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
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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 |
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.
Keywords:
Research and development,
investment decision,
technical uncertainty,
dynamic programming,
variational inequality.
Mathematics Subject Classification:
91B06, 91B38, 93C15, 93E20, 49J40.
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
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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. |
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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.
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| [10] |
J. B. Jou,
R&D investment and patent renewal decisions, The Quarterly Review of Economics and Finance, 69 (2018), 144-154.
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| [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. |
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. |
| [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. |
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 |
| 0.1 | 0.06 | 0.04 | 0.03 | |
| 0.8 | 0.5 | 0.3 | 0.1 | |
| 0.06 | 0.04 | 0.02 | 0.01 | |
| 0.8 | 0.5 | 0.2 | 0.1 | |
| 3 | 1 | 0.5 | 0.1 | |
| 0.5 | 0.1 | 0.05 | 0.05 | |
| 80 | 40 | 20 | 10 | |
| 5 | 2 | 1 | 0.5 |
| Parameters | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 |
| 0.1 | 0.06 | 0.04 | 0.03 | |
| 0.8 | 0.5 | 0.3 | 0.1 | |
| 0.06 | 0.04 | 0.02 | 0.01 | |
| 0.8 | 0.5 | 0.2 | 0.1 | |
| 3 | 1 | 0.5 | 0.1 | |
| 0.5 | 0.1 | 0.05 | 0.05 | |
| 80 | 40 | 20 | 10 | |
| 5 | 2 | 1 | 0.5 |
| Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | |
| X* | 73.0839 | 32.4965 | 15.9099 | 8.247 |
| 1.8797 | 0.9948 | 0.4836 | 1 |
| Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | |
| X* | 73.0839 | 32.4965 | 15.9099 | 8.247 |
| 1.8797 | 0.9948 | 0.4836 | 1 |
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