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Optimal product release time for a new high-tech startup firm under technical uncertainty

 1 School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan 611130, China 2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 3 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway H91 TK33, Ireland

* Corresponding author: Nan-Jing Huang

Received  September 2020 Revised  June 2021 Early access November 2021

Decision makers of new high-tech startup firms always want to choose an optimal time to launch their products which are under research and development (R&D) to obtain the maximum net income of these firms. However, existing models fail to consider the optimal release time of products for these new high-tech startup firms. In this paper, the optimal time to launch the product of the R&D project is assumed to be the first time when the product of the R&D project is released to the market. Based on this assumption, we develop a continuous-time model to find the optimal time at which the startup firm launches its product of the R&D project by considering the price of the similar product. Employing the methods of dynamic programming and variational inequalities, we also provide a closed form solution to our model. We also find that these high-tech startup firms prefer to delay their product release time when the price of the similar product is at a phase of rapid growth or the price has considerable uncertainty. Moreover, some numerical examples are provided to investigate the properties of our model.

Citation: Ming-hui Wang, Nan-jing Huang, Donal O'Regan. Optimal product release time for a new high-tech startup firm under technical uncertainty. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021186
References:
 [1] A. Azevedo and D. Paxson, Developing real option game models, European J. Oper. Res., 237 (2014), 909-920.  doi: 10.1016/j.ejor.2014.02.002. [2] A. Bensoussan, J. D. Diltz and S. R. Hoe, Real options games in complete and incomplete markets with several decision makers, SIAM J. Financial Math., 1 (2010), 666-728.  doi: 10.1137/090768060. [3] A. Bensoussan and J.-L. Lions,, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, 1982. [4] M. L. Bart, Real options in finance, Journal of Banking and Finance, 81 (2017), 166-171. [5] B. Cassiman and M. Ueda, Optimal project rejection and new firm start-ups, Management Science, 52 (2006), 262-275. [6] F. Gavazzoni and A. M. Santacreu, International R&D spillovers and asset prices, Journal of Financial Economics, 136 (2020), 330-354. [7] Z. Griliches, Market value, R&D, and patents, Economics Letters, 7 (1981), 183-187. [8] A. Huchzermeier and C. H. Loch, Project management under risk: Using the real options approach to evaluate flexibility in R&D, Management Science, 47 (2001), 85-101. [9] B. H. Hall, A. Jaffe and M. Trajtenberg, Market value and patent citations, The RAND Journal of Economics, 36 (2005), 16-38. [10] J. B. Jou, R&D investment and patent renewal decisions, The Quarterly Review of Economics and Finance, 69 (2018), 144-154. [11] P. M. Kort, Optimal R&D investment of the firm, OR Spektrum, 20 (1998), 155-164.  doi: 10.1007/BF01539764. [12] A. Moawia, A note on the theory of the firm under multiple uncertainties, European J. Oper. Res., 251 (2016), 341-343.  doi: 10.1016/j.ejor.2015.12.003. [13] M. Nishihara, Valuation of R&D investment under technological, market, and rival preemption uncertainty, Managerial and Decision Economics, 39 (2018), 200-212. [14] K. Osamu, Public R&D and commercialization of energy-efficient technology: A case study of Japanese projects, Energy Policy, 38 (2010), 7358-7369. [15] E. Pennings and O. Lint, The option value of advanced R&D, European Journal of Operational Research, 103 (1997), 83-94. [16] E. Pennings and L. Sereno, Evaluating pharmaceutical R&D under technical and economic uncertainty, European Journal of Operational Research, 212 (2011), 374-385. [17] R. S. Pindyck, Investments of uncertain cost, J. Financial Economics, 34 (1993), 53-76. [18] C. J. Serrano, The dynamics of the transfer and renewal of patents, The RAND Journal of Economics, 41 (2010), 686-708. [19] M. Serena, M. Federico, O. Raffaele and R. Gaetan, Commercialization Strategy and IPO Underpricing, Research Policy, 46 (2010), 1133-1141. [20] P. G. Sandner and J. H. Block, The market value of R&D, patents and trademarks, Research Policy, 40 (2011), 969-985. [21] M. H. Wang and N. J. Huang, Optimal consumption and R&D investment for a risk-averse entrepreneur, J. Nonlinear Convex Anal., 20 (2019), 1837-1852. [22] M. H. Wang and N. J. Huang, Robust optimal R&D investment under technical uncertainty in a regime-switching environment, Optimization, preprint. doi: 10.1080/02331934.2020.1818745. [23] A. E. Whalley, Optimal R&D investment for a risk-averse entrepreneur, J. Econom. Dynam. Control, 35 (2011), 413-429.  doi: 10.1016/j.jedc.2009.11.009. [24] X. N. Yu, Y. F. Lan and R. Q. Zhao, Cooperation royalty contract design in research and development alliances: Help vs. knowledge-sharing, European J. Oper. Res., 268 (2018), 740-754.  doi: 10.1016/j.ejor.2018.01.053.

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References:
 [1] A. Azevedo and D. Paxson, Developing real option game models, European J. Oper. Res., 237 (2014), 909-920.  doi: 10.1016/j.ejor.2014.02.002. [2] A. Bensoussan, J. D. Diltz and S. R. Hoe, Real options games in complete and incomplete markets with several decision makers, SIAM J. Financial Math., 1 (2010), 666-728.  doi: 10.1137/090768060. [3] A. Bensoussan and J.-L. Lions,, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, 1982. [4] M. L. Bart, Real options in finance, Journal of Banking and Finance, 81 (2017), 166-171. [5] B. Cassiman and M. Ueda, Optimal project rejection and new firm start-ups, Management Science, 52 (2006), 262-275. [6] F. Gavazzoni and A. M. Santacreu, International R&D spillovers and asset prices, Journal of Financial Economics, 136 (2020), 330-354. [7] Z. Griliches, Market value, R&D, and patents, Economics Letters, 7 (1981), 183-187. [8] A. Huchzermeier and C. H. Loch, Project management under risk: Using the real options approach to evaluate flexibility in R&D, Management Science, 47 (2001), 85-101. [9] B. H. Hall, A. Jaffe and M. Trajtenberg, Market value and patent citations, The RAND Journal of Economics, 36 (2005), 16-38. [10] J. B. Jou, R&D investment and patent renewal decisions, The Quarterly Review of Economics and Finance, 69 (2018), 144-154. [11] P. M. Kort, Optimal R&D investment of the firm, OR Spektrum, 20 (1998), 155-164.  doi: 10.1007/BF01539764. [12] A. Moawia, A note on the theory of the firm under multiple uncertainties, European J. Oper. Res., 251 (2016), 341-343.  doi: 10.1016/j.ejor.2015.12.003. [13] M. Nishihara, Valuation of R&D investment under technological, market, and rival preemption uncertainty, Managerial and Decision Economics, 39 (2018), 200-212. [14] K. Osamu, Public R&D and commercialization of energy-efficient technology: A case study of Japanese projects, Energy Policy, 38 (2010), 7358-7369. [15] E. Pennings and O. Lint, The option value of advanced R&D, European Journal of Operational Research, 103 (1997), 83-94. [16] E. Pennings and L. Sereno, Evaluating pharmaceutical R&D under technical and economic uncertainty, European Journal of Operational Research, 212 (2011), 374-385. [17] R. S. Pindyck, Investments of uncertain cost, J. Financial Economics, 34 (1993), 53-76. [18] C. J. Serrano, The dynamics of the transfer and renewal of patents, The RAND Journal of Economics, 41 (2010), 686-708. [19] M. Serena, M. Federico, O. Raffaele and R. Gaetan, Commercialization Strategy and IPO Underpricing, Research Policy, 46 (2010), 1133-1141. [20] P. G. Sandner and J. H. Block, The market value of R&D, patents and trademarks, Research Policy, 40 (2011), 969-985. [21] M. H. Wang and N. J. Huang, Optimal consumption and R&D investment for a risk-averse entrepreneur, J. Nonlinear Convex Anal., 20 (2019), 1837-1852. [22] M. H. Wang and N. J. Huang, Robust optimal R&D investment under technical uncertainty in a regime-switching environment, Optimization, preprint. doi: 10.1080/02331934.2020.1818745. [23] A. E. Whalley, Optimal R&D investment for a risk-averse entrepreneur, J. Econom. Dynam. Control, 35 (2011), 413-429.  doi: 10.1016/j.jedc.2009.11.009. [24] X. N. Yu, Y. F. Lan and R. Q. Zhao, Cooperation royalty contract design in research and development alliances: Help vs. knowledge-sharing, European J. Oper. Res., 268 (2018), 740-754.  doi: 10.1016/j.ejor.2018.01.053.
the behaviors of the startup firm's net value J(x, y) as a function of the expected cost of the R&D project $x$ and the price of the similar product $y$ in different scenarios
The behaviors of the value of the R&D project $F(x)$ as a function of the expected cost $x$ in different $I^*$ with $\beta = 0.5$, where $I^* = 2$, $I^* = 6$ and $I^* = 10$ and the corresponding boundaries $X^*$ are $29.5129$, $32.4965$ and $39.5507$, respectively.
The behaviors of the value of the R&D project $F(x)$ as a function of the expected cost $x$ in different $\beta$ with $I^* = 2$, where $\beta = 0.3$, $\beta = 0.5$ and $\beta = 0.8$ and the corresponding boundaries $X^*$ are $29.5129$, $32.4965$ and $39.5507$, respectively.
The behaviors of the value of $g(y)$ as a function of the price of the similar product $y$ in different $\alpha$ with $\alpha = 0.04$, $\alpha = 0.02$ and $\alpha = 0.01$, where the corresponding thresholds $\tilde{y}$ are $0.9984$, $0.5344$ and $0.4429$, respectively.
the behaviors of the value of $g(y)$ as a function of the price of the similar product $y$ in different $\sigma$ with $\sigma = 0.3$, $\sigma = 0.5$ and $\sigma = 0.8$, where the corresponding thresholds $\tilde{y}$ are $0.5726$, $0.9984$ and $1.9849$, respectively.
the behaviors of the value of $g(y)$ as a function of the price of the similar product $y$ in different $\delta_1$ with $\delta_1 = 3$, $\delta_1 = 1$ and $\delta_1 = 0.5$, where the corresponding thresholds $\tilde{y}$ are $0.3316$, $0.9948$ and $1.9897$, respectively.
the behaviors of the value of $g(y)$ as a function of the price of the similar product $y$ in different $\delta_2$ with $\delta_2 = 0.1$, $\delta_2 = 0.3$ and $\delta_2 = 0.5$, where the corresponding thresholds $\tilde{y}$ are $0.9948$, $2.9845$ and $4.9742$, respectively.
The behaviors of the threshold $\tilde{y}$ as a function of $\alpha$ with different $\sigma$.
The behaviors of the threshold $\tilde{y}$ as a function of $\sigma$ with different $\alpha$.
Parameters for four different scenarios
 Parameters Scenario 1 Scenario 2 Scenario 3 Scenario 4 $r$ 0.1 0.06 0.04 0.03 $\sigma$ 0.8 0.5 0.3 0.1 $\alpha$ 0.06 0.04 0.02 0.01 $\beta$ 0.8 0.5 0.2 0.1 $\delta_1$ 3 1 0.5 0.1 $\delta_2$ 0.5 0.1 0.05 0.05 $V$ 80 40 20 10 $I^*$ 5 2 1 0.5
 Parameters Scenario 1 Scenario 2 Scenario 3 Scenario 4 $r$ 0.1 0.06 0.04 0.03 $\sigma$ 0.8 0.5 0.3 0.1 $\alpha$ 0.06 0.04 0.02 0.01 $\beta$ 0.8 0.5 0.2 0.1 $\delta_1$ 3 1 0.5 0.1 $\delta_2$ 0.5 0.1 0.05 0.05 $V$ 80 40 20 10 $I^*$ 5 2 1 0.5
Simulated results for $X^*$ and $\tilde{y}$
 Scenario 1 Scenario 2 Scenario 3 Scenario 4 X* 73.0839 32.4965 15.9099 8.247 $\tilde{y}$ 1.8797 0.9948 0.4836 1
 Scenario 1 Scenario 2 Scenario 3 Scenario 4 X* 73.0839 32.4965 15.9099 8.247 $\tilde{y}$ 1.8797 0.9948 0.4836 1
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