本研究以製藥產業為應用背景，探討競爭進入、產品生命週期、技術創新如何影響企業的最適投資時點與等待選擇權價值。在連續時間實質選擇權架構下，本文延伸 Cheng et al. (2015) 的單一企業架構，引入對稱雙占競爭模型，涵蓋獨占基準及雙占市場下的三種投資情境：雙方同時等待、領導者與跟隨者先後進入，以及雙方同時進入。本文運用動態規劃推導等待投資區間的價值方程，據以將企業的投資時點表述為最適停止問題，並進行數值求解與比較靜態分析。
研究結果顯示：波動度與成長率提高、產品生命週期延長、技術創新到達頻率提高、企業定價能力增強，以及折現率與研發支出占比降低，均會使企業提高最適投資門檻，反映其等待投資傾向增強。在基準參數下，本研究的最適投資門檻高於 Dixit 與 Pindyck (1994) 及 Cheng et al. (2015)；而等待選擇權價值因企業在市場中不同策略而部分低於Dixit 與 Pindyck (1994) 且部分高於 Cheng et al. (2015)，顯示雙占競爭會侵蝕等待投資的價值。此外，搶先進入市場的領導企業具有較高的等待選擇權價值；跟隨企業則具有較低的等待選擇權價值，顯示先發制人優勢存在。
本研究說明競爭進入、產品生命週期、技術創新會共同改變企業的投資門檻及價值分配，並將單一企業的實質選擇權架構延伸至具有同時等待、依序進入與同時進入三種策略的競爭市場，為研發密集且產品生命週期有限之產業提供分析投資時點的理論架構。

Built on a real option game framework, the model captures competitive entry effects and provides a basis for analyzing investment timing in R&D-intensive industries with finite product life cycles. Extending Cheng et al.’s (2015) continuous-time single-firm real options framework, this study compares a monopoly benchmark with three investment scenarios in a duopoly market: simultaneous waiting by both firms, sequential entry with leader–follower roles, and simultaneous entry. Using dynamic programming, the study formulates firms’ investment-timing decisions as optimal stopping problems, derives the continuation-region value functions, numerically solves for the optimal investment thresholds, and conducts comparative statics.
Greater volatility, a higher growth rate, a longer product life cycle, a higher arrival rate of technological innovation, greater pricing power, a lower discount rate, and a lower R&D expenditure ratio raise firms’ optimal investment thresholds, thereby strengthening their incentives to delay investment. Under the baseline parameterization, the model’s optimal investment thresholds exceed those obtained from numerical implementations of the models of Dixit and Pindyck (1994) and Cheng et al. (2015). Across duopoly equilibria, however, the option value of waiting is strategy-dependent: it falls below the corresponding value implied by Dixit and Pindyck (1994) under some strategies but exceeds that implied by Cheng et al. (2015) under others. The option value of waiting is higher for the leader than for the follower, indicating a first-mover advantage.

中文部分
廖四郎、陳坤銘、鄭宗松（2003）。最適投資決策與產品生命週期：實質選擇權分析法。《中山管理評論》，11（3），571–596。
西文部分
Abadie, L. M., & Chamorro, J. M. (2014). Valuation of wind energy projects: A real options approach. Energies, 7(5), 3218–3255.
Adkins, R., & Paxson, D. (2016). Subsidies for renewable energy facilities under uncertainty. The Manchester School, 84(2), 222–250.
Alvarez, L. H. R., & Stenbacka, R. (2001). Adoption of uncertain multi-stage technology projects: A real options approach. Journal of Mathematical Economics, 35(1), 71–97.
Alvarez, L. H. R., & Stenbacka, R. (2004). Optimal risk adoption: A real options approach. Economic Theory, 23, 123–147.
Balcer, Y., & Lippman, S. A. (1984). Technological expectations and adoption of improved technology. Journal of Economic Theory, 34(2), 292–318.
Baldwin, C. Y., & Meyer, R. F. (1979). Liquidity preference under uncertainty: A model of dynamic investment in illiquid opportunities. Journal of Financial Economics, 7(4), 347–374.
Balter, A., Huisman, K., & Kort, P. M. (2023). Product life cycles and investment: A real options analysis (CentER Discussion Paper No. 2023-005). CentER, Center for Economic Research.
Bass, F. M. (1969). A new product growth for model consumer durables. Management Science, 15(5), 215–227.
Berk, J. B., Green, R. C., & Naik, V. (2004). Valuation and return dynamics of new ventures. The Review of Financial Studies, 17(1), 1–35.
Bernanke, B. S. (1983). Irreversibility, uncertainty, and cyclical investment. The Quarterly Journal of Economics, 98(1), 85–106.
Bøckman, T., Fleten, S.-E., Juliussen, E., Langhammer, H. J., & Revdal, I. (2008). Investment timing and optimal capacity choice for small hydropower projects. European Journal of Operational Research, 190(1), 255–267.
Bollen, N. P. B. (1999). Real options and product life cycles. Management Science, 45(5), 670–684.
Boomsma, T. K., Meade, N., & Fleten, S.-E. (2012). Renewable energy investments under different support schemes: A real options approach. European Journal of Operational Research, 220(1), 225–237.
Bradley, J. R., & Guerrero, H. H. (2008). Product design for life-cycle mismatch. Production and Operations Management, 17(5), 497–512.
Caballero, R. J. (1991). On the sign of the investment-uncertainty relationship. American Economic Review, 81(1), 279–288.
Carruth, A., Dickerson, A., & Henley, A. (2000). What do we know about investment under uncertainty? Journal of Economic Surveys, 14(2), 119–154.
Cheng, J. T. S., Jiang, I.-M., & Liu, Y.-H. (2015). Technological innovation, product life cycle and market power: A real options approach. International Journal of Information Technology & Decision Making, 14(1), 93–113.
Chevalier-Roignant, B., Flath, C. M., Huchzermeier, A., & Trigeorgis, L. (2011). Strategic investment under uncertainty: A synthesis. European Journal of Operational Research, 215(3), 639–650.
Chronopoulos, M., & Lumbreras, S. (2017). Optimal regime switching under risk aversion and uncertainty. European Journal of Operational Research, 256(2), 543–555.
Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1–2), 145–166.
Delaney, L., & Driouchi, T. (2025). Coordination through committees and markets with option timing games. Economic Theory, 80(4), 1177–1201.
DiMasi, J. A., Grabowski, H. G., & Vernon, J. (1995). R&D costs, innovative output and firm size in the pharmaceutical industry. International Journal of the Economics of Business, 2(2), 201–219.
Dixit, A. K. (1993). Choosing among alternative discrete investment projects under uncertainty. Economics Letters, 41(3), 265–268.
Dixit, A. K., & Pindyck, R. S. (1994). Investment under uncertainty. Princeton University Press.
Fama, E. F. (1970). Efficient capital markets: A review of theory and empirical work. The Journal of Finance, 25(2), 383–417.
Farzin, Y. H., Huisman, K. J. M., & Kort, P. M. (1998). Optimal timing of technology adoption. Journal of Economic Dynamics and Control, 22(5), 779–799.
Fleten, S.-E., Maribu, K. M., & Wangensteen, I. (2007). Optimal investment strategies in decentralized renewable power generation under uncertainty. Energy, 32(5), 803–815.
Fudenberg, D., & Tirole, J. (1985). Preemption and rent equalization in the adoption of new technology. The Review of Economic Studies, 52(3), 383–401.
Garlappi, L. (2002). Preemption risk and the valuation of R&D ventures [Working paper]. University of Texas at Austin.
Grenadier, S. R. (1996). The strategic exercise of options: Development cascades and overbuilding in real estate markets. The Journal of Finance, 51(5), 1653–1679.
Grenadier, S. R. (2002). Option exercise games: An application to the equilibrium investment strategies of firms. The Review of Financial Studies, 15(3), 691–721.
Grenadier, S. R., & Weiss, A. M. (1997). Investment in technological innovations: An option pricing approach. Journal of Financial Economics, 44(3), 397–416.
Gutiérrez, Ó., & Ruiz-Aliseda, F. (2011). Real options with unknown-date events. Annals of Finance, 7(2), 171–198.
Hagspiel, V., Huisman, K. J. M., Kort, P. M., & Nunes, C. (2016). How to escape a declining market: Capacity investment or exit? European Journal of Operational Research, 254(1), 40–50.
Hoppe, H. C. (2000). Second-mover advantages in the strategic adoption of new technology under uncertainty. International Journal of Industrial Organization, 18(2), 315–338.
Huisman, K. J. M., & Kort, P. M. (1999). Effects of strategic interactions on the option value of waiting (CentER Discussion Paper No. 1999-92). CentER, Center for Economic Research.
Huisman, K. J. M., & Kort, P. M. (1999). Effects of strategic interactions on the option value of waiting (CentER Discussion Paper No. 1999-92). Tilburg University, Center for Economic Research.
Huisman, K. J. M., & Kort, P. M. (2003). Strategic investment in technological innovations. European Journal of Operational Research, 144(1), 209–223.
Huisman, K. J. M., & Kort, P. M. (2004). Strategic technology adoption taking into account future technological improvements: A real options approach. European Journal of Operational Research, 159(3), 705–728.
Huisman, K. J. M., & Kort, P. M. (2015). Strategic capacity investment under uncertainty. The RAND Journal of Economics, 46(2), 376–408.
Jeon, H., & Nishihara, M. (2015). The effects of business cycle and debt maturity on a firm’s investment and default decisions. International Review of Economics & Finance, 38, 326–351.
Joaquin, D. C., & Butler, K. C. (2000). Competitive investment decisions: A synthesis. In M. J. Brennan & L. Trigeorgis (Eds.), Project flexibility, agency, and competition: New developments in the theory and application of real options (pp. 324–339). Oxford University Press.
Jou, J.-B., & Lee, T. (2008). Irreversible investment, financing, and bankruptcy decisions in an oligopoly. Journal of Financial and Quantitative Analysis, 43(3), 769–786.
Kanniainen, V. (1993). Optimal production of innovations under uncertainty. Journal of Economics, 57(2), 147–168.
Kester, W. C. (1984). Today’s options for tomorrow’s growth. Harvard Business Review, 62(2), 153–160.
Kort, P. M. (1998). Optimal R&D investments of the firm. OR Spectrum, 20(3), 155–164.
Koussis, N., Martzoukos, S. H., & Trigeorgis, L. (2013). Multi-stage product development with exploration, value-enhancing, preemptive and innovation options. Journal of Banking & Finance, 37(1), 174–190.
Kulatilaka, N., & Perotti, E. C. (1998). Strategic growth options. Management Science, 44(8), 1021–1031.
Lambrecht, B. M. (2004). The timing and terms of mergers motivated by economies of scale. Journal of Financial Economics, 72(1), 41–62.
Lukas, E., Spengler, T. S., Kupfer, S., & Kieckhäfer, K. (2017). When and how much to invest? Investment and capacity choice under product life cycle uncertainty. European Journal of Operational Research, 260(3), 1105–1114.
Malliaris, A. G., & Brock, W. A. (1982). Stochastic methods in economics and finance. North-Holland.
Martzoukos, S. H. (2003). Real R&D options with endogenous and exogenous learning. In D. A. Paxson (Ed.), Real R&D options (pp. 111–129). Butterworth-Heinemann.
Mason, R., & Weeds, H. (2010). Investment, uncertainty and pre-emption. International Journal of Industrial Organization, 28(3), 278–287.
Matros, A., Smirnov, V., & Wait, A. (2026). General timing games with multiple players. Journal of Economic Dynamics and Control, 187, Article 105319.
Mauer, D. C., & Ott, S. H. (2000). Agency costs, underinvestment, and optimal capital structure: The effect of growth options to expand. In M. J. Brennan & L. Trigeorgis (Eds.), Project flexibility, agency, and competition: New developments in the theory and application of real options (pp. 151–179). Oxford University Press.
McDonald, R. L., & Siegel, D. (1986). The value of waiting to invest. The Quarterly Journal of Economics, 101(4), 707–727.
Miltersen, K. R., & Schwartz, E. S. (2004). R&D investments with competitive interactions. Review of Finance, 8(3), 355–401.
Mombello, B., Olsina, F., & Pringles, R. (2026). Investments under strategic competition and uncertainty: A literature review on real option games. Computers & Operations Research, 186, Article 107328.
Mosiño, A. (2012). Producing energy in a stochastic environment: Switching from non-renewable to renewable resources. Resource and Energy Economics, 34(4), 413–430.
Murto, P. (2004). Exit in duopoly under uncertainty. The RAND Journal of Economics, 35(1), 111–127.
Myers, S. C., & Howe, C. D. (1997). A life-cycle financial model of pharmaceutical R&D (Working Paper No. 41-97). MIT Program on the Pharmaceutical Industry.
Nielsen, M. J. (2002). Competition and irreversible investments. International Journal of Industrial Organization, 20(5), 731–743.
Nishihara, M. (2021). Preemptive competition between two firms with different discount rates. Managerial and Decision Economics, 42(3), 675–687.
Norton, J. A., & Bass, F. M. (1987). A diffusion theory model of adoption and substitution for successive generations of high-technology products. Management Science, 33(9), 1069–1086.
Pawlina, G., & Kort, P. M. (2001). Real options in an asymmetric duopoly: Who benefits from your competitive disadvantage? (CentER Discussion Paper No. 2001-95). CentER, Center for Economic Research.
Pawlina, G., & Kort, P. M. (2006). Real options in an asymmetric duopoly: Who benefits from your competitive disadvantage? Journal of Economics & Management Strategy, 15(1), 1–35.
Pennings, E., & Lint, O. (1997). The option value of advanced R&D. European Journal of Operational Research, 103(1), 83–94.
Pindyck, R. S. (1993). Investments of uncertain cost. Journal of Financial Economics, 34(1), 53–76.
Pindyck, R. S. (2002). Optimal timing problems in environmental economics. Journal of Economic Dynamics and Control, 26(9–10), 1677–1697.
Pindyck, R. S. (2007). Uncertainty in environmental economics. Review of Environmental Economics and Policy, 1(1), 45–65.
Pirayesh Neghab, D., Restrepo Diaz, J., Cevik, M., & Wahab, M. I. M. (2024). Game-theoretic approaches to product introduction strategies for durable products. Managerial and Decision Economics, 45(5), 2700–2726.
Reinganum, J. F. (1981a). Dynamic games of innovation. Journal of Economic Theory, 25(1), 21–41.
Reinganum, J. F. (1981b). Market structure and the diffusion of new technology. The Bell Journal of Economics, 12(2), 618–624.
Reinganum, J. F. (1981c). On the diffusion of new technology: A game theoretic approach. The Review of Economic Studies, 48(3), 395–405.
Reinganum, J. F. (1982). A dynamic game of R&D: Patent protection and competitive behavior. Econometrica, 50(3), 671–688.
Reiss, A. (1998). Investment in innovations and competition: An option pricing approach. The Quarterly Review of Economics and Finance, 38(3, Part 2), 635–650.
Riedel, F., & Steg, J.-H. (2017). Subgame-perfect equilibria in stochastic timing games. Journal of Mathematical Economics, 72, 36–50.
Samuelson, P. A. (1965). Proof that properly anticipated prices fluctuate randomly. Industrial Management Review, 6(2), 41–49.
Saravanan, G., & Vikkraman, P. (2013). A study on the new product introduction process and competitive benchmarking with special reference to Caterpillar 40/50T off-highway trucks manufactured in India. Asian Journal of Research in Business Economics and Management, 3(11), 173–185.
Sarkar, S. (2011). Optimal size, optimal timing and optimal financing of an investment. Journal of Macroeconomics, 33(4), 681–689.
Schonberger, R. J. (1987). Frugal manufacturing. Harvard Business Review, 65(5), 95–100.
Schwartz, E. S., & Moon, M. (2000). Rational pricing of Internet companies revisited. The Financial Review, 36(4), 7–26.
Sendstad, L. H., Chronopoulos, M., & Hagspiel, V. (2021). Optimal risk adoption and capacity investment in technological innovations. IEEE Transactions on Engineering Management, 70(2), 576–589.
Siddiqui, A. S., & Fleten, S.-E. (2010). How to proceed with competing alternative energy technologies: A real options analysis. Energy Economics, 32(4), 817–830.
Smets, F. (1991). Exporting versus FDI: The effect of uncertainty, irreversibilities, and strategic interactions [Unpublished manuscript]. Yale University.
Smets, F. R. (1993). Essays on foreign direct investment (Publication No. 9331566) [Doctoral dissertation, Yale University]. ProQuest Dissertations and Theses Global.
Smit, H. T. J., & Trigeorgis, L. (2001). Flexibility and commitment in strategic investments. In E. S. Schwartz & L. Trigeorgis (Eds.), Real options and investment under uncertainty: Classical readings and recent contributions (pp. 451–498). MIT Press.
Smit, H. T. J., & Trigeorgis, L. (2006). Real options and games: Competition, alliances and other applications of valuation and strategy. Review of Financial Economics, 15(2), 95–112.
Smit, H. T. J., & Trigeorgis, L. (2017). Strategic NPV: Real options and strategic games under different information structures. Strategic Management Journal, 38(13), 2555–2578.
Steg, J.-H., & Thijssen, J. J. J. (2023). Strategic investment with positive externalities. Games and Economic Behavior, 138, 1–21.
Stenbacka, R., & Tombak, M. M. (1994). Strategic timing of adoption of new technologies under uncertainty. International Journal of Industrial Organization, 12(3), 387–411.
Sun, C.-H. (2025). To commit or not to commit in product-innovation timing games. The B.E. Journal of Economic Analysis & Policy, 25(2), 177–208.
Takalo, T., & Kanniainen, V. (2000). Do patents slow down technological progress? Real options in research, patenting, and market introduction. International Journal of Industrial Organization, 18(7), 1105–1127.
Terrazas-Santamaria, D. (2021). Strategic introduction of a new product under uncertainty: A duopoly case. Managerial and Decision Economics, 42(4), 796–807.
Thijssen, J. J. J., Huisman, K. J. M., & Kort, P. M. (2006). The effects of information on strategic investment and welfare. Economic Theory, 28(2), 399–424.
Thijssen, J. J. J., Huisman, K. J. M., & Kort, P. M. (2012). Symmetric equilibrium strategies in game theoretic real option models. Journal of Mathematical Economics, 48(4), 219–225.
Triantis, A. J., & Hodder, J. E. (1990). Valuing flexibility as a complex option. The Journal of Finance, 45(2), 549–565.
Trigeorgis, L. (1996). Real options: Managerial flexibility and strategy in resource allocation. MIT Press.
Trigeorgis, L., & Mason, S. P. (1987). Valuing managerial flexibility. Midland Corporate Finance Journal, 5(1), 14–21.
Trigeorgis, L., Baldi, F., & Makadok, R. (2022). Compete, cooperate, or both? Integrating the demand side into patent deployment strategies for the commercialization and licensing of technology. Academy of Management Review, 47(1), 31–58.
Wahab, M. I. M., Lee, C.-G., & Sarkar, P. (2023). A real options approach to value manufacturing flexibilities with regime-switching product demand. Flexible Services and Manufacturing Journal, 35(3), 864–895.
Weeds, H. (2002). Strategic delay in a real options model of R&D competition. The Review of Economic Studies, 69(3), 729–747.
Woodruff, R., Parekh, F., Lamb, K., Mekkaoui, L., Allen, C., Smetanova, K., Huang, J., Williams, A., Toledo, G. S., Lilova, K., Roddie, C., Sillibourne, J., & Pule, M. (2023). Large-scale manufacturing of base-edited chimeric antigen receptor T cells. Molecular Therapy: Methods & Clinical Development, 31, Article 101123.