University invention disclosure: balancing the optimal stage and type

Abstract

Invention disclosure is a complex problem faced by many university faculty members. Existing studies have investigated related issues, but few researchers have considered disclosure from the perspectives of both disclosure stage (early or late) and disclosure type (university or firm). This paper seeks to address this gap by investigating the influence of economic benefit, reputation, competition and collaboration on invention disclosure. We build several theoretical models, finding that reputation and competition have opposite effects on university disclosure, whereas economic benefit and collaboration are positively related to firm disclosure. We also find these four influencing factors have close relationships with disclosure stage. The simulation results further illustrate that initial reputation and additional reputation have different impacts on disclosure stage. An increase in a faculty and firm’s capability incentivizes firm disclosure in the early stage, while the competitor’s capability has the opposite impact. However, entry cost relates negatively to the disclosure stage and to the multi-stage disclosure. This paper provides insights for faculty on invention disclosure, as well as implications for university management.

Appendix

1.1 Proof of Eq. (6)

In this study, we believe faculty’s disclosure decisions depend on their current and future expected payoffs. Therefore, if the current expected payoff is larger than the future one, and the current expected payoff is the larger one, faculty disclosure their inventions in the current stage and then choose either university disclosure or firm disclosure. Based on this principle, we derive the following:

$$ \prod_{I} \left( {r_{t} } \right) - \prod_{I} \left( {r_{t + 1} } \right) > 0 $$

In order to find the largest expected payoff, we use

$$ \mathop {\hbox{max} }\limits_{{r_{t} }} F\left( {r_{t} } \right) = q\pi_{I}^{I - F} \left( {r_{t} } \right) + \left( {1 - q} \right)\pi_{I}^{I - U} \left( {r_{t} } \right) $$

Then, according to Eq. (5), we find that

$$ \mathop {\hbox{max} }\limits_{{r_{t} }} F\left( {r_{t} } \right) = \frac{1}{p}\left[ {\prod_{I} \left( {r_{t} } \right) - \prod_{I} \left( {r_{t + 1} } \right)} \right] + \prod_{I} \left( {r_{t + 1} } \right) $$

The maximization problem of disclosure stage is then written as:

$$ \begin{aligned} \mathop {\hbox{max} }\limits_{{r_{t} }} F\left( {r_{t} } \right) = \frac{1}{p}\left[ {\prod_{I} \left( {r_{t} } \right) - \prod_{I} \left( {r_{t + 1} } \right)} \right] + \prod_{I} \left( {r_{t + 1} } \right) \hfill \\ {\text{S}} . {\text{t}}.\,\prod_{I} \left( {r_{t} } \right) - \prod_{I} \left( {r_{t + 1} } \right) > 0,\,0 < r_{t} < r_{t + 1} \le 1 \hfill \\ \end{aligned} $$

1.2 Proof of Eq. (8)

When we do not consider risk preference, faculty disclose their inventions at the same level of probability for each stage, i.e., p = 0.5. In addition, we also assume faculty would choose university or firm disclosure at the same level of probability, i.e., q = 0.5. Therefore, Eq. (5) is as below:

$$ \begin{aligned} \prod_{I} \left( {r_{t} } \right) = \frac{1}{2}\left[ {rV\left( {1 + \alpha + n\mu } \right) + \frac{{\left( {1 - r} \right)(1 + nb)q_{i}^{2} V}}{{q_{i} + nq_{c} }} - q_{i} K} \right] \\ + \frac{1}{2}\left[ {(1 - r)\left( {(1 + q_{f} )q_{i} V - [(1 - q_{i} q_{f} ) + (q_{i} - q_{f} )^{2} ]q_{i} K} \right)} \right] \\ \end{aligned} $$

In order to make this equation more clear and concise, we denote

$$ M = \frac{{(1 + nb)q_{i}^{2} V}}{{q_{i} + nq_{c} }},\,H = (1 + q_{f} )q_{i} V - [(1 - q_{i} q_{f} ) + (q_{i} - q_{f} )^{2} ]q_{i} K,\,Z = V + \alpha V + n\mu V $$

Then, we rewrite F(r _t ), yielding:

$$ F\left( {r_{t} } \right) = \frac{1}{2}\left[ {r(Z - M - H) + M + H - q_{i} K} \right] $$

F(r _t ) is a function of r _t . Then, according to Eq. (5), a recurrence equation, we find the following:

$$ \prod_{I} \left( {r_{t} } \right) = \left( {\frac{1}{2}} \right)^{t + 1} Z + \frac{1}{2}\left[ {(Z - M - H)r_{t} + Z - q_{i} K} \right] $$

1.3 Simulation detail

In Sect. 3.3, the optimization problems in four scenarios are too complex to obtain analytical solutions. It is difficult to analyze the marginal effect of each parameters, such as μ, b, q _i , q _f and q _c . Therefore, we create simulation cases to examine each parameter’s influence.

First, according to Eqs. (14) and (15), we identify faculty’s research capability. All cases are then assigned to one of four scenarios: S1:$q_{i} < \hbox{min} \left\{ {q_{c} ,q_{f} } \right\}$, S2:$q_{f} \le q_{i} \le q_{c}$, S3:$q_{c} \le q_{i} \le q_{f}$, S4:$q_{i} > \hbox{max} \left\{ {q_{c} ,q_{f} } \right\}$.

Second, we assume the technology absorptivity rate is 0.85, according to Haeussler et al. (2014). The other parameters, such as μ, b, q _i , q _f , q _c and $k \equiv {K \mathord{\left/ {\vphantom {K V}} \right. \kern-0pt} V}$, vary over (0,1).

Based on these settings, each parameter has multiple values. We then solved the problem using MATLAB 2007b. The basic program file is as below:

Run the above program by MATLAB 2007b and then 16,384 cases are created.