Competing or coordinating: IT R&D investment decision making subject to information time lag

Abstract

In this paper, we apply a dynamic option-game framework to examine the impact of information time lag on Research and Development (R&D) investment in innovative information technology (IT) industry. We focus on incentives of competition and coordination in R&D. Our results show that shorter information time lag may induce firms to coordinate their investments and avoid over-investment. The threshold of information time lag developed in this paper can facilitate managerial decisions on whether to compete or coordinate R&D investments.

Appendix

Proof of Proposition 1

Consider a suitably smooth function $J^{i} ,i \in \{ A,B\}$, satisfying the Hamiltonian-Jacobian-Bellman (HJB) equation:

$$ {\mathbb{D}J^{i} (t,x,n_{t}^{i} ,n_{t}^{ - i} ) + \zeta_{t}^{i} X_{t} + \mathop {\sup }\limits_{{u_{t}^{i} }} \{ u_{t}^{i} [\pi (J^{i} (t,x,n_{t}^{i} + 1,n_{t}^{ - i} ) - J^{i} (t,x,n_{t}^{i} ,n_{t}^{ - i} )) - I]\} + \phi^{ - i*} \pi [J^{i} (t,x,n_{t}^{i} ,n_{t}^{ - i} + 1) - J^{i} (t,x,n_{t}^{i} ,n_{t}^{ - i} )] = 0} $$

(23)

where

$$ \begin{gathered} \mathbb{D}J^{i} (t,x,n_{t}^{i} ,n_{t}^{ - i} ) = 12\sigma^{2} x^{2} J_{xx}^{i} + \mu xJ_{x}^{i} + J_{t}^{i} - rJ^{i} \hfill \\ \end{gathered} $$

where the subscript J refers to the partial derivative.

The expected payoffs follow from the above HJB equation (23). The parameters can be solved by applying boundary conditions, value matching conditions, smooth pasting conditions, and transitional boundary conditions. For discussions of the above conditions, see Dixit [4], Berk et al. [1], and Yao [23].

Proof for Corollary 1

From Eqs. 6 and 7, $x^{L*} = x^{F*} = x^{N*}$, we have Eq. 9. It can be shown that the threshold $x^{N*}$as in Eq. 9a and coefficients as follows:

$$ \begin{aligned} c_{3N} = & \pi (r + \pi + \pi - \mu )(r - \mu ) \\ c_{4N} = & - Ir + \pi + \pi \\ a_{1N} = & (1 + \gamma_{2,r + \pi + \pi } )c_{3N} x^{N*} + c_{4N} \gamma_{2,r + \pi + \pi } \gamma_{2,r + \pi + \pi } - \gamma_{1,r} (x^{N*} )^{{ - \gamma_{1,r} }} \\ c_{2N} = & (1 + \gamma_{1,r} )c_{3N} x^{N*} + c_{4N} \gamma_{1,r} \gamma_{2,r + \pi + \pi } - \gamma_{1,r} (x^{N*} )^{{ - \gamma_{2,r + \pi + \pi } }} \\ \end{aligned} $$

Proof of Proposition 2

Similar as the proof of Proposition 1. Consider a suitably smooth function J ^C, satisfying the Hamiltonian-Jacobian-Bellman (HJB) equation:

$$ \begin{gathered} \mathbb{D}J^{C} (t,x,n_{t} ) + (\zeta_{t}^{A} + \zeta_{t}^{B} )X_{t} + \mathop {\sup }\limits_{{u_{t} = (u_{t}^{i} ,u_{t}^{ - i} )}} \{ u_{t}^{i} [\pi^{i} (n_{t}^{i} )(J^{C} (t,x,n_{t}^{i} + 1,n_{t}^{ - i} ) - J^{C} (t,x,n_{t}^{i} ,n_{t}^{ - i} )) - I] \hfill \\ + u_{t}^{ - i} [\pi^{ - i} (n_{t}^{ - i} )(J^{C} (t,x,n_{t}^{i} ,n_{t}^{ - i} + 1) - J^{C} (t,x,n_{t}^{i} ,n_{t}^{ - i} )) - I]\} = 0 \hfill \\ \end{gathered} $$

(24)

where

$$ \begin{gathered} \mathbb{D}J^{C} (t,x,n_{t} ) = 12\sigma^{2} x^{2} J_{xx}^{C} + \mu xJ_{x}^{C} + J_{t}^{C} - rJ^{C} \hfill \\ \end{gathered} $$

where the subscript parameter of J refers to the partial derivative.

The expected payoffs follow from the above HJB equation (24). The parameters can be solved by applying boundary conditions, value matching conditions, smooth pasting conditions, and transitional boundary conditions.

Proof for Corollary 2

From Eq. 10, $x^{CL*} = x^{CF*} = x^{C*}$, we have Eq. 12. It can be shown that the threshold $x^{N*}$ as in Eq. 12a and coefficients as follows:

$$ \begin{aligned} c_{3C} = & 2\pi (r + \pi + \pi - \mu )(r - \mu ) \\ c_{4C} = & - 2Ir + \pi + \pi \\ a_{1C} = & (1 + \gamma_{2,r + \pi + \pi } )c_{3C} x^{C*} + c_{4C} \gamma_{2,r + \pi + \pi } \gamma_{2,r + \pi + \pi } - \gamma_{1,r} (x^{C*} )^{{ - \gamma_{1,r} }} \\ c_{2C} = & (1 + \gamma_{1,r} )c_{3C} x^{C*} + c_{4N} \gamma_{1,r} \gamma_{2,r + \pi + \pi } - \gamma_{1,r} (x^{C*} )^{{ - \gamma_{2,r + \pi + \pi } }} \\ \end{aligned} $$

Proof of Theorem 1

First note that

$$ 1 \ge \Pr (n_{t + \delta } = (0,0)|n_{t} = (0,0),\phi^{i} ,\tilde{\phi }^{ - i} ) \ge e^{ - (\pi + \pi )\delta } $$

(25)

If $\delta \le \delta^{L*}$, then $\delta \le \delta^{L} (x)$ for all x. From (25), it can be shown that $f_{1} (x,\delta ) < 0$, then the corresponding strategy profile $\psi = (\psi^{A} ,\psi^{B} )$ defined in (4) constitutes a Nash equilibrium for the history-dependent game.

Similarly, if $\delta > \delta^{H*}$, then $\delta \ge \delta^{H} (x)$ at least for some x. From (25), $f_{2} (x,\delta ) > 0$, then the corresponding strategy profile $\psi = (\psi^{A} ,\psi^{B} )$ defined in (4) fails to constitute a Nash equilibrium for the history-dependent game.