The effects of market size, wealth, and network effects on digital piracy and profit

Abstract

The effect of digital piracy is often framed as a creator having to compete against unauthorized copies of their own creation, despite intellectual property rights laws. This framing has empirical and theoretical support, but the empirical findings often suggest that the magnitude and even sign of piracy effects depend on the characteristics of the software and the market. For example, piracy seems to have a larger negative effect on sales of high-profile works, but a smaller and perhaps even positive effect on lesser-known works. This paper seeks a theoretical explanation of differential piracy effects. It is unique in that it gives considerable focus to the market size, and also to budgetary limitations of the consumer base, motivated by high piracy rates in developing countries. The models imply that piracy is more likely to help developers when the market for the software is smaller; when network effects are neither too weak nor too strong; when piracy is neither to accessible nor too inaccessible; when the cost of piracy is relatively homogeneous; and when the consumer base is not too poor. All things considered, the inclusion of market size, consumer budgets, and heterogeneous piracy costs suggest that piracy is less likely to be beneficial to developers than previous literature suggest. Developer profit may be higher or lower with piracy, but buyer welfare is no worse, and is sometimes better, with piracy.

Appendices

Appendix 1: Proofs for Section 2 (a world without piracy)

Proposition 1

If $\eta < \mu /N$ and the threat of piracy does not exist, then price is higher when market size, consumer budgets, or network effects are larger. Furthermore, development expenditure is also larger when consumers budgets are larger.

Proof

Applying the quotient rule on $p^*$ gives respective numerators of

$$\begin{aligned}&\mu ^2(N^2\eta ^3 + N^2\eta ^2 + 2N\eta ^2\mu + \eta \mu ^2 + \mu ^2),\\&\quad N^2(N^2\eta ^4 + 2N\eta ^3\mu + 2N\eta ^2\mu + \eta ^2\mu ^2 + 4\eta \mu ^2 + 2\mu ^2),\\&\quad N\mu ^2(N^2\eta ^2 + 2N^2\eta + 2N\eta \mu + \mu ^2), \end{aligned}$$

all of which are unambiguously positive.

With respect to quality and consumer budgets, the derivative is

$$\begin{aligned} \frac{\partial q^*}{\partial \mu } = \frac{N^2( 2N\eta ^2\mu + 3\eta \mu ^2 + 2\mu ^2- N^2\eta ^3)}{(N^2\eta ^2 + 2N\eta \mu + 2N\mu + \mu ^2)^2}, \end{aligned}$$

which is positive when $2N\eta ^2\mu + 3\eta \mu ^2 + 2\mu ^2> N^2\eta ^3$. When $\eta = \mu /N$, the condition can be written as $4\mu /N + 2 > 0$, which is clearly true. Now increase $\mu$ by any amount. The case requirement $\eta < \mu /N$ holds; the left-hand side of the positivity requirement increases; whereas the right-hand side of the positivity requirement is constant; together establishing that the positivity requirement does in fact hold whenever $\eta < \mu /N$ holds. $\square$

Proposition 2

If $\eta < \mu /N$ and the threat of piracy does not exist, then development expenditure is decreasing in network effects. Furthermore, development expenditure approaches zero when network effects approach $\mu /N$ and market size approaches $\mu /\eta$.

Proof

Simply plug in $\eta = \mu /N$ and $N = \mu /\eta$ into Eq. (5), in which case the numerators are zero and denominators non-zero. $\square$

Proposition 3

If $\eta < \mu /N$ and the threat of piracy does not exist, then expected profit is higher when market size, consumer budgets, or network effects are larger.

Proof

Using the quotient rule, the respective numerators are

$$\begin{aligned}&N\mu ^3(1 + 2\eta )(N + \mu + N\eta ),\\&\quad N^3\mu (1 + 2\eta )(N\eta ^2 + \mu \eta + \mu )\\&\quad N^2\mu ^2( N\mu + \mu ^2- N^2\eta ^2 - N^2\eta ). \end{aligned}$$

The first two for market size and consumer budgets are clearly positive. The third for $\eta$ is positive as long as $N\mu + \mu ^2 > N^2\eta ^2 + N^2\eta$. Suppose $\eta = \mu /N$. Then the condition for positivity becomes an equality; but for any smaller $\eta$, the left-hand side will remain constant whereas the right-hand side will decrease, thereby satisfying the constraint. $\square$

Proposition 4

Buyer welfare is weakly increasing with network effects when there is no piracy.

Proof

Buyer welfare is zero when $\eta < \mu /N$. Buyer welfare is $(N\eta - \mu )/2$ when $\eta \ge \mu /N$, which is positive over the interval and increasing linearly with $\eta$. $\square$

Appendix 2: Proofs for Section 3 (a world with piracy)

Proposition 5

Piracy cannot be beneficial to developers when network effects are very weak or very strong.

Proof

When $\eta =0$, there is no benefit to adding pirate users, but there is the cost of increasing quality to attract pirate users; therefore the developer will not bother incentivizing budget-constrained pirates when $\eta =0$. By continuity, this is also true of sufficiently small $\eta >0$: the benefit of adding pirate users is small, and adding them requires a relatively large cost in increasing quality to incentivize budget-constrained pirates. It can be concluded that piracy is harmful for developers when network effects are weak.

Now suppose $\kappa < \mu /2$. Large $\eta \ge \mu /N$ implies that piracy-exploiting profit becomes $(1 - \kappa /\mu )N\kappa$, which is strictly less than piracy-free profit $N\mu /4$. Because the inequality is strict, it also holds for some smaller $\eta$ as well.

Finally, suppose $\kappa \ge \mu /2$. Large $\eta \ge \max \{\kappa /N, \mu /N \}$ implies that profit is $N\mu /4$ with or without piracy. $\square$

Proposition 6

If a consumer base is moderately budget-poor, then a range of network effects exists such that piracy is beneficial to developers. But if a consumer base is extremely budget-poor, then no strength of network effects exists such that piracy is beneficial to developers.

Proof

Suppose $\mu /2 \le \kappa < \mu$. It follows that $\kappa /N < \mu /N$. Let $\eta = \mu /N$, so that piracy-free profit and piracy-exploiting profit are both $N\mu /4$. Now reduce $\eta$ by $\epsilon$. It is no longer the case that $\eta \ge \mu /N$, so piracy-free profit falls; but it is still true that $\eta \ge \kappa /N$, so piracy-exploiting profit remains at $N\mu /4$.

Now suppose $\mu \le \kappa$. It follows that $\mu /N \le \kappa /N$. Let $\eta = \kappa /N$ so that piracy-free profit and piracy-exploiting profit are both $N\mu /4$. Now reduce $\eta$ by $\epsilon$. It is no longer the case that $\eta \ge \kappa /N$, so piracy-exploiting profit falls; but it is still true that $\eta \ge \mu /N$, so piracy-free profit remains at $N\mu /4$.

Piracy-exploiting profit falls at a faster or equal pace than piracy-free profit when

$$\begin{aligned} N\mu ^2(- N^2\eta ^2 - N^2\eta + N\mu + \mu ^2) \le (\kappa - N\eta )(N^2\eta ^2 + 2N\eta \mu + 2N\mu + \mu ^2)^2. \end{aligned}$$

The left-hand side is zero when $\eta = \mu /N$, whereas the right-hand side is non-negative since $\kappa /N \ge \mu /N$. When $\eta =0$, the left-hand side is less than the right-hand side as an implication of $\kappa \ge \mu$. The left-hand side is monotonically decreasing in $\eta$ over this range, whereas the polynomial behavior of the right-hand side implies that the larger (i.e. relevant) vertex is concave, demonstrating continued dominance of piracy-free profit. $\square$

Proposition 7

Suppose the consumer base is budget-rich. If either the cost of piracy is sufficiently small or the market size is sufficiently large, then no range of network effects exists that makes piracy beneficial to developers. But if the cost of piracy is sufficiently large and the size of the market is sufficiently small, then a range of network effects exists that makes piracy beneficial to developers.

Proof

First consider a low cost of piracy. Specifically, suppose $\mu /2 > \kappa$, and let $\kappa =0$. Piracy-exploiting profit as given in Eq. (11) is either negative or zero, which is less than piracy-free profit as given by Eqs. (6) or (8). By continuity, this must also hold for small but positive $\kappa$. This establishes the first part of the proposition.

Now consider market size. Suppose $\kappa < \mu /2$ and take $N \rightarrow \infty$. It follows that $\eta \ge \mu /N$. This also implies that $\eta \ge \kappa /N$ since $\mu > \kappa$. In this scenario, piracy-free and piracy-exploiting profits are respectively $N\mu /4$ and $(1-\kappa /\mu )N\kappa$. Denied piracy has higher profit because

$$\begin{aligned} \left( 1 - \dfrac{\kappa }{\mu } \right) N\kappa < \frac{N\mu }{4} \quad \implies \quad (\mu - 2\kappa )^2 > 0, \end{aligned}$$

the strict inequality justified by $\kappa < \mu /2$. Because the inequality is strict, there is room for the preceding inequality to be maintained even for finite but large N by continuity.

Finally, consider a larger value of $\kappa$ and a smaller value of N. Specifically, suppose $\kappa = \mu /2$ and $\eta = \kappa /N$. Piracy-exploiting profit is $N\mu /4$, whereas piracy-free profit is given by Eq. (6). Piracy-exploiting profit is higher when $0 < (N \eta - \mu )^2$, which is clearly true since $\eta = \kappa /N < \mu /N$. By strictness of the inequality and by continuity, this conclusion must also hold for a range of $\kappa$ around $\mu /2$ and a range of $\eta$ around $\kappa /N$. $\square$

Proposition 8

Buyer welfare is weakly increasing with network effects when there is piracy.

Proof

First consider denied piracy. If piracy is denied and non-binding, then the outcome is exactly the same as with no piracy, so $W_B=0$. If piracy is denied and binding, then p and q are constrained such that $q = \kappa - N\eta$ and $p = \mu \kappa /(\mu + N\eta )$, which also generates $W_B=0$.

Now consider exploited piracy with a budget-rich consumer base. If $\mu /2> \kappa$ and $\eta < \kappa /N$, then $W_B=0$. If $\mu /2> \kappa$ and $\eta \ge \kappa /N$, then $W_B = N\eta - \kappa$, which is clearly increasing in $\eta$.

Finally, consider exploited piracy with a budget-poor consumer base. If $\mu /2 \le \kappa$ and $\eta < \kappa /N$, then $W_B = \kappa - \mu /2$. If $\mu /2 \le \kappa$ and $\eta \ge \kappa /N$, then $W_B = N\eta - \mu /2$, which is clearly increasing in $\eta$. $\square$

Proposition 9

Buyer welfare is no worse, and is sometimes better, with exploited piracy relative to a world with no piracy.

Proof

Let us first give expressions for buyer welfare. In a world without piracy, buyer welfare is zero when $\eta \le \mu /N$ and is $(N\eta - \mu )/2$ otherwise. If the consumer base is budget-rich and $\eta < \kappa /N$, then exploiting piracy gives zero buyer welfare; if $\eta \ge \kappa /N$, then $W_B = N\eta - \kappa$. If the consumer base is budget-poor and $\eta < \kappa /N$, then exploiting piracy gives $W_B = \kappa - \mu /2$; if $\eta \ge \kappa /N$, then $W_B = N\eta - \mu /2$.

Suppose the consumer base is budget-rich, and that $\eta \ge \kappa /N$. Then $W_B = N\eta - \kappa$ with exploited piracy, and is either $(N\eta - \mu )/2$ or zero without piracy. Exploited piracy gives weakly higher buyer welfare because $N\eta - \kappa \ge 0$ and because $\kappa < \mu$ implies that $N\eta - \kappa > (N\eta - \mu )/2$.

Now suppose that the consumer base is budget-rich, and that $\eta < \kappa /N$. There are three subcases. if $\eta< \kappa /N < \mu /N$, then buyer welfare is zero in either case. If $\kappa /N \le \eta \le \mu /N$, then buyer welfare with no piracy is zero but with exploited piracy is $\max \{N\eta - \kappa ,0\}$. If $\kappa /N < \mu /N \le \eta$, then buyer welfare with exploited piracy is $N\eta - \kappa$ and without piracy is $(N\eta - \mu )/2$. Suppose no piracy gives higher welfare, which is equivalent to $N\eta < 2\kappa - \mu$. The consumer base is assumed budget-rich, which implies that $N\eta< 2\kappa - \mu < 0$, which cannot be the case since $N\eta \ge 0$. It follows that buyer welfare is again highest with exploited piracy.

Now suppose that the consumer base is budget-poor, and that $\eta < \kappa /N$. When $\eta \le \min \{\kappa /N, \mu /N \}$, buyer welfare with exploited piracy is $\kappa - \mu /2>0$ but is zero with no piracy. When $\mu /N \le \eta < \kappa /N$, buyer welfare with exploited piracy is $\kappa - \mu /2>0$, which is larger than buyer welfare of $(N\eta - \mu )/2$ without piracy whenever $\eta< 2\kappa <N$, which is already satisfied as an implication of $\eta < \kappa /N$.

Finally, suppose that the consumer base is budget poor, and that $\eta \ge \kappa /N$. If $\eta \ge \max \{\kappa /N, \mu /N\}$, then buyer welfare with exploited piracy is $N\eta - \mu /2$, which is clearly larger than the no-piracy counter part of $(N\eta - \mu )/2$. When $\kappa /N \le \eta < \mu /N$, buyer welfare with exploited piracy is $N\eta - \mu /2 > 0$, which is greater than no-piracy buyer welfare of zero.

So in all possible cases, exploiting piracy leads to equal or greater, and sometimes strictly greater, buyer welfare than the case where piracy does not exist at all. $\square$

Appendix 3: Heterogeneous piracy cost

Let $\Phi (\cdot )$ denote the truncated normal cumulative distribution function, and consider the market with a budget-rich consumer base that satisfies $\sigma > p/\mu$. Budgets are still assumed uniformly distributed, so proportion $1 - p/\mu$ of the market is under consideration. Of those users, the proportion who will rather purchase than pirate is given by $1 - \Phi (p/\kappa )$, provided $p \le q + C(p,q)N\eta$. This gives a proportion of buyers of

$$\begin{aligned} B(p,q) = \left[ 1 - \Phi \left( \frac{p}{\kappa } \right) \right] \left[ 1 - \frac{p}{\mu } \right] . \end{aligned}$$

(13)

For those with $\gamma < p/\kappa$, piracy is so cheap that they’d rather pirate; and piracy is better than abstaining if $\gamma \le [q + C(p,q)N\eta ]/\kappa$. Of these two constraints, $\gamma < p/\kappa$ is tighter as an implication of $p \le q + C(p,q)N\eta$, so it can be concluded that the proportion of pirates in this segment of the market is

$$\begin{aligned} \Phi \left( \frac{p}{\kappa }\right) \left[ 1 - \frac{p}{\mu }\right] . \end{aligned}$$

And now of the $p/\mu$ proportion of users who cannot afford the software, those who satisfy $\gamma \le [q + C(p,q)N\eta ]/\kappa$ will pirate. So in this segment of the market the proportion of pirates is

$$\begin{aligned} \Phi \left( \frac{q + CN\eta }{\kappa }\right) \left[ \frac{p}{\mu }\right] . \end{aligned}$$

Ergo the total proportion of pirates is

$$\begin{aligned} P(p,q) = \Phi \left( \frac{p}{\kappa }\right) \left[ 1 - \frac{p}{\mu }\right] + \Phi \left( \frac{q + C(p,q)N\eta }{\kappa }\right) \left[ \frac{p}{\mu }\right] , \end{aligned}$$

(14)

and the total proportion of users C(p, q) is given by the sum of Eqs. (13) and (14), which can be simplified into

$$\begin{aligned} C(p,q) = \left[ 1 - \frac{p}{\mu } \right] + \Phi \left( \frac{q + C(p,q)N\eta }{\kappa }\right) \left[ \frac{p}{\mu }\right] . \end{aligned}$$

(15)

The preceding equation pins down C, which in turn allows B and P to be pinned down.