When Should Software Firms Commercialize New Products ... - MISRC

and αb +δb ≥ 0.75. Then π ∗ CE ≤ 0.25c, π∗ TLF = 0.25c, and π∗ FLF
≥ 0.3c. By switching to
FLF, the firm can achieve at least 20% profit increase. Thus, it is important for the firm
not just to consider implementing any freemium model, but to carefully choose between the
available freemium options.
8 Social Welfare Analysis
It is straightforward to show that, under a central planner economy, the optimal price
under each of the five models is zero due to sunk costs. In this section we focus on the more
interestingsocial welfareimplications whenthefirmchoosesitsoptimal pricingstrategy. We
denote by SW ∗ M
the social welfare corresponding to the profit maximizing strategy under
model M, where M ∈ {CE,FLF,TLF,SS,CS}. Define function φ : [0,13 −4 √ 10] → R,
1+2x+2x φ(x) = 1− 2
2(1+x)( √ 1+x+ √ 2x) 2. We show in the Appendix in Lemma A6 that φ is strictly
increasing over the interval [0,13−4 √ 10], with φ(0) = 1
2 and φ(13−4√10) ≈ 0.82 < 7
8 < 1.
Then, we have the following social welfare ranking:
Theorem 2. The following hold true:
(a) SW∗ TLF > max{SW∗ CE ,SW∗ SS ,SW∗ CS } for any α > 0;
(b) SW ∗ FLF ≥ SW∗ TLF
(i) γ ≤ 1
3 ; or
(ii) 1
3 < γ ≤ 1 and φ−1
(iii) 1 < γ <
if and only if one of the following conditions is satisfied:
7γ−1
8γ
≤ αb +δb; or
1
7−8φ(13−4 √ ≈ 2.262 and φ−1
10)
7γ−1
8γ
≤ αb +δb < 13−4 √ 10.
The results in Theorem 2 are illustrated in Figure 3. Note that, when the firm max-
imizes profit, social welfare under CE, SS, and CS models is always smaller than under
TLF model. Thus, highest social welfare will be attained under either FLF or TLF. In
other words, freemium, in one form or another, is always preferred from the society’s per-
spective. The adoption dynamics for FLF can be explained through Figure 1 with αb +δb
instead of α on the horizontal axis. When αb + δb ≥ 13 − 4 √ 10, then, under FLF, we
have only period 1 adoption and exactly half of the potential customers purchase module
B. Therefore, under both FLF and TLF, high-type customers (with type θ ≥ 1
2 ) get to
use both modules A and B for two periods. The difference is that, under FLF, low-type
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