Optimal timing and proportion in two stages learning investment

Abstract

This article introduces a two-stage real option approach with a learning effect to examine the optimal timing and proportion of investment for a firm entering a new market. Numerical findings illustrate that firms with different learning speeds exhibit distinct investment strategies: those with slower learning speeds tend to invest large proportion in the early time of first stage and invest the rest of small proportion in the later time of second stage, whereas firms with faster learning speeds invest small proportion in the early time of first stage and invest the rest of large proportion in the later time of second stage, compared to traditional one-stage investments. Leveraging the flexibility provided by two-stage learning investment, firms can effectively utilize timing and scale options, as emphasized in previous research. Furthermore, the proposed model addresses instances of learning investments with losses that cannot be accounted for by one-stage approaches.

Appendices

Appendix 1. The proof of the formula for one-stage investment strategy

According to the value-matching condition, it is optimal for the firm to invest when the shift of the market demand parameter, $X_{t}$, is at least as large as an investment threshold $X_{{}}^{ * }$ that also exceeds the total investment expense $I$, which means the value of the investment opportunity equals the net present value of the project, i.e., $F\left( {X_{{t^{ * } }} ,K} \right) = V\left( {X_{{t^{ * } }} ,K,Q} \right)$, or

$$F\left( {X_{t} ,K} \right) \equiv \alpha_{1} \left( {X_{t}^{*} } \right){}^{{\beta_{1} }} = \frac{{X_{t} K}}{\rho - \mu } - \frac{{\varphi K^{2} }}{\rho } - \frac{{cKe^{ - \gamma K} }}{\rho + \gamma K} - I \equiv V\left( {X_{{t^{ * } }} ,K,Q} \right).$$

(40)

To ensure that investment occurs along the optimal path and the value of the equity satisfies this condition at the endogenous investment threshold (Dixit and Pindyck 1994), we need the smooth-pasting condition as follows: $F_{X} \left( {X_{{t^{ * } }} ,K} \right) = V_{X} \left( {X_{{t^{ * } }} ,K,Q} \right)$ or

$$F_{X} \left( {X_{t} ,K} \right) \equiv \alpha_{1} \beta_{1} \left( {X_{{}}^{*} } \right){}^{{\beta_{1} - 1}} = \frac{K}{\rho - \mu } \equiv V_{X} \left( {X_{{t^{ * } }} ,K,Q} \right).$$

(41)

By using Eqs. (40) and (41), we can obtain the optimal threshold value of the shift of the market demand parameter $X_{{}}^{ * }$ and the coefficient $\alpha_{{1}} = \tfrac{{KX_{{}}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}\left( {\tfrac{1}{{X_{{}}^{*} }}} \right)^{{\beta_{1} }}$.

Finally, we can get the corresponding value for the firm to wait to invest in one-stage investment as follows:

$$F\left( {X_{t} ,K} \right) = \left\{ \begin{gathered} \frac{K}{{\beta_{1} - 1}}\left( {\frac{\varphi K}{\rho } + \frac{{ce^{ - \gamma K} }}{\rho + \gamma K} + i} \right)\left( {\frac{{X_{t} }}{{X^{ * } }}} \right)^{{\beta_{1} }} \quad \quad \quad {\text{if}}\;X_{t} < X^{*} \hfill \\ \frac{{X_{t} K}}{\rho - \mu } - \frac{{\varphi K^{2} }}{\rho } - \frac{{cKe^{ - \gamma K} }}{\rho + \gamma K} - I\quad \quad \quad \quad \quad \quad \;\,{\text{if}}\;X_{t} \ge X^{*} \hfill \\ \end{gathered} \right.,$$

with the optimal threshold value of the shift of the market demand parameter

$$X_{{}}^{*} = \frac{{\beta_{1} \left( {\rho - \mu } \right)}}{{\beta_{1} - 1}}\left( {\frac{\varphi K}{\rho } + \frac{{ce^{ - \gamma K} }}{\rho + \gamma K} + i} \right).$$

Appendix 2. The proof of the formula for two-stage investment strategy

The backward induction method is applied to derive the optimal threshold value of the shift of the market demand parameter, the option value of waiting to invest, and the optimal investment proportion for a two-stage investment strategy. Let's first start with investment in the second stage.

2.1 The option value in the second stage for two-stage investment strategy

Thanks to Dixit and Pindyck (1994), the similarity procedure can be well executed from Appendix 1 with one-stage investment. The value-matching condition for the second stage is given by Eq. (20) and can be expressed as follows:

$$\begin{gathered} N_{1} \left( {X_{2}^{ * } } \right)^{{\beta_{1} }} + \frac{{X_{2}^{ * } \eta K}}{\rho - \mu } - \frac{{\varphi \eta^{2} K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - i\eta K \hfill \\ \quad = \frac{{X_{2}^{ * } K}}{\rho - \mu } - \frac{{\varphi (2\eta^{2} - 2\eta + 1)K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - \frac{{ce^{{ - \gamma \left( {1 - \eta } \right)K}} \left( {1 - \eta } \right)K}}{{\rho + \gamma \left( {1 - \eta } \right)K}} - iK, \hfill \\ \end{gathered}$$

(43)

The smooth-pasting condition for the second stage, represented by Eq. (21), can also be shown as:

$$\beta_{1} N_{1} \left( {X_{2}^{*} } \right)^{{\beta_{1} - 1}} + \frac{\eta K}{{\rho - \mu }} = \frac{K}{\rho - \mu }.$$

(43)

By using Eqs. (42) and (43), we can obtain the optimal threshold value of the shift of the market demand parameter $X_{2}^{ * }$ in the second stage investment and the coefficient $N_{1} = \frac{1}{{\beta_{1} }}\left( {\frac{{K\left( {1 - \eta } \right)X_{2}^{ * } }}{\rho - \mu }} \right)\left( {\frac{1}{{X_{2}^{*} }}} \right)^{{\beta_{1} }}$.

Hence, we can obtain the optimal threshold value of the shift of the market demand parameter $X_{2}^{ * }$ and the corresponding option value for waiting to invest in the second stage investment as follows:

$$F_{2} \left( {X_{t} ,\left( {1 - \eta } \right)K} \right) = \left\{ \begin{gathered} \left[ {\frac{{\left( {1 - \eta } \right)KX_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{{X_{t} }}{{X_{{^{2} }}^{*} }}} \right)^{{\beta_{1} }} + \frac{{X_{t} \eta K}}{\rho - \mu } - \frac{{\varphi \eta^{2} K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - i\eta K\quad \quad \quad \quad \quad {\text{if}}\;X_{1}^{ * } \le X_{t} < X_{2}^{*} \hfill \\ \frac{{X_{2}^{*} K}}{\rho - \mu } - \frac{{\varphi (2\eta^{2} - 2\eta + 1)K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - \frac{{ce^{{ - \gamma \left( {1 - \eta } \right)K}} \left( {1 - \eta } \right)K}}{{\rho + \gamma \left( {1 - \eta } \right)K}} - iK\quad \quad {\text{if}}\;X_{t} \ge X_{2}^{*} \, \hfill \\ \end{gathered} \right.$$

and

$$X_{2}^{*} = \frac{{\beta_{1} \left( {\rho - \mu } \right)}}{{\beta_{1} - 1}}\left[ {\frac{{\varphi \left( {1 - \eta } \right)K}}{\rho } + \frac{{ce^{{ - \gamma \left( {1 - \eta } \right)K}} }}{{\left( {\rho + \gamma \left( {1 - \eta } \right)K} \right)}} + i} \right].$$

2.2 The option value in the first stage for two-stage investment strategy

After finding the option to invest in the second stage, we use the backward induction method to find the value of the option to invest in the first stage. Before the firm starts to invest, the value of the option to invest is assumed to be $F_{1} \left( {X_{t} ,\eta K} \right)$ as follows:

$$F_{1} \left( {X_{t} ,\eta K} \right) = M_{1} \left( {X_{t} } \right){}^{{\beta_{1} }}.$$

The net present value of the project in the first stage if the firm invests immediately equals the corresponding option value for waiting to invest in the second stage investment, and it can be shown as follows:

$$V_{1} \left( {X_{{t^{ * } }} ,K,Q} \right) = \left[ {\frac{{K\left( {1 - \eta } \right)X_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{{X_{1}^{ * } }}{{X_{2}^{*} }}} \right)^{{\beta_{1} }} + \frac{{X_{1}^{*} \eta K}}{\rho - \mu } - \frac{{\varphi \eta^{2} K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - i\eta K.$$

It is optimal for the firm to invest in the first stage when the shift of the market demand parameter, $X_{t}$, is at least as large as an investment threshold, $X_{1}^{ * }$, (i.e.,$X_{t} \ge X_{1}^{*}$), that exceeds the investment expense $i\eta K$ in the first stage, which means the value of the investment opportunity in the first stage equals the net present value of the project in the first stage. Hence, the value-matching condition for the first stage is given by Eq. (26) and can be expressed as:

$$M_{1} \left( {X_{1}^{*} } \right){}^{{\beta_{1} }} = \left[ {\frac{{\left( {1 - \eta } \right)KX_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{{X_{1}^{ * } }}{{X_{2}^{*} }}} \right)^{{\beta_{1} }} + \frac{{X_{1}^{*} \eta K}}{\rho - \mu } - \frac{{\varphi \eta^{2} K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - i\eta K.$$

(44)

The smooth-pasting condition for the first stage, Eq. (27), can also be shown as:

$$\beta_{1} M_{1} \left( {X_{1}^{*} } \right){}^{{\beta_{1} - 1}} = \frac{{\beta_{1} }}{{X_{1}^{ * } }}\left[ {\frac{{\left( {1 - \eta } \right)KX_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{{X_{1}^{ * } }}{{X_{2}^{*} }}} \right)^{{\beta_{1} }} + \frac{\eta K}{{\rho - \mu }}.$$

(45)

After some computation using Eqs. (44), (45), we can derive the optimal threshold value of the shift of the market demand parameter, $X_{1}^{ * }$, and the coefficient $M_{1} = \left[ {\frac{{K\left( {1 - \eta } \right)X_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{1}{{X_{2}^{*} }}} \right)^{{\beta_{1} }} + \frac{\eta K}{{\beta_{1} \left( {\rho - \mu } \right)}}\left( {X_{1}^{*} } \right){}^{{1 - \beta_{1} }}.$

Finally, we obtain the optimal threshold value of the shift of the market demand parameter, $X_{1}^{ * }$, and the corresponding option value for waiting to invest in the first stage as follows:

$$F_{1} \left( {X_{t} ,\eta K} \right) = \left\{ \begin{gathered} \frac{K}{{\beta_{1} \left( {\rho - \mu } \right)}}\left[ {\eta \left( {X_{1}^{*} } \right)^{{1 - \beta_{1} }} + \left( {1 - \eta } \right)\left( {X_{2}^{*} } \right)^{{1 - \beta_{1} }} } \right]\left( {X_{t} } \right)^{{\beta_{1} }} \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{if}}\;X_{t} < X_{1}^{*} \hfill \\ \left[ {\frac{{K\left( {1 - \eta } \right)X_{2}^{*} }}{{\beta_{1} \left( {\rho - \mu } \right)}}} \right]\left( {\frac{{X_{1}^{ * } }}{{X_{2}^{*} }}} \right)^{{\beta_{1} }} + \frac{{X_{1}^{*} \eta K}}{\rho - \mu } - \frac{{\varphi \eta^{2} K^{2} }}{\rho } - \frac{{ce^{ - \gamma \eta K} \eta K}}{\rho + \gamma \eta K} - i\eta K\quad \quad {\text{if}}\;X_{t} \ge X_{1}^{*} \, \hfill \\ \end{gathered} \right.,$$

and

$$X_{1}^{*} = \frac{{\beta_{1} \left( {\rho - \mu } \right)}}{{\beta_{1} - 1}}\left( {\frac{\varphi \eta K}{\rho } + \frac{{ce^{ - \gamma \eta K} }}{\rho + \gamma \eta K} + i} \right).$$