1. Cournot Oligopoly. Supposetherearetwo identicalfirmsthat facethefollowing inversedemand:P=a−QwhereQ=q1+q2;costfunctionsforbothfirmsarecqi for i=1,2.
A. Findeachfirm’sreactioncurveandsolvefortheCournot-Nashequilibrium. B.Findthetotalmarketprofit(sumprofitsfrombothfirms)
B. Nowsupposethereisonlyonefirminthemarket;findthemonopolist’soutput andprofit.
C. Supposeinsteadthemarketiscompetitiveandthefirmssetp=MC;whatis thetotalmarketoutputandtotalprofit?
D. Reflectonthedifferencesbetweenyouranswerstopart(b),(c),and(d)
2. Cournot Oligopoly. SupposetwoCournotfirmsfacethefollowinginversedemandP=12−2q1−q2 wherecostfunctionsforfirm1is3q1 andforfirm2is4q2.
A. Findeachfirm’sreactioncurveandsolvefortheCournot-Nashequilibrium. B.Findthetotalmarketprofit(sumprofitsfrombothfirms)
B. Characterizeanequilibriuminwhichonlyonefirmproducesbutwherethere arepositiveeconomicprofits.Findtheprice,theoutputlevel,andtotalprofits. [Hint: thinkcarefullyabout which firmoptimallyremainsinthemarketand howtheotherfirmneedstobecompensatedtostayoutof themarket]
3. SwitchingCosts.Considerthefollowingsituation:Gobackto1996.Supposetherearetworivalonlinedatingsites.Thinkofthestrategicinteractionbetweentherespective firms. Thetypical businessstrategyinvolves allowingmemberstomakeprofilesfor free,butthenchargingvarying amountstobeabletoviewotherprofiles,contactother members,etc.Firmsmayalsochoosetoallowadvertisersaccesstotheirsubscribers. Datingsitesareinsomesenseanexperiencegoodi.e.consumersdonottrulyknowthevalue,highorlow,untilafterpurchase.
A. Writedownthisgameusingalocationmodelofproductdifferentiation.You willuseyourmodeltoassistyourreasoningovertherestoftheexercise. Specifythingslikefirmlocationandconsumertransactioncosts.Writedown asimplepayofffunctionforeachfirmandfora typicalconsumer.
B. Whathappens intheinitial periodwhenthefirstbatchofconsumersare makingtheiradoptiondecisionandfirmsaretrying toattract consumers? Whatsort offirmbehaviordoyouexpecttoobserve?
C. Ascompetitionintensifies,firmsbecomeworriedaboutconsumersleavingtheir networkfortherival.Whatcanfirmsdotoavoidthis?
D. Nowitis2006,supposeultimatelyonefirmbuysout theother.Basedonyour answerin(a)and(c),shouldthefirmcontinuetooperatetwoseparatesites orshouldtheymerge? Whatcanthemergedfirmdotoalleviateswitching costsintheeventthatitwishestopoolconsumers?
95-710 ProblemSet#3,Page2of2 Spring2014
4. Network Demand. Inlecture,wediscussedthedemandcurve foranetworkgood andshowedthatmultipleequilibriacouldarise.Usethismodeltopresent asimilarcharacterizationandanalysisofthefollowingsituation:Bitcoin,avirtualcurrency,hasbeenrisinginusageoverthepastfouryearsorso.Itishighlydecentralizedbutcanbebothtradedonexchangesandusedprivately. SupposeuponleavingHeinzyouare offeredawonderful job.Butyourprospectiveemployerinsistsonpayingyour(lucrative) salaryinBitcoin.Willyouaccepttheoffer?
5. Networks Effects. Considerthefollowingsituation: assumewehave anewgood with networkdemand. Sonoonehas joinedthenetworkjustyet. Thereare100potentialadoptersandtherereallyisactuallyonlyabenefitifatleast50ofthem join. Let’sassumethenetworkexternalityisapproximatelythesameforallvaluesof n1 <50andhigherforallvaluesofn2 ≥50wheren1 isanynumberofadoptions lowerthan50andn2 isanynumberofadoptionsequalto orabove50. Let’smodel thisasatwoplayergamewhereeachplayerhasthestrategiesofJoinandNot. Assumethatthepayoffscorrespondingtothefourstrategyprofilesarerankedasfollows: (Join,Join),(Not,Join),(Join,Not),(Not,Not).Writedownyourgameandsolveforall equilibria(i.e.pureandmixed).
6. NetworkEffects.Therearenineconsumers.Eachneedstodeterminewhetherornot tojoinanewsocialnetworkingsite.Consumerscanbeorderedintermsof theirtypes, sothefirstconsumeristype1,thesecondconsumeristype2,andsoon.Eachperson’s willingnesstopaytojointhesiteisproportionaltothenumberof othermembers.Ifnisthenumberofconsumersthathavebecomemembers,thewillingnesstopayoftypetisjustnt. Findthelargestpricesuchthatthere are7consumersthatareatleast indifferenttojoining.
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