For the last strategy, the institution will now offer both the certificate price and the monthly subscription fee for learners in the mixed model. We have proved that most of the time the mixed model will outperform the other two revenue models in the previous chapter. By properly implementing the pricing strategy, the institution can not only pre-vent learners from pretending other types, but eliminate the potential determinant cause by irrational efficient learners. In this section, we will also focus on the differentiating strategy, and further consider the effect of externality.
To differentiate learners, the institution now has two choice. One is to induce different type of learners choosing the different revenue model, and the other is to induce different type of learners choosing the different learning attitude. For the first condition, the institution will induce the efficient learner to choose the subscription model so that they can paying less amount then choosing the certificate model, and let inefficient learners choose the certificate model. That is to say, the two price should compliance with the restriction that fM ≤ pM ≤ tfM. For the second condition, since the threshold for efficient learners to be assiduous is higher than for inefficient learners, the institution then can induce only the inefficient one to differentiate the learners. In this case, she just need to
Figure 5.3: Number of consumers and the paying amounts of the mixed model.
set the monthly subscription fee fM within the range that the efficient learners will not exert effort and the inefficient learners will, which can be characterized as thk ≤ fM ≤ kh. It is worth mentioning that, there will be positive externality if diligent learners exist in this particular ecosystem. So, sometimes for the external benefits, the institution may not want to differentiate learner by their learning attitude, but induce all of them to be diligent. To meet this, the monthly subscription fee fM should be greater than kh.
Given the certificate price pM and the monthly subscription fee fM, learners first need to compare the three options (purchasing the certificate, subscribing to the course, and paying nothing), and decide whether to exert effort if choosing the subscription model to obtain the highest utility. The sequence and the following paying amount can be shown in Figure 5.3.
Again, by setting different certificate price and monthly subscription fee, learners will consequently make different choice, thus separates the market into several regions.
Figure 5.4 depicts the above phenomenon. The institution can determine the target segment after solving the profit maximization problems of each region. Collectively, the institution’s profit maximum function are:
Region A (Both of the learners choose the subscription model and exert effort):
max β(1− fM)fM + (1− β)(1 − tfM)tfM s.t. fM ≤ k
th
(5.11)
Region B (Both of the learners choose the subscription model. Efficient learners are undiligent, and Inefficient learners are diligent):
max β(1− fM + ηx)fM + (1− β)[1 − (1 − h)tfS− k + ηx](1 − h)tfM
s.t. k
th ≤ fM ≤ k h
(5.12)
, Region C (Both of the learners choose the subscription model and exert no effort.):
max β[1− (1 − h)fM − k + ηx](1 − h)fM
+ (1− β)[1 − (1 − h)tfM − k + ηx](1 − h)tfM
s.t. k h ≤ fM
(5.13)
Region D (The efficient learners choose the subscription model and exert no effort. The inefficient learners choose the certificate model):
max β(1− fM)fM + (1− β)(1 − pM)pM
Region E (The efficient learners choose the subscription model and exert effort. The
inefficient learners choose the certificate model):
Region F (Both the efficient learners and the inefficient learners choose the certificate model and exert no effort):
max (1− pM)pM
s.t. pM ≤ fM
(1− h)fM + k ≤ pM ≤ (1 − h)tfM + k.
(5.16)
The optimal price corresponding to each region are presented in Proposition 6.
Proposition 6. The optimal certificate price, optimal monthly subscription fee, and
equilibrium profit under the mixed model are:
Region A:
Figure 5.4: Market segment for the mixed model. (t = 1.2, η = 0.8, h = 0.6, k = 0.3, and β = 0.7)
and
k(h(β+(1−β)(t(1+k(2t−1))+k(t−1)βη))−h2(1+k(t−1))t(1−β)−k(β+(1−β)(t2+(t−1)βη)))
h2(1−(1−β)η)) , if fSB∗ = kh
p∗M E =
1. If there exists no externality (η = 0), the mixed model and the certificate model are equally good. Moreover, they both strictly outperforms the subscription model (πM∗ = π∗C > πS∗).
2. If externality exists (η > 0), then sometimes the subscription model is better than the certificate model (πS∗ > πC∗), and sometimes the mixed model strictly outperforms them (πM∗ ≥ π∗C and πM∗ ≥ π∗S).
Proposition 7 also brings some insights to us. First, if there’s no external profit come with the learners’ effort, the institution will have no reason to introduce the subscription model, because the increasing amount of learner is not high enough to compensate the
total reduced subscription fee. In this way, the certificate model can bring the maximum profit for the institution.
However, we can observe that, if there exists positive externality, which means the effort of a learner can benefit all the other learners, then it will attract more learners willing to be diligent and form a virtuous cycle, eventually brings higher profit for the whole ecosystem.
This can somehow explains why there are usually forums and group works on Coursera or other major MOOC platforms. By these mechanisms, platforms can enhance the communication between learners, and broaden the externality of efforts, end up increasing the total social welfare. As we mentioned before, to make more profit is not the only reason why institutions offer MOOCs. Same reasons can apply to the pricing strategies.
From the above discussion we can find that, the introduction of the subscription model not only increasing the profitability, but also encourage more learners to be diligent and create a better learning environment, which is a win-win strategy.
Figure 5.5 shows the optimal pricing strategy with different h, k, and η. We concludes our observations in Observation 1.
Observation 1. From Figure 5.5a we can observe that,
1. When k is large, region F is the most profitable.
2. When k is small and h is large, region E becomes the most favorable.
3. When η gets larger, it is more suitable for the institution to adopt the subscription model, which we can see the range for region C and region E also become larger.
We can further extend Observation 1 that, when k is small, the institution should encourage the efficient learners to be diligent, on the other hand, when k is large, it is not worth to encourage learners to be diligent. Besides, as externality gets larger and larger, it will be more profitable for the institution to differentiate the learners.
(a) η = 0.2 (b) η = 0.5
(c) η = 0.8 (d) η = 1
Figure 5.5: Profits comparison
Chapter 6
Conclusion
In this paper, we present a game-theoretic model featuring learner’s rationality to study MOOCs pricing strategy. We introduce three revenue models including the certificate model, the subscription model, and the mixed model. We also discuss two important factors follow from learner heterogeneity that may effect the probability of institution, which are rationality and diligence.
In general, if learners all are fully rational, the platform never needs to consider the subscription option. However, the bounded rationality of learners makes the subscription option more attractive. Though a pure subscription model is never optimal, the mixed model combining subscription and certificate is the most favorable. Interestingly, while the existence of irrational inefficient learners benefits the institution and platform, it does not matter whether irrational efficient learners exist or not. The potential detriment brought by the irrational efficient learners can somehow be eliminated by the optimal mixed pricing strategy.
Beside, if learner all refuse to exert effort while taking the course, the subscription model will be helpless in increasing the institution’s profit. But as long as there exists diligent learners, their power will spread into the whole ecosystem and trigger external benefit, and make the subscription model more profitable.
Both of the discussions provide strong evidence for the introduction of the subscrip-tion model, but there are still several possible ways to extend this study. For instance, learners may stop subscribing to the course. Including this will broaden our analysis about the subscription model. Moreover, we assume that a learner is either fully ratio-nal or completely irratioratio-nal in this study. If the heterogeneity on irratioratio-nality extends to be presented like a spectrum between full rationality and full irrationality, the op-timal pricing strategy may also change. These extensions of our study call for future investigation.
Appendix A
Proof of Propositions
Proof of Proposition 1 By completing the problem of πC(pC), we get πC(pC) = pC(1− pC). The first- and second-order derivatives of πC(pC) with respect to pC are
∂πC(pC)
∂pC = 1− 2pC and ∂2πC(pC)
∂p2C =−2, Because ∂2π∂pC(p2 C)
C
< 0, the function is concave. For the concavity, the optimal solution must satisfy ∂π∂pC(pC)
C = 0, i.e., p∗C = 12. We plug in this into πc∗ , and we can obtain π∗c = 14.
Proof of Proposition 2 The first- and second-order derivatives of πS(fS) with respect to fS are
∂πS(fS)
∂fS = β[(1−αE)(1−2fS) + αE(1−2tfS)] + (1−β)t[α(1−2fS) + (1−αI)(1−2tfS), ] and
∂2πS(fS)
∂fS2 =−2{β[1 + (t − 1)αE] + (1− β)t[t − (t − 1)αI]}, Because ∂2π∂fS(f2S)
S
< 0, the function is also concave. For the concavity, the optimal solution must satisfy ∂πS(fS) = 0, i.e., fS∗ = (β+t(1−β))/(2[β(1−(t−1)αE)+t(1−β)(t−(t−1)αI)]).
The optimal profit under subscription model πS∗ is therefore πS(fS∗). Note that
fS∗ = β + t(1− β)
2[β + t(1− β)] + 2[β(t − 1)αE + t(1− β)(t − 1)(1 − αI)] < 1/2.
fs∗ may also be shown to be no greater than 1t as long as t < 2. Finally, the derivatives of fS∗ with respect to β, αI,and αE may be easily shown to be negative, positive, and negative, respectively.
Proof of Proposition 3 By reorganizing the profit maximization problem of the mixed
model, the problem become
max
pM,fM
fM(1− fM)[β(1− αE) + t(1− β)αI] + pM(1− pM)[βαE+ (1− β)(1 − αI)]
s.t. fM ≤ pM ≤ tfM
If we ignore the constraints, both the optimal price fM∗ and p∗M are obviously 12. As they satisfies all the constraints, they are indeed optimal. We then substitute the two optimal price into the problem, and obtain the maximum profit of the mixed model πM∗ = 14[1 + αI(1− β)(t − 1)].
Proof of Proposition 4 To show that the certificate model is out performance the
subscription model, we simply subtract πS∗f romπC∗, and get π∗C− π∗S = −14 [β(β− 1)(1 − t)2 + αE(1− t)β + αIt(1− t)(β − 1)]. Note that πC∗ − πS∗ decreases in αI and is that minimized at alphaI = 1. As πC∗ − πS∗ can be shown to be positive when αI = 1, it is confirmed that πC∗ > πS∗ in all cases.
As for the profitability of the mixed model and the certificate model, recall that α = 0 represents all inefficient learners are rational. When this happenes, the maximum profit of the mixed model becomes 14, which is the same as the maximum profit of the certificate model, while when αI > 0, πM∗ is obviously greater than πC∗.
Recall that αE represents the proportion of irrational in efficient learners. When alphaE = 0, πC∗ − π(S)∗ still positive, and apparently αE has no influence on πM∗ . Hence it is proved that the existence of αE does not affect the outcome.
Proof of Proposition 5 For region A: The first- and second-order derivatives of
Equa-tion 5.2 respects to fSA are
∂πSA(fSA)
< 0, the function is concave. For the concavity, the interior optimal solution must satisfy ∂πSA∂ ¯f( ¯fSA)
SA = 0, i.e., ¯fSA = 2(β+(1β+(1−β)t−β)t2). If the interior optimal price exceeds the boundary of region A, which is thk, we then set the boundary price thk as the optimal price. Hence, the optimal price for πSA is either 2(β+(1β+(1−β)t−β)t2) or thk. By plugging the optimal price, we can obtain the optimal equilibrium profit π∗SA
For region B: We first need to discuss the external benefit. Since in this case, only inefficient learners will exert effort, let u∗I be the threshold for inefficient learners to subscribe, the external benefit ηx will become η(1 − β)(1 − u∗I). Substitute this into the inefficient learners’ utility function, we can obtain u∗I = (1−h)tf1−(1−β)ηSB+k−(1−β)η. Following the same rule, let u∗E be the threshold for efficient learners to subscribe, replace ηx by η(1− β)(1 − u∗I), we can obtain u∗E = f +η((1−β)(1−h)tf1−(1−β)ηSB+k−fSB−1).
Now, we can completes the profit function of region B, and obtain its first- and second-order derivatives respects to fSB. Because ∂2π∂fSB2(fSB)
SB
= −2(β+(1−β)((1−h)2t2−(1−(1−h)t)βη))
1−(1−β)η <
0, the function is concave. For the concavity, the interior optimal solution must satisfy
∂πSB( ¯fSB)
∂ ¯fSB = 0, i.e., ¯fSB = β+(1−β)[(1−h)(1−k)t−βηk]
2{β+(1−β)[(1−h)2t2+βη((1−h)t−1)]}. If the interior optimal price is within the feasible region, which means thk ≤ ¯fSB ≤ kh, we set it the optimal price fSB∗ , if f¯SB < thk, we set fSB∗ = thk, and if ¯fSB ≥ hk, set fSB∗ = kh. By plugging the optimal price, we can obtain the corresponding optimal equilibrium profit πSB∗
For Region C: Again, we need to find out the external benefit first. In this case, all of the learners will be diligent while taking the course, the external benefit thus is ηx = β(1−u∗E) + (1−β)(1−u∗I). Substitute this into the utility function of both efficient learners and inefficient learners, we can obtain a simultaneous program
1−η . Then, we just replace these two objects into the profit maximize function of region C, and we can easily find the first-, and second-order derivatives respects to fSC. Because ∂2π∂fSC2(fSC)
SC
= −2(1−h)2(β−(1−β)(−t1−η 2+(t−1)2βη)) < 0, the function is concave.
For the concavity, the interior optimal solution must satisfy ∂πSC∂ ¯f( ¯fSC)
SC = 0, i.e., ¯fSC =
(1−k)(β+(1−β)t)
2(1−h)(β−(1−β)((t−1)2βη−t2)). If the interior optimal price is within the feasible region, which means kh ≤ ¯fSC, we set it the optimal price fSC∗ , otherwise, we set the boundary price
k
h = fSC∗ . By plugging the optimal price, we can obtain the corresponding optimal equilibrium profit πSC∗
And for the last part of the proposition, we can find that in the condition β = 0.5, η = 0.8, h = 0.6, k = 0.35, and t = 1.2, the optimal interior subscription fee of region B and C are ¯fSB = 0.573161 and ¯fSC = 0.737417, since thk = 0.175, and kh = 0.583333, both the interior price are within the feasible region. Therefore, the optimal equilibrium profit
for these two region are πSB∗ = (β+(1−β)((1−h−k+hk)t+βηk))2
4((1−(1−β)η)(β−(1−β)((1−(1−h)t)βη−(1−h)2t2))) = 0.279894, and πSC∗ = 4(1−η)(β−(1−β)((t−1)(1−k)2(β+(1−β)t)2βη2 −t2)) = 0.527254, respectively. We can find that both of the optimal profits are greater than 14 , which is the optimal profit of the certificate model.
Since the interior optimal profit must be greater than the boundary optimal profit, we can thus prove that the optimal profit of region B and region C is greater than the optimal profit of the certificate model in certain cases.
Proof of Proposition 6 Region A, B, and C have been proved in Proof of Proposition
5, and region E has been proved in Proof of Proposition 1. Now we use the same approach to prove the rest region.
For region D: The first- and second-order derivatives of Equation 5.14 respects to fM D
are
The first- and second-order derivatives of Equation 5.14 respects to pM D are
∂πM D(pM D)
∂pM D = (1− β)(1 − 2pM D) ,and
∂2πM D(pM D)
∂p2M D =−2.
Because −2 < 0, the function is concave. For the concavity, the interior optimal solution must satisfy ∂πM D∂ ¯f( ¯fM D)
M D = 0 and ∂πM D∂ ¯p( ¯pM D)
M D = 0, i.e., ¯fM D = 12, and ¯pM D = 12. By plugging the optimal price, we can obtain the interior optimal profit is πM D∗ = 1.
For Region E: We need to find out the external benefit first. In this case, all of the learners will be diligent while taking the course, the external benefit thus is ηx = β(1−u∗E)+(1−β)(1−u∗I). Substitute this into the utility function of both efficient learners and inefficient learners, we can obtain u∗E = k+f (11−βη−h)−βη, and u∗I = p−βη+βη(k+f (11−βη−h)−βη). Then, we just replace these two objects into the profit maximize function of region E, and we can easily find the first-, and second-order derivatives respects to fM E. Because
∂2πM E(fM E)
∂fM E2 < 0 and ∂2πM E∂p2(pM E)
M E < 0, the function is concave. For the concavity, the interior optimal solution must satisfy ∂πM E∂ ¯f( ¯fM E) boundary price to be the optimal price. By plugging the optimal price, we can obtain the corresponding optimal equilibrium profit πM E∗ .
Proof of Proposition 7 Before proving the proposition, one thing should be know is
that the profit of the mixed model will be either the same or greater than the certificate model and the subscription model. So, for the first part, when η = 0, the profits of region A, region B, and region C are 4(β+(1β+(1−β)t−β)t2), (β+(1−β)(1−h)(1−k)t)2
4(β+(1−β)(1−h)2t2) , and (1−k)4(β+(12(β+(1−β)t−β)t)2) 2
respectively. Because all of them < 14, we can say that the certificate model strictly outperforms the subscription model.
And the second part, we can find that when t = 1.2, β = 0.5, η = 0.5, h = 0.6, k = 0.2, the profit of region E is 0.2706, which is greater than region F = 0.25, region A = 0.2114, region B = 0.2058, and region C = 0.1915, which are the same as the profit of the pure certificate model and the pure subscription model A, B, C respectively. So, we can say sometimes the mixed model strictly outperforms them.
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