demands.

Inthecasesofunitdemandsorperfectcomplements,themonopoly rmneednottyinginordertoreachthemonopolyoutcomeonbothmarkets.Ontheotherhand,ifthedemandsforthetwoproductsareindependentanddownward-sloping,andifthemonopolistisrestrictedtouseuniform(orlinear)pricing,tyingispro table.Thebasicinsightisthatifthemonopolistisunabletoextractfullytheconsumer’ssurplusinthemonopolymarket(e.g.,duetothelimitationoflinearpricing),tyingprovidesaninstrument(ormeteringdevice)forthe rmtodiscriminateacrossconsumersandextractmoreconsumersurplus.Inadeterministicenvironmentwithvariabledemandsandlinearpricing,MathewsonandWinter(1997)provideanecessaryandsu cientconditionfortyingtobepro table.Inparticular,theyshowthattyingisnotpro tableforalllevelsofconstantmarginalcostsifandonlyifthetwogoodsarestronglycomplementary,i.e.,theratioofthedemandsforthetwoproductsisindependentofoneoftheprices.Thisconditionissatis edwithperfectcomplementsbutviolatedwithindependentdemands.Their ndingsummarizespreciselythepreviousliteratureontying.

Ifthepro tabilityoftyingisduetothemonopolist’sinabilitytoextractfullytheconsumer’ssurplusinthemonopolymarketbyusinglinearpricing,thequestionisthenwhethertyingisusefulwhenthe rmcanimplementnon-linearpricingsuchastwo-parttari s.ThesecondpartoftheMathewsonandWinter(1997)paperaddressesthisquestion.Intheirsetting,demandsareuncertainandarerepresentedbyarepresentativeconsumer’s(quasi-linear)indirectutilitywithtwoparametersthatmeasuretherelativewillingnesstopayforthesetwoproductsandthatarenotobservedbythe rm,butthe rmknowstheirjointdistribution.5Ifthemonopolistcanimplementtwo-parttari s,MathewsonandWintershowthattyingispro tablewhenthetwoparametersarepositivelycorrelated.Thatis,evenifthe rmcanusetwo-parttari stoextractmoreconsumersurplusthanlinearpricingdoes,tyingcanfurtherimprovepro tability.Notethattwo-parttari smaynotbethebestnon-linearpricingschemeforthemonopolistwhentyingisnotallowed.Thequestionstillremainsifthemonopoly rmprefersthesecond-bestnon-linearpricinginthemonopolymarkettotying.

Whiletheearlyliteratureontyingmostlyassumethatthesecondproductmarketisperfectlycompetitive,Whinston(1990)isthe rsttorelaxthisassumption.Inhismodel,themonopoly rmandonerivalo erdi erentiatedproductsinthesecond

Theadvantageofusingthequasi-linearutilityfunctionisthatthereisnoincomee ect.Thisisoftenusedinapplicationssinceonecanconvenientlycalculateconsumers’surplusandconductpar-tialequilibriumwelfareanalysis.Inarecentpaper,Duhamel(2004)providesageneralequilibriumfoundationforarbitrarypartialequilibriumwelfareanalysisinsecond-besteconomies.5