Economics of Open Source Technology: A Dynamic Approach

Abstract

We analyze open source licensing and its effects on firms’ decisions whether to use the open source technology or not and on the incentives for innovation, through a dynamic model of innovation and competition in an environment with a ladder-type technology. We model the basic features of the General Public License (GPL), one of the most popular open source licenses and study how firms behave under this license when competition is present. Under the GPL, any innovative findings using open source technology must also be open source in the subsequent periods, and this obligation creates a trade-off. We focus on how this trade-off affects incentives to use and build up the open source technology.

Appendix

1.1 Finite Horizon

Proof of Lemma 2

Recall that the expected inverse demand with $K_{i}$ is ${\mathbf {P}}_{K_{i}^{t}}\equiv P({\mathbf {Q}}_{K_{i}^{t}})=A-\mathbf {Q }_{K_{i}^{t}}$. The expected total quantity can be decomposed; ${\mathbf {Q}} _{K_{i}^{t}}=q_{K_{i}^{t}}+{\mathbf {Q}}_{-i}^{t}$ where ${\mathbf {Q}} _{-i}^{t}=\sum _{K\ne K_{i}^{t}} \mathbf {N(K,t)}q_{K^{t}} + (\mathbf {\ N(K_{i}^{t},t)}-1)q_{K_{i}^{t}}$. Under Cournot competition, a firm i with $K_{i}^{t}$ solves the following problem

$$\begin{aligned} \max _{q}E\left[ \pi _{K_{i}}^{t}\right] =\left( {\mathbf {P}} _{K_{i}^{t}}-c\left( K_{i}^{t}\right) \right) q \end{aligned}$$

The first-order condition is

$$\begin{aligned} \frac{\partial E\left[ \pi _{K_{i}^{t}}\right] }{\partial q}= & {} P\left( {\mathbf {Q}} _{K_{i}^{t}}\right) -c\left( K_{i}^{t}\right) +\frac{\partial P\left( {\mathbf {Q}}_{K_{i}^{t}}\right) }{ \partial q}q \\= & {} A-{\mathbf {Q}}_{K_{i}^{t}}-c\left( K_{i}^{t}\right) -q \\= & {} A-c\left( K_{i}^{t}\right) -{\mathbf {Q}}_{-i}^{t}-2q=0 \end{aligned}$$

Using symmetry, we get the following equilibrium condition;

$$\begin{aligned} 2q_{K_{i}^{t}}=A-c(K_{i}^{t}) - \left( \sum _{K\ne K_{i}^{t}} \mathbf {N(K,t)} q_{K} + (\mathbf {N(K_{i}^{t},t)}-1)q_{K_{i}^{t}} \right) \end{aligned}$$

That is,

$$\begin{aligned} (\mathbf {N(K_{i}^{t},t)}+1)q_{K_{i}^{t}}=A-c(K_{i}^{t}) - \sum _{K\ne K_{i}^{t}} \mathbf {N(K,t)}q_{K} \end{aligned}$$

To see $q_{K}-q_{{\hat{K}}} =c({\hat{K}})-c(K)$ for any $K,{\hat{K}}$, first note that

$$\begin{aligned} (\mathbf {N(K,t)}+1)q_{K}&=A-c(K) - \sum _{ k \ne K} \mathbf {N(k,t)} q_{k}\\&=A-c(K) - \mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})q_{{\hat{K}}} - \sum _{ k \ne K,{\hat{K}}} \mathbf {N(k,t)}q_{k} \end{aligned}$$

and similarly,

$$\begin{aligned} (\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})+1)q_{{\hat{K}}}&=A-c({\hat{K}}) - \sum _{ k \ne {\hat{K}}} \mathbf {N(k,t)}q_{k}\\&=A-c({\hat{K}}) - \mathbf {N(K,t)}q_{K} - \sum _{ k \ne K, {\hat{K}}} \mathbf {N(k,t)}q_{k} \end{aligned}$$

Subtracting the two equations we get,

$$\begin{aligned} (\mathbf {N(K,t)}+1)q_{K}-(\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})+1)q_{{\hat{K}}}=c({\hat{K}}) -c(K) + \mathbf {N(K,t)}q_{K}-\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})q_{{\hat{K}}} \end{aligned}$$

Thus, we get, $q_{K}-q_{{\hat{K}}}=c({\hat{K}}) - c(K)$, that is, for any $K, {\hat{K}}$, we have $q_{{\hat{K}}}=q_{K}+c(K)-c({\hat{K}})$. Now, substituting $q_{ {\hat{K}}}=q_{K_{i}^{t}}+c(K_{i}^{t})-c({\hat{K}})$ into

$$\begin{aligned} \left( \mathbf {N(K_{i}^{t},t)}+1\right) q_{K_{i}^{t}}=A-c(K_{i}^{t})-\sum _{ {\hat{K}}\ne K_{i}^{t}}\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})q_{{\hat{K}}} \end{aligned}$$

(3)

we get

$$\begin{aligned} q_{K_{i}^{t}}=\frac{1}{\mathbf {N(K_{i}^{t},t)+}\sum _{{\hat{K}}\ne K_{i}^{t}} \mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})+1}\left[ A-c(K_{i}^{t}) - \sum _{{\hat{K}}\ne K_{i}^{t}} \mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}}) \left[ c(K_{i}^{t}) - c({\hat{K}}) \right] \right] \end{aligned}$$

Since $\mathbf {N(K_{i}^{t},t)+}\sum _{{\hat{K}}\ne K_{i}^{t}}\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})=M$, we have

$$\begin{aligned} q_{K_{i}^{t}}= & {} \frac{1}{M+1} \left[ A-c(K_{i}^{t}) - \sum _{{\hat{K}}\ne K_{i}^{t}}\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}}) \left[ c(K_{i}^{t}) - c({\hat{K}}) \right] \right] \\= & {} \frac{1}{M+1} \left[ A-c(K_{i}^{t}) \left( 1+\sum _{{\hat{K}}\ne K_{i}^{t}} \mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}})\right) + \sum _{{\hat{K}}\ne K_{i}^{t}}\mathbf {N}({\hat{{\mathbf {K}}}},{\mathbf {t}}) c({\hat{K}})\right] \end{aligned}$$

Note that this is a function of the technology distribution over firms at each period. Now, we calculate the equilibrium profit levels. First, note that by Eq. 3, in the equilibrium, we have

$$\begin{aligned} q_{K_{i}^{t}}=A-c(K_{i}^{t}) - \sum _{K\ne K_{i}^{t}} \mathbf {N(K,t)}q_{K} - \mathbf {N(K_{i}^{t},t)}q_{K_{i}^{t}} = A-c(K_{i}^{t}) - \sum _{K} \mathbf { N(K,t)}q_{K} \end{aligned}$$

Using ${\mathbf {Q}}_{K} = \sum _{ k} \mathbf {N(k,t)}q_{k}$ and ${\mathbf {P}}_{K} = A-{\mathbf {Q}}_{K}=A-\sum _{k} \mathbf {N(k,t)}q_{k}$, we get

$$\begin{aligned} E[\pi _{K_{i}}^{t}] = \left[ {\mathbf {P}}_{K_{i}^{t}}-c(K_{i}^{t})\right] q_{K_{i}^{t}} = \left[ A- c(K_{i}^{t}) - \sum _{k} \mathbf {N(k,t)}q_{k}\right] q_{K_{i}^{t}} = (q_{K_{i}^{t}})^{2} \end{aligned}$$

$\square$

Proof of Proposition 1

We first show this result for $k_{i}^{T-1}=k_\mathrm{os}^{T-1}$. Then, for $k_{i}^{T-1}>k_\mathrm{os}^{T-1}$, it will be straightforward. If a firm with $k_{i}^{T-1}=k_\mathrm{os}^{T-1}$ chooses not to use the open source technology, at $T-1$, then the firm will have the following expected payoff.

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}\right) \left[ E\left[ \pi _{k_\mathrm{os}+1}^{T-1}\right] +\delta W_{S}\left( 0,k_\mathrm{os}^{T-1}+1 \right) \right] \\&+ \left( 1-p\left( 0,k^{T-1}_\mathrm{os}\right) \right) \left[ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1} \right) \right] - C\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) \end{aligned}$$

where $W_{S}(d^{t},k^{t})$ denotes the expected continuation payoff from period $t+1$ on, when at t the open source technology use decision is $d^{t}$ and the technology level at the end of period t is $k^{t}$, with a success in innovation in the investment stage, and likewise, $W_{F}(d^{t},k^{t})$ denotes the expected continuation payoff from period $t+1$ on, when at t the open source technology use decision is $d^{t}$ and the technology level at the end of period t is $k^{t}$, with a failure in innovation in the investment stage. Arranging this, we get

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}\right) \left[ E\left[ \pi _{k_\mathrm{os}+1}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta \left[ W_{S}\left( 0,k_\mathrm{os}^{T-1}+1\right) - W_{F}\left( 0,k_\mathrm{os}^{T-1}\right) \right] \right] \\&+ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1}\right) -C\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) \end{aligned}$$

Similarly, a firm with $k_{i}^{T-1}=k_\mathrm{os}^{T-1}$ chooses to use the open source, at $T-1$, then the firm will have the following expected payoff.

$$\begin{aligned} V\left( 1,k^{T-1}_\mathrm{os}\right)= & {} p\left( 1,k^{T-1}_\mathrm{os}\right) \left[ E\left[ \pi _{k_\mathrm{os}+1}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta \left[ W_{S}\left( 1,k_\mathrm{os}^{T-1}+1\right) - W_{F}\left( 1,k_\mathrm{os}^{T-1}\right) \right] \right] \\&+ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 1,k_\mathrm{os}^{T-1}\right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \end{aligned}$$

Recall that the optimal investment levels are

$$\begin{aligned} C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right)= & {} E\left[ \pi _{k_\mathrm{os}+1}^{T-1}\right] -E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta \left[ W_{S}\left( 0,k_\mathrm{os}^{T-1}+1\right) - W_{F}\left( 0,k_\mathrm{os}^{T-1}\right) \right] \\ C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right)= & {} E\left[ \pi _{k_\mathrm{os}+1}^{T-1}\right] -E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta \left[ W_{S}\left( 1,k_\mathrm{os}^{T-1}+1\right) - W_{F}\left( 1,k_\mathrm{os}^{T-1}\right) \right] \end{aligned}$$

Thus, we have

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) - C\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) + E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1}\right) \\ V\left( 1,k^{T-1}_\mathrm{os}\right)= & {} p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) + E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 1,k_\mathrm{os}^{T-1}\right) \end{aligned}$$

Note also that $W_{F}(0,k_\mathrm{os}^{T-1})=W_{F}(1,k_\mathrm{os}^{T-1})$. Then, we get

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}\right) - V\left( 1,k^{T-1}_\mathrm{os}\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) - C\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) \\&-\,\left[ p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \right] \end{aligned}$$

Note that, the function $F(p)=pC^{\prime }(p)-C(p)$ is an increasing function. This is because, the first derivative is $F^{\prime }(p)=C^{ \prime }(p)+pC^{\prime \prime }(p)-C^{\prime }(p)=pC^{\prime \prime }(p)>0$ since $C^{\prime \prime }>0$ and $p>0$. Thus, if $p(0,k^{T-1}_\mathrm{os})>p(1,k^{T-1}_\mathrm{os})$, then $V(0,k^{T-1}_\mathrm{os}) >V(1,k^{T-1}_\mathrm{os})$. Also, note that

$$\begin{aligned} C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}\right) \right) -C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right)= & {} \delta \left[ W_{S}\left( 0,k_\mathrm{os}^{T-1}+1\right) - W_{S}\left( 1,k_\mathrm{os}^{T-1}+1\right) \right] \end{aligned}$$

Now, we have $W_{S}(0,k_\mathrm{os}^{T-1}+1) > W_{S}(1,k_\mathrm{os}^{T-1}+1)$. To see this, note that in period T, if firm i has used the open source technology in period $T-1$, then it will be at the same technology level as the user firms by Lemma 1, because its success in period $T-1$ will be public for each user firm at T. Denote the expected payoff of this case by $W^{eq}$. That is, $W^{eq}=W_{S}(1,k_\mathrm{os}^{T-1}+1)$. However, if firm i has not used the open source in period $T-1$, then its technology level will be the same as all the user firms with some probability $\nu$ and will be one step higher than all the user firms with probability $1-\nu$, where $\nu$ is the probability that at least one user firm is successful in period $T-1$. Denote the latter case’s expected payoff with $W^{sup}$. Then, $W_{S}(0,k_\mathrm{os}^{T-1}+1) > W_{S}(1,k_\mathrm{os}^{T-1}+1)$ if $(1-\nu ) W^{sup}+ \nu W^{eq} > W^{eq}$. However, $W^{sup} > W^{eq}$ since in the former expected payoff firm i has a higher technology level than the latter. Thus, $W_{S}(0,k_\mathrm{os}^{T-1}+1) > W_{S}(1,k_\mathrm{os}^{T-1}+1)$, which in turn implies that $C^{\prime }(p(0,k^{T-1}_\mathrm{os})) >C^{\prime }(p(1,k^{T-1}_\mathrm{os}))$. Since $C^{\prime \prime }>0$, we get $p(0,k^{T-1}_\mathrm{os})>p(1,k^{T-1}_\mathrm{os})$, which proves $V(0,k^{T-1}_\mathrm{os}) >V(1,k^{T-1}_\mathrm{os})$. When $k^{T-1}_{i}>k^{T-1}_\mathrm{os}$, the proof works just the same. $\square$

Proof of Proposition 2

If a firm with $k_{i}^{T-1}=k_\mathrm{os}^{T-1}-1$ chooses not to use the open source, at $T-1$, then the firm will have the following expected payoff.

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}-1\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}-1\right) \left[ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{S}\left( 0,k_\mathrm{os}^{T-1} \right) \right] \\&+ \left( 1-p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \left[ E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1} -1 \right) \right] \\&-\, C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \end{aligned}$$

Arranging this, we get

$$\begin{aligned}&V\left( 0,k^{T-1}_\mathrm{os}-1\right) \\&\quad = p\left( 0,k^{T-1}_\mathrm{os}-1\right) \left[ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] +\delta W_{S}\left( 0,k_\mathrm{os}^{T-1}\right) - \delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \right] \\&\qquad + \,E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \\&\qquad -\,C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \end{aligned}$$

Recall the equilibrium investment probability of firm i with $k^{T-1}_{i}$ and $d^{T-1}_{i}=0$ is given by

$$\begin{aligned} C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right)= & {} E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \\&+ \left[ \delta W_{S}\left( 0,k_\mathrm{os}^{T-1}\right) - \delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \right] \end{aligned}$$

Plugging this into $V(0,k^{T-1}_\mathrm{os}-1)$, we get

$$\begin{aligned} V\left( 0,k^{T-1}_\mathrm{os}-1\right)= & {} p\left( 0,k^{T-1}_\mathrm{os}-1\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \\&+\, E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] +\delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \\ V\left( 1,k^{T-1}_\mathrm{os}-1\right)= & {} p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&+\,E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] +\delta W_{F}\left( 1,k_\mathrm{os}^{T-1}\right) \end{aligned}$$

Note that $W_{F}(1,k_\mathrm{os}^{T-1})=W_{F}(0,k_\mathrm{os}^{T-1}-1)$, since in both cases the firm enters the second stage of the last period at the same technology level by Lemma 1, thus will have the same expected period T payoff. Thus, we get

$$\begin{aligned} V\left( 1,k^{T-1}_\mathrm{os}-1\right) - V\left( 0,k^{T-1}_\mathrm{os}-1\right)= & {} p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\,\left[ p\left( 0,k^{T-1}_\mathrm{os}-1\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \right. \\&\left. -\,C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \right] \\&+\, E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \end{aligned}$$

Note that $E[\pi _{k_\mathrm{os}-1}^{T-1}]= q_{k^{T-1}_\mathrm{os}-1}^{2}$ and $E[\pi _{k_\mathrm{os}}^{T-1}]= q_{k^{T-1}_\mathrm{os}}^{2}$. Note also that $q_{k^{T-1}_\mathrm{os}-1}<q_{k^{T-1}_\mathrm{os}}$. Thus, $E[\pi _{k_\mathrm{os}}^{T-1}] - E[\pi _{k_\mathrm{os}-1}^{T-1}]>0$. Now, if $p(1,k^{T-1}_\mathrm{os})>p(0,k^{T-1}_\mathrm{os}-1)$, then, since $pC^{\prime }(p)-C(p)$ is an increasing function,

$$\begin{aligned}&p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&\quad > p\left( 0,k^{T-1}_\mathrm{os}-1\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) . \end{aligned}$$

This, together with $E[\pi _{k_\mathrm{os}}^{T-1}] - E[\pi _{k_\mathrm{os}-1}^{T-1}]>0$, implies that $V(1,k^{T-1}_\mathrm{os}-1)>V(0,k^{T-1}_\mathrm{os}-1)$. If, however, $p(1,k^{T-1}_\mathrm{os})<p(0,k^{T-1}_\mathrm{os}-1)$, then we have

$$\begin{aligned} V\left( 1,k^{T-1}_\mathrm{os}-1\right) - V\left( 0,k^{T-1}_\mathrm{os}-1\right)= & {} p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\,\left[ p\left( 0,k^{T-1}_\mathrm{os}-1\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \right. \\&\left. -C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \right] \\&+\, E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \\= & {} C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&+\,p\left( 1,k^{T-1}_\mathrm{os}\right) C^{\prime }\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\, p\left( 0,k^{T-1}_\mathrm{os}-1\right) C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) \\&+\,E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \\> & {} C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\, C^{\prime }\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) + E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \\= & {} C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\, \left[ E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] + \delta W_{S}\left( 0,k_\mathrm{os}^{T-1}\right) \right. \\&\left. - \,\delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \right] \\&+\,E\left[ \pi _{k_\mathrm{os}}^{T-1}\right] - E\left[ \pi _{k_\mathrm{os}-1}^{T-1}\right] \\= & {} C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) \\&-\, \left[ \delta W_{S}\left( 0,k_\mathrm{os}^{T-1}\right) - \delta W_{F}\left( 0,k_\mathrm{os}^{T-1}-1\right) \right] \end{aligned}$$

The inequality follows since $C^{\prime }>0$ and $p(0,k^{T-1}_\mathrm{os}-1)$ can be at most 1. Note that $C(p(0,k^{T-1}_\mathrm{os}-1))-C(p(1,k^{T-1}_\mathrm{os}))>0$ since $C^{\prime }>0$. Also note that, $W_{S}(0,k^{T-1}_\mathrm{os}) = W_{F}(0,k^{T-1}_\mathrm{os}-1)$. This is because at the end of the first stage of period T (after open source technology use decisions are made), the technology level of the firm will be the same under the both cases (again by Lemma 1), and since the firm is not using the open source technology in period $T-1$, its own success at period $T-1$ will not be available for the other firms in period T, thus the two expected payoffs are the same. Thus,

$$\begin{aligned} V\left( 1,k^{T-1}_\mathrm{os}-1\right) - V\left( 0,k^{T-1}_\mathrm{os}-1\right)> C\left( p\left( 0,k^{T-1}_\mathrm{os}-1\right) \right) -C\left( p\left( 1,k^{T-1}_\mathrm{os}\right) \right) >0 \end{aligned}$$

which completes the proof. $\square$

1.2 Infinite Horizon

Here, we layout the infinite horizon version of our model. When there are infinitely many periods, we need to conduct an analysis using the value function and use steady state dynamics. At the beginning of period t, a nonuser firm i with technology level $k_{i}^{t}$, makes a decision $d_{i}^{t}$ whether to use the open source or not. After $d_{i}^{t}$ is realized, the technology level of firm i is then $\kappa _{_{i}}^{t}$. Then in the second stage of the same period, investment in cost reducing (or technology level advancement) is made, $p_{i}^{t}$. After the success/ failure outcome is realized, the new technology level of firm i is $K_{i}^{t}$, which is also equal to $k_{i}^{t+1}$. Thus, the cycle is represented as $k_{i}^{t}\Rightarrow _{d_{i}^{t}}\kappa _{_{i}}^{t}\Rightarrow _{p_{i}^{t}}K_{i}^{t}$, where $K_{i}^{t}=k_{i}^{t+1}$. Given $K_{i}^{t}$, in the last stage of period t, firm i picks its quantity level, $q_{K_{i}^{t}}$ in the Cournot competition.

At each period t, firm i chooses a triple of $\left( d_{i}^{t},p_{i}^{t},q_{i}^{t}\right)$ where each decision also depends on the firm’s technology level right before the decision made, the distribution of technology levels of other firms, and the innovation probability depends also on the usage decision of the firm. More precisely, $d_{i}^{t}=d(k_{i}^{t},N^{t})$, $p_{i}^{t}=p(d_{_{i}}^{t},\kappa _{i}^{t}, N^{t})$ and $q_{i}^{t}=q(K_{i}^{t}, N^{t})$, where $N^{t}=\left( N_{k_{1}}^{t},\ldots ,N_{k_{M}}^{t}\right)$ with $\sum _{k^{t}=k_{1}^{t}}^{k_{M}^{t}}N_{k}^{t}=M$ for each t, with with $N_{k}^{t}$ denoting the number of firms with technology level k at period t.¹² An equilibrium is characterized by $\left\{ d_{i}^{t},p_{i}^{t},q_{i}^{t}\right\} _{i,t}$. The value function of a firm i at period t is

$$\begin{aligned} V\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right)= & {} p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) \left[ E\left[ \pi _{k_{i}+1}^{t}\right] +\delta V\left( d_{i}^{t+1},\kappa _{i}^{t}+1, N^{t+1}\right) \right] \\&+\, \left( 1-p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) \right) \left[ E\left[ \pi _{k_{i}}^{t}\right] +\delta V\left( d_{i}^{t+1},\kappa _{i}^{t}, N^{t+1}\right) \right] -C\left( p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) \right) \end{aligned}$$

According to the open source license, $d_{i}^{t^{\prime }}=1$ if $d_{i}^{t=1}$ for any $t, t^{\prime }$ where $t^{\prime }>t$. Thus, it is enough to know $d_{_{i}}^{t}$ to define the value function at period t, rather than ${\mathbf {d}}_{i}^{t}=(d_{i}^{1},d_{i}^{2},\ldots ,d_{i}^{t})$. Also note that the distribution of others’ decisions ${\mathbf {d}}^{t}=({\mathbf {d}}_{1}^{t}, {\mathbf {d}}_{2}^{t},\ldots ,{\mathbf {d}}_{M}^{t})$ is already taken into account through $N^{t}$.

In Stage 3, firm i’s period-specific profit maximization problem is $\max _{q_{i}^{t}}E[\pi _{k_{i}}^{t}]=({\mathbf {P}} _{K^{t}}-c(k_{i}^{t}))q_{i}^{t}$. The optimal production level will be

$$\begin{aligned} q_{k_{i}}^{t}=\frac{1}{M+1}\left[ A-c(k_{i})-\sum _{k_{-i}}{\mathbf {N}} _{k_{-i}}\left( {\mathbf {c}}(k_{i}^{t}) -{\mathbf {c}}(k_{-i}) \right) \right] . \end{aligned}$$

In Stage 2, the optimal investment level $p_{i}^{t}=p(d_{i}^{t},\kappa _{_{i}}^{t}, N^{t})$ satisfies

$$\begin{aligned} C^{\prime }\left( p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) \right)&=E\left[ \pi _{\kappa _{i}+1}^{t}\right] -E\left[ \pi _{\kappa _{i}}^{t}\right] \nonumber \\&\quad +\,\delta \left[ V\left( d_{i}^{t+1},\kappa _{i}^{t}+1, N^{t+1}\right) -V\left( d_{i}^{t+1},\kappa _{i}^{t},N^{t+1}\right) \right] \end{aligned}$$

(4)

In Stage 1, the optimal use decision, $d_{i}^{t}=d(k_{i}^{t},N^{t})$, is determined by the comparison of $V(1,\kappa _{i}^{t},N^{t})$ and $V(0,\kappa _{i}^{t},N^{t})$. Arranging terms, we can write $V(d_{i}^{t},\kappa _{i}^{t},N^{t})$ as

$$\begin{aligned} V\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right)= & {} p\left( d_{i}^{t},\kappa _{i}^{t}, N^{t}\right) \left[ E\left[ \pi _{\kappa _{i}+1}^{t}\right] -E\left[ \pi _{\kappa _{i}}^{t}\right] \right. \\&\left. +\,\delta \left\{ V\left( d_{i}^{t+1},\kappa _{i}^{t}+1, N^{t+1}\right) -V\left( d_{i}^{t+1},\kappa _{i}^{t}, N^{t+1}\right) \right\} \right] \\&+\,E\left[ \pi _{\kappa _{i}}^{t}\right] +\delta V\left( d_{i}^{t+1},\kappa _{i}^{t}, N^{t+1}\right) -C\left( p\left( d_{i}^{t},\kappa _{i}^{t}, N^{t}\right) \right) \end{aligned}$$

Proof of Proposition 6

By using Eq. 4, we can rewrite $V(d_{i}^{t},\kappa _{i}^{t},N^{t})$ as

$$\begin{aligned} V\left( d_{i}^{t},\kappa _{i}^{t}, N^{t}\right)&=p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) C^{\prime }\left( p\left( d_{i}^{t},\kappa _{i}^{t},N^{t}\right) \right) -C\left( p\left( d_{i}^{t},\kappa _{i}^{t}, N^{t}\right) \right) +E\left[ \pi _{k_{i}}^{t}\right] \\&\quad +\,\delta V\left( d_{i}^{t+1},\kappa _{i}^{t},N^{t+1}\right) \end{aligned}$$

The difference of value functions between not use and use, simplifying the notation, is given by $V(0,\kappa _{i}^{t})-V(1,\max \{\kappa _{i}^{t},\kappa _\mathrm{os}^{t}\})$. Denoting $\max \{\kappa _{i}^{t},\kappa _\mathrm{os}^{t}\}=\kappa _{max}^{t}$, we can write

$$\begin{aligned} V\left( 0,\kappa _{i}^{t}\right) -V\left( 1,\max \{\kappa _{i}^{t},\kappa _\mathrm{os}^{t}\}\right)= & {} V\left( 0,\kappa _{i}^{t}\right) -V\left( 1,\kappa _{\max }^{t}\right) \\= & {} p\left( 0,\kappa _{i}^{t}\right) C^{\prime }\left( p\left( 0,\kappa _{i}^{t}\right) \right) -C\left( p\left( 0,\kappa _{i}^{t}\right) \right) \\&-\,\left[ p\left( 1,\kappa _{\max }^{t}\right) C^{\prime }\left( p\left( 1,\kappa _{\max }^{t}\right) \right) -C\left( p\left( 1,\kappa _{\max }^{t}\right) \right) \right] \\&+\,\delta \left( V\left( 0,\kappa _{i}^{t}\right) -V\left( 1,\kappa _{\max }^{t}\right) \right) \end{aligned}$$

Rearranging the terms, we have

$$\begin{aligned} \left( 1-\delta \right) \left[ V\left( 0,\kappa _{i}^{t}\right) -V\left( 1,\kappa _{\max }^{t}\right) \right]= & {} p\left( 0,\kappa _{i}^{t}\right) C^{\prime }\left( p\left( 0,\kappa _{i}^{t}\right) \right) -C\left( p\left( 0,\kappa _{i}^{t}\right) \right) \\&-\,\left[ p\left( 1,\kappa _{\max }^{t}\right) C^{\prime }\left( p\left( 1,\kappa _{\max }^{t}\right) \right) -C\left( p\left( 1,\kappa _{\max }^{t}\right) \right) \right] \end{aligned}$$

Note that the function $F(p)\equiv pC^{\prime }(p)-C(p)$ is increasing in p, since its first derivative is given by $F^{\prime }(p)=C^{\prime }(p)+pC^{\prime \prime }(p)-C^{\prime }(p)=pC^{\prime \prime }(p)>0$ where $C^{\prime \prime }>0$ and $p>0$. Thus, if $p(0,\kappa _{i}^{t})>p(1,\kappa _{\max }^{t})$, then $V(0,\kappa _{i}^{t})>V(1,\kappa _{\max }^{t})$, and it is optimal to not use the open source technology. If $p(0,\kappa _{i}^{t})<p(1, \kappa _{\max }^{t})$, then $V(0,\kappa _{i}^{t})<V(1,\kappa _{\max }^{t})$, and it is optimal to use the open source technology.

To see whether $p(0,\kappa _{i}^{t})>p(1,\kappa _{\max }^{t})$ or $p(0,\kappa _{i}^{t})<p(1,\kappa _{\max }^{t})$, we use Eq. 4 again to get

$$\begin{aligned} C^{\prime }\left( p\left( 0,\kappa _{i}^{t}\right) \right) -C^{\prime }\left( p\left( 1,\kappa _{\max }^{t}\right) \right)&=\delta \left[ V\left( 0,\kappa _{i}^{t}+1\right) -V\left( 0,\kappa _{i}^{t}\right) \right. \nonumber \\&\quad \left. - \left( V\left( 1,\kappa _{\max }^{t}+1\right) -V\left( 1,\kappa _{\max }^{t}\right) \right) \right] \end{aligned}$$

(5)

Also note that for any firm i, the period-specific profit $\pi _{k_{i}}^{t}$ is increasing in the number of competitors with an inferior technology to its own, and it is decreasing in the number of competitors with a superior (or same) technology level, that is, $\frac{\partial \pi _{k_{i}}^{t}}{ \partial N_{k-}^{t}}>0$ and $\frac{\partial \pi _{k_{i}}^{t}}{\partial N_{k+}^{t}}<0$, where $k_{-}<k_{i}\le k_{+}$. $\square$

(i) A firm that is ahead of the open source technology level Now, suppose firm i is a nonuser firm, at the end of $t-1$, with $k_{i}^{t} > k_\mathrm{os}^{t}$, that is, i is a firm that is ahead of the open source technology level at the start of period t. For such a firm, using the open source does not change own technology level, $\kappa _{\max }^{t}=\kappa _{i}^{t}$. Also, the marginal benefit of a jump in the technology for a nonuser firm is larger than the one for a user firm, because the nonuser firm will get the cost advantage on its own starting right after the jump in technology level, but if it starts using the open source technology, other open source firms will adopt this firm’s technology level and the firm will need to share that jump with all these other user firms from next period on (due to GPL), that is, $N_{k-}^{t}$ will decrease if the firm joins the open source community. Thus, the marginal benefit of the jump for this firm will be smaller than the one for the nonuser firm. Thus, we have $V(0,\kappa _{i}^{t}+1)-V(0,\kappa _{i}^{t})>V(1,\kappa _{i}^{t}+1)-V(1,\kappa _{i}^{t})$. This implies that the right-hand side of Eq. 5 is positive, thus, $C^{\prime }(p(0,\kappa _{i}^{t}))>C^{\prime }(p(1,\kappa _{i}^{t}))$. Since $C^{\prime \prime }>0$, we get $p(0,\kappa _{i}^{t})>p(1,\kappa _{i}^{t})$, implying $V(0,\kappa _{i}^{t})>V(1,\kappa _{i}^{t})$. This establishes that a nonuser firm, with $\kappa _{i}^{t} > \kappa _\mathrm{os}^{t}$, does not use the open source technology. Thus, for a firm the steady state decision, when it is ahead of the open source technology level, is $d^{*}(k)=0$ for $k>k_\mathrm{os}$. A firm that is ahead of the open source technology level in the first period will adopt this decision, and for any other period $t>1$, it will not use the open source technology level as long as it is ahead of it.

(ii) A firm that is behind the open source technology level Now suppose, firm i is a nonuser firm at the end of $t-1$, with $k_{i}^{t} < k_\mathrm{os}^{t}$, that is, i is a firm that is behind the open source technology level at the start of period t. For such a firm, using the open source technology changes its technology level, $\kappa _{\max }^{t}=\kappa _\mathrm{os}^{t}$.¹³ To show that using the open source technology level is optimal for this firm, first we make two observations.

First, if it is optimal for a firm with $k_{i}^{t} = k_\mathrm{os}^{t}-1$ to use the open source technology, then it is also optimal to use it for a firm with $k_{i}^{t} <k_\mathrm{os}^{t}-1$. This is clearly the case since when for the firm with $k_{i}^{t} =k_\mathrm{os}^{t}-1$, the net benefit of jumping one step up in the technology ladder dominates the net benefit of not using the open source, jumping more than one step must also dominate the net benefit of not using the open source. Thus, we can restrict attention to a firm that is behind by only one step: $k_{i}^{t} = k_\mathrm{os}^{t}-1$.

Secondly, if it is optimal to use the open source technology for a firm with $k_{i}^{t}=k_\mathrm{os}^{t}-1$ when the distribution of firms’ technology levels is $\{n(k_\mathrm{os}-1, t), n(k_\mathrm{os}, t),n(k_\mathrm{os}+1, t)\}$ with $M=n(k_\mathrm{os}-1, t) +n(k_\mathrm{os}, t) +n(k_\mathrm{os}+1, t)$, then it is also optimal for this firm to use the open source when the distribution of firms’ technology levels is $\{n(k_\mathrm{os}-1, t), n(k_\mathrm{os}, t),{\hat{n}}(k_\mathrm{os}+1, t),{\hat{n}}(k_\mathrm{os}+2, t),\ldots ,{\hat{n}}(k_\mathrm{os}+K, t)\}$ with $M=n(k_\mathrm{os}-1, t) +n(k_\mathrm{os}, t) +\sum _{s=1}^K {\hat{n}}(k_\mathrm{os}+s, t)$. The reason for this observation is that the benefit from using the open source under the former distribution is smaller than the benefit under the latter one. This is because the latter is a relatively worse distribution for this firm, and therefore, a jump from $k_\mathrm{os}^{t}-1$ to $k_\mathrm{os}^{t}$ has a higher benefit under the latter than under the former. Thus, if it is optimal to use the open source under the former distribution, it must also be optimal under the latter distribution.

By these two above observations, it suffices to show that it is optimal to use the open source for a firm when it is behind the open source technology by one step and the distribution of firms’ technology levels is given by $\{n(k_\mathrm{os}-1, t), n(k_\mathrm{os}, t),n(k_\mathrm{os}+1, t)\}$ with $M=n(k_\mathrm{os}-1, t) +n(k_\mathrm{os}, t) +n(k_\mathrm{os}+1, t)$.

Assuming that every other firm follows the usage decision rule, $d^{*}(k_{j}) =1$ for $k_{j} < k_\mathrm{os}$ and $d^{*}(k_{j}) =0$ for $k_{j} >k_\mathrm{os}$, we will show that the usage decision rule $d^{*}$ is a best response for this firm. To see this, consider the possible success/fail realization of this firm in the period t, which is behind the open source technology by one step, at the first stage of period t.

First, consider the realization where the firm is successful in innovation. If the firm had chosen to use the open source, the firm will have a technology level $k_\mathrm{os}+1$ at the competition stage of the current period and the within period profit will be $\pi _{k_\mathrm{os}+1}$. Let the continuation expected profit level be $V_{k_\mathrm{os}+1}^{d=1}$. If this firm had chosen not to use the open source technology, then with the success in innovation its technology level will be $k_\mathrm{os}$ at the competition stage of the current period and the within period profit will be $\pi _{k_\mathrm{os}}$. Let the continuation expected profit level be $V_{k_\mathrm{os}}^{d=0}$.¹⁴ Using the open source has an immediate marginal benefit: $\pi _{k_\mathrm{os}+1} -\pi _{k_\mathrm{os}}$. However, the effect of using the open source technology in terms of the expected continuation profit can be negative: $V_{k_\mathrm{os}}^{d=0}$ may be larger than $V_{k_\mathrm{os}+1}^{d=1}$ even though the starting technology level is one step behind that of the latter. This is because, regarding the former continuation profit, the firm does not need to share its successes with other open source firms, but in the latter it has to share them with one period lag, which may worsen the distribution of the technology levels against the favor of this firm. However, the continuation profit is discounted by $\delta$, and if $\delta$ is small enough, this effect will be smaller relative to the positive effect due to the immediate profit difference. Or, if the current size of the open source community, $n(k_\mathrm{os},t)$, is large enough (which will be the case if the initial size of the open source community, $n(k_\mathrm{os},1)$, is large enough), the likelihood of the open source technology to advance in the technology will more likely and this will decrease the negative effect of the obligation to share the success since the open source technology will advance regardless. Thus, again the negative effect will be smaller relative to the positive effect of the larger immediate profit. Notice that as long as a nonuser firm’s innovation outcome at period t is made independently, we may ignore its effect on the expected technology distribution. Hence, if either one of these two above conditions is met, we will have $\pi _{k_\mathrm{os}+1} -\pi _{k_\mathrm{os}} > \delta [V_{k_\mathrm{os}}^{d=0}-V_{k_\mathrm{os}+1}^{d=1}]$, and it will be optimal to use the open source technology, when the firm is behind it.

Now, we look at the other realizations, that is, the firm’s investment in innovation fails at the current period. Once again using the open source technology level now has an immediate positive effect: $\pi _{k_\mathrm{os}} -\pi _{k_\mathrm{os}-1}$. And, regarding the continuation profit, the effect of using the open source may be negative: $V_{k_\mathrm{os}-1}^{d=0}-V_{k_\mathrm{os}}^{d=1}$. However, since the firm has failed, this effect (potentially negative) is weaker than the one when the firm has succeeded (using the open source does not affect the distribution of technology levels in the next period, since the firm has no success to share). Thus, if either one of the conditions given above holds, then we will once again have $\pi _{k_\mathrm{os}} -\pi _{k_\mathrm{os}-1} > \delta [V_{k_\mathrm{os}-1}^{d=0}-V_{k_\mathrm{os}}^{d=1}]$, and it again will be optimal to use the open source technology, when the firm is behind it.

The optimal decision whether to use or not to use the open source technology is based on the overall expected profit from using and the expected profit from not using. The net effect of using the open source technology on the overall expected profit is a linear combination of the effects depicted for the two success/fail realizations, weighted by the probabilities of innovation. Assuming at least one of the two conditions is met (either $\delta$ is small enough, or $n(k_\mathrm{os},1)$ is large enough, or both), since using the open source technology makes the firm better off under each realization, the expected overall effect of using the open source technology will be positive relative to not using it. It follows that it is optimal for this firm i to use the open source technology, if it is behind it.

Therefore, the usage decision function will be $d^{*}(k_{i})=1$ for $k_{i} < k_\mathrm{os}$ and $d^{*}(k_{i}) =0$ for $k_{i} > k_\mathrm{os}$ for any i. In the equilibrium, each firm will use this decision rule $d^{*}(k_{i})$ in every period. $\square$

1.3 The Definitions for All Parameters and the Decision Variables

See Tables 1, 2 and 3.

Table 1 Model parameters

Table 2 Decision variables

Table 3 Flow variables