Knowledge licensing in a model of R&D-driven endogenous growth

2.3.1 In-house R&D

High-tech firms can engage in R&D for accumulating knowledge and increasing λ. This can be interpreted as a process innovation that increases productivity (the firms are able to produce more of x), or as a quality upgrade (the firms are able to produce the same amount of higher quality x). Knowledge is not a rival input so that, potentially, it can be used at the same time in multiple places/firms.

In this section, I offer two different settings for the R&D process. The differences stem from how knowledge is exchanged among high-tech firms. In both cases, I assume that the firms cannot collude and stop innovating (e.g. because of antitrust regulation or non-sustainability of collusion).^[3]

Hereafter, when appropriate, for ease of exposition, I describe the properties of the high-tech industry, taking as an example a high-tech firm j. In order to improve its knowledge λ_j the firm needs to hire researchers/labor Lrj. Researchers use the current knowledge of the firm in order to create better knowledge.

Knowledge licensing: This is the benchmark setup, and I call it S.1. Knowledge in this setup can be licensed. If high-tech firm j decides to license knowledge from other high-tech firms, its researchers combine that knowledge with the knowledge available in the firm, in order to produce new knowledge. The knowledge available in the firm is an essential input in the R&D of the firm. Moreover, it is the only essential input. This implies that the high-tech firm does not need to acquire knowledge from other firms in order to advance its own. However, it needs to have its own knowledge in order to build on it. This is in line with that high-tech firms produce distinct goods.

I assume that the in-house R&D process is given by

where ξ is an exogenous efficiency level (ξ>0), u_i,_j is the share of knowledge of firm i that firm j licenses, u_j,_j≡1, and 1>α>0.

Intellectual property regulation facilitates excludability of knowledge and grants bargaining power to the licensors in the sense that they can make a ‘take it or leave it’ offer to licensees. In turn, license contracts do not allow for sub-licensing.^[4]

It can be shown that in (10) the elasticity of substitution between the different types of knowledge that the high-tech firm licenses is equal to 1/(1–α). It can also be shown that the elasticity of substitution between the high-tech firm’s knowledge and any knowledge that it licenses is lower than 1/(1–α). This restates the importance of the firm’s knowledge for its R&D process.

In this R&D process, the productivity of researchers increases linearly with knowledge licensed from an additional high-tech firm because of summation. Such a formulation can be justified if there are significant complementarities among the knowledge of high-tech firms.^[5] Further, it might seem brave to assume that R&D in a single firm can have non-decreasing returns.^[6] This assumption allows the focus to be on the effect of market structure of the high-tech industry on innovation in that industry through competitive pressure. It can be relaxed by setting u_j,_j≡0 in square brackets in (10). In such a case, in this model knowledge licensing (or exchange of knowledge) is a necessary condition to ensure non-decreasing returns to R&D and positive growth in the long-run (for further details, see online Appendix E.2). The R&D process (10) can also be viewed as a simplification leading to tractable results. It ensures that there is a balanced growth path, for example.^[7]

One way to think about this setup is that each high-tech firm can license the patented knowledge of other firms in order to generate its own patented knowledge, which helps to improve its production or output. The firm does not use the knowledge that it licensed directly in the production of its good, because that knowledge needs to be combined with its own knowledge, and that requires investments in terms of hiring researchers and time. The latter seems plausible for technologically sophisticated (e.g. high-tech) goods.

Knowledge spillovers: In this setup, S.2, there are knowledge spillovers among high-tech firms. In high-tech firm j, the researchers combine the knowledge that spills over from other high-tech firms with the knowledge available in the firm, while generating new knowledge. In order to maintain symmetry, I also assume that the researchers do not fully internalize the use of the current knowledge available in the firm and have external benefits from it. Similar to the previous setup, this assumption allows the focus to be on the effect of market structure of the high-tech industry on innovation through competitive pressure.

The R&D process is given by

where I assume that in equilibrium Λ˜ is equal to

An interpretation for this case is that intellectual property regulation does not enforce excludability and firms cannot maintain secrecy. Another possible interpretation is that there is a market for knowledge, and intellectual property regulation grants the bargaining power to the potential licensees, so that they have the right to make a ‘take it or leave it’ offer. The licensees under this assumption receive the knowledge at no cost (i.e. there are spillovers) if the supply of knowledge is not elastic. The supply is necessarily inelastic if licensors do not have trade-offs associated with licensing knowledge. It seems natural to assume that once knowledge is created its supply entails virtually no costs. There would then be no trade-offs if licensors do not take into account that the knowledge they license is used for stealing business: the licensees use it in order to reduce their prices and steal market share. In this line of literature, it is common to assume that the originators of knowledge spillovers do not internalize the effect of spillovers on others’ R&D and production processes. In the frames of this model, this assumption is necessary in order to give such a market-based interpretation to knowledge spillovers. The choice of the interpretation is a matter of taste.

In practice, licensing and spillovers tend to coexist in high-tech industries. These setups are then polar cases. This sharpens the comparison of inference (see online Appendix E.3 for a setup where there is knowledge licensing and spillovers).

Similar to λ, the design of a high-tech good can be viewed as knowledge/patent. It needs to be assumed that (at least for some time) the knowledge about the design of high-tech goods cannot be used by other firms without appropriate compensation, in order to guarantee that high-tech firms have incentives to innovate. Any high-tech firm, nevertheless, could sell the design of its good at market value: the discounted sum of profit streams. Therefore, the assumption that intellectual property regulation enforces excludability of knowledge λ and grants the bargaining power to the licensors seems to be more consistent in such a setup.^[8]

2.3.2 Optimal problem

The revenues of high-tech firm j are gathered from the supply of its good (xj) and when there is knowledge licensing from the supply of its knowledge (u_j,_iλ_j; ∀i≠j). The costs are the labor compensations and license fees, case when there is knowledge licensing. The profits of high-tech firm j are given by

where the term in square brackets appears when there is knowledge licensing, and puj,iλj and pui,jλi are the license fees for u_j,_iλ_j and u_i,_jλ_i.

The high-tech firm maximizes the present discounted value V_j of its profit streams. For brevity, I assume that high-tech firms choose the prices of their products and in that sense engage in a generalized type of Bertrand competition see for Cournot competition (see Jerbashian 2014 for Cournot competition). Therefore, the problem of high-tech firm j is

Vj(t¯)=maxpxj,Lrj,{uj,i,ui,j}i=1;(i≠j)N{∫t¯+∞πj(t)exp[−∫t¯tr(s)ds]dt}s.t.(6), (9), (13), and either (10) or (11),

where t̅ is the entry date and λ_j(t̅)>0 is given.

The solution of the optimal problem implies that the supply of high-tech good x_j and the demand for labor for R&D are given by

where e_j is the elasticity of substitution between high-tech goods perceived by the high-tech firm and qλj is the shadow value of knowledge accumulation.

It can be shown that

The term in square brackets in (16) measures the extent of strategic interactions among high-tech firms, which create a wedge between e and the actual elasticity of substitution ε. Therefore, it measures some of the distortions in the economy which stem from imperfect competition with a finite number of high-tech firms. The term in square brackets and these distortions tend to zero when the number of firms increases.

In the case when there is knowledge licensing, the returns on R&D are given by

where the first term in brackets is the benefit from accumulating knowledge in terms of increased output. The second term is the benefit in terms of higher amount of knowledge available for subsequent R&D:

The third term is the benefit in terms of increased amount of knowledge that can be licensed.

The demand for, and the supply of knowledge in this case are given by

which means that the firm has a downward sloping demand for knowledge and licenses (supplies) all its knowledge.

In the case when there are knowledge spillovers among high-tech firms, the returns on R&D are given by (17) but

and

The first expression means that the licensees receive knowledge (patents) for free. In turn, there is a difference between (18) and (22) because when there is knowledge licensing the returns to R&D are fully appropriated within high-tech firms.

The expression for the price of knowledge (19) indicates that the licensees pay a fixed fee for it. The fee is equal to their marginal valuation of the knowledge that they acquire. This valuation includes all future benefits from using that knowledge for augmenting their current knowledge. Therefore, the licensors appropriate all the benefit from licensing knowledge (i.e. they make the ‘take it or leave it’ offer). With a continuous accumulation of knowledge, as given by (10), at each and every instant the licensees acquire new knowledge at a fixed fee.

From (19) it follows that puj,iλj declines with λ_j. It is clear from (17) that I have assumed that the firm treats puj,iλj as exogenous and does not take into account this effect while accumulating knowledge. In this sense, I focus on a perfect market for knowledge where the price of knowledge is equal to its marginal product and the licensors appropriate all benefits.

In the frames of this model the assumption that the licensors of knowledge do not take into account that their knowledge is used for stealing business amounts to assuming that firm j takes qλi as exogenous for any i different than j. This is in line with assuming that it takes puj,iλj as exogenous (see online Appendix E.4 for the case when it takes into account puj,iλj).

Finally, in equilibrium there is no difference if high-tech firms license their knowledge in return to wealth transfer or knowledge of other firms. Therefore, knowledge licensing among high-tech firms can also be thought to resemble patent consortia and cross-licensing.^[9]

2.3.3 Firm entry

I focus on two regimes of entry into the high-tech industry. In the first regime there are exogenous barriers to entry (i.e. there is no entry) and all firms in the market are assumed to have entered at t=0. In the second regime there are no barriers to entry. Moreover, entry entails no costs see for a setup with endogenous sunk costs (see for a setup with endogenous sunk costs Jerbashian 2014).

3.1.1 Social optimum

The hypothetical Social Planner selects the paths of quantities so as to maximize the lifetime utility of the household. The Social Planner solves the following problem:

maxLx,Lr{∫0+∞Ct1−θ−11−θexp(−ρt)dt}

s.t.

λ(0)>0.

The following proposition offers the Social Planner’s allocations and growth rates, where SP superscript is used to make a distinction between these outcomes and the decentralized equilibrium outcomes.

Proposition 3:The Social Planner chooses labor force allocations so that the economy, where there is “no entry”, makes a discrete jump to a balanced growth path, where

and

The Social Planner necessarily innovates if ξσL>ρ, which holds as long as (31) holds since σ>D. As in decentralized equilibrium, this inequality states that the benefit from R&D outweighs its cost.

Corollary 2:In decentralized equilibrium, the economy innovates less than is socially optimal and therefore grows at a lower rate: gλNE,SP>gλNE,S.1. Moreover, it fails to have socially optimal labor force allocations.

The drivers behind these results are the relative price distortions and knowledge spillovers in the S.2 case. Due to these distortions final goods producers substitute labor for high-tech goods, which lowers the output of high-tech firms and the number of researchers that these firms hire. The spillovers in R&D have an effect of similar direction. If such spillovers are present then high-tech firms do not fully internalize the returns on R&D, which reduces their incentives to invest in R&D.

The deviation of D from σ summarizes the differences among socially optimal and decentralized equilibrium growth rates and labor force allocations which stem from the relative price distortions. It is easy to notice that for sufficiently high N

limε→+∞D=σ.

This equality holds because for sufficiently high N the limiting case ε=+∞ would imply perfect competition in the high-tech industry. According to (32)–(36) and (39)–(43), socially optimal and decentralized equilibrium allocations and growth rates would then coincide when there is knowledge licensing. This is because high-tech firms fully appropriate the benefits from accumulation of knowledge when there is knowledge licensing and there are no distortions in the market for knowledge.^[12]

In such a limiting case, however, in decentralized equilibrium high-tech firms make no profits from the sale of their goods and have no market incentives to innovate. In this respect, if there are no subsidies that keep the profits of high-tech firms non-negative, the positive relationship between innovation and ε holds, as long as high-tech firms have sufficient profits to cover the costs of R&D. Profits of high-tech firms and ε are inversely related. Once profits net of R&D expenditures are equal to zero, increasing ε further reduces innovation to zero. The relationship between intensity of product market competition and innovation then resembles an “inverted-U” shape because of the concavity of the relationship between innovation and ε and this discontinuity. Such a relation is consistent with Schumpeter’s argument that firms need to be sufficiently large in order to innovate. Moreover, it is in line with the empirical findings of Aghion et al. (2005), and provides an alternative explanation for those findings.