doi: 10.3934/jimo.2021186
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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 & Management Optimization, doi: 10.3934/jimo.2021186
References:
[1]

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A. BensoussanJ. 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.  Google Scholar

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B. Cassiman and M. Ueda, Optimal project rejection and new firm start-ups, Management Science, 52 (2006), 262-275.   Google Scholar

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F. Gavazzoni and A. M. Santacreu, International R&D spillovers and asset prices, Journal of Financial Economics, 136 (2020), 330-354.   Google Scholar

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Z. Griliches, Market value, R&D, and patents, Economics Letters, 7 (1981), 183-187.   Google Scholar

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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.   Google Scholar

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B. H. HallA. Jaffe and M. Trajtenberg, Market value and patent citations, The RAND Journal of Economics, 36 (2005), 16-38.   Google Scholar

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J. B. Jou, R&D investment and patent renewal decisions, The Quarterly Review of Economics and Finance, 69 (2018), 144-154.   Google Scholar

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P. M. Kort, Optimal R&D investment of the firm, OR Spektrum, 20 (1998), 155-164.  doi: 10.1007/BF01539764.  Google Scholar

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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.  Google Scholar

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M. Nishihara, Valuation of R&D investment under technological, market, and rival preemption uncertainty, Managerial and Decision Economics, 39 (2018), 200-212.   Google Scholar

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K. Osamu, Public R&D and commercialization of energy-efficient technology: A case study of Japanese projects, Energy Policy, 38 (2010), 7358-7369.   Google Scholar

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E. Pennings and O. Lint, The option value of advanced R&D, European Journal of Operational Research, 103 (1997), 83-94.   Google Scholar

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E. Pennings and L. Sereno, Evaluating pharmaceutical R&D under technical and economic uncertainty, European Journal of Operational Research, 212 (2011), 374-385.   Google Scholar

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R. S. Pindyck, Investments of uncertain cost, J. Financial Economics, 34 (1993), 53-76.   Google Scholar

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C. J. Serrano, The dynamics of the transfer and renewal of patents, The RAND Journal of Economics, 41 (2010), 686-708.   Google Scholar

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M. SerenaM. FedericoO. Raffaele and R. Gaetan, Commercialization Strategy and IPO Underpricing, Research Policy, 46 (2010), 1133-1141.   Google Scholar

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P. G. Sandner and J. H. Block, The market value of R&D, patents and trademarks, Research Policy, 40 (2011), 969-985.   Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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X. N. YuY. 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.  Google Scholar

show all references

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.  Google Scholar

[2]

A. BensoussanJ. 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.  Google Scholar

[3]

A. Bensoussan and J.-L. Lions,, Applications of Variational Inequalities in Stochastic Control, North-Holland, Amsterdam, 1982.  Google Scholar

[4]

M. L. Bart, Real options in finance, Journal of Banking and Finance, 81 (2017), 166-171.   Google Scholar

[5]

B. Cassiman and M. Ueda, Optimal project rejection and new firm start-ups, Management Science, 52 (2006), 262-275.   Google Scholar

[6]

F. Gavazzoni and A. M. Santacreu, International R&D spillovers and asset prices, Journal of Financial Economics, 136 (2020), 330-354.   Google Scholar

[7]

Z. Griliches, Market value, R&D, and patents, Economics Letters, 7 (1981), 183-187.   Google Scholar

[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.   Google Scholar

[9]

B. H. HallA. Jaffe and M. Trajtenberg, Market value and patent citations, The RAND Journal of Economics, 36 (2005), 16-38.   Google Scholar

[10]

J. B. Jou, R&D investment and patent renewal decisions, The Quarterly Review of Economics and Finance, 69 (2018), 144-154.   Google Scholar

[11]

P. M. Kort, Optimal R&D investment of the firm, OR Spektrum, 20 (1998), 155-164.  doi: 10.1007/BF01539764.  Google Scholar

[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.  Google Scholar

[13]

M. Nishihara, Valuation of R&D investment under technological, market, and rival preemption uncertainty, Managerial and Decision Economics, 39 (2018), 200-212.   Google Scholar

[14]

K. Osamu, Public R&D and commercialization of energy-efficient technology: A case study of Japanese projects, Energy Policy, 38 (2010), 7358-7369.   Google Scholar

[15]

E. Pennings and O. Lint, The option value of advanced R&D, European Journal of Operational Research, 103 (1997), 83-94.   Google Scholar

[16]

E. Pennings and L. Sereno, Evaluating pharmaceutical R&D under technical and economic uncertainty, European Journal of Operational Research, 212 (2011), 374-385.   Google Scholar

[17]

R. S. Pindyck, Investments of uncertain cost, J. Financial Economics, 34 (1993), 53-76.   Google Scholar

[18]

C. J. Serrano, The dynamics of the transfer and renewal of patents, The RAND Journal of Economics, 41 (2010), 686-708.   Google Scholar

[19]

M. SerenaM. FedericoO. Raffaele and R. Gaetan, Commercialization Strategy and IPO Underpricing, Research Policy, 46 (2010), 1133-1141.   Google Scholar

[20]

P. G. Sandner and J. H. Block, The market value of R&D, patents and trademarks, Research Policy, 40 (2011), 969-985.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

X. N. YuY. 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.  Google Scholar

Figure 1.  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
Figure 2.  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.
Figure 3.  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.
Figure 4.  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.
Figure 5.  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.
Figure 6.  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.
Figure 7.  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.
Figure 8.  The behaviors of the threshold $ \tilde{y} $ as a function of $ \alpha $ with different $ \sigma $.
Figure 9.  The behaviors of the threshold $ \tilde{y} $ as a function of $ \sigma $ with different $ \alpha $.
Table 1.  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
Table 2.  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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