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Q1. (20 points): A group of 12 consumers are considering whether to connect to...
Sep 13, 2024
*** Provide detailed explanation for your answers to get full credit. *** Q1. (20 points): A group of 12 consumers are considering whether to connect to a new social network. Index them as consumer 1, 2, 3, 4, …, 12. Let n denote the size of the network and vi is value or willingness-to-pay. • For consumers 1, 2, 3, 4, their willingness-to-pay (WTP) is: vi=2*n. • For consumers 5, 6, 7, 8, their WTP is: vi=5*n. • For consumers 9, 10, 11, 12, their WTP is: vi=7*n. For all subparts below, assume that a consumer will join the network if and only if his/her WTP after joining is higher than or equal to the price. (a) The network subsidizes consumers 7, 8, 11, 12 to join for free. After that the price to join the network is $32. Which consumers will join the network? Describe the order in which these consumers decide to join the network. Assume that for any given number of people who are already in the social network, the potential participant with the highest WTP will join first. You need to show your calculation (you can construct a table similar to the one in class) to justify your answer in order to receive full credit. (10 points) (b) In this subpart, the network does not subsidize any consumers. Suppose that the network can charge each consumer a different price, and that it can arrange their entry order. Assume that marginal cost = 0 and fixed cost = 0. Explain how the network will arrange the entry order and set prices (different consumers may be charged at different prices) to maximize its profits. What are the maximum profits that the network can achieve? You need to show your calculation (you can construct a table similar to the one in class) to justify your answer in order to receive full credit. (10 points) [Note that because there is no production cost, revenue is the same as profits in this question. That is, maximizing profits is the same as maximizing revenue.]
Solution
a
Willingness-to-Pay Calculation: For consumers 1-4, vi=2n v_i = 2n ; for consumers 5-8, vi=5n v_i = 5n ; for consumers 9-12, vi=7n v_i = 7n . With n=12 n = 12 , we calculate: v1=v2=v3=v4=2×12=24 v_1 = v_2 = v_3 = v_4 = 2 \times 12 = 24 , v5=v6=v7=v8=5×12=60 v_5 = v_6 = v_7 = v_8 = 5 \times 12 = 60 , v9=v10=v11=v12=7×12=84 v_9 = v_{10} = v_{11} = v_{12} = 7 \times 12 = 84 . The price to join is $32. Consumers 7, 8, 11, and 12 join for free. The remaining consumers will join based on their WTP: Consumers 9, 10, 11, and 12 will join first as their WTP is higher than the price. Thus, the order is: 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1
b
Price Setting and Entry Order: The network will charge each consumer based on their WTP. The entry order will be from the highest WTP to the lowest. The prices will be set just below each consumer's WTP: Consumer 12 pays 83,Consumer11pays83, Consumer 11 pays 83, Consumer 10 pays 83,Consumer9pays83, Consumer 9 pays 83, Consumer 8 pays 59,Consumer7pays59, Consumer 7 pays 59, Consumer 6 pays 59,Consumer5pays59, Consumer 5 pays 59, Consumers 4, 3, 2, and 1 pay $24. The total revenue (and profit) is \( 4 \times 83 + 4 \times 59 + 4 \times 24 = 332 + 236 + 96 = 664 \)
Answer
Consumers 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, and 1 will join the network in that order, with maximum profits of $664.
Key Concept
Willingness-to-pay (WTP) determines consumer participation in a network.
Explanation
The network maximizes profits by charging each consumer based on their WTP and arranging entry from highest to lowest WTP.
Q2. (15 points): A group of 12 consumers are considering whether to connect to a new social network. Consumer 1 has an intrinsic value (k) of 1forconnectingtothenetwork,consumer2hasanintrinsicvalue(k)of1 for connecting to the network, consumer 2 has an intrinsic value (k) of 2, consumer 3 has an intrinsic value (k) of 3,andsoonuptoconsumer12.Eachconsumer’swillingness−to−pay(WTP)toconnecttothenetworkdependsonthetotalnumberofpersonswhoareconnectedtoit.Theexactrelationshipisdescribedasfollows:consumeri’swillingness−to−paytoconnecttothenetworkisitimesthetotalnumberofpersonsconnected.Forinstance,consumer2’sWTPiffivepeopleareconnectedtothenetwork,includingconsumer2,is3, and so on up to consumer 12. Each consumer’s willingness-to-pay (WTP) to connect to the network depends on the total number of persons who are connected to it. The exact relationship is described as follows: consumer i’s willingness-to-pay to connect to the network is i times the total number of persons connected. For instance, consumer 2’s WTP if five people are connected to the network, including consumer 2, is 10 (=2*5). For all subparts below, assume that: - for any given number of people who are already in the social network, the potential participant with the highest WTP will sign up first; - a consumer will join the network if and only if his/her WTP after joining is higher than or equal to the price. a. (3 points) Use a table to show the relationship among the total number of people in the network, the intrinsic value of the consumer next in line (marginal consumer), and the willingness-to-pay for that consumer. The table should have three columns: (i) the intrinsic value of the marginal consumer, (ii) # participants in the network assuming the marginal consumer joins; (iii) WTP of the marginal consumer. b. (2 points) Suppose we set the price of joining the network to be 35.Whatisthecriticalmassofthisnetworkgivenp=35. What is the critical mass of this network given p=35? Explain your answer. c. (5 points) Suppose the network has decided to subsidize consumers 1 and 2 (i.e., the consumers with intrinsic value = 1 and 2) to join the network for free, and then set the price at 35.Whatisthetotalnumberofconsumerswhowillsignupthenetwork?[Createatablesimilartopart(a)anduseittohelpyouanswerthequestion.Toreceivefullcredit,youneedtoshowthistableinyouranswer.]d.(5points)Supposethenetworkhasdecidedtosubsidizeconsumers7,11and12(i.e.,theconsumerswithintrinsicvalue=7,11,and12)tojointhenetworkforfree,andthensetthepriceat35. What is the total number of consumers who will sign up the network? [Create a table similar to part (a) and use it to help you answer the question. To receive full credit, you need to show this table in your answer.] d. (5 points) Suppose the network has decided to subsidize consumers 7, 11 and 12 (i.e., the consumers with intrinsic value = 7, 11, and 12) to join the network for free, and then set the price at 40. What is the total number of consumers who will sign up the network? [Create a table similar to part (a) and use it to help you answer the question. To receive full credit, you need to show this table in your answer.]
Solution
a
To analyze the relationship among the total number of people in the network, the intrinsic value of the marginal consumer, and the willingness-to-pay (WTP), we can create a table. The intrinsic value for consumer ii is ki=ik_i = i, and the WTP for consumer ii when there are nn participants is given by WTPi=ki×nWTP_i = k_i \times n. The table will show the intrinsic value, the number of participants, and the WTP for the marginal consumer. Here’s the table: | Intrinsic Value (k) | # Participants (n) | WTP (WTP) | |----------------------|--------------------|------------------| | 1 | 1 | 1×1=11 \times 1 = 1 | | 2 | 2 | 2×2=42 \times 2 = 4 | | 3 | 3 | 3×3=93 \times 3 = 9 | | 4 | 4 | 4×4=164 \times 4 = 16 | | 5 | 5 | 5×5=255 \times 5 = 25 | | 6 | 6 | 6×6=366 \times 6 = 36 | | 7 | 7 | 7×7=497 \times 7 = 49 | | 8 | 8 | 8×8=648 \times 8 = 64 | | 9 | 9 | 9×9=819 \times 9 = 81 | | 10 | 10 | 10×10=10010 \times 10 = 100 | | 11 | 11 | 11×11=12111 \times 11 = 121 | | 12 | 12 | 12×12=14412 \times 12 = 144 |
b ⋮ To find the critical mass of the network given a price p=35p = 35, we need to determine the maximum number of participants nn such that the WTP of the marginal consumer is at least equal to the price. From the table, we see that the WTP for consumer 6 (when n=6n = 6) is 3636, which is greater than 3535, while for consumer 5 (when n=5n = 5), it is 2525, which is less than 3535. Therefore, the critical mass is n=6n = 6.
c
If consumers 1 and 2 are subsidized to join for free, we can analyze the situation again. The WTP for consumer 3 when there are 2 participants (1 and 2) is 3×2=63 \times 2 = 6. The table will look like this: | Intrinsic Value (k) | # Participants (n) | WTP (WTP) | |----------------------|--------------------|------------------| | 1 | 1 | 1×1=11 \times 1 = 1 | | 2 | 2 | 2×2=42 \times 2 = 4 | | 3 | 3 | 3×3=93 \times 3 = 9 | Since consumers 1 and 2 join for free, consumer 3 will also join because 9≥359 \geq 35. Continuing this process, we find that consumers 1, 2, 3, 4, 5, and 6 will join, totaling 6 consumers
d ⋮ If consumers 7, 11, and 12 are subsidized to join for free, we analyze the situation again. The WTP for consumer 6 when there are 3 participants (7, 11, and 12) is 6×3=186 \times 3 = 18. The table will look like this: | Intrinsic Value (k) | # Participants (n) | WTP (WTP) | |----------------------|--------------------|------------------| | 1 | 1 | 1×1=11 \times 1 = 1 | | 2 | 2 | 2×2=42 \times 2 = 4 | | 3 | 3 | 3×3=93 \times 3 = 9 | | 4 | 4 | 4×4=164 \times 4 = 16 | | 5 | 5 | 5×5=255 \times 5 = 25 | | 6 | 6 | 6×6=366 \times 6 = 36 | | 7 | 7 | 7×7=497 \times 7 = 49 | Since consumers 7, 11, and 12 join for free, consumer 6 will also join because 36≥4036 \geq 40. Continuing this process, we find that consumers 1, 2, 3, 4, 5, 6, and 7 will join, totaling 7 consumers.
Answer
a: 6 consumers will join the network when subsidized consumers 1 and 2 join for free.
b: The critical mass of the network is 6 participants.
c: 7 consumers will join the network when subsidized consumers 7, 11, and 12 join for free.
Key Concept
Willingness-to-pay (WTP) and consumer subsidies impact network participation.
Explanation
The analysis shows how intrinsic values and WTP determine consumer decisions to join a network, especially under different pricing and subsidy scenarios.
Johns Hopkins Carey Business School is organizing its inaugural conference in healthcare marketing. The main purpose of this conference is to share cutting-edge research topics and methodologies among academics. Another purpose is to show healthcare providers the latest research in marketing so that they can apply it to improve adoption of new treatments and patient experiences. The conference provides a platform for professional networking and exchange of ideas, learning and knowledge. Conference attendees will include both academicians and industry practitioners. If someone would like to present his/her research in the conference, he/she needs to submit a research paper to the conference by a deadline, and the conference committee will select a subset of the submissions for presentation. In order to attend the conference, attendees must register. The registration fees are: i. Presenters 0ii.Academicparticipants0 ii. Academic participants 300 iii. Industry participants $600 Note that Industry participants are the people who work for companies. Explain your answers in detail based on network externality and two-sided market theory. (a) Why does the conference charge the industry participants more than academic participants? (2 points) (b) Why does the conference decide NOT to charge presenters anything? (3 points)
Answer
a
Pricing Strategy: The conference charges industry participants more than academic participants because industry participants typically derive greater economic benefits from attending. The higher registration fee for industry participants (600)reflectstheirhigherwillingness−to−pay(WTP)duetothepotentialfordirectapplicationofmarketingresearchtoimprovetheirbusinessoutcomes.Incontrast,academicparticipants(600) reflects their higher willingness-to-pay (WTP) due to the potential for direct application of marketing research to improve their business outcomes. In contrast, academic participants (300) may have a lower WTP as their primary goal is knowledge acquisition rather than immediate economic gain
b
Incentivizing Participation: The conference does not charge presenters anything to encourage high-quality submissions and participation. By waiving the registration fee for presenters, the conference attracts valuable research contributions, enhancing the overall quality and appeal of the event. This strategy aligns with the two-sided market theory, where the value of the conference increases with the number of quality presenters, thereby attracting more attendees and creating a network effect
Answer
Industry participants are charged more due to their higher willingness-to-pay and potential economic benefits.
Presenters are not charged to incentivize quality research contributions, enhancing the conference's value.
Q4. (10 points) A group of 12 consumers are considering whether to connect to a new social network. Index them as consumer 1, 2, 3, 4, …, 12. Let n denote the size of the network. • For consumers 1, 2, 3, 4, their willingness-to-pay (WTP) is: vi=5*n. • For consumers 5, 6, 7, 8, their WTP is: vi=8*n. • For consumers 9, 10, 11, 12, their WTP is: vi=4*n. Assume that a consumer will join the network if and only if his/her WTP after joining is higher than or equal to the price. (a) The network does not subsidize any consumers. Suppose that the network can charge each consumer a different price, and that it can arrange their entry order. Assume that marginal cost = 0 and fixed cost = 0. Explain how the network will arrange the entry order and set prices (different consumers may be charged at different prices) to maximize its profits. What is the maximum profit that the network can achieve? You need to show your calculation (you can construct a table similar to the one in class) to justify your answer in order to receive full credits. (10 points) [Note that because there is no production cost, revenue is the same as profits in this question. That is, maximizing profits is the same as maximizing revenue.]
Solution
a
Willingness-to-Pay Calculation: For each consumer, we calculate their WTP based on the size of the network n n . The WTP for each group is as follows: v1=v2=v3=v4=5n v_1 = v_2 = v_3 = v_4 = 5n v5=v6=v7=v8=8n v_5 = v_6 = v_7 = v_8 = 8n v9=v10=v11=v12=4n v_9 = v_{10} = v_{11} = v_{12} = 4n
b
Entry Order Arrangement: To maximize profits, the network should arrange consumers in descending order of their WTP. The order will be: Consumers 5, 6, 7, 8 (WTP = 8n 8n ), followed by 1, 2, 3, 4 (WTP = 5n 5n ), and finally 9, 10, 11, 12 (WTP = 4n 4n )
c
Pricing Strategy: The network can charge each consumer their maximum WTP. Thus, the prices will be set as follows: P5=P6=P7=P8=8n P_5 = P_6 = P_7 = P_8 = 8n P1=P2=P3=P4=5n P_1 = P_2 = P_3 = P_4 = 5n P9=P10=P11=P12=4n P_9 = P_{10} = P_{11} = P_{12} = 4n
d
Profit Calculation: The total revenue (and profit) can be calculated as follows: Total Revenue=4×8n+4×5n+4×4n=32n+20n+16n=68n \text{Total Revenue} = 4 \times 8n + 4 \times 5n + 4 \times 4n = 32n + 20n + 16n = 68n Thus, the maximum profit the network can achieve is 68n 68n
Answer
The maximum profit that the network can achieve is 68n 68n .
Key Concept
Profit maximization through consumer pricing and entry order arrangement
Explanation
The network maximizes profit by charging consumers based on their willingness-to-pay and arranging them in order of highest to lowest WTP.
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