2.5 Conclusion
3.2.4 The R&D sector
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“unconstrained”markup (Jones and Williams, 2000). Another scenario is that the markup is subject to an adoption constraint, which may happen if the new designs are linked together in the innovation cluster. This involves the property of the research process, which we will discuss in more detail in the next subsection.
3.2.4 The R&D sector
R&D creates new varieties of intermediate goods for …nal-good production. In line with Romer (1990) and Jones (1995), we assume that new varieties are developed by labor input (scientists). The production technology is given as:
(1 + ) _At= ~&tLA;t; 0; (14) where LA;t is the labor input used in the R&D sector, ~&t is the productivity of R&D which the innovators take as given. The meaning of the parameter will be explained later.
We follow Jones (1995) to specify that the productivity takes the following function form:
~&t = &LA;t1At; & > 0; 1 > 0; 1 > > 0; (15) where & is a constant productivity parameter. In addition to &, eqs (14) and (15) contain three parameters , and . These parameters capture salient features of R&D proposed by Jones and Williams (1998). We then discuss each of them.
First, the parameter 1 > 0 re‡ects a (negative) duplication externality or a congestion e¤ect of R&D. It implies that the social marginal product of research labor can be less than the private marginal product. This may happen because of, for example, a patent race, or if two researchers accidentally work out a similar
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idea. Jones and Williams (1998) coin this negative duplication externality as the stepping on toes e¤ect. Notice that this e¤ect is stronger with a smaller , and it vanishes when = 1.
Second, the parameter 1 > > 0 re‡ects a (positive) knowledge spillover e¤ect due to the fact that richer existing ideas are helpful to the development of new ideas. A higher means that the spillover e¤ect is greater. In his pioneering article, Romer (1990) speci…es = 1; however, Jones (1995) argues that = 1 exhibits a scale e¤ect which is inconsistent with the empirical evidence. We thus follow Jones (1995) to assume that < 1 to escape from the scale e¤ect. The knowledge spillover e¤ect is dubbed by Jones and Williams (1998) as the standing on shoulders e¤ect.
Finally, the parameter 0measures the size of innovation clusters, which is associated with the concept of creative destruction pointed out by Grossman and Helpman (1991) and Aghion and Howitt (1992). The basic idea is that innovations must come together in clusters, some of which are new, while others simply build on old fashions. More speci…cally, suppose that an innovation cluster, which contains (1 + ) varieties, has been invented. Out of these (1 + ) varieties, only one unit of variety is entirely new and thus increases the mass of the variety of intermediate goods. The remaining portion, of size , simply replaces the old versions. This portion captures the spirit of creative destruction since new versions are created with the elimination of old versions. However this part does not contribute to the increase of existing varieties. In other words, for (1 + ) intermediate goods
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underlying this result can be understood as follows. Consider that the current number of varieties is At. Now an innovation cluster with size (1 + ) is developed.
This increases the mass of varieties to At+1; at the same time it also replaces old-version varieties by units. Subsequently, the …nal-good …rm faces two choices.
It can either adopt the new innovation cluster and then use At+1 intermediate goods priced at a markup, or part with the new innovation cluster and still use At intermediate goods to produce. If the …nal-good …rm chooses the latter, since now varieties have been displaced, the …nal-good …rm needs only to purchase At units of intermediate goods at a markup price, and purchase units of displaced intermediate goods at a lower (competitive) price. When the size of innovation cluster is high (a larger value of ), the …nal-good …rm tends not to adopt the new innovation cluster because sticking to old clusters is cheaper. As a result, the intermediate-good …rms have to decrease the markup so as to attract the …nal-good …rm to adopt the new innovation cluster. This adoption constraint explains why an increase in the size of innovation clusters reduces the markup.
Jones and Williams (2000) show that the constrained markup is negatively related to both the size of innovation clusters and the elasticity of substitution be-tween capital goods. Speci…cally, they demonstrate that the constrained markup is limited not to exceed the value [(1 + )= ]1= 1. Together with the unconstrained markup we discussed in subsection 3.2.3., the …nally realized markup is:
= min ( 1
; 1 + 1 1 1)
; (16)
which is independent of i and t. Combining eqs (10) and (16) implies that all intermediate-good …rms are symmetric. Therefore, notation i in subsection 3.2.3 can be dropped from here.
Given ~&t, the R&D sector hires LA;t to create (1 + ) varieties. Thus, the pro…t
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function is A;t = PA;t(1 + ) _At wtLA;t. By assuming free entry in the R&D sector, we can obtain:
PA;t = st 1 st
(1 )Yt
(1 + ) _At; (17)
where st LA;t=Ltis the ratio of research labor to total labor supply Lt. Moreover, the no-arbitrage condition for the value of a variety is:
rtPA;t= x;t+ _PA;t A_t
AtPA;t: (18a)
Without creative destruction ( = 0), the familiar no-arbitrage condition reports that, for each variety, the return of the equity shares rtPA;t will be equal to the sum of the ‡ow of the monopolistic pro…t x;t plus the capital gain or loss _PA;t. When creative destruction is present, existing goods are replaced. Accompanied with new varieties _At being invented, the amount of _At existing varieties will be replaced. Therefore, for each variety, the expected probability of being replaced is _At=At, which gives rise to the expected capital loss expressed by the last term in eq. (18a).