命題 11 の証明

証明. x^N_i > x^∗_i を示す. x^∗_i は

x^∗_i = 2(2−β)(a−¯c)θ

2β²κ+ [9κ−2(2−β)(1 +β)]θ と書ける. 右辺の分子と分母とをそれぞれθで割り

x^∗_i = 2(2−β)(a−c)¯

2β²κ/θ+ [9κ−2(2−β)(1 +β)]

を得る. x^N_i は

x^N_i = 2(2−β)(a−¯c) 9κ−2(2−β)(1 +β)

であり2β²κ/θ >0である. よって, x^N_i の分母はx^∗_i の分母よりも小さな値となる. したがっ て, x^N_i > x^∗_i となり命題11は成立する.

関連図書

Amir, R., J.Y. Jin, and M. Troege (2008) “On additive spillovers and returns to scale in R&D,”International Journal of Industrial Organization, Vol. 26, No. 3, pp. 695–703.

Amir, R. and J. Wooders (1998) “Cooperation vs. Competition in R&D: the Role of Sta-bility of Equilibrium,”Journal of Economics, Vol. 67, No. 1, pp. 63–73.

Amir, Rabah, Igor Evstigneev, and John Wooders (2003) “Noncooperative versus cooper-ative R&D with endogenous spillover rates,” Games and Economic Behavior, Vol. 42, No. 2, pp. 183–207.

Amir, Rabah, Christine Halmenschlager, and Jim Jin (2011) “R&D-induced industry po-larization and shake-outs,”International Journal of Industrial Organization, Vol. 29, No.

4, pp. 386–398.

Amir, Rabah and John Wooders (2000) “One-way spillovers, endogenous innova-tor/imitator roles, and research joint ventures,” Games and Economic Behavior, Vol.

31, No. 1, pp. 1–25.

Anderson, R. (2001) “Why Information Security is Hard – An Economic Perspective,” An Economic Perspective. London.

Anderson, R. and T. Moore (2006) “The economics of information security,” Science, Vol.

314, No. 5799, p. 610.

Anton, James J and Dennis A Yao (2003) “Patents, invalidity, and the strategic transmis-sion of enabling information,” Journal of Economics & Management Strategy, Vol. 12, No. 2, pp. 151–178.

Arrow, K. (1962) “Economic welfare and the allocation of resources for invention,” 1962, pp. 609–626.

Atallah, Gamal (2005) “R&D cooperation with asymmetric spillovers,” Canadian Journal of Economics/Revue canadienne d’´economique, Vol. 38, No. 3, pp. 919–936.

Brander, James A and Barbara J Spencer (1983) “Strategic commitment with R&D: the symmetric case,” The Bell Journal of Economics, pp. 225–235.

Bulow, Jeremy I, John D Geanakoplos, and Paul D Klemperer (1985) “Multimarket oligopoly: Strategic substitutes and complements,” The Journal of Political Economy, pp. 488–511.

Dasgupta, Partha and Joseph Stiglitz (1980) “Industrial structure and the nature of inno-vative activity,” The Economic Journal, pp. 266–293.

d’Aspremont, C. and A. Jacquemin (1988) “Cooperative and noncooperative R & D in duopoly with spillovers,” The American Economic Review, Vol. 78, No. 5, pp. 1133–

1137.

De Bondt, Raymond (1997) “Spillovers and innovative activities,”International Journal of Industrial Organization, Vol. 15, No. 1, pp. 1–28.

Etro, Federico (2006) “Aggressive leaders,”The RAND Journal of Economics, Vol. 37, No.

1, pp. 146–154.

Friedman, James W (1988) “On the strategic importance of prices versus quantities,” The RAND Journal of Economics, pp. 607–622.

Fudenberg, Drew and Jean Tirole (1984) “The fat-cat effect, the puppy-dog ploy, and the lean and hungry look,”The American Economic Review, pp. 361–366.

Gil Molt´o, Mar´ıa Jos´e, Nikolaos Georgantzis, and Vicente Orts (2005) “Cooperative R&D with endogenous technology differentiation,”Journal of Economics & Management Strat-egy, Vol. 14, No. 2, pp. 461–476.

Gill, David (2008) “Strategic disclosure of intermediate research results,” Journal of Eco-nomics & Management Strategy, Vol. 17, No. 3, pp. 733–758.

Goodchild, J. (2009) “5 Tips for Managing Security in a Reces-sion,” CSOonline.com. ”http://www.csoonline.com/article/480175/

5-tips-for-managing-security-in-a-recession”.

Gordon, L.A. and M.P. Loeb (2002) “The economics of information security investment,”

ACM transactions on information and system security, Vol. 5, No. 4, pp. 438–457.

(2006)Managing Cybersecurity Resources: A Cost-Benefit Analysis: McGraw-Hill.

Griliches, Z. (1992) “The Search for R&D Spillovers,”Scandinavian Journal of Economics, Vol. 94, pp. 29–47.

Henriques, I. (1990) “Cooperative and noncooperative R&D in duopoly with spillovers:

Comment,” The American Economic Review, Vol. 80, No. 3, pp. 638–640.

Ijichi, T., T. Iwasa, H. Odagiri, H. Keira, T. Koga, A. Goto, Y. Tawara, A. Nagata, and Y. Hirano (2004) “Statistics on Innovation in Japan Report on the Japanese National Innovation Survey 2003 (J-NIS 2003),”Statistics, p. 440P.

Kamien, M.I., E. Muller, and I. Zang (1992) “Research joint ventures and R&D cartels,”

The American Economic Review, Vol. 82, No. 5, pp. 1293–1306.

Kamien, M.I. and I. Zang (2000) “Meet me halfway: research joint ventures and absorptive capacity,”International Journal of Industrial Organization, Vol. 18, No. 7, pp. 995–1012.

Kawashige, R. (2011) “Strategic investment in information security of firms: R&D spillovers and information management.” Tokyo Metropolitan University, Graduate School of So-cial Sciences, Research Paper Series, Number 92.

Kultti, K. and T. Takalo (1998) “R&D spillovers and information exchange,” Economics letters, Vol. 61, No. 1, pp. 121–123.

Lambertini, Luca, Francesca Lotti, and Enrico Santarelli (2004) “Infra-industry spillovers and R&D cooperation: theory and evidence,” Economics of Innovation and New Tech-nology, Vol. 13, No. 4, pp. 311–328.

Lee, Kong Rae (1996) “The role of user firms in the innovation of machine tools: The Japanese case,” Research Policy, Vol. 25, No. 4, pp. 491–507.

Mansfield, Edwin (1985) “How rapidly does new industrial technology leak out?” The journal of industrial economics, pp. 217–223.

Mansfield, Edwin, Mark Schwartz, and Samuel Wagner (1981) “Imitation costs and patents:

an empirical study,” The Economic Journal, pp. 907–918.

Milliou, Chrysovalantou (2009) “Endogenous protection of R&D investments,” Canadian Journal of Economics/Revue canadienne d’´economique, Vol. 42, No. 1, pp. 184–205.

Noda, T. (2009) “IBM exec warns against budget cuts on IT security,” NETWORK-WORLD. ”http://www.networkworld.com/news/2009/020309-ibm-exec-security.

html”.

Novshek, William (1993) Mathematics for economists: Academic press.

Piga, C. and J. Poyago-Theotoky (2005) “Endogenous R&D spillovers and locational choice,”Regional Science and Urban Economics, Vol. 35, No. 2, pp. 127–139.

Poyago-Theotoky, J. (1999) “A note on endogenous spillovers in a non-tournament R & D duopoly,” Review of Industrial Organization, Vol. 15, No. 3, pp. 253–262.

Qiu, L.D. (1997) “On the dynamic efficiency of Bertrand and Cournot equilibria,” Journal of Economic Theory, Vol. 75, No. 1, pp. 213–229.

Riggs, William and Eric Von Hippel (1994) “Incentives to innovate and the sources of innovation: the case of scientific instruments,”Research policy, Vol. 23, No. 4, pp. 459–

469.

Shirey, Robert (2013) 「インターネットセキュリティ小事典 (Internet Security Glos-sary)」．[訳]宮川 寧夫, 立石 尋太郎, IPA, ”http://www.ipa.go.jp/security/rfc/

RFC2828-00JA.html”．

Shy, O. (1996) Industrial organization: theory and applications: The MIT press.

Simon and Blume (1994) Mathematics for Economists: W.W.Norton & Company.

Spence, Michael (1984) “Cost reduction, competition, and industry performance,” Econo-metrica: Journal of the Econometric Society, pp. 101–121.

Steurs, Geert (1995) “Inter-industry R&D spillovers: what difference do they make?” In-ternational Journal of Industrial Organization, Vol. 13, No. 2, pp. 249–276.

Suzumura, K. (1992) “Cooperative and Noncooperative R&D in an Oligopoly with Spillovers,” The American Economic Review, Vol. 82, No. 5, pp. 1307–1320.

Suzumura, Kotaro and Noriyuki Yanagawa (1993) “Cooperative and Non-Cooperative R&D in an Oligopoly with Spillovers: Strategic Substitutes versus Strategic Comple-ments,” Hitotsubashi Journal of Economics, Vol. 34, No. 1, pp. 1–11.

Tirole, J. (1988) The theory of industrial organization: MIT Press.

Vives, X. (2001) Oligopoly pricing: old ideas and new tools, pp. 51–58: The MIT press.

伊丹敬之・加護野忠男(1997) 『ゼミナール 経営学入門 第2版』，有斐閣．

奥野正寛 (2008)『ミクロ経済学』，東京大学出版会．

岡田章 (1996)『ゲーム理論』，有斐閣．

河重隆一郎 (2013) 「非負制約を考慮した研究開発の協力と競争」，『経営と制度』，第11 巻，19–34頁．