Acs' and Audretsch's findings for birthrates of large and small US firms .... Acs, Zoltan J.and David B. Audretsch, 1988, `Innovation in Large and Small Firms'.
Multi-Plant Firms, Entry Barriers and Innovation Maria Brouwer University of Amsterdam Department of Economics Abstract: Industries that are characterized by firms operating many plants were found to spend more on intangibles, such as R&D and advertising than industries that operated only a few plants. The paper provides a theoretical explanation for the positive relationship between the multi plant phenomenon and intangible investment intensity. Intangible investments are sunk costs and constitute a barrier to entry. Empirical research found that economies of scale due to large optimal plant size pose less of a hurdle for small entrant firms than intangible investments in advertising and R&D. Another stylized fact involves that R&D intensity does not differ for firms of various sizes. Consequently, small entrant firms can spend less on R&D than their larger counterparts in proportion to their relative size. Entry is, however, an essential element of the innovation process. Start-ups have introduced many innovations. The question, therefore, arises how small firms can compete with large ones in R&D intensive industries? A small firm can only compete in innovation, if it succeeds in conquering the disadvantages inherent to its small relative size. Small firms can either be more efficient in R&D or be expected to grow at a more rapid rate than their large rivals. Gibrat’s law, however, predicts that a firm’s expected growth is independent of its size. Several explanations have been put forward to explain the existence and success of small entrant firms, ranging from pure luck to superior R&D efficiency. The paper argues that entry investment should be looked at from the perspective of the collectivity of entrants and not through the lenses of one firm. Entry investment share was found to vary among industries and countries. The paper argues that Gibrat’s law will hold for the collectivity of entrants in an industry or country in the absence of shocks that disturb equilibrium. Opening up a country to foreign investment, new technologies that favor small firms and changes in the regulatory regime are examples of such shocks. The paper compares the results drawn from the multi-plant model with theoretical and empirical studies on R&D intensity and entry investment shares.
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Key Words: Entrepreneurship, Economies of Scale, Competition and Firm Size, Innovation
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Multi Plant Firms, Entry Barriers and Innovation 1. Introduction
The questions involving the relative innovative performance of large and small, new and incumbent firms have become classical in economics. The questions date back to Schumpeter’s contention that large firms prevail in innovation, if R&D spending is involved (Schumpeter 1974). This position differs from his earlier model of competitive capitalism, in which entrant entrepreneurs feature predominantly as innovators (Schumpeter, 1934). Both models seem to miss the mark, since both large and small, new and old firms generate innovations. The unpredictability of successful investment requires that no single category of firms is expected to prevail in innovation. Otherwise, investors would flock to one category of firms and eschew others. Arguably, investment projects in all categories of firms should attract funding in order to incite vigorous competition for innovation. The shares of investments in firms of different size classes can vary across industries and countries. Stylized facts on R&D intensity and firm size indicate that R&D intensities do not differ for firms of different size within an industry. Another stylized fact involves the small size of newly established firms (greenfield entry). As a consequence, small entrant firms face a competitive disadvantage vis á vis their larger rivals, as they can spend less on R&D. Small firm entry is particularly impeded in highly concentrated industries. These findings seem to corroborate Schumpeter’s contention that R&D spending prohibits small firm innovation. However, small firms are vital to innovation in many industries. Small firms, therefore, need to overcome the disadvantages inherent to their small relative size.
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The literature on R&D intensity, firm size and entry size is discussed and a multi-plant model is developed to explain the constancy of R&D intensities for firms of various sizes within an industry. The model is expanded to trace the effect of entry on R&D intensity. Empirical evidence on entry shares for countries and industries is presented in the next section. The question how small entrant firms can overcome the innate competitive disadvantages due to their relative small size is subsequently discussed. A clue can be found in studies, which indicate that small firms produce more innovations per employee than large firms. Their higher efficiency in R&D thus counteracts the disadva ntage of small firm size. Another finding involves that young and small firms achieve higher growth rates than large incumbent firms, violating Gibrat’s law of proportionate growth. The question is raised, whether small firms as a group need to be better to match their large rivals, or whether this only applies to individual firms?
2. A multi-plant approach to R&D Intensity, Firm Size and Market Structure Empirical Findings Research on R&D intensity, firm size and concentration has led to several `stylized facts’ (Cohen & Klepper, 1996). The first `stylized fact’ involves that large firms are more likely to conduct R&D (Bound et al, 1984; Cohen et al, 1987). The second is that within industries; R&D spending among R&D performers rises monotonically with firm size (Scherer, 1990, 1991, 645-5, Link et al, 1988). The relationship is closer for industries with high R&D intensity, as the coefficient of variation declines as R&D intensity increases (Scherer, 1965, Grabowski & Baxter, 1973). A third `stylized fact’ is that small
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firms spend less on R&D per patent and innovation than large firms (Bound et al, 1984, Acs & Audretsch, 1988). Another `stylized fact’ involves that industry R&D intensity is positively related to industry concentration (Horowitz, 1962, Mansfield, 1968). These `stylized’ facts indicate that the competitive disadvantage of small firms can be measured by their relative size. These findings are interesting for an analysis of entry barriers, since entrant firms are usually of small size. Newly created firms (greenfield / entrepreneurial entrants) were one fifth of average firm size in Canadian manufacturing (Baldwin, 1995, 16). The size differential is larger, if we compare average sizes of acquisition and entrepreneurial entry. Canadian acquisitions were almost 13 times larger than their entrepreneurial counterparts, indicating that acquisition entry occurs in the larger size classes (Baldwin, 1995, 42). Small relative entrant firm size can be due to either small plant size or to differences in the number of plants operated by entrepreneurial and incumbent firms. Almost all `greenfield’ entry (98,6%) occurred by one-plant firms (Baldwin, 1995, 42), indicating that differences in the number of plants operated by entrepreneurial and incumbent firms are an important explanation for relative size differences. Small relative plant size was found to have a negative impact on productivity, which is compensated by the lower wages small plants pay their employees (Baldwin et al, 2004). The competitive disadvantage of firms of relative small size can be due either to smaller price-cost margins or to lower absolute profits. Price cost margins can be equal for firms of different sizes, if small firms operate plants of efficient scale. The main competitive disadvantage of small firms would then derive from their lower absolute profits due to operating fewer plants.
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A Multi-Plant Model We assume that firms operate in a market with a demand curve that can be depicted by P = a- bQ. Price is identical for all firms within the industry. Cost functions for each firm have the following form C(qi) = cqi +Fi + Ai + R&Di. Gross profits of each firm are defined as price minus marginal costs times output (P-ci).qi. Gross profits can be used to pay for tangible and intangible investments and for remittances to stakeholders. Tangible investment in buildings and equipment (F) can constitute a sunk cost or be retrievable on second hand markets. Intangible investments in R&D and advertising (A) are considered sunk, if they are non-retrievable in disembodied form. The annual amount of tangible fixed costs Fi is determined by depreciation allowances and interest rates. Firms are engaged in oligopoly competition. Firms in industries, which are characterized 2
by quantity setting (Cournot) competition, will have gross profits of bqi (appendix 1). Small firms have smaller profits than large firms, which derive from both smaller price cost- margins and smaller size. Price cost margins (P-ci) equal bqi in Cournot competition. More efficient firms thus have both higher price cost margins and higher output than their less efficient counterparts in Cournot competition. Market shares would equal 1/n, if marginal costs ci were identical for all n firms in the industry. Cournot (quantity setting) competition allows less efficient firms to survive. This does not apply to Bertrand (price setting) competition, where less efficient firms risk to be wiped from the industry by undercutting.
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All firms in an industry can survive, if gross profits equal or exceed their fixed costs Fi. Fixed costs per plant are assumed identical for all plants of efficient size (qef). Efficient plant size for one-plant firms in Cournot competition equals:
q ef = qi π ef = bqef2 = F → qef =
F b
Fixed costs for a multi-plant firm are xiF. The share of fixed costs in gross profits in the case of a multi-plant industry (x > 1), where x indicates the number of plants per firm (of equal size) can be calculated in the following way: Multi-Plant Firm Industry’s fixed costs’ share of profits:
1 q x i 1 F 1 qi = → 2 bqi2 = F x b x 1 π F = 2 bqi2 = 2i x x xF 1 → = πi x q ef =
(1)
We can conclude from (1) that firms in a two-plant industry will spend half of their gross profits on fixed costs. The other half can be spent either on intangible investments or be distributed among stakeholders. Firms in one-plant industries -on the other hand- cannot spend money on either R&D or advertising in long run equilibrium.
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The price-cost margin amounts to
1 in an industry composed of n equal sized firms ne
operating under Cournot conjectures. Tangible cost intensity in equilibrium can be calculated as follows: Tangible investment Intensity:
x.F x.F π 1 1 1 = . = . = PQ π PQ x ne x.n.e
(2)
More can thus be spent on intangibles, if the number of plants per firm increases: Upper bound of Intangible Investment Intensity:
R&D + A 1 1 x −1 = − = PQ ne x n e x n e
(3)
Multi plant industries can spend on intangibles because they have higher price-cost margins than one-plant industries. Higher price–cost margins occur, when multi-plant industries have either fewer firms and/or lower marginal costs than one-plant industries.
Firms of Unequal Sizes within an industry It was mentioned above that firms are of equal size, if they have equal marginal costs. Size differences are due to (marginal) cost differences in a Cournot model. We could argue that the large number of small firms that are not engaged in R&D can be explained by their smaller price cost margins due to inefficient plant size. But, firms of small relative size whose marginal costs equal those of their large rivals would have identical cost margins. A firm’s relative small size, which is not due to cost inefficiencies, could be a result of history. Smaller firms might be newly founded, whereas larger firms might be of older date. The close relationship found between R&D outlays (Cohen & Klepper,
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1992, 774) and firm size within industries could be explained, if firms that conduct R&D are not handicapped by low price-cost margins. It is demonstrated below that firms of different sizes within an industry will have identical R&D intensities, if all firms within an industry operate plants of identical size and have equal marginal costs. This implies that price cost margins P-ci are also identical for all firms in the industry, and equal: bq = bxqef (see appendix 2). Differences in gross profits across firms of various sizes can then be completely explained by differences in the number of plants they operate: spend on tangible investment
2 Fi = xbq i ef
π i = b( xqef ).xi .qef . Each firm will
and the share of tangible investment equals:
2 xi bqeff Fi 1 = = π i x .xi bqeff2 x
(4)
Hence, a one-plant firm in an industry, in which the average number of firms equals 2 can spend half of its gross profits on intangibles, if it is of efficient size. This model can thus explain the stylized empirical fact that small firms conducting R&D are as R&D intensive as large firms within an industry. One- plant firms are thus not disadvantaged with respect to R&D intensity, if they are cost efficient. But, the absolute amount small firms can spend on R&D is proportionate to the number of plants they operate. An industry that is composed of 10 firms; one with 11 plants and 9 with 1 plant will have an industry average of 2 plants per firm. Each firm can spend half of its gross profits on intangibles, but the larger firm can spend 11 times as much as each small firm.
3. Multi-Plant Firm Industries and Entry Barriers
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Entry barriers can – in accordance with Bain- be defined as the unit cost disadvantages of small-scale entrants (Bain, 1956). Bain distinguished three entry barriers; economies of scale, product differentiation and financial cost disadvantages. The latter stem from capital markets, which do not supply finance to entrants and incumbents on equal terms. Plant specific economies of scale are present, if plant and equipment are of a lumpy nature (F is large). Product differentiation constitutes an entry barrier, if advertising expenditures are lumpy. Nationwide advertising fits this description. Firms that want to build and sustain brand images through nationwide advertising need to spend large sums to achieve this. R&D spending also constitutes a barrier to entry, identical to advertising, if large amounts need to be spent on R&D to obtain an innovation. Relative small entry size will give entrants a cost disadvantage vis á vis incumbents in the realms of advertising and R&D that is proportional to the size differential. But, Stigler (and with him what has become known as the Chicago School in Economics) disagree with Bain on the issue of entry barriers. Large incumbent size does not pose an entry barrier, in their view, since capital markets will finance entrants to enter at a large scale. The relationships between relative incumbent and entry size and the way capital markets function are thus central to the debate between Bainians and Chicagoans. Each `greenfield’ entrant adds to industry capacity and large entrants add more than small ones. Incumbents could react to large-scale entry by shrinking output, so that a new Cournot equilibrium of equal-sized firms might ensue. However, the fact that `greenfield’ entry occurs at a small relative size, as was indicated above, does not support the view that investors anticipate such accommodating behavior.
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We assume that all incumbent firms in an industry operate either an equal number of plants (x) or an average number of efficient plants
x ≥ 2 , whereas an entrant operates
only one plant of efficient size. An entrant will thus supply
output under these assumptions: q e =
1 of average incumbent x
qi . An entrant will thus only be of half (average) x
incumbent size in a two-plant industry and one third of incumbent size in a 3-plant industry. Price and therewith price cost margins drop by bqe upon each entry of efficient size. Such entry would exactly cut incumbent profits in half in a two plant industry, as is demonstrated below:
qe = qef =
qi x
( P − c) afterentry = b(qi − qe ) = b( x − 1)qef =
x −1 bqi x
(5)
π i / afterentry = b ( x − 1)qef 2 x ; π e = b ( x − 1)qef2 An entrant would reduce intangible investment intensity to zero, if it entered an industry, where the (average) number of plants equaled 2. Tangible Investments’ share of profits:
xi F π i x 1 = . = π iae x ( x − 1).π i x − 1
(6)
Upper bound of Intangible investments’ share of profits:
Int xF x − 2 = 1− = π iae π iae x − 1
(7)
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Both incumbents and entrants could thus spend half of their profits on intangibles, if the (average) number of incumbent plants amounted to 3. A second entrant of efficient size would, however, reduce intangible investment intensity to zero in a 3-plant industry. Small entrant firms can thus spend less on R&D than their large rivals, but also have a depressing effect on R&D intensity for all firms in the industry. We could argue that entry barriers are required to keep incentives to innovate intact. Easy entry would reduce price-cost margins and therewith both the funds and the incentives for conducting R&D. Gross profits would be totally absorbed by tangible fixed costs, if x-1 plants of efficient size would enter the industry, while all incumbent firms maintained output. However, the absence of entry would also not stimulate innovation. Leading, large firms could become complacent, if newcomers do not challenge them.
4. Empirical Evidence on Entry Shares
National Differences It was noted above that entrepreneurial or `greenfield’ entrants are usually of a small size. Klapper et al found that more than 90 percent of Western European entry employed less than 50 employees (Klapper et al, 2004. A main reason for relative small entry size given by Klapper et al. are the financial limitations posed on entrants. Start-ups cannot finance their investments out of cash flows, but need to finance through personal savings or attract external capital. The Chicagoan question; why entrants cannot raise enough capital to compete with incumbents on equal terms could thus be answered by pointing at capital market imperfections. A new firm that is as good as an incumbent should have equal
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access to capital. The share of entry investment in a country would then depend on the relative attractiveness of both categories of investments. Entry investment will be stimulated, if bureaucratic impediments to entry are low and when external capital is widely available. Entry investment is relatively unattractive, if bureaucratic entry costs are high, as is the case in Italy. It was also found that entry investment is higher in countries with more developed financial markets (Klapper et al, 2004). Large entry investment shares are not due to larger chunks of entry investment, but to the launching of more ventures in entry intensive countries (Klapper et al, 2004). We could argue that entry investments constitute a distinct investment category. Entrants lack a track record and are usually not diversified. This feature distinguishes entrants from incumbents, which all have histories of some sort and have spread their risks through diversification. Small relative entry size can be explained by the great uncertainty that surrounds entry investments. It will not be clear at the outset, whether an entrant firm constitutes a good investment opportunity or not. It seems plausible that external investors invest little capital at the start of an entrepreneurial venture and revise their investments periodically, as more information becomes available. This depicts the way the venture capital industry operates; venture capitalists invest in entrants in staged commitments (of roughly one year), which are contingent on expected outcomes. Capital is thus periodically re-allocated among entrant cohorts; some stop getting funding, whereas others expand rapidly. As a consequence, many entrants do not survive, whereas others thrive. Half of Canadian entrants had exited within 10 years, indicating that entrepreneurial entry investment is subject to great uncertainty. Half of all acquisition
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entrants had also exited within a 10- year period, but acquisition exit is not identical to failure (Baldwin, 1995, 55-6). Large differences exist between entry shares of employment across nations. Some transition economies, such as the Baltic countries have large entry shares. We would expect a country’s entry share in innovative industries to be positively related to the degree of development of its capital markets. The finding that developed (external) capital markets especially promotes entry R&D investments supports this hypothesis (Klapper et al, 2004). Differences in regulation and access to finance between countries can thus explain differences in entry investment shares. R&D intensive entry is also positively related to the protection of intellectual property rights (Klapper et al., 2004).
Industry Differences It is also useful to distinguish between `greenfield’ and acquisition entry in interpreting empirical research results on industry determinants of entry investment share. Acquisition entry does not add to industry capacity, whereas `greenfield’ entry does. Acquiring firms might be attracted to highly concentrated industries due to high price-cost margins, whereas `greenfield’ entrants could avoid them due to high entry barriers. Baldwin’s research on Canadian manufacturing firms points out that the effects of the concentration variable for these two groups are –indeed- diametrically opposed. Greenfield entry is negatively related to concentration, while acquisition entry is positively related to it (Baldwin, 1995, 50-51). Higher concentration hardly affected entry size, but exerted its influence mainly by depressing the number of entrants (Baldwin, 371). Advertising intensity severely hampered `greenfield’ entry share (Baldwin, 1995, 371-2). Economies
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of scale also impeded entry, but were drowned out by the concentration variable (Baldwin, 1995, 375). R&D intensity was positively related to `greenfield’ entry (Baldwin, 1995, 372 & 396). R&D intensity constitutes thus less of an entry impediment than advertising intensity. This could indicate that large firms’ comparative advantage with respect to R&D spending can more easily be overcome than their advantage in the realm of advertising. The nature of the industries involved could explain this difference. Advertising intensive industries are usually consumer goods industries, in contrast to R&D intensive industries. `High tech’ industries such as communications and computer services had relative high entry shares in both Europe and the US (Klapper, 2004). Acs’ and Audretsch’s findings for birthrates of large and small US firms corroborate Baldwin’s and Klapper’s results. Birth rates are (similarly to Baldwin and Klapper) defined as the share of new firms in industry employment. Acs and Audretsch conclude that concentration inhibits small and medium sized entry, but stimulates large firm births (> 500 employees). They also found that birth rates are negative ly related to advertising intensity. Capital intensity, by contrast had no significant effect on birth rates (Acs and Audretsch, 1989). Advertising intensity thus poses a greater barrier to entry than economies of scale and advertising intensity.
5. Overcoming the Disadvantages of Small Relative Size
Efficiency in R&D and Growth Rates We have assumed this far that a firm’s R&D spending is determined by its actual profits. It was demonstrated above how a multi-plant model can explain the constancy of price
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cost margins of firms of different size under this condition. We could, however, object to the above construct that a firm’s R&D spending is not determined by present, but by expected profits. This is a valid objection, which applies with the greatest force to entrant firms, which lack present profits at their start. A firm’s R&D investment is not determined by its present profits, if it can obtain external capital to fund its projects. External investors will be guided by their expectations of future profits in making their decisions. Few investors will invest in entrants, if average entry chance of success is considered small compared to incumbents. The opposite occurs, when the average chance of entry success is considered to be relatively high. High entry rates in transition economies can be explained by the absence of entry investment before transition took off and the opening up of these countries to foreign investment. It was pointed out above that entry investment occurs in small chunks, which put s entrants at a comparative disadvantage vis a vis their large rivals in an industry. Entrants of small relative size can overcome their innate disadvantage in two ways. They can either be more efficient in R&D (spend less R&D per innovation), or they can dissipate their disadvantage by growing more rapidly than incumbents. Cohen & Klepper depict a model, in which chances to succeed in innovation are independent of size. Small firms have equal chances to succeed at innovation, but will receive less from innovation than large firms, since innovation revenues are assumed to be proportional to initial size (Cohen & Klepper, 1992, 781). This would imply that small firms could spend less on an innovation in proportion to their size and need to be
q s / ql as efficient in R&D as a large rival. Their model could explain the higher efficiency in R&D found in small firms. Small firms were found to produce 1,43 times as
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many innovations per employee as large firms (Acs & Audretsch, 1988). The efficiency difference was much larger for a sub sample of highly innovative industries, where small firms produced 6,64 more innovations per employee than large firms. The finding that small firms are more efficient in innovation has been explained in several ways. R&D spending would be subject to declining returns to scale, or large firms might undertake more risky innovations (Cohen & Klepper, 1996). However, decreasing returns to scale would put large firms at a comparative disadvantage compared to their smaller counterparts; and would increase entry share. Innovations of large firms could be subject to greater uncertainty, if their innovations would be more radical. But, the SBA, which tested innovations of large and small firms, found no quality differences between the two groups (Acs & Audretsch, 1988).
Firm Growth and Gibrat’s Law The relative efficiency requirement ( q s / ql ) can be eased, if small firms are expected to grow more rapidly than large firms. A certain cohort of small entrant firms will receive more investment and increase its employment share, if its profit potential is more positively revised during its lifetime than the prospects of incumbents. There is some theoretical justification for the idea that small innovative firms grow at a more rapid rate than their large counterparts. A certain marginal cost decrease e caused by innovation would cause an innovative firm to grow more rapidly, if its relative size is smaller and the number of firms in the industry (n) is larger in Cournot competition (see appendix 3). These results apply with even greater force to industries where Bertrand competition
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prevails. The innovative firm –invariant of its size- could then undercut all noninnovative incumbents and establish a monopoly. Expectations on entry growth could apply both to a whole cohort of entry firms or to individual entrants. Entry investment is subject to greater uncertainty than incumbent investment. That is why entry investment occurs in small chunks. Re-allocation of investment causes many entrant firms to exit rapidly, while some grow to large size. We might expect that low chances of entry survival need to be compensated by higher entry growth rates of surviving firms to prompt investment. A negative relationship between entry growth and entry survival was indeed found (Audretsch, 1995, 83), which supports our intuition that lower survival rates require higher expected rewards for the survivors in order to sustain entry investment. Small entrant firms cannot overcome their size disadvantage by growing more rapidly than incumbents, if Gibrat’s law of proportionate effect (1931) would hold. This law contends that a firm’s growth rate is independent of its initial size (Mansfield, 1962). But, Gibrat’s law could still hold for entry cohorts, if higher growth rates of survivors compensated for higher exit rates of small entrant firms. Testing Gibrat’s law then depends crucially on the population studied. Small firms would show higher growth rates, if only surviving firms are taken onto account (Sutton, 1997, 44), since small entrant firms have a high hazard rate. Countries and industries that are not subject to Gibrat’s law of proportionate effect would either increase or decrease their share of entry investment. Entry investment share would remain stable in countries, in which Gibrat’s law is upheld. Gibrats’s law was not upheld in a number of cases even if this statistical caveat was taken into account. Bronwyn Hall (1987) found a negative relationship between size and
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growth for two samples of incumbent US manufacturing firms receiving external capital. Evans (1987) found a negative relationship between both growth and age for a sample of US manufacturing firms. Baldwin’s research also points out that cohorts of surviving entrants in Canadian manufacturing grew at a rate of 8 percent per year and increased their employment share from 17 till 33 percent in the 1970s (Baldwin, 1995, ch 5). There are thus indications that cohorts of young firms grew more rapidly than older cohorts; indicating that investment was re-allocated from old to young firms. We could argue that the 1970s constituted a transition period, in which entry finance became more amply available in both the US and Canada. We could also argue that large firms had become complacent and young firms were more vigorous. We could argue that the average entrant firm does not need to be more efficient at innovation than the average incumbent, if capital markets are well developed. The average entrant’s expected rate of return would then equal that of the average incumbent.
6. Conclusions The relationships between entry, innovation and firm size are analyzed within a multiplant framework. (Greenfield) entrants usually arrive at a small size and either survive and prosper or exit quickly. Entrant firms are usually one-plant firms, whereas incumbents operate several plants. Investment in new ventures is assumed to be subject to greater uncertainty than investments in established firms. A multi-plant model can explain the absence of a relationship between firm size and R&D intensity. Entry barriers such as concentration and advertising intensity can explain differences in entry shares across industries. Expansion of a country’s entry share can be explained by changes in
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the relative attractiveness of entry investment due to a lessening of bureaucracy and the development of capital markets.
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Cohen, W. M., R. Levin R and D. Mowery, 1987. `Firm Size and R&D Intensity; a Reexamination’, Journal of Industrial Economics, vol. 35, pp. 543-65.
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Cohen , Wesley M., Steven Klepper, 1996. `A Reprise of Size and R&D’, The Economic Journal, vol. 106, no 437, 925-951.
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Mansfield, Edwin, 1968. `Industrial Research and Technological Innovation; an Econometric Analysis’. New York: Norton.
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Appendix 1 Price-Cost Margins in Cournot Competition P = a –bQ; Cqi = ciqi + Fi + Ai ;
MRi = a − 2bqi − bQ− i MRi = ci a − ci − bqi − bQ = 0 P − ci − bqi = 0 P − ci = bqi 22
Appendix 2 Price cost margins for firms with an unequal number of efficient plants (c i = c j = c)
P − ci = bqi n
b∑ qi = nP − nc 1
P−c =
bQ = bq n
Appendix 3 Output expansion of a an innovative firm in Cournot competition Innovation reduces ci of innovative firm i by e. The addition to output of firm i is negatively related to industry concentration and to a firm’s relative size.
qi1 = a − ci1 − bQ1 qi 2 = a − ci1 + ε − bQ2 ∆ qi = ε − ∆Q n
Q1 =
na − ∑ ci 1
n +1 n
Q2 =
na − ∑ ci + ε 1
n +1
ε n +1 ε n ∆ qi = ε − = ε n +1 n +1 ∆Q =
A firm’s size is indicated by its number of efficient plants xi qef
23
qi = siQ =
xi Q nx
∆qi nε nx n2 ε x = . = . qi n + 1 xiQ ( n + 1)Q xi
24
