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CORNELL ECON 2040 - ps7

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Networks: Fall 2013 Homework 7David Easley and´Eva Tardos Due at 11:15am, Friday, November 15, 2013As noted on the course home page, homework solutions must be sub mi t te d by upload to course’sBlackboard site. The file you upload must be in PDF format. It is fine to write the homeworkin another format such as Word; from Word, you can save the file ou t as PDF for uploading.You can choose type ”pdf” when you save the file, or print it and choose ”Adobe pdf” as yourprinter. (Changing the file extension from doc, or docx to pdf does not change the forma t , onlymakes the file unreadable)Blackboard will stop accepting homework uploads after the posted due date. We cannot ac-cept late homework except for University -a p p r oved excuses (which include illness, a familyemergency, or travel as part of a University sports team or other University activity).Reading: The qu est i on s below are primarily b ased on t h e m at erial in Chapters 17 and 18 ofthe book on Network E↵ects and the Rich gets Richer Phenomena.(1) )(20 points) Consider a good that has network e↵ects in the sense of our model fromChapter 17. Consumers are named using real numbers between 0 and 1. The reser vation pricefor consumer x when a z fraction of the population uses the product is given by the formular(x)f(z), where r ( x )=2(1 x)andf(z)=2z.(a) Suppose that the good is sold at price of 1/2. What are the possible equilibrium fractionsof the population purchasing the good?Solution. z=0 is an equilib r iu m , as at that point no user has any value of the good. Otherequilibria are solutions to 2z ⇥ 2(1  z)=1/2whichisz2 z +1/8=0,whichsolvestoz =(1+/ q1  1/2)/2whichsolvedto0.15and0.85.(b) Which of the equilibria you found in part (a) are stable? Explain.(c) Suppose this is a new product, and you discover that the fraction of potential user sinitially using the product is 0.25. What do you expect the trend i n users to be? Is the productlikely to b eco m e more popular over time, or do you expect that after the initial excitement oftrying the new product dies o↵,theproductwillbecomelesspopular?Explain.(d) Would your answer to (c) change if you found that the fraction of potential users initiallyusing the pr oduct was 0.10? Explain.(e) How does your answer to (a) through (d) change if the pr i ce changes to 1.10 ? Explainbriefly the n ew answers.1(2) (10 points) Two companies each o↵er a very similar service, sharing a market that hasnetwork e↵ects. The two services ar e a bit di↵erent, and some customers prefer (have highervalue for) the serv i ce of company A, others h ave higher value for the version of the serviceo↵ered by company B, but everyone agrees that both services get better if more people use theservice. Typically, com pany A is a bit more popular, and has 60 percent of the market, whilecompany B has 40 percent of the market. Occasionally the number of users of one service versusthe other temporarily rises or falls, usually based on favorable or unfavorable press coverage ofthe respective companies. In the past the market share of company A varied between 70 and50 percent, but the share of consumers quickly reverted to 60 percent.Recently there was unfavorable press coverage of company A’s service and the s h ar e ofpurchasers fell to 40 percent. Initially th e co m p a ny was not concerned about this fall in salesas they assumed that it would revert to 60 percent again. In fact, the share of purchasershas continued to fall a n d now the company is puzzled about why th i s is occurring and whatthey can do to stop it. The service both companies provide is unchanged, there are no newcompetitors, and they have not changed the price of the service.(a) Using your knowledge of network e↵ects provide the company with a possible explanationof what has happened.(b) Again using your kn owledge of network e↵ects suggest at least one action that thecompany could take to help sales recover to its old sh are of 60 percent. Explain why yoursuggestion should work.(3) (10 points) There’s some informal conventional wisdom among people who work onsocial media concerning how these social-media platforms have a↵ected the way events areorganized. Here’s the contrast that get s drawn in t h es e discussions.In the 1980’s, you would generally invite people to a plan n ed event like a party by individ-ually inviting each person and then getting a response from each o f them. The analogous eventtoday is often organized using a site like Evite, where all the invitees get an automated messageto check the invitation on the site; when they visit the on-line invitation, they see the full listof the people who have been invited, and if they check back repeatedly over time they can seewho — at any given point — has already accepted the invitation, who has already decli n e d theinvitation, and who is stil l deciding.The informal argument is that th i s has caused certai n structural changes in the kinds ofattendance you expect at these events — nowadays, you have much more of a “boom-or-bust”e↵ect, where some parties get n earl y 100% participation, whil e others have almost everyon edeclining the invitation. In the past, the argu m ent goes, there would have been more part i eswhere roughly half the people were able to come, rather than almost all or almost none.It’s not clear how genuine this e↵ect is, but there’s some sense that it may be hap pening.It’s hard to collect empirical data, since even with data from Evite, we’d need to be able tocompare to relatively det a i l ed statistics on how party attendance looked in th e 1980’s. Buteven without empi r i cal data, we can a sk a plausibility question — are there natural hypothesesfor why this shi ft might be occurring?2(a) Using the information-based principles from Chapter 16, ca n you suggest why the changein format (from separate personal invitations to public Web si t es) mi ght have led to this changein the way people respond to party invitations?(b) Alt er n a t el y, using the di r ec t- benefit principles from Chapter 17, can you su g ge st whythe change in format (from separate personal invitations to public Web sites) might have ledto this change in the way people respond to party invitations?Note: (a) and (b) are asking, essentially, for two di↵erent hypotheses as to why this mightbe happening, using two di↵erent sets of p r i n ci p l es. So in your answer, you should make surethat the


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