Optimal Piracy Control: Should a Firm Implement Digital Rights Management?

Abstract

Many firms offer digital rights management (DRM) technologies to increase the piracy cost, thereby protecting illegal copy and distribution. However, many industrial cases contradict the speculation that using DRM technologies reduces legal users’ net value. Therefore, we develop a theoretical model to examine the trade-off based on consumer heterogeneity in piracy cost sensitivity to explore under what conditions a monopoly implements DRM policy. In the model, we categorize pirates into two types according to their behaviours in the piracy process. We find that adopting DRM restriction is profitable to the firm when the level of DRM restriction is high. The optimal restriction level is the maximal technological level when piracy cost is high and the maximal technological level is higher than a threshold. In addition, we derive that the firm should implement policies to prevent downloading unauthorized products shared by other legal users from a P2P network. When both piracy cost and DRM restriction level are low, a lower piracy cost increases the firm’s profit.

Appendix

1.1 Proof of Proposition 1

Differentiating ${D}_P^{\ast }$ with respect to f, we derive that $\frac{\partial {D}_P^{\ast }}{\partial f}=\frac{-\gamma \left(\delta -1\right)\beta -\left(h-1\right)\beta v}{2\delta \Big){f}^2}$. Therefore, we obtain that $\frac{\partial {D}_P^{\ast }}{\partial f}>0$ when $\updelta <1+\frac{v- hv}{\gamma }$, while we derive $\frac{\partial {D}_P^{\ast }}{\partial f}<0$ when $\updelta >1+\frac{v- hv}{\gamma }$.

1.2 Proof of Proposition 2

Differentiating π^∗ with respect to f, we derive that

$$ \frac{\partial {\pi}^{\ast }}{\partial f}=-\frac{\left(\left(\updelta -1\right)\upgamma \upbeta +\updelta \mathrm{f}-\upbeta \mathrm{v}\left(\mathrm{h}-1\right)\right)\left(\left(\updelta -1\right)\upgamma \upbeta +\updelta \mathrm{f}+\upbeta \mathrm{v}\left(\mathrm{h}-1\right)\right)}{4\delta {f}^2}. $$

We divide the results into several parts as follows:

• (a) when $\updelta <1+\frac{v- hv}{\gamma }$, we have

• (1) if $\mathrm{f}<-\frac{\beta \left(\delta \gamma + hv-\gamma -v\right)}{\delta }$, we derive $\frac{\partial {\pi}^{\ast }}{\partial f}<0$.

• (2) if $\mathrm{f}>-\frac{\beta \left(\delta \gamma + hv-\gamma -v\right)}{\delta }$, we derive $\frac{\partial {\pi}^{\ast }}{\partial f}>0$.

• (b) when $\updelta >1+\frac{v- hv}{\gamma }$, we have $\frac{\partial {\pi}^{\ast }}{\partial f}>0$.

1.3 Proof of Proposition 3

Differentiating ${D}_P^{\ast }$ with respect to β, we derive that $\frac{\partial {D}_P^{\ast }}{\partial \beta }=\frac{\updelta \upgamma -\upgamma +\mathrm{v}\left(\mathrm{h}-1\right)}{2\delta f}.$ Therefore, we obtain that $\frac{\partial {D}_P^{\ast }}{\partial \beta }>0$ when $\updelta >1+\frac{v- hv}{\gamma }$, while we derive $\frac{\partial {D}_P^{\ast }}{\partial \beta }<0$ when $\updelta <1+\frac{v- hv}{\gamma }$.

1.4 Proof of Proposition 4

Differentiating π^∗ with respect to β, we derive that

$$ \frac{\partial {\pi}^{\ast }}{\partial \beta }=\frac{\left(\gamma -\delta \gamma +v\left(1-h\right)\right)\left(\delta f+\beta \left(v\left(1-h\right)+\gamma \left(1-\delta \right)\right)\right)}{2\delta f}>0. $$

1.5 Proof of Proposition 5

Differentiating ${D}_P^{\ast }$ with respect to δ, we derive that $\frac{\partial {D}_P^{\ast }}{\partial \delta }=-\frac{\beta \left(\left(h-1\right)v-\gamma \right)}{2f{\delta}^2}>0$.

1.6 Proof of Proposition 6

Differentiating π^∗ with respect to δ, we derive that

$$ \frac{\partial {\pi}^{\ast }}{\partial \delta }=\frac{\left(\updelta \mathrm{f}+\upbeta \left(\left(\mathrm{h}-1\right)\mathrm{v}-\upgamma -\updelta \upgamma \right)\right)\left(\updelta \mathrm{f}+\upbeta \left(\mathrm{v}\left(1-\mathrm{h}\right)+\upgamma -\updelta \upgamma \right)\right)}{4f{\delta}^2}. $$

We divide the results into several parts as follows:

• (a) when βγ > f, we derive $\frac{\partial {\pi}^{\ast }}{\partial \delta }<0$.

• (b) when γβ < f, we have $\frac{\partial {\pi}^{\ast }}{\partial \delta }<0$ if and only if $\updelta <-\frac{\upbeta \left(\mathrm{v}+\upgamma -\mathrm{hv}\right)}{\beta \gamma -f}$, while we derive $\frac{\partial {\pi}^{\ast }}{\partial \delta }>0$ if and only if $\updelta >-\frac{\upbeta \left(\mathrm{v}+\upgamma -\mathrm{hv}\right)}{\beta \gamma -f}$.

1.7 Proof of Proposition 7

Note that ${\pi}^{\ast }-{\pi}_N^{\ast }=\frac{\left(\updelta -1\right)\left(\left({\gamma}^2\updelta -{\left(\left(h-1\right)v-\gamma \right)}^2\right)\right){\beta}^2-2 f\beta \gamma \delta +\delta {f}^2\Big)}{4\delta f}$. Thus, we have ${\pi}^{\ast }>{\pi}_N^{\ast }$ if $\delta >\frac{\beta^2\left({\left( hv-\gamma \right)}^2+v\left(v-2 hv+2\gamma \right)\right)}{{\left(f-\beta \gamma \right)}^2}$, while we have ${\pi}^{\ast }<{\pi}_N^{\ast }$ if $\delta <\frac{\beta^2\left({\left( hv-\gamma \right)}^2+v\left(v-2 hv+2\gamma \right)\right)}{{\left(f-\beta \gamma \right)}^2}$.

1.8 Proof of Proposition 8

The firm optimize its profit $\pi =p\left(1-\frac{\beta \delta \gamma + h\beta v-\beta \gamma +p-\beta v}{f\delta}\right)$ by determining the optimal price. The optimal price is characterized by the first-order conditions of π and we can derive the price p(δ) = (fδ + βv + βγ − βδγ − βvh)/2. Substituting this price into the profit function, we derive the following results:

• (a) when βγ > f, we derive δ^∗ = 1.

• (b) when βγ < f, we have δ^∗ = 1 if and only if $\overline{\delta}<\frac{\beta^2\left({\left( hv-\gamma \right)}^2+v\left(v-2 hv+2\gamma \right)\right)}{{\left(f-\beta \gamma \right)}^2}$, while we derive ${\delta}^{\ast }=\overline{\delta}$ if and only if $\overline{\delta}>\frac{\beta^2\left({\left( hv-\gamma \right)}^2+v\left(v-2 hv+2\gamma \right)\right)}{{\left(f-\beta \gamma \right)}^2}$.