) of freely available usable knowledge whichcan be transformed in patentable market driven applications (see next subsection for furtherdetails). We assume that this relation is increasing and concave, i.e. dK/dr > 0, d2K/dr2 ≤0 and K(0) = 0.

Even if we do not explicitly model the behavior of the researchers, we neverthelessassume that the amount r̄ of research which is produced corresponds to an optimal runningof the institution in the sense that each researcher has chosen its optimal strategy given thesaid peer validation rule. This, therefore, implies that these optimal behaviors can only bechanged at some costs. This may happen, for instance, if a ‘regulator’ wants to increase by�, the amount of research, in order to increase the stock of freely available usable ideasto y(�) = K(r̄ + �). This is why we introduce a function C(�) which measures the costsupported by the university if one decides to encourage its members to implicitly increase

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the set of applicable ideas by improving the amount of research. If, for instance, an optimaltime allocation of a representative researcher is introduced, this last function captures thecost induced by the deviation from a standard research activity, based on pure academicresearch, to a more applied one. But this does not necessarily imply that these activitiesact as substitutes. If they are, for instance, complementary,11 and if an optimal mixture ischosen under a specific institutional arrangement, we simply claim that any change in thismixture leads to a worse solution and hence induces a cost of change.

Moreover, if this cost comes from a deviation from an optimal solution of a con-cave program, it is immediate that C(�) is increasing and convex, i.e. dC/d� > 0,d2C/d�2 > 0 and that C(0) = 0. We also assume, in the same vein, that very small devia-tions are not too costly while very large ones are prohibitive, i.e. lim�→0 dC/d� = 0 andlim�→+∞ dC/d� = +∞.

2.2. The private R&D labsLet us now move to the behavior of the j = 1, . . . , m private R&D labs. Each lab transformsa certain amount xj of freely available knowledge K(r) into private patented knowledge.To be more precise, we interpret the set [0, K(r̄)] as the set of potential patents whichare induced by the academic research. A part xj of them is transformed by each lab intoreal patents and becomes by this transformation mutually exclusive, so that X := ∑m

j=1 xjdenotes the total amount of patented ideas.

From that point of view, we do not directly model the process of nurturing an ideathrough to a viable patent. We take a more axiomatic point of view by taking as given theset of potential patents,12 imposing some restrictions on the profit function of the differentR&D labs which reflect, at least partially, this process, and trying to keep the functionalforms as general as possible.

We first assume that this patenting activity, even if it is done at some R&D costs (in orderto adapt the free ideas), is assumed to increase the profitability of firm j. Several standardindustrial organization arguments can be recalled in order to justify such an assumption,going from the construction of a dominant position on the existing market to the developmentof new profit opportunities by creating new products. We take all these arguments for grantedand simply assume that the profit πj is increasing in the number of patented ideas xj butnevertheless at a decreasing rate.

We also want to capture the idea that the set [0, K(r̄)] of potential patents is heteroge-neous in the sense that some ideas are less costly to accommodate than some others. Thedifferent labs will, of course, compete for the ideas which are easier to accommodate. If oneassumes that the different potential patents within the set [0, K(r̄)] are ranked by increasingaccommodation cost, one typically has in mind that the marginal cost, say for lab j, ofincreasing its number of patents must be at least related to the number of unused potentialpatents given by S := K(r̄) − X . One would even expect that these costs are decreasingwith S, seeing that a lower residual stock of free ideas indicates a greater adoption of lessproductive ideas. By introducing a standard law of decreasing return, one would even saythat these costs are convex in S. To spare notations, we however do not explicitly introducethis cost function. We simply say that the profit πj(xj , S) of each private lab is a functionof both the quantity of patents it deposes and the remaining stock S of potential patentsinduced by academic research – this profit function is increasing, at a decreasing rate, in itssecond argument.

Due to this negative externality, it is a matter of fact to observe that these m labs playa game in which they simultaneously decide on the amount xj of potential patents they

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transform into a private one by having in mind that they receive a payoff of πj(xj , S). So,in order to make sure that the competition between firms is not too severe, we introducestrategic substitutability by controlling the effect of the residual stock S of free potentialpatents on the marginal productivity of the deposed patents xj .

Finally, we also need more technical assumptions which guarantee the interiority of oursolutions. We first say that a firm, who patents as the remaining stock of potential patentsbecomes small, always has an incentive to decrease its patenting activity xj in order tobenefit from the gain induced by a larger remaining stock of free potential patents. Wealso make sure that each firm at least patents something since for a small level of patents,the marginal gain of an increase of xj covers the aggregate costs induced by the negativeexternality and we finally control the behavior at infinite.

To summarize, we assume that each private R&D lab j is characterized by a profitfunction πj(xj , S) satisfying the following restrictions.

(1) This function is increasing, i.e. ∂xj πj > 0 and ∂Sπj > 0, globally strictly concave(which implies that ∂2

S,Sπj and ∂2xj ,xj

πj < 0) and satisfies ∂2xj ,Sπj ≥ 0 (i.e. the best

responses are strategic substitutes).13

(2) Inactivity is allowed, i.e. ∀S, πj(0, S) = 0.(3) The following boundary conditions14 are met: (i) ∀xj > 0, limS→0 ∂xj πj/∂Sπj < 1,

(ii) ∀S > 0, limxj→0 ∂xj πj/∑m

k=1 ∂Sπk > 1 and (iii) ∀S > 0, limxj→∞ ∂xj πj/∂Sπk < 1.

3. The inefficient patent allocationThe existence of external costs between the different users of this amount y := K(r̄) ofpotential patents leads immediately to the following definition of a patent allocation.

Definition 1 Given any amount y := K(r̄) of potential patents induced by the academicresearch, a Nash equilibrium allocation of patents, also called a decentralized or aninefficient one, is a vector (xd

j (y))mj=1 ∈ R

m+ given by

∀j = 1, . . . , m xdj (y) ∈ arg max

xj∈R+πj

⎛⎜⎝xj , y − xj −

m∑k=1k �=j

xdk

⎞⎟⎠ .

And in order to check consistency, we can prove, under our assumptions, that:

Proposition 1 For any strictly positive stock of potential patents y := K(r̄), this gameadmits a unique interior Nash equilibrium which has the property that the marginal privatebenefit of patenting, ∂xj πj , is equal to the marginal private costs of patenting, ∂Sπj , whichis captured by the decrease of the remaining stock of free ideas. More formally we verifythat

∀j = 1, . . . , m ∂xj πj

⎛⎜⎝xd

j , y − xdj −

m∑k=1k �=j

xdk

⎞⎟⎠ − ∂Sπj

⎛⎜⎝xd

j , y − xdj −

m∑k=1k �=j

xdk

⎞⎟⎠ = 0. (1)

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It now remains to understand how the stock y := K(r̄) of potential patentsimpacts the strategies (xd

j (y))mj=1 and the equilibrium payoffs (πd

j (y))mj=1 := (πj(xd

j (y), y −∑mj=1 xd

j (y)))mj=1. We expect that both functions are increasing. This is immediate when all

labs are symmetric. In this case, condition (1) can be reduced to a single equation given by

∂xπ(xd , y − mxd) − ∂Sπ(xd , y − mxd) = 0 (2)

and a standard computation leads to

dxd

dy= ∂2

x,Sπ − ∂2S,Sπ

(∂2S,xπ − ∂2

x,xπ) + m(∂2x,Sπ − ∂2

S,Sπ)≤ ∂2

x,Sπ − ∂2S,Sπ

m(∂2x,Sπ − ∂2

S,Sπ)

and we even observe that dxd/dy ∈ [0, 1/m]. The equilibrium profits πdj (y) accruing to

each firm are also increasing with the stock y of usable ideas since

dπd

dy= (∂xπ − m∂Sπ)

dxd

dy+ ∂Sπ

=[

1 − (m − 1)dxd

dy

]∂Sπ (by the first-order equilibrium conditions)

≥ 0(

sincedxd

dy<

1m

).

Using symmetry again, we also obtain that the aggregate equilibrium profits accruing to theprivate sector,

�d(y) :=m∑

j=1

πdj (y) = mπd(y),

and the total patent allocation,

X d(y) =m∑

j=1

xdj (y) = mxd(y),

are also increasing with y.By making use of a more elaborated argument presented in the appendix and based on

the proof of Proposition 1, we can even extend these results to the non-symmetric case andstate that:

Proposition 2 For any strictly positive free potential patents stock y := K(r̄), the indi-vidual allocation of patents xd

j (y) and the profits accruing to each firm πdj (y) are

increasing with the stock y of usable ideas. And so are the aggregate allocation X d(y) andprofits �d(y).

4. Overuse of free knowledge and optimal patent taxationIt is a matter of fact to observe that the private labs, at a Nash equilibrium, do not care aboutthe external accommodation costs. In fact, when a lab increases its patents, it knows thatit modifies its own costs of exploiting an additional free idea, but it does not realize thatit also increases the adoption costs for all the other labs. This Tragedy of the Commons

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usually leads to an inefficient patents allocation characterized by a global over-investmentin patents. This is why we suggest in the second part of this section to use the patent feesin order to regulate this over-investment. We even show that the regulator can implement ataxation scheme (i) which is acceptable from the point of view of the private labs and (ii)which leaves them some additional resources.

4.1. The optimal patent allocationIn order to present this over-investment problem, let us, as in the previous section, take asgiven the amount y := K(r̄) of potential patents and let us look at a central planer allocation.Such an allocation is characterized in Definition 2.

Definition 2 Given any amount y := K(r̄) of potential patents induced by the academicresearch, a centralized patent allocation is a vector (xc

j (y))mj=1 ∈ R

m+ with the property that

(xcj (y))

mj=1 ∈ arg max

(xj)mj=1∈R

m+

m∑j=1

πj

⎛⎝xj , y −

m∑j=1

xj

⎞⎠ .

The reader immediately observes what we can state in the following remark.

Remark 1 This problem admits a unique solution since:

(i) the objective function is strictly concave because ∀j, πj(xj , S) has this property,(ii) as y is given, the domain of the problem is compact (remember that S ≥ 0) and

convex.

Moreover, our restrictions on the marginal rates of substitution between xj and S on theboundaries ensure that this solution is an interior one. The following first-order conditionsare therefore necessary and sufficient:

∀j = 1, . . . , m ∂xj πj

⎛⎝xc

j , y −m∑

j=1

xcj

⎞⎠ −

m∑k=1

∂Sπk

⎛⎝xc

j , y −m∑

j=1

xcj

⎞⎠ = 0. (3)

By comparing Equations (1) and (3), we observe that, in the centralized case, the con-gestion effect induced by the patenting activity is internalized by the central planner, i.e.the negative effect of a new patent on the stock of free potential patents and, so, on theprofits is considered in aggregate. In other words, it is the social accommodation costs of anidea which matter here, instead of the private costs like in a decentralized allocation. Thissuggests, if we have in mind the symmetric case, that each lab should optimally invest lessin patents. But this intuition does not extend to the asymmetrical case since one lab shouldperhaps increase its activity while another should be refrained. We nevertheless prove thatthere is, at the aggregate level, an overuse of the free resource y. In fact we say that:

Proposition 3 Given any amount y := K(r̄) of potential patents induced by the academicresearch, the centralized allocation problem induces a lower aggregate level of patents

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than the decentralized one, i.e.

X c(y) :=m∑

j=1

xcj (y) < X d(y) :=

m∑j=1

xdj (y)

while the aggregate profit is higher in the centralized case than in the decentralized one,i.e.

�c(y) :=m∑

j=1

πj(xcj (y), y − X c(y)) ≥ �d(y) :=

m∑j=1

πj(xdj (y), y − X d(y)).

Let us, finally, spell out some additional properties of the centralized solution. We areparticularly interested in the effect of a change of the level r̄ of academic research on thataggregate profit level and on the total amount of patents at an optimal patent allocation.These results will help us to conduct the discussion in our next section, in which we linkthe academic research sector to a regulated private lab sector.

Proposition 4 If the level r̄ of academic research changes, we observe that:

(i) The aggregate profit �c(K(r̄)) at a centralized solution is increasing and strictlyconcave with respect to r̄.

(ii) At an efficient solution, the total amount X c(K(r̄)) of patents as well as the residualstock Sc(K(r̄)) of unused free potential patents are increasing with respect to r̄.

4.2. An optimal patent taxation schemeProposition 4 clearly suggests that there is some room for a public policy since there is again, in aggregate, from the implementation of the centralized solution. This task typicallyinvolves a public patent agency which has the ability to modulate the patent fees andtherefore contributes to lowering the over-investment in patents. In this subsection, wewill essentially show that such a taxation scheme exists and that it can be designed in away which leaves some money to the regulator. Of course, we take here a pure optimaltaxation point of view, that is a normative approach which prescribes a way to limit theover-investment in patents.15

Let us first illustrate this argument in the symmetric case before moving to the moregeneral case. In this simplified set-up, Equation (3) becomes

∂xπ(xc, y − mxc) − m∂Sπ(xc, y − mxc) = 0. (4)

By comparing Equation (4) with Equation (2), we observe that a patent agency whichintroduces a pigovian tax on each patent, given by t = (m − 1)∂Sπ(xc, yd − mxc), slowsdown the patents race and implements the optimal allocation. But this also increases thegross profits (before taxation) accruing to each lab. The patent agency has therefore theopportunity to even associate a lump-sum transfer, which leaves each firm at the sameprofit level as in the decentralized case and which is given by

F = π(xd , y − mxd) − π(xc, y − mxc) + Ix>0txc,

where Ix>0 = 1 if x > 0 and 0 otherwise. By acting in this way, the patent agency will besure that (i) the unique outcome of the decentralized patent allocation coincides with the

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efficient solution, (ii) the firms are willing to participate since they maintain their profits attheir initial level, and (iii) all additional profits are collected for further use, as we will seelater.

This intuition straightforwardly extends to the asymmetric case. We simply need tocheck that the set of Nash equilibria of the game modified by the fiscal scheme coincideswith the set of optimal allocations. Since we also know from Remark 1 that every centralizedsolution is unique, we can therefore conclude that the decentralized equilibrium of themodified game exists and is unique.

Proposition 5 Let the amount y := K(r̄) of potential patents induced by the academicresearch be taken as given, if we introduce, in a decentralized patent allocation mechanism(see Definition 1), a distortionary tax system on patents and a lump-sum transfer scheme,given, respectively, by

tj(y) =m∑

k=1k �=j

∂Sπk(xck(y), y − X c(y))

and

Fj(y) = [πj(xdj , y − X d) − πj(xc

j , y − X c)] + Ixj>0tjxcj ,

where Ixj>0 = 0 if xj = 0 and 1 if xj > 0, we can assert that the unique decentralizedpatents allocation of the modified game corresponds to the efficient allocation introducedin Definition 2. We can even observe that the implementation of this fiscal scheme leaves tothe patents agency an amount of money �c(y) − �d(y) ≥ 0, this inequality holding strictlywhen the decentralized allocation introduced in Definition 1 is inefficient.

So even if this optimal taxation scheme has a strong normative content, it tells us thata public patent agency always has an incentive to collect fees in order to limit the over-appropriation of the set of potential patents indirectly induced by academic research, aslong as one accepts the idea that they are globally increasing accommodation costs. Thisprinciple is even independent of the level of academic research albeit this level influencesthe optimal tax scheme. By doing so, this public agency implements an optimal allocationwhich leaves it some money back since it collects all the additional gains of the private labswith respect to a situation without regulation.

5. Toward a partial reallocation of the patent fees to the universityNow let us bring to mind that:

• the total amount X c(r̄) of innovations increases with the amount of academic research(see (ii) of Proposition 4). We can therefore expect, even if this is not a part of ourmodel, that the welfare also increases with r̄ since a larger amount X c(r̄) of scientificresults can then be transformed into new demand driven technical solutions.

• The residual stock Sc(r̄) of unused potential patents is also increasing with r̄ (seeagain (iii) of Proposition 4). This suggests, when the optimal patent allocation isimplemented, that the social accommodation costs of new ideas decrease as r̄ grows.This will again be welfare improving.

• The profits left to the private labs after taxation, which correspond to the one theyobtain at a Nash equilibrium, are increasing with the level of academic research r̄

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since this increases the level K(r̄) of potential patents (see Proposition 2). Privatelabs are also better off when r̄ increases.

These three observations therefore suggest that a public patent agency must not onlycollect fees in order to limit the over-appropriation of free potential patents: it also oughtto support research programs whose results meet the peer validation criterion, so that theseresults stay in the public domain and increase indirectly the set K(r̄) of potential patents. Thismeans, in our setting, that the regulator may have an incentive to increase the optimallychosen research level from r̄ to r̄ + �. We, however, know that the realization of suchprograms requires a deviation from the optimal research strategy dictated by the institutionalarrangement being at work in the academic sector. This is why we have introduced, at theuniversity level, a cost to change C(�) from the optimal strategy r̄.

If we now take the position of a central planner or, here, of our public patent agency, wehave to answer two questions. What is the optimal change � that we should impose to theacademic sector by having in mind that this change increases the set of potential patents?And how can we implement this optimal solution in a decentralized setting, in a way thatmeets the agreement of the university and the private labs?

A natural way to answer the first question consists of comparing the marginal costsupported by the university to the marginal gains of an increase of the set of potentialpatents K(r̄ + �) when an optimal patent allocation is implemented. In other words, weshould set � to �∗ given by

�∗ ∈ arg max�≥0

�c(K(r̄ + �)) − C(�). (5)

Moreover, it is a matter of fact to observe that:

Remark 2 There always exists a unique strictly positive solution to the previous pro-gram because (i) we know by Proposition 4 that �c is increasing and strictly concaveand (ii) we have assumed that C(�) is concave and verifies lim�→0 dC/d� = 0 andlim�→+∞ dC/d� = +∞.

But this gives us no indication on the implementation of this optimal solution. In thiscase, the patent agency has to design a mechanism including simultaneously a fiscal schemefor the private labs and a bundle of research contracts for the university which has theproperties that:

(1) the decentralized patent allocation coincides with the optimal one by having in mindthat the total amount of potential patents is given by K(r̄ + �∗);

(2) the private labs are ready to accept this mechanism. This implies that the profit theyobtain under this new fiscal scheme is at least equal to the one they earn when thismechanism is not implemented, i.e. when there are no taxes and no transfer to theuniversity in order to improve the amount of potential patents;

(3) the academic researchers also accept the deal. This occurs if the transfer inducedby the bundle of research contracts compensates the cost C(�∗) from moving awayfrom an optimal research strategy r̄.

Properties 1 and 2 can be obtained easily. In fact, if one looks at Proposition 5, oneimmediately notices that this fiscal scheme can be spelled out for any amount y of potentialpatents, in particular for y = K(r̄ + �∗). So, if we also want to meet the third property, we

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have to check that the amount of money which is obtained from the lump-sum transfersenforced by the patent agency covers the cost of change C(�∗) in the academic sector.

In order to check that point, let us first consider the symmetric case. In this case, weknow that xd , the symmetric Nash patent allocation when no additional effort is asked tothe university, solves Equation (2) and we denote by xc∗ the symmetric centralized patentallocation when the amount of potential patents is of y = K(r̄ + �∗), i.e. the quantity whichverifies16 Equation (4) for y = K(r̄ + �∗). From Proposition 5, we know that the pigoviantax rate which decentralizes xc∗ is given by t = (m − 1)∂Sπ(xc∗, K(r̄ + �∗) − mxc∗) andthe lump-sum (negative) transfer which leaves the profit of each private lab unchanged isdefined by F = π(xd , K(r̄) − mxd) − π(xc∗, K(r̄ + �∗) − mxc∗) + txc∗. From that point ofview, the global maximal amount of money that can be extracted by the patent agency isgiven by

T = −mF + mtxc∗ = mπ(xc∗, K(r̄ + �∗) − mxc∗)︸︷︷︸�c(r̄+�∗)

− mπ(xd , K(r̄) − mxd)︸︷︷︸�d (r̄)

.

It therefore remains to check that T ≥ C(�∗). So let us first remember that the aggregateprofit level of the private labs is always greater in the centralized case compared with thedecentralized one, in particular �c(K(r̄)) ≥ �d(K(r̄)). We can therefore say that

T = �c(r̄ + �∗) − �d(r̄) ≥ �c(r̄ + �∗) − �c(r̄).

Now let us remember (see Proposition 4) that the aggregate profit level �c(r) of the privatelabs at the centralized solution is strictly concave. By a standard concavity argument, wecan therefore say that

T ≥ �c(r̄ + �∗) − �c(r̄) > (�c)′|r̄+�∗ × �∗.

But the optimal change �∗ solves the program given by Equation (5). By the first-ordercondition, we know that (�c)′|r̄+�∗ = C ′|�∗ . Finally, let us recall that the cost functionC(�) is convex and has the property that C(0) = 0. By a standard convexity argument, wethen know that the marginal cost is always greater then the average cost. We can thereforeconclude that:

T > C ′|�∗ × �∗ ≥ C(�∗).

But the reader has surely observed that this last argument does not rely on symmetry.This is why we can generalize to what is stated in Proposition 6.

Proposition 6 Within our setting, a public patent agency has the ability, on the one hand,to control by a suitable fiscal scheme the over-appropriation of the freely available usableideas in order to reduce the social appropriation costs and, on the other hand, to finance,with these fees, some research projects in order to reach an optimal transfer of resultsfrom the universities to the private labs. In our specific model, a budget-balanced fiscalscheme associated with an optimal transfer of C(�∗) to the university, which reaches theagreement of all actors, is given by a patent tax rate of tj = ∑m

k=1k �=j

∂Sπk(xc∗k , y + �∗ − X c∗)

and a (negative) lump-sum transfer to the labs of Fj = [πj(xdj , y − X d) − πj(xc∗

j , y + �∗ −X c∗)] + Ixj>0tjxc∗

j .

Of course, we do not claim that this policy mechanism is directly applicable since itrelies on an optimal taxation mechanism and requires, in order to be implemented, a huge set

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of information. We simply want to point out that the patent agencies are not only registrationchambers which protect the innovators but are also a relevant instrument of an innovationpolicy and can help to restore, at least partially, efficiency in the mechanism that transformsfree academic results into usable innovations.

6. Concluding remarksWe essentially looked at our initial question by considering the relation between scienceand technology and more precisely the relation between public fundamental research andprivate heterogeneous R&D labs. Following Dasgupta and David (1987), we assumed thatthese two institutions work with opposite institutional arrangements so that there is, on theone hand, an over-appropriation of the free applicable results produced by the university,while there is, on the other hand, a relative scarcity of the amount of such applicable ideassince their evaluation by academic standards is less efficient. This over-appropriation issuewas related to the existence of accommodation costs: an idea issued from the academicsector must be transformed in order to meet a social demand but all ideas do not have thesame transformation costs.

This brings us to the conclusion that a public patent agency can be a suitable instrumentof this relation regulation. This agency has the ability to control the over-appropriation ofthis knowledge by setting suitable patents fees and therefore to contribute to the reductionof the accommodation costs. But it can also contribute to an optimal provision of free usableideas by sponsoring, thanks to the returns of the patents fees, research programs that meetthe academic standards and that therefore increase the set of freely available ideas.

We must nevertheless concede that this paper can be too caricatural in several aspects.First of all, we essentially concentrate on patents which are induced by free academic

knowledge, as we said we only looked upstream, at the interaction between science andtechnology. We therefore neglected (i) the patents induced by innovations generated by theapplied knowledge developed by the private labs and (ii) the question of the renewal, breadthor length of a patent. The first point is linked to the sequential aspect of the innovationprocess and the problem related to the Anticommon Tragedy introduced by Heller andEisenberg (1998). The second point refers to the kind of market protection from a privateand a social point of view. This point has induced a lot of works since Nordhaus’s (1969)seminal argument in favor of the finiteness of the patents length.17 But many of these studiesoften detect other sources of inefficiency that can, from our point of view, be corrected byan appropriate patents fees policy.

Second, we do not really describe the process which transforms a free idea into apatentable knowledge. We rather adopt an ‘axiomatic’ approach by imposing some prop-erties on this process. It could therefore be interesting to open this ‘black box’ in orderto provide some justifications of the properties that we have imposed. This process canhowever be very intricate and surely relies on uncertainty. In this way we decided to leavethis question for further work.

Furthermore, the reader surely observed that our argument is worked out in a completeinformation setting. From that point of view, neither the university nor the private labs areable to capture an information rent. In other words, the introduction of imperfect informationwould lead to a second best solution since the patents agency has, on the one hand, to controla Bayesian game between the private labs and, on the other hand, to manage a standardprincipal-agent problem concerning its relation with the university.

Finally, we have also observed that the regulation introduced by the patents agency isglobally improving. We can therefore, especially in the context of incomplete information,

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ask the question of the existence of another institutional arrangement that also rendersthe system more efficient. However and at least in the short term, we hope that we haveconvinced the reader that we should see public patent agencies not as profit centers, but asagencies in charge of the different aspects of the innovation policy.

AcknowledgementsWe thank Emeric Henry and two anonymous referees for their comments on an early version of thispaper. We also acknowledge the financial support of the Agence National pour la Recherche (ANRNT_NV_46). But the remaining errors are, of course, ours.

Notes1. In the line of this paper which emphasizes the impact of the renewal fees, the reader is also

referred to Cornelli and Schankerman (1999). For more general recommendations concerningthe innovation policy, see Encaoua, Guellec, and Martinez (2005).

2. See also Barba Navaretti et al. (1996) or Carraro and Siniscalco (2003).3. See, for instance, Siegel et al. (2008) or Bulut and Moschini (2009) who bring to the fore the role

of university technology transfer offices in the increase of universities’ involvement in patentingand licensing activities.

4. For other empirical studies going in the same direction, see also Adams (1990) or Narin, Hamilton,and Olivastro (1997).

5. One often opposes informal local exchange to worldwide published results, see for instance,respectively, Audretsch, Lehmann, and Warning (2005) who argue that geographic considerationsmatter, while Cohen, Nelson, and Walsh (2002) put forward open science channels such aspublications.

6. See for instance, respectively, Acs, Audretsch, and Feldman (1994) who emphasize comparativeadvantages of small startups, versus Link and Rees (1990) who put forward the existence of R&Ddepartments in large firms.

7. By doing so, we implicitly assume that a part of the academic knowledge is immediately valuable.We therefore neglect the question of the progressive maturation of an idea and the debate aroundthe relative contribution of the public and the private research sectors (see, for instance, Aghion,Dewatripont, and Stein 2005). This is left for a later work.

8. Even if, in general, patents can be based on either basic or applied research, recent survey evidencesuggests that potential R&D solutions must often be adjusted, or even totally rethought, in orderto meet the taste of their consumers (Thursby, Jensen, and Thursby 2002).

9. The reader is referred, for instance, to Levin and Stephan’s (1991) early theoretical contributionor, more recently, to Jensen, Thursby, and Thursby (2003) or Thursby, Thursby, and Gupta-Mukherjee (2007).

10. This is, of course, a strong simplifying assumption since an academic researcher usually allocatesher time between these different activities (see the previous footnote). The introduction of suchbehaviors could be an interesting extension.

11. See, for instance, Zucker, Darby, and Brewer (1998), Murray (2002) or Stephan et al. (2003) whorecently claimed that basic and applied researches are complementary.

12. One can perhaps think of introducing some uncertainty on the set of potential patents. This is, ofcourse, an important issue. But this question is largely related to the observability issue. In fact,if everybody observes, ex post, the realization of the random variable, each player, including theregulator, simply chooses a contingent action. The picture changes if asymmetric information isintroduced since the regulator faces an agency problem. We leave this extension for future works.

13. Note also that we implicitly assume that ∂2S,Sπj∂

2xj ,xj

πj − (∂2xj ,Sπj)

2 > 0, otherwise the Hessianof πj(xj , S) is not negative definite, i.e. this function is not strictly concave.

14. The reader can be surprised by this asymmetrical treatment, but it is a matter of fact to observethat the first boundary assumption implies that ∀xj > 0, limS→0 ∂xj πj/(

∑mk=1 ∂Sπj) < 1 while

the second induces that ∀S > 0, limxj→0 ∂xj πj/∂Sπk > 1. Interiority is therefore guaranteed inboth the centralized and the decentralized problems.

15. In other words, we do not claim that this taxation rule is directly applicable. This mechanism, forinstance, supposes that the regulator is totally informed on the behavior of the different actors. It

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underlines, however, the importance of the patent fees as an instrument to correct inefficienciesin the appropriation mechanism of freely available potential patents.

16. This quantity, of course, exists and is unique because we have, in the last section, worked out ourargument whatever y is. So it is especially true for y = y0 + �∗.

17. In order to reach such a conclusion, Nordhaus considered decreasing returns to scale concerningR&D activities. Moreover, by defining the breadth of a patent according to the monopoly powerit gives to its owner, Gilbert and Shapiro (1990) concluded that a long and narrow patent ismore likely to insure a given level of incentives. On the contrary, Gallini (1992) proposed to alsomeasure a patent breadth with the R&D costs needed to imitate a patented innovation outside thepatent domain and proved that, within such a context, a short and large patent is better.

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universities. International Journal of Industrial Organization 21, no. 9: 1271–300.Klevorick, A.K., R. Levin, R.R. Nelson, and S. Winter 1995. On the source and significance of

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AppendixProof of Proposition 1

Step 1: (xdj )n

j=1 is a NE if and only if ∀j = 1, . . . , n

∂xj πj

⎛⎜⎜⎝xd

j , y − xdj −

m∑k=1k �=j

xdk

⎞⎟⎟⎠ − ∂Sπj

⎛⎜⎜⎝xd

j , y − xdj −

m∑k=1k �=j

xdk

⎞⎟⎟⎠ = 0.

Since the profit functions πj(xj , S) are concave, standard concavity inequalities allow us to checkthat the profit function of each player is concave with respect to its own strategic variable. It thereforeremains to verify that the constrains are not binding. If at equilibrium S = y − xd

j − ∑mk=1k �=j

xdk = 0,

each player j has an incentive to decrease xdj since we have assumed that limS→0 ∂xj πj/∂Sπj < 1.

Now let us assume that ∃j at equilibrium such that xdj = 0. In this case, this player has an incentive

to increase xdj because one of our boundary conditions says that ∀S > 0, limxj→0 ∂xj πj/∂Sπj > 1.

Step 2: An application of the implicit function theorem (IFT).Let us define Hj(xj , X , y) := ∂xj πj(xj , y − X ) − ∂Sπj(xj , y − X ) and let us look at Hj(xj , X , y) = 0

for any fixed y (it will be omitted for the moment in order to spare notations). Since we haveassumed that ∀S > 0, limxj→0 ∂xj πj/∂Sπj > 1 and ∀S > 0, limxj→+∞ ∂xj πj/∂Sπj < 1, we, respec-tively, observe that ∀X < y, limxj→0 Hj(xj , X ) > 0 and limxj→+∞ Hj(xj , X ) < 0. But we also knowthat ∂xj Hj(xj , X ) = ∂xj ,xj πj − ∂S,xj πj < 0. The IFT therefore says that

∃φj :]0, y[→ R++, Hj(φj(X ), X ) = 0

and that

∂X φj = −∂X Hj

∂xj Hj= −

−∂2xj ,Sπj + ∂2

S,Sπj

∂2xj ,xj

πj − ∂2S,xj

πj< 0.

This argument also implies that ∀X < y, φj(X ) > 0, but we can even observe limX →y φj(X ) = 0.The last point is immediate. We have assumed that ∀xj > 0, limS→0 ∂xj πj/∂Sπj < 1, so that ∀xj > 0,limX →y Hj(xj , X ) < 0. If we have in mind that ∂xj Hj(xj , X ) < 0, the equation Hj(xj , y) = 0 cannotadmit a strictly positive solution. But we know that xj = φj(X ) > 0 for all X < y. We can thereforeconclude by continuity that limX →y φj(X ) = 0.

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Step 3: The computation of an equilibrium.Let us now construct �(X ) := ∑m

j=1 φj(X ) − X . By step 2, we know that limX →y φj(X ) = 0, andthat ∀X < y, φj(X ) > 0, we deduce, respectively, that limX →y �(X ) < 0 and that limX →0 �(X ) > 0.By computation, we also observe that ∂X � = ∑m

j=1 ∂X φj − 1 < 0. We can therefore assert that ∃X d ∈]0, y[ a unique scalar satisfying �(X d) = 0. So by moving back to step 2, we can say that there existsa unique vector (xd

j )mj=1 = (φj(X d))m

j=1 which verifies the condition stated in Equation (1). Moreover

it is immediate, by construction, that xdj > 0 and that

∑mj=1 xd

j = X d < y for all y > 0. �

Proof of Proposition 2 Let us now remark that the result of Proposition 1 is true for each y and let usremember that the function φj introduced in step 2 of the previous proof also takes y as an argument.By applying the IFT, we even observe that ∂yφj(X , y) = −∂X φj(X , y). So if we move to step 3 of theprevious proof, we can, by using again the IFT, observe that

dX d

dy= −

∑mj=1 ∂yφj(X , y)

(∑m

j=1 ∂X φj(X , y)) − 1=

∑mj=1 ∂X φj(X , y)

(∑m

j=1 ∂X φj(X , y)) − 1> 0 since ∀j, ∂X φj(X , y) < 0.

Moreover,

dxdj

dy= ∂X φj(X d(y), y)

dX d

dy+ ∂yφj(X d(y), y) = ∂X φj(X d(y), y)

[ ∑mj=1 ∂X φj(X d(y), y)

(∑m

j=1 ∂X φj(X d(y), y)) − 1− 1

]

= ∂X φj(X d(y), y)(∑m

j=1 ∂X φj(X d(y), y)) − 1> 0.

It follows that

dπdj

dy= ∂xπj

dxdj

dy+

(1 − dX d

dy

)∂Sπj = ∂xπj

dxdj

dy+ ∂Sπj

1 − (∑m

j=1 ∂X φj)> 0

which implies that

d�d

dy=

m∑j=1

dπdj

dy> 0 �

Proof of Proposition 3 This proof is going to be very closed to the one of Proposition 1 and we willdirectly use some of its elements, contained in steps 2 and 3.

Step 1: A preliminary observation.Let us define, for a fixed y,⎧⎪⎪⎨

⎪⎪⎩H d

j (xj , X , y) := ∂xj πj(xj , y − X ) − ∂Sπj(xj , y − X )

H cj (xj , X , y) := ∂xj πj(xj , y − X ) −

m∑k=1

∂Sπk (xk , y − X ).

We know from the proof of Proposition 1 that ∃φdj (X , y) with the property that H d

j (φdj (X , y),

X , y) = 0. Since we have assumed that ∀S > 0, limxj→0 ∂xj πj/∑m

j=1 ∂Sπj > 1 and limxj→+∞ ∂xj πj/

∂Sπj < 1, we can, by a similar argument, also prove that ∃φcj (X , y) such that H c

j (φcj (X , y), X , y) = 0

and that ∂X φcj (X , y) < 0. Now remember that ∀j, ∂Sπj > 0. It follows that ∀(xj , X , y), H c

j (xj , X ; y) <

H dj (xj , X ; y). But we also know that ∂xj H

dj < 0 and ∂xj H

cj < 0. By combining these two observations,

we obtain∀j, ∀(X , y), φc

j (X , y) < φdj (X , y).

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Step 2: ∀y, X c(y) < X d(y)Let us now construct�c(X , y) := ∑m

j=1 φcj (X , y) − X and�d(X , y) := ∑m

j=1 φdj (X , y) − X . The

proof of Proposition 1 tells us that ∃X d(y) ∈]0, y[ verifying �d(X d , y) = 0. With a similar argumentrequiring the boundary condition associated to the centralized problem, it is also easy to prove that∃X c(y) ∈]0, y[, �c(X c, y) = 0. Now remember, from step 1, that ∀(X , y), φc

j (X , y) < φdj (X , y). It

follows that ∀(X , y), �c(X , y) < �d(X , y). But we know that ∂X �c < 0 and ∂X �d < 0. We cantherefore conclude that X c(y) < X d(y).

Step 3: ∀y, �c(y) ≥ �d(y)This follows directly from the definition of a maximum. �

Proof of Proposition 4 (i) Let us first show that �c(r̄) is increasing with r̄. If we apply the enveloptheorem to the program that is solved by the planner (see Definition 2), we obtain

d�c(y)dy

=m∑

j=1

∂Sπj(xcj (y), y − X c(y)) > 0.

Since dK/dr̄ > 0, it directly follows that d�c(K(r̄))/dr̄ > 0.It remains to check that �c(K(r̄)) is concave. So let us choose r̄1, r̄2 ∈ R and λ ∈]0, 1[. Moreover,

let us denote by (x1j )m

j=1 and (x2j )m

j=1 the optimal patents allocation under, respectively, r̄1 and r̄2. Itis a matter of fact to observe that λx1

j + (1 − λ)x2j is a feasible allocation when the stock is K(λr̄1 +

(1 − λ)r̄2). To see this, remember that (x1j )m

j=1 and (x2j )m

j=1 are feasible under, respectively, r̄1 and r̄2.By combining both constrains we have

m∑j=1

(λx1j + (1 − λ)x2

j ) ≤ λK(r̄1) + (1 − λ)K(r̄2)

≤ K(λr̄1 + (1 − λ)r̄2) since K(r) is concave.

By definition of a maximum, we can therefore claim that

�c(K(λr̄1 + (1 − λ)r̄2)) ≥m∑

j=1

πj

⎛⎝λx1

j + (1 − λ)x2j , K(λr̄1 + (1 − λ)r̄2) −

m∑j=1

(λx1j + (1 − λ)x2

j )

⎞⎠.

Now remember that K(r̄) is concave and that ∀j, ∂Sπj(x,, S) > 0, it follows that

�c(K(λr̄1 + (1 − λ)r̄2))

≥m∑

j=1

πj

⎛⎝λx1

j + (1 − λ)x2j , λK(r̄1) + (1 − λ)K(r̄2) −

m∑j=1

(λx1j + (1 − λ)x2

j )

⎞⎠

=m∑

j=1

πj

⎛⎝λx1

j + (1 − λ)x2j , λ(K(r̄1) −

m∑j=1

x1j ) + (1 − λ)

⎛⎝K(r̄2) −

m∑j=1

x2j

⎞⎠

⎞⎠ .

Finally since ∀j, πj(x,, S) is strictly concave, we conclude that

�c(K(λr̄1 + (1 − λ)r̄2)) > λ

m∑j=1

πj

⎛⎝x1

j , y1 −m∑

j=1

x1j

⎞⎠ + (1 − λ)

m∑j=1

πj

⎛⎝x2

j , y2 −m∑

j=1

x2j

⎞⎠

= λ�c(K(r̄1)) + (1 − λ)�c(K(r̄2)).

(ii) Let us come back to the proof of Proposition 3. We had observed that X c(y) wasobtained by solving �c(X , y) := ∑m

j=1 φcj (X , y) − X = 0 with the property that φc

j (X , y) is,

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j = 1, . . . , m, given by H cj (φc

j (X , y), X , y) = 0 and where the function H cj (xj , X , y) := ∂xj πj(xj , y −

X ) − ∑mk=1 ∂Sπk (xk , y − X ). By making us of the IFT on the first and second equations, we can say

that

dX c

dy= −

∑mj=1 ∂yφ

cj∑m

j=1 ∂X φcj − 1

, ∂X φcj (X , y) = −

−∂2xj ,Sπj + ∑m

k=1 ∂2S,Sπk

∂2xj ,xj

πj − ∂2S;xj

πjand

∂yφcj (X , y) = −∂X φc

j (X , y).

So that

dX c

dy=

∑mj=1 ∂X φc

j∑mj=1 ∂X φc

j − 1=

∑mj=1((−∂2

xj ,Sπj + ∑mk=1 ∂2

S,Sπk )/(∂2xj ,xj

πj − ∂2S;xj

πj))∑mj=1((−∂2

xj ,Sπj + ∑mk=1 ∂2

S,Sπk )/(∂2xj ,xj

πj − ∂2S;xj

πj)) − 1.

But we have assumed that for all j = 1, . . . , m, ∂2xj ,Sπj > 0, ∂2

xj ,xjπj < 0 and ∂2

S,Sπj < 0, it followsthat dX c/dy > 0. It should be remembered that K(r̄) is increasing in r̄, so that dX cdr̄ > 0.

It remains to check that Sc(K(r̄)) = K(r̄) − X c(K(r̄)) is increasing in r̄. So let us first observeby using the previous results that

dSc

dy= 1 − dX c

dy= 1 −

∑mj=1 ∂X φc

j∑mj=1 ∂X φc

j − 1= −1∑m

j=1 ∂X φcj − 1

> 0.

It again remains to remember that K(r̄) is increasing in r̄, so that dSc/dr̄ > 0. �

Proof of Proposition 5 Let us first recall that a decentralized patent equilibrium associated to thefiscal scheme

(tj , Fj)mj=1 =

⎛⎜⎜⎝

m∑k=1k �=j

∂Sπk (xck , y − X c), [πj(xd

j , y − X d) − πj(xcj , y − X c)] + Ixj>0tjxc

j

⎞⎟⎟⎠

is a vector (x̃j)mj=1 of patents allocation with the property that

∀j = 1, . . . , m, x̃j ∈ arg maxxj∈R

m+πj

⎛⎜⎜⎝xj , y − xj −

m∑k=1k �=j

x̃k

⎞⎟⎟⎠ − tjxj + Fj s.t. y − xj −

m∑k=1k �=j

x̃k ≥ 0.

(A1)

Step 1: Every equilibrium of the modified game is an interior one.Let us first verify that ∀j, x̃j > 0. The definition of the fiscal scheme (tj , Fj) matters. In fact,

we know that tjxcj > 0 since (i) xc

j > 0, i.e. efficient allocations are interior solutions and (ii) ∀j,∂Sπk (xk , S) > 0. So, if ∃j0, x̃j0 = 0, this player obtains a strictly smaller transfer Fj than in a situationwhere it plays ε > 0. Because of this jump and since πj0(xj0 , S) is continuous, we can choose an ε0with the property that j0 is better off when xj0 = ε0 is played.

Now assume that ∃j0, y − x̃j0 − ∑mk=1k �=j

x̃k = 0. Since y > 0, we can choose j0 such that x̃j0 > 0.

But we have assumed that ∀xj > 0, lims→0 ∂xj πj/∂Sπj < 1 and we know that tj > 0. Player j0 hastherefore an incentive to decrease xd

j because he increases its gross profit πj0(xj0 , S) and decreases thetax it pays.

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Step 2: The modified game has at least an equilibrium.By step 1, we know that an equilibrium verifies:

∀j = 1, . . . , m ∂xj πj(x̃j , y − X̃ ) − ∂Sπj(x̃j , y − X̃ ) −m∑

k=1k �=j

∂Sπk (x̃ck , y − X̃ c)

︸︷︷︸:=Hj(x̃j ,X̃ ) with X̃ :=∑m

j=1 x̃j

= 0

and since πj(xj , S) is concave we can argue as in the proof of Proposition 1 that these conditionsare not only necessary but also sufficient. We can even observe that the centralized solution obtainedin Definition 2 solves this system of equations. This modified game admits therefore at least oneequilibrium given by (x̃c

j )mj=1.

Step 3: The modified game has a unique equilibrium.Assume that ∃(x̃j)

mj=1 �= (x̃c

j )mj=1 another equilibrium. First, let us assume that X̃ = X̃ c, and

let us choose j0 such that x̃j0 �= x̃cj0 . Since ∂xHj0(xj0 , X̃ ) < 0 it is impossible that Hj(x̃j , X̃ ) =

Hj(x̃cj , X̃ ) = 0. Now assume that X̃ > X̃ c, there exists therefore at least one j1,such that x̃j1 > x̃c

j1 .

Since ∂X Hj0(xj0 , X ) > 0, we have Hj0(x̃j0 , X̃ ) > Hj0(x̃j0 , X̃ c) and since ∂xHj0(xj0 , X̃ c) < 0 we observethat Hj0(x̃j0 , X̃ ) > Hj0(x̃j0 , X̃ ) = 0, a contradiction. Finally, observe that symmetric argument workswhen X̃ < X̃ c. �

Proof of Proposition 6 This proof is the same as in the symmetric case. In fact the reader surelynotices that we essentially use aggregated quantities in order to check that T ≥ C(�∗). From that pointof view, the argument does not rely on the symmetry of the private labs and extends straightforwardly.

�

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Should we reallocate patent fees to the universities?

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